Phonon runaway in carbon nanotube quantum dots

We explore electronic transport in a nanotube quantum dot strongly coupled with vibrations and weakly with leads and the thermal environment. We show that the recent observation of anomalous conductance signatures in single-walled carbon nanotube quantum dots [B. J. LeRoy et al., Nature (London) 395, 371 (2004) and B. J. LeRoy et al., Phys. Rev. B 72, 075413 (2005)] can be understood quantitatively in terms of current driven “hot phonons” that are strongly correlated with electrons. Using rate equations in the many-body configuration space for the joint electron-phonon distribution, we argue that the variations are indicative of strong electron-phonon coupling requiring an analysis beyond the traditional uncorrelated phonon-assisted transport (Tien-Gordon) approach.

Figure 1

(Color online) (a) Schematic of scanning tunnel microscope (STM) measurement on CNT QDs; (b) and (c) Experimental observation and theoretical calculation for CNT QDs at lower and higher current level [solid line: conductance; dashed line: $⟨N_{ph}⟩$ in (c)]. $1,2,\dots ,6$ denotes the main Coulomb peaks. The parameters for the calculation are ${\widetilde{ϵ}}_1={\widetilde{ϵ}}_2=24\mathrm{meV}$, ${\widetilde{ϵ}}_3={\widetilde{ϵ}}_4={\widetilde{ϵ}}_5={\widetilde{ϵ}}_6=54\mathrm{meV}$, $\tilde{U}=27\mathrm{meV}$, $E_F=0$, $\hslash {\omega }_1=11.5\mathrm{meV}$, ${\lambda }_{11}={\lambda }_{21}=\cdots ={\lambda }_{61}=1.6$, ${\Gamma }_{L1}+{\Gamma }_{L2}=7.5\times {10}^{-6}\mathrm{eV}$, $\beta =5\times {10}^{-8}\mathrm{eV}$, $\eta =0.6$, and $T=5\mathrm{K}$.

Figure 2

The dot is electrically connected to the left (right) contact (with electron tunneling rates ${\Gamma }_{L,R}∕\hslash$) and mechanically to the phonon bath (with a phonon escape rate of $\beta ∕\hslash$). The dot has electronic degrees of freedom ${ϵ}_i$ and phonon degrees of freedom $\hslash {\omega }_j$ $(i,j=1,2,3,\dots )$ with electron-phonon coupling ${\lambda }_{ij}$.

Figure 3

(Color online) Variation of conductance with the current at the second Coulomb peak (at $V_{appl}=-0.1\mathrm{V}$) and its associated first phonon sidepeak (at $V_{appl}=-0.125\mathrm{V}$). Inset: Variation of the number of phonons with current at the above-mentioned conductance peaks. The parameters used to generate these results are ${\widetilde{ϵ}}_1={\widetilde{ϵ}}_2=\widetilde{ϵ}=8\mathrm{meV}$, $\tilde{U}=12\mathrm{meV}$, $E_F=0$, $\hslash {\omega }_1=12.5\mathrm{meV}$, ${\lambda }_{11}={\lambda }_{21}=2.1$, ${\Gamma }_L=5\times {10}^{-6}\mathrm{eV}$, ${\Gamma }_R=0.00005{\Gamma }_L$–$0.5{\Gamma }_L$, $\beta =5\times {10}^{-10}\mathrm{eV}$, $\eta =0.2$, and $T=5\mathrm{K}$.

Figure 4

(Color online) Simple one single electron level model does not match the experimental result at all. The values of the parameters used are ${\widetilde{ϵ}}_1=0.05\mathrm{eV}$, $\hslash {\omega }_1=12.5\mathrm{meV}$, ${\lambda }_{11}=1.7$, ${\Gamma }_L=1\times {10}^{-5}\mathrm{eV}$, ${\Gamma }_R=0.00005{\Gamma }_L$–$0.5{\Gamma }_L$, $\beta =5\times {10}^{-10}\mathrm{eV}$, $\eta =0.5$, and $T=5\mathrm{K}$.

Figure 5

(Color online) Variation of conductance with current for current values higher than the experimental range. Inset: The variation of conductance at the Coulomb peak in the absence of electron-phonon coupling. The values of the parameters used are the same as those of Fig. 3 (except that there is no coupling between electrons and phonons for the results in the inset). The region inside the dotted box is the range of experimental observation in Fig. 3.

Figure 6

(Color online) Variation of conductance with current for four degenerate single electron levels at the four Coulomb peaks (solid lines) and their associated first phonon sidepeaks (dashed lines). The values of the parameters used are ${\widetilde{ϵ}}_1={\widetilde{ϵ}}_2={\widetilde{ϵ}}_3={\widetilde{ϵ}}_4=\widetilde{ϵ}=20\mathrm{meV}$, $\tilde{U}=30\mathrm{meV}$, $E_F=0$, $\hslash {\omega }_1=12.5\mathrm{meV}$, ${\lambda }_{11}={\lambda }_{21}={\lambda }_{31}={\lambda }_{41}=2.1$, ${\Gamma }_L=5\times {10}^{-6}\mathrm{eV}$, ${\Gamma }_R=0.00005{\Gamma }_L$–$0.5{\Gamma }_L$, $\beta =5\times {10}^{-10}\mathrm{eV}$, $\eta =0.5$, and $T=5\mathrm{K}$.