Mathematical Model Template
Title: "Model for Electric Potential for Gate Electrodes in a Quantum Bus"
Authors:
- family-names: Koprucki
given-names: Thomas
orcid: https://orcid.org/0000-0001-6235-9412
- family-names: Shehu
given-names: Aurela
orcid: https://orcid.org/0000-0002-1994-0612
Date-Released: 2024-04-05
Version: 1.0.0
Mathematical Model MM1: Electron Shuttling Model
Description: The gate electrodes form an electric potential landscape that generates an array of QDs in the QW. Suitable pulsing allows to propagate the QDs along the channel and thus enables conveyor-mode shuttling. As the device is operated at deep cryogenic temperature (50 mK), there exist no thermally activated electrons in the conduction band and space charge regions can be safely neglected. In this case, the electric potential $\Phi(r, t)$ obeys the homogeneous Poisson equation.
Properties: Is Deterministic, Is Space-Continous, Is Time-Continous, Is Linear
List of Mathematical Formulations
F1: Poisson's equation
Description: homogeneous Poisson's equation for electric potential
Defining formulation:
| Symbol | Quantity | Quantity Id | Quantity Kind | Quant. Kind Id | Description |
|---|---|---|---|---|---|
| - | - | Electric Potential | Q55451 | time-dependent profile of the electric potential in the quantum bus |
| | -| - | | | | | $\varepsilon$ | - | - | Permittivity | Q211569 | static dielectric permittivity of a material | | $\boldsymbol{r}$ | - | - | Position | Q192388 | position vector used for description of fields | | $t$ | - | - | Time | Q11471 | time |
F2: Permittivity law
Description: definition of static dielectric permittivity of a material by the relative permittivity
DefiningFormulation:
$\varepsilon(\boldsymbol{r}) = \varepsilon_0 \varepsilon_r(\boldsymbol{r})$
| Symbol | Quantity | Quantity Id | Quantity Kind | Quant. Kind Id | Description | |:-------:|:--------:|:-----------:|:-------------:|:--------------:|:-----------:| | $\varepsilon_0$ | Vacuum Permittivity| Q6158 | Permittivity | Q211569 | absolute dielectric permittivity of classical vacuum | | $\varepsilon_r$ | Relative Permittivity | Q4027242 | Dimensionless quantity | Q126818 | relative permittivity of a material |
Relations to other Mathematical Formulations:
F2 Contained as Definition In F1
F3: Boundary condition for electrode interfaces
Description: Dirichlet boundary conditions to apply gate voltages
Defining formulation:
$\Phi(\boldsymbol{r},t)|_{\Gamma_k} = U_k^{tot}(t)$
| Symbol | Quantity | Quantity Id | Quantity Kind | Quant. Kind Id | Description | |:-------:|:--------:|:-----------:|:-------------:|:--------------:|:-----------:| | $\Gamma_k$ | Electrode interface | TBD | Physical Surface | Q3783831 | Interface between gate electrode $k$ and device | | $U_k^{tot}$ | Gate Voltage | TBD | Voltage | Q25428 | time-dependent applied voltage at gate electrode $k$ | | $k$ | Electrode Index | TBD | - | - | Index of the set of electrodes |
Relations to other Mathematical Formulations:
F3 Contained as Boundary Condition In F1
F4: Boundary condition for artificial boundary
Description: homogenoues Neumann boundary conditions at artificial boundary
Defining formulation:
$\boldsymbol{n} \cdot \nabla \Phi(\boldsymbol{r},t)|_{\Gamma_N} = 0$
| Symbol | Quantity | Quantity Id | Quantity Kind | Quant. Kind Id | Description | |:-------:|:--------:|:-----------:|:-------------:|:--------------:|:-----------:| | $\boldsymbol{n}$ | Normal vector | Q56353263 | Normal | Q273176 | Normal to boundary surface | | $\Gamma_N$ | Artificial Boundary | TBD | Physical Surface | Q3783831 | Remaining artificial boundary |
Relations to other Mathematical Formulations:
F4 Contained as Boundary Condition In F1
Computational Task CT1: Calculation of the electric potential
Description:
For a given set of gate voltages entering the boundary condition F3, solve the Poisson equation F1 with the material law F2 together with the boundary conditions F3 and F4. The device structure enters the material law F2 by the spatial profile of the relative permittivity $\varepsilon(\boldsymbol{r})$.
Formulations: F1, F2, F3, F4
Input: $U^{tot}_k$, k = 1:6, F2
Output: $\Phi$
Relations between Mathematical Formulations and Computational Tasks:
F2 Contained As Assumption In CT1.
F3 Contained As Boundary Condition In CT1.
F4 Contained As Boundary Condition In CT1.
Publication
P1: WIAS-Preprint 3082
DOI: 10.20347/WIAS.PREPRINT.3082
Relations between Mathematical Model and Publication:
MM1 Used In P1
Relations between Computational Task and Publication:
CT1 Documented In P1
Research Field
RF1: Semiconductor Physics
WikiData: Q4483523
Research Problem
RP1: Electrostatics in a Si/SiGe quantum bus
Description: Simulation of the electrostatics in a Si/SiGe quantum bus
