The anomalous Hall effect in ferromagnets and the Berry phase
N. P. Ong, Wei-Li Lee, and
R. J. Cava
In non-magnetic metals, the flow of electrons in crossed E (electric) and H (magnetic) fields is deflected sideways by the well-known Lorentz force. This produces the familiar Hall current which flows parallel to E x H. It has been known for a long time that, in ferromagnets such as Fe or Ni, a different kind of Hall current is observed (the first observation was made in 1880 by E.H. Hall who called it 'pressing electricity'). The application of E alone leads to a transverse Hall current . Since H is not necessary, the Hall current in ferromagnets is called spontaneous or anomalous (in practice a small H serves to align the magnetic domains). The origin of the anomalous Hall effect (AHE) has been vigorously debated for many decades. In this note, we summarize recent experimental progress from our group .
Roughly 50 years ago (1954), Karplus and Luttinger  theorized that the AHE current arises from a general property of how electrons move in a periodic lattice. In all textbooks the group velocity of electrons in the semiclassical limit is given as v(k) = grad[e(k)]. This is now known to be incomplete (we set \hbar to 1 throughout). Karplus and Luttinger calculated that the position operator x in a periodic lattice actually fails to commute with itself (see below).
This has 2 immediate and important consequences. First, the general expression
for group velocity in all lattices is amended to
v(k) = grad[e(k)] - eE x W(k), (1)
where W(k) is the "
Secondly, the AHE current in a ferromagnet is dissipationless, i.e. it is
independent of the scattering rate of the electrons. (In a ferromagnet,
time-reversal invariance is spectacularly broken, which allows the second term
in Eq. 1 to be observed.) The proposed dissipationless current met with
considerable skepticism (discovery of the
Since the Hall resistivity is rxy = sH r2 (in general), the Karplus-Luttinger prediction that the Hall conductivity sH is independent of electron lifetime implies that rxy should vary as the square of r as the latter is increased by say implanting impurities. In contrast, skew scattering predicts that it should grow linearly with r. Experiments were unable to resolve this controversy because, at 4 K where the predictions are valid, the AHE current is usually barely detectable. Moreover, changing the carrier density to vary r over a wide range usually kills the goose (the magnetic state) altogether.
In the 70's when experimentalists actively investigated the 'Kondo problem', numerous Hall measurements were performed on nonmagnetic metals (e.g. Cu) with a dilute concentration of magnetic impurities, e.g. Mn. The weak AHE observed was consistent with rxy being linear in r (skew scattering). These systems are paramagnetic rather than ferromagnetic (hence not really relevant). Nonetheless, the results fostered a collective, if unjustified, tilt towards the skew scattering mechanism and away from Luttinger's theory.
Figure 1 The crystal structure of the spinel CuCr2Se4. Se ions are in yellow, Cu in white and Cr in red. To a good approximation, the Cr ions have a large local moment. The nearest path between nearest-neighbor Cr ions goes through a Se ion to form a 90-degree bond. By a process called superexchange, the 90-degree bond gives rise to a ferromagnetic exchange between Cr ions.
Figure 2 The observed Hall resistivity rxy in two samples of CuCr2Se4-xBrx,
with the Br content x = 0.25 (Panel A) and x = 1.0 (Panel B). In Panel A,
the AHE signal is negative at low temperatures and saturates to the value ~ 40
nWm at 5 K. The sample with x =
1.0 has about 25 times smaller carrier concentration (Panel B). Its AHE
signal is positive and rises to 7 mWm
at 5 K (possibly the largest AHE signals ever recorded in any ferromagnet)
[from Wei Li Lee et al. (Ref. 1)].
Figure 3 Plot of the Hall signal rxy/n (normalized to per hole)
versus the resistivity r (both measured
at 5 K). The Karplus-Luttinger theory predicts a slope of 2 while the
skew scattering theory has a slope of 1. The observed slope over 3
decades of the independent variable r
is a = 1.95. [Wei-Li Lee et al.
AHE in Spinels The spinel represents a broad class of
transition-metal (or rare earth) oxides and chalcogenides with the chemical
formula AB2X4 where A and B
are metallic 3d, 4d or 4f elements and X is O, S or Se (Fig. 1).
Among the many members which are magnetic, the most familiar is the black iron
oxide magnetite (Fe3O4). The spinel CuCr2Se4
is a conducting ferromagnet with a fairly high Curie temperature (450 K).
The ferromagnetic state is predominantly stabilized by the role of
superexchange between local moments on the Cr ions. In this member, the
charge carriers (holes) are quite irrelevant to formation of the ferromagnetic
state. Recently, a
discovery of the Berry phase
(geometric phase) in 1983 spurred intense theoretical effort to work out its
implications in many areas of physics. The Hall conductivity with its
peculiar sensitivity to symmetry effects has attracted particular
attention. In the late 90's Qian Niu (U. Texas, Austin) and colleagues
 and Naoto Nagaosa (U. Tokyo) and colleagues  re-opened the theoretical
investigation of the Karplus-Luttinger theory. The new Berry-phase
perspective provides a better understanding of the anomalous velocity term.
In a periodic lattice, the position operator may be written as the sum x
= R + X, where R = id/dk (the Wannier co-ordinate)
locates the electron in one of the unit cells. The
[ xi, xj ] = iWk eijk, (2)
where eijk is the Levi-Civita symbol and W = curl X(k) is known as the
The velocity is calculated as the commutator of x and the Hamiltonian
H, which has the form
H = e(k) - eE.x (3)
in an applied weak E-field. Applying the result Eq. (2), we obtain directly the velocity given in Eq. (1).
For more details, see the review Ref. 5.
1. Wei-Li Lee, Satoshi Watauchi, R. J. Cava and N. P. Ong, “Dissipationless anomalous Hall current in the ferromagnetic spinel CuCr2Se4-xBrx”, Science 303, 1647 (2004).
2. R. Karplus, J. M. Luttinger, Phys. Rev. 95, 1154 (1954).
3. G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999).
4. M. Onoda and
5. For a review, see N. P. Ong and Wei-Li Lee, “Geometry and the Anomalous Hall Effect in Ferromagnets”, Foundations of Quantum Mechanics in the Light of New Technology (Proceedings of ISQM-Tokyo’05), ed. Sachio Ishioka and Kazuo Fujikawa (World Scientific 2006)p. 121, cond-mat/0508236.