Quantum Theory of the Seebeck Coefficient in YBCO

Shigeji Fujita and Akira Suzuki

### Abstract

The measured in-plane thermoelectric power (Seebeck coefficient) Sab in YBCO below the superconducting temperature T<sup>c</sup> (�94 K) Sab 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<sup>c</sup> 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 Tc. The center of mass of pairons moves as bosons. The resistivity ρab above T<sup>c</sup> 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-T<sup>c</sup> cuprate superconductors (La-Ba-Cu-O, T<sup>c</sup> >30 K). Since then many investigations [2, 3] have been carried out on high-T<sup>c</sup> superconductors (HTSC) including Y-Ba-Cu-O (YBCO) with T<sup>c</sup> � 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<sup>0</sup> (� 10Å), (iii) high critical temperature T<sup>c</sup> (� 100 K), and (iv) two energy gaps. The transport behaviors above T<sup>c</sup> are significantly different from those of a normal metal.

YBCO has a critical (superconducting) temperature T<sup>c</sup> � 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 RH, and the Seebeck coefficient (thermoelectric power) S in YBCO above the critical temperature Tc. A summary of the data is shown in Figure 1. In-plane Hall coefficient RH ab is positive and temperature ð Þ T -independent, while in-plane Seebeck coefficient Sab 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<sup>H</sup> ab and Sab suggests that "electrons" and "holes" are responsible for the behaviors. We shall explain this behavior,

#### Figure 1.

Normal-state transport of highly oxygenated YBa2Cu3O<sup>7</sup>�<sup>δ</sup> after 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.

by assuming "electrons" and "holes" as carriers and using statistical mechanical calculations. Out-of-plane Hall coefficient R<sup>H</sup> <sup>c</sup> is negative and temperatureindependent, while out-of-plane Seebeck coefficient S<sup>c</sup> is roughly temperature ð Þ 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 ρ above the critical temperature T<sup>c</sup> in Section 6.

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

Since electric currents flow in the copper planes, there are continuous k-vectors and Fermi energy εF. Many experiments [1–3, 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

may be assumed, where c<sup>s</sup> is the sound speed. The attraction generated by the exchange of longitudinal acoustic phonons is long-ranged. This mechanism is good

ε ¼ csℏk, (1)

in dimension, we must assume that the charge state Qn of the nth layer can change without limits: Qn ¼ …, � 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 Qn may not be large. At any rate if Qn 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

Figure 2.

35

Arrangement of atoms in a crystal of YBa2Cu3O7.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

with its axis along the c-axis. Hence a 2D conduction is established.

have long wavelengths <sup>λ</sup> and a linear energy-momentum (ε‐ℏk) relation:

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).

#### 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., 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–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 c-axis than perpendicular to it [7]. This appears to contradict the prediction based on the

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

#### Figure 2. Arrangement of atoms in a crystal of YBa2Cu3O7.

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 Qn of the nth layer can change without limits: Qn ¼ …, � 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 Qn may not be large. At any rate if Qn 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 εF. Many experiments [1–3, 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 <sup>λ</sup> and a linear energy-momentum (ε‐ℏk) relation:

$$
\varepsilon = \mathfrak{c}\_{\mathfrak{s}} \hbar k,\tag{1}
$$

may be assumed, where c<sup>s</sup> is the sound speed. The attraction generated by the exchange of longitudinal acoustic phonons is long-ranged. This mechanism is good

by assuming "electrons" and "holes" as carriers and using statistical mechanical

Normal-state transport of highly oxygenated YBa2Cu3O<sup>7</sup>�<sup>δ</sup> after Terasaki et al.'s [5, 6]. Resistivities (top panel); Hall coefficients (middle panel); Seebeck coefficient (bottom panel). The subscripts ab and c denote

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independent, while out-of-plane Seebeck coefficient S<sup>c</sup> is roughly temperature ð Þ 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

In this paper we are mainly interested in the sign and the temperature behavior

completeness. There are no Seebeck currents in the superconducting state below the

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–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 responsi-

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

discuss the resistivity ρ above the critical temperature T<sup>c</sup> in Section 6.

of the Seebeck coefficient in YBCO. But we discuss the related matter for

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

<sup>c</sup> is negative and temperature-

calculations. Out-of-plane Hall coefficient R<sup>H</sup>

in-copper plane and out-of-plane directions, respectively.

critical temperature (S ¼ 0).

Figure 1.

ble for superconductivity.

34

for a type I superconductor whose pairon size is of the order of 10<sup>4</sup> 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:

$$
\varepsilon = \varepsilon\_0 + A\_1 \left( k\_1 - \frac{\pi}{a\_1} \right)^2 + A\_2 \left( k\_2 - \frac{\pi}{a\_2} \right)^2,\tag{2}
$$

where ε0, A1, and A<sup>2</sup> 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 ð Þ¼ a1; a<sup>2</sup> ð Þ 3:88; 3:82 Å, and the limit wavelengths ð Þ λmin at the Brillouin boundary are twice these values. The observed coherence length ξ<sup>0</sup> is of the same order as λmin:

$$
\xi\_0 \sim \lambda\_{\text{min}} \simeq 8\text{Å}.\tag{3}
$$

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 A in the O-Fermi sheet to B in the Cu-Fermi sheet and another electron

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 at B; B<sup>0</sup> ð Þ and the hole pairon at A; A<sup>0</sup> ð Þ. The phonon must have a momentum p � ℏk, with k being greater than

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 exchang-

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:

> 1 2

> > 2

where w<sup>0</sup> is the ground-state energy of the Cooper pair (pairon) and v<sup>F</sup> is the Fermi speed. This relation was obtained for a three-dimensional (3D) system. For a

ε ¼ w<sup>0</sup> þ

ε ¼ w<sup>0</sup> þ

The center of mass (CM) motion of a composite is bosonic (fermionic) according to whether the composite contains an even (odd) number of elementary

3. Quantum statistical theory of superconductivity

, creating the �pairon at B; B<sup>0</sup> ð Þ and the +pairon at A; A<sup>0</sup> ð Þ.

vFp, (4)

<sup>π</sup> <sup>v</sup>Fp: (5)

transition from A<sup>0</sup> to B<sup>0</sup>

the distance between the electron circle and the hole pockets.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

Figure 4.

ing an optical phonon.

2D system, we obtain

37

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<sup>0</sup> 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 <sup>x</sup><sup>1</sup>‐x<sup>2</sup> 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 <sup>k</sup>-vectors with bases taken along 110 ½ � and 110 . 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 i because the Cu-sublattice of a uniform charge

Figure 3. The idealized copper plane contains twice as many O's as Cu's.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

#### Figure 4.

for a type I superconductor whose pairon size is of the order of 10<sup>4</sup> Å. This attraction is in action also for a HTSC, but it alone is unlikely to account for the much

> a1 <sup>2</sup>

where ε0, A1, and A<sup>2</sup> 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 ð Þ¼ a1; a<sup>2</sup> ð Þ 3:88; 3:82 Å, and the limit wavelengths ð Þ λmin at the Brillouin boundary are twice these values. The observed

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<sup>0</sup> ¼ 3:85 Å, as shown in Figure 3. Twice as many oxygen (O) atoms as copper (Cu) atoms occupy midpoints

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 two-

dimensional <sup>k</sup>-vectors with bases taken along 110 ½ � and 110 . If the number density of electrons is small, the Fermi surfaces should then be a small circle as shown in the

Second, we consider a hole wave packet that extends over more than one O-site.

It may move easily in 100 h i because the Cu-sublattice of a uniform charge

of the lattice constants, and they have a quadratic dispersion relation:

<sup>ε</sup> <sup>¼</sup> <sup>ε</sup><sup>0</sup> <sup>þ</sup> <sup>A</sup><sup>1</sup> <sup>k</sup><sup>1</sup> � <sup>π</sup>

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coherence length ξ<sup>0</sup> is of the same order as λmin:

of the nearest neighbors (Cu, Cu) in the <sup>x</sup><sup>1</sup>‐x<sup>2</sup> plane.

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

central part in Figure 4.

Figure 3.

36

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

<sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>k</sup><sup>2</sup> � <sup>π</sup>

a2 <sup>2</sup>

ξ<sup>0</sup> � λmin ≃8Å: (3)

, (2)

smaller pairon size.

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 at B; B<sup>0</sup> ð Þ and the hole pairon at A; A<sup>0</sup> ð Þ. The phonon must have a momentum p � ℏk, with k being greater than the distance between the electron circle and the hole pockets.

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 A in the O-Fermi sheet to B in the Cu-Fermi sheet and another electron transition from A<sup>0</sup> to B<sup>0</sup> , creating the �pairon at B; B<sup>0</sup> ð Þ and the +pairon at A; A<sup>0</sup> 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.

#### 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:

$$
\varepsilon = w\_0 + \frac{1}{2} v\_{\text{F}} p,\tag{4}
$$

where w<sup>0</sup> is the ground-state energy of the Cooper pair (pairon) and v<sup>F</sup> is the Fermi speed. This relation was obtained for a three-dimensional (3D) system. For a 2D system, we obtain

$$
\varepsilon = w\_0 + \frac{2}{\pi} \nu\_{\rm F} p.\tag{5}
$$

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<sup>c</sup> 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<sup>c</sup> in 2D is given by

$$k\_{\rm B}T\_{\rm c} = 1.24 \,\hbar v\_{\rm F} n^{1/2},\tag{6}$$

condensation of free massless bosons in 2D, a peak with no jump at T<sup>c</sup> with the T<sup>2</sup>

law decline on the low-temperature side. The maximum heat capacity at T<sup>c</sup> 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<sup>c</sup> as the pair dissociation point and predicts no features

The molar heat capacity C for a 2D massless bosons rises like T<sup>2</sup> in the condensed region and reaches 4:38<sup>R</sup> at <sup>T</sup> <sup>¼</sup> <sup>T</sup>c; its temperature derivative <sup>∂</sup>C Tð Þ ; <sup>n</sup> <sup>=</sup>∂<sup>T</sup> 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. Since CV ¼

<sup>F</sup>=∂T<sup>2</sup> , <sup>∂</sup>CV=∂<sup>T</sup> ¼ �<sup>T</sup> <sup>∂</sup><sup>3</sup>F=∂T<sup>3</sup> � <sup>∂</sup><sup>2</sup>

condensation is a third-order phase transition. The temperature behavior of the heat capacity C in Figure 6 is remarkably similar to that of YBa2Cu3O6:<sup>92</sup> (optimal sample) in Figure 5. This is an important support for the quantum statistical theory.

Our quantum statistical theory can be applied to 3D superconductors as well. The linear dispersion relation (4) holds. The superconducting temperature T<sup>c</sup> in 3D

kBT<sup>c</sup> ¼ 1:01ℏvFn

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.

which is identified as the BEC point. The molar heat capacity C for 3D bosons with the linear dispersion relation <sup>ε</sup> <sup>¼</sup> cp rises like <sup>T</sup><sup>3</sup> and reaches 10:8R, <sup>R</sup> <sup>¼</sup> gas

1

<sup>0</sup> . It then drops abruptly by 6:57R and approaches 3R in

above Tc.

is given by

Figure 6.

39

<sup>T</sup>ð Þ <sup>∂</sup>S=∂<sup>T</sup> <sup>V</sup> ¼ �<sup>T</sup> <sup>∂</sup><sup>2</sup>

constant, at T<sup>c</sup> ¼ 2:02ℏcn

Other support is discussed in Sections 5 and 6.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

1=3

The molar heat capacity C for 2D massless bosons rise like T<sup>2</sup>

then decreases to 2R in the high-temperature limit.


F=∂T<sup>2</sup> , the B-E

<sup>3</sup>, (10)

, reaches 4:38R at the critical temperatureTc, and

where n is the pairon density. The inter-pairon distance

$$
\sigma\_0 \equiv n^{-1/2} = \mathbf{1.24} \hbar \upsilon\_\mathbf{F} (k\_\mathbf{B} T\_\mathbf{c})^{-1} \tag{7}
$$

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

$$
\xi\_0 \equiv 0.181 \hbar v\_{\rm F} (k\_{\rm B} T\_{\rm c})^{-1}. \tag{8}
$$

Hence the BEC occurs without the pairon overlap. Phonon exchange can be repeated and can generate a pairon-binding energy ε<sup>b</sup> of the order of kBTb:

$$k\_{\rm b} \equiv k\_{\rm B} T\_{\rm b}, \quad T\_{\rm b} \sim 1000 \text{ K}. \tag{9}$$

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 YBa2CuO6þ<sup>δ</sup> 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)

Figure 5. Electronic heat capacity Cel plotted as Cel=T vs. temperature T after Loram et al. [15] for YBa2Cu3O<sup>6</sup>þ<sup>δ</sup> with the δ values shown.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

fermions. The Cooper pairs, each having two electrons, move as bosons. In our quantum statistical theory of superconductivity [13], the superconducting temperature T<sup>c</sup> 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

<sup>k</sup>BT<sup>c</sup> <sup>¼</sup> <sup>1</sup>:24ℏvFn1=<sup>2</sup>

is several times greater than the BCS pairon size represented by the BCS

<sup>ξ</sup><sup>0</sup> � <sup>0</sup>:181ℏvFð Þ <sup>k</sup>BT<sup>c</sup> �<sup>1</sup>

Hence the BEC occurs without the pairon overlap. Phonon exchange can be repeated and can generate a pairon-binding energy ε<sup>b</sup> of 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

Loram et al. [15] extensively studied the electronic heat capacity of YBa2CuO6þ<sup>δ</sup> 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 Cel plotted as Cel=T vs. temperature T after Loram et al. [15] for YBa2Cu3O<sup>6</sup>þ<sup>δ</sup> with

lattice, the Fermi surface and the Hamiltonian and calculate everything, using

statistical mechanical methods. The details are given in Ref. [15].

, (6)

: (8)

<sup>r</sup><sup>0</sup> � <sup>n</sup>�1=<sup>2</sup> <sup>¼</sup> <sup>1</sup>:24ℏvFð Þ <sup>k</sup>BT<sup>c</sup> �<sup>1</sup> (7)

ε<sup>b</sup> � kBTb, T<sup>b</sup> � 1000 K: (9)

completeness. The critical temperature T<sup>c</sup> in 2D is given by

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where n is the pairon density. The inter-pairon distance

coherence length:

Figure 5.

38

the δ values shown.

condensation of free massless bosons in 2D, a peak with no jump at T<sup>c</sup> with the T<sup>2</sup> law decline on the low-temperature side. The maximum heat capacity at T<sup>c</sup> 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<sup>c</sup> as the pair dissociation point and predicts no features above Tc.

The molar heat capacity C for a 2D massless bosons rises like T<sup>2</sup> in the condensed region and reaches 4:38<sup>R</sup> at <sup>T</sup> <sup>¼</sup> <sup>T</sup>c; its temperature derivative <sup>∂</sup>C Tð Þ ; <sup>n</sup> <sup>=</sup>∂<sup>T</sup> 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. Since CV ¼ <sup>T</sup>ð Þ <sup>∂</sup>S=∂<sup>T</sup> <sup>V</sup> ¼ �<sup>T</sup> <sup>∂</sup><sup>2</sup> <sup>F</sup>=∂T<sup>2</sup> , <sup>∂</sup>CV=∂<sup>T</sup> ¼ �<sup>T</sup> <sup>∂</sup><sup>3</sup>F=∂T<sup>3</sup> � <sup>∂</sup><sup>2</sup> F=∂T<sup>2</sup> , 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 YBa2Cu3O6:<sup>92</sup> (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<sup>c</sup> in 3D is given by

$$k\_{\rm B}T\_{\rm c} = \mathbf{1.01}\hbar v\_{\rm F}n^{\frac{1}{3}},\tag{10}$$

which is identified as the BEC point. The molar heat capacity C for 3D bosons with the linear dispersion relation <sup>ε</sup> <sup>¼</sup> cp rises like <sup>T</sup><sup>3</sup> and reaches 10:8R, <sup>R</sup> <sup>¼</sup> gas constant, at T<sup>c</sup> ¼ 2:02ℏcn 1=3 <sup>0</sup> . It then drops abruptly by 6:57R and approaches 3R 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<sup>2</sup> , reaches 4:38R at the critical temperatureTc, and then decreases to 2R in the high-temperature limit.

Figure 7.

The molar heat capacity C for 3D massless bosons rises like T<sup>3</sup> and reaches 10:8R at the critical temperature <sup>T</sup><sup>c</sup> <sup>¼</sup> <sup>2</sup>:02ℏcn1=<sup>3</sup> <sup>0</sup> . It then drops abruptly by 6:57 R and approaches the high-temperature limit 3R.

### 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

$$\mathbf{j} = \sigma(-\nabla V) + A(-\nabla T) = \sigma \mathbf{E} - A \nabla T \tag{11}$$

holds, where E ¼ �∇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

$$
\sigma \mathbf{E}\_{\rm S} - A \nabla T = \mathbf{0},
\tag{12}
$$

"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

> ð<sup>∞</sup> εF

nex ¼ Nex=A, A ¼ planer area, (15)

<sup>d</sup><sup>ε</sup> <sup>1</sup> eβ εð Þ �<sup>μ</sup> þ 1

particle ¼ �qD∇nex, (17)

<sup>F</sup>τ, v ¼ vF, ℓ ¼ vτ, (18)

<sup>A</sup><sup>d</sup> <sup>k</sup>BDð Þ <sup>ε</sup><sup>F</sup> <sup>∇</sup>T, (19)

<sup>F</sup> kBDð Þ ε<sup>F</sup> τ: (20)

τ=m<sup>∗</sup> , (21)

<sup>F</sup>: (22)

<sup>2</sup> <sup>m</sup><sup>∗</sup> <sup>v</sup><sup>2</sup>

(14)

(16)

energies greater than the Fermi energy εF, is defined and calculated as

dεDð Þε fð Þ ε; T; μ ≈ Dð Þ ε<sup>F</sup>

where Dð Þε is the density of states. This formula holds for 2D and 3D in high

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.

> <sup>S</sup> <sup>¼</sup> <sup>&</sup>lt; 0 for "electrons" > 0 for "holes"

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<sup>F</sup> is the Fermi velocity and τ the relaxation time of the charged particles. The symbol d denotes the dimension. The density gradient ∇nex is generated by the

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

Nex �

After using Eqs. (13) and (14), we obtain

ð<sup>∞</sup> εF

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

≃ ln 2 kBTDð Þ ε<sup>F</sup> ,

degeneracy. The density of thermally excited "electrons,"

�

j ¼ qj

<sup>d</sup> <sup>v</sup><sup>ℓ</sup> <sup>¼</sup> <sup>1</sup>

d v2

<sup>∇</sup>nex <sup>¼</sup> ln 2

<sup>A</sup> <sup>¼</sup> ln 2 <sup>2</sup><sup>A</sup> qv<sup>2</sup>

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

<sup>S</sup> � <sup>A</sup>=<sup>σ</sup> <sup>¼</sup> ln 2 <sup>k</sup>Bε<sup>F</sup>

<sup>σ</sup> <sup>¼</sup> nq<sup>2</sup>

nq

Dð Þ ε<sup>F</sup>

The relaxation time τ cancels out from numerator and denominator. This result

<sup>A</sup> , <sup>ε</sup><sup>F</sup> � <sup>1</sup>

<sup>D</sup> <sup>¼</sup> <sup>1</sup>

temperature gradient ∇T and is given by

We divide A by the conductivity

is independent of the temperature T.

41

diffusion coefficient A as

where E<sup>S</sup> is the field generated by the thermal emf. The Seebeck coefficient S, also called the thermoelectric power or the thermopower, is defined through

$$E\_{\mathbb{S}} = \mathbb{S}\nabla T, \quad \mathbb{S} \equiv \mathbb{A}/\sigma. \tag{13}$$

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 YBa2Cu3O7�<sup>δ</sup>, reproduced in Figure 1.

We assume that the carriers are conduction electrons ("electron," "hole") with charge <sup>q</sup> (�<sup>e</sup> for "electron," <sup>þ</sup><sup>e</sup> for "hole") and effective mass <sup>m</sup><sup>∗</sup> . 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<sup>F</sup> and which circulates clockwise (counterclockwise) viewed from the tip of the applied magnetic field vector.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

"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

$$\begin{split} N\_{\rm ex} & \equiv \int\_{\rm rF}^{\infty} \mathrm{d}\varepsilon \mathcal{D}(\varepsilon) f(\varepsilon, T, \mu) \approx \mathcal{D}(\varepsilon\_{\rm F}) \int\_{\rm rF}^{\infty} \mathrm{d}\varepsilon \frac{1}{e^{\beta(\varepsilon-\mu)} + 1} \\ & \simeq \ln 2 \, k\_{\rm B} T \mathcal{D}(\varepsilon\_{\rm F}), \end{split} \tag{14}$$

where Dð Þε is the density of states. This formula holds for 2D and 3D in high degeneracy. The density of thermally excited "electrons,"

$$m\_{\rm ex} = N\_{\rm ex} / \mathcal{A}\_{\rm \bullet} \quad \mathcal{A} = \text{planer} \quad \text{area},\tag{15}$$

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

$$S = \begin{cases} <0 & \text{for "electrons"}\\ >0 & \text{for "holes"} \end{cases} \tag{16}$$

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

$$
\dot{\mathbf{j}} = q \mathbf{j}\_{\text{particle}} = -qD \nabla n\_{\text{ex}}.\tag{17}
$$

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

$$D = \frac{1}{\mathbf{d}} \boldsymbol{\nu} \ell = \frac{1}{\mathbf{d}} \boldsymbol{\nu}\_{\mathbb{F}}^2 \boldsymbol{\tau}, \quad \boldsymbol{\nu} = \boldsymbol{\nu}\_{\mathbb{F}}, \quad \ell = \boldsymbol{\nu} \boldsymbol{\tau}, \tag{18}$$

where v<sup>F</sup> is the Fermi velocity and τ the relaxation time of the charged particles. The symbol d denotes the dimension. The density gradient ∇nex is generated by the temperature gradient ∇T and is given by

$$
\nabla n\_{\rm ex} = \frac{\ln 2}{\mathcal{A} \mathbf{d}} k\_{\rm B} \mathcal{D}(\varepsilon\_{\rm F}) \nabla T,\tag{19}
$$

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

$$A = \frac{\ln 2}{2\mathcal{A}} q v\_{\rm F}^2 k\_{\rm B} \mathcal{D}(\varepsilon\_{\rm F}) \,\text{r.}\tag{20}$$

We divide A by the conductivity

$$
\sigma = \mathfrak{n} q^2 \mathfrak{r} / m^\*,\tag{21}
$$

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

$$S \equiv A/\sigma = \ln 2 \frac{k\_{\rm B} \varepsilon\_{\rm F}}{nq} \frac{\mathcal{D}(\varepsilon\_{\rm F})}{\mathcal{A}}, \quad \varepsilon\_{\rm F} \equiv \frac{1}{2} m^\* v\_{\rm F}^2. \tag{22}$$

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

4. In-plane Seebeck coefficient above the critical temperature

Advanced Thermoelectric Materials for Energy Harvesting Applications

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

The molar heat capacity C for 3D massless bosons rises like T<sup>3</sup> and reaches 10:8R at the critical temperature

<sup>0</sup> . It then drops abruptly by 6:57 R and approaches the high-temperature limit 3R.

holds, where E ¼ �∇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

where E<sup>S</sup> 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

We assume that the carriers are conduction electrons ("electron," "hole") with charge <sup>q</sup> (�<sup>e</sup> for "electron," <sup>þ</sup><sup>e</sup> for "hole") and effective mass <sup>m</sup><sup>∗</sup> . 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<sup>F</sup> and which circulates clockwise (counterclockwise) viewed from the tip of the applied magnetic field vector.

j ¼ σð Þþ �∇V Að Þ¼ �∇T σE � A∇T (11)

σE<sup>S</sup> � A∇T ¼ 0, (12)

E<sup>S</sup> ¼ S∇T, S � A=σ: (13)

4.1 Seebeck coefficient for conduction electrons

j and the gradients

40

Figure 7.

<sup>T</sup><sup>c</sup> <sup>¼</sup> <sup>2</sup>:02ℏcn1=<sup>3</sup>

flows. Thus from Eq. (11), we obtain

YBa2Cu3O7�<sup>δ</sup>, reproduced in Figure 1.

#### 4.2 In-plane thermopower for YBCO

We apply our theory to explain the in-plane thermopower data for YBCO. For YBa2Cu3O7�<sup>δ</sup> (composite), there are "electrons" and "holes". The "holes", having smaller <sup>m</sup><sup>∗</sup> and higher <sup>v</sup><sup>F</sup> � <sup>2</sup>εF=m<sup>∗</sup> ð Þ1=<sup>2</sup> , dominate in the Ohmic conduction and also in the Hall voltage VH, yielding a positive Hall coefficient R<sup>H</sup> ab (see Figure 1). But the experiments indicate that the in-plane thermopower Sab is negative. This puzzle may be solved as follows.

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

$$
\varepsilon = \left( p\_x^2 + P\_\chi^2 \right) / 2m^\* \,. \tag{23}
$$

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

temperature as

<sup>S</sup>ab � <sup>A</sup>ab σab

5. Out-of-plane thermopower

tive charge (see Figure 1).

where p<sup>i</sup> p<sup>f</sup>

43

squared matrix-elements M<sup>2</sup>

<sup>w</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

j ð Þi <sup>c</sup> ¼ j ð Þi <sup>c</sup>,<sup>H</sup> � j ð Þi

¼ � ln 2 <sup>k</sup>Bε<sup>F</sup>

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

experiments in YBa2Cu3O7�<sup>δ</sup>, reproduced in Figure 1.

ε ¼

2

8 < :

<sup>ℏ</sup> <sup>p</sup><sup>f</sup> <sup>j</sup>Ujp<sup>i</sup> �� � � �

πeℏ<sup>2</sup>

1 v ð Þ1 F

� 1 v ð Þ2 F

The factors nphs drop out from numerator and denominator. The obtained Seebeck coefficient Sab is negative and T-independent, in agreement with

Terasaki et al. [17, 18] and Takenaka et al. [19] measured the out-of-plane resistivity ρ<sup>c</sup> in YBa2Cu3Ox. In the range 6:6<x<6:92, the data for ρ<sup>c</sup> can be fitted with

where C<sup>1</sup> and C<sup>2</sup> are constants and ρab is the in-plane resistivity. The first term C1ρab arises from the in-plane conduction due to the (predominant) "holes" and þ pairons. The second term C2=T arises from the � pairons' quantum tunneling between the copper planes [20]. Pairons move with a linear dispersion relation [21]:

> <sup>π</sup> <sup>v</sup>F<sup>p</sup> � cp, p<sup>&</sup>lt; <sup>p</sup><sup>0</sup> � <sup>∣</sup>w0∣=<sup>c</sup> 0, otherwise

with ∣w0∣ being the binding energy of a pairon. The Hall coefficient R<sup>H</sup>

boring layer with the jump rate given by the Dirac-Fermi golden rule

� 2

energy and U is the imperfection-perturbation. We assume a constant absolute

. The current density j

of particles <sup>i</sup> having charge <sup>q</sup>ð Þ<sup>i</sup> and momentum-energy ð Þ <sup>p</sup>; <sup>ε</sup> is calculated from

where nð Þ<sup>i</sup> is the 2D number density, a<sup>0</sup> the interlayer distance, and j

with the same speed c ¼ ð Þ 2=π vF, but the velocity component vx is

vx <sup>¼</sup> <sup>∂</sup><sup>ε</sup> ∂px

<sup>c</sup>,<sup>L</sup> <sup>¼</sup> <sup>q</sup>ð Þ<sup>i</sup> <sup>a</sup>0wnð Þ<sup>i</sup> <sup>v</sup>

represents the current density from the high (low)-temperature end. Pairons move

<sup>¼</sup> cpx <sup>p</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup>

along the c-axis) is observed to be negative, indicating that the carriers have nega-

The tunneling current is calculated as follows. A pairon arrives at a certain lattice-imperfection (impurity, lattice defect, etc.) and quantum-jumps to a neigh-

δ εð Þ� <sup>f</sup> � ε<sup>i</sup>

� � and <sup>ε</sup>ið Þ <sup>ε</sup><sup>f</sup> are, respectively, the initial (final) momentum and

2π <sup>ℏ</sup> <sup>M</sup><sup>2</sup>

ð Þi

ð Þi <sup>H</sup> � v ð Þi L

! <sup>n</sup><sup>1</sup>

m<sup>∗</sup> <sup>1</sup> v ð Þ1 F þ

n2 m<sup>∗</sup> <sup>2</sup> v ð Þ2 F

: (30)

(32)

<sup>c</sup> (current

δ εð Þ <sup>f</sup> � ε<sup>i</sup> , (33)

<sup>c</sup> along the c-axis due to a group

ð Þi c,<sup>H</sup> j ð Þi c,L � �

� �, (34)

<sup>ε</sup> px: (35)

!�<sup>1</sup>

ρ<sup>c</sup> ¼ C1ρab þ C2=T, (31)

The 2D density of states including the spin degeneracy is

$$\mathcal{D} = \mathfrak{m}^\* A / \left(\mathfrak{m}\hbar^2\right),\tag{24}$$

which is independent of energy. The "electrons" (minority carriers), having heavier mass m<sup>∗</sup> <sup>1</sup> , contribute more to A, and hence the thermopower Sab 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

$$\begin{split} A\_{\rm ab} &= -e \ln 2 \frac{k\_{\rm B}}{2\pi \hbar^{2}} \left( v\_{\rm F}^{(1)} \right)^{2} m\_{1}^{\ast} \, \tau\_{1} + e \ln 2 \frac{k\_{\rm B}}{2\pi \hbar^{2}} \left( v\_{\rm F}^{(2)} \right)^{2} m\_{2}^{\ast} \, \tau\_{2} \\ &= -e \ln 2 \frac{k\_{\rm B} \varepsilon\_{\rm F}}{\pi \hbar^{2}} (\tau\_{1} - \tau\_{2}). \end{split} \tag{25}$$

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

$$
\Gamma \equiv \mathfrak{r}^{-1} = n\_{\text{ph}} \upsilon\_{\text{F}} \mathfrak{s}, \tag{26}
$$

where s is the scattering diameter and nph denotes the phonon population given by the Planck distribution function:

$$n\_{\rm ph} = \left[ \exp \left( \varepsilon\_{\rm ph} / k\_{\rm B} T \right) - 1 \right]^{-1}, \tag{27}$$

where εph is a phonon energy. We then obtain

$$\begin{split} \tau\_1 - \tau\_2 &= \mathbf{1} / \Gamma\_\mathbf{I} - \mathbf{1} / \Gamma\_\mathbf{I} = \left( n\_{\text{ph}} v\_\mathbf{F}^{(1)} s \right)^{-1} - \left( n\_{\text{ph}} v\_\mathbf{F}^{(2)} s \right)^{-1} \\ &= \frac{\mathbf{1}}{n\_{\text{ph}} s} \left( \frac{\mathbf{1}}{v\_\mathbf{F}^{(1)}} - \frac{\mathbf{1}}{v\_\mathbf{F}^{(2)}} \right) > \mathbf{0}, \qquad \left( v\_\mathbf{F}^{(1)} < v\_\mathbf{F}^{(2)} \right). \end{split} \tag{28}$$

The total conductivity is

$$\begin{split} \sigma = \sigma\_1 + \sigma\_2 &= \frac{e^2 n\_1}{m\_1^\*} \tau\_1 + \frac{e^2 n\_2}{m\_2^\*} \tau\_2 \\ &= \frac{e^2 n\_1}{m\_1^\* \ v\_{\rm F}^{(1)} n\_{\rm ph} s} + \frac{e^2 n\_2}{m\_2^\* \ v\_{\rm F}^{(2)} n\_{\rm ph} s} . \end{split} \tag{29}$$

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

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

$$\mathcal{S}\_{\rm ab} \equiv \frac{A\_{\rm ab}}{\sigma\_{\rm ab}} = -\ln 2 \frac{k\_{\rm B} \varepsilon\_{\rm F}}{\pi e \hbar^2} \left( \frac{1}{\upsilon\_{\rm F}^{(1)}} - \frac{1}{\upsilon\_{\rm F}^{(2)}} \right) \left( \frac{n\_1}{m\_1^{\*}} \upsilon\_{\rm F}^{(1)} + \frac{n\_2}{m\_2^{\*}} \upsilon\_{\rm F}^{(2)} \right)^{-1}. \tag{30}$$

The factors nphs drop out from numerator and denominator. The obtained Seebeck coefficient Sab is negative and T-independent, in agreement with experiments in YBa2Cu3O7�<sup>δ</sup>, reproduced in Figure 1.

### 5. Out-of-plane thermopower

4.2 In-plane thermopower for YBCO

smaller <sup>m</sup><sup>∗</sup> and higher <sup>v</sup><sup>F</sup> � <sup>2</sup>εF=m<sup>∗</sup> ð Þ1=<sup>2</sup>

puzzle may be solved as follows.

heavier mass m<sup>∗</sup>

negative as shown below.

<sup>A</sup>ab ¼ �eln 2 <sup>k</sup><sup>B</sup>

by the Planck distribution function:

The total conductivity is

42

¼ �eln 2 <sup>k</sup>Bε<sup>F</sup>

where εph is a phonon energy. We then obtain

<sup>¼</sup> <sup>1</sup> nph s

τ<sup>1</sup> � τ<sup>2</sup> ¼ 1=Γ<sup>1</sup> � 1=Γ<sup>2</sup> ¼ nph v

<sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>1</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>n<sup>1</sup>

1 v ð Þ1 F

� 1 v ð Þ2 F

m<sup>∗</sup> 1 τ<sup>1</sup> þ

<sup>¼</sup> <sup>e</sup><sup>2</sup>n<sup>1</sup> m<sup>∗</sup> <sup>1</sup> v ð Þ1 <sup>F</sup> nph s

!

We apply our theory to explain the in-plane thermopower data for YBCO. For YBa2Cu3O7�<sup>δ</sup> (composite), there are "electrons" and "holes". The "holes", having

But the experiments indicate that the in-plane thermopower Sab is negative. This

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

<sup>x</sup> <sup>þ</sup> <sup>P</sup><sup>2</sup> y � �

which is independent of energy. The "electrons" (minority carriers), having

When both "electrons" (1) and "holes" (2) exist, their contributions to the

m<sup>∗</sup>

where s is the scattering diameter and nph denotes the phonon population given

ð Þ1 <sup>F</sup> s � ��<sup>1</sup>

> e<sup>2</sup>n<sup>2</sup> m<sup>∗</sup> 2 τ2

> > þ

m<sup>∗</sup> <sup>2</sup> v ð Þ2 <sup>F</sup> nph s :

>0, vð Þ<sup>1</sup>

<sup>n</sup>ph <sup>¼</sup> exp <sup>ε</sup>ph=kB<sup>T</sup> � � � <sup>1</sup> � ��<sup>1</sup>

<sup>1</sup> , contribute more to A, and hence the thermopower Sab can be

<sup>1</sup> <sup>τ</sup><sup>1</sup> <sup>þ</sup> <sup>e</sup>ln 2 <sup>k</sup><sup>B</sup>

also in the Hall voltage VH, yielding a positive Hall coefficient R<sup>H</sup>

Advanced Thermoelectric Materials for Energy Harvesting Applications

<sup>ε</sup> <sup>¼</sup> <sup>p</sup><sup>2</sup>

The 2D density of states including the spin degeneracy is

thermal diffusion are additive. Using Eqs. (20) and (24), we obtain

ð Þ1 F � �<sup>2</sup>

<sup>π</sup>ℏ<sup>2</sup> ð Þ <sup>τ</sup><sup>1</sup> � <sup>τ</sup><sup>2</sup> :

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

<sup>2</sup>πℏ<sup>2</sup> <sup>v</sup>

, dominate in the Ohmic conduction and

=2m<sup>∗</sup> : (23)

<sup>D</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup> <sup>A</sup><sup>=</sup> <sup>π</sup>ℏ<sup>2</sup> � �, (24)

<sup>2</sup>πℏ<sup>2</sup> <sup>v</sup>

<sup>Γ</sup> � <sup>τ</sup>�<sup>1</sup> <sup>¼</sup> <sup>n</sup>ph <sup>v</sup><sup>F</sup> s, (26)

� nph v

<sup>F</sup> <v ð Þ2 F � �

e<sup>2</sup>n<sup>2</sup>

ð Þ2 <sup>F</sup> s � ��<sup>1</sup>

:

ð Þ2 F � �<sup>2</sup>

m<sup>∗</sup> <sup>2</sup> τ<sup>2</sup>

, (27)

(25)

(28)

(29)

ab (see Figure 1).

Terasaki et al. [17, 18] and Takenaka et al. [19] measured the out-of-plane resistivity ρ<sup>c</sup> in YBa2Cu3Ox. In the range 6:6<x<6:92, the data for ρ<sup>c</sup> can be fitted with

$$
\rho\_{\rm c} = \mathbf{C\_1} \rho\_{\rm ab} + \mathbf{C\_2}/T,\tag{31}
$$

where C<sup>1</sup> and C<sup>2</sup> are constants and ρab is the in-plane resistivity. The first term C1ρab arises from the in-plane conduction due to the (predominant) "holes" and þ pairons. The second term C2=T arises from the � pairons' quantum tunneling between the copper planes [20]. Pairons move with a linear dispersion relation [21]:

$$\varepsilon = \begin{cases} \frac{2}{\pi} v\_{\text{F}} p \equiv cp, & p < p\_0 \equiv |w\_0|/c \\ 0, & \text{otherwise} \end{cases} \tag{32}$$

with ∣w0∣ being the binding energy of a pairon. The Hall coefficient R<sup>H</sup> <sup>c</sup> (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

$$\mathcal{L}w = \frac{2\pi}{\hbar} \left| \left< \mathbf{p}\_{\rm f} \left| U \right| \mathbf{p}\_{\rm i} \right> \right|^{2} \delta(\varepsilon\_{\rm f} - \varepsilon\_{\rm i}) \equiv \frac{2\pi}{\hbar} \mathcal{M}^{2} \delta(\varepsilon\_{\rm f} - \varepsilon\_{\rm i}),\tag{33}$$

where p<sup>i</sup> p<sup>f</sup> � � and <sup>ε</sup>ið Þ <sup>ε</sup><sup>f</sup> are, respectively, the initial (final) momentum and energy and U is the imperfection-perturbation. We assume a constant absolute squared matrix-elements M<sup>2</sup> . The current density j ð Þi <sup>c</sup> along the c-axis due to a group of particles <sup>i</sup> having charge <sup>q</sup>ð Þ<sup>i</sup> and momentum-energy ð Þ <sup>p</sup>; <sup>ε</sup> is calculated from

$$j\_{\mathbf{c}}^{(i)} = j\_{\mathbf{c},\mathbf{H}}^{(i)} - j\_{\mathbf{c},\mathbf{L}}^{(i)} = q^{(i)} \mathfrak{a}\_0 \mathfrak{w} n^{(i)} \left( \boldsymbol{\nu}\_{\mathbf{H}}^{(i)} - \boldsymbol{\nu}\_{\mathbf{L}}^{(i)} \right), \tag{34}$$

where nð Þ<sup>i</sup> is the 2D number density, a<sup>0</sup> the interlayer distance, and j ð Þi c,<sup>H</sup> j ð Þi c,L � � represents the current density from the high (low)-temperature end. Pairons move with the same speed c ¼ ð Þ 2=π vF, but the velocity component vx is

$$
\sigma\_{\mathfrak{x}} = \frac{\partial \varepsilon}{\partial p\_{\mathfrak{x}}} = \frac{cp\_{\mathfrak{x}}}{p} = \frac{c^2}{\varepsilon} p\_{\mathfrak{x}}.\tag{35}
$$

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 ΔTð Þ ¼ T<sup>H</sup> � T<sup>L</sup> causes a change in the B-E distribution F:

$$F(\varepsilon) \equiv \left[e^{(\varepsilon-\mu)\beta} + 1\right]^{-1}, \quad \beta \equiv (k\_B T)^{-1}, \tag{36}$$

where μ is the chemical potential. We compute the current density j <sup>c</sup> from

$$j\_c = 2\varepsilon \frac{\mathcal{M}^2 B}{\hbar^3 c^2} \frac{a\_0 \Delta T}{k\_B T^2} \int\_0^{p\_0} \mathrm{d}\varepsilon \, \frac{\mathrm{d} F(\varepsilon)}{\mathrm{d}\beta},\tag{37}$$

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

At the BEC temperature ð Þ T<sup>c</sup> , the chemical potential μ vanishes:

$$
\mu(T\_c) = \mathbf{0},\tag{38}
$$

Hence at x ¼ 7, we have an expression for the out-of-plane Seebeck coefficient

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<sup>c</sup> > 0, which is

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

YBCO is a "hole"-type HTSC in which "holes" are the majority carriers above Tc,

¼ AcC1ρab ∝T >0: ð Þ ρab ∝ T : (46)

<sup>2</sup> and charge þe,

<sup>d</sup><sup>t</sup> <sup>¼</sup> eE: (47)

τ<sup>2</sup> (48)

E, (49)

j ¼ σE, (50)

S<sup>c</sup> above the critical temperature:

<sup>S</sup><sup>c</sup> � <sup>A</sup><sup>c</sup> σc

in accord with experiments (see Figure 1).

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

6. Resistivity above the critical temperature

phonons, which makes the kinetic theory applicable.

while Nd1:84Ce0:<sup>16</sup> CuO4 is an "electron"-type HTSC.

Consider a system of "holes," each having effective mass m<sup>∗</sup>

m<sup>∗</sup> 2

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

mdvx

Solving it for vx and assuming that the acceleration persists in the mean-free

<sup>v</sup><sup>d</sup> <sup>¼</sup> eE m<sup>∗</sup> 2

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

e<sup>2</sup> τ<sup>2</sup> m<sup>∗</sup> 2

j ¼ en2v<sup>d</sup> ¼ n<sup>2</sup>

where n<sup>2</sup> is the "hole" density. Assuming Ohm's law

we obtain an expression for the electrical conductivity:

6.1 In-plane resistivity

time τ2, we obtain

45

and

$$
\beta \mu \equiv \mu / k\_{\text{B}} T \tag{39}
$$

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

$$F(\varepsilon) \approx f\_{\text{\tiny o}}(\varepsilon) = \exp\left(\mu - \varepsilon\right)\beta,\tag{40}$$

which is normalized such that

$$\frac{1}{\left(2\pi\hbar\right)^{2}}\Big|\mathbf{d}^{2}pf\_{0}(\varepsilon) = n\_{0}\left(\text{pairon density}\right).\tag{41}$$

All integrals in (37) and (41) can be evaluated simply by using Ð <sup>∞</sup> <sup>0</sup> <sup>d</sup>x e�xxn <sup>¼</sup> <sup>n</sup>!. Hence we obtain

$$\int \mathrm{d}^2 p f\_0(\varepsilon) = 2\pi m \int\_0^\infty \mathrm{d}\varepsilon e^{\beta \mu} e^{-\beta \varepsilon} = 2\pi m \, e^{\mu \beta} \beta^{-1}. \tag{42}$$

The integral in (37) is then calculated as

$$\int\_0^{p\_0} \mathrm{d}\varepsilon \, \frac{\mathrm{d}F(\varepsilon)}{\mathrm{d}\beta} \approx \int\_0^{p\_0} \mathrm{d}\varepsilon \, \frac{\mathrm{d}}{\mathrm{d}\beta} f\_0(\varepsilon) = \varepsilon^{\mu\beta} \int\_0^{\infty} \mathrm{d}\varepsilon \, e^{-\beta\varepsilon} = \varepsilon^{\beta\mu} \beta^{-2}.\tag{43}$$

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

$$A\_{\rm c} \sim 2e\mathcal{M}^2 \mathcal{B}k\_{\rm B}a\_0 \left(\hbar^3 c^2\right)^{-1},\tag{44}$$

which is T-independent.

Experiments [5] indicate that the first term C1ρab in (31) is dominant for x>6:8:

$$
\rho\_{\rm c} \sim C\_1 \rho\_{\rm ab} \propto T. \tag{45}
$$

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

, <sup>β</sup> � ð Þ <sup>k</sup>B<sup>T</sup> �<sup>1</sup>

<sup>d</sup><sup>ε</sup> <sup>d</sup>Fð Þ<sup>ε</sup>

μð Þ¼ T<sup>c</sup> 0, (38)

βμ � μ=kBT (39)

� �: (41)

<sup>0</sup> <sup>d</sup>x e�xxn <sup>¼</sup> <sup>n</sup>!.

: (42)

: (43)

Fð Þε ≈f <sup>0</sup>ð Þ¼ ε exp ð Þ μ � ε β, (40)

�βε <sup>¼</sup> <sup>2</sup>πm eμββ�<sup>1</sup>

�βε <sup>¼</sup> <sup>e</sup>

ρ<sup>c</sup> � C1ρab ∝T: (45)

βμβ�<sup>2</sup>

, (44)

dεεe

, (36)

<sup>d</sup><sup>β</sup> , (37)

<sup>c</sup> from

ΔTð Þ ¼ T<sup>H</sup> � T<sup>L</sup> causes a change in the B-E distribution F:

Advanced Thermoelectric Materials for Energy Harvesting Applications

ð Þ <sup>ε</sup>�<sup>μ</sup> <sup>β</sup> <sup>þ</sup> <sup>1</sup> h i�<sup>1</sup>

where μ is the chemical potential. We compute the current density j

a0ΔT kBT<sup>2</sup>

ðcp<sup>0</sup> 0

assuming a small ΔT. Not all pairons reaching an imperfection are triggered into

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

p f <sup>0</sup>ð Þ¼ ε n<sup>0</sup> pairon density

M<sup>2</sup> B ℏ3 c2

At the BEC temperature ð Þ T<sup>c</sup> , the chemical potential μ vanishes:

Fð Þ� ε e

j <sup>c</sup> ¼ 2e

tunneling. The factor B contains this correction.

and

distribution function:

Hence we obtain

ðcp<sup>0</sup> 0

44

which is normalized such that

ð d2

<sup>d</sup><sup>ε</sup> <sup>d</sup>Fð Þ<sup>ε</sup> <sup>d</sup><sup>β</sup> <sup>≈</sup>

which is T-independent.

The integral in (37) is then calculated as

ðcp<sup>0</sup> 0

1 ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>2</sup> ð d2

p f <sup>0</sup>ð Þ¼ ε 2πm

<sup>d</sup><sup>ε</sup> <sup>d</sup> dβ

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

<sup>A</sup><sup>c</sup> � <sup>2</sup>eM<sup>2</sup>

All integrals in (37) and (41) can be evaluated simply by using Ð <sup>∞</sup>

ð<sup>∞</sup> 0 dεe βμ e

f <sup>0</sup>ð Þ¼ ε e

μβ ð<sup>∞</sup> 0

BkBa<sup>0</sup> ℏ<sup>3</sup>

Experiments [5] indicate that the first term C1ρab in (31) is dominant for x>6:8:

c <sup>2</sup> � ��<sup>1</sup>

Hence at x ¼ 7, we have an expression for the out-of-plane Seebeck coefficient S<sup>c</sup> above the critical temperature:

$$\mathcal{S}\_{\mathsf{c}} \equiv \frac{A\_{\mathsf{c}}}{\sigma\_{\mathsf{c}}} = A\_{\mathsf{c}} \mathcal{C}\_{\mathsf{l}} \rho\_{\mathsf{ab}} \propto T > \mathbf{0}. \ (\rho\_{\mathsf{ab}} \propto T). \tag{46}$$

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<sup>c</sup> > 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:<sup>16</sup> CuO4 is an "electron"-type HTSC.

#### 6.1 In-plane resistivity

Consider a system of "holes," each having effective mass m<sup>∗</sup> <sup>2</sup> 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

$$m\_2^\* \frac{\text{m}dv\_x}{\text{dt}} = eE.\tag{47}$$

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

$$
\sigma\_{\rm d} = \frac{eE}{m\_2^\*} \,\tau\_2\tag{48}
$$

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

$$j = e n\_2 v\_d = n\_2 \frac{e^2 \tau\_2}{m\_2^\*} E,\tag{49}$$

where n<sup>2</sup> is the "hole" density. Assuming Ohm's law

$$j = \sigma E,\tag{50}$$

we obtain an expression for the electrical conductivity:

Advanced Thermoelectric Materials for Energy Harvesting Applications

$$
\sigma\_2 = \frac{n\_2 e^2}{m\_2^\*} \frac{1}{\Gamma\_2},
\tag{51}
$$

where <sup>Γ</sup><sup>2</sup> � <sup>τ</sup>�<sup>1</sup> <sup>2</sup> is the scattering rate. The phonon scattering rate can be computed, using

$$
\Gamma\_2 = n\_{\rm ph} \upsilon\_{\rm F} A\_{\rm 2} \tag{52}
$$

where A<sup>2</sup> is the scattering diameter. If acoustic phonons having average energies

$$
\langle \hbar a\_q \rangle \equiv a\_0 \hbar a\_\mathcal{D} \ll k\_\mathcal{B} T, \quad a\_0 \sim 0.20 \tag{53}
$$

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

$$n\_{\rm ph} = n\_{\rm a} \left[ \exp \left( a\_0 \hbar a o\_{\rm D} / k\_{\rm B} T \right) - 1 \right]^{-1} \simeq n\_{\rm a} \frac{k\_{\rm B} T}{a\_0 \hbar a o\_{\rm D}},\tag{54}$$

where

$$m\_{\mathbf{a}} \equiv \left(2\pi\right)^{-2} \int \mathbf{d}^{2}k \tag{55}$$

is the small k-space area where the acoustic phonons are located. Using Eqs. (51), (52), and (54), we obtain

$$
\sigma\_2 = \frac{C\_2 n\_2 e^2}{T}, \quad C\_2 \equiv \frac{a\_0 \hbar a\_\mathrm{D}}{n\_\mathrm{a} m\_2^\* \, k\_\mathrm{B} v\_\mathrm{F} A\_2}. \tag{56}
$$

Similar calculations apply to "electrons." We obtain

$$
\sigma\_1 = \frac{\mathbf{C}\_1 \mathbf{n}\_1 e^2}{T}, \quad \mathbf{C}\_1 \equiv \frac{a\_0 \hbar a \mathbf{o}\_\mathbf{D}}{n\_\text{a} m\_1^\* k\_\text{B} v\_\text{F} A\_2}. \tag{57}
$$

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

$$
\rho \equiv \frac{1}{\sigma} \propto T.\tag{58}
$$

v ð Þ3 <sup>d</sup> <sup>¼</sup> <sup>2</sup>ec<sup>2</sup>

2

Boltzmann distribution for bosonic pairons above Tc, then we obtain

<sup>¼</sup> <sup>2</sup>n2e<sup>2</sup>C<sup>3</sup>

σ<sup>3</sup> ¼ ð Þ 2e

ð<sup>∞</sup> 0 dpp 1 ε e <sup>β</sup>cp ! <sup>2</sup><sup>π</sup>

<sup>¼</sup> ð Þ <sup>k</sup>B<sup>T</sup> �<sup>1</sup> for <sup>T</sup> <sup>&</sup>gt;Tc:

average. Using this and Ohm's law, we obtain

dashed lines data for highly overdosed samples, after Iye [24].

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>2</sup>

<sup>σ</sup><sup>3</sup> <sup>¼</sup> <sup>4</sup>n3e<sup>2</sup>c<sup>2</sup> kBTΓ<sup>3</sup>

obtain, by using the results (56) and (65):

<sup>ε</sup>�<sup>1</sup> � � � <sup>2</sup><sup>π</sup>

obtain

47

Figure 8.

where τ<sup>3</sup> is the pairon mean free time and the angular brackets denote a thermal

Resistivity in the ab plane, ρab vs. temperature T. Solid lines represent data for HTSC at optimum doping and

where n<sup>3</sup> is the pairon density and Γ<sup>3</sup> is the pairon scattering rate. If we assume a

The rate Γ<sup>3</sup> is calculated with the assumption of a phonon scattering. We then

The total conductivity σ for YBCO is σ<sup>2</sup> þ σ3. Thus taking the inverse of σ, we

<sup>T</sup><sup>2</sup> , C<sup>3</sup> � <sup>8</sup>

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>2</sup>

<sup>3</sup> , <sup>Γ</sup><sup>3</sup> � <sup>τ</sup>�<sup>1</sup>

ð<sup>∞</sup> 0 dppe<sup>β</sup>cp !�<sup>1</sup>

π2

α0ℏωDv<sup>F</sup> nak<sup>2</sup> <sup>B</sup>A<sup>3</sup>

c ε�<sup>1</sup> � �n3Γ�<sup>1</sup>

τ3E ε�<sup>1</sup> � �, (62)

<sup>3</sup> , (63)

(64)

: (65)

Let us now consider a system of + pairons, each having charge þ2e and moving with the linear dispersion relation:

$$
\varepsilon = c p.\tag{59}
$$

Since

$$
\rho\_x = (\text{d}\boldsymbol{\varepsilon}/\text{d}\boldsymbol{p})(\partial\boldsymbol{p}/\partial\boldsymbol{p}\_x) = \boldsymbol{\varepsilon}\left(\boldsymbol{p}\_x/\boldsymbol{p}\right),
\tag{60}
$$

Newton's equation of motion is

$$\frac{p}{c}\frac{\text{d}\upsilon\_{\text{x}}}{\text{d}t} = \frac{e}{c^2}\frac{\text{d}\upsilon\_{\text{x}}}{\text{d}t} = 2eE,\tag{61}$$

yielding vx <sup>¼</sup> <sup>2</sup>e c<sup>2</sup> ð Þ <sup>=</sup><sup>ε</sup> Etþinitial velocity. After averaging over the angles, we obtain

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

#### Figure 8.

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup>2e<sup>2</sup> m<sup>∗</sup> 2

where <sup>Γ</sup><sup>2</sup> � <sup>τ</sup>�<sup>1</sup>

computed, using

where

1 Γ2

<sup>2</sup> is the scattering rate. The phonon scattering rate can be

where A<sup>2</sup> is the scattering diameter. If acoustic phonons having average energies

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

<sup>n</sup>ph <sup>¼</sup> <sup>n</sup>a½ � exp ð Þ� <sup>α</sup>0ℏωD=kB<sup>T</sup> <sup>1</sup> �<sup>1</sup> <sup>≃</sup> <sup>n</sup><sup>a</sup>

<sup>n</sup><sup>a</sup> � ð Þ <sup>2</sup><sup>π</sup> �<sup>2</sup>

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

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup>2n2e<sup>2</sup>

Similar calculations apply to "electrons." We obtain

<sup>σ</sup><sup>1</sup> <sup>¼</sup> <sup>C</sup>1n1e<sup>2</sup>

ð d2

<sup>T</sup> , C<sup>2</sup> � <sup>α</sup>0ℏω<sup>D</sup>

<sup>T</sup> , C<sup>1</sup> � <sup>α</sup>0ℏω<sup>D</sup>

The resistivity ρ is the inverse of the conductivity σ. Hence the resistivity for

Let us now consider a system of + pairons, each having charge þ2e and moving

yielding vx <sup>¼</sup> <sup>2</sup>e c<sup>2</sup> ð Þ <sup>=</sup><sup>ε</sup> Etþinitial velocity. After averaging over the angles, we obtain

<sup>ρ</sup> � <sup>1</sup>

vx <sup>¼</sup> ð Þ <sup>d</sup>ε=d<sup>p</sup> <sup>∂</sup>p=∂px

p c dvx <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>ε</sup> c2 dvx nam<sup>∗</sup>

nam<sup>∗</sup>

<sup>2</sup> kBvFA<sup>2</sup>

<sup>1</sup> kBvFA<sup>2</sup>

<sup>σ</sup> <sup>∝</sup> <sup>T</sup>: (58)

ε ¼ cp: (59)

� � <sup>¼</sup> c px=<sup>p</sup> � �, (60)

<sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup>eE, (61)

ℏω<sup>q</sup>

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

YBCO is proportional to the temperature T:

with the linear dispersion relation:

Newton's equation of motion is

Since

46

, (51)

Γ<sup>2</sup> ¼ nph vFA2, (52)

kBT α0ℏω<sup>D</sup>

k (55)

: (56)

: (57)

, (54)

� � � <sup>α</sup>0ℏω<sup>D</sup> <sup>≪</sup> <sup>k</sup>BT, <sup>α</sup><sup>0</sup> � <sup>0</sup>:<sup>20</sup> (53)

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

$$
\sigma\_{\rm d}^{(3)} = 2 \text{ec}^2 \tau\_3 E \langle \epsilon^{-1} \rangle,\tag{62}
$$

where τ<sup>3</sup> is the pairon mean free time and the angular brackets denote a thermal average. Using this and Ohm's law, we obtain

$$
\sigma\_3 = (2e)^2 c \langle e^{-1} \rangle n\_3 \Gamma\_3^{-1}, \quad \Gamma\_3 \equiv \tau\_3^{-1}, \tag{63}
$$

where n<sup>3</sup> is the pairon density and Γ<sup>3</sup> is the pairon scattering rate. If we assume a Boltzmann distribution for bosonic pairons above Tc, then we obtain

$$
\langle \epsilon^{-1} \rangle \equiv \left( \frac{2\pi}{\left( 2\pi\hbar \right)^{2}} \int\_{0}^{\infty} \mathrm{d}p \, p \, \frac{1}{\epsilon} \, e^{\beta \varepsilon p} \right) \left( \frac{2\pi}{\left( 2\pi\hbar \right)^{2}} \int\_{0}^{\infty} \mathrm{d}p \, p \, e^{\beta p} \right)^{-1} \tag{64}
$$

$$
= \left( k\_{\mathrm{B}} T \right)^{-1} \quad \text{for} \quad T > T\_{\mathrm{c}}.
$$

The rate Γ<sup>3</sup> is calculated with the assumption of a phonon scattering. We then obtain

$$\sigma\_3 = \frac{4n\_3e^2c^2}{k\_\text{B}T\Gamma\_3} = \frac{2n\_2e^2\mathcal{C}\_3}{T^2}, \qquad \mathcal{C}\_3 \equiv \frac{8}{\pi^2} \frac{a\_0\hbar a\_\text{D}v\_\text{F}}{n\_\text{a}k\_\text{B}^2A\_3}.\tag{65}$$

The total conductivity σ for YBCO is σ<sup>2</sup> þ σ3. Thus taking the inverse of σ, we obtain, by using the results (56) and (65):

Advanced Thermoelectric Materials for Energy Harvesting Applications

$$\rho\_{\rm ab} \equiv \frac{1}{\sigma} = \left(\frac{\text{C}\_2 \text{\textit{n}} \text{\textit{e}}^2}{T} + \frac{\text{C}\_3 \text{\textit{n}} \text{\textit{3}} \text{\textit{2}}^2}{T^2}\right)^{-1} = \frac{T^2}{n\_2 \text{\textit{e}}^2 (\text{C}\_2 T + 2 \text{C}\_3)}\tag{66}$$

while the conductivity for Nd1:84Ce0:16CuO4 is given by σ<sup>1</sup> þ σ3, and hence the resistivity is similarly given by

$$\rho\_{\rm ab} = \frac{T^2}{n\_1 e^2 (C\_1 T + 2C\_3)}.\tag{67}$$

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Scientific; 1988

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49

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Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378

> [14] Lanzara A, Bogdanov PV, Zhou XJ, Kellar SA, Feng DL, Lu ED, et al.

[15] Loram JW, Mirza KA, Cooper JR,

Superconductivity. 1994;7:347

[16] Fujita S, Godoy S, Nguyen D. Foundations of Physics. 1995;25:1209

[17] Terasaki I, Sato Y, Terajima S. Physical Review B. 1997;55:15300

[18] Terasak I, Sato Y, Tajima S. Journal of the Korean Physical Society. 1994;

[19] Takenaka K, Mizuhashi K, Takagi H, Uchida S. Physical Review B. 1994;50:

[20] Fujita S, Tamura Y, Suzuki A. Modern Physics Letters B. 2000;14:30

[21] Fujita S, Godoy S. Theory of High Temperature Superconductivity. The Netherlands: Kluwer Academic; 2001.

[22] Fujita S, Kim SY-G, Okamura Y. Modern Physics Letters B. 2000;14:495

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In Nd1:84Ce0:<sup>16</sup> while in YCuO4 system, "electrons" and � pairons play an essential role for the conduction. In YBa2Cu3O7�<sup>δ</sup> the "holes" and þ 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.

At higher temperature ð Þ >160K , the resistivity ρab is linear (see formula (58)):

$$
\rho\_{\text{ab}} \propto T, \quad T > 160 \text{ K}, \tag{68}
$$

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 ρab shows a T-quadratic behavior [see formula (66)]:

$$
\rho\_{\text{ab}} \propto T^2 \quad (\text{near} \text{ and } \text{above} \, T\_{\text{c}}). \tag{69}
$$

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.

### Author details

Shigeji Fujita<sup>1</sup> and Akira Suzuki<sup>2</sup> \*

1 Department of Physics, University at Buffalo, SUNY, Buffalo, NY, USA

2 Department of Physics, Tokyo University of Science, Tokyo, Japan

\*Address all correspondence to: asuzuki@rs.kagu.tus.ac.jp

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quantum Theory of the Seebeck Coefficient in YBCO DOI: http://dx.doi.org/10.5772/intechopen.86378
