Surface Science 58 (1976) 60-70

0 North-Holland Publishing Company

CONDUCTION MECHANISMS IN BANDTAILS AT THE Si-SiO, INTERFACE

Emil ARNOLD *

Mullard Research Laboratories, Redhill, Surrey, England

Hall effect and conductivity data are analysed to elucidate the conduction mechanisms

in the bandtails in silicon surface-inversion layers. The experimental galvanomagnetic pro-

perties are interpreted as a combination of percolation around, scattering from, and ther-

mal emission over random potential barriers. The results indicate that pseudometallic

transport extends into the low-energy region inside the bandtail and, simultaneously, the

thermally activated conductance persists above the percolation threshold. At low tem-

peratures the contribution due to tunnelling becomes increasingly important. The rms

potential fluctuations are found to increase at low carrier density because of reduced

screening as the density of states at the Fermi energy decreases.

1. Introduction

Inversion layers at the Si-SiO, interface appear to undergo a metal-insulator transition when the concentration of carriers induced in the channel decreases below a certain critical value [l-4] . At high carrier concentrations (n 2 1 X 1012 cme2) the behaviour is typically metallic, and both the conductivity g and the Hall mobility ,+ are independent of temperature. At lower concentrations the conductance be- comes thermally activated, and the activation energy increases with decreasing con- centration, the effect being especially pronounced at low temperatures. Several au- thors have interpreted the observed behaviour in terms of hopping between localized bandtail states [ 1,2] , and by thermal excitation of carriers into states above a mobil- ity edge [l] . Other work, however, indicates a possible coexistence of metallic and thermally-activated transport [3,4].

The arguments in favour of the hopping model [ 1,2] are based mainly on the ob- served decrease in the activation energy with decreasing temperature from a log g 0: T-l behaviour to logg 0: Tv113. A complete disappearance of activation energy at temperatures below 1 K, however, has also been reported [4]. Hall effect measure- ments [3] indicate that, in the range where the conductance is thermally activated, the Hall mobility is equal to the effective mobility, peff -g/en, and the Hall carrier

* On leave from Philips Laboratories, Briarcliff Manor, New York 10510, USA.

60

E. Arnold / Conduction mechanisms in bandtails at Si-,902 interface 61

concentration nH -g/epI, equals the carrier concentration n deduced from the gate

voltage and the conductance threshold at 80 K [5] . Since all carriers in the inversion layer contribute to the observed Hall effect and

conductance, it may be concluded that the temperature dependence is not due to thermal excitation from localized to extended states or to thermally-assisted hopping between localized states [3,7]. The results are thus irreconcilable with the postulate of a sharp mobility edge in this system and suggest that, at least in some samples, one has to deal with a gradual mobility transition. In this paper we interpret the observed

electrical properties of inversion layers in terms of a semiclassical percolation system and show that it can account for the basic transport properties of carriers at the Si-SiOz interface.

2. Density of states in the bandtail

It was first suggested by Ziman [8] that the transport of particles in a random potential could be treated as a problem in percolation theory. This approach has been applied to various three-dimensional disordered systems [9,10] and, more recently, to carrier localization in surface inversion layers [3,6]. In the latter system a spatially fluctuating potential is presumed to arise as a result of random variations in the den- sity of the futed interfacial charge located near the semiconductor-insulator inter- face. As a result of such fluctuations the classical particles are excluded from those regions of space where their energy E is lower than the local band edge I’. The local potential is assumed to remain approximately uniform within a correlation length L. Accordingly, for each energy, the two-dimensional space is divided conceptually into allowed regions where V < E and prohibited regions where V > E. Two possible si- tuations that arise are illustrated in fig. 1. At high energies nearly all space is allowed, and the prohibited regions may be regarded as islands in an allowed sea [9,11], in which the particle can propagate freely, with occasional scattering off the prohibited islands. At low energies the allowed regions form isolated lakes in a continent of prohibited space; a particle placed in one such region will be localized at T = 0. The density of states in the lowest subband is reduced by the fraction p(E) of allowed space [3] :

WE) = Q@) , (1)

where D,, = m*/?rIi2 is the “metallic” density of states in the lowest subband and m* is the effective mass.

For a random Gaussian potential with a standard deviation u and a probability distributionP(V) = (2n02)-1/2exp(-V2/202), the fraction of allowed space is [3] :

p(E)= y P(v) dV= i erfc(-E/&a) , _m

(2)

62 E. Arnold / Conduction mechanisms in bandtails at Si-SiDz interface

Fig. 1. Percolation in a two-dimensional potential. The water level represents the regions which

are allowed to a particle of energy E; the land regions are prohibited. In the top picture the high

energy particle can propagate freely around the prohibited regions. For the low-energy particle

in the bottom picture the allowed regions form isolated lakes, and the particle becomes localized.

where erfc is the complementary error function and the energy is measured from the edge of the unperturbed subband. As is shown in Appendix A, the standard deviation is given by [3]

u2 = 2rN+(Ze2~)2/9(Ks + Ku)2 ) (3)

where I@ is the density of the fixed ions at the interface, each carrying a charge Ze, h is the screening length for the two-dimensional potential, K, and K~ are the dielec-

Fig. 2. Normalized density of states in the bendtail (i.e. fraction of allowed space) versus energy in units of E/g? o. The dashed line represents the unperturbed density of states Da in the lowest subband. The classical percolation threshold is at E = 0, the subband edge. The shaded areas in both inserts represent the elassicafly prohibited regions to a particle of energy E.

“Eric constants of the semiconduc~~r and the insulator, and Z is the ~~e~age separation of the carriers from the interface. The normalized density of states, or p(E), given by eqs. (1) and (2), is shown in fig. 2 as a function of energy in units of E/6 CT. The two inserts in the figure illustrate the re~atianshi~ between the fraction of allowed space and the density of states at an energy E.

In general, u will be a function af the carrier density, because the screening length increases with decreasing density of states at the Fermi energy f12,13] :

X(EF) = Ks[27re%(E~)] --I . (4)

The density of states at the Fermi energy, cahxlated by a serf-consistent solution of eqs. (l)-(4) is shown in fig. 3, where we have neglected the weaker dependence of H on carrier density [ 121. The f~~~~i~g quantities were used in the c~culation: D, = 2 x 1014 cm -2 eV_l , Z = 25 A, @= 6 X 10” f cmW2. The standard deviation

_l”c, -8 -6 1 -4 8 -2 L 0 2 1 4 I 6 8 8 -1 10

E, imeVi

Fig. 5. Normatied density of states D(I+)/Do at the Fermi energy. The iong tair at Iow energies is due to reduced screening as I)&?$$ decreases.

64 E Arnold / Conduction mechanisms in bandtails at Si-SiOz interface

was found to vary between 4 meV at E, > 0, and 20 meV at E, = --lo meV. It can be seen that, as a result of reduced screening, D(EkT) in the bandtail falls off more slowly and persists farther below the band edge. It is not clear whether these tail states are related to the continuous energy distribution of surface states which is normally observed within the bandgap in the Si-Sio2 system.

The Fermi energy is related to the average carrier concentration through:

n= s

@O erfc[-E/fla(Et.)l f(E, EF) dE , (9 _m

where fl.E, EI:) is the Fermi-Dirac distribution function. At low temperatures the states are filled just up to E,, and eq. (5) simplifies to

n = 2-1/2q)a[n-1/2 exp(-E;/202) + (EF/fi a) erfc(-EF-/fl o)] . (6)

It can be seen that, when E,

at 0 K.

3. Electrical conductance

% 0, ii + DOE,, as expected for an ideal inversion layer

To obtain an estimate of the effect of disorder on the electrical properties of sur- face inversion layers, we begin with the Kubo-Greenwood formula for the conductivi-

ty 171:

g= +@I (WW do -d&I. (7)

The approximate equality is valid at low temperature because the bandtail states are filled just up to the Fermi level. The qualitative behaviour at T = 0 can be inferred from figs. 1 and 2. At high carrier concentrations, when E, > 0, nearly all space is allowed to the particle at E,, and the conductivity of the system approaches the value go characteristic of the uniform material. As the carrier concentration decreases and E, + 0, the prohibited regions occupy increasingly larger proportion of space, the conducting channels become more tortuous with many blind alleys, and the con- ductance thus decreases. As the concentration and E, decrease further, a critical point is reached at which a continuous unbroken conducting channel no longer exists, and the conductivity vanishes at T = 0. In a two-dimensional system a transi- tion from a conducting to a non-conducting state, or percolation threshold, occurs [3,1 l] when p(E) = l/2. It can be seen from eq. (2) that at this point E = 0, and we thus expect the conductivity at T = 0 to vanish at a concentration for which E, = 0.

At T > 0 the conductivity at all carrier concentrations will be higher as a result of thermal occupation of states at the top of potential barriers. The contribution g(EF, V’) to the conductance from a region where the local potential energy is V

E. Arnold / Conduction mechanisms in bandtails at Si-SiO2 interface 65

should thus be proportional to the Fermi factor { 1 + exp [( L’ - EF)/kT] }-l. If

gl (EF) is the average conductance of the pseudometallic regions, in which V < E,, and we neglect the energy dependence of mobility, then

AE,, v) %gr(EF) exp](EF - YJ?lkTl y V>E,,

=g#F) 9 V<EF. (8)

The average conductance gfJ(EF) of the pseudoinsulating regions is, therefore,

&‘(EF) = e-l /- g(E,, v)p(v) dV

EF

= (g1/2f) exp(02/2k2T2 + EF/kT) erfc(EF/fi CJ + o/G kT) , (9)

where E(EF) = 1 - p(EF) = $ erfc(EF/fi a) is the fraction of space occupied by the pseudoinsulating regions.

When quantum effects are included, tunnelling through the prohibited regions will also contribute to the current. The additional conductance by carriers with

energy E is approximately

g$)(E) = e-l sg 0 exp {(-24,/t?) [2m*(V - E)] 1/2} P(v) dV, E

(10)

where lb(E) is the average barrier width. It is shown in Appendix B that Z,,(E) = L [ 1 -p(E)] /p(E). The tunnelling contribution is expected to become progressively more important at low concentrations and temperatures where the thermal emission over high barriers becomes less probable. A calculation using L = 300 A, Z = 25 A, shows that the tunnelling and emission currents become comparable at 12 N 4 X 1011 cmP2, T = 1.7 K. At higher concentrations and temperatures the emission current will predominate and, in the following discussion, will be assumed to constitute the predominant transport mechanism across the prohibited regions. We mention, inci- dentally, that tunnelling may also be responsible for the peaked structure in the field effect mobility that has been observed in MOS devices at low temperatures [ 141. Such oscillating behaviour could arise from resonances in the transmission coefficient due to the partly-reflecting boundaries at the edges of the potential barriers. The energy separation between such peaks is -(7rfi/lt,)2(2m*)-1, so that barrier widths of several hundred A are consistent with the reported peak separation of several hundred PeV in Fermi energy [ 141.

The conductance gr(EF) of the pseudometallic allowed regions will be somewhat smaller than go because the mean free path is reduced by the presence of prohibited regions. An estimate of this effect is given in Appendix B; in p-type channels the re- duction in mobility is found to be small for L 2 100 A.

66 I?. Arnold / Conduction mechanisms in bandtails at Si-SiOz interface

The overall conductivity g(E,) in the inversion layer may be estimated with the aid of the effective medium theory of conductance in mixtures [ 15,161 . In this treat- ment of the problem the average effect of the random spatial conductance variations is approximated by choosing a single value g such that the effect of changing the conductance of any arbitrary region back to its true value will, on the average, cancel out. The result for a two-dimensional binary mixture of components with conducti- vitiesgl andg2 may be written as [ 151 :

g = 4(&J - 1)&l - g2) + [$Gv - Vk1 ~ QY +8&l u2 ; (11)

here p(EF) is the fraction of space occupied by the pseudometallic (allowed) regions with conductance gl(EF), and g2(EF) represents the thermally assisted conductance across the pseudoinsulating (prohibited) regions. It may be seen from eq. (11) that for impenetrable barriers (gl S g2) the conductivity becomes

&F)=gJ2P@F)- 11 ; Q-+0. (12)

Thus, if tunnelling is neglected, the current at T = 0 should vanish when E, = 0, in agreement with percolation theory.

The Hall coefficient in the case of impenetrable barriers in two dimensions is given by [ 161:

R =R, , (13)

where R 1 is the Hall coefficient of the pseudometallic regions. The carrier concen- tration in the pseudometallic regions is determined only by the gate voltage and is independent of the presence of the prohibited regions. Thus R 1 = l/ne and the Hall mobility is

P&‘F) =Rl g@F) =g/ne , (14)

and is, therefore, identical with the effective mobility peff. The carrier concentration determined from a Hall effect measurement is

nH =g/epH = n . (15)

The effective medium theory thus predicts that (unlike in the case of hopping con- duction [7] ) the Hall effect measurement should give the correct free carrier con-

centration in the bandtail. Eq. (12) gives the approximate conductivity above the percolation threshold at

very low temperatures (say below 1 K) where thermal emission becomes vanishingly small. At these temperatures the current is determined mainly by the percolation paths, so that g(EF) increases with increasing E, but shows very little temperature dependence. The decrease in the apparent activation energy with temperature at low temperatures may resemble the transition to a variable-range hopping conductivity although, in this case, a proportionality of log g to Th1j3 over a limited temperature range would be purely fortuitous.

IT. Arnold / Conduction mechanisms in bandtails at Si-SiOz interface 61

At somewhat higher temperatures the thermal emission becomes important, and

the conductivity as well as the apparent activation energy at all carrier concentra- tions increases with temperature. The thermal emission across the prohibited regions continues to contribute to the current when E, lies above the percolation threshold as long as the prohibited regions at E, still occupy an appreciable fraction of space. The temperature-dependent conductivity thus persists up to rather high values of carrier concentration, where a large portion of the current already flows around the

prohibited regions. The apparent activation energy shows no discontinuity at E, = 0

but approaches zero with zero slope. The actual values of activation energy thus bear no simple relationship to the position of the Fermi level; an attempt to estimate the standard deviation of the fluctuations from the carrier concentration at which the conductivity becomes independent of temperature will result in a value that is too large [3].

All the above-mentioned characteristics are evident in fig. 4 which shows the cal-

culated and experimental effective mobility in an MOS device as a function of reci- procal temperature. The value of the oxide charge for this sample was 6 X 1011 cm-*

other quantities used in the calculation were m* = 0.5 m. and Z = 25 A. The cal- culated carrier concentration for which E,(O K) = 0 is - 4 X 1011 cmP2, although the temperature-dependent conductance persists to much higher concentrations. The mobility transition is smooth and it is difficult to estimate the location of the percolation threshold from the mobility data even at temperatures as low as 1.7 K.

T (K)

1o-3 1 0.5

a 0 02 04 0.6

+ (K-l)

Fig. 4. Experimental points: effective mobility in a p-channel MOS sample, relative to the maxi-

mum value fig, versus reciprocal temperature. The value of cco is 400 cm2 V-’ s-l, at a carrier

concentration n r 1.6 X 1012 cme2. Solid curves: calculated from effective medium theory.

68 E. Arnold / Conduction mechanisms in bandtails at Si-SiO2 interface

4. Summary

A semiclassical percolation model has been used to interpret the observed Hall effect and conductance in silicon inversion layers. Random potential fluctuations give rise to a thermally activated conductance, although the Hall effect is essentially metallic in nature. The observed activation energy is expected to persist above the percolation threshold and to decrease with decreasing temperature. At very low tem- peratures tunnelling through the classically prohibited regions becomes increasingly important, and should dominate the current near the threshold of inversion. The density of states in the bandtail falls off rather slowly with energy as a result of re- duced screening, while the screening itself becomes less effective with decreasing density of states. The experimentally-observed characteristics of inversion layers ap-

pear to bear out the predictions of the model.

Acknowledgement

The author is indebted to Mullard Research Laboratories for their hospitality during the course of this work. Valuable discussions with Dr. J.M. Shannon and

Dr. F. Berz are gratefully acknowledged.

Appendix A: Standard deviation for potential fluctuations

To obtain an expression for the standard deviation in eq. (3) we use an adaptation to two dimensions of a procedure due to Morgan [ 17,181. We assume that ions with- in the oxide, each carrying a charge Ze, are located in a plane separated by a distance 5 from the inversion layer. Let N’ be the average density of the ions distributed at

random among N, available sites, and let Pi(ni) denote the probability of finding ex- actly ni ions in a region i of area A,, containing mi ionic sites. This probability may be expressed as the coefficient of s ni in the expansion of its generating function G,(s) which, for a binomial distribution, is given by

Gi(S) = (PS + 1 - p)mi = CPi(lli)Sni , ni

(A.11

where p E p/N,. If each ion in the region i contributes an energy -& to the poten- tial energy of an electron in the inversion layer, and the regions are small enough, so that c#+ may be considered constant in each, then the total energy contributed by ni ions is -nisi and each configuration of ions contributes an energy T/= -~ini~i. The regions are chosen to be concentric rings of width dx at the plane of the interfacial charge, so that mi = 27rN8 dx, & = G(r), r * = x2 + F*. The complete generating func- tion for the probability density at a point, chosen as the origin, may then be written

E. Arnold / Conduction mechanisms in bandtails at Si-SiOz interface 69

as

G(s) = rI(,sPi + 1 - p)mi i

= exp[ 2nN, Jx dx ln(ps-@ + 1 - p)]

0

= exp[27rN+J(~-Q(~) - 1)x dx] =JP(V)s’dV, (A.2)

where we have neglected the terms in p 2. The standard deviation o* = ((I/ - q2) is

given by

u* = [d2G/ds2]S,l = 27rN+ Jx $20) dw . (A.3) 0

For the two-dimensional screened Coulomb potential we take [ 121 :

N(r) = 2Ze23i/(K s + Ko) r3 .

Carrying out the integration in eq. (A.3) we obtain

(A.4)

U* = 2&(Ze2X)*/;i2(Ks + Ko)* . (A.5)

Appendix B: Scattering off prohibited regions

We outline here an estimate, given by &garter [9], of the reduction in carrier mobility as a result of scattering from prohibited regions. Let L denote the correla- tion length for potential fluctuations, so that V(r) varies significantly only over dis- tances > L. The fraction p(E) of allowed space is also equal to the probability that

a point chosen at random is allowed. Thus the probability that, along a given path, the particle finds k consecutive allowed cells of length L, the (k + 1)th being prohi- bited, is

pk =p(E)k[l - ?@)I

This leads to a mean free path

(B.1)

1, = Fk& =Lp(E)[l -p(E)]-1 . VW

It can also be seen from a similar argument that the effective barrier width at energy E is

I, = L [ 1 - PWI /P(E) 3 03.3)

70 E. Arnold / Conduction mechanisms in bandtails at Si-SiOz interface

so that at the percolation threshold, where p(E) = l/2, I, = I,, = L.

The scattering time corresponding to the mean free path la is

Ta C-Q = l, @M u) ,

where (u) is the average velocity over the allowed regions:

(u) =p(E)-l / [(E - v)/‘m*] 1/2 P(v) dV. -cc

(B.4)

WI

If 7. is the relaxation time due to all other scattering processes in the allowed regions, then the combined scattering time is

71(E) = 7a7*(7a + 70)-l :

and the mobility in the allowed regions is

P&W&) = Q(E)l 70 = [ 1 + rg(u)/l,(E)] -l ,

where p. is the mobility in an allowed region of infinite extent.

(B.6)

(3.7)

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111 r21 131 [41 [51

161 171 181 [91

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[ISI [161

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