Dynamical Coulomb blockade under a temperature bias

We observe and comprehend the dynamical Coulomb blockade suppression of the electrical conductance across an electronic quantum channel subjected to a temperature difference. A broadly tunable, spin-polarized Ga(Al)As quantum channel is connected on-chip, through a micron-scale metallic node, to a linear $RC$ circuit. The latter is made up of the node's geometrical capacitance $C$ in parallel with an adjustable resistance $R\in \{1/2,1/3,1/4\}\times h/e^2$ formed by 2–4 quantum Hall channels. The system is characterized by three temperatures: Temperatures of the electrons in the large electrodes ($T$) and in the node ($T_{\mathrm{node}}$), and a temperature of the electromagnetic modes of the $RC$ circuit ($T_{\mathrm{env}}$). The temperature in the node is selectively increased by local Joule dissipation, and characterized from current fluctuations. For a quantum channel in the tunnel regime, a close match is found between conductance measurements and tunnel dynamical Coulomb blockade theory. In the opposite near ballistic regime, we develop a theory that accounts for different electronic and electromagnetic bath temperatures, again in very good agreement with experimental data. Beyond these regimes, for an arbitrary quantum channel set in the far out-of-equilibrium situation where the temperature in the node significantly exceeds the one in the large electrodes, the equilibrium (uniform temperature) prediction for the conductance is recovered, albeit at a rescaled temperature $\alpha T_{\mathrm{node}}$.

Figure 1

(a) Device $e$-beam micrograph. A single generic channel of electron transmission probability $\tau \in [0,1]$ (with $\tau /R_{\mathrm{K}}$ its conductance; $R_{\mathrm{K}}\equiv h/e^2$), as well as $N_1$ and $N_2$ fully transmitted channels ($N_1=N_2=1$ shown) are separately connected to a small metallic island (light gray). The two quantum Hall edge channels are represented as lines. A Joule heating of the island is realized by applying balanced voltages across the fully transmitted channels ($N_1{V}_1+N_2{V}_2=0$), such that the island's dc voltage $〈V_{\mathrm{node}}〉$ remains zero. The heated-up island temperature $T_{\mathrm{node}}$ is monitored through noise measurements. (b) Schematic representation. The total number of fully transmitted channels controls the series resistance $R=R_{\mathrm{K}}/(N_1+N_2)$. The applied Joule power $P_{\mathrm{J}}$ results in a temperature bias $T_{\mathrm{node}}-T$ across the generic channel and also across $R$. In contrast with the thermally biased generic channel, the $RC$ circuit is here approximately modeled as a heated up electromagnetic environment at a uniform temperature $T_{\mathrm{env}}$ generally taken as $(T+T_{\mathrm{node}})/2$.

Figure 2

Renormalized transmission probability versus bias voltage in the tunnel regime. Symbols represent, in a log-log scale versus $V-V_{\mathrm{node}}$ and for different uniform temperatures $T\simeq T_{\mathrm{node}}\simeq T_{\mathrm{env}}$, the experimental values of $\tau /{\tau }_{\infty }$. Those are obtained from $R_{\mathrm{K}}/\tau =1/G-R$, with $G$ the measured differential conductance of the sample including both tunnel contact and series resistance $R=R_{\mathrm{K}}/3$ (see Appendix Fig. 9 for similar data at $R=R_{\mathrm{K}}/2$ and $R_{\mathrm{K}}/4$). The intrinsic (unrenormalized) transmission probability ${\tau }_{\infty }$ depends on applied gate voltages, and constitutes the only adjustable parameter in the data-theory comparison. Black continuous lines: Full quantitative predictions of the tunnel DCB theory [see Eq. (1) and Appendix pp2-s1]. Red dashed line: Power law $\eta {\left(V-V_{\mathrm{node}}\right)}^{2R/R_{\mathrm{K}}}$ predicted for $k_{\mathrm{B}}T\ll e(V-V_{\mathrm{node}})\ll h/2\pi RC$, using the quantitative theoretical value of $\eta$ [see Appendix Eq. (B4)].

Figure 3

Renormalized transmission probability versus node temperature in the tunnel regime (${\tau }_{\infty }\ll 1$) at low bias voltage ($V\to 0$). Full symbols: Experimental $\tau /{\tau }_{\infty }$ at a uniform temperature ($T_{\mathrm{node}}\simeq T\simeq T_{\mathrm{env}}$) for $R=R_{\mathrm{K}}/3$ (purple) and $R_{\mathrm{K}}/4$ (green). Gray dash-dotted lines and black dashed lines: Full and asymptotic ($k_{\mathrm{B}}T\ll h/2\pi RC$) predictions, respectively, both of them for a uniform temperature $T=T_{\mathrm{node}}=T_{\mathrm{env}}$. Open symbols: Renormalized transmission probability data obtained in the presence of the series resistances $R=R_{\mathrm{K}}/2, R_{\mathrm{K}}/3$ and $R_{\mathrm{K}}/4$, with a temperature bias $T_{\mathrm{node}}-T\ge 0$ developing from different fixed values of the temperature $T$. Black continuous lines and red dashed lines: Full and asymptotic ($k_{\mathrm{B}}T\ll k_{\mathrm{B}}{T}_{\mathrm{node}}\ll h/2\pi RC$) predictions, respectively, both of them versus $T_{\mathrm{node}}$ at fixed $T$ and assuming an electromagnetic environment at $T_{\mathrm{env}}=(T+T_{\mathrm{node}})/2$. For clarity, the data and theory results are shifted vertically in this log-log plot by a factor of 2 (4) for $R=R_{\mathrm{K}}/3$ ($R_{\mathrm{K}}/4$).

Figure 4

Conductance versus node temperature in the near ballistic regime ($1-\tau \ll 1$). The difference $G_{\max }-G$, between ballistic limit $G_{\max }=\frac{N}{N+1}{R}_{\mathrm{K}}^{-1}$ and sample linear conductance $G(V\to 0)$, is plotted in a log-log scale as a function of the rescaled node temperature $T_{\mathrm{node}}/T_{\mathrm{I}}$ for $R=R_{\mathrm{K}}/2,R_{\mathrm{K}}/3$, and $R_{\mathrm{K}}/4$. Gray dash-dotted lines: Full universal predictions at uniform temperatures ($T_{\mathrm{node}}=T=T_{\mathrm{env}}$) [35]. Black and red dashed lines: Asymptotic ($G_{\max }-G\to 0$) power law predictions at uniform temperatures (black) and in the limit $T_{\mathrm{node}}\gg T$ with $T_{\mathrm{env}}=T_{\mathrm{node}}/2$ (red). Black continuous lines: Predictions for a near ballistic junction temperature biased at $T_{\mathrm{node}}\ge T$ with $T=8\mathrm{mK}$ and $T_{\mathrm{env}}=(T+T_{\mathrm{node}})/2$. Full and open symbols: Measurements at a uniform temperature $T\simeq T_{\mathrm{node}}$ (full) and in the presence of a temperature bias at fixed $T\simeq 8$ mK (open), respectively. The scaling temperature $T_{\mathrm{I}}$ is set by adjusting to the universal theory the lowest temperature point $T_{\mathrm{node}}\simeq 8$ mK.

Figure 5

Conductance for arbitrary channel tunings and $R=R_{\mathrm{K}}/3$ versus node temperature. Open symbols: Measured sample conductance $G$ at $T\simeq 8\mathrm{mK}$ versus heated-up $T_{\mathrm{node}}$. Different tunings (${\tau }_{\infty }$) of the channel are shown using different symbols and colors. For each tuning a unique value of $T_{\mathrm{I}}$ is determined by matching the lowest temperature data point at $T_{\mathrm{node}}\simeq T\simeq 8$ mK with $G_{1/3}^{\mathrm{eq}}(T/T_{\mathrm{I}})$, the universal prediction at uniform temperatures (gray dash-dotted line). The black line corresponds to the universal prediction with the temperature reduction factor $T_{\mathrm{node}}\to \alpha T_{\mathrm{node}}$ predicted to apply for large temperature bias in the tunnel regime [$\alpha \simeq 0.648$, see Appendix Eq. (B17)]. Insets: Magnified view for intermediate channel tunings.

Figure 6

Insets: Symbols show illustrative measurements at ${\tau }_{\infty }\sim 0.98$ of $G-G_{\max }$ for $R=R_{\mathrm{K}}/N$ (top and bottom panels: $N=4$ and 3, resp.) versus equilibrium temperature $T$ at $V=0$, with $G_{\max }=\frac{1}{R_{\mathrm{K}}}\frac{N}{N+1}$ (data also shown in Fig. 4). Continuous lines are quantitative theoretical predictions of Eq. (C4), using ${\tau }_{\infty }=\tau (V=-58\mu \mathrm{V}$) without any fit parameter. Dashed lines are fits using ${\tau }_{\infty }$ as a free parameter adjusted by matching the data point at $T=90$ mK. Main panel: Symbols represent the fitted values of ${\tau }_{\infty }$ versus the corresponding large bias voltage measurements $\tau (V=-58\mu \mathrm{V}$).

Figure 7

Symbols: Relative difference between the calculated node temperature $T_{\mathrm{node}}^{\mathrm{calc}}$ and the measured one $T_{\mathrm{node}}^{\mathrm{meas}}$, shown at base temperature $T\sim 8$ mK for both $R=R_{\mathrm{K}}/3$ and $R=R_{\mathrm{K}}/4$, over the full range of explored $\tau$ values.

Figure 8

Open symbols: Renormalized transmission probability $\tau /{\tau }_{\infty }$ of the generic channel in series with resistances $R_{\mathrm{K}}/2, R_{\mathrm{K}}/3$, and $R_{\mathrm{K}}/4$ versus the node temperature at $V=0$ at the base temperatures $T\simeq 8$ mK in a log-log scale. Black lines: Predictions of the full tunnel DCB theory for different temperatures (see Appendix pp2-s3) calculated using $T=8$ mK, $T_{\mathrm{node}}$, and $T_{\mathrm{env}}=(T+T_{\mathrm{node}})/2$. ${\tau }_{\infty }$ is the only adjustable parameter per value of $R/R_{\mathrm{K}}$ in the data-theory comparison. The gray areas correspond to the predicted range of conductance for $T_{\mathrm{env}}\in [T,T_{\mathrm{node}}]$, using $T=8$ mK and $T_{\mathrm{node}}$.

Figure 9

Tunnel DCB theory-data comparison under a bias voltage at $R_{\mathrm{K}}/2$ and $R_{\mathrm{K}}/4$. Symbols: Measured renormalized transmission probability across the generic channel in series with a resistance $R_{\mathrm{K}}/2$ ($R_{\mathrm{K}}/4$) plotted versus the channel bias voltage $V-V_{\mathrm{node}}$ at different temperatures $T$ in a log-log scale. Black lines: Full tunnel DCB theory calculated with the parameters $C=3.1$ fF ($C=2.5$ fF) and the measured temperature $T$. Red dashed line: Asymptotic power law predictions at zero temperature with no fit parameter.

Figure 10

Universal renormalization flow of the conductance at equilibrium. Colored continuous lines represent the experimental curves (green for $R_{\mathrm{K}}/4$ is shifted vertically by 0.1, purple shows $R_{\mathrm{K}}/3$) obtained by averaging the ensemble of data measured from $T=8$ mK to $T=90$ mK for each ${\tau }_{\infty }$ configuration (see [33] for the detailed procedure). The exact theoretical predictions $G_{R/R_{\mathrm{K}}}^{\mathrm{eq}}(T/T_{\mathrm{I}})$ derived in [35] are shown as black dashed lines. The full renormalization curve has not been measured for $R=R_{\mathrm{K}}/2$.

Figure 11

Conductance of a generic channel in series with $R=R_{\mathrm{K}}/2$ [panel (a)] and $R_{\mathrm{K}}/4$ [panel (b)] under a temperature bias ($T_{\mathrm{node}}\ge T$) at base temperature $T\simeq 8$ mK. Open symbols: Measured sample conductance $G$ for different channel settings of ${\tau }_{\infty }$, each shown using a different color and symbol shape. For a given channel setting, a unique value of the renormalization temperature $T_{\mathrm{I}}$ is determined by matching the first data point at equilibrium ($T_{\mathrm{node}}\simeq T$) with the predicted universal conductance curve at equilibrium $G_{R/R_{\mathrm{K}}}^{\mathrm{eq}}(T_{\mathrm{node}}/T_{\mathrm{I}})$ (gray dash-dotted line). Black lines: Universal conductance curve at equilibrium with the same effective reduction in temperature expected at large $T_{\mathrm{node}}/T$ from the tunnel DCB theory, namely $G_{R/R_{\mathrm{K}}}^{\mathrm{eq}}(\alpha T_{\mathrm{node}}/T_{\mathrm{I}})$ with $\alpha$ given in Eq. (B17).