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Infrared Physics & Technology 51 (2008) 259–262

Unified carrier density approximation for non-parabolic andhighly degenerate HgCdTe semiconductors covering

SWIR, MWIR and LWIR bands

Sudha Gupta *, R.K. Bhan, V. Dhar

Solid State Physics Laboratory, Lucknow Road, Delhi 110054, India

Received 24 April 2007Available online 27 July 2007

Abstract

We propose a unified approximate analytical expression for the carrier density of HgCdTe semiconductors that have non-parabolicenergy bands and are highly degenerate. The proposed expression is without any adjustable parameter. It can be applied to HgCdTe forthe case where electron densities are very high and the material is strongly degenerate, e.g., a highly accumulated surface due to passi-vant-induced negative fixed charge density in an n-HgCdTe photoconductor device. The proposed expression is simultaneously valid forSWIR, MWIR and LWIR bands.� 2007 Elsevier B.V. All rights reserved.

Keywords: Carrier density; Narrow gap semiconductor; Degenerate; Non-parabolic energy bands; HgCdTe

1. Introduction

The traditional parabolic Boltzmann’s approximation ofcarrier statistics is not applicable to the majority of narrowgap semiconductors (NGS), for example HgCdTe, due to anon-quadratic band structure. Boltzmann’s approximationalso fails when the carriers are degenerate i.e. (Ef � EC)/kT > 0. To estimate electron density, n in these NGS onehas to use the Fermi Dirac integral valid for non-parabolicbands. The detailed numerical tabulation of this integralwas done by Bebb and Ratliff [1] and simple analyticalapproximations for a variety of special cases were givenby them. However, in the case of strong degeneracy i.e.the normalized Fermi energy, / > 10 was not treated bythem because it is difficult to handle. It was stated by theauthors that this case is not of much use [1]. However, itis well known that infrared (IR) photoconductive detectorsbased on HgCdTe material use anodic oxide as a passivant,which yields positive fixed charges (Qf) of the order of

1350-4495/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.infrared.2007.07.008

* Corresponding author.E-mail address: 117@ssplnet.org (S. Gupta).

1012 cm�2 at the surface of n-HgCdTe [2]. This makes thesurface highly n+ with carrier density of the order of1017–1018 cm�3. As a result, the normalized Fermi energy,/, at 77 K is very high at around 40–50 [3,4]. Hence, it isdesirable to have an approximation of the Fermi integralfor / > 10 for the case of passivated HgCdTe detectors.Bhan and Dhar presented a detailed model for the calcula-tion of Qf including degeneracy and non-parabolicityinvolving the use of very high values of / up to 50 [5].However, the error in the approximate expressions of nused by these authors was very high (100% near / � 30).Therefore, more general and accurate expressions of n aredesirable for such calculations.

Recently, detailed analysis of various approximationsavailable in the literature were reviewed by Bhan and Dhar[6]. Additionally, in Ref. [6], the authors presented two sim-ple approximations for carrier density, n, applicable toMWIR and LWIR bands separately. In the present paperwe propose a single analytical approximation applicableto SWIR, MWIR and LWIR bands simultaneously. Itretains the basic advantages as mentioned in Ref. [5],besides being applicable to a larger range of wavelengths.

Fig. 1. Comparison of the present approximation with the Bebb’s solutionof the Fermi integral for typical SWIR, MWIR and LWIR cases.

Fig. 2. Comparison of the present approximation with the Bebb’s solutionof the Fermi integral as a function of cut-off wavelength (range 1–17 lm)for / taking values between 10 and 50.

260 S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262

This will be useful for modeling the moderate and highlydegenerate surfaces in n-HgCdTe semiconductor devices.

2. Results and discussion

The carrier density, n, including band non-parabolicity,valid for both NGS and wide band semiconductors is givenby [1,7,8]:

n ¼ 2N Cffiffiffipp

Z 1

0

e1=2ð1þ aeÞ1=2ð1þ 2aeÞde1þ expðe� /Þ ð1Þ

where e = (E � EC)/kT is the normalized energy. Theabove integral is not analytically solvable. Its variousapproximations for some specific situations were given inRef. [1]. In the above relation, a is the coefficient of non-parabolicity that can be calculated for the conduction bandfrom the k Æ p model [8] as follows:

a ¼ 1

eg

ð1� m�Þ2 ð2Þ

where eg = (EC � EV)/kT is the normalized band gap andm* = me/m0, me being the electron effective mass and m0

the free electron rest mass. There are only two basic mate-rial parameters in Eqs. (1) and (2) namely, eg and me.

In the present analysis, me is calculated following Weil-er’s relation (Eq. (5) of Ref. [9]) taking band gap depen-dence into consideration as follows:

me

m0

¼ 1þ 2F þ 0:33Ep

2

EG

� �þ 1

ðEG þ DÞ

� �ð3Þ

where F = �0.8, Ep = 19 eV, EG is the band gap in eV andD = 1 eV. The temperature T has been assumed to be 77 K.

2.1. Single approximation for SWIR, MWIR and LWIR

case

Following Eqs. (4) and (7) given in Ref. [6], in this sec-tion we propose a new modified version of the Fermienergy, /, for applications in SWIR (1–2.5 lm), MWIR(3–5 lm) and LWIR (8–14 lm). This has been obtainedby further tuning the power of the exponents, as suggestedin Ref. [6] and completely eliminating the additive con-stant. The new modified relation is given below:

/ ¼ log

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6:143aþ 0:01ð1� m�Þ2

ð1� m�Þ2B0

sn=N Cð Þ0:29�0:0045ð1�m�Þ2

a

" #

þ 0:86B1 150n=N Cð Þ0:45þ0:0002ð1�m�Þ2a

� �ð4Þ

Additionally, for a < 0.5, B0 = 1 + 3.75a + 3.218a2 �2.461a3 and the factor B1 = (1/B3

0)(0.3245 + 5.115a +3.5166a2 � 2.3243 a3) [8].

The value of n (or n/NC) from Eq. (4) is estimated byusing iterative procedure and the values calculated arecompared with those from Bebb’s numerical solution of

Eq. (1) for 10 < / < 50. The advantage of Eq. (4) is thatit is in the form useful for efficient device modeling becausenumerical integration has been avoided without the addi-tion of any adjustable parameters. Further, if one wantsto calculate the Fermi energy for a given normalized carrierdensity, it can be used directly without any iterations.

As will be shown, this equation is valid for kc in therange of 1–17 lm. Fig. 1 shows the comparison for the plotof n/NC vs. / using Eqs. (1) and (4) for SWIR, MWIR andLWIR bands. From Fig. 1 it is seen that agreementbetween our proposed approximation and Bebb’s numeri-cal solution is very good for all the cases. The quantitativeerror will be presented later. Furthermore, it can be seenfrom this figure that our approximation shows excellentagreement for / > 10 i.e. for moderately and stronglydegenerate cases. Compared to approximations given inRefs. [5,6] the proposed new relation (Eq. (4)) tries to coverlarger range of wavelengths with additional advantages ofbeing a single unified relation.

Fig. 2 shows the plot of n/NC vs. kc for Eqs. (1) and (4)for / taking values of 10, 20, 30, 40, and 50. It can be seenfrom this figure that comparison is reasonable for / > 10 in

Fig. 4. The comparison of Gauss quadrature (15 points) method withBebb’s results for two different values of upper limit of integration beingequal to 5eg and 3eg.

S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262 261

the range 1 lm < k < 17 lm. However, at lower kc’s there isincreasing disagreement between the Bebb’s numericalsolution and our relation, particularly for very low (<10)and very high values of / (>50). This is because we havetuned our empirical fitting about the centre of the SWIR,MWIR and LWIR range (1–17 lm). This is the range oftenused by IR detectors based on HgCdTe material and coversall the practical cases.

In Fig. 3 we show the percentage error of our approxi-mation to the Bebb’s numerical solution as a function ofcut-off wavelength. This figure shows that for / = 10, 20,30, 40, and 50, the respective average errors are 10.3%,6.3%, 3.0%, 6.5% and 11% in the cut-off wavelength regionof 1–17 lm. This error is much lower than in Ref. [5] andits validity is for a larger range than in Ref. [6]. For the caseof weak degeneracy (/ 6 10), there are many approxima-tions that give reasonably accurate values as discussed inRef. [6]. It may be stated here that there is a scarcity ofapproximations for strongly degenerate cases (/ > 30) [6].We have tried to tune our approximations specifically forthis range of / with increased wavelength range. In thisregard, it may be seen from this figure that error for /> 30 shows oscillatory behavior and decreases beyond therange of 1–17 lm. Hence our approximation for this caseis valid even for wider range of wavelengths.

In general, the average percentage error for our approx-imation varies between 3% and 11% for the 1–17 lm wave-length range. The corresponding standard deviation of thepercentage error varies from 2.6% to 6.3%.

At this point, it may be pointed out that for the major-ity of numerical solution methods, when the upper limitof integration is infinity (as in the present case, Eq. (1)),then it is replaced by a suitable finite number and theaccuracy of the solution depends upon the adjustmentand optimization of this number. This number changeswith the value of kc in the present case. For illustration,we have chosen the most widely used Gauss’s 15 pointquadrature method to evaluate the integral (Eq. (1)).

Fig. 3. Percentage error vs. cut-off wavelength in calculating n/NC usingour proposed approximation.

The results are depicted in Fig. 4. It can be seen thatfor higher kc (>8 lm) an upper integration limit of �5eg

gives a good accuracy whereas the same value is not ade-quate for lower kc (<8 lm). The error w.r.t. Bebb’s calcu-lations is as high as 46% for / = 10 at 2 lm (see symboln in the figure). However, if one wants to improve the fit-ting in the lower cut-off wavelength region, then the upperintegration limit should be <3eg. However, the results inthat case degrade at higher values of / (>30) (see symbol� in the figure), particularly for higher kc. The maximumerror being 68% for / = 50 at 17 lm. In short there is nounique upper integration limit valid for all wavelengths.Thus, for modeling where evaluation of the Fermi integralis only a small part of the calculation, this is impractical.On the other hand, our proposed Eq. (4) may be used assuch.

In the case of PC detectors, our approximation couldbe used to calculate the areal surface electron density incm�2 used in Eq. (15) of Ref. [10] (Qf in that work).Furthermore, our approximation can be used to modelthe capacitance–voltage characteristics of a metal-semi-conductor–insulator devices in the case of large gate volt-ages that involve the use of very high values of / and tocalculate space charge density as a function of the Fermienergy (see Fig. 3 of Ref. [7]). In the case of PV detec-tors, the approximation can be used to estimate the car-rier density in the heavily doped donor region of pn+

diodes.

3. Conclusions

In summary, we have proposed an analytical approxi-mation of the Fermi integral (valid for 10 < / < 50) to esti-mate the carrier density in an NGS, including the non-parabolicity of the energy bands and the degeneracy ofthe carriers. This approximation is useful for efficient mod-eling of IR detectors based on n-HgCdTe material in theSWIR, MWIR and LWIR range.

262 S. Gupta et al. / Infrared Physics & Technology 51 (2008) 259–262

Acknowledgement

The authors are thankful to the Director, SSPL for hiskind permission to publish this work.

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