Shallow donor states near a semiconductor-insulator-metal interface

The lowest energy electronic states of a donor located near a semiconductor-insulator-metal interface are investigated within the effective mass approach. The effect of the finite thickness of the insulator between the semiconductor and the metallic gate on the energy levels is studied. The lowest energy states are obtained through a variational approach, which takes into account the influence of all image charges that arise due to the presence of the metallic and the dielectric interfaces. We compare our results with a numerical exact calculation using the finite element technique.

Figure 1

(Color online) The dependence of the ground state energy (in units of $2R^{\ast }$) vs the donor position $d/a_B$ with respect to the metal interface (for the semiconductor/metal interface problem). We compare the variational results with two parameters (circles), with the three variational parameter trial function (solid line) and “exact solution” (stars) found in Ref. 8. The thin dashed line is the result for $d=\infty$. For the comparison the ground state energy for the system in the absence of the image effects is also presented (dot-dashed curve). Inset: the configuration of the semiconductor/metal system.

Figure 2

(Color online) The average value of the electron position in $\rho$ direction in the $1s$ state vs the donor position in units of $a_B$ for the semiconductor/metal interface. Inset: the dependence of the average distance between the electron and the donor along the $z$ direction.

Figure 3

(Color online) The dependence of the excited state energies $2s$ (solid curve) and $2p_x$ (dashed curve) calculated variationally (with three parameters), as well as the numerical exact result indicated by stars for $2s$ and by circles for $2p_x$ state (in units of $2R^{\ast }$) vs the donor position $d/a_B$ from the metal gate (for the system with an infinitely thin insulator layer). The excited state energies $2s$ (dot-dot-dashed curve) and $2p_x$ (dot-dashed curve) for the system in the absence of the image effects.

Figure 4

(Color online) The dependence of the ground state energy (in units of $2R^{\ast }$) as a function of $d/a_B$ with respect to the semiconductor/dielectric interface using the trial function with three variational parameters (solid curve). As a comparison we indicate by circles the results obtained with the two parameters trial function as well as by stars the exact results. The result for $d\to \infty$ is indicated by the dashed horizontal line. We present also the ground state energy in the absence of the image effects (dot-dashed curve). Inset: the configuration of the semiconductor/dielectric system.

Figure 5

(Color online) The average value of the electron position in the lateral direction for the $1s$ state vs the donor position in units of $a_B$ for the case of semiconductor/dielectric interface. Inset: the dependence of the average distance between the electron and the donor along the $z$ direction.

Figure 6

(Color online) The dependence of the excited state energies $2s$ (solid curve) and $2p_x$ (dashed curve) calculated variationally and numerical exact indicated by stars for $2s$ and by circles for $2p_x$ (in units of $2R^{\ast }$) as a function $d/a_B$ for a semiconductor/dielectric system. The excited state energies for $2s$ (dot-dot-dashed curve) and $2p_x$ (dot-dashed curve) for the system in the absence of the image effects.

Figure 7

(Color online) The potential energy (in units of $2R^{\ast }$) of the electron along the $z$ axis (with $\rho =0$) for a semiconductor/dielectric/metal interface with the oxide layer thickness $l/a_B=0.5$; solid line (dashed line) corresponds to the case when the impurity is located at the distance $d/a_B=1$ $\left(d/a_B=3\right)$. Inset: the configuration of the semiconductor/insulator/metal system.

Figure 8

(Color online) The dependence of the ground state energy (in units of $2R^{\ast }$) as a function of $d/a_B$ with respect to the semiconductor/dielectric/metal interface, with the oxide layer thickness $l/a_B=0.5$, using the finite element techniques (stars) also from the trial function with three variational parameters (solid curve) and two parameters (circles). The thin dashed line is the result for $d=\infty$. The ground state energy $\left(l/a_B=0.5\right)$ for the case when for each of the donor and the electron we take only two images (dot-dot-dashed curve).

Figure 9

(Color online) The dependence of the excited state energies $2s$ (solid curve) and $2p_x$ (dashed curve) calculated variationally and exactly indicated by stars for $2s$ and by circles for $2p_x$ state (in units of $2R^{\ast }$) as a function of $d/a_B$ for a semiconductor/dielectric/metal system with $l/a_B=0.5$. The excited state energies for $2s$ (dot-dot-dashed curve) and $2p_x$ (dot-dashed curve) for the system in the absence of the image effects, as well as $2s$ (dotted curve) and $2p_x$ (dashed curve) when for each of the donor and the electron we take only one image with respect to each of the metal and the dielectric interfaces.

Figure 10

(Color online) The average value of the electron position on the $\rho$ direction in the $1s$ state vs the donor position in units of $a_B$ for the semiconductor/dielectric/metal interface. Inset: the dependence of the average distance between the electron and the donor along the $z$ direction.

Figure 11

(Color online) The ground state energy as a function of $d/a_B$ for a semiconductor/dielectric/metal system with different thicknesses of the oxide layer: $l/a_B=0.1$, 0.3, 0.5, and 1. Inset: the ground state energy as a function of $l/a_B$ for fixed $d/a_B=0.5$, 1, and 2.

Figure 12

(Color online) The dependence of the ground state energy (in units of $2R^{\ast }$) as a function of $d/a_B$ with respect to the semiconductor/vacuum/metal interface $\left({\epsilon }_{ox}=1\right)$, with a vacuum layer thickness $l/a_B=0.5$, using the finite element techniques (stars) also from the trial function with three variational parameters (solid curve). The ground state energy in the absence of the image effects (dot-dashed curve), as well as for the case when for the donor and the electron we take only two images (dot-dot-dashed curve). Inset: the ground state energy as a function of $l/a_B$ for fixed $d/a_B=0.5$, 1, and 2.

Figure 13

(Color online) The dependence of the excited state energies $2s$ (solid curve) and $2p_x$ (dashed curve) calculated variationally and exactly indicated by stars for $2s$ and by circles for $2p_x$ state (in units of $2R^{\ast }$) as a function $d/a_B$ for a semiconductor/vacuum/metal system $\left({\epsilon }_{ox}=1\right)$ with $l/a_B=0.5$. For the system in the absence of the image effects $2s$ (dot-dot-dashed curve) and $2p_x$ (dot-dashed curve), as well as $2s$ (dotted curve) and $2p_x$ (dashed curve) when for each of the donor and the electron we take only two images.