METAL-INSULATOR TRANSITIONS IN DISORDERED SYSTEMS

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METAL-INSULATOR TRANSITIONS IN DISORDERED SYSTEMS

F. Yonezawa

To cite this version:

F. Yonezawa. METAL-INSULATOR TRANSITIONS IN DISORDERED SYSTEMS. Journal de

Physique Colloques, 1974, 35 (C4), pp.C4-115-C4-122. �10.1051/jphyscol:1974420�. �jpa-00215611�

METAL-INSULATOR TRANSITIONS IN DISORDERED SYSTEMS (*) F. YONEZAWA (**)

Belfer Graduate School of Science Yeshiva University 2495, Amsterdam Avenue, New York, New York 10033, USA

RCsumB.

Les effets des correlations electroniques et d'une distribution aleatoire des atomes sont incorpores dans une th6orie de la transition metal-isolant de certains systemes desordonnes, bases sur le mod&le de Hubbard. Les criteres de Economou-Cohen et Mattis-Yonezawa sont utilises dans l'ttude de la localisation des electrons a la maniere de Anderson. La densit6 critique pour la transition varie selon la nature du desordre. On etudie I'energie $activation du cBtC isolant et la mobilite du cBte metallique. Quelques commentaires sur les etudes futures sont inclus.

Abstract. - Both electron correlations and the effect of random atomic distributions are taken into account to describe metal-insulator transitions observed in some disordered systems.

The Hubbard model is employed. Criteria due to Economou-Cohen and due to Mattis-Yonezawa are made use of to discuss the Anderson type localization of electrons. It is shown how the critical density at which the transition takes place is affected by the type of disorder. The behaviour of the activation energy on the insulator side and the behaviour of the mobility on the metallic side are examined. Some comments are given on the future study.

1. Introduction. - Some topics about metal-non- metal (M-NM) transitions induced by electron corre- lations in several disordered systems such as doped semiconductors, disordered alloys, metal rare-gas systems and dense metallic vapours are going to be a central interest of today's talk. A particular attention is paid to the effects of interaction between electrons and of disordered atomic configurations on the occur- rence of M-NM transitions, the behaviours of these systems near the transition points and so on.

As for the first theme of electron correlations, it is some twenty years since Mott has suggested that a crystalline array of hydrogen-like atoms, or more generally atoms with an incomplete shell, may not necessarily show metallic conduction [I], if the Coulomb repulsion between electrons are properly taken into account. Thus, contrary to the prediction of the Bloch-Wilson theorem that a crystalline solid composed of atoms with odd number of electrons is always metallic, it will undergo a transition from metallic to insulating behaviour when the lattice is expanded continuously. In spite of a considerable amount of theoretical work proposed ever since, there is still no satisfactory theory to study this transition except a simplified model of Hubbard [2].

(*) This work is supported in part under AFOSR Grant No.

73-2430 and 72-21 53B.

(**)

On leave from Department of Applied Physics, Tokyo Institute of Technology, Meguroku, Tokyo 152, Japan.

Accordingly, the essential problem of how such a transition takes place is left unsolved even for a regular system where the situation must be easier.

The second theme is what is the bearing of random atomic distributions on M-NM transitions in disor- dered systems. This problem has been tried to under- stand in the light of Anderson's principle that, given sufficient disorder, all states in a band are localized [3].

On the basis of this Anderson's concept, combined with a well-known conclusion that states in the tails of the electronic bands for a disordered system are localized, it is argued that, when the system is less random than the Anderson's critical disorder for the absence of diffusion, there exist energies, now widely called mobility edges, which devide localized states from extended states [4]. This problem too suffers from the lack of any firm theoretical explanation, and there are several questions still controversial, although the Anderson's localization due to random fluctua- tions is essentially a property of non-interacting electrons.

Turning to our topic of M-NM transitions due to electron correlations in disordered systems, we are going to confront both of the above-described diffi- culties, neither of which has as yet been settled. Thus, we must be prepared for a situation which is more than just twice as much difficult.

An important point therefore is that no theory could be perfect. Every theory has merits and demerits,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974420

C4-116 F. YONEZAWA and it is easy to hypercriticize other theories. But

what is required of us in the presence of various approximate theories is to discern what is essentia from what is not. Besides, it is expected under this situation of the theoretical work that experiments can give a break-through to the controversial pro- blems.

Another point which I would like to emphasize here is that, when we are talking about disorder, it is not just one same kind of disorder. Type of disor- der varies from system to system. It is rather mea- ningless to discuss a matter based upon one model of disorder and try to compare the results thereof with experimental data of a system containing another type of disorder. It must be remembered that, when we construct a theory, we should know what kind of disorder characterizes the system we are looking at.

In the present talk, we shall discuss several quan- tities which can be estimated from experiments. Parti- cularly, we try to see how the theoretical results are influenced by the choice of model about the type of disorder.

2. Examples of M-NM transitions in practical disordered systems. - Insulator-metal transitions are phenomena which have been observed in a wide variety of systems. Here, let us confine ourselves to the case in which both the effect of electron corre- lations and the localization bf electrons are considered to take part. Some examples of disordered systems which show the M-NM transition may be roughly classified according to the type of disorder [5] as follows :

1) Compositional disorder : transition metal chal- cogenides (V20,-Cr20,, etc.) [6], doped or reduced titanates (S,TiO,) [7], tungsten bronzes (Na,WO,) [8].

2) Structural disorder : doped semiconductors (Sip, SiB, etc.) [9], supercritical fluids of alkali metals (K, Cs, etc.) [lo], metal ammonia solutions [Ill, metal rare-gas systems (NaAr, CuAr) [12, 131.

3) Topological disorder : amorphous Ge [14].

It must be noted that this is not a comprehensive list, but includes only those work that have come to the author's knowledge. The M-NM transitions are attained by concentration changes in two-component systems and by density changes in one-component systems.

In category I), systems are characterized by regular crystalline arrays of atoms in which the components on the lattice sites are either substituted or removed.

The systems in category 2) have no fixed lattice points, and positions of atoms therein are random.

Topologically disordered systems in category 3) are regarded to have short range order but not long range order. In the case of amorphous germanium, for instance, short range order persists in the way that a tetrahedrally bonded structure retains, although

bond lengths and angles are slightly distorted from their ideal values in the crystalline state ; but there is no long range order owing to the existence of missing atoms, voids, unpaired bonds and so on [15].

Pressure induced M-NM transitions in amorphous germanium have been reported [14]. The transitions are interesting in the sense that they are of pressure induced type and that the system is in a different category from those in 1) and 2) about which theore- tical considerations and discussions have already been given to some extent. Thus, the problem of amorphous germanium is worthy of theoretical investigations.

3. Points of discussion. - Now, let me describe briefly what we are going to discuss with special interest. The following three topics mainly about the states either near the transition point or the mobility edges have been picked up both from aca- demic and practical points of view. Academically, the nature of the states with eigenvalue close to the transition energy or the mobility edges are of great interest because they give us information of the mechanism how localized states change to extended states in a disordered system. Practically speaking, on the other hand, the states in this region play an important role in determining the electronic properties of real physical systems.

The three topics are :

1) How is the density of the M-NM transition influenced by the type of disorder included in the system ?

2) What is the behaviour of the activation energy on the insulating side of a mobility edge ? Or what is the size of the localized states near the mobility edge ?

3) What is the behaviour of the mobility p(E) on the metallic side ?

The results presented by several authors will be introduced and compared with those obtained by the present author and the coworkers [16-181.

4. Hubbard model in disordered systems. - In this section, let me explain the model and approximate methods we (Yonezawa and Watabe [16] and Yoneza- wa, Watabe, Nakamura and Ishida [17]) have employed in the actual calculation of the electronic structure in disordered systems.

We start with the Hubbard Hamiltonian of the form

where a& and a,,, are respectively the creation and

destruction operators for an electron of spin a in

the atomic orbital on the i-th site and ni,u

is the corresponding number operator ;

is an

atomic level of an atom at site i and U i is the at T ⁼ 0 ; or ii) to the systems at temperatures higher Coulomb interaction between electrons with opposite than the NCel temperature.

spins on the same site i. The summation i is either

over fixed lattice points or over the random distri- 5. Criteria of localization. - It is not enough to bution of atoms, depending on whether we are dealing calculate just the Hubbard density-of-states in order with the substitutionally disordered case or with to discuss the M-NM transition because, even when the structurally disordered case. Here, we are concern- the two Hubbard bands are overlapping, the states ed with two kinds of disorder ; that is, the random near the minimum of the spectrum are localized in configuration of spins and the random distribution the Anderson's sense and the system remains insulating of atoms. if the Fermi level is in this localized

mobility gap

~ ~ l the ~ l ~ ~ b ~ approach, b i ~ ~ ~ we calculate d ~ where the mobility is zero. To check the localization the quasi-particle density-of-states which is defined of electrons, we make use of criterion due to Economou by the average one-particle Green's function, where and Cohen [25] and due to Mattis and Yonezawa [18].

we confine ourselves to the nonmagnetic case. As for

and [251 have a the average over the atomic distributions, the coherent localization function which is expressed in a very potential approximation (CPA) ~ ~ 9 1 is employed for simple form within the framework of single-site theo- substitutionally disordered systems, while the renor- ries including the coherent potential approximation malized single-site approximation [20] is made use (CPA). In a single-site approximation, the ensemble- of for structurally disordered systems. averaged one-particle Green's function for a given

~h~ reason why we have chosen the ~ ~ b b energy E is described by a momentum-independent ~ ~ d Hamiltonian is that this is the only tractable model self-energy C(E) as < ^Gk(E) > ⁼ EE ^-

^{- C(E)l-l}

which presents a m~croscop~c description of the impor- where Ek is the band-energy. Their localization func- tant physical effects due to electron correlations and tion in this case is written as

which has been studied most extensively. However,

F(E)

E

we must mention that the Hubbard's mathematical ] E - C(E) 1 ^{(5 - 1)}

solution has been questioned [21]. The main objection

is that the Hubbard result does not properly describe in which Eb denotes the half-width of the band of the the metallic phase since no true Fermi surface seems corresponding regular system. They argue that the to exist and the Fermi volume violates the Luttinger's states corresponding t o energy E are localized if theorem [22]. Another objection is that the Hubbard F(E) < 1. Since their criterion for localization is model fails to give a reasonable explanation of the originally intended for studying the electronic proper- magnetic properties. Especially, the question whether ties in substitutionally or cellularly disordered systems, the Hubbard splitting, which yields two overlapping it is not possible to discuss the problem of structurally Hubbard bands, is possible when there is no long- disordered systems with eq. (5.1). Note that the situa- range antiferromagnetic ordering, has long been a tion as for the self-energy does not change in that the controversy. According to the understanding due to appropriate single-site theory yields the self-energy Mott [23], we consider that the most serious aspects which is independent of momentum and dependent of these objections have been resolved by the idea of only on energy even when atoms in the system are Brinkman and Rice [24] about a highly correlated gas. distributed in a structurally or t o ~ o l o g i c a l l ~ random They describe a state of the Hubbard system where manner. The trouble is that no corresponding regular most of atomic sites are occupied by one electron systems can be defined in the structurally disordered only, but the moments resonate between their two case and accordingly we do not have a clear-cut idea positions at zero temperature. Current is carried by about Eb in eq. (5.1). But, it seems still worth while to (<an electron

on a doubly occupied site or by

a estimate Eb as obtained from a reasonable limiting hole

The Hubbard density-of-states corresponds to procedure. A detailed discussion is given in these current-carrying while the true density- along this line and the modified localization function of-states is much higher so that the Luttinger's theorem is given as

is not violated. On the other hand, the system on the

metallic side is regarded as a highly correlated gas and N [g

dRij

antiferro-magnetic ordering has disappeared because )= E

Z(E) I ^(5.2)

of the resonance of spins. On the insulating side

where the

carriers

are localized in the Anderson's where is the number density of atomic centers, and sense, Mott argues [23] that, if localization is weak, g (Rij) is the pair correlation function of atoms.

the wave functions extend over many atoms and the As for the second criterion proposed by Mattis and moments still resonate so that at T = 0 the time Yonezawa [18], I do not intend to go further with a average of the moments is zero. long explanation because a full notion of the criterion Therefore, it must be noted that our theory is is reported under separate paper in the present pro- effective either i) some way on the metallic side of a ceedings. Let me give a few words about this criterion.

highly-correlated gas and just on the insulating side We have shown that, if there is a region of anomalous

C4-118 F. YONEZAWA

dispersion, ordinary wavepackets made with states in this region cannot propagate and therefore this region of energy may be identified as a mobility gap.

6. Hubbard bands for disordered systems. - Now, let me show our numerical results of the density-of- states in disordered systems.

1) First, we treat an AB binary alloy in which only the site diagonal terms are random. (Full papers on this problem are found in [17]). Referring to eq. (5. I), c i = EA or EB according as the site i is occupied by an A or B atom. The Coulomb interaction U i _{is taken} to be U, ⁼ U between electrons on an A atom while U, is assumed to be zero. This model simulates a system in which the effect of electron correlations in the lower band is important and the existence of B atoms serves as a sort of scattering centers. The band energy E,, obtained from the Fourier transfer of t i j , is assumed as a semicircular band p(E) for an unperturbed system as

= 0 otherwise , (6.1)

where A is the band width.

As actual systems to which the concept of the model is applicable, we have in mind two examples ; they are mixed crystals of monovalent metallic elements with rare-gas atoms such as NaAr or CuAr, and heavily doped semiconductors. In the first example, the Hubbard bands of the metallic elements are considered to be in the forbidden gap of a rare-gas system. Electrons from metallic elements propagate partly via excited orbitals of rare-gas atoms and accordingly the Hubbard bands are modified by the existence of the vacant band composed of excited states of the rare-gas. In the second example of a heavily doped semiconductor, the activation energy

in the intermediate concentration region is regarded to be a Hubbard gap in the impurity band. The effect of the conductio~l band to the properties of this impurity band becomes important, especially when the concentration of donors or acceptors are very high.

In figure 1, the calculated density-of-states is shown.

The energy is measured by choosing U as an energy unit, and EB - EA is taken to be 3 U. The concen- tration x of A atoms is chosen to be 0.4. The self- consistent calculation is performed for various values of A/U. The shadowed region corresponds to localized states according to the Economou-Cohen criterion.

As is expected, the mobility gap persists even after the density-of-states gap vanishes and therefore the actual transition from an insulator to a conductor takes place at a larger ratio (A/U), which corresponds to a disappearance of the mobility gap. In figure 2, the region in which the density-of-states is non-vanish- ing is shown in the (AlU, E/U) plane. A similar beha-

FIG. 1. - The calculated density-of-states D(E) for a disordered binary system. The density-of-states is normalized by 1/U.

The concentration of A atoms is

x

0.4.

FIG. 2. - The dependence of the quasi-particle density-of- states upon the A/U ratio. The concentration of A atoms is

CA =

x

0.4. The shadowed region corresponds to localized states. The density-of-states is non-vanishing in the hatched region of the (EIU, A / U ) plane. The states in the double-hatching

are occupied.

viour of the density-of-states as in a regular lattice case studied by Hubbard is observed. The difference is in the asymmetric band structure in our case while the original bands of a regular lattice is symmetric.

This is reasonable because the existence of the upper (B) band works to lower the impurity (A) bands.

2) In the next place, we consider a system composed of atoms distributed randomly. (A detailed study of this problem is found in [17]). The Hamiltonian (6.1) in this case is characterized by e i = 0, U i = U and the transfer matrix t i j is determined on the basis of the hydrogen-like (Is) orbital with an effective Bohr radius a* ; i. e.

Vo being twice the ionization energy. The modified localization function (5.2) is used to check the loca- lization of the states in the spectra.

Some doped semiconductors and supercritical alkali

fluids are the candidates for which this model of

structurally disordered systems are applicable.Although doped semiconductors such as Sip, GeAs are, by nature, substitutionally disordered, it sometimes seems adequate to treat them as randomly distributed systems. This is because of large effective Bohr radii in these systems. Therefore, when the effect of the conduction bands is not very important, the present model serves as a good first approach. Another example is alkali metals under high temperature and high pressure, in which the averge interatomic distance is large enough for the tight-binding representation to be appropriate.

In figure 3, the numerical results for the density-of- states are shown, where u is U/Vo and p is the dimen- sionless density defined by p = 32 z N(u*)~. The value of u is decreased from top to bottom. Appa- rently, a smaller u corresponds to a smaller interatomic distance and the transition from an insulator to a metal is observed. The similar behaviour is obtained when we change p with keeping u fixed.

FIG. 3. - The density-of-states of a random liquid with p

0.2 and four values of u

0.3, 0.36, 0.4 and 0.5.

A marked difference in the numerical results of this case from those of a binary alloy is that the Hubbard bands are very asymmetric for a structurally disordered system. This is due to the larger density fluctuations in a randomly distributed system compared with a substitutional alloy. Thus, the two Hubbard bands are easy to overlap. It must be noted however that the states in the tails are localized due to the selfsame fluctuations and the mobility gap persists even for larger densities. This problem will be discussed again in the next section.

7. Behaviour near the critical energy. - In this section, we discuss three topics proposed in section 3.

7 . 1 TRANSITION

TYPE

DISORDER. -

First, we see how the transition density is influenced

by the type of disorder included in the system under consideration. For this purpose, we study the following three models.

i) A disordered AB alloy in which ^{c i} in eq. (4.1) is v or - v according as the site i is occupied by an A or B atom. The concentration is chosen such that

c, = c, ⁼ 4. The Coulomb interaction is U for both atoms (U, = U, = U) and the zero of energy is taken to be at E = U/2. The larger value of v corres- ponds to the higher degree of disorder in Anderson's sense.

Employing a semicircular band (6.1) for the unper- turbed system and making use of the CPA, the Hubbard bands are evaluated. The two critical values 6, = (AlU), and 60 = (A/U), are obtained as ratios at which the density-of-states gap and the mobility gap, respectively, vanish. The mean interatomic distances corresponding to (AlU), and (A/U), are derived as a function of v by assuming a simple cubic lattice and a nearest-neighbour transfer matrix between hydrogen-like orbitals ; that is

ii) An amorphous metal analogy of a binary alloy for which .ci in eq. (4.1) is 0 or co depending on whether an A or B atom is sitting on the site i with respective probabilities c, = 1 - ^x and cB = x. Since EB is brought to infinity in the formulation, the model simulates an amorphous system where electrons cannot enter sites B and paths are formed by A atoms alone.

As actual systems to which the above idea is appli- cable, metal rare-gas mixtures and supercritical fluids are considered. A semicircular band assumption given in eq. (4.1) is employed, and two critical ratios 6, ⁼ (A/U), and 60 ⁼ (A/U), for the disappearance of density-of-states gap and mobility gap are obtained.

The lattice constants d, = R,/a* and do = Ro/a*

are estimated from 6, and 6, by the same idea as in the case i).

The average interatomic distance ;between A atoms (metallic atoms) is related to the lattice constant d by ^{c f =} d/x1I3. By making use of this relation, the critical interatomic distances 4 and d;, are evaluated from d, a_nd do. We can show that, for 0 < x < 3 - 2 J2, the states at the Fermi level are always localized no matter how large 6 = A/U may be. It is - interesting to note that the value x, = 3 - 2 4 2 is nearly the same as the percolation concentrations for several typical three dimensional lattices.

iii) A structurally disordered system as discussed in section 6 . 2 . When u is taken to be 1, the density-of- states gap disappears at p = 817 which corresponds to d, ⁼ 4.44. The mobility gap disappears at the mean atomic distance do = 3.51. The corresponding densi- ties are ~ 2 ' ~ ^{a* = 0.225 and}

^{a* = 0.285.}

For the purpose of studying how the behaviours

of the electronic states near the transition points are influenced by the type of disorder, let us compare the transition densities in the above-described three systems. On noting that, in the model No iii) of a structurally disordered case, the gap between two Hubbard bands vanishes at the mean interatomic distance dc = Rc/a* = 4.44, we choose the degree of disorder in models No. i) and ii) such that the density-of-states gap in these systems also may disappear at the same mean distance d = 4.44. In model No. i) of a binary alloy, this is performed by picking up a suitable value for v, the difference between atomic levels of two constituent atoms. On the other hand, model No. ii) of an amorphous analogy is characterized by the concentration x of the metallic elements, and the above requirement for the band- overlapping density is fulfilled by an appropriate choice of x. Practically, v is approximately 0.163 and x is 0.26.

The calculated results are summarized in Table I.

This results tell us that the two critical values dc and do (or N, and No) do not differ too much from each other in the compositionally disordered case.

This can be ascribed to the fact that, as discussed at the end of the preceding section, the density fluc- tuations in structurally disordered systems are larger than those in binary alloys. In the former case, the probabilities for several atoms to come closer and form a cluster are non-zero. Clusters of various sizes give a wide range of allowed states, some of which contribute to the formation of the band tails, and the density-of-states gap is easily filled. At the same time, the states coming from the clusters are localized because of the density fluctuations and localization in Anderson's sense remains for larger densities.

The results in Table 1 suggests that the electronic states in the random distributed systems behave in a quite different manner from those in substitutional alloys.

Critical atomic distances do = Ro/a* and densities N:l3 a* ,for i) a binary alloy, ii) an amorphous analogy and iii) a structurally disordered system.

Note that dc = 4.44 and a*

0.225.

do ⁼ Ro/aH N:l3a*

- -

i) Binary 4.25 0.235

alloy

ii) Amorphous 4.22 0.237

analogy iii) Structurally

disordered 3.51 0.285

system

7 . 2 SIZE

GY. - In the next place, we discuss what have been suggested about the size of localization and activation

energy in the insulating side of the mobility edge.

From the dimensional consideration, Mott and Davis [26] have conjectured that the radial extension RL of the wave function near the mobility edge behaves as RL cc I E - Ec I-'/'. Abram and Edwards [27] treat an electron in a random array of dense, weak scatterers as a model which is expected to exhibit the typical features of a disordered system.

An average Green's function for the electron is defined and derived as a Feynmann path integral.

They have found that the extent of the wave function near the edge has the form RL = I E - Ec This model of dense, weak random scatterers have also been studied by Freed [28], who has attained that, below the mobility edge, the size of the localized states is found to vary RL cc I E - Ec Recently, Anderson [29] has shown that RL cc 1 E - Ec

It is always pointed out that the problem of electron localization can rigorously been investigated only through the averages of a product of two Green's functions such as the localization probability. In this context, it is interesting to mention what Freed argues in his paper [28]. He has shown that, in the study of the above-mentioned model, the self-consistent field theory obtained from the average Green's function alone gives rise to the same analytic structure as the self-consistent field theory which is based upon the localization probability. He suggests that this fact indicates a general possibility of extracting information concerning electron localization from the simpler average Green's function.

Now, let us turn to the problem of the activation energy

which corresponds to the half of the mobility gap in the minimum central region of two overlapping Hubbard bands. From the relation RL cc I E - Ec Mott and Davis [26] have predicted that

tends 'to zero as (d - in which do is the mean atomic distance at the transition. Based upon the Anderson's argument of RL cc I E - E, Mott [30] has given a revised form of

c2 (d - d,J9I5. We have attained the conclusion that, in the framework of the CPA for substitutionally disordered systems, both the Economou-Cohen criterion [25] and the Mattis- Yonezawa criterion [18] predict the behaviour of

proportional to (d - d0)'l2 near the M-NM transition.

Although we have not obtained any analytic formu- lation about

for structurally disordered systems, the numerical calculations show that cZ tend to zero more sharply in this case.

Some experimental data of the activation energy

have been reported for low-compensated n-type germanium (which is effectively composed of randomly distributed centers) as a function of mean distance between centers, but we must be very careful when discussing those data in comparison with the predic- tions from various models and approximations.

7.3. MOBILITY

- Another

intersting point of still controversy is the behaviour

of the mobility p(E) on the metallic side of the mobility edge Ec when conduction takes up. Mott has long been putting forward that the transition is discontinuous in the sense that p(E) becomes from zero to a finite value discontinuously when E crosses E,. He has introduced a concept of minimum metallic conduc- tivity

1301 corresponding to this finite value of p(E) at E 2 E, and argues that this idea is supported by several experimental results. Cohen and Eggater [32, 331 have studied an electron moving through a random assembly of hard spheres and calculated the mobility based upon the semiclassical percolation theory. They have shown that the transition is conti- nuous since the mobility approaches zero when the density of spheres are decreased. According to the Mattis-Yonezawa criterion, we have proved that the mobility behaves p(E) cc I E - Ec I just aboveEc [18].

Note added in proof: Recent re-examination of this approach (Mattis-Yonezawa theory) yields, however, a discontinuity at the mobility edge.

8. Discussion. - In conclusion, let me give some remarks on this field of research. The present stage of development in this field is identified as the end of the very first step at which no clues at all are given to attack the problem. Now, we know, to some extent, what are the most significant points to be considered.

But, it still seems to be almost impossible to solve the problem with sufficient mathematical rigor. No theory could be perfect and free from ambiguity owing to the limited scope of the assumptions and approximations. It is expected that experiments play

an important role in resolving various controversial problems and may go ahead of theoretical work.

In this respect as well, theoreticians are required to give prospective suggestions as to what should be investigated and made clear from experiments. Let me emphasize here again that, both in theoretical and experimental study, we must know first of all the characteristic features of the systems under consi- deration such as type of disorder. We cannot discuss everything all together in the lump. If a prediction based upon one type of theoretical model of disorder gives a good agreement with the experimental results about a system containing another type of disorder, it is either just accidental or may have a deeper mean- ing. When the latter is the case, this

deeper meaning >>

must be clarified.

Since no theory is rigorous, any theory or proposal

- if it suggests something new

must be appreciated in its own right, even though it has unsatisfactory aspects. It is a good thing that several different schools are trying to solve the problem through different approaches with different philosophy and faith in a different style. It is not necessary to bring different attempts in one line when a field is young and serious mathematical work has hardly started. Just as the Ziman's theory of liquid metals [34] and Anderson's idea about localization of electrons have made comple- mentary contributions to the understanding of conduction in disordered systems, there are good chances that completely different concepts turn a new phase to the problem, and I believe, before long, we can see the matter in a better frame.

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