1 Material Model 4 Single Electron Band Structure in Strained Layers

2 Single Electron in Strained Layer Structures Strain changes the semiconductor lattice structure. If we adopt r’ as the new coordinate system for the strained lattice, Schr Ö dinger’s equation takes the same form under r’: All material and structural parameters in the Hamiltonians (for both bulk semiconductors and QW structures) are still given in the original (unstrained) lattice structure under coordinate system r. Therefore, all we need to do is to project the new spatial variables (x’, y’, z’) in the equation to the original space coordinate system (x, y, z). As such, the governing equation for strained layer structures will be obtained.

3 Single Electron in Strained Layer Structures Coordinate projection from r’ to r: Dimensionless relative shift in the space domain due to the lattice deformation Since the lattice deformation must be small to avoid relaxation through defects generation:

4 Single Electron in Strained Layer Structures Or: Hence we find:

5 Single Electron in Strained Layer Structures Consequently, the Hamiltonian for the strained layer structure becomes: where: Since the extra term introduced in the Hamiltonian is a Perturbation, following the same approach in dealing with the k-p term in the L-K model, we can readily find the solution directly through a mapping from k αβ to e αβ, and from the Luttinger constants to the deformation (hydrostatic and shear) energies.

6 Single Electron in Strained Layer Structures Strained bulk semiconductor Conduction bands: Valence bands: in the same (unstrained bulk) L-K Hamiltonian form, with extra terms added to its parameters

7 Single Electron in Strained Layer Structures Strained QW structure Conduction bands: Valence bands: in the same (unstrained QW structure) L-K Hamiltonian form, with extra terms added to its parameters

8 Single Electron in Strained Layer Structures For zinc blende structure

9 Single Electron in Strained Layer Structures Effect of strain for bulk semiconductors: For compressive strain: e 0 0, Q e >0 For tensile strain: e 0 >0, P c e >0, P v e <0, Q e <0

10 E c =E g EcEc E hh =E lh =0 E hh E lh QeQe QeQe EcEc E hh QeQe QeQe Unstrained bulk semiconductor Tensile strained bulk semiconductor Compressive strained bulk semiconductor

11 Single Electron in Strained Layer Structures Effect of strain for QW structures: similar to the effect for bulk semiconductors, with more features due to the non-degeneracy of the heavy hole and light hole energies at k=0 and the band mixings. For compressive strain: HH and LH further separate apart at k=0, which reduces the mixing and bring in smaller effective mass in the neighborhood of k=0; consequently, the density of states reduces, which raises the differential gain and lowers the transparent carrier density

12 Single Electron in Strained Layer Structures For tensile strain: HH and LH get closer or even reversed at k=0; if not reversed, the tensile strain enhances the band mixing and bring in larger effective mass, which may not be desired; however, if HH and LH get reversed with LH on top, we may still obtain higher differential gain and lower transparent carrier density, due to the larger overlap between the conduction band electron and valence band light hole wave functions (i.e., larger dipole matrix element). This effect is often more significant than the effective mass reduction.

13 E c =E g EcEc E hh =0 E lh <0 E hh E lh QeQe QeQe EcEc E hh QeQe QeQe Unstrained QW structure Tensile strained QW structure Compressive strained QW structure