Is Dp-gap-only superconductivity possible
in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys?
P. Jiji Thomas Joseph, Prabhakar P. Singh
*
Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India
Abstract
Using density-functional-based method, we study the k-resolved r- and p-band holes in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys. We
find that the calculated profiles of the loss of r- and p-band holes in these two systems as a function of impurity concentration are
in qualitative agreement with experiments, as expected. We also describe its implications vis-a-vis superconductivity in Mg1 xAlxB2
and Mg(B1 yCy)2. Based on our results, a Dp-gap-only superconductivity seems unlikely in Mg1 xAlxB2 alloys.
PACS: 71.20. b; 71.20.Be; 74.25.Jb; 71.15.Nc
Keywords: Superconductivity; Disorder; Local-density; Approximation
1. Introduction
The surprising discovery of superconductivity at 40 K in
a simple metal MgB2 [1] a few years ago continues to test
the details of the microscopic picture offered by Bardeen–
Cooper–Schriefer theory of superconductivity and its
extensions thereof. Generally, it is agreed that in MgB2:
(i) there are two superconducting gaps [2–8] Dr and Dp of
7 MeV and 2 MeV, respectively, (ii) the two gaps arise from
the holes in the B pr and pp bands [8–14], (iii) the holes in
the B pr and pp bands couple strongly to the in-plane E2g
phonon mode [14–17], and (iv) the coupling is particularly
strong along C–A direction and around M point in the hexagonal Brillouin zone [18].
To gain further insight into the role played by the holes
of the B pr and pp bands in deciding the superconducting
properties as well as the relative strengths of the two gaps
Dr and Dp in MgB2, recently, there has been a concerted
effort to examine the various superconducting properties
of doped MgB2. In particular, the Al doping at the Mg site
and the C doping at the B site in MgB2 [19–29] are expected
to fill up the holes in the B pr and pp bands which, in turn,
would affect the two gaps Dr and Dp. Understanding the
changes in the superconducting properties of Mg1 xAlxB2
as a function of Al concentration x and of Mg(B1 yCy)2
as a function of C concentration y would provide a more
detailed information about the electron–phonon and the
electron–electron interactions in these alloys.
Experimentally, in Mg1 xAlxB2 the superconducting
transition temperature Tc as well as the gap Dr are found
to decrease with increasing Al concentration [19,22,31,
30,21]. The change in the gap Dp is relatively small up to
x = 0.7 at which the superconductivity vanishes in
Mg1 xAlxB2. However, in Mg(B1 yCy)2 the Tc [34,32,27,
35,33,36,40,39,38,37] and the Dr decrease rapidly as a function of C concentration with superconductivity vanishing
at around y = 0.15. The change in Dp, once again, is found
to be minimal in Mg(B1 yCy)2 [42,28,40,41,38].
Most of the previous theoretical studies [43–52] of
Mg1 xAlxB2 and Mg(B1 yCy)2 used virtual-crystal approximation [44,48,49,51,52] or supercell approach [43,47], having limited predictive capability, to describe the chemical
44
disorder in the Mg sub-lattice and the B sub-lattice, respectively. An accurate and reliable description of chemical disorder, provided by coherent-potential approximation, has
been used previously in the study of Mg1 xAlxB2 and
Mg(B1 yCy)2 [45,46,50] but these studies provided either
k-integrated information or were inconclusive.
Since many of the superconducting properties of MgB2
and its alloys Mg1 xAlxB2 and Mg(B1 yCy)2 are crucially
dependent on their normal state electronic structure
[15,17,54,8,53,55], especially along C–A direction and M
point of the hexagonal Brillouin zone (BZ), we study the
changes in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys as a function of Al and C concentrations respectively, using kresolved Bloch spectral function A(k, E) [56,57] and B
and C px(y) and pz densities of states. It is expected that
with increasing Al and C concentrations the holes in the
respective alloys would get gradually filled up. However,
not surprisingly, the way the holes are filled up in
Mg1 xAlxB2 and Mg(B1 yCy)2 alloys make the two systems quite different from each other as detailed below.
In this work we show that in Mg1 xAlxB2: (i) the rband holes along C–A get annihilated by x = 0.8 with the
holes at C vanishing first at x = 0.5, and (ii) the p holes
at M vanish by x = 0.7. However, in Mg(B1 yCy)2: (i) the
r-band holes along C–A deplete rapidly and vanish beyond
y 0.25 with the holes at A vanishing first, and (ii) the pband holes at M as well as r-band holes at A reduce substantially beyond y 0.1. Before we discuss our results, we
briefly describe the details of our calculations.
2. Computational details
The normal metal electronic structure of the disordered
alloys Mg1 xAlxB2 and Mg(B1 yCy)2 are calculated using
the Korringa–Kohn–Rostoker (KKR) Green’s function
method formulated in the atomic sphere approximation
(ASA) [58,59], which has been corrected by the use of both
the muffin-tin correction for the Madelung energy, needed
for obtaining an accurate description of ground state properties in the ASA [60], and the multipole moment correction
to the Madelung potential and energy which significantly
improves the accuracy by taking into consideration the
non-spherical part of polarization effects [61]. Chemical
disorder was taken into account by means of coherentpotential approximation (CPA) [62]. The exchange and
correlation were included within the local-density approximation (LDA) using the Perdew–Wang parameterization
of the many-body calculations of Ceperley and Alder [63].
During the self-consistent procedure the reciprocal space
integrals were calculated by means of 2299 k-points in the
irreducible part of the hexagonal BZ, while the energy integrals were evaluated on a semicircular contour in the complex energy plane using 24 energy points distributed in such
a way that the sampling near the Fermi energy was
increased. When calculating the density of states the number of k-points were increased to 6075. The atomic sphere
radii of Mg and B were kept as 1.294 and 0.747, respec-
tively, of the Wigner–Seitz radius. The sphere radii of the
substituted Al and C were kept the same as that of Mg
and B, respectively. The overlap volume resulting from
the blow up of the muffin-tin spheres was approximately
10%, which is reasonable within the accuracy of the approximation [64]. The charge self-consistency iterations were
accelerated using the Broyden’s mixing scheme [65], the
CPA self-consistency was accelerated using the prescription
of Abrikosov et al. [66]. The calculations for Mg1 xAlxB2
for 0 6 x 6 1 and Mg(B1 yCy)2 for 0 6 y 6 0.3 were carried
out at experimentally obtained lattice parameters of these
alloys [67,68], assuming a rigid underlying lattice. The
effects of local lattice-relaxation as well as any possible
short-range ordering effects are not considered. Our results
for Mg1 xAlxB2 and Mg(B1 yCy)2 are analyzed in terms of
Bloch spectral density, A(k, E) [56] and the B and C p densities of states. A k-resolved understanding of how the addition of Al and C in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys
affect the hole states, which are responsible for superconductivity in these alloys, over the entire BZ is essential. Such
changes in the states are reliably described by the Bloch
spectral density.
3. Results and discussion
In general, electron doping of MgB2 by substituting Al
at the Mg sub-lattice or C at the B sub-lattice can have
three important effects: (i) the outward movement of the
Fermi energy due to addition of electrons, (ii) disorderinduced broadening of peaks in A(k, E), and (iii) redistribution of states due to hybridization. If the disorder and
hybridization-related effects are negligible then electron
doping can be well described by a rigid-band-like picture,
where EF is shifted up using MgB2 densities of states until
the added electrons are accommodated. In such a case, the
r- and p-band holes would gradually get filled up as a function of electron doping. Thus, in the rigid-band picture the
r- and p-band-hole-dependent superconducting properties
of MgB2 alloys would be similar as a function of electron
doping. However, if the disorder and the hybridization lead
to significant modifications of the r- and p-band holes then
the electronic properties of both the normal and the superconducting states of MgB2 alloys would depend on the
details of electron doping rather than just the electron
count. Experimentally, as outlined in earlier paragraphs,
the r- and p-band-hole-dependent properties of the two
alloys, Mg1 xAlxB2 and Mg(B1 yCy)2, are found to differ
significantly from each other as a function of Al and C concentrations, underscoring the importance of disorder and
hybridization effects in these alloys, in particular on rand p-band holes. Therefore, in the following, we primarily
focus on the changes in the states close to EF, which happen to be the pr states at C and A points and pp state at
M point of the hexagonal BZ.
The most important result of the present work for
Mg1 xAlxB2 alloys is shown in Fig. 1, where we show the
Bloch spectral function A(k, E) as a function of x for
45
100
Γ
80 Γ
Mg1-xAlxB2
40
50
0
Α
A(k,E)
A(k,E)
0
30
0
Α
30
0
Μ
Μ
15
20
0
Mg(B1-yCy)2
-0.05
0
0
-0.05
0.05
0
0 6 x 6 1 at C, A and M points of the hexagonal BZ in a
limited energy window (±0.1 Ry) around Fermi energy
EF. Along C–A, the doubly degenerate B pr states in
Mg1 xAlxB2 alloys move closer to EF with increasing Al
concentration from x = 0.1 to x = 0.70, indicating a
decrease in r-band holes and subsequently, weakening of
electron–phonon coupling along this direction. We also
find that for x 0.5, the peak in A(k, E) is at EF indicating
some topological changes in the Fermi surface of the alloy.
By x = 0.90, the B pr states at A are well inside EF, resulting in the annihilation of all the pr hole states along C–A.
Thus, we should not expect r-band-hole-based superconductivity beyond x = 0.80. The present result should be
compared with the supercell-based results of Ref. [43],
which predict annihilation of r-band holes by x = 0.6.
We also find that the Mg-derived antibonding s-state (C6)
moves closer to the pr state for intermediate concentrations but separates out for x = 1.0.
The B pp hole states at M in Mg1 xAlxB2 alloys get
gradually annihilated with increasing Al concentration
from x = 0.1 to x = 0.50. We find that the A(k, E) peak corresponding to pp states crosses EF at a lower x than the
pr states at A point. As the contribution to the electron–
phonon coupling responsible for p-band superconductivity
comes from M and other similar points in the BZ, a Dpgap-only superconductivity seems unlikely in Mg1 xAlxB2
alloys. However, it does not preclude a crossover from a
r to p dominated superconductivity [52].
In Fig. 2 we show the Bloch spectral function A(k, E) of
Mg(B1 yCy)2 alloys as a function of y for 0 6 y 6 0.3 at C,
A and M points around Fermi energy. We find that with
increasing C concentration from y = 0.05 to y = 0.2 the
pr states along C–A move towards EF as well as get redistributed on the energy scale due to disorder and hybridization. In fact, in the range y = 0.1–0.15, we find substantial
reduction in pr states around A point. Since the electron–
phonon coupling is stronger near A point, the loss of pr
Fig. 2. The Bloch spectral function A(k, E) of Mg(B1 yCy)2 alloys
calculated at C (top panel), A (middle panel) and M (bottom panel)
points of the hexagonal Brillouin zone using the KKR–ASA–CPA method
for y equal to 0 (solid line), 0.05 (dotted line), 0.1 (short-dashed line), 0.15
(dot–dashed line), 0.2 (long-dashed line), 0.25 (dot–dot–dashed line) and
0.3 (dot–dash–dashed line), respectively. The vertical line through the
energy zero represents the Fermi energy.
1.5
Mg(B1-yCy)2
Mg1-xAlxB2
B px
B px
C px
B pz
B pz
C pz
1
DOS (st/Ry)
Fig. 1. The Bloch spectral function A(k, E) of Mg1 xAlxB2 alloys
calculated at C (top panel), A (middle panel) and M (bottom panel)
points of the hexagonal Brillouin zone using the KKR–ASA–CPA method
for x equal to 0 (solid line), 0.1 (dotted line), 0.3 (short-dashed line), 0.5
(dot–dashed line), 0.7 (long-dashed line), 0.9 (dot–dot–dashed line) and 1
(dot–dash–dashed line), respectively. The vertical line through the energy
zero represents the Fermi energy.
0.05
E(Ry)
E(Ry)
0.5
0
1
0.5
0
-0.1
0
0.1-0.1
0
E (Ry)
0.1-0.1
0
0.1
Fig. 3. The atom (B and C) and orbital (px(y) and pz) projected density of
states (DOS) in Mg1 xAlxB2 (left panels) and Mg(B1 yCy)2 (middle and
right panels) alloys calculated using the KKR–ASA–CPA method for x(y)
equal to 0 (0) (solid line), 0.1 (0.05) (dotted line), 0.3 (0.1) (short-dashed
line), 0.5 (0.15) (dot–dashed line), 0.7 (0.2) (long-dashed line), 0.9 (0.25)
(dot–dot–dashed line) and 1 (0.3) (dot–dash–dashed line), respectively.
The vertical line through the energy zero represents the Fermi energy.
states should dramatically affect the r-band superconductivity. Similar to the reduction of pr states along C–A,
the pp hole states at M point in Mg(B1 yCy)2 alloys are
drastically reduced by y = 0.1. Here, it must be noted that
the reduction in r- and p-band holes in Mg1 xAlxB2 with
increasing Al concentration is mostly due to the upward
movement of the Fermi energy. In contrast, disorder and
hybridization play a dominant role in reducing the rand p-band holes in Mg(B1 yCy)2 alloys.
The changes in the B pr and pp hole states in
Mg1 xAlxB2 as a function of Al concentration and in
Mg(B1 yCy)2 as a function of C concentration can be seen
more clearly in their respective partial densities of states as
shown in Fig. 3. In Mg1 xAlxB2 alloys the B px(y) hole
states decrease steadily as a function of x, vanishing for
46
x 0.8 as shown in Fig. 3. Thus, as stated earlier in the
context of A(k, E), we should not expect r-band-hole-based
superconductivity beyond x = 0.80 Mg1 xAlxB2. However,
the decrease in B pz hole states with increasing x is not as
much. Even for x = 1.0, there are B pz hole states available
in Mg1 xAlxB2 as shown in Fig. 3 but it does not sustain
Dp-gap-only superconductivity.
For Mg(B1 yCy)2 alloys, our results show that px(y) and
pz densities of states of both B and C behave similarly with
increasing C concentration, as can be seen from Fig. 3. By
y 0.25, most of the px(y) hole states of B and C have
moved inside EF. As pointed out above, in Mg(B1 yCy)2
alloys with the addition of C those r-band holes are lost
first which couple to the phonons more strongly which
may lead to a loss of superconductivity in spite of having
some px(y) hole states of B and C. The changes in the pz
hole states are minimal in Mg(B1 yCy)2, with a slight
increase in the hole states for higher C concentrations.
We like to point out that the approximations used in the
present work, namely the use of spherically symmetric
potential to describe the effective one-electron interaction
and the overlapping atomic spheres to fill the space, are
expected to affect some numerical aspects of the present
work. The use of optimized lattice constants is expected
to similarly affect the present work, especially in
Mg(B1 yCy)2 alloys for y P 0.2 because for these concentrations we have used the lattice constants corresponding
to y = 0.15. However, our results for MgB2 and AlB2 are
consistent with the more accurate, full-potential results,
and since the coherent-potential approximation is known
to describe the effects of disorder reliably, we expect our
intermediate-concentration-results to be robust and qualitatively correct.
4. Conclusions
In conclusion, we have studied the k-resolved r- and pband holes in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys. The
calculated profiles of the loss of r- and p-band holes in
these two systems as a function of impurity concentration
are expected to be in qualitative agreement with the experiments. We have also described its implications vis-a-vis
superconductivity in Mg1 xAlxB2 and Mg(B1 yCy)2 alloys.
In particular, we have shown that a Dp-gap-only superconductivity was unlikely in Mg1 xAlxB2 alloys.
Acknowledgements
One of us (P.J.T.J.) thank Dr. Andrei Ruban for providing the KKR–ASA code. Discussions with Dr. Igor Mazin
on the electronic structure properties of MgB2 is gratefully
acknowledged.
References
[1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu,
Nature 410 (2001) 63.
[2] S. Souma, Y. Machida, T. Sato, T. Takahashi, H. Matsui, S.-C.
Wang, H. Ding, A. Kaminski, J.C. Campuzano, S. Sasaki, K.
Kadowaki, Nature 423 (2003) 65.
[3] M. Iavarone, G. Karapetrov, A.E. Koshelev, W.K. Kwok, G.W.
Crabtree, D.G. Hinks, W.N. Kang, Eun-Mi Choi, Hyun Jung Kim,
Hyeong-Jin Kim, S.I. Lee, Phys. Rev. Lett. 89 (2002) 187002.
[4] F. Bouquet, Y. Wang, I. Sheikin, T. Plackowski, A. Junod, S. Lee,
S. Tajima, Phys. Rev. Lett. 89 (2002) 257001.
[5] S. Tsuda, T. Yokoya, Y. Takano, H. Kito, A. Matsushita, F. Yin, J.
Itoh, H. Harima, S. Shin, Phys. Rev. Lett. 91 (2003) 127001.
[6] P. Szabo, P. Samuely, J. Kacmarck, T. Klein, J. Marcus, D. Fruchart,
S. Miraglia, C. Marcenat, A.G.M. Jansen, Phys. Rev. Lett. 87 (2001)
137005.
[7] H. Schmidt, J.F. Zasadzinski, K.E. Gray, D.G. Hinks, Phys. Rev.
Lett. 88 (2002) 127002.
[8] A.Y. Liu, I.I. Mazin, J. Kortus, Phys. Rev. Lett. 87 (2001) 087005.
[9] H. Schmidt, J.F. Zasadzinski, K.E. Gray, D.G. Hinks, Phys. Rev. B
63 (2001) 220504.
[10] G. Karapetrov, M. Iavarone, W.K. Kwok, G.W. Crabtree, D.G.
Hinks, Phys. Rev. Lett. 86 (2001) 4374.
[11] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Nature 418
(2002) 758.
[12] W.L. McMillan, Phys. Rev. 175 (1968) 537.
[13] P. Seneor, C.-T. Chen, N.-C. Yeh, R.P. Vasquez, L.D. Bell, C.U.
Jung, Min-Seok Park, Heon-Jung Kim, W.N. Kang, Sung-Ik Lee,
Phys. Rev. B 65 (2001) 012505.
[14] C. Joas, I. Eremin, D. Manske, K.H. Bennemann, Phys. Rev. B 65
(2002) 132518.
[15] J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Antropov, L.L.
Boyer, Phys. Rev. Lett. 86 (2001) 4656.
[16] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Phys. Rev.
B 66 (2002) 020513.
[17] K.-P. Bohnen, R. Heid, B. Renker, Phys. Rev. Lett. 86 (2001) 5771.
[18] P.P. Singh, Phys. Rev. B 67 (2003) 132511.
[19] J.S. Slusky, N. Rogado, K.A. Regan, M.A. Hayward, P. Khalifah,
T. He, K. Inumaru, S.M. Loureiro, M.K. Haas, H.W. Zandbergen,
R.J. Cava, Nature 410 (2001) 343.
[20] S. Agrestini, D. Di Castro, M. Sansoni, N.L. Saini, A. Saccone, S. De
Negri, M. Giovannini, M. Colapietro, A. Bianconi, J. Phys.:
Condens. Matter 13 (2001) 11689.
[21] J.Y. Xiang, D.N. Zheng, J.Q. Li, L. Li, L. Lang, H. Chen, C. Dong,
G.C. Che, Z.A. Ren, H.H. Qi, Y. Tian, Y.M. Ni, Z.X. Zhao, Phys.
Rev. B 65 (2002) 214536.
[22] J.Q. Li, L. Li, F.M. Liu, C. Dong, J.Y. Ziang, Z.X. Zhao, Phys. Rev.
B 65 (2002) 132505.
[23] P. Postorino, A. Congeduti, P. Dore, A. Nucara, A. Bianconi, D. Di
Castro, S. De Negri, A. Saccone, Phys. Rev. B 65 (2001) 020507.
[24] S. Margadonna, K. Prassides, I. Arvanitidis, M. Pissas, G. Papavassiliou, A.N. Fitch, Phys. Rev. B 66 (2002) 014518.
[25] G. Papavassiliou, M. Pissas, M. Karayanni, M. Fardis, S. Koutandos, K. Prassides, Phys. Rev. B 66 (2002) 140514.
[26] A. Bianconi, S. Agrestini, D. Di Castro, G. Campi, Z. Zangari, N.L.
Saini, A. Saccone, S. De Negri, M. Giovannini, G. Profeta, A.
Continenza, G. Satta, S. Massidda, A. Cassetta, A. Pifferi, M.
Colapietro, Phys. Rev. B 65 (2002) 174515.
[27] R.A. Ribeiro, S.L. Budko, C. Petrovic, P.C. Canfield, Physica C 385
(2003) 16.
[28] H. Schmidt, K.E. Gray, D.G. Hinks, J.F. Zasadzinski, M. Avdeev,
J.D. Jorgensen, J.C. Burlea, Phys. Rev. B 68 (2003) 060508.
[29] R.A. Ribeiro, S.L. Bud’ko, C. Petrovic, P.C. Canfield, Physica C 384
(2003) 227.
[30] J. Karpinski, N.D. Zhigadlo, G. Schuck, S.M. Kazakov, B. Batlogg,
K. Rogacki, R. Puzniak, J. Jun, E. Muller, P. Wagli, R. Gonnelli, D.
Daghero, G.A. Ummarino, V.A. Stepanov, Phys. Rev. B 71 (2005)
174506.
[31] M. Putti, C. Ferdeghini, M. Monni, I. Pallecchi, C. Tarantini, P.
Manfrinetti, A. Palenzona, D. Daghero, R.S. Gonnelli, V.A. Stepanov, Phys. Rev. B 71 (2005) 144505.
47
[32] M. Paranthaman, J.R. Thompson, D.K. Christen, Physica C 355
(2001) 1.
[33] W. Mickelson, J. Cumings, W.Q. Han, A. Zettl, Phys. Rev. B 65
(2002) 052505.
[34] S.M. Kazakov, R. Puzniak, K. Rogacki, A.V. Mironov, N.D.
Zhigadlo, J. Jun, Ch. Soltmann, B. Batlogg, J. Karpinski. condmat/0405060.
[35] T. Takenobu, T. Ito, D.H. Chi, K. Prassides, Y. Iwasa, Phys. Rev. B
64 (2001) 134513.
[36] T. Masui, S. Lee, S. Tajima, Phys. Rev. B 70 (2004) 024504.
[37] R.H.T. Wilke, S.L. Bud’ko, P.C. Canfield, D.K. Finnemore, R.J.
Suplinskas, S.T. Hannah, Phys. Rev. Lett. 92 (2004) 217003.
[38] G.A. Ummarino, D. Daghero, R.S. Gonnelli, A.H. Moudden, Phys.
Rev. B 71 (2005) 134511.
[39] A.V. Sologubenko, N.D. Zhigadlo, S.M. Kazakov, J. Karpinski,
H.R. Ott, Phys. Rev. B 71 (2005) 020501.
[40] Z. Holanova, P. Szabo, P. Samuely, R.H.T. Wilke, S.L. Bud’ko, P.C.
Canfield, Phys. Rev. B 70 (2004) 064520.
[41] S.M. Kazakov, R. Puzniak, K. Rogacki, A.V. Mironov, N.D.
Zhigadlo, J. Jun, Ch. Soltmann, B. Batlogg, J. Karpinski, Phys.
Rev. B 71 (2005) 024533.
[42] P. Samuely, Z. Holanova, P. Szabo, J. Kacmarck, R.A. Ribeiro, S.L.
Bud’ko, P.C. Canfield, Phys. Rev. B 68 (2003) 020505.
[43] S. Suzuki, S. Higai, K. Nakao, J. Phys. Soc. Japan 70 (2001) 1206.
[44] O. de la Pena, A. Aguayo, R. de Coss, Phys. Rev. B 66 (2002) 012511.
[45] P.P. Singh, Physica C 382 (2002) 381.
[46] P.P. Singh, Solid State Commun. 127 (2003) 271.
[47] A. Bussmann-Holder, A. Bianconi, Phys. Rev. B 67 (2003) 132509.
[48] G. Profeta et al., Phys. Rev. B 68 (2003) 144508.
[49] G.A. Ummarino et al., Physica C 407 (2004) 121.
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
D. Kasinathan, et al. cond-mat/0409563.
J. Kortus et al., Phys. Rev. Lett. 94 (2005) 027002.
L.D. Cooley et al., Phys. Rev. Lett. 95 (2005) 267002.
I.I. Mazin, O.K. Andersen, O. Jepsen, O.V. Dolgov, J. Kortus, A.A.
Golubov, A.B. Kuz’menko, D. van der Marel, Phys. Rev. Lett. 89
(2002) 107002.
T. Yildirim, O. Gulseren, J.W. Lynn, C.M. Brown, T.J. Udovic, Q.
Huang, N. Rogado, K.A. Regan, M.A. Hayward, J.S. Slusky, T. He,
M.K. Haas, P. Khalifah, K. Inumaru, R.J. Cava, Phys. Rev. Lett. 87
(2001) 037001.
P.P. Singh, Phys. Rev. Lett. 87 (2001) 087004.
J.S. Faulkner, G.M. Stocks, Phys. Rev. B 21 (1980) 3222.
J.S. Faulkner, Prog. Mater. Sci. 27 (1982) 1.
I. Turek, V. Drchal, J. Kudrnovsky, M. Sob, P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces,
Kluwer Academic Publishers, Boston, 1997.
P. Phariseau, B.L. Gyorffy, L. Scheire (Eds.), Electrons in Disordered
Metals and at Metallic Surfaces, Plenum Press, New York, 1979.
N.E. Christensen, S. Satpathy, Phys. Rev. Lett. 55 (1985) 600.
A.V. Ruban, H.L. Skriver, Comput. Mater. Sci. 15 (1999) 119.
P. Soven, Phys. Rev. 156 (1967) 809.
J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244.
Hans L. Skriver, The LMTO Method, Muffin Tin Orbitals and
Electronic Structure, Springer-Verlag, 1984.
D.D. Johnson, Phys. Rev. B 38 (1988) 12807.
I. Abrikosov et al., Phys. Rev. B 56 (1997) 9319.
A. Bharathi, S.J. Balaselvi, S. Kalavathi, G.L.N. Reddy, V.S. Sastry,
Y. Hariharan, T.S. Radhakrishnan, Physica C 370 (2002) 211.
M. Avdeev, J.D. Jorgensen, R.A. Ribeiro, S.L. Bud’ko, P.C. Canfield,
Physica C 387 (2003) 301.