The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space

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Science  03 Oct 2003:
Vol. 302, Issue 5642, pp. 92-95
DOI: 10.1126/science.1089408


Efforts to find the magnetic monopole in real space have been made in cosmic rays and in particle accelerators, but there has not yet been any firm evidence for its existence because of its very heavy mass, ∼1016 giga–electron volts. We show that the magnetic monopole can appear in the crystal momentum space of solids in the accessible low-energy region (∼0.1 to 1 electron volts) in the context of the anomalous Hall effect. We report experimental results together with first-principles calculations on the ferromagnetic crystal SrRuO3 that provide evidence for the magnetic monopole in the crystal momentum space.

Dirac (1) postulated in 1931 the existence of a magnetic monopole (MM), searching for the symmetry between the electric and magnetic fields in the law of electromagnetism. A singularity in the vector potential is needed for this Dirac MM to exist. Theoretically, the MM was found (2, 3) as the soliton solution to the equation of the non-Abelian gauge theory for grand unification. However, its energy is estimated to be extremely large, ∼1016 GeV, which makes its experimental observation difficult. In contrast to this MM in real space, one can consider its dual space, namely, the crystal momentum (k-) space of solids, and the Berry phase connection (4) of Bloch wave functions. This MM in momentum space (5) is closely related to the physical phenomenon of the anomalous Hall effect (AHE) observed in ferromagnetic metals.

The AHE is a phenomenon in which the transverse resistivity (ρxy) in ferromagnets contains a contribution from the magnetization (M) in addition to the usual Hall effect. The conventional expression for ρxy is Math(1) where B is the magnetic field, R0 is the usual Hall coefficient, and Rs is the anomalous Hall coefficient. This expression implicitly assumes that the additional contribution is proportional to M, and it is used as an experimental tool to measure M as a function of temperature. This analysis is extensively used in studies of ferromagnetic semiconductors with dilute magnetic impurities, which are the most promising materials for applications in spintronics (6). However, the mechanism of AHE has long been controversial (711). The key issues are whether the effect is intrinsic or extrinsic and how to treat the impurity and phonon scatterings. Karplus and Luttinger (7) were the first to propose the intrinsic mechanism of AHE, in which the matrix elements of the current operators are essential. Other theories (8, 11) attribute the AHE to the impurity scattering modified by the spin-orbit interaction, namely, the skew scattering (8) and/or the side-jump mechanism (11). These extrinsic mechanisms are rather complicated and depend on the details of the impurities as well as on the band structure of the materials. Nevertheless, all these conventional theories (711) for the AHE derive Eq. 1, as they are based on the perturbative expansion of the spin-orbit coupling (SOC) λ and M; i.e., Rs ∝ λ.

Recently, the geometrical meaning of the intrinsic-origin AHE (6) has been recognized (1215). The transverse conductivity (σxy) can be written as the integral of the Berry phase curvature (the gauge field) over the occupied electronic states in crystal momentum space (Eq. 5). The MM corresponds to the source or sink of the gauge field or curvature defined by this Berry phase connection. Therefore, the AHE can be a direct fingerprint of the MM in crystal momentum space. The presence of time-reversal symmetry results in σxy = ρxy = 0 in the dc limit from the generic argument, and the group theoretical condition for the nonzero σxy is equivalent to that of finite ferromagnetic moment (supporting online text). Therefore, ferromagnets are necessary to study σxy, even though the Berry phase connection is more universal and exists even in nonmagnetic materials.

We show by detailed first-principles band calculations combined with transport, optical, and magnetic measurements that the observed unconventional behavior of the AHE and Kerr rotation in the metallic ferromagnet SrRuO3 is of intrinsic origin and is determined by the existence of a MM in k-space. The conventional expression (Eq. 1) is not supported by our experimental data, which show a nonmonotonous temperature dependence that even includes a sign change.

SrRuO3 with perovskite structure is an itinerant (metallic) ferromagnet. Ru4+ has four t2g electrons with low spin configurations. The 4d orbitals of SrRuO3 are relatively extended and the bandwidth is large compared with its Coulombic interaction. The relativistic SOC is also large in its 4d electrons, because of the heavy atomic mass (on the order of 0.3 eV for the Ru atom). High-quality single crystal is now available with a residual resistivity on the order of 10 μΩcm. These aspects make this system an ideal candidate in which to observe the AHE due to the k-space gauge field. Stoichiometric SrRuO3 (bulk and thin film) and Ca-doped (Sr0.8Ca0.2RuO3 thin film) single crystals were prepared (16). The M curve (Fig. 1) of SrRuO3 film is quite similar to that of bulk single crystal, except that the Curie temperature (Tc, ∼150 K) is slightly lower than that of bulk (∼160 K), because of the strain effects. However, the isovalent Ca-doping markedly suppresses Tc and M. All of the samples were used for transport measurements in the dc limit. As seen from Fig. 1C, the ρxy changes nonmonotonously with temperature and even includes a sign change. Such behavior is far beyond the expectation based on the conventional expression in Eq. 1. In addition to the transport measurement, the frequency (ω)–dependent conductivities (Fig. 2) were measured for SrRuO3 film by an optical method (16). In addition to the strong structures around 3.0 eV, which are mostly due to the charge transfer from O-2p to Ru-4d, sharp structures were also observed for both the real and the imaginary part of σxy(ω) below 0.5 eV. These low-energy sharp structures cannot be fitted by extended Drude analysis. The lower the energy is, the stronger the deviations from fitting are. We will show, by combination with first-principles calculations, that these unconventional behaviors actually originate from the singular behavior of MMs in momentum space.

Fig. 1.

Measured temperature dependence of the (A) magnetization M, (B) longitudinal resistivity ρxx, and (C) transverse resistivity ρxy for the single-crystal and thin-film SrRuO3, as well as for the Ca-doped Sr0.8Ca0.2RuO3 thin film. μB, Bohr magneton. (D) The corresponding transverse conductivity σxy is shown as a function of M, together with the results of first-principles calculations for cubic and orthorhombic structures (25). In our calculations, the change of magnetization is taken into account by the rigid splitting of up and down spin bands. As σxy should vanish with M at high temperatures, the calculated σxy is multiplied by the additional M/M0 (where M0 = 1.5μB) factor, which does not affect its behavior except in the vicinity of Tc.

Fig. 2.

Calculated (Calc., left panels) and experimental (Exp., right panels) longitudinal (σxx) and transverse (σxy) optical conductivity of SrRuO3 film. Measurements were performed at low temperature (10 K). Calculations for both the orthorhombic single-crystal structure and the hypothetical cubic structure kept the average Ru-O bond length. The experimental σxy values shown are multiplied by a factor of 4. A quantitative comparison of the absolute values of σxy between the experiments and the calculations would require more accurate structure information. Nevertheless, the clear peak structures for the low-energy σxy demonstrate monopoles associated with bz(k).

We now turn to some details of the theoretical analysis. The Berry phase is the quantal phase acquired by the wave function that is associated with the adiabatic change of the Hamiltonian (4, 17). Let |n(α)〉 be the nth eigenstate of the Hamiltonian H(α), with α = (α1,....,αm) being the set of parameters. The Berry's connection is the overlap of the two wave functions infinitesimally separated in α-space, i.e. Math Math(2) where the vector potential an(α) is defined by an(α) = in(α)|∇α|n(α)〉. Although the concept of the Berry phase has broad applications in physics (17), its relevance to the band structure in solids has been recognized only recently and in limited situations, such as the quantum Hall effect under a strong magnetic field (18) and the calculation of electronic polarization in ferroelectrics (19, 20). In this case, the parameter α is the crystal momentum k. For the Bloch wave function ψnk(r) = eikrunk(r), where n denotes the band index and unk is the periodic part, the vector potential for the Berry phase a (k) is Math(3) where μ is the chemical potential. With this vector potential, the gauge covariant position operator xμ for the wave packet made out of the band n is given by xμ = ikμa(k). Therefore, the commutation relation between xμ and xν includes the gauge field Fμν = ∂kμ ak a, as Math(4) which leads to the additional (anomalous) velocity –i[xμ, V(x)] = –FμνV(x)/∂xν, being transverse to the electric field Eν = –∂V(x)/∂xν. Therefore, the transverse conductivity σxy is given by the sum of this anomalous velocity over the occupied states as (18) Math(5) where bz(k) = Fxy(k), ϵ is ϵn(k) is the eigen energy of ψnk(r), β is the inverse temperature, and nF[ϵ] = 1/(eβ(ϵ–μ) + 1) is the Fermi distribution function. Hence, the behavior of gauge field bz(k) in k-space (21) determines that of σxy. One might imagine that it is a slowly varying function of k, but that is not the case. Fig. 3B is the calculated result for bz(k) in the real system SrRuO3. It has a very sharp peak near the Γ-point and ridges along the diagonals. The origin for this sharp structure is the (near) degeneracy and/or the band crossing, which act as MMs. Consider the general case where the two-band Hamiltonian matrix H(k) at k can be written as H(k) = Σμ = 0,1,2,3 fμ(kμ where σ1,2,3 are the Pauli matrices and σ0 is the unit matrix. When k is mapped to the vector f(k) = [f1(k), f2(k), f3(k)], then the contribution to σxy from the neighborhood of this degeneracy region is given by the solid angle dΩf for the infinitesimal dkxdky integrated over k. Therefore, the gauge flux near the MM, namely, the degeneracy point f = 0, is singularly enhanced (Fig. 3) (supporting online text).

Fig. 3.

(A) Geometrical meaning of the contribution to σxy when the two bands are nearly degenerate. The two-dimensional Hamiltonian matrix H(k) can be generally written as H(k) = Σμ = 0,1,2,3 fμ (kμ where σ1,2,3 are the Pauli matrices and σ0 is the unit matrix. When Embedded Image is mapped to the vector Embedded Image, the contribution to σxy from the neighborhood of this degeneracy region can be given by the f -space solid angle dΩf = |δ(f, φf)/ δ(kx, ky) sin f, dkx, dky = dφfsin d f for the infinitesimal dkxdky integrated over k. The solid angle corresponds to the flux from the monopole at Embedded Image (supporting online text). (B) Calculated flux distribution in Embedded Image space for t2g bands as a function of (kx, ky) with kz being fixed at 0 for SrRuO3 with cubic structure. The sharp peak around kx = ky = 0 and the ridges along kx = ±ky are due to the near degeneracy of dyz and dzx bands because of symmetry (supporting online text).

We studied the behavior of σxy by first-principles calculations (16). The calculated density of states is not so different between the cases with and without SOC (Fig. 4A), whereas the σxy should be very sensitive to the Bloch wave functions and depends on the Fermi-level position and the spin-splitting (magnetization) substantially, as predicted by the discussion above. We determined the behavior of σxy as a function of the Fermi-level position by using a small broadening parameter for the lifetime δ (70 meV) (Fig. 4B). When the Fermi level was shifted, not only the absolute value but also the sign of σxy was found to change. The sharp and spiky structures are the natural results of the singular behavior of the MM (Fig. 3). For the case without any shift of Fermi level, we obtained a value of σxy = –60 Ω–1 cm–1, which has the same sign as and is comparable with the experimental value (about –100 Ω–1 cm–1). Such a spiky behavior should also be reflected in the ω-dependent σxy, especially for the low-energy range with longer lifetime, whereas it should be suppressed at higher activation energies with shorter lifetime. As shown in Fig. 2 for the ω dependence of optical conductivity, the high-energy (>0.5 eV) part, which is dominated by the pd charge transfer peak, is usual and can be well reproduced by our calculations, whereas the observed peak structure of σxy(ω) below 0.5 eV is a clear demonstration of the predicted spiky behavior. The spectra below 0.2 eV were not measured because of the technical difficulty, but structure there should be even sharper, because the dc limit Rexy) ≈ –100 Ω–1 cm–1 has the opposite sign [the Imxy) at the dc limit should go back to zero]. Such low-energy behavior is well represented by our calculations, providing further evidence for the existence of MMs.

Fig. 4.

The calculated (A) density of states (DOS) and (B) σxy as functions of Fermi-level position for the orthorhombic structure of single-crystal SrRuO3. The Fermi level is shifted rigidly relative to the converged solution, which is specified as the zero point here. The sharp and spiky structure of σxy demonstrates the singular behavior of MMs. f.u., the formula unit SrRuO3.

It is straightforward to understand the results of our transport measurement for σxy. We attribute the temperature (T) dependence of σxy to that of the magnetization M(MT). As the result of k-space integration over occupied states, the calculated σxy is nonmonotonous as a function of M (Fig. 1D). With the reduction of spin-splitting, the calculated σxy, after the initial increase, decreases sharply, then increases and changes sign (becoming positive), and finally decreases again, capturing the basic features of the experimental results. Even more surprisingly, when the measured ρxy versus T curves shown in Fig. 1C are converted into the σxy versus M curves shown in Fig. 1D, they now all follow the same trend and match with our calculations. The curves are measured for different samples (with different saturation moments), but all follow the same rule qualitatively and could be simply explained by the reduction of M (22) (Fig. 1A). However, the comparison between the experiments and the calculations should be semi-quantitative, because the results are sensitive to the lattice structures. The calculated σxy for the fictitious cubic structure shows a strong deviation from that obtained for orthorhombic structure, and it changes the sign to be positive at low temperature (at large M). Therefore, more accurate information on the structure is needed to obtain the quantitative result. However, such a sensitivity does not affect our main results, i.e., the nonmonotonous behavior of σxy. Even the calculations for cubic structure show such behavior and may be used as a guide of possible deviation.

The results and analysis presented here should stimulate and urge the reconsideration of the electronic states in magnetic materials from a very fundamental viewpoint. For example, the MM is accompanied by the singularity of the vector potential, i.e., the Dirac string (1). As shown by Wu and Yang (23) this means that more than two overlapping regions have to be introduced, in each of which the gauge of the wave function is defined smoothly. This means that one cannot define the phase of the Bloch wave functions in a single-gauge choice when the MM is present in the crystal momentum space. This leads to some nontrivial consequences, such as the vortex in the superconducting order parameter as a function of k (24), and many others are left for future studies.

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