Nonlinear thermoelectricity in point contacts at pinch off: A catastrophe aids cooling

I consider refrigeration and heat engine circuits based on the nonlinear thermoelectric response of point contacts at pinch off, allowing for electrostatic interaction effects. I show that a refrigerator can cool to much lower temperatures than predicted by the thermoelectric figure of merit $ZT$ (which is based on linear-response arguments). The lowest achievable temperature has a discontinuity, called a fold catastrophe in mathematics, at a critical driving current $I=I_{\mathrm{c}}$. For $I>I_{\mathrm{c}}$ one can in principle cool to absolute zero, when for $I<I_{\mathrm{c}}$ the lowest temperature is about half the ambient temperature. Heat backflow due to phonons and photons stops cooling at a temperature above absolute zero, and above a certain threshold turns the discontinuity into a sharp cusp. I also give a heuristic condition for when an arbitrary system's nonlinear response means that its $ZT$ ceases to indicate (even qualitatively) the lowest temperature to which the system can refrigerate.

Figure 1

Heat current $J(T_{\mathrm{isl}},I)$ through a point contact when driven with a current $I$, for negligible phonon or photon heating. Blue indicates cooling of the island in Fig. 2a, while red indicates heating. The solid curve is the steady state ($J=0$), with the catastrophe at $I_{\mathrm{c}}$. The straight line is the maximum current, $I_{\max }$, corresponding to infinite bias.

Figure 2

Thermoelectric circuits made with point contacts shown in (a), (b); “e” (“h”) means the point contact is in a material whose charge carriers are electrons (holes). One should minimize the heat current carried by phonons and photons, $J_{\mathrm{ph}}$, by suspending the island.[29, 30] The temperature of the island in similar setups (albeit not suspended) has been probed experimentally using a quantum dot as thermometer,[31] although not yet in the regime of refrigeration. (c) Motion of charges in the gates (arrows) caused by making $V_{\mathrm{L}}$ positive, which partially screens $V_{\mathrm{L}}$ at some distance from the point contact. (d) Point contact 1 tuned to pinch off ($E_{\mathrm{pc}}$ equals the island's chemical potential) by adjusting $V_{\mathrm{g}}$.

Figure 3

Heat current as a function of $V$ and $T_{\mathrm{isl}}$, with the steady state, $J=0$, marked by the black curve [note the darkest color is used for all $hJ/{\left(k_{\mathrm{B}}{T}_0\right)}^2>0.18$ and white for all $hJ/{\left(k_{\mathrm{B}}{T}_0\right)}^2<-0.25$]. Superimposed are lines of constant current (dashed); these are $hI/\left(e{k}_{\mathrm{B}}{T}_0\right)=0$, 0.08, 0.16, 0.24, 0.32 from bottom to top.

Figure 4

(a) Steady-state refrigeration curves, $J(T_{\mathrm{isl}},I)=0$, for increasing heat flow due to phonons and photons; $\alpha /{\alpha }_0=0,0.02,0.06,0.12,0.3$. (b) The solid v-shaped curve is the steady-state refrigeration curve, $J(T_{\mathrm{isl}},I)=0$ for $\alpha =10{\alpha }_0$, i.e., about 30 times stronger phonon backflow than (a)'s uppermost curve. It is very different from the nearly linear theory for the same $\alpha$ (dashed parabola).

Figure 5

Steady-state refrigeration curves for a composite system; point contact in parallel with a barrier of conductance $G_{\mathrm{barrier}}=e^{2}g/h$ for $g=0,0.001,0.005,0.025,0.1,0.25,1$ (solid curves). The parabolas (dashed) are the nearly linear approximation for $g=0,1/4$; for $g=1$ the difference from the exact curve is not visible.

Figure 6

Heat-engine efficiency curves for $\alpha /{\alpha }_0=0,0.02,0.06,0.12,0.3,1$. Dashed lines are the linear-response predictions, Eq. (1), for $\alpha /{\alpha }_0=0,1$.

Figure 7

Heat current when the barrier peak is at $E_{\mathrm{pc}}=k_{\mathrm{B}}{T}_0/4$. The results which give this curve will be discussed elsewhere. For comparison, the thin-dashed curves are the steady state for $E_{\mathrm{pc}}=0$ in Fig. 1.