In this appendix the semiclassical approach to simulate the scattering rate of Ref. [108] in Fig. 3.2 is described. The simulation employed Boltzmann’s transport equation (BTE) and Fermi’s golden rule (FGR). Both expressions will be derived and explained in the following. It will be shown that the results of Fermi’s golden rule are only valid on time scales long compared to the time between two scattering processes.
The understanding of electric conductivity is closely linked to L. Boltzmann and his kinetic theory from 1872. The Boltzmann Equation sets the framework to describe many particle pro-cesses, e.g., in atomic gases or, that is interesting in our context, for electric currents. Under a single-particle approximation, Boltzmann described an ensemble of carriers in phase space un-ambiguously with the non-equilibrium distribution functiong(r,k,t). If all scattering processes are neglected, an externally applied electric field would modify the coordinates in phase space according to the classical equations of motion [225]:
˙
Thus electrons at(v,k,t)had at earlier times(t−dt)the coordinates(r−v(k)dt,k−Fdt/¯h).
Furthermore, electrons may have scattered out or into the phase segment of(r,k,t), which is regarded in the expression
∂g
∂t
col. The combination of both terms determine the temporal evolution of the non-equilibrium distribution function
g(r,k,t) =g(r−v(k)dt,k−Fdt/¯h,t−dt) +
Evolving the left hand side to the linear order indt, reduces Eq. (3) fordt→0 to the Boltz-mann equation:
The drift terms on the left hand side express Newton’s law and the collision term on the right hand side stands for all scattering processes of the ensemble. The validity of Boltzmann’s transport equation depends only on the description of scattering processes within the collision term. The first and most basic assumption for the scattering term stems from Paul Drude and it was surprisingly successful.
He assumed that scattering occurs. Free carriers inside a conductor are adapted as an ideal electron gas with an average time of flight between two scattering processes. For most semi-conductors this empirical relaxation time τ amounts to approximatively τSC=200 fs. The relaxation time approximation does not distinguish between electrons and holes as carriers or between scattering with phonons, electrons or impurities. Just the simple change of the distri-bution function g(k)in scattering events is stated [Eq. (5)]. However, the results are relative accurate and, e.g., ohmic currents, are well-explained by this model.
dg(k) The next refinement [Eq. (6)] considers already the fermionic character of electrons. Pauli blocking prevents scattering fromk0 into an occupiedkposition and vice versa. The transition probabilitiesPk,k0 andPk0,kfor such a scattering event are calculated with Fermi’s golden rule derived in the next section. Deduced from quantum mechanical perturbation theory, it regards the actual microscopical situation including wave function overlap and density of states.
The Validity of Fermi’s Golden Rule
The expression was derived originally 1927 by G. Wentzel from the quantum mechanical per-turbation theory [226]. It describes the transition probability between eigenstates caused by an external perturbation potential V(t). V is assumed to be small compared to the unperturbed Hamilton operatorH0. It is switched on at a timet0and remains constant afterwards. Using the Heaviside functionΘ(t), the perturbation potential is written asV(t) =VΘ(t). This situation is described fort≥t0in the Schrödinger equation
i¯h∂
∂t |ψ,ti= [H0+V(t)]|ψ,ti. (7) The Dirac formalism separates out the known part ofH0and the remaining perturbation term is evolved into a Neumann sequence. Its first order termPmnexpresses the transition probability from one energy eigenstate|miinto another orthogonal eigenstate|ni:
Pmn=
nm/2 from Eq. (10) is equivalent to the Dirac functionδ on long time scales. We substitute the expression with
δt(α) =sin2αt
π α2t . (11)
Its functional characteristics can be classified as follows
δt(α)
t/π forα=0
≤1/π α2t forα6=0 (12)
Integration over t of Eq. (11) yields for any functionF(α)fort→∞
t→∞lim Z ∞
−∞
dα δt(α)F(α) =F(0). (13) This is exactly the behavior of aδ function. That means only on long time scalest→∞the equation holds
tlim→∞δt(α) =δ(α). (14)
This substitution limits the validity of Fermi’s Golden Rule only to long times scales. Insert-ing Eq. (14) into Eq. (10), yields for the transition probabilityPmn
Pmn(t) =t2π
h¯ δ(En−Em)|hn|V|mi|2. (15) It results in a transition rate, i.e., the transition probability per time, of
Γmn=2π
¯
h δ(En−Em)|hn|V |mi|2. (16) Electronic states within a periodic semiconductor form continuous valence and conduction bands. The electron lifetime of the conduction band eigenstate|mi, for one distinct momentum kand one distinct energyE, needs to be calculated. Scattering into all other conduction band eigenstates|niis allowed, if energy and momentum is conserved. The number of possible states to scatter into is given by the density of statesρ(E). In this case the dependence ofρ on the momenta can be ignored, because LO phonons have nearly the same energy for allqmomenta (see Fig. 3.9). Since the energy is conserved one LO phonon will be addressed, which fulfils the momentum conservation as well. Thus we obtain for the transition rate out of the eigenstate
|mi
∑
nΓmn=ρ(En)2π
¯
h |hn|V |mi|2. (17)
Most important in our context is the validity range of Fermi’s Golden Rule. It is illustrated in Fig. 1. The expression for the transition probability in Eq. (15) is based on the approximation of sinω2ωnmt/2
nm/2 2
in Eq. (10) as aδ function. This approximation is only valid for energetic widths 2πh/t¯ much smaller than the width of the energy distribution of the final states∆E[113]. The second condition requires a sufficient number of final states within theδ-like function. Trans-lated into the time domain, the following condition must hold:
2πh¯
∆E t 2πh¯
δ ε (18)
For continuous energy bands of semiconductors (δ ε ≈0) the r.h.s. of the inequation 18 is certainly fulfilled. To clarify the validity of Fermi’s Golden Rule on short times, we consider Heisenberg’s uncertainty relation∆E∆t≥h. The minimal energetic width of the transition (∆E)
Figure 1: The approximation of δt ≈δ remains valid if ∆E is much larger and δ ε is much smaller thanδt, i.e., for 2π∆Eh¯ t 2π¯h
δ ε. This chart is taken from Ref. [113].
is proportional to the inverse interaction time (∆t−1, given by phonon energy to scatter with).
Inserting Heisenberg’s uncertainty relation into Eq. (18) yields
∆t t. (19)
Thus Fermi’s golden rule applies only to scattering timestlong compared to the interaction time∆t, i.e., inverse oscillation period of the phonon to scatter with.