Dec 4, 1994  by determining the shift in T, between a 100% 160 sam ple and a 65% ..... Zheng, J. M. Newsam, S. K. Sinha, D. Vaknin and. A. J. Jacobson ...
Brazilian Journal of Physics, vol. 24, no. 4, December, 1994
Phonons, Isotope Effect, and Superconductivity in Bai,KXBiO3:a Molecular Dynamics Simulation Marcos H. Degani Departamento de Fisica Geml e Aplicada Universidade São Francisco Rua Alexandre Rodrigues Barbosa, 45, 13251900, Itatiba, SP, Brazil Received August 20, 1994
The phonon densityofstates (DOS) of insulating BaBi03 in orthorhombic phase and superconducting Bal,K,Bi03 in cubic phase are studied using the molecular dynamics (MD) method. The MD simulations are carried out with an effective interaction potential which includes Coulomb interactions, the chargedipole interactions due to the electronic polarizability of O, and steric effects. Partia1 DOS of Ba, K, Bi and O in BaBi03 and Bal,K,Bi03 are also determined from MD simulations and reveal that phonons above 20 meV are due to oxygen vibrations. The reference oxygenisotopeeffect exponent, a,, = dln < w > /dln Mo, of Bal,K,Bi03 is determined to be a,, = 0.42 k 0.05 from the mass (Mo) variation of the first moment of the phonon DOS, l6 < w > and w >. This value is in very good agreement with the oxygen isotopeeffect exponent a o , determined experimentally from the variation of Tc,and suggests that Bal,KxBi03 is a weaktomoderate coupling BCSlike superconductor and that the high Tc (N 30K) results from large electronphonon matrix elements involving highenergy oxygen phonons.
'v
I. Introduction
n ~ n m a ~ n e t i c [ l " 'while ~ ] the other highTc related mater i a l ~in , the parent nonsuperconducting phases, display
Since 1986 the physical mechanism responsible for
antiferr~ma~netism[~~I.
hightemperature superconductivity in the oxide materials has been the focus of research in condensed mat
A wide range of experimental investigations have
ter physics[l]. In general, there are two kinds of ox
been carried out this system. According to neutron
ide superconductors, one containing copper and the
diffraction m e a s ~ r e m e n t s [ ~potassium ~ ~ ~ ~ ~ ~atoms ~]
other without any transition metal e~ements[~]. Super
are randomly distributed over the barium sites.
conductivity in Bal,K,Bi03
Structural properties, electric, magnetic, thermal,
was first discovered by
~ ]1988. The strucMattheiss, Gyorgy, and ~ o h n s o n [in
and optical responses of the Bal,K,BiOa
ture of the superconducting material was determined by
have been studied.
Cava et al.[4] and a detailed account of synthesis, struc
r e f l e c t i ~ i t ~ [ ~e ~l el ~ t~r Io,n  t u n n e l i n ~ [ ~photoemis~~~],
ture, and transition temperature as a function of x was
sion, and inverse photoemission[35~361, inelastic neutron
given by Hinks et a ~ . [ ~  For ~ ] . x N 0.4, Bal,K,Bi03
 d i f f r a ~ t i o n [ ~ ~the  ~ ~oxygen ], isotope e f f e ~ t [ ~ ~ ~ " ~ ~ ] ,
exhibits superconductivity at Tc N 3 0 K , which is the
specific heat[4143] upper and lower critical magnetic
highest transition temperature reported for any oxide
f i e l d ~ [ " ~ ~ thermal  ~ ~ ] , c o n d u c t i ~ i t ~ and [ ~ ~ thermoelecI,
material not containing copper[216]. The supercon
tric power[1a14" have been measured. The crystalline
(0.37 < x < 0.5) forms
a cubic perovskite cristal structure[l71 which shows none
structure and the phase diagram of Bal,K,BiO3, as a function of temperature for the range x = O  0.5
of the planar structures observed in other highTc com
have been investigated by Pei et al.[l71.
pounds. As the potassium concentration is reduced, su
fect measurements indicate that the carriers are
perconductivity disappears when the structure changes
e l e c t r o n ~ [ ~ "whereas ~~I in the cuprates, with exception
from cubic to orthorhombic. Furthermore, BaBi03 is
of Nd2,Ce,Cu04 and Pr2,Ce,Cu04,
ducting phase of Bal,K,Bi03
system
~ a m a n  s c a t t e r i n ~ [ infrared~~],
Hall ef
the carriers are
Marcos H. Degani
(J 30K) results from large electronphonon matrix el
holes[']. Batlogg et al.1" measured the oxygen isotopeeffect
ements involving highenergy oxygen modes. The su
by determining the shift in T, between a 100% 160 sam
perconducting properties of Bal,K,Bi03
ple and a 65%
finite temperatures have been calculated within the
exchanged sample of Ba0.6K0.4Bi03
at zero and
and they found an exponent a o = 0.22 f 0.03 in the
framework of Eliashberg theory and compared with ex
T, Mõa"relation, where Mo is the mass of the oxygen isotope. Measurements by Hinks et using a 100% 160 and a 96% '9 exchanged sample of
periments by Jin, Degani, Kalia and ~ a s h i s h t a [ ~ ~ ] .

Ba0.625K0.375Bi03 indicate a substantial oxygen isotope effect, ao = 0.41 & 0.03. Kondoh et a1.L'"
have deter
mined ao = 0.35 f 0.05. This value is larger than the isotopeeffect exponents in highT, cuprate superconductors.
11. Molecular dynamics simulations of phonon density of states The molecular dynamics m e t h ~ d [was ~ ~ ]used to obtain the partia1 and total phonon densities of states. The MD simulations on BaBi03 were performed on a 540particle system in the orthorhombic phase a t
Attempts to carry out electron tunneling experi
the experimental density of 7.88 g/cm3 with the lat
ments in superconductorinsulatorsuperconductor (S
tice parameters a = 6.2000A,
IS) junctions in YBa2Cu307a have not been very
c = 8.6948A, and 625 particles for Bao.sKo.4Bi03. In
successful due to the very small coherence lenght (J
this system the substitution of 40% of the Ba atoms
10A). In Bal,KxBiOs,
however, electron tunneling
by I< atoms was done randomly at the experimental
experiments on SIS junctions have recently been car
density of 7.33 g/cm3 in a cubic phase with lattice pa
ried out by Zasadzinski et al.[27128]and have revealed
rameter a = 4.3160A. The unit cells of the two systems
well resolved structures in the highenergy range 30
are shown in Fig. 1. Effective interparticle interactions
60 meV in agreement with the phonon DOS obtained
were used in the MD simulations. The Newton equa
Sato ~ ] . et al.PII by neutron scattering e ~ ~ e r i m e n t s [ ~
tions of motion are integrated by Beeman's m e t h ~ d [ ~ ~ ]
have performed a tunneling experiments on thin films of
using a time step of At = 5 x 1015sec, which conserves
Bao.6Ko,4Bi03and obtained the ratio 2A(0)/KBTc =
energy to better than 1 part in 104 over severa1 thou
3.7 f 0.5, where A(0) is the superconducting energy
sand time steps. The longrange nature of the Coulomb
gap at zero temperature, which is in agreement with
interaction is taken into account by Ewald's summation meth~d[~~].
the optically derived gap ratio by Schlesinger et
b = 6.1561A, and
In this paper we report the calculation of the phonon
I
In this simulation we have used effective pairwise densities of states of superconducting ~ a ~  , K , B i ~ ~ 0 3 interactions. The potentials include chargedipole inand BalxK,Bi1"3 (x = 0.4), and of insulating teractions due to large electronic polarizability of O~ a B i ' ~ 0Mie ~ .find a significant softening of the oxy
ions, steric repulsion between ions, and Coulomb inter
gen phonon modes around 30 and 60 meV, in the su
actions due t o chargetransfer effects. The total poten
perconducting material. In order to characterize the
tia1 has the form
nature of superconductivity within the framework of BCSEliashberg theory[50151]we investigate the correlation between the isotope shifts in the phonon DOS and
Ej(r)= ( a i ~ ~ + ~ j ~ ~ ) e  P I ' s c / 2 ~ 4 + ~, i j / ~ ~ i ~ + ~
in T, of
and B ~ ~ . ~ K O . ~We B ~ob'~O~. (1) where Zi is the effective charge and ai is the electronic tain a value of a,, = 0.42 it 0.05 for Ba0.6K0.4Bi03
which is very close t o the isotope effect exponent in Te.
polarizability of the ith ion and H i j and qij are the
The results of our study indicate that Bal,KxBi03
strengths and exponents of the steric repulsion between
is a weaktomoderate coupling BCS superconductor,
the ions i and j , respectively. The screening length, r,,
and the high superconducting transition temperature
is chosen so that chargedipole interaction does not have
Braziliar? Journal of Physics, vol. 24, no. 4, December, 1994
958
a long tail. The steric repulsion balances the attractive interactions between cations and anions at short distances so as to give the correct bond lengths. The parameters for the interaction potentials, used in Eq. (1) for BaBi03 and Ba0.6K0.4Bi03,are summarized in Tables I and 11, respectively.
(a)
BaBiO,
Table I  Constants in the effective potentials for BaBiOs. Units of length and energy are A and e 2 /A = 14.39 eV respectively. Z is the effective charge (in 3 units of /e/),a the electronic polarizability (A ), q the repulsive exponent, and H the repulsive strength.
Ba Bi
O
Z 0.800 1.600 0.800 4.430
7's c
BaBa BaBi BaO BiBi BiO
00
a
0.00 0.00 2.40
77 11 11 9 11 9 7
H 1186.8 157.3 281.5 13.2
60.8 49.2
Table I1  Constants in the effective potentials for Bao.6Ko.3Bi03. Units of length and energy are A and e2/A = 14.39 eV respectively. Z is the effective charge 3 (in units of lei), a the electronic polarizability (A ), q the repulsive exponent, and H the repulsive strength.
Figure 1: Crystal structures of (a) orthorhombic BaBiOs and (b) cubic Ba0.6K0.4Bi03. To establish the dynamical stability of the BaBi03, the system was put in the orthorhombic structure j n an MD cell of fixed volume. The partia1 pair distribution functions and bond angle distribution functions were calcuiated to verify the bond lengths and coordi
BaBa BaK BaBi BaO KK KBi KO BiBi BiO 00
11
1007.0
Marcos H. Degani
nation numbers. The system was slowly heated to 600K
The second method to calculate the phonon DOS
and thermalized, for severa1 thousand time steps. Af
involves the displacement autocorrelation function and
ter this it was run uninterruptedly for more than 30,000
~ ] implement . this the equationofmotion m e t h ~ d [ ~To
time steps and various structural correlations were calculated to examine the symmetry. The system at 600K
method it is essential to bring the system to a local minimum energy by carrying out the steepest descent
was slowly cooled, thermalized, and then subjected to
quench which guarantees that the force and the velocity
a steepest descent quench[55] (SDQ) which is a mathe
for each particle is zero. Each particle is then given a random displacement,
matically well defined method of examining the underlying mechanically stable structures. The partial pair correlation funcbions and bond angle distribution func
Sij (O) = So cos(6ij ) ,
tions were calculated again to ascertain the symmetry
where So is the amplitude of an initial displacement and
of the MD system. After performing the above mentioned procedure on the BaBi03 system, it was determined that the resulting final symmetry was the same
as that of the starting orthorhombic structure. The cubic Bao 6Ko 4Bi03 system was also subjected to the
Bij
(4)
are random variables distributed uniformly between
O and 27r. The system is allowed to evolve according to the classical equations of motion and the time variation of rij( t )is obtained. The displacement autocorrelation function is given by
same procedure t o ensure dynamic stability. The phonon density of states was calculated using two differents methods. We find t h < ~the t results of a11 these two calculations are in agreement with one
where S ~ i j ( t )=
another. The first method involves calculating the ve
the Fourier transforms of this autocorrelation functions
locity autocorrelation function for each species and the
give the density of states.
partial phonon DOS Fi(w) is obtained by the Fourier transforms of this autocorrelation functions.
rij (t)  rij (0).
In the harmonic limit,
111. R e s u l t s and discussion
The normalized velocityvelocity autocorrelation
To identify the physical origin of the peaks in the to
function for Pth species (P=Ba, Bi, K, or O) is given
tal DOS, we first examine the MD partial DOS for insu
by
lating BaBi03 and superconducting Bao 6Ko 4Bi03. In Fig. 2, we show the MD partial DOS FB,(w), FBz(w), Fo(w), and the total DOS F(w) for BaBiOs. The partia1 DOS is normalized to 3N,, where N, is the total particle number for the ith species in the MD system.
< ... > is an
It can be seen that a11 the peaks in Fo(w) are located
average over MD configurations. The frequency spectrum of the r P ( t ) gives the partial phonon density of
between 20 and 80 meV, FBa(w) exhibits a single peak ) two peaks at 1 2 and 17 at 11 meV and F B ~ ( Wshows
states
meV. Clearly, in the total DOS the peak a t 11 meV is
where vi is the velocity of particle i and
due to both Ba and Bi and the peak a t 16 meV is due to Bi alone. Above 20 meV the entire spectrum arises from oxy gen vibrations. The MD results for the partial DOS, FBa(w), where a Gaussian window enction with y = 1 and
= 3ps is used. The total density of states is obtained by T
summing the concentration weighted partial densities of states. Additional weighting with the coherent neutron crosssections is required to obtain the neutron density of states.
FK(w), FB%(w),Fo(w) and the total DOS F(w) for superconducting Bao sKo 4Bi03 are presented in Fig. 3. Also in this system, a11 peaks located above 20 meV are due to oxygen vibrations. In contrast to BaBi03, FK(w) for Bao 6Ko 4Bi03 shows an additional peak at 20 meV and the twopeak feature in FB2(w) is
Brazilian Journal of Physics, vol. 24, no. 4, December, 1994
960
less pronounced. Fo(w) of BaBiO3 shows sharp pea.ks around 26, 32, 37, 40, 44, 51, 60, 66, and 74 meV. In Bao.6Ko.4Bi03the peaks between 20 to 40 meV merge into a band, and those between 60 and 80 meV broaden and show a slight shift to lower energies.
BaoãKo,Bi03 Total
Energy (meV) Figure 3: Moleculardynamics results of partial and total phonon DOS for cubic Bao.6 Ko.4 Bi160>
In order to compare the neutron data[37] with MD simulation, we have calculated the neutronweighted DOS, G ( w ) using the partial DOS. The results are shown in Figs. 4 and 5 for BaBi03 and Bao.6Ko.4BiOa1
O
20
40
60
80
Energy (meV) Figure 2: Moleculardynamics results of partial and total phonon DOS for orthorhombic BaBiOs.
repectively. In general, there is an overall semiquantitative agreement between the MD results and neutron spectrum. In the case of BaBi03, the lowenergy peaks at l1 and l7 meV cannOt be resO1ved in the neutrOn data due to the relatively poor resolution in this
uncertainties of multiplescattering background in the inelastic neutron scattering experiments. Otherwise, the peaks a t 25, 3040, 50, 60, and 6575 meV of the MD DOS are identifiable with similar structures in the . ~ B MD ~ O phonon ~ measured DOS. For B ~ o , ~ K ~ the DOS shows a threeband structure with intensities centered around 14, 35, and 65 meV. It is clear from INS measurements and MD simulation that the oxygen phonon modes soften by 510 meV with 40% K doping of BaBiOs. Higherresolution neutron measurements may reveal the additional features .observed in the simulation.
IV.Isotope effect due to
Figure 4: Neutronweighted ~hononDOS for BaBiOa. Upper panel: INS experimental values (the solid line is a guide to the eye), and lower panel: moleculardynamics simulation results.
160to
leO substitution
The isotopic substitution of a particular atomic species will affect the superconducting transition temperature for a BCS superconductor as well as the phonon spectrum. The variation of Tc upon oxygen isotopic substitution is characterized by the oxygen isotopeeffect exponent,
where Mo is the mass of the oxygen isotope. When superconductivity is due to electronphonon coupling and the strong coupling effects are included, the isotope effect of the lattice is reflected through the superconducting transition temperature,
where f (A, ...,p*)is an unknown functional determined from the solution of the Eliashberg gap equations without any weakcoupling approximation, X is a dimensionless electronphonon coupling constant, and p* is the Coulomb pseudopotential. The characteristic phonon frequency < w > is defined as the first frequency moment , Figure 5: Neutronweighted phonon DOS for Bao.6 K0.4 Bir603. Upper panel: INS experimental values (the solid line is a guide to the eye), and lower panel: moleculardynamics simulation results. energy region. The difference between the MD and neutron G ( w ) in the relative magnitude of the lowenergy DOS is probably due to resolution effects and
The oxygen isotopeeffect exponent in Eq.(6) can be written as a sum of two terms obtained by differentiating Eq.(6):
Brazilian Journal of Physics, vol. 24, no. 4, December, 1994
962
where a,, is the reference isotopeeffect exponent defined by
a,, = dln
< w > /dlnMo .
(10)
The reference isotopeeffect exponent reflects the mass variation of the phonon DOS in a material whereas the oxygen mass variation of T, is given by the isotopeeffect exponent, ao. 6ao is a measure of the contribution arising from the strongcoupling effects. Clearly, for a monoatomic weakcoupling superconductors ao is 112. In the presence of strongcoupling effects, ao will deviate from a,, due to a significant contribution from the factor exp[ f (A, ...,p * ) ] . For multicomponent systems such as Bal,KxBi03, a partia1 isotopeeffect exponent a,i may be quite different[56]from 112 for isotopic substitution of the ith atomic species, e. g., l80 for 160. A low value of ao does not necessarily mean that strongcoupling effects are important. A large 6a0, on the other hand, implies that the strongcoupling effects are important.
B a 0 . 6 K ~ . ~ Bobtained i0~ from INS and from MD simulations. The overall shape of the phonon DOS for the and l600s similar, except that above 20meV the phonon spectrum is shifted to lower energies by 34 meV. A similar behavior is observed from the MD results. The reference isotopeeffect exponent a,, is found to be 0.42, which is in good agreement with the experimental values of a measured by Hinks et al.i40] (O.4lf 0.03) and Kondoh et ai.['" (0.35%0.05) fromT,, but significantly different from the results of Batlogg et al.r8l (0.22 5 0.03).
.V. Conclusion In conclusion, this paper describes the MD simulations of isotopically substituted samples of an oxide superconductor. The comparison of the phonon DOS of the insulating BaBiO3 with Ba0.6K0.4Bi03provides evidence for the importance of electronphonon interaction in the superconducting Bai,K,BiO3. The reference isotopeeffect exponent of oxygen, a,, , is estimated to be 0.42, only slightly higher than the isotopeeffect exponent for T,, a o = 0.41. This result indicates that Bal,KXBiO3 is a weaktomoderate coupling BCSlike superconductor and the high T, results from large electronphonon matrix elements involving highenergy oxygen phonons. Acknowledgment s
I would like to thank my collaborators in this work, Dr. ChunKeung Loong, Professor Priya Vashishta, Dr. Rajiv K. Kalia, and Dr. Wei Jin for valuable discussions. This work was partially supported by the Brazilian Agency CNPq. References
O
20
40
60 Energy (meV)
80
Figure 6: Neutronweighted phonon DOS for ~ a o . 6 ~ 0Bi16 . 4 o3and B a o . 6 ~ o . 4 B i ~Upper ~ 0 ~ panel: INS experimental values (the solid line is a guide to the eye), and lower panel: moleculardynamics simulation results. In Fig. 6 we show the neutronweighted phonon samples of DOS, G(E), for the 160 and
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