Imaging the valley and orbital Hall effect in monolayer MoS₂

Abstract

The topological properties of a material’s electronic structure are encoded in its Berry curvature, a quantity which is intimately related to the transverse electrical conductivity. In transition metal dichalcogenides with broken inversion symmetry, the nonzero Berry curvature results in a valley Hall effect. In this paper we identify a previously unrecognized consequence of Berry curvature in these materials: an electric field-induced change in the electrons’ charge density orientation. We use first principles calculations to show that measurements of the electric field-induced change in the charge density or local density of states in MoS₂ can be used to measure its energy-dependent valley and orbital Hall conductivity.

Introduction –

Transition metal dichalcogenides (TMDs) exhibit a wide array of novel phenomena related to their valley degree of freedom [1]. The valley index labels one of multiple degenerate energy extrema in the conduction or valence band. Phases of TMDs which lack inversion symmetry possess nontrivial Berry curvature a quantity that determines the transverse electrical conductivity and which describes the topological properties of the electronic structure [2]. States in different valleys have opposite sign of Berry curvature and therefore flow in opposite directions transverse to an applied electric field. This results in vanishing net transverse charge current (or Hall current), but a nonzero valley Hall current [1]. The valley Hall effect is accompanied by accumulation of orbital angular momentum at sample boundaries, which has been measured with the magneto-optical Kerr effect [3]. Other experimental probes of the valley Hall effect use an applied magnetic field [4] or circularly polarized light to generate a valley-polarized state [5–7], enabling a nonzero transverse Hall current. The valley Hall effect has also been detected with nonlocal voltage measurements [8, 9].

In this work we introduce a new observable associated with the valley Hall effect in TMD’s: an electric field-induced change in the bulk charge density distribution. More generally, we demonstrate that in a specified class of lattices, the electric field-induced change in a state’s charge density distribution is directly related to its Berry curvature. Using first principles calculations of MoS₂ as a representative example, we show that measurements of the electric field-induced change in density provide a quantitative estimate of the valley Hall conductivity. We also propose that spectroscopic measurements of the electric field-induced change in the local density of states enables a measurement of the energy-resolved valley Hall conductivity.

Formalism –

We first present the general relation between a state’s Berry curvature and the electric field-induced change in its real space density. This analysis applies to lattices composed of a monoatomic unit cell with orbitals of the same character (e.g., p-like or d-like). Although these are restrictive assumptions, they pertain to cases of practical interest, such as near-band edge states of TMDs in the 2H phase [1]. We begin with the relation between a state’s real space density ρ(r) and its velocity. The periodic lattice Hamiltonian is represented in a basis of real localized orbitals, such as Wannier functions or atomic orbitals, which are a product of a radial function and angular function: ${\varphi }_M(r)=R_{n,\mathcal{l}}(r){\mathcal{Y}}_{\mathcal{l}}^M(\theta ,\varphi )$, where R_n,ℓ(r) is a radial function for states in the n^th shell with angular momentum ℓ. ${\mathcal{Y}}_{\mathcal{l}}^M$ is the real-valued spherical harmonic, with 2ℓ + 1 distinct M labels (e.g., for ℓ = 1, M = {x, y, z}). Note that M is sufficient to label the basis states since we assume all orbitals have the same value for n and ℓ. In this basis, the tight-binding crystal field Hamiltonian is:

$$H_0(k)=\sum _{R,M,M^{\prime }}\text{cos}(k\cdot R)t_{M,M^{\prime }}^{\sigma }(R)\left(c_M^{†}{c}_{M^{\prime }}+\mathit{\text{h.c.}}\right),$$ (1)

where k is the crystal momentum, and the sum R includes the primary unit cell (R = 0) and all other Bravais lattice vectors. Typically the sum is truncated up to the n^th nearest-neighbor. $c_M^{†}\left(c_{M^{\prime }}\right)$ is the electron creation (annihilation) operator for orbital M(M′) located in the primary unit cell. $t_{M,M^{\prime }}^{\sigma }$ (R) is the σ-hopping parameter between orbital M of the primary unit cell and orbital M′ centered at lattice site R. Eq. 1 includes only σ-hopping, however generally π and δ hopping amplitudes are smaller, so that the above model is typically adequate (see Supp. Info for a comprehensive assessment of the model’s validity).

The velocity operator v = dH/dk derived from this Hamiltonian is:

$$v(k)=-\sum _{R,M,M^{\prime }}R\text{sin}(k\cdot R)t_{M,M^{\prime }}^{\sigma }(R)\left(c_M^{†}{c}_{M^{\prime }}+h.c.\right).$$ (2)

Each term in the sum over R represents the current flowing between the atom at R = 0 and the atom at R. Note that a current between sites separated by R requires nonzero overlap between k and R, and overlap between orbitals. The orbital overlap is closely related to the shape of the wave function density ρ₀(r):

$${\rho }_0(r)=\sum _{M,M^{\prime }}\left(c_M^{†}{c}_{M^{\prime }}+\mathit{\text{h.c.}}\right){\varphi }_M(r){\varphi }_{M^{\prime }}(r).$$ (3)

ρ₀ corresponds to the density contribution from “on-site” orbitals centered at R = 0, and omits contributions from overlap between neighboring sites in the lattice. In the Supp. Info we show that the differences between the total and “on-site” densities are small in the regions of interest for this work. The form of the orbital basis implies that ρ₀ is the product of a radial function and an angular function: ${\rho }_0(r)={\rho }_0^{\text{rad}}(r){\rho }_0^{\text{ang }}(\widehat{r})$, where the atom center is at r = 0, and ${\rho }_0^{\text{rad}}(r)$ is normalized such that ${\int }_0^{\infty }{r}^2{\rho }^{\text{rad}}(r)dr=1$ (so that ${\rho }_0^{\text{ang }}(\widehat{r})$ is dimensionless).

In the Supp. Info, we show that the angular part of the density ${\rho }_0^{\text{ang }}(\widehat{r})$ is related to the hopping parameters and wave function coefficients of Eq. 2 (the last two factors in that equation). This leads to the following relation between a state’s velocity and density:

$$v(k)=-{\sum }_{R}R\text{sin}(k\cdot R){\tilde{t}}^{\sigma }(R){\rho }_0^{\text{ang}}(k,\widehat{R}).$$ (4)

where ${\tilde{t}}^{\sigma }(R)=t^{\sigma }(R)/Y_{\mathcal{l}}^{m=0}{(0,0)}^2$ is the σ-hopping parameter normalized by a (known) constant factor. Note that we added a k label for ρ; we explicitly include this argument for all k-dependent quantities in the paper. t^σ(R) is the R-dependent σ-hopping function whose form is semi-universal and taken as known a priori [10]. In the sum over neighboring atom positions R, a positive current corresponds to a state propagating outward from r = 0 to r = R, while a negative current corresponds to a state propagating inward from r = R to r = 0.

Eq. 4 is one of our primary results, and provides an intuitive relationship between a state’s charge density and current: the velocity along $\widehat{R}$ is the product of the wave function phase change along $\widehat{R}$ (given by the factor sin(k · R)) and the density along $\widehat{R}$ (given by the factor ${\rho }_0^{\text{ang }}(k,\widehat{R})$). This representation of velocity is a substantial simplification of the general form given in Eq. 2, which requires knowledge of the full hopping elements (encoded in Slater-Koster tight-binding tables[11]) and the complex wave function coefficients.

Armed with this density-velocity connection, we next turn to the relation between a state’s Berry curvature and its velocity. The z-component of the Berry curvature ${\Omega }_z^n$ of an eigenstate ψ_n is [12]:

$${\Omega }_n^z(k)=-\mathrm{2 }\text{Im}\sum _{m\ne n}\frac{\langle {\psi }_{nk}\left|v_x\right|{\psi }_{mk}\rangle \langle {\psi }_{mk}\left|v_y\right|{\psi }_{nk}\rangle }{{\left({ϵ}_m-{ϵ}_n\right)}^2},$$ (5)

where ϵ_n is the energy of the n^th eigenstate |ψ_nk〉, and v_x,y = dH/dk_x,y. The intrinsic contribution to the anomalous Hall conductivity σ_AHE in the clean limit is given by the sum of the occupied states’ Berry curvature. In this work, we also consider the valley Hall and orbital Hall effects. The expression for the valley Hall conductivity σ_OHE is the same as Eq. 5 with an additional factor of ±1 depending on the state’s valley index (e.g., the location of the k-point in the Brillouin zone, see Sec. II). We also compute the orbital Hall conductivity, which describes the electric field-induced flow of orbital angular momentum, oriented in the z-direction and with velocity transverse to the applied field [13–16]. The orbital Hall conductivity is also given by a Kubo formula expression like Eq. 5, with the replacement v_y → (v_yL_z + L_zv_y)/2, where L_z is the atomic orbital angular momentum operator. Finally, we note that the Berry curvature may be nonzero only in the presence of time reversal and/or inversion symmetry breaking [17].

A state’s Berry curvature can also be understood in terms of perturbation theory. An applied electric field perturbs the eigenstates and may change their velocity. The linear-in-E change in velocity δv determines the Berry curvature: ${\Omega }_n(k)=\widehat{E}\times \langle \delta v_n(k)\rangle$. We invoke Eq. 4 to relate the change in velocity to a change in the charge density. Using this expression for the Berry curvature, we conclude that the real space density response of the n^th state is related to its Berry curvature:

$${\Omega }_n(k)=-\widehat{E}\times \sum _{R}R\text{sin}(k\cdot R){\tilde{t}}^{\sigma }(R)\langle \delta {\rho }_{0,n}^{\text{ang }}(k,\widehat{R})\rangle .$$ (6)

Eq. 6 is another primary result, which provides the connection between a state’s Berry curvature and the E-field-induced change in its density distribution. We emphasize that this relation is valid for lattices with monoatomic unit cells and orbitals with the same character.

It’s straightforward to show that the net changes in velocity and density are derived from mutually exclusive sets of states. The net change of any observable is obtained by summing over k. The sin(k · R) factor on the right hand side of Eq. 4 indicates that v(k) and δρ₀(k, R) have opposite parity under k → −k. The anomalous Hall conductivity is derived from states with even-ink Ω(k). However these states’ δρ₀(k) is odd-in-k, and therefore make no contribution to $\delta {\rho }_0^{\text{tot }}$. The converse also holds: states which contribute to $\delta {\rho }_0^{\text{tot }}$ do not contribute to σ_AHE.

We finally note recent work which formulates the anomalous Hall conductivity as a local property without reference to Bloch wave vector k [18–20]. The formal structure of this theory includes the response of the second order cumulant of the charge density, resembling the picture we describe of a change in the charge density distribution. The present work straddles between the limiting cases of the more standard formulation of Berry curvature strictly in k-space and the work [18, 19] which resides entirely in real space.

Imaging the valley Hall effect in insulating TMD –

As an application of our formalism, we consider a monolayer of MoS₂ in the 2H phase. We’ll show the relation between the electric field-induced change in density and the valley Hall effect for three cases of increasing complexity: the response at a single k point, the net response in the insulating phase, and the energy-resolved response. MoS₂ is a nonmagnetic direct band-gap semiconductor [21], and its crystal and electronic structure are shown Fig. 1(a) and (b), respectively. It has a direct band gap of Δ = 1.7 eV (computed value) located at k = ±K = ±4π/3a(1, 0), where a is the in-plane lattice constant. (Note that −K is often labeled K′.) Figs. 1 (c) and (d) show the k-dependent Berry curvature and orbital Hall conductivity, respectively. For energies away from the band edge, the valley index of a state can be associated with the nearest valley, but the label becomes less well-defined.

FIG. 1.

(a) depicts the 2H-MoS₂ monoloayer. (b) shows the band structure and Berry curvature along high symmetry lines. (c) and (d) show the Berry curvature and the orbital Hall conductivity in the Brillouin zone (units of nm²).

The unit cell contains 2 S atoms and 1 Mo atom, and there is significant p-d orbital hybridization in some regions of the band structure. The states near the band gap at K and −K exhibit the largest contribution to the Berry curvature. These states are composed primarily of d-orbitals localized on the Mo atoms, which form a hexagonal lattice, and which do approximately satisfy the assumptions of our analysis. The conduction band is composed of L_z = 0 states, corresponding to $|d_{z^2}\rangle$, while the valence band is L_z = ±2 states at k = ±K, corresponding to $\left(|d_{x^2-y^2}\rangle \pm i|d_{xy}\rangle \right)/\sqrt{2}$ [1]. These essential features are shared by other 2H TMD’s [1], so that our formalism is applicable to this family of materials.

In the vicinity of ±K, the conduction and valence bands are described by a massive Dirac Hamiltonian [1]. Letting q = k ∓ K, we have:

$$H_{\nu }=ta\left(\nu q_x{\tau }_x+q_y{\tau }_y\right)+\frac{\Delta }{2}{\tau }_z$$ (7)

where τ is the Pauli matrix in the space of conduction and valence band states, and ν = ±1 is the valley index. We ignore the spin degree of freedom, as it doesn’t play an essential role and makes quantitative contributions on the order of λ/Δ, where λ is the atomic spin-orbit parameter [15]. In MoS₂ this ratio is approximately 10⁻². For TMD’s composed of heavier elements, the larger spinorbit splitting will change some details of the analysis that we present here.

We first consider the currents and charge densities of the equilibrium states at q = 0 (or k = K). The eigen-states are ${\psi }_c=\left(\begin{array}{l}1\hfill \\ 0\hfill \end{array}\right)$ and ${\psi }_v=\left(\begin{array}{l}0\hfill \\ 1\hfill \end{array}\right)$. The R sin(K·R) factor of Eq. 4 contributes to the bond currents as indicated by the bond-aligned arrows Fig. 2(a). The charge density of both conduction and valence band equilibrium states is isotropic in the x-y plane, so that the bond currents are weighted equally and the net current vanishes.

FIG. 2.

Depiction of how intrinsic charge current is formed at K point. The green arrows along the 3 unique nearest-neighbor bonds show the sign of sin(K·R). In (a), the valence band unperturbed charge density is isotropic. The bond currents add equally and sum to zero. (b) With an E-field in the y-direction, the change in density (with red (blue) is positive (negative)) leads to a net current in the x-direction. (c) With an E-field in the x-direction, the change in density is along the 45° direction, leading to a net current in the y-direction.

Applying an electric field in the y-direction perturbs the valence band wave function: ${\psi }_v^{\prime }={\psi }_v+\left(\mathit{\text{eEta}}/{\Delta }^2\right){\psi }_c$, where e is the magnitude of the electric charge. The mixing of valence and conduction band states leads to an anisotropic ρ(r) which overlaps unevenly with the nearest neighbor bonds. This in turn breaks the balance of bond currents between nearest neighbors and enables a nonzero net current, corresponding to the transverse conductivity or Berry curvature Ω(K). The modification of the density for electric fields along x and y-directions is depicted schematically in Fig. 2(b) and (c), and provides an intuitive rationale for the formation of Hall current at the K point [3, 7].

To test the quantitative validity of Eq. 4 for MoS₂, we compute the electric field-induced change in charge density of the conduction band state at K using the first principles tight-binding Hamiltonian and real space orbitals obtained with Quantum Espresso [22] and Wannier90 [23]. We can independently compute both left and right-hand sides of Eq. 6 in a model with no approximations beyond those found in density functional theory. This provides a stringent test on the applicability of our formalism to real materials, and yields a prediction for the observable quantity δρ(r). Fig. 3(a) shows isosurface contours of the electric field-induced change in density for the conduction band state at k = K, for an applied electric field along the y-direction. Fig. 3(b) shows δρ(K, r) in the plane of the Mo atoms, and Fig. 3(c) shows the density δρ(K, x) along a line connecting nearest neighbors. From the δρ(K, x) profile, we extract the (normalized) radial part δρ^rad(K, r) to obtain $\delta {\rho }^{\text{ang }}(K,\widehat{R})$. Applying Eq. 4 leads to

$$\Omega (K)=-a{\tilde{t}}^{\sigma }\frac{\delta {\rho }^{\text{ang }}(K,\widehat{x})}{eE}$$ (8)

We use a value of t^σ = 1.09 eV obtained by fitting the conduction/valence bands to the effective model near K. Plugging in numbers we obtain a predicted value Ω(K) = 0.074 nm² from the real space density analysis, compared to the directly computed value Ω(K) = 0.094 nm². We find semi-quantitative agreement between the two values, validating the applicability of our approach to real materials. The difference between the two quantities is traced back to the nonzero contribution of the p orbitals of the wave function at K.

FIG. 3.

(a) Isovalue contours of change in density per electric field of ψ_v(K) (units of (nm² · eV)⁻¹), for an applied field in the y-direction. Change of density integrated along x direction is projected along the yz plane (similarly for cyclic permutation of direction indices). (b) Change of density in plane of Mo atoms. (c) Change of density along line connecting nearest neighbors along the x-direction. The normalized radial function obtained from the total change of density is also shown. (d) Change of total density of the insulating state, along the line y = z = 0.

We next extend our analysis to the full, k-integrated response for the insulating system. The Berry curvature at finite q is given by:

$${\Omega }_{\nu }(q)=\nu \frac{2\Delta t^2{a}^2}{{\left(4q^2{t}^2+{\Delta }^2\right)}^{3/2}}.$$ (9)

The valley Hall conductivity σ_VHE of the effective model is obtained by integrating over q and summing over ν, which results in σ_VHE = σ_OHE = 1/(2π). (We omit factors of e²/ℏ in reporting conductivity values.) The values for σ_VHE and σ_OHE obtained with first principles calculations, which are given by 0.71/(2π) and 1.05/(2π), respectively. The deviations between conductivities obtain with the effective model and first principles reflect the difference between the two models’ band structure. In the Supp. Info, we show that the q-dependent perturbed density is proportional to the Berry curvature:

$$\frac{\delta {\rho }_{\nu }(q,r)}{eE}=\nu \frac{{\Omega }_{\nu }(q)}{\sqrt{2}ta}{\varphi }_{z^2}(r){\varphi }_{x^2-y^2}(r).$$ (10)

where ${\varphi }_{z^2\left(x^2-y^2\right)}$ is the d-like atomic wave function for the conduction (valence) band state. Importantly, the change in charge density and the Berry curvature have the same q-dependence. Integrating Eq. 10 over q relates the total change in charge density δρ^tot to the valley Hall conductivity. Fig. 3(d) shows the line cut of the total change in charge density along x. We can derive the following relation between the extrema of the change in charge density along x and valley Hall conductivity:

$$-f{\mathit{\text{tax}}}_{\text{min}}^3\frac{\delta {\rho }^{\text{tot}}\left(x_{\text{min}}\right)}{eE}={\sigma }_{\text{VHE}}$$ (11)

where $f=3\pi /32\sqrt{3/2}\text{exp}(4)$. Using the change in density obtained from first principles and Eq. 11, we obtain an estimated value for σ_VHE of 0.95/(2π). This compares well with the directly computed value of valley Hall conductivity. The comparison with the orbital Hall conductivity is more favorable, which is a consequence of the close relationship between the valley and orbital Hall conductivity in this material, and because the orbital Hall conductivity is more strongly concentrated near the ±K points (see Fig. 1(d)). The good agreement between the directly computed σ_VHE and the value predicted with our real space analysis again demon-strates the applicability of our approach to real materials.

Valley/orbital Hall spectroscopy –

We next consider the energy-resolved valley/orbital Hall conductivities and change in density. We again focus on a linecut along the x-direction, and assume that the measured local density of states is proportional the integral of the density along the z direction (e.g., the integral of Eq. 10 along z). In the Supp. info we show that:

$$-\int dz\frac{\delta {\rho }_{\nu }\left(q,x_{\text{min}},0,z\right)}{eE}\times \frac{{\mathit{\text{tax}}}_{\text{min}}^2}{0.065}=\nu {\Omega }_{\nu }(q)$$ (12)

where x_min is the position of the minimum in the change of density (see Fig. 4(a)). The proportionality between the q-dependent change in charge density and the Berry curvature implies that the two quantities are also proportional when integrating over q. In particular, the energy-resolved quantities are proportional. Denoting the energy-resolved density, or local density of states, as LDOS, we find:

$$-\frac{{\mathit{\text{tax}}}_{\text{min}}^2}{0.065}\left(\frac{\delta \text{LDOS}\left(\epsilon ,x_{\text{min}}\right)}{eE}\right)={\sigma }_{\text{VHE}}(\epsilon )$$ (13)

FIG. 4.

(a) The change in local density of states δLDOS(ε, r)/eE integrated along the z-direction for three energy values. The important features are the position of x_min and δLDOS(ε, x)/eE at this position. (b) Energy-resolved valley Hall (black dashed line) and orbital Hall (blue line) conductivity near the valence and conduction band edges. The red line shows the value inferred from the local density of states measurement using Eq. 13 (note break in the x-axis).

Fig. 4(a) shows the energy-resolved change in density for three energies below the valence band edge. Fig. 4(b) shows a comparison between the directly computed valley Hall/orbital conductivity and the predicted value based on Eq. 13. We see semi-quantitative agreement up to 500 meV away from conduction and valence band edges. We observe that the change in density adheres more closely to the orbital Hall effect at energies away from the band edges. This is due to the fact that the change in density and orbital Hall effect are dominated by states near ±K for most energies, whereas the valley Hall conductivity acquires contributions from states near Γ (k = 0) at energies away from the band edges.

To summarize, we show that the real space density response of TMD’s such as MoS₂ provides a direct estimate of the intrinsic valley Hall conductivity. This is a consequence of a relation we derive between the Berry curvature and the charge density response of a specific class of lattices. Experimentally, various probes attain the required sub-Angstrom spatial resolution, such as X-ray diffraction [24], electron beam diffraction [25], and scanning tunneling microscopy [26]. We estimate a fractional change of the local density of states on the order of 10⁻⁴ at the breakdown electric field of MoS₂ [27]. The measurement can be calibrated with the ground state density or LDOS, as described in the Supp. Info, enabling the quantitative estimate of the valley Hall conductivity.