Band-edge alignment of $\mathrm{SiGe}∕\mathrm{Si}$ quantum wells and $\mathrm{SiGe}∕\mathrm{Si}$ self-assembled islands

We report on the energy band gap and band lineup of $\mathrm{SiGe}∕\mathrm{Si}$ heterostructures either in the case of coherently strained quantum wells or in the case of $\mathrm{SiGe}∕\mathrm{Si}$ self-assembled islands. We take into account the strain field and the quantum confinement effects through an accurate description of the conduction band including the $\Delta$ and $L$ bands. The strain field is calculated using a microscopic valence force field theory. The conduction-band diagram and energies are obtained from a 30-band $\mathbf{k}\bullet \mathbf{p}$ Hamiltonian accounting for the strain through the Bir-Pikus Hamiltonian. The band-edge description is first given for biaxially strained pseudomorphic $\mathrm{SiGe}$ layers. In $\mathrm{SiGe}$ quantum wells grown on relaxed silicon, the band line-up switches from type I to type II depending on the value of the average valence band offset. Applying the 30-band formalism to the case of heterostructures grown on relaxed silicon germanium buffer layers indicates that a better agreement with experimental data is obtained for a valence-band offset value $\mathrm{\Delta }{E}_v=0.54x$ where $x$ is the Ge composition. For this parameter, a type-II band lineup is thus expected for all compositions of pseudomorphic $\mathrm{SiGe}$/relaxed Si heterostructures. For $\mathrm{GeSi}∕\mathrm{Si}$ islands, we take into account the strain relaxation in the surrounding Si matrix. A type-II band lineup is predicted for all Ge compositions. The near-infrared interband recombination energy of the islands is calculated as a function of their $\mathrm{SiGe}$ composition.

Figure 1

$O_h$ group bands involved in the 15-band $\mathbf{k}\bullet \mathbf{p}$ model for simple group used by Cardona and Pollak (Ref. 15) and for double group used in the 30-band model at $\mathbf{k}=\mathbf{0}$ (center of the Brillouin zone) in the Koster notation. (Ref. 31). The momentum matrix elements $P_i$ which are used in the 30-band model are defined in Ref. 11.

Figure 2

Fundamental band-gap energy of strained $\mathrm{SiGe}∕\mathrm{Si}$ calculated within the framework of 30-band $\mathbf{k}\bullet \mathbf{p}$ model (full line). The data reported in Ref. 8 (triangles) and in Ref. 23 (full dots) are given as comparison.

Figure 3

(a) Conduction-band edges in the $\Delta$ and $L$ valleys. The lower part of the figure is a zoom at low energy to highlight the difference between the split $\mathrm{\Delta }2$ and $\mathrm{\Delta }4$ valleys. (b) Valence-band edges for pseudomorphically strained ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ on unstrained Si calculated using the $\mathbf{k}\bullet \mathbf{p}$ formalism uncluding strain via the Bir-Pikus Hamiltonian. The band edges at $x=0$ correspond to those of the unstrained Si substrate with the valence-band edge taken as energy origin.

Figure 4

Band lineup and resulting fundamental electron and hole confined levels and wave functions in strained ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}∕\mathrm{Si}$ quantum well with a $5\mathrm{nm}$ height, calculated within the framework of the 30-band model. The calculation uses an average valence-band offset given by Colombo et al. (Ref. 20). Note that the $\mathrm{\Delta }4$ valley lineup (in the upper part of the figure) is not represented on the same scale as the hh valley lineup (in the lower part of the figure).

Figure 5

Calculated strain components along the growth direction through the middle of a single island embedded in Si with 50% of Ge average composition. An island with a height of $3\mathrm{nm}$ and a base diameter of $50\mathrm{nm}$ is considered. The profile is obtained from a 3D finite-element calculation using a valence force field theory (Ref. 29).

Figure 6

(a) Band-edge alignment and confined states in the conduction band. A zoom of the conduction-band lineups is shown in the lower part of the figure. (b) Band-edge alignment in the valence band calculated using the 30-band $\mathbf{k}\bullet \mathbf{p}$ model introducing strain profile of Fig. 5, i.e., for a single ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ island embedded in a silicon matrix. $\mathrm{hh}\_1$ is the ground state energy level of the heavy hole.

Figure 7

(a) Calculated fundamental band gap at $0\mathrm{K}$ taking into account the confinement energy $E_G^{e\text{−}h}=E_{\mathrm{\Delta }2\_1}-E_{hh\_1}$ (squares) and without taking into account the confinement energy $E_G=E_{\mathrm{\Delta }2}^{Si}-E_{hh}^{SiGe}$ (full line). The values obtained in the case of an average band offset of $0.54x\left(\mathrm{eV}\right)$ are shown in dotted lines for comparison. (b) Spatial overlap $\theta$ (see text for definition) between electrons in the $\mathrm{\Delta }2\_1$ level and holes in the $\mathrm{hh}\_1$ level for various Ge average compositions in the island.

Figure 8

Calculated interband recombination energy as a function of the island height for a 50% Ge composition. The line is a guide to the eye. A valence-band offset of $0.47x$ is considered in the calculation.