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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.

In 2000, Yao, Kane and Dekker [

SWNTs can be produced by rolling graphene sheets into circular cylinders of about one nanometer (nm) in diameter and microns in length [5,6]. The electrical conduction in SWNTs depends on the circumference and pitch, and can be classified into two groups: either semiconducting or metallic [

follows. A SWNT is likely to have an integral number of carbon hexagons around the circumference. If each pitch contains an integral number of hexagons, then the system is periodic along the tube axis, and “holes” (not “electrons”) can move along the tube length. Such a system is semiconducting and its electrical conductivity increases with the temperature (semiconductor-like), and is characterized by an activation energy [

In the present work, we present a unified microscopic theory of both lowand high-field conductivities. We primarily deal with the metallic SWNTs in the present work. Before dealing with high-field transports, we briefly discuss the micro-field (~mV) transports. Tans et al. [

1) A non-Ohmic behavior is observed for all, that is, the currents are not proportional to the bias voltage except for high bias. The gate voltage charges the tube. The Coulomb (charging) energy of the system having charge

is represented by

where is the total capacitance of the tube.

2) The current near the origin is nearly constant for different gate voltages, (A)-(C). This feature was confirmed by later experiments [9,11]. The current does not change for small varying gate voltage in a metallic SWNT (while the current (magnitude) decreases in a semiconducting SWNT).

3) The current at gate voltage (A) reverts to the normal resistive behavior after passing the critical bias voltages on both (positive and negative) sides. Similar behaviors are observed for (B) and (C).

4) The flat current is destroyed for higher bias voltages (magnitudes). The critical bias voltage becomes smaller for higher gate voltages.

5) There is a restricted -range (view window) in which the horizontal stretch can be observed.

Tan et al. [

We propose a different interpretation of the data in

The supercurrents run without obeying Ohm’s law. This explains the feature (1). The supercurrents can run with no resistance due to the phonon and impurity scattering and with no bias voltage. Bachtold et al. [

where is the kinetic energy of the conduction electrons and the pairons, and

is the Coulomb field energy, then the system is stable. Experiments in

If the bias voltage is high enough so that the inequality symbol in Equation (2) is reversed, then normal currents revert and exhibit the Ohmic behavior, which explains the feature (3).

The feature (4) can be explained as follows. For higher there is more amount of charge, and hence the charges, , for the three cases (A, B, C) satisfy the inequalities:

The horizontal stretches are longer for smaller bias voltages. At the end of the stretch the system energy equals the condensation energy. Hence, we obtain from Equation (1.2) after dropping the kinetic energy

Using Equation (1.4), we then obtain

which explains the feature (iv).

The horizontal stretch becomes shorter as the gate voltage is raised; it vanishes when is a little over 120.0 mV. The limit is given by

If the charging energy exceeds the condensation energy, then there are no more supercurrents, which explains the feature (v). Clearly the important physical property in our pairon model is the condensation energy.

In the currently prevailing theory [

Yao et al. [

where and are constants. Strikingly, the I-V curves at great bias measured at different temperatures between 4 K and room temperatures overlap with each other. From the shape of the I-V curves, it is clear that the trend of decreasing conductance continues to high bias. Extrapolating the measured I-V curves to higher voltage would lead to a current saturation, that is, a vanishing conductance.

In the present work, we present a quantum statistical theory of the transports, starting with the crystal structure, establishing the electron energy bands, electron-phonon interaction, the BCS-like Hamiltonian and calculating everything steps by steps.

If the SWNT is unrolled, then we have a graphene sheet, which can be superconducting at a finite temperature. We first study the conduction behavior of graphene in Section 2, starting with the honeycomb lattice and

introducing “electrons” and “holes” based on the orthogonal unit cell. Phonons are generated based on the same orthogonal unit cell. In Section 3, we treat phonons and phonon-exchange attraction. In Section 4, we construct a Hamiltonian suitable for the formation of the Cooper pairs. We derive, in Section 5, the linear dispersion relation for the center-of-mass motion of the pairons. The pairons moving with a linear dispersion relation undergoes a Bose-Einstein condensation (BEC) in 2D, which is shown in Section 6. Low-bias anomaly is discussed in Section 7. Current saturation and the temperature behavior are discussed in Section 8. Summary and discussion are given in Section 9.

Following Ashcroft and Mermin [

where, and are Cartesian orthonormal vectors since the -vectors are involved in the semiclassical equation of motion:

where and are the electric and magnetic fields, respectively, and the vector

is the electron velocity where is the energy. The 2D crystals such as graphene can also be treated similarly, only the -components being dropped. The choice of the Cartesian axes and the unit cell is obvious for the cubic crystals. We must choose an orthogonal unit cell also for the honeycomb lattice, as shown below.

Graphene forms a 2D honeycomb lattice. The WS unit cell is a rhombus shown in

where

are Bravais vectors with the primitive vectors and integers. In the field theoretical formulation, the field point is given by

where is the point defined within the standard unit cell. Equation (12) describes the 2D lattice periodicity but does not establish the -space, which is explained below.

To see this clearly, we first consider an electron in a simple square (sq) lattice. The Schrödinger wave equation is

where is the effective electron mass. The Bravais vector for the sq lattice, , is

The system is lattice periodic:

If we choose a set of Cartesian coordinates along the sq lattice, then the Laplacian term in Equation (15) is given by

If we choose a periodic square boundary with the side length, integer, then there are 2D Fourier transforms and (2D) -vectors.

We now go back to the original graphene system. If we choose the -axis along either or, then the potential energy field is periodic in the x-direction, but it is aperiodic in the -direction. For an infinite lattice the periodic boundary is the only acceptable boundary condition for the Fourier transformation. Then, there is no 2D -space spanned by 2D -vectors. If we omit the kinetic energy term, then we can still use Equation (12) and obtain the ground state energy (except the zero point energy).

We now choose the orthogonal unit cell shown in

where is the nearest neighbor distance between two C’s. The unit cell contains 4 C’s. The system is lattice-periodic in the - and -directions, and hence there are 2D -space.

The “electron” (“hole”) is defined as a quasi-electron that has an energy higher (lower) than the Fermi energy and “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 “electron” (wave packet) may move up or down along the -axis to the neighboring hexagon sites passing over one C^{+}. The positively charged C^{+} acts as a welcoming (favorable) potential valley for the negatively charged “electron”, while the same C^{+} acts as a hindering potential hill for the positively charged “hole”. The “hole”, however, can move horizontally along the x-axis without meeting the hindering potential hills. Thus the easy channel directions for the “electrons” (“holes”) are along the y-(x-)axes.

Let us consider the system (graphene) at 0 K. If we put an electron in the crystal, then the electron should occupy the center O of the Brillouin zone, where the lowest energy lies. Additional electrons occupy points neighboring the center O in consideration of Pauli’s exclusion principle. The electron distribution is lattice-periodic over the entire crystal in accordance with the Bloch theorem [

Carbon (C) is a quadrivalent metal. The first few lowlying energy bands are completely filled. The uppermost partially filled bands are important for the transport properties discussion. We consider such a band. The Fermi surface, which defines the boundary between the filled and unfilled k-spaces (area) is not a circle since the x-y symmetry is broken. The “electron” effective mass is lighter in the y-direction than perpendicular to it. Hence the electron motion is intrinsically angle-dependent (anisotropic). The negatively charged “electron” is near the positive ions C^{+} and the “hole” is farther away from C^{+}. Hence, the gain in the Coulomb interaction is greater for the “electron”. That is, the “electron” is more easily activated. Thus, the “electrons” are the majority carriers at zero gate voltage.

We may represent the activation energy difference by [

The thermally-activated (or excited) electron densities are given by

where and 2 denote the “electron” and “hole”, respectively. The prefactor is the density at the high-temperature limit.

Phonons are bosons corresponding to the running normal modes of the lattice vibrations. They are characterized by the energy, where is the angular frequency, and the momentum vector, whose magnitude is times the wave numbers. The q-vector for phonons is similar to the k-vector for the conduction electrons. The phonon with represents a plane-wave proceeding in the -direction. The frequency is connected with the q-vector through the dispersion relation:

The excitation of the phonons can be discussed based on the same rectangular unit cell introduced for the conduction electrons. We note that phonons can be discussed naturally based on the orthogonal unit cells. [It is difficult to describe phonons in the WS cell model.] For example, longitudinal (transverse) phonons proceeding upwards are generated by imagining a set of plates each containing a number of rectangular cells executing small oscillations vertically (horizontally). A longitudinal wave proceeding in the crystal axis, is represented by

where is the displacement in the x-direction. If we imagine a set of parallel plates containing a great number of ions fixed in each plate, then we have a realistic picture of the lattice vibration mode. The density of ions changes in the x-direction. Hence, the longitudinal modes are also called the density-wave modes. The transverse wave mode can also be pictured by imagining a set of parallel plates containing a great number of ions fixed in each plate executing the transverse displacements. Notice that this mode generates no charge-density variation.

The Fermi velocity in a typical metal is of the order while the speed of sound is of the order. The electrons are likely to move quickly to negate any electric field generated by the density variations associated with the lattice wave. In other words, the electrons may follow the lattice waves instantly. Given a traveling normal wave mode in Equation (21), we may assume an electron density variation of the form:

Since electrons follow phonons immediately for all, the coefficient can be regarded as independent of. If we further assume that the deviation is linear in the scalar product and again in the electron density, we then obtain

This is called the deformation potential approximation [

Each normal mode corresponds to a harmonic oscillator characterized by. The displacements can be expressed as

where are operators satisfying the Bose commutation rules:

Let us now construct an interaction Hamiltonian, which has the dimensions of an energy and which is Hermitian. Using Equations (22) and (23), we obtain

where h.c. denotes the Hermitian conjugate. This Hamiltonian can be expressed as

where, and, are electron operators satisfying the Fermi anticommutation rules:

The in Equation (30) is the Fröhlich Hamiltonian [

Bardeen, Cooper and Schrieffer (BCS) published a historic theory of superconductivity in 1957 [

In the ground state there are no currents for any system. To describe a supercurrent, we must introduce moving pairons, that is, pairons with finite center-of-mass (CM) momenta. Creation operators for “electron” (1) and “hole” (2) pairons are defined by

We calculate the commutators among and, and obtain

Pairon operators of different types always commute:

and

represent the number operators for “electrons” and “holes”.

Let us now introduce the relative and net momenta such that

Alternatively we can represent pairon annihilation operators by

The prime on will be dropped hereafter. In the k-q representation the commutation relations in Equations (33) and (34) are re-expressed as

Using the new notations, we can write the full Hamiltonian as

This is the full Hamiltonian for the system, which can describe moving pairons as well as stationary pairons. Here, the prime on the summations indicates the restriction arising from the phonon exchange attraction, see below. The connection with BCS Hamiltonian [

The phonon exchange attraction is in action for any pair of electrons near the Fermi surface. In general the bound pair has a net momentum, and hence it moves. The energy of a moving pairon can be obtained from:

which is Cooper’s equation, Equation (1) of his 1956 Physical Review Letter [

Equation (42) can be solved as follows. We assume that the energy is negative:

Then,

Rearranging the terms in Equation (1.42) and dividing by, we obtain

where

which is k-independent.

Introducing Equation (45) in Equation (42), and dropping the common factor, we obtain

We now assume a free electron moving in 3D. The Fermi surface is a sphere of the radius (momentum)

where represents the effective mass of an electron. The energy is given by

The prime on the -integral in Equation (47) means the restriction:

We may choose the polar axis along q as shown in

After performing the integration and taking the small-q and small- limits, we obtain

where the pairon ground-state energy is given by

As expected, the zero-momentum pairon has the lowest energy. The excitation energy is continuous with no energy gap. Equation (53) was first obtained by Cooper (unpublished), and it is recorded in Schrieffer’s book [

Such a linear energy-momentum relation is valid for pairons moving in any dimension (D). However, the coefficients slightly depend on the dimensions; in fact

and for 3D and 2D, respectively.

In Section 4, we saw that the pair operators appearing in the full Hamiltonian in Equation (41) satisfy rather complicated commutator relations in Equations (39) and (40). In particular part of Equation (39)

reflect the fermionic natures of the constituting electrons. Here, represents creation operator for zero momentum pairons. BCS [

Here, we expressed the “electron” and “hole” kinetic energies in terms of pairon operators. The reduced Hamiltonian is bilinear in pairon operators, and can be diagonalized exactly. BCS obtained the groundstate energy as

where is the density of states at the Fermi energy. The is the ground-state energy of the pairon, see Equation (54). Equation (58) means simply that the ground state energy equals the numbers of pairons times the ground-state energy of the pairon. Our Hamiltonian in Equation (41) is reduced to the original BCS Hamiltonian (see Ref. [

To describe a supercurrent, we must introduce moving pairons. We now show that the center-of-masses of the pairons move as bosons. That is, the number operator of pairons having net momentum

have the eigenvalues

The number operator for the pairons in the state is

where we omitted the spin indices. Its eigenvalues are limited to zero or one:

To explicitly see this property in Equation (60), we introduce

and obtain

Although the occupation number is not connected with as, the eigenvalues of satisfying Equation (64) can be shown straightforwardly to yield [

with the eigenstates

In summary, pairons with both and specified are subject to the Pauli exclusion principle, see Equation (62). Yet, the occupation numbers of pairons having a CM momentum are.

The most important signature of many bosons is the Bose-Einstein Condensation (BEC). Earlier we showed that the pairon moves in 2D with the linear dispersion relation, see (53):

where we designated the pairon net momentum (magnitude) by the more familiar rather than.

Let us consider a 2D system of free bosons having a linear dispersion relation:,. The number of bosons, , and the Bose distribution function

are related by

where is the chemical potential, , and

is the number of zero-momentum bosons. The prime on the summation in Equation (69) indicates the omission of the zero-momentum state. For notational convenience, we write

We divide Equation (69) by the normalization area, and take the bulk limit:

We then obtain

where is the number density of zero-momentum bosons and the total boson density. After performing the angular integration and changing integration variables, we obtain from Equation (73):

where the fugacity

is less than unity for all temperatures. After expanding the integrand in Equation (74) in powers of, and carrying out the x-integration, we obtain

Equation (76) gives a relation among, , and.

The function monotonically increases from zero to the maximum value

as is raised from zero to one. In the low-temperature limit, , , and the density of excited bosons, , varies as as seen from Equation (76). This temperature behavior of persists as long as the right-hand-side (r.h.s.) of Equation (76) is smaller than; the critical temperature occurs at. Solving this, we obtain

The BEC of pairons moving in 2D occurs at a finite temperature. This appears to contradict with Hohenberg’s theorem (no long range order in 2D). But this theorem is proved under the assumption of the f-sum rule arising from the mass conservation. The pairons move massless with the linear dispersion relation [see Equation (71)], and hence they are not subject to Hohenberg’s theorem [

If the temperature is raised beyond, the density of zero momentum bosons, , becomes vanishingly small, and the fugacity can be determined from

In summary, the fugacity is equal to unity in the condensed region:, and it becomes smaller than unity for, where its value is determined from Equation (80).

Formula (79) for the critical temperature is distinct from the famous BCS formula

where is the zero temperature electron energy gap in the weak coupling limit. The electron energy gap and the pairon ground-state energy both depend on the phonon-exchange coupling energy parameter, which appears in the starting Hamiltonian H in Equation (41). The energy is negative (bound-state energy). Hence, this cannot be obtained by the perturbation theory. The connection between and is very complicated. This makes it difficult to discuss the critical temperature based on the BCS relation (81). Unlike the BCS formula, formula (79) is directly connected with the measurable quantities: the pairon density and the Fermi speed.

We emphasize here that both formulas (79) and (81) were derived, starting with the Hamiltonian in Equation (41) and following statistical mechanical calculations, see the reference [

The unusual current-dips at zero bias in

With no bias, the nanotube’s wall below 150 K is in a superconducting state. If a small bias is applied, then the system is charged, positively or negatively depending on the polarity of the external bias. The applied bias field will not affect the neutral supercurrent but can accelerate the charges at the outer side of the carbon wall. The resulting normal currents carried by conduction electrons are scattered by impurities and phonons. The phonon population changes with temperature, and hence the phonon scattering is temperature-dependent. The normal electric currents along the tube length generate circulating magnetic fields, which eventually destroy the supercurrent running in the wall at a high enough bias. Thus, the current versus the voltage (mV) is non-linear near the origin because of the supercurrents running in the wall. The differential conductance is very small and nearly constant (superconducting) for in the HRC sample, see the lower panel in

The high-bias I-V curves in

The current saturation observed in

must saturate to the maximum number as the bias is raised further.

The unusual non-Ohmic transport behaviors at low (zero) bias anomaly, observed in metallic SWNT are explained in terms of a two-currents model. Supercurrents run in the graphene wall below 150 K. The normal “hole” currents on the outer-side of the tube are subject to scattering by phonons and impurities. The currents along the tube length generate circulating magnetic fields, which eventually destroy the supercurrent in the wall at high enough bias, and restore the Ohmic behavior. The lowcurrent anomaly is temperature-dependent since the phonon population changes with the temperature.

The I-V curves for the high bias (0.3 - 5 V) are temperature-independent (4 - 150 K), which arises from the fact that the neutral supercurrent running in the tube wall is not accelerated by the bias below the superconducting (critical) temperature. It is highly desirable to find the critical temperature by performing experiments above 150 K (experimental temperature).

The current saturation above 0.5 V observed arises from the limitation of the quantum-state sites for the “holes” running on the outer surface of the tube. If the tube’s circumference size is raised, then the saturation current should increase.

In the course of our calculations we uncover several significant facts as follows:

• To establish a 2D -space for graphene we must introduce a rectangular unit cell distinct from the WS unit cell (rhombus).

• Electrons and phonons run anisotropically in graphene.

• Electrons and phonons are generated based on the same rectangular unit cells. This is important when dealing with the electron-phonon scattering and the phonon-exchange attraction.

• Phonons (bosonic quanta) representing the running plane-wave modes of the lattice oscillations are generated.

• The so-called Fröhlich interaction Hamiltonian was derived, with the assumption that the electrons move in the perturbing density waves generated by the longitudinal phonons. The transverse phonons do not contribute to the Fröhlich interaction.

• “Electrons” and “holes” move as wave packets whose sizes are of the orthogonal unit cells.

• Phonons’ average size is much greater than the electron size. The average phonon energy is much smaller than the conduction electron energy.

• The BEC temperature for moving pairons is regarded as the superconducting temperature. Finding for metallic SWNT which is greater than 150 K from the studied experiments [