The magnetoelectric effect due to a semispherical capacitor surrounded by a spherical topologically insulating shell
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2020-07-13 |
| Journal | Physica Scripta |
| Authors | Daniel G. VelĂĄzquez, Luis F. Urrutia |
| Institutions | Universidad Nacional AutĂłnoma de MĂ©xico |
| Citations | 1 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive Summaryâ- Core Achievement: Theoretical demonstration of a detectable magnetoelectric effect (MEE) induced by a semispherical capacitor embedded within a spherical shell of a Topological Insulator (TI).
- Mechanism: An applied potential difference (2V) across the capacitor plates generates an electric field (E), which, due to the TIâs magnetoelectric polarizability ($\theta$), induces a measurable magnetic field (B).
- Field Magnitude: Numerical optimization identified configurations capable of generating magnetic fields up to 4.17 G at the external interface ($r_2 = 1$ ”m).
- Optimal Geometry: The configuration where the TI shell directly contacts the capacitor plates ($a = r_1$) and the shell is thin (e.g., $a = r_1 = 0.75$ ”m, $r_2 = 1$ ”m) maximizes the anisotropic magnetic field (0.60 G at $\theta = \pi/2$).
- Detectability (Field): The calculated magnetic field strengths (0.1 G to 4 G) are well within the sensitivity range of state-of-the-art Nitrogen-Vacancy (NV) center magnetometers (sensitivity > 10-2 G).
- Detectability (Flux): The calculated magnetic flux is approximately 10-9 Gcm2, which is comfortably detectable by modern scanning SQUID magnetometry devices (sensitivity > 10-14 Gcm2).
- Methodology: The analysis employed a first-order perturbative expansion in the effective magnetoelectric coupling parameter ($\tilde{\alpha}$), solving modified Maxwell equations using coupled electric and magnetic scalar potentials.
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| Applied Potential ($V$) | 3 | V | Potential used in numerical calculations (Total difference 2V). |
| TI Material (Simulated) | TlBiSe2 | N/A | Material choice for numerical estimates. |
| Relative Permittivity ($\epsilon_2$) | 4 | Dimensionless | Permittivity of the TI shell (Region 2). |
| Effective Coupling ($\tilde{\alpha}$) | $\approx 1/137$ | Dimensionless | Approximated as the fine structure constant ($\alpha$). |
| External Radius ($r_2$) | 1 | ”m | Fixed radius of the external TI interface for measurements. |
| Optimal Inner Radius ($a = r_1$) | 0.75 | ”m | Configuration maximizing anisotropic B field at $\theta = \pi/2$. |
| Maximum Induced B Field | 4.17 | G | Calculated magnitude at $r_2=1$ ”m, $\theta = \pi/2$, for $a=r_1=0.95$ ”m. |
| Isotropic B Field (Example) | 0.20 | G | Calculated magnitude at $a=r_1=0.50$ ”m, highly isotropic across $\theta$. |
| Calculated Magnetic Flux ($\Phi$) | 10-9 | Gcm2 | Average flux through a 10 ”m radius pickup loop. |
| SQUID Sensitivity (Flux) | 10-14 | Gcm2 | Current measurement capability of SQUID devices. |
| NV Center Sensitivity (Field) | 10-2 | G(Hz)-1/2 | Sensitivity threshold for diamond-based magnetometers. |
Key Methodologies
Section titled âKey Methodologiesâ- System Definition: The geometry consists of a semispherical capacitor (radius $a$) held at $\pm V$, surrounded by a thick spherical shell of a Topological Insulator (TI) defined between radii $r_1$ and $r_2$. Regions 1 and 3 (outside the TI) are treated as vacuum ($\epsilon_1 = 1, \theta_1 = 0$).
- Governing Equations: The electromagnetic response is governed by modified Maxwell equations incorporating the magnetoelectric coupling term $L_{\theta} \propto \theta \mathbf{E} \cdot \mathbf{B}$.
- Potential Formulation: Since the fields are static ($\nabla \cdot \mathbf{E} = 0, \nabla \cdot \mathbf{B} = 0$), the fields are derived from coupled electric ($\Phi$) and magnetic ($\Psi$) scalar potentials: $\mathbf{E} = -\nabla \Phi$ and $\mathbf{B} = -\nabla \Psi$. These potentials satisfy Laplaceâs equation in the bulk regions.
- Perturbative Solution: Due to the complexity of the full solution, a perturbative expansion in the effective coupling parameter $\tilde{\alpha} = (\theta_2 - \theta_1)\alpha/\pi$ was used. The solution retained the zeroth order for the electric potential and the first order for the magnetic potential, justified by $\tilde{\alpha}$ being of the order of the fine structure constant.
- Boundary Conditions: Twelve linear equations were established for the coefficients of the potentials (for every odd Legendre polynomial $l$), derived from boundary conditions at the interfaces ($r=r_1$ and $r=r_2$):
- Continuity of $\Phi$ and $\Psi$.
- Continuity of the tangential components of $\mathbf{E}$ and $\mathbf{H}$.
- Continuity of the normal components of $\mathbf{D}$ and $\mathbf{B}$.
- Specific conditions at the capacitor surface ($r=a$): $\Phi(a, \theta) = \pm V$ and the normal component of $\mathbf{B}$ is zero.
- Numerical Optimization: Calculations focused on the configuration where the TI shell touches the capacitor ($a=r_1$) and the magnetic field is measured at the external interface ($r=r_2$). Optimization was performed by varying $r_1$ to find the maximum magnetic field magnitude for different angular directions ($\theta$).
Commercial Applications
Section titled âCommercial Applicationsâ- Nanoscale Quantum Sensing: The ability to generate localized magnetic fields up to 4 G using only a small voltage across a micro-capacitor provides a crucial mechanism for operating and calibrating solid-state quantum sensors, particularly those based on NV centers in diamond.
- Topological Spintronics: This geometry offers a pathway for manufacturing active components in spintronic circuits, where the electric control of magnetic properties (MEE) can be utilized for energy-efficient data storage and logic operations.
- Micro-Transducers and Actuators: The system functions as a highly efficient electric-to-magnetic transducer at the microscale, potentially useful in miniaturized medical devices or micro-electromechanical systems (MEMS) requiring magnetic actuation.
- SQUID Calibration Standards: The calculated magnetic flux (10-9 Gcm2) provides a precise, theoretically derived standard for testing and validating the sensitivity and accuracy of scanning SQUID magnetometers.
- Composite Magnetoelectric Materials Research: The study validates the use of TI shells in composite structures to enhance magnetoelectric coupling, guiding the design of next-generation multiferroic materials with stronger coupling coefficients than monophasic TIs.
View Original Abstract
We consider the magnetoelectric effect produced by a capacitor formed by two semispherical perfectly conducting plates subjected to a potential difference and surrounded by a spherical shell of a topologically insulating material. The modified Maxwell equations are solved in terms of coupled electric and magnetic scalar potentials using spherical coordinates and in the approximation where the effective magnetoelectric coupling is of the order of the fine structure constant. The emphasis is placed in the calculation of the magnetic field for several relevant configurations designed to enhance the possibility of measuring this field. The magnitudes we obtain fall within the sensitivities of magnetometers based upon nitrogen-vacancy centers in diamond as well as of devices using scanning SQUID magnetometry.