Phonon thermoelectric transistors and rectifiers

We describe nonlinear phonon-thermoelectric devices where charge current and electronic and phononic heat currents are coupled, driven by voltage and temperature biases, when phonon-assisted inelastic processes dominate the transport. Our thermoelectric transistors and rectifiers can be realized in a gate-tunable double quantum-dot system embedded in a nanowire which is realizable within current technology. The inelastic electron-phonon scattering processes are found to induce pronounced charge, heat, and cross rectification effects, as well as a thermal transistor effect that, remarkably, can appear in the present model even in the linear-response regime without relying on the onset of negative differential thermal conductance.

Figure 1

(a) Scheme of a DQD that can serve as a phonon-thermoelectric diode and transistor. The QDs are embedded in the nanowire and are controlled by gate voltages: $l$ and $r$ control the local potentials, and $t$ tunes the tunneling between the QDs. The two electrodes, $L$ and $R$, apply voltage and temperature biases across the QDs. The insulation layer suppresses the thermal contact between the metal electrodes and the substrate, which provides thermal energy to phonons. (b) Elastic and inelastic contributions to the charge current as a function of level detuning; microscopic processes are illustrated in panels (c) and (d). (c) Inelastic thermoelectric transport assisted by a phonon of frequency ${\omega }_q$. (d) Elastic thermoelectric transport. The shaded green area represents the broadening of the left and right QDs including the hybridization between them. The dashed arrows display the two main tunneling paths, of different energies. In (b) we used $t=15\mu \mathrm{eV}, {\xi }_0=1\mu \mathrm{eV}, {\omega }_0=100\mu \mathrm{eV}, {\gamma }_{\ell }={\gamma }_r=5\mu \mathrm{eV}, k_{B}T=20\mu \mathrm{eV}$ (for $T_L=T_R=T_{ph}=T$), $E_{\ell }=0$, and ${\mu }_L=-{\mu }_R=40\mu \mathrm{eV}$.

Figure 2

Charge, heat, and cross-rectification effects. (a) Charge current $I_e$ for $E_{\ell }=-E_r=-2k_{B}T$ and (b) optimization of charge rectification $R_e$ with QD energies $E_{\ell }$ and $E_r$ at $\delta \mu =20\mu \mathrm{eV}$. (c) Electronic heat current $I_Q^e$ (“aW” denoting attowatts) for $E_{\ell }=-5k_{B}T$ and $E_r=-k_{B}T$, and (d) optimization of the associated rectification strength $R_t$, by tuning $E_{\ell }$ and $E_r$ using $\delta T=0.5T$. (e) Electronic heat current as a function of applied voltage for $E_{\ell }=-k_{B}T$ and $E_r=2k_{B}T$, and (f) optimization of the relevant rectification $R_{te}$ at the bias $\delta \mu =20\mu \mathrm{eV}$. (g) Charge current against the (metals) temperature difference for $E_{\ell }=-5k_{B}T$ and $E_r=-k_{B}T$, and (h) optimization of the rectification strength $R_{et}$ by controlling the DQDs at $\delta T=0.5T$. We used $t=15\mu \mathrm{eV}, {\xi }_0=1\mu \mathrm{eV}, {\omega }_0=100\mu \mathrm{eV}, {\gamma }_{\ell }={\gamma }_r=5\mu \mathrm{eV}, k_{B}T=20\mu \mathrm{eV}$, and $T_{ph}=T_0$ for all figures.

Figure 3

(a) Linear thermal transistor. We plot the electronic heat currents $I_Q^L$ and $I_Q^e$ (the former describes the current leaving the $L$ terminal) and the phonon heat current $I_Q^{ph}$ (“aW” denoting attowatts), against the temperature of the phonon bath. The two electronic heat currents vary significantly, whereas the phonon current changes very little with $T_{ph}$. $E_{\ell }=4.5k_{B}T, E_r=5k_{B}T, \delta T=0.02T, k_{B}T=20\mu \mathrm{eV}$, and $\delta \mu =0$. (b) Phonon transistor. The charge current $I_e$, plotted as a function of $\delta \mu$, is largely controlled by the temperature of the phonon bath. $E_{\ell }=-4k_{B}T, E_r=2k_{B}T$, and $\delta T=0$. Other parameters are the same as in Fig. 2.

Figure 4

(a) Electron-phonon coupling strength ${\xi }_{\lambda }$ vs quantum dot radius $l_{qd}$ for longitudinal phonon with deformation potential coupling mechanism (“LDP”), transverse phonons with piezoelectric coupling mechanism (“TPE”), and longitudinal phonons with piezoelectric coupling mechanism (“LPE”). Calculated from Eqs. (A7, A8, A9) using material parameters of GaAs. (b) Characteristic phonon frequency ${\omega }_{\lambda }$ as functions of quantum dot radius $l_{qd}$ (“L” and “T” stand for longitudinal and transverse phonons, respectively).

Figure 5

$I_e-V$ curves for different phonon temperatures calculated from the rate equation method (denoted as “RATE” in the figure) and the Fermi-golden-rule (denoted as “FGR”) approximation. The parameters are the same as in Fig. 3.