PH Y
SICAL RE VIE%
VO LUME
B
19,
N
UMBER
15 JUNE 1979
12
Electronic energy bands in P-cerium
Department
Rajanikanth S. Rao
of Physics, Indian Institute of Technology, Bombay-400 076, India
Chanchal
Physics Department,
K. Majurndar
92 Acharya Prafulla Chandra Road, Calcutta 700 009, India
I
Department
of Physics,
Rudra P. Singh
Indian Institute of Technology, Bombay-400 076, India
(Received 15 May 1978)
(KKR) method for Pcalculation has been done by the Korringa-Kohn-Rostoker
Electronic-energy-band
cerium with the Kohn-Sham-Gaspar (KSG) exchange term in the crystal potential. The energy eigenvalues
have been calculated at 57 points uniformly distributed over three symmetry planes in the Brillouin zone.
They have also been obtained in fair detail along a few major symmetry directions. The electronic states
method, as achieving this by KKR method
have been classified using the symmetrized-augmented-plane-wave
requires much more computational effort. It is found that the band in P-cerium lies above the Fermi level,
and the s-d complex shows good qualitative agreement with the band pattern in a-lanthanum. The density of
states at the Fermi level has been found to be 1.71 (e/at. )/eV. However, it appears that the conventional
band picture may not describe the level in metallic cerium very accurately. Within its limitation, the bandstructure calculation, where strong correlations are not involved, explains the situation in cerium better when
we use KSG exchange rather than Slater exchange. It is suggested that x-ray absorption experiments should
be done to check if the, band lies above the Fermi level as found in this calculation.
f
-
f
f
I. INTRODUCTION
Koskimaki and Gschneidner have developed
techniques to prepare pure p-cerium. ' It is no
longer necessary to estimate its properties by
extrapolation from data measured at several alloy
compositions to 100% pure P-cerium. Experiments on pure P-cerium have revealed many interesting features. It occurs in double hexagonal
close-packed structure with c/2a ratio of 1.611,
which is close to the ideal value. There are two
ordering at 12.45
types of antiferromagnetic
and 13.7'K. The electrical resistivity is characterized by a relatively small decrease upon
cooling from 300 to 50'K and a large drop below
20'K. ' Gschneidner et al. have pointed out that
the high-temperature
resistivity could be fitted
to a Kondo-type model for the scattering of conduction electrons, and the rapid drop was due to
the quenching of such a scattering by magnetic
ordering which brought in strong spin correla-
'
tions.
In other allotropic forms of cerium, the a, e',
and y phases, interest centered around the continuous y- n transition, high Pauli susceptibility,
of the e' phase. The
and the superconductivity
band structures of the y and a phases have already
been reported, ' and several models have been
proposed' to discuss the continuous transition.
In this paper, we report the band structure of
P-cerium calculated by the nonrelativistic
l9
Korringa-Kohn-Rostoker
(KKR) method. ' There
has been a relativistic band-structure calculation'
on cy-Ce, and the spin-orbit coupling is expected
to show some finer effects in Fermi-surface
topology. But the Fermi-surface data on the
cerium Phases are not available. Properties
like the electronic specific-heat constant can be
estimated with confidence by nonrelativistic calculations.
Section II discusses the computation of the potential and the bands. Section III discusses the
results and our conclusions from all the bandstructure calculations of cerium.
.
II. CALCULATION OF THE POTENTIAL AND BANDS
Since the KKR method of band-structure calculation is well known, ' we shall only give the
important details pertinent to the calculation.
The atomic charge densities of the 4f'5d'6s'
configuration of cerium" were kindly supplied
to us by Liberman (the computationai method is
described in Liberman et aI.
These have been
used for the construction of the muffin-tin potential by the method suggested by Mattheiss. '2 The
Coulomb part of the atomic potentials have been
overlapped by a technique similar to Lowdin's
n expansion, and discussed by Loucks.
For
the exchange potential, the Kohn-Sham-Gaspar
Here the
(KSG) expression has been chosen.
").
"
"
6274
1979
The American Physical Society
19
ELECTRONIC ENERGY BANDS
atomic charge densities are overlapped first by
the same method, and then with this overlapped
charge density we use the KSG prescription for
the exchange potential. Twenty-nine nearest
shells of atoms have been used to get the overlapped quantities. One can, however, limit to the
13 nearest shells with negligible loss in accuracy.
The radius of the muffin-tin sphere is fixed at
3.4463 Bohr units with the lattice constants
a = 6.9562 Bohr units and c =22.407 Bohr units.
With these values the muffin-tin spheres of
nearest-neighbor atoms just fail to touch one
another, and this ensures rapid convergence. The
average potential in the interstitial region is
found to be -1.2464 Ry, and the discontinuity at
the boundary of the muffin-tin sphere is 0.01860
Ry. The relative smallness of this discontinuity
shows that the muffin-tin potential obtained is a
good approximation to the actual potential. This
aspect is expected from the fact that the c/2a
ratio for P-cerium is very close to the ideal
value.
We have used the terms up to l = 3 in the KKR
expansion of the one-electron wave function in
our calculation. This gives the energy eigenvalues accurate to 0.001 Ry. However, if one
is interested in very accurate eigenvalues of the
band, one may have to go to f =4 terms, because
the f-band levels are densely clustered together.
(For a gross property such as density of states,
the higher accuracy is not necessary. ) In our
calculation with the KSG exchange, the band
lies about 0.09 Ry above the Fermi level (Fig. 1),
and we could limit ourselves to within l = 3
f
f
for satisfactory results.
Since there are four atoms per unit cell, we get
energy levels quite close to one another even
in the s-d hybridized region. Hence it is difficult
to label the electronic states and also to trace
them accurately whenever they cross by the KKR
method by a reasonably limited expenditure of
computational time. Symmetrization is much
simpler in the augmented plane wave method,
and we have therefore employed the symmetrized
augmented plane wave method" for this purpose.
One may have to take terms up to l = 12 in this
method, and it is computationally convenient to
fit the logarithmic derivatives of the radial functions in the form of polynomials in energy variables. Following Loucks, the derivatives can be
put in the form
[R,'(r„, E))/R, (r„, E) = Q, (E)/P, (E) +1/r„,
(1)
where P, (E) and Q, (E) are separately fitted for
each l into polynomials in E and their degrees
may be different for different l values. Here r&
IN
P-CERIUM
6275
3Logarithmic
derivative
--- Phase
dR3
~
Lg=
shift ( ~3
R
3
df
)
2-
«D
0
I
I
I
-1.0 -0-9 -0.8 -0.7
-0-6
-0-5 -0.4
-0.3
ENERGY ( RY)
FIG. 1. Phase shifts and the logarithmic derivatives
of the radial functions of p-cerium for /=3. The bottom
of the band lies near the point & indicated in the figure,
which is about 0.09 By abov'e the Fermi level E~.
f
is the muffin-tin radius. For higher l values the
polynomial fit matches very well with the actual
value mainly because the I/r„ term dominates.
Even the fifth-degree polynomials give logarithmic derivatives accurate up to as many as ten
significant figures. For small l values we have
found that we have to go to the ninth-degree
polynomials for a satisfactory fit.
The energy eigenvalues below the Fermi level.
are calculated at 57 points in the 1/24th part of
the Brillouin zone for the construction of the
histogram density of states. The energy eigenvalues have been calculated at larger number of
points along the major symmetry directions in
order to trace the curves accurately. One has
to have the k points. as uniformly distributed as
possible while constructing the histogram, so
these additional points were not utilized in it.
An averaged histogram of bar width 0.004 Ry has
been obtained by averaging over 5 histograms,
each with energy bar width of 0.02 Ry but different starting energy. The structure of the
histogram depends on the starting point, and an
averaged histogram hides the detailed fine
structure of the density of states,
a small
spike in its shape.
e~,
RAO,
6276
MA JUlg
BAR,
AND
SING
H
3'
-07
-07t
1
CL
QJ
FIG. 2. Energy band of
P-cerium along I'M, I"A,
3
CD
C3
~ -08
and AL,
.
2
Q
UJ
UJ
+
3'
—09
for the density of states of n-lanthanum are
available. The s-d band pattern of P-cerium
III. RESULTS AND DISCUSSIONS
The energy bands of P-cerium along the seven
major symmetry directions are depicted in Figs.
2, 3, and 4. The density of states is shown in
Fig. 5. The Fermi level is found to be 0.216 Ry
above the bottom of the conduction band when one
assumes three conduction electrons per atom.
The density of states 2V(Ez) at the Fermi level
is 1. I1 electronic states per (atom) eV. There
is no direct experimentally measured number,
but Burgardt et al. suggested that the heat capacity
of P-cerium should be comparable to that of o. lanthanum. ' Except for the absence of the 4f
electrons, n- lanthanum has physical properties,
such as crystal structure, Debye temperature,
and the valence comparable to those of Pcerium. Theoretical and experimental estimates
from our calculation shows good qualitative
similarity with the detailed band-structure calculations of n-l. antha'nuum by Ghosh, Kumar, and
Das. Table I compares the density of states.
Our value for P-cerium compares well indeed
with the theoretical estimate for n-lanthanum.
The discrepancy between the experimental and
theoretical values in n-La is attributed partially
to the electron phonon mass enhancement in
"
a-La.
It is generally agreed that P-Ce is trivalent or
nearly trivalent
Burgardt et aL. assign a
valency of 3.04. In our band pictures, the 4f band
lies about 0.09 Ry above the Fermi level and
has a width of about 0.08 Ry. This position is
obtained with the KSQ exchange in our potential;
—
-2+
2
'
JE
I
1
3
-07'
)
-0 70
- 071
-0-71
CL
LU
CD
4J
K
C)
C3
—-0.8
g
z
Ul
2
1+
1+'
~-OS
IX
x
1
UJ
-09
3
-09
1
t
K
T'
FIG. 3. Energy bands of P-cerium along I'K and EM.
A
H
S'
L
FIG. 4. Energy bands of P-cerium along AH and HL.
ELECTRONIC ENERGY BANDS
6277
f
f
situation implies that the level does not play
a prominent role in the phase transitions of
cerium, whereas one generally attributes the different allotropic forms in cerium, at least partially, to the changes in the relative occupancy
of the 4f and 5d6s states. Many of the partially
successful theoretical models in a y transition
require the band to lie above the Fermi level
with some amount of hybridization.
Experimentally, the situation is not cl.ear. Older optical
data indicated that the band should lie just below
the Fermi level. The x-ray photoemission data
of Baer and Busch (now repeated by Steiner,
Hochst, and Hiifner)" were interpreted similarly
to indicate that the band lies lower than the
Fermi level, and Johansson's idea" that the
n- y transition is a Mott transition is based on
this interpretation.
as Baer and
Unfortunately,
Busch themselves state, the separation of the
structure from the broad featureless data is too
arbitrary. If one puts the narrow band below
the Fermi level one could expect a sharp feature;
alternatively, one must have strong hybridization
of the bands. The latter does not seem to be
true of any band calculation done so far the
levels remain in a fairly narrow group and move
as a whole, and hybridization is weak. Another
difficulty in trying to push the level down below
the Fermi level is that the Fermi level will immediately get attracted down into the states because of their very large density of states, unless one invokes strong correlations to prevent
electrons from going into these levels. In that
case the band picture should not describe the behavior of metallic cerium. Since the KSG bandstructure calculations give the band higher than
the Fermi level, experiments in soft x-ray absorption that can detect the density of unoccupied
~
O
I-10I
I
l/l
Z
I
O
I
I
I
I
O
I
LU
I
~0.5-
IEF
I
I
I
I
-0 85
I
- 0 80
07S
-0 71-070
ENERGY (RY)
FIG. 5. Density of electronic states of P-cerium.
energy bar width is 0.004 Ry.
'
f
I
I
I
- 0 90
-
f
LLJ
0
P-CERIUM
Fermi level would be difficult to settle theoretically. However, the Slater-exchange potential
seems to be inadequate for cerium in a bandstructure calculation because it puts the band
several electron volts (= 10 eV) below the Fermi
level in P as well as in u and y phases. ' Such a
15-
I
IN
The
f
the atomic charge-density calculations of Liberman et a/. have the same exchange term. Recall
that this KSG exchange utilizes the exchange potential applicable to electrons at the Fermi level
for all the electrons in the band. The position
of the 4f band with respect to the s-d complex is
very strongly dependent on the exchange term.
As one goes to the Slater exchange, which uses
the exchange potential averaged over all the occupied electron states in the band, the 4f band
rapidly shifts to lower energy, narrowing in
width at the same time. This behavior of the
band can be estimated by the behavior of the
logarithmic derivatives as a function of energy
for the l =3 radial wave function occurring in the
band-structure calculation. Figure 1 displays
the variations of the logarithmic derivative and
the phase shift with energy for the l =3 radial
wave function in P-cerium in the case of KSG
exchange potential.
The position of the band with respect to the
f
f
f
— f
f
f
f
/
f
f
TABLE I. Density of states N(E~) at the Fermi level.
& —lanthanum
Present
Theoretical
calculation
fol
Non-
P-cerium
relativistic
Relativistic
capacity
Susceptibility
1.71
1.53
1.43
3.98
3.15
Reference
Reference
Reference
Reference
16.
20.
21.
22.
Experimental
Heat
RAO,
MA
JUMDAR, AND SINGH
states should be performed. We must note that
the Fermi-surface measurements have not been
done on cerium. Theoretical calculations are
available for the n and y phases and can be worked
out for the P phase; the details are naturally very
sensitive to the levels if we put them very near
(above or below) the Fermi level, and to the
relativistic spin-orbit splitting. Because of the
large uncertainty, there is little point in reporting
the present Fermi-surface calculation.
Apart from the position of the band with respect to the Fermi level, the character of the
electron, whether localized or itinerant, would
be important in the computation of several physical properties measured by Burgardt et al, .
Ottewell, Stewardson, and Wilson" have stated
that the localized 4f level in y or P phase does
not become a 4f band in the o. phase. They indicate that the electron is localized in the y
and P phases, but the position of the level remained undetermined.
We may thus summarize the conclusions of our
f
f
f
f
f
C. Koskimaki and K. A. Gschneidner, Jr. , Phys.
Rev. B 10, 2055 (1974).
P. Burgardt, K. A. Gschneidner, Jr. , D. C. Koskenmaki, D. K. Finnemore, J. O. Moorman, S. Legvold,
G. Stassis, and T. A. Vyrostek, Phys. Rev. B 14,
2995 (1976).
3D. C. Koskenmaki and K. A. Gschneidner, Jr. , in Handbook on the Physics and Chemistry of Raze Earths,
edited by K. A. Gschneidner, Jr. , and L. Eyring
(North-Holland, Amsterdam, 1976), Chap. IV.
4K. A. Gschneidner, Jr. , P. Burgardt, S. Legvold,
J. O. Moorman, T. A. Vyrostek, and C. Stassis, J.
Phys. F 6, L49 (1976).
5G. Mukhopadhyay and C. K. Majumdar, J. Phys. C 2,
J. Phys. F 2, 450
924 (1969); G. Mukhopadhyay,
(1972); B. S. Bao, C. K. Majumdar, B. S. Shastry, and
R.P. Singh, Pramana 4, 45 (1975).
R. Ra~irez and L. M. Falicov, Phys. Rev. B 3, 2425
(1971); see Ref. 2 for a critical review.
YJ. Korringa, Physica 13, 392 (1947); W. Kohn and
N. Bostoker, Phys. Bev. 94, 1111 (1954).
B. Segall, Phys. Bev. 105, 108 (1957); B. Segall and
F. S. Ham, Methods in Computational Physics, edited
by B. Alder, S. Fernbach, and M. Botenberg (Academic, New York, 1968), Vol. 8, p. 251.
~D.
19
band-structure calculations of the P-cerium.
The level of metallic cerium cannot be described wel. l by the conventional band-structure
calculation; this failure in the case of a pure
element is somewhat surprising. Within the
limitations of the band picture, we believe that
the situation is described better by the KSG exchange than by the Slater exchange, and that the
bands, weakly hybridized, should lie above the
Fermi level. X-ray absorption experiments and
the Fermi-surface data would be valuable in
clarifying the position of the band.
f
f
f
ACKNOWLEDGMENTS
One of us (R. S.R. ) is grateful to the Department of Atomic Energy, Government of India for
the award of a Senior Research Fellowship during
the tenure of which this work has been done. He
is also grateful to the Director of the Indian
Institute of Technology, Bombay, for providing
all facilities during the progress of this work.
B. Johansson, J. Phys. F 7, 877 (1977).
L. Brewer, J. Opt. Soc. Am. 61, 1101 (1971).
"D. Liberman, J. T. Waber, and D. T. Cromer, Phys.
~
Rev. 137, A27 (1965).
L. F. Mattheiss, Phys. Bev. 133, A1399 (1964).
'3T. L. Loucks, Au gm, ented Plane Wav e Method (Benjamin, New York, 1967).
'4W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)-.
'5L. F. Mattheiss, J. H. Wood, and A. C. Switendick,
Methods in ComPutational Physics, edited by B. Alder,
S. Fernbach, and M. Rotenberg (Academic, New York,
1968), Vol. 8, 63.
A. Ghosh, S. Kumar, and K. C. Das, Indian J. Pure
Appl. Phys. 8, 685 (1970).
~~Y. Baer and G. Busch, Phys. Rev. Lett. 31, 35 (1973);
P. Steiner, H. Hochst, and S. Hufner, J. Phys. F 7,
L145 (1977).
~
B. Johansson, Philos. Mag. 30, 469 (1974).
~~D. Ottewell, E. A. Stewardson, and J. E. Wilson, J.
Phys. B 6, 2184 (1973).
G. S. Fleming, S. H. Liu, and T. L. Loucks, Phys.
Bev. Lett. 21, 1524 (1968).
2~D. L. Johnson and D. K. Finnemore, Phys. Rev. 158,
376 (1967).
J. M. Lock, Proc. Phys. Soc. B 70, 566 (1957).
~