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Submitted on 1 Jan 1981
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RENORMALIZED RESPONSE THEORY WITH
APPLICATIONS TO PHONON ANOMALIES
C. Falter, Wolfgang Ludwig, M. Selmke
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n o 12, Tome 42, dgeembre 1981 page c6-516
RENORMALIZED RESPONSE THEORY WITH APPLICATIONS TO PHONON ANOKALIES
C. Falter, W. Ludwig and M. Selmke
I n s t i t u t fiir Theoretisehe Physik 11 der Universitat Muenster, Domagkstr. 7 5 ,
0-4400 Muenster, F. R. G.
Abstract.-A recently developed renormalization method for the electron respon- se problem in effective ion-interactions is applied to phonon anomalies. A mi- croscopic mechanism for the anomalies is proposed and its relation to the electron-phonon-coupling parameter X and thus to T is discussed.
The renormalization method proposed in [1,2] for various applications allows for a
division of the electronic density response function D into a part
6,
which renor-malizes the potential (and other quantities) thus leading to screened interactions and a part which acts in the renormalized system containing the relevant degrees of
freedom in a special situation. This method is able to isolate certain features of
the electronic structure like the Fermi surface effect in metallic compounds, the local field effect in covalent crystals or special many body effects. The characte- ristic items of a definite phenomenon are prepared by decomposing the polarizabili-
ty function n , which is defined in terms of quasiparticles, into a part 7 describ-
ing the screening (renormalization) and a complementary part A, containing the rele-
vant aspects of the response in a special case: a = F + A
.
(1)This division can be shown to achieve a renormalization of D, which is related to the interactions of bare particles. With Coulomb-interaction v and dielectric func-
tion E we obtain
D =
ij
+
(z-l)+~~;-~
; E-I
.=.
E -1 .E ,-I ; D-
=---I
T E (2a)-
1E = (l+cA)-' :
-
Dr = AE;' ; E = ~ + V ? ;
-
v = ,-l E v. (2b)In lowest order we can replace Dr + A
,
D g D+
(?-')+A?-'
.
( 3 )The effective ion-interaction between two ions at
2
Lz
[1,2] is given byI1 + -+
~(l,$) = V ( A , B ) - V1 D V$
,
(4)where V" is the direct ion-ion-interaction and V+ the ion-(pseudo-)~otential;
A
1
= (:,a) contains indices for unit cell and basis, resp.. The Fourier-transformedeffective interaction splits according to (4,2a) into additive parts, the anomalous one being 131 (Q1=
<+E'
; Q"=G+?'
etc.)@r a
B
-
1, =
-
Q
)
DrQtrQ18a (' ) 0 ~ 1v8(Q7)
9 (5)so that the renormalized part of the dynamical matrix can be written as (VZ: volume
+
of unit cell; G: reciprocal lattice vector)
-1/2 {*raB +
i j ij
(s)
-aaB
Y1
nrq;(b)
1
(6)with
*a
+,+B
,,rolB q -+ V -1
(;+8)i(G+?i1)j
ei(GR-GR).@r+9*-+B++- -+ q+G
,
qiG'
G.G1
The problem of phonon anomalies and especially the correlation of soft-phonon modes to high Tc is often dealt with but it is not solved in a unique way (see
@j
for a review). In [3,5,6] a general explanation for the phonon anomalies is proposed. As a minimal assumption for the use of the renormalization method in A we separate off the Fermi surface effect as the relevant part of the metallic binding component which we assume to be dominant in screening. The only characteristic property of a special-f
material which enters the theory are the small critical wave vectors q (nesting cr
vectors), which occur in Fermi surfaces with a special geometry (high density of states at the Fermi level E ~ , / 7 ] ) . We define A (eq. (l)) to be the difference in the polarizabilities of the anomalous and the normal system (e.g. TaC and HfC), one hav- ing Fermi energy + 6 ~ , the other one A = IT(E +&E)
-
T(E~).
F (8)
A contains all the effects originating from the different topologies of the Fermi surfaces in the two materials. In our model calculations [3,5,6] we use eq. (5) with a diagonal renormalization for the potentials and obtain, linear in A,
A can be shobm to be essentially the difference of the free electron polarizabili- ties IT FE
..,
a
~k A aFE ( E ~ + ~ E )-
T (E ) = -(vFE) .6k =-
FE F a k ~1
;+z
+In general A will have a pronounced q-space resonance structure at the q (or at cr
2k in the model, resp.) of the Fermi surface (Fig. 1). This resonance then occurs F
in the effective ion-interaction (9) and finally in the dynamical matrix (6,7). The magnitude of the anomaly effect is determined by the competition between the strength of the renormalized electron-ion-interaction and the density of states Z(E ), eqs.
-+ +
+
F(9,Io). If the resonance condition
lq+~ls
2kF is satisfied for G = 0 (i.e. small0
=
k~ Q2. Anomalous part of the dynamical matrix
C6-5 18 JOURNAL DE PHYSIQUE
critical wave vectors), we have strong potentials
7
[3,5,61 and thus an observableanomaly effect if Z(cF) is high (Fig.2). The corresponding frequency shift is Aw C
+
+
2
6
being the frequency of the reference system. The factor qi.qj in (7) tellsus, that the anomaly only occurs in LA-modes. In the case of a second sublattice (NaC1-structure) there is a coupling between LAILO- and TAITO-modes; then the fre-
quency-shift in (I ,o,o) is approximately (linearly in tr) given by
C -1
Aw, = (4;+) ;M+ tic
'
[ ~ ~ ( ~ k ~
-
+ Y2t:Cl}
9-
where tr denotes the anomalous components of the dynamical matrix (6), M stands for
metal, C for carbon, resp, and
+
denote optical or acoustical modes, resp.. Eq.(ll)shows the scaling of Aw, with the inverse frequency
jj,
of the reference system; this- -
explains the suppression of the anomaly in the optical modes relatively to the acoustic ones
We will now trace how the anomalies due to the Fermi surface effect enter the elec-
tron-phonon-coupling parameter A and thus the critical temperature T of a supercon-
ductor. According to [ 4 ) the (isotropically) averaged expression forCX is
2 <J >
2 -t
a
s c + +
+ 2X =
1
z(E~)+ ; <J > =<I
<kll V" ('-Ra) k' >l
>FS ; (Tc),cssODexp (-1 /A).(] 2) a Ma<w >aRa
The gradient of the total self-consistent crystal potential V: (harmonic theory) in
(12) can be calculated in linear response theory. Thus the same effects which pro- 2
duce the phonon anomalies enter <J > quadratically via A, eq.(8); the anomalous con-
-1 -1
tribution is given by [3,6] €-lv = E (F
(l-7A)
7
a r C1 (13)
This indicates clearly, that the squared matrix element <.TL> in (12) is reduced in
general compared to the reference system. On the other hand the high Z(sF)values of
the special Fermi-surfaces under consideration'produce an increase in X. Thus the 2
product Z(cF).<J > to a certain extent is limited or at least, it should vary less
2 2
than either Z(E~) or <J > ; but <W > likewise involves the electron-phonon-coupling
"
2and varies similarly as <J > (phonon-softening), so that we can expect an approxima-
tive proportionality between
X
and Z(e ) . These theoretical predictions are support-F
ed empiricallyon the basis of experimental data for the transition metals and com- pounds [8]
.
References.
-
1.C.Falter and M.Selmke, Phys.Rev. E, 2078(1980); 2.C.Falter, W.Lud-wig and M.Selmke, Phys.Lett.
g,
195(1981); 3.C.Falter, Eabil.Schrift (to be pub-lished); 4.S.K.Sinha, P.B.Allen, Dynamical Properties of Solids
2,
North Holland,Amsterdam (1980); 5.C.Falter and P!.Selmke, Phys.Rev.
B24
(to appear 1981); 6.C.Fal-ter, W.~udwig and M.Selmke, Phys.Lett.
A
(to be published); ?.B.M.Klein, L.L.Boyerand D.A.Papaconstantopoulos, Solid State Corn.
