Vibrational cooling and thermoelectric response of nanoelectromechanical systems

An important goal in nanoelectromechanics is to cool the vibrational motion, ideally to its quantum ground state. Cooling by an applied charge current is a particularly simple and hence attractive strategy to this effect. Here we explore this phenomenon in the context of the general theory of thermoelectrics. In linear response, this theory describes thermoelectric refrigerators in terms of their cooling efficiency $\eta$ and figure of merit $ZT$. We show that both concepts carry over to phonon cooling in nanoelectromechanical systems. As an important consequence, this allows us to discuss the efficiency of phonon refrigerators in relation to the fundamental Carnot efficiency. We illustrate these general concepts by thoroughly investigating a simple double-quantum-dot model with the dual advantage of being quite realistic experimentally and amenable to a largely analytical analysis theoretically. Specifically, we obtain results for the efficiency, the figure of merit, and the effective temperature of the vibrational motion in two regimes. In the quantum regime in which the vibrational motion is fast compared to the electronic degrees of freedom, we can describe the electronic and phononic dynamics of the model in terms of master equations. In the complementary classical regime of slow vibrational motion, the dynamics is described in terms of an appropriate Langevin equation. Remarkably, we find that the efficiency can approach the maximal Carnot value in the quantum regime, with large associated figures of merit. In contrast, the efficiencies are typically far from the Carnot limit in the classical regime. Our theoretical results should provide guidance to implementing efficient vibrational cooling of nanoelectromechanical systems in the laboratory.

Figure 1

Top: Sketch of the setup under consideration. A suspended nanostructure (e.g., a carbon nanotube) is contacted by left and right electrodes. The setup consists of two gate-tunable quantum dots. Tunneling between the quantum dots couples to vibrational motion of the suspended part of the structure. Bottom: Configuration for vibrational cooling. The electronic levels ${\epsilon }_L$ and ${\epsilon }_R$ of the quantum dots are arranged such that electron tunneling between the dots is preferentially accompanied by phonon absorption, enabled by the dependence of the tunneling amplitude $t\left(\widehat{x}\right)$ on the vibrational mode coordinate $\widehat{x}$. We also consider the situation when the system is in contact with an additional phonon bath of temperature $T_{\mathrm{ph}}$.

Figure 2

Thermoelectric response coefficients $L_{11}$ (solid line), $L_{12}=L_{21}$ (circles), and $L_{22}$ (squares) as function of temperature for weak electron phonon interaction $\lambda =0.1$. The upper (lower) panel corresponds to a coupling to the electron reservoirs of $\Gamma =0.01$ $\left(\Gamma =0.1\right)$. The quantum-dot levels are in the resonance configuration ${\epsilon }_R-{\epsilon }_L=\omega$ with ${\epsilon }_L=0.025$ and ${\epsilon }_R=1.025$ with chemical potential $\mu =0$. Energies and temperatures are expressed in units of $\omega$. Differences in units between the response coefficients are also compensated by factors of $\omega$.

Figure 3

Sketch of the phonon-absorption process in the resonant situation ${\epsilon }_R={\epsilon }_L+\omega$.

Figure 4

Figure of merit $ZT$ and efficiency $\eta$ of the refrigerator for different hybridization strengths ${\Gamma }_L={\Gamma }_R=\Gamma =0.01,0.1$ (left and right panels, respectively). Solid (dashed) lines correspond to electron-phonon interaction $\lambda =1$ and $\lambda =0.1$, respectively. Other parameters are as in Fig. 2.

Figure 5

Points connected with solid lines: Probability distribution function $P_n$ calculated from the numerical solution of Eq. (15) in the stationary nonequilibrium case by considering up to 13 phonon modes. Circles: Equilibrium probability distribution function $P_n^{\mathrm{eq}}$ [see Eq. (16)] corresponding to a phonon temperature $T_{\mathrm{ph}}=T^{\mathrm{eff}}$, where the effective temperature is defined in Eq. (26). Dashed lines: $P_n^{\mathrm{eq}}$ for $T_{\mathrm{ph}}=T$. Left (right) panel corresponds to $T=1,V=0.5$ $\left(T=1,V=2.5\right)$ in which case $T_{\mathrm{eff}}/T=0.67$ $\left(T_{\mathrm{eff}}/T=0.3\right)$. We are considering a small relaxation rate $1/\tau$ $\left(\tau |t_0{|}^2{\lambda }^2/\omega =10\right)$, $t_0=0.1$, and weak coupling to the reservoirs $\Gamma =0.01$. All other parameters are as in Fig. 2 and all the energies are expressed in units of $\omega$.

Figure 6

Phonon-emission processes which limit the phonon refrigeration in the resonant situation ${\epsilon }_R={\epsilon }_L+\omega$. The process on the left (right) is dominant for $eV<\omega$ $\left(eV>\omega \right)$.

Figure 7

$T^{\mathrm{eff}}/T$ vs detuning of the electronic energy levels and bias voltage, for weak electron-phonon coupling $\lambda =0.1$, small relaxation rate $1/\tau$ $\left(\tau |t_0{|}^2{\lambda }^2/\omega =10\right)$, $\Gamma /\omega =0.01$, and $T/\omega =1$.

Figure 8

${\Lambda }_{11}\left(\epsilon \right),{\Lambda }_{12}\left(\epsilon \right)/{\left(t_0\lambda \right)}^2,{\Lambda }_{22}\left(\epsilon \right)/{\left(t_0\lambda \right)}^2$ de-fined in Eq. (53), corresponding, respectively, to solid, dashed, and dot-dashed lines, for ${\Gamma }_L={\Gamma }_R=0.5$. The amplitude of the interdot hopping is $t_0=1$ and sets the energy scale. The energies of the levels of the dots are ${\epsilon }_L=0.025$ and ${\epsilon }_R=1.025$.

Figure 9

Efficiency and figure of merit as a function of the temperature $T$ in the classical regime. We chose $M=1,\mu =-0.8$, and $\lambda =0.1$. Solid and dashed lines correspond to $\Gamma =0.5$ and $\Gamma =1$, respectively. The unit of energy is set by $t_0=1$, while the energies of the levels of the dots are ${\epsilon }_L=0.025$ and ${\epsilon }_R=1.025$.

Figure 10

Figure of merit vs temperature $T$ and chemical potential $\mu$ in the classical regime for $\Gamma =0.5$. The mass of the oscillator is $M=1$, the energies of the levels of the dots are ${\epsilon }_L=0.025$ and ${\epsilon }_R=1.025$, while $t_0=1$ defines the scale for the energies.

Figure 11

$T^{\mathrm{eff}}/T$ as a function of the temperature $T$ of the electron reservoirs. Different plots correspond to different voltages $V=0.5,1,1.5,2$. The chemical potentials are ${\mu }_R=-0.8$ and ${\mu }_L={\mu }_R\pm V$, where the upper (lower) sign corresponds to the left (right) panel. The energies of the levels of the dots are ${\epsilon }_L=0.025$ and ${\epsilon }_R=1.025$, while $t_0=1$ defines the scale for the energies as in the previous figures.