Page 102 - Principles and Applications of NanoMEMS Physics

90 Chapter 3

unit volume, v is the velocity of sound, and l is the phonon mean free path,
p
i.e., the typical device dimension L >> l . At nanoscale dimensions,
p
however, L < l and the phonons propagate ballistically. In this case, theory
p
developed by Rego and Kirczenow [127], and experiments performed by
Schwab, Henriksen, Worlock, and Roukes [128], have shown that the
thermal conductance between isolated right and left temperature reservoirs,
which are only interconnected through the device, is given by Landauer’s
theory as,

N
′
½
α
¸ ωζ
κ = 1 ­ N α ∞ d ω= ω §n R ()− ωω n L ()· α ()+ ¦³ ∞ d ω= ω §n R ()− ωω n L ()· ¸ζ ′ α () ω , (29)
®¦³
¨
¨
¾
π 2 ¯ α 0 © ∆T ¹ α ′ ′ α ω () © ∆ T ¹ ¿
0
where ω α () k and ζ α () k are the frequency and phonon transmission
probability of normal mode α , respectively, and () (en ω = = ω k B T i − ) 1 − 1
i
represents the thermal distribution of phonons in reservoir with temperature
T . While, it has been demonstrated in the works of Angelescu, Cross, and
i
Roukes [129], and of Rego and Kirczenow [127], that the transmission
probability is sensitive to the geometrical features of the nanoscopic systems,
in particular, to phonon scattering due to surface roughness and transitions
(non-adiabatic mode coupling), the main conclusion from (29 ) was that at
low temperatures heat transport is mediated by a universal constant, namely,
the quantum of thermal conductance due to phonons, k π 2 h 3 [128]. This
2
B
has serious implications pertaining to the maximum rate at which power can
be dissipated in NanoMEMS, and indeed nanoscale thermal transport is a
very active area of current research [130].

3.1.4.2 Fermi Liquids and Lüttinger Liquids

As suggested at the beginning of this chapter, transmission lines (TLs) are
ubiquitous in circuits and systems at all length scales. Since TLs should
simply transfer or guide signals from one location to another, without
decreasing their amplitude or power, it is imperative that they exhibit the
lowest possible loss. This is the reason why metals, due to their lowest
resistivity, are preferably utilized to implement interconnects (TLs).
The resistivity of conventional (large-dimension) TLs reflects the
dimensionality of electron motion. For instance, in TLs of rectangular cross-
sectional area A, as dimensions shrink electron motion may become
quantized in certain directions, thus giving rise the to the creation of energy