# Hydrodynamic equations for an electron gas in graphene

- Luigi Barletti
^{1}Email author

**Received: **14 December 2015

**Accepted: **12 May 2016

**Published: **8 June 2016

## Abstract

In this paper we review, and extend to the non-isothermal case, some results concerning the application of the maximum entropy closure technique to the derivation of hydrodynamic equations for particles with spin-orbit interaction and Fermi-Dirac statistics. In the second part of the paper we treat in more details the case of electrons on a graphene sheet and investigate various asymptotic regimes.

## Keywords

## MSC

## 1 Introduction

This paper is devoted to present some results on the derivation of hydrodynamic equations describing electrons subject to spin-orbit-like interactions. Systems of this kind, of particular interest for applications to microelectronics, include electrons undergoing the so-called Rashba effect [1], the Kane’s two-band K⋅P model [2] and electrons in single-layer graphene [3]. The diffusive and hydrodynamic descriptions of such systems are extensively treated in Refs. [4–10]. Here, we summarize the results contained in Refs. [7–10], concerning the hydrodynamic description, and extend them to the non-isothermal case.

In comparison with kinetic models, the advantages of fluid models for applications are evident. In fact, from the numerical point of view, a system of PDEs for a set of macroscopic quantities is much more desirable than a single equation for a density in phase-space, where also the components of momentum are independent variables. Moreover, from the point of view of mathematical modeling, they offer more flexibility, as various kind of boundary conditions and coupling terms (*e.g.* with a self-consistent potential or with various types of scattering mechanisms) can be very naturally embodied in the mathematical model.

On the other hand, the derivation of fluid equations for the systems under consideration, which possess spinorial degrees of freedom and non-parabolic dispersion relations (energy bands), is far from being a trivial extension of the techniques employed for standard (*i.e.* scalar and parabolic) particles. The best strategy to obtain hydrodynamic (or, more in general, fluid-dynamic) equations in this case, seems to be their systematic derivation from an underlying kinetic description by means of the Maximum Entropy Principle (MEP) and its quantum extensions [11–14]. The MEP basically stipulates that the microscopic (kinetic) state of the system is the most probable among all states sharing the same macroscopic moments of interest, providing therefore a formal closure of the system of moment equations. This is a very general principle which finds a variety of applications to different fields, ranging from statistical mechanics to signal theory [15]. For quantum systems it can be used in combination with the quantum kinetic framework provided by the phase-space formulation of quantum mechanics due to Wigner [16, 17].

In the present work the Wigner formalism is used ‘semiclassically’, which means that some quantum features are retained (namely, the peculiar energy-band dispersion relations and the Fermi-Dirac statistics) while others are neglected (namely, the quantum coherence between different bands). Consequently, the obtained hydrodynamic description misses some interesting physics when quantum interference between bands becomes important (*e.g.* close to abrupt potential variations [18, 19]). Nevertheless, the derived equations possess an interesting mathematical structure and reveal some interesting physics, still occurring in absence of such interference phenomena (see, in particular, Section 5).

The paper is organized as follows. In Section 2 the kinetic-level formalism, based on a semiclassical Wigner description, is introduced for a fairly general spin-orbit Hamiltonian that includes all cases of interest. In Section 3 we write the moment equations for density, velocity and energy, and perform their formal closure by means of the MEP. Then, the second part of the paper is focused on the case of graphene. In Section 4 the general theory exposed in the first part is specialized for the Dirac-like Hamiltonian describing electrons on a single-layer graphene sheet. In Section 5 we obtain the asymptotic form of the hydrodynamic equations derived in Section 4 in some physically relevant limits (namely, the high temperature, zero temperature, collimation and diffusive limits). Finally, Section 6 is devoted to conclusions and perspectives.

## 2 Phase-space description of spin-orbit particles

*α*is the Rashba constant.

*m*is the electron (bare) mass, \(E_{g}\) is the band-gap and

**K**is the matrix element of the gradient operator between conduction and valence Bloch functions.

We remark that the variable **p** has to be interpreted as the crystal pseudo-momentum, rather than the ordinary momentum. The interpretation of the vector variable σ depends on the cases: it is (proportional to) the spin vector in the case of Rashba Hamiltonian, while it is a pseudo-spin in the other two examples [2, 3]. For graphene, in particular, the pseudo-spin is related to the decomposition of the honeycomb lattice into two inequivalent sublattices, which reflects the presence of two carbon atoms in the fundamental cell of the lattice [3, 22].

- 1.the two energy bands($$ E_{\pm}(\mathbf {p}) = h_{0}(\mathbf {p}) \pm{\bigl\vert { \mathbf {h}(\mathbf {p})} \bigr\vert } $$(2)
*i.e.*, the eigenvalues of*H*with \(V=0\)); - 2.the projectors on the eigenspaces corresponding to \(E_{\pm}(\mathbf {p})\),where$$ P_{\pm}(\mathbf {p}) = \frac{1}{2} \bigl(\sigma_{0} \pm \boldsymbol {\nu }( \mathbf {p})\cdot \boldsymbol {\sigma }\bigr), $$(3)is the pseudo-momentum direction;$$ \boldsymbol {\nu }(\mathbf {p}) = \frac{\mathbf {h}(\mathbf {p})}{{\vert{ \mathbf {h}(\mathbf {p})} \vert}} $$(4)
- 3.the semiclassical velocities$$ \mathbf {v}_{\pm}(\mathbf {p}) = \nabla_{\mathbf {p}} E_{\pm}( \mathbf {p}); $$(5)
- 4.the effective-mass tensor [20]$$ \mathbb {M}_{\pm}^{-1}(\mathbf {p}) = \nabla_{\mathbf {p}} \otimes \mathbf {v}_{\pm}(\mathbf {p}) = \nabla_{\mathbf {p}}\otimes\nabla_{\mathbf {p}} E_{\pm}(\mathbf {p}). $$(6)

## 3 Maximum entropy closure

In order to obtain from (12) a closed system of equations for the moments (10), we assume that the system is in a state \(F^{ \mathrm {me}}\) of maximum entropy, according to the so-called *Maximum Entropy Principle (MEP)* [11–15] which in the present case reads as follows:

### MEP

\(F^{ \mathrm {me}}\) is the most probable microscopic state with the observed macroscopic moments \(n_{\pm}\), \(\mathbf {u}_{\pm}\) and \(e_{\pm}\).

**x**and

*t*), and, moreover,

*s*is given by (15) and \(f^{ \mathrm {me}}\) by (16). It is not difficult to show that

## 4 The case of graphene

*c*and direction ν, and the effective-mass tensor is

*i.e.*

*holes*. This is achieved by means of the substitution

*c*has been absorbed in the Lagrange multiplier

*C*). Moreover, we slightly change the definition of

**u**to be the average

*direction*

*c*. The inequality \({\vert{ \mathbf {u}} \vert} \leq1\) is an obvious consequence of the fact that

**u**is an average of directions.

*A*, \(\mathbf {B}= (B_{1},B_{2})\) and

*C*in terms of the moments

*n*, \(\mathbf {u}= (u_{1},u_{2})\) and

*e*, as resulting from the constraint equations

*T*has the physical meaning of the electron gas temperature. From (36) and the constraint equations (32), we obtain that

**B**has the same direction as

**u**and that \((n,{\vert{ \mathbf {u}} \vert},e)\) are related to the scalar Lagrange multipliers \((A,B,T)\) by

*n*,

**u**,

*T*and the functions \(\mathcal {I}_{N}^{s}= \mathcal {I}_{N}^{s}(A,B)\):

## 5 Asymptotic regimes

The expressions (38) of \(P_{ij}\), \(Q_{ij}\) and \(S_{j}\) are still not explicit, as functions of *n* and **u**. In fact, these expressions depend, through the functions \(\mathcal {I}_{N}^{s}(A,B)\), on the two scalar Lagrange multipliers *A* and *B*, which are related to *n* and \({\vert{ \mathbf {u}} \vert}\) via the relations (37). In Ref. [7] it has been proven that the correspondence between \((A,B)\) and \((n,{\vert{ \mathbf {u}} \vert})\) is 1-1 but, as far as we know, it is not possible to give an explicit, analytic, expression of the former as functions of the latter.

- 1.
the asymptotic region \(A^{2}+B^{2} \to\infty\) with \(A < -B\) (

*i.e.*\((A,B)\) below the ‘critical line’ \(A+B=0\)), corresponds to a regime of high temperatures, where the Fermi-Dirac distribution is well approximated by a Maxwell-Boltzmann distribution; - 2.
the asymptotic region \(A^{2}+B^{2} \to\infty\) with \(A > -B\) corresponds to the limit \(T\to0\), in which case we speak of ‘degenerate fermion gas’;

- 3.
the asymptotic region \(A^{2}+B^{2} \to\infty\) with \(A \sim B\) (

*i.e.*\((A,B)\) approaches the critical line \(A+B=0\)) corresponds to a ‘collimation regime’, \({\vert{ \mathbf {u}} \vert} \to1\), where the velocities of the electrons are all aligned along a (\((x, t)\)-dependent) direction in the**p**-space (the direction determined by**u**); there are two types of collimation, depending on whether the critical line is approached from below (Maxwell-Boltzmann collimation) or from above (degenerate gas collimation); - 4.
opposite to the collimation limit, the asymptotic region \(B \to0\) corresponds to the diffusive limit \({\vert{ \mathbf {u}} \vert} \to0\), where the velocities are randomly spread over all directions.

The asymptotic analysis of Eqs. (30) in these regimes is based on the following result, which has been proven in Ref. [7].

### Theorem

*The functions*\(\mathcal {I}_{N}^{s}\)

*have the following asymptotic behavior*:

- 1.
*in the Maxwell*-*Boltzmann limit*, \(A^{2}+B^{2} \to\infty\),*with*\(A < -B\),$$ \mathcal {I}_{N}^{s}(A,B) \sim \mathrm {e}^{A} I_{N}(B), $$(39)*where*\(I_{N}\)*are the modified Bessel functions of the first kind*; - 2.
*in the degenerate gas limit*, \(A^{2}+B^{2} \to\infty\),*with*\(A > -B\),$$ \mathcal {I}_{N}^{s}(A,B) \sim \frac{1}{\pi\Gamma(s+1)} \int_{0}^{\mathrm{C}(A,B)} \cos(N\theta) (A+B\cos \theta)^{s} \,d\theta, $$(40)*where*$$ \mathrm{C}(A,B) = \Re \biggl[\cos^{-1} \biggl(- {\frac{A}{B}} \biggr) \biggr] = \textstyle\begin{cases} \arccos (-{\frac{A}{B}} ),& \textit{if }-B < A < B, \\ \pi, &\textit{if }A \geq B. \end{cases} $$(41)

### 5.1 Maxwell-Boltzmann regime

*T*and corresponds to \(A^{2}+B^{2} \to\infty\) with \(A < -B\) in the \((A,B)\) half plane. In this case, we can use the approximation (39). Note, in particular, that in such limit the functions \(\mathcal {I}_{N}^{s}\) become factorized and independent on the index

*s*. Then, the constraint equations (37) become

*B*is given by (44).

*i.e.*the component of the force field parallel to the velocity field must vanish. In this case, the pseudo-momentum balance equation reduces to

### 5.2 Degenerate gas regime

- 1.\({\vert{ \mathbf {u}} \vert}\) only depends on
*ψ*and we can write$$ \psi= \biggl( \frac{\mathcal {F}_{1}^{2}}{\mathcal {F}_{0}^{2}} \biggr)^{-1}\bigl({\vert{ \mathbf {u}} \vert}\bigr); $$(54) - 2.
recalling (35), from the first of the above equations we have \(R \sim1/T\) and, then, the third equation shows that

*e*remains positive even though \(T\to0\).

*θ*denotes the Heaviside function and \(\psi(\mathbf {u})\) is given by (54). Using (38), (50) and (53) we obtain the following expressions of \(P_{ij}\) and \(Q_{ij}\) and \(S_{j}\) for a degenerate electron gas:

By using the techniques developed in Ref. [7], it is not difficult to calculate the asymptotic behavior of the functions \(Y({\vert{ \mathbf {u}} \vert})\), \(Z({\vert{ \mathbf {u}} \vert})\), \(Z_{\perp} ({\vert{ \mathbf {u}} \vert})\) and \(W({\vert{ \mathbf {u}} \vert})\) in the two limits \({\vert{ \mathbf {u}} \vert} \to0\) (diffusion) and \({\vert{ \mathbf {u}} \vert} \to1\) (collimation).

### 5.3 Collimation regime

The collimation limit corresponds to the absence of spread in the particle directions, *i.e.* to \({\vert{ \mathbf {u}} \vert} \to1\). It can be shown [7, 8] that this limit is equivalent to \(A^{2}+B^{2} \to\infty\) with \(A/B \to-1\). However, there is a completely different behavior when the critical line \(A = - B\) is approached from below (Maxwell-Boltzmann collimation) of from above (degenerate gas collimation).

**p**and

**u**, namely

*n*. As pointed out in Refs. [7, 8], this equation reveals that collimated electrons in graphene have the properties of a geometrical-optics system, with ‘refractive index’

*Q*.

### 5.4 Diffusion regime

We remark that, owing to the conical dispersion relation (29), the drift-diffusion equations (68), (69) and (70) have a ‘specular’ structure with respect to the drift-diffusion equations for Fermions with the usual parabolic dispersion relation [13, 24, 25]. Indeed, the diffusion coefficient (which is proportional to the variance of the velocity distribution), is here independent of the temperature *T*, because the particles move with constant speed *c*, while it is proportional to *T* in the parabolic case. On the other hand, the mobility coefficient (which is related to the distribution of the second derivative of the energy, *i.e.* to the effective-mass tensor) is here temperature-dependent while in the parabolic case is constant. Also the nonlinearity, which in the parabolic case affects the diffusive term, in Eqs. (68) and (70) is found in the drift term.

## 6 Conclusions

We have presented the systematic derivation from the Maximum Entropy Principle of hydrodynamic equations describing a population of electrons subject to spin-orbit interactions. In the second part of the paper we have treated more extensively the case of electrons on a single-layer graphene sheet.

The hydrodynamic equations have the form of a Euler-like system of conservation laws for density, *n*, momentum, **u**, and energy, *e*, in each of the two bands (the band indices are here omitted). Such system is of hyperbolic character, which ensures its (at least) local well-posedness. It is worth to remark that the full nonlinear structure of the MEP-state is retained, so that no assumptions of linear response or quasi-isotropic distribution are needed.

The system, in general, is not explicitly closed, *i.e.* no explicit constitutive relations, expressing the higher-order moments \(P_{ij}\), \(Q_{ij}\) and \(S_{j}\) as functions of *n*, **u** and *e*, can be given. However, in the case of graphene and for particular asymptotic regimes (namely, the limits of high and zero temperature, the limit of collimated directions and the diffusive limit), the closure is fully explicit.

As already mentioned in the Introduction, our results are not able to capture the physics of the system when the semiclassical approximation is not valid, that is when the quantum coherence becomes important. This typically happens in presence of rapid potential variations, such as potential steps or barriers. In these cases one expects the equations derived here to be a good approximation in a ‘semiclassical region’, far enough from the potential steps (constituting instead the ‘quantum region’). The semiclassical regions could be coupled to the quantum ones by means of quantum-classical interface conditions, analogous to those developed for standard, *i.e.* scalar and parabolic, particles (see Ref. [26] and references therein).

## Declarations

### Acknowledgements

This work has been partially supported by INdAM-GNFM, Progetto Giovani Ricercatori 2013 *Quantum fluid dynamics of identical particles: analytical and numerical study*.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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