The Hubbard model: useful for stretchable nano-materials?

The aim of this lecture is to test the consequences of stretching and doping on the electrical conductivity of the Hubbard model. It is motivated by a recent experiment in which conductivity of stretchable nanomaterials was studied,and which showed that conductivity tends to zero but remains finite for considerable values of the strain. The Hubbard model reproduces such behavior for certain values of the parameters,and for the hopping,band filling and lattice ''constant" dependent on the strain.


Introduction
According to standard condensed matter physics, some materials are electrical conductors because of the overlap of electronic orbitals on adjacent atoms. As a consequence, diminishing the overlap (for example by stretching a material) leads to a decreasing overlap of the orbitals and ultimately to a decrease of the conductivity.
On the other hand, certain technological issues, as varied as flexible conductors,t-shirts and cardiostimulating implants [1], [2] require materials which are conducting and stretchable, and whose conductivity remains as high as possible with increasing applied strain. The interest which the scientific community and the industry have in such materials is illustrated by a singular detail: a search on http://www. sciencemag.org gave 6749 papers published on the topic "stretchable electronics" from 2006 to the end of August 2014.
The aim of this lecture is to try to discuss the problem of "stretchable conductors" (SC) within a particular well known model in condensed matter physics -the so called Hubbard model (HM ). No attempt will be made to fit any particular set experimental data,such as [1]. Practically speaking, it will be attempted to discuss the consequences of stretching and the dependence on stretching of various parameters of the HM on its electrical conductivity. The lecture is divided into several sections. As the calculations pertain to nanomaterials, the next section starts with a definition of nanomaterials and a brief description of the experiment which motivated this work [1]. The third part is devoted to a brief reminder of the HM , and the fourth part to a discussion of the influence of stretching and doping on the conductivity of the HM . The final part contains the conclusions.

Definition
According to [3] nanomaterials are in the simplest way defined as materials with at least one external dimension in the size range from approximately 1-100 nanometers. This is clear and simple, but it does not take into account materials which have only a size fraction in the nano scale, or which contain agregates of nano particles [4]. In the present paper the definition proposed in [2] will be considered as satisfactory.

A recent experiment
In the experiment analyzed in [1], the possibility of making a stretchable conductor consisting of polyurethane containing spherical gold nanoparticles deposited in two different ways was demonstrated. The nanoparticles had a mean diameter of 13.0 ± 0.3 nm, and they were citrate stabilized which made them negatively charged. Polyurethane, which is positively charged, was used as a polymeric partner for these particles, and thin films were made by two different techniques : layer-by layer deposition (LBL) and vacuum-assisted flocculation (VAF). The thickness of the films thus obtained was 2.0 ± 0.2 µm. Films obtained by the two methods had different properties, but lamination led to an increase of the conductivity, and improved their stretchability. Strain applied to an object of initial length l 0 so that it achieves length l is defined by ǫ = (l − l 0 )/l 0 . For example, a strain of ǫ = 0.6 applied uniaxialy on a 5LBL film led to a decrease of conductivity by two orders of magnitude. This is illustrated on figure 1, taken from [1]. However,as noted in [1], the values of the conductivity obtained in this experiment for high strain were higher than the corresponding values obtained for carbon-nanotube-based materials. Full details of the experiment are avaliable in [1], but it clearly indicates the problem: how do these materials manage to stay in their conducting phase at such high values of strain? In the remainder of this paper it will be attempted to analyze this problem by using the (HM ).

The Hubbard model
The HM has been treated in considerable detail in the lectures given by the present author at two earlier sessions of this school [7,8].Only the essential detail will be repeated here. The HM was proposed by the British physicist John Hubbard during his work on the microscopic theory of the metal to insulator (M I) transition [9]. The Hamiltonian by which the HM is defined contains only two terms: the "free" kinetic term H 0 and the "interaction" term H I Complexity quickly becomes visible when this Hamiltonian is written completely. In two dimensions,using the second quantisation formalism, it has the form: In spite of simplifying assumptions introduced by Hubbard, and the fact that it was proposed nearly 60 years ago, the HM has so far been solved analytically only in the 1D case [10]. Extensive work is going on aiming at solving the so called extended HM in 2D [11]. Knowing the Hamiltonian of a system opens the possibility of calculating its electrical conductivity. When the Hamiltonian commutes with the current operator j, this calculation is not complicated. Problems can arise when [j, H] = 0, which is the usual case in realistic calculations. There is no general "prescription" for such situations.
One of the methods by which it can be done is the so called "memory function" method developed in the early 1970s as a consequence of previous work by Kubo. It was recently reviewed in [5]. The algorithm for the calculation of the conductivity within this method consists of the following equations [6]: where A = B = [j, H], j denotes the current operator, and where σ denotes the electrical conductivity, ω is the complex frequency, ω P is the plasma frequency, χ(ω) is the frequency dependent susceptibility and χ(0) the static susceptibility. Some details of the derivation of the expression for the conductivity are given in [7][8] and references given there. The result, for the case of the 1D HM is and the symbol S denotes the following function S = 42.49916 × (1 + exp(β(−µ − 2t))) −2 + 78.2557 × (1 + exp(β(−µ + 2t cos(1 + π)))) −2 + (bt/(ω + bt)) × (4.53316 × (1 + exp(β(−µ − 2t))) −2 + 24.6448(1 + exp(β(−µ + 2t cos(1 + π))))) −2 ) where U is the Hubbard U, b = −4 × [1 + cos(1 − π)] = −1.83879, N is the number of lattice sites, t is the hopping and µ denotes the chemical potential of the electron gas on a 1D lattice given by [7] µ = (βt) 6 (ns − 1) |t| 1.1029 + .1694(βt) 2 + .0654(βt) 4 In eq.(7) the symbol s denotes the lattice constant and n the band filling. The immaginary part of the conductivity is given by where χ R (ω) denotes the real part of the susceptibility,given by The plasma frequency ω P is given by ω 2 P = 4πe 2 n e /m, and e, n e , m are the electron charge, number density of the background electron gas approximated as a jellium [6] and the electron mass respectively, while β is the inverse temperature . This calculation pertains to the 1D HM .
The simplest 2D case,a square lattice of side length a can be treated as an extension of these results. If the lattice sides are denoted by x and y,the electric current flowing through such a system can be expressed as where e x and e y are unitary vectors of the two lattice axes.The total current is given by As by definition j = σE where σ is the electrical conductivity and E the electric field,and assuming that E x = E y = E, it finally follows that It is assumed here that the conductivities along the two axes are mutually independent.  Figure 2 shows the electrical conductivity as a function of strain for T = 116K and two arbitrarily chosen values of the band filling: n = 0.9 and n = 1.2. It was assumed that t = 0.01eV and s 0 = 1. The conductivity was normalized to 1 at T = 116K,n = 0.8,s 0 = 1,ǫ = 0. Data presented on the figure refer to the 1D HM . The conductivity of a 2D HM , assuming that the lattice is square, can be calculated by eq.(10). In the simplest case of a square lattice and equal conductivities along the two lattice axes, the total lattice conductivity is where σ x denotes the conductivity along one of the lattice axes.  Figure 2 shows the electrical conductivity of a 1D HM as a function of the applied strain for two values of the band filling Clearly, the function σ(ǫ) is an increasing function of the strain, which could have highly interesting practical implications. However, it was assumed in the calculation that t is a constant, while in physical reality t depends on the strain. In the remainder of this paper the dependence of t on the strain will be taken into account.

Strain dependence of the hopping and the band filling
In order to make a physically realistic calculation of the electrical conductivity of a 1D HM as a function of the applied strain, one must take into account the strain dependence of the hopping energy t and of the band filling n. The physical origin of t is the overlap of the electronic wave functions in atoms on adjacent lattice sites,so it can be expected that it depends on the lattice constant. The band filling is defined as n = N e /L, where N e is the number of electrons and L is the length of the lattice. Due to strain, L(ǫ) = L 0 (1 + ǫ),so it follows that n(ǫ) = n 0 /(1 + ǫ). The mathematically simplest possible example of the dependence of t on the applied strain has the form where t 0 denotes the value of the hopping under strain zero. Performing the calculation of the conductivity for n = 0.9,T = 116K, and two values of the hopping t, normalized to 1 at the same point as for figure 2, one gets figure 3. Note that a larger value of t leads to a bigger value of the conductivity.
The conductivity is clearly a decreasing function of the applied strain,which is in agreement with the experimental results presented in figure 1. Calculating the overlap of electronic wave functions in arbitrarily complex molecules (or atoms) under arbitrary conditions would be a complicated task. However, a realistic approximate solution to the problem can be found by standard quantum mechanics. Using results on the overlap of atomic wave functions (for example [13]), it follows that the dependence of t on the interatomic distance r can be expressed as where the distance r is measured in the units of the Bohr radius a 0 . This relation can be reformulated as a function of the form t(ǫ), where ǫ denotes the strain. Using the definition of strain it follows that Developing into series, and retaining only the first order terms in ǫ one gets which is approximately similar to the mathematical form of eq.(14). The conductivity of a 1D HM as a function of the applied strain, but in a low temperature case, is presented on figure 4 18th  Figure 5 shows a comparison of conductivities for two values of the temperature. Note the difference of nearly two orders of magnitude between the values of conductivity with no strain applied and at the highest values of strain for which the calculation was performed. Figure 6 represents the conductivity as a function of applied strain for a square lattice.

The two dimensional case
All the calculations discussed so far are relevant to the 1D HM . This is a good illustration of the procedure,but the real "challenge" is the conductivity of a 2D model,and its dependence on the applied strain. For the case of a square lattice this can be determined using eq.(12). In the case of a material having a small value of the ratio σ y /σ x , it can easily be shown that the conductivity reduces just to σ x . However, in the case of material characteristics along the two axes being similar, eq.(12) can give interesting results. One such example is shown on figure 7, obtained with n 0x = 1.1,n 0y = 1.3, T = 150K and t 0 = 0.01eV along both lattice axes. Note that the ratio given on the vertical axis is σ 2D (ǫ)/σ x .

Discussion and conclusions
The basic aim of this paper was to test the applicability of the Hubbard model to the nanomaterials used for stretchable electronics. The question from the title does not have a simple answer.
In the simplest case, assuming that all material parameters are constant in spite of the fact that it is being stretched, one gets a strange result -the conductivity increases with the applied strain. This is obviously unphysical, and can be traced back to the assumption that material parameters are independent of strain.
Taking into account the strain dependence of t and n (the hopping and the band filling) gives results in agreement with the experimental data. This agreement means that the HM remains conductive for high values of the strain applied. However, more experimental data should be taken into account before stating that the HM is applicable to stretchable electronic materials. The HM in 2D behaves in the same way.
Although the general behavior of σ(ǫ) agrees with the experimental data, the shape this function for small values of ǫ does not. This will be investigated in the future. A preliminary possibility is that the strain dependence of some parameter has not been taken into account.