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

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