SolidStare Ekcrronics Vol. 35, No. 12, pp. 17051708, 1992 Printed in Great Britain. All rights reserved
0038I 101/92 $5.00 + 0.00 Copyright c 1992 Pergamon Press Ltd
THE USE OF GENERALISED MODELS BEHAVIOUR OF OHMIC CONTACTS
TO EXPLAIN THE TO nTYPE GaAs
PATRICK J. MCNALLY School
of Electronic
Engineering,
Dublin
City University,
Dublin
9, Ireland
(Received 23 April 1992; in revised form 2 July 1992) AbstractThe use of two generalised carrier transport models to account for the NE’ dependence of the specific contact resistance (p,) of metalsemiconductor Ohmic contacts to ntype GaAs is proposed. Both models include the effects of thermionic emission and diffusion across the highlow 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
years
Ohmic
many
contacts
efforts
to fabricate
to compound
lowresist
semiconductors
have been quite successful[l31. However, attempts to model the behaviour of such contacts have proved more ambiguous [481. Early studies focused on tunnelling theory to account for the basic operation of such contacts. More recently, studies have attempted to account for the highlow barrier which follows the metalsemiconductor 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. nGaAs, 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 highlow 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 highlow 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 highlow 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 closedform 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 metalsemiconductor 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
tunnelling
R = hm
the height of the barrier, where:
*xW)* h’
’
metal
(2)
and
where 2. NEW MODELS
The problem with all previous models lies in deciding a priori which transport mechanism governs the highlow 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
PATRICKJ. MCNALLY
I706
where z = v/(N,q’/2ckT), low junction and:
IV = width
*i, s
= exp(  r’n,‘)
F(w)
of the high
cxp(_,,‘) d,,.
(8)
0
Fig. 1. Band diagram of the Ohmic contact structure showing the metalsemiconductor tunnelling barrier and the n +n (highlow) junction barrier.
the highlow junction the barrier height is given by &, where we use N,, = N,exp( &,/kT) with NC = effective density of states.
3. THE CROWELLSZE
APPROACH
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:
(5)
with
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 highlow 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 comingling 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)
F(w)
(7)
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 highlob 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 highlow junction barrier. levels high degenerate doping For very (N, 3 10” cm~ ‘) & disappears and 11~= p,, , given b> eqn (1). This is calculated for metalsemiconductor barriers c$,,,,,~= 0.3 and 0.4 eV and iv, = 5 x IO” to
.
10E 1
’
lllllbl’ “i1liII’
x 10’4
Carrier
1 “JlIIl’
1 x 10’6
concentration
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 I1 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 ntype GaAs 5 x 1020cm~3, which approximates most of the situations encountered in practice. These are seen as the horizontal components of lines 14 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 SIMMONSTAYLOR
1707
1 x 10Z 67 k
1o4
L 5
1o5
?.
10e
i
107
r% 10a 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 closedform formulation for
APPROACH
Pc2.
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 nonequilibrium carrier density to obtain a generalised expression for the JV 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

11
(13)
i,’ O,h
I_!,=
ve
2P(hJ_ VI
(14)
Here, fl = q/kT, vlh= V = average thermal velocity of electrons in the semiconductor, p = mobility and
67
E ”
104
E 6
10S
i!
106
(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.
5. CONCLUSION
Carrier concentration Fig. 3. Specific contact lated
PC2
analytically
No (atoms / cm3)
resistance vs substrate using the SimmonsTaylor 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 highlow 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.
PATKICR J. MCNALLY
1708
In this study new numerical calculations were performed and exact closedform expressions were found for calculating the impact of the highlow junction on the specific contact resistance of a metalsemiconductor 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 highlow 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.
3.
4. 5. 6.
7. H. 9. IO.
I I. 12.
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I?. 14. IS.
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