**Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption and Multiplication Regions**

Viacheslav Kholodnov and Mikhail Nikitin

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50778

## **1. Introduction**

[16] Takahashi, Y., Neogi, A., & Kawaguchi, H. (1998). Polarization-Dependent Nonlinear

Gain in Semiconductor Lasers. *IEEE J. Quantum Electron.*, 34(9), 1660-1672.

26 Photodiodes - From Fundamentals to Applications

Minimal value of dark current in reverse biased *p* −*n* junctions at avalanche breakdown is determined by interband tunneling. For example, tunnel component of dark current be‐ comes dominant in reverse biased *p* −*n* junctions formed in a number semiconductor ma‐ terials with relatively wide gap *Eg* already at room temperature when bias *Vb* is close to avalanche breakdown voltage *VBD* (Sze, 1981), (Tsang, 1981). The above statement is ap‐ plicable, for example, to *p* −*n* junctions formed in semiconductor structures based on ter‐ nary alloy *I n*0.53*Ga*0.47*As* which is one of the most important material for optical communication technology in wavelength range *λ* up to 1.7 μm (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981). Significant decreasing of tunnel current can be achieved in avalanche photo‐ diode (APD) formed on multilayer heterostructure (Fig. 1) with built-in *p* −*n* junction when metallurgical boundary of *p* −*n* junction (*x* =0) lies in wide-gap layer of heterostruc‐ ture (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011). Design and specification of heterostructure for creation high performance APD must be such that in operation mode the following two conditions are satisfied. First, space charge region (SCR) penetrates into narrow-gap light absorbing layer (absorber) and second, due to decrease of electric field *E*(*x*) into depth from *x* =0 (Fig. 1), process of avalanche multiplication of charge carriers could only develop in wide-gap layer. This concept is known as APD with separate absorption and multiplication regions (SAM-

© 2012 Kholodnov and Nikitin; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Kholodnov and Nikitin; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

APD). Suppression of tunnel current is caused by the fact that higher value of *E* corre‐ sponds to wider gap *Eg*. Electric field in narrow-gap layer is not high enough to produce high tunnel current in this layer. Dark current component due to thermal generation of charge carriers in SCR (thermal generation current with density *JG*) is proportional to in‐ trinsic concentration of charge carriers *ni* ∝exp(−*Eg* / 2*kBT* ), here *kB* – Boltzmann constant, *T* – temperature (Sze, 1981), (Stillman, 1981). Tunnel current density *JT* grows considera‐ bly stronger with narrowing *Eg* than *ni* and depends weakly on *T* (Stillman, 1981), (Bur‐ stein & Lundqvist, 1969). Therefore, component *JT* will prevail over *JG* in semiconductor structures with reasonably narrow gap *Eg* even at room temperature. Another dark cur‐ rent component − diffusion-drift current caused by inflow of minority charge carriers into SCR from quasi-neutral regions of heterostructure is proportional to *ni* <sup>2</sup> × *N* <sup>−</sup>1 (Sze, 1981), (Stillman, 1981) (where *N* is dopant concentration). To eliminate it one side of *p* −*n* junc‐ tion is doped heavily and narrow-gap layer is grown on wide-gap isotype heavily doped substrate (Tsang, 1981). Thus heterostructure like as *pwg* <sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg* <sup>+</sup> is the most opti‐ mal, where subscript ‹*wg*› means wide-gap and ‹*ng*› − narrow-gap, properly. To ensure tunnel current's density not exceeding preset value is important to know exactly allowa‐ ble variation intervals of dopants concentrations and thicknesses of heterostructure's lay‐ ers. Thickness of narrow-gap layer *W*2 is defined mainly by light absorption coefficient *γ* and speed-of-response. But as it will be shown further tunnel current's density depends strongly on thickness of wide-gap layer *W*1 and dopant concentrations in wide-gap *N*<sup>1</sup> and narrow-gap *N*2 layers. Approach to optimize SAM-APD structure was proposed in articles (Kim et al, 1981), (Forrest et al, 1983) (see also (Tsang, 1981)). Authors have devel‐ oped diagram for physical design of SAM-APD based on heterostructure including *I n*0.53*Ga*0.47*As* layer. However, diagram is not enough informative, even incorrect signifi‐ cantly, and cannot be reliably used for determining allowable variation intervals of heter‐ ostructure's parameters. The matter is that diagram was developed under assumption that when electric field *E*(*x*) (see Fig. 1b) at metallurgical boundary of *pwg* <sup>+</sup> <sup>−</sup>*nwg* junction *E*(0)≡*E*1 is higher than 4.5×105 V/cm then avalanche multiplication of charge carriers oc‐ curs in *InP* layer where *pwg* <sup>+</sup> <sup>−</sup>*nwg* junction lies at any dopants concentrations and thick‐ nesses of heterostructure's layers. However, electric field *E*<sup>1</sup> =*E*1*BD* at which avalanche breakdown of *p* −*n* junction occurs depends on both doping and thicknesses of layers (Sze, 1981), (Tsang, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1988), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholodnov, 1998), (Kholodnov & Kurochkin, 1998). As a consequence, avalanche multiplication of charge carriers in considered heterostructure can either does not occur at electric field value *E*1=4.5×105 V/cm or occurs in narrow-gap layer (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989). Value of electric field re‐ quired to initialize avalanche multiplication of charge carriers can even exceed *E*1*BD* (Sze, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholod‐

nov, 1998), (Kholodnov & Kurochkin, 1998) that has physical meaning in the case of tran‐ sient process only (Groves et al, 2005), (Kholodnov, 2009). Further, in development of diagram was assumed that maximal allowable value of electric field in absorber at heterointerface with multiplication layer *E*2 (see Fig. 1b) is equal to 1.5×105 V/cm. But tunnel current density *JT* in narrow-gap absorber *I n*0.53*Ga*0.47*As* (Osipov & Kholodnov, 1989) is much smaller at that value of electric field than density of thermal generation current *JG* which in the best samples of *InP* − *I n*0.53*Ga*0.47*As* − *InP* heterostructures (Tsang, 1981), (Tar‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

into account the fact that tunnel current in wide-gap multiplication layer can be much greater than in narrow-gap absorber (Osipov & Kholodnov, 1989). Therefore, total tunnel

In present chapter is done systematic analysis of interband tunnel current in avalanche het‐ erophotodiode (AHPD) and its dependence on dopants concentrations *N*1 in *nwg*wide-gap and *N*2 in *nng*narrow-gap layers of heterostructure and thicknesses *W*1 and *W*2, respectively (Fig. 1) and fundamental parameters of semiconductor materials also. Performance limits of AHPDs are analyzed (Kholodnov, 1996). Formula for quantum efficiency *η* of heterostruc‐ ture is derived taking into account multiple internal reflections from hetero-interfaces. Con‐ centration-thickness nomograms were developed to determine allowable variation intervals of dopants concentrations and thicknesses of heterostructure layers in order to match preset noise density and avalanche multiplication gain of photocurrent. It was found that maximal possible AHPD's speed-of-response depends on photocurrent's gain due to avalanche mul‐ tiplication, as it is well known and permissible noise density for preset value of photocur‐ rent's gain also. Detailed calculations for heterostructure *InP* − *I n*0.53*Ga*0.47*As* − *InP* are performed. The following values of fundamental parameters of *InР* (I, Fig. 1) and *I n*0.53*Ga*0.47*As* (II, Fig. 1) materials (Tsang, 1981), (Stillman, 1981), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Braer et al, 1990), (Stillman et al, 1983), (Bur‐ khard et al, 1982), (Casey & Panish, 1978) are used in calculations: band-gaps *Eg*1= 1.35 eV

relative dielectric constants *ε*<sup>1</sup> = 12.4 and *ε*2=13.9; light absorption coefficient in *I n*0.53*Ga*0.47*As γ*=104 сm-1; specific effective masses *m* \* =2*mc* ×*mv* / (*mc* + *mv*) of light carriers *m*1= 0.06*m*<sup>0</sup> and *m*2= 0.045*m*0, where *m*0 – free electron mass. The chapter material is presented in analytical form. For this purpose simple formulas for avalanche breakdown electric field *EBD* and volt‐ age *VBD* of *p* −*n* junction are derived taking into account finite thickness of layer. Analytical expression for exponent in well-known Miller's relation was obtained (Sze, 1981), (Tsang, 1981), (Miller, 1955) which describes dependence of charge carriers' avalanche multiplica‐ tion factors on applied bias voltage *Vb*. It is shown in final section that Geiger mode (Groves et al, 2005) of APD operation can be described by elementary functions (Kholodnov, 2009).

(1) =108

сm-3 and *ni*

(2)

=5.4×1011 сm-3;

. However, diagram does not take

http://dx.doi.org/10.5772/50778

29

of et al, 1990), (Braer et al, 1990) can be up to 10-6 A/cm2

and *Eg*2= 0.73 eV; intrinsic charge carriers concentrations *ni*

current can exceed thermal generation current.

nov, 1998), (Kholodnov & Kurochkin, 1998) that has physical meaning in the case of tran‐ sient process only (Groves et al, 2005), (Kholodnov, 2009). Further, in development of diagram was assumed that maximal allowable value of electric field in absorber at heterointerface with multiplication layer *E*2 (see Fig. 1b) is equal to 1.5×105 V/cm. But tunnel current density *JT* in narrow-gap absorber *I n*0.53*Ga*0.47*As* (Osipov & Kholodnov, 1989) is much smaller at that value of electric field than density of thermal generation current *JG* which in the best samples of *InP* − *I n*0.53*Ga*0.47*As* − *InP* heterostructures (Tsang, 1981), (Tar‐ of et al, 1990), (Braer et al, 1990) can be up to 10-6 A/cm2 . However, diagram does not take into account the fact that tunnel current in wide-gap multiplication layer can be much greater than in narrow-gap absorber (Osipov & Kholodnov, 1989). Therefore, total tunnel current can exceed thermal generation current.

APD). Suppression of tunnel current is caused by the fact that higher value of *E* corre‐ sponds to wider gap *Eg*. Electric field in narrow-gap layer is not high enough to produce high tunnel current in this layer. Dark current component due to thermal generation of charge carriers in SCR (thermal generation current with density *JG*) is proportional to in‐ trinsic concentration of charge carriers *ni* ∝exp(−*Eg* / 2*kBT* ), here *kB* – Boltzmann constant, *T* – temperature (Sze, 1981), (Stillman, 1981). Tunnel current density *JT* grows considera‐ bly stronger with narrowing *Eg* than *ni* and depends weakly on *T* (Stillman, 1981), (Bur‐ stein & Lundqvist, 1969). Therefore, component *JT* will prevail over *JG* in semiconductor structures with reasonably narrow gap *Eg* even at room temperature. Another dark cur‐ rent component − diffusion-drift current caused by inflow of minority charge carriers into

(Stillman, 1981) (where *N* is dopant concentration). To eliminate it one side of *p* −*n* junc‐ tion is doped heavily and narrow-gap layer is grown on wide-gap isotype heavily doped

mal, where subscript ‹*wg*› means wide-gap and ‹*ng*› − narrow-gap, properly. To ensure tunnel current's density not exceeding preset value is important to know exactly allowa‐ ble variation intervals of dopants concentrations and thicknesses of heterostructure's lay‐ ers. Thickness of narrow-gap layer *W*2 is defined mainly by light absorption coefficient *γ* and speed-of-response. But as it will be shown further tunnel current's density depends strongly on thickness of wide-gap layer *W*1 and dopant concentrations in wide-gap *N*<sup>1</sup> and narrow-gap *N*2 layers. Approach to optimize SAM-APD structure was proposed in articles (Kim et al, 1981), (Forrest et al, 1983) (see also (Tsang, 1981)). Authors have devel‐ oped diagram for physical design of SAM-APD based on heterostructure including *I n*0.53*Ga*0.47*As* layer. However, diagram is not enough informative, even incorrect signifi‐ cantly, and cannot be reliably used for determining allowable variation intervals of heter‐ ostructure's parameters. The matter is that diagram was developed under assumption

<sup>2</sup> × *N* <sup>−</sup>1 (Sze, 1981),

<sup>+</sup> is the most opti‐

<sup>+</sup> <sup>−</sup>*nwg* junction

<sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg*

SCR from quasi-neutral regions of heterostructure is proportional to *ni*

that when electric field *E*(*x*) (see Fig. 1b) at metallurgical boundary of *pwg*

*E*(0)≡*E*1 is higher than 4.5×105 V/cm then avalanche multiplication of charge carriers oc‐

nesses of heterostructure's layers. However, electric field *E*<sup>1</sup> =*E*1*BD* at which avalanche breakdown of *p* −*n* junction occurs depends on both doping and thicknesses of layers (Sze, 1981), (Tsang, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1988), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholodnov, 1998), (Kholodnov & Kurochkin, 1998). As a consequence, avalanche multiplication of charge carriers in considered heterostructure can either does not occur at electric field value *E*1=4.5×105 V/cm or occurs in narrow-gap layer (Osipov & Kholodnov, 1987), (Osipov &, Kholodnov, 1989). Value of electric field re‐ quired to initialize avalanche multiplication of charge carriers can even exceed *E*1*BD* (Sze, 1981), (Osipov & Kholodnov, 1987), (Kholodnov, 1996-2), (Kholodnov, 1996-3), (Kholod‐

<sup>+</sup> <sup>−</sup>*nwg* junction lies at any dopants concentrations and thick‐

substrate (Tsang, 1981). Thus heterostructure like as *pwg*

28 Photodiodes - From Fundamentals to Applications

curs in *InP* layer where *pwg*

In present chapter is done systematic analysis of interband tunnel current in avalanche het‐ erophotodiode (AHPD) and its dependence on dopants concentrations *N*1 in *nwg*wide-gap and *N*2 in *nng*narrow-gap layers of heterostructure and thicknesses *W*1 and *W*2, respectively (Fig. 1) and fundamental parameters of semiconductor materials also. Performance limits of AHPDs are analyzed (Kholodnov, 1996). Formula for quantum efficiency *η* of heterostruc‐ ture is derived taking into account multiple internal reflections from hetero-interfaces. Con‐ centration-thickness nomograms were developed to determine allowable variation intervals of dopants concentrations and thicknesses of heterostructure layers in order to match preset noise density and avalanche multiplication gain of photocurrent. It was found that maximal possible AHPD's speed-of-response depends on photocurrent's gain due to avalanche mul‐ tiplication, as it is well known and permissible noise density for preset value of photocur‐ rent's gain also. Detailed calculations for heterostructure *InP* − *I n*0.53*Ga*0.47*As* − *InP* are performed. The following values of fundamental parameters of *InР* (I, Fig. 1) and *I n*0.53*Ga*0.47*As* (II, Fig. 1) materials (Tsang, 1981), (Stillman, 1981), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Braer et al, 1990), (Stillman et al, 1983), (Bur‐ khard et al, 1982), (Casey & Panish, 1978) are used in calculations: band-gaps *Eg*1= 1.35 eV and *Eg*2= 0.73 eV; intrinsic charge carriers concentrations *ni* (1) =108 сm-3 and *ni* (2) =5.4×1011 сm-3; relative dielectric constants *ε*<sup>1</sup> = 12.4 and *ε*2=13.9; light absorption coefficient in *I n*0.53*Ga*0.47*As γ*=104 сm-1; specific effective masses *m* \* =2*mc* ×*mv* / (*mc* + *mv*) of light carriers *m*1= 0.06*m*<sup>0</sup> and *m*2= 0.045*m*0, where *m*0 – free electron mass. The chapter material is presented in analytical form. For this purpose simple formulas for avalanche breakdown electric field *EBD* and volt‐ age *VBD* of *p* −*n* junction are derived taking into account finite thickness of layer. Analytical expression for exponent in well-known Miller's relation was obtained (Sze, 1981), (Tsang, 1981), (Miller, 1955) which describes dependence of charge carriers' avalanche multiplica‐ tion factors on applied bias voltage *Vb*. It is shown in final section that Geiger mode (Groves et al, 2005) of APD operation can be described by elementary functions (Kholodnov, 2009).

From relations (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Grekhov & Serezhkin, 1980),

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

( ), ( ), ( , ) ( ) ( ) ( ) ,

*M M L M M L M L L g x M x dx g x dx*

<sup>0</sup> ( ) ( , ) (1 ), ( , ) ( ) ( , ) , ( , ) exp ( ) *n*

= - - - =a - = b-a ê ú¢ ê ú ë û

can be determined, in principal, dependences of multiplication factors *M* in *p* −*n* structures on *Vb*, where *Mn* and *Mp* – multiplication factors of electrons and holes inflow into space charge region (SCR); value of multiplication factor of charge carriers generated in SCR *<sup>M</sup>*˜

lies between *Mn* and *Mр*; specific rate of charge carriers' generation in SCR *g* = *gd* + *gph* con‐ sists of dark *gd* and photogenerated *gph* components; *L <sup>p</sup>* and *L <sup>n</sup>* – thicknesses of SCR in *p* and *n* sides of structure; *α*(*E*) and *β*(*E*)= *K*(*E*)×*α*(*E*) – impact ionization coefficients of elec‐ trons *α*(*E*) and holes *β*(*E*); *Е*(*х*) – electric field. Let's denote by *N*<sup>1</sup> *pt* dopant concentration *N*<sup>1</sup> so that for *N*<sup>1</sup> < *N*<sup>1</sup> *pt* "punch-through" (depletion) of *nwg* layer occurs that means penetration of non-equilibrium SCR into *nng* layer (Fig. 1). Optical radiation passing through wide-gap window is absorbed in *nng* layer and generates electron-holes pairs in it. When *N*<sup>1</sup> < *N*<sup>1</sup> *pt* then photo-holes appearing near *nwg*/*nng* heterojunction (*х* =*W*1) are heated in electric field of non-equilibrium SCR and, at moderate discontinuities in valence band top *Ev* at *х* =*W*1, pho‐ to-holes penetrate into *nwg* layer (layer I) due to emission and tunneling. If *W*<sup>1</sup> is larger than some value *W*1min(*N*1, *N*2, *W*2) (Osipov & Kholodnov, 1989), which is calculated below, then avalanche multiplication of charge carriers occurs only in *nwg* layer, i.e. photo-holes fly through whole region of multiplication. In this case photocurrent's gain (Tsang, 1981), (Art‐

tion of charge carriers in it can be neglected (Kholodnov, 1996-2), (Kholodnov & Kurochkin, 1998). Under these conditions thicknesses in relations (1) and (2) can be put *L <sup>p</sup>* =0 and

It is remarkable that responsivity *SI* (*λ*) (where *λ* – is wavelength) of heterostructure increas‐ es dramatically once SCR reaches absorber *nng* (layer II on Fig. 1) and then depends weakly on bias *Vb* till avalanche breakdown voltage value *VBD* (Stillman, 1981). This effect is caused by potential barrier for photo-holes on *nwg*/*nng* heterojunction and heating of photo-holes in

*M x Y x L m m L L x Y x W dx Y x x dx* -

*p*

*p p n p*

*n n*

*L L*

*p p*

0

ò ò (2)

<sup>+</sup> layer is doped so heavy that avalanche multiplica‐

1 1 ( ,0) / [1 (0, )] *M YW m W ph* = - (3)

é ù

http://dx.doi.org/10.5772/50778

31

*L L*


*L x*

*L x*

=- = - = ò ò % (1)

(Artsis & Kholodnov, 1984)

*n p p n pn*

sis & Kholodnov, 1984) *M ph* =*Mр*. Let *pwg*

*L <sup>n</sup>* =*W*1, i.e.

**Figure 1.** Energy diagram of heterostructure in operation mode (**a**) and electric field distribution in it (**b**). Ec and Ev − energy of conduction band bottom and valence band top. Solid lines − *N*<sup>2</sup> = *N*<sup>2</sup> (1), dashed − *N*<sup>2</sup> <sup>&</sup>gt; *<sup>N</sup>*<sup>2</sup> (1)

#### **2. Formulation of the problem: Basic relations**

Let's consider *pwg* <sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg* + heterostructure at reverse bias *Vb* sufficient to initialize avalanche multiplication of charge carries. This structure is basic for fabrication of AHPDs. From relations (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Grekhov & Serezhkin, 1980), (Artsis & Kholodnov, 1984)

$$M\_n = M(-L\_p), \\ M\_p = M(L\_n), \\ \tilde{M}(-L\_{p'}, L\_n) = \int\_{-L\_p}^{L\_p} \mathbf{g}(\mathbf{x}) M(\mathbf{x}) d\mathbf{x} \/ \left/ \int\_{-L\_p}^{L\_p} \mathbf{g}(\mathbf{x}) d\mathbf{x} \right. \tag{1}$$

$$M(\mathbf{x}) = Y(\mathbf{x}, -\mathbf{L}\_p) \Big/ \langle 1 - m \rangle,\\ m(-\mathbf{L}\_p, \mathbf{L}\_n) = \int\_{-\mathbf{L}\_p}^{\mathbf{L}\_p} \alpha(\mathbf{x}) Y(\mathbf{x}, -\mathbf{W}\_p) d\mathbf{x},\\ Y(\mathbf{x}, \mathbf{x}\_0) = \exp\left[\int\_{\mathbf{x}\_0}^{\mathbf{x}} (\beta - \alpha) d\mathbf{x}'\right] \tag{2}$$

can be determined, in principal, dependences of multiplication factors *M* in *p* −*n* structures on *Vb*, where *Mn* and *Mp* – multiplication factors of electrons and holes inflow into space charge region (SCR); value of multiplication factor of charge carriers generated in SCR *<sup>M</sup>*˜ lies between *Mn* and *Mр*; specific rate of charge carriers' generation in SCR *g* = *gd* + *gph* con‐ sists of dark *gd* and photogenerated *gph* components; *L <sup>p</sup>* and *L <sup>n</sup>* – thicknesses of SCR in *p* and *n* sides of structure; *α*(*E*) and *β*(*E*)= *K*(*E*)×*α*(*E*) – impact ionization coefficients of elec‐ trons *α*(*E*) and holes *β*(*E*); *Е*(*х*) – electric field. Let's denote by *N*<sup>1</sup> *pt* dopant concentration *N*<sup>1</sup> so that for *N*<sup>1</sup> < *N*<sup>1</sup> *pt* "punch-through" (depletion) of *nwg* layer occurs that means penetration of non-equilibrium SCR into *nng* layer (Fig. 1). Optical radiation passing through wide-gap window is absorbed in *nng* layer and generates electron-holes pairs in it. When *N*<sup>1</sup> < *N*<sup>1</sup> *pt* then photo-holes appearing near *nwg*/*nng* heterojunction (*х* =*W*1) are heated in electric field of non-equilibrium SCR and, at moderate discontinuities in valence band top *Ev* at *х* =*W*1, pho‐ to-holes penetrate into *nwg* layer (layer I) due to emission and tunneling. If *W*<sup>1</sup> is larger than some value *W*1min(*N*1, *N*2, *W*2) (Osipov & Kholodnov, 1989), which is calculated below, then avalanche multiplication of charge carriers occurs only in *nwg* layer, i.e. photo-holes fly through whole region of multiplication. In this case photocurrent's gain (Tsang, 1981), (Art‐ sis & Kholodnov, 1984) *M ph* =*Mр*. Let *pwg* <sup>+</sup> layer is doped so heavy that avalanche multiplica‐ tion of charge carriers in it can be neglected (Kholodnov, 1996-2), (Kholodnov & Kurochkin, 1998). Under these conditions thicknesses in relations (1) and (2) can be put *L <sup>p</sup>* =0 and *L <sup>n</sup>* =*W*1, i.e.

$$M\_{ph} = \mathcal{Y}(\mathcal{W}\_1, \mathcal{O}) / \left[1 - m(\mathcal{O}, \mathcal{W}\_1)\right] \tag{3}$$

It is remarkable that responsivity *SI* (*λ*) (where *λ* – is wavelength) of heterostructure increas‐ es dramatically once SCR reaches absorber *nng* (layer II on Fig. 1) and then depends weakly on bias *Vb* till avalanche breakdown voltage value *VBD* (Stillman, 1981). This effect is caused by potential barrier for photo-holes on *nwg*/*nng* heterojunction and heating of photo-holes in

**Figure 1.** Energy diagram of heterostructure in operation mode (**a**) and electric field distribution in it (**b**). Ec and Ev −

avalanche multiplication of charge carries. This structure is basic for fabrication of AHPDs.

(1), dashed − *N*<sup>2</sup> <sup>&</sup>gt; *<sup>N</sup>*<sup>2</sup>

+ heterostructure at reverse bias *Vb* sufficient to initialize

(1)

energy of conduction band bottom and valence band top. Solid lines − *N*<sup>2</sup> = *N*<sup>2</sup>

**2. Formulation of the problem: Basic relations**

<sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg*

30 Photodiodes - From Fundamentals to Applications

Let's consider *pwg*

electric field of non-equilibrium SCR. If losses due to recombination are negligible (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Forrest et al, 1983), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), for example, at punch-through of absorber, then *SI* (*λ*) in operation mode is determined by well-known expression (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2011):

$$S\_I(\lambda) = \mathfrak{n}(\lambda) \times \frac{\lambda}{1.24} \times M\_{ph} \tag{4}$$

mary tunnel currents *IT* <sup>1</sup> = *JT* <sup>1</sup> × *AS* and *IT* <sup>2</sup> = *JT* <sup>2</sup> × *AS* (total primary tunnel current *IT* = *IT* <sup>1</sup>+ *IT* <sup>2</sup>). Distribution of electric field *Е*(*х*) that should be known to calculate parameters (4) and (5) of AHPD is obtained from Poisson equation and in layers I and II is determined by ex‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

1 1 1 0 1 ( ) \_( ), *qN x Ex E U l x* æ ö =- ´ - ç ÷ e e è ø

2 1 12

( ) ( ) \_( ), *qN Ex E x W U W l x* é ù = - - ´ +- ê ú

\_( ), *qN W E E Ul W* æ ö <sup>e</sup> = - ´- ç ÷

<sup>0</sup> ( ) \_( ), *<sup>i</sup> i i ii i i i*

*U*−(*x*) and *U*+(*x*) – asymmetric unit stepwise functions (Korn G. & Korn T., 2000), *ε*0 – dielectric constant of vacuum, *ε*1 and *ε*2 – relative dielectric permittivity of *nwg* and *nng* layers (Fig. 1).

**3. Avalanche multiplication factors of charge carriers in p-n structures**

For successful development of semiconductor devices using effects of impact ionization and avalanche multiplication of charge carriers is necessary to know dependences of avalanche multiplication factors *M* (*V* ) of charge carriers in *p* −*n* structures on applied bias *Vb*. We need to know among them dependence of avalanche breakdown voltage *VBD* on parameters of *p* −*n* structure and distribution of electric field *E*(*x*) related to *VBD* dependence. Usual way to compute required dependencies is based on numerical processing of integral rela‐ tions (1) and (2) in each case. Impact ionization coefficients of electrons *α*(*E*) and holes *β*(*E*) depend drastically on electric field *E*. At the same time theoretical expressions for *α*(*E*) and *β*(*E*) include usually some adjustable parameters. Therefore, to avoid large errors in calcu‐ lating of multiplication factors, in computation of (1) and (2) are commonly used experimen‐ tal dependences for *α*(*E*) and *β*(*E*). Avalanche breakdown voltage *VBD* is defined as applied

= ´ -+ ´ - (9)

*l E UW l W U l W*

2

e e ë û

1 11 2 1 1 1

2 02

*i*

e e

**3.1. Preliminary remarks: Avalanche breakdown field**

*qN* <sup>+</sup>

e ee è ø

0 2

(6)

33

http://dx.doi.org/10.5772/50778

(7)

(8)

pressions:

Where

where *λ* in μm and value of quantum efficiency *η* is considered below. Photocurrent gaining and large drift velocity of charge carriers in SCR allow creating high-speed high-perform‐ ance photo-receivers with APDs as sensitive elements (Sze, 1981), (Tsang, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Woul, 1980). Reason is high noise density of external elec‐ tronics circuit at high frequencies or large leakage currents that results in decrease in Noise Equivalent Power (NEP) of photo-receiver with increase of *Мph* despite of growth APD's noise-to-signal ratio (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966). De‐ crease in NEP takes place until *Мph* becomes higher then certain value *Мph opt* above which noise of APD becomes dominant in photo-receiver (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980). Even at low leakage current and low noise density of external electron‐ ics circuit, avalanche multiplication of charge carriers may lead to degradation in NEP of photo-receiver due to decreasing tendency of signal-to-noise ratio dependence on APD's *Мph* under certain conditions (Artsis & Kholodnov, 1984). Moreover, excess factor of avalan‐ che noise (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966) may decrease with powering of avalanche process as, for example, in metal-dielectric-semiconductor ava‐ lanche structures, due to screening of electric field by free charge carriers (Kurochkin & Kholodnov 1999), (Kurochkin & Kholodnov 1999-2). Using results obtained in (Artsis & Kholodnov, 1984), (McIntyre, 1966), noise spectral density *SN* of *pwg* <sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg* <sup>+</sup> hetero‐ structure which performance is limited by tunnel current can be written as:

$$S\_N = 2 \times q \times A\_S \times M\_{ph}^2 \times \sum\_{i=1}^2 J\_{T,i}(V) F\_{\text{eff},i}(M\_{ph})\_\prime \tag{5}$$

where *q* – electron charge; *АS* – cross-section area of APD's structure; *Fef* ,*<sup>i</sup>* (*M ph* ) – effective noise factors (Artsis & Kholodnov, 1984) in wide-gap multiplication layer (*i* =1) and in ab‐ sorber (*i* =2); *JT* ,*<sup>i</sup>* (*V* ) – densities of primary tunnel currents in those layers, i.e. tunnel cur‐ rents which would exist in layers I and II in absence of multiplication of charge carriers due to avalanche impact generation. Comparison of two different APDs in order to determine which one is of better performance is reasonable only at same value of *Мph* . Expression (5) shows, that for preset gain of photocurrent, noise density is determined by values of pri‐ mary tunnel currents *IT* <sup>1</sup> = *JT* <sup>1</sup> × *AS* and *IT* <sup>2</sup> = *JT* <sup>2</sup> × *AS* (total primary tunnel current *IT* = *IT* <sup>1</sup>+ *IT* <sup>2</sup>). Distribution of electric field *Е*(*х*) that should be known to calculate parameters (4) and (5) of AHPD is obtained from Poisson equation and in layers I and II is determined by ex‐ pressions:

$$E(\mathbf{x}) = \left(E\_1 - \frac{qN\_1\mathbf{x}}{\varepsilon\_0\varepsilon\_1}\right) \times II\\_{\left(l\_1 - \infty\right)}\tag{6}$$

$$E(\mathbf{x}) = \left[ E\_2 - \frac{qN\_2}{\varepsilon\_0 \varepsilon\_2} (\mathbf{x} - W\_1) \right] \times \mathcal{U}\_- (W\_1 + l\_2 - \mathbf{x})\_\prime \tag{7}$$

Where

electric field of non-equilibrium SCR. If losses due to recombination are negligible (Sze, 1981), (Tsang, 1981), (Stillman, 1981), (Forrest et al, 1983), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), for example, at punch-through of absorber, then *SI* (*λ*) in operation mode is determined by well-known expression (Sze, 1981), (Tsang, 1981), (Stillman, 1981),

where *λ* in μm and value of quantum efficiency *η* is considered below. Photocurrent gaining and large drift velocity of charge carriers in SCR allow creating high-speed high-perform‐ ance photo-receivers with APDs as sensitive elements (Sze, 1981), (Tsang, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Woul, 1980). Reason is high noise density of external elec‐ tronics circuit at high frequencies or large leakage currents that results in decrease in Noise Equivalent Power (NEP) of photo-receiver with increase of *Мph* despite of growth APD's noise-to-signal ratio (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966). De‐

noise of APD becomes dominant in photo-receiver (Sze, 1981), (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980). Even at low leakage current and low noise density of external electron‐ ics circuit, avalanche multiplication of charge carriers may lead to degradation in NEP of photo-receiver due to decreasing tendency of signal-to-noise ratio dependence on APD's *Мph* under certain conditions (Artsis & Kholodnov, 1984). Moreover, excess factor of avalan‐ che noise (Tsang, 1981), (Filachev et al, 2011), (Woul, 1980), (McIntyre, 1966) may decrease with powering of avalanche process as, for example, in metal-dielectric-semiconductor ava‐ lanche structures, due to screening of electric field by free charge carriers (Kurochkin & Kholodnov 1999), (Kurochkin & Kholodnov 1999-2). Using results obtained in (Artsis &

crease in NEP takes place until *Мph* becomes higher then certain value *Мph*

Kholodnov, 1984), (McIntyre, 1966), noise spectral density *SN* of *pwg*

structure which performance is limited by tunnel current can be written as:

where *q* – electron charge; *АS* – cross-section area of APD's structure; *Fef* ,*<sup>i</sup>*

2 2

1 2 ( ) ( ), *<sup>N</sup> S ph T i ef i ph i S q A M J VF M*

noise factors (Artsis & Kholodnov, 1984) in wide-gap multiplication layer (*i* =1) and in ab‐

rents which would exist in layers I and II in absence of multiplication of charge carriers due to avalanche impact generation. Comparison of two different APDs in order to determine which one is of better performance is reasonable only at same value of *Мph* . Expression (5) shows, that for preset gain of photocurrent, noise density is determined by values of pri‐

=

, ,

(*V* ) – densities of primary tunnel currents in those layers, i.e. tunnel cur‐

= ´´ ´ ´å (5)

() () 1.24 *<sup>I</sup> ph S M* <sup>l</sup> l =hl ´ ´ (4)

*opt* above which

<sup>+</sup> hetero‐

(*M ph* ) – effective

<sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg*

(Filachev et al, 2011):

32 Photodiodes - From Fundamentals to Applications

sorber (*i* =2); *JT* ,*<sup>i</sup>*

$$E\_2 = \left(\frac{\varepsilon\_1}{\varepsilon\_2} E\_1 - \frac{qN\_1W\_1}{\varepsilon\_0\varepsilon\_2}\right) \times LI\_{-}(l\_1 - W\_1)\_{\prime} \tag{8}$$

$$\mathcal{U}\_{i} = \frac{\mathbb{E}\_{i}\mathbb{E}\_{0}}{q\mathcal{N}\_{i}}\boldsymbol{E}\_{i} \times \boldsymbol{\mathcal{U}}\_{+}(\mathcal{W}\_{i} - \boldsymbol{l}\_{i}) + \mathcal{W}\_{i} \times \boldsymbol{\mathcal{U}}\\_{\left(\boldsymbol{l}\_{i} - \boldsymbol{W}\_{i}\right)}\boldsymbol{\iota}\tag{9}$$

*U*−(*x*) and *U*+(*x*) – asymmetric unit stepwise functions (Korn G. & Korn T., 2000), *ε*0 – dielectric constant of vacuum, *ε*1 and *ε*2 – relative dielectric permittivity of *nwg* and *nng* layers (Fig. 1).

#### **3. Avalanche multiplication factors of charge carriers in p-n structures**

#### **3.1. Preliminary remarks: Avalanche breakdown field**

For successful development of semiconductor devices using effects of impact ionization and avalanche multiplication of charge carriers is necessary to know dependences of avalanche multiplication factors *M* (*V* ) of charge carriers in *p* −*n* structures on applied bias *Vb*. We need to know among them dependence of avalanche breakdown voltage *VBD* on parameters of *p* −*n* structure and distribution of electric field *E*(*x*) related to *VBD* dependence. Usual way to compute required dependencies is based on numerical processing of integral rela‐ tions (1) and (2) in each case. Impact ionization coefficients of electrons *α*(*E*) and holes *β*(*E*) depend drastically on electric field *E*. At the same time theoretical expressions for *α*(*E*) and *β*(*E*) include usually some adjustable parameters. Therefore, to avoid large errors in calcu‐ lating of multiplication factors, in computation of (1) and (2) are commonly used experimen‐ tal dependences for *α*(*E*) and *β*(*E*). Avalanche breakdown voltage *VBD* is defined as applied bias voltage at which multiplication factor of charge carriers tends to infinity (Sze, 1981), (Tsang, 1981), (Miller, 1955), (Grekhov & Serezhkin, 1980). Therefore, as seen from (2), breakdown condition is reduced to integral equation with *m*=1 where field distribution *E*(*x*) is determined by solving Poisson equation. Bias voltage at which breakdown condition *V* =*VBD* is satisfied can be calculated by method of successive approximations on computer. Thus, this method of determining *VBD* and, hence, *E*(*x*) at *V* =*VBD* requires time-consuming numerical calculations. The same applies to dependence *M* on *V* . Similar calculations were performed for a number of semiconductor structures for certain thicknesses of diode's base by which is meant high-resistivity side of *p* <sup>+</sup> −*n* homojunction or narrow-gap region of het‐ erojunction (Kim et al, 1981), (Stillman et al, 1983), (Vanyushin et al, 2007). In addition to great complexity, there are other drawbacks of this method of *M* (*V* ) and *VBD* determination – difficulties in application and lack of illustrative presentation of working results. Availa‐ bility of analytical, more or less universal expressions would be very helpful to analyze dif‐ ferent characteristics of devices with avalanche multiplication of charge carriers, for example, expression describing *E*(*x*), when we estimate tunnel currents in AHPDs. In this section are presented required analytical dependences (Osipov & Kholodnov, 1987), (Kho‐ lodnov, 1988), (Kholodnov, 1996-3). For quick estimate of breakdown voltage in abrupt *p* <sup>+</sup> −*n* homojunction or heterojunction is often used well-known Sze-Gibbons approximate expression (Sze, 1981), (Sze & Gibbons, 1966):

$$\mathbf{V}\_{\rm BD} = A\_V \times \mathbf{N}^{-(s-2)/s} \text{ \(\mathbf{V}\)}\tag{10}$$

*ε*<sup>0</sup> and *ε* − dielectric constant of vacuum and relative dielectric permittivity of base material; *q* − electron charge. Unless otherwise stated, in formulas (12) and (13) and below in sections

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

http://dx.doi.org/10.5772/50778

35

3.1-3.3 is used SI system of measurement units.

**Figure 2.** Schematic drawing of diode based on *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

(0), 2<sup>−</sup> *<sup>N</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*<sup>1</sup>

Formulas (10) and (11) cannot be used for reliable estimates of *VBD* and *EBD* in semiconduc‐ tor structures with thin enough base. Indeed, dependence of *VBD* on *N* is due to two factors.

avalanche breakdown voltage (**b**); 1− *N*<sup>1</sup> = *N*<sup>1</sup>

heterostructure (**a**) and distribution of electric field in it at

(0); *N*1 − dopant concentration *nwg* in wide-gap layer I

where

$$\mathbf{s} = \mathbf{8}\_{\prime} A\_{V} = \mathbf{6} \times 10^{13} \times \left( E\_{\mathcal{g}} / 1.1 \right)^{3/2} \text{ ,} \tag{11}$$

Gap *Eg* of semiconductor material forming diode's base and dopant concentration *N* in it are measured in eV and cm-3, properly. As follows from Poisson equation, voltage value giv‐ en by (10) corresponds to value of electric field at metallurgical boundary (*x* =0, Fig. 2) of *p* <sup>+</sup> −*n* junction:

$$E(0) = E\_{BD} = A \times N^{1/s} \tag{12}$$

where at *s* =8

$$A = \sqrt{\frac{1.2 \times q}{\varepsilon \varepsilon\_0}} \times \left(\frac{E\_\text{g}}{\varepsilon \varepsilon\_0}\right)^{3/4} \times 10^{10} \text{ }^\circ \text{C} \tag{13}$$

*ε*<sup>0</sup> and *ε* − dielectric constant of vacuum and relative dielectric permittivity of base material; *q* − electron charge. Unless otherwise stated, in formulas (12) and (13) and below in sections 3.1-3.3 is used SI system of measurement units.

bias voltage at which multiplication factor of charge carriers tends to infinity (Sze, 1981), (Tsang, 1981), (Miller, 1955), (Grekhov & Serezhkin, 1980). Therefore, as seen from (2), breakdown condition is reduced to integral equation with *m*=1 where field distribution *E*(*x*) is determined by solving Poisson equation. Bias voltage at which breakdown condition *V* =*VBD* is satisfied can be calculated by method of successive approximations on computer. Thus, this method of determining *VBD* and, hence, *E*(*x*) at *V* =*VBD* requires time-consuming numerical calculations. The same applies to dependence *M* on *V* . Similar calculations were performed for a number of semiconductor structures for certain thicknesses of diode's base by which is meant high-resistivity side of *p* <sup>+</sup> −*n* homojunction or narrow-gap region of het‐ erojunction (Kim et al, 1981), (Stillman et al, 1983), (Vanyushin et al, 2007). In addition to great complexity, there are other drawbacks of this method of *M* (*V* ) and *VBD* determination – difficulties in application and lack of illustrative presentation of working results. Availa‐ bility of analytical, more or less universal expressions would be very helpful to analyze dif‐ ferent characteristics of devices with avalanche multiplication of charge carriers, for example, expression describing *E*(*x*), when we estimate tunnel currents in AHPDs. In this section are presented required analytical dependences (Osipov & Kholodnov, 1987), (Kho‐ lodnov, 1988), (Kholodnov, 1996-3). For quick estimate of breakdown voltage in abrupt *p* <sup>+</sup> −*n* homojunction or heterojunction is often used well-known Sze-Gibbons approximate

( 2)/ , V, *s s V AN BD V*

Gap *Eg* of semiconductor material forming diode's base and dopant concentration *N* in it are measured in eV and cm-3, properly. As follows from Poisson equation, voltage value giv‐ en by (10) corresponds to value of electric field at metallurgical boundary (*x* =0, Fig. 2) of

1/ (0) ,

0 0

*<sup>A</sup>* ´ æ ö =´ ´ ç ÷ ee ee ç ÷ è ø

*s*

3/4

1.2 10 , *<sup>g</sup> <sup>q</sup> <sup>E</sup>*

10


( )3/2 <sup>13</sup> 8, 6 10 / 1.1 , *V g sA E* = =´ ´ (11)

*BD E E AN* = =´ (12)

(13)

expression (Sze, 1981), (Sze & Gibbons, 1966):

34 Photodiodes - From Fundamentals to Applications

where

*p* <sup>+</sup> −*n* junction:

where at *s* =8

**Figure 2.** Schematic drawing of diode based on *p* <sup>+</sup> −*n* −*n* <sup>+</sup> heterostructure (**a**) and distribution of electric field in it at avalanche breakdown voltage (**b**); 1− *N*<sup>1</sup> = *N*<sup>1</sup> (0), 2<sup>−</sup> *<sup>N</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*<sup>1</sup> (0); *N*1 − dopant concentration *nwg* in wide-gap layer I

Formulas (10) and (11) cannot be used for reliable estimates of *VBD* and *EBD* in semiconduc‐ tor structures with thin enough base. Indeed, dependence of *VBD* on *N* is due to two factors. First, as follows from Poisson equation, the larger *N* the steeper the field *E*(*x*) decreases into the depth from *x* =0 comparing to value *E*<sup>1</sup> =*E*(0) (Fig. 1b). Second, value of electric field *E*<sup>1</sup> =*E*(0) at *V* =*VBD* falls with decreasing of *N* due to decreasing of |∇*E* | in SCR. Drop of *E*(*x*) becomes more weaker with decreasing of *N* (Fig. 1b), therefore, at preset base's thick‐ ness *W* , initiation of avalanche process will require fewer and fewer field intensity *E*1. At sufficiently low concentration *N* , the lower the thicker *W* will be, variation of electric field *E*(*x*) on the length of base is so insignificant that probability of impact ionization becomes practically the same in any point of base. It means that breakdown voltage *VBD* and field *EBD* are independent on *N* and at the same time are dependent on *W* , moreover, the thinner *W* then, evidently, the higher *EBD*. So using of formulas (10) and (11) at any values of *W* , that done in many publications, contradicts with above conclusion. In next section 3.2 will be shown that value of breakdown field of stepwise *p* <sup>+</sup> −*n* junction in a number of semicon‐ ductor structures can be estimated by following formula:

$$E\_{BD}(N\_\prime \mathcal{W}) = E\_{BD}(0, \mathcal{W}) \times \left[1 + \frac{N}{\tilde{N}(\mathcal{W})}\right]^{1/s} \tag{14}$$

*Vb* applied to heterostructure, electric field *E* =*E*<sup>2</sup> in narrow-gap layer on *nwg*/*nng* heterojunc‐ tion (Fig. 2) reaches avalanche breakdown field *E*2*BD* in this layer earlier than electric field *E*<sup>1</sup> on metallurgical boundary (*<sup>х</sup>* =0) of *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction becomes equal to breakdown field *E*1*BD* in wide-gap *nwg* layer. This is due to the fact that at small values of *W*1 and *N*1 variation of field *E*(*х*) within wide-gap layer is insignificant and probability of impact ionization in nar‐ row-gap layer is much higher than in wide-gap. If, however, *W*1 and *N*1, *N*2 are large enough and *W*<sup>2</sup> thin enough, then avalanche process is developed in wide-gap *nwg* layer on‐ ly. For these values of thicknesses and concentrations electric field *E*<sup>1</sup> reaches value *E*1*BD* earlier than *E*2 – value *E*2*BD*. Because of significant decreasing of electric field *E*(*х*) in *nwg* layer with increasing distance from *х* =0, field *E*2 remains smaller *E*2*BD* despite the fact that band-gap *Eg*1 in *nwg* layer is wider than band-gap *Eg*2 in *nng* layer. Distribution of electric field *E*(*x*) in *nwg* and *nng* layers of considered heterostructure is obtained by solving Poisson equation as defined by (6)-(9). When avalanche breakdown voltage *VBD* is applied to struc‐ ture, then either *E*<sup>1</sup> =*E*1*ВD*(*N*1, *W*1) or *E*<sup>2</sup> =*E*2*ВD*(*N*2, *W*2). In section 3.1 is noted that at low

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

avalanche breakdown fields *EiBD*(*Ni*

mula (12) should be modified so that when *N* →0 then breakdown field *EBD* tends to some non-zero value. It would seem that it is enough to add some independent on *N* constant to right side of (12). It is easy to see that such modification of formula (12) leads to contradic‐ tion. To verify that let's consider situation when avalanche multiplication of charge carriers occurs in *nwg* layer, i.e. *E*1 is close to *E*1*ВD* and multiplication factor of holes *Мр* (1) is fixed. Then, with increasing concentration *N*1, field *EI* (*W*1) (Fig. 2b) shall be monotonically falling function of *N*1. Indeed, with increasing *N*1, field *E*1*ВD* and |∇*EI* (*x*)| are increasing also. In‐ creasing |∇*EI* | must be such that when *x* became larger some value *x*¯ then value *EI* (*x*) has decreased (Fig. 2b). Otherwise, field *E*(*x*) would increase throughout SCR that reasonably would lead to growth of *Мр*. This is evident from (1) and (2). On the other hand, adding con‐ stant to right side of expression (12) does not change ∂*E*1*BD* / ∂ *N*<sup>1</sup> and therefore results in, as follows from (6) and (9), non-monotonic dependence *EI* (*W*1) on *N*1. Equation (14) which can

1/ ( , ) [ ( )] *<sup>s</sup>*

(0, ) ( )

*E W N W*

*iBD i*

*A* é ù <sup>=</sup> ê ú ê ú ë û

*i*

does not lead to that and other contradictions, From (17) follows that:

*i i*

*iBD i i i i i i E NW A N NW* =´ + % (17)

% (18)

*s*

, *Wi*

, where *i* =1, 2. To account for this effect, for‐

) should not depend on

http://dx.doi.org/10.5772/50778

37

enough concentrations *Ni*

and have definite value depending on *Wi*

be rewritten for each of *nwg* and *nng* layers as:

*Ni*

where

$$E\_{BD}(0, W) = A \times \left(\frac{\varepsilon \varepsilon\_0 \times A}{s \times q \times W}\right)^{1/(s-1)},\tag{15}$$

$$
\tilde{N}(\mathcal{W}) = \left(\frac{\varepsilon \varepsilon\_0 \times A}{s \times q \times \mathcal{W}}\right)^{s/(s-1)}.\tag{16}
$$

It seen from expression (14) that at *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*˜(*<sup>W</sup>* ) electric field of avalanche breakdown *EBD* is practically independent on dopant concentration *N* in diode's base.

#### **3.2. Avalanche breakdown field**

Consider *pwg* <sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg* + heterostructure (Fig. 2). Symbols *nwg* and *nng* indicate to un‐ equal, in general, doping of high-resistivity layers of structure. Denote as *W*1, *W*2 and *N*1, *N*2 thicknesses of *nwg* and *nng* layers and dopant concentrations in them, properly. Case *<sup>W</sup>*<sup>2</sup> =0 corresponds to diode formed on homogeneous *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup> structure. Let values *N*1 and *W*<sup>1</sup> such that upon applying avalanche breakdown voltage *VBD* to structure, SCR penetrates into narrow-gap *nng* layer (Fig. 2). When *W*1 and *N*1, *N*2 are small enough and *W*2 is thick enough then avalanche process develops in *nng* layer. In other words, with increasing bias First, as follows from Poisson equation, the larger *N* the steeper the field *E*(*x*) decreases into the depth from *x* =0 comparing to value *E*<sup>1</sup> =*E*(0) (Fig. 1b). Second, value of electric field *E*<sup>1</sup> =*E*(0) at *V* =*VBD* falls with decreasing of *N* due to decreasing of |∇*E* | in SCR. Drop of *E*(*x*) becomes more weaker with decreasing of *N* (Fig. 1b), therefore, at preset base's thick‐ ness *W* , initiation of avalanche process will require fewer and fewer field intensity *E*1. At sufficiently low concentration *N* , the lower the thicker *W* will be, variation of electric field *E*(*x*) on the length of base is so insignificant that probability of impact ionization becomes practically the same in any point of base. It means that breakdown voltage *VBD* and field *EBD* are independent on *N* and at the same time are dependent on *W* , moreover, the thinner *W* then, evidently, the higher *EBD*. So using of formulas (10) and (11) at any values of *W* , that done in many publications, contradicts with above conclusion. In next section 3.2 will be shown that value of breakdown field of stepwise *p* <sup>+</sup> −*n* junction in a number of semicon‐

( , ) (0, ) 1 , ( )

<sup>0</sup> (0, ) ,

<sup>0</sup> ( ) . *s s <sup>A</sup> N W sqW* - æ ö ee ´ <sup>=</sup> ç ÷ ´ ´ è ø

It seen from expression (14) that at *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*˜(*<sup>W</sup>* ) electric field of avalanche breakdown *EBD* is

equal, in general, doping of high-resistivity layers of structure. Denote as *W*1, *W*2 and *N*1, *N*2 thicknesses of *nwg* and *nng* layers and dopant concentrations in them, properly. Case

*W*<sup>1</sup> such that upon applying avalanche breakdown voltage *VBD* to structure, SCR penetrates into narrow-gap *nng* layer (Fig. 2). When *W*1 and *N*1, *N*2 are small enough and *W*2 is thick enough then avalanche process develops in *nng* layer. In other words, with increasing bias

*sqW* - æ ö ee ´ = ´ ç ÷ ´ ´ è ø

= ´+ ê ú

*<sup>N</sup> E NW E W*

*<sup>A</sup> E WA*

1/

*N W* é ù

1/( 1)

*s*

/( 1)

*s*

% (16)

+ heterostructure (Fig. 2). Symbols *nwg* and *nng* indicate to un‐

structure. Let values *N*1 and

ë û % (14)

(15)

ductor structures can be estimated by following formula:

36 Photodiodes - From Fundamentals to Applications

*BD*

practically independent on dopant concentration *N* in diode's base.

*<sup>W</sup>*<sup>2</sup> =0 corresponds to diode formed on homogeneous *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup>

**3.2. Avalanche breakdown field**

<sup>+</sup> <sup>−</sup>*nwg* <sup>−</sup>*nng* <sup>−</sup>*nwg*

Consider *pwg*

where

*BD BD*

*Vb* applied to heterostructure, electric field *E* =*E*<sup>2</sup> in narrow-gap layer on *nwg*/*nng* heterojunc‐ tion (Fig. 2) reaches avalanche breakdown field *E*2*BD* in this layer earlier than electric field *E*<sup>1</sup> on metallurgical boundary (*<sup>х</sup>* =0) of *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction becomes equal to breakdown field *E*1*BD* in wide-gap *nwg* layer. This is due to the fact that at small values of *W*1 and *N*1 variation of field *E*(*х*) within wide-gap layer is insignificant and probability of impact ionization in nar‐ row-gap layer is much higher than in wide-gap. If, however, *W*1 and *N*1, *N*2 are large enough and *W*<sup>2</sup> thin enough, then avalanche process is developed in wide-gap *nwg* layer on‐ ly. For these values of thicknesses and concentrations electric field *E*<sup>1</sup> reaches value *E*1*BD* earlier than *E*2 – value *E*2*BD*. Because of significant decreasing of electric field *E*(*х*) in *nwg* layer with increasing distance from *х* =0, field *E*2 remains smaller *E*2*BD* despite the fact that band-gap *Eg*1 in *nwg* layer is wider than band-gap *Eg*2 in *nng* layer. Distribution of electric field *E*(*x*) in *nwg* and *nng* layers of considered heterostructure is obtained by solving Poisson equation as defined by (6)-(9). When avalanche breakdown voltage *VBD* is applied to struc‐ ture, then either *E*<sup>1</sup> =*E*1*ВD*(*N*1, *W*1) or *E*<sup>2</sup> =*E*2*ВD*(*N*2, *W*2). In section 3.1 is noted that at low enough concentrations *Ni* avalanche breakdown fields *EiBD*(*Ni* , *Wi* ) should not depend on *Ni* and have definite value depending on *Wi* , where *i* =1, 2. To account for this effect, for‐ mula (12) should be modified so that when *N* →0 then breakdown field *EBD* tends to some non-zero value. It would seem that it is enough to add some independent on *N* constant to right side of (12). It is easy to see that such modification of formula (12) leads to contradic‐ tion. To verify that let's consider situation when avalanche multiplication of charge carriers occurs in *nwg* layer, i.e. *E*1 is close to *E*1*ВD* and multiplication factor of holes *Мр* (1) is fixed. Then, with increasing concentration *N*1, field *EI* (*W*1) (Fig. 2b) shall be monotonically falling function of *N*1. Indeed, with increasing *N*1, field *E*1*ВD* and |∇*EI* (*x*)| are increasing also. In‐ creasing |∇*EI* | must be such that when *x* became larger some value *x*¯ then value *EI* (*x*) has decreased (Fig. 2b). Otherwise, field *E*(*x*) would increase throughout SCR that reasonably would lead to growth of *Мр*. This is evident from (1) and (2). On the other hand, adding con‐ stant to right side of expression (12) does not change ∂*E*1*BD* / ∂ *N*<sup>1</sup> and therefore results in, as follows from (6) and (9), non-monotonic dependence *EI* (*W*1) on *N*1. Equation (14) which can be rewritten for each of *nwg* and *nng* layers as:

$$E\_{i\text{BD}}(\text{N}\_{i\text{'}}\text{W}\_{i}) = A\_{i} \times \text{[N}\_{i} + \tilde{\text{N}}\_{i}(\text{W}\_{i})]^{1/s} \tag{17}$$

does not lead to that and other contradictions, From (17) follows that:

$$
\tilde{N}\_i(\mathcal{W}\_i) = \left[\frac{E\_{iBD}(0, \mathcal{W}\_i)}{A\_i}\right]^s \tag{18}
$$

To determine dependences *EiBD*(0, *Wi* ), let's consider behavior of *EI* (*W*1) when parameters of heterostructure *N*1, *N*2 and *W*2 are varying. From (6)-(9), (17) and (18) we find that when value

$$
\Delta = N\_2 + \tilde{N}\_2(\mathcal{W}\_2) - \left(\frac{\varepsilon\_1 \times A\_1}{\varepsilon\_2 \times A\_2}\right)^s \times \tilde{N}\_1(\mathcal{W}\_1) > 0 \tag{19}
$$

where

*EВ<sup>D</sup>* (*InP*)

– in μm, respectively.

**3.3. Avalanche breakdown voltage**

structure is given by expressions

i.e. when diode's base is not punch-through and

i.e. when diode base is punch-through. In expression (28)

*BD BD*

namely when

*EBD*(0, *W* )= *X<sup>ε</sup>*

3/7 <sup>×</sup> *Xg*

(0, *<sup>W</sup>* )=4.3×10<sup>5</sup> <sup>×</sup>*<sup>W</sup>* <sup>−</sup>1/7, *<sup>V</sup>* / *cm*; *<sup>N</sup>*˜

<sup>−</sup>6/7 <sup>×</sup>*EВ<sup>D</sup>* (*InP*)

(Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011) are as follows:

(0, *<sup>W</sup>* ); *<sup>N</sup>*˜(*<sup>W</sup>* )= *<sup>X</sup><sup>ε</sup>*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

And values for *InP* semiconductor widely used for manufacturing of AHPDs (Tsang, 1981),

*X<sup>ε</sup>* =12.4 / *ε*, *Xg* =1.35 / *Eg* and gap *Eg* in diode's base is measured in eV and its thickness *W*

It follows from expressions (6)-(9) and (14)-(16) that breakdown voltage *VBD* for *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup>

<sup>0</sup> <sup>2</sup> ( ) ( ) <sup>1</sup> <sup>1</sup> <sup>1</sup> , if <sup>2</sup>

*q N N N*

1/ <sup>1</sup> ( , ) (0, ) 1 () 2 () , if

<sup>2</sup> <sup>1</sup> <sup>0</sup> <sup>1</sup> (0, ) *<sup>s</sup> <sup>s</sup> <sup>s</sup> BD <sup>A</sup> V WA <sup>W</sup>*

*s q* - - - æ ö ee ´ =´ ´ ç ÷ ´ è ø

Value of parameter *θ* is defined from equation *θ* =*s* ×(1 + *θ*)1/*s* and with good degree of accu‐ racy it equals to *ss*/(*s*−1). Because *θ* >>1, therefore expression (27) practically coincides with formula (10), i.e. *VВ<sup>D</sup>* of diode with thick base is independent on its thickness *W* . For diodes with thin base formed on semiconductors with parameters satisfying relations (11) and (14),

*N NN V NW V W*

= ´+ ´ º ´+ ´ < êú êú <sup>q</sup> ëû ëû

*s s BD <sup>V</sup> N W N W <sup>N</sup> V A N A <sup>N</sup>*


2/ 2 2 2/

% %% (27)

*s s s s*

*s*

ì ü ï ï é ù = ´+ - í ý ê ú >

1

*NW s NW N*

ë û ´ <sup>q</sup> ï ï î þ

<sup>−</sup>4/7 <sup>×</sup> *Xg*

−6/7*N*˜

*InP*(*<sup>W</sup>* )=3.4×10<sup>15</sup> <sup>×</sup>*<sup>W</sup>* <sup>−</sup>8/7, *cm*−<sup>3</sup> (26)

% % % (28)

(29)

*InP*(*W* ) (25)

http://dx.doi.org/10.5772/50778

39

then avalanche breakdown is controlled by *nwg* layer. It means that

$$E\_I(\mathcal{W}\_1) = E\_{1BD}(\mathcal{N}\_{1'}, \mathcal{W}\_1) - \frac{qN\_1 \times \mathcal{W}\_1}{\varepsilon\_0 \varepsilon\_1} \tag{20}$$

If, however, *Δ* <0 then avalanche breakdown is controlled by *nwg*/*nng* heterojunction, i.e.

$$E\_I(\mathcal{W}\_1) = \frac{\varepsilon\_2}{\varepsilon\_1} E\_{2BD}(\mathcal{N}\_{2'}, \mathcal{W}\_2) \tag{21}$$

From (17)-(21) we obtain that

$$\frac{\partial \mathbb{E}\_I(W\_I)}{\partial \mathbf{N}\_1} \Big|\_{N\_1 \to 0} = \begin{cases} \frac{A\_1}{s} \times \tilde{N}\_1^{(1/s)-1} - \frac{q \times W\_1}{\mathbb{E}\_0 \mathbb{E}\_1}, \text{ at } \Delta > 0\\ 0 & \text{, at } \Delta < 0 \end{cases} \tag{22}$$

Formulas (15) and (16) follow from expressions (18), (19) and requirement (23)

$$\lim\_{\Delta \to -0} \left\{ \frac{\partial E\_I(\mathcal{W}\_I)}{\partial \mathcal{N}\_1} \Big|\_{\mathcal{N}\_1 \to 0} \right\} = \lim\_{\Delta \to +0} \left\{ \frac{\partial E\_I(\mathcal{W}\_I)}{\partial \mathcal{N}\_1} \Big|\_{\mathcal{N}\_1 \to 0} \right\} \tag{23}$$

which means smoothness of field dependence *E*(*x*) in real heterostructures, where parame‐ ters are varying continuously. Particularly, in semiconductors for which relations (11) and (13) are valid, breakdown field at metallurgical boundary of *p* <sup>+</sup> −*n* junction (or at heterojunc‐ tion boundary, in narrow-gap layer of heterojunction, including isotype) can be described by formula

$$E\_{BD}(N\_\prime \mathcal{W}) = E\_{BD}(0, \mathcal{W}) \times \left[1 + \frac{N}{\tilde{N}(\mathcal{W})}\right]^{1/8} \tag{24}$$

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 39

where

To determine dependences *EiBD*(0, *Wi*

38 Photodiodes - From Fundamentals to Applications

From (17)-(21) we obtain that

by formula

value

), let's consider behavior of *EI* (*W*1) when parameters

% % (19)

*E*2*ВD*(*N*2, *W*2) (21)

(20)

(22)

(23)

of heterostructure *N*1, *N*2 and *W*2 are varying. From (6)-(9), (17) and (18) we find that when

1 1 2 22 1 1 2 2 ( ) ( )0 *s*

*A*

*qN*<sup>1</sup> ×*W*<sup>1</sup> *ε*0*ε*<sup>1</sup>

*<sup>A</sup> N NW N W*

If, however, *Δ* <0 then avalanche breakdown is controlled by *nwg*/*nng* heterojunction, i.e.

1 1 (1/ ) 1 1 0 0 1


0 , at 0

( ) , at 0

1 1 0 0 0 0 1 1

*E W E W N N* ® ® D®- D®+ ì üì ü ï ïï ï ¶ ¶ í ýí ý <sup>=</sup> ï ïï ï ¶ ¶ î þî þ

( , ) (0, ) 1 ( ) *BD BD <sup>N</sup> E NW E W*

= ´+ ê ú

*N N*

which means smoothness of field dependence *E*(*x*) in real heterostructures, where parame‐ ters are varying continuously. Particularly, in semiconductors for which relations (11) and (13) are valid, breakdown field at metallurgical boundary of *p* <sup>+</sup> −*n* junction (or at heterojunc‐ tion boundary, in narrow-gap layer of heterojunction, including isotype) can be described

1/8

ë û % (24)

*N W* é ù

<sup>ì</sup> ´ ¶ <sup>ï</sup> ´ - D> <sup>=</sup> <sup>í</sup> e e ¶ <sup>ï</sup> D < <sup>î</sup>

%

*s*

*ε*2 *ε*1

*A qW E W <sup>N</sup> s*

Formulas (15) and (16) follow from expressions (18), (19) and requirement (23)

( ) ( ) lim *I I* lim *I I*

æ ö e ´ D= + - ´ > ç ÷ e ´ è ø

then avalanche breakdown is controlled by *nwg* layer. It means that

*EI* (*W*1)=

1

®

*N*

1

*I I*

*N*

*EI* (*W*1)=*E*1*ВD*(*N*1, *W*1)−

$$\mathbb{E}\_{\rm BD}(\mathbf{0}, \mathcal{W}) = \mathbf{X}\_{\varepsilon}^{\ 37} \times \mathbf{X}\_{\mathcal{g}}^{-67} \times \mathbb{E}\_{\rm BD}^{(\rm ImP)}(\mathbf{0}, \mathcal{W});\\\widetilde{\mathcal{N}}(\mathcal{W}) = \mathbf{X}\_{\varepsilon}^{-47} \times \mathbf{X}\_{\mathcal{g}}^{-67} \widetilde{\mathcal{N}}\_{\rm ImP}(\mathcal{W}) \tag{25}$$

And values for *InP* semiconductor widely used for manufacturing of AHPDs (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011) are as follows:

$$E\_{BD}^{(\text{InP})}(0,\text{W}) = 4.3 \times 10^5 \times \text{W}^{-1/7}, \text{ V/cm;} \,\,\, \widetilde{\text{N}}\_{\text{InP}}(\text{W}) = 3.4 \times 10^{15} \times \text{W}^{-8/7}, \text{ cm}^{-3} \tag{26}$$

*X<sup>ε</sup>* =12.4 / *ε*, *Xg* =1.35 / *Eg* and gap *Eg* in diode's base is measured in eV and its thickness *W* – in μm, respectively.

#### **3.3. Avalanche breakdown voltage**

It follows from expressions (6)-(9) and (14)-(16) that breakdown voltage *VBD* for *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup> structure is given by expressions

$$V\_{BD} = \frac{\varepsilon \varepsilon\_0}{2q} A^2 \times \left[1 + \frac{\tilde{N}(W)}{N}\right]^{2/s} \times N^{-\frac{s-2}{s}} = A\_V \times \left[1 + \frac{\tilde{N}(W)}{N}\right]^{2/s} \times N^{-\frac{s-2}{s}}, \text{ if } \frac{\tilde{N}}{N} < \frac{1}{\Theta} \tag{27}$$

i.e. when diode's base is not punch-through and

$$V\_{BD}(N, W) = V\_{BD}(0, W) \times \left[ \left[ 1 + \frac{N}{\tilde{N}(W)} \right]^{1/s} - \frac{N}{2s \times \tilde{N}(W)} \right], \text{ if } \frac{\tilde{N}}{N} > \frac{1}{\Theta} \tag{28}$$

i.e. when diode base is punch-through. In expression (28)

$$\mathcal{W}\_{BD}(0, W) = A \times \left(\frac{\varepsilon \varepsilon\_0 \times A}{s \times q}\right)^{\frac{1}{s-1}} \times \mathcal{W}^{\frac{s-2}{s-1}} \tag{29}$$

Value of parameter *θ* is defined from equation *θ* =*s* ×(1 + *θ*)1/*s* and with good degree of accu‐ racy it equals to *ss*/(*s*−1). Because *θ* >>1, therefore expression (27) practically coincides with formula (10), i.e. *VВ<sup>D</sup>* of diode with thick base is independent on its thickness *W* . For diodes with thin base formed on semiconductors with parameters satisfying relations (11) and (14), namely when

$$\mathcal{W} \le \tilde{\mathcal{W}}(\mathcal{N}) = 9 \times X\_{\varepsilon}^{-1/2} \times X\_{\mathcal{g}}^{-3/4} \times \left(\frac{3 \times 10^{15}}{N}\right)^{7/8} \tag{30}$$

example, owing to big role of phonons in formation of distribution functions. It is shown in this section that for number of semiconductors the following approximate relation is satis‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

10 ( ) ( ) ( , ( ), ( )) 9 10 ( ) , ( ) ln

æ ö a -b <sup>e</sup> a b º´ ´ ´ = ´ » º ç ÷ ç ÷ é ù <sup>a</sup> è ø ê ú ë û <sup>b</sup>

Where: *ε* – relative dielectric permittivity, and gap *Eg*, electric field *E*, *α* and *β* are measured

0

where *ε*<sup>0</sup> – dielectric constant of vacuum; *ε* – relative dielectric permittivity of base material; *<sup>q</sup>* – electron charge; *s* and *A* – constants defining dependence of electric field *EBD* <sup>≈</sup> *<sup>A</sup>*<sup>×</sup> *<sup>N</sup>* 1/*<sup>s</sup>* at metallurgical boundary (*x* =0) of abrupt *p* <sup>+</sup> −*n* junction on dopant concentration *N* in base for avalanche breakdown in thick *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (Sections 3.1-3.3, (Sze, 1981), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966)). When condition (34) is satisfied then avalanche

<sup>1</sup> <sup>0</sup> ( ) *<sup>s</sup>*

*sqW* æ ö ee - » ´ ç ÷ è ø

3/4

*q*

10

And, under these conditions, variation of electric field *Е*(*x*) along length of base *W* is so in‐ significant that probability of impact ionization is practically the same in any point of base of considered structure. For many semiconductors including *Ge*, *Si*, *GaAs*, *InP*, *GaP* rela‐ tions given below are valid (Sze, 1981), (Kholodnov, 1988-2), (Kholodnov, 1996), (Sze & Gib‐

*<sup>A</sup> EWA*

0 1.2 8, 10 , <sup>11</sup> *<sup>g</sup> q E*

æ ö == ´ ´ ç ÷ ee ç ÷ è ø

*BD*

*s A*

In this case as it follows from (34) and (35)

0

*<sup>s</sup> <sup>A</sup> WW N qs*

<= ´ ee

*E E ZE E E CE Z Z*

2

To derive relation (33) let's consider thin *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

sistivity base layer *W* satisfies to inequality

<sup>7</sup> 5 3

*E E E*

( )

*E*

1


*s*

1

0 0 6

*g*

http://dx.doi.org/10.5772/50778

structure in which thickness of high-re‐

(34)

(33)

41

(35)

(36)

fied (Kholodnov, 1988)

in eV, V/cm and 1/cm, properly.

breakdown field can be written as

bons, 1966)

breakdown voltage of diode depends on *W* and *N* as follows

$$\begin{aligned} V\_{BD}(\mathbf{N}\_{\prime}\mathbf{W}) &= V\_{BD}(\mathbf{0}, \mathbf{W}) \times \\ \times \left[ \left( 1 + X\_{\varepsilon}^{4/7} \times X\_{\mathcal{g}}^{6/7} \times W^{8/7} \times \frac{\mathbf{N}}{2.65 \times 10^{15}} \right)^{1/8} - X\_{\varepsilon}^{4/7} \times X\_{\mathcal{g}}^{6/7} \times W^{8/7} \times \frac{\mathbf{N}}{4.24 \times 10^{18}} \right] \end{aligned} \tag{31}$$

where

$$V\_{BD}(0, W) = 43.1 \times X\_{\varepsilon}^{3/7} \times X\_{g}^{-6/7} \times \mathcal{W}^{6/7} \tag{32}$$

In expressions (30)-(32) *X<sup>ε</sup>* =12.4 / *ε*, *Xg* =1.35 / *Eg* and gap *Eg* in base, dopant concentration in it *N* and thickness *W* is measured in eV, cm-3 and μm, respectively.

Avalanche breakdown voltage of double heterostructure discussed in Section 4 (Fig. 1) de‐ pends on relations between fundamental parameters of materials of *nwg* and *nng* layers, their thicknesses and doping, and is determined, as follows from (6)-(9) and (14)-(16), by different combinations (with slight modification) of expressions (27)-(29) for these layers of hetero‐ structure.

#### **3.4. About correlation between impact ionization coefficients of electrons and holes**

One of main goals of many experimental and theoretical studies of impact ionization phe‐ nomenon in semiconductors is to determine impact ionization coefficients of electrons *α*(*E*) and holes *β*(*E*) as functions of electric field *E* (Sze, 1981), (Tsang, 1985), (Grekhov & Serezh‐ kin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Parameters of some semiconductor devices, for example, APDs (Sze, 1981), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (Stillman & Wolf, 1977) depend significantly on ratio *K*(*E*)=*β*(*E*) / *α*(*E*). Performance of APD can be calculated on computer if *α*(*E*) and *β*(*E*) are known (Sze, 1981), (Tsang, 1985), (Fila‐ chev et al, 2011), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Dependences *α*(*E*) and *β*(*E*) are known, with greater or lesser degree of accuracy, for a number of semiconductors (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Still‐ man & Wolf, 1977), (Dmitriev et al, 1987). However in works concerned determination of impact ionization coefficients the problem of interrelation between *α*(*E*) and *β*(*E*) has never been put. Even so, laws of conservation of energy and quasi-momentum in the act of impact ionization are maintained mainly by electron-hole subsystem of semiconductor (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Dmitriev et al, 1987). Therefore, there is a reason to hy‐ pothesize some correlation between *α*(*E*) and *β*(*E*), although perhaps not quite unique, for example, owing to big role of phonons in formation of distribution functions. It is shown in this section that for number of semiconductors the following approximate relation is satis‐ fied (Kholodnov, 1988)

7/8 <sup>15</sup>

è ø % (30)

15 18

<sup>e</sup> =´ ´ ´ (32)

(31)

2.65 10 4.24 10


1/2 3/4 3 10 ()9 *W WN X Xg <sup>N</sup>* - e æ ö ´ £ =´ ´ ´ ç ÷

4/7 6/7 8/7 4/7 6/7 8/7

*g g*

in it *N* and thickness *W* is measured in eV, cm-3 and μm, respectively.

1/8

*N N XXW* e e *XXW*

3/7 6/7 6/7 (0, ) 43.1 *V W XX W BD <sup>g</sup>*

In expressions (30)-(32) *X<sup>ε</sup>* =12.4 / *ε*, *Xg* =1.35 / *Eg* and gap *Eg* in base, dopant concentration

Avalanche breakdown voltage of double heterostructure discussed in Section 4 (Fig. 1) de‐ pends on relations between fundamental parameters of materials of *nwg* and *nng* layers, their thicknesses and doping, and is determined, as follows from (6)-(9) and (14)-(16), by different combinations (with slight modification) of expressions (27)-(29) for these layers of hetero‐

**3.4. About correlation between impact ionization coefficients of electrons and holes**

One of main goals of many experimental and theoretical studies of impact ionization phe‐ nomenon in semiconductors is to determine impact ionization coefficients of electrons *α*(*E*) and holes *β*(*E*) as functions of electric field *E* (Sze, 1981), (Tsang, 1985), (Grekhov & Serezh‐ kin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Parameters of some semiconductor devices, for example, APDs (Sze, 1981), (Filachev et al, 2011), (Artsis & Kholodnov, 1984), (Stillman & Wolf, 1977) depend significantly on ratio *K*(*E*)=*β*(*E*) / *α*(*E*). Performance of APD can be calculated on computer if *α*(*E*) and *β*(*E*) are known (Sze, 1981), (Tsang, 1985), (Fila‐ chev et al, 2011), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev et al, 1987). Dependences *α*(*E*) and *β*(*E*) are known, with greater or lesser degree of accuracy, for a number of semiconductors (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Still‐ man & Wolf, 1977), (Dmitriev et al, 1987). However in works concerned determination of impact ionization coefficients the problem of interrelation between *α*(*E*) and *β*(*E*) has never been put. Even so, laws of conservation of energy and quasi-momentum in the act of impact ionization are maintained mainly by electron-hole subsystem of semiconductor (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Dmitriev et al, 1987). Therefore, there is a reason to hy‐ pothesize some correlation between *α*(*E*) and *β*(*E*), although perhaps not quite unique, for

é ù æ ö ´+ ´ ´ ´ - ´ ´ ´ ê ú ç ÷ è ø ´ ´ ë û

breakdown voltage of diode depends on *W* and *N* as follows

( , ) (0, )

= ´

*BD BD*

*V NW V W*

40 Photodiodes - From Fundamentals to Applications

1

where

structure.

$$\mathbb{E}\left(\mathbb{E},\alpha(E),\mathbb{B}(E)\right) = 9 \times 10^2 \times \left(\frac{10^5}{E}\right)^7 \times \frac{\alpha(E) - \mathbb{B}(E)}{\ln\left[\frac{\alpha(E)}{\mathbb{B}(E)}\right]} = \mathbb{C}(E) \times Z\_0 \approx Z\_0 \equiv \frac{\mathbb{E}^3}{E\_g^6} \tag{33}$$

Where: *ε* – relative dielectric permittivity, and gap *Eg*, electric field *E*, *α* and *β* are measured in eV, V/cm and 1/cm, properly.

To derive relation (33) let's consider thin *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure in which thickness of high-re‐ sistivity base layer *W* satisfies to inequality

$$\mathcal{W} < \mathcal{W}\_0 = \frac{A\varepsilon\varepsilon\_0}{qs} \times \mathcal{N}^{\frac{1-s}{s}} \tag{34}$$

where *ε*<sup>0</sup> – dielectric constant of vacuum; *ε* – relative dielectric permittivity of base material; *<sup>q</sup>* – electron charge; *s* and *A* – constants defining dependence of electric field *EBD* <sup>≈</sup> *<sup>A</sup>*<sup>×</sup> *<sup>N</sup>* 1/*<sup>s</sup>* at metallurgical boundary (*x* =0) of abrupt *p* <sup>+</sup> −*n* junction on dopant concentration *N* in base for avalanche breakdown in thick *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (Sections 3.1-3.3, (Sze, 1981), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966)). When condition (34) is satisfied then avalanche breakdown field can be written as

$$E\_{BD}(\mathcal{W}) \approx A \times \left(\frac{A \varepsilon \varepsilon\_0}{s \eta \mathcal{W}}\right)^{\frac{1}{s-1}} \tag{35}$$

And, under these conditions, variation of electric field *Е*(*x*) along length of base *W* is so in‐ significant that probability of impact ionization is practically the same in any point of base of considered structure. For many semiconductors including *Ge*, *Si*, *GaAs*, *InP*, *GaP* rela‐ tions given below are valid (Sze, 1981), (Kholodnov, 1988-2), (Kholodnov, 1996), (Sze & Gib‐ bons, 1966)

$$s = 8, A = \sqrt{\frac{1.2\eta}{\varepsilon\varepsilon\_0}} \times \left(\frac{E\_{\frac{\alpha}{\xi}}}{11\eta}\right)^{3/4} \times 10^{10} \text{ }\tag{36}$$

In this case as it follows from (34) and (35)

$$\mathcal{W}\_0 = \frac{1}{4} \times \sqrt{\varepsilon} \times E\_{\mathcal{g}}^{3/4} \times \left(\frac{3 \times 10^{15}}{N}\right)^{7/8} \text{ \AA} \tag{37}$$

7 6

e

(42)

43

(44)

*<sup>ε</sup>* <sup>×</sup> *Eion* <sup>6</sup> . (43)

http://dx.doi.org/10.5772/50778

< *x* < *L <sup>n</sup>* (see inset in Fig. 3), is accepted

3 7/6 14 min max 7 3 5 10 10 , *R g ion*

*E W W*

*g R R*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

*E E E* - e l ´ ´ » << l << » ´

<sup>≈</sup> *EBD*(*W*max)≈*E*min <sup>&</sup>lt;*<sup>E</sup>* <sup>&</sup>lt;*E*max <sup>≈</sup> *EBD*(*W*min)≈2×10<sup>6</sup> <sup>×</sup> *Eg*

Interval of electric field (43) is most often realized in experimental studies (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Ratio *W*min / *λR* is usually not more than a few units. Therefore, when *W* <*W*min then

and hence when *E* >*E*max instead of (33) must be valid relation

() () () 1 ( ) ln

where *Еion* to be understood by largest in value threshold ionization energy of electrons and holes. On basis of relations (33) (or its upgraded version, if parameters *s* and *A* differ from values of (36)) and (44)) can be obtained although approximate but relatively simple and universal analytical dependences of charge carriers multiplication factors and excess noise factors (Tsang, 1985), (Stillman et al, 1983), (Artsis & Kholodnov, 1984), (Woul, 1980), (McIn‐ tyre 1966), (Stillman & Wolf, 1977) on voltage as well as analytical expressions for avalanche breakdown voltage at different spatial distributions of dopant concentration in *p* −*n* struc‐

a -b ´ =» é ù <sup>a</sup>

( )

ê ú ë û <sup>b</sup>

**3.5. Miller's relation for multiplication factors of charge carriers in p-n structures**

tion *g*(*x*) in space charge region (SCR), i.e. when − *L <sup>p</sup>*

Usual way to calculate dependences of avalanche multiplication factors of charge carriers *M* (Section 2) in *p* −*n* structures on applied voltage *Vb* is based on numerical processing of inte‐ gral relations (1) and (2) in each case. Distribution of specific rate of charge carriers' genera‐

in this Section 3.5 as exponential (and as special case − uniform). It is valuable for practical applications to have analytical, more or less universal, dependences *M* on *Vb*. In article (Sze & Gibbons, 1966) was proposed analytical expressions for avalanche breakdown voltage *VBD*, i.e. applied voltage value at which *M* =*∞*, in asymmetric abrupt and linear *p* −*n* junc‐ tions. Expression for *VBD* (Sze & Gibbons, 1966) in the case of asymmetric abrupt *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junc‐ tion was generalized in (Osipov & Kholodnov, 1987) for the case of thin *p* <sup>+</sup> −*n*(*p*)−*n* <sup>+</sup>

*E*

*ion <sup>E</sup> E E c E E E*

*E E*

*ion R*

From (38) and (41) we find desired interval of electric field:

10<sup>4</sup> × *ER λR*

*EBD* ≈

tures.

*Eion <sup>W</sup>* ×10<sup>4</sup>

And avalanche breakdown electric field for thin *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure is defined by approxi‐ mate universal formula

$$E\_{BD}(\mathcal{W}) \approx \left(\frac{E\_g^2}{\varepsilon}\right)^{3/7} \times \frac{10^6}{\mathcal{W}^{1/7}}\tag{38}$$

In expressions (37) and (38) and below in this Section 3.4 concentration is measured in cm-3, energy – in eV, length – in μm, electric field – in V/cm. On the other hand condition of ava‐ lanche breakdown of *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (Sections 2, 3.1 and (Sze, 1981), (Tsang, 1985), (Gre‐ khov & Serezhkin, 1980), (Stillman & Wolf, 1977))

$$m(0, \mathcal{W}) = \left| \int\_0^\mathcal{W} \alpha(E(\mathbf{x})) \times \exp\left| \int\_0^\mathbf{x} (E(\mathbf{x'})) - \alpha(E(\mathbf{x'})) \right] d\mathbf{x'} \right| d\mathbf{x'} = 1,\tag{39}$$

takes the form

$$\ln W \times \left[ \alpha(E\_{BD}) - \left[ \beta(E\_{BD}) \right] \right] = \ln \left[ \frac{\alpha(E\_{BD})}{\beta(E\_{BD})} \right]\_{\text{'}} \tag{40}$$

That means the same probability of impact ionization in any point of diode's base. And rela‐ tion (33) follows from expressions (38) and (40). Let's estimate applicable electric field inter‐ val for this relation. Expression (38) will be valid when inequality (41) is satisfied both for electrons and for holes

$$E\_{BD}(\mathcal{W}) \times \mathcal{W} > \left(\frac{\mathcal{W}}{\lambda\_R} \times E\_R + E\_{\text{ion}}\right) \times 10^4 \tag{41}$$

where *λR*, *Еion*, *ЕR* – mean free path for charge carriers scattered by optical phonons, thresh‐ old ionization energy of electrons or holes and energy of Raman phonon, respectively (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Taking into account that for many semiconductors

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 43

$$25 \times 10^{-3} \times \frac{\sqrt{\varepsilon}}{E\_{\text{g}}} E\_{\text{ion}}^{7/6} \approx W\_{\text{min}} < \frac{E\_{\text{ion}}}{E\_R} \lambda\_R < \cdot \le W\_{\text{max}} \approx 10^{14} \times \frac{\lambda\_R^7 E\_{\text{g}}^6}{E\_R^7 \varepsilon^3} \tag{42}$$

From (38) and (41) we find desired interval of electric field:

7/8 <sup>15</sup>

(37)

(38)

(40)

(41)

structure is defined by approxi‐

}*dx* =1, (39)

è ø

3/7 <sup>2</sup> <sup>6</sup>

<sup>10</sup> ( ) , *<sup>g</sup>*

In expressions (37) and (38) and below in this Section 3.4 concentration is measured in cm-3, energy – in eV, length – in μm, electric field – in V/cm. On the other hand condition of ava‐ lanche breakdown of *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (Sections 2, 3.1 and (Sze, 1981), (Tsang, 1985), (Gre‐

*E*

æ ö » ´ ç ÷ ç ÷ <sup>e</sup> è ø

1/7

))−*α*(*E*(*x* ′

*BD*

*BD*

*E*

bë û

)) *d x* ′

*W*

3/4

*W Eg <sup>N</sup>* æ ö ´ = ´ e´ ´ ç ÷

<sup>1</sup> 3 10 , <sup>4</sup>

0

And avalanche breakdown electric field for thin *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

*BD*

khov & Serezhkin, 1980), (Stillman & Wolf, 1977))

0

Taking into account that for many semiconductors

*W*

*m*(0, *W* )= *∫*

takes the form

electrons and for holes

*E W*

*α*(*E*(*x*))×exp{*∫*

0

*BD BD*

*<sup>E</sup> WE E*

*β*(*E*(*x* ′

( ) [ ( ) ( )] ln , ( )

That means the same probability of impact ionization in any point of diode's base. And rela‐ tion (33) follows from expressions (38) and (40). Let's estimate applicable electric field inter‐ val for this relation. Expression (38) will be valid when inequality (41) is satisfied both for

é ù <sup>a</sup> ´ a -b = ê ú

<sup>4</sup> ( ) <sup>10</sup> *BD R ion R <sup>W</sup> E WW E E* æ ö

´> ´ + ´ ç ÷ lè ø

where *λR*, *Еion*, *ЕR* – mean free path for charge carriers scattered by optical phonons, thresh‐ old ionization energy of electrons or holes and energy of Raman phonon, respectively (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987).

*x*

mate universal formula

42 Photodiodes - From Fundamentals to Applications

$$10^4 \times \frac{E\_R}{\lambda\_R} \approx E\_{BD}(W\_{\text{max}}) \approx E\_{\text{min}} < E < E\_{\text{max}} \approx E\_{BD}(W\_{\text{min}}) \approx 2 \times 10^6 \times \frac{E\_g}{\sqrt{\varepsilon} \sqrt{E\_{\text{min}}}}.\tag{43}$$

Interval of electric field (43) is most often realized in experimental studies (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Dmitriev, 1987). Ratio *W*min / *λR* is usually not more than a few units. Therefore, when *W* <*W*min then *EBD* ≈ *Eion <sup>W</sup>* ×10<sup>4</sup> and hence when *E* >*E*max instead of (33) must be valid relation

$$\frac{E\_{ion}}{E} \times \frac{\alpha(E) - \beta(E)}{\ln\left[\frac{\alpha(E)}{\beta(E)}\right]} = c(E) \approx 1\tag{44}$$

where *Еion* to be understood by largest in value threshold ionization energy of electrons and holes. On basis of relations (33) (or its upgraded version, if parameters *s* and *A* differ from values of (36)) and (44)) can be obtained although approximate but relatively simple and universal analytical dependences of charge carriers multiplication factors and excess noise factors (Tsang, 1985), (Stillman et al, 1983), (Artsis & Kholodnov, 1984), (Woul, 1980), (McIn‐ tyre 1966), (Stillman & Wolf, 1977) on voltage as well as analytical expressions for avalanche breakdown voltage at different spatial distributions of dopant concentration in *p* −*n* struc‐ tures.

#### **3.5. Miller's relation for multiplication factors of charge carriers in p-n structures**

Usual way to calculate dependences of avalanche multiplication factors of charge carriers *M* (Section 2) in *p* −*n* structures on applied voltage *Vb* is based on numerical processing of inte‐ gral relations (1) and (2) in each case. Distribution of specific rate of charge carriers' genera‐ tion *g*(*x*) in space charge region (SCR), i.e. when − *L <sup>p</sup>* < *x* < *L <sup>n</sup>* (see inset in Fig. 3), is accepted in this Section 3.5 as exponential (and as special case − uniform). It is valuable for practical applications to have analytical, more or less universal, dependences *M* on *Vb*. In article (Sze & Gibbons, 1966) was proposed analytical expressions for avalanche breakdown voltage *VBD*, i.e. applied voltage value at which *M* =*∞*, in asymmetric abrupt and linear *p* −*n* junc‐ tions. Expression for *VBD* (Sze & Gibbons, 1966) in the case of asymmetric abrupt *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junc‐ tion was generalized in (Osipov & Kholodnov, 1987) for the case of thin *p* <sup>+</sup> −*n*(*p*)−*n* <sup>+</sup> structure (like as *p* −*i* −*n*) as discussed in Section 3.3. Using as model abrupt (stepwise) *p* −*n* junction under assumption that *K*(*E*)=*β* / *α* =*const* (Kholodnov, 1988-2) has been shown that from (1), (2) and approximate relation (33), which is valid for number of semiconductors in‐ cluding *Ge*, *Si*, *GaAs*, *InP*, *GaP*, can be obtained analytical dependences of multiplication factors of charge carriers on voltage.

Rewrite (33) in the form

possible because ratio *K*(*E*)−<sup>1</sup>

tional common form

*3.5.1. Stepwise p − n junction*

*α*(*E*)

*K*(*E*)−1 ln*K*(*E*) <sup>=</sup> <sup>5</sup>

<sup>6</sup> ×( *εε*<sup>0</sup> 6×10<sup>8</sup> ×*q* ) 3 ×( 1.1 *Еg* ) 6 ×( *<sup>E</sup>* <sup>10</sup><sup>5</sup> ) 7

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

In (45) and below in this Section 3.5 is accepted (unless otherwise specified) the following, convenient for this study, system of symbols and units (Sze, 1981): gap *Eg* and threshold ionization energy *Еion* in eV; electric field *E* in V/cm; bias *Vb* in V; multiplication factors *α* and *β* in cm-1, electron charge *q* in C; dielectric constant of vacuum *ε*<sup>0</sup> in F/m; concentration including shallow donors *ND* and acceptors *NA* in cm-3; concentration gradient *a* in cm-4; width of SCR *L <sup>р</sup>* and *L <sup>n</sup>* in *p* and *n* layers and thicknesses of these layers (inset in Figure 3) in μm, light absorption coefficient *γ* in cm-1. In this section, analytical dependences *M* (*V* ) in *p* −*n* structures have been calculated under no *K*(*E*)=*const* condition. Such calculations are

ln*K*(*E*) varies, typically, much slighter than *<sup>E</sup>* <sup>7</sup>

lows using relation (45) to integrate analytically (in some cases – approximately) expressions (1) and (2) and, thus, get analytical, more or less universal, relatively simple dependences *M* (*V* ). The most typical cases are considered: abrupt (stepwise) and gradual (linear) *p* −*n* junctions like as in model given in (Sze, 1987), (Sze & Gibbons, 1966) and thin *p* <sup>+</sup> −*n*(*p*)−*n* <sup>+</sup> structure (like as *p* −*i* −*n*) with stepwise doping profile as in model presented in (Osipov & Kholodnov, 1987). For purposes of discussion and comparison of obtained results with nu‐ merical calculations and experimental data, multiplication factors will be written in tradi‐

> 1 11 , ,, <sup>1</sup> *n p* <sup>1</sup> <sup>1</sup> *MMM n p n nn <sup>v</sup> <sup>v</sup> <sup>v</sup>* = == - - - %

where *v* =*V* / *VBD*. This form was first proposed by Miller in 1955 (Miller, 1955) and then, de‐ spite lack of analytical expressions for exponents *nn*, *np*, *n*˜, has been widely used as "Miller's relation" (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986). It was found that values of these exponents depend on many factors including, in general, voltage as well (Kholodnov, 1988-2), (Grekhov & Serezhkin, 1980), Fig.

3. Form of writing (46) clearly shows that *M* (*V* )→*∞* when *V* →*VBD*.

*dE*

In this case from relations (1), (2) and (45) and Poisson equation (SI units)

0

*A*

*qN*

*dx qN*

<sup>ï</sup> ee <sup>=</sup> <sup>í</sup>

ì

0

*D*

ï- > <sup>ï</sup> ee <sup>î</sup>

ï <

, x 0

, x 0

(45)

45

(47)

. In some cases it al‐

http://dx.doi.org/10.5772/50778

% (46)

**Figure 3.** Dependences of exponents in Miller's relation for electron *nn* and holes *np* for "thick" abrupt р −*n* junction on applied voltage *V* at different values *K* =β / α equal to 1, 2, 3, and 4

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 45

Rewrite (33) in the form

structure (like as *p* −*i* −*n*) as discussed in Section 3.3. Using as model abrupt (stepwise) *p* −*n* junction under assumption that *K*(*E*)=*β* / *α* =*const* (Kholodnov, 1988-2) has been shown that from (1), (2) and approximate relation (33), which is valid for number of semiconductors in‐ cluding *Ge*, *Si*, *GaAs*, *InP*, *GaP*, can be obtained analytical dependences of multiplication

**Figure 3.** Dependences of exponents in Miller's relation for electron *nn* and holes *np* for "thick" abrupt р −*n* junction

on applied voltage *V* at different values *K* =β / α equal to 1, 2, 3, and 4

factors of charge carriers on voltage.

44 Photodiodes - From Fundamentals to Applications

$$
\alpha(E) \frac{K(E) - 1}{\ln K(E)} = \frac{5}{6} \times \left(\frac{\varepsilon \varepsilon\_0}{6 \times 10^8 \times q}\right)^3 \times \left(\frac{1.1}{E\_\odot}\right)^6 \times \left(\frac{E}{10^5}\right)^7 \tag{45}
$$

In (45) and below in this Section 3.5 is accepted (unless otherwise specified) the following, convenient for this study, system of symbols and units (Sze, 1981): gap *Eg* and threshold ionization energy *Еion* in eV; electric field *E* in V/cm; bias *Vb* in V; multiplication factors *α* and *β* in cm-1, electron charge *q* in C; dielectric constant of vacuum *ε*<sup>0</sup> in F/m; concentration including shallow donors *ND* and acceptors *NA* in cm-3; concentration gradient *a* in cm-4; width of SCR *L <sup>р</sup>* and *L <sup>n</sup>* in *p* and *n* layers and thicknesses of these layers (inset in Figure 3) in μm, light absorption coefficient *γ* in cm-1. In this section, analytical dependences *M* (*V* ) in *p* −*n* structures have been calculated under no *K*(*E*)=*const* condition. Such calculations are possible because ratio *K*(*E*)−<sup>1</sup> ln*K*(*E*) varies, typically, much slighter than *<sup>E</sup>* <sup>7</sup> . In some cases it al‐ lows using relation (45) to integrate analytically (in some cases – approximately) expressions (1) and (2) and, thus, get analytical, more or less universal, relatively simple dependences *M* (*V* ). The most typical cases are considered: abrupt (stepwise) and gradual (linear) *p* −*n* junctions like as in model given in (Sze, 1987), (Sze & Gibbons, 1966) and thin *p* <sup>+</sup> −*n*(*p*)−*n* <sup>+</sup> structure (like as *p* −*i* −*n*) with stepwise doping profile as in model presented in (Osipov & Kholodnov, 1987). For purposes of discussion and comparison of obtained results with nu‐ merical calculations and experimental data, multiplication factors will be written in tradi‐ tional common form

$$M\_n = \frac{1}{1 - \upsilon^{n\_n}},\\ M\_p = \frac{1}{1 - \upsilon^{n\_p}},\\ \tilde{M} = \frac{1}{1 - \upsilon^{\tilde{n}}},\tag{46}$$

where *v* =*V* / *VBD*. This form was first proposed by Miller in 1955 (Miller, 1955) and then, de‐ spite lack of analytical expressions for exponents *nn*, *np*, *n*˜, has been widely used as "Miller's relation" (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986). It was found that values of these exponents depend on many factors including, in general, voltage as well (Kholodnov, 1988-2), (Grekhov & Serezhkin, 1980), Fig. 3. Form of writing (46) clearly shows that *M* (*V* )→*∞* when *V* →*VBD*.

#### *3.5.1. Stepwise p − n junction*

In this case from relations (1), (2) and (45) and Poisson equation (SI units)

$$\frac{dE}{d\mathbf{x}} = \begin{cases} \frac{qN\_A}{\varepsilon\varepsilon\_0}, & \mathbf{x} < \mathbf{0} \\\\ -\frac{qN\_D}{\varepsilon\varepsilon\_0}, & \mathbf{x} > \mathbf{0} \end{cases} \tag{47}$$

follow that

$$M\_n = (K\_0 - 1) / (K\_0 - K\_0^{v^4}) \text{, } M\_p = K\_0^{v^4} \times M\_n \tag{48}$$

*3.5.2. Gradual (linear) p − n junction*

and therefore

hyk, 1963)

*3.5.3. Thin p <sup>+</sup> − n(p) − n <sup>+</sup>*

*<sup>W</sup>* <sup>&</sup>gt;*<sup>W</sup>*˜= <sup>6</sup>*εε*<sup>0</sup>

In this case Poisson equation can be written as (SI units):

where *σ*- slope of linear concentration profile

*BD*

*V*

0 *dE <sup>q</sup> <sup>x</sup>*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

2/5 6/5 1/5 <sup>20</sup> 3 10 17.7 <sup>60</sup> . 1.1 *g*

In derivation of relations (55) and (56) was used known expression for voltage distribution on linear *p* −*n* junction (Sze, 1981) and was also taken into account that (Gradstein & Ryz‐

6435

Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear *p* −*n* junction (Sze, 1981), (Sze & Gibbons, 1966) by last multiplicand, which for typical

*<sup>N</sup>* 7/8 <sup>≈</sup><sup>2</sup> *<sup>ε</sup>* <sup>×</sup>*Е<sup>g</sup>*

where *N* – dopant concentration (for example, donor) in base, and when *Vb* =*VBD* then SCR does not extend to entire thickness of base ((Osipov & Kholodnov, 1987), Sections 3.1-3.3, in‐ set in Fig. 4). In this case, expressions (48)-(53) remain apparently valid. In opposite case, base is depleted by free charge carriers when *Vb* <*VBD* that gives in the result substantially

æ ö æ ö ´ æ ö =´ ´ ´ ç ÷ ç ÷ ç ÷ s e ç ÷ è ø è ø è ø

7

*x*

( ) 4096

*<sup>y</sup> y x dx y*

0

values of *ε* ≈10 (Sze, 1981), (Casey & Panish, 1978) is close to unity.

 *structure (p − i − n)*

<sup>5</sup>*<sup>q</sup>* ×( *<sup>Е</sup><sup>g</sup>* 1.1 ) 3/4 <sup>×</sup> <sup>10</sup><sup>10</sup>

When thickness of high-resistivity region (base) of considered structure

*E*

5 4

15/2


3/4 ×( 3×10<sup>15</sup> *<sup>N</sup>* )

7/8

, (58)

0 00 0 ( 1) / ( ), *v v M K KK MK M <sup>n</sup> p n* =- - =´ (55)

´ s =- ´ ee (54)

http://dx.doi.org/10.5772/50778

(56)

47

*dx*

$$\mathbf{V}\_{BD} = \mathbf{6} \times \mathbf{10}^{13} \times \left(\frac{E\_g}{1.1}\right)^{3/2} \times \mathbf{N}\_{eff}^{-3/4} \text{ }^\circ \text{N}\_{eff} = \frac{\mathbf{N}\_A \times \mathbf{N}\_D}{\mathbf{N}\_A + \mathbf{N}\_D} \tag{49}$$

where *K*0 – value *K*(*х*) when *Е*(*х*)=*Е*(0)=*Е*0, i.e. value of *K* at metallurgical boundary of *p* −*n* junction (see inset in Fig. 3). Formula (49) for *VBD* at *ND* < < *NA*or *NA* < < *ND* becomes wellknown Sze-Gibbons relation (Sze, 1981), (Sze & Gibbons, 1966). If charge carriers are gener‐ ated uniformly in SCR then computations lead to following expressions:

$$\frac{\tilde{M}}{M\_n} = \frac{N\_A \times \exp[\xi\_A \times e(K\_0 - 1) + \xi\_D \,/\, g] + N\_D \times \exp[\xi\_D \times e(1 - K\_0) + \xi\_A \,/\, g]}{N\_A + N\_D} \tag{50}$$

when

$$\left| \xi\_{A,D}(v) \times (N\_{\rm eff} \,/ \, N\_{A,D}) \times v^4 \times \left| \ln K\_0 \right| \le 1; \tag{51}$$

$$
\tilde{\mathcal{M}} = \left( 1 - K\_{\rm eff}^{\*4} \frac{1 - K\_{\rm eff}^{4-\vartheta}}{K\_{\rm eff} - 1}^{4} \right)^{-1} \times M\_{\rm n\prime} \tag{52}
$$

when

$$K\_{\rm eff}^{v^4 \times \frac{N\_{\rm eff}}{N\_{A,D}}} \gg 1,\tag{53}$$

*e*(*x*) – unity function (Zeldovich & Myshkis, 1972), *Keff* = *K*<sup>0</sup> + *K*<sup>0</sup> −1 . Expression (50) is ob‐ tained by expanding the function *Y* (*x*, − *L <sup>p</sup>*) as a power series in

*∫* −*L <sup>p</sup> x* (*β* −*α*)*d x* ′ ,

and expression (52) was derived by standard method of integrating fast-changing functions (Zeldovich & Myshkis, 1972).

#### *3.5.2. Gradual (linear) p − n junction*

follow that

46 Photodiodes - From Fundamentals to Applications

%

when

when

*∫* −*L <sup>p</sup>* (*β* −*α*)*d x* ′

,

(Zeldovich & Myshkis, 1972).

*x*

4 4

where *K*0 – value *K*(*х*) when *Е*(*х*)=*Е*(0)=*Е*0, i.e. value of *K* at metallurgical boundary of *p* −*n* junction (see inset in Fig. 3). Formula (49) for *VBD* at *ND* < < *NA*or *NA* < < *ND* becomes wellknown Sze-Gibbons relation (Sze, 1981), (Sze & Gibbons, 1966). If charge carriers are gener‐

0 0 exp[ ( 1) / ] exp[ (1 ) / ], *AA D DD A*

4

1

<sup>4</sup> <sup>1</sup>


<sup>1</sup> 1 , *v eff eff <sup>K</sup> <sup>v</sup> <sup>K</sup> MK M eff <sup>n</sup>*

> , 1, *eff A D N v N Keff* ´

and expression (52) was derived by standard method of integrating fast-changing functions

4

4

*e*(*x*) – unity function (Zeldovich & Myshkis, 1972), *Keff* = *K*<sup>0</sup> + *K*<sup>0</sup>

tained by expanding the function *Y* (*x*, − *L <sup>p</sup>*) as a power series in

1 8


, ,0 ( ) ( / ) ln 1; *A D eff A D* x ´ ´´ £ *vN N v K* (51)

% (52)

>> (53)

−1

. Expression (50) is ob‐

*M N eK g N eK g*

´ x ´ - +x + ´ x ´ - +x <sup>=</sup> <sup>+</sup>

*VBD* =6×10<sup>13</sup> ×( *<sup>Е</sup><sup>g</sup>*

*n A D*

*M N N*

1.1 ) 3/2 × *Neff*

ated uniformly in SCR then computations lead to following expressions:

0 00 0 ( 1) / ( ), *v v M K KK MK M <sup>n</sup> p n* =- - =´ (48)

*NA* + *ND*

(49)

(50)

<sup>−</sup>3/4, *Neff* <sup>=</sup> *NA* <sup>×</sup> *ND*

In this case Poisson equation can be written as (SI units):

$$\frac{dE}{d\mathbf{x}} = -\frac{q \times \sigma}{\varepsilon \varepsilon\_0} \times \mathbf{x} \tag{54}$$

where *σ*- slope of linear concentration profile and therefore

$$M\_n = (K\_0 - 1) / (K\_0 - K\_0^{v^5}) , M\_p = K\_0^{v^4} \times M\_n \tag{55}$$

$$V\_{BD} = 60 \times \left(\frac{3 \times 10^{20}}{\sigma}\right)^{2/5} \times \left(\frac{E\_g}{1.1}\right)^{6/5} \times \left(\frac{17.7}{\varepsilon}\right)^{1/5}.\tag{56}$$

In derivation of relations (55) and (56) was used known expression for voltage distribution on linear *p* −*n* junction (Sze, 1981) and was also taken into account that (Gradstein & Ryz‐ hyk, 1963)

$$\int\_{0}^{y} \frac{(y-x)^7}{\sqrt{x}} dx = \frac{4096}{6435} y^{15/2} \tag{57}$$

Formula (56) differs from known formula Sze-Gibbons for avalanche breakdown voltage of linear *p* −*n* junction (Sze, 1981), (Sze & Gibbons, 1966) by last multiplicand, which for typical values of *ε* ≈10 (Sze, 1981), (Casey & Panish, 1978) is close to unity.

$$3.5.3. \text{ This } p^{+}-n(p)-n^{+} \text{ structure } (p-i-n)^{+}$$

When thickness of high-resistivity region (base) of considered structure

$$\Delta W > \overline{W} = 4 \frac{\overline{6 \varepsilon \varepsilon\_0}}{5q} \times \left(\frac{E\_g}{1.1}\right)^{3/4} \times \frac{10^{10}}{N} \approx 2 \sqrt{\varepsilon} \times E\_g^{3/4} \times \left(\frac{3 \times 10^{15}}{N}\right)^{7/8} \tag{58}$$

where *N* – dopant concentration (for example, donor) in base, and when *Vb* =*VBD* then SCR does not extend to entire thickness of base ((Osipov & Kholodnov, 1987), Sections 3.1-3.3, in‐ set in Fig. 4). In this case, expressions (48)-(53) remain apparently valid. In opposite case, base is depleted by free charge carriers when *Vb* <*VBD* that gives in the result substantially other expressions for avalanche multiplication factors of charge carriers and avalanche breakdown voltage. When *<sup>W</sup>* <sup>&</sup>lt;*<sup>W</sup>*˜ then from relations (1), (2) and (45) and Poisson equation

$$\frac{dE}{d\alpha} = -\frac{qN}{\varepsilon\varepsilon\_0} \tag{59}$$

we find that

where

8 8

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

8 8

6

4 6

*qW E N*

<sup>6</sup> 1.1 1 . 5 10 *<sup>g</sup>*

voltage drop on them is considered negligible. This is justified because of significant de‐ creasing of electric field *E*(*х*) deep into high-doped layers of the structure (Sze, 1981), (Kho‐ lodnov 1996-1), (Kholodnov 1998), (Leguerre & Urgell, 1976). Admissibility of such neglect is confirmed also by formula (49) when *NA* < < *ND* or when *ND* < < *NA*. Avalanche break‐ down voltage is determined by equation *v*˜ = 1 which has no exact analytical solution. How‐ ever, till *W* surpasses *<sup>W</sup>*˜/ 8, then value of field at *<sup>x</sup>* <sup>=</sup>*W* is much less than value of field at

> 8 1 2 12 1 *<sup>V</sup> V* æ ö ç ÷ - ´ << è ø

8 1 1 8 2 ( )( ) , *VV VV*

*V*

2

0 10 , <sup>2</sup> *qNW <sup>V</sup>* - = ´ ee

æ ö æ ö ee = ´´ ç ÷ ç ÷ ´ ç ÷ è ø è ø

*v*

1

8 0 2 8 2

In deriving expressions (60)-(63), multiplication of charge carriers in *p* <sup>+</sup>

we find that in zeroth-order approximation with respect to this parameter

0

1 , 8

*V*

*x* =0. In this case, using smallness parameter

In the case of very thin base when

0 00 0 ( 1) / ( ), , *v v M K KK MK M <sup>n</sup> p n* =- - =´ % %(60)

+ -- % <sup>=</sup> (61)

2 1. *V VV BD* = - (65)

*WW W* £ = % (66)

(62)

49

(63)

(64)

layers and

and *n* <sup>+</sup>

http://dx.doi.org/10.5772/50778

**Figure 4.** Dependences of analytical (solid lines) and numerical (dashed lines) (Leguerre & Urgell, 1976) limiting values of exponent *nB* = lim *<sup>V</sup>* <sup>→</sup>*VBD n*(*V* ) in Miller's relation (46) on concentration of donor dopant *ND* in "thick" high-resistivity layer of stepwise *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure, 1 − Si, 2 − Ge, 3 − GaAs, 4 − GaP. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are

taken from (Sze & Gibbons, 1966). In inset − scheme of "thick" *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 49

we find that

other expressions for avalanche multiplication factors of charge carriers and avalanche breakdown voltage. When *<sup>W</sup>* <sup>&</sup>lt;*<sup>W</sup>*˜ then from relations (1), (2) and (45) and Poisson equation

0

**Figure 4.** Dependences of analytical (solid lines) and numerical (dashed lines) (Leguerre & Urgell, 1976) limiting values

*n*(*V* ) in Miller's relation (46) on concentration of donor dopant *ND* in "thick" high-resistivity layer

structure, 1 − Si, 2 − Ge, 3 − GaAs, 4 − GaP. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are

structure

of exponent *nB* = lim

of stepwise *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

*<sup>V</sup>* <sup>→</sup>*VBD*

48 Photodiodes - From Fundamentals to Applications

taken from (Sze & Gibbons, 1966). In inset − scheme of "thick" *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

*dx* = - ee (59)

*dE qN*

$$\boldsymbol{M}\_{n} = (\boldsymbol{K}\_{0} - \boldsymbol{1}) / (\boldsymbol{K}\_{0} - \boldsymbol{K}\_{0}^{\vec{v}^{8}}) , \boldsymbol{M}\_{p} = \boldsymbol{K}\_{0}^{\vec{v}^{8}} \times \boldsymbol{M}\_{n'} \tag{60}$$

where

$$
\tilde{\upsilon}^8 = \frac{(V + V\_1)^8 - (V - V\_1)^8}{V\_2^{\frac{8}{8}}},\tag{61}
$$

$$V\_1 = \frac{qNW^2}{2\varepsilon\varepsilon\_0} \times 10^{-6} \,\text{\AA} \tag{62}$$

$$V\_2^8 = \left(\frac{6\varepsilon\varepsilon\_0}{5 \times 10^8 \,\mathrm{q}\,\mathrm{W}^2}\right)^4 \times \left(\frac{1.1}{E\_\mathrm{g}}\right)^6 \times \frac{1}{N}.\tag{63}$$

In deriving expressions (60)-(63), multiplication of charge carriers in *p* <sup>+</sup> and *n* <sup>+</sup> layers and voltage drop on them is considered negligible. This is justified because of significant de‐ creasing of electric field *E*(*х*) deep into high-doped layers of the structure (Sze, 1981), (Kho‐ lodnov 1996-1), (Kholodnov 1998), (Leguerre & Urgell, 1976). Admissibility of such neglect is confirmed also by formula (49) when *NA* < < *ND* or when *ND* < < *NA*. Avalanche break‐ down voltage is determined by equation *v*˜ = 1 which has no exact analytical solution. How‐ ever, till *W* surpasses *<sup>W</sup>*˜/ 8, then value of field at *<sup>x</sup>* <sup>=</sup>*W* is much less than value of field at *x* =0. In this case, using smallness parameter

$$\left(1 - 2 \times \frac{V\_1}{V\_2}\right)^8 << 1\tag{64}$$

we find that in zeroth-order approximation with respect to this parameter

$$V\_{BD} = V\_2 - V\_1.\tag{65}$$

In the case of very thin base when

$$
\mathcal{W} \le \mathcal{W}\_0 = \frac{1}{8}\tilde{\mathcal{W}}\_{\prime} \tag{66}
$$

electric field varies so slightly along base that probability of impact ionization is practically the same in any point of it ((Osipov & Kholodnov, 1987), (Kholodnov 1988-1), Sections 3.2 and 3.4). As a result

$$M\_n = (K - 1) / (K - K^{v^?}) , \\ M\_p = K^{v^?} \times M\_{n'} \tag{67}$$

<sup>4</sup> (0, ) 10 , V/cm *ion*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

*<sup>W</sup>* = ´ (71)

http://dx.doi.org/10.5772/50778

51

(а)

(0, *W* ). Curves 1 and 1' − Si, 2 − InP, 3 − Ge. Values α(*E*) and β(*E*) are taken: for curves

(0, *W* ). Dashed curve − analytical value of effective avalanche breakdown field *EBD*

1 and 3 − from Table 1 of monograph (Grekhov & Serezhkin, 1980), for curve 1' − from (Kuzmin et al, 1975), for curve

If assume that in *Si* threshold energy of impact ionization *Еion* of holes is higher than elec‐ trons, and it equals to 5 eV (Sze, 1981), then from (70) we find for *Si W*min ≈0.1 μm. Estimates based on data from studies (Sze, 1981), (Tsang, 1985), (Stillman et al, 1983), (Grekhov & Ser‐ ezhkin, 1980), (Stillman & Wolf, 1977) show that for *Ge* and *InP* value *W*min is 2-3 times

Therefore curve 1 in Fig. 5 starts to fall significantly below unity at larger values *W* than curves 2 and 3. Analytical and computed dependences *EBD* on *N* for *InP* used in high-per‐ formance APDs for wavelength range *λ* =(1÷1, 7) μm as wide-gap layers in double hetero‐

(0, *W* ) obtained by formulas (25) and

\* (0, *W* )

*<sup>E</sup> E W*

*BD*

**Figure 5.** Dependence of ratio between analytical value of breakdown field *EBD*

(*c*)

(*InP*)

6/7 <sup>×</sup> *EBD*(0, *<sup>W</sup>* )≡*E*В*<sup>D</sup>*

(26) and numerical value *EBD*

2 − from (Cook et al, 1982)

= *X*<sup>ε</sup> <sup>−</sup>3/7 <sup>×</sup> *Xg*

smaller.

$$\tilde{M} = \frac{\gamma W}{\gamma W + v^{\gamma} \times \ln K} \times \frac{K^{v^{\gamma}} \times \exp(\gamma W) - 1}{\exp(\gamma W) - 1} \times M\_{n^{\prime}} \tag{68}$$

$$V\_{BD} = 7 \times \sqrt{\frac{3}{25} \times \left(\frac{3q}{50\varepsilon\varepsilon\_0}\right)^3 \times \left(\frac{E\_g}{1.1} W\right)^6} \times 10^6 \approx 98 \times \left(\frac{W \times E\_g}{\sqrt{\varepsilon}}\right)^{6/7},\tag{69}$$

where *γ*< 0, if structure is illuminated through *p* <sup>+</sup> region (front-side illuminated) and *γ*> 0 if structure is illuminated through *n* <sup>+</sup> region (back-side illuminated).

#### **3.6. Discussion of the results. Comparison with computed and experimental data**

#### *3.6.1. To formulas for avalanche breakdown electric field and voltage for abrupt p+ - n junction*

In sections 3.1-3.3 were derived approximate universal formulas for avalanche breakdown field *EBD* and voltage *VBD* for abrupt *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction taking into account finite thickness of high-resistivity layer *W* . Comparative values of breakdown field *EBD*(0, *W* ) for *Si*, *Ge* and *InP* most often used for fabrication of APDs computed by formulas (25) and (26) and found from numerical solution of breakdown integral equation *m*=1, where *m* is defined by (2) are shown on Fig. 5 (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Groves et al, 2005), (Stillman et al, 1983), (Trommer, 1984), (Woul, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986), (Gasanov et al, 1988), (Brain, 1981), (Tager & Vald-Perlov, 1968). It is seen that in the most practically interesting range *W* ≈(0.2÷10) μm for all a.m. semiconductors analytical *EBD* (*a*) (0, *W* ) and calculated *EBD* (*c*) (0, *W* ) values of breakdown field differ by less than 20 %. Relatively drastic fall of ratio *EBD* (*a*) (0, *W* )/*EBD* (*с*) (0, *W* ) in compari‐ son to unity with decrease of *W* (for thin enough *W* ) is due to the fact that, as shown in Sec. 3.4, if

$$\mathcal{W} \le \mathcal{W}\_{\text{min}} \approx \dots \approx 10^{-3} \times \frac{\sqrt{\varepsilon}}{E\_{\text{g}}} \times E\_{\text{ion}}^{7/6} \quad \text{\mu m.} \tag{70}$$

then formulas (25) and (26) are not true. To estimate breakdown field *EBD*(0, *W* ) at values *W* defined by (70) can be used the following formula

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 51

electric field varies so slightly along base that probability of impact ionization is practically the same in any point of it ((Osipov & Kholodnov, 1987), (Kholodnov 1988-1), Sections 3.2

7

where *γ*< 0, if structure is illuminated through *p* <sup>+</sup> region (front-side illuminated) and *γ*> 0 if

In sections 3.1-3.3 were derived approximate universal formulas for avalanche breakdown field *EBD* and voltage *VBD* for abrupt *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction taking into account finite thickness of high-resistivity layer *W* . Comparative values of breakdown field *EBD*(0, *W* ) for *Si*, *Ge* and *InP* most often used for fabrication of APDs computed by formulas (25) and (26) and found from numerical solution of breakdown integral equation *m*=1, where *m* is defined by (2) are shown on Fig. 5 (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Filachev et al, 2011), (Groves et al, 2005), (Stillman et al, 1983), (Trommer, 1984), (Woul, 1980), (Leguerre & Urgell, 1976), (Bogdanov et al, 1986), (Gasanov et al, 1988), (Brain, 1981), (Tager & Vald-Perlov, 1968). It is seen that in the most practically interesting range *W* ≈(0.2÷10) μm for all

(0, *W* ) and calculated *EBD*

son to unity with decrease of *W* (for thin enough *W* ) is due to the fact that, as shown in Sec.

*Еg* ×*Еion*

then formulas (25) and (26) are not true. To estimate breakdown field *EBD*(0, *W* ) at values

(*c*)

(*a*)

(0, *W* )/*EBD* (*с*)

7/6, μm, (70)

region (back-side illuminated).

*v*

*<sup>W</sup> K W M M Wv K W* <sup>g</sup> ´ g- =´ ´ g+´ g -

**3.6. Discussion of the results. Comparison with computed and experimental data**

*3.6.1. To formulas for avalanche breakdown electric field and voltage for abrupt p+*

(*a*)

*<sup>W</sup>* <sup>&</sup>lt;*W*min <sup>≈</sup>5×10−<sup>3</sup> <sup>×</sup> *<sup>ε</sup>*

field differ by less than 20 %. Relatively drastic fall of ratio *EBD*

*W* defined by (70) can be used the following formula

7

7 7

exp( ) 1 , ln exp( ) 1

( 1) / ( ), , *v v M K KK M K M <sup>n</sup> p n* =- - = ´ (67)

% (68)

×10<sup>6</sup> <sup>≈</sup>98×( *<sup>W</sup>* <sup>×</sup>*Е<sup>g</sup>*

*n*

*<sup>ε</sup>* )

6/7

, (69)

 *- n junction*

(0, *W* ) values of breakdown

(0, *W* ) in compari‐

and 3.4). As a result

*VBD* =7×

50 Photodiodes - From Fundamentals to Applications

structure is illuminated through *n* <sup>+</sup>

a.m. semiconductors analytical *EBD*

3.4, if

3 <sup>25</sup> ×( <sup>3</sup>*<sup>q</sup>* 50*εε*<sup>0</sup> ) 3 ×( *<sup>Е</sup><sup>g</sup>* 1.1 *<sup>W</sup>* ) 6

$$E\_{BD}(0, \mathcal{W}) = 10^4 \times \frac{E\_{ion}}{\mathcal{W}} \text{ , V/cm} \tag{71}$$

**Figure 5.** Dependence of ratio between analytical value of breakdown field *EBD* (а) (0, *W* ) obtained by formulas (25) and (26) and numerical value *EBD* (*c*) (0, *W* ). Dashed curve − analytical value of effective avalanche breakdown field *EBD* \* (0, *W* ) = *X*<sup>ε</sup> <sup>−</sup>3/7 <sup>×</sup> *Xg* 6/7 <sup>×</sup> *EBD*(0, *<sup>W</sup>* )≡*E*В*<sup>D</sup>* (*InP*) (0, *W* ). Curves 1 and 1' − Si, 2 − InP, 3 − Ge. Values α(*E*) and β(*E*) are taken: for curves 1 and 3 − from Table 1 of monograph (Grekhov & Serezhkin, 1980), for curve 1' − from (Kuzmin et al, 1975), for curve 2 − from (Cook et al, 1982)

If assume that in *Si* threshold energy of impact ionization *Еion* of holes is higher than elec‐ trons, and it equals to 5 eV (Sze, 1981), then from (70) we find for *Si W*min ≈0.1 μm. Estimates based on data from studies (Sze, 1981), (Tsang, 1985), (Stillman et al, 1983), (Grekhov & Ser‐ ezhkin, 1980), (Stillman & Wolf, 1977) show that for *Ge* and *InP* value *W*min is 2-3 times smaller.

Therefore curve 1 in Fig. 5 starts to fall significantly below unity at larger values *W* than curves 2 and 3. Analytical and computed dependences *EBD* on *N* for *InP* used in high-per‐ formance APDs for wavelength range *λ* =(1÷1, 7) μm as wide-gap layers in double hetero‐ structures (Fig. 1, 2) are shown on Fig. 6 (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Forrest et al, 1983), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984). It is seen that *EBD* (*a*) (*N* , *W* ) and *EBD* (*c*) (*N* , *W* ) differ from each other by less than 10 %. In Fig. 7 and 8 are shown universal dependences of breakdown voltage *VBD* (*a*) on *N* and *W* cal‐ culated by formulas (11), (27)-(29). It is seen from Fig. 7 that Sze-Gibbons relations (10) and (11) (Sze, 1981), (Sze & Gibbons, 1966) can be used to determine *VBD* when *<sup>N</sup>* <sup>&</sup>gt; *<sup>N</sup>*min <sup>≈</sup>10× *<sup>N</sup>*˜(*<sup>W</sup>* ) only. Value of this minimal concentration, for example, for classic sem‐ iconductors *Si*, *Ge*, *GaAs*, *GaP* and *InP* at *W* = (1-2) μm equals to (1÷5)×1016cm-3. As shown on lower inset in Fig. 7, dependence *VBD* on *N* is in the strict sense non-monotonic. Such kind of dependence *VBD* on *N* is due to the fact that for small enough *N* breakdown field *EBD* is growing faster with increasing *N* than |∇*E*|∝ *N* in diode's base. Maximum *VBD* is reached, as it follows from (28), at

$$N = N\_{\text{max}} = \left( 2^{\frac{s}{s-1}} - 1 \right) \times \tilde{N}(W) \tag{72}$$

and expressed as

$$V\_{BD}^{(\text{max})} = \left[ \left( \text{s} - 1 \right) \times \overline{\mathbf{2}^{s-1}} + 1 \right] \times \left( \text{2s} \right)^{-1} \times V\_{BD}(\text{0,W}) \tag{73}$$

**Figure 6.** Dependence of field *EBD* on *N* for *InP*: 1 − *W* = 0.5 μm, 2 − *W* = 2 μm, 3 − *W* = 8 μm. Solid lines – formulas (24)-(26), dashed curves − numerical calculation. Values α(*E*) and β(*E*) are taken from (Cook et al, 1982). In inset is

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

http://dx.doi.org/10.5772/50778

53

4/7 <sup>×</sup> *Xg*

6/7 <sup>×</sup> *<sup>N</sup>*˜ <sup>≡</sup> *<sup>N</sup>*˜*InP* on *<sup>W</sup>* . Concentration is measured in cm-3,

shown dependence of effective concentration *N*˜\*= *<sup>X</sup>*<sup>ε</sup>

field − in V/cm and thickness *W* − in μm.

when *<sup>s</sup>* =8, value *N*max <sup>≈</sup>1.2× *<sup>N</sup>*˜, *ΔV*max (*rel*) <sup>≈</sup>2.86×10−<sup>2</sup> <<1 and absolute value *ΔV*max can reach tens Volts, and even more (see Fig. 7). The analytical dependences *VBD* (*a*) (*N* , *W* ) (Fig. 7 and 8) for a number of semiconductors are in good agreement with *VBD* (*c*) (*N* , *W* ) computed on the basis of integral equations (1) and (2) (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976). Note that results of com‐ parison *VBD* (*a*) (*N* , *W* ) with *VBD* (*c*) (*N* , *W* ) and *ЕBD* (*a*) (*N* , *W* ) with *ЕBD* (*c*) (*N* , *W* ) depend on accuracy of determination of impact ionization coefficients of electrons *α*(*Е*) and holes *β*(*Е*) which are sharp functions of electric field *Е*. As a rule, different authors obtain different results (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) (see, for example, curves 1 and 1' in Fig. 5). In ad‐ dition, deducing of relations (1) and (2) is based on local relation between *α* and *β* (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2011), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Ta‐ ger & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) which is not always valid (McIntyre, 1972), (Gribnikov et al, 1981), (Okuto & Crowell, 1974), (McIntyre, 1999).

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 53

structures (Fig. 1, 2) are shown on Fig. 6 (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2010), (Forrest et al, 1983), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer,

culated by formulas (11), (27)-(29). It is seen from Fig. 7 that Sze-Gibbons relations (10) and (11) (Sze, 1981), (Sze & Gibbons, 1966) can be used to determine *VBD* when *<sup>N</sup>* <sup>&</sup>gt; *<sup>N</sup>*min <sup>≈</sup>10× *<sup>N</sup>*˜(*<sup>W</sup>* ) only. Value of this minimal concentration, for example, for classic sem‐

iconductors *Si*, *Ge*, *GaAs*, *GaP* and *InP* at *W* = (1-2) μm equals to (1÷5)×1016cm-3. As shown on lower inset in Fig. 7, dependence *VBD* on *N* is in the strict sense non-monotonic. Such kind of dependence *VBD* on *N* is due to the fact that for small enough *N* breakdown field *EBD* is growing faster with increasing *N* than |∇*E*|∝ *N* in diode's base. Maximum *VBD* is

> <sup>1</sup> max 2 1 () *s N N <sup>s</sup>*- *N W* æ ö = = -´ ç ÷ ç ÷ è ø

(max) <sup>1</sup> <sup>1</sup> ( 1) 2 1 (2 ) (0, ) *s <sup>s</sup> V s BD BD* - *sV W* - é ù = -´ + ´ ´ ê ú ê ú ë û

basis of integral equations (1) and (2) (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Leguerre & Urgell, 1976). Note that results of com‐

(*a*)

of determination of impact ionization coefficients of electrons *α*(*Е*) and holes *β*(*Е*) which are sharp functions of electric field *Е*. As a rule, different authors obtain different results (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) (see, for example, curves 1 and 1' in Fig. 5). In ad‐ dition, deducing of relations (1) and (2) is based on local relation between *α* and *β* (Sze, 1981), (Tsang, 1985), (Stillman, 1981), (Filachev et al, 2011), (Stillman et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Ta‐ ger & Vald-Perlov, 1968), (McIntyre, 1972), (Cook et al, 1982) which is not always valid

(*N* , *W* ) with *ЕBD*

(*rel*)

(*N* , *W* ) and *ЕBD*

(McIntyre, 1972), (Gribnikov et al, 1981), (Okuto & Crowell, 1974), (McIntyre, 1999).

tens Volts, and even more (see Fig. 7). The analytical dependences *VBD*

for a number of semiconductors are in good agreement with *VBD*

(*c*)

(*N* , *W* ) differ from each other by less than 10 %. In

(*a*)

% (72)

<sup>≈</sup>2.86×10−<sup>2</sup> <<1 and absolute value *ΔV*max can reach

(*c*)

(*c*)

(*a*)

on *N* and *W* cal‐

(73)

(*N* , *W* ) (Fig. 7 and 8)

(*N* , *W* ) computed on the

(*N* , *W* ) depend on accuracy

1984). It is seen that *EBD*

52 Photodiodes - From Fundamentals to Applications

reached, as it follows from (28), at

when *<sup>s</sup>* =8, value *N*max <sup>≈</sup>1.2× *<sup>N</sup>*˜, *ΔV*max

(*N* , *W* ) with *VBD*

and expressed as

parison *VBD*

(*a*)

(*a*)

(*N* , *W* ) and *EBD*

(*c*)

Fig. 7 and 8 are shown universal dependences of breakdown voltage *VBD*

**Figure 6.** Dependence of field *EBD* on *N* for *InP*: 1 − *W* = 0.5 μm, 2 − *W* = 2 μm, 3 − *W* = 8 μm. Solid lines – formulas (24)-(26), dashed curves − numerical calculation. Values α(*E*) and β(*E*) are taken from (Cook et al, 1982). In inset is shown dependence of effective concentration *N*˜\*= *<sup>X</sup>*<sup>ε</sup> 4/7 <sup>×</sup> *Xg* 6/7 <sup>×</sup> *<sup>N</sup>*˜ <sup>≡</sup> *<sup>N</sup>*˜*InP* on *<sup>W</sup>* . Concentration is measured in cm-3, field − in V/cm and thickness *W* − in μm.

**Figure 7.** Dependence of avalanche breakdown voltage *VBD* of homogeneous *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup> structure on dopant concen‐ tration *N* in base: solid line − (31) and (32), dotted line − expressions (10) and (11). In lower inset: dependence of relative voltage Δ*V* (*rel*) <sup>=</sup> *VBD* / *VBD*(0, *<sup>W</sup>* ) <sup>−</sup>1 normalized to concentration *N*˜(*<sup>W</sup>* ) at *<sup>N</sup>* <sup>≤</sup>4 × *<sup>N</sup>*˜(*<sup>W</sup>* ). In upper inset: de‐ pendence of effective Δ*V*max \* <sup>=</sup> *<sup>X</sup>*<sup>ε</sup> <sup>−</sup>3/7 <sup>×</sup> *Xg* 6/7 <sup>×</sup> *VBD* <sup>−</sup>*VBD*(0, *<sup>W</sup>* ) max≡Δ*V*max (*InP*) on base thickness *W* . Voltage is measured in V, thickness *W* − in μm.

**Figure 8.** Dependence of effective avalanche breakdown voltage *VBD*

tion is measured in cm-3, voltage − in V, thickness *W* − in μm

structure on thickness of its base *W* for three values of effective concentration *<sup>N</sup>* \*= *<sup>X</sup>*<sup>ε</sup>

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

<sup>1</sup><sup>−</sup> *<sup>N</sup>* \*= 3 × 1016cm-3, 2 <sup>−</sup> *<sup>N</sup>* \*= 3 × 1014cm-3, 3 <sup>−</sup> *<sup>N</sup>* \*= 3 × 1012cm-3. In inset is shown dependence *W*˜ on *<sup>N</sup>* \*

*p* <sup>+</sup> −*n* −*n* <sup>+</sup>

\* <sup>=</sup> *<sup>X</sup>*<sup>ε</sup>

<sup>−</sup>3/7 <sup>×</sup> *Xg*

6/7 <sup>×</sup>*VBD*<sup>≡</sup> *VBD*

(*InP*)

4/7 <sup>×</sup> *Xg*

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55

of homogeneous

6/7 <sup>×</sup> *<sup>N</sup>* <sup>≡</sup> *NInP*:

. Concentra‐

**Figure 8.** Dependence of effective avalanche breakdown voltage *VBD* \* <sup>=</sup> *<sup>X</sup>*<sup>ε</sup> <sup>−</sup>3/7 <sup>×</sup> *Xg* 6/7 <sup>×</sup>*VBD*<sup>≡</sup> *VBD* (*InP*) of homogeneous *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure on thickness of its base *W* for three values of effective concentration *<sup>N</sup>* \*= *<sup>X</sup>*<sup>ε</sup> 4/7 <sup>×</sup> *Xg* 6/7 <sup>×</sup> *<sup>N</sup>* <sup>≡</sup> *NInP*: <sup>1</sup><sup>−</sup> *<sup>N</sup>* \*= 3 × 1016cm-3, 2 <sup>−</sup> *<sup>N</sup>* \*= 3 × 1014cm-3, 3 <sup>−</sup> *<sup>N</sup>* \*= 3 × 1012cm-3. In inset is shown dependence *W*˜ on *<sup>N</sup>* \* . Concentra‐ tion is measured in cm-3, voltage − in V, thickness *W* − in μm

**Figure 7.** Dependence of avalanche breakdown voltage *VBD* of homogeneous *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*<sup>n</sup>* <sup>−</sup>*<sup>n</sup>* <sup>+</sup>

relative voltage Δ*V* (*rel*)

pendence of effective Δ*V*max

in V, thickness *W* − in μm.

\* <sup>=</sup> *<sup>X</sup>*<sup>ε</sup>

54 Photodiodes - From Fundamentals to Applications

<sup>−</sup>3/7 <sup>×</sup> *Xg*

tration *N* in base: solid line − (31) and (32), dotted line − expressions (10) and (11). In lower inset: dependence of

6/7 <sup>×</sup> *VBD* <sup>−</sup>*VBD*(0, *<sup>W</sup>* ) max≡Δ*V*max

<sup>=</sup> *VBD* / *VBD*(0, *<sup>W</sup>* ) <sup>−</sup>1 normalized to concentration *N*˜(*<sup>W</sup>* ) at *<sup>N</sup>* <sup>≤</sup>4 × *<sup>N</sup>*˜(*<sup>W</sup>* ). In upper inset: de‐

structure on dopant concen‐

(*InP*) on base thickness *W* . Voltage is measured

#### *3.6.2. To сorrelation between values of impact ionization coefficients of electrons and holes*

In Section 3.4 is shown that there is reason to suppose existence of some correlation between values of impact ionization coefficients of electrons *α*(*Е*) and holes *β*(*Е*), and form of required relation (expression (33) and (45)) is proposed. It is obvious from Fig. 9 that values *Z*<sup>0</sup> <sup>≡</sup>*<sup>ε</sup>* <sup>3</sup> / *Eg* 6 may differ by many orders of magnitude in different semiconductors. At the same time, for presented in Fig. 9 *Ge*, *Si* and *GaP*, function *c*(*Е*) (see relations (33) and (45)) in range of fields where *α*(*Е*) and *β*(*Е*) vary in several orders of magnitude (Okuto & Crowell, 1975), remains, as it follows from (33) and (45), of the order of unity. Calculations based on experimental de‐ pendences *α*(*Е*) and *β*(*Е*) (Cook et al, 1982) show that in *InP* value *c*(*Е*) is some more closely to 1. It is evident from Fig. 10 that for *GaAs*, regardless of orientation of crystal with respect to electric field, function *c*(*Е*) depends weakly on *Е* in comparison with impact ionization coeffi‐ cients of charge carriers (which values are taken from (Lee & Sze, 1980)), and differs from uni‐ ty by no more than 2-3 times. A similar situation takes place in *Ge* (Fig. 11, according to (Mikawa et al, 1980)). As shown in (Kobajashi et al, 1969) dependences *α*(*Е*) and *β*(*Е*) meas‐ ured in (Miller, 1955), (McKay & McAfee, 1953) in the range of fields *Е* =(1.5÷2.7)×10<sup>5</sup> V/cm can be described in *Ge* by formulas *α*(*Е*)=7.81×10−<sup>34</sup> ×*Е* <sup>7</sup> , *β*(*Е*)≅2*α*(*Е*). This result agrees well with expression (33). Note that, *c*(*Е*) differs from unity approximately by the same factor, as values *α*(*Е*) and *β*(*Е*) for the same material obtained by different authors differ, respectively, from each other (Sze, 1981), (Tsang, 1985), (Forrest et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (Cook et al, 1982), (Okuto & Crowell, 1974), (Okuto & Crowell, 1975), (Lee & Sze, 1980), (Mikawa et al, 1980), (Kuzmin et al, 1975). Using procedure described in Section 3.4, we can al‐ so determine relation between *α*(*Е*) and *β*(*E*)= *K*(*E*)×*α*(*E*) in the case when relations (11) and (13) are not satisfied (Grekhov & Serezhkin, 1980). It seems, relation required for such case, i.e. under assumption of power dependence *α* on *Е* and *K*(*E*)=*const*, was obtained for the first time in (Shotov, 1958).

#### *3.6.3. To Miller's relation*

From (48), (55) and (67) follow that, exponents in Miller's relation (46) for multiplication fac‐ tors of electrons and holes are given by

$$m\_n \times \ln \upsilon = \ln \left[ \left( K\_0^{\upsilon^\circ} - 1 \right) / \left( K\_0 - 1 \right) \right] \tag{74}$$

ponents *nn* and *n<sup>р</sup>* are also expressed by (74) and (75) but in right side of those expressions *v*˜ substitutes *v* and *<sup>ξ</sup>* =8. Value of exponent *n*˜ lies between values *nn* and *nр*. From (1) and (2) apparent that when *<sup>α</sup>* <sup>=</sup>*<sup>β</sup>* then factors *Mn*, *Mp* and *<sup>M</sup>*˜ coincide with each other, i.e., *nn*=*nр*=*n*˜=

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57

**Figure 9.** Dependence *Z*(*E*) [relation (33)] in Ge, Si, and *GaP* for α(*E*) and β(*E*) from (Okuto & Crowell, 1975)

*n*, and, as it follows from expressions (74) and (75), regardless of bias voltage applied, *n*= 4, 5 and 7 for situations 1, 2 and 3, respectively. Exponents in Miller's relation have the same val‐ ues when *V* < <*VBD*, more exactly, when |ln*K*<sup>0</sup> / ln*v*|<<*ξ*, regardless of ratio

0 0

*K K* = ºx´ ´ = ºx´ - - (76)

*K K*

0 0

ln ln , . ( 1) ( 1) *n nB p pB*

*K*<sup>0</sup> =*β*(*E*0)/ *α*(*E*0). When *V* →*VBD* or more exactly, if

1

*<sup>ξ</sup>* }, *<sup>M</sup>* >>1

0

*nn K nn*

*<sup>ξ</sup>* |ln*K*<sup>0</sup> <sup>|</sup> ;

*<sup>Δ</sup><sup>v</sup>* =1−*<sup>v</sup>* < <min{ <sup>1</sup>

Then for these situations

$$\ln n\_p \times \ln v = \ln \left[ \frac{K\_0}{K\_0 - 1} (1 - K\_0^{-v^{\tilde{\varsigma}}}) \right],\tag{75}$$

where *ξ*= 4, 5 and 7 for stepwise *p* −*n* junction, linear *p* −*n* junction and very thin (66) *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (situation 1, 2 and 3, respectively). If thickness of base in *p* <sup>+</sup> −*n* −*n* <sup>+</sup> struc‐ ture is not very small, i.e., *W*<sup>0</sup> <sup>&</sup>lt;*<sup>W</sup>* <sup>&</sup>lt;*<sup>W</sup>*˜ (situation 4) then as it follows from formula (60), ex‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 57

**Figure 9.** Dependence *Z*(*E*) [relation (33)] in Ge, Si, and *GaP* for α(*E*) and β(*E*) from (Okuto & Crowell, 1975)

ponents *nn* and *n<sup>р</sup>* are also expressed by (74) and (75) but in right side of those expressions *v*˜ substitutes *v* and *<sup>ξ</sup>* =8. Value of exponent *n*˜ lies between values *nn* and *nр*. From (1) and (2) apparent that when *<sup>α</sup>* <sup>=</sup>*<sup>β</sup>* then factors *Mn*, *Mp* and *<sup>M</sup>*˜ coincide with each other, i.e., *nn*=*nр*=*n*˜= *n*, and, as it follows from expressions (74) and (75), regardless of bias voltage applied, *n*= 4, 5 and 7 for situations 1, 2 and 3, respectively. Exponents in Miller's relation have the same val‐ ues when *V* < <*VBD*, more exactly, when |ln*K*<sup>0</sup> / ln*v*|<<*ξ*, regardless of ratio *K*<sup>0</sup> =*β*(*E*0)/ *α*(*E*0). When *V* →*VBD* or more exactly, if

$$
\Delta v = 1 - v < \text{min}\left\{ \frac{1}{\xi \mid \text{Im} K\_0 \mid \ ' \frac{\xi}{\xi} \}} ; \frac{1}{\xi} \right\} , M > 1
$$

Then for these situations

*3.6.2. To сorrelation between values of impact ionization coefficients of electrons and holes*

can be described in *Ge* by formulas *α*(*Е*)=7.81×10−<sup>34</sup> ×*Е* <sup>7</sup>

56 Photodiodes - From Fundamentals to Applications

time in (Shotov, 1958).

*3.6.3. To Miller's relation*

tors of electrons and holes are given by

*p*

In Section 3.4 is shown that there is reason to suppose existence of some correlation between values of impact ionization coefficients of electrons *α*(*Е*) and holes *β*(*Е*), and form of required relation (expression (33) and (45)) is proposed. It is obvious from Fig. 9 that values *Z*<sup>0</sup> <sup>≡</sup>*<sup>ε</sup>* <sup>3</sup> / *Eg*

may differ by many orders of magnitude in different semiconductors. At the same time, for presented in Fig. 9 *Ge*, *Si* and *GaP*, function *c*(*Е*) (see relations (33) and (45)) in range of fields where *α*(*Е*) and *β*(*Е*) vary in several orders of magnitude (Okuto & Crowell, 1975), remains, as it follows from (33) and (45), of the order of unity. Calculations based on experimental de‐ pendences *α*(*Е*) and *β*(*Е*) (Cook et al, 1982) show that in *InP* value *c*(*Е*) is some more closely to 1. It is evident from Fig. 10 that for *GaAs*, regardless of orientation of crystal with respect to electric field, function *c*(*Е*) depends weakly on *Е* in comparison with impact ionization coeffi‐ cients of charge carriers (which values are taken from (Lee & Sze, 1980)), and differs from uni‐ ty by no more than 2-3 times. A similar situation takes place in *Ge* (Fig. 11, according to (Mikawa et al, 1980)). As shown in (Kobajashi et al, 1969) dependences *α*(*Е*) and *β*(*Е*) meas‐ ured in (Miller, 1955), (McKay & McAfee, 1953) in the range of fields *Е* =(1.5÷2.7)×10<sup>5</sup>

with expression (33). Note that, *c*(*Е*) differs from unity approximately by the same factor, as values *α*(*Е*) and *β*(*Е*) for the same material obtained by different authors differ, respectively, from each other (Sze, 1981), (Tsang, 1985), (Forrest et al, 1983), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf 1977), (Dmitriev et al, 1987), (Tager & Vald-Perlov, 1968), (Cook et al, 1982), (Okuto & Crowell, 1974), (Okuto & Crowell, 1975), (Lee & Sze, 1980), (Mikawa et al, 1980), (Kuzmin et al, 1975). Using procedure described in Section 3.4, we can al‐ so determine relation between *α*(*Е*) and *β*(*E*)= *K*(*E*)×*α*(*E*) in the case when relations (11) and (13) are not satisfied (Grekhov & Serezhkin, 1980). It seems, relation required for such case, i.e. under assumption of power dependence *α* on *Е* and *K*(*E*)=*const*, was obtained for the first

From (48), (55) and (67) follow that, exponents in Miller's relation (46) for multiplication fac‐

0 0 ln ln ( 1) / ( 1) , *<sup>v</sup>*

0

where *ξ*= 4, 5 and 7 for stepwise *p* −*n* junction, linear *p* −*n* junction and very thin (66) *p* <sup>+</sup> −*n* −*n* <sup>+</sup> structure (situation 1, 2 and 3, respectively). If thickness of base in *p* <sup>+</sup> −*n* −*n* <sup>+</sup>

ture is not very small, i.e., *W*<sup>0</sup> <sup>&</sup>lt;*<sup>W</sup>* <sup>&</sup>lt;*<sup>W</sup>*˜ (situation 4) then as it follows from formula (60), ex‐

0 ln ln (1 ) , <sup>1</sup>

*<sup>K</sup> nv K K* <sup>x</sup> - é ù ´= - ê ú - ë û

0

*v*

*<sup>n</sup> nv K K* <sup>x</sup> é ù ´= - - ê ú ë û (74)

6

V/cm

(75)

struc‐

, *β*(*Е*)≅2*α*(*Е*). This result agrees well

$$m\_n = n\_{n\text{B}} \equiv \xi \times K\_0 \times \frac{\ln K\_0}{(K\_0 - 1)},\\ n\_p = n\_{p\text{B}} \equiv \xi \times \frac{\ln K\_0}{(K\_0 - 1)}.\tag{76}$$

Graphs in Fig. 4 allow comparing numerical values of exponents *nnB* and *nрB* calculated in (Le‐ guerre & Urgell, 1976) *nB* (*c*) and analytical *nB* (*a*) computed by formulas (76) for asymmetrical stepwise *p* −*n* junction. Like as in (Leguerre & Urgell, 1976), experimentally determined func‐ tional dependencies *α*(*E*0) and *β*(*E*0) (Sze & Gibbons, 1966) were used in calculations of de‐ pendences *nB* (*a*) . As follows from (46), when *M* >>1, then ratio of analytical value of multiplication factor *M* (*a*) to calculated *M* (*c*) equals to ratio *nB* (*c*) to *nB* (*a*) (Fig. 11-13). It obviously from Fig. 11-13 that for all considered semiconductors (with curves *α*(*Е*) and *β*(*Е*) taken from (Sze & Gibbons, 1966)), dependences *M* (*a*) (*V* ) and *M* (*c*) (*V* ) do not differ by more than 50 %. De‐ pendences of exponents *nn* (*a*) and *np* (*a*) on voltage and *nnB* (*a*) and *npB* (*a*) on ratio *K* =*β* / *α* are illustrat‐ ed in Fig. 3 and 14, respectively. It should be noted that numerical values of exponent in Miller's relation, as well as, value *VBD* depend, obviously, on what functions *α*(*Е*) and *β*(*Е*) are used in (1) and (2) in calculations. Let's take the simplest case when *α*(*E*)=*β*(*E*) and *p* −*n* junc‐ tion is stepwise. Varying expressions (1) and (2), we find that under considered conditions

$$m\_{B} = \frac{\varepsilon \varepsilon\_{0}}{500 \times q \times N\_{eff}} \times a(E\_{BD}) \times E\_{BD'} \tag{77}$$

where *EBD* =*E*(0) at *V* =*VBD* is determined from condition

$$\int\_{0}^{E\_{\rm E}} \alpha(E) dE = \frac{100}{\varepsilon \varepsilon\_{0}} \times N\_{eff} \tag{78}$$

In Fig. 15a are shown dependences *nB*(*Neff* ) calculated from relations (77) and (78) for four values *α*(*E*)=*β*(*E*) obtained for *GaAs* by different authors (Grekhov & Serezhkin, 1980), (Okuto & Crowell, 1975), (Kressel & Kupsky, 1966), (Nuttall & Nield, 1974). It is seen that analytical value *nnB* =*npB* =4 calculated by formulas (76) approximately equals to mean value with respect to curves 1-4 in Fig. 15a. According to obtained above results expressions (48)- (53) are not valid when concentration

$$N\_{\rm eff} \geqslant (N\_{\rm eff})\_{\rm max} \cong 2 \times 10^{17} \times (E\_g)^2 \times E\_{ion}^{-4/3} \tag{79}$$

which for many semiconductors is of the order of 1017 cm-3. At such high concentrations, as it follows from Section 3.4 and (Kholodnov, 1988-1) and relations (1) and (2), for stepwise *p* −*n* junction

$$n\_n = \frac{\ln\left(\left(K\_0^v - 1\right) / \left(K\_0 - 1\right)\right)}{\ln v}, n\_p = n\_n + \frac{1 - v}{\ln v} \ln K\_{0^\prime} \tag{80}$$

**Figure 10.** Dependence С(*E*) at different orientations of *GaAs* crystal with respect to electric field for values α(*E*) and

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β(*E*) from (Lee & Sze, 1980)

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 59

Graphs in Fig. 4 allow comparing numerical values of exponents *nnB* and *nрB* calculated in (Le‐

stepwise *p* −*n* junction. Like as in (Leguerre & Urgell, 1976), experimentally determined func‐ tional dependencies *α*(*E*0) and *β*(*E*0) (Sze & Gibbons, 1966) were used in calculations of de‐

from Fig. 11-13 that for all considered semiconductors (with curves *α*(*Е*) and *β*(*Е*) taken from

on voltage and *nnB*

<sup>0</sup> () , <sup>500</sup> *<sup>B</sup> BD BD eff n E E q N*

> 0 0 <sup>100</sup> ( )

*eff* a =´ *E dE N*

In Fig. 15a are shown dependences *nB*(*Neff* ) calculated from relations (77) and (78) for four values *α*(*E*)=*β*(*E*) obtained for *GaAs* by different authors (Grekhov & Serezhkin, 1980), (Okuto & Crowell, 1975), (Kressel & Kupsky, 1966), (Nuttall & Nield, 1974). It is seen that analytical value *nnB* =*npB* =4 calculated by formulas (76) approximately equals to mean value with respect to curves 1-4 in Fig. 15a. According to obtained above results expressions (48)-

max≅2×10<sup>17</sup> ×(*Еg*)<sup>2</sup> <sup>×</sup>*Еion*

which for many semiconductors is of the order of 1017 cm-3. At such high concentrations, as it follows from Section 3.4 and (Kholodnov, 1988-1) and relations (1) and (2), for stepwise *p* −*n*

ln ( 1) / ( 1) <sup>1</sup> , ln , ln ln

*v v*

*K K v n nn K*

ed in Fig. 3 and 14, respectively. It should be noted that numerical values of exponent in Miller's relation, as well as, value *VBD* depend, obviously, on what functions *α*(*Е*) and *β*(*Е*) are used in (1) and (2) in calculations. Let's take the simplest case when *α*(*E*)=*β*(*E*) and *p* −*n* junc‐ tion is stepwise. Varying expressions (1) and (2), we find that under considered conditions

(*V* ) and *M* (*c*)

. As follows from (46), when *M* >>1, then ratio of analytical value of

(*a*)

(*c*) to *nB* (*a*)

 and *npB* (*a*)

ee <sup>=</sup> ´a ´ ´ ´ (77)

ee ò (78)

0


equals to ratio *nB*

computed by formulas (76) for asymmetrical

(*V* ) do not differ by more than 50 %. De‐

(Fig. 11-13). It obviously

on ratio *K* =*β* / *α* are illustrat‐

<sup>−</sup>4/3 (79)

(*a*)

guerre & Urgell, 1976) *nB*

58 Photodiodes - From Fundamentals to Applications

(*a*)

(Sze & Gibbons, 1966)), dependences *M* (*a*)

(53) are not valid when concentration

junction

multiplication factor *M* (*a*)

pendences of exponents *nn*

pendences *nB*

(*c*)

(*a*) and *np* (*a*)

where *EBD* =*E*(0) at *V* =*VBD* is determined from condition

*BD E*

*Neff* >(*Neff* )

*v*

( 0 0 )

*n p n*

and analytical *nB*

to calculated *M* (*c*)

**Figure 10.** Dependence С(*E*) at different orientations of *GaAs* crystal with respect to electric field for values α(*E*) and β(*E*) from (Lee & Sze, 1980)

**Figure 11.** Dependence of ratio between analytical values of avalanche multiplication factors М (а) of electrons and holes and numerical values М (с) (Leguerre & Urgell, 1976) in stepwise asymmetric *Ge p* −*n* junction on value of multi‐ plication factor М =М (а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resis‐ tivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 − 6 × 1016. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

**Figure 12.** Dependence of ratio between analytical values of avalanche multiplication factors М (а)

cation factor М =М (а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 − 6 × 1016. Values *K*(*E*), as in (Leguerre &

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(Leguerre & Urgell, 1976) in stepwise asymmetric *Si p* −*n* junction on value of multipli‐

holes and numerical values М (с)

Urgell, 1976), are taken from (Sze & Gibbons, 1966)

of electrons and

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 61

**Figure 12.** Dependence of ratio between analytical values of avalanche multiplication factors М (а) of electrons and holes and numerical values М (с) (Leguerre & Urgell, 1976) in stepwise asymmetric *Si p* −*n* junction on value of multipli‐ cation factor М =М (а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 − 6 × 1016. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

**Figure 11.** Dependence of ratio between analytical values of avalanche multiplication factors М (а)

tivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 − 6 × 1016. Values *K*(*E*), as in (Leguerre

(Leguerre & Urgell, 1976) in stepwise asymmetric *Ge p* −*n* junction on value of multi‐

of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resis‐

holes and numerical values М (с)

& Urgell, 1976), are taken from (Sze & Gibbons, 1966)

60 Photodiodes - From Fundamentals to Applications

plication factor М =М (а)

of electrons and

**Figure 13.** Dependence of ratio between analytical values of avalanche multiplication factors М (а) of electrons and holes and numerical values М (с) (Leguerre & Urgell, 1976) in stepwise asymmetric *GaAs* (solid lines) and *GaP* (dashed lines) *p* −*n* junctions on value of multiplication factor М =М (а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 − 6 × 1016. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

**Figure 14.** Dependence of limiting values *nB* = lim

for "thick" abrupt р −*n* junction on *K* =β / α

*<sup>V</sup>* <sup>→</sup>*VBD*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

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63

*n*(*V* ) of exponents in Miller's relation for electron *nn* and holes *np*

**Figure 14.** Dependence of limiting values *nB* = lim *<sup>V</sup>* <sup>→</sup>*VBD n*(*V* ) of exponents in Miller's relation for electron *nn* and holes *np* for "thick" abrupt р −*n* junction on *K* =β / α

**Figure 13.** Dependence of ratio between analytical values of avalanche multiplication factors М (а)

6 × 1016. Values *K*(*E*), as in (Leguerre & Urgell, 1976), are taken from (Sze & Gibbons, 1966)

lines) *p* −*n* junctions on value of multiplication factor М =М (а) of charge carriers. Solid lines – electrons, dashed – holes. Dopant concentration in high-resistivity part of *p* −*n* junction *N* , cm-3: 1 − 1015, 2 − 3 × 1015, 3 − 1016, 4 − 3 × 1016, 5 −

(Leguerre & Urgell, 1976) in stepwise asymmetric *GaAs* (solid lines) and *GaP* (dashed

holes and numerical values М (с)

62 Photodiodes - From Fundamentals to Applications

of electrons and

moreover

$$
\boldsymbol{m}\_{\boldsymbol{n}\boldsymbol{B}} = \boldsymbol{K}\_0 \ln \left( \boldsymbol{K}\_0 \,/\left( \boldsymbol{K}\_0 - 1 \right) \right) = \boldsymbol{K}\_0 \times \boldsymbol{n}\_{p\boldsymbol{B}}.\tag{81}
$$

value *VBD* =*Eion* / *q*. This conclusion accords with results of studies (Grekhov & Serezhkin, 1980), (Nuttall & Nield, 1974). Obtained results agree well with experimental results for a num‐ ber of *p* −*n* structures, including based on *Ge*, *Si*, *GaAs*, *GaP* (Sze, 1981), (Tsang, 1985), (Still‐ man et al, 1983), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Bogdanov et al, 1986), (Cook et al, 1982), (Shotov, 1958). We present here three cases of studies. In experimental study (Miller, 1955) of breakdown in *Ge* stepwise *p* −*n* junc‐ tion was found that measured values of exponents in Miller's relation were lying in range from 3 to 6.6. The same values of exponents are obtained from expressions (74) and (75) with *ξ* =4 if we take into account that in *Ge* with doping levels used in (Miller, 1955) *К*0≅2÷3 (Sze, 1981), (Tsang, 1985), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Shotov, 1958). In experimental study (Bogdanov et al, 1986) of APD based on MIS structure (metal-in‐ sulator-semiconductor APD) multiplication of charge carriers occurs in thick *p* −*Si* substrate. From point of view of avalanche process this structure is similar to asymmetric stepwise *n* <sup>+</sup> − *p* junction. Therefore, avalanche process in MIS APD can be described by expressions (74)-(76) with *ξ* =4. Concentration of shallow acceptors in substrate of investigated structure was 1015 cm-3. At this doping avalanche breakdown in *Si* occurs when electric field near insulator-semi‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

(Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Kuzmin et al, 1975). Measured in (Bogdanov et al, 1986) value *nn* at *VBD* −*V* < <*VBD* was found equal to 0.2.

perimental (Shotov, 1958) and calculated by formulas (48) and (55) values of multiplication fac‐ tors of electrons *Mn*(*V* ) and holes *Mp*(*V* ) in *Ge* stepwise and linear *p* −*n* junctions. Obviously, for these *p* −*n* junctions, experimental and analytical values of multiplication factors differ

from each other by less than 20 % in whole voltage *V* range used in measurements.

**Experiment**

0.65 1.35 1.30 0.70 1.50 1.44 0.75 1.75 1.65 0.80 2.10 1.98 0.85 2.65 2.55 0.90 3.70 3.71 0.95 7.00 7.30

**Table 1.** Experimental (Shotov, 1958) and computed [from Equation (48)] hole avalanche multiplication factor *Mp* in step-wise *p* −*n* junction in *p* −*Ge* for different ratios of applied voltage to avalanche breakdown voltage *V* / *VBD*. It is

V/cm (Sections 3.1 and 3.2, (Sze, 1981), (Osipov

follows that *nn<sup>В</sup>* =0.186. In Tables 1 and 2 are presented ex‐

**Mp**

**(Shotov, 1958) Theory**

(Sze, 1981), (Tsang, 1985),

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65

conductor interface reaches value *EBD* <sup>≅</sup>3×10<sup>5</sup>

From formulas (76) with *К*0≅10−<sup>2</sup>

**V/VBD**

assumed that *K*<sup>0</sup> = 2 (Shotov, 1958)

& Kholodnov, 1987), (Sze & Gibbons, 1966)), and therefore *К*0≅10−<sup>2</sup>

**Figure 15.** Dependences *nB*(*Neff* ) in *GaAs* calculated on the base of different dependences α(*E*)=β(*E*), taken from: 1 − (Shabde & Yeh, 1970), 2 − (Grekhov & Serezhkin, 1980), 3 − (Okuto & Crowell, 1975), 4 − (Kressel & Kupsky, 1966), 5 − (Sze & Gibbons, 1966). Dashed lines − analytical values

For comparison, in Fig. 15b are presented dependences of *nB* (*c*) (*Neff* ) and *nB* (*a*) (*Neff* )=1 for the case *α* =*β*, when *nn<sup>В</sup>* =*nрВ* =*nВ*. It is seen that value *nB* (*a*) (*Neff* )= 1 is approximately equal to mean value with reference to curves 2, 3 and 5 in Fig. 15b plotted on the base of numerical data. Note that starting from *Neff* ≅(*Neff* ) max breakdown voltage *VBD* dependence on *Neff* becomes, with growth *Neff* , more and more weaker than that described by equation (49), and in limit tends to moreover

64 Photodiodes - From Fundamentals to Applications

( ) 0 00 0 ln / ( 1) . *nB pB n K K K Kn* = -=´ (81)

**Figure 15.** Dependences *nB*(*Neff* ) in *GaAs* calculated on the base of different dependences α(*E*)=β(*E*), taken from: 1 − (Shabde & Yeh, 1970), 2 − (Grekhov & Serezhkin, 1980), 3 − (Okuto & Crowell, 1975), 4 − (Kressel & Kupsky, 1966), 5 −

value with reference to curves 2, 3 and 5 in Fig. 15b plotted on the base of numerical data. Note

growth *Neff* , more and more weaker than that described by equation (49), and in limit tends to

(*a*)

(*c*)

max breakdown voltage *VBD* dependence on *Neff* becomes, with

(*Neff* ) and *nB*

(*Neff* )= 1 is approximately equal to mean

(*a*)

(*Neff* )=1 for the

(Sze & Gibbons, 1966). Dashed lines − analytical values

that starting from *Neff* ≅(*Neff* )

For comparison, in Fig. 15b are presented dependences of *nB*

case *α* =*β*, when *nn<sup>В</sup>* =*nрВ* =*nВ*. It is seen that value *nB*

value *VBD* =*Eion* / *q*. This conclusion accords with results of studies (Grekhov & Serezhkin, 1980), (Nuttall & Nield, 1974). Obtained results agree well with experimental results for a num‐ ber of *p* −*n* structures, including based on *Ge*, *Si*, *GaAs*, *GaP* (Sze, 1981), (Tsang, 1985), (Still‐ man et al, 1983), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Bogdanov et al, 1986), (Cook et al, 1982), (Shotov, 1958). We present here three cases of studies. In experimental study (Miller, 1955) of breakdown in *Ge* stepwise *p* −*n* junc‐ tion was found that measured values of exponents in Miller's relation were lying in range from 3 to 6.6. The same values of exponents are obtained from expressions (74) and (75) with *ξ* =4 if we take into account that in *Ge* with doping levels used in (Miller, 1955) *К*0≅2÷3 (Sze, 1981), (Tsang, 1985), (Miller, 1955), (Grekhov & Serezhkin, 1980), (Stillman & Wolf, 1977), (Shotov, 1958). In experimental study (Bogdanov et al, 1986) of APD based on MIS structure (metal-in‐ sulator-semiconductor APD) multiplication of charge carriers occurs in thick *p* −*Si* substrate. From point of view of avalanche process this structure is similar to asymmetric stepwise *n* <sup>+</sup> − *p* junction. Therefore, avalanche process in MIS APD can be described by expressions (74)-(76) with *ξ* =4. Concentration of shallow acceptors in substrate of investigated structure was 1015 cm-3. At this doping avalanche breakdown in *Si* occurs when electric field near insulator-semi‐ conductor interface reaches value *EBD* <sup>≅</sup>3×10<sup>5</sup> V/cm (Sections 3.1 and 3.2, (Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966)), and therefore *К*0≅10−<sup>2</sup> (Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), (Sze & Gibbons, 1966), (Stillman & Wolf, 1977), (Kuzmin et al, 1975). Measured in (Bogdanov et al, 1986) value *nn* at *VBD* −*V* < <*VBD* was found equal to 0.2. From formulas (76) with *К*0≅10−<sup>2</sup> follows that *nn<sup>В</sup>* =0.186. In Tables 1 and 2 are presented ex‐ perimental (Shotov, 1958) and calculated by formulas (48) and (55) values of multiplication fac‐ tors of electrons *Mn*(*V* ) and holes *Mp*(*V* ) in *Ge* stepwise and linear *p* −*n* junctions. Obviously, for these *p* −*n* junctions, experimental and analytical values of multiplication factors differ from each other by less than 20 % in whole voltage *V* range used in measurements.


**Table 1.** Experimental (Shotov, 1958) and computed [from Equation (48)] hole avalanche multiplication factor *Mp* in step-wise *p* −*n* junction in *p* −*Ge* for different ratios of applied voltage to avalanche breakdown voltage *V* / *VBD*. It is assumed that *K*<sup>0</sup> = 2 (Shotov, 1958)


( )

If variation of electric field within length of tunneling *ΔЕ* < <*Е*, i.e. specific length of varia‐

( ) [1 ( / 2 ) / ] *g*

then true ABC barrier coincides to high degree of accuracy with triangle ABC′ to which cor‐

2 0 ( ) ( ) 1, 2 () *Nx Eg <sup>x</sup>*

´ d º << ee ´

*E x*

*Eg*

As shown below, due to large values of field *Е* at avalanche breakdown of *p* −*n* structures, inequality (85) is valid for almost all materials up to concentration *N* =1017 cm-3 and even

Under these conditions specific rates of charge carriers' tunnel generation *gTi*(*x*) in layers I

<sup>1</sup> <sup>2</sup> ( ) ( ) exp , ( ) *Ti i*

*q x E x*

¶ é ù º´ = ´ ´ -ê ú ¶ ë û (87)

*J a g x A Ex*

*T*

When *N* (*x*)=*const* then equation (83) is exact. As can be seen from Fig. 16a, if

It follows from (83) and Poisson equation that inequalities (84) are satisfied if

*l*

*Ti Ti*

and II of structure can be described by expression

*<sup>T</sup>* (*Eg*, *E*)=

*E*

*<sup>T</sup> xl x*

*x E q E x dx* +

*g*

*<sup>T</sup>* , then expanding function *Е*(*<sup>x</sup>* ′

find that in the first order of parameter of smallness *l*

*T*

*l*

tion of field *l*

at that

high.

*<sup>E</sup>* > >*l*

responds uniform field *Е*(*х*) (Fig. 16b).

( ') '

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

*<sup>T</sup>* / *l*

*qE x l E E x* <sup>=</sup> ´ - ´¶ ¶ (83)

, , *CC l l CB E E TT c g* ¢ º D << º D << ¢ ¢ (84)

= ´ ò (82)

) in Taylor series around point *x* ′

*<sup>q</sup>* <sup>×</sup>*E*(*x*) (86)

*<sup>E</sup>* equation (82) takes the form

http://dx.doi.org/10.5772/50778

= *x*, we

67

(85)

**Table 2.** Experimental (\*) (Shotov, 1958) and computed [from Equation (55)] avalanche multiplication factors *Mp* and *Mn* for holes and electrons in *Ge* linear *p* −*n* junction for different ratios of applied voltage to avalanche breakdown voltage *V* / *VBD* (Shotov, 1958)

Finally, it is interesting to analyze application of expressions (45) and (76) to describe ava‐ lanche process in *InSb*. The fact is that dependence *α*(*Е*) in *InSb* was quite well known al‐ ready in 1967 (Baertsch, 1967), but no one could obtain information about dependence *β*(*Е*) (Dmitriev et al, 1987), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968). Substituting in (45) dependence *α*(*Е*) for *InSb* (Baertsch, 1967), (Dmitriev et al, 1983), (Dmi‐ triev et al, 1982), (Gavrjushko et al, 1968), we find that ratio *K* =*β*(*E*) / *α*(*E*) is vanishingly small up to electric field *E* ≅4×10<sup>4</sup> V/cm resulting in extremely high value *nрВ* when at the same time value *nn<sup>В</sup>* is extremely small. It means that *Mn*(*V* ) becomes much larger than uni‐ ty, even at voltages *Vb* noticeably lower avalanche breakdown voltage *VBD*, and value *Mp*(*V* ) remains equal to unity up to values *Vb* very close to *VBD*. Effect obtained from appli‐ cation of relations (45) and (76) accords very well with experimental data (Baertsch, 1967), (Dmitriev et al, 1983) and explains why multiplication of holes in *InSb* is extremely hard to observe (Dmitriev et al, 1987), (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968).

#### **4. Tunnel currents in avalanche heterophotodiodes**

#### **4.1. Calculation of tunnel currents in approximation of quasi-uniform electric field and conditions of its applicability**

In act of interband tunneling electron from valence band overcomes potential barrier ABC (Fig. 16a). The length of tunneling *l <sup>T</sup>* , i.e. length on which energy of bottom of conduction band *Ес*(*x*) changes by value equal to *Еg* is found by solving integral equation

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 67

$$E\_{\mathcal{g}} = q \times \int\_{\infty}^{x + l\_{\mathcal{T}}(\mathbf{x})} E(\mathbf{x}') d\mathbf{x}' \tag{82}$$

If variation of electric field within length of tunneling *ΔЕ* < <*Е*, i.e. specific length of varia‐ tion of field *l <sup>E</sup>* > >*l <sup>T</sup>* , then expanding function *Е*(*<sup>x</sup>* ′ ) in Taylor series around point *x* ′ = *x*, we find that in the first order of parameter of smallness *l <sup>T</sup>* / *l <sup>E</sup>* equation (82) takes the form

$$l\_T = \frac{E\_g}{qE(\mathbf{x}) \times \left[1 - (l\_T / 2E) \times \left| \partial E / \partial \mathbf{x} \right| \right]}\tag{83}$$

When *N* (*x*)=*const* then equation (83) is exact. As can be seen from Fig. 16a, if

$$\left| \mathbf{C}' \mathbf{C} \right| \equiv \Delta l\_T << l\_{T'} \left| \mathbf{C}' \mathbf{B}' \right| \equiv \Delta E\_c << E\_{g'} \tag{84}$$

then true ABC barrier coincides to high degree of accuracy with triangle ABC′ to which cor‐ responds uniform field *Е*(*х*) (Fig. 16b).

It follows from (83) and Poisson equation that inequalities (84) are satisfied if

$$\mathcal{S}(\mathbf{x}) \equiv \frac{N(\mathbf{x}) \times E\_{\mathcal{g}}}{2\varepsilon\varepsilon\_{0} \times E^{2}(\mathbf{x})} << 1,\tag{85}$$

at that

**V/VBD**

voltage *V* / *VBD* (Shotov, 1958)

(Gavrjushko et al, 1968).

**conditions of its applicability**

(Fig. 16a). The length of tunneling *l*

small up to electric field *E* ≅4×10<sup>4</sup>

**Mp Mn**

66 Photodiodes - From Fundamentals to Applications

**K0 (\*) Experiment (\*) Theory Experiment (\*) Theory**

0.65 1.25 1.19 1.12 1.09 2.10 0.70 1.40 1.28 1.20 1.14 2.00 0.75 1.60 1.44 1.30 1.22 2.00 0.80 1.85 1.70 1.40 1.33 2.10 0.85 2.40 2.13 1.70 1.56 2.00 0.90 3.50 3.10 2.20 2.00 2.10 0.95 6.80 5.89 3.90 3.45 2.00 0.975 13.00 11.64 7.00 6.32 2.00 0.98 - 14.52 - 7.76 2.00 0.985 - 19.33 - 10.16 2.00 0.99 30.00 28.90 - 14.97 2.00

**Table 2.** Experimental (\*) (Shotov, 1958) and computed [from Equation (55)] avalanche multiplication factors *Mp* and *Mn* for holes and electrons in *Ge* linear *p* −*n* junction for different ratios of applied voltage to avalanche breakdown

Finally, it is interesting to analyze application of expressions (45) and (76) to describe ava‐ lanche process in *InSb*. The fact is that dependence *α*(*Е*) in *InSb* was quite well known al‐ ready in 1967 (Baertsch, 1967), but no one could obtain information about dependence *β*(*Е*) (Dmitriev et al, 1987), (Dmitriev et al, 1983), (Dmitriev et al, 1982), (Gavrjushko et al, 1968). Substituting in (45) dependence *α*(*Е*) for *InSb* (Baertsch, 1967), (Dmitriev et al, 1983), (Dmi‐ triev et al, 1982), (Gavrjushko et al, 1968), we find that ratio *K* =*β*(*E*) / *α*(*E*) is vanishingly

same time value *nn<sup>В</sup>* is extremely small. It means that *Mn*(*V* ) becomes much larger than uni‐ ty, even at voltages *Vb* noticeably lower avalanche breakdown voltage *VBD*, and value *Mp*(*V* ) remains equal to unity up to values *Vb* very close to *VBD*. Effect obtained from appli‐ cation of relations (45) and (76) accords very well with experimental data (Baertsch, 1967), (Dmitriev et al, 1983) and explains why multiplication of holes in *InSb* is extremely hard to observe (Dmitriev et al, 1987), (Baertsch, 1967), (Dmitriev et al, 1983), (Dmitriev et al, 1982),

**4.1. Calculation of tunnel currents in approximation of quasi-uniform electric field and**

In act of interband tunneling electron from valence band overcomes potential barrier ABC

band *Ес*(*x*) changes by value equal to *Еg* is found by solving integral equation

**4. Tunnel currents in avalanche heterophotodiodes**

V/cm resulting in extremely high value *nрВ* when at the

*<sup>T</sup>* , i.e. length on which energy of bottom of conduction

$$\mathcal{U}\_{\rm T}(E\_{\mathcal{S}'}E) = \frac{E\_{\rm g}}{q \star E(\infty)}\tag{86}$$

As shown below, due to large values of field *Е* at avalanche breakdown of *p* −*n* structures, inequality (85) is valid for almost all materials up to concentration *N* =1017 cm-3 and even high.

Under these conditions specific rates of charge carriers' tunnel generation *gTi*(*x*) in layers I and II of structure can be described by expression

$$\log\_{\text{Ti}}(\mathbf{x}) \equiv \frac{1}{q} \times \frac{\partial \mathbf{J}\_{\text{Ti}}}{\partial \mathbf{x}} = A\_{\text{Ti}} \times E^2(\mathbf{x}) \times \exp\left[-\frac{a\_i}{E(\mathbf{x})}\right],\tag{87}$$

and expressions (6)-(9) result in convenient formula for analysis of primary interband tunnel

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

where characteristic dimensions of areas of charge carriers' tunnel generation in layers I and II

)=min{*WTi* <sup>≡</sup> *<sup>ε</sup>*0*ε<sup>i</sup>* <sup>×</sup> *Ei*

2

)=

These conditions mean the following. If inequalities (91) for *gTi*(*E*) are satisfied then expres‐

equalities (91) is satisfied then tunnel generation drops sharply with decreasing *E*, and

Fulfillment of conditions (92) is necessary at punch-through of proper layers of structure for neglecting tunneling through its hetero-interfaces which is not accounted for by formula (89). We show further, that at avalanche breakdown, inequalities (91) and (92) are valid for

breakdown field *EiBD* of proper layer of structure ((Sze, 1981), (Tsang, 1985), (Grekhov &

Breakdown fields *EiBD* can be obtained by formula (14) (Osipov & Kholodnov, 1987), (Osi‐

( , ) (0, ) 1 , ( )

*i i*

*Egi q* ×*Ei*

*i gi i*

*N E E E a* ´ d º < << ee ´

0

*<sup>T</sup>* (*Egi*, *Ei*

2

sion (87) is valid, at least in the neighborhood of field value *E* =*Ei*

almost all real values of material parameters, concentrations *Ni*

*iBD i i iBD i*

*<sup>N</sup> E NW E W*

of heterostructure. Avalanche breakdown occurs when one of fields *Ei*

*i*

*l Ti* ≡ *l* × *L Ti* ×*Ei*

*q* ×*ai* × *Ni*

1,

< <*l i*

is mainly determined by tunneling in areas 0≤ *x* ≤*WT* <sup>1</sup> and

1/

*i*

*i i*

*N W* é ù *s*

= ´+ ê ú ê ú ë û % (93)

2

, *Wi*

<sup>2</sup> ×exp( <sup>−</sup> *ai*

*Ei*

), (89)

http://dx.doi.org/10.5772/50778

69

(91)

}. (90)

. (92)

. When right side of in‐

and layers' thicknesses *Wi*

becomes close to

<sup>2</sup> *mi* \* *Egi*

current density

*JT* <sup>=</sup>∑ *i*=1 2

Equation (89) is valid under conditions

therefore *ITi* at *WTi* <*Wi*

Serezhkin, 1980), Sections 3.1-3.3).

pov &, Kholodnov, 1989), i.e.,

*W*<sup>1</sup> ≤ *x* ≤*W*<sup>1</sup> + *WT* 2.

*JTi* <sup>=</sup>2×*<sup>q</sup>* <sup>3</sup> (2*π*)

*L Ti*(*Ei*

, *Wi*

<sup>3</sup> <sup>×</sup>ℏ<sup>2</sup> <sup>×</sup><sup>∑</sup> *i*=1

**Figure 16.** Physical meaning of quasi-uniform field approximation: a − band diagram, b − field distribution on length of tunneling. ABC − true potential barrier, ABC' − potential barrier used de facto. Dashed lines − Е(х) =*const*

obtained in (Kane, 1960) (see also (Burstein & Lundqvist, 1969)) for *Е*(*х*)=*const*, in which

$$A\_{Ti} = \frac{q^2}{(2\pi)^3 \times \hbar^2} \times \sqrt{\frac{2m\_i^\*}{E\_{gi}}},\\a\_i = \frac{\pi}{4q \times \hbar} \times \sqrt{2m\_i^\* \times E\_{gi}^{-3}}.\tag{88}$$

Here ℏ, *Еgi* and *mi* \* =2*mc* ×*mv* / (*mc* + *mv*) – crossed Plank constant, gaps and specific effective masses of light charge carriers in proper layers. Approximation of quasi-uniform field (87) Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 69

and expressions (6)-(9) result in convenient formula for analysis of primary interband tunnel current density

$$J\_{T} = \sum\_{i=1}^{2} J\_{Ti} = \frac{\sqrt{2} \times q^{3}}{(2\pi)^{3} \times \hbar^{2}} \times \sum\_{i=1}^{2} \hbar \left| \frac{m\_{i}^{\*}}{E\_{g^{i}}} \times L\_{Ti} \times E\_{i} \times \exp\left(-\frac{a\_{i}}{E\_{i}}\right) \right| \tag{89}$$

where characteristic dimensions of areas of charge carriers' tunnel generation in layers I and II

$$L\_{\rm Ti}(E\_{i\prime}, W\_i) = \min \left\{ W\_{\rm Ti} \equiv \frac{\varepsilon\_0 \varepsilon\_i \times E\_i \, 2}{q \times a\_i \times \mathbf{N}\_i \, ^\prime} \, \vert \, W\_i \right\}. \tag{90}$$

Equation (89) is valid under conditions

obtained in (Kane, 1960) (see also (Burstein & Lundqvist, 1969)) for *Е*(*х*)=*const*, in which

of tunneling. ABC − true potential barrier, ABC' − potential barrier used de facto. Dashed lines − Е(х) =*const*

**Figure 16.** Physical meaning of quasi-uniform field approximation: a − band diagram, b − field distribution on length

masses of light charge carriers in proper layers. Approximation of quasi-uniform field (87)

\* 3

, 2.

<sup>p</sup> = ´ = ´´ p ´ <sup>h</sup> ´ <sup>h</sup> (88)

=2*mc* ×*mv* / (*mc* + *mv*) – crossed Plank constant, gaps and specific effective

2 \*

(2 ) 4

2

*i Ti i i gi gi <sup>q</sup> <sup>m</sup> <sup>A</sup> a mE E q*

3 2

Here ℏ, *Еgi* and *mi*

\*

68 Photodiodes - From Fundamentals to Applications

$$\delta\_i \equiv \frac{N\_i \times E\_{gi}}{2\varepsilon\varepsilon\_0 \times E\_i^2} < \frac{E\_i}{a\_i} << 1,\tag{91}$$

$$I\_{Ti} \equiv I\_T(E\_{g^{j\prime}}, E\_i) = \frac{E\_{g^{j\prime}}}{q \times E\_i} \le l\_i. \tag{92}$$

These conditions mean the following. If inequalities (91) for *gTi*(*E*) are satisfied then expres‐ sion (87) is valid, at least in the neighborhood of field value *E* =*Ei* . When right side of in‐ equalities (91) is satisfied then tunnel generation drops sharply with decreasing *E*, and therefore *ITi* at *WTi* <*Wi* is mainly determined by tunneling in areas 0≤ *x* ≤*WT* <sup>1</sup> and *W*<sup>1</sup> ≤ *x* ≤*W*<sup>1</sup> + *WT* 2.

Fulfillment of conditions (92) is necessary at punch-through of proper layers of structure for neglecting tunneling through its hetero-interfaces which is not accounted for by formula (89). We show further, that at avalanche breakdown, inequalities (91) and (92) are valid for almost all real values of material parameters, concentrations *Ni* and layers' thicknesses *Wi* of heterostructure. Avalanche breakdown occurs when one of fields *Ei* becomes close to breakdown field *EiBD* of proper layer of structure ((Sze, 1981), (Tsang, 1985), (Grekhov & Serezhkin, 1980), Sections 3.1-3.3).

Breakdown fields *EiBD* can be obtained by formula (14) (Osipov & Kholodnov, 1987), (Osi‐ pov &, Kholodnov, 1989), i.e.,

$$E\_{iBD}(N\_{i'}W\_i) = E\_{iBD}(0, W\_i) \times \left[1 + \frac{N\_i}{\tilde{N}\_i(W\_i)}\right]^{1/s} \tag{93}$$

where

$$E\_{iBD}(\mathbf{O}\_{\prime}\mathcal{W}) = A\_{i} \times \left(\frac{A\_{i} \times \varepsilon\_{i}\varepsilon\_{0}}{sq\mathcal{W}\_{i}}\right)^{1/(s-1)}, \quad \tilde{N}\_{i}(\mathcal{W}\_{i}\ ) = \left(\frac{A\_{i} \times \varepsilon\_{i}\varepsilon\_{0}}{sq\mathcal{W}\_{i}}\right)^{s/(s-1)}\tag{94}$$

**3.** When

doping of *рwg*

**breakdown**

**I.**

situations are possible.

then length of tunneling *l*

dopant's distribution profile *Ni*

*Wi* >>

1.8×10−<sup>2</sup>

down of heterostructure inequities (91) and (92) are satisfied for real values of *Ni*

*Ti* at *Ei* =*EiBD* is much shorter than thickness *Wi*

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

<sup>+</sup> layer, tunnel current in it can be ignored; this is situation similar to MIS struc‐

In expressions (96)-(99) *Еgi* is measured in eV. Analysis shows that under avalanche break‐

and *Ei* <*EiBD*, i.e. in layer which does not control avalanche breakdown also. As can be seen from Fig. 17, when punch-through of layer *nwg* stops then, obviously, conditions (91) and (92) become no longer valid. Note that calculations of tunnel currents in approximation of quasi-uniform field lead to some overestimation of actually available. In fact, due to high

tures (Anderson, 1977). In *n* type layers electric field decreases with increasing distance from metallurgical boundary of *p* <sup>+</sup> −*n* junction (Fig. 1b), and because gradient of potential is ex‐ pressed as *dφ* / *dx* = − *E* then slope of zones *Еc*(*x*) and *Еv*(*x*) decreases with increasing *x*. It is shown from Figure 16a that use of quasi-uniform field approximation means underestimat‐ ing of thickness of actual barrier ABC. As expected, numerical calculations in WKB approxi‐ mation (Anderson, 1977) give a somewhat smaller value of tunnel currents than formula (89). Since tunnel currents are strongly dependent on parameters of material, which in real samples, usually, more or less different from those used in calculations (moreover, exact

then slight overestimation of tunnel currents values provides some technological margin

Analysis of expression (89) under avalanche breakdown of *p* <sup>+</sup> −*n* heterostructure, i.e., when either *E*<sup>1</sup> =*E*1*BD* or *E*<sup>2</sup> =*E*2*BD*, shows that in contrast to homogeneous *p* −*n* junction (Stillman, 1981), (Ando et al, 1980) density of initial tunnel current *JT* , as a rule, is not a monotonic

rent and vice versa – decrease of tunnel current when *N*1 and *Wi* have different values. De‐

**4.2. Features of interband tunnel currents in** *p* **<sup>+</sup> −***n* **heterostructures under avalanche**

that is needed for development of devices with required specifications.

function *N*1. An increase in *N*2 cause, for some values of *N*1 and *Wi*

*4.2.1. Independent doping levels of wide-gap and narrow-gap n type layers*

1/2 1/2

pending on gap *Egi* of heterostructure's layers and their thicknesses *Wi*

1 \* 1 1 22

*<sup>W</sup> A NW W W W A N NW*

2 2 2 2 22

º³= ´ ç ÷ ê ú

*<sup>X</sup>ε<sup>i</sup>* <sup>×</sup> *Xgi* <sup>6</sup> , μ*m*, (99)

(*x*) and hence shape of barrier ABC are usually unknown),

( 1)/

% (100)

( ) . ( )

*s s s*


%

e ´ + è ø ë û

of this layer.

http://dx.doi.org/10.5772/50778

, the rise of tunnel cur‐

the following

and *Wi*

71

(*s* and *Ai* – some constants).

For many semiconductors including *I nxGa*1−*<sup>x</sup>ASyP*1−*<sup>y</sup>* alloy which is one of the main materi‐ als for avalanche heterophotodiodes fabrication (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), (Woul, 1980)

$$s = 8, A\_i = \sqrt{\frac{1.2 \times q}{\varepsilon\_i \varepsilon\_0}} \times \left(\frac{E\_{\text{gd}}}{11q}\right)^{3/4} \times 10^{10}. \tag{95}$$

From expressions (93) and (94) when relations (95) are satisfied we find the following.

**1.** When

$$N\_i \le N\_i^{(1)} = \frac{8.9 \times 10^{19}}{X\_{\text{mi}}^4 \times X\_{\text{ci}}^4 \times X\_{\text{gi}}^6}, \text{ cm}^{\circ 3},\\W\_i \ge W\_i^{(1)} = X\_{\text{mi}}^{3.5} \times X\_{\text{ci}}^3 \times X\_{\text{gi}}^6 \times 1.4 \times 10^{-4}, \text{ } \text{\textmu m,} \tag{96}$$

then ratio *Е<sup>i</sup>* to *ai* is less than 0.1, where *Χmi* =0.06 / *mi*<sup>0</sup> \* , *Χε<sup>i</sup>* =12.4 / *ε<sup>i</sup>* , *Χgi* =1.35 / *Egi* (for *InP* which is often used for growing of wide-gap layers of heterostructure (Tsang, 1981), (Still‐ man, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al, 1980), (Trommer, 1984), (Woul, 1980)), *Xmi* = *Xε<sup>i</sup>* = *Xgi* =1, *mi*<sup>0</sup> \* <sup>=</sup>*mi* \* / *m*<sup>0</sup> (*m*0– free-electron mass)

#### **2.** When

$$N\_i \le N\_i^{\langle \rangle 2} = X\_{\rm mi}^{\langle 0 \rangle 2} \times X\_{\rm ci}^{1.6} \times X\_{\rm gi}^{0.4} \times 3.3 \times 10^{17}, \text{ cm}^3, \; W\_i \ge W\_i^{\langle 2 \rangle} \tag{7} \\ = \frac{X\_{\rm gi}^{0.4} \times 1.8 \times 10^{-2}}{X\_{\rm mi}^{0.7} \times X\_{\rm ci}^{1.9}}, \; \mu\text{m} \tag{97}$$

then under avalanche breakdown of proper layer of structure ratio *δ<sup>i</sup>* to *ЕiBD* / *ai* is not exceed unity, moreover, even when *Ni* = *Ni* (2)

$$
\delta\_i \le X\_{mi}^{0.6} \times X\_{el}^{1.2} \times X\_{gi}^{0.8} \times 10^{-1} \,\text{.}\tag{98}
$$

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 71

**3.** When

where

(*s* and *Ai*

**1.** When

then ratio *Е<sup>i</sup>*

**2.** When

*Ni* ≤ *Ni*

(*m*0– free-electron mass)

*Ni* ≤ *Ni*

(2)= *Χmi*

unity, moreover, even when *Ni* = *Ni*

0.2 <sup>×</sup>*Χε<sup>i</sup>*

1.6 <sup>×</sup>*Χgi*

to *ai*

– some constants).

70 Photodiodes - From Fundamentals to Applications

1980), (Trommer, 1984), (Woul, 1980)

(1)= 8.9×10<sup>19</sup> *Xmi* <sup>4</sup> <sup>×</sup> *<sup>X</sup>ε<sup>i</sup>*

1/( 1) /( 1)

% (94)

(95)

, μm, (96)

, *Χgi* =1.35 / *Egi* (for *InP*

1.9 , μm (97)

\* <sup>=</sup>*mi* \* / *m*<sup>0</sup>

is not exceed

*s s s*

0 0 (0, ) , ()

0 1.2 8, 10 . 11 *gi*

From expressions (93) and (94) when relations (95) are satisfied we find the following.

´ æ ö == ´ ´ ç ÷ e e ç ÷ è ø

*q E*

*i*

<sup>6</sup> , cm-3, *Wi* <sup>≥</sup>*Wi*

is less than 0.1, where *Χmi* =0.06 / *mi*<sup>0</sup>

*iBD i i i*

*i*

*s A*

<sup>4</sup> <sup>×</sup> *Xgi*

*A A E WA N W*

*i i i i*

*sqW sqW* - - æö æö ´e e ´ee = ´ ç÷ ç÷ <sup>=</sup> èø èø

For many semiconductors including *I nxGa*1−*<sup>x</sup>ASyP*1−*<sup>y</sup>* alloy which is one of the main materi‐ als for avalanche heterophotodiodes fabrication (Tsang, 1981), (Stillman, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman et al, 1983), (Ando et al,

*i i*

3/4

*q*

(1)=*Χmi*

which is often used for growing of wide-gap layers of heterostructure (Tsang, 1981), (Still‐ man, 1981), (Filachev et al, 2010), (Kim et al, 1981), (Forrest et al, 1983), (Tarof et al, 1990), (Ito et al, 1981), (Clark et al, 2007), (Hayat & Ramirez, 2012), (Filachev et al, 2011), (Stillman

et al, 1983), (Ando et al, 1980), (Trommer, 1984), (Woul, 1980)), *Xmi* = *Xε<sup>i</sup>* = *Xgi* =1, *mi*<sup>0</sup>

0.4 ×3.3×1017, cm3

then under avalanche breakdown of proper layer of structure ratio *δ<sup>i</sup>*

*δ<sup>i</sup>* < *Χmi*

(2)

0.6 <sup>×</sup>*Χε<sup>i</sup>*

1.2 <sup>×</sup>*Χgi*

3.5 <sup>×</sup>*Χε<sup>i</sup>*

\*

, *Wi* ≥*Wi*

0.8 ×10−<sup>1</sup>

(2)= *Xgi*

<sup>3</sup> <sup>×</sup>*Χgi*

, *Χε<sup>i</sup>* =12.4 / *ε<sup>i</sup>*

<sup>6</sup> ×1.4×10−<sup>4</sup>

0.4 ×1.8×10−<sup>2</sup>

to *ЕiBD* / *ai*

. (98)

*Xmi* 0.7 <sup>×</sup> *<sup>X</sup>ε<sup>i</sup>*

10

$$\mathcal{W}\_i \stackrel{>}{>} \frac{1.8 \times 10^{-2}}{\sqrt{X\_{\varepsilon i}} \times \sqrt{X\_{gi}}}, \text{ } \mu m \text{.} \tag{99}$$

then length of tunneling *l Ti* at *Ei* =*EiBD* is much shorter than thickness *Wi* of this layer.

In expressions (96)-(99) *Еgi* is measured in eV. Analysis shows that under avalanche break‐ down of heterostructure inequities (91) and (92) are satisfied for real values of *Ni* and *Wi* and *Ei* <*EiBD*, i.e. in layer which does not control avalanche breakdown also. As can be seen from Fig. 17, when punch-through of layer *nwg* stops then, obviously, conditions (91) and (92) become no longer valid. Note that calculations of tunnel currents in approximation of quasi-uniform field lead to some overestimation of actually available. In fact, due to high doping of *рwg* <sup>+</sup> layer, tunnel current in it can be ignored; this is situation similar to MIS struc‐ tures (Anderson, 1977). In *n* type layers electric field decreases with increasing distance from metallurgical boundary of *p* <sup>+</sup> −*n* junction (Fig. 1b), and because gradient of potential is ex‐ pressed as *dφ* / *dx* = − *E* then slope of zones *Еc*(*x*) and *Еv*(*x*) decreases with increasing *x*. It is shown from Figure 16a that use of quasi-uniform field approximation means underestimat‐ ing of thickness of actual barrier ABC. As expected, numerical calculations in WKB approxi‐ mation (Anderson, 1977) give a somewhat smaller value of tunnel currents than formula (89). Since tunnel currents are strongly dependent on parameters of material, which in real samples, usually, more or less different from those used in calculations (moreover, exact dopant's distribution profile *Ni* (*x*) and hence shape of barrier ABC are usually unknown), then slight overestimation of tunnel currents values provides some technological margin that is needed for development of devices with required specifications.

#### **4.2. Features of interband tunnel currents in** *p* **<sup>+</sup> −***n* **heterostructures under avalanche breakdown**

Analysis of expression (89) under avalanche breakdown of *p* <sup>+</sup> −*n* heterostructure, i.e., when either *E*<sup>1</sup> =*E*1*BD* or *E*<sup>2</sup> =*E*2*BD*, shows that in contrast to homogeneous *p* −*n* junction (Stillman, 1981), (Ando et al, 1980) density of initial tunnel current *JT* , as a rule, is not a monotonic function *N*1. An increase in *N*2 cause, for some values of *N*1 and *Wi* , the rise of tunnel cur‐ rent and vice versa – decrease of tunnel current when *N*1 and *Wi* have different values. De‐ pending on gap *Egi* of heterostructure's layers and their thicknesses *Wi* the following situations are possible.

*4.2.1. Independent doping levels of wide-gap and narrow-gap n type layers*

**I.**

$$\frac{W\_1}{W\_2} \equiv W\_{1/2} \ge W\_{1/2}^\* = \left(\frac{\varepsilon\_1 \times A\_1}{\varepsilon\_2 \times A\_2}\right)^s \times \left[\frac{\tilde{N}\_2(W\_2)}{N\_2 + \tilde{N}\_2(W\_2)}\right]^{(s-1)/s} \tag{100}$$

1/2

% % (102)

1 (*T* ) 73

, (103)

http://dx.doi.org/10.5772/50778

(*<sup>T</sup>* ) at which *JT*


Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

( ) 1 ( 0.5)/( 1)

If *s* sufficiently large ((Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966), Sec‐ tions 3.1-3.3), then with further increase of *N*1 tunnel current is monotonically falling. How‐ ever, in most real cases, for example, when relations (95) is valid, tunnel current at *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜

One can see that at minimum of tunnel current, as a rule, the following inequality is valid

*κ* (*s*−2)/(*s*−1) *<sup>s</sup>* 1/(*s*−1) <sup>×</sup> *<sup>y</sup>*

> , *<sup>y</sup>* <sup>=</sup> *<sup>N</sup>*<sup>1</sup> *r* × *N*˜ 1

1 (1/ ) 1 (2/ ) ( ) ln[ ( ; )] 1, ( ) *s*


*f* 2 (*y*)

<sup>x</sup> + ´ ´ - ´ ´ L x= é ù ë û (104)

1−*ξ* ×*r* 1/*<sup>s</sup>* × *f* (*y*)×(*s* −4 +

æ ö æ ö - k =ç ÷ ´ ´ ´ ç ÷ ç ÷ ç ÷ è ø è ø % (105)

(1−*κ*)×*<sup>κ</sup>* <sup>×</sup>*<sup>s</sup>* <sup>×</sup> *<sup>f</sup>* (*y*)−*<sup>κ</sup>* <sup>×</sup> *<sup>y</sup>*

1 + *<sup>ξ</sup>* <sup>×</sup> <sup>4</sup>*<sup>r</sup>*

(1 ) . *<sup>g</sup>*

*s <sup>r</sup>* <sup>×</sup> *<sup>y</sup>* )

,

<

*a*<sup>2</sup> ×*ε*<sup>2</sup> *a*<sup>1</sup> ×*ε*<sup>1</sup>

Therefore, as it follows from (6)-(9), (89), (90) and (93), concentration *N*<sup>1</sup> = *N*1min

*<sup>y</sup> s fy r y y f y <sup>r</sup>*

(*y*) <sup>×</sup>

1 1 2

2 2 1

*m E r N*

*g m N E*

3/2 5/2 \* <sup>2</sup>

Expression (105) is valid when inequality *WT* <sup>2</sup> <*W*<sup>2</sup> is fulfilled. This inequality and inequali‐ ty (103) also are fulfilled at minimum of tunnel current in the most practically interesting cases. Below is explained difference between situations *WT* <sup>2</sup> >*W*2 and *WT* <sup>2</sup> <*W*2 at

<sup>2</sup> (0, ) ( ) <sup>1</sup> *<sup>T</sup> s s NN s BD E W NW W*

1 1 1 1 11 1

< @´ ´ ´ ç ÷ ´ µ - e´ è ø

2 2

*E*1*BD*(0, *W*1) *a*1

*<sup>s</sup>*/(*s*−1), *κ* =1−

*s*

(*y*)

\*

*y* × *f* (*y*)−*κ* × *y* <sup>2</sup> × *κ* ×*s* − *f* <sup>1</sup>−*<sup>s</sup>*

*B*


*s a*

then tunnel current is almost independent on *N*1.

*ξ* ≡

first decreases and then increases.

where

where

*f* (*y*)=(*y* + *r* <sup>−</sup><sup>1</sup>

) 1/*s*

*<sup>Λ</sup>*(*y*;*ξ*)= *<sup>B</sup>* <sup>×</sup> *<sup>f</sup>* <sup>3</sup>−*<sup>s</sup>*

, *r* =(*κ* ×*s*)

When (103) is fulfilled then *WT* <sup>1</sup> <*W*1.

reaches minimum is defined by equation

**Figure 17.** Dependence of generalized parameters of smallness δ<sup>2</sup> \* and *l T* 2 \* in quasi-uniform field approximation on concentration *N*1, at М*ph* = 100, in case, when charge carriers multiplication occurs in *nwg* : *InP* layer. Solid lines − δ<sup>2</sup> \* , dashed − *l T* 2 \* . Values *W*1, μm: 1 − 0.5, 2 – 2, 3 − 8. *N*1*pt* <sup>−</sup> maximal concentration *N*<sup>1</sup> at which punch-through of *nwg* layer is possible; δ<sup>2</sup> = (*N*<sup>2</sup> / <sup>10</sup>16)×(ε<sup>2</sup> / <sup>ε</sup>1) × *Eg*<sup>2</sup> <sup>×</sup>δ<sup>2</sup> \* ; *l <sup>T</sup>* <sup>2</sup> = (ε<sup>2</sup> / ε1) × *Eg*<sup>2</sup> ×*l T* 2 \* ; *Eg*2- eV, concentration − cm-3.

In this case, at any concentration *N*1, field *E*<sup>1</sup> =*E*1*BD*(*N*1, *W*1), and *E*<sup>2</sup> <*E*2*BD*, i.e., avalanche breakdown is controlled by *nwg* layer.

As follows from (6)-(9), (89) and (93), if

$$\exp\left[-\frac{a\_1}{E\_{1BD}(0,\ W\_1)} \times \left(1 - \frac{\varepsilon\_2 \times a\_2}{\varepsilon\_1 \times a\_1}\right)\right] < -1,\tag{101}$$

which is fulfilled with large margin at *a*2*ε*<sup>2</sup> <*a*1*ε*1 due to large ratio of *a*1 to *E*1*BD*(0, *W*1) (1-2 orders of magnitude) while

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 73

$$N\_1 < \tilde{N}\_1^{(\Gamma)} \equiv s \times \left(\frac{2}{s-1} \times \frac{\varepsilon\_1}{\varepsilon\_2 \times a\_2} \times E\_{1\text{RD}}(0, \mathcal{W}\_1)\right)^{1/2} \times \tilde{N}\_1(\mathcal{W}\_1) \propto \mathcal{W}\_1^{-(s+0.5)/(s-1)}\tag{102}$$

then tunnel current is almost independent on *N*1.

If *s* sufficiently large ((Sze, 1981), (Osipov & Kholodnov, 1987), (Sze & Gibbons, 1966), Sec‐ tions 3.1-3.3), then with further increase of *N*1 tunnel current is monotonically falling. How‐ ever, in most real cases, for example, when relations (95) is valid, tunnel current at *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜ 1 (*T* ) first decreases and then increases.

One can see that at minimum of tunnel current, as a rule, the following inequality is valid

$$\xi \equiv \frac{E\_{1BD}(0, \,\,\,W\_1)}{a\_1} \prec \frac{\kappa^{(s-2)/(s-1)}}{s^{1/(s-1)}} \times \frac{y}{f^{-2}(y)}\tag{103}$$

where

$$f\left(y\right) = \left(y + r^{-1}\right)^{1/s}, \; r = \left(\kappa \times s\right)^{s/(s-1)}, \; \kappa = 1 - \frac{a\_2 \times \varepsilon\_2}{a\_1 \times \varepsilon\_1}, \; y = \frac{N\_1}{r \times N\_{1,1}}$$

When (103) is fulfilled then *WT* <sup>1</sup> <*W*1.

Therefore, as it follows from (6)-(9), (89), (90) and (93), concentration *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) at which *JT* reaches minimum is defined by equation

$$\frac{y}{f(y)} + \frac{\xi}{r^{1 - (2/s)}} \times \left[ s \times f(y) - r^{1 - (1/s)} \times y \right] \times \ln[\Lambda(y; \xi)] = 1,\tag{104}$$

where

In this case, at any concentration *N*1, field *E*<sup>1</sup> =*E*1*BD*(*N*1, *W*1), and *E*<sup>2</sup> <*E*2*BD*, i.e., avalanche

*<sup>T</sup>* <sup>2</sup> = (ε<sup>2</sup> / ε1) × *Eg*<sup>2</sup> ×*l*

concentration *N*1, at М*ph* = 100, in case, when charge carriers multiplication occurs in *nwg* : *InP* layer. Solid lines − δ<sup>2</sup>

which is fulfilled with large margin at *a*2*ε*<sup>2</sup> <*a*1*ε*1 due to large ratio of *a*1 to *E*1*BD*(0, *W*1) (1-2

*ε*<sup>2</sup> ×*а*<sup>2</sup> *ε*<sup>1</sup> ×*а*<sup>1</sup>

\* and *l T* 2

*T* 2

\* . Values *W*1, μm: 1 − 0.5, 2 – 2, 3 − 8. *N*1*pt* <sup>−</sup> maximal concentration *N*<sup>1</sup> at which punch-through of *nwg*

) < <1, (101)

\* in quasi-uniform field approximation on

\* ; *Eg*2- eV, concentration − cm-3.

\* ,

breakdown is controlled by *nwg* layer. As follows from (6)-(9), (89) and (93), if

layer is possible; δ<sup>2</sup> = (*N*<sup>2</sup> / <sup>10</sup>16)×(ε<sup>2</sup> / <sup>ε</sup>1) × *Eg*<sup>2</sup> <sup>×</sup>δ<sup>2</sup>

72 Photodiodes - From Fundamentals to Applications

orders of magnitude) while

dashed − *l*

*T* 2

exp <sup>−</sup> *<sup>a</sup>*<sup>1</sup>

**Figure 17.** Dependence of generalized parameters of smallness δ<sup>2</sup>

*<sup>E</sup>*1*BD*(0, *<sup>W</sup>*1) ×(1<sup>−</sup>

\* ; *l*

$$\Lambda(y;\boldsymbol{\xi}) = \mathbf{B} \times \frac{f^{\ast \ast}(y)}{y \times \mathbb{L}f(y) - \kappa \times y \mathbf{J}^2 \times \mathbb{L} \kappa \times s - f^{\ast \ast}(y)} \times \frac{1 - \boldsymbol{\xi} \times r^{1/s} \times f(y) \times \left(s - 4 + \frac{s}{r \times y}\right)}{1 + \boldsymbol{\xi} \times \frac{4r}{(1 - \boldsymbol{\kappa}) \times \kappa \times s} \times \mathbf{f}(f(y) - \kappa \times y)},$$

$$B = \left(\frac{m\_1^\*}{m\_2^\*}\right)^{3/2} \times \left(\frac{E\_{\mathbb{g}1}}{E\_{\mathbb{g}2}}\right)^{5/2} \times \frac{N\_2}{\tilde{N}\_1} \times \frac{(1-\kappa)^2}{r}.\tag{105}$$

Expression (105) is valid when inequality *WT* <sup>2</sup> <*W*<sup>2</sup> is fulfilled. This inequality and inequali‐ ty (103) also are fulfilled at minimum of tunnel current in the most practically interesting cases. Below is explained difference between situations *WT* <sup>2</sup> >*W*2 and *WT* <sup>2</sup> <*W*2 at *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) . Equation (104) can be solved by successive approximations using parameters of smallness *ξ* and 1 /*s*.

As a result we find

$$N\_{\rm Imin}^{(\Gamma)} = \left[\frac{\varepsilon\_{1}\varepsilon\_{0}\times A\_{1}}{q\times W\_{1}}\times\left(1-\frac{\varepsilon\_{2}\times a\_{2}}{\varepsilon\_{1}\times a\_{1}}\right)\right]^{\frac{s}{s-1}}\times y\_{0}\times\left\{1-\xi\times\frac{1-\kappa}{\kappa}\times r^{1/s}\times\ln\left[\Lambda(y\_{0};0)\right]\times\frac{y\_{0}\times\{\kappa\times s\times y\_{0}+1\}}{(s-1)\times\kappa\times y\_{0}+1}+0(\xi)\right\},\tag{106}$$

where

$$y\_0 = 1 + \frac{1}{\kappa \times s^2} + 0\left(\frac{1}{s^2}\right). \tag{107}$$

drops sharply, same as *JT* min, with increase *W*1, but it increases with increasing *N*2. Value of this ratio is usually several orders of magnitude less than unity. For example, for combina‐ tion of layers *nwg* :*InP* / *nng* :*I n*0.53*Ga*0.47*As*, differential of currents, as can be shown, does not

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

22 1 ( ) 2 1min 11 22 2 *<sup>T</sup> a W N N a aW*

then in minimum of *JT* (*N*1) takes place punch-through of narrow-gap layer, i.e. non-equili‐

*N*1, and at the same time, non-equilibrium SCR will penetrate into narrow-gap layer until

Nature of above dependence *JT* on *N*1 is competition between tunnel currents in wide-gap

*nwg* layer at its heterojunction (Fig. 1b) coincide with very high accuracy with *E*1*BD*. Due to relatively large field *E*<sup>2</sup> =(*ε*<sup>1</sup> / *ε*2)×*E*1*BD*, current density *JT* is determined by tunneling of charge carriers in narrow-gap layer, i.e. *JT* ≈ *JT* 2 (Fig. 1a). With increasing *N*1, field *E*2 and therefore current *JT* <sup>2</sup> decrease due to fall *EI* (*W*1) (Fig. 18). Decrease *EI* (*W*1) with increase *N*<sup>1</sup> is caused by requirement (1) of constancy of photocurrent gain *M ph* =*Mр*. Indeed, increase *N*1 for given *M ph* should lead to growth *E*1. Otherwise, due to growth |∇*Е*(*х*)| with in‐ creasing *N*1, field would be reduced everywhere in SCR, which in turn would lead to a de‐ crease *M ph* . However, increase *E*<sup>1</sup> should not be too large, and it should be such that *Е*(*х*) at *х* greater than some value in interval 0< *x* <*W*1 is decreased. In other words, *Е*(*х*) anywhere in SCR would increase, that, evidently, would increase *M ph* . It can be seen directly from (1) and (2). Note that for sufficiently large values of multiplication factors *M ph* , field *E*1 is prac‐ tically independent on *M ph* and very close to breakdown field *E*1*BD*(*N*1, *W*1) when value of integral *m* (2) is equal to unity. This allows to use value *E*<sup>1</sup> =*E*1*BD*(*N*1, *W*1) (93) instead of

> 1 (*T* )

*nwg* layer is still very insignificant, but it is enough to affect value *JT* <sup>2</sup>. Due to decrease *E*<sup>2</sup>

× 1 + 0(1) > *N*1min

e ´ < ´´

> 0 *qW*<sup>1</sup>

) *s*/(*s*−1)

<sup>+</sup> layer. When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*1min

*<sup>N</sup>*<sup>1</sup> <sup>=</sup> *<sup>N</sup>*<sup>1</sup> *pt* =( *<sup>A</sup>*<sup>1</sup> <sup>×</sup>*ε*1*<sup>ε</sup>*

and narrow-gap layers of heterostructure (Fig. 1a). When *N*<sup>1</sup> <sup>≤</sup> *<sup>N</sup>*˜

, where *N*2 is measured in cm-3.

e ´ -e ´ (110)

(*<sup>T</sup>* ) then tunnel current increases with increasing

1 (*T* )

, then variation of field *Е*(*х*) at distance *W*1 in

(*<sup>T</sup>* ) , current density *JT* <sup>≈</sup> *JT* <sup>1</sup> in‐

<sup>1</sup>), current is more and more determined by tunnel‐

(*T* ) (111)

http://dx.doi.org/10.5772/50778

75

then field *E* =*EI* (*W*1) in

exceed values (*N*<sup>2</sup> / <sup>10</sup>18)

When concentrations

brium SCR reaches *nwg*

concentration *N*1 reaches value

true value *Е*1(*N*1, *<sup>W</sup>*1, *<sup>M</sup> ph* ). When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜

with growth *N*<sup>1</sup> (especially when *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜

ing of charge carriers in *nwg* layer, therefore when *N*<sup>1</sup> > *N*1min

0.9 ×2×10−<sup>4</sup>

It is shown from (105) and (106) that *N*1min (*<sup>T</sup>* ) is decreased with growth *W*<sup>1</sup> and, also, although weakly, with increase *N*2.

When *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) then density of tunnel current

$$\mathbb{E}\left[J\_T(\mathcal{N}\_1) = J\_{T\min} = \mathbb{C}\_0 \times \frac{\mathbb{s}\_1 \mathbb{s}\_0 \times q^3}{2\pi^4 \times \hbar \times E\_{\mathbb{s}1}^2} \times \frac{E\_{1\text{BD}}^4(\mathbf{0}, \mathcal{W}\_1)}{\tilde{N}\_1(\mathcal{W}\_1)} \times \Lambda^{-\mathbb{N}\_1}(y\_0; \mathbf{0}) \times \exp\left[-\frac{\mathbb{C}\_1 \times \boldsymbol{a}\_1}{E\_{1\text{BD}}(\mathbf{0}, \mathcal{W}\_1)}\right] \times \left[1 + \mathbb{O}(1)\right],\tag{108}$$

Where

$$\begin{aligned} \mathbf{C}\_{0} &= \boldsymbol{y}\_{0}^{3} \times \frac{\boldsymbol{y}\_{0} \times \boldsymbol{\kappa} \times (\boldsymbol{s} - 1) + 1}{(\boldsymbol{s} \times \boldsymbol{\kappa} - 1) \times \boldsymbol{y}\_{0} + 1} \times \{\boldsymbol{\kappa} \times \boldsymbol{s}\}^{(4-\kappa)/(s-1)} \; \boldsymbol{C}\_{1} = \left[\boldsymbol{y}\_{0} \times \{\boldsymbol{\kappa} \times \boldsymbol{s}\}^{1/(s-1)}\right]^{-1} \; \boldsymbol{\kappa} \\\\ \boldsymbol{m}\_{1} &= \frac{\boldsymbol{y}\_{0} \times (1 - \boldsymbol{\kappa})}{(\boldsymbol{s} - 1) \times \boldsymbol{\kappa} \times \boldsymbol{y}\_{0} + 1} \; \boldsymbol{A} \end{aligned}$$

From (94), (105) and (108) follow that *JT* min decreases sharply with increasing *W*1. Value *JT* min decreases also, although weakly, with increasing *N*2. Ratio

$$\frac{J\_{T\min}}{J\_{T}(N\_{1})\Big|\_{N\_{1}\leq N\_{1}^{(T)}}}\propto \left[\frac{N\_{2}}{\tilde{N}\_{1}(W\_{1})}\right]^{n\_{2}}\times\exp\left[-(\mathbf{C}\_{1}+\mathbf{x}-1)\times\frac{a\_{1}}{E\_{1\text{BD}}(0,W\_{1})}\right],\tag{109}$$

Where

*n*<sup>2</sup> = *y*<sup>0</sup> ×(*κ* ×*s* −1) + 1 (*<sup>s</sup>* <sup>−</sup>1)×*<sup>κ</sup>* <sup>×</sup> *<sup>y</sup>*<sup>0</sup> + 1 , drops sharply, same as *JT* min, with increase *W*1, but it increases with increasing *N*2. Value of this ratio is usually several orders of magnitude less than unity. For example, for combina‐ tion of layers *nwg* :*InP* / *nng* :*I n*0.53*Ga*0.47*As*, differential of currents, as can be shown, does not exceed values (*N*<sup>2</sup> / <sup>10</sup>18) 0.9 ×2×10−<sup>4</sup> , where *N*2 is measured in cm-3.

When concentrations

*N*<sup>1</sup> = *N*1min

where

smallness *ξ* and 1 /*s*.

74 Photodiodes - From Fundamentals to Applications

As a result we find

(*<sup>T</sup>* ) . Equation (104) can be solved by successive approximations using parameters of

<sup>1</sup> ( 1) <sup>1</sup> <sup>1</sup> ln ( ;0) 0( ) , ( 1) 1

æ ö

1

*g BD*

1 1 1 1 1

ë û p´ ´ ë û % <sup>h</sup> (108)

1/(*s*−1) −1 ,

1 0 1 1 1 1


(4−*s*)/(*s*−1), *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>y</sup>*<sup>0</sup> ×(*<sup>κ</sup>* <sup>×</sup>*s*)

From (94), (105) and (108) follow that *JT* min decreases sharply with increasing *W*1. Value

1

% ê ú ë û ë û % (109)

é ù é ù <sup>µ</sup> ê ú ´ - +k- ´ ê ú

(0, ) ( ) ( ;0) exp 1 0(1) , <sup>2</sup> ( ) (0, ) *BD n*

k ´ è ø (107)

(*<sup>T</sup>* ) is decreased with growth *W*<sup>1</sup> and, also, although

(106)

<sup>1</sup> ( ) 10 1 2 2 1/ 0 0

*<sup>s</sup> <sup>T</sup> <sup>A</sup> <sup>s</sup> <sup>a</sup> y sy <sup>N</sup> y ry qW a s y*

1 11 0

0 2 2

=+ + ç ÷

1 1 *y* 1 0. *s s*

é ù ee ´ æ ö e ´ - ì ü ï ï - k ´ k´ ´ + = ´- ê ú ç ÷ ´ ´ -x´ ´ ´ L ´ í ý é ù + x ë û ê ú ´ e´ ï ï <sup>k</sup> - ´k´ + ë û è ø î þ

1min 0 0

(*<sup>T</sup>* ) then density of tunnel current

1 min 0 4 2 0

3 4

*JT* min decreases also, although weakly, with increasing *N*2. Ratio

2

*T N N BD J N <sup>a</sup> <sup>C</sup> J N N W E W* £

*n*

min 2 1

1 1 1 1 1 exp ( 1) , ( ) ( ) (0, ) *<sup>T</sup>*

*<sup>q</sup> E W C a JN J C <sup>y</sup> <sup>E</sup> N W E W*

*s*

It is shown from (105) and (106) that *N*1min

*y*<sup>0</sup> ×*κ* ×(*s* −1) + 1

*T*

*y*<sup>0</sup> ×(*κ* ×*s* −1) + 1 (*<sup>s</sup>* <sup>−</sup>1)×*<sup>κ</sup>* <sup>×</sup> *<sup>y</sup>*<sup>0</sup> + 1 ,

(*<sup>s</sup>* <sup>×</sup>*<sup>κ</sup>* <sup>−</sup>1)× *<sup>y</sup>*<sup>0</sup> + 1 ×(*<sup>κ</sup>* <sup>×</sup>*s*)

( ) 1 1

weakly, with increase *N*2.

When *N*<sup>1</sup> = *N*1min

*T T*

3 ×

*<sup>n</sup>*<sup>1</sup> <sup>=</sup> *<sup>y</sup>*<sup>0</sup> ×(1−*κ*) (*<sup>s</sup>* <sup>−</sup>1)×*<sup>κ</sup>* <sup>×</sup> *<sup>y</sup>*<sup>0</sup> + 1 .

Where

*C*<sup>0</sup> = *y*<sup>0</sup>

Where

*n*<sup>2</sup> =

$$N\_2 < \frac{\varepsilon\_2 \times a\_2}{\varepsilon\_1 \times a\_1 - \varepsilon\_2 \times a\_2} \times \frac{W\_1}{W\_2} \times N\_{1\text{min}}^{(T)}\tag{110}$$

then in minimum of *JT* (*N*1) takes place punch-through of narrow-gap layer, i.e. non-equili‐ brium SCR reaches *nwg* <sup>+</sup> layer. When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*1min (*<sup>T</sup>* ) then tunnel current increases with increasing *N*1, and at the same time, non-equilibrium SCR will penetrate into narrow-gap layer until concentration *N*1 reaches value

$$N\_1 = N\_{1 \cdot pt} = \left(\frac{A\_1 \times \varepsilon\_1 \varepsilon\_0}{q \mathcal{W}\_1}\right)^{s/(s-1)} \times \mathbf{[1} + 0(1)\mathbf{j} > N\_1 \mathbf{j}\_{1 \text{min}}^{(T)}\tag{111}$$

Nature of above dependence *JT* on *N*1 is competition between tunnel currents in wide-gap and narrow-gap layers of heterostructure (Fig. 1a). When *N*<sup>1</sup> <sup>≤</sup> *<sup>N</sup>*˜ 1 (*T* ) then field *E* =*EI* (*W*1) in *nwg* layer at its heterojunction (Fig. 1b) coincide with very high accuracy with *E*1*BD*. Due to relatively large field *E*<sup>2</sup> =(*ε*<sup>1</sup> / *ε*2)×*E*1*BD*, current density *JT* is determined by tunneling of charge carriers in narrow-gap layer, i.e. *JT* ≈ *JT* 2 (Fig. 1a). With increasing *N*1, field *E*2 and therefore current *JT* <sup>2</sup> decrease due to fall *EI* (*W*1) (Fig. 18). Decrease *EI* (*W*1) with increase *N*<sup>1</sup> is caused by requirement (1) of constancy of photocurrent gain *M ph* =*Mр*. Indeed, increase *N*1 for given *M ph* should lead to growth *E*1. Otherwise, due to growth |∇*Е*(*х*)| with in‐ creasing *N*1, field would be reduced everywhere in SCR, which in turn would lead to a de‐ crease *M ph* . However, increase *E*<sup>1</sup> should not be too large, and it should be such that *Е*(*х*) at *х* greater than some value in interval 0< *x* <*W*1 is decreased. In other words, *Е*(*х*) anywhere in SCR would increase, that, evidently, would increase *M ph* . It can be seen directly from (1) and (2). Note that for sufficiently large values of multiplication factors *M ph* , field *E*1 is prac‐ tically independent on *M ph* and very close to breakdown field *E*1*BD*(*N*1, *W*1) when value of integral *m* (2) is equal to unity. This allows to use value *E*<sup>1</sup> =*E*1*BD*(*N*1, *W*1) (93) instead of true value *Е*1(*N*1, *<sup>W</sup>*1, *<sup>M</sup> ph* ). When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜ 1 (*T* ) , then variation of field *Е*(*х*) at distance *W*1 in *nwg* layer is still very insignificant, but it is enough to affect value *JT* <sup>2</sup>. Due to decrease *E*<sup>2</sup> with growth *N*<sup>1</sup> (especially when *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*˜ <sup>1</sup>), current is more and more determined by tunnel‐ ing of charge carriers in *nwg* layer, therefore when *N*<sup>1</sup> > *N*1min (*<sup>T</sup>* ) , current density *JT* <sup>≈</sup> *JT* <sup>1</sup> in‐ creases with increase *N*1 because *E*1*BD* grows with increase *N*1. Initial plateau (Fig. 18a) on the graph *JT* (*N*1) is caused by extremely weak dependences *E*1*BD* on *N*1 (93) and *Е* on *х* in *nwg* layer when *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*˜ 1 (*T* ) . Reducing of value *JT* min (108) with growth *N*<sup>2</sup> is due to increasing length of tunneling in narrow-gap *nng* layer (Fig. 1). Indeed, in this layer ∇*Е* ~ − *N*<sup>2</sup> <0, and *E*2 under these conditions does not depend on *N*2. It means, that *Е*(*х*) everywhere in *nng* layer, except of point *x* =*W*1, falls with increase *N*2 (1b). Since *d Ec dx* <sup>=</sup> *d Ev dx* <sup>=</sup> *<sup>d</sup><sup>φ</sup> dx* <sup>=</sup> <sup>−</sup> *<sup>E</sup>* <0, then slopes of *Ec*(*x*) and *Ev*(*x*) everywhere in *nng* layer, except of point *x* =*W*1, decrease also with increasing *N*2, that leads to increase length of tunneling. Reducing of *JT* is more significant with growth *N*2 when *N*<sup>1</sup> < *N*1min (*<sup>T</sup>* ) (Fig.18b), because current density *JT* <sup>2</sup> increases with de‐ crease *N*1 while *JT* <sup>1</sup> decreases. When *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*˜ 1 (*T* ) then current density *JT* <sup>1</sup> ≤ *JT* 2, and if *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) it exceeds *JT* 2. Therefore, ratio of *JT* min to *JT* <sup>|</sup> *<sup>N</sup>*1<*<sup>N</sup>*˜ 1 (*T* ) (109) increases with increas‐ ing *N*2. Because at *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) value *JT* <sup>1</sup> <sup>&</sup>gt; *JT* <sup>2</sup>, then, naturally, concentration *N*1min (*<sup>T</sup>* ) (106) slightly decreases with increasing *N*2 (Fig. 18b). For small values *N*2, when *WT* <sup>2</sup> >*W*2, *Е*(*х*) in *nng* layer coincides with *E*2 with high accuracy. Therefore, length of tunneling in this layer, and hence *JT* also, do not depend on *N*2. Reducing of values *N*1min (*<sup>T</sup>* ) (106) and *JT* min (108) with increasing *W*1 (Figure 18a) is due to the fact that the more is *W*1 then the less is *E*1*BD* and the greater is fall of field *Е*(*х*) in depth of *nwg*layer.

#### **II.**

Condition (100) is not satisfied. For example, for combination of layers *nwg*:*InP* / *nng* :*I n*0.53*Ga*0.47*As* such situation takes place when

$$\frac{W\_1}{W\_2} \times \left(1 + \frac{N\_2}{2.2 \times 10^{15}} \times W\_2^{8/7}\right)^{7/8} < 21.5,\tag{112}$$

**Figure 18.** Dependence of tunnel current density on concentration *N*1 in case of independent doping levels of *nwg* : *InP* and *nng* : *I n*0.53*Ga*0.47*As* layers at *W*<sup>2</sup> = 2 μm. **<sup>a</sup>** <sup>−</sup> *<sup>N</sup>*<sup>2</sup> = 1014cm-3; *W*1, μm: 1 − 0.1, 2 − 0.2, 3 − 0.5, 4 − 1. **<sup>b</sup>** <sup>−</sup>

(*<sup>T</sup>* ) ; *W*<sup>2</sup> = 2 μm; *N*2, cm-3: 1 − 1014, 2 − 1015, 3 − 1016, 4 − 10<sup>17</sup>

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

http://dx.doi.org/10.5772/50778

77

neighborhood of value *N*<sup>1</sup> = *N*1min

where *N*2 and *Wi* are measured in cm-3 and μm, respectively. Under this condition, when *<sup>N</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯ 1, where *<sup>N</sup>*¯ <sup>1</sup> satisfies equation

$$\frac{\varepsilon\_2}{\varepsilon\_1} \times A\_2 \times \left[N\_2 + \tilde{N}\_2(W\_2)\right]^{1/s} + \frac{q \times \tilde{N}\_1 \times W\_1}{\varepsilon\_1 \varepsilon\_0} = A\_1 \times \left[N\_1 + \tilde{N}\_1(W\_1)\right]^{1/s} \tag{113}$$

avalanche breakdown is controlled by *nng* layer, i.e. *E*<sup>2</sup> =*E*2*BD*(*N*2, *W*2), and *E*1<*E*1*BD* and it increases linearly with *N*1. Therefore, strictly speaking, when *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯ <sup>1</sup> then tunnel current in‐ creases with increasing *N*1. At the same time, *JT* <sup>2</sup> does not depend on *N*1 under following conditions.

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 77

creases with increase *N*1 because *E*1*BD* grows with increase *N*1. Initial plateau (Fig. 18a) on the graph *JT* (*N*1) is caused by extremely weak dependences *E*1*BD* on *N*1 (93) and *Е* on *х* in

length of tunneling in narrow-gap *nng* layer (Fig. 1). Indeed, in this layer ∇*Е* ~ − *N*<sup>2</sup> <0, and *E*2 under these conditions does not depend on *N*2. It means, that *Е*(*х*) everywhere in *nng*

slopes of *Ec*(*x*) and *Ev*(*x*) everywhere in *nng* layer, except of point *x* =*W*1, decrease also with increasing *N*2, that leads to increase length of tunneling. Reducing of *JT* is more significant

slightly decreases with increasing *N*2 (Fig. 18b). For small values *N*2, when *WT* <sup>2</sup> >*W*2, *Е*(*х*) in *nng* layer coincides with *E*2 with high accuracy. Therefore, length of tunneling in this layer,

with increasing *W*1 (Figure 18a) is due to the fact that the more is *W*1 then the less is *E*1*BD*

Condition (100) is not satisfied. For example, for combination of layers *nwg*:*InP* /

15 2

1/ 1/ <sup>2</sup> 1 1 2 2 22 1 1 11

( ) () , *s s qN W A N NW A N NW* <sup>e</sup> ´ ´ ´´ + + =´ + é ù éù ë û ëû e e e

avalanche breakdown is controlled by *nng* layer, i.e. *E*<sup>2</sup> =*E*2*BD*(*N*2, *W*2), and *E*1<*E*1*BD* and it

creases with increasing *N*1. At the same time, *JT* <sup>2</sup> does not depend on *N*1 under following

1 21.5,

7/8

are measured in cm-3 and μm, respectively. Under this condition, when

è ø ´ (112)

% % (113)

1 (*T* )

. Reducing of value *JT* min (108) with growth *N*<sup>2</sup> is due to increasing

*d Ec dx* <sup>=</sup>

(*<sup>T</sup>* ) (Fig.18b), because current density *JT* <sup>2</sup> increases with de‐

*<sup>N</sup>*1<*<sup>N</sup>*˜ 1 (*T* )

(*<sup>T</sup>* ) value *JT* <sup>1</sup> <sup>&</sup>gt; *JT* <sup>2</sup>, then, naturally, concentration *N*1min

*d Ev dx* <sup>=</sup> *<sup>d</sup><sup>φ</sup>*

then current density *JT* <sup>1</sup> ≤ *JT* 2, and if

(109) increases with increas‐

(*<sup>T</sup>* ) (106) and *JT* min (108)

<sup>1</sup> then tunnel current in‐

*dx* <sup>=</sup> <sup>−</sup> *<sup>E</sup>* <0, then

(*<sup>T</sup>* ) (106)

*nwg* layer when *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*˜

*N*<sup>1</sup> = *N*1min

**II.**

where *N*2 and *Wi*

1, where *<sup>N</sup>*¯

*<sup>N</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯

conditions.

1 (*T* )

76 Photodiodes - From Fundamentals to Applications

crease *N*1 while *JT* <sup>1</sup> decreases. When *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*˜

and the greater is fall of field *Е*(*х*) in depth of *nwg*layer.

*nng* :*I n*0.53*Ga*0.47*As* such situation takes place when

2

<sup>1</sup> satisfies equation

1 1 0

increases linearly with *N*1. Therefore, strictly speaking, when *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯

*W*

with growth *N*2 when *N*<sup>1</sup> < *N*1min

ing *N*2. Because at *N*<sup>1</sup> = *N*1min

layer, except of point *x* =*W*1, falls with increase *N*2 (1b). Since

(*<sup>T</sup>* ) it exceeds *JT* 2. Therefore, ratio of *JT* min to *JT* <sup>|</sup>

and hence *JT* also, do not depend on *N*2. Reducing of values *N*1min

1 2 8/7

æ ö ´+ ´ < ç ÷

2.2 10 *W N <sup>W</sup>*

**Figure 18.** Dependence of tunnel current density on concentration *N*1 in case of independent doping levels of *nwg* : *InP* and *nng* : *I n*0.53*Ga*0.47*As* layers at *W*<sup>2</sup> = 2 μm. **<sup>a</sup>** <sup>−</sup> *<sup>N</sup>*<sup>2</sup> = 1014cm-3; *W*1, μm: 1 − 0.1, 2 − 0.2, 3 − 0.5, 4 − 1. **<sup>b</sup>** <sup>−</sup> neighborhood of value *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) ; *W*<sup>2</sup> = 2 μm; *N*2, cm-3: 1 − 1014, 2 − 1015, 3 − 1016, 4 − 10<sup>17</sup>

**1.** If

$$\left(\frac{W\_{1/2}}{W\_{1/2}^\*}\right)^{1/(s-1)} > 1 - \frac{s-1}{2s^2},\tag{114}$$

cal character with *JT* (*N*1)| *<sup>N</sup>*2=*const* in the case of 4.2.1. **I**, and is caused by the same physical grounds. The only difference is that when *N* < *N*2, then curves *JT* (*N* ) lie higher on plotting

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

This occurs because at given value *E*2 length of tunneling in narrow-gap layer is the greater

breakdown is controlled by *nng* layer, i.e. *E*2=*E*2*BD*(*N* , *W*2), and *E*1<*E*1*BD*(*N* , *W*1) and increas‐ es linearly with *N* . Dependence *JT* (*N* ) has, in contrast to situation 4.2.1, not only deep mini‐ mum, but high maximum also (Fig. 19a). This is due to the fact that when *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯ then *<sup>E</sup>*<sup>1</sup> grows and *E*2 grows also reaching at *<sup>N</sup>* <sup>=</sup> *<sup>N</sup>*¯ maximal value (Fig. 19b). As a result, when *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯ then *JT* 1 grows with increase *N* and *JT* <sup>2</sup> grows also. Note that when doping of *nwg*

minimal value, is determined by formula (106) with accuracy up to small corrections of or‐ der *ξ* =*E*1*BD*(0, *W*1) / *a*<sup>1</sup> <<1, as in the case of independent doping of *nwg* and *nng* layers. For‐

In punch-through conditions of absorber *nng*, current responsivity *SI* (*λ*) of heterostructure under study can be described by relation (4). In calculating quantum efficiency *η* of hetero‐ structure, we take into account that optical radiation is not absorbed in its wide-gap layers. Let's assume that light beam falls perpendicularly to front surface of heterostructure (Fig. 1), and absorption coefficient in narrow-gap layer *γ*(*λ*) does not depend on electric field. Quan‐ tum efficiency is ratio of number of electron-hole pairs generated in sample by absorbed

1 2

1 2 (1 ) (1 ) , <sup>1</sup> *R R R R*

where reflection coefficient of light from illuminated surface *R*<sup>1</sup> = ( *εex* − *ε*1)

1

2


/( *ε*<sup>2</sup> + *ε*1)

2

mula for *JT* minmay be obtained from expression (108), if we replace *N*2 by *N*min

**5. Basic performance of avalanche heterophotodiode**

(*T* )

<sup>1</sup>=*N*2=*<sup>N</sup>*¯) avalanche

http://dx.doi.org/10.5772/50778

79

, at which tunnel current density *JT* has

(*T* ) in it.

2

; *εex*– relative dielectric

/ ( *εex* + *ε*1)

2

area, and when *<sup>N</sup>* <sup>&</sup>gt; *<sup>N</sup>*2 – lower, than curves *JT* (*N*1)| *<sup>N</sup>*2=*const* in the case of 4.2.1. **I**.

Then, till *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯, (where *N*¯ is determined by equation (113), where *N*¯

the higher is level of doping of this layer.

and *nng* layers are equal then concentration *N* = *N*min

photons per unit time to incident flux of photons.

and from interfaces of heterostructure *R*<sup>2</sup> = ( *ε*<sup>2</sup> − *ε*1)

ii. Condition (117) is not satisfied.

**5.1. Responsivity**

Therefore, (Fig. 20a)

then at *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯ 1,*JT* <sup>2</sup> > > *JT* 1 with margin of several orders of magnitude, and therefore with very high accuracy *JT* (*N*1)=*const*. If *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*¯ 1 then due to decrease *E*2 and hence *JT* <sup>2</sup> also, den‐ sity of tunnel current *JT* (*N*1) begins drop sharply and, reaching minimum value (108) at concentration (106), then starts to grow again due to growth *JT* <sup>1</sup>(*N*1).

**2.** If

$$\left(\frac{W\_{1/2}}{W\_{1/2}^\*}\right)^{1/(s-1)} << 1 - \frac{s-1}{2s^2} \,\,\,\,\tag{115}$$

then after initial plateau *JT* (*N*1) grows monotonically. It is due to monotonic increase in com‐ ponent of tunnel current density *JT* (*N*1), which at *N*<sup>1</sup> <sup>≥</sup> *<sup>N</sup>*¯ 1 is considerably superior to *JT* <sup>2</sup>.

**3.** If

$$\left(\frac{W\_{1/2}}{W\_{1/2}^\*}\right)^{1/(s-1)} \approx 1 - \frac{s-1}{2s^2},\tag{116}$$

then for small enough thicknesses *W*1 of layer *nwg* dependence *JT* (*N*1) has distinct maximum at *N*<sup>1</sup> <sup>=</sup> *<sup>N</sup>*¯ 1, however, at least in this case minimum is not deep. This is due to the fact that components of tunnel current density *JT* 1 and *JT* <sup>2</sup> are equal to each other in order of magni‐ tude at small enough *W*1. Characteristics of tunnel currents in heterostructure with inde‐ pendent doping of *nwg* and *nng* layers are illustrated in Fig. 18. Note that if in case **I** increase *N*2 leads to decrease *JT* at all values *N*1, then in case **II**, increase *N*2, when *N*1 is small enough, leads to increase of tunnel current, but at sufficiently large *N*<sup>1</sup> tunnel current de‐ creases, particularly, in the vicinity of concentration *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) .

#### *4.2.2. Equal doping levels of wide-gap and narrow-gap n type layers*

Under this condition density of tunnel current is given by expression (89), where *N*1=*N*2=*N*

i.

$$\frac{\partial W\_1}{\partial W\_2} \ge \left(\frac{\varepsilon\_1 \times A\_1}{\varepsilon\_2 \times A\_2}\right)^s \tag{117}$$

At this relation of parameters avalanche breakdown is controlled by *nwg* layer, i.e. *E*1= *E*1*BD*(*N*1, *W*1), and *E*2<*E*2*BD*(*N*2, *W*2) regardless of doping. Dependence *JT* on *N* has identi‐ cal character with *JT* (*N*1)| *<sup>N</sup>*2=*const* in the case of 4.2.1. **I**, and is caused by the same physical grounds. The only difference is that when *N* < *N*2, then curves *JT* (*N* ) lie higher on plotting area, and when *<sup>N</sup>* <sup>&</sup>gt; *<sup>N</sup>*2 – lower, than curves *JT* (*N*1)| *<sup>N</sup>*2=*const* in the case of 4.2.1. **I**.

This occurs because at given value *E*2 length of tunneling in narrow-gap layer is the greater the higher is level of doping of this layer.

ii. Condition (117) is not satisfied.

**1.** If

**2.** If

**3.** If

at *N*<sup>1</sup> <sup>=</sup> *<sup>N</sup>*¯

i.

then at *N*<sup>1</sup> <sup>&</sup>lt; *<sup>N</sup>*¯

very high accuracy *JT* (*N*1)=*const*. If *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*¯

78 Photodiodes - From Fundamentals to Applications

1/( 1)

*s W s W s* - æ ö - ç ÷ > - ç ÷

1/( 1)

1/(*s*−1)

*s W s W s* - æ ö - ç ÷ << - ç ÷

\* 2

then after initial plateau *JT* (*N*1) grows monotonically. It is due to monotonic increase in com‐

<sup>≈</sup> 1<sup>−</sup> *<sup>s</sup>* <sup>−</sup><sup>1</sup>

then for small enough thicknesses *W*1 of layer *nwg* dependence *JT* (*N*1) has distinct maximum

components of tunnel current density *JT* 1 and *JT* <sup>2</sup> are equal to each other in order of magni‐ tude at small enough *W*1. Characteristics of tunnel currents in heterostructure with inde‐ pendent doping of *nwg* and *nng* layers are illustrated in Fig. 18. Note that if in case **I** increase *N*2 leads to decrease *JT* at all values *N*1, then in case **II**, increase *N*2, when *N*1 is small enough, leads to increase of tunnel current, but at sufficiently large *N*<sup>1</sup> tunnel current de‐

Under this condition density of tunnel current is given by expression (89), where *N*1=*N*2=*N*

æ ö e ´ ³ ç ÷ e ´ è ø

At this relation of parameters avalanche breakdown is controlled by *nwg* layer, i.e. *E*1= *E*1*BD*(*N*1, *W*1), and *E*2<*E*2*BD*(*N*2, *W*2) regardless of doping. Dependence *JT* on *N* has identi‐

*s*

1 11 2 22

*W A W A*

1, however, at least in this case minimum is not deep. This is due to the fact that

(*<sup>T</sup>* ) .

\* 2

sity of tunnel current *JT* (*N*1) begins drop sharply and, reaching minimum value (108) at

<sup>1</sup> 1 , 2

<sup>1</sup> 1 , 2

1,*JT* <sup>2</sup> > > *JT* 1 with margin of several orders of magnitude, and therefore with

1 then due to decrease *E*2 and hence *JT* <sup>2</sup> also, den‐

1 is considerably superior to *JT* <sup>2</sup>.

<sup>2</sup>*<sup>s</sup>* <sup>2</sup> , (116)

(114)

(115)

(117)

1/2

1/2

è ø

concentration (106), then starts to grow again due to growth *JT* 1(*N*1).

1/2

1/2

è ø

( *<sup>W</sup>*1/2 *W*1/2 \* )

creases, particularly, in the vicinity of concentration *N*<sup>1</sup> = *N*1min

*4.2.2. Equal doping levels of wide-gap and narrow-gap n type layers*

ponent of tunnel current density *JT* (*N*1), which at *N*<sup>1</sup> <sup>≥</sup> *<sup>N</sup>*¯

Then, till *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯, (where *N*¯ is determined by equation (113), where *N*¯ <sup>1</sup>=*N*2=*<sup>N</sup>*¯) avalanche breakdown is controlled by *nng* layer, i.e. *E*2=*E*2*BD*(*N* , *W*2), and *E*1<*E*1*BD*(*N* , *W*1) and increas‐ es linearly with *N* . Dependence *JT* (*N* ) has, in contrast to situation 4.2.1, not only deep mini‐ mum, but high maximum also (Fig. 19a). This is due to the fact that when *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯ then *<sup>E</sup>*<sup>1</sup> grows and *E*2 grows also reaching at *<sup>N</sup>* <sup>=</sup> *<sup>N</sup>*¯ maximal value (Fig. 19b). As a result, when *<sup>N</sup>* <sup>&</sup>lt; *<sup>N</sup>*¯ then *JT* 1 grows with increase *N* and *JT* <sup>2</sup> grows also. Note that when doping of *nwg* and *nng* layers are equal then concentration *N* = *N*min (*T* ) , at which tunnel current density *JT* has minimal value, is determined by formula (106) with accuracy up to small corrections of or‐ der *ξ* =*E*1*BD*(0, *W*1) / *a*<sup>1</sup> <<1, as in the case of independent doping of *nwg* and *nng* layers. For‐ mula for *JT* minmay be obtained from expression (108), if we replace *N*2 by *N*min (*T* ) in it.

## **5. Basic performance of avalanche heterophotodiode**

#### **5.1. Responsivity**

In punch-through conditions of absorber *nng*, current responsivity *SI* (*λ*) of heterostructure under study can be described by relation (4). In calculating quantum efficiency *η* of hetero‐ structure, we take into account that optical radiation is not absorbed in its wide-gap layers. Let's assume that light beam falls perpendicularly to front surface of heterostructure (Fig. 1), and absorption coefficient in narrow-gap layer *γ*(*λ*) does not depend on electric field. Quan‐ tum efficiency is ratio of number of electron-hole pairs generated in sample by absorbed photons per unit time to incident flux of photons.

Therefore, (Fig. 20a)

$$\eta = \frac{(1 - R\_1) \times (1 - R\_2)}{1 - R\_1 \times R\_2} \times \eta\_{1'} \tag{118}$$

where reflection coefficient of light from illuminated surface *R*<sup>1</sup> = ( *εex* − *ε*1) 2 / ( *εex* + *ε*1) 2 and from interfaces of heterostructure *R*<sup>2</sup> = ( *ε*<sup>2</sup> − *ε*1) 2 /( *ε*<sup>2</sup> + *ε*1) 2 ; *εex*– relative dielectric

constant of environment; and quantum efficiency *η*1 with respect to light ray which has pe‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

quantum efficiency *η*<sup>2</sup> with respect to light ray which has reached to second interface of het‐

*R*1*R*2(1−*R*2)2*ζ* 1−*R*1*R*<sup>2</sup>

*ζ* =exp(−*γW*2), *R*3 – reflection coefficient of light from not illuminated (backside) surface.

1 2 1 2

(1 ) (1 ) , , 1,2,3. <sup>1</sup>

2

*<sup>W</sup> at R R*

*<sup>W</sup> at R*

2

12 2

<sup>ì</sup> - -g ïh ¥ <sup>=</sup> <sup>ï</sup> - -g

1 exp( ) ( ) , , 1 exp( ) ( ) 1 exp( 2 ) ( ) , 1. 1 exp( 2 )

*R W*

12 2


Dependence *η* on *W*2 for heterostructure *InP* / *I n*0.53*Ga*0.47*As* / *InP* is shown in Fig. 20b. It should be noted that since in operation, electric field is high even in absorption layer, then,

*R W*

1 exp( ) ( ) ( ) 1 exp( ) 1 exp( 2 ) *R W W W*

> (1 )(1 ) ( ) , <sup>1</sup> *R R R R*

*i jj i*

*R RR R <sup>R</sup> i j R R*

*i j*

*R*1(1−*R*2)2*ζ* 1−*R*1*R*<sup>2</sup>

1 2 h = -z+h ´z 1 ; (119)

(1−*<sup>ζ</sup>* <sup>+</sup> *<sup>η</sup>*2*ζ*) + (1−*R*2)2*R*<sup>3</sup>

(1−*ζ* + *η*2*ζ*) ,

23 2

12 23 2



3 1

3

*RR W* + -g h g = h ¥ ´ - -g ´ é ù ë û - -g (121)

1−*R*2*R*<sup>3</sup>

×

http://dx.doi.org/10.5772/50778

(120)

81

(124)

netrated into narrow-gap layer is written

2

From expressions (118)-(120) follow, that

*ζ*(1−*ζ*) + *η*2(*ζR*2)2 +

<sup>×</sup> (1−*ζ*)(1 + *<sup>R</sup>*2*ζ*) + *<sup>η</sup>*2*R*2*<sup>ζ</sup>* <sup>2</sup> <sup>+</sup>

2 2

*ij*

2

hg = í

*W*

erostructure,

where

Particularly,

*η*<sup>2</sup> =*R*2(1−*ζ*) + *R*<sup>2</sup>

**Figure 19.** Dependences of tunnel current density (**a**) and fields Е (**b**) on dopant concentration *N* in case of equal doping levels of *nwg* : *InP* and *nng* : *I n*0.53*Ga*0.47*As* layers, at *W*<sup>2</sup> = 2 μm. *W*1, μm: 1 – 10, 2 – 1, 3 – 0.1, Curves 1', 2', 3' – Е2(*N* ); curve 4 − Е1(*N* ), weakly dependent on *W*<sup>1</sup>

constant of environment; and quantum efficiency *η*1 with respect to light ray which has pe‐ netrated into narrow-gap layer is written

$$
\hbar \eta\_1 = 1 - \zeta + \eta\_2 \times \zeta;\tag{119}
$$

quantum efficiency *η*<sup>2</sup> with respect to light ray which has reached to second interface of het‐ erostructure,

$$\begin{split} \eta\_{2} = R\_{2}(1-\zeta) + R\_{2}^{2}\zeta(1-\zeta) + \eta\_{2}(\zeta R\_{2})^{2} + \frac{R\_{1}R\_{2}(1-R\_{2})^{2}\zeta}{1-R\_{1}R\_{2}}(1-\zeta+\eta\_{2}\zeta) + \frac{(1-R\_{2})^{2}R\_{3}}{1-R\_{2}R\_{3}}\times\\ \times \left[ (1-\zeta)(1+R\_{2}\zeta) + \eta\_{2}R\_{2}\zeta^{2} + \frac{R\_{1}(1-R\_{2})^{2}\zeta}{1-R\_{1}R\_{2}}(1-\zeta+\eta\_{2}\zeta) \right] \end{split} \tag{120}$$

*ζ* =exp(−*γW*2), *R*3 – reflection coefficient of light from not illuminated (backside) surface. From expressions (118)-(120) follow, that

$$\mathfrak{n}(\gamma W\_2) = \mathfrak{n}(\infty) \times \left[1 - \exp(-\gamma W\_2)\right] \times \frac{1 + R\_{23} \exp(-\gamma W\_2)}{1 - R\_{12} R\_{23} \exp(-2\gamma W\_2)}\tag{121}$$

where

$$\eta(\infty) = \frac{(1 - R\_1)(1 - R\_2)}{1 - R\_1 R\_2},\tag{122}$$

$$R\_{ij} = \frac{R\_i(1 - R\_j) + R\_j(1 - R\_i)}{1 - R\_i R\_j}, i, j = 1, 2, 3. \tag{123}$$

Particularly,

**Figure 19.** Dependences of tunnel current density (**a**) and fields Е (**b**) on dopant concentration *N* in case of equal doping levels of *nwg* : *InP* and *nng* : *I n*0.53*Ga*0.47*As* layers, at *W*<sup>2</sup> = 2 μm. *W*1, μm: 1 – 10, 2 – 1, 3 – 0.1, Curves 1', 2', 3' –

Е2(*N* ); curve 4 − Е1(*N* ), weakly dependent on *W*<sup>1</sup>

80 Photodiodes - From Fundamentals to Applications

$$\mathfrak{n}(\gamma\mathcal{W}\_2) = \begin{cases} \mathfrak{n}(\infty) \frac{1 - \exp(-\gamma\mathcal{W}\_2)}{1 - R\_{12}\exp(-\gamma\mathcal{W}\_2)}, & \text{at } R\_3 = R\_{1\prime} \\\\ \mathfrak{n}(\infty) \frac{1 - \exp(-2\gamma\mathcal{W}\_2)}{1 - R\_{12}\exp(-2\gamma\mathcal{W}\_2)}, & \text{at } R\_3 = 1. \end{cases} \tag{124}$$

Dependence *η* on *W*2 for heterostructure *InP* / *I n*0.53*Ga*0.47*As* / *InP* is shown in Fig. 20b. It should be noted that since in operation, electric field is high even in absorption layer, then, due to Franz-Keldysh effect, quantum efficiency is slightly higher than given in Fig. 20b. This is especially true when absorbing layer *W*2 is very thin.

**5.2. Noise**

fective noise factor *Fef* ,*<sup>i</sup>*

Fig. 21), in which *<sup>N</sup>*˜

values *s* and *Ai*

mum

is given by

It was noted above that in order to achieve the best performance of SAM-APD special dop‐ ing profile is formed in heterostructure which facilitates penetration of photogenerated charge carriers with higher impact ionization coefficient into multiplication layer. In this case, at given voltage bias on heterostructure, current responsivity *SI* (*λ*) is maximal, and ef‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

lodnov, 1984), (McIntyre 1966), and hence, as it is evident from expression (5), noise spectral density *SN* is also minimal. If *α* =*β*, then (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kho‐

2.3 to 1.4 (Tsang, 1985), (Filachev et al, 2011), (Cook et al, 1982). Therefore, noise spectral density of heterostructure with *InP* multiplication layer and optimal doping is slightly less

tion of charge carriers in narrow-gap layer does not occur. Under these conditions, field val‐ ue at metallurgical boundary of *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction (*<sup>х</sup>* =0, Fig. 1) equals to *E*<sup>1</sup> <sup>=</sup>*E*1*B*(*N*1, *<sup>W</sup>*1) (see (93) and (94)). For many semiconductors (see Sections 3.1-3.2) including *I nхGa*1−*<sup>х</sup>AsyP*1−*y*,

lodnov, 1984), (McIntyre 1966) *Fef* (*M ph* )=*M ph* , and therefore

than value given by formula (125). When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*¯

*i* (*Wi*

In *InP* ratio *K*(*E*)=*β* / *α* in interval of fields of interest *E* =(3.3÷7.7)×10<sup>5</sup>

*I n*0.53*Ga*0.47*As* / *InP*, in first approximation in parameters of smallness

1 2 1 7 2 1 1

0,49

cm *<sup>T</sup> N W N WN*

22 1 2 7

<sup>5</sup> *N W QW*( ) 10 ,

1 2 0,07 1 2 <sup>2</sup> ( , ) 2.19 10 exp( 27.88 ), / , *<sup>T</sup> <sup>W</sup> <sup>J</sup> <sup>W</sup> <sup>W</sup> <sup>A</sup> cmN*

( ) 2 1 1min 1 2 8 7 1 7 15 1 1

*W W*

1 1

*BD E W*

we find that value of concentration *N*<sup>1</sup> = *N*1min

min

(*M ph* ) is minimal (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kho‐

1, (where *<sup>N</sup>*¯

are defined by relations (95). In the case of heterostructure *InP* /


*<sup>N</sup>* = ´´ ´ - ´ (127)

3.69 10

14

´³ = ´ (129)

(*<sup>T</sup>* ) at which function *JT* (*N*1) reaches its mini‐

2

8 1 1 7

16 2 8 7

1

*W*

´ ´ é ù - æ ö ´ = ´- ´ ê ú ç ÷ - ´ ë û è ø

2.33 10 2.52 10 -3 (,) <sup>1</sup> ln 1 , .41

Formulas (127) and (128) are valid when *WT* <sup>2</sup> ≤*W*2, i.e., as follows from Section 4.2.1, when

(0, ) 2.786 10 1 1 , <sup>64</sup>

*a W s*

) is defined by formula (94) for *i* =1, 2) then avalanche multiplica‐

<sup>3</sup> 2 . *<sup>N</sup> T ph S qAJ M* = ´´ ´ (125)

V / cm varies from

http://dx.doi.org/10.5772/50778

83

(128)

<sup>1</sup> satisfies equation (113) (see

**Figure 20.** Layout view of multiple internal reflections and absorptions of light beam in heterostructure (**a**) and de‐ pendence of quantum efficiency η of structure *InP* / *I n*0.53*Ga*0.47*As* / *InP* on absorption layer thickness *W*2, μm (**b**): 1 − *R*<sup>3</sup> = *R*1 2 − *R*<sup>3</sup> = 1. It is assumed that relative dielectric permittivity of environment ε*ex* = 1

#### **5.2. Noise**

due to Franz-Keldysh effect, quantum efficiency is slightly higher than given in Fig. 20b.

**Figure 20.** Layout view of multiple internal reflections and absorptions of light beam in heterostructure (**a**) and de‐ pendence of quantum efficiency η of structure *InP* / *I n*0.53*Ga*0.47*As* / *InP* on absorption layer thickness *W*2, μm (**b**): 1 −

*R*<sup>3</sup> = *R*1 2 − *R*<sup>3</sup> = 1. It is assumed that relative dielectric permittivity of environment ε*ex* = 1

This is especially true when absorbing layer *W*2 is very thin.

82 Photodiodes - From Fundamentals to Applications

It was noted above that in order to achieve the best performance of SAM-APD special dop‐ ing profile is formed in heterostructure which facilitates penetration of photogenerated charge carriers with higher impact ionization coefficient into multiplication layer. In this case, at given voltage bias on heterostructure, current responsivity *SI* (*λ*) is maximal, and ef‐ fective noise factor *Fef* ,*<sup>i</sup>* (*M ph* ) is minimal (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kho‐ lodnov, 1984), (McIntyre 1966), and hence, as it is evident from expression (5), noise spectral density *SN* is also minimal. If *α* =*β*, then (Tsang, 1985), (Filachev et al, 2011), (Artsis & Kho‐ lodnov, 1984), (McIntyre 1966) *Fef* (*M ph* )=*M ph* , and therefore

$$S\_N = 2q \times A \times J\_T \times M\_{ph}^3. \tag{125}$$

In *InP* ratio *K*(*E*)=*β* / *α* in interval of fields of interest *E* =(3.3÷7.7)×10<sup>5</sup> V / cm varies from 2.3 to 1.4 (Tsang, 1985), (Filachev et al, 2011), (Cook et al, 1982). Therefore, noise spectral density of heterostructure with *InP* multiplication layer and optimal doping is slightly less than value given by formula (125). When *N*<sup>1</sup> <sup>&</sup>gt; *<sup>N</sup>*¯ 1, (where *<sup>N</sup>*¯ <sup>1</sup> satisfies equation (113) (see Fig. 21), in which *<sup>N</sup>*˜ *i* (*Wi* ) is defined by formula (94) for *i* =1, 2) then avalanche multiplica‐ tion of charge carriers in narrow-gap layer does not occur. Under these conditions, field val‐ ue at metallurgical boundary of *<sup>p</sup>* <sup>+</sup> <sup>−</sup>*n* junction (*<sup>х</sup>* =0, Fig. 1) equals to *E*<sup>1</sup> <sup>=</sup>*E*1*B*(*N*1, *<sup>W</sup>*1) (see (93) and (94)). For many semiconductors (see Sections 3.1-3.2) including *I nхGa*1−*<sup>х</sup>AsyP*1−*y*, values *s* and *Ai* are defined by relations (95). In the case of heterostructure *InP* / *I n*0.53*Ga*0.47*As* / *InP*, in first approximation in parameters of smallness

$$\delta\_1 \equiv \frac{E\_{1BD}(0, W\_1)}{a\_1} = \frac{2.786 \times 10^{-2}}{W\_1^{1/7}}, \; \delta\_2 = \frac{1}{s^2} = \frac{1}{64} \tag{126}$$

we find that value of concentration *N*<sup>1</sup> = *N*1min (*<sup>T</sup>* ) at which function *JT* (*N*1) reaches its mini‐ mum

$$J\_{T\_{\min}}(W\_1, N\_2) = 2.19 \times 10^8 \times \frac{W\_1^{0.49}}{N\_2^{0.07}} \times \exp(-27.88 \times W\_1^{1\prime\prime}) \text{, A } / \text{cm}^2 \text{ } \tag{127}$$

is given by

$$N\_{1\text{min}}^{(T)}\left(W\_1, N\_2\right) = \frac{2.33 \times 10^{16}}{W\_1^{8f7}} \times \left[1 - \frac{2.52 \times 10^{-2}}{W\_1^{1f7}} \times \left(\ln\frac{N\_2 \times W\_1^{8f7}}{3.69 \times 10^{15}} - 1.41\right)\right] \text{ } \text{cm}^{-3} \tag{128}$$

Formulas (127) and (128) are valid when *WT* <sup>2</sup> ≤*W*2, i.e., as follows from Section 4.2.1, when

$$\mathcal{N}\_2 \times \mathcal{W}\_2 \ge \mathcal{Q}(\mathcal{W}\_1) = \frac{5}{W\_1^{2\frac{27}{7}}} \times 10^{14} \,\mathrm{\,}\,\tag{129}$$

where concentration and thicknesses, as in (127) and (128), are measured in cm-3 and μm, respectively.

If inequality (129) is not satisfied, then values *N*1min (*<sup>T</sup>* ) and *JT* min will be again determined by (127) and (128), in which *N*<sup>2</sup> is replaced by *Q*(*W*1) / *W*2. It is shown from (127) and (128) that *N*1min (*<sup>T</sup>* ) and *JT* min are decreasing, moreover *JT* min sharply, with increase *W*1 (see Fig. 21, 22), and, also, although weakly, with increase *N*2. Decrease of values *N*1min (*<sup>T</sup>* ) and *JT* min with in‐ crease *W*1 is caused by situation when the thicker *W*1 the less *E*1*BD* and the greater fall of field *E*(*x*) on *nwg* layer thickness. Slight decrease *N*1min (*<sup>T</sup>* ) and *JT* min with growth *N*<sup>2</sup> is due to increasing of length of interband tunneling *l Tng* in narrow-gap *nng* layer with increase *N*<sup>2</sup> and the fact that at minimum *JT* <sup>1</sup> > *JT* 2. For small values either *N*2 or *W*2, field *E*(*x*) is so weakly dependent on *x* in *nng* layer, that value *l Tng* in it is almost constant. Therefore, when *N*2*W*<sup>2</sup> <*Q*(*W*1) then values *N*1min (*<sup>T</sup>* ) and *JT* min do no longer depend on *N*2 and slightly decrease with increase *W*2 due to reducing the length of tunneling generation region in narrow-gap material. In high performance diode, absorber should be punched-through when voltage bias *Vb* on heterostructure is less than voltage of avalanche breakdown *VBD*. This eliminates dark diffusion current from narrow-gap layer and increases operational speed. Condition of punch-through of absorber, as follows from 4.1 and 4.2 is given by:

$$N\_1 \times W\_1 + N\_2 \times W\_2 < \frac{\varepsilon\_0 \varepsilon\_1}{q} \times E\_{1BD}(N\_{1'}W\_1). \tag{130}$$

Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when *R*<sup>1</sup> =*R*3 quantum efficiency reaches almost its maximal value when *W*<sup>2</sup> =2 μm. Therefore, for development of concentration – thickness nomogram in Fig. 21, namely this value *W*<sup>2</sup> was selected. Note that decrease in dispersion in *N*<sup>2</sup> results in increase in dispersion *N*1 and *W*1, while increase gives the opposite result. Val‐ ue of noise current density *IN* <sup>≤</sup>10<sup>−</sup>12A/Hz1/2 corresponds to *JT* <sup>≤</sup>1.8×10−<sup>5</sup> А/сm<sup>2</sup> , and value *IN* <sup>≤</sup>10−13 А/ Hz1/2 corresponds to *JT* <sup>≤</sup>1.8×10−<sup>7</sup> А/сm<sup>2</sup> .

#### **5.3. Operational speed**

Minimal possible time-of-response of this class of devices

$$\pi = 2 \times \left( \tau\_{tr1} \times f(M\_{ph}) + \tau\_{tr2} \right) \tag{131}$$

**Figure 21.** Concentration-thickness nomogram for avalanche *InP* / *I n*0.53*Ga*0.47*As* / *InP* heterophotodiode when *<sup>N</sup>*<sup>2</sup> = (1 ÷ 5) × 1015cm-3, *W*<sup>2</sup> = 2 μm, М*ph* = 15, cross-section area *A*= 5 × 10<sup>3</sup> μm2. When noise current *IN* <sup>=</sup> *SN* <sup>≤</sup>10−<sup>13</sup> A/Hz1/2, then allowable set of points in space (*N*1, *<sup>W</sup>*1) lies inside figure a-b-c-d; when *IN* <sup>=</sup> *SN* <sup>≤</sup>10<sup>−</sup>12 A/Hz1/2 − inside

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

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85

(*<sup>T</sup>* ) (*W*1) and *N*¯1(*<sup>W</sup>*1), respectively: 1 <sup>−</sup> *<sup>N</sup>*<sup>2</sup> = 1015cm-3, 2 <sup>−</sup>

figure a-e-f-g. Dashed and dash-dot curves − dependences *N*1min

*<sup>N</sup>*<sup>2</sup> = 5 × 1015cm-3. *N*1 is measured in units of 1016cm-3, *W*1 − in μm

is determined by time-of-flight of charge carriers through multiplication layer *τtr*1 and ab‐ sorber *τtr*2, and also by value of function *f* (*M ph* ), which is close to 1 when *K* >>1, and is where concentration and thicknesses, as in (127) and (128), are measured in cm-3 and μm,

(127) and (128), in which *N*<sup>2</sup> is replaced by *Q*(*W*1) / *W*2. It is shown from (127) and (128) that

crease *W*1 is caused by situation when the thicker *W*1 the less *E*1*BD* and the greater fall of

and the fact that at minimum *JT* <sup>1</sup> > *JT* 2. For small values either *N*2 or *W*2, field *E*(*x*) is so

with increase *W*2 due to reducing the length of tunneling generation region in narrow-gap material. In high performance diode, absorber should be punched-through when voltage bias *Vb* on heterostructure is less than voltage of avalanche breakdown *VBD*. This eliminates dark diffusion current from narrow-gap layer and increases operational speed. Condition of

> 0 1 1 1 2 2 1 11 ( , ). *N W N W E NW BD q* e e

Allowable intervals of concentrations and thicknesses of heterostructure layers are shown in Fig. 21. As can be seen from Fig. 20b, even, when *R*<sup>1</sup> =*R*3 quantum efficiency reaches almost its maximal value when *W*<sup>2</sup> =2 μm. Therefore, for development of concentration – thickness nomogram in Fig. 21, namely this value *W*<sup>2</sup> was selected. Note that decrease in dispersion in *N*<sup>2</sup> results in increase in dispersion *N*1 and *W*1, while increase gives the opposite result. Val‐

> А/сm<sup>2</sup> .

is determined by time-of-flight of charge carriers through multiplication layer *τtr*1 and ab‐ sorber *τtr*2, and also by value of function *f* (*M ph* ), which is close to 1 when *K* >>1, and is

and, also, although weakly, with increase *N*2. Decrease of values *N*1min

punch-through of absorber, as follows from 4.1 and 4.2 is given by:

ue of noise current density *IN* <sup>≤</sup>10<sup>−</sup>12A/Hz1/2 corresponds to *JT* <sup>≤</sup>1.8×10−<sup>5</sup>

(*<sup>T</sup>* ) and *JT* min are decreasing, moreover *JT* min sharply, with increase *W*1 (see Fig. 21, 22),

(*<sup>T</sup>* ) and *JT* min will be again determined by

(*<sup>T</sup>* ) and *JT* min with growth *N*<sup>2</sup> is due to

*Tng* in narrow-gap *nng* layer with increase *N*<sup>2</sup>

*Tng* in it is almost constant. Therefore, when

(*<sup>T</sup>* ) and *JT* min do no longer depend on *N*2 and slightly decrease

´+´< ´ (130)

t= ´ t ´ +t 2 () ( *tr ph tr* 1 2 *f M* ) (131)

(*<sup>T</sup>* ) and *JT* min with in‐

А/сm<sup>2</sup>

, and value

respectively.

*N*1min

If inequality (129) is not satisfied, then values *N*1min

field *E*(*x*) on *nwg* layer thickness. Slight decrease *N*1min

increasing of length of interband tunneling *l*

weakly dependent on *x* in *nng* layer, that value *l*

*IN* <sup>≤</sup>10−13 А/ Hz1/2 corresponds to *JT* <sup>≤</sup>1.8×10−<sup>7</sup>

Minimal possible time-of-response of this class of devices

**5.3. Operational speed**

*N*2*W*<sup>2</sup> <*Q*(*W*1) then values *N*1min

84 Photodiodes - From Fundamentals to Applications

**Figure 21.** Concentration-thickness nomogram for avalanche *InP* / *I n*0.53*Ga*0.47*As* / *InP* heterophotodiode when *<sup>N</sup>*<sup>2</sup> = (1 ÷ 5) × 1015cm-3, *W*<sup>2</sup> = 2 μm, М*ph* = 15, cross-section area *A*= 5 × 10<sup>3</sup> μm2. When noise current *IN* <sup>=</sup> *SN* <sup>≤</sup>10−<sup>13</sup> A/Hz1/2, then allowable set of points in space (*N*1, *<sup>W</sup>*1) lies inside figure a-b-c-d; when *IN* <sup>=</sup> *SN* <sup>≤</sup>10<sup>−</sup>12 A/Hz1/2 − inside figure a-e-f-g. Dashed and dash-dot curves − dependences *N*1min (*<sup>T</sup>* ) (*W*1) and *N*¯1(*<sup>W</sup>*1), respectively: 1 <sup>−</sup> *<sup>N</sup>*<sup>2</sup> = 1015cm-3, 2 <sup>−</sup> *<sup>N</sup>*<sup>2</sup> = 5 × 1015cm-3. *N*1 is measured in units of 1016cm-3, *W*1 − in μm

**Figure 22.** Dependences of minimal tunnel current *JT* min, A/cm2 of avalanche heterophotodiode *InP* / *I n*0.53*Ga*0.47*As* / *InP* on multiplication layer thickness *W*1, μm: 1 − *N*<sup>2</sup> = 1013cm-3, 2 − *N*<sup>2</sup> = 1015cm-3, 3 − *N*<sup>2</sup> = 1017cm-3

equal to *M ph* when *K* =1 (Tsang, 1985), (Filachev et al, 2011), (Emmons, 1967), (Kurochkin & Kholodnov, 1996). It was noted above that in *InP* 1< *K* ≤2.3. Therefore, in *InP* / *I nxGa*1−*<sup>x</sup>AsyP*1−*<sup>y</sup>* / *InP* SAM-APD

$$
\pi \equiv 2 \times \left( \tau\_{tr1} \times M\_{ph} + \tau\_{tr2} \right). \tag{132}
$$

*W*<sup>2</sup> =0.5 μm and fully reflecting backside surface. Minimal value *τtr*<sup>1</sup> is determined by maxi‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

**6. Analytical model of avalanche photodiodes operation in Geiger mode**

We consider possibility to describe transient phenomena in *p* −*i* −*n* APDs by elementary functions, first of all, when initially applied voltage *V*<sup>0</sup> is greater than avalanche breakdown voltage *VBD*. Formulation of the problem is caused by need to know specific conditions of APDs operation in Geiger mode. Simple expression describing dynamics of avalanche Gei‐ ger process is derived. Formula for total time of Geiger process is obtained. Explicit analyti‐ cal expression for realization of Geiger mode is presented. Applicability of obtained results is defined. APDs in Geiger mode (pulsed photoelectric signals) make possible detection of single photons (Groves et al, 2005), (Spinelli & Lacaita, 1997), (Zheleznykh et al, 2011), (Stop‐ pa et al, 2005), (Gulakov et al, 2007). It is worked at reverse bias voltages *Vb* >*VBD*. Different types of devices are realized on APDs in Geiger mode (Groves et al, 2005), (Spinelli & Lacai‐ ta, 1997), (Zheleznykh et al, 2011), (Stoppa et al, 2005), (Gulakov et al, 2007). At the same time, review of publications shows that theoretical studies have tendency to carry out in‐ creasingly sophisticated numerical simulations. In (Vanyushin et al, 2007) was proposed dis‐ crete model of Geiger avalanche process in *p* −*i* −*n* structure. Obtained iterative relations allow to determine, although fairly easy, but only by numerical method, options for realiza‐

tion of Geiger mode when ratio *K* ≡*β* / *α* differs very much from unity, where *α*(*Ei*

section admits value *К* =1. Considered below approach allows also to describe conditions of realization of Geiger mode and its characteristics by mathematically simple, graphically il‐ lustrative relations. It is adopted that photogeneration (PhG) is uniform over sample crosssection area *S* transverse to axis *x* (Fig. 23). Then, in the most important single-photon process, area *S*, according to uncertainty principle, shall not exceed in the order of magni‐ tude, square of wavelength of light *λ*. Under these conditions, it is allowably to consider problem as one-dimensional (axis *x*, Fig. 23). There are grounds to suppose that go beyond one-dimensional model at local illumination make no sense. Single-photon case arises itself

estimates below, has no time to spread significantly over cross section area. Consider serial circuit: *p* −*i* −*n* diode – load resistance *R* – power supply source providing bias *Vb* >*VBD*. Let *p* and *n* regions are heavily doped, so that prevailing share of bias falls across base *i*. Then after charging process voltage on it can be considered equals to *V*<sup>0</sup> =*Vb*. When electron-hole

pairs appear in the base then occurs their multiplication that results in decrease *Vi*

*GE*, Fig. 23). "Continuous" model (Kholodnov, 2009) developed in this

. The matter is that charge, during Geiger avalanche process, as show

*GE* in base by major charge carriers inflowing into *p* and *n* regions (Fig.

*GE*) – impact ionization coefficients of electrons and holes and *Ei*

ns.

A/cm2, then as follows from Fig. 22,

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87

*GE*) and

due to

*GE* – electric field in *i*–

mum allowable minimal value *W*1min. When *JT* <sup>≤</sup>10−<sup>6</sup>

*<sup>W</sup>*1min≅2 μm, and therefore *τ*min≅(4*<sup>M</sup> ph* + 1)×10−<sup>2</sup>

*β*(*Ei*

layer (base 0< *x* <*Wi*

when *<sup>S</sup>* > >*S*<sup>1</sup> <sup>≈</sup>*<sup>π</sup>* <sup>×</sup>*<sup>λ</sup>* <sup>2</sup>

screening of field *Ei*

As is evident from Fig. 20b, in *InP* / *I n*0.53*Ga*0.47*As* / *InP* heterostructure quantum efficiency value *η* lies in interval 0.5≤*η* ≤0.686 when *R*<sup>3</sup> =1 and *W*<sup>2</sup> ≥0.5 μm. It means that, because of not so much loss in quantum efficiency *η* compared to maximal possible (only 27 % less), time-of-response value *τtr*<sup>2</sup> =5 ps can be achieved by forming absorber with thickness *W*<sup>2</sup> =0.5 μm and fully reflecting backside surface. Minimal value *τtr*<sup>1</sup> is determined by maxi‐ mum allowable minimal value *W*1min. When *JT* <sup>≤</sup>10−<sup>6</sup> A/cm2, then as follows from Fig. 22, *<sup>W</sup>*1min≅2 μm, and therefore *τ*min≅(4*<sup>M</sup> ph* + 1)×10−<sup>2</sup> ns.

#### **6. Analytical model of avalanche photodiodes operation in Geiger mode**

We consider possibility to describe transient phenomena in *p* −*i* −*n* APDs by elementary functions, first of all, when initially applied voltage *V*<sup>0</sup> is greater than avalanche breakdown voltage *VBD*. Formulation of the problem is caused by need to know specific conditions of APDs operation in Geiger mode. Simple expression describing dynamics of avalanche Gei‐ ger process is derived. Formula for total time of Geiger process is obtained. Explicit analyti‐ cal expression for realization of Geiger mode is presented. Applicability of obtained results is defined. APDs in Geiger mode (pulsed photoelectric signals) make possible detection of single photons (Groves et al, 2005), (Spinelli & Lacaita, 1997), (Zheleznykh et al, 2011), (Stop‐ pa et al, 2005), (Gulakov et al, 2007). It is worked at reverse bias voltages *Vb* >*VBD*. Different types of devices are realized on APDs in Geiger mode (Groves et al, 2005), (Spinelli & Lacai‐ ta, 1997), (Zheleznykh et al, 2011), (Stoppa et al, 2005), (Gulakov et al, 2007). At the same time, review of publications shows that theoretical studies have tendency to carry out in‐ creasingly sophisticated numerical simulations. In (Vanyushin et al, 2007) was proposed dis‐ crete model of Geiger avalanche process in *p* −*i* −*n* structure. Obtained iterative relations allow to determine, although fairly easy, but only by numerical method, options for realiza‐ tion of Geiger mode when ratio *K* ≡*β* / *α* differs very much from unity, where *α*(*Ei GE*) and *β*(*Ei GE*) – impact ionization coefficients of electrons and holes and *Ei GE* – electric field in *i*– layer (base 0< *x* <*Wi GE*, Fig. 23). "Continuous" model (Kholodnov, 2009) developed in this section admits value *К* =1. Considered below approach allows also to describe conditions of realization of Geiger mode and its characteristics by mathematically simple, graphically il‐ lustrative relations. It is adopted that photogeneration (PhG) is uniform over sample crosssection area *S* transverse to axis *x* (Fig. 23). Then, in the most important single-photon process, area *S*, according to uncertainty principle, shall not exceed in the order of magni‐ tude, square of wavelength of light *λ*. Under these conditions, it is allowably to consider problem as one-dimensional (axis *x*, Fig. 23). There are grounds to suppose that go beyond one-dimensional model at local illumination make no sense. Single-photon case arises itself when *<sup>S</sup>* > >*S*<sup>1</sup> <sup>≈</sup>*<sup>π</sup>* <sup>×</sup>*<sup>λ</sup>* <sup>2</sup> . The matter is that charge, during Geiger avalanche process, as show estimates below, has no time to spread significantly over cross section area. Consider serial circuit: *p* −*i* −*n* diode – load resistance *R* – power supply source providing bias *Vb* >*VBD*. Let *p* and *n* regions are heavily doped, so that prevailing share of bias falls across base *i*. Then after charging process voltage on it can be considered equals to *V*<sup>0</sup> =*Vb*. When electron-hole pairs appear in the base then occurs their multiplication that results in decrease *Vi* due to screening of field *Ei GE* in base by major charge carriers inflowing into *p* and *n* regions (Fig.

equal to *M ph* when *K* =1 (Tsang, 1985), (Filachev et al, 2011), (Emmons, 1967), (Kurochkin & Kholodnov, 1996). It was noted above that in *InP* 1< *K* ≤2.3. Therefore, in *InP* /

**Figure 22.** Dependences of minimal tunnel current *JT* min, A/cm2 of avalanche heterophotodiode *InP* / *I n*0.53*Ga*0.47*As*

/ *InP* on multiplication layer thickness *W*1, μm: 1 − *N*<sup>2</sup> = 1013cm-3, 2 − *N*<sup>2</sup> = 1015cm-3, 3 − *N*<sup>2</sup> = 1017cm-3

As is evident from Fig. 20b, in *InP* / *I n*0.53*Ga*0.47*As* / *InP* heterostructure quantum efficiency value *η* lies in interval 0.5≤*η* ≤0.686 when *R*<sup>3</sup> =1 and *W*<sup>2</sup> ≥0.5 μm. It means that, because of not so much loss in quantum efficiency *η* compared to maximal possible (only 27 % less), time-of-response value *τtr*<sup>2</sup> =5 ps can be achieved by forming absorber with thickness

t@ ´ t ´ +t 2 . ( *tr ph tr* 1 2 *M* ) (132)

*I nxGa*1−*<sup>x</sup>AsyP*1−*<sup>y</sup>* / *InP* SAM-APD

86 Photodiodes - From Fundamentals to Applications

23) in quantity *Nn* and *Pp* and voltage drop across load resistor *VR* and, hence, current in external circuit arise

$$I\_R = \frac{V\_R}{R} = \frac{V\_b - V\_i}{R}.\tag{133}$$

sumed uniform. Numerical value *Ei*

in quasi-neutral parts of structure

*N t*

current, *q* – absolute value of electron charge, *t* - time.

*d*

¶

*Rd d d GE*

= ´ + < > < >= ´

*<sup>V</sup> IC I I I x t dx t W*

where *Vd* – voltage on DL, *Cd* =*εε*<sup>0</sup> ×*S* / *Wd* and *Wd* – DL capacity and thickness, *ε*0 –

dielectric constant of vacuum, *ε* – dielectric permittivity, < *Id* > let's call avalanche current

Relation (135) generalizes well-known theorem of Rameau (Spinelli & Lacaita, 1997), it takes into account key feature of Geiger mode – variation over time of voltage across DL, and it is valid for any distribution profile of dopant. In our formulation of the problem (in *p* −*i* −*n* structure) *i* – layer can be considered as DL, i.e., *d* in (135) and below should be replaced by

*i*. By integrating continuity equation for *IN* and *IP* with respect to *x* from 0 to *Wi*

*<sup>q</sup> I t I t I W t I t q Gt <sup>t</sup>* ¶ ´ =a´ +b´ + - + ´ ¶

00 0 0

*GE GE GE GE WW W W ii i i*

( ) ( ) ( ) ( , ) (0, ) ( ), *<sup>i</sup> GE N P Ni N*

( ) ( ) ( ) ( , ) (0, ) ( ), *<sup>i</sup> GE N P Pi P P t q I t I t I W t I t q Gt <sup>t</sup>* ¶ ´ =a´ +b´ - + + ´ ¶

, ,

*N N x t dx P P x t dx I t I x t dx G t G x t dx <sup>i</sup> <sup>i</sup> N P N P* == = = òò ò ò % % (138)

( ,) , ( ,) , () ( ,) , () ( ,)

marking linear density of photogeneration rate as *G*(*x*, *t*) we obtain equations

of saturation *vs*, i.e.,

*Iav*.

*GE* <sup>=</sup>*EBD* when *Vi* <sup>=</sup>*VBD* for a number of materials can be

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89

quickly determined by formulas given in Section 3. As in (Vanyushin et al, 2007), we restrict consideration by PhG in base only, we neglect recombination in it, and we assume that cur‐ rents of electrons *IN* and holes *IP* are determined by their drift in electric field with velocity

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

where *N* and *P* – linear density (per unit length) of electrons and holes, *I* – full conductive

Substituting volume charge density from Poisson equation in continuity equation for *I* and integrating over depletion layer (DL) we obtain that, in approximation of zero-bias current,

( , ) ( , ), ( , ) ( , ) ( , ) ( , ), *N sP N s I xt q v Nxt I xt Ixt I xt q v Pxt* =´ ´ º - =´ ´ (134)

0

¶ ò (135)

% % % (136)

% % % (137)

*GE* and

*Wd*

<sup>1</sup> , ( ,)

*i*

**Figure 23.** Avalanche process in*p* −*i* −*n* structure: "-" − acceptors charge in boundary *i* − *p* layer (cathode plate − Cath‐ ode); "+" − donors charge in boundary *i* −*n* layer (anode plate − Anode); ⊖ and ⊕ − generated in *i* − region avalanche photoelectrons and photoholes; *Nn* and *Pp* − inflowing in *n* − and *p* − regions avalanche photoelectrons and photo‐ holes; Ес and Еv – energy of conduction band bottom and valence band top; hν − photon energy

In present structure charge is mainly concentrated in thin near border *n* −*i* and *p* −*i* layers (let's call them plates, Fig. 23). Therefore, as in (Vanyushin et al, 2007), field *Ei GE* will be as‐ sumed uniform. Numerical value *Ei GE* <sup>=</sup>*EBD* when *Vi* <sup>=</sup>*VBD* for a number of materials can be quickly determined by formulas given in Section 3. As in (Vanyushin et al, 2007), we restrict consideration by PhG in base only, we neglect recombination in it, and we assume that cur‐ rents of electrons *IN* and holes *IP* are determined by their drift in electric field with velocity of saturation *vs*, i.e.,

23) in quantity *Nn* and *Pp* and voltage drop across load resistor *VR* and, hence, current in

*Vb* −*Vi*

**Figure 23.** Avalanche process in*p* −*i* −*n* structure: "-" − acceptors charge in boundary *i* − *p* layer (cathode plate − Cath‐ ode); "+" − donors charge in boundary *i* −*n* layer (anode plate − Anode); ⊖ and ⊕ − generated in *i* − region avalanche photoelectrons and photoholes; *Nn* and *Pp* − inflowing in *n* − and *p* − regions avalanche photoelectrons and photo‐

In present structure charge is mainly concentrated in thin near border *n* −*i* and *p* −*i* layers

holes; Ес and Еv – energy of conduction band bottom and valence band top; hν − photon energy

(let's call them plates, Fig. 23). Therefore, as in (Vanyushin et al, 2007), field *Ei*

*<sup>R</sup>* . (133)

*GE* will be as‐

*IR*<sup>=</sup> *VR <sup>R</sup>* <sup>≡</sup>

external circuit arise

88 Photodiodes - From Fundamentals to Applications

$$I\_N(\mathbf{x}, t) = q \times \upsilon\_s \times N(\mathbf{x}, t),\\ I\_P(\mathbf{x}, t) \equiv I(\mathbf{x}, t) - I\_N(\mathbf{x}, t) = q \times \upsilon\_s \times P(\mathbf{x}, t), \tag{134}$$

where *N* and *P* – linear density (per unit length) of electrons and holes, *I* – full conductive current, *q* – absolute value of electron charge, *t* - time.

Substituting volume charge density from Poisson equation in continuity equation for *I* and integrating over depletion layer (DL) we obtain that, in approximation of zero-bias current, in quasi-neutral parts of structure

$$I\_R = C\_d \times \frac{\partial V\_d}{\partial t} + < I\_d > ,< I\_d > = \frac{1}{W\_i^{GE}} \times \int\_0^{W\_d} I(\mathbf{x}, t)d\mathbf{x} \tag{135}$$

where *Vd* – voltage on DL, *Cd* =*εε*<sup>0</sup> ×*S* / *Wd* and *Wd* – DL capacity and thickness, *ε*0 –

dielectric constant of vacuum, *ε* – dielectric permittivity, < *Id* > let's call avalanche current *Iav*.

Relation (135) generalizes well-known theorem of Rameau (Spinelli & Lacaita, 1997), it takes into account key feature of Geiger mode – variation over time of voltage across DL, and it is valid for any distribution profile of dopant. In our formulation of the problem (in *p* −*i* −*n* structure) *i* – layer can be considered as DL, i.e., *d* in (135) and below should be replaced by *i*. By integrating continuity equation for *IN* and *IP* with respect to *x* from 0 to *Wi GE* and marking linear density of photogeneration rate as *G*(*x*, *t*) we obtain equations

$$\alpha \times \frac{\partial N\_i(t)}{\partial t} = \alpha \times \tilde{I}\_N(t) + \emptyset \times \tilde{I}\_P(t) + I\_N(W\_i^{\text{CE}}, t) - I\_N(0, t) + q \times \tilde{G}(t), \tag{136}$$

$$\mathbf{q} \times \frac{\partial P\_i(\mathbf{t})}{\partial \mathbf{t}} = \mathbf{a} \times \tilde{I}\_N(\mathbf{t}) + \mathbb{B} \times \tilde{I}\_P(\mathbf{t}) - I\_P(\mathbf{W}\_i^{\text{GE}}, \mathbf{t}) + I\_P(\mathbf{0}, \mathbf{t}) + \mathbf{q} \times \tilde{\mathbf{G}}(\mathbf{t}),\tag{137}$$

$$N\_i = \int\_0^{W\_i^{\mathcal{CE}}} N(\mathbf{x}, t) d\mathbf{x},\\P\_i = \int\_0^{W\_i^{\mathcal{CE}}} P(\mathbf{x}, t) d\mathbf{x},\\\tilde{I}\_{N, P}(t) = \int\_0^{W\_i^{\mathcal{CE}}} I\_{N, P}(\mathbf{x}, t) d\mathbf{x},\\\tilde{G}(t) = \int\_0^{W\_i^{\mathcal{CE}}} G(\mathbf{x}, t) d\mathbf{x} \tag{138}$$

Because plates are very thin, then generation and recombination in them can be neglected. Now by integrating same equations with respect to thickness of plates, we find that in ap‐ proximation of absence of minority carriers in *p* and *n* regions

$$I\_N(0, \ t) = I\_R + q \times \frac{\partial N\_n}{\partial t} = I\_R - \mathcal{C}\_i \times \frac{\partial V\_i}{\partial t} = I\_P(\mathcal{W}\_i^{\ominus \to}, t), \ I\_P(0, \ t) = I\_N(\mathcal{W}\_i^{\ominus \to}, t) = 0 \tag{139}$$

Strictly speaking, equations (139) are valid when *r*<sup>1</sup> ≡ *Pp* / *Nn* =1, from which *r*<sup>2</sup> ≡ | *Pi* − *Ni* | / *Nn* =0. Therefore, let's assume uniform PhG along *x*. Then, at *К* =1, symmetry requires *r*<sup>1</sup> =1. Equations (139) are correct in concern of the order of magnitude both when *К* is not too big and when small also. This follows from quasi-discrete computer iterations in uniform static field. Computer iterations are performed in several evenly spaced points of PhG *xg* succeeded by averaging with respect to *xg* and take into account much more number acts of impact ionization by holes than similar iterations in (Vanyushin et al, 2007). Iteration procedure performed in interval equals to several time-of-flight of charge carriers through base *ttr* gives 0.6 <*r*1< 1, and *r*2< 0.4 (Fig. 24a), which corresponds to approximation of uni‐ form field. Note that smallness *r*2 does not mean smallness *Pi* + *Ni* (curve 3 in Fig. 24a).

Relations (133)-(139) allow obtaining equations

$$\mathbb{E}\left[V\_{R'}\middle|\left(1\;\tau\_{i}\right)\right] = \frac{\hat{\sigma}^{2}V\_{R}}{\hat{\sigma}t^{2}} + \left|\frac{1}{\tau\_{i}} - \upsilon\_{s} \times \mathbb{Y}\left[\mathbb{E}\_{i}^{\mathrm{GE}}\left(V\_{R}\right)\right]\right| \times \frac{\hat{\sigma}V\_{R}}{\hat{\sigma}t} - \frac{\upsilon\_{s}}{\tau\_{i}} \times \mathbb{Y}\left[\mathbb{E}\_{i}^{\mathrm{GE}}\left(V\_{R}\right)\right] \times V\_{R} = q \times \tilde{\mathbb{G}}(t) \times \frac{2 \times \upsilon\_{s}}{\mathbb{C}\_{i} \times \mathcal{W}\_{i}^{\mathrm{GE}}},\tag{140}$$

with initial conditions

$$\left. V\_R(0) = 0, \left. \frac{\partial V\_R}{\partial t} \right|\_{t=0} = \frac{2v\_s}{\mathbb{C}\_i \times \mathcal{W}\_i^{GE}} \times \lim\_{t \to 0} q \times \int\_{-t}^t \tilde{\mathcal{G}}(t')dt' \tag{141}$$

where

$$\text{Y } \mathbf{Y}(\mathbf{E}\_{i}^{GE}) = \mathbf{X}(\mathbf{E}\_{i}^{GE}) - \{\mathbf{Z} \,/\,\mathbf{W}\_{i}^{GE}\},\\\mathbf{X} = \mathbf{c}(\mathbf{E}\_{i}^{GE}) + \emptyset \{\mathbf{E}\_{i}^{GE}\},\\\mathbf{E}\_{i}^{GE} = \{\mathbf{V}\_{b} - \mathbf{V}\_{R}\} / \,\mathbf{W}\_{i}^{GE},\\\mathbf{\tau}\_{i} = \mathbf{R}\mathbf{C}\_{i},\\\mathbf{C}\_{i} = \mathbf{s}\mathbf{x}\_{0} \times \{\mathbf{S} /\,\mathbf{W}\_{i}^{GE}\} \tag{142}$$

At delta-shaped time-evolving illumination *<sup>G</sup>*˜ *i* (*t*)= *N ph* ×*δ*(*t*) relations (140) and (141) are converted into

$$F[V\_R; (1/\tau\_i)] = 0, V\_R(0) = 0, \left. \frac{\partial V\_R}{\partial t} \right|\_{t=0} = A^{GE} \equiv \frac{q \times \mathcal{D}v\_s \times N\_{ph}}{\varepsilon \varepsilon\_0 \times S} \tag{143}$$

**Figure 24.** Evaluation of applicability of quasi-uniform field approximation. (**a**) − Results of quasi-discrete computer

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

mination of breakdown field on *K* =β / α; accepted (Tsang, 1985), (Grekhov & Serezhkin, 1980)

). (**b**) − Dependence of error ER during deter‐

http://dx.doi.org/10.5772/50778

91

(*K*): *r*1=*Pp*/*Nn*, *r*2=| *Pi* − *Ni* | / *Nn*, *r*3=(*Pp* + *Nn*) / (*Pi* + *Ni*

α(*E*) = *AGE* × exp(− *B* / *E*), where *AGE* , 1/μm: 1 − 200, 2 − 400, 3 − 800, 4 − 2000, 5 − 5000

iterative procedure *rj*

where *N ph* – number of absorbed photons.

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 91

Because plates are very thin, then generation and recombination in them can be neglected. Now by integrating same equations with respect to thickness of plates, we find that in ap‐

Strictly speaking, equations (139) are valid when *r*<sup>1</sup> ≡ *Pp* / *Nn* =1, from which *r*<sup>2</sup> ≡ | *Pi* − *Ni* | / *Nn* =0. Therefore, let's assume uniform PhG along *x*. Then, at *К* =1, symmetry requires *r*<sup>1</sup> =1. Equations (139) are correct in concern of the order of magnitude both when *К* is not too big and when small also. This follows from quasi-discrete computer iterations in uniform static field. Computer iterations are performed in several evenly spaced points of PhG *xg* succeeded by averaging with respect to *xg* and take into account much more number acts of impact ionization by holes than similar iterations in (Vanyushin et al, 2007). Iteration procedure performed in interval equals to several time-of-flight of charge carriers through base *ttr* gives 0.6 <*r*1< 1, and *r*2< 0.4 (Fig. 24a), which corresponds to approximation of uni‐

<sup>1</sup> <sup>2</sup> [ ;(1 / )] [ ( )] [ ( )] () , *R R GE s s GE R i s iR iR R GE*

<sup>2</sup> (0) 0, lim ( ') '

<sup>0</sup> ( ) ( ) (2 / ), ( ) ( ), ( ) / , , ( / ) *GE GE GE GE GE GE GE GE Y E E W X E E E V V W RC C S W ii i i i i b R i i ii <sup>i</sup>* =C - = a + b = - t = = ee ´ (142)

*i*

[ ;(1 / )] 0, (0) 0, *<sup>R</sup> GE s ph*

¶ ´ ´ t= = = º

*<sup>V</sup> q vN F V <sup>V</sup> <sup>A</sup>*

*t*

*V V v v F V v YE V YE V V q Gt <sup>t</sup> <sup>t</sup> C W* ¶ ¶ ì ü ï ï ´ t º + - ´ ´ - ´ ´ =´ ´ í ý ¶ ´ ï ï <sup>t</sup> ¶ t î þ

0

*R s <sup>R</sup> GE <sup>t</sup> <sup>t</sup> i i <sup>t</sup> <sup>V</sup> <sup>v</sup> <sup>V</sup> q G t dt <sup>t</sup> C W* ® <sup>=</sup> -

*i i i i*

0

0 0

*t S* <sup>=</sup>

2

*t*

¶ = = ´´ ¶ ´ <sup>ò</sup> % (141)

(*t*)= *N ph* ×*δ*(*t*) relations (140) and (141) are

¶ ee ´ (143)

*GE*, *<sup>t</sup>*), *IP*(0, *<sup>t</sup>*) = *IN* (*Wi*

*GE*, *t*) =0 (139)

(curve 3 in Fig. 24a).

% (140)

proximation of absence of minority carriers in *p* and *n* regions

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *IR* <sup>−</sup>*Ci* <sup>×</sup>

form field. Note that smallness *r*2 does not mean smallness *Pi* + *Ni*

Relations (133)-(139) allow obtaining equations

At delta-shaped time-evolving illumination *<sup>G</sup>*˜

where *N ph* – number of absorbed photons.

*Ri R*

2 2

with initial conditions

where

converted into

∂*Vi* <sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *IP*(*Wi*

∂ *Nn*

*IN* (0, *t*)= *IR* + *q* ×

90 Photodiodes - From Fundamentals to Applications

**Figure 24.** Evaluation of applicability of quasi-uniform field approximation. (**a**) − Results of quasi-discrete computer iterative procedure *rj* (*K*): *r*1=*Pp*/*Nn*, *r*2=| *Pi* − *Ni* | / *Nn*, *r*3=(*Pp* + *Nn*) / (*Pi* + *Ni* ). (**b**) − Dependence of error ER during deter‐ mination of breakdown field on *K* =β / α; accepted (Tsang, 1985), (Grekhov & Serezhkin, 1980) α(*E*) = *AGE* × exp(− *B* / *E*), where *AGE* , 1/μm: 1 − 200, 2 − 400, 3 − 800, 4 − 2000, 5 − 5000

If we take *R* =0 and lim *t*→*∞ G*˜ *i* (*t*)=*const* ≠0 then we find that breakdown is determined by condi‐ tion *Wi GE* <sup>×</sup>*Χ*(*Ei GE*)=2, which at *<sup>К</sup>* <sup>≠</sup>1 gives another value for breakdown field *Ei GE* <sup>=</sup>*Eav* than *Ei GE* <sup>=</sup>*EBD* obtained directly from solving of stationary problem in Section 3. However, dis‐ crepancy between *Eav* and *EBD* is no more than 20 %, if *К* is different from 1 by no more than two orders of magnitude (Fig. 24b). Equation (140) admits only numerical solution. Howev‐ er, Geiger mode can be described without solving this equation, by using physical grounds and limit *R* →*∞*, when

$$\mathbf{U}\_{\rm av} = \mathbf{C}\_{i} \times \frac{\partial \Delta V\_{i}}{\partial t}, \mathbf{F}[\Delta V; 0] = 0, \mathbf{E}\_{i}^{\rm CE} = V\_{i} / \mathbf{W}\_{i}^{\rm CE} = \left[V\_{b} - \Delta V\_{i}(t)\right] / \mathbf{W}\_{i}^{\rm CE}, \Delta V\_{i}(0) = 0, \frac{\partial \Delta V\_{i}}{\partial t}\bigg|\_{t=0} = \mathbf{A}^{\rm CE} \tag{144}$$

and problem is solved in quadratures. To solve in elementary functions let's approximate exact dependence *Y Ei GE*(*ΔVi* ) by piecewise-linear function passing through principal point *ΔVi* =*Dav* ≡*Vb* −*Vav* =*Vb* −*Eav* ×*Wi GE* (Fig. 25 and 26), where *<sup>Y</sup>* =0, and *Iav* reaches its peak during *tav*.

**Figure 26.** Ratio of approximate dependence *Y*˜(Δ*<sup>V</sup>* ) to exact *<sup>Y</sup>* (Δ*<sup>V</sup>* ) for *Ge* with orientation <100>; <sup>δ</sup> <sup>=</sup>Δ*<sup>V</sup>* / (Δ*<sup>V</sup>* )max;

*GE*=2 μm; *Dav*, V: 1 − 0.25, 2 − 0.5; 3 − 1

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

*GE*, where *Eb* <sup>=</sup>*Vb* / *Wi*


more than value of break point *ΔVk* of piecewise-linear approximation (Fig. 25). Under

<sup>1</sup> ln ( ) , 2, , 40, *av*

*Z Z v Y qN*

lanche diode, avalanche is able to develop and cancel itself in full. As seen from (145) it is

Since *ttr*<<*tav*, then results of computer evaluation of uniform field approximation applicabil‐ ity can be considered reasonable. To evaluate transverse charge spreading let's use expres‐

*Z Z SY D Vt V V Dt Z*

D =D ´ D = = º >>

*GE*. Then *ΔVi*max <sup>≡</sup>lim

*b av*

0

. Maximal voltage drop on load equals to *V <sup>R</sup>*max =*ΔVi*max.

*s b ph*

<sup>+</sup> ´ ´ (145)

*t*→*∞ ΔVi*

http://dx.doi.org/10.5772/50778

93

of inverse recharge of ava‐

(*t*) is not

− *Wi*

these conditions

*GE*=1 μm, − *Wi*

/

max / max

*i i t t i av av*

where *Yb* =*Yb*(*Eb*). Geiger mode occurs when during time *R* ×*Ci*

*av t t*

Suppose, for simplicity *X* (*Eb*)≤4 / *Wi*

happened when *R* ≥*R*min≅*tav* / *Ci*

**Figure 25.** Form of approximation of function *Y* (Δ*V* ). Dependences (1 - exact, 2 - approximate) are plotted for *Ge* with orientation <100> (Tsang, 1985) taken *Wi GE*= 1 μm, *Dav* = 4 V

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption… http://dx.doi.org/10.5772/50778 93

If we take *R* =0 and lim

*GE* <sup>×</sup>*Χ*(*Ei*

and limit *R* →*∞*, when

exact dependence *Y Ei*

during *tav*.

point *ΔVi* =*Dav* ≡*Vb* −*Vav* =*Vb* −*Eav* ×*Wi*

with orientation <100> (Tsang, 1985) taken *Wi*

tion *Wi*

*Ei*

*t*→*∞ G*˜ *i*

92 Photodiodes - From Fundamentals to Applications

(*t*)=*const* ≠0 then we find that breakdown is determined by condi‐

) by piecewise-linear function passing through principal

*GE* (Fig. 25 and 26), where *<sup>Y</sup>* =0, and *Iav* reaches its peak

*GE* <sup>=</sup>*Eav* than

0

*t*

*GE*)=2, which at *<sup>К</sup>* <sup>≠</sup>1 gives another value for breakdown field *Ei*

*GE* <sup>=</sup>*EBD* obtained directly from solving of stationary problem in Section 3. However, dis‐ crepancy between *Eav* and *EBD* is no more than 20 %, if *К* is different from 1 by no more than two orders of magnitude (Fig. 24b). Equation (140) admits only numerical solution. Howev‐ er, Geiger mode can be described without solving this equation, by using physical grounds

, [ ;0] 0, / [ ( )] / , (0) 0, *<sup>i</sup> GE GE GE <sup>i</sup> GE*

and problem is solved in quadratures. To solve in elementary functions let's approximate

**Figure 25.** Form of approximation of function *Y* (Δ*V* ). Dependences (1 - exact, 2 - approximate) are plotted for *Ge*

*GE*= 1 μm, *Dav* = 4 V

¶ ¶ (144)

*V V I C F V E V W V Vt W V <sup>A</sup> t t* <sup>=</sup>

¶D ¶D = ´ D = = º -D D= =

*av i i ii b i i i*

*GE*(*ΔVi*

**Figure 26.** Ratio of approximate dependence *Y*˜(Δ*<sup>V</sup>* ) to exact *<sup>Y</sup>* (Δ*<sup>V</sup>* ) for *Ge* with orientation <100>; <sup>δ</sup> <sup>=</sup>Δ*<sup>V</sup>* / (Δ*<sup>V</sup>* )max; − *Wi GE*=1 μm, − *Wi GE*=2 μm; *Dav*, V: 1 − 0.25, 2 − 0.5; 3 − 1

Suppose, for simplicity *X* (*Eb*)≤4 / *Wi GE*, where *Eb* <sup>=</sup>*Vb* / *Wi GE*. Then *ΔVi*max <sup>≡</sup>lim *t*→*∞ ΔVi* (*t*) is not more than value of break point *ΔVk* of piecewise-linear approximation (Fig. 25). Under these conditions

$$
\Delta V\_i(t) = \Delta V\_{i\max} \times \frac{Z^{t/t\_w} - 1}{Z^{t/t\_w} + Z},
\Delta V\_{i\max} = 2D\_{av},
t\_w = \frac{\ln Z}{\upsilon\_s \times Y\_b},
Z \equiv \frac{\varepsilon \mathbf{x}\_0 \times \mathbf{S} \times Y\_b \times D\_{av}}{q \times N\_{ph}} \gg 40,
\tag{145}
$$

where *Yb* =*Yb*(*Eb*). Geiger mode occurs when during time *R* ×*Ci* of inverse recharge of ava‐ lanche diode, avalanche is able to develop and cancel itself in full. As seen from (145) it is happened when *R* ≥*R*min≅*tav* / *Ci* . Maximal voltage drop on load equals to *V <sup>R</sup>*max =*ΔVi*max. Since *ttr*<<*tav*, then results of computer evaluation of uniform field approximation applicabil‐ ity can be considered reasonable. To evaluate transverse charge spreading let's use expres‐ sion (21) from (Pospelov et al, 1974). It determines dependence *χ*(*t*)≡*r* /*r*0, where *r*(*t*) and *r*<sup>0</sup> – current and initial radii of charge "drop" of parabolic type. Implying under capacity in (Pospelov et al, 1974) value *Ci* and putting *Wi GE* =1 μm, *r*<sup>0</sup> <sup>=</sup>*<sup>λ</sup>* =1 μm, in the case of singlephoton process we get *χ*(*tav*)< ˜ 21/4≅1.2. This justifies our assumption that charge spreading over sample cross-section during avalanche Geiger process is not intensive.

**Author details**

Viacheslav Kholodnov1

ces, Moscow, Russia

**References**

1350-4495

and Mikhail Nikitin2

2 Science & Production Association ALPHA, Moscow, Russia

6, (June 1980) p. L277-L280, ISSN 0021-4922

svyaz, Moscow [in Russian]

p. 361 - 368. ISSN 1862-6300

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ary 1982) p. 655-662, ISSN 0003-6951

151-159, ISSN 0033-8494 [in Russian]

1 V.A. Kotelnikov Institute of Radio Engineering and Electronics Russian Academy of Scien‐

Physical Design Fundamentals of High-Performance Avalanche Heterophotodiodes with Separate Absorption…

http://dx.doi.org/10.5772/50778

95

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Tekh. Fiz., Vol. 16, No. 8, (August 1990) p. 8-11, ISSN 0320-0116 [in Russian]

#### **7. Conclusions**

The above analysis shows that to create high performance SAM-APD (in particular, based on widely used *InP* / *I nxGa*1−*<sup>x</sup>AsyP*1−*<sup>y</sup>* / *InP* heterostructures) it is necessary to maintain close tolerances on dopants concentration in wide-gap multiplication layer I – *N*1 and in narrowgap absorption layer II – *N*2, and also on thickness *W*<sup>1</sup> of wide-gap multiplication layer (Fig. 1). This is due to strong dependence of interband tunnel current in such heterostructures on *N*1,*N*2 and *W*1. Allowable variation intervals of values *N*1,*N*2 and *W*1, and, optimal thick‐ ness of absorber also, can be determined using results obtained in Sections 4 and 5. Value of minimal possible time-of-response *τ*min depends not only on photocurrent's gain *Мph* but on allowable noise density at preset value of photocurrent's gain also. The lower noise density, the larger is value *τ*min. For example, for heterostructure *InP* / *I n*0.53*Ga*0.47*As* / *InP* minimal time-of-response equals to *τ*min <sup>≈</sup>0.6 ns, when noise current equals to 3.3×10−<sup>11</sup> А/Hz1/2 and current responsivity 10.3 A/W. Analysis shows that operational speed can be slightly in‐ creased by means of inhomogeneous doping of wide-gap multiplication layer. To ensure op‐ erational speed in picosecond range it is necessary to use as multiplication layer semiconductor layer with low tunnel current and impact ionization coefficients of electrons and holes much different from each other, for example, indirect-gap semiconductor silicon. As has long been known maximal operational speed is achieved by APD if light is absorbed in space-charge region. In this case, as it was shown in Section 6, when bias voltage *Vb* ex‐ ceeds breakdown voltage *VBD* of no more than a few volts, then, for *K* ≡*β* / *α* values lying in interval from a few hundredths to a few tens, elementary relations (145) can be used for ap‐ proximate description of Geiger mode in *p* −*i* −*n* APD. Moreover if cross-section area *<sup>S</sup>* <sup>&</sup>gt;*S*<sup>1</sup> <sup>≈</sup>*<sup>π</sup>* <sup>×</sup>*<sup>λ</sup>* <sup>2</sup> , then we can expect that in single-photon case under *S* in (145) should imply value of order *S*1. This is due to finite size of single-photon spot *S*<sup>1</sup> and not intensive spread‐ ing of charge during time of avalanche Geiger process *tav* when photogeneration of charge carriers occurs in *i* – region of *p* −*i* −*n* structure depleted by charge carriers. Proposed ap‐ proach allows describing Geiger mode by elementary functions at voltages higher *Vb* as well. Note that equation (140) and physical grownds allow to expect three possible process modes at pulse illumination under *Vb* >*VBD*. When *RC* < <*tav* then generated photocurrent will tend to reach some constant and flow indefinitely (unless, of course, ignore energy loss‐ es). When *RC* =*tav* then generated photocurrent will be of infinitely long oscillatory charac‐ ter. When *RC* > >*tav* then Geiger mode is realized.
