Problem 1: Wire with a cross-sectional area (40 points) Consider a wire with a cross-sectional area as shown in the figure below and answer the...

Problem 1: Wire with a cross-sectional area (40 points)

Consider a wire with a cross-sectional area as shown in the figure below and answer the

questions that follow.

a) On a log-log scale plot the number of conduction channels or modes (M) of a wire with a

square cross-section versus width (W) for widths between 1 nm to 1 μm. Assume a Fermi

energy of 1 eV and a temperature of 300 K. Show that for large widths M becomes linearly

proportional to cross-sectional area.

b) Plot conductivity versus length for lengths between 100 nm to 100 μm for widths of 2 nm,

20 nm, and 100 nm for two mean free paths, ???? of 40 nm and 1 μm. The Fermi energy and

temperature are the same as those in part (a).

Hint: Conductivity is simply defined as ???? = $

%&, where ???? is the length, ???? is the cross-sectional area,

and ???? is the resistance. Use ???? = +,../0

1 1 + $

4 and M is the number of modes.

Problem 2: Density of states in a general 1D system (10 points)

Find the density of states in a quantum wire as a function of electron velocity without making

any assumption for the relationship between E and k.

Problem 3: Density of states in a 1D nanotube (20 points)

Consider that the energy-dispersion relationship of an arbitrary 1D nanotube is given as

???? = ℎ????8 ????:

, +

2????????

????

,

where ????@ = 8×10D m/s, h is Planck’s constant, D is the tube diameter, m is an integer.

Assume m can start from zero, which means the first subband corresponds to m = 0. Plot density

of states in this tube as a function of energy for E < 2 eV for diameters of 1 nm and 5 nm. Hint:

you may use the results of the problem 2.

Problem 4: Using experimental data to find an average mean free path for backscattering in

Landauer transport theory (15 points)

To find ????EFGthat can be used to obtain transmission ???? = 4IJK

4IJKL$ in Landauer transport theory,

two experimental measurements are often used: (i) Diffusive conductivity, ????M and (ii) Carrier

concentration, ????O.

Consider a 2D semiconductor material with parabolic energy-dispersion relationship. Derive ????EFG

in terms of ????M and ????O in the semiconductor under (i) degenerate conditions (FD statistics) and (ii)

non-degenerate conditions (Boltzmann statistics).

Problem 5: Diffusive mobility in 2D system (15 points)

Consider a perfectly diffusive 2D conductor with parabolic energy-dispersion relationship. For

this semiconductor, the mean-free-path of electrons in a sub-band is given per the following

power-law relationship:

???? ???? = ????P

???? − ????R

S

????????

T

,

????R

S = ????RP + ????OUV,

????OUV = Sub − band energy,

????P = energy − independent constant, ???? = characteristic exponent.

a. Derive an expression for the diffusive mobility, μdiff, using Landauer’s transmission

theory under FD statistics.

b. From the general result obtained in (a) above, find the limiting diffusive mobility under

(i) non-degenerate conditions and (ii) strongly degenerate conditions.

c. Does the diffusive mobility increase or decrease with carrier concentration?

Hint: The general relationship between diffusive conductance, diffusive conductivity, and

diffusive mobility is given as

???? ???????????????????????????????????? = ????M,qr88

????

????

????M,qr88 = ????????O????qr88

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