Highly persistent spin textures with giant tunable spin splitting in the two-dimensional germanium monochalcogenides
Moh. Adhib Ulil Absor, Yusuf Faisho, Muhammad Anshory, Iman Santoso, Sholihun, Harsojo, Fumiyuki Ishii
HHighly persistent spin textures with giant tunablespin splitting in the two-dimensional germaniummonochalcogenides
Moh. Adhib Ulil Absor , ∗ , Yusuf Faishol , MuhammadAnshory , Iman Santoso , Sholihun , Harsojo , and FumiyukiIshii Department of Physics, Universitas Gadjah Mada, Sekip Utara BLS 21 Yogyakarta55281, Indonesia. Nanomaterials Research Institute, Kanazawa University, 920-1192, Kanazawa,Japan.E-mail: ∗ [email protected] February 2021
Abstract.
The ability to control the spin textures in semiconductors is a fundamentalstep toward novel spintronic devices, while seeking desirable materials exhibitingpersistent spin texture (PST) remains a key challenge. The PST is the propertyof materials preserving a unidirectional spin orientation in the momentum space,which has been predicted to support an extraordinarily long spin lifetime of carriers.Herein, by using first-principles density functional theory calculations, we report theemergence of the PST in the two-dimensional (2D) germanium monochalcogenides(GeMC). By considering two stable formation of the 2D GeMC, namely the pureGe X and Janus Ge XY monolayers ( X, Y = S, Se, and Te), we observed the PSTaround the valence band maximum where the spin orientation is enforced by the lowerpoint group symmetry of the crystal. In the case of the pure Ge X monolayers, wefound that the PST is characterized by fully out-of-plane spin orientation protectedby C v point group, while the canted PST in the y − z plane is observed in the caseof the Janus Ge XY monolayers due to the lowering symmetry into C s point group.More importantly, we find large spin-orbit coupling (SOC) parameter in which the PSTsustains, which could be effectively tuned by in-plane strain. The large SOC parameterobserved in the present systems leads to the small wavelength of the spatially periodicmode of the spin polarization, which is promising for short spin channel in the spinHall transistor devices.
1. Introduction
Finding novel materials with strong spin-orbit coupling (SOC) has been one of theimportant research theme in the field of spintronics [1]. The large SOC effect isindispensable in spintronics since it could effectively manipulate the spin of electronelectrically, which plays a central role in many intriguing phenomena such as spin a r X i v : . [ c ond - m a t . s t r- e l ] F e b relaxation [2, 3], hidden spin polarization effect [4, 5, 6, 7], spin Hall effect [8], spingalvanic effect [9], and spin ballistic transport [10]. In a system lacking an inversioncenter, the SOC induces momentum-dependent spin-orbit field (SOF) lifting Kramer’sspin degeneracy and leading to a complex k -dependent spin texture of the electronicbands through the so-called Rashba[11] and Dresselhauss [12] effects. In particular,the Rashba spin texture can be manipulated electrically to produce non equilibriumspin polarization [13, 14], which has potential application as spin-field effect transistor(SFET) [15, 16]. Although the large SOC is beneficial for spintronic devices, it is alsoknown to induces the undesired effect of causing the spin decoherency. In a diffusivetransport regime, the momentum-dependent SOF induces spin randomization by aprocess known as the Dyakonov-Perel spin relaxation [17], which significantly reducesthe spin lifetime, and hence limits the performance of the spintronics functionality.The problem of the spin dephasing by the SOC can be eliminated by designing thematerials to exhibit unidirectional SOF. Here, the spin texture is enforced to be uniformand independent from the electron momentum, called as the persistent spin texture(PST) [18, 19, 20], arising when the linear Rashba and Dresselhauss contributionscompensate each other. Such a peculiar spin textures leads to a spatially periodicmode of the spin polarization in the crystal known as persistent spin helix (PSH) mode,enabling long-range spin transport without dissipation [19, 21]. The PST has beenpreviously observed on [001]-oriented semiconductors quantum well (QW) having anequal the Rashba and Dresselhauss SOC parameters [22, 23, 24, 25], or on [110]-orientedsemiconductor QW [26] described by the [110] Dreseelhauss model [19]. Similar to the[110]-oriented QW, the PST has also been reported on strained LaAlO /SrTiO interface[27]. Although achieving the PST requires to control the Rashba and DresselhaussSOC parameters, it is technically non-trivial since both the parameters are materialdependent. This has triggered much attention to find novel systems where the PST canbe observed intrinsically.Recently, the concept of the PST has been developed in more general way byenforcing the symmetry of the crystal rather than fine tuning the SOC parameters.For instant, the PST protected by nonsymmorphic space group has been proposed, asrecently reported on various three-dimensional (3D) bulk systems such as BiInO with P na space group [28], CsBiNb O with P am space group [29], and Ag Se with P space group [30]. Moreover, the symmetry-protected PST with purely cubic spinsplitting has been predicted in the 3D bulk materials crystallizing in the ¯6 m Pb O , Pb Br F , and Pb Cl F compounds[31]. Furthermore, the canted PST has been reported in wurtzite ZnO [10¯10] surfacehaving C s point group symmetry impossed by the non-polar direction of the surface[32]. In addition, the symmetry-enforced PST has also been observed in several two-dimensional (2D) systems, although it is still very rarely discovered. For the best of ourknowledge, only few classes of the 2D materials that has been reported to support thePST including WO Cl monolayer with P mc space group [33] and various group-IVmonochalcogenide monolayers with P mn space group such as SnSe [34, 35] and SnTemonolayers [36, 37]. More recently, the PST induced by the lower symmetry of thestructure has been predicted on several 2D transition metal dichalcogenides with theline defect [38, 39].Although the PST has been widely studied in the 3D bulk and QW systems,the search for the ultra-thin 2D materials supporting the PST has lately been verydemanding of attention because of their potential for miniaturization spintronic devices[36, 35, 37]. In this work, we predict the emergence of the PST in the 2D germaniummonochalcogenides (GeMC) monolayers by using first-principles density functionaltheory calculations. We have considered two stable formations of the 2D GeMCmonolayers, namely the pure Ge X and Janus Ge XY monolayers ( X, Y = S, Se, andTe), and found that the PST is observed around the valence band maximum where thespin orientation is enforced by the lower point group symmetry of the crystal. In thecase of the pure Ge X monolayers, we found that the PST is characterized by the fullyout-of-plane spin orientation, which is protected by C v point group. On the other hand,the PST is canted in the y − z plane for the case of the Janus Ge XY monolayers, whichis due to the lowering symmetry into C s point group. More interestingly, we identifiedlarge SOC parameter in the spin-split bands where the PST maintains, which could beeffectively regulated by applying in-plane strain. The observed large SOC parameter inthe present systems results in that the small wavelength of the spatially periodic modeof the spin polarization is achieved. Thus, we proposed the present system as a shortspin channel in the spin Hall transistor devices. Figure 1.
Crystal structure corresponding to the symmetry operations for: (a) pure2D Ge X and (b) Janus Ge XY monolayers. The unit cell of the crystal is indicatedby blue lines characterized by a and b lattice parameters in the x and y directions,respectively.
2. Model and Computational Details
The 2D GeMC monolayers crystallize in a black phosphorene-type structures [40],forming two different stable structures, called as the pure Ge X and Janus Ge XY monolayers [Fig. 1(a)-(b)]. The pure Ge X monolayers have C v or P mn space groupcharacterized by four symmetry operations in the crystal lattice [Fig. 1(a)]: (i) identityoperation E ; (ii) twofold screw rotation ¯ C y (twofold rotation around the y axis, C y ,followed by translation of τ = a/ , b/ a and b is the lattice parameters along (cid:126)a and (cid:126)b directions, respectively; (iii) glide reflection ¯ M xy (reflection with respect to the xy plane followed by translation τ ); and (iv) reflection M yz with respect to the yz plane.By replacing one of the X atoms in the Ge X monolayers with the Y atoms, we get thejanus Ge XY monolayers [Fig. 1(b)]. Here, only the M yz mirror symmetry survives,thus lowering the symmetry into C s point group.We performed first-principles DFT calculations using the generalized gradientapproximation (GGA) [41] implemented in the OpenMX code [42]. Here, we adoptednorm-conserving pseudo potentials [43] with an energy cutoff of 350 Ry for chargedensity. The 12 × × k -point mesh was used. The wave functions were expandedby linear combination of multiple pseudo atomic orbitals generated using a confinementscheme [44, 45], where two s -, two p -, two d -character numerical pseudo atomic orbitalswere used. The SOC interaction was included self consistently in all calculations byusing j -dependent pseudo potentials [46].We calculated the spin textures by deducing the spin vector component ( S x , S y , S z ) of the spin polarization in the reciprocal lattice vector (cid:126)k obtained from the spindensity matrix of the spinor wavefunctions [47]. The spin density matrix, P σσ (cid:48) ( (cid:126)k, µ ), arecalculated using the following relation, P σσ (cid:48) ( (cid:126)k, µ ) = (cid:90) Ψ σµ ( (cid:126)r, (cid:126)k )Ψ σ (cid:48) µ ( (cid:126)r, (cid:126)k ) d(cid:126)r = (cid:88) n (cid:88) i,j [ c ∗ σµi c σ (cid:48) µj S i,j ] e (cid:126)R n · (cid:126)k (1)where S ij is the overlap integral of the i -th and j -th localized orbitals, c σµi ( j ) is expansioncoefficient, σ ( σ (cid:48) ) is the spin index ( ↑ or ↓ ), µ is the band index, and (cid:126)R n is the n -thlattice vector. Here, Ψ σµ ( (cid:126)r, (cid:126)k ) is the spinor Bloch wave function, which is obtained fromthe OpenMX calculations after self-consistent is achieved. Table 1.
The optimized lattice parameters [ a (˚A), b (˚A)], structure anisotropy factor, κ , defined as κ = ( a − b ) / ( a + b ), the formation energy, E f (eV), and the band gap, E g (eV), obtained for the 2D GeMC monolayers. The star (*) indicates the semiconductorwith direct bandgap.GeC Monolayers a (˚A) b (˚A) κ E f (eV) E g (eV)GeS 3.68 4.40 0.09 1.45GeSe 3.99 4.39 0.05 1.10 ∗ GeTe 4.27 4.47 0.02 0.92Ge SSe 3.84 4.47 0.08 0.03 1.32Ge STe 4.03 4.53 0.06 0.15 0.97Ge SeTe 4.14 4.47 0.04 0.05 0.88
Figure 2.
Electronic band structure corresponding to the density of states (DOS)projected to the atomic orbitals for: (a) GeS, (b) GeSe, (c) GeTe, (d) Ge SSe, (e)Ge STe, and (f) Ge SeTe monolayers. For the band structure, the black and pinkcolours indicate the calculated bands without and with the SOC, respectively. Theinsert of Fig. 2(a) shows the first Brillouin zone used in the band structure calculations.
We used a periodic slab to model the pure Ge X and Janus Ge XY monolayers,where a sufficiently large vacuum layer (20 ˚A) is applied in order to avoid interactionbetween adjacent layers. We used the axes system where the layer is chosen to sit onthe x − y plane [Fig. 1(a)-(b)]. During the structural relaxation, the energy convergencecriterion was set to 10 − eV. The lattice and positions of the atoms were optimized untilthe Hellmann-Feynman force components acting on each atom was less than 1 meV/˚A.In the case of the Janus Ge XY monolayers, the energetic stability of the structure isconfirmed by calculating the formation energy, E f , through the following relation, E f = E tot [ Ge XY ] − E tot [ Ge X ] − E tot [ Ge Y ] , (2)where E tot [ Ge XY ], E tot [ Ge X ], and E tot [ Ge Y ] are the total energy of Ge XY , Ge X ,and Ge Y , respectively. The optimized structural-related parameters of the Ge X andJanus Ge XY monolayers, and the formation energy of the Janus Ge XY monolayersare summarized in Table 1, and are in a good agreement overall with previously reporteddata [36, 48, 49].
3. Results and discussion
Fig. 2 shows the band structure of various 2D GeMC monolayers calculated along thefirst Brillouin zone (FBZ) [see the insert of Fig. 2(a)] corresponding to their density ofstates (DOS) projected to the atomic orbitals. Without the SOC, the GeCM monolayersare semiconductors with an indirect band gap, except for the GeSe monolayer having adirect band gap. The calculated band gap is smaller for the heavier materials (GeTe,Ge SeTe) and larger for the lighter materials (GeS, Ge SSe) [see Table 1], which isconsistent with previous reports [36, 50, 49]. The valence band maximum (VBM) islocated at the Γ − Y line, while the conduction band minimum (CBM) show a differentlocation for the different chalcogen atoms. In the case of the pure Ge X monolayers, theCBM is located at the Γ − X line for both the GeS and GeTe monolayers [Figs. 2(a) and2(c)], while it is located at the Γ − Y line for the GeSe monolayer [Fig. 2(b)]. In contrast,for the case of the Janus Ge XY monolayers, the CBM is located at the Γ − X line, Γpoint, and M − Y line for the Ge SSe, Ge STe, and Ge SeTe monolayers, respectively[Figs. 2(d)-(f)]. Our calculated DOS projected to the atomic orbitals confirmed thatthe CBM is mainly originated from the contribution of the Ge- p and X ( Y )- s orbitals,while the VBM is dominated by the Ge- s and X ( Y )- p orbitals.Turning the SOC leads to a sizable splitting of the bands due to the broken ofthe inversion symmetry, which is especially pronounced in the bands around the Y point at the VBM [Fig 2(a)-(f)]. Here, the larger band splitting is identified for themonolayers with heavier elements such as Se and Te atoms. However, we observe theband degeneracy for the (cid:126)k along the Γ − Y line in the case of the pure Ge X monolayers,which is protected by the M yz and ¯ M xy symmetry operations [see the Appendix Afor detail symmetry analysis]. Conversely, this degeneracy is lifted for the case ofthe Ge XY monolayers, which is due to the broken of the ¯ C y and ¯ M xy symmetryoperations. Since both the the GeTe and Ge SeTe monolayers have the largest bandsplitting among the members of 2D GeMC monolayers, in the following discussion, wefocused on these monolayers as a representative example of the pure Ge X and Ge XY monolayers, respectively.Figs. 3(a)-(b) show the map of the spin-splitting energy along the entire FBZcalculated at near the VBM for GeTe and Ge SeTe monolayers, respectively. We cansee that both the monolayers exhibit a strongly anisotropic spin splitting around the
Figure 3. (a)-(b) Spin-splitting energy of the bands around the VBM mappedon the k -space in the first Brillouin zone for the GeTe and Ge SeTe monolayers,respectively, are shown. The magnitude of the spin-splitting energy, ∆ E , definedas ∆ E = | E ( k, ↑ ) − E ( k, ↓ ) | , which is representated by the color scales. (c)-(d)Orbital-resolved electronic band structures calculated around the VBM for the GeTeand Ge SeTe monolayers, respectively, are given. The radii of the circles reflect themagnitudes of spectral weight of the particular orbitals to the band. Y point, where the large different splitting energy between the spin-split bands alongthe M − Y and Y − Γ lines. Here, the maximum splitting energy up to 0.28 eV (0.23eV) is achieved along the Y − M line at the GeTe (Ge SeTe) monolayer. By calculatingorbitals-resolved projected to bands at near the VBM around the Y point, we clarifiedthat the large splitting along the Y − M line is mainly contributed from the stronghybridization between the s orbital of the Ge atom and the p y orbital of the chalcogenatoms (Te atom in the GeTe monolayer; and the mixing between Se and Te atoms inthe Ge SeTe monolayer) [Fig. 3(c)-(d)]. Remarkably, the calculated splitting energies inboth the GeTe and Ge SeTe monolayers are comparable with that observed on several2D materials including the pure and Janus transition metal dichalcogenide [0.15 eV -0.55 eV] [51, 52, 53, 54].To further demonstrate the nature of the observed anisotropic spin splitting, weshow in Figs. 4(a) and 4(b) the calculated results of the spin textures around the Y point at near the VBM for the GeTe and Ge SeTe monolayers, respectively. We alsohighlight the spin textures by providing the spin-resolved projected to the bands asdepicted in Figs. 4(c) and 4(d). It is clearly seen that the spin textures of the GeTemonolayer are characterized by fully out-of-plane spin components S z , while the in-plane spin components ( S x , S y ) are almost zero [Figs. 4(a) and 4(c)]. These featuresof the spin textures are strongly different either from the Rashba [11] and Dresselhaus[12] spin textures. Moreover, these typical spin textures gives rise to the so-calledpersistent spin textures (PST) [18, 19, 20], which is consistent with that described bythe [110] Dresselhaus model in a [110]-oriented semiconductor QW [19, 26] and similarto that recently reported on various 2D materials such as WO Cl [33], SnSe [34, 35]and SnTe monolayers [36, 37]. The emergence of the PST preserves in the case of theGe SeTe monolayer but it shows the different features, i.e., the spin textures are mainlycharacterized by S z and S y spin components, except at k x = 0 (along the Γ − Y line)where the S x spin component retains [Figs. 4(b) and 4(d)]. Accordingly, quasi-onedimensional spin textures are observed, which is uniformly tilted from the out-of-plane z - to the in-plane y -direction at k x (cid:54) = 0, forming a canted PST in the y − z plane similarto that observed on ZnO (10¯10) surface [32]. The existence of the PST in both the GeTeand Ge SeTe monolayers is expected to induce a unidirectional SOF, protecting the spinsfrom decoherence through suppressing the Dyakonov-perel spin-relaxation mechanism[17]. Therefore, an ultimately long spin lifetime is achieved [21], which is promising forrealization of an efficient spintronics device.The physical origin of the band splitting and spin textures observed around the Y point can be clarified by using a simple Hamiltonian model in the presence of SOCderived from the symmetry of the wave vector k . Since the GeTe monolayer possesses C v point group symmetry at Y point, the symmetry adapted (cid:126)k · (cid:126)p Hamiltonian can bewritten in the linear term of the k as H C v Y = E ( k ) + αk x σ z , (3)where E ( k ) = ¯ h [( k x / m ∗ x ) + ( k y / m ∗ y )] is the nearly-free-hole energy, k x and k y are thewave vectors in the x - and y -directions, respectively, m ∗ x and m ∗ y are the hole effectivemass in the x - and y -directions, respectively, σ z is the z -component of the Pauli matrices,and α is the SOC parameter in the spin-split bands along the k x direction [detailsderivation of the Hamiltonian H C v Y , see Appendix B].We can see clearly that the derived H C v Y in Eq. (3) is only coupled with k x σ z term,justifying that the spin textures around the Y point being oriented in the fully out-of-plane z -direction. As expected, the out-of-plane PST is achieved, which is consistent-well with the calculated spin textures shown in Figs. 4(a) and 4(c). Moreover, by solvingthe eigenvalue problem involving the H C v Y , we obtain that the energy dispersion can Figure 4.
The spin textures around the Y point at the outer branch of the spin-splitbands at the VBM calculated for (a) GeTe and (b) Ge SeTe monolayers. In Fig. 4(a)-(b), the arrows show the in-plane components of the spin, while the colour indicatethe out-of-plane component of the spins, S z . Spin-resolved projected to the spin-splitbands at the VBM around the Y point for (c) GeTe and (d) Ge SeTe monolayers. Thecolor bars in Fig. 4(c)-(d) represents the expected values of the S x , S y , and S z spincomponents. be expressed as E C v Y, ± ( k ) = E ( k ) ± αk x . This fact implies that the bands are splittedalong the Y − M line ( k x ) but are degenerated along the Y − Γ line ( k y ), which is inagreement with the calculated band dispersion of the VBM around the Y point shownin Fig. 2(c). By fitting E C v Y, ± ( k ) to the DFT band dispersion of the GeTe monolayeraround the Y point at the VBM, we find that the calculated value of α is 3.93 eV˚A,which supports the large spin-splitting energy along the Y − M line ( k x ) in agreementwith the DFT results provided in Fig. 3(a).A similar analysis can also be applied to explain the band splitting and spin texurein the Ge SeTe monolayer. Here, the lowering symmetry into C s point group leads tothe fact that the (cid:126)k · (cid:126)p Hamiltonian around the Y point can be expressed as H C s Y = E ( k ) + ( α σ z + α σ y ) k x + βk y σ x , (4)where σ x , σ y , σ z are the x -, y -, and z -components of the Pauli matrices, respectively, α and α are the SOC parameters in the spin-split bands along the k x direction, and β is0the SOC parameter in the spin-split bands along the k y direction [details derivation ofthe Hamiltonian H C s Y , see the Appendix B].The first and second terms of H C s Y in Eq. (4) imposed the spin textures to exhibitthe S z and S y spin components at k x (cid:54) = 0, while the S x spin component retains at k x = 0 due to the third term of the H C s Y . As a result, the canted PST in the y − z planeis expected at k x (cid:54) = 0, which matches well with the calculated spin textures obtainedfrom the DFT calculation shown in Figs. 4(b) and 4(d). Moreover, the H C s Y leads tothe energy dispersion, E C s Y, ± ( k ) = E ( k ) ± (cid:113) α k x + β k y , where α = (cid:113) α + α is thetotal SOC parameter defined in the spin-split bands along the k x direction. Our fittingcalculation to the DFT band dispersion of the Ge SeTe monolayer around the Y pointat the VBM found that the calculated value of α parameter (3.10 eV˚A) is much largerthan that of the β parameter (0.008 eV˚A). The large different value between α and β indicates that the band splitting is strongly anisotropic between Y − M ( k x ) and Γ − Y ( k y ) lines, which is consistent with the band dispersion shown in Fig. 2(d). In addition,the large value of α parameter implies that the large spin-splitting energy is achievedalong Y − M ( k x ) line, which is also consistent with the spin-splitting energy obtainedfrom the DFT calculations described in Fig. 3(b). Table 2.
The calculated SOC parameter ( α ) [in eV˚A] and the wavelength of the PSH( l PSH ) [in nm] for the GeTe and Ge SeTe monolayers at the VBM around the Y pointcompared with those observed on various PST materials.PST Systems α (eV˚A) l PSH (nm) Reference
GeMC
GeTe 3.93 6.53 This workGe SeTe 3.10 8.52 This work
Semiconductor QW
GaAs/AlGaAs (3.5 - 4.9) × − (5.0 - 10) × Ref.[22, 23]InAlAs/InGaAs 1.0 - 2.0 × − Ref.[24, 25]Strained LaAlO /SrTiO × − Semiconductor Surface
ZnO(10-10) surface 34.78 × − × Ref.[32]
3D Bulk systems
BiInO O
2D systems
SnTe 1.2 - 2.85 1.82 - 8.8 Ref.[36, 37]SnSe 0.76 - 1.15 Ref.[35]Doped SnSe 1.6 - 1.76 1.2 - 1.41 Ref.[34]WO Cl with line defect 0.2 - 1.14 6.33 - 28.19 Ref.[39]WS with line defect 0.14 - 0.26 8.56 - 10.18 Ref.[38] We emphasized here that among the member of the 2D GeMC monolayers, the GeTeand Ge SeTe monolayers have the largest SOC parameter ( α ). Importantly, comparedto the other PST systems, the calculated value of α in both the monolayers is also thelargest of all known PST materials so far [see Table 2]. Moreover, the emergence of the1PST in these monolayers leads to the spatially periodic mode of the spin polarization,forming the persistent spin helix (PSH) mode with the wavelength of l PSH = π ¯ h / ( m ∗ x α )[19]. We can estimate l PSH by considering m ∗ x and α parameters obtained from the bandsalong the Y − M line, and find that the calculated value of l PSH is about 6.53 nm and8.52 nm for the GeTe and Ge SeTe monolayers, respectively. These values of l PSH arethree-orders of magnitude smaller than that reported on semiconductor QW systemsand comparable with that predicted on the 3D bulk PST system such as BiInO (2.0nm) [28] and 2D PST systems including the doped SnSe monolayer (1.2 nm - 1.41 nm)[34] and the SnTe monolayers (1.82 nm - 8.8 nm) [36, 37] [see Tabel 2]. Remarkably,the large SOC parameter ( α ) and the small wavelength of the PSH mode ( l PSH ) foundin the present systems are important for miniaturization spintronic devices operating atroom temperatures.Next, we discuss the tunability of the observed PST in the GeTe and Ge SeTemonolayers by introducing in-plane strain. Here, we consider the strain created bychanging the lattice parameters along (cid:126)a and (cid:126)b directions [see Fig. 1(a)]. We definedegree of the strain as (cid:15) i = (cid:126)a,(cid:126)b = ( a i − a i ) /a i × a i and a i are the latticeparameters of equilibrium and strained structures, respectively, calculated along theselected i = ( (cid:126)a,(cid:126)b ) direction. Although the electronic properties of of the monolayers aresensitive to the strain, the PST preserves under large strain (up to ± α ) and the strain, which is plotted in Fig. 5(a). We find that stretching orcompressing the monolayers significantly changes the magnitude of α , i.e., the value of α sensitively increases (decreases) under the tensile (compressive) strains [Fig. 5(a)]. Forinstant, in the case of the GeTe monolayer, when the tensile strain of +6% is appliedalong the (cid:126)a ( (cid:126)b ) direction, the value of α increases up to 6.65 eV˚A (5.75 eV˚A), which ismuch larger than the original value of 3.93 eV˚A. The similar trend also holds for the caseof the Ge SeTe monolayer where the increasing value of α up to 4.57 eV˚A (4.02 eV˚A) isachieved under the tensile strain of +6% along the (cid:126)a ( (cid:126)b ) direction. Benefiting from thestrong enhancement of α by the tensile strains, we should ensure that the wavelength l PSH of the PSH mode becomes significantly smaller than that of the original systems[see Fig. 5(b)], which is important for miniaturization spintronic devices.Finally, based on the highly PST found in the present monolayers, we propose a spinHall transistor (SHT) device as illustrated in Fig. 5(c). Motivated from the previouswork reported by Choi, et. al.[55], we design the SHT device consisted of three parts,namely, region I, II, and III, representing an injection, gate-controlled, and detectorregions, respectively. In the region I, the pure spin currents polarized along the out-of-plane ( z ) orientation can move to the x -direction and are efficiently injected into themiddle part of the device (region II) through direct spin Hall effect (SHE). Subsequently,in the region II in which the GeTe (or Ge SeTe) monolayer is take placed, the out-of-plane spin polarizations effectively induce the PSH mode in the crystal, which could bemodulated by the out-of-plane electric filed ( E z ) driven by the gate electrode depositedon the top of the monolayers. Here, as a spin channel of the SHT, the minimum length2 Figure 5. (a) The calculated SOC parameter, α , and (b) the wavelength of thepersistent spin helix (PSH) mode, l PSH , as a function of strain (cid:15) i = (cid:126)a,(cid:126)b for the GeTe andGe SeTe monolayers are shown. (c) Schematic view of the the spin Hall transistordevice consisted of three regions: the left part (Region I) is the spin Hall injectorregion, the middle part (Region II) is the gate-controlled region, and the right part(Region III) is the inverse spin Hall detector region. of the spin channel in the region II should be proportional to the wavelength of the PSHmode, l PSH . For instant, a very short channel length of about 3.81 nm (5.62 nm) isachieved when the GeTe (Ge SeTe) monolayer with +6% tensile strain along the the (cid:126)a direction is used as a spin channel in the region II, which is much smaller than the channellength of the SHT reported on InAs QW system [55]. Moreover, the presence of E z willdetermine the on/off logical functionality based on the preserving of the out-of-planespin polarizations in the PSH mode. When E z = 0, the out-of-plane spin polarizationsare robust to maintain the PSH mode, thus the spin polarizations are efficiently injectedinto the region III without losing the spin-information. By utilizing the inverse SHEeffect in the region III, the out-of-plane spin polarization is converted into the electriccurrents along the y direction, which generates the Hall voltage. In contrast, the PSHmode is broken when E z (cid:54) = 0 is applied [see Appendix C for the calculated spin-resolved3projected to bands for the systems with an external out-of-plane electric field E z ], whichresults in the spin decoherency and significantly decreases the Hall voltage detected inthe region III.
4. Conclusion
In summary, we have predicted the emergence of the PST in the 2D GeMC monolayerby performing first-principles DFT calculations combined with the symmetry analysis.We have studied the two stable formations of 2D GeMC monolayers, namely the pureGe X and Janus Ge XY germanium mochalcogenides ( X, Y : S, Se, and Te) monolayersand confirmed that the PST is observed around the VBM where the spin orientationis imposed by the lower point group symmetry of the crystal. In the case of the pureGe X monolayers, we have found that the PST is characterized by the fully out-of-plane spin polarization, which is protected by C v point group symmetry, while thePST is canted in the y − z plane for the case of the Janus Ge XY monolayers dueto the lowering symmetry into C s point group. More importantly, we have found thatthe PST sustains in the spin-split bands exhibiting large SOC parameter, which couldbe effectively controlled by applying the in-plane strains. The larger SOC parameterobserved in the present systems leads to the fact that the smaller wavelength of the PSHmode is achieved, which is useful for miniaturization of the spin channel in the spin Halltransistor devices.Since the PST found in the present study is solely dictated by the C v and C s pointgroup symmetries of the crystal, we expect that the similar features are also sharedby other materials having the similar symmetry. Recently, there are a few number ofother 2D materials that are predicted to maintain the similar symmetry of the crystals,including the 2D ferroelectric Ga XY ( X = S, Se, Te; Y = Cl, Br, I) family[7] and 2Dsingle-elemental multiferroic materials such as Te, and Bi [56, 57]. Therefore, we haveexpected that our predictions will stimulate further theoretical and experimental studiesin the exploration of the PST systems in the 2D-based materials, thus broadening therange of the 2D materials for future spintronic applications. Appendix A. Symmetry analysis for the band degeneracy along the Γ − Y line in the Ge X monolayers In this appendix, we discuss the origin of the double degeneracy in the band structuresof the Ge X monolayers along the Γ − Y line in the presence of SOC by considering thesymmetry of the wave vector, (cid:126)k . The wave vector (cid:126)k along the Γ − Y line is invariantunder ¯ C y screw rotation and ¯ M xy glide mirror reflection. Both the symmetry operationshold the following algebra:¯ M xy ¯ C y = − e − ik x ¯ C y ¯ M xy (A.1)where the minus sign is originated from the fact that two spin rotation operators σ y and σ z are anti-commutative, { σ y , σ z } = 0, so that both ¯ C y and ¯ M xy operators are also4anti-commutative, (cid:110) ¯ C y , ¯ M xy (cid:111) = 0, on the Γ − Y line. As a result, for an eigenvector | ψ m (cid:105) of ¯ M xy operator with the eigenvalue of m , we obtain the following relation,¯ M xy ( ¯ C y | ψ m (cid:105) ) = − m ( ¯ C y | ψ m (cid:105) ) . (A.2)The Eq. (A.2) shows that | ψ m (cid:105) and ¯ C y | ψ m (cid:105) ) are different states with the sameeigenvalue, thus ensures the double degeneracy along the Γ − Y line. Appendix B. Derivation of (cid:126)k · (cid:126)p Hamiltonian
To clarify the observed spin-splitting and the spin textures around the Y point, we derivean effective (cid:126)k · (cid:126)p Hamiltonian, which can be deduced from the symmetry considerations.Here, we assume that only the linear term with respect to wave vector (cid:126)k contributingto the SOC Hamiltonian. We construct the Hamiltonian by identifying all symmetry-allowed term such that the following relation is obtained[58]:ˆ O † ˆ H ( k ) ˆ O = ˆ H ( k ) , (B.1)where ˆ O denotes all symmetry operations belonging to the little group of the wave vectoraround the Y point supplemented by time reversal symmetry. Table B1.
Transformation rules for crystal momentum k and spin operator σ underconsidered point-group symmetry operations. Time-reversal symmetry defined as iσ y K , where K is complex conjugation and σ denotes Pauli matrices, revers both themomentum and spin. The point-group operations are defined as C y = iσ y , M yz = iσ x ,and M xy = iσ z .Symmetry operations ( k x , k y ) ( σ x , σ y , σ z ) C y ( − k x , k y ) ( − σ x , σ y , − σ z ) M yz ( − k x , k y ) ( σ x , − σ y , − σ z ) M xy ( k x , k y ) ( − σ x , − σ y , σ z ) For the case of the GeTe monolayer, the little group of
Y k -point is C v , comprisingtwo mirror symmetry operations, M yz and M xy , and one two-fold rotation C y around the y -axis. Taking into account the transformation role listed in Table B1, the symmetry-allowed linear spin-momentum coupling can be expressed asˆ H C v Y ( k ) = E ( k ) + αk x σ z . (B.2)In contrast, both M xy and C y symmetry operations are broken for the case ofthe Ge SeTe monolayer. Therefore, the crystal structure of the Ge SeTe monolayerbelongs to the C s point group. Accordingly, the little point group at the Y k -pointalso belongs to C s point group, which comprises only M yz mirror symmetry operation.The effective low-energy Hamiltonian around the Y point in the Ge SeTe monolayer canagain be deduced by considering only the M yz mirror symmetry operation. By usingtransformation role given in Table B1 for M yz operation, we obtain the following (cid:126)k · (cid:126)p Hamiltonian, ˆ H C s Y ( k ) = E ( k ) + ( α σ z + α σ y ) k x + βk y σ x . (B.3)5 Figure C1.
Spin-resolved projected to the bands at the VBM around the Y pointunder an out-of-plane external electric field of 0.1 V/˚A for: (a) GeTe and (b) Ge SeTemonolayers
Appendix C. Spin-resolved of the bands under an external out-of-planeelectric field
Figs. C1(a)-(b) show the calculated results of the spin-resolved projected to the bands atthe VBM around the Y point for the GeTe and Ge SeTe monolayers, respectively, underan external out-of-plane electric field of 0.1 V/˚A. In the case of the GeTe monolayer,we find that the PST is broken by the electric field as indicated by the appearance ofthe in-plane S x and S y spin components in the spin-split bands [Fig. C1(a)]. Similarly,the canted PST in the case of the Ge SeTe monolayer is also broken by the electricfield since the in-plane S x and S y spin components appear in the spin-split bands [Fig.C1(b)]. Acknowledgments
This work was partly supported by PD Research Grant (2021) and PDUPT researchgrant (2021) funded by RISTEK-BRIN, Republic of Indonesia. Part of this research wassupported by the BPPTNBH Research grant (2021) funded by Faculty of Mathematicsand Natural Sciences, Universitas Gadjah Mada, Republic of Indonesia. This work waspartly supported by Grants-in-Aid for Scientific Research (Grant No. 16K04875) fromJSPS and Grant-in-Aid for Scientific Research on Innovative Areas Discrete GeometricAnalysis for Materials Design (Grant No. 18H04481) from MEXT Japan. Thecomputation in this research was partly performed using the supercomputer facilities at6RIIT, Kyushu University, Japan. Part of the computation in this research was performedusing the computer facilities at Universitas Gadjah Mada, Republic of Indonesia.
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