Phonon spectroscopy by electric measurements of coupled quantum dots

We propose phonon spectroscopy by electric measurements of the low-temperature conductance of coupled quantum dots, specifically employing dephasing of the quantum electronic transport by the phonons. The setup we consider consists of a T-shaped double-quantum-dot (DQD) system in which only one of the dots (dot 1) is connected to external leads and the other (dot 2) is coupled solely to the first one. For noninteracting electrons, the differential conductance of such a system vanishes at a voltage located in-between the energies of the bonding and the antibonding states, due to destructive interference. When electron phonon (e-ph) on the DQD is invoked, we find that, at low temperatures, phonon emission taking place on dot 1 does not affect the interference, while phonon emission from dot 2 suppresses it. The amount of this suppression, as a function of the bias voltage, follows the effective e-ph coupling reflecting the phonon density of states and can be used for phonon spectroscopy.

Figure 1

The T-shaped double-dot system. The upper dot, dot 1, is connected to the external leads by the tunneling matrix elements $t_{\text{L}}$ and $t_{\text{R}}$; dot 2 is connected, by the tunneling matrix element $t_{\text{C}}$, solely to dot 1. Both dots are represented by a single-energy level each, ${\epsilon }_1$ and ${\epsilon }_2$.

Figure 2

The differential conductance $dI/dV$ of a symmetric ($\epsilon ={\epsilon }_1={\epsilon }_2$ and $\alpha =1$) DQD structure as a function of the gate voltage $\epsilon$. In the absence of the e-ph interaction, the conductance follows the dotted curve. The solid line is the differential conductance when the transport electrons are coupled to acoustic phonons on both dots. Here $t_{\text{C}}=0.5\Gamma$.

Figure 3

The differential conductance $dI/dV$ (solid lines) as a function of the gate voltage $\epsilon$ of a symmetric DQD ($\epsilon ={\epsilon }_1={\epsilon }_2$ and $\alpha =1$), for e-ph coupling with acoustic phonons. The bias voltage is $eV=2\Gamma$ and $t_{\text{C}}=0.5\Gamma$ ($\Gamma$ is the level broadening on dot 1). Panel (a) the e-ph interaction is on dot 1 alone; panel (b) the e-ph interaction is on dot 2 alone. The dotted curves are the differential conductance when the e-ph interactions are absent.

Figure 4

The differential conductance $dI/dV$ as a function of the gate voltage $\epsilon$ of a symmetric DQD ($\epsilon ={\epsilon }_1={\epsilon }_2$ and $\alpha =1$), for e-ph coupling with acoustic phonons. The bias voltage is $eV=2\Gamma$ and $t_{\text{C}}=0.5\Gamma$ ($\Gamma$ is the level broadening on dot 1). Panel (a) changing the size of dot 2. The solid line is the differential conductance when both dot sizes are equal, $L_1=L_2=c_{\text{S}}/\left(2.0\Gamma \right)$ and the dotted line is for the case $L_1=c_{\text{S}}/\left(2.0\Gamma \right)$ and $L_2=c_{\text{S}}/\left(0.5\Gamma \right)$. Panel (b) changing the size of dot 1. The solid line is the differential conductance when both dot sizes are equal, $L_1=L_2=c_{\text{S}}/\left(2.0\Gamma \right)$ and the dotted line is for the case $L_1=c_{\text{S}}/\left(0.5\Gamma \right)$ and $L_2=c_{\text{S}}/\left(2.0\Gamma \right)$.

Figure 5

The differential conductance $dI/dV$ as a function of the gate voltage $\epsilon$ of a symmetric DQD ($\epsilon ={\epsilon }_1={\epsilon }_2$ and $\alpha =1$), for e-ph coupling with optical phonons. Here $t_{\text{C}}=0.5\Gamma$ ($\Gamma$ is the level broadening on dot 1), and the strengths of e-ph coupling on the dots are ${\zeta }_1={\zeta }_2=0.3\Gamma$. $\delta$ introduced in Eqs. (A14, A15) is $\delta =0.5\Gamma$. Panel (a) the phonon energy in dot 1 (dot 2) is ${\omega }_1=\Gamma$ $\left({\omega }_2=3\Gamma \right)$, and the bias voltage is $eV=\Gamma$. The dotted line indicates the conductance in the absence of e-ph interaction. Panel (b) the same as in (a) with $eV=3\Gamma$.

Figure 6

The differential conductance $dI/dV$ as a function of the bias voltage $V$ for a symmetric DQD ($\epsilon ={\epsilon }_1={\epsilon }_2$ and $\alpha =1$), at the conductance dip $\left(\epsilon -eV=0\right)$. The tunnel coupling between the dots is $t_{\text{C}}=0.3\Gamma$ (solid lines) and $t_{\text{C}}=0.5\Gamma$ (dashed lines). Panel (a) coupling with acoustic phonons; the dotted line is $\chi \left(\omega \right)$ [Eq. (23)] as a function of $\omega$ (marked on the right-side vertical axis). Panel (b) coupling with optical phonons. The vibration energy in dot 1 is ${\omega }_1=\Gamma$ and on dot 2 is ${\omega }_2=3\Gamma$.