Ground State thermodynamic and response properties of electron gas in a strong magnetic and electric field: Exact analytical solutions for a conventional semiconductor and for Graphene

Consequences of an exceedingly strong electric field (E field) on the ground state energetics and transport properties of a 2D spinless electron gas in a perpendicular magnetic field (a Quantum Hall Effect (QHE) configuration) are investigated to all orders in the fields. For a conventional semiconductor, we find fractional values of the Hall conductivity and some magnetoelectric coefficients for certain values of E and B fields that do not result from interactions or impurities, but are a pure consequence of a strong enough in-plane E field. We also determine analytically the ground state energy, and response properties such as magnetization and polarization as functions of the electromagnetic field in the strong E field limit. In the case of Graphene, we obtain more complex behaviors leading to the possibility of irrational Hall values. The results are also qualitatively discussed in connection to various mechanisms for the QHE-breakdown.


Conventional system 1.1 Low E-field strength
Consider an ideal nonrelativistic two-dimensional spinless electron gas in a perpendicular and homogeneous magnetic field B  directed along the positive z-axis. The dimensions of the plane are taken to be macroscopically large there is also an in-plane, homogeneous electric field E  pointing in the y-direction that is very weak, i.e. it doesn't cause any overlap of different Landau Levels (L.Ls). In this manner, the Fermi energy can always be situated in the interior of an energy gap (i.e. for certain areal density), causing, as is well-known, the universality of quantization of Hall conductance   . If the electric field is further increased from its first critical value (to be determined below) then the gap closes and the system is highly nonlinear, the general consensus being that this results to destruction of the quantization of   , although from what we will see below several refined behavioral patterns remain (in this strong electric field case), that can even lead to the survival or even a different type of integer quantization in the nonlinear regime. This strong electric field case will be separately studied in the next section. 2 We choose to work in the Landau gauge By 3). Therefore, this case of no-overlap involves a limitation of drift velocity's values that depends on both field strengths and is equivalent to either of two criteria: (i) Dy mV L  (the angular momentum of an electron in one edge with respect to a point in the other edge is  ), or (ii) 2 2 2 / 2 / 2 Dy mV mL  (the drift kinetic energy is  a confinement energy along the y-direction (due to the uncertainty principle)). In this case all L.Ls can be filled independently: according to the least energy principle, at zero temperature T=0, all electrons occupy states labeled by small quantum numbers   , nl , (with n the above mentioned Landau Level index, a nonzero positive integer, and with l another integer specifying the eigenvalue of

Thermodynamic properties
The total internal energy of the system at T=0 is a sum over all occupied quantum numbers n and 0 Y : In the macroscopic (continuum) limit    4 We then have   Fig. 1.2 see the graphs of energy and magnetization per electron, for 2 1 Ay nL  (a good value so that the internal structure of these quantities are shown in sufficient detail). Using the second thermodynamic law we can also determine analytically the equilibrium magnetization and polarization per electron, which turn out to be with h e 2     the Hall conductivity (see below). It is interesting to note a characteristic half of the Hall conductance in the coefficient that connects the socalled magnetoelectric effects with the fields that cause them (that in a problem with chiral properties, usually in Topological Insulator materials, correspond to an extra magnetization caused by a parallel electric field (as in (1.11) above) and an extra polarization caused by an extra magnetic field (see [2] and references therein); here we note such a trend even in a Quantum Hall system (which is not unexpected, and is actually justified based on general Physics arguments [2]). One can actually see such a trend in the polarization as well, for a general B, in (1.10), where in the last linear term we can again see σ H /2 appearing. [It should be added that all these behaviors originate from the final term of (1.9) that describes an EB-coupling, something that we will also see later for the relativistic case.] In addition to all this, note also that for i.e. the total energy per electron reduces to 2D energy of free electron gas plus a drift kinetic term. These last simple results are not quite unexpected and may be justified with proper semiclassical considerations.

Hall Conductivity
The Hall conductivity is defined as In the following we show that, in the very strong E-field case, fractional filling factors may also occur, by variation of electron number N with E and B fixed. [In that case, the analogous width of the corresponding plateau is expected to be reduced, although a serious consideration of plateau-observability should include a study in the presence of disorder, a subject that is beyond the scope of the present article, or in the presence of edge states (see Section 2.5 for a qualitative discussion).]

High E-field strength
When the electric field exceeds its first critical value, , inter L.L. overlap occurs. As E gets stronger, more and more L.Ls overlap and degenerate states that belong to different values of the quantum number n appear (in the previous case of a weak electric case, the standard Landau degeneracy had been completely lifted). Energy gap closes, and Fermi energy is always located on a single quantum state, with a significant number of available nearby states. Although Fermi energy is no longer in an energy gap, it will make jumps from one L.L to another by varying the magnetic field (or the particle number N).
i.e. the electric field has a strength such that the single particle energy of the last in L.L. 0  n is greater than single particle energy of the first in L.L. j n  and lower than the single particle energy of the first particle in L.L. 1   j n . Then, which is indeed (2.1).
with m Y the guiding center position of the last electron (with highest single particle energy) located at n=ρ-1 (see the isolated dot in Fig. 2.1). In this way, we ensure that variations of N will result in moving , the left edge of n=ρ-1 L.L. and R Y is a critical guiding center above which L.L. n=ρ is occupied. This can be determined by equating the Fermi energy to the single particle energy   , or, using eq. (2.1): . Any number N that takes R Y above this critical value will result in a nonzero occupation of n=ρ L.L. Therefore, to ensure that exactly ρ L.Ls are occupied, m Y must vary in the following window: guaranteeing that exactly ρ L.Ls are occupied. Naturally this equation becomes pathological when z is smaller than unitythe weak electric field case - This of course happens because the above equation is not valid in this case, where no inter-L.L overlaps occur. Limitations must therefore be imposed here. The smallest value that z can take, must be above 1, with z=1 defining the non-overlap limit. If 1  z the above relation just becomes: as earlier. Having now defined the Fermi level's exact location analytically, we proceed with our method by counting how many L.Ls are fully occupied and how many are partially occupied. An L.L. is completely filled with electrons only when the Fermi energy is greater than the single electron energy located at 2 / for fixed E and B) according to eq. (2.5) result in variations in F i as follows: are the numbers of completely filled L.Ls calculated at the edges of eq. (2.6). Let's see now some examples:

Fig. 2.2 Fig. 2.3
In Fig. 2  [This is actually consistent with the wellknown corresponding result in the case we fold the system (in the x-direction) into a cylinder, that is a key result in the Laughlin argument [4] that gives the integral quantization of the Hall conductivity from general gauge argumentsor equivalently to the charge-pumping picture of Thouless [5] or a more general property of the socalled spectral flow [6] in topologically nontrivial systems, such as topological insulators [7].] It is interesting to see that when z reaches its lowest possible value, , meaning that the maximum number of partially occupied L.Ls is always 1. Needless to say that z remains a constant only when E and B fields are constant too, (or in a special case that both fields vary in the exactly same rate) and the only variable is the electron number. This happens because of eq. (2.1), is the number of partially occupied L.Ls.

Number of states under Fermi energy
This number, in a canonical ensemble, is exactly equal to the constant particle number N: which results in the following window of values for N: . The above relation defines windows of values for B (if it is considered as a variable) for constant E field, and constant electron number N. Equivalently, one may prepare an experiment, where electron number is not conserved (i.e. an electric circuit) and keep E and B fixed. Note that the above relation shrinks to eq. (1.14) when 1  z :

Thermodynamics
We now proceed to ground state energy calculation, which is a sum over all single electron states, until reaching the Fermi energy:

 
The left term describes the energy due to fully occupied L.Ls, while the right term describes the energy of partially occupied L.Ls. After a number of algebraic manipulations, and using eq. (2.7) we conclude to the following result for the total energy per electron: that consists of a sum of several terms which are not all symmetric (with respect to E and B). The last term vanishes in weak E field limit (   A few remarks are in order about these figures: It is clear that when the E field is exceedingly strong, in contrast to the case of a weak E field, there appears a global minimum with respect to particle number variations. This minimum is a consequence of the E field effect on the thermodynamic properties; as E gets stronger, more and more states will gain negative energy, (see for example Fig. 2.3). As these states get occupied by electrons, the total energy will become negative at first, then will rise up to positive values, because states with positive energy will begin to fill up. The result of this competition is this minimum, which can be fully controlled by examining the corresponding z-value. The greater z gets, the more negative-energy states will be occupied, and the minimum will move to greater particle numbers (or larger particle densities). In comparison with the low E field case, where the positive states are more, in this case the total energy may be even lower.
We now plot the total energy as a function of variable z, for a constant E field with / where z lies in the following window: Fig. 2 for a 2D system [9].
In addition, it should be pointed out that the correctness of our results is witnessed by the fact that the total energy turns out to be a smooth (continuous and differentiable) function of z for every z (i.e. the positions of the windows match with the corresponding expressions)this smoothness of the figures (in their joining through different window-values) strictly testifying for the analytical correctness of the overall expressions that we derived for the total energy.

Hall conductivity
The Hall conductivity is defined as: we will show that by varying N, fractional filling factors appear, with no interactions and no impurities taken into account. E and B fields are considered as constants (and therefore so is z) throughout all N (and ρ) variations. Substituting eq. (2.1) into (2.17) we may eliminate E: Let's examine now some cases regarding z-values: z is an integer: We re-emphasize here that the above fractional values in Hall conductivity do not result from any interactions (i.e. they are not related to the Physics of the Fractional Quantum Hall Effect) but are a genuine consequence of a high E-field strength that causes the L.L overlaps.

Plateaux formation
When the external E field is relatively small, i.e. given by eq. (1.3), the Hall conductance has a plateau-like structure. To see this, imagine that we vary in a continuous matter the particle number, keeping B fixed. As a result, the Fermi energy is continuously moved from the beginning of a Landau Level to the right end, after passing through all states in that L.L. And when Fermi energy is in the interior of a L.L., Hall conductivity remains a constant and it only changes at the critical N values ( 0 / N     ), namely when Fermi energy has a transition between adjacent L.Ls.
In the strong E field case however, this is not the case. When Fermi energy lies in the interior of the ρ th L.L, plateaux are destroyed, because the energy spectrum is no longer discretized. It is as if one has a single L.L. band (all L.Ls form a single band), with an infinite number of neighboring states to accommodate all electrons (a metallic character) and Hall conductivity is more likely to behave as in the classical case, with an eternal increasing (or decreasing) character by varying either N or B.
Inclusion of impurity potentials [10] however may change this expectation in the sense that, there might be a case where Fermi energy will lie in a mobility gap (which is the true criterion for the existence of QHE), and some of the extended states per L.L might be occupied, resulting in a blurred quantization of Hall conductance. This is a matter that needs to be investigated more carefully. In this case, the previous   values will correspond to certain plateaux by varying B or N.

Inclusion of edge states
In real-life samples, however, electronic systems are confined, and the confinement and impurity potential must be seriously taken into account. QHE manifestation is accomplished by considering both effects on transport properties (the confinement and disorder). In this case, a thermodynamic approach may be of low importance and insufficient to explain the robust quantization of Hall conductivity. Here, we will try to visualize a special case, where the electric field is strong enough, so that it causes an inter-L.L overlap and the system is at the same time confined in y-direction. The key to solve this problem lies in the range of influence of the confinement potential. As one starts from y=0, and moving towards lets say, the right end ( /2 y L ) he will experience the confinement potential influence only on a certain distance from the end (a few multiples of magnetic length B l ). From that point on, L.Ls will start to rise abruptly.

Fig. 2.8. Impact of the edges on Landau Levels. (a) low-E field, (b) strong-E field case
Now, if the electric field is low enough (Fig. 2.8a) there is no inter-L.L overlap and the Fermi energy will always intersect the same number of edge states at the left end and right end respectively. But in the case of a strong E field (Fig. 2.8b), things may change dramatically; if (for a certain E field) the overlap starts at a point before the confinement potential influence, there might be a case where the Fermi energy intersects two points (two L.Ls) at the left end and only one point (L.L) at the right end (where the edge states are located). This is possible only if there is an additional intersection in the bulk of a L.L. (see the middle L.L at Fig. 2.8b). Varying then the Fermi energy will not change the number of intersections, until the Fermi energy reaches the bottom of the top L.L. and so on. In other words, the occupations of edge states will remain the same, until another L.L will be occupied. Therefore, there might be some robustness of the Hall values (found above) before this crossover occurs.

Graphene -Low E-field strength
Our results can also be applied to Relativistic solid state systems that obey a Diractype of equation, such as Graphene [11] and topological insulators [7]. In this Section we give a first analytical study of strong electric field effects in the ground state energetics and also in transport properties of Graphene, i.e. Hall conductivity. We find again that, when the E-field is strong enough, L.Ls with different quantum numbers overlap, and the Hall conductivity becomes electric and magnetic field-dependent, indicating the non-linearity of Hall conductivity in relativistic systems, this having more refined consequences compared to the previous nonrelativistic case; we obtain, for example, the possibility of irrational values (the observability of which is of course up to the uncertainties related to the lack of disorder, already mentioned in the last section).
We start with Graphene's energy spectrum when placed in homogeneous crossed  This energy gap depends on E-field, in contrast to conventional semiconductor systems (of last section); in addition, the greater L.L. index, the smaller the gap. This means that in some L.L. range there will be unavoidable overlap accompanied with a zero energy gap (and the system is then metallic). This is exactly the case we will examine later. Now, for adjacent L.Ls, no overlap is observed only when the following condition holds:

(3.6).
For small enough indexes n, and small enough E-fields, inequality (3.6) will be satisfied. But, for a given E-field, there will be a critical L.L. index  gap and with always available neighboring states for an electron to be scattered in. It is useful to define a new continuous number z, which obeys: This number, when put in (3.7), results in the following equality:   which will be used later below. We now proceed to our calculation of thermodynamic properties by considering that Fermi energy is always located at an L.L. whose index where the Hall conductivity is quantized in half integer multiples of the quantity 2 4/ eh . We now calculate the total internal energy of the system, which has to be minimalized when the temperature is zero (T=0):   This is the result we would have gotten had we solved the problem from the beginning without the E-field.

Stronger E-field
The inequality   field induced character and Hall conductivity is modified (analogous to the nonrelativistic case) by a term containing both electric and magnetic field.
Recall from previous work [13] that the following relation also holds: (so that no collapse of L.Ls occurs), which can always be made true as long as we treat particle number as our variable, and keep E and B-fields constant. Or, equivalently, we may keep N and B fixed and treat E as our main variable, starting from zero until reaching maximum value . We remind here the reader of the limitations of this problem regarding also the magnetic field, which has to be such that the magnetic length is much greater than lattice constant (i.e. we will not study any Hofstadter effects in this work). Either way, our results and predictions can be obtained by satisfying all possible limitations. For example, we may prepare an experiment where E is strong enough for inter L.L. overlap to occur, and at the same time B is also strong enough to satisfy the above inequality.    We examine now the case appearing in Fig. 3 To determine the number of states in L.L. n=3, we examine its intersection with Fermi energy (see Fig. (3.2)):

Considering the QHE-breakdown in high injected currents
At this point one may wonder about the relevance of the above results, as these do not seem to take into account mechanisms associated with the breakdown of QHE under non-equilibrium conditions, i.e. due to high injected current densities [14][15][16]. Indeed, in the case of hundreds of μΑ (0.6-0.9 mΑ, taken from ref. [16], the exact value depending on the filling factor) flowing through the sample in the y-direction, the longitudinal resistivity is different from zero by orders of magnitude, resulting in an inevitable dissipation of energy, and the Hall conductivity deviates from its exact quantized values.
One of the reasons behind this deviation is the thermal instability that happens due to large current densities flowing continuously for a period of time through the material. The amount of heat gained by the electrons per unit time per unit area due to the electric field is given by This amount of heat is then passed to the lattice structure, ending up in the Helium coolant. This means that there is a gradient of temperature between the electrons and the atoms forming the crystal that relates the heat transferred to the atoms through the following equation: When the electric field exceeds a critical value, the amount of heat gained by the electrons ( ) is always greater than the amount of heat transferred to the atoms ( ) and the excess in thermal energy results in an energy excitation in the order of the cyclotronic energy, destroying the stability of the quantum Hall conductivity plateaus.
It should be noted that the present work takes into account the applied E-field directly within the energy spectrum, and there is no need to investigate the heating of the electron gas separately. It is contained a priori in the thermodynamic formulation that we have used to derive our results. Furthermore, the results shown in this paper can be relevant at even relatively low E-fields, or low injected current densities as well. To see this, we may assign some numbers to our mathematical relations, starting from eq. (2.1) that gives the limit of strong E-field: Above this critical E-field (and therefore for low Hall E-field) all the inter-L.L. overlaps that we considered can occur and the need for introducing breakdown effects is therefore eliminated. In the non-equilibrium breakdown regime of the QHE state in Graphene, it has been verified that not all L.Ls experience the breakdown [17]. For example, the n=0 L.L. remains robust under the influence of a relatively strong Efield. Therefore, the breakdown in Graphene is a filling factor-dependent phenomenon, with different critical electric fields for each filling factor in a way that we may always impose a low enough E-field to achieve the results derived in this paper. The stronger the B-field is, the weaker the E-field gets, much lower than the breakdown limit.
Apart from the above mentioned electron-heating mechanism we point out, for completeness, that percolation of incompressible regions [18] has also been invoked for the QHE-breakdown, as well as the possible existence of compressible regions in the bulk [19], which however do not seem to offer insights relevant to the present calculation.
Finally, there exists a small number of publications in the literature that seem to be close to our exact solution (although they deviate from it); they also find fractional quantization of the Hall conductivity by seriously taking into account a conceptual difference between the externally applied E-field and the Hall-electric field (the one that is formed in the (classical) steady state, as opposed to the field being applied). They take into account the non-negligible effect of the Hall E-field on the electronic density of states, which is further broadened by the presence of this Hall E-field, see i.e. ref. [20]. Just like interactions lead to a splitting of L.Ls (as is well-known in the Fractional QHE), the Hall electric field succeeds in a similar manner to divide each Landau Level into many sub-levels without any introduction of impurities or interactions. Fractional quantization is then observed in higher L.Ls (due to the density of states-broadening), but not in the lowest L.L. (where interactions are dominant, and a Composite Fermion approachhence a passage to the so-called Λ-Levelsmust be used instead (see [9] for a practical use of Λ-Levels)). However, strong anisotropies have been observed at these fractions [17] which suggests that they need to be considered more carefully as functions of the Hall E-field.
By way of comparison with the above, we must state that in the present work we have avoided the notion of a Hall electric field, as this is a built-in consequence of the presence of the B-field in the (classical) steady state (or in a stationary state in the corresponding quantum problem), or, alternatively, as this can be viewed as a response of the system to an external voltage (or the externally applied E-field) which is already fully taken into account in our physical picture (Hamiltonian) and in our associated calculations. We use, instead, the -physical broadening‖ given by the presence of the external E-field in order to describe the fractional quantization. Furthermore, with respect to the possible robustness of our results (such as eq. (2.18) for a conventional system, that gives the Hall conductivity value for a certain magnetic field and fixed particle number), we recall that topological protection and exact quantization of in a high E-field is not guaranteed for our system in the thermodynamic limit. This is better clarified at least in a visual manner in Figures 2.8a and 2.8b (where the number of edge states plays a role), but it further needs inclusion of other factors not taken into account here, like the impurity potential; this, however, is actually expected to have a positive effectit is expected to eventually broaden the electronic density of states, and, at least, assist in establishing the topological stability of the system (in a similar manner as this occurs in the ordinary Integer-QHE).

Conclusions
We have shown that by finding the optimal energy at zero temperature in the case of conventional semiconductors, fractional Hall values appear at the points of jumps of the Fermi energy. This does not necessarily mean that plateaux appear. Inclusion of impurity potential may lead to quantized plateau structure even in the case of strong E field, due to further broadening of L.Ls, isolating extended from localized states. We should point out that it is not the gap closing that causes plateau disappearance, but rather the fact that we have ignored the impurity potential in our calculations. We have also calculated analytically, using ground state energy considerations the total internal energy, magnetization and polarization as functions of the electromagnetic field. The associated de Haas-van Alphen oscillation periods are also influenced by the presence of the electric field in a specific quantitative manner. A corresponding exact calculation in a pseudo-relativistic system, such as Graphene, is more involved but it has also been carried out here in detail. An immediate result of our toy model is