Valley degeneracy in biaxially strained aluminum arsenide quantum wells

This paper describes a complete analytical formalism for calculating electron subband energy and degeneracy in strained multivalley quantum wells grown along any orientation with explicit results for AlAs quantum wells (QWs). In analogy to the spin index, the valley degree of freedom is justified as a pseudospin index due to the vanishing intervalley exchange integral. A standardized coordinate transformation matrix is defined to transform between the conventional-cubic-cell basis and the QW transport basis whereby effective mass tensors, valley vectors, strain matrices, anisotropic strain ratios, piezoelectric fields, and scattering vectors are all defined in their respective bases. The specific cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) QWs are examined, as is the unconventional (411) facet, which is of particular importance in AlAs literature. Calculations of electron confinement and strain for the (001), (110), and (411) facets determine the critical well width for crossover from double- to single-valley degeneracy in each system. The biaxial Poisson ratio is calculated for the high-symmetry lower Miller index (001)-, (110)-, and (111)-oriented QWs. An additional shear-strain component arises in the higher Miller index (411)-oriented QWs and we define and solve for a shear-to-biaxial strain ratio. The notation is generalized to address non-Miller-indexed planes so that miscut substrates can also be treated, and the treatment can be adapted to other multivalley biaxially strained systems. To help classify anisotropic intervalley scattering, a valley scattering primitive unit cell is defined in momentum space, which allows one to distinguish purely in-plane momentum scattering events from those that require an out-of-plane momentum component.

Figure 1

(Left) Real-space depiction of AlAs QW structures defining the transport basis axes: $a,b,\text{and}c$. (Right) Brillouin zone for bulk AlAs defining the conventional-cubic-cell basis $x,y,\text{and}z$. Note the three degenerate valleys that are occupied with electrons as indicated by ellipsoidal equienergy contours.

Figure 2

Depiction of the valley degeneracies for various QW orientations. Each row is labeled with its Miller index. Left column: 3D representation of the X valleys showing both the CCC $k_x,k_y,k_z$ and the transport basis $k_a,k_b,k_c$. In all diagrams, the double-degenerate $X_{x,y}$ valleys are red and the singly degenerate $X_z$ valley is blue. For the (111)-oriented AlAs QW, the three degenerate $X_{x,y,z}$ valleys are purple. We note that the red and purple valleys exhibit robust valley-degeneracy condition whilst the intersection point of the red and blue traces in the rightmost column identifies the crossover-degeneracy condition. The purple valleys satisfy the robust valley-degeneracy condition as well as the crossover-degeneracy condition trivially for all well widths. Center column: 2D projection and 3D representation of the valley-scattering unit cell described in the text. Right column: Valley-degenerate ground energy as a function of QW width for a QW barrier of Al${}_{0.45}$Ga${}_{0.55}$As at T = 4 K. Dashed lines represent the calculated strain-energy shift of the respective electron valley relative to unstrained AlAs. The results of the heterostructure simulation software nextnano are depicted with scatter plots and verified with the analytical calculations shown as continuous lines. The (111) and (411) calculations assume that the piezoelectric field has been canceled by a top gate to result in a flat QW.

Figure 3

Illustration of shear strain for the (411)-oriented QWs due to the nonzero strain-to-shear ratio $D_1^{411}$. Parameters $\Delta a$, $\Delta b$, and $\Delta c$ represent strain displacements along the respective vectors of the transport $\mathbf{a}$-basis, and $\mathit{\alpha }$ represents the shear vector in the plane of the QW, which for the (411)-oriented QWs is parallel to the $\mathbf{b}$ axis.

Figure 4

3D representation of the valley-scattering cell for (111)AlAs as well as 2D lattice of such cells. In AlAs, the valleys are located at the $\mathbf{q}=(2\pi /a)(\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}})$. The standard depiction of 2D Brillouin zone (small shaded hexagon)[21] projects together three identical but laterally and vertically displaced planar hexagonal lattices, translated according to the representative hexagonal cells A, B, and C. The area of standard 2D Brillouin zone is one third that of the valley-scattering cell since its height is three times that of the valley-scattering cell. Arrows illustrate the planar intervalley-scattering vectors ${\mathbf{Q}}^{\tau \overline{\tau }}$ for (111)AlAs.