Table Of ContentQuantum Kinetic Theory of Current-Induced Torques in Rashba Ferromagnets
D. A. Pesin and A. H. MacDonald
Department of Physics, The University of Texas at Austin, Austin TX 78712 USA
(Dated: January 5, 2012)
Motivated by recent experimental studies of thin-film devices containing a single ferromagnetic
layer, we develop a quantum kinetic theory of current-induced magnetic torques in Rashba-model
ferromagnets. We find that current-induced spin-densities that are responsible for the switching
behavior are due most essentially to spin-dependent quasiparticle lifetimes and derive analytic ex-
2 pressions for relevant limits of a simple model. Quantitative model parameter estimates suggest
1 that spin-orbit coupling in the adjacent metal normal magnetic layer plays an essential role in the
0 strength of the switching effect.
2
PACSnumbers: 75.60.Jk,72.25.-b
n
a
J
4
Introduction—The central goal of spintronics is the dis-
covery of efficient mechanisms for electrical control of
]
l nanomagnet orientation. The past fifteen years have
l
a witnessed many advances (see reviews [1–3], and refer-
h
ences therein) that are potentially important for infor-
-
s mation storage technologies and, because they depend
e
on improved understanding of non-equilibrium collective
m
properties, also interesting from a fundamental point of
.
t view. A breakthrough has recently been achieved with
a
thedemonstration[4,5]ofreliableelectrically-controlled
m
magnetization switching in trilayer thin-film devices, il-
-
d lustrated schematically in Fig. 1, that have a single FIG. 1: (color online) Schematic view of trilayer devices for
n perpendicular-anisotropy magnetic layer. (Magnetiza- electrical magnetic switching. A perpendicular-anisotropy
o tioncontrolwithweakcurrentinducedtorqueshadbeen ferromagnetic layer (Co) is sandwiched between a metallic
c observedpreviously[6]in(Ga,Mn)As.) Inthedemonstra- layer(forexamplePt)andaninsulating(forexampleAl2O3)
[
capping layer. The transport current, j , flows in the itiner-
tion devices the ferromagnetic layer was Co, the normal tr
1 metal layer Pt, and the insulator Al O . (Ref. 5 also antferromagnetandnormalmetalinthedirectionparallelor
v 2 3 antiparallel to the external magnetic field, Bext.
studiedotherinsulatorstopartiallydemonstratetheuni-
0
versality of the effect.)
9
9 Although they appear to be observing the same effect,
0
Refs. [4] and [5] ascribe the current induced torques to therein). ForRashbaferromagnetswithzˆdirectioninver-
.
1 different mechanisms. Ref. [4] speculates that the effect sion symmetry breaking, out-of plane spin-densities can
0
isduetoRashbaspin-orbitinteractionsofcarriersinthe be explained [8] only by a full solution of the two-band
2
ferromagnet and that it is analogous to the surprising quantum kinetic equation. With this motivation we re-
1
: semiconductor two-dimensional electron gases (2DEGs) port on a full quantum kinetic theory of current-induced
v
zˆdirectionspin-densitiesinducedbycurrentflowparallel torques in a ferromagnetic conductor with structural in-
i
X to an applied magnetic field [7, 8]. On the other hand, version asymmetry.
r Ref. [5] argued that the torques are due to a large spin Rashba Ferromagnets— Quantitative microscopic theo-
a Hallcurrent inPtthatflows verticallyintothe magnetic
ries of transport in transition metal ferromagnets are
layer. The two effects are identical from the symmetry
complicated by the presence at the Fermi energy of sev-
point of view, as already observed in Ref. [4].
eral 3d- and 4s-derived electronic bands that have com-
The spin-Hall-effect point of view has been elaborated plicated Fermi surfaces. Most transport properties of
on rather extensively in Ref. [5]. The model we study in collinear ferromagnetic conductors, including the impor-
thispaperisthereforemostcloselyrelatedtotheRashba tantgiantandtunnelmagnetoresistanceeffects,cannev-
effectinterpretation. Ingeneral,current-inducedtorques erthelessbeunderstoodqualitativelyintermsofasimple
in ferromagnets have for several years been recognized two-channelmodel[13–16]inwhich↑-spinand↓-spincar-
theoretically as a general consequence of spin-orbit cou- riers conduct in parallel with conductances that depend
pling(seeRefs.[9–11],aswellasreview[3],andreferences stronglyonthemagnetizationdirectionanditsspacede-
2
where the index i labels the impurity positions r , P0 =
i ±
pendence. The direction dependence is due to a combi- (1±ß·m)/2areprojectorsontothelocaldirectionofthe
nation of spin-split band structure and spin-dependent exchange field, and u and u characterize the strengths
↑ ↓
scattering effects. We will see that the spin-dependent of majority and minority spin scattering, respectively.
transport is also important for the present problem. We Because majority-spin electrons have a higher mobility
take the point of view that qualitative insights can be we must chose u > u . Our model neglects inelastic
↓ ↑
achieved by studying a simplified model that accounts phononandmagnonscatteringwhichshouldberelatively
for the interplay between structural inversion asymme- unimportant in highly disordered ultrathin films even at
try and spin-dependent transport. room temperature. Below we refer to the high-velocity
Cobalt is a strong ferromagnet [17] in which the electrons described by the Rashba ferromagnet model as
majority-spin Fermi surface intersects only a high- the transport electrons, and to the spin-density of the
velocity free-electron-like (s) band with a substantial d-electrons as the magnetization.
conductance, while the minority-spin Fermi surface in- Current Induced Torques— Because the dynamics of the
tersects several complex bands with generally small ve- itinerantelectronsinmagneticconductorsismuchfaster
locities, large densities-of-states, and dominant atomic- thanthatofthemagnetization,itissufficienttoconsider
like d-character. Minority-spin states at the Fermi en- the kinetics of transport electrons in a static exchange
ergyhavestronglyhybridizedfree-electron-likeanddam- field.
plitudes. Even high-velocity minority-spin states con- The effective magnetic field exerted on the magnetiza-
tribute less effectively to transport because they scat- tionbythetransportelectronsisduetotheirmutualex-
ter rapidly into the many available d-states. To cap- changeinteraction[10]andthereforeproportionalto∆ :
xc
ture the essential physics we employ a phenomenolog-
ical two-band model with short-lifetime minority spins 1 δ(cid:104)H(cid:105) ∆
H =− s =− xc (cid:104)ˆs(cid:105) . (3)
and long-lifetime majority spins, neglecting the d-orbital eff M δm M s
S S
contribution to transport as is customary in theories of
GMR [13–16]. Scattering is spin-dependent because any The subscript ‘s’ on the angle brackets indicates the
crystaldefectwillproducedifferentpotentialsformajor- expected value in the transport steady state. Here
ityandminorityelectronswhenstrongexchangeistaken ˆs is the transport electron spin-density operator ˆs =
into account, and because minority s-electrons can scat- (cid:126)Ψ†σßσσ(cid:48)Ψσ(cid:48)/2 and MS is the d-electron spin-density.
ter into empty minority d-electron states. As is common Thetheoryofcurrentinducedtorquesisthereforeequiv-
inmodelstudies,weaddspin-orbitcouplingandvertical alent to a theory of (cid:104)ˆs(cid:105)s. The required transport theory
structural inversion asymmetry to the two-band model differs essentially from the 2DEG problem considered in
by adding a Rashba SO term to the Hamiltonian. Ref.[8],sincethespindependenceoftransportcomespri-
These considerations lead to a Rashba-model ferro- marily from the spin-dependence of impurity scattering
magnet with band Hamiltonian: in an itinerant ferromagnet and the presence of a minor-
ity Fermi surface for d-like electronic bands, rather than
(cid:90) (cid:18) 1 (cid:19)
H = drΨ† (cid:15)ˆ − B ß+Uˆ Ψ . (1) solelyfromband-structurespin-splittingandangularde-
s σ p 2 p dis σ
pendence of scattering as in a 2DEG [8]. (Below we will
Here(cid:15)ˆ denotestheoperatorforthebandenergyofspin- alsopresenttheleading-orderresultsforthelattercase.)
p
orbit and exchange uncoupled itinerant electrons. Be- The component of (cid:104)ˆs(cid:105)s perpendicular to the magnetiza-
cause the Co layer is thick enough (∼ 10˚A) to support tion, which is responsible for magnetization switching in
several 2D subbands, we take p to be a 3D vector. The the devices of interest, depends essentially on current-
momentum-dependent effective magnetic field includes induced interband coherence and must be evaluated us-
an exchange contribution in the direction m of the d- ing a two-band quantum kinetic theory.
electronmagnetizationandaspin-orbitfieldinthezˆ×p The kinetic equation can be obtained along the route
direction: B =−∆ m+2αzˆ×p,αbeingthestrength outlined in Ref.[12]. In this paper we do not, however,
p xc
oftheRashbacoupling. (WeneglectZeemancouplingin switch to the energy representation, choosing to work in
comparison, assuming that it is important only in help- the more intuitive p-representation. The final form of
ingtofixthedirectionofmˆ.) Toavoidconfusioninwhat the equations we use is therefore slightly different.
follows,wedefinemtobethedirectionoflocalmagneti- To facilitate the discussion that follows, we first es-
zation. Thusthemajority(minority)electronshavetheir tablish some notation. In the absence of disorder, the
spins antialigned (aligned) with the local magnetization, eigenvalues of Hamiltonian (1) are given by
which corresponds to ∆ > 0. Uˆ is the disorder po-
xc dis
1
tential. We adopt a short-range-disorder model with a (cid:15) =(cid:15) − νB , (4)
spin-dependence that is dictated by m: pν p 2 p
Uˆ =(cid:88)δ(r−r )(u P0 +u P0), (2) where the index ν = ± distinquishes majority (+) and
dis i ↓ + ↑ −
minority (−) bands. The projection operators onto the
i
3
where the check accent denotes matrices in Keldysh
space [18].
corresponding eigenstates are
(cid:18) (cid:19)
1 ß·B
P = 1+ν p . (5)
pν 2 B
p
For weak disorder of the form specified in Eq. (1), the
Thequantumkineticequationforthetwo-banddensity
disorder self-energy is
matrix fˆ , linearized with respect to a time-independent
p
Σˇ(r)=n γˆGˇ(r,r)γˆ, γˆ =u P0 +u P0, (6) uniform external electric field E, is
i ↓ + ↑ −
(cid:26) (cid:27)
1 1 i
∂ fˆ + ∂ (cid:15) − ∂ B ß,∂ fˆ − [B ß,fˆ ]+eE∂ fˆeq =Iˆ . (7)
t p 2 p p 2 p p r p 2 p p p p st
Herefˆ is2×2matrixinspin-spaceandfˆeq =(cid:80) P f ((cid:15) ),wheref (E)istheFermifunction,istheequilibrium
p p ν pν th pν th
distribution function. Disorder effects appear in the collision integral Iˆ which takes the form
st
Iˆ =−πn (cid:88)(cid:90) d3p(cid:48) δ((cid:15) −(cid:15) )(cid:16)P fˆ γˆP γˆ+γˆP γˆfˆ P −P γˆfˆ P γˆ−γˆP fˆ γˆP (cid:17).
st i (2π)3 pν p(cid:48)ν(cid:48) pν p p(cid:48)ν(cid:48) p(cid:48)ν(cid:48) p pν pν p(cid:48) p(cid:48)ν(cid:48) p(cid:48)ν(cid:48) p(cid:48) pν
ν,ν(cid:48)
It is easy to check that the collision integral vanishes where 1/τ = πn N (u2 ±u2) > 0, N being the den-
s,d i 0 ↓ ↑ 0
when fˆ is replaced by fˆeq. sity of states at the Fermi level, and the overbar accent
p p
In what follows, we assume that the spin splitting of denotes the average over the directions of p. The spin-
the transport electrons Bp is small compared to their decoherence rate, 1/τ⊥, i. e. the rate of local relax-
FermienergyE . Intheanalogoussemiconductor2DEG ation of spin-components perpendicular to m, is equal
F
kineticequation[8]onehastokeepatleastthetermslin- to πniN0(u↓ −u↑)2 in this model, and thus is a func-
ear in Bp/EF in the kinetic equation in order to con- tion of 1/τs,d. We keep it as an independent parameter
sistently describe the coupling between charge currents to recognize that this property does not hold for general
andspindensities. Inthepresentcase,however,thespin spin-dependent disorder models [22]. Longitudinal spin
and charge densities are already coupled at (B /E )0 density (aligned with m) is not generated to linear order
p F
level because of spin-dependent scattering. In order to in the spin-orbit interaction in the present case and we
isolate the essential physics as transparently as possible thereforedonotneedtotheintroducethecorresponding
we initially neglect Fermi surface splitting in the field- relaxation time.
generation and collision terms. It is legal to keep the
The solution of Eq. (8) is straightforward. The scalar
precession term, which is linear in B , since it competes
p equationisusedtoeliminatethechargeresponsefromthe
with relaxation rates only. We comment further on the
vector equation, thereby introducing a generation term
influence of spin-split bands below.
and an additional relaxation term in the equation for
We now specialize to the stationary and uniform case.
f . To linear order in α, we first obtain an equation for
(cid:80) p
After straightforward manipulations, using P =1,
ν pν the longitudinal spin response (the component of f par-
performing the integrals over p(cid:48) in the collision integral, p
allel to m), which does not directly produce a torque.
and representing the distribution function as a sum of
When this solution is substituted into the equation for
scalar and vector parts using fˆ =n +ß·f , we obtain
p p p the transverse response it adds a generation term to the
the following equations:
equation for the perpendicular component because of its
precession in the spin-orbit field. The current-induced
1 m
eE∂ f = − (n −n )− ·(f −f ), spin-densities and currents are then obtained by sum-
p th τ p p τ p p
s d ming the scalar and vector responses over p.
m 1
B ×f = − (n −n )− (f −f )
p p τ p p τ p p Using Eq. (3), we finally obtain the following expres-
d s
1 sions for the effective magnetic fields that act on the fer-
− m×f ×m, (8)
τ p romagnet’s magnetization, and for the current density
⊥
4
are proportional to γ but even in m since they can-
0 z
flowing the in the film: not lead to switching. Terms with the symmetry of HS
eff
fromEq.(9)doappearatO(α3)order, andarealsopro-
αmj τ −τ ∆2 τ2
HReff = |e|Mtrτ↑+τ↓1+x∆c2⊥τ2m×yˆ×m, portional to γ0, but the corresponding expressions are
S ↑ ↓ xc ⊥ cumbersome and are not shown.
HS = αmjtrτ↑−τ↓ ∆xcτ⊥ yˆ×m, By examining these two analytically accessible lim-
eff |e|M τ +τ 1+∆2 τ2 its, we have concluded that the crucial ingredient in the
S ↑ ↓ xc ⊥
n e2 switchingbehaviorisspin-dependentscattering. Itseems
jtr = tmr (τ↑+τ↓)E. (9) likely to us that the most important source of this spin-
dependence in Co is the presence at the Fermi energy of
wherejtristhetransportcurrentdensity,ntristhetrans- minority spin d electrons which is captured in Eqs. 9.
port electron density per spin; we assumed for simplicity
Discussion - The physical origin of the effects described
a parabolic spectrum of transport electrons, and intro-
here is the fact that when an electric field is applied to
duced m, the correpsonding band mass (the same for
a ferromagnet with different majority and minority mo-
majorityandminorityelectrons). Thelatterassumption
bilities, the two spin species react to it differently. This
is not crucial, and has been invoked only for reasons of
creates spin polarization in momentum space, whose an-
clarity of the expressions. Finally, τ−1 = τ−1±τ−1 are
↓,↑ s d gular dependence “matches” the angular dependence of
the minority (1/τ ) and majority (1/τ ) spin scattering
↓ ↑ the y component of Rashba SO field. Thus a compo-
rates. These expression are the main result of this work.
nent of spin polarization perpendicular to the ferromag-
We identify the field HR, which produces a torque iden-
eff net’s magnetization is generated. This component then
tical to a field in the yˆ direction and cannot switch a
precesses around the exchange field, and gets relaxed ei-
perpendicular film as the Rashba field. We note that
therbymagneticscatteringortheDyakonov-Perelmech-
the expression for the Rashba field given here is para-
anism. A similar scenario was outlined previously [8] in
metrically larger than the one in existing literature [9],
connectionwiththeexplanationforout-of-planespinpo-
which contains an additional ∆ /E smallness factor.
xc F larizations in in-plane fields in 2DEGs with Rashba in-
If Ref. [19] was analyzed using our expression it would
teractions.
decrease the experimental estimate of the Rashba cou-
Despitethesimilarunderlyingmechanism,theleading
plingparameter. ThefieldHS hasthesamedependence
eff order results in the two models we considered, Eqs. (9)
on magnetization as the effective field produced by a yˆ-
and (10) have different form. In particular, Eqs. (9) cor-
polarized spin-current flowing into the magnetic layer,
respond to the case of spin-dependent impurity scatter-
and can switch perpendicular currents [4, 5].
ing, and Eqs. (10) are those for the case of a spin-split
In the general case of spin-dependent scattering and
band structure and spin-independent impurity scatter-
exchange fields of arbitrary strength the two-band quan-
ing. The difference is thus not surprising: since the ori-
tum kinetic equation cannot be solved analytically.
gin of the spin-dependent scattering is different in the
There are however other limits in which instructive an-
two models, generation and relaxation terms have differ-
alytic results can be obtained. One interesting limit is
ent dependence on the Rashba SO strength.
that of a 2D Rashba ferromagnet with spin-independent
There is some correspondence between the current-
disorderandaspin-splittingthatislargerthantheBloch
induced fields and torques that we have evaluated for
state lifetime. Results for the effective field, h , to the
eff a Rashba ferromagnet model, and those found experi-
leading order in α and in the clean limit, B τ (cid:29)1, are
p mentally. The torques exerted on the magnetization by
HS ,Eqs.(9),havethesamesymmetrypropertiesasthe
hR = αmjtr ∆xcyˆ, exepfferimental current-induced torques in Refs.[4] and [5].
eff 2|e|M E
S F In addition, the field has the same sign as observed ex-
αmj ∆ 1 m m
hS = −γ tr xc x z m. (10) perimentally, since it implies switching from up to down
eff 02|e|M E ∆ τ 1+m2
S F xc z direction for current flowing in the direction of the ap-
(cid:16) (cid:17) plied external in-plain magnetic field, if α is positive.
In the above expressions γ0 = pvFF ∂∂pvFF −1 is the The fact that it is positive in the experiments of Miron
nonparabolicity parameter, introduced in Ref. [8]. In et al. follows from their observation that HR points in
eff
Eqs. (10), the field hR is the Rashba field found earlier the positive yˆ direction (the coordinate system used in
eff
in Ref. [9]. The field hS , despite being aligned with the this work is identical to theirs). This universality is en-
eff
magnetization, can in principle lead to switching since sured by the transport properties of Co, in which the
it is odd in m . The factor 1+m2 in the denominator majority electrons have a much larger lifetime. In the
z z
of the expression for hS can be traced to the fact that case of a 2DEG, Eqs. (10), the direction of the switching
eff
for a 2DEG with a Rashba SO, the Dyakonov-Perel [20] field is less universal, in the sense that it depends on the
in-plane spin-relaxation time is twice as long as the out- detailsofthebandstructurethroughthenonparabolicity
of-planeone. Wehavedroppedcontributionstoh that parameter γ .
eff 0
5
[2] T. Jungwirth, Jairo Sinova, J. Maek, J. Kucera, and A.
The magnitudes of the effective fields in a Rashba fer- H. MacDonald, Rev. Mod. Phys. 78, 809 (2006).
romagnet, Eqs. (9), depend on two parameters, α and
∆ τ , which could be estimated by detailed first prin- [3] P. Gambardella and I. M. Miron, Philos. Trans. R. Soc.
xc ⊥
ciples studies of Co/Pt multilayers in the presence of a London A 369, 3175 (2011).
[4] I. M. Miron, et al., Nature 476, 189 (2011)
structuralasymmetry. Bycomparingtheratioofswitch-
[5] L. Liu and O. J. Lee and T. J. Gudmundsen and D. C.
ing and Rashba fields in Eqs. (9), we observe that the
Ralph and R. A. Buhrman, arXiv:1110.6846v2 (2011).
values of HR /j ≈ 10−8Tcm2A−1 claimed in Ref. [19],
eff tr [6] A. Chernyshov and M. Overby and X. Y. Liu and J. K.
and HSeff/jtr ≈ 10−9Tcm2A−1, claimed in Ref. [4] in a Furdyna,Y.Lyanda-Geller,andL.P.Rokhinson,Nature
similar setup, yield a plausible value of ∆ τ ∼10. Phys. 5, 656 (2009).
xc ⊥
For a large ∆ τ our results for the Rashba spin- [7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.
xc ⊥
density correspond to Awschalom, Phys. Rev. Lett. 93, 176601 (2004).
[8] H.-A. Engel, E. I. Rashba, B. I. Halperin, Phys. Rev.
sR αp v Lett. 98, 036602 (2007).
∼ F D (11) [9] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422
n ∆ v
tr xc F (2009).
[10] Ion Garate and A. H. MacDonald, Phys. Rev. B 80,
where v ∼ eEτ/m is the transport drift velocity. In
D 134403 (2009).
Co we estimate that v ∼ 100m/s at a current density
D [11] A. S. Nu´n˜ez and A. H. MacDonald, Solid State. Comm.
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parameter of Co on Pt we note that the majority-spin Halperin, Phys. Rev. B 73, 075316 (2006).
Fermi level is ∼ 0.5eV above the top of the d-bands in [13] I.A.Campbell,A.Fert,andA.R.Pomeroy,Philos.Mag.
15, 977 (1967).
Co and and below the top of the band in Pt. This im-
[14] A. Fert and I. A. Campbell, J. Phys. F 6, 849 (1976).
plies an electric potential drop comparable to the Fermi
[15] I.A.Campbell,A.Fert,andO.Jaoul,J.Phys.C3,S95
energy over the thickness t of the Co layer and a
Co (1970).
Rashba coupling strength at the Fermi energy αpF that [16] O. Jaoul, I. A. Campbell, and A. Fert, J. Magn. Magn.
is ∼ EF2/(mc2pFtCo) which should be around 105 times Mater. 5, 23 (1977).
smaller than ∆ . We conclude that it is hard to ex- [17] E. P. Wohlfarth, J. Appl. Phys. 41, 1205 (1970).
xc
plain dimensionless current-induced spin-densities much [18] E. M. Lifshitz, and L. P. Pitaevskii, Physical kinetics,
largerthan∼10−9 at1012A/m2 inCobasedonthisbulk (Butterworth-Heinemann, Oxford, 1999).
[19] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A.
mechanism. On the other hand the quantitative analysis
Schuhl,S.Pizzini,J.Vogel,andP.Gambardella,Nature
inRef.[5]suggestsaswitchingdimensionlessspin-density
Mat. 9, 230 (2010).
at this current level that is ∼ 10−6. We thus note that [20] M. I. Dyakonov and V. I. Perel, Sov. Phys. Solid State
a substantial enhancement of SO interactions is needed 13, 3023 (1971).
to explain the experimental switching current densities [21] C. R. Ast, Phys. Rev. Lett. 98, 186807 (2007).
based on the Co Rashba mechanism. This enhancement [22] Since Pt atoms are effective spin scatterers, and Co and
Pt are quite miscible, we should have included Elliot-
could come from the hybridization of Co transport elec-
Yafet-type of spin-orbit scattering by impurities. This
trons with Pt d-orbitals, analogous to SO enhancement
would contribute to the spin decoherence rate, as well
by surface alloying with heavy elements [21]. This hy-
as introduce relaxation rate for the diagonal component
bridization must play an essential role in the switching of the local in space spin polarization (L. Berger, Phys.
effect in both Rashba coupling and spin-Hall scenarios. Rev. B 83, 054410 (2011)). However, relaxation of non-
We therefore propose that the effect is strongest when zeroangularharmonicsofthedistributionfunctionwould
these bands are closely aligned. If so, this may explain bestilldominatedbythetermsalreadypresentinequa-
tion(8).Thuswewouldliketokeepτ asanindependent
the efficacy of Co/Pt structures and also suggest strate- ⊥
parameter, instead of complicating the kinetic equation
gies for optimizing the effect.
unnecessarily.
TheAuthorsacknowledgestimulatingdiscussionswith
G.E.W. Bauer. This work was supported by Welch
Foundation grant TBF1473 and by the ARO MURI on
bioassembled nanoparticle arrays.
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