The use of generalised models to explain the behaviour of ohmic contacts to n-type GaAs

The use of generalised models to explain the behaviour of ohmic contacts to n-type GaAs

Solid-Stare Ekcrronics Vol. 35, No. 12, pp. 1705-1708, 1992 Printed in Great Britain. All rights reserved 0038-I 101/92 $5.00 + 0.00 Copyright c 1992...

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Solid-Stare Ekcrronics Vol. 35, No. 12, pp. 1705-1708, 1992 Printed in Great Britain. All rights reserved

0038-I 101/92 $5.00 + 0.00 Copyright c 1992 Pergamon Press Ltd




of Electronic



City University,


9, Ireland

(Received 23 April 1992; in revised form 2 July 1992) Abstract-The use of two generalised carrier transport models to account for the NE’ dependence of the specific contact resistance (p,) of metal-semiconductor Ohmic contacts to n-type GaAs is proposed. Both models include the effects of thermionic emission and diffusion across the high-low barrier junction a priori. Calculations of pc, and comparison with experimental data, show conclusively that thermionic emission is the dominant transport mechanism across the barrier. It is stressed that these models do not rely on prior choices of either of the transport processes. These conclusions are arrived at a posteriori.

1. INTRODUCTlON In recent ance






to fabricate

to compound



have been quite successful[l-31. However, attempts to model the behaviour of such contacts have proved more ambiguous [4-81. Early studies focused on tunnelling theory to account for the basic operation of such contacts. More recently, studies have attempted to account for the high-low barrier which follows the metal-semiconductor tunnelling junction in an Ohmic contact structure. This was necessary to account for the inverse proportionality between the specific contact resistance (p,) and the carrier concentration (No) of, e.g. n-GaAs, observed experimentally. A number of theories succeed in describing this NE’ dependence qualitatively but they differ in the estimated quantitative values of pc by up to two orders of magnitude[&8]. The chief difficulty lies in the assumption of a transport mechanism across the high-low junction u priori. As an illustration of this we will consider two such prominent theories, namely those of Dinfgen et a1.[7] and Gupta et aL[8]. Dinfgen et al. assumed that the dominant transport mechanism across the high-low junction was best described by diffusion theory, while Gupta et al. assumed the dominant mechanism was thermionic emission over the junction barrier. Their results, while accounting for the NE] dependence, differed by greater than an order of magnitude in an estimation of pc as a function of bulk doping. Furthermore it was not shown conclusively which transport mechanism, if either, dominates.

transport process. The relevant pc information is then easily extracted. The use of two models is proposed. The first utilises Crowell and Sze’s (CZ) combined thermionic emission/diffusion model applied to the high-low junction barrier[9]. The second model uses the SimmonsTaylor (ST) generalised approach applied to the same n +-PI barrier[ lo]. Though in many respects both are equivalent models, the ST model allows an evaluation of a closed-form expression for pc, while the CZ model requires numerical analysis to calculate pc. Figure 1 is a diagram of the Ohmic contact structure under investigation. The barrier to conduction at the metal-semiconductor junction is assumed to be thin enough to be described by tunnelling theory. The specific contact resistance produced by this barrier (p,,) has previously been described[l l] and is given by:

with hms representing semiconductor


R = hm

the height of the barrier, where:

*xW)* h’




where 2. NEW MODELS

The problem with all previous models lies in deciding a priori which transport mechanism governs the high-low barrier. We present a new formulation which avoids this problem by incorporation of both mechanisms into a generalised representation of the

and m * is the effective electron mass, c is the dielectric constant of the semiconductor and z+ = E, - E,, the separation between the conduction band and Fermi level in the degenerately doped semiconductor. For 1705



where z = v/(N,q’/2ckT), low junction and:

IV = width

*i, s

= exp( - r’n,‘)


of the high-

cxp(_,,‘) d,,.



Fig. 1. Band diagram of the Ohmic contact structure showing the metal-semiconductor tunnelling barrier and the n +-n (high-low) junction barrier.

the high-low junction the barrier height is given by &, where we use N,, = N,exp( -&,/kT) with NC = effective density of states.



This model, which is a synthesis of both Bethe’s thermionic emissionI 121 and diffusion theories, relies on the concept of there being a “recombination velocity”, v,, for carriers at the top of the barrier. The net electron current density across the barrier is given by the difference between v,n and v,n,, where n is the electron density at the top of the barrier and n, is the value at zero bias. An excellent review of this theory is given in Rhoderick and Williams’ book[l3]. An expression for the current density across the barrier is given by:



where t’,,[exp(q V/kT) - I] would be the mean velocity due to drift and diffusion of electrons at the top of the barrier if the diffusion process were dominant. Here, NC is the density of states, C& is the height of the high-low barrier, and v,, the recombination velocity ZZL!= average thermal velocity of electrons in the semiconductor = ,/@kT/rrm *), ECis the energy level at the bottom of the conduction band and V is applied voltage. The salient feature of the model is that it succeeds in consistently co-mingling both transport mechanisms into the one formula. If ~3~$ r, eqn (5) reverts back to Bethe’s thermionic emission formulation. If cd <<27,then it is seen that the diffusion mechanism dominates. Equation (6) presents one minor difficulty, namely the evaluation of the integral. Rhoderick[l4] has shown that this integral reduces to: !I

-’ exp(q&/kT)



is known as Dawson’s integral and has been tabulated previously[ 151. The author has retabulated Dawson’s integral with an aim to providing an appropriate approximation for all values of aa’. It was found that the following approximation suffices: F(U)

0 < Yll‘ < I

Z ZM’!2,

F(w) z li(21M.).

(9) (10)

?I 3 I.

The greatest error occurs in the range 0 < XL{’c I. but the error is still no more than 10% which is more than acceptable in order to facilitate ease of computation. In any case, it turns out that ~11’ti I for most practical situations. The specific contact resistance (pC2) of the high--lob junction is given by:

(II) and this was numerically evaluated Figure 2 is a plot of the specific contact of the Ohmic contact structure of Fig. of bulk carrier concentration N,. In

from eqn (5). resistance (pC) 1 as a function actuality: (12)

0, = P,, + PC

where pC, = the contribution due to the metal semiconductor tunnelling barrier and I’,: = the contribution due to the high-low junction barrier. levels high degenerate doping For very (N, 3 10” cm~ ‘) & disappears and 11~= p,, , given b> eqn (1). This is calculated for metal-semiconductor barriers c$,,,,,~= 0.3 and 0.4 eV and iv, = 5 x IO” to


10-E 1

lllllbl’ “i1liII’

x 10’4


1 “JlIIl’

1 x 10’6


8‘lIJ;ld / ‘I= 1 x 10’8

N, (atoms

! cm3)

Fig. 2. Specific contact resistance vs substrate doplng WICLIlated numerically from the CrowellkSze (CZ) model. Lines I-1 are calculated using the method described in the text. Line 5 represents the approach of Gupta e/ u/.[X] while line 6 represents that of Dingfen CI u1.[7]. The experimental data of [8] are also shown

Behaviour of Ohmic contacts to n-type GaAs 5 x 1020cm~3, which approximates most of the situations encountered in practice. These are seen as the horizontal components of lines 1-4 in the plot. As the doping decreases, pc, given by (11) begins to dominate and displays the required NE’ dependence. Line 5 represents the approach of Gupta et al. and line 6 represents that of Dingfen et al. For comparison, the experimental data used by Gupta et al. in Ref. [6] are also shown. The agreement with experiment is excellent showing a definite closer agreement with the thermionic emission model. This comes as no surprise as our calculations show L’~% v,, therefore directly implying that thermionic emission is the dominant transport mechanism. The essential point is that the assumption of a relevant transport mechanism is arrived at u posteriori, an assumption verified by both the experimental data and a close agreement with the pure thermionic emission model. 4. THE SIMMONS-TAYLOR


1 x 10-Z 67 k


L 5






r% 10-a 3 1 x 10’4

1 x 10’6

Carrier concentration

1 x 10’8

No (atoms / cm3)

Fig. 4. Numerical calculation of specific contact resistance vs substrate doping evaluated using the ST model. Details as in Fig. 2.

A = 2c/qN,. Using (11) on the two equations above allows one to calculate a closed-form formulation for



This approach turns out to be a more powerful and indeed a more elegant approach to our problem. Their original work[lO] relies on the usage of an analytical expression for non-equilibrium carrier density to obtain a generalised expression for the J-V characteristics of a diode (both Schottky and pn cases). This technique also includes a combination of thermionic emission and diffusion processes. Employing their methods the current density can be expressed as: J = qN,exp(-P&)[exp(PV




i,’ O,h



2P(hJ_- VI


Here, fl = q/kT, vlh= V = average thermal velocity of electrons in the semiconductor, p = mobility and


E ”


E 6




(15) where we again have used v,~ = V = J(8kT/m*), with m * = effective electron mass. Figure 3 is a plot similar to Fig 2 of pc = pc, + pc2 vs ND using eqn (15). The agreement here is excellent also, with the estimation of pc2 being slightly smaller than that of Gupta et al. (pure thermionic emission model). Once again if v, ti v,,, then thermionic emission would be the dominant transport mechanism, and indeed, our calculations show this to be ture. Figure 4 is a plot of ps vs ND calculated numerically using the ST theory, shown for consistency with the CZ calculations on Fig. 2. As expected the agreement is very close, the slight differences between them possibly arising due to the simplifications employed in the evaluation of Dawson’s integral in eqn 8.


Carrier concentration Fig. 3. Specific contact lated



No (atoms / cm3)

resistance vs substrate using the Simmons-Taylor Details as in Fig. 2.

doping calcu(ST) model.

It is more heartening that in all cases (and most clearly so for the ST model) the experimental values are larger than the high-low junction barrier resistance predicted in the above theories. As was perspicaciously pointed out by Gupta et af.[8] this is indicative of the fact that the substrate doping sets a lower bound on pc, and it can never be less than pc2 even if the tunnelling junction resistance were set to zero by extremely high doping of the surface layer.



In this study new numerical calculations were performed and exact closed-form expressions were found for calculating the impact of the high-low junction on the specific contact resistance of a metal-semiconductor Ohmic contact. The two models proposed included the effects of both thermionic emission and diffusion processes II ~riori and a comparison with experimental data, allied to an extraction of relevant parameters (I postwiori. draws one to the conclusion that thermionic emission over the high-low junction is the predominant transport mechanism over the barrier. This barrier represents the limiting factor in producing even lower specific contact resistance Ohmic contacts in any future technology.


4. 5. 6.

7. H. 9. IO.

I I. 12.

REFERENCES I. A Piotrowska. A. Guivich and G. Pelous. So/i&S’I. &c/ron. 26, I79 (I 983). 2. J. E. E. Baglin. H. B. Harrison, J. L. Tandon and J. S. William, Ion Implmtrrtion md Berm Processing (Edited

I?. 14. IS.

by J. J. Williams and J. M. Poate), Chap. I I. Academic Press. New York (1984). B. L. Sharma, Semiconducrors und Seminwtul.s (Edited by R. K. Willardson and A. C. Beer). Vol. 15. Chap I. Academic Press, New York (1981). W. Dinfgen and K. Heine, Electron Lrrt. 18,940 (1982). C. Y. Chang, F. K. Fang and S. M. Sze. So/id-S/. E/ec,tron 14, 541 (1971). R. P. Gupta and W. S. Khokle, IEEE Electron Lkicc Let/. EDL-6, 300 (1985). W. Dingfen. W. Dening and K Heime. So/id-S/. Ekctron. 29, 489 (1986). R. P. Gupta and W. S. Khokle. Solid-S/. Ektron. 29, 672 (1986). C. R. Crowell and S. M. Sze. Solid-St. Electron 9, 1035 ( 1966). J. G. Simmons and G. W. Taylor, Solid-St. Electron. 26, 705 (1983). A. Y. C. Yu. Solid-St. Ekrron. 13, 239 (1970). H. A Bethe, MIT Radiation Lab. R~P.~ 43-12 (1942). E. H. Rhoderick and R. H. Williams, Metal-Semiconductor Contar/s. Clarendon Press, Oxford (1988). E. H. Rhoderick. J. Phj,.s. D. Appl. Phj:r. 5, 1920 (1972). W. L. Miller and A. R. Gordon. J. Ph!,.v. Chem. 35, 2785 (1931).