Determination of band offsets in strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ quantum wells by capacitance-voltage profiling and Schrödinger-Poisson self-consistent simulation

The results of capacitance-voltage profiling and numerical simulation of charge carrier distribution and energy states for strained quantum wells ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ $\left(0.06⩽x⩽0.29\right)$ are presented. Precise values of conduction band offsets for these pseudomorphic QWs have been obtained by means of self-consistent solution of Schrödinger and Poisson equations and following fitting to experimental data. For the conduction band offsets in strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$-QWs the expression $\Delta E_C\left(x\right)=0.85x-0.3x^2$ has been obtained.

Figure 1

The layer sequence of the grown samples with ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ quantum wells.

Figure 2

$C\text{−}V$-characteristics of ${\mathrm{In}}_{0.225}{\mathrm{Ga}}_{0.775}\mathrm{As}∕\mathrm{GaAs}\text{−}\mathrm{QW}$ at different temperatures (sample #307).

Figure 3

$C\text{−}V$-profiling of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ quantum wells. Samples #296–#309. (a) Common view of apparent carrier concentrations for several samples; (b) apparent concentration peak values as a function of $x$. Note that near $x=0.14$ there are 4 samples with different width of QW.

Figure 4

Scheme of computations. The “quantum box” is unshaded. 1, Schrödinger’s concentration; 2, concentration derived from Eq. (7); 3, resulting calculated “apparent” carrier concentration (matched the experiment).

Figure 5

Electron concentration in first 16 energy subbands (relatively to the concentration in the first subband) at different reverse biases. Sample #300 $\left(\Delta E_C=175\mathrm{meV}\right)$, $T=300\mathrm{K}$.

Figure 6

Results of numerical calculations for sample #300 ($x=0.23$; $\text{well}\text{width}=7.2\mathrm{nm}$; offset $\Delta E_C=175\mathrm{meV}$); $T=300\mathrm{K}$: (a) first 8 energy levels as function of $U_{\mathrm{rev}}$; (b) the conduction band bottom near the QW at different $U_{\mathrm{rev}}$.

Figure 7

Simulated concentration profiles of electrons in the region of QW at $U=-1\mathrm{V}$ (solid) and $-2.5\mathrm{V}$ (dashed line), $T=300\mathrm{K}$. Sample #300.

Figure 8

Experimental (dots) and fitted (solid) apparent concentration profiles of ${\mathrm{In}}_{0.23}{\mathrm{Ga}}_{0.77}\mathrm{As}∕\mathrm{GaAs}$ quantum well (sample #300). Best fit with $\Delta E_C=175\mathrm{meV}$.

Figure 9

Resolution of the fitting. The results for the best fit ($\Delta E_C=120\mathrm{meV}$, solid line) and for $\Delta E_C=130\mathrm{meV}$ (dashed line) are presented. Dotted curve, the experimental apparent profile for sample #303 $\left(x=0.145\right)$.

Figure 10

Apparent concentration profile of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ quantum well with low In-content $x_{\mathrm{In}}=0.065$ (dotted) and fitted curve (solid) with $\Delta {\mathrm{E}}_C=55\mathrm{meV}$ (sample #298).

Figure 11

Position of bound energy level $E_1$ in strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$-QWs (triangles up for $w⩽7.2\mathrm{nm}$ and triangles down for $w>7.2\mathrm{nm}$). Solid line, calculated dependence $E_1$ of $x_{\mathrm{In}}$ in the assumption $w=7.2\mathrm{nm}$ and for the impurity doping as in the sample #300. Dashed-dotted line, the corresponding position of the Fermi level.

Figure 12

Conduction band offsets in strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$ QWs as function of composition. Errors bars correspond to the 95% confidence interval. Dashed line, as recommended in Ref. 9. Open circles, the last decade results of capacitance and optical investigations on strained ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{GaAs}$-QWs: 1, Ref. 38; 2, Ref. 14; 3, Ref. 19; 4, Ref. 18; 5, Ref. 13; 6, Ref. 11.