The magnetic anisotropy energy of Ni n ͞Cu͑100͒ is calculated in terms of the spin-polarized fully relativistic Korringa-Kohn-Rostoker method including surface relaxation by using 2D structure constants originally described for low-energy electron diffraction calculations. Investigating different relaxations, an explanation for the reorientation transition from in-plane to perpendicular can be given. For a relaxation of 25.5% (c͞a 0.945) this reorientation occurs at about seven layers of Ni and yields second order terms to the magnetic anisotropy energy that are in excellent agreement with experiment. [ S0031-9007(98)08322-7]    PACS numbers: 75.30. Gw, 75.70.Ak, 75.70.Cn    Thin films of Ni on Cu(100) show an unexpected behavior of magnetic phase transitions  [2] and references therein) can phenomenologically be described by where K 2 refers to the second order term of the magnetic anisotropy energy (MAE) and u denotes the angles of M with respect to the surface normal. As indicated in Eq.  In the present paper the fully relativistic spin-polarized screened Korringa-Kohn-Rostoker (KKR) method  The magnetic anisotropy energy DE a , defined as the energy difference between a uniform in-plane (perpendicular to the surface normal in all planes of atoms) and a uniform perpendicular (along the surface normal in all planes of atoms) orientation of the magnetization of the system was obtained  In order to evaluate DE b 990 k k points in the ISBZ were used, guaranteeing well converged quantities. 0031-9007͞99͞82(6)͞1289(4)$15.0

C Sommers

C Uiberacker

J Zabloudil

L Szunyogh

P Weinberger

CiteSeerX

VOLUME 82, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 8 FEBRUARY 1999
ustriaLattice Relaxation Driven Reorientation Transition in NinyCus100d
C. Uiberacker, J. Zabloudil, and P. Weinberger
Institut für Technische Elektrochemie and Center of Computational Materials Science,
Technical University Vienna, Getreidemarkt 9/158, A-1060 Wien, Austria
L. Szunyogh
Department of Theoretical Physics, Technical University Budapest, Budafoki út. 8, H-1521 Budapest, Hungary
and Center of Computational Materials Science, Technical University Vienna, Getreidemarkt 9/158, A-1060 Wien, A
C. Sommers
Laboratoire de Physique des Solides, Batiment 510, Campus d’Orsay, 91405 Orsay, France
(Received 16 July 1998)
The magnetic anisotropy energy of NinyCus100d is calculated in terms of the spin-polarized fully
relativistic Korringa-Kohn-Rostoker method including surface relaxation by using 2D structure constants
originally described for low-energy electron diffraction calculations. Investigating different relaxations,
an explanation for the reorientation transition from in-plane to perpendicular can be given. For a
relaxation of25.5% (cya ­ 0.945) this reorientation occurs at about seven layers of Ni and yields
second order terms to the magnetic anisotropy energy that are in excellent agreement with experiment.
[S0031-9007(98)08322-7]
PACS numbers: 75.30.Gw, 75.70.Ak, 75.70.Cny-
m-
e
o-
i-
45
as
-
ur-
ly
d
m
rm
ll
g
a-
d
a-
gy
y
Thin films of Ni on Cu(100) show an unexpected be
havior of magnetic phase transitions [1–12]. In contra
to quite a few other magnetic thin films on noble met
surfaces, where with increasing film thickness the m
ments first line up perpendicular to the surface and th
reorientate to an in-plane direction, forn , 7 monolayers
of Ni on Cu(100) an in-plane direction of the magnetiza
tion M is observed [5,6,10] that reorientates [5,6,10,11]
aboutn ­ 7 to perpendicular to plane, switching eventu
ally back to in plane forn . 37 [5]. The anisotropic part
of the free energy for magnetic multilayer systems (se
e.g., Ref. [2] and references therein) can phenomenolo
cally be described by
E ­ 2pM2 cos2 u 2 K2 cos
2 u , (1)
K2 ­ K
y
2 1 K
s
2yd , (2)
where K2 refers to the second order term of the mag
netic anisotropy energy (MAE) andu denotes the angles
of M with respect to the surface normal. As indicated
Eq. (2), whered refers to the film thickness,K2 is thought
to consist of two parts, namely, a thickness-independe
“volume”-like contributionKy2 and a thickness-dependen
“interface-surface”-like contributionKs2 . It is more or less
the interplay of these two constants that together with t
shape anisotropy (magnetic dipole-dipole interaction) d
termines the unique features of the reorientation propert
of thin films of Ni on Cu(100).
In the present paper the fully relativistic spin-polarize
screened Korringa-Kohn-Rostoker (KKR) method [13,1
was applied using the spin-polarized local density fun
tional as given by Voskoet al. [15]. In order to be able
to treat layer relaxation, the occurring screened structu0031-9007y99y82(6)y1289(4)$15.00-
st
al
o-
en
-
at
-
e,
gi-
-
in
nt
t
he
e-
ies
d
4]
c-
re
constants [16,17] have been derived for a system of la
ers which have only the same in-plane translational sy
metry [18–20], but otherwise can differ in the respectiv
interlayer distance. Self-consistency for the effective p
tentials and effective exchange fields (with a uniform or
entation perpendicular to plane) was obtained using
kk points in the irreducible part of the surface Brillouin
zone (ISBZ). In all cases three layers of Cu served
“buffer” at the CuyNi interface and three “empty” layers
as buffer to the Niyvacuum interface; i.e., for a given num
ber n of Ni monolayers the total numberL of atomic lay-
ers investigated self-consistently isL ­ n 1 6. It should
be noted that the present calculations refer to free s
faces of Ni on Cu(100), which implies that in here actual
relaxation effects for a semi-infinite system are include
and not bulklike relaxations as in the paper by Hjortsta
et al. [8].
The magnetic anisotropy energyDEa,
DEa ­ Eskd 2 Es'd , (3)
defined as the energy difference between a unifo
in-plane (perpendicular to the surface normal in a
planes of atoms) and a uniform perpendicular (alon
the surface normal in all planes of atoms) orient
tion of the magnetization of the system was obtaine
[13,14] by making use of the force theorem approxim
tion, namely, as a sum of the respective band ener
difference DEb and the magnetic dipole-dipole energ
contributionDEdd,
DEa ­ DEb 1 DEdd . (4)
In order to evaluateDEb 990 kk points in the ISBZ were
used, guaranteeing well converged quantities.© 1999 The American Physical Society 1289
VOLUME 82, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 8 FEBRUARY 1999
r-
,
e
es
r
the
ur-
e-
f
).
m
In Fig. 1 the magnetic anisotropy energyDEa is
shown together with the corresponding band energy d
ferenceDEb and magnetic dipole-dipole energy differ-
enceDEdd versus the number of Ni layers. Three case
are displayed, namely: (a) The values correspondi
to an fcc “parent lattice” [21] with respect to the ex
perimental lattice spacing of fcc Cu (a0 ­ 6.8309 a.u.);
(b) those corresponding to a (uniform) relaxation of th
interlayer distance by22.5%; and (c) for an interlayer
relaxation of25.5%. It should be noted that case (b
refers to a cya ratio of 0.975 and (c) to acya ra-
tio of 0.945 (experimental value [4,8]), since the (two
dimensional) lattice spacing within the planes of atom
remains unchanged. As can be seen from Fig. 1, t
shape of theDEa curve arises mainly from the band
0 2 4 6 8 10 12 14 16
-0.20
-0.15
-0.10
-0.05
0.00
 unrelaxed
 relaxed -2.5 %
 relaxed -5.5 %
∆ E
dd
 [m
eV
]
n, number of Ni layers
0 2 4 6 8 10 12 14 16
-1.0
-0.5
0.0
0.5
1.0
∆ E
b 
[m
eV
]
0 2 4 6 8 10 12 14 16
-1.0
-0.5
0.0
0.5
perpendicular
in-plane
∆ E
a 
[m
eV
]
FIG. 1. Magnetic anisotropy energyDEa (top), band energy
difference DEb (middle), and magnetic dipole-dipole energy
differenceDEdd (bottom) versus the number of Ni layers on
Cu(100). Triangles, squares, and circles refer in turn to
uniform relaxation by0%, 22.5%, and 25.5%, i.e., to acya
ratio of 1, 0.975, and 0.945.1290if-
s
ng
-
e
)
-
s
he
a
energy contributionDEb, sinceDEdd scales linearly with
the volume and is always negative (DEdd tends to turn
the magnetization in plane). Relaxation of the inte
layer distances leavesDEdd nearly unchanged, however
as can be seen from Fig. 1, it is of crucial importanc
for DEb.
In Fig. 2 the layer resolved band energy differenc
DE
p
b for six, nine, and twelve Ni layers are shown fo
the three cases mentioned above. One can see that
surface and interface contributions are negative, the s
face giving the larger contribution, and that relaxation pr
dominantly increases the contribution from the interior o
the Ni films. Defining therefore the following quantities:
0 2 4 6 8 10 12 14 16 18
-0.3
-0.2
-0.1
0.0
0.1
0.2
 unrelaxed
 relaxed -2.5 %
 relaxed -5.5 %
∆ E
b 
[m
e
V
]
layer
0 2 4 6 8 10 12 14 16
-0.2
-0.1
0.0
0.1
0.2
∆ E
b 
[m
e
V
]
layer
0 2 4 6 8 10 12 14
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
∆ E
b 
[m
e
V
]
layer
FIG. 2. Layer resolved band energy differencesDEnb for six
(top), nine (middle), and twelve (bottom) Ni layers on Cu(100
Triangles, squares, and circles refer in turn to a unifor
relaxation by0%, 22.5%, and25.5%, i.e., to acya ratio of 1,
0.975, and 0.945.
VOLUME 82, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 8 FEBRUARY 1999
gy
i-
en
n-
b-
py
es
lent
un-
al
ch
be-
c
e
n-
of
Z
on
en-
d
k
ns
ion
lf-K
S1
2 ­
X
p­1,4
DE
p
b , K
S2
2 ­
X
p­L23,L
DE
p
b , (5)
KI2 ­
1
sL 2 8d
X
p­5,L24
DE
p
b , (6)
DEb ­ K
S1
2 1 K
S2
2 1 sL 2 8dKI2 , (7)
namely, a contribution toDEb from the CuyNi interface
(K
S1
2 ), from the Niyvacuum interface (K
S2
2 ) and an aver-
aged contribution from the interior (KI2), then one easily
can identify these quantities with the second order co
stants in Eq. (2),
Ky2 ­ K
I
2 , K
s
2 ­ K
S1
2 1 K
S2
2 . (8)
Quite clearly, the only relevant quantity in Eqs. (5)–(8
is DEb; the grouping of terms in the sum over layer de
pendent contributions toDEb serves merely purposes o
interpretation. For this very reason a comparison to t
approach used by Cinalet al. [22] in calculating the mag-
netocrystalline anisotropy in ferromagnetic films does n
seem to be very useful.
In Table I the present values for a relaxation of25.5%
(cya ­ 0.945) are compared to the corresponding con
stants obtained experimentally. It should be noted that
“error bars” in this table for the present values arise fro
the fact that the theoretical data for both nine and twel
layers of Ni, shown in Fig. 2, were used. It is evident from
this table that the present values agree very well with t
experimental data. Furthermore, it is interesting to no
that in comparison to Co and Fe on Cu(100), the sign
K
S1
2 andK
S2
2 is reverse, which in turn explains the in-plan
orientation in the regime of3 # n # 7 Ni layers.
From a theoretical standpoint of view it is temptin
to ask whether a kind of frozen potential approximatio
can be applied to this particular problem of relaxatio
i.e., whether from the self-consistent potentials for th
unrelaxed geometry one can calculateDEb for a relaxed
geometry. In Fig. 3 the band energy differences f
nine layers of Ni on Cu(100) as calculated with thi
frozen potential approximation are compared to tho
corresponding to self-consistently determined potentia
As one can see from this figure only up to rather sma
relaxations (,2%) this kind of approximation seems to
be useful. It is worthwhile to mention that up to25.5%
DEb , as obtained from self-consistent potentials, vari
almost linearly with the relaxation.
In conclusion, we have investigated the reorientatio
transition of the magnetization for free surfaces of Ni o
TABLE I. Volume and surface anisotropy constants (meV)
for free surfaces of Ni on Cu(100) atT ­ 0 K.
Ky2 K
s
2 cya
Experiment [6,11] 70 6 20 2100 ,0.945
Ref. [4] 140136290 0.945
Present calculation 80 6 20 2100 6 20 0.945n-
)
-
f
he
ot
-
the
m
ve
he
te
of
e
g
n
n,
e
or
s
se
ls.
ll
es
n
n
Cu(100) by calculating the magnetic anisotropy ener
within the fully relativistic spin-polarized KKR method
including relaxation of the Ni layers. We found a reor
entation transition from in-plane to perpendicular at t
and seven layers of Ni for22.5% and25.5% relaxation,
respectively. This compares very well to the experime
tal results. Furthermore, not only the experimental o
servation of a negative surface and interface anisotro
constant could be confirmed, but also the actual valu
for the second order anisotropy constants are in excel
agreement with experiment. We can state therefore
ambiguously that atT ­ 0 the reorientation transition of
free surfaces of Ni on Cu(100) is driven by tetragon
distortion and that the actual number of layers at whi
this transition occurs is caused by a delicate balance
tween the interface (K
S1
2 ), the surface (K
S2
2 ), the interior
(KI2) contributions toK2, and, of course, the magneti
dipole-dipole contributionDEdd to the magnetic anisotrop
y energyDEa.
This paper resulted from a collaboration within th
TMR Network on “Ab initio calculations of magnetic
properties of surfaces, interfaces and multilayers” (Co
tract No. ERB4061PL951423). Financial support
the Center for Computational Materials Science (G
308.941), Vienna, the Austrian Science Foundati
(P12146, P12352), and the Hungarian National Sci
tific Research Foundation (OTKA No. T024137 an
No. T021228) is kindly acknowledged. We also than
the computing center at Orsay as part of the calculatio
was performed on their Cray T3E.
0 1 2 3 4 5 6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
∆Eb
∆Eb (nscf)
∆  
E
 [m
eV
]
relaxation (%)
FIG. 3. Band energy difference (DEb) for nine layers of Ni
on Cu(100) versus relaxation. Circles denote a calculat
using unrelaxed potentials, while squares refer to se
consistently relaxed potentials.1291
VOLUME 82, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 8 FEBRUARY 1999
.
v.
d
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Lattice relaxation driven reorientation transition

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Lattice relaxation driven reorientation transition

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