Magneto-mechanical interplay in spin-polarized point contacts

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by  Maria Stamenova

Magneto-mechanical interplay in spin-polarized point contacts
Maria Stamenova,1 Sudhakar Sahoo,2 Cristi´an G. Sa´nchez,2,3 Tchavdar N. Todorov,2 and Stefano Sanvito1,∗
1School of Physics, Trinity College, Dublin 2, Ireland
2School of Mathematics and Physics, Queen’s University of Belfast, Belfast BT7 INN, UK
3Unidad de Matem´atica y F´ısica, Faculatad de Ciencias Qu´ımicas, INFIQC,
6
Universidad Nacional de C´ordoba, Ciudad Universitaria, 5000 C´ordoba, Argentina
0
0
We investigate the interplay between magnetic and structural dynamics in ferromagnetic atomic
2
pointcontacts. Inparticular,welookattheeffectoftheatomicrelaxationontheenergybarrierfor

n magnetic domain wall migration and, reversely, at the effect of the magnetic state on the mechan-
a ical forces and structural relaxation. We observe changes of the barrier height due to the atomic
J relaxation up to 200%, suggesting a very strong coupling between the structural and the magnetic

0 degrees of freedom. The reverse interplay is weak, i.e. the magnetic state has little effect on the
1 structural relaxation at equilibrium or undernon-equilibrium,current-carryingconditions.

] PACSnumbers: 75.75.+a,73.63.Rt,75.60.Jk,72.70.+m
l
l
a
h
Existing experimental techniques are capable of con- ble metals andthe basic electronic structure of a generic
-
s structinglow-dimensionalmagneticconstrictionsconsist- magneticmetal. Thishasthebenefitofbeingreasonably
e
ingofatinynumberofatomsandmeasuringtheirtrans- realistic while keeping the computational overheads to a
m
port properties [1, 2, 3]. In this rapidly evolving field minimum.
t. experiments and theoretical simulations are closely en- The magnetic and structural degrees of freedom both
a
tangled and the latter can provide important informa- enter our model as classical variables. We associate a
m
tion on the fundamental dynamics at the atomic scale. classical magnetic dipole moment (MM) and a set of
-
d The modeling of atomic-sized ferromagnetic devices un- Cartesian coordinates to each atom in the system. The
n der bias requires the combined description of electron magneticstate(MS)isdefinedbyasetofangularcoordi-
o transport and of the local magnetization and structural nates Φ ≡ {φ }. These represent the angles of the MMs
i
c dynamics at the atomic level [4]. In fact, since the spin- of the “live”atoms (those contributing to the dynamics)
[
polarizedcurrenttransfersbothspinandcharge,itexerts with respect to a given direction. The structural state

1 a twofold effect on the current-carrying structure, inter- (SS) is defined by a set of spatial coordinates R≡ {RRR }
i
v
linking its structural and magnetic degrees of freedom. for these atoms. The interplay between MS and SS is
0
0 In order to investigate the mutual interplay between then investigated by either fixing Φ and evolving R or
2 magnetic and structural properties, we have developed by fixing R and evolving Φ. In the first case we study
1 a computational scheme for evaluating spin-polarized how the MS affects the structural relaxation, and in the
0 currents and the associated current-induced forces and second how the SS modifies the magnetic dynamics.
6
torques. On one hand, we are able to examine the ener- Ourmixedquantum-classicalHamiltonianforthissys-
0
/ geticsofmagnetizationdynamicalprocessesasafunction tem reads
t
a ofthe atomic rearrangementsinmagneticpointcontacts
m under bias. On the other hand, we can ravel out the H(Φ,R) = [HiTjB(R)+Viσj(Φ)]c†iσcjσ+
- effect of the magnetic configuration itself on the struc- iX,j,σ
d
tural relaxation. Our main finding is that this interplay +Ω(R)+W(Φ) (1)
n
is strong only in one direction. While the atomic re-
o
c arrangements can modify drastically the point contact wherec†iσ andciσ arecreationandannihilationoperators
: magneto-dynamics,the magnetic configuration has little for electrons with spin σ at atomic site i. H(Φ,R) is
v
i effect on the atomic configuration. separatedinto severalquantum and classicalterms. The
X
Our computational approach combines the non- first term reads
r
equilibrium Green’s function (NEGF) method for eval-
a q
ǫc a
uating the charge density and the current [5, 6] with a HTB(R)= E +VTB(R) δ − f (2)
description of the magnetization dynamics in terms of ij (cid:2) 0 i (cid:3) ij 2 (cid:18)Rij(cid:19) (cid:12)(cid:12)j6=i
(cid:12)
quasistatic thermally-activated transitions between sta- (cid:12)
tionary configurations [7]. This scheme is general and where E0 is the on-site energy for anisolatedatom, ViTB
is conceptually transferable [7] to first-principles Hamil- represents an empirical approximation to the second-
tonians, for instance [6], within density functional the- order Coulomb energy at the i-th site
ory. However,itiscurrentlyimplementedinanempirical
κ ∆q
single-orbitaltight-binding (TB) model[8]with parame- VTB(R)= f (R)∆q = el k (3)
ters chosen to simulate the mechanical properties of no- i Xk ik k Xk Ri2k+κ2el/U2
p
2
where we have defined f (R), κ = e2/4πε = 14.4 The remaining terms in Eq. (1) involve only classical
ik el 0
eV˚A, ∆q = ρσ −(ρ ) is the deviation of the elec- variables. We have adopted a repulsive pair potential
j σ jj 0 jj
tron occupanPcy of site j (ρσjj being a diagonal term of which decays as an inverse power law [8], [16]
the density matrix for spin σ) from its equilibrium non-
11
interacting value ρ . The last term in Eq. (2) is the 1 ǫ a
0 f
Ω(R)= Ω (R)= , (5)
intersitehoppingintegral,dependingasaninversepower 2 ij 2 (cid:18)R (cid:19)
q =4 on the interatomic distance R =|RRR −RRR |. Here iX,j6=i iX,j6=i ij
ij i j
we use ǫ = 7.8680 meV, c = 139.07, a = 4.08 ˚A, which
f and a Heisenberg-type nearest-neighborsspin-spin inter-
are parameters fitted to elastic properties of bulk gold
action between the localized spins
[8]. The value of the on-site Coulomb repulsion is U =7
eV and the band filling is ρ = 0.7 electrons per atom. J J
0 W(Φ)=− dd SSS ·SSS =− dd cos(φ −φ ),
This non-integer band filling effectively accounts for the 2 i j 2 i j
s-p-d hybridization. iX,j6=i iX,j6=i
(6)
The structure of our atomic point-contact system is
where J > 0 is a parameter describing the strength of
dd
presented in Figure 1. It consists of a linear chain of
the intersite exchangeinteraction. Typicalvaluesfor the
three atoms attached to two semi-infinite leads with a
exchange parameters are J = 1 eV and J = 50 meV,
dd
simple cubicstructureand3×3atomcross-section. The
which are of the same order of magnitude as those for
lead magnetizations are polarized along the z-axis and
bulk ferromagnetic metals within the s−d model [13],
they are anti-parallel (AP) to each other. The MMs
andthosederivedfromtheCurietemperaturewithinthe
of the atoms in the chain are allowed to rotate in the
Heisenberg model [14]. Clearly, the low coordination al-
(x−z)-plane, and are described by the set of polar co-
ters the values for the exchange parameters [2], though
ordinates Φ = {φ } (i = 1,2,3) about the z-axis. In
i this effect was not explicitly included here. However, we
this setup an abrupt domain wall (DW) is formed in the
haverepeatedthecalculationforJ from0.8eVupto2eV
constriction. Note that our model does not include spa-
as well as for other force parameterizations (for Cu [8])
tial spin anisotropy, therefore the direction of the spin-
andfoundno qualitativedifferencesinthe mainphysical
quantization axis z is arbitrary. In this sense a Ne´el and
results.
a Bloch wall are physically identical within our model.
Forces and torques, associated with the classical
degrees of freedom, are derived from the Hellmann-
z Feynman theorem, generalized for non-equilibrium sys-
tems. [9, 10] In our model the corresponding forces
a
φ φ φ read[11]
1 2 3
(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1) y(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) Fi =− 2(∇iHiTjB)Re[ρij]+∆qi∆qj∇ifij +∇iΩij ,
0 1 2 3 4 Xj6=i(cid:2) (cid:3)
(7)
a x a
wherethe liveatoms(i=0,...,4)arethoseinthe chain
and their leftmost and rightmost neighbors in the leads
µ
1 a (i = 0,4). Index j runs over all the atoms in the self-
V µ consistent region, which apart from the chain includes
2 two atomic planes from each lead (see Fig. 1). The
torques T read [7]
FIG.1: (Color online) Schematicatomic configuration of the
magnetic point contact investigated in this work. Some dis- J Jdd
T =− s sinφ − [sin(φ −φ )+sin(φ −φ )] ,
tances are magnified for clarity. i 2 i i 2 i i−1 i+1 i
(8)
The spin-dependent term in the electronic part of Eq. where s = ρ↑ −ρ↓ is the on-site spin-polarization (i =
i ii ii
(1) is approximatedby a classicalHeisenberg-type inter-
1,2,3), φ = 0 and φ = π are considered frozen and
0 4
action as follows
aligned along the same direction as in the corresponding
J lead.
Viσj(Φ)=−Jσσσ·SSSiδij =−σz2 cos(φi)δij, (4) Thequantumtransportproblemissolvedbyusingthe
NEGFmethod [5, 6,7]. An analyticalexpressionis used
where σσσ = (0,0,1σ ) is the spin of the current-carrying forthesurfaceGreen’sfunctionoftheleadsandapositive
2 z
electrons, collinear and polarized along the z axis (σ = external bias V is introduced as a rigid shift of V/2 of
z
±1) and J > 0 is the exchange integral. We assume theon-siteenergiesoftheleft-handsideelectronreservoir
that the local MMs, associated with the local spins, are andof−V/2fortheright-handsidewithrespecttotheir
constant in magnitude [4], and impose |SSS |=1. equilibrium value. The non-integer band-filling of the
i
3
spin-split s band in the leads generates an asymmetry motion induced displacements ∆y = y (φ )−y (00π)
φ V 2 V
in the transportpropertiesof the spin-up andspin-down are monotonic functions of φ [see Fig. 2(b), where
2
conduction channels upon left-to-right inversion of the V = 1V]. They represent nearly rigid translations of
system in its AP state. The relative spin-polarization the whole atomic chain in the direction of the electron
Pσ =(n↑−n↓)/(n↑+n↓) of the density-of-states at the flow. However, these displacements are very small and
Fermi level, nσ(E ), of the left lead in equilibrium is constitute about 3% of the displacements from the uni-
f
about-19%(-17%)for J =1 eV(2 eV). The net current form structure ∆y = y ({φ(1,2)})−y [Fig. 2(a)].
uni 0 uni
is calculated as in Ref. [7] . The self-consistent density- The former depend weakly on the bias [Fig. 2(d)].
matrixisusedintoEqs. (7)and(8),andbothforcesand The effect ofthe bias voltageon the atomic relaxation
torques are determined. ∆y =y (φ)−y (φ)forφ={φ(1,2)}isdemonstratedin
V V 0
Figure 2(c). The relative current-induced displacements
) are smooth functions of V and promote a tendency to-
Å
-2φφ) - y () (10uni 1-0055 V=0 φφ((12))==(00a0π)ππ φ)φ)y( - y(V0-011 (c) amsbwhiriataeaevirnvsid-otendssfgetd∆tpomhiemyetunφhledst.ereaiipnmzMIlnaceeetStieaooorb{nrefφrds.u∆(te1pir|n,y∆t2{g)oDφ}ylfy(,VW2,i)nw|}ts/hham[[esiecea.<hfeggo.nriF0smi(i.tπg7aue,.%rd0re2,ea((πsrtudue)op)l]t,a]tt.olahomrOfeVeottdsfhhot=ieesuiprnnw2ldMsaeecVantSeko)--,
( -10
0 produce an effect of similar magnitude on the atomic re-
y 0 1 2 3 4 -2 laxation (not shown in this work). This establishes that
atomic index
0.3
0.3 ) (d) the structural properties of a magnetic nano-device un-
) π derbiasaretoalargeextentindependentofthemagnetic
00π( 0.2 V=1V 00(V aattoomm ""12"" state.
V y atom "3" Next we address the converse, that is whether the
φ) - y210.1 0ππ) - 0.2 smtrigurcattuiroanl.rTelhaixsaitsiocnalacffuelacttesdthfreomenetrhgeytobraqrurieerT2foarsDtWhe
(V (b) (V MM of atom “2” is quasistatically rotated. The other
1 y
y 0 0.1 two live MMs are continually relaxed so that their asso-
0 0.5 1 0 0.5 1 1.5 2 ciatedtorques arekept atzero[7]. The net workfor this
φ/π V (V)
2 rotation is then
FIG. 2: (Color online) Displacements (in picometers) of the φ2
W(φ )=− T (φ′)dφ′ (9)
atomsinthechain: (a)fromtheuniformgeometryatV =0; 2 Z 2 2 2
0
(b) as function of the DW migration reaction-coordinate φ2
at V =1 V; (c,d) as function of the bias voltage V. See text Our calculated DW migration barriers (DWMB) for a
for details. Here J =1 eV, Jdd =50 meV. few values of J are shown in Figure 3. A key feature of
these profiles is the tilt of the barrier at any finite bias
The “live” atoms in the constriction are relaxed at a even for a uniform atomic arrangement. This tilt results
givenfinitebiasforafixedMS.Forsymmetryreasonsthe from the fact that in the absence of the special symme-
atomic relaxation always results in displacements only try in the density of states that characterize the present
along the longitudinal direction (y-axis). Thus, we de- simplecubicstructureinthecaseofahalf-filledband[7],
note the relaxed atomic positions at a bias voltage of V the leading contributions to current-induced forces and
and a magnetic state φ during relaxationby yV(φ). The torquesare linear in the bias. This asymmetry results in
initial geometry, denoted by yuni, is that of equidistant a preferential spatial localization of the DW at a given
atoms with a nearest-neighbor distance of a=2.5 ˚A(see bias: underthepresent(positive)biasthe(0,0,π)config-
Fig. 1). This is near the equilibrium bond length of a urationisstable,whilethe(0,π,π)isatmostmetastable.
periodic 1D chain within our model [12]. However, such The DW can then be driven back and forth in the con-
a bond length produces a compressive stress in the bulk striction by an alternating current. That is an explicit
leads, as a result of which the leftmost and rightmost realization of current-driven DW motion.
atoms are pushed slightly outof the leads and the whole We have then investigated the contribution of atomic
chainatequilibriumshrinksby about2%[seeFig. 2(a)]. relaxation to the DW migration barrier. The bottom
Firstweinvestigatehowthe MSoftheconstrictionaf- panels of Figure 3 show ∆W = W −W and ∆W =
0 0 uni
fects its structural relaxation. We relax the live atoms W −W as functions ofφ atdifferent bias voltagesV,
uni 2
in {φ(1)} = (0,0,π) or {φ(2)} = (0,π,π) configuration, whereW(φ )istheDWmigrationwork(seeEq. (9))for
2
whichrepresenttwopossiblespatialpositionsofthe DW atomicstructurerelaxedatthegivenbiasandMS,while
inside the constriction, as well as for intermediate MSs W (φ ) andW (φ )correspondto atomicstructure re-
0 2 uni 2
with φ ∈ [0,π] and φ such that T = 0. The DW- laxed at V = 0 and to a uniform (unrelaxed) structure
2 1,3 1,3
4
J=1eV J=1.5eV J=2eV J=1eV J=2eV
7 70 47.5
6 30 60 100 47.4 V=1V
5 50 47
V) A) A)
e 4 20 40 µ µ
(m 3 30 80 47.2I ( 46.5I (
W V=1V
2 10 20
46
1 10 47
0 0 VV==00V.5V 0 A) 60 0 φ0./5π 1 0 φ0./5π 1
1.5 V=1V µ 2 2
W00.6 1 2 I ( r aetl a φxed
∆ 0.4 40 2
1
eV) 0.200 0.005 00 Α) 02 V=0 Α) V=0 02
m µ µ
(uni-2 -4 -1-05 20 ∆Ι (--42 uni ∆Ι ( uni --42
W -4 -8
∆ -15 0 V 1 (V) 2 0 V 1 (V) 2
-6 -12 -20 0
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 1.5 20 0.5 1 1.5 2
φ/π φ/π φ/π
2 2 2 V (V) V (V)
FIG. 3: (Color online) DW-migration energy barriers at dif- FIG. 4: (Color online) I −V curves for a geometry relaxed
ferent bias V for J = 50 meV and J = 1,1.5,2 eV (panels at V. Solid (dashed) lines represent (0,0,π) [(0,π,π)] state.
dd
from left to right). The solid (dashed) lines are for struc- Topinset: dependenceofnetcurrentforV =1VontheDW-
ture relaxed at (0,0,π) [(0,π,π)]. W(V,φ2) corresponds to migration reaction coordinate φ2, circles represent structure,
geometry relaxed at the given bias V; ∆W0 = W −W0 and relaxed at φ2 [see Fig. 2(b)]. Bottom inset: ∆I =I−I0/uni,
∆Wuni=W −Wuni, where W0 refers to geometry relaxed at whereI0 (Iuni)refertorelaxedatV =0(uniform)structure.
V =0 and Wuni to uniform geometry.
In conclusion we have developed a method to investi-
respectively. The small current-inducedatomic displace-
gate the interplay between the magnetic and structural
ments (see Fig. 2) systematically increase the DWMB
height(∆W >0)by afew percent. ∆W doesnotshow degrees of freedom of a ferromagnetic atomic point con-
0 0
tact under bias. We have used it to assess the effects of
a dependence on the MS.
the structural relaxation on the magnetic DWMB and
However,therelaxationatV =0fromtheuniformar-
reversely the effect of the magnetic configuration on the
rangement, which shortens the interatomic distance be-
structural relaxation. Our main finding is that the in-
tween the live MMs by about 4% [Fig. 2 (a)], reduces
terplay is only in one direction, that is the structural
theheightofDWMBapproximatelyby∆W /W(φ )=
uni 2
relaxation strongly modifies the DW migration barrier.
25−30% (see Fig. 3) for J = 1.5,2 eV. This effect can
increase dramatically (to about 200%) for exchange pa- In particular, we have found that the DWMB shows
rameters close to magnetic phase transitions. [7], [17] a substantial asymmetry, which increases with the ex-
Thus the magnetic properties are strongly affected by ternal bias even for a spatially symmetric system. That
the geometry of the contact, especially in the region of opens the possibility of voltage-controlled DW motion
parameterswhere the J couplingmechanismstartscom- in such systems. The current-induced displacements
peting with the direct exchange mechanism. (∆y /a < 0.7%) from the relaxed at V = 0 struc-
V
Finally, the effect of the structural relaxation on the ture produce a small positive shift in the DWMB height
conductivity of the system is found to be small (Fig. 4). (∼3%). Thisissmallcomparedtotheeffectoftherelax-
The current is clearly insensitive to the DW migration ationfromtheinitialuniformatomicconfigurationatthe
within the constriction. The overall variation of the net given bias. The latter is ∆y /a < 4% and has a much
uni
current for the rotation of φ for fixed geometry, which moredramaticeffectontheDWMBprofile,reducingthe
2
is in itself a small quantity, is further substantially com- barrier height by upto 2/3 or even making the alterna-
pensated by structural rearrangement,i.e. the structure tive MS unstable, i.e. blocking the DW migration, for
is found to respond to the magnetic rearrangement by some exchange parameters. That is a signiture of the
structural adjustment, which minimizes the variation in strongnon-lineardependenceofspin-polarizedtransport
the conductivity (see inset of Fig. 4). We have also propertiesonstructuralrearrangements. However,struc-
found a decreaseinconductivity due to relaxationofthe ture is not affected by the DW migration under bias.
structure from the uniform geometry. This agrees quali- Thisissobecauseinteratomicforcesdependonthetotal
tativelywiththefindingsinRef. [15],althoughtheeffect charge density of the current-carrying electrons, not on
we observe is much smaller in magnitude. their spin polarization.
5
This work has been sponsoredby the Irish Higher Ed- 72, 134407 (2005)
ucational Authority under the North South Programme [8] A. P. Sutton, T. N. Todorov, M. J. Cawkwell and J.
for Collaborative Research. The authors wish to ac- Hoekstra, Phil. Mag. A 81, 1833 (2001)
[9] T.N.Todorov,J.HoekstraandA.P.Sutton,Phil.Mag.
knowledgetheSFI/HEAIrishCentreforHigh-EndCom-
B 80, 421 (2000)
puting(ICHEC)fortheprovisionofcomputationalfacil-
[10] M. Di Ventra, S. T. Pantelides, Phys. Rev. B 61, 16207
ities and support.
(2000)
[11] T. N. Todorov, J. Phys.: Condens. Matter 13, 10125
(2001)
[12] T.N.Todorov,J.HoekstraandA.P.Sutton,Phys.Rev.
Lett. 86, 3606 (2001)
∗ Contact email address: [email protected] [13] C. Zener, Phys.Rev. 81, 440 (1951)
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dd
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dd
[7] M. Stamenova, S.Sanvito, T. N. Todorov, Phys. Rev.B occurs for J between 0.8 eV and 0.9 eV.