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The unattainability of the absolute zero of temperature is proved by using Carnot’s theorem. Hence this unattainability is distinct from the Planck-Fer-mi statement of the Third Law of Thermodynamics that the entropy vanishes at T=0. It is shown that the isothermal compressibility KT is in general larger than the adiabatic compressibility Ks and the difference KT − Ks vanishes in the low temperature limit.

The Wigner-Seitz unit cell (rhombus) for a honeycomb lattice fails to establish a k-vector in the 2D space, which is required for the Bloch electron dynamics. Phonon motion cannot be discussed in the triangular coordinates, either. In this paper, we propose a rectangular 4-atom unit cell model, which allows us to discuss the electron and phonon (wave packets) motion in the k-space. The present paper discusses the band structure of graphene based on the rectangular 4-atom unit cell model to establish an appropriate k-vector for the Bloch electron dynamics. To obtain the band energy of a Bloch electron in graphene, we extend the tight-binding calculations for the Wigner-Seitz (2-atom unit cell) model of Reich et al. (Physical Review B, 66, Article ID: 035412 (2002)) to the rectangular 4-atom unit cell model. It is shown that the graphene band structure based on the rectangular 4-atom unit cell model reveals the same band structure of the graphene based on the Wigner-Seitz 2-atom unit cell model; the π-band energy holds a linear dispersion (e−k ) relations near the Fermi energy (crossing points of the valence and the conduction bands) in the first Brillouin zone of the rectangular reciprocal lattice. We then confirm the suitability of the proposed rectangular (orthogonal) unit cell model for graphene in order to establish a 2D k-vector responsible for the Bloch electron (wave packet) dynamics in graphene.

The unusual quantum Hall effect (QHE) in graphene is described in terms of the composite (c-) bosons, which move with a linear dispersion relation. The “electron” (wave packet) moves easier in the direction [1 1 0 c-axis] ≡ [1 1 0] of the honeycomb lattice than perpendicular to it, while the “hole” moves easier in [0 0 1]. Since “electrons” and “holes” move in different channels, the particle densities can be high especially when the Fermi surface has “necks”. The strong QHE arises from the phonon exchange attraction in the neighborhood of the “neck” surfaces. The plateau observed for the Hall conductivity and the accompanied resistivity drop is due to the superconducting energy gap caused by the Bose-Einstein condensation of the c-bosons, each forming from a pair of one-electron–two-fluxons c-fermions by phonon-exchange attraction. The half-integer quantization rule for the Hall conductivity: (1/2)(2P−1)(4e 2/h), P=1,2,..., is derived.

Reflection symmetry properties play important roles for the stability of crystal lattices in which electrons and phonons move. Based on the reflection symmetry properties, cubic, tetragonal, orthorhombic, rhombohedral (trigonal) and hexagonal crystal systems are shown to have three-dimensional (3D) k-spaces for the conduction electrons (“electrons”, “holes”). The basic stability condition for a general crystal is the availability of parallel material planes. The monoclinic crystal has a 1D k-space. The triclinic has no k-vectors for electrons, whence it is a true insulator. The monoclinic (triclinic) crystal has one (three) disjoint sets of 1D phonons, which stabilizes the lattice. Phonons’ motion is highly directional; no spherical phonon distributions are generated for monoclinic and triclinic crystal systems.

Following Ashcroft and Mermin, the conduction electrons (“electrons” or “holes”) are assumed to move as wave packets. Dirac’s theorem states that the quantum wave packets representing massive particles always move, following the classical mechanical laws of motion. It is shown here that the conduction electron in an orthorhombic crystal moves classical mechanically if the primitive rectangular-box unit cell is chosen as the wave packet, the condition requiring that the particle density is constant within the cell. All crystal systems except the triclinic system have k-vectors and energy bands. Materials are conducting if the Fermi energy falls on the energy bands. Energy bands and gaps are calculated by using the Kronig-Penny model and its 3D extension. The metal-insulator transition in VO2 is a transition between conductors having three-dimensional and one-dimensional k-vectors.

A quantum theory for a one-electron system can be developed in either Heisenberg picture or Schrodinger picture. For a many-electron system, a theory must be developed in the Heisenberg picture, and the indistinguishability and Pauli’s exclusion principle must be incorporated. The hydrogen atom energy levels are obtained by solving the Schrodinger energy eigenvalue equation, which is the most significant result obtained in the Schrodinger picture. Both boson and fermion field equations are nonlinear in the presence of a pair interaction.

Individual metallic single-wall carbon nanotubes show unusual non-Ohmic transport behaviors at low and high bias fields. For low-resistance contact samples, the differential conductance increases with increasing bias, reaching a maximum at ~100 mV. As the bias increases further, drops dramatically [1]. The higher the bias, the system behaves in a more normal (Ohmic) manner. This low-bias anomaly is temperature-dependent (50 - 150 K). We propose a new interpretation. Supercurrents run in the graphene wall below ~150 K. The normal hole currents run on the outer surface of the wall, which are subject to the scattering by phonons and impurities. The currents along the tube length generate circulating magnetic fields and eventually destroy the supercurrent in the wall at high enough bias, and restore the Ohmic behavior. If the prevalent ballistic electron model is adopted, then the temperature-dependent scattering effects cannot be discussed. For the high bias (0.3 - 5 V), (a) the I-V curves are temperature-independent (4 - 150 K), and (b) the currents (magnitudes) saturate. The behavior (a) arises from the fact that the neutral supercurrent below the critical temperature is not accelerated by the electric field. The behavior (b) is caused by the limitation of the number of quantum-states for the “holes” running outside of the tube.

Based on the idea that different temperatures generate different conduction electron densities and the resulting carrier diffusion generates the thermal electromotive force (emf), a new formula for the Seebeck coefficient (thermopower) S is obtained: S=(2/3)ln2(qn)-1eFkBD0, where kB is the Boltzmann constant, and q, n, eF, D0 are charge, carrier density, Fermi energy, density of states at eF, respectively. Ohmic and Seebeck currents are fundamentally different in nature, and hence, cause significantly different behaviors. For example, the Seebeck coefficient S in copper (Cu) is positive, while the Hall coefficient is negative. In general, the Einstein relation between the conductivity and the diffusion coefficient does not hold for a multicarrier metal. Multi-walled carbon nanotubes are superconductors. The Seebeck coefficient S is shown to be proportional to the temperature T above the superconducting temperature Tc based on the model of Cooper pairs as carriers. The S follows a temperature behavior, , where Tg ’= constant, at the lowest temperatures.

The conduction of a single-wall carbon nanotube depends on the pitch. If there are an integral number of carbon hexagons per pitch, then the system is periodic along the tube axis and allows “holes” (not “electrons”) to move inside the tube. This case accounts for a semiconducting behavior with the activation energy of the order of around 3 meV. There is a distribution of the activation energy since the pitch and the circumference can vary. Otherwise nanotubes show metallic behaviors (significantly higher conductivity). “Electrons” and “holes” can move in the graphene wall (two dimensions). The conduction in the wall is the same as in graphene if the finiteness of the circumference is disregarded. Cooper pairs formed by the phonon exchange attraction moving in the wall is shown to generate a temperature-independent conduction at low temperature (3 - 20 K).

Mani observed zero-registance states similar to those quantum-Hall-effect states in GaAs/AlGaAs but without the Hall resistance plateaus upon the application of radiations [R. G. Mani, Physica E 22, 1 (2004)]. An interpretation is presented. The applied radiation excites “holes”. The condensed composite (c)-bosons formed in the excited channel create a superconducting state with an energy gap. The supercondensate suppresses the non-condensed c-bosons at the higher energy, but it cannot suppress the c-fermions in the base channel, and the small normal current accompanied by the Hall field yeilds a B-linear Hall resistivity.

A metallic (semiconducting) single-wall nanotube contains an irrational (integral) number of carbon hexagons in the pitch. The room-temperature conductivity is higher by two to three orders of magnitude in metallic nanotubes than in semiconducting nanotubes. Tans et al. [Nature386 (1997) 474] measured the electrical currents in metallic single-wall carbon nanotubes under bias and gate voltages, and observed non-Ohmic behaviors. The original authors interpreted their data in terms of a ballistic transport due to the Coulomb blockage on the electron-carrier model. The mystery of why a ballistic electron is not scattered by impurities and phonons is unexplained, however. An alternate interpretation is presented based on the Cooper pair (pairon)–carrier model. Superconducting states are generated by the Bose–Einstein condensation of the ± pairons at momenta 2πℏn/L, where L is the tube length and n a small integer. As the gate voltage changes the charging state of the tube, the superconducting states jump between different n. The normal current peak shapes appearing in the transition are found to be temperature-dependent, which is caused by the electron–optical phonon scattering.

The motion of the guiding center of magnetic circulation generates a charge transport. The application of kinetic theory to the motion gives a modified Drude formula for the magnetoconductivity: σ = e 2 n c τ / M * , where M⁎ is the magnetotransport mass distinct from the cyclotron mass, nc the density of the conduction electrons, and τ the relaxation time. The density nc depends on the applied magnetic field direction relative to copper's face-centered-cubic lattice, when the Fermi surface of copper is nonspherical with necks. The anisotropic magnetoresistance of copper is calculated with the assumption of the necks representing by spheres of radius a centered at the eight singular points on the ideal Fermi surface. A good fit with experiments is obtained.

The motion of the guiding center of magnetic circulation generates a charge transport. By applying kinetic theory to the guiding center motion, an expression for the magnetoconductivity σ is obtained: σ = e2ncτ/M*, where M* is the magnetotransport mass distinct from the cyclotron mass, nc the density of the conduction electrons, and τ the relaxation time. The density nc depends on the magnetic field direction relative to copper's fcc lattice, when Cu's Fermi surface is nonspherical with “necks”. The anisotropic magnetoresistance is analyzed based on a one-parameter model, and compared with experiments. A good fit is obtained.

At the even-denominator filling factor ν=P/Q, even Q, a fermionic Quantum Hall (QH) state is shown to be developed at the lowest temperatures. In this state, formed by the composite (c-) fermions, each with an electron and the even number Q of flux quanta (fluxons), the conductivity σ≡J/E(J= current density , E= applied field ) becomes a universal constant (e2/h)Q-2 as the temperature T approaches zero while the Hall conductivity σH≡J/EH(EH= Hall field ) becomes approximately equal to (e2/h)(P/Q).

We use Boltzmann’s transport equation to calculate the 3D incoherent Landauer equation for neutrons and photons diffusing in a slab of amorphous material. We use the P1 approximation to calculate the multiple-scattering transmission coefficient as a function of the diffusion coefficient. The 3D transmission coefficient obtained is smaller than the 1D case. The P1 result, valid only for thick slabs, is proved to be an excellent approximation to the exact solution.

The phonon exchange between the electron and the elementary magnetic flux (fluxon) induces an attractive transition in the degenerate Landau states. This attraction bounds an electron–fluxon complex. The center-of-mass of the complex moves as a boson with a linear dispersion relation (∊ = cp). The 2D system of free massless bosons undergoes a Bose–Einstein condensation at k B T c = 1.954ℏcn1/2, where n is the boson density. For GaAs/AlGaAs, T c ~ 1 K at the principal Landau-level occupation ratio ν = 1, where the electron number equals the fluxon number. Below T c , there is an energy gap, which stabilizes the Hall resistivity plateau. The plateau value (j/N)(h/e2) at the fractional occupation ratio ν = N/j, for odd j, indicates that the composite boson containing an electron and j fluxons carries the fractional charge (magnitude) e/j due to the magnetic confinement.

The out-of-plane resistivity, ρc, in a single-crystal YBa 2 Cu 3 O y follows the experimental law: ρc=C1ρab+C2/T, where C1, C2 are the constants and ρab is in-plane resistivity in the concentration range 6.6 c , 90 K, occurring at y=6.88 and in the temperature range T c 2/T arises from the quantum tunneling between copper planes of the – pairons, moving with a linear dispersion relation, and the first term C1ρab from the in-plane currents due to the "holes" and +pairons.

The magnetic susceptibility χ in La 2-x Sr x CuO 4 shows unusual concentration x- and temperature T-behaviors. The χ at 400 K increases with x in the range 0.040+B0/T. The Cooper pair (pairon) has no net spin, and hence its spin contribution to χ is zero. But its motion with the linear dispersion relation: ∊=(2/π)vFp, where vF=Fermi speed, can generate a T-dependent contribution -B1/T. These two contributions generate a χ-maximum at Tm in the range 0.15

Based on the idea that different temperatures generate different carrier densities and the resulting diffusion causes the thermal emf, a new formula for the Seebeck coefficient S is obtained: [Formula: see text], where q, n, εF, [Formula: see text]. and [Formula: see text]. are respectively charge, carrier density, Fermi energy, density of states at ∊F and volume. Ohmic and Seebeck currents are fundamentally different in nature. This difference can cause significantly different transport behaviors. For a multi-carrier metal the Einstein relation between the conductivity and the diffusion coefficient does not hold in general. Seebeck (S) and Hall (RH) coefficients in noble metals have opposite signs. This is shown to arise from the Fermi surface having "necks" at the Brillouin boundary.

A kinetic theory is developed for the infrared (IR) Hall effect. The dynamic transport coefficients including the conductivity σ, cot θ H (θ H = Hall angle ) and the Hall coefficient R H for a system of conduction electrons ("electrons" or "holes") are shown to be obtained by applying the conversion rule: γ0 → γ(ω) -iω to the expressions for the static coefficients, where γ0 [γ(ω)] are static (dynamic) scattering rates which depend on the frequency ω and temperature T. If the real (Re) and imaginary (Im) parts of σ(ω) are measured, the ratio Re [σ(ω)]/ Im [σ(ω)] is equal to γ/ω, which directly gives the dynamic rate γ(ω, T). The ratio Re [ cot θ H ]/ Im [ cot θ H ] = -γ H (ω, T)/ω yields the dynamic Hall rate γ H (ω, T). The IR Hall effect experiments give a remarkable result: γ H (ω, T) = γ H , 0(T), that is, the dynamic Hall scattering rate is equal to the static rate up to mid-IR ~1000 cm -1.

A kinetic theory is developed for the infrared Hall effect. Expressions for the dynamic conductivity σ(ω), ω = laser frequency, and other transport coefficients are shown to be obtained by applying the conversion rule: γ0 → γ -iω to those for the static coefficients, where γ0 (γ) are the static (dynamic) scattering rates. The observed T2-law of [Formula: see text] in YBa 2 Cu 3 O 7 are shown to arise from the drift velocity of the low-energy Cooper pairs (pairons) moving with the linear dispersion relation: ε = (2/π) v F p, where v F is the Fermi velocity.

A new formula for the Seebeck coefficient S is obtained: [Formula: see text], where q, n, ε F , [Formula: see text] and [Formula: see text] are, respectively, charge, carrier density, Fermi energy, density of states at ε F and volume. Seebeck (S) and Hall (R H ) coefficients in alkali metals are both negative while these coefficients in noble metals have opposite signs. This difference is shown to arise from the different shapes of the Fermi surface.

The d-wave Cooper pairs (pairons) in cuprates with a strong binding along the a- and b-axes are shown to arise from the optical-phonon-exchange attraction.

We show that the eigenvalues of the number operator [Formula: see text] in the pair state (k1, k2), where c's and c†'s are fermion operators, are limited to 0 or 1, while the eigenvalues of the pair-number operator with the net momentum q≡ k1+k2 fixed, [Formula: see text], are 0, 1, 2,…. Both fermionic and bosonic natures of the Cooper pairs (pairons) must be used in the superconductivity theory. The Bose–Einstein condensation (BEC) of the pairons moving with a linear energy–momentum relation occurs at [Formula: see text] (3D), [Formula: see text] (2D), where n0 is the pairon density and v F the Fermi velocity. The connection between this [Formula: see text] and the BCS superconducting temperature is discussed.

The quantum statistics of a composite is studied by looking at its Center-of-Mass (CM), which moves, following general principles of quantum theory and relativity. With respect to the CM motion the Ehrenfest–Oppenheimer–Bethe rule is shown to hold: a composite is fermionic if it contains an odd number of elementary fermions, bosonic if the number of fermions in it is even.

The Cooper pair (pairon) field operator ψ†(r,t) changes, following Heisenberg's equation of motion. If the Hamiltonian ℋ contains pairon kinetic energies h0, a condensation energy α(1-r2), β>0, the evolution equation for ψ is nonlinear, from which we obtain the Ginzburg–Landau (GL) equation: [Formula: see text] for the GL wave function Ψσ(r)≡ 1/2|σ>, where σ denotes the state of the condensed pairons, and n the density operator. The GL equation with α=-εg(T) is shown to hold for all temperatures (T) below Tc, where εg is the pairon energy gap. Its equilibrium solution yields that the condensed pairon density n0(T)=|Ψσ(r)|2 is proportional to εg(T). The original GL T-dependence of the expansion parameters near Tc:α=-b(Tc-T), β= constant is justified. With the assumption of h0, a new formula for the penetration depth is obtained.

The electron-pair density matrix \gr(k 1, k 2; k 3, k 4, t) =: \gr(1, 2; 3, 4, t) changes in time, following a quantum Liouville equation with in the presence of a Coulomb interaction {itv}{in{itc}}, where \gW is the volume. If the virtual phonon exchange is in action, the density matrix \gr is shown to change similarly with an effective interaction {itv}{in{ite}}, by using a time-dependent perturbation theory and a Markoffian approximation. The dominant longitudinal-acoustic-(optical)-phonon-exchange attraction at 0K is shown to be q-independent (-dependent). The results are used to discuss the Cooper pair size, the origin of type II superconductivity and the formation of d-wave Cooper pairs in the cuprates.

The Cooper pair (pairon) field operator ψ(r,t) changes in time, following Heisenberg’ s equation of motion. If the system Hamiltonian \(\mathcal{H}\) contains the pairon kinetic energies h0, the condensation energy per pairon α( 0, the evolution equation for ψ is non-linear, from which we obtain the Ginzburg-Landau equation: Open image in new window for the complex order parameter Ψσ(r) := | n1/2 | σ >, where σ denotes the state of the condensed pairons, and n the pairon density operator. The total kinetic energy h0 for “electron” (1) and “hole” (2) pairons is Open image in new window where vF(j) ≡ (2 eF / mj1/2 are Fermi velocities, and A the vector potential. A new expression for the penetration depth λ is obtained: $$\lambda = \frac{c} {e}\left[ {\frac{p} {{4\pi n_0 \left( {v_F^{(2)} + v_F^{(1)} } \right)}}} \right]^{1/2} ,$$ where p and n0 are respectively the momentum and density of condensed pairons.

The binding energy of a Cooper pair formed with the BCS model interaction potential is obtained numerically for all coupling in two and three dimensions for all non-zero center-of-mass momentum (CMM) of the pair. The pair breaks up for very small CMM, at most about four orders of magnitude smaller than the maximum CMM allowed by the BCS model interaction, and its binding energy is remarkably linear over the entire range of the CMM up to breakup.

An exactly solvable spatio-temporal model (time series) is proposed. The solution of this model is obtained as the most probable path determined by counting the number of all possible paths in the cells obtained by dividing a spatio-temporal space. The model is characterized by two parameters, a maximum jump distance pa (a = unit step and p = integer) and a fractal geometrical exponent v (ato = constant, to = unit time).

A quantum statistical theory is developed for a fractional quantum Hall effects in terms of composite bosons (fermions) each of which contains a conduction electron and an odd (even) number of fluxons. The cause of the QHE is by assumption the phonon exchange attraction between the conduction electron ("electron", "hole") and fluxons (quanta of magnetic fluxes). We postulate that c-fermions with \emph{any} even number of fluxons have an effective charge (magnitude) equal to the electron charge $e$. The density of c-fermions with $m$ fluxons, $n_\phi^{(m)}$, is connected with the electron density $n_{\mathrm e}$ by $n_\phi^{(m)}=n_{\mathrm e}/m$, which implies a more difficult formation for higher $m$, generating correct values $me^2/h$ for the Hall conductivity $\sigma_{\mathrm H}\equiv j/E_{\mathrm H}$. For condensed c-bosons the density of c-bosons-with-$m$ fluxons, $n_\phi^{(m)}$, is connected with the boson density $n_0$ by $n_\phi^{(m)}=n_0/m$. This yields $\sigma_{\mathrm H}=m\,e^2/h$ for the magnetoconductivity, the value observed of the QHE at filling factor $\nu=1/m$ ($m=$odd numbers). Laughlin's theory and results about the fractional charge are not borrowed in the present work., Comment: 11 pages, 1 figure

Newtonian equations of motion for the Bloch electron are derived and discussed in this chapter. “Electrons” (“holes”), which appear in the Hall coefficient mesurements, are generated near the Fermi surface on the negative (positive) curvature side of the surface.

The conductivity σ is positive, but the Seebeck coefficient S can be positive or negative. We see that in Fig. 1, the measured Seebeck coefficient S in Al at high temperatures (400 – 670 ◦C) is negative, while the S in noble metals (Cu, Ag, Au) are positive (Rossiter & Bass, 1994). Based on the classical statistical idea that different temperatures generate different electron drift velocities, we obtain S = − cV 3ne , (1.4)

An energy eigenvalue equation for a quasi-particle is derived, starting with the Heisenberg equation of motion for an annihilation operator. An elementary derivation of the Fermi liquid model having a sharply defined Fermi surface in the k-space is given, starting with a realistic model of a metal including the Coulomb interaction among and between electrons and lattice-ions. The Ginzburg-Landau wave function [Psi][sub [sigma]](r), where [sigma] represents the superconducting pairon (Cooper-pair) state, is shown to be connected with the one-pairon density operator n by [Psi][sub [sigma]](r) = [l angle]r[vert bar]n[sup 1/2][vert bar][sigma][r angle]. A close analogy between supercurrent and laser is indicated. 31 refs.

The relationship between superconductivity and band structures of electrons and phonons is established on the basis of a generalized Bardeen-Cooper-Schrieffer model in which the interaction strengths (V11,V12,V12) among and between “electron” (1) and “hole” (2) Cooper pairs are differentiated. Elemental superconductors must have local hyperboloidal Fermi surfaces called “necks” or “inverted double caps.”

With the help of quantum‐scattering‐theory methods and the approximation of stationary phase, a one‐dimensional coherent random‐walk model which describes both tunneling and scattering above the potential diffusion of particles in a periodic one‐dimensional lattice is proposed. The walk describes for each lattice cell, the time evolution of modulating amplitudes of two opposite‐moving Gaussian wave packets as they are scattered by the potential barriers. Since we have a coherent process, interference contributions in the probabilities bring about strong departures from classical results. In the near‐equilibrium limit, Fick’s law and its associated Landauer diffusion coefficient are obtained as the incoherent contribution to the quantum current density along with a novel coherent contribution which depends on the lattice properties as [(1−R)/R]1/2.

The unusual quantum Hall effect (QHE) in graphene is often discussed in terms of Dirac fermions moving with a linear dispersion. A new theory describing the same phenomena is presented in terms of the more traditional composite bosons. The "electron" (wave packet) is shown to move easier in the direction [110] ≡ [110 c- axis ] of the honeycomb lattice than perpendicular to it, while the "hole" moves easier in [001]. Since "electrons" and "holes" move in different channels, the number densities can be very high especially when the Fermi surface has "necks". The strong QHE at filling factor ν = 2 arises from the "neck" Fermi surfaces.