# Thermoelectric Properties of the Corbino Disk in Graphene

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## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

#### 2.1. Scattering of Dirac Fermions

#### 2.2. Thermoelectric Characteristics

## 3. Results and Discussion

#### 3.1. Zero-Temperature Conductance

#### 3.2. Thermopower and the Lorentz Number

#### 3.3. Smooth Potential Barriers

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Incoherent Transport at the Magnetic Field

**Figure A1.**(

**a**) Propagation between subsequent scatterings on interfaces at $r={R}_{\mathrm{i}}$ and $r={R}_{\mathrm{o}}$ (with incident angles ${\theta}_{1}$ and ${\theta}_{2}$, respectively) in a uniform magnetic field, defining the cyclotron orbit centred at $r={r}_{x}$ with its radii ${r}_{c}$. (

**b**) Incoherent conductance calculated from Equation (A1) for $B=0$ (dashed line) and $B=0.5\phantom{\rule{0.166667em}{0ex}}$ T (solid line). The disk radii are ${R}_{\mathrm{o}}=2{R}_{\mathrm{i}}=1000\phantom{\rule{0.166667em}{0ex}}$ nm; a rectangular potential barrier is considered. Characteristic Fermi energies ${E}_{c,1}$ and ${E}_{c,2}$ marked with vertical lines correspond to ${r}_{c}=({R}_{\mathrm{o}}-{R}_{\mathrm{i}})/2$ and ${r}_{c}=({R}_{\mathrm{i}}+{R}_{\mathrm{o}})/2$, respectively.

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**Figure 1.**(

**a**) The Corbino setup in graphene. Voltage source passes the current between the circular leads (yellow areas) via a disk-shaped sample with inner radii ${R}_{\mathrm{i}}$ and outer radii ${R}_{\mathrm{o}}$. The uniform magnetic field $\mathbf{B}=(0,0,B)$ is perpendicular to the sample. Additionally, the gate electrode (not shown) tunes the doping in the disk area. (

**b**–

**d**) Transport regimes for different fields and carrier concentrations ${n}_{E}$. At high fields, if doping is adjusted to a Landau level ($E={E}_{n\mathrm{LL}}$, $n=0,\phantom{\rule{3.33333pt}{0ex}}\pm 1,\phantom{\rule{3.33333pt}{0ex}}\dots $) resonance occurs (

**b**). At low field but high doping (such that the cyclotron diameter $2{r}_{c}>{R}_{\mathrm{o}}-{R}_{\mathrm{i}}$), incoherent scattering along the classical trajectory governs the transport (

**c**).

**Figure 2.**(

**a**,

**b**) Zero-temperature conductance, (

**c**) the Seebeck coefficient and (

**d**) the Lorentz number, both for $T=5\phantom{\rule{0.166667em}{0ex}}$ K, for the system of Figure 1 with ${R}_{\mathrm{o}}=2{R}_{\mathrm{i}}=1000\phantom{\rule{0.166667em}{0ex}}$ nm and the rectangular potential barrier (${V}_{0},m\to \infty $; see Equation (3)) displayed as functions of the chemical potential. The magnetic field is varied from $B=0$ (red solid lines in all plots) to $B=0.4\phantom{\rule{0.166667em}{0ex}}$ T (blue solid lines) with steps of $0.1\phantom{\rule{0.166667em}{0ex}}$ T. Inset in (

**a**) is zoomed-in, with black dashed lines depicting the incoherent conductance (see Appendix A). (

**b**) shows the same data as (

**a**), but using a semi-logarithmic scale, with the inset presenting positions of the actual transmission maxima for $B>0$ (${E}_{n\mathrm{LL}}$) versus the values for bulk graphene (see Equation (30)). A setup for the thermoelectric measurements is also depicted (see inset in (

**d**)).

**Figure 3.**(

**a**) Maximum amplitude of the Seebeck coefficient and (

**b**) maximum Lorentz number for same system as in Figure 2 at $T=5$ K (open symbols) and $T=10$ K (closed symbols) as functions of the maximum interval between the Landau levels of $\mathsf{\Delta}{E}_{\mathrm{max}}\equiv {E}_{1\mathrm{LL}}=|{E}_{0\mathrm{LL}}-{E}_{\pm 1\mathrm{LL}}|$. Solid lines depict the asymptotic expressions given in Equations (28) and (29). Dashed line in (

**a**) corresponds to the Goldsmid–Sharp relation, see Equation (1), with ${E}_{g}={E}_{1\mathrm{LL}}$.

**Figure 4.**Zero-temperature conductance at (

**a**) $B=0$ and (

**b**) $B=0.2\phantom{\rule{0.166667em}{0ex}}$ T versus the Fermi energy. The disk radii are the same as in Figure 2, the barrier height (see Equation (3)) is fixed at ${V}_{0}={t}_{0}/2=1.35\phantom{\rule{0.166667em}{0ex}}$ eV, and the parameter m is specified for each line. Inset in (

**a**) shows the selected potential profiles. (

**c**) Zoom-in, for low energies, with the same datasets as in (

**b**) displayed on a semi-logarithmic scale.

**Figure 5.**(

**a**–

**d**) Maximum amplitude of the Seebeck coefficient $\mathsf{\Delta}S=({S}_{\mathrm{max}}-{S}_{\mathrm{min}})/2$ and (

**e**–

**h**) Lorentz number for the same system in Figure 4 and selected values of the exponent m defining the potential profile (see Equation (3)) displayed as the bulk Landau-level energy ${E}_{1\mathrm{LL}}$ obtained from Equation (30) with $n=1$. The line/colour encoding is same as in Figure 3.

**Figure 6.**(

**a**–

**c**) Thermoelectric characteristics of the disk with smooth potential barriers in the quantum Hall regime displayed versus ${L}_{\mathrm{eff}}$ given by Equation (33). (

**a**) The minimum zero-temperature conductance reached for $-{E}_{1\mathrm{LL}}<E<0$, with the inset depicting the scaling according to ${G}_{\mathrm{min}}\propto exp(-\frac{1}{2}{L}_{\mathrm{eff}}^{2}/{l}_{B}^{2})$ with ${l}_{B}=\sqrt{\hslash /eB}$ for the magnetic length. (

**b**) The maximum thermopower amplitude. (

**c**) The maximum Lorentz number. Insets in (

**b**,

**c**) show the same data as functions of the rescaled length ${L}_{\mathrm{eff}}/{l}_{B}$. The magnetic field is varied between the datasets (see data points, dashed lines are only a guide). Additionally, in (

**b**,

**c**) the temperature is varied to keep the constant ${E}_{1\mathrm{LL}}/\left(2{k}_{B}T\right)\simeq 11.6356$. Solid horizontal lines in (

**b**,

**c**) mark the values from Equations (28) and (29).

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**MDPI and ACS Style**

Rycerz, A.; Rycerz, K.; Witkowski, P.
Thermoelectric Properties of the Corbino Disk in Graphene. *Materials* **2023**, *16*, 4250.
https://doi.org/10.3390/ma16124250

**AMA Style**

Rycerz A, Rycerz K, Witkowski P.
Thermoelectric Properties of the Corbino Disk in Graphene. *Materials*. 2023; 16(12):4250.
https://doi.org/10.3390/ma16124250

**Chicago/Turabian Style**

Rycerz, Adam, Katarzyna Rycerz, and Piotr Witkowski.
2023. "Thermoelectric Properties of the Corbino Disk in Graphene" *Materials* 16, no. 12: 4250.
https://doi.org/10.3390/ma16124250