Spin Hall effects due to phonon skew scattering

0.41 MB·English

by  Cosimo Gorini

Spin Hall effects due to phonon skew scattering
Cosimo Gorini
Institut fu¨r Theoretische Physik, Universita¨t Regensburg, 93040 Regensburg, Germany
Ulrich Eckern
Institut fu¨r Physik, Universita¨t Augsburg, 86135 Augsburg, Germany
Roberto Raimondi
Dipartimento di Matematica e Fisica, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy
5
1
A diversity of spin Hall effects in metallic systems is known to rely on Mott skew scattering.
0
In this work its high-temperature counterpart, phonon skew scattering, which is expected to be of
2
foremostexperimentalrelevance,isinvestigated. Inparticular,thephononskewscatteringspinHall

g conductivityisfoundtobepracticallyT-independentfortemperaturesabovetheDebyetemperature
u T . As a consequence, in Rashba-like systems a high-T linear behavior of the spin Hall angle
D
A demonstrates the dominance of extrinsic spin-orbit scattering only if the intrinsic spin splitting is
smaller than the temperature.

8
1
The spin Hall effect (SHE) [1–5] is the generation of a at T = 0, where they arise from electron scattering at
]
l transverse spin current by an applied electric field, the static impurities. In this case one has (explicitly in 2D)
l
a spin current polarization being perpendicular to both [21]:
h current and field directions. Indeed, a family or related
- effects exists [6]. The SHE and its inverse are routinely en(cid:18)λ(cid:19)2 (cid:18)λk (cid:19)2 en
s σsH = , σsH = F 2πN v τ (1)
e employed in spin injection or extraction experiments in sj,0 (cid:126) 2 ss,0 4 m 0 0 0
m
a variety of systems [7–13], and their potential for spin-
. tronics applications is becoming ever more evident [14]. with n the electron density, λ the effective Compton
t
a Acrucialissueisthedeterminationofthedominantspin- wavelength of extrinsic spin-orbit coupling, v the scat-
0
m
orbit mechanism responsible for such effects. In partic- tering amplitude, k the Fermi wave vector, N =
F 0
- ular, whether this is of intrinsic origin, i.e., connected m/2π(cid:126)2 the density of states, τ the elastic scattering
d 0
with the band and/or device structure or geometry, or time,ande>0theunitcharge. Equation(1)showsthat
n
o extrinsic, i.e., due to impurities. Spin-orbit phenomena the side-jump conductivity is independent of the scat-
c are typically complex in their own right, mixing charge tering mechanism (at least in simple parabolic bands),
[ and spin (magnetic) degrees of freedom in a plethora of whereas the skew scattering one is proportional to τ ,
0
2 ways, and standard experimental setups add to such a i.e., to the Drude conductivity σD = e2nτ0/m = −enµ
v complexity [15]. We will see that one of the main phe- (µ=−eτ /misthemobility). FromtheseT =0results,
0
2 nomenological arguments employed to discern the dom- theT (cid:54)=0spinHallconductivitybehaviorisextrapolated
9
inant spin-orbit mechanism misses the central aspect of arguing that the skew scattering conductivity should be-
0
dynamicalspin-orbitinteraction. Thelatterdescribesin- have as σsH ∝ µ, with the proportionality constant de-
3 ss
0 ter aliathedirectinteractionbetweentheelectrons’spin pending on microscopic details (impurity concentration,
. and phonons, and, though it will be the leading process k , etc.), but not on the temperature. Hence the argu-
1 F
0 at experimental temperatures T ≈ 300 K, it has been mentgoesasfollows[8,10,13,22,23]: (i)inhighmobil-
5 mostly neglected until now [16]. itysamplesskewscatteringshoulddominate,and(ii)the
1 spinHallsignalshouldscaleasthemobilitywithrespect
v: At T = 0 in metallic systems there are three main to its T-dependence. On the contrary, the same signal
extrinsic spin-orbit mechanisms: (i) side-jump [17], (ii)
i shouldbeT-independentinsampleswheretheside-jump
X skew scattering [18], and (iii) Elliott-Yafet spin relax-
mechanism is the leading one.
r ation [19]. When a charge current is driven through a
However, we will see that this simple and appealing
a sample, (i) and (ii) give rise to a transverse spin current
phenomenological extrapolation from T = 0 to T (cid:54)= 0
via the side-jump and skew scattering spin Hall conduc-
misses a critical feature of high-T skew scattering—
tivities, denoted σsH and σsH, respectively. Elliott-Yafet
sj ss which, following Ref. [13], we call “phonon skew scat-
spin relaxation is typically weak, but is needed to ensure
tering”. NamelythatfortemperaturesT (cid:38)T , withT
the proper analytical behavior of the full spin Hall con- D D
the Debye temperature, σsH does not scale as the mobil-
ductivityσsH whenalsointrinsicspin-orbitinteractionis ss
ityandratherbecomesT-independent. Sincetypicalspin
present—as is the case in thin films or two-dimensional
Hall experiments are performed at room temperature in
(2D) electron or hole gases [20].
“soft”metalssuchasAu(T =165K),Pt(T =240K)
D D
The above mechanisms have been extensively studied orTa(T =240K)[8,11,24],thismakesdistinguishing
D
2
between side-jump and skew scattering contributions an
even more complicated issue.
The Hamiltonian is (compare Ref. [16])
(a1) (a2)
λ2
H =H +δVph(r,t)− σ×∇δVph(r,t)·p+Hph, (2)
0 4(cid:126) 1
whereσisthevectorofPaulimatrices. HereH contains
0
the static electronic part of the Hamiltonian, in first- (b1) (b2) (b3)
quantized notation given by
p2 α λ2
Hel = − σ×zˆ·p+V (r)− σ×∇V (r)·p, (3)
0 2m (cid:126) imp 4(cid:126) imp
(c1) (c2) (c3)
as well as the standard harmonic phonon contribution:
H0 =H0el+H0ph. Thesecondtermonther.h.s.ofEq.(3) FIG. 1. (a) Self-energy in the standard self-consistent
is a Bychkov-Rashba-like intrinsic spin-orbit term [25], Born approximation for electron-impurity (a1) and electron-
whichappearsattheinterfacebetweentransitionmetals phonon (a2) scattering. (b) Diagrams describing skew-
andinsulators/vacuumwhereinversionsymmetryisbro- scatteringfromimpurities,and(c)diagramsdescribingskew-
scatteringfromphonons. Dashedandwigglylinesindicatethe
ken [26]. The potential from static impurities is denoted
impurity average and phonon propagator, respectively. The
V (r),andδVph(r,t)stands—classicallyspeaking—for
imp square box is the spin-orbit insertion due to both impurity
the time-dependent potential due to lattice vibrations at
and phonon potential.
T (cid:54)=0.
The actual calculations employ well-known quantum
fieldtheoreticaltechniques,seebelow. Hereweonlymen-
tion that it is convenient to introduce the phonon field Just as for the second order self-energies [Fig. 1(a)],
operator [27] there is a direct correspondence between the diagrams
due to impurities [Fig. 1(b)] and those due to phonon
(cid:114)
ϕˆ(r)=i(cid:88) vsk (cid:16)ˆb eik·r−c.c.(cid:17) , (4) scattering [Fig. 1(c)]. Such a correspondence appears
2V k in the high-temperature limit, where the phonon dy-
k
namics become irrelevant, roughly speaking ϕˆ(r,t) →
where ˆb and ˆb† are annihilation and creation operators ϕ(r) [29]. To illustrate this further, consider the dia-
k k
for longitudinal Debye phonons of momentum (cid:126)k, v is grams of Fig. 1(b), with the impurity potentials—before
s
thesoundvelocity,andV thevolume(ortheareain2D). averaging—replaced by the classical phonon field gϕ(r).
√
Notethatϕˆ(r)correspondstov ρtimesthedivergence The average is then performed using the equipartition
s
of the ionic displacement, where ρ is the ionic mass den- law
sity. As usual, the electron-phonon coupling constant
(cid:104)ϕ(r )ϕ(r )(cid:105) =k T δ(r −r ), (6)
will be denoted by g [27]. Finally, the anharmonic term 1 2 T B 1 2
(3-phonon processes) reads
where (cid:104)...(cid:105) denotes the classical average, and
T
Λ (cid:90)
Hph = drϕˆ3(r). (5) (cid:104)ϕ(r )ϕ(r )ϕ(r )(cid:105) =−Λ(k T)2δ(r −r )δ(r −r )
1 3! 1 2 3 T B 1 2 1 3
(7)
In its most general form, there appears a tensor arising which follows from expanding the Boltzmann factor to
fromthethirdderivativesofthecrystalpotentialwithre- first order with respect to the anharmonic term. In the
specttosmalldisplacements[28]. Forourpurposes,how- case of impurity scattering, the equivalent of the r.h.s.
ever, it is sufficient to characterize the anharmonicity by of Eq. (6), assuming “white-noise” disorder, is given by
thesingleparameterΛ,whichisrelatedtotheGru¨neisen n v2δ(r − r ), while the three-field average results in
i 0 1 2
parameterγ byΛ=−γ/ρ1/2v ;typicallyγ ≈2...3[28]. n v3δ(r −r )δ(r −r ). Thissuggeststhatonecanobtain
s i 0 1 2 1 3
The T = 0 processes, as well as the dynamical side- thehigh-T resultsthroughthefollowingcorrespondence:
jump and Elliott-Yafet, have been discussed in Ref. [16].
1
Inparticular,skewscatteringfromimpuritiesisdescribed niv02 →g2kBT = 2πN τ (8)
by the self-energy diagrams of Fig. 1(b), together with 0
1
the self-energy (a1) yielding the self-consistent Born ap- n v3 →−Λg3(k T)2 = (−k TgΛ). (9)
i 0 B 2πN τ B
proximation for the elastic scattering time. In order to 0
study finite (high) temperatures, the self-energy (a2) as Note that τ here denotes the T-dependent electron-
well as the skew-scattering from phonons via the self- phonon scattering time, in contrast to τ due to elas-
0
energy diagrams of Fig. 1(c) have to be taken into ac- tic scattering from impurities; −1/2τ corresponds to the
count. imaginarypartoftheretardedself-energyasderivedfrom
3
(a2), with the result given in (8) [28]. Using (8), (9) in low-T conductivities we thus obtain
Eq. (1), the skew scattering conductivity reads
(cid:126)Λ γ
∼ ∼0.1, (11)
gτ (cid:15) τ
F 0
(cid:18)λk (cid:19)2 en(cid:126)Λ where, to be explicit, we assumed (cid:15)Fτ0/(cid:126) ≈ 20. Note,
σsH =− F , (10) however, that there might be a sign change as a function
ss 4 m g
of temperature, depending on the nature of the impuri-
ties, i.e., the sign of v , as well as on the sign of g.
0
Aquantumfieldtheoretical(Keldysh)calculationcon-
which is, in particular, T-independent. Thus the T- firms the above results up to a numerical prefactor in
dependence of the spin Hall conductivity must inter- Eq. (9). Besides providing a solid basis for what has
polate between the two limiting expressions at low [see been obtained through simple and intuitive arguments,
Eq. (1)] and high [see Eq. (10)] temperature. we stress that such a calculation is necessary in order to
study the full temperature range 0<T <T .
D
Inordertocomparetheorderofmagnitudeofthetwo We now outline the Keldyh calculation in the high-
limits,weusethestandardrelationsvalidforanisotropic T regime, where the self-energy diagrams of Fig. 1(c)
metal, v ∼1/2N (screened Coulomb impurities), g2 ∼ acquire a transparent form [30]. In fact it is sufficient to
0 0
1/2N , ρv2 ∼N (cid:15)2. Fortheratiobetweenthehigh-and consider the first one (now (cid:126)=k =1):
0 s 0 F B
λ2 (cid:88) (cid:90)
(c1): ΣT =− Λg3 (cid:15) σ (−i∇G) (∇D) G D D D G . (12)
ss,13 4 ijk j 1 i 1 k 12 14 24 34 23
i,j,k 2,4
Here the G’s are SU(2)-covariant electron propagators acts only on the following G-function, and similarly for
[16, 31, 32], while the D’s are free phonon propagators, ∇D. After analytical continuation [33], the t-integrals
1
both defined on the Keldysh contour. The arguments, run from −∞ to +∞, and the Keldysh structure is car-
written as subscripts, include both space and time, e.g., riedbytheR,A,K propagatorcomponents. Inthehigh-
1=(r ,t ). Thenotation∇G indicatesthatthegradient T regime, T (cid:38) T , we use D< ≈ D> ≈ 1DK [34], with
1 1 1 D 2
the result
(c1): (cid:2)ΣT (cid:3)<(>) =−λ2 (cid:88)(cid:15) σ (−i∇G) (∇D) (cid:90) (cid:16)GR G<(>)+G<(>)GA(cid:17)D , (13)
ss,13 4 ijk j 1 i 1 k 12 23 12 23 123
i,j,k 2
where 0 → T > T correspondence for skew scattering thus
D
reads
Λg3 (cid:90)
D = [DR DK DK (14)
123 4 14 24 34 n v3 →−3Λg3(k T)2. (16)
4 i 0 B
+DK DR DK +DK DK DR].
14 24 34 14 24 34 This yields at once
Equation(13)hasthestandardformduetothecoupling (cid:18)λk (cid:19)2 en(cid:126)Λ
to an external field, whose role is here played by D. Ex- σsssH =−3 4F m g , (17)
ploiting the fact that the phonon frequencies (∼ω ) are
D
small compared to ω ∼ T, which physically means that whichisthecentralresultofourwork. Apartfromtheal-
electron-phonon scattering is elastic, we obtain ready mentioned factor of 3, it confirms the heuristically
obtained Eq. (10), and shows that the skew scattering
D ≈−3Λg3(k T)2, (15) conductivity at high temperatures does not scale as the
123 B
mobility, being rather T-independent.
havingrestoredherek foreasycomparisonwithEq.(9). We stress that the current interpretation of (inverse)
B
Theonlydifferencewiththelatterisafactorof3,missed spinHallexperimentsis,however,basedonthe“scaling-
by the simple introductory argument. The correct T = as-mobility”assumption[8,10,13,22,23]. Equation(17)
4
shows that a more careful analysis seems to be required,
and has important consequences for the spin Hall an- 0.20
gle θsH ≡ eσsH/σ . As shown in Fig. 2, the spin mo- 1
D 0.1
tion becomes diffusive for ∆ < (cid:126)/τ ∼ k T ∼ 10−2
B 0.01
eV and ballistic for ∆ > kBT, with ∆ = 2αkF the 0.15 0.001
intrinsic splitting. If a nonlinear or decreasing behav- 0.0001
ior of θsH is observed, we deduce that kBT > ∆ and u.)
(wσessajHke+r tσhssasHn)/t(hee/8inπt(cid:126)r)ins(cid:28)ic o1ne(se)x.tIrfi,nsoinc tehffeecottshearrehamndu,cha H (a.0.10
sθ
linearbehaviorisobserved,noconclusioncanbereached
by simply looking at the T-dependence, since there are
0.05
two possibilities: (i) k T > ∆ and the extrinsic and in- Diffusive
B
trinsic effects are comparable (light blue curves in Fig.
Ballistic
2); (ii) k T < ∆ and nothing can be said about the
B 0.00
relative strength of extrinsic and intrinsic mechanisms 0 5 10 15 20 25
k T/∆
B
(all curves, i.e., for different parameter values, look the
same). FIG.2. QualitativeplotoftheT >T spinHallangleθsH as
D
The relative importance of phonon vs. impurity skew afunctionofk T,measuredinunitsoftheintrinsicspin-orbit
B
scatteringisobtainedbycomparingtheself-energies(b1) splitting, for the paradigmatic case of a Rashba-like system.
and (c1), yielding The spin Hall conductivity is given by Eq. (19), and we set
λ/λ ≈ 10−1 [16]. Darker (lighter) curves are for weaker
F
ΣTss/Σ0ss ∼−γ(τ0/τ)(kBT/(cid:15)F). (18) 1(s0t−ro4n.g.e.r1). extrinsic conductivities, (σssjH + σsssH)/(e/8π(cid:126)) =
Inametalatroomtemperaturewehavek T/(cid:15) ∼10−2,
B F
setting as threshold for the dominance of phonon skew
scattering τ (cid:38)102τ.
0 term(5)washandledviaan“s”-waveapproximation,ig-
In general, the T = 0 → T > T correspondence lets
D noring the tensor structure of Λ as well as any details
us immediately turn known T =0 results into their T >
of the generally anisotropic phonon-phonon coupling:
T counterparts. For example, the full expression for
D these, however, are not expected to qualitatively modify
the high-T spin Hall conductivity and current-induced
ourconclusionsconcerningthe T-dependence. Thesame
spin polarization [35] due to intrinsic Bychkov-Rashba
is true when other phonon modes are included, provided
couplingandextrinsicdynamicalspin-orbitinteractionis their typical frequencies are smaller than k T/(cid:126).
B
structurallyidenticaltotheT =0expressionsappearing Second,φ4 (andhigher)anharmonicities,formallynec-
inRef.[32]. Explicitlyfora2Dhomogeneousbulksystem
essary to stabilize the system, could also be considered.
These have their T = 0 parallel in the T-matrix resum-
σsH = 1 (cid:0)σsH+σsH+σsH(cid:1) (19) mation of skew scattering. However, whereas the latter
1+τ /τ int sj ss
s DP does not add qualitative new features to the physics de-
scribedbythediagramsofFig.1(b),higheranharmonici-
where σsH =(e/8π(cid:126))(2τ/τ ) is the intrinsic part of the
int DP tiescould. Roughlyspeaking,anyadditionalphononline
spin Hall conductivity, and
connected to the anharmonic vertex in the diagrams of
1 1 (cid:18)λk (cid:19)4 1 1 (∆τ/(cid:126))2 Fig. 1(c) should contribute a further kBT factor in the
= F , = (20) T > T regime, as well as modifying the prefactor of
τ τ 2 τ 2τ [(∆τ/(cid:126))2+1] D
s DP “3” missed by the simple introductory arguments. This
would further increase the importance of phonon skew
are, respectively, the Elliott-Yafet and Dyakonov-Perel
scattering at high T’s, possibly implying a T-behavior of
spin relaxation rates. Furthermore, the current-induced
σsH opposite to that of the mobility. Indeed, it would
spin polarization “conductivity”, P, is given by ss
be highly desirable to develop a more detailed theory of
P =−2mα 1 (cid:0)σsH+σsH+σsH(cid:1) , (21) phonon scattering, in analogy with the T =0 treatment
(cid:126)2 1/τ +1/τ int sj ss by Fert and Levy [40], as well as to elucidate the role of
s DP
Umklapp processes.
This phenomenon, together with its inverse [36–38], is Third, band nonparabolicities could be relevant since
intimately related to the spin Hall effect [38, 39] and can theymodify,inparticular,theside-jumpmechanism,and
be similarly exploited for spin-to-charge conversion [36– thus possibly its T-dependence. Finally, and probably
38]. most importantly, the intermediate temperature regime,
Weconcludebydiscussingfutureperspectivesandcer- 0<T <T ,needstobeproperlyinvestigated. Westress
D
tain limitations of our approach. First, the anharmonic that our Keldysh approach gives an expression for the
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