Heterointerface effects on the charging energy of the shallow $D^{-}$ ground state in silicon: Role of dielectric mismatch

Donor states in Si nanodevices can be strongly modified by nearby insulating barriers and metallic gates. Experimental results indicate a strong reduction in the charging energy of isolated As dopants in Si nonplanar field effect transistors relative to the bulk value. By studying the problem of two electrons bound to a shallow donor within the effective mass approach, we find that the measured reduction in the charging energy (measurements also presented here) may be due to a combined effect of the insulator screening and the proximity of metallic gates.

Figure 1

(Color online) Schematic representation of a negatively charged donor in Si (solid circles) located a distance $d$ from an interface. The open circles in the barrier (left) represent the image charges. The sign and magnitude of these charges depend on the relation between the dielectric constants of Si and the barrier given by $Q=\left({ϵ}_{\text{barrier}}-{ϵ}_{\text{Si}}\right)/\left({ϵ}_{\text{barrier}}+{ϵ}_{\text{Si}}\right)$. For the electrons, $Q<0$ corresponds to repulsive electron image potentials and a positive donor image potential [opposite signs of potentials and image charges for $Q>0$, see Eq. (6)].

Figure 2

(Color online) Energy of the neutral donor versus its distance $d$ from an interface for different values of $Q=\left({ϵ}_{\text{barrier}}-{ϵ}_{\text{Si}}\right)/\left({ϵ}_{\text{barrier}}+{ϵ}_{\text{Si}}\right)$.

Figure 3

(Color online) Results for $Q=-0.5$. (a) Energy for a neutral donor $D^0$, and for the ground $D^{-}|1s,1s,s⟩$ and first excited $D^{-}|1s,2s,t⟩$ negatively charged donor. (b) Binding energies of the $D^{-}$ states. (c) Value of the variational parameter $b$ for the $D^{-}$ ground state. For $d<4$, $D^{-}|1s,1s,s⟩$ is not stable and the energy is minimized with $b\to \infty$. Bulk values are represented by short line segments on the right.

Figure 4

(Color online) Same as Fig. 3 for $Q=0.5$.

Figure 5

(Color online) Differential conductance stability diagram showing the transport characteristics of a single As donor in a FinFET device (Ref. 16). The differential conductance is obtained by a numerical differentiation of the current with respect to $V_b$ at a temperature of 0.3 K. Extracting the charging energy from the stability diagram can be done by determining the gate voltage for which Coulomb blockade of a given charge state (the $D^0$ charge state in this case) is lifted for all $V_g$. The transition point is indicated by the horizontal arrow, leading to a charging energy $U=36\text{meV}$, as given by the vertical double arrow. The inset shows the electrical circuit used for the measurement and a top view scanning electron microscopy image of the device.

Figure 6

(Color online) Charging energy $U$ of the $D^{-}$ ground state for three different values of $Q$. For $Q\le 0$, the charging energy is nearly constant with $d$. For these cases, the negatively charged donor is not bound for small $d$. For $Q>0$ the charging energy decreases as the donor gets closer to the interface, at relatively small distances $d$. The latter is consistent with the experimental observation.