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

VM Ayres, ECE , F13 Lecture 28, 05 Dec 13 Examples: |sp 3 > hybrid orbitals Setting up a an E-k factor in graphene (HW05: in polyacetylene) CNT as a 1-D system: review of Lec Oct 13

VM Ayres, ECE , F13

4 bonds, arranged in a tetrahedron

VM Ayres, ECE , F13

Lecture 28, 05 Dec 13 Examples: |sp 3 > hybrid orbitals Setting up a an E-k factor in graphene (HW05: in polyacetylene) CNT as a 1-D system

VM Ayres, ECE , F13 second

VM Ayres, ECE , F13

Lecture 28, 05 Dec 13 Examples: |sp 3 > hybrid orbitals Setting up a an E-k factor in graphene (HW05: in polyacetylene) CNT as a 1-D system

VM Ayres, ECE , F13 Four terminal Two terminal Same as Datta Chp. 02

VM Ayres, ECE , F13 Coherent: Same as Datta Chp. 03

VM Ayres, ECE , F13 Incoherent: Same as Datta Chp. 03

VM Ayres, ECE , F13 Same as Datta Chp. 02

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

VM Ayres, ECE , F13 Lecture 13, 10 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV 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 2.6: Occupied states (q,p) as scatterers 3.1 scattering/S matrix In Chapter 02 in Datta: How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out Example: 4-point probe Pr. 2.4 set-up Example: 4-point probe R Hall Pr. 2.3: Roukes article

VM Ayres, ECE , F13 In Section 2.5: 2-t example: with broadened Fermi f 0 In Section 2.6: 3-t example with reflections Lec12: Reflections can be introduced in several ways:

VM Ayres, ECE , F13 Lec12: Landauer-Buttiker so far does not accommodate reflections: qqq p = 1, 2, 3, 4 q = 1, 2, 3, 4 and q = 4

VM Ayres, ECE , F13 In Section 2.5: 2-t example: with broadened Fermi f 0 Steps to current I in Lec08:

VM Ayres, ECE , F13 Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to e-s that carry the current as shown, just above and below the Fermi level (M = 1 parabola shown here, can have more parabolas): Also called  1 Also called  2

VM Ayres, ECE , F13 Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to contacts and currents as shown when a potential difference V/chemical potential difference  1 =  2 is present: Fermi probabilities in this picture are shown next slide

VM Ayres, ECE , F13

Contained an assumption about the Fermi probability f 0 : it’s a step function O (E-E f ). This led to transport as shown: EvEv ESES E gap E f-eq F+ =  1

VM Ayres, ECE , F13 But with local heating due to probes at contacts: f 0 in (b) looks like f 0 in (c) you can activate e-s in contact 2 that can populate –k x states

VM Ayres, ECE , F13 Hot:

VM Ayres, ECE , F13 With thermal broadening of f 0 Without thermal broadening of f 0 : With local heating due to probes at contacts: There are back-currents.

VM Ayres, ECE , F13

This part of I comes from e- that turned around (not in picture) at the barrier when approaching from the right. When e-transmits (next slide) it can use a different channel: T’

VM Ayres, ECE , F13 = ?

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

VM Ayres, ECE , F13 If T(E) = T(E)’, can get to Landauer-Buttiker: If T(E) = T(E f ), can take it outside the integral: Quasi-Fermi level Contact01 Quasi-Fermi level Contact02 you recover: But today we are dealing with high temps

VM Ayres, ECE , F13 Consider: if energies from probes don’t take e- far from equilibrium: f eq f 1 (E) = f eq +  f 1 f 2 (E) = f eq +  f 2

VM Ayres, ECE , F13 New useful G:

VM Ayres, ECE , F13 Check: Good, recovered ‘cold’ result for G.

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

VM Ayres, ECE , F13 Even without channel M changes, the transmission probability e- gets for each transmission depends on how much energy the e- had. Lots of possibilities so T(E) jumps a lot. New conductivity G^ will multiple the jumpiness by the part of sech 2 between  1 and  2, then sum up the combo.

VM Ayres, ECE , F13 I = GV = G (  1 -  2 ) = Area e

VM Ayres, ECE , F13 I = GV = G (  1 -  2 ) = Area e

VM Ayres, ECE , F13 So it depends: 6 values of T (in red  1 –  2 box), but only 1-2 multiplied by something big (in blue box). How different are the values of these two? Summing up will also average them. An average of two close values is representative, an average of two far away values is not.

VM Ayres, ECE , F13 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: does the figure shown appear to meet the linear (I = G^ V) regime criteria? Criteria is:  1 -  2 << k B T FWHM shown on sech 2 is k B T

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.

VM Ayres, ECE , F13 Lecture 13, 10 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV 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 2.6: Occupied states (q,p) as scatterers 3.1 scattering/S matrix In Chapter 02 in Datta: How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out Example: 4-point probe Pr. 2.4 set-up Example: 4-point probe R Hall Pr. 2.3: Roukes article

VM Ayres, ECE , F13 HW02: Pr. 2.4: test circuit for 1-DEG resistances in series:

VM Ayres, ECE , F13 HW02: Pr. 2.4: test circuit for 1-DEG resistances in series: Now: B out of page. Solve this problem. Find R.

VM Ayres, ECE , F13 Landauer-Buttiker treat places and e- can go equally to start set-up: Show some work.

VM Ayres, ECE , F13 This is a 2-terminal circuit: R 2-t. 1 and 3 have probes, but 2 and 4 do not: they are just real estate but e- can be deflected into them by a B-field. Which of 1, 2, 3, 4, will you set to V = 0?

VM Ayres, ECE , F13 This is a 2-terminal circuit: R 2-t. 1 and 3 have probes, but 2 and 4 do not: they are just real estate but e- can be deflected into them by a B-field. Answer: ordinary choice: V 3 = 0.

VM Ayres, ECE , F13 V 3 = 0: 3x3 G matrix Switch to a T p q description: 3x3 T 14, … etc. matrix Switch to a T L, T R, and T F (2 choices) description.

VM Ayres, ECE , F13 Pr. 2.3Pr. 2.4