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Two-band model

We consider the two-dimensional two-band Hamiltonian

where \(k=| {{{{{{{\bf{k}}}}}}}}| =\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\) is the modulus of the electron momentum, σ_i is the Pauli matrix, and d_i(k) is an arbitrary real function. The two eigenenergies of this Hamiltonian are given by ε_±,k = ℏ²k²/2m ± d(k), where \(d({{{{{{{\bf{k}}}}}}}})=\sqrt{{d}_{1}^{2}({{{{{{{\bf{k}}}}}}}})+{d}_{2}^{2}({{{{{{{\bf{k}}}}}}}})}\). The corresponding Bloch wave functions are obtained as \({e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{\bf{r}}}}}}}}}{(\pm ({d}_{1}({{{{{{{\bf{k}}}}}}}})-i{d}_{2}({{{{{{{\bf{k}}}}}}}}))/d,1)}^{{{{{{{{\rm{T}}}}}}}}}/\sqrt{2}\). We impose the restriction d_i(k) = − d_i(−k) to reflect time-reversal symmetry of our system. The Rashba and Dresselhaus Hamiltonians, representative time-reversal symmetric models, satisfy this relationship. Under such a restriction, the eigenvalues of the system are even functions of the momentum k.

Spin Hall conductivity

In this paper, we consider in-plane electric field polarized along the x-direction and propagating along the y-direction as illustrated in Fig. 1a, i.e., \({{{{{{{\bf{E}}}}}}}}={E}_{x}\widehat{x}{e}^{iqy-i\omega t}\)+c.c.³⁵. Then the induced transverse spin Hall conductivity in the static limit ω → 0 is given by the Kubo–Greenwood formula

where −e is the charge of the electron, S is the area of the system. We define \({\langle \widehat{O}\rangle }_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{q}}}}}}}}}\equiv \langle {u}_{-,{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}/2}| \widehat{O}| {u}_{+,{{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}}/2}\rangle\) and Δε_k,q ≡ ε_−,k−q/2 − ε_+,k+q/2, where \(|{u}_{\pm ,{{{{{{{\bf{k}}}}}}}}}\rangle\) are the periodic part of Bloch wave functions for the upper unoccupied band (+) and the lower occupied band (−) respectively, as shown in Fig. 1b. Here \({{{{{{{\bf{q}}}}}}}}=q\widehat{y}\) represents the momentum transfer due to the modulation of the electric field, \({\widehat{v}}_{x}={\partial }_{{k}_{x}}{H}_{0}/\hslash\) is the velocity operator along x-direction, and \({\widehat{j}}_{y}^{z}=\hslash \{{\sigma }_{3},{\widehat{v}}_{y}\}/4=({\hslash }^{2}{k}_{y}/2m){\sigma }_{3}\) is the spin current operator polarized perpendicular to the xy-plane and flowing alone the y-direction.

a A two-dimensional system under consideration. The inhomogeneous in-plane electric field E is applied. Spin Hall current \({j}_{y}^{z}\) flows perpendicular to the polarization direction of the applied field. b Energy bands model in one quadrant with the Fermi energy ε_F higher than the nodal energy. The temperate is taken as zero. \(|{u}_{{\pm}\!,{{{{{{{\bf{k}}}}}}}}}\rangle\) are the periodic part of Bloch wave functions for the upper unoccupied band and the lower occupied band, respectively.

In the uniform field limit (q → 0), we have the conventional intrinsic SHC

where \({A}_{+-}^{x}=i\langle {u}_{+,{{{{{{{\bf{k}}}}}}}}}|{\partial }_{{k}_{x}}|{u}_{-,{{{{{{{\bf{k}}}}}}}}}\rangle\) is the interband or cross-gap Berry connection^36,37 along x-axis and Δε_k ≡ Δε_k,0 = ε_−,k − ε_+,k denotes the energy difference between two bands at a given momentum. Here the relation \(\langle {u}_{-,{{{{{{{\bf{k}}}}}}}}}|{\sigma }_{3}|{u}_{+,{{{{{{{\bf{k}}}}}}}}}\rangle =-1\) is used.

For nonuniform external electric field with long wavelength excitation, the SHC can be expanded as \({\sigma }_{{{{{{{{\rm{SH}}}}}}}}}(q)={\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{(0)}+q{\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{(1)}+{q}^{2}{\sigma }_{{{{{{{{\rm{SH}}}}}}}}}^{(2)}+O({q}^{3})\). Due to the time-reversal symmetry of our system, the band velocity and interband Berry connection are odd functions of k. As a result, the q-linear term vanishes after integration over k. Then the q² term becomes the leading term for the deviation of the SHC from its value under uniform electric field, which is given by (see Supplementary Method 1)

where \({v}_{\pm ,i}={\partial }_{{k}_{i}}{\varepsilon }_{\pm ,{{{{{{{\bf{k}}}}}}}}}/\hslash\) is the group velocity along i-axis (i = x, y), \({A}_{+-}^{y}=i\langle {u}_{+,{{{{{{{\bf{k}}}}}}}}}|{\partial }_{{k}_{y}}|{u}_{-,{{{{{{{\bf{k}}}}}}}}}\rangle\) is the interband Berry connection with respective to k_y, and g_yy is the Fubini-Study quantum metric³⁸ for the filled band given by

Here we have used the relation \(\langle {u}_{\pm ,{{{{{{{\bf{k}}}}}}}}}|{\sigma }_{3}|{u}_{\pm ,{{{{{{{\bf{k}}}}}}}}}\rangle =0\), which means that spin-up and -down are equally distributed.

Under the gauge transformation \(|{u}_{\pm ,{{{{{{{\bf{k}}}}}}}}}\rangle \to {e}^{i{\phi }_{{{{{{{{\bf{k}}}}}}}}}}|{u}_{\pm ,{{{{{{{\bf{k}}}}}}}}}\rangle\), both the interband Berry connection and quantum metric are invariant. Therefore, the correction in (4) is also gauge-invariant. If we impose a further restriction for our system: d_j(k) is a linear function of the momentum, i.e., \({\partial }_{{k}_{i}}^{2}{d}_{j}({{{{{{{\bf{k}}}}}}}})=0\), our result (4) can be further simplified because \({\partial }_{{k}_{i}}({v}_{-,i}-{v}_{+,i})=4{{\Delta }}{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}{g}_{ii}\) in such a case³⁹.

The theory can be extended to more general case. When the term ℏ²k²/2m in the Hamiltonian (1) is replaced by d₀(k) which satisfies the restriction d₀(k) = d₀(−k), the spin current operator will become \({\widehat{j}}_{y}^{z}={\partial }_{{k}_{y}}{d}_{0}({{{{{{{\bf{k}}}}}}}}){\sigma }_{3}/2\). As a consequence, the factor ℏ²k_y/m in the results (3) and (4) is simply changed to \({\partial }_{{k}_{y}}{d}_{0}({{{{{{{\bf{k}}}}}}}})\).

Rashba and Dresselhaus models

Let us consider the Rashba model given by \({H}_{0}^{{{{{{{{\rm{R}}}}}}}}}={\hslash }^{2}{k}^{2}/2m+\alpha {k}_{y}{\sigma }_{x}-\alpha {k}_{x}{\sigma }_{y}\), where α is the strength of the Rashba SOC. For this system, we have interband Berry connection \({A}_{+-}^{x}={k}_{y}/2{k}^{2}\), \({A}_{+-}^{y}=-{k}_{x}/2{k}^{2}\), and quantum metric \({g}_{yy}={k}_{x}^{2}/4{k}^{4}\). Using (3) and (4), the SHC up to q²-term is evaluated as

where ε_F is the Fermi energy. As shown in Fig. 2a, one can adjust the intrinsic SHC by tuning the Fermi energy, which is unlike the uniform electric field case. This provides us with a new way to manipulate spin current. It is worth noting that in the homogeneous field case, SHC in other systems can be tunable^40,41,42. The SHC (6) obtained from the geometric formula (4) matches well with the exact result from (2) as ε_F grows and q decreases as plotted in Fig. 2a, b. This is due to the fact that the perturbation method for expanding the conductivity is valid for small \(q/{{\Delta }}{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{2}\).

a The green and red solid lines are the correction \({q}^{2}{\sigma }_{yx}^{(2)}\) with the dimensionless wave vector ℏ²q/(mα) = 0.4 and ℏ²q/(mα) = 0.2, respectively. b The green and red solid lines are the correction \({q}^{2}{\sigma }_{yx}^{(2)}\) with the dimensionless Fermi energy ℏ²ε_F/(mα²) = 0.4 and ℏ²ε_F/(mα²) = 0.2, respectively. The correction is in the unit of e/(8π). The solid lines are drawn from Eq. (6). The dotted lines are drawn from the Kubo–Greenwood formula (Eq. (2)) with the corresponding parameters.

If we perform the transformation: d₁(k) ↔ d₂(k) and α → β, the Rashba system changes to the Dresselhaus model with the Hamiltonian \({H}_{0}^{{{{{{{{\rm{D}}}}}}}}}={\hslash }^{2}{k}^{2}/2m-\beta {k}_{x}{\sigma }_{x}+\beta {k}_{y}{\sigma }_{y}\), where β is the Dresselhaus coupling strength. Under such transformation, the band structure and quantum metric remain the same, while the interband Berry connection changes its sign: \({A}_{+-}^{j}\to -{A}_{+-}^{j}\). As a result, the SHC of the Dresselhaus system becomes

There is a sign different from the result of the Rashba model.

There are many materials, whose electronic structures are descried by the Rashba or Dresselhaus model, such as (i) 2D interfaces of InAlAs/InGaAs⁴³ and LaAlO₃/SrTiO₃⁴⁴, (ii) Si metal-oxide-semiconductor (MOS) heterostructure⁴⁵, and (iii) surfaces of heavy materials like Au⁴⁶ and BiAg(111)^47,48. By using the well-known experimental techniques probing intrinsic spin Hall effect^4,7,9,49,50, we expect to detect the intriguing Fermi level-dependence of the q²-term of the SHC under the spatially modulating electric field by tuning its wavelength.

Apart from Rashba and Dresselhaus system, the theory is also applicable to the two-dimensional heavy holes in III-V semiconductor quantum wells with the cubic Rashba coupling^51,52: \({H}_{0}^{{{{{{{{\rm{C}}}}}}}}}={\hslash }^{2}{k}^{2}/2m+i\alpha ({k}_{-}^{3}{\sigma }_{+}-{k}_{+}^{3}{\sigma }_{-})/2\), where k_± = k_x ± ik_y and σ_± = σ_x ± iσ_y. In such a material system, d₁(k) and d₂(k) are respectively \(\alpha {k}_{y}(3{k}_{x}^{2}-{k}_{y}^{2})\) and \(\alpha {k}_{x}(3{k}_{y}^{2}-{k}_{x}^{2})\), which can reflect time-reversal symmetry. Here we should note that the angular momentum quantum numbers of the heavy holes is 3/2, so the spin current operator is \({\widehat{j}}_{y}^{z}=3({\hslash }^{2}{k}_{y}/2m){\sigma }_{3}\)⁵² and the results (3) and (4) should be multiplied by 3.

Wave packet approach

Conventionally, people get physical intuition for transport phenomena from the semiclassical analysis conducted based on the wave packet dynamics. However, it was mentioned that the AHC calculated from the single-band wave packet method could be inconsistent with the one predicted by the Kubo–Greenwood formula³³. We show that we should construct the wave packet from two bands for the AHC or SHC to be consistent with the Kubo–Greenwood formula.

Under the presence of the external field, the Hamiltonian of the system is \(H={H}_{0}+{H}^{\prime}\). Here \({H}^{\prime}\) is the perturbative coupling term with the field, which is given by

where A_x = (E_x/iω)e^iqy−iωt+c.c. is the vector potential. It’s worth noting that in order to get the transverse conductivity, we expand the perturbed Hamiltonian in terms of the vector potential rather than scalar potential⁵³.

A wave packet is usually constructed from the unperturbed Bloch wave function within a single band. However, this conventional way gives inconsistent results with the Kubo–Greenwood formula when calculating AHC³³. Besides, as can be seen from (3) and (4), the SHC results from the interband coupling. The single-band wave packet method will yield a null result for the SHC. Therefore, we use both the upper and lower bands to construct the wave packet as

where \({H}_{+-}^{\prime}=\langle {u}_{+,{{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}}/2}|{H}^{\prime}|{u}_{-,{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}/2}\rangle\) is the transition matrix element, and the amplitude a(k, t) satisfies the normalization condition ∫dk∣a(k, t)∣² = 1. One can note that the Bloch wave functions in the upper band are involved in constructing the wave packet in the same manner as the first-order stationary perturbation scheme for the lower band. A similar wave packet structure can be found in Ref. ^34,54 and in the non-Abelian formulation⁵⁵.

The spin current flowing alone y-direction can be evaluated as

where k_c is the momentum of the wave packet, and \({R}_{y\sigma }=\langle {\psi }_{-\sigma }(t)|{\widehat{r}}_{y}|{\psi }_{-\sigma }(t)\rangle\) is the average position of the spin-σ part of the wave packet given by \(|{\psi }_{-\uparrow }(t)\rangle =(1,0)|{\psi }_{-}(t)\rangle\) and \(|{\psi }_{-\downarrow }(t)\rangle =(0,1)|{\psi }_{-}(t)\rangle\) for spin-up and -down, respectively. Since the expression \(\langle {\psi }_{-\sigma }(t)|\widehat{O}|{\psi }_{-\sigma }(t)\rangle\) is actually \(\langle {\psi }_{-}(t)|2\widehat{O}+{{{{{{{\rm{sgn}}}}}}}}(\sigma )\{{\sigma }_{3},\widehat{O}\}|{\psi }_{-}(t)\rangle /4\), we have

where \({{{{{{{\rm{sgn}}}}}}}}(\sigma )=1(-1)\) for spin-up(down). It is worth to note that in this paper, we use the classical form of the spin current operator \({\widehat{j}}_{y}^{z}=\hslash \{{\sigma }_{3},{\widehat{v}}_{y}\}/4\)^9,25 to substitute the effective spin current operator \((\hslash /2){{{{{\rm{d}}}}}}({\widehat{r}}_{y}{\sigma }_{3})/{{{{{\rm{d}}}}}}t\)⁵⁶. Besides, for time-reversal symmetric system, there is no anomalous Hall effect, i.e., \(\langle {\psi }_{-}(t)|{\widehat{v}}_{y}|{\psi }_{-}(t)\rangle =0\). Therefore in such a case, the velocities in spin-up and -down basis satisfy the relation: \({\dot{R}}_{y\uparrow }=-{\dot{R}}_{y\downarrow }=\langle {\psi }_{-}(t)|{\widehat{j}}_{y}^{z}|{\psi }_{-}(t)\rangle /\hslash\), which indicates that the spin-up and -down part of wave packet have opposite velocity. On the other hand, for time-reversal symmetry breaking system, the charge current along y-direction is

where \({R}_{y}=\langle {\psi }_{-}(t)|{\widehat{r}}_{y}|{\psi }_{-}(t)\rangle ={R}_{y\uparrow }+{R}_{y\downarrow }\) is the position of the wave packet.

The spin current and charge current can be written in the same form \(j=\frac{1}{S}{\sum }_{{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\bf{c}}}}}}}}}}\langle {\psi }_{-}(t)|\widehat{j}|{\psi }_{-}(t)\rangle\), where \(\widehat{j}\) is \({\widehat{j}}_{y}^{z}\) for spin current or \({\widehat{j}}_{y}=-e{\widehat{v}}_{y}\) for charge current. Substituting (9) into it, we have

According to Fermi’s golden rule, we take ω as − Δε_k,q/ℏ for the transition matrix element \({H}_{+-}^{\prime}\) and Δε_k,q/ℏ for the transition matrix element \({H}_{-+}^{\prime}\). Then from (8) and (13), the conductivity j/(E_xe^iqy−iωt) can be obtained as

where σ(q) represents the SHC or AHC depending on the choice of \(\widehat{j}\). Here we have taken ∣a(k, t)∣² ≈ δ(k − k_c), i.e., the wave packet is sharply peaked at the momentum k_c. One can note that the Kubo–Greenwood formula is reproduced from the wave packet approach, i.e., the two-band wave packet approach is compatible with the Kubo–Greenwood formula.