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The anomalous Hall effect in ferromagnets and the Berry phase
N. P. Ong, Wei-Li Lee, and R. J. Cava
Princeton University


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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 [1].

Roughly 50 years ago (1954), Karplus and Luttinger [2] 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 "Berry curvature" in modern terms [3,4] (see below).  The extra term eE x W(k) is called the Luttinger anomalous velocity term.

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 Berry phase lay 30 years in the future).   A rival, more conventional, mechanism called skew scattering, proposed by Jan Smit in 1955, seemed more plausible to many.  In skew scattering the AHE current arises because the impurity cross-section seen by a beam of electrons possesses right-left asymmetry in a ferromagnet.  However, calculations by Kondo, Maranzana, Smit and others showed that the skew-scattered AHE current is 2-3 orders of magnitude too feeble to square with observation.  Luttinger and collaborators published further clarification of the KL theory, and in the process laid the foundations of quantum-transport theory in solids.  However, the original AHE controversy was not resolved theoretically.  Most calculations showed that both contributions could exist in principle, but their magnitudes are difficult to pin down.

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. (Ref. 1)].
 

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 Princeton collaboration (Wei-Li Lee et al. [Ref. 1]) showed that, by introducing Br as a dopant, the hole density n can be reduced by a factor of 30 without degrading the magnetization.  The steep change in the hole concentration and the increased disorder leads to a 1000-fold increase in the resistivity r.  Further, the AHE signal remains very large at 5 Kelvin (Fig. 2).  The 3-decade change in r allows a clean test of the AHE theories.  (A slight modification is needed to compensate for the change in n).  By plotting r_xy/n against r, Wei-Li Lee et al. found that the former increases by 6 decades, as the square of r (Fig. 3).  This provides (finally) hard, experimental evidence in favor of the Karplus-Luttinger theory.  The experimental result confirms that the anomalous term in Eq. 1 has physical reality, with observable consequences.

  The 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 [3] and Naoto Nagaosa (U. Tokyo) and colleagues [4] 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 Adams operator X(k), which locates where the electron is within a cell, is the integral of the matrix element (uk|id/dk |uk) over the cell.  This is the Berry vector gauge [3,4].  Its existence implies that the position operator fails to commute with itself.  Instead, we have [5]
 

[ xi, xj ] = iWk eijk,                    (2)


where eijk is the Levi-Civita symbol and W = curl X(k) is known as the Berry curvature.  One may regard W(k) as an effective 'magnetic field' that lives in reciprocal space to produce a k-space Lorentz force, as suggested by Eq. (1).  It arises because an itinerant electron (or hole), moving in the conduction band, spends part of its life exploring the valence band (or nearest higher bands), an effect described by the intracell operator X(k).  If the lattice has both time-reversal invariance and inversion symmetry, W(k) vanishes identically. 

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. 

References
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 N. Nagaosa, J. Phys. Soc. Jpn. 71, 19 (2002).
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.