PH 671 - Spring 2016, Due 5pm on Friday, Week 3

Under construction

Show that the number of occupied 1d subbands in a metal wire is approximately equal to the number of atoms in the cross-section of the wire. Assume a free-electron Sommerfeld model when deciding which states will be occupied.

Show that the conductance of a ballistic (no scattering), one-dimensional wire is independent of temperature when the wire has sufficient charge carriers (μ and μ + eV » k_BT). Your answer should include a calculation of density of states D(E) in a one-dimensional system. Note that he Fermi-Dirac function f(E) can used to determine the occupancy of quantum state in the left and right reservoirs (chemical potentials μ + eV and μ respectively).

a) Starting from the expression for probability of transmission through a 1d channel with two inelastic scattering sites,

$$T = {{{{\left| {{t_1}} \right|}^2}{{\left| {{t_2}} \right|}^2}} \over {1 - {{\left| {{r_1}} \right|}^2}{{\left| {{r_2}} \right|}^2}}}$$

use the Landauer formula to show that the resistance of this 1d channel is $$R = {h \over {2{e^2}}}\left( {1 + {{{{\left| {{r_1}} \right|}^2}} \over {{{\left| {{t_1}} \right|}^2}}} + {{{{\left| {{r_2}} \right|}^2}} \over {{{\left| {{t_2}} \right|}^2}}}} \right)$$

b) Consider a 1d wire where transport is characterized by a inelastic scattering length ${L \over {{\ell _{ph}}}}$. Interpret ${\ell _{ph}}$ as the spacing between scattering events, and each scattering event having a 50% transmission probability. Assuming the length of the 1d wire L is much greater than ${\ell _{ph}}$ show that

$$R = {h \over {2{e^2}}}{L \over {{\ell _{ph}}}}$$

c) Show that the the above result is equivalent to the Drude formula, within a factor of 2, if we assume a free electron dispersion relation $E = {{{\hbar ^2}{k^2}} \over {2m}}$.

$${\rho _{1d}} = {m \over {{n_{1d}}{e^2}{\tau _{e - ph}}}}$$

${\tau _{e - ph}}$ is the time between phonon scattering events. ρ_1d is one-dimensional resistivity (units Ωm^-1).

This question explores the “60 mV/decade limit” for the subthreshold slope of field-effect transistors.

To reduce the power consumption of microprocessors, transistors should switch off completely (infinite resistance) by application of a minimal voltage. This is not possible at room temperature. For a standard transistor design, it takes at least 60 mV of gate voltage to increase the channel resistance by a factor of 10 (factor 10 = one decade).

Set up an integral to calculate the number of free electrons in silicon when the chemical potential is 200 meV below the conduction band edge. Do the same thing when the chemical potential is 260 meV below the conduction band edge. What is the ratio of electron densities? (work out the number, not just an expression).

• Note: On page 572 of A&M is a section called “Number of carriers in thermal equilibrium”.
• Factoid: The 60 mV/decade limit was beaten in 2004 by a team at IBM (pdf) using a new type of tunnel diode.

Write a one-paragraph summary (4 or 5 sentences) about an experimental or theoretical solid state physics paper from 2010 or later that contains one or more of the following:

• Measurement of current (or resistance) in a nanoscale system
• Measurements of current (or resistance) in a magnetic field
• Measurement of current (or resistance) near a phase transition (for example, normal metal to superconductor).
• An STM spectroscopy measurement

Include full bibliographic information (journal name, volume number, page number, article title). Please limit yourself to the following journals (this list can be augmented with class consensus; and you may need to request an interlibrary loan to access some):

• Science
• Nature
• Proceedings of the National Academy of Sciences (PNAS)
• Nature Physics
• Physical Review Letters
• Nano Letters
• Applied Physics Letters
• Physical Review X