Full-band envelope-function approach for type-II broken-gap superlattices

We present a charge self-consistent mesoscopic electronic-structure method for type-II broken-gap superlattices that is based on the multiband $k\cdot p$ envelope-function method. This scheme avoids the separate classification and occupation of electron and hole states that causes the standard effective-mass theory to fail once conduction- and valence-band states strongly intermix. The computational efficiency of envelope-function methods is maintained. Free or bound charge-carrier redistributions can be taken into account self-consistently. With this method that we term as full-band envelope-function approach, we calculate effective band gaps, effective masses, and optical transition energies of InAs/GaSb superlattices as a function of the layer width. Good agreement with experiment is obtained. We also discuss semiconductor to semimetal transitions in wide layer structures. We find the charge carriers to form a two-dimensional gas of approximately massless Dirac particles at a critical layer width.

Figure 1

Schematic subband dispersion and state occupation for a narrow gap superlattice, as a function of the in-plane wave vector ${\mathbf{k}}_{\parallel }$. Regions of fully and partly occupied states are marked in dark and light gray, respectively. The shaded region indicates the required amount of background charge density ${\rho }_b$, when the $k$-space integration is restricted to $k_{\parallel }^{\text{max}}$ as explained in the text.

Figure 2

(Color online) Schematic band structure of InAs/GaSb superlattice with InAs and GaSb layer widths $w_1$ and $w_2$, respectively. (a) Sketch of the broken bulk band edges (dashed: conduction bands, full: heavy-hole bands) for narrow layer widths with negative gap $E_G$ in position space. The lowest unoccupied (labeled by $e_0$) and highest occupied (labeled by $h_0$) superlattice subbands are also shown and lead to a positive effective band gap $E_g^{\text{eff}}$. (b) Same as in (a) for larger layer widths. The character of the lowest unoccupied and highest occupied subbands is reversed. The effective band gap is still positive but there is a charge transfer between the adjacent layers that leads to a band bending as indicated schematically.

Figure 3

(Color online) Calculated subband dispersions of an InAs/GaSb superlattice as a function of ${\mathbf{k}}_{\parallel }$ along the [010] direction. The InAs and GaSb layer widths are 16 and 8 nm, respectively. The lateral lattice constant $a$ is the unstrained GaSb lattice constant. The insets show probability densities of subband eigenstates at different values of ${\mathbf{k}}_{\parallel }$. The InAs and GaSb layer is indicated by the light and dark gray shading, respectively. All energies are given in meV relative to the chemical potential $\mu$.

Figure 4

(a) Calculated local charge densities close to the interface of an intrinsic InAs/GaSb superlattice with equal layer widths of 9 nm. The results are shown for two different values of $k_{\parallel }^{\text{max}}$ (full line: 0.05, dashed line: 0.1 in units of $2\pi /a$). (b) Corresponding potential profile $-e\varphi \left(z\right)$ for charge density shown in (a). The interface dipoles modify the offset between the band edges.

Figure 5

(a) Calculated local charge density for a wide layer superlattice with 18 and 9 nm for the InAs and GaSb layer width, respectively. The charge density very close to the interface depends on the chosen maximum wave vector $k_{\parallel }^{\text{max}}$ (full line: 0.05, dashed line: 0.1 in units of $2\pi /a$) in contrast to the charge near the well centers. (b) Calculated electrostatic potential corresponding to (a), superimposed on the bulk conduction $\left(E_C\right)$ and valence-band $\left(E_V\right)$ edges.

Figure 6

(Color online) (a) Calculated effective band gap $E_G^{\text{eff}}$ of InAs/GaSb superlattices as a function of the InAs layer width for a fixed GaSb layer width of 10 nm. The vertical dotted lines indicate regimes where the band structure corresponds to a direct semiconductor, an indirect semiconductor, and a semimetal, respectively. (b) Calculated in-plane effective masses of LU and HO subband extrema for the same structure. The Dirac point indicates a situation where the in-plane subband dispersion becomes linear.

Figure 7

Calculated effective band gap of InAs/GaSb superlattices of equal individual layer widths, as a function of the layer width. For comparison, we show experimental data from Ref. 18 (full circles) and from Ref. 14 (open circles).

Figure 8

(Color online) Calculated transition energies between occupied and unoccupied band-edge states of InAs/GaSb superlattices with fixed GaSb layer width of 10 nm, as a function of InAs layer width. The experimental results are from Ref. 4 (circles).