Quantum eletrical phenomena at low temperatures

Quantum mechanism of migration polarization in layered crystals in an alternating electric field in the low-temperature region. Calculation of the density matrix for an ensemble of noninteracting protons moving in a one-dimensional multi-well relief.

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Карагандинский государственный технический университет, Караганда, Казахстан

Quantum eletrical phenomena at low temperatures

Калытка В.А.

Оспанов Б.С.

The investigating of nonlinear electrophysical, magnetic and optical properties of instrumental and constructional materials with compound structure of the crystal lattice (layered crystals, ceramics, MDS - structures) is an actual scientific - technical task, which decision should be built in a complex, on a combination of the experimental and theoretical results directed on detection of features of polarizing, magnetic and electrooptical effects in the field [1,2] of phase transition.

In the last 10 years considerable interest for physical materials science represents research of effects of a nanocrystalline state of dielectrics [3,4], semiconductors and a ferroelectric material in the range of low and ultralow temperatures that is important for space and medical technologies.

The kinetic phenomena excited by electric field, in various crystal structures, are reduced to the through movement of free electrons (conductivity current in conductors), diffusion -relaxation movement of interstitial ions [5, 6] (ionic conductivity and polarization in dielectrics), electronically - hole conductivity in semiconductors.

The studies of the kinetic phenomena at polarization and magnetization of a crystal, in case of distribution of carriers of a charge (or magnetic atoms) on levels of energy of a continuous range (classical statistics), has to be based on the solution of the kinetic equation of Boltzmann together with system of the equations of Maxwell under the set boundary conditions [1]. In a case distribution of particles on levels of energy of a diskette range research of statistical properties of system, in lack of degeneration, has to rely on quantum statistics of Boltzmann [2], and kinetics of polarizing and magnetic processes on the quantum kinetic equation of Liouville allowing to calculate the statistical operator of system depending on structure of its Hamiltonian revolted with an external field. From this point of view electrophysical and magnetic properties of various crystals in the wide range of tension of electric and magnetic fields and temperatures can be investigated on the basis of the uniform kinetic theory allowing to calculate experimentally measured macroscopic characteristics (polarization, magnetization) and, on this basis parameters of a crystal lattice and molecular parameters [2] the relaxing particles.

1. Unbalanced density matrix of many-particle system in external perturbations

We will consider a system of interacting particles (relaxation oscillator) moving in a steady crystal potential field . Unperturbed Hamiltonian of separate particle [1] says

The particles are distributed over the energy levels of the discrete spectrum , according to the [3] Liouville equation

describes by the unperturbed stationary statistical operator

where statistic sum of system; NF - full quantity of particles in system. Unperturbed equilibrium (balanced) density matrix of the system takes the form

When imposing on a crystal of external perturbation , system of particles with Hamiltonian , described by the equation [3] of Liouville

The solution of the operator equation (5), in the first approximation of perturbation theory is constructed in the form [7], in the steady indignant state, going to asymptotic approach for the revolting amendment , write indignant stationary statistical system operator

In (6) the statistical sum of indignant states. The indignant equilibrium (balanced) density matrix of systems has the form

In (7), power range of particles En it is calculated taking into account stationary indignation , in the linear approximation of the perturbation theory , where . For large periods of oscillation of the external field, accepting [1, 2], we receive the quasi stationary indignant range of energy

In (8) the expression is calculated as the slowly changed in time function.

According to (8), transform (7) into the type

owing in the field of low indignations to and taking into account (4), we receive approximate expression

In (10) accepted the designation

According to [1], getting an approximate density matrix

2. Quantum effects under polarization of dielectrics at low temperatures

The analysis of the experimental spectra of specific volumetric electric conductivity and tangent of dielectric losses of solid dielectrics with compound crystalline structure (layered minerals, ceramics) showed. That at high temperatures (T = 100 - 450 K) mechanism of dielectric relaxation is reduced to a thermo - activated threw relaxators (ions, dipoles) at the fastening (lattice sites) in the direction power lines (towards the electric lines) of the polarizing field (for cations; cation vacancies) or against the field (for anions, anion vacancies) [2].

The most effective relaxation polarization, is shown in hydrogen bonded crystals (HBC), classified by electrophysical properties in the voltage range of the polarizing field and temperatures K, as proton semiconductors and dielectrics [2] and characterized by proton conductivity - diffusive transfer of hydrogen ions [H+] (protons) along the hydrogen links towards the electric lines of polarizing field [2]. The totality of the polarization processes associated with relaxation - diffusive motion of the protons in the HBC is determined as a proton relaxation. The molecular mechanism of the polarization of the hydrogen sublattice in the HBC [2] allows classifying it as a migratory polarization. quantum polarization electric proton

The kinetics of a proton relaxation in the field of high temperatures is rather well investigated, both experimentally, and theoretically [1]. We measured the temperature spectra of thermally stimulated depolarization currents (TCDP) and frequency - temperature spectra of the tangent of dielectric losses in crystals of talc [1], mica and in plaster.

A linear theory of dielectric losses, in good agreement with measurements of the spectra of density of TCDP J(T) in the HBC with a compound crystalline structure (mica, chalcanthite, phlogopite) [1]. For theoretical research of a low-temperature proton relaxation, in HBC, it is not enough linear approach on the polarizing field [2]. In the range of low temperatures (70 - 100 K), as showed experiment, the dominating contribution to migratory polarization in layered crystals is made by tunneling of protons inside and among [2] the ions of an anion sublattice.

So, studies of the quantum properties of an ensemble of relaxing protons conducted on the basis of the unperturbed (non - disturbed) Hamiltonian of whole the system (crystal) , without taking into account a proton - proton and a proton - phonon interaction , in adiabatic approximation [1]. Then, we [2] accept to approach

where total number (full quantity) of protons relaxing at a predetermined activation energy U0 [1]; - unperturbed potential image for the proton [2]. By analogy with (12) with respecting (11), in the area of low fields, write the perturbed quasi stationary proton density matrix in the linear approximation of external indignation (perturbation)

In the (14) accepted the designations: unperturbed balanced proton density matrix; - the statistical sum of system of the non - interacting protons distributed of levels of unperturbed energy spectra. The parameter determined for the matrix element of dimensionless coordinate ; d - crystal thickness. Upon based [4], write the operator of the equilibrium concentration of excess

The wave functions of stationary states were calculated in [1]. The operator of polarization of a proton subsystem, in a variation field [2] taking into account (15) takes the form

Full quantum mechanical averaging of the operator with the help of the wave functions of mixed states [4], with respecting of (14), in approximation of unperturbed density matrix [1], finally gives

The expression (17) can be used for the calculating of theoretical temperature spectra of stationary (at optic frequency) dielectric constant (SDC) and frequency spectra of complex dielectric permittivity (CDP) of HBC and ferroelectrics (KDP, DKDP). Use of the device of a matrix of density to research of tunneling of protons in a hydrogen sublattice of KDP, will allow, to open, at the molecular level, the quantum nature of electro-voltage effect and to predict nonlinear electrooptical properties [5,6].of a ferroelectric material near a point of phase transition.


1. Калытка В.А. “Аналитическое исследование термостимулированных токов деполяризации в кристаллах с водородными связями при низких температурах” // Диссертация на соискание ученой степени кандидата физико - математических наук, по специальности 01.04.07 “физика конденсированного состояния”, г. Томск,-2012. - Государственная публичная научно - техническая библиотека России. http://library.gpntb.ru.

2. Анненков Ю.М., Калытка В.А., Коровкин М.В. Квантовые эффекты при миграционной поляризации в нанометровых слоях протонных полупроводников и диэлектриков при сверхнизких температурах // Известия Вузов. Физика. - 2015 г. - Т. 58, № 1. -С. 35 - 41.

3. Ландау Л.Д., Лифшиц Е.М. Статистическая физика. -М.: Наука, 1989. - Т.9. - 186 с.

4. Ландау Л.Д., Лифшиц Е.М. Квантовая механика. -М:Наука.1974. -Т. 3, -34 с .

5. Лайнс M., Гласс А.Сегнетоэлектрики. M: Мир, -1981. -436 с.

6. Левин А. А., Долин С.П., Зайцев A.P. Химическая физика. -1996. -Т. 15. - 84 с.


Quantum mechanism of interlayer polarization in layered crystals in alternating electric field in the limits of low temperatures is studied. Unbalanced density matrix is calculated for the ensemble of noninteracting protons, moving in one-dimension multipit potential image of rectangular shape disturbed by variable polarizing field. Results of quantum-mechanical investigation of migratory polarization may be used in the study of tunnel mechanism of spontaneous polarization of ferrielectrics (KDP, DKDP).

Keywords: hydrogen - bonded crystals (HBC), interlayer polarization, proton relaxation, unbalanced proton density matrix.

Исследуется квантовый механизм миграционной поляризации в слоистых кристаллах в переменном электрическом поле в области низких температур. Вычисляется неравновесная матрица плотности для ансамбля невзаимодействующих протонов двигающихся в одномерном многоямном прямоугольном потенциальном рельефе возмущенном переменным поляризующим полем. Результаты квантово - механического исследования миграционной поляризации могут быть использованы при изучении туннельного механизма спонтанной поляризации сегнетоэлектриков (KDP,DKDP).

Ключевые слова: кристаллы с водородными связями (КВС), миграционная поляризация, протонная релаксация, неравновесная матрица плотности для протонов.

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