Charge current driven by spin dynamics in disordered Rashba spin-orbit system

0.18 MB·English

by  Jun-ichiro Ohe

Charge current driven by spin dynamics in disordered Rashba spin-orbit system
Jun-ichiro Ohe∗
I. Institut fu¨r Theoretische Physik, Universtit¨at Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
Akihito Takeuchi
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
Gen Tatara
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Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
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PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
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(Dated: February 1, 2008)
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n PumpingofchargecurrentbyspindynamicsinthepresenceoftheRashbaspin-orbitinteraction
a is theoretically studied. Considering disordered electron, the exchange coupling and spin-orbit
J interactions are treated perturbatively. It is found that dominant current induced by the spin

8 dynamics is interpreted as a consequence of the conversion from spin current via the inverse spin
2 Halleffect. Wealsofoundthatthecurrenthasanadditionalcomponentfromafictitiousconservative
field. Results are applied tothe case of moving domain wall.

]
l
l
a Recent spintronics studies aim at manipulation both of charge and spin degrees of freedom [1]. Central roles
h
are played by the spin-orbit interaction and the exchange interaction between the conduction electrons and local
-
s spins [2, 3, 4, 5, 6]. It has been shown that the exchange coupling is useful for electrical control of magnetization
e dynamics via spin transfer torque [2, 3]. It can also be used to pump spin current from the precession of the
m
magnetization [7, 8, 9]. The spin-orbit interaction has been recently found to induce a magnetism by the application
t. of electric voltage (the spin Hall effect) [10, 11, 12].
a
By combining the exchange and the spin-orbit interactions, various phenomena are expected, and the subject
m
discussedinthispaperis oneofthem; pumping ofchargecurrentbydynamicalmagnetization. The ideaisto convert
-
the pumped spin currentinto a chargecurrentby using of the spin-orbitinteractionas proposedby Saitohet al.[13].
d
n ThiscurrentduetotheinversespinHalleffectwasindeedobserved[13,14,15]inmetallicsystemswherethespin-orbit
o interaction is induced by Pt atom.
c Theoretically, generation of the electric field or voltage due to the dynamical spin structure was discussed by
[
Stern [16]. The field is not a real electric field, but an effective one due to a spin Berry phase acting only on charge

3 degrees of freedom with spin (like the electron). The mechanism is similar to the Faraday’s law, but a magnetic flux
v isreplacedbyafictitious fieldfromthe spinBerryphase. Theinducedfieldis describedby∇×E =−b˙,wherebis
eff
7
field of spin Berry phase, and current is divergenceless; ∇·j =0. The theory was recently applied to a domain wall
7
by Barnes and Maekawa [17], and the effect of the spin relaxation was studied by Duine [18], where the relaxation
2
0 was introduced by a phenomenologicalterm (β-term). All these studies have been done in the adiabatic limit, where
. the exchange coupling between the local spin and the conduction electrons is strong.
5
0 ChargecurrentasaresultofinversespinHalleffectwastheoreticallystudiedbyZhangandNiu[19]andHankiewicz
7 et al. [20] (the effect was called reciprocal spin Hall effect). The calculation was done as a response to applied spin-
0 dependent chemical potential, which would not be easy to control experimentally. In contrast, our study tries to
:
v derive direct relation between physically accessible quantities, current and magnetization (local spin).
i Another type ofa voltagegeneratedataninterface ofa ferromagnet-nonmagnetcontactwaspredicted by Wang et
X
al. [21] in order to explain the experimental observation [22]. They have pointed out that a net charging is due to a
r
a back flow of spin current at the interface, and the spin accumulation at the interface is essential.
In this letter, we theoretically predict another mechanism of a spin-induced charge battery realized in disordered
conductors. We consider the perturbative regime of the exchange coupling, that is in the opposite limit of adiabatic
cases [16, 17, 18]. Proposed mechanism does not rely on the interface, and the pumped current arises simply when
the exchange interaction and the spin-orbit interaction exist simultaneously.
Weconsiderthetwo-dimensionalelectrongassystem(2DEGs)withtheRashbaspin-orbitinteractionthatoriginates
fromthelackoftheinversionsymmetry[5,6,23]. Theconductionelectronscouplestothelocalmagneticmomentvia
theexchangecoupling. Suchasystemcanbeachievedbythe2DEGsattachedtotheferromagneticcontact[24,25,26],
or the 2DEGs in magnetic semiconductors (e.g., CdTe/CdMnTe) [27, 28].
Let us consider the current representation by the simple argument. The current j , proportional to the average
µ
trhk i of the electron wave vector with respect to electron states (tr is trace over spin indices), vanishes if the
µ
system is spatially symmetric. The Rashba interaction, proportional to (k×σˆ) , breaks the spatial symmetry, but
z
2
S
j
s-o
FIG.1: Diagrammatic representation ofcurrentsatthefirstorderin Jex andthelowest orderinΩ, whichturnsouttovanish.
DottedlinesandwavylinesdenotelocalspinsS andtheRashbainteraction,respectively,andthicklinerepresentsthediffusion
ladder, Dq.
charge current does not arise since trhk (k×σˆ) i = 0 (only spin current arises [7]). Charge current appears when
µ z
we introduce the exchange coupling, proportional to S · σˆ where S is a local spin. We would have the current
j ∝trhk (k×σˆ) S·σˆi∝ǫ Sν to the firstorderexchangeinteraction,andj ∝ǫ (S×S′)ν atthe secondoder
µ µ z µνz µ µνz
interaction with different spins, S and S′.
Inordertoobtainthe preciseexpressionofthecurrent,weperformanalyticalcalculationsbyusingadiagrammatic
technique. Although the argument in the previous paragraph gives the qualitative idea of pumping charge current,
it turns out that the linear term in S vanishes identically in the Rashba case, and the second order term needs a
dynamical part as S ×S˙. We will demonstrate that the pumped current has two components. One describes the
inversespinHalleffect[13],andtheotherisaconservativecurrentwhichwrittenasadivergenceofascalarpotential.
We will show that the current due to the inverse spin Hall effect is dominant in various cases, and the conservative
currentis relativelysmall. However,the conservativeoneis alsoimportantfromthe view ofthe fundamentalphysics,
because it provides the fictitious scalar potential which acts only on the particle having both charge and spin.
We consider a disordered 2DEGs with the Rashba spin-orbit interaction. The 2DEGs also interact with the local
spin, S (t), via the exchange coupling. The local spin is treated as classical, and slowly varying in space and time.
x
The system is represented by a Hamiltonian H = H +H (t)+H +H , where H ≡ ε c†c describes free
0 ex so imp 0 k k k k
electrons with εk =~2k2/2m (m being the effective mass). The exchange and the Rashba sPpin-orbit interactions are
given by
H (t) = −J S (Ω)eiΩtc† σˆc ,
ex ex q k+q k
kX,q,Ω
H = −α ǫ k (c†σˆνc ), (1)
so µνz µ k k
Xk
where J is the exchange coupling constant, S (Ω) denotes the Fourier transform of the local spin structure
ex q
and α is the strength of the Rashba spin-orbit interaction. Spin-independent disorder is represented by H =
imp
ni uei(k−k′)·ric† c , which gives rise an elastic electron lifetime τ =(2πρnu2/V)−1, where ρ is the density
i=1 k,k′ V k′ k i
Pof statPes, ni is the number of the impurities, u is the strength of the impurity scattering and V is the volume of the
system.
The charge current density of this system is given by
j (x,t) = ie ei(k−k′)·xtr (k+k′)µ −αǫ σˆν G< (t,t) . (2)
µ V (cid:20)(cid:18) 2m µνz (cid:19) k,k′ (cid:21)
kX,k′
G< (t,t′) is a lesser Green function which is a 2 × 2 matrix in spin space with components G< (t,t′) =
k,k′ kσ,k′σ′
ihc† (t′)c (t)i (σ,σ′ =±), where h···i is the expectation value estimated by the total Hamiltonian H.
k′σ′ kσ
WecalculatethecurrentbytreatingbothRashba(tothefirstorder)andexchangeinteractionsperturbatively,which
isvalidifJ τ ≪1andαk τ ≪1[29]. Successiveimpurityscatteringsaredenotedbyladderapproximation,resulting
ex F
inadiffusionpropagatoratsmallmomentumtransfer(q),D ≡(Dq2τ)−1,whereD ≡k2τ/2m2isadiffusionconstant.
q F
Dominant contributions are from diagrams that include a maximal number of diffusion propagators. Contributions
from the first order in J are shown in Fig. 1, that turn out to vanish identically. The leading contribution coming
ex
from the second order in J are shown in Fig. 2.
ex
3
(a) (b)
FIG. 2: Leading contribution from the second order in Jex. Contributions from (a) and (b) are denoted by jµφ and jµISH in
Eq. (4), respectively. Contributions from otherdiagrams cancel out or are smaller byO(1/εFτ).
After straightforwardcalculations, the current in the slowly varying limit (Ωτ ≪1) is obtained as
j (x,t) = 4eαJex2ǫ (niu2) e−iQ·xD (S ×S˙ )
µ iπmV νηz V q Q−q q η
Xq,Q
× (niu2)D D AµBνC +D Eµ Bν −D AµFν , (3)
V q Q Q q qQ q qQ q Q Q qQ
h i
where Aµ ≡ k gr ga , Bν ≡ k (gr)2ga , C ≡ Re grga ga , Eµ ≡
Q k µ k−Q2 k+Q2 q k ν k k+q qQ k k k+q k+Q qQ
P P P
iIm k+ Q grga ga and Fν ≡ Re (k+q) gr(ga )2ga (gr = (ga)∗ = (ε −ε + i )−1 (ε being
k 2 k k+q k+Q qQ k ν k k+q k+Q k k F k 2τ F
(cid:16) (cid:17)µ
thePFermienergy)). ThesecondandthirdtermsPinEq.(3)(proportionaltoEµ andFν ,respectively)arethesecond
qQ qQ
leading term with a long range limit (q,Q → 0). It is, however, physically the most essential term as we see below.
For a spatially smooth structure of spins, i.e., q,Q ≪ ℓ−1 (ℓ is the electron mean free path), we can approximate
Aµ ∼ 2πiρτmDQ , Bν ∼ 4πρτ3ε q , C ∼ −ρ/2ε2, Eµ ∼ −πiρτ2q and Fν ∼ πρτ3Q . Then, the current is
Q µ q F ν qQ F qQ µ qQ ν
obtained as [30] j (x,t)=jISH(x,t)+jφ(x,t), where
µ µ µ
3emαJ 2τ2 d2x
jISH(x,t) = − ex ǫ 1D (S ×S˙ ) ,
µ π µηzZ a2 x−x1 x x1 η
2eαJ 2τ3 ∂ d2x d2x ∂D(2)
jφ(x,t) = ex ǫ 1 2D x1−x2(S ×S˙ ) . (4)
µ π2 νηz∂xµ Z a2 Z a2 x−x1 ∂x1ν x1 x2 η
Here, D ≡ a2 e−iq·xD , D(2) ≡ a2 e−iq·x(D )2 anda is the lattice constant. (Note that singularbehaviorat
x V q q x V q q
q →0 is cut offPat q ∼L−1, where L is Pa system size.)
The first term in Eq. (4), jISH, describes a current whose direction is correlated with the magnetization direction,
perpendicular to S×S˙, and represents inverse spin Hall effect [13]. In order to make clear the physical meaning of
this current, we compare with the spin currents pumped by magnetization. In the absence of spin-orbit interaction,
we obtain the pumped spin current at the lowest order in J as
ex
J ε τ2 ∂ d2x d2x
j (x,t) = ex F 1D S˙ −2J τ 2D (S ×S˙ ) . (5)
sµ 2π ∂xµ Z a2 x−x1(cid:20) x1 ex Z a2 x1−x2 x1 x2 (cid:21)
Thepolarizationofthespincurrentisinbothdirections,∝hS˙iandhS×S˙i,whereh···idenotesaverageoverdiffusive
electron motion. This spin current is a gradient of a certain spin potential, j = −∇ φ . The result of Eq. (5) is
sµ µ s
consistent with the observation by Tserkovnyak et al. [8], where the spin current appears at the interface between
ferromagnetandnormalmetalsassociatedwiththephenomenologicalparameterofspin-mixingconductance. (Wenote
thatthereisalsoanequilibriumcomponentofspincurrent[31]. Thiscomponent,j (eq)(x)= Jex2 (∇ S )×S ,is
sµ 24π2ε2Fτ µ x x
freefromdiffusionpoles. Hence,itislocalandthereforesmallcomparedwithdynamicalcontributions.) Themeaning
of pumped spin current is understood by taking a divergence:
mJ d2x
∇·j (x,t) = − ex S˙ −2J τ 1D (S ×S˙ ) . (6)
s 2π (cid:20) x ex Z a2 x−x1 x x1 (cid:21)
4
FIG. 3: (Color online) Two typical geometries with ferromagnets attached to two-dimensional electron system. Left: Datta-
Das geometry [24] and Right: perpendicular geometry like in Ref. [13]. In both cases, inverse spin Hall current is given by
jµISH ∝ǫµνz(S×S˙)ν (µ=x,y).
Comparing the second term (∇·j (2)ν) to jISH, we see that jISH = γ ǫ (∇·j (2)ν), where γ = −3eατ. This
s µ ISH µνz s ISH
expression is the Rashba-version of inverse spin Hall effect, j ∝ j ×σˆ, proposed in Ref. [13]. (Note that the spin
s
current considered in Ref. [13] is the one flowing through the interface that enables the spin current to enter without
divergence.) Eq.(6)representsaconservationlawofspin,andcorrectlydescribesafactthattheGilbert-typedamping
(hS×S˙i) results in a flow of spin current or S˙.
In contrast, the second term in Eq. (4), jφ, is a gradient of a scalar quantity. It can be interpreted as a current
arisingfromapotentialoraconservedforce. Thefictitiouselectricfield,definedbyE =jφ/σ ,whereσ =e2nτ/m
ind 0 0
is Boltzmann conductivity, is written as E =−∇φ . The scalar potential is obtained as
ind ind
2αJ 2τ2 d2x d2x ∂D(2)
φ (x,t) = − ex ǫ 1 2D x1−x2(S ×S˙ ) . (7)
ind πeεF νηzZ a2 Z a2 x−x1 ∂x1ν x1 x2 η
(Note that these scalar potential and field are fictitious ones, acting only on charge having spin degrees of freedom.)
The current jφ is in the direction where magnetization changes. It contributes to the perpendicular current in the
Datta-Das spin transistor geometry [24]. However, it is not in-plane current in the layer geometry [13] as shown in
Fig. 3. It does not contribute to the case of moving domain walls as we will show below.
Let us apply our results to a case of a moving domain wall (as would be realized by using magnetic semiconduc-
tors [27, 28]). We define polar angles (θ,φ) with respect to the easy and hard axis of spin as cosθ =S /S,tanφ=
easy
S /S , where S , S and S denote spin components in easy-, hard- and medium-anisotropy directions.
hard mid easy hard mid
Rigid and one-dimensional (in the x-direction) domain wall solution is represented by two dynamical variables, X(t)
and φ(t) [32], as cosθ(x,t) = tanhx−X(t) and sinθ(x,t) = [coshx−X(t)]−1, where λ is wall thickness. By assuming
λ λ
a dirty case λ ≫ℓ (ℓ being the mean free path), and noting that Sx×S˙x′ vanishes if |x−x′|≫λ, we can approx-
imate Sx ×S˙x′ by local value as Sx ×S˙x ∼ sinθ(x,t)(φ˙eθ − Xλ˙eφ). Here, eθ = (cosθcosφ,cosθsinφ,−sinθ) and
e = (−sinφ,cosφ,0), indicating that the damping (S ×S˙) on the translational motion (X˙) is in the φ-direction,
φ
whileitiswithinthewallplanewhenφvaries. ThepumpedinversespinHallcurrent,jISH ∼−j ǫ hS×S˙i ,where
µ 0 µηz η
j ≡ 3emαJ 2τ2D (D ≡ L λdxD ) depends muchonthe wallgeometry. We considerthree types ofthe domain
0 π ex 0 0 a2 ℓ x
wallas showninFig.4, a NeelwRall(N) andtwoBlochwalls((B-i) and(B-ii)). By estimatinghS×S˙iinside the wall
(by using hcosθi=htanhxi=0 etc.), we obtain
λ
jISH(N) ∼−j X˙ sinφ, jISH(B−i) ∼j X˙ cosφ, (8)
x 0L x 0L
which are driven by translational motion, and
jISH(B−ii) ∼j λφ˙, (9)
x 0L
which is driven by tilt of the wall. In the case of the Neel and the out-of-plane Bloch wall, the current is a constant
(if wall velocity is constant) at small speed and shows an oscillation (sinφ or cosφ) when the domain wall is above
Walker’s break down. In contrast, no current is induced for the in-plane Bloch wall at small velocity and finite but
steady current arises above breakdown (as long as φ˙ is more or less constant). The gradient part of the current, jφ,
on the other hand, is averagedout for any type of the domain wall.
Let us briefly see the magnitude of current (Eqs. (8)(9)). We use ε = 10meV, J = 10meV, a = 10nm and
F ex
α = 0.3×10−11eVm [6]. Using D ∼ (L3/a2ℓ2λ)ln(L/λ) (our diffusive result depends much on sample size L), the
0
currentis estimatedas j ∼4×10−14[C/m]×L2λ2 lnL×Ω[Hz]. If we chooseΩ=100MHz, L=1µm andλ∼10a, we
a4 λ
obtain j ∼10[A/m], i.e., current is I ≡jL∼10µA, which would be detectable experimentally.
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FIG. 4: (Color online) Three domain walls in the xy-plane,Neel, Bloch-i and Bloch-ii.
We haveconsideredacaseofanuniformRashbainteraction. Ifweallowαtobe positiondependent, α(x), wehave
other contributions. For instance, charge current linear in S arises proportional to h(∇α)S˙i. Thus, various currents
are expected in the case of finite-size Rashba system (finite size is always the case in experiments).
Inconclusion,wehavetheoreticallyshownthatchargecurrentispumpedbymagnetizationdynamicsinthepresence
of the Rashba spin-orbit interaction. The dominant part was found to be due to the inverse spin Hall effect, i.e.,
conversion of spin current into charge current by spin-orbit interaction. In addition to the inverse spin Hall current,
we found a conservative current flowing basically along the gradient of the magnetization damping. This current
is rotation free, and should be distinguished from the inverse spin Hall current and from the divergenceless current
predicted by Stern and others [16, 17, 18]. It would be extremely interesting if one could experimentally determine
the type of the pumped current.
The authors are grateful to E. Saitoh, B. Kramer, R. Raimondi, M. Yamamoto, S. Kettemann, J. Shibata, H.
Kohno, S. Murakami, and T. Ohtsuki for valuable discussions. This work has been supported by the Deutsche
Forschungsgemeinschaftvia SFBs 508 and 668 of the Universit¨at Hamburg.
Noted added in proof. —After finishing the manuscript, we found optically induced inverse spin Hall effect was
observed in GaAs[33].
∗ Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan; Electronic address:
[email protected]
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6
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