Strong Electric Field in 2D Graphene: The Integer Quantum Hall regime from a different (many-body) perspective

We investigate the emerging consequences of an applied strong in-plane electric field on a macroscopically large graphene sheet subjected to a perpendicular magnetic field, by determining in exact analytical form various many-body thermodynamic properties and the Hall coefficient. The results suggest exotic possibilities that necessitate very careful experimental investigation. In this alternate form of Quantum Hall Effect, non-linear phenomena related to the global magnetization, energy and Hall conductivity (the latter depending on the strengths of magnetic B- and electric E-fields) emerge without using perturbation methods, to all orders of E-field and B-field strengths. Interestingly enough, when the value of the electric field is sufficiently strong, fractional quantization also emerges, whose topological stability has to be verified.


Introduction
Dirac-type materials, such as Topological Insulators, monolayer graphene, and three-dimensional (3D) Dirac and Weyl semimetals, appear nowadays as stable (actually very robust) topological phases of matter, displaying behavioral patterns that produce new physics at a very fundamental level and at the same time give the possibility of exotic future applications [1] [2] [3]. What make their fundamental properties so fascinating are the well-known dissipationless surface states that can propagate without any resistance and give rise to nontrivial topo-Advances in Materials Physics and Chemistry logical properties that are currently under intense investigation. When certain types of such materials (i.e. 2D Topological Insulators) are subject to a perpendicular magnetic field, they may as well undergo a phase transition to Quantum Hall Insulators [4], violating the time reversal symmetry that controls the topology of the surface states. Normally, there is a transverse (to B) small electric field E, which-to first order in E-is responsible for the macroscopic quantization of the Hall conductivity [5], and which is the central quantity in the present paper.
Interestingly enough, the strong E-field regime has not been investigated in sufficient detail so far, in particular with respect to the role of the E-field on thermodynamic many-body properties (see however [6], and for some earlier attempts see [7]- [14]), as these properties are determined in the noninteracting electrons framework (the one that, in any case, pertains to the Integer Hall Effect regime).
In this work, we present potential consequences (on thermodynamic and transport properties) of a strong electric field applied tangentially to a macroscopic 2D graphene sheet, when also subjected to a perpendicular magnetic field of arbitrary strength.
Let us start with the graphene energy spectrum when a monolayer is subjected to an in-plane electric E (taken along the x-direction) and a perpendicular magnetic field B in the z-direction (and let us focus on the positive branch, and also ignore the Zeeman interaction term), and take the Landau gauge A = (0, xB, 0) in which the energy spectrum turns out to be (through a Dirac equation procedure similar to the one in [15] a dimensionless parameter (always supposed to be lower than unity, 1 β < ), n = 0, 1, 2, 3. The Landau Level index for the positive branch, f u is the Fermi velocity, and y k is the wave vector along the y-direction. We also find that the guiding center operator's eigenvalue (projected on the x-axis) X 0 reads ( ) ( ) with B l c eB =  the magnetic length and ( ) sgn n is the sign function. Due to the spatial confinement in the x-direction, the guiding center operator 0 X may acquire any value in the following range: with x L being the x-direction size of the system, which is here supposed to be Unlike conventional semiconductors, the energy gap has an E-field dependence. As can easily be seen from (1.5), the larger the E-field gets, the larger the energy gap becomes. On the other hand, the larger the L.L. index, the smaller the energy gap. This interplay will play a major role later on, when we consider the thermodynamic occupations of the energy levels. One can always prove that, for a given L.L. index and a value of E-field (such that 1 β < ) there will always be an unavoidable overlap (states of greater n values have lower energy than states with lower n values). We can set conditions (for arbitrary E and n) for which this kind of overlap is avoided as:  That is, in words, when the work performed by the electric field is smaller than the energy gap at a certain 0 X , no overlap is observed between the L.Ls n and n + 1. Generally, for strong enough electric field or for a small enough L.L. index, the above inequality will be true (see Figure 2); but as the L.L. index of occupied levels gets larger (hence for a large number of electrons N) the energy gaps between adjacent L.Ls will become lower, until an inevitable gap closing occurs ( Figure 2 providing a concrete example).
We define   The red lines indicate the top-most energetic state in a given L.L., in comparison with the next adjacent L.L. lower state. The stronger the E-field gets, the larger the energy gap becomes, and the overlap will occur at an energetically high L.L. In the above example, overlaps occur between (n = 2, n = 3), (n = 3, n = 4), (n = 4, n = 5, n = 6 (not shown)). Levels n = 0, n = 1 do not overlap and they provide independent energy states when following an occupation procedure. In this case, 1 The above is a generic case for an arbitrary value of E and n. Of course there might be cases where overlaps start from n = 0 (for a low enough E-field), in

The Strong E-Field Regime
After the above discussion and definitions, we proceed to thermodynamic occupations of the graphene's energy levels at zero temperature. For this purpose, we consider a collection of N electrons at T = 0, which fill the lowest energy levels until the Fermi energy denoted by F ε . In reality, the Fermi energy is not constant when there is an electric field running through the system; what we then mean by Fermi energy is actually a Fermi point, which is the topmost occupied state in the energy diagram. We also make the supposition that this Fermi point is located on a L.L. indexed with 1 , so that there are always ρ L.Ls occupied at any time (the last level 1 n ρ = − being generally partially occupied). First, we will focus on the special case where all ρ Landau Levels are not overlapping, and can be occupied independently by the N electrons. In this case the following relation must be satisfied: and for strong enough magnetic fields, it is guaranteed that Equation (2.1) will always be satisfied, and no overlap between L.Ls with different quantum numbers will be observed. In what follows, we will consider a constant, strong E-field, while the magnetic field may vary, but always in a way that satisfies Equation Considering that the L.L. with n = 0 only has a capacity for 2Φ/Φ 0 electrons, and that all the other 0 n ≠ L.Ls may host up to 4Φ/Φ 0 electrons, we find that in order to have ρ L.Ls occupied, the following inequality must hold: Note that when 1 ρ = then h .
In the case we are considering the total energy of the system (minimized at T In the thermodynamic limit x L → ∞ , we may approximate the sum with respect to 0 X as follows: full part In what follows, we will consider the case To determine part E , we must first determine the 0 X value at the Fermi point  ( ) ( ) Finally, adding Equations (2.14) and (2.8) we arrive at the following result: i.e. the proportional constant is equal to half Hall conductivity, similar to the corresponding result in a conventional semiconductor case [3]. (Plots of the field-free Energy and Magnetization are given in Figure 3).

The Weak E-Field Regime
We now proceed to a considerably more difficult case: the low E-field regime. In this regime, as the electric field becomes weaker, the energy gap gets smaller. As a result, there will be an unavoidable point where some L.Ls will overlap ( Figure   1(b)), and occupational patterns turn out to be more complex.
The number of states in L.L. n = 3 is then given by the difference of the initial and final values of l, namely tion of localized and extended states that lie inside this gap. When the gap is destroyed, the system becomes metallic; it may thus not be unreasonable to find a Hall conductivity that is electric/magnetic field-dependent, destroying the plateaux formation. This is something that needs to be further investigated experimentally.

Conclusion
In this work, a thermodynamic study has been conducted with respect to a 2D Graphene monolayer subjected to crossed electric and magnetic fields. Thermodynamic quantities like the global energy and magnetization have been exactly and analytically determined, in combination with transport properties, i.e. the Hall conductivity. The results suggest exotic possibilities that are here pointed out as a natural outcome of an exact and careful calculation in the noninteracting electrons many-body framework, and these necessitate exceedingly careful experimental investigation. Finally, with respect to the range of validity, although our results involve no approximations whatsoever, the general role of disorder in combination with the above physics of the strong fields is certainly something that needs further study.