Single-dopant resonance in a single-electron transistor

Single dopants in semiconductor nanostructures have been studied in great detail recently as they are good candidates for quantum bits, provided they are coupled to a detector. Here we report the coupling of a single As donor atom to a single-electron transistor (SET) in a silicon nanowire field-effect transistor. Both capacitive and tunnel coupling are achieved, the latter resulting in a dramatic increase of the conductance through the SET, by up to one order of magnitude. The experimental results are well explained by the rate-equation theory developed in parallel with the experiment.

Figure 1

Gated nanowire device. (a) Geometrical arrangement of the main components of the device. The Si nanowire channel ($20\hspace{0.5em}\text{nm}$ thick and $60\hspace{0.5em}\text{nm}$ wide) bridges the constriction between the source and drain. The gate (40-nm-wide polysilicon finger) is deposited on top of the nanowire, and it is surrounded by silicon nitride (${\text{Si}}_3{\text{N}}_4$), thus forming two spacers (each $40\hspace{0.5em}\text{nm}$ wide) on both sides of the gate. The red spheres illustrate the doping with As atoms inside the Si material; dopants on the surface of Si are omitted for clarity. (b) Cross section along the nanowire. In the experiment, the implantation with As atoms is made after the deposition of the gate and spacers. Therefore, during the implantation, a small number of As atoms may diffuse into the region under the spacer, forming isolated donors. The SET island is formed by electron accumulation when a positive voltage is applied to the gate. (c) Transmission-electron-microscopy image of a cross section along the nanowire in an actual device. The nanowire (Si) is separated from the polysilicon finger (gate) by a 4-nm-thick oxide layer (${\text{SiO}}_2$); the spacers are hard to discern on the micrograph. (d) High-resolution image showing the atomic lattice of the nanowire and its interface with the buried oxide (BOX).

Figure 2

Experimental signatures of tunnel coupling between a single-dopant atom and a single-electron transistor. (a) Linear conductance through the silicon nanowire as a function of the gate voltage $V_g$ at various temperatures. The strong, resonance-like modulation of the CB oscillations is attributed to the presence of a dopant atom in the SET barrier. (b) Color plot of the differential conductance of the device versus the gate and bias voltages. The small Coulomb diamonds of the SET are strongly modulated in intensity, revealing, on a larger scale, resonant tunneling associated with the dopant atom.

Figure 3

Anomalies in transport through SETs caused by single-dopant atoms. (a) Linear conductance measured in a sample in which the dopant atom couples mostly capacitively to the SET. At $T=1\hspace{0.5em}\text{K}$, the CB peaks are suppressed in an interval of $V_g$, marking the extension of the anomaly. At $T=4.2\hspace{0.5em}\text{K}$, the CB oscillations are weakly modulated. (b) Modulation of CB oscillations in a different device, in which the dopant atom is tunnel coupled to both the SET island and the lead. The envelope of CB oscillations reveals resonant tunneling through the dopant atom. At $T=0.9\hspace{0.5em}\text{K}$, two resonances, separated by a suppression region, are visible in the envelope function. At $T=4.2\hspace{0.5em}\text{K}$, the resonances overlap and form a single broad resonance due to tunneling via the dopant atom. Note the factor of $5$ difference in the scale on the ordinate axis as compared to (a). (c) Linear conductance calculated using a circuit approach; see Sec. 3b4. With the donor coupling parameters $U_{12}=1.2\hspace{0.5em}\text{meV}$ and ${\Gamma }_L={\Gamma }_R=0$, the experiment in (a) is reproduced almost quantitatively. (d) The same as in (c) but plotted using the coupling parameters $U_{12}=1.15\hspace{0.5em}\text{meV}$ and ${\Gamma }_L={\Gamma }_R=40\hspace{0.5em}\mu \text{eV}$, which approaches the result of (b). The SET parameters used in (c) and (d) are tabulated in Table .

Figure 4

Setup, stability diagram, energy gap, phase shift, and envelope of CB oscillations. (a) Schematic view of the device. The SET island, with charge $N$, is tunnel coupled to two leads (source and drain). The donor, with charge $n$, is situated in the tunnel barrier between the source and the SET island. The top gate, at voltage $V_g$, couples capacitively to the SET island and donor; the coupling to the latter is the weaker of the two. (b) Stability diagram of the donor-SET system. Solid lines separate regions with different ground-state charge configurations. Sweeping the gate voltage $V_g$ in the experiment corresponds to traversing the plane of the stability diagram along the dash-dotted line, the “working line” of the device. The anomaly occurs in the region between the upper and lower rows of triple points. For small coupling of the donor to the gate ($\alpha ⩽1$), the working line is nearly horizontal; the anomaly region comprises then a large number of charge configurations. (c) The energy gap, governing the activated conductance at CB peaks, as a function of the gate voltage $N_g$. The solid line corresponds to the case $g_L\ne 0$, whereas the dashed line corresponds to $g_L=0$. (d) The phase shift $\phi$, which gives the position of the CB peaks across the anomaly region and outside. At low temperatures, $\phi$ changes stepwise in the center of the anomaly. At high temperatures, $\phi$ changes gradually over an interval that grows proportionally to the temperature. (e) The envelope of the CB oscillations versus the gate voltage. At low temperatures, two resonances occur on the sides of the anomaly. At high temperatures, the resonances overlap, forming a single broad resonance.

Figure 5

Activation energy $\Delta$ around a sequential-tunneling peak. (a) Top: fragment of the stability diagram [see Fig. 4b] showing the working line (dash-dotted line) outside the anomaly region. Bottom: the $\Delta$ vs $N_g$ dependence, corresponding to the top panel. (b) Same as in (a) but with the working line in the anomaly region. The smallest value of $\Delta$ is $\delta$.

Figure 6

The width of the CB peak in the anomaly region as a function of temperature.

Figure 7

Circuit of resistors, representing the donor-SET system at low temperatures.

Figure 8

Comparison of the circuit approach against an exact numerical evaluation of the conductance. The solid lines represent the result of the circuit approach, whereas the dots show the conductance evaluated by solving the Pauli master equation numerically. (a) Conductance calculated for the same parameters as in Fig. 3c. The two approaches coincide identically in this case because the circuit approach is exact for the donor-SET model in the limit ${\Gamma }_l\to 0$, ($l=L,R$). In particular, Eq. (38) can be obtained from Eqs. (48), (56), (58), and (59). (b) Conductance calculated for the same parameters as in Fig. 3d. For the low-temperature trace ($T=0.9\hspace{0.5em}\mathrm{K}$), the circuit approach deviates from the numerically exact result by a relative error $\varepsilon ⩽36\%$. However, the deviation takes place far in the tails of the peaks, where the conductance is small; hence, no visible difference appears on the scale of the plot. For the high-temperature trace ($T=4.2\hspace{0.5em}\mathrm{K}$), the relative error is $\varepsilon ⩽7\%$, and the deviation is visible in several CB valleys.