Zhen Zhan

Quantum dissipation with conditional wave functions: Application to the realistic simulation of nanoscale electron devices

Without access to the full quantum state, modelling dissipation in an open system requires approximations. The physical soundness of such approximations relies on using realistic microscopic models of dissipation that satisfy completely positive dynamical maps. Here we present an approach based on the use of the Bohmian conditional wave function that, by construction, ensures a completely positive dynamical map for either Markovian or non-Markovian scenarios, while allowing the implementation of realistic dissipation sources. Our approach is applied to compute the current-voltage characteristic of a resonant tunnelling device with a parabolic-band structure, including electron-lattice interactions. A stochastic Schrodinger equation is solved for the conditional wave function of each simulated electron. We also extend our approach to (graphene-like) materials with a linear band-structure using Bohmian conditional spinors for a stochastic Dirac equation.

Time-dependent simulation of particle and displacement currents in THz graphene transistors

Although time-independent models provide very useful dynamical information with a reduced computational burden, going beyond the quasi-static approximation provides enriched information when dealing with terahertz (THz) frequencies. In this work, the THz noise of dual-gate graphene transistors with DC polarization is analyzed from a careful simulation of the time-dependent particle and displacement currents. From such currents, the power spectral density (PSD) of the total current fluctuations are computed at the source, drain and gate contacts. The role of the lateral dimensions of the transistors, the Klein tunneling and the positive–negative energy injection on the PSD are analyzed. Through the comparison of the PSD with and without band-to-band tunneling and graphene injection, it is shown that the unavoidable Klein tunneling and positive–negative energy injection in graphene structures imply an increment of noise without similar increment on the current, degrading the (either low or high frequency) signal-to-noise ratio. Finally, it is shown that the shorter the vertical height (in comparison with the length of the active region in the transport direction), the larger the maximum frequency of the PSD. As a byproduct of this result, an alternative strategy (without length scaling) to optimize the intrinsic cut-off frequency of graphene transistors is envisioned.

Dissipative quantum transport using one-particle time-dependent (conditional) wave functions

An effective single-particle Schrodinger equation to include dissipation into quantum devices is presented. This effective equation is fully understood in the context of Bohmian mechanics, a theory of particles and waves, where it is possible to define unambiguously the wave function of a subsystem, the so-called conditional wave function. In particular the change in energy and momentum of an electron when interacting with a phonon is presented, both theoretically and numerically. This work is a first step to include dissipation into the fully-quantum simulator BITLLES.

Unphysical features in the application of the Boltzmann collision operator in the time-dependent modeling of quantum transport

In this work, the use of the Boltzmann collision operator for dissipative quantum transport is analyzed. Its mathematical role on the description of the time-evolution of the density matrix during a collision can be understood as processes of adding and subtracting states. We show that unphysical results can be present in quantum simulations when the old states (that built the density matrix associated to an open system before the collision) are different from the additional states generated by the Boltzmann collision operator. As a consequence of the Fermi Golden rule, the new generated sates are usually eigenstates of the momentum or kinetic energy. Then, negative values of the charge density may appear after the collision. This fact is originated by the different time-evolutions of the old and new states. This unphysical feature disappears when the Boltzmann collision operator generates states that were already present in the density matrix of the quantum system before the collision. Following these ideas, in this paper, we introduce an algorithm that models phonon–electron interactions through the Boltzmann collision operator without negative values of the charge density. The model only requires the exact knowledge, at all times, of the states that build the density matrix of the open system.

Comparing Wigner, Husimi and Bohmian distributions: which one is a true probability distribution in phase space?

The Wigner distribution function is a quasi-probability distribution. When properly integrated, it provides the correct charge and current densities, but it gives negative probabilities in some points and regions of the phase space. Alternatively, the Husimi distribution function is positive-defined everywhere, but it does not provide the correct charge and current densities. The origin of all these difficulties is the attempt to construct a phase space within a quantum theory that does not allow well-defined (i.e. simultaneous) values of the position and momentum of an electron. In contrast, within the (de Broglie–Bohm) Bohmian theory of quantum mechanics, an electron has well-defined position and momentum. Therefore, such theory provides a natural definition of the phase space probability distribution and by construction, it is positive-defined and it exactly reproduces the charge and current densities. The Bohmian distribution function has many potentialities for quantum problems, in general, and for quantum transport, in particular, that remains unexplored.

Absorption and Injection Models for Open Time-Dependent Quantum Systems

In the time-dependent simulation of pure states dealing with transport in open quantum systems, the initial state is located outside of the active region of interest. Using the superposition principle and the analytical knowledge of the free time evolution of such a state outside the active region, together with absorbing layers and remapping, a model for a very significant reduction of the computational burden associated with the numerical simulation of open time-dependent quantum systems is presented. The model is specially suited to study (many-particle and high-frequency effects) quantum transport, but it can also be applied to any other research field where the initial time-dependent pure state is located outside of the active region. From numerical simulations of open quantum systems described by the (effective mass) Schrödinger and (atomistic) tight-binding equations, a reduction of the computational burden of about two orders of magnitude for each spatial dimension of the domain with a negligible error is presented.

Implications of the Klein tunneling times on high frequency graphene devices using Bohmian trajectories

Because of its large Fermi velocity, leading to a great mobility, graphene is expected to play an important role in (small signal) radio frequency electronics. Among other, graphene devices based on Klein tunneling phenomena are already envisioned. The connection between the Klein tunneling times of electrons and cut-off frequencies of graphene devices is not obvious. We argue in this paper that the trajectory-based Bohmian approach gives a very natural framework to quantify Klein tunneling times in linear band graphene devices because of its ability to distinguish, not only between transmitted and reflected electrons, but also between reflected electrons that spend time in the barrier and those that do not. Without such distinction, typical expressions found in the literature to compute dwell times can give unphysical results when applied to predict cut-off frequencies. In particular, we study Klein tunneling times for electrons in a two-terminal graphene device constituted by a potential barrier between two metallic contacts. We show that for a zero incident angle (and positive or negative kinetic energy), the transmission coefficient is equal to one, and the dwell time is roughly equal to the barrier distance divided by the Fermi velocity. For electrons incident with a non-zero angle smaller than the critical angle, the transmission coefficient decreases and dwell time can still be easily predicted in the Bohmian framework. The main conclusion of this work is that, contrary to tunneling devices with parabolic bands, the high graphene mobility is roughly independent of the presence of Klein tunneling phenomena in the active device region.

The relationship of the field effect transistor geometry and its cutoff frequency

In this work we tackle two different issues. We first study the effects of the size of the cross section of the channel of silicon-based Gate-All-Around Field Effect Transistors (GAAFETs) on their performance. By computing the admittance parameters, |Y12| and |Y11|, and the hybrid parameter |H21|, we demonstrate a strategy to improve the intrinsic cutoff frequency of GAAFETs without reducing their channel length nor employing any novel high-electron-mobility material. Second, we discuss the intrinsic cutoff frequency of a graphene-based Dual Gate Field Effect Transistor (DGFET). We compare numerical results for the admittance parameters computed using its exact expressions and by means of two different approximations, i.e. the Quasi-Static (QS) and Non-Quasi-Static (NQS). The exact admittance parameters are obtained through the total time-dependent currents computed by means of the self-consistent Monte Carlo solution of the Boltzmann and Poisson equations, which are implemented in the BITLLES simulator.

The shortest simulation-box for time-dependent computation of wave packets in open system

A novel algorithm for a reduction of the computational burden associated to the time-dependent simulation of quantum transport with pure states is presented. The algorithm is based on using the superposition principle and the analytical knowledge of the free time-evolution of an initial state outside of the active region, together with absorbing layers. It is specially suited to study (many-particle and high-frequency effects) quantum transport, but it can also be applied to any other research field where the initial time-dependent pure state is located outside of the active region. Numerical results for a 1D system with the shortest simulation box imply a reduction of the computational burden of more than one order of magnitude with a negligible error.