Study of intrinsic spin and orbital Hall effects in Pt based on a (6s, 6p, 5d) tight-binding model

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by  H. Kontani

Study of intrinsic spin and orbital Hall effects in Pt
based on a (6s,6p,5d) tight-binding model
H. Kontani1, M. Naito1, D.S. Hirashima1, K. Yamada2 and J. Inoue3
1Department of Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan.
2College of Science and Engineering, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan.
3Department of Applied Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan.
(Dated: February 5, 2008)
We study the origin of the intrinsic spin Hall conductivity (SHC) and thed-orbital Hall conduc-
8
tivity (OHC) in Pt based on a multiorbital tight-binding model with spin-orbit interaction. We
0 findthat theSHCexceeds 1000 ~e−1·Ω−1cm−1 when theresistivity ρ is smaller than ∼10 µΩ cm,
0 whereas it decreases to 300 ~e−1·Ω−1cm−1 when ρ ∼ 100 µΩ cm. In addition, the OHC is still
2
larger than the SHC. The origin of the huge SHE and OHE in Pt is the large “effective magnetic

n flux”thatisinducedbytheinterorbitaltransition betweendxy-anddx2−y2-orbitalswiththeaid of
a thestrong spin-orbit interaction.
J

8 PACSnumbers:

]
l Recently, the spin Hall effect (SHE) has attracted orbital Hall effect (OHE) in Pt by analyzing a realistic
l
a much attention due to its fundamental interest and its multiorbital tight-binding model. In the low-resistivity
h
potentialapplicationinspintronics. TheSHEhasaclose regime where ρ < 10 µΩcm, both the SHC and or-
-
s relationtotheanomalousHalleffect(AHE)inferromag- bital Hall conductivity (OHC) are constant of order
e nets: In 1954, Karplus and Luttinger (KL) [1] studied 1000∼3000 ~e−1·Ω−1cm−1, whereas they are strongly
m
the Hall effect in multiband systems and found that an suppressed in the high-resistivity regime where ρ ≫
.
t electricfieldinduces aspin-dependenttransversecurrent 10µΩcm. Thederivedcoherent-incoherentcrossoverisa
a
m inthepresenceofspin-orbit(SO)interaction. Thiseffect universal property of intrinsic Hall effects [15, 17]. Both
causesthe AHE (transversechargecurrent)in ferromag- the SHE andOHEoriginatefromakindofPeierlsphase
-
d netic metals and the SHE (transverse spin current) in factor due to the “effective magnetic flux” [14] that is
n paramagnetic metals. These phenomena are fundamen- induced by a combination of the angular dependence of
o tal issues in recent condensed matter physics [2–17]. In d-orbital wave functions and SO interaction. In Pt, the
c
theseyears,greatprogressontheSHEinsemiconductors dominant contribution to the SHE is given by the d -
[ xy
has beenmade. Murakamiet al. [2]andSinova etal. [3] orbital(int2g)andthedx2−y2-orbital(ineg). Therefore,
3
have studied the intrinsic (impurity-independent) SHE boththet -ande -orbitalsshouldbetakenintoaccount
2g g
v
in semiconductors by developing the theory of KL. Now, to explain the huge SHE in Pt.
5
3 the SHE in two-dimensionalelectrongas (2DEG) with a
5 Rashba-typeSOinteractioniswellunderstood[4–7]. Al-
3 thoughtheSHEinsemiconductorswasrecognizedbythe Pt has a face-centered cubic (FCC) structure with
5. optical detection of spin accumulation [8, 9], it is unfor- a = 3.9 ˚A, and the nearest inter-atomic distance is 2.8
0 tunately too small for quantitative analysis. Therefore, ˚A. In the present study, we use the Naval ResearchLab-
7 materials that show a large SHE are highly desirable. oratory tight-binding (NRL-TB) model [18, 19] to de-
0 scribe the bandstructure in Pt. The NRL-TB model
v: Recent experiments have revealed that the SHE also employs the scheme of the two-center, non-orthogonal
exists in metals such as Al [10] and Cl-doped ZnSe [11].
i Slater-Koster (SK) Hamiltonian [20]. The SK param-
X In particular, the huge spin Hall conductivity (SHC) in
eters are represented with distance- and environment-
r Ptatroomtemperature[240~e−1·Ω−1cm−1][13],which
dependent parameters that are determined so that the
a is104timeslargerthantheSHCreportedinsemiconduc-
obtainedtotalenergyandthe bandstructuresagreewell
tors, has attracted great attention. Simple 2DEG mod-
with those obtained by the first-principles calculations.
els cannot explain this experimental fact. Recently, the
Here, we take into account 6s, 6p, and 5d orbitals (in
present authors have studied the SHE in Sr RuO that
2 4 total, nine) and hopping integrals up to the sixth near-
is described by the t -orbital tight-binding model [14]
2g est neighbor sites. The electron number per Pt atom is
andfoundthattheanomalousvelocityduetointerorbital
ten. The NRL-TBmodelusesnon-orthogonalbases,but
hoppinggivesrisetohugeSHCintransitionmetals. This
we neglect the overlap integrals between different sites
mechanism also causes the large AHE [15–17]. To reveal
in this study for simplicity [21]. This simplification sig-
the origin of the huge SHE in Pt, we have to investigate
nificantly changes the dispersion of the s-band far from
the anomalous velocity due to the multiorbital effect by
theFermilevel,whilethatofthebandstructurenearthe
considering all the d-orbitals (t +e orbitals).
2g g Fermi energy is little affected. In the presence of the SO
In this letter, we study the intrinsic SHE and the d- interaction for 5d electrons H = λ (l·s) , the total
SO i i
P
2
Γ X W L Γ K X by local impurities (or inelastic scattering by phonons).
0.2
The charge current in the present model is
y]
(cid:10)(cid:67)(cid:11)R 0
gy [ ˆjC 0
er JˆC = µ . (2)
En 0.2 µ (cid:18) 0 ˆjµC (cid:19)
30]0.4 Here, ˆjC = −e∂Hˆ0, where −e is the electron charge and
(cid:10)(cid:68)(cid:11) x1 5 µ ∂kµ
k) [ µ = x,y. In this case, the atomic SO interaction is not
zIσ(xy 0 involved in the charge current since it is k-independent.
σ z I(k) Then, the s -spin current JˆS = {JˆC,sˆ }/2 is expressed
2000 xy k =0.19π z µ µ z
z as
1500
(cid:10)(cid:69)(cid:11)
1000
500 ˆjC 0
(π,00) JˆµS = (−~/e) 0µ −ˆjC . (3)
(cid:18) µ (cid:19)
k
y
Here, we discuss the current vertex correction (CVC)
(0,π)
k due to the local impurity potentials in the Born ap-
(π,π) x proximation, which is given by ∆JˆC ∝ GˆAJˆCGˆR.
µ k µ
(cid:10)(cid:70)(cid:11)E [Ry]k-0.01π ∆ k k y=k z=0.19π0 W(ssp,ehdce)tn-otroαbikitsa↔lso,ne−(Hˆko0f,(kathn))edα,ptβh-oeirsrbeifatonarlesotdahdnedf(uαβn,cPβits)io-cononmewpiotohfnterhneet-
x
0 64 of (∂/∂k )Gˆ = GˆJˆCGˆ is an even function. Note that
µ µ
|p i → −|p i (ν = x,y,z) under the parity transforma-
ν ν
FIG. 1: (Color online) (a) Band structure of a (6s,6p,5d) tion. Thus, (∆JˆC) is finite only when either α or β
µ α,β
tight-binding model for Pt (λ = 0.04). Γ = (0,0,0), is a p-orbital. In Pt, however, we have verified that the
X = (π,0,0), W = (π,π/2,0), L = (π/2,π/2,π/2), and
CVC affects the SHE only slightly [less than 5%] since
K = (3π/4,3π/4,0). (b) σ¯z (k) for γ = 0.002 along Γ-X.
(c) σ¯xzy(k) on the plane kz x=y 0.19π, which is plotted only the 6p-level is 20 eV higher than the Fermi level µ and
for σz (k) > 25. The crosses represent the Fermi surfaces. the p-electrondensity ofstates (DOS)atµ is verysmall.
xy
(d) kx-dependence of Ekl for ky = kz =0.19π; the minimum For this reason, we disregard the CVC hereafter.
bandsplitting near theFermi level ∆ is ∼0.035. According to the linear response theory [26], the SHC
is given by σz =σzI +σzII, where
xy xy xy
Hamiltonian becomes
1
Hˆ = λH(ˆˆl0++λiˆlˆlz)//22 λH(ˆˆlx−−λiˆlˆly)//22 , (1) σxzIy = 2πN Xk TrhJˆxSGˆRJˆyCGˆAiω=0, (4)
(cid:18) x y 0 z (cid:19) −1 0 GˆR
σzII = dωTr JˆS JˆCGˆR
where Hˆ0 is a 9×9matrixgivenby the NRL-TB model. xy 4πN k Z−∞ " x ∂ω y
The matrix elements of l are given in ref. [22]. The X
GˆR
bandstructure obtained for the (6s,6p,5d) tight-binding −JˆSGˆRJˆC −hR→Ai . (5)
model with λ=0.04 Ry is shown in Fig. 1 (a), which is x y ∂ω #
in good agreement with the result of a relativistic first-
principles calculation [23, 24] near the Fermi level. Ac-
Here, I and II represent the “Fermi surface term” and
cording to optical spectroscopy, λ = 0.03 Ry for a 5d
the “Fermi sea term”, respectively. In the same way,
electron in Pt, and λ = 0.013 Ry for a 4d electron in the OHC of the Fermi surface term OzI and that of
xy
Pd [22]. Hereafter, we set the unit of energy Ry; 1 Ry
the Fermi sea term OzII are given by eqs. (4) and
= 13.6 eV. Based on the NRL-TB model, spin wave ex- xy
(5), respectively,by replacing JˆS with the l -orbitalcur-
citations and the electron self-energy corrections in the x z
rent JˆO = {JˆC,ˆl }/2. Because of the cubic symmetry
ferromagneticFearestudiedusingtherandom-phaseap- x x z
of Pt, σδ = σz · ǫ and Oδ = Oz · ǫ , where
proximation [25]. µν xy µνδ µν xy µνδ
µ,ν,δ = x,y,z and ǫ is the antisymmetrized tensor
The18×18matrixformoftheretardedGreenfunction µνδ
isgivenbyGˆR(k,ω)=(ω+µ−Hˆ+iΓˆ)−1,whereµisthe with ǫxyz =1.
chemical potential and Γˆ is the imaginary part of the k- When Γ = γδ (constant γ approximation), ω-
αβ αβ
independentself-energy(dampingrate)duetoscattering integration in eq. (5) can be performed analytically as
3
shown in ref. [15]: Then, σzII =σzIIa+σzIIb, where (a) 1.5
xy xy xy γ=0.02 σz: λls
σz z z
σzIIa = −1 Im (JS)ml(JC)lm 1 isotropic xy
xy 2πN kX,l6=m (cid:8) x y (cid:9)Ekl −Ekm πa] 1 ~1000 +h/eΩcm
×Im(cid:26)(EEklkl−+iγE)(kmE−km2−iγiγ)(cid:27), (6) C [|e|/2 0.5 σonz:l yλ flozrs (zxy, x2−y2)
H
σzIIb = 1 Im (JS)ml(JC)lm 1 S σz: λ(lxsx+lysy)
xy πN x y (El −Em)2
kX,l6=m (cid:8) (cid:9) k k 0
El −iγ σzII
k
×Im ln , (7)
Em−iγ 0 0.01 0.02 0.03 0.04
(cid:26) (cid:18) k (cid:19)(cid:27) λ [Ry]
where l,m represent the band indices. El is the lth
k (b)
eigenenergy of Hˆ measured from the chemical potential OHC Berr|yσ czIuIbr|vature term
µ; αβUl†αH0αβUβm =Eklδlm,whereα,β arethe orbital a] 100 SHC 100
indices and U is a k-dependent unitary matrix. (JS)ml π2
gilnoivcIePaenqnlsti.bmhye(p6nBu)iromairtpnnyIdp2a(poN17tp)ePrnoistxkiigaImilmvaae(ntnGiˆdobAnny(,i0mΓˆP)p−iαnisGβˆttURhhel(†eα0G(i)mJ)r,xSepwe)uαnhrβieftUuryeβndmcIet.iinsoxsntithyies. SHC, OHC [|e|/1100−−21 ρ = σ−γ1+1 γ−n2egative110012 [2a/|e|]ρπ
Whenλ/EF ≪1,Γˆisalmostdiagonalwithrespecttothe λ=0.03
orbitalindex;Γα,β =γαδα,β [14]. InthecaseofSr2RuO4, 10−310−3 10−2 10−1 100 101
theSHCintheBornapproximationisnearlythreetimes γ [Ry]
greater than that in the constant γ approximation [14].
In Pt, in contrast, we have verified that both approxi-
mations give a similar SHC in the clean limit. For this FIG.2: (Color online) (a) λ-dependenceof theSHC.There-
reason, we use the constant γ approximation hereafter. lation σxzy ≈ σxzIy ≫ σxzIyI is realized. The matrix element of
Here, we determine the partof the Fermi surface from the SO interaction between the dxy-orbital and the dx2-y2-
orbital gives the dominant contribution to the SHC. Note
whichtheSHCoriginates: InFig. 1(b)and(c),weshow
that 1 [|e|/2πa] ≈ 1000 ~e−1 ·Ω−1cm−1 for a = 4 ˚A. (b)
1
σ¯zI(k) ≡ σzI(k′), where σzI(k) Crossover behaviors of theSHCand OHC at γ ∼∆≈0.035.
xy 8 xy xy
kx′X=±kxky′X=±kykz′X=±kz ρ is approximately proportional to γ. ρ = 1 corresponds to
is the integrand in eq. (4). [Apparently, 1 σ¯zI(k)= 1000 µΩcm.
N k xy
σzI.] σ¯z (k) is finite only on the Fermi surface, and
xy xy P
it takes huge values at (0.73π,0,0) (on Γ-X) and at
(0.42π,0.42π,0.42π) (on L-Γ) since two bands are very is concluded that the z-componentof the SO interaction
close on the Fermi level in the present model. However, givesthedecisivecontributiontotheSHC.Thematrixel-
the contribution of these two points to the SHC is small ement of lz is finite only for hyz|lz|zxi=−hzx|lz|yzi=i
after taking the k-summation. The dominant contribu- and hxy|lz|x2-y2i = −hx2-y2|lz|xyi = 2i. Among them,
tioncomesfromawideareaaround(0.19π,0.19π,0.57π) the dxy- and dx2-y2-orbitals, both of which are given by
as shown in Fig. 1 (d). Here, the bandsplitting ∆ near the linear combinations of lz = ±2, cause a dominant
the Fermi level is 0.035. contribution to the SHC as shown in Fig. 2 (a).
Now,weperformthenumericalcalculationoftheSHC, Here, we discuss the γ-dependence of the SHC and
using 1283 ∼ 5123 k-meshes. Figure 2 (a) shows the OHC: When γ is sufficiently small, these intrinsic Hall
λ-dependence of the total SHC σz = σzI + σzII for conductivities are proportional to the lifetime of the in-
xy xy xy
γ = 0.02, which is smaller than ∆ = 0.035. σzII rep- terbandparticle-holeexcitation: ~/∆[1–3,14,15,17]. In
resents the Fermi sea term in eq. (5). Apparently, fact, Fig. 2 (b) shows that both the SHC and OHC for
σz ≈σzI ≫σzII isrealized. σz increaseswithλmono- λ=0.03[22]areindependentofγ forγ ≪∆∼0.035. In
xy xy xy xy
tonically,anditreaches1000~e−1·Ω−1cm−1atλ=0.03. the high-resistivity regime where γ ≫∆, both SHC and
ToclarifytheoriginoftheSHE,westudytheSHCwhen OHC decrease drastically with γ since the interband ex-
the SO interaction is anisotropic: As shown in Fig. 2 citationissuppressedwhenthequasiparticlelifetime~/γ
(a), σz for H = λ (l s +l s ) is much smaller is shorter than ~/∆. This coherent-incoherent crossover
xy SO i x x y y i
than that in the isotropic case whereH =λ (l·s) ]. of the intrinsic Hall conductivities (σ = const. for
On the other hand, σzPfor H = λSO(l s ) ialmoist γ ≪ ∆ and σ ∝ ρ−2 for γ ≫ ∆) haxys been analyzed
xy SO i z zPi xy
coincides with that in the isotropic case. Therefore, it theoretically in refs. [15, 17]. In Pt, the SHC decreases
P
4
much faster than ρ−2 in the high-resistivity regime and By considering the sign of the interorbital hopping in-
the SHC becomes negative for γ >1, which may be due tegral (±t and ±t′) and matrix elements of the SO in-
to a complex multiband structure. If we put γ ∼ 0.07, teraction, we can verify that a clockwise (anticlockwise)
ρ ∼0.1 [|e|/2πa]∼100 µΩcm. Then, the obtained SHC motion along any triangle path with the SO interac-
is∼300~e−1·Ω−1cm−1,whichisclosetotheexperimen- tion causes the factor +i (−i). This factor can be in-
tal SHC of Pt [13]. In the experimental situation, γ in terpreted as the Aharonov-Bohm phase factor e2πiφ/φ0
Fig. 2 (b) corresponds to ~/2τ within the spin diffusion [φ =hc/|e|], where φ represents the “effective magnetic
0
lengthinPt(∼10nm)fromtheinterfaceofthejunction, flux” [14] φ = Adr = ±φ /4. This effective magnetic
0
which might be larger than the bulk value of γ. flux gives rise to the SHC of order O(λ).
H
We comment on σzIIb, which is frequently called the We also discuss the origin of the OHE by considering
xy
“Berry curvature term”. When Γαβ = γδαβ and γ → 0, the motion of an electron with |lz = +2i ∝ |x2-y2i +
σz =σzIIb ≈σzI [15]. However,σzIIb istotallydifferent i|xyi. We can show that an electron with |l = ±2i in
xy xy xy xy z
fromσz inthehigh-resistivityregimeasshowninFig. 2 Pt acquires the Aharonov-Bohm phase, which gives rise
xy
(b),sincethecancellationbetweenσzI andσzIIabecomes to the OHC of order O(λ0) [14].
xy xy
worse when γ is large. In many systems including Pt, In summary, we have studied the origin of huge SHC
σxzy ≈σxzIy is realizedfor awiderangeofparameters[15]. and OHC in Pt using a (6s,6p,5d) tight-binding model,
We briefly discuss the SHC using the Born approx- and found that the SHC reaches 1000 ~e−1 ·Ω−1cm−1
imation, where γα is proportional to the DOS for the in the low-resistivity regime where ρ <10 µΩcm. Other
α-orbital, ρα(0). When γα is α-dependent, σxzIyIb 6= σxzy significant findings of the present study are that (i) the
eveninthecleanlimit[14]. Infact,theSHCinSr2RuO4 OHC is still largerthan the SHC in Pt, which will cause
given by the Born approximation is much larger than largesurface magnetizationofPt; (ii) the huge SHC and
that given by the constant γ approximation since the α- OHC originate from the effective magnetic flux created
dependence of ρα(0) is large [14]. In Pt, however, both by the dxy- and dx2-y2-orbitals; and (iii) the coherent-
approximations give similar results. For this reason, we incoherent crossoverbehaviors of the SHC and OHC are
use the constant γ approximation. derived by taking both the I-term and II-term into ac-
count correctly. When ρ ∼ 100 µΩcm, the obtained
SHC becomes comparable with the experimental value
240 ~e−1·Ω−1cm−1. Note that ρ in the present calcula-
tion corresponds to the resistivity within the spin diffu-
sion length (∼10 nm) from the interface of the junction.
We comment that the effect of the overlap inregral re-
duces the magnitude of the SHC for Pt to some extent
[21]. Finally, we discuss the role of the Coulomb inter-
action: Although the SHC is independent of the renor-
malization factor z = (1−∂Σ(ω)/∂ω)−1| (= m/m∗)
ω=0
[14, 15], it will depend on the ω-dependence of γ(ω) as
well as the CVC due to the Coulomb interaction. They
are important future issues.
During the preparation of this paper, we found a pa-
FIG. 3: (Color online) Effective magnetic flux for ↑-electron
perwheretheSHC wascalculatedbasedonarelativistic
in the two-dimensional Pt model. This is the origin of the
first-principles calculation [27]. Only the Berry curva-
hugeSHC and AHCin Pt.
ture term given by eq. (7) was calculated, which is not
justified in the high-resistivity regime (γ ≫∆) [15].
Figure 3 shows the FCC crystal structure of Pt on
The authors acknowledge fruitful discussions with Y.
the xy-plane. Based on this two-dimensional model, we
Otani, T. Kimura, M. Sato, and T. Tanaka. This work
explain an intuitive reason why the huge SHC appears
was supported by the Next Generation supercomputing
in Pt, by considering only dxy-, dx2-y2-, and s-orbitals.
Project,NanoscienceProgram,Grant-in-Aidforthe21st
±t represents the hopping integrals between the nearest
neighbord -orbitalands-orbital,and±t′isusedforthe CenturyCOE“FrontiersofComputationalScience”,and
xy
Grant-in-AidforScientificResearchfromtheMinistryof
next nearest neighbor dx2-y2-orbital and s-orbital. Both
Education, Science, Sports and Culture of Japan.
the hopping integrals change their signs by rotation by
π/2. Here, we consider the motion of a ↑-spin electron
on the left side of Fig. 3 along a triangle of a half unit
cell: An electron in the d -orbital can transfer to the
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