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DOI : 10.22661/AAPPSBL.2013.23.3.03
From the Anomalous Hall Effect...
ZHONG FANG
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From the Anomalous Hall Effect
to the Quantum Anomalous Hall Effect

HONGMING WENG, XI DAI, AND WHONG FANG
BEIJING NATIONAL LABORATORY FOR CONDENSED MATTER PHYSICS
AND THE INSTITUTE OF PHYSICS, CHINESE ACADEMY OF SCIENCES

In 1879, Edwin H. Hall discovered that when a conductor carrying longitudinal current was placed in a vertical magnetic field, the carrier would be pressed against the transverse side of the conductor, which led to an observed transverse voltage. This is called the Hall effect (HE) [1], and it was a remarkable discovery. The understanding of HE was difficult at that time since the electron would only be discovered 18 years later. After the discovery of the electron, HE became well understood; as we know, it is the Lorentz force experienced by the moving electrons in magnetic field that presses them against the transverse side. HE is now widely used as to measure the carrier density or the strength of magnetic fields.

Fig. 1: (Color online) (a) Hall effect: The longitudinal current Ix under vertical external magnetic filed Hz contributes to the transversal voltage Vy due to the Lorentz force experienced by carriers. (b) Quantum Hall effect: The strong magnetic field Hz enforces electrons into Landau level with cyclotron motion and become localized in bulk while conducting at edges. (c) Anomalous Hall effect: The electrons with majority and minority spin (due to spontaneous magnetization Mz) having opposite "anomalous velocity" due to spin-orbit coupling, then causes unbalanced electron concentration at two transversal sides and leads to finite voltage Vy. (d) Quantum Anomalous Hall effect: The nontrivial quantum state satisfies all necessary conditions and leads to insulating bulk while having a topologically protected conducting edge state with spontaneous magnetization. (e) Spin Hall effect: In a nonmagnetic conductor, equivalent currents in both spin channels with opposite "anomalous velocity" leads to balanced electron concentration at both sides with net spin current in the transversal direction. (f) Quantum Spin Hall effect: The 2D nontrivial Z2 insulator has conducting edge states with opposite spin in different directions, which can be viewed as two time reversal symmetrical copies of QAHE.

In 1880, Edwin H. Hall further found that this "pressing electricity" effect can be larger in ferromagnetic (FM) iron than in non-magnetic conductors. The enhanced Hall effect comes from the additional contribution of spontaneous long range magnetic ordering, which can be observed even without applying an external magnetic field. To be distinguished from the former effect, this effect has been termed as the anomalous Hall effect (AHE) [2]. Though HE and AHE are quite similar to each other, the underlying physics are much different since there is no orbital effect (Lorentz force) when an external magnetic field is absent in AHE.

The mechanism of AHE has been an enigmatic problem since its discovery, and it lasted for almost a century. The AHE problem involves concepts deeply related with topology and geometry that have been formulated only in recent years after the Berry phase was recognized in 1984 [3]. In hindsight, in 1954 Karplus and Luttinger [4] provided an initial step in unraveling this problem. They showed that moving electrons can have an additional contribution to its group velocity when an external electric field is applied. This additional term, dubbed "anomalous velocity," which is contributed by all occupied band states in FM conductors with spinorbit coupling (SOC), can be non-zero and leads to AHE. Therefore, this contribution depends only on the band structure of perfect periodic Hamiltonian and is completely independent of scattering from impurities or defects (therefore called intrinsic AHE). This made it hard to be widely accepted before the concept of the Berry phase was well established. For a long time, two other "extrinsic" contributions had been thought to be the dominant mechanisms that give rise to AHE. One contribution was the skew scattering [5] from impurities caused by effective SOC, and the other contribution was the side jump [6] of carriers due to different electric fields experienced when approaching and leaving an impurity. Controversy also arose because it is hard to make quantitative comparisons with experimental measurements. The unavoidable defects or domains in samples are complex and hard to be treated quantitatively within any extrinsic model and the contributions from both "intrinsic" and "extrinsic" mechanisms co-exist.

In 1980, K. von Klitzing, et al. made the remarkable discovery of the quantum Hall effect (QHE) [7]. He found that with the increase of an external magnetic field, Hall conductivity exhibits a series of quantized plateaus, and at the same time, the longitudinal conductivity becomes zero, i. e., the sample bulk becomes insulating when Hall conductivity is quantized. In QHE, electrons constrained in two-dimensional (2D) samples are enforced to change their quantum states into new ones, namely Landau energy levels, under a highly intensive magnetic field. The originally free-like conducting electrons start to make cyclotron motion. If the magnetic field is strong enough, such cyclotron motions will form full circles, which makes electrons localized in the bulk and the sample becomes insulating. While along the edges of the 2D system, the circular motion enforced by the magnetic field cannot be completed due to the presence of the edges, which enforce the electrons to travel in one way, forming so called edge states. The electrons in such a state can "smartly" circumambulate defects or impurities in their way. Therefore, current carried by these electrons is dissipationless and conductance is quantized into units of e2/짱h with the quantum number corresponding to the number of edge states. Such fascinating quantum states and physical phenomena are highly interesting and impacted the whole field of physics, because these were new states of condensed matter and could be characterized by the topology of the electronic wave-function [8]. This topological number was given by D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs [9] in 1982 and is called the TKNN number or the first Chern number, which has direct physical meaning as the number of edge states or the quantum number of Hall conductance.

After reaching this point, one may immediately ask the following question: can we have the quantum version of AHE, similar to the QHE? Unfortunately, this question was irrelevant at that historical moment, because at that point we did not even know the fundamental mechanism of AHE. Nevertheless, in 1988, F. D. M. Haldane proposed [10] that QHE without any external magnetic field is in principle possible. Although this proposal was very simple and had nothing to do with either AHE or SOC, his idea played important roles for many of our present studies, such as the topological insulators and the quantum anomalous Hall effect (QAHE), because he pointed out the possibility of having a novel electronic band structure of perfect crystal, which carries a nonzero Chern number even in the absence of an external magnetic field.

The underlying physics of AHE, in particular, the topological nature of intrinsic AHE was not fully ap-preciated until the early years of the 21st century. In a series of papers, (i.e, Jungwirth, et al. (2002) [11]; Onoda, et al. (2002) [12]; Fang, et al. (2003) [13]; Yao, et al. (2004) [14]), it was discovered that the intrinsic AHE can be related to the Berry phase of the occupied Bloch states. The so-called "anomalous velocity" is originated from the Berry-phase curvature, which can be regarded as an effectively magnetic field in the momentum space, and thus modifies the equation of motion of electrons, leading to AHE. This effective magnetic field can be also traced back to the band crossings and the magnetic monopoles in the momentum space [13], which are now called Weyl nodes in the Weyl semimetals [15, 16]. The quantitative first-principles calculations for SrRuO3 [13] and Fe [14], in comparison with experiments, suggested that intrinsic AHE actually dom-inates over extrinsic ones. The presence of SOC and the breaking of time reversal symmetry (due to FM ordering) are crucially important, because otherwise the Berry phase contribution will be prohibited by symmetry. What makes those understandings unique is that, like QHE, the intrinsic AHE is now directly linked to the topological properties of the Bloch states. It is now proper to ask the following question: can AHE also be quantized like its cousin HE? If it is realized, we will have a kind of QHE without a magnetic field, although it is already much different from Haldane?占퐏 original speculation in the sense that SOC must play important roles here. Nevertheless, from the applicational point of view, the realization of QAHE will certainly stimulate the wide usage of such novel quantum phenomena in future technologies, in particular, dissipationless edge transport without a magnetic field.

To reach this goal, there is still a big step to be overcome. QAHE requires four necessary conditions: (1) The system must be 2D; (2) insulating bulk; (3) FM ordering; and (4) a non-zero Chern number. It may be easy to satisfy one or two conditions, but it is hard to realize all of them simultaneously. Fortunately, the recent rapid progress in the studies of topological insulators (TI) [17, 18] make the challenge of realizing QAHE possible. TI is a new state of quantum matter, which is characterized by the Z2 topological number [19] and is protected by time-reversal symmetry (TRS), in contrast to QHE and QAHE, where TRS must be broken. However, an important view is that the 2D topological insulator and the quantum spin Hall effect (QSHE) [20, 21] can be effectively viewed as two copies of distinct QAHE states which are related by TRS; in other words, the 2D QSHE state can be viewed as the time-reversal invariant version of QAHE state. Given the known material realizations of 2D and 3D TI [17, 21??5], it is natural to start from those systems and try to break TRS in order to achieve QAHE. Following this strategy, several possibilities are proposed theoretically. Qi, et al. [26] first pointed out that gapping the Dirac type surface states of 3D TI by FM insulating cap-layers may produce QAHE, although their arguments are not concrete. Later on, the "band inversion" picture and the experimental observations of QSHE [21, 22] inspired the idea of realizing QAHE by transition metal (Mn) doped HgTe quantum well structures [27]. Unfortunately, in such cases, the Mn local moments do not order ferromagnetically. In 2010, Yu, et al., [28] predicted that when a topological insulator Bi2Se3 or Bi2Te3 is made thin and magnetically doped (by Cr or Fe), the system should order ferromagnetically through the van Vleck mechanism, and exhibit QAHE with a quantized Hall resistance of h/e2 ?? a proposal that was finally achieved experimentally by Chang, et al. [29, 30] after great effort.

While this is not the end of the story, we believe this success will inspire more extensive research on QAHE. The following two important issues are natural directions for future studies: (1) to increase the temperature range of QAHE (it is now observed only in the tens mK range), and; (2) to realize a higher plateau with a Chern number larger than 1. There are several other proposals worth trying. HgCr2Se4 has been predicted to be a Weyl semimetal, and its quantum well structure with proper thickness would exhibit QAHE with a Chern number of 2 [16], different from the Cr-doped Bi2Te3 family thin film. The advantage of this proposal is that HgCr2Se4 is a chemically pure and stable compound with a known bulk Curie temperature higher than 100K. One similar proposal is GdBiTe3 [31], the thin film of which has one edge state contributing to QAHE. Garrity and Vanderbilt [32] proposed that Au, Pb, Bi, Tl, I, and etc heavyelement layers on the surface of magnetic insulators may contribute to large AHE and even QAHE. With further material breakthroughs, we have strong reason to expect that QAHE may someday find its place in our electronic devices.
This work was supported by the National Science Foundation of China and by the 973 program of China.

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Zhong FANGreceived his Ph.D degree from the Hua-Zhong University of Sci. & Tech. of China in 1996. He visited the National Institute of Advanced Industrial Science and Technology (AIST) of Japan and the Oak Ridge National Laboratory (ORNL) of USA from 1996 to 2003. He joined the Institute of Physics (IoP), Chinese Academy of Sciences (CAS), in 2003. He is now a professor and serving as the deputy director of IoP of CAS. He was selected as a APS fellow in 2011. He was the recipient of the ICTP prize in 2008, the Qiu-Shi Outstanding Achievement Awards in 2011, and the OCPA Achievement in Asia Award in 2012.

 
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