Crystal field, spin-orbit coupling and magnetism in a ferromagnet YTiO3

0.2 MB·English

by  R. J. Radwanski

Crystal field, spin-orbit coupling and magnetism in a ferromagnet YTiO ♠
3
R. J. Radwanski
Center of Solid State Physics, SntFilip 5, 31-150 Krakow, Poland,
Institute of Physics, Pedagogical University, 30-084 Krakow, Poland∗
Z. Ropka
Center of Solid State Physics, SntFilip 5, 31-150 Krakow, Poland
6
0 Magnetic properties of stechiometric YTiO3 has been calculated within the single-ion-based
0 paradigm taking into account the low-symmetry crystal field and the intra-atomic spin-orbit cou-
2 pling of the Ti3+ ion. Despite of the very simplified approach the calculations reproduce perfectly

n thevalueof themagnetic moment and itsdirection as well as temperaturedependenceof themag-
a netic susceptibility χ(T). It turns out that the spin-orbit coupling is fundamentally important for
J 3d magnetism and magnetic properties are determined bylattice distortions.

2
PACSnumbers: 75.25.+z,75.10.Dg
] Keywords: CrystallineElectricField,3doxides,magnetism,spin-orbitcoupling,YTiO3
l
e
-
r INTRODUCTION tion Y3+Ti3+O2−. The relevant charge transfer occurs
st YTiO3 is a unique 3d ferromagnet [1, 2, 3] - the most duringtheforma3tionofthecompound. Intheperovskite-
t. of oxides are antiferromagnetic. In combination with based structure of YTiO3 the Ti3+ ion is surrounded by
a a rather simple perovskite structure and Ti3+ ions ex- six oxygenions forming distorted octahedron. There are
m
pected to have one electron in the incomplete 3d shell, still some Ti ions at the surface, for instance, with a
d- YTiO3 is regarded to be very good examplary sys- reduced symmetry, but they are generally neglected, be-
n tem for studying basic interactions in 3d oxides. De- cause we are interested in intrinsic properties of YTiO3.
o spite this simplicity its properties are not understood The local octahedra in YTiO3 are tilted and rotated,
c yet. In fact, there is going on at present a hot debate what causes the need for consideration a larger elemen-
[
on description of its magnetic and electronic properties tarycell,withfourTiionsinsteadofoneasinthesimple

1 [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. perovskite structure. Thanks it other three Ti ions get
v AtimewhensuchsystemwastreatedasaS=1/2system a crystallographicfreedom. As a resultof rotations,tilts
5
i.e. withthespin-onlymagnetismandthefullyquenched and other atom displacements there are four short Ti-O
0
0 orbitalmagnetismisalreadygone. Alsothesimplestver- bonds and two long Ti-O bonds.
1 sionsofthebandpictureturnedouttobecompletelyuse- TheorthorhombiclatticeinthePbnmstructureresults
0 less giving already at the start disagreementwith exper-
fromthatofanidealcubic perovskitebysetting a =√2
o
6 iment predicting a metallic groundstate whereasYTiO3 a , b =√2 a and c =2a , where a denotes the lattice
0 c o c o c c
is in the reality a good insulator.
/ parameter of the simple cubic perovskite. ac is of order
at YTiO3, when stoichiometric, is very good insulator. of 400 pm.
m Its resistivity at room temperature amounts to 5·10−2 Inthe Pnma structure usedby some authorsthe dou-
Ωcm [11]. The resistivity rapidly growsup with decreas-
- blingoccursalongthebdirection. Thec,a,andbaxisin
d ingtemperature. ItisferromagneticbelowTc of30-35K the Pnma structure becomes the a, b and c axis in the
n [3,4]. Themacroscopicmagnetisation,ifrecalculatedper
Pbnm structure,respectively. Independently of the used
o
theformulaunit,pointstoamomentof0.84µ [3]. The
c B Pbnm or Pnma space group the derivation of the local
paramagneticsusceptibility has been found to follow the
: surroundings must be, of course, the same.
v
Curie-Weiss lawwith θ = 39 K,if substracteda diamag-
i Here we use the Pbnm space group. Then the easy
X netic andtemperature independent orbitalcontributions
magnetic axis is the c axis, [3]. The lattice parameters
of a total size of 0.35110−3 emu/mol.
ar Theaimofthispape·ristopresentresultsofsingle-ion at T = 293 K, according to Ref. 2, cited by [5, 15], are:
a =531.6 pm, b =567.9 pm and c = 761.1 pm. These
based calculations of properties of YTiO3. Our single- poarameters haveobeen confirmed boy detailed structural
ion results seem to be quite remarkable. We took into
measurements of Loa et al. [19].
accountlow-symmetryoff-octahedralcrystal-field(CEF)
Cwiketal. [5]havederivedtherespectivebonds: 207.7
interactions and the intra-atomic spin-orbit (s-o) cou-
pm (Ti-O(b)), 201.6 pm (Ti-O(a)) and the apical bond
pling, that turn out to be of the comparable strength.
202.3pm (Ti-O(c)). Loa et al. [19] have measuredinflu-
THEORETICAL OUTLINE enceofthe externalpressureonthesebonds. Fromthese
We consider exactly stechiometric YTiO3. From this crystallographic studies we get an input to our theoreti-
and the insulating ground state we infer that all Ti ions calconsiderationsthatthe lattice surroundingsof the Ti
areinthetrivalentstateaccordingtothechargedistribu- ion in YTiO3 is predominantly octahedral with a slight
2
orthorhombic distortion.
2Eg 1.03mB2 1.00mB2
K) 2 . 2 4 1.00mB 2 1.03mB 2
RESULTS AND DISCUSSION 40
The electronic structure of the Ti3+ ion with one d- E (1 1 3do 1 cstyastem
electronunder the action of the crystal-fieldinteractions 120B4 B = +240K
2.5 eV 4
H and in the presence of the intra-atomic spin-orbit l= +220K
CF
cioonu-plilkinegHHams−ioltowneiacnalocuftleanteudsewditihntdheescursipetoiofnthoef 3sidngimle-- 0 25D . 2 tetragonal distortion
purity states in the Electron Paramagnetic Resonance _ac>1 _ac<1
[22, 23, 24], which we accept also for a solid, where 3d 1.00mB2 0.60mB2 1.31mB2
2T
atom is the full part of the crystallographicstructure: -1 2g
Hd =HCF +Hs−o = 3 . 2 00..0030mmBB4 00..4030mmBB22 00..2080mmBB22
B4(O40+5O44)+λs−oL S+B20O20+µB(L+geS) Bext.(1) B 20 = +10K B 20 = -10K
· · a) b) c) d) e)
The calculated electronic structure is presented in Fig. FIG. 1: The calculated fine electronic structure of the 3d1
1. The crystal field has been divided into the cubic electronic system(Ti3+,V4+ ions) intheparamagnetic state
part, usually dominant in case of compounds containing under the action of the crystal field and spin-orbit interac-
tions: a) the 10-fold degenerated 2D term realized in theab-
3d ions, and the off-octahedral distortion written by the
sence of theCEF and thes-o interactions; b) the splitting of
second-orderleadingtermB02O02. The lastterm, Zeeman the 2D term by the octahedral CEF surrounding B4=+240
term, allows calculations of the influence of the external K (λs−o =0) yielding the 2T2g cubic subterm as the ground
magnetic field. ge amounts to 2.0023. The Zeeman term state;c)thesplittingofthelowest2T2g cubicsubtermbythe
is necessary for calculations, for instance, of the param- combined octahedral CEF and spin-orbit interactions (B4=
agneticsusceptibility-infacttheparamagneticsuscepti- +240 K and λs−o= +220 K); the degeneracy and the asso-
bility is customarily calculated[25]as the magnetization ciated magnetic moments are shown; d) the splitting due to
theelongatedtetragonaloff-octahedraldistortionofB0=+10
in an external field of, say, 0.1 T applied along different 2
K (c/a>1); e) the splitting due to the compressing tetrago-
crystallographic directions. nal distortion of B0= -10 K (c/a<1, apical oxygens become
2
The detailed form of the Hamiltonian (1) is written
closer). Figs c, d and e are not to the left hand energy scale
down in the LS space that is the 10 dimensional spin- - the splitting of the three lowest states on Figs c, d and e
orbitalspace LSL S . The L andS quantumnumbers amounts to333 K, 368 K and 372 K, respectively.
| z zi
forone3delectronareequaltoL=2andS=1/2(here,for -11300
1
3d1 configuration,lowerl andscouldbe alsoused). The 3d
octa
Hmaemthioltdosnoiawnin(g1)toisthceuswteoamkanreisllsyotfrtehaetesd-obcyouppelrintugrfboarttiohne --11E (K)11540000 2T2gtetra 2 aBBzc 402=> == 1[ 0++021140]0 K K
3d ions in comparisonto the strength of the crystal-field 2
interactions. We have accepted the weakness of the s-o -11600 2
coupling, what is reflected by the sequence of terms in
-11300
the Hamiltonian (1), but we have performed direct cal- 1
3d
octa
counlatthieonssamtreeaftoinotginagll.shTohwensetepramrastienfitghuerHesa,mifilptorensieannt(e1d), -11E (K)400 tetra 2 BzB 402= == [ 0+-102014 ]0K K
areDsiahgoownnalfiozrattihoenihllauvsetrbaeteionnpererfaosromnse.dforphysicallyrel- -11500 2T2g 2 ac<1
evant values of λs−o of +220 K (= 150 cm−1) found for -11600 2
the Ti3+ ion[21]. The cubic CEF parameterB4 is taken 0 50 100 150 200
(K)
as +240 K (results are not sensitive to its exact value
provided B4 > +50 K). The positive sign of B4 comes FIG. 2: The calculated spin-orbit coupling λ dependence of
from ab initio point charge calculations of octupolar in- the three lowest states t2g (=2T2g) of the 3d1 configuration
teractions of the Ti3+ ion with the octahedral oxygen for the elongated (c/a>1) and stretched (c/a<1) tetragonal
off-octahedral distortion. The right hand states correspond
(negative charges)surroundings. Such value of B4 yields to thethreelowest states shown in Fig. 1d and 1e.
the T2g-Eg splitting of 2.5 eV. A splitting of 2.15-2.5
eV has been observed for Ti3+ ions in Al2O3, where the (c/a>1)and stretched(c/a<1)tetragonaloff-octahedral
similaroxygenoctahedronexists. In LaCoO3 wederived distortion, realized for B02 > 0 and B02 < 0, respectively.
value of B4 of 280-320 K [26]. Analyzing the effect of the sign of the tetragonal distor-
In Fig. 2 we show detailed calculations of the influ- tion lead us to a conclusion that the magnetic moment
ence of the spin-orbitcoupling onthe three loweststates lies along the tetragonal axis for the z-axis stretching.
t2g (=2T2g) of the 3d1 configuration for the elongated For the elongation case the moment lies perpendicularly
3

500 -1
S=1/2 c
-11100 -1.84 B Ti3+/ 3d1 YTiO3 )B YTiO3 - c1-ndd
-11200 –1.79 B 2 ( T/ 400 Ti3+ 3d1
-E (K)11300 +1.69 B 2 300 CEF
T2g
-11400 b B4 = +240 K
-0.01 B 0.02 B 200 a B02 = -50 K
-11500 +0.03 B 2 c B22 = -10 K
= +220 K
100
-11600 ndd = 69 T/ B
z = [001]
-11700 0
0 Tc=31K 100 200 T (K) 300 400
0.78 B FIG.4: Calculated temperaturedependenceof theparamag-
-11800 -0.68 B = (2•0.44-0.1) netic susceptibility χ (T) for the 3d1 system Ti3+ in YTiO3
2
forthreedifferentcrystallographicdirections(a,b,andc)cal-
-11900
0.84 B=2•0.45-0.06 TC a) culated for B4= +240 K, B02= -50 K, B22= -10 K and λs−o
= +220 K; these curves are denoted with CEF. The lowest
1,0
solidlineistheχ (T)dependencecalculatedwithtakinginto
0.84 B=2•0.45-0.06 c
on)0,8 account the ferromagnetic interactions with nd−d= 69 T/µB
/iB -thiscurveshouldbecomparedwithexperimentaldata. Ex-
m (0,6 perimental data, taking after Ref. [4], are shown by thelow-
estdashedline. Theshadowareaindicatestheferromagnetic
0,4
state. ThestraightlinedenotedwithS =1/2showstheCurie
0,2 TC law expected for thefree S =1/2 spin.
b)
0,0
12 calculated ground-state eigenfunction (the z component
B4 = +240 K
1mol)0 B02 = -50 K of L and S are shown)
(J/Kcd68 B22 == +-21200 K K ψGS± = 0.690(cid:12)(cid:12)±2,∓21(cid:11) - 0.678(cid:12)(cid:12)∓2,∓21(cid:11)
ndd = 69 T/ B - 0.253(cid:12) 1, 1(cid:11) - 0.020(cid:12) 1, 1(cid:11) (2)
4 (cid:12)± ±2 (cid:12)∓ ±2
2 where the sign refers to two conjugate Kramers
c)
±
0 states. This state has S = 0.44 and L = 0.10. The
0 10 20 T (K)30 40 50 z ∓ z ±
FIG. 3: The calculated temp erature dependence of some resultant moment mz = 0.78 µB cancels each other in
±
properties of YTiO3. a) the temperature dependence of the the paramagnetic state as is denoted in Fig. 3a.
three lowest states (t2g states) of the Ti3+-ion in YTiO3 in Making use of the xy, function ex-
the magnetically-ordered state below Tc of 30.6 K; in the tended for the spin comp|onen∓t,i as xy, =
paramagnetic state the electronic structure is temperature p1/2 (cid:0)(cid:12)2, 1(cid:11) (cid:12) 2, 1(cid:11)(cid:1) (also functio|ns ∓ixz ,
independent unless we do consider a changing of the CEF (cid:12) ∓2 −(cid:12)− ∓2 | i
yz , (cid:12)x2 y2(cid:11)and(cid:12)z2(cid:11) )one canwrite the groundstate
parameters, for instance, due to the thermal lattice expan- | i (cid:12) − (cid:12)
sion. The used parameters: B4= +240 K, B02= -50 K, B22= ψGS± function approximately as:
-10 K, λs−o = +220 K and nd−d= 69 T/µB. The splitting
of the Kramers doublets should be noticed in the ferromag- ψGS± = 0.967xy, + 0.0085(cid:12)x2 y2, (cid:11)
| ∓i (cid:12) − ∓
netic state. Excited states are at 377, 645, 28843 and 29447 - 0.186xz, - 0.172yz, +....(3)
K.(b)thetemperaturedependenceof theTi3+-ionmagnetic | ±i | ±i
moment inYTiO3. At0 Kthetotalmoment mTi of 0.84 µB Inthe magnetic state the groundstate ψGS± Kramers
is built up from theorbital m and spin m moment of -0.06
o s doublet function becomes polarized as a molecular field
and0.90 µ ,respectively. c) Thecalculated temperaturede-
B is self-consistently settled down and the function
pendenceofthe3dcontributionc (T)totheheatcapacityof
d
YTiO3. The λ-typepeak marks Tc. ψGS+ = 0.695(cid:12)(cid:12)+2,−21(cid:11) - 0.686(cid:12)(cid:12)−2,−21(cid:11)
- 0.215(cid:12)+1,+1(cid:11) - 0.016(cid:12) 1,+1(cid:11)(4)
tothetetragonalaxis. TakingintoaccountthatinYTiO3 (cid:12) 2 (cid:12)− 2
the ordered moment lies along the c direction (in the is obtained as the ground state. The higher conjugate
Pbnm structure) andour long-lastingstudies of CEFef- state is calculated to be described by
fects [16, 17, 27, 28] we came to values for B0 = -50 K
2
and B22 = -10 K which reproduce the magnetic moment ψGS− = 0.665(cid:12)(cid:12)+2,+21(cid:11) - 0.682(cid:12)(cid:12)−2,+21(cid:11) -
value, its direction and (some) thermodynamics. The + 0.303(cid:12) 1, 1(cid:11) + 0.026(cid:12)+1, 1(cid:11)(5).
(cid:12)− −2 (cid:12) −2
4
Below Tc there opens, as is seen in Fig. 3a, a spin- prove that magnetic properties of YTiO3 are predomi-
like gap that amounts at T = 0 K to 59.7 K (= 41.4 nantlydeterminedbytheatomic-scalelatticedistortions,
cm−1 = 5.15 meV). The spin-like gap is associated with crystal-fieldandthe spin-orbitcouplingofthe Ti3+ ions,
the splitting of the Kramers doublet ground state in the whereas charge fluctuations are negligible. An interplay
ferromagnetic state. The magnetic ground state ψGS+ of the spin-orbit coupling, lattice distortions and mag-
has S = -0.45 and L = +0.06 and the resultant mo- netic orderis verysubtle involvingrathersmallenergies,
z z
ment amounts to 0.84 µ . The appearance of the mag- smallerthan 5 meV making theoreticalstudies quite dif-
B
netic state is calculated self-consistently. It appears in ficult.
the instability temperature (Tc) in the temperature de- We point out that all discussed by us parameters are
pendence of the CEF paramagnetic susceptibility when physicalmeasurableparameters. The B0 and B2 param-
2 2
χ−CF1(Tc) = nd−d (6) eroteurnsdainregsr.elAatendegtaotitvheevoabluseervoefdthoertBh0orphaormambiectleorcraelssuultrs-
2
as is illustrated in Fig. 4 for different crystallographic from the stretching of the apical bonds with respect to
directions. With decreasing temperature this equality is the average bond length within the a b plane. The B22
−
reachedthefirstforthecdirectionpointingthepreferred parameterisrelatedtothedifferenceinthea bplane. A
−
magneticarrangementaxis. Themagneticstateiscalcu- success of our ionic approach, called due to extension to
lated self-consistently by adding to the Hamiltonian Eq. the magnetic state a quantum atomistic solid-state the-
(1) the inter-site magnetic (spin-dependent) interactions ory(QUASST)[32],ifappliedtoYTiO3,isrelatedtoour
instead of the last Zeeman term [29] long-lastingsystematic studies, despite of discrimination
Hd−d = nd−d(cid:0)−md·md+ 12(cid:10)m2d(cid:11)(cid:1) (7) afinelddiinnqtuerisaicttioionn,soifnt3hde cspominp-oorubnitdscoaunpdliningsatnuddycroyfsttahle-
where nd−d is the molecular-field coefficient. region, where the spin-orbit coupling and off-octahedral
Having eigenvalues in a function of temperature we lattice distortions are of the comparable strength.
calculate the free energy F(T). From F(T) using well- Acknowledgements. We are very grateful to all our
known statistical formulae we calculate all thermody- opponents. Although we do not think that the discrimi-
namicsliketemperaturedependenceofthemagneticmo- nationandinquisitionshouldtakeplaceinPhysicsatthe
ment,oftheadditionalc heatcapacity,ofparamagnetic XXI century their critics largely stimulated these, very
d
susceptibility, of the 3d-shell quadrupolar moment and natural for us studies. This discrimination is the best
many others. The present calculations are similar to proof that the knowledge about the CEF and the role of
those performed for FeBr2 [30] and CoO [31]. the spin-orbit coupling in 3d magnetism is rather poor.
We are thankful to a numerous members of the Inter-
CONCLUSIONS
We have calculated consistently a value of the mag- national Committee of the Strongly-CorrelatedElectron
netic moment of 0.84 µ and its direction (along the c Systems Conference in Vienna 2005 for the friendly sup-
B
axisinthe Pbnmstructure)inYTiO3 aswellastemper- port. We are convinced that Physics can develop only
ature dependence of the paramagnetic susceptibility in in the friendly atmosphere and in the open exchange of
very good agreement with experimental observations. information.
This remarkable reproduction of so many physi- ♠ dedicated to John Van Vleck and Hans Bethe, pio-
cal properties has been obtained within the localized- neers of the crystal-field theory, to the 75th anniversary
electron approach taking into account lattice off- of the crystal-field theory, and to Pope John Paul II, a
octahedral distortions and the intra-atomic spin-orbit man of freedom and honesty in life and in Science.
coupling [25]. Although the spin-orbit coupling is weak,
itamountstoonly1.2%oftheoctahedralCEFsplitting,
it has enormous influence on the low-energy electronic
structure and low-temperature magnetic and electronic
∗ URL: http://www.css-physics.edu.pl; Electronic ad-
properties. Thespin-orbitcouplingbindstheorbitalmo-
dress: [email protected]
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