**1. Introduction**

high-resolution (640×480) uncooled infrared focal plane arrays. *Optical Engineering*,

[71] Blackwell, R., Lacroix, D., & Bach, T. (2009). micron microbolometer FPA Technology

[72] Li, C., Skidmore, G., Howard, C., Clarke, E., & Han, J. (2009). Advancement in 17 mi‐ cron pixel pitch uncooled focal plane arrays. *Proceedings of SPIE*, 7298, 72980S.

[73] Schimert, T., Hanson, C., & Brady, J. (2009). Advanced in small-pixel, large-format al‐

[74] Trouilleau, C., Fieque, B., Noblet, S., & Giner, F. (2009). High Performance uncooledamorphous silicon TEC less XGA IRFPA with 17 micron pixel. *Proceedings of SPIE*,

[75] Dhar, N. K., & Dat, R. (2012). Advanced Imaging Research and Development at

[76] D'Souza, A. I., Robinson, E., Ionescu, A. C., Okerlund, D., de Lyon, T. J., Rajavel, R. D., Sharifi, H., Yap, D., Dhar, N., Wijewarnasuriya, P. S., & Grein, C. (2012). MWIR InAs1-xSbx nCBn Detectors Data and Analysis. *Proceedings of SPIE*, 8353, 835333.

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[78] Logeeswaran, V. J., Jinyong, O., Nayak, A. P., Katzenmeyer, A. M., Gilchrist, K. H., Grego, S., Kobayashi, N. P., Wang, S. Y., Talin, A. A., Dhar, N. K., & Islam, M. S. (2011). A perspective on Nanowire Photodetectors: Current Status, Future Challeng‐ es, and Opportunities. *IEEE Journal of selected topics in quantum electronics*, 17(4),

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*Proceedings of SPIE*, 8353, 8353 2R.

45, 014001.

190 Optoelectronics - Advanced Materials and Devices

7298, 72980Q.

1002-1023.

Measurements of semiconductor photocurrent (PC) spectra have a long and rich history. During the 1960s and 1970s, the topic became one of the most studied phenomena in semi‐ conductor research so that entire textbooks were dedicated to the subject [1-4]. In spite these considerable activities, only a few theoretical efforts were published in order to fit PC spec‐ tra. The first attempt is attributed to DeVore [5], who, with the purpose to find an explana‐ tion for the typically measured PC peak in the vicinity of the band gap, presumed enhanced carrier recombinations at the semiconductor surface with respect to the bulk. In other words, along the light propagating coordinate the carrier lifetime (and therefore the recom‐ bination rate) is changing. However, instead to transfer this idea directly into a mathemati‐ cal model, DeVore used the non-measurable parameter carrier surface recombination velocity in order to achieve PC fits with a peak.

Much more recently, investigating the PC of thin-film Bi2S3, Kebbab et al. and Pejova report‐ ed that the photoconductivity peak cannot be explained using the DeVore model [6-8], while, on the other hand, Pejova noticed that the PC formula published by Bouchenaki, Ull‐ rich et al. (BU model hereafter) in 1991 [9] fits the measured spectra very well. The expres‐ sion in Ref. [9] was intuitively derived and does not use surface recombination velocity but indeed different recombination rates at the surface and the bulk by introducing different carrier lifetimes along the propagation coordinate of the impinging light. Besides the above mentioned references, Ullrich's formula was successfully employed to fit PC spectra of thinfilm CdS [9,10], GaAs [11], ZnS [12] and of the non-common semiconductor YBCO6 [13].

Considering its correctness for a vast variety of semiconductors, we present here a precise derivation of the BU model and, using bulk CdS as representative semiconducting material, the work reveals the identical excellent agreement between PC experiments and theory for

both: Experimental absorption data and absorption coefficient calculations by combination of density of states and modified Urbach rule [14]. The work further stresses the correct link between the here promoted PC model and the actually measured surface to bulk lifetime ra‐ tio, and presents an extended and more detailed manuscript based on our recently publish‐ ed paper [15].

#### **2. General Theory**

#### **2.1. Fundamental Equations**

The sample geometry for the experiments and theoretical analysis is shown in Fig. 1. The impinging light intensity along the *z*-axis is *I* 0 [in J/(s cm2 )], and *n*(*x*, *y*, *z*, *t*), *p*(*x*, *y*, *z*, *t*) are the nonequilibrium electron and hole densities [in 1/cm3 ], respectively, generated by the in‐ coming photons. The continuity equations are expressed by their general form,

$$\frac{\partial n}{\partial t} = + \frac{1}{e} \nabla \bullet \vec{j}\_n - \frac{n}{\tau\_n} + \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) \tag{1}$$

*NN n* = +<sup>0</sup> (5)

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

*PP p* = +<sup>0</sup> (6)

⇀

vv v (8)

<sup>v</sup> <sup>v</sup> (9)

<sup>v</sup> <sup>v</sup> (10)

⇀ in terms of dark current densi‐

*ph* , which is the current

http://dx.doi.org/10.5772/51412

193

⇀ =0 under

*<sup>p</sup>* into Eqs. (1)

⇀ *<sup>n</sup>*, *j* ⇀

where N0 and P0 are the electron and hole equilibrium uniform densities. Hence, the total

( ) vv v <sup>v</sup> (7)

*j j j eN P E e D n D p* = + = + + Ñ- Ñ *np n p e h* ( )

 m

dark ph *jj j* = +

0 0 ( ) *dark n p j eN P E* = + m

n p ( ) ph ( ) *e h j en p E e D n D p* = + + Ñ- Ñ

uniform and time independent, i.e., ∂N0/∂t=∂P0/∂t = ∇N0=∇P0=0, and (iii) ∇•*E*

<sup>2</sup> <sup>0</sup> exp( ) *e n n n <sup>I</sup> D n z En*

 a

> a

 m

> m

v

v

<sup>h</sup> (11)

<sup>h</sup> (12)

ha

¶ = + Ñ - + - + ·Ñ

 w

<sup>0</sup> exp( ) *<sup>2</sup> h p p p <sup>I</sup> D p z Ep*

ha

¶ = + Ñ - + - - ·Ñ

 w

local neutrality condition. With the substitution of the current density *j*

t

t

We now assume (i) local neutrality condition, i.e., n=p, which implies equal lifetimes of electrons and holes, τn=τp=τ. (ii) the equilibrium electron and hole density N0 and P0 are

 m

m

 m

m

*dark* , i.e., current in absence of illumination, and the PC density *j*

current density is given by the sum of Esq. (3) and (4),

ty *j* ⇀

where,

and

and (2), we obtain:

*t*

¶

*t*

¶

We may decompose now Eq. (7) in the total current density *j*

generated by the impinging light on the semiconductor,

$$\frac{\partial \hat{p}}{\partial t} = -\frac{1}{e} \nabla \bullet \vec{j}\_{\rho} - \frac{p}{\tau\_p} + \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) \tag{2}$$

where the terms *n*/*τ* n and *p*/*τ* <sup>p</sup> represent the recombination rates for the electrons and holes, and *τ* n and *τ* p as their respective lifetimes. The term *I* 0/(ℏ *ω*) α exp(α*z*) is the decay of the generation rate [in 1/(s cm2 )] along the penetration of light of the nonequilibrium carriers, where ℏ*ω* is the impinging light energy and α is the absorption coefficient [in 1/cm], η is the unit less conversion efficiency coefficient, and e the elementary charge. The vectors *j* → *<sup>n</sup>*and *j* → *p* are the electron and hole components of the current density and are given by,

$$
\vec{j}\_n = \vec{j}\_{n,E} + \vec{j}\_{n,\mathcal{D}} = eN\,\mu\_n \vec{E} + \mathbf{e}D\_e \nabla n \tag{3}
$$

$$
\vec{j}\_{\text{p}} = \vec{j}\_{\text{p},\text{E}} + \vec{j}\_{\text{p},\text{D}} = \text{e}P\mu\_{\text{p}}\vec{E} - \text{e}D\_{h}\nabla p\tag{4}
$$

where *j* → *<sup>n</sup>*,*D*, *j* → *<sup>p</sup>*,*D* stands for the diffusion current driven by the density gradient and *j* → *<sup>n</sup>*,*E*, *j* → *p*,*E* represents the conduction current driven by an external electric field*E* <sup>→</sup> . The terms D*e*∇n and D*h*∇p refers to the diffusion of the non-equilibrium carriers, whereas D*e* and D*h*, is the diffu‐ sion constant of electrons and holes, respectively. The drift mobility of electrons and holes is μn and μp, and the total electron and hole densities are N and P, which are given by

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors http://dx.doi.org/10.5772/51412 193

$$N = N\_0 + n \tag{5}$$

$$P = P\_0 + p\tag{6}$$

where N0 and P0 are the electron and hole equilibrium uniform densities. Hence, the total current density is given by the sum of Esq. (3) and (4),

$$\bar{j} = \bar{j}\_n + \bar{j}\_p = e(N\mu\_n + P\mu\_p)\bar{E} + e\left(D\_e\nabla n - D\_h\nabla p\right) \tag{7}$$

We may decompose now Eq. (7) in the total current density *j* ⇀ in terms of dark current densi‐ ty *j* ⇀ *dark* , i.e., current in absence of illumination, and the PC density *j* ⇀ *ph* , which is the current generated by the impinging light on the semiconductor,

$$
\bar{j} = \bar{j}\_{\text{dark}} + \bar{j}\_{\text{ph}} \tag{8}
$$

where,

both: Experimental absorption data and absorption coefficient calculations by combination of density of states and modified Urbach rule [14]. The work further stresses the correct link between the here promoted PC model and the actually measured surface to bulk lifetime ra‐ tio, and presents an extended and more detailed manuscript based on our recently publish‐

The sample geometry for the experiments and theoretical analysis is shown in Fig. 1. The

0

*j z*

 a

 a

)] along the penetration of light of the nonequilibrium carriers,

rr r <sup>r</sup> (3)

rr r v (4)

ha

 w

0

*j z*

ha

 w

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

<sup>1</sup> exp( ) *<sup>p</sup>*

where the terms *n*/*τ* n and *p*/*τ* <sup>p</sup> represent the recombination rates for the electrons and holes,

where ℏ*ω* is the impinging light energy and α is the absorption coefficient [in 1/cm], η is the

coming photons. The continuity equations are expressed by their general form,

*n n I*

*p p I*

and *τ* n and *τ* p as their respective lifetimes. The term *I* 0/(ℏ *ω*) α exp(*-*

represents the conduction current driven by an external electric field*E*

r

r

n

p

unit less conversion efficiency coefficient, and e the elementary charge. The vectors *j*

, n,D <sup>n</sup> e *n nE <sup>e</sup> j j j eN E D n* = + = +Ñ m

p p,E p,D p e e *<sup>h</sup> j j j PE Dp* = + = -Ñ m

μn and μp, and the total electron and hole densities are N and P, which are given by

*<sup>p</sup>*,*D* stands for the diffusion current driven by the density gradient and *j*

D*h*∇p refers to the diffusion of the non-equilibrium carriers, whereas D*e* and D*h*, is the diffu‐ sion constant of electrons and holes, respectively. The drift mobility of electrons and holes is

are the electron and hole components of the current density and are given by,

t

¶ =- Ñ· - + -

t

¶ =+ Ñ· - + -

)], and *n*(*x*, *y*, *z*, *t*), *p*(*x*, *y*, *z*, *t*) are

], respectively, generated by the in‐

<sup>h</sup> (1)

<sup>h</sup> (2)

α

*z*) is the decay of the

→ *<sup>n</sup>*and *j* → *p*

→ *<sup>n</sup>*,*E*, *j* → *p*,*E*

<sup>→</sup> . The terms D*e*∇n and

ed paper [15].

**2. General Theory**

**2.1. Fundamental Equations**

192 Optoelectronics - Advanced Materials and Devices

generation rate [in 1/(s cm2

where *j* → *<sup>n</sup>*,*D*, *j* →

impinging light intensity along the *z*-axis is *I* 0 [in J/(s cm2

the nonequilibrium electron and hole densities [in 1/cm3

*t e*

*t e*

¶

¶

$$
\bar{j}\_{duk} = \mathbf{e}(N\_0 \mu\_n + P\_0 \mu\_p)\bar{E} \tag{9}
$$

and

$$\bar{J}\_{\rm ph} = e(n\mu\_{\rm n} + p\mu\_{\rm p})\bar{E} + e\left(D\_{\rm e}\nabla n - D\_{\rm h}\nabla p\right) \tag{10}$$

We now assume (i) local neutrality condition, i.e., n=p, which implies equal lifetimes of electrons and holes, τn=τp=τ. (ii) the equilibrium electron and hole density N0 and P0 are uniform and time independent, i.e., ∂N0/∂t=∂P0/∂t = ∇N0=∇P0=0, and (iii) ∇•*E* ⇀ =0 under local neutrality condition. With the substitution of the current density *j* ⇀ *<sup>n</sup>*, *j* ⇀ *<sup>p</sup>* into Eqs. (1) and (2), we obtain:

$$\frac{\partial n}{\partial t} = +D\_\varepsilon \nabla^2 n - \frac{n}{\tau} + \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) + \mu\_n \vec{E} \bullet \nabla n \tag{11}$$

$$\frac{\partial p}{\partial t} = +D\_h \nabla^2 p - \frac{p}{\tau} + \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) - \mu\_p \bar{E} \bullet \nabla p \tag{12}$$

We multiply now Eq. (11) and Eq. (12) with the hole and electron conductivity, i.e., σ<sup>p</sup> and σn, respectively, and, by adding both equations and simultaneously replacing p with n (n=p), we obtain the following relationship:

$$\frac{\partial \mathcal{U}}{\partial t} = \frac{\sigma\_p D\_e + \sigma\_n D\_h}{\sigma\_n + \sigma\_p} \nabla^2 n - \frac{n}{\tau} + \eta \frac{I\_0}{\hbar \omega} n \exp(-az) + \frac{\mu\_n \sigma\_p - \mu\_p \sigma\_n}{\sigma\_n + \sigma\_p} \tilde{E} \bullet \nabla n \tag{13}$$

Furthermore, we define the bipolar diffusion coefficient as,

$$D = \frac{\sigma\_{\text{p}} D\_{\text{e}} + \sigma\_{\text{n}} D\_{\text{h}}}{\sigma\_{\text{n}} + \sigma\_{\text{p}}} \tag{14}$$

**2.2. Photocurrent spectra for experimental setup condition**

by lx×ly×lz, where lz=d is the thickness of the sample,

ering the following general conditions:

**i.** stationary state, i.e., ∂n/∂t=0,

**iv.** (iv) the external electric field *E*

In Eq. (19), we have dropped the term *E*

pendicular to each other.

the vector *d A*

equation (Eq. 17), we get,

dent light and along x-direction, i.e.,*E*

2

*dz*

t

the z-direction. In other words, no physical carrier diffusion current *j*

Here, the electrical current cross-section area vector *d A*

diffusion current term ∇*n* =(*dn* / *dz*)*e*

boundary z=0 and z=d. From Eq. (18), we arrive at the boundary conditions,

*dn dz and dn dz*

0 0

=

*z d*

=

By using Eq. (18), we can obtain the PC passing through the sample as*I ph* = *∬ j*

⇀

ute to the PC. Finally, we obtain the PC for the specific experimental setup as,

*z*

=

0

⇀ is perpendicular to the cross-section area vector*e*(*De* <sup>−</sup>*Dh* )∇*n*. Therefore, the

=

So far, we have made some general assumption regarding the process of nonequilibrium carriers. We shall now restrict ourselves to the detailed determination of PC spectra, consid‐

**ii.** the light illuminates uniformly the entire semiconductor sample, whereas the direc‐

**iii.** the sample possesses a uniform nonequilibrium carrier density in the x-y plane, and diffusion takes place along the z-axis only, i.e., n(x, y, z)=n(z), and

> 0 <sup>2</sup> exp( ) 0 *dn n <sup>I</sup> D z*


In the following step we introduce the boundary condition for solving Eq. (19). Since the PC is measured along the x-axis, i.e., the external electric filed is perpendicularly to the direc‐ tion of the impinging light as shown in Fig. 1, consequently, no closed current loop exists in

ha

 w ⇀ <sup>=</sup>*Exe* ⇀

> a

tion of the impinging light is along the z-axis and the volume of the sample is given

⇀ applied to the sample is perpendicular to the inci‐

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

<sup>h</sup> (19)

⇀ •∇*<sup>n</sup>* =0due to the fact that the two vectors are per‐

⇀ is given by*<sup>d</sup> <sup>A</sup>*

*<sup>z</sup>* in Eq. (18) which is along z-direction does not contrib‐

*<sup>x</sup>*. Hence, with the stationary continuity

http://dx.doi.org/10.5772/51412

195

*<sup>D</sup>* will take place at the

⇀ *ph* •*d A* ⇀ .

*<sup>x</sup>*. Notice that

⇀ <sup>=</sup>*dzdye* ⇀

(20)

and the bipolar drift mobility μE as,

$$
\mu\_E = \frac{\mu\_n \sigma\_p - \mu\_p \sigma\_n}{\sigma\_n + \sigma\_p} \tag{15}
$$

Note that the bipolar drift mobility μE is different from the bipolar diffusion mobility μE, which is defined as

$$
\mu\_D = \frac{\mu\_n \sigma\_\rho + \mu\_\rho \sigma\_n}{\sigma\_n + \sigma\_\rho} \tag{16}
$$

Thus, we have the continuity equation for n (and consequently for p, because n=p),

$$\frac{\partial \boldsymbol{n}}{\partial t} = \boldsymbol{D} \nabla^2 \boldsymbol{n} - \frac{\boldsymbol{n}}{\tau} + \frac{\eta \boldsymbol{I}\_0}{\hbar \omega} \boldsymbol{a} \exp(-a\boldsymbol{z}) + \mu\_E \vec{\boldsymbol{E}} \bullet \nabla \boldsymbol{n} \tag{17}$$

and for the PC density:

$$
\vec{j}\_{\text{ph}} = e(\mu\_n + \mu\_p) n \vec{E} + e(D\_e - D\_h) \nabla n \tag{18}
$$

The Eqs. (17) and (18) are the fundamental equations that allow us, in case the energy dis‐ persion of α in known, to calculate the distribution of the carrier density, carrier current, and the PC spectra.

#### **2.2. Photocurrent spectra for experimental setup condition**

So far, we have made some general assumption regarding the process of nonequilibrium carriers. We shall now restrict ourselves to the detailed determination of PC spectra, consid‐ ering the following general conditions:

**i.** stationary state, i.e., ∂n/∂t=0,

We multiply now Eq. (11) and Eq. (12) with the hole and electron conductivity, i.e., σ<sup>p</sup> and σn, respectively, and, by adding both equations and simultaneously replacing p with n

*α*exp(−*αz*) +

*μnσ<sup>p</sup>* −*μpσ<sup>n</sup> σ<sup>n</sup>* + *σ<sup>p</sup>*

<sup>+</sup> <sup>=</sup> <sup>+</sup> (14)


<sup>+</sup> <sup>=</sup> <sup>+</sup> (16)

⇀ <sup>+</sup> *<sup>e</sup>*(*De* <sup>−</sup>*Dh* )∇*<sup>n</sup>* (18)

⇀ •∇*<sup>n</sup>* (17)

*E*

⇀ •∇*<sup>n</sup>* (13)

(n=p), we obtain the following relationship:

194 Optoelectronics - Advanced Materials and Devices

*σpDe* + *σnDh σ<sup>n</sup>* + *σ<sup>p</sup>*

and the bipolar drift mobility μE as,

∂*n*

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup>*D*∇2*<sup>n</sup>* <sup>−</sup> *<sup>n</sup>*

*j* ⇀

which is defined as

and for the PC density:

the PC spectra.

<sup>∇</sup>2*<sup>n</sup>* <sup>−</sup> *<sup>n</sup> τ* + *η I*0 ℏ*ω*

p

*E*

*D*

m

*τ* + *ηΙ*<sup>0</sup> ℏ*ω*

*ph* =*e*(*μn* + *μp*)*nE*

m

*D D e nh <sup>D</sup>* s

> s s

ms

s s

n p

*n p pn*

 ms

*n p*

Note that the bipolar drift mobility μE is different from the bipolar diffusion mobility μE,

*n p pn*

 ms

ms

Thus, we have the continuity equation for n (and consequently for p, because n=p),

s s

*n p*

The Eqs. (17) and (18) are the fundamental equations that allow us, in case the energy dis‐ persion of α in known, to calculate the distribution of the carrier density, carrier current, and

*α*exp(−*αz*) + *μE E*

 s

Furthermore, we define the bipolar diffusion coefficient as,

∂*n* <sup>∂</sup>*<sup>t</sup>* <sup>=</sup>


$$D\frac{d^2\eta}{d\tau^2} - \frac{\eta}{\tau} + \eta \frac{I\_0}{\hbar \phi} \alpha \exp(-\alpha z) = 0\tag{19}$$

In Eq. (19), we have dropped the term *E* ⇀ •∇*<sup>n</sup>* =0due to the fact that the two vectors are per‐ pendicular to each other.

In the following step we introduce the boundary condition for solving Eq. (19). Since the PC is measured along the x-axis, i.e., the external electric filed is perpendicularly to the direc‐ tion of the impinging light as shown in Fig. 1, consequently, no closed current loop exists in the z-direction. In other words, no physical carrier diffusion current *j <sup>D</sup>* will take place at the boundary z=0 and z=d. From Eq. (18), we arrive at the boundary conditions,

$$\left.\frac{dn}{dz}\right|\_{z=0} = 0$$
 and 
$$\left.\frac{dn}{dz}\right|\_{z=d} = 0$$

By using Eq. (18), we can obtain the PC passing through the sample as*I ph* = *∬ j* ⇀ *ph* •*d A* ⇀ . Here, the electrical current cross-section area vector *d A* ⇀ is given by*<sup>d</sup> <sup>A</sup>* ⇀ <sup>=</sup>*dzdye* ⇀ *<sup>x</sup>*. Notice that the vector *d A* ⇀ is perpendicular to the cross-section area vector*e*(*De* <sup>−</sup>*Dh* )∇*n*. Therefore, the diffusion current term ∇*n* =(*dn* / *dz*)*e* ⇀ *<sup>z</sup>* in Eq. (18) which is along z-direction does not contrib‐ ute to the PC. Finally, we obtain the PC for the specific experimental setup as,

$$I\_{\rm ph} = (\frac{l\_y}{l\_x})(\Delta U e \mu) \Big|\_{0}^{d} n(z) dz \tag{21}$$

Where the constant S is called surface recombination velocity. By solving the standard sec‐ ond order differential equation n(z) with the boundary conditions [Eq. (22)], and substitute

0

<sup>0</sup> <sup>0</sup> ( )( )( ) *<sup>y</sup>*

<sup>1</sup> (1 exp( ) (1 exp( ) 1 exp( ) 1 / 1 coth( / 2) *Sd d F d*

( ) <sup>0</sup> 1 exp *ph PC II d* = -- é ù

*F d* = - 1-exp ( )

**3.2. PC theory based on the spatial non-uniform recombination rate (BU model)**

a

Consequently, DeVore's formula is not able to explain the PC maximum in the vicinity of the band gap for the geometry shown in Fig. 1. The boundary conditions expressed by Eq.

The ansatz of the original BU model in Ref. [9] led to an intuitive and straightforward theory that implemented DeVore's assumption directly. It simply assumed that the spatial distribu‐

a

where α is the energy dependent absorption coefficient. We argue that the boundary conditions [Eq. (22)] are not consistent with the experimental setup, i.e. the PC is driv‐ en by an applied electric field, which is perpendicular to the impinging light. There‐ fore, at the surface along the incident light there is an open circuit. The correct boundary condition is that there will be no particle current across the sample sur‐

m t

w

 a

é ù + -+ - - = - -- ê ú - + ë û (26)

tb

*<sup>l</sup> <sup>I</sup> I Ue l*

*x*

at

= D

*ph PC I IF* = (24)

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

2 2

=0. Under this boundary condition, the DeVore's

ë û (27)

(28)

 b

*S d*

 ab

<sup>h</sup> (25)

http://dx.doi.org/10.5772/51412

197

 a

n(z) into the expression of Iph [Eq. (21)], we arrive at DeVore's formula,

*Ph*

a

∂*n* ∂ *z* <sup>|</sup> *<sup>z</sup>*=*<sup>d</sup>*

i.e., the spectral factor F in Eq. (25) is reduced to the absorption,

(22) are artificially acting as source of non-equilibrium carriers.

and the dimensionless spectrum factor F has the form,

<sup>=</sup> <sup>−</sup> <sup>1</sup> *τβ* <sup>2</sup>

where the PC magnitude is given by,

2 2

a b

face, i.e.,<sup>−</sup> <sup>1</sup>

PC becomes,

*τβ* <sup>2</sup>

∂*n* <sup>∂</sup> *<sup>z</sup>* <sup>|</sup> *<sup>z</sup>*=0

where μ=μn+μp is the mobility and ΔU=Exlx and the voltage drop across the sample. It is worthwhile noting that Iph has the physical unit of Ampere (A). Equations (19-21), the ener‐ gy dependence of the absorption coefficient *α*(ℏ*ω*) , and the spatial lifetime distribution of carriers (τ) form the complete set needed for the fit of PC spectra measured with the config‐ uration displayed in Fig. 1.

### **3. DeVore's approch vs. BU model**

In this section, we present the comparison of the two existing theoretical PC models, which were initially introduced in Refs. [5] and [9]. The PC spectra are based on the common ex‐ perimental setup shown in Fig.1, where the direction of the applied electric field is perpen‐ dicular to the direction of illumination.

#### **3.1. The DeVore's photocurrent spectral theory**

It has been the conventional way of engage DeVore's formula to fit PC spectra. However, as presented below, for the standard PC experiments displayed in Fig 1, DeVore's formula is actually not the correct one to use. We now briefly outline DeVore's early work [5], in which th calculation is based on the following equation,

$$\frac{1}{\pi \beta^2} \frac{d^2 n}{d\varepsilon^2} - \frac{n}{\pi} + \eta \frac{I\_0}{\hbar \alpha \nu} \alpha \exp(-\alpha z) = 0 \tag{22}$$

This formulation is based on four assumptions:


$$-\frac{1}{\pi \beta^2} \frac{\partial n}{\partial z}\bigg|\_{z=0} = -\left.n\right|\_{z=0} S \qquad \text{and} \qquad -\frac{1}{\pi \beta^2} \frac{\partial n}{\partial z}\bigg|\_{z=d} = +n\Big|\_{z=d} S \tag{23}$$

Where the constant S is called surface recombination velocity. By solving the standard sec‐ ond order differential equation n(z) with the boundary conditions [Eq. (22)], and substitute n(z) into the expression of Iph [Eq. (21)], we arrive at DeVore's formula,

$$I\_{ph} = I\_{PC}^0 F \tag{24}$$

where the PC magnitude is given by,

ph

uration displayed in Fig. 1.

196 Optoelectronics - Advanced Materials and Devices

**3. DeVore's approch vs. BU model**

dicular to the direction of illumination.

**3.1. The DeVore's photocurrent spectral theory**

th calculation is based on the following equation,

This formulation is based on four assumptions:

**ii.** Assuming steady state, i.e., ∂n/∂t=0.

0

dary conditions:

tb

2

point will be presented in a forthcoming paper).

**iii.** The recombination rate 1/τ and diffusion length β-1 are a constant.

2 2 0

*z z*

¶ ¶ - =- - =+

 t

2 2

tb

*y x*

*I Ue n z dz*

*l*

*l* = D

0 ( )( ) ( ) *d*

where μ=μn+μp is the mobility and ΔU=Exlx and the voltage drop across the sample. It is worthwhile noting that Iph has the physical unit of Ampere (A). Equations (19-21), the ener‐ gy dependence of the absorption coefficient *α*(ℏ*ω*) , and the spatial lifetime distribution of carriers (τ) form the complete set needed for the fit of PC spectra measured with the config‐

In this section, we present the comparison of the two existing theoretical PC models, which were initially introduced in Refs. [5] and [9]. The PC spectra are based on the common ex‐ perimental setup shown in Fig.1, where the direction of the applied electric field is perpen‐

It has been the conventional way of engage DeVore's formula to fit PC spectra. However, as presented below, for the standard PC experiments displayed in Fig 1, DeVore's formula is actually not the correct one to use. We now briefly outline DeVore's early work [5], in which

0

<sup>1</sup> exp( ) 0 *dn n <sup>I</sup> <sup>z</sup> dz* ha


**i.** the external electric field is perpendicular to the incident light direction, which cor‐

**iv.** The recombination current at the surface is given by setting the following boun‐

*n n n S n S*

tb= =

¶ ¶ (23)

1 1 and *<sup>z</sup> z d z z d*

= =

 w  a

responds the experimental configuration in Fig. 1 (additional discussions of this

<sup>h</sup> (22)

ò (21)

m

$$I\_{p\_{\hbar}}^{0} = (\frac{l\_{\wp}}{l\_{x}}) (\Delta U e \mu)(\tau \frac{I\_{0}}{\hbar o}) \tag{25}$$

and the dimensionless spectrum factor F has the form,

$$F = \frac{1}{1 - \alpha^2/\beta^2} \left[ 1 - \exp(-\alpha d) - \frac{\alpha S \tau (1 + \exp(-\alpha d) + \alpha^2 \beta^2 (1 - \exp(-\alpha d))}{1 + S \tau \beta \coth(\beta d / 2)} \right] \tag{26}$$

where α is the energy dependent absorption coefficient. We argue that the boundary conditions [Eq. (22)] are not consistent with the experimental setup, i.e. the PC is driv‐ en by an applied electric field, which is perpendicular to the impinging light. There‐ fore, at the surface along the incident light there is an open circuit. The correct boundary condition is that there will be no particle current across the sample sur‐ face, i.e.,<sup>−</sup> <sup>1</sup> *τβ* <sup>2</sup> ∂*n* <sup>∂</sup> *<sup>z</sup>* <sup>|</sup> *<sup>z</sup>*=0 <sup>=</sup> <sup>−</sup> <sup>1</sup> *τβ* <sup>2</sup> ∂*n* ∂ *z* <sup>|</sup> *<sup>z</sup>*=*<sup>d</sup>* =0. Under this boundary condition, the DeVore's PC becomes,

$$I\_{ph} = I\_{PC}^0 \left[ 1 - \exp\left(-ad\right) \right] \tag{27}$$

i.e., the spectral factor F in Eq. (25) is reduced to the absorption,

$$F = 1\text{-exp}\,(-\alpha d)\tag{28}$$

Consequently, DeVore's formula is not able to explain the PC maximum in the vicinity of the band gap for the geometry shown in Fig. 1. The boundary conditions expressed by Eq. (22) are artificially acting as source of non-equilibrium carriers.

#### **3.2. PC theory based on the spatial non-uniform recombination rate (BU model)**

The ansatz of the original BU model in Ref. [9] led to an intuitive and straightforward theory that implemented DeVore's assumption directly. It simply assumed that the spatial distribu‐ tion recombination rates along the light propagation - i.e., at the surface and in the bulk re‐ gion - are different. At the surface, due to the increased density of recombination centers with respect to the bulk, the carrier lifetime is much shorter than in the bulk region. Specifi‐ cally, it was assumed that the spatial lifetime decay τ (z) takes the simple form:

$$\tau(z) = \tau\_b \left[ 1 - \left( 1 - \frac{\tau\_s}{\tau\_b} \right) \exp(-z \mid L) \right] \tag{29}$$

**4. Experimental PC Results and Fits**

diode in order to express the PC in terms of responsivity.

**Figure 1.** Experimental arrangement employed for the experiments. Note that d=lz.

**4.1. PC fit using Dutton's experimentally determined absorption coefficient**

sion was found by analyzing the photoluminescence of thin-film CdS [17].

The comparison of the experiment (symbols) with the fit (dotted line) using Eq. (31-32) is shown in Fig. 2(a). The best fit between the experimental data and theoretical formula was achieved with d=3.60×10-3 cm, L=8.65×10-5 cm, and τs/τb=0.068, while the used α(ωћ) values, which are shown in Fig. 2 (b), have been extracted from Dutton's paper [16]. The fit reveals that only an effective thickness of the sample, which does not necessarily correspond to the physical thickness, contribute to the formation of the PC signal. The corresponding conclu‐

The spectral PC dispersion was investigated by using the experimental arrangement in Fig. 1: The x, y plane of the CdS bulk material faces the incoming monochromatic light, while the z coordinate is parallel to the light propagation. The electric field of the light is perpendicu‐ larly oriented to the c-axis of the CdS sample, which was an industrially produced single crystal with d=1 mm. Two vacuum evaporated Al contacts with a gap of 0.5 mm between them were used for the electrical connection. The applied electric field driving the PC was 200 V/cm and the optical excitation was carried out with intensities typically in μW/cm2 range. The PC was recorded at room temperature with lock-in technique by chopping the impinging light at 25 Hz. The measured spectrum was corrected by a calibrated Si photo‐

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

http://dx.doi.org/10.5772/51412

199

Where τs is the lifetime of the nonequilibrium carriers at the surface, τb is the lifetime of the nonequilibrium carriers in the bulk, and L is the length scale beyond which the recombina‐ tion rate is predominately ruled by τb. Thus we have a multi-time scale relaxation model. We rearrange Eq. (19), resulting in,

$$n = \frac{1}{\beta^2} \frac{d^2 n}{dz^2} + \tau \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) \tag{30}$$

Here, we have used the mean field approximation for the term Dτ≈D<τ>=1/β2 and assumed that diffusion length β-1 is a constant. We now substitute Eq. (30) into the general PC expres‐ sion Eq. (21), we find,

$$I\_{ph} = (\frac{l\_y}{l\_x})(\Delta U e \mu) \Big|\_{0}^{d} \Big| \left( \frac{1}{\beta^2} \frac{d^2 n}{dz^2} + \tau(z) \eta \frac{I\_0}{\hbar \alpha} \alpha \exp(-\alpha z) \right) \mathrm{d}z = I\_{Ph}^0 F \tag{31}$$

The first integral over the diffusion term is zero because of the zero flux boundary condition [Eq. (20)]. Thus, the dimensionless spectrum factor F takes the following form:

$$F = \int\_{0}^{l} \frac{\tau(z)}{\tau\_{b}} \alpha \exp(-\alpha z) dz = 1 - \exp(-\alpha d) - (1 - \frac{\tau\_{s}}{\tau\_{b}}) \frac{\alpha L}{1 + \alpha L} [1 - \exp(-\alpha d - d / L)] \tag{32}$$

where, the PC magnitude is given by,

$$I\_{ph}^0 = (\frac{l\_\nu}{l\_x})(\Delta U e \mu)(\frac{I\_0}{\hbar \alpha} \tau\_b)\eta \tag{33}$$

Equations (31-33) basically correspond to the intuitively deduced PC dispersion Ref. [9]. We not that by expressing the PC in terms of the responsivity (R ) in [Ampere/Watt], i.e., for constant impinge light power, the proportionality R∝F holds.
