Ground-state-cooling vibrations of suspended carbon nanotubes with constant electron current

We investigate the efficiency of cooling the vibrations of a nanomechanical resonator, constituted by a partially suspended carbon nanotube and operating as double-quantum dot. The motion is brought to lower temperatures by tailoring the energy exchange via electromechanical coupling with single electrons, constantly flowing through the nanotube when a constant potential difference is applied at its extremes in the Coulomb-blockade regime. Ground-state cooling is possible at sufficiently high-quality factors, provided that the dephasing rate of electron transport within the double dot does not exceed the resonator frequency. For large values of the dephasing rates cooling can still be achieved by appropriately setting the tunable parameters.

Figure 1

(Color online) (a) Sketch of the considered setup. The double quantum dot is a CNT operating as single-electron transistor. In the picture, the dot on the right is suspended and mechanically vibrates. Its vibrations couple to the electronic current via electron-phonon interaction. In this paper we will also consider the situation in which it is the dot on the left which is suspended. (b) Energy diagram for the DQD. States $|L⟩$ and $|R⟩$ denote one excess electron in the left and right dot, respectively. The two levels are at tunable energy difference $\hslash \Delta$ and are coupled by a tunneling barrier with tunneling matrix element $T_c$. Left (right) electrode act as source (drain) such that electron tunnels from left to right at rates ${\Gamma }_L$ and ${\Gamma }_R$.

Figure 2

Contour plots of the average phonon number at steady state ${\overline{n}}_{\text{St}}$, Eq. (36), and of the cooling rate ${\gamma }_{\text{tot}}$, Eq. (34) as a function of $\Delta$ and $T_c$, in absence of dephasing. The gray scale is such, that the darkest (brightest) regions indicate the smallest (largest) values, with the exception of the region labeled with “no steady state,” where the resonator is heated by electromechanical processes. The dashed line in the plots (b) and (d) indicates the curve ${\Delta }^2+4T_c^2={\omega }^2$. The parameters are ${\Gamma }_L=100\omega$, ${\Gamma }_R=0.2\omega$, $\alpha =0.1\omega$, ${\Gamma }_d=0$, and [(a) and (b)] ${\gamma }_p=0$ and [(c) and (d)] ${\gamma }_p={10}^{-5}\omega$ with ${\overline{n}}_p=50$.

Figure 3

Same as in Fig. 2 but taking $Q={10}^5$ $\left({\gamma }_p={10}^{-5}\omega \right)$ and ${\overline{n}}_p=50$ everywhere and finite dephasing rate: In (a) and (b) ${\Gamma }_d=0.1\omega$ and (c) and (d) ${\Gamma }_d=5\omega$.

Figure 4

(a) Average excitation number ${\overline{n}}_{\text{St}}$ and (b) cooling rate ${\gamma }_{\text{tot}}$ as a function of ${\Gamma }_R$ for ${\gamma }_p=0$. The solid (dash-dotted) curves are found for ${\gamma }_p=0$ $\left({\gamma }_p={10}^{-5}\omega \right)$ by maximizing the values of ${\gamma }_0$ as a function of $\Delta$ and $T_c$ in Eqs. (35, 36). The other parameters are $\alpha =0.1\omega$, ${\Gamma }_L=100\omega$, ${\Gamma }_d=0$, and ${\overline{n}}_p=50$. The dots and the crosses correspond to ${\gamma }_p=0$ and ${\gamma }_p={10}^{-5}\omega$, respectively, and are extracted from numerical simulation, where the evolution of the resonator mean phonon number is calculated by numerical integration of master Eq. (10), for the same parameters as in the analytical case. The cooling rate is then determined by fitting the curve of the numerical integration with an exponential decay, minimizing the sum of the squares of the offsets of the numerical points from the exponential fit. The value of ${\overline{n}}_{\text{St}}$ is extracted from the behavior of the curve at sufficiently long evolution times. The dashed line in (b), visible at small ${\Gamma }_R$, indicates the regime in which the analytical results are not valid.

Figure 5

Same as in Fig. 2, but when the injection rate into the DQD is small, ${\Gamma }_L=0.1\omega$ and ${\Gamma }_R=10\omega$, such that DQD is essentially empty. The other parameters are ${\Gamma }_R=10\omega$, $\alpha =0.1\omega$, ${\Gamma }_d=0.1\omega$, ${\gamma }_p={10}^{-5}\omega$, and ${\overline{n}}_p=50$. In (a) and (b) the resonator is coupled to the right dot, whereas in (c) and (d) the resonator is coupled to the left dot.