Large distance limit of the Casimir energy between carbon nanotubes

1. Introduction

The quantum-thermal fluctuations of the electromagnetic field cause the appearance of forces between neutral objects, as was predicted by Casimir.¹ Later, Lifshitz generalized the study for finite temperature plates.² Today, the Lifshitz formula has been generalized to objects with arbitrary geometry by the Multiscattering formalism.^{7,8,9,10,11,12}

The Casimir force is relevant in the micromechanical range of scales, therefore, it is expected that it can have an important effect in the interaction between single wall carbon nanotubes (SWCNs). In the large distance limit, the Casimir interaction between SWCNs has been computed numerically,³ but any analytical result is shown, also, the Casimir force of one SWCN inside another was obtained in Ref. 4. By using the modern Multiscattering formalism,⁹ the Casimir energy between parallel cylinders can be easily and systematically obtained,⁹ even the case of inclined cylinders is obtained with this formalism.⁵

Here we are going to apply the Multiscattering formalism to compute the large distance limit (when the distance d between the SWCNs is much greater than their respective radius R) between parallel SWCNs, and we are going to obtain analytical results for asymptotic Casimir energies in the zero and high-temperature limits.

This paper is organized as follows. In Sec. 2, we introduce the Multiscattering formalism that we are going to use for objects with cylindrical symmetry, then we define the cylindrical vector multipoles and the translation and T scattering matrices we are going to use. In Sec. 3, we review the axial electric conductivity of SWCNs, needed to obtain the T-matrix. We show the main results of this paper in Sec. 4 and we summarize our results in Sec. 5.

2. Formalism for Casimir Effect

Here we are going to use the multiscattering formalism for the Casimir effect.^{7,8,9,10,11,12} The Casimir energy $E_T$ between two objects at any finite temperature T is calculated by

where $\left|A\right|$ denotes the determinant of A, ${\kappa }_n=2\pi \frac{k_{B}T}{\hslash c}n$ are the imaginary Matsubara frequencies ($\omega =\mathrm{\text{i}}c{\kappa }_n$), $k_B$ is the Boltzmann constant, $\hslash$ the Planck constant, c the light velocity, $𝕀$ is the identity matrix, ${𝕋}_a$ is the T-scattering matrix of the object a placed at $x_a$, and ${𝕌}_{ab}$ is the translation matrix in open space of multipole waves from $x_a$ to $x_b$ and the prime in the summatory means that the $n=0$ case has a $1/2$ weight.

In the case of parallel infinite cylinders, all matrices are diagonal in the cylindrical axes (labeled by z here) and the formula is simplified into

where L is the length of the cylinders. This formula can be simplified in the zero and high-temperature (or classical) limit as and respectively.

2.1. Vector multipoles

We need a basis to obtain the $𝕋$ scattering matrices for SWCN and to define the translations matrices $𝕌$ between the objects. We use the following basis for vector multipoles⁹ :

for imaginary frequencies $\omega =\mathrm{\text{i}}\xi =\mathrm{\text{i}}c\kappa$ and ${\kappa }_{\rho }=\sqrt{{\xi }^2+k_z^2}\ge 0$, ${\varphi }_{n{k}_z}^{\mathrm{\text{reg}}}(x,\kappa )=I_n({\kappa }_{\rho }\rho )e^{\mathrm{\text{i}}n\theta }{e}^{\mathrm{\text{i}}{k}_{z}z}$ and ${\varphi }_{n{k}_z}^{\mathrm{\text{out}}}(x,\kappa )=K_n({\kappa }_{\rho }\rho )e^{\mathrm{\text{i}}n\theta }{e}^{\mathrm{\text{i}}{k}_{z}z}$.

2.2. Translation U-matrices

Using the results of Ref. 9, the translation matrix ${𝕌}_{ab}$ in open space of cylindrical multipole waves from $x_a$ to $x_b$ is a matrix with the following entries :

where $X_{ij}=x_i-x_j=\left(\right.X_{\perp }^{ij}cos({\theta }^{ij}),X_{\perp }^{ij}sin({\theta }^{ij}),X_z^{ij})$, P and Q represent the vectorial multipoles $M(\kappa )$ and $N(\kappa )$ and for vectorial multipoles defined in systems of coordinates with parallel axes but displaced by $X_{ij}$.

2.3. T-matrix for SWCN

By using the Waterman formalism,¹⁶ we obtain the T-matrix for a cylindrical surface of radius R with only axial conductivity ${\sigma }_{zz}(\xi )\ne 0$ (for compact dielectric cylinders see¹⁹) for imaginary frequencies $\omega =\mathrm{\text{i}}\xi$ as

This result coincides with Ref. 17, but differs from Ref. 18 because in our case we are considering that the azimuthal conductivity ${\sigma }_{\varphi \varphi }$ is zero, while in Ref. 18, it is assumed that ${\sigma }_{\varphi \varphi }={\sigma }_{zz}$. In the following section will be clear why we only need the contribution of the axial conductivity for the large distance limit of the Casimir effect.

Note that this T-matrix depends only on the electric axial conductivity of the SWCN ${\sigma }_{zz}$, that we are going to discuss in the following section.

Note also that this T-matrix is very different from the result shown in Ref. 19 for compact homogeneous dielectric cylinders. One considers only the dielectric properties of the bulk, while the other takes into account the conductivity of the SWCN only at the surface as surface conductivities. In addition to that, in Eq. (9) we show the T-matrix of a cylinder with inhomogeneous surface conductivity (${\sigma }_{\varphi \varphi }=0$ while ${\sigma }_{zz}\ne 0$), which results in a different structure of the T-matrix in the polarization space, with only one diagonal term of the T-matrix different from zero (the $NN$ term), while in the homogeneous cases,^18,19 the T-matrix is nonzero for the four components of the polarization space.

3. Conductivity of SWCNs

As we are interested in the large distance limit ($d\gg R$) of the Casimir energy between carbon nanotubes, we only need to know the behavior of the conductivity in a small neighborhood around $\omega =0$. For those small frequencies, the azimuthal conductivity is much smaller than the axial conductivity ${\sigma }_{\varphi \varphi }\ll {\sigma }_{zz}$, then, we can safely approach ${\sigma }_{\varphi \varphi }=0$ in our analysis.¹³ In particular, in Ref. 14, it is shown that the azimuthal conductivity ${\sigma }_{\varphi \varphi }$ becomes different from zero for frequencies $\omega >\frac{2\hslash v_F}{3R}$, therefore, this term behaves like a strong dielectric, and it is expected that its contribution to the large distance limit of the Casimir energy will be sub-dominant to the contribution of the axial conductivity ${\sigma }_{zz}$. In addition to that, for very small frequencies the contribution of exitonic peaks is negligible.^4,13 Therefore, we characterize the carbon nanotubes only by their axial conductivity ${\sigma }_{zz}(\omega )$.

The basic properties of each SWCN are captured by their chirality index $(n,m)$, which also determines the SWCN radius $R=\frac{\sqrt{3}a}{2\pi }\sqrt{m^2+nm+n^2}$($a=1.42\mathrm{\text{ Å}}$ is the $C-C$ interatomic distance) and the electronic modulation index $\nu =\{-1,0,+1\}$. The Hamiltonian for the electronic quasiparticle with spin s of an SWCN is approximated by¹⁴

where $k=(k_1,k_2)=(k_x,k_z)$ is the quasi-momentum of the electronic quasiparticle, ${\mathrm{\Delta }}_s^{\eta }$ is the effective Dirac mass, $v_F\approx \frac{c}{300}$ is the Fermi velocity of electronic quasiparticles in the SWCN, ${\tau }_i$ is the ith Pauli matrix of the sublattice pseudo-spin for the A and B sites, $\eta =\pm 1$ is the valley index, and $k_n^{\nu }=\frac{1}{R}(n-\frac{\nu }{3})$ with $n\in \mathbb{Z}$ and $\nu =\{-1,0,+1\}$ is the quantized azimuthal momentum due to the cylindrical symmetry of the SWCN. The effective Dirac mass of the line is given by ${\mathrm{\Delta }}_{n\nu }^{\eta s}=\sqrt{{({\mathrm{\Delta }}_s^{\eta })}^2+{(\hslash v_F{k}_n^{\nu })}^2}$.

The local conductivity ${lim}_{q\to 0}{\sigma }_{zz}(\xi ,q)={\sigma }_{zz}(\xi )$ of a 1D Dirac hyperbola for imaginary frequencies $\omega =\mathrm{\text{i}}\xi$ at zero temperature and with chemical potential $\mu$ consists on the sum of its intraband and interband contributions

where, by using the Kubo formula, we obtain where $S=|\mathrm{\text{M}}|-\sqrt{{\mathrm{\text{M}}}^2-{\mathrm{\Delta }}^2}$, $M=\mathrm{\text{Max}}[|\mu |,|\mathrm{\Delta }|]$, $\alpha \approx \frac{1}{137}$ is the fine structure constant, c is the light velocity, $\hslash$ is the Planck constant, $\mathrm{\Xi }=\hslash \xi +\hslash \mathrm{\Gamma }$ and $\mathrm{\Gamma }=1/\tau$ are the unavoidable losses of the electronic quasiparticles, written in terms of $\tau$, the mean lifetime of the electronic quasiparticle.

Note that, when $\mathrm{\Delta }=0$, the SWCN is always metallic, because there is at least one band crossed by the chemical potential $\mu$, and that, for very small frequencies, we have

For finite temperatures, we can apply the Maldague formula¹⁵ to obtain the conductivity, it is easy to see that, for any temperature $T>0$, the intraband terms make ${\sigma }_{zz}(0,\mathrm{\Delta },\mu )>0$.

The complete axial conductivity is given by

where N ($-N$) is the maximum (minimum) allowed index for $k_n^{\nu }$.

4. Results

Using the multiscattering formula for the Casimir energy for parallel cylinders given in Eq. (2), the translation matrix given in Eq. (8) and the T-matrix given in Eq. (9), we are going to obtain the large distance behavior of the Casimir energy between SWCNs. As the $\mathcal{𝒯}$ operator is only different from zero for the $NN$ polarization term, the problem is effectively reduced to the study of a scalar Casimir effect. In the large distance limit, only the conductivity at very small frequencies is relevant, the electric conductivity can be expanded as

Using the results of Sec. 3, we can obtain analytical results for any coefficient ${\sigma }_n$ of the expansion given in Eq. (16). Only in the dissipation-less limit, when $\mathrm{\Gamma }=0$, we have ${\sigma }_{-1}>0$ from Eq. (12) with $\mathrm{\Xi }=\hslash \xi$ ($\hslash \mathrm{\Gamma }=0$). In the rest of cases when dissipation is taken into account ($\mathrm{\Gamma }>0$), when $T>0$ we have ${\sigma }_0>0$ and, in the zero temperature limit, we have ${\sigma }_0>0$ for metallic SWCN (from Eq. (12)) with $\mathrm{\Gamma }>0$ and ${\sigma }_0=0$ and ${\sigma }_1>0$ (from Eq. (14)) for dielectric SWCN. Note also that, from Eq. (16), it is expected that SWCNs with different electronic behavior will have different power laws in the large distance limit. We are going to study each one of those cases separately. In several experiments, it has been observed that the use of the plasma model (the use of the conductivity in the dissipation less limit $\mathrm{\Gamma }\to 0$) provides a much better fit of the experimental results,²⁰ therefore, it is worth to study this limiting case here.

Using $r=\frac{R}{d}$, the Casimir energy in the large distance limit at $T=0$ for two equal dissipation-less ($\mathrm{\Gamma }=0$, therefore, we have ${\sigma }_{-1}>0$) SWCNs of the same radius R is

for two equal metallic SWCNs with dissipation ($\mathrm{\Gamma }>0$, therefore, ${\sigma }_{-1}=0$ and ${\sigma }_0>0$) of the same radius R and for two equal dielectric SWCNs (${\sigma }_{-1}={\sigma }_0=0$ and ${\sigma }_1>0$) of the same radius R As the azimuthal conductivity behaves as a dielectric, it is expected that considering ${\sigma }_{\varphi \varphi }\ne 0$ will only provide a small correction term to ${\mathcal{ℰ}}_0^1$. It is clear that the large distance limit of the Casimir energy depends on the conductivity properties of the SWCN, those results are corrected at smaller distances by terms with a larger power decay in distance d that are function of the material properties of the SWCNs codified in the terms ${\sigma }_n$ defined in Eq. (16).

In the high-temperature limit, the Casimir energy in the large distance limit for two equal metallic SWCNs (with and without dissipation $\mathrm{\Gamma }$) of the same radius R is

In this case, the intraband conductivity of dielectric SWCN becomes greater than zero (something easy to probe by using the Maldague formula¹⁵). As a consequence, in this classical limit, the Casimir energy between dielectrics is also given by Eq. (20), being that result an universal high T limit for SWCNs.

5. Conclusions

We have obtained the far distance limit ($d\gg R$) of the Casimir energy between two equal SWCNs for the quantum ($T=0$) and classical limits $(d{k}_{B}T\ll \hslash c)$. We obtain that, depending of the electronic properties of the SWCNs (dielectric, or metallic with or without dissipation), the power law decay of the energy is modified. In the high-temperature limit, the Casimir energy for any pair of equal SWCNs is given by the universal result shown in Eq. (20).

Acknowledgments

P. R.-L. acknowledges support from Ministerio de Ciencia, Innovación y Universidades (Spain), Agencia Estatal de Investigación, under project NAUTILUS (PID2022-139524NB-I00).

ORCID

Pablo Rodriguez-Lopez  https://orcid.org/0000-0003-0625-2682