PHYSICAL REVIEW B 67, 104507 共2003兲
Raman- and infrared-active phonons in superconducting and nonsuperconducting rare-earth
transition-metal borocarbides from full-potential calculations
P. Ravindran,* A. Kjekshus, and H. Fjellvåg
Department of Chemistry, University of Oslo, Box 1033, Blindern, N-0315, Oslo, Norway
P. Puschnig and C. Ambrosch-Draxl
Institut für Theoretische Physik, University Graz, Universitätsplatz 5, A-8010 Graz, Austria
L. Nordström and B. Johansson
Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, 75121, Uppsala, Sweden
共Received 24 July 2002; published 14 March 2003兲
Ab initio frozen-phonon calculations were performed for superconducting YNi2 B2 C and LuNi2 B2 C and
nonsuperconducting LaNi2 B2 C and YCo2 B2 C to establish the phonon frequencies for all Raman- and 共IR-兲
infrared-active optical q⫽0 modes using the generalized-gradient-corrected full-potential linear augmented
plane-wave method. From a series of atomic force calculations the shape of the phonon potential is established.
Our calculated Raman-active phonon frequencies are found to be in very good agreement with the available
Raman-scattering measurements. In the case of IR-active phonons, to our best knowledge, there are no experimental frequencies available and our theoretical study is the first report for these series. The Raman-scattering
intensities for all Raman-active modes of YNi2 B2 C are determined allowing a detailed comparison between
theoretical and experimental Raman spectra. The changes in the electronic structure introduced by the phonon
modes are also analyzed. Although the calculated electronic structure of these materials has three-dimensional
character we found a large anisotropy in the optical dielectric function due to the layered nature of the crystal
structure.
DOI: 10.1103/PhysRevB.67.104507
PACS number共s兲: 74.70.Ad, 74.25.Gz, 74.25.Jb
I. INTRODUCTION
The discovery1 of superconductivity in the Y-Ni-B-C system and its coexistence with magnetism in quaternary
transition-metal 共T兲 borocarbides with the formula RNi2 B2 C
(R⫽Y, Ho, Er, Tm or Lu兲 共Ref. 2兲 has stimulated considerable research activity. The layered R-T boride carbides
RT 2 B2 C show a rich variety of phenomena: superconductivity with elevated T c ; 1–3 exotic magnetic behavior such as
helical magnetism;4 heavy Fermion behavior;5,6 cascades of
field-induced magnetic phase transitions;7 mixed-valence
states;8,9 a striking manifestation of the interplay between
magnetism and superconductivity, such as reentrance and
coexistence;2– 4,7,10,11 anisotropic upper critical fields; angular
dependence of in-plane magnetization;12 hexagonal-tosquare vortex lattice transition;13–15 and phonon softening at
a finite wave vector where strong Fermi-surface nesting is
suggested.16 –20
The superconducting mechanism and the pairing symmetry of RT 2 B2 C have been extensively studied over the past
few years. There are many experimental observations of
strong electron-phonon interaction, while considerable doubt
has been expressed regarding the adequacy of a solely conventional electron-phonon 共EP兲 mechanism to account for
such high T c . The presence of the light elements boron and
carbon, which is expected to result in high phonon frequencies, opened the door for speculation that some exotic pairing mechanism is responsible for the relatively high transition temperatures. For example, recent measurements of the
microwave surface impedance in the vortex state suggest that
0163-1829/2003/67共10兲/104507共11兲/$20.00
low-energy quasiparticles may have d-wave dispersion.21
The residual absorption in the superconducting gap 共SG兲
seen by Raman spectroscopy22 and the upward curvature of
the critical field vs temperature23,24 are difficult to explain
within BCS theory. The absence of a Hebel-Slichter peak,
the temperature to the third power dependence of the nuclear
relaxation rate,25 and the direct measurement of the SG by
photoemission spectroscopy26 suggest the existence of nodes
in the SG.
However, band-structure calculations,27–30 the T c vs ␥
relation,31 and the variation in superconducting transition
temperature with nonmagnetic impurities32,33 suggest strong
EP interaction as a possible origin for the superconductivity.
Also, the good agreement between transport and critical-field
properties based on Eliashberg theory24,34 as well as s-wavelike tunneling spectra35 support the above viewpoint. Furthermore, the observation36 of the de Haas-van Alphen effect
indicates that the superconducting state evolves from the
Fermi-liquid ground state. While it is generally believed that
EP interaction is the underlying mechanism for superconductivity in these systems, the responsible phonons are claimed
to be either of a high-frequency (B-A 1g near 830 cm⫺1 )
共Ref. 37兲 or low-energy soft-mode nature.18,27,38 Hence, it is
important to fully characterize the lattice-dynamical properties of superconducting as well as nonsuperconducting borocarbides for understanding the superconducting mechanism.
Among the RNi2 B2 C superconductors the nonmagnetic
compounds with R⫽Lu and Y exhibit2,3 the highest transition temperatures, T c ⫽16.5 and 15.5 K, respectively. In addition, electronic band-structure calculations27,29,39 suggest
that these materials are conventional superconductors with a
67 104507-1
©2003 The American Physical Society
PHYSICAL REVIEW B 67, 104507 共2003兲
P. RAVINDRAN et al.
relatively high electronic density of states 共DOS兲 at the
Fermi level (E F ) and that there is a complex set of bands
crossing at E F which are strongly coupled to the phonons
and may be responsible for the superconducting properties.
No detectable superconductivity has been observed for isoelectronic and isostructural LaNi2 B2 C. The latter finding is
believed to be due to the larger size of the La ion which
decreases the DOS and the electron-phonon coupling at
E F . 37,39 The ac-susceptibility data show40 that YCo2 B2 C is a
Pauli paramagnetic metal and no superconductivity is observed down to low temperatures. From studies of the electronic structure and lattice-dynamical properties of these materials one can gain knowledge about the mechanism for the
superconductivity in the RT 2 B2 C phases. Usually the infrared 共IR兲-active phonon frequencies are measured by IR ellipsometry. Windt et al.41 studied the IR and optical properties
of LuNi2 B2 C and suggested weak to moderately strong coupling in this material. However, the IR-active phonon frequencies for these materials have not yet been determined
experimentally and hence a theoretical study is highly relevant.
Raman scattering provides valuable information about the
phonons, and in particular, highlight the interaction between
the motion of the ions and the electronic subsystem. It plays
an especially important role in the study of unconventional
superconductors owing to the strong dependence of the Raman response on the symmetry of superconducting order parameters. In recent years several experimental and theoretical
studies have been made on Raman spectra for superconducting RNi2 B2 C. From Raman-scattering measurements on
polycrystalline YNi2 B2 C Hadjiev et al.42 assigned all four
Raman-active modes (B-A 1g , Ni-B 1g , Ni-E g , and B-E g ).
On the other hand, Park et al.43 did not observe the Ni-E g
and B-E g modes in RNi2 B2 C (R⫽Lu, Ho or Y兲 by measurements on single crystals. So, calculations on Raman-active
phonons are needed to settle this issue. Dervenagas et al.16
measured the low-lying phonon-dispersion curves of
LuNi2 B2 C along the ( ␰ 0 0兲 and 共0 0 ␰ ) symmetry directions. They found that the frequencies of these modes near
共0.55 0 0兲 decrease with decreasing temperature, and, as a
result, the corresponding branches exhibit strong dips in the
vicinity of this point at low temperatures. The softening of
these modes is so significant that it was observed in phonon
DOS measurements by Gompf et al.44 as well as in pointcontact spectra by Yanson et al.38 A similar study by Kawano
et al.17 on the low-lying excitations in the Y compound as a
function of temperature and magnetic field uncovered dramatic changes in the spectrum associated with the onset of
superconductivity that includes a sharp feature at approximately 4 meV. Temperature-dependent single-crystal elasticconstant measurements on YNi2 B2 C also showed a softening
around T c . 45 The lattice-dynamical properties of RT 2 B2 C
are accordingly of considerable interest.
For the theoretical investigation of the Raman spectra a
recently developed method46 can be used which allows to
take into account the anharmonicity of the lattice-dynamical
potential. Thereby all the required quantities can be obtained
by first-principles band-structure methods within the frozenphonon approach, where the total energy and dielectric ten-
TABLE I. Experimental 共Refs. 33 and 48兲 lattice parameters
used for the calculations and optimized z B parameters for RT 2 B2 C
compounds.
zB
Compound
a(Å)
c(Å)
c/a
Theory
Experiment
YCo2 B2 C
YNi2 B2 C
LaNi2 B2 C
LuNi2 B2 C
3.5065
3.5330
3.7941
3.4639
10.6130
10.5660
9.8224
10.6313
3.0266
2.9906
2.5888
3.0619
0.3571
0.3576
0.3477
0.3600
0.3588
0.3492
0.3618
sor as a function of the atomic coordinates are the main
ingredients.
The rest of the paper is organized as follows. Structural
aspects and the computational details are given in Sec. II.
The results are presented and discussed in Sec. III. Finally
we summarize the important findings of the present study in
Sec. IV.
II. STRUCTURAL ASPECTS AND COMPUTATIONAL
DETAILS
A. Structural information
47
Siegrist et al. found that the crystal structure of
RNi2 B2 C represents a filled-up variant of the well-known
tetragonal ThCr2 Si2 -type structure 共space group I4/mmm),
where a carbon atom occupies the vacant 2b position in the
R plane. It is highly anisotropic (c/a⬇3) with alternate
stacking of NaCl-type (RC) and inverse PbO-type (Ni2 B2 )
layers. Although the structure of RNi2 B2 C is layered, like the
high-T c copper oxides, several publications provide strong
evidence that these compounds can be considered classical
quasi-three-dimensional superconductors which owe their
relatively high superconducting temperatures to strong coupling between phonons and electrons. The experimental lattice parameters used for the phonon calculations along with
the optimized boron positional parameter (z B ) are given in
Table I.
B. Computational details for the full-potential linear
augmented plane-wave FLAPW calculations
For the frozen-phonon calculations we employed the fullpotential linear augmented plane-wave method as implemented in the WIEN97 code49 in a scalar-relativistic version
without spin-orbit coupling. The radii of the muffin-tin
spheres are chosen such that they stay fixed and remain nonoverlapping under distortions. In the calculations we have
used atomic sphere radii of 2.3 a.u. for Y, La, and Lu; 2.1 a.u.
for Ni and Co; and 1.35 a.u. for B and C. The charge density
and the potentials are expanded into lattice harmonics up to
l⫽6 inside the spheres and into Fourier series in the interstitial regions. The initial basis set included 5s, 5p, and 4d
valence and 4s and 4p semicore functions for Y; 6s, 6p,
and 5d valence and 5s and 5p semicore functions for La;
6s, 6 p, 5d, and 4 f valence and 5s and 5p semicore functions for Lu; 4s, 4p, and 3d valence and 3p semicore func-
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PHYSICAL REVIEW B 67, 104507 共2003兲
RAMAN- AND INFRARED-ACTIVE PHONONS IN . . .
tions for Ni and Co; and 2s, 2 p, and 3d valence functions
for B and C. The set of basis functions was supplemented
with local orbitals for additional flexibility in representing
the semicore states and for relaxing the linearization errors in
general. The cutoff angular momentum was l⫽10 for the
partial waves used inside the atomic spheres and 4 for the
partial waves in the computation of non-muffin-tin matrix
elements. The number of augmented plane waves included
was about 388 per atom. The effects of exchange and correlation are treated within the generalized-gradient-corrected
local-density approximation 共LDA兲 using the parametrization scheme of Perdew et al.50 To ensure convergence for the
Brillouin-zone 共BZ兲 integration, we have used 99 k points in
the irreducible wedge of the first BZ of the body-centered
tetragonal lattice for the A 1g mode. The other phonon displacements lower the symmetry of the lattice. For the same
density of k points as for the A 1g mode, 282 k points are
involved in the calculation of E g -mode frequencies. Selfconsistency was achieved by demanding the convergence of
the total energy to be smaller than 10⫺5 Ry/cell. This corresponds to a convergence of the charge below 10⫺4 electrons/
atom. The phonon frequencies are derived from the evaluation of the total energy as a function of the atomic
displacements, whereas all the other parameters related to the
calculation have been kept fixed.
C. Calculation of phonon frequencies
Phonon frequencies are studied at the ⌫ point in the BZ
within the harmonic approximation. In the treatment of lattice dynamics, the frequencies at the ⌫ point are calculated
from the dynamical matrix obtained using forces corresponding to different sets of linearly independent displacements of
the atoms in the unit cell. The frozen-phonon calculations
described below require knowledge of the phonondisplacement patterns. We thus start with the symmetry
analysis of the vibrational spectra. At the zone center (⌫),
the acoustic modes with symmetry ⌫ ac ⫽A 2u ⫹E u have zero
frequencies. The irreducible representations of the optical
modes are
Raman
Raman
IR
⫹B 1g
⫹2E Raman
⫹3A 2u
⫹3E IR
⌫ op ⫽A 1g
g
u .
共1兲
Here, the subscripts g and u represent symmetric and antisymmetric modes with respect to the center of inversion.
Since the unit cell has a center of inversion, there are only
four Raman-active modes with a nonzero Raman-tensor element. The A 1g , B 1g , and E 2u modes have inversion symmetry and are thus Raman active. A schematic representation of
the atomic displacements in the Raman-active E g , B 1g , and
A 1g modes is given in Fig. 1. The B-A 1g mode, where the
light boron atoms move along c while the centered Ni atom
and the adjacent C atoms are at rest, is depicted in Fig. 1共c兲.
In the Ni-B 1g mode, the B and C atoms are at rest and the Ni
atoms move along c as shown in Fig. 1共b兲. The E g mode
关Fig. 1共a兲兴 involves displacements of both B and Ni atoms
along the a and b directions. Although all these modes
change the Ni-B and B-C bond lengths they differ significantly in frequency because Ni is much heavier than B; the
mode involving primarily B displacements has higher fre-
FIG. 1. Schematic representation of Raman-active phonon
modes in RNi2 B2 C. Legends to the atom symbols are given on the
left illustration.
quencies than the other Raman-active modes. There are three
E 2u and A 2u modes, which have no inversion symmetry and
hence are IR active. The three A u modes stem essentially
from the vibration of the rigid Ni-B-C cluster along c entirely against the heavy R atom. The three E 2u excitations
come from the vibration of the Ni-B-C cluster within the ab
plane towards R.
The phonon frequencies are derived from ‘‘frozenphonon’’ calculations, i.e., distortions 共u兲 consistent with the
symmetry of the mode in question are introduced and the
corresponding total energy is calculated. If we introduce the
off-equilibrium displacements u s of the atoms under consideration, the Taylor expansion of the total energy around the
equilibrium positions in terms of the displacements can be
written as
E 共 u s 兲 ⫽E (0) ⫹
1
2
1
E (3)
兺i j E (2)
i j u iu j⫹
i jk u i u j u k ⫹ . . . .
6 兺
i jk
共2兲
Here, the summations run from i⫽1 to 3n, where n denotes
the number of atoms in the unit cell. Note that we have
assumed q⫽0 phonons and are therefore able to drop the
unit-cell index for the displacements and expansion coefficients. By making use of the undistorted crystal symmetry,
the above expansion for the 3n degrees of freedom breaks
down into several separate equations for each irreducible
representation. The total energies were calculated for five
different distortions for the A 1g mode, five for B 1g , nine for
E g , and twelve for A 2u and E u modes, the maximum amplitude of displacement being approximately 0.04 Å.
In the special case where the block size of the irreducible
representation is 1, the calculated total energy as a function
of u can be fitted with a simple polynomial
E 共 u 兲 ⫽E (0) ⫹a 2
冉冊 冉冊
u
L
2
⫹a 3
u
L
3
⫹ ...,
共3兲
where L is the lattice parameter which is a for E g and E u
modes and c for A 1g , B 1g , and A 2u modes. As an example
we show in Fig. 2 E(u) for the A 1g mode. In the present
computations, the second derivatives required for the forceconstant matrix elements were obtained by calculating the
forces exerted on all atoms when one or two of the atoms are
104507-3
PHYSICAL REVIEW B 67, 104507 共2003兲
P. RAVINDRAN et al.
dynamical matrix D i j (q) for q⫽0 phonons and the energyexpansion coefficients of Eq. 共2兲 is given by
D i j 共 q⫽0 兲 ⫽
E (2)
ij
共4兲
,
冑M i M j
where M i denotes the atomic mass associated with the atom
with index i. The diagonalization of the dynamical matrix
yields the eigenvectors and the phonon frequencies. In order
to increase the accuracy and reduce the number of necessary
total-energy and force calculations the coefficients E (2)
i j were
determined from a fit of the calculated atomic forces rather
than from a fit to the total energies.
D. Calculation of Raman-active phonon intensities
FIG. 2. The variation of total energy and force with displacement of B in the A 1g mode for YNi2 B2 C.
displaced in the a, b or c direction. Both positive and negative displacements were considered to take into account possible anharmonic effects. For all the frozen-phonon calculations we have used the experimental structural 共equilibrium兲
parameters given in Table I except for z B which was taken
from our force minimization. The A 1g -mode frequency can
be obtained by varying z B . The thus optimized B positions
are compared with experimental neutron-diffraction data in
Table I. These optimized z B values are, in turn, used for the
calculations of the B 1g -, E g -, E u -, and A 2u -mode frequencies.
The coefficient a 2 in the second-order term of Eq. 共3兲 is
the harmonic contribution to the total energy, naturally referred to as a force constant. Knowing this, we then obtain
the phonon frequency ␻ ph as
␻ ph ⫽ 共 2 ␲ L 兲
⫺1
冋 册
a2
2␮
The key feature of the present study is the ability to compute Raman intensities for each of the vibrational modes
directly within the generalized-gradient approximation
共GGA兲. The Raman-scattering activity associated with a
given vibrational mode is related to the change in the electrical polarizability of the material due to the normal-mode
displacements of the involved atoms.52 A formalism53 has
been developed recently to derive the Raman spectrum from
parameters which can be directly obtained from frozenphonon ab initio band-structure calculations 共Raman intensity expressed by the fluctuations in the dielectric functions兲.
The components of the Raman tensor ␹ for the RT 2 B2 C
compounds are given by
冉
冉
冉
0
0
0
xx
0
0
0
zz
xx
0
0
0
⫺xx
0
0
␹ 共 A 1g 兲 ⫽
␹ 共 B 1g 兲 ⫽
␹共 Eg兲⫽
1/2
,
where ␮ is the mass of the atom which is involved in a given
phonon mode. From the second derivative of the total energy
or the first derivative of the force with respect to the displacement we have calculated the A 1g -phonon frequencies. A
similar procedure can be adopted for the Ni- and Co-B 1g
modes since they also involve only one degree of freedom as
can be seen from the irreducible representation of the optical
modes given in Eq. 共1兲. Since the frequencies of the A 1g and
B 1g modes are easy to derive from this kind of calculation,
we also determined these frequencies using the full-potential
linear muffin-tin orbital51 共FLMTO兲 method. The computation of the two Raman-active E g modes and the IR-active
A 2u and E u modes, however, involves the diagonalization of
dynamical matrices with sizes 2⫻2 and 3⫻3, respectively,
as can be seen from Eq. 共1兲. The connection between the
冊
冊
冊
xx
,
0
,
zz
0
0
xz
0
0
0
zx
0
0
.
We have determined the band structure and the dielectric
function for different displacements of the atoms employing
the FLAPW method. From these data we have evaluated the
derivatives of the dielectric function with respect to the displacements. These derivatives are used to determine the corresponding Raman spectra using the following formalism.
The Raman-scattering intensity I( ␻ ) of a given mode is related to the derivative of the dielectric function ( ⑀ ) with
respect to the normal coordinate q i of the vibration46
104507-4
I共 ␻ 兲⫽
兺i exp
冉 冊兺
⫺ប ␻ i
k BT
兺i
f
兩 M i f 兩2L共 ␻ L , ␻ i , ␻ f , ␥ 兲
exp
冉 冊
⫺ប ␻ i
k BT
. 共5兲
PHYSICAL REVIEW B 67, 104507 共2003兲
RAMAN- AND INFRARED-ACTIVE PHONONS IN . . .
TABLE II. Raman-active phonon frequencies 共in cm⫺1 ) for RT 2 B2 C compounds obtained from FLMTO
and FLAPW methods in comparison with available experimental data.
Mode
YCo2 B2 C
YNi2 B2 C
LaNi2 B2 C
LuNi2 B2 C
766
799
821
830,893a
813,b 832,c 847,d 823e
825f
184
200,199a
199,b 198,c193e
201f
271
287,b 282,c 273f
447
460,b 470,c 439f
826
890
833b
818f
185
189
181b
180f
207
208,b 203f
449
441,b 419f
869
875
863e
B-A 1g 共FLAPW兲
B-A 1g 共FLMTO兲
B-A 1g 共experiment兲
B-A 1g 共theory兲
Ni- or Co-B 1g 共FLAPW兲
Ni- or Co-B 1g 共FLMTO兲
Ni-B 1g 共experiment兲
Ni-B 1g 共theory兲
Ni- or Co-E g 共FLAPW兲
Ni-E g 共experiment兲
B-E g 共FLAPW兲
B-E g 共experiment兲
232
237
251
448
a
d
b
e
Reference 60.
Reference 59.
c
Reference 42.
⳵ ⑀ jl
具 f 兩 q j兩 i 典 ,
⳵qi
共6兲
where 兩 i 典 and 具 f 兩 denote the initial and final vibrational
states.
Once the energies ⑀ kn and functions 兩 kn 典 for the n bands
are obtained self-consistently, the interband contribution to
the imaginary part of the dielectric functions ⑀ 2 ( ␻ ) can be
calculated by summing transitions from occupied to unoccupied states 共with fixed k vector兲 over the BZ, weighted with
the appropriate matrix element for the probability of the transition. To be specific, the components of ⑀ 2 ( ␻ ) are given by
⑀ 2jl 共 ␻ 兲 ⫽
Ve 2
2 ␲ បm 2 ␻ 2
冕
d 3k
300
461
Reference 58.
Reference 43.
f
Reference 44.
Here, ␻ i and ␻ f denote the phononic eigenvalues, ␻ L
represents the incident laser photon energy, and
L( ␻ L , ␻ i , ␻ f , ␥ ) accounts for the Lorentzian term of lifetime
broadening in one-phonon processes with the phonon linewidth ␥ . The matrix element M i f is given by
Mif⫽
188
193
190e
具 kn 兩 p j 兩 kn ⬘ 典具 kn ⬘ 兩 p l 兩 kn 典
兺
nn
⑀ jl 共 ␻ 兲 ⫽b ojl 共 ␻ 兲 ⫹b 1jl 共 ␻ 兲 q i ⫹b 2jl 共 ␻ 兲 q 2i ⫹ . . . .
In this case the derivatives of the dielectric functions appearing in Eq. 共6兲 are represented by the coefficients of the
polynomial at the incident photon energy. The pure
vibrionic-matrix elements are calculated numerically after
solving the harmonic 共or anharmonic兲 oscillator problem for
the potential given in Eq. 共2兲. Equations 共3兲 and 共8兲 thus
contain all the information necessary from the band-structure
calculations to determine the Raman spectra for a certain
eigenmode. We have computed ⑀ jl for five points along each
normal coordinate and fitted the q i dependence of ⑀ jl at the
chosen frequency of incident light ␻ . All modes are almost
harmonic, therefore, the vibrational states 兩 i 典 and 具 f 兩 are
eigenstates of the harmonic oscillator. More detailed discussions about the approach used here to calculate Raman spectra from experimentally observable parameters 共which can be
directly obtained theoretically from frozen-phonon ab initio
band-structure calculations兲 can be found elsewhere.53,55
⬘
⫻ f kn 共 1⫺ f kn ⬘ 兲 ␦ 共 ⑀ kn ⬘ ⫺ ⑀ kn ⫺ប ␻ 兲 .
共8兲
III. RESULTS AND DISCUSSION
共7兲
Here (p x ,p y ,p z )⫽p is the momentum operator and f kn is
the Fermi distribution. The evaluation of the matrix elements
in Eq. 共7兲 is done over the muffin-tin and interstitial regions
separately. Further details about the evaluation of the matrix
elements are given elsewhere.54 Due to the tetragonal structure of these compounds the dielectric function is a tensor.
The symmetry gets further reduced due to the phonon-mode
displacements. The real part of the components of the dielectric tensor ⑀ 1 ( ␻ ) is then calculated using the KramersKronig transformation. Like the phonon potential in Eq. 共3兲,
the frequency-dependent dielectric tensor ⑀ jl can be expanded as a polynomial in terms of the phonon eigenvector
with coefficients obtained by fitting to the first-principles
results:
A. Calculated IR- and Raman-active phonons
Earlier studies show56,57 that calculated vibrational frequencies by GGA are in better agreement with the experimental data than those obtained by LDA. So all the results
given in this paper are obtained from GGA calculations. In
order to determine the equilibrium position of B we have
calculated the total energy and forces as a function of B
displacement along c 共see the example in Fig. 2兲. In general
our calculated equilibrium z B parameters are in good agreement with the experimental values 共Table I兲. The calculated
Raman-active phonon frequencies for the RT 2 B2 C compounds are given in Table II along with the available experimental values from Raman-scattering measurements42,43,58,59
and inelastic neutron-scattering experiments with Born-von
Karman 共BvK兲 model evaluation.44 The experimental results
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PHYSICAL REVIEW B 67, 104507 共2003兲
P. RAVINDRAN et al.
show43 that the full width at half maximum of the B-A 1g
mode is about 100 cm⫺1 , i.e., the B modes are extremely
broad, and accordingly a comparison with Raman frequencies is problematic. Taking this fact into account when comparing our calculated values with experimental data the
agreement is as good as it can be. The calculated A 1g - and
B 1g -phonon frequencies are also comparable 共see Table II兲
with the frequencies calculated by Weht et al.60 from the
FLMTO method. The overall very good agreement between
the experimental and present theoretical values indicates the
reliability of the predicted values for the other modes and
compounds. The polarization-dependent Raman-mode
measurements43 on LuNi2 B2 C with the boron isotope 10B
substituted for 11B indicated that the mode near 830 cm⫺1 is
due to the vibration of the B atoms which is confirmed by
our theoretical study. The A 1g mode involves vertical B displacements that stretch/compress the linear B-C-B bonds
while bending the B-Ni-B bond angles in the NiB4 tetrahedron. The stiffness of the A 1g frequency reflects the strong
B-Ni and B-C bonds consistent with an earlier chemicalbonding analysis.32 Our calculated Raman-active phonons
are found to be in good agreement with the experimental
values indicating that GGA calculations can accurately predict the static density response and that phonons are not
heavily dressed by spin or excitonic fluctuations in the borocarbides.
Park et al.43 could not identify the Ni-E g and B-E g modes
from polarization-dependent Raman measurements for
RNi2 B2 C single crystals. However, our calculated phonon
frequencies of the Ni-E g and B-E g modes for YNi2 B2 C are
found to be in excellent agreement with the phonon frequencies obtained by Raman-scattering measurements of Hadjiev
et al.42 and Hartmann et al.59 which should clarify the controversy between the three reports. Even though YNi2 B2 C
and LaNi2 B2 C are found to be isostructural and isoelectronic, the former is superconducting with T c ⫽15.5 K and
the latter is nonsuperconducting down to 1 K. So, it is interesting to identify the difference in the phonon behavior between these two compounds. The Raman-active phonons
given in Table II show that there is no significant difference
in frequencies between these two compounds except for the
Ni-E g mode.
Experimental studies show61 that the electronic specificheat coefficient ␥ of YCo2 B2 C is comparable to that of isostructural superconducting YNi2 B2 C. But the former possesses
Pauli
paramagnetic
character
and
no
superconductivity is observed down to 30 mK. So it is interesting to establish its phonon behavior. Comparison of our
calculated B 1g -phonon frequency for YCo2 B2 C with other
RT 2 B2 C compounds show that YCo2 B2 C takes the largest
value. This is consistent with the experimental findings in the
sense that inelastic neutron-scattering measurements44 on
Co-doped YNi2 B2 C found shifts 共between 40 and 70 meV兲
towards higher energy in the vibrational frequency. Stiffening of the phonon mode could explain why no superconductivity is observed in YCo2 B2 C.
Mattheiss et al.37 proposed that the mechanism for the
superconductivity of RNi2 B2 C is based on high-frequency
B-A 1g optical phonons that dynamically modulate the
B-Ni-B bond angles of the NiB4 tetrahedra. Contrary to this
viewpoint, Pickett and Singh27 point out that to obtain T c
⬇17 K, the phonon frequency has to be 145 cm⫺1 provided
the electron-phonon coupling is strong (␭⫽2.6). Thus the
phonons responsible for the superconductivity are expected
to be of much lower frequency than the highest frequency of
the B vibrations. So, either soft modes or large contributions
from the heavier atoms must be responsible for the large EP
coupling in the RT 2 B2 C compounds. The present results
show that the frequency of the B-A 1g mode does not change
significantly between the superconducting and nonsuperconducting members. However, the in-plane transition-metal vibration 共Ni- or Co-E g mode兲 is shifted to higher frequencies
for the superconducting phases compared with the nonsuperconducting phases. On the other hand, the electronic structure does not change significantly with the E g -mode displacement indicating that the interaction between the
conduction electrons and the E g mode is weak. This indirectly supports the conclusion derived as a result of the
point-contact measurement that strong interaction of the conduction electrons with soft phonons is the distinguishing feature between the EP interaction in the superconducting and
nonsuperconducting materials.38 Experimental high-pressure
studies62 show that T c for these materials decreases with increasing pressure. In order to gain insight into the effect of
pressure on the phonon behavior we have calculated the
changes in the phonon frequencies for the A 1g and B 1g
modes for YNi2 B2 C with up to 8% volume compression.
The thus obtained phonon frequencies are found to increase
linearly with increasing pressure indicating that stiffening of
the phonons at high pressures reduces T c . Further the highpressure electronic structure studies show that the DOS at E F
decreases with increasing pressure. Hence, it seems that the
nature of the superconductivity in these materials can be explained through the conventional EP mechanism.
The phonon frequencies ( ␻ ) and eigenvectors obtained
from the diagonalization of the dynamical matrix for the IRactive modes of the RT 2 B2 C compounds are given in Table
III. The eigenvectors convey information about the contribution from various atoms to the phonon frequencies. Indirect
experimental information on the phonon eigenvectors can be
obtained from the polarization dependence of the IR and
Raman intensities.
From BvK processing of data, Gompf et al.44 found that
1
excithe highest-frequency phonon mode should be the A 2u
tation which is associated with antisymmetric bond1
has the higheststretching vibrations. We also found that A 2u
phonon frequency among the phonon modes in these
RT 2 B2 C compounds 共Table III兲. The inelastic neutronscattering experiments44 indicated a spectral component at
1209 cm⫺1 for YNi2 B2 C which is in excellent agreement
1
-phonon frequency of 1215 cm⫺1 .
with our calculated A 2u
3
; see Table
The lowest-frequency IR-active mode (A 2u
III兲 in these compounds is 60% contributed by Ni or Co with
the remaining 40% coming equally from B and C. The
1
; see above and
highest-frequency IR-active mode (A 2u
Table III兲 originates from the vibration of C and B along c
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PHYSICAL REVIEW B 67, 104507 共2003兲
RAMAN- AND INFRARED-ACTIVE PHONONS IN . . .
TABLE III. Calculated IR-active phonon frequencies ( ␻ in cm⫺1 ) and eigenvectors of the dynamical
matrix. Note that relative atomic displacements can be obtained by dividing the eigenvector components by
the square root of the atomic mass.
Eigenvector
Compound
Mode
␻
Ni or Co
B
C
YCo2 B2 C
E 1u
E 2u
E 3u
A 12u
A 22u
A 32u
E 1u
E 2u
E 3u
A 12u
A 22u
A 32u
E 1u
E 2u
E 3u
A 12u
A 22u
A 32u
E 1u
E 2u
E 3u
A 12u
A 22u
A 32u
484
429
134
1193
407
89
475
380
99
1215
370
67
447
367
142
1266
332
105
449
399
115
1290
410
112
0.235
⫺0.205
⫺0.717
0.017
0.540
0.841
0.130
⫺0.309
0.941
⫺0.012
0.574
⫺0.818
0.193
⫺0.241
0.570
⫺0.014
⫺0.579
⫺0.814
⫺0.246
0.233
⫺0.775
0.013
⫺0.508
0.860
0.851
⫺0.418
0.166
0.333
⫺0.443
0.284
0.690
⫺0.653
⫺0.310
⫺0.429
⫺0.427
⫺0.315
0.831
⫺0.467
⫺0.241
⫺0.420
0.466
⫺0.324
⫺0.818
0.468
0.245
0.404
0.518
0.330
0.464
0.834
0.140
⫺0.790
⫺0.504
0.340
0.710
0.690
0.129
0.798
⫺0.354
⫺0.354
0.519
0.820
0.234
0.803
0.477
⫺0.354
⫺0.514
⫺0.827
0.061
⫺0.801
0.507
0.310
YNi2 B2 C
LaNi2 B2 C
LuNi2 B2 C
with two-third of the contribution from C and one-third from
B. The highest-frequency E u mode (E 1u ) is up to 55% contributed from B mixed with around 30% from C and 15%
from Ni or Co. The E 2u mode which mainly originates from C
takes the largest frequency for YCo2 B2 C (429 cm⫺1 ) compared with all the other compounds considered here. This
mode has a 55% contribution from C, approximately 30%
from B, and approximately 15% from Ni or Co. There are
quite significant differences between the nonsuperconducting
LaNi2 B2 C and the superconducting Y and Lu analogs concerning the lowest-energy E u mode (E 3u ). First, the frequency of this phonon mode is larger in nonsuperconducting
LaNi2 B2 C than in the superconducting Y and Lu analogs.
Second, some 50% of the contribution comes from Ni in
LaNi2 B2 C as compared with some 70% from Ni in the Y and
Lu variants. This indicates the significance of Ni for the occurrence of superconductivity in these compounds.
B. Raman-active phonons and superconductivity
Theoretical findings39 have suggested that the highfrequency Raman modes are responsible for the enhanced T c
of these borocarbides. Hadjiev et al.42 advanced the attractive suggestion that it is the Ni vibrations that modulate the
Ni-B bonds which are of importance for the paired state, and
that renormalization effects may appear below T c for the
Ni-B 1g mode at 198 cm⫺1 and the B-A 1g mode at
820 cm⫺1 . Hence, it is interesting to examine the changes in
the electronic structure imposed by different phonon modes.
Earlier band-structure calculations27,29,39 for RNi2 B2 C have
shown that the position of the E F is close to a peak 共formed
mainly by Ni-3d electrons兲 in the DOS. Our calculated band
structures for superconducting YNi2 B2 C are given in Figs. 3
and 4 where the continuous lines refer to the equilibrium
structure. The key feature of these band structures is the
presence of a flat band along the ⌫-X direction 共110兲 which
is believed to be important for the superconductivity in these
compounds. In order to gain insight into the role of the
Raman-active phonon modes on the changes of the electronic structure of YNi2 B2 C we have calculated the band
structure with displacements of ⫾0.10566 Å 共viz. ⫾1%兲 for
each mode. The changes in the band structure of superconducting YNi2 B2 C by the A 1g and B 1g modes 共with the said
displacements from the equilibrium structure兲 are shown in
Figs. 3 and 4. The large changes in the band structure of
YNi2 B2 C by A 1g and to a certain degree B 1g phonons indicate that these modes are strongly coupled with the electrons
and hence participate in the superconductivity. This observation is consistent with the experimental study in the sense
that an appreciable boron isotope effect has been observed
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PHYSICAL REVIEW B 67, 104507 共2003兲
P. RAVINDRAN et al.
FIG. 3. Band structure of YNi2 B2 C in the A 1g mode. The continuous lines represent the band structure of the equilibrium structure and circles represent that of the distorted lattice with the B
atom displaced 0.105 66 Å along c 共a兲 above and 共b兲 below the
equilibrium position.
for YNi2 B2 C. 63,64 The A 1g -optical phonons involve vertical
displacements of the B atom which stretch/compress the B-C
bonds and change the B-Ni-B bond angle. The corresponding
in-plane displacements produce A 2u phonons which have
negligible coupling with band states near E F compared with
the A 1g mode.37
Since the A 1g - and B 1g -phonon-mode motions of B and
Ni change the B(p)-Ni(d) orbital overlap, significant EP
coupling is expected for both B(p) and Ni(d) electrons, and
this plays an important role in the superconducting pairing.
This can be most easily seen by comparing the band structure of the undistorted lattice to that of the distorted for the
zone-center phonons. For the phonon-mode-involved inplane vibrations we did not see any significant changes near
the E F . However, for the A 1g and B 1g modes, there is significant splitting of the bands as well as shifts near the E F as
shown in Figs. 3 and 4, indicating strong EP coupling. Since
the value of the deformation potential is a measure of the EP
coupling, we have calculated the deformation potential for
all the Raman-active phonons at the high-symmetry points of
the BZ. For the in-plane vibrations from the Ni-E g and B-E g
phonons we obtained insignificant deformation potentials
FIG. 4. Band structure of YNi2 B2 C in the B 1g mode. The continues line represent the band structure of the equilibrium structure
and circles represent that of the distorted lattice with the Ni atom
displaced 0.105 66 Å along c 共a兲 above and 共b兲 below the equilibrium position.
FIG. 5. Calculated imaginary part of the optical dielectric tensor
for superconducting YNi2 B2 C. The continuous lines represent the
⑀ 2 ( ␻ ) spectra for the equilibrium lattice and the dashed line that of
the distorted lattice with the B atom displaced 0.105 66 Å along c
above the equilibrium position.
and conclude that the EP coupling is negligible for these
modes. The deformation potentials obtained for the A 1g
phonons at the ⌫, 110, and 111 points are 3.154, 1.583, and
1.559 eV/Å, respectively. In the case of the B 1g mode we
have obtained a much smaller deformation potential than for
the A 1g mode at the ⌫, 110, and 111 points with values of
1.014, 0.102, and 0.077 eV/Å, respectively. If one also considers the smaller mass of the B atoms which are involved in
the A 1g -phonon modes, one could expect larger EP coupling
for the A 1g phonons compared with the other Raman-active
phonons. It should be noted that the band degeneracy along
⌫ to 110 of the BZ is split by the B 1g -mode phonon displacements 共Fig. 4兲. The deformation potential at the banddegeneracy-split point in the BZ is found to take a considerably increased value of 2.731 eV/Å and hence the splitting
of the degeneracy by the B 1g mode significantly contributes
to the EP coupling. It is worth recalling that temperaturedependent Raman-scattering measurements58 on YNi2 B2 C
show a remarkable redistribution in the low-frequency region
( ␻ ⭐250 cm⫺1 ) upon lowering of the temperature. This supports the just advocated viewpoint that Ni-B 1g phonons are
of importance for the superconductivity in these materials.
Among the Raman-active modes, the T-E g mode is distinctly different for the superconducting and nonsuperconducting materials and hence one may suspect that the Ni-E g
mode plays an important role in deciding the superconducting properties. Therefore, we have also calculated the electronic structure of YNi2 B2 C with E g -mode perturbations.
However, our calculated band structure for different phonon
modes shows that the bands at the E F are much less sensitive
to the T-E g modes than the B-A 1g and T-B 1g modes. So, we
believe that apart from the B-A 1g and T-B 1g modes, phonon
softening and/or other low-energy phonon excitations may
be of importance.
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PHYSICAL REVIEW B 67, 104507 共2003兲
RAMAN- AND INFRARED-ACTIVE PHONONS IN . . .
FIG. 6. Calculated intensity for Raman-active phonons as a
function of laser energy for different polarizations in YNi2 B2 C.
C. Calculated Raman spectrum for YNi2 B2 C
Several studies of the optical conductivity of these borocarbides have been published.65– 67 The calculated imaginary
part of the optical dielectric tensor ⑀ 2 ( ␻ ) for YNi2 B2 C is
shown in Fig. 5 where E储 a and E储 c represent the electric
field parallel to the crystallographic axes a and c, respectively. Even though the electronic structure studies show
three-dimensional behavior, we found a huge anisotropy in
the optical spectra which reflects the layered nature of the
structure. One can derive68 all linear optical properties from
the ⑀ 2 ( ␻ ) spectra given in Fig. 5, but a more detailed account of our findings will be published later. From the derivative of the optical dielectric function with normal coordinates for phonon modes we have calculated the Raman
intensities of all Raman-active modes using the formalism
described in Sec. II C. The calculated Raman intensity as a
function of laser energy is given in Fig. 6 for different polarizations.
The calculated intensity for the Ni-B 1g phonon for the zz
polarization is negligibly small and hence it is not included
in Fig. 6. The overall Raman intensity is found to be high for
the laser energy around 2.5 eV. So, with a proper selection of
a laser source operating in this energy range one can obtain
better yield of Raman-scattering intensity from these materials. Further we have detected a huge difference in the Raman
intensity between different polarizations. In particular for the
A 1g and B 1g modes the Raman intensity in the zz polarization is much smaller than that in the xx polarization.
FIG. 7. The calculated Raman spectra for superconducting
YNi2 B2 C.
The calculated Raman spectrum for superconducting
YNi2 B2 C with a laser wavelength of 514.5 nm is shown in
Fig. 7. From this figure it is clear that the intensity of the E g
modes is much weaker than that of the A 1g mode. This may
be a reason for the problem of observing the E g mode in
some of the experimental investigations.43 The phonon frequency of the heavier Ni atom is expected to be lower than
that of B and C. Figure 7 is consistent with this expectation
that the Ni-B 1g mode has the lowest frequency.
IV. SUMMARY
We have performed first-principles frozen-phonon calculations for superconducting LuNi2 B2 C and YNi2 B2 C and
nonsuperconducting YCo2 B2 C and LaNi2 B2 C to obtain all
Raman- and IR-active phonon frequencies and eigenvectors.
The overall agreement of our calculated Raman-active phonon frequencies compared to reported experimental values is
found to be very good. We predict all the IR-active phonons
for these compounds, since to our best knowledge there are
no experimental data available, and we hope the present results will motivate such studies. Among the Raman-active
phonon frequencies of the superconducting and nonsuperconducting compounds, only the T-E g mode shows a significant distinction with a smaller phonon frequency for the nonsuperconducting compounds. However, our electronic
structure studies show that the energy bands near E F are not
significantly distorted by the T-E g mode displacement. We
have also found that the B-A 1g and Ni-B 1g modes are
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PHYSICAL REVIEW B 67, 104507 共2003兲
P. RAVINDRAN et al.
strongly coupled to the electrons at E F in the superconducting materials, whereas such strong coupling is found far
away from E F in the nonsuperconducting materials. This indicates that the origin of the superconductivity in RT 2 B2 C
materials can be explained in terms of a conventional
electron-phonon mechanism. We have also predicted the Raman spectrum for YNi2 B2 C and here shown that if one
chooses the laser source with a wavelength around 2.5 eV
one can get maximum intensity in the Raman spectra for
these compounds.
We have analyzed the electron-phonon coupling and
hence superconductivity in these materials based on the
changes in the phonon modes imposed by the zone-boundary
phonons alone. However, one cannot conclude that the coupling is weak along the entire zone on the basis of nonobservations at the ⌫ point alone. So, it would be interesting to
map the complete phonon band structure and the electron-
phonon coupling strength within the entire BZ to obtain a
better understanding of the superconducting behavior of
these materials. This can be achieved by a complete set of
linear-response calculations, but owing to the volume of
computations involved in such a calculation it was regarded
as outside the scope of the present study.
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ACKNOWLEDGMENTS
P.R. is grateful for the financial support from the Research
Council of Norway. Part of these calculations were carried
out on Norwegian supercomputer facilities 共Program for Supercomputing兲. C.A.D. appreciates the financial support
from the Austrian Science Fund 共Project No. P13430-PHY兲
and C.A.D., L.N., and B.J. acknowledge support from the
Swedish Research Council 共VR兲.
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