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hw3

Homework #3

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

1D Subbands (5 pts)

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.

Quantum of conductance at finite temperature (5 pts)

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 » kBT). 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).

Landauer to Drude (10 pts)

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 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. ρ1d is one-dimensional resistivity (units Ωm-1).

Field-effect transistor fundamental limits (5 pts)

This question explores the “60 mV/decade limit” for the sub-threshold 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).

  • Factoid: The 60 mV/decade limit was beaten in 2004 by a team at IBM using a new type of tunnel diode. See Appenzeller et al., PRL 93, 196805 (2004).

Journal reading (5 pts)

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

Articles

  • Monte-Carlo simulation of nano-collected current from a silicon sample containing a linear arrangement of uncapped nanocrystals , Mohammed Ledra and Abdelillah El Hdiy J. Appl. Phys. 118, 115705 (2015);
  • Picosecond photoresponse in van der Waals heterostructures, M. Massicotte et al. Nature Nanotechnology 11, 42–46 (2016) doi:10.1038/nnano.2015.227.
  • Determination of band alignment in the single-layer MoS2/WSe2 heterojunction, Ming-Hui Chiu, et al., Nature Communications 6, 7666, doi:10.1038/ncomms8666
  • Electron-Phonon Coupling and Energy Flow in a Simple Metal beyond the Two-Temperature Approximation, Lutz Waldecker, Roman Bertoni, Ralph Ernstorfer, and Jan Vorberger, PHYSICAL REVIEW X 6, 021003; (2016)
  • The Valley Hall effect in MoS2 transistors. K. F. Mak et al Science 344, pp. 1489-1492 (2014) DOI: 10.1126/science.1250140
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