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:

Schrödinger Equation (Time Dependent)	$\mathrm{i}\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
Laplace Equation For The Electric Potential	$-\mathrm{\nabla }\cdot \left({ϵ}_s\mathrm{\nabla }\varphi \right)=0$
Quantum Hamiltonian (Electric Charge)	$H=H_0+q{\varphi }$
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
Neumann Boundary Condition For Electric Potential	$n\cdot \mathrm{\nabla }\varphi (r,t){|}_{{\mathrm{\Gamma }}_N}=0$
Periodic Boundary Condition For Electric Potential	$\varphi (r,t)=\varphi (r+L,t)$

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:
Schrödinger Equation (Time Dependent)	$\mathrm{i}\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$
$\psi (t)$	symbol represents:
Schrödinger Equation (Time Dependent)	$\mathrm{i}\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$
$\hslash$	symbol represents:
Schrödinger Equation (Time Dependent)	$\mathrm{i}\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$
$t$	symbol represents:
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
$H$	symbol represents:
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
${\psi }_n$	symbol represents:
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
$E_n$	symbol represents:
Schrödinger Equation (Time Independent)	$\hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩$
$n$	symbol represents:
Laplace Equation For The Electric Potential	$-\mathrm{\nabla }\cdot \left({ϵ}_s\mathrm{\nabla }\varphi \right)=0$
$\varphi$	symbol represents:
Laplace Equation For The Electric Potential	$-\mathrm{\nabla }\cdot \left({ϵ}_s\mathrm{\nabla }\varphi \right)=0$
${ϵ}_s$	symbol represents:
Quantum Hamiltonian (Electric Charge)	$H=H_0+q{\varphi }$
${\varphi }$	symbol represents:
Quantum Hamiltonian (Electric Charge)	$H=H_0+q{\varphi }$
$q$	symbol represents:
Quantum Hamiltonian (Electric Charge)	$H=H_0+q{\varphi }$
$H_0$	symbol represents:
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
$\varphi$	symbol represents:
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
$U_k$	symbol represents:
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
$t$	symbol represents:
Dirichlet Boundary Condition For Electric Potential	$\varphi (r,t){|}_{{\mathrm{\Gamma }}_k}={\varphi }_0+U_k(t)$
${\mathrm{\Gamma }}_k$	symbol represents:
Neumann Boundary Condition For Electric Potential	$n\cdot \mathrm{\nabla }\varphi (r,t){|}_{{\mathrm{\Gamma }}_N}=0$
${\mathrm{\Gamma }}_N$	symbol represents:
Neumann Boundary Condition For Electric Potential	$n\cdot \mathrm{\nabla }\varphi (r,t){|}_{{\mathrm{\Gamma }}_N}=0$
$\varphi$	symbol represents:
Neumann Boundary Condition For Electric Potential	$n\cdot \mathrm{\nabla }\varphi (r,t){|}_{{\mathrm{\Gamma }}_N}=0$
$t$	symbol represents:
Periodic Boundary Condition For Electric Potential	$\varphi (r,t)=\varphi (r+L,t)$
$\varphi$	symbol represents:
Periodic Boundary Condition For Electric Potential	$\varphi (r,t)=\varphi (r+L,t)$
$L$	symbol represents:
Periodic Boundary Condition For Electric Potential	$\varphi (r,t)=\varphi (r+L,t)$
$t$	symbol represents:

Test quantity:: Quantum Hamiltonian Operator

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