Bipolaron Mechanism of High-Temperature Superconductivity of Ammonia Systems

In the present paper, we propose an explanation of the ambiguity of the results of experiments on the study of high-temperature superconductivity of ammonia systems. At the heart of the theoretical interpretation of the experiments, we put the bipolaron model. In this study, we have shown mathematically that the barrier of repulsion between polarons can be effectively reduced if the polarons are in the macroscopic dielectric layers, or capillaries. We constructed the theory of polaron states in the macroscopic dielectric layers. We specify the conditions under which the polarons are hold in the layer between dielectrics. It was found that the electrostatic image forces lead to the appearance of additional forces of attraction between polarons. These forces are conditioned by oscillations of polarons around the position of their fixation. Derivations are given of the upper and lower limits on the width of the gap in which the polaron oscillations are not suppressed. In this case take place disappearance Coulomb repulsion of the polarons. A long-range resonant interaction of two oscillators resulting in the appearance of effective attraction between polarons is discussed. This leads to the formation of diamagnetic singlet bipolarons due to quantum exchange interactions and the effects of electronelectron correlations. For glass capillaries (quasi-one-dimensional bipolaron) and for gap between glass plates (quasi-two-dimensional bipolaron) we give quantitative estimates of the gap width and the critical temperature at which there is a barrier-free formation of the bipolaron in ammonia. Numerical estimates are obtained for a case of the bipolaron in ammonia. We got a quantitative evaluation, which indicate that the barrier-free formation of singlet bipolaron in ammonia begins at temperatures below 80K. As the experiment showed the electrical resistance of ammonia systems decreases abruptly by 10-12 orders of magnitude in this temperature range. At the same time, experiments have shown that for the bulk superconductivity superconducting phase is only ~ 0.01%.


Introduction
Repeated attempts to identify by experimentally the superconductivity of metal-ammonia systems have led to mixed results.Some experimental studies confidently asserts the existence of superconductivity at temperatures K T 50  .For the first time Ogg [1,2] experimentally observed electron transition to the superconducting state.Later, these experimental results were confirmed [3,4].The most detailed results of successful experiments presented in [4].At the same time in other studies (see., Eg, [5,6]) denied the existence of the superconductivity in metal-ammonia systems.Recently the problem of high-temperature superconductivity of ammonia systems was discussed in papers [7,8].In present paper we discuss the possible reasons that lead to negative experimental results.
In this paper we will use the bipolaron model to identify conditions appearance of superconductivity of electrons in ammonia.The states of electrons are described using a continuous model, which presupposes interaction between the electrons and the longitudinal branch of polarization oscillations of the medium.The criteria for validity of the theory reduce to the following inequality:


eV is the energy of the longitudinal polarization oscillations of the medium.The orientational oscillations of molecules about their equilibrium position in a polar liquid form elastic waves which may be treated as in a crystal.Within the framework of this model, we were

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able to interpret the different electronic properties of ammonia systems [9][10][11].It was noted that a singlet paired state of two electrons may be formed in ammonia.However, bound states of electrons with metal cations do not form, as was demonstrated experimentally.The valence electrons of dissociated metal atoms and metal cations are clearly spatially separated in the solution.This conclusion also agrees with the results of investigations of the optical properties of electrons in ammonia, which proved to be independent of the nature of the dissolved alkalineearth metals, including bivalent ones.
In the present paper we discuss the possibility of a lowering of the potential barrier and hence of an effective shift of the dynamic equilibrium between polarons and bipolarons in the singlet state in the direction of polaron pairing.

Mathematical Methods
Consider the motion of an electron in ammonia, placed into between two massive parallel plate insulators.The gap can be regarded as macroscopic between insulators in the sense that it is much larger than the size of the polaron.We shall write down the Hamiltonian of an electron coupled to the scalar field of phonons using the effective mass approximation: Here, r is the radius vector of the electron; m* is the effective mass of the electron;  is the limiting ) frequency of the longitudinal branch of optical phonons;  f f are the Fourier coefficients in the expression describing the interaction of an electron with long-wavelength phonons; describes the effective permittivity of the polar medium; Φ j is the electrostatic potential of the j-th external source.
In the subsequent analysis of the Hamiltonian (1) we shall use the Bogolyubov-Tyablikov adiabatic perturbation theory [12,13] in which the kinetic energy of the phonon field is used as a small quantity.Following Refs.[12] and [13] we shall use the relationship     0 to introduce a dimensionless small parameter 1 


. We shall replace the Bose operators f b and  f b with complex coordinates of the f-th field oscillator and canonically conjugate momentum using the relationships: Using the equations (2), we can rewrite the Hamiltonian (1) as follows: The electrostatic potential ) (r U will be determined later.It has been taken into account here that the motion of an electron interacting with the longitudinal branch of the oscillations of a polar medium is complex: the fluctuation motion of the electron in a polarization potential well is superimposed on the translational motion of its center of inertia.Using the method of [12,13], we write the radius vector r as a sum of vectors: a translation-invariant variable ρ and variable R, which characterizes the translational motion of the center gravity of the polaron: , ( ) exp( ).
It follows that we replace a single variable r by two independent variables R and ρ .Vector ρ describes the electron oscillations in the polarization potential well.Here, f Q are new coordinate of the f-th field oscillator describing its small quantum fluctuations near the classical value f u .The ground state of the phonon field is determined by a set of c-numbers f u ; the c-numbers f u are determined from the requirement that the total energy should be minimum.Since ρ and R are independent variables and we have three more new variables, it is necessary to remove three excess degrees of freedom by imposing three coupling conditions on f Q which can be chosen in the following form: Without loss of generality, we may assume that the c-numbers f v satisfy the orthogonality condition: * v , , , , .
Vladimir K. Mukhomorov.Bipolaron Mechanism of High-Temperature Superconductivity of Ammonia Systems All the c-numbers we have introduced satisfy the following reality conditions: We consider the situation in which the electron is placed in a polar medium (j = 0), which is located between two dielectrics (j = 1, 2).For all three dielectric mediums (j = 0, 1, 2) … electrostatic potentials satisfy the following relations: Here the subscripts 1 and 2 refer to the outer flat dielectrics which has a thickness H; the index 0 refers to thin layer of ammonia between 1 and 2 dielectrics.The thickness of the gap layer is equal to 2h << H. zaxis is perpendicular to the surface of dielectrics.For a symmetric system we have: Using the image method [14,15] we obtained the following equations for the potentials In accordance with image method we can write the electrostatic potentials 1  and 2  in infinitely extended mediums 1 and 2 as follows: q are the image charges in the mediums 1 and 2, respectively.The virtual image charges are found the boundary conditions (5).We shall need only the charges  2q can either be negative or positive.
If they are negative 0 , the electrostatic forces acting on an electron are directed into the medium 0. The electron may become pinned in the region of the medium 0 at some equilibrium distance along the z axis.In the symmetric case 1 Using the definition (4), and the following relation for derivatives we shall represent the Hamiltonian (1) in terms of new variables (4) as a series in powers of  : The following notation is used here: , and an allowance is also made for the fact that for the assumed gap layer thickness h the energy can be expanded as a Taylor series near rz = 0. Following Ref. [13] we can find the undetermined complex quantities . We shall assume that the terms linear in f  and in H1 vanish.Then the condition for vanishing of the coefficients in front of f  becomes: where Utilizing the freedom of selection of k a we shall assume that ' k a = 0. Then Eq. ( 13) becomes Using the orthogonality condition we find from the system (6) that Eq. ( 14) is satisfied if we assume the where V is the velocity vector operator of a polaron gravity center.Using the definition of the momentum operator and multiplying Eq. ( 15) from the left and from the right by * k ku and then summing over k , we obtain the following expression for the momentum of the polaron as a whole where  n is an unit vector parallel to the axis  and diagonal components of the translational polaron mass tensor can be determined as follow: Using the definition (16) and Eq. ( 17) we can determine the complex quantities k v from Eq. ( 15): is the component of the translation mass of a polaron in the  -th direction.
The wave equation with the Hamiltonian (12) can be solved by the perturbation theory method.We shall do this by writing down the complete wave function as a series ...
and the corresponding total energy ...
Then the equation 0 reduces to an infinite system of equations 0 0 0 ) The principal term of the expansion ( 12) which carries nontrivial information on the system is 0 H . Since the operator 0 H acts only on the variables ρ and R and does not include the field variables Qf the complete wave function in the zeroth approximation can be approximated by the following multiplicative form: 00 ( , ) (... ...).

   
Here, ...) (... where Taking into account the definition of quantity vf in Eq. ( 18), we can rewrite Eq. ( 23) as follows: where   For the electronic ground state we can be neglected by the operator . The complete system depending on the coordinates ρ and R can be separated into two conservative subsystems, one of which depends on the variable ρ and the other on R. Then the full wave function in the zeroth approximation can be described by a product of wave functions: ...) We have thus in the adiabatic approximation separated coordinates of the particle and those of the phonon field.A discrete spectrum of eigenvalues and the wave functions is described in detail in Ref. [16] without allowance for the image forces.Bearing in mind that where ) , ( Bearing in mind the above equality we find from Eq. ( 20) that The operator H1 averaged over the wave function 0  is a linear form in terms of Qf and ' f P so that the equation   cannot have a regular solution if the operator is not identically zero.We shall select the functions k u occurring in the operator H1 in such a way as to satisfy Using the definition of H1 in Eq. ( 12) we obtain the unknown quantities f u   The Schrödinger equation for free motion in the xy plane is no interest and will not be given here.Therefore, when we speak of a system of electrons, we mean a system which forms a two-dimensional energy band with a continuum of translational states in a plane parallel to the surface.
Equation (32) describing the motion of the center of inertia of a polaron along the z axis can be rewritten conveniently as follows: where the eigenfrequency of normal oscillations is equal to: The q in Eq. (34).
We shall now assume that in the same plane xy at a large distance R0 > Rp that is greater then effective size of the polaron (we shall consider specifically that the direction of the axis joining the polarons coincides with the x axis) from the selected electron there is one further polaron in its ground state.The functional for the self-consistent total energy of a two-electron formation in which the variables ρ and R have already been separated, can be written as follows [10,17]:  36) corresponds to direct electron-electron interaction, whereas the last two terms in the Hamiltonian allow for the cross interaction of the first electron e1 with the image charge of the second electron and of the second electron e2 with the image charge of the first electron.
The functional of the relative motion of the two electrons (36) (first and second lines) had been analyzed in detail allowing for the influence of the electron-electron correlations [10,[17][18][19].We have been found that the total energy of bipolaron has a minimum at a finite equilibrium distance between the gravity centers of polarons (see Figure 1).Importantly, the pairing of the electrons occurs in the coordinate space, but is not momentum space.
 we can readily see from Eq. (10)   that the charges are 0 , and the last two terms in Eq. (36) give rise to an additional weak attraction (~ h -3 ) to the total energy resulting in some in some reduction of the barrier height.In the limiting case when 1 / ) ( 0 *     even such a weak attraction may increases the binding energy of bipolaron. A much greater polaron attraction effect appears because of a resonant interaction between the two oscillators in Eq. (33).Within the framework of macroscopic electrodynamics the operator of the longrange interaction (considered in the dipole approximation) of two oscillators in an isotropic homogeneous insulating medium can be written in the form [20]: are the dipole moment operators for the first and second oscillators; n is an unit vector directed along the x axis; ) ( 0   is the permittivity of the medium 0; the oscillation frequency of the oscillators obtained on the assumption that this frequency does not coincide with the resonances ) ( 0   .The last factor in Eq. (37) allows for correction to the effective value of D due to the internal field in the medium.
The equation describing two one-dimensional harmonic oscillators of Eq. (37) belongs to the class of exactly solvable problems.Using the linear transformations ) 2 ( ) 1 ( we can reduce to one equation for coupled oscillators (33) but with new eigenfrequences: where the eigenfrequencies of oscillations polarized along the vector n are (1 ) , The change in the total energy as a result of the interaction between two oscillators is  40) in terms of the parameter  and limiting the series to just the quadratic terms we obtain the following estimate of the energy of a resonant interaction between two oscillators: Therefore, F  is a negative quantity, which implies the existence of an effective attraction between polarons.

Discussion and Conclusion
We shall consider a numerical example in the form of the states of a self-trapped electron in ammonia when the use of the methods of the theory of a large bipolaron is fully permissible [21][22][23].The effective mass of an electron at the bottom of the conduction band is m m 73 . 1 *  [10,11].It is found by comparing the theoretical and experimental positions of the maximum of the optical absorption band of a polaron.
The limiting frequency  can be estimated from the width 2 / 1

W
of the optical absorption spectrum of polaron at its half-height.At sufficiently low temperatures we can use the equation of Pekar [24]: The energy p s A of reorganization of a polar medium applies to the strongest phototransition p s 2 1  [24]: Here ) (r i  is the wave function of i-th state of polaron.Index s refers to the ground state of the polaron, and the subscript p refers to the first excited state of the polaron. The experimental value of the half-width 2 / 1 W = 0.46 eV [25] can be used in Eq. (42) to estimate the limiting frequency at which the polaron-polaron interaction is practically identical with its Coulomb asymptote 0 0 2 / R e   [10,17].The continuous approximation we have adopted imposes restrictions on the equilibrium distance R0.It must be greater than the effective radius of a polaron Rp: . Comparing Coulomb asymptote with Eq. (41), we can find the conditions for complete compensation of the forces of the Coulomb repulsion of polarons at the distance R0: We shall further assume that mediums 1 and 2 are a glass.Such a theoretical formulation is qualitative close to the system studied experimentally in Refs.[1,4] where the conductivity of ammonia systems was measured.The average values of the high-frequency and dc permittivities for most cover glasses lie in the vicinity of 6 .and assuming that the absolute temperature is T = 50K, and also using the value given by Eq. (35) for the longitudinal effective mass, we find that the thickness of the medium 0 gap satisfies the inequality: On the other hand, h should be such that Hence, we obtain the upper limit for h: The relevant task is discussed in the experimental work [4].In this work were carried out experimental studies of the electrical resistance of the metalammonia solutions.Ammonia is placed between cover glasses, as well as in microscopic capillaries.The measurements were performed at the temperature range 20-240 K.It was found that at temperatures below T = 80K there is a sharp drop in electrical resistivity of the metal-ammonia solution at 12-13 orders.Typically cover slip static dielectric constant is in the range 3.75-5.5.To quantify the binding energy of the two oscillators, we select the following static permeability .That is, in this case there is full compensation of repulsive forces of the polarons and there is no obstacle to the formation of the bipolarons.At the same time experiments [5,6] are carried out in a bulk of ammonia.In these cases the content of the electronic superconducting phase ~ 0.01% [5] and ~ 0.1% [6].These results are close to the results of the magnetic measurements of ammonia solutions (see references in paper [9]).The barrier can be overcome as a result of fluctuation of the polarization of the dielectric, or by Debye screening of the long-range repulsive Coulomb forces.
Thus, for realistic values of h and Essentially similar effects can be deduced also by replacing the action of balanced electrostatic image of balanced electrostatic image forces, for example in the medium 2, with the action of an external homogeneous electric field.It is possible that precisely such oscillations of electrons are related to changes in the temperature of the superconducting transition in films subjected to an external transverse electric field and found in the experiments reported in [29].
hf permittivity of the medium 0, 0  is a static permittivity of the medium 0. The quantum amplitudes of the scalar field f b and  f b satisfy the commutation relationships for the Bose amplitudes: of a polaron in the xy plane.Let us assume that E0 and ) (ρ  are the energy and the wave function of the ground nondegenerate discrete level of Eq. (20 derivative, we can represent the zeroth approximation Hamiltonian in the form: fact that * * z m occurs explicitly in the expression for the oscillation frequency  allows us, in principle, to use Eq.(34) for direct experimental determination of the longitudinal translational mass of a polaron by resonance methods.It is easy to find the numerical value of the translation mass of the polaron in the z-we find from Eq. (34) that in the case of the assumed values of h we have


represent the Fourier transform of the electron distribution.The third term in Eq. ( of the potential well and the height of the corresponding barrier depends strongly on the ratio

Figure 1 .
Figure 1.The bipolaron binding energy in the ground state versus of the distance R.

If
which lies in the experimentally accessible range of long-wavelength librational vibrations (5.1-6.3)10 13 s -1 of the ammonia molecules[26].Using the permittivities of ammonia 8

T
formation of the bipolarons takes place without any barrier.With the increase of the ratio 1 increases.It is important to note that the polaron-polaron pair potential (see.Figure) leads to a periodic distribution of electrons.For the concentration of electrons of the experiment[4] the periodicity in the distribution of electrons in ammonia appears at temperature K T 70 [27,28].At this temperature the ammonia passes into the crystalline state. )