Thermoelectric efficiency of nanoscale devices in the linear regime

We study quantum transport through two-terminal nanoscale devices in contact with two particle reservoirs at different temperatures and chemical potentials. We discuss the general expressions controlling the electric charge current, heat currents, and the efficiency of energy transmutation in steady conditions in the linear regime. With focus in the parameter domain where the electron system acts as a power generator, we elaborate workable expressions for optimal efficiency and thermoelectric parameters of nanoscale devices. The general concepts are set at work in the paradigmatic cases of Lorentzian resonances and antiresonances, and the encompassing Fano transmission function: the treatments are fully analytic, in terms of the trigamma functions and Bernoulli numbers. From the general curves here reported describing transport through the above model transmission functions, useful guidelines for optimal efficiency and thermopower can be inferred for engineering nanoscale devices in energy regions where they show similar transmission functions.

Figure 1

Schematic representation of a two-terminal power-generator device, with electron transmission function $\mathcal{T}\left(E\right)$, and $T_L>T_R$. In a steady situation, the charge current is conserved in the left and right electrodes. Heat current is not conserved, and heat flowing from the hot source is partially transmitted to the cold one. (For refrigerators, maintaining $T_L>T_R$, the arrows must be oriented in the opposite direction.)

Figure 2

Universal curves for (a) the efficiency $\eta /{\eta }_c$, and (b) the figure of merit ${\left(ZT\right)}_{\mathrm{el}}$, of the thermal machine with Lorentzian line shape as a function of the dimensionless energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$, for fixed values of the dimensionless broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$. Notice the logarithmic scale on the vertical axis of (b).

Figure 3

Universal curves for the thermoelectric power (in units $k_B/e)$ of Lorentzian transmission functions vs the dimensionless energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$ for fixed values of the dimensionless broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$.

Figure 4

(a) Universal family of curves for the Lorenz number (in units $k_B^2/e^2)$ vs the energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$ for fixed values of $\gamma =\mathrm{\Gamma }/k_{B}T$ of the resonance transmission function, and (b) vs the broadening parameter $\gamma$ for fixed values of $\varepsilon$. The straight line drawn at ${\pi }^2/3$ represents the common asymptotic value of the family of curves for large values of $\gamma$.

Figure 5

Universal curves for (a) the efficiency $\eta /{\eta }_c$ and (b) the figure of merit ${\left(ZT\right)}_{\mathrm{el}}$ of the thermal machine with antiresonance line shape as a function of the energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$, for fixed values of the broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$.

Figure 6

Universal curves of thermoelectric power (in units $k_B/e)$ of antiresonance transmission functions, versus the energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$, for fixed values of the broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$.

Figure 7

Universal curves for the the Lorenz number of antiresonance transmission functions vs the energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$, for fixed values of the broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$.

Figure 8

Universal curves for (a) the efficiency $\eta /{\eta }_c$ and (b) the figure of merit ${\left(ZT\right)}_{\mathrm{el}}$ of the thermal machine with Fano line shape as a function of the dimensionless energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$ for fixed values of the broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$. The asymmetry parameter $q$ has been set equal to 1.

Figure 9

Thermoelectric power (in units $k_B/e)$ for Fano transmission functions vs the $\varepsilon =(E_d-\mu )/k_{B}T$ energy parameter, for different values of the asymmetry parameter $q$. The broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$ has been chosen equal to unity.

Figure 10

(a) Thermoelectric power (in units $k_B/e)$ for Fano transmission functions vs the energy parameter $\varepsilon =(E_d-\mu )/k_{B}T$, for fixed values of the broadening parameter $\gamma =\mathrm{\Gamma }/k_{B}T$. The asymmetry parameter $q$ has been set equal to 1 in (a) and equal to 5 in (b).