Chapter 5 (HgCd)Te Photoconductive Detectors

SEMICONDUCTORS AND SEMIMETALS. VOL. 18

CHAPTER 5

(HgCd)Te Photoconductive Detectors R. M . Broudy and V . J . Mazurczyk I. INTRODUCTION..

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157 159 159

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11. PERFORMANCE PARAMETERS. . . . . . . . . . . . . . 1. Figures of Merit. . . . . . . . . . . . . . . . . . 111.

IV.

V.

VI.

2. Fundamental Limit of Performance f o r Photoconductive Detectors . . . . . . . . . . . SIMPLEPHOTOCONDUCTIVITY . . . . . . . . . . . 3. Responsivity . . . . . . . . . . . . . . . . . 4.Noise . . . . . . . . . . . . . . . . . . . . 5 . Detectivity . . . . . . . . . . . . . . . . . . 6. Temperature and Background Dependence . . . . PHOTOCONDUCTIVE DEVICEANALYSES . . . . . . . 7. Power Dissipation . . . . . . . . . . . . . . . 8. Surface Recombination . . . . . . . . . . . . 9. I/f Noise in (HgCd)Te Photoconductors . . . . . 10. Transport Effects-Dr$t and Dgfusion . . . . . 11. Summary of Equations f o r the One-Dimensional Approximation with Ohmic Contacts . . . . . . PHOTOCONDUCTIVE DEVICEDESIGN. . . . . . . . 12. Extended Contacts . . . . . . . . . . . . . . 13. Complex Conjigurations-Geometry and Contacts 14. Transverse Field Effects-Accumulation Layers . 15. Transverse Field Effect--“Trapping Photoconductivity” . . . . . . . . . . . . . . TECHNOLOGY OF (HgCd)Te DETECTORS. . . . . . REFERENCES.. . . . . . . . . . . . . . . . . .

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161 162 163 164 167 168 170 170 171 173 175

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I. Introduction

The emergence of practical and widely useful infrared systems has been made possible by advances in infrared sensing components. Because photon detectors approaching theoretical behavior are becoming available, thermal imaging, surveillance, and other military, space, medical, and commercial systems are achieving performance levels that open up new and broader opportunities. The possibility of improved system behavior has in turn supported and encouraged the development of practical infrared detectors. 157

Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0- 12-7521 18-6

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R. M . BROUDY A N D V. J . MAZURCZYK

(HgCd)Te was recognized early as a promising material for infrared detectors due to a number of uniquely suitable physical, thermal, and electronic characteristics (Long and Schmit, 1970; Levinstein, 1970; Dornhaus and Nimtz, 1976). Since the initial reports of its crystal growth in I959 and infrared detection behavior in 1962, (HgCd)Te has become the most widely used infrared detector material today, and photoconductive detectors of (HgCd)Te have become the accepted standard in the 8-12 pm region. Photon detectors may be made for operation either in the photovoltaic (PV) or photoconductive (PC) mode (Long, 1977; Kinch and Borello, 1975), and the semiconductor properties of (HgCd)Te are suitable for both modes. The present generation of (HgCd)Te detectors are PC, possibly because initial effort was devoted almost exclusively to this mode, for which material was available with the desired low carrier concentration. During the decade since the appearance of the first article in this series on (HgCd)Te (Long and Schmit, 1970), extensive research and development on PC devices has produced a high degree of maturity for this technology. Material characteristics have improved in crystallinity, carrier mobility, and lifetime. Producible device fabrication processes have been developed. Sophisticated device designs have been implemented, based on the recognition that photoconductivity in (HgCd)Te is a majoritycarrier phenomenon that is controlled by minority-carrier properties. In all respects, the behavior of PC (HgCd)Te photodetectors has been found to be qualitatively and quantitatively consistent with fundamental semiconductor theory and technology. The actual PC (HgCd)Te detector used in systems is a complex multielement array requiring a high level of sophistication in analysis, design, and processing. In fact, the fabrication of modern PC detector array components requires techniques similar to those applied in integrated circuit technology. The purpose of this chapter is to present an up-to-date description of the theory and principles of PC (HgCd)Te detectors in a form most suitable for design and application. It is our objective to consider all relevant mechanisms affecting performance in actual devices, making use of the maximum available rigor. In the interests of conciseness we will rely on information already accessible in several excellent treatises and papers (Long and Schmit, 1970; Levinstein, 1970; Dornhaus and Nimtz, 1976; Long, 1977; Kinch and Borello, 1975; Kruse et al., 1962; Kingston, 1978; Eisenman et al., 1977; Kruse, 1979). Most of the original references may also be obtained from these works. The discussion must be limited for the following reasons: first, many details are not openly available due to classification and proprietary limitations; and second, a rigorous theory has

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

159

not yet appeared for some of the fundamental features of device behavior. We begin in Part I1 with a summary of well-known performance parameters for evaluation of photodetectors, concentrating on those aspects most useful for 8- 12 pm PC (HgCd)Te. For heuristic purposes, and to aid in correlation with much of the literature, we have chosen to initiate the detector analysis with Part 111, which is limited to simple photoconductivity. In this section only conductivity modulation in the bulk of the material is considered, and none of the effects associated with the drift, diffusion, and thermal properties have been included. The behavior of actual devices, however, is considerably modified by additional phenomena, including minority and majority-carrier transport effects and other behavior related to practical device structures and application requirements. Part IV includes these effects. Part V presents a brief summary of the principles used in modern photoconductor design, whereby special structures for control of minoritycarrier transport may greatly improve performance. We conclude with a summary of the principal aspects of practical device fabrication.

11. Performance Parameters

The following subsections relate material parameters and external variables (i.e., temperature, background radiation, bias, and frequency) to the detector performance. As a background for these discussions, it is useful to present a brief survey of the principles by which photodetectors are evaluated. Although there are several well-recognized works on infrared detectors which treat this subject in detail and provide references to the original literature, some issues still remain to be resolved. Much of the discussion in this section follows Kruse et al. (1962). 1. FIGURES OF MERIT

Widely used figures of merit for describing detector performance are the spectral detectivity D,*(h,f,Af)and the responsivity 9iA(f).They are defined by

D f = D*(X,f,Af) = (%&/VN)(AAf)”*,

% = V,/P,,

(1) (2)

where V , is the rms signal voltage at the amplifier input, VN is the rms noise voltage at the amplifier input measured within the electrical band-

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R . M . BROUDY AND V. J . MAZURCZYK

width Af, and PAis the rms optical power in watts incident on the detector active area, A . The optical power considered is in the spectral region , of 3 are between A and A + AA. The units of D* are cm H Z ” ~ / W and

v/w.

The value of ’?PA must be known absolutely. One technique is to use a blackbody source and some means of limiting the optical bandwidth, such as a narrow pass optical filter. Alternatively, Df may be determined by first measuring D&(Tz ,f,Afl, the wideband response due to blackbody radiation from a source at temperature T2. This quantity is defined by the relationship

where P B B ( T 2 ) is the total optical power incident on the detector from a blackbody at temperature T2 and is equal to the integral of PAover all wavelength. The spectral distribution of the blackbody is known, so that the two types of detectivity can be related when the spectral distribution of the quantum efficiency, y ~ ,is known. The ratio g = D?/D&B

(4)

is given by

where v, is the frequency of the signal power ’?PA, and where M(v,TZ)/BB, is the fraction of power per unit optical frequency interval emitted by a radiating body, and %!BB is the total power emitted by an ideal blackbody at temperature T2. The quantity ~ ( v is) the rate at which incident photons (of frequency v and wavelength A = c/v) are converted to electron-hole pairs. For photon detectors the signal is proportional to the absorbed number of photons, and D* increases with wavelength until the incident photon energy, hv, is less than the bandgap of the material, E g ;below this energy the response falls rapidly to zero. The wavelength at which the peak response occurs is denoted by A,. and it is the value at which D* is usually quoted (with the designation DL). The cutoff wavelength, A,, is generally defined by the wavelength or frequency at which the response has fallen to 50% of the peak value. In (HgCd)Te detectors a reasonable approximation is that h,/Ap .- 1.1. For wavelengths greater than A, (i.e., hv < Eg),71 falls rapidly to zero. A useful approximation is to assume that 7) is constant for all wavelengths less than A c , and zero otherwise. The value of g then depends only on T2 and A,. For example, with T2 = 500 K and A, = 12 pm, g = 3.5. When

5 . (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

161

greater accuracy is required, the measured spectral dependence of ~ ( h ) must be used in Eq. ( 5 ) . The electrical spectrum of the detectivity is dominated at low frequencies by llfnoise and at high frequencies by amplifier and Johnson noise. It is therefore useful to specify D* at a mid-range value along with parameters which indicate the frequencies at which D* has decreased by 3 dB at each end of the spectrum. The low end parameter is usually referred to as the llfcorner frequency and the high end parameter is denoted by f*.Both parameters give the frequency at which the total noise power has doubled relative to the signal. The responsivity rolls off at high frequency with the time constant T = f C / 2 7 r wheref, is the frequency at which the signal has decreased by 3 dB. These parameters will be discussed in the following subsections. 2. FUNDAMENTAL LIMITOF PERFORMANCE PHOTOCONDUCTIVE DETECTORS

FOR

The total noise that determines detector performance originates not only from contributions related to the absorption of photons, but also from a number of other factors, including thermal effects in the detector (Johnson noise), thermal and other effects in the associated circuitry, and llfnoise both in the detector and elsewhere. This section is concerned only with ultimate performance possibilities, and therefore does not consider the second class of effects, although, as will be discussed below, the latter often determines practical performance possibilities. The value of D* ( h , f , A f ) under photon limited conditions is given by

where only fluctuations in the arrival rate of the photons have been considered. Exact and approximate solutions for Eq. (6) are presented in Kruse et al. (1962). For detectors operating below 50 pm and for TB not too far from 300 K, the solution reduces to the well-known form

where QB is the total background photon flux incident on the detector and is assumed to be constant for frequencies above vc. Derivations of ultimate D* proceed from apparently quite different viewpoints in theories that are concerned with fluctuations in excess carriers due to absorption of radiation. One well-known approach uses the

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R. M . BROUDY A N D V. J . MAZUKCZYK

“g-r theorem” of Burgess and Van Vliet (Van Vliet, 1958, 1967; Burgess, 1954, 1955, 1956; Van Vliet and Fassett, 1965; Long, 1970) to derive carrier fluctuations due to both generation and recombination of excess carriers. Theories using this viewpoint result in the familiar expression for D* given by Eq. (6) except for a factor of 2 instead o f d in the denominator: 0%=

& [$I1’”

In spite of the extensive treatments of the subject in the literature, there does not appear to be a completely rigorous description of noise effects in photoconductive detectors. It seems clear that ultimate performance limits must be determined by fluctuations in the arrival rate of photons, as discussed by Kruse et ul. (1962). It is striking that whichever point of view is used, the ultimate performance limit D* (D*BLIP, for background limited infrared photodetector) differs at most by a factor of 4 between the two approaches (perhaps because of the fundamental statistical nature of fluctuations of any type). It is tempting to join both approaches in a conceptually attractive manner which is not rigorous, but may be physically correct: That is, to separate recombination from generation noise and to equate carrier generation fluctuations with those of the photon flux. If this may be done, then the generation and recombination fluctuations contribute equally, and the limiting L)* of Eq. (7) is, in fact, equal to that of a photovoltaic detector, which has no recombination, while a factor of I / d multiplies the ultimate D” of a photoconductor which has both generation and recombination. This approach may conceptually clarify the limiting D* of a photoconductor operating in the sweepout mode (which has been speculated (Milton, 1973; Williams, 1968) to follow Eq. (7) as discussed below) for which recombination noise is negligible since the minority carriers recombine at or beyond the contact regions. 111. Simple Photoconductivity

A typical photoconductive device is thin and rectangular in shape and is bonded to a much thicker block of material, called the substrate. Electrical leads are bonded to pads of metal as shown in Fig. 1 . Initially, it is instructive to consider only photoconductivity due to the influence of bulk properties. Fundamental principles can be illustrated readily, correspondence is more readily achieved with much of the literature, and useful descriptions of device performance may be obtained. Moreover, a semirigorous theory of noise is available only for this condition. The basic parameters of responsivity, noise, and D* are derived in this section from this point of view.

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

163

ELECTRICAL

SUBSTRATE

RESERVOR

FIG.1. Principal elements of a typical photoconductive device.

3. RESPONSIVITY

The conductance G of the detector is

G

=

( q / L 2 ) ( / d+ php),

(9)

where pe is the electron mobility p h is the hole mobility, N is the total number of electrons, and P is the total number of holes. The detector length is L , and q is the electronic charge. The absorption of photons generated by a constant background will change the conductance from its value at thermal equilibrium, G o , to G by changing No to N and Po to P , where thermal equilibrium values have the 0 subscript. For the majority of applications, changes in N produced by background flux will be small compared to No for the typical n-type PC (HgCd)Te photoconductor. The rms signal photon flux per unit area at wavelength A, namely Q,(A), produces a change in the conductance given by

AG = (q/L*)(peAN

+ ph AP),

(10)

where AN and AP are the total number of excess carriers in steady state. Whether or not AN and AP are equal depends on the recombination mechanism involved. However, it appears that the dominant recombination mechanism for good quality (HgCd)Te is Auger, and thus AN = AP. In the remainder of this work it will be assumed that only Auger recombination is important. At sufficiently low background and temperature apparent or real minority-carrier trapping effects are important (see Part V). Defining the quantity 7,which will subsequently be identified with the excess carrier lifetime, by the expression 7

=

AN/[Q,(A)r)(A)Al,

(11)

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R. M. BROUDY A N D V. J . MAZURCZYK

where Q s is the signal photon flux, the change in conductance becomes

AG

=

( q / ~ ’ ) ~ , ~ [ Q , ( ~ ) ~ r ( ~+ ) A61, ][l

(12)

where

b = Pe/ph * (13) For n-type (HgCd)Te, b >> 1, and in the remainder of the discussion the approximation I + b = b will be used. The device is usually placed in a simple series circuit with a load resistor whose conductance is much smaller than that of the device. A change in the latter’s conductance, G, produced by a change in the arrival rate of photons, will result in a signal voltage across the load resistor given by AVL = Vb AGIG,

(14)

where V , is the dc bias voltage on the detector. Combining Eqs. (2), (9), (12), and (14), and recognizing that PA= Q,Ahv, gives the responsivity in steady state: %A

=

[71(A)/Lwd)(A/hc)v,T/n,,

115)

where no is the average thermal equilibrium carrier density and d is the thickness of the photoconductor. Under the assumptions of no drift or diffusion and equal recombination, the time dependence of the average excess carriers is described by

aP/at = Q J ~= - A P / t .

(16)

Solving for M ( t )gives AP(r) = Q&qTe-‘’’, or in the frequency domain

+ (27”7)’]-’’*.

A P ( f ) = QsAr)T[l

(18)

The frequency dependent responsivity can therefore be written

+ (2flfT)2]-”2,

%!~(f) = %!~(o)[l

(19)

where WA(0)is the steady-state value given by Eq. (15).

4. Noise

(I. Johnson Noisc The conductance of the detector defined by Eq. (9) is of course an average value. The total current in the detector is made up of many individual vector components whose contributions are constantly changing

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

165

even if the number of charge carriers remain constant. The rms noise per unit bandwidth Af, due to thermal fluctuations, is independent of the material and device characteristics, and is given by the well-known expression for Johnson noise =

( 4 k T / @ Af.

(20)

b . Generation -Recombination Noise

The average number of carriers in the detector is determined by the balance between the generation and recombination processes. Fluctuations in these quantities results in another type of noise appropriately called generation-recombination noise. For a semiconductor where direct recombination dominates, the noise per unit bandwidth Af is (Van Vliet, 1958, 1967; Long, 1970)

where ( A N 2 )is the variance in the number of majority carriers and no is the average density. For a two-level system the variance has been evaluated in general and is given by (Long, 1970)

(22)

( A N 2 ) = g7,

where g is the generation rate and 7 is the relaxation time constant defined by Eq. (11). Equation (21) is correct both for steady-state as well as thermal equilibrium conditions. A key and simplifying assumption is often made that thermal and optical generation-recombination processes are independent. The variances are therefore uncorrelated and combine linearly so that (AN2) = (g&lerrn+

(g7)0,t.

(23)

The first variance is given by (Van Vliet, 1958, 1967; Long, 1970)

and the second by (g7)opt =

Ph =

PhLwd,

(25)

where P b represents the optically generated background hole density. Using the above relations, the g-r noise is found to be

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R. M . B R O U D Y A N D V . J . M A Z U R C Z Y K

The full equation is practically required only over a small range of temperature, where the semiconductor changes from intrinsic to extrinsic. Outside of that range either optical or thermal generation dominates, and Eq. (26) reduces to the g-r noise for one or the other single generation process. Holes generated by the background flux are relatively independent of temperature when the detector is an extrinsic semiconductor and po may often be made negligible by reducing the temperature so that P b dominates. Under these conditions the noise is photon limited since it is determined by fluctuations in the photon arrival rate; then, as discussed in Part 11, computation of D* using Eqs. (l), (2), and (26) yields the BLIP limit given by Eq. (8). c. l / f Noise

The phenomenon of l/f noise will always determine the low frequency limit of device performance, although it is not related to the same fundamental principles that control the other noise mechanisms. At present, there is no rigorous theory of llfnoise. Nevertheless, it is well recognized that it is related to practical device structures, and may be reduced by improved device design and fabrication techniques. For convenience, the existence of l/f noise may be modeled in terms of the “corner frequency” f o , which is defined as the frequency at which the I/f noise power equals the g-r noise power. llfnoise at any frequency can then be modeled from the relation:

G,f= (fo/f)%-r(O),

(27)

where V&O) is the generation-recombination noise at the plateau region at frequencies below the high rolloff and well above the llfcorner frequency. Both V,,, and Vg-r depend on bias, temperature, and background flux in possibly different ways, so that fo may be a function of these parameters. llfnoise is treated in more detail in Part IV.

d . Ampl$er Noise The noise contribution of the amplifier can be modeled by the presence of a voltage noise generator of rms magnitude e, and a current noise generator i, . The voltage generator appears in series with the amplifier input while the current noise generator is in parallel with the input. These are usually considered to be white noise sources. When a detector of resistance r d is placed across the input of the amplifier, the noise (aside from detector Johnson noise) generated will be

Vf; = ef; + if;r:.

(28)

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

167

With typical (HgCd)Te photoconductive detectors the value of rd is usually less than 100 a. This low resistance, combined with an upper usable frequency range of perhaps 10 MHz allows one to ignore capacitance at the input. However, individual cases should be examined carefully to insure the validity of this assumption. e . Total Noise

The individual noise contributions from the detector and amplifier (V,) are uncorrelated and therefore add in quadrature. The total noise of the detector is given by

v: = vq + v;-*+ (Vl,f)Z + v:.

(29)

A characteristic noise spectrum is shown in Fig 2 to illustrate the contribution of the various noise components.

5. DETECTIVITY Combining equations of the previous sections gives the result,

The expression for D* has been put into this form to emphasize the fact that the maximum performance is given by DgUpand to isolate those factors which cause the performance to deviate from that value.

lif NOISE

g-r CORNER FREQUENCY g-r

NOISE

I/f CORNER F R E O U E N C Y

AMPLl F IE R [NOISE

FREQUENCY

FIG. 2. Characteristic noise spectrum of a photoconductive detector.

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R. M. BROUDY A N D V. J. MAZURCZYK

At higher temperature where the detector is an intrinsic semiconductor (i.e., no = po) the g-r noise is sufficiently large that the condition V&,(f>

>>

v; + v;

(3 1)

can be achieved with moderate bias levels (except at high frequencies). Performance in the midrange frequencies, therefore, is determined entirely by the factor

which is strongly temperature dependent. Since the performance of the detector in this region is relatively poor, it is used under these conditions only when cooling capacity is limited. With decreasing temperature no becomes constant and dominates p o . The detector then becomes an extrinsic semiconductor, and D* is given by

D” = D&,p(l + ~ o / p b ) ~ ” ’ [ l

+ ( f o / f ) + (Vj +

V~)/VPr(f)]-”’.

(33)

It is obvious that maximum performance is achieved when po/pb<< 1

and

(V;

+

V:)/V:-,

<< 1.

(34) The detector performance at frequencies higher than (27TT)-l is affected by the decrease of V&,(f> relative to V; + Vi . Increasing the bias will help maintain performance but eventually (even for an ideal detector) the amplifier and Johnson noise will dominate and the detectivity will be given by [D*]hf

(f)

+ V;)”’.

= %L(A Af)”’/(V;

(35)

The frequency at which the detectivity has decreased by 3 dB from its midband is often designated asf”. Since it is the value at which the noise power has doubled, that frequency is given by 1

f* = ?&

v;-,(o) [v: + v2,

+

,

as derived from Eqs. ( l ) , (19), and (35). Heating effects can cause a superlinear dependence of Vg-r (0) on bias so that f* no longer corresponds to a decrease of fiin D*.Equation (36) must therefore be used with caution. 6. TEMPERATURE A N D BACKGROUND DEPENDENCE

The temperature and background dependence of the detectivity, responsivity and g-r noise is determined by the variation of the majority and

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

169

minority-carrier concentrations as well as the time constant, as discussed by Kinch and Borello (1975). The majority- and minority-carrier mobilities which affect the above performance parameters indirectly do not depend on background flux and have a relatively weak dependence on temperature. For (HgCd)Te, or any n-type semiconductor where the donor activation energy is negligible, the magnitude and temperature dependence of the quantities of interest are given by

where ND - N A is the concentration of uncompensated donors. The minority-carrier concentration is obtained from (38)

Po = $ / n o ,

and the bulk time constant for Auger recombination (Kinch et al., 1973) is given by 5-

= 2d/(no

+ po)(no).

(39)

The value of ri is 71

= CO(Eg/kT)3/2 exp(EglkT),

(40)

where Eg is the energy gap and the constant Co cannot be calculated without making some gross assumptions. Experimentally (Kinch et al., 1973), the value of T has been found to be approximately 1 x sec at 77 K for x = 0.195 and sec for x = 0.205. The temperature dependence and magnitude of these quantities is determined (Kittel, 1961) by the intrinsic carrier concentration n i , which can be determined from empirical relations (Mazurczyk et af., 1974; Finkman and Nemirovsky, 1979; Schmit and Stelzer, 1969; Schmit, 1970) ni = (A

+ B7)C1P8.75T1.5 exp(-

Eg/2kT),

(41)

with A

=

1.093 - 0.296x,

B = 4.42 x lop4,

C = 4.293 x 1014,

and x is the stoichiometric ratio. The energy gap Egis also dependent on composition and temperature and is given with some accuracy for x = 0.2 by (Finkman and Nemirovsky, 1979; Schmit and Stelzer, 1969; Schmit, 1970)

Eg = 1 . 5 9 ~- 0.25

+ 0 . 3 2 7 ~+~5.23 x

10-4(1 - 2.08x)T.

(42)

Incorporation of these equations into the expression for the detectivity

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R. M . BROUDY A N D V. J. M A Z U R C Z Y K

will give a good description of the temperature dependence when all the parameters are known and the detector is of uniform composition. The background dependence of the detectivity and other quantities depends on the number of steady-state excess carriers Pb that are optically generated. When P b << no in n-type (HgCd)Te, the background dependence of the detectivity is determined in a straightforward manner from Eqs. (38) and (7). At higher levels of Q B the value of the lifetime is affected. If drift and diffusion effects are neglected then the value of 7 is determined from the simultaneous solution of 7

=

2Tinf/(no 4- Po

P d n o 4- Po),

(43)

where background and all minority carrier effects are considered (Kinch et al., 1973) and Pb

= 77Q/d.

(44)

In real devices, drift and diffusion effects are important and must be considered. The complete analyses of photoconductive device behavior including these effects is discussed in Part IV, where expressions will be derived for the case of A p << no and P b << no including the influence of drift and diffusion. IV. Photoconductive Device Analyses In Part I11 only the influence of volume photoresponse was considered. Real photodetectors, however, are subject to a number of additional constraints related to the practical necessity of holding down the photosensitive material and shaping it to a specified configuration in a working device including provision for contacts. The following subsections modify the simple theory to include these effects. Specifically treated are resistive heating effects, surface recombination, l/f noise, and transport effects.

7. POWERDISSIPATION Generally, photodetectors must be maintained at reduced temperatures to achieve optimum performance by eliminating thermally generated noise. Since there are a number of thermal interfaces between the detector and the cooling reservoir, the Joule heating due to bias current will produce a rise in detector temperature. Assuming edge effects can be neglected and that no heat is lost by radiation or thermal shorts, the rise in detector temperature is closely given by the one-dimensional approximation

5. (HgCd)Te

171

PHOTOCONDUCTIVE DETECTORS

where the thickness and thermal conductivities of the various layers are I, and Z , , respectively. The power dissipation is H, and A, is the contact area between layers. However, achieving good thermal interfaces between the various layers is always a problem. Even with the use of materials that have high thermal conductivities, the effective contact area can be very low due to poor physical contact, air bubbles, and other problems. The epoxy bonding layer between detector and substrate might typically have a thickness of 3 X cm and a thermal conductivity near W/cm K. Similar values can exist for a layer between substrate and reservoir. For example, a substrate of sintered alumina has --- 1 W/cm K at 80 K . For a substrate thickness of 0.05 cm, the thermal resistance is negligible and A T is determined by the thermal resistance of the epoxy layers. For a device such as shown in Fig. 1, these values give

AT = 0,6H/A,,

(46)

where A, is the total effective contact area. It includes the area under the bonding pads of the detector. The thermal interface will limit detector performance when heating causes the term (1 + p o / p b ) to increase. The responsivity may also decrease with increasing temperature, but it is usually less important, except in wideband and other applications where the detectivity is limited by amplifier noise. In many applications, however, the thermal resistance between detector and reservoir would be less important if not for the additional complications of surface recombination and sweepout. 8. SURFACE RECOMBINATION

The simple model also does not account for surface recombination which is perhaps the single most important mechanism that limits photoconductor performance. It is well known that the surfaces of a semiconductor are regions where recombination can proceed at a higher rate than in the bulk. Surface recombination reduces the total number of steadystate excess carriers by effectively reducing the average recombination time, T . In modeling this effect, the bulk recombination time T can be re~ (Kruse ef al., 1962, p. 330) placed by T A where A, =

(1

sinh(d/l,) + S2[cosh(d/L,) + S1S2)sinh(d/l,) + (S, + S,)

The parameters S, , S2are defined by

Sf= s ~ T / L , , i

=

1,2

- l)] cosh(d/L,)

'

(47)

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R. M . BROUDY A N D V. J . MAZURCZYK

where SI, Sz are the surface recombination velocities of the front and rear surfaces, and where La is given by La

=

[Do~]1'2.

(49)

The equilibrium value of the diffusion constant, Do,is appropriate when assuming low excess carrier generation. Since the parameters S, and Sz are difficult to measure directly, Eq. (47) is not immediately useful in modeling performance. It is useful, however, in establishing the degree to which performance can deteriorate. An appreciation of this effect can be obtained by considering a specific example. The parameters of the device to be considered are given in Table I. For convenience in modeling we assume that Sz = 00 at the back surface of the detector (i.e., Ap = 0) and Sl= 0 at the front surface, in keeping with the general recognition that back surfaces may be more difficult to control in real detectors. From the values in Table I, A = 0.026 and T = 80 K. A calculation of the D* for this detector and an ideal detector is shown in Fig. 3. The performance of the ideal detector drops for temperatures above 80 K because thermally generated minority carriers are beginning to dominate. The low bias level of 45 mV, however, is sufficient for the detector g-r noise to dominate the white noise generated by the amplifier and detector resistance. The addition of surface recombination considerably reduces the D*.To restore performance, the bias must be increased to 150 mV [to satisfy Eq. (34)] and the temperature reduced to 60 K [to satisfy Eq. (3311. For large arrays the increased bias and cooling capacity may not be available.

TABLE I -

PARAMETERS OF A TYPICAL (HgCd)Te DEVICE

Composition (n-type) Quantum efficiency Cutoff wavelength Carrier concentrations Intrinsic Auger time constant Mobilities Physical dimensions Thermal interface Background flux

x = 0.200 7)=1 A, = 14.2 pm at 80 K N , - N , = 9 x 10" n, = 3.08 x lOI3 ern-$ at 80 T~ =

0.6

X

K

sec at 80 K

p e = 1.6 x 105 crnz/V sec at 80 K p h = 800 cm2/V sec at 80 K

length = 0.015 crn thickness = 0.001 cm (IIZA,) = 3 K/W QB = 1.08 x I017/cm2at 80 K

"1 y\

5 . (HgCd)Te PHOTOCONDUCTIVE DETECTORS

--r

'%

2

* 4

a

173

DETECTOR IDEAL

45-mV BIAS

DETECTOR WITH SURFACE RECOMBINATION 150 mV 45 mV

TEMPERATURE (K)

FIG.3. The influence of surface recombination on detector performance.

9. l/f NOISEI N (HGCD)TEPHOTOCONDUCTORS

The existence of fluctuations varying inversely with frequency has been widely recognized in both electronic and nonelectronic phenomena. Although llfnoise ultimately limits the low-frequency performance of all electronic devices, no satisfactory or rigorous theory exists. a. The Classical Theory

In photoconductors, most of the literature and theory of llfnoise has taken what might be called the classical approach, in which llfnoise is expected to be independent of other sources of noise but behaves similarly except for the inverse frequency dependence. Thus, for this approach, dimensional relations can be derived from the usual approaches which assume that the noise originates from microscopic noise generators distributed throughout the device. Then, from fairly general noise theory arguments (Kruse et al., 1962, p. 255), the Ilfnoise voltage can be written as

where L, w ,and d a r e length, width, and thickness of the detector, E is the dc-bias electric field, Af is the noise bandwidth,fis the frequency, and C , is a coefficient which gives the strength of the llfnoise. In particular, C , is found to depend on carrier concentration but is independent of detector dimensions even though it possesses the dimensions of cm3. The classical corner frequency can be calculated from Eqs. (26), (27), and (50). With the usual assumptions for n-type (HgCd)Te that no >> p o and p B , fo

174

R. M. BROUDY A N D V. J . MAZURCZYK

becomes

fo = C I ~ % / ~-k@PBO

~ Y

(5 1)

where we have assumed that (07)~ << 1. If the temperature is low enough that thermally generated carriers become negligible

fgB‘.IP

= C1n;d/4r2qQB.

(52)

Thus, f o depends on the detector thickness but not on the active area dimensions. In keeping with the classical approach, the coefficient C1 has been calculated from approaches which assume either that the noise generators originate at the surface, or throughout the bulk. McWhorter’s model (Van der Ziel, 1959) for l/f noise in semiconductors is based on the existence of surface traps. Fluctuations in the capture and release of electrons at these surface traps produce fluctuations in the bulk electron concentration and consequently in the conductivity. In this theory, the characteristic l/f frequency dependence of the noise power due to these fluctuations is the result of assuming that electrons are released from the surface traps by tunneling to the bulk. The tunneling probability depends exponentially on the distance over which tunneling must occur. Consequently, a distribution of lifetimes is obtained which leads to the characteristic llffrequency dependence for the noise power. This model leads to the following expression for C,:

C1 = (NT/4ni)l/ad,

(53)

where NT = the concentration of traps in the surface layer and a is a characteristic tunneling constant. Hooge’s (1969) model for llfnoise assumes that it is a universal bulk phenomenon. His theory leads to the following empirical expression for the coefficient C,:

C , = 2 x 10-3/no.

(54)

It is notable that although these theories have received much attention there has been little if any experimental verification of their validity or even usefulness in photoconductors. b. New Phenomenological Theory A useful phenomenological theory (Broudy , 1974) has recently been developed for l/f noise in (HgCd)Te photoconductive detectors. It was arrived at as the result of a survey and analysis of available data from a large number of photoconductive (HgCd)Te detectors operating in many conditions.

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

175

Consideration of these and other data have led to the notable observation that the l/f noise voltage, Vl,f, bears a simple relation to the generation-recombination voltage, Vg-r. Specifically, the following empirical relation has been found to apply:

V/f (Kl/fl VZ-r

9

(55)

where K1 is a constant. Note that this empirical relation holds whatever the source of VgPr(possibly only the recombination noise may cause the effect). In this viewpoint, llfnoise is “current noise” simply because VgVrvaries with the current. Also, l/fnoise decreases inversely with detector resistance simply because Vg-r behaves in this manner. The corner frequency, fo, is readily determined to be Equations (55) and (56) state that l/f noise increases with g-r noise. Therefore, to reduce 1/fnoise, everything else being equal, material with low g-r noise should be selected. This conclusion is a key and new consequence of the empirical theory which does not appear in the classical theory where Vg.-r and Vlv are essentially independent and are related solely through their current dependences. In terms of detector and material parameters criteria for reduction of l/fnoise can be established from Eq. (26) for g-r noise:

(a) (b) (c) (d) (e)

Use lowest feasible bias current. Keep detector temperature below thermal generation range. Reduce detector background as far as possible. Choose semiconductor material with large donor concentration. Develop improved processing methods (to reduce Kl).

This effect is practically illustrated in Fig. 4 which shows the background dependence offo we have measured for a typical (HgCd)Te photoconduction detector. Note that the detector follows a Qi1’2dependence at the higher background but that dependence diminishes for QB below 10’‘ as thermal g-r noise becomes appreciable. Borello et al. (1977) have also reported background dependences of 1/f noise. EFFECTS-DRIFTA N D DIFFUSION 10. TRANSPORT a . The Fundamental Approach

In real devices, drift and diffusion play a significant role and must be considered. In a classic publication Rittner (1956) has provided a unified and broad treatment of photoconductivity. He derived the basic photoconductivity equations beginning from the first principles of continuity

176

R . M . B R O U D Y A N D V. J . MAZURCZYK

10'

10"

1Ol8

BACKGROUND FLUX ( p h o t o n s / c m 2 s e c i

FIG. 4. l/f knee versus background for a (HgCd)Te photoconductor. Ad = 0.004 in. 0.004 in. A,, = 12.1 pm. T = 78 K.

X

equations, carrier currents, and the Poisson equation. His complete analyses included trapping effects and make the physically reasonable assumption for low impedance photoconductors of the (HgCd)Te type that space-charge neutrality exists and that there are negligible ionic currents. Neglecting trapping, the following equation is obtained (Rittner, 1956):

+ + D div grad Ap + p E

aAp/at = - (Ap/Tg) j

*

grad Ap,

(57)

where f is the generation rate ( f = v Q s / d cm-* set)-*, T~ is the generalized lifetime for all levels of carrier excitation, and the generalized diffusion constant D and generalized mobility p are given by

-n

=

n I L + PIP.'

where Dh and D, are the hole and electron diffusion coefficients, respectively. This equation is nonlinear in general, since the coefficients D, p , and T~ depend on n and p . However, it may evolve into a usable form for the physically realistic case that Ap << n, which applies for low light intensities: ahplat

=

-(Ap/T)

+ f + Do div grad A p + mE - grad A p ,

(58)

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

177

where T is the lifetime for low excitation levels and Do and poare given by

Do = (no -k PO)/[(nO/Dh)+ (PO/De)l,

(58a)

po = ( P o - no)/[(nO/ph) + (PO/pe)l.

(58b)

Using the Einstein relation D = k T p / q to describe diffusion coefficients in terms of mobilities, these become

Do = (kT/q)PePh(& Po)/(Peno + P h P o ) r

(58c)

Po = (PO - nO)peph/(penO+ phP0)-

(584

Useful solutions may be obtained in terms of the parameters in Eqs. (58), since T, D o , and p,, are constant coefficients. For purposes of comparison with literature, it is useful to recognize that parameters closely related to Do and po have been similarly derived by authors other than Rittner, and are often denoted by the terminology “ambipolar diffusion coefficient” and “ambipolar mobility.” These parameters differ from Do and po only in signs, as would be expected, since they have been correctly derived for the identical physical phenomena. In any case it is required merely to be self-consistent with whatever approach is chosen. In this chapter, we follow Rittner’s analyses, using Do and p,,, and in order to eliminate confusion with other work, we will avoid the ambipolar expressions, while recognizing that one is dealing with the same device principles. In the following section, Rittner’s equations are applied to n-type (HgCd)Te photoconductors under the assumptions of no trapping and of low excess carrier levels. In Rittner’s work the symbol T is meant to imply a composite lifetime that is the product of the bulk lifetime and the surface recombination factor. In the remainder of this work we will also use this same convention.

6 . One-Dimensional Formulation with Ohmic Contacts Specific solutions of the previous equations were obtained by Rittner for the one-dimensional case with the further simplification that the contacts are ohmic. The electric field is in the x direction [-(L/2) < x < (L/2)]. The low-light-level approximation permits n = no + An, and p = po + Ap, and the ohmic contact assumption requires that An = A p = 0 at x = - L/2 and at x = +L/2. The excess minority-carrier density at the position x on a detector extending from L/2 to L/2 is given by ealr sinh(azL/2)

- emx sinh(a,l/2) sinh(al - a2)L/2

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R. M. BROUDY

A N D V . J . MAZURCZYK

where -

a1.2 =

2 0 0 -e

[(& 2Do)2+

(594

and

Do = (kT/dPo. The drift length I, of an excess carrier is is It is sometimes convenient to write

a.

=

a1.2

-1 & [($

+

(60) and the diffusion length l2 terms of ll and 12:

al,zin

$1

1/2

Using these results, Rittner (1956) calculated the steady-state photocurrent of a detector

AJ = qhh(b + l)vQsTE5/d,

(62)

(az- a,)sinh(a,L/2) sinh(a21/2) a1az(L/2) sinh(a, - a2)L/2 *

(63)

where 5=1+

Then for b >> 1, the voltage responsivity becomes

9tk = XqqrDp,&&/hcd.

(64)

At high fields where the drift length 1, is greater than the detector length L or the diffusion length 1 2 , the following first-order approximations can be made 1

^-P

then

a L L sinh-?- =2 211’

sinh a 4 / 2 L z - 1 + -. (65b) then sinh(a, - az)L/2 211 Substitution of these approximations into (63) gives the high field limit for -az =

+ LL >> 1, 21;

5:

The high field responsivity is therefore %If

= (Uhc)(qqPe/2Po)rd.

(65c)

In (HgCd)Te, the mobility of the electrons are field dependent; thus, to be completely rigorous, the field dependent resistance should be used in Eqs. (65).

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

179

Note that the drift length I , depends on the mobility p o ,which for extrinsic n-type material reduces to the hole mobility. In an intrinsic semiconductor po is zero and there are no sweepout effects. A rigorous calculation of the g-r noise that includes the effects of drift and diffusion is not yet published, although such a theory is reported soon to become available (Smith, 1981). Many discussions of detector performance in the literature (Williams, 1968; Kinch et al., 1977) assume that g-r noise has the same field dependence as the responsivity, and indeed, experimental results verify the accuracy of this assumption. For modeling purposes, a useful approximation for the g-r noise may be obtained from the simple formula by making the substitutions T*&

Pb’Pb,

pO+pO*

(66)

Lacking a rigorous theory, we shall include a strictly empirical field dependent sweepout factor:

F = F(E), which has been introduced from the viewpoint discussed in Part I1 to account for the possible effect of sweepout on minority carrier fluctuations. This parameter, which also has not been rigorously derived, would have a value between l/& and 1. At low fields, sweepout is small, most of the carriers recombine within the bulk, and there is the full complement of g-r noise due to fluctuations in the bulk recombination; therefore F I1. As the field increases, the number of carriers that recombine in the bulk becomes smaller as more holes (for n-type material) are swept to the contact regions to recombine; thus F becomes smaller, and would saturate to the value of l / d at the maximum field strength where recombination noise disappears. In detectors where the ambipolar diffusion length is comparable to the detector length, one would expect F to be less than 1 even at zero field, since an appreciable number of carriers reach the contacts by thermal diffusion. It is important to recognize explicitly that application of the parameter F depends on two hypotheses: First, as discussed in Part 11, in the absence of a fully regorous theory it is assumed that generation and recombination noises may be partially or totally independent and are equal at low field (in the absence of diffusion to the contacts), such that in full sweepout, only half the maximum g-r noise remains (Milton, 1973; Williams, 1968). Second, it is assumed that there is negligible noise due to recombination of minority carriers at the contact regions. The applicability of the latter assumption will depend on the specific type of contactpossibilities include high-low junctions, true metal-semiconductor inter-

180

R. M. BROUDY A N D

V. J .

MAZURCZYK

faces, and low field regions in transverse field devices, such as discussed in Part V. Using the substitutions indicated in Eq. (66) and the F factor, the g-r noise becomes (for frequencies less than 3m)

Comparison with the responsivity shows that for the extrinsic case both are proportional to (r and the ratio can be expected to be at least roughly constant, except for the factor F ( E ) . As implied above, the electric field cannot distinguish between optically and thermally generated holes. But the latter depend on T and (. But because Vg-r as well as the responsivity saturates, there will be a limit to the achievable detectivity even in the absence of heating and surface recombigation effects. This limit occurs when the g-r noise is background dominated and has saturated at a value less than the white noise generated by the detector resistance and the amplifier. The responsivity at this bias has saturated to the maximum value given by Eq. (65), and the sweepout limited detectivity is therefore

ELECTRIC FIELD

FIG. 5 . Field dependent respoiisivity for an (HgCd)Te detector. 0, data; -, theory using the following parameters: 7bulk = 3.4 psec at 80 K; pn = 1.3 x 105 cmP/V sec; pa = 166 cm2/V sec; rd = 59 a.

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

181

20 19 18

-3

77

-

c1

I

1.6

'

1 5

n

14

5

0 I

13

12 1 1

10

0

2

4

6

8

10 12 14 16 18 20

22 24 26 28 30

ELECTRIC FIELD ( V i c r n l

FIG.6. Field-dependentD* for an (HgCd)Te detector.

This limit is independent of the time constant and is therefore also independent of the effects of surface recombination. The value of the field needed to reach this limit, however, increases with surface recombination. Experimental evidence for the reasonableness of this empirical approach to sweepout and diffuse-out may be seen from the data presented in Figs, 5 and 6, which show the field dependent responsivity and D* for an (HgCd)Te photoconductor (Broudy er af., 1975; Broudy, 1976). However, other reported data (Kinch er af., 1977) have not shown this effect. It may be speculated that experimental disagreements are due to differences in contact characteristics for various detectors (see the following discussion on complex device configurations). To illustrate the effect of sweepout on the temperature and field dependences of actual photoconductors, we have calculated the responsivity and g-r noise for conditions which may occur in practice. For convenience in exposition, the sweepout factor, F, has been set to 1. In Figs. 7 and 8 the g-r noise and the responsivity are shown as a function of bias for the following conditions: (1) An ideal detector with the parameters listed in Table 1 (see curve

4.

(2)

eters:

The same detector as in (1) but with surface recombination paramsz

(see curve b).

=

00,

s,

=0

182

R . M . BROUDY A N D V. J . MAZURCZYK

15

CURVE ( a ) IDEAL DETECTOR CURVE ( d ) SURFACE RECOMBINATION, PLUS SWEEPOUT PLUS THERMAL RESISTAKE

\

c-

I 0

SURFACE

r

N

P 3 >b

-

5

Q5

v,

1.0 (V)

FIG. 7. Calculated bias-dependent g-r noise for high and low thermal conductivity bonding layers. cr = 1.

(3) The same detector as in (2) but with sweepout present (see curve

C)

.

(4) The same detector as in (3) but with a thermal interface having ~- -

ZA,

3 ~ / ~ c m

(see curve d). Curves b and c in Fig. 7 illustrate how surface recombination and sweepout dramatically reduce g-r noise and so make it much more difficult to satisfy Eq. (34). Increasing the bias will be effective only until it begins to increase the detector temperature. At this point the g-r noise will show a rapid increase. (Fig. 7b). This increase will be beneficial as far as Eq. (34) is concerned but will cause an overall reduction in the detectivity because of Eq. (33). This is illustrated in Fig. 9 where the detectivity as a function of temperature is shown. For an ideal detector a bias of 45 mV is

CURVE ( a 1 IDEAL DETECTOR

PLUS SWEEPUJT

THERMAL RESISTANCE

I

I

0.5

J

1.0 Vb ( V )

FIG. 8. Calculated bias-dependent responsivity for high and low thermal conductivity bonding layers. u = 1 .

r

.-

.-

45- mV BIAS

0.5 Y

DETECTOR WITH DRIFT, DIFFUSION, ANDTHERMAL EFFECTS

* r D

I

I

I

60

70

I

I

I

1

80 90 400 TEMPERATURE (K)

FIG.9. Calculated temperature-dependent D" for high and low thermal conductivity bonding layers, with and without sweepout, at bias voltage levels of 45 and 150 mV. F = 1. QB = 1.08 x 10" cmz.

184

R. M . BROUDY A N D V . J . MAZURCZYK

sufficient to satisfy Eq. (34). But with the addition of surface recombination, sweepout and a thermal interface, the performance drops drastically if the bias is maintained at 45 mV. Increasing the bias to 150 mV might be expected to produce a substantial improvement in the detectivity , but actually the higher levels of bias result in a decrease of performance because of detector heating. A consequence of minority-carrier sweepout is that the excess carriers and therefore the photoresponse are nonuniformly distributed along the length of the detector as shown in Eq. (59). An experimental example (Mazurczyk, unpublished data) is shown in Fig. I0 which is quantitatively in agreement with Rittner's theory as determined from Eq. (59). The detector has been scanned with a 0.001-in. spot for several different fields. The triangular shape of the response at high fields occurs because the life-

3.3

66 13.1 19 7

33 8

i

I

I

//////A

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

185

time of the carrier is determined by the time to reach the negative contact rather than by bulk or surface recombination. Although the responsivity saturates with increasing bias, the time constant does not and continues to decrease; therefore, the detector upper frequency limit f * increases until heating effects or impact ionization cause an abrupt decrease in the responsivity. Again, from the work of Rittner (1956), the time dependence of the excess carrier density may be determined from the following expressions for a detector of length L:

AP(E,f)= APo @X,f),

(69)

where

+ sin2(nr/2)]

(- l)n+'n2 exp( - Dlf/~)[sinh2(DiL/2) n=O

&(Dl

-

The quantities f, and f 2 are the drift and diffusion lengths defined previously. We have performed a sample computer calculation of Eq. (69) as shown in Fig. 11 . A short detector of L = 25 pm was considered for maximizing the influence of drift and diffusion. It is notable that for field strengths below 30 V/cm the theoretical time dependence is almost exponential as is commonly observed experimentally. The values of T calculated using the T[ approximation agree closely with the rigorous calculation as shown in Fig. 12. At higher fields one must use Eqs. (67) and (70). The preceeding discussions have illustrated several important points. ( 1 ) A consideration of the simple model for detectors leads to the requirements for maximum performance, i.e., Eqs. (33) and (34). These conditions remain true when the effects of sweepout, surface recombination, and heating are introduced. (2) Drift and diffusion of minority carriers, surface recombination, and thermal resistance effects cause the actual performance of detectors to seriously deviate from the predictions of the simple model. However, these effects can be approximated by relatively simple substitutions. (3) The drifted diffusion of minority carriers in the detector generally results in the need for more bias power and refrigeration to achieve the same performance as the ideal detector.

10-2 9

'

I

I

30 V/cm

5 0 V/cm

0.5

1.0

1.5

T (fO-'SW)

FIG.11. Calculation of the transient decay of photoconductivity for varying field strength, using Rittner's theory (1956).

T~~

2

I

roc ( M O D I F I E D EXPONE

z

g '

lRlGOflCUS C A L C U L A T I O N )

~-

10

'ri

t-

1

10

50

FIG. 12. Time constant as determined from Rittner's theory and from the exponential approximation.

5 . (HgCd)Te

187

PHOTOCONDUCTIVE DETECTORS

11. SUMMARY OF EQUATIONS FOR THE ONE-DIMENSIONAL APPROXIMATION WITH OHMICCONTACTS

The equations developed in the preceding sections for the onedimensional case are summarized in Table 11. The parameters must be TABLE I1 ANALYTICAL EXPRESSIONS O F DETECTOR PERFORMANCE FOR T H E ONE-DIMENSIONAL CASE W I T H OHMICCONTACTS

=@%L

Responsivity (without sweepout):

Rh

Generation - recombination noise (without sweepout):

"-' = (Lwd)1f2no

fo

Amplifier noise associated with a detector of resistance rd :

=

=

7

~

)

~

'

~

K1Vg-r

VZ, = e',

+ :z

D'L'p

ri - A 11 "= - 2hc (Q,) -112

(without sweepout):

where

~

(4kTrd)lI2Afl"

Maximum D* at wavelength A, (without sweepout): Detectivity

Sweepout factor:

+~

[[1+PO nO ][PbTAf]]1'2 p b p o + no 1 + w2?

2vh

Johnson noise associated with V, the detector resistance: Ilfnoise with comer frequency f,:

hc no Lwd (1

Pb P O

( = 1'+ al,8

=

+

120

(az - a l )sinh(alL/2) sinh(a&/2)

alaa(L/2) sinh(a, - az)L/2

-E f [(A)* + &]I1* 2kT/q

2kT/q

kTpor

carefully chosen for use in these equations, since many of them are interdependent. For example, variations in the fabrication process can modify the thermal resistance, detector thickness, surface recombination velocity, and l/f noise. In addition, calculations of hypothetical detector performance, especially for arrays, must take into account the possible range of variations. For example, one approach would be to assume a distribution of values for each of the parameters and then calculate the range of expected performance using Monte Carlo techniques. V. Photoconductive Device Design Extensive development of (HgCd)Te photoconductive detectors during recent years has led to semiconductor devices of considerable sophistica-

188

R. M . B R O U D Y A N D V. J. MAZURCZYK

tion and complexity. Designs and process techniques have been developed based on recognition and utilization of the fact that photoconductivity is due to majority carriers that are controlled by the behavior of minority carriers. In most cases, the objective has been to increase the minority carrier lifetime and therefore the device responsivity by reduction of minority-carrier recombination in the appropriate area of the device. This part describes several implementations of these principles.

12. EXTENDED CONTACTS Rittner's theory (1956) and the experimental verifications thereof show that the ultimately limiting lifetime of a photoconductor is the transit time of the minority carrier to the opposite polarity contact. Thus, the upper limit for the minority-carrier lifetime of an n-type (HgCd)Te photoconductor is the time required for a hole generated just inside the positive contact to reach the negative contact, a distance away, T = L / p h E . Carriers generated closer to the negative contact will recombine in a shorter time. Thus, the device lifetime may be seriously diminished for a small geometry detector arrays, as shown in Part IV. For example, for (HgCd)Te detector elements in the 0.001-0.002 in. range, the effective lifetime may never be longer than 700 nsec, whatever the bulk lifetime. Kinch et al. (1977) have proposed and demonstrated a device structure which increases the effective lifetime by extending the contacts away from the active area. In this overlap structure, as shown in Fig. 13, the optically active area is defined by a metal overlay of the insulating antireflection coating and not by the contacts. Although there is additional power dissipation from the overlap regions of the detector, Kinch et al. (1977) show that for detector lengths up to about a factor of 5 greater than the hole diffusion length, an increase in

LY

4 I

I

'I' I

I

r*T I

1

i

i

I

I

INSULATOR (ANTIRFFLFCTION FILM) MFTAL

i

/

/

\ (Hqf d j i e

FIG. 13. Contact overlap structure for geometrical enhancement. (From Kinch et ul., 1977.)

(HgCd)Te

10

189

PHOTOCONDUCTIVE DETECTORS

I02

Io3

I 0“

I05

OETECTOI~BIAS PowEri (UW)

FIG.14. Responsivity versus detector bias power for overlap and standard devices 0.1-eV (HgCd)Te at 77 K and a 55” FOV. (From Kinch et al., 1977.)

length results in an enhancement in responsivity for equal bias power (assuming that the detector lifetime between the contacts is sufficiently long). Figure 14 illustrates the effect for a standard detector and an overlap structure for 0.1-eV n-type (HgCd)Te measured at 77 K and a 55” FOV. The nominal width of the active areas is 0.002 in. and the overlap length is 0.0010 in. 13. COMPLEX CONFIGURATIONS-GEOMETRY A N D CONTACTS

Actual devices are not one dimensional and may have contacts that deviate from ohmicity, either due to intentional or coincidental processing procedures. In this section we present an abbreviated discussion of the influence of these important effects. Inspection of actual device configurations, such as sketched in Fig. 1 to approximate scale, makes it clear that significant deviations from onedimensional field and current distributions are to be expected. Rigorous multidimensional analyses just becoming available (Kolodny and Kidron, 1981) have shown that departures from the predictions of one-dimensional theory for typical devices may lead to an increase in responsivity and time constant by as much as 80%. This occurs because device resistance is greater and because reduction in effective device thickness originates from redirection and compression of field lines. Moreover, detector noise may be reduced when carriers diffuse away and recombine under the contacts.

190

R . M . BROUDY A N D V. J . MAZURCZYK

Contacts may be either ohmic or “blocking” in nature. In the latter case, a more intensely doped region at the contact (n+ for n-type devices) causes a built-in electric field that repels minority carriers, thereby reducing recombination and increasing the effective lifetime and the responsivity (Long, 1977). Shacham-Diamand and Kidron (1981) have presented an analysis of the influence of blocking or partially blocking contacts on photoconductive detectors. They modified the basic theory of Rittner to include the effects of built-in electric fields due to n+-n regions at the contacts and then applied their model to the calculation of current responsivity and time constant of (HgCd)Te photoconductors as a function of a parameter 2 = n + / n , the doping ratio. Comparison with experimental results showed excellent agreement with observed responsivities, which may be as much as 5 times greater than predicted by the simple Rittner theory with ohmic contacts. It is instructive to note that the field dependence and magnitude of responsivity and noise for the more complex configurations follow closely the same functional behavior predicted by the simple Rittner analysis, but the magnitude of a key parameter must be modified from the actual material values to achieve correspondence with theory and experiment. Specifically, use of Rittner’s one-dimensional ohmic contact theory to match experimental responsivities shows excellent agreement over the entire range of electric field if the magnitude of the ambipolar mobility, p a , which is the only adjustable parameter, is reduced by a factor of 2-5 below the known material value (as applied in Fig. 5). 14. TRANSVERSE FIELDEFFECTS-ACCUMULATION LAYERS

As discussed in Part IV, Section 8, a common practical limitation of performance is caused by the reduction of lifetime due to recombination at the surfaces of the device. In fact, much of the technology of present-day device fabrication is designed to minimize the surface recombination of minority carriers that reach the surface by using appropriate chemical and mechanical preparation procedures to reduce the surface recombination velocity. There is, however, a different approach to reducing surface recombination that has been quite successful and may be more controllable. In this method recombination is reduced by preventing the minority carriers from reaching the surface at all by means of a built-in electric field. In the case of n-type (HgCd)Te, a transverse field would be provided by an n+ accumulation layer extending inward from the surface. An accumulation layer in a semiconductor may exist or be intentionally introduced in two ways: The first utilizes external (positive) charge (for

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

191

n-type devices). This effect can result from charge at three locations: (1) in donor or acceptor surface states; (2) the presence of positive or negative ions in the insulator outside the semiconductor; or (3) charge may be produced by metal field plates above the insulator. In the second method for introducing accumulation layers, n-type doping is provided, extending inward from the surface to form an n+n high-low junction (for n-type devices in this approach, electric charge is provided internally by occupied acceptors). Although both methods have been demonstrated in actual devices, detailed information is limited due to the proprietary nature of the techniques. It is probably accurate to estimate that the majority of devices within recent years have some form of surface accumulation layer. It is also noteworthy that the principle of accumulation for reflection of minority carriers is equally applicable at the contacts of semiconductor detectors to increase minority carrier lifetime and reduce sweepout effects as discussed in the preceding section.

15. TRANSVERSE FIELDEFFECTS“TRAPPING PHOTOCONDUCTIVITY’’ a . Background und Older Theories It has been recognized for some time that the lifetime (and therefore responsivity) of photoconductors could be significantly increased at reduced temperature and background if there were some mechanism for trapping of minority carriers in sites with low recombination probability. Evidence for such phenomena had apparently been seen from experimental results on (HgCd)Te photoconductors at low temperature and low background. Typically, there is a strong increase in responsivity as the temperature is reduced, followed by a plateau region, where the responsivity becomes nearly temperature independent, but varies as I/Q” with n varying between 0.5-1.0. The experimental results indicate clearly that some additional mechanism must be operating for reduction of carrier recombination, since photoconductor lifetimes under low temperature and low background conditions have been observed to be much longer than would be possible even for the smallest known bulk recombination process. It was recognized that such behavior could not be explained by any simple trapping mechanism, since for any typical density and distribution of traps, the traps will become fully occupied at reduced temperature leading to a strong reduction in responsivity. To circumvent this limitation, it was proposed (Broudy and Beck, 1976) that a continuum of minority-carrier traps exists within the lower half of the forbidden gap. Theoret-

192

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ical analyses using this model gave promising agreement with experimentation; best results were found for a variable continuum which diminished exponentially with energy separation from the valence band. However, to validate this model, it was found necessary (Broudy, unpublished analyses) to assume recombination probabilities for the trapped holes that were so small that they appeared physically unreasonable and inconsistent with the very high photoconductive gains measured on the best devices. For this and other reasons, an alternate theory was proposed (Broudy, unpublished analyses; Beck, unpublished analyses; Beck and Sanborn, 1979)for trapping photoconductivity that is based on a quite different mechanism in which traps, per se, do not exist.

b . Chlirg e SPp a ru t ion ( I nd 1ra tisve rs e Field Devices It is now clear that the very high responsivities and photoconductive gain observed in (HgCd)Te photoconductors as well as (probably) other semiconductors can be ascribed to the mechanism of charge separation due to built in transverse electric fields. The principle is essentially an expansion of that utilized in the accumulation device. Minority carriers are physically separated from majority carriers by the presence of a transverse electric field which may be generated by any of the mechanisms described above for the accumulation device. The charge separation, or transverse field effect may also be analyzed from a different, but equivalent, point of view, by recognizing that the photoconductor is essentially a field effect device with a floating gate. If the transverse field originates in the surface or insulator, the analog is a MOSFET device, whereas if it originates from a transverse junction, the analog is a JFET. The effect may be modeled as conductivity modulation due to changes of depletion layer width with background illumination. Since the analysis is less complex for the transverse junction case, only this device is discussed below. The results are similar for the transverse MOSFET device. The structural elements of these devices are illustrated in Fig. 15 (Beck and Sanborn, 1979). c. Charge Sepurution Analysis

The correlation with trapping behavior and general photoconductivity may best be illustrated from direct consideration of charge separation. From this viewpoint, greater photoconductive gain occurs because the minority-carrier lifetime is increased, being ultimately limited by thermally generated current across the transverse junction. The basic principles may be illustrated by a simple analysis (Broudy , unpublished analyses) of the basic concepts, beginning with Rbtthe transfer rate of minority carriers (usually holes) from the bulk into the p-type region across the

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

193

,/-

NEGAIIVE CHARGE

h

I

N CHANNEL I

vDP--Q I\ CHANNEL

(b)

FIG. 15. Structural elements of transverse field photoconductors (a) with negative charge and (b) with p + .

junction Rbt

=

K , AP,

(73)

where Ap is the excess photogenerated minority-carrier concentration and K , is a constant. It is assumed that the bulk lifetime of the excess minority carriers is long enough so that eventually they all diffuse to the junction region. The transfer rate of excess (due to illumination) carriers across the junction into the bulk is obtained from the standard diode current equation: =

Zo(eqVlmkT

- 1)

(74)

We now make the simple assumption that the voltage across the (floating) junction is proportional to Apt, the total density of excess holes in the p-type region (note that these holes reside at the depletion region across the junction). Rtb,the transfer rate of holes back into the main body of the photoconductor, is given by

Rtb =

J&-L;/”(@

@I

lmkT

- 1)

(75)

where K2 is another constant. In the steady state Rtb = Rbt,leading to the

194

R.

M. BROUDY

A N D V . J . MAZURCZYK

expression: An = Ap

+ Apj = & + __ log d lnkT K3

(1 +

@./kT

where K4 is a different constant, and where we have included the dependence of A P on the photon flux Q (qTQ/d = Ap);the first term being simply the bulk photoconductive gain. Then, in the small signal case, the photoconductive gain Gt is given by

where QBis the background photon flux. It can be seen at once that Eq. (77) has the qualitative features of the experimental results. At high background and low temperatures, the second term in the denominator predominates and the photoconductivity goes as QG’. At lower background and higher temperature, the first term predominates and the photoconductive gain goes exponentially with temperature with the activation energy El3

d . JFET Analysis Further physical insight may be gained from the JFET approach (Beck, unpublished analyses; Beck and Sanborn, 1979) which leads to the identical result to the charge transfer analysis. In this approach, the simple expression for the source-to-drain conductance Gsd is given by Gsd

= qpe(d

- xD)nw/L,

(78)

where w = device width, L, the distance between electrodes, and xDis the depletion width. From this viewpoint the change in excess charge, A p t , is directly determined by the change in depletion width. The depletion width can be calculated from the well known expression:

where ND is the donor concentration and where the built in potential = V,, - V,,, the difference between the built-in across the junction T,,, and open circuit potential. The photoconductive gain and voltage responsivity may be calculated from the current responsivity:

w, = arD/as,

(80)

where Inis the detector current (which from the JFET point of view may be considered to flow from drain to source to the device). Then, as in Part

5. (HgCd)Te

I, the expression for

PHOTOCONDUCTIVE DETECTORS

195

becomes

Since Z, = GsdVb,the depletion modulation determined responsivity becomes

It is convenient to relate the latter term to u,, , the open circuit potential of the floating junction:

(ax~/a Qs)

=

(ax~/a vd(avda Qs).

(83)

Thus the performance may be readily calculated by reference to standard photodiode derivations. Under the assumption of realistic bias levels, V,/kT >> 1 , and Vo, takes the simple well-known form for an illuminated floating junction (Beck and Sanborn, 1979)

+ sqQsA/Zo>.

Voc = fkT/q) Using Eqs. (79), (82), (83), and (84),

(84)

takes the convenient form

Alternately, the photoconductive gain G,, may be used in addition to, or in place of %N from Eq. (85) by noting that %N =

(A/hc)sqGw.

(86)

It should be noted that Eq. (85) has essentially the same form as Eq. (77). Beck and Sanborn (1979) have calculated the temperature and background dependence of & and G, from Eq. (85) for the diffusion limited junction case in the low voltage bias limit for x = 0.39 (HgCd)Te detectors with the following set of typical device parameters and operating conditions: QB = lo", 10l2,and 1013 photons/cm2 sec, N A = 6 x 10l6 in transverse junction, N D = 6 x 1014 cm-3 in body of photoconductor, q = 1.0, d =7 x cm, device thickness, E, = 17.2, dielectric constant, 7, = 1 x sec, bulk lifetime.

196

R . M . BROUDY A N D V. J . MAZURCZYK

,-OB =! x 1 0 t 2 photons/sec ,in2

og

photons/sec cmZ

t

t.0

0

"

'

2

1

4

"

"

6

1

"

8

"

~

1 0 4 2 t 4

tOOO/T ( K - 0 FIG. 16. Calculated gain of the transverse junction device versus temperature and background for the diffusion limited case at low bias for an x = 0.39 (HgCd)Te photoconductor.

The results are shown in Fig. 16. It should be pointed out that the above derivations represent only a first-order evaluation, since the unavoidable effects have not been considered of contacts and longitudinal potential variation along the JFET. Inclusion of these effects (Beck and Sanborn, 1979) will modify the background dependences as well as bias dependences. Similar approaches (Beck and Sanborn, 1979) for the externally initiated transverse field devices may be based on MOSFET theory. VI. Technology of (HgCd)Te Detectors

Modern (HgCd)Te photoconductive detectors are generally fabricated in arrays by methods of which many are quite similar to processes used for silicon integrated circuits, including photolithography, etching, vacuum and sputtering metallization, insulator deposition, and wire bonding. In addition, special techniques applicable to (HgCd)Te may be used, such as slab bonding and subsequent lapping and etching. Array performance close to the theoretical limit has been achieved in many cases.

5. (HgCd)Te

PHOTOCONDUCTIVE DETECTORS

197

A typical set of key steps for array fabrication are the following: (1) Start with good material: Develop methods for material evaluation and selection. (2) Obtain the material in slices sufficiently thick for ease of handling: Prepare the backside. (3) Bond the slab to a substrate: Epoxy is generally used. (4) Lap, polish, and etch the material almost to final thickness (usually close to cm) chosen to be thick enough to absorb almost all of the optical radiation and thin enough to minimize bias current. (5) Delineate the array using a photolithographic process with etching. (6) Accumulate the surface: This process will often also be performed between (2) and (3) above. (7) Metallize after further photolithography for contact and possibly active area delineation. (8) Wire bond for external electrical contact.

For similar reasons to those well known in integrated circuit pro-

FIG. 17. Section of a linear (HgCd)Te photoconductive array. Active area dimensions are 0.00125 x 0.002 in. (L x W) for each element.

198

-

4

R. M. BROUDY A N D V. J. MAZURCZYK

-

0

-

0

X

i.“

3 -

2 -

I

1

.. . FIG.18. D*performance under reduced background levels for a 60-element array with the configuration shown in Fig. 17.

cessing, scrupulous attention must be given to process technique, procedures, and cleanliness for all of these steps. Fig. 17 shows a photograph of section of a modern multielement PC array that has been prepared according to this procedure. The D* of this array is presented in Fig. 18, which shows values approaching the theoretical limit for the reduced background of this measurement.

REFERENCES Beck, J. D., and Sanborn, G. S . (1979). Air Force Materials Laboratory Rep. AFML-TR-79. Borello, S., Kinch, M., and Lamont, D. (1977). fnfrared Phys. 17, 21. Broudy, R. M. (1974). NASA Rep. CR-132512. Broudy, R. M. (1976). Frequency characteristics of high performance (HgCd)Te detectors, Proc. Infrared Informar. Symp. Detector Specidly Group. Broudy, R. M., and Beck, J. D. (1976). Pror. Infrured Informar. Symp. Detector Speciulty Group. Broudy, R. M., Mazurczyk, V. J., Aldrich N. C., and Lorenze, R. V. (1975). Advanced (HgCd)Te array technology, Proc. Infrured Informat. Symp. Defector Sperialty Group. Burgess, R. E. (1954). Physica 20, 1007. Burgess, R. E. (1955). Proc. Phys. Soc. London B68, 661. Burgess, R. E. (1956). Proc. Phys. Soc. London B69, 1020. Domhaus, R., and Nimtz, G. (1976). The properties and applications of the Hg,-,Cd,Te alloy system, in “Springer Tracts in Modem Physics,” Vol. 78, pp. 1-119. SpringerVerlag, Berlin and New York. Eisenman, W. L., Meniam, J. D., and Potter, R. F., (1977). Operational characteristics of

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PHOTOCONDUCTIVE DETECTORS

199

infrared photodetectors, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 12, Chapter 2. Academic Press, New York. Finkman, E., and Nemirovsky, Y. (1979). J. Appl. Phys. 50, 4356. Hooge, F. N. (1969). Phys. Lett. 29A, 129. Kinch, M. A., and Borello, S. R. (1975). Infrared Phys. 15, 11 1. Kinch, M. A., Brau, M. J., and Simmons, A. (1973). J. Appl. Phys. 44, 1649. Kinch, M. A., Borello, S. R., Breazale, B. H., and Simmons, A. (1977). Infrared Phys. 17, 137. Kinch, M. A., Borrello, S. R., and Simmons, A. (1977). Infrared Phys. 17, 127. Kingston, R. H . (1978). “Detection of Optical and Infrared Radiation.” Springer-Verlag, Berlin and New York. Kittel, C. (1961). “Elementary Statistical Physics,” p. 145. Wiley, New York. Kolodny, A., and Kidron, I. (1981). Infrured Phys. (to be published). Kruse, P. W. (1979). The photon detection process, in “Infrared and Optical Detectors,” Chapter 1. Springer-Verlag, Berlin and New York. Kruse, P. W., McGlauchlin, L. D., and McQuistan, R. B. (1962). “Elements of Infrared Technology: Generation, Transmission and Detection.” Wiley, New York. Levinstein, H. (1970). Characterization of infrared detectors, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 5 , Chapter I . Academic Press, New York. Long, D. L. (1970). Infrared Phys. 7 , 169. Long, D. (1977). Private communication, who refers to the work of J. R. Hauser and P. M. Dunbar, Solid State Electron. 18, 716 (1975). Long, D. (1977). Photovoltaic and photoconductive infrared detectors, in “Topics in Applied Physics” (R. J. Keyes, ed.), Vol. 19, Optical and Infrared Detectors. SpringerVerlag, Berlin and New York. Long, D., and Schmit, J. L. (1970). Mercury-cadmium telluride and closely related alloys, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 5 , Chapter 5 . Academic Press, New York. Mazurczyk, V. J., Graney, R. N., and McCullough, J. B. (1974). High performance, wide bandwidth (Hg,Cd)Te detectors, Opt. Eng. 13, 307. Milton, A. F. (1973). Proc. Infrared Informat. Symp. Detector Specialty Group. Rittner, E. S . (1956). In Photoconduct. Conf. (R. Breckenridge, B. Russell, and E. Hautz, eds.), p. 215ff. Wiley, New York. Shacham-Diamand, Y. J., and Kidron, I. (1981). Infrared Phys. 21, 105. Schmit, J. L. (1970). J . Appl. Phyc. 41, 2867. Schmit, J. L., and Stelzer, E. L . (1969). J . Appl. Phys. 40, 4865. Smith, D. (1981). Submitted for publication. Van der Ziel, A. (1959). “Fluctuation Phenomena in Semiconductors.” Butterworth, London. Van Vliet, K. M. (1958). Proc. IRE 46, 1004. Van Vliet, K. M. (1967). Appl. Opt. 6 , 1145. Van Vliet, K. M., and Fassett, J. R. (1965). Fluctuations due to electronic transistions and transport in solids, in “Fluctuation Phenomena in Solids” (R. E. Burgess, ed). Academic Press, New York. Williams, R. L. (1968). Infrared Phys. 8, 337.