2503
J. Phys. Chem. 1994, 98, 2503-2507
FEATURE ARTICLE Models of Anderson Localization J. L. Skinner Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706 Received: October 20, 1993; In Final Form: December 20, 1993'
Anderson localization is the phenomenon whereby static disorder in a single-particle Hamiltonian for an infinite system causes eigenstates to be localized in space. The localization threshold is defined as the amount of disorder at which all states become localized. In this article we consider a variety of disordered tight-binding models, where the disorder is in the site energies or in the positions of the sites. These include lattice and continuum models with and without correlated disorder. For each model we use the finite-size scaling/quantum connectivity approach to calculate the localization threshold. The results for different models are contrasted and compared.
1. Introduction
Many of our ideas about the properties of condensed matter systems come from the solid-state physics of perfectly ordered crystals.' Examplesthat come quickly tomindare the descriptions of the collective crystalline vibrations by phonons and electronic states by band theory. Small amounts of structural or thermal disorder can then be treated by perturbation theory, leading, for example, to an approximate description of electrical or thermal conductivity.' For sufficiently large amounts of disorder perturbation theory breaks down, and it is perhaps not surprising that the properties of condensed phase systems in this regime are not correctly described even qualitatively by standard solid-state physics theories. The phenomenon of Anderson localization,24 which is the subject of this Feature Article, involves one such instance. A quantum-mechanical system with a spatially periodic Hamiltonian has, according to Bloch's theorem, single-particle eigenstates that are uniformly extended (delocalized)throughout space.' The presence of static disorder breaks the translational invariance of the original Hamiltonian and therefore has the possibility of causing some of the eigenstates to be localized in space. In general, the larger the amount of disorder, the greater the number of localized states. In some instances, when the disorder is sufficiently large, all of the extended states disappear. This phenomenon is called an Anderson transition, and the point at which this occurs is called the localization threshold. The extended or localized nature of the eigenstates is intimately connected with the presence or absence of particle transport on a macroscopic length scale-extended states give rise to a meansquared displacement that grows with time for long times, while localized states do not. Certain spectroscopic observables are also sensitive to the nature of the eigenstates. Thus Anderson localization has been used to explain a wide variety of transport or spectroscopic phenomena including the metal-insulator transition in doped semiconductors,5-7 the metallic state of certain polymers,s transport of excitons in mixed organic crystals,9J0the spectroscopy of polysilanes' and J-aggregates,lz-I4the phononfracton crossover in amorphous systems,15J6 energy flow within molecules,I7J8and the propagation of light in disordered media.1930 In this article I will not focus on these various applications but rather will be concerned with different theoretical models of Abstract published in Advance ACS Abstracts, February 1, 1994.
localization. Even though these models are in all cases idealizations of real physicalsystems,interestinginsight into localization phenomena can be obtained from their study. While the field of Anderson localization includes many important topics such as scaling,2*critical exponents,22-2S multifractal p r o p e r t i e s , ~ . and ~~7 the effects of lower dimensionality,2*this article will focus solely on understanding localization thresholds in three dimensions. In addition, I will not attempt to review the theoretical results of others (since this has already been done in our previous paper~~~2~J~-33) but rather will survey our own results. The various models to be discussed herein can all be written in terms of the tight-binding Hamiltonian:
i
i#j
In the above the index i labels the 'sites" of the model, and the ith site is located at position ii. Centered at each site is a single localizedquantum-mechanical state li),and for most of the models the set of these states are taken to be orthonormal. ti is the "site energy" of the ith state. Jij,which depends on the distance ril = IFi - FA, is the transfer matrix element between pairs of states i andj. If the site energies are identical and the site positions are on a regular lattice, then thesystemis perfectly ordered. Ifinstead one allows the site energies and/or positions of the sites to be randomvariables, then the system is disordered. Different models of disorder correspond to different choices for the probability distributions of the random variables, and for the distance dependence of the transfer matrix elements. Several of these models will be introduced briefly below. Perhaps the simplest model takes the positions of the sites to lie on a regular lattice in three dimensions, the transfer matrix elements to be nonzero (and constant) only for nearest-neighbor pairs of sites, and the site energies to be uncorrelated random variables, each with a variance 2 2 . The limit 2 = 0 corresponds to a perfectly ordered Hamiltonian, and increasing 2 increases the disorder. For finite 2 , random site energy differences tend to inhibit mixing and to produce localized states; the localization threshold occurs at a critical value 2,. A physically more reasonable variation on this theme takes the site energies to be random variables that are correlated over a length scale S: The problem then is to determine how the critical threshold depends on {.
0022-3654/94/2098-2503%04.50/0 0 1994 American Chemical Society
2504 The Journal of Physical Chemistry, Vol. 98, No. 10, 1994 Another class of problems is identicalto the uncorrelated model above except that each site energy is characterized by a binary distribution, with energy CA occurring with probability p and CB with probability 1 - p. In the limit - CA the eigenstates belong to one of two completely separate subbands and one can ask about the localizationproperties of either band. For example, focusing on the lower subband, an Anderson transition occurs as p is decreased from the value of p = 1 (perfect order). This is known as the quantum (site) percolation problem, and there is significant interest in understanding the relation of the quantum percolation threshold to the classical percolation threshold for the same lattice. An interesting variation of this problem incorporates a probability distribution with variance Z2 for the site energies in the lower band. Then the problem becomes one of finding the localization phase diagram in the Zp plane. A third class of problems arises when the site energies are all identical, but the positions of the sites are random variables. In this case the system is said to be topologically disordered. If the positionsof the sites are uncorrelated and described by a uniform distribution, and the distance dependence of the transfer matrix elements is characterized by a length scale a, then the-localization threshold occurs at a dimensionless critical density pc = p 4 3 ( p c is the critical number density). If the transfer matrix elements are in fact cut off after a distanced, then this becomes the problem of continuum quantum percolation, and again, it is quite interesting to try to understand the relation of this problem to the corresponding classical percolation problem. An interesting variation on this problem obtains when the positions of the sites are correlated in that each site is centered on a hard sphere of diameter u. In this case the two length scales u and d produce a fascinating interplay. In this paper I will define the models described above more precisely, and in each case will present results for the localization threshold that are based on the finite-size scaling/quantum connectivity approach to localization that we introduced about five years ag0.2~ I hope to show that by comparing the results from a sequence of related models of disorder we can learn somethingqualitative about the nature of the Anderson transition. In section 2 I will very briefly review the theoretical method, in section 3 I will discuss the results for the individual models, and in section 4 I conclude.
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2. Finite-Size Scaling and Quantum Connectivity There are several different possible theoretical approaches to determining the localization threshold. Perhaps the most straightforward of these involves simply generating random configurations of the Hamiltonian for a large but finite system, diagonalizing to find the eigenstates, repeating for many configurations, repeating the whole process for several values of the relevant disorder parameter and then ascertaining at which point (on the average) all of the states become localized. In fact it is this last step that is ambiguous. For a finite system in which all sites are connected directly or indirectly by transfer matrix elements, every eigenstate will have an amplitude on every site, nomatter how large thedisorder. This makes it difficult todecide when states are truly localized, although severalpossible schemes have been proposed. A more powerful approach makes use of the apparatus of renormalizationgroup theory. Since it appears that the Anderson transition is indeed a second-order phase tran~ition,2~.2Z2~J~ it is reasonable that renormalizatioh groupor similar scaling methods would be applicable. A popular method involves setting up the Hamiltonian matrix for a “bar” of sites that is finite in two dimensions but infinite in the third, and then calculating the “localization length” in the direction of this third dimen~ion.~&3~ By determining how the localizationlengthvaries with the length of the finite dimensions,one can infer the properties of an infinite system. Thus this approach is based on the technique of finite-
Skinner size scaling.38 This method has been used successfully to obtain accurate results for the three-dimensional Anderson model on a simple cubiclattice with nearest-neighbortransfer matrix elements andvariousmodelsfor thesiteenergydistribution (thesiteenergies are uncorrelated). A drawback of this method is that straightforward application to more complicated problems (for example, where the sites are not on a regular lattice, the site energies are correlated, the transfer matrix elements are longer range than nearest neighbor, the transfer matrix elements are possibly zero, or when one does not know at what energy the mobility edges merge) is not possible. Several years ago my group developed a related method, also involving finite-size scaling, that does not suffer from the limitations mentioned above. The method is based on calculating the “quantum connectedness length” for a system that is finite in all threedimensions.29 Onediagonalim a particular realization of the Hamiltonian matrix and expresses the eigenstates Ip) as
From the expansion coefficients c~,,one can then define the Yquantumconnectivity” of two sites i and j by24929
Aij = Pu(PiiPjj)-’12
(3) (4)
Ir
Averaging over many configurations then gives the expression for the quantum connectedness length24 5:
Note that unlike the more usual definition of the localization length, the quantum connectedness length involves an average over all energies. To determine the localization threshold for an infinite system from the results of calculations o f t for finite-size systems, we employthe method of finitesize scaling,’sspecifically the phenomenological renormalization group e q ~ a t i o n . This ~~,~ involves calculating 5 for a sequence of finite sizes and several values of the relevant disorder parameter for each size. Since our finite-size scaling/quantum connectivity method has been developedat some length in previous publications,7.24.29-33 we will not discuss it further here and will simply focus on the results of calculations for several different models. 3. Results and Discussion As mentioned earlier, all of the models we have studied can be defined in terms of eq 1. Thus for each of the models, I will first define the model and then present the principal results. 1. Anderson Model with Uncorrelated Site-Energy Disorder. Inthismodelthesitesiineq 1 arelocatedonaregularlattice-the usual (and our) choice is a simple cubic lattice. The transfer matrix elements are taken to be nearest neighbor only, so that Jfj = J if sites i and j are nearest neighbors on the lattice and 0 otherwise. The site energies cf are uncorrelated random variables described by the probability distribution P(c). The two most popular choices for P(E)are the rectangular distribution
+
P ~ ( ~ q)w=/ 2 - E)e(E w / 2 ) / w
(6)
and the Gaussian distribution ~ , ( c )=
(I/&Z) exp(-e2/2~)
(7)
Thus in the two cases respectively the disorder parameters are
The Journal of Physical Chemistry, Vol. 98, No. 10, 1994 2505
Feature Article
above does not hold. In fact just the opposite seems to be true: since the site energy correlation is of finite range, for an infinite system nearly all of the site energy differences are not affected by the presence of this correlation, and therefore neither is the localization threshold. We expect this conclusion to remain true for any finite value of 5. It should be noted, however, that in one-dimensional systems certain types of site-energy correlation can dramatically affect localization properties.28 3. Quantum Percolation Model. This model follows from q 1 by again assuming that the sites i define a simple cubic lattice and the transfer matrix elements are nearest neighbor only, but now the site energies (which are uncorrelated) are described by the binary distribution:
P(c) = p S ( c - c * ) + ( l - p ) 6 ( € - € , )
0
0.1
0.2
0.3
Y
(9)
-
0.4
-
Figure 1. Localization threshold 4 a8 a function of the correlation parameter y as discussed in section 3.2. The point at y 0 corresponds to the uncorrelated value of E c / J = 6.00 f 0.17. Wand 2. We have found29 that for the rectangular distribution all states are localized for W > W,,and Wc/J = 15.95f 0.25. (In what follows all error bars are 2 standard deviations.) In comparison we might note that for W = 0 (perfect order) the bandwidth in this model is 12J. Thus all states become localized when the spread in site energies is on the order of the bandwidth. For the Gaussian distribution we find24 that the critical threshold is at Z,/J = 6.00 f 0.17. To compare these two results quantitatively, we note that the variance of the rectangular distribution is wZ/12. Therefore the square root of the critical variance for this distribution is (in units of J) 4.60 f 0.07. Inasmuch as this does not agree with 2, for the Gaussian distribution, we can conclude that details of the shape of the probability distribution are important for determining the localization threshold.)' 2. Anderson Model with Correlated Site-Energy Disorder. In real systems site energy disorder is often due to the presence of latticedefects, which are dilutein concentration but whoseeffects are long range. It follows that the energieson nearby sites should be correlated. To investigate the effects of this correlation, we have studied a model identical to the above, except that the site energies (which each have zero mean) are described by a multivariate Gaussian distribution. The covariances are chosen to b e 3 3
where ru is the distance between sites t and j , a is the lattice spacing, and y Z 0. The correlation length (in units of the lattice spacing) is therefore given by 5 = l/ln(l/y), Because of the relatively small lattices used in our numerical calculations we were only able to consider values of y up to 0.4or values of t u p to slightly larger than one lattice site. Our results33 for the localization threshold as a function of y are shown in Figure 1 The point at 7 = 0 corresponds to the uncorrelated situation discussedabove. Weseethatwithin theerrorbars, thelocalization threshold is independent of the amount of correlation. The explanation of this is as follows: At first sight one might have thought that since localization is in some sense a competition between site energy mismatches and transfer matrix elements, the mismatches between sites that are directly coupled (nearest neighbors) would be most important, and therefore the localization threshold would be sensitive to local correlation. However, at the Anderson transition the eigenstates are of infinite extent, and thus so strongly mixed that a perturbative argument such as the I
Such a distribution could quite naturally model a binary substitutionally disordered mixed crystal. In the limit CB - EA =, sites with different site energies are not mixed by the transfer matrix elements, and the eigenstata form two completelyseparate subbands. If we focus on the lower band, for example, the sites with energy EB might just as well be removed from the problem. In this sense this defines a percolation problem, where only a fraction p of the sites are "occupied". When p = 1 the lattice is perfectly ordered and all states are extended. For p < 1 we have the possibility of obtaining an Anderson transition; we find30 that the localization threshold occurs at pc = 0.477 f 0.01 1. It is interesting to compare this to the classical percolation threshold, which I will call, hopefully without confusion,pel. For a simple cubic lattice with nearest-neighbor connections, this is4' pol = 0.3117 f 0.0003. It necessarily follows from the definitions of the two problems that pc 1pol,since for p pel, meaning that even though there is an infinite cluster at the classical percolation threshold, there is sufficient disorder to preclude the existence of delocalized states. We see that the ratio of the quantum to classical percolation thresholds is pc/pcl 1-53, For the fcc lattice it is now known42 that pc = 0.265 f 0.005 (and also that41pcl= 0.198),giving a ratio ofp,/pd = 1.34. Thus it would be fascinating to understand how the ratio po/pd depends on lattice topology. For the discussion that follows we note that for site percolation problems this ratio is identical to the ratio of the average coordination numbers (average number of nearest neighbors) at the quantum and classical thresholds. 4. Quantum Percolation with Site-Energy Disorder. For a binary mixed crystal in the separated band limit, even within one subband there is additional energetic disorder due to other crystalline defects. Therefore, for example, one can generalize sections 3.1 and 3.3 with the following site energy distribution:
-
where P R ( ~is) given by q 6. Again, in the limit €6 = there will be two separate subbands. Now, however, the lower subband is characterized by two disorder parameters, p and W,and therefore when p gets too small or Wgets too large, one would expect all states to be localized. We have calculated the "localization phase diagram" for this model, and the results30 are shown in Figure 2. From the shape of the phase boundary one concludes that for a given value of one type of disorder, increasing the other type of disorder only causes the states to be more localized. While this behavior is certainly reasonable, it would also have been possible for the two types of disorder to "interactw in a nonlinear manner, 5. Topologically Disordered Models. For some physical systems (amorphous solids and liquids) a lattice model is completely inappropriate. Thus in "topologically disordered" models the disorder comes about from the random positions of +
2506 The Journal of Physical Chemistry, Vol. 98, No,10, 1994
12
-
Skinner I
I
I
I
t
I
I
I
0.4
0.6
1
I
quantum locolized
W
claaslcally Iocali red
r
-0 0
0.2
0.4
P
0.6 0.4 extended
0.6
08
1.o
Figure 2. Localization phase diagram for the model discussed in section 3.4. The region marked "classically localized"corrmponds to thosevalucs ofp where an infinite cluster of connected sites does not exist. The phase boundary (solid line) between "quantum localized" and "extended" is a guide to the eye. Reprinted with permission from ref 30. Copyright 1988 American Institute of Physics.
thesitesratherthan thesiteenergies. Weconsidered thesituation where the site positions are distributed completely randomly in some volume such that the number density is p, which in this case is the disorder parameter. We took all the site energies to be identical and studied two models for the distance dependence of JIJ:~' JI.,I-Je-'da
J~,= -J(I
+ r,,/a)e+u'a
(1 1) (12)
These functional forms arise from considering the interaction of hydrogenic impurities in semiconductors, and as seen, both are characterized by a range u. Note that with these models even p = = does not correspond to perfect order. Nonetheless, lower density means moredisorder in that the fluctuations in therelevant off-diagonal matrix elements increase. Thus we expect that for p < pcall states are localized. Because the Hamiltonian involves only one_length scale a, one can define a dimensionless crit_ical density pc pcu3that is independent of a. We find31 that pc = 0.0170 f 0.0020 and 0.0106 f 0.0038 for the two models, respectively. Since the typical interparticle separation is given by p-V, the Anderson transition occurs (in both models) when this is roughly 4a. 6. Topologically Disordered Model with Orbital Overlap. We have also considered7 a more realistic model of interacting hydrogenic impurities by including the overlap of orbitals on different sites. Thus for this model the transfer matrix elements are given by q 12, but the site states li) are not orthogonal;their overlap matrix elements are given by
S,,= 6, + (1 - 6,){l
+ 2 + 3a2
(13)
The finite-size scaling/quantum connectivity approach can be generalized to includ? this possibility. For this model we find7 a critical density of po = 0.0156 i 0.0021,in good agreement with experiments on th_e metal-insulator transition in doped semiconductors, where pc = 0.018 f O.OlO.5 7. Continuum Quantum Percolation. A straightforward generalization of the model in section 3.5 obtains when
o,2
"
1
I
bCl
t 0
0.2
0.8
1
dd Npurc3. p,andidvs aldfor thecontinuumpercolationmodellrdilourrsd in section 3.8.
which defines a continuum quant_um percolation problem. Defining a dimensionless density by p = pB, we fmd32 that the localization threshold occurs at pc= 1.OSf 0.03.At this density the average interparticle separationjs approximatelyq u a l to d. It is fascinating that for this model pcis 60 or a 100 times larger than the localization thresholds for the exponential models discussed in section 3.5! Thus the exponentialtails have a profound effect. It is also interesting to compare this result to that for classical continuJm percolation. Our best estimate for that problem yields32 pol = 0.646 f 0.007. As in the lattice percolation problem we find that pF> pol; in this case their ratio is p o / p ~ 1.67. 8. Continuum Quantum Percolation dtb Hard-core Interaction~. This last model generalizes the immediately preceding one by assuming that the positions of the sites are correlated in the following manner: each site is centered on a hard sphere of diameter u < d, and their positions are chosen from equilibrium configurationsat a given density. Thus the limit u = 0 rccovers the continuum quantum percolation model discussed It is of interest to determine how the localization threshold pc= p# varies with the hard-core diameter u. This study was motivated by a corresponding study'3-45of classical continuum percolation with hard-core interactions that showed an interesting nonmonotonic dependence of the classical percolation threshold on u. Our results32 for both the quantum and classical problems are shown in Figure 3. One sew that for each value of u, pc> pd, and also that the nonmonoticity of the quantum results is particularly pronounced. One also ObSCr~es~~ that the ratio pC/pcldecreases monotonically from about 1.67 toward 1 as u goes from 0 to d. More interesting is the ratio of the averagecoordination numbers (average number of particles within a distance d of a given particle), since this is analogous to the ratio of the percolation probabilities in lattice problems, We find32 that the ratio of the average coordination numbers at the quantum and classical thresholds decreases slowly from about 1.7 to 1.5 (albeit with large error bars) as u increases from 0 to d.
-
see.
4. Conclusion
In this Feature Article I have briefly discussed a versatile approach to calculatinglocalization thresholds and have reviewed our results for several different models of disorder. The principal conclusionsare as follows: For models with siteenergy (diagonal) disorder, the localization threshold is sensitive to the shape of the probability distribution but not to theamount of correlation among site energies. For models with site positional disorder, the
Feature Atticle localization threshold occurs when the average interparticle separation is about 1 or 4 times the range of the transfer matrix element,depcndingonwhether it has a sharpcutoff or exponential tails. For lattice and continuum percolation models the ratio of the average coordination numbers at the quantum and classical thresholds varies from 1.3 to 1.7. This means that one needs on the order of 50% more local connections than for classical percolation in order to produceextended states. An understanding of the magnitude of this number and of its variation for different models is needed. In summary, it seems that we do now have a reasonable qualitative understanding of some aspects of the Anderson localization threshold, While it would not be difficult to calculate thresholds for other models, at this point it is perhaps more important to understand under what circumstances the various models can be applied to real systems.
Acknowledgment. I am indebted to my former and current students Leslie Root, Joe Bauer, Tsun-Mei Chang, Jeff Saven, and Matt Stephens for their unflagging energy in trying to understand localization. I am grateful to Matt Stephens, Dale Orth, and Ned Sibert for their critical readingsof the manuscript and to the National Science Fbundation (Grant CHE92-19474) for support of this research. References and Notes (1) Aahcroft, N. W.; Mermin, N. D. SolldState Physlcs; Holt, Rinehart and Winston: New York, 1976. Anderaon, P. W. Phys. Rev. 19511,109, 1492. Lee, P. A.; Ramakrirhnan, T. V. Reu. Mod.Phys. 1985, 57, 287. Phillip, P. Annu. Rw. Phy8. Chem. 1993, 44, 115. Edwards, P. P.; Sienko, M. J. Phys. Reu. B 1978, 17, 2575. Milligan, R. F.; Thomag, G. A. Annu. Reu. Phys. Chem. 1985,36, Bauer, J. D.; Logovintky, V.;Skinner, J. L. J . Chem. Phys. 1989,90, (8) Phillip, P.; Wu, H. L. Sclener 1991, 252, 1805. (9) Klafter, J.; Jortner, J. J. Chem. Phys. 1979, 71, 1961. (10) Klafter, J.; Jortner, J. J. Chem, Phys. 1979, 71, 2210. (1 1) Tilgner, A.; Tromdorff, H. P.; Zcigler, J. M.; Hochstraswr, R. M. J. Chem. Phys. 1991,96,781. (12) Fidder, H.; Knoecter, J.; Wiersma, D. A. J. Chem. Phys. 1991,95, 7880.
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d
