Analysis of the R0A product in n+-p Hg1−xCdxTe photodiodes

Infrared Phys. Vol. 28, No. 3, pp. 139453,

1988

0@20-089lj88 $3.00 + 0.00 Copyright 0 1988 Pergamon Press plc

Printed in Great Britain. All rights reserved

ANALYSIS

OF THE R,A PRODUCT PHOTODIODES

IN n +-p

Hg, _.Cd,Te

A. ROGALSKI Institute of Technical Physics, WAT, 01-489 Warsaw 49, Poland (Received 28 Augusf 1987) Abstract-The influence of different junction current components (diffusion current for radiative and Auger 7 recombination mechanisms, tunneling and depletion layer currents) on the &,A product of n+-p -Hg,_,Cd,Te photodiodes is considered. The considerations are carried out for the 77-300K temperature region and 1-15 pm cutoff wavelength. Optimum doping concentrations in the p-type region of n +-p abrupt junctions are determined, taking into account the influence of the tunneling current and of a fixed surface charge density of the junction passivation layer. Results of calculations are compared with experimental data reported by many authors. An attempt is made to explain the discrepancy between theoretical calculations and experimental data.

INTRODUCTION

The emergence of Hg, _,Cd,Te as the most important intrinsic semiconductor alloy system for IR detectors is well established. (l-4)Second generation systems with electronic scanning on the focal plane are under intensive research and development efforts. At present 2-D arrays contained above IO4of elements are developed.“) Hg, _ .Cd,Te j unction photodiodes based on p-type material have found important applications in hybrid 2-D focal plane arrays (FPAs) as well as in heterodyne detection systems. (4)Also, p-type Hg, _ .Cd,Te has potential applications in monolithic n-channel IR charge coupled devices.@) The realization of Hg, _,Cd,Te photodiodes is usually based on an n +-p structure. Theoretical reasons for the choice of the n +-p configuration includes the significantly longer minority-carrier diffusion length in p-type Hg, _,Cd,Te and the possibility of achieving longer minority-carrier lifetimes in p-type material rather than in an n-type one of comparable carrier concentration. The most common technique for producing n+-p junctions is by ion implantation into a p-type substrate.c2) At present boron implantation becomes the main technique of preparation of n +-p photodiodes, perhaps due to the fact, that boron being a well-behaved shallow donor impurity in Hg, _ ,Cd,Te, is also a standard implant for silicon. The most significant parameter which characterizes the photodiodes is the zero bias resistancejunction area, &A product. There have been numerous theoretical discussions of &A product for Hg, _,Cd,Te photodiodes. Up till now the most comprehensive analysis has been done by Reine et ~1.c~)Also strongly asymetric hi-lo n+-p junctions have been considered. However these considerations have not included the effect of the Auger 7 recombination mechanism in p-type material, the influence of interband and trap-assisted tunneling transitions on R,,A product of n +-p junctions. In this paper the influence of different junction current components on R,,A product of n+-p Hg, _,Cd,Te photodiodes is considered. The considerations are carried out for the 77-300 K temperature range and I-15pm spectral cutoff wavelength. Also included is the analysis of influence a positive fixed surface charge density of passivation junction layer on optimum doping concentrations in p-type region. Results of calculations are compared with the experimental data reported by many authors. This paper completed the results of Reine et ~1.‘~)and submit a certain data into estimation of performance limits for n +-p Hg, _ .Cd,Te photodiodes. DARK

CURRENT

IN

p-n

JUNCTION

Several physical mechanisms are involved in determining the dark current-voltage 139

character-

A. ROGALSKI

140

istics of the photodiode. The dark current is superposition of current contributions from three diode regions: bulk, depletion region and surface. Between them we can distinguish:“) 1. Thermally generated current in the bulk and depletion region: (a) diffusion current in the bulk p and n regions, (b) generation-recombination current in the depletion region, (c) band to band tunneling current, (d) intertrap and trap to band tunneling, (e) anomalous avalanche current, (f) ohmic leakage across the depletion region. 2. Surface leakage current: (a) surface generation current from surface states, (b) generation current in a field induced surface depletion region, (c) tunneling induced near the surface, (d) ohmic (or nonohmic) shunt leakage, (e) avalanche multiplication in a field induced surface region. Each of the above components has its own individual relationship to voltage and temperature. Below, we will be concerned with the current contribution of high-quality photodiodes with high &A products limited by: (a) diffusion outside of the depletion region, (b) generationrecombination within the depletion region, (c) tunneling through the depletion region. Surface leakage effects will not be discussed here since they may be minimized by proper surface passivation or guard ring structures. Avalanche multiplication will not be involved because it is not observed over the small bias voltage range. Diflusion

current

Diffusion current is the fundamental current mechanism in a p-n junction photodiode. The R,,A product determined by this current is:(*)

+D,p,y,ch[(t L,

+d-x,-w)/L,J+sh[(t

Y2Sh[(f+

d -x,,

+d-x,-w)/L,]

- w)/L,] + ch[(t + d -x,

- w)/L,]

(1)

where y, = s,L,,/Dh, y2 = s2L,/D,, p,,,, and n,,,, are the concentrations of minority carriers on both sides of the junction, s, and s, are the surface recombination velocity at the illuminated and back photodiode surface. This relation is fulfilled for the 1-D photodiode model with an abrupt junction where the spatial charge of width w surrounds the metallographic junction boundary x = t, and two quasi-neutral regions (0, x,) and (x, + w, t + d) are homogeneously doped. For a junction with thick quasineutral regions (x, >>Lh , t + d - x, - w >>L,) and when the Boltzmann statistic is valid, expression (1) assumes the form: (%A),

=

(kT)“z &(y+j$y]-

(2)

where n, is the intrinsic carrier concentrations, ppoand n, are the hole and electron majority carrier concentrations, pLpand ph are the electron and hole mobilities, and re and f,, the electron and hole lifetimes in the p- and n-type regions, respectively. Analysis of the effect of structure of a clasical photodiode (of thick p-type region) has shown (‘) that the ROA product for junction depths 0 < t < 0.2& and surface recombination velocities 0 < s, < lo6 cm/s differs from product (R,A), for a photodiode with thick regions on both sides of the junction by a factor of 0.3-2. For n +-p diode structure, the junction resistance is limited by diffusion of minority carriers from the p side into the depletion region. In the case of conventional bulk diodes, where d >>L,: (3)

Analysis of the &A product n +-p Hg, _ .Cd,Te photodiodes

141

By thinning the substrate to a thickness smaller than the minority-carrier diffusion length (thus reducing the volume in which diffusion current is generated) the corresponding bA product increases, provided that the back surface is properly passivated to reduce surface recombination. From eqns (1) follows that in the case n +-p junction if the thickness of the p type region is such that d<. L,, we obtain: (4) As a result, &A can increase by a factor of L,/d. Generation-recombination

current

It appears that the space-charge region generation-recombination current could be more important than the diffusion current especially at low temperatures, although the width of the space-charge region is much less than a minority carrier diffusion length.(g) The generationrecombination current density under reverse-bias voltage and for forward-bias voltage values that are less than the bulk-in voltage V,, by several (kT/q)was derived as:

where td), rhoare the carrier lifetimes within the depletion region. The functionf(b) is a complicated expression involving the trap level E,, the two lifetimes and applied voltage V. For small applied bias, the functionf(b) may be taken as independent of V. We can also neglect the bias dependence of the depletion width w for small bias. The zero bias resistance-area product is then equal:

For simplicity, we further assume r M)= Q, = ~~ and f(b) = 1. Then, eqn (6) becomes:

where vb = kT ln(N,N,/nf).

In evaluating eqn (7) the term of greatest uncertainty

is rO.

Tunneling current

A third type of dark current component that can exist is the tunneling current caused by electrons directly tunneling across the junction from the valance band to the conduction band (direct tunneling) or they indirectly tunnel across the junction by way of intermediate trap sites in the junction region (indirect tunneling or trap-assisted tunneling). The usual direct tunneling calculations assumes a particle of constant effective mass incident on a triangular(‘O~l’) or parabolic(lO*“) potential barrier. Assuming approximation the triangular potential barrier for the n+-p junction we obtain:

(8) where m* = mtmf/(mf + mh+) is the effective mass characterizing the edges of the conduction and light-hole bands. The more physically realistic tunnel calculation was developed by Anderson who based it on the k-p theory, developed expressions for direct interband tunneling for asymmetrical abrupt p-n junctions in the InSb-like semiconductors:(‘3)

CR,&A =

(3ncoEs/qN,) “*E;”

w~3Y’2(~/q)(P/q) q(3qN,/t,~,)“*E;‘*

exp

4*2”yP/q)

1’

(9)

where P is the interband matrix element. Trap-assisted tunneling in Hg, _,Cd,Te p-n junctions at low temperatures has been reported. Small but finite acceptor activation energies are generally observed in p-type material.(‘4) Recently the trap-assisted tunneling theory with specific application to Hg, _ XCdXTehas been developed by

142

A. ROGALSKI

Wow, (“) Kinch et uZ.,(‘~“‘)Anderson and Hoffman. (I*)The general expression for the trap-assisted tunneling R,A product takes the form of:c2) 1

(10)

(& AL1 = exp 4 qBo

where: B

0

=4xqm*E I;

8312=

Assuming approximation

(RON,,=

h3 qEh 271(2m*)l” ’

0 312 EL = (E, _ Et)1/2’

for the electric field in n +-p junction as E = (2qNaVb/~oc,)“2, we obtain:

h2(~Ot,)“2(Eg - Et)“’ 3 2q7’2(m*NaV,,)‘12

N, C

1+

s Bt 1

q3’2h(NaVb)“2

2n (m* co6 )‘I2 (E - E )‘I2 kT x exp

n2(m*~o~3)‘~2(Eg - Et)“* q3’2h(2NaVb)“2

’

(11)

In the above expression N, is the valence-band effective density of states, and Et is the activation of acceptor center. In comparison with direct tunneling, indirect tunneling is critically dependent not only on doping concentration but also on location of acceptor centers in the band-gap. energy

EFFECT

OF FIXED

CHARGE

OF PASSIVATION

LAYER

The surface of actual devices is passivated in order to stabilize surface against chemical and heat-induced changes as well as to control surface recombination, leakage and related noise. A native oxide and overlying insulator introduces fixed charge states which next induce accumulation or depletion at the semiconductor-insulator surface. The effect of a fixed insulator charge on the effective junction space-charge region is discussed in Refs (2, 19). When sufficient fixed charge is present, accumulated and inverted regions as well as n-type and p -type surface channels are formed, what degrade the device performance. For the case of a p-type semiconductor, a good approximate criterion for the onset of strong inversion is that the electron concentration near the surface should exceed the concentration of the substrate impurity ions. It appears that such strong inversion conditions are fulfilled if the fixed positive insulator charge per unit area is given by:(3*‘9) N i/2 ns=z E,c,,kTN,ln” (12) 4 > ( It has been found that native oxides on Hg, _,Cd,Te possess positive fixed charges which invert the surface of p-type material. Although they form high-quality surface passivation for n-type photoconductors, they are inadequate for devices on p-type material. The problem of finding a suitable dielectric film with proper interface properties to passivate the surface of p-type material has been rather difficult. Deposited dielectrics include evaporated ZnS and Si02 photochemically deposited at low temperatures. The deposited ZnS forms inconsistent interfaces on surfaces which have been exposed to chemicals during the processing cycle. (“) SiO, deposited by low temperature CVD method exhibits improved interface properties. However the best results with SiOz have been obtained when a few layers of “natural” or native oxide are present on the Hg, _,Cd,Te before the Si02 deposition.“‘) Recently a novel anodic sulfidization process for forming native sulfide films on Hg, _.Cd,Te has been described. c2’) This process is particularly suitable for photodiodes implemented on p-type material. The fixed surface charge density of the order of 5 x 10” cm-’ are observed.“‘) Band bending at the p-n junction surface can be controlled by a gate electrode overlaid around the junction perimeter on the insulating film. cz3)Obviously, the ideal passivation would be a wide gap insulator with no fixed charge at the interface (flat-band conditions).

143

Analysis of the &A product n +-p Hg, _,Cd,Te photodiodes R,,A

PRODUCT

ANALYSIS

In this section the R,,A product for the one-sided abrupt n+-p Hg, _,Cd,Te junctions in dependence of the doping concentration and cutoff wavelength is analysed. The influence of the diffusion current (for radiative and Auger recombination), of the tunneling and depletion layer currents are considered using equations (3), (7)-(9) and (11). For numerical calculations we have used the values of parameters described in the Appendix. The results of calculations are compared with experimental data reported by other authors. As a result of the utilization of high-quality Hg, _,Cd,Te material with good surface passivation, the experimental data presented here show diodes with high R,,A products. Since the bandgap of Hg,_,Cd,Te is tunable compositionally, the cutoff wavelength of the photodiodes can be tailored from 1 to 14 pm. For convenience, we divide the spectral regions into three categories: short wavelength IR (SWIR) for l-3 pm; medium wavelength IR (MWIR) for 3-5 pm; and long wavelength IR (LWIR) for 8-14 pm. L WIR photodiodes

The dependence of the %A product components on the dopant concentrations for n+-p Hg,,,, Cd,,2Te abrupt junctions at 77 K is shown in Fig. 1.(21-26) We can see that the performance of the junctions at zero bias is limited by the junction tunneling current if the substrate doping is too high. The doping concentration of lO”jcm-’ or less is required to produce implanted photodiodes with high R,,A products. Below this concentration the &A product is determined by diffusion current with minority carrier lifetime conditioned by the Auger 7 process. Numerical calculations carried out according to equations (8) and (9) indicate that formula (8) is a good approximation to the more physically realistic tunnel calculations developed by Anderson.(13)

1

4

77K

10-l

1o14

x=0.22

1o16

1o'5

10"

N, [cm? Fig. 1. The dependence of the R,,A product on acceptor concentration for n +-p H&,,,,,C&,,,Te photodiodes at 77 K. The theoretical lines are calculated from equations (3), (8), (9) and (11). The experimental values are taken from Refs (23) (O), (24) (A), (25) (0) and (26) ( x ).

A. ROGALSKI

144

In the calculation of the trap tunneling limited R,A product, the acceptor activation energy E, = 10 meV and the built-in voltage I$, x E,/q have been assumed. The acceptor activation energy of about 10 meV has been reported for p-type Hg, _,Cd,Te. 0.‘4)A comparison of our calculations with the experimental data as seen from Fig. 1 thus suggests that the trap assisted tunneling may be one of the possible reasons limiting the ultimate achievable &A product of the implanted n +-P Hg, _.Cd,Te photodiodes. The presented (R,A),, calculations should be treated as approximative, because the parameters taken in calculations are also approximative and owing to uncertainty in the space-charge region electric field E, which enters into the exponential via term A, in equation (10) and drastically affects the magnitude of (&A),,. Furthermore, the @,,A),, product is critically dependent on Et value [see equation (ll)]. The effect of the radiative recombination is significant for N, < 10” cmm3. However to obtain the possibly highest value of the R,A product and avoid the effect of a fixed insulator charge of the passivation layer the technological process of photodiode preparation should be performed in the manner to obtain a dopant concentration of about lOI cmm3. Figure 2(a) illustrates the relation of the fixed positive insulator charge per unit area to the acceptor substrate concentration at which strong inversion with n-type surface channel take place. The lowest fixed surface charge density of Hg, _,Cd,Te passivation layers are about 5 x 10” cm-* this means that the dopant concentration should not be smaller than 3 x 10” cme3 in Hg, _,Cd,Te photodiodes (see Fig. 2a). The similar dependences for others n +-P Hg, _,Cd,Te photodiodes operated at 193 and 300 K are shown in Figs 2(b) and 2(c). From the comparison of the theoretical curves with the experimental data it may be seen that a satisfactory consistence has not been achieved. The main reason for these discrepancies is probably due to inconsistence of the abrupt junction model with the experimental junction profiles. The junction grading cannot be determined accurately experimentally. The Anderson tunneling theory assumes a uniform charge model for the potential barrier and nonparabolicity of the band structure. Smaller values of tunnel current (higher experimental R,,A products for acceptor concentration lOI < N, < 2 x lOI cmP3-see Fig. 1) can be associated with a decrease of electric field away from the metallographic junction boundary. Furthermore, the tunnel current calculations are strictly valid only for the approximately empty well condition. As the well fills, the tunnel current tends to decrease due to the collapse of the well.

IO6

1o12

1o13

1o15

lo'& N, I cni’ 1 Fig. 2(a)

lo=

1o17

145

Analysis of the &A product n +-p Hg, _,Cd,Te photodiodes

10’2

I

I

/

I

I

193 K 10" -

nE 2 10'0 -

x- 0.28

2 x=0.32

c" x=036 109

I

._

lfl6

I

I

I

I

I

10'3

10'4

10'5

1o16

10'7

I

I

10'8

N, [cm-‘1

(b) 10'3

I

1

300 K

log 1

10'3

10“

10'5

10'6 N, [cm-'1

10'7

10'8

(4 Fig. 2. The relation of the tixed positive insulator charge density to acceptor concentration at which strong inversion takes place for n +-p Hg, _,Cd,Te photodiodes at 77 K (a), 193 K (b) and 300 K (c). In the calculations equation (12) was used.

Experimental data from Ref. (26) gives the value 200Rcm* which is situated above (&A),, curve. This improvement in &A is probably due to a diffusion-limited regime. With a p-type epilayer thickness.of 5-20 pm, R,,A can increase by a factor of 2-10 [see equation (4)]. Following the above considerations the optimum doping substrate concentration for n +-p Hg, _,Cd,Te junctions operated at 77 K is about lOI6cme3. This concentration can also be assumed as optimal for n +-p photodiodes at 77 K with cutoff wavelength between 3 and 14 pm.‘*‘) On account of this the dependence of the @,,A), product for radiative and Auger recombination on the long wavelength spectral cutoff at 77 and 145 K, was calculated using equation (3) and A satisfactory consistence between the theoretical curves and IV, = 10’6cm-3 (Figs 3 and 4). (28*29) the experimental data has been achieved for both temperatures. The experimental data at 77 K show greater spread, probably due to additional currents in the p-n junctions, such as the surface leakage current. The Auger 7 recombination contribution &A is decisive and increases with & increasing. Moreover, in comparing with radiative recombination the effect of Auger 7 recombination comes to the fore with temperature increasing (compare Figs 3 and 4).

Fig. 3. The dependence of the l[bA product on the long wavelength cutoff for n $-p LWIR Hg, _ $d,Te photodiodes at 77 K. The theoretical lines are calcufated from equation (3) for radiative and Auger 7 r~ombination mechanisms, and N, = 10’6cm-3. The dashed curve refers to the R,,A effective product. The experimental values are taken from Refs (2) (o), (28) (0) and (29) (i).

Fig. 4. Tbe dependence of the R+,Aproduct on the long wavelength cutoff for n +-p LWIR Hg, -.Cd,Te photodiodes at 145 K. The theoretical lmes are calcuIated from equation (3) for radiative and Auger 7 recombination mechanisms, and N, = 1Ou’cmm3. The dashed curve refers to the %A effective product. The expe~mental values are taken from Ref. (2).

Analysis of the &A product n +-p Hg, _,Cd,Te photodiodes

147

In hybrid FPA technology the photodiode impedence must be greater than that of the charge coupled device input to achieve efficient injection of charge. For example, to achieve an 80% injection efficiency from a diode with 2, = 10.5 pm in a background flux of 2.5 x 10i6cm2s-r, the value of &A > 5 Q cm2 is required. f30)This value is presently achieved in fab~cation of 2-D diode arrays. However requirement of high injection efficiency becomes progressively more difficult as the cutoff wavelength is increased. Consistently 2-D arrays have currently been restricted to cutoff wavelengths below 10 pm.(3’) A4 WZR p~otodiod~s

MWIR FPAs were the first to be developed, and many mature technologies have been used to demonstrate FPAs.(~-“) The dependence of the R,A product components on the dopant concentrations for n+-p Hgo,,,Cd,,,Te abrupt junctions at 77 and 195 K is shown in Fig. 5. (2*26*32,33) At 77 K the RoA product is determined by the generation-recombination current of the junction depletion layer. The theoretical estimates yield for the radiative and Auger 7 r~ombinations values of the &A product by three orders of magnitude larger. Space-charge region generation-recombination current is reduced to the point where the diffusion current generally limits the &A product of 3-5 ,um photodiodes at temperatures around 170-200 K. For MWIR photodiodes in the range of higher temperatures, unlike LWIR photodiodes, contributions of the radiative and Auger 7 recombination processes are comparable. It can be seen that for photodiodes at 195 K the region of optimum concentrations is shifted in the direction of higher concentrations in comparison with photodiodes operated at 77 K. As the optimal doping concentration N, = lOi6cmw3 can be assumed. We observe sufficiently good agreement of experimental data with expected theoretical values. The data in Fig. 5(b) roughly follow the (R,A), line which has a slope of l/2 (Na” dependence) and confirmed that minority carrier lifetimes data obtained on these photodiodes indeed are determined by radiative recombination mechanism [see equations (3) and (A@]. Also the experimental values for n +-p Hg,,, Cdo,3sTe photodiodes in Fig. 6 follow the dashed line which has a slope of l/2. However in this case a satisfactory consistence has been achieved for the time of recombination via Shockley-Read centres in the depletion layer [see equation (7) z. = lo-* s]. The theoretical evaluation of radiative recombination gives R,A values which are greater by two orders of magnitude. It should be noticed that the influence of space-charge region generation-recombination current increases in the shorter wavelength photodiodes. Data for the &A products of MWIR n +-p Hg, _,Cd,Te photodiodes vs cutoff wavelength at 77 and 193 K are shown in Figs 7(24~26~29.32~34-37) and 8. (2*33,34,38*39) The experimental values at 77 K are more scatter probably due to the onset of leakage current mechanisms. The main contribution to the current flow through the junctions comes from the generation-recombination current in the depletion layer. At 193 K the contributions of the radiative recombination and the Auger 7 r~ombination in determining of R,A product are comparable. S WZR photodiodes

Among the first applications considered for Hg, _,Cd,Te photodiodes was room temperature detection of radiation in the I--3 pm spectral region. c40)Recently, hybrid FPA architecture with SWTR Hg, _ J Cd,Te photodiodes has been rapidly developed. (3’)Also there has been an increasing interest in the use of SWTR photodiodes for fiber optic applications.(4’,4z) n +-p SWIR Hg, _ .Cd,Te photodiodes are formed either by a standard ion implantation into as-grown p-type material or by doubly grown (liquid phase epitaxy technique) p- and n +-layers. The carrier concentration of the p-type material is usually between 5 x lOI cme3 and 10’6cm-3. The R,A products of SWIR n +-p Hg, _ XCd,Te photodiodes are shown in Fig. 9(ao,4’A3) as a function of cutoff wavelength. The theoretical lines are calculated from equation (4) for radiative and Auger 7 recombination mechanisms and from equation (7) for T,, = lo-* s, N, = lOI crnw3. From analysis of data presented in Fig. 9 we can see that the generation-recombination process in the space-charge region is dominant and determined the R,,A product. Especially the experimental data according to Ref. (41) are in very good agreement with theoretical calculation for the generation-recombination case with z. = IO-* s. The influence of diffusion current on R,A product is essential in the region of cutoff wavelength above 3 pm.

A.

148

i

ROGALSKI

IO9

ii

a

2 IO'

10'

IO6

lo5 1o15

IO"

10‘6 N,

IO”

lcrnm3 1

(4 10):

,

lo-*

I

IO’&

, ,,I,,

I!lI/Ill

,

I 1o15

, ,,/,/

/lIllill

,

I 10’6

I /lIlllI

I 1o17

I IIIIII 1o18

No [ cme3I

Cb) Fig. 5. The dependence of the R,A product on acceptor concentration for n+-p H&,,C&,,,Te photodiodes at 77 K (a) and 195 K (b). The theoretical lines are calculated from equations (3). (7)-(9) and (11). The experimental values are taken from Refs (2) (O), (26) (O), (32) (A) and (33) (+).

Analysis of the %A product n ‘-p Hg, _ .%Cd,Tephotodiodes

I

8 -I 0

149

150

A. ROGALSKI

I

Analysisof the &A product n +-p Hg, _,Cd,Te photodiodes THERMAL

DETECTIVITY

I51

LIMITS

The detectivity of photodiode determined by the Johnson-Nyquist noise of the zero bias resistance & and the shot noise of the current generated by the background photon flux 0, is given by?*)

1 -l/2

(13)

where q is the quantum efficiency, c is the speed of light, and A is the wavelength of the incident radiation. This formula is valid for an ideal detector. Additional noise cont~butions, such as l/f noise, are not included in equation (13). If the noise conditioned by the background photon flux is greater than the thermal noise of & (i.e. when 4kT/&A
author wishes to thank Dr Z. Djuric for helpful suggestions and useful discussions during the

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APPENDIX Energy Gap

Hansen et uI.(~) made a critical analysis of the experimental data concerning E&x, T) and derived the following expression: E,(x, T) = -0.302 + 1.93x + 5.35 x 1O-4 T (1 - 2x) -0.810x2 + 0.832x3.

(AlI

This empirical expression has been fitted to energy gap data from a variety of sources covering the ranges 0 Q x < 0.6 @lusx=1)and4.2~T~3OOK. Intrinsic Carrier Concentration

Hansen and Schmit’45)calculated the intrinsic carrier concentration n,(x, T) using equation (Al) and the most recent available values of fundamental parameters. They also took into account the nonparabolicity of the bands by using the k*p method and the average value of the heavy hole mass m ,$,= 0.443 m. The momentum matrix element was assumed to be P = 8.49 x lo-* eVcm. The intrinsic carrier concentration is approximated by the expression: n, = (5.585 - 3.820x + 1.753 x 10e3 T - 1.364 x 10s3xT) lOI Ei’4 T’12 exp (-EJ2

kT)cxxm3.

(A2)

This expression has been fitted within 1% of the calculated n, over most of the fitting range (x < 0.7; 50 < T $300 K). Effective Masses

The energy-dependent

masses rn: and rn$ were calculated within the Kane band model: “=1+ m: “=l__ mZ

(A3)

?$+$-A) 4mP2

(A4)

3h 2E,

where P = 8.49 x IO-‘eV cm and A = 0.9 eV. The heavy-hold effective mass is rn& = 0.443 m. Carrier Mobility The carrier mobility is evaluated by generating a formula to fit the available Hall data. An expression that approximates

the data for 0.2
9x lO*b with Z2”

b = (O.~/X)‘.~

a = (0.2/x)O.6

where for T > 50 K

Z = T

for T < 50 K

Z = 1.18 x 105/[2600- (T - 25)*O’].

The hole mobility may be estimated at -0.01 times the electron mobility of equation (A5). The electron mobility in p-type material are substantially lower than those in n-type material of similar composition ratio. The dominant scattering mechanisms for electrons are by optical phonons, by disorder, and by charged centers.(46) Since p-type material is heavily compensated,‘47A8)one can expect a decrease in the electron mobility by a factor of p/(p f 2N,) in P-type material of similar majority concentration. (49)A most recent work shows a donor concentration comparable to the hole concentration. (501 Thus a reduction in mobility by a factor of 2-3 can be expected. In calculations the factor of 3 was assumed.

Analysis of the &A product n +-p Hg, _ .Cd,Te photodiodes Radiative The band-to-band

radiative recombination

Recombination

153

Lifetime

lifetime is given by: nf 7a =

W)

G,(%+Po)

where?‘) Ga = 2.8 x 10” cotm/?T’/* exp(--E,/kT)

[Ei + 3kTE, + 3.75(kT)2]

647)

For j?, the following expression is obtained:o’) /I = 2.109 x 10’ GT

“*(cmr~2eV-r~z) (. > It should be noticed that Humphreyso2) re-examined the Van Roosbroeck and Shockley radiative recombination theory. cJ3)Due to reabsorption effects, this lifetime is an underestimate of what would be measured in the bulk and further improvements in the state of the art for IR detectors are possible. Auger Recombination

Lifetime

Petersen and Casselman(“*55) argued that the Auger 7 process is the dominating Auger mechanism for p-type Hg, _,Cd,Te in the extrinsic region. The lifetime associated with the Auger 7 process is: 2nT7’

7A7 =&$gFQ7

(A9)

where rk, is the Auger lifetimes for intrinsic material and can be approximated by r A, = y7i,. y is the ratio between the intrinsic lifetimes, and is given by!“) M&)(1 - 5&/4kT) m:(l - 3E,,/2kT) For the Auger 7 process the threshold energy Et,, u E,. From Kane’s nonparabolic approximation m:(&,)/m~ over the range 0.16~x~0.40 and 50K<7’$3OOK, 36~66. The intrinsic lifetime for the Auger 1 process is equal:(“) y=2

WO) z 3, and

115W’

I+ 2m:/m: Ep (Al 1) 1 l +mt/mf kT where F, and Fz are the overlap integrals of the periodic part of Bloch’s function. In practice the value of (F, F, 1is taken as a constant equal to anywhere between 0.1 and 0.3. We have chosen the value of 0.2.o’) Schacham and Finkman have presented experimental evidence that the Auger 7 process is a dominating recombination mechanism in p-type material. However, for the ratio between the intrinsic Auger lifetimes y, they used the same functional dependence as Casselman and Petersen@‘) [see equation (AIO)] but ten times larger values. TX,= 3.8 x IO-‘* (c,,e,)2(m/m:)(l

+ m:/rn~)“* (1 + 2m:/m,‘)

Static

(E,/kT)3’2exp

and High Frequency

Dielectric

Constants

The dielectric constants are not a linear function of x and that a temperature dependence was not observed within the experimental resolution. The dependence of t, and L, on x and T can be approximated by:@‘)

INF

28/3-B

t, = 20.5 - 15.5x + 5.7x2

(Af2)

c, = 15.2 - 13.7.x +6.4x=.

(At3)