Table Of Content2 x x 4
Anisotropi
Sus
eptibility of La − Sr CoO related to
8
the Spin States of Cobalt
0
0
2 1 1 1 1
N Hollmann , M W Haverkort , M Cwik , M Benomar ,
n 1 2 1
a M Reuther , A Tanaka , T Lorenz
J 1
II. Physikalis
hes Institut, University of Cologne,Zülpi
herstrasse 77, D-50937Köln,
4
1 G2 ermany
Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima
]
l 739-8530,Japan
e
- E-mail: hollmannph2.uni-koeln.de
r
t
s
.
t Abstra
t.
a 2−x x 4
We presenta study of the magneti
sus
eptibility of La Sr CoO single
rystals
m 0.3 x 0.8
inadopingrange ≤ ≤ . Ourdatashowsapronoun
edmagneti
anisotropyfor
- S =0
d all
ompounds. This anisotropy is in agreement with a low-spin ground state ( )
3+ x 0.4 S = 3/2 2+
n
of Co for ≥ and a high-spin ground state ( ) of Co . We
ompare
o
our data with a
rystal-(cid:28)eld model
al
ulation assuming lo
al moments and (cid:28)nd a
c x 0.5
[ good des
ription of the magneti
behavior for ≥ . This in
ludes the pronoun
ed
kinksobservedin the inversemagneti
sus
eptibility, whi
hresult from theanisotropy
1 2+
v and low-energy ex
ited states of Co and are not related to magneti
ordering or
7 temperature-dependent spin-state transitions.
8
0
2
.
1 PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gw
0
8
0
:
v
i
X
r
a
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 2
Transition-metal oxides are known for their
omplex interplay between di(cid:27)erent
degrees of freedom like spin,
harge and orbitals. In some of these systems another
3+
interesting property is found: ions like Co in a
rystal-(cid:28)eld environment
an o
ur in
di(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possible
3
for some
ompounds. A prominent example showing this phenomenon is LaCoO ,
whi
h has been examined and dis
ussed sin
e the middle of the last
entury (see e.g.
Refs. [1(cid:21)11℄).
The existen
e of di(cid:27)erent spin states arises from the
ompetition between
rystal-
3d
(cid:28)eld e(cid:27)e
ts and on-site Coulomb intera
tion. Crystal (cid:28)elds lift the degenera
y of the
states. If the
rystal (cid:28)eld is strong, this
an lead to a violation of Hund's rules. In the
3d
ase of
ubi
symmetry and in a one-ele
tron pi
ture, the (cid:28)ve-fold degenerate states
t e
2g g
are splitintoa three-folddegenerate and a two-fold degenerate level. The splitting
t e 10Dq
2g g
between and states is
alled . With a strong
rystal (cid:28)eld, the ele
trons will
3d6
be for
ed into the low-spin state (LS). Regarding a system, this state
onsists of an
t6 e0 S = 0
2g g
antiparallel alignment of spins with and . On the other hand, the Coulomb
intera
tion manifests itself in the e(cid:27)e
t of Hund's
oupling. A parallel arrangement of
spins minimizes the ele
tron-ele
tron repulsion be
ause the Pauli prin
iple for
es the
ele
trons to o
upy di(cid:27)erent orbitals. In a weak
ubi
rystal (cid:28)eld, this e(cid:27)e
t will
dominate and lead to the high-spin state (HS), whi
h is the
on(cid:28)guration with the
3d6
highest total spin possible in a
ordan
e with the Pauli prin
iple. For a system, this
t4 e2 S = 2 3+
2g g
on(cid:28)guration is with . For Co the
rossover between these two di(cid:27)erent
10Dq = 2.2
spin states o
urs roughly at an energy di(cid:27)eren
e of eV.
3+
3
Inthe
ase ofLaCoO , alow-spingroundstateforCo wasfound[2℄. Interestingly,
the di(cid:27)eren
e between the
rystal (cid:28)eld energies and the promotional energies is so small
thatanex
itedstatewithdi(cid:27)erentspinstate
anbe rea
hedby thermalex
itation. This
ex
ited state has been a subje
t of debate for a long time. Apart from the HS state
t5 e1 S = 1
2g g
des
ribed above, the intermediate-spin state (IS, ) was also dis
ussed [4℄ and
reported in many experiments, but it was shown that it might have been
onfused with
a spin-orbit
oupled HS state [11℄.
2−x x 4
For the layered
obaltates La Sr CoO , mu
h less is known about the spin state
3+ x 0.7
of Co . Based on magneti
measurements [12℄ a HS ground state for ≤ and a
x > 0.7
spin-state transition to an IS ground state for was proposed. This
on
lusion
was based on a Curie-Weiss analysis of the sus
eptibility in a temperature range of
100K to300K and NMR measurements [13℄. Unrestri
ted Hartree-Fo
k
al
ulations[14℄
showed aslightlydi(cid:27)erentpi
ture. Here, threedi(cid:27)erent magneti
phases were found. An
x < 0.39 0.39 x 0.52
antiferromagneti
HS phase ( ), a ferromagneti
HS phase ( ≤ ≤ ) and
x > 0.52
anantiferromagneti
LS-HS-orderedphase ( )wereproposed. These
al
ulations
were also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking into
2−x x 4
a
ount that La Sr CoO is an anisotropi
material with rather strong spin-orbit
oupling
ompared to the tetragonal
rystal (cid:28)eld splitting, the validity of the Curie-
Weisslawisquestionable. Theaimofthispaperistoanalyzethemagneti
sus
eptibility
3+
from a di(cid:27)erent perspe
tive,
on
entrating on the spin state of Co .
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 3
2−x x 4
Figure 1: Magneti
sus
eptibility of La Sr CoO for two di(cid:27)erent dire
tions of the
magneti
(cid:28)eld. The insert is an expanded view of the low-temperature region to show
0.3 x 0.6
the di(cid:27)eren
e between FC and ZFC measurements for ≤ ≤ . The
urves with
the lower sus
eptibility refer to the ZFC measurements.
The single
rystals used for the magneti
measurements have been grown using the
0.3 x 0.8
(cid:29)oating-zone te
hnique in an image furna
e. A strontium doping range of ≤ ≤
2−x x 4
was
overed. Resistivitymeasurementsrevealed thatLa Sr CoO isa stronginsulator
x = 0.5
for all Sr doping
on
entrations. The sample turned out to possess the highest
2+ 3+ 750
resistivity, whi
h is in a
ordan
e with the
harge ordering of Co and Co at ≈ K
that has already been reported [15℄.
The magnetization was measured with a Quantum Design vibrating sample
2
magnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well as
c
perpendi
ular to these planes (the
rystallographi
dire
tion). The
orresponding
χ χ
ab c
omponents and are plotted in Fig. 1. A short-range antiferromagneti
order
0.3 x 0.6
has been found in the
ompounds ≤ ≤ by neutron measurements [16℄. This
frustrated short-range order is also re(cid:29)e
ted by the di(cid:27)eren
e between (cid:28)eld
ooled (FC)
and zero-(cid:28)eld
ooled (ZFC) measurements at low temperatures. The sus
eptibility is
smaller for ZFC than FC below a
ertain freezing temperature.
Figure 2 shows the inverse magneti
sus
eptibility for both orientations of (cid:28)eld.
The main feature of the sus
eptibility in the paramagneti
regime is the pronoun
ed
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 4
2−x x 4
Figure 2: Inverse sus
eptibility of La Sr CoO for two di(cid:27)erent dire
tions of the
magneti
(cid:28)eld. The form of the
urves and the magneti
anisotropy strongly deviates
from Curie-Weiss behavior.
magneti
anisotropy, both in magnitude and form of the
urves. The dire
tion of the
χ χ
ab c
anisotropy is the same for all
rystals, (cid:28)nding to be bigger than .
Here, the magneti
anisotropy arises from band stru
ture and spin-orbit
oupling.
For Mott and
harge-transfer insulators with well-lo
alized moments, band-stru
ture
and
ovalen
y e(cid:27)e
ts
an be approximated by an e(cid:27)e
tive
rystal (cid:28)eld. The
rystal (cid:28)eld
3d
re(cid:29)e
ts the anisotropy of the latti
e and lifts the degenera
y of the states, resulting
in a new set of linear
ombinations of the unperturbed wave fun
tions as a basis. The
expe
tation values of the
omponents of the orbitalmoment result from these new linear
ombinations. Thus, the
rystal's anisotropy may result in an anisotropy of the orbital
moment. Spin-orbit
oupling ties the spin moment to this anisotropy. The spin-orbit
ζP l s i
oupling Hamiltonian will be written as i i · i where the sum over runs over all
ele
trons. The dot produ
t between spin and orbital momentum tends to align these
moments antiparallel for ea
h ele
tron. For an anisotropi
orbital momentum, the spin
is aligned in the dire
tion of maximum orbital momentum [17℄.
d6 d7
The full many-body ground-state for a or
on(cid:28)guration in a
rystal-(cid:28)eld
al
ulation in
luding spin-orbit
oupling and a tetragonal distortions is not simple [18℄
but well known. In order to get an intuitive pi
ture one would like to fall ba
k to a
single ele
tron des
ription. In the limit of full spin polarization this
an be done and
gives important results. In the following we will (cid:28)rst dis
uss the magneti
anisotropy of
2−x x 4
La Sr CoO in terms of a one-ele
tron pi
ture. We will show that ea
h spin state has
a di(cid:27)erent magneti
anisotropy from whi
h, by
omparison to the experiment, the spin
states of the Co ion
an be
on
luded. In order to obtain also a quantitative des
ription
and to verify our simple argumentation we will present a full many-body
rystal-(cid:28)eld
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 5
3d
Figure 3: Splitting of the levels in a tetragonal
rystal (cid:28)eld arising from an elongated
oxygen o
tahedron. The two sket
hes on the left show the o
upation for the LS and
3+ 2+
HS state of Co in a one-ele
tron pi
ture. The right sket
h refers to HS Co . The
real orbitals on the right
an be used as a basis.
al
ulation afterwards.
3d e t
g 2g
Within a
ubi
rystal stru
ture, the states split into orbitals and orbitals.
3z2 r2,x2 y2 xy,xz,yz
The basis
an be
hosen as { − − } and { }, respe
tively. A partially
t L˜ = 1
2g
(cid:28)lled shell
an produ
e a pseudo orbital moment of . Though the individual
d d d
xy yz xz
real wave fun
tions , and of the basis themselves have a
ompletely quen
hed
dx dy
orbital moment, their linear
ombinations are in general
omplex. Writing ml, ml and
dz x y
ml as the orbital wave fun
tion with the orbital moment quantized along the axes ,
z
and , respe
tively, one (cid:28)nds
1
dx = ( d +id )
±1 xy xz
√2 ± (1)
1
dy = ( d +id )
±1 yz xy
√2 ± (2)
1
dz = ( d +id ).
±1 xz yz
√2 ± (3)
10Dq
In the limit of very large
rystal (cid:28)eld splittings and with a partially (cid:28)lled
t t
2g 2g
shell, the moment is isotropi
, despite the large orbital moment. In fa
t, the
p
ele
trons are sometimes
ompared to ele
trons [19℄.
Introdu
ing a tetragonal distortion, the orbital moment be
omes anisotropi
. The
e a b
g 1g 1g
tetragonal
rystal (cid:28)eld splits the
ubi
states into non-degenerate and levels,
t b e
2g 2g g
while the states split into a non-degenerate state and a two-fold degenerate
z
level. As a basis for these states, the real orbital fun
tions
an also be used. The
c
axis of the system is taken to be identi
al with the axis of the
rystal. In the
ase
c
2−x x 4
of La Sr CoO , the oxygen o
tahedron is elongated in the dire
tion. The order of
levels ando
upationsforthe ground states of the two
obaltions isillustratedinFig.3.
3+
TheCo low-spinstateis, ex
ept forasmallVanVle
ksus
eptibility,nonmagneti
3+
sin
e it does not
arry any moment. In the high-spin state, Co has spin and orbital
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 6
xz
moment. The orbital degenera
y is not
ompletely quen
hed: the degenerate and
yz
are o
upied by three ele
trons. These real orbitals
an thus be re
ombined to form
z
omplexorbitals
arryingorbitalmomentinthe dire
tionasdes
ribedinEq. (3). Note
xy
that the linear
ombinations in equations (1) and (2)
annot be formed be
ause the
xz yz
orbital is not degenerate with the and orbitals. Therefore the orbital moment is
z
larger in the dire
tion making this the easy axis. This anisotropy should be re(cid:29)e
ted
χ >χ
c ab
in the sus
eptibility with , whi
h obviously
ontradi
ts the anisotropy found in
3+
the measurements. A Co HS system shows the wrong magneti
anisotropy.
3+
Next, we dis
uss the IS state of Co . Due to the elongation of the oxygen
e 3z2 r2
g
o
tahedra, the ele
tron o
upies the − orbital. This also e(cid:27)e
ts the splitting of
t xz yz
2g
the states. The and orbitalsremaindegenerate but arenotdegenerate withthe
xy 3z2 r2
orbital. This arises from the di(cid:27)erent
harge distributions in relation to the −
t
2g
orbital. We have (cid:28)ve ele
trons to (cid:28)ll in the states, whi
h
an also be treated as one
t e 3z2 r2
2g g
hole in the states. This hole is attra
ted to the ele
tron in the − orbital.
3z2 r2
Fig. 4 shows the
harge distribution of this hole and the − orbital. In the left
3z2 r2 xy
sket
h, both the ele
tron in the − orbital and a hole in the orbital is drawn.
3z2 r2
The sket
h in the
enter shows the − orbital and the hole in a linear
ombination
xz yz
of the degenerate and orbitals. Regarding the distan
es between the ele
tron and
xz yz
the hole, on
an
on
lude that the state with the hole in the and orbitals is lower
in energy. As the order of levels is reversed when we are speaking about holes instead
xy
of ele
trons, this means that the orbital is lowered in energy. Thus, the order and
o
upation of levels is the one shown in Fig. 4. The magneti
anisotropy is the same as
3+
in the
ase of Co HS whi
h we treated in the last paragraph. With three ele
trons in
xz yz
the degenerate level of the and orbitals, we
an also make use of Eq. (3) to show
2−x x 4
that the dire
tion of anisotropy does not (cid:28)t the measurements for La Sr CoO .
3+
Neither the HS nor the IS state of Co show the
orre
t magneti
anisotropy. But
2+ 2+
before we draw further
on
lusions, Co should be dis
ussed. Although Co has also
been found in the LS state in some intramole
ular
ompounds [21℄, for bulk
rystals it
2+
an be safely assumed that Co is in the HS state [22℄.
Regardingthe ground state produ
ed by the
rystal (cid:28)eld in Fig. 3, where the lowest
level is (cid:28)lled, no linear
ombinations like in Eq. (3)
an be formed. The orbital moment
would be
ompletely quen
hed and the magneti
moment would be determined by an
isotropi
spin. If, however, spin-orbit
oupling is of the same magnitude as the splitting
t
2g
of the orbitals, it will mix these states. Mixing in a state with orbital moment in
z dz
±1
the dire
tion would require pla
ing a hole in the orbital (see Eq. 3). This would
t
2g
ost an energy equal to the splitting within the orbitals. It is more favorable to mix
x y dx
±1
in a state with orbital moment in the (or ) dire
tion by pla
ing a hole in the (or
dy
±1
the ) orbital (see Eq. 1). This
osts only half of the
rystal-(cid:28)eld splitting within the
t x y z
2g
orbitals. An orbital moment in the (or ) dire
tion
osts less energy than in the
xy 2+
dire
tion. So, the plane will be the easy plane for Co in this elongated tetragonal
xy
stru
ture [24℄. The spin moment will also be found in the plane. Comparing these
2+
(cid:28)ndings to the measurement, we see that Co shows the
orre
t dire
tion of magneti
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 7
Figure 4: The pi
ture on the left shows the
harge distribution for an ele
tron in the
3z2 r2 xy
− orbital (bla
k) and a hole in the orbital (white). The
harge distribution in
3z2+r2
the
enter refers to an ele
tron in the orbital and a hole in a linear
ombination
xz yz 3d 3+
of the and orbitals. The splitting and the o
upation of the levels for Co in
the IS state is shown on the right.
3+ x 0.5
anisotropy. Nowa
on
lusionaboutthespin stateofCo
an be drawn. For ≥ , at
3+ 2+
least half of the
obalt ions in the
rystal are Co . Taking into a
ount that Co has
3+
a smaller spin moment than HS Co , the latter would dominate the anisotropy. The
opposite is, however, the
ase in the data. Thus, for this doping range, the spin state of
3+ 2+
Co is identi(cid:28)ed with the LS state and the magnetization must be assigned to Co ,
x = 0.4 3+
whi
h shows the
orre
t anisotropy. For , we still have 40% Co and, be
ause
of the pronoun
ed anisotropy, we
an follow the same argumentation here. Regarding
x = 0.3
in Fig. 2, the anisotropy is found to be rather small
ompared to the other
ompounds.
It should be stressed that the dire
tionof anisotropy inlayered perovskites depends
on the dire
tion in whi
h the oxygen o
tahedron is distorted. In some systems
c
2 4
like K CoF the o
tahedra are
ompressed in the dire
tion, resulting in an easy-
2+
axis anisotropy [23℄; Co in almost
ubi
symmetry also tends to have an easy-axis
2−x x 4
anisotropy like in CoO. However, in our
ase of La Sr CoO , the oxygen o
tahedron
surrounding the
obalt ion is elongated by the inherent tetragonal stru
ture. This
situation is
omparable to the
hange of anisotropy in CoO (cid:28)lms on di(cid:27)erent substrates
[24℄.
For a quantitative analysis of the data and a
on(cid:28)rmation of the results found
in the qualitative dis
ussion of the anisotropy, a full-multiplet
al
ulation was
arried
3d
out within the
rystal (cid:28)eld approximation [25℄. A Hamiltonian for the shell of the
2+
Co ion was set up, in
luding
rystal (cid:28)eld, spin-orbit
oupling and magneti
(cid:28)eld.
F F F
0 2 4
The Slater integrals , and
hara
terizing the Coulomb intera
tion and the
spin-orbit
oupling
onstant were taken from a Hartree-Fo
k approximation [20℄. Any
2p
ex
itations from the oxygen levels and mixing with higher states were negle
ted,
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 8
Figure 5: Measured magnetization (
olored lines)
ompared with the
al
ulation for
2+
Co (bla
k lines), plotted inverse. A mean-(cid:28)eld magneti
ex
hange was added to the
al
ulation. Note that the magnetization of the ion was weighted with the
ontent of
2+ 1 x
Co ( − ).
as these e(cid:27)e
ts are not signi(cid:28)
ant when
onsidering thermal energies. The parameters
for the
rystal (cid:28)eld were treated semi-empiri
ally. Regarding the tetragonal distortion
3d
as a perturbation of the
ubi
symmetry, the e(cid:27)e
t of the hybridization of the Co
2p
shell with the surrounding oxygen shells only in
reases the splitting of the
ubi
t e
2g g
and states. Hybridization
an thus be in
luded into the
rystal (cid:28)eld parameters
and need not be treated separately. The Hamiltonian was diagonalized and Eigenstates
and Eigenvalues were used to obtain the temperature dependen
e of the magnetization.
Magneti
ex
hange was in
luded in the
al
ulation on a mean-(cid:28)eld level. The results
an be seen in Fig. 5, where the inverse magnetization is plotted.
10Dq
The
rystal-(cid:28)eld parameters were
hosen as follows: the splitting and the
e
g
splittingwithinthe levelswere(cid:28)xedat1.5eVand0.2eV,respe
tively. Thesetwovalues
areof minorimportan
eforthe
al
ulation,be
ause the anisotropyof the magnetization
t
2g
arises from the ele
trons, as des
ribed in the dis
ussion above. The splitting of the
t
2g
states showed up to be
ru
ial for the form and the anisotropy of the
al
ulated
t
2g
urves. The values of the splitting within the states were
hosen as 80meV, 60meV,
x =
68meV, and 50meV for the
ompounds 0.5, 0.6, 0.7, and 0.8, respe
tively. These
t
2g
rystal (cid:28)eld splittings for Co are in good agreement with previous measurements
on strained CoO thin (cid:28)lms [24℄. They might appear to be surprisingly small
ompared
to the values found for the Titanates [26,27℄ in the order of 200 meV. The di(cid:27)eren
e
2−x x 4
Anisotropi
Sus
eptibility of La Sr CoO related to the Spin States of Cobalt 9
d
between Co and Ti
an be understood by
onsidering the radial extent of the wave
d
fun
tion and the transition metal - oxygen (TM-O) distan
e. The Co wave fun
tion
d
is mu
h smaller than the Ti wave fun
tion, redu
ing the
ovalen
y and thereby the
size of the
rystal-(cid:28)eld splitting. At the same time, the average TM-O distan
es of
2−x x 4 3
La Sr CoO and LaTiO are almost equal (≈2.02Å for Co-O [16℄ and ≈2.04Å for
e
g
Ti-O [28℄). This is related to the o
upation of the anti-bonding orbitals in a HS
2+
Co
ompound whi
h pushes the O atoms further away.
x = 0.6
It
an be seen in Fig. 5 that the
al
ulation gives a good des
ription for
x = 0.7 x = 0.5
and . The magnetization of the half-doped sample is reprodu
ed well for
higher temperatures. At temperatures below the freezing temperature, the
al
ulation
annot des
ribe the system be
ause magneti
order is not in
luded in the model. The
x = 0.8 T 350
sampleis(cid:28)ttedwellforlowertemperatures. Above ≈ K,themagnetization
in the measurement is enhan
ed by another e(cid:27)e
t and results in a deviation from the
al
ulation. This e(cid:27)e
t
an be seen in all
urves in Fig. 2, but at signi(cid:28)
antly higher
temperatures. This in
rease of magnetization
ould arise from the thermal population
3+
of the Co HS or IS state. But surely further measurements at higher temperatures
are needed to
on(cid:28)rm this assumption. Nevertheless, the full-multiplet
al
ulation of a
2+
pure Co system already su
eeds in des
ribing the main features of the magnetization
x 0.5 3+
for ≥ . This
on(cid:28)rms that Co is a LS system in this doping range.
2−x x 4
Insummary,themagneti
sus
eptibilityofLa Sr CoO hasbeenanalyzedfortwo
di(cid:27)erent dire
tions of the external magneti
(cid:28)eld. A de(cid:28)nite anisotropy of the magneti
χ >χ
ab c
moment is found experimentally with . This dire
tion of anisotropy does not
3+ 2+
mat
h with the behavior of HS or IS Co , whereas Co in the HS state shows the
3+ x 0.4
orre
t single-ion anisotropy. Thus, the spin state of Co for ≥ must be the LS
state. This
on
lusion is also
on(cid:28)rmed by a full-multiplet
al
ulation.
Wea
knowledge(cid:28)nan
ialsupportbytheDeuts
he Fors
hungsgemeins
haftthrough
SFB608.
Referen
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