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Phonons are the lattice vibrations, but those can be imagined as particles which carry vibrational energy in a similar manner to photons, i.e. they are discrete and quantized [117]. The energy of a phonon is characterized by its own intrinsic frequency. In a lattice with a basis of more than one atom in the primitive cell (which may or may not be different), the allowed frequencies of a propagating wave can be split into an upper branch known as the optical branch, and a lower branch called the acoustical branch. Acoustic phonons are coherent movements of atoms of the lattice out of their equilibrium positions. In contrast for optic phonons, the center of mass of the cell during oscillations does not move [117, 128, 155] (i.e. one atom moving to the left, and its neighbour to the right). The nature of the vibrations are sketched in Figure 4.9. The acoustic branch has its name because it gives rise to long wavelength vibrations, and the speed of its propagation is the speed of a sound wave in the lattice. The optical branch is a higher energy vibration, and one can excite these modes with the electromagnetic radiation [117]. For both of acoustic and optical modes, the vibration is restricted to the direction of propagation in the longitudinal mode, whereas in the transversal mode the vibration occurs in the perpendicular planes.

In the three-dimensional lattice system the number of optical modes for the primitive cell that contains p atoms is given by the expression 3(p-1), while the number of acoustic modes is always three. Each of the modes has three components: two transversal (TA1, TA2, TO1, TO2) and one longitudinal (LA, LO).

In the analysis of intrasubband electron-phonon scattering, the explicit forms of the polarization vectors of phonons are needed. Following [156, 53] the phonons polarization vectors can be written as

(4.17)

(4.18)

(4.19)

where q = is related to the momentum transfer in the scattering [53].

The formulation for the transition rate from one energy eigenstate of a quantum system into the other energy eigenstates in a continuum is given by Fermi’s golden rule [157]. The electron-phonon mediated momentum and the spin relaxation rates, where m is the relaxation mechanism signifying contributions from the acoustic and the optical phonon for transitions from band i to band j are calculated by using Fermi’s second Golden Rule as described in [53, 157]

(4.20)

where M represents the momentum scattering (spin relaxation) matrix element, and the material volume is chosen as the unit volume. The value 4 (2) for spin (momentum) relaxation accounts for the fact that the net number of spin polarized electrons changes by two with each spin flip [53].

4.3.3 Intravalley and g-Intervalley Relaxation Processes Rates

The spin relaxation rate for the wave vector k1 in subband i can be written as [69]

(4.21)

Here, the momentum transfer vector q can be realized by q = (k2 -k1,q_z), D_α1α2 is the deformation potential, (α₁,α₂)=(x,y), ϱ_α2(q)=(ϱ_LA(q),ϱ_TA1(q),ϱ_TA2(q)) is the polarization vector.

By applying Fubini’s theorem the modulus in the above equation can be replaced by a repeated integral,

(4.22)

M^AC is the deformation potential matrix, the exact form of the matrix depends on the phonon mode and is shown later. The L_AC term depends on the spin-flip process and the phonon mode [53]. By replacing the order of the integration in Equation 4.22 the acoustic phonon mediated relaxation rate can be written as

(4.23)

Intervalley g-Process Spin Relaxation

The g-process describes the electron intervalley scattering between opposite valleys, which includes only the [001] valley pair in the Brillouin zone. The f-process involves scattering between valleys that reside on perpendicular axes, which will be treated later. For intervalley scattering [53], L_AC=1. By introducing the Dirac delta function, 2πδ(z - z^′) = ∫ _-∞^∞ exp(-iq _z|z - z^′|)dq _z. Again,

(4.24)

This simplifies Equation 4.23 to

(4.25)

Here, ν_LA=8700, M^AC contains the Elliott and Yafet contributions [53], and can be written as (M^′)

(4.26)

(4.27)

(4.28)

where (r_y,r_x)=k1 + k2, D_SO=15meV/k₀, Ξ=12eV as the acoustic deformation potential, and other parameters as in Table 3.2. Thus, the expression for M^′ can be reformulated as

(4.29)

Intravalley Transversal Acoustic Phonons Spin Relaxation

Intrasubband transitions are important for the contributions determined by the shear deformation potential. The term L_AC due to transversal acoustic phonons is [53]

(4.30)

Applying the theory of residues with Q² = q _x² + q _y², one can perform the following integral,

(4.31)

The matrix M^AC for the intrasubband transversal acoustic phonons (M) can be expressed like

(4.32)

where D is the shear deformation potential as mentioned in Table 3.2. Indeed the intrasubband transitions are important for the contributions determined by D.

Following Equation 4.23 the intrasubband transversal acoustic phonons is

(4.33)

Here, ν_TA=5300 is the transversal phonon velocity, and ρ=2329 is the silicon density [158].

Intravalley Longitudinal Acoustic Phonon Spin Relaxation

The term L_AC due to the longitudinal acoustic phonons is [53]

(4.34)

Applying the theory of residues with Q² = q _x² + q _y², one can perform the following integral,

(4.35)

The matrix M^AC for the intrasubband longitudinal acoustic phonons is the same as in Equation 4.32. Then, the intravalley spin relaxation rate due to longitudinal acoustic phonons can be expressed as

(4.36)

When both the surface roughness and the acoustic phonon mediated components are calculated, the total spin lifetime is calculated by the Matthiessen rule.