Elsevier

Annals of Physics

Volume 321, Issue 11, November 2006, Pages 2566-2603
Annals of Physics

Inelastic electron transport in granular arrays

https://doi.org/10.1016/j.aop.2005.12.012Get rights and content

Abstract

Transport properties of granular systems are governed by Coulomb blockade effects caused by the discreteness of the electron charge. We show that, in the limit of vanishing mean level spacing on the grains, the low-temperature behavior of 1d and 2d arrays is insulating at any inter-grain coupling (characterized by a dimensionless conductance g). In 2d and g  1, there is a sharp Berezinskii-Kosterlitz-Thouless crossover to the conducting phase at a certain temperature, TBKT. These results are obtained by applying an instanton analysis to map the conventional ‘phase’ description of granular arrays onto the dual ‘charge’ representation.

Introduction

It has been long appreciated that the low-temperature physics of generic disordered metals is characterized by a subtle interplay of electron–electron interactions and coherent disorder scattering. While both effects are of crucial importance, their unified treatment still evades a complete theoretical description. It is useful, thus, to approach them separately. The limiting case of “coherence without interactions” has been studied intensely. It is well understood that the coherent multiple scattering off impurities leads to Anderson localization: in one and two dimensions all states are localized [1]. While in homogeneously disordered systems this phenomenon has always to be taken into account, granular systems admit for a parameter regime where the physics is entirely controlled by interaction effects. It is the purpose of this paper to explore the regime of “interactions without coherence” accessible in metallic granular arrays.

Metallic granular arrays are also of great interest in their own right [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In particular, strongly coupled arrays (g  1, where g is the dimensionless inter-grain conductance) have become a subject of increased theoretical attention in recent years [8], [9], [10], [11], [12], [13], [14]. An isolated grain or quantum dot is characterized by three energy scales: the Thouless energy ETh, the charging energy Ec, and the mean level spacing δ. The tunneling coupling between the grains adds another parameter: the dimensionless conductance g of a contact between two neighboring grains. Throughout this paper, we focus on the regime, where the Thouless energy ETh is the largest energy scale. This allows one to treat each grain as a zero-dimensional object and disregard the intra-grain in comparison with the inter-grain resistance. The interaction effects are controlled by the charging energy Ec (in the simplest model Ec = e2/(2C), where C is the self-capacitance of a grain). In our studies, it is the next largest energy scale in the system. Finally, quantum coherence effects are governed by the energy scale(s)  δ. If such a scale is much smaller than all relevant temperatures, one may treat each grain as having a continuous spectrum. This assumption allows one to disregard phase coherence. In essence: an electron exiting a grain is never the same electron that has previously entered it.

The parameter regime specified above justifies the “interactions without coherence” program. It is clear though, that such a simplification cannot work down to the very smallest temperatures. At low enough temperature, coherent propagation through multiple grains will become important and our approximation is bound to fail. It is, thus, important to realize that the subject of our considerations is a transient (though possibly wide) temperature range. In this range, coherence may be disregarded while interactions (and inter-grain tunneling) are crucially important in determining the electrical properties of the array.

At small inter-grain conductance g  1, an electron is completely localized within a single grain. Therefore, the problem is reduced to the description of classical charges moving on the lattice (which is a simple limiting case of the considerations given below). In the simplest case of on-site interactions only, there is an energy barrier Ec impeding the transition of electrons between two neighboring grains. It is thus natural to expect activation behavior of the conductivity, with the activation temperature T = Ec.

The present paper is devoted to the more intricate case of large inter-grain conductances, g  1. In this case, the charge may spread over many grains to decrease its charging energy. The interplay of interactions and tunneling dictates that this spreading involves an (exponentially) large, but finite number of grains. As a result, the lowest energy excitations of the system are large single-charge solitons. The activation energy for creating such an extended charge carrier is substantially reduced, leading to the low-temperature conductivity of the form:σ(T)=gexp-TT,where T(1d)gEcexp[-g/4] and T(2d)g2Ecexp[-g]. The important consequence of Eq. (1) is that one- and two-dimensional arrays are insulating at arbitrarily large inter-grain conductance, g. This is a pure interaction effect; Anderson localization physics is not included in the model. Switching off the interactions, one obtains Ohmic metallic behavior with a temperature-independent conductivity.

The solitons interact with each other up to distances comparable to their (exponentially large) radius, even if the initial model possesses on-site interactions only. Once they start to overlap, Eq. (1) is not valid anymore. In 1d this happens at TT(1d), where the conductivity smoothly crosses over to its high-temperature behavior [9]σ(1d)(T) = g  2ln (gEc/T). In two dimensions, the solitons interact logarithmically over a large range of distances. This leads to a Berezinskii-Kosterlitz-Thouless (BKT) unbinding of soliton–anti-soliton pairs [16], [17] at the temperatureTBKT=T(2d)/gT(2d).Around this temperature, the conductivity undergoes a sharp crossover from the exponentially small value given by σ (TBKT), cf. Eq. (1), up to the high-temperature asymptotics, σ(2d)(T) = g  ln (gEc/T). In the model with only mutual capacitances between neighboring grains the Coulomb interaction is logarithmic at arbitrarily large distances. This results in a true BKT phase transition with zero conductivity below the transition point. The g  1 version of the latter model was previously considered in [3], [18]. The introduction of on-grain Coulomb interactions transforms the transition into a crossover. Interestingly, for g  1 the BKT remains sharp even for the pure on-grain (self-capacitance only) Coulomb interactions.

Technically, we approach the problem from two complimentary perspectives: the phase and the charge representations. The former is straightforwardly derived from the microscopic fermionic model [19]. It is commonly employed in the study of both homogeneous and granular interacting systems. While being effective in the high-temperature regime, it becomes increasingly difficult to handle at lower temperatures. To treat this latter regime, we employ the charge model, introduced previously within the context of quantum dot physics [20], [21]. Our main technical achievement is the proof of equivalence of these two approaches over a parametrically wide range of temperatures. For these temperatures, both models may be handled in a controlled way. We thus conclude that the charge model, although not directly deduced from the microscopic Hamiltonian, is indeed the proper description of the low-temperature phase of the system. The results mentioned above (as well as others discussed below) then follow in an almost straightforward manner from the charge description.

The equivalence of the two approaches is based on a very important observation. The charge discreteness (crucial in the low-temperature insulating phase) manifests itself in the phase model through the 2π-periodicity of the phase field (the internal space of the field is the circle S1). The latter results in the existence of topologically distinct stationary-point field configurations, classified by the integer winding numbers Wl (where the vector index l numerates the grains on the lattice). In strongly connected arrays, g  1, the action cost for configurations with non-zero winding numbers (so-called Korshunov instantons [22]) is exponentially large. However, one has to take into account Gaussian fluctuations around the topologically non-trivial stationary points, which yield a factor (gEc/T)1/d for each winding number mismatch between neighboring grains. This factor suggests that the instanton configurations are increasingly important at low temperatures [23]. Summation of the instanton “gas” along with the corresponding Gaussian fluctuations and the phase-volume factors results exactly in the classical (low-frequency) limit of the d-dimensional charge model. Specifically (see below), the instanton expansion of the phase model coincides term by term with the perturbative expansion in backscattering amplitudes of the charge model. Therefore, we are convinced that the explicit account for instantons in the phase-like models is imperative to restore the charge discreteness and, thus, to describe the insulating phase.

One may justifiably worry about the role of non-Gaussian fluctuations. The latter are known to become large at a low enough temperature T0  Ec edg/2, violating the validity of the instanton gas picture. Crucially, however, (in 1d and 2d) the corresponding charge model predicts an activation gap which is parametrically larger (exponential (in g) in 1d and algebraic in 2d) than T0. As a result, there is a wide range of temperatures, where the fluctuations are well under control, while the physics is completely dominated by the proliferation of instantons. The latter results in the appearance of the unit-charge extended solitons as low-energy charged excitations and, thus, in activation insulating behavior, Eq. (1). In 3d, proliferation of instantons and the onset of strong non-Gaussian fluctuations, respectively, take place at comparable temperatures. As a result the phase–charge equivalence cannot be reliably established. It seems plausible, however, that the instanton gas—and thus the corresponding charge representation—provide a qualitatively correct description of the 3d insulator as well.

This paper is an extension of two previous shorter publications [10], [11]. Its intent is twofold. First, we present some new results. In particular, we extend calculations beyond the tunneling limit, accounting for arbitrary transmission amplitudes between neighboring grains. Furthermore, in addition to an evaluation of the transport properties, we discuss the behavior of the single-particle density of states (DoS). Second, we bring out the philosophy of our approach and expose extensive technical details of the calculations. Our main message is that charge quantization is crucial in describing the low-temperature physics of the array—and, therefore, a description in terms of charge degrees of freedom is appropriate. As mentioned above, this description is obtained by accounting for topologically non-trivial field configurations in the phase picture. This goes beyond the commonly used perturbative treatment of the phase model. In 1d and 2d arrays, the latter completely misses the appearance of a new temperature scale T marking the crossover to insulating behavior.

The paper is organized as follows: in Section 2, we introduce the phase and charge models. Before coming to the main part, namely quantum dot arrays, in Section 3, we discuss the physics of a single dot connected to two leads. Section 4 discusses one-dimensional arrays whereas Section 5 contains the two-dimensional arrays. The conclusion and open questions are discussed in Section 6.

Section snippets

Phase and charge representations

In this section, we introduce two effective models used to describe d-dimensional quantum dot arrays. As mentioned in Section 1, the two descriptions are optimally adjusted to the nominally metallic and the nearly insulating regime, respectively. The application of the two models to the computation of observables, and the mapping of one onto the other will be discussed in later sections.

Widely used in the literature is the so-called Ambegoakar–Eckern–Schön (AES) model [19], [6]—a description of

Single quantum dot

The simplest setup on which the impact of interactions on transport through an almost open system can be studied is a single quantum dot coupled to two leads [26], [21]. Interesting in its own right, the discussion of the quantum dot will facilitate the development of the formalism required to describe arrays. We follow a three-step program: in Section 3.1 the AES phase model is investigated, in Section 3.2 the alternative charge description is used, and in Section 3.3 the two procedures are

1d array

We next advance to the 1d array geometry. As in the previous section, we start with the AES phase model (4.1) and then continue to the charge model (4.2).

2d arrays

We next extend our discussion to two-dimensional quantum dot arrays. Our strategy will parallel that of the previous sections: starting from the phase representation, we will establish a connection to the complementary charge representation, and then discuss the physics of the system in terms of the latter.

Conclusions

We have studied transport properties of inelastic granular arrays. We find that charge quantization and the localization of a unit charge within a finite area play the crucial role in the low-temperature behavior of these systems. This is the case even in systems where due to a large bare conductance g  1 charge behaves like a fluid at high temperatures. At large g, the elementary charged excitations are solitons spreading over (exponentially) many grains. In contrast, for weakly coupled arrays (

Acknowledgment

We are grateful to K.B. Efetov, M. Fogler, and A.I. Larkin for valuable discussions. Work at the University of Minnesota was supported by the A.P. Sloan foundation and the NSF Grant DMR04-05212 (AK) and by NSF Grants DMR02-37296, and EIA02-10736 (LG). J.S.M. was partially supported by a Feodor Lynen fellowship of the Humboldt Foundation as well as by the US Department of Energy, Office of Science, under Contract No. W-31-109-ENG-38. Work at the University of Cologne was supported by

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