Revision as of 13:49, 12 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

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 $Φ (r, t)$ 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:
$- \nabla \cdot (ε (r) \nabla Φ (r, t)) = 0$

Symbol	Quantity	Quantity Id	Quantity Kind	Quant. Kind Id	Description
$Φ$	-	-	Electric Potential	Q55451	time-dependent profile of the electric potential in the quantum bus
$ε$	-	-	Permittivity	Q211569	static dielectric permittivity of a material
$r$	-	-	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:
$ε (r) = ε_{0} ε_{r} (r)$

Symbol	Quantity	Quantity Id	Quantity Kind	Quant. Kind Id	Description
$ε_{0}$	Vacuum Permittivity	Q6158	Permittivity	Q211569	absolute dielectric permittivity of classical vacuum
$ε_{r}$	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:
$Φ (r, t) |_{Γ_{k}} = U_{k}^{t o t} (t)$

Symbol	Quantity	Quantity Id	Quantity Kind	Quant. Kind Id	Description
$Γ_{k}$	Electrode interface	TBD	Physical Surface	Q3783831	Interface between gate electrode $k$ and device	$U_{k}^{t o t}$	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:
$n \cdot \nabla Φ (r, t) |_{Γ_{N}} = 0$

Symbol	Quantity	Quantity Id	Quantity Kind	Quant. Kind Id	Description
$n$	Normal vector	Q56353263	Normal	Q273176	Normal to boundary surface	$Γ_{N}$	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 $ε (r)$ .
Formulations: F1, F2, F3, F4
Input: $U_{k}^{t o t}$ , k = 1:6, F2
Output: $Φ$

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.

@@ Line 14: / Line 14: @@
 = 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.
+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
@@ Line 25: / Line 25: @@
 Description: homogeneous Poisson's equation for electric potential<br />
 Defining formulation:<br />
-$-\nabla \cdot ( \varepsilon(\boldsymbol{r}) \nabla \Phi(\boldsymbol{r},t)) = 0$
+<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 | |:-------:|:--------:|:-----------:|:-------------:|:--------------:|:-----------:| | $\Phi$ | -| - | 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 |
+! 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 ===
@@ Line 33: / Line 68: @@
 Description: definition of static dielectric permittivity of a material by the relative permittivity<br />
 DefiningFormulation:<br />
-$\varepsilon(\boldsymbol{r}) = \varepsilon_0 \varepsilon_r(\boldsymbol{r})$
+<math>\varepsilon(\boldsymbol{r}) = \varepsilon_0 \varepsilon_r(\boldsymbol{r})</math>
+{| class="wikitable"
-| 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 |
+! 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 />
@@ Line 44: / Line 99: @@
 Description: Dirichlet boundary conditions to apply gate voltages<br />
 Defining formulation:<br />
-$\Phi(\boldsymbol{r},t)|_{\Gamma_k} = U_k^{tot}(t)$
+<math>\Phi(\boldsymbol{r},t)|_{\Gamma_k} = U_k^{tot}(t)</math>
+{| class="wikitable"
-| 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 |
+! 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 />
@@ Line 55: / Line 136: @@
 Description: homogenoues Neumann boundary conditions at artificial boundary<br />
 Defining formulation:<br />
-$\boldsymbol{n} \cdot \nabla \Phi(\boldsymbol{r},t)|_{\Gamma_N} = 0$
+<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 | |:-------:|:--------:|:-----------:|:-------------:|:--------------:|:-----------:| | $\boldsymbol{n}$ | Normal vector | Q56353263 | Normal | Q273176 | Normal to boundary surface | | $\Gamma_N$ | Artificial Boundary | TBD | Physical Surface | Q3783831 | Remaining artificial boundary |
+! 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 $\varepsilon(\boldsymbol{r})$.<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: $U^{tot}_k$, k = 1:6, F2<br />
+Input: <math>U^{tot}_k</math>, k = 1:6, F2<br />
-Output: $\Phi$
+Output: <math>\Phi</math>
--->
 Relations between Mathematical Formulations and Computational Tasks:<br />
 F2 Contained As Assumption In CT1.<br />

Mathematical Model Collection Template: Difference between revisions

Revision as of 13:49, 12 April 2024

Contents

Title: "Model for Electric Potential for Gate Electrodes in a Quantum Bus"

Mathematical Model MM1: Electron Shuttling Model

List of Mathematical Formulations

F1: Poisson's equation

F2: Permittivity law

F3: Boundary condition for electrode interfaces

F4: Boundary condition for artificial boundary

Computational Task CT1: Calculation of the electric potential

Publication

P1: WIAS-Preprint 3082

Relations between Mathematical Model and Publication:

Relations between Computational Task and Publication:

Research Field

RF1: Semiconductor Physics

Research Problem

RP1: Electrostatics in a Si/SiGe quantum bus