Thermopower and dynamical Coulomb blockade in nonclassical environments

Charge and heat transfer through a nanoscale conductor is not only determined by the transmission properties of the electrons, but can also be strongly impacted by coupling to other degrees of freedom in the environment of the conductor. Here, we analyze the influence of the electromagnetic environment on a simple, yet significant, thermoelectric property, the thermopower, in the simple transport scenario of single-charge transfer across a tunnel junction. Considering both thermal and out-of-equilibrium steady-state environments, we find that the thermopower can be strongly affected by the environmental state and can, in turn, act as a sensitive probe of environmental properties.

Figure 1

A tunnel junction with an applied temperature gradient is placed in series to an electromagnetic environment at temperature $k_{B}T$ and driven by a dc voltage $V$. The electromagnetic environment can consist of arbitrary circuit elements and is modeled as a collection of $LC$ circuits. Energy dependent tunneling is modeled by a resonant level in the junction, which is strongly tunnel coupled to one side (inset).

Figure 2

Thermopower of a cold junction ($\gamma =0.01, {\theta }_{\mathrm{el}}=0.001$) coupled to a hot ohmic impedance ($\theta =1$) vs the level position $\epsilon$. The hot environment enables exchange processes over a wide range of energies. Nonetheless, the inelastic thermopower [red, Eq. (12)] retains the Mott form (7) of elastic transport without such exchange processes (black). Units are scaled with $E_C$.

Figure 3

In a low-temperature ohmic environment the charging energy $E_C$ appears as an additional scale in the thermopower. In the ${\theta }_{\mathrm{el}}\ll 1$ and $\theta \to 0$ limit of Eq. (14) a broad feature of width $E_C(=1)$ and additional copies of the sharp Mott feature at $\epsilon \approx \pm 1$ combine (magenta curve). The crossover from the ballistic Mott result (black) for intermediate environmental temperatures is illustrated by the red curve and the limit of vanishing electronic and environmental temperature is shown (blue dashed line). Units are scaled with $E_C$ and $\gamma =0.01$.

Figure 4

Thermopower $S_0^{\mathrm{inel}}$ in Sommerfeld approximation (solid lines) within an electromagnetic environment consisting of a high quality cavity in its ground state. Strong Mott features in the thermopower at multiples of the cavity resonance, $\epsilon =\pm k (k\in \mathbb{N})$, signal inelastic transport channels with $k$-photon emission into the cavity. Electronic temperature ${\theta }_{\mathrm{el}}$ and coupling strength $\rho$ determine whether transport through an inelastic channel becomes dominant over the ballistic background (dashed lines). If so, there will be a large thermopower signal, even in cases where the total conductance (inset) may be small. Units are scaled with $\hslash \mathrm{\Omega }$ and $\gamma =0.1$.

Figure 5

Thermopower $S_0^{\mathrm{inel}}$ for a cavity in a Fock state $|n_0=2\rangle$. The nonclassical environment opens new transport channels with strong Mott features appearing at energies, $\epsilon =\pm k(k\in \mathbb{N},k\le n_0)$. The low-temperature electronic system absorbs energy quanta from the environment lifting electrons from far below to the edge of the Fermi sea, where they yield a strong thermoelectric signal. Units are scaled by $\hslash \mathrm{\Omega }$ and $\gamma =0.1$; dashed curves are ballistic results.

Figure 6

Thermopower $S_0^{\mathrm{inel}}$ for a cavity in a squeezed state ($r\ne 0$). A squeezed state provides enhanced probabilities for emission and absorption of energy over a range increasing with the strength of squeezing. The even-odd asymmetry of a squeezed state is reflected in the relative height of the Mott features at energies $\epsilon =\pm 1$ and $\epsilon =\pm 2$. Units are scaled with $\hslash \mathrm{\Omega }$ and $\gamma =0.1$. Results for $\rho =1$ (solid) and ballistic ones for $\rho =0$ (dashed) are shown.

Figure 7

Left: Comparison of the thermopower for the three considered environments: empty cavity (red), Fock state (magenta), and squeezed state (blue) at ${\theta }_{\mathrm{el}}=0.2, \rho =1, \gamma =0.1$. The upper panels show the coupling strength dependence of $S(\epsilon =k\mathrm{\Omega }-\gamma )$ for $k=-3, k=-2$, and $k=+1$ respectively. For weak coupling $\rho$ a generic scaling with ${\rho }^{\left|k\right|}$ is observed for $k$-photon emission and absorption for all environments. It can be traced back to a corresponding scaling of $p_{\mathrm{tot}}\left[k\right]$. Units are scaled with $\hslash \mathrm{\Omega }$. Right: Weights $p_{\mathrm{tot}}\left[k\right]$ entering the energy-exchange function $P_{\mathrm{tot}}\left(E\right)$ for a squeezed state cavity, $|r=1\rangle$. Fock state (dots) and empty cavity (dot-dash) show the same generic scaling (shown for $k=1$).