Influence of conduction-band nonparabolicity on electron confinement and effective mass in ${\text{GaN}}_x{\text{As}}_{1-x}∕\text{GaAs}$ quantum wells

We derive an analytical model to describe the conduction-band states of GaNAs-based quantum well structures, including the band anticrossing effect between $N$ resonant states and the conduction-band edge. The predictions of the model are compared to those obtained using a full ten-band $\mathbf{k}·\mathbf{p}$ model based on the same set of parameters. Both methods are then tested by comparison with the experimentally determined ground- and excited-state interband transition energies of ${\text{GaN}}_x{\text{As}}_{1-x}$ quantum wells of different well widths and $N$ composition $x$ obtained at $300\mathrm{K}$ and under hydrostatic pressures up to $2.0\text{GPa}$. We show that the transition energies can be described by a consistent set of material parameters in all the samples studied, and present how the conduction to valence-band offset ratio varies strongly with $x$ in ${\text{GaN}}_x{\text{As}}_{1-x}∕\text{GaAs}$ quantum well structures. We conclude that the model presented can be used to predict the transition energies and electron subband structure of any ${\text{GaN}}_x{\text{As}}_{1-x}∕\text{GaAs}$ quantum well with well width between 2 and $25\text{nm}$, and $N$ composition $x$ between 1 and $4\%$, although further work is still required to confirm the optimum choice for the variation of band offset ratio with composition.

Figure 1

(Color online) Well width dependence of the transition energies of ${\text{GaN}}_{0.02}{\text{As}}_{0.98}∕\text{GaAs}$ quantum wells calculated by ten-band $\mathbf{k}·\mathbf{p}$ Hamiltonian (dashed line), two-band BAC Hamiltonian (open circles) and Eq. (10) (solid line).

Figure 2

(Color online) $N$ dependence of the squared amplitude of the conduction-band character $\mid {\alpha }_{c-}{\mid }^2$ and of the $N$ character $\mid {\alpha }_{N-}{\mid }^2$ at the band edge $(k_z=0)$, of the mixed state $E_{-}$ of bulk ${\text{GaN}}_{0.02}{\text{As}}_{0.98}$, at room and at low temperature, for unstrained structures, and for layers biaxially strained to the GaAs lattice constant.

Figure 3

(Color online) Conduction-band character ${\mid f_c\left(z\right)⟩}^2$ (solid line) and nitrogen-band character ${\mid f_N\left(z\right)⟩}^2$ (dashed line) of the first electron state in the conduction-band of a $7\text{nm}$ wide ${\text{GaAs}}_{0.98}{\mathrm{N}}_{0.02}∕\text{GaAs}$ quantum well.

Figure 4

(Color online) Series of photomodulated reflectance spectra of a $8\text{nm}$ and a $4\text{nm}$ ${\text{GaN}}_{0.018}{\text{As}}_{0.982}∕\text{GaAs}$ QW obtained under hydrostatic pressure at $300\mathrm{K}$. The solid curves are fits to the experimental data (open circles). The straight lines are a guide to the eye and indicate the pressure shifts of the $eihhi$ QW transitions and the GaAs barrier.

Figure 5

(Color online) Comparison of transition energies in ${\text{GaN}}_{0.018}{\text{As}}_{0.982}∕\text{GaAs}$ QW’s of different width extracted from photomodulated reflectance spectra with those calculated using the ten-band $\mathbf{k}·\mathbf{p}$ model (dashed lines) and the analytical model (solid lines). Ambient pressure and $T=300\mathrm{K}$.

Figure 6

(Color online) Comparison of the hydrostatic pressure dependence of the transit-ion energies of $9\text{nm}$ ${\text{GaN}}_{0.0133}{\text{As}}_{0.9887}∕\text{GaAs}$, $8\text{nm}$ ${\text{GaN}}_{0.018}{\text{As}}_{0.982}∕\text{GaAs}$, and $7\text{nm}$ ${\text{GaN}}_{0.024}{\text{As}}_{0.976}∕\text{GaAs}$ QWs extracted from photomodulated reflectance spectra with those calculated in the framework of the ten-band $\mathbf{k}·\mathbf{p}$ model (dashed lines) and the analytical model (solid lines) all at $T=300\mathrm{K}$.

Figure 7

(Color online) Calculated variation of the conduction, $\Delta E_c$, heavy-hole $\Delta E_{\text{hh}}$ and light-hole $\Delta E_{\text{lh}}$ band offsets as well as nitrogen resonant level $E_N$ and conduction-band edge of host material $E_c$, as a function of $N$ composition, $x$, for a ${\text{GaN}}_x{\text{As}}_{1-x}$ layer grown pseudomorphically on GaAs. Solid lines present band offsets obtained numerically using parameters from Tables , while the open triangles are obtained using the analytical expressions of Eqs. (25, 26, 27, 28, 29). Inset: dashed lines present the band offset ratios $\Delta E_{\text{hh}}∕\Delta E_g$ and $\Delta E_{\text{lh}}∕\Delta E_g$.

Figure 8

(Color online) Calculated variation of the perpendicular and in-plane bulk conduction-band edge masses as a function of nitrogen concentration at low and room temperature. Dashed lines: obtained numerically using the ten-band $\mathbf{k}·\mathbf{p}$ Hamiltonian; solid triangles: the room temperature mass values obtained using Eq. (30) in the two-band BAC model.

Figure 9

(Color online) Calculated variation of in-plane effective mass of the CB ground and first excited QW states as a function of QW width $2L$ for $x=1\%$ and $x=2\%$ at low and room temperature. Dashed line: ten-band $\mathbf{k}·\mathbf{p}$ Hamiltonian; open circles: numerical solution of two-band BAC model; solid line: analytical solution obtained using Eq. (18).