Exploitation of the Maximum Entropy Principle in Mathematical Modeling of Charge Transport in Semiconductors

Mascali, Giovanni; Romano, Vittorio

doi:10.3390/e19010036

Cited by 30 publications

(17 citation statements)

References 57 publications

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“…Before going into details, we emphasize that the Kelvin formula is of course merely a convenient approximation and by no means exact. More accurate and microscopically better justified approaches to calculate the Seebeck coefficient are based on advanced kinetic models such as the semi-classical Boltzmann transport equation beyond the relaxation time approximation (retaining the full form of the collision operator [71,72]) or fully quantum mechanical methods [36,37,39,42].…”

Section: Kelvin Formula For the Seebeck Coefficientmentioning

confidence: 99%

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Kantner

2020

Journal of Computational Physics

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Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson-Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter-Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter-Gummel scheme for degenerate semiconductors (Fermi-Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

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Section: Kelvin Formula For the Seebeck Coefficientmentioning

confidence: 99%

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Kantner

2020

Journal of Computational Physics

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“…In the last two decades, the Maximum Entropy Principle (MEP) has been successfully employed to close this hierarchy of balance equations. Important results have also been obtained for the description of charge/thermal transport in devices made both of elemental and compound semiconductors, in cases where charge confinement is present and the carrier flow is two-or one-dimensional (see [31] for a review). By multiplying both sides of the MBTE equation 3by the weight functions…”

Section: Extended Hydrodynamic Modelmentioning

confidence: 99%

Low-field electron mobility evaluation in silicon nanowire transistors using an extended hydrodynamic model

et al. 2018

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Silicon nanowires (SiNWs) are quasi-one-dimensional structures in which electrons are spatially confined in two directions and they are free to move in the orthogonal direction. The subband decomposition and the electrostatic force field are obtained by solving the Schrödinger-Poisson coupled system. The electron transport along the free direction can be tackled using a hydrodynamic model, formulated by taking the moments of the multisubband Boltzmann equation. We shall introduce an extended hydrodynamic model where closure relations for the fluxes and production terms have been obtained by means of the Maximum Entropy Principle of Extended Thermodynamics, and in which the main scattering mechanisms such as those with phonons and surface roughness have been considered. By using this model, the low-field mobility of a Gate-All-Around SiNW transistor has been evaluated.

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“…The equilibrium density matrix can be obtained by employing a generalisation of the Maximum Entropy Principle (hereafter MEP) in a quantum context [ 10 , 17 , 18 ] (for the semiclassical case see [ 6 , 9 , 12 , 19 , 20 , 21 , 22 ]). According to the quantum version of MEP the equilibrium density matrix is obtained by maximising the quantum entropy

under suitable constraints on the expectation values.…”

Section: Equilibrium Density Functionmentioning

confidence: 99%

“…A starting point is the determination of the equilibrium Wigner function. It can be obtained by using the Jaynes approach [ 6 , 10 , 11 , 12 ] of maximizing the entropy under suitable constraints on the expectation values. A crucial issue is the expression of the entropy in the quantum case.…”

Section: Introductionmentioning

confidence: 99%

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

Camiola¹,

Luca²,

Romano³

2020

Entropy

Self Cite

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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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Exploitation of the Maximum Entropy Principle in Mathematical Modeling of Charge Transport in Semiconductors

Cited by 30 publications

References 57 publications

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Low-field electron mobility evaluation in silicon nanowire transistors using an extended hydrodynamic model

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

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