Physica C 408–410 (2004) 104–106

www.elsevier.com/locate/physc

Electronic structure and anisotropic superconductivity

in diborides and borocarbides
a,*
S.-L. Drechsler , H. Rosner b, I. Opahle a, S.V. Shulga a, H. Eschrig a

a
Leibniz-Institut f. Festk€orper- u. Werkstoffforschung Dresden, P.O. Box 270116, D-01171 Dresden, Germany
b
Max-Planck-Institut f. Chemische Physik fester Stoffe, Dresden, Germany

Abstract
We compare calculated Fermi surface sheet (FSS) areas F and masses of MgB2 and ZrB2 with dHvA data. Devi-
ations in F in MgB2 can be removed by a small mutual shift of r- and p bands. The dHvA masses lead to orbit averaged
el–ph coupling constants kr
1:03 and kp
0:32, while for ZrB2 only  k < 0:1 holds. The anisotropy of the Fermi
velocities and the orbital character of various FSS of rare earth (R) transition metal (T) borocarbides RT2 B2 C, their
relationship to the Hc2 -anisotropy, and the coexistence of magnetism and superconductivity are discussed.
Ó 2004 Elsevier B.V. All rights reserved.

PACS: 74.20; 74.25)q; 74.70

Keywords: Electronic structure; Multiband superconductivity; Upper critical field anisotropy

Although MgB2 and borocarbides appear to be well systems with Y, Lu (sometimes partially) replaced by
described as Fermi liquids based on the band structure rare earths elements R has been ascribed to crystal field
described in the local density approximation (LDA), effects (CFE) [8]. Here, we address the questions: How is
only recently detailed comparisons with de Haas van cH 6 1=cv related to the vF -anisotropy (defined as
Alphen (dHvA) [1] and angle resolved photoemission pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cv ¼ 2vz = v2x þ v2y ) averaged on disjoint Fermi surface
spectroscopy (ARPES) data [2] became possible. In the
mean time exotic scenarios have been proposed [3,4]. sheets (FSS)? What is their specific role in the coexis-
Lacking/low-Tc superconductivity in other diborides tence of superconductivity and magnetism?
should be explained too [5–7]. With this aim we compare We have performed relativistic and scalar relativistic
first normale state properties of MgB2 with nonsuper- (pseudo-core for 4f electrons where necessary) band
conducting ZrB2 . As a basic property of a clean limit structure calculations for most diborides and borocar-
type-II superconductor the upper critical field Hc2 ðT Þ bides using the FPLO code [5,9–11]. To ensure high
provides insight into the relationship of electronic precision and consistency of the calculations, we have
structure, e.g. by the Fermi velocities vF , and super- applied for diborides also the FLAPW method as
conductivity. implemented in the WIEN97 code which has produced
A marked out-of-plane (OPA, cH ðT Þ ¼ Hc2 ab c
=Hc2 > 1) equivalent results. ARPES was the first check to probe
and no in-plane (IPA) Hc2 -anisotropies have been re- the LDA electronic structure of MgB2 [2]. The inspection
ported for MgB2 whereas Y(Lu)Ni2 B2 C exhibit weak of the data revealed some unexpected features which
(10%) OPA and IPA as well. A sizable reversed OPA could not be ascribed to bulk states. Applying the layered
(cH < 1 up to 0.5) at low T in related (nearly)magnetic KKR, we were able to resolve assignment problems in
terms of surface states [12]. Let us turn to the FSS areas F
and electronic masses of MgB2 (see Table 1). Basically,
we find good agreement for the F similarly as Ref. [13].
*
Corresponding author. Tel.: +49-351-4659-384; fax: +49- However, at variance with Ref. [13], the remaining
351-4659-490. deviations between our LDA and dHvA F can be
E-mail address: drechsler@ifw-dresden.de (S.-L. Drechsler). removed by a slight downshift d  100 meV of the

0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.physc.2004.02.053
 S.-L. Drechsler et al. / Physica C 408–410 (2004) 104–106 105

Table 1
LDA parameters of MgB2 compared to dHvA data1 , (F in kT)
Orbit Fcalc Fexp mb jm j k ¼ jm j=jmb j 1
rSC 0.78 (0.79) 0.551 )0.25 0.548 1.19
rLC 1.65 (1.67) 1.18 )0.57 1.2 1.11
pC 34.5 1.87
rSA 1.83 (1.81) 1.534 )0.31 0.61 0.97
rLA 3.45 (3.46) 2.971 )0.64 1.18 0.84
pA 30.6 )0.93
pM 0.45 0.576 )0.25 0.31 0.26
pL 3.03 2.705 0.32 0.439 0.37
The FPLO values, using 16221 k points in the irreducible BZ; values in parentheses––FLAPW using the GGA exchange-correlation;
band masses in me . Orbit notation see Ref. [11].

r-bands relative to the p-bands. Its microscopic reason since the magnetism is transferred from the R 4f states to
(beyond the LDA) is still unclear. Several scenarios such the conducting electrons via the local polarization of the
as weak polaronic corrections generic for multiband R 5d states. Thus the pillow FSS derived electrons are of
systems with markedly different el–ph interaction in main interest for the coexistence of magnetism and
various bands and high-frequency phonons xph involved superconductivity. Naturally, the significant opposite
[5]: d / ðkr kp Þhxph or electron–electron self-energy vF -anisotropy cv  2–2.4 is comparable with the re-
effects will be considered elsewhere. Here, we note only versed value cH
0.5 observed for Y0:8 Tb0:2 Ni2 B2 C and
that the search of B isotope effects for the F might be RNi2 B2 C, where R ¼ Ho, Dy, Er. Further improve-
helpful to distinguish between them. Turning to ZrB2 , we ments should result taking into account the ignored
note that the agreement between dHvA and LDA is even CFE [8] which are relevant for R ¼ Tm, Gd with an
better than in MgB2 . The value of the orbit averaged el– enhanced normal cH -value.
ph coupling constant k 6 0:1 is in accord with the FS To summarize, the nice description of the LDA
averaged k from specific heat [6], point-contact mea- FSCS of MgB2 and ZrB2 as well as the ARPES data for
surements [7], and the lack of superconductivity. MgB2 emphasizes our quite reasonable understanding of
All considered RTBC exhibit a complex FS with up diborides. Most of them differ from MgB2 mainly by
to five sheets. The largest FSS shows small nested parts, lacking r-derived holes with strong el–ph interaction.
most pronounced for RTBC with T ¼ Ni with a partial Thus, our study yields strong support for multiband
weight of about 5% of the total density of states (DOS) Eliashberg models [16] (possibly with small many-body
at EF . These slow electrons with strong electron–phonon corrections). Hc2 is governed mainly by the Fermi
interaction prove to cause the Hc2 -pecularities in velocity anisotropy. Less space is left for exotic scenarios
Y(Lu)Ni2 B2 C. To a first approximation their main [3].
contribution to Hc2 ð0Þ and their interplay with FSS’s of
weakly interacting fast electrons in determining the up-
ward curvature near Tc have been explained using a two- Acknowledgements
band model [14]. The vF -anisotropy of the nested parts
on the large FSS is huge 1=cv
10. It dominates the The Deutsche Forschungsgemeinschaft, the Emmy-
value of cH for nonmagnetic RTBC. For isolated nesting Noether-Programm, and the Sonderforschungsbereich
electrons we would arrive at a huge cH like in MgB2 . But 463 are gratefully acknowledged for financial support.
the net effect is reduced by the admixture of opposite We are indebted to A.I. Morosov for drawing our
contributions from other FSS with cv > 1. The orbital attention to the potential importance of self-energy ef-
analysis of states near EF shows that these nested parts fects for shifts of the chemical potential (see also Refs.
of the large FSS contain marked admixtures of R 5d [4,5]).
states whereas the central, second-largest, FSS inside the
lemon-like features is formed almost only by Ni 3dx2 y 2
and 3dxy derived states. This FSS (called pillow-like in
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