Ab initio calculation of intrinsic spin Hall conductivity of Pd and Au

0.14 MB·English

by  G. Y. Guo

Ab initio calculation of intrinsic spin Hall conductivity of Pd and Au
G. Y. Guo∗
Department of Physics and Center for Theoretical Sciences,
National Taiwan University, Taipei 106, Taiwan
(Dated: January 21, 2009)
An ab initio relativistic band structure calculation of spin Hall conductivity (SHC) (σz ) in Pd
xy
andAumetalshasbeenperformed. Itisfoundthatatlowtemperatures,intrinsicSHCsforPdand
Au are, respectively, ∼ 1400(~/e)(Ωcm)−1 and ∼ 400(~/e)(Ωcm)−1. The large SHC in Pd comes
9
from the resonant contribution from the spin-orbit splitting of the doubly degenerated 4d bands
0
near the Fermi level at symmetry Γ and X points, and the smaller SHC in Au is due to the broad
0
free electron like 6s6p-bands. However, as the temperature increases, the SHC in Pd decreases
2
monotonicallyandreducesto∼330(~/e)(Ωcm)−1 at300K,whiletheSHCinAuincreasessteadily

n and reaches ∼ 750(~/e)(Ωcm)−1 at room temperature. This indicates that the gigantic spin Hall
a effect [σz ≈ 105(~/e)(Ωcm)−1] observed recently in the Au/FePt system [T. Seki, et al., Nature
J Materialxsy7, 125 (2008)] is due to the extrinsic mechanisms such as the skew scattering by the

1 impurities in Au.
2
PACSnumbers: 71.15.Rf,72.15.Eb,72.25.Ba
]
l
l
a
Spin transport electronics (spintronics) has recently acts as the Yang-monopole, which enhances the SU(2)
h
- become a very active research field in condensed mat- non-Abelian Berry curvature [14].
s
ter and materials physics because of its potential appli- On the other hand, the SHE in metallic systems is
e
m cations in information storage and processing and other currently attracting interest, stimulated by latest exper-
electronic technologies [1] and also because of many fun- imental reports on the SHE or inverse spin Hall effect
.
at damental questions on the physics of electron spin [2]. (ISHE), i.e., the transverse voltage drop due to the spin
m Spin current generation is an important issue in the current [15, 16, 17]. The spin Hall conductivity (SHC)
emerging spintronics. Recent proposals of the intrinsic in metals is much larger than that in the semiconduc-
-
d spin Hall effect (SHE) are therefore remarkable [3, 4]. tors. Naively, this may be due to the large number of
n
In the SHE, a transverse spin current is generated in carriers, but the band structure is very important.[18]
o
response to an electric field in a metal with relativis- Furthermore,theFermidegeneracytemperatureismuch
c
[ tic electron interaction (spin-orbit coupling). This effect higher than room temperature, and hence the quantum

wasfirstconsideredtoariseextrinsically,i.e.,byimpurity coherence is more robust against the thermal agitations
1
scattering[5, 6]. The scatteringbecomesspin-dependent compared with the semiconductors systems.
v
7 in the presence of spin-orbit coupling (SOC), and this In recent experiments for metallic systems[15, 16, 17],
8 givesrisetotheSHE.Intherecentproposals,incontrast, platinumshowsthe prominentSHEsurvivingevenupto
2 the spin Hall effect can arise intrinsically in hole-doped room temperature, whereas aluminum and copper show
3
(p-type) bulk semiconductors [3] and also in electron- relativelytinySHE.However,themechanismoftheSHE
.
1 doped (n-type) semiconductor heterostructures [4] due in metals has been assumed to be extrinsic. In Ref. [16]
0 to intrinsic SOC in the band structure. The intrinsic this difference is attributed to a magnitude of spin-orbit
9
SHE can be calculated and controlled, whereas the ex- coupling for each metal. However, platinum seems to
0
: trinsicSHEdependssensitivelyondetailsoftheimpurity be special even among heavy elements. This material
v scattering. Therefore, the intrinsic SHE is more impor- dependence strongly suggests a crucial role of intrinsic
i
X tant for applicational purposes, in comparison with the contributions. Thereforeit is highly desiredto study the
r extrinsic SHE. intrinsic SHE of platinum as a representative material
a
In semiconductors, there have been experimental re- for metallic SHE. If this analysis successfully explains
ports on the SHE in n-type GaAs [7], p-type GaAs [8] the experiment, it will open up the possibility to theo-
andn-type InGaN/GaNsuperlattices[9]in recentyears. retically design the SHE in metallic systems. Therefore,
These experimental results have been discussed theoret- werecentlycarriedoutab initiocalculationsfortheSHC
ically, and it is now recognized that the SHE in n-type in platinum.[18] We found that the intrinsic SHC is as
GaAs is due to the extrinsic mechanisms,i.e., skew scat- large as ∼ 2000(~/e)(Ωcm)−1 at low temperature, and
teringandside-jumpcontributions[10,11],while thatin decreases down to ∼ 200(~/e)(Ωcm)−1 at room temper-
p-typeGaAsismainlycausedbytheintrinsicmechanism ature. It is due to the resonant contribution from the
[12,13]. Thisconclusionissupportedbytheoreticalanal- spin-orbitsplitting ofthe doublydegeneratedd-bandsat
ysisoftheimpurityeffectandvertexcorrectiontoSHEin high-symmetryLandX pointsnearthe Fermilevel. We
theRashbaandLuttingermodels[10,12]. Inp-GaAsthe showed, by modeling these near-degeneracies by an ef-
fourfold degeneracy at the Γ-point of the valence bands fective Hamiltonian, that the SHC has a peak near the
2
Fermi energy and that the vertex correction due to im- andfinallybecomesverysmallbelow−5.0eV.Notethat
purity scatteringvanishes,indicatingthat the largeSHE the bands below −5.5 eV and also above 0.5 eV are pre-
observedexperimentallyinplatinum[16]isofintrinsicna- dominantly of 5s character and the effect of the SOC is
ture. small.
Inthispaper,westudytheintrinsicSHEinPdandAu
metals with first-principles band structure calculations.
The band structures of Pd and Au have been calculated 2 2
using a fully relativistic extension [19] of the all-electron (a) (b)
linearmuffin-tinorbitalmethod[20]basedonthedensity 0 0
fcc Au
functional theory with local density approximation [21].
The experimental lattice constants for Pd and Au used -2 -2
are 3.89 and 4.08 ˚A, respectively. The basis functions eV)
usedares,p,dandf muffin-tinorbitals[20]. Intheself- gy ( -4 -4
consistent electronic structure calculations, 89 k-points ner
E -6 -6
in the fcc irreducible wedge (IW) of the BZ were used in
thetetrahedronBZintegration. TheSHCisevaluatedby
-8 -8
the Kubo formula, as described in Ref. 22. A fine mesh
of60288k-pointsonalargerIW(threetimesthefccIW) -10
-10
is used. Thesenumbers correspondto the divisionofthe W L Γ X W Γ -2000 0 2000
ΓX line into 60 segments (see Fig. 1). σ z (Ωcm)-1
xy
(a) (b) FIG.2: (coloronline) (a)Relativisticbandstructureand(b)
2 SOC 2
spin Hall conductivity of fcc Au. The zero energy and the
noSOC
dotted line is the Fermilevel.
0 0
V)
e
y ( -2 -2
g We notice thata peak in the SHC appearsatthe dou-
er
n ble degeneracies on the L and X points near E in the
E -4 -4 F
fcc Pd scalar-relativisitcband structure (i.e., without the SOC)
while the other peak at −3.0 eV occurs near the double
-6 -6
degeneraciesattheLandΓpoints(seeFig.1). Thedou-
ble degeneracy (bands 5 and 6) at L is made mostly of
-8 -8
W L Γ X W Γ -2000 0 2000 dx′z′ and dy′z′ (z′: threefold axis), being consistent with
σxyz (Ωcm)-1 thepointgroupD3d atL. Thedoubledegeneracy(bands
4and5)atX consistsmainlyofdx′z′ anddy′z′ (z′: four-
FIG.1: (coloronline) (a)Relativisticbandstructureand(b) fold axis), being consistent with the point group D4h.
spin Hall conductivity of fcc Pd. The zero energy and the These double degeneracies are lifted by the SOC, with a
dotted line is the Fermi level. The dashed curves in (a) are
ratherlargespin-orbitsplitting. As inPt,the largeSHC
thescalar-relativistic band structure.
inPdmaybeattributedtothesedoubledegeneracies.[18]
Fig. 1 shows the relativistic band structure of Pd, TherelativisticbandstructureofAu,andalsotheSHC
and also the SHC (σxzy) as a function of EF. Clearly, (σxzy) as a function of EF are displayed in Fig. 2. It
the SHC peaks just below at the true Fermi level (0 is clear that both the shape and amplitude of the SHC
eV), with a large value of ∼1400 (~/e)(Ωcm)−1. This versus E curve (Fig. 2b) of Au are very similar to that
F
large value of the SHC is smaller than that of Pt [18]. of Pd (Fig. 1b) and Pt [18]. This is because the band
Nonetheless, it is still orders of magnitude larger than structure of Au (Fig. 2a) is rather similar to that of
thecorrespondingvalueinp-typesemiconductorsSi,Ge, Pd (Fig. 1a) and Pt[18]. However, because Au has an
GaAs and AlAs [22, 23]. Note that the SHC in Pd de- extravalenceelectronandhenceitsd-bandiscompletely
creasesmonotonically as the EF is artificially raisedand filled, the EF falls in the broad 6s6p band where the
becomes rather small above 3.0 eV. When the EF is ar- SOC is relatively small. As a result, the SHC in Au is
tificially lowered, the SHC first peaks just below the EF relatively small at low temperatures [σxy(T = 0K) ≈
(-0.3 eV), then decreases considerably, and changes its 400(~/e)(Ωcm)−1],beingconsistentwiththe previousab
sign at −1.2 eV. As the EF is further lowered, the SHC initiocalculation[23]. Nonetheless,itisafewtimeslarger
increases in magnitude again, and becomes peaked at than the SHC in semiconductors [22, 23].
−3.0eVwith alargevalue of−2600(~/e)(Ωcm)−1. The
SHC decreases again when the E is further lowered, The SHC can be written in terms of Berry curvature
F
3
investigation.[25]
2000
The author thanks N. Nagaosa, S. Murakami, T.-W.
fcc Pd
ChenandS.Maekawaforstimulatingdiscussionsandcol-
-1m) 1500 fcc Au laborationon spin Hall effect in metals. The author also
c
Ω thanks National Science Council of Taiwan for support,
zσ (xy1000 and also NCHC of Taiwan for the CPU time.
500
0
0 100 200 300
T (K) ∗ Electronic address: [email protected]
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