Is there ever a minimum metallic conductivity?

SohASrure Elecrronrcr Vol. 28, Nos. l/2. Printed in Great Britain

pp. 57-59,

1985

0038-1101/x5 $3.00 + .xJ Pergamon Press Ltd.

IS THERE EVER A MINIMUM METALLIC CONDUCTIVITY? N. F. MOTT Cavendish Laboratory, Cambridge, U.K. Abstract-When the metallic conductivity of a noncrystalline system goes to zero through change of composition or stress, it is observed to go continuously to zero, as predicted by scaling theories. When the transition is induced by a magnetic field, this appears not to be so. In the case of InP investigated by Long and Pepper and Biskupski, the discontinuity in the conductivity corresponds numerically to the author’s minimum metallic conductivity 0.03e2/Aa. A theory is given which accounts for this. It is suggested that in Si:P the metal-non-metal transition takes place in the conduction band, and that this can only occur in many-valley

bands.

I want to discuss the metal-insulator transition in disordered materials, in the approximation of noninteracting electrons, and based on Anderson’s [l] concept of localization and the idea of a mobility edge [2,3], which I denote by EC (Fig. 1). I refer to a degenerate electron gas with Fermi energy EF, and only to the limit of low temperatures. In Fig. 1, if EF lies above E,, the material is a metal; if EF lies below EC, it is a “Fermi glass;” the conductivity at high temperatures is of the form u=uoexp{

-(EC-E,)/kT},

NIE)

(bl

Fig. 1. Density of states in an impurity band. EC is the mobility edge, and localized states are shaded. (a) is for a nonmetal (Fermi glass) and (b) for a metallic state.

(1)

I follow Vollhardt and Wiiifle[8] in taking s = 1. At a finite temperature L = L, , where

L, is the inelastic

diffusion

length

L, = ( oqy2;

(2)

where a is the mean distance between the centres. I want to ask how this is to be explained? I believe that many lines of investigation lead to the conclusion (again in a theory of noninteracting electrons) that for an energy A E above EC, u = 0.03e2/AL;

E

EC

lol

and at lower temperatures, if N( EF) is finite, by hopping. We now know, following the scaling theory of Abrahams et al. [4] and much experimental work, that if EF is moved downwards as a consequence of change of composition or stress, o(T = 0) goes continuously to zero. However the work of Long and Pepper[S] in Cambridge, which was reported at a 1984 meeting in Santa Cruz and that of Biskupski et al. [6] in Lille, shows that in doped InP where the transition is induced by a magnetic field H, o(T = 0) drops continuously to the value given by the present author as u m,n= 0.03e2/tra,

E

EC

(5)

D is the diffusion coefficient and ~~ the inelastic lifetime. This expression should also give the preexponential in (1). I have examined [9] the situation for a-Si-H, and find that, because of the large value of l/7,, L, is not much changed from previous estimates. In a magnetic field H, however, L = L, where L, = ( ch/eH)1’2 and when, as in the work L, 5 a, one should take

of Long

(6) and

Pepper,

(3) L=a.

the numerical factor differs from that of some other investigators; I have tried to justify this in a recent paper [7]. Here at zero temperature and in the absence of a magnetic field L is equal to the IocaIization length .$ at an equal energy A E below EC, and behaves like [=a/(AE)“. SSE28:1,2-E

and so (2) is valid, both for the smallest nonactivated conductivity and for the pre-exponential in (1). If at zero temperature and in the absence of a field u = const e2/h.$,

(7)

I have to ask myself what the wave functions look likej7). Below the mobility edge wave functions of

(4) 57

N. F. Mom

58 localized

states are of the form 4---R

%Oc = Re [ qeX, 1exp( -

Ir- r,I/C>

+

(8)

where

Here the +,,, are random phases and the r, are random points in space. The c,, may fluctuate strongly. Within a volume c3 there will be N(E)t3 states per unit energy range, so that the interval between them is 1/N(E)c3. I argue (Kaveh and Mott[lO]) that they never coincide; the argument was used by Mott[ll] to show that a(w)-&?

J0

mexp(-R/t)R4dR,

where a(w) is the conductivity at frequency they could overlap, we should have a(o)-&?

J0

Fig. 2. Suggested envelopes of the wave functions for energies (a) below and (b) above E, w. If

mexp(-R/[)R4dR,

giving 6’ without the logarithm. In the discussion by Shapiro[l2] and by Ortuno and Kaveh[l3] on u(w), a distinction is drawn between the range w < Ok when the w* law applies and o > w,. when conduction is essentially metallic, varying as w113; w< is equal to D/t’, which is the same as l/ftN( E)t3, as may easily be verified. If then I take an energy AE below the mobility edge and if E < A q, I argue that two localised states cannot overlap much; their minimum distance apart cannot be less than R=[ln(E/AE), where E is of the order of the band width. Then at an energy A E above EC the wave functions cannot have changed much; their envelopes must be as I illustrate in Fig. 2. Near the maxima they are (a) real (b) all identical, while near the minima they are both different and complex. They are thus strongly correlated, both as regards amplitudes and phases. In the Kubo-Greenwood formula for the conductivity,

it will be seen that, if \kE,\IIE. have independent phases, then x-integration over a range - .$ will give a term in (t/a) ‘j2; if the 9’s are real and identical, however, this term is absent. Calculation shows then that the author’s u,,,~,, has to be multiplied by a/[, giving 0.03eL/A[.

In a magnetic field, however, the localized wave functions (8) are not real. Indeed, if L,, - u, there is a large phase change in going from one centre to another. This is why the phase correlation breaks down, and u,,,,,, reappears. Another way of looking at this problem is to take the expression valid far from the transition, obtained by many authors,

u=

c

usg2 1 i

gZ( k,/)”

‘1-i 1

(10) II

Here I is the elastic mean free path, nH the Boltzmann conductivity and g is the drop in the density of states due to disorder. Without the factor g the expression was given for instance by Kawabata[l4], and obeys scaling theory with j? a function of Lu( L). In an earlier work by the author l/g was written instead of l/g*, but calculations show that the term is proportional to l/N(E) D, and since u is equal to eN( E) D, g’ is appropriate. Moreover equation (10) now obeys scaling theory. The constant C should lie between 1 and 3, depending on the cut-off taken for an integration over a wave number g(l/l or m/l). If we take C = 1 and kJ= n at the Ioffe-Regel limit and extrapolate to the transition, this occurs when g( = g, ) = l/3, which is approximately correct using recent numerical work (Elyutin et ul[15]) for the Anderson transition based on the 1958 model of Anderson [l]. If behaviour of the form (10) is valid near the transition and if C = 1, we find for the p-function ,l3=1-l/&G where

G is the dimensionless

conductance

G = ALu/e’.

Minimum metallic conductivity so that at the transition G=G,., where

u = (e*/ti)G,/L,

and where g,. = 1/n*.

The factor 0.03 is the same as obtained above, but we note that scaling theory implies that this should be a universal constant; this is perhaps surprising, since if we write the constant 1/3g5, g,. should depend on co-ordination number if one uses localization criteria worked out for one model of Anderson [ 11. All I have said up till now applies in principle to a compensated impurity band, where intra-atomic interaction (the Hubbard U) should not make much difference, though long-range interaction leads to the famous expression

that in the specific heat yT, y conforms fairly closely to the value for the many-valley conduction band right down to the transition, with only some minor interaction effects. This is hardly compatible with an impurity band, and suggests that the transition is in a conduction band with g = 1. I have argued that this is only possible in a manyvalley band. However, it now seems that the work of Long and Pepper [17] on Si:Sb suggests that in this material a transition between a situation when electrons are in an impurity band and one in which they are in a conduction band can occur very easily. I wish I could say something now about the origin of eqn (13), and the corresponding behaviour of the dielectric constant of Si:P. At first sight it looks as if, with electron-electron interactions, l/t is proportional to (E - EC)1/2, a result I am very reluctant to believe for reasons given earlier [18].

REFERENCES 1. P. W. Anderson, Phys. Reu. 109, 1492 (1958). 2. N. F. Mott, Adv. Phys. 16, 49 (1977).

a = u0 + mTl/*, observed in many amorphous metals as well as in highly doped semiconductors. For an uncompensated impurity band, one would in principle expect a discontinuous “Mott” transition, though as I have argued elsewhere disorder mq be great enough to broaden this and give it the properties of an Anderson transition. What then are we to think of the properties of Si:P as shown by the remarkable results of Gordon Thomas’s group? In a paper with Kaveh [16] I argued that the very sharp drop of o(T = 0) to zero (in a range an/n - 0.01) might indicate a slightly broadened Mott transition. I do not believe this now, partly because of Thomas’ stress tuning work which shows that

now

aa(n-n,.)1’2,

59

(13)

which I fail to account for on this hypothesis, and even more because his work and that of Sasaki show

3. M. H. Cohen, H. Fritzsche and S. R. Ovshinsky, Rev. Left 22, 1065 (1969).

Phvs.

4. E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramaktishnan, Phys. Rev. Lett 42, 673 (1979). 5. A. R. Long and M. Pepper, J. Phys. C 17 3391 (1984). 6. G. Biskupski, H. Dubois, 5. L. Wojkiewin, H. Briggs and G. Remenyi, J. Phys. C. 17, L411 (1984). 7. N. F. Mott, Phi/ Mug. B 49, L75 (1984). 8. D. Vollhardt and P. Wolfle, Phys. Rev. B 22, 466 (1981). 9. N. F. Mott, Phil. Mug. B (1984) (in press). 10. M. Kaveh and N. F. Mott Phil. Mug. B, (1985) (in press). 11. N. F. Mott Phi/. Mug. 22, 7 (1970). 12. B. Shapiro Phys. Rev. B 25, 4266 (1982). 13. M. Orbuno and M. Kaveh J. Phys. C. 17, L487 (1984). 14. A. Kawabata, Solid St. Commun. 34, 431 (1980). 15. T. V. Elyutin, B. Hickey, G. J. Morgan and W. F. Weir, Phys. Stut. Solidi (1984). 16. N. F. Mott and M. Kaveh, Phil. Mug. B 47, 537 (1983). 17. A. P. Long and M. Pepper, J. Phys. C 17, L425 (1984). 18. N. F. Mott, Commun. in Phys. 1, 203 (1978) and Phil. Mug. B 44, 265 (1981).