Model for Electric Potential for Gate Electrodes in a Quantum Bus (1): Difference between revisions
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= Mathematical Model MM1: Electron Shuttling Model = | = Mathematical Model MM1: Electron Shuttling Model = | ||
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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 | 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 <math>\Phi(r, t)</math> obeys the homogeneous Poisson equation. | ||
potential | |||
Properties: Is Deterministic, Is Space-Continous, Is Time-Continous, Is Linear | Properties: Is Deterministic, Is Space-Continous, Is Time-Continous, Is Linear | ||
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Description: homogeneous Poisson's equation for electric potential<br /> | Description: homogeneous Poisson's equation for electric potential<br /> | ||
Defining formulation:<br /> | Defining formulation:<br /> | ||
<math>-\nabla \cdot ( \varepsilon(\boldsymbol{r}) \nabla \Phi(\boldsymbol{r},t)) = 0</math> | |||
{| class="wikitable" | {| class="wikitable" | ||
! Symbol | ! Symbol | ||
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! Description | ! Description | ||
|- | |- | ||
| | | <math>\Phi</math> | ||
| - | | - | ||
| - | | - | ||
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| time-dependent profile of the electric potential in the quantum bus | | time-dependent profile of the electric potential in the quantum bus | ||
|- | |- | ||
| | | <math>\varepsilon</math> | ||
| - | | - | ||
| - | | - | ||
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| static dielectric permittivity of a material | | static dielectric permittivity of a material | ||
|- | |- | ||
| | | <math>\boldsymbol{r}</math> | ||
| - | | - | ||
| - | | - | ||
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| position vector used for description of fields | | position vector used for description of fields | ||
|- | |- | ||
| | | <math>t</math> | ||
| - | | - | ||
| - | | - | ||
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Description: definition of static dielectric permittivity of a material by the relative permittivity<br /> | Description: definition of static dielectric permittivity of a material by the relative permittivity<br /> | ||
DefiningFormulation:<br /> | DefiningFormulation:<br /> | ||
<math>\varepsilon(\boldsymbol{r}) = \varepsilon_0 \varepsilon_r(\boldsymbol{r})</math> | |||
{| class="wikitable" | {| class="wikitable" | ||
! Symbol | ! Symbol | ||
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! Description | ! Description | ||
|- | |- | ||
| | | <math>\varepsilon_0</math> | ||
| Vacuum Permittivity | | Vacuum Permittivity | ||
| Q6158 | | Q6158 | ||
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| absolute dielectric permittivity of classical vacuum | | absolute dielectric permittivity of classical vacuum | ||
|- | |- | ||
| | | <math>\varepsilon_r</math> | ||
| Relative Permittivity | | Relative Permittivity | ||
| Q4027242 | | Q4027242 | ||
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Description: Dirichlet boundary conditions to apply gate voltages<br /> | Description: Dirichlet boundary conditions to apply gate voltages<br /> | ||
Defining formulation:<br /> | Defining formulation:<br /> | ||
<math>\Phi(\boldsymbol{r},t)|_{\Gamma_k} = U_k^{tot}(t)</math> | |||
{| class="wikitable" | {| class="wikitable" | ||
! Symbol | ! Symbol | ||
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! Description | ! Description | ||
|- | |- | ||
| | | <math>\Gamma_k</math> | ||
| Electrode interface | | Electrode interface | ||
| TBD | | TBD | ||
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| Interface between gate electrode <math>k</math> and device | | Interface between gate electrode <math>k</math> and device | ||
|- | |- | ||
| | | <math>U_k^{tot}</math> | ||
| Gate Voltage | | Gate Voltage | ||
| TBD | | TBD | ||
| Voltage | | Voltage | ||
| Q25428 | | Q25428 | ||
| time-dependent applied voltage at gate electrode | | time-dependent applied voltage at gate electrode <math>k</math> | ||
|- | |- | ||
| | | <math>k</math> | ||
| Electrode Index | | Electrode Index | ||
| TBD | | TBD | ||
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Description: homogenoues Neumann boundary conditions at artificial boundary<br /> | Description: homogenoues Neumann boundary conditions at artificial boundary<br /> | ||
Defining formulation:<br /> | Defining formulation:<br /> | ||
<math>\boldsymbol{n} \cdot \nabla \Phi(\boldsymbol{r},t)|_{\Gamma_N} = 0</math> | |||
{| class="wikitable" | {| class="wikitable" | ||
! Symbol | ! Symbol | ||
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! Description | ! Description | ||
|- | |- | ||
| | | <math>\boldsymbol{n}</math> | ||
| Normal vector | | Normal vector | ||
| Q56353263 | | Q56353263 | ||
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| Normal to boundary surface | | Normal to boundary surface | ||
|- | |- | ||
| | | <math>\Gamma_N</math> | ||
| Artificial Boundary | | Artificial Boundary | ||
| TBD | | TBD | ||
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Description:<br /> | Description:<br /> | ||
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 | 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 <math>\varepsilon(\boldsymbol{r})</math>.<br /> | ||
Formulations: F1, F2, F3, F4<br /> | Formulations: F1, F2, F3, F4<br /> | ||
Input: | Input: <math>U^{tot}_k</math>, k = 1:6, F2<br /> | ||
Output: | Output: <math>\Phi</math> | ||
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Relations between Mathematical Formulations and Computational Tasks:<br /> | Relations between Mathematical Formulations and Computational Tasks:<br /> | ||
F2 Contained As Assumption In CT1.<br /> | F2 Contained As Assumption In CT1.<br /> | ||
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DOI: 10.20347/WIAS.PREPRINT.3082 | DOI: 10.20347/WIAS.PREPRINT.3082 | ||
== Relations between Mathematical Model and Publication: == | == Relations between Mathematical Model and Publication: == | ||
Revision as of 10:28, 15 April 2024
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
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
