HAL Id: jpa-00246996
https://hal.archives-ouvertes.fr/jpa-00246996
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Fluxes in incommensurate crystals
A. Levanyuk
To cite this version:
A. Levanyuk. Fluxes in incommensurate crystals. Journal de Physique I, EDP Sciences, 1994, 4 (9), pp.1353-1364. �10.1051/jp1:1994113�. �jpa-00246996�
Classification Physi<.s Abstia<.ts
61.60M 64.70K 64.90
Fluxes in incommensurate crystals
A. P. Levanyuk
Departamento de Fisica de la Matena Condensada, Universidad Autonoma de Madrid, 28049 Madrid, Spain
(Received 2J Maic.h J994, ieceived in final foi-m ii May J994, a<.cepted J8 May J994)
Abstract.-The interaction between varions fluxes and the motion of incommensurate(1) modulation is treated within a phenomenological approach. Three types of fluxes are considered : the polarization current, the conductivity current and the heat flux. It is shown that in the I phase ail
currents are associated with the motion of the I modulation and thus phenomena qualitatively
similar to those well known for the electric current in the charge-density-wave systems (low~
frequency~ and electnc~field~dependent conductivity) are predicted for ail the I systems and
varions fluxes. Some anomalies of the relevant kinetic coefficients near the normal I transition
are aise expected. The observability of the predicted phenomena is discussed for I dielectncs.
1. Introduction.
According to the general philosophy of the Landau theory of phase transitions the new properties which arise due to a lowering of symmetry at a transition can be understood, at least
qualitatively, starting from the symmetry properties of the order parameter only. This seems not to be the case for the incommensurate (I) systems : a special class of the I-systems, the
charge-density~wave (CDW) ones (see, e.g., Il ), exhibits such phenomena as the motion of the I-modulation by an extemal electnc field and non-linear conductivity (due to depinning of the CDW by the field) which were never discussed, to the best of my knowledge, for I dielectrics which are mdistinguishable, as to the symmetry properties of the order parameter for normal (N)-I transition, from the CDW-systems. I am also not aware of a discussion of the
above-mentioned properties of CDW systems m a truly Landau manner, I,e. startmg with an
expression of the free energy (or other relevant functions, see below) for the N phase. The interaction energy of the externat electnc field and the CDW-modulation, let it be one-k one
with the wave vector k along the x-axis, is wntten Il as ~E,, where ~ is the phase of the
modulation wave and as to the N phase the form of this term remains unspecified:
~ loses its meanmg there.
The motivation of this paper is the observation that if the interaction term is wntten as
#j, where j is a vector with properties of a current (1.e, it changes its sign at the time reversai) the interaction can be treated in a conventional Landau manner, i e. it can be denved from
some interaction relevant to the N phase. Indeed, if
1~~ and 1~~ are the order parameter components, and for the I phase
1~ = p cas q~, 1~~ = p sin ~, the above expression can be written as (l~j12 ii 'J2)j., " P~ #j,, the scalar character of this term follows from the
existence of the Lifshitz invariant :
1~ (61~2/à>. (ôl~ j/ôx 1~ ~. In other words some anharmon-
ic interaction in the high~symmetry phase becomes harmonic in the low-symmetry phase as in many other cases.
The above interaction is relevant to all the I-systems, including, of course, the CDW-ones but I am not in a position to discuss the latter and shall restrict the analysis to I-dielectrics where the interaction term q~E, is clearly absent. The aim of the paper is to discuss the
phenomena which can be predicted starting from the knowledge of the existence of the above interaction. It will be shown that phenomena similar to the above mentioned for CDW-systems
can be observed, in principle, in I-dielectrics if the concentration of free charge carriers is non-
zero. More importantly, the interactions of the above type bring about some phenomena which
are observable, at least in pnnciple, in the absence of the free charge carriers, it deals with dielectric fosses and thermal conductivity or diffusion.
Let us emphasize that, as the interaction term introduced above contains lime-derivatives, it
is clearly not present in the Landau free energy but rather in something like a dissipative
function which provides dissipative terms in equations of motion. It is convenient to begin the discussion of this question with the case where j,
m P~ where P is the electric polarization
vector; this case is considered in section 2. In section 3 the interaction between electric conduction current and the I-modulation is treated. Section 4 is devoted to the discussion of the thermal conductivity in the I-phase and in the N-phase close to the N-I phase transition (T
= T,). In section 5 the results of the paper are summarized with an emphasis on the
observability of the predicted phenomena, and the relevant estimates being made in the respective sections as well as in Appendix1.
2. Dielectric fosses in the I~phase and close to (.
To obtain phenomenologically the kinetic equations for the order parameter and other variables of interest one can well apply the symmetry arguments to these equations or address these
arguments to the Landau free energy (which will then be considered as the potential energy) as
well as to the kinetic energy and the dissipation function. The coupling between the order parameter and other (« non-cntical ») variables is descnbed usually by terms originating from the free energy. One can refer, e,g. to the 8tudy of acoustic anomalies near phase transitions
(for recent references see, e.g. [2, 3]), though coupling terms origmating from the dissipation
function are possible, of course, if allowed by symmetry. Let us recall that the latter
« dissipative » or « viscosity » coupling had been repeatedly considered within the so called
mode-couphng theory [4, 5].
The coupling which is considered in this paper is of the same type with the only difference that it is anharmonic, in the N phase, unlike that treated in the mode-coupling theory. This
creates a semantic difficulty the dissipation function is quadratic in velocities but the term
introduced above contains in addition the order parameter components. Let us call the
corresponding function the « generahzed dissipation » one allowing it to contain the terms of the above type and having in mind that when the order parameter acquires a spontaneous value the « generalized » dissipation function converts into a conventional one. It is important to
realize that the introduction of the new function is a matter of convenience and does not imply
any additional assumptions : one could wnte the anharmonic terms in the kinetic equations
using the symmetry arguments only and then take mto account the Onsager relations
considenng the lineanzed equations for the low-temperature phase (see Sect. 4).
To write the kinetic equations for the order parameter and the polarization component P, m P we use the density of the Landau free energy :
and of the generalized dissipation function
At the moment, we ignore the kinetic energy because we are interested in the low-frequency properties.
Neglecting first the fluctuations one has for the I-phase
oeP + VF + bp~ qi
=
E (3)
y# + bP
=
0. (4)
Thus, due to the movement of the I modulation wave, the coefficient v becomes renorrnalized and for the renormalized coefficient, P, one has :
P
= v b~ ~~
= v f ~/ (5)
Y a
where a is the lattice constant, f is a dimensionless coefficient which is expected to oscillate from material to matenal and is, on average, of the order of magnitude of unity. The estimate of the coefficients which were used to obtain the last equality are adduced and discussed in
Appendix I.
As p~-a~(T,-T)/T~, and a~(T,-T)/T,, for displacive and order -disorder systems respectively (T~~ is the « atomic » temperature (10~-10~ K)) one sees that although it does not
seem to be impossible to observe the phase-movement renormalization of the coefficient
v one has to be careful m choosmg the appropnate materai to be sure that f » 1, which seems to be possible (see Appendix I), Wé also see that, due to the effect of the phase movement, the
coefficient v acquires an additional temperature dependence. Actually, a similar effect may be due to a more general reason : existence of the term (1~ + l~j) P~ in the generalized dissipation
function but it is possible to discriminate between the two contributions. Indeed, the specific
feature of the phase movement contribution to any properties is the phase-pinning effect, e
the renormalization due to the phase movement is observable m perfect crystals of large
enough sizes (because the boundaries pin the phase as well) or at large enough amplitudes of an extemal field due to depinning of the modulation wave while the renormalization due to the term (1~) + ~Jj)P~ is far less sensitive to the real structure of the crystal and does not imply
nonlineanties. To describe the effect of pinning in somewhat more detail we generalize equation (4) in the same way as for the CDW-systems [1], and we have
p#+y#+pw/v~=-bÉ (6)
where pinning is represented by an average pinning frequency w~ and the inertial term is taken into account, p is the « phason » or, rather « optical » mass density which is of the same order of magnitude as the conventional mass density. One sees that the inverse electrical
susceptibihty of the I-phase acquires the forrn :
ce + v w +
~~ ~~~°~
(7
P (wo w + iyw
This form is oversimplified, of course, because it is a clear oversimplification to represent the
pmning by a smgle frequency. Moreover, the pinning force may become anharmonic at
relatively small amplitudes so that the polarization response of the I-phase to an harmonic electric field is expected to contain frequencies other than that of the field, m obvious analogy
with the current oscillations in the CDW systems Il I am not aware of relevant experimental
data on I-dielectrics.
The interaction described by the second term in equation (2) brings about some anomaly,
near T,, of the complex dielectnc constant due to the order parameter fluctuations. To get a
general idea about them it is quite enough to treat them within the theory of the first fluctuation corrections to the mean-field results. For a recent example of the use of this approach to treat kinetic phenomena see references [2, 3]. As this approach is quite simple it is presented beiow.
To consider the N phase one has to start from the equations which are obtainable from
equations (1) and (2)
OE~ + "~ + b(lJj 1J2 Ill 1Î2)
~
E (8)
P41+ Y41+A~~ -Dv2~j -b~2P
=
0 (9)
Pi~ + Yi~ + Ay~~ D v2~~ + b~j P
=
o. (io)
The higher order terms m equations (9) and (10) are neglected as long as we are considering the
N-phase and restncting our consideration to the first order fluctuation corrections.
Equations (9) and (10) being applied to fluctuations (their right-hand sides contain the
Langevin forces in this case) allow us to find the change m fluctuations due to the time-
dependent polanzation which is considered here to be space-homogeneous. From (9) one finds (for the Fourier components)
b~ )°1(k, w fl 1RP
n
1~ (k, w
= V
~
I I -pw +A(k)+iyw
where V is the volume of the system, A(k)
= A +Dk~ and the superindices (0) and (1) designate the unperturbed fluctuations and the first order perturbation. An analogous
expression (with the opposite sign) is also valid for the perturbation of 1~('1(k, w ) as one can see from equation (10).
Substituting the expressions for the perturbed fluctuations into equation(8) which is
convenient to rewrite m the Fourier components as well, perforrnmg the statistical averaging
taking mto account that
~~~~~ ~~ ~~~~~ ~~~~
"~ (Pw~-A(k))~ + y~w~~~~ ~ ~~~~~~~' ~~~~
where ( ) designate the statistical averaging, and integrating over w one obtains for the order parameter fluctuation renormalization of the coefficient
v :
" "
Y
~~Ln ~ VÎk)
y /Î~Ln ~'~ ~'~~~IOC ~~~~
where ( )
j~~
designate the local thermodynamic fluctuation. One sees that the renormalization
is mdependent of frequency if the order parameter dynamics is purely relaxational
(p
=
0) but diminishes at fl
~ rm y/p for displacive systems. Let us recall that for the
displacive systems r is expected to be much less than the characteristic atomic frequencies.
In fact, equation (13) is also valid, for T
~ T,, if the averaging is understood as that over the statistical ensemble the « local fluctuation » is equal to p~ plus the thermal fluctuation, i-e-
equation (13) proves to be a generalization of equation (5). The local fluctuation as defined above depends on temperature monotonically just as the entropy or the refractive index (for a
recent discussion see [6]) and one could mention that a similar (« trivial ») anomaly may be
due to the term (~ + ~j) P ~ in the generalized dissipation function. However this conclusion
is valid in the absence of the phase pinning only. It has been already mentioned that due to the
pinning the p~-contribution due to the interaction given by equation (2) is suppressed and it does not happen with the «trivial» one. Thus the anomalous part of the «polanzation viscosity » v calculated above is maximum at T
= T, in real systems the form of the maximum
being, to the first order approximation, Ci -C± )T-T,)"~ where C± =,fiC_, the
subindices + and refer to T
~ T, and to T
~ T,, respectively. The relative magnitude of the anomaly is the same as that of the
« tails » of the refractive index if f - 1, e. it is of order of
magnitude of T,/T~, (several percents) for displacive systems, but as long as it is expected that
in some systems f »1 (see Appendix I) this anomaly might be more pronounced there.
A question may arise about the defect contribution to the anomaly. Indeed, it is known that the random field defects cause anomalies of thermodynamic quantities and kinetic coefficients
which, for the symmetrical phase, are qualitatively similar to those due to the order parameter fluctuations [7, 8]. Thus one could expect that m the presence of the random-field defects the
interaction given by equation (2) leads to some precursor phenomena in the N phase even when
the thermal fluctuations can be neglected. However, it can be easily seen that the defect contribution to the anomaly of the polarization viscosity coefficient is zero at fl
=
0 and thus it
can be neglected in the low-frequency experiments. Indeed, at P
= const the changes in the
order-parameter field do not depend on time and the last term in the left-hand side of equation (8) disappears.
3. Carriers mobility in the I~phase.
We assume that the system under consideration contains free charge carriers, with a
concentration n « n~~ m a~ ~ as long as we consider dielectncs. Using the results of section 2 it
is straightforward to see that the mobility of a carrier increases due to the existence of the I-
modulation. Indeed, considering the system of mobile carriers one can introduce the
poJanzation of the system, P
= qn-r, where q is the charge and x is the displacement of the carriers, i-e- é
= j, where j is the current density. In this case
ce =
0 and v has the meaning of
inverse conductivity. One sees that, due to the interaction with the phase motion, this quantity
diminishes, ie. the conductivity increases. One has, however, to be careful with the
dependence on the carrier concentration of the results on the carrier concentration and it is why
it is convenient to perform the treatment anew, especialJy since it is quite elementary.
For the system crystal plus mobile carriers the density of the generalized dissipation function contains the term :
~jd(~/j ~/2 ~l2 ~ll)V,, à(r- r,) (14)
where v, is the velocity of the carrier along the modulation axis and r, designates the point of location of the i-th carrier. For a carrier m the I-phase one has
ôv, + dp~ qi
= qE, (15)
where q is the electrical charge of the carrier and à is its viscosity coefficient. Instead of
equation (4) one obtains :
y# +n du, =o (16)
and the renormalized carrier viscosity coefficient proves to be :
à
=
à
=
nd~ p9y
= à (na~ ~~~ (17)
The estimates of the coefficients used to obtain the latter equality are adduced in Appendix I.
One sees that due to interaction with the I-modulation the mobility of the carrier increases and
the excess is proportional to p ~. Let us recall that in the CDW case it is found to be proportional
to p ii j.
Comparing equations (17) and (5) which are fairly similar we see that in the present case
there is an additional small factor, na~, and this creates additional difficulties to observe the effect. It is clear that one has to deal either with a superionic conductor or with a crystal with very high electron concentration. The latter is not impossible for semiconducting I-systems which are known [9, 10] albeit a few.
The most interesting effect in a conducting I-system is the motion of the modulation wave m the extemal permanent electric field. This motion is possible when the field is strong enough.
To estimate the threshold field we rewrite equation (6) for our case in the form :
p# + yqi + pw(q~ +ndv, =0 (18)
and demand the third and the fourth terms to be comparable. Using once more the estimate of the coefficients from Appendix I we obtain :
~ (l ÷10~~)~ d ~
(19)
at ~ Wat
For the I insulators the reported expenmental values of wo/w~~ are typically 10~ ~ 10~ ~ II
and therefore for n/n~, ~10~~ the value of E~ may be not too high for the experimental
observation of the unpinning although it is quite a difficult task to obtain a sample with a high concentration of carriers and small enough phason gap. Let us mention as well that the above estimate of the threshold electric field refers to the start of the homogeneous movement of the modulation wave. Taking into account the clear analogy with the plasticity one can expect that
an inhomogeneous movement due to the creation and travelling of the dislocations in the I- superstructure will be appreciable at much smaller fields. This question, however, deserves a
special discussion.
As well as in the case of the dielectric fosses (Sect. 2) one can expect some precursor phenomena in the N-phase. It is shown in Appendix II that the fluctuation contribution to the
mobility of a carrier govemed by the interaction term of equation (14) does not depend on the carrier concentration, unlike the renorrnalization due to the phase movement in the I-phase, cf.
equation (17). This seems to give a possibility to estimate the constant d in equation (14) by studying the temperature dependence of the mobility at T~ T, in a system with a small
concentration of carriers. However a consideration within the long-wave approximation
(Appendix II) shows that the anomaly can hardly be observed : it consists m a discontinmty of
the temperature derivative of the mobility at T
= T, the order of magnitude of the discontinuity being (po/T~~)(T,/T~,1'+ ' for displacive systems (Mo is the mobility far from the transition,
