Giant Intrinsic Spin and Orbital Hall Effects in Sr2MO4 (M=Ru,Rh,Mo)

0.19 MB·English

by  H. Kontani

M M
Giant Intrinsic Spin and Orbital Hall Effects in Sr O ( =Ru,Rh,Mo)
2 4
H. Kontani1, T. Tanaka1, D.S. Hirashima1, K. Yamada2 and J. Inoue3
(Dated: February 4, 2008)
We investigate the intrinsic spin Hall conductivity (SHC) and the d-orbital Hall conductivity
(OHC)inmetallicd-electronsystems,byfocusingonthet2g-orbitaltight-bindingmodelforSr2MO4
(M=Ru,Rh,Mo). The conductivities obtained are one or two orders of magnitude larger than
8 predicted values for p-type semiconductors with ∼ 5% hole doping. The origin of these giant Hall
0 effectsisthe“effectiveAharonov-Bohmphase”thatisinducedbythed-atomicangularmomentum
0 inconnectionwiththespin-orbitinteractionandtheinter-orbitalhoppingintegrals. ThehugeSHC
2 andOHCgeneratedbythismechanismareexpectedtobeubiquitousinmultiorbitaltransitionmetal
n complexes, which opens thepossibility of realizing spintronics as well as “orbitronics” devices.
a
J PACSnumbers: 72.25.Ba,74.70.Pq,85.75.-d

6
1
Recently, the spin Hall effect (SHE) has been attract- SHE in d-electron systems. We stress that not only the

] ing considerable interest since controlling spin by apply- SOI play a significant role in the SHE, but so does the
l
e ing electric field is one of the key issues in the field of phaseofhoppingintegralscharacteristicofd-electronsys-
- spintronics [1]. The SHE was originally predicted based tems. In contrast, the “Dirac monopole mechanism” [4]
r
t on the assumption of extrinsic electron-scattering [2, 3]. is appropriate when massless Dirac cone dispersion ex-
s
. Intrinsic SHE in semiconductors is caused by uniform ists in close proximity to the Fermi level, as is the case
t
a spin-orbit interaction (SOI) in systems and is indepen- in semiconductors. The present mechanism of SHE also
m
dent of the presence of impurities. It existence was differsfromthatintheRashba-type2DEGmodeldueto
- predicted by Murakami et al.[4] for p-type semiconduc- themomentum-dependentSOI[5]. Wefurthershowthat
d
tors, and by Sinova et al.[5] for a two-dimensional elec- the calculatedSHC depends stronglyonthe longitudinal
n
trongas (2DEG) with a Rashba-typeSOI. Motivated by resistivity in the high-resistivity region, even though the
o
c these predictions, many studies have been performed to SHE is of intrinsic origin.
[ demonstrate the prominent disorder effects on the spin To demonstrate the mechanism of the intrinsic

4 Hallconductivity(SHC)[6–9],thepossibilityofquantum SHC and OHC due to d-orbital degrees of free-
v SHE in insulating phase of semiconductors [10–12], and dom, we adopt the square-lattice tight-binding model
7 the existence ofthe orbitalHalleffect(OHE).According with (d ,d ,d )-orbitals, which was first proposed
xz yz xy
4 totheresultoftheabinitio calculation[13],SHCandor- to describe the realistic electron-structure of Sr MO
2 4
4
bitalHallconductivity(OHC)forp-typesemiconductors (M=Ru,Rh,Mo) [17], and atomic LS coupling as the
2
increase with hole number n , and they reachabout 100 SOI. This model is subsequently referred to as the t -
0 h 2g
7 and 10 [~e−1·Ω−1cm−1] respectively, for nh =0.05/cell. model. We show that the calculated SHC and OHC are
0 one or two orders of magnitude greater than theoretical
Very recently, the so-called inverse SHE has been ob-
t/ servedforspincurrentsinjectedintoconventionalmetals values in highly doped p-type semiconductors [13]. The
a
presentstudystronglypredictsthatgiantSHEandOHE
m suchasAl andPtfromaferromagnet[14,15]. The SHC
observedinPtis∼240[~e−1·Ω−1cm−1],whichisabout areubiquitous in d-electronsystems,and suggestsa new
-
d 104 times larger than the experimental SHC in n-type method to analyze them in various metals.
n semiconductors reported in ref. [16]. This observation The Hamiltonian for the t2g-model in a square lattice
o isgivenbyH =H +H ,whereH isthekineticterm
raises the question of whether a novel mechanism may K SO K
:c be responsible for the giant SHE in d-electron systems. andHSO = ı2λlı·si is the SOI for M 4d-orbitals. The
v Moreover, a simple theoretical model that is capable of bandparametersofthet2g-modelare: (i)theintraorbital
i nearest neigPhbor (NN) hopping integrals t = t (≡ t)
X describing the giant SHE inherent in d-electron systems 1 2
and t for the xz,yz and xy orbitals, respectively, (ii)
r is highly desirable. 3 ′
a intraorbital second NN hopping t for xy orbital, and
In this Letter, we propose a new mechanism for the 3 ′
(iii) interorbital second NN hopping t between the xz
giant intrinsic SHC and OHC that originate from the
and yz orbitals. It will be shown later that interorbital
d-orbitaldegreesof freedom, whichis absentin semicon- ′
hopping t is indispensable for SHE and OHE.
ductors. In a multiorbital system, a conduction electron
The matrix elements of H are then easily given by
acquiresan“effectiveAharonov-Bohmphasefactor”due K
to d-atomic angular momentum in conjunction with the ξ g 0
1
SOI and the inter-orbital hoppings. This mechanism is Hˆ = g ξ 0 , (1)
K 2
responsible for the large SHE. An intuitive explanation  0 0 ξ 
3
based on the tight-binding model, which is given in Fig.
 
3, appears to successfully predict the characteristics of where ξ = −2tcosk , ξ = −2tcosk , ξ =
1 x 2 y 3
2
−2t3(coskx + cosky) − 4t′3coskxcosky + ξ30, and g = 2 (0,π) (π,π)
′
−4t sink sink . Here, the first, second, and third rows
x y α
(columns) of Hˆ correspond to |xzi, |yzi, and |xyi, re- SHC γ
spectively, andK1, 2, and 3 denote orbitals xz, yz, and πa] 1 (-1)xOHC ky β
2
xy, respectively. We set ~ = 1 during the calculation. e|/
A constant ξ30 is included in ξ3 to allow the number of HC [| 0 (0,0) kx (π,0)
electrons n in the l-orbital to be adjusted. O
l C,
The total Hamiltonian is given by H
S -1
Hˆ +λˆl λ(ˆl −iˆl )
Hˆ = K z x y , (2)
λ(ˆl +iˆl ) Hˆ −λˆl -2
(cid:18) x y K z (cid:19) 0 0.1 0.2 0.3 0.4
λ
wherethefirstandthesecondrows(columns)correspond
t(ζo=|↑xi,ayn,dz)|is↓is,perecsifipeedctiinveqlyu.anTtuhme mmaetcrhiaxneiclesmteexnttboofokˆlζs FtyIpGic.a1l:vλa-ludeepoefnλdefnocreRouf4t+h-eioSnHiCs ∼an0d.2O.HWCeinseStrγ21Ru=Oγ42. =A
as, for example, (ˆl ) = i(δ δ −δ δ ). In ac- 0.005 and γ3/γ1 = 3. Inset: Fermi surfaces of Sr2RuO4 for
z l,m l,2 m,1 l,1 m,2 λ=0.2.
cordance with Refs. [17–19], we set t = 1 (≈ 0.2 eV),
′ ′
t = 0.1, t = 0.8, t = 0.35, and λ ∼ 0.2. The band
3 3
structure for Sr RuO is reproduced by selecting the the currentvertex correction(CVC) due to the localim-
2 4
chemicalpotentialµandξ0 sothatn =n =n =n/3. purity potentials in the (extended) Born approximation,
3 1 2 3
The Fermi surfaces for λ = 0.2 are shown in the inset which is given by ∆JˆM ∝ GˆAJˆMGˆR (M=S,C or O),
ζ k ζ
of Fig. 1. (α,β)-bands [γ-band] are mainly composed of vanishes identically. (In contrast, the CVC is crucial
P
(d ,d )-orbitals[d -orbital]and,duetothesymmetry in the Rashba 2DEG model since the SOI, which is an
xz yz xy
of Hˆ under k ↔ −k, each band is two-fold degenerate odd-function with respect to k, gives a k-independent
even if λ > 0, in contrast to the Rashba 2DEG model. anomalous velocity [6].) For this reason, the SHC at
TheshapeoftheFermisurfacesagreeswiththatobtained T =0 is given by σz =σzI +σzII [21], where
xy xy xy
in the ARPES measurements, LDA band calculations,
1
and dHvA oscillations [20]. σzI = Tr JˆSGˆRJˆCGˆA , (4)
The 6×6 matrix retarded Green function is given by xy 2πN x y ω=0
k
GˆR( k,ω)=(ω+µ−Hˆ+iΓˆ)−1,whereΓˆ istheimaginary X h i
partofthe k-independentself-energyduetolocalimpu- σzII = −1 0 dωTr JˆSGˆRJˆCGˆR
rities. Since a tiny residual resistivity in Sr2RuO4 sug- xy 4πN k Z−∞ " x ∂ω y
X
geststhattheimpuritypotentialsaresmall,weapplythe
GˆR
Born approximation: Γˆ ∝ −Im kGˆR(k,0). If λ = 0, −JˆxSGˆRJˆyC∂ω −hR→Ai . (5)
thenΓˆ isdiagonal;Γˆ ∝γδ (l,m=1,2,3). Theoff- #
l,m l l,mP
diagonal terms will then be negligible if λ ≪ W . γ
band l Here, I and II represent the “Fermi surface term” and
is the quasiparticledamping rate for the l-orbital,and is
the“Fermiseaterm”. Takingthe traceineq. (4),σzI in
proportionaltothelocaldensityofstates(LDOS)forthe xy
zI(1)
l-orbital,ρl(0). InSr2RuO4,1≪ρ3(0)/ρ1(0)∼2.5since the present model is straightforwardly given by σxy +
zI(2) zI(3)
the γ-band Fermi surface is very close to the van-Hove σxy +σxy , where
singular point at (π,0) [17].
2eλ
The charge current for ζ-direction (ζ = x,y) is given σzI(1) = γ vav ((µ−ξ )2+γ2)/|DR|2, (6)
by JˆC = −e·dHˆ/dk (−e is the electron charge), and xy πN 1 x y 3 3
ζ ζ k
X
the σz-spin current is given by the Hermitian operator: eλ2
JˆζS = (−1/e){JˆζC,sˆz}/2 [4–13]. Since the SOI is k- σxzIy(2) = πN [2γ1vxavy(µ−ξ3)
independent, they are given by k
X
+γ (gv v +2vav (µ−ξ ))]/|DR|2, (7)
vˆ 0 1 vˆ 0 3 x y x y 3
JˆC =−e ζ , JˆS = ζ , (3) 2eλ3
ζ (cid:18) 0 vˆζ (cid:19) ζ 2(cid:18) 0 −vˆζ (cid:19) σxzIy(3) = πN γ3vxavy/|DR|2, (8)
k
where vˆ = dHˆ /dk . Also, the l -orbital current is X
ζ K ζ z
given by JˆζO ={JˆζC,ˆlz}/2 [13]. where vζa = dg/dkζ (ζ = x,y), vx = dξ1/dkx, vy =
We now calculate the SHC and OHC in the presence dξ /dk ,andDR =det(µ−Hˆs+iΓˆ). σzI(m) (m=1,2,3)
2 y xy
of local impurities using the linear-response theory. In isproportionaltoλm exceptfortheλ-dependenceofDR.
the current model, vˆ and G are respectively odd and The “Fermi sea term” can also be obtained by inserting
ζ
even functions with respect to k ↔ − k. Therefore, JˆC,S ineq. (3)intoeq. (5). Wedonotgiveanexpression
x
3
for it since its contribution is very small in the metallic γ = γ = 0.005 and γ /γ = 3. The Fermi sea terms
1 2 3 1
state [23] (see Fig. 2 (b)). (II) of both the SHC and OHC are negligible [23] since
In the same way, the OHC of the Fermi surface term they are about 30 times smaller than the Fermi surface
OzI andthatofthe FermiseatermOzII arerespectively terms (I) given in eqs. (6)-(10) (see Fig. 2 (b)). Here,
xy xy
given by eqs. (4) and (5) by replacing JˆS with JˆO. In 1.0 [e/2πa] corresponds to ≈ 670 [~e−1 · Ω−1cm−1] if
thepresentmodel,OzI isgivenbyOzI(0)+xOzI(2),xwhere we put the interlayer distance of Sr2MO4, a ≈ 6˚A. The
xy xy xy obtainedSHC andOHC for typical values of λ∼0.2 are
e muchlargerthanthoseinsemiconductors[4,13]. InFig.
OzI(0) = γ v gv +va(ξ −ξ ) (9)
xy πN 1 x y y 1 2 1, the SHC increases approximately in proportion to λ2
×((µX−k ξ )2+(cid:2)γ2)/|DR|2, (cid:3) since σxzIy(2) in eq. (7) gives the dominant contribution
3 3 for 0.05 < λ < 0.3. On the other hand, the OHC exists
eλ2 even if λ = 0 since the orbital current is independent of
OzI(2) = γ v gv +va(ξ −ξ ) /|DR|2.(10)
xy πN 3 x y y 1 2 spin. Note that the OHC vanishes if λ = 0 in Rashba
k
X (cid:2) (cid:3) 2DEG model since the anomalous velocity arises from
The interorbital velocity va = dg/dk is termed the the k-dependent SOI.
ζ ζ
“anomalous velocity”, which is the origin of the Hall Figure2(a)showstheγ -dependencesoftheSHCand
1
effect [23]. Since vxa = −4t′sinkycoskx has the same OHC for γ3/γ1 = 3. They are approximately indepen-
symmetry as ky, eqs. (6)-(8) are finite even after the k- dent of γ1 in the low-resistivity region where γ1 < 0.1.
′
summations. Sincet ismuchlargerthantheRashbaSOI This behavior is typical of the intrinsic Hall effect found
in semiconductors,a largeSHC is realizedin the present by Karplus and Luttinger [22]. However, both the SHC
′
model. Although the sign of t changes due to the gauge andOHCstarttodecreasewithγ inthehigh-resistivity
1
transformation |xzi → −|xzi, both the SHC and OHC region where γ > 0.1. This crossover behavior is also
1
are invariant if the sign change of (ˆl ) is taken into realized in the intrinsic anomalous Hall effect (AHE)
ζ l,1
account [23]. We stress that eqs. (6)-(10) are indepen- [23,24]: SincetheintrinsicHalleffectoriginatesfromthe
dent of the renormalization factor due to the Coulomb interbandparticle-holeexcitationinducedbythe electric
interaction: z =(1−∂Σ(ω)/∂ω)−1| [23]. field [22], the Hall conductivity is proportional to the
ω=0
lifetime of the excitation ~/∆, where ∆ is the minimum
(a) 101 (b) 0 band-splittingaroundtheFermilevel[23]. Inthismodel,
|OHC| −σzIIxy ∆ ∼ 0.2. In the high-resistivity region, the SHC de-
100 SHC λ=0.2 creases drastically since the interband excitation is sup-
pressed when the quasiparticle lifetime ~/γ is shorter
|SHC|
a] a] than ~/∆. Interestingly, the SHC becomes positive for
πe|/210−1 πe|/2 −1 γ1 ≥0.3 since σxzIy(1) (>0) is dominant over σxzIy(2).
[| [|
Even in the low-resistivity region, the SHC and OHC
10−2 λ=0.2 SHC>0 t’=0.05 sensitivelydependonthevalueofγ3/γ1 asshowninFig.
γ=3γ (−1)xOHC 2 (b). |σz | increases significantly with γ /γ since the
3 1 xy 3 1
10−3 −2 numerator of the second term in eq. (7) and that in
10−3 10−2 10−1 100 0 2 4 6 8 10
γ γ/γ [γ=0.005] eq. (8) are proportional to γ . On the other hand, Oz
1 3 1 1 3 xy
is approximately independent of γ /γ . Here, we con-
(c) 0 3 1
siderhowtoincreaseγ /γ inrealsystems. Accordingto
3 1
SHC ref. [25],γ-bandFermisurfacetouches(π,0)inLa-doped
π2a] −1 Sr2−yLayRuO4fory ∼0.23(n∼4.23). Then,γ3/γ1 ≫1
e|/ isexpectedwithintheBornapproximation. Wealsocon-
[|
(−1)xOHC sider Ca-doped Ca2−ySryRuO4 (n = 4): According to
λ=0.2
optical measurements [26], a huge mass-enhancement is
−2
0 1 2 3 4 5 6 realized in γ-band, whereas α,β-bands remain light. In
n
this situation, γ /γ ≫ 1 due to inelastic scattering is
3 1
expected at finite temperatures.
FIG. 2: (a) γ1-dependence and (b) γ3/γ1-dependence of the
|SHC| and |OHC| in Sr2RuO4. |σxzy(t′=0.05)| is larger than Figure 2 (c) shows the electron filling (n) dependence
|σxzy(t′ =0.1)|, which isunderstoodfrom therelation |σxzy|∝ of the SHC and OHC in the t2g-model. For Sr2MO4,
t′∆−2 in the present model. (c) n-dependence of the SHC M=Ru, Mo and Rh correspond to n = 4, 2 and 5, re-
and OHC in the t2g-model [Sr2MO4]. spectively [27, 28]. Here, we correctly account for the
n-dependence of γ /γ =ρ (0)/ρ (0). It is thus possible
3 1 3 1
Figure1showstheλ-dependenceoftheSHCandOHC to control the values of SHC and OHC by varying the
for the t -model with M=Ru (n = 4). Here, we put electron filling.
2g
4
yz t -model. TheSHCobtainedisaboutthreetimeslarger
2g
t’ -t’ 2λlzsz:sz=−1/2 than the recently observed “giant SHC” in Pt [15], and
it can be further increased by controlling γ /γ . The
yz 3 1
φ/4 φ/4
xz 0 0 large SHE in our model originates from the “effective
-iℏλ iℏλ Aharonov-Bohmphase” induced by the atomic SOI and
φ/4 φ/4 the interorbital hopping integral. Moreover, huge OHC
0 0
xz
-t’ t’ inthismodelwillenableustocontroltheatomicd-orbital
state by applying an electric filed. The present calcula-
tion predicts that “giant SHE and OHE” will be ubiq-
uitous in d-electron systems. Using the theoretical tech-
nique developed here, SHC in various transition metals
FIG. 3: “Effective magnetic flux” for ↓-electron through tri-
such as Pt had recently been investigated [33].
angles of half unit cells in the(dxz,dyz)-model. Theeffective
magnetic fluxfor ↑-electron is −φ0/4. The authors acknowledge fruitful discussions with
Y. Tanaka, S. Onari and T. Nomura. This work
was supported by the Next Generation Super Comput-
WenowpresentanintuitiveexplanationfortheSHEin ing Project, Nanoscience Program, MEXT, Japan and
d-electronsystemsbasedonthe(d ,d )-model[23,31], Grant-in-Aid for the 21st Century COE ”Frontiers of
xz yz
which is a simplified version of the present model. Here, Computational Science”.
all the matrix elements of ˆl and ˆl vanish identically.
x y
In Fig. 3, we consider the motion of a ↓-spin elec-
tron along a triangle of a half unit cell. An electron
inthe d -orbitalcantransfertothed -orbitalandvice
xz yz
versa using the SOI for ↓-electron −~λˆl , hyz|ˆl |xzi = [1] For an overview see e.g., Semiconductor Spintronics and
z z Quantum Computation, ed. by D.D. Awschalom et al.
−hxz|ˆlz|yzi = i. These relations are derived from the (Springer, Berlin, 2002).
fact that the (π/2)-rotation operator about the z-axis is [2] M.I.D’yakonovetal., ZhETFPis.Red.13,657(1971).
given by exp(−iπˆl /2) = cos(π/2)−iˆl sin(π/2) = −iˆl [3] J. E. Hirsh, Phys.Rev. Lett. 83, 1834 (1999).
z z z
for ˆl2 = 1. By considering the sign of the interorbital [4] S. Murakami et al., Science 301, 1348 (2003).
z ′ [5] J. Sinovaet al., Phys.Rev.Lett. 92, 126603 (2004).
hopping integral (±t) and the matrix elements of the
[6] J. Inoueet al., Phys.Rev.B70, 041303 (2004).
SOI, it can be shown that a clockwise (anti-clockwise) [7] S. Murakami, Phys. Rev.B 69, 241202(R)(2004).
motion along any triangle of the half unit cell causes the [8] B.A.Bernevigetal.,Phys.Rev.Lett.95,16801(2005).
factor +i (−i). [9] J. Inoueet al., Phys.Rev.Lett., 97, 46604 (2006).
This factor can be interpreted as the Aharonov-Bohm [10] S. Murakami et al, Phys. Rev.Lett., 93, 156804 (2004).
phase factor e2πiφ/φ0 [φ = hc/e], where φ = Adr = [11] C.L.KaneandE.J.Mele, Phys.Rev.Lett., 95,146802
0 (2005).
φ0/4representsthe“effectivemagneticflux”thrHoughthe [12] B. A.Bernevig et al., Science 314, 1757 (2006).
triangleofthe halfunitcell. Since theeffective magnetic [13] G.Y. Guoet al., Phys. Rev.Lett. 94, 226601 (2005).
flux for ↑-electron is opposite in sign, ↑-electrons and ↓- [14] S. Valenzuela et al, Nature 442, 176 (2006).
electronsmoveinoppositedirections. Thus,theeffective [15] T. Kimuraet al., Phys. Rev.Lett. 98, 156601 (2007).
magnetic flux in Fig. 3 gives rise to a SHC of order [16] J.Wunderlichetal.,Phys.Rev.Lett.94,047204(2005).
O(λ) [eq. (6)], which is analogous to the AHE due to [17] K.K.NgandM.Sigrist.,Europhys.Lett.49,473(2000).
[18] T.NomuraandK.Yamada.,J.Phys.Soc.Jpn.71,1993
spin chirality [29, 30]. (In the t -model, the SHC of
2g (2002); Y.YanaseandM.Ogata.,J.Phys.Soc.Jpn.72,
order O(λ2) in eq. (7) is derived from the interorbital
673 (2003).
transition between (xy ↑) and (xz ↓) [or (yz ↓)] using [19] T. Mizokawa et al., Phys.Rev.Lett. 87, 077202 (2001).
the SOI twice.) We can also discuss the origin of the [20] A.P.MackenzieandY.Maeno,Rev.Mod.Phys.75,657
OHEbyconsideringthe motionofanelectronwith|l = (2003).
z
±1i∝∓|xzi+i|yziintheabsenceoftheSOI.Therefore, [21] P.Streda: J.Phys.C:SolidStatePhys.15(1982) L717.
large SHE and OHE due to such an effective flux will be [22] R. Karplus and J.M. Luttinger, Phys. Rev. 95, 1154
(1954).
ubiquitous in multiorbital systems.
[23] H. Kontaniet al., Phys. Rev.B 75 (2007) 184416.
We comment that the (dxz,dyz)-model has been ap- [24] H.KontaniandK.Yamada,J.Phys.Soc.Jpn. 63, 2627
plied to calculate the large AHE in Ru-oxides [23, 31]. (1994).
Consistent with the theory, paramagnetic compound [25] N. Kikugawa et al., Phys.Rev.B 70, 134520 (2004).
Ca Sr RuO exhibits large AHE under the magnetic [26] J. S.Lee et al., Phys.Rev. Lett. 96, 057401 (2006).
1.7 0.3 4 [27] S. Ikedaet al., J. Phys. Soc. Jpn. 69 (2000) 3162.
field [32]. We find that the same mechanism,that is, the
[28] F.Baumbergeretal.,Phys.Rev.Lett.96,246402(2006).
effective Aharonov-Bohm phase due to d-orbital degrees
[29] M. Taillefumier et al., Phys. Rev.B 74, 085105 (2006).
of freedom, gives rise to the large SHE and AHE. [30] K. Ohgushiet al., Phys. Rev.B 62, R6065 (2000).
In summary, we have studied SHC and OHC in the [31] M. Miyazawa et al., J. Phys. Soc. Jpn. 68 (1999) 1625.
5
[32] L.M. Galvin et al., Phys. Rev.B 63, 161102(R) (2001). T. Tanaka et al., arXiv: 0711.1263.
[33] H. Kontani et al., J. Phys. Soc. Jpn. 76, 074707 (2007);