Modeling of dark characteristics of mercury cadmium telluride n+-p junctions

Modeling of dark characteristics of mercury cadmiumtelluride nþ–p junctions

Vishnu Gopal *, Sudha Gupta, R.K. Bhan, R. Pal, P.K. Chaudhary, V. Kumar

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

Received 7 May 2002

Abstract

Dark dynamic impedance versus applied bias voltage characteristics of HgCdTe nþ–p junctions has been modelled

here using a recently proposed [Infrared Phys. Technol. 43 (6) (2002) 317–326] analytical approach. The results of the

analysis on two pixels from the same array are discussed to illustrate the advantage of this approach in identifying the

possible source of variation of diode impedance among the diodes in an array. It has been shown that the contribution

of dark current contributing mechanisms can be separated from each other in each diode. An idea of the dominant

mechanism of the two sources of the ohmic current, namely surface leakage currents and contribution of dislocations

intersecting the junction can also be had from temperature dependent study of shunt impedance.

� 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

In a previous paper [1] we have proposed a

systematic analytical approach and used it to es-

timate the contribution of each of the dark current

mechanisms from the analysis of dark current–

voltage (I–V ) and dynamic impedance–voltage(Rd–V ) characteristics of pþ–n HgCdTe junction

diodes. The object of this paper is now to extend

the same approach to model the dark (I–V ) and(Rd–V ) characteristics of nþ–p diodes. The diodes

were fabricated in our laboratory in linear array

configuration by Bþ implantation on p-type bulk

grown wafers using ZnS passivation layers. The

results of analysis on two diodes with wide differ-

ence in zero-bias diode impedances have been

presented to illustrate the advantage of our ana-

lytical approach in identifying the possible source

of variation of diode impedance among the diodes

in an array. It has been also shown that a tem-

perature dependent study of shunt impedance can

lead to the identification of the dominating ohmiccurrent sources, viz., the surface leakage and the

dislocation contributions.

2. Modeling of dark currents and dynamic resis-

tances

The dark current of a HgCdTe junction will bemodelled [1] here as a combination of the current

contributions due to the thermal diffusion of mi-

nority carriers from the quasi-neutral regions, g–r

current in the depletion region, band-to-band

*Corresponding author.

E-mail address: vishnu_gopal/[email protected] (V. Gopal).

1350-4495/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S1350-4495 (02 )00185-8

Infrared Physics & Technology 44 (2003) 143–152

www.elsevier.com/locate/infrared

mail to: vishnu_gopal/[email protected]

(BTB) tunnelling current, trap assisted tunnelling

(TAT) current and an ohmic component respon-

sible for shunt like behaviour. The contribution

due to avalanche multiplication (AVM) will be

omitted here as the diodes to be analysed in the

present paper belong to case I described in Ref. [1].In this section we will first briefly summarise the

relevant expressions for calculating the dark cur-

rents and the associated dynamic impedance con-

tribution due to each of these mechanisms in an

nþ–p junction.

2.1. Diffusion current (Idif )

In a one-sided step junction the thermal diffu-

sion of the minority carriers from the heavily

doped nþ-side may be assumed negligible. The

minority carrier diffusion current in an nþ–pjunction may therefore be described by the fol-

lowing well known equation [2]:

Idif ¼qAn2iNa

kTq

lnsn

� �1=2

C expqVkT

� �� 1

�ð1Þ

where Na is the acceptor concentration on the

lightly doped p-side of the junction, ni the intrinsiccarrier concentration, A the junction area, V the

diode bias voltage, sn the electron lifetime and lnthe electron mobility. C, the multiplication factor,which accounts for the effect of the surface re-

combination velocity ðSÞ of the carriers at the backsurface in a diode of finite thickness ðdÞ of the basep-region, is given by:

C ¼Ssn cosh d

Ln

� �þ Ln sinh d

Ln

� �Ssn sinh d

Ln

� �þ Ln cosh d

Ln

� �

where Ln is the diffusion length of the minority

carriers, i.e. electrons.

The associated dynamic resistance and its de-

rivatives are given by:

R�1dif ¼

qkT

� � qAn2iNa

kTq

lnsn

� �1=2

C

" #exp

qVkT

� �

ð2Þ

o2IdifoV 2

¼ qkT

� �2 qAn2iNa

kTq

lnsn

� �1=2

C

" #exp

qVkT

� �

ð3ÞThe above Eq. (3) and similar expressions for g–r,

TAT and BTB mechanisms will be required lateron to determine the trap density Nt from the ex-

perimental data.

2.2. Generation–recombination current (Ig–r):

In this type of current mechanism, defects

within the depletion region act as intermediate

states (usually referred to as Shockley–Read orsimply S–R centres) for the thermal generation

and recombination of carriers. Sah, Noyce and

Shockley (SNS) analysed these currents and de-

veloped equations for the idealized case [3]. Un-

fortunately these do not yield a closed form

solution for describing the complete I–V charac-

teristics. In the literature, several simplifications of

their theory has been used [2,4–8]. In the previouspaper [1], an approximate expression with limited

range of validity was used for calculating the g–r

contribution. In the present paper, we will, how-

ever, use Schoolar�s [4] simplification of original

SNS approach as applicable to mercury cadmium

telluride diodes. These simplified expressions are

given below:

½Ig–r�V<0 ¼qAniWdepV

Vtsg–rð4aÞ

and

½Ig–r�V>0 ¼2AniWdepkT

Vtsg–rsinh

qV2kT

� �ð4bÞ

where sg–r is the g–r lifetime and Wdep is the voltage

dependent depletion region width given by:

Wdep ¼2eoesVtðNa þ NdÞ

qNaNd

1=2ð5Þ

where Vt ¼ Vbi � V is the total junction potential,

Vbi is the built-in junction voltage and V is theexternal bias applied to the junction, eo is the

permittivity of free space, es is the static dielectricconstant of HgCdTe and Nd the electron concen-

tration on the heavily doped n-side of the junction.

144 V. Gopal et al. / Infrared Physics & Technology 44 (2003) 143–152

It may be noted here that Eq. (4b) is valid for

forward bias voltage values less than Vbi by severalkT=q. Also, though, Eq. (4a) appears to be small

voltage approximation of Eq. (4b), it is valid

throughout the reverse bias region [4]. For the ease

of application, the g–r current can be rewritten as:

½Ig–r�V<0 ¼ 2Ag–rV

V 1=2t

ð6aÞ

½Ig–r�V>0 ¼4Ag–rkT

qV 1=2t

sinhqV2kT

� �ð6bÞ

Ag–r ¼qniA2sg–r

2eoesðNa þ NdÞqNaNd

1=2

where Ag–r is voltage independent term in Eqs. (6a)

and (6b). The associated dynamic resistance and

its derivatives are given by:

½R�1g–r�V<0 ¼ 2Ag–r

1

V 1=2t

"þ V

2V 3=2t

#ð7aÞ

½R�1g–r�V>0 ¼ 2Ag–r

kT

qV 3=2t

sinhqV2kT

� �"

þ 1

V 1=2t

coshqV2kT

� �#ð7bÞ

o2IoV 2

V<0

¼ 2Ag–r1

V 3=2t

"þ 3V

4V 5=2t

#ð8aÞ

o2IoV 2

V>0

¼ 2Ag–r

3kT

2qV 5=2t

"þ q

2kTV 1=2t

!sinh

qV2kT

� �

þ 1

V 3=2t

!cosh

qV2kT

� �#ð8bÞ

2.3. Trap assisted tunnelling current (Itat):

In this type of current, minority carriers may

tunnel from the occupied trap states on the quasi-

neutral side to the empty band states on the other

side of the junction or through trap sites present in

the depletion region of the junction [9–12]. In nþ–p

junctions, the major contribution of the TAT may

be due to tunnelling of electrons via trap levels to

conduction band on the n-side. This current is

being calculated using a modified simple one-di-

mensional model [13]:

Itat ¼ qANtWdepWcNc ð9Þ

where WcNc, the tunnelling rate of electrons is gi-

ven by [11]:

WcNc ¼p2qmenM2

h3ðEg � EtÞ

� exp

(� 8pð2meÞ1=2ðEg � EtÞ3=2

ð3qhnÞ

)ð10Þ

where me is the effective mass of electrons in the

conduction band, Eg is the band gap, M is the

matrix element associated with the trap potential,h the Planck�s constant, n the electric field strengthacross the depletion region and Et is the position oftrap levels in the band gap measured from the top

of the valence band on the p-side. As reported by

the earlier authors, we assume the value of the

quantity M2ðme=mÞ as 10�23 V cm3 for HgCdTe

[9,11].

The associated dynamic resistance and its de-rivatives are given by:

ðRtatÞ�1 ¼2q3Ap2meM2

h3ðEg � EtÞNt exp

� B

V 1=2t

!

� 1

þ B

2V 1=2t

!ð11Þ

o2ItatoV 2

¼ 2q3Ap2meM2

h3ðEg � EtÞNt

� exp

� B

V 1=2t

!B

4V 3=2t

þ B2

4V 2t

!ð12Þ

B ¼ 8pð2meÞ1=2ðEg � EtÞ3=2

3qh 2qNa=eoesð Þ1=2n o ð13Þ

where B, depends on the shape of the barrier andfor a triangular barrier it�s value is given in Eq.

(13) and Nt is the density of traps occupied by

holes.

V. Gopal et al. / Infrared Physics & Technology 44 (2003) 143–152 145

2.4. Band to band tunnelling current (Ibtb):

At relatively higher bias voltages, the electrons

directly tunnelling from the valance band on the p-

side to the conduction band on the nþ-side areresponsible for the BTB current [11,14,15]. For

modelling this current we use the simple approach

presented in Ref. [14]. The BTB tunnelling current

is given by:

Ibtb ¼qA4�hp2

EgkTP 2

� �1=2 Z 0

�Emax

TpE2

� �dE ð14Þ

where E is the energy of the carriers in eV and

Emax ¼ �qVt � Ef

where P is the momentum matrix element and Efthe Fermi energy. Tp is the tunnelling probabilityassociated with the BTB tunnelling and is given by:

Tp ¼ exp

"� 2me

�h2

� �1=22eoesq2Na

� �1=2

Eg 1

�(� EEg

�

� p2

"� sin�1

�EEg � E

� �1=2#�� EEg

�1=2)#

ð15Þ

The associated dynamic resistance and its deriva-

tives are given by:

ðRbtbÞ�1 ¼q2A4�hp2

� �EgkTP 2

� �Tp

Emax2

� �ð16Þ

o2IbtboV 2

¼ q3A4�hp2

� �EgkTP 2

� �2me

�h2

� �1=22eoesq2Na

� �1=2

� EgTpEmax2

� �264� p2Eg

þsin�1 �E

Eg�E

� �1=2Eg

375

ð17Þ

2.5. Ohmic current (Ish):

The current–voltage characteristics of a mercury

cadmium telluride junction often exhibit an excess

current component, a part of which can be mod-

elled as an ohmic current component given by

[1,9,10]:

Ish ¼VRsh

ð18Þ

where V is the applied voltage across the junction

and Rsh is the diode shunt resistance. The surfaceleakage currents [16] and the dislocations [17,18] inthe material that intersect the junction are gener-

ally held responsible as a possible source for this

part of excess current.

3. Analytical approach

Based on the physical and analytical consider-ations discussed in the previous paper [1], the

contribution of the various dark current contrib-

uting mechanisms to HgCdTe diodes can be

grouped in the following two cases:

Case I: If a diode has been fabricated in a ma-

terial which has sizeable number of defect levels in

the band gap contributing to g–r and TAT cur-

rents, then the experimental data of such a diodecould be analysed by assuming contribution from

diffusion, g–r, TAT, BTB and ohmic currents.

Case II: If a diode has been fabricated in a good

quality material with negligible defect levels re-

sponsible for g–r and TAT contributions, then the

AVM, diffusion and ohmic surface leakage cur-

rents can be considered as the possible dark cur-

rent contributing mechanisms to the diode current.The contribution due to BTB can be assumed

negligible in this case as the condition favourable

to AVM (e.g. relatively large depletion layer

width) is unfavorable to BTB.

In practice, an initial guess about the type of the

diode i.e. either belonging to case I or case II can

be had from the analysis of forward I–V charac-

teristic of the given diode for it�s ideality factor. Aunity ideality factor that indicates diffusion limited

diode performance will support the cause of ana-

lyzing the experimental data for a type II diode.

Whereas the cases belonging to ideality factor

g > 1, could be first analyzed for the case of type I

diode.

As will be seen later, the experimental data an-

alyzed in this paper is that of type I diode. We willtherefore limit our description of analytical ap-

proach to this case only. It is well known that


Rd–V characteristic of this type of diode exhibit a

peak in the low or medium reverse bias region due

to TAT contribution. Recently we have shown

[1,9,10] that the position of this peak can be ex-

ploited to estimate Nt as well as Rsh as explainedbelow. The peak position in Rd–V curve corre-sponds to the condition:

oRdoV

¼ 0 ð19Þ

where Rd is the resultant dynamic resistance given

by:

1

Rd¼ 1

Rdifþ 1

Rg–rþ 1

Rtat

þ 1

Rbtbþ 1

Rshð20Þ

An expression for the trap density (Nt), obtained

from mathematical manipulation of Eqs. (3), (8a),

(8b), (12), (17), (19) and (20) is given by

Nt ¼o2IdifoV 2

� �þ o2Ig–r

oV 2

� �þ o2Ibtb

oV 2

� �n oV¼Vm

2q3Ap2meM2

h3ðEg�EtÞ exp � BV 1=2t

� �V¼Vm

B4V 1=2

t

þ B2

4V 2t

� �V¼Vm

ð21Þwhere Vm is the value of V at the peak position.

Since all the parameters on the right hand side of

the above equation are generally known for thegiven diode, the density of traps responsible for

TAT can be estimated in a straightforward manner

from the observed position of the maximum. It

may be noted here that the above estimate of Nt is

practically independent of ohmic current contri-

bution, since oRsh=oV ¼ 0 (shunt resistance Rsh is

independent of applied voltage V ). This situationallows us to further obtain an estimate of shuntresistance Rsh from a comparison of experimental

peak dynamic resistance and theoretically calcu-

lated resultant dynamic resistance due to TAT

(making use of estimated Nt), g–r current, BTB

tunnelling and thermal diffusion current contri-

butions.

4. Experimental

Hg1�xCdxTe diodes used in this study were

fabricated on p-type wafer obtained from crystals

grown in our laboratory by solid-state re-crystal-

lization. The composition of the material

(Hg1�xCdxTe) was x ¼ 0:27 and the measured

values of carrier concentration and mobility at 77

K were 1.5E16 cm�3 and 300 cm2 V�1 s�1. The nþ–

p diodes were fabricated in a linear array config-

uration by Bþ implantation with the active diodearea as 48 lm� 82 lm and base p-region thick-

ness of 10 lm glued to a sapphire substrate by a

low temperature adhesive. The concentration of

carriers in nþ-region was 1E18 cm�3. The diodes

were passivated on both front and back surfaces

by ZnS layers deposited by thermal evaporation.

The array was mounted in a dewar with facility to

vary the temperature of the device from 77 K toroom temperature. The DC current–voltage dark

characteristics were measured with zero FOV as a

function of temperature using a laboratory as-

sembled set up on IEEE-488 bus, HP4140B pico-

ammeter and Keithley 263 voltage source. In the

low current measurement range (6 1� 10�12 A),

long integration time of 2.56 s was used to achieve

an accuracy of about 5% (along with zero offset)for the current sampling. The noise of the system

was less than 8� 10�13 A. The temperature was

measured by a pre-calibrated resistance tempera-

ture detector (RTD) from British Electricals

mounted on the cold finger of the dewar. The

temperature resolution of this detector is limited

by resolution of resistance measurement which was

0.75 X (corresponding to 0.25 �C) with full reli-ability. The accuracy of the temperature controller

is indicated by the horizontal bars in Figs. 3 and 4.

5. Results and discussion

The measured 77 K I–V and the corresponding

Rd–V characteristics of the two diodes 103/43 and103/56 from the same array are shown in Figs. 1

and 2. These two diodes were chosen because of

the two orders of magnitude difference in their

zero bias impedance. To begin with, the ide-

ality factor was determined by subjecting the

forward I–V characteristics of these diodes to

the curve fitting by an expression of the form:

I ¼ I0 expðqV =gkT Þ, where g is the ideality factor.While determining g, the correction was made forthe voltage drop across the series resistance of the


diode. Note that forward Rd–V characteristics of

these diodes indicate a finite series resistance. The

values of ideality factor thus determined are shown

in Table 1. There is also an evidence of TAT

contribution as Rd–V curve of each diode exhibits

a peak in the reverse bias region at low bias. Both

Fig. 1. I–V and Rd–V characteristics of diode #103/43.Fig. 2. I–V and Rd–V characteristics of diode #103/56.

Table 1

Comparison of dark current contributions to the zero bias impedance of the diodes

Sample no. Temp. (K) g Rdif0 (X) Rg–r0 (X) Rsh (X) R0 (X) NT (cm�3) ET (meV)

103/43 77 1.4 4E10 3.5E7 3.4E6 3.4E6 8.6E14 18

90 1.4 8.7E8 2.0E7 3.2E6 2.3E6 5.6E14 21

103 1.3 67E6 1.2E7 3E6 2.2E6 1.6E13 61

160 1 1.2E5 – 1.5E6 1.4E5 – –

172 1 34E3 – 1.7E6 72E3 – –

208 1 5E3 – 1E6 30E3 – –

103/56 77 1.3 5E10 1.9E9 4.9E8 3.8E8 4.6E12 24.8

120 1 6.4E6 – 1E8 5.8E6 – –

153 1 3.6E5 – 6.8E7 3.6E5 – –

185 1 34E3 – – 52E3 – –


of these observations, i.e. an ideality factor of more

than unity and evidence of TAT contribution led

us to conclude that these diodes were of type I and

can be analysed by assuming diffusion, g–r, TAT,

BTB and ohmic contributions.

Before proceeding to estimate Nt and Rsh fromthe reverse bias Rd–V characteristics, one needs to

know the parameters sn and sg–r as these were notknown from independent measurements. Here

we propose to determine these parameters from

the fitting of forward I–V and Rd–V characteris-

tics by treating both sn and sg–r as variable fit-

ting parameters. An intermediate ideality factor

(1 < g < 2) suggests comparable g–r and diffusioncontributions. Thus to account for both g–r and

diffusion contributions, the forward I–V and Rd–Vcharacteristics of these diodes were fitted by using

expressions I ¼ Idif þ Ig–r and 1=R ¼ 1=Rdifþ1=Rg–r, where Idif , Rdif , Ig–r and Rg–r are respectively

defined by Eqs. (1), (2), (6b) and (7b). The trap

density Nt, contributing to TAT and ohmic shunt

resistance Rsh were next estimated from the peakposition (10 mV in diode 103/43 and 20 mV in

diode 103/56) in reverse bias Rd–V characteristics

and are shown in Table 1.

Using the estimated and fitted values of the

various parameters from Table 1, the calculated

I–V and Rd–V characteristics of both the diodes at

77 K are compared with the experimental data in

Figs. 1 and 2. It can be clearly seen that there is avery good agreement between theory and experi-

ment. To conclude the dominant contributions we

have also shown in these figures the individual

contributions due to each of the contributing

mechanisms. It is observed that ohmic shunt re-

sistance, whose source of origin could be either the

surface leakage currents or dislocations intersect-

ing the junction or both, limits the impedance ofboth the diodes. Interestingly it can also be seen

from Table 1 that ohmic shunt is approximately

two orders of magnitude lower in diode 103/43

than in diode 103/56. At the same time the esti-

mated trap density contributing to TAT in these

diodes also differ by two orders of magnitude,

though this contribution is not limiting the zero

bias impedance of these diodes. Similarly g–rcontribution also differ approximately by two or-

ders. Estimated sn and sg–r from forward charac-

teristics are also smaller in diode 103/43 (1.5

and 0.8 ns at 77 K) than in diode 103/56 (3 and

44 ns at 77 K). All of these observations put

together suggest that the material beneath the

diode 103/43 may have a higher defect density.

In this analysis though the ohmic shunt contri-bution could be isolated from other dark cur-

rent contributions, it is however not possible to

distinguish between the two different sources

(surface leakage and dislocations) of ohmic con-

tribution. As an attempt towards this end,

we present in the following paragraph an investi-

gation of the observed temperature depen-

dence of shunt and zero bias impedance of thesediodes.

In the past, a temperature dependent study of

zero bias impedance (R0) or its product with the

diode area (R0A) has been used by a number of

workers [2,17,19,20], to assess the limiting current

mechanism in HgCdTe diodes. In this case mi-

nority carrier lifetime is assumed to be tempera-

ture independent and variation of R0 or R0Aproportional to 1=n2i indicates diffusion limited

performance, variations proportional to 1=ni cor-responds to g–r limited performance while high

dislocation density samples exhibit variations [17]

proportional to 1=n1=2i . We have also carried out a

temperature dependent study of R0 of our diodes.

The results obtained by our detailed analysis are

shown in Table 1. Figs. 3 and 4 also show theseresults in the form of plots of R0 and Rsh versus

1000=T . Circles in these figures depict the observedvariation of corresponding ohmic shunt resistance

as a function of temperature. Some of the notice-

able features, which are clearly visible from this

study, are summarized below:

Only the three mechanisms namely thermal

diffusion, g–r and ohmic currents influence thezero bias impedance of HgCdTe diodes at low

temperatures normally used in the operation of

these diodes. TAT and BTB contribute to the

diode impedance in the low-to-medium and high

reverse bias regions respectively.

Zero bias impedance of both the diodes indi-

cates diffusion-limited performance in the higher

temperature region. This is also corroborated bythe observed unity ideality factor at these tem-

peratures.


Shunt resistance limits low temperature zero

bias impedance of the diodes. In the intermediate

temperature range, depending upon the relativemagnitudes of diffusion, g–r and shunt contribu-

tions, the zero bias impedance may appear to fol-

low g–r trend even if it was not so. For example

the results of the detailed analysis given in Table 1

show the absence of g–r contribution to the im-

pedance of diode 103/56 at 120 K, but from Fig. 4

one may conclude it as g–r limited performance.

The most interesting and new result is in respectof temperature dependence of shunt resistance.

Note from Figs. 3 and 4 that the shunt resistance is

a slowly varying function of temperature. Our

experimental results of shunt resistance shown by

circles in Figs. 3 and 4 follow approximately pro-

portionality of 1=n1=4i , whereas reported [17] vari-

ation in high dislocation density samples is

proportional to 1=n1=2i . The evidence of excesscomponent of ohmic current over and above the

contribution of dislocations is interpreted here as

the contribution of surface leakage currents. These

results have thus led us to conclude that the two

orders of magnitude difference in the zero bias

impedance of these diodes is due to higher surface

leakage currents in diode 103/43. However thepossibility of degraded passivant-material inter-

face due to the higher defect density (as concluded

above) in the base material of diode 103/43 leading

to higher surface leakage currents can also be not

ruled out.

6. Summary and conclusions

In this paper we have used our recently pro-

posed [1] analytical method to model dark char-

acteristics of nþ–p HgCdTe junctions. For a given

diode it has been possible to isolate the contribu-

tions of each of the dark current contributing

mechanism operating in the diode. The method

has been further extended to show that an idea ofthe dominant contributing mechanism among the

two sources of ohmic current, namely surface

leakage currents and contribution of dislocations

intersecting the junction can also be had from the

h

Fig. 3. Zero bias impedance (R0) and shunt resistance (Rsh) of diode 103/43 as a function of temperature. Error bars parallel to x-axisindicate the uncertainty in the measured temperature of the sample.


temperature dependent study of shunt impedanceof the diode. By taking the example of two diodes

from the same array but with large difference in

their zero bias impedances, it has been possible to

show that this approach can be used to identify the

possible source of variation of diode impedance

among the diodes in an array.

Acknowledgement

The authors are grateful to the Director, Solid

State Physics Laboratory, Delhi, for grantingpermission to publish this work.

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Modeling of dark characteristics of mercury cadmium telluride n+-p junctions

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