Electronic structure and electric-field gradients analysis in $CeIn_3$

0.36 MB·English

by  S. Jalali Asadabadi

Electronic structure and electric-field gradients analysis in CeIn
3
S. Jalali Asadabadi1,2,∗
1Department of Physics, Faculty of Science, University of Isfahan (UI)
Hezar Gerib Avenue, Isfahan 81744, Iran
2Research Center for Nano Sciences and Nano Technology
University of Isfahan (UI), Isfahan 81744, Iran
(Dated: February 3, 2008)
7
0 Electric field gradients (EFG’s) were calculated for the CeIn3 compound at both 115In and
0 140Ce sites. The calculations were performed within the density functional theory (DFT) using
2
the augmented plane waves plus local orbital (APW+lo) method employing the so-called LDA+U

n scheme. TheCeIn3 compound were treated as nonmagnetic, ferromagnetic, and antiferromagnetic
a cases. Our result shows that the calculated EFG’s are dominated at the 140Ce site by the Ce-4f
J states. An approximately linear relation is intuited between the main component of the EFG’s

4 and total density of states (DOS) at Fermi level. The EFG’s from our LDA+U calculations are
2 in better agreement with experiment than previous EFG results, where appropriate correlations
had not been taken into account among 4f-electrons. Our result indicates that correlations among

] 4f-electrons play an important role in thiscompound and must be taken into account.
l
e
- PACSnumbers: 71.20.-b,71.28.+d,76.80.+y,71.27.+a,71.20.Eh,75.30.Mb,75.25.+z,31.30.Gs,31.5.Ar
r
t
s
t. I. INTRODUCTION waves (FP-LAPW)12.
a
In this paper, we have examined whether one can, us-
m
ing an intermediate way, improve the previous results of
Hyperfineinteractionsprovidesensitivephysicalquan-
-
d tities such as electric field gradients (EFG’s), which thelocalizedanddelocalizedlimits. Forthis purpose,we
n can be used to shed light experimentally1,2 and have employed the LDA+U scheme13,14,15 and then cal-
o theoretically3,4 into the electronic states of materials. In culated the EFG’s at both 115In and 140Ce sites. The
c more advanced method of augmented plane waves plus
thispaperwehavefocusedontheCeIn asaninterested
[ system composed of strongly correlated3 4f-electrons. It localorbital(APW+lo)16,17wereusedtolinearizetheen-

2 is a cubic heavy-fermion (HF) local moment antiferro- ergies. According to our knowledge, this is a first-report
v of the EFG calculations within the LDA+U scheme
magnetic (LMAF) system at ambient pressure with a
6 N´eel temperature of 10.1 K5. This concentrated Kondo employing APW+lo method for this compound. Our
8
4 compound exhibits6 various fascinating and unexpected s14p0inC-eposlaitreiziend cthaelcuplraetsioenncsedeomf ospnisntr-aotrebintocno-uzperliongE.FGThaet
1 physicalproperties. The various properties ofthis heavy
EFG’s, according to our knowledge, had not been previ-
0 fermion originate from the fact that one cannot assign a
7 definite localization to the 4f-states, irrespective of the ouslycalculatedatthecubic140Cesiteinthiscompound.
0 The cubic symmetry of the Ce site explains why the
applied conditions to the compound. The unexpected
/ EFG’swerelesssignificanttobepreviouslyreported. We
t physical behavior may be then attributed to the degree
a have emerged, nevertheless, a new result that the EFG’s
oflocalizationofthe4f-electrons,i.e. thepositionsofthe
m are dominated at the 140Ce site by 4f-states and not as
4f-density of states (4f-DOS) with respect to the Fermi
- usual by p-states. The goal of this work is to illustrate
level. The position of the 4f-DOS demonstrates the de-
d
anapproximatelylinearrelationshipbetweenthevaluesof
n greeofhybridizationbetweenlocalized4fandconduction
EFG and density of states (DOS) at Fermi level (E ),
o bands. The EFG quantity is extremely sensitive to the F
c anisotropicchargedistributionsofthe coreelectrons7,as viz. EFG ∝ DOS(EF). According to our knowledge,
: such a linear relationship had not been previously ob-
v well as to the aspherical electron density distribution of
i valance electrons8, and as a result to the valance elec- served. We also aim to justify about the tendency of the
X 4f-electrons in the ground state of the antiferromagnetic
tronic structure. The EFG, thereby, can serve as a pow-
r erful gauge for measuring such a degree of localization. CeIn3 compound to show their degree of localization.
a
We have also found that correlations among 4f-electrons
J. Rusz et al.9 very recently calculated Fermi sur-
influence semicore states of 5p-Ce.
faces of the CeIn regardless its antiferromagnetic or-
3
dering. Their calculations were performed in the local-
ized extreme limit within the open core treatment10,11.
II. DETAILS OF THE CALCULATIONS
On the other delocalized extreme limit two individual
groups3,4 calculated the EFG at the 115In site in this
compound. The calculations of the former3 and later4 Allthecalculationsinthisworkwereperformedinthe
groups were performed, respectively, in the antiferro- frame work of the density functional theory18,19 (DFT).
magnetic and nonmagnetic phases employing a similar We havetakenthe generalizedgradientapproximation20
method of the full-potential linearized augmented plane (GGA) into account for the exchange-correlation func-
2
tional. We have employed the full-potential augmented
plane wavesplus localorbital(APW+lo) method16,17 as
embodied in the WIEN2k code21. The muffin-tin radii
(R ) were chosento be 2.2 ˚A and 2.8 ˚A for the In and
MT
Ce atoms, respectively. It has been allowed to be the
4s 4p 4f orbitals of the Ce and 5d 6s orbitals of In in
the valance states. The expansion of the wave functions
inside the spheres in lattice harmonics and in the inter-
stitialregioninplanewaveswerecutoffbythemaximum
eigenvalue of l =10 and the R K =7, respec-
max MT max
tively. ThecutofffortheFourierexpansionofthecharge
density and potential was taken to be G = 16√Ry.
max
We used a mixing parameter of 0.001 in the Broyden’s
scheme to reduce the probability of occurrence of the
spurious ghostbands. A mesh of 165 special k points
was taken in the irreducible wedge of the first Brillioun
zone, which corresponds to the grids of 18 18 18
in the scheme of Monkhorst-Pack22. In ord×er to×per- FIG. 1: ”(color online)”(a) Chemical unit cell of CeIn3 in
form the calculations nearly in the same accuracy for AuCu3 prototype having the lattice parameter of a=4.69 A.
the case of magnetic super cell compared to the non- (b)Constructedconventionalsupercellwiththesymmetryof
magnetic unit cell, the mesh of k points was reduced to face-centered cubic (Pm¯3m space group) from the chemical
121 corresponding to 9 9 9 grids. We diagonalized unit cell by a factor of 2 as a number of chemical unit cells
× × drawn in three directions of xyz Cartesian coordinates. (c)
spin-orbit coupling (SOC) Hamiltonian in the space of
scalarrelativistic23eigenstatesusingasecond-variational Primitive rhombohedral unit cell of the constructed conven-
tional supercell. (d) Magnetic supercell imposing spin order-
procedure24imposing(111)directionontheCemagnetic
ing of (↑↓) to the Ce moments along (111) axis.
moments within a cut-off energy of 3 Ry. In order to
take into account strong correlations of 4f Ce states, we
have used the LDA+U method13,14,15. In one side, it is
generallybelievedthat, the on-siteCoulombrepulsionU Finally, since the factor of (U-J) appears in the total
integral, as an input parameter for the LDA+U calcula- energy of the LDA+U method instead of U and J in-
tions, depends on the system under study. This means dividually, we set J to zero, and let U be equal to the
that the U parameter can vary for an atom, i.e. Ce in effective value27 of U = U - J = 5.5 eV.
eff
this work, from one case to the others. Here, we have
used a value of 6.2 eV for the U parameter of Ce in
CeIn3. Fortunately, this value, U = 6.2 eV, had been III. CHEMICAL, MAGNETIC AND
before calculated25 for our own case of CeIn . Further- ELECTRONIC STRUCTURES
3
more, on the other hand, the value obtained for the U
parameterdependsonthemethodofcalculations. Unfor- A. Chemical Structure
tunately, the calculations of the U value were performed
within the linear-muffin-tin-orbital (LMTO) method25.
As shown in Fig. 1(a), CeIn crystallizes in the space
3
Although the method of our calculations, i.e. APW+lo, groupofPm¯3mwiththebinary-fccprototypeofAuCu .
3
isdifferentfromtheLMTO,butwehavenoticedthatfor
Itis a cubic unit cell,where Ce atomsarelocatedonthe
another case of face-centered-cubic γ Ce the value of
− corners, and the In atoms on the middle of the surfaces.
6.1 eV could give satisfactory results in agreement with
ThepointgroupsofCeandInatomsarethecubicm-3m,
experiment26. The U=6.1 eV value of fcc-γ Ce is not
− and the non-cubic 4/mmm, respectively. The lattice pa-
sofar fromthe U=6.2eVvalue oftheCeIn3 compound. rameter of the CeIn were measured28 to be 4.69˚A. We
3
Therefore, we did neither more elaboration to calculate
have used this chemical structure to simulate the non-
the Coulomb repulsion U within our APW+lo method,
magnetic and ferromagnetic phases.
nor change this value to optimize it. Another input pa-
rameterfortheLDA+Ucalculationistheexchangeinte-
gralJ.Here, we have used 0.7 eV for the J value. This J
B. Magnetic Structure
parametercanbederivedusingthevaluesof8.34eV,5.57
eV, and 4.12 eV for the F2,F4,F6 Slater integrals26, re-
It has been experimentally observed that the cerium
spectively,withinthefollowingexpression,whichisvalid
moments in this compound are aligned antiferromagnet-
for the f electrons:
ically in adjacent (111) magnetic planes5. Therefore, in
order to simulate the antiferromagnetic situation, first
1 2 1 50 the sides of the non-magnetic unit cell were doubled as
J = F2+ F4+ F6 0.69eV. (1)
3 15 11 429 ≈ depicted in Fig. 1(b) in all the 3 Cartesian xyz direc-
(cid:18) (cid:19)
3
tions. We have preserved the space group of Pm¯3m for ingSOCgivesrisetoareductionof14.60states/Ryfrom
thenewmagneticsupercell. Thepreservedfccsymmetry FM to FM+SOC. The DOS(E ) shows a total reduc-
F
of the magnetic supercell causes to be reduced the num- tion of 47.06 states/Ry in going from NM to FM+SOC.
berofatomsintheprimitiverhombohedralmagneticunit Hence itseemsthatthe SP andSOC caninfluence phys-
cell,see Fig.1(c), comparedto the conventionalunitcell ical quantities in a similar direction in this compound.
showninFig.1(b). Second,wehaveimposedtheantifer-
The DOS’s were calculated for the AFM state in the
romagnetic ordering on the magnetic moments of 140Ce
lackofSOC.The laterstate iscalled”AFM(001)”phase
sites. The direction of the 4f spin in the ground state
in this paper. In the absenceof the SOC there is no pre-
electronconfigurationofCeatom,i.e. [Xe].4f↑.5d↑.6s↑↓, ferred spatial direction at the cubic 140Ce site. In this
were exchanged, i.e. [Xe].4f↓.5d↑.6s↑↓, alternatively
case all the planes, for example (111), (001) and so on,
with the orderingof ( ) in the (111)direction as shown
are identical. Thus, for simplicity, the antiferromagnetic
↑↓
in Fig. 1(d).
ordering, ( ), were aligned along (001) axis. The SOC
↑↓
interactions were then included in the AFM(001) phase.
One could introduce, at this step, a preferable direction
C. Electronic Structure in the presence of SOC. However, we would postpone
setting up the correct direction of the cerium moments.
We have calculated the density of states (DOS) in the The cerium moments were then kept still along (001)
lack of both spin-polarization (SP) and spin orbit cou- axis, which we call it ”AFM(001)+SOC” phase. The
pling (SOC). This calculation has been performed using cerium moments in the AFM+SOC(001)phase were not
the chemical structure; Fig. 1(a). This is what we call it changed to the more natural (111) direction, in order to
nonmagnetic (NM) phase from later on. The SOC was avoid mixing the effects of SOC with the effects of spin
thenincludedintwoindividualsteps. Thecalculationsin directions in the AFM phase. The result of AFM(001)
the first step were performed including spin-orbit inter- or AFM(001)+SOC shows that the picks of the semi-
actions among only Ce-electrons. We refer to the results core Ce 5p and 4d In DOS’s were nearly doubled in the
ofthiscalculationbythenameof”NM+SOC(onlyCe)”. AFM compared to the NM and FM phases. This is in
The SOC was then, as the second step, included among consistentwith the fact that the number ofelectrons are
the In-electrons as well. We call it ”NM+SOC” phase doubled in going from the chemical cell to the magnetic
form now on. For sure, the SOC is included in both Ce supercell. We have then directed the 4f Ce spins from
and In atoms in the ”NM+SOC” phase. Our calculated (001) to (111) direction, and recalculated the density
DOS’s are in agreement with the previous calculations3, of states self-consistently including spin-orbit coupling.
and we avoid repeating them here. The SOC not only The DOS’s qualitatively were similar to those for the
influences the semicore states of 5P Ce and 4d In, but lastcaseof(001)direction. However,quantitativelyonly
also changes the density of states at the Fermi level changing the direction changes the values of DOS(EF)
(DOS(E )). The SOC splitting in the valance states from167.47sates/(spin.Ry)to127.98sates/(spin.Ry)for
F
is much smaller than the SOC splitting in the semi core the spin up, and from 167.24 sates/(Ry.spin) to 128.24
states. However,the effectsofthe SOCwouldnotbe ne- states/(spin.Ry) for spin down. Such changes can affect
glectedonthe resultsdue to the factthat manyphysical the sensitive physical quantities.
quantities, e.g. EFG, are very sensitive to the value of We haveincluded anappropriatecorrelationamong4f
the DOS(EF). The later point seems to be more signifi- Ce electrons with employing LDA+U calculations. The
cant for the Ce based compounds. A typical LDA/GGA 4fspinorientationswerepreservedantiferromagnetically
calculation produces a sharp density of states for the 4f along (111) direction, and spin-orbit interactions were
Ce,andsituatesitattheFermilevel. Thereby,anysmall included in the LDA+U calculations. This constitutes
changes in the sharply located 4f Ce DOS at the Fermi our ”AFM(111)+SOC+LDA+U” phase. Total DOS’s
level can change more significantly the DOS(EF), and of the AFM(111)+SOC+LDA+U phase are shown in
consequently the physical results. Figs. 2. The result shows a significant reduction of the
The DOS’s were also calculated for the ferromag- DOS(E ) to 28.11 states/(spin.Ry) for both up and
F
netic phase in the lack of SOC, which we call it ”FM” down spins. Total DOS’s are separately illustrated in
phase. The result shows that imposing spin polarization semicore, Fig. 2(b), and valance, Fig. 2(c), regions. The
(SP) causes to be reduced the DOS(E ) from 141.82 DOSofthesemicoreregionshowsfurthersplittinginone
F
states/(spin.Ry) in NM phase to 94.46 states/(spin.Ry) of the branches of 5p Ce DOS. This occurs in compari-
for spin up and to 14.90 states/(spin.Ry) for spin down son with the phase of AFM(111)+SOC. Our calculated
in the FM phase. We have added up and down DOS’s total DOS of the later phase, which is not shown here,
of the FM phase. Therefore, a total reduction of 32.46 is in complete accord with previous calculations3. One
states/Ry in DOS(E ) is occurredin going fromNM to may confirm the above mentioned further splitting by
F
FM phase. Such a reduction affects the physical quanti- comparing Fig. 1 given in Ref. 3 with our Fig. 2(a) or
tiesthatweshalldiscusstheminsubsequentsections. We (b). One would more clearly observe the further split-
have then included the SOC in the FM phase, which we ting in Fig. 2(d). The semicore region in Fig. 2(d) is
call it ”FM+SOC” phase. The result shows that includ- restricted to an appropriate interval, where the split-
4
1500 up 1200 up
LDA+U LDA+U
dn dn
1000 800
500 400
0 0
y
R
S 2
O -500 9 -400
D (a) 0.5 (b)
T -1000 = -800
O E f
T
-1500 -1200
c
ti -1.0 -0.5 0.0 0.5 1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
e
n
g
a 200 up 160 up
m LDA+U LDA+U
o dn dn
r
r
e 100 80
f
ti
n
A
0 0
-100 -80
(c) (d)
Ef = 0.592 Ry
-200 -160
0.0 0.2 0.4 0.6 0.8 1.0 -0.68 -0.66 -0.64 -0.62 -0.60 -0.58

Energy (Ry)
FIG. 2: ”(color online)” Total density of states of the AFM(111)+SOC+LDA+U phase presenting (a) semicore and valance
states, b) semicore states, (c) valance states, (d) selected interval of semicore states exhibiting further splitting in 5p Ce DOS
from LDA+U interaction. Fermi level is shown using dashed lines
ting is occurred. This shows that correlations among cant reduction in the DOS(E ) originates mainly from
F
4f electrons influence not only 4f Ce states directly, but the splitting betweenoccupiedandunoccupied4fbands;
also 5p Ce semicore states indirectly in this compound. seeFig.3(b). SuchasplittingwithinLDA+Utreatment
There are two nonequivalent Ce atoms in the magnetic isdue to the effective CoulombrepulsionHubbardUpa-
cell with opposite 4f spin directions, which are indexed rameter. The4fCeDOSpilesupinthevicinityofFermi
as Ce1 and Ce2. The 4f DOS’s of Ce1 and Ce2 are level within our GGA calculations. The later point is
showninFigs.3forbothphasesofAFM(111)+SOCand what one can clearly see for the AFM(111)+SOC phase
AFM(111)+SOC+LDA+U. The up and down 4f-DOS’s inFig.3(a). ThisisinthecasethattheLDA+Ucalcula-
of Ce1 are asymmetric with respect to eachother,which tionsgiverisetobeconstitutedthe4fCeDOS’sfarfrom
is the case for Ce2 as well. This is in the case that in Fermilevel. AsshowninFig.3(b),includingrepulsionU
overall Ce 4f DOS is entirely symmetric regardless in- potential energy causes to be shifted down occupied and
dexes 1 and 2. Therefore, in spite of the fact that Ce1 shifted up unoccupied 4f DOS’s with respect to Fermi
and Ce2 can individually impose magnetic moment, no level. The separation energy between upper and lower
net magnetic moment is imposed in the whole of the su- 4f bands is calculated to be 0.43 Ry. Thus contributions
percell. The4fDOSoftheAFM(111)+SOCphaseisalso of 4f Ce to the value of DOS(E ) are highly decreased.
F
showninFig.3(a). Inthiscaseonecanmoreeasilycom- Therecedingofthe4fCeDOS,fromthevicinityofFermi
pareitwiththelaterphaseofAFM(111)+SOC+LDA+U level,indicatesthatthe4felectronsaregoingtolosetheir
shown in Fig. 3 (b). The result shows that the signifi- itinerantcharacter,andasaresulttogaintheirlocalized
5
of putting 4f states right at the Fermi level. Thus the
band-like treatment overestimates hybridization of the
4f Ce states with the other valance states. The later
treatmentwouldnotbe also welltrustedbecause ofcon-
(a) up&dn Ce2 fining 4f states in the core region. Thus the open-core
225
up&dn Ce1 treatment usually underestimates hybridization of 4f Ce
150 states with the other valance states.
75
0 IV. ELECTRONIC SPECIFIC HEAT
y
S R
O -75 84
D 0.5 In this section, to fix ideas it seems advisable to com-
4f -150 E = f parethebehavioroftheelectronicspecificheatswiththe
e behavior of the DOS(E ) through introduced phases in
C-225 F
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 the preceding sec. IIIC. The findings of this comparison
c
eti in the next sec. V will be of relevance to the goal of this
n (b) up&dn Ce2 paper to realize whether or not such a comparison can be
g 80 LDA+U up&dn Ce1
a generalized to the EFG quantity as well. Therefore, we
m
have plotted in Fig. 4(a) the calculated total and 4f Ce
o 0.43 Ry
err 40 DOS′s(EF) versus the discussedphases. The number of
f atoms in all the AFM phases is two times the number
Anti 0 y of other phases. The later point is also the case for the
2 R DOS′s(EF). We have then divided the DOS′s(EF) by
9
5 2 for all the AFM phases. In this case, one can compare
-40 0.
E = f tphheasAesFMre-gDarOdSle′sss(EthFe)nwuimthbethreoDf aOtoSm′ss(EinF)thoef tmhaegontehteicr
-80 and chemical cells. One performing such a comparison
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
can focus only on the magnetic ordering effects through
Energy (Ry) allthedefinedvariousphases. The behavior of this curve
constitutes the backbone of this paper.
Wehavecalculatedtheelectronicspecificheats,C ,in
V
the lakeofbothphonon-phononandelectron-phononin-
FIG. 3: ”(color online)” 4f-DOS of the (a) AFM(111)+SOC teractions,andplottedtheSommerfeldlinearcoefficient,
phase and (b) AFM(111)+SOC+LDA+U phase. Solid (dot- γ = C /T, in Fig. 4(b) versus all the phases. The re-
V
ted)curvesshowupanddownDOS’softheCe1(Ce2),where
sult nicely represent the behavior of the total DOS(E )
F
label1(2)referstotheattributed↑(↓)ordering. Fermilevels
shown in Fig. 4(a), provided that the calculated specific
are shown using dashed lines for each of cases.
heatspercellwerealsodividedby2fortheAFMphases.
Onewouldnotbesurprisedobservingsuchaperfectcon-
sistency between in one side the behavior of the specific
heatsandonthe otherhandthe behaviorofDOS′s(E )
F
character. In order to justify about the tendency of the going through our defined phases. One would not be so,
4f-electronstoclarifythedegreeoftheirlocalization,one because the specific heats calculations were performed
can compare Fig. 2(c) with Fig. 3(b). The comparison taking only electron-electron interactions into account.
shows that the 4f Ce states play an important role and One can analytically omitting phonon interactions eas-
must not be ignored. The later point may be deduced ily prove29 the formula of C /T = 1/3π2k2DOS(E ).
V B F
from two subsequent facts. First in energy space, the This formula demonstrates a linear relation between γ
4f states as shown in Fig. 3(b) are distributed over an and DOS(E ). According to our knowledge, it had
F
energy interval for which other conduction bands, e.g. not been analytically established such a relation between
In-states, as shown in Fig. 2(c) are distributed as well. electric filed gradients (EFG’s) and DOS(E ). There-
F
Second in real space, each In atom is surrounded by 4 fore, the above sketched strategy will numerically make
Ce atoms as one can see in Fig. 1(a). Consequently, the an opportunity in the next section trying to demon-
4f states are well hybridized with the other conduction strateanapproximatelylinearrelationbetweenEFGand
bands even after including LDA+U. Thus, the localiza- DOS(E ). Weclosethissectionbyreportingthevalueof
F
tion is reduced but not vanished by LDA+U. Therefore, 9.74 mJ/(mol.cell.K2) within our LDA+U calculations.
we conclude that not only band-like treatment, but also This calculated γ value is almost one order of magni-
open-core treatment cannot provide satisfactory results tude less than the experimentally measured30 value of
for thecaseof localized 4f statesin this CeIn3compound. 130mJoule/(mol.cell.K2). The discrepancy is in agree-
The formertreatmentwould notbe welltrusted because ment with all the other ab initio calculations for other
6
tensorhasbeencalculatedusingthe followingformula34:
5 V
20
V = lim , (2)
zz r→0 4π r2
r
200
where radial potential coefficient, V , within LAPW
-1 ell (a) method has been calculated as follows280,34:
c
-1 Ry 150 RMT 5
es V (r =0) = 1 ρ20 1 r d3r
OS(E) statF10500 T4fO CTe 20 + 45πZ0 Vr(3K()j2−(K(cid:18)RRMMTT)Y(cid:19)20()Kˆ). (3)
D Divided by 2 in XK
AFM phases
0 The integral yields the EFG contribution of the elec-
tronsinsideandoverthesurfaceofthemuffin-thinsphere
-2 K 60 withradiusofR . ThesummationyieldstheEFGcon-
-1 cell 50 Divided by 2 in tribution of theMelTectrons entirely outside of the spheres.
-1 mol 40 AFM phases (b) TcahlleedcothnetrvibaulatniocneEoFfGth,ewheliecchtrwoenhseirnesdideenottheeitsbpyheVrzevzails.
e The contribution of the electrons over the surface and
ul 30
o outsideofthespheresiscalledthelatticeEFG,whichwe
J
m 20 here denote it by Vlat.
-1 T Electric field grazzdients, V ’s, and their respective
C V 10 valance,Vval’s,andlattice,Vzlazt’s,componentswerecal-
zz zz
0 culated for all the introduced phases in sec. IIIC. Our
NM+SNOMC(only Ce)NM+SOC FMFM+SOCAFMA F(0M0 1(0)01)A +FASMFOM C(1 (1111)1 +) S+OSCOC+LDA+U rotliiersosetnuetsidlctasoilfna∆craeTplca(cuEbol.mFat)IpiIoaanrnfesoddrin∆wadliTlt(haEtbhFe.e)xIpw.peherraiTmeseheasenl.stoanaTcinashdolecturoorltaephtsyeeurdlftu,tahnniecnd--

Tab.I, showsthatourcalculatedEFGwithinthephaseof
AFM(111)+LDA+U is in better agreement with experi-
ment than previous calculations1,3. The better agree-
FIG. 4: ”(color online)” (a) Total and 4f-Ce DOS’s versus mentconfirmsthatwehavetakenmoreproperlycorrela-
a variety of phases discussed in the text at their respective tionsamong4felectronsintoaccountwithinourLDA+U
Fermi levels. Thevaluesof DOS(EF) are dividedby2 for all calculations. One observes, from Tab. II, that at In site
tinhemAJFoMulep.hmaosel−s.1.(cbe)llS−o1m.Km−e2rfeolfdtlhineeaerleccoteroffinciicenstpγec=ificCVhe/aTt gthreeaatebrsotlhuatne vthaleuaebosfol∆utpe(EvaFl)ueisoofn∆edo(rEder),owf mhiachgniistundoet
F
shown by crossed-squares versus all the phases. For compar-
the caseatCe site. Forthe latercase,the absolutevalue
ison the γ coefficients, as shown by filled circles, are divided
of ∆p(E ) is one order of magnitude smaller than the
by 2 for theAFM phases. F
absolute value of ∆d(E ), i.e. ∆p(E ) < ∆d(E ).
F F F
| | | |
We will back to this point soon. What important for
us here is to study the variation of the EFG and ap-
cases in the lack of phonon interactions31,32,33. propriateanisotropyfunctionsversusdiscussedphasesto
obtain a relation between EFG and DOS(E ). There-
F
fore, we perform more comparisons throughout Tab. I
and Tab. II illustratively in Figs. 5. We have plotted
V. ELECTRIC FIELD GRADIENT
V in Fig. 5(a) and ∆p in Fig. 5(b), both evaluated at
zz
115In site, versus all the phases. It is generally believed
The electric fieldgradient(EFG) is a tensor ofrank 2. that, contributions to the EFG originating from p states
The EFG tensor has only 2 independent components in dominate34. Consequently, one expects to find a linear
the principle axes system (PAS). The axes of the system relation between EFG and ∆p, i.e. EFG ∆p. This
∝
were chosen such that Vzz Vyy Vxx . One can is what one expects to observe looking at Fig. 5(a) and
| | ≥ | | ≥ | |
then only evaluate the main Vzz component of the EFG Fig. 5(b). It is hard to realize, however, the linear rela-
andtheasymmetryparameterη = Vxx−Vyy todetermine tion of EFG ∆p by comparing these two figures with
Vzz ∝
thetwoindependentcomponentsoftheEFGinthePAS. each other. Therefore, in order to exhibit such a linear
In this paper we only focus on the V as our calculated relationship, we have separated 3 nonmagnetic phases
zz
electric field gradients, since the asymmetry parameters fromthe other 6 magnetic phases. The V and∆p were
zz
are zero for our case. The main component of the EFG respectively shown in Fig. 5(c) and Fig. 5(d) versus 3
7
TABLE I: The main component of the EFG, Vzz, and its decomposition to valance, Vzvzal, and lattice, Vzlzat, components given
intheunitsof1021V/m2 atbothInand140Cesitesforallthephasesdiscussedinthetexttogetherwiththecalculatedresults
within LAPW method bythe others as well as experimental EFG.
Inte. Vzz Vzvzal Vzlzat
−→
Phase SOC LDA+U M|| In Ce In Ce In Ce
NM No No 13.209 0 13.263 0 -0.054 0
NM Only Ce No 12.466 0 12.515 0 -0.049 0
NM Yes No 12.847 0 12.538 0 -0.051 0
FM No No 13.018 0 13.024 0 -0.006 0
FM Yes No (001) 12.843 0.082 12.867 0.105 -0.024 -0.023
AFM No No (001) 13.027 0.089 13.032 0.086 -0.005 -0.003
AFM Yes No (001) 12.966 0.086 12.968 0.066 -0.002 -0.002
AFM Yes No (111) 12.857 0.029 12.961 0.040 -0.004 -0.011
AFM Yes Yes (111) 12.431 -2.863 12.442 -2.889 -0.011 0.026
⊙AFM Yes No (111) 12.490 - 12.540 - -0.050 -
⊗AFMexp unknown 11.6 - - - - -
⊙Ref.3
( ⊗Ref.1
TABLEII:Valancepanddanisotropyfunctions,∆p(EF)and∆d(EF),evaluatedatInand140Cesitesforthediscussedphases
in thetext.
Inte. ∆p ∆d
−→
Phase SOC LDA+U M|| In Ce In Ce
NM No No 0.1066 0 -0.0060 0
NM Only Ce No 0.1040 0 -0.0061 0
NM Yes No 0.1043 0 -0.0061 0
FM No No 0.0373 0 -0.0050 0
FM Yes No (001) 0.0369 0 -0.0052 0
AFM No No (001) 0.0373 0.00015 0.0025 0.00130
AFM Yes No (001) 0.0372 0.00005 0.0024 0.00130
AFM Yes No (111) 0.0370 -0.00010 0.0024 -0.00085
AFM Yes Yes (111) 0.0361 0.00715 0.0026 0.01880
nonmagnetic phases. Similar results, V and ∆p, for The comparison yields the result that the EFG is ap-
zz
the magnetic phases were separately shown in Fig. 5(e) proximately proportional to the DOS(E ). The result is
F
and Fig. 5(f). Now one can among non magnetic phases obtained, because the V and the DOS(E ) vary simi-
zz F
clearly observe the linear relation between EFG and ∆p larly, going through all the phases. This is analogous to
by comparing the behavior of V , Fig. 5(c), with the the used strategy in sec. IV to realize the linear relation
zz
behavior of ∆p shown in Fig. 5(d). They behave simi- of C /T = 1/3π2k2DOS(E ). Our result shows that
V B F
lar to each other trough non magnetic phases. This is the proportional constant is negative for the non mag-
also the case for the magnetic phases as well, since the netic phases,while it is positivefor the magnetic phases.
V shown in Fig. 5(e) behaves similar to ∆p shown in The former constant is negative, because the V and
zz zz
Fig. 5(f) through the magnetic phases. Now time seems DOS(E ) inversely vary through non magnetic phases.
F
apttointuitthatthereisanapproximatelylinearrelation They, however, vary directly versus magnetic phases re-
between EFG and DOS(E ), i.e. EFG DOS(E ). sulting to the positive constant.
F F
∝
The relation can be realized, if the behavior of the to- Furthermore, one can also conclude that the value
tal DOS(EF) shown in Fig. 4(a) is compared with the of 4f-DOS(EF) can significantly influence the value of
behavior of the Vzz shown in Fig. 5(c) and Fig. 5(e). EFG. One may confirm this conclusion due to a simi-
8
)
2
- m 13.2 (a) 13.2 (c) 13.0 (e)
V
21 0 13.0 13.0
1 12.8
G (x 12.8 12.8
F 12.6 12.6
E 12.6
n
I 12.4 12.4
12.4

y
rop 0.105 (b) 0.107 (d) 0.0372 (f)
ot
s 0.090 0.106 0.0368
ni 0.075
a
p 0.060 0.105 0.0364
5
n 0.045 0.104 0.0360
I 0.030
NM+SONCM(only CeN)M+SOC FMFM+SOA ACFFMM ( 0(00A01A1F)F)M M+ (S 1(1O111C1) )+ +SSOOCC+LDA+U NM NM+SOC(only Ce ) NM+SOC FM FM+SOCAFMA (F00M1 )(001) +ASFOAMFC (M11 (11)1 +1S) +OSCOC+LDA+U

FIG. 5: ”(color online)” (a) The calculated EFG at the 115In site and (b) the In 5p anisotropy function, ∆p(EF), for all the
nonmagnetic and magnetic phases discussed in the text, which they are also represented in (c) and (d) for only nonmagnetic
phases as well as in (e) and (f) for only magnetic phases.
larity between the behavior of 4f DOS(E ) and total at this site. The non zero values for the EFG’s origi-
F
DOS(E ) shown in Fig. 4 (a). The foregoing result of nate from the fact that SOC or magnetic ordering can
F
EFG DOS(E )ensuresthattheEFGcanbealsopro- give rise to a little bit deviation from cubic symmetry.
F
portion∝al to the 4f DOS(E ); i.e. EFG DOS4f(E ). One expects that the small deviation gives rise to small
F F
∝
EFG’s. Our result, in Tab. I, confirms the later point
The later proportionality between EFG and 4f
apartfrom the last phase of AFM(111)+SOC+LDA+U.
DOS(E ) makes more crucial the method of treatment
F
For the later AFM(111)+SOC+LDA+U phase the EFG
with 4f-Ce electrons. The later point describes why the
valueofEFGat115InsitewithintheLDA+Ucalculation at Ce site is also practically small, but it is 2 order of
magnitude larger than the other phases. To find the
is smaller than those obtained within all the other cal-
source of such a discrepancy, we follow our last strategy
culations performed in the lack of LDA+U interactions.
lookingatthe behaviorofEFGandanisotropyfunctions
The description can be provided taking the splitting of
through corresponded phases. The calculated EFG at
0.43 Ry shown in Fig. 3(b) into account between occu-
Ce site can be compared with the ∆p versus magnetic
pied and unoccupied bands due to the LDA+U treat-
phases (apart from FM phase for which EFG is exactly
ment. The splitting gives rise to be shifted downwards
zero) using Fig. 6 (a) and (b). The comparisondoes not
the 4f DOS from the vicinity of Fermi level, and as a
show similar behavior for the EFG and ∆p along the
result to be reduced the value of 4f DOS(E ). The
F
magnetic phases. To find the reason why they do not
later reduction together with the discussed relation of
EFG DOS4f(E ) provides the satisfactory descrip- show similar behavior, now we come back to the men-
F
∝ tioned point that ∆d(E ) > ∆p(E ) at the Ce site;
tion why the EFG is reduced within LDA+U calcula- | F | | F |
see Tab. II. One first suspects, due to the larger value
tions. All these support our previously concluded result
of ∆d(E ) than ∆p(E ), that the behavior of EFG
insec.IIIC concerningthe importantroleof4felectrons | F | | F |
mightbe similartothe behaviorof∆d(E ). The behav-
in this compound. F
iors of EFG and ∆d(E ) are comparedin Fig. 6 (a) and
F
At Ce site due to its cubic point group the EFG’s (c). However, they also do not show similar behavior.
are zero, as listed in Tab. I, for all non magnetic phases Therefore, we could not reproduce the behavior of the
and FM phase. For FM+SOC phase and all the AFM EFG using either ∆p(E ) or ∆d(E ). For more realiza-
F F
phases, however, our result shows non zero EFG values
9
12 up+dn
0.5 6 8

4
21-2 0 Vm)-00..05 4 0 (a)
G (x1-1.0 2 p-p up
EF-1.5 (a) dn
Ce -2.0 0
-2.5 0.8
-3.0 1.2 up+dn
0.008 0.6 0.8
0.4
) 0.0
py 0.006 -2 m 0.4 d-d (b)
o
otr V 0.2 up
nis 0.004 (b) 1 dn
a 2
p 0 0.0
e 1
C 0.002 x
(
s 0
0.000 n
o
nisotropy00..001250 contributi -1--284 f-f udpn --11--62084 up+ dn (c)
a0.010
Ce d 0.005 (c) val zz -16
V
8
0.000
4
FM+SOC AFM (001) AFM (001) +SOC AFM (111) +ASFOMC (111) +SOC+LDA+U -04 tot -01 up+dn (d)
up -2
-8 dn -3

-12

12
FIG. 6: ”(color online)” (a) The calculated EFG at the
140Ce site, (b) the Ce p anisotropy function, ∆p(EF), and 8
(c)theCedanisotropyfunction,∆d(EF),forthephasesdis- tot - f (e)
cussed in the text for which nonzero EFG’s have been cal- 4
culated at the 140Ce site, viz. ranging from FM+SOC to
0
AFM(111)+SOC+LDA+U.
tion, we have decomposed the EFG at Ce site into its FM+SOC AFM (0A0F1M) (001) +SAOFCM (11M1 )( 1+1S1O) C+SOC+LDA+U
AF
valance contributions. The results are shown in Figs. 7
for spin up and spin down. In the Figs. 7, we have inset
a figure to show sum of up and down spins for each of
FIG. 7: ”(color online)” (a) Decomposed up, ↑, down,
the valance contributions individually. The result, com- ↓, and up plus down, ↑ + ↓(see insets), valance contri-
paring Fig. 7 (a) and (b), shows that contributions of butions of EFG evaluated at 140Ce site for the magnetic
d-states to the EFG is one order of magnitude less than phases discussed in the text ranging from FM+SOC to
contributions of p-states. This is in the case that, as AFM(111)+SOC+LDA+U.(NB:TherearenoanyEFGcon-
expressed before, ∆d(E ) is one order of magnitude tributions at Ce site for FM phase, in the lack of SOC, due
F
larger than ∆p(E| ). The|refore, even for the case of its cubicsymmetry.)
F
| |
∆d(E ) > ∆p(E ), contributions to the EFG origi-
F F
| | | |
10
nating from p-states compared to the d-states dominate. appliedcircumstancesmaybeausefulstrategytoemerge
This result is in complete accord with the calculations physical properties. We have found within such a strat-
performed in Ref. 34 for transition metals. However, a egy that the main component of the electric field gradi-
differentandimportantresultcanbeemergedfromcom- ent,V ,isapproximatelyproportionaltothevalueofto-
zz
paring Fig. 7 (c) with Fig. 6 (a). The result is this that taldensityofstatesatFermilevel(DOS(E ))aswellas
F
the EFG mainly originates from f-states at Ce site. We 4f DOS(E ). Despite anisotropy function of d-states
F
−
have added all the valance contributions to the EFG in islargerthantheoneofp-statesatCesite,contributions
Fig. 7 (d). The result shows that the behavior of EFG totheEFGoriginatingfromp-statesdominatecompared
shown in Fig. 6 (a) is similar to total valance EFG con- to d-states. This is in the case that, however,the EFG’s
tributions shown in Fig. 7 (c). Lattice contributions to are dominated at the 140Ce site by the 4f-states, while
the EFG also cannot change the similarity,because they they are as usual dominated by p-states at the In site.
are negligible as listed in Tab. I. Thereforecontributions The result shows that 4f Ce states are hybridized with
totheEFG originating from f states dominate at Ce site. conductionbandsandplayanimportantrolewhichmust
For sure in Fig. 7 (e) we have subtracted contributions notbeignored. Thuswehavewithinourelectronicstruc-
to the EFG of f-states from total valance EFG contribu- ture calculations predicted that neither band-like treat-
tions. The result, Fig. 7 (e), represents the behavior of ment, nor open-core treatment can provide satisfactory
p-anisotropy function shown in Fig. 6 (b). Such a repre- resultsforthiscase. OurLDA+Ucalculationsareinbet-
sentation reconfirms our last conclusion. ter agreement with experiment indicating the fact that
correlations among 4f electrons are taken more properly
into account. The correlations among 4f electrons in-
VI. CONCLUSION fluence not only 4f Ce states directly, but also 5p Ce
semicore states indirectly in this compound.
Wehaveinvestigatedthevariationsofelectricfieldgra-
dient (EFG’s) and their valance contributions as well as
their anisotropy functions at both In and Ce sites in the Acknowledgments
CeIn foravarietyofcircumstances. Foreachofthecir-
3
cumstances, density of states at Fermi level (DOS(E )) This work has been performed based on the research
F
is calculated. We have found that comparing the behav- project number 821235, University of Isfahan (UI), Isfa-
ioroftheEFGwiththebehaviorofDOS(E )versusthe han, Iran.
F
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