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Quantum Theory of the Seebeck Coefficient in YBCO

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Shigeji Fujita and Akira Suzuki

Submitted: November 5th, 2018 Reviewed: April 16th, 2019 Published: May 27th, 2019

DOI: 10.5772/intechopen.86378

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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
  • supercurrent
  • YBCO

1. Introduction

In 1986, Bednorz and Müller [1] reported their discovery of the first of the high-Tccuprate superconductors (La-Ba-Cu-O, Tc>30K). Since then many investigations [2, 3] have been carried out on high-Tcsuperconductors (HTSC) including Y-Ba-Cu-O (YBCO) with Tc94K [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 ξ0(10Å), (iii) high critical temperature Tc(100 K), and (iv) two energy gaps. The transport behaviors above Tcare significantly different from those of a normal metal.

YBCO has a critical (superconducting) temperature Tc94K, 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 RH, and the Seebeck coefficient (thermoelectric power) Sin YBCO above the critical temperature Tc. A summary of the data is shown in Figure 1. In-plane Hall coefficient RabHis positive and temperature T-independent, while in-plane Seebeck coefficient Sabis 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 Tindependence of RabHand Sabsuggests 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 RcHis negative and temperature-independent, while out-of-plane Seebeck coefficient Scis roughly temperature T-linear. We shall show that the pairons, whose Bose condensation generates the supercurrents below Tc, 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 ρabove the critical temperature Tcin Section 6.

Figure 1.

Normal-state transport of highly oxygenated YBa2Cu3O7δafter Terasaki et al.’s [5,6]. Resistivities (top panel); Hall coefficients (middle panel); Seebeck coefficient (bottom panel). The subscripts ab and c denote in-copper plane and out-of-plane directions, respectively.

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=0).


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

HTSC have layered structuressuch that the copper planes comprising Cu and O are periodically separated by a great distance (e.g., a=3.88Å, b=3.82Å, c=11.68Å 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–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.

Figure 2.

Arrangement of atoms in a crystal of YBa2Cu3O7.

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 Qnof the nth layer can change without limits: Qn=,2,1, 0, 1, 2,in units of the electron charge (magnitude) e. Because of unavoidable short circuits between layers due to lattice imperfections, these Qnmay not be large. At any rate if Qnare distributed at randomover 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 nok-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 thec-axis. Hence a 2D conduction is established.

Since electric currents flow in the copper planes, there are continuous k-vectors and Fermi energy εF. Many experiments [1, 2, 3, 9] indicate that a singlet pairs with antiparallel spins called Cooper pairs(pairons) form a supercondensate below Tc.

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 λand a linear energy-momentum (εk) relation:


may be assumed, where csis 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 104Å. 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 ε0, A1, and A2are 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 a1a2=3.88,3.82Å, and the limit wavelengths λminat the Brillouin boundary are twice these values. The observed coherence length ξ0is of the same order as λmin:


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 a0=3.85Å, 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 x1x2plane.

Figure 3.

The idealized copper plane contains twice as many O’s as Cu’s.

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 110rather than the first neighbor directions 100and 010. The Bloch wave packets are superposable; therefore, the electron can move in any direction characterized by the two-dimensional k-vectors with bases taken along 110and 11¯0. 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.

Figure 4.

The two-dimensional Fermi surface of a cuprate model has a small circle (electrons) at the center and a set of four small pockets (holes) at the Brillouin boundary. Exchange of a phonon can create the electron pairon atBBand the hole pairon atAA. The phonon must have a momentumpk, withkbeing greater than the distance between the electron circle and the hole pockets.

Second, we consider a hole wave packet that extends over more than one O-site. It may move easily in 100because 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 Ain the O-Fermi sheet to Bin the Cu-Fermi sheet and another electron transition from Ato B, creating the pairon at BBand the +pairon at AA. From momentum conservation the momentum (magnitude) of a phonon must be equal to 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.


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 w0is the ground-state energy of the Cooper pair (pairon) and vFis 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 Tcis 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 Tcin 2D is given by


where nis 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 εbof the order of kBTb:


Thus, the pairons are there above the superconducting temperature Tc. 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 YBa2CuO6+δwith varying oxygen concentrations 6+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) condensation of free massless bosons in 2D, a peak with no jump at Tcwith the T2-law decline on the low-temperature side. The maximum heat capacity at Tcwith 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 Tcas the pair dissociation point and predicts no features above Tc.

Figure 5.

Electronic heat capacityCelplotted asCel/Tvs. temperatureTafter Loram et al. [15] for YBa2Cu3O6+δwith theδvalues shown.

The molar heat capacity Cfor a 2D massless bosons rises like T2in the condensed region and reaches 4.38Rat T=Tc; its temperature derivative CTn/Tjumps at this point. The order of phase transitionis defined to be that order of the derivative of the free energy Fwhose discontinuity appears for the first time. Since CV=TS/TV=T2F/T2, CV/T=T3F/T32F/T2, the B-E condensation is a third-order phase transition. The temperature behavior of the heat capacity Cin Figure 6 is remarkably similar to that of YBa2Cu3O6.92(optimal sample) in Figure 5. This is an important support for the quantum statistical theory. Other support is discussed in Sections 5 and 6.

Figure 6.

The molar heat capacityCfor 2D massless bosons rise likeT2, reaches4.38Rat the critical temperatureTc, and then decreases to2Rin the high-temperature limit.

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


which is identified as the BEC point. The molar heat capacity Cfor 3D bosons with the linear dispersion relation ε=cprises like T3and reaches 10.8R, R=gas constant, at Tc=2.02cn01/3. It then drops abruptly by 6.57Rand approaches 3Rin the high-temperature limit. This temperature behavior of Cis 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 7.

The molar heat capacityCfor 3D massless bosons rises likeT3and reaches10.8Rat the critical temperatureTc=2.02cn01/3. It then drops abruptly by6.57Rand approaches the high-temperature limit3R.


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 Eis applied across a conductor, an electromotive force (emf) is generated. For small potential and temperature gradients, the linear relation between the electric current density jand the gradients


holds, where E=Vis the electric field and σis 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 ESis the field generated by the thermal emf. The Seebeck coefficient S, also called the thermoelectric power or the thermopower, is defined through


The conductivity σis always positive, but the Seebeck coefficient Scan 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 YBa2Cu3O7δ, reproduced in Figure 1.

We assume that the carriers are conduction electrons (“electron,” “hole”) with charge q(efor “electron,” +efor “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-electronwhich has an energy higher lower than the Fermi energy εFandwhich 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” Nex, having energies greater than the Fermi energy εF, is defined and calculated as


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


where vFis the Fermi velocity and τthe relaxation time of the charged particles. The symbol ddenotes the dimension. The density gradient nexis generated by the temperature gradient Tand is given by


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


We divide Aby the conductivity


and obtain the Seebeck coefficient S[see Eq. (13)]:


The relaxation time τcancels out from numerator and denominator. This result is independent of the temperature T.

4.2 In-plane thermopower for YBCO

We apply our theory to explain the in-plane thermopower data for YBCO. For YBa2Cu3O7δ(composite), there are “electrons” and “holes”. The “holes”, having smaller mand higher vF2εF/m1/2, dominate in the Ohmic conduction and also in the Hall voltage VH, yielding a positive Hall coefficient RabH(see Figure 1). But the experiments indicate that the in-plane thermopower Sabis 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 m1, contribute more to A, and hence the thermopower Sabcan 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 sis the scattering diameter and nphdenotes the phonon population given by the Planck distribution function:


where εphis a phonon energy. We then obtain


The total conductivity is


Using Eqs. (25)(29), we obtain the in-plane thermopower Sababove the critical temperature as


The factors nphsdrop out from numerator and denominator. The obtained Seebeck coefficient Sabis negative and T-independent, in agreement with experiments in YBa2Cu3O7δ, reproduced in Figure 1.


5. Out-of-plane thermopower

Terasaki et al. [17, 18] and Takenaka et al. [19] measured the out-of-plane resistivity ρcin YBa2Cu3Ox. In the range 6.6<x<6.92, the data for ρccan be fitted with


where C1and C2are constants and ρabis the in-plane resistivity. The first term C1ρabarises from the in-plane conduction due to the (predominant) “holes” and +pairons. The second term C2/Tarises from the pairons’ quantum tunneling between the copper planes [20]. Pairons move with a linear dispersion relation [21]:


with w0being the binding energy of a pairon. The Hall coefficient RcH(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 pipfand εiεfare, respectively, the initial (final) momentum and energy and Uis the imperfection-perturbation. We assume a constant absolute squared matrix-elements M2. The current density jcialong the c-axis due to a group of particles ihaving charge qiand momentum-energy pεis calculated from


where niis the 2D number density, a0the interlayer distance, and jc,Hijc,Lirepresents the current density from the high (low)-temperature end. Pairons move with the same speed c=2/πvF, but the velocity component vxis


Lower-energy (smaller p) pairons are more likely to get trapped by the imperfection and going into tunneling. We represent this tendency by K=B/ε, where Bis a constant having the dimension of energy/length. Since the thermal average of the vis different, a steady current is generated. The temperature difference ΔT=THTLcauses a change in the B-E distribution F:


where μis the chemical potential. We compute the current density jcfrom


assuming a small ΔT. Not all pairons reaching an imperfection are triggered into tunneling. The factor Bcontains this correction.

At the BEC temperature Tc, the chemical potential μvanishes:




is negative and small in magnitude for T>Tc. For high temperature and low density, the B-E distribution function Fcan be approximated by the Boltzmann distribution function:


which is normalized such that


All integrals in (37) and (41) can be evaluated simply by using 0dxexxn=n!. Hence we obtain


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 C1ρabin (31) is dominant for x>6.8:


Hence at x=7, we have an expression for the out-of-plane Seebeck coefficient Scabove 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 Sc>0, 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 [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 Tc, while Nd1.84Ce0.16CuO4is an “electron”-type HTSC.

6.1 In-plane resistivity

Consider a system of “holes,” each having effective mass m2and charge +e, scattered by phonons. Assume a weak electric field Eapplied along the x-axis. Newton’s equation of motion for the “hole” with the neglect of the scattering is


Solving it for vxand assuming that the acceleration persists in the mean-free time τ2, we obtain


for the drift velocity vd. The current density (x-component) jis given by


where n2is the “hole” density. Assuming Ohm’s law


we obtain an expression for the electrical conductivity:


where Γ2τ21is the scattering rate. The phonon scattering rate can be computed, using


where A2is the scattering diameter. If acoustic phonons having average energies


are assumed, then the phonon number density nphis given by [23].




is the small k-space area where the acoustic phonons are located.

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


Similar calculations apply to “electrons.” We obtain


The resistivity ρis the inverse of the conductivity σ. Hence the resistivity for YBCO is proportional to the temperature T:


Let us now consider a system of + pairons, each having charge +2eand moving with the linear dispersion relation:




Newton’s equation of motion is


yielding vx=2ec2/εEt+initial velocity. After averaging over the angles, we obtain


where τ3is the pairon mean free time and the angular brackets denote a thermal average. Using this and Ohm’s law, we obtain


where n3is the pairon density and Γ3is the pairon scattering rate. If we assume a Boltzmann distribution for bosonic pairons above Tc, then we obtain


The rate Γ3is calculated with the assumption of a phonon scattering. We then obtain


The total conductivity σfor YBCO is σ2+σ3. Thus taking the inverse of σ, we obtain, by using the results (56) and (65):


while the conductivity for Nd1.84Ce0.16CuO4is given by σ1+σ3, and hence the resistivity is similarly given by


In Nd1.84Ce0.16while in YCuO4 system, “electrons” and pairons play an essential role for the conduction. In YBa2Cu3O7δthe “holes” and +pairons are the major carriers in the in-plane resistivity. The resistivity in the plane (ρab) 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.

Figure 8.

Resistivity in the ab plane,ρabvs. temperatureT. Solid lines represent data for HTSC at optimum doping and dashed lines data for highly overdosed samples, after Iye [24].

At higher temperature >160K, the resistivity ρabis 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 Tc, the in-plane resistivity ρabshows a T-quadratic behavior [see formula (66)]:

ρabT2near  and  aboveTc.E69

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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Written By

Shigeji Fujita and Akira Suzuki

Submitted: November 5th, 2018 Reviewed: April 16th, 2019 Published: May 27th, 2019