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THE INHOMOGENEOUS TRANSPORT REGIME AND METAL-NONMETAL TRANSITIONS IN
DISORDERED MATERIAL
Morrel Cohen, J. Jortner
To cite this version:
Morrel Cohen, J. Jortner. THE INHOMOGENEOUS TRANSPORT REGIME AND METAL- NONMETAL TRANSITIONS IN DISORDERED MATERIAL. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-345-C4-366. �10.1051/jphyscol:1974467�. �jpa-00215658�
THE INHOMOGENEOUS TRANSPORT REGIME AND METAL-NONMETAL TRANSITIONS IN DISORDERED MATERIALS (*)
Morrel W. C O H E N
The James Franck Institute and Department of Physics
The University of Chicago, 5640 Ellis Avenue, Chicago, Illinois 60637, U. S. A.
a n d J. JORTNER
Department of Chemistry, Tel-Aviv University Tel-Aviv, Israel
Rbum6. - Nous avanqons une representation physique pour les changements apparemment continus dans la structure electronique et les proprietks de transport, observes au cours des tran- sitions metal-non metal, se produisant dans les nombreux materiaux dksordonnes. Des deforma- tions structurales ayant pour origine des fluctuations de densite, des modifications de liaisons, la formation de composes ou d'agglomerats, peuvent se traduire par une non-homogen6it6 micro- scopique locale dans la structure Clectronique de tels mattriaux.
Quand la courte distance de correlation - de Debye - pour les fluctuations est suffisamment grande, celles-ci peuvent Ctre considerees comme statistiquement indkpendantes. De plus en prenant les phases electroniques au hasard, a 1'6chelle de variation de la configuration locale, on peut definir une structure 6lectronique locale et des fonctions locales approprikes. Finalement quand les effets quantiques produits par effet tunnel, et quand les corrections d'energie cinetique sont petits, I'image semi-classique est applicable. En consequence nous pouvons considerer un regiment de transport non homogene dans lequel des effets de percolation se traduisent par un changement continu des proprietes de transport.
Une version generaliske de la thCorie du milieu effectif pour la conductivite thermique, l'effet Hall, et le pouvoir thermoelectrique, a BtB utilisCe pour analyser plusieurs classes de materiaux subissant une transition continue metal-non metal. Une application detaillee de la thkorie est presentee pour des systemes a un constituant tels que Hg liquide ktendu et Te liquide, ainsi que pour des systemes binaires tels que alliages mktalliques et solutions metal-ammoniaque.
Abstract. - We advance a physical picture for the apparently continuous changes in theelectro- nic structure and transport properties observed during the course of the metal-nonmetal transitions occurring in many disordered materials. Structural nonuniformities originating from density fluctuations, bonding modifications, compound formation or clustering may result in local micro- scopic inhomogeneities in the electronic structure in such materials.
When the Debye short correlation length for the fluctuations is sufficiently large, these can be considered as statistically independent. Furthermore, given that the electronic phases are random on the scale of variation of the local configuration, one can define a local electronic structure and local response functions. Finally, when quantum effects originating from tunnelling and kinetic energy corrections are small, the semiclassical picture is applicable. Consequently, we can consider an inhomogeneous transport regime within which percolation effects result in a continuous varia- tion in the transport properties.
A generalized version of the effective medium theory for the conductivity, Hall effect, and thermal conductivity and thermoelectric power was used to analyze several classes of materials undergoing a continuous metal-nonmetal transition. A detailed application of the theory is presented for one component systems such as expanded liquid Hg and liquid Te and for two component systems such as some metallic alloys and metal-ammonia solutions.
1. Introduction. - Many apparently continuous collected in Table I. I n particular, the primary variables metal-nonmetal transitions, induced by changes i n of state appear t o be density or temperature in a density, temperature o r composition, are exhibited one component system, such a s expanded liquid i n disordered materials [I]. A number of examples is H g [2], o r liquid Te [3], respectively, and composition (*) Based on research supported in part by the U. S. Army i n two component systems, such a s binary metallic Research Office (Durham) and the National Science Foundation alloys and metal-ammonia-solutions- I n all the cases at the University of Chicago. listed in Table I, there is n o evidence for a n abrupt
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974467
MORREL H. COHEN AND J. JORTNER
TABLE I
Examples of the Continuovs Metal-Nonmetal Transition in Disordered Materials
Substance Phase Variable
- -
Hg ( a ) liq
Cs ( b ) liq
Te (9 liq
Me-NH3 (&) liq T I X T ~ I - ~ liq GazTes (f) liq G a ~ ( T e ~ S e l - ~ ) ( 8 ) liq Mgx-Bi r-x (h) liq
1-9 MPM X 0-0.66 T 700-1 200 K
X 0-0.7
X 0-1.0
Applicability of Inhomogeneous Regime Breakdown of Correlation Thermo-
Nonmetallic 0 < 2 500 Friedman's of transport dynamic Structural b state (SZ cm)-1 relations data by EMT evidence evidence 19
- - - - (i) - - -
semiconductor + + + 15
semiconductor +
semiconductor + f + (MI
electrolyte + + + (MI + f 30
semiconductor + + i- f
semiconductor + + (M)
semiconductor 4- + (M)
- for
semiconductor X < 0.58 inapplicable inapplicable -
X r 0.63
+ for + - -
X > 0.58 X < 0.63
(a) Transport data from references [2, 171 and [24].
( b ) Transport data from reference [25].
(3 Transport data from references [26a]-[26f]. Magnetic data reference [26g].
(6) Transport data from reference [30] and [58]. Magnetic data from reference [57]. Thermodynamic data references I361 and [37].
Structural data references [38] and [39].
(3 Transport data reference [27]. Thermodynamic data reference [35].
(f) Transport and magnetic data reference [56].
(9) Transport and magnetic data from reference [56].
(") Transport data reference from reference [32] and from SOMMER N., and EVEN U., to be published.
(i) (M) refers to analysis of transport data by EMT where C is extracted from magnetic data.
transition in the transport properties at any value of the relevant state variable. The problem we face is understanding the nature of the apparently continuous changes in the electronic structure and transport properties during the course of a transition from clearly metallic to clearly semiconducting behavior.
In this paper we advance a physical picture recently developed by us [4-81 for the continuous metal- nonmetal transitions in disordered materials and review its applications.
2. Conventional transport regimes in disordered materials. - There are four electronic transport regimes which have been identified so far in disordered materials :
2.1 METALLIC. - 2.1.1 Propagation [9-111. - When the mean free path, I, of the conduction elec- trons considerably ex~eeds the reciprocal Fermi wave number, i. e.
k F l > 1 (2.1) there is practically (( free )> propagation of the electrons
a perturbation expansion in powers of the potential converges well. Ziman [9, 101 and Faber [ I l l have shown that in the lowest order corrections to the Born approximation there is a cancellation of those changes in the conductivity resulting from changes in the density of states. The conductivity is well represented in terms of Ziman's nearly free electron theory [9, 101. Although the theoretical situation concerning the Hall coefficient is not yet completely clarified [lo, 121, we can conclude that in this regime there is no special relation between the conductivity and the Hall coefficient and further assert that on the basis of experimental data the latter quantity is close to its free electron value. Correspondingly, the transport properties must satisfy the conditions
where o is the conductivity, R the Hall coefficient, and RFE the nearly free electron value of R.
Consequently the Hall mobility
between distinct scattering events. In this regime p = Ro (2.3)
is dominated in its dependence on the state of the material by changes in o.
2.1 . 2 DifSusion 113-1 51. - As disorder increases, the mean free path can decrease to the point where the condition (2.1) no longer holds. It makes no sense then to think of I as the distance between collisions ; the electrons are continuously in collision. However, when (2.1) does hold, I is the distance over which the electronic wavefunctions retain phase coherence.
We can generalize I into a phase coherence length, which can become arbitrarily small. For 1 significantly smaller than the interatomic separation, the phase of the electronic wavefunctions becomes effectively random. Propagation and interference effects are unimportant. The motion is a kind of diffusion or Brownian motion.
Friedman [16] has studied transport in a crystal with a tight binding s-band. He assumes that the wavefunction amplitudes are everywhere constant but the phases on different sites are uncorrelated.
He then inserts these wavefunctions into the Kubo- Greenwood formulae and obtains
where z is the number of nearest neighbours, Z is the number of triangular closed paths around each lattice site, y -- Q is a geometrical factor, a is the internuclear separation, and the parameter
contains J, the nearest neighbour electron transfer integral, and the density of states n(EF) at the Fermi energy EF. Since all parameters except X can be readily estimated in particular situations, it is more convenient [I71 to relate o and p to R F E / R
where a, is the internuclear separation at the density p,.
Thus in this diffusion or Brownian motion, i. e.
strong scattering, regime
where the proportionality factors contain parameters which either depend explicitly or weakly on the variatles of state. The third member of (2.7) is suggest
by considerations on the Hall effect as the diffusion regime is approached like those given by Kubo [18].
Friedman [16] has related in a rough way the trans- port coefficients he derived to g, Mott's ratio [19] of density of states to the free electron density of states ( R ~ ~ / R ) O1g (2.8a)
oag2 (2.8b)
where
g = n ( E ~ ) / n ~ ~ ( E ~ ) . (2.9) Together with Mott [20] we assume the qualitative features of Friedman's results [16], eq. (2.7), have general validity for disordered materials, provided that there is sufficient microscopic homogeneity.
2 . 2 SEMICONDUCTING. - 2.2.1 Pseudointrinsic. -
In the current picture of disordered semiconductors, the crystalline band gap is replaced by a mobility gap [21], E,, within which all states are localized and of low mobility, falling to zero at T = 0. Conduction at sufficiently elevated temperatures is by thermal excitation of carriers across the mobility gap with a consequent correlation between conductivity and thermoelectric power S [22]
provided near compensation does not occur in S.
2 . 2 . 2 Pseudoextrinsic. - In general, the Fermi energy lies within the mobility gap and the density of states there is finite and continuous. As a consequence, electrons near the Fermi energy contribute to the transport via Mott hopping [23], which can dominate the conductivity at sufficiently low temperature, when
o = a, exp[- (T,/T)'/~] . (2.11) 3. Inapplicability of conventional transport regimes.
- We shall now confront the transport data on several typical disordered materials undergoing metal- nonmetal transition with those characteristics of the various conventional transport regimes discussed in section 2. We confine our explicit remarks to several systems which were studied by us in detail.
For expanded liquid Hg [2, 17, 241 several distinct transport regimes can be identified by combining conductivity, Hall effect and thermoelectric power data. For 11.0 gm cm-3 < p < 13.6 gm cm-3 the Hall coefficient (Fig. 1) has the free electron value, R/RFE = 1 , and the conductivity, which varies in the range lo4-2 600 (a cm)-', exhibits no particular relation to R. Thus on the basis of eq. (2.2) we assert that the propagation regime holds. For
C4-348 MORREL H. COHEN AND J. JORTNER
FIG. 1. - Density dependence of the electrical conductivity and of the Hall coefficient normalized to its free electron value for expanded liquid Hg, after Even and Jortner (ref. [17]).
FIG. 3. - The dependence of the Hall mobility on (RFEIR) for expanded liquid Hg, after Even and Jortner (ref. [17]). In the diffusion regime 9.3 gm cm-3 < p < 11.0 gm cm-3 ~ ~ ( R F E I R )
in agreement with eq. ( 2 . 7 ) .
both and (Fig. and 3) are in quantitative positive temperature dependence as is evident from agreement with eq. (2.7), and the diffusion regime 'gure 4. At lower densities
applies. In the diffusion regime o exhibits a weak 5.0 gm cm-3 < p < 7.8 gm
1 , , , , , , , ,
1
o and S are related by eq. (2. lo), figure 5, whereuponIo0o I 0.3 0 5 1.0 the pseudointrinsic semiconducting regime fits in the
( R F E / R ) range p < 7.8 gm ~ m - ~ . In the range
FIG. 2. - The dependence of the electrical conductivity on 7.8 gm cm-3 < p < 9.2 gm ~ m - ~ ,
RFE/R for expanded liquid Hg, after Even and Jortner (ref. [17]).
The diffusion regime spans the density range 9.3 grn cm-3 < p < (which was measured only to 8.6 gm cm-3)
11.0 gm cm-3 a a r ( R ~ ~ / R ) z in accordance with eq. (2.7). exhibits only a weak variation with p while both
o and R vary by about a factor of 10, the o vs S rela- tion (2.10) does not hold, and the volume and tempe- rature coefficients of the conductivity (Fig. 5) reveal a fast increase with decreasing p. None of the conven- tional transport regimes appears to be consistent with the data in this density range.
FIG. 5. - The dependence of In a on S for expanded liquid Hg, after Schmutzler and Hensel (ref. [24e]). The straight line corresponds to a slope of (kle). According to eq. (2.10) the pseu-
dointrinsic semiconducting regime holds for p < 7.8 gm cm-3.
For expanded liquid cesium recently studied by Freyland and Hensel [25] three conventional transport regimes can be tentatively identified by the combi- nation of electrical conauctivity and thermopower data. In the density range p > 1.2 gm cmP3,
holds for the conductivity (see Fig. 6), and the tem- perature coefficient of the conductivity, figure 7, is small and negative. In the density region 1.2 gm cmP3 > p > 1.0 gm ~ m - ~ , , o varies between
FIG. 6 . - The density dependence of the conductivity of expanded Cs, after Freyland et al. (ref. [25]).
CES
3 ! n G
-- 31"" IT
8 -
I U M '-
4 -
2 -
FIG. 7. - The temperature dependence of the conductivity of expanded Cs, after Freyland et al. (ref. [25]).
2 500 (Q cm) - ' and 1 500 (Q cm)-' ana the tempe- rature coefficient of a is still low, so that the diffusion regime may apply. For p < 0.5 gm cm-3 (where
o < 300 (Q cm)-') the conductivity-thermoelectric
power relation (2.10) holds, figure 8, and the pseudo- intrinsic semiconductor regime applies. In the range 0.5 gm cm-3 < p < 1.0 gm cmP3 the temperature coefficient of o is positive and increases fast with decreasing p (see Fig. 8) while the conductivity is still high, 300 (Q cm)-' < o < 1 500 (Q an)-' and does not obey relation (2. lo), whereupon none of the conventional transport regimes is applicable.
FIG. 8. - The thermopower-conductivity correlation for expan- ded Cs, after Freyland et al. (ref. 1251). Eq. (2.10) holds for
p < 0.5 gm cm-3.
In liquid Te [3, 261 a metallic regime characterized by the conductivity o
-
3 000 (Q cm)-' which is on the borderline between propagation and diffusion fits the data above about 1 250 K. In the temperature range 670 K < T < 1 250 K in which the liquid has been studied, the conductivity varies in the rangeC4-350 MORREL H. COHEN AND J. JORTNER
2 700 (R cm)-' to 1 150 (R cm)-' while the Hall mobility remains constant within 5 % (see Fig. 9).
FIG. 9. - The temperature dependence of the electrical conductivity and the Hall coefficient (ref. [26a]-[26fl) and of the
Knight shift (ref. [26g]) for liquid Te.
Mott [22] and Warren [26g] have utilized the Knight shift data, K, figure 8, to derive the g parameter (eq. (2.9)) and conclude that relation (2.8a) in the form o = A' K2 is obeyed, providing evidence for the applicability of the diffusion regime. We note, however, that an alternative, linear relation a = A" K - B"
also holds over the relevant T range. More important, the applicability of the diffusion regime woula require ,u to vary by about 60 %, according to eq. (2.8b), while the propagation regime would require ,u to vary by about a factor of - 2.6. Both predicted variations are at least an order of magnitude larger than is observed. We thus conclude that again none of the conventional regimes is consistent with the transport data.
In Tl,'Iel-x liquid alloys 1271, the resistivity and the Hall coefficient decrease with decreasing X, exhibiting a maximum at the composition correspon- ding to the T1,Te compound. While o and R both vary by about a factor of - 10 over the range
0 < X < 0.66, ,u changes only by about 30 % and none
of the conventional transport regimes is applicable.
In the high temperature materials such as liquid Hg, liquid Te and liquid T1,Tel-,, p remains constant within 30 % while a varies by about a factor of
- 10 in the range where the conventional regimes do not fit. The diffusion regime would require ,u to vary by about one order of magnitude more than it actually does for each of these systems. Similarly, the propa- gation regime would require a high conductivity o 2 3 000 (R cm)-' and far less variation in R than is actually observed. In the foregoing discussion we have relied heavily on the Hall effect data for esta- blishing transport regimes in disordered materials.
We are, of course, familiar with the fact that small deviations of the Hall coefficient of binary alloys [28]
in the propagation regime, e. g. 15 % for Hg/In alloys are difficult to interpret because of uncertainties about the organization of higher order perturbation theory as the strong scattering limit is approached.
This can by no means be used as an argument [29]
that the Hall effect is not suitable as a diagnostic tool for establishing transport regimes as we are concerned with changes of 100 %-I 000 % in this quantity. We conclude that the serious discrepancies of the transport data in the high temperature materials from what is expected for the conventional metallic propagation and diffusion regimes rule out the appli- cability of these homogeneous transport regimes.
Finally, we shall turn our attention to a <( low temperature )) two-component system. Metal-ammonia solutions [30] are fairly well understood in the extreme concentration limits. Most dilute solutions (< MPM) [3 11 exhibit the features of electrolytes containing localized solvated electrons and solvated cations. Ionic association involving the formation of solvated electron-cation pairs and subsequent spin pairing occurs in moderately dilute (10-3-0.5 MPM) solutions without gross modification of the structure of the solvated electron. Concentrated solutions (10 MPM to saturation) consist of solvated cations, unbound ammonia molecules and solvated electrons, constituting a homogeneous amorphous system whose local structure resembles that of a molten salt. Near saturation the electron cavities overlap over most of their surface, so that the nearly free electron approxi- mation is appropriate. Consequently, in this high concentration range the conductivity is high a = lo3 - 5 x lo3 (R cm)- l, the mean free path is long 12-70 A, the Hall coefficient is close to RFE, and the system corresponds to the propagation regime.
In the intermediate range, 1-10 MPM, the conductivity drops by about 4 orders of magnitude while R/RFE does not go above 2 for the lithium solutions. One could attempt instead to derive g from Knight shift or paramagnetic susceptibility data both being proportional to n(E,) in a homogeneous metallic regime. Eq. (2.2) and (2.8) do not hold whereupon both homogeneous metallic transport regime do not apply.
We conclude that some new kind of transport regime is clearly interposing itself between the metallic and the nonmetallic regimes in disordered materials.
Elucidating it is the main concern of our work.
4. Microscopic inhomogeneities in disordered mate- rials. - The electronic properties of materials which are undergoing a metal-nonmetal transition must be exceedingly sensitive to small changes in state or condition. Now, microscopic fluctuations in local condition are present in all disordered materials.
Indeed, the cumulative consequences of such fluctua- tions may be regarded as responsible for the break- down of long-range order. Microscopic inhomogenei- ties in disordered materials may originate from several causes :
1) Density fluctuations in a one component system.
Thus, for expanded liquid Hg and Cs in thermody- namic equilibrium there are density fluctuations throughout the material at any given instant. The
regions of higher density exhibit metallic behavior, while those of lower density tend towards semiconducting behavior.
2) Bonding modifications in a one component system. In liquid Te, Cabane and Friedel [3] have already proposed a state of mixed coordination in order to explain structural and magnetic resonance data. They suggest the existence of regions of 3-fold coordination which are metallic mixed with regions of 2-fold coordination which are semiconducting.
Using crude chemical terms, in the 2-fold coordinated state a lone pair does not participate in bonding.
In the 3-fold coordinated state, the lone pair electrons become bonding electrons, one being tied up in the adaitional bond, leaving one free.
3) Compound formation and clustering in a multi- component system. Such effects may be exhibitea in some amorphous and liquid binary metallic alloys where inter.metallic, molecule or compound for- mation [32] followed by subsequent aggrega- tion [33, 341 results in semiconducting molecular or compound clusters. For liquid alloys concentration fluctuation determination through chemical potential measurements were used to establish the binding strength of the molecule or compound and the cluster size [35, 361. In Te-TI liquid alloys evidence exists for extensive clustering on the TI rich side [35], while for the Te rich side the thermodynamic data indicate marginal clustering [35] (i. e. cluster size consisting of - 4 molecules).
4) Clustering in a two component system. For metal-ammonia solutions, there is substantial indirect evidence [30a], and only now some direct evidence from chemical potential determinations [36, 371 and small angle X-ray [38] and neutron scattering [39], for the existence of clusters of solvated electrons and cations in the concentration region 1-10 MPM.
Within the clusters, the concentration is in the metallic range.
In order to apprehend how such local non-unifor- mity of structure, condition or configuration can affect the electronic structure or properties, we consider first a centrally important local microscopic confi- gurational parameter, X(r). Define the autocorrelation function A ( R ) as
We know from the Ornstein-Zernike theory of fluctuations [40] that asymptotically
where 5 is the fluctuation decay length. On the other hand, condensed systems are stiff and tend to resist rapid change of configuration so that, very roughly speaking, X(r) varies significantly only over distances greater than b, the Debye short correlation length [41].
The consequences for A ( R ) are shown in figure 10.
FIG. 10a. - Autocorrelation function A(R) for a local configu- rational variable X(r) at low temperatures.
FIG. 106. - Autocorrelation function C(R) for a local confi- gurational variable X(r) at high temperature.
FIG. 1 0 ~ . - Local configurational variable at low temperatures.
FIG. 10d. - Local configurational variable at high ternpe- ratures.
In general, we shall be dealing with systems for which values of X at two points separated by more than b are therefore statistically independent. Within a radius b we can imagine two limiting cases of
C4-352 MORREL H. COHEN AND J. JORTNER variation of A(R). One, shown in figure IOU, corres-
ponds to low temperatures. A(R) remains nearly constant up to b ana then drops abruptly to zero.
The other, shown in figure 106, corresponds to high temperature. A(R) can vary appreciably within b.
In the first case, neighbourhoods of the material have differing, locally constant values of X with fairly abrupt switches from one value to another figure 10c.
In the second, values of X vary gradually over dis- tances of order 2 b, figure 10d. In the first case, a step function of radius b is the natural approximation to A(R). In the second case, no great error is made by dividing space up into a complete set of Wigner- Seitz spheres of radius b and replacing the values of X within each sphere by their average, the dotted line in figure 10d. In other words, we once again make step function approximation to A(R).
In any event b marks the distance scale of local variations in the configuration of the material.
Accordingly, it is the distance over which a kind of generalized deformation potential varies. Consequent- ly the amplitude of the electronic wavefunctions, vary over that same distance scale. Provided the phase-coherence length 1 is smaller than b,
the electronic structure can be treated semiclassically within this highly nonuniform deformation potential, boundary reflection can be ignored, and local response functions can be used.
The correlation length b is generally of order several interatomic separations. Thus, if the electrons are in the diffusion regime, the condition (4.3) is met.
We can then consider the medium as a submacro- scopically inhomogeneous random mixture of regions of radius b which can be treated semiclassically as locally uniform. The electron wavefunction does not actually go to zero within the excluded region. Never- theless, it can become small enough to be unimportant there in a variety of contexts. Accordingly we define an allowed volume fraction C(E) as that fraction of the total volume of the material actually allowed to electrons of energy E. Now, the Weyl theorem tells us that as long as the de Broglie wavelength is sufficiently small compared to the dimensions of the allowed regions, the density of states will be independent of the geometry of the allowed regions and of the boundary conaitions presented by the forbidden regions and proportional to the allowed volume. Thus we may take [34] as an approximate definition of C(E),
tunnelling across the excluded regions (see section 5) must be quantitatively unimportant for the physical properties.
The actual inhomogeneities can be expected to be of order several interatomic separations in radii.
Consider now an electron at the Fermi energy EF in a metallic region. EF will correspond to an energy in the middle of the semiconducting gap in certain circumstances. The electron will then be Bragg reflect- ed at the boundary, penetrating only a short distance into the semiconducting region. More generally, electrons with energies within the semiconducting gap are effectively excluded from the regions of the material in the semiconducting configuration.
There are some interesting immediate consequences of these simplications. If the configurational relaxa- tion is much shorter than the uncertainty time for the metallic Knight shift, the nuclear resonance line will be motionally narrowed. The corresponding Knight shift in the nonmetallic region is
provided the Chemical shift or Knight shift in the nonmetallic region is unimportant. Otherwise it is given by
K = KO C(EF) + K,[I - C(EF)] (4.5b) where KO and K , are the nuclear resonance shifts in the metallic and nonmetallic regions. Either of eq. (4.5) provides the basis for a direct experimental determination of C(EF). A term in (1 - C) must also be added to the density of states if the contributions of the nonmetallic regions to the density of states are not negligible in the energy region where C(E) differs appreciably from unity. This changes the definition of C somewhat from that of eq. (4.4), but we shall not dwell on this point here.
5. The continuous metal-nonmetal transition. -
We shall now consider some important features relating to the nature of the electronic states in the metallic regions [42-45, 341. If C(E) falls below the critical value C* for classical percolation [46, 471, percolation theory tells us that a continuous extended path through metallic regions does not exist. The metallic wavefunctions are therefore localized at that energy. In the vicinity of a local minimum in C(E) which lies below C*, there will be localized states and a pseudogap over the energy range Ev < E < Ec, where the mobility edges Ev,, are defined through
where n,(E) is the density of states per unit volume Our best estimate of the value of C* is 0.2.
of a metallic region of macroscopic extent and n(E) The condition for pseudogap formation is now is the actual density of states of the microscopically
min, ( C(E) ) = C* . (5 .2a) inhomogenous material. Defined in this way C(E)
allows properly for penetration into the excluded Provided that the Fermi energy in the metallic regions. However, for it to be a useful concept, regions lies close to the middle of the semiconducting
gap, the condition for a metal-nonmetal transition in the inhomogenous system is obtained from eq. (4.4) and (5.1) in the form
This result is distinct from
as proposed by Mott [1 5, 19, 221 for a microscopically homogenous system. The principal difference between (5.2b) and (5.3) is that n,(E,) need bear no relation whatsoever to the free-electron density of states.
C* is not sufficiently different from 3, which was only an estimate by Mott, to be concerned with.
The overall pictures of the metal-nonmetal tran- sition and the various transport regimes thus arrived at is summarized in figure 11. As X, an appropriate
THE INHOMOGENEOUS TRANSPORT REGIME
FIG. 11. - The Inhomogeneous Transport Regime. Transport regimes and shown as a function of the mean value of an
appropriate state variable of configurational parameter.
mean configurational parameter, varies from right to left in the metallic regime, the phase-coherence length 1 decreases. The system passes from the pro- pagation to the diffusion regimes. At some value of
X which could be reached either within the propa- gation or the diffusion regime, the system becomes microscopically inhomogeneous to a sufficient degree to affect the electronic properties in the manner described above. An inhomogeneous transport regime occurs within which C(EF) decreases mono- tonically from 1 to 0 as Xvaries from right to left.
Within the inhomogeneous regime, the material is an admixture of metallic regions of volume fraction C(EF) and nonmetallic regions of volume fraction U - C(EF). For 1 > C(EF) > C* the metallic regions are connected ; we term this portion of the inhomo- geneous transport regime the pseudometallic regime.
For C* > C(EF) > 0 the metallic regions are discon- nected and we have a pseudosemiconducting, or pseudoelectrolytic regime, as the case may be. As X
changes further from right to left, the inhomogeneities within the material again become insignificant for electronic structure and transport, and the material becomes semiconducting, electrolytic, or what have you. In the semiconducting case at elevated tempe- ratures the correlation (2.10) need not be established immediately at the value of X where C = 0 when the
energy gap is a continuous function of F. The energy gap may reach the value of about 4 KT required for that correlation to bold at a value of X only beyond the limit of the inhomogeneous regime.
The validity of the semiclassical picture has been challenged by Mott [29]. His main point appears to be that tunnelling effects are effective in erasing classical inhomogeneities in the electron motion, except where potential fluctuations are very slowly varying, such as near a critical point. In Mott's view, one can consider the wavefunction amplitudes as everywhere comparable, which implies the existence of a minimum metallic conductivity. In our view, the energy range in which percolation effects within the semiclassical picture are important depends on the magnitude of b, and when b is sufficiently long the inhomogeneities in the potential are smooth enough, within a deformation type of formalism, to make the semiclassical picture qualitatively valid. If, further, the potential fluctuations have e. g. a Gaussian dis- tribution of amplitudes with r. m. s. width a,, perco- lation effeets are important within an energy range 2 6, inside the mobility edge. An important test for the qualitative accuracy of the percolation picture is the amplitude < T2 > of the mean tunnelling probability. For a probability distribution, P(V), of the potential V, we get
m
7 = J dVP(V) exp ( - 4 b[2 m(V - ~ ) ] ' ' ~ / h 1 .
E
When P(V) is Gaussian, we obtain at the mobility edge E = E, where C(E) = C*, T~ N 1.1 C*/P provided P > 1.3 where P = 8(m6,)112 b. For tunnel- ling effects to be unimportant we require that
< T2 > = 0.01, which implies a value of P = 4.5
and 6,b2 = 4 eV (A)2. A typical value of 6,, 0.1 eV, would require a correlation length of b -- 6 A for tunnelling effects to be negligible. Thus the semiclassi- cal picture is valid for a physically realistic fluctuation correlation length.
6. Microscopic conductivity. - We are clearly dealing with an immensely complicated problem.
We have simplified it somewhat by introducing this concept of the allowed volume fraction. The latter is sufficient for dealing with such gross averages as density of states, Knight shift, and spin susceptibility.
However, to obtain quantitative results for transport properties, drastic further simplifications are necessary.
Linear response theory states that in a microsco- pically inhomogeneous medium there is a nonlocal conductivity tensor relating the microscopic current density, j(s1, to the microscopic electric field, E(r), j(r) = [ d3rr a(r, rf).E(rl) . (6.1)
C4-354 MORREL H. COHEN AND J. JORTNER The range of o(r, r') in I r - r' 1 over which it is non-
local is the phase coherence length I. On the other hand, the variation a(r, r') with mean position ($) (r + r') is on the distance scale of b, if both the Fermi wavelength and I are smaller than b. The Fermi wavelength is definitely smaller than b under our assumption that b exceeds the internuclear spacing.
Moreover, in the diffusion regime, the phase coherence length is substantially smaller than the Fermi wave- length. Phase incoherence of the wavefunction over the distance b permits the use of a local relation between the microscopic current density and micro- scopic electric field. Examination of (6.1) and the inverse relation for E(r) shows that both j(r) and E(r) vary on the scale of b. Thus, the integration over r' in (6.1) can be carried out by setting E(rl) equal to E(r) and taking the latter outside the integral
where
o(r) = J d3r' o(r, r') . (6.3)
As a further approximation, we shall ignore any tensorial character in ~ ( r ) , except when it is induced by a magnetic field, and set o(r) -+ o(r).
The position dependence of a(r) is analogous to that of X(r'l in figure 10. We can make the correspond- ing approximation of replacing a(r) by a set of constant values within a set of space filling spheres of radius b. Of course, the Debye short correlation length for o(r) need not be precisely the same as that for X(r), but after the many stages of simplification carried out here, there is little point in maintaining the distinction.
Finally, we note that it is the variation in local configuration or constitution which ultimately gives rise to the variation in local conductivity o(r). As all the other lengths in the problem are shorter than b, it is a good approximation simply to put
Values of a(r) at points separated by more than 2 b are statistically independent. The probability distribution of the values o(r) at any point r can be obtained from that of the values of X(r) once the functional relationship (6.4) is known explicitly.
The macroscopic current density j and field E are just the spatial averages of the corresponding microscopic quantities
Eq. (6.2)-(6.5) together with Maxwell's equations and the equation of continuity define the macroscopic conductivity tensor
Our problem now is to find the functional (6.6) and evaluate it for typical geometries and distributions of fluctuations. We must do this not only for the electrical conductivity a but also for the Hall constant R, the Hall mobility p, the thermoelectric power S, the optical dielectric function ~ ( w ) , etc. We now address the problem of calculating the transpor properties of a material with nonuniform local transport coefficients.
7. Effective medium theory. - We are concerned with a material which behaves like a random macro- scopic medium with regions of randomly varying transport coefficients. Within each region of radius b the transport coefficients are constant. Their values in different regions are statistically independent. Our problem is to calculate the macroscopic transport coefficients of such a medium. Even in this reduced form, exact solution of the problem is not possible.
The simplest and most useful approximation now available is the effective medium theory (EMT) of Bruggeman [48] introduced by him in 1935, redis- covered by Olelevskii [50] in 1951 and by Landauer [49]
in 1952, and which was recently revived by Kirkpatrick [51].
Kirkpatrick [51] has shown that the effective medium theory for the conductivity is remarkably accurate for a simple cubic array of resistors which randomly take on one of two values providing the conductance ratio is greater than about Even when the ratio vanishes, the worst possible case, the theory is in error oniy in the range of fraction of resistors present between the percolation threshold and 0.4. We assume here that this range of accuracy of the effective medium theory is more general than the simple system for which it was found and go on to develop and use an effective medium theory appropriate to the systems we are considering.
The effective medium theory has been developed in detail for the electrical conductivity only. The corres- ponding theory for R, p, the thermoelectric power S, the optical dielectric constant &(a), etc., had not been done before we undertook our investigations.
In the presence of a magnetic field the local conductivity becomes a tensor of the form
where a: is independent of magnetic field H and a; is antisymmetric and such that
all to first order in H. Here B is any arbitrary vector, and cc is a constant.
We have carried out [52] an effective medium theory for the full magneto-conductivity tensor for arbitrary distribution of conductivity componenst.
Here we give the essential results. The effective medium condition for determining the macroscopic zero-field conductivity is
where o' is the random value of the local conductivity.
The averaging process in (7.3) is defined by
< f (a') > = ) do' p(o') f (GO (7.4) where P(ol) is the probability distribution of values of a' and f (o') is any function of a'. Correspondingly, the effective medium condition for the full magneto- conductivity tensor is
where the average is over the joint probability dis- tribution of all the components of a'. Eq. (7.5) can be solved to first order in H for a, by expanding and inserting (7.3). We get
Let us now simplify the problem by ignoring the possible continuous change of the conductivity tensor and assume that there are two types of regions only, metallic regions characterized by the conductivity o,, the Hall constant R, and the Hall mobility p, with the probability C(EF), and nonmetallic regions specified by the values a , , R 1 and hi, with the pro- bability 1 - C(EF). The averaging prescription (7.4) then reduces to the simple form
The results of the effective medium theory for a can be put into the convenient form [49, 51, 521
o = fa,
Carrying out the average (7.7) for a, in eq. (7.6) yields [52]
We have evaluated [52] eq. (7.8) and (7.9) nume- rically for the conductivity ratio ,f, the mobility
ratio g, and the Hall constant ratio h. The results are shown in figures 12-14. A number of interesting features emerge. Those most relevant to the central issue of this paper fall into two patterns :
0 0.2 0.4 0.6 0.8 1 .O C
FIG. 12. - The electrical conductivity of a microscopically inhomogeneous material obtained from the effective-medium
theory.
1) For small x and y, i. e. x -- and y - l o T 2
as are appropriate to metal-ammonia solutions, both a and p drop rapidly for 0.4 < C < 1 whereas R is relatively weakly varying.
2) For larger x and y, i. e. < x < lo-', 1 < y < 3 as is appropriate for high temperature materials such as liquid Hg and Te, o and R can vary strongly for 0.4 < C < 1 while p exhitits only weak variation, e. g. 30 % in p and 10-fold in o, R.
Pattern (1) is precisely what is observed in metal- NH, solutions and pattern (2) in Hg and Te. This strongly suggest that we are on the right track and that it is worth proceeding with quantitative fits to the data. These are described in the next section.
Before going on, however, we point out that, at least within the effective medium theory, our results are somewhat more general than the use of only two distinct microscopic values of the conductivity and of a conductivity tensor constant within b would appear to imply. First we note that the Debye short
