View source for Mathematical Model Collection Template

= Title: &quot;Model for Electric Potential for Gate Electrodes in a Quantum Bus&quot; =

Authors:

* family-names: Koprucki<br />
given-names: Thomas<br />
orcid: https://orcid.org/0000-0001-6235-9412
* family-names: Shehu<br />
given-names: Aurela<br />
orcid: https://orcid.org/0000-0002-1994-0612

Date-Released: 2024-04-05<br />
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 <math>\Phi(r, t)</math> obeys the homogeneous Poisson equation.

Properties: Is Deterministic, Is Space-Continous, Is Time-Continous, Is Linear
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== List of Mathematical Formulations ==

=== F1: Poisson's equation ===

Description: homogeneous Poisson's equation for electric potential<br />
Defining formulation:<br />
<math>-\nabla \cdot ( \varepsilon(\boldsymbol{r}) \nabla \Phi(\boldsymbol{r},t)) = 0</math>
{| class="wikitable"
! Symbol
! Quantity
! Quantity Id
! Quantity Kind
! Quant. Kind Id
! Description
|-
| <math>\Phi</math>
| -
| - 
| Electric Potential
| Q55451
| time-dependent profile of the electric potential in the quantum bus
|-
| <math>\varepsilon</math>
| -
| -
| Permittivity
| Q211569
| static dielectric permittivity of a material
|-
| <math>\boldsymbol{r}</math>
| -
| -
| Position
| Q192388
| position vector used for description of fields
|-
| <math>t</math>
| -
| -
| Time
| Q11471
| time
|}

=== F2: Permittivity law ===

Description: definition of static dielectric permittivity of a material by the relative permittivity<br />
DefiningFormulation:<br />
<math>\varepsilon(\boldsymbol{r}) = \varepsilon_0 \varepsilon_r(\boldsymbol{r})</math>
{| class="wikitable"
! Symbol
! Quantity
! Quantity Id
! Quantity Kind
! Quant. Kind Id
! Description
|-
| <math>\varepsilon_0</math>
| Vacuum Permittivity
| Q6158
| Permittivity
| Q211569
| absolute dielectric permittivity of classical vacuum
|-
| <math>\varepsilon_r</math>
| Relative Permittivity
| Q4027242
| Dimensionless quantity
| Q126818
| relative permittivity of a material 
|}

Relations to other Mathematical Formulations:<br />
F2 Contained as Definition In F1

=== F3: Boundary condition for electrode interfaces ===

Description: Dirichlet boundary conditions to apply gate voltages<br />
Defining formulation:<br />
<math>\Phi(\boldsymbol{r},t)|_{\Gamma_k} = U_k^{tot}(t)</math>
{| class="wikitable"
! Symbol
! Quantity
! Quantity Id
! Quantity Kind
! Quant. Kind Id
! Description
|-
| <math>\Gamma_k</math>
| Electrode interface
| TBD
| Physical Surface
| Q3783831
| Interface between gate electrode <math>k</math> and device
|-
| <math>U_k^{tot}</math>
| Gate Voltage
| TBD
| Voltage
| Q25428
| time-dependent applied voltage at gate electrode <math>k</math>
|-
| <math>k</math>
| Electrode Index 
| TBD 
| - 
| -
|Index of the set of electrodes 
|}

Relations to other Mathematical Formulations:<br />
F3 Contained as Boundary Condition In F1

=== F4: Boundary condition for artificial boundary ===

Description: homogenoues Neumann boundary conditions at artificial boundary<br />
Defining formulation:<br />
<math>\boldsymbol{n} \cdot \nabla \Phi(\boldsymbol{r},t)|_{\Gamma_N} = 0</math>
{| class="wikitable"
! Symbol
! Quantity
! Quantity Id
! Quantity Kind
! Quant. Kind Id
! Description
|-
| <math>\boldsymbol{n}</math>
| Normal vector 
| Q56353263
| Normal
| Q273176
| Normal to boundary surface
|-
| <math>\Gamma_N</math>
| Artificial Boundary
| TBD
| Physical Surface
| Q3783831
| Remaining artificial boundary
|}

Relations to other Mathematical Formulations:<br />
F4 Contained as Boundary Condition In F1
-->

= Computational Task CT1: Calculation of the electric potential =

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 <math>\varepsilon(\boldsymbol{r})</math>.<br />
Formulations: F1, F2, F3, F4<br />
Input: <math>U^{tot}_k</math>, k = 1:6, F2<br />
Output: <math>\Phi</math>

Relations between Mathematical Formulations and Computational Tasks:<br />
F2 Contained As Assumption In CT1.<br />
F3 Contained As Boundary Condition In CT1.<br />
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