Page 112 - Principles and Applications of NanoMEMS Physics

100 Chapter 3

1 2π 2 ε ω ) εd ′ ∞ δ (ε − ω − ε + ε ′′ ) εd ′′
′
τ ε ~ = V ³ 0 D (E F ) ω Dd ³ 0 (E F − ³ ∞ ) (ED F . (61)
π 2
~ V D 3 (E F )ε 2
=

This results suggests that, the smaller the quasi-particle (excitation) energy
ε
0
ε , the longer will its lifetime be, in particular, as → , the lifetime tends
to infinity. An interesting result that relates the validity of the quasiparticle
concept to the dimensionality d of the system was derived by Schofield
[134], by making a change of variables to express (61) in terms of the
momentum and energy transferred. His result was the expression,

1 = 2π ε ) dωω 2 k F q d −1 dq D (q,ω ) 2
τ ε = ³ D (E F ³ (2π L ) ( v= F ) q 2 . (62)
d
ω
0
F v =
The integral (62) is interpreted as follows [136]: 1) The integral over ω
accounts for the number of possible hole excitations that can be created; 2)
The lower limit of the momentum integral, over q, signifies that a minimum
momentum must be transferred to give a change in energy of ω ; 3) The
2
denominator ( v= ) q in the integrand embodies the fact of already having
F
performed integration over the direction of the momentum and it reflects that
there is an increased time available for small deflections; 4) The numerator,
D ( ω ,q ) is the matrix element for the scattering process. Examination of the
impact of setting the dimension to d = 1 reveals that, if one assumes
D ( ω ,q ) to be constant, then due to the singularity of the q integral, the
projected quasiparticle lifetime τ , is not much greater than, but in fact is it
ε
close to, = ε . Therefore, (62) is violated as the quasiparticle, in principle,
can never have enough time to form. The importance of this result is that
Fermi liquid theory breaks down when applied to one-dimensional metallic
systems, such as are typical at nanoscales. The new situation is described by
the concept of the Lüttinger liquid.

3.1.4.2.3 Lüttinger Liquids

The term Lüttinger liquid is used to denote the behavior of interacting
electrons confined to one-dimensional transport [137]. Such behavior is