ECE 802-604: Nanoelectronics

ECE 802-604:Nanoelectronics

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]

VM Ayres, ECE802-604, F13

Lecture 14, 15 Oct 13

In Chapters 02 and 03 in Datta:

How to correctly measure I = GVBrief discussion: Roukes articleTwo typo correction HW02 Pr. 2.3

Add scattering to Landauer-ButtikerCaveat: when Landauer-Buttiker doesn’t workWhen it does: Sections 2.5 and 2.6: motivation: why:

2.5: Probes as scatterers especially at high bias/tempsButtiker approach for dealing with incoherent scatterers

2.6 occupied states as scatterers

3.1 scattering/S matrix


Roukes article:


Roukes article:


Roukes article: TR possibilities:

F = q(v x B) = -|e| (v X B)

rebound directrebound direct

+ seems to be X B


Roukes article: TL possibilities:

F = q(v x B) = -|e| (v X B)

rebounddirect

rebounddirect


Roukes article:

HW01 VA Pr.01: x = 0.4, here it is x = 0.3

HW01 Datta E1.1Find f

HW01 Datta E1.2Find n and


Two typos Pr. 2.3:

Roukes article: TR

rebound direct

Roukes article: TL

rebounddirect


Two typos Pr. 2.3, marked in magenta:

Datta Pr. 2.3, p. 113: TL Datta Pr. 2.3, p. 113: TR

T2

1 X B




How to correctly measure I = GVBrief discussion: Roukes article: HW02 Pr. 2.3:


2.5: Probes as scatterers especially at high bias/tempsButtiker approach for dealing with incoherent scatterers



In Section 2.5:2-t example: with broadened Fermi f0

Lec13:

Practical example: Roukes


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:


Can expect T = T’ at equilibrium.Consider: if energies from probes don’t take e- far from equilibrium:

Lec13:


New useful G:

Lec13:


E

Ef )(0

Basically I = G^V = G^ (1-2) ethat works when probes hotted things up but not too far from equilibrium

Lec13:


Example: does the figure shown appear to meet the linear (I = G^ V) regime criteria?

Criteria is: 1-2 << kBT

FWHM shown is kBT

Answer: No, they appear to be about the same (red and blue).

However, part of FT(E) is low value. Comparing an ‘effective’ 1-2 (green)

maybe it’s OK.

Lec13:


Lec13: If T(E) changes rapidly with energy, the “correlation energy” c is said to be small.

E

T(E)

0.85

0.09

5 eV 5.001 eV

A very minimal change in e- energy and you are getting a different and much worse transmission probability.


2-DEG


Example: in HW01 Pr. 1.1 you solved for for a 2-DEG in GaAs @ 1 K using the graph shown in Figure 1.3.2 .

Estimate the corresponding correlation energy c


Answer:

Estimate the corresponding correlation energy c




How to correctly measure I = GVBrief discussion: Roukes articleTwo typo correction HW02 Pr. 2.3


2.5: Probes as scatterers especially at high bias/temps Buttiker approach for dealing with incoherent scatterers



Lec10: Scattering: Landauer formula for R for 1 coherent scatterer X:

Reflection = resistance


E > barrier height V0

E < barrier height V0

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


Lec 10: Transmission probability for 2 scatterers: T => T12:

That’s interesting. That Ratio is additive:

Assuming that the scatterers are identical:


Therefore: Resistance for two coherent scatterers is:






Resistance is due to partially coherent /partially incoherent transmission


1 Deg, M = 1

Example:probe

LR

X XO


probe

LR

1 Deg, M = 1

X XO


Model the phase destroying impurity as two channels attached to an energy reservoir


Influence of the incoherent impurity can be described using a Landauer approach as:


L R


L R


L R


L R


L R

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


L R

Landauer-Buttiker treats all “probes” equally: what is going into “probe” 4:


Outline of the solution:

Goal: V = IR, solve for R

What is V: A – B

What is I: I = I1 = I2


probe

LR

A B

1 Deg, M = 1

X XO





Condition: net current I3 + I4 = 0









LR

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ECE 802-604: Nanoelectronics