Abstract
Owing to unique fundamental physics and device applications, twisted moiré physics in twodimensional (2D) van der Waals (vdW) layered magnetic materials has recently received particular attention. We investigate magnetic vdW Fe_{3}GeTe_{2} (FGT)/CrGeTe_{3} (CGT) moiré heterobilayers with twist angles of 11° and 30° from firstprinciples. We show that the moiré heterobilayer is a ferromagnetic metal with an ntype CGT layer due to the dominant spinmajority electron transfer from the FGT layer to the CGT layer, regardless of various stacked structures. The spinmajority hybridized bands between Cr and Fe bands crossing the Fermi level are found regardless of stacking. The band alignment of the CGT layer depends on the effective potential difference at the interface. We show that an external electric field perpendicular to the inplane direction modulates the interface dipole and band edges. Our study reveals a deeper understanding of the effects of stacking, spin alignment, spin transfer, and electrostatic gating on the 2D vdW magnetic metal/semiconductor heterostructure interface.
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Introduction
2D vdW layered metal–semiconductor nonmagnetic heterostructures have been studied for developing highperformance devices with the contact of metal electrodes and semiconductors^{1,2,3,4,5,6,7}. Recently, heterostructures consisting of 2D ferromagnetic (FM) vdW layered materials have attracted considerable interest^{8,9,10}. In particular, research on the 2D vdW heterobilayer composed of magnetic metals and magnetic semiconductors is still early, as it was only recently revealed that vdW monolayer (ML) or fewlayer materials retain magnetism^{11,12,13}. Furthermore, the recent discovery of novel properties of twisted vdW nonmagnetic homo and heterobilayers^{14,15,16} further prompts the investigation of twisted vdW magnetic homo and heterobilayers.
Recent experiments have demonstrated the longrange 2D FM order along the outofplane direction in the metallic ML FGT^{17,18,19,20} and the semiconducting CGT bilayer samples^{21,22}. Thus, metallic Fe_{3}GeTe_{2} (FGT) and semiconducting CrGeTe_{3} (CGT) materials are promising candidates for forming lowdimensional heterobilayers. Several experimental works have reported that bulk FGT is an itinerant FM metal with a high Curie temperature, T_{c}, of approximately 230 K; T_{c} can be increased to room temperature by electrostatic gating and decreased by decreasing the layer thickness^{17,18,19,20}. Theoretical studies have reported changes in the magnetic anisotropy energy (MAE), magnetic order from the bulk to ML, the anomalous Hall effect, and spindependent transport in tunnel junctions^{10,23,24,25,26}. On the other hand, some experimental studies have revealed that bulk CGT is an intrinsic FM semiconductor with a T_{c} of 61 K^{27} and an experimental bandgap, E_{g}, in the range 0.2–0.7 eV^{28,29,30,31}. Several theoretical works have examined variations in T_{c}, magnetic order, MAE, and E_{g} from the bulk to ML^{28,32,33,34,35,36,37,38,39}. Although each FGT and CGT has been thoroughly investigated, the basic properties of the FGT/CGT heterobilayer have not been fully explored. Moreover, there has not been much research on the twisted moiré heterobilayer of vdW magnetic metal/semiconductor materials^{11}. Hence, it is worth examining the promising FGT/CGT twisted moiré heterobilayer.
The Schottky barrier heights at the interface between a metal and a semiconductor are crucial properties for device applications. They classify the heterostructure interface into a Schottky contact (n or ptype) or an Ohmic contact. In general, the additional interface states caused by extra bonds at the interface facilitate the formation of Schottky contact^{40,41}, whereas they complicate that of Ohmic contact. Thus, the Schottky barrier heights at the metal–semiconductor interface with Schottky contact have been extensively studied. Experimental studies have reported converting a Schottky contact to an Ohmic contact by large gating voltages in 3D metal/2D vdW semiconductor heterostructures^{42,43}. Meanwhile, the formation of extra bonds at the interface between 2D vdW materials can be prevented due to the vdW interlayer distance, and thus the heterostructure has an abrupt interface. Because of this advantage, the band edges of heterostructures combined with 2D vdW metals and semiconductors have been investigated with and without electrostatic gating for device applications. Several theoretical studies have shown the electrostatic gating effect on the band edges of nonmagnetic 2D vdW metal/semiconductor heterobilayers^{1,5}. However, firstprinciples studies on the band edges of 2D vdW magnetic twisted moiré metal/semiconductor heterobilayers have rarely been performed^{11}.
In this work, we conducted a firstprinciples study on the structural, electronic, and magnetic properties of the FGT/CGT moiré heterobilayer with twist angles of 11° and 30°. First, we examined the atomic distortion and unfolded band structures in a large moiré supercell with a twist angle of 11°, and then investigated the band edges depending on stacking in a small moiré supercell with a twist angle of 30° with and without electrostatic gating. Each small moiré heterobilayer having one of six possible stacked structures was an FM metal with an ntype CGT layer, induced by the effective spinmajority electron transfer from FGT to CGT. We also compared the band structures of the FM and AFM heterobilayers, having FM and antiferromagnetic (AFM) spin alignments between FGT and CGT. We found that only the FM heterobilayer had the hybridized bands between the Cr and Fe atoms. Further, we explained the stackingdependent band edges of the CGT layer in terms of the effective potential difference. Finally, we addressed the external electric field effect on the band edges of the CGT layer in the FM heterobilayer.
Methods
We conducted firstprinciples electronic structure computations using the noncollinear density functional theory (DFT) with the Hubbard repulsion, U, and exchange interaction, J, as well as the spin–orbit coupling (SOC). We used the normconserving pseudopotentials^{44} with the SIESTA code^{45}. We adopted the generalizedgradient approximation, parameterized by the Perdew–Burke–Ernzerhof (PBE) formula. Further, we used the rotationally invariant U approach^{46} to describe the electron correlation for Cr 3d orbitals. The atomic structures were optimized using the spinpolarized optB88 vdW method until all atomic forces were less than 0.001 eV Å^{−1} except for a huge moiré supercell with a twist angle of 11°, where the force criteria were 0.036 eV Å^{−1}. In addition, we generated a semicore pseudopotential for Cr using the (3s^{2}, 3p^{6}, 3d^{5}, 4s^{0}) valence configuration; the valence configurations for Fe, Ge, and Te pseudopotentials were (3d^{7}, 4s^{1}, 4p^{0}), (3d^{10}, 4s^{2}, 4p^{2}), and (4d^{10}, 5s^{2}, 5p^{4}), respectively. We generated realspace grids with a cutoff energy of 400 Ry and expanded the electronic wave functions using pseudoatomic orbitals (PAOs) of the doublezeta polarization basis set. We utilized a \(16\times 16\times 1\) kgrid for all structure relaxation and electronic structure calculations except for a large moiré supercell with a twist angle of 11°, for which we applied a \(2\times 2\times 1\) kgrid. For ML CGT, we obtained U = 5.97 eV and J = 1.00 eV for Cr 3d orbitals with a cutoff radius of 1.58 Å by using a PAObased constrained DFT (cDFT) method^{45,47}. However, our previous study confirmed that the suitable theoretical bandgap close to the experimental one was obtained by the PBE + U + SOC method using a U in the range of 3–4 eV and J = 1 eV^{48}. Therefore, we selected U = 3.5 eV and J = 1 eV for our calculations. To remove the interactions between FGT and CGT, we applied a vacuum space of 16 Å.
Results and discussion
Twisted FGT/CGT moiré heterobilayer
Bulk FGT has a layered hexagonal structure with space group P6_{3}/mmc and experimental hexagonal cell parameters a = b = 4.030 and 3.991 Å and c = 16.343 and 16.336 Å from refs. 47 and 48, respectively. In ML FGT, each unit cell contains three Fe atoms occupying two nonequivalent Fe positions; Fe1 and Fe2 denote the relevant atoms. ML FGT consists of five sublayers: the first and fifth are occupied by Te atoms, the second and fourth by Fe1 atoms, and the third by Fe2 and Ge atoms. Our relaxed lattice constants for pristine ML FGT that are obtained using the spinpolarized optB88 vdW method are a = b = 4.091 Å. Bulk CGT has a rhombohedral symmetry with space group R\(\overline{3 }\) and experimental hexagonal cell parameters a = b = 6.8196 Å and c = 20.3710 Å at 5 K^{27}. ML CGT has two Cr atoms in a unit cell. ML CGT consists of five sublayers: the first and fifth are occupied by Te atoms, the second and fourth by Ge atoms, and the third by Cr atoms. ML CGT has a honeycomb lattice of edgesharing CrTe_{6} octahedrons, and Ge dimers located at the center of the hexagon. Our relaxed lattice constants for pristine ML CGT that are obtained using the spinpolarized optB88 vdW method are a = b = 6.940 Å. The point group for ML FGT and ML CGT is D_{3h} and C_{1}, respectively. Thus the heterobilayer built by ML FGT and ML CGT has the C_{1} symmetry.
The heterobilayer of ML FGT and ML CGT has a moiré pattern due to the lattice mismatch. In principle, the twist angle \(\phi\) between the supercell lattice vectors of the FGT and CGT layers for a moiré supercell (i.e., \(\phi\) between \({\overrightarrow{L}}_{\mathrm{CGT},1} \mathrm{and} {\overrightarrow{L}}_{\mathrm{FGT},1}\) in Fig. 1a) is the same as that between the unit vectors of ML FGT and ML CGT. Hence, the moiré reciprocal lattice vector, \(\overrightarrow{k}\)_{moiré}, and the moiré wavelength, λ, can be easily obtained as \(\vec{k}_{{{\text{moir}}\mathop {\text{e}}\limits^{\prime } }} = \vec{k}_{{{\text{FGT}}}}  \vec{k}_{{{\text{CGT}}}}\) and \({\uplambda } = \frac{{2{\uppi }}}{{\left {\vec{k}_{{{\text{moir}}\mathop {\text{e}}\limits^{\prime } }} } \right}}\), respectively, using reciprocal lattice vectors corresponding to the unit cell vectors. Concretely, when the unit vectors of ML CGT are \({\overrightarrow{a}}_{1}=a\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\) and \({\overrightarrow{a}}_{2}=a\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\), and the unit vectors of ML FGT are rotated by \(\phi\) relative to those of ML CGT, the reciprocal lattice vectors become \({\overrightarrow{k}}_{\mathrm{CGT}}=\frac{2\pi }{a}\left(\mathrm{1,0}\right)\) for ML CGT and \({\overrightarrow{k}}_{\mathrm{FGT}}=\frac{2\pi }{a(1\delta )}\left(cos\phi ,sin\phi \right)\) for ML FGT, where \(\delta =1{a}_{\mathrm{FGT}}/{a}_{\mathrm{CGT}}\) as shown in the left panel in Fig. 1a. Thus, the moiré wavelength, λ, becomes \({\uplambda } = \frac{{2{\uppi }}}{{\left {\vec{k}_{{{\text{moir}}\mathop {\text{e}}\limits^{\prime } }} } \right}} = \left( {1  \delta } \right)a/\sqrt {2\left( {1  \delta } \right)\left( {1  cos\phi } \right) + \delta^{2} }\)^{49,50}. The right panel in Fig. 1a shows λ (red lines) with respect to the twist angle \(\phi\) between ML CGT and ML FGT optimized using the spinpolarized optB88 vdW method. The moiré superlattice vectors can be expressed as \({\overrightarrow{L}}_{\mathrm{CGT},1}=m{\overrightarrow{a}}_{1}+n{\overrightarrow{a}}_{2}\) and \({\overrightarrow{L}}_{\mathrm{CGT},2}=n{\overrightarrow{a}}_{1}+(m+n){\overrightarrow{a}}_{2}\) for ML CGT and \({\overrightarrow{L}}_{\mathrm{FGT},1}=p{\overrightarrow{b}}_{1}+q{\overrightarrow{b}}_{2}\) and \({\overrightarrow{L}}_{\mathrm{FGT},2}=q{\overrightarrow{b}}_{1}+(p+q){\overrightarrow{b}}_{2}\) for ML FGT, where m, n, p, and q are integers. We can obtain the commensurate moiré lattice vectors by applying a small biaxial strain △ between ML CGT and ML FGT layers. △ is estimated by \(\Delta =a\sqrt{{m}^{2}+{n}^{2}+mn}b\sqrt{{p}^{2}+{q}^{2}+pq}{(b\sqrt{{p}^{2}+{q}^{2}+pq})}^{1}\). The moiré wavelengths within the constrained condition of △ being less than 1.0% are shown as blue squares in the right panel in Fig. 1a. Here the twist angle \(\phi\) is determined as \(cos\phi =\left({\left{\overrightarrow{L}}_{CGT,1}\right}^{2}+{\left{\overrightarrow{L}}_{FGT,1}\right}^{2}{\left\overline{AB}\right}^{2}\right){\left(2\left{\overrightarrow{L}}_{CGT,1}\right\left{\overrightarrow{L}}_{FGT,1}\right\right)}^{1}\), where \(\left\overline{AB}\right=\left{\overrightarrow{L}}_{CGT,1}{\overrightarrow{L}}_{FGT,1}\right=m{\overrightarrow{a}}_{1}+n{\overrightarrow{a}}_{2}p{\overrightarrow{b}}_{1}q{\overrightarrow{b}}_{2}\)^{51}.
Figure 1b displays the top and side views of the moiré superlattice with λ = 25.02 Å, △ = 0.55%, \(\phi\) = 11.39°, (m, n, p, q) = (1, 3, 3, 4), and the total number of atoms being 352. Only Fe1, Te_{FGT}, Te_{CGT}, Ge_{CGT}, and Cr atoms near the interface are depicted in the top view of Fig. 1b. The moiré supercell has CrTe_{6} hexagonal networks on top of FGT and various stacked structures. The inplane and outofplane distortions obtained by the spinpolarized optB88 vdW method are displayed in Fig. 1c,d, respectively. The inplane displacement vectors indicated by red arrows show more inplane distortion in CGT than in FGT and are largest at the Te_{CGT} layer. In particular, the inplane displacement vectors of all Te_{CGT} atoms have consistent patterns regardless of stacking, as depicted by black circles, but those of other atoms vary depending on stacking. The zcoordinates of Te_{CGT} and Te_{FGT} atoms inside the green shaded region in (b) vary depending on stacking, as shown in Fig. 1d. The outofplane distortion of the Te_{CGT} layer is much larger than that of the Te_{FGT} layer. The variation of interlayer distance between two interface Te layers is largest in the area where Te_{CGT} atoms are closely located above Te_{FGT} atoms.
The unfolded band structures of the moiré heterobilayer with a twist angle of 11.39° reveal some distinct features compared with the unitcell band structures of a single layer, as shown in Fig. 1e. All the band structures were obtained using the noncollinear PBE + U + SOC method when U = 3.5 eV and J = 1.0 eV for Cr 3d orbitals, and an FM spin alignment between FGT and CGT was considered. The average magnetic moment is 3.64 \({\upmu }_{\mathrm{B}}\)/atom for Cr, 2.61 \({\upmu }_{\mathrm{B}}\)/atom for Fe1, and 1.65 \({\upmu }_{\mathrm{B}}\)/atom for Fe2. The unfolded band structures of FGT (HB FGT) near the Fermi level are similar to the unitcell band structures of ML FGT except for bands shifted higher in energy denoted by red arrows. On the other hand, the unfolded band structures of CGT (HB CGT) exhibit a reduced bandgap compared with that of ML CGT, and the conduction bands of HB CGT cross the Fermi level, generating an ntype CGT layer. Also, unexpected bands of HB CGT are observed as indicated by a blue arrow compared to ML CGT. These intriguing electronic features of moiré heterobilayer motivate us to examine the critical elements that influence band structures.
To investigate the electronic structures in terms of stacked structures, we constructed small moiré supercells of the FGT/CGT heterobilayer forming only one stacked structure with a twist angle of 30° and △ = 1.36%, as shown in Fig. 2a. The inplane supercell indicated by a solid green diamond is equal to the \(1\times 1\) inplane unit cell of ML CGT and also corresponds to the 30° rotated \(\sqrt{3}\times \sqrt{3}\) inplane supercell of ML FGT, drawn by an open black diamond. Inside the supercell, there are two CrTe_{6} octahedrons.
ML FGT is an Isingtype FM metal, whereas ML CGT is an Isingtype FM semiconductor. To predict the magnetic structures of the heterobilayer, we compared the total energy of the heterobilayer with the FM and antiferromagnetic (AFM) spin alignments between ML FGT and ML CGT along the direction perpendicular to the interface. As shown in the left panel in Fig. 2a, we considered 15 different stacked structures while fixing ML FGT and sliding ML CGT. Concretely, we considered six different stacked structures in two diagonal directions of the inplane supercell and three different stacked structures in the direction of the inplane lattice. All the atoms were optimized during the relaxation in the x, y, and zdirections. The FM spin alignment between ML FGT and ML CGT always has lower energy than the AFM spin alignment at each stacked structure, as shown in the right panel in Fig. 2a.
We categorized the FM energy into three classes: the lowest, E_{L}, intermediate, E_{I}, and the highest energy, E_{H}. The corresponding stacked structures are depicted in the top panel in Fig. 2b. The background lattice shows the FGT top view. The red and purple triangles represent the interface Te triangles in the FGT and CGT layers, respectively. We drew two purple triangles on the bottom Te atoms in CGT related to two CrTe_{6} octahedrons inside a supercell. In the E_{I} structure, the interface Te_{CGT} atom is above the Fe1_{FGT} atom; in the E_{L} structure, it is above the Ge_{FGT} atom; and, in the E_{H} structure, it is located above the interface Te_{FGT} atom. Three more possible stacked structures illustrated in the bottom panel were examined; the purple triangle consisting of interface Te_{CGT} atoms was rotated 180° relative to those in the top panel. Two more intermediate energy structures, E_{I2} and E_{I3}, and another highest energy structure, E_{H2}, were found. In E_{I2}, E_{I3}, and E_{H2} structures, the interface Te_{CGT} atom is above the Fe1_{FGT}, Ge_{FGT}, and Te_{FGT} atoms, respectively.
Figure 2c presents the total energy difference with the FM (red) and AFM (blue) spin alignments, as well as the interlayer distance (brown) between the interface Te atoms of the FM structures. All the atoms and axes were optimized during the relaxation in the x, y, and zdirections using the spinpolarized optB88 vdW method. The total energy difference changes gradually from E_{I} through E_{L} to E_{H} (left panel) and from E_{I2} through E_{I3} to E_{H2} (right panel). The total energy difference and the interlayer distance are the smallest for E_{L} and the largest for E_{H} and E_{H2}. E_{I}, E_{L}, E_{H}, E_{I2}, E_{I3}, and E_{H2} structures have interlayer distances of 3.39 Å, 3.22 Å, 3.78 Å, 3.33 Å, 3.31 Å, and 3.71 Å, respectively. At each FM heterobilayer, the lattice constants of the CGT and FGT layers become tensile strained and compressive strained, respectively, as summarized in Table 1. The total energy of FM spin alignment is lower than that of AFM for E_{I}, E_{L}, E_{H}, E_{I2}, E_{I3}, and E_{H2} by 0.02 eV, 0.04 eV, 0.02 eV, 0.02 eV, 0.05 eV, and 0.03 eV, respectively. At the FM spin alignment, the total energy of the E_{L} structure is lower than that of E_{I}, E_{H}, E_{I2}, E_{I3}, and E_{H2} structures by 0.08 eV, 0.30 eV, 0.06 eV, 0.03 eV, and 0.29 eV, respectively. Figure 2d shows the binding energy, E_{b}, per Cr atom along the zdirection regarding the small displacement. At the equilibrium position, the binding energy per Cr atom obtained using the spinpolarized optB88 vdW method has a negative value of − 0.44 eV, − 0.58 eV, and − 0.54 eV for E_{H}, E_{L}, and E_{I}, respectively. The lower the total energy is, the lower the binding energy is. In the PBE + U + SOC method, the FM spin alignment also has lower energy than the AFM spin alignment when U is 3–4 eV and J = 1 eV for all the stacked structures, as shown in Fig. 2e.
The band alignment of FM and AFM heterobilayers was investigated. Figure 3 shows the magnetic moment (left panels), spinmajority (up) and spinminority (down) band structures (middle panels), and atom projected density of states (PDOS, right panels) of (a) the FM CGTonly system, (b) FM FGTonly system, (c) FM FGT/CGT heterobilayer for E_{H}, and (d) AFM FGT/CGT heterobilayer for E_{H} in the supercell band lines. The band structures were calculated by using the noncollinear PBE + U + SOC method on Cr 3d orbitals with U = 3.5 eV and J = 1.0 eV. It should be noted that the CGT (FGT)only system refers to only the CGT (FGT) layer and no FGT (CGT) layer while maintaining the relaxed heterobilayer structures. The band projections onto the spinup and spindown bands of Cr or Fe atoms are distinguished by red and blue colors, respectively, with various color intensities proportional to the projection magnitude to examine the band alignment. The semiconducting bandgap of 0.43 eV is shown for a Cr atom in the FM CGTonly system (Fig. 3a), while the metallic band structure is shown for a Fe2 atom in the FM FGTonly system (Fig. 3b). The magnetic moment is 3.69 \({\upmu }_{\mathrm{B}}\)/atom for Cr, 2.62 \({\upmu }_{\mathrm{B}}\)/atom for Fe1, and 1.67 \({\upmu }_{\mathrm{B}}\)/atom for Fe2. The atom PDOS of a Fe2 atom in (b) indicates an equivalent magnitude of spinup and spindown PDOS near the Fermi level. The CGTonly system features a conduction band minimum (CBM) at a k point between the \(\Gamma\) and K points dominated by spinup empty Cr 3d e_{g} bands. Its valence band maximum (VBM) at the \(\Gamma\) point is dominated by Te 5p orbital characteristics, similar to the pristine ML CGT.
On the other hand, the heterobilayer formed by CGT and FGTonly systems is an FM metal, as shown in Fig. 3c. The bands projected on a Cr atom show the initial Cr bands observed in the CGTonly system as well as unique spinup hybridized bands with Fe2 bands crossing the Fermi level, as indicated by green and blue arrows, respectively. The hybridization is not related to Fe1 bands, as seen by the band projected on a Fe1 atom. Furthermore, when the hybridized spinup bands are ignored, the CBM of Cr bands appears to be at the spindown bands, located at an energy close to the Fermi level, generating an ntype CGT layer. The ntype CGT means that electrons are transferred from the FGT to CGT layers. In the right panel of Fig. 3c, the downwardband shifts of a Cr atom in the heterobilayer can be observed in the atom PDOS. The spinup Fe2 bands shift higher in energy near the \(\Gamma\) point when the system switches from an FGTonly system to a heterobilayer, as indicated by the black arrows in (b) and (c). These upward shifted spinup Fe2 bands might also originate from the electron transfer from the FGT to CGT layers. Unlike the FM heterobilayer, the AFMheterobilayerband structures do not exhibit hybridized bands between FGT and CGT bands but do expose an ntype CGT layer via the spinup and spindown CBM of the original Cr bands crossing the Fermi level. AFM heterobilayer, like FM heterobilayer, features shifted spindown Fe2 bands near the \(\Gamma\) point that are higher in energy than those of the FGTonly system (black arrow in (d)), indicating electron transfer from the FGT to CGT layers.
Figure 4 shows the spinup (red) and spindown (blue) band structures projected on a Cr atom of all the stacked structures with the FM spin alignment. All the band structures exhibit spinup hybridized bands crossing the Fermi level. However, depending on stacking, the position of the spindown CBM of Cr bands close to the Fermi level varies. The origin of the band alignment based on stacked structures is shown in Fig. 5. The relative band alignment between the FGT and CGT layers before and after stacking is shown in Fig. 5a. Before stacking, the calculated work function of pristine ML CGT, 4.33 eV, is slightly larger than that of pristine ML FGT, 4.08 eV. After stacking, atomic and electronic rearrangements occur, especially near the interface. The electron transfer from the FGT to CGT layers causes an ntype CGT layer to be formed. When the work function of the metal is less than that of the ntype semiconductor, band bending at the interface generates an Ohmic contact between the metal and the ntype semiconductor. Our vdW heterobilayer, on the other hand, has the vdW interlayer space without any extra bonds between the two constituent layers, resulting in an abrupt interface. There is only the interface dipole at the interface, which is dictated by electron distributions. The interface dipole is defined by the associated effective potential difference at the interface, \(\updelta {V}_{\mathrm{eff}}\). The electrostatic potential difference between the vacuum levels of the FGT and CGT layers in the heterobilayer can be used to determine \(\updelta {V}_{\mathrm{eff}}\), which is reported in Table 2. In addition, \(\updelta {V}_{\mathrm{eff}}\) is compared with that calculated using the electron density difference, and the relationship between \(\updelta {V}_{\mathrm{eff}}\) and electron transfer is explored in Fig. 6.
The band edges of the ntype CGT layer in the heterobilayer were computed to quantify the band alignment. The spinup (red) and down (blue) bands of E_{L} projected on a Cr atom are shown in the left panel of Fig. 5b, with color intensity according to the projection magnitude. Colored arrows indicate the CBM (\({E}_{\mathrm{C},\uparrow ,\downarrow }^{\mathrm{CGT}}\)) and VBM (\({E}_{\mathrm{V},\uparrow ,\downarrow }^{\mathrm{CGT}}\)) of the original Cr bands relative to the Fermi level. The CGT layer’s schematic band edges are shown in the right panel of Fig. 5b. In metal/ntype semiconductor heterostructures, the ntype spinup and spindown band edges are specified as \({\Delta }_{n,\uparrow ,\downarrow }= {E}_{\mathrm{C},\uparrow ,\downarrow }^{\mathrm{CGT}}{E}_{\mathrm{F}}\). In metal/ptype semiconductor heterostructures, the ptype spinup and spindown band edges are expressed as \({\Delta }_{p,\uparrow ,\downarrow }={E}_{\mathrm{F}}{E}_{\mathrm{V},\uparrow ,\downarrow }^{\mathrm{CGT}}\). Because \({E}_{\mathrm{V},\uparrow }^{\mathrm{CGT}}={E}_{\mathrm{V},\downarrow }^{\mathrm{CGT}}\) in our case, \({\Delta }_{p,\uparrow }={\Delta }_{p,\downarrow }\). The effective bandgap in the CGT layer can be described as \({E}_{\mathrm{g}}^{\mathrm{CGT}}={\Delta }_{n,\downarrow }+{\Delta }_{p,\uparrow ,\downarrow }\) even if it is not a genuine bandgap owing to the metal wave function tail of the FGT layer up to the Fermi level. The band edges of the CGT layer and \(\updelta {V}_{\mathrm{eff}}\) with respect to the various stacked structures are shown in Fig. 5c, and the estimated values for the band edges and \({E}_{\mathrm{g}}^{\mathrm{CGT}}\) are summarized in Table 2. According to Fig. 5c, the variations in band edges and \(\updelta {V}_{\mathrm{eff}}\) across different stacked structures are similar, suggesting a significant correlation. For example, the magnitude order in \({\Delta }_{n,\uparrow ,\downarrow }\) is E_{I} > E_{L} > E_{H}, which is the same as in \(\updelta {V}_{\mathrm{eff}}\). In all the stacked structures, \({\Delta }_{p,\uparrow ,\downarrow }\) is larger than \({\Delta }_{n,\downarrow }\), confirming the ntype CGT, and the biggest variation in band edges across different structures is roughly 0.1 eV.
To study the spin/electron transfer and \(\updelta {V}_{\mathrm{eff}}\) derived from the electron density difference, the spinresolved valence electron density difference \(\updelta {\uprho }_{\uparrow ,\downarrow }\left(z\right)={\rho }_{\mathrm{HB}\uparrow ,\downarrow }\left(z\right){\rho }_{\mathrm{FGT}\uparrow ,\downarrow }(z){\rho }_{\mathrm{CGT}\uparrow ,\downarrow }(z)\) was computed. Here, \({\rho }_{\mathrm{HB}\uparrow ,\downarrow }(z)\),\({\rho }_{\mathrm{FGT}\uparrow ,\downarrow }(z)\), and \({\rho }_{\mathrm{CGT}\uparrow ,\downarrow }(z)\) are the spindependent inplaneaveraged electron densities of the heterobilayer, FGTonly, and CGTonly systems. In the case of FM heterobilayer, \(\updelta {\uprho }_{\uparrow ,\downarrow }\) (top panel in Fig. 6b) illustrates that the spinup density transfers from the FGT to CGT layers, and the spindown density transfers from the CGT to FGT layers, resulting in the effective electron density difference (black dotted lines,\({\delta \rho }_{\mathrm{eff}}\left(z\right)={\delta \rho }_{\uparrow }\left(z\right)+{\delta \rho }_{\downarrow }(z)\)). A considerable electron transfer from Fe2 atoms to Cr atoms is observed in the atomicstructure figure via the bonding between Fe2 atoms and interface Te_{FGT} atoms, via the vdW spacing between interface Te_{FGT} atoms and interface Te_{CGT} atoms, and via the bonding between interface Te_{CGT} atoms and Cr atoms. Whereas the electron transfer from Fe1 atoms to Cr atoms is absent due to no direct bonding between Fe1 atoms and interface Te_{FGT} atoms. This electrontransfer pathway explains the hybridized band between Cr atoms and Fe2 atoms. In the interface region, the interface dipole appears as electron depletion and accumulation, represented by + and − signs, respectively, resulting in an effective electric field and a potential difference at the interface.
Specifically, the onedimensional Gauss law can be used to compute the effective electric field, \({E}_{\mathrm{eff}}\left(z\right)\), and effective potential difference, \(\updelta {V}_{\mathrm{eff}}(z)\), at the interface, as illustrated in the middle panel of Fig. 6b. \({Q}_{in}={\int }_{z0}^{z}\delta {\rho }_{\mathrm{eff}}\left(z\right)dz\) gives the electron density inside the gray dashed box in the top panel, where z_{0} is an arbitrary point z at the left side vacuum outside the FGT layer. Because the electric field at the left side vacuum is zero, \({E}_{\mathrm{eff}}\left(z\right)={Q}_{\mathrm{in}}(z){({\varepsilon }_{r}{\varepsilon }_{o})}^{1}\) gives the electric field at location z, where \({\varepsilon }_{o}\) is the vacuum permittivity, and \({\varepsilon }_{r}\) is the positiondependent relative permittivity. Since \({\varepsilon }_{r}\) is around 1.0, 13.50, and \(\infty\) in the vacuum, CGT, and FGT layers, respectively, the effective electric field exists only at the interface between the two layers. Finally, \(\updelta {V}_{\mathrm{eff}}(z)\) for electrons at the interface may be calculated using \(\updelta {V}_{\mathrm{eff}}\left(z\right)={\int }_{0}^{z}{E}_{\mathrm{eff}}\left(z\right)dz\); further, it is verified that \(\updelta {V}_{\mathrm{eff}}(z)\) equals the sum of \(\updelta {V}_{\mathrm{eff},\uparrow }(z)\) and \(\updelta {V}_{\mathrm{eff},\downarrow }(z)\). The Gauss law yields \(\updelta {V}_{\mathrm{eff}}\) values (V_{o} in Fig. 6b) of 0.28 eV, 0.32 eV, and 0.37 eV for E_{H}, E_{L}, and E_{I}, respectively, which are extremely similar to the electrostatic potential differences of 0.28 eV, 0.31 eV, and 0.35 eV, respectively. Notably, the general signs for \({E}_{\mathrm{eff}}\) and \(\updelta {V}_{\mathrm{eff}}\) defined for a positive charge are the inverse of those defined for the electron in this work.
The spin variation (\({\Delta \rho }_{\uparrow ,\downarrow }={\rho }_{\mathrm{HB},\uparrow ,\downarrow }{\rho }_{1\mathrm{L},\uparrow ,\downarrow })\) and the charge variation (\(\Delta\uprho ={\Delta \rho }_{\uparrow }+{\Delta \rho }_{\downarrow })\) per single layer were quantitatively investigated using the spindependent Mulliken electron differences at the atomic sites, as shown in the bottom panel of Fig. 6b. The spinup electron in the FGT layer moves to the spinup electron in the CGT layer by 0.10 e. In contrast, the spindown electron in the CGT layer moves to the spindown electron in the FGT layer by 0.05 e, resulting in a 0.05 e effective spinup electron transfer from FGT to CGT. Also, the magnetic moment variation per single layer, i.e., \({\Delta \rho }_{\uparrow }{\Delta \rho }_{\downarrow }\) is \(\Delta {m}_{\mathrm{FGT}}=\) 0.15 \({\mu }_{\mathrm{B}}\) and \(\Delta {m}_{\mathrm{CGT}}=\) 0.15 \({\mu }_{\mathrm{B}}\). In addition, the magnetic moment variation by the atomic site (\(\updelta m\), solid black square) demonstrates that magnetic moments are slightly larger at Cr sites and slightly smaller at Fe2 sites than those of a single layer. The spindown variation (blue open circle) contributes more to the magnetic moment changes at the Cr and Fe2 sites than the spinup variation (red open circle).
On the other hand, the FGT layer’s \(\updelta {\uprho }_{\uparrow ,\downarrow }\) in the AFM heterobilayer, as illustrated in the top panel of Fig. 6a, is reversed compared to that in the FM heterobilayer due to the FGT layer’s reversed spinup and spindown bands in the AFM heterobilayer. However, the electron distribution at the interface in the AFM heterobilayer is identical to that in the FM heterobilayer, resulting in similar \({E}_{\mathrm{eff}}\) and \(\updelta {V}_{\mathrm{eff}}\). As demonstrated in the atomicstructure figure in Fig. 6a, the practical electrontransfer pathway in the AFM heterobilayer is comparable to that in the FM heterobilayer. According to the quantitative spin variation per single layer, complex spin transfer between FGT and CGT results in a 0.05 e effective electron transfer from FGT to CGT. Also, the magnetic moment variation per single layer is \(\Delta {m}_{\mathrm{FGT}}=\) 0.11 \({\mu }_{\mathrm{B}}\) and \(\Delta {m}_{\mathrm{CGT}}=\) 0.07 \({\mu }_{\mathrm{B}}\). The magnetic moments of both the Cr and Fe2 sites are slightly larger than those of a single layer.
Electrostatic gating effect
This study considers positive and negative electric fields within 0.5 VÅ^{−1}, perpendicular to the inplane direction. When utilizing the PBE + U + SOC method with U = 3.5 eV and J = 1.0 eV to compare the total energies of the FM and AFM E_{H}, E_{I}, and E_{L} structures concerning the external electric field, FM structures are always more stable than AFM structures, as shown in Fig. 7a. Figure 6c,d show the spin/electron transfer for the FM E_{H} structure. We show only E_{H} results because the main effect of electrostatic gating on all stacked structures is almost comparable. Our results show that the external electric field mainly modulates the spinmajority spin transfer between FGT and CGT. As shown in Fig. 6c, the variation of \(\updelta {\rho }_{\uparrow ,\downarrow }\) increases when a negative electric field of − 0.5 VÅ^{−1} is introduced. Quantitatively, the spinup electron moves from FGT to CGT by 0.21 e, whereas the spindown electron moves from CGT to FGT by 0.03 e, resulting in a 0.17 e effective spinup electron transfer from FGT to CGT. As a result, the magnetic moment variation per single layer increases, with \(\Delta {m}_{\mathrm{FGT}}= \) 0.25 \({\mu }_{\mathrm{B}}\) and \(\Delta {m}_{\mathrm{CGT}}=\) 0.23 \({\mu }_{\mathrm{B}}\), respectively. More enhanced \(\updelta m\) at Cr sites and more reduced \(\updelta m\) at Fe2 sites are also induced by the negative electric field. In the interface region, the negative electric field promotes electron depletion (++) and accumulation (−−), resulting in larger \({E}_{\mathrm{eff}}\) and \(\updelta {V}_{\mathrm{eff}}\) (V_{o} + V_{a}, where V_{a} = 0.76 eV in Fig. 6c).
In contrast, the variation of \(\updelta {\rho }_{\uparrow ,\downarrow }\) decreases under a positive electric field of 0.5 VÅ^{−1}, as shown in Fig. 6d Consequently, the spinup electron transfer is relatively tiny, whereas the spindown electron transfer from CGT to FGT is 0.06 e, resulting in a 0.06 e effective spindown electron transfer from the CGT to FGT layers. This spin transfer results in less increased and decreased \(\updelta m\) at Cr and Fe2 sites, respectively, and reduces the magnetic moment variation per single layer. The positive electric field, especially in the interface region, generates even the opposite electron distribution, revealing the opposite action of the negative electric field. In particular, \({E}_{\mathrm{eff}}\) and \(\updelta {V}_{\mathrm{eff}}\) (V_{o} − V_{a}, where V_{a} = 0.70 eV in Fig. 6d) at the interface have the opposite sign.
The FGT/CGT heterobilayer is analogous to a Schottky diode in device performance, except for the opposite electron distribution in the interface. In a Schottky diode depletion zone, electron accumulation occurs near the metal, and electron depletion occurs near the ntype semiconductor. However, the electron distribution in our heterobilayer is reversed. A negative electric field to the heterobilayer, i.e., reverse bias, is achieved by connecting the battery’s positive terminal to CGT and the negative terminal to FGT. The positive terminal to CGT allows FGT’s electrons to migrate more to CGT, resulting in a stronger interface dipole (middle panel in Fig. 6c) and CGT’s lower energyband shift relative to the Fermi level. Furthermore, as illustrated in the left panel of Fig. 7b, the CGT’s lower energyband shift defines narrower ntype band edges but wider ptype band edges.
In contrast, a positive electric field, i.e., forward bias, is achieved by connecting the battery positive terminal to FGT and the negative terminal to CGT. The positive terminal to FGT causes CGT’s electrons to move to FGT, resulting in a smaller or inverted interface dipole (middle panel in Fig. 6d). On the other hand, the negative terminal to CGT leads to the CGT’s higher energyband shift relative to the Fermi level. Furthermore, as shown in the right panel of Fig. 7b, the CGT’s higher energyband shift sets wider ntype band edges but narrower ptype band edges.
Figure 7c illustrates how the band edges change with the external electric field. As the external electric field increases, ntype band edges widen, and ptype band edges narrow, regardless of the stacked structures of E_{H}, E_{I}, and E_{L}. However, slightly different bandedge values depending on the stacked structures cause differences in the contact type as the external electric field changes. In the E_{L} heterobilayer, the ntype contact is maintained in the range of − 0.5 VÅ^{−1} to 0.5 VÅ^{−1}. In the E_{H} heterobilayer, the ntype contact changes to a particular contact (\({\Delta }_{n,\downarrow }<0\)) below − 0.3 VÅ^{−1}. In the E_{I} heterobilayer, the ntype contact changes to ptype contact above 0.36 VÅ^{−1}. These differences, depending on the different stacked structures, originate from the different initial potential differences (\(\updelta {V}_{\mathrm{eff}}\) when there is no external electric field) due to different asymmetric interface structures (Supplementary Information).
Conclusions
We presented the structural, electronic, and magnetic properties of layered FGT/CGT twisted moiré heterobilayers and the electrostatic gating effect. Our results show that the moiré heterobilayer with a twist angle of 30° is an FM metal with an ntype CGT layer regardless of various stacked structures. In the spin transfer of the FM heterobilayer, an effective spinup electron transfer occurs from FGT to CGT. Furthermore, the spinup hybridized bands between Cr and Fe2 atoms crossing the Fermi level are found only in the FM heterobilayer, not in the AFM heterobilayer, independent of stacking. It also turns out that the band alignment of the CGT layer is explained by the effective potential difference at the interface. Our findings further reveal that, in the FM heterobilayer, the external electric field regulates the spinup electron transfer between the FGT and CGT layers and affects the band edges of the CGT layer. Our study could provide helpful information for understanding the effects of stacking, spin alignment, spin transfer, and electrostatic gating in the magnetic 2D vdW metal/semiconductor heterobilayer.
Data availability
All data generated or analyzed during this study are included in this published article.
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Acknowledgements
E. Ko was supported by an individual Grant (CG075002) at Korea Institute for Advanced Study (KIAS). E. Ko thanks YoungWoo Son and Beom Hyun Kim for useful discussions. E. Ko expresses gratitude to Sehoon Oh for assisting with developing the SIESTA code used to calculate the unfolded band structure. This work was supported by the Center for Advanced Computation at KIAS and by the KISTI Supercomputing Center with supercomputing resources, including technical support (KSC2019CRE0208).
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Ko, E. Hybridized bands and stackingdependent band edges in ferromagnetic Fe_{3}GeTe_{2}/CrGeTe_{3} moiré heterobilayer. Sci Rep 12, 5101 (2022). https://doi.org/10.1038/s4159802208785x
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DOI: https://doi.org/10.1038/s4159802208785x
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