Emergence of electromotive force in precession-less rigid motion of deformed domain wall

0.37 MB·English

by  T. Farajollahpour

EPJ manuscript No.
(will be inserted by the editor)
Emergence of electromotive force in precession-less rigid motion
of deformed domain wall
T. Farajollahpour1,2, N. Darmiani1,2, and A. Phirouznia1,2
1 Department of Physics, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran
6
2 Condensed Matter Computational Research Lab. Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran
1
0
2 Received: date / Revised version: date

p
e Abstract. Recentlyithasbeenrecognizedthattheelectromotiveforce(emf)canbeinducedjustbythespin
S precession where the generation of the electromotive force has been considered as a real-space topological
pumping effect. It has been shown that the amount of the electromotive force is independent of the
7
functionalityofthelocalizedmoments.Itwasalsodemonstratedthattherigiddomainwall(DW)motion
1
cannot generate electromotive force in the system. Based on real-space topological pumping approach in

] the current study we show that the electromotive force can be induced by rigid motion of a deformed
l DW. We also demonstrate that the generated electromotive force strongly depends on the DW bulging.
l
a MeanwhileresultsshowthattheDWbulgingleadstogenerationoftheelectromotiveforcebothalongthe
h axis of the DW motion and normal to the direction of motion.
-
s
e PACS. 75.60.Ch Domain walls magnetic properties and materials – 75.78.Fg Magnetization dynamics of
m domain structures
.
t
a
1 Introduction ↔
m where the energy of the wave packet is E(r ,k ) and Ω
c c
istheBerrycurvaturetensor.TheBerrycurvaturetensor
-
d Oneofthepowerfultechniquesincondensedmatterphysics in terms of for (r,t) is defined by
n is based on the manifestation of Berry phase in electronic
o systems [1,2,3,4]. Berry curvature in momentum space ∂ ∂
Ω = A − A (3)
c Ω give rise to the anomalous Hall effect [5], in which rt ∂r t ∂t r
[ kk
theBerryphaseconceptbehaveasalinkbetweenanoma-
where
4 lousHalleffectandtopologicalnatureoftheHallcurrents.
v This effect quantitatively has been calculated by abinitio ∂
9 methods in Refs. [6,7,8,9,10]. As reported in Ref. [11] At =i(cid:104)uk|∂t|uk(cid:105) (4)
0
topological Hall effect is described by real space Berry
6 curvature in which emergent real space Berry curvature and
2
acts like a real magnetic field. This effect has also been
0 ∂
1. observedexperimentallyinseveralstudiesofthisfield[12, Ar =i(cid:104)uk|∂r |uk(cid:105) (5)
11,13,14].
0
Inphasespace,anadiabaticevolutionofawavepacket knownasBerryconnections(gaugepotentials),|u (cid:105)ispe-
5 k
1 generates various interesting Berry phase effects [15]. The riodic part of the Bloch eigenstate of selected band. We
: evolutionofthewavepacketcenter(rc,kc)inphasespace have focused on space-time Berry curvature, Ωrt, which
v is describe by a set of semi-classical equations, determines the emf of the sample. The emf in terms of
i
X Berrycurvature isdescribed in Refs.[16,17,18]wherethe
(cid:18) (cid:19)
r ∂rc = ∂E(rc,kc) − Ω↔ · ∂rc +Ω↔ · ∂kc −Ω emf along a path l can be calculated by
a ∂t (cid:126)∂k kr ∂t kk ∂t kt
c (cid:90)
(1) emf =−(cid:126)/e dr·Ω . (6)
rt
l
(cid:18) (cid:19)
∂kc =−∂E(rc,kc) + Ω↔ · ∂rc +Ω↔ · ∂kc −Ω , It has been shown that DWs could be driven by high
∂t (cid:126)∂r rr ∂t rk ∂t rt electriccurrents.Thiseffectcanbeexplainedbyspincur-
c
(2) rentinducedspintorquewhichresultsinDWdynamics.In
theotherwordsDWcouldbedrivenbyspin-transferpro-
Send offprint requests to: cessoriginatesformthespinpolarizedcurrent[19,20].Itis
2 T. Farajollahpour et al.: Emergence of electromotive force in precession-less rigid motion of deformed domain wall
interestingtoknowtheinverseofthementionedeffecthas 2 Electromotive force due to the rigid domain
also been predicted [21]. In the inverse effect DW motion wall motion
couldgenerateaneffectiveelectromotiveforce.Generation
ofelectromotiveforce(emf)bymovingDWinaferromag- The localized spin texture of the DW configuration could
netic material, known as a ferro-Josephson effect, which be described by the following Hamiltonian
wasreportedinexperimentsperformedbyYangetal.[22].
(cid:88) (cid:88)
Theferro-JosephsoneffecthasbeenpredictedbyBergerin H =−J n(r )·n(r )− γ n ·n(r )
Loc loc i j z m i
1986 [21], meanwhile alternative approaches reformulated
<i,j> i
this effect [17,23,24].
(cid:88)
− (K n(r )·eˆ−K n(r )·eˆ(cid:48))+V (8)
(cid:107) i ⊥ i pin
Electromotive force could be induced by Berry cur- i
vature fields in a time-dependent topologically nontrivial
In which the first term describes the Heisenberg Hamil-
spin texture, for example in a moving domain wall (DW).
tonian of the first neighbor spins where J denotes the
loc
In this case it was shown that the space-time Berry cur-
exchange coupling constant between the localized spins e
vature measures density of the emf in the real space [18]. denotes the easy axis while e(cid:48) being the direction of the
It has been demonstrated that the rigid motion of the
hardaxis.K ,K standforanisotropicconstantsforeasy
(cid:107) ⊥
DWwithoutprecessioncannotinduceelectromotiveforce
andhardaxisrespectively.n(r )describesthedirectionof
i
within the topological pumping approach [18]. It should
the localized spins as mentioned before. γ is the Zeeman
z
be noted that the precession-less DW translation takes
splitting, V stands for pinning potential and n de-
pin m
placewhentheDWdrivingmagneticfieldislessthanthe
notestheunitvectordescribingthedirectionoftheexter-
Walkerbreakdownfield[25].Thereforethespinprecession
nalmagneticfield.Thecompetitionbetweentheexchange
hasacentralroleingenerationoftheelectromotiveforce.
andanisotropiccouplingsdeterminestheDWwidthwhich
This implies that even precessing of pinned DWs could (cid:112)
is given by d∼ K /J .
(cid:107) loc
generate electromotive force along the system when the
Meanwhile the dynamics of the local spins have been
magnetic field is higher than the Walker breakdown field
determined with the Landau and Lifshitz equation which
[18].
could be expressed as
In the current work we have obtained the influence of dn(r ,t) dn(r ,t)
i =γn(r ,t)×n +αn(r ,t)× i , (9)
the DW bulging on real space topological pumping effect dt i m i dt
during the rigid DW motion. Results indicate that the
rigid motion of a deformed DW could generate an effec- where α is the damping constant.
tive electromotive force regardless of the amount of the DW motion can be considered as a successive spin
magnetic filed. switching of the local precessing moments as a result of
thedampingprocedure.Whenthemagneticfieldishigher
than depinning field spin falls along the direction of effec-
If the generic Hamiltonian takes the form like H =
tive field during the damping process. This was known as
h(R)·σ,whereσ isthePaulimatrices,andh(R)depends
(cid:98) (cid:98)
spin switching process.
on parameters R, then the gauge independent Berry cur-
DWmotionitselfcouldtakeplaceindifferentregimes.
vature in parametric space R is defined as [26].
When the magnetic field is less than the Walker break-
down field [25] DW moves actually in a shape preserved
manner. In this case DW makes a rigid movement since
the spin switching process takes place almost abruptly.
Therefore rigid DW motion rests largely on how fast the
1∂(β,cosγ) spin switching takes place. On the other hand when the
ΩRiRj = 2 ∂(R ,R ) (7) magneticfieldishigherthanWalkerbreakdownfield,DW
i j will precess during its translational motion. As shown by
(cid:18) (cid:19)
= 1det ∂β/∂R1 ∂cosγ/∂R1 . Yang and et al for a moving DW without precession i.e
2 ∂β/∂R2 ∂cosγ/∂R2 forrigidDWmotionthereisnoelectromotiveforceinthe
system [18]. This could be demonstrated for both Bloch
andN´eeltypeDWs.Intheabsenceoftheprecession,when
the spin switching takes place abruptly, rigid DW motion
In which the orientation of h(R) in the parametric space could be described by the following profiles
hasbeenaspecifiedbythesphericalanglesgivenbycosγ =
h (R)/|h(R)|,andβ =tan−1h (R)/h (R).Inthepresent φ=const
z y x
case in which the exchange interaction of the moving and θ(x,t) =π/2+(π/2)tanh((x−v t)/d) (10)
D
localizedspinsisgivenbythefollowingHamiltonianH =
ex
Jσˆ.n(r) one can obtain cosγ = n (θ,φ) and also β = for Bloch type DWs and
(cid:98) z
tan−1(n (θ,φ)/n (θ,φ)).
y x
Inwhichtheunitvectorn(r)=sinθcosφx+sinθsinφy+ φ(x,t)=(π/2)tanh((x−vDt)/d)
(cid:98) (cid:98) (cid:98)
cosθz describes the direction of the localized spins. θ =const (11)
(cid:98)
T. Farajollahpour et al.: Emergence of electromotive force in precession-less rigid motion of deformed domain wall 3
for N´eel type DWs.
Thespaceandtimedependenceofβ andcosγ arejust
given by the spherical angles (θ and φ). Therefore it can
be easily shown that
(cid:18) (cid:19)
1 ∂β/∂x∂cosγ/∂x
Ω = det
xt 2 ∂β/∂t ∂cosγ/∂t
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∂ θ ∂ φ ∂ β ∂ cosγ
= det x x det θ θ
2 ∂tθ ∂tφ ∂φβ ∂φcosγ
(cid:18) (cid:19)
∂θ/∂x∂φ/∂x
=Ω det . (12)
θφ ∂θ/∂t ∂φ/∂t
Using the Eq. 12 it can be inferred that electromo-
tive force generated by DW motion identically vanishes
for rigid translation of the DW in both of the N´eel and
Bloch-type configurations. This can be easily obtained if
we consider that for either of the profiles given for the
rigid motion of the DW we have
Fig. 1. A typical form of the deformed DW in the real space.
(cid:18)∂θ/∂x∂φ/∂x(cid:19) ∂(θ,φ) DW has been placed between two oppositely directed mag-
det ∂θ/∂t ∂φ/∂t = ∂(x, t) (13) neticdomains.ABloch-typeDWhasbeenconsideredinwhich
the DW rotation axis located in the plane of the system. The
=0
bulging of the DW can be formulated if we consider that the
DW rotation axis is always vertically oriented to the DW
and therefore Ωxt =0. boundaries. nD denotes the DW rotation axis while vD in-
dicates the speed of the DW and d is the DW width.
3 Electromotive force generated by the rigid
R isthespinrotationoperatorandX (r,t)=nˆ (r)·r(t)
θ D D
motion of a deformed domain wall
is the projection of position vector along the direction of
the DW rotation axis. Accordingly we can obtain
If we consider the bulging effect in the DW it could be
demonstrated that the rigid motion of the DW generates σ·n=(cos(θ(XD))+2n2Dzsin2(θ(XD)/2))σz (16)
an effective electromotive force inside the system. DW +(2n n sin2(θ(X )/2)−n sin(θ(X )))σ
Dx Dz D Dy D x
bulging could be characterizes by a position dependent
+(2n n sin2(θ(X )/2)+n sin(θ(X )))σ .
rotation axis nˆ (r) and the boundary curve of the DW, Dy Dz D Dx D y
D
wherewehaveassumedthattheDWrotationaxisshould
Thereforetheunitvectorofthelocalmagnetizationcould
be directed vertically to the DW boundary surface. This
be extracted as follows
meansthatwhentheDWboundaryhasbeenspecifiedby
asurfacedescribedbyF(r )=cthenDWrotationaxisis
0 n(r) = (17)
givenbynˆ (r)=∇F/|∇F|ateachpointoftheboundary
curve. D  cos(θ(XD)+2n2Dzsin2(θ(XD)/2)) 
InthecurrentcasewehaveassumedadeformedBloch 2nDxnDzsin2(θ(XD)/2)−nDysin(θ(XD)).
typeDW.Ifthedirectionofthelocalmagnetizationisup- 2n n sin2(θ(X )/2)+n sin(θ(X ))
Dy Dz D Dx D
ward at the left boundary of the DW then local direction
of the magnetization could be obtained by the spin rota-
If we restrict ourself to the case of the two-dimensional
tion operator as follows
DWs i.e. when (n = 0) we obtain the simplest case of
Dz
two dimensional bulging as follows
σ·n=R† σ R
(θ) z (θ)
=cos2(θ(X )/2)σ n(r)=−nDysin(θ(XD))xˆ
D z
+sin2(θ(X )/2)cos2(θ(X )/2)[σ·n ,σ ] +nDxsin(θ(XD))yˆ+cos(θ(XD))zˆ. (18)
D D D z
+sin2(θ(X )/2)(σ·n )σ (σ·n), (14)
D D z DWwidthhasbeenassumedtobelocallyconstantduring
the DW rigid motion. The DW motion has been consid-
in which
ered as a successive switching of the local spins that as
mentioned before. In the case of the rigid DW motion it
Rθ(XD) =exp(−iσ·nD(r)θ(XD)) (15) was assumed that the spin switching time is very short
=1cos(θ(X )/2)−iσ·n (r)sin(θ(X )/2), andalocalizedspinswitchesabruptlywithoutprecession.
D D D
4 T. Farajollahpour et al.: Emergence of electromotive force in precession-less rigid motion of deformed domain wall
Meanwhile it was assumed that the DW rigidly moves andn (x,−y)=−n (x,y).ThereforeX remainsun-
Dy Dy D
along the x axis. Since in this case we can write changed under the coordinate change: y →−y and Y →
0
−Y .InthiscasesinceΩ isanoddfunctionofy thespa-
0 xt
φ(y)=tan−1(−n (y)/n (y))=−φ (19) tial average of the electromotive force along the direction
Dy Dx D
of the DW motion, (−(cid:126)/eL)(cid:82) dxdyΩ , vanishes iden-
S xt
and tically. Where the integration has been performed over
the DW area which has been specified by S and L is
XD(r,t)=nD·r (20) the length of the DW. Meanwhile in the normal direction
=n (x−X (Y )−v t)+n (y−Y ) Ω is an even function and the averaged electromotive
Dx 0 0 D Dy 0 foyrcte, (−(cid:126)/eL)(cid:82) dxdyΩ , cannot vanish simultaneously.
S yt
whereX (Y )istheleftboundaryoftheDWattheinitial Finally we can conclude that electromotive force induced
0 0
time. This curved boundary preserves its shape during by the rigid DW motion cannot go all the way to zero for
the DW motion which sweeps and removes the downward any arbitrary open curved DWs.
ferromagnetic region (right region in Fig. 1) during the It should be noted that in the previous studies in this
motion of the DW. filed[27,28,29]spin-motiveforcewasshowntobepropor-
The space time Berry curvature is given by tional to |∂m/∂t×∂m/∂x|. Since in the precession-less
limit ∂m/∂t identically vanishes, this means that the ze-
Ωyt =Ωθφ(∂yφ)(∂tθ(XD)) (21) roth order spin-motive force is zero for rigid motion of
=Ω n v ∂ θ(X ))(n ∂ n −n ∂ n ). the DW. Meanwhile if we consider the perturbative in-
θφ Dx D XD D Dx y Dy Dy y Dx duced effects in the effective field of the system, pertur-
Since we can write n2 +n2 =1, therefore bative terms of ∂m/∂t could be non-zero and account-
Dx Dy
able. Spatially non-homogeneous magnetization results in
n (n ∂ n −n ∂ n )=∂ n (22) effective magnetic field on conduction electrons given by
Dx Dx y Dy Dy y Dx y Dy
B ∼ (cid:15)ijk(∂ m×∂ m)·m. Since the spin torque on the
i j k
Meanwhile in the absence of the Rashba interaction we local magnetization is the opposite of the torque exerted
haveβ =φandcosγ =cosθandthereforeΩ =1/2sin(θ). on the conduction electrons. Therefore local magnetiza-
θφ
Accordingly tion even in the precession-less limit has time evolution
[28].
π
Ω = sin(θ(X ))v cosh−2(X /d)∂ n (23)
yt 4d D D D y Dy
4 Conclusion
and similarly we can obtain
In the current study we have developed the Berry curva-
π
Ω = sin(θ(X ))v cosh−2(X /d)∂ n . (24) ture based theory of the DW electromotive force for the
xt 4d D D D x Dy
deformedDWs.Inthepreviousstudiesithasbeendemon-
Unlike the straight DWs in which the electromotive force strated that for straight DWs rigid motion of the DW
could only generated as a result of the precession, regard- could not contribute in the ferro-Josephson effect [18]. In
less of the DW motion, in the current case for deformed the other words this type of domain wall dynamics which
DWswehaveshownthattheelectromotiveforcecouldbe takesplacebelowtheWalkerfieldcouldnotgenerateelec-
generated along the direction of motion and even normal tromotive force and the spin precession plays the central
to this direction. roleintheelectromotiveforcegeneration[18].Howeverin
It should be considered that the averaged electromo- the current study we have demonstrated that this is not
tive force along the direction of the DW motion could thecaseforthedeformedDWs.Ithasbeenshownthatthe
vanish identically when the DW curve has mirror sym- DW bulging results in effective electromotive force in the
metry and the direction of the DW speed is parallel to caseofrigidthemotionofthedomainwall.Inadditionwe
the plane of mirror symmetry. Meanwhile, even in this have shown that the generated electromotive force could
case the normal electromotive force still survives and the be either along the DW motion axis or in the transverse
rigid DW motion could generate an electromotive force direction.
normal to the direction of motion i.e. perpendicular to
the mirror symmetry plane. Therefore the electromotive
Acknowledgment
force induced by the rigid motion of the domain wall can
identically vanish when the domain wall has two vertical
This research has been supported by Azarbaijan Shahid
mirrorsymmetryplanes.Howeverinthiscasethedomain
Madani university.
wall should be formed as a closed symmetric curve i.e.
one of the magnetic domains should be surrounded by its
opposite counterpart.
References
Without loss of generality we can assume that the
domain wall motion is along the x direction. If we as- 1. M.V.Berry,inProceedings of the Royal Society of London
sume that the xz plane is the mirror plane of the DW A: Mathematical, Physical and Engineering Sciences, vol.
therefore X (−Y ) = X (Y ), n (x,−y) = n (x,y) 392 (1984), vol. 392, pp. 45–57
0 0 0 0 Dx Dx
T. Farajollahpour et al.: Emergence of electromotive force in precession-less rigid motion of deformed domain wall 5
2. D. Xiao, M.C. Chang, Q. Niu, Rev. Mod. Phys. 82(3),
1959 (2010)
3. F.Wilczek,A.Shapere,Geometricphasesinphysics,vol.5
(World Scientific Singapore, 1989)
4. A. Bo¨hm, H. Koizumi, Q. Niu, J. Zwanziger,
A. Mostafazadeh, The geometric phase in quantum
systems (Springer, 2003)
5. N.Nagaosa,J.Sinova,S.Onoda,A.MacDonald,N.Ong,
Rev. Mod. Phys. 82(2), 1539 (2010)
6. I. Turek, J. Kudrnovsky`, V. Drchal, Phys. Rev. B 86(1),
014405 (2012)
7. J. Weischenberg, F. Freimuth, J. Sinova, S. Blu¨gel,
Y. Mokrousov, Phys. Rev. Lett. 107(10), 106601 (2011)
8. S.Lowitzer,D.Koedderitzsch,H.Ebert,Phys.Rev.Lett.
105(26), 266604 (2010)
9. X. Wang, J.R. Yates, I. Souza, D. Vanderbilt, Phys. Rev.
B 74(19), 195118 (2006)
10. Y. Yao, L. Kleinman, A. MacDonald, J. Sinova, T. Jung-
wirth,D.s.Wang,E.Wang,Q.Niu,Phys.Rev.Lett.92(3),
037204 (2004)
11. A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz,
P. Niklowitz, P. Bo¨ni, Phys. Rev. Lett. 102(18), 186602
(2009)
12. N. Kanazawa, Y. Onose, T. Arima, D. Okuyama,
K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata,
Y. Tokura, Phys. Rev. Lett. 106(15), 156603 (2011)
13. S.Mu¨hlbauer,B.Binz,F.Jonietz,C.Pfleiderer,A.Rosch,
A. Neubauer, R. Georgii, P. Bo¨ni, Science 323(5916), 915
(2009)
14. S. Huang, C. Chien, Phys. Rev. Lett. 108(26), 267201
(2012)
15. G. Sundaram, Q. Niu, Physical Review B 59(23), 14915
(1999)
16. A. Stern, Phys. Rev. Lett. 68(7), 1022 (1992)
17. S. Barnes, S. Maekawa, Phys. Rev. Lett. 98(24), 246601
(2007)
18. S.A.Yang,G.S.D.Beach,C.Knutson,D.Xiao,Z.Zhang,
M. Tsoi, Q. Niu, A.H. MacDonald, J.L. Erskine, Phys.
Rev. B 82, 054410 (2010)
19. Y. A, O. T, N. S, M. K, M. K, S. T, Phys. Rev. Lett. 92,
077205 (2004)
20. T. Gen, K. Hiroshi, Phys. Rev. Lett. 92, 086601 (2004)
21. L. Berger, Phys. Rev. B 33, 1572 (1986)
22. S.A. Yang, G.S. Beach, C. Knutson, D. Xiao, Q. Niu,
M. Tsoi, J.L. Erskine, Phys. Rev. Lett. 102(6), 067201
(2009)
23. W.M. Saslow, Phys. Rev. B 76, 184434 (2007)
24. R.A. Duine, Phys. Rev. B 77, 014409 (2008)
25. N.L.Schryer,L.R.Walker,J.Appl.Phys.45,5406(1974)
26. M.Nakahara,Geometry,topologyandphysics(CRCPress,
2003)
27. G.E. Volovik, Journal of Physics C: Solid State Physics
20(7), L83
28. S. Zhang, S.S.L. Zhang, Phys. Rev. Lett. 102, 086601
(2009)
29. Y. Yamane, K. Sasage, T. An, K. Harii, J. Ohe, J. Ieda,
S.E.Barnes,E.Saitoh,S.Maekawa,Phys.Rev.Lett.107,
236602 (2011)