Intershell resistance in multiwall carbon nanotubes: A Coulomb drag study

We calculate the intershell resistance $R_{21}$ in a multiwall carbon nanotube as a function of temperature $T$ and Fermi level ${\epsilon }_{\mathrm{F}}$ (e.g., a gate voltage), varying the chirality of the inner and outer tubes. This is done in a so-called Coulomb drag setup, where a current $I_1$ in one shell induces a voltage drop $V_2$ in another shell by the screened Coulomb interaction between the shells neglecting the intershell tunneling. We provide benchmark results for $R_{21}=V_2∕I_1$ within the Fermi liquid theory using Boltzmann equations. The band structure gives rise to strongly chirality-dependent suppression effects for the Coulomb drag between different tubes due to selection rules combined with mismatching of wave vector and crystal angular momentum conservation near the Fermi level. This gives rise to orders of magnitude changes in $R_{21}$ and even the sign of $R_{21}$ can change depending on the chirality of the inner and outer tube and misalignment of inner and outer tube Fermi levels. However for any tube combination, we predict a dip (or peak) in $R_{21}$ as a function of gate voltage, since $R_{21}$ vanishes at the electron-hole symmetry point. As a by-product, we classified all metallic tubes into either zigzaglike or armchairlike, which have two different nonzero crystal angular momenta ${\mathfrak{m}}_a$, ${\mathfrak{m}}_b$ and only zero angular momentum, respectively.

Figure 1

(Left): The experimental setup to directly measure the Coulomb drag effect in a MWCNT. The intershell resistance is $R_{21}=V_2∕I_1$. (Right): The basic mechanism in the intershell resistance in a drag configuration: the intershell $e\text{−}e$ interaction and thereby momentum transfer to induce the voltage drop $V_2$.

Figure 2

The two categories of metallic nanotubes: armchairlike (AL, left) and zigzaglike (ZL, right). The AL bands near ${\epsilon }_{\mathrm{F}}=0$ have zero crystal angular momentum $\mathfrak{m}=0$ and $\Pi =\pm 1$, where $k_0\equiv 2\pi ∕3\mid \mathbf{T}\mid$. The ZL tubes have doubly degenerate bands crossing ${\epsilon }_{\mathrm{F}}=0$, i.e., for each $\xi =\pm 1$ we have either ${\mathfrak{m}}_a=(2n+m)∕3\left(\mathrm{mod}\mathfrak{n}\right)$ or ${\mathfrak{m}}_b=(2m+n)∕3\left(\mathrm{mod}\mathfrak{n}\right)$, where ${\mathfrak{m}}_a\ne {\mathfrak{m}}_b$ $[\mathfrak{n}=\gcd (n,m)]$. The thin lines are the tight-binding bands near ${\epsilon }_{\mathrm{F}}=0$ for a $(n,n)$ tube (with $\mid \mathbf{T}\mid =a$) and a $(3m,0)$ tube.

Figure 3

The possible backscattering processes in any metallic tube with a slightly raised Fermi level ${\epsilon }_{\mathrm{F}}$. Left: Backscattering in a zigzaglike tube without crystal angular momentum change $\Delta \mathfrak{m}=0$ (i.e., $\varsigma ={\varsigma }^{\prime }$) and a small wave vector $\mid k^{\prime }-k\mid \sim 2{\epsilon }_{\mathrm{F}}∕\hslash v_0$ change, which is suppressed by $g\lesssim {10}^{-2}$ from Eq. (15). Center: Backscattering in a zigzaglike tube with crystal angular momentum change, which have $g\sim 1$ from Eq. (16). Here $\overline{\mathfrak{m}}$ denotes the opposite of $\mathfrak{m}$ in the set $\{{\mathfrak{m}}_a,{\mathfrak{m}}_b\}$. Right: Two types of backscattering in armchairlike tubes: (i) A large wave vector transfer (for $\varsigma =-{\varsigma }^{\prime }$) between states with the same crystal angular momentum $(\mathfrak{m}=0)$ and $g\sim 1$ [Eq. (16)] and (ii) a small wave vector transfer $q\simeq 2{\epsilon }_{\mathrm{F}}∕\hslash v_0$ suppressed by $g\lesssim {10}^{-2}$. Note that the distance between the points $\pm 2\pi ∕3\mid \mathbf{T}\mid$ are not to scale (i.e., $2{\epsilon }_{\mathrm{F}}∕\hslash v_0⪡4\pi ∕3\mid \mathbf{T}\mid$) and that the armchairlike bands are connected as in Fig. 2.

Figure 4

The transresistance per length $R_{21}∕L$ (in $\Omega ∕\mu \mathrm{m}$) as a function of the Fermi level ${\epsilon }_{\mathrm{F}}$ (in eV) (e.g., a gate voltage) for equal Fermi levels of the two tubes, ${\epsilon }_{\mathrm{F}}={\epsilon }_{\mathrm{F}}^{\left(1\right)}={\epsilon }_{\mathrm{F}}^{\left(2\right)}$. The temperature is $T=80\mathrm{K}$ (dotted), $T=150\mathrm{K}$ (dashed), and $T=300\mathrm{K}$ (full line). The dip in $R_{21}$ at ${\epsilon }_{\mathrm{F}}=0$ reflects the electron-hole symmetry at this point. Inset: A sketch of the situation for misaligned Fermi levels ${\epsilon }_{\mathrm{F}}^{\left(1\right)}\ne {\epsilon }_{\mathrm{F}}^{\left(2\right)}$ as a function of gate voltage (see text for details).

Figure 5

Contour plot of the $F$ functions for the intraband scattering for $0<q<\pi ∕\mid \mathbf{T}\mid$, ${\epsilon }_{\mathrm{F}}>0$ and the temperature $T=0.1T_{\mathrm{F}}$. Note the smearing by the Fermi functions due to the temperature on some edges and the sharp edge at $\omega =v_{0}q$ from the step function $\theta (-\omega +v_{0}q)$.

Figure 6

Contour plot of the $F_{\mathrm{intra}}$ function. $F_{\mathrm{intra}}$ gives the phase space for intraband scattering in (real) armchair tubes. $F_{\mathrm{intra}}$ is seen for $0<q<\pi ∕a$ and is odd in $q$ and should be repeated periodically with $2\pi ∕a$ as a function of $q$.

Figure 7

The transresistance per length $R_{21}∕L$ (in units of $\Omega ∕\mu \mathrm{m}$) versus temperature $T$ [or $T∕T_{\mathrm{F}}$ (left)]. The curves are obtained from a numerical integration of Eq. (31) for a (5,5) in a (10,10) tube. Curves for four different Fermi levels ${\epsilon }_{\mathrm{F}}={\epsilon }_{\mathrm{F}}^{\left(1\right)}={\epsilon }_{\mathrm{F}}^{\left(2\right)}$ (i.e., gate voltages or dopings) are seen: ${\epsilon }_{\mathrm{F}}=0.006\mathrm{eV}$ $(T_{\mathrm{F}}=69\mathrm{K})$ (left, dashed line), ${\epsilon }_{\mathrm{F}}=0.015\mathrm{eV}$ $(T_{\mathrm{F}}=174\mathrm{K})$ (left, solid line), ${\epsilon }_{\mathrm{F}}=0.15\mathrm{eV}$ $(T_{\mathrm{F}}=1740\mathrm{K}$) (right, dashed line) and ${\epsilon }_{\mathrm{F}}=0.3\mathrm{eV}$ (right, solid line). Note the difference in magnitude between the transresistances $R_{21}$.

Figure 8

The transresistance per length $R_{21}∕L$ versus radius $(r\propto n)$ for armchair tubes. The different outer and inner armchair tubes are: A (5,5) in a $(n,n)$ (dots), a (6,6) in a $(n,n)$ (triangles) and a (9,9) in a $(n,n)$ (stars). The radius of the outer tube is: $r=\sqrt{3}a∕2\pi n$ for a $(n,n)$ tube. Here $T=300\mathrm{K}$ and ${\epsilon }_{\mathrm{F}}^{\left(1\right)}={\epsilon }_{\mathrm{F}}^{\left(2\right)}={\epsilon }_{\mathrm{F}}=0.3\mathrm{eV}$ is used. Note the logarithmic scale.

Figure 9

The transresistance per length $R_{21}∕L$ as a function of the ratio of the translational vectors length $\mid {\mathbf{T}}_1\mid ∕\mid {\mathbf{T}}_2\mid$ for two armchairlike tubes. The peaks corresponding to different scattering processes are seen as explained in the text. Numerically, we use $\mid {\mathbf{T}}_2\mid =a$, radii as for a (5,5) in a (10,10) tube, $T=300\mathrm{K}$ and ${\epsilon }_{\mathrm{F}}^{\left(1\right)}={\epsilon }_{\mathrm{F}}^{\left(2\right)}={\epsilon }_{\mathrm{F}}=0.3\mathrm{eV}$. If the tubes have a different radius, only the magnitude of the peak is changed (see Fig. 8). Inset (a): The scattering processes in tube 1 and 2 leading to the peak at $\mid {\mathbf{T}}_1\mid ∕\mid {\mathbf{T}}_2\mid \simeq 1.28$. Note that the backscattering processes are electronlike and holelike, respectively, so $R_{21}<0$. Inset (b): Peaks around $\mid {\mathbf{T}}_1\mid ∕\mid {\mathbf{T}}_2\mid =1∕2$. Note the difference in scale.

Figure 10

Left: The $\mathfrak{m}=1$ band for a (5, 5) tube in the FBZ of the primitive unit cell as a function of $\kappa ∊]-\pi ,\pi ]$. Center: The $\mathfrak{m}=1$ band is pushed into the smaller FBZ of the translational unit cell by using $\kappa =ka∕2+n_c\pi ∕5$, with $n_c=1$ and $n_c=6$, since $\mathfrak{m}=1$. Note that the band is symmetrical around $\pi ∕5$, since $\mathfrak{m}=1$. Right: The band structure for the translational unit cell. Both bands have crystal angular momentum $\mathfrak{m}=1$, but indices $n_c=1$ and $n_c=6$.