Anomalous Hall Effect due to Non-collinearity in Pyrochlore Compounds: Role of Orbital Aharonov-Bohm Effect

Anomalous Hall Effect due to Non-collinearity
in Pyrochlore Compounds: Role of Orbital Aharonov-Bohm Effect

Takeshi Tomizawa    Hirhoshi Kontani Department of Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan.
July 13, 2019

To elucidate the origin of spin structure-driven anomalous Hall effect (AHE) in pyrochlore compounds, we construct the -orbital kagome lattice model and analyze the anomalous Hall conductivity (AHC). We reveal that a conduction electron acquires a Berry phase due to the complex -orbital wavefunction in the presence of spin-orbit interaction. This “orbital Aharonov-Bohm (AB) effect” produces the AHC that is drastically changed in the presence of non-collinear spin structure. In both ferromagnetic compound and paramagnetic compound , the AHC given by the orbital AB effect totally dominates the spin chirality mechanism, and succeeds in explaining the experimental relation between the spin structure and the AHC. Especially, “finite AHC in the absence of magnetization” observed in can be explained in terms of the orbital mechanism by assuming small magnetic order of Ir -electrons.

72.10.-d, 72.80.Ga, 72.25.Ba

I Introduction

Figure 1: Tilted ferromagnetic state in the kagome lattice. Blue circles are Mo ions. Arrows at Mo sites are the tilted ferromagnetic exchange field.

Recently, theory of intrinsic anomalous Hall effect (AHE) in multiband ferromagnetic metals has been developed intensively from the original work by Karplus and Luttinger (KL) KarplusLuttinger (). The anomalous Hall conductivity (AHC) due to intrinsic AHE shows the almost material-specific value that is independent of the relaxation time. The intrinsic AHE in heavy fermion compounds KontaniYamada (), Fe Yao (), and Ru-oxides Miyazawa (); Fang (); KontaniTanakaYamada () had been studied intensively based on realistic multiband models. Also, large spin Hall effect (SHE) observed in Pt and other paramagnetic transition metals Kimura (), which is analog to the AHE in ferromagnets, is also reproduced well in terms of the intrinsic Hall effect KontaniTanakaHirashima (); Guo (); TanakaKontani (). The intrinsic AHE and SHE in transition metals originate from the Berry phase given by the -orbital angular momentum induced by the spin-orbit interaction (SOI), which we call the “orbital Aharonov-Bohm (AB) effect” KontaniTanaka2 ().

In particular, AHE due to nontrivial spin structure attracts increasing attention, such as Mn oxides Ye () and spin glass systems Tatara (). The most famous example would be the pyrochlore compound Yoshii (); Kageyama (); Yasui (); Taguchi (). Here, Mo 4 electrons are in the ferromagnetic state below K, and the tilted ferromagnetic state in Fig. 1 is realized by the non-coplanar Nd 4 magnetic order below K, due to the - exchange interaction. Below , the AHC is drastically changed by the small change in the tilting angle of Mo spin; in the neutron-diffraction study Yasui (). This behavior strongly deviates from the KL-type conventional behavior . Moreover, the AHC given by the spin chirality mechanism Ohgushi (); Taillefumier (), which is proportional to the solid angle subtended by three spins, is also too small to explain experiments. Moreover, takes the minimum value under Tesla according to the neutron-diffraction study Yasui (), whereas the AHC monotonically decreases with . Thus, the origin of the unconventional AHE in had been an open problem for a long time.

Very recently, this problem was revisited by the present authors by considering the -orbital degree of freedom and the atomic SOI TomizawaKontani (), and found that a drastic spin structure-driven AHE emerges due to the orbital AB effect, in the presence of non-collinear spin order. Since the obtained AHC is linear in , it is much larger than the spin chirality term for . In Ref. TomizawaKontani (), we constructed the orbital kagome lattice model based on the spinel structure (): Although Mo atoms in and are equivalent in position and forms the pyrochlore lattice, positions of O atoms in are much complicated.

In this paper, we construct the kagome lattice tight-biding model based on the pyrochlore structure, by taking the crystalline electric field into account. We find that the orbital AB effect causes large -linear AHC, resulting from the combination of the non-collinear spin order (including orders with zero scalar chirality) and atomic SOI. The realized AHC is much larger than the spin chirality term due to non-coplanar spin order, and it explains the salient features of spin structure-driven AHE in . We also study another pyrochlore compound , and find that the orbital AB effect also gives the dominant contribution: We show that important features of the unconventional AHE in , such as highly non-monotonic field dependence and residual AHC in the absence of magnetization, are well reproduced by the orbital AB effect.

The paper is organized as follows: In Sec. II, we introduce the pyrochlore-type orbital tight-binding model and the Hamiltonian. We give the general expressions for the intrinsic AHC in Sec. III, and explain the orbital Aharonov-Bohm effect in Sec. IV. The numerical results for and are presented in Sec. V and VI, respectively. In Sec. VII, we make comparison between theory and experiment.

Ii Model and Hamiltonian

Figure 2: Pyrochlore structure. Blue (white) circles are Mo (O) ions. The Mo ions on the [111] plane form the kagome lattice.
Figure 3: Configurations of Mo and O. The -coordinate is defined by the surrounding O tetrahedron.
Ion Site Coordinate
Mo A (1/4 ,0, 0)
B (0, 1/4, 0)
C (0, 0, 1/4)
D (1/4, 1/4, 1/4)
O 1 (1/8, 1/8, -1/16)
2 (1/8, -1/16, 1/8)
3 (-1/16, 1/8, 1/8)
4 (5/16, 1/8, 1/8)
5 (1/8, 5/16, 1/8)
6 (1/8, 1/8, 5/16)
Table 1: Coordinates of Mo and O in pyrochlore structure as shown in Fig. 3 in the -coordinate.

First, we introduce the crystal structure of the pyrochlore oxide : It has the face centered cubic structure, in which two individual 3-dimensional networks of the corner-sharing A and B tetrahedron are formed. In this paper, we mainly discuss the AHE in , and is also discussed in section VI. Figure 2 represents the Mo ions (Blue circles) and O ions (White circles) in the pyrochlore structure. The [111] Mo layer forms the kagome lattice. The Mo -electrons give itinerant carriers while the Nd -electrons form local moments.

We construct pyrochlore type -orbital tight binding model in the kagome lattice for Mo electrons, where the unit cell contains three sites A, B and C in Fig. 3. The coordinates of Mo and O are shown in Table 1 Subramanian (), and the quantization axis for the Mo -orbital is fixed by the surrounding O octahedron. To describe the -orbital state, we introduce the -coordinate for sites shown by Fig.3. The -coordinate is defined by the surrounding O ions. In the case of the -coordinate, we choose , and axes as MoO, MoO and MoO direction, respectively, in Fig. 3. We also choose the - and -coordinates in the same way.

Figure 4: (a) Umbrella like locale exchange field at Mo sites represented by arrows. A unit cell contains sites A, B, C. (b) Kagome lattice. is a half Bravais vector. (c) First Brillouin zone in a hexagon shape.

Moreover, we introduce the -coordinate on the kagome layer shown in Fig. 4(b). We choose axis as MoMo direction and axis is perpendicular to axis on the kagome layer. A vector in the -coordinate is transformed into in the -coordinate as , where the coordinate transformation matrix is given by


Arrows in Fig. 4(a) represents the local effective magnetic field at Mo sites, which is composed of the ferromagnetic exchange field for Mo 4-electrons and the exchange field from Nd 4 electrons. Under the magnetic field parallel to direction below , the direction of the local exchange fields at sites A, B and C in the -coordinate are , and , respectively. In NdMoO, the tilting angle changes from negative to positive as increases from Tesla, corresponding to the change in the spin-ice state at Nd sites Yasui (); Sato ().

Now, we explain the Hamiltonian. The Hamiltonian for the -orbital kagome lattice model is given by


where is a creation operator for -electron on Mo ions while the field arises from the ordered Nd moments, which are treated as a static, classical background. , and represent the sites, -orbitals and spins, respectively. Hereafter, we denote the -orbitals as for simplicity. The first term in eq. (2) describes electrons hopping. is the hopping integrals between and . The direct - hopping integrals are given by the Slater-Koster (SK) parameters , and SlaterKoster (). In the present model, however, SK parameter table given in Ref. SlaterKoster () is not available since the -orbitals at each site are described in the different coordinate as shown in Fig. 3. In Appendix A, we will derive the hopping integral between the sites with different coordinates. The second term in eq. (2) represents the Zeeman term, where is the local exchange field at site . is the magnetic moment of an electron. Here, we put =1. The third term represents the SOI, where is the spin-orbit coupling constant, and and are the -orbital and spin operators, respectively.

The Hamiltonian in Eq. (2) is rewritten in the momentum space as


where summation is over the first Brillouin zone in Fig. 4(c), and


Here and hereafter, we denote the creation operators at sites A, B and C as and , respectively. is given by 1818 matrix:


where is a 66 matrix with respect to .

Here, we divide the Hamiltonian (3) into four parts:


where we added the crystalline electric field potential term to Eq. (2). The kinetic term is given by


where and is a half Bravais vector in Fig. 4(b).

The Zeeman term is given by Taillefumier ()


where =, and is a Kronecker’s delta.

Now, we consider the SOI term in Eq. (6). For convenience in calculating the AHC, we take the -axis for the spin quantization axis. Then, is given by the Pauli matrix vector in the -coordinate. To derive the , however, we have to express the spin operator in the -coordinate, which is given by the relationship and Eqs. (1a)-(1c). In the -coordinate, the nonzero matrix elements of are given as and their Hermite conjugates Friedel (); TanakaKontani (). Thus, the matrix elements for are given as

Thus, the - component of the third term in Eq. (6) becomes


The - and - components are calculated in a similar way. The obtained results are given by


Finally, we consider the crystalline electric field Hamiltonian , which describes the splitting of level into two levels (non-degeneracy) and (two-fold degeneracy) by the trigonal deformation of MoO octahedron. The crystalline electric field Hamiltonian in this case is given by


The eigenvalues of at each site are for state; , and for states; and . Thus, the crystalline electric field splitting between and is .

Iii Anomalous Hall conductivity

In this section, we propose the general expressions for the intrinsic AHC based on the linear-response theory. The Green function is given by a matrix: , where is the chemical potential. According to the linear response theory, the AHC is given by Streda ():


Here, is the retarded (advanced) Green function, where is the quasiparticle damping rate. is the charge current, where is the electron charge. Since all the matrix is odd with respect to , the current vertex correction due to local impurities vanishes identically TanakaKontani (); KontaniTanakaHirashima (). Thus, we can safely neglect the current vertex correction in calculating AHC in the present model. In the band-diagonal representation, eqs. (15) and (16) are transformed into


at zero temperature. Here, and are the band indices, and we dropped the diagonal terms since their contribution vanishes identically. We perform the numerical calculation for the AHC using Eqs. (17) and (18) in later section.

and are called the Fermi surface term and the Fermi sea term, respectively. According to Refs. TanakaKontani (); Streda (), can be uniquely divided into and the Berry curvature term . The intrinsic AHC is given by when since . In general cases, however, the total AHC is not simply given by since is finite when or . Therefore, we calculate the total AHC in this paper.

Iv Orbital Aharonov-Bohm effect

Before proceeding to the numerical calculation for the AHE, we present an intuitive explanation for the unconventional AHE induced by the non-collinear local exchange field . For this purpose, we assume the strong coupling limit where the Zeeman energy is much larger than the kinetic energy and the SOI TomizawaKontani (). The energy levels are split into the two triply-degenerate states by the Zeeman effect, as shown in Fig. 6. Its eigenstate for are given by


where . In addition, we assume the SOI is much larger than the kinetic energy. Since , the SOI term at site is replaced with , where . Its eigenenergies in the space are and , as shown in Fig. 6. The corresponding eigenstates are given by TomizawaKontani ()


where in the -coordinate is given by . In the complex wavefunction , the phase of each -orbital within the -linear term is given in Table II.

=A,B,C 0
Table 2: Phases for orbitals

Here, we explain that the -dependence of the -orbital wavefunction gives rise to a prominent spin structure-driven AHE TomizawaKontani (). Figure 6 shows the motion of an electron: (a) moving from to , (b) transferring form to at the same site, and (c) moving from to . Here, we assume that the electron is in the eigenstate at each site. The total orbital phase factor for the triangle path along is given by the phase of the following amplitude:


where is the kinetic term in the Hamiltonian. For simplicity, we take only the following largest hopping , and assume that is real. Considering that for =A, B, and C, the hopping amplitude is expressed as


where is the flux quantum, and is “the effective AB phase” induced by the complex -orbital wavefunction. The large -linear term in gives rise to the large spin structure-driven AHE in .

Note that is not actually a real number if , since the rotation of the spin axis induces the phase factor; see Eq. (19). This fact gives rise to the effective flux due to the spin rotation ; is given by the solid angle subtended by , and Ohgushi (). Thus, the total flux is given by . However, is negligible for TomizawaKontani (). Since all the upward and downward ABC triangles in the kagome lattice are penetrated by , the orbital AB effect induces prominent spin structure-driven AHE in .

Figure 5: Eigenenergies for electron under the exchange field and the SOI ; see Ref. TomizawaKontani ().
Figure 6: Orbital AB phase given by the complex orbital wavefunction at site C. The electron acquires the phase difference between - and -orbitals (orbital AB phase) via the movement (a)(b)(c).
Figure 5: Eigenenergies for electron under the exchange field and the SOI ; see Ref. TomizawaKontani ().

V Numerical Study

In this section, we perform numerical calculation for the AHC using Eqs. (17) and (18), using realistic model parameters. We use two SK parameters between the nearest neighbor Mo sites as and where we represent the set of SK parameters as . Hereafter, we put the unit of energy , which corresponds to 2000K in real compound. The spin-orbit coupling constant for Mo is TanakaKontani (). The number of electrons per unit cell is (1/3-filling) for since the valence of Mo ion is . We choose to reproduce the magnetization of Mo ion in Yasui ().

Figure 7: Total and partial DOS for (a) and (b)

Figure 7 shows the total and partial density of states (DOS) for at (a) and (b) , with the damping rate . For () we set () to reproduce the magnetization of Mo ion Yasui (). The crystalline electric field splitting for in Fig. 7 (b) corresponds to , consistently with the band calculation Solovyev (). In both cases, the states and gives large partial DOS near the Fermi level.

Figure 8: 3rd-12th bands from lowest for and .
Figure 9: Fermi surface for , (a) , and (b) .

Figure 8 shows the 3rd-12th bands from the lowest. Nine bands near the Fermi level() are composed of and as understood from Fig. 7(b). As shown in Fig. 9, the band structure and the Fermi surface are hardly changed by varying by 3 degrees.

Here, we present the numerical results of the AHC for two SK parameter sets; and . We set or ; each parameter set reproduces the magnetization of Mo ion Yasui (). We also put the damping rate (clean limit) unless otherwise noted. Hereafter, the unit of the conductivity is , where is the Plank constant and is the lattice constant. If we assume Å, . In the numerical study, we use -meshes.

Figure 10: -dependence of AHC for , for (a) and (b) .
Figure 11: -dependence of AHC for and .

Figure 10 shows the obtained AHC for at and , for (a) a wide range of and (b) a narrow range of . Since the present spin structure-driven AHE is linear in , a very small causes a prominent change in the AHC although the Fermi surfaces are hardly changed (see Fig. 9). The -dependence of the AHC for other SK parameter is shown by Fig. 11 for . A remarkable change of the AHC is also caused by small change in . Therefore, large -linear term in the AHC is obtained by using general SK parameters.

The finite AHC at is nothing but the conventional KL-term. However, obtained -linear AHC deviates from the conventional KL-term that is proportional to the magnetization . We stress that the large -linear term in Figs. 10 and 11 cannot be simply understood as the movement of Dirac points (or band crossing points) across the Fermi level, since the change in the band structure by is very tiny as recognized in Fig. 9. Thus, the origin of the -linear term should be ascribed to the orbital AB phase TomizawaKontani () discussed in Sec. IV.

Figure 12: -dependence of AHCs for and . The AHC starts to decrease for , which corresponds to in the present parameters.

Next, we discuss the -dependence of the AHC. As increases from , spike-like fine structure in Figs. 10 and 11 becomes moderate as recognized in Ref. TomizawaKontani (). Moreover, the intrinsic AHC shows a crossover behavior, that is, the AHC starts to decrease when exceeds the band-splitting , proved by using tight-binding models KontaniYamada (); KontaniTanakaYamada (); Streda2 () or local orbitals approach Streda2 (). Figure 12 shows the -dependence of the AHC in the present model. Line (i) represents the total AHC for ; , and line (ii) represents the variation of the AHC from to ; . We also calculate the AHC for , which represents the spin chirality driven AHC . Note that is an even function of , and =0. In Fig. 12, we plot as line (iii). The variation of the AHC from to due to the orbital mechanism is 100 times larger than the spin chirality term in the clean limit. Note that the intrinsic AHC follows an approximate scaling relation KontaniYamada (); KontaniTanakaYamada (); Streda2 () in the “high-resistivity regime”. In Fig. 12, we see that also follows the relation similarly.

Figure 13: -dependence of AHCs for , and , in cases of , , and .
Figure 14: -dependence of AHCs for at . The spin chirality term () is very small.

Next, we analyze the overall -dependence of the AHC, by ignoring the experimental condition . Figure 13 shows the AHCs as functions of . Solid and dashed lines represent the AHCs for and , respectively. They have large -linear terms for , and they take finite values even if (coplanar order). Note that obtained -dependence of the AHC is insensitive to the value of . The AHCs for corresponds to the conventional KL-type AHE. Dotted line in Fig. 13 shows the AHC for , which gives the spin chirality term . It is proportional to for small , and becomes zero when . Finally, we analyze the -dependence of the AHC more in detail for in Fig. 14. In the case of , the AHC for changes the sign at due to the orbital AB effect, and it is more that 100 times larger than the AHC for the spin chirality term ().


In the previous section, we discussed the unconventional AHE in the pyrochlore . Here, we discuss other pyrochlore . Unlike Mo electrons in , Ir 5 electrons are in the paramagnetic state. Below K, localized Pr 4 electrons form non-coplanar spin-ice magnetic order. Under the magnetic field along [111], the non-coplanar structure of Pr Ising moments are expected to change from “2in 2out” to “3in 1out”. The AHC increases in proportion to the magnetization with field from 0 to 0.7 Tesla, whereas it rapidly decreases as the spins of Pr tetrahedron change from “2in 2out” to “3in 1out” for Tesla.

On Ir sites in , the tilted ferromagnet state shown in Fig. 1 is also realized. In , however, the ferromagnetic exchange interaction is absent, and the local exchange field on Ir ion is composed of only the exchange field from the Pr moment; . Since is parallel to the sum of the nearest Pr momenta, of Ir spin is much larger than the of Mo spin in . Therefore, the tilted ferromagnetic state with large and small is realized in .

Now, we explain the local exchange field on Ir sites given by Pr tetrahedron. Details of the derivation of these local exchange field are presented in Appendix B. In the strong magnetic field along [111] , the spins of Pr tetrahedron have “3in 1out” structure, and the realized local exchange fields at Ir sites are and in Fig. 15. We denote this Ir spin structure as . In this section, we promise that and . In the intermediate field (), the spin of Pr tetrahedron can take three types of “2in 2out” structures with negative Zeeman energy. If we take one “2in 2out” structure among three, the exchange fields at Ir sites are , , and in Fig. 15. We denote this Ir spin structure as . In real compounds, domain structures of three “2in 2out” structures are expected to be formed, and the total magnetization is parallel to -axis. The total AHC will be insensitive to the domain structure since ’s due to three structures are equivalent. If we take average of three “2in 2out” structures, the local exchange fields belongs to the -structure with , as shown in Fig. 15. We denote this Ir spin structure as . As a result, the Ir spin structure changes as (or ) with increasing the field from 0.7 Tesla gradually.

Figure 15: The spin configurations in Pr tetrahedron are shown in the first line. Three “2in 2out” states are realized under Tesla, and one “3in 1out” state is realized under the higher field. These Pr spin configurations induce the local exchange fields at Ir sites as shown in the last line. In the -structure, . In the -structure, .
Figure 16: -dependence of AHC in for (a) and (b) where . Each line represents the AHC for the -structure in Fig. 15. (c) Ir spin structure below under the weak exchange field due to “2in 2out” Pr spin-ice order; . The total magnetization of the Ir tetrahedron is zero since CD, where site D is the apical Ir site. Note that spin structure is the reverse of .

Here, we perform the numerical calculation for . We put the atomic SOI as , which is slightly smaller than the atomic value for Ir TanakaKontani (). The number of electrons per unit cell is for . We set since is estimated to be larger than 14K experimentally Machida (). We also put the damping rate (clean limit). Figure 16 (a) shows the AHC in with . Each line represents the AHC for the -structure in Fig. 15. The line with “” represents the spin chirality term. In the case of , the AHC in the present model is 10 times larger than the AHC for . Thus, the orbital AB effect dominates the chirality mechanism. The variation of the AHC for (or ) can explain the experimental results, ignoring the sign of the AHC. For example, the sign of the AHC is changed if is negative.

In Fig. 16 (b), we put with and . Although the KL term at decreases from negative to positive with , the overall -dependence of the AHC is not very sensitive to .

Recently, Ref. MachidaNature () reports that the AHC in shows a hysteresis behavior under the magnetic field below K. That is, the AHC shows the “residual AHE with zero magnetization” in . In terms of the spin chirality mechanism, the authors claimed the existence of a long-period magnetic (or chirality) order of Pr sites with 12 original unit cells MachidaNature (). However, there is no theoretical justification for this complex state. Even if it is justified, the origin of the hysteresis behavior is unclear. In addition, the magnetic susceptibility does not show anomaly at experimentally.

Here, we propose an alternative explanation for the residual AHE based on the orbital AB effect: In IrO with =Nd, Sm, and Eu, the Ir 5-electrons show magnetic order at 36 K, 117 K, and 120 K, respectively Matsuhira (). Thus, monotonically decreases as the radius of ion increases. Since Pr is on the left-hand-side of Nd in the periodic table, one may expect a finite in PrIrO. We stress that small amount of impurities could induce the magnetic order in the vicinity of magnetic quantum-critical-point Kontani-ROP (). Here, we analyze the Ir spin structure below , considering the classical Heisenberg model for Ir tetrahedron under the exchange field by Pr spins (see in Appendix B):


where is the -th Ir spin, and the positive is the antiferromagnetic interaction between Ir spins. When , then is parallel to . When , we have to find the spin configuration to minimize eq. (23) under the constraint .

Under the exchange field by one of ‘2in 2out” Pr order, the obtained Ir spin structure for is shown in Fig. 16. For under Tesla, the Ir spin structure is changed to the -structure with in Fig. 15 (c), which we denote structure. The obtained AHC under this spin structure is , as denoted in Fig. 15 (a): The experimental residual AHC is smaller, since the Ir ordered moment is expected to be smaller experimentally. The AHC is reversed under Tesla since the Ir spin structure is reversed. As a result, we can naturally explain the “hysteresis behavior of the AHC” below