Valley drift and valley current modulation in strained monolayer $\mathrm{Mo}{\mathrm{S}}_2$

Elastic-mechanical deformations are found to dramatically alter the electronic properties of monolayer (ML) $\mathrm{Mo}{\mathrm{S}}_2$; particularly, the low-energy Bloch bands are responsive to a directional strain. In this study, in-plane uniaxial deformation is found to drift the low-energy electron/hole valleys of strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ far away from $K/K’$ points in the Brillouin zone (BZ). The amount of drift differs notably from hole to electron bands, where the conduction band minimum (CBM) drifts nearly 2 times more than the valence band maximum (VBM) in response to a progressively increasing strain field (0–10%). The resulting strain-induced valley asymmetry/decoherence can lift the momentum degeneracy of valley carriers at the $K$ point, thereby affecting the low-energy valley excitations ($K$-valley polarization) in a strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ lattice. The quantum origin of this decoherent valley arises from the differences in the Bloch orbital wave functions of electron and hole states at the exciton band edges and their deformation under strain. A higher drift (>1.5 times) is noticed when strain is along the zigzag (ZZ) axis relative to the armchair (AC) axis, which is attributed to a faster decline in Young's modulus and Poisson's ratio (PR) along the ZZ direction. A similar valley drift only in the VBM of uniaxially strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ was reported in an earlier local density approximation (LDA) based density functional theory (DFT) study [Q. Zhang et al., Phys. Rev. B 88, 245447 (2013)], where a massive valley drift occurring at the CBM was fully overlooked. Moreover, the giant VBM drift reported therein is 6 times the drift observed in our DFT studies based on spin-orbit coupling (SOC) and Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) functionals. The physical origin of valley drift has been ascertained in our thorough investigations. The robustness of our approach is substantiated as follows. With progressive increase in strain magnitude (0–10%), the band gap remains direct up to 2% uniaxial tensile strain, under SOC, which accurately reproduces the experimental strain-induced direct-to-indirect band gap transitions occurring at ∼2% strain. Based on LDA-DFT [Q. Zhang et al., Phys. Rev. B 88, 245447 (2013)], this crossover in band gap has been incorrectly reported to occur at a higher value of uniaxial strain of 4%. Moreover, the direct SOC band gap shows a linear redshift at a rate of 51–53 meV/(% of strain), under uniaxial tensile strain, which is in excellent quantitative agreement with experimentally observed rates in the redshift of direct excitonic transitions measured in several optical absorption and photoluminescence (PL) spectroscopy experiments. In addition, the Berry curvature Ω($k$) of electron/hole bands gets significantly modulated in strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$, where the intensity of the flux profile increases as a function of the magnitude of strain with an opposite drift around $K/K’$, when strained along the ZZ/AC direction. A strong strain-valley coupling leads to an enhancement in the strength of spin-orbit induced spin splitting of bands at VBM/CBM, which is sizably enhanced (∼7 meV) simply by the strain-controlled orbital motions. Our findings are of prime importance in the valley physics of $\mathrm{Mo}{\mathrm{S}}_2$. Besides, the important theoretical insights emerging from this work will trigger further experimental investigations on $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ to realize its novel technological potential in nanoelectronics, spintronics, and valleytronics.

Figure 1

(a) A top view schematic of monolayer (ML) $\mathrm{Mo}{\mathrm{S}}_2$: (i) optimized geometry in primitive hexagonal unit cell, and (ii) rectangular supercell with lattice parameters presented in panels (iii) and (iv), respectively. The lattice vectors for hexagonal and rectangular cells are ${\mathbf{a}}_h$, ${\mathbf{b}}_h$ and ${\mathbf{a}}_r$, ${\mathbf{b}}_r$, respectively, whereas, ${\mathbf{e}}_1$, ${\mathbf{e}}_2$, ${\mathbf{e}}_3$ are the nearest neighbor vectors about a $C_3$-rotation axis centered over the S atom in a Mo-S trigonal-prismatic coordination. (b) A side view schematic of $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ with an A-B-A type trilayer S-Mo-S atomic packing. The dashed blue line in (b) over the central Mo layer is an indicator of the plane of mirror symmetry (${\sigma }_h$) in broken inversion symmetric $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$. (c)–(d) A schematic representation of the 2D unfolded hexagonal Brillouin zone (BZ) (yellow filled region) and folded rectangular BZ (white filled region) for a primitive hexagonal and orthorhombic supercell superimposed with relevant high-symmetry $k$ points.

Figure 2

(a) The band structure of strain-free $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ in an orthorhombic supercell shows a direct semiconducting band gap, $E_g=1.69\mathrm{eV}$ (without SOC), along the X-Γ high-symmetry line. (b) Evolution in the band structure of unstrained (0%) and uniaxially strained (5%, 9%) $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ along the ZZ/AC direction. The CBM valleys near K/K' show a strong strain-dependent momentum drift over the valence band hills. The band structure shown in (b) includes the effect of SOC. (c)–(d) Low-energy spin-split conduction band minima (CB1, CB2) and valence band maxima (VB1, VB2) around the K point, when $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ is strained by 1%. The CBM/VBM valleys drift along the opposite direction in response to lattice strain along the ZZ/AC direction. The vertical color bars represent the position of energy band top or bottom.

Figure 3

Isoenergy contours of low-energy valence and conduction bands in a 2D $k$ plane for unstrained (0%) and 5% strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ along the ZZ/AC direction. The dashed line and arrows illustrate the energy valley drift near the K point. This energy contour feature is also true for its symmetry inequivalent K′ point due to time-reversal symmetry. The VBM/CBM constant energy contours have been scaled with respect to the energy of the valence band top and conduction band bottom at the K point. Under strain we see a strong effect of trigonal warping (TW) of energy bands near the K point, where the energy valleys drift along opposite directions under ZZ/AC strain.

Figure 4

(a) Momentum drift in the lowermost spin-split conduction band (CB1) and uppermost valence band (VB1) at K/K' points as a function of uniaxial strain along the ZZ/AC direction in a strain range of 0–10%. Variation in Young's modulus (b) and Poisson's ratio (c) with strain. A strong strain-dependent anisotropy in elastic parameters can be seen in (b) and (c), when strain is along its ZZ/AC direction. (d)–(e) Functional relationship of band-edge deformation energy with uniaxial strain applied along the ZZ/AC direction. CB1, CB2 are the spin-split conduction band minima (CBM) and VB1, VB2 are the spin-split valence band maxima (VBM) near the K/K' point, while VB is the spin-degenerate valence band top at the Γ point. The bracketed numbers are their variation rates in units of meV/strain%, with energy referenced to the absolute vacuum energy $\left(E_{\mathrm{vac}.}\right)$. (f)–(g) Variation in the direct, $E_g\left(K\text{−}K\right)$ and indirect, $E_g\left(\mathrm{\Gamma }\text{−}K\right)$ band gaps with uniaxial tensile strain (0–10%) along the ZZ/AC direction, with (w/) and without (w/o) explicit inclusion of spin-orbit coupling (SOC) effects into the band structure calculation. The red and green shading in (f) and(g) is the strain range for direct-to-indirect band-gap crossover w/o and w/ SOC effects turned on.

Figure 5

Bloch states at various $k$ points, orbital wavefunction of electronic states at the vertices of the conduction band valley (CBM_K), and valence band hills (VBM) located at K and Г points for strain-free (a), and 5% strained $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ along the zigzag (b) and armchair (c) directions. The orbital structure of Bloch states is projected from different crystallographic viewpoints with their orbital character labeled therein. The charge density isosurfaces have the same isolevel along a particular column in (a)–(c) in units of $e/{\AA }^3$. The table summarizes the orbital composition of Bloch states at the band edges of CBM/VBM. (d)–(e) Difference in Bloch wavefunction along the ZZ/AC direction for relevant edges at an isosurface level of $0.6\times {10}^{-3}e/{\AA }^3$. A strong delocalization effect can be seen when strain is along the ZZ direction.

Figure 6

(a) Spatial distribution of Bloch wave functions at the band edges, CBM (at K), and (b)–(c) VBM at K and Г, respectively, when strained along the ZZ/AC direction. The planar-average squared wave functions $\left(\right|\psi {|}^2)$ have been projected along the direction perpendicular to the basal surface of $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ (width direction). The atomic positions have been marked therein with solid vertical lines on S-Mo-S.

Figure 7

(a) Berry curvature distribution Ω($k$) over all the occupied Bloch bands in units of ${\AA }^2$ along the high-symmetry $k$ line X'-K'-Γ-K-X, under varying degrees of strain (5%, 9%) along the ZZ/AC direction. (b) Berry curvature distribution of two lowermost unoccupied bands under strain. Inset in (a): contour map of Berry curvature distribution in a 2D $k$ plane for occupied bands.

Figure 8

(a)–(c) Spin-resolved band structure of strain-free (0%) and uniaxially strained (9%) $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ along the ZZ/AC direction. The spin projection is along the out-of-plane spin-projection operator, $S_z$ (dimensionless spin Pauli matrix), and perpendicular to the basal plane of $\mathrm{ML}\text{−}\mathrm{Mo}{\mathrm{S}}_2$. The green and blue colors indicate spin-up and spin-down polarization states. (d) Spin-orbit induced spin splittings of conduction band (CB1, CB2) and (e) valence band (VB1, VB2) under varying degrees of uniaxial strain along the ZZ/AC direction. Insets in (d) and (e) show the CBM and VBM band dispersions for strain-free (0%) and 9% strained cases along the ZZ direction.