Single grain heating due to inelastic cotunneling

We study heating effects of a single metallic quantum dot weakly coupled to two leads. The dominant mechanism for heating at low temperatures is due to inelastic electron cotunneling processes. We calculate the grain temperature profile as a function of grain parameters, bias voltage, and time and show that for nanoscale size grains the heating effects are pronounced and easily measurable in experiments.

Figure 1

(Color online) Sketch of a single grain weakly coupled to two leads, $g_t⪡1$. Temperatures $T$ and $T_g$ refer to the leads and grain temperatures, respectively. Arrows indicate the inelastic cotunneling process through a single grain with a typical energy of the electron-hole pair $T_g$. The dashed line corresponds to the Fermi level in the grain. Here only the electron propagation process responsible for the heating of the grain is shown $\left({\sigma }_{\text{in}}\right)$. The heat is removed from the grain by a similar process but with the electron starting and ending in the same lead (inelastic cotunneling loop, ${\kappa }_{\text{in}}$) and by phonons $\left({\kappa }_{\text{ph}}\right)$.

Figure 2

(Color online) Voltage dependence of the dimensionless steady state grain temperature ${\mathfrak{T}}_g^{\ast }$ [solution of Eq. (2)] for different values of the tunneling conductance $g_t$, shown next to the graphs. All graphs are plotted for the typical parameters given in the text and at temperature $T=10\text{K}$. The values for $\mathcal{V}$ are valid for $\mathcal{V}<E_c/T$.

Figure 3

(Color online) Plots of the grain temperature $T_g$ vs dimensionless time $\tau =t\delta$. The characteristic scale $T_g^{\ast }$ is given by Eq. (2) and ${\tau }^{\ast }$ approximately by Eq. (7). The $y$ axis is shifted by $T$ and scaled to 1 for better comparison. The graph shows plots for three different values of the tunneling conductance $g_t$ written next to each group of two curves which represent different values for ambient temperature $T:T=10\text{K}$ (solid) and $T=15\text{K}$ (dashed). The inset shows the process of heating and cooling (with the same axis as the main plot): in the left part $\left(\tau <150\right)$ a finite voltage is applied [$V=5\text{mV}$ in Eq. (7)] which is switched off at $\tau =150\left[V=0\right]$.