Thermoelectric efficiency at maximum power in low-dimensional systems

Low-dimensional electronic systems in thermoelectrics have the potential to achieve high thermal-to-electric energy conversion efficiency. A key measure of performance is the efficiency when the device is operated under maximum power conditions. Here we study the efficiency at maximum power, in the absence of phonon-mediated heat flow, of three low-dimensional, thermoelectric systems: a zero-dimensional quantum dot with a Lorentzian transmission resonance of finite width, a one-dimensional (1D) ballistic conductor, and a thermionic (TI) power generator formed by a two-dimensional energy barrier. In all three systems, the efficiency at maximum power is independent of temperature, and in each case a careful tuning of relevant energies is required to achieve maximal performance. We find that quantum dots perform relatively poorly under maximum power conditions, with relatively low efficiency and small power throughput. Ideal one-dimensional conductors offer the highest efficiency at maximum power (36% of the Carnot efficiency). Whether 1D or TI systems achieve the larger maximum power output depends on temperature and area filling factor. These results are also discussed in the context of the traditional figure of merit $ZT$.

Figure 1

(Color online) The basic setup considered here consists of a device described by its transmission function $\tau \left(E\right)$ (${\tau }_{\text{QD}}$ and ${\tau }_{1\text{D}}$ are sketched as examples), with contact leads that act as the hot and cold electron reservoirs. A bias voltage, $V$, is applied symmetrically with respect to the average chemical potential $\mu$, which can be tuned relative to the transmission function, using a gate voltage.

Figure 2

(Color online) (a) Power and (b) efficiency normalized by Carnot efficiency, of a QD as a function of bias voltage $V$ and average chemical potential $\mu$, for $T_C=300\text{K}$, $T_H=330\text{K}$ $\left(\Delta T/T_C=0.1\right)$, and $\Gamma =0.01kT$. The open-circuit voltage, $V_{oc}$ is highlighted in (red) dashed line (peak $V_{oc}$ corresponds to $S\approx 2\text{meV}/\text{K}$). The system works as a generator when the bias is between zero and $V_{oc}$. The vertical green line indicates the $\mu$ where maximum power occurs. (c) Current through a QD is the integral over the product of ${\tau }_{\text{QD}}$ (green) [Eq. (3)] and $\Delta f=\left(f_H-f_C\right)$, shown here in blue, using the $\mu$ and $V$ that result in $P_{\text{max}}$. Two transmission widths, $\Gamma =0.5kT$ and $5kT$ are plotted here in the approximate position where maximum power would be achieved.

Figure 3

(Color online) Power versus normalized efficiency of a quantum dot for various $\Gamma$ as indicated. The left vertical axis shows data for $T_C=300\text{K}$ and $T_H=330\text{K}$ whereas the right vertical axis shows the power for $T_C=1\text{K}$ and $T_H=1.1\text{K}$. The shape of the curves does not depend on temperature.

Figure 4

(Color online) ${\eta }_{\text{max}P}$ (red, crosses) and ${\eta }_{\text{max}}$ (blue, open dots), both normalized by Carnot efficiency, and maximum power (green, full dots) of a quantum dot as a function of $\Gamma /kT$ for $T_C=300\text{K}$ and $T_H=330\text{K}$. Maximum power peaks around $\Gamma /kT=2.25$. Efficiency at maximum power ${\eta }_{\text{max}P}$ approaches ${\eta }_{\text{CA}}=51\mathrm{\%}$ for small $\Gamma$ and is always smaller than ${\eta }_{\text{max}}$.

Figure 5

(Color online) Thermoelectric performance of a 1D conductor. (a) Power (nW) and (b) normalized efficiency, $\eta /{\eta }_C$, near the first $\left(E_1\right)$ and second $\left(E_2\right)$ subband edges for $T_H=330\text{K}$ and $T_C=300\text{K}$. The full green line indicates the $\mu$ for maximum power output. (c) Transmission function ${\tau }_{1\text{D}}$ [Eq. (7), blue line] and $\Delta f=f_H-f_C$ near the onset of the lowest (green) and second subbands (brown) with $V=-2\text{mV}$ and $\left(E_2-E_1\right)=0.5\text{eV}$.

Figure 6

(Color online) (a) Power density (in $\text{MW}/{\text{m}}^2$) and (b) normalized efficiency $\eta /{\eta }_C$ for a thermionic (TI) energy barrier as a function of $V$ and $\mu$ for $T_H=330\text{K}$ and $T_C=300\text{K}$. (c) Transmission function ${\tau }_{\text{TI}}\left(E\right)$ [Eq. (10)] and $\Delta \zeta ={\zeta }_H-{\zeta }_C>0$ [Eq. (9)] for $V=0$ (green) and for $V=-2\text{mV}$ (brown) and for $\mu -E_1=0.05\text{eV}$ where maximum power is produced. The choice of $\mu >E_1$ ensures that all of the region where $\Delta \zeta ={\zeta }_H-{\zeta }_C>0$ contributes to current flow.

Figure 7

(Color online) (a)–(d) Loops along constant $\mu$ chosen at $P_{\text{max}}$ of each system [i.e., along the green line of Figs. 2b, 5b, 6b] show that efficiency at maximum power is independent of temperature. Note that the power values of the TI system depend on $A_0$ (see main text), whereas the efficiency values are independent of this choice. The QDs values depend on $\Gamma$.

Figure 8

(Color online) Maximum power as a function of temperature with $m^{\ast }=0.7m_e$ and $\Delta T/T_C=0.1$. $T_{\times }$ is the temperature where 1D and TI systems yield the same power. $T_{\times }$ depends on $A_0$ (see main text) and on the electron effective mass [Eq. (8). see also Fig. 9].

Figure 9

(Color online) $T_{\times }$ as defined in Fig. 8 as a function of effective area $A_0$ of a 1D system for different effective mass: InAs $\left(0.023m_e\right)$, GaAs $\left(0.07m_e\right)$, and PbTe $\left(0.17m_e\right)$.

Figure 10

(Color online) (a) Power factor $S^{2}G$ (blue) and thermal conductance (green), (b) ${\left(ZT\right)}_{el}$ (brown) and $P_{\text{max}}$ (cyan) as a function of $\mu -E_0$ for a quantum dot with $\Gamma =0.01kT$ and $T_C=300\text{K}$ and $T_H=330\text{K}$.

Figure 11

(Color online) Plot of $\eta /{\eta }_C$ vs ${\left(ZT\right)}_{el}$ for various values of $\Gamma$ as indicated in the legend. The blue solid line is Eq. (15).

Figure 12

(Color online) (a) Power factor and thermal conductance, (b) ${\left(ZT\right)}_{el}$ and $P_{\text{max}}$ of 1D, as a function of $\left(\mu -E_1\right)/kT$ with $T_C=300\text{K}$ and $T_H=330\text{K}$. (c) and (d) power factor, thermal conductance, ${\left(ZT\right)}_{el}$, and $P_{\text{max}}$ of the same condition for TI system. Note that $G$ and $K$ are obtained by assuming the effective area to be $100{\text{nm}}^2$. ${\left(ZT\right)}_{el}$ of 1D is independent of temperature.