Stabilization of single-electron pumps by high magnetic fields

We demonstrate theoretically and experimentally how magnetic fields influence the single-electron tunneling dynamics in electron pumps, giving a massively enhanced quantization accuracy and providing a route to a quantum current standard based on the elementary charge. The field dependence is explained by two effects: field-induced changes in the sensitivity of tunneling rates to the barrier potential and the suppression of nonadiabatic excitations due to a reduced sensitivity of the Fock-Darwin states to the electrostatic potential. These effects lead to a continued improvement in quantisation accuracy at high field which is important for applications in metrology.

Figure 1

(a) Scanning electron microscope image of a typical device. (b) Schematic electrical connections. Electrons are pumped from left to right. (c) Potential profile during the pumping cycle (offset vertically): (i) loading, (ii) back-tunneling, (iii) trapping, and (iv) ejection. (d) Pump current at $f=$ 0.4 GHz as a function of $V_{G2}$. Curves are offset vertically by a fixed amount as the magnetic field is stepped in intervals of 2 T from 0 to 14 T. Data for $B<14$ T have been shifted horizontally to align the $I=1ef$ to $2ef$ transition. (e) High-resolution scans at $f=0.4$ GHz at $B=$ 10,11, . . . ,14 T. Scans are offset by 10 fA. Dashed line is $ef$ for each field. (f)–(h) $dI/d{V}_{G2}$ on a color scale as a function of $V_{G2}$ and B for $f=$ 0.1, 0.4, and 1 GHz. The structure arising from the first excited state is labeled $L_1$.

Figure 2

Model calculations: (a) Contour plot of the model 2D potential well with a line cut along the pumping direction. (b) Tunneling rate $\Gamma$ from the potential well as a function of $V_1$, the entrance barrier for different values of B for a constant $V_2=-50$ mV. Solid lines are fits to an exponential function. (c) Schematic diagram indicating the relative tunneling rate for $n=1$ and 2 (solid and dashed lines, respectively) for states in zero and large field. ${\Gamma }_2=e^{\delta \left(B\right)}{\Gamma }_1$ and ${\Gamma }_2\left(e{V}_1\right)={\Gamma }_1(e{V}_1+{\Delta }_c)$. See text for definition of $\delta$. (d) Calculated pump current as a function of control voltage in the back-tunneling model. (e) Fit of experimental data to determine $\delta \left(B\right)$ at 100 MHz (sample A). (f) Comparison of field-induced changes in $\delta$ found experimentally in sample A (as in Fig. 1) alongside model predictions (symbols). The shaded region indicates where nonadiabatic effects influence plateau flatness at $f=0.4$ GHz. (g) Similar data for sample B.

Figure 3

Field dependence of $\delta \left(B\right)$ assuming that it is determined by the effective barrier thickness.

Figure 4

(a) Current plateaus at 1 GHz for fields of 6, 10, and 14 T along with $dI/d{V}_{G2}$ (offset vertically, $V_{G1}=-0.4$ V). Voltage changes $\Delta V_{G2}$ are measured from the rising edge of the first plateau. All data in this figure are from sample B. The arrows indicate excitation features. (b) Maps of $dI/d{V}_{G2}$ (color scale) as a function of both $V_{G2}$ and $V_{G1}$ for 6, 10, and 14 T at 1 GHz. (c) Color map of $dI/d{V}_{G2}\left(\Delta V_{G2}\right)$ as a function of field up to 14 T. The excitation gap $\Delta \varepsilon ={\varepsilon }_1-{\varepsilon }_0$ is indicated. (d) Fock-Darwin state probability density for different combinations of B and $\hslash {\omega }_0$. The open curves are for $\hslash {\omega }_0=8$ meV and the filled curves are for $\hslash {\omega }_0=2$ meV. (e) Eigenenergy of the ground-state solution of the Fock-Darwin confinement potential (Ref. 7) for different combinations of $B$ and electrostatic confinement energy $\hslash {\omega }_0$. (f) Field dependence of the first excitation gap between $(n,l)=(0,0)$ and $(n,l)=(0,1)$ states, where $n,l$ are the radial quantum number and orbital angular momentum, respectively, for $\hslash {\omega }_0$ = 2,4,8 meV.