Quantum Theory of the Seebeck Coefficient in YBCO

The measured in-plane thermoelectric power (Seebeck coefficient) S ab in YBCO below the superconducting temperature T c ( (cid:1) 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.


Introduction
In 1986, Bednorz and Müller [1] reported their discovery of the first of the high-T c cuprate superconductors (La-Ba-Cu-O, T c > 30 K). Since then many investigations [2,3] have been carried out on high-T c superconductors (HTSC) including Y-Ba-Cu-O (YBCO) with T c $ 94 K [4]. 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 ξ 0 ($ 10Å), (iii) high critical temperature T c ($ 100 K), and (iv) two energy gaps. The transport behaviors above T c are significantly different from those of a normal metal.
YBCO has a critical (superconducting) temperature T c $ 94 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 ρ, the Hall coefficient R H , and the Seebeck coefficient (thermoelectric power) S in YBCO above the critical temperature T c . A summary of the data is shown in Figure 1. In-plane Hall coefficient R H ab is positive and temperature T ð Þ-independent, while in-plane Seebeck coefficient S ab is negative and T-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 T independence of R H ab and S ab 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 R H c is negative and temperatureindependent, while out-of-plane Seebeck coefficient S c is roughly temperature T ð Þ-linear. We shall show that the pairons, whose Bose condensation generates the supercurrents below T c , are responsible for this strange T-linear behavior. The in-plane resistivity appears to have T-linear and T-quadratic components. We discuss the resistivity ρ above the critical temperature T c 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 (S ¼ 0).

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., a ¼ 3:88 Å, b ¼ 3:82 Å, c ¼ 11:68 Å for YBCO). The lattice structure of YBCO is shown in Figure 2. The succession of layers along the c-axis can be represented by CuO-BaO-CuO 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 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 c-axis than perpendicular to it [7]. This appears to contradict the prediction based on the naive application of the Bloch theorem. This puzzle may be solved as follows [8]. 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 Q n of the nth layer can change without limits: Q n ¼ …, À 2, À 1, 0, 1, 2, … in units of the electron charge (magnitude) e. Because of unavoidable short circuits between layers due to lattice imperfections, these Q n may not be large. At any rate if Q n are distributed at random over all layers, then the periodicity of the potential for electron along the c-axis is destroyed. The Bloch theorem based on the electron potential periodicity does not apply even though the lattice is periodic along the c-axis. As a result there are no k-vectors along the c-axis. This means that the effective mass in the c-axis direction is infinity, so that the Fermi surface for a layered conductor is a right cylinder with its axis along the c-axis. Hence a 2D conduction is established.
Since electric currents flow in the copper planes, there are continuous k-vectors and Fermi energy ε F . Many experiments [1][2][3]9] indicate that a singlet pairs with antiparallel spins called Cooper pairs (pairons) form a supercondensate below T c .
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 λ and a linear energy-momentum (ε-ℏk) relation: may be assumed, where c s 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 10 4 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 ε 0 , A 1 , and A 2 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 [10]. Lattice constants for YBCO are given by a 1 ; a 2 ð Þ¼ 3:88; 3:82 ð ÞÅ, and the limit wavelengths λ min ð Þat the Brillouin boundary are twice these values. The observed coherence length ξ 0 is of the same order as λ min : 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 a-and b-axes, Cu atoms form a square lattice of a lattice constant a 0 ¼ 3:85 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 x 1 -x 2 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 110 h i rather than the first neighbor directions 100 ½ and 010 ½ . The Bloch wave packets are superposable; therefore, the electron can move in any direction characterized by the twodimensional k-vectors with bases taken along 110 ½ and 110 Â Ã . 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 100 h ibecause the Cu-sublattice of a uniform charge Þand the +pairon at A; A 0 ð Þ. From momentum conservation the momentum (magnitude) of a phonon must be equal to ℏ times the k-distance AB, 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.

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 w 0 is the ground-state energy of the Cooper pair (pairon) and v F 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 [13], the superconducting temperature T c is regarded as the Bose-Einstein condensation (BEC) point of pairons. The center of mass of a pairon moves as a boson [13]. Its proof is given in Appendix for completeness. The critical temperature T c in 2D is given by where n 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 ε b of the order of k B T b : Thus, the pairons are there above the superconducting temperature T c . The angle-resolved photoemission spectroscopy (ARPES) [14] 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. [15].
Loram et al. [15] extensively studied the electronic heat capacity of YBa 2 CuO 6þδ with varying oxygen concentrations 6 þ δ. 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) Electronic heat capacity C el plotted as C el =T vs. temperature T after Loram et al. [15] for YBa 2 Cu 3 O 6þδ with the δ values shown.
condensation of free massless bosons in 2D, a peak with no jump at T c with the T 2law decline on the low-temperature side. The maximum heat capacity at T c 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 T c as the pair dissociation point and predicts no features above T c .
The molar heat capacity C for a 2D massless bosons rises like T 2 in the condensed region and reaches 4:38 R at T ¼ T c ; its temperature derivative ∂C T; n ð Þ=∂T jumps at this point. The order of phase transition is defined to be that order of the derivative of the free energy F whose discontinuity appears for the first time.
, the B-E condensation is a third-order phase transition. The temperature behavior of the heat capacity C in Figure 6 is remarkably similar to that of YBa 2 Cu 3 O 6:92 (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 T c in 3D is given by which is identified as the BEC point. The molar heat capacity C for 3D bosons with the linear dispersion relation ε ¼ cp rises like T 3 and reaches 10:8 R, R ¼ gas constant, at T c ¼ 2:02 ℏcn 1=3 0 . It then drops abruptly by 6:57 R and approaches 3 R in the high-temperature limit. This temperature behavior of C 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. Figure 6. The molar heat capacity C for 2D massless bosons rise like T 2 , reaches 4:38 R at the critical temperatureT c , and then decreases to 2R in the high-temperature limit.
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 E 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 E ¼ À∇V is the electric field and σ 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 E S is the field generated by the thermal emf. The Seebeck coefficient S, also called the thermoelectric power or the thermopower, is defined through The conductivity σ is always positive, but the Seebeck coefficient S 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 YBa 2 Cu 3 O 7Àδ , reproduced in Figure 1.
We assume that the carriers are conduction electrons ("electron," "hole") with charge q (Àe for "electron," þe for "hole") and effective mass m * . At a finite temperature T > 0, "electrons" ("holes") are excited near the Fermi surface if the surface curvature is negative (positive) [16]. The "electron" ("hole") is a quasi-electron which has an energy higher lower than the Fermi energy ε F and which circulates clockwise (counterclockwise) viewed from the tip of the applied magnetic field vector. The molar heat capacity C for 3D massless bosons rises like T 3 and reaches 10:8R at the critical temperature T c ¼ 2:02ℏcn 1=3 0 . It then drops abruptly by 6:57 R and approaches the high-temperature limit 3R.
"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" N ex , having energies greater than the Fermi energy ε F , is defined and calculated as where D ε ð Þ 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 D is the diffusion constant, which is computed from the standard formula: where v F is the Fermi velocity and τ the relaxation time of the charged particles. The symbol d denotes the dimension. The density gradient ∇n ex is generated by the temperature gradient ∇T and is given by where Eq. (14) is used. Using Eqs. (17)- (19) and (11), we obtain the thermal diffusion coefficient A as We divide A by the conductivity and obtain the Seebeck coefficient S [see Eq. (13)]: The relaxation time τ cancels out from numerator and denominator. This result is independent of the temperature T.

In-plane thermopower for YBCO
We apply our theory to explain the in-plane thermopower data for YBCO. For YBa 2 Cu 3 O 7Àδ (composite), there are "electrons" and "holes". The "holes", having smaller m * and higher v F 2ε F =m * ð Þ 1=2 , dominate in the Ohmic conduction and also in the Hall voltage V H , yielding a positive Hall coefficient R H ab (see Figure 1). But the experiments indicate that the in-plane thermopower S ab 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 m * 1 , contribute more to A, and hence the thermopower S ab can be negative as shown below.
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 s is the scattering diameter and n ph denotes the phonon population given by the Planck distribution function: where ε ph is a phonon energy. We then obtain The total conductivity is Using Eqs. (25)-(29), we obtain the in-plane thermopower S ab above the critical temperature as The factors n ph s drop out from numerator and denominator. The obtained Seebeck coefficient S ab is negative and T-independent, in agreement with experiments in YBa 2 Cu 3 O 7Àδ , reproduced in Figure 1.

Out-of-plane thermopower
Terasaki et al. [17,18] and Takenaka et al. [19] measured the out-of-plane resistivity ρ c in YBa 2 Cu 3 O x . In the range 6:6 < x < 6:92, the data for ρ c can be fitted with where C 1 and C 2 are constants and ρ ab is the in-plane resistivity. The first term C 1 ρ ab arises from the in-plane conduction due to the (predominant) "holes" and þ pairons. The second term C 2 =T arises from the À pairons' quantum tunneling between the copper planes [20]. Pairons move with a linear dispersion relation [21]: with |w 0 | being the binding energy of a pairon. The Hall coefficient R H c (current along the c-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 p i p f À Á and ε i ε f ð Þ are, respectively, the initial (final) momentum and energy and U is the imperfection-perturbation. We assume a constant absolute squared matrix-elements M 2 . The current density j i ð Þ c along the c-axis due to a group of particles i having charge q i ð Þ and momentum-energy p; ε ð Þ is calculated from where n i ð Þ is the 2D number density, a 0 the interlayer distance, and j i ð Þ c, H j i ð Þ c, L represents the current density from the high (low)-temperature end. Pairons move with the same speed c ¼ 2=π Lower-energy (smaller p) pairons are more likely to get trapped by the imperfection and going into tunneling. We represent this tendency by K ¼ B=ε, where B is a constant having the dimension of energy/length. Since the thermal average of the v is different, a steady current is generated. The temperature difference where μ is the chemical potential. We compute the current density j c from assuming a small ΔT. Not all pairons reaching an imperfection are triggered into tunneling. The factor B contains this correction.
At the BEC temperature T c ð Þ, the chemical potential μ vanishes: and is negative and small in magnitude for T > T c . For high temperature and low density, the B-E distribution function F can be approximated by the Boltzmann distribution function: 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 T-independent. Experiments [5] indicate that the first term C 1 ρ ab in (31) is dominant for x > 6:8: Hence at x ¼ 7, we have an expression for the out-of-plane Seebeck coefficient S c 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 ab-plane. Hence S c > 0, which is in accord with experiments (see Figure 1).

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 T c , while Nd 1:84 Ce 0:16 CuO 4 is an "electron"-type HTSC.

In-plane resistivity
Consider a system of "holes," each having effective mass m * 2 and charge þe, scattered by phonons. Assume a weak electric field E applied along the x-axis. Newton's equation of motion for the "hole" with the neglect of the scattering is Solving it for v x and assuming that the acceleration persists in the mean-free time τ 2 , we obtain for the drift velocity v d . The current density (x-component) j is given by where n 2 is the "hole" density. Assuming Ohm's law we obtain an expression for the electrical conductivity: where τ 3 is the pairon mean free time and the angular brackets denote a thermal average. Using this and Ohm's law, we obtain where n 3 is the pairon density and Γ 3 is the pairon scattering rate. If we assume a Boltzmann distribution for bosonic pairons above T c , then we obtain The rate Γ 3 is calculated with the assumption of a phonon scattering. We then obtain The total conductivity σ for YBCO is σ 2 þ σ 3 . Thus taking the inverse of σ, we obtain, by using the results (56) and (65): Figure 8. Resistivity in the ab plane, ρ ab vs. temperature T. Solid lines represent data for HTSC at optimum doping and dashed lines data for highly overdosed samples, after Iye [24]. ρ ab 1 σ ¼ C 2 n 2 e 2 T þ C 3 n 3 2e 2 T 2 À1 ¼ T 2 n 2 e 2 C 2 T þ 2C 3 ð Þ (66) while the conductivity for Nd 1:84 Ce 0:16 CuO 4 is given by σ 1 þ σ 3 , and hence the resistivity is similarly given by In Nd 1:84 Ce 0:16 while in YCuO 4 system, "electrons" and À pairons play an essential role for the conduction. In YBa 2 Cu 3 O 7Àδ the "holes" and þ pairons are the major carriers in the in-plane resistivity. The resistivity in the plane (ρ ab ) vs. temperature (T) in various samples at optimum doping after Iye [24] is shown in Figure 8. The overall data are consistent with our formula.
At higher temperature > 160 K ð Þ , the resistivity ρ ab 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 T c , the in-plane resistivity ρ ab shows a T-quadratic behavior [see formula (66)]: ρ ab ∝ T 2 near and above T c ð Þ : 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.