Current induced distortion of a magnetic domain wall

5 Pages·0.13 MB·English

by  Xavier Waintal

Current induced distortion of a magnetic domain wall
Xavier Waintal and Michel Viret
CEA, Service de physique de l’´etat condens´e,
Centre d’´etude de Saclay F-91191 Gif-sur-Yvette cedex, France
3 (Dated: February 2, 2008)
0
0 Weconsiderthespintorqueinducedbyacurrentflowingballisticallythroughamagneticdomain
2 wall. Inadditiontoaglobalpressureinthedirectionoftheelectronicflow,thetorquehasaninternal
structure of comparable magnitude due to the precession of the electrons’ spins at the ”Larmor”
n
frequency. As a result, the profile of the domain wall is expected to get distorted by the current
a
and acquires a periodic sur-structure.
J

6 PACSnumbers:
1

] With the adventof ”spintronics”,which aims at using magnetization under the effect of the ”Larmor” preces-
l
l the spin of charge carriers in devices, electronic trans- sion[6]. There,noMRisobserved[5,7]buteachelectron
a
h port in ferromagnets is being revisited from a different going through the DW will flip its spin and give a quan-
- viewpoint. The focus has been for a long time on the ef- tum~ofangularmomentumtothewallinducingaglobal
s
e fect ofmagnetismontransportproperties(e.g. Magneto pressure.
m Resistance, MR) [1], but it is now realized that the elec- Experimentally, the wall resistance (DWR) has been
. tronic currentcanbe a toolto change the magnetization measuredinmacroscopicsystemswhereitisofthe order
at direction. The relevant effect, known as spin torque has of a few percent per wall. This resistance results from
m attracted considerable interest recently in the context of a slight mistracking of the conduction electrons’ spins
ferromagnetic–normal metal–ferromagnetic trilayers [2]. which mixes the majority and minority channels within
-
d There, the first magnetic layer acts as a spin filter, and the wall[6, 8, 9]. Numerousmeasurementshavebeenre-
n theincidentpolarizedelectronsexertatorqueonthesec- ported in the last few years finding both positive [6, 14]
o ond layer. At the heart of this physical effect is the fact or negative [10] effects. The spread in the experimen-
c
[ thatspincurrentsarenotpreservedwhenelectronscross tal results probably reflects how difficult it is to extract
a magnetic layer, and as a result some angular momen- the DWR among other contributions in series coming
1 tum is transferred to the magnetization. It was shown fromdomains(liketheAnisotropicMR).Other(ballistic)
v experimentally that for strong enough current densities modelsweredeveloped[11,12]toexplainnegativeeffects,
3
this mechanism can lead to magnetic reversal [3]. This but it was later recognized that proper band structure
9
2 demonstrates the feasability of current controlled mag- calculations [13] are needed to get reliable quantitative
1 netic memory cells, but the current needed for complete results. Nevertheless, recent clean experiments [14] in
0 reversalmightbetoohighforindustrialimplementation. nanostructures of perpendicularly magnetized materials
3 Analternativewouldbe tousethecurrenttomoveado- (FePd)havedemonstratedthatthe spin scatteringmod-
0 main wall (DW) in between two stable positions. The els developedin[6,8]canaccountfor the MRin domain
/
t idea that a current can apply a force on a DW is due to walls in 3d metals. Concerning torque effects, very re-
a
Berger [4] in the seventies. It is the aim of this letter to cently several groups have been able to push DWs with
m
study in detail the spin torque exerted on a DW in the a current [15].
- presence of an electric current. Our main finding is that
d In this letter, we proceed as follows. First, an heuris-
inadditiontoaglobalpressure,thetorquehasaspatially
n tic argument is given to explain the origin of the spatial
o dependant component that will lead to a deformation of structure of the torque. Then, we introduce our (ballis-
c the DW in a periodic sub-structure.
tic) model and point out that some key features of the
:
v AttherootofunderstandingthespintorqueinaDWis band structure must be taken into account. Finally we
Xi thequestionofwhathappenstothespinofaconducting calculatethe spintorque andevaluate the corresponding
electron when going through a DW. Two extreme cases distortion of the DW profile.
r
a can be considered called ”interface” and ”adiabatic”. A Heuristic argument: Let us follow an electron going
verysharpdomainwall(expectedinconstrictionsforex- through the (Neel) DW of size λ sketched in Fig.1 (a
w
ample), canbe treatedas aninterface on whichthe elec- Bloch wall is completely equivalent for MR and torque).
trons can be specularly reflected [5]. This leads to the The process is described in Fig.2: Before the wall, the
giant magneto resistance effect as a magnetic field will electron’s spin is alignedwith the localmagnetizationm~
remove the DW and the associatedextra resistance. Be- (a). Then, when entering the DW, m~ begins to rotate,
causetheelectronicspinisconservedduringthisprocess, and a small angle α starts to build up between the spin
nospintorqueisexertedontheDW.Ontheotherhand, andthelocalmagnetization(b). Assoonasαisnotzero,
in a very long domain wall, the electron’s spin will adi- the electron’s spin starts to precess around the direction
abatically adapt itself to remain aligned with the local of an effective magnetization with a period equal to the
2
low, it gives rise to the spatially dependant part of the
w
torque inducing a deformation with a period λ .
L
Model. Let us introduce a simple model for the con-
z
v ducting electrons through the DW of Fig.1. The system
u is a ferromagnetic wire running along x and of typical
transverse dimension d. We approximate the wall by a
linear rotation of the local magnetization on a scale λ .
m(x) w
θ(x) The unit vector mˆ(x) lies inside the xz-plane and makes
an angle θ(x) = πx/λ with the z axis inside the DW.
w
y The Hamiltonian reads,
x
0 λw H =− ~2 ∆− Jexc mˆ(x)·~σ. (1)
2m∗ 2
FIG. 1: Schematic of the domain wall of length λw and Here Jexc is the exchange energy, m∗ the effective mass,
transverse dimension d. The arrows stand for the magne- ~σ = (σx,σy,σz) the vector of pauli matrices and the
tization direction mˆ that makes an angle θ(x) with the z- Fermi energy is noted E . The spin dependant part of
F
axis. The same x, y and z axis are used for both the H reads,
space and spin basis for convenience. The rotating frame
(u,v,w)=(dmˆ/dθ,dmˆ/dθ×mˆ,mˆ)followsthemagnetization. σz, x<0
mˆ(x)·~σ = R σ R , 0<x<λ (2)
 θ(x) z −θ(x) w
−σ , λ <x
 z w
”Larmor”precessionlengthλL ((c)and(d)). Attheend with the rotationmatrix defined as Rθ(x) = e−iσyθ(x)/2.
of everyperiod, the electron’s spin is back onto the local
In the region outside the wall, the eigenstates of H are
magnetization direction [6]. At x = λ /2, α reaches a
L plane waves in both the transverse yz-plane, with to-
maximum. Overthisdistance,m~ hasrotatedbyanangle tal momentum k⊥, and in the x-direction, with mo-
πλ /2λ whichthusgivesanupperboundforα. Hence,
L w mentum kk (kk) respectively for majority and minor-
η = λ /λ is the parameter that controls the crossover ↑ ↓
L w
from the adiabatic limit (η ≪ 1) to the interface one ity electrons. In the transverse direction, k⊥ is quan-
(η ≫1). tized in units of 2π/d and for the corresponding eigen-
state to be propagating in the x-direction, one needs
kk = 2m∗/~2(E ±J /2)−(k⊥)2 > 0 which leads
↑,↓ F exc
z toN (N )propagatingchannelsforthemajority(minor-
↑ p↓
ity) spin. Once these propagating channels are defined,
the natural way to calculate physical quantities such as
(a) (b) (c) (d) (e) the current or the spin current is to use the Landauer-
Buttikertheoryandfillupthedifferenteigenstatesofthe
system. This has been widely used in the literature and
α
eventuallyleadstotheLandauerformulafortheconduc-
tance g =e2/h Tr tt† where t is the transmission matrix
of the system.
x
0 λ E E
L
(a) (b)
FIG. 2: Cartoon of the effect of the Larmor precession al-
lowing theelectron spin tofollow thelocal magnetization m~. Na Na
The thin arrow represents m~ while the thick one stands for
theconducting electron spin.
Nb Nb
The (almost)adiabatic case is the mostinteresting re-
gardingspintorquebecausetheincidentspinupelectron Maj min Maj min
endsupwiththe(almost)oppositespinaftercrossingthe
wall, losing an angular momentum ~ in the process. In FIG.3: Cartoonofthepositionofthemajorityandminority
return, since the total angular momentum is conserved, channelsasafunctionoftheperpendicularenergy. Thenum-
a torqueofequalmagnitude andopposite signis exerted berofmajority channelsisN↑ =Na+Nb. (a)simplepicture
on the magnetization within the wall. This is the global where Nb =N↓. (b) “realistic” magnet where Nb 6=N↓.
pressure mentioned earlier. The spin precession around
the slowlyrotatingeffective magnetizationdoes notcon- In this simple picture, the N down channels are ex-
↓
servethe angularmomentum either,and, as detailedbe- actly matched by the first N = N up channels with
b ↓
3
equal perpendicular energy, and the polarization of the Fromthe lastequation,wederiveforthe torqueper unit
currentis entirelydue to the remainingN ≡N −N = voltage:,
a ↑ b
N −N up channels that have large perpendicular en-
↑ ↓
ergy (see Fig.3 (a)). Hence their longitudinal energy ∂~τ =− e ∂ Re Tr 1+ PI −PNσ t(x)~σt†(x) (7)
(∼ Jexc/m∗), is low and the corresponding minority ∂V 4π∂x (cid:20) 1−PN z(cid:21)
channels may not propagate. This stems from the as-
p where the generalizedtransmissionmatrix t(x) gives the
sumption that the up and down bands have exactly the
amplitudes of the different modes inside the wall.
sameshape,whichisobviouslyerroneous. Itwasrecently
Domain wall close to the adiabatic limit. We now pro-
clearly underlined by Mazin [16] that by oversimplifying
ceedwiththetreatmentofEq.(1)andEq.(2)inthelimit
the band structure of magnetic metals, one introduces
of a long wall. The first step consists in writing the
symmetriesinthesystemthatoughtnottobethere. In-
schr¨odinger equation in the basis aligned with the local
deed, when the polarization of the current P is taken
I
magnetization. An eigenstate Ψ(x) is then written as,
equal to the polarization P i.e.,
N
Ψ(x)=R Φ(x) (8)
I −I N J θ(x)
P ≡ ↑ ↓ =P ≡ a = exc (3)
I N
I↑+I↓ N↑+N↓ Ef An effective equation is obtained for Φ(x) (see [8] for a
similar treatment, the exact solution of the model can
this artificial symmetry can lead to erroneous physics.
be found using the “spin spiral” state, see [12]). The
For instance, after an “interface” wall, all the channels
solutionisthenformallyexpandedinseriesof(1/λ )and
w
with high perpendicular energy are blocked (the N =
a we match Ψ(x) and its derivative at x = 0 and x = λ .
w
(N −N )inFig.3(a))andthepolarizationofthecurrent
dro↑psto↓almostzero. Inarealsystemhowever(sketched In first order in (1/Qkλw) where Qk = k↑kk↓k, we get
in Fig. 3 (b)), the number Na of blockedchannels would for incoming majority electrons in the waqll:
not be equal to (N −N ) and the system would retain
↑ ↓
somepolarizationPI ≈(Nb−N↓)/(N↑−N↓). Acomplete eik↑kx iπ
Φ (x)= and Φ (x)= × (9)
treatment of the problem would require ab-initio calcu- ↑ ↓
kk 4Q λ kk
lations as described in [16]. Here, we break this artificial ↑ k w ↓
symmetry by giving a different weight to the different q q
channels in the Landauer formula, i.e. allowing P and
PN to be independent numbers. This can be viewIed as 1 [eik↑kx−eik↓kx]+Pk[−eik↑kx+ξe−ik↓kx]
P
replacingtheferromagnetbyanon-magneticreservoirin (cid:18) k (cid:19)
series with a normal-ferromagneticinterface that acts as
and similar solutions for incoming minority spins. The
a spin filter. It leads to the following modified Landauer
wave functions are normalized to carry unit fluxes and
formula,
only the longitudinal part has been written. We have
g = eh2Tr 1+ P1I−−PPNσz tt† (4) pdeofiinntedouPtkth=at(kt↑kh−eko↓ksc)/il(lakt↑ko+ryk↓kfi)r,satntderξm=ineit(hk↑ke+bkr↓ka)λcwke.tWsies
(cid:20) N (cid:21) theexpressionoftheLarmorprecessionwhilethe second
where the σ matrix applies in spin space and only to one is a reflected wave. In the region x<0 the reflected
z
thosechannelsthatcanpropagateforbothmajorityand wave takes the form,
minority spins. We emphasize that we introduce this
modification to take into account the band structure of Φ (x)= iπ Pk (ξ−1)e−ik↓kx (10)
the magnet and do not question the Landauer-Buttiker ↓ 4 Qkλw kk
formalism itself. This correction is crucial here since, as ↓
will be seen later, by symmetry the spin torque vanishes q
Spintorqueinamacroscopic DW.Toproceedwiththe
when P =P .
N I calculation of the conductance and torque, one incorpo-
ForagivenspinorwavefunctionΨ(x),thespincurrent
ratestheexpressionforthewavefunctionsintoEq.(4)and
flowing along the x-direction is defined as [17],
Eq.(7)andperformsthe averageovertransversemomen-
~2 ∂ tum. Thisaverageisdoneintwosteps,firstoverξ,which
J~ (x)= Im dydzΨ†(x)~σ Ψ(x) (5) is taken to be a random phase and then over k . Also,
s 2m∗ ∂x ⊥
Z the Fermi wave length λF and Larmor precession length
λ are defined as,
This is not a conserved quantity, and the corresponding L
lossofspincurrentisidentifiedwiththetorque[17]. The
gthloeblaolctaolrtqourequiset(hpeenr~τutontit=dJi~sst(axnc=e)0)is−, J~s(x=λw)while λF =2πs2m~∗2EF, λL =πrJEexfcs2m~∗2Jexc (11)
∂ We get for the correction to the conductivity due to
~τ =−∂xJ~s(x) (6) the presence of the DW,∆g/g = −PNλ2F/(64λ2w). This
4
correctionis (too)smallbecauseina ballisticmodel, the , λ = 3nm, and P − P ≈ 0.5. A current density
L I N
calculatedquantity is the reflectedpartof the wavefunc- j =1010A/m2 wouldthenleaveinaNiwall10−5J/m−2,
tion due to the potential step. Diffusive models, on the which is one tenth of the wall surface energy. Hence the
other hand, neglect this contribution and estimate the distortion in angular gradient should be of the order of
resistance due to spin mixing between the up and down several%. Thisisasignificanteffectconsideringthatthe
electrons. It turns out that in the macroscopic case, the chosen current density is one order of magnitude below
latter dominates. Inthe interfacelimit however,the bal- that used in tri-layer spin torque experiments.
listic contribution becomes important. We calculate the
Constriction ”interface” limit. Inaconstriction(small
torque per current along the local frame (u,v,w) from
transverse direction d), the DW width is expected to
the wavefunctions expressions as,
scale with d [18, 20] and the system could be driven to
the interface limit. As explained earlier, the spin of the
∂τ (x) ~ π x
u = P +(P −P )cos 2π , transferredelectronswouldremainmostlyun-flippedand
I I N
∂I e2λ λ
w (cid:20) (cid:18) L(cid:19)(cid:21) the torque would decrease as a result. Hence, while in
this regime the MR gets larger, the torque described in
this paper would drop. In addition, since in this limit
∂τ (x) ~ π x
v =− (P −P )sin 2π . (12) the reflection is higher, another contribution to the to-
I N
∂I e2λw (cid:18) λL(cid:19) tal pressure could come from the transfer of momentum
from the reflected electrons to the DW. Although it is
Equation (12) is the central result of this letter. The
not clear to us whether this change of momentum would
torque consists of two terms. The first one, proportional
actuallypushonthewallorontheatoms,itseffectwould
to P pushes the wall in the direction of the electrons
I be much smaller than that in a long wall.
anddoesnotdependonx. The secondpartofEq.(12)is
Conclusion. The torque generated by a current on a
much more interesting since it leads to a deformation of
domain wall in a ferromagnetic metal has been studied
the wall on the scale of λ . Note that although its net
L
in unconstrained DWs. It is composed of two contribu-
contributionto the globaltorque is small, its intensity is
tions of comparable magnitude. The first one is a global
ofthesamemagnitudeasthefirstterm. However,theN
a
pressure resulting from the loss of angular momentum
channels where the perpendicular energy is high (hence
of electrons crossing the wall. The second is due to the
the down electrons cannot propagate) do not contribute
precession of electrons’ spins inside the wall which gen-
to the torque since no precession is possible inside those
erate a periodic torque. The first effect can be used to
channels. As a result, this component of the torque van-
move the DW with a current. Depending on pinning,
ishes when P = P , hence the necessity of taking into
I N
the current density necessary to dislodge the DW could
account the fact that P 6= P in real materials. We
I N
be smaller than that necessary to reverse a full magne-
pointoutthatalthoughourcalculationisdoneforabal-
tized layer in a spin valve. The effect can then poten-
listic model, the result should hold for realistic domain
tially be useful in spin electronic devices where a DW
walls. Indeed, both the spin diffusion length (about 50
switched between two stable positions could be used in
nm for Ni) andthe mean free path(a few nm) arelarger
the gate of a three terminal device. Moreover, the pe-
than λ (of the order of 3 nm for Ni) and at the Lar-
L
riodic torque will distort the wall’s internal structure in
mor length scale, the electrons can thus be considered
a significant manner when current densities of the order
ballistic. In a system where the mean free path would
of 1010A/m2 are driven through it. This may help the
be smaller than λ but still with a (rather) large spin
L
depinning process and might also be able to switch the
diffusion length, our conclusions would remain qualita-
wallbetweendifferenttypes(BlochandNeelforexample)
tively correct,thougha quantitative treatment might be
in a similar manner to the predicted temperature effect
needed.
in a constriction [20]. The distortion may be measured
In order to quantify the expected deformation of the
by several techniques including polarized neutron reflec-
wall,one wouldhaveto solveself-consistently the torque
tivity (PNR) and domain wall resonance. We also infer
equation together with the reaction from mˆ(x) linked to
thata currentflowingparallelto the wallwouldproduce
the”stiffness”ofthewall(usingforinstancetheLandau-
a surstructure of the same period, which may be easier
Lifshitz-Gilbert equation). Here we simply evaluate the
to measure with PNR.
magnitude of the deformation by comparing the energy
given to the wall via the torque to the total wall energy. Acknowledgment. Itis apleasure to thank R.Jalabert
Typical quantities for Ni are λ = 100nm, λ = 0.2nm and D. Weinmann for interesting discussions.
w f
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