Mechanical generation of spin current by spin-rotation coupling

0.43 MB·English

by  Mamoru Matsuo

Mechanical generation of spin current by spin-rotation coupling
Mamoru Matsuo1,2, Jun’ichi Ieda1,2, Kazuya Harii1,2, Eiji Saitoh1,2,3,4, and Sadamichi Maekawa1,2
1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan
3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
4WPI, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
(Dated: January 17, 2013)
Spin-rotation coupling, which is responsible for angular momentum conversion between the elec-
tronspinandrotationaldeformationsofelasticmedia,isexploitedforgeneratingspincurrent. This
method requires neither magnetic moments nor spin-orbit interaction. The spin current generated
3 innonmagnetsiscalculatedinpresenceofsurfaceacousticwaves. Wesolvethespindiffusionequa-
1 tion,extendedtoincludespin-rotationcoupling,andfindthatlargerspincurrentscanbeobtained
0 inmaterialswithlongerspinlifetimes. Spinaccumulationinducedonthesurfaceispredictedtobe
2 detectable by time-resolved Kerr spectroscopy.

n
a PACSnumbers: 72.25.-b,85.75.-d,71.70.Ej,62.25.-g
J

6
Introduction.— Spincurrent, aflowofspins, isakey
1
concept in the field of spintronics[1, 2]. It can be gen-

] erated from non-equilibrium spin states, i.e., spin accu-
l
l mulation and spin dynamics. The former is routinely
a
h produced in nonlocal spin valves[3]. In ferromagnets,
- the latter is excited by ferromagnetic resonance[4], tem-
s
perature gradient[5], and sound waves in the magnetic
e
m insulator[6]. Alternatively, spin currents in nonmagnets
havebeengeneratedbythespinHalleffect[7],inwhicha
.
t
a strong spin-orbit interaction (SOI) is utilized. All these
m existingmethodsrelyonexchangecouplingofspinswith
- local magnetization or on SOI.
d In this Letter, we pursue a new route for generating
n
spin currents by considering spin-rotation coupling[8]:
o
c ¯h
[ HS =−2σ·Ω, (1)

1 where ¯hσ/2 is the electron spin angular momentum and
v
Ω is the mechanical rotation frequency. The method re-
6
quires neither magnetic moments nor SOI. In this sense,
9
5 the mechanism proposed here is particularly relevant in
3 nonmagnets with longer spin lifetime.
. Nonuniform rotational motion—. Here, we consider
1
0 rotational motion of the lattice:
FIG. 1. Snapshot of mechanical generation of spin current
3 1
induced by SAW. (a) In presence of a SAW propagating in
1 Ω= ∇×u˙, (2)
2 thex-direction,agradientofmechanicalrotationaroundthe
:
v whereuisthedisplacementvectorofthelattice[9]. When z-axisisinduced. Therotationcouplestoelectronspins,and
i thenthez-polarizedspincurrentflowsinthey-direction. (b)
X the lattice vibration has transverse modes, Eq. (2) does
Spinaccumulationinducedonthesurface. Becausespinsare
not vanish. In such a case, the mechanical angular mo-
r polarized parallel to the rotation axis (±z), the striped pat-
a mentumofthelatticecanbeconvertedintospinangular
tern of spin accumulation arises at the surface.
momentum via H [10]. However, as shown later, that
S
Ω is finite is insufficient to generate spin currents elas-
tically. Both the time derivative and the gradient of ro-
spin diffusion equation to include the coupling between
tational modes are necessary for the generation of spin
elasticrotationandspin. Bysolvingtheequationinpres-
current. For this purpose, we focus on surface acous-
ence of SAW, we can evaluate the induced spin current
tic waves (SAWs), which induce rotational deformations
for metals and semiconductors.
that vary in space and time (Fig. 1).
In presence of SAW, a gradient of mechanical rotation Spin diffusion equation with spin-rotation coupling.—
is induced in the attenuation direction. We extend the First of all, we examine effects of spin-rotation coupling
2
onspindensity. Whenthemechanicalrotation,Ω,whose theratioofvelocitiesas(c /c )2 =(1−2ν)/2(1−ν),and
t l
axis is in the z-direction, is applied, the electron spins ν and ξ are related as ξ ≈(0.875+1.12ν)/(1+ν).
alignparalleltotheaxisofrotation. Thisisknownasthe SpinaccumulationgeneratedbytheSAWcanbeeval-
Barnetteffect[11]. Inthiscase, thebottomoftheenergy uated by solving Eq. (5). By inserting Eq. (7) and
band of the electron is shifted by ¯hΩ/2. The number δµ = δµ (y,t)eikx into Eq. (5), the spin diffusion equa-
y
density of up(down) spin electrons is then given by tion can be rewritten as
(cid:90) µ↑(↓) (cid:0)∂ −D∂2+τ˜−1(cid:1)δµ (y,t)=−iω¯hΩ e−kty−iωt, (8)
n = dεN (ε), (3) t y sf y 0
↑(↓) 0
±h¯Ω/2 whereτ˜ =τ (1+λ2k2)−1 withthespindiffusionlength
√sf sf s
whereN isthedensityofstatesforelectrons,andµ and λ = Dτ and Ω = ω2u /2c . With the boundary
0 ↑ s sf 0 0 t
µ arechemicalpotentialsforupanddownspinelectrons, condition ∂ δµ = 0 on the surface y = 0, the solution is
↓ y
respectively. The z-direction is selected as the quantiza- given by
tion axis. Then, spin density can be estimated as
(cid:90) ∞ (cid:90) ∞ θ(t−t(cid:48))e−(t−t(cid:48))/τ˜sf
δµ (y,t)=−iω¯hΩ dt(cid:48) dy(cid:48)
n↑−n↓ ≈N0(δµ−¯hΩ), (4) y 0 0 0 (cid:112)4πD(t−t(cid:48))
where δµ = µ↑ −µ↓ is spin accumulation. Here, a con- ×(e−4(Dy−(ty−(cid:48)t)2(cid:48)) +e−4(Dy+(ty−(cid:48))t2(cid:48)))e−kty(cid:48)−iωt(cid:48). (9)
stant density of state is assumed for simplicity. Spin re-
laxation occurs in two processes: one is on-site spin flip Here, let us consider the time evolution of spin accu-
with the spin lifetime, τ , and the other is spin diffusion mulation at the surface, y =0. Because each spin aligns
sf
with thediffusion constant, D. Equatingthese processes parallel to the rotation axis, i.e., the ±z-axis, a striped
leadsto∂ (n −n )=τ−1N δµ+D∇2(N δµ). Next,we pattern of spin accumulation [shown in Fig. 1 (b)] arises
t ↑ ↓ sf 0 0
obtain the extended spin diffusion equation in presence at the surface. The period of spatial pattern is the same
of spin-rotation coupling: as the wavelength of SAW, 2π/k.
Recently, spin precession controlled by SAW was ob-
(cid:0)∂ −D∇2+τ−1(cid:1)δµ=h¯∂ Ω, (5)
t sf t served by using the time-resolved polar megneto-optic
Kerr effect (MOKE)[12, 13]. In our case, in-plane spin
The R.H.S. of Eq. (5) is a source term originating from
polarization is induced. Therefore, transversal or longi-
spin-rotation coupling. Z-polarized spin current can be
tudinalMOKEcanbeusedtoobservepatternsshownin
calculated from the solution of Eq. (5) as
Fig. 1 (b).
σ
Jz = 0∇δµ, (6) Spin current from SAW.— From Eqs. (6) and (9)
s e we obtain z-polarized spin current in the y-direction. In
with conductivity σ . If the mechanical rotation is con- Fig. 2,theSAW-inducedspincurrentisshown. Thespin
0
stant with time, the source term vanishes. Moreover, current, Jsz, is plotted as a function of kty and ωt in Fig.
evenifmechanicalrotationdependsontime,theuniform 2(a). Thespincurrentoscillateswiththesamefrequency
rotation in space cannot generate spin currents because as that of the mechanical resonator, ω. The maximum
spin accumulation is independent of space. amplitude, JsMax, is found near the surface, kty ≈ 1. In
Spin accumulation induced by SAW.— Let us con- Fig. 2 (b), maximum amplitude scaled by ω3 is plotted
sidergenerationofspincurrentduetospin-rotationcou- as a function of ωτsf. The amplitude increases linearly
plingofSAWsinnonmagneticmetalsorsemiconductors. when ωτsf (cid:28)1, whereas it saturates when τsf (cid:29)ω−1. In
Our setup is shown in Fig. 1 (a). SAWs are generated otherwords,JsMax ∝ω4 forωτsf (cid:28)1whereasJsMax ∝ω3
in the xz-plane and penetrates a nonmagnetic material for ωτsf (cid:29)1.
along the y-direction. They then induce mechanical ro- To clarify material dependence of spin current, we use
tation around the z-axis, whose frequency Ω = (0,0,Ω) an asymptotic solution of Eq. (5) for kty (cid:29)1:
is given by[9]
iωτ˜
δµ≈ sf ¯hΩ e−kty+i(kx−ωt), (10)
Ω(x,y,t)= ω2u0 exp{−k y+i(kx−ωt)}, (7) iωτ˜sf +λ2skt2−1 0
2c t
t which leads to
where ω and u0 are the frequency and amplitude of −iωτ˜ ¯hσ ω3u (cid:112)1−ξ2
the mechanical resonator, k is wave number, c is the Jz ≈ sf 0 0 e−kty+i(kx−ωt)
t s iωτ˜ +λ2k2−1 2e c2 ξ
transverse sound velocity, and k is the transverse wave sf s t t
t
(11)
number. The frequency ω is related to the wave num-
ber as ω = ctkξ and the transverse wave number as where Ω =ω2u /2c .
k =k(cid:112)1−ξ2, where ξ satisfies the equation ξ6−8ξ4+ 0 0 t
t When spin relaxation is absent, ωτ˜ (cid:29)1, one obtains
8ξ3(3−2c2/c2)−16(1−c2/c2)=0 and c is the longitu- sf
t l t l l
dinal sound velocity. The Poisson ratio, ν, is related to δµ≈¯hΩ e−kty+i(kx−ωt) (12)
0
3
Let us examine the SAW-induced spin current in typ-
ical nonmagnetic materials. Using Eqs. (6) and (9), the
maximum value of the spin current for Al, Cu, Ag, Au,
and n-doped GaAs normalized by that of Pt, JMax,Pt, is
s
computed as listed in Table 1. The ratio of the maxi-
mum amplitude of the spin current to that of Pt, J¯, is
s
definedasJ¯ =JMax/JMax,Pt. Theratiodependsmainly
s s s
on the conductivity, σ , and spin lifetime, τ . The or-
0 sf
der, J¯Cu > J¯Al > J¯Ag > J¯Au > J¯Pt = 1, is unchanged,
s s s s s
since these materials well satisfy ωτ (cid:28) 1. For GaAs,
sf
theratio,J¯GaAs,isgreaterthan1forω <1GHz,whereas
s
it becomes smaller than 1 for ω >1GHz. This happens
because the spin lifetime of GaAs is much longer than
that of Pt; i,e., the dimensionless parameter, ωτ , be-
sf
comes much greater than 1 when ω > 1GHz. In such a
case, JMax/ω3 for GaAs saturates, whereas that for Pt
s
linearly increases, as shown in Fig. 2 (b).
It is worth noting that the spin current generated in a
metalwithweakspin-orbitinteractionsuchasthoseofAl
and Cu is much larger than that in Pt. In addition, the
spin current in n-doped GaAs is comparable to that in
Pt. Although conductivities of semiconductors are much
smaller than those of metals, the spin lifetime is much
longer. Hence,theamplitudeoftheinducedspincurrent
in GaAs is comparable to that in Pt for ωτGaAs <1.
sf
Recently,SAWsintheGHzfrequencyrangehavebeen
used for spin manipulation[13, 14]. Here, we evaluate
spin current at such high frequencies. In case of u =
0
10−9m,ω/2π =10GHz,Pthasthemaximumamplitude,
Jz,Pt ≈4×106A/m2.
s
Conventionally, generation of spin current in nonmag-
netic materials has required strong SOI because the spin
FIG. 2. (a) Spin current induced by SAW Jsz plotted as a Hall effect has been utilized. In other words, nonmag-
function of k y and ωt for fixed x and z. The spin current
t netic materials with short spin lifetimes have been used.
oscillateswithtime. Maximumamplitudeislocatednearthe
On the contrary, the mechanism proposed here requires
surface,k y≈1. (b)Maximumamplitudeofthespincurrent
t
scaledbyω3plottedasafunctionofωτ . Whenωτ (cid:28)1,the longer spin lifetimes to generate larger spin currents.
sf sf
scaled amplitude, JMaxω−3, increases linearly. On the other Therefore, more options are available for spin-current
s
hand,itsaturateswhenωτ (cid:29)1. Accordingly,themaximum generation in nonmagnets than ever before.
sf
amplitude, JMax, is proportional to ω4 in the former case, Enhancement of the SAW-induced spin current.—
s
whereas JsMax ∝ω3 in the latter case. Very recently, it has been predicted that spin-rotation
coupling can be enhanced by an interband mixing of
solids[20]:
and
¯h
H(cid:48) =−(1+δg) σ·Ω. (15)
S 2
Jz ≈−(σ /e)k ¯hΩ e−kty+i(kx−ωt). (13)
s 0 t 0
Here, δg is given by δg = g −g where g = 2 and g
0 0
are electron g factors in vacuum and solids, respectively.
When ωτ˜ (cid:28) 1 and λ k (cid:28) 1, the spin current be-
sf s t Considering enhancement, the mechanical rotation, Ω,
comes
inserted into the extended spin diffusion equation, Eq.
(5), is replaced by (1 + δg)Ω. Consequently, the spin
(cid:112)
¯hσ ω3u 1−ξ2
Jz ≈ωτ · 0 0 e−kty+i(kx−ωt+π/2).(14) accumulation, δµ, is modified as δµ → (1+δg)δµ, and
s sf 2e c2 ξ accordingly,theinducedspincurrentasJz →(1+δg)Jz.
t s s
For lightly doped n-InSb at low temperature, g ≈ −49
As seen in Eq. (14), the larger spin current can be has been employed in a recent experiment[21]. In this
obtained from materials with the longer spin lifetime, case, one obtains δg ≈−51. Therefore, the amplitude of
namely, weaker SOI. the spin current can be 50 times larger.
4
TABLE I. SAW-induced spin current for Pt, Al, Cu, Ag, Au, and GaAs. The ratio is given by J¯ = JMax/JMax,Pt, where
s s s
JMax,Pt is the maximum amplitude of the spin current for Pt. The ratio J¯ depends on the Poisson’s ratio, ν, the transverse
s s
velocity, c , conductivity, σ , and spin lifetime, τ .
t 0 sf
ν c [m/s] σ [107(Ωm)−1] τ [ps] J¯(0.1GHz) J¯(1GHz) J¯(2.5GHz) J¯(10GHz) Ref.
t 0 sf s s s s
Pt 0.377 1730 0.96 0.3 1 1 1 1 [15]
Al 0.345 3040 1.7 100 390 290 210 62 [16]
Cu 0.343 2270 7.0 42 950 700 650 330 [3]
Ag 0.367 1660 2.9 3.5 44 38 34 32 [17]
Au 0.44 1220 2.5 2.8 42 35 33 30 [18]
GaAs 0.31 2486 3.3×10−4 105 1.6 0.13 0.050 0.013 [19]
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