Page 111 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 99

function of the interaction parameter V and turn-on time τ is given by
V
[135],
H = H + Ve t V τ / , (58)
0

then the time it takes a quasiparticle of excitation energy ε to decay, τ , must be

ε
much greater than the interaction turn-on time, τ , and also much greater
V
than the time it takes the quasiparticle of to absorb the excitation energy,
given by Heisenberg’s uncertainty principle ε = form,

τ ε >> τ V >> = . (59)
ε

Obviously, at large excitation energies ∆E = ε , the associated time during
which this energy is absorbed = ε may become much smaller than the
lifetime τ , which means that no quasiparticle has a chance to form and,
ε
thus, the quasiparticle concept breaks down. An estimate of this lifetime is
given in [134] by calculating the decay rate of a quasi-particle with energy
ε above the Fermi energy E , at absolute zero. Using Fermi's golden rule,
F
which describes the transition between initial states i and final states f
elicited by a scattering potential V ,
if

1 = 2 π 2 δ ( − εε ) , (60)
τ ε = ¦ V if f
f

assuming V is constant and enforcing conservation of energy and Pauli
if
exclusion principles, see Figure 3-10, one obtains,
loses ω
loses ω
loses ω
ε ε ε
gains ω
gains ω
k,ε’
k,ε’
k,ε’ gains ω
k+q,ε”
k+q,ε” quasi-particle
k+q,ε”
quasi-particle
quasi-particle
q,ω
q,ω
q,ω
ε ε’ ε’ ’
p-q,ε-ω
p,ε
p-q,ε-ω
p,ε p-q,ε-ω
p,ε
filled Fermi sea
filled Fermi sea
filled Fermi sea
Figure 3-10. Energy relationship of quasi-particle scattering process. The energyω lost in a
scattering event by the quasi-particle must be lower than its initial energy ε , and there must
be an electron state at an energy ε ′ capable of absorbing at this energy ω .