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.
- Seebeck coefficient
- in-plane thermoelectric power
- out-of-plane thermoelectric power
- moving bosonic pairons (Cooper pairs)
- Bose-Einstein condensation
In 1986, Bednorz and Müller  reported their discovery of the first of the high-cuprate superconductors (La-Ba-Cu-O, K). Since then many investigations [2, 3] have been carried out on high-superconductors (HTSC) including Y-Ba-Cu-O (YBCO) with K . These compounds possess all of the main superconducting properties, including zero resistance, Meissner effect, flux quantization, Josephson effect, gaps in the excitation energy spectra, and sharp phase transition. In addition these HTSC are characterized by (i) two-dimensional (2D) conduction, (ii) short zero-temperature coherence length (Å), (iii) high critical temperature (100 K), and (iv) two energy gaps. The transport behaviors above are significantly different from those of a normal metal.
YBCO has a critical (superconducting) temperature K, which is higher than the liquid nitrogen temperature (77 K). This makes it a very useful superconductor. Terasaki et al. [5, 6] measured the resistivity , the Hall coefficient , and the Seebeck coefficient (thermoelectric power) in YBCO above the critical temperature . A summary of the data is shown in Figure 1. In-plane Hall coefficient is positive and temperature -independent, while in-plane Seebeck coefficient is negative and -independent (anomaly). Thus, there are different charge carriers for the Ohmic conduction and the thermal diffusion. We know that the carrier’s mass is important in the Ohmic currents. Lighter mass particles contribute more to the conductivity. The independence of and suggests that “electrons” and “holes” are responsible for the behaviors. We shall explain this behavior, by assuming “electrons” and “holes” as carriers and using statistical mechanical calculations. Out-of-plane Hall coefficient is negative and temperature-independent, while out-of-plane Seebeck coefficient is roughly temperature -linear. We shall show that the pairons, whose Bose condensation generates the supercurrents below , are responsible for this strange -linear behavior. The in-plane resistivity appears to have -linear and -quadratic components. We discuss the resistivity above the critical temperature in Section 6.
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., Å, Å, Å for YBCO). The lattice structure of YBCO is shown in Figure 2. The succession of layers along the -axis can be represented by CuO–BaO–CuO2–Y-CuO2–BaO-CuO–[CuO–BaO–…]. The buckled CuO2 plane where Cu-plane and O-plane are separated by a short distance as shown is called the copper planes. The two copper planes separated by yttrium (Y) are about 3 Å apart, and they are believed to be responsible for superconductivity.
The conductivity measured is a few orders of magnitude smaller along the -axis than perpendicular to it . This appears to contradict the prediction based on the naive application of the Bloch theorem. This puzzle may be solved as follows . Suppose an electron jumps from one conducting layer to its neighbor. This generates a change in the charge states of the layers involved. If each layer is macroscopic in dimension, we must assume that the charge state of the th layer can change without limits: in units of the electron charge (magnitude) . Because of unavoidable short circuits between layers due to lattice imperfections, these may not be large. At any rate if are distributed at random over all layers, then the periodicity of the potential for electron along the -axis is destroyed. The Bloch theorem based on the electron potential periodicity does not apply even though the lattice is periodic along the -axis. As a result there are no -vectors along the -axis. This means that the effective mass in the -axis direction is infinity, so that the Fermi surface for a layered conductor is a right cylinder with its axis along the -axis. Hence a 2D conduction is established.
Since electric currents flow in the copper planes, there are continuous -vectors and Fermi energy . Many experiments [1, 2, 3, 9] indicate that a singlet pairs with antiparallel spins called 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 and a linear energy-momentum () relation:
may be assumed, where is the sound speed. The attraction generated by the exchange of longitudinal acoustic phonons is long-ranged. This mechanism is good for a type I superconductor whose pairon size is of the order of Å. This attraction is in action also for a HTSC, but it alone is unlikely to account for the much smaller pairon size.
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 , , and are constants. The attraction generated by the exchange of a massive boson is short-ranged just as the short-ranged nuclear force between two nucleons generated by the exchange of massive pions, first shown by Yukawa . Lattice constants for YBCO are given by Å, and the limit wavelengths at the Brillouin boundary are twice these values. The observed coherence length is of the same order as :
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 - and -axes, Cu atoms form a square lattice of a lattice constant Å, as shown in Figure 3. Twice as many oxygen (O) atoms as copper (Cu) atoms occupy midpoints of the nearest neighbors (Cu, Cu) in the plane.
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 rather than the first neighbor directions and . The Bloch wave packets are superposable; therefore, the electron can move in any direction characterized by the two-dimensional -vectors with bases taken along and . If the number density of electrons is small, the Fermi surfaces should then be a small circle as shown in the central part in Figure 4.
Second, we consider a hole wave packet that extends over more than one O-site. It may move easily in because the Cu-sublattice of a uniform charge distribution favors such a motion. If the number of holes is small, the Fermi surface should consist of the four small pockets shown in Figure 4. Under the assumption of such a Fermi surface, pair creation of pairons via an optical phonon may occur as shown in the figure. Here a single-phonon exchange generates an electron transition from in the O-Fermi sheet to in the Cu-Fermi sheet and another electron transition from to , creating the pairon at and the +pairon at . From momentum conservation the momentum (magnitude) of a phonon must be equal to times the -distance , which is approximately equal to the momentum of an optical phonon of the smallest energy. Thus an almost insulator-like layered conductor should have a Fermi surface comprising a small electron circle and small hole pockets, which are quite favorable for forming a supercondensate by exchanging an optical phonon.
3. Quantum statistical theory of superconductivity
Following the Bardeen, Cooper, and Schrieffer (BCS) theory , we regard the phonon-exchange attraction as the cause of superconductivity. Cooper  solved Cooper’s equation and obtained a linear dispersion relation for a moving pairon:
where is the ground-state energy of the Cooper pair (pairon) and is the Fermi speed. This relation was obtained for a three-dimensional (3D) system. For a 2D system, we obtain
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 , the superconducting temperature is regarded as the Bose-Einstein condensation (BEC) point of pairons. The center of mass of a pairon moves as a boson . Its proof is given in Appendix for completeness. The critical temperature in 2D is given by
where is the pairon density. The inter-pairon distance
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 of the order of :
Thus, the pairons are there above the superconducting temperature . The angle-resolved photoemission spectroscopy (ARPES)  confirms this picture.
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. .
Loram et al.  extensively studied the electronic heat capacity of YBa2Cuwith varying oxygen concentrations . A summary of their data is shown in Figure 5. The data are in agreement with what is expected of a Bose-Einstein (B-E) condensation of free massless bosons in 2D, a peak with no jump at with the -law decline on the low-temperature side. The maximum heat capacity at with a shoulder on the high-temperature side can only be explained naturally from the view that the superconducting transition is a macroscopic change of state generated by the participation of a great number of pairons with no dissociation. The standard BCS model regards their as the pair dissociation point and predicts no features above .
The molar heat capacity for a 2D massless bosons rises like in the condensed region and reaches at ; its temperature derivative jumps at this point. The order of phase transition is defined to be that order of the derivative of the free energy whose discontinuity appears for the first time. Since , , the B-E condensation is a third-order phase transition. The temperature behavior of the heat capacity in Figure 6 is remarkably similar to that of YBa2Cu3(optimal sample) in Figure 5. This is an important support for the quantum statistical theory. Other support is discussed in Sections 5 and 6.
Our quantum statistical theory can be applied to 3D superconductors as well. The linear dispersion relation (4) holds. The superconducting temperature in 3D is given by
which is identified as the BEC point. The molar heat capacity for 3D bosons with the linear dispersion relation rises like and reaches , gas constant, at . It then drops abruptly by and approaches in the high-temperature limit. This temperature behavior of is shown in Figure 7. The phase transition is of second order. This behavior is good agreement with experiments, which supports the BEC picture of superconductivity.
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 is applied across a conductor, an electromotive force (emf) is generated. For small potential and temperature gradients, the linear relation between the electric current density j and the gradients
holds, where is the electric field and is the conductivity. If the ends of the conducting bar are maintained at different temperatures, no electric current flows. Thus from Eq. (11), we obtain
where is the field generated by the thermal emf. The Seebeck coefficient , also called the thermoelectric power or the thermopower, is defined through
The conductivity is always positive, but the Seebeck coefficient can be positive or negative depending on the materials. We present a kinetic theory to explain Terasaki et al.’s experimental results [5, 6] for the Seebeck coefficient in YBa2Cu3, reproduced in Figure 1.
We assume that the carriers are conduction electrons (“electron,” “hole”) with charge (for “electron,” for “hole”) and effective mass . At a finite temperature , “electrons” (“holes”) are excited near the Fermi surface if the surface curvature is negative (positive) . The “electron” (“hole”) is a quasi-electron which has an energy higher lower than the Fermi energy and 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” , having energies greater than the Fermi energy , is defined and calculated as
where is the density of states. This formula holds for 2D and 3D in high degeneracy. The density of thermally excited “electrons,”
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 is the diffusion constant, which is computed from the standard formula:
where is the Fermi velocity and the relaxation time of the charged particles. The symbol denotes the dimension. The density gradient is generated by the temperature gradient and is given by
We divide by the conductivity
and obtain the Seebeck coefficient [see Eq. (13)]:
The relaxation time cancels out from numerator and denominator. This result is independent of the temperature .
4.2 In-plane thermopower for YBCO
We apply our theory to explain the in-plane thermopower data for YBCO. For YBa2Cu3(composite), there are “electrons” and “holes”. The “holes”, having smaller and higher , dominate in the Ohmic conduction and also in the Hall voltage , yielding a positive Hall coefficient (see Figure 1). But the experiments indicate that the in-plane thermopower is negative. This puzzle may be solved as follows.
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 , contribute more to , and hence the thermopower can be negative as shown below.
If phonon scattering is assumed, then the scattering rate is given by
where is the scattering diameter and denotes the phonon population given by the Planck distribution function:
where is a phonon energy. We then obtain
The total conductivity is
The factors drop out from numerator and denominator. The obtained Seebeck coefficient is negative and -independent, in agreement with experiments in YBa2Cu3, reproduced in Figure 1.
5. Out-of-plane thermopower
where and are constants and is the in-plane resistivity. The first term arises from the in-plane conduction due to the (predominant) “holes” and pairons. The second term arises from the pairons’ quantum tunneling between the copper planes . Pairons move with a linear dispersion relation :
with being the binding energy of a pairon. The Hall coefficient (current along the -axis) is observed to be negative, indicating that the carriers have negative charge (see Figure 1).
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 and are, respectively, the initial (final) momentum and energy and is the imperfection-perturbation. We assume a constant absolute squared matrix-elements . The current density along the -axis due to a group of particles having charge and momentum-energy is calculated from
where is the 2D number density, the interlayer distance, and represents the current density from the high (low)-temperature end. Pairons move with the same speed , but the velocity component is
Lower-energy (smaller ) pairons are more likely to get trapped by the imperfection and going into tunneling. We represent this tendency by , where is a constant having the dimension of energy/length. Since the thermal average of the is different, a steady current is generated. The temperature difference causes a change in the B-E distribution :
where is the chemical potential. We compute the current density from
assuming a small . Not all pairons reaching an imperfection are triggered into tunneling. The factor contains this correction.
At the BEC temperature , the chemical potential vanishes:
is negative and small in magnitude for . For high temperature and low density, the B-E distribution function can be approximated by the Boltzmann distribution function:
which is normalized such that
The integral in (37) is then calculated as
which is -independent.
Hence at , we have an expression for the out-of-plane Seebeck coefficient above the critical temperature:
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 -plane. Hence , which is in accord with experiments (see Figure 1).
6. Resistivity above the critical temperature
We use simple kinetic theory to describe the transport properties . 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 , while is an “electron”-type HTSC.
6.1 In-plane resistivity
Consider a system of “holes,” each having effective mass and charge , scattered by phonons. Assume a weak electric field applied along the -axis. Newton’s equation of motion for the “hole” with the neglect of the scattering is
Solving it for and assuming that the acceleration persists in the mean-free time , we obtain
for the drift velocity . The current density (-component) is given by
where is the “hole” density. Assuming Ohm’s law
we obtain an expression for the electrical conductivity:
where is the scattering rate. The phonon scattering rate can be computed, using
where is the scattering diameter. If acoustic phonons having average energies
are assumed, then the phonon number density is given by .
is the small -space area where the acoustic phonons are located.
Similar calculations apply to “electrons.” We obtain
The resistivity is the inverse of the conductivity . Hence the resistivity for YBCO is proportional to the temperature :
Let us now consider a system of + pairons, each having charge and moving with the linear dispersion relation:
Newton’s equation of motion is
yielding initial velocity. After averaging over the angles, we obtain
where is the pairon mean free time and the angular brackets denote a thermal average. Using this and Ohm’s law, we obtain
where is the pairon density and is the pairon scattering rate. If we assume a Boltzmann distribution for bosonic pairons above , then we obtain
The rate is calculated with the assumption of a phonon scattering. We then obtain
while the conductivity for is given by , and hence the resistivity is similarly given by
In while in YCuO4 system, “electrons” and pairons play an essential role for the conduction. In YBa2Cu3the “holes” and pairons are the major carriers in the in-plane resistivity. The resistivity in the plane () vs. temperature () in various samples at optimum doping after Iye  is shown in Figure 8. The overall data are consistent with our formula.
At higher temperature , the resistivity is linear (see formula (58)):
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 , the in-plane resistivity shows a -quadratic behavior [see formula (66)]:
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.