ME381R Homework #2 (Due 10/7 before class. Late homework will not be accepted):
1. (i) Using a log-log scale, sketch the specific heat C, phonon mean free path l, and
thermal conductivity K of an one-dimensional semiconductor nanowire as a
function of temperature in the range of 4-400 K. Specify the slope of the thermal
conductivity in the log-log plot.
(ii) In the same plots, sketch the corresponding bulk properties (C, l, K) assuming
that the bulk material is three dimensional. Specify the slope of the thermal
conductivity in the log-log plot.
(iii) Which material has a higher thermal conductivity, the nanowire or the bulk
one? Why?
(iv) Does the maximum thermal conductivity of the nanowire occur at a higher or
lower temperature than that of the bulk counterpart? Why?
(v) In the bulk material and at room temperature, Umklapp phonon-phonon
scattering is found to be the dominant scattering mechanism and its mean free
path is  = 100 nm. The bulk thermal conductivity is measured to be 150 W/m-K.
Ignore the fact that the nanowire and the bulk material can have different phonon
dispersion and assume that they have the same specific heat and phonon group
velocity, (roughly) estimate the thermal conductivity of a 10-nm-diameter
nanowire.
2. Derive the specific heat of optical phonons using the Einstein model of density of
states, i.e., D() = N(-0), where N is the number of primitive cells, and (0) is the delta function centered at 0.
3. The Wiedemann-Franz law states that for metals at not too low temperatures the
ratio of the thermal conductivity to the electrical conductivity is directly
proportional to the temperature, with the value of the constant of proportionality
independent of the particular metal. This is expressed as K/ = LT, where K, ,
and T are thermal conductivity, electrical conductivity, and temperature, and L is
a constant called the Lorenz number. Prove the Wiedemann-Franz law and derive
an expression for L. Note that electrons dominate over phonons for heat
conduction in a metal, and the electrical conductivity is a function of electron
density, charge, mean free time, and mass.
4. Using the parameter in Table 1-10, estimate the electron heat capacity of Al as a
function of temperature. Compare its value with the actual heat capacity at room
temperature. Are they very different and if so why? From the experimental value
of thermal conductivity of Al, calculate the electron mean free path and time at
room temperature assuming that electrons dominate heat conduction in Al. From
the experimental value of electrical conductivity of Al, calculate the electron
mean free path and time at room temperature, and compare the results with those
estimated from the thermal conductivity.