Russian Physics Journal, Vol. 45, No. 10, 2002
NONLINEAR THEORY OF SPECTRA OF THERMALLY STIMULATED CURRENTS IN
COMPLEX CRYSTALS WITH HYDROGEN BONDS
M. P. Tonkonogov, Zh. T. Ismailov, V. M. Timokhin,
UDC 539.2 + 537.226
K. K. Fazylov, V. A. Kalitka, and Z. K. Baimukhanov
The nonlinear theory of thermally stimulated depolarization currents is developed. The theory explains the processes of hetero- and
homocharge relaxation in complex crystals with hydrogen bonds and allows the relaxation oscillator parameters to be calculated
using the quadratic approximation for the external electric field.
INTRODUCTION
Crystals with hydrogen bonds are very promising for the production of nonlinear elements used in electronics, optoelectronics,
and laser technique [1]. The possibilities of their practical application are determined by the degree of comprehension of the process of
proton migration over hydrogen bonds.
The methods of empirical description of dielectric relaxation with the help of Cole-Davidson, Havriliak-Negami, and FurthKirkwood equations are widely used in the literature [2-4]. Theoretical studies are phenomenological in character and, as a rule, they
disregard the dielectric relaxation mechanism at the molecular level [5].
Previously in [6] we suggested the dielectric relaxation model that took into account proton tunneling through the potential
barrier. The model is based on solving the system of Fokker-Planck and Poisson equations and describes the creation and annihilation
of the electret state [7]. It can be used to calculate thermally stimulated depolarization currents caused by relaxation of hetero- and
homocharges using the linear approximation for the external field. In the present paper, we develop a theory that studies the kinetics of
proton migration at the molecular level using the quadratic approximation for the external field. This allows tunneling transitions to be
described more correctly.
1. NONLINEAR PROTON RELAXATION MODEL
The processes of creation and annihilation of homo- and heterocharges are studied by the mathematical modeling of the electret
state of crystals with hydrogen bonds based on the nonlinear Fokker-Planck equation:
which is solved together with the Poisson equation [7]:
Here p = N- N0 is the excess concentration of relaxing agents compared to the equilibrium value N0,
Karaganda State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 76-84, October, 2002.
Original article submitted February 1, 2002; revision submitted May 7, 2002.
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1064-8887/02/4510-1008$27.00 ©2002 Plenum Publishing Corporation
£, = x/a, a is the lattice parameter of a one-dimensional crystal, x is the coordinate, x = Dt/a2, D is the diffusion coefficient, t is time, 0
= Noy%, y = a\\En/D, % = qa/ssoEn, q and p are the charge and mobility of the relaxing agents, Ef is the strength of the field in which
the thermoelecteret was produced, and the expression for yn can be found in [7]. By solving the Poisson equation in the linear
approximation, we obtain the function p in the quadratic approximation in the small parameter y :
(4)
(5)
The expansion of Eqs. (4) and (5) into series yields the following system of equations:
(6)
(3)
(7)
( 8)
(9)
(io)
The boundary conditions for the blocking and ohmic electrodes can be written as follows:
(ii)
The boundary condition of the Poisson equation is specified by the voltage applied to the electrodes. For the thermal polarization in the
absence of fields, it has the form
(12)
2.
KINETICS OF THERMAL DEPOLARIZATION WITH THE BLOCKING ELECTRODES
A solution of kinetic equation (1) even in the linear approximation obtained with the help of series (3) is a cumbersome expression [7].
However, an analytical form of the nonlinear solution of the boundary-value problem with the
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blocking electrodes is simpler compared to the ohmic electrodes. As in [7], the initial homocharge distribution was specified by the
expression
(13)
that describes an exponential decay of the homocharge concentration with increasing crystal depth. To obtain a heterocharge
distribution, the system of equations (7)—(10) for the unpolarized crystal in the constant electric field Ef must be solved. Then Eq. (7)
is simplified (p0 = 0):
(14)
In this case,
(15)
Taking the Laplace transforms of Eqs. (14) and (15), we obtain
(16)
After calculations, we obtain
(20)
(17)
The inverse transform of the function R1 is
(18)
where
Let us write down Eq. (8) considering that y 0 (£,, x) = 1:
The corresponding boundary and initial conditions have the form
(21)
The function y1(^, x) entering into Eq. (20) can be found by solving Eq. (10). The boundary conditions imposed on this equation
follow from equality (12) and the second equation of system (3). The correctness of the equality
(22)
can be easily demonstrated. As a result, we obtain the solutions of Eq. (10)
(23)
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(24)
and Eq. (19)
Taking advantage of Eq. (4) and proceeding to the limit at x
crystal caused by the polarization in the constant electric field Ef :
, we
Phet X>) = YP1 W) + Y2p2
W)
obtain the heterocharge
distribution in the
.
(25)
Thus, the heterocharge distribution at the start of the thermal polarization process is given by the expression
Error!
(26)
The electret charge at the beginning of thermal polarization is
(27)
The kinetics of proton migration is studied with the help of the system of equations (6)-(10). Initial conditions can be specified by
equating terms with the same powers of the small parameter in the left and right sides of Eq. (27), namely:
Error!
(30)
Error!
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(28)
(29)
T )aEf
Let us introduce a small parameter r = —— . It should be emphasized that r is calculated at temperatures close to the
temperature of the monorelaxation maximum in the thermally stimulated depolarization current. Rather cumbersome formulas for
(20)
calculating the functions p0 (£,, x), p1 (£,, x), and p2 (£,, x) entering into Eq. (4) are given in the Appendix.
Finally, the solution of the kinetic equation in the quadratic approximation has the form
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Differentiating the polarization averaged over the sample thickness with respect to time and taking into account that heating occurs
with a constant rate, we obtain the formula determining the spectrum of the thermally stimulated depolarization current in the vicinity of
the experimentally observed maxima:
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where
TABLE 1. Parameters of Relaxation Oscillators in CuSO4 5H2O Calculated in the Quadratic Approximation for the Blocking Electrodes
Serial number and the
temperature of the
maximum, K
Concentration of
Potential barrier
relaxation
width, A
oscillators, m-3
Activation energy, eV
Measured
Calculated
Proton transition inside and
between protonated anions [8 ]
H3 O+ ions
2 - 94
1.84 •1 o17
0.85
0.07 ± 0.01
0.13
3 - 138
2 .oo -10 18
0.90
0.11 ± 0.01
0.21
Water of crystallization
Adsorbed water
4 - 17o
5 - 2 o6
7.oo -10 18
1 .oo •1 o19
0.90
0.85
0.23 ± 0.02
0.35 ± 0.05
0.27
0.32
OH _ ions
Z-defects
6 - 23o
4.oo -10 19
2 .oo ■10 2°
0.70
0.80
0.70 ± 0.08
0.46
0.62
Relaxation oscillator type
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7 - 246
Fig. 1. Experimental (1) and theoretical dependences (2-7) of the thermally stimulated depolarization current in the CuSO4 • 5H2O single
crystal grown from chemically pure materials. Both theoretical and experimental
monorelaxation maxima correspond to the temperatures T = 94, 138, 17o, 2o6, 23o, and 246 K.
charge, that is, it was not monorelaxation one. Its analytical study within the framework of the developed theory is impossible.
After thermal decontamination and separation of thermally stimulated depolarization current maxima, the relaxation times, activation
energies, and pre-exponential factors of dipoles were determined. The calculated parameters of the relaxation oscillators are given in
Table 1. The theory and experiment agree quite satisfactorily; it is important that not only the activation energy but also the concentration
of relaxation oscillators can be determined in the quadratic approximation. The significant discrepancy between the theoretical and
experimental maxima at higher temperatures is caused by the through conductivity; however, the theory allows the relaxation component
of the thermally stimulated depolarization current to be selected, thereby providing acceptable accuracy of calculations of the relaxation
oscillator parameters. The observed discrepancy between the calculated and measured activation energies of relaxation oscillators at low
temperatures (94 and 138 K) is due to the use of the quasi-classical approximation when calculating proton tunneling [ 6 ].
The lack of experimental investigations of the nonlinear proton relaxation phenomena [1] hampers a more detailed analysis of the
theory that describes analytically the relaxation processes of homo- and heterocharge annihilation.
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CONCLUSIONS
1. The kinetics of dielectric relaxation in crystals with hydrogen bonds in external electric fields has been investigated analytically
in the quadratic approximation.
2. The formulas have been derived that can be used to calculate the spectra of thermally stimulated depolarization currents in
complex crystals with hydrogen bonds.
3. The relaxation oscillator parameters in CuSO4 • 5H2O have been calculated. The agreement between the
calculated and experimentally measured parameters was satisfactory.
4. The developed mathematical model can be used to analyze the nonlinear phenomena accompanying the homo- and
heterocharge annihilation.
APPENDIX
Taking the Laplace transform of Eq. (6), we obtain
Error!
(A1)
with the boundary conditions
(A2)
(A3)
A solution of Eq. (A1) has the form
Its inverse transform is
In Eq. (A4), we have used the following designations:
The specific feature of the function p0 (£,, x) is the presence of terms explicitly independent of time. This demonstrates the creation of
a residual charge retained by the crystal for a long time, that is, a homocharge. Its appearance at x ^ <» is detected already in the zero
approximation [6].
Taking the Laplace transform of Eq. (7), we obtain
(A4)
1016
(A5)
Here
and the boundary conditions are
(A6)
It is easy to derive the inverse transform of the solution. It has the form
(A7)
The function p2(£„ x) can be found from Eq. (8). Taking the Laplace transform, we obtain
(A8)
In this case, the boundary conditions assume the form
(A9)
and the inverse transform of the solution is the function
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REFERENCES
1.
M. P. Tonkonogov, Usp. Fiz. Nauk, No. 1, 168, 29-54 (1998).
2. A. A. Potapov and M. S. Metsik, in: Dielectric Polarization [in Russian], Publishing House of Irkutsk State University, Irkutsk
(1986), pp. 1-263.
3. V. I. Suslyaev, P. F. Tarasenko, A. V. Zhuravlev, and V. A. Zhuravlev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 11, 15-22 (1999).
4.
V. K. Yarmarkin and S. P. Teslenko, Fiz. Tverd. Tela, 10, No. 40, 1915-1918 (1998).
5.
A. A. Pozdnyakov, R. M. Sultanaev, and V. I. Kiselev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 1, 44-48 (1992).
6. M. P. Tonkonogov, T. A. Kuketaev, Zh. T. Ismailov, and K. K. Fazylov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 2, 80-85 (1998).
7. M. P. Tonkonogov, T. A. Kuketaev, Zh. T. Ismailov, and K. K. Fazylov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 6, 77-83 (1998).
8. M. P. Tonkonogov, T. A. Kuketaev, Zh. T. Ismailov, and K. K. Fazylov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 10, 97-99 (2000).
9.
S. M. Usmanov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 10, 102-107 (1991).
10.
M. P. Tonkonogov, T. A. Kuketaev, Zh. T. Ismailov, and K. K. Fazylov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 5,
87-89 (2001).
3.
COMPARISON WITH THE EXPERIMENT
An analysis of experimental spectra of thermally stimulated depolarization currents is a complicated problem [9]. We succeeded
in the development of a simple and efficient procedure of comparing theoretical and experimental results based on the construction
and use of a comparison function [8, 10].
Figure 1 shows the experimental (curve 1) and theoretical dependences (curves 2-7) of the thermally stimulated depolarization
current density in a CuSO4 • 5H2O single crystal on the temperature. We performed measurements for the
polarizing field strength Ef = 2 - 1 0 5 V / m, the polarization temperature Tp = 3 0 0 K , and the rate of linear heating
c = 5.5K/min .
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According to the procedure of computer exhaustive search for the relaxation oscillator parameters [10], the experimental and
theoretical plots of thermally stimulated depolarization currents in CuSO4 • 5H2O are compared in the
vicinities of the monorelaxation maxima. The last experimental maximum at T = 290 K (Fig. 1) was caused by the space
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