**Axion electrodynamics and axion insulators**

(page under construction)

Axion electrodynamics is a new fascinating playground in the field of topological insulators (TIs). It has been predicted in theory [1] that under slow variations of electromagnetic fields, i.e. when the frequencies are << E_{g}/ℏ (E_{g} – topological gap), the cross induction takes place when external electric (magnetic) field induces magnetic (electric) polarization, the phenomena hereby called *direct* and *inverse* topological magnetoelectric (TME) effects. The proportionality coefficient for direct and inverse TME effects is odd-multiple [1] of fine structure constant. Experimental observation of such cross induction was made using Kerr and Faraday effects at THz frequencies. It still remains a question how strong the static and dynamic polarizations induced by TME effects can be in real experimental situations. For direct TME effect, the magnetic polarization can be conveniently expressed in numbers of Bohr magnetons per unit cell. For layered van der Waals (vdW) material with unit cell volume* a ^{2}h* (

*a*∼ 3Å; layer height

*h∼*1 nm) the number of induced Bohr magnetons (

*N*) can be thus determined, and after substitution of all constants we find:

*N* ∼ (*h* /*a** _{B}*) × E(V/Å),

where *a*_{B} is classical Bohr radius, and electric field E is normalized to V/Å. Thus, significant *N*∼1 can be obtained in external fields ∼0.1 V/Å. Such fields (∼V nm^{-1}) can be easily obtained using STM setup with vacuum gap or using properly designed dielectric gates. The estimations for inverse TME effect are less optimistic. Even at B=10 T the induced electric dipole moment per unit cell ∼0.001 Å × electron charge.

Another important development in axion electrodynamics was introduction of a new class of materials, axion insulators. These are materials exhibiting both TME effect and quantum anomalous Hall (QAH) effect, a quantized Hall conductance without Landau levels. Characterized by a quantized Hall conductance and vanishing longitudinal conductance at zero magnetic field, the QAH effect requires combining topology with magnetism. TIs with intrinsic magnetism represent the most promising class of materials because the ideal magnetic-TI interface in these materials naturally appears as a result of crystal growth. First TI with intrinsic antiferromagnetism MnBi_{2}Te_{4} was theoretically predicted and experimentally discovered in 2018 [2]. MnBi_{2}Te_{4} crystallizes in a rhombohedral structure, built of the stacking of Te-Bi-Te-Mn-Te-Bi-Te septuple layers (SLs) along the *c*-axis as shown in Fig. 1a. The antiferromagnetism of this material is produced by the Mn sublattice located in the middle of SL, while its nontrivial surface state is formed by inverted Bi and Te *p _{z}* bands at the Γ-point due to strong spin-orbital coupling. Its magnetic state is characterized by an A-type order formed by ferromagnetic Mn layers antiferromagnetically stacked along the

*c*-axis with bulk T

_{N}= 25 K [3]. The FM layer near the surface is anticipated to break time-reversal symmetry, thus opening a large gap E

_{g }∼ 100 meV in the topological surface state. MnBi

_{2}Te

_{4}is expected to be an ideal quantum material for hosting interesting topological phases, including a high-temperature QAH insulator in thin films with odd numbers of SLs, an axion insulator state in thin films with even numbers of SLs, an ideal Weyl semimetal state with a single pair of Weyl nodes near the Fermi level, and possibly chiral Majorana modes.

Figure 1. (a) Unit cell of MnBi_{2}Te_{4} built of the stacking of Te-Bi-Te-Mn-Te-Bi-Te septuple layers (SL) along the c-axis. SL height is 1.36 nm. (b) SEM micrograph of MnBi_{2}Te_{4} crystal from NIST courtesy of A. Davydov.

Current STM research in the laboratory is conducted using the samples from PSU collaborator Prof. Zhiqiang Mao [4] and is focused on

- STM imaging and spectroscopy of differently cleaved MnBi
_{2}Te_{4}samples - STM imaging mesoscopic states.

**Literature**

5. Room temperature spin-polarized scanning tunneling microscopy.