# Metadata **Title:** Interface-induced sign reversal of the anomalous Hall effect in magnetic topological insulator heterostructures **Authors:** Wang, Fei; Wang, Xuepeng; Zhao, Yi-Fan; Xiao, Di; Zhou, Ling-Jie; Liu, Wei; Zhang, Zhidong; Zhao, Weiwei; Chan, Moses H. W.; Samarth, Nitin; Liu, Chaoxing; Zhang, Haijun; Chang, Cui-Zu **Year:** 2021 **Tags:** #TI [URL](http://dx.doi.org/10.1038/s41467-020-20349-z)|[Bookends](bookends://sonnysoftware.com/71961) --- # Introduction The [[Berry Phase]] encodes the [[Adiabatic]] evolution of [[Eigenstate|eigen]] [[wavefunction|wavefunctions]] around the [[Fermi surface]] in the first [[Brillouin Zone]] of [[momentum space]]. It is important for understanding many physical phenomena in [[quantum material|quantum materials]] such as the intrinsic [[anomalous Hall]] (AH) effect. The [[Berry Curvature]] of the occupied [[Bloch's Theorem|Bloch wavefunctions]] is equivalent to an effective [[magnetic field]] in [[momentum space]], possibly affecting the motion of electrons and giving rise to an [[anomalous Hall|AH]] effect in [[ferromagnetic]] (FM) materials. The [[Hall conductance]] of the intrinsic [[anomalous Hall|AH]] effect in a two-dimensional (2D) [[ferromagnetic|FM]] material can be calculated from integrating the [[Berry Curvature]] $\Omega$ in the first [[Brillouin Zone]] (BZ): $ \sigma_{xy} = \frac{-e^2}{2 \pi h} \int_{\text{BZ}} \Omega(\pmb{k}) \text{d}^2\pmb{k} = - \frac{e^2}{h} C, $ where $e$ is the [[elementary charge]], $h$ is [[Planck's constant]]. Here $C$ is the [[Chern number]] a [[topological invariant]]. We can study the origin and mechanism of the [[anomalous Hall|AH]] effect in the metallic regiome via the scaling relationship between the [[Hall conductance|AH resistance]] $\rho_{yx}$ and the longitudinal resistance $\rho_{xx}$. There are two possibilities. 1. $\rho_{yx} \propto \rho_{xx}^2$: the AH effect is induced by scattering independent mechanisms, i.e., intrinsic or extrinsic [[side-jump]]. 2. $\rho_{yx} \propto \rho_{xx}^2$: the AH effect is induced by scattering dependent mechanisms, i.e., only extrinsic skew-[[scattering]]. In metallic [[diluted magnetic semiconductors]] (DMSs), it has been proven that the AH effect is dominated by intrinsic mechanisms involving the [[Berry Phase]]. This is expected to hold for magnetically doped [[Time-Reversal Symmetric Topological Insulator|TIs]] since they fall in this category of DMSs. In this work, the authors do the following: 1. They deposit TI films of different thicknesses on top of magnetic TI films to form TI/magnetic TI bilayers, and carry out Hall measurements. They observe a quadratic dependence between $\rho_{yx}$ and $\rho_{xx}$ in the metallic regime, i.e., the [[anomalous Hall|AH]] effect is of scattering independent origin. 2. By tuning the TI thickness and/or the gate voltage, the magnitude and sign of the AH effect for the magnetic TI heterostructures can be changed. 3. They perform *ab initio* calculations and attribute this AH effect change to the interface induced [[Berry Curvature]] reconstruction. This occurs due to the [[Band Structure]] modulation of the magnetic TI layers due to the built-in electric fields. 4. They fabricate magnetic TI/TI/magnetic TI sandwiches with different dopants and find a [[topological Hall]] (TH) effect-like behavior. Analysis of this data shows that the effect is not a result of [[chiral spin textures]], but due to the superposition of two [[anomalous Hall|AH]] effects with opposite signs. # Experimental Results ## Magnetic TI The magnetic TI considered in this section is V-doped Sb$_2$Te$_3$. Considering the [[Berry Curvature]], we have two cases: 1. $\int_{\text{BZ}} \Omega(\pmb{k}) \text{d}^2\pmb{k} < 0 \implies \sigma_{xy} > 0$ 2. $\int_{\text{BZ}} \Omega(\pmb{k}) \text{d}^2\pmb{k} > 0 \implies \sigma_{xy} < 0$ ![[Pasted image 20210622134338.png]] In the figure above, we have an external [[magnetic field]] applied perpendicular to the sample. The spontaneously ordered magnetic moments deflect the motion of the electrons and the [[anomalous Hall|AH]] effect arises. When the internal magnetization points upwards,the mobile electrons move to the high-potential side for $\int \Omega < 0$ and vice-versa for $\int \Omega > 0$. Thus, we have $\sigma_{xy} > 0$ in the first case and $\sigma_{xy} < 0$ in the second case. In the sample under study, they found $\rho_{yx} \propto \rho_{xx}^\alpha$ where $\alpha \approx 2$, indicating that [[scattering]]-independent mechanisms are responsible for the AH effect. However, since [[Time-Reversal Symmetric Topological Insulator|TRS-TIs]] have strongly [[Spin-Orbit Coupling|spin-orbit coupled]] [[Band Structure|bands]], the intrinsic contribution is expected to be dominant, similar to the AH effect in magnetic [[Heusler metal|Heusler]] and half-Heusler metals. ## TI/Magnetic TI Heterostructures In this section, the authors study undoped TI (Sb$_2$Te$_3$)/magnetic TI (V-doped Sb$_2$Te$_3$) heterostructures. The hall resistance $\sigma_{xy}$ is much lower than the quantized values, indicating that the [[Fermi energy|Fermi level]] crosses the bulk [[valence band|valence bands]] and that the sample is metallic. This is seen in the DFT calculations in the next section as well. They also observe that $\rho_{yx}(H = 0) \approx 0$ at $T = 40 \text{K}$ suggesting that is the [[Curie temperature]]. They also study the dependence on [[gate voltage]], as shown in the figure below. Negative $V_g$ introduces [[hole]] [[charge carrier|carriers]] and positive $V_g$ introduces [[electron]] carriers. ![[Pasted image 20210622135402.png]] ![[Pasted image 20210622135418.png]] # Theoretical Results We can interpret the sign and changes in $\rho_{yx}$ induced by varying the thickness of the top undoped [[Time-Reversal Symmetric Topological Insulator|TI]] layer based on the [[Berry Curvature]] distribution change in the bottom V-doped TI layer. It is modulated by the built-in electric fields at the interface between Sb$_2$Te$_3$ ([[Sb2Te3]]) and V-doped Sb$_2$Te$_3$. It is known that V dopants in Sb$_2$Te$_3$ introduce an excess of hole carriers, i.e., the [[chemical potential]] of the V-doped Sb$_2$Te$_3$ layer compared to that of Sb$_2$Te$_3$ is deeper in the bulk [[valence band|valence bands]]. When an additional layer of Sb$_2$Te$_3$ is deposited on top of the V-doped Sb$_2$Te$_3$, the chemical potential of form the heterostructure under study, holes move from the V-doped system to the undoped system, and a built in [[electric field]] is created at the interface. The chemical potentials of the two systems are pulled to the same level. The thicker the layer of undoped Sb$_2$Te$_3$, the more holes we can expect it to accept, thus the stronger the electric field. ![[Pasted image 20210622125945.png]] The authors perform DFT calculations where they only include the V-doped Sb$_2$Te$_3$ and an external electric field to simulated the effect of the undoped Sb$_2$Te$_3$ on top. They calculate the [[Hall conductance]] and [[Berry Curvature]] for multiple values of the electric field, thus simulating different thicknesses of the deposited Sb$_2$Te$_3$ films. ![[Pasted image 20210622130301.png]] From the figure, we observe the following: 1. There is no [[quantum anomalous Hall]] effect in any of these systems. This is expected because one surface state of the undoped TI is always ungapped. 2. A trival gap is formed between the bulk conduction bands and valence bands, where the [[Hall conductance]] $\sigma_{xy}$ is zero. This gap is likely due to [[hybridization]] between the top and bottom surface states. 3. The [[Hall conductance]] $\sigma_{xy}$ can be positive or negative at different values of the [[electric field]] depending on where the [[chemical potential]] is located. 4. The calculations show that band crossings and/or anti-crossings (via [[Spin-Orbit Coupling]]) in the bulk bands can contribute to large [[Berry Curvature]]. The [[Berry Curvature]] changes significantly once the chemical potential shifts away from the crossings and/or anti-crossings, which can correspondingly change the sign and/or magnitude of $\sigma_{xy}$ significantly. 5. At 0 [[electric field]] ($E = 0$), the [[Inversion Operator|inversion symmetry]] is preserved and thus we see a hexagonal distribution of the [[Berry Curvature]] in the [[Brillouin Zone]]. This does not hold for non-zero electric fields. 6. The [[electric field]] can tune the position of the band crossings and anti-crossings. 7. There is a narrow energy region with positive $\sigma_{xy}$ near the Fermi level originating from large negative [[Berry Curvature]] integral, due to the band anti-crossing *(b, d)*. Within this small energy region, the gap of band anti-crossing along the $\Gamma-K'$ direction closes then reopens as the electric fields are tuned, and larger electric field fields tune $\sigma_{xy}$ from positive to negative *(f, h)*. 8. Note that the $m=0$ sample corresponds to no electric field, and in the measurements of the real sample, it had a positive $\sigma_{xy}$, in agreement with the theoretical calculations. They made the following approximations in the DFT calculations: 1. They do not include the top layer of the undoped [[Time-Reversal Symmetric Topological Insulator|TI]] and simulate its presence using an [[electric field]]. 2. They use a supercell with uniformly distributed V atoms, whereas in the actual sample the atoms are randomly distributed. In the DFT calculations, due to the uniform distribution of V atoms, they form bands near the [[Fermi energy]]. Many of the bulk [[valence band|valence bands]] come from the V atoms' $d$ orbitals, while in the real samples these orbitals cannot form bands due to the random distribution. Thus, some hole carries in the real sample that occupy the V atom orbitals should be localized and cannot contribute to the transport measurements. Thus, the authors argue that in the real samples, the chemical potential is located within or near the [[anomalous Hall|AH]] sign reversal region. Note that, as seen in the experimental results, $\rho_{xy}$ reaches a maximum value with $m=4$, meaning that any thicker undoped TI layer does not contribute to the formation of the built-in electric field and shunts the current going through the bottom magnetic TI layer. This leads to a decrease in the [[anomalous Hall|AH]] effect until it eventually disappears. In conclusion, varying the thickness of the undoped layer affects the magnitude of the built-in [[electric field|electric fields]] and thus the electronic [[Band Structure]] and the [[Berry Curvature]] distribution, while the application of the [[gate voltage]] primarily induces a [[chemical potential]] shift. # Magnetic TI/TI/Magnetic TI