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

Lecture 08, 24 Sep 13 In Chapter 02 in Datta: Transport: current I = GV V = IR => I = GV Velocity Energy levels M  M(E) Conductance G in a 1-DEG Scaling back up to 3-DEG using M Transmission probability: the new ‘resistance’ How to evaluate the Transmission/Reflection probability How to correctly measure I = GV Landauer-Buttiker: all things equal 4-point probe experiments: set-up and read out VM Ayres, ECE802-604, F13

Current I: basic idea: VM Ayres, ECE802-604, F13

Current I: basic idea: VM Ayres, ECE802-604, F13

Current I: basic idea: VM Ayres, ECE802-604, F13

1-Deg (no B) experiments: transport direction along kx: z y px2 + ny2 ħ2p2 + nz2 ħ2p2 + Ec 2m* 2m*Ly2 2m*Lz2 E = x Transport in x direction Infinite square well (top) in y direction OR Parabolic well (bottom) in y direction Infinite square well in z direction Well separated energy levels => nz = 1 px2 + (ny + 1 )ħw0 + nz2 ħ2p2 + Ec 2m* 2 2m*Lz2 E = VM Ayres, ECE802-604, F13

1 Deg from parabolic U(y): energy eigenvalues: x x x x Group velocity is ready VM Ayres, ECE802-604, F13

Group velocity relationship: where from: Conservation of Energy (special Sch eq’n): VM Ayres, ECE802-604, F13

Current I: basic idea: wrong: VM Ayres, ECE802-604, F13

1-DEG NT(E): just write down: VM Ayres, ECE802-604, F13

1-DEG NT(E): just write down: Answer: VM Ayres, ECE802-604, F13

1-DEG NT(E) for parabolic U(y), B = 0: VM Ayres, ECE802-604, F13

1-DEG N(E) for parabolic U(y), B = 0: VM Ayres, ECE802-604, F13

A finesse on N(E) in a transport situation: VM Ayres, ECE802-604, F13

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

Calculate I: check Units first: VM Ayres, ECE802-604, F13

Calculate I: VM Ayres, ECE802-604, F13

Calculate I: VM Ayres, ECE802-604, F13

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

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. Sum Conductance G, just like nL in Pr. 1.3, and the result is: VM Ayres, ECE802-604, F13

W Vsource to drain keep constant Vgate changes W VM Ayres, ECE802-604, F13

What are m1 and m2: Units: these are energies related to the applied potential. note: eV / e = Volts VM Ayres, ECE802-604, F13

1 Deg + contacts and VDS: introduces a E-field vector potential in x: effect on transport px z y (px+pxE)2 + ny2 ħ2p2 + nz2 ħ2p2 + Ec 2m* 2m*Ly2 2m*Lz2 E = x Transport in x direction Infinite square well (top) in y direction OR Parabolic well (bottom) in y direction Infinite square well in z direction Well separated energy levels => nz = 1 (px+pxE)2 + (ny + 1 )ħw0 + nz2 ħ2p2 + Ec 2m* 2 2m*Lz2 E = VM Ayres, ECE802-604, F13