ECE 802 604 Nanoelectronics Prof Virginia Ayres Electrical

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

Lecture 06, 17 Sep 13 In Chapter 01 in Datta: Two dimensional electron gas (2 -DEG) DEG goes down, mobility goes up Define mobility Proportional to momentum relaxation time tm Count carriers n. S available for current – Pr. 1. 3 (1 -DEG) How n. S influences scattering in unexpected ways – Pr 1. 1 (2 -DEG) One dimensional electron gas (1 -DEG) Special Schrödinger eqn (Con E) that accommodates: Electronic confinement: band bending due to space charge Useful external B-field Experimental measure for mobility Examples VM Ayres, ECE 802 -604, F 13

Lec 05: Example: write down the wave function for a 1 -DEG VM Ayres, ECE 802 -604, F 13

Lec 05: Example: write down the energy eigenvalues for a 1 DEG assuming an infinite square well potential in the quantized directions VM Ayres, ECE 802 -604, F 13

Example: draw a diagram of this 1 -DEG kz ky W t kx z Width W in y y Thickness t in z x VM Ayres, ECE 802 -604, F 13

Example: write down the energy eigenvalues for a 1 -DEG assuming an infinite square well potential in the quantized directions. Assume nz = 1 st and Ly W VM Ayres, ECE 802 -604, F 13

Example: write down the energy eigenvalues for a 1 -DEG assuming an infinite square well potential in the quantized directions. Assume nz = 1 st and Ly W Answer: VM Ayres, ECE 802 -604, F 13

Example: find the number of energy levels NT(E) for a 1 -DEG assuming an infinite square well potential in the quantized directions VM Ayres, ECE 802 -604, F 13

Example: Generally what is the relation of the N(E) to NT(E)? Write this down for both a 2 -DEG and a 1 -DEG. VM Ayres, ECE 802 -604, F 13

Answer: VM Ayres, ECE 802 -604, F 13

Example: Generally what is the relation of concentration ns to N(E)? VM Ayres, ECE 802 -604, F 13

Example: Generally what is the relation of concentration n to N(E)? Answer: n= N(E)n-DEG f 0(E) d. E Key for correct nanotechnology VM Ayres, ECE 802 -604, F 13

Example: How do you define “hot” versus “cold” for the Fermi probability f 0(E)? VM Ayres, ECE 802 -604, F 13

Answer: The definitions are what the denominator is doing: Hot: Cold: You can’t meet the Cold condition by any change in T. The only way to do it is with Ef > E: Cold means the semiconductor is degenerate. VM Ayres, ECE 802 -604, F 13

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Hint: Plot n versus (Ef – ES )/ E 1 not versus Ef VM Ayres, ECE 802 -604, F 13

Parabolic potential is new. Why interesting: this is the single electron transistor (SET) Kastner article, MIT VM Ayres, ECE 802 -604, F 13

2 -DEG: Before: U(x, y) = 0 and A = 0: no E or B Now: U(x, y) = U(y) = ½ mw 02 y 2. Still have A = 0: no E or B but let’s get ready for B anyway VM Ayres, ECE 802 -604, F 13

U(x, y) = U(y), and B is possible: z B y x IDS e- Like Hall effect: expect: the x motion is disturbed by the B-field VM Ayres, ECE 802 -604, F 13

2 -DEG -> 1 -DEG: x x Now put in: B=0 U(y) = ½ mw 02 y 2 VM Ayres, ECE 802 -604, F 13

x Wavefunction: VM Ayres, ECE 802 -604, F 13

Energy eigenvalues are: USE this in your HW VM Ayres, ECE 802 -604, F 13

Now find N(E) Now find n. L VM Ayres, ECE 802 -604, F 13

Useful B-field: experimental measures: In real life, electron densities and mobilities do not come printed on nanowires, nanotubes or graphene sheets! VM Ayres, ECE 802 -604, F 13

Useful B-field: experimental measures: What happens when you run a Hall effect measurement in a 2 DEG? Measurement set-up: VM Ayres, ECE 802 -604, F 13

Expectation: Drude model: wrong: 2 -DEG: VM Ayres, ECE 802 -604, F 13

Expectation: Drude model: wrong: Write in terms of something you can measure: J: VM Ayres, ECE 802 -604, F 13

Expectation: Drude model: wrong: Dig out your resistivities and then do V = IR VM Ayres, ECE 802 -604, F 13

Expectation: Drude model: wrong: VHall Vx Dotted lines are fictitious VM Ayres, ECE 802 -604, F 13

Expectation: Drude model: wrong: Any low-field place where the measurement is actually doing this, life is good. VM Ayres, ECE 802 -604, F 13

Any low-field place where the measurement is actually doing this, life is good. VM Ayres, ECE 802 -604, F 13

What happens as you increase B: VHall develops a staircase Vx develops oscillations VM Ayres, ECE 802 -604, F 13

What happens when you run a Hall effect measurement in a 2 DEG? Stated without proof: The density of states used to be a constant: Now it’s a bunch (n + ½) of spikes (delta function). Each n = 0, 1, … is called a Landau level. VM Ayres, ECE 802 -604, F 13

2 -DEG density of states + B-field: d[E – (ES + En)] 2 nd: n = 1 1 st: n = 0 VM Ayres, ECE 802 -604, F 13

Spikes in N(E) => spikes in n. S => spikes/troughs in current Which can be interpreted as an oscillation in resistivity. Resistivities are proportional to the measured voltages VM Ayres, ECE 802 -604, F 13

High B-field measurement of carrier density: number of occupied Landau levels Changes by 1 between any two levels VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13