Band alignments and strain effects in $\text{PbTe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Te}$ and $\text{PbSe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Se}$ quantum-well heterostructures

The optical transitions and band alignments of $\text{PbTe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Te}$ and $\text{PbSe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Se}$ multi-quantum-well heterostructures were studied using temperature-modulated absorption spectroscopy and envelope function calculations. By taking advantage of the strain-induced splitting of conduction- and valence-band states at the different $L$ points of the Brillouin zone, a reliable determination of the band offsets is obtained, taking into account the biaxial strain of the quantum wells derived by x-ray diffraction. Through this analysis, the normalized conduction-band offsets are determined as $\Delta E_c/\Delta E_g=0.45\pm 0.05$ for the $\text{PbTe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Te}$ system and $0.6\pm 0.1$ for the $\text{PbSe}/{\mathrm{Pb}}_{1-x}{\mathrm{Sr}}_x\text{Se}$ system for Sr contents up to 13%. Within the experimental precision, the band offsets are independent of temperatures from 20–300 K. With these parameters, precise modeling of the energy levels and optical transitions is achieved as required for optoelectronic device applications.

Figure 1

High-resolution (222) x-ray diffraction spectra of a (a) PbTe/PbSrTe and (b) PbSe/PbSrSe MQW with QW thicknesses of $d_{\mathrm{QW}}=72$ and 84 Å, respectively. The peaks from the buffer layer, substrate, and the MQW stack are labeled by $B$, $S$, SL0, and $\pm i$. (c), (d) Reciprocal space maps of the samples around the asymmetric ($15\overline{3})$ Bragg reflection, with the reciprocal space coordinates $q_x$ and $q_z$ parallel and perpendicular to the surface. The nominal position of bulk ($15\overline{3})$ PbTe and PbSe reciprocal lattice points is indicated by (). The structural parameters derived from these measurements are listed in Table 1.

Figure 2

Differential transmission spectra $\Delta \mathcal{T}/\mathcal{T}$ of three PbTe/PbSrTe MQWs with $d_{\mathrm{QW}}$ = (a) 72 Å, (b) 147 Å, and (c) 190Å measured at $T=80$ K. The peaks arising from the $i\text{−}i$ QW transitions in the longitudinal $\left(l\right)$ and oblique $\left(o\right)$ valleys are indicated by the blue and red arrows, respectively, and the shaded regions mark the energy range above the barrier band gap. (a) Inset: Orientation of the effective mass ellipsoids of the longitudinal and three oblique $L$ valleys with respect to the (111) QW plane. (f),(g) Calculated transition energies (dashed lines) in dependence of the normalized unstrained conduction-band offset $\Delta E_c/\Delta E_g$ for $d_{\mathrm{QW}}=72$ and 190 Å, respectively. The corresponding transitions are indicated by the boxes in (a) and (b). The transition energies were calculated for four different QW strain values ${\epsilon }_{\left|\right|,\mathrm{QW}}=0,0.15,0.3$, and 0.45%, where ${\epsilon }^{\exp }=0.3\%$ (dotted line) is the experimental value. The measured transition energies $E_{ii}^{\left(\exp \right)}$ are represented by the horizontal lines and the squares $\left(\blacksquare \right)$ mark the intersection with the theoretical curves. This yields the unstrained band alignment as $\Delta E_c/\Delta E_g=0.45\pm 0.05$. (d), (e) Derived strained band profiles for each valley type.

Figure 3

Differential transmission spectra $\Delta \mathcal{T}/\mathcal{T}$ of two PbSe/PbSrSe MQW samples with $d_{\mathrm{QW}}=$ (a) 84 Å and (b) 138 Å measured at $T=80$ K. The peaks arising from the $E_{ii}^{(l,o)}$ quantum-well transitions in the longitudinal $\left(l\right)$ and oblique $\left(o\right)$ valleys are indicated by the blue and red arrows, respectively, and the shaded regions mark the energy range above the barrier band gap. (e), (f) Calculated transition energies plotted in dependence of the normalized unstrained conduction band offset $\Delta E_c/\Delta E_g$ (dashed/dotted lines) for the highest QW transitions $E_{ii}^{(l,o)}$ in both samples. The lines were calculated for four different tensile QW strain values ${\epsilon }_{\left|\right|,\mathrm{QW}}=0,0.15,0.3$, and 0.45%, where ${\epsilon }^{\exp }=0.1\%$ (dotted line) is the value determined by x-ray diffraction. The measured transition energies $E_{ii}^{\left(\exp \right)}$ are represented by the horizontal solid lines and the squares $\left(\blacksquare \right)$ mark the intersection with the calculated values curves. This yields $\Delta E_c/\Delta E_g$ = $0.6\pm 0.1$ for this system. (c),(d) Derived strained band profiles for each valley type.

Figure 4

Transition energies of (a) PbTe/PbSrTe and (b) PbSe/PbSrSe MQWs as a function of quantum-well width at 80 K. Symbols and represent the measured data of our samples (see Table 1) and the solid lines represent the calculations using the envelope function model for $x_{\mathrm{Sr}}=13\%$ and the band offsets and strain values ${\epsilon }_{\left|\right|,\mathrm{QW}}=0.3$ and 0.1% derived from Figs. 1–3.

Figure 5

Transition energies of the QWs (symbols $•)$ calculated as a function of the unstrained relative band offset $\Delta E_c/\Delta E_g$ at 20 and 300 K (left- and right-hand side, respectively). (a),(b) PbTe/PbSrTe MQW with $d_{\mathrm{QW}}=72$ Å; (c),(d) PbSe/PbSrSe MQW with $d_{\mathrm{QW}}=84$ Å. The measured transition energies are represented by the solid horizontal lines and the sample parameters are listed in Table 1. The transitions are labeled by $E_{ii}^{l,o}$ where the indices $i$ denotes the $i\text{th}$ initial and final state in the QWs, and $l$ and $o$ denote the longitudinal and oblique valleys, respectively. The shaded regions mark the range of the band offset values within the error bounds, giving $\Delta E_c/\Delta E_g=0.45\pm 0.05$ and $0.6\pm 0.1$ for the PbTe/PbSrTe and PbSe/PbSrSe systems independent of temperature.

Figure 6

Measured (symbols and ) and calculated (solid lines) temperature-dependent transition energies $E_{ii}^{l,o}$ of two (a),(b) PbTe/PbSrTe and (c),(d) PbSe/PbSrSe MQWs with QW thicknesses of 72, 147, 84, and 138 Å, respectively, for a constant band offset of $\Delta E_c/\Delta E_g=0.45$ for PbTe/PbSrTe and $\Delta E_c/\Delta E_g=0.6$ for PbSe/PbSrSe, demonstrating an excellent agreement between theory and experiments at all temperatures. The shaded regions indicate the PbTe, PbSe and PbSrTe, PbSrSe band gaps for the given Sr contents $x_{\mathrm{Sr}}$ of 11.6, 14.1, 13.0, and 12.6% as listed in Table 1.