Tunable dynamical magnetoelectric effect in antiferromagnetic topological insulator MnBi₂Te₄ films

Abstract

Axion was postulated as an elementary particle to solve the strong charge conjugation and parity puzzle, and later axion was also considered to be a possible component of dark matter in the universe. However, the existence of axions in nature has not been confirmed. Interestingly, axions arise out of pseudoscalar fields derived from the Chern–Simons theory in condensed matter physics. In antiferromagnetic insulators, the axion field can become dynamical due to spin-wave excitations and exhibits rich exotic phenomena, such as axion polariton. However, antiferromagnetic dynamical axion insulator has yet been experimentally identified in realistic materials. Very recently, MnBi₂Te₄ was discovered to be an antiferromagnetic topological insulator with a quantized static axion field protected by inversion symmetry ${\mathcal{P}}$ and magnetic-crystalline symmetry ${\mathcal{S}}$. Here, we studied MnBi₂Te₄ films in which both the ${\mathcal{P}}$ and ${\mathcal{S}}$ symmetries are spontaneously broken and found that substantially enhanced dynamical magnetoelectric effects could be realized through tuning the thickness of MnBi₂Te₄ films, temperature, or element substitutions. Our results show that thin films of MnBi₂Te₄ and related compounds could provide a promising material platform to experimentally study axion electrodynamics.

Introduction

In condensed matter physics, an effective axion field can be derived from the (4+1)-dimensional Chern-Simons theory of topological insulators (TIs)^1,2, described by an axion action ${S}_{\theta }=\frac{\theta }{2\pi }\frac{{e}^{2}}{h}\int {{\rm{d}}}^{3}x{\rm{d}}t{\bf{E}}\cdot {\bf{B}}$, in which E and B are the electromagnetic fields inside the insulators, e is the charge of an electron, h is Planck’s constant, θ is the dimensionless pseudoscalar parameter as the axion field³. The axion field θ is odd under the time reversal symmetry ${\mathcal{T}}$ or spatial inversion symmetry ${\mathcal{P}}$, so it is quantized to π (mod 2π) for TIs and 0 (mod 2π) for normal insulators (NIs) if ${\mathcal{T}}$ or ${\mathcal{P}}$ is preserved. Such a quantized θ can lead to the image magnetic monopole effect⁴, the quantized magneto-optical Faraday/Kerr effect^{5,6,7,8,9,10,11,12}, the topological magnetoelectric effect (TME)^{2,13,14,15,16,17,18} and the half-integer quantum Hall effect on the ${\mathcal{T}}$-breaking surface of TIs². Recently, the axion field in condensed matters was studied in magnetic topological insulators^{19,20,21,22,23,24,25,26,27}, topological-material heterostructures^28[,29, and charge-density-wave Weyl semimetals^30,31. An overall review of the axion physics in condensed matters can be found in refs. ^32,33.

In antiferromagnetic (AFM) insulators, the axion field θ can be an unquantized value, once both ${\mathcal{T}}$ and ${\mathcal{P}}$ are broken. The spin-wave excitations can induce fluctuations of the axion field^34,35,36, called as the dynamical axion field (DAF) θ(r, t) which has spatial and temporal dependence. θ(r, t) can lead to rich dynamical magnetoelectric (ME) effects, for example, the dynamical chiral magnetic effect^37,38,39,40, anomalous Hall effect³⁷, axionic polariton^34,41 and unconventional electromagnetic effects^42,43,44. In principle, generic AFM insulators may exhibit the DAFs, such as Cr₂O₃^15,45,46, but the DAFs turn out to be too weak to be detected³⁶. Large DAF requires the topologically nontrivial AFM insulators described by nonzero spin Chern numbers⁴⁷. Recently, layered material Mn₂Bi₂Te₅ and the superlattice (MnBi₂Te₄)₂(Bi₂Te₃) were predicted to host large DAFs^36,47, denoted as DAF insulators. However, the experimental synthesis of them has not been reported. Therefore, the challenge is to discover realistic DAF insulators.

Recently, bulk MnBi₂Te₄ was discovered to be an AFM TI and it hosts a quantized axion field θ = π protected by the inversion symmetry ${\mathcal{P}}$ and the magnetic-crystalline symmetry ${\mathcal{S}}={\mathcal{T}}{\tau }_{1/2}$, where τ_1/2 is the half-translation operator^{22,23,25,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62}. Interestingly, both ${\mathcal{P}}$ and ${\mathcal{S}}$ are spontaneously broken in MnBi₂Te₄ films with even septuple layers (SLs). In this article, our results indicate that the fluctuations of AFM order can lead to substantially enhanced dynamical ME effects in MnBi₂Te₄ thin films, induced by the ${\mathcal{P}}$ and ${\mathcal{S}}$ breaking.

As shown in Fig. 1a, the AFM TI MnBi₂Te₄²² consists of SLs coupled to each other by a Van der Waals-type interaction. It features an A-type AFM magnetic ground state with the out-of-plane easy axis. Although the magnetic order explicitly breaks the ${\mathcal{T}}$, the bulk MnBi₂Te₄ possesses the ${\mathcal{S}}$, marked in Fig. 1a. In addition, the ${\mathcal{P}}$ is preserved for the bulk MnBi₂Te₄ with the inversion center at the Mn layer in the middle of each SL. Interestingly, both the ${\mathcal{S}}$ and ${\mathcal{P}}$ are explicitly broken in even-SL MnBi₂Te₄ thin films due to the finite size, schematically shown in Fig. 1b. As a result, the restriction for the quantization of the axion field θ is removed, which will lead to a tunable dynamical ME effect.

Fig. 1: Schematic of ${\mathcal{P}}$, ${\mathcal{S}}$-breaking mass in MnBi₂Te₄ thin films.

a Crystal structure of MnBi₂Te₄. The red arrows denote the spin moment of Mn atoms and the green arrows represent the half-translation operator τ_1/2 for the ${\mathcal{S}}$ symmetry. b Schematic of MnBi₂Te₄ thin films with even septuple layers grown along the easy axis (the z-direction). c Schematic of the ${\mathcal{P}}$, ${\mathcal{S}}$-breaking mass m₅ in MnBi₂Te₄ thin films. m₅ expects to increase when the film thickness decreases due to the ${\mathcal{P}}$, ${\mathcal{S}}$-breaking.

Results

Effective model analysis

We start from the low-energy effective Hamiltonian of bulk MnBi₂Te₄. Taking the relevant symmetries ${\mathcal{S}}$, ${\mathcal{P}}$ and ${{\mathcal{C}}}_{3z}$ into account, the four-band k ⋅ p model up to the third order of k can be obtained as²² H = H₀ + H₃, where

$${H}_{0}=\epsilon ({\bf{k}}){{{\Gamma }}}_{0}+\mathop{\sum }\limits_{i=1}^{5}{d}_{i}({\bf{k}}){{{\Gamma }}}^{i},$$

(1)

with ${d}_{i}({\bf{k}})=\left(\right.{A}_{2}{k}_{y},-{A}_{2}{k}_{x},{A}_{1}{k}_{z},M({\bf{k}}),0\left)\right.$, $\epsilon ({\bf{k}})={C}_{0}+{D}_{1}{k}_{z}^{2}+{D}_{2}({k}_{x}^{2}+{k}_{y}^{2})$, and $M({\bf{k}})={m}_{4}+{B}_{1}{k}_{z}^{2}+{B}_{2}({k}_{x}^{2}+{k}_{y}^{2})$. Here, m₄ is the mass term with m₄ < 0 (m₄ > 0) depicting the inverted (normal) band structure. The Dirac matrices are given by Γ_0,1,...,5 = (τ₀ ⊗ σ₀, τ_x ⊗ σ_x, τ_x ⊗ σ_y, τ_y ⊗ σ₀, τ_z ⊗ σ₀, τ_x ⊗ σ_z), where τ_x,y,z and σ_x,y,z denote the Pauli matrices for the orbital and spin, respectively. The third order term H₃ is given explicitly by

$${H}_{3}={R}_{1}\left({k}_{x}^{3}-3{k}_{x}{k}_{y}^{2}\right){{{\Gamma }}}_{5}+{R}_{2}\left(3{k}_{x}^{2}{k}_{y}-{k}_{y}^{3}\right){{{\Gamma }}}_{3}.$$

(2)

The detailed parameters can be obtained by fitting with first-principles band structure and have already been provided in ref. ²².

For MnBi₂Te₄ thin films, both ${\mathcal{S}}$ and ${\mathcal{P}}$ are broken, but the combined symmetry ${\mathcal{PT}}$ is preserved. Based on the symmetry analysis, a mass term m₅Γ₅ to the leading order adds to the effective Hamiltonian of MnBi₂Te₄ thin films and significantly changes the axion field θ from the quantized value (π). Actually, the mass term m₅Γ₅ indicates an effective staggered Zeeman field generated by the AFM order³⁴. Due to the finite-size effect, it expects to decrease with increasing thickness of the films, and finally vanishes for the bulk limit, as illustrated in Fig. 1c. This is in contrast to the three-dimensional (3D) DAF insulator^34,36,47 with a non-vanishing m₅ in the bulk limit. Based on the effective Hamiltonian, the energy gap at the Γ point of MnBi₂Te₄ thin films is found to be ${E}_{\text{gap}}=2\sqrt{{m}_{4}^{2}+{m}_{5}^{2}}$. When the strength of spin-orbit coupling (SOC) λ is tuned from zero to the realistic value λ₀, the band inversion is induced along with the mass term m₄ changing from m₄ > 0 to m₄ < 0, and the energy gap reaches the minimum E_gap = 2∣m₅∣ when the band inversion happens (m₄ = 0). Based on this picture, we can quantitatively characterize the mass m₅ for MnBi₂Te₄ thin films.

As an example, the first-principles band structures of 12-SL MnBi₂Te₄ with different SOC are calculated, shown in Fig. 2a. The SOC dependence of E_gap is further extracted and shown in Fig. 2b. When SOC is weak, the band structures indicate an energy gap E_gap without a band inversion. As expected, when increasing SOC, E_gap decreases to a minimum around SOC (λ ~ 0.9λ₀) without gap closing, and then increases with a band inversion. In Fig. 2c, it explicitly shows the mass m₅ as a function of the thickness of N bi-SL MnBi₂Te₄ thin films (red line), which indeed gradually decreases with increasing N. For comparison, we also present the result of m₅ for Mn₂Bi₂Te₅ thin films (blue line). It is obvious that for large N towards the bulk limit, m₅ approaches zero (a finite value) for MnBi₂Te₄ (Mn₂Bi₂Te₅), as expected. Note that m₅ of MnBi₂Te₄ films is tunable to zero within a larger range than that of Mn₂Bi₂Te₅ films, as shown in Fig. 2c. Furthermore, it is numerically found that m₅ exhibits an approximately linear behavior against 1/N, namely, the inverse of the thickness for both MnBi₂Te₄ and Mn₂Bi₂Te₅ films, as shown in Fig. 2d.

Fig. 2: Finite-size effect of MnBi₂Te₄ thin films.

a The evolution of the band structure of the 12-SL MnBi₂Te₄ thin film as a function of the strength of the SOC λ/λ₀. b The evolution of the energy gap with the SOC strength. The energy gap first goes smaller and then goes larger without exactly closing. It is clear to see a minimum energy gap when increasing the SOC, which corresponds to 2∣m₅∣ (the ${\mathcal{P}}$, ${\mathcal{S}}$-breaking mass). c,d The mass m₅ versus the thickness of MnBi₂Te₄ (red) and Mn₂Bi₂Te₅ (blue) thin films. N presents the number of bi-SLs of MnBi₂Te₄ films or (nonuple layers)NLs of Mn₂Bi₂Te₅ films.

Dynamical magnetoelectric effect

The application of an electric field can induce a magnetization, and reversely, the application of a magnetic field can lead to a polarization, which is the typical ME effect. In 3D TIs, such as Bi₂Se₃⁶³, the ME response is isotropic, which can be described by the effective axion action S_θ with the quantized axion field θ = π, protected by the time reversal symmetry ${\mathcal{T}}$. In AFM TI MnBi₂Te₄, both ${\mathcal{P}}$ and ${\mathcal{S}}$ constrain the axion field θ to the quantized value of π. However, for even-SL MnBi₂Te₄ thin films, all ${\mathcal{P}}$, ${\mathcal{T}}$ and ${\mathcal{S}}$ symmetries are broken and the mass m₅ is induced, thus rendering an unquantized θ. It should be pointed out that due to the finite-size effect in MnBi₂Te₄ thin films, the ME response is generically anisotropic⁶⁴. Moreover, since m₅ is related to the inherently fluctuating AFM order, the axion field will be time dependent ∂_tθ(t) ≠ 0, leading to the dynamical ME effect. In the following, we will show that the dynamical ME effect can be realized and substantially tuned by adjusting the film thickness, temperature or element substitutions.

Here, we consider applying a magnetic field B in the plane of MnBi₂Te₄ thin films (e.g. the x-axis in Fig. 1a), which will induce a charge polarization P in the same direction. To describe such a ME response, we need to calculate the ME coefficient α defined as P = − αB and it can be obtained from the quantity $\gamma \equiv (1/d)\mathop{\int}\nolimits_{-d/2}^{d/2}{\rm{d}}z\eta (z)$ through α = γ(e²/2h)^17,64. Here, η(z) is a dimensionless function defined as

$$\eta (z)=2\mathop{\int }\limits_{-d/2}^{d/2}{\rm{d}}z^{\prime} z^{\prime} {{{\Pi }}}_{xy}(z,z^{\prime} ),$$

(3)

where the current correlation function ${{{\Pi }}}_{xy}(z,z^{\prime} )$ is given by the Kubo formula,

$$\begin{array}{lll}{{{\Pi }}}_{xy}(z,z^{\prime} )&=&\frac{{\hslash }^{2}}{2\pi e}\int {d}^{2}{\bf{k}}\mathop{\sum}\limits_{n\ne m}f({\epsilon }_{n{\bf{k}}})\\ &&\times 2{\rm{Im}}\left[\frac{\left\langle n{\bf{k}}\right|{j}_{x}({\bf{k}},z)\left|m{\bf{k}}\right\rangle \left\langle m{\bf{k}}\right|{j}_{y}({\bf{k}},z^{\prime} )\left|n{\bf{k}}\right\rangle }{{({\epsilon }_{n{\bf{k}}}-{\epsilon }_{m{\bf{k}}})}^{2}}\right].\end{array}$$

(4)

Here, $\left|n{\bf{k}}\right\rangle$ is the normalized Bloch wavefunction of the n-th electron subband, and j(k, z) = (e/ℏ)∂_kH(k, z) represents the x-y plane current density operator, where H(k, z) is obtained by discretizing the system into a tight-binding model along the z-direction and fixing the periodic boundary condition in the x-y plane to preserve k_x and k_y as good quantum numbers.

In the bulk limit, the quantity γ is related to the axion field by γ = θ/π. Considering the inherent AFM fluctuations with δm₅, θ can be divided into a static part θ₀ and a dynamical part δθ(t). Similarly, γ is also comprised of a static part γ₀ and a dynamical part δγ(t) given by δγ = δm₅/g with the coefficient 1/g = ∂γ/∂m₅ to the linear order of δm₅. We can see that 1/g is a key quantity for a large dynamical ME effect under AFM fluctuations δm₅.

In Fig. 3a, we present the quantity γ₀ as a function of the thickness of N bi-SLs MnBi₂Te₄ thin films (red line) and we take Mn₂Bi₂Te₅ films (yellow line) for comparison. It can be seen that with increasing N, γ₀ gradually increases and approaches 1 at the bulk limit with N → ∞ for MnBi₂Te₄ while it approaches a finite value ( < 1) for Mn₂Bi₂Te₅ films at the bulk limit. The dependence of γ₀ on the mass term m₅ for MnBi₂Te₄ films with different thicknesses (N = 4, 6, 8, 16) are shown in Fig. 3b. When N is large, γ₀ quickly increases when m₅ is small. With increasing m₅, γ₀ exhibits a linear decrease for all films, which indicates a large m₅ tends to suppress γ₀. Based on the dependence of γ₀ on m₅, the dependence of the the quantity 1/g on m₅ can be further obtained, shown in Fig. 3d. The generic feature is that 1/g increases with decreasing m₅, and approaches the maximum value when m₅ → 0. It is notable that there is a broad range of m₅ to a large 1/g for thin films (e.g. N = 4, 6, 8), while that range of m₅ is extremely reduced for thick films (e.g. N = 16). We then investigate the tunability of 1/g from the film thickness, temperature and element substitutions.

Fig. 3: Dynamical magnetoelectic effect.

a, c The ME quantity γ₀ (a) and 1/g (c) versus the thickness with N bi-SLs for MnBi₂Te₄ (red) and NLs Mn₂Bi₂Te₅ (yellow) thin films, where values of m₅ are obtained from first-principles calculations. b, d γ₀ (b) and 1/g (d) as a function of m₅ for films with different thickness (N = 4, 6, 8, 16). The shaded regions in (d) denote the substantially tunable range 1/g of realistic MnBi₂Te₄ thin films by decreasing m₅ with increasing temperatures up to the Néel temperature. e, f γ₀ (e) and 1/g (f) as a function of the thickness (N bi-SLs) of films for three representative values of the mass term m₄, where negative (positive) m₄ represents topologically nontrivial (trivial) band structure with (without) the band inversion. The mass term m₄ can be tuned through the element substitution of Mn(Bi_1−xSb_x)₂(Te_1−ySe_y)₄.

We first inspect the finite-size effect. Based on first-principles calculations of m₅ presented in Fig. 2c, the dependence of 1/g on varying N is shown in Fig. 3c. We can see that 1/g is large for the ultrathin limit, and quickly decreases with increasing the film thickness (N), and then crosses zero from positive to negative, and gradually saturates. Secondly, we discuss the temperature effect. M⁻ = 1/2(〈S_iA〉 − 〈S_iB〉) is taken as the AFM order parameter, where i denotes the unit cell and A/B denotes the magnetic sites. The AFM order is in the $\hat{z}$ direction (${{\bf{M}}}^{-}={M}^{-}\hat{z}$) for MnBi₂Te₄ films and it leads to ${m}_{5}\propto {M}_{z}^{-}$³⁴. With increasing the temperature from zero to the Néel temperature, ${M}_{z}^{-}$ inevitably decreases from the maximum value to zero. Consequently, for MnBi₂Te₄ thin films, the m₅ can be continuously tuned by the temperature, from the maximum value at zero temperature to zero near the Néel temperature T_N. Therefore, a tunable range of substantially enhanced 1/g is obtained through adjusting the temperature from zero to T_N (see the shaded regions). According to Fig. 3a, d, the ultrathin films (e.g. N = 4) have a broad tunable range of m₅ to large 1/g, but a small static γ₀. The bulk limit films have a tiny range of m₅ to large 1/g (e.g. N = 16). Most interestingly, the proper thin films (e.g. N = 6, 8) have both a broad tunable range of m₅ to large 1/g and a large static γ₀, which should be a promising choice for experiments.

At last, we investigate the effect of element substitutions. The mass term m₄, with m₄ < 0 (m₄ > 0) indicates a topologically nontrivial (trivial) electronic structure. Interestingly, m₄ can be experimentally tuned through element substitutions in Mn(Bi_1−xSb_x)₂(Te_1−ySe_y)₄ materials. Here, we choose three representative m₄, namely, two in the band-inverted nontrivial regime [m₄ = − 116.7 meV (red lines) is taken from ref. ²² and m₄ = − 50 meV (blue lines) presents a moderate band inversion], and one in the trivial regime [m₄ = 50 meV (yellow lines)], to show γ₀ and 1/g for different thick films in Fig. 3e, f. In the topologically trivial case, γ₀ ~ 0. In the topologically nontrivial case, it can be seen that a larger ∣m₄∣ exhibits a larger γ₀, which approaches 1 for N → ∞. According to Fig. 3f, 1/g in the topologically trivial case is obviously negligible compared to that in the topologically nontrivial case. As for the nontrivial case, overall speaking, a weaker band inversion with smaller ∣m₄∣ (m₄ < 0) leads to a larger magnitude of 1/g, shown in Fig. 3f. Notably, the band-inversion magnitude could be modified by the finite-size effect⁶⁵, as reflected by the non-monotonic behavior (blue line) of 1/g on N in Fig. 3f.

Experimental detection

Many interesting experimental designs of axion electrodynamics have been proposed, including the instability in an external electric field⁴², the recent proposal of unconventional level attraction in cavity axion polariton⁴¹ and the proposal to detect dark matter axions⁶⁶, but the experimental progress is quite slow due to the lack of realistic DAF materials. The tunable dynamical ME effect in MnBi₂Te₄ thin films expects to be used for these proposals. Here, we quantitatively estimate that it is experimentally accessible to detect the dynamical ME effect in MnBi₂Te₄ thin films through the current transport techniques. The basic idea is to measure the in-plane response current induced by the chiral magnetic effect, given by ${j}_{\text{CME}}=-\frac{{e}^{2}}{2h}\frac{\partial \delta \gamma }{\partial t}B$, where B is the uniform inplane magnetic field. Since the dynamical ME coefficient δγ is proportional to the change of the AFM order δm₅, a time-varying δγ can be driven by AFM resonance³⁷, in the presence of both a uniform magnetic field along the out-of-plane (z) direction and a microwave magnetic field along the in-plane direction. Therefore, as schematically illustrated in Fig. 4, we apply a tilted static magnetic field B, with components B_x and B_z along x and z directions, respectively, and a microwave magnetic field B_ω with a frequency of ω along the inplane direction. Taking $\delta \gamma (t)\approx \delta {m}_{5}^{0}\sin (\omega t)/g$, where $\delta {m}_{5}^{0}$ is the amplitude of the driven AFM fluctuation, the response current can be obtained as

$${j}_{\text{CME}}=-\frac{{e}^{2}}{2h}\frac{\delta {m}_{5}^{0}\omega {B}_{x}}{g}\cos (\omega t).$$

(5)

Note that since 1/g can be tuned by the film thickness, temperature, and element substitutions, the magnitude of j_CME can thus be tunable. As a concrete example, we choose N = 6 bi-SL MnBi₂Te₄ thin film with γ₀ = 0.6, m₅ = 3 meV, and 1/g = 0.1 meV⁻¹. The other parameters are chosen typically as B_x = 0.1 T, ω/2π = 20 GHz, and $\delta {m}_{5}^{0}=0.5$ meV. Finally, the amplitude of j_CME is approximated as 1.2Acm⁻², which can be further much enhanced through tuning 1/g (see the shade region in Fig. 3d), therefore it is accessible by current transport experimental techniques.

Fig. 4: Experimental setup for detecting the tunable dynamical ME effect.

A tilted static magnetic is applied with components along both x and z directions. A microwave magnetic field with a frequency of ω is applied along the in-plane x-direction to drive the AFM resonance. The chiral magnetic effect due to the time-varying dynamical ME coefficient will lead to a measurable inplane response current.

Discussion

To summarize, our work presents that AFM TI films MnBi₂Te₄ family, which have been successfully synthesized and well studied, provide a promising material platform to realize the dynamical ME effects. More interestingly, the dynamical ME effect is tunable and even substantially enhanced through engineering the thickness of films, temperature or element substitutions. It expects to be experimentally detectable and should be used to realize those exotic axion electrodynamics previously proposed.

Methods

Computational details

We performed the first-principles calculations for bulk MnBi₂Te₄ and Mn₂Bi₂Te₅ with different SOC through employing the Vienna ab-initio simulation package(VASP)^67,68 and the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof(PBE)^69,70 type exchange-correlation potential was adopted, with the energy cutoff fixed to 420 eV. By considering the transition metal Mn, GGA+U functional with U = 3 eV for Mn d orbitals for all the results in this work. The k-point sampling grid of Brillouin zone in the self-consistent process was a Γ-centered Monkhorst–Pack k-point mesh of 8 × 8 × 4 for the bulk systems, and a total energy tolerance 10⁻⁷eV was adopted for self-consistent convergence. For MnBi₂Te₄ and Mn₂Bi₂Te₅ films, we employed the maximally localized Wanneir functions (MLWF) from the first-principles calculations^71,72 to construct tight-binding Hamiltonians for different thickness. The Mn − d, Te − p and Bi − p orbitals were initialized for MLWFs by Wannier90⁷³.