Intraband absorption and Stark effect in silicon nanocrystals

A theoretical investigation of the combined effect of confinement geometry, valley degeneracy, crystallographic orientations, and anisotropy of the effective mass on the intraband absorption of the conduction band of $\mathrm{Si}$ nanocrystals is presented. We show that the coupling between the nanocrystal shape and arbitrary orientations of the crystalline $\mathrm{Si}$ core plays a dramatic role in the intraband optical properties through interesting changes in the electronic structure and intra- and inter-valley degeneracy. Different orientations of the crystalline $\mathrm{Si}$ also affect the shape and orientation of the orbital wave functions, thereby modifying transition rules and fine structure of the intraband absorption. The anisotropy of the effective mass is also responsible for the emergence of absorption peaks that are not present in isotropic nanocrystals. Additional intraband transitions are also induced by changes in the nanocrystal shape with respect to spherical dots. Moreover, the quantum confined Stark effect blueshifts the absorption peaks and induces the appearance of peak structures as a consequence of the relaxation of selection rules caused by the displacement of the wave functions in response to external electric fields. This effect was found to be only appreciable in large nanocrystals.

Figure 1

(a) Schematics of the simulation cell studied in this work. We define a truncation parameter $\Delta$ which controls the NC shape, which is (i) spherical for $\Delta =D∕2$ ($D$ represents the NC diameter), (ii) hemispherical for $\Delta =0$ and (iii) truncated for intermediate values. (b) Sixfold $\mathrm{Si}$ band energy valleys. (c) Top view (xy plane) of the orientation of the $\mathrm{Si}$ energy valleys with respect to truncation of the NC for $\left[010\right]$, $\left[110\right]$, and $\left[111\right]$ crystallographic planes of the $\mathrm{Si}$ bulk perpendicular to the $y$ direction of the simulation cell.

Figure 2

Size dependence of the lowest four bound states of a spherical NC for the $\left[010\right]$, $\left[110\right]$, and $\left[111\right]$ orientations perpendicular to the $y$ direction of the simulation cell. The inset graph shows the wave functions of the $E_2\left(X{X}^{\prime }\right)$ state for three crystal orientations.

Figure 3

Dependence of the lowest four electron states (${\mathrm{E}}_j$, where $j=0,\dots ,3$) on the truncation parameter $\Delta$ of a $10\mathrm{nm}$ wide NC $\left[010\right]$ crystal orientation. The energy states within the $X{X}^{\prime }$, $Y{Y}^{\prime }$, and $Z{Z}^{\prime }$ valleys are represented by the symbols $\times$, $•$, and $◻$, respectively.

Figure 4

Size dependence of the lowest four electron states ($E_0$-solid, $E_1$-dashed, $E_2$-dotted, and $E_3$-dash-dotted lines, respectively) in a hemispherical NC with $\left[010\right]$, $\left[110\right]$, and $\left[111\right]$ crystallographic orientations.

Figure 5

Schematics of intravalley intraband transitions within the conduction band of $\mathrm{Si}$ NC’s. Only the ground state of each valley is occupied (only one electron), and the transitions occur from this to the upper excited states. The possibility of intervalley transitions is disregarded. Due to the computational cost, we arbitrary chose $N=20$ in this work.

Figure 6

(Left axis) Intraband absorption coefficient of spherical NC’s with 3, 5, and $10\mathrm{nm}$ of diameter and $\left[010\right]$ orientation of crystalline bulk. (Right axis) Oscillator strength of the most important transitions $(f_{ul}>{10}^{-1})$ for each valley. Transitions within the $X{X}^{\prime }$, $Y{Y}^{\prime }$, and $Z{Z}^{\prime }$ valleys are represented by the symbols $\times$, $•$, and $◻$, respectively.

Figure 7

(Left axis) Intraband absorption coefficient of a hemispherical NC with $10\mathrm{nm}$ of diameter for the $\left[010\right]$, $\left[110\right]$, and $\left[111\right]$ orientations of crystalline bulk. (Right axis) Oscillator strength of the most important transitions $(>{10}^{-1})$ for each valley. Transitions within the $X{X}^{\prime }$, $Y{Y}^{\prime }$, and $Z{Z}^{\prime }$ valleys are represented the symbols $\times$, $•$, and $◻$, respectively.

Figure 8

Intraband absorption coefficient of a biased $10\mathrm{nm}$ wide spherical NC with the $\left[010\right]$ crystallographic orientation for $F_{\mathrm{EXT}}$ pointing to positive $y$ direction with intensities values of (a) $0\mathrm{kV}∕\mathrm{cm}$, (b) $300\mathrm{kV}∕\mathrm{cm}$, and (c) $600\mathrm{kV}∕\mathrm{cm}$. In the inset graph, the intraband absorption is depicted for $F_{\mathrm{EXT}}$ varying from $-600$ to $600\mathrm{kV}∕\mathrm{cm}$ from top to bottom. (Right axis) Oscillator strength of the most important transitions $(>{10}^{-1})$ for each valley. Transitions within the $X{X}^{\prime }$, $Y{Y}^{\prime }$, and $Z{Z}^{\prime }$ valleys are represented the symbols $\times$, $•$, and $◻$, respectively.

Figure 9

Comparison between spherical NC’s ($10\mathrm{nm}$ of diameter) with isotropic (ISO) and anisotropic effective masses ($X{X}^{\prime }$, $Y{Y}^{\prime }$, and $Z{Z}^{\prime }$ valleys of a $\left[010\right]$ oriented NC) for the following properties: (a) absorption coefficient with and without bias $(600\mathrm{kV}∕\mathrm{cm})$, (b) oscillator strengths at zero bias, (c) energy spectrum for the lowest 20 states, where the state indexes $j$ are displayed in the horizontal axis, and (d) orbital wave functions of the lowest four confined states, where the state index $j$ grows from top to bottom. The multivalley feature is not present in isotropic NC’s.

Figure 10

Intraband absorption coefficient of a biased $5\mathrm{nm}$ (left column) and $10\mathrm{nm}$ (right column) wide hemispherical NC with the $\left[010\right]$ crystallographic orientation for $F_{\mathrm{EXT}}$ varying from $-600$ to $600\mathrm{kV}∕\mathrm{cm}$ from top to bottom pointing to $x$ (top row) and $y$ (bottom row) directions.

Figure 11

Comparison of the intraband absorption coefficient of (a) unbiased and biased [panels (b) and (c)] hemispherical NC’s with $10\mathrm{nm}$ of diameter and $\left[010\right]$ crystal orientation. In this figure, only $F_{\mathrm{EXT}}=\pm 600\mathrm{kV}∕\mathrm{cm}$ pointing to both $x$ and $y$ directions are used.

Figure 12

Comparison of the intraband absorption coefficient of $10\mathrm{nm}$ wide hemispherical NC’s with $\left[110\right]$ (left column) and $\left[111\right]$ (right column) crystalline orientations. $F_{\mathrm{EXT}}$ varies from $-600$ to $600\mathrm{kV}∕\mathrm{cm}$ from top to bottom pointing to x (top row) and y (bottom row) directions.