Size scaling of the addition spectra in silicon quantum dots

We investigate small artificial quantum dots obtained by geometrically controlled resistive confinement in low mobility silicon-on-insulator nanowires. Addition spectra were recorded at low temperature for various dot areas fixed by lithography. We compare the standard deviation of the addition spectra with theory in the high electron concentration regime. We find that the standard deviation scales as the inverse area of the dot and its absolute value are comparable to the energy spacing of the one-particle spectrum.

Figure 1

(Left) SEM picture of a specific morphological characterization structure: Cross section of a $12\text{−}\mathrm{nm}$-thick and $60\text{−}\mathrm{nm}$-wide silicon nanowire. (Right) Layout of a sample showing the 2D electron gas (dash lines) of surface area $A=W\times L+2\times d\times L$.

Figure 2

Peak spacing $\delta V_g$ vs gate voltage $V_g$ of a Si-wire transistor (sample S6) at $T=4.2\mathrm{K}$. The inset shows typical conductance resonances as a function of $V_g$ at $T=1\mathrm{K}$ and $T=4.2\mathrm{K}$. For the statistics only the high $V_g$ regime (many electrons) was considered, where $\delta V_g$ is approximately constant [slope of linear fit (line): $-{10}^{-4}$].

Figure 3

Ratio of the measured $\left(\delta V_g\right)$ and calculated $\left(\delta V_{\mathit{cal}}\right)$ peak-to-peak distance for a flat capacitance $C={ϵ}_0{ϵ}_{\mathrm{r}}A∕t_{\mathit{ox}}$ (${ϵ}_{\mathrm{r}}=3.9$, area $A$, oxide thickness $t_{\mathit{ox}}=4\mathrm{nm}$). The inset shows measured oscillations after subtraction of the background in units of $e^2∕h$ as a function of the gate voltage $V_g$ for three different areas (lowest curve: sample S6; middle curve: sample S4; highest curve: sample S3). Curves for S4 and S3 were shifted for clarity.

Figure 4

Fluctuations of the peak spacing distributions ${\sigma }_2$, normalized to the mean one-particle energy-level spacing ${\Delta }_1$ vs area $A$ (filled dots: $4.2\mathrm{K}$, open dots: $1\mathrm{K}$). The inset shows the typical $V_{\mathit{DS}}-V_g$ diamonds of the firstconductance peak in $A=3300{\mathrm{nm}}^2$ sample. The one-particle levels appear as additional lines in the diamonds. Projecting along the $V_g$ axis and taking $\alpha =0.4$ gives the measured excitation spectrum, with a mean value ${\Delta }_1=0.6\pm 0.2\mathrm{meV}$, in good agreement with the calculated values of Table (see text).

Figure 5

Relative peak spacing distribution $P\left(\delta V_g\right)$ for three different dots’ areas at $4.2\mathrm{K}$. The solid lines are the Gaussian fits used to extract the standard deviation. The shift towards larger values of $\delta V_g$ for decreasing areas is clearly visible. The distribution becomes larger when the sample area decreases. This enhancement of the fluctuations in smaller samples is shown in the inset, where the standard deviation $\sigma$ of the normalized fluctuation $(\delta V_g-⟨\delta V_g⟩)∕⟨\delta V_g⟩$ is plotted against the surface area of the sample.