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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.
Department of Physics, University at Buffalo, SUNY, USA
Akira Suzuki*
Department of Physics, Tokyo University of Science, Japan
*Address all correspondence to: asuzuki@rs.kagu.tus.ac.jp
1. Introduction
In 1986, Bednorz and Müller [1] reported their discovery of the first of the high-Tc cuprate superconductors (La-Ba-Cu-O, Tc>30 K). Since then many investigations [2, 3] have been carried out on high-Tc superconductors (HTSC) including Y-Ba-Cu-O (YBCO) with Tc∼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 ξ0 (∼10Å), (iii) high critical temperature Tc (∼ 100 K), and (iv) two energy gaps. The transport behaviors above Tc are significantly different from those of a normal metal.
YBCO has a critical (superconducting) temperature Tc∼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 RabH is positive and temperature 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 RabH and Sab suggests that “electrons” and “holes” are responsible for the behaviors. We shall explain this behavior, by assuming “electrons” and “holes” as carriers and using statistical mechanical calculations. Out-of-plane Hall coefficient RcH is negative and temperature-independent, while out-of-plane Seebeck coefficient Sc is 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 ρ above the critical temperature Tc in Section 6.
In this paper we are mainly interested in the sign and the temperature behavior of the Seebeck coefficient in YBCO. But we discuss the related matter for completeness. There are no Seebeck currents in the superconducting state below the critical temperature (S=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=3.88 Å, b=3.82 Å, c=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 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=…,−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 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 λ and a linear energy-momentum (ε‐ℏk) relation:
ε=csℏk,E1
may be assumed, where cs is the sound speed. The attraction generated by the exchange of longitudinal acoustic phonons is long-ranged. This mechanism is good for a type I superconductor whose pairon size is of the order of 104 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:
ε=ε0+A1k1−πa12+A2k2−πa22,E2
where ε0, A1, and A2 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 a1a2=3.88,3.82 Å, and the limit wavelengths λmin at the Brillouin boundary are twice these values. The observed coherence length ξ0 is of the same order as λmin:
ξ0∼λmin≃8Å.E3
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=3.85 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 x1‐x2 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 rather than the first neighbor directions 100 and 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 110 and 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.
Second, we consider a hole wave packet that extends over more than one O-site. It may move easily in 100 because the Cu-sublattice of a uniform charge distribution favors such a motion. If the number of holes is small, the Fermi surface should consist of the four small pockets shown in Figure 4. Under the assumption of such a Fermi surface, pair creation of ± pairons via an optical phonon may occur as shown in the figure. Here a single-phonon exchange generates an electron transition from A in the O-Fermi sheet to B in the Cu-Fermi sheet and another electron transition from A′ to B′, creating the −pairon at BB′ and the +pairon at AA′. 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:
ε=w0+12vFp,E4
where w0 is the ground-state energy of the Cooper pair (pairon) and vF is the Fermi speed. This relation was obtained for a three-dimensional (3D) system. For a 2D system, we obtain
ε=w0+2πvFp.E5
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 Tc 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 Tc in 2D is given by
kBTc=1.24ℏvFn1/2,E6
where n is the pairon density. The inter-pairon distance
r0≡n−1/2=1.24ℏvFkBTc−1E7
is several times greater than the BCS pairon size represented by the BCS coherence length:
ξ0≡0.181ℏvFkBTc−1.E8
Hence the BEC occurs without the pairon overlap. Phonon exchange can be repeated and can generate a pairon-binding energy εb of the order of kBTb:
εb≡kBTb,Tb∼1000K.E9
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+δ with 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 Tc with the T2-law decline on the low-temperature side. The maximum heat capacity at Tc 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 Tc as the pair dissociation point and predicts no features above Tc.
The molar heat capacity C for a 2D massless bosons rises like T2 in the condensed region and reaches 4.38R at T=Tc; its temperature derivative ∂CTn/∂T jumps at this point. The order of phase transition is defined to be that order of the derivative of the free energy F whose discontinuity appears for the first time. Since CV=T∂S/∂TV=−T∂2F/∂T2, ∂CV/∂T=−T∂3F/∂T3−∂2F/∂T2, 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 YBa2Cu3O6.92 (optimal sample) in Figure 5. This is an important support for the quantum statistical theory. Other support is discussed in Sections 5 and 6.
Our quantum statistical theory can be applied to 3D superconductors as well. The linear dispersion relation (4) holds. The superconducting temperature Tc in 3D is given by
kBTc=1.01ℏvFn13,E10
which is identified as the BEC point. The molar heat capacity C for 3D bosons with the linear dispersion relation ε=cp rises like T3 and reaches 10.8R, R= gas constant, at Tc=2.02ℏcn01/3. 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.
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
j=σ−∇V+A−∇T=σE−A∇TE11
holds, where E=−∇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
σES−A∇T=0,E12
where ES is the field generated by the thermal emf. The Seebeck coefficient S, also called the thermoelectric power or the thermopower, is defined through
ES=S∇T,S≡A/σ.E13
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−δ, reproduced in Figure 1.
We assume that the carriers are conduction electrons (“electron,” “hole”) with charge q (−e for “electron,” +e for “hole”) and effective mass m∗. At a finite temperature T>0, “electrons” (“holes”) are excited near the Fermi surface if the surface curvature is negative (positive) [16]. The “electron” (“hole”) is a quasi-electron which has an energy higher lower than the Fermi energy εFand which circulates clockwise (counterclockwise) viewed from the tip of the applied magnetic field vector. “Electrons” (“holes”) are excited on the positive (negative) side of the Fermi surface with the convention that the positive normal vector at the surface points in the energy-increasing direction. The number of thermally excited “electrons” Nex, having energies greater than the Fermi energy εF, is defined and calculated as
Nex≡∫εF∞dεDεfεTμ≈DεF∫εF∞dε1eβε−μ+1≃ln2kBTDεF,E14
where Dε is the density of states. This formula holds for 2D and 3D in high degeneracy. The density of thermally excited “electrons,”
nex=Nex/A,A=planerarea,E15
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=<0for“electrons”>0for“holes”E16
The Seebeck current arises from the thermal diffusion. We assume Fick’s law:
j=qjparticle=−qD∇nex,E17
where D is the diffusion constant, which is computed from the standard formula:
D=1dvℓ=1dvF2τ,v=vF,ℓ=vτ,E18
where vF 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
∇nex=ln2AdkBDεF∇T,E19
where Eq. (14) is used. Using Eqs. (17)–(19) and (11), we obtain the thermal diffusion coefficient A as
A=ln22AqvF2kBDεFτ.E20
We divide A by the conductivity
σ=nq2τ/m∗,E21
and obtain the Seebeck coefficient S [see Eq. (13)]:
S≡A/σ=ln2kBεFnqDεFA,εF≡12m∗vF2.E22
The relaxation time τ cancels 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 YBa2Cu3O7−δ (composite), there are “electrons” and “holes”. The “holes”, having smaller m∗ and higher vF≡2εF/m∗1/2, 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 Sab is negative. This puzzle may be solved as follows.
We assume an effective mass approximation for the in-plane “electrons”:
ε=px2+Py2/2m∗.E23
The 2D density of states including the spin degeneracy is
D=m∗A/πℏ2,E24
which is independent of energy. The “electrons” (minority carriers), having heavier mass m1∗, 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
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−δ, reproduced in Figure 1.
Terasaki et al. [17, 18] and Takenaka et al. [19] measured the out-of-plane resistivity ρc in YBa2Cu3Ox. In the range 6.6<x<6.92, the data for ρc can be fitted with
ρc=C1ρab+C2/T,E31
where C1 and C2 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 + 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]:
ε=2πvFp≡cp,p<p0≡∣w0∣/c0,otherwiseE32
with ∣w0∣ being 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
w=2πℏpfUpi2δεf−εi≡2πℏM2δεf−εi,E33
where pipf and εiεf are, respectively, the initial (final) momentum and energy and U is the imperfection-perturbation. We assume a constant absolute squared matrix-elements M2. The current density jci along the c-axis due to a group of particles i having charge qi and momentum-energy pε is calculated from
jci=jc,Hi−jc,Li=qia0wnivHi−vLi,E34
where ni is the 2D number density, a0 the interlayer distance, and jc,Hijc,Li represents the current density from the high (low)-temperature end. Pairons move with the same speed c=2/πvF, but the velocity component vx is
vx=∂ε∂px=cpxp=c2εpx.E35
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 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 ΔT=TH−TL causes a change in the B-E distribution F:
Fε≡eε−μβ+1−1,β≡kBT−1,E36
where μ is the chemical potential. We compute the current density jc from
jc=2eM2Bℏ3c2a0ΔTkBT2∫0cp0dεdFεdβ,E37
assuming a small ΔT. Not all pairons reaching an imperfection are triggered into tunneling. The factor B contains this correction.
At the BEC temperature Tc, the chemical potential μ vanishes:
μTc=0,E38
and
βμ≡μ/kBTE39
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:
Fε≈f0ε=expμ−εβ,E40
which is normalized such that
12πℏ2∫d2pf0ε=n0pairondensity.E41
All integrals in (37) and (41) can be evaluated simply by using ∫0∞dxe−xxn=n!. Hence we obtain
Experiments [5] indicate that the first term C1ρab in (31) is dominant for x>6.8:
ρc∼C1ρab∝T.E45
Hence at x=7, we have an expression for the out-of-plane Seebeck coefficient Sc above the critical temperature:
Sc≡Acσc=AcC1ρab∝T>0.ρab∝T.E46
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).
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 Nd1.84Ce0.16CuO4 is an “electron”-type HTSC.
6.1 In-plane resistivity
Consider a system of “holes,” each having effective mass m2∗ and charge +e, 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
m2∗mdvxdt=eE.E47
Solving it for vx and assuming that the acceleration persists in the mean-free time τ2, we obtain
vd=eEm2∗τ2E48
for the drift velocity vd. The current density (x-component) j is given by
j=en2vd=n2e2τ2m2∗E,E49
where n2 is the “hole” density. Assuming Ohm’s law
j=σE,E50
we obtain an expression for the electrical conductivity:
σ2=n2e2m2∗1Γ2,E51
where Γ2≡τ2−1 is the scattering rate. The phonon scattering rate can be computed, using
Γ2=nphvFA2,E52
where A2 is the scattering diameter. If acoustic phonons having average energies
ℏωq≡α0ℏωD≪kBT,α0∼0.20E53
are assumed, then the phonon number density nph is given by [23].
nph=naexpα0ℏωD/kBT−1−1≃nakBTα0ℏωD,E54
where
na≡2π−2∫d2kE55
is the small k-space area where the acoustic phonons are located.
Similar calculations apply to “electrons.” We obtain
σ1=C1n1e2T,C1≡α0ℏωDnam1∗kBvFA2.E57
The resistivity ρ is the inverse of the conductivity σ. Hence the resistivity for YBCO is proportional to the temperature T:
ρ≡1σ∝T.E58
Let us now consider a system of + pairons, each having charge +2e and moving with the linear dispersion relation:
ε=cp.E59
Since
vx=dε/dp∂p/∂px=cpx/p,E60
Newton’s equation of motion is
pcdvxdt=εc2dvxdt=2eE,E61
yielding vx=2ec2/εEt+initial velocity. After averaging over the angles, we obtain
vd3=2ec2τ3Eε−1,E62
where τ3 is the pairon mean free time and the angular brackets denote a thermal average. Using this and Ohm’s law, we obtain
σ3=2e2cε−1n3Γ3−1,Γ3≡τ3−1,E63
where n3 is the pairon density and Γ3 is the pairon scattering rate. If we assume a Boltzmann distribution for bosonic pairons above Tc, then we obtain
The total conductivity σ for YBCO is σ2+σ3. Thus taking the inverse of σ, we obtain, by using the results (56) and (65):
ρab≡1σ=C2n2e2T+C3n32e2T2−1=T2n2e2C2T+2C3E66
while the conductivity for Nd1.84Ce0.16CuO4 is given by σ1+σ3, and hence the resistivity is similarly given by
ρab=T2n1e2C1T+2C3.E67
In Nd1.84Ce0.16 while in YCuO4 system, “electrons” and − pairons play an essential role for the conduction. In YBa2Cu3O7−δ the “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.
At higher temperature >160K, the resistivity ρab is linear (see formula (58)):
ρab∝T,T>160K,E68
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)]:
ρab∝T2near 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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