ECE 802-604: Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

Lecture 12, 08 Oct 13 In Chapter 02 in Datta: Transport: current I = GV V = IR => I = GV Velocity Energy levels M  M(E) Conductance G = GC in a 1-DEG Example Pr. 2.1: 2-DEG-1-DEG-2-DEG Example: 3-DEG-1-DEG-3-DEG Transmission probability: the new ‘resistance’ How to evaluate the Transmission/Reflection probability T(N) for multiple scatterers T(L) in terms of a “how far do you get length” L0 How to correctly measure I = GV Landauer-Buttiker: all things equal 3-, 4-point probe experiments: set-up and read out Example: 3-point probe (a) and (b) Example: 4-point probe (a) and (b) VM Ayres, ECE802-604, F13

Lecture 12, 08 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Add scattering to Landauer-Buttiker In Dresselhaus: Carbon bond hybridizations Graphene Carbon nanotubes VM Ayres, ECE802-604, F13

Lec11: Example: 3 point probe-a Question: How would you find R for this configuration? Answer: a/ Choose: V3 = 0 b/ Choose: ideal I2 = 0 c/ From I-V 2x2 matrix equations, identify the R that you want as: = VM Ayres, ECE802-604, F13

Lec 11:Example: 3 point probe-b Question: How would you find R for this configuration? Answer: a/ Choose: V3 = 0 b/ Choose: ideal I1 = 0 c/ From I-V 2x2 matrix equations, identify the R that you want as: For HW 02 VM Ayres, ECE802-604, F13

Shown: 4-point configurations for conductor resistance: VM Ayres, ECE802-604, F13

HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance: 1 3 2 B is present I VM Ayres, ECE802-604, F13

From Lec 11: how to start: q q q VM Ayres, ECE802-604, F13

how to start: Landauer-Buttiker for 4-point: q = 1, 2, 3, 4 q q q and q = 4 VM Ayres, ECE802-604, F13

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HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance: with B: Set one of the V’s to 0. Choice: V3 or V4 Choose V4 since the focus is on the Hall resistance. 4 1 3 2 B out of page thinking I VM Ayres, ECE802-604, F13

= 0 VM Ayres, ECE802-604, F13

Same matrix as p. 83 but not same electronic configuration as either (a) or (b). VM Ayres, ECE802-604, F13

In solutions: “See eq’n 2.4.9” VM Ayres, ECE802-604, F13

New: Switch G to Tbar description New: Switch G to Tbar description. Why: because there is a clever experimental way to evaluate Tq p: Typo: discuss Lec13. Roukes talks about e- steering, I think Datta tried to re-write as I steering T2 1 B into of page B small B out of page As the B-field is changed the changing transmission probability can be used to “direct” the current VM Ayres, ECE802-604, F13

From the reference: VM Ayres, ECE802-604, F13

left 4 forward 1 3 2 right B present I VM Ayres, ECE802-604, F13

Tforward T2 1 Tleft Tright VM Ayres, ECE802-604, F13

VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: 4 1  O for origin 3 2 Will consistently use: Blue TL Black  TF Red  TR VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: VM Ayres, ECE802-604, F13

So far: TO TO TO VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T1 2  TL 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T1 3  TF 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T2 1  TR 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T2 3  TL 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T1 3  TF 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Go right, left or forward defined w.r.t. the starting point: Remaining terms: T3 2  TR 4 1  O for origin 3 2 VM Ayres, ECE802-604, F13

Result: Easier matrix to invert and more physically meaningful. VM Ayres, ECE802-604, F13

If you check, all remaining combinations will be as shown: VM Ayres, ECE802-604, F13

HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance: Previously set: V4 = 0 Which I is ideally = 0? 4 1 3 2 B is present I VM Ayres, ECE802-604, F13

HW02: Pr 2.3: Shown: 4-point configurations for Hall resistance: Previously set: V4 = 0 I2 is ideally = 0 Also: I3 = -I1 4 1 3 2 B is present I VM Ayres, ECE802-604, F13

Fill this info into eq’n 2.4.9” Goal: Find RHall associated with VHall. VM Ayres, ECE802-604, F13

May be an unimportant typo in answer in some editions: VM Ayres, ECE802-604, F13

Lecture 12, 08 Oct 13 In Chapters 02 and 03 in Datta: How to correctly measure I = GV Add scattering to Landauer-Buttiker 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 VM Ayres, ECE802-604, F13

Reflections can be introduced in several ways: In Section 2.5: 2-t example: with broadened Fermi f0 In Section 2.6: 3-t example with reflections VM Ayres, ECE802-604, F13

Landauer-Buttiker so far does not accommodate reflections: p = 1, 2, 3, 4 q = 1, 2, 3, 4 q q q and q = 4 VM Ayres, ECE802-604, F13

2-t example: with broadened Fermi f0 In Section 2.5: 2-t example: with broadened Fermi f0 Steps to current I in Lec08: VM Ayres, ECE802-604, F13

Lec08: And can fill in the kx blank for group velocity: VM Ayres, ECE802-604, F13

Lec08: Calculate I: VM Ayres, ECE802-604, F13

Lec08: What does ny do? It means e- has a choice of available energy levels, all of which have conductance G that become available as e- get more energy from applied V VM Ayres, ECE802-604, F13

Lec08: Calculate I: VM Ayres, ECE802-604, F13

Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). 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 m1 Also called m2 VM Ayres, ECE802-604, F13

Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). This led to contacts and currents as shown when a potential difference V/chemical potential difference m1=m2 is present: VM Ayres, ECE802-604, F13

Contained an assumption about the Fermi probability f0: it’s a step function O (E-Ef). This led to transport as shown: F+ = m1 Ef ES Egap VM Ayres, ECE802-604, F13 Ev

But with local heating due to probes at contacts: f0 in (b) looks like f0 in (c) you can activate e-s in contact 2 that can populate –kx states VM Ayres, ECE802-604, F13

With local heating due to probes at contacts: There are back-currents. Without thermal broadening of f0: With thermal broadening of f0 VM Ayres, ECE802-604, F13