Electronic, optical, and structural properties of quantum wire superlattices on vicinal (111) GaAs substrates

We have studied corrugated quantum wells as a semiconductor quantum wire structure. The quantum well corrugation results from a step bunching effect during epitaxial growth on vicinal (111) GaAs substrates and might be used to form a quantum wire superlattice. The strain and piezoelectric effects were studied both by an atomistic valence force field method and by an elastic continuum model. Within the elastic continuum model we also studied the electromechanical coupling. The electronic band structure was calculated with the eight-band $\mathbf{k}\mathbf{\bullet }\mathbf{p}$ model. The nonphysical oscillating solutions were eliminated by appropriate fine tuning of the material parameters. We have also studied the density of states and the polarization dependence of interband light absorption in the electric dipole approximation.

Figure 1

The QWRS geometry. (a) 3D CC structure, (b) CC structure (strained), (c) CP1 structure (strained), (d) CP2 structure (lattice matched). The shaded region between the corrugated surfaces is the quantum well, $P_{\mathit{QW}}=13.2\mathrm{nm}$ is the period of the corrugation, $T_{\mathit{QW}}=8\mathrm{nm}$ is the thickness of the quantum well, and $\theta$ is the offset angle of the vicinal (111) substrate. The corrugated material interfaces are formed from (111) and (110) lattice planes.

Figure 2

The strain component ${\epsilon }_{xx}$ in the crystal coordinate system along the vertical line shown in Fig. 3. In (a) the thickness is $T_{\mathit{QW}}=3\mathrm{nm}$, and in (b) $T_{\mathit{QW}}=6\mathrm{nm}$. The VFF and FEM results are shown by solid and dashed lines, respectively.

Figure 3

The strain component ${\epsilon }_{xx}$ calculated with VFF method for $T_{\mathit{QW}}=6\mathrm{nm}$. The results of Fig. 2 were plotted along the dashed line shown here.

Figure 4

The probability density of the electron ground states for the (a) strain CC, (c) strained CP1, (e) lattice-matched CP2 structure and (b), (d), and (f) show the corresponding probability density of the hole ground states, for $\mathbf{k}=0$. The contour scale is linear (arbitrary units) and the black colour represents the maximum amplitude of each plot. The black dashed lines show the geometry used in the calculation.

Figure 5

Schematic picture of the band edge for the strained CC structure. The solid lines represent the band edges excluding strain effects. The dashed line shows the effect of the piezoelectric field. The piezoelectric potential energy difference across the well is $\Delta E_{\mathit{piezo}}=0.20\mathrm{eV}$. The probability density of the ground states of the conduction and valence bands are shown by dotted lines and the definition of the confinement energies $E_{C0}$ and $E_{V0}$ are shown by arrows.

Figure 6

The band diagram (solid lines) and DOS (dotted lines) for the strained CC structure. The band energy is given with respect to the strain-free valence band edge of bulk ${\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$. The axis below the panels correspond to the wave vector of the energy bands and the axis at the top correspond to the DOS. The energy scale to the left is common to all curves.

Figure 7

The band diagram for the strained CP1 structure. The band energy is given with respect to the unstrained valence band edge of bulk ${\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$.

Figure 8

The band diagram for the strained CP2 structure. The band energy is given with respect to the valence band edge of bulk ${\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$.

Figure 9

Absorption coefficient for the strained CC structure of Fig. 1b. The solid and dotted lines correspond to the absorption of light polarized in $x$ and $z$ directions, respectively ($z$ is parallel to the QWRS). The absorption coefficient was calculated without any linewidth broadening in order not to smear out the fine structure. The vertical lines indicate the energies of the diagonal transitions at $\mathbf{k}=0$.

Figure 10

Effective potential energy of an electron (solid line) and a hole (dashed line) as a function of $x$ coordinate, in (a) the strained CC, (b) the strained CP1, and (c) the lattice-matched CP2 structure. The shaded areas and the $y$ axis to the right show the corresponding geometry.