Electron Shuttling Model

quantum dynamical model of an electron to be shuttled in a silicon quBus device

The clavier gate electrodes on the top surface generate a moving array of QD potentials

Top view on the Si-QuBus with the four different clavier gate sets highlighted in color

Quantum dynamical modeling of an electron to be shuttled, governed by the electric potential generated by the clavier (and other) gates in a Silicon QuBus device. Spin and valley states as well as the respective interactions are neglected. Moreover, the current model is limited to the coherent wave packet evolution and disregards the effects of noise and dissipation.

The Electron Shuttling Model contains the following mathematical expressions with quantities:

Schrödinger Equation (Time Dependent)	$\mathrm{i}\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$
$H$ symbol represents Quantum Hamiltonian Operator
$\hslash$ symbol represents Planck Constant
$\psi (t)$ symbol represents Quantum State Vector (Dynamic)
$t$ symbol represents Time
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
$E_n$ symbol represents Quantum Eigen Energy
$H$ symbol represents Quantum Hamiltonian Operator
${\psi }_n$ symbol represents Quantum State Vector (Stationary)
$n$ symbol represents Quantum Number
Laplace Equation For The Electric Potential	$-\mathrm{\nabla }\cdot \left({ϵ}_s\mathrm{\nabla }\varphi \right)=0$
${ϵ}_s$ symbol represents Permittivity (Dielectric)
$\varphi$ symbol represents Electric Potential
Quantum Hamiltonian (Electric Charge)	$H=H_0+q{\varphi }$
$H_0$ symbol represents Quantum Hamiltonian Operator
${\varphi }$ symbol represents Electric Potential
$q$ symbol represents Electric Charge
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
$U_k$ symbol represents Applied External Voltage
${\mathrm{\Gamma }}_k$ symbol represents Electrode Interfaces
$\varphi$ symbol represents Electric Potential
$t$ symbol represents Time
Neumann Boundary Condition For Electric Potential	$n\cdot \mathrm{\nabla }\varphi (r,t){|}_{{\mathrm{\Gamma }}_N}=0$
${\mathrm{\Gamma }}_N$ symbol represents Electrode Interfaces
$\varphi$ symbol represents Electric Potential
$t$ symbol represents Time
Periodic Boundary Condition For Electric Potential	$\varphi (r,t)=\varphi (r+L,t)$
$L$ symbol represents Length Of Unit Cell
$\varphi$ symbol represents Electric Potential
$t$ symbol represents Time

The Electron Shuttling Model is applied by the following computational tasks:

Optimal Control

Quantum Time Evolution

Quantum Stationary States

Semiconductor Charge Neutrality