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An Effective Medium Approach to Electron Waves: Graphene Superlattices

We develop an effective medium approach to characterize the propagation of matter waves in periodic structures, such as graphene or semiconductor superlattices. It is proven that the time evolution of the states that are not more localized in space than the characteristic period of the structure can be described exactly through an effective Hamiltonian, and that the electronic band structure of the system can be exactly determined from the effective Hamiltonian. As an illustration of the application of the method, we characterize the mesoscopic response of graphene superlattices. It is shown that these structures may be described using simply two effective parameters: a dispersive potential, and an anisotropy tensor that characterizes the pseudospin. Our model predicts that a graphene superlattice characterized by an indefinite anisotropy tensor - such that the eigenvalues of the tensor have opposite signs - may permit the perfect tunneling of all the stationary states with a specific value of the energy when it is paired with a dual graphene superlattice with positive definite anisotropy tensor.


I. Introduction
Effective medium semi-empirical theories, such as the k⋅p method [1] or Bastard's envelope function approximation [2], have become invaluable tools to characterize the electronic properties of bulk semiconductors and related heterostructures, and associated devices. Such treatments are made possible by the fact that the most relevant physical phenomena in semiconductors are determined by the form of the electronic structure in the vicinity of some high symmetry points in the momentum space. The k⋅p methods are based on perturbation theory [3], and enable the calculation of the band structure and effective masses of different types of semiconductors (e.g. with zincblende structure).
On the other hand, stimulated by the development of electromagnetic metamaterials [4]- [6], recently there has been an intense research of methods that permit characterizing periodic structures from an effective medium perspective, typically through the introduction of an effective permittivity and an effective permeability [7]- [13]. It is thus natural to wonder if some of these ideas and methods can be extended to the characterization of matter waves, in the context of the Schrödinger equation.
The objective of this work is precisely, by generalizing our previous studies in the context of electromagnetic metamaterials [12,13], to bridge two fields and to develop from first principles a systematic approach that enables the computation of an effective Hamiltonian that describes within some approximations the time evolution of a quantum system, and that reduces drastically the complexity of the problem. Our analysis neglects electron-electron interactions, and thus many body effects. In some cases, these effects may be modeled by an effective potential.
We apply the developed theory to the case of one-dimensional graphene superlattices. It is demonstrated that the low-energy physics in these structures can be described simply in terms of an energy dependent effective potential and an anisotropy tensor that characterizes the pseudospin. Based on this effective medium model, we predict a novel perfect tunneling effect in graphene superlattices, showing that electron waves with a specific energy can be perfectly tunneled through a nanomaterial with specific properties.

II. Effective Medium Approach
The starting point of our analysis is the one-body Schrödinger equation for an electron in some periodic structure (e.g. a crystalline material or a superlattice), Let us suppose that the initial state is described by Thus, only the spatial harmonics with k within the first Brillouin zone are retained after the spatial filtering. This type of spatial filtering is usually designated by ideal low-pass filtering.
By definition, a "macroscopic" state has the property macroscopic state is unaffected by the spatial averaging. In particular, a "macroscopic state" cannot be more localized than the characteristic period of the material (or superlattice). As proven in Appendix A, a macroscopic state ψ is a superposition of states of the form σ k , with k in the first Brillouin zone. Similar definitions can be introduced in the context of electromagnetic metamaterials [12,13], where the role of the macroscopic state is played by the macroscopic electromagnetic fields.
In Appendix A, it is shown that provided the initial state ( ) 0 ψ is macroscopic, then the smooth part of the wavefunction in the σ r representation, satisfies exactly an homogenized Schrödinger equation of the form, where ˆe f H is the effective Hamiltonian of the system, defined in such a way that Hamiltonian can be written explicitly in terms of matrix elements , h σ σ ′ , as follows where N is the dimension of the system (e.g. N=2 for a graphene sheet). Notice that the "microscopic" wavefunction is denoted by ( ) ,t ψ r , whereas the macroscopic wavefunction (after spatial averaging) is denoted by ( ) ,t Ψ r . The indicated properties of ( ) , , h t σ σ′ r imply that its Fourier transform (unilateral in time and bilateral in space), is such that    considered here in the context of matter waves is the analogue of the spatially dispersive effective dielectric function introduced in Refs. [12,13] in the context of electromagnetic metamaterials. The explicit relation between the two formalisms is given in Appendix B.
An important property of the effective Hamiltonian is that the corresponding energy eigenstates n E , which satisfy for some non-trivial Ψ ( ) are exactly coincident with the energy eigenstates of the microscopic Hamiltonian.

III. Graphene Superlattices
In the rest of the paper, we illustrate the application of the proposed homogenization method to the case of a graphene superlattice. Graphene is a one-atom-thick material whose low-energy excitations are massless, chiral, Dirac fermions [15]. Its unusual electronic properties make it a unique platform for the development of novel electronic devices with superior characteristics [16,17,18] and for flatland transformation optics [19]. Here, we will obtain an effective medium model for a graphene superlattice characterized by a 1D-electrostatic periodic potential, whose geometry is sketched in Fig. 1. Some recent works have shown that these graphene superlattices may be used to collimate an electron beam with virtually no spatial spreading or diffraction [20,21]; quite differently here we concentrate on the effective medium description of electron waves in the superlattices, and highlight how by tailoring the microscopic potential one can control the "macroscopic" Hamiltonian of the quantum system.
The charge carriers in a graphene superlattice are described by the massless Dirac-type equation, where 6 10 / x y σ σ are the Pauli matrices, and V is the external periodic electrostatic potential. Here, without loss of generality, we restrict the analysis to the study of electrons whose transport properties are determined by the Dirac K point [15]. The wavefunction ψ has two components (it is a pseudo-spinor), each component, Notice that Eq. (11) Hence, for fixed ( ) ,ω k the simplest way to determine the effective Hamiltonian is to solve Eq. (11) for l=1,2, and after this to compute the matrix where the symbol ";" separates the columns of the 2×2 matrices. Evidently, in general, Eq. (11) needs to be solved numerically.
It is clear from the previous discussions, that in general ef V exhibits spatial dispersion (dependence on i ↔ − ∇ k ) and time dispersion (dependence on ω , or equivalently on the energy E ω = h ), and this introduces some complexity in the effective medium model. This is fully analogous to electromagnetic metamaterials, where in general the effective dielectric response depends on frequency and wave vector [12,13]. In order to further simplify the model, we consider the case where the spatial dispersion is weak so that it is possible to approximate ef V by its Taylor series of first order in k , Taking into account the symmetries of the microscopic Hamiltonian, it is simple to verify that is a scalar. Within this approximation, using Eqs. (4) and (12), it is seen that the effective Hamiltonian of the superlattice is such that: where Ψ is a pseudo-spinor and In order to understand how the effective parameters vary with the energy, so that the model can be further simplified, in the next section we consider a numerical example.

IV. Numerical Example
In Fig. 2, we depict the numerically calculated effective parameters as a function of analogous to what is done when solving a scattering problem [22]. Very interestingly, the numerical results show that to an excellent approximation (Fig. 2b) yy . We shall refer to r v as the anisotropy tensor of the superlattice, -12-and neglect its dependence on E (due to its weak dependence on energy). On the other hand, the effective potential ef V (at 0 k = ) for low-energy excitations varies linearly with where α is a dimensionless positive constant. Notice that in pristine graphene . On the other hand, because of the Klein tunneling effect [15], it is evident that , - 13-We have indeed verified that the numerically calculated effective parameters satisfy exactly this relation. In particular, it follows that energy dispersion of the superlattice may be written simply as where , , is by definition the anisotropy ratio. The "exact" energy dispersion of a graphene superlattice with 1 / 6.0 Fig. 3a, and is compared with our effective medium theory in Fig. 3b, revealing a very good agreement between both theories. As seen in Fig. 3a, consistent with the results of Ref. [21], the graphene superlattice is strongly anisotropic, and the usual Dirac cone of pristine graphene is stretched along the y-direction.
The pseudo-spinor associated with a state of energy E and wave vector k is, where θ q is the angle between the vector ( ) and the x-axis (Fig. 4b). Thus, in the case where the superlattice is characterized by strong anisotropy with 1 χ << the pseudospin is such that either 0 θ ≈ q or θ π ≈ q . A similar result was reported in Ref. [21], based on direct band structure calculations. It should be noted that, unlike in pristine graphene, θ θ ≠ q k , where θ k is the angle between k and the x-axis (Fig. 4b). Evidently, the angle θ q determines the relative phase of the wavefunction in the two sublattices of graphene.
In Fig. 2c-d we plot , r ii v and χ (evaluated for 0 E ≈ ), respectively, as a function of the normalized potential 1 V . As seen, the anisotropy ratio can be quite large if 1 which is when the parameter , r yy v crosses zero. This extreme regime of operation may permit a supercollimation of an electron beam [21].

V. Perfect Tunneling
One of the most striking features of Fig. 2d is the fact that χ can be negative for an applied potential with normalized amplitude 1 / 6.3 . At a microscopic level these nanomaterials may be regarded as superlattices characterized by a suitable microscopic potential, consistent with the previous discussions. Moreover, we assume that there is a static potential offset V δ between the nanomaterials, so that the energy dispersion of the stationary modes in the first material is , whereas the dispersion in the second material is . We say that the pseudospin of a material is positive when 0 i χ > and that the pseudospin is negative if 0 i χ < . Consider now the geometry depicted in Fig. 4d, which shows a rectangular slab of the second material embedded in the first material. Using the developed effective medium theory, it is straightforward to compute the transmissivity for a stationary state (plane wave) that describes an electron incident from the left region, 0 x < , on the slab of the material with negative anisotropy. This is done by expanding the wave function in the different regions into plane wave modes, and ensuring the continuity of the pseudospinors at the interfaces (see Appendix D). In Fig. 4c the transmissivity is shown as a function of the transverse wave number ( y k ) of the incoming particle, for different values of the energy. Quite interestingly, it is seen that there is a perfect tunneling -independent of the direction of incidence of the incoming particle -when the energy is / 2 E V δ = (green curve). This is distinctively different from Klein tunneling [15], which only occurs  . 4b). This implies perfect matching at the interfaces, and hence perfect transmission independent of the angle of incidence. It is interesting to point out that there -16-is an evident parallelism between our graphene-based nanomaterial with negative pseudospin and double negative (DNG) electromagnetic metamaterials, as proposed in [4]. Indeed, similar to electromagnetic metamaterials, our graphene-based nanomaterial may provide perfect focusing and negative refraction of electrons with the energy , when the source of electrons is at a distance / 2 W from the slab of the second material (Fig. 4d). A graphene analogue of Veselago-Pendry's lens has been proposed in an earlier publication [17], however very different from the configuration Unfortunately, it can be checked in Fig. 2c that the minimum value for 1/ χ , with 0 χ < , for a superlattice characterized by a Kronig-Penney type potential is about 4.6 .
However, this does not preclude that for a different profile of the microscopic potential, 1/ χ cannot be made smaller. For example, by solving the effective medium problem numerically using a finite-difference method, we calculated 1/ χ as a function of 1 V for a microscopic potential of the form ( ) 1 array of nodes and the derivatives are replaced by finite differences; then the problem is reduced to the solution of a linear system). The obtained result is represented in Fig. 2d with a dashed line. As seen, for this alternative potential the minimum value of 1/ χ (with 0 χ < ) is reduced to 2.5 . This is still far from 2 1 χ = − , but indicates that by tailoring the shape of the microscopic potential it may at least be possible to better approximate the value 2 1

VI. Conclusion
In conclusion, we have described a completely general self-consistent approach (many body effects are however neglected) to characterize electron waves in periodic systems from an effective medium perspective, which extends our previous work on electromagnetic metamaterials to matter waves [12,13]. The proposed theory may be -18-instrumental to establish novel analogies between electromagnetics and electronics, as further pursued in Ref. [24] in case of semiconductor superlattices. Here, we applied the formalism to the case of a graphene superlattice characterized by a 1D-periodic potential, showing that the low energy excitations can be described in terms of an effective scalar potential and an anisotropy tensor

Appendix A: The effective Schrödinger equation
The spatial averaging operator defined in the main text in the coordinate representation, { } av , can more generally be described by the projection operator: represents the first Brillouin zone, and the normalization It is also useful to introduce the (unilateral) Fourier transform of ( ) Denoting the propagator of the system by ( ) U t , we can write: In the Fourier domain these relations are equivalent to: If the initial state is macroscopic, i.e.

Appendix B: Effective medium description of electromagnetic metamaterials
In this Appendix, we discuss the explicit connection between the formalism developed in the main text and that of our previous work on electromagnetic metamaterials, highlighting the equivalence between the two [12,13]. The starting point is to note that the Maxwell's equations in a continuous medium can be written as, where ψ ↔ g is a six-component vector, and the operator Ĥ is given by: Obviously, the above Hamiltonian does not represent the energy of the system, but it is rather an operator that describes the dynamics of the classical electromagnetic field. For simplicity, from now on in this Appendix we set 1 = h . It is interesting to notice that Ĥ is Hermitian with respect to the inner product, In other words, the standard normalization of the wavefunction does not apply to the averaged state vector.   Fig. 1. Sketch of the geometry of a graphene superlattice characterized by a step-like periodic electrostatic potential.