Quantum Monte Carlo simulation of dissipative transport using Bohmian trajectories

Abolfazl Abedi; Mohammad Javad Sharifi

doi:10.1007/s10825-017-1117-1

Journal of Computational Electronics

March 2018, Volume 17, Issue 1, pp 68–75 | Cite as

Quantum Monte Carlo simulation of dissipative transport using Bohmian trajectories

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Abolfazl Abedi
Mohammad Javad SharifiEmail author

Article

First Online: 14 December 2017

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Abstract

In this paper, a Monte Carlo method is proposed, which utilizes Bohmian trajectories to simulate dissipative transport in one-dimensional quantum devices. The proposed method, similar to the classical Monte Carlo method, is capable of simulating both elastic and inelastic scattering effects, with the distinction that quantum effects such as tunneling are also included. At first, the Bohmian trajectories for the wave packets injected from the right and the left contacts are obtained by solving the time-dependent Schrodinger equation, and then scattering effects are included via stochastic changes applied on the electron trajectories. We have shown that the results of the proposed model agree well with those of NEGF formalism.

Keywords

Dissipative transport Bohmian trajectories Monte Carlo method Quantum transport Elastic and inelastic scatterings

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Appendix

In classical MC approach for solving the Boltzmann transport equation, the local density of states is calculated by the distance from the conduction band as:

$$\begin{aligned} \mathrm{LDOS}\left( {x,E} \right) =\frac{\sqrt{2m_{\mathrm{eff}} }}{\pi \hslash }\frac{1}{\sqrt{E-E_c \left( x \right) }} \end{aligned}$$

(A1)

Equation (A1) works well when the electrostatic potential does not vary too rapidly [21], but it is a semi-classical model and cannot be used for the classical forbidden regions. Here, we propose another approach for evaluating LDOS from Bohmian trajectories, which can be used for quantum systems.

The local density of states at energy E for a quantum phase coherent system can be described

as [21]:

$$\begin{aligned} \mathrm{LDOS}\left( {x,E} \right) =\mathrm{LDOS}_L \left( {x,E} \right) +\mathrm{LDOS}_R \left( {x,E} \right) \end{aligned}$$

(A2)

where $\mathrm{LDOS}_{L,R} $ is the local density of states due to the incident electron waves from the left and the right leads. If we combine Eqs. A1 and A2 and describe in term of velocity instead of energy we have:

$$\begin{aligned} \mathrm{LDOS}\left( {x,E} \right) =\frac{1}{\pi \hslash }\left( {1/{V^{+}}+1/{V^{-}}} \right) \end{aligned}$$

(A3)

$V^{+}$ is due to electrons coming from the left and $V^{-}$ is due to electrons coming from the right. We can estimate these two contributions from Bohmian trajectories as:

$$\begin{aligned} \mathrm{LDOS}_{L,R} \left( {x,E} \right) =\frac{1}{\pi \hslash }\frac{\left( {\varDelta t_{L,R} /N} \right) }{\varDelta x} \end{aligned}$$

(A4)

where ${\Delta }t_{L,R} $ is the total time spent by right-going/left-going (related to wave packets injected from right or left contact) Bohmian trajectories at the interval x to $x+\varDelta x$, and N is the total number of Bohmian trajectories chosen proportional to the density of the squared Schrödinger wave function, $\left| \psi \right| ^{2}$. Then the initial electron density in the channel of the device versus position can be evaluated as:

$$\begin{aligned} n\left( x \right)= & {} \smallint \mathrm{d}E[ \mathrm{LDOS}_L \left( {x,E} \right) f_L \left( E \right) \nonumber \\&+\,\mathrm{LDOS}_R \left( {x,E} \right) f_R \left( E \right) ] \end{aligned}$$

(A5)

where $f_{L,R} $ is the Fermi function for the left and right contacts. The local density of state evaluated from the Bohmian trajectories regarding the potential shown in Fig. 4a is depicted in Fig. 6 (simulation parameters are same as Fig. 4). It is obvious from Fig. 6 that the evaluated LDOS from Bohmian trajectories, unlike the semi-classical picture, is nonzero for the regions under the conduction band. It is also seen that the oscillations of Bohmian LDOS is larger near the tunneling barrier and becomes smother far from the barrier.

Fig. 6
LDOS evaluating from Bohmian trajectories

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Abolfazl Abedi
- 1
Mohammad Javad Sharifi
- 1
Email author

1.Department of Electrical EngineeringShahid Beheshti UniversityTehranIran

Quantum Monte Carlo simulation of dissipative transport using Bohmian trajectories

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