ECE : Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE , F13 Lecture 14, 15 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Brief discussion: Roukes article Two typo correction HW02 Pr. 2.3 Add scattering to Landauer-Buttiker Caveat: when Landauer-Buttiker doesn’t work When it does: Sections 2.5 and 2.6: motivation: why: 2.5: Probes as scatterers especially at high bias/temps Buttiker approach for dealing with incoherent scatterers 2.6 occupied states as scatterers 3.1 scattering/S matrix

VM Ayres, ECE , F13 Roukes article:

VM Ayres, ECE , F13 Roukes article:

VM Ayres, ECE , F13 Roukes article: T R possibilities: F = q(v x B) = -|e| (v X B) rebounddirect rebounddirect + seems to be X B

VM Ayres, ECE , F13 Roukes article: T L possibilities: F = q(v x B) = -|e| (v X B) rebound direct rebound direct

VM Ayres, ECE , F13 Roukes article: HW01 VA Pr.01: x = 0.4, here it is x = 0.3 HW01 Datta E1.1 Find f HW01 Datta E1.2 Find n and 

VM Ayres, ECE , F13 Two typos Pr. 2.3: Roukes article: T R rebounddirect Roukes article: T L rebound direct

VM Ayres, ECE , F13 Two typos Pr. 2.3, marked in magenta: Datta Pr. 2.3, p. 113 : T L Datta Pr. 2.3, p. 113: T R T 2 1 X B

VM Ayres, ECE , F13 Lecture 14, 15 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Brief discussion: Roukes article: HW02 Pr. 2.3: Add scattering to Landauer-Buttiker Caveat: when Landauer-Buttiker doesn’t work When it does: Sections 2.5 and 2.6: motivation: why: 2.5: Probes as scatterers especially at high bias/temps Buttiker approach for dealing with incoherent scatterers 3.1 scattering/S matrix

VM Ayres, ECE , F13 In Section 2.5: 2-t example: with broadened Fermi f 0 Lec13: Practical example: Roukes

VM Ayres, ECE , F13 If T = T’, can get to Landauer-Buttiker but no reason why T should = T’. Especially if energies from probes took e- far from equilibrium. i as a function of how much energy E/what channel M the e- is in Lec13:

VM Ayres, ECE , F13 Can expect T = T’ at equilibrium. Consider: if energies from probes don’t take e- far from equilibrium: Lec13:

VM Ayres, ECE , F13 New useful G: Lec13:

VM Ayres, ECE , F13 Basically I = G^V = G^ (  1 -  2 ) e that works when probes hotted things up but not too far from equilibrium Lec13:

VM Ayres, ECE , F13 Example: does the figure shown appear to meet the linear (I = G^ V) regime criteria? Criteria is:  1 -  2 << k B T FWHM shown is k B T Answer: No, they appear to be about the same (red and blue). However, part of F T (E) is low value. Comparing an ‘effective’  1 -  2 (green) maybe it’s OK. Lec13:

VM Ayres, ECE , F13 Lec13: If T(E) changes rapidly with energy, the “correlation energy”  c is said to be small. E T(E) eV5.001 eV A very minimal change in e- energy and you are getting a different and much worse transmission probability.

VM Ayres, ECE , F13 2-DEG

VM Ayres, ECE , F13 Example: in HW01 Pr. 1.1 you solved for   for a 2-DEG in 1 K using the graph shown in Figure Estimate the corresponding correlation energy  c

VM Ayres, ECE , F13 Answer: Estimate the corresponding correlation energy  c

VM Ayres, ECE , F13 Lecture 14, 15 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Brief discussion: Roukes article Two typo correction HW02 Pr. 2.3 Add scattering to Landauer-Buttiker Caveat: when Landauer-Buttiker doesn’t work When it does: Sections 2.5 and 2.6: motivation: why: 2.5: Probes as scatterers especially at high bias/temps Buttiker approach for dealing with incoherent scatterers 3.1 scattering/S matrix

VM Ayres, ECE , F13 Lec10: Scattering: Landauer formula for R for 1 coherent scatterer X: Reflection = resistance

VM Ayres, ECE , F13 E > barrier height V 0 E < barrier height V 0 Lec 10: Coherent scattering means that phases of both transmitted and reflected e- waves are related to the incoming e- wave in a known manner

VM Ayres, ECE , F13 Lec 10: Transmission probability for 2 scatterers: T => T 12 : That’s interesting. That Ratio is additive: Assuming that the scatterers are identical:

VM Ayres, ECE , F13 Therefore: Resistance for two coherent scatterers is:

VM Ayres, ECE , F13

Resistance is due to partially coherent /partially incoherent transmission

VM Ayres, ECE , F13 1 Deg, M = 1 Example: probe LL RR  X X O

VM Ayres, ECE , F13 probe LL RR   1 Deg, M = 1 X X O

VM Ayres, ECE , F13 Model the phase destroying impurity as two channels attached to an energy reservoir 

VM Ayres, ECE , F13 Influence of the incoherent impurity can be described using a Landauer approach as:

VM Ayres, ECE , F13 LL RR 

LL RR 

LL RR 

LL RR 

LL RR  Landauer-Buttiker treats all “probes” equally: what is going into “probe” 3:

VM Ayres, ECE , F13 LL RR  Landauer-Buttiker treats all “probes” equally: what is going into “probe” 4:

VM Ayres, ECE , F13 Outline of the solution: Goal: V = IR, solve for R What is V:  A –  B What is I: I = I 1 = I 2

VM Ayres, ECE , F13 probe LL RR   AA BB 1 Deg, M = 1 X X O

VM Ayres, ECE , F13

Condition: net current I 3 + I 4 = 0

VM Ayres, ECE , F13

Condition: net current I 3 + I 4 = 0

VM Ayres, ECE , F13 Condition: net current I 3 + I 4 = 0

VM Ayres, ECE , F13

LL RR