The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections These quantum corrections are $O( {\hbar ^2 } )$ The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms These quantum terms allow particles to tunnel through potential barriers and to build up in potential wellsThe three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind methodSimulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v

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The quantum hydrodynamic model for semiconductor devices

Photon-assisted transport in semiconductor nanostructures

Traditional plasma physics has mainly focused on regimes characterized by high temperatures and low densities, for which quantum-mechanical effects have virtually no impact. However, recent technological advances (particularly on miniaturized semiconductor devices and nanoscale objects) have made it possible to envisage practical applications of plasma physics where the quantum nature of the particles plays a crucial role. Here, I shall review different approaches to the modeling of quantum effects in electrostatic collisionless plasmas. The full kinetic model is provided by the Wigner equation, which is the quantum analog of the Vlasov equation. The Wigner formalism is particularly attractive, as it recasts quantum mechanics in the familiar classical phase space, although this comes at the cost of dealing with negative distribution functions. Equivalently, the Wigner model can be expressed in terms of $N$ one-particle Schr{o}dinger equations, coupled by Poisson's equation: this is the Hartree formalism, which is related to the `multi-stream' approach of classical plasma physics. In order to reduce the complexity of the above approaches, it is possible to develop a quantum fluid model by taking velocity-space moments of the Wigner equation. Finally, certain regimes at large excitation energies can be described by semiclassical kinetic models (Vlasov-Poisson), provided that the initial ground-state equilibrium is treated quantum-mechanically. The above models are validated and compared both in the linear and nonlinear regimes.

How to model quantum plasmas

The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

Recent advances in Wigner function approaches

Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.

Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices

Quantum transport in the resonant-tunneling diode (RTD) is modeled here with the Wigner formalism including self-consistent potentials for the first time. We examine the computational aspects of the Wigner-function approach and the boundary conditions for the model. The calculated I-V characteristics show an intrinsic bistability in the negative-differential-conductivity region of the curve. Intrinsic bistability results from charge storage and the subsequent shifting of the internal potential of the device. The cathode region of the RTD shows a strong depletion and quantization of electrons in a deep triangular potential well, which reduces the barrier height to a ballistic electron injected from the cathode, enhancing the valley current and reducing the peak-to-valley ratio. Undoped spacer layers prevent the formation of a deep quantum well at the cathode barrier, and the distribution does not deplete as sharply as without the spacer layer. The I-V curve with the spacer layers shows a much lower negative resistance, and a sharper bistable region. A finite relaxation time for the electrons increases the negative resistance, reduces the peak-to-valley ratio of the current, and causes a ``soft'' hysteresis in the bistable region. A zero-bias anomaly is found to result from high-momentum tails in the distribution at the barrier interface. These high-momentum tails contribute a small high-conductance current. The transient current during switching from the peak to the valley of the I-V curve shows inductive behavior and negative resistance for frequencies below 2 THz.

Self-consistent study of the resonant-tunneling diode

Orthonormality and completeness relations are found for scattering states, the states used in treatments of resonant-tunneling double barriers and other short microstructures. Explicit normalized states are used to study the density matrix and Wigner distribution function of some simple but nontrivial structures. Analytic expressions describe the quantum repulsion of a sharp barrier at distances comparable to a de Broglie wavelength. As a function of distance from the step, the zero-current density approaches its constant-potential value asymptotically as a Gaussian. We also observe that, by virtue of the exact orthogonality of scattering states, a proposed correction to Esaki and Tsu's calculation of resonant tunneling current is rigorously zero.

Scattering states and distribution functions for microstructures.

Tunneling by an electron packet with an initially sharp wavefront

A quantum mechanical analysis is used to treat the transient behavior of the resonant-tunneling diode (RTD). The use of the Wigner formalism permits inclusion of the quantum mechanics inherent in the device, while offering a Boltzmann-like equation that is rather easily implemented. Self-consistent treatment of the potential introduces plasma oscillations in the distribution, which leads to the oscillatory current transient. Fourier analysis of this transient indicates that the RTD behaves inductively at frequencies under 2 THz, consistent with the ballistic nature of the carriers. At higher frequencies, the dominant mechanism is the capacitive charging and discharging of the quantum well, which leads to capacitive behavior of the device. The real part of the conductance is negative for frequencies under 1.5 THz, and positive for higher frequencies. The critical frequencies are shown to be independent of the relaxation time used to model dissipation, although the magnitude of the conductance decreases as the dissipation increases. >

A.M. Kriman

Papers

Self-consistent study of the resonant-tunneling diode

Scattering states and distribution functions for microstructures.

Tunneling by an electron packet with an initially sharp wavefront

Transient switching behavior of the resonant-tunneling diode

Wigner function simulation of quantum tunneling