# Electrical Manipulation of Spin-Dependent Anisotropy of a Dirac Cone in a Graphene Superlattice with Alternating Periodic Electrostatic and Exchange Fields

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

## 3. Results and Discussion

#### 3.1. Spin-Dependent Anisotropy Miniband

#### 3.2. Spin-Polarized Transport and Extra Dirac Points

#### 3.3. Electrical Controllable Spin-Dependent Band Structure

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of PEP-PExP graphene superlattice produced by a series of magnetic insulator and electrostatic gates. Here, L is the superlattice constant separated into two regions, labeled by A and B, where their widths are ${w}_{A}$ and ${w}_{B}$, respectively. This periodic field profile is approximated as a series of square barriers, in which each part contains the exchange field and the electrostatic with barrier heights of ${u}_{A}$(${u}_{B}$) and ${m}_{A}$ (${m}_{B}$) for region A (B), respectively.

**Figure 2.**Model of PEP-PExP graphene superlattice that can locate original miniband Dirac point at $({k}_{x},{k}_{y})=(0,0)$. (

**a**) In-phase and (

**b**) out-of-phase coupling patterns between PEP and PExP, where the PEP pattern of each region is ${u}_{A}=-{u}_{B}=0.5\pi $, and the PExP of the in-phase (out-of-phase) pattern is ${m}_{A}=-{m}_{B}=1\pi $ ($-{m}_{A}={m}_{B}=1\pi $). The corresponding miniband structures for (

**c**) the in-phase and (

**d**) out-of-phase patterns are shown for each spin index as a function of ${k}_{y}$.

**Figure 3.**Miniband structure of zero-averaged PEP-PExP graphene superlattice as a function of ${k}_{y}$ with $m=1\pi $ when varying magnitude of u for (

**a**) spin-up (and (

**c**) spin-down) and (

**b**) its corresponding approximation with the superlattice Dirac point for (

**b**) spin-up (and (

**d**) spin-down).

**Figure 4.**Contour plot of group velocity of the electron in zero-averaged PEP-PExP graphene superlattice versus u and m for (

**a**) spin-up and (

**b**) spin-down. (

**c**) The corresponding line plot of the group velocity with varying of u with $m=1.0\pi $, $3.0\pi $, and $5.0\pi $. (

**d**) Contour plot of the group velocity difference between spin-up and spin-down with spots (black dots) that indicate max difference points of the group velocity difference on $(u,m)$.

**Figure 5.**Energy contour of PEP-PExP graphene superlattice miniband structure with fixed $m=1.0\pi $ for spin-up (left) and spin-down (right) for (

**a**) $u=1.0\pi $, (

**b**) $u=2.0\pi $, and (

**c**) $u=3.0\pi $. (

**d**) The demonstration of the electron trajectory propagates across the PEP-PExP graphene superlattice of spin-up (red line) and spin-down (dashed blue line).

**Figure 6.**Contour plot of extra Dirac point number at zero energy of zero-averaged PEP-PExP graphene superlattice with variations of u and m for (

**a**) spin-up, (

**b**) spin-down, and (

**c**) different numbers of the extra Dirac points of spin-up and spin-down.

**Figure 7.**Conductance of spin-up (red line) and spin-down (dashed blue line) electron in x direction of finite zero-sum fields of PEP-PExP graphene superlattice at zero energy with variation of u and $N=10,50\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}100$ with (

**a**) $m=1.0\pi $, (

**b**) $m=1.0\pi $, and (

**c**) $m=5.0\pi $.

**Figure 8.**(

**a**) Configuration of supercell profiles in PEP-PExP graphene superlattice, where ${u}_{A}=-{u}_{B}=1\pi $ coupled with ${m}_{A}=1\pi $ and ${m}_{B}=0$. (

**b**) Its corresponding miniband structure at ${k}_{x}=0$.

**Figure 9.**Supercell profiles and its corresponding miniband structure of the PEP-PExP graphene superlattice that fixed ${m}_{A}=1\pi $ and ${m}_{B}=0\pi $. (

**a**) Pattern (1): superlattice profile in which $|{u}_{A,B}|=u\ge {m}_{A}$ ($u=1$), and corresponding (

**b**) spin-up and (

**c**) spin-down minibands with varying u such that $u\ge {m}_{A}$. (

**d**) Pattern (2): superlattice profile where $|{u}_{A,B}|=u\le {m}_{A}$ ($u=1$) and corresponding (

**e**) spin-up and (

**f**) spin-down minibands with varying of u such that $u\le {m}_{A}$.

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**MDPI and ACS Style**

Somroob, P.; Liewrian, W.
Electrical Manipulation of Spin-Dependent Anisotropy of a Dirac Cone in a Graphene Superlattice with Alternating Periodic Electrostatic and Exchange Fields. *Condens. Matter* **2023**, *8*, 28.
https://doi.org/10.3390/condmat8010028

**AMA Style**

Somroob P, Liewrian W.
Electrical Manipulation of Spin-Dependent Anisotropy of a Dirac Cone in a Graphene Superlattice with Alternating Periodic Electrostatic and Exchange Fields. *Condensed Matter*. 2023; 8(1):28.
https://doi.org/10.3390/condmat8010028

**Chicago/Turabian Style**

Somroob, Pattana, and Watchara Liewrian.
2023. "Electrical Manipulation of Spin-Dependent Anisotropy of a Dirac Cone in a Graphene Superlattice with Alternating Periodic Electrostatic and Exchange Fields" *Condensed Matter* 8, no. 1: 28.
https://doi.org/10.3390/condmat8010028