Introduction

In recent years, new responses leading to orbital magnetization have been proposed in systems without inversion symmetry1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. One of the focuses is the conversion of electron current and magnetization on crystal structure with low symmetry. Among such proposals are kinetic magnetoelectric effect (KME)1,2,3,10,11,13,14,15,18, also called orbital Edelstein effect, i.e., current-induced orbital magnetization, and the gyrotropic magnetic effect4,5,6. These effects have similar response coefficients. In particular, KME is an orbital analog of the Edelstein effect19,20,21. KME emerges even in systems without spin-orbit interactions1,3. In particular, the KME emerges in crystals with a chiral structure1,2,3, similar to the phenomenon in which the solenoid creates a magnetic field when a current flows. In a similar context, the recent finding of spin-selective electron transport through chiral molecules, the so-called chirality-induced spin selectivity (CISS) effect22,23,24,25,26, suggests an alternative method of using organic materials as spin filters for spintronics applications.

In the KME, the electric field induces magnetization, which may look similar to the magnetoelectric (ME) effect27,28,29,30,31,32,33,34,35. Nonetheless, in the spin Edelstein effect and KME, metallic systems are considered, and nonequilibrium electron distribution by the electric field is a key to generate magnetization. On the other hand, the ME effect originates from the change of the electronic band structure by the electric field, while the electron distribution is assumed to stay in equilibrium. Thus the ME effect is mainly considered in insulators, but can also exist in metals36,37. In accordance with these differences in mechanisms between the KME and the ME effects, their symmetry requirements are different. In the ME effect, inversion and time-reversal symmetries should be broken. On the other hand, in the Edelstein effect, the inversion symmetry should be broken, but the time-reversal symmetry can either be preserved or broken. It is seen in chiral systems1,2,3,10,11,13,14, and in polar systems15.

Magnetoelectric tensors may require careful consideration of boundaries. While orbital magnetization is independent of the boundary38,39,40,41,42,43, the orbital magnetization when an electric field is applied may not have such properties. The general orbital magnetoelectric response32,33,34,35 depends on the boundary41. Therefore, it is important to study the effect of the boundary of the response of orbital magnetization.

In this paper, we investigate KME in topological insulators such as three-dimensional Chern insulators and Z2 topological insulators (Z2-TIs)44,45,46 in which the currents are localized on the surface. First, we calculate the KME in three-dimensional topological insulators with the chiral crystal structures. Second, we derive the KME based on the surface Hamiltonian, and we show that this effect depends on surface states. Finally, we propose candidate materials for this effect and estimate the values of the KME. By comparing the results with the results in a chiral semiconductor tellurium5, we show that in topological insulators the orbital magnetization as a response to the current is much larger than metals by many orders of magnitude.

Results

Formulation for KME

We consider a crystal in a shape of a cylinder along the z-axis and calculate its orbital magnetization along the z-axis generated by the current along the z-axis. Let c be the lattice constant along the z-axis. We introduce the velocity operator v as \({{{{{{{\bf{v}}}}}}}}=-\frac{i}{\hslash }[{{{{{{{\bf{r}}}}}}}},H],\) where r is the position operator and H is the Hamiltonian. In the limit of the system length along the z-axis to be infinity, the orbital magnetization at zero temperature is

$$\begin{array}{ll}{M}_{z} {\hskip-75pt}=\frac{1}{2\pi }\int\nolimits_{-\pi /c}^{\pi /c}{{{{{\mathrm{d}}}}}}{k}_{z}\frac{1}{S}\mathop{\sum }\limits_{n}^{N}f({E}_{n}({k}_{z}))\\ \times \left(-\frac{e}{2}\right)\int {{{{{\mathrm{d}}}}}}x{{{{{\mathrm{d}}}}}}y{\psi }_{n,{k}_{z}}^{{{{\dagger}}} }(x,y){({{{{{{{\bf{r}}}}}}}}\times {{{{{{{\bf{v}}}}}}}})}_{z}{\psi }_{n,{k}_{z}}(x,y),\end{array}$$
(1)

where −e is the electron charge, \({\psi }_{n,{k}_{z}}(x,y)\) and En is the nth occupied eigenstates and energy eigenvalues of H at the Bloch wavenumber kz, respectively. f(E) is the distribution function at the energy E, S is the cross-section of the crystal along the xy-plane and N is the number of occupied states. We note that the position operator r is unbounded and problematic if a system is infinite in some directions. Nonetheless, in the present case, the system is infinite only along the z-direction, while (r × v)z does not involve z, which means Eq. (1) is well defined. The unit of the orbital magnetization is [A/m] in the SI unit.

Then, within the Boltzman approximation4, the applied electric field Ez changes f(E) from f0(E) into \(f(E)={f}^{0}(E)+\frac{e\tau {E}_{z}}{\hslash }\frac{\partial {f}^{0}(E)}{\partial {k}_{z}}\) in a linear order in Ez, where τ is the relaxation time assumed to be constant and f0(E) is the Fermi distribution function \({f}^{0}(E)={({e}^{\beta (E-\mu )}+1)}^{-1},\) β = 1/kBT, kB is the Boltzman constant and μ is the chemical potential. Then the orbital magnetization is generated as

$$\begin{array}{ll}{M}_{z}^{{{{{{{{\rm{KME}}}}}}}}} {\hskip-62pt}= \frac{1}{2\pi }\int\nolimits_{-\pi /c}^{\pi /c}{{{{{\mathrm{d}}}}}}{k}_{z}\frac{1}{S}\mathop{\sum }\limits_{n}^{N}\frac{e\tau {E}_{z}}{\hslash }\frac{\partial {f}^{0}({E}_{n}({k}_{z}))}{\partial {k}_{z}}\\ \left.\times \left(-\frac{e}{2}\right)\int {{{{{\mathrm{d}}}}}}x{{{{{\mathrm{d}}}}}}y{\psi }_{n,{k}_{z}}^{{{{\dagger}}} }(x,y){({{{{{{{\bf{r}}}}}}}}\times {{{{{{{\bf{v}}}}}}}})}_{z}{\psi }_{n,{k}_{z}}(x,y)\right .\end{array}$$
(2)

This is the KME. In Chern insulators, because the TRS is broken, an orbital magnetization is nonzero in equilibrium, and a current leads to a change in an orbital magnetization due to the KME. On the other hand, in Z2-TIs, an orbital magnetization vanishes in equilibrium due to TRS, and the KME leads to the appearance of an orbital magnetization. We note that in addition to the off-equilibrium electron distribution discussed above, the electric field also modifies the electronic states. In Z2-TIs, where TRS is preserved, this modulation of the electronic states does not lead to the orbital magnetization in the linear order in Ez, because the TRS is preserved within this order. On the other hand, in topological insulators without TRS, such as Chern insulators, this may also contribute to the orbital magnetization. This mechanism is similar to the conventional ME effect in insulators and has been discussed also in metals36,37

This calculation method is different from that in the previous study on metals1,2,4, where bulk contribution in a system infinite along x and y directions are calculated.

Model calculation on a Chern insulator

As an example of a topological insulator, we consider an orbital magnetization in a Chern insulator with a chiral crystal structure. For this purpose, we introduce a three-dimensional tight-binding model of a layered Chern insulator, as shown in Fig. 1a, connected via right-handed interlayer chiral hoppings (Fig. 1b). Each layer forms a square lattice within the xy-plane, with a lattice constant a, and they are stacked along the z-axis with a spacing c. as shown in detail in the “Methods” section. The Brillouin zone and the band structure are shown in Fig. 1c–e. We set the Fermi energy in the energy gap.

Fig. 1: The model of a Chern insulator with a chiral structure and model calculation of the kinetic magnetoelectric effect (KME).
figure 1

a Individual layer of the model forming a square lattice. The blue regions surrounded by the broken line are the unit cells consisting of four sublattices. b Schematic picture of the chiral hopping (red) between the two neighboring layers. These hoppings form a structure similar to right-handed solenoids. c Brillouin zone of our model with high-symmetry points. d, e Energy bands for the Hamiltonian H with parameters (d) tx = ty = m = bx = by, t3 = 0.1ty and t4 = 0.15ty and (e) tx = ty = m = bx = by, t3 = 0.001ty and t4 = 0.15ty. The symbols such as Γ and X represent high-symmetry points in the wavevector space. f A one-dimensional model with periodic boundary conditions in z-direction. It has a rectangle shape of a size Lx × Ly within the xy plane. The dots are the sites, and th blue squares denote the unit cells, as specified in a. In order to see the boundary effect on KME, the outermost layers on the xz surface have no chiral hoppings and those on the yz surface have chiral hoppings. g, h KME calculated with parameters (g) tx = ty = m = bx = by, t3 = 0.1ty and t4 = 0.15ty and (h) tx = bx = ty, ty = by = m, t4 = 0.15ty, and t3 = 0.001ty. The blue, orange, and green curves represent the data for Lx = 10a, Lx = 20a, and Lx = 30a, respectively, while Ly is fixed as Ly = 30a.

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We calculate KME in a one-dimensional quadrangular prism with xz and yz surfaces shown in Fig. 1f (see “Methods” section), with its results in Fig. 1g,h with the interlayer hopping t3 = 0.1ty and t3 = 0.001ty, respectively, for several values of the system size, Lx and Ly, representing the lengths of the crystal in the x and y directions. Thus, the KME is affected by boundaries and system size, and this size dependence remains even when the system size is much larger than the penetration depth of topological surface states. Therefore, KME cannot be defined as a bulk quantity. In contrast, the orbital magnetization in equilibrium is shown to be a bulk quantity in crystals38,39,40 in 2005, which is nontrivial, because the operator for the orbital magnetization involves the position operator r. Later, we give an interpretation on this characteristic size dependence.

Surface theory of KME for a slab

In topological insulators such as Chern insulators, only the topological surface states can carry a current. Here we calculate the KME using an effective Hamiltonian for the crystal surface. Thereby, we can capture the natures of KME through this surface theory. We consider slab systems, with their surfaces on y = y±(y+ > y−). The slab is sufficiently long along with the x and z directions and we impose periodic boundary conditions in these directions. To induce the orbital magnetization \({M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{KME}}}}}}}}}\), we apply an electric field Ez in the z-direction. Due to the interlayer chiral hoppings, the surface current acquires a nonzero z-component (Fig. 2a).

Fig. 2: Chern insulator with a chiral structure and its kinetic magnetoelectric effect (KME) in slab systems.
figure 2

a Surface velocities (red arrow) of the topological surface states under an electric field E (green arrow). b, c Band structure of the topological chiral surface states on the y = y− surface. Their (b) dispersion and (c) Fermi surface are shown. In b, the valence and conduction bands are represented by orange and blue boxes, and the topological surface states are drawn in purple in between the two bands. c is the Fermi surface, which is a section of b at the energy E equal to the chemical potential μ. d Slab models I and with chiral hopping on the surface. Only the section along the xy plane is shown. The dots are the sites of the tight-binding model, and the blue boxes represent the unit cells, as shown in Fig. 1a. e Slab model II with no chiral hopping on the surface. The symbols are the same as (d). f, g The Fermi surfaces for the slab models (f) I and (g) II with parameters tx = ty = m = bx = by, t3 = 0.1ty, t4 = 0.15ty, and μ = 0. The red and blue curves represent Fermi surfaces on the y = y− and y = y+ planes, respectively. h, i KME for the slab models I and II, shown as red solid lines and as blue broken lines, respectively, with parameters (h) tx = ty = m = bx = by, t3 = 0.1ty and t4 = 0.15ty and (i) tx = ty = m = bx = by, t3 = 0.001ty and t4 = 0.15ty.

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Let \({E}_{-}={E}_{-}({k}_{x},{k}_{z})(={E}_{-}({k}_{x},{k}_{z}+\frac{\pi }{c}))\) be the surface-state dispersion on the y = y− surface as shown in Fig. 2b,c. For simplicity, we assume C2z symmetry of the system. Then the surface state dispersion on the y = y+ surface is given by E+ = E+(kx, kz) = E−(−kx, kz). Here, we assume that the surface states are sharply localized at y = y±, namely, we ignore finite-size effects due to a finite penetration depth. Then, we rewrite Eq. (2) to

$$\begin{array}{l}{M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{KME}}}}}}}}} {\hskip-80pt}=\frac{{e}^{2}\tau {E}_{z}}{2{\hslash }^{2}}\int\nolimits_{-\pi /c}^{\pi /c}\frac{{{{{{\mathrm{d}}}}}}{k}_{z}}{{(2\pi )}^{2}}\\ \times \frac{\partial E({k}_{x}^{\prime},{k}_{z})}{\partial {k}_{z}}{{{{{{{\rm{sgn}}}}}}}}\left(\frac{\partial E({k}_{x}^{\prime},{k}_{z})}{\partial {k}_{x}^{\prime}}\right){\left|\right.}_{E({k}_{x}^{\prime},{k}_{z}) = \mu },\end{array}$$
(3)

(see Supplementary Note 1 for details). We note that the Fermi surface depends on the surface termination, and so does the KME. We also confirm the surface dependence from numerical calculations as shown in Fig. 2d–i. This formula applies to any topological insulators such as Z2-TIs44,45,46 (see Supplementary Note 2).

Surface theory of KME for a cylinder

From this slab calculation, we calculate the KME for a cylinder geometry. We consider a current along the z-direction in a one-dimensional quadrangular prism with xz and yz surfaces (surfaces I–IV in Fig. 3a) through its surface Hamiltonian. Let Lx and Ly denote the system sizes along the x and y directions, respectively. Because the KME is sensitive to differences in crystal surfaces, as shown in-slab systems, we consider the individual surfaces separately.

Fig. 3: One-dimensional prism of a Chern insulator.
figure 3

a Schematic figure of the one-dimensional prism of a Chern insulator, extended along z-axis, with its size Lx × Ly within the xy plane. We call the four side surfaces I, II, III and IV. b The schematic figure for current conservation at the corners around the one-dimensional prism in a. Here, \({j}_{x}^{{{{{{{{\rm{I}}}}}}}}}\) and \({j}_{y}^{{{{{{{{\rm{II}}}}}}}}}\) are current densities in the circumferential direction of the prism on the surfaces I and II, respectively, and we impose them to be equal. \({v}_{x}^{{{{{{{{\rm{I}}}}}}}}}\) and \({v}_{y}^{{{{{{{{\rm{II}}}}}}}}}\) are corresponding velocities of electrons, and \({u}_{xz}^{{{{{{{{\rm{I}}}}}}}}}\) and \({u}_{yz}^{{{{{{{{\rm{II}}}}}}}}}\) are amplitudes of the wavefunctions in the respective surfaces.

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In particular, in Chern insulators, we can calculate the energy eigenstates for the whole system from those for the surface Hamiltonians. For simplicity, we assume twofold rotation symmetry C2z of the system, which relates between I and III, and between II and IV. Then, only the surface I and II are independent. We write down the eigenequations for these surfaces as

$${H}^{{{{{{{{\rm{I}}}}}}}}}{\psi }_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}(x,z)={E}_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}{\psi }_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}(x,z),$$
(4)
$${H}^{{{{{{{{\rm{II}}}}}}}}}{\psi }_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}(y,z)={E}_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}{\psi }_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}(y,z),$$
(5)

where HI and HII are the surface Hamiltonians for the surfaces I and II, respectively, and \({\psi }_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}={u}_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}({k}_{x},{k}_{z}){e}^{i{k}_{x}x}{e}^{i{k}_{z}z}\) and \({\psi }_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}={u}_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}({k}_{y},{k}_{z}){e}^{i{k}_{y}y}{e}^{i{k}_{z}z}\) are Bloch eigenstates on the surfaces I and II, respectively. We can determine these eigenstates from four conditions, equality of the energy eigenvalues, current conservation at the corner (Fig. 3b)47,48,49, periodic boundary condition on the crystal surface, and the normalization condition (see Supplementary Note 3).

Thus we obtain a formula for KME in a one-dimensional prism of a three-dimensional Chern insulator

$${M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}=-\frac{{e}^{2}\tau {E}_{z}}{\hslash }\frac{1}{{(2\pi )}^{2}}\int\nolimits_{-\pi /c}^{\pi /c}{{{{{\mathrm{d}}}}}}{k}_{z}\frac{{L}_{x}\frac{\partial {k}_{x}^{{{{{{{{\rm{I}}}}}}}}}}{\partial {k}_{z}}+{L}_{y}\frac{\partial {k}_{y}^{{{{{{{{\rm{II}}}}}}}}}}{\partial {k}_{z}}}{\frac{{L}_{x}}{{v}_{x}^{{{{{{{{\rm{I}}}}}}}}}}+\frac{{L}_{y}}{{v}_{y}^{{{{{{{{\rm{II}}}}}}}}}}}{\left|\right.}_{E = \mu },$$
(6)

where \({v}_{x}^{{{{{{{{\rm{I}}}}}}}}}=\frac{1}{\hslash }\frac{\partial {E}^{{{{{{{{\rm{I}}}}}}}}}}{\partial {k}_{x}}\) and \({v}_{y}^{{{{{{{{\rm{II}}}}}}}}}=\frac{1}{\hslash }\frac{\partial {E}^{{{{{{{{\rm{II}}}}}}}}}}{\partial {k}_{y}}\) and \({k}_{x}^{{{{{{{{\rm{I}}}}}}}}}({k}_{z},E)\) and \({k}_{y}^{{{{{{{{\rm{II}}}}}}}}}({k}_{z},E)\) are functions obtained from \(E={E}_{{k}_{x}{k}_{z}}^{{{{{{{{\rm{I}}}}}}}}}\) and \(E={E}_{{k}_{y}{k}_{z}}^{{{{{{{{\rm{II}}}}}}}}}\), respectively. When vx and vy are almost independent of kz, we approximate Eq. (6):

$${M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}=2\frac{\frac{{L}_{x}}{\langle {v}_{x}^{{{{{{{{\rm{I}}}}}}}}}\rangle }{M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{I,KME}}}}}}}}}+\frac{{L}_{y}}{\langle {v}_{y}^{{{{{{{{\rm{II}}}}}}}}}\rangle }{M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{II}}}}}}}},{{{{{\mathrm{KME}}}}}}}}{\frac{{L}_{x}}{\langle {v}_{x}^{{{{{{{{\rm{I}}}}}}}}}\rangle }+\frac{{L}_{y}}{\langle {v}_{y}^{{{{{{{{\rm{II}}}}}}}}}\rangle }},$$
(7)

where \({M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{I,KME}}}}}}}}}\) and \({M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{II}}}}}}}},{{{{{\mathrm{KME}}}}}}}\) represent the KME for a slab (Eq. (3)) with the surface I and that with the surface II, respectively. Thus, the KME of the one-dimensional system can be well approximated by Eq. (7) expressed in terms of that for the slabs along xz and along yz planes.

In general topological insulators, we can also derive KME in terms of a simple picture of a combined circuit, consisting of four surfaces I–IV with anisotropic transport coefficients. We obtain

$${M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}={j}_{{{{{{{{\rm{circ}}}}}}}}}=\frac{\frac{{L}_{x}}{{\sigma }_{xx}^{{{{{{{{\rm{I}}}}}}}}}}{\sigma }_{xz}^{{{{{{{{\rm{I}}}}}}}}}+\frac{{L}_{y}}{{\sigma }_{yy}^{{{{{{{{\rm{II}}}}}}}}}}{\sigma }_{yz}^{{{{{{{{\rm{II}}}}}}}}}}{\frac{{L}_{x}}{{\sigma }_{xx}^{{{{{{{{\rm{I}}}}}}}}}}+\frac{{L}_{y}}{{\sigma }_{yy}^{{{{{{{{\rm{II}}}}}}}}}}}{E}_{z},$$
(8)

where jcirc is the circulating current density within the xy plane around the prism per unit length along the z-direction. \({{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{ij}^{{{{{{{{\rm{I,II}}}}}}}}}\) is the electric conductivity tensor for the surfaces I and II (see Supplementary Note 4). On the other hand, we can also show \({M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{I,KME}}}}}}}}}=\frac{1}{2}{\sigma }_{xz}^{{{{{{{{\rm{I}}}}}}}}}{E}_{z},{M}_{z,{{{{{{{\rm{slab}}}}}}}}}^{{{{{{{{\rm{II}}}}}}}},{{{{{\mathrm{KME}}}}}}}=\frac{1}{2}{\sigma }_{yz}^{{{{{{{{\rm{II}}}}}}}}}{E}_{z}.\) In Chern insulators, by using \({\sigma }_{xx}^{{{{{{{{\rm{I}}}}}}}}}\propto \langle {v}_{x}\rangle\) and \({\sigma }_{yy}^{{{{{{{{\rm{II}}}}}}}}}\propto \langle {v}_{y}\rangle ,\) we arrive at Eq. (7). Thus, we can calculate the KME from the surface electrical conductivity from Eq. (8), which depends on the aspect ratio Lx/Ly.

We numerically confirm that the results of direct calculation by Eq. (2) and those for surface calculation by Eq. (7) agree well (Fig. 4a–c). When the interlayer hopping is large (Fig. 4c), they slightly deviate from each other. This is because we cannot ignore the kz dependence of vx and vy and they are out of the scope of the approximate expression (Eq. 7).

Fig. 4: Kinetic magnetoelectric effect (KME) in one-dimensional systems.
figure 4

a–c KME is calculated from two different methods; one is a direct calculation by Eq. (2) (solid lines) and the other is by a combination of calculation results for surfaces along xz and yz planes based on Eq. (7) (dashed lines). Parameter values are (a) tx = ty = m = bx = by, t3 = 0.001ty and t4 = 0.15ty, (b) tx = bx = 1.1ty, ty = by = m, t4 = 0.15ty and t3 = 0.001ty, and (c) tx = ty = m = bx = by, t4 = 0.15ty and t3 = 0.1ty. The results are shown for Lx = 10a (blue), Lx = 20a (orange), and Lx = 30a (green), while Ly is fixed to be 30a. d Dependence of the KME on the aspect ratio within the xy plane with parameters tx = bx = 1.1ty, ty = by = m, t4 = 0.15ty, t3 = 0.001ty, and μ = 0. Blue points represent the result of direct calculation by Eq. (2) for various system sizes. With parameter values tx = bx = 1.1ty, ty = by = m, t4 = 0.15ty, t3 = 0.001ty, and μ = 0, we calculate the KME with various system sizes from Eq. (2). The system sizes are (Lx, Ly) ∈ {10a, 12a, …, 38a} × {10a, 12a, …, 38a}. Red points represent the result of fitting the numerical results of Eq. (2) with the fitting function in Eq. (13), and its limit for Lx, Ly → ∞ is shown as the dashed line. The solid line represents the results of the surface theory in Eq. (7).

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Finite-size effect

In our approximation theory, we assumed that the surface current is localized at the outermost sites and ignored a finite penetration depth. In fact, we can fit well the data with various system sizes with a trial fitting function which includes a finite-size effect in Eq. (7) (see “Methods” section and Supplementary Note 5). From these results, the finite-size effect is of the order 1/L in the leading order, coming from the finite penetration depth. When the system size is much larger than the penetration depth, the result is well described by the surface theory as shown in Fig. 4d.

Materials

Topological insulators without inversion symmetry can be a good platform for obtaining large KME because the current flows on the surface. Therefore, the closed-loop created by the current is macroscopic and it efficiently induces orbital magnetization. In contrast, in the conventional KME in metals, a bulk current generates microscopic current loops in the bulk, which leads to a much smaller effect than a surface current in topological insulators. One can also regard this set of current loops as a macroscopic current loop along the surface, but the current, in this case, is of a microscopic amount, determined by the current per bulk unit cell. Therefore, the resulting effect in bulk metals is much smaller than the KME in topological insulators, where the current along the surface is of macroscopic size. Moreover, the surface states of topological materials are robust against perturbations caused by impurities.

Under the non-inversion-symmetry constraint, we cannot diagnose Z2-TIs easily because the Z2 topological invariant is expressed in terms of k-space integrals. Our idea here is to use S4 symmetry to diagnose Z2-TIs, where we only need to calculate wavefunctions at four momenta according to the symmetry-based indicator theories50,51,52. After searching in the topological material database53, we notice that Cu2ZnSnSe454 with 82 and CdGeAs2 with 122 are two ideal candidates of Z2-TIs with a direct gap for obtaining a large KME (see Supplementary Note 6 for details). In the following, we will use Cu2ZnSnSe4, which only has S4 symmetry as shown in Fig. 5a, as an example to show the magnitude of the KME with different surfaces and different surfaces terminations.

Fig. 5: First-principle calculations on Cu2ZnSnSe4.
figure 5

a Crystal structure of Cu2ZnSnSe4. b Brillouin zone and surface Brillouin zone along [001] direction. c Electronic structure with spin-orbit coupling for the bulk. d–e Surface states and Fermi arcs calculation on the [001] surface with Cu-Sn layer termination (surface A). f–g Surface states and Fermi arcs calculation on the [001] surface with Se layer termination (surface B).

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Since the magnetoelectric tensor for the space group 82, defined by M = αE, is \({\alpha }_{{{{{{{{\boldsymbol{82}}}}}}}}}=\left[\begin{array}{lll}{\alpha }_{11}&{\alpha }_{12}&0\\ {\alpha }_{12}&-{\alpha }_{11}&0\\ 0&0&0\end{array}\right]\), we can obtain an orbital magnetization \({M}_{1}^{{{{{{{{\rm{KME}}}}}}}}}\) by adding an external electric field E1, through the surface currents both on the [001] surface and on the [010] surface thanks to the nonzero α11 (see Supplementary Note 7 for details). In our discussion of the KME effect, we set the current direction to be along z-axis. Therefore we will set the 1-axis in the above magnetoelectric tensor to be the z-axis in our theory.

Figure 5b, c is the Brillouin zone and the band structure of Cu2ZnSnSe4 with a gap, through the first-principle calculations whose details are explained in “Methods” section. On the [001] surface, terminations with Cu-Sn layer (surface A) and with Se layer (surface B) have different surface energies and Fermi surfaces, as shown in Fig. 5d–g, which contribute to a magnetoelectric susceptibility of \({\alpha }_{11}^{{{{{{{{\rm{A}}}}}}}}}=-1.804\times 1{0}^{8}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ and \({\alpha }_{11}^{{{{{{{{\rm{B}}}}}}}}}=-2.565\times 1{0}^{9}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ, respectively. On the A surface, there is a single surface Dirac cone at Γ point, forming an electron-like Fermi surface. On the B surface, the Dirac cone at Γ point forms an almost zero Fermi surface, but two surface Dirac cones at two \(\bar{X}\) momenta form two hole-like Femi surfaces. Because the Fermi surfaces on the B surface are much larger than those on the A surface, the magnetoelectric susceptibility on the B surface is one order of magnitude larger than that on the A surface. Similar calculations on the [010] surface area in Supplementary Note 8, and the result is \({\alpha }_{11}^{{{{{{{{\rm{C}}}}}}}}}=-2.324\times 1{0}^{9}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ for surface C.

Let us compare the results with metallic materials in the bulk. For simplicity, we focus on the cases with the electric field E and the resulting magnetization MKME along the z-direction. This kinetic magnetoelectric response is expressed as \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}={\alpha }_{zz}{E}_{z}\), and the conductivity is jz = σzzEz. Thus the magnetization in response to the current is \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}=({\alpha }_{zz}/{\sigma }_{zz}){j}_{z}\). In the relaxation time approximation, both αzz and σzz are proportional to the relaxation time τ. For simplicity we consider the system to be a cube with its size L × L × L. In the bulk metallic systems, the current is carried by the bulk states, and αzz ∝ L0, σzz ∝ L0. On the other hand, in topological systems, only the surface conducts the current, and the conductivity σzz scales as σzz ∝ L−1. On the other hand, we have shown αzz ∝ L0, which means that αzz is an intensive quantity. Thus, in topological insulators, the scaling of the KME as a response to the electric field is represented by the response coefficient αzz ∝ L0. Meanwhile, the response coefficient αzz/σzz to the current is proportional to L. It means that as a response to the current, topological materials will generate a large amount of orbital magnetization as compared to metals.

We compare our results with KME in p-doped tellurium, which has a chiral crystal structure5. For the acceptor concentration Na = 4 ⋅ 1014cm−3 at 50 K, the induced orbital magnetizations is \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}=7.0\cdot 1{0}^{-8}\)μB/atom ~ 1.85 × 10−2A/m by a current density jz = 1000 A/cm2. Thus the response coefficient of the orbital magnetization \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}\) to the current density jz is αzz/σzz = 1.85 × 10−9m. If we approxima1te σzz by \({\sigma }_{zz} \sim \frac{{N}_{a}{e}^{2}\tau }{m}\) with the electronic charge e and mass m, we get α = 2.1 × 104 s−1Ω−1 ⋅ τ. For other acceptor concentrations Na = 4 ⋅ 1016cm−3 and Na = 1 ⋅ 1018cm−3, one can similarly get αzz = 2.1 × 106 s−1Ω−1 ⋅ τ and αzz = 2.3 × 107 s−1Ω−1 ⋅ τ. Thus, the size of αzz for the topological insulator Cu2ZnSnSe4 is larger than that of Te by two to five orders of magnitude.

On the other hand, in topological insulators, the induced orbital magnetization as a response to the current becomes huge compared with metals. To show this, we consider a system with surfaces having anisotropic transport coefficients with sheet resistance σxz,□ and σzz,□, where □ denotes a sheet conductance. Then we can define an angle θ by \({\sigma }_{xz,\square }/{\sigma }_{zz,\square }=\tan \theta\), where θ describes an angle between the electric field along the z-direction and the surface current density jsurf. For example, for the [001] surface of Cu2ZnSnSe4, we get \({\sigma }_{21,\square }^{{{{{{{{\rm{A}}}}}}}}}=2{\alpha }_{11}^{{{{{{{{\rm{A}}}}}}}}}=-3.6\times 1{0}^{8}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ, \({\sigma }_{11,\square }^{{{{{{{{\rm{A}}}}}}}}}=7.3\times 1{0}^{8}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ, \({\sigma }_{21,\square }^{{{{{{{{\rm{B}}}}}}}}}=2{\alpha }_{11}^{{{{{{{{\rm{B}}}}}}}}}=-5.1\times 1{0}^{9}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ, \({\sigma }_{11,\square }^{{{{{{{{\rm{B}}}}}}}}}=2.0\times 1{0}^{10}{s}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ, which yield \(\tan {\theta }^{{{{{{{{\rm{A}}}}}}}}}=-0.49\) and \(\tan {\theta }^{{{{{{{{\rm{B}}}}}}}}}=-0.26\) by identifying x = 2 and z = 1. Then the total current along the z direction is 4Ljz while the circulating current is \({j}_{{{{{{{{\rm{circ}}}}}}}}}={j}_{x}^{\,{{\mbox{surf}}}}={j}_{z}^{{{\mbox{surf}}}\,}\tan \theta\). Thus the magnetization response \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}\) to the current density \({j}_{z}(=4L{j}_{z}^{\,{{\mbox{surf}}}\,}/{L}^{2})\) is \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}/j=({j}_{{{{{{{{\rm{circ}}}}}}}}}/{j}_{z}^{\,{{\mbox{surf}}}\,})(L/4)=(L/4)\tan \theta\), which is proportional to the system size. It is the response coefficient α/σ shown in Table I. Thus for the macroscopic system size, the response \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}/{j}_{z}\) is also of the macroscopic size such as millimeters, and it is much order of magnitude larger than that in tellurium, where \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}/{j}_{z}\) is evaluated to be \({M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}/{j}_{z}=1.85\times 1{0}^{-9}\)m.

This scaling to the system size shows a prominent difference in the KME in topological insulators from similar effects. In bulk metals studied in previous works, the KME is always independent of the system size L. In topological insulators, the current induces both the spin and the orbital magnetizations, and we propose that only the orbital magnetization shows a different scaling behavior. As a comparison, we calculate the spin magnetization induced by the electric field in Cu2ZnSnSe4, with calculation details and spin textures presented in Supplementary Note 9, and obtain \({\alpha }_{11}^{\,{{\mbox{A,spin}}}}=2.798\times 1{0}^{-2}{{\mbox{m}}}\cdot {{{\mbox{s}}}}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ/Ly, and \({\alpha }_{11}^{\,{{\mbox{B,spin}}}}=-2.575{{\mbox{m}}}\cdot {{{\mbox{s}}}}^{-1}{{{\Omega }}}^{-1}\) ⋅ τ/Ly. Thus it inversely scales with the system size along the y-direction. Thus suppose the system size of 1mm, they are of the order 101 − 103s−1Ω−1 ⋅ τ, and the orbital magnetization from the KME, having 108 − 109s−1Ω−1 ⋅ τ, is larger than the spin counterpart by 5–8 orders of magnitude. Thus, while the spin and orbital magnetization behave similarly in bulk metals, they are quite different in topological insulators, which is the main point of the present paper.

Conclusion

In summary, we propose KME in topological insulators with chiral structures. This KME is carried by the surface current due to the asymmetric crystal structure of the surface. Therefore, the KME is sensitive to surface terminations, and it cannot be defined as a bulk quantity. We derive a formula for the KME as a surface quantity using the surface Hamiltonian and show that it fits with numerical results.

In theoretical treatments, atomic orbitals can classify the orbital magnetization into intraatom and interatom contributions. Some atomic orbitals such as px ± ipy have orbital angular momentum, which leads to corresponding intraatomic orbital magnetization. On the other hand, the hopping between atoms leads, to interatomic orbital magnetization. In tight-binding models with atomic orbitals, they are separately calculated. In some papers10,11, the intraatomic orbital magnetization is studied, while the interatomic one is studied in other papers1,2,15 In real materials, these two contributions are not separable, and in the ab initio calculation5, their sum is calculated. In this paper, we found that in topological materials, the interatomic contribution is much larger due to the macroscopic current loop. We show that the response to the current in topological insulators is much larger than in metals. In Table 1, we show scaling behaviors of the spin and intraatom/interatom orbital magnetizations for metals and TIs. The KME is shown as responses to an electric field, α, and to a current, α/σ. In particular, in the response to a current, α/σ, it scales as L1 only in the interatom orbital magnetization, while other entities scale with L0. This shows a particular feature of the interatom orbital magnetization generated by a current proposed in the present paper.

Table 1 Scaling behaviors of interatom and intraatom orbital and spin magnetizations versus the system size L, as a response to the electric field represented by α, and to the current α/σ in metals and in topological insulators (TIs).
Full size table

In this paper, we put two Z2 TIs as candidates for the topological KME. In addition to Chern insulators and Z2 TIs, other classes of topological insulators such as various classes of topological crystalline insulators, and topological semimetals will also show the topological KME. In topological semimetals, topological surface states coexist with bulk metallic states, but the contribution from the former overwhelms that from the latter, and the topological KME is expected.

Methods

Details of the first-principle calculations

First-principle calculations of Cu2ZnSnSe4 are implemented in the Vienna ab initio simulation package (VASP)55,56,57 with Perdew-Burke-Ernzerhof exchange-correlation. A Γ-centered Monkhorst-Pack grid with 10 × 10 × 10 k-points and 460.8 eV for the cut-off energy of the plane wave basis set is used for the self-consistent calculation. Surface states and Fermi surfaces calculations are performed by the tight-binding model obtained by the maximally localized Wannier functions58.

Details of the model Hamiltonian

We consider a Chern insulating system with a chiral crystal structure. The model is composed of infinite layers of the two-dimensional Wilson-Dirac model59,60. The lattice sites are expressed by (i, j, l), with i, j, l being integers, specifying the x, y, and z-coordinates. At each lattice site, we consider two orbitals 1 and 2. Let ci,j,l,σ denote the annihilation operator of electrons at the (i, j, l)-site with orbital σ( = 1, 2), and we write \({c}_{i,j,l}={({c}_{i,j,l,1},{c}_{i,j,l,2})}^{T}\). The model Hamiltonian is H = HWD(m, tx, ty, bx, by) + Hinterlayer(t3, t4), where HWD is an in-plane Wilson-Dirac Hamiltonian, and Hinterlayer is an interlayer Hamiltonian representing a structure similar to right-handed solenoids. The in-plane Wilson-Dirac Hamiltonian is

$${H}_{{{{{{{{\rm{WD}}}}}}}}}= \, m\mathop{\sum}\limits_{i,j,l}{c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{z}{c}_{i,j,l}\\ -\frac{i{t}_{x}}{2}\mathop{\sum}\limits_{i,j,l}({c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{x}{c}_{i+1,j,l}-H.c.)\\ -\frac{i{t}_{y}}{2}\mathop{\sum}\limits_{i,j,l}({c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{y}{c}_{i,j+1,l}-H.c.)\\ +\frac{{b}_{x}}{2}\mathop{\sum}\limits_{i,j,l}({c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{z}{c}_{i+1,j,l}+H.c.-2{c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{z}{c}_{i,j,l})\\ +\frac{{b}_{y}}{2}\mathop{\sum}\limits_{i,j,l}({c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{z}{c}_{i,j+1,l}+H.c.-2{c}_{i,j,l}^{{{{\dagger}}} }{\sigma }_{z}{c}_{i,j,l}),$$
(9)

where H.c. stands for Hermitian conjugate of the preceding terms, † represents Hermitian conjugate, and m, tx, ty, bx, and by are real parameters. This Hamiltonian HWD can be rewritten in the momentum space as

$$\begin{array}{lll}{\tilde{H}}_{{{{{{{{\rm{WD}}}}}}}}}({{{{{{{\bf{k}}}}}}}}) {\hskip-80pt}= {t}_{x}\sin {k}_{x}a{\sigma }_{x}+{t}_{y}\sin {k}_{y}a{\sigma }_{y}\\ +(m-{b}_{x}(1-\cos {k}_{x}a)-{b}_{y}(1-\cos {k}_{y}a)){\sigma }_{z},\\ \end{array}$$
(10)

where k is the Bloch wavenumber. An isotropic version of the two-dimensional Wilson-Dirac model with b ≡ bx = by and tx = ty exhibits the Chern insulating phase when 0 < m/b < 2 and 2 < m/b < 460. Next, we add interlayer hoppings, including a direct hopping t4 along the z-axis and a chiral hopping t3, where t3 and t4 are real parameters. To describe the chiral hopping t3, the lattice sites in the square lattice in each layer into groups of four sites, (2i − 1, 2j − 1), (2i − 1, 2j), (2i, 2j − 1), and (2i, 2j) where i and j are integers, and we introduce chiral hoppings between the groups on the neighboring layers. Then the total Hamiltonian for this model on a tetragonal lattice is given by

$$H={H}_{{{{{{{{\rm{WD}}}}}}}}}+{H}_{{{{{{{{\rm{interlayer}}}}}}}}},$$
(11)

where

$$\begin{array}{ll}{H}_{{{{{{{{\rm{interlayer}}}}}}}}}= {t}_{3}\mathop{\sum}\limits_{i,j,l}({c}_{2i-1,2j-1,l}^{{{{\dagger}}} }{c}_{2i,2j-1,l+1}+H.c.)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;+{t}_{3}\mathop{\sum}\limits_{i,j,l}({c}_{2i,2j-1,l}^{{{{\dagger}}} }{c}_{2i,2j,l+1}+H.c.)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;+{t}_{3}\mathop{\sum}\limits_{i,j,l}({c}_{2i,2j,l}^{{{{\dagger}}} }{c}_{2i-1,2j,l+1}+H.c.)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{t}_{3}\mathop{\sum}\limits_{i,j,l}({c}_{2i-1,2j,l}^{{{{\dagger}}} }{c}_{2i-1,2j-1,l+1}+H.c.)\\ \;\;\;\;\;\;\;+{t}_{4}\mathop{\sum}\limits_{i,j,l}({c}_{i,j,l}^{{{{\dagger}}} }{c}_{i,j,l+1}+H.c.).\end{array}$$
(12)

These hoppings in Hinterlayer form structures similar to right-handed solenoids. When HWD is in the Chern insulator phase, even if HWD is perturbed by Hinterlayer, the system remains in the Chern insulator with the Chern number within the xy plane equal to −1 as long as t3 and t4 are small. In the main text, we are interested in the KME due to the topological surface states in the topological Chern insulating phase, in which the Fermi energy is in the energy gap.

Fitting function for KME

By taking into account the finite-size effect in Eq. (7), we give a fitting function. The finite penetration depth of the surface states will lead to O(1/L) correction to the KME, and that around the corner will lead to O(1/L2) correction38. Thus, the fitting function is

$${M}_{z}^{{{{{{{{\rm{KME}}}}}}}}}=\frac{{w}_{1}{L}_{x}+{w}_{2}{L}_{y}+{w}_{3}+\frac{{w}_{4}}{{L}_{x}}+\frac{{w}_{5}}{{L}_{y}}}{{w}_{6}{L}_{x}+{w}_{7}{L}_{y}},$$
(13)

where wi(i = 1, 2, …, 7) are real constants.