## Abstract

The measured in-plane thermoelectric power (Seebeck coefficient) S ab in YBCO below the superconducting temperature T c ( ∼ 94 K) S ab is negative and T -independent. This is shown to arise from the fact that the “electrons” (minority carriers) having heavier mass contribute more to the thermoelectric power. The measured out-of-plane thermoelectric power S c rises linearly with the temperature T . This arises from moving bosonic pairons (Cooper pairs), the Bose-Einstein condensation (BEC) of which generates a supercurrent below T c . The center of mass of pairons moves as bosons. The resistivity ρ ab above T c has T -linear and T -quadratic components, the latter arising from the Cooper pairs being scattered by phonons.

### Keywords

- Seebeck coefficient
- in-plane thermoelectric power
- out-of-plane thermoelectric power
- moving bosonic pairons (Cooper pairs)
- Bose-Einstein condensation
- supercurrent
- YBCO

## 1. Introduction

In 1986, Bednorz and Müller [1] reported their discovery of the first of the high-

YBCO has a critical (superconducting) temperature

In this paper we are mainly interested in the sign and the temperature behavior of the Seebeck coefficient in YBCO. But we discuss the related matter for completeness. There are no Seebeck currents in the superconducting state below the critical temperature (

## 2. The crystal structure of YBCO: two-dimensional conduction

HTSC have * layered structures* such that the copper planes comprising Cu and O are periodically separated by a great distance (e.g.,

_{2}–Y-CuO

_{2}–BaO-CuO–[CuO–BaO–…]. The buckled CuO

_{2}plane where Cu-plane and O-plane are separated by a short distance as shown is called the

*. The two copper planes separated by yttrium (Y) are about 3 Å apart, and they are believed to be responsible for superconductivity.*copper planes

The conductivity measured is a few orders of magnitude smaller along the
* at random* over all layers, then the periodicity of the potential for electron along the

no

the Fermi surface for a layered conductor is a right cylinder with its axis along the

*. Hence a 2D conduction is established.*-axis

Since electric currents flow in the copper planes, there are continuous
* Cooper pairs* (pairons) form a supercondensate below

Let us first examine the cause of electron pairing. We first consider attraction via the longitudinal acoustic phonon exchange. Acoustic phonons of lowest energies have long wavelengths

may be assumed, where

Second we consider the optical phonon exchange. Roughly speaking each copper plane has Cu and O, and 2D lattice vibrations of optical modes are expected to be important. Optical phonons of lowest energies have short wavelengths of the order of the lattice constants, and they have a quadratic dispersion relation:

where

Thus an electron-optical phonon interaction is a viable candidate for the cause of the electron pairing. To see this in more detail, let us consider the copper plane. With the neglect of a small difference in lattice constants along the

First, let us look at the motion of an electron wave packet that extends over more than one Cu-site. This wave packet may move easily in

Second, we consider a hole wave packet that extends over more than one O-site. It may move easily in

## 3. Quantum statistical theory of superconductivity

Following the Bardeen, Cooper, and Schrieffer (BCS) theory [11], we regard the phonon-exchange attraction as the cause of superconductivity. Cooper [12] solved Cooper’s equation and obtained a linear dispersion relation for a moving pairon:

where

The center of mass (CM) motion of a composite is bosonic (fermionic) according to whether the composite contains an even (odd) number of elementary fermions. The Cooper pairs, each having two electrons, move as bosons. In our quantum statistical theory of superconductivity [13], the superconducting temperature

where

is several times greater than the BCS pairon size represented by the BCS coherence length:

Hence the BEC occurs without the pairon overlap. Phonon exchange can be repeated and can generate a pairon-binding energy

Thus, the pairons are there above the superconducting temperature

In the quantum statistical theory of superconductivity, we start with the crystal lattice, the Fermi surface and the Hamiltonian and calculate everything, using statistical mechanical methods. The details are given in Ref. [15].

Loram et al. [15] extensively studied the electronic heat capacity of YBa_{2}Cu

The molar heat capacity
* order of phase transition* is defined to be that order of the derivative of the free energy

*. The temperature behavior of the heat capacity*third-order phase transition

_{2}Cu

_{3}

Our quantum statistical theory can be applied to 3D superconductors as well. The linear dispersion relation (4) holds. The superconducting temperature

which is identified as the BEC point. The molar heat capacity

## 4. In-plane Seebeck coefficient above the critical temperature

### 4.1 Seebeck coefficient for conduction electrons

When a temperature difference is generated and/or an electric field
and the gradients

holds, where

where

The conductivity
_{2}Cu_{3}

We assume that the carriers are conduction electrons (“electron,” “hole”) with charge
* quasi-electron* which has an energy higher lower than the Fermi energy

*which circulates clockwise (counterclockwise) viewed from the tip of the applied magnetic field vector. “Electrons” (“holes”) are excited on the positive (negative) side of the Fermi surface with the convention that the positive normal vector at the surface points in the energy-increasing direction. The number of thermally excited “electrons”*and

where

is higher at the high-temperature end, and the particle current runs from the high- to the low-temperature end. This means that the electric current runs toward (away from) the high-temperature end in an “electron” (“hole”)-rich material. After using Eqs. (13) and (14), we obtain

The Seebeck current arises from the thermal diffusion. We assume Fick’s law:

where
* diffusion constant*, which is computed from the standard formula:

where

where Eq. (14) is used. Using Eqs. (17)–(19) and (11), we obtain the thermal diffusion coefficient

We divide

and obtain the Seebeck coefficient

The relaxation time

### 4.2 In-plane thermopower for YBCO

We apply our theory to explain the in-plane thermopower data for YBCO. For YBa_{2}Cu_{3}

We assume an effective mass approximation for the in-plane “electrons”:

The 2D density of states including the spin degeneracy is

which is independent of energy. The “electrons” (minority carriers), having heavier mass

When both “electrons” (1) and “holes” (2) exist, their contributions to the thermal diffusion are additive. Using Eqs. (20) and (24), we obtain

If phonon scattering is assumed, then the scattering rate is given by

where

where

The total conductivity is

Using Eqs. (25)–(29), we obtain the in-plane thermopower

The factors
_{2}Cu_{3}

## 5. Out-of-plane thermopower

Terasaki et al. [17, 18] and Takenaka et al. [19] measured the out-of-plane resistivity
_{2}Cu_{3}O_{x}. In the range

where

with

The tunneling current is calculated as follows. A pairon arrives at a certain lattice-imperfection (impurity, lattice defect, etc.) and quantum-jumps to a neighboring layer with the jump rate given by the Dirac-Fermi golden rule

where

where

Lower-energy (smaller

where

assuming a small

At the BEC temperature

and

is negative and small in magnitude for

which is normalized such that

All integrals in (37) and (41) can be evaluated simply by using

The integral in (37) is then calculated as

From Eqs. (11) and (37) along with Eq. (43), we obtain

which is

Experiments [5] indicate that the first term

Hence at

The lower the temperature of the initial state, the tunneling occurs more frequently. The particle current runs from the low- to the high-temperature end, the opposite direction to that of the conduction in the

## 6. Resistivity above the critical temperature

We use simple kinetic theory to describe the transport properties [22]. Kinetic theory was originally developed for a dilute gas. Since a conductor is far from being the gas, we shall discuss the applicability of kinetic theory. The Bloch wave packet in a crystal lattice extends over one unit cell, and the lattice-ion force averaged over a unit cell vanishes. Hence the conduction electron (“electron,” “hole”) runs straight and changes direction if it hits an impurity or phonon (wave packet). The electron–electron collision conserves the net momentum, and hence, its contribution to the conductivity is zero. Upon the application of a magnetic field, the system develops a Hall electric field so as to balance out the Lorentz magnetic force on the average. Thus, the electron still move straight and is scattered by impurities and phonons, which makes the kinetic theory applicable.

YBCO is a “hole”-type HTSC in which “holes” are the majority carriers above

### 6.1 In-plane resistivity

Consider a system of “holes,” each having effective mass

Solving it for

for the drift velocity

where

we obtain an expression for the electrical conductivity:

where

where

are assumed, then the phonon number density

where

is the small

Using Eqs. (51), (52), and (54), we obtain

Similar calculations apply to “electrons.” We obtain

The resistivity

Let us now consider a system of + pairons, each having charge

Since

Newton’s equation of motion is

yielding

where

where

The rate

The total conductivity

while the conductivity for

In
_{4} system, “electrons” and
_{2}Cu_{3}

At higher temperature

in agreement with experiments (Figure 8). This part arises mainly from the conduction electrons scattered by phonon. At the low temperatures close to the critical temperature

This behavior arises mainly from the pairons scattered by phonons. The agreement with the data represents one of the most important experimental supports for the BEC picture of superconductivity.

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