LOCALIZATION BY CORRELATION AND BY DISORDER

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LOCALIZATION BY CORRELATION AND BY

DISORDER

D. Thouless

To cite this version:

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Abstract. —• Various aspects of the interplay between electron correlation and localization by disorder are discussed. Disordered electron spins in the Hubbard model should make the system less metallic before there is a band gap. Correlation in the Anderson model may not alter the qualita-tive properties significantly and the dielectric constant should become very large just below the mobility edge. The electron-phonon interaction will distort the density of states for localized electrons but should not be expected to produce a real gap at the Fermi energy. When the equilibrium involves doubly occupied localized states, the existence of singly occupied localized states has an important effect on the photoelectric current.

Several different mechanisms can be responsible for nisms influence one another. Much of what I say will be a metal-insulator transition, and in real situations they inconclusive, and we do not yet understand these may all operate to some degree, so that we have to things in sufficient detail. I shall summarize our consider not only their relative importance, but also knowledge of the Anderson mechanism of localization the extent to which one mechanism may enhance or by disorder so that at least the problems can be diminish the effect of another. The most obvious reason understood even if the solutions are wrong. Where I for a metal-insulator transition is a change in band want to consider a definite example I shall use the structure, so that in the insulating state the Fermi inversion layer descussed by Pollitt, Adkins and surface of the electrons coincides with a Brillouin zone Pepper [2] in the previous paper, as this is a case where boundary, and the exclusion principle prevents elec- orders of magnitude can be estimated from the expe-trons from moving in response to a weak electric field, rimental results fairly easily.

Secondly the electrons may be held in localized states Electron correlation can produce an insulator if the by their correlation energy, due to Coulomb interac- increase in Coulomb energy needed to remove electrons tion between the electrons or to strong interaction from atomic or molecular orbital sites is greater than with the ions in their neighbourhood, and a substance the reduction in kinetic energy in going from these will be insulating if the correlation energy in the atomic or molecular states to extended Bloch states, insulating state is more negative than the band energy The simplest model to illustrate this behaviour is the in the metallic state. This is the Mott transition, and is Hubbard model for tightly bound s-electrons, in which probably less obvious to physicists than the effect of the Coulomb interaction is represented by a repulsive band structure, although no-one would expect liquid interaction between electrons on the same atom, which nitrogen to be metallic despite its lack of sharp Bloch must be of opposite spin. The spin- up electrons move levels. Thirdly there is the possibility of localized states in the potential produced by the spin-down electrons produced by disorder, which was suggested by Ander- and vice versa, and so the spin-up states form two son [1]. This has proved much harder to understand energy bands as a result of the higher energy on those than the other two mechanisms, and some basic atoms occupied by spin-down electrons. If there is one features of this mechanism are still controversial or electron per atom the system will be insulating when the poorly understood. In this paper I shall attempt to bands become separated by an energy gap, since the assess the extent to which the last two of these mecha- lower bands will be full and the upper bands empty.

LOCALIZATION BY CORRELATION AND BY DISORDER

D. J. THOULESS

Department of Mathematical Physics, University of Birmingham Birmingham B15 2TT, U. K.

JOURNAL DE PHYSIQUE Colloque C4, supplément au n° 10, Tome 37, Octobre 1976, page C4-349

Résumé. — On discute divers aspects présentés par l'interaction des corrélations entre électrons et la localisation par le désordre.

Des spins électroniques désordonnés dans le modèle d'Hubbard devraient rendre le système moins métallique avant l'apparition d'un gap à un électron. Les corrélations dans le modèle d'Ander-son ne devraient pas changer substantiellement les propriétés qualitatives, et la constante diélectri-que devrait augmenter très fortement juste au-dessous du seuil de mobilité. L'interaction électron-phonon produira une distorsion de la densité d'états pour les électrons localisés, mais on ne s'attend pas à ce qu'elle crée un véritable gap au niveau de Fermi. Quand l'équilibre fait intervenir des états localisés doublement occupés, l'existence d'états localisés occupés par un seul électron a un effet important sur le courant photoélectrique.

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C4-350 D. J. THOULESS

Any excess carriers should, according to this argument, go into the empty bands and be free to move, so the conductivity should be very sensitive to doping by impurities or imperfections in the lattice.

A transition of this sort can also occur even without the aid of a lattice, as in the Wigner transition of an electron gas, but it goes much less readily than in the Hubbard model, so that correlations can be quite strong in the disordered (conducting) state before a transition occurs to the ordered (presumably insulating) state. In the Monte-Carlo calculations of Pollock and Hansen [3] for the three-dimensional electron gas the ratio of the correlation energy at zero temperature to the transition temperature is found to be 140 K, while in two dimensions Hockney and Brown [4] find 25 K. For the case of an &version layer with an electron density of 2 X 1015 me2 in a material with dielectric constant 12 the maximum available correlation energy is about 2.5 meV, which is comparable with other energies, but the temperature for the Wigner transition is about 1.2 K, which is considerably smaller than the degeneracy temperature, so one would not expect a Wigner transition, but would expect the correlation to be fairly important.

Anderson's work on the effect of disorder showed that disorder. could produce a continuum of localized states. If the disorder is weak these localized states will only occur in the band tails, but with increasing disorder the mobility edges separating the localized states from the extended states will move into the centre of the band until for sufficient disorder they merge and the whole band is localized. The localized states can only contribute to the conductivity when energy is supplied so that the electrons can hop from one level to a near one of slightly different energy, and so the system is only metallic if the Fermi surface lies in the region of extended states. If the Fermi surface is in the region of localized states the conductivity is insensitive to impurities and imperfections, in sharp contrast to the simple picture of localization due to electron correlation presented earlier, because these impurities simply shift the Fermi-surface slightly in the continuous region of IocaIized states instead of moving it up to the conduction band or down to the valence band.

This mechanism for localization has proved to be hard to understand in simple theoretical terms, and much of what we have learnt has come from the inter- pretation of experimental situations which are often ambiguous (see Mott, Pepper, Pollitt, Wallis and Adkins [5] for a recent summary) and from the numerical simulation of disordered systems. It has become clear from numerical and theoretical work that the density of states does not show up the mobility edge. For example Watabe (private communication), has shown that the coherent potential approximation, which gives an analytic form for the density of states, is a good approximation right through the region in which the mobility edge occurs. One complication of this

problem is that at least two length scales are involved in a fundamental way. Extended states for electrons are characterized by a wavelength and a mean free path. Mott [6] has emphasized the importance of the argu-

ment made by Ioffe and Regel [7] that half the wavelength

-

the distance in which the phase of the wavefunction changes by n: - cannot be less than the mean free path - the distance in which the wavefunction loses phase coherence - and has said that the approximate equality of these two lengths should occur at the mobility edge. Our numerical work on two- dimensional systems (Licciardello pnd Thouless [8],

Debney [9]) has shown that this criterion gives a

reasonably good guide to where the mobility edge occurs, in some cases much better than criteria based on the ratio of the density of states to the unperturbed band width. Localized states are characterized not only by the length of their exponential fall-off, which is very small in the band tail and becomes infinite at the mobility edge, but also by other lengths such as the distance between energy levels and the dimension of the volume needed to sustain such a state. For some types of potential fluctuations the states of lowest energy only occur in a fluctuation of large volume, so that they occupy a large volume but decay in a short distance outside that volume. It is not clear what other length is most relevant near the mobility edge, but we have argued (Last and Thouless [10]) that one might expect the wave function to fall off like r - q e-"' at large distances where a tends to zero at the mobility edge and y is an exponent depending on dimensionality. If, as seems likely, y is greater than 1

4

in three dimensions the wave function is normalizable at the mobility edge, and

may be an important length scale for the system, since it measures the region in which the wave function is concentrated.

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LOCALIZATION BY CORREJ ,ATION AND BY DISORDER C4-351 localized, since there the wavelength will be greater

than the mean free path. Further, as has been argued by Cyrot [Ill, we might expect electrons in the centre of the band, where the Fermi surface lies, to be localized by the disorder before the two sub-bands completely separate, since there is a low density of states there (Fig. 1). This should lead to a low mobility, since such electrons can only move in combination with the electrons which provide the potential for the localized state, and so form spin polarons.

FIG. 1. - Density of states for Hubbard model before two sub- bands have separated. States between the two energies marked EL

are localized.

FIG. 2. -Density of states when localized states are shifted down by a finite amount at the mobility edge by lattice defor-

mation.

Similarly when the Anderson model is applied to real electron systems the Coulomb interaction has to be taken into account, and may result in important mo$- fications of the theory. We have already remarked that for the inversion layer the correlation energy is quite large in magnitude, but this does not necessarily mean it changes the system in a directl$ observable manner

- a similar situation exists for metals. When the Fermi surface lies sufficiently far below the mobility edge that the localized wave functions fall off in a distance small compared with the separation of occupied levels the correlation effect must ensure that the occupied levels are more or less uniformly distributed in space, since the increase in energy due to any particularly close pair of electrons is likely to be of the order of 5 meV, and this is large compared with the energies of the localized states in the absence of interaction. The Coulomb energy should also produce a large shift in the energies of unoccupied states relative to occupied states, since the unoccupied states are in a sense interstitial sites in the irregular lattice of occupied states and are therefore high in energy. Efros and Shlovskii [l21 have argued similarly that the density of states should go to zero at the Fermi surface, but Mott 1131 disagrees. When electrons move under the influence of a field, as in the mea~ured'hoppin~ conduc-

tivity, they must move in a collective manner so that they do not acquire a high Coulomb energy. The ana- logy of electrons in a metal suggests that this may occur without making results obtained from a theory which ignores the Coulomb interaction invalid, but I do not think we have any detailed picture of how it occurs. In some sense the electrons must adjust themselves among the localized states to screen the main part of the Coulomb interaction when an electron makes a hop in the variable-range hopping process.

This shows that we must expect electrons in localized states to be able to respond to the strong electric field produced by an electron by moving between localized states. Similarly we can expect localized electrons to respond to a weak electric field and so change the dielectric constant. The expression for the polarizability given by perturbation theory is

where

2

is the position operator, E, and f, are the energies and occupation numbers of the state p, E is

the dielectric constant and E, the permittivity of free space. If the state p is strongly localized the main matrix elements of

2

may be to states well up in the extended region, but close to the mobility edge there should be substantial matrix elements between the localized states and the bottom of the extended region. Thus the denominator gets small as the mobility edge is approached, and the numerator gets larger, with of the order of the range a- l of the localized state, so the dielectric constant may diverge as (EL

-

where s is the exponent with which a goes to zero. If this is correct the long range effect of charges is partially screened by the localized states, and the screening gets more and more effective as the mobility edge is approached.

Anderson [l41 has pointed out that while electrons in extended states are not usually shifted much by their interaction with the phonons, electrons in localized states may be. In a simple model of the electron-phonon interaction the energy shift is proportional to

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C4-352 D . J. THOULESS

Thouless [10], however, suggested that

~

may be normalized right up to the mobility edge in three dimensions, and in that case the expression (3) is dis- continuous at the mobility edge. This situation was envisaged by Anderson as a consequence of strong coupling, but we are suggesting that it can happen however weak the electron-phonon interaction. This problem has been studied by Mr. El Zain at Bir- mingham, and the results quoted are partly from his work. It would seem that when the Fermi surface is placed at the mobility edge the occupied localized states will shift down leaving a gap to the extended states, and this might lead to a change in the variable- range hopping behaviour. However, although only extended states are empty, it would be possible to deform the lattice locally in such a way that an additional localized state is formed, and the binding energy of the electron can more than compensate for the deformation energy of the lattice. Thus although the static potential has a gap between the localized and extended states, this gap can be completely filled by the additional states brought down by a self-consistent deformation of the lattice, and so the gap will have no effect unless the production of such a state with its corresponding deformation potential is very slow. In fact we find that the deformation can occur very readily provided the energy shift is much less than the phonon energy.

Anderson [l61 recently made the suggestion that electrons in chalcogenide glasses interact so strongly with the lattice that the extra binding energy produced by a second electron in a localized state, which should double the deformation of the lattice and so double the

[l] ANDERSON, P. W., Phys. Rev. 109 (1958) 1492-505. [2] P O L L I ~ , S., ADKINS, C. J. and PEPPER, M,, J. Physique

Collog. 37 (1976) C4-341.

[~];POLLOCK, E. L. and HANSEN, J. P., Phys. Rev. A 8 (1973)

3110-22.

[4] HOCKNEY, R. W. and BROWN, T. R., J. Phys. C 8 (1975) 1813-22.

(51 MOTT, N. F., PEPPER, M., POLLITT, S., WALLIS, R. H. and ADKINS, C. J., Proc. R. SOC. 345 (1975) 169-205. [6] MOTT, N. F., Metal-Insulator Transitions (Taylor and

Francis, London) 1974, p. 26.

IOFFE, A. F. and REGEL, A. R., Progr. Semicond. 4 (1960)

energy shift per electron, more than compensates for the Coulomb repulsion between two electrons in the same state. In this case the localized states are occupied either by two electrons or none, as the states can only be singly occupied if dissociation energy is supplied. It is even suggesteh that singly occupied states may not be bound at all, but may all be extended states. These doubly occupied states do not contribute to spin para- magnetism or to paramagnetic resonance, and the absence of such effects is good evidence for this mechanism. Hopping conductivity by means of the pairs should be possible, although the mobility should be considerabIy reduced and so it might be hard to see. It should be possible to find something about the nature of the singly occupied states by studying the decay of photoelectric currents. If an electron is excited into the conduction band above the mobility edge it may be trapped by a localized state if there are localized states which can be occupied by a single electron, and this singly occupied localized state can serve as a trap for a second conduction electron, so that the pair returns to the doubly occuped equilibrium state. In this case the decay rate of the current should be inde- pendent of the concentration of carriers. If, however there are no localized one-electron states the electrons in the conduction band can only return to localized states in pairs, and so the photoelectric current should decay at a rate proportional to the number of carriers, and so the current should be proportional to the square root of the light intensity.

I wish to acknowledge that most of what is contained in this paper has been influenced by numerous conver- sations with Dr. D. C. Licciardello. '

[S] LICCIARDELLO, D. C. and THOULESS, D . J., J. Phys. C 8

(1 975) 41 57-70.

[9] DEBNEY, B., J. Phys. C 9 (1976) to be published.

1101 LAST, B. J. and THOULESS, D. J., J. Phys. C 7 (1974) 715-31.

[l11 CYROT, M., Eilat Conference on Low Conductivity Mate- rials (Tannhauser 1971).

[l21 EFROS, A. L. and SHLOVSKII, B. I., J. Phys. C 8 (1975) L 49- 51.

[l31 MOTT, N. F., J. Phys. C 8 (1975) L 239-40.

[l41 ANDERSON, P. W., Nature, Phys. Sci. 235 (1972) 163-5. [l51 M o n , N. F., to be published in Comm. Phys.