Charge avalanches and depinning in the Coulomb glass: The role of long-range interactions
We explore the stability of far-from-equilibrium metastable states of a three-dimensional Coulomb glass at zero temperature by studying charge avalanches triggered by a slowly varying external electric field. Surprisingly, we identify a sharply defined dynamical (“depinning”) phase transition from stationary to nonstationary charge displacement at a critical value of the external electric field. Using particle-conserving dynamics, scale-free system-spanning avalanches are observed only at the critical field. We show that the qualitative features of this depinning transition are completely different for an equivalent short-range model, highlighting the key importance of long-range interactions for nonequilibrium dynamics of Coulomb glasses.
pacs:75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.-i
The long-range nature of the Coulomb interaction plays only a secondary role in metals, where it remains screened by mobile electrons down to atomic length scales. The situation is, however, far more interesting on the insulating side of disorder-driven metal-insulator transitions Dobrosavljević et al. (2012), where screening is suppressed due to charge localization. Here, the unscreened Coulomb interaction leads to the opening of the “Coulomb gap” in the electronic density of states, as first pointed out in pioneering works of Pollack Pollak (1970), as well as Efros and Shklovskii (ES). The ES theory Efros and Shklovskii (1975); Shklovskii and Efros (1988) predicts a universal form of the Coulomb gap, and explains how its existence modifies hopping transport Shklovskii and Efros (1988) in disordered insulators, consistent with numerous experiments Lee and Ramakrishnan (1985). Early work also revealed that Coulomb interactions in disordered insulators generally contribute to the formation of an extensive number of metastable states, i.e., the formation of the Coulomb glass (CG) Davies et al. (1982, 1984); Xue and Lee (1988). In subsequent work, various aspects of glassy behavior of the CG were explored theoretically Grannan and Yu (1993); Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Grempel (2004); Pankov and Dobrosavljević (2005); Kolton et al. (2005); Müller and Pankov (2007); Surer et al. (2009) and experimentally Ben-Chorin et al. (1993); Ovadyahu and Pollak (1997); Vaknin et al. (2000); Bogdanovich and Popovic (2002); Vaknin et al. (2002); Orlyanchik and Ovadyahu (2004); Jaroszyński et al. (2004); Ovadyahu (2006); Jaroszyński and Popović (2006, 2007); Raičević et al. (2008, 2011); Lin et al. (2012).
More recent progress followed with the formulation of analytical theories of the CG Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Pankov and Dobrosavljević (2005); Müller and Pankov (2007) which adapted Parisi’s replica methods Parisi (1983); Rammal et al. (1986); Mézard et al. (1987); Young (1998) for spin glasses to disordered Coulomb systems. These theories find a Coulomb gap of the same universal form as predicted by the ES theory, but this behavior emerges only within the low-temperature glassy phase (displaying replica symmetry breaking). Within this mean-field picture, the universality of the Coulomb gap, as well as the saturation of the appropriate stability bound, can be directly traced back to the “marginal stability” of the entire glassy phase Pastor and Dobrosavljević (1999). In physical terms, the marginal stability reflects the emergence of “replicons,” soft (gapless) collective excitations involving simultaneous rearrangements of many electrons. If such soft excitations indeed characterize the Coulomb glass, they should also govern the physical response to various weak perturbations (e.g.m the external electric fields), perhaps leading to large-scale avalanches. Precisely such behavior has already been established Pázmándi et al. (1999); Andresen et al. (2013) for infinite-range spin-glass models, leading to scale-free avalanches characterizing an entire manifold of metastable states. Despite the successes of the mean-field approach, its applicability to finite space dimensions remains the subject of much controversy and debate Young and Katzgraber (2004); Katzgraber and Young (2005); Boettcher et al. (2008); Jörg et al. (2008); Katzgraber et al. (2009); Larson et al. (2013). Furthermore, a computational search for a finite-temperature glass transition in the CG in two and three space dimensions has remained inconclusive Grempel (2004); Kolton et al. (2005); Surer et al. (2009). To shed additional light on the nature of excitations in the CG, and further test the mean-field ideas, it is therefore useful to examine the stability of the low-lying metastable states by external electric fields.
In this work, we investigate the out-of-equilibrium behavior of a three-dimensional Coulomb glass at zero temperature and study the hopping and total charge displacement avalanches triggered by increasing an externally-applied electric field. Previous work on avalanches in the CG in three space dimensions done by Palassini and Goethe Palassini and Goethe (2012), which trigger avalanches via dipole excitations or charge insertions, find scale-free behavior for long-range hopping dynamics, but when hopping is bounded by a finite fixed range they do not find any scale-free avalanches. Because physical electrons rearrange themselves by finite-range hopping it is of interest to search for a scale-free behavior in the CG for bounded hopping dynamics by other means.
Here we study the CG with particle-number-conserving short-range hopping, by “adiabatically” increasing an external electric field up to a depinning electric field that separates the steady current state from just finite electron rearrangements as a reaction to the external field. We find that scale-free avalanches arise in the Coulomb glass when the electric field is close to . To emphasize the role played by the long-range Coulomb interactions we repeat our simulations for an equivalent short-range interacting model. In this case we still find a sharply defined depinning transition, but a completely different form for the critical behavior. Here we do not find any scale-free avalanches, in dramatic contrast to the behavior of the CG model.
The Coulomb glass Hamiltonian (in dimensionless units) is given by Efros and Shklovskii (1975)
where is the electron number at site , is the filling factor, is the coordinate of site , and a randomly-distributed on-site energy. For a charge neutral system, i.e., , in a constant external electric field in -direction, Eq. (1) can be rewritten in an Ising spin formulation by setting Davies et al. (1982) ( an Ising spin variable)
where the electric potential is and is the -position of spin . This form of the Hamiltonian with is of a random-field Ising model with long-range antiferromagnetic interactions given by
The site energy is sampled from a Gaussian distribution with zero mean and standard deviation . To keep the dynamics of the two models identical it is necessary to constrain the Ising-like Hamiltonian in Eq. (2) to have a constant magnetization ( for ) at all times. This is accomplished by using magnetization-conserving Kawasaki dynamics Newman and Barkema (1999).
The corresponding short-range model (SR) is given by the same Hamiltonian in Eq. (2), but with long-range interactions replaced by nearest-neighbor interactions (on a cubic lattice) of the form
ii.1 Determination of the initial configurations
In our simulations, we need to generate stable initial configurations of the system. In this context, “stable” refers to stable towards single nearest-neighbor electron hopping. We implement this procedure for both the CG and the SR model. In order to have an initial configuration with a Coulomb gap and track its dependence on the electric field, we compute pseudo-ground-state configurations using jaded extremal optimization (JEO) Middleton (2004).
The single-particle density of states (DOS) of a classical Coulomb system is given by
where the local single-particle energy is given by
and the average is performed both over thermal fluctuations and disorder instances. The ground state of the CG is well known to display a Coulomb gap Efros and Shklovskii (1975) in the DOS at the Fermi energy, which gradually fills up when temperature is increased Davies et al. (1982, 1984); Grannan and Yu (1993); Sarvestani et al. (1995); Surer et al. (2009).
For the CG, we can empirically check how “far” or “close” a given configuration is from the ground state by examining the form of the DOS. Depending on the depth of the Coulomb gap, we can argue whether the configurations are close or far from their respective ground state. The SR ground states do not have a Coulomb gap Boettcher et al. (2008), but have a “dip” at the Fermi energy that converges to a finite value in the thermodynamic limit. Again, we can empirically check if we have a good approximation of the ground state by studying at the DOS distribution. In Fig. 1(a), we show the DOS of the CG using the pseudo ground states for all simulated linear system sizes (the systems have spins). The occupation at is very close to zero, showing that the configurations found using JEO are not far from the true ground state. In Fig. 1(b), we show the DOS of the CG at electric fields . The data suggest that we are further away from a ground-state configuration, however, a pronounced gap in the DOS is still visible. The configurations for the SR model found by the JEO algorithm are likewise not far from the ground state (not shown).
Iii Numerical Details
For the description of the algorithm, we introduce a stability criterion, which for an electron () or a vacancy () at a given site is given by
For each pseudo-ground state generated via JEO [see Fig. 1(a)], we proceed as follows.
Select the least stable electron with one nearest-neighbor hole in the opposite direction of the electric field.
Apply an electric field just strong enough to destabilize the selected electron, such that it will hop to the neighboring hole.
Perform the electron-hole hopping that minimizes the energy; go to step 3.
The careful reader will have noticed that the above procedure is in fact an infinite loop stuck between steps 3 and 4 when a certain electric field threshold is reached. This electric field threshold is the depinning field of the system, which separates two regions: Below there are only short charge displacement pulses due to the rearrangement of the electrons as a response to the external electric field, and above it there is a steady current. A sketch of the different scenarios is shown in Fig. 2. The infinite loop between step 3 and step 4 is the steady current flowing through the system. Since we are interested in the number of times step 3 and step 4 are repeated at each -field (this, in turn, yields the avalanche size ) before we reach the depinning field, we artificially stop the process if the avalanche size surpasses a given number , where is the total number of sites of the system. Note that is much larger than the maximal avalanche size measured for for a given system size .
To cope with the long-range Coulomb interactions between the electrons we use the Ewald summation method Wang and Holm (2001). Furthermore, the applied electric field is periodic to avoid an electron pileup at the edge of the system. The simulation parameters are listed in Tab. 1.
iii.2 Measured observables and statistical data analysis
At each increase of we count the number of electrons that hopped and the total charge displacement in the direction of the applied electric field. Using these data, we compute their distributions and , respectively (see, for example, Fig. 3). To determine the depinning field we compute the cumulative distribution function of the depinning distributions which gives the probability whether a randomly picked sample is in the pinned or depinned state for a given system size and at a given field. We perform a finite-size scaling assuming that the function has a universal form Dong et al. (1993); Newman and Ziff (2000); Xi et al. (2015)
[see Fig. 5(b) and Fig. 8(b) for the CG model and the SR model, respectively], which gives us an estimate of the depinning field. Note that the depinning field is defined as the typical electric field necessary to induce a continuous current for a given system size, i.e, for , the system just rearranges its electron configuration by electron hopping, whereas for , the field induces a steady current.
In addition, we define the characteristic avalanche size of the system by fitting the exponential tail of the avalanche distributions to an exponential function . For each system size , we thus obtain a characteristic avalanche size . To estimate the value of in the thermodynamic limit, we do an extrapolation of by using the following functional ansatz:
where , , and are fitting parameters.
Finally, we also monitor the DOS as a function of the applied electric field . For example, Fig. 1(b) shows the density of states at an electric field range of .
Different finite-size scaling Ansätze have been attempted Pázmándi et al. (1999) to scale the and data without yielding any satisfactory results. We therefore empirically re-sized the avalanche curves without making any a priori assumptions. Interestingly, the following scaling ansatz showed good results:
Figure 3 shows electron hop, as well as total charge displacement avalanche distributions for the CG for different ranges of the electric field . The field is increased in the different panels from top to bottom. Figures 3(a) – 3(c) show how the avalanche sizes progressively become system spanning, i.e., when [as is the case in Fig. 3(c)] avalanche size distributions become power laws. As the field reaches a hunch in the curves emerges separating a power-law region from an exponential cutoff, for the measured avalanches distribution . Figure 6 shows the dependence of the inverse of the characteristic avalanche size as a function of the electric field . We can extract from the figure that the depinning field lies somewhere around . A precise estimate of the depinning field can be obtained by analyzing the cumulative distribution function as shown in Fig. 5(b). For the CG model we obtain . In Fig. 7 we show an example of the estimation of using Eq. (10) for a given field window . Similar qualitative results are obtained for the charge displacement distribution , as shown in Fig. 3(d). We attempt to scale the data for the distributions and in Fig. 4. The data scale well with no adjustable parameters (especially for the larger system sizes) according to Eqs. (11) and (12).
In addition, we study the total charge displacement distribution and electron hop distribution as a function of the applied field for the SR model, where the estimated depinning field is as seen in Fig. 8. Electron avalanche distributions are shown in Fig. 9. For low fields, i.e. , the characteristic avalanche size can be estimated analogously as for the CG model, i.e., fitting the tail to an exponential function and using Eq. (10) to extrapolate to the thermodynamic limit. As for the CG model at low fields, no system-spanning avalanches were found, moreover no emergent avalanche size dependence is observed [Fig. 9(a)-(b)]. For fields closer to the depinning field, i.e. , the exponential fitting function [Eq. (10)] gives unsatisfactory fitting results, therefore we additionally fitted the distribution to a stretched exponential function
The characteristic avalanche size defined through the stretched exponential function is bounded in the thermodynamic limit for all fields, especially close to the depinning field: the inset of Fig. 9(d) shows the values of for the field window . The stretched exponential exponent has a strong field dependence as seen in Fig. 10. At low fields and as the field increases it monotonically decreases to com ().
We observe that the CG model and the SR model have a well defined depinning field transition, but that they differ in the way they behave close to . The CG model total charge displacement and electron hop avalanche distributions close to the depinning field have a power-law shape (with power-law exponent ) with a system-size dependent exponential cutoff. This finite-size effect vanishes in the thermodynamic limit, revealing its scale-free behavior at . In clear contrast the SR model total charge displacement and electron hop avalanche distributions show no signs of scale-free avalanche behavior (power-law shape) close to and are best described by a stretched exponential function, which is defined by the exponent and the parameter . The exponent shows a strong field dependence; it decreases monotonically as the field is increased, while does not show any systematic system-size dependence at any field, not even close to the depinning field. The different avalanche distributions in the SR and CG models hint towards a different mechanism behind the depinning transition.
Our large-scale computational study of the Coulomb glass has demonstrated that, under external electric fields and nearest-neighbor particle-conserving hopping dynamics, scale-free avalanches only occur in the vicinity of a characteristic depinning field . For small external electric fields, no large avalanches are present, in agreement with the results of Palassini and Goethe Palassini and Goethe (2012). For a short-range variation of the Coulomb glass model we do not find any sign of scale-free avalanches, not even close to the depinning electric field. Furthermore, we find that the initial Coulomb gap vanishes as the field is ramped up, suggesting that it is not a generic feature on the hysteresis loop formed in an external electric field. We empirically find a simple scaling ansatz to collapse the avalanche and charge displacement distributions, reinforcing the notion that the scale-free behavior of the CG emerges close to the depinning electric field.
The scale-free behavior found in the CG is not a self-organized critical (SOC) state, because an external parameter has to be tuned Bak et al. (1987); Drossel and Schwabl (1992); Schenk et al. (2002); Andresen et al. (2013), namely the electric field . Nevertheless, it is interesting to note the difference between the CG and the SR model: in the former the combination of the diverging number of neighbors and disorder results in power-law distributions, which is not the case in the latter. This behavior is very similar to that found for the three-dimensional random-field Ising model Sethna et al. (1993); Perkovic et al. (1995, 1999); Kuntz et al. (1998); Sethna et al. (2004), where scale-free avalanches have been observed at a critical field strength . These unexpected results for the Coulomb glass show that a diverging number of neighbors is necessary but not sufficient in a model Hamiltonian to show SOC behavior, and that the dynamics of a model might play an important role for showing SOC (i.e., the order-parameter conserving Kawasaki dynamics used here vs single-spin flip dynamics used for the random-field Ising model).
Our results bring into question the validity of the mean-field picture of the Coulomb glass Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Pankov and Dobrosavljević (2005); Müller and Pankov (2007), predicting extreme fragility of the ground state to external perturbations. However, the generic absence of SOC for avalanches driven by a uniform electric field may be related to the fact that such large avalanches locally violate charge neutrality. Other dynamical perturbations may couple differently to the elementary excitations and may perhaps serve as a more sensitive probe to the proposed SOC nature of the CG ground state. This could be achieved by applying external fields that do not directly couple to the uniform charge density, such as varying the amplitude of the disorder potential. Such or similar studies represent an opportunity to further elucidate the long-standing mystery of the Coulomb glass, however, exploring this exciting research direction remains a challenge for future work.
Acknowledgements.We would like to thank R. S. Andrist for many discussions, as well as Mauricio Andresen for providing the necessary motivation to complete this project. V.D. and Y.P. were supported by the NSF (Grant No. DMR-1005751). G.T.Z. was supported by the NSF (Grant No. DMR-1035468). H.G.K. acknowledges support from the NSF (Grant No. DMR-1151387) and would like to thank ETH Zurich for CPU time on the Brutus cluster, as well as Aspall (Suffolk) for inspiration. Part of H.G.K.’s research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA8721-05-C-0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.
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