**Mathematical Modeling of Multi-Element Infrared Focal Plane Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit'**

I.I. Lee and V.G. Polovinkin

*A.V. Rzhanov Institute of Semiconductor Physics Siberian Branch of Russian Academy of Sciences Russia* 

### **1. Introduction**

250 Photodetectors

Zakhleniuk, N.A. (2007). Theory and Modelling of High-Field Carrier Transport in High-

24-28, 2007

Speed Photodetectors. *Proceedings of NUSOD'07 Numerical Simulation of Optoelectronic Devices*, pp. 77-78, ISBN 978-1-4244-1431-4, Newark, DE, September

> The direct-injection readout circuit was proposed in 1973 by A. Steckl and T. Koehler (Steckl & Koehler, 1973). With such readout circuits, first hybrid multi-element InfraRed Focal Plane Arrays (IR FPAs) were implemented (Steckl, 1976; Iwasa, 1977). The direct-injection readout circuits are being used in the majority of hybrid multi-element Long-Wave InfraRed (LWIR) FPAs, which hold more-than-70% a share in the world market of thermography systems (Rogalski, 2000).

> A diagram of FET-based readout circuit is shown in Fig.1a, the most popular design involving a charge-coupled device, Fig.1b (Longo, 1978; Felix,1980; Takigawa, 1980; Rogalski, 2000*).* On application of a dc voltage UG to the input gate, a certain voltage, defined by the surface potential under the input gate of the photodetector channel, sets across the photodiode, and the current generated in the IR detector is integrated at the storage capacitor Сint. On applying of a transfer pulse Ft (Fig.1a), the charge accumulated in the storage capacitor transfer to the column read bus.

The equivalent circuit of the direct injection readout circuit is shown in Fig. 2.

Normally, the current through the photodiode is in the range of 10-7÷10-10A, the input FET being therefore operated in subthreshold mode. The transconductance of the input FET channel is:

$$\mathbf{q}\_{\rm in} = \frac{\partial \mathbf{I}\_{\rm in}}{\partial \mathbf{V}\_{\rm G}} = \frac{\mathbf{q} \mathbf{I}\_{\rm in}}{\mathbf{N}^\* \mathbf{k} \mathbf{T}} \tag{1}$$

Here, q is the electron charge, k is the Boltzmann constant, N\*= (CОХ+СD\*+СSS\*)/CОХ (CОХ is the specific capacitance of gate dielectric, СSS\*=qNss\* is the specific capacitance of fast surface states, CD\* is the specific capacitance of the depletion region, Т is temperature, and the asterisk \* indicates that the parameter value is taken for the conditions with surface Fermi level equal to 3/2 φFB.

Mathematical Modeling of Multi-Element Infrared Focal Plane

comparison with readout circuits of other types.

LWIR FPAs;

input FET gate.

mainly used.

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 253



J. Longo *et al.* (Longo, 1978) proposed a design of direct injection readout circuit capable of diminishing the spread of photodiode bias voltages to the typical difference of threshold voltages of two closely spaced FETs. Nonetheless, being capable of improving certain characteristics of direct injection readouts, the additional components (the amplifier normally involves 4 to 6 FETs) take off a substantial fraction of surface area from the photodetector cell, doing simultaneously the direct injection readout circuits out of their main advantageous features, low energy consumption and a large storage capacity, high in

The advances in silicon technology and in the technology of IR photodiodes based on narrow bandgap semiconductors have allowed a substantial increase of photodiode dynamic resistance and a reduced variation of photodiode bias voltages across FPA from the values ~ 20-50 mV, typical of the technology level of the 1970s, to ~ 1-10 mV. For this reason, as a rule, the simplest design of direct injection readout circuit shown in Fig.1 is

An analysis of the system 'photodiode – direct injection readout circuit' given in (Steckl & Koehler 1973; Steckl, 1976; Iwasa, 1977; Rogalski, 2000; Longo, 1978; Felix,1980; Takigawa, 1980; Gopal, 1996) was carried out versus photodiode biasing, although the bias voltage across the photodiode is defined by the surface potential in the input-FET channel and can be adjusted through the proper choice of the potential UG applied to the input-FET gate. That is why such estimates do not allow the prediction of responsitivity of multi-element IR FPAs and evaluation of fixed-pattern noise level and Noise-Equivalent Temperature Difference (NETD) figures of thermography systems based on such IR FPAs. In other words,

In the present article, we propose a mathematical model of the system 'photodiode – direct injection readout circuit', a computer program, and an analysis procedure for thermography systems based on multi-element IR FPAs; these model and analysis procedure were developed to solve the above-indicated problems. We describe a procedure for revealing the effects due to the spread of electrophysical parameters of photodiodes and Si readouts based on plotting calculated fixed-pattern noise, detectivity, and NETD histograms of thermography systems based on multi-element IR FPAs. It is significant that all

such estimates do not permit numerical modeling of multi-element IR FPAs.

Fig. 1. Readout circuit with charge direct injection.

Fig. 2. The equivalent circuit of the photodetector channel of a hybrid IR FPA based on the system 'photodiode – direct-injection readout circuits'. Here, IPD is the current that flows through the photodiode, InPD is the noise current that flows through the photodiode, RPD is the dynamic resistance of the photodiode, Iin is the current integrated under the storage gate of the readout circuit, Inin is the noise current integrated under the storage gate, and Сin is the capacitance of the input direct injection.

From the equivalent circuit, all basic relations for the readout circuit of interest can be inferred. In particular, for injection efficiency ηI we obtain (Longo, 1978; Felix,1980):

$$\mathfrak{m}\_{\rm I} = \frac{\mathbf{g}\_{\rm in}}{\mathbf{g}\_{\rm PD} + \mathbf{g}\_{\rm in} + \mathrm{j} \mathrm{o} \mathbf{C}\_{\rm in}} \tag{2}$$

where gPD=1/RPD, RPD is the dynamic resistance of the photodiode.

Analysis of transfer characteristics of the system 'photodiode – direct-injection readout circuit' has revealed several drawbacks inherent to such readout circuits (Longo, 1978; Felix,1980; Takigawa, 1980; Rogalski, 2000*)*:

**U Ft**

**Column bus**

**Frst**

**Ud Ft**

**I nin**

**Iin**

**in**

(2)

**1/g Cin**

Fig. 2. The equivalent circuit of the photodetector channel of a hybrid IR FPA based on the system 'photodiode – direct-injection readout circuits'. Here, IPD is the current that flows through the photodiode, InPD is the noise current that flows through the photodiode, RPD is the dynamic resistance of the photodiode, Iin is the current integrated under the storage gate of the readout circuit, Inin is the noise current integrated under the storage gate, and Сin is

From the equivalent circuit, all basic relations for the readout circuit of interest can be

Analysis of transfer characteristics of the system 'photodiode – direct-injection readout circuit' has revealed several drawbacks inherent to such readout circuits (Longo, 1978;

in PD in in g <sup>g</sup> +g j <sup>C</sup>

inferred. In particular, for injection efficiency ηI we obtain (Longo, 1978; Felix,1980):

**U Ud**

**UG**

**Сint**

**G**

**a**

**PD**

**b**

**RPD**

the capacitance of the input direct injection.

Felix,1980; Takigawa, 1980; Rogalski, 2000*)*:

**UPD**

Fig. 1. Readout circuit with charge direct injection.

**PD nPD**

where gPD=1/RPD, RPD is the dynamic resistance of the photodiode.

**I I**


Several circuit designs were proposed to eliminate the above drawbacks. For instance, R. Bluzer and R. Stehlik (Bluzer & Stehlik,1978) proposed to use, with the aim of reducing the input impedance of readouts, the so-called buffered direct injection, implemented through introduction of an inverting amplifier in between the input direct injection and input FET gate.

J. Longo *et al.* (Longo, 1978) proposed a design of direct injection readout circuit capable of diminishing the spread of photodiode bias voltages to the typical difference of threshold voltages of two closely spaced FETs. Nonetheless, being capable of improving certain characteristics of direct injection readouts, the additional components (the amplifier normally involves 4 to 6 FETs) take off a substantial fraction of surface area from the photodetector cell, doing simultaneously the direct injection readout circuits out of their main advantageous features, low energy consumption and a large storage capacity, high in comparison with readout circuits of other types.

The advances in silicon technology and in the technology of IR photodiodes based on narrow bandgap semiconductors have allowed a substantial increase of photodiode dynamic resistance and a reduced variation of photodiode bias voltages across FPA from the values ~ 20-50 mV, typical of the technology level of the 1970s, to ~ 1-10 mV. For this reason, as a rule, the simplest design of direct injection readout circuit shown in Fig.1 is mainly used.

An analysis of the system 'photodiode – direct injection readout circuit' given in (Steckl & Koehler 1973; Steckl, 1976; Iwasa, 1977; Rogalski, 2000; Longo, 1978; Felix,1980; Takigawa, 1980; Gopal, 1996) was carried out versus photodiode biasing, although the bias voltage across the photodiode is defined by the surface potential in the input-FET channel and can be adjusted through the proper choice of the potential UG applied to the input-FET gate. That is why such estimates do not allow the prediction of responsitivity of multi-element IR FPAs and evaluation of fixed-pattern noise level and Noise-Equivalent Temperature Difference (NETD) figures of thermography systems based on such IR FPAs. In other words, such estimates do not permit numerical modeling of multi-element IR FPAs.

In the present article, we propose a mathematical model of the system 'photodiode – direct injection readout circuit', a computer program, and an analysis procedure for thermography systems based on multi-element IR FPAs; these model and analysis procedure were developed to solve the above-indicated problems. We describe a procedure for revealing the effects due to the spread of electrophysical parameters of photodiodes and Si readouts based on plotting calculated fixed-pattern noise, detectivity, and NETD histograms of thermography systems based on multi-element IR FPAs. It is significant that all

Mathematical Modeling of Multi-Element Infrared Focal Plane

expressed as the amount of noise electrons:

for instance, the measuring channel induced noise.

can be described using the simple IR photodiode model:

**3. Examples of model calculations** 

FPA, which all are listed in Table. 1.

**circuits** 

<sup>2</sup>

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 255

2 2

2 2 2 2 2 2 OX D 0 in PD in PD 0 2KI 1 CR 1 T <sup>Q</sup> sin d qWL(C C ) 1 gR CR ( ) 2 

3 2 2 PD in in

The forth term in (4) stands to allow for the 1/f noise of the photodiode:

2 2 2 in PD 4 3 PD K P 2

g R Q2I I

 

(6)

2 in

(7)

(8)

2 in in PD 2 in

We substitute the third term of (4) into (3) and obtain, after integration, the following analytical expression for Q3 - the noise component due to photodiode current noise

2 in PD 2 PD in in PD

 

2

<sup>2</sup> 1gR CR

The detectivity D\* of the photodetector channel in which photosignals from the IR

c QQQQQ 

where АPD is the photodiode area and Qother stands for all other noise components such as,

The modeling procedure for the system 'IR photodiode – direct injection readout circuits'

Here, ηк is photodiode quantum efficiency, I0 is the photodiode saturation current, RS is the

The calculation starts with specifying the values of electrophysical and design parameters of readout circuits, photoelectrical parameters of photodiodes, and operating conditions of IR

At a given input gate voltage VG, the voltage across the photodiode can be determined from the condition Iin = IPD. We identify the points of intersection of the current-voltage

\* PD in I K

**3.1 Modeling procedure for multi-element IR FPAs with direct injection readout** 

PD K P 0 PD

photodiode shunt resistance, and IP is the current induced by background radiation.

1/2 <sup>22222</sup> 1 2 3 4 other

1 1T sin d

1/2

PD

S <sup>V</sup> I I I 1 exp V <sup>R</sup> (10)

(9)

g R 2kT R C 1 gR <sup>Q</sup> qI <sup>T</sup> 1 exp T q (1 g R ) R 1 gR <sup>R</sup> <sup>С</sup>

q

2 2 2 2 <sup>0</sup> <sup>0</sup> in PD in PD

photodiode are read into the direct injection input is given by (Rogalski, 2000):

A T /2 <sup>D</sup>

in PD PD in PD PD in

dependences were obtained as dependent on the input-gate voltage of readouts; this enables a comparison with the experiment and formulation of requirements to electrophysical and design parameters of photodiodes and Si readouts necessary for implementation of design characteristics.

#### **2. A mathematical model for the system 'IR photodiode - direct-injection readout circuit'**

Basic assumptions adopted in the proposed mathematical model of the system 'IR photodiode – direct injection readout circuit' were first formulated in (Kunakbaeva, Lee & Cherepov, 1993) and further developed in (Kunakbaeva & Lee, 1996; Karnaushenko; Lee et al., 2010; Gumenjuk-Sichevska, Karnaushenko, Lee & Polovinkin, 2011). In calculating the transfer characteristics of the system, for the input FET we employ the model of longchannel transistor in weak inversion (Overstraeten, 1975). The noise charge Q(t) integrated under the storage gate of the direct injection readout circuit is calculated as a McDonalds function expressed in terms of spectral current density Si(ω) (Buckingham, 1983):

$$\overline{\mathbf{Q}^2}(\mathbf{t}) = \frac{1}{\pi} \overline{\int\_0^\pi \frac{\mathbf{S}\_i(\alpha)}{\alpha^2} (1 - \cos \alpha t) d\alpha} \tag{3}$$

The spectral density of the noise current in the input-gate modulated channel is given by

$$\mathbf{S}\_{i}(\alpha) = 4\mathbf{k}\,\mathrm{T}\mathbf{g}\_{\mathrm{in}}\alpha\_{1}\left|1-\eta\_{\mathrm{l}}\right|^{2} + \frac{2\pi\mathbf{B}\_{\mathrm{in}}\mathbf{I}^{2}\_{\mathrm{in}}}{\alpha}\left|1-\eta\_{\mathrm{l}}\right|^{2} + \left(2\mathbf{q}\mathbf{I}\_{\mathrm{PD}} + \frac{4\mathbf{k}\,\mathrm{T}\alpha\_{2}}{\mathbf{R}\_{\mathrm{PD}}}\left|\eta\_{\mathrm{l}}\right|^{2} + \frac{2\pi\mathbf{B}\_{\mathrm{PD}}}{\alpha}\left|\eta\_{\mathrm{l}}\right|^{2}\tag{4}$$

The first and second terms in (4) stand for the thermal noise of the input FET and for the 1/f noise induced by this FET (here, α1 and α2 are coefficients) and the third and forth terms stand for the thermal noise and 1/f noise of the photodiode, here, Bin= <sup>2</sup>πKIin2/WL(CОХ+СD\*)2, K is a coefficient (D'Souza, 2002), and <sup>2</sup> B II PD 3 PD K P , α3 is a coefficient (Tobin, 1980). Substituting the first term of (4) into (3), after integration we obtain the following analytical expression for Q1 - the thermal noise of the gate-modulated channel expressed as the amount of noise electrons:

$$\begin{aligned} \mathbf{Q}\_{1}^{2} &= \frac{2\mathbf{k}\mathbf{T}\mathbf{C}\_{\text{in}\text{f}\_{\text{in}}\text{R}\_{\text{PD}}}\mathbf{R}\_{\text{PD}}\mathbf{a}\_{1}}{\mathbf{q}^{2}(1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}})} \left| 1 - \exp\left(-\mathbf{T}\_{\text{in}}\frac{1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}}{\mathbf{R}\_{\text{PD}}\mathbf{C}\_{\text{in}}}\right) \right| + \\ &+ \frac{2\mathbf{k}\mathbf{T}\mathbf{g}\_{\text{in}}\mathbf{a}\_{1}}{\mathbf{q}^{2}(1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}})^{2}} \left[ \mathbf{T}\_{\text{in}} - \frac{\mathbf{R}\_{\text{PD}}\mathbf{C}\_{\text{in}}}{1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}} \left( 1 - \exp\left(-\mathbf{T}\_{\text{in}}\frac{1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}}{\mathbf{R}\_{\text{PD}}\mathbf{C}\_{\text{in}}}\right) \right) \right] \end{aligned} \tag{5}$$

where Tin is the integration time.

In estimating the 1/f noise of the input FET Q2, also expressed as the amount of noise electrons, while performing integration, we have to take the fact into account that the readout regime involves a high-pass filter with transfer characteristic ω2/ (ω<sup>0</sup> 2 + ω2), where ω0 = /Тin.

dependences were obtained as dependent on the input-gate voltage of readouts; this enables a comparison with the experiment and formulation of requirements to electrophysical and design parameters of photodiodes and Si readouts necessary for implementation of design

Basic assumptions adopted in the proposed mathematical model of the system 'IR photodiode – direct injection readout circuit' were first formulated in (Kunakbaeva, Lee & Cherepov, 1993) and further developed in (Kunakbaeva & Lee, 1996; Karnaushenko; Lee et al., 2010; Gumenjuk-Sichevska, Karnaushenko, Lee & Polovinkin, 2011). In calculating the transfer characteristics of the system, for the input FET we employ the model of longchannel transistor in weak inversion (Overstraeten, 1975). The noise charge Q(t) integrated under the storage gate of the direct injection readout circuit is calculated as a McDonalds function expressed in terms of spectral current density Si(ω) (Buckingham,

**2. A mathematical model for the system 'IR photodiode - direct-injection** 

<sup>i</sup> <sup>2</sup>

2

2 in in PD 1 in PD

 

2kTC g R 1 gR <sup>Q</sup> 1 exp <sup>Т</sup> q (1 g R ) R С

2 2 in in

in PD PD in

2kТ<sup>g</sup> <sup>R</sup> <sup>С</sup> 1 gR <sup>Т</sup> 1 exp <sup>Т</sup> q (1 g R ) 1 gR R С

 

In estimating the 1/f noise of the input FET Q2, also expressed as the amount of noise electrons, while performing integration, we have to take the fact into account that the readout regime involves a high-pass filter with transfer characteristic ω2/ (ω02 + ω2), where

in 1 PD in in PD

in PD in PD PD in

1 2 in

expressed as the amount of noise electrons:

where Tin is the integration time.

ω0 = /Тin.

2 0

The spectral density of the noise current in the input-gate modulated channel is given by

2BI 4kT 2 B S ( ) 4kTg 1 1 2qI <sup>R</sup>

i in 1 I I PD I I

The first and second terms in (4) stand for the thermal noise of the input FET and for the 1/f noise induced by this FET (here, α1 and α2 are coefficients) and the third and forth terms stand for the thermal noise and 1/f noise of the photodiode, here, Bin= <sup>2</sup>πKIin2/WL(CОХ+СD\*)2, K is a coefficient (D'Souza, 2002), and <sup>2</sup> B II PD 3 PD K P , α3 is a coefficient (Tobin, 1980). Substituting the first term of (4) into (3), after integration we obtain the following analytical expression for Q1 - the thermal noise of the gate-modulated channel

<sup>1</sup> <sup>S</sup> Q (t) (1 cos t)d

(3)

PD

(5)

(4)

2 2 2 2 in in 2 PD

characteristics.

**readout circuit'** 

1983):

$$\text{Q}\_{2}^{2} = \frac{2\text{KL}\_{\text{in}}^{2}}{\text{qWL}(\text{C}\_{\text{OX}} + \text{C}\_{\text{D}})^{2}} \Big[ \frac{1 + \left(\text{C}\_{\text{in}}\text{R}\_{\text{PD}}\text{o}\right)^{2}}{\left(1 + \text{g}\_{\text{in}}\text{R}\_{\text{PD}}\right)^{2} + \left(\text{C}\_{\text{in}}\text{R}\_{\text{PD}}\text{o}\right)^{2}} \frac{1}{\text{o}\left(\text{o}\_{0}^{2} + \text{o}^{2}\right)} \sin^{2}\frac{\text{o}\,\text{T}\_{\text{in}}}{2} \text{d}\alpha\tag{6}$$

We substitute the third term of (4) into (3) and obtain, after integration, the following analytical expression for Q3 - the noise component due to photodiode current noise expressed as the amount of noise electrons:

$$\mathbf{Q}\_{3}^{2} = \frac{\left(\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}\right)^{2}}{\mathbf{q}^{2}(1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}})^{2}} \left(\mathbf{q}\mathbf{I}\_{\text{PD}} + \frac{2\mathbf{k}\mathbf{T}\mathbf{a}\_{2}}{\mathbf{R}\_{\text{PD}}}\right) \left(\mathbf{T}\_{\text{in}} - \frac{\mathbf{R}\_{\text{PD}}\mathbf{C}\_{\text{in}}}{1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}} \left(1-\exp\left(-\mathbf{T}\_{\text{in}}\frac{1+\mathbf{g}\_{\text{in}}\mathbf{R}\_{\text{PD}}}{\mathbf{R}\_{\text{PD}}\mathbf{C}\_{\text{in}}}\right)\right)\right) \tag{7}$$

The forth term in (4) stands to allow for the 1/f noise of the photodiode:

$$\begin{aligned} \mathbf{Q}\_4^2 &= 2\alpha\_3^2 \left(\mathbf{I}\_{\rm PD} - \eta\_\mathbf{K} \mathbf{I}\_\mathbf{P}\right)^2 \frac{\left(\mathbf{g}\_{\rm in} \mathbf{R}\_{\rm PD}\right)^2}{\mathbf{q}^2} \cdot \\ &\cdot \int\_0^\alpha \frac{1}{\left(1 + \mathbf{g}\_{\rm in} \mathbf{R}\_{\rm PD}\right)^2 + \left(\mathbf{C}\_{\rm in} \mathbf{R}\_{\rm PD}\right)^2} \frac{1}{\alpha\left(\alpha\_0^2 + \alpha^2\right)} \sin^2\frac{\alpha \mathbf{T}\_{\rm in}}{2} d\alpha \end{aligned} \tag{8}$$

The detectivity D\* of the photodetector channel in which photosignals from the IR photodiode are read into the direct injection input is given by (Rogalski, 2000):

$$\mathbf{D}^\* = \frac{\lambda}{\hbar \mathbf{c}} \frac{\left(\mathbf{A}\_{\rm PD} \mathbf{T}\_{\rm in} / \mathbf{2}\right)^{1/2} \eta\_{\rm l} \eta\_{\rm K}}{\left(\mathbf{Q}\_1^2 + \mathbf{Q}\_2^2 + \mathbf{Q}\_3^2 + \mathbf{Q}\_4^2 + \mathbf{Q}\_{\rm other}^2\right)^{1/2}} \tag{9}$$

where АPD is the photodiode area and Qother stands for all other noise components such as, for instance, the measuring channel induced noise.

### **3. Examples of model calculations**

#### **3.1 Modeling procedure for multi-element IR FPAs with direct injection readout circuits**

The modeling procedure for the system 'IR photodiode – direct injection readout circuits' can be described using the simple IR photodiode model:

$$\mathbf{I}\_{\rm PD} = \eta\_{\rm K} \mathbf{I}\_{\rm P} + \mathbf{I}\_{0} \left[ \mathbf{1} - \exp\left( -\beta \mathbf{V}\_{\rm PD} \right) \right] + \frac{\mathbf{V}\_{\rm PD}}{\mathbf{R}\_{\rm S}} \tag{10}$$

Here, ηк is photodiode quantum efficiency, I0 is the photodiode saturation current, RS is the photodiode shunt resistance, and IP is the current induced by background radiation.

The calculation starts with specifying the values of electrophysical and design parameters of readout circuits, photoelectrical parameters of photodiodes, and operating conditions of IR FPA, which all are listed in Table. 1.

At a given input gate voltage VG, the voltage across the photodiode can be determined from the condition Iin = IPD. We identify the points of intersection of the current-voltage

Mathematical Modeling of Multi-Element Infrared Focal Plane

(VG) (Fig.4d) for those cases differ substantially.

**I, A**


Ohm, IP = 1·10-8 A, and ŋК =0.8.

0,0

1,0x10-8

2,0x10-8

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 257

characteristics of IR FPAs based on direct injection readout circuits can be evaluated using standard simplifications (Longo, 1978; Felix,1980). In "non-ideal" systems, in which the relation ginRPD >>1 is fulfilled not too strict (see curves 2-5 in Figs. 4c and 4d), curves Iin (VG) and D\*(VG) show a different behavior. The current Iin integrated in the readout circuit increases with increasing VG. The dependences D\*(VG) exhibit a pronounced maximum. Being considered as a function of photodiode electrophysical parameters and background illumination current, the maximum detectivity is attained at a voltage VG at which the photodiode gets driven by 5-30 mV in reverse direction (curve 4 in Fig. 4); this detectivity rather weakly depends on the value of ginRPD. For instance, as the value of ginRPD decreases from 794 to 3.4 (curves 1 and 4 in Fig. 4) the detectivity D\* falls in value from 2.87·1011 cm·Hz1/2·Wt-1 to 1.96·1011 cm· Hz1/2·Wt-1, i.e. within a factor of 1.5. Note that the values of ginRPD for curves 2 and 3 are identical; nonetheless, the dependences Iin (VG) (Fig. 4c) and D\*


Fig. 3. Current-voltage characteristics of photodiode (curve 1) and input FET in weak inversion (curves 2-4) for three values of VG, 1.17, 1.198, and 1.22 V. I0=5·10-9 A, RS=3·107

The factors causing these differences can be clarified if one considers the calculated dependences of Fig. 5; in this figure, in addition to curves D\* (VG), also dependences Q1(VG), Q2(VG), Q3(VG), and Q4(VG) are shown. In the "ideal" system (see Fig.5 a), with increasing the voltage VG, when the injection efficiency approaches unity, the total noise charge Q5 becomes defined just by the photodiode noise current Q3, having almost the same magnitude as the noise induced by background radiation fluctuations (9) since Iin ≈ IP and η<sup>I</sup> ≈ 1. In "non-ideal" systems (see calculated dependences in Fig. 5 b-e), the charge Q5 grows in value with increasing VG. The increase of the noise and the related reduction of D\* is primarily defined by the growth of Q3 and Q4. For the dependences shown in Fig. 5 b-e the difference between the curves D\*(VG) is primarily defined by the growth of the 1/f noise of photodiode (component Q4). The difference of the dependences Q4(VG) is due to the higher current Iin and a lower value of ηI for curves 4. With identical values of ginRPD at zero voltage drop across the photodiode (compare Fig. 5b and Fig. 5c), a better IR FPA responsivity can be reached with diodes exhibiting a larger dynamic resistance on their biasing in reverse direction by 10-30 mV. For state-of-the-art level of silicon technology (Cох ~ (0.5-1.2)·10-7 F·cm-2, N\*≈ 1), the noise induced by the input FET (cp. curves 1 and 2) is only substantial at

2 3 4

VPD, V

1


characteristic of the photodiode and the transfer characteristics of the input FET to determine the photodiode bias voltage.

Table 1. Design and electrophysical parameters of readout circuits

Figure 3 shows the curves of currents Iin and IPD for the photodiode model (10). With the parameters of photodiode, input FET, and radiation environment adopted in Fig. 3, the voltage drop across the photodiode is zero at VG0 = 1.198 V. Given the voltage VG, the quantities ŋI, Iin, and D\* can be calculated. In the next cycle, a new value is assigned to VG, and all the characteristics are to be recalculated. In this way, we obtain the main performance characteristics of the system 'photodiode-direct injection readout' as a function of the gate voltage at the input gate. The dependences in Fig. 3 can be used in a joint analysis of the effect due to noise characteristics of photodiodes, measured versus photodiode bias voltage, and the responsivity of photodetector channels based on the system 'photodiode – direct injection readout circuits'.

Figures 4a, b, c, and d shows the calculated curves of photodiode bias voltage VPD(VG), injection efficiency ηI(VG), input current Iin(VG), and detectivity D\*(VG), respectively.

At VPD = 0 V, the product ginRPD for curves 1-5 in Fig. 4 equals respectively 794, 11.1, 11.1, 3.4, and 1.4. As it is seen from Figs. 4a and 4b, the dependences VPD(VG) and ηI(VG) small informative in evaluating the uniformity level of transfer characteristics and responsivities of photodetector channels, more helpful here being the dependences Iin(VG) and D\*(VG) (see Figs 4c and 4d). As it is seen from Fig. 4, curves 1, at ginRPD >100 the photodiode-direct injection readout system can be considered "ideal". With increasing the gate voltage VG, as the photodiode bias voltage approaches zero, the injection efficiency ŋI tends to unity, the current integrated in the readout circuit becomes roughly equal to IPD, Iin ≈ IPD, and the detectivity reaches D\* ≈ 2.87·1011 cm·Hz1/2·Wt-1, this value being comparable with the theoretical limit of D\* for a photodiode with ηK=0.8 operating in photovoltaic regime in BLIP mode. With further increase of VG, for the "ideal" system the performance characteristics of the photodetector channel based on the system 'photodiode – direct injection readout circuit' (namely, ŋI, Iin, and D\*) become almost independent of both VG and electrophysical and design parameters of readout circuits. In the latter case, performance

characteristic of the photodiode and the transfer characteristics of the input FET to

Designation Parameter Numerical value μ Mobility of minorities in the inversion channel of FET 500 cm2 V-1·c-1 ND Donor concentration in the substrate 7·1014 cm-3 СOX Specific capacitance of gate dielectric 1.24·10-7 F·cm-2

VFB Flat-band voltage 0 V

Table 1. Design and electrophysical parameters of readout circuits

Nss\*- Surface-state density 1·109 cm-2·eV-1 W, L Length and width of input-gate modulated channel 30 µm, 3 µm Сin Input capacitance of FPA cell 0.5 pF α<sup>1</sup> Numerical coefficients 2 α<sup>2</sup> Numerical coefficients 2 α<sup>3</sup> Numerical coefficients 10-3

K Numerical coefficients 1.5·10-24 F2·cm-2 АPD Photodiode area 9·10-5 cm2 η<sup>K</sup> Photodiode quantum efficiency 0.8

Figure 3 shows the curves of currents Iin and IPD for the photodiode model (10). With the parameters of photodiode, input FET, and radiation environment adopted in Fig. 3, the voltage drop across the photodiode is zero at VG0 = 1.198 V. Given the voltage VG, the quantities ŋI, Iin, and D\* can be calculated. In the next cycle, a new value is assigned to VG, and all the characteristics are to be recalculated. In this way, we obtain the main performance characteristics of the system 'photodiode-direct injection readout' as a function of the gate voltage at the input gate. The dependences in Fig. 3 can be used in a joint analysis of the effect due to noise characteristics of photodiodes, measured versus photodiode bias voltage, and the responsivity of photodetector channels based on the system 'photodiode –

Figures 4a, b, c, and d shows the calculated curves of photodiode bias voltage VPD(VG),

At VPD = 0 V, the product ginRPD for curves 1-5 in Fig. 4 equals respectively 794, 11.1, 11.1, 3.4, and 1.4. As it is seen from Figs. 4a and 4b, the dependences VPD(VG) and ηI(VG) small informative in evaluating the uniformity level of transfer characteristics and responsivities of photodetector channels, more helpful here being the dependences Iin(VG) and D\*(VG) (see Figs 4c and 4d). As it is seen from Fig. 4, curves 1, at ginRPD >100 the photodiode-direct injection readout system can be considered "ideal". With increasing the gate voltage VG, as the photodiode bias voltage approaches zero, the injection efficiency ŋI tends to unity, the current integrated in the readout circuit becomes roughly equal to IPD, Iin ≈ IPD, and the detectivity reaches D\* ≈ 2.87·1011 cm·Hz1/2·Wt-1, this value being comparable with the theoretical limit of D\* for a photodiode with ηK=0.8 operating in photovoltaic regime in BLIP mode. With further increase of VG, for the "ideal" system the performance characteristics of the photodetector channel based on the system 'photodiode – direct injection readout circuit' (namely, ŋI, Iin, and D\*) become almost independent of both VG and electrophysical and design parameters of readout circuits. In the latter case, performance

injection efficiency ηI(VG), input current Iin(VG), and detectivity D\*(VG), respectively.

determine the photodiode bias voltage.

direct injection readout circuits'.

characteristics of IR FPAs based on direct injection readout circuits can be evaluated using standard simplifications (Longo, 1978; Felix,1980). In "non-ideal" systems, in which the relation ginRPD >>1 is fulfilled not too strict (see curves 2-5 in Figs. 4c and 4d), curves Iin (VG) and D\*(VG) show a different behavior. The current Iin integrated in the readout circuit increases with increasing VG. The dependences D\*(VG) exhibit a pronounced maximum. Being considered as a function of photodiode electrophysical parameters and background illumination current, the maximum detectivity is attained at a voltage VG at which the photodiode gets driven by 5-30 mV in reverse direction (curve 4 in Fig. 4); this detectivity rather weakly depends on the value of ginRPD. For instance, as the value of ginRPD decreases from 794 to 3.4 (curves 1 and 4 in Fig. 4) the detectivity D\* falls in value from 2.87·1011 cm·Hz1/2·Wt-1 to 1.96·1011 cm· Hz1/2·Wt-1, i.e. within a factor of 1.5. Note that the values of ginRPD for curves 2 and 3 are identical; nonetheless, the dependences Iin (VG) (Fig. 4c) and D\* (VG) (Fig.4d) for those cases differ substantially.

Fig. 3. Current-voltage characteristics of photodiode (curve 1) and input FET in weak inversion (curves 2-4) for three values of VG, 1.17, 1.198, and 1.22 V. I0=5·10-9 A, RS=3·107 Ohm, IP = 1·10-8 A, and ŋК =0.8.

The factors causing these differences can be clarified if one considers the calculated dependences of Fig. 5; in this figure, in addition to curves D\* (VG), also dependences Q1(VG), Q2(VG), Q3(VG), and Q4(VG) are shown. In the "ideal" system (see Fig.5 a), with increasing the voltage VG, when the injection efficiency approaches unity, the total noise charge Q5 becomes defined just by the photodiode noise current Q3, having almost the same magnitude as the noise induced by background radiation fluctuations (9) since Iin ≈ IP and η<sup>I</sup> ≈ 1. In "non-ideal" systems (see calculated dependences in Fig. 5 b-e), the charge Q5 grows in value with increasing VG. The increase of the noise and the related reduction of D\* is primarily defined by the growth of Q3 and Q4. For the dependences shown in Fig. 5 b-e the difference between the curves D\*(VG) is primarily defined by the growth of the 1/f noise of photodiode (component Q4). The difference of the dependences Q4(VG) is due to the higher current Iin and a lower value of ηI for curves 4. With identical values of ginRPD at zero voltage drop across the photodiode (compare Fig. 5b and Fig. 5c), a better IR FPA responsivity can be reached with diodes exhibiting a larger dynamic resistance on their biasing in reverse direction by 10-30 mV. For state-of-the-art level of silicon technology (Cох ~ (0.5-1.2)·10-7 F·cm-2, N\*≈ 1), the noise induced by the input FET (cp. curves 1 and 2) is only substantial at

Mathematical Modeling of Multi-Element Infrared Focal Plane

'photodiode-direct injection readout' system.

1,10 1,15 1,20 1,25 1,30

1,10 1,15 1,20 1,25 1,30

Q, number electrons

e

for curves 1-5, respectively.

0,0 5,0x103 1,0x104 1,5x104 2,0x104 2,5x104

3

4

3 4 0,0 5,0x1010 1,0x1011 1,5x1011 2,0x1011 2,5x1011 3,0x1011

0,0 5,0x1010 1,0x1011 1,5x1011 2,0x1011 2,5x1011 3,0x1011

VG, V

VG, V

6

6

0,0 2,0x103 4,0x103 6,0x103 8,0x103 1,0x104

0,0 2,0x103 4,0x103 6,0x103 8,0x103 1,0x104 1

5

2

1

5

2

Q, number electrons

Q, number electrons

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 259

injection efficiencies ηI and, as a consequence, to variation of transfer characteristics of the

Q, number electrons

b

Q, number electrons

d

1,10 1,15 1,20 1,25 1,30

2

Fig. 5. Calculated dependences of D\*(VG) (curve 6, right axis) and Q1(VG), Q2(VG), Q3(VG),

photodiode electrophysical parameters adopted in the calculations are the same as in Fig. 4

Figure 7 shows the calculated curve Iin (VG) that illustrate the influence of the dispersion of threshold voltages with σ(Vth) = 10 mV on the characteristics of a multi-element IR FPA (the values of photoelectrical parameters for IR photodiodes are the same as those adopted in Fig. 4 for curves 2). It is seen from Fig. 7 a-b that for a "non-ideal" system the dispersion of threshold voltages brings about a fixed-pattern noise and a spread of currents integrated in the readouts, Iin (VG). For multi-element IR FPAs, with allowance for the spread of threshold voltages, at a voltage VG =1.205 V (Fig.5b), for which the detectivity of a single channel D\*

Q4(VG), and Q5(VG) (curves 1-5, left axis). Q5=(Q12+Q22+Q32+Q42)1/2. The values of

5

3 <sup>4</sup> <sup>6</sup>

1

D\*, cm Hz1/2

а 1,10 1,15 1,20 1,25 1,30

D\*, cmHz1/2

c 1,10 1,15 1,20 1,25 1,30

W


0,0 2,0x10<sup>3</sup> 4,0x10<sup>3</sup> 6,0x10<sup>3</sup> 8,0x10<sup>3</sup> 1,0x10<sup>4</sup>

0,0 2,0x103 4,0x103 6,0x103 8,0x103 1,0x104 1,2x104 1,4x104 1

5

2

1 2

0,0 5,0x10<sup>10</sup> 1,0x10<sup>11</sup> 1,5x10<sup>11</sup> 2,0x10<sup>11</sup> 2,5x10<sup>11</sup> 3,0x10<sup>11</sup>

VG, V

3

4

3 4 <sup>5</sup> <sup>6</sup>

D\*, cmHz1/2

W


0,0 5,0x1010 1,0x1011 1,5x1011 2,0x1011 2,5x1011 3,0x1011

0,0 5,0x1010 1,0x1011 1,5x1011 2,0x1011 2,5x1011 3,0x1011

VG, V

VG, V

6

D\*, cmHz1/2

D\*, cmHz1/2

W


W


W


voltages VG < VG0 even if the product ginRPD has a value close to unity, and this noise does not limit the IR FPA detectivity. Knowing of the values of individual noise components enables goal-directed optimization of electrophysical and design parameters of photodiodes and readout circuits.

Fig. 4. Calculated dependences of performance characteristics of a photodiode-direct injection system on input-gate voltage. a – photodiode bias voltage, b – injection efficiency ηI, c – current Iin integrated in the readout circuit, d - detectivity D\*. Integration time Тin =5·10-4 s, IP =1·10-8 A, ŋК =0.8, for curve 1 - I0 =1·10-11 A, RP=1·1011 Ohm; for curve 2 - I0 =5·10-10 A, RP =3·107 Ohm; for curve 3 - I0 =7·10-10 A, RP =3·108 Ohm; for curve 4 - I0 = 2·10-9 A, RP = 2·107 Ohm; for curve 5 - I0 = 5·10-9 A, RP =1·107 Ohm.

Important figures of merit of multi-element IR FPAs are the fixed-pattern noise and the responsivity uniformity of photodetector channels. The spread of photoelectric parameters is defined by the variation of electrophysical parameters of readouts and by the variation of photoelectrical parameters of photodiodes. For direct injection readouts operated in weak inversion regime, the main effect is due to input-FET threshold variations rather than due to non-uniformity of geometric dimensions of input-FET gates.

Histograms of photodiode bias voltages and injection efficiencies calculated with allowance for FET threshold variations are shown in Fig. 6. In performing the calculations, it was assumed that the spread of threshold voltages obeys a normal distribution law, and a total of 1000 realizations were considered. As it is seen from Fig. 6, the spread of FET threshold voltages leads to a spread of photodiode bias voltages (Fig. 6 a), and also to a spread of

voltages VG < VG0 even if the product ginRPD has a value close to unity, and this noise does not limit the IR FPA detectivity. Knowing of the values of individual noise components enables goal-directed optimization of electrophysical and design parameters of photodiodes

VG , V

VG, V

Fig. 4. Calculated dependences of performance characteristics of a photodiode-direct injection system on input-gate voltage. a – photodiode bias voltage, b – injection efficiency ηI, c – current Iin integrated in the readout circuit, d - detectivity D\*. Integration time Тin =5·10-4 s, IP =1·10-8 A, ŋК =0.8, for curve 1 - I0 =1·10-11 A, RP=1·1011 Ohm; for curve 2 - I0 =5·10-10 A, RP =3·107 Ohm; for curve 3 - I0 =7·10-10 A, RP =3·108 Ohm; for curve 4 - I0 = 2·10-9

b 1,10 1,15 1,20 1,25 1,30

d

Important figures of merit of multi-element IR FPAs are the fixed-pattern noise and the responsivity uniformity of photodetector channels. The spread of photoelectric parameters is defined by the variation of electrophysical parameters of readouts and by the variation of photoelectrical parameters of photodiodes. For direct injection readouts operated in weak inversion regime, the main effect is due to input-FET threshold variations rather than due to

Histograms of photodiode bias voltages and injection efficiencies calculated with allowance for FET threshold variations are shown in Fig. 6. In performing the calculations, it was assumed that the spread of threshold voltages obeys a normal distribution law, and a total of 1000 realizations were considered. As it is seen from Fig. 6, the spread of FET threshold voltages leads to a spread of photodiode bias voltages (Fig. 6 a), and also to a spread of

0,0 5,0x1010 1,0x1011 1,5x1011 2,0x1011 2,5x1011 3,0x1011


D\*, cm Hz1/2

W

c

0,0 4,0x10-9 8,0x10-9 1,2x10-8 1,6x10-8 2,0x10-8 2,4x10-8 Iin, A

1,10 1,15 1,20 1,25 1,30

1 2

1

VG, V

2 3

4

5

VG, V

3

4 5

and readout circuits.


0,0

0,2

0,4

0,6

0,8

1,0

V**PD ,** 

à

a

h**I**

**V**

1,10 1,15 1,20 1,25 1,30

1,10 1,15 1,20 1,25 1,30

A, RP = 2·107 Ohm; for curve 5 - I0 = 5·10-9 A, RP =1·107 Ohm.

non-uniformity of geometric dimensions of input-FET gates.

5

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup>

injection efficiencies ηI and, as a consequence, to variation of transfer characteristics of the 'photodiode-direct injection readout' system.

Fig. 5. Calculated dependences of D\*(VG) (curve 6, right axis) and Q1(VG), Q2(VG), Q3(VG), Q4(VG), and Q5(VG) (curves 1-5, left axis). Q5=(Q12+Q2 2+Q3 2+Q42)1/2. The values of photodiode electrophysical parameters adopted in the calculations are the same as in Fig. 4 for curves 1-5, respectively.

Figure 7 shows the calculated curve Iin (VG) that illustrate the influence of the dispersion of threshold voltages with σ(Vth) = 10 mV on the characteristics of a multi-element IR FPA (the values of photoelectrical parameters for IR photodiodes are the same as those adopted in Fig. 4 for curves 2). It is seen from Fig. 7 a-b that for a "non-ideal" system the dispersion of threshold voltages brings about a fixed-pattern noise and a spread of currents integrated in the readouts, Iin (VG). For multi-element IR FPAs, with allowance for the spread of threshold voltages, at a voltage VG =1.205 V (Fig.5b), for which the detectivity of a single channel D\*

Mathematical Modeling of Multi-Element Infrared Focal Plane

8,0x10<sup>10</sup> 1,2x10<sup>11</sup> 1,6x10<sup>11</sup> 2,0x10<sup>11</sup> 2,4x10<sup>11</sup>

0

distributions of RPD and I0 obey normal distribution laws.

c

of the single photodetector channel.

50

100

150

200

250

D\* , cmHz 1/2 W -1

0

Fig. 4.

200

a

400

600

800

detectivity of the majority of photodetector channels decreases (Fig. 8c).

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 261

of multi-element IR FPAs because of the spread of threshold voltages, there arises a necessity to adjust the gate voltage (for the values of electrophysical parameters of photodiodes adopted in the calculations, Fig. 8b); the optimum value here is VG =1.225 V. The increase of VG results in a substantial reduction of fixed-pattern noise (see Fig.7b). In the latter situation, the minimal detectivity is not lower than 1.85·1011 cm·Hz1/2·Wt-1, and more than 98% pixels have a detectivity D\* greater than 2.0·1011 cm·Hz1/2·Wt-1. With further increase of VG, "dark" parasitic photodiode current components grow in value and the

,

8,0x10<sup>10</sup> 1,2x10<sup>11</sup> 1,6x1011 2,0x10<sup>11</sup> 2,4x10<sup>11</sup>

Fig. 8. Histograms D\*(VG) calculated for FET threshold voltages normally distributed with σ(Vth) = 10 mV for VG = 1.201, 1.225, and 1.249 V (respectively Figs. 8a, 8b, and 8.6c). The calculations were performed for photodiodes with parameter values adopted for curves 2 in

The effect due to dispersion of threshold voltages in fabricated readout circuits is illustrated by a comparison of calculated histograms Iin(VG) and D\*(VG) in Fig. 8 with histograms calculated for σ(Vth) = 2 mV, Fig. 9. As it is seen from the calculated dependences in Fig. 9, in the latter case the detectivity of the multi-element IR FPA is at the level of about 90% of D\*

Along with non-uniformity of input-FET threshold voltages, the photoelectric parameters of photodiodes, ŋК, RPD, and I0, also inevitably display some scatter of values. Figure 10 shows histograms of Iin (Fig. 10a) and D\* (Fig.10b) calculated on the assumption that the

D\* , cmHz 1/2 W -1

б)

b

8,0x10<sup>10</sup> 1,2x10<sup>11</sup> 1,6x10<sup>11</sup> 2,0x10<sup>11</sup> 2,4x10<sup>11</sup>

D\* , cmHz 1/2 <sup>W</sup>-1

attains a maximum, the spread of currents Iin integrated in the readouts falls in the range from 4·10-9 to 1.15·10-8 A (Fig. 7a), whereas the current integrated in the "ideal" system is Iin = 0.8·10-8 A.

Fig. 6. Effect due to the spread of FET threshold voltages on the characteristics of multielement IR FPAs; a – histogram of photodiode voltages, b – histogram of injection efficiencies ηI for I0=5·10-10 A, RР =3·107 Ohm, IP =1·10-8A, ŋК =0.8, and VG=1.225 V. The standard deviation of input-FET threshold voltages is σ(Vth) = 10 mV.

Fig. 7. Calculated histograms Iin (VG) for a normal distribution of input-FET threshold voltages with σ(Vth) = 10 mV. For Figs. 7a and 7b the voltages VG are respectively 1.205 Vand 1.225 V. The calculations were performed for photodiodes with parameter values the same as those for curves 2 in Fig. 4.

Figure 8 show histograms D\*(VG) calculated with allowance for the spread of threshold voltages. In the calculations of detectivity, the accumulation time was defined by the charge capacity of the readout circuit and by the maximal (over 1000 realizations) current Iin (VG).

In multi-element IR FPAs, in view of non-uniformity of threshold voltages, at VG =1.205 V about 1% of photodetector channels have a detectivity D\* lower than 1·1011 cm·Hz1/2·Wt-1 (see Fig. 8b), whereas for a single channel we have D\* = 2.48·1011 cm·Hz1/2·Wt-1. In the case

attains a maximum, the spread of currents Iin integrated in the readouts falls in the range from 4·10-9 to 1.15·10-8 A (Fig. 7a), whereas the current integrated in the "ideal" system is Iin

VPD, V

standard deviation of input-FET threshold voltages is σ(Vth) = 10 mV.

,

Fig. 6. Effect due to the spread of FET threshold voltages on the characteristics of multielement IR FPAs; a – histogram of photodiode voltages, b – histogram of injection efficiencies ηI for I0=5·10-10 A, RР =3·107 Ohm, IP =1·10-8A, ŋК =0.8, and VG=1.225 V. The

Iin, А

Fig. 7. Calculated histograms Iin (VG) for a normal distribution of input-FET threshold voltages with σ(Vth) = 10 mV. For Figs. 7a and 7b the voltages VG are respectively 1.205 Vand 1.225 V. The calculations were performed for photodiodes with parameter values the

,

Figure 8 show histograms D\*(VG) calculated with allowance for the spread of threshold voltages. In the calculations of detectivity, the accumulation time was defined by the charge capacity of the readout circuit and by the maximal (over 1000 realizations) current Iin (VG). In multi-element IR FPAs, in view of non-uniformity of threshold voltages, at VG =1.205 V about 1% of photodetector channels have a detectivity D\* lower than 1·1011 cm·Hz1/2·Wt-1 (see Fig. 8b), whereas for a single channel we have D\* = 2.48·1011 cm·Hz1/2·Wt-1. In the case

b

0

100

200

300

400

0

200

b

400

600

800

0,80 0,84 0,88 0,92 0,96 1,00

4,0x10-9 6,0x10-9 8,0x10-9 1,0x10-8 1,2x10-8

I

Iin, А

= 0.8·10-8 A.

0

0

а

100

200

300

400

500

à

a

50

100

150

200


4,0x10-9 6,0x10-9 8,0x10-9 1,0x10-8 1,2x10-8

same as those for curves 2 in Fig. 4.

of multi-element IR FPAs because of the spread of threshold voltages, there arises a necessity to adjust the gate voltage (for the values of electrophysical parameters of photodiodes adopted in the calculations, Fig. 8b); the optimum value here is VG =1.225 V. The increase of VG results in a substantial reduction of fixed-pattern noise (see Fig.7b). In the latter situation, the minimal detectivity is not lower than 1.85·1011 cm·Hz1/2·Wt-1, and more than 98% pixels have a detectivity D\* greater than 2.0·1011 cm·Hz1/2·Wt-1. With further increase of VG, "dark" parasitic photodiode current components grow in value and the detectivity of the majority of photodetector channels decreases (Fig. 8c).

Fig. 8. Histograms D\*(VG) calculated for FET threshold voltages normally distributed with σ(Vth) = 10 mV for VG = 1.201, 1.225, and 1.249 V (respectively Figs. 8a, 8b, and 8.6c). The calculations were performed for photodiodes with parameter values adopted for curves 2 in Fig. 4.

The effect due to dispersion of threshold voltages in fabricated readout circuits is illustrated by a comparison of calculated histograms Iin(VG) and D\*(VG) in Fig. 8 with histograms calculated for σ(Vth) = 2 mV, Fig. 9. As it is seen from the calculated dependences in Fig. 9, in the latter case the detectivity of the multi-element IR FPA is at the level of about 90% of D\* of the single photodetector channel.

Along with non-uniformity of input-FET threshold voltages, the photoelectric parameters of photodiodes, ŋК, RPD, and I0, also inevitably display some scatter of values. Figure 10 shows histograms of Iin (Fig. 10a) and D\* (Fig.10b) calculated on the assumption that the distributions of RPD and I0 obey normal distribution laws.

Mathematical Modeling of Multi-Element Infrared Focal Plane

approximated with the expressions:

photodiode quantum efficiency (Vasilyev, 2010).


0

5

10

15


20

Iin(VG), and D\*(VG).

FPAs.

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 263

photodiodes were fabricated on the basis of a Hg1-хCdхTe variband heteroepitaxial structure. The stoichiometric composition of the photosensitive layer was x=0.225 (Vasilyev, 2010).

In performing the calculations, the experimental current-voltage characteristics were

The characteristics of multi-element IR FPA were calculated by the procedure, described in section 3.1, that allows calculation of all photodetector-channel characteristics, ηI(VG),

Figure 12 shows the histograms D\*(VG) and Iin(VG) calculated for the standard deviation value of threshold voltages of the readouts σ(Vth) =10 mV. An analysis of dependences calculated with lower values of σ(Vth) suggests that the use of silicon technology permitting values σ(Vth) < 2–3 mV will allow a substantial (by 20–30%) reduction of the fixed-pattern noise level. The changes in the calculated detectivity histograms proved to be less substantial since the spread of detectivity values is primarily defined by the variations of

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40

Fig. 11. A family of experimental current-voltage characteristics of Hg1-хCdхTe photodiodes; "dark" characteristics are shown with blue lines, and characteristics measured under room-

It should be noted that the numerical values of coefficients α1, α2, α3, and K were borrowed from literature sources. Experimentally measured values of these coefficients or more elaborated model for photodiode noises can easily be incorporated into the computer problem, allowing an improved accuracy in predicting the characteristics of IR

temperature radiation background with red lines are shown with red lines.


where the values of coefficients С0-С5 were chosen individually for each photodiode.

I (V ) C C exp C V PD PD 0 1 2 PD 3 4 PD 5 PD С exp C V С V (11)

Fig. 9. Influence of the threshold voltage spread on the characteristics of multi-element IR FPAs; a – histogram of Iin, b – histogram of D\*. The standard deviation of input-FET threshold voltages is σ(Vth) = 2 mV, I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG =1.213 V.

Fig. 10. Influence of the spread of photoelectric parameters of photodiodes on the performance characteristics of multi-element IR FPAs; а – histogram of Iin, b – histogram of D\*. Parameter values adopted in the calculations: σ(RPD) = 3·106 Ohm, σ(I0) = 1·10-10 A, I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG =1.225 V.

Plotting histograms of performance characteristics of multi-element IR FPAs, namely, currents integrated in the readout circuits and detectivity) versus the input-gate voltage VG allows one to carry out numerical experiments and compare their results with experimental data. Such histograms can be considered generalized characteristics of multi-element IR FPAs indicative of their quality.

#### **3.2 Modeling procedure for multi-element IR FPAs with direct injection readout circuits using the experimental-current voltage characteristics of photodiodes**

The developed procedure enables performing an analysis of the system 'photodiode –directinjection readout' using the experimental current-voltage characteristics of photodiodes. Figure 11 shows a family of "dark" and "light" current-voltage characteristics of one hundred photodiodes measured under room-temperature background conditions. The

Iin, A ,

Iin, A

Fig. 10. Influence of the spread of photoelectric parameters of photodiodes on the

performance characteristics of multi-element IR FPAs; а – histogram of Iin, b – histogram of D\*. Parameter values adopted in the calculations: σ(RPD) = 3·106 Ohm, σ(I0) = 1·10-10 A,

Plotting histograms of performance characteristics of multi-element IR FPAs, namely, currents integrated in the readout circuits and detectivity) versus the input-gate voltage VG allows one to carry out numerical experiments and compare their results with experimental data. Such histograms can be considered generalized characteristics of multi-element IR

**3.2 Modeling procedure for multi-element IR FPAs with direct injection readout circuits using the experimental-current voltage characteristics of photodiodes** 

The developed procedure enables performing an analysis of the system 'photodiode –directinjection readout' using the experimental current-voltage characteristics of photodiodes. Figure 11 shows a family of "dark" and "light" current-voltage characteristics of one hundred photodiodes measured under room-temperature background conditions. The

Fig. 9. Influence of the threshold voltage spread on the characteristics of multi-element IR FPAs; a – histogram of Iin, b – histogram of D\*. The standard deviation of input-FET threshold voltages is σ(Vth) = 2 mV, I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG

0

b

100

b

200

300

400

2,0x10<sup>11</sup> 2,2x10<sup>11</sup> 2,4x10<sup>11</sup>

2,0x1011 2,2x1011 2,4x1011

D\*

, cmHz1/2W-1

D\*

, cmHz1/2W-1

a

à

a

0

100

200

0

=1.213 V.

300

50

100

150

200

8,0x10-9 1,0x10-8 1,2x10-8

8,0x10-9 1,0x10-8 1,2x10-8

FPAs indicative of their quality.

I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG =1.225 V.

photodiodes were fabricated on the basis of a Hg1-хCdхTe variband heteroepitaxial structure. The stoichiometric composition of the photosensitive layer was x=0.225 (Vasilyev, 2010).

In performing the calculations, the experimental current-voltage characteristics were approximated with the expressions:

$$\mathbf{I}\_{\rm PD}(\mathbf{V}\_{\rm PD}) = \mathbf{C}\_0 + \mathbf{C}\_1 \exp(\mathbf{C}\_2 \mathbf{V}\_{\rm PD}) + \mathbf{C}\_3 \exp(\mathbf{C}\_4 \mathbf{V}\_{\rm PD}) + \mathbf{C}\_5 \mathbf{V}\_{\rm PD} \tag{11}$$

where the values of coefficients С0-С5 were chosen individually for each photodiode.

The characteristics of multi-element IR FPA were calculated by the procedure, described in section 3.1, that allows calculation of all photodetector-channel characteristics, ηI(VG), Iin(VG), and D\*(VG).

Figure 12 shows the histograms D\*(VG) and Iin(VG) calculated for the standard deviation value of threshold voltages of the readouts σ(Vth) =10 mV. An analysis of dependences calculated with lower values of σ(Vth) suggests that the use of silicon technology permitting values σ(Vth) < 2–3 mV will allow a substantial (by 20–30%) reduction of the fixed-pattern noise level. The changes in the calculated detectivity histograms proved to be less substantial since the spread of detectivity values is primarily defined by the variations of photodiode quantum efficiency (Vasilyev, 2010).

Fig. 11. A family of experimental current-voltage characteristics of Hg1-хCdхTe photodiodes; "dark" characteristics are shown with blue lines, and characteristics measured under roomtemperature radiation background with red lines are shown with red lines.

It should be noted that the numerical values of coefficients α1, α2, α3, and K were borrowed from literature sources. Experimentally measured values of these coefficients or more elaborated model for photodiode noises can easily be incorporated into the computer problem, allowing an improved accuracy in predicting the characteristics of IR FPAs.

Mathematical Modeling of Multi-Element Infrared Focal Plane

photodiodes.

τn, τp, τnV, τpV

Wc

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 265

By way of example, Figure 13 shows the current-voltage characteristics of Hg1-xCdxTe photodiodes for different long-wave spectral-response cutoffs, and Figure 14 shows the calculated dependences of RPD АPD on the cutoff wavelength of the spectral response of the

As it was shown in (Gumenjuk-Sichevska, 1999; Sizov, 2006), the photodiode model that was used in the present calculations permits calculation of the current-voltage characteristics as a function of the stoichiometric composition of Hg1-xCdxTe, temperature, and electrophysical parameters of substrate material, the obtained dependences being

Figure 15 shows the calculated dependences of the maximum values of D\*(VG) of a photodetector channel based on the system 'HgCdTe photodiode – direct injection readout circuit' on the cutoff wavelength λ2. In calculating the background radiation flux reaching the photodiodes in the spectral region from λ1 to λ2 (λ1 =4 μm and λ2 is the cutoff wavelength), for the blackbody temperature a value 300 K was adopted; the aperture angle was assumed defined by the relative aperture of the optical system F/f = 0.5, and for the

Case 1 Case 2

2.0·1022 m-3, 2.0·1021 m-3

3.0·1021 m-3, 1.0·1021 m-3

0.2·10-6 s, 0.2·10-6 s, 1.0·10-7 s, 1.0·10-7 s

1.0·1022 m-3, 2.0·1021 m-3

6.0·1021 m-3, 2.0·1021 m-3

3·10-67 J2·m3

0.96 eV

0.2·10-6 s, 0.2·10-6 s, 6.0·10-6 s, 6.0·10-6 s

consistent with those reported in the literature (Rogalski, 2000; Yoshino,1999).

Designation Parameter Numerical value

transmission of the optical system, a value 0.9 was adopted.

n junction and in the quasi-

Electron and hole lifetimes in the p-n junction and in the quasi-neutral regions

Et Energy position of trap level Et=0.7 Eg eV

P Interband matrix element 8.3·10-10 eV·m

tunneling of charge carriers from the trap level into the band

Na, Nd Acceptor and donor

concentrations

Nt, NtV Concentration of traps in the p-

neutral regions

<sup>2</sup> Squared matrix element for the

ε<sup>r</sup> Static dielectric permittivity 17.5

АPD Photodiode area 9·10-5 cm2

Table 2. Electrophysical parameters of Hg1-xCdxTe photodiodes.

Δ Spin – orbital interaction constant

Fig. 12. Influence of the spread of photoelectric parameters of photodiodes on the performance characteristics of multi-element IR FPAs; а – histogram of Iin, b – histogram of D\*. Parameter values adopted in the calculations: σ(RPD) = 3·106 Ohm, σ(I0) = 1·10-10 A, I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG =1.225 V.

#### **3.3 Analysis of LWIR thermography systems based on Hg1-xCdxTe photodiodes**

The simple photodiode model (10) proved to be insufficient for adequately predicting the current-voltage characteristics of Hg1-xCdxTe photodiodes in the spectral region 8 to 14 µm. In modeling current-voltage characteristics of Hg1-xCdxTe photodiodes, all main mechanisms of charge transport in *p-n* junctions must be taken into account, including the background radiation current ηКIP, the diffusion current, the thermal generation/recombination current, the tunneling current through localized levels within the bandgap of the depleted region of the *p-n* junction, the interband tunneling current (Anderson,1981), the Shockley-Reed-Hall current through traps in quasi-neutral *n* and *p* regions, the Auger current, and the radiative generation/recombination current in the *p-n* junction and in the quasi-neutral regions (Anderson, 1981; Rogalski, 2000). All the currents listed above are independent currents to be taken into account additively, except for the thermal generation-recombination and trap-assisted tunneling through traps of the Shockley-Reed-Hall type in the depleted region of the *р-n* junction, since the rates of the latter processes is determined by the trap occupation. The two latter mechanisms were modeled in the approximation of balance equations for charge carriers at trap levels (Anderson, 1981; Gumenjuk-Sichevska, 1999; Sizov, 2006; Yoshino,1999). We assumed the presence of localized donor-type centers in the bandgap with energy Et=0.6-0.7Eg over the valence-band edge (Krishnamurthy, 2006). In calculating the band-gap energy of the material as a function of stoichiometric composition and temperature, we used the expression (Rogalski, 2000).

$$\mathbf{E}(\mathbf{x},\mathbf{T}) = \left[ -0.302 + \mathbf{x} \left[ 1.93 + \mathbf{x} \left( -0.81 + 0.832 \mathbf{x} \right) \right] + 5.32 \cdot 10^{-4} \cdot \left( 1 - 2 \mathbf{x} \right) \frac{-1822.0 + \mathbf{T}^3}{255.2 + \mathbf{T}^2} \right] \tag{12}$$

The main electrophysical parameters of Hg1-xCdxTe photodiodes adopted in the calculation of their current-voltage characteristics are listed in Table 2.

Iin, nÀ ,

**3.3 Analysis of LWIR thermography systems based on Hg1-xCdxTe photodiodes** 

of their current-voltage characteristics are listed in Table 2.

1822.0 T E x,T 0.302 x 1.93 x 0.81 0.832x 5.32 10 1 2x

The main electrophysical parameters of Hg1-xCdxTe photodiodes adopted in the calculation

performance characteristics of multi-element IR FPAs; а – histogram of Iin, b – histogram of D\*. Parameter values adopted in the calculations: σ(RPD) = 3·106 Ohm, σ(I0) = 1·10-10 A,

The simple photodiode model (10) proved to be insufficient for adequately predicting the current-voltage characteristics of Hg1-xCdxTe photodiodes in the spectral region 8 to 14 µm. In modeling current-voltage characteristics of Hg1-xCdxTe photodiodes, all main mechanisms of charge transport in *p-n* junctions must be taken into account, including the background radiation current ηКIP, the diffusion current, the thermal generation/recombination current, the tunneling current through localized levels within the bandgap of the depleted region of the *p-n* junction, the interband tunneling current (Anderson,1981), the Shockley-Reed-Hall current through traps in quasi-neutral *n* and *p* regions, the Auger current, and the radiative generation/recombination current in the *p-n* junction and in the quasi-neutral regions (Anderson, 1981; Rogalski, 2000). All the currents listed above are independent currents to be taken into account additively, except for the thermal generation-recombination and trap-assisted tunneling through traps of the Shockley-Reed-Hall type in the depleted region of the *р-n* junction, since the rates of the latter processes is determined by the trap occupation. The two latter mechanisms were modeled in the approximation of balance equations for charge carriers at trap levels (Anderson, 1981; Gumenjuk-Sichevska, 1999; Sizov, 2006; Yoshino,1999). We assumed the presence of localized donor-type centers in the bandgap with energy Et=0.6-0.7Eg over the valence-band edge (Krishnamurthy, 2006). In calculating the band-gap energy of the material as a function of stoichiometric composition and temperature, we used the

Fig. 12. Influence of the spread of photoelectric parameters of photodiodes on the

, Iin, A 0

b

50

100

150

200

250

0 1x10<sup>11</sup> 2x1011 3x10<sup>11</sup>

D\*, cmHz1/2W-1

3

(12)

2

255.2 T

4

0 4 8 12 16 20

I0=5·10-10 A, RPD =3·107 Ohm, IP =1·10-8 A, ŋК =0.8, VG =1.225 V.

0

expression (Rogalski, 2000).

a

50

100

150

200

250

By way of example, Figure 13 shows the current-voltage characteristics of Hg1-xCdxTe photodiodes for different long-wave spectral-response cutoffs, and Figure 14 shows the calculated dependences of RPD АPD on the cutoff wavelength of the spectral response of the photodiodes.

As it was shown in (Gumenjuk-Sichevska, 1999; Sizov, 2006), the photodiode model that was used in the present calculations permits calculation of the current-voltage characteristics as a function of the stoichiometric composition of Hg1-xCdxTe, temperature, and electrophysical parameters of substrate material, the obtained dependences being consistent with those reported in the literature (Rogalski, 2000; Yoshino,1999).

Figure 15 shows the calculated dependences of the maximum values of D\*(VG) of a photodetector channel based on the system 'HgCdTe photodiode – direct injection readout circuit' on the cutoff wavelength λ2. In calculating the background radiation flux reaching the photodiodes in the spectral region from λ1 to λ2 (λ1 =4 μm and λ2 is the cutoff wavelength), for the blackbody temperature a value 300 K was adopted; the aperture angle was assumed defined by the relative aperture of the optical system F/f = 0.5, and for the transmission of the optical system, a value 0.9 was adopted.


Table 2. Electrophysical parameters of Hg1-xCdxTe photodiodes.

Mathematical Modeling of Multi-Element Infrared Focal Plane

0,0

5,0x10<sup>10</sup>

photodiode 1/f-noise level (Q4 in Fig. 16b, c).

threshold voltages become more stringent.

electrons.

1,0x10<sup>11</sup>

1,5x10<sup>11</sup>

2,0x10<sup>11</sup>

D\*, cm Hz1/2

W-1

2,5x10<sup>11</sup>

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 267

7 8 9 10 11 12 13 14 15

Fig. 15. Calculated curves D\*(λ2) for a photodetector channel based on the system 'Hg1 xCdxTe photodiode-direct injection readout'; 1, 2 – photodiode temperature 77 K, 3, 4 – photodiode temperature 60 К. Curves 1, 3 and 2, 4 were calculated for parameter values indicated in Table 1 as cases 1 and 2, respectively, for direct injection charge capacity

Figure 16 shows the curves D\*(VG) and the dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG) and Q5(VG) calculated for λ2 = 10, 12, and 13 µm. The calculations were performed for

The detectivity of the photodetector channel (Fig. 16а) at λ2 = 10 µm is close to BLIP detectivity D\*; this detectivity is limited by the photodiode current noise Q3 and at voltages VG >1.2 V it is almost independent of VG. For λ2 = 12 µm (Fig. 16b) the maximum detectivity of the photodetector channel is D\* ≈ 8.34·1010 cm·Hz1/2·Wt-1 (BLIP detectivity is D\*= 1.77·1011 cm·Hz1/2·Wt-1). For λ2 = 13 µm (Fig. 16c) the maximum detectivity of the photodetector channel is D\* ≈ 4.7·1010 cm·Hz1/2·Wt-1 (BLIP detectivity is D\*= 1.74·1011 cm·Hz1/2·Wt-1). The degradation of detectivity at voltages VG > 1.225 V is primarily defined by the increase in the

An increase of the direct injection storage capacitance and the related increase of the accumulation time both lead to a greater contribution made by the 1/f-noise. Figure 17 shows the curves D\*(VG) and the dependences Q1(VG), Q2(VG), Q3(VG), and Q4(VG) similar to those shown in Fig. 16b yet calculated for direct injection storage capacity Qin = 2·108

A comparison between the dependences in Figs. 16 and 17 shows that the maximal detectivity rather weakly depends on the storage capacity of the readout circuit. Yet, because of the increased contribution due to the 1/f noise of photodiodes, with increasing the input-gate voltage the dependence of D\* on VG becomes more clearly manifested. That is why with increasing the storage capacity of direct injection readouts and, hence, with increasing the accumulation time, the requirements imposed on the standard deviation of

Qin=5·107 electrons. Curve 5 – calculated dependences for BLIP detectivity.

photodiodes with parameter values indicated in Table 2 as case 1.

<sup>1</sup> <sup>2</sup> <sup>3</sup>

, m

4

5

Fig. 13. Calculated current-voltage characteristics of Hg1-xCdxTe photodiodes at temperatures 77 K (a) and 60 K (b). For curves 1 – 4 (1\* - 4\*) the cutoff wavelength is respectively λ2 = 11, 12, 13, and 14 μm. The electrophysical parameters of photodiodes are listed in Table 2; case 1 – curves 1-4, case 2 - curves 1\* - 4\*.

The detectivity of the photodetector channel at temperature T=77 K for photodiodes with electrophysical-parameter values indicated as case 1 in Table 2 (curve 1 in Fig. 15) at λ<sup>2</sup> > 10 µm becomes lower than Background Limited Performance (BLIP) detectivity (curve 3). On cooling the photodiodes to temperature 60 K, the detectivity of photodetector channel approaches BLIP detectivity at wavelengths below ~ 13 μm (curve 3). For photodiodes with electrophysical-parameter values indicated as case 2 in Table 2.2, due to a larger dynamic resistance, the long-wave spectral-response cutoff at which the FPA detectivity attains its maximum value shifts towards longer wavelengths (see curves 2 and 4 in Fig. 15).

Fig. 14. Calculated dependences of RPD АPD on the long-wave cutoff of photodiodes; curves 1 and 2 refer to temperature T=77 K, and curves 3 and 4, to temperature T=60 K. Electrophysical-parameter values: curves 1 and 3 – case 1, curves 2 and 4 – case 2.

IPD, A

VPD, V

Fig. 13. Calculated current-voltage characteristics of Hg1-xCdxTe photodiodes at temperatures 77 K (a) and 60 K (b). For curves 1 – 4 (1\* - 4\*) the cutoff wavelength is respectively λ2 = 11, 12, 13, and 14 μm. The electrophysical parameters of photodiodes are

maximum value shifts towards longer wavelengths (see curves 2 and 4 in Fig. 15).

1

1 and 2 refer to temperature T=77 K, and curves 3 and 4, to temperature T=60 K. Electrophysical-parameter values: curves 1 and 3 – case 1, curves 2 and 4 – case 2.

,

The detectivity of the photodetector channel at temperature T=77 K for photodiodes with electrophysical-parameter values indicated as case 1 in Table 2 (curve 1 in Fig. 15) at λ<sup>2</sup> > 10 µm becomes lower than Background Limited Performance (BLIP) detectivity (curve 3). On cooling the photodiodes to temperature 60 K, the detectivity of photodetector channel approaches BLIP detectivity at wavelengths below ~ 13 μm (curve 3). For photodiodes with electrophysical-parameter values indicated as case 2 in Table 2.2, due to a larger dynamic resistance, the long-wave spectral-response cutoff at which the FPA detectivity attains its

8 9 10 11 12 13 14

Fig. 14. Calculated dependences of RPD АPD on the long-wave cutoff of photodiodes; curves

2

4

m

3

b





5,0x10-9

0,0

2

1

1\*

2\*


3

3\*

VPD, V

4

4\*


listed in Table 2; case 1 – curves 1-4, case 2 - curves 1\* - 4\*.

10-1

100

101

102

103

104

RPD

APD, Omcm2

105

106

4\* 4





2,0x10-7

IPD, A

a

0,0

2

3 3\* 1

1\* 2\*

Fig. 15. Calculated curves D\*(λ2) for a photodetector channel based on the system 'Hg1 xCdxTe photodiode-direct injection readout'; 1, 2 – photodiode temperature 77 K, 3, 4 – photodiode temperature 60 К. Curves 1, 3 and 2, 4 were calculated for parameter values indicated in Table 1 as cases 1 and 2, respectively, for direct injection charge capacity Qin=5·107 electrons. Curve 5 – calculated dependences for BLIP detectivity.

Figure 16 shows the curves D\*(VG) and the dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG) and Q5(VG) calculated for λ2 = 10, 12, and 13 µm. The calculations were performed for photodiodes with parameter values indicated in Table 2 as case 1.

The detectivity of the photodetector channel (Fig. 16а) at λ2 = 10 µm is close to BLIP detectivity D\*; this detectivity is limited by the photodiode current noise Q3 and at voltages VG >1.2 V it is almost independent of VG. For λ2 = 12 µm (Fig. 16b) the maximum detectivity of the photodetector channel is D\* ≈ 8.34·1010 cm·Hz1/2·Wt-1 (BLIP detectivity is D\*= 1.77·1011 cm·Hz1/2·Wt-1). For λ2 = 13 µm (Fig. 16c) the maximum detectivity of the photodetector channel is D\* ≈ 4.7·1010 cm·Hz1/2·Wt-1 (BLIP detectivity is D\*= 1.74·1011 cm·Hz1/2·Wt-1). The degradation of detectivity at voltages VG > 1.225 V is primarily defined by the increase in the photodiode 1/f-noise level (Q4 in Fig. 16b, c).

An increase of the direct injection storage capacitance and the related increase of the accumulation time both lead to a greater contribution made by the 1/f-noise. Figure 17 shows the curves D\*(VG) and the dependences Q1(VG), Q2(VG), Q3(VG), and Q4(VG) similar to those shown in Fig. 16b yet calculated for direct injection storage capacity Qin = 2·108 electrons.

A comparison between the dependences in Figs. 16 and 17 shows that the maximal detectivity rather weakly depends on the storage capacity of the readout circuit. Yet, because of the increased contribution due to the 1/f noise of photodiodes, with increasing the input-gate voltage the dependence of D\* on VG becomes more clearly manifested. That is why with increasing the storage capacity of direct injection readouts and, hence, with increasing the accumulation time, the requirements imposed on the standard deviation of threshold voltages become more stringent.

Mathematical Modeling of Multi-Element Infrared Focal Plane

value can be calculated by the formula (Taubkin, 1993).

1/2T NETD

**IR FPAs with direct injection readouts** 

is the atmospheric transmission.

circuit and detectivity are calculated.

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 269

**4. Temperature resolution of thermography systems based on multi-element** 

For thermography systems based on multi-element photodetectors, an important figure of merit is the noise-equivalent temperature difference (NETD) (Rogalski, 2000). The NEDT

> <sup>2</sup> <sup>2</sup> 1/2 \* 2 PD Op 1 2

Here, dR/dλ is the blackbody spectral luminous exitance, F/f is the relative aperture of the optical system, f is the focal distance, GOp is the optical transmission of the system, and S(λ)

An analysis of NETD starts with calculating the current-voltage characteristics of photodiodes and the background radiation level. Then, currents integrated in the readout

For an "ideal" thermography system (curve 1), in which the integration time is equal to the frame time, Тin =20 ms in the present calculations, and the detectivity of photodetector channels is close to BLIP detectivity, we obtain the well-known dependences NETD(λ) that show that the temperature resolution improves with increasing the wavelength λ2. The dependencies in Fig. 18 reveal the causes of the losses in temperature resolution owing to insufficiently high dynamic resistance of photodiodes or limited storage capacitance of silicon direct injection readout circuits of thermography systems in comparison with the theoretical limit. It can be inferred from the dependences Dλ\*(λ2) (Fig. 15) and NETD(λ2) (Fig. 18) that the temperature resolution of thermography systems based on the photodiode – direct injection readout system attains a maximum value at some wavelength λ2 and, then, decreases (see curves 2 and 3 in Fig. 18). The position of NEDT maximum is defined by electrophysical parameters of HgCdTe photodiodes, by the optical transmission of the system, and by the photodiode temperature. Yet, as it follows from the calculated curves of NETD in Fig. 18 (curves 4-7) the main factor limiting the temperature resolution of the

With increasing the cutoff wavelength λ2, the intensity of background radiation coming to photodetector also increases and, as a result, the integration time ТН decreases, with simultaneous shift of the maximum temperature resolution of the thermography system towards short wavelengths in comparison with curves 2 and 3 in Fig. 18. The charge capacity of readout circuits Qin = 5·107 electrons presents a maximal value reported in literature for multi-element matrix IR FPAs (at 30-µm photocell pitch) (Rogalski, 2000). At such values of integration capacitance of readout circuits the temperature resolution of thermography systems (curve 5) already with λ2 = 8 µm is more than one order of magnitude and with λ2 = 14 µm, two orders of magnitude lower than that of the "ideal" thermography system (curve 1). The storage capacitance can be increased through implementing a new architecture of readout circuits (curves 6 and 7 in Fig. 18) (Lee, 2010). A

1

Figure 18 shows the calculated curves of NETD versus the cutoff wavelength λ2.

thermography systems is the storage capacitance of silicon readout circuits.

<sup>F</sup> d dR /d A G S D , ,T d 4f dT

in

(13)

Fig. 16. Curves D\*(VG) (right axis) and dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG), and Q5(VG) (left axis). Temperature 77 K, direct injection charge capacity Qin=5·107 electrons; a – λ2 =10 µm, b – λ2 = 12 µm, c – λ2 = 13 µm.

Fig. 17. Calculated curves D\*(VG) (right axis) and dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG), and Q5(VG) (left axis). The photodiode temperature is 77 K; the storage capacity of direct injection readout is Qin=2·108 electrons; λ2 = 12 µm. The values of electrophysical parameters of photodiodes are given in Table 2.2, case 1.

D\* cm Hz1/2

Q, number electrons

0,0 3,0x10<sup>3</sup> 6,0x10<sup>3</sup> 9,0x10<sup>3</sup> 1,2x10<sup>4</sup> 1,5x10<sup>4</sup>

Q4

Q3 Q1 Q2

Q2

b

1,19 1,20 1,21 1,22 1,23 1,24 1,25

1,19 1,20 1,21 1,22 1,23 1,24 1,25

Fig. 17. Calculated curves D\*(VG) (right axis) and dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG), and Q5(VG) (left axis). The photodiode temperature is 77 K; the storage capacity of direct injection readout is Qin=2·108 electrons; λ2 = 12 µm. The values of electrophysical

Q5

Q2

D\*

Q4

Q3

Q2

Fig. 16. Curves D\*(VG) (right axis) and dependences Q1(VG), Q2(VG), Q3(VG), Q4(VG), and Q5(VG) (left axis). Temperature 77 K, direct injection charge capacity Qin=5·107 electrons; a –

Q5

1,19 1,20 1,21 1,22 1,23 1,24 1,25

D\*, cmHz1/2

W-1

Q3 Q4

Q5 D\*

Q1

0

VG, V

1x10<sup>10</sup>

VG, V

0

1x1010 2x1010 3x1010 4x1010 5x1010 6x1010 7x1010 8x1010

2x10<sup>10</sup>

3x10<sup>10</sup>

4x10<sup>10</sup>

5x10<sup>10</sup>

0,0

VG, V

D\*, cmHz1/2

W-1

2,0x1010 4,0x1010 6,0x1010 8,0x1010 1,0x1011

D\*, cmHz1/2

W-1

W-1

0,0 4,0x10<sup>10</sup> 8,0x10<sup>10</sup> 1,2x10<sup>11</sup> 1,6x10<sup>11</sup> 2,0x10<sup>11</sup>

D\*

VG, V

1,19 1,20 1,21 1,22 1,23 1,24 1,25

0,0

5,0x10<sup>3</sup>

1,0x10<sup>4</sup>

1,5x10<sup>4</sup>

2,0x10<sup>4</sup>

Q, number electrons

c

λ2 =10 µm, b – λ2 = 12 µm, c – λ2 = 13 µm.

0

parameters of photodiodes are given in Table 2.2, case 1.

1x104

2x104

3x104

4x104

Q, number electrons

5x104

6x104

2,5x10<sup>4</sup>

Q4

Q5 D\*

0,0 2,0x10<sup>3</sup> 4,0x10<sup>3</sup> 6,0x10<sup>3</sup> 8,0x10<sup>3</sup> 1,0x10<sup>4</sup>

Q2 Q1

Q3

à

Q, number electrons

### **4. Temperature resolution of thermography systems based on multi-element IR FPAs with direct injection readouts**

For thermography systems based on multi-element photodetectors, an important figure of merit is the noise-equivalent temperature difference (NETD) (Rogalski, 2000). The NEDT value can be calculated by the formula (Taubkin, 1993).

$$\frac{\text{NETD} = \frac{\sqrt{1/2\text{T}\_{\text{in}}}}{\text{F}^2} \text{(A}\_{\text{PD}}\text{)}^{1/2}\text{G}\_{\text{Op}}\int\_{\lambda1}^{\lambda2} \frac{\text{d}\left(\text{dR} \,/\text{ d}\lambda\right)}{\text{dT}}\text{S}(\lambda)\text{D}\_{\lambda}^{\*}\left(\lambda\_1, \lambda\_2, \text{T}\_{\text{L}}\right)\text{d}\lambda} \tag{13}$$

Here, dR/dλ is the blackbody spectral luminous exitance, F/f is the relative aperture of the optical system, f is the focal distance, GOp is the optical transmission of the system, and S(λ) is the atmospheric transmission.

An analysis of NETD starts with calculating the current-voltage characteristics of photodiodes and the background radiation level. Then, currents integrated in the readout circuit and detectivity are calculated.

Figure 18 shows the calculated curves of NETD versus the cutoff wavelength λ2.

For an "ideal" thermography system (curve 1), in which the integration time is equal to the frame time, Тin =20 ms in the present calculations, and the detectivity of photodetector channels is close to BLIP detectivity, we obtain the well-known dependences NETD(λ) that show that the temperature resolution improves with increasing the wavelength λ2. The dependencies in Fig. 18 reveal the causes of the losses in temperature resolution owing to insufficiently high dynamic resistance of photodiodes or limited storage capacitance of silicon direct injection readout circuits of thermography systems in comparison with the theoretical limit. It can be inferred from the dependences Dλ\*(λ2) (Fig. 15) and NETD(λ2) (Fig. 18) that the temperature resolution of thermography systems based on the photodiode – direct injection readout system attains a maximum value at some wavelength λ2 and, then, decreases (see curves 2 and 3 in Fig. 18). The position of NEDT maximum is defined by electrophysical parameters of HgCdTe photodiodes, by the optical transmission of the system, and by the photodiode temperature. Yet, as it follows from the calculated curves of NETD in Fig. 18 (curves 4-7) the main factor limiting the temperature resolution of the thermography systems is the storage capacitance of silicon readout circuits.

With increasing the cutoff wavelength λ2, the intensity of background radiation coming to photodetector also increases and, as a result, the integration time ТН decreases, with simultaneous shift of the maximum temperature resolution of the thermography system towards short wavelengths in comparison with curves 2 and 3 in Fig. 18. The charge capacity of readout circuits Qin = 5·107 electrons presents a maximal value reported in literature for multi-element matrix IR FPAs (at 30-µm photocell pitch) (Rogalski, 2000). At such values of integration capacitance of readout circuits the temperature resolution of thermography systems (curve 5) already with λ2 = 8 µm is more than one order of magnitude and with λ2 = 14 µm, two orders of magnitude lower than that of the "ideal" thermography system (curve 1). The storage capacitance can be increased through implementing a new architecture of readout circuits (curves 6 and 7 in Fig. 18) (Lee, 2010). A

Mathematical Modeling of Multi-Element Infrared Focal Plane

realizations in the calculations was 200).

5,0x1010 1,0x10<sup>11</sup> 1,5x1011 2,0x10<sup>11</sup>

0

c

level (Rogalski, 2000).

10

20

30

40

50

D\*, cmHz1/2W-1

0

b

0,000 0,005 0,010 0,015 0,020 0,025 0,030

Fig. 19. Calculated histograms of performance characteristics of multi-element IR FPAs; a detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth)=5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate - σ(х) = 0.1%. Average stoichiometric composition - х=0.2167, Т=77 К, storage capacitance of readout circuit Qin=5·107 electrons.

The photocurrent level was calculated with allowance for the long-wave cutoff of photosensitivity individually for each photodiode. The integration time was defined by the magnitude of the storage capacitance of the readout circuit and by the maximum level of photodiode current over 200 realizations. At Т=77 K, the mean stoichiometric composition х=0.2167 ensures a long-wave photosensitivity cutoff λ2=11 µm at standard dispersion of stoichiometric composition σ(х) = 0.1%, typical of the present-day state-of-the-art technology

For a single channel with с λ2=11 µm, at the same values of electrophysical and design parameters of photodiodes and readout circuits the maximum sensitivity D\* is 1.4·1011

10

20

30

40

50

0,0 2,0x10-8 4,0x10-8 6,0x10-8 8,0x10-8 1,0x10-7

NETD, K

Iin, A

0

a

10

20

30

40

50

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 271

Figure 19 shows calculated histograms of D\*, Iin and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters given in Table 2, case 1. In the calculations, it was assumed that the dispersion of threshold voltages and the nonuniformity of the stoichiometric composition of substrate material obey normal distribution laws with parameter values indicated in the caption to the figure (the total number of

cardinal solution here is implementation of ADC in each pixel of the readout circuit (Zhou, 1996; Martijn, 2000; Fowler, 2000). In (Bisotto, 2010), each pixel was provided with a 15-bit ADC, and the effective storage capacitance was in excess of 3·109 electrons. This has enabled an increase in integration time up to the frame time and, in this way, allowed reaching a NETD of LWIR thermography systems based on multi-element IR FPAs amounting to 2 mK, the latter value being close to the maximum theoretically possible NETD.

Fig. 18. Calculated curves NETD(λ2). In the calculations, the values Fp/f= 0.5 and λ1 =5 µm were adopted. Curve 1 was calculated for D\* <sup>λ</sup> = D\* <sup>λ</sup>BLIP and Тн=20 ms; curves 2 and 3 were calculated with allowance for the dependence of D\* <sup>λ</sup> on the long-wave spectral-response cutoff, integration time Тin=20 ms; curves 4-7 were calculated for conditions with limited storage capacitance of direct injection readouts (curves 4 and 5 - Qin = 5·107 electrons, curves 6 and 7 - Qin = 2·108 electrons); curves 2, 4, and 6 – photodiode temperature 60 K, curves 3, 5, and 7 – photodiode temperature 77 К. The electrophysical parameters of photodiodes are given in Table 2, case 1.

The dependences D\*(VG) and NETD(λ2) calculated for different values of the storage capacity of readout circuits on the input-gate voltage VG and shown in Figs. 15-18 define the ultimate performance characteristics of thermography systems in the spectral range 8-14 µm. In fact, these dependences are will also be observed for a single-element FPA based on the system 'Hg1-xCdxTe photodiode – direct injection readout circuit' since they were plotted using just the maximum values of D\*. For multi-element IR FPAs, the non-uniformity of input-FET threshold voltages leads to an increased fixed-pattern noise level and, for some part of photodetector channels, to a considerable reduction of D\* and NETD in comparison with the dependences shown in Figs. 15 and 18. An additional factor causing an increase in the fixed-pattern noise, a decrease of detectivity, and worsened temperature resolution of multi-element IR FPAs is non-uniformity of the stoichiometric composition of the Hg1 xCdxTe substrate.

For multi-element IR FPAs with λ2 ≤ 10 µm, detectivity rather weakly depends on the gate voltage VG and, hence, on the dispersion of threshold voltages (see Fig.16a). The fixedpattern noise level and the spread of NETD values of multi-channel IR FPAs are primarily defined by the scatter of the long-wave photosensitivity cutoff owing to non-uniform stoichiometric composition of the Hg1-xCdxTe substrate.

cardinal solution here is implementation of ADC in each pixel of the readout circuit (Zhou, 1996; Martijn, 2000; Fowler, 2000). In (Bisotto, 2010), each pixel was provided with a 15-bit ADC, and the effective storage capacitance was in excess of 3·109 electrons. This has enabled an increase in integration time up to the frame time and, in this way, allowed reaching a NETD of LWIR thermography systems based on multi-element IR FPAs amounting to 2 mK,

8 9 10 11 12 13 14 15

Fig. 18. Calculated curves NETD(λ2). In the calculations, the values Fp/f= 0.5 and λ1 =5 µm

cutoff, integration time Тin=20 ms; curves 4-7 were calculated for conditions with limited storage capacitance of direct injection readouts (curves 4 and 5 - Qin = 5·107 electrons, curves 6 and 7 - Qin = 2·108 electrons); curves 2, 4, and 6 – photodiode temperature 60 K, curves 3, 5, and 7 – photodiode temperature 77 К. The electrophysical parameters of photodiodes are

<sup>λ</sup> = D\*

The dependences D\*(VG) and NETD(λ2) calculated for different values of the storage capacity of readout circuits on the input-gate voltage VG and shown in Figs. 15-18 define the ultimate performance characteristics of thermography systems in the spectral range 8-14 µm. In fact, these dependences are will also be observed for a single-element FPA based on the system 'Hg1-xCdxTe photodiode – direct injection readout circuit' since they were plotted using just the maximum values of D\*. For multi-element IR FPAs, the non-uniformity of input-FET threshold voltages leads to an increased fixed-pattern noise level and, for some part of photodetector channels, to a considerable reduction of D\* and NETD in comparison with the dependences shown in Figs. 15 and 18. An additional factor causing an increase in the fixed-pattern noise, a decrease of detectivity, and worsened temperature resolution of multi-element IR FPAs is non-uniformity of the stoichiometric composition of the Hg1-

For multi-element IR FPAs with λ2 ≤ 10 µm, detectivity rather weakly depends on the gate voltage VG and, hence, on the dispersion of threshold voltages (see Fig.16a). The fixedpattern noise level and the spread of NETD values of multi-channel IR FPAs are primarily defined by the scatter of the long-wave photosensitivity cutoff owing to non-uniform

5

, m

<sup>λ</sup>BLIP and Тн=20 ms; curves 2 and 3 were

<sup>λ</sup> on the long-wave spectral-response

2

3

7

1

6

4

the latter value being close to the maximum theoretically possible NETD.

10-3

were adopted. Curve 1 was calculated for D\*

given in Table 2, case 1.

xCdxTe substrate.

calculated with allowance for the dependence of D\*

stoichiometric composition of the Hg1-xCdxTe substrate.

10-2

10-1

NETD, K

Figure 19 shows calculated histograms of D\*, Iin and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters given in Table 2, case 1. In the calculations, it was assumed that the dispersion of threshold voltages and the nonuniformity of the stoichiometric composition of substrate material obey normal distribution laws with parameter values indicated in the caption to the figure (the total number of realizations in the calculations was 200).

Fig. 19. Calculated histograms of performance characteristics of multi-element IR FPAs; a detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth)=5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate - σ(х) = 0.1%. Average stoichiometric composition - х=0.2167, Т=77 К, storage capacitance of readout circuit Qin=5·107 electrons.

The photocurrent level was calculated with allowance for the long-wave cutoff of photosensitivity individually for each photodiode. The integration time was defined by the magnitude of the storage capacitance of the readout circuit and by the maximum level of photodiode current over 200 realizations. At Т=77 K, the mean stoichiometric composition х=0.2167 ensures a long-wave photosensitivity cutoff λ2=11 µm at standard dispersion of stoichiometric composition σ(х) = 0.1%, typical of the present-day state-of-the-art technology level (Rogalski, 2000).

For a single channel with с λ2=11 µm, at the same values of electrophysical and design parameters of photodiodes and readout circuits the maximum sensitivity D\* is 1.4·1011

Mathematical Modeling of Multi-Element Infrared Focal Plane

2,0x1010 4,0x1010 6,0x1010 8,0x1010 1,0x1011

0

c

wavelength interval from 12.6 to 14 µm.

10

20

30

40

50

60

0

a

10

20

30

40

50

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 273

D\*, cmHz1/2W-1 0,0 5,0x10-8 1,0x10-7 1,5x10-7 2,0x10-7

NETD, K

Iin, A

b

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07

Fig. 20. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.1%. Average stoichiometric composition х=0.2105, Т=77 K. Storage capacitance of readout circuit - Qin=5·107 electrons.

Figure 22 shows calculated histograms of currents detectivities D\*, Iin and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters indicated in Table 2 as case 1. The parameter values used in the calculations were the same as those in

At mean stoichiometric composition х=0.2105, 0.1% dispersion of stoichiometric composition and temperature Т=60 K, the long cutoff wavelength will fall into the

Figs. 19 and 20, and the photodiode temperature was assumed to be 60 K.

cm·Hz1/2·W-1, DBLIP\* =1.8·1011 cm·Hz1/2·W-1, and NETD=10.8 mK (see curve 5 in Fig. 18). The dispersion of the stoichiometric composition of substrate σ(х) = 0.1% results in a spread of long-wave cutoff wavelengths in the interval from 11 to 11.8 µm, this being the main factor causing NETD degradation.

With increasing the long-wave cutoff wavelength λ2, requirements to the uniformity of threshold voltages under the input gates of direct injection readout circuits and requirements to the uniformity of the stoichiometric composition of substrate both become more stringent. Figure 20 shows calculated histograms of currents Iin, detectivities D\* and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters indicated in Table 2 as case 1. At mean stoichiometric composition х=0.21055, σ(х) = 0.1% and Т=77 K the non-uniformity of the stoichiometric composition of substrate material results in a spread of long-wave photosensitivity cutoffs in the interval from 12 to 13.2 µm. The input-gate voltage value VG=1.23 V appears to be optimal for the radiation environment conditions and values of electrophysical and design parameters of photodiodes and readout circuits adopted in the calculations.

Note that for a single channel at λ2=12 µm the maximum detectivity D\* is 8.4·1010 cm·Hz1/2·W-1 and NETD = 0.022 K (Fig. 18, curve 5). Taking the dispersion of threshold voltages with σ(Vth) = 5 mV and substrate stoichiometric composition with σ(х) = 0.1% into account results to two-three-fold degradation of NETD in a considerable fraction of photodetector channels.

Figure 21 shows calculated histograms of currents Iin, detectivities D\* and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric-parameter values adopted in Fig. 20 yet under more stringent conditions in terms of the dispersion of FET threshold voltages and stoichiometric composition of Hg1-xCdxTe substrate, σ(Vth) = 2 mV and σ(х) = 0.03%. A comparison between the histograms in Figs. 20 and 21 shows that more stringent requirements imposed on the uniformity of threshold voltages and stoichiometric composition of substrate allow a substantial reduction of the fixed-pattern noise and a considerable improvement of NEDT values of thermography systems.

With the adopted values of electrophysical parameters of Si readout circuits and Hg1-xCdxTe photodiodes (Tables 1 and 2), thermography systems based on multi-element IR FPAs intended for operation at liquid-nitrogen temperature in the spectral range up to 13-14 µm with maximum possible NETD values (Fig. 18, curve 5) can be implemented at an acceptable level of fixed-pattern noise by:


However, requirements to σ(Vth) and σ(х) more stringent than the requirements that were adopted in calculating data in Fig. 21 presently cannot be met by silicon CMOS technology and synthesis processes of epitaxial Hg1-xCdxTe layers (Phillips, 2002).

Achieving photosensitivity in the spectral region 12-14 µm necessitates cooling the hybrid IR FPA assembly to a temperature below liquid-nitrogen temperature, see curves 4 and 6 in Fig. 19.

cm·Hz1/2·W-1, DBLIP\* =1.8·1011 cm·Hz1/2·W-1, and NETD=10.8 mK (see curve 5 in Fig. 18). The dispersion of the stoichiometric composition of substrate σ(х) = 0.1% results in a spread of long-wave cutoff wavelengths in the interval from 11 to 11.8 µm, this being the main

With increasing the long-wave cutoff wavelength λ2, requirements to the uniformity of threshold voltages under the input gates of direct injection readout circuits and requirements to the uniformity of the stoichiometric composition of substrate both become more stringent. Figure 20 shows calculated histograms of currents Iin, detectivities D\* and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters indicated in Table 2 as case 1. At mean stoichiometric composition х=0.21055, σ(х) = 0.1% and Т=77 K the non-uniformity of the stoichiometric composition of substrate material results in a spread of long-wave photosensitivity cutoffs in the interval from 12 to 13.2 µm. The input-gate voltage value VG=1.23 V appears to be optimal for the radiation environment conditions and values of electrophysical and design parameters of

Note that for a single channel at λ2=12 µm the maximum detectivity D\* is 8.4·1010 cm·Hz1/2·W-1 and NETD = 0.022 K (Fig. 18, curve 5). Taking the dispersion of threshold voltages with σ(Vth) = 5 mV and substrate stoichiometric composition with σ(х) = 0.1% into account results to two-three-fold degradation of NETD in a considerable fraction of

Figure 21 shows calculated histograms of currents Iin, detectivities D\* and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric-parameter values adopted in Fig. 20 yet under more stringent conditions in terms of the dispersion of FET threshold voltages and stoichiometric composition of Hg1-xCdxTe substrate, σ(Vth) = 2 mV and σ(х) = 0.03%. A comparison between the histograms in Figs. 20 and 21 shows that more stringent requirements imposed on the uniformity of threshold voltages and stoichiometric composition of substrate allow a substantial reduction of the fixed-pattern noise and a

With the adopted values of electrophysical parameters of Si readout circuits and Hg1-xCdxTe photodiodes (Tables 1 and 2), thermography systems based on multi-element IR FPAs intended for operation at liquid-nitrogen temperature in the spectral range up to 13-14 µm with maximum possible NETD values (Fig. 18, curve 5) can be implemented at an acceptable


However, requirements to σ(Vth) and σ(х) more stringent than the requirements that were adopted in calculating data in Fig. 21 presently cannot be met by silicon CMOS technology

Achieving photosensitivity in the spectral region 12-14 µm necessitates cooling the hybrid IR FPA assembly to a temperature below liquid-nitrogen temperature, see curves 4 and 6 in

factor causing NETD degradation.

photodetector channels.

level of fixed-pattern noise by:

Fig. 19.

photodiodes and readout circuits adopted in the calculations.

considerable improvement of NEDT values of thermography systems.


and synthesis processes of epitaxial Hg1-xCdxTe layers (Phillips, 2002).

voltages and stoichiometric-composition uniformity of the substrate.

Fig. 20. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.1%. Average stoichiometric composition х=0.2105, Т=77 K. Storage capacitance of readout circuit - Qin=5·107 electrons.

Figure 22 shows calculated histograms of currents detectivities D\*, Iin and NETD values of thermography systems for Hg1-xCdxTe photodiodes with photoelectric parameters indicated in Table 2 as case 1. The parameter values used in the calculations were the same as those in Figs. 19 and 20, and the photodiode temperature was assumed to be 60 K.

At mean stoichiometric composition х=0.2105, 0.1% dispersion of stoichiometric composition and temperature Т=60 K, the long cutoff wavelength will fall into the wavelength interval from 12.6 to 14 µm.

Mathematical Modeling of Multi-Element Infrared Focal Plane

2,0x10<sup>10</sup> 4,0x10<sup>10</sup> 6,0x10<sup>10</sup> 8,0x10<sup>10</sup> 1,0x10<sup>11</sup>

0

c

stoichiometric composition х=0.2105, Т=60 K.

**5. Conclusion** 

results.

10

20

30

40

50

0

a

10

20

30

40

50

Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 275

0

b

0,006 0,007 0,008 0,009 0,010

A mathematical model was developed to analyze the performance characteristics of IR FPAs based on the system 'IR photodiode – direct injection readout circuit'. The proposed mathematical model is based on the solution of the self-consistent problem for the current through photodiode and for the current integrated in the readout circuit, and also on the calculation of the noise charge Q(t) in terms of noise-current spectral density Si(ω). Such an approach allows one to determine the main performance characteristics of thermography systems based on multi-element IR FPAs versus the gate voltage of photodetector channels, to calculate histograms D\*(VG), Iin(VG), and NETD as functions of non-uniformity of photoelectric parameters of photodiodes and silicon readout circuits, and to compare predicted performance characteristics of IR FPAs with experimental

The analysis of performance characteristic of multi-element FPAs can be performed with setting photodiode current-voltage characteristics either in analytical form ("classical"

Fig. 22. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.1%. VG = 1.26 V. Average

10

20

30

40

50

5,0x10-8 6,0x10-8 7,0x10-8 8,0x10-8 9,0x10-8 1,0x10-7

NETD, K

Iin, A

D\*, cmHz1/2W-1

Fig. 21. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 2 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.03%. Average stoichiometric composition х=0.2126, Т=77 K.

As it follows from the calculated dependencies shown in Fig. 22, the cooling of the hybrid assembly down to temperature 60 K will allow implementation of thermography systems with NEDT values close to maximum possible figures, limited only by the value of the storage capacitance of silicon readout circuits at an acceptable level of fixed-pattern noise. The possibility of variation of the cooling temperature of hybrid assembly in the calculations allows formulation of requirements to required accuracy in maintaining the temperature of cooled hybrid assembly, i.e. requirements to be imposed on the cryostat and cooling system.

Mathematical Modeling of Multi-Element Infrared Focal Plane Arrays Based on the System 'Photodiode – Direct-Injection Readout Circuit' 275

Fig. 22. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 5 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.1%. VG = 1.26 V. Average stoichiometric composition х=0.2105, Т=60 K.

### **5. Conclusion**

274 Photodetectors

2,0x10<sup>10</sup> 4,0x10<sup>10</sup> 6,0x10<sup>10</sup> 8,0x10<sup>10</sup> 1,0x10<sup>11</sup>

0

c

composition х=0.2126, Т=77 K.

and cooling system.

10

20

30

40

50

D\*, cmHz1/2W-1

0

0,00 0,01 0,02 0,03 0,04 0,05

Fig. 21. Calculated histograms of performance characteristics of multi-element IR FPAs; a - detectivity D\*, b – currents Iin integrated in the readout circuits, c - NETD. Standard deviation of input-FET threshold voltages - σ(Vth) = 2 mV. Standard deviation of the stoichiometric composition of Hg1-xCdxTe substrate σ(х) = 0.03%. Average stoichiometric

As it follows from the calculated dependencies shown in Fig. 22, the cooling of the hybrid assembly down to temperature 60 K will allow implementation of thermography systems with NEDT values close to maximum possible figures, limited only by the value of the storage capacitance of silicon readout circuits at an acceptable level of fixed-pattern noise. The possibility of variation of the cooling temperature of hybrid assembly in the calculations allows formulation of requirements to required accuracy in maintaining the temperature of cooled hybrid assembly, i.e. requirements to be imposed on the cryostat

b

10

20

30

40

50

60

0,0 5,0x10-8 1,0x10-7 1,5x10-7 2,0x10-7

NETD, K

Iin, A

0

a

10

20

30

40

50

A mathematical model was developed to analyze the performance characteristics of IR FPAs based on the system 'IR photodiode – direct injection readout circuit'. The proposed mathematical model is based on the solution of the self-consistent problem for the current through photodiode and for the current integrated in the readout circuit, and also on the calculation of the noise charge Q(t) in terms of noise-current spectral density Si(ω). Such an approach allows one to determine the main performance characteristics of thermography systems based on multi-element IR FPAs versus the gate voltage of photodetector channels, to calculate histograms D\*(VG), Iin(VG), and NETD as functions of non-uniformity of photoelectric parameters of photodiodes and silicon readout circuits, and to compare predicted performance characteristics of IR FPAs with experimental results.

The analysis of performance characteristic of multi-element FPAs can be performed with setting photodiode current-voltage characteristics either in analytical form ("classical"

Mathematical Modeling of Multi-Element Infrared Focal Plane

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thermography system based on such FPAs.

**6. References** 

pp.9130-9145

pp.160-167

(2010), pp.30-36

(1983)


**Photodetection Systems** 

Zhou Z., Pain B., et al. (1996). On-focal-plane ADC: Recent progress at JPL. *Proc. SPIE, "Infrared readout electronics III*", Vol. 2745, (1996), pp.111-122 **Part 3**  *"Infrared readout electronics III*", Vol. 2745, (1996), pp.111-122 **Part 3** 

**Photodetection Systems** 

278 Photodetectors

Zhou Z., Pain B., et al. (1996). On-focal-plane ADC: Recent progress at JPL. *Proc. SPIE,* 

**13** 

*1France 2Switzerland* 

**Ultrafast Imaging in Standard** 

Around 1822, the French inventor Niépce made the first photographic image by the use of a *camera obscura*. He formed the image passing through the hole on a metal plate with a bitumen coating. After 8 hours of exposure, the bitumen on the illuminated sections of the plate was hardened. By washing the unhardened regions, a print of the observed scene appeared. After Niépce's death in 1833, Daguerre worked on the improvement of the chemical process involving interaction of the plate with light. In 1839 he announced the invention of a new process using silver on a copper plate. This invention reduced the exposure time to 30 minutes and denotes the birth of modern photography. During the following years, improvement on the photographic processes led to increased sensitivity and allowed shorter exposure times. In 1878, Muybridge gave an answer to a popular question at this time: whether all four hooves of a horse are off the ground at the same time during a gallop. By taking the first high-speed sequence of 12 pictures, each picture spaced about 400 ms from the neighbouring one with an exposure time of less than 500 µs. In 1882, George Eastman patented the roll film, which led to the acquisition of the first motion pictures. Four years later, a student of Daguerre, Le Prince, patented a *Method of, and apparatus for, producing animated pictures*. Through its *16 lens receiver*, as he called his camera, and by the use of an Eastman Kodak paper film, Le Prince filmed the first moving picture sequences known as the *Roundhay Garden Scene*, which was shot at 12 frames per second (fps) and lasted less than 2 seconds. Two years later, Edison presented the *Kinetoscope*, a motion picture device capable of acquiring sequences at up to 40 fps. It creates the illusion of movement by conveying a strip of perforated film filled with sequential images over a light source through a mechanical shutter. In 1904, the Austrian physicist Musger patented the *Kinematograph mit Optischem Ausgleich der Bildwanderung* which is capable of recording fast transients and projecting them in slow motion. In acquisition mode, the light is turned off and a rotating mirror, projecting them in slow motion. In acquisition mode, a rotating mirror mechanically coupled to the film shifting mechanism, reflects images onto the film. During the projection, the light is turned on and the same operation is carried out, but at a much slower rate. This high-speed photographing principle was used during the First World War by the German company Ernemann Werke AG to develop the *Zeitlupe*, a 500 fps camera used mainly for ballistic purposes. In 1926, Heape and Grylls constructed *Heape and* 

**1. Introduction** 

**(Bi)CMOS Technology** 

*1University of Strasbourg and CNRS* 

*2ABB Switzerland Ltd.* 

Wilfried Uhring1 and Martin Zlatanski2
