Interlayer coupling enhanced by the interface roughness

0.19 MB·English

by  Ching-Hao Chang

Interlayer coupling enhanced by the interface roughness
Ching-Hao Chang and Tzay-Ming Hong
Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan, Republic of China
(Dated: January 8, 2009)
Previous experiment on Fe/Cr/Fe(001) trilayers reported a surprising observation that the in-
terlayer exchange coupling could be enhanced drastically by the bombardment of irradiation even
at low fluences. We propose that it is due to the resonant states in the spacer, made possible
when the topography of both interfaces is correlated and exhibits prominent Fourier components.
A systematic procedure is developed tohandle theinterface roughness and predict on how to opti-
9 mizetheinterlayercoupling. This methodcan beextendedto bridgethegap between theories and
0 experimentsin otherheterojunctions.
0
2 PACSnumbers: 43.30.Hw,75.70.-i,68.35.Ct,79.60.Jv

n
a
I. INTRODUCTION based on a perturbative treatment of the IR. This mod-
J
ifies the wavefunction of the conduction electrons in the
8
spacerandthe interlayercouplingtheymediate. Mainly,
The interlayer exchange coupling has been studied for

] more than twenty years[1, 2, 3, 4, 5, 6] with applica- we found two additional terms to the scattering wave-
n
functions due to the topography of IR besides the usual
tions in phenomena such as Giant Magnetoresistance[7,
n
static average results. How and when these extra terms
- 8] (GMR) and Tunnelling Magnetoresistance (TMR).
s When it comes to estimate the effect of interface rough- canleadtoenhancementofthe interlayercouplingis ad-
i
d ness (IR)onthe coupling,thereare few microscopicthe- dressed.
t. ories other than resorting to static average[4, 9, 10, 11]. In section II, we discuss the energy spectrum of quan-
a Arecentexperiment[12]reportedthatthestrengthofthe tum well states (QWS) in a trilayer system, which
m coupling in Fe/Cr/Fe(001)trilayerscouldbe modified in is shown to be broadened by the rough interface, as
- a controlled manner and enhanced even at low fluences expected[20] from taking the static average. In addi-
d
by the ion beam irradiation. The ion bombardment is tion, whenthe two interfacesare assigneda commonpe-
n
o known to be an excellent tool for patterning magnetic riodic variation, the spectrum will get shifted upwards.
c areas on interfaces of multilayer system without chang- In section III, a resonance is predicted to occur when
[ ing the sample structure[13]. the length scale of IR matches the spacer width or the

1 This enhancement of interlayer coupling is not ex- Fermi wavelength. We propose that the latter matching
is responsible for the enhancement of interlayer coupling
v pected by the conventional approach to statistically av-
by the ion-beam bombardment. Conclusions and discus-
5 erage over the IR. Many experiments[12, 14, 15] have
6 been dedicated to address this surprising finding. They sionsare arrangedin sectionIV, inwhich other coupling
9 mechanisms,suchasthemagneticdipoleinteractionand
tailored the interface into exhibiting certain patterns by
0 thebiquadraticcoupling,andtheeffectofalloyingatthe
the ion beam irradiation, but so far there is no conclu-
1. sive result[14, 15]. Among the many possible coupling interface are addressed. To preserve the conciseness of
0 mechanisms in the trilayer systems that might give rise the main text, detailed calculations are all arranged in
9 the Appendices.
to this enhancement, the authors of Ref.[12] have esti-
0
mated the percentage of having the magnetic bridges
:
v and concluded that their contribution to the coupling
Xi wasminor. The spin density wavein bulk chromiumhas II. ROUGH INTERFACE AND CORRECTIONS
been known[16, 17] to be crucial at understanding how TO QUANTUM WELL STATES
r
a the electronic structure and the topology of Fermi sur-
faceinfluencethemagneticpropertiesofFe/Crmultilay- We start by studying a simple 2-D case: an interface
ers. However, since the enhanced coupling was observed describedbyx=Asin(py)andassumetheeffectivemass
with a spacer of only 8 monolayers[12], the indirect ex- to be the same for carriers at different sides of the inter-
changeis morelikelytobe mediatedbythe conventional face. As a plane wave comes in with momentum (k ,k )
x y
Ruderman-Kittel-Kasuya-Yosidainteraction[6]insteadof from the left, we assume the amplitude A of the rough-
spin density waveswhich occur[17, 18] only abovea crit- ness to be much smaller than both 1/k and 1/p. The
x
ical thickness of 30 monolayers. Another possible candi- k A 1 allows us to treat the roughness as a perturba-
x
≪
date for the coupling is magnetic dipole interactions[19], tion, while pA 1 precludes the possible emergence of
≪
madepossiblebyIR.Nevertheless,itisnotlikelytodom- local states at the interface, as in water waves along the
inate the coupling because the analysis of STM images coastline. Underthis limit,the actionofstaticaveraging
showed that the thickness fluctuations of the spacer in overoneroughinterfacecanbeshowntobeequivalentto
Ref.[12] was only one monolayer. thescatteringoftwoeffectivesmoothinterfaces,thefirst
In this article, we present the first microscopic theory and second terms in Eq.(1). Whiles, the topography at
2
theinterfacecreatesthethirdandfourthterms,ofwhich
the strength is proportionalto A and ky is shifted by p: 12
Φin ≈12heik√x2AΦ0in(x− √A2,y)+e−ik√x2AΦ0in(x+ √A2,y)i its) 180
n
+aq(~11)e−iq1xx+i(ky−p)y+aq(~12)e−iq2xx+i(ky+p)y (1) . u 6
b
r
where Φ0in(x,y) denotes the wave function scattered by (a 4
a smooth interface and a(1) are defined in Appendix A I 2
q~1,2
whichcontainsdetailsofthefirst-orderperturbationcal-
culations.
0.1 0.2 0.3 0.4 0.5 0.6
Inatrilayersystem,QWSdeterminetheinterlayerex-
E
change coupling. It is crucial to study the variation of
QWSduetotheIR,whichbroadenstheresonancepeaks,
shifts the energy spectrum, and sometimes reduces the FIG.1: Photoemissionintensityisplottedasafunctionofthe
number of bound states. We find that the energy shift energy of quantum well states when both interfaces exhibit
thesame topography x=2Asin(py). The results for smooth
becomes large when (1) 1/p is comparable to the thick-
interfacesareplottedindashedlineforcomparison. Thegray
ness of the spacer D and (2) the topographies on these
line takes care of the IR by static average, while the solid
two interfaces are correlated, which feature could not be
line further considers the corrections due to the first-order
obtained by the static average[4, 9, 10] or another ap- perturbation. Theparameterschosenforthisplotarem⋆ =1,
proach which combined it with the Green’s function[21]. the depth of quantum well is V0 = 0.5 in unit of Ry/π2,
An example is shown in Fig.1, assuming both interfaces 2A = 0.5 in unit of the Bohr radius, p = 0.5, broadening
exhibit the same topography x = Asin(py). From Ap- factor definedin AppendixC is set to be1, and D=6.8.
pendix C, it is shown that the only effect of static av-
erage is to broaden the peaks in the energy spectrum
that represent bound states. When the first-order per- the QWS in the interface Brillouinzone[4, 6] (IBZ). The
turbation corrections due to the IR are added, the spec- existence of interfaces without the IR affects the total
trumbecomes shifted andthe originalbound statesnear electron energies of the spacer, the shift of which equals
the barrier edge can become unbounded. This shift has
beenobservedinthephotoemissionexperiment[20]. Note 1 ∞ 1 EF
E = − Im dE d2~k rn rn e2ink D
that both features are more pronounced at high energy △ 4π3 nZ Z k L,k R,k ⊥
states because the shorter wavelength of QWS renders nX=1 −∞ IBZ ⊥ ⊥
(2)
them more susceptible to the topography of the inter-
perunitareaofinterface. Thereflectivecoefficientsr
L/R
face. We shalldiscusslater ourfinding thatthe p 1/D
≈ atthe left/rightinterfacearefunctions ofthe wavenum-
thatcausesthebiggestspectrumshiftoftenbringsabout
beralongthe normaldirectionoftheinterfacek ,which
⊥
thegreatestenhancementorsuppressionoftheinterlayer
dependsontheenergyE and~k ~k ~k . Theintegern
coupling. Otherdetails,suchastheinterfaceintermixing k ≡ − ⊥
denotes the number of round-trip reflections by the two
andBrillouinzonedilation[22],seemnottobecrucialfor
interfaces. In the presence of IR, two corrections are in-
qualitative predictions.
troduced to Eq.(2): static average terms in Eq.(1) come
A similar feature is found in the scattering states,
in through the r , while the last two terms due to the
namely, interlayer coupling can be enhanced or sup- L/R
perturbation contribute additional close loops when the
pressed by the IR when (1) the topographies on both
topography of both interfaces are correlated. By use of
interface are correlated and (2) p happens to be compa-
the T-matrix and perturbation methods, we derive the
rable to one of the wavelengths k that satisfy the condi-
energy shift in Appendix B, which takes the form:
tion for the Fano resonance. Although the shift of these
k by IR is small, they somehow collaborate to creat a 2 EF
profound effect on the coupling. Er Im dE d2~k
△ ≈ (2π)3 Z Z k
−∞ IBZ
1
− rn (1 d2k2)nrn (1 d2k2)ne2ink D
III. INTERLAYER EXCHANGE COUPLING hXn n R,k⊥ − L ⊥ L,k⊥ − R ⊥ ⊥
+ b(1) b(1) ei(q +k )D (3)
In this section, we will study the effect of IR on the X L,kyqy R,qyky ⊥ ⊥ i
interlayer exchange coupling for a 3-D trilayer system, qy
whichconsistsofaspacerofwidthDandpotentialV =0
where d are corrections after averaging the IR on
sandwiched between layers L and R with potential V , L/R
L (1) (1)
V , respectively. Assuming the lattice structure along the left/right interface, and b (b ) are addi-
R L,kyqy R,qyky
theplanardirectionsoftheinterfaceisideal,thecoupling tionaltransmissioncoefficientsfrommomentum k ,k
⊥ y
{ }
strength is a superposition of the contributions from all to q ,q (vice versa) induced by the tomography on
⊥ y
{ }
3
interface L (R). We checked that the above equation
indeed reduced to Eq.(2) at the limit of p 0. 6
→
An important application of trilayers is the GMR and
4
TMR systems where the two side layers exhibit perma-
nent magnetization. When they are ferromagnetically 2
coupled, the energy shift by Eq.(2) can be calculated
8 10 12
by evaluating the reflective coefficients under the corre-
J 14
sponding Zeeman field. It is found to differ from that
of the antiferromagnetic case. The magnetic coupling -2
strength J(D) can then be defined as the difference be-
-4
tween these two energies[6]. For simplicity, we assume
the two magnetic layers to be of the same material and -6 D
their interface topographies are correlated. After some
routine calculations in Appendix B, the magnetic cou-
pling is found to be: FIG. 2: Coupling strength in unit 10−5Ry/π2a0 between
roughinterfaceswithrandomtopographyin2-Discalculated
J(D) 2 Im EF dE d2~k by Eq.(4) (in dashed line) and the static average Eq.(5) (in
≈ (2π)3 Z Z k grayline)asafunctionoftheaveragewidthDinBohrradius
−∞ IBZ
nXn −n1(cid:2)rpn(1−d2k⊥2)n−ranp(1−d2k⊥2)n(cid:3)2e2ink⊥D aVm0pa.x=Timh0ue.m0p1aaramanmpdleitVtueardpse=woef0tu.h1seed(IeRnareoern:gybthouetnhFitienritmseriRfaeync/eeπri2gs)y,aE0a/nF3d.=Ftho1er,
comparison,theresultforsmoothinterfacesisplottedinsolid
+ b(1) b(1) b(1) b(1) ei(q +k )D
Xqy (cid:2) p,kyqy − ap,kyqy(cid:3)(cid:2) p,qyky − ap,qyky(cid:3) ⊥ ⊥ o line.
(4)
whererp/rap denotethereflectivecoefficientsforcarriers 6
withspinparallel/antiparalleltothemagnetizationofthe
4
layer they reflect from.
2
A. Uncorrelated interfaces
J 8 10 12 14
-2
An intuitive equation, which resorts to static average -4
over the IR, has been widely used by experimentalists
-6
and theoretists alike[4, 9, 10, 11]:
-8
D
J¯(D)= w(n)J(nd) (5)
X
n
wherew(n)istheweightingofn-layersamongthevarious FIG. 3: Coupling strength is plotted as a function of the
widths bewteen two rough interfaces, d is the width of average film width D as Fig.2 with the same parameters ex-
one unit layer in the spacer, and D is defined as the cept that the two interfaces are now assumed to share the
mean width w(n)nd. This weighting w(n) is usually same topography A(y) = 1/3sin(py). Again, the result for
n
assumed to oPbey the Guassian distribution[4, 9, 10, 11]. smooth interfaces is shown in solid line for comparison. The
gray/dashed lines represent p=0.05/1.5, respectively.
When the IR on the two interfaces are uncorrelated,
thelastterminEq.(4)duetotheirinterferencevanishes.
TheremainingtermisshowninFig.2tobeequivalentto
thestaticaverage. Itcanbeseenfromthefigurethatun- survive and give rise to new signatures. The correla-
correlated IR always reduces the coupling strength, but tioncanbeintroduced[22]bytheionirradiationthrough
retains the same period as the exact result for smooth the trilayer, which has the capacity of rearranging the
interfaces, which turns out to be roughly π/kF as pre- atomsonbothinterfaces. AlthoughRef.[12]usedanion-
dicted by the Ruderman-Kittel-Kasuya-Yosidaformula - ized helium of 5 keV, smaller than the 2-MeV Au+2 in
a second-order result. Ref.[22], we believe this correlation is still possible be-
cause the penetration depth is inversely proportional to
the cross section of the incoming ions. Consider the ex-
B. Correlated interfaces treme case that both interfaces share the same topog-
raphy, A(y) = sin(py)/3 in unit of a . The coupling
0
When the topography of both interfaces is correlated, strength is plotted in Fig.3 with the same parameters as
the close loops in Eq.(4) formed by the interference will Fig.2.
4
7
8
6
6
5
J
J 4 4
2 3
0.01 0.02 0.05 0.1 0.2 0.5 1
n
0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4
p
FIG. 5: Absolute value of the coupling strength (in unit of
10−5Ry/(2π3a20)) is plotted as a function of ion dosage (in
FIG. 4: Absolutevalueof thecoupling strength for thesame unit of ions/a20) for a 3-D trilayer. Same parameters as the
2-DtrilayerinFig.3 isreplotted asafunctionofpfor afixed
last three figures. The average spacer width is D = 7a0 and
D. Thesolid/graylinesareforD=6/7a0,respectively. The the topography is A(y,z) = 1/3sin(√n y)sin(√n z). The
resonance peaksappear near p 1/D.
≈ resonance peak appears near “n = 0.0025”, consistent with
theconditionfortheboundstatetobecomeresonant,andso
is hard to discern in the figure. This peak is sharper than
that of Fig.4 for a 2-D case with a magnitude close to zero.
Since the gray and dashed lines are for different p, it
isnotsurprisingthattheyoscillatewithdifferentperiod.
However, when D is much larger than the amplitude of IV. CONCLUSIONS AND DISCUSSIONS
bothtopography,the roughnessbecomesimmaterialand
both lines start to merge with the solid line for smooth In conclusion, we have developed a perturbative
interfaces. This is consistent with the experimental ob- method to handle the interface roughness. The broad-
servation that the effect of irradiation diminishes when ening of the peaks in the spectrum and the usual trend
the spacer gets thicker[12]. The important feature to of suppressing the interlayer coupling can be calculated
note here isthat, unlike Fig.2 whereuncorrelatedrough- quantitatively by our procedures, but they are already
ness always diminishes the coupling strength, the cou- capturedbythepreviousmethodofstaticaverage. What
plingherecanbeenhancedatcertainD,aswasobserved is special about our method is that it can predict an
experimentally[12]. enhancement in the interlayer coupling, as has been re-
ported expermentally, when the Fourier transformation
The coupling constant J is plotted in Fig.4 as a func-
ofthetopographyonbothinterfacesisdominatedbythe
tion of p for two fixed values of D. Two signatures are
sameprominentcomponentpwhichiscomparabletok .
worth noticing: (1) both lines exhibit two peaks - the F
We ascribe it to the scattering states under such a cor-
sharp one has been discussed in section II to occur at
relation. In the mean time, if p is close to the inverse
p 1/D and is caused by the bound state, while the
≈ of the average spacer width, we show that the bound
broad one at p k comes from the scattering state
≈ F state, namely, the peaks in the spectrum will be shifted
where we chose k 1; (2) the coupling strength de-
F ≈ and give rise to another sharper anomaly in the inter-
creases when p>1.
layer coupling, which can be either an enhancement or
Let us now generalize to a 3-D trilayer. Again, as- suppression depending on the specific value of D and p.
sume the two interfaces to share the same topography, Under the classicalmechanismofdipole interactions[19],
A(y,z)=1/3sin(√n y)sin(√n z),after being irradiated correlations between the topography on both interfaces
by ion beams with fluence n. The variation of the cou- have been studied to give rise to an enhanced coupling.
plingstrengthwithrespecttonisplottedinFig.5,which However, it is precipitated by the largeness of IR ampli-
resembles the gray line for the 2-D case in Fig.4. The tude instead of when p 1/D or k .
F
≈
position of the first peak is shifted down to such a small Biquadratic coupling[4, 23] has been known to coexist
valuethatitis hardto discernitinthefigure. Notethat in the trilayers with the bilinear one that we calculated
both the qualitative behavior and position of the second in this work. For completeness, we have also checked
peak in Fig.5 agree excellently with the solid-circle line the effect of IR on the intrinsic biquadratic coupling by
inFig.3ofRef.[12]byDemokritovet al. Thereasonwhy the perturbative approach. It is found to be enhanced
we didnotextendthe iondosageto higher valuesis that as well under the same conditions as the bilinear term.
they violate the requirement pA 1 mentioned in the However, the enhanced strength remains three orders of
≪
beginning of section II for the perturbation method to magnitude smaller than the latter, which conclusion is
work. similar to that reported[6] for a smooth interface and
5
without the intralayer coupling. where Φ (x,y) and Ψ (x,y) are the original scattering
0 0
Anothercommonfeaturelinkedtotheion-beamirradi- states for a smooth interface, and Ψ (x,y) carries mo-
0
ationisthealloyingeffect. Althoughthisismorelikelyto mentum(k′,k ). Foranelasticscattering,thedispersion
x y
take place at the interface, the possibility of its happen- relation in Eq.(A2) is:
ing in the whole spacer can not be excluded. When this
occursatsomespecialcomposition,thenestingeffect[24] k2+k2 k′2+k2
mightcomeaboutandgiverisetoanenhancedinterlayer E = x2m∗y = x2m∗ y +V0
coupling. Theeffectofalloyingattheinterfaceisconsid-
q2+q2 q′2+q2
ered in Appendix D. It is found to always suppresse the = x y = x y +V (A3)
interlayer coupling, consistent with the conclusion of ab 2m⋆ 2m⋆ 0
initio calculations[25]. When the interdiffusion is severe,
where m⋆ denotes the effective mass of the charge carri-
the reduction is exponential. If the extent of alloying is
ers. InsertEq.(A3)intoEq.(A2),andusek A(y),k′A(y)
narrow,the correctionis small and the relative enhance- x x
as the perturbation to expand the boundary conditions
mentduetothecorrelationsisthesameasintheabsence
in Eq.(A1). We shall retain only up to the second order
of the alloying.
in perturbation for a and b :
Theexcellentagreementofourconclusionsandfigures ky,qy ky,qy
did not rely on any fitting parameter, and the micro-
scopic topography of the interface roughness is our only a(k1y),qy =−i(qx′ −qx)Tkx,kx′F[A(y),qy −ky]
input. We thus conclude that the enhancement of inter-
=b(1) (A4)
layer coupling strength, the prime interest of this work, ky,qy
does not depend critically onthe anisotropicfield due to a(2) = 1 [(k′ +q′)α+β] (A5)
the interface roughness,northe possiblenew channelfor ky,qy qx+qx′ x x
direct exchange between the magnetic side layers which
b(2) =a(2) α (A6)
is made possible by the irradiation[12]. ky,qy ky,qy −
We thank Professors John Cheng-Chung Chi and
Raynien Kwo and Mr. C. H. Chen for useful comments,
and acknowledgethe support by NationalScience Coun- α=T m∗V [A2(y),q k ] (A7)
kx,kx′ 0F y− y
cil in Taiwan under grants 95-2120-M007-008 and 96-
2120-M-007-002. β =−i2m∗V0F[Xa(k1y),qy′eiqy′y,qy] (A8)
qy′
APPENDIX where the superscripts (1/2)denotes the first/secondor-
der result, the subscript “k ,q ” denotes scattering from
y y
APPENDIX A: PERTURBATIVE METHOD OF ky to qy state, Tkx,kx′ is the transmissioncoefficient for a
SCATTERING smooth interface, and “ A(y),q k ” represents the
y y
F −
Fourier Transformation of(cid:2)function A(y(cid:3)) to the momen-
In order to realize the IR in a microcsopic way, we tum space in qy ky .
−
start from the microscopic Hamiltonian for a 2-D het- When [A(y),p] contains many p components, their
F
erojunction with an irregular topography, A(y), and the contributions to the third and fourth terms in Eq.(1)
potential onthe rightside is set to be higher byV . The tend to cancel each other at the level of first order per-
0
wavefunctions on the left/right sides are respectively de- turbation, unless one or two p dominants. When the
noted by Φ(x,y) and Ψ(x,y). The boundary conditions cancellation happens, we are forced to go up to the sec-
are: ond order. Except when the scattered wave happens to
exhibit the same momentum as the incoming wave, the
Φ(A(y),y)=Ψ(A(y),y) different components againcanceleach other and we are
∂Φ(x,y) ∂Ψ(x,y) leftwiththesameresultasthatofthestaticaverage: For
= (A1)
example,A(y)= B sin(p )+C cos(p y) contains
∂x (cid:12)x=A(y) ∂x (cid:12)x=A(y) n n n n n
(cid:12) (cid:12) many modes, thePpert(cid:2)urbative wavefunctions in(cid:3) Eq.(A2)
(cid:12) (cid:12)
In order to find the scattering wavefunctions, let us are reduced to:
consider a plane wave with momentum (k ,k ) moving
x y
from the left towards the interface. In the limit that Φ(x,y) eikxx+ikyy+R (1 k2d2)e−ikxx+ikyy
A 1/k ,1/p, the scattering states can be obtained by ≈ kx,kx′ − x 0
≪ x (k k′)2d2
treatingtheIRasaperturbationtothesmoothinterface: Ψ(x,y)≈Tkx,kx′h1− x−4x 0ieikx′x+ikyy (A9)
Φ(x,y)=Φ (x,y)+ a e−iqxx+iqyy
0 ky,qy
X where k2d2 =k2 B2 +C2 signifies that they origi-
qy x 0 x n n n
nate from the secPond(cid:0)-order per(cid:1)turbation. The above re-
Ψ(x,y)=Ψ0(x,y)+ bky,qyeiqx′x+iqyy (A2) sults can be shown to be equivalent to the effect of suc-
X
qy cessive scattering from two smooth interfaces with √2d0
6
distance apart. of Rˆ+−. The notation Kˆ± denotes
R
ΨΦ((xx,,yy))≈≈ 1212(cid:2)eeiikk√x√xdd2200ΦΨ00((xx−−√√dd0022,,yy))++ee−−iikk√x√xd2d200ΦΨ00((xx++√√dd0202,,yy))(cid:3) Kˆ+ =Kˆ− =k00⊥ .0.. q0⊥ ···.
(cid:2) (A10) (cid:3)  ... ... 
 
(B3)
When the topography A (y)/A (y) at the left/right
L R
interfaces is considered, the second-order perturbation
result in Appendix A will introduce corrections to the
reflective matrices (Rˆ−++εˆ )/(Rˆ+−+εˆ ):
L L R R
0 b
··· L,qyky ···
+  ... ... 
εˆ−+ =
L  b 0 
 L,kyqy 
 .. .. 
 . . 
 
r d2k2 0 0
 L,k⊥ L ⊥ .. ···
0 .
A(y) (B4)
− 0 rL,q d2Lq⊥2 
FIG. 6: As a plane wave is scattered by a rough interface,  ... ⊥ ... 
 
the second-order effect on the wavefunction can be shown to
be equivalent to the successive scattering from two smooth andsimilar expressionfor εˆ after replacingthe L by R.
R
interfaces. By use of Appendix A, the weighting b(1) and d can be
copied as:
APPENDIX B: CORRECTION TO THE b(L1,)kyqy =i(q⊥−qL′⊥)TL,~kk~′F[AL(y),qy−ky]
INTERLAYER COUPLING
b(1) =i(k k′ )T [A (y),k q ]
R,qyky ⊥− R⊥ R,q~q~′F R y − y
Interlayer exchange coupling in Eq.(2) can be written d = B2 +C2
L Ln Ln
in a more general way by the T Matrix[6]: Xn (cid:0) (cid:1)
d = B2 +C2 (B5)
1 R Rn Rn
△E = 4π3ImZ dE Xn (cid:0) (cid:1)
E<EF
Tr ln(I Rˆ−+eiKˆ+DRˆ+−eiKˆ−D) (B1) Insertthe full reflectivematrices with IRinto Eq.(B1)
× − L R
(cid:2) (cid:3) andexpandthe tracetermto the secondorderinεˆL/εˆR:
where I is the unit matrix, Rˆ−+/Rˆ−+ are the reflective
L R
matrices fromthe left/rightsmooth interfaces whichis a 1
Er Im dE
functionofenergyE andarenonzeroonlyinthediagonal △ ≈ 4π3 Z
E<EF
components, and the superscript + denotes that the
1 1 1 1
left-going wave is turned into right−-going and vice versa Tr ln(I Mˆ)+εˆ εˆ εˆ (B6)
for +−. Matrix RˆL−+ takes the form[6]: × h − I−Mˆ − 2 I −Mˆ I −Mˆ i
where
r 0 0
L,k
 0⊥ ... ··· Mˆ Rˆ−+eiKˆ+DRˆ+−eiKˆ−D (B7)
Rˆ−+ = ≡ L R
L  0 r 
 L,q 
 ... ⊥ ...  and
 
(B2) εˆ εˆ eiKˆ+DRˆ+−eiKˆ−D+Rˆ−+eiKˆ+Dεˆ eiKˆ−D
≡ L R L R
where r /r are the reflective coefficients of mo- +εˆ eiKˆ+Dεˆ eiKˆ−D (B8)
L,k L,q L R
mentums k⊥ /q ⊥from the left interface. Similar expres-
⊥ ⊥
sion by replacing the subscript L by R is the definition Substituting the above two definitions into Eq.(B6), the
7
interlayer exchange coupling can be obtained as take the limit A (y) 0 or p 0.
L,R
→ →
2 EF
Er Im dE d2~k
△ ≈(2π)3 Z Z k
−∞ IBZ
APPENDIX D: ALLOYING
1
− rn (1 d2k2)nrn (1 d2k2)ne2ink D
hXn n R,k⊥ − L ⊥ L,k⊥ − R ⊥ ⊥ Besides the topography that we have discussed in de-
+ b(1) b(1) ei[q +k ]D . (B9) tail, the reallignment of atoms at the interface due to
X L,kyqy R,qyky ⊥ ⊥ i ionbeamirradiationshouldalsoinclude alloying. Wees-
qy
timate its effect on the coupling by softening the sharp
boundary at the interface into a smooth Eckart poten-
APPENDIX C: ENERGY SPECTRUM IN ROUGH tial, V(x) = V [1 + tanh(x/a)]/2 where a depends on
0
THIN FILMS the extent of alloying. The reflective probability for this
potential can be found in the literature[26] as:
The photoemission intensity I(E) can be used to de-
termine the energy spectrum[20] of a smooth thin film, r 2 = cosh[2πa(kx−kx′)]−1 (D1)
twhheiclhefti/sraighfutnicnttieornfaocfesthaendretflheectfiivlme cwoiedffithcieDn:ts rL/R on (cid:12)(cid:12) kx,kx′(cid:12)(cid:12) cosh[2πa(kx+kx′)]−1
where k and k′ are the longitudinal momenta in either
2 x x
1 side of the interface. Comparing with the sharp barrier,
I(E) (C1)
∝(cid:12)(cid:12)(cid:12)1−rL,k⊥rR,k⊥ei2k⊥D+iΛ(cid:12)(cid:12)(cid:12) tithyi.smWohdeunlaatioisnmaluwcahysladrigmerintihshaenstthheerFeeflremcitiwveapverolebnagbtihl-,
(cid:12) (cid:12)
where Λ is the broadening factor[20] which depends on Eq.(D1) is suppressed exponentially as exp 8πk′a .
the film thickness and the mean free path. Bound states According to Eq.(2), the magnetic coupling(cid:2)s−trengtxh (cid:3)is
satisfy rL,k rR,k ei2k⊥D = 1 and correspond to the proportional to the reflective probability, and so is ex-
peaksinI(E⊥). W⊥henIRareconsidered,moreclose-loops pected to be reduced drastically by the alloying as con-
from successfive reflections and the action of static aver- cluded by the ab initio calculations[25].
aging will modify both r . Eq.(C1) becomes
L/R If the interdiffusive process is weak, namely, k a
x
≪
1, the alloying only introduces a small correction to the
I(E) 1 r r (1 d2k2)(1 d2k2)ei2k D
∝(cid:12) − L,k⊥ R,k⊥ − L ⊥ − R ⊥ ⊥ interlayer coupling strength:
(cid:12)+ b(1) b(1) ei[q +k ]D+iΛ −2 (C2)
Xqy L,kyqy R,qyky ⊥ ⊥ (cid:12)(cid:12) △Ea ≈△Ee−34(πa)2kFkF′ (D2)
where dL,R, b(L1,)kyqy, and b(R1,)qyky are defined in Eq.(B5). This means that the relative enhancement due to the
It is checked that Eq.(C2) reduces to Eq.(C1) when we correlationsisthesameasbeforeconsideringthealloying.
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