Dynamic electrocaloric effect in ftrroelectrics icon

Dynamic electrocaloric effect in ftrroelectrics

НазваниеDynamic electrocaloric effect in ftrroelectrics
Дата конвертации10.08.2012
Размер117.86 Kb.


Мокеjеv А.А., Мокеjеv Аn. A.

Private laboratory, 210035, Vitebsk, street. Smolensk, 5, cоr.1, sq. 18, Byelorussia.

AO Тreigon, Moscow, str. Matrosskaia teeshina, structure 23, коr.2. of. 8, Russia.


The phase transition in ferroelectrics of displacement type represents the relaxation of structures of crystal lattice ( relaxation of sublattices elasticity) of an initial meta-stable 0f polarised phase to elasticity of steady final phase and simultaneously avalanche disintegration of polarization under action of a feedback between polarization and Lorentz - Wisse field. The feedback is expressed as negative friction in the equation of movement of ions of soft sublattice. With the nonlinear dependence of its elasticity on displacement it results in occurence autooscillations of polarization. The power tacked away by oscillations of polarization from a sample is allocated in the shunting resistor, the sample is cooled. The decrease of transition temperature due to amplification of a depolarized field stabilizes cooling so, that in a significant part of a period of oscillations abnormal thermal capacity appears small, and decrease of temperature - significant.


The disperce of the dielectric permeability of ferroelectrics near to phase transition [1-6], has both Debay relaxation properties and Tomsons resonant properties, and specifies, that the ferroelectric phase transition is the relaxation of elasticity of soft sublattice and is the oscillations of polarization, then its complete decay is prevented by increase of the stabilization elasticity of other sublattices. The phase transition occurs before the sublattice becomes absolute soft. These phenomena do not have explanation in phenomenological thermodynamic and in microscopic theory [1-7].

According to works [12,13] the smooth decrease of soft sublattice elasticity of a crystal is interrupted by the jump-figurative relaxation up to elasticity of stabilization with temperature of phase transition Tt < To, smaller than temperature of stability loss.

This process is owing to quantum transitions of ions of soft sublattice from a stationary states with the energy which is equal to thermal energy to the states with deltafigurative peak of number of microstates, which arise as imposing of the rich site of a spectrum of stationary states of these ions at bottom of a wide rectangular potential hole of stabilization on a rich site of its spectrum in top of Gaussian hole of soft sublattice.

There is an avalanche disintegration of polarization due to an establishment of a positive feedback between it and Lorentz - Wise field . With it the depolarizing field is broken up, and similarly to external field it reduces the temperature of transition.

The disintegration of polarization is replaced by growth and there are autooscillations of polarization. The compression of the oscillation spectrum of soft sublattice with heating to the phase transition temperature leads to the abnormal fast heat capacity growth and to its jumping decrease. This jump is not Landau jump. The examination of the temperature dependence of the heat capacity in experience, shows that the Landau jump is not observed. The abnormal fast growth of heat capacity caused by fluctuations of polarization is not significant. The electrical current of polarization with negative fading of autooscillations absorbs the heat, but does not give off.

The increasing of depolarizing field and of polarization decreases the temperature of phase transition. The voltage on the shunting resistor makes the average deviation of temperature from the phase transition temperature so as to the heat capacity being enough small to the significant electrocalorical effect.


The phase transition in ferroelectric and the spontaneous polarization of it are the irreversible large displacement of soft sublattice ions of a crystal with charges q and concentration N from equilibrium states which are the poles of pceudospin Sn, which is moved from neutral centres of elementary cells with lattice vectors n on a shoulder h, so the returning force disappears, the potential energy becomes constant Uo,

Potential energy Uo of collective interaction g>>1 of ions and electrons in each volume Vc of correlation of their movements and of screening of their electrical field with radius of correlation D, which is equal to radius of screening of their electrical field , is the energy with coherent interaction of g(g-1) pairs. With a dielectric susceptibility of vacuum o, with concentration of ions and electrons Ne eith distance between them r, with Boltzmann constant kB. This energy is decreased in comparison with interaction energy in infinite crystal on value of energy of comprehensive compression of condensate. That is expressed through the screening coefficient of the charge .

Electrical field of this interaction is quasistatic. Polarization is determined by density of electric current j (r, t) which is determined by distance x' of charges and by the delay t' of perturbation propagation to a point of observation with coordinates x in the moment t,

x = x - x', t = t - t' and by the intensity of a molecular field E (x, t), created in the retarded moments of time t' by the charges from the removed points x', through the function f of causing .


This field does not destroy the crystal and is weak in comparison with the forces, which hold the structure of crystal [7, 12], so the causal function f is represented by first components of decomposition in a series in the small ratios:

1) of a distance x, which is near to the correlation volume size l>> lat and is greater than atomic size , to wave length of a molecular field l, 2) of time of delay t to relaxation time, which is equal to plasma frequency p, 3) of energy in a field Е to energy of coupling

/<<, t p<<1, /Ug <<1.

The density of current is equal to the sum of current density of a polarization jp and current density of internal screening jc [9]

. (2)

Density of a polarization current is expressed through locally equilibrium density of the connected charges o(x) as a polarization velocity


The charge density o(x') is determined by the probability density W(x) of its states with local normalizing factor No(x),

, No(x) = No(z), z = /

With z=0 volume of correlation is completely neutralised No(0) = 0

With z = 1 volume of correlation is completely electrolised No(1) = N

With 0 < z < 1

No(z ) = No(0) +(dNo/dz) z +... = N z (3)

With small displacement dx the return force is increased, and displacement probability is decreased. In the first approximation of a method of successive approximations

the charge density in correlation volume Vc is equal


Current density of internal screening is expressed through a charge dq which creates the polarization P about the coordinates X and Р ' about the coordinates X '

It is proportional to density of screening charge



Quasi - ctatic processes in a crystal are determined by the Macswell equations for a locally equilibrium subsystem for vector and scalar potentials


The transfer to the left of component with scalar potential,

with a modified Lorentz condition

results this equation in a individual kind

The Gauss theorem

by substitution of expression for intencity

and by substitution (7) is resulted in a kind

of the equations for scalar potential with divided variable


With const = 0 for quasistatic processes it is equivalent to two equations

The solution of the second of them is searched as

It is the valid solution, if is a solution of the dispersion equation

It determines relaxation processes for parameters of a crystal.

The Gauss theorem (the first of these equations) after substitution in it of complete density of charge gives the equation for the potential of electrical field in the following approximation

which with

by substitution is resulted in the equation with constant factors


The solution of this equation determines the individual solution of the equations for scalar potential of the screening self-coordinated field acting on a soft ion.

h® 0 (10)

In this force field of indirect interaction the ion moves with potential energy U(x))

in Gaussian potential hole of width D, depth Uo. In quasistatic processes the velcity of relaxation is indefinitely small, but determines the parameters of a crystal. With emperature T = 300 K, concentration of ions .

Potential energy Ugo of collective indirect interaction much more than energy of its screening interaction Uo with distance between particles r

so the plasma zero approximation is applied.

The reduction of factor of screening results in increase of screening radius D, and with it - to increase g and Ugo. That reduces still the screening factor, while Ugo will not reach value of bond energy of a crystal appropriate to temperature its transformations into plasma .

The interaction of ions in a ferroelectric crystal in average field approximation is divided on four parts: the long-range action part summering from all-round compression and creating a crystal potential hole part the long-range action of the dipole moments of elementary cells, creating a molecular Lorentz -Wise field , and the part which is the screening field inside cells, creating the turning forces acting on a soft sublattice ion, and part of the weak interaction forming dense statistical system in a locally equilibrum state of nonequilibrum thermo-dynamics [18, 19].

The energy of soft sublattice is imposition these potential holes of elementary cells


With the energy  < Uo the soft sublattice ion moves in one of potential holes about one of the pseudo-spin poles Sn = 1 and forms the dipole moment dn and the polarization Р, which creates a Lorentz field with intensity

El = P/o l

(l - the Lorentz factor ) and the depolarizing field with intensity

Ed = -P/o d .

In this field the soft sublattice gets energy.


If polarizability of an elementary cell is rather great then between polarization and intensity of a Lorentz field arises positive feedback. The polarization avalanchely disappears before stabilization by the interaction with rigid sublattices. With the energy  > Uo the movement of a soft ion becomes infinite and should be included the stabilization interaction with rigid sublattices with energy of stabilization.

D1 >> D

The external electrical field of intensity Eo together with depolarization field with intensity Ed, E = Eo - Ed imparts to soft sublattice the energy which is equal to the sum of energy ordering of pseudo-spins and of energy of elastic deformation by a field inside of dielectric.


Energy of elastic interaction between soft sublattice ions [3]


develops from from energy of pseudo-spin interaction


and energy of elastic deformation of a lattice


and energy of interaction between spin - systems and oscillatory system, which in approximation of an average field is zero, [3], Uxs = 0 .

In Hamiltonian of soft sublattice is selected the parts: the kinetic energy, the energy of system of effective linear oscillators in a Lorentz - Wisse field and in external electrical field and energy of pceudo-spin system Hs .


The ions of soft sublattice with energy  = kB T < Uc smaller than the average energy in potential holes of soft ions oscillate in one of them near to one pceudo-spin pole Sn = 1, between points of turn with coordinates Xon (amplitudes), determined by the energy conservation law

Xon infinitely is increased if the energy approachs to a threshold of stability loss of movement Uo. The effective oscillator has the rigidity a(Xn) and the own frequency wo, which decreases with growth of displacement Xn from balance position at centre of a potential hole with approach of energy  to a threshold of stability Uo.


In the zero approximation their period is equal to a period of garmonic oscillations of displacement [16], which is the solution of movement equation in the moments of passage of equilibrium and turn points.

The frequency coincides with own frequency of nonlinear oscillator of a soft sublattice .

In temperature units Uo = kB To, its elasticity and potential energy are equal


The energy of a soft ion differs from energy of effective linear oscillator with the same frequency by the latent energy with the value Wn(po) which is equal to a difference of complete and maximal kinetic energy of equivalent linear oscillator.

This difference exists as potential energy. It does not transfer with thermal collisions. It is latent and is determined by a pulse amplitude of a soft ion pon. Complete energy of soft ion is equal to the sum of its kinetic, potential and latent energy

With energies  = kB T>Uc the movement of soft sublattice ions is the free oscillations between collisions with the walls of the almost rectangular potential hole of interaction of stabilization with kinetic energy

with the average velocity V and with the frequency of collisions c, and with the effective elasticity b


The ions of soft sublattice in an elementary cell of a crystal such as oxygen оctaeder in the barium titanate form the locally equilibrium subsystem of vibrons [7].

The electromagnetic interaction of vibrons of soft sublattice and the optical oscillations causes the transitions between their stationary states /n >, /m > with probability [8].


are determined by matrix elements of interaction energy with the operator

and with the operator of generation of polaritons Ckl

The basic contribution to it give resonant components which are determined by transition frequencies


With energy from the sum on q get out the components which correspond to transitions from stationary states with the energy, which is equal to a thermal energy, to the states with the threshold energy and to the states with energy, appropriate to transition in to the new phase .

With energy from the sum on q get out the components which correspond to transitions from a stationary states with the energy, which is equal to a thermal energy, to the states with the average energy . Probability of such transitions

is proportional to the number of final states. The number of such states is more larger than more larger the energy and displacement Xn. So average displacement is proportional to

. These transitions are the causes displacement of ions

and create the effective dissipative force Ff

The dissipative force Ff creates the resistance to an internal current in vibrone

which determines a relaxation parameter in the equation for scalar potential. The modules of elasticity of vibrones and of system of stabilization, of Lorentz - Wisse field and of fluctuations are included by function of relaxation [12,13]


This is a function of adiabatic inclusion with -  < t < 0, which with t> 0 and T> To is transformed to the law of relaxing switching off interaction and of modules of elasticity of soft sublattice . In quasistatic process, t - > , this function becomes the generalised Heaviside function of jump.

With T = Tt the time of relaxation is infinite, that corresponds to critical delay and occurrence of frozen spatial fluctuations of polarization, of the depolarized field and of the temperatures of stability loss, Curie temperatures Tc and phase transition temperatures Tt, in absence of structure fluctuations.

The elasticities which are found in [12,13]


switched of by the relaxation funktions Ф.

The ion goes under action of a Lorentz - Wise field El and of external field Е (t) and of molecular field which creates the turn force with the rigidity a (T-Tt), which is ncluded adiabatically with t = - by the function ~ exp (wt (Tt - T) (t-t ') of relaxation

and is switched off with Tt - T < 0 by this function.

This relaxation is stabilized by the interaction with rigid sublattices with rigidity b ~ exp (wt (T - Tc) (t-t '), included with Curie temperature Tc < Tt.

Because of weak interaction of soft vibrons with system of optical phonons they depend on the time with different relaxation speeds for a subsystem of soft vibrons and for subsystem of stabilization through the relaxation function  (Tt-T).

The interaction of ions of soft sublattice transforms it in oscillator system with many dimensions . Their displacement is superposition of the plane waves with wave vectors k and amplitudes Xk and frequencies wk, [7,12],


which together with component of pceudo-spin with k = 0 is coherent displacement of all ions of sublattice.

Hamiltonian of soft sublattice in normal coordinates is equal


where - kinetic energy of ions of soft sublattice, - the latent energy of system,

the energy of oscillators system


in which energy of ordering is selected


and fluctuations energy,


which contains energy of influence of fluctuations on coherent displacement Xo with coefficient 


energy of stabilization


energy of soft sublattice in a Lorentz field


energy in external and depolarisation fields


energy of soft sublattice oscillations


energy of pceudo-spin system.



Polarizations of a ferroelectric crystal P (in agreement with the theorem of the fluctua-

tion - dissipation ) is a special case of the locally average value on locally equilibrium distribution f (Po, P, t) of the statistical system and represents the partially determined casual process in dense system of the poorly cooperating soft vibrones [7]. The electrical and thermodynamic properties of ferroelectrics are determined by the statistical sum Z of its soft sublattice [8], which is calculated in phase space with coordinates : displacement of soft sublattice ions rn, their pulses pn, and pseudo-spin components sn. Excess of energy of a movement along the of coherent displacement above thermal leads to selection of product of number of microstates with this energy (n) = exp(N n /kB T) and of the factor, containing latent and kinetic energies, depending from amplitude of pulses of ions of soft sublattice .


According to the theorem of affinity of canonical distribution to microcanonical their product is proportional to Dirac -function from average energy of soft sublattice ions

e(Xo) = < ( xon>)> = kB T, which corresponds to their average displacement

Xo=< Xon >.



The statistical sum becomes the product of the pseudo-spin sum Zs and of the sum of system of effective linear oscillators Ze. Along coordinate q = Xn the energy is greater than thermal, so the statistical sum and free energy become the functions of dynamical variable Xo (incomplete thermodynamic potential [3])

The soft sublattice breaks up to soft vibron system and pseudo-spin subsystem. The free energy in a locally equilibrium state is equal to the sum of free energy of nonlinear oscillators system and of energy of pseudo-spin system.


In it get out energy of ordering eo and the fluctuation energy Yf. From f is selected the component, which is express the influence of polarization fluctuations on the ordering of displacement of soft sublattice ions. It is transferred in the free energy of this ordering .


The probability of locally equilibrium fluctuations near to transition W (Xo) ( deviation of displacement from equilibrum) is equal [10, 12]


The polarization of a crystal is equal to the average dipole moment of nonlinear oscillators system of all soft sublattice ions.


It is represented by the product of the average pceudo-spin and of the vibron moment.

It is determined by average displacement of ions soft sublattice Xo.

Coherent displacement Xo of soft sublattice ions and average pceudo-spin S in a local thermo - dynamic balance or in quasistatic process are determined by the equations of local balance through free energy,

P = q S N Xo



Temperature of pseudo-spin ordering is determined by expression [3]

Tk (T) = q h El /kB

With temperatures Т < Tc which smaller than Curie temperature, when stabilization is switched off the fluctuations of polarization are small, free energy and the module of polarising elasticity are decreased linearly with growth of temperature .


With T> Tc the stabilization is switched on and diagram of temperature dependence of free energy and of elasticity module demonstrates break, that is equivalent to shift of temperature of loss of stability .

With the further heating a free energy of a crystal in a Lorentz-Wisse field El > Ed and in the external field Eo and in depoplarizating field Ed = -P/o d, [4],


is compared to free energy of para - phase, where a = 0, E = Ep = Eo-Epd

The phase transition occurs, when free energy of para- phase equal to energy of ferro-phase [4]. Temperature of transition is determined by this equation. In the first approximation of method of successive approximations the depolarizing field intensity Ed and the intensity of Lorenz - Wise field El are accepted of equal intensity created by the polarization P with transition temperature Tt (T, E) = 200 K .

The depolarizing field intensity Ed is determined by the equation

In Lorentz - Wise field is selected the part fl, which not vanishes in para - phase

The intensity of the Lorentz - Wise field is determined


The substitution in it of expressions for a, b Xo and Xop in ferro- and para - phases gives the equation of temperature of phase transition in next approximation


Where in a denominator approximately


Tt(150,0)=138 K , Tt(150,20) = 141 K, Tt(150,40) = 144.5 K.

Tt(150,60) = 148 K, Tt(150,89) 150 K, Tt(150,100) = 153 K

Temperature of phase transition is moved from a position between the Curie temperature Tc and the temperature of stability loss To by the Lorentz field and by the external field to large value, and by a fluctuations and the depolarizing field to smaller value. Known intensity of a Lorentz field determines the temperature of pseudo-spins ordering Tk. [3]


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