Tunability of a $2e$ periodic single Cooper pair box

We have measured the fully $2e$ periodic Coulomb staircase of a single Cooper pair box (SCB) in superconducting quantum interference design geometry, using a radio-frequency single-electron transistor. We have determined the energies of the SCB with microwave spectroscopy and compared the calculated shape of the Coulomb staircases to the measured staircases. We find excellent agreement as the Josephson coupling energy is tuned by an external magnetic field.

Figure 1

(a) The energy for a SCB with $E_J=0$ (dotted) and $E_J\ne 0$ (solid). The dashed and gray lines indicate the odd charge state with ${\tilde{\Delta }}_1$ and ${\tilde{\Delta }}_2$, respectively. (b) The corresponding average charge $⟨n⟩$ for the energy diagram in (a). The effect on the Coulomb staircase when the odd charge state is energetically favorable, ${\tilde{\Delta }}_2<E_C-E_J∕2$, is the appearance of the $1e$ step (gray solid line).

Figure 2

(a) The Coulomb staircase when continuous microwaves are applied to the SCB for sample A. (b) The filled circles indicate the energy separation $\Delta E$ versus $\Delta n_g$ for the applied frequencies ${\nu }_1=22.8\mathrm{GHz}$, ${\nu }_2=28.4\mathrm{GHz}$, ${\nu }_3=32\mathrm{GHz}$, ${\nu }_4=37.1\mathrm{GHz}$, and ${\nu }_5=44\mathrm{GHz}$. These points are fitted to $\Delta E$ with $E_C∕k_B=1.25\mathrm{K}$ and $E_J∕k_B=0.94\mathrm{K}$ (solid line). Data obtained from sample A. (c) The Coulomb staircase obtained at the same applied $B$ as above (solid line) together with the calculated Coulomb staircase for $E_C∕k_B=1.25\mathrm{K}$ and $E_J∕k_B=0.94\mathrm{K}$ (dotted line). (d) Same as (c), but with the derivative of the Coulomb staircase.

Figure 3

An electron micrograph of the layout for the integrated SCB and SET.

Figure 4

(a) $E_J$ vs $B_{\Vert }$ for sample A. The filled circles are measured values of $E_J$ obtained from spectroscopy. They are fitted to Eq. (2) with decreasing $E_J^{\max }\left(\Delta \right)$ according to Eq. (3). (b) Measurements of sample A with zero and maximum (positive and negative) flux applied from the Helmholtz coils at $B_{\Vert }=460\mathrm{mT}$. They are fitted to Eq. (2), with $E_J^{\max }∕k_B=0.92\mathrm{K}$. (c) $E_J$ vs flux for sample B. The filled circles indicate measurements of $E_J$ from spectroscopy with maximum (positive and negative) flux generated by $B_{\perp }$. They are fitted to Eq. (2) (solid line).

Figure 5

The derivative of Coulomb staircases (solid lines) that corresponds to $E_J∕k_B=0.15\mathrm{K}$ up to $E_J∕k_B=1.05\mathrm{K}$ together with the derivative of the calculated Coulomb staircases obtained from the spectroscopic data (dotted lines). The curves are measured for sample A, and are ordered with increasing $B_{\Vert }$, from $B_{\Vert }=370\mathrm{mT}$ (bottom) to $B_{\Vert }=540\mathrm{mT}$ (top). At $E_J∕k_B=0.35\mathrm{K}$ $(B_{\Vert }=490\mathrm{mT})$ the thermal excitation to the odd charge state broadens the staircase, since $(E_C-E_J∕2)\approx \tilde{\Delta }$. For $E_J∕k_B=0.15\mathrm{K}$ $(B_{\Vert }=540\mathrm{mT})$ a clear separation into two peaks (the $1e$ step) is visible. Here $(E_C-E_J∕2)>\tilde{\Delta }$ and the odd charge state is energetically favorable around the charge degeneracy [see Fig. 1].

Figure 6

(a) FWHM of the peak in $d⟨n⟩∕d{n}_g$ vs $E_J$, where diamonds are measured values for sample A and circles for sample B. Dotted and solid lines are calculated FWHMS for samples A and B, respectively. (b) Peak height vs $E_J$, with same legend as in (a). The squares are measurements where the staircase has lost its full $2e$ periodicity. The peak heights are retrieved along the dotted line in the inserted image. The image shows $d⟨n⟩∕d{n}_g$ vs $n_g$ and $E_J$ where $\tilde{\Delta }$ is a function of $E_J$. In both (a) and (b), the fact that samples A and B have different $E_C$’s is evident in the slopes and zero crossings of the curves. (c) $d⟨n⟩∕d{n}_g$ vs $n_g$ for sample B, where the measured trace for $E_J∕k_B=0.16\mathrm{K}$ is the solid line together with the corresponding calculated trace as the dotted line. The asymmetric behavior is visible as a short shoulder on the right side of the peak. (d) $⟨n⟩$ vs $n_g$ for the same data as in (c) (same legend). The shoulder feature in (c), is clearly visible as a rounding off of the top edge.