Cooling of cryogenic electron bilayers via the Coulomb interaction

Heat dissipation in current-carrying cryogenic nanostructures is problematic because the phonon density of states decreases strongly as energy decreases. We show that the Coulomb interaction can prove a valuable resource for carrier cooling via coupling to a nearby cold electron reservoir. Specifically, we consider the geometry of an electron bilayer in a silicon-based heterostructure and analyze the power transfer. We show that, across a range of temperatures, separations, and sheet densities, the electron-electron interaction dominates the phonon heat-dissipation modes as the main cooling mechanism. Coulomb cooling is most effective at low densities, when phonon cooling is least effective in silicon, making it especially relevant for experiments attempting to perform coherent manipulations of single spins.

Figure 1

Illustration of our proposed cooling scheme. Two 2DEGs, an active layer and an auxiliary layer, are separated by a distance $d$. The active layer is involved in an electrical experiment and is heated by hot electrons that enter the sample through the measurement leads. The auxiliary layer is wired to the dilution refrigerator and is held at its base temperature. Heat is transferred from the active to the auxiliary layer via the Coulomb interaction. For mathematical simplicity, in this work we take the sample material, spacer, and both layers to be Si. Further, we approximate the two 2DEGs to have zero thickness.

Figure 2

The dimensionless function $Y(\eta ,{\zeta }_0)$ [Eq. (3)] plotted vs $\eta$ for various values of ${\zeta }_0$ and with layer densities $n_1=n_2$. Here, $\eta =k/k_F$ is the scaled momentum and ${\zeta }_0=k_{B}T/E_F$ is the characteristic magnitude of energy fluctuations due to temperature. The function follows a power law for both low and high $\eta$, with a maximum occurring for an intermediate $\eta$, which we denote as ${\eta }^{*}$.

Figure 3

Calculated values for the power per unit area $P/A$ transferred via the Coulomb interaction versus separation [Eq. (2)]. Here, the sheet density is $4\times {10}^{11}$ cm${}^{-2}$ in both layers, the temperature of the active layer is $T=100$ mK, and the temperature of the auxiliary layer is 0 K. As can be seen by the dashed lines, the power transfer varies as approximately $1/d^2$ at small distances and $1/d^5$ at large distances. The crossover length scale occurs when $d\approx E_F/\left(k_F{k}_{B}T\right)$.

Figure 4

Calculated values for the power per unit area $P/A$ transferred between an active ($T=50$ mK) and a heat sink ($T=0$ K) 2DEG via the Coulomb interaction as a function of the layer density at three different values for the separation between layers. Here, the densities of both layers are identical. For comparison, $P/A$ due to phonons from experimental data in Ref. 6, scaled for changing density, is shown as a dotted line.

Figure 5

Values for the power per unit area scaled by $T^4$, $P/\left(A{T}^4\right)$, transferred via the Coulomb interaction between two 2DEGs versus the temperature of the active layer at three different values for the separation between the layers. The auxiliary layer temperature is set to absolute zero. For comparison, the scaled $P/\left(A{T}^4\right)$ due to phonons from experimental data in Ref. 6 is shown. The density of both layers is $1\times {10}^{11}$/cm${}^2$.

Figure 6

The rescaled function $\upsilon \left(\theta \right)$ [Eq. (D7)], plotted versus the scaled coordinate $\theta =\eta /{\zeta }_0$. Here, $\eta =k/k_F$, the momentum transfer scaled by the Fermi momentum, and ${\zeta }_0=k_{B}T/E_F$. For small $\theta$, $\upsilon$ takes on the expected asymptotic value of 1. The curves essentially coincide until around $\theta \gtrsim 1$. This enables us to treat $\upsilon$ as approximately independent of ${\zeta }_0$ before that point.