Electronic transport through defective semiconducting carbon nanotubes

The transmission function ${ \mathcal T }(E)$ for one single CNT with randomly distributed defects depends strongly on their exact positions and alignments. It is intuitive that the transmission should lower with increasing number of defects. In fact the defect states introduce resonances into the system due to quantum interference. The more defects the system has, the more and the sharper the resonances are, leading to an accumulation of random peaks, preventing a reasonable analysis of the results [51]. In the following we always analyze ensemble averages, which leads to smooth curves. We omit the averaging symbol for simplicity.

Figure 3 shows the (average) transmission function for the (9,0)-CNT with 1 up to 20 defects. A systematically decreasing transmission can be seen as well as a defect-induced resonance for the MV around 140 meV below the Fermi energy, which leads to a dip. Figure 4 shows the dependence on the number of defects N, normalized to the bulk transmission ${{ \mathcal T }}_{0}$, which equals the number of conductance channels. Transmissions calculated at four different energies are depicted: the valence band edge, the conduction band edge and two energies within the bands. A clear exponential decrease can be seen for sufficiently large values of N. This can be explained by the strong localization regime, where the electronic states are exponentially localized due to destructive interference with a characteristic localization length ℓ^loc. For Anderson-like disorder it was shown that 1D systems are always in the strong localization regime [52–54]. As a consequence the transport is exponentially suppressed. For a fixed defect probability and varying system length L the transport in the limit of large L resp. small ${ \mathcal T }$ can be described by

$$\begin{eqnarray}&&{ \mathcal T }\propto {{\rm{e}}}^{-L/{{\ell }}^{\mathrm{loc}}}.\end{eqnarray} \tag{ 6 }$$

The same exponential dependence holds for fixed length and varying defect probability or—in our case—number of defects, as shown in [30, 32]:

$$\begin{eqnarray}&&{ \mathcal T }\propto {{\rm{e}}}^{-N/{N}^{\mathrm{loc}}}.\end{eqnarray} \tag{ 7 }$$

The localization exponent N^loc is a material constant, which describes the number of additional defects needed to lower the transmission by a factor of e. The fluctuations visible in figure 4 for ${ \mathcal T }/{{ \mathcal T }}_{0}\lt {10}^{-3}$ are merely caused by the ensemble not being large enough for sampling a very high number of defects.

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The systematic deviations from the exponential behavior for few defects at high transmission ${ \mathcal T }/{{ \mathcal T }}_{0}\gt {10}^{-1}$ are caused by an increasing localization length ℓ^loc. For decreasing N and fixed system length L of the central region, the defect probability per cell decreases. Thus, ℓ^loc becomes of the same order or larger than L at small N and the system is no more in the strong localization regime. This range of high transmission can be described by the diffusive regime, where only elastic scattering occurs, but without the long-range destructive interference effects limiting the transport. In the diffusive regime the transport can be described by a resistance, which increases linear with the system length for fixed defect density [55]. For our case, where we fix the system length, we get for the transmission

$$\begin{eqnarray}&&\displaystyle \frac{{ \mathcal T }}{{{ \mathcal T }}_{0}}={\left(1+\displaystyle \frac{L}{{{\ell }}^{\mathrm{mfp}}}\right)}^{-1}={\left(1+\displaystyle \frac{N}{{N}^{\mathrm{mfp}}}\right)}^{-1}.\end{eqnarray} \tag{ 8 }$$

ℓ^mfp is the elastic mean free path, while N^mfp is the dimensionless elastic mean free path.

Both the diffusive and the localized regime, can be described phenomenologically by solving the steady-state diffusion equation with an additional sink term, which traps the diffusing electrons and hinders them to pass lengths larger than ℓ^loc: $0=n^{\prime\prime} (x)-n(x)/{{\ell }}^{\mathrm{loc}}$, where x is the position along the 1D system and n is the 1D electron density. The local current density can be calculated using $j(x)=-{Dn}^{\prime} (x)$ with the diffusion constant $D={{\ell }}^{\mathrm{mfp}}v/2$ and the average particle velocity v. The trapped fraction of the current is ${j}_{\mathrm{trap}}\,=D/{({{\ell }}^{\mathrm{loc}})}^{2}\cdot {\int }_{0}^{L}n(x){\rm{d}}x=j(0)-j(L)$. With the appropriate boundary conditions n(L)v = j_out and $n(0)v={j}_{\mathrm{in}}+{j}_{\mathrm{ref}}=2{j}_{\mathrm{in}}-{j}_{\mathrm{out}}-{j}_{\mathrm{trap}}$ for the ingoing/outgoing/reflected current density ${j}_{\mathrm{in}/\mathrm{out}/\mathrm{ref}}$ of a device configuration and ${ \mathcal T }={j}_{\mathrm{out}}/{j}_{\mathrm{in}}$, the transmission can be obtained [56]:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{ \mathcal T }}{{{ \mathcal T }}_{0}} & = & {\left(\cosh \displaystyle \frac{L}{{{\ell }}^{\mathrm{loc}}}+\displaystyle \frac{{{\ell }}^{\mathrm{loc}}}{{{\ell }}^{\mathrm{mfp}}}\sinh \displaystyle \frac{L}{{{\ell }}^{\mathrm{loc}}}\right)}^{-1}\\ & = & {\left(\cosh \displaystyle \frac{N}{{N}^{\mathrm{loc}}}+\displaystyle \frac{{N}^{\mathrm{loc}}}{{N}^{\mathrm{mfp}}}\sinh \displaystyle \frac{N}{{N}^{\mathrm{loc}}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 9 }$$

In the limit $L\gg {{\ell }}^{\mathrm{loc}}$ this simplifies to (6), in the limit $L\ll {{\ell }}^{\mathrm{loc}}$ to (8). The strong localization regime is approximately valid for N > N^loc, the diffusive regime for N < N^loc/2⁵ .

An example for the transmission ${ \mathcal T }(N)$ at $E={E}_{{\rm{F}}}+0.2\,\mathrm{eV}$ is shown in figure 4. Equation (9) (red dash-dotted line) describes the general shape of the data very well. A corresponding regression gives N^loc = 23 and N^mfp = 4.2. Both regimes, localization (7) and diffusion (8), are also depicted (red dashed and dotted lines) and fit well in the appropriate limit (see footnote 5). The regressions yield N^loc = 24 and N^mfp = 6.4. The discrepancy concerning N^mfp comes from the fact, that (9) underestimates the transition regime and gives a systematically overestimated derivative at N = 0. This can be seen in the inset of figure 4. Because of this we determine N^loc and N^mfp separately using (7) and (8). In detail, we obtain N^loc for each energy with a linear regression of the logarithmic data in the range ${10}^{-3}\lt { \mathcal T }/{{ \mathcal T }}_{0}\lt {10}^{-1}$. N^mfp is the slope of the linear dependence ${{ \mathcal T }}^{-1}(N)$ and is calculated in the limit $N\to 0$.

Like in the preceding example the localization exponent is calculated for each energy and for all the different CNTs of figure 2. The results are shown in the left column of figure 5 for the MV (first three rows) and the DV (last row). Here, a distinction between different subsets concerning (m − n)mod 3 must be made to qualify and quantify further dependencies. The CNTs with (m − n)mod 3 = 0 are the semi-metallic ones with very small bandgaps in the range (50...130)meV. The CNTs with (m − n)mod 3 = 1 and (m − n)mod 3 = 2 are the true semiconducting CNTs, the former ones with bandgaps in the range (550...1540)meV and the latter ones with bandgaps in the range (340...1040) meV.

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For each subset clear trends can be seen. The qualitative shape of N^loc(E) is mostly the same: small values at the valence/conduction band edge and increasing values with decreasing/increasing energy up to the next band edge with singularity-like shapes there. Furthermore, the curves follow a clear trend with varying diameter for most of the energies. There are two exceptions: (a) near the band edges, where N^loc diverges, and (b) near defect states, which lead to peaks/dips in the transmission spectrum and in N^loc(E). The latter is the case for (m − n)mod 3 = 0: for the MV at energies below 0.1 eV and for the DV at energies between 0.0 eV and 0.2 eV. The right column of figure 5 depicts the diameter dependence of N^loc at selected energies not fulfilling (a) or (b). For all the cases a very clear linear dependence can be seen with only small deviations from the linear regressions (solid lines). This is in good agreement with [32] ⁶ . The DV and (m − n)mod3 = 0 case shows that there is a lower limit, here approximately d = 9 nm and N^loc = 6. These deviations for smaller tube diameters can be explained by strong curvature effects and the resulting changes of the π-bonds. Also the fact that the defect occupies a large part of the CNT circumference can result in additionally distorted structures and changed transport properties. Furthermore, it is important to mention that many different chiralities are included, especially for the (m − n)mod 3 = 0 case. As the diameter dependence matches very well, a chirality dependence can be excluded. In conclusion, the slope of N^loc(d) only depends on the subset, energy, and defect type, but not on the chirality. The prefactor in (7) is between 0.4 and 1.2⁷ .

N^mfp is calculated in dependence on the energy in the same way as done before for N^loc. The result is shown exemplarily for the (16,0)-CNT in figure 6 in comparison to N^loc. It can be seen that N^mfp follows the general trend of N^loc. This is in agreement with former general studies, in which relations between the localization length and the elastic mean free path were derived [29, 55]. Furthermore, a linear diameter dependence at fixed energy holds for N^mfp, too. We checked this for all studied CNTs with the same general result.

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A way for comparing N^mfp and N^loc is (9) in the limit N ≫ N^loc. This gives the prefactor $2/(1+{N}^{\mathrm{loc}}/{N}^{\mathrm{mfp}})$. With this, the three regression parameters N^mfp from (8), N^loc from (7), and the corresponding prefactor from (7) can be compared. For all studied CNTs and energies, except near the band edges, where the regressions are less trustable, we get a good agreement. The maximum deviations of $2/(1+{N}^{\mathrm{loc}}/{N}^{\mathrm{mfp}})$ compared to the regression prefactor are in the range of ±40%. Details are listed in table 1. This comparison shows that the determination of the three parameters is consistent, (9) can be used to describe both regimes, and the prefactor of the localization regime can be estimated using N^loc and N^mfp with acceptable errors.

The linear dependencies ${N}^{\mathrm{loc}}(d)$ and N^mfp(d) can be used to predict the transmission of large diameter CNTs of any chirality. But as ${ \mathcal T }$ is energy-dependent this is not very practicable.

In the mesoscopic range, the zero-bias conductance of an arbitrary scattering region between two reservoirs can be calculated using the Landauer-Büttiker formula [57]

$$\begin{eqnarray}&&G=-{{\rm{G}}}_{0}{\int }_{-\infty }^{\infty }{ \mathcal T }(E)\displaystyle \frac{{\rm{d}}f(E)}{{\rm{d}}E}{\rm{d}}E.\end{eqnarray} \tag{ 10 }$$

${{\rm{G}}}_{0}=2{{\rm{e}}}^{2}/{\rm{h}}$ is the conductance quantum and f(E) is the Fermi distribution, whereby the effect of temperature is included.

An example for the conductance as a function of the number of defects is shown in figure 7 for the (9, 0)-CNT for different temperatures. For better comparison, all the data are normalized to the conductance of the (9, 0)-CNT with one defect G(N = 1). A similar picture as shown in figure 4 with a different effective localization exponent ${\tilde{N}}^{\mathrm{loc}}$ and elastic mean free path ${\tilde{N}}^{\mathrm{mfp}}$ can be seen. In contrast to the transmission in the strong localization regime, the dependence on the defect number is not strictly exponential. If the localization exponent N^loc is not energy-dependent it is clear that ${\tilde{N}}^{\mathrm{loc}}={N}^{\mathrm{loc}}$, as it is for metallic CNTs [31]. But as previously shown, the very large variations in N^loc(E) of the semiconducting CNTs lead to a complicated summation of conductance contributions with different exponential dependence. In the limit of large N, the largest N^loc would dominate, but this is not a useful description, as it would be much above typical numbers of defects of experimental relevance. With several simplifications for the Landauer-Büttiker formula and N^loc(E), solving the integral (10) yields an additional prefactor 1/(1 + N). But in all cases it varies not very much compared to the exponential dependence and can be treated as a slight correction of the localization exponent. Consequently, (6) can be used as a good approximation for estimating and predicting the conductance in the strong localization regime.

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The effective localization exponent ${\tilde{N}}^{\mathrm{loc}}$ for the conductance is calculated for T = 300 K and for all the different CNTs of figure 2. The regressions are done around G/G(N = 1) = 10⁻⁴, where the range is extended as far as possible, while keeping the regression inaccuracy within certain limits. The results are shown in the left column of figure 8 for the MV (first three rows) and the DV (last row), where ${\tilde{N}}^{\mathrm{loc}}$ is depicted as a function of the tube diameter. It shows a very good linear dependence for the four cases. Linear regressions are depicted in figure 8 with straight black lines. The results are listed in table 2. As the data includes CNTs with many different chiral angles, a chirality dependence of the localization exponent can be excluded. These result can be used as an approximate guideline to estimate and predict the conductance for CNTs with arbitrary diameters and chiral angles. There are deviations for smaller tube diameters, which can be explained by strong curvature effects and the resulting changes of the π-bonds. The DV data for the $(m-n)\mathrm{mod}3=0$ case (lower row) show that ${\tilde{N}}^{\mathrm{loc}}$ is limited by a lower bound, here approximately at d = 9 nm and ${\tilde{N}}^{\mathrm{loc}}=5$.

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As done before for the localization regime, the diffusive regime can be described by an effective elastic mean free path ${\tilde{N}}^{\mathrm{mfp}}$ at a given temperature, e.g. at 300 K. For the four studied cases, ${\tilde{N}}^{\mathrm{mfp}}$ is depicted as a function of the CNT diameter in the right column of figure 8. A linear trend can be seen here, too. Linear regressions are depicted in figure 8 with straight black lines. The results are listed in table 2, too. The deviations for small diameter tubes are likely due to curvature effects. For the MV defect with (m − n)mod 3 = 0 the larger deviations are caused by the much stronger defect features near the band gap, which are not present in the other three cases.

Figure 9 shows some examples for the temperature dependence of ${\tilde{N}}^{\mathrm{loc}}$ for CNTs with MV defects. The complete data can be found in the supplementary material. The general shape of the dependence ${\tilde{N}}^{\mathrm{loc}}(T)$ can be derived under some rough assumptions and simplifications by the analytical integration of (10). For this, the diverging shape of N^loc(E) is assumed to be ${N}_{0}^{\mathrm{loc}}/(1-E/{E}_{2})$. The integral is restricted to the interval [E₁, E₂]. The derivative of the Fermi distribution is approximated for large energies by ${{\rm{e}}}^{-(E-{E}_{{\rm{F}}})/{k}_{{\rm{B}}}T}(1-2{{\rm{e}}}^{-(E-{E}_{{\rm{F}}})/{k}_{{\rm{B}}}T})$. The effective localization exponent is calculated by ${\tilde{N}}^{\mathrm{loc}}=-\tfrac{\partial \mathrm{ln}G}{\partial N}$. Alltogether this yields a function with four free parameters, which can be used for a regression:

$$\begin{eqnarray}&&{\tilde{N}}^{\mathrm{loc}}(T)=a+b\displaystyle \frac{(1-{{\rm{e}}}^{c-d/T})(d/T-c)}{d/T-c-(1-{{\rm{e}}}^{c-d/T})}\end{eqnarray} \tag{ 11 }$$

The parameters are related to the bandgap, the band edges, and the magnitude of the derivative of ${N}^{\mathrm{loc}}(E)$. For the examples in figure 9, the corresponding regressions are shown as solid lines. For different shapes, the regressions agree with the data for sufficiently high temperatures. There are some deviations for low temperatures, where the defect-induced features in N^loc(E) are more dominant compared to the general trend.

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