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/¯h 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?�s 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 FANG**received 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.