Controlling interlayer magnetic coupling in the two-dimensional magnet ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$

Controlling interlayer magnetic coupling in the two-dimensional magnet ${Fe}_{3} {GeTe}_{2}$

In Kee Park, Cheng Gong, Kyoo Kim, and Geunsik Lee

Phys. Rev. B 105, 014406 – Published 5 January 2022

Abstract

Interlayer magnetic coupling in emerging two-dimensional layered magnets holds great potential for manipulating layered magnetic structures for cross-layer transport or tunneling phenomena. In this paper, we employed first-principles calculations to show enhanced ferromagnetic (FM) interlayer exchange coupling for ${Fe}_{3} {GeTe}_{2}$ by reducing stacking symmetry or reducing the layer number. Electronic structure analysis reveals that the former is mainly due to low-symmetry enhanced interlayer orbital hopping, and the latter originates from reduced Pauli potential for out-of-plane metallic electrons with respect to thicker layers. Interlayer FM coupling could also be weakened by substrates due to the screened Coulomb interactions, simulated by reducing the onsite Coulomb repulsion for Fe $d$ electrons. In this paper, we provide guidance to rationally control interlayer magnetic coupling via engineering stacking configuration and dielectric environment.

Received 1 June 2021
Revised 1 October 2021
Accepted 14 December 2021

DOI:https://doi.org/10.1103/PhysRevB.105.014406

Physics Subject Headings (PhySH)

Exchange interaction Magnetic coupling

Bilayer films Van der Waals systems

Density functional calculations First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

In Kee Park¹, Cheng Gong^2,*, Kyoo Kim^3,†, and Geunsik Lee ^1,‡

¹Department of Chemistry, Center for Superfunctional Materials, Center for Wave Energy Materials, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Republic of Korea
²Department of Electrical and Computer Engineering, Quantum Technology Center (QTC), University of Maryland, College Park, Maryland 20742, USA
³Korea Atomic Energy Research Institute, Daejeon 34057, Republic of Korea

^*gongc@umd.edu
^†kyoo@kaeri.re.kr
^‡gslee@unist.ac.kr

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Vol. 105, Iss. 1 — 1 January 2022

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Figure 1
Stacking types and interlayer binding energies for ${Fe}_{3} {GeTe}_{2}$ (FGT). (a) Side view for FGT unit cell for three possible van der Waals (vdW) stacking types and associated top views of four interfacial atomic layers ${Fe}^{I} Te − Te {Fe}^{I}$ with lower and upper two atomic layers shaded and unshaded, respectively, to show different in-plane coordinates. Crosses in the vdW gap for $H_{0}$ and $H_{1}$ indicate the horizontal $C_{2}$ rotation axis. (b) Interlayer binding energies with respect to vertical distance $h$ [as shown in (a)] for three stacking types, where the different symbols indicate different density functional theory (DFT) functionals (LDA + $U$ = square, optB86b = circle, and PBE + TS = triangle) employed to calculate $E_{b} = E_{bulk} − 2 E_{ML}$ , where $E_{bulk}$ and $E_{ML}$ are total energies for bulk per unit cell and monolayer, respectively, and dashed lines are only for guidance.
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Figure 2
Stacking-dependent interlayer magnetic coupling for bulk and bilayer ${Fe}_{3} {GeTe}_{2}$ (FGT). Total energy difference between antiferromagnetic (AFM) and ferromagnetic (FM) interlayer exchange coupling configurations for (a) bulk and (b) bilayer FGT for three stacking types by LDA + $U$ , varying Fe $d$ Hubbard $U_{eff} = 0 - 4 eV$ . AFM and FM total energies, $E_{AFM}$ and $E_{FM}$ , respectively, per unit cell, are used to obtain $Δ E = (E_{AFM} − E_{FM}) / 2$ for bulk and $Δ E = E_{AFM} − E_{FM}$ for bilayer to compare interlayer exchange coupling in millielectronvolts per van der Waals (vdW) interface (meV/int.) from (a) and (b). The open circles in (a) and (b) indicate the results by HSE06 with a small horizontal offset for clarity, where $Δ E =$ 14.2, 16.4, and 16.0 meV/int. for $H_{0}, H_{1}$ , and $H_{2}$ , respectively, in bulk and 24.4, 30.6, and 27.8 meV/int. in bilayer. (c) $Δ E$ for bulk and bilayer FGT for $H_{0}$ type by $LDA + U_{eff} = 3 eV$ with varying $h$ . Experimental distance $(h^{\exp} = 2.95 Å)$ is indicated by the vertical line. Dashed lines are included for guidance to calculated data trends.
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Figure 3
Exchange constants for interlayer Fe atom pairs. Fe atoms associated with interlayer exchange coupling $J_{m}$ ( $m = 1$ , 2, 3, 4) across the van der Waals (vdW) gap, indicated by two-sided arrows with four different colors, where each Fe pair involves either ${Fe}^{I}$ or ${Fe}^{II}$ atoms in the lower or upper layer, respectively. Side views are the [110] direction with only two nearest atomic layers shown by unshaded (front) and shaded (rear) scheme. Red, green, and light blue indicate $J_{1}$ for ${Fe}^{I} − {Fe}^{I}, J_{2}$ for ${Fe}^{II} − {Fe}^{II}$ , and $J_{3}$ for ${Fe}^{I} − {Fe}^{II}$ or ${Fe}^{II} − {Fe}^{I}$ , respectively, with blue color indicating $J_{4}$ for ${Fe}^{II} − {Fe}^{I}$ since $H_{2}$ lacks inversion symmetry. Dashed arrows for $m = 1$ , 3, and 4 correspond to pairs associated with ${Fe}^{I}$ at further vertical distance, denoted by $J_{1}^{'}, J_{1}^{''}, J_{3}^{'}$ , and $J_{4}^{'}$ . Left-most rectangular boxes indicate interlayer exchange constants significantly affected by stacking type ( $J_{1}$ ) or termination ( $J_{2}$ ; see Table 1 and above for details).
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Figure 4
Out-of-plane metallic band for bulk ${Fe}_{3} {GeTe}_{2}$ (FGT) and itinerant-type interlayer exchange coupling for FGT quartet layer. (a) Spin-up band structure $(LDA + U_{eff} = 3 eV)$ for bulk FGT with $H_{0}$ type, with ${Fe}^{II} d_{z}^{2}$ and Te $p_{z}$ orbital contributions indicated by red and blue circles sizes, respectively. Right panel shows crystal orbital Hamilton population (COHP) curves for Te $p_{z}$ orbitals with respect to energy for bulk and bilayer, where positive and negative values indicate bonding and antibonding interlayer orbital hybridizations, respectively. (b) Isosurfaces (isolevel $3 \times 10^{- 5}$ ) for four Bloch orbitals at Γ, indicated by numbered arrows in (a) band structure, with yellow and blue colors indicating positive and negative, respectively. (c) Calculated $J_{2}$ for quartet layer FGT with $H_{0}$ type with respect to vertical interatomic distance for ${Fe}^{II} − {Fe}^{II}$ varying as $z_{i j} = c / 2, c$ , and $3 c$ /2, where $c = 16.33 Å$ is the bulk lattice parameter. $z_{i j} = c / 2$ exhibits two equivalent outer pairs and one inner pair with $J_{2} (c / 2) = 2$ .0 and 1.5 meV, respectively. Similarly, there are two equivalent pairs for $z_{i j} = c$ and one pair for $z_{i j} = 3 c / 2$ .
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Figure 5
Stacking-dependent interlayer orbital hopping. (a) Spin-up band structures $(LDA + U_{eff} = 3 eV)$ for bulk ${Fe}_{3} {GeTe}_{2}$ (FGT) for three stacking types with contributions from ${Fe}^{I} d_{x z} / d_{y z}$ and Te $p_{x} / p_{y}$ orbitals indicated by red and blue circles, respectively. $Λ_{1, 2}$ and $Λ_{3, 4}$ have even or odd parity along the Γ- $K$ direction under $C_{2 y}$ (twofold rotation about [110]), whereas $Λ_{1}$ and $Λ_{2}$ for $H_{2}$ have even or odd parity by mirror operation $M_{z}$ about the central atomic layer ( ${Fe}^{II} Ge$ ). Insets show $H_{0}$ and $H_{1}$ isocharges for Bloch orbital at $K$ with side views of adjacent four atomic layers ${Fe}^{I} Te − Te {Fe}^{I}$ at the van der Waals (vdW) gap. The Brillouin zone is given at the most right. (b) Stacking-dependent ( $H_{0}$ or $H_{1}$ ) interlayer hopping magnitudes between two monolayer Bloch orbitals at $K$ . Blue circles with $L$ and $U$ represent lateral positions for interfacial Te atoms from lower and upper layers, respectively, at the vdW contact, with relative lateral positions $τ_{j} (j = 1, 2, 3)$ . (c) Spin-up band structures for bilayer ${CrI}_{3}$ with two stacking symmetries, where dashed circle indicates stacking-dependent band splitting similar to $H_{0}$ vs $H_{1}$ in bulk FGT. Right panel shows calculated interlayer exchange energy for bilayer ${CrI}_{3}$ with increasing number of electrons ( $n$ ) per unit cell. We used the PBEsol density functional theory (DFT) functional for ${CrI}_{3}$ with Hubbard $U_{eff} = 3.0 eV$ [14].
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Physical Review B

covering condensed matter and materials physics

Controlling interlayer magnetic coupling in the two-dimensional magnet ${Fe}_{3} {GeTe}_{2}$

In Kee Park, Cheng Gong, Kyoo Kim, and Geunsik Lee

Phys. Rev. B 105, 014406 – Published 5 January 2022

Abstract

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Physical Review B

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Controlling interlayer magnetic coupling in the two-dimensional magnet Fe3GeTe2

In Kee Park, Cheng Gong, Kyoo Kim, and Geunsik Lee

Phys. Rev. B 105, 014406 – Published 5 January 2022

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Controlling interlayer magnetic coupling in the two-dimensional magnet ${Fe}_{3} {GeTe}_{2}$