Magnetic quantum oscillations in borocarbide superconductors
We report systematic de Haas–van Alphen (dHvA) investigations in the normal and superconducting state of RNi₂B₂C (R = Y and Lu). The observed rich frequency spectrum of the dHvA signals results from a rather complex electronic band structure with different open and closed Fermi-surface sheets. Fro...
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Bergk, B. Wosnitza, J. 2017-05-22T15:25:44Z 2017-05-22T15:25:44Z 2009 Magnetic quantum oscillations in borocarbide superconductors / B. Bergk, J. Wosnitza // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 872-878. — Бібліогр.: 45 назв. — англ. 0132-6414 PACS: 74.70.Dd, 71.18.+y, 74.25.Jb https://nasplib.isofts.kiev.ua/handle/123456789/117357 We report systematic de Haas–van Alphen (dHvA) investigations in the normal and superconducting state of RNi₂B₂C (R = Y and Lu). The observed rich frequency spectrum of the dHvA signals results from a rather complex electronic band structure with different open and closed Fermi-surface sheets. From our data in combination with full-potential local-orbital calculations we are able to extract the angular-resolved mass-enhancement factors, λ, for different bands. We find a strong anisotropy and band dependence of λ, clearly reflecting the multiband character of the superconductivity in RNi₂B₂C. We further were able to resolve dHvA oscillations deep into the superconducting state. The observed additional damping of the dHvA amplitudes is much less than expected from most theories. This hints at a reduced or even zero superconducting gap for the detected Fermi surface. We acknowledge P.C. Canfield and G. Behr for supplying the high-quality samples, H. Rosner, V. Petzold, and S.-L. Drechsler for their theoretical input, A. D. Bianchi, M. Bartkowiak, O. Ignatchik, I. Sheikin, and J. Perenboom for experimental support. We thank all of them for helpful discussions. The work was supported by the DFG through SFB 463. Part of this work has been supported by EuroMagNET under the EU contract RII3-CT-2004-506239 of FP6. The work at GHMFL was supported by the EC program Transnational Access — Specific Support Action (Contract No. RITA-CT-2003-505474) .Work at the Ames Laboratory was supported by the Department of Energy, Basic Energy Sciences under Contract No. DE-AC02-07CH11358. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Электронные свойства проводящих систем Magnetic quantum oscillations in borocarbide superconductors Article published earlier |
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Magnetic quantum oscillations in borocarbide superconductors |
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Magnetic quantum oscillations in borocarbide superconductors Bergk, B. Wosnitza, J. Электронные свойства проводящих систем |
| title_short |
Magnetic quantum oscillations in borocarbide superconductors |
| title_full |
Magnetic quantum oscillations in borocarbide superconductors |
| title_fullStr |
Magnetic quantum oscillations in borocarbide superconductors |
| title_full_unstemmed |
Magnetic quantum oscillations in borocarbide superconductors |
| title_sort |
magnetic quantum oscillations in borocarbide superconductors |
| author |
Bergk, B. Wosnitza, J. |
| author_facet |
Bergk, B. Wosnitza, J. |
| topic |
Электронные свойства проводящих систем |
| topic_facet |
Электронные свойства проводящих систем |
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2009 |
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English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We report systematic de Haas–van Alphen (dHvA) investigations in the normal and superconducting
state of RNi₂B₂C (R = Y and Lu). The observed rich frequency spectrum of the dHvA signals results from a
rather complex electronic band structure with different open and closed Fermi-surface sheets. From our data
in combination with full-potential local-orbital calculations we are able to extract the angular-resolved
mass-enhancement factors, λ, for different bands. We find a strong anisotropy and band dependence of λ,
clearly reflecting the multiband character of the superconductivity in RNi₂B₂C. We further were able to
resolve dHvA oscillations deep into the superconducting state. The observed additional damping of the
dHvA amplitudes is much less than expected from most theories. This hints at a reduced or even zero superconducting
gap for the detected Fermi surface.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117357 |
| citation_txt |
Magnetic quantum oscillations in borocarbide superconductors / B. Bergk, J. Wosnitza // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 872-878. — Бібліогр.: 45 назв. — англ. |
| work_keys_str_mv |
AT bergkb magneticquantumoscillationsinborocarbidesuperconductors AT wosnitzaj magneticquantumoscillationsinborocarbidesuperconductors |
| first_indexed |
2025-11-27T01:28:26Z |
| last_indexed |
2025-11-27T01:28:26Z |
| _version_ |
1850790972050374656 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9, p. 872–878
Magnetic quantum oscillations in borocarbide
superconductors
B. Bergk and J. Wosnitza
Hochfeld-Magnetlabor Dresden (HLD), Forschungszentrum Dresden-Rossendorf, Dresden D-01314, Germany
E-mail: j.wosnitza@fzd.de
Received April 7, 2009
We report systematic de Haas–van Alphen (dHvA) investigations in the normal and superconducting
state of RNi2B2C (R = Y and Lu). The observed rich frequency spectrum of the dHvA signals results from a
rather complex electronic band structure with different open and closed Fermi-surface sheets. From our data
in combination with full-potential local-orbital calculations we are able to extract the angular-resolved
mass-enhancement factors, �, for different bands. We find a strong anisotropy and band dependence of �,
clearly reflecting the multiband character of the superconductivity in RNi2B2C. We further were able to
resolve dHvA oscillations deep into the superconducting state. The observed additional damping of the
dHvA amplitudes is much less than expected from most theories. This hints at a reduced or even zero super-
conducting gap for the detected Fermi surface.
PACS: 74.70.Dd Ternary, quaternary, and multinary compounds;
71.18.+y Fermi surface: calculations and measurements; effective mass, g factor;
74.25.Jb Electronic structure.
Keywords: superconducting state, borocarbide, oscillations, Fermi surface.
Introduction
One of the most powerful methods to determine bulk
electronic band-structure properties of metals and super-
conductors is the measurement of magnetic quantum
oscillations. From that, the Fermi-surface topology,
band-resolved effective masses, scattering times, and
other band-structure parameters are obtainable. In combi-
nation with state-of-the-art band-structure calculations
the electronic properties including many-body effects can
be investigated in great detail.
The mostly used experimental technique is the detec-
tion of the de Haas–van Alphen (dHvA) effect, i.e., mea-
suring the field-dependent oscillations in the magneti-
zation. For that, a well developed and highly accurate
semi-classical theoretical description, the so-called Lif-
shitz–Kosevich (LK) theory, is available [1,2]. By use of
this technique important information on materials even
with rather involved band structures, such as in RNi2B2C,
can be gained. R stands for a rare-earth ion and here in es-
pecially for Y and Lu. For that, highly sophisticated ef-
forts both from the experimental as well as from the theo-
retical side are needed.
Even 15 year after the discovery of superconductivity
in the quaternary borocarbides [3,4] this material class re-
mains fascinating because of the possible coexistence of
magnetism and superconductivity for some rare-earth
ions. But even for the nonmagnetic superconductors
YNi2B2C and LuNi2B2C certain features of the supercon-
ducting state are still unclear and, therefore, a matter of
debate. Although many results point to a simple s-wave
phonon-mediated superconductivity (see e.g. [5,6]),
recent thermodynamic [7], tunneling [8], ultrasound [9],
and point-contact measurements [10] suggest other sce-
narios, such as anisotropic s- or d-wave, or (s g� )-wave
pairing. Furthermore, there is an increasing evidence for
multiband superconductivity in these materials with dif-
ferent and even anisotropic gaps for the various bands
[11–18]. The rather involved crystal structure of
YNi2B2C and LuNi2B2C (Fig. 1) leads to a complicated
band structure [17,19–22] which supports this notion. In
order to gain more insight in which bands are mainly in-
volved in the superconductivity, dHvA experiments in
combination with modern band-structure calculations are
a preferable choice. Here, we summarize our results re-
cently obtained for YNi2B2C and LuNi2B2C.
© B. Bergk and J. Wosnitza, 2009
Experimental
We investigated YNi2B2C single crystals prepared by
two different methods. Part of the crystals was grown by a
zone-melting technique as described in Ref. 23, improved
by optical heating. This resulted in large crystals of a few
cm 3 size from which smaller parts were cut for our inves-
tigations. The high crystal quality was evident from a
high residual resistivity ratio [24] as well as from x-ray
diffraction and metallographic methods, such as elec-
tron-beam microanalysis. The other part of the YNi2B2C
crystals was prepared by a flux-growth technique as de-
scribed in [25]. Here, we show only data for a crystal
grown by the latter method. Both types of crystals show
the same superconducting transition temperature at Tc �
= 15.1(1) K. The dHvA signals of both crystal types in the
normal state agrees nicely with each other verifying simi-
lar crystal qualities as reflected by comparable scattering
rates. Only the damping of the dHvA oscillations in the
superconducting state differs largely. Whereas the
zone-melted crystals revealed an abrupt vanishing of the
dHvA signal below the upper critical field, Bc2 [24], the
dHvA oscillations in the flux-grown crystals persisted
down to rather low fields, very much as observed for the
LuNi2B2C crystals discussed below. Although we can
only speculate on the origin of this different damping
behavior, it might be related to different flux-pinning
mechanisms and, therefore, different internal magne-
tic-field inhomogeneities (see the discussion below). The
LuNi2B2C single crystals were as well flux grown result-
ing in platelets of a few mm 3 size (onset Tc �16.5 K) [25].
The dHvA oscillations were measured either by use of
a capacitive cantilever torquemeter or using the modula-
tion-field technique. The latter method was mainly uti-
lized to study the dHvA signals in the superconducting
state. These measurements were performed at the HLD in
Dresden using a 3He cryostat equipped with a 15 T super-
conducting magnet. The torque measurements were also
done at the HLD up to 15 T and up to about 32 T at the
Grenoble High Magnetic Field Laboratory (GHMFL) and
at the High Field Magnetic Laboratory in Nijmegen. A di-
lution refrigerator as well as 3He cryostats have been used
to cover the temperature range from 20 mK up to about
10 K. All samples were placed on cantilever platforms
which could be rotated in situ.
Results and Discussion
A typical torque signal measured at T � 50 mK at the
GHMFL is shown in Fig. 2,a. In the superconducting
state, below Bc2 � 8.3 T a large hysteresis between up and
down sweep is apparent. Just before the normal state is
reached, a pronounced peak appears in the up-sweep
torque signal. This feature, known as the peak effect [26],
has been observed with different magnitudes for all an-
gles and for all investigated RNi2B2C crystals [24,27,28].
Although the details of this phenomenon are not fully un-
derstood, it has been observed for a number of type-II su-
perconductors and is attributed to distinct vortex-matter
phases with intrinsically different pinning strengths [26].
Magnetic quantum oscillations in borocarbide superconductors
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 873
[100]
Lu
Ni
B
C
[010]
[001]
Fig .1. Crystal structure of LuNi2B2C.
0
5
10
15
20
T
o
rq
u
e,
ar
b
.
u
n
it
s
T
o
rq
u
e,
ar
b
.
u
n
it
s
2
0
–2
12 15 18 21 24
0 5 10 15 20 25
B, T
B, T
0 1000 2000 3000 4000 5000
dHvA Frequency, T
0.05
0.10
0.15
0.20
A
,
ar
b
.
u
n
it
s
F
T
a
b
F�
2F� F�
F� 2F�2F�
LuNi B C2 2
� 100 = 63
T = 50 mK
Fig. 2. Field dependence of the torque signal of LuNi2B2C
with the magnetic field rotated 63° from the c axis towards
[100]. The arrows indicate the field-sweep directions. The in-
set shows the torque signal after background subtraction (a).
Fourier spectrum of the signal shown in the inset (b).
Above the upper critical field, Bc2, clear dHvA oscilla-
tions become visible, which can be better resolved in the
inset of Fig. 2,a, where the background torque signal has
been subtracted. The dominant oscillation frequency,
F� � 440 T (see the Fourier transformation in Fig. 2,b),
originates from a spheroidal Fermi surface. The
quasiparticles of the band forming this Fermi surface
have a reduced effective mass, meff
� , between 0.3 and 0.45
me depending on the field orientation, where me is the
free-electron mass [17]. dHvA oscillations originating
from this Fermi surface were clearly observed for all crys-
tallographic orientations. The found angular dependence
of F� reflects the spheroidal shape and agrees with band-
structure calculations (Figs. 3 and 4). For details on the
sophisticated fully relativistic full-potential local-orbital
band-structure calculations see Ref.17. The calculated F�
is, however, shifted upwards by about 400 T. Although
that appears to be a large deviation, this can be explained
by the very small contribution (less than 1 procent) of the
corresponding band to the total density of states at the
Fermi energy. By shifting the band filling by a tiny
amount as low as 0.012 e
, excellent agreement with the
dHvA data can be reached. The change of the dHvA fre-
quencies of the other Fermi surfaces would be negligible.
The small density of states together with a small mass
renormalization on the spheroidal Fermi surface (see be-
low) [17] reflects the minor relevance of this band for su-
perconductivity in LuNi2B2C.
Besides the frequency F� , the Fourier-transformed
dHvA signal shows a rich frequency spectrum (Fig. 2). For
the shown angle, two other independent oscillation fre-
quencies appear, F� and F� , which can be ascribed to a
cube-like and cushion-like Fermi surface originating from
one other band (Fig. 3) [17]. The other peaks in the Fourier
transformation are the second harmonics of the mentioned
dHvA frequencies. The effective masses of the � and �orbit
are larger (between about 1 and 2 free-electron masses)
than for the � orbit. Consequently, much higher magnetic
fields are needed to resolve these oscillations.
The dHvA-frequency spectra at other field orienta-
tions become even richer (Fig. 4). As a further example,
the background- subtracted torque signal for the magnetic
field aligned along the [100] direction together with the
Fourier-transformed spectrum is shown in Fig. 5,a for
LuNi2B2C. For this angle some smaller peaks in the Fou-
rier transformation appear next to the frequency F� .
Their origin is unclear, but they most likely result from
more evolved, e.g., corrugated, Fermi-surface topologies
for the different bands.
Besides the already introduced dHvA frequencies, F� ,
F� , and F� , a new orbit gives rise to the strong peak
at about 5260 T, labeled Fx . This orbit could not be
assigned directly to an extremal orbit of the calculated
Fermi surface, but is very likely caused by the compli-
cated branched Fermi surface (Fig. 3) [17]. Isshiki et al.
have observed this frequency, too [29]. They called the
frequency F� and assigned it to an extremal orbit of the
mentioned branched Fermi surface.
For direct comparison, the dHvA signal of YNi2B2C
measured at the same temperature and for the same field
orientation is shown in Fig. 5,b, together with its Fourier
transformation. A very similar dHvA signal and spectrum
are observed; the peaks in the Fourier spectrum at F� , F� ,
and Fx can be ascribed to equivalent orbits as for
874 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
B. Bergk and J. Wosnitza
3
2
1
0
–1
–2
–3
� n
(k
),
eV
Ã Õ Ð Ã3 Z à Ã1 Ã2
ZÃ3
Ð Ã
Ã1
Ã2X
Fig. 3. Calculated band structure (upper panel) and corre-
sponding Fermi surfaces of LuNi2B2C. The bands are labeled
according to the resulting Fermi surfaces: the band highlighted
by circles (�) results in the spheroidal, diamonds (�) give the
cube-like, rhombs (�) give rise to the cushion-like, and crosses
(�) result in the branched Fermi-surface. The cube-like and
cushion-like Fermi surfaces in the third panel are shifted by (�/a,
�/a,0) to present simply connected sheets. The high-symmetry
points are given in the inset of the upper panel. Reproduced
from Ref. 17.
LuNi2B2C. The first double peak at about 900 T in the
Fourier spectrum, labeled F /F� � , presumably comprises
besides the � orbit also the � orbit. This assignment is,
however, not unambiguously possible, since the � orbit in
YNi2B2C is somewhat larger and the � orbit seems to be
somewhat smaller compared to the orbits in LuNi2B2C.
Altogether, the extremal areas of the corresponding Fermi
surfaces differ only slightly for the two materials. For
YNi2B2C, the frequency Fx has also been labeled F� in
Ref. 24 and presumably originates in the � orbit as shown
in the calculated band structure for YNi2B2C (Fig. 2 in
Ref. 30). Our experimental results for YNi2B2C fit well
with data reported previously [31–36].
The apparent similarity of the Fermi-surface topolo-
gies of the two nonmagnetic borocarbides YNi2B2C and
LuNi2B2C is reflected among other properties in their
comparable Tc values of 15.1 and 16.5 K, respectively.
This hints at very similar interaction strengths for the
Cooper pairing in these superconductors. These coupling
strengths should be reflected in the mass enhancements of
the different bands. For LuNi2B2C, we have investigated
this mass enhancement in great detail by use of tempera-
ture-dependent dHvA studies in combination with
band-structure calculations applying a full-potential lo-
cal-orbital code [37] in its scalar-relativistic version
within the local density approximation [17,18].
From the temperature dependences of the dHvA oscil-
lation amplitudes, AFT , the effective masses, meff , includ-
ing all many-body renormalizations can be extracted for
each orbit separately [1,2]. This is shown in Fig. 6 for the
orbits F
1 and F
2. These orbits belong to the branched
Fermi surface (Fig. 3). F
1 could be ascribed to a wind-
mill-shaped orbit and F
2 to a lemon-like orbit [17]. The
temperature-dependent dHvA-amplitude reduction is de-
scribed by the LK damping factor R X/ XT � sinh( ), with
X m T/B� � eff and � �� �2 2k m /eB e � 14.69 T/K. The solid
lines in Fig. 6 are fits using this function. From this
two-parameter fit for each curve, we obtain similar effec-
tive masses of 3.5(2) and 3.6(3) me for both orbits of the
branched Fermi surface. Since in the calculated effective
masses, mcalc , no electron–phonon effects are included,
the angular- reso lved mass enhancement fac tor,
� �
m /meff calc 1, can be extracted directly for each as-
cribed orbit. For the main crystallographic directions, the
results are summarized in Table 1.
For the nonmagnetic borocarbides renormalizations of
nonphononic origin are expected to be weak. This is evi-
denced by point-contact spectroscopy measurements
[38,39] as discussed in more detail in Ref. 17. Conse-
Magnetic quantum oscillations in borocarbide superconductors
Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 875
10
4
10
3
10
2
[110] [100] [001]30 60 30 30 60 [110]
�001 �100 �110
d
H
v
A
fr
eq
u
en
cy
,
T
F�
F
2
F�
F�
F�
F�
F�
F�
F�
F
1
Fig. 4. Angular dependence of the dHvA frequencies in
LuNi2B2C. The solid lines are obtained by use of band-struc-
ture calculations [17]. The calculated dHvA frequency for the
spherical Fermi surface, F�, is somewhat larger than measured,
whereas for the other Fermi surfaces excellent agreement be-
tween theory and experiment is found. For the open circles no
corresponding Fermi surface could be assigned unambiguously.
0.5
1.0
1.5
2.0
T
o
rq
u
e,
ar
b
.
u
n
it
s
5
0
–5
20 22 24 26 28
0 5 10 15 20 25
0
2 4 6 8
dHvA frequency, 10 T
3
0.1
0.2
A
,
ar
b
.
u
n
it
s
F
T
a
b
F�
Fx
F�
F�
2F�2F�
LuNi B C2 2
B || [100]
T = 50 mK
2Fx
B, T
A
,
ar
b
.
u
n
it
s
F
T
dHvA frequency, 10 T
3
10
F /� F�
Fx
F�
T
o
rq
u
e,
ar
b
.
u
n
it
s
1
0
–1
20 24 28
YNi B C2 2
B || [100]
T = 50 mK
B, T
Fig. 5. The inset shows the field dependence of the torque sig-
nal of LuNi2B2C after background subtraction with the mag-
netic field aligned along [100]. The main panel shows the Fou-
rier spectrum of the signal (a). The same for YNi2B2C (b).
quently, the extracted � values are most likely directly re-
lated to the Cooper-pair coupling parameter. For all ob-
served bands, we find prominent anisotropies and largely
different coupling strengths. The weakest coupling is
found for the spherical Fermi surface. This and the men-
tioned low contribution of this band to the total density of
states proves the minor relevance of this band for super-
conductivity. Interestingly, the dHvA signal of the spheri-
cal Fermi surface is resolvable well below Bc2 deep into
the mixed state (see below). For the � dHvA frequency,
originating from the cushion-like Fermi surface, � is
about 1 in the (100) plane and decreases continuously to
about 0.23 towards [ ]110 . This Fermi surface is believed to
allow superconductivity together with commensurate
antiferromagnetism in the magnetic members of the
borocarbide family. The 5d states of the rare-earth atoms,
that mediate the magnetic interaction between the local-
ized 4 f magnetic moments, are not affecting this Fer-
mi-surface, so that superconductivity may survive in the
magnetic borocarbides [40].
For the dHvA frequency �, resulting from the cube-like
Fermi surface, the coupling strength is highly anisotropic
reaching � � 2.7 for B aligned along the [100] direction
down to � �1 for other directions. For both identified
parts of the branched Fermi surface, we determine inter-
mediate-strength coupling constants of 0.8 and 0.9. Our
data compare well with other results stating Fermi-sur-
face-averaged coupling constants between 0.5 and 1.2
[8,15,41]. Our dHvA results give strong support for mul-
tiband superconductivity in the borocarbides as has been
suggested in earlier work [11–16].
A direct way to study the field-dependent evolution of
the superconducting energy gap is the measurement of
dHvA oscillations below Bc2. By opening of a gap, an ad-
ditional damping of the dHvA oscillations is expected
[42]. This has been evidenced for a number of type II su-
perconductors [42,43]. For the nonmagnetic borocarbides
YNi2B2C and LuNi2B2C, however, highly controversial
results have been reported [24,28,29,31–36]. This has
been discussed as a consequence of the highly anisotropic
and band-dependent coupling strengths (see above) lead-
ing to anisotropic multiple gaps. Another explanation,
however, would be different internal field inhomoge-
neities due to flux-pinning effects. This would explain the
mentioned different additional damping for crystals pre-
pared via different routes as well as our recent findings
for LuNi2B2C discussed below.
In earlier work, we have measured the dHvA oscilla-
tions by use of the torque method [17,18,24,27,28]. This
technique is advantageous at higher magnetic field, but
looses sensitivity towards low fields, that are of interest
here. We, therefore, measured the dHvA signal of
LuNi2B2C by use of the modulation-field technique which
gains in sensitivity at lower magnetic fields [2]. By care-
fully using this method, we were able to observe the dHvA
signal of the spherical Fermi surface for all crystallo-
graphic directions deep into the superconducting state. For
some magnetic field orientations close to the [100] direc-
tion, we additionally could resolve the � dHvA frequency
of the cushion-like Fermi surface in the mixed state.
As an example, Fig. 7 shows the measured modula-
tion-field signal for up and down sweeps of the magnetic
field. The dHvA oscillations become clearly visible after
careful stepwise background subtractions (two upper pa-
nels of Fig. 7). The grey lines (shone in the upper panels)
876 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9
B. Bergk and J. Wosnitza
0 0.2 0.4 0.6 0.8
0.3
0.4
A
,
ar
b
.
u
n
it
s
F
T
0.3
LuNi B C2 2
0.5
0.4
0.6
0.7
0.5
0.6
A
,
ar
b
.
u
n
it
s
F
T
T, K
F
2
F
1
m = (3.6 0.3) m
2 e�
m = (3.5 0.2) m
1 e�
Fig. 6. Temperature dependence of the oscillation amplitudes
of the lemon and windmill extremal areas of the branched
Fermi surface with the respective fits by use of the Lif-
shitz–Kosevich theory.
Table 1. Band structure parameters, such as dHvA frequency F, ef-
fective mass meff , in units of the free-electron mass me, calculated ef-
fective mass without renormalization mcalc, and the mass-enhance-
ment factor, �, for the high-symmetry directions of LuNi2B2C
Direction Orbits F(T) meff(me) mcalc(me) �
[100] � 530 0.47 0.3 0.6
� 1075 0.8 0.4 1
� 1480 1.68 0.9 0.9
[110] � 520 0.44 0.26 0.7
� 1440 2 0.62 2.2
� 1820 1.9 1.55 0.23
[001] � 300 0.29 0.22 0.32
� 1230 2.3 0.62 2.7
� 5440 4.25 2.0 1.1
1 2440 3.5 2.0 0.8
2 12500 3.6 1.9 0.9
depict the dHvA signal without opening of a superconduct-
ing energy gap extrapolated from the normal state. In the
Dingle plot (lower panel of Fig. 7, right axis) this extrapo-
lated behavior is shown by the dashed line.
Just below Bc2, we observe clearly an additional
damping of the dHvA signal which is more or less re-
stricted to the peak-effect region. We have described that
field dependence of the dHvA amplitude by use of the
well-accepted theory [44] taking into account fluctua-
tions close to the superconducting transition [45]. With
Bc2 � 8.2 T, we obtain a zero-field energy gap of � �
= (3.0±0.5) meV. The relevance of this energy gap is,
however, highly questionable since the dHvA signals
clearly recovers towards lower magnetic fields below the
peak effect (Fig. 7). Such kind of behavior cannot be de-
scribed by use of the usual theories. These theoretical
treatments are anyway mostly only applicable close to
Bc2. For LuNi2B2C, we can resolve dHvA signals in some
cases down to 2.5 T, way beyond any expectation for a
gapped Fermi surface deep in the superconducting state.
It is, therefore, highly likely that the energy gap for the
spherical Fermi surface in LuNi2B2C is tiny or even not
existing. The additional damping observed in the region
of the prominent peak effect may be caused by large mag-
netic-field inhomogeneities. In this region a strongly
changing flux-line lattice is expected which, therefore,
should lead to an inhomogeneous internal magnetic field.
Towards lower fields, a more equally distributed flux-line
lattice is expected leading to a less inhomogeneous inter-
nal field. Towards very low fields the flux lines become
more and more dilute and lead to an increasing field mod-
ulation of the internal magnetic field. Whether that is the
reason for the observed small additional damping at low
fields is so far unclear. Consequently, our results point to
a very small or zero gap on one of the bands in LuNi2B2C.
Summary
We carefully measured the magnetic quantum oscilla-
tions of the nonmagnetic borocarbide superconductors
YNi2B2C and LuNi2B2C. In our comprehensive dHvA
study we could resolve the rather involved band structure
in combination with state-of-the-art band-structure calcu-
lations. By that, we obtained clear evidence for highly
anisotropic band-dependent superconducting coupling
strengths. For one band, our dHvA measurements in the
superconducting state point to a tiny or non-existing
energy gap.
Acknowledgments
We acknowledge P.C. Canfield and G. Behr for supply-
ing the high-quality samples, H. Rosner, V. Petzold, and
S.-L. Drechsler for their theoretical input, A. D. Bianchi, M.
Bartkowiak, O. Ignatchik, I. Sheikin, and J. Perenboom for
experimental support. We thank all of them for helpful dis-
cussions. The work was supported by the DFG through SFB
463. Part of this work has been supported by EuroMagNET
under the EU contract RII3-CT-2004-506239 of FP6. The
work at GHMFL was supported by the EC program Trans-
national Access — Specific Support Action (Contract No.
RITA-CT-2003-505474) .Work at the Ames Laboratory was
supported by the Department of Energy, Basic Energy Sci-
ences under Contract No. DE-AC02-07CH11358.
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�
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/R
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/B
)
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