**2. Fundamental HgCdTe properties**

The mercury cadmium telluride (Hg1–*x*Cd*x*Te) (MCT) is a practically perfect IR detector material system. Its distinctive position depends on three key features:


**•** favorable inherent recombination mechanisms that lead to long carrier lifetime and high operating temperature.

Additionally, extremely small change of lattice constant versus composition makes it possible to grow high-quality layers and heterostructures.

#### **2.1. Energy bandgap**

information about their position in space, temperature, surface properties, as well as informa‐ tion about the chemical composition of the atmosphere through which the radiation is trans‐

All information carried by the infrared radiation can be read and processed by suitable sensors (detectors) that transform infrared energy into other forms, directly and easy to measure. The sensors used to detect infrared radiation are usually equipped with two types of detectors:

At present, the technology of the mid- (MWIR, 3–8 μm) and long-wave (LWIR, 8–14 μm) infrared radiation is mainly connected with photon detectors, designed on the basis of complex semiconductor materials, such as mercury cadmium telluride (HgCdTe) or indium gallium arsenide (InGaAs). The incident radiation is absorbed within the material by interaction with electrons, and the detector signal is caused by changes of the electric energy distribution. They exhibit both perfect signal-to-noise performance and a very fast response. But to achieve this, the present photon detectors require cryogenic cooling. Cryogenic cooling creates the cost and

Thus, higher operation temperature (HOT) condition is one of the most important research areas in infrared technology. The development of a new detector's architecture has been driven by applications requiring multispectral detection, high-frequency response, high detectivity, small size, low weight and power consumption (SWaP), and finally HOT condition. Significant improvements in the reduction of the dark current leading to HOT condition have been achieved by the suppression of Auger thermal generation [1]. In practice, most of the HgCdTe Auger suppressed photodiodes are based on complex graded gap and doping multilayer

demonstrated the suppression of Auger mechanisms by reducing the absorber carrier density below thermal equilibrium in reverse bias condition. A recent strategy to achieve HOT

This chapter exhibits the fundamental properties of HgCdTe semiconductors and relates those material parameters that have successful applications as an IR barrier detector alloy. It presents different barrier HgCdTe structures in terms of dark current. The intent of this chapter is to concentrate on a barrier device approach having the greatest impact on IR industry develop‐

The mercury cadmium telluride (Hg1–*x*Cd*x*Te) (MCT) is a practically perfect IR detector

**•** cadmium (Cd) composition-dependent energy bandgap (sensing wavelength can be tuned

πN<sup>+</sup>

/ν) and extraction (N<sup>+</sup>

 or N<sup>+</sup> νP<sup>+</sup>

/π or P+

device structures

/ν) junctions have

mitted.

thermal detectors and photon detectors.

72 Modeling and Simulation in Engineering Sciences

with a combination of exclusion (P+

**2. Fundamental HgCdTe properties**

ment today.

inconvenient limitations, especially in civil applications.

structures, complicated to grow in terms of technology. The P+

detectors includes simple nBn (B: barrier layer) barrier structures [2].

material system. Its distinctive position depends on three key features:

by varying its alloy composition *x* over the 1–30 μm range),

**•** large optical coefficients that enable high quantum efficiency, and

/π or N<sup>+</sup>

The electrical and optical properties of Hg1–*x*Cd*x*Te are determined by energy gap *E*g. Energy gap of this compound ranges from −0.30 eV for semimetallic HgTe goes through 0 eV for approximately Cd composition *x* = 0.15 and finally up to 1.608 eV for CdTe. A number of expressions approximating *E*g dependence on composition and temperature are available at present. The most widely used formula is given by Hansen et al. [3]:

$$E\_g(\mathbf{x}, T) = -0.302 + 1.93\mathbf{x} - 0.81\mathbf{x}^2 + 0.832\mathbf{x}^3 + 5.35 \times 10^{-4}T(\mathbf{l} - 2\mathbf{x})\tag{1}$$

where *E*g is the energy gap in eV, *T* is the temperature in K, and *x* is the Cd molar composition.

**Figure 1** shows the empirical fit of the Hg1–*x*Cd*x*Te bandgap according to Hansen et al. [3] versus the Cd molar composition, *x* at temperature 77 and 300 K. The cutoff wavelength *λ*c, defined as that wavelength at which the response drops to 50% of its peak value, is also plotted.

**Figure 1.** The bandgap structure of Hg1–*x*Cd*x*Te according to Hansen et al. [3] as a function of Cd molar composition, *x* at temperature of 77 K (solid line) and 300 K (dashed line).

#### **2.2. Electron affinity**

For a semiconductor-vacuum interface, electron affinity is defined as the energy obtained by moving an electron from the vacuum just outside the semiconductor to the bottom of the conduction band just inside the semiconductor. For HgCdTe, electron affinity *X* can be calculated from [4]:

$$X = 4.23 - 0.813(E\_g - 0.083) \tag{2}$$

#### **2.3. Intrinsic concentration**

The most widely used expression for intrinsic carrier concentration *n*<sup>i</sup> is that of Hansen and Schmit [5]:

$$n\_i = (5.585 - 3.82x + 1.753 \times 10^{-3}T + 1.364 \times 10^{-3}) \times 10^{14} E\_{\text{g}}^{3/4} T^{3/2} \exp\left(-\frac{E\_{\text{g}}}{2k\_B T}\right) \tag{3}$$

where *kB* is the Boltzmann constant in eV/K.

The effective mass of electron, *m*e, in the narrow gap HgCdTe can be expressed by Weiler's formula that can be approximated by *m*e/*m*0 ≈ 0.071*E*g, while the effective mass of heavy hole, *mh*, is often assumed in modeling of IR detectors according to *mh* = 0.55*m*0.

#### **2.4. Mobility**

Due to small effective masses the electron mobility in HgCdTe is high. Mobility Cd molar composition dependence results primarily from the *x*-dependence of the bandgap and the temperature dependence of the scattering mechanisms. The electron mobility in Hg1−*<sup>x</sup>*Cd*x*Te in composition range 0.2 < *x* < 0.6 and temperature *T* > 50 K can be approximated as [6, 7]

$$
\mu\_e = \frac{9 \times 10^8 \,\mathrm{s}}{T^{2r}} \tag{4}
$$

where *μe* is the electron mobility in m2 /V s, *s* = (0.2/*x*) 7.5, and *r* = (0.2/*x*) 0.6.

For modeling IR HgCdTe photodetectors, the hole mobility is usually calculated assuming that the electron-to-hole mobility ratio, *μe*/*μh* is constant and equal to 100.

#### **2.5. Absorption coefficient**

High-quality HgCdTe samples exhibit absorption coefficient *α* in the short-wavelength region to be in proper agreement with the Kane model. The problems appear to be complicated in the LWIR region by the appearance of an absorption tail extending at energies lower than the energy gap attributed to the composition inhomogeneity. In simulations, a modified Urbach's rule is implemented [8]:

HgCdTe Mid- and Long-Wave Barrier Infrared Detectors for Higher Operating Temperature Condition http://dx.doi.org/10.5772/63943 75

$$\alpha = \alpha\_0 \exp\left[\frac{\sigma (E - E\_0)}{T + T\_0}\right] \tag{5}$$

where *α* is the absorption coefficient in cm−1, *α*0 = exp(53.61*x* − 18.88), *E* is energy in eV, *T* is the temperature in K, *T*0 = 81.9 in K, *E*0 = −0.3424 + 1.838*x* + 0.148*x*<sup>2</sup> , and *σ* = 3.267 × 104 (1 + *x*).

An empirical formula was thereby employed in the Kane region [9]:

$$
\alpha = \alpha\_s \exp[\beta(E - E\_s)]^{1/2} \tag{6}
$$

where the *β* parameter after Chu et al. [9] is *β* = −1 + 0.083*T* + (21 − 0.13*T*)*x*.

#### **2.6. Dielectric constant**

conduction band just inside the semiconductor. For HgCdTe, electron affinity *X* can be

The most widely used expression for intrinsic carrier concentration *n*<sup>i</sup> is that of Hansen and

<sup>3</sup> 3 14 3/ 4 3/ 2 (5.585 3.82 1.753 10 1.364 10 ) 10 exp <sup>2</sup>

= - + ´ + ´´ -ç ÷

The effective mass of electron, *m*e, in the narrow gap HgCdTe can be expressed by Weiler's formula that can be approximated by *m*e/*m*0 ≈ 0.071*E*g, while the effective mass of heavy hole,

Due to small effective masses the electron mobility in HgCdTe is high. Mobility Cd molar composition dependence results primarily from the *x*-dependence of the bandgap and the temperature dependence of the scattering mechanisms. The electron mobility in Hg1−*<sup>x</sup>*Cd*x*Te in composition range 0.2 < *x* < 0.6 and temperature *T* > 50 K can be approximated as [6, 7]

> 8 2 9 10 *e r*

*T*

/V s, *s* = (0.2/*x*)

For modeling IR HgCdTe photodetectors, the hole mobility is usually calculated assuming that

High-quality HgCdTe samples exhibit absorption coefficient *α* in the short-wavelength region to be in proper agreement with the Kane model. The problems appear to be complicated in the LWIR region by the appearance of an absorption tail extending at energies lower than the energy gap attributed to the composition inhomogeneity. In simulations, a modified Urbach's

m

the electron-to-hole mobility ratio, *μe*/*μh* is constant and equal to 100.

*s*

´ <sup>=</sup> (4)

0.6.

7.5, and *r* = (0.2/*x*)

*<sup>E</sup> <sup>n</sup> x T E T*

*i g*

*mh*, is often assumed in modeling of IR detectors according to *mh* = 0.55*m*0.

4.23 0.813( 0.083) *X E* =- -*<sup>g</sup>* (2)


*g*

*B*

*k T*

è ø (3)

calculated from [4]:

Schmit [5]:

**2.4. Mobility**

**2.3. Intrinsic concentration**

74 Modeling and Simulation in Engineering Sciences

where *kB* is the Boltzmann constant in eV/K.

where *μe* is the electron mobility in m2

**2.5. Absorption coefficient**

rule is implemented [8]:

The dielectric constants *ε* are not a linear function of *x* and temperature dependence was not observed within the experimental resolution. These dependences can be described by the following relations [10]:

$$
\varepsilon\_{\circ} = 1\,\mathrm{5.2} - \mathrm{5.6x} + \mathrm{8.2x^2} \tag{7}
$$

$$
\sigma\_S = 20.5 - 15.6x + 5.7x^2 \tag{8}
$$

where *ε*∞ is the high-frequency dielectric constant and *ε*s is the static dielectric constant.

#### **3. Numerical procedure and generation-recombination mechanisms**

Theoretical modeling of the HgCdTe barrier detectors has been performed using our original numerical program developed at the Institute of Applied Physics, Military University of Technology (MUT), and the commercially available APSYS platform (Crosslight Inc.). Both programs are based on numerical solution of the Poisson's and the electron/hole current continuity Eqs. (9)–(11) [11, 12]. In addition, both programs include the energy balance Eq. (12):

$$\frac{\partial \mathbf{n}}{\partial t} = \frac{1}{q} \nabla \vec{j}\_{\mu} + G - R \tag{9}$$

$$\frac{\partial \hat{p}}{\partial t} = -\frac{1}{q} \nabla \vec{j}\_{\rho} + G - R \tag{10}$$

$$
\nabla^2 \psi = -\frac{q}{\varepsilon \varepsilon\_0} \rho - \frac{1}{\varepsilon} \nabla \psi \nabla \varepsilon \tag{11}
$$

$$C\_V \frac{\partial T}{\partial t} - H = -\nabla(\boldsymbol{\chi} \cdot \nabla T) \tag{12}$$

where *q* is the elementary charge, *j* is the current density, *G* is the generation rate, and *R* is the recombination rate. Indices *n* and *p* denote electron and hole, respectively. In Poisson's Eq. (11) *Ψ* is the electrostatic potential and *ρ* is the electrical charge. In the last term, *CV* is the specific heat, *χ* is the thermal conductivity coefficient, and *H* is the heat generation term. A Joule heat is introduced as the heat generation in order to include the thermoelectric effect and heat balance.

Current density is usually expressed as functions of quasi-Fermi levels:

$$
\vec{j}\_n = q\,\mu\_\mu\eta \nabla \Phi\_n\tag{13}
$$

$$
\vec{j}\_p = q\mu\_h p \nabla \Phi\_p \tag{14}
$$

where *Φn* and *Φp* denote the quasi-Fermi levels.

The difference *G* – *R* (Eqs. (9) and (10)) is the net generation of electron-hole pairs and depends on all generation recombination (GR) effects including influence of thermal mechanisms as well as tunneling mechanisms.

Depending on this approach, either two or three important carrier thermal GR processes, Shockley-Read-Hall (SRH), Auger, and radiative mechanisms were included in simulations. A radiative GR process could be ignored because photon recycling restricts the influence of that process on the performance of HgCdTe photodiodes [13, 14]. Tunneling mechanisms were considered due to band-to-band tunneling (BTB) and trap-assisted tunneling (TAT).

Thermal generation could be given as a sum of radiative, Auger 1, Auger 7, and SRH mecha‐ nisms. BTB and TAT effects can also be included as a GR process:

$$\left(\left(G-R\right) = \left(G-R\right)\_{\text{R}\text{D}} + \left(G-R\right)\_{\text{A}\text{U}\text{G}} + \left(G-R\right)\_{\text{SRH}} + \left(G-R\right)\_{\text{BTB}} + \left(G-R\right)\_{\text{TAT}}\tag{15}$$

The set of transport Eqs. (9)–(11) is commonly known; however, their solution consists of serious mathematical and numerical problems. The equations are nonlinear, and are complex functions of electrical potential, quasi-Fermi levels, and temperature. The details concerning the solutions of Poisson's equation are presented in Appendix A.

#### **3.1. Radiative process/photon recycling**

2

76 Modeling and Simulation in Engineering Sciences

balance.

y

*t*

Current density is usually expressed as functions of quasi-Fermi levels:

where *Φn* and *Φp* denote the quasi-Fermi levels.

well as tunneling mechanisms.

0 *q* 1

ee

 r

( ) *<sup>V</sup> <sup>T</sup> CH T*

¶ - = -Ñ × Ñ

*<sup>n</sup> e n j qn* = Ñ m F

*<sup>p</sup> h p j qp* = Ñ m F

The difference *G* – *R* (Eqs. (9) and (10)) is the net generation of electron-hole pairs and depends on all generation recombination (GR) effects including influence of thermal mechanisms as

Depending on this approach, either two or three important carrier thermal GR processes, Shockley-Read-Hall (SRH), Auger, and radiative mechanisms were included in simulations. A radiative GR process could be ignored because photon recycling restricts the influence of that process on the performance of HgCdTe photodiodes [13, 14]. Tunneling mechanisms were

Thermal generation could be given as a sum of radiative, Auger 1, Auger 7, and SRH mecha‐

The set of transport Eqs. (9)–(11) is commonly known; however, their solution consists of serious mathematical and numerical problems. The equations are nonlinear, and are complex functions of electrical potential, quasi-Fermi levels, and temperature. The details concerning

( )( ) ( ) ( ) ( ) ( ) *GR GR GR GR GR GR* -=- +- +- +- +- *RAD AUG SRH BTB TAT* (15)

considered due to band-to-band tunneling (BTB) and trap-assisted tunneling (TAT).

nisms. BTB and TAT effects can also be included as a GR process:

the solutions of Poisson's equation are presented in Appendix A.

 ye

Ñ =- - Ñ Ñ (11)

¶ (12)

<sup>r</sup> (13)

<sup>r</sup> (14)

 e

c

where *q* is the elementary charge, *j* is the current density, *G* is the generation rate, and *R* is the recombination rate. Indices *n* and *p* denote electron and hole, respectively. In Poisson's Eq. (11) *Ψ* is the electrostatic potential and *ρ* is the electrical charge. In the last term, *CV* is the specific heat, *χ* is the thermal conductivity coefficient, and *H* is the heat generation term. A Joule heat is introduced as the heat generation in order to include the thermoelectric effect and heat

For a long time, internal radiative GR processes have been considered to be the main funda‐ mental limit to detector performance and the performance of practical devices have been compared to that limit. The following relation can be used to estimate radiative (*G* – *R*)*RAD* contribution [15]:

$$(G - R)\_{R \land D} = B(np - n\_i^2) \tag{16}$$

$$B = 5.9052 \times 10^{18} n\_i^{-2} \varepsilon T^{3/2} \sqrt{\frac{1+\mathbf{x}}{(81.9+T)}} \exp\left(-\frac{E\_g}{k\_B T}\right) (E\_g^2 + 3k\_B T E\_g + 3.75k\_B^2 T^2) \tag{17}$$

Due to photon recycling effect, the radiative lifetime is highly extended. According to Hum‐ preys [13], most of the photons emitted in photodetectors as a result of radiative decay are immediately reabsorbed and the observed radiative lifetime is only a measure of how well photons can escape from the volume of the detector. In many cases, especially in the case of semiconductors with high reflective index, radiative mechanism can be omitted in numerical modeling.

#### **3.2. Auger processes**

There are several types of Auger processes, and among them Auger 1 and Auger 7 are the most dominant due to the smallest threshold energies. The Auger 1 generation is the impact ionization by an electron, generating an electron-hole pair and is dominant in n-type material while Auger 7 is the impact generation of electron-hole pair by a light hole and dominates in p-type material.

The Auger generation and recombination rates strongly depend on temperature via the dependence of carrier concentration and intrinsic time on temperature. Therefore, cooling is a natural and a very effective way to suppress Auger processes according to the following relation [16]:

$$((G - A)\_{\
u \cup \bar{\imath}} = (C\_{\bar{\imath}}n + C\_{\bar{\jmath}}p)(np - n\_i^2) \tag{18}$$

$$\begin{split} C\_{s} &= 5 \times 10^{-22} \, | \, F\_{1}F\_{2} \, | \left[ \left( \frac{E\_{\rm g}}{k\_{\rm g}T} \right)^{\circ} \exp \left( 1 + 2 \frac{m\_{e}^{\star}}{m\_{h}^{\star}} \right) \left( \frac{E\_{\rm g}}{k\_{\rm g}T \left( m\_{e}^{\star} / m\_{h}^{\star} \right)} \right) \right]^{-1/2} \\ &\times n\_{i}^{-2} \, [3.8 \times 10^{-18} \, \varepsilon^{2}] \frac{1}{m\_{e}^{\star}} \Big( 1 + 2 \frac{m\_{e}^{\star}}{m\_{h}^{\star}} \right) \sqrt{1 + \frac{m\_{e}^{\star}}{m\_{h}^{\star}}} \, \mathrm{I}^{-1} \end{split} \tag{19}$$

$$\mathbf{C}\_{\rho} = \mathbf{y} \cdot \mathbf{C}\_{\mathbf{u}} \tag{20}$$

where | *F*1*F*<sup>2</sup> | is the overlap integrals for Bloch functions. The ratio *γ* has been calculated as a function of composition and temperature and is assumed to be ranging from 3 to 60 de‐ pending on temperature [17, 18].

#### **3.3. Shockley-Read-Hall processes**

The Shockley-Read-Hall (SRH) mechanism is not an intrinsic process because it occurs via levels in the forbidden energy gap. Metal site vacancies (mercury vacancies) are considered as an SRH centers in HgCdTe. The reported position of SRH centers for both n- and p-type material is assumed as 1/3 *Eg* or 3/4 *Eg* from the conduction band. The (*G* – *R*)*SRH* rate could be calculated according to the following formula [19–21]:

$$\tau\_i(G-R)\_{SRH} = \frac{np - n\_i^2}{\tau\_{p0}(n\_0 + n\_1) + \tau\_{n0}(p\_0 + p\_1)}\tag{21}$$

where

$$n\_{\!} = n\_{\!} \exp\left(\frac{E\_{\!} - E\_{\! \! \! \! \! \! \! /}}{k\_B T}\right) \tag{22}$$

$$p\_1 = n\_i \exp\left(\frac{E\_{fl} - E\_r}{k\_B T}\right) \tag{23}$$

*n*1 and *p*<sup>1</sup> mean concentrations in the case in which the trap level *ET* coincides with the Fermi level *EFi*. The terms *τn*0 = (*cnNT*) −1 and *τp*0 = (*cpNT*) −1 are the shortest possible time constants for the electron and hole capture coefficients (*cn* and *cp*), respectively. *NT* denotes the mercury vacancy concentration.

#### **3.4. Tunneling processes**

BTB tunneling is calculated as a function of the applied electric field *F* [22–24]:

$$(G - R)\_{\text{BTB}} = \frac{qFm^\ast}{2\pi^2\hbar^3} P\_0 \overline{E} \tag{24}$$

where ℏ is the reduced Planck constant and *m*\* is the electron effective mass related to the tunneling mechanism defined as

HgCdTe Mid- and Long-Wave Barrier Infrared Detectors for Higher Operating Temperature Condition http://dx.doi.org/10.5772/63943 79

$$m^\* = \frac{m\_\varepsilon m\_\nu}{m\_\varepsilon + m\_\nu} \tag{25}$$

where *mc* and *mv* are the effective masses in conduction and valence bands, respectively.

*C C p n* = × g

pending on temperature [17, 18].

78 Modeling and Simulation in Engineering Sciences

**3.3. Shockley-Read-Hall processes**

level *EFi*. The terms *τn*0 = (*cnNT*)

where ℏ is the reduced Planck constant and *m*\*

tunneling mechanism defined as

vacancy concentration.

**3.4. Tunneling processes**

where

calculated according to the following formula [19–21]:

*SRH*

where | *F*1*F*<sup>2</sup> | is the overlap integrals for Bloch functions. The ratio *γ* has been calculated as a function of composition and temperature and is assumed to be ranging from 3 to 60 de‐

The Shockley-Read-Hall (SRH) mechanism is not an intrinsic process because it occurs via levels in the forbidden energy gap. Metal site vacancies (mercury vacancies) are considered as an SRH centers in HgCdTe. The reported position of SRH centers for both n- and p-type material is assumed as 1/3 *Eg* or 3/4 *Eg* from the conduction band. The (*G* – *R*)*SRH* rate could be

2

*i*


è ø (22)

è ø (23)


−1 are the shortest possible time constants for

is the electron effective mass related to the

00 1 0 0 1

 t*nn p p*

( ) ( )( )

<sup>1</sup> exp *T Fi i*

<sup>1</sup> exp *Fi T i*

*p n k T* æ ö - <sup>=</sup> ç ÷

−1 and *τp*0 = (*cpNT*)

BTB tunneling is calculated as a function of the applied electric field *F* [22–24]:

2 3 <sup>0</sup> ( ) <sup>2</sup> *BTB qFm G R P E* p

*E E n n k T* æ ö - <sup>=</sup> ç ÷

*B*

*B E E*

*n*1 and *p*<sup>1</sup> mean concentrations in the case in which the trap level *ET* coincides with the Fermi

the electron and hole capture coefficients (*cn* and *cp*), respectively. *NT* denotes the mercury

\*

*np n G R* t

*p n*

(20)

Tunneling probability *P*0 with a zero perpendicular (to the *x*-direction) momentum can be estimated using the following relations:

$$P\_0 = \exp\left(\frac{\pi (m^\*)^{1/2} E\_g^{3/2}}{2\sqrt{2}qF\hbar}\right) \tag{26}$$

$$\overline{E} = \frac{\mathcal{Q}2qF\hbar}{2\pi (m^\*)^{1/2}E\_g^{1/2}}\tag{27}$$

where *E*¯ is a measure of significance of perpendicular momentum range. In other words, tunneling probability decreases with increasing value of *E*¯. If is *E*¯ small, the only electrons with perpendicular momentum near zero can tunnel through the energy barrier. Typically *E*¯ is in a range of 5–100 meV.

TAT contribution was estimated according to formula similar to the SRH process described by Eq. (21) [25]. However, the trapping rate strongly depends on the ionization energy *ET*; the electric field can essentially influence *τn*0 and *τp*0 values that are inversely proportional to trapping rates. This means that the electric field enhances a GR process around the mercury vacancy by decreasing the energy required both for the emission of an electron with energy level *ET* to the conduction band as well as for the emission of a hole with the empty level *ET* to the valence band. Taking into account this mechanism, the *cn* and *cp* parameters are modified. For this purpose, Eq. (21) is modified with *δn* and *δp* parameters that determine the relative changes in the size of the emission factors of electrons and holes caused by the effects associated with the electric field. The TAT effect expressed by GR process is

$$(G - R)\_{TAT} = \frac{np - n\_i^2}{\frac{\pi\_{p0}}{\delta\_p}(n\_0 + n\_1) + \frac{\pi\_{n0}}{\delta\_n}(p\_0 + p\_1)}\tag{28}$$
