3.3 Two-Band k ⋅ p Hamiltonian of [001] Valley at the X-Point

As the lowest two conduction bands Δ₁ and Δ_2′ (c.f. Figure 3.3) have their minima just k₀ (c.f. Table 3.2) away from the X-point in the Brillouin zone, a two-band perturbation theory considering only these two bands developed near the X-point describes the band dispersion and subband wave functions very well [72]. The two-band k ⋅ p Hamiltonian accurately describes the bulk structure up to energies of 0.5-0.8eV [72]. However, this approach is in contrast to [144, 145] where the model has been developed around the Γ-symmetry point which is far away from the conduction band minimum and therefore requires a significant increase in considered bands.

The two-band k ⋅ p Hamiltonian of a [001] valley in the vicinity of the X-point of the Brillouin zone along the quantization OZ-axis including the shear strain (εxy) must be of the form [72, 146]

(3.1)

where k_i with i ∈ x,y,z are the projections of the wave vector on the coordinate axes, Ũ(z) is the confinement potential, and εxy is the shear strain in [110] direction. Ũ(z) arises as the UTB silicon film is supposed to be sandwiched between two oxide layers. The diagonal terms of the (2x2) Hamiltonian correspond to the Hamiltonian of the individual bands, and the off-diagonal term signifies the coupling between those two [72].

Hamiltonian including the Spin Degree of Freedom

The corresponding k ⋅ p Hamiltonian including the spin degree of freedom considering only the relevant [001] oriented valleys written in the vicinity of the X-point along the OZ-axis in the Brillouin zone can also be derived from Equation 3.1 by introducing the intrinsic spin-orbit term △SO [71, 129]. △SO couples the states with opposite spin projections to their respective opposite valleys. The basis is conveniently chosen as [(X₁, ↑) , (X₁, ↓) , (X_2^′, ↑) , (X_2^′, ↓)], where ↑ and ↓ indicate the spin projection at the quantization OZ-axis, X₁ and X_2^′ are the basis functions corresponding to the two valleys. Thus, the effective (4x4) Hamiltonian with the spin degree of freedom reads [129, 147]

(3.2)

where H₁, H₂, and H₃ are written as,

(3.3)

(3.4)

The spin-orbit field (SOF) acts along (k_x, -k_y) direction. For a zero value of the confinement potential Ũ(z) the energy dispersion of the lowest conduction bands is given by [129]

(3.5)

where

(3.6)

The ± sign signifies the two bands. This expression generalizes the corresponding dispersion relation from [71] by including shear strain.

In order to evaluate the effective spin-orbit interaction △SO term one can use the dispersion relation Equation 3.5. If one evaluates the dispersion for k_x≠0 but k_y=k_z=0, the gap between the lowest two conduction bands can be opened by △SO alone in an unstrained sample. The band splitting along the OX-axis is then equal to 2|△SOk_x| and thus linearly related to k_x. This splitting can also be evaluated numerically by the empirical pseudopotential method, and thereby one can obtain the value for △SO using the linear fitting technique as described in [129]. △SO is reported to be 1.27meVnm.