Floating Electron States in Covalent Semiconductors

We report first-principles electronic-structure calculations that clarify the floating nature of electron states in covalent semiconductors. It is found that wave functions of several conduction- and valence-band states, including the conduction-band minima, do not distribute near atomic sites, as was taken for granted, but float in interstitial channels in most semiconductors. The directions and shapes of the interstitial channels depend on the crystal symmetry so that mysterious variation of the energy gaps in SiC polymorphs is naturally explained by considering the floating nature.

Figure 1

Calculated energy bands [(a)–(c)] and wave functions (KS orbitals) [(d),(e)] of SiC. Energy bands for $2H$ (a), $3C$ (b), and 6 $H$ (c) structures represented along symmetry lines in hexagonal BZ are shown. The energy of the valence-band top is set 0. Contour plots of the KS orbitals of the conduction band minimum at $M$ (d) and of the fourth lowest conduction-band state at $\Gamma$ (e) on $(0\overline{1}1)$ plane for $3C\mathrm{\text{−}}\mathrm{SiC}$. This $M$ point corersponds to a point $X=(0,0,2\pi /a_0)$ in the cubic BZ. The value for each contour color is relative to the corresponding maximum absolute value. White and burgundy balls depict C and Si atoms, respectively.

Figure 2

Energy analyses of KS orbitals in SiC. The kinetic-energy contribution ${\epsilon }_{\mathrm{kin}}=⟨{\phi }_i|-{\nabla }^2/2|{\phi }_i⟩$ and the Hartree-energy contribution ${\epsilon }_{\mathrm{H}}=⟨{\phi }_i|\int \rho ({\mathit{r}}^{\prime })/|\mathit{r}-{\mathit{r}}^{\prime }|d{\mathit{r}}^{\prime }|{\phi }_i⟩$ to the orbital energy of each KS state for $M$ point in $3C\mathrm{\text{−}}\mathrm{SiC}$ (a) and for $K$ point in $2H\mathrm{\text{−}}\mathrm{SiC}$ (b). The abscissa represents the $i$th KS state from the valence-band bottom and the 25th state is the conduction-band minimum.

Figure 3

KS orbital of the conduction-band minimum at $K$ point of $2H\mathrm{\text{−}}\mathrm{SiC}$. Isovalue surface (left panel) and contour plot (right panel). The burgundy and the white balls depict Si and C atoms, respectively. In the left panel, the isovalue surface of the squared KS orbital viewed from [0001] direction is shown at its value of 80% of the maximum value. In the right panel, the contour plot of the squared KS orbital on ($1\overline{1}00$) plane is shown.

Figure 4

Calculated energy bands and the conduction-band-minimum state of AlN in $2H$ and $3C$ structures. (a) Energy bands in a $2H$ structure and the isovalue surface of the squared KS orbital at $K$ viewed from [0001] direction. The isovalue is taken as 80% of the maximum value. (b) Energy bands in a $3C$ structure and the contour plot of the KS orbital at $M$ on $(0\overline{1}1)$ plane. (c) The isovalue surface (left panel) along [0001] direction and the contour plot (right panel) on $(11\overline{2}0)$ plane of the KS orbital at $\Gamma$ in $2H\mathrm{\text{−}}\mathrm{AlN}$. In the left panel, the green surface corresponds to the value of 85% of the positive maximum value, whereas the blue surface corresponds to the negative value with the same absolute value. The value for each contour color in (b) and (c) is relative to the corresponding maximum absolute value. Burgundy and blue balls depict Al and N atoms, respectively.

Figure 5

Contour plots of the KS orbitals in $3C\mathrm{\text{−}}\mathrm{Si}$. (a) Contour plot on (110) plane of the CBM state at $M$. The $sp$ orbitals mixed with the floating state distributed in $⟨110⟩$ channel. (b) Contour plot on (110) plane of the valence-band minimum state at $\Gamma$. The state is mainly of $s$ bonding character but extends in the interstitial channel region. (c) Contour plot on (110) plane of the conduction-band state at $\Gamma$ located at 7.81 eV above the valence-band top.