H. Kontani et al- Giant Intrinsic Spin Hall Effect in Transition Metal Compounds

8/3/2019 H. Kontani et al- Giant Intrinsic Spin Hall Effect in Transition Metal Compounds

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Giant Intrinsic Spin Hall Effect

in Transition Metal Compounds

H. KONTANI1, T. TANAKA1, D.S. HIRASHIMA1, K. YAMADA2 and J. INOUE3

1

Department of Physics, Nagoya University,Furo-cho, Chikusa-ku, Nagoya 464-86022Engineering, Ritsumeikan University,

1-1-1 Noji Higashi, Kusatsu, Shiga 525-85773Department of Applied Physics, Nagoya University,

Furo-cho, Nagoya 464-8602

Abstract

We study the intrinsic spin Hall conductiv-

ity (SHC) and the d-orbital Hall conductivity(OHC) in metallic d-electron systems, by fo-cusing on the t2g-orbital tight-binding modelfor Sr2MO4 (M=Ru,Rh,Mo). This is thefirst theoretical study of the SHE in transitionmetal compounds based on a realistic tight-binding model. The obtained SHC is muchlarger than that in semiconductors. The ori-gin of these huge Hall effects in paramagneticstate is the effective Aharonov-Bohm flux in-duced by signs of inter-orbital hopping inte-

grals as well as atomic spin-orbit interaction.Huge SHC and OHC due to this mechanismis expected to be ubiquitous in multiorbitaltransition metal complexes, which will openthe possibility of realizing spintronics as wellas orbitronics devices.

1 Introduction

In these years, spin Hall effect (SHE) attracts

much attention due to its potential applicationin spintronics. The spin Hall effect is the phe-nomenon that an electric field induces a spincurrent (not a charge current) in a tranversedirection. The SHE occurs even if the time re-versal symmetry is not violated since the spincurrent does not change by time reversal. Byusing the spin Hall effect, one can create a purespin current by applying an electric field to aparamagnetic metal. Inversely, spin current is

converted to the electric field in the perpendic-ular direction via inverse SHE. Therefore, onecan detect the spin current by using the inverse

spin Hall effect.

The SHE in paramagnetic metals has aclose relation to the anomalous Hall effect(AHE) in ferromagnets [1], which is the phe-nomenon that an electric field induces a trans-verse charge current. In 1954, Karplus andLuttinger [2] showed that in multiband ferro-magnetic systems with spin-orbit interaction, aconduction electron feels a force perpendicularto the electric field even in the absence of mag-

netic field: For an electron with sz = h/2,the spin-orbit interaction 2l s is reduced tobe hlz. This term affects the motion of theelectron like a magnetic field in a multior-bital system where the atomic angular momen-tum is not quenched; we will discuss in detailin later sections. This is the origin of AHEas shown in Fig. 1 (a). When the magnetiza-tion is reversed, the charge current due to theAHE, jAHEy , also changes its sign. In a para-magnetic state where N = N, jy = jy

as shown in Fig. 1 (b). Then, spin currentjsy = (h/2e)(jyjy) flows along y-directionwhere e (< 0) is the electron charge, whereascharge current jy = jy+jy cancels out identi-cally. Therefore, origin of the SHE and that ofthe AHE are the same. [In above explanation,we disregarded the fact that the the electronspin can be reversed by the x, y componentsof the spin-orbit interaction 2(lxsx + lysy) tosimplify the explanation.]


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Ey

Jxs (spin current)

(b) paramagnetic

Ey

Jxs (spin current)

(b) paramagnetic

Ey

Jx (charge current)

(a) ferromagnetic

Ey

Jx (charge current)

(a) ferromagnetic

Figure 1: (a) AHE in a ferromagnetic metal.(b) SHE in a paramagnetic metal. In (b), fi-nite AHE appears when the magnetic field isapplied along z-axis .

Recently, Murakami et al. [3] and Sinovaet al. [4] studied the SHE in semiconductorsby developing the theory of Karplus and Lut-tinger. Experimentally, existence of the SHEin semiconductors was reported by optical de-tection of spin accumulation at the sampleedges [5, 6]. The observed SHE is, however, toosmall to perform quantitative study. There-fore, materials with large SHE has been highlyrequired. Parallel to these studies for semi-conductors, recent experimental efforts have

revealed that SHE also exists in conventionalmetals such as Al [7] and in Pt [8]. Especially,SHE in Pt is 104 times larger than the SHC on-served in n-type semiconductors. Nowadays,there is increasing interest in the mechanismof SHE in d-electron systems, for possible tech-nological applications.

In this article, we present the first reporton the theoretical study of the SHE in transi-tion metal, based on a realistic tight-binding

model [9]. We study the intrinsic SHE as wellas d-OHC in Sr2MO4 (M=Ru,Rh,Mo) basedon t2g-orbital tight-binding model. Sr2RuO4 isa famous triplet superconductor at Tc = 1.5 K.The obtained SHC is a few times larger thanSHC in Pt. The present study strongly sug-gests that giant SHE and OHE are ubiquitous

in multiorbital d-electron systems. The ori-gin of huge SHE and OHE is the anomalousvelocity due to the atomic orbital degrees offreedom, which also causes large AHE in d, f-electron systems [10, 11, 12]. In fact, param-agnetic compound Ca1.7Sr0.3RuO4 shows largeAHE under the magnetic field. This exper-imental fact is consistent with the presentstudy.

2 Anomalous velocity due to

interorbital hopping

Here, we study the SHC in Sr2MO4(M=Ru,Rh,Mo), where the metalicity appearsin two-dimensional MO2 planes, and the Fermisurface is composed mainly oft2g (dxz, dyz , dxy)orbitals. The tight-binding model for Sr2MO4,which we call the t2g-model, is introduced inref. [13]. Figure 2 shows the hopping inte-grals: (a) and (b) show the intraorbital hop-pings. According to ref. [13], we put t = 1,t = 0.1, t3 = 0.8, t3 = 0.35, and assume thatt 0.2eV and 0.2t. (c) represents the in-terorbitalhopping (dxz-dyz) between next near-est neighbor sites, t.

Hereafter, we denote xz = 1, yz = 2, xy = 3.Using this presentation, the matrix element ofthe Hamiltonian without spin-orbit coupling isgiven by [13]

H0 = 1(k) g(k) 0g(k) 2(k) 0

0 0 3(k) , (1)

where the first, the second and the third row(column) correspond to xz, yz and xy, respec-tively. 1 = 2t cos kx, 2 = 2t cos ky, and3 = 2t3(cos kx + cos ky) 4t

3 cos kx cos ky +

03 are interorbital kinetic energies. The in-terorbital kinetic energy g = 4t sin kx sin ky,which breaks the mirror symmetry with re-spect to kx- and ky-axes, causes the large

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x

y

d(xz)

d(yz)

-t

-t

x

y

d(xz)

d(yz)

-t

-t

t -t

(a) intra-orbita l (xz, yz)

(c) inter-orbita l (xzyz)

(b) intra-orbital (xy)

t3t3

t3

t3

-t t

3.1416 1e05 3.1416

kx

3.1416

1e05

3.1416

ky

(,)(,)

(,)(,)

=0.2

Figure 2: (a) and (b) show the intraorbitalhoppings, whereas (c) represents the interor-bital hopping between dxz and dyz . (d) Fermisurfaces of Sr2RuO4. (,)-bands are com-posed of (dxz, dyz)-orbitals, and -band is com-posed of dxy-orbital, respectively.

anomalous velocity [10]. This is the origin ofhuge SHE.

Then, charge current operator for -direction ( = x, y) is given by [10]

jCx = eH

kx= e

vx v

ax 0

vax 0 00 0 vz

x

,(2)

where vx = 1/kx and vzx = 3/kx.

The interorbital velocity vax = g/kx =

4t sin ky cos kx is called the anomalous ve-locity because it has the same symmetry asky. Since v

axvyFS = 0, v

a is the origin of the

AHE, SHE, and OHE [9]. Next, the z-spincurrent JSx = {J

Cx , sz}/2 and the lz-orbital cur-

rent JOx = {JCx , lz}/2 [14, 9] are given by

jSx = (sz/e)jCx , (3)

jOx = h

0 ivx 0ivx 0 0

0 0 0

. (4)

where sz = h/2. We note that eqs. (3) and(4) are also obtained from the kinetic equation;

jS

(r) = i[H, rsz(r)] and jO(r) = i[H, rlz(r)].

Next, we consider the spin-orbit interactionHSO =

i 2lisi. Since the spin-orbit interac-

tion mixes electrons with different spins, HSOis given by 6 6 matrix:

HSO = h2

0 i 0 0 0 i+i 0 0 0 0 10 0 0 i 1 00 0 i 0 i 00 0 1 i 0 0

i 1 0 0 0 0

, (5)

where the first three rows (columns) corre-spond to xz , yz and xy , and the sec-ond three rows (columns) correspond to xz ,

yz and xy , respectively. As a result, thetotal Hamiltonian Htot = H0+HSO is given by6 6 matrix. The Fermi surfaces for Sr2RuO4is shown in Fig. 2 (d). In the present model,the Fermi surface does not split by the spin-orbit interaction. When = 0, - and -bands weakly hybridize due to t, whereas -band does not hybridizes with others. When = 0, all the band hybridizes due to the spin-orbit interaction.

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The current operators in this 6 6 Hilbertspace are given by

JCx =

(jCx 0

0 jCx

), (6)

JSx = (h/e)

(jCx 0

0 jCx

), (7)

JOx =

(jOx 0

0 jOx

), (8)

The 6 6 matrix form of the retarded Greenfunction is given by GR(k, ) = ( + H+i)1, where is the chemical potential and is the imaginary part of the k-independentself-energy (damping rate) due to scattering bylocal impurities. If is much smaller thanthe bandwidth, is approximately diagonal

with respect to orbital; ,(0) , (, =1, 2, , 6) and = +3 for = 1, 2, 3.

3 Spin and orbital Hall con-

ductivities

From now on, we calculate the intrinsic SHCand OHC in the presence of local impurities us-ing the linear-response theory. Intrinsic SHCand OHC originate from the interband process,

in contrast to the ordinary Hall conductivitydue to Lorentz force comes from the intraor-bital process. The former is independent ofquasiparticle damping rate , whereas the lat-ter is proportional to 2.

It is known that the extrinsic SHC and OHC,which are intraband terms and are derivedfrom the current vertex correction due to lo-cal impurities, are proportional to 1. Thisfact suggests that the extrinsic SHC will exceedthe intrinsic SHC in high-conductivity regime.

However, in various transition metal ferro-magnets with low resistivity, intrinsic anoma-lous Hall effect dominates the extrinsic one.This experimental fact can be explained inthe present model: If the impurity potentialis local, the current vertex correction van-ishes identically because JC,S,Ox FS = 0 i nthe present model. (In contrast, current ver-tex correction remains in the Rashba modelwithin the Born approximation; the intrinsic

SHC cancels out due to the current vertex cor-rection [14].) It is noteworthy that current ver-tex correction due to Coulomb interaction doesnot cause the extrinsic conductivities [12].

According to the linear response theory, theSHC at T = 0 is given by zxy =

zIxy +

zIIxy ,

where [9]

zIxy =1

2N

k

Tr

JSx GRJCy G

A=0

, (9)

zIIxy =1

4N

k

0

dTr

[JSx

GR

JCy G

R

JSx GRJCy

GR

R A

].(10)

Here, I and II represent the Fermi surfaceterm and the Fermi sea term. The diagram-matic expression for the SHE is given in fig. 3.

Jx Jy

'

'+

CS

Figure 3: Diagrammatic expression for zxy.

Hereafter, we use a unit h = c = 1 for sim-plicity. Inserting eqs. (2) and (3) into eq. (9),the Fermi surface term of the SHC in t2g-model

is given by zIxy = zI(1)xy +

zI(2)xy +

zI(3)xy , where

zI(1)xy =2e

N

k

1vaxvy(( 3)

2 + 23)

/|DR|2, (11)

zI(2)xy =e2

Nk[21v

axvy( 3)

+ 3(gvxvy + 2vaxvy( 3))]

/|DR|2, (12)

zI(3)xy =2e3

N

k

3vaxvy/|D

R|2, (13)

where DR = det( H + i). Here, zI(m)xy

(m = 1, 2, 3) is proportional to m except forthe -dependence of DR=0. The Fermi seaterm is also obtained by inserting eqs. (2)

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and (3) into eq. (10). We do not show itsexpression since its contribution is very smallin the metallic state [10].

According to the linear response theory, theOHC of the Fermi surface term OzIxy and that of

the Fermi sea term OzIIxy are given by eqs. (9)

and (10) by replacing JSx with JOx . The Fermi

surface term is given by OzIxy = OzI(0)xy + O

zI(2)xy ,

where

OzI(0)xy =e

N

k

1vx

gvy + vay(1 2)

(( 3)2 + 23)/|D

R|2, (14)

OzI(2)xy =e2

N

k

3vx

gvy + vay(1 2)

/|DR|2. (15)

The total OHC is given by Ozxy

= OzIxy

+ OzIIxy

.The obtained intrinsic SHC and OHC are inde-pendent of damping if3/1 is fixed. Also, therenormalization factor z = m/m due to manybody effect does not modify SHC nor OHC.

4 Numerical results

In this section, we show obtained numericalresults. Figure 4 shows the -dependence ofthe SHC and OHC for t2g-model (M=Ru;

n1 = n2 = n3 = 4/3). We use the Born ap-proximation since a tiny residual resistivity inSr2RuO4 suggests that the impurity potentialsare small. Then, is proportional to the localdensity of states (LDOS) (0). In Sr2RuO4,1 3(0)/1(0) 2.5 because the -band(dxy-orbital) Fermi surface is very close to thevan-Hove singular point (, 0). Therefore, weput 3/1 = 3 and 1 = 0.005. The total SHC(OHC) is given by the summation of Fermi sur-face term (I) and Fermi sea term (II). The

Fermi sea terms of both the SHC and OHCare much smaller than the Fermi surface termsgiven in eqs. (11)-(15) [10]. Here, 1.0 [|e|/2a]corresponds to 670 [h/|e|]1cm1 if we putthe interlayer distance of Sr2MO4; a 6A.The obtained SHC and OHC for a typical val-ues of 0.2 are much larger than values insemiconductors [3], because of the large Fermisurfaces and the large SO interaction in tran-sition metal atoms.

0 0.1 0.2 0.3 0.4

1.5

1

0.5

0

0.5

S

HE[

|e|/2a]

SHE(I+II)

SHE(II)

SHE(I)

SHE(I, 2)

0 0.1 0.2 0.3 0.4

2

1

0

OHE[|e|/2a]

OHE(I+II)

OHE(II)

OHE(I)

OHE(I, 2)

Figure 4: -dependence of the (a) SHC and(b) OHE in t2g-model for Sr2RuO4 [n = 4]. Atypical value of for Ru4+-ion is 0.2.

As shown in Fig. 4 (a), zI(2)xy given by

eq. (12), whose leading order is O(2), causesthe dominant contribution to the Fermi sur-face term of the SHC for 0.02 < < 0.3. Wesee that |Ixy| < |

zI(2)xy | since both

zI(1)xy and

zI(3)xy are positive. The total SHC zI

xyshows a

maximin at 0.25. As for the OHE, |OzI(2)xy |

increases drastically with increase as shown

in Fig. 4 (b), whereas |OzI(0)xy | decreases mono-

tonically with . As a result, OzIxy is approxi-mately constant for 0 < < 0.2.

We stress that the OHC in the present modelis finite even if = 0. In contrast, OHC van-ishes in previously studied electron gas mod-els, where anomalous velocity is caused by

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spin-orbit interaction [3, 4]. In the presentmodel, anomalous velocity vax in eqs. (2) and(3) is caused by interorbital (xz yz) hop-ping t. Now, we briefly discuss the reasonwhy the OHC is finite even if = 0 in thepresent model: In the {|xz, |yz}-basis, theoff-diagonal kinetic term is given by gx as

shown in eq. (1), where is the Pauli ma-trix. After the unitary transformation, gx istransformed to be gz in the {|lz = +1, |lz =1}-basis. Then, x-component of the anoma-lous velocity is given by vaxz. In an equi-librium state, the average of anomalous ve-locity over the Fermi surface vanishes sincevax(kx, ky) = v

ax(kx, ky) and v

ax(kx, ky) =

vax(kx, ky). However, when the electric fieldis applied parallel to +y direction, the Fermisea shifts toward y direction. Then, the

average of va

xz over the shifted Fermi sur-face becomes finite, and the (1,1)-component[(lz = +1)-orbital] and the (2,2)-component[(lz = 1)-orbital] take opposite values. Forthis reason, finite orbital current along x-axisappears when the electric field is applied alongy-axis. This is nothing but the orbital Halleffect.

0 1 2 3 4 5

n

2

1

0

1

SHC,O

HC[

|e|/2a]

SHC

OHC

=0.2

6

Figure 5: n-dependence of the SHC and OHCin t2g-model for Sr2MO4. n = 2, 3, 5 corre-sponds to M=Mo,Ru,Rh, respectively.

Figure 5 shows the filling-dependence of theSHC and OHC in t2g-model. n = 6 corre-sponds to the band insulator. For Sr2MO4,M=Ru, Mo and Rh correspond to n = 4, 2 and

5, respectively. In the present calculation, wetake the n-dependence of 3/1 = 3(0)/1(0)into account. Therefore, we can control val-ues of SHC and OHC by changing the electronfilling n. Here, consider the AHE in paramag-netic metals, which is caused by the magneti-zation induced by the magnetic field Bz.

0 2 4 6 8 10

3/

1[

1=0.005]

2

1

0

SHC,O

HC

[|e|/2a]

OHC

zII

xy (=0.2)

SHC

=0.2, t=0.05

=0.1

=0.2

=0.2

=0.1

Figure 6: 3/1-dependence of the SHC andOHC in Sr2RuO4. |

zxy(t

= 0.05)| is largerthan |zxy(t

= 0.1)|, which is understood bythe relation |zxy| t

2 in the presentmodel.

Figure 6 shows the 3/1-dependence of theSHC and OHC in Sr2RuO4 for = 0.1 0.3.|zxy| significantly increases with 3/1 since

numerators ofzI(2)xy and

zI(3)xy contain 3. On

the other hand, Ozxy is approximately indepen-

dent of 3/1 since the increment of OzI(2)xy

is approximately canceled by the decrement

of OzI(0)xy . If 3/1 is fixed, the SHC (and

OHC) is independent of damping rate whenmax{1, 3} ; is the minimum band-

splitting around the Fermi level [10], and 0.2 in the present t2g-model. It is noteworthythat |zxy| for t

= 0.05 is larger than that fort = 0.1, although the anomalous velocity va

is proportional to t. This result is understoodby the relation |zxy| v

a2 in the presentmodel, and increases with t since the hy-bridization between , -bands is caused by t.

We propose how to increase 3/1 in real sys-tems: According to ref. [15], -band Fermi sur-

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face touches (, 0) in La-doped Sr2yLayRuO4for y 0.23 (n 4.23) since the electron fillingin the -band increases. Then, 3/1 1 isexpected within the Born approximation. Wealso discuss Ca-doped Ca2ySryRuO4 (n = 4),where electron-electron correlation is enhancedbecause Ca-doping reduces hopping integrals

associated with the tilting of RuO6-octahedra.Recent optical measurement [16] observed ahuge mass-enhancement due to strong corre-lation effect in -band, whereas electrons in, bands remain light. In this situation,3/1 1 will be realized due to inelasticscattering at finite temperatures. Therefore,we expect that SHC takes a large value inSr2yLayRuO4 and in Ca2ySryRuO4.

5 Effective Aharonov-Bohmphase in the present model

In previous sections, we derived the SHC int2g-model using the Green function method.In this section, we explain an intuitive reasonwhy SHC appears in the present multi-orbitalmodel. For simplicity, we disregard the dxy-orbital, and consider only (dxz, dyz)-orbitals.Hereafter, we consider the motion of a -spinelectron along a triangle of half unit cell as

shown in Fig. 7: Let us consider to trans-fer an electron in the dxz-orbital at the centersite 1 to the dyz-orbital at site 2 through theinterorbital hopping integral +t, and to thesame orbital at site 3 through the intraor-bital hopping t. In succession, let us trans-fer it to the dxz-orbital at the same site us-ing the spin-orbit interaction for -spin elec-tron; hlz. Then, the electron aquire the fac-tor +ih since the matrix element of lz is givenby yz|lz|xz = xz|lz|yz = i. Finally, let

us transfer it to the original site 1 throught. By this clockwise movement, an electronaquire the Berry phase +i. If the electronmoves in the anti-clockwise direction, the re-sultant Berry phase is i.

By considering the signs of interorbital hop-ping integral and matrix elements of spin-orbitinteraction, it is shown that a clockwise (anti-clockwise) movement along any triangle of halfunit cell causes the factor +i (i). This fac-

2lzsz

i

yz

xz

-i

=1/2sz:

t -t

t-t

0/4

0/40/4

0/4

yz

xz11

22

33

Figure 7: Effective magnetic flux in thepresent model for -electron.

tor can be interpreted as the Aharonov-Bohmphase factor e2i/0 [0 = hc/|e|], where represents the effective magnetic flux =Adr = 0/4. Since the effective flux for

-spin electron is opposite in sign, electronswith different spins move to opposite direction.Threfore, the effective magnetic flux gives riseto the SHC of order O() given by eq. (11).Note that the SHC of order O(2), which iscaused by eq. (12), is given by the interorbital

transition between (xy ) and (xz ) [or (yz )]due to spin-orbit interaction. As for OHE, lz inthe orbital current JOx = {J

Cx , lz}/2 works as

the SO interaction. Large SHE and OHE dueto such effective flux will be realized in variousmultiorbital transition metal complexes.

In the ferromagnetic state, the effective mag-netic flux causes the AHE [10]. Here, we notethat the intrinsic AHE and SHE due to theeffective magnetic flux is independent of reis-

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itivity, although the ordinary Hall conductiv-ity due to real magnetic field is proportionalto 1/2 2. This difference can be under-stood as follows: In the present model, anyclosed loop that causes the Aharonov-Bohmphase, whose sign changes if the direction ofrotation is reversed, should use the interorbital

hopping t

at least once. Since the conduc-tivity due to interband particle-hole excitationis independent of , the present intrinsic SHCis independent of resistivity.

6 Summary

In summary, we derived the SHC and OHC int2g-model, which is generalized to include theeffects of multi-orbital and band-dependence of. The obtained SHC is much larger than thatin semiconductors, and it can be controlled bychanging the filling and the band-dependenceof . Large SHE in our model originates fromthe spin-dependcent effective magnetic fluxthat is induced by sign of hopping integralsand atomic SO interaction. Moreover, hugeOHC in this model will enable us to controlthe atomic d-orbital state by applying electricfiled. The present study strongly suggests thatgiant SHE and OHE are ubiquitous in mul-tiorbital d-electron systems, because of largeanomalous velocity due to d-orbitals and bylarge SO interaction. Recently, we studied theSHE in Pt based on a (6s, 6p, 5d)-tight-bindingmodel [17], and succeeded in explaining thehuge SHC in Pt ( 240 [he1 1cm1]) [8].The origin of large SHC in Pt is the effec-tive magnetic flux that is induced by the in-terorbital transition between dxy- and dx2y2-orbitals due to strong 5d-SO interaction.

References

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[7] S. Valenzuela et al, Nature 442, 176(2006).

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H. Kontani et al- Giant Intrinsic Spin Hall Effect in Transition Metal Compounds