CHAPTER 4
ELECTRONIC STRUCTURE OF THE RHODIUM COMPOUNDS
ARh2 (A = Ca and Sr)
4.1
INTRODUCTION
The binary phase diagram of rhodium (Rh) with group II elements
comprises mostly of the Laves phase compounds. The element with a low atomic
number such as beryllium (Be) forms two compounds BeRh and Be^Rh with
rhodium (Hansen et al 1958 and Johnson et al 1970), and in the case of
magnesium (Mg), there are two compounds: MgRh and MgjRh2 crystallising in
CsCl and Al3Co2 structures respectively (Shunk 1970). Since Be and Mg are
relatively smaller in size compared with rhodium, no Laves phase compounds are
identified in the binary phase diagram of Be and Mg with rhodium. But in the case
of Ca,Sr and Ba whose atomic sizes are larger compared to that of rhodium form
only Laves phase compounds all crystallising in the same C15 type structure and
in addition SrRhj is also formed in the phase diagram of Sr with rhodium (Elliott
1958).
It is interesting to note that, they not only crystallise in the same structure,
but also their lattice constants are very nearly the same. The change in the lattice
constant is only 0.15
A in going from Ca - Sr -Ba. Further all of them are found
to have their superconducting transition temperatures around 6K. The changes in
the A atom size in these compounds does not seem to even influence the structure
type which is supported by many examples like ACo2 and ANi2 ( A= Sc, Y and
Lu) compounds in the C15 type and AMg2 (A= Ca, Sr and Ba) compounds in the
C14 type structure.
68
The identification of superconductivity in these compounds are themselves
based on Matthias et al’s (1963) qualitative arguments. It states that
superconductivity is normally observed in metals with 2 to 8 valence electrons. It
should be noted that neither pure A nor B in these systems is superconducting.
It was expected that, if compounds are formed out of them thereby increasing or
decreasing the average valence electron per atom ratio, they might become
superconductors. It turns out that the average electron per atom ratio for these
compounds is 6.67 which lies within the range. A thorough and systematic study
of their electronic structures are undertaken mainly to understand their
superconducting behaviour.
Before performing the electronic structure calculation on these compounds,
one can anticipate certain systematics in these compounds from the earlier studies
which have been made on the Laves phase transition metal compounds and they
are listed below.
(i)
It has been generally observed that all Fe compounds of the type T-Fe2
exhibit ferromagnetic behaviour like elemental Fe (Yamada 1988). Hence
for the rhodium compounds which are presently investigated, the different
physical properties might be expected to resemble that of rhodium.
(ii)
Even in the elemental case, all the divalent atoms Ca, Sr and Ba have
strikingly different behaviour. Sr is a semimetal whereas Ca becomes a
semimetal under high pressure (Skriver 1982). Inspite of the fact that the
elemental solids exhibit striking differences in their behaviour, their
influence on the electronic structure of these compounds (Ca,Sr,Ba)Rh2 will
be negligible as has been observed by Yamada et al (1988) in his
systematic study of the transition metal compounds. Of course in some of
the Laves phase compounds, the A atoms too play a crucial role in deciding
the structure and magnetism. For instance the crystal structure of ScFe2 and
YFe2 are different. ScFe2 stabilises in the C14F state (Ikeda et al 1974),
while YFe2 is found to be in the C15F state (Buschow and Stapele 1970).
69
(iii)
The decreasing lattice constant from BaRh2 to CaRh2, should result in an
increased hybridisation of the bands of A atoms with that of B atoms and
hence the bandwidth can be expected to be large in the latter case.
The electronic structures of the compounds CaRh2 and SrRh2 are
performed here to make quantitative estimates of the various physical properties.
4.2
ELECTRONIC STRUCTURE OF CaRh2
CaRh2 crystallises in the fee lattice with C15 type structure and its lattice
constant is 7.525
A
(Wood and Compton 1958). The details regarding the
arrangement of atoms in C15 type structure were discussed in chapter 2. The
Brillouin zone for the fee lattice is shown in Figure 4.1. The Wigner-Seitz sphere
radii used for the electronic structure calculation are determined by equating the
volume of the spheres to that of the cell with ideal ratio of 1.225 for
The
trial potential parameters are constructed with the following valence electron
configurations for Ca and Rh: Ca - Core + 4s2, Rh - Core + 4d8 5s3. The eigen
values were obtained for 89 K-points distributed uniformly within the irreducible
wedge of the fee Brillouin zone. The self- consistent iterations were continued till
the eigen value accuracy of ImRy was achieved. The self-consistent potential
parameters are given in Table 4.1 along with the Wigner-Seitz radii of the
individual atoms. The electronic structure and the density of states histogram are
shown in Figures 4.2 and 4.3 respectively.
The energy bands crossing the Fermi energy are 99% Rh d- like. The
s-bands of both the A and B atoms form the low lying bands, while the d-bands
of Ca is above the Fermi level. The cluster of bands present immediately below
the Fermi energy gives rise to the first peak in the DOS curves close to the Fermi
energy. The two characteristic peaks in the DOS curves are as expected for any
transition metal Laves phase compound, with the B atom dominance near the
Fermi level (Inoue et al 1985). From the paramagnetic bandstructures and the
position of the Fermi level, Yamada (1988) successfully explained the different
70
*<Z
71
Table 4.1
Wigner-Seitz sphere radii and self-consistent potential parameters
of CaRhi
Ca
s
(a.u)
a
(Ry)
S^-)
(Ry)
*(-)/*(+)
«pl>%
(Ry)
V(S)
(Ry)
Rh
3.4510
2.8171
s
p
d
-0.0982
0.5846
0.3409
-0.2789
0.7433
-0.2117
s
P
d
0.2541
0.2136
0.0569
0.3525
0.3112
0.0454
s
P
d
0.8654
0.6412
-0.0026
0.8596
0.6277
0.0642
s
P
d
0.2964
0.3113
1.1710
0.2107
0.1992
0.9198
-0.6580
-0.7835
ENERGY (Ry)
72
Ef
Figure 4.2 Band structure of CaRh2
NOS
DOS
(E LE C T R O N S /C E LL)
(S T A T E S /R y /C E L L )
73
Figure 4.3 Density of States of CaRh2
kinds of magnetism exhibited by compounds of magnetic B atoms. According to
Yamada (1988), if the Fermi energy is located to the left of the first peak, then
mostly the compound will be antiferromagnetic. If EF lies on the second peak, the
compound will be ferromagnetic. On the otherhand, if EF lies to the right of the
second peak, then it will be paramagnetic. These observations have in general
been found to be true in most of the transition metal Laves phase compounds. In
the present case, the Fermi energy is to the right of the second peak and hence
paramagnetic. Nothing interesting is there with regard to the magnetism in these
compounds as the magnetism in most of the Laves phase compounds is either due
to the magnetic B atoms or the magnetic A atoms like the rare earth elements.
The partial s,p and d contributions to the number of states and the density
of states of the individual atoms in the unit cell are given in Figure 4.4. The major
contributions to the density of states at the Fermi energy as seen from Figure 4.4
is only from the rhodium d-bands. The total density of states at the Fermi energy
can be used to calculate the ground state properties like the low temperature
electronic specific heat coefficient and also the Pauli paramagnetic susceptibility.
Table 4.2 gives the Fermi energy, the number of states, the density of states at the
Fermi energy and the calculated values of the electronic specific heat coefficient
and Pauli paramagnetic susceptibility along with the conduction band width.
43
ELECTRONIC STRUCTURE OF SrRh2
SrRh2 also crystallises in the cubic C15 type structure whose lattice constant
is 7.706
A (Wood and Compton 1958). As in the earlier case, the initial potential
parameters were obtained from the atomic charge densities with the following
valence electron configurations for the constitutent atoms: Sr-Core + 5S2 ;
Rh-Core + 4d8 5s1. As the structure is the same as that of CaRh2, the structure
dependent part remains the same in the LMTO calculation, but the potential
dependent part gets changed. Eigen values in this case also were obtained
self-consistently for 89 K-points as before with an eigen value accuracy of ImRy.
The electronic structure of SrRh2 is shown in Figure 4.5 and Figure 4.6 gives the
DOS
(STATES/Ry/CELL)
NOS (ELECTRONS/CELL)
75
Figure 4.4 Angular aoaentua resolved
density of states of CaRh2
76
Table 4.2
The Fermi energy (EF), the number of states, the density of states
at the Fermi energy DOS(EF) and the other calculated properties
of CaRhj
Ca
Ef
(Ry)
Rh
-0.0949
NOS(Ef)
(electrons/cell)
s
P
d
1.0160
1.4163
1.9292
2.7003
2.2529
30.6853
DOS(Ef)
s
0.1358
1.4525
(States/Ry-cell)
P
d
0.4822
2.2058
6.2777
103.1895
Total DOS(Ef)
(States/Ry-cell)
113.7433
Conduction
bandwidth (Ry)
0.4910
y
(mJ/mole-K2)
9.8672
XP
(10-* emu/mole)
135.1000
ENERGY (Ry)
77
Ef
Figure 4.5 Band structure of SrRh2
DOS
NOS
(STATES/Ry/CELL)
(ELECTRONS/CELL)
78
Figure 4.6 Density of States of SrRh2
79
total and the component resolved density of states. The self-consistent potential
parameters are listed in Table 4.3.
A comparison of the electronic structure and the density of states curves
of CaRh2 and SrRh2 shows that they are very similar to each other. This is in
concurrence with the earlier observation that neither the change in the atomic size
nor the screening of the different A atoms due to the core electrons leads to any
significant difference in the DOS curves. The fact that the electronic behaviour of
SrRh2 resembles that of CaRh2, implies the dominant role played by the rhodium
electrons. The low lying bands which are Sr s-like are pushed up compared to that
of CaRh2. The major structures in the DOS curve are very similar to CaRh2. The
second peak in the DOS curve due to the electron pocket around L point is closer
to the Fermi energy in the case of SrRh2 than in CaRh2. The partial contributions
by the different kinds of atoms to the number of states and the density of states
are shown in Figure 4.7. The value of the DOS at EF is also nearly the same as
that of CaRh2. The calculated electronic properties are given in Table 4.4.
4.4
SUPERCONDUCTING PROPERTIES OF LAVES PHASE COMPOUNDS
Except for the compounds discussed in the previous chapter, all the other
Laves phase compounds studied in this thesis are found to be superconducting.
Hence a brief summary of the superconducting property of the Laves phase
compounds are discussed here.
Superconductivity has been detected in seventy five out of the hundred
Laves phase compounds investigated (Rapp et al 1974 and Inoue et al 1973).
Almost all the superconducting Laves phase compounds belong to two structural
types C15 and C14 and there is only one compound HfMo2 (Rapp et al 1974)
found to be superconducting in the C36 type structure. Till the late seventies, only
HfV2 and ZrV2 were known to have the highest Tc among the Laves phase
compounds: ZrV2 with a Tc of 9.62 K (Matthias 1961) and HfV2 with a Tc of 9.4K
(Rapp and Vieland 1971). But what is more interesting in the study of
80
Table 43
Wigner Seitz sphere radii and self-consistent potential parameters
of SrRhj
Sr
(a.u)
s
Ey
S$2(-)
m
s
(Ry)
s
p
d
P
d
0(-)/0(+)
s
P
d
sA
V
(Ry)
s
P
d
V(S)
(Ry)
Rh
3.5340
2.8848
0.0166
0.8098
0.3953
-0.2863
0.6821
-0.1921
0.2579
0.2176
0.0797
0.3300
0.2929
0.0419
0.8759
0.6879
0.1591
0.8575
0.6232
0.0560
0.2811
0.3123
0.7733
0.2326
0.2150
0.9910
-0.6293
-0.7362
NOS
DOS
(STATES/Ry/CELL)
(ELECTRONS/CELL)
81
Figure 4.7 Angular loieniun resolved
density of states of SrRh2
82
Table 4.4
The Fermi energy (EK), the number of states, the density of states
at the Fermi energy (DOS(EF)) and the other calculated properties
of SrRh2
Sr
Ef
(Ry)
Rh
-0.0860
NOS
(electrons/cell)
s
P
d
0.8458
1.0280
1.9026
2.8505
2.2356
31.1375
DOS(Ep)
(States/Ry-cell)
s
P
d
0.1243
0.4360
2.1978
1.5730
6.7384
119.2044
Total DOS(Ep)
(States/Ry-cell)
Conduction
bandwidth (Ry)
y (mJ/mole-K2)
XP (10-6 emu/mole)
130.2739
0.4677
11.3013
154.6000
83
superconductivity of the Laves phase compounds is that, these materials have low
hardness and are less brittle compared to the other high-field superconductors and
thus are technologically more interesting (Inoue and Tachikawa 1975 ). Apart
from a sufficiently high transition temperature (Tc), these materials possess high
upper critical fields (H^) (Tc = 10.1 K,
= 240 KOe for (Zr03Hfo.j)V2).
Composites from ZrV2 and HfV2 like (ZrxHf].x)V2 compounds possessing higher
critical parameters (Inoue and Tachikawa 1975 ) than those of V3Ga and Nb3Sn
have already been obtained. There were other experimental studies on C15 type
compounds like ZrV2, HfV2, LaRu2 and TaV2
to know more about the
superconducting property (Kimura 1974). It is only in the beginning of the eighties
that fairly high transition temperatures were reported in C14 type compounds
(ScTcj with a Tc of 10.9 K).
Only after the development of the BCS (Bardeen et al 1957) formalism in
understanding the basic concept of the superconducting state, much of progress
was made in arriving at more exact numerical solutions for calculation of the
superconducting transition temperature. According to the strong coupling theory
(McMillan 1968), the superconducting transition temperature is uniquely
determined by the quantity A which is called the electron-phonon coupling
constant It was hoped that if the parameters determining the quantity X are
known exactly, the superconducting transition temperature can be calculated with
an accuracy of 10'2 K. The electron-phonon coupling constant can be determined
by any one of the following techniques.
(a)
by measuring a number of normal state properties of a metal (Eliashberg
1960).
(b)
by measuring physical quantities from the tunneling effect in the
superconducting state and with the subsequent use of the Eliashberg
Equation
(c)
by calculating it (Butler 1977) according to the formula
A = 2 / mAa2 F(a>) da)
(4.1)
84
where a2F(«) can be calculated from the first principles by using the phonon
frequencies determined by neutron scattering experiments.
Knowing X, one can calculate Tc according to the more often used
McMillan’s (1968) formula which is given below.
rp
@d
Tc = ___ exp
1.45
(4.2)
0D is the Debye temperature and //* is the electron-electron interaction parameter.
However, it is difficult to determine the values of this parameter X in an accurate
way either experimentally or theoretically. Both experimental and theoretical
attempts have been made successfully in the case of A15 type superconductors.
Unfortunately such studies have not been carried out for the Laves phase
compounds (Radousky 1982). The theories for calculating X in metals have been
extended to ordered compounds also (Mattheiss 1972). Such an attempt was made
by Gomersall and Gyorffy (1974) for compounds NbC and NbN.
4.5
ELECTRON-PHONON COUPLING CONSTANT FOR METALS
Most of the existing theories to determine the strength of the electron-phonon
coupling rely on the rigid-ion approximation (Gaspari and Gyorffy 1972). In the
case of materials with one atom per unit cell, all the electronic and phononic
contributions in this formulation factorise as
(4.3)
M<a>2>
where M is the atomic mass and <(o2> is the average of the square of the phonon
frequency and rj depends only on output quantities from our bandstructure
calculations. In the present calculation atomic spheres are used to evaluate
electron-phonon matrix elements rather than the muffin tin spheres. Within the
85
RMTA, the spherically averaged part of this Hopfield parameter (Skriver and
Mertig 1985) may be obtained from (in atomic rydberg units )
,„ = 2N(Ef)
2 (1 + 1) M,V,
1
f<
f'"'
(21 + 1)
(21+3)
(4.4)
where N(EF) is the total state density per spin at the Fermi level, f, the relative
partial state density
f, =
N|(Ef)
(4.5)
N(EF)
and M| ,+1 the electron-phonon matrix element which can be written in terms of
phase shifts as follows
dV
Ri __
?
0
dr
R/+i r2 dr
(4.6)
However, in the atomic-sphere approximation the usual phase-shift notation
becomes meaningless and instead, Mj f+1 is expressed in terms of logarithmic
derivative D* = rRt'/Rt evaluated at the sphere boundary. So
= - 0i(Ef) ^i+i(Ef)
{ [D,(Ef) - i] [D/+j(Ef) + l +2] + [Ep-V(S)] S2 >
(4.7)
where V(S) is the one electron potential and <pt the sphere boundary amplitude
of the I partial wave evaluated at the Fermi level.
As the spherical contributions to the Hopfield parameter is small compared
to rj, the calculation of is restricted to rjQ.
86
The electron-electron interaction parameter // is generally assumed to be
0.13 (McMillan 1968) for transition metals . But it can also be estimated from the
empirical expression
U
N(Ef)
0.26
(4.8)
1 + N(Ef)
given by Bennemann and Garland (1971). The corresponding theoretical A can be
obtained by means of the following assumption
«o2>* = 0.69 eD
(4.9)
of the average phonon frequency
4.6
ESTIMATION OF A FOR COMPOUNDS
In the case of compounds, several formulae were used to determine A, few
of which are given below.
A =
2
Vi n‘
t
M,<cu,2>
2 >7, n,
(4.10)
(4.11)
M<w2>
A
Z
7. n,
t
M,<co2>
(4.12)
87
The subscript denotes the atom type, and n, is the number of atoms of
type t M, is the atomic mass at the i,h site, while M denotes the average atomic
mass. <d)2> is the average of the square of the phonon frequency for the
compounds.
These approximations are valid under different conditions. Equation (4.10)
is valid for those cases where A is metallic and B is non-metallic with the chemical
formula AB. Such separation of A is valid only when the difference in masses is
large, which is the case in carbides (MA > > MB ). If the A and B atoms are not
vibrating in different modes, but behave as coupled oscillators, then Equation
(4.12) can be used. So, we have made use of (4.12) to evaluate A for CaRh2 and
SrRh2 and for the rest of the superconducting Laves phase compounds studied in
this thesis in the absence of any other better approximation.
4.7
CALCULATION
OF
SUPERCONDUCTING
TRANSITION
TEMPERATURE OF CaRh2 AND SrRh2
The parameters for the calculation of superconducting transition
temperatures are evaluated at the Fermi energy and at the corresponding sphere
radii and the parameters that enter into the calculation of Tc are given in
Tables 4.5 and 4.6 for CaRh2 and SrRh2 respectively. As far as the electronic part
is concerned, we infer from the above tables that excepting for the difference in
the logarithmic derivatives and the d-density of states of rhodium, all the other
parameters are nearly the same for both the compounds. The contribution to rj
from A atom is very very small. Having calculated rj from band structure
calculation, a good estimate of the electron-phonon coupling constant is possible
if the phonon spectrum is available. In the case of alkaline earth compounds with
rhodium, no attempt has so far been undertaken to study the phonon spectrum.
In the absence of the phonon-spectrum, the mean square of the phonon
frequency is approximated by the relation <co2> = 0.5 0D2 (Papaconstantopolous
1977). Unfortunately even the Debye temperatures are not available for these
compounds. Joseph and Gschneidner (1968) has prescribed a method for
88
Table 4.5
Partial wave functions (<p), logarthimic derivatives D(EF), electronstiffness constant (rj), electron-phonon coupling constant (A) and the
superconducting transition temperature (Tc) for CaRhz
Ca
Rh
ef
(Ry)
0(Ef)
(Ry)
s
P
d
-0.9337
1.1304
0.9554
-0.9214
-1.2101
-0.1676
D(Ef)
(Ry)
s
P
d
-1.0203
0.3718
0.2810
-1.5649
0.0573
-8.1807
0.0060
0.6789
rj(eV/A2)
-0.0949
234
^d(K)
0.170
A
expt.
TC(K)
Theory
6.00
89
Table 4.6
Partial wave functions (<p), logarthimic derivatives D(EF), electronstiffness constant (rj), electron-phonon coupling constant (X) and the
superconducting transition temperature (Tc) for SrRh2
Sr
Rh
ef
(Ry)
0(Ef)
(Ry)
s
P
d
0.9997
-1.0993
-0.9499
0.8892
-1.1812
-0.1446
D(Ef)
(Ry)
s
P
d
-0.6266
1.0669
0.1721
-1.6662
0.0085
-8.0253
0.0090
0.4768
V (eV/A2)
-0.0855
MK)
147
X
10.258i
Expt.
TC(K)
Theory
6.20
90
evaluating Debye temperatures of the Laves phase compounds from the
knowledge of the radii ratio and the Debye temperature of the component atoms.
If rA“'/rB (here rA“‘ is the A-A distance in the compound) is less than the ideal
ratio, then the Debye temperature will be very close to that of the A atom and if
it is greater than the ideal ratio, then it will be that of the B atom. The validity of
the prescription was checked for many Laves phase compounds for which the
experimental Debye temperatures are available and it is given in Table 4.7.
Though there are certain exceptions like TiBe2 etc., the agreement is fairly good
for most of the transition metal compounds. Following the above procedure, the
Debye temperatures of CaRh* SrRh2 and BaRh2 should be those of Ca, Sr and
Ba respectively.
But with these Debye temperatures, the calculated A values turned out to
be very low. If the Debye temperature assumed for the calculation of A are correct
atleast approximately, then the possible reason why A is underestimated may be
because of the neglect of the f-contribution in the calculation of ijt inspite of the
fact that the B atom is d-band metal which has a major contribution to rj. More
detailed discussions on the effect of the inclusion of f-contribution in the
calculation of rj are given in the following chapters. The reason why the electronic
structure calculations were not repeated including f-states is that Ca,Sr and Ba
have only s-electron configuration and hence the inclusion of f-states will only lead
to unphysical values of rjat for the A atoms.
4.8
SUMMARY
The bandstructures and the DOS curves of CaRh2 and SrRh2 as well as the
density of states histograms exhibit a overall similarity as expected. Neither the
experimental photo-emission spectroscopic data nor XPS data are available for a
direct comparison with the bandstructures or the DOS curves. The shorter B-B
distance in CaRh2 when compared with SrRh2 increases the mixing of the Rh
bands in CaRh2 and the bandwidth is slightly more in CaRh2 when compared to
SrRh* This is reflected in the first band shift near the T point. In both the
compounds, the Fermi energy is located on the second peak and they are nearly
91
Table 4.7
Comparison of the Debye temperatures of the Laves phases ($&)
and their component metals (0A and 0B)
Compound
6>jj.
TyPe
Ta/ib
MgCu2
C15
1.25
1.19
332
396
342
ScCo2
C15
1.32
1.20
344
360
452
LuCo2
C15
1.4
1.25
222
280
452
YCo2
C15
1.4
1.23
238
183
452
ZrCo2
C15
1.28
1.20
420
289
452
HfCo2
C15
1.26
1.20
368
256
452
CeRu2
C15
1.38
1.22
171
239
600
LaRu2
C15
1.40
1.24
160
142
600
HfV2
C15
1.22
1.22
255
256
380
ZrV2
C15
1.23
1.23
290
291
380
LaPt2
C15
1.35
1.21
236
142
234
LaOs2
C15
1.38
1.23
163
142
ZrZn2
C15
1.11
1.10
291
TiBe2
C15
1.29
1.22
370
650
500
327
420
CaMg2
C14
1.23
1.19
426
234
1440
396
ScT^
C14
1.21
1.18
310
360
350
ScRe2
C14
1.20
1.17
345
360
430
rA(cal)/rB
eA
92
the same. Even the small change in the DOS at Fermi energy are mainly due to
the Rh d-electrons. The other properties calculated, the electronic specific heat
coefficient and Pauli paramagnetic susceptibility show only small differences which
are obviously due to the small change in the density of states of CaRh2 and SrRh2.
The very small values of A calculated theoretically will lead to negligibly small
values of the transition temperatures because of the fact that A occurs inside the
exponential and any little change will drastically alter the value of T<. Since it is
the rhodium d-bands that cross the Fermi level in these compounds, the inclusion
of f-states is very essential for better estimates of ri and hence T<-