ECE453Fall2010Notes4-2[1] - Fall 2010 ECE 453 Lecture Notes...

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Fall 2010 ECE 453 Lecture Notes #4 For Purdue ECE 453 students ONLY NOT FOR CIRCULATION ECE 453 Lecture Notes#4 Excerpted from (LNE) Lessons from nanoelectronics: A new perspective on transport 1. The bottom-up approach 4 Introductory concepts 2. Why electrons flow 6 3. The elastic resistor 10 4. The new Ohm's law 14 Notes #1 5. Where is the resistance? 19 ***************************************************************** 6. Transverse modes 27 7. Drude formula 33 8. Kubo formula 38 Notes # 2 9. How realistic is an elastic resistor? 41 ***************************************************************** Semiclassical and quantum transport 10. Beyond low bias: The nanotransistor 44 Notes#3 11. Semiclassical Transport and the Scf method 50 ***************************************************************** 12. Resistance and uncertainty 56 13. Quantum Transport: Schrodinger to NEGF 63 Notes#4 14. Resonant tunneling and Anderson localization 68 ***************************************************************** 15. Coulomb blockade and Mott transition 72 16. Hall effect / QHE xx Beyond voltages and currents 17. Thermoelectricity xx 18. Heat flow xx 19. Spin flow xx 20. Spin transistor xx 21. Electronic Maxwell's demon xx 22. Physics in a grain of sand xx Supriyo Datta, Purdue University World Scientific (2011), to be published
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Lessons from nanoelectronics: Copyright S. Datta All Rights Reserved 56 12. Resistance and uncertainty In Chapter 4 we saw that the conductance of a short ballistic conductor is given by (q 2 /h)*M, where M is the number of independent conduction channels. What about a really short "molecular" conductor? There cannot be more than a few conduction channels at best, and so we would expect a maximum conductance ~ a few (q 2 /h). Indeed experimentalists have found the conductance of a hydrogen molecule to be ~ 2 (q 2 /h). Would our theoretical treatment predict this? The answer is no. Let me explain why. For something like a hydrogen molecule the energy level diagram should look just like the one-level resistor that we discussed in Section 3 where we obtained the following expression for the conductance G one level q 2 t F T ( 0 ) (same as Eq.(3.2)) This suggests a maximum conductance of ( q 2 / t )*(1/4 kT ) if the energy level sits right on the electrochemical potential μ 0 , dropping down to zero as the level moves away from μ 0 . Clearly this is not the quantized conductance q 2 / h that we are looking for, since there is no reason to expect that 4kT*t < h. After all, kT and t are two independent physical quantities: kT is related to the temperature, while t is the transfer time related to the quality of the contacts if the transit time through the channel is negligible. For any given t, we can always find a temperature below which the conductance would exceed our "maximum." What Eq.(3.2) misses is the broadening of the energy level: Quantum mechanically the process of coupling to a level, effectively spreads it out giving a picture more like the one shown in Fig.6.2. The one-level resistor then effectively becomes a resistor with a density of states, D ~1/ , being the broadening of the energy level. The conductance even for a one-level resistor should then be obtained from the multilevel result G q 2 D / t (see Eq.(3.3c)) which gives G one level q 2 t
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ECE453Fall2010Notes4-2[1] - Fall 2010 ECE 453 Lecture Notes...

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