Phononless Hopping in the Coulomb Glass

 bius: Phononless Hopping in the Coulomb Glass J. Talamantes and A. Mo 45 phys. stat. sol. (b) 205, 45 (1998) Subject classification: 71.23.An; 71.30.+h; 71.45.±±d Phononless Hopping in the Coulomb Glass  bius (b) J. Talamantes (a) and A. Mo (a) Department of Physics, California State University, Bakersfield, CA 93311, USA Tel.: 8056642335; Fax: 8056642040; e-mail: [email protected] (b) Institute for Solid State and Materials Research, D-01171 Dresden, Germany Tel.: +493514659534; Fax: +493514659537; e-mail: [email protected] (Received August 19, 1997) We studied the effects of coherent hopping on localization in disordered systems of interacting localized electrons. Using a computational approach, we found that, within the parameters of our study, the inter-site Coulomb interactions between electrons enhance delocalization. We suggest that this is because short low-energy few-electron coherent hops are more effective in delocalizing electrons in the presence of those interactions. These transitions are relatively rare in the non-interacting case. As the system size is increased the effect is reduced, although it remains important. The reason seems to be that, for small systems, our approach magnifies coherent hopping due to our choice of boundary conditions. Introduction. We attempt to evaluate the importance of quantum effects (QE) in disordered systems interacting via their Coulomb potential. The main question of interest here is whether the inter-site electron±electron interaction acts to promote localization. This work is based on [1], where only 2d systems were considered (d is the dimensionality of the system). The new contributions are that we now address 3d systems, investigate the effects of system size on our results, and present a method to investigate the mobility of electrons as they tunnel coherently from site to site. We start from the tight-binding Hamiltonian P P P ÿ1 Hˆ 1† Jij ayi aj : rij ni nj ‡ ei n i ‡ i j<i j<i In this expressison, ei is the random energy of site i; ni its occupation number, rij the inter-site distance (the dielectric constant and the electronic charge have been set equal to unity), ayi ai † the creation (annihilation) operator for an electron at site i, and Jij is the elastic hopping energy. We adopt the common practice of accounting somewhat crudely for a strong intra-site (Hubbard) energy by precluding double occupation of sites. This is appropriate here because we only investigate low-energy states. Equation (1) without the third term on the right-hand side describes what is commonly termed the Coulomb glass. Here we investigate the eigenvalue±eigenstate pairs (eigenpairs) of eq. (1) in full, i.e. Coulomb glasses with elastic hopping, and refer to this case as CI because it contains all Coulomb interactions. Also, for comparison, we investigate the eigenpairs of eq. (1) without the second term, and refer to this case as HI (because it contains only Hubbard interactions). We begin the search for the eigenpairs of H by writing the wavefunctions of the electrons as hydrogenic functions centered at the sites. We let Ns be the number of sites in 46  bius J. Talamantes and A. Mo the system and n  Ns † the number of electrons. The electronic states of the Coulomb glass can be written as Slater determinants SI constructed from the local wavefunctions of electrons at sites which are occupied in that state, i.e. one generally writes SI ˆ jn1I n2I . . . niI . . . nNs I i, where niI is the occupation number of site i in configuration I. (By a configuration one means a particular arrangement of electrons among the sites.) Most studies assume that every state of the system can be represented by a single SI . Although our approach is different, for the purpose of shedding light on the conditions under which this (classical) approximation is valid, we examine the eigenstates of (1) in the representation of the SI for small Jij , perform a perturbation expansion in the SI , and write the eigenfunctions of (1) as P Y I ˆ SI ‡ aIJ SJ ; J 6ˆ I with aIJ ˆ HIJ ; EI ÿ EJ I 6ˆ J : 2† HIJ is the matrix element that allows for the resonant tunneling between configurations I and J, and EI is the energy of I. The classical approximation is valid only if the jaIJ j  1. When this condition is unmet, the process of coherent, or ªphononlessº hopping takes place. It delocalizes the states because it allows for ªsharingº of electrons among the sites. The role of the inter-site interaction is twofold, so there is a competition between two effects [2]: (i) The transfer integral HIJ in CI as compared to HI systems. One can show [1] that HIJ  exp ÿs IJ †, where sIJ is the total distance (normalized to the Bohr radius a0 of the localized electrons) that electrons must move for the system to go from SI to SJ . In the CI system, the low-lying configurations tend to differ from each other by one- and many-electron hops [3]. Also, these hops tend to be short because the interaction is repulsive, which tends to ªclear outº the vicinity of occupied sites. Thus, electrons are more likely to tunnel to nearby sites since they are more likely to be unoccupied. This electron±electron correlation is obviously absent in the HI case, where the low-energy transitions can involve longer hops, and thus relatively large s IJ s. So jHIJ j might be enhanced by the interaction. This tends to increase jaIJ j, and thus enhance delocalization in the CI as compared to the HI case. (ii) The system density of states (DOS) in the CI as opposed to the HI case. The effect of the introduction of interactions in the random potential problem is to deplete the DOS at low energies [3]. This effect has been known for a long time [4, 5] in the one-particle density of states. It results from the fact that an electron in an interacting system must overcome the Coulomb gap. Thus, the magnitude of EI ÿ EJ in eq. (2) is increased by the interaction. This tends to decrease jaIJ j, and so to suppress delocalization in the CI as compared to the HI case. Epperlein et al. [6] use a Hartree-Fock approach to address the present issue. That method neglects correlations, and thus effect (i) above. We aim in this work to investigate whether the effect in (i) is strong enough to overcome the depletion of the DOS and so to enhance delocalization. Phononless Hopping in the Coulomb Glass 47 Procedure. Our systems are defined by placing L3 sites on a cubic lattice, and assigning a random energy to each site with a uniform distribution in the range jei j  B=2. Each electron carries a unit charge, and each site a charge of minus half. We find the N lowest-energy configurations SI of the system in the classical limit, and their energies EI by the method in [7]. We use periodic boundary conditions. Then, we compute H in the representation of the SI by setting HII ˆ EI and calculating the HIJ as in [1]. For a 1=d particular system, one such matrix is calculated for every D is the P value of a  a0 D density of sites). We then find the eigenpairs Ea , a ˆ AaI SI of (1). In some cases we use the Lanczos algorithm, the program is due to Cullum and Willoughby [8], and in others QR factorization, see e.g. [9]. Finally, we analyze the result (as described below) concerning the localization degree of the electronic states. There is a limit for investigating QE in this way: one only knows a subset of N lowest-energy configurations which span some energy range D. As a is increased, level repulsion effects a drop in the ground state energy. Eventually the magnitude of this drop is of O D†. One must not increase a further because that would bring higher-energy configurations into the states a. We use this criterion to stop our procedure. Analyzing the eigenstates of (1) for localization is difficult because there is no unambiguous single parameter which characterizes localization in the interacting system. (In the one-particle case, an inverse participation ratio, see e.g. [10], has proven to be very helpful, but a straight-forward extension to the present case is not possible because one is dealing with many indistinguishable particles.) We have used the following tools to address the question of localization: (i) The study of level spacing statistics has been shown to provide significant insight into the problem of localization [1, 11]. The idea is that as the overlaps between configurations increase (and thus configuration mixing, which leads to delocalization, is enhanced), so does level respulsion. This is reflected in the distribution p s† of nearestneighbor spacing s. For negligible level repulsion, p s† is a Poisson distribution. As configuration mixing becomes significant level repulsion increases, the relative number of small spacings is reduced, andP p s† tends toward a Wigner distribution. (ii) For an eigenstate a ˆ AaI SI of (1), one can define [12] a configuration space P inverse participation ratio Ra ˆ jAaI j4 . Roughly, Ra decreases from 1 in the localized case to 0 in the case of a delocalized state for a macroscopic sample. (iii) When the system is in some state a, clusters of sites are formed as the system tunnels coherently among the SI that contribute to that state, i.e. electrons are shared in some way among the sites belonging to these clusters. We investigate the nature of these clusters by allowing the computer to perform a random walk in configuration space. We first put the system in the SI which contributes the most to a. We then pick a configuration SJ with probability jAaJ j2 , and make the transition SI ! SJ in such a way as to minimize s IJ . We keep track of the location of the electrons, and of the frequency with which every site-to-site link is activated in the process of coherent hopping. This procedure is continued while keeping track of the fraction fI of the time the system has spent in SI . We stop when j fI ÿ jAaJ j2 †=jAaJ j2 j  e, for all I, and for some small fractional error e (we used e ˆ 0:05†. We thus obtain a map of the clusters of sites spanned by the electrons, as well as the frequency with which every site-to-site link has been activated by an electron. This is a measure of the zero-temperature conductance of that link. We also know the number ck of electrons which traverse the system in the k-direction k ˆ x; y; z† per step in our random walk. 48  bius J. Talamantes and A. Mo Fig. 1. Level spacing distributions for a ˆ 0:15 in the CI case, and (a) only nearest-neighbor one-electron hops; (b) only nearest-, next-nearest, and next-to-nextnearest neighbor one-electron hops; (c) effectively no restrictions on the hops. Reference distributions are also shown None of these tools alone presents a complete picture of the physical situation. It is hoped that together they give enough detail to understand the QE in Coulomb glasses. Results. We analyzed systems with L ˆ 5 n ˆ 63† and L ˆ 7 n ˆ 171†. B was set equal to the nearest-neighbor Coulomb interaction, which we use as the unit of energy. We investigated mixing of N ˆ 5000 configurations. (To be safe, we report only results for the first 3000 eigenstates of (1). We disregarded the higher eigenstates obtained from diagonalization because they probably require mixing of configurations I > 5000:† We consider many-electron hops of arbitrary complexity; however HIJ decreases exponentially with s IJ , which increases with the order of the transition SI ! SJ . Thus, for computational convenience, one can impose a cutoff nM on the number nIJ of electron transfers that need be considered. HIJ is set to zero if nIJ > nM . Also, for the purpose of investigating the details of the nature of the transitions which contribute to QE, we have used sometimes a cutoff sM on s IJ , and set HIJ ˆ 0 if sIJ > sM . Thus, setting sM ˆ 1 corresponds to considering only transitions for which one electron is transferred between nearest neighbor sites. We present in Fig. 1 the results for p s† for a CI system with L ˆ 5 and a ˆ 0:15. Plot a was obtained by setting s M ˆ 1; plot b corresponds to nM ˆ 1 and sM ˆ 2; plot c corresponds to nM ˆ 5 and no restriction on s IJ . The result for nM ˆ 3; sM ˆ 3 (not plotted) is indistinguishable from Fig. 1, curve c. The results which follow correspond to nM ˆ 5 and s IJ unrestricted. We observed consistently that our criterion for stopping this approach is met for a  0:15. Fig. 2. Level spacing distributions for a ˆ 0:15 and (a) L ˆ 7, HI case; (b) L ˆ 7, CI case; (c) L ˆ 5, HI case; (d) L ˆ 5, CI case. Reference distributions are also shown Phononless Hopping in the Coulomb Glass 49 Fig. 3. Configuration-space inverse participation ratio Ra for eigenstates a of the Coulomb glass Hamiltonian with quantum effects. a ˆ 0:15 in all graphs. a) L ˆ 7, HI case; b) L ˆ 7, CI case; c) L ˆ 5, HI case; d) L ˆ 5, CI case The plots of the DOS (not shown here) for HI and CI show that it is reduced at low energies due to level repulsion. The results are qualitatively similar to those shown in [1] for 2d. Figs. 2 and 3 are results for a ˆ 0:15 and (a) L ˆ 7, HI; (b) L ˆ 7, CI; (c) L ˆ 5, HI; and (d) L ˆ 5, CI. We found p s† for a different set of ei as those used for Fig. 2. The results were the same as those shown here, so we do not include here plots for those systems. In Fig. 4 we present graphs to elucidate the mobility of electrons for the ground states at a ˆ 0:15: a) L ˆ 7, HI R0 ˆ 0:342; cx ˆ cy ˆ cz ˆ 0†; b) L ˆ 7, CI R0 ˆ 0:180; cx ˆ cy ˆ cz ˆ 0†; c) L ˆ 5, HI R0 ˆ 0:083; cx ˆ 4:5  10ÿ4 ; cy ˆ 2:9  10ÿ3 , cz ˆ 8:7  10ÿ4 †; and d) L ˆ 5, CI R0 ˆ 0:176; cx ˆ 7:4  10ÿ4 ; cy ˆ 1:0  10ÿ6 , cz ˆ 3:0  10ÿ5 †. The bonds shown differ in conductance, but such information has been left out of the graphs for clarity. In a) and b) we show all bonds activated in the configuration-space random walk. In c) and d) we started from highly conducting bonds, and decreased the conductance of bonds allowed in the graph until a continuous path across the system was established. We show those bonds only. Discussion. In view of Fig. 1 we argue that it is not enough to consider only oneelectron nearest-neighbor hops in the CI case. The contributions to level repulsion arising from short-, few-electron transitions can be seen to be important. From Fig. 4c we see that long hops can contribute to QE in the HI case. Fig. 4a does not show long 50  bius J. Talamantes and A. Mo Fig. 4. Graphs showing the bonds activated as electrons hop coherently between sites. All graphs correspond to the ground state. a) L ˆ 7, HI case; b) L ˆ 7, CI case; c) L ˆ 5, HI case; d) L ˆ 5, CI case. In c) and d) we show only those bonds whose conductance must be included to establish a continuous path across the system hops, but other graphs we produced (which we cannot present here for reasons of space) for L ˆ 7 in the HI case do show long hops. This is in agreement with previous results [13] for 2d, and follows from the fact that low-energy transitions are longer one-electron hops in the absence of inter-site electron±electron interactions. One can readily see in Fig. 2 that: (i) for B ˆ 1 inter-site electron±electron interactions increase the level repulsion, and suppress the localization of electronic states; (ii) size effects are important, at least for L ˆ 5. The reason seems to be that, to make a transition from one specific configuration to another, there are fewer favorable choices (from the point of view of minimizing s IJ † for large systems than for smaller ones. This is due to the use of periodic boundary conditions [2]. In Fig. 5 we present a 2d illustration of how this occurs. There, an L ˆ 5 system is imbedded in an L ˆ 7 system. The arrows show how a particular two-electron transition is made differently in the two systems ±± the smaller system is able to ªchooseº to make the transition with a smaller sIJ . Thus, the number of nonvanishing HIJ is reduced with increasing system size. Evidently, the increase in the DOS with increasing L is insufficient to offset that effect. For L ˆ 7, however, our results are probably relatively free of such problems. This is clear from graphs such as Fig. 4a, b, where the number of links which cross the boundary is small relative to the total number of links. In Fig. 3 we see that there is little correlation in the degree of localization between states neighboring in energy. This is in agreement with the results in [12]. The trend can be seen in that figure to be that the degree of delocalization decreases with system size, and is enhanced by the inter-site interactions. This is consistent with our previous discussion. Phononless Hopping in the Coulomb Glass 51 Fig. 5. Illustration of how small systems take advantage of periodic boundary conditions. The arrows labeled 7 (5) indicate how the L ˆ 7 L ˆ 5† system makes a specific 2-electron transition while minimizing s IJ . The L ˆ 5 system is able to ªchooseº a smaller s IJ because there the two sites labeled X are in fact the same. Those sites are different in the larger system Our results here were obtained for B ˆ 1. One can argue [2] that both the numerator and the denominator in eq. (2) are affected by the inter-site interactions. As B is varied, the effect can be different in the two terms. Thus, our results could be different for B 6ˆ 1. Acknowledgements. J. T. wishes to acknowledge very fruitful discussions with J. Cullum. He also wishes to thank the San Diego Supercomputer Center and the Institute for Solid State and Materials Research Dresden for their partial support of this work. Special thanks are due to Prof. M. Pollak for sharing with the authors his vast insight in the subject of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] J. Talamantes, M. Pollak, and L. Elam, Europhys. Lett. 35, 511 (1996). M. Pollak, private communication (1997). K. Tenelsen and M. Schreiber, Europhys. Lett. 21, 697 (1993). M. Pollak, Disc. Faraday Soc. 50, 13 (1970). A. L. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975). F. Epperlein, M. Schreiber, and T. Vojta, preprint (1997).  bius and M. Pollak, Phys. Rev. B 53, 16197 (1996). A. Mo J. Cullum and R. Willoughby, see: Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1, 2, Birkhauser, Basel 1985. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes. The Art of Scientific Computing, 2n ed., Cambridge University Press, New York 1992. F. Wegener, Z. Phys. B 36, 209 (1980). E. Hofstetter and M. Schreiber, Phys. Rev. B 48, 16 979 (1993), and references therein. J. Talamantes and M. Pollak, in: Hopping and Related Phenomena, Eds. O. Millo and Z. Ovadyahu, Racah Institute of Physics, Jerusalem 1995 (p. 56). J. Talamantes and M. Pollak, in: The Physics of Semiconductors, Ed. D. J. Lockwood, World Scientific Publ. Co., Singapore 1995 (p. 97). 6 physica (b) 205/1