Quantum corrected hydrodynamic models for charge transport in graphene

Luca, Liliana; Romano, Vittorio

doi:10.1016/j.aop.2019.03.018

Cited by 28 publications

(28 citation statements)

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“…The equilibrium Wigner function can be used into the relaxation time approximation of the collision term in the Wigner–Boltzmann equation or to introduce quantum corrections in the description of quantum fluids, e.g., to existing drift-diffusion, energy-transport and hydrodynamical models for charge transport. A possible application could be a generalization of the model for electron transport in graphene presented in [ 27 ]. We mention also that in view of the enhancement of the importance of the thermal effects in nanoscale devices, the developed formalism can be also adopted for a quantum description of phonon transport based on the Wigner equation.…”

Section: Discussionmentioning

confidence: 99%

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

Camiola¹,

Luca²,

Romano³

2020

Entropy

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The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.

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Section: Discussionmentioning

confidence: 99%

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

Camiola¹,

Luca²,

Romano³

2020

Entropy

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“…Further developments could include the effects of crystal heating and quantum effects in the mobility models. Although the subject is still in an early stage, attempts in this direction can be found in [10,11] regarding the thermal influence on the electric performance, and in [12][13][14][15] regarding the inclusion of quantum corrections.…”

Section: Discussionmentioning

confidence: 99%

Improved mobility models for charge transport in graphene

Nastasi

Romano

2019

Communications in Applied and Industrial Mathematics

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Charge transport in graphene is crucial for the design of a new generation of nanoscale electron devices. A reasonable model is represented by the semiclassical Boltzmann equations for electrons in the valence and conduction bands. As shown by Romano et al. (J. Comput. Phys., 2015), the discontinuous Galerkin methods are a viable way to tackle the problem of the numerical integration of these equations, even if efficient DSMC with a proper inclusion of the Pauli principle have been also devised. One of the advantages of the solutions obtained with deterministic approach is of course the absence of statistical noise. This fact is crucial for an accurate estimation of the low field mobility as proved by Majorana et al. (J. Math. Industry, 2016) in the case of a unipolar charge transport in a suspended graphene sheet under a constant electric field. The mobility expressions are essential for the drift-diffusion equations which constitute the most adopted models for charge transport in CAD. Here the analysis by Majorana et al. (J. Math. Industry, 2016) is improved in two ways: by including the charge transport both in the valence and conduction bands; by taking into account the presence of an oxide as substrate for the graphene sheet. New models of mobility are obtained and, in particular, relevant improvements of the low field mobility are achieved.

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“…As it is shown in Ref. [4], the four equations (21) are coupled at ξ = 0 by the following nonhomogeneous version of the KTC:…”

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“…We now need to relate the asymptotic densities n i,∞ s to the quantities X ss . In order to do this, let us consider any function θ i s that satisfies the half-space equation (21). Integrating in p yields…”

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confidence: 99%

“…However, as remarked above, since the aim of this paper is mainly to illustrate the mathematical method of quantum transmission conditions, rather than to simulate a particular device, we preferred to use a simple model for the bulk transport and to focus on the treatment of the quantum interface. Of course, a more detailed description of the electron scattering [11,20,23], more refined expressions for the mobility [24,26], a self-consistent potential model [25,26] or even quantum drift-diffusion equations [21,29,32], could be used to improve the model (see also Ref. [7] for a general reference).…”

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Mathematical modelling of charge transport in graphene heterojunctions

Barletti¹,

Nastasi²,

Negulescu³

et al. 2021

KRM

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A typical graphene heterojunction device can be divided into two classical zones, where the transport is basically diffusive, separated by a "quantum active region" (e.g., a locally gated region), where the charge carriers are scattered according to the laws of quantum mechanics. In this paper we derive a mathematical model of such a device, where the classical regions are described by drift-diffusion equations and the quantum zone is seen as an interface where suitable transmission conditions are imposed that take into account the quantum scattering process. Numerical simulations show good agreement with experimental data.

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Quantum corrected hydrodynamic models for charge transport in graphene

Cited by 28 publications

References 32 publications

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

Improved mobility models for charge transport in graphene

Mathematical modelling of charge transport in graphene heterojunctions

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