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--- hw3 [2016/04/06 09:06] – [Landauer to Drude (10 pts)] janet
+++ hw3 [2016/04/06 09:31] – [Homework #3] janet
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 ====== Homework #3  ======
 PH 671 - Spring 2016, //Due 5pm on Friday, Week 3//
-**Under construction**
 ===== 1D Subbands (5 pts) =====
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 $$T = {{{{\left| {{t_1}} \right|}^2}{{\left| {{t_2}} \right|}^2}} \over {1 - {{\left| {{r_1}} \right|}^2}{{\left| {{r_2}} \right|}^2}}}$$
-use the Landauer formula to show that the resistance of this 1d channel is (oops, I'm missing a factor of 2)
+use the Landauer formula to show that the resistance of this 1d channel is
+$$R = {h \over {2{e^2}}}\left( {1 + {{{{\left| {{r_1}} \right|}^2}} \over {{{\left| {{t_1}} \right|}^2}}} + {{{{\left| {{r_2}} \right|}^2}} \over {{{\left| {{t_2}} \right|}^2}}}} \right)$$
-{{:hw2eq2.png|}}
-**b)** Consider a 1d wire where transport is characterized by a inelastic scattering length //l//_phonon. (You can interpret //l//_phonon 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 //l//_phonon show that
+**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
-{{:hw2eq3.png?180|}}
+$$R = {h \over {2{e^2}}}{L \over {{\ell _{ph}}}}$$
-**c)** Show that the the above result is equivalent to the Drude formula if we assume a free electron dispersion relation (//E// = //ħ//²//k//²/2//m//).
+**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 $E = {{{\hbar ^2}{k^2}} \over {2m}}$.
-{{:hw2eq4.png?220|}}
+$${\rho _{1d}} = {m \over {{n_{1d}}{e^2}{\tau _{e - ph}}}}$$
-within a factor of 2. τ_phonon is the time between phonon scattering events. ρ_1D is one-dimensional resistivity (units Ωm^-1).  The factor 2 difference is explained in Kittel.
+${\tau _{e - ph}}$ is the time between phonon scattering events. ρ<sub>1d</sub> is one-dimensional resistivity (units Ωm<sup>-1</sup>).
 ===== Field-effect transistor fundamental limits (5 pts) =====
-This question explores the "60 mV/decade limit" for the subthreshold slope of field-effect transistors.
+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 [[http://en.wikipedia.org/wiki/MOSFET|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).
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 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 using a new type of tunnel diode. See Appenzeller //et al.//, PRL 93, 196805 (2004).
-  *Factoid: The 60 mV/decade limit was beaten in 2004 by a team at IBM ({{:2004-prl_band-to-band_tunneling.pdf|pdf}}) using a new type of tunnel diode.
 ===== Journal reading (5 pts) =====