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Calculated effects of charge fluctuations on the phonon dispersion of YBa2Cu3O6 and YBa2Cu3O7. I. The model

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Ann. Physik 3 (1994) 225-241
Annalen
der Physik
@ Johann Ambrosius Barth 1994
Calculated effects of charge fluctuations on the phonon
dispersion of YBa2Cu306and YBa2Cu307
I. The model
M. Klenner, C. Falter, and Q. Chen*
Institut fiir Theoretische Physik I1 - Festktjrperphysik
Klemm-StraRe 10, D-48149Munster, Germany
-, University of Munster, Wilhelm-
Received 2 February 1994,in revised form 1 1 March 1994, accepted 18 March 1994
Abstract. In this paper we present a microscopic model that allows us to study the effects of charge
fluctuations on the phonon dispersion of the high-temperature superconductor YBa,Cu307 and its insulating counterpart, YBa,Cu,O,. An ab-initio rigid-ion model with pair potentials calculated by the
Gordon-Kim method from the free-ion charge densities is used as a reference system. Starting from
this reference system, charge fluctuations at the copper- and oxygen ions are introduced into the model.
The charge fluctuations are treated as adiabatic degrees of freedom in a non-phenomenological way.
The parameters entering the model are estimated consistently with the reference system from first principles rather than refering to the experimentally determined phonon dispersion. In addition to the
metallic behavior (appropriate to YBa,Cu307) obtained in this way, insulating behavior (appropriate
to YBa,Cu,06) is simulated by requiring the polarizability function to fulfill a corresponding longwavelength sum rule. Screened site-potential changes are defined that (besides the charge fluctuations)
constitute a qualitative measure of the electron-phonon-interaction potential. Furthermore we investigate the long-wavelength limiting behavior of the most important quantities occurring in our formalism. We derive formulae that allow us to calculate the contribution of the charge fluctuations to
the macroscopic dielectric constant and the transverse effective charges in the insulating phase.
Keywords: High-T, cuprates; Phonons; Electron-phonon interaction.
1 Introduction
Since the discovery of high-temperature superconductivity in the copper oxides [11,
various explanations for its microscopic origin have been proposed. Many of the
theories concentrate on purely electronic coupling mechanisms rather than the classical
source of superconductivity, i.e. the electron-phonon interaction (EPI). Indeed,
theoretical estimates of the electron-phonon-coupling constant arrive at values that
seem to be too small to be compatible with critical temperatures as high as 100 K when
based on the conventional rigid-muffin-tin or, more generally, rigid-ion approximations
[2]. On the other hand, there are several experimental indications of an intimate relationship between phonons and the superconducting transition in the high-temperature
* Permanent address: Department of Applied Mathematics and Physics, Beijing University of Aeronautics and Astronautics, 100083 Beijing, PR China.
226
Ann. Physik 3 (1994)
superconductors (HTSC); for a review of experimental studies on the lattice properties
and the electron-lattice interaction in the HTSC see, e.g., Ref. [3]. Moreover, it was
recognized quite early, that the usual muffin-tin (rigid-ion) approximations might be inappropriate for the description of the EPI in the HTSC because of the ionic nature of
these compounds [4,5]. In contrast to the situation in conventional high-density metals,
the changes of the single-electron effective potential associated with displacements of
the ions are not screened locally and thus are not confined to the sites of the ions
displaced in the cuprates but are long-ranged and nonlocal in the sense of acting even
at the sites of the non-displaced ions (“nonlocal electron-phonon interaction”). The
rigid muffin-tin approximation (RMTA), on the other hand, takes into account local,
short-range potential changes only. Nonlocal electron-phonon interaction yields considerable extra contributions to the coupling constant, A, as compared to the RMTA,
that might be strong enough to account for high T, [6,7]. Unfortunately, complete
self-consistent calculations within the local-density approximation are computationally
very intensive. Therefore, microscopic model calculations (like ours) can help to find
phonon modes exhibiting indications of large nonlocal electron-phonon interaction
(that then eventually could be studied in more detail theoretically and experimentally).
The layered copper oxides are characterized by a common general structural principle: there are Cu02 planes with covalent/metallic character separated by (more or
less) purely ionic layers. This structural principle is varied within the different classes
of high-T, compounds, primarily with respect to the number of Cu02 layers contained
in the unit cell. The simplest realization of this structural principle is represented by
the so-called infinite-layer- or parent structure, CaCuOz [8], consisting of Cu02
planes alternating with Ca layers. Several infinite-layer compounds (with Ca [partially]
replaced by Sr or Ba) have been made superconducting recently [9,10].
In the structures under discussion, nonlocal electron-phonon interaction effects are
associated, in particular, with axial displacements of the ions in the ionic layers, inducing potential changes in the Cu02 planes. (Actually, relative displacements of the ionic
layers and CuO, planes are important only.) Effects of that kind seem to be particularly efficient for diplacements of the apical oxygen ions present in many of the HTSC.
In contrast to the cations separating the Cu02 planes, the apical oxygens are, however,
covalently bound to the planes to some extent, and may therefore be actively involved
in the electron dynamics instead of staying passive as do the cations. We would like to
remark that the alkali fullerides [ll], having attracted much interest recently, fit into
this picture as well: the carbon balls characteristic for these compounds correspond to
the Cu02 layers and the alkali ions to the ions in the separating layers. The alkali ions
induce potential changes on the carbon balls when vibrating against the latter or, conversely, when the carbon balls perform breathing vibrations against the alkali ions (that
stay at rest) where the latter situation is expected to be more efficient.
The nonlocal potential changes lead to charge fluctuations (CF) at the non-closedshell ions, i,e. the copper- and oxygen ions (having significant spectral weight at the
Fermi level). This was found in frozen-phonon calculations by Cohen et al. [4] where
an axial, symmetric displacement of the apical oxygens in LazCuOl causes a charge
redistribution between the copper- and the in-plane-oxygen ions.
In a preceding paper [12] we have modeled such CF in LazCu04 starting from an
ab initio ionic model and treating the CF as adiabatic degrees of freedom. Here we exand YBa2Cu307(referred to as 07)
tend our studies to YBa2Cu306(referred to as 0,)
applying the same formalism used in Ref. [12]. The present paper reviews the underlying theory and gives the adaptation of the model to the compounds under considera-
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
227
tion. In an Appendix we investigate the long-wavelength asymptotic behavior of the
relevant quantities in our formalism and show how to calculate macroscopic dielectric
constants and transverse effective charges from our model. Results will be presented
in the following paper where we also will give the conclusions.
First, let us fix some conventions. Following common notation, we designate the
chain-copper- and -oxygen ions as Cul and 01, respectively, the copper- and oxygen
ions in the planes as Cu2, 0 2 , and 0 3 , respectively, (where 0 3 is in chain direction)
and the chain-plane-bridging-oxygenions as 04. In the tetragonal phase (0,)0 2 and
0 3 are equivalent, and we designate them together as 0 2 3 if we do not wish to distinguish between the individual sites. The unit cell is chosen centered at the chain with
respect to the c direction (i.e. it does not contain neighboring CuOz pianes).
2 Theory and modeling
In what follows, we review the essential steps leading to the model we have applied to
calculate the phonon dispersion of O6 and 0, in the present work. For a more complete presentation, see Ref. 1121. Part A deals with the ionic reference model. Here, we
use the method of Gordon and Kim I131 in order to obtain pair potentials between the
ions. The equilibrium structural parameters are determined by minimization of the
crystal energy as calculated from the pair potentials. In the next step, the dynamical
matrix is set up.
In part B, the CF degrees of freedom are introduced into the model. This can be accomplished formally by parametrizing the electron density appropriately and then using the variational principle of density-functional theory in order to (adiabatically)
eliminate the electronic degrees of freedom. An extra electronic contribution to the dynamical matrix results. The different sorts of coupling coefficients entering the formalism (CF-ion coupling and CF-CF coupling) can be expressed in terms of well-defined quantities of density functional theory. However, a main obstacle to the practical
application of the formulas derived, is related with the proper choice of the kinetic(single-particle-) energy functional. Thus, some model will be introduced in our
calculations. Simplifying assumptions will be made at other places as well.
2A Ab-initio ionic model
We base our calculations on an ab initio ionic model as a reference system. In this
model, the crystal energy E (which is a function of the configuration (R)of the ions)
is given by a sum of pair potentials:
The constant energy Eo comprises the self-energies of the rigid ions. In Eq. (I), 6
numbers the unit cells in the crystal, a and p are sublattice indices, and R! = R6 +
The pair potentials Gap (that depend on the distance of the ions only, see below) are
calculated by the method of Gordon and Kim [13]. In the Gordon-Kim method one
(i) assumes that the electron density of an ion pair is given by the superposition of the
individual ionic densities (i.e., no distortion occurs) and (ii) calculates the correspond-
228
Ann. Physik 3 (1994)
ing energy within the local-densitiy approximation (LDA). In particular, the kinetic(single-particle-) energy functional Ts [ p ] is treated in the Thomas-Fermi approximation (i.e., is not evaluated by solving the Kohn-Sham equations). Using, moreover,
spherical ionic densities leads to central forces between the ions.
Two problems arise immediately when we wish to apply this model to the cuprates.
The first concerns the ionic charges [14]. In La2Cu04,the use of the nominal charges
)
to be reasonable, though it is by no means necessary. Ac(La3+, Cu2+,0 2 - appears
tually, the covalent character of the C u - 0 bonding suggests to use somewhat dethe situation is worse: assigning the nominal charges
creased charges. In 0 6 and 07,
to the ions (Y3', Ba2+, Cu2+, 0 2 - )conflicts with charge neutrality. When we look
for a set of ion charges in accordance with charge neutrality, the copper charge is most
' . 07,the restriction to inlikely to be at disposal. In 06,we take Cul" and C U ~ ~ In
teger values would suggest the assignment Cu13+ and C U ~ ~but
+ ,Cu3+ is unlikely to
occur in the superconducting oxides [2]. Hence, we choose Cul and Cu2 to have identical (noninteger) charges 7/3, as did Cohen et al. [6] in their so-called potential-infor
duced-breathing (PIB) model. We have also tried a reduced copper charge, 1.61 iCul and Cu2 in O7 as well as in O6 (requiring Oi.69- in O7 and Oi.972-in O6 with the
Y- and Ba-charges fixed), according to the findings of Krakauer et al. [2] who get the
result that C U ' * ~ leads
~ + to the best representation of their self-consistent LAPW
charge density by overlapping spherical ion densities. However, the use of these reduced charges does not significantly improve our results for the phonon dispersion, but
does, on the contrary, increase the number of unstable phonon branches, as compared
to the calculations with the values of the charges mentioned first.
The second problem that arises is that a free 02-ion is unstable and is stabilized
by the environment in the crystal only, However, this problem is easily overcome by
a standard procedure which consists in enclosing the 02-ion in a so-called Watson
sphere (151 of opposite charge and a radius chosen to give (inside the sphere) the
Madelung potential at the ion site. For simplicity, a Watson-sphere potential of 1.5 Ry
has been used for all oxygen sites.
The (spherical) ionic densities are calculated with a modified version of the HermanSkillman program [ 161 including, in particular, averaged self-interaction corrections
[171.
The long-range Coulomb part is split off from the pair potentials and is treated
exactly using the Ewald method. The remaining short-range part, d, is calculated
numerically on a mesh of ion distances, R , and is fitted by the generalized BornMayer-type expression
d ( R )= cy+e-fi+R-.-e-fi-R .
(2)
The mean relative error in the fit is usually quite small (typically of the order 0.1 - 1%
for an R range of 1 a.u. around the equilibrium ion distances).
The crystal energy, Eq. (l), is then minimized with respect to the structural parameters, yielding the equilibrium structure for the subsequent phonon calculations.
The dynamical matrix is easily set up using the Ewald method for the long-range
Coulomb contribution to the pair potentials and employing Eq. (2) for the short-range
interactions.
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
229
2 B Charge fluctuations as adiabatic degrees of freedom
In the adiabatic approximation the electron density in a crystal depends parametrically
on the locations of the atoms. This dependence or, more concretely, the change in density upon a (unit) displacement of an atom, is of central interest in microscopic lattice
dynamics. In view of the mixed ionic-covalent (metallic) nature of the HTSC and
remembering the findings of Cohen et al. [4], see Section 1, we expect that in the HTSC
there will be, in particular, density changes of CF type. In a physical picture, one can
imagine that electrons are added to or are taken away from the copper-d- and oxygenp-shells, respectively, of the overlapping ions making up the crystal. Thus, density
changes having the shape of the corresponding orbital densities will result. The same
picture arises when we start from a tight-binding description and ignore all contributions to the density except the self-overlap contributions of the tight-binding functions.
Formally we can account for this change in density by parametrizing the electron density as follows:
Here the first argument, R , that is a shorthand notation for the complete set of the
locations of the nuclei, [I?:], denotes an explicit dependence of p on the coordinates
of the nuclei. In the present case, this comprises the density change corresponding to
the rigid displacement of the ion densities with the nuclei what just constitutes the rigidion model (RIM). The second argument, = (<:), denotes a set of “generalized coordinates” corresponding to some appropriate degrees of freedom of the density (such
as, e.g., CF), the changes of which are not explicitly prescribed with respect to the motion of the nuclei, but are to be determined implicitly from the variational principle
for the energy. 2 denotes the unit cell the effective electronic degree of freedom (EDF)
(: is associated with, and K numbers the different EDF within the unit cell. The
dependence of p on the space point i is suppressed in the notation of Eq. (3). According to Eq. (3), the displacement-induced change in density, p:(fl 1181, is given by
<
[P:(J)IRIM
denotes the explicit change in density associated with the RIM. The densities pK(F) that describe the shape of the change in density associated with the EDF
are referred to as “form factors” in what follows. As indicated above, we identify the
form factors with the copper-d- and oxygen-p-orbital densities in our model. The quantity X that expresses the reaction of the EDF upon an ion displacement is given, in a
shorthand notation, by
Here, the coefficients CZtl describe the mutual interaction between the EDF. They
are defined by
230
Ann. Physik 3 (1994)
where E(C,R) is the function that results when Eq. (3) is inserted into the HohenbergKohn energy functional.
The quantities B Y (A=Za) describe the interaction between the EDF on the one
hand and the ions on the other hand, according to their definition:
A parametrization of the eIectron density of the above kind is obtained, e.g., by making the ionic charges variable in our ab initio ionic model, i.e., C = (Zz),where 2: is
the ionic charge of the Za-ion. Like in the ionic model it is supposed that the total electron density in the crystaI is given by the superposition of the individual spherical ionic
electron densities where the charge of the latter may now vary in response to the
displacements of the ions in a lattice vibration. The energy of the system in the pairpotential approximation, Eq. (l),
then depends on the ionic charges, and we may, in
principle, calculate the quantities C and B according to Eq. (6) and (7), respectively,
from Eq. (1). Note that also the ionic self-energies (i.e. the energies of the individual
ions), that have been ignored up to now, have to be considered in the present case since
they depend on the ionic charges. In practice, the kinetic energy is excluded from the
pair potentials and the ionic self-energies in calculating the interaction C, and is treated
in a different manner (see below). In varying the ionic charges we ignore relaxation effects, i.e., we do not re-calculate self-consistently the ionic wave functions for each
value of 2, but do simply put electrons in or take electrons away from the fixed outmost shell. The form factors (shape of the density variation) are then given by the
(spherically averaged) orbital densities (wave function squared) of the corresponding
shells. We allow for charge fluctuations at the copper- and oxygen ions. The density
changes thus will have the shape of the (spherically averaged) Cu-3 d- and 0-2p-orbital
densities, respectively. Y and Ba, on the other hand, have fixed charges in our model,
i.e., no charge-fluctuation degrees of freedom are associated with these ions.
It is convenient to write Eq. ( 5 ) in a form familiar from ordinary density response
theory:
X =RE-’B
with
&=l+Vn.
In Eqs. (8) and (9) we have introduced a “polarizability”, n, describing the kineticenergy contribution to C.The Coulomb- and exchange-correlation (XC) contributions,
on the other hand, are contained in the effective electron-electron interaction, V, that
is defined analogously to C in Eq. (6), but with E replaced by its Hartree- and XC part
only. This separation is especially useful if we want to discriminate between thescreening properties of a metal and those of an insulator, the differences being contained in
n. Instead of refering to the direct space representation (where, e.g., n is a matrix with
respect to the indices ZK and 6~‘),
Eqs. (8) and (9) are valid for the Fourier-transformed quantities (depending on the wave vector 4) equally well. The quantities n, E ,
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
23 1
and Pare then 9 x 9- or 10X 10-matrices,respectively, (9 CF degrees of freedom in the
unit cell in 06,10 in 0,) and the quantities B and X are 9 x 12 (10 x 13) matrices (12
atoms in the unit cell in 06,13 in 07).
Let us now exemplify, using the zone-corner oxygen breathing mode in La2Cu04as
an example, how the charge-fluctuation degrees of freedom work. In the mentioned
mode, the plane (and, to a lesser extent, the apical) oxygens are vibrating symmetrically
against Cu. Suppose, the oxygens are moving towards a certain Cu site. Since the oxygens are negatively charged, the potential seen by an electron will be raised at this site.
Consequently, electrons will leave the corresponding Cu-3d orbital and go to the
neighboring Cu where the oxygens are vibrating away and hence the potential is
lowered. This relaxes the rigid-ion Coulomb energy, and hence a decrease in the vibration frequency results. The situation just described should occur for a dominant,
positive on-site polarizability of the Cu ion. In general, ( - x:$) relates the CF at ZK,
&:', to the screened change in potential at 6 ~ ' &$,
,
(Sc = - n.64;6c<O means that
electrons flow onto the ion, & > 0 means that the change in potential is attractive for
electrons; note that the d@include, in addition to the usual Kohn-Sham potential
(Hartree- and XC contribution), a kinetic-energy contribution as well, see also
Eq. (23); the same is true for the "bare" potential changes related to the quantity B ,
see below). Most intuitive are the on-site polarizabilities which relate the CF to the
screened potential change on the same ion. Inter-site polarizabilities must necessarily
be taken into account in an insulator: the CF have to vanish asymptotically for a longwavelength screened potential in this case (see Eq. (1 l)), so that the inter-site contributions from the neighboring ions have to compensate the on-site contribution for each
ion. The quantity P:, describes the Coulomb- and XC-interaction energy of a pair
~ 6~'.
Its inter-site contribution is dominated by the long-range
of CF excited at i i and
Coulomb interaction. In our example there will be a gain in Coulomb energy due to
CF of opposite sign at neighboring Cu ions. The on-site contribution,
U,, is
repulsive and counter-acts the occurrence of CF. It is mainly due to the Coulomb
energy associated with the CF form-factor densities. U, determines (besides n ) to a
large extent the size of the CF and hence the renormalization of the phonon frequencies. The larger the on-site interaction is, the smaller are the CF. Finally, the quantity
BF-describes essentially the (bare) change in potential at the 6~ CF site when an ion
at A is displaced in i direction. (More presisely, it is the change in potential averaged
over the 6~ CF form factor.) Thus, B P A gives the driving force for the 6~CF due
to a unit displacement of the ion at A.E - ~ isB then the screened change of the site
potentials in response to a unit ion displacement. The dominant contribution to B
comes from the long-range Coulomb interaction. In our example, the potential at those
Cu sites, where the oxygens move towards Cu, is increased, and at those sites, where
they move away, is decreased. At the displaced plane-oxygen ions themselves, the
change in potential is vanishing because of symmetry, and no CF occur at these ions.
The long-range Coulomb contributions to C and B are easily extracted from Eqs. (6)
and (7). Their contribution to the dynamical matrix (see below) is treated exactly using
the Ewald method. The short-range (overlap) contribution to B is estimated by evaluating the short-range part of E(c,R)in the pair-potential (Gordon-Kim) approximation
for different values of c and R. In the case of only the self-terms, P:t, are significant
cant and are estimated by calculating the Hartree- and XC-energy of the ions as a function of the outmost-shell (Cu-d, 0 - p ) occupation number (keeping the orbital densities
fixed). Actually, we multiply the results taking into account the Hartree contribution only
by a factor 3/4 in order to get our final parameters. We thereby interpolate between the
ct",..
<
232
Ann. Physik 3 (1994)
two extremes of the pure Hartree approximation on the one hand and the Hartree-Fock
approximation on the other hand, where the latter would yield exactly half the Hartree
results. The LDA-XC contribution, on the other hand, only slightly reduces the pure
Hartree results.
The kinetic-energy contribution, contained in the quantity n in Eqs. (8) and (9),is
rather hard to treat on a first-principles basis if one wants to go beyond the simple
Thomas-Fermi approximation. We therefore identify n with the polarizability obtained
in a tight-binding approach [ 181 restricting to the self-overlap contributions [12].
In Ref. [12], we employed a simple tight-binding band structure in order to calculate
an estimate for n(4f)in La2Cu04. Here, we make use of a formula relating the tightbinding partial densities of states (PDOS) at the Fermi level to the tight-binding polarizability at zero wave vector:
We assume n to be constant (as a function of the wave vector) and diagonal (with
respect to K and K'), i.e., only the self-terms n z are considered non-zero in the directspace representation. Using the LAPW PDOS values of Ref. [2], we then derive
estimates for nzc in the metallic phase, 0,.
In view of the approximations made especially for the quantity n, but also for the
short-range contributions to B and P (where, in the case of B, the Thomas-Fermi approximation has been used for the kinetic-energy contribution) it should be emphasized, however, that the renormalization of the phonon dispersion curves is, at least
qualitatively, rather independent of the exact parameter values in the case of the
metallic phase, and our main findings, such as the drastic renormalization of the Ag
(023; 2) mode (see the following paper), are obtained even using different parameter
sets. For example, we have also calculated n within the Thomas-Fermi approximation
for the metallic phase and have not found any significant qualitative differences in the
phonon dispersion. In the case of the insulating phase (see below) the detailed renormalization features depend on how the metallic polarizability is generalized by adding
intersite couplings in order to satisfy the insulator sum rule (Eq. (11)). Due to the interference with the intersite polarizabilities the renormalization is more sensitive to the
parameter values chosen for the on-site polarizabilities here.
Next, we show how to discriminate between metallic and insulating behavior within
our approach. Noting that the density of states at the Fermi-level vanishes in an insulator, we see that Eq. (10) holds putting ZK(EF)formally to zero on the left-hand
side in this case. More precisely, the proper long-wavelength limiting behavior of the
density response requires
as 4f+6 in an insulator (see Appendix). This means that there are necessarily some
of different EDF, Z K # 6 ~ ! ,in an insulator.
couplings, ,z:
In our model, we introduce a Cu2 - 0 2 3 coupling, n (Cu2 - 023), a Cu2 - 0 4 coupIing, n (Cu2 - 04), and a Cut - 0 4 coupling, n(Cu1- 04), in order to describe the inhaving thus coupled all EDF in the unit cell, at least indirectly.
sulating phase, 06,
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
233
The following relations must hold according to Eq. (1 1a):
Z(CU2) + 4 ~ ( C u -2 023) + Z(CU2 - 04) = 0 ,
(12a)
~ ( 0 2 3 ) + 2 * n ( C ~ 2 - 0 2=
3 )0 ,
(12b)
~ ( C u l ) + 2 . ~ ( C u l - 0 4=) 0 ,
(1W
+
~ ( 0 4 ) n(Cu1- 04) + ~ ( C u 2 04) = 0
.
(124
z ( C U ~ )z, (023), z (Cul), and z (04) denote the self-terms. There are totally 7 parameters, 3 of which are independent in view of the 4 equations, (12a) - (12d). We take over
O7 and fix the remaining parameters by the above
n(Cu2), n(023), and ~ ( 0 4 from
)
conditions.
The Fourier-transformed quantities B y ( 4 ) and
( 4 ) are defined as follows:
vKK,
. long-range
The expression for nKK8(Q)is analogous to that one for v K K t ( d )The
Coulomb contributions to B and are given by
v
and are dealt with in Eq. (13) and (14), respectively, by the Ewald method. Z , is the
ionic charge of the a t h ion. The short-range (overlap) contribution to B, as derived
from the pair potentials, has the form
with
Here, p denotes the ion type the CF degree of freedom K belongs to. Only B,,(R)
for some neighbors considered most important are taken into account, namely
Cu2 - 02/03, Cu2 - 04, Cul- 04, interplane 0 2 - 0 2 / 0 3 - 0 3 , and, in the case of
07,
Cul- 01. The derivatives occurring in Eq. (18) are evaluated numerically. In the
case of the short-range contribution to only the on-site contributions are taken into
account. The Hartree contribution of the latter, which is multiplied by 3/4 to get the
final value, is given by
v,
234
Ann. Physik 3 (1994)
where (Ea)His the Hartree energy of the electron density of the a th ion, and where
a is the ion type associated with the CF degree of freedom K . Although ( I Y ~
can) be
~
calculated directly from the Hartree energy of the orbital density of the CF shell,
Eq. (19) has the advantage that it is appropriate also for taking the XC contribution
into account. The derivative is performed numerically. As already mentioned above,
are taken into account in xKK.(Q)in the metallic case, and
only the self-terms, n:,
only a few additional inter-site terms in the case of an insulator. The values of :~7 for
the metallic phase are given in the caption of Table 6 in the following paper.
Including the contribution from the EDF, the dynamical matrix takes the form
The first term on the right-hand side of Eq.(20) is the contribution of the reference
system, i.e. the RIM.
The CF amplitudes are obtained from the quantity X (Eq. (8)) by multiplying it by
the displacement amplitudes u4 (Go) (a : branch index):
sl;? = - c X~(Q).ua(-a),iq(R"-da)
1 4
ai
The displacement amplitudes have been normalized as follows:
o&) and P(Qa) are the phonon frequency and corresponding eigenvector for
branch a and wave vector 4, respectively.
Analogously to the CF amplitudes, Bl;?, we may define screened site-potential
)
of X in Eq. (21). It may be shown that
changes, B@?, by using ( - - E - ~ Binstead
where p , ( f ) is the form factor of the EDF K , and sVef,(CQa) is the change in the selfconsistent potential (Kohn-Sham potential) associated with the phonon mode ija. The
second term on the right-hand side of Eq. (23) is a shorthand notation for an expression
of the form of Eq. (21) with Xreplaced by ( -Bkin),Bkinbeing the kinetic-energy contribution to B . Apart from this kinetic-energy contribution, S@? thus describes the
weighted average of the change in the self-consistent potential, with the form factor
of the EDF d~ as a weighting function.
Before actually calculating the renormalization of the phonon dispersion curves by
the CF we may ask first, which of the phonon modes are (potentially) affected by the
CF in view of symmetry, and which are not. In more detail, we may ask, which of the
phonon modes do couple to a definite CF degree of freedom. In order to settle this
question we have to determine the irreducible representations of the group of the wave
vector Q being contained in the representation spanned by the different CF degrees of
freedom. The results of such considerations are summarized in Tables 1 - 3.
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
235
Table 1 Selection rules for the charge fluctuations in several directions in the Brillouin zone of O6
and 0,. The entry ‘‘all’’ means that all CF allowed for (i.e. CF at the copper- and oxygen-ions) do
couple to phonons of the corresponding symmetry. Phonons of other symmetries than included in the
table do not couple to the CF, nor do phonons of symmetries with missing entries. The numbers 1 - 4
in the top panel denote the irreducible representations. The row labeled “A (Z)” refers to the A-directhe row labeled “Z/r- S” refers to the Z-direction in 0,and to the A- as well as Z-direction in 0,;
tion in 0, and to the r-S-direction in 07,respectively.
~~
O6
1
0 7
2
A (Z)
all
A
all
023
.E/F-S
all
023
3
4
1
all except
Cul
3
all
all except
cu 1
2
023
all
all
4
all except
chain
all except
chain
-
-
Table 2 Selection rules for the charge fluctuations at several high-symmetry points in the Brillouin
zone of 0,. The upper top panel refers to the r-,Z-, and M point, the lower one to the X point. E i
and E i ( E i and EY,) denote the x- or y-row, respectively, of the Eg(Eu)irreducible representation.
See remarks in Table 1.
A,,
A,
Bl g
r
all
02,03
X
all except
02
Z
all
M
Cul, Cu2,
04
E:
E:
A2u
B2g
BlU
all except
c u1
02
Ei
B2u
EY,
B3 u
02, 0 3
02
Cu2, 0 3 ,
04
all except
c u1
02,03
03
02
02, 0 3
Cu2,04
03
02
Table 3 Selection rules for the charge fluctuations at several high-symmetry points in the Brillouin
zone of 0,. See the remarks in Table 1.
A,
r
all
X
all except 0 2
Y
Cut, Cu2, 02. 0 4
2
all
S
Cul, Cu2, 0 4
B2r
4,
Bl u
B2U
B3 U
all except chain
02
Cu2, 0 3 , 0 4
02
03
Cu2, 02, 0 4
01, 0 3
all except chain
02
03
Cu2, 0 4
01,03
02
236
Ann. Physik 3 (1994)
Appendix
In this Appendix we will investigate the long-wavelength asymptotic behavior of some
important quantities entering our formalism. This is of interest because it is just the
limiting behavior for small wave vectors that distinguishes the metallic from the insulating phase.
Let us first consider the quantity
In order to determine its asymptotic behavior as q-+O we have to separate out the
singular component of the Coulomb interaction. This is given by the G = 6 term in the
Fourier-space expression for the Coulomb-XC interaction, P@). Here, and in what
follows, G denotes a reciprocal-lattice vector. We thus split into a contribution P
that is regular as 4-+6and the singular G = 6 term:
v
p,' is the row vector made up from the
PA4+
a,and
G = d components of the EDF form factors.
where V, is the unit-cell volume.
With Eq. (A2) 8 , as defined in Eq. (9), can be inverted to yield
c-l=
C-l-C-1 poo(l +up,+ C - l p o ) - * p , '
C-l
.
Let us define
pe-g
Eq. (A4) then can be written as
0
x=R-Rc-Rr
*
&W
By multiplying Eq. (A8) from the left by p,' and/or from the right by po we obtain
xi=fi/&, , i=o,r,c ,
(A 9)
M. Klenner et al., Charge fluctuations in high-7''-cuprates. I. The model
where xi is defined analogously to f i (Eq. (A6)) in terms of
Eq. (A9) for i = o in Eq. (A7), E , may be represented as
237
x instead of f . Using
In order to show that, in the limit d - 6 (and in the case of an insulator), this is indeed
the macroscopic dielectric constant, we first recall from Ref. [12] that (the EDF-derived
part of) the density response function (DRF) .is given by
(Eq. (A 11) is actually the Fourier-transformed version of Eq. (11) in Ref. [12].) In particular, we have
D,'D(ij,d)
= - v,-'pOfpo= - V,-'X,
.
(A 12)
Now, we know [19, 201 that the macroscopic dielectric constant may be expressed, in
the limit d - 6 , by
1 / ( 1 + 4 ? ' ~ q - ~ D, ~ )
(A 13)
which is, by Eq. (A12), identical with the right-hand side of Eq. (AlO).
In order to yie1d.a finite value for the macroscopic dielectric constant, E,, in the
case of an insulator, the quantity xo or, equivalently, f o clearly has to be O ( q 2 )as
ij-6 according to Eq. (A 10) or (A7), respectively. In order to prove that f is indeed
0 (q2) we have to distinguish between CF-type EDF (monopoles) and dipole degrees
of freedom. Higher multipoles do not contribute to E-. Although not actually included in our calculations we consider dipol EDF here. We decompose the vector po
into its monopole components, pio), being 0(1) as 4-6, and its dipol components,
phi), being O ( q ) . We assume the CF components to be normalized to unity,
pK(@+6) = 1, for K a CFEDF. Thus we have
20 = ( P
6"' +P 6")
+
f ( P bO) +p 6")
= [P6O'I
+
ad0)+ [P6"1+ fPhO'
+ [PbO'I + fP6') + [P6''1 + fP6"
*
(A 14)
The last term on the right-hand side of Eq. (A14) is clearly O ( q 2 )since p;') is O ( q ) .
In the case of the mixed terms we need
The latter equation follows from Eq. (1 1 a) in the case only CF degrees of freedom are
taken into account ,(this is the case considered in Section 2 B). It will, moreover, remain
valid even in the more general case considered here (including dipol EDF) if the summation in Eq. ( l l a ) is understood to run over the CF degrees of freedom only. E is
calculated with P instead of by definition.
Because f 'is hermitian the two mixed terms are simply complex conjugate to each
other, and 'because pL1)is already O ( q ) , it follows that they are O ( q 2 ) ,as required.
23 8
Ann. Physik 3 (1994)
In the case of the first term we make use of
and hence
Again, we have used Eq. (1 1 a). It follows that
where we now have applied Eq. (1 1b) additionally, generalizing it in such a way that
the summations again are understood to run over the CF degrees of freedom only. This
completes the proof that f , is O(q2).Using Eq. ( l l a ) and the fact that p f ) is O ( q ) it
can easily be shown in an analogous manner that f c and f , are O ( q ) as @-)a.In summary, we have
in an insulator as @+a.
In a metal, on the other hand, the left-hand sides of Eqs. ( l l a ) and (11b) tend to
finite (non-zero) values and f,, f c , and f , tend to finite constants as 4-6. Moreover,
we have E ,
q - 2 for small q in this case according to Eq. (A7), and thus &,+a as
@-+a.
-
Let us now examine the long-wavelength behavior of the quantity X = ne - B zs X B
(Eq. (8)). To this end we have first to separate out the singular (G= 6 ) term in B :
B is regular as a-6, and we have defined a vector b according to
bl = i q i Z ,
(A211
where 2, is the (bare) ionic charge of the a th ion and I = ia. bT denotes the transpose
of b.
Using Eqs. (A@, (A6a), and (A9) we find
x=f B + f , - (Ub T - f , B )
&a,
Defining
.
M. Klenner et al., Charge fluctuations in high-Tc-cuprates. I. The model
239
this can be written as
In the case of an insulator P T is O ( q ) (because f r is O ( q ) ; cf. Eq. (A19b)) and we
may define the transverse-effective-charge tensor 2,' by
The second term on the right-hand side of Eq. (A25) is then finite but non-analytic,
i.e., it depends on the direction of the wave vector 4. Eq. (A25) is analogous to the
equation describing the long-wavelength asymptotic behavior of the G 6 components
of the displacement-induced change in density, pa(4 + G) [21].
In the case of a metal, P T is finite and we clearly cannot define transverse charges.
The second term on the right-hand side of Eq. (A25) is analytic in this case because
the factor U / E , is.
Multiplying Eq. (A25) by p: from the left we get
+
X,=p;X
= bT-
&;lPT
(A271
where we have used Eqs. (A23), (A6b, c), (A24), and (A7). The second term on the
right-hand side of Eq. (A27) defines, via an equation analogous to Eq. (A26), the
longitudinal effective charge tensor Z: that.is, however, dependent on the direction of
the wave vector in general, because E , is.
From Eq. (A7) we see that X, is O ( q ) both in the insulating as well as in the
metallic case: in the insulating case E , is finite but P T is O(q);in the metallic case P T
is finite but E,'
q2 for small q; bTis clearly O ( q ) by definition (Eq. (A21)) in each
case. This just expresses the fact that (regardless which phase is concerned) the CF amplitudes sum up to zero in the unit cell as @+a, in accordance with charge neutrality.
In order to see this, note that the first equality in Eq. (A27) reads in full notation
-
(A 28)
and that the CF form factors are normalized to unity, i.e., p , ( g ) - , l as d + 6 .
Moreover, remember that X yields the CF amplitudes Sc by multiplication by the
displacements u (Eq. (21)).
For tetragonal symmetry the transverse-effective-charge tensor is diagonal and has
two independent components. These can be extracted, in view of Eq. (A26), from
Eq. (A27) by solving the latter equation for P T and multiplying it by (- iq')/q2in the
limit Q-+ 6 :
42x4 = q-a
lim [ E , ( d )[z, + i i j * 2 : ( ~ ) / q ~ ].)
(A 29)
4 is the unit vector in the direction of d .
In order to get the two independent components of 2,' we have to evaluate
Eq. (A29) for I c and Qllc.
240
Ann. Physik 3 (1994)
Next, let us show that the Zf obey the acoustic sum rule
c zf=g.
U
To this end we rewrite Eq. (A24) in a more explicit form:
NOW,we sum this equation over a. The bare ionic charges sum up to zero because of
charge neutrality,
C zu=o.
U
In the case of the quantity B we apply the translational-invariance condition
c B;a(Q=d)=O.
U
This equation states that there are no EDF excited if the crystal is displaced as a whole,
i.e., the change in density is then entirely due to the displacements of the rigid ions.
In our model Eq. (A33) is used to determine the on-site EDF-ion couplings.
Remembering that f r = O ( q ) and that B ( d ) is analytic as Q+6, we are left with
From this the acoustic sum rule, Eq. (A30), follows in view of Eq. (A26).
In Section 2B we stated that the quantity & - ' B ydenoted as Win what follows, is
related to the screened site-potential changes. Its long-wavelength asymptotic behavior
is given by
W=
U
@+(El-l),-p*
8
,
where J@ is the regular part of Wand (El-')= is defined analogously to f c (Eq. (A6a)).
Eq.( A 3 9 may be obtained from the identity
that follows from Eqs. (8) and (9), and using the decompositions given by Eqs. (A2),
(A20), and (A25).
From Eq. (A 17) it follows that
(E"-1),=pp+O(q)
(A 37)
in the case of an insulator, because pfp) may be replaced by po on the left-hand side
of Eq.(A 17), pL1)being 0 (4).
M. Klenner et al., Charge fluctuations in high-T,-cuprates. I. The model
24 1
Eq. (A35) then says that for an insulator there is a macroscopic potential, i.e. a potential being constant with respect to the different CF sites in the unit cell, that is set up
by the displacements of the transverse effective charges and is screened by the
macroscopic dielectric constant, E , , exactly, as is familar from conventional theory.
In a metal, on the other hand, the second term in Eq. ( A 3 3 is finite for q+O and
depends from the particular CF site in the unit cell.
Finally, we may convince ourselves that 2,'is indeed the usual transverse-effective-charge tensor. To this end we notice that Xo(Eq. (A27)) is related to the = 6
component of the displacement-induced change in density, Po,in the following way:
e
Po = PO,
RIM - Xo
(A381
where Po,RIM
is the rigid-ion contribution to Po and is given to leading order by
PO,
RIM)
7 = - iqi(z:- za
-t0 (q2) *
(A 39)
ZC, denotes the charge of the ion core of the Q th ion (i.e. the nuclear charge in an allelectron treatment). Thus we get with the help of Eq. (A27)
This is in accordance with the usual definition of 2
: [20, 211.
Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References
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