Intersubband infrared absorption spectra of $\mathrm{Si}/{\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ quantum wells grown in the [110] direction

Matrix elements of the $6\times 6$ $\mathbf{k}\mathbf{\cdot }\mathbf{p}$ Hamiltonian, $H_{\mathbf{k}\mathbf{\cdot }\mathbf{p}},$ suitable for describing quantum wells grown in the [110] crystallographic orientation, are introduced. The $\mathbf{k}\mathbf{\cdot }\mathbf{p}$ Hamiltonian is combined with a strain $H_{\mathrm{st}},$ and a spin-orbit $H_{\mathrm{so}}$ Hamiltonian to yield a total Hamiltonian ${H=H}_{\mathbf{k}\mathbf{\cdot }\mathbf{p}}{+H}_{\mathrm{st}}{+H}_{\mathrm{so}}$ that describes the bulk barrier and well regions of a single-quantum-well heterostructure. Eigenfunctions and eigenvalues of the total Hamiltonian are used together with the envelope-function approximation to calculate quantum-well dispersion relations and momentum matrix elements for B-doped $\mathrm{Si}/{\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ quantum wells grown in the [110] direction. The momentum matrix elements are used together with a simple Lorentzian model for line broadening and an exchange energy correction to calculate the intersubband absorption spectrum. The results are in reasonable agreement with experimental observation, and confirm that quantum wells grown in the [110] direction exhibit strong normal incidence absorption.