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NON-LINEAR DYNAMICAL EXCITATIONS IN SOLIDS
A. Maradudin, A. Martin, H. Bilz, R. Wallis
To cite this version:
A. Maradudin, A. Martin, H. Bilz, R. Wallis. NON-LINEAR DYNAMICAL EXCITA- TIONS IN SOLIDS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-137-C6-139.
�10.1051/jphyscol:1981641�. �jpa-00221577�
JOURNAL D E PHYSIQUE
CoZZoque C6, suppZ6ment au nO1 2, Tome 42, dgcembre 1981 page C6-137
NON-LINEAR DYNAMICAL EXCITATIONS IN SOLIDS
A.A. ~aradudin*, A.J. Martin, H. Bilz and 'R.F. wallis*
w ax-PZanck-Institut fiir FestkBrperforschung, Heisenbergstrasse 1, 7000 Stuttgart 80, F. R.G.
Abstract.- Vibrational excitations can exist in non-linear sy- stems which have no counterpart in the corresponding linear sy- stems. In this paper we investigate such excitations in systems where the non-linearity is of an on-site character. In particu- lar, we analyze the existence conditions for localized (and de- localized) excitations, and their interaction with external fields.
In this paper we discuss the properties of non-linear periodic lattice waves in models of perfect and imperfect crystals. Our model Hamiltonians contain a fourth-order on-site electron-ion potential+
reminiscent of I$~-models. This approach is motivated by the success of the self-consistent phonon approximation (SPA) for the investigat- ions of ferroelectric-type phase transitions in terms of such models
(refer to Bussmann et al., this Conference).
2. General Solution to the Dynamical Monomer Problem
The monomer (Fig. 1 ) consists of one anion and one cation. If the cen- ter of mass motion is factored out, the sy- stem is described by two coupled variables, which may be chosen as the displacement, u, of the cation relative to the anion core and the displacement, w, of the anion shell rela- tive to its core. The oscillation frequency of the monomer is displayed
-
in Fig. 2 as a function of wg, where wo is the maximum dis-2 2 placement from the origin. The range w >wcrit
0
Fig. 1 The monaner unit.g4 belongs to anharmonic oscillations which tra-
IS the fourth Order verse the origin w=o (the solution at wo=O. 2 4 constant.
is the only harmonic one). The range w: describes oscillations which are localized in one or another of two equivalent potential wells. The minimum squared displacement from the origin increases from zero to the left of wCrit until it coincides 2 with :W at -g2/g4 (notice that the frequency does not go to zero at this point).
"Permanent address : Department of Physics, University of California, Irvine, CA 92717, U.S.A.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981641
JOURNAL DE PHYSIQUE
3. A Nonlinearly Polarizable Impurity in a Linear Chain
1 .o We now turn to the consideration
t of a problem in which a nonlinearly po-
w larizable atom is present as a substi-
tutional impurity in a monatomic linear chain (Fig. 3). We seek a solution for the displacement field in the chain that decays exponentially with increasing di- stance from the impurity site, i.e. a w:- localized mode.
Fig. 2 Frequency vs. squared ma- We can obtain an exact solution to xh m n p l i ~ e . ~ / ~ . 2 , g2/g4 = the equations of motion of the chain -0.2, w is inunits of (2/3) x with the
Ansatz
( f / p ~ ~ / ~ , where~ is thereduced = A coswt
0 (3.2a)
mass, and a =3g4>o, B-f+g2, f >o.
u 0 = ~ ( ' ) c o s ~ t + ~ ( ~ ) c o s 3 ~ t (3.2b)
u (-1
ln[c(l
)e-nhcosUt+~(3)e-np cos3wtJn>l
- ,
(3.2~)where wo is the relative shell-core dis- placement of the impurity atom, and un is the displacementment of the nth core, when we assume in addition that u - ~ = u ~ . In Eq. (3.2~) the quantities A and p
are the real quantities given by A
-2 -1 0 1 2 =2cosh-I (w/oL)
,
p=2cosh (3w/wL)-
1,
wherewL=(4y/M) is the maximum normal mode frequency of the unperturbed chain. The Fig. 3 A nonlinearly polarizable frequency of the localized mode must impurity in am0na-c therefore be larger than wL for the so- chain. lution (3.2) to exist. The equation de-
termining w is
With w determined from this equation the amplitude A is found from the relation
where X =[Mu2-f-Y (l+e-')l/
[
Mu 2 y (1+e4 ).
It should be pointed out that for g2<o, Eqs. (3.3)- (3.4) have a solution even if M'=M and f=y provided that
1
g21/y> 2.8. In this case the localized mode has no counterpart in the harmonic approximation.4. Response to External Electric Fields
If the periodon solutions to the nonlinear equation of motion of a crystal containing atoms or ions possessing nonlinear polarizabilities are to be observed experimentally, it is likely to be through features they contribute to the functions characterizing the response of such crystals to external probes. We have therefore begun a study of the pe- riodon-type solutions of the equations of motion of a monatomic linear crystal of this type (Fig. 4), in the presence of an external spatially and temporally varying electric field.
We consider an electric field given by En (t) =Eocos (n$ - w t )
.
Inthis case the equations of motion of the crystal have the exact so- lution
w (t) = A cos (n@-ut)
n (4.1)
provided that the frequency and wave vector
1 ' 1 ' 1 ' 2
are related through u = 4 (f+f I ) /9M sin2
$$.
We then have that the amplitude A is obtained in terms of the electric field amplitude Eo from the equation
Fig. 4 A mnatcmic d i n e &
cham.
while
3 3 - Eo-g2 A
-
g4A- g4 A3
B ( ~ ) MU 2 -4f1sin2
f
4 B(3)=-
16f sin 2 3 3$ 'where Z is the charge on each core and -Z is the charge on each shell.
Generalizations of this solution can be obtained by adding higher odd harmonics to the right hand sides of Eqs. (4.1). However, in that case a solution in finite terms, like that given by Eqs. (4.1 )
