Valley polarization caused by crystalline symmetry breaking

Abstract

In two-dimensional (2D) hexagonal lattices with inversion asymmetry, time-reversal (T) connected valleys are at the center of current valleytronic research. In order to trigger valley polarization, dynamical processes and/or magnetism have been considered. In this work, we propose a new mechanism, valley-contrasting sublattice polarization (VCSP), to polarize valleys by reducing the crystalline symmetry that connects the valleys. In our mechanism, significant valley polarization could be readily generated without magnetism, an electric field, or an optical process. Based on tight-binding model analysis and first-principle calculations, the control of valley polarization via crystalline symmetry can be successfully realized in concrete LaOBiS₂ polytypes with Peierls-like structure distortion. Our results provide an unprecedented possibility for exploring valley-contrasting physics.

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New concepts

Based on our proposed valley-contrasting sublattice polarization mechanism, we report for the first time the role played by crystallographic symmetry, rather than time-reversal asymmetry, in introducing valley polarization. Using this new mechanism, evident valley polarization could be readily generated without magnetism, electric fields, or optical processes. Tight-binding model analysis and first-principles calculations confirm that LaOBiS₂ is a concrete and ideal carrier to translate the new valley polarization paradigm to practical applications, suggesting high experimental possibility. Therefore, we believe that these intriguing findings will spark great attention in the field of valleytronics.

Introduction

In addition to the charge and spin degrees of freedom, valleys, i.e., the multiple energy extrema of the conduction and valence bands in momentum space, represent an emergent degree of freedom with which low-energy electrons can also be endowed. Valleytronics is analogous to spintronics, which utilizes the spin degree of freedom, as the valley degree of freedom can also be manipulated to encode, process and store information.^1–7 In recent years, the study of valleytronics has flourished with the accessibility of group-VI transition metal dichalcogenides (TMDCs, such as MoS₂, WS₂, and WSe₂)^1,8–12 in which two inequivalent valleys, K and K′, are connected by time-reversal symmetry (T).

In two-dimensional (2D) hexagonal lattices with inversion symmetry breaking, the functions of the K and K′ valleys are odd under T, and, therefore, opposite Berry curvatures and orbital magnetic moments are assigned to them.¹³ Thus, in order to use the valley index as an information carrier, several strategies have been proposed, such as dynamical processes,^12,14–19 magnetic fields, magnetic doping and the magnetic proximity effect.^20–28 From a device applications point of view, static methods are preferred over dynamical processes, and magnetic regulation usually weakens the device performance due to the unavailability of a large external magnetic field, clustering effect, and impurity scattering. In this respect, 2D ferrovalley materials with spontaneous valley polarization due to the intrinsic magnetism caused by T breaking have been proposed to overcome these challenges.^29–31 However, the few existing materials still suffer from many drawbacks, including insufficient valley polarization,^29,30 and the ferromagnetic ground state of 2D structures is still controversial.³¹

Compared to magnetic and optical control, the use of fully electric field control with existing semiconductor technology to achieve valley polarization is more promising. However, it is evident that under the existing spin–valley coupling paradigm, an electric field cannot couple to valleys since it cannot break the T. In order to realize electric control of valley polarization, a so-called valley-layer coupling (VLC) mechanism was proposed, in which direct interactions between valleys and external gate electric fields were introduced.³² In this case, crystalline symmetry, instead of T, connects the inequivalent valleys. The application of a gate electric field for external regulation still presents the problem of power consumption, and adequate materials are scarce. The electric control of valley polarization thus remains greatly challenging, and it is still highly desirable to search for an unpresented way to realize valley polarization in easily accessible materials.

In this work, we propose a tantalizing design principle for valley polarization deprecating the use of an external field, magnetism, or circularly polarized light to tackle these challenges. A new coupling mechanism, which we refer to as valley-contrasting sublattice polarization (VCSP), is proposed. In the paradigm in which valleys are connected by crystalline symmetry instead of T, p4mm (LG 55) can be added to the set of layer groups that meet the requirements, and model analysis confirms the feasibility of our VCSP mechanism to achieve valley polarization. In accordance with the symmetry requirement for our VCSP mechanism, specific materials were used to prove our idea via model analysis and first-principles calculations (calculation details can be found in the ESI†). The results shown that robust valley polarization can be achieved by VCSP, and our results thus enrich the significant new arena of symmetry-controlled valley-contrasting physics.

Results and discussion

In order to gain deep insight into the symmetry requirements for systems with symmetry-controlled valley polarization, it is necessary to clarify the difference between the patterns of valleys connected by T and by crystalline symmetry. In the 2D hexagonal Brillouin zone of nonmagnetic TMDCs with D_3h symmetry, for example, band edges are located at the K and K′ points. In the light of the C₃ symmetry operation of D_3h, the time-reversal point of K (i.e., −K) is equivalent to K′. The T and inversion asymmetry guarantee opposite Berry curvature and energy degeneracy between the K and K′ points. In this case, valley polarization can be achieved by T breaking. On the other hand, in order to remove the energy degeneracy of two inequivalent valleys by reducing the crystalline symmetry rather than T breaking, two inequivalent valleys must be located at the same time-reversal invariant momentum (TRIM) point, Fig. 1(a). In this situation, the crystalline symmetry operation U guarantees that they are degenerate in energy. As shown in Fig. 1(b), we find that 2D materials with a square lattice consisting of two sublattices can meet the basic symmetry requirements, provided that X and X′ occur at the requested TRIM point.³² It is evident that 2D materials with the layer group p4mm would be a rational choice, where X and X′ are connected by {C_4z|000} or {m_1–10|000}, rather than T.

Fig. 1 (a) The Brillouin zone of 2D square lattices. (b) A top view of a crystal framework with a checkerboard lattice; α and β represent two sublattices in the unit cell. Band structures calculated via the tight-binding model (c) with and (d) without {C_4z|000} and {m_1–10|000} symmetries. The system achieves valley polarization when {C_4z|000} and {m_1–10|000} are broken.

Based on the above analysis, an effective model was developed to discuss our new design principle to realize valley polarization. As Fig. 1(b) shows, a single-layer checkerboard lattice with p4mm symmetry was considered as a prototype structure. Here, it is assumed that low-energy physics occurs at the point X (X′) with little group C_2v, with the conduction band minimum (CBM) and valence band maximum (VBM) corresponding to the one-dimensional representation B₁ or B₂, respectively. Therefore, the p_x (p_y) orbitals of the different sublattices (α and β) contribute to the CBM and VBM at the point X (X′), and the basis states are chosen to be |μ,p_i,k〉, where μ ∈ {α,β} and i ∈ {x,y} represent the different sublattices and atomic orbitals, respectively. By considering the orbital interaction without spin–orbit coupling (SOC), the Hamiltonian can be written as

where the wave vector k is measured from the Brillouin zone center; ε_α(β), α₁, , β₁, , γ₁ and γ₂ are material-dependent parameters. [small sigma, Greek, circumflex] represents the Pauli matrices for the sublattice index. As shown in Fig. 1(c), this model gives rise to a pair of valleys with energy degeneracy at the points X and X′. The two inequivalent valleys are connected by {C_4z|000} and {m_1–10|000}. For valley polarization to occur, a perturbation H′ = Δ(I_2×2⊗[small sigma, Greek, circumflex]_z) is necessary to break these symmetries and lift the energy degeneracy, as indicated in Fig. 1(d). In materials with heavy elements, the large band splitting caused by SOC should be considered. In the ESI,† the Hamiltonian with SOC is shown, and the relevant band structures are presented in Fig. S1 (ESI†). As a consequence of the time-reversal-invariant momenta (TRIM), however, the Kramer degeneracy of the point X (X′) cannot be broken. As a result, SOC introduces mainly Rashba-type band splitting, and the two-fold degeneracy at the point X (X′) is still present. Without loss of generality, the influence of SOC at the point X (X′) will not be discussed in the main text.

In order to better understand the physics of valley polarization, we proposed the VCSP mechanism for the material lattices shown in Fig. 1(b). The sublattice polarization is defined as P_n(k)

where n is the band index and k is the wave vector in the 2D Brillouin zone; thus, P_n(k) reflects the polarization of the sublattice states |α(β),p_i,k〉. As shown in Fig. 2(a), values of P > 0 (P < 0) mean that the CBM and VBM are mainly contributed by the terms 〈α,p_x,k|α,p_x,k〉 (〈α,p_y,k|α,p_y,k〉) and 〈β,p_y,k|β,p_y,k〉 (〈β,p_x,k|β,p_x,k〉), respectively. As a consequence, the sublattice polarizations of the CBM and VBM at the point X (X′) are P_c(X) = 1 (P_c(X′) = −1) and P_v(X) = −1 (P_v(X′) = 1), respectively, and the value of P_n(k) is independent of the parameters of the tight-binding Hamiltonian. In analogy to the valley-contrasting spin polarization in 2D TMDC hexagonal lattices, VCSP corresponds to valley-contrasting sublattice atomic orbital polarization. In VCSP, the crystalline symmetry operation U connecting the two inequivalent valleys (U⁻¹P(X)U = −P(X′)) should be removed in order to generate the valley polarization, since this crystalline symmetry protects the energy degeneracy of the X and X′ valleys, see Fig. 2(b). Therefore, the quantity P_n(k) offers a feasible way to inspect the band behavior, since two inequivalent valleys with opposite P would show opposite energy shifts.

Fig. 2 The band alignment of valley-contrasting sublattice polarization (VCSP) in the presence of (a) U and (b) U breaking. U corresponds to {C_4z|000} and {m_1–10|000} symmetries. Solid (dashed) curves denote the contributions of the p_x (p_y) orbitals, and purple and green represent the contributions from the α and β sublattices, respectively.

After demonstrating that the crystalline symmetry can control valley polarization, we now focus on material development. In the past few years, layered LaOBiS₂ has been widely studied due to its novel properties such as charge density instability,^33,34 hidden spin polarization,^35,36 electrically tunable Rashba effect, and quantum anomalous Hall effect.^37,38 However, the valleytronic properties of LaOBiS₂ and similar structures (ZrCuSiAs-type) have not been explored so far. As illustrated in Fig. 3(a) and (b), LaOBiS₂ adopts a square lattice with the layer group p4/nmm, and its unit cell consists of a triple-layer (TL) structure with the La₂O₂ layer sandwiched between two BiS₂ layers. The Bi (S) atom in the upper layer is vertically aligned with the S (Bi) atom in the lower layer, and the two BiS₂ layers are related by the glide operation ĝ_z = {m̂_z|τ}, which consists of a reflection in the x–y plane, followed by a non-primitive translation . In Fig. 4a–c, the electronic properties of TL LaOBiS₂ are presented. In the absence of SOC, the energy dispersion relation shows a semiconductor nature with direct band gaps located at the points X and X′. By projecting the bands onto different atomic orbitals, we find that the low-energy spectra near the Fermi level originate mainly from the BiS₂ layers, with the CBM and VBM near X (X′) points being dominated by the p_x (p_y) orbitals of Bi and S atoms, respectively. This indicates that the La₂O₂ layer acts as a barrier to prevent the interactions between the upper and lower BiS₂ layers, and, therefore, the electric properties of the checkerboard lattice can be well preserved. TL LaOBiS₂ thus satisfies the requirements that we put forward in the model analysis.

Fig. 3 (a) Side and (b) top views of the undistorted TL LaOBiS₂ (DS₀) crystal structure. Schematic diagrams of the distortions (green) of (c) DS₁, (d) DS₂, and (e) DS₃ with respect to the undistorted lattice (purple). Bi₁ () and Bi₂ () represent the upper and lower layers before (after) distortion, respectively.

Fig. 4 Energy contours of the (a) conduction and (b) valence bands of TL LaOBiS₂ in reciprocal space. Orbital-resolved band structures of TL LaOBiS₂ (c) without and (d) with an out-of-plane asymmetric potential. In order to introduce the out-of-plane asymmetric potential using a field, the atom position relaxed under zero field is adopted. The color bar denotes the energy scale in eV.

To realize valley polarization, an out-of-plane external electric field is usually applied to break the symmetry. In the case of TL LaOBiS₂, as expected, an out-of-plane electric field breaks the {S_4z|000} and symmetries, which connect the X and X′ valleys. As Fig. 4(d) shows, however, the intervalley energy degeneracy cannot be lifted by a vertical field, but intravalley band splitting occurs due to the asymmetric potential. This is a consequence of the still-extant symmetry little group at the point X (X′), which protects the intervalley energy degeneracy. In other words, the equivalence between p_x and p_y has not been removed. In addition, the heavy atom (Bi) will cause a large SOC; the energy dispersion taking the SOC into account is shown in Fig. S2(f) (ESI†). It can be found that the SOC breaks the two-fold degeneracy (except at the TRIM), but the band dispersion is still symmetric around the Γ point because of the Kramer degeneracy. In the absence of structural distortion, therefore, SOC cannot remove the intervalley energy degeneracy, and the sublattice polarization is not influenced.

In the phonon spectrum of pristine TL LaOBiS₂ (DS₀) in Fig. 5(a), it is of interest that the branches of the optical vibrational modes drop around the Γ point with a negative frequency, which is an indication of a potential structural phase transition. The two doubly degenerate E_u and E_g phonon modes at the Γ point indicate that the Bi and S atoms in each BiS₂ layer tend to move oppositely in x–y plane, thus forming three polytypes of LaOBiS₂. In this first case, the initial displacements were imposed on the Bi atoms in the top (Bi₁) and bottom (Bi₂) of BiS₂ layers along the −x and +x directions, respectively, as shown in Fig. 3(c). In Fig. 5(b), the phonon dispersion confirms the dynamic stability of the distorted structure (DS₁); this distortion is similar to the Peierls-like structural distortion. As shown in Fig. 6(a) and Fig. S2(b) (ESI†), it is greatly significant that the energy degeneracy of the X and X′ valleys is lifted because of the breaking of the {S_4z|000}, and symmetries. In particular, the intervalley energy splittings for the CBM and VBM are 175.1 and 31.8 meV, respectively. These energy values are greater than the thermal energy at room temperature, suggesting potential applications in room-temperature valleytronic devices. In the second case, a low-energy distorted structure in which both the Bi₁ and Bi₂ atoms moved slightly along the −x direction (DS₂) was obtained, and is dynamically stable (see Fig. 3(d) and 5(c)). If the three symmetry operations are removed, symmetry analysis finds that valley polarization may occur. As shown in Fig. 6(b) and Fig. S2(c) (ESI†), as expected, DS₂ lifts the intervalley energy degeneracy, and the energy splittings are 135.8 and 26.7 meV for the CBM and VBM, respectively. As the two BiS₂ layers show C_4z rotation symmetry, the atom distortions in the x and y directions are equivalent; thus, the displacement of both the Bi₁ and Bi₂ atoms along the y direction gives rise to similar results to that in the x direction.

Fig. 5 Phonon band dispersions of (a) pristine (DS₀), (b) DS₁, (c) DS₂ and (d) DS₃ LaOBiS₂ polytypes. For distorted structures, the phonon branches are almost free of imaginary frequencies, except around the Γ point, indicating the dynamic stabilities of these structures. The small imaginary frequencies in the out-of-plane acoustic mode near the Γ point arise from numerical artifacts caused by the inaccuracy of the fast Fourier transform grid.

Fig. 6 Energy contours for the conduction (top panel) and valence bands (bottom panel) of distorted TL LaOBiS₂ in reciprocal space for (a) DS₁, (b) DS₂, and DS₃ (c) without and (d) with an out-of-plane electrical field acting on it. As discussed above, the sublattice polarizations of the CBM and VBM at the point X (X′) are P_c(X) = 1 (P_c(X′) = −1) and P_v(X) = −1 (P_v(X′) = 1), respectively. In this case, the value of P_n(k) is independent of the structural parameters of DS₀–DS₃. The color bars denote the energy scales in eV.

If the Bi₁ atom of the top BiS₂ layer deforms along the −x direction while the Bi₂ atom of the bottom layer moves toward the −y direction, we obtain the third energetically favorable distorted structure (DS₃, Fig. 3(e)); its dynamic stability is confirmed by the phonon spectrum in Fig. 5(d). As shown in Fig. 6(c) and Fig. S2(d) (ESI†), DS₃ retains the energy degeneracy of the X and X′ valleys due to the preserved symmetry. In order to prove our analysis, an out-of-plane external electric field of 0.1 V Å⁻¹ (achievable with current experimental techniques³⁹) was applied to DS₃ to break the symmetry. From Fig. 6(d) and Fig. S2(e) (ESI†), we can see that the field can induce a valley energy splitting of 131.2 and 28.4 meV for the CBM and VBM, respectively. Both the CBM and VBM move upward in the X′ valley and down in the X valley, successfully generating valley polarization in good agreement with our analysis. As listed in Table S1 (ESI†), the polytypes DS₁, DS₂ and DS₃ are likely to coexist due to the relatively small energy difference among them (0.005 meV per atom), and are about −2.2 meV per atom lower in energy than DS₀. In addition, the relevant structural parameters are also summarized in Table S1 (ESI†). It should be emphasized that, for DS₁ and DS₂, valley polarization can be achieved spontaneously due to the breakage of the {S_4z|000}, and symmetries by spontaneous structural distortion, and such structural distortion occurs without the need for an out-of-plane external electric field. In case of DS₃, however, spontaneous structure distortion still preserves the symmetry, which protects the intervalley energy degeneracy. In order to break the symmetry, approaches such as an experimentally achievable out-of-plane electric field could be employed, and our results confirm that the energy degeneracy between X and X′ can be lifted after removing the symmetry.

As shown in Fig. S1 and S2 (ESI†), the model analysis and first-principle calculation results indicate that the valley polarization is retained in the distorted structures when the SOC is considered. In addition, for the CBM in DS₁, DS₂ and DS₃ the energy splitting is consistently larger than that of the VBM, which can be attributed to the structural distortion. In particular, for DS₁, DS₂ and DS₃ the distortion mainly changes the interaction strength between the p_y orbitals of neighboring Bi atoms rather than S atoms. It has been demonstrated that subtle electronic markers of the different polytypes of the layered compound LaOBiS₂ can be distinguished by ARPES;⁴⁰ therefore, valley polarization is observable due to the large energy difference between X and X′ in TL LaOBiS₂. As a result of the C_2v little group, the VBM at the valley centers X and X′ corresponds to the 1D representations B₁ and B₂, respectively; thus, the VBM and additional CB couple only to the B₁ (B₂) state at X (X′) by x-linearly (y-linearly) polarized light excitation. This is analogous to the 2D rectangular-lattice group-IV monochalcogenides SnS and GeSe^41,42 and the 2D square-lattice TiSiCO.³² TL LaOBiS₂ polytypes could be potential candidates for polarizers by using special linearly polarized light to selectively excite electrons from the valence to the conduction band for individual valleys. Additionally, polarization switching could also be detected optically by combining a polarized light detector.⁴³ It should be pointed out that the middle La₂O₂ layer produces a strong polar field, which causes Rashba splitting for the BiS₂ layer at X and X′ points when SOC is considered. As a result, coupling between valley and helical Rashba spin polarization could introduce other fascinating properties, which require further study in future works.

Conclusions

In conclusion, we proposed a new design principle, i.e., valley-contrasting sublattice polarization, to realize the symmetry control of valley polarization. The feasibility of this principle was confirmed via effective model analysis. Remarkably, this new mechanism for valley polarization is applicable to ZrCuSiAs-type structures, particularly layered LaOBiS₂ polytypes. The large intervalley energy splitting indicates great promise for device applications, and we look forward to experimental confirmation of our results.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 51872170), Young Scholars Program of Shandong University (YSPSDU), Shandong Provincial Key Research and Development Program (Major Scientific and Technological Innovation Project) (No. 2019JZZY010302), Natural Science Foundation of Shandong Province (No. ZR2019MEM013), and Taishan Scholar Program of Shandong Province.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0mh01441a

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