University of Kurdistan logoNanoelectronics: A New Perspective on TransportWeek 8 · Quasi-Fermi levels, Hall effect · C4
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WEEK 8 · QUASI-FERMI LEVELS AND THE HALL EFFECT · C4

Nanoelectronics:
A New Perspective on Transport

Landauer and Büttiker formulas, what probes measure, Hall effect and edge states; mid-course review
Prof. Mohammad RazaghiUniversity of Kurdistan · Graduate course in Electronics Engineering
Based on S. Datta, Lessons from Nanoelectronics, Part A: Basic Concepts, Chapters 10–11
Part 1 is the fixed lecture (Ch. 10–11). Part 2 is circular session C4: review of Weeks 1–8 and a practice midterm.

RecapWhere we are

Week 7 gave us

  • µ⁺ and µ⁻ for the two streams; I = (q/h)M(µ⁺ − µ⁻)
  • The voltage drops at the interfaces: qIRB/2 each
  • µ = (µ − U) + U; equilibrium is constant µ

Today's questions

Can potentials inside a conductor be measured? What does a voltage probe really read? And what happens to µ when a magnetic field pushes electrons sideways?

Week 8Today's goals

  1. Find µ⁺ and µ⁻ across a single defect; derive G2t = (q²/h)MT and G4t = (q²/h)MT/(1 − T).
  2. Explain what a voltage probe measures, and how a negative resistance can appear.
  3. Use Büttiker's formula and its sum rules for multi-terminal conductors.
  4. Derive the Hall resistance, explain its sign for n and p, and Datta's Fermi-surface view.
  5. Describe edge states and the quantized Hall resistance h/(q²i).

Physics refresherRefresher: Lorentz force and cyclotron motion

  • F = −qv × B: perpendicular to v, changes direction not speed.
  • The velocity turns at ωc = qB/m on a circle rc = p/qB.
  • Full circles only if ωcτ ≫ 1; note ωcτ = mobility × B.
GaAs at 1 T: ωc = 2.6 × 10¹² s⁻¹; with mobility 10⁶ cm²/Vs, ωcτ = 100.

Physics refresherRefresher: the textbook Hall effect

I = q(N/L)vd,   VH/W = vdBEqs. 11.2, 11.5
RH = B/(qN/LW)Eq. 11.6
  • Slope of RH(B) gives the sheet density: the standard way to measure n.
  • It changes sign for p-type, usually "explained" by positive holes.

Datta Sec. 10.1Measuring inside: four-terminal versus two-terminal

G4t = I/[(µ₁* − µ₂*)/q] = (q²/h)M·T/(1 − T)Eq. 10.1 · Landauer's original
G2t = I/[(µ₁ − µ₂)/q] = (q²/h)M·TEq. 10.2 · "the" Landauer formula
1/G2t − 1/G4t = h/q²Mthe interface (Sharvin) resistance

Early 1980s: much confusion between the two; Imry identified the difference as the quantized interface resistance.

Datta Sec. 10.2µ⁺ and µ⁻ across a defect

µ⁺(right) = Tµ₁ + (1 − T)µ₂Eq. 10.5
µ⁻(left) = (1 − T)µ₁ + Tµ₂Eq. 10.6
  • µ⁺ − µ⁻ = T(µ₁ − µ₂) on both sides ⇒ I = (q/h)MT(µ₁ − µ₂).
  • Each of µ⁺, µ⁻ drops by (1 − T)(µ₁ − µ₂) at the defect (Eq. 10.7).
  • An IR drop at the defect, but no I²R heat there: elastic resistor.

Datta Sec. 10.1What does a voltage probe measure?

µprobe = αµ⁺ + (1 − α)µ⁻,  α = g⁺/(g⁺ + g⁻)Eq. 10.4
  • Atomic-scale (STM) probe: α ≈ ½, reads (µ⁺ + µ⁻)/2 (Fig. 10.6).
  • Same α on both probes: drop = (1 − T)(µ₁ − µ₂), Eq. 10.1 holds (Eq. 10.8).
  • Different α: anything goes. α = 0 left, 1 right: (1 − 2T), negative for T > ½.

Knowing the internal state predicts what a probe reads; the reverse is not true.

InteractiveExplore: one defect and two probes

A single defect: two Landauer formulas and what a probe measures  (Datta Ch. 10, Figs. 10.1–10.8)

An otherwise ballistic wire with one defect in the middle that transmits a fraction T of the electrons. As in Week 7, the density of each stream (counted) is its potential: μ+ for right-movers, μ− for left-movers (μ₁ = 1, μ₂ = 0). Two weakly coupled probes read μprobe = αμ+ + (1 − α)μ− (Eq. 10.4).

Two-terminal G2t / (q²M/h) = T (Eq. 10.2)–
Four-terminal G4t / (q²M/h) = T/(1−T) (Eq. 10.1)–
Probe readings μL*, μR*–
Measured R4t / RB–

Try this

  1. T = 0.4, fair probes: predict the readings (0.8, 0.2) and R4t/RB (1.5).
  2. Move both α sliders together: readings shift, their difference does not.
  3. Press Biased probes (T = 0.7): a negative four-terminal resistance.
  4. Check that the counted transmission equals T and G2t = (q²M/h)T.

Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.

Fair probes reproduce Büttiker: 1 − T/2 and T/2.

Datta Sec. 10.3Büttiker's formula

Im = (1/q) Σn Gm,n(µm − µn)Eq. 10.3
Gm,m = Σn≠mGm,n = Σn≠mGn,mEq. 10.13 · no current at equilibrium; currents add to zero
  • At each terminal know µ (driven) or I (probe: I = 0); solve for the rest.
  • Only terminal quantities: no arguments about internal potentials.

Datta Sec. 10.3Büttiker applied, and non-reciprocal circuits

Weak probes around a defect (click)
t₁ = τ(2 − T), t₂ = Tτ (Eqs. 10.23–10.24) ⇒ µ₁* = 1 − T/2, µ₂* = T/2 (Eq. 10.25): the same as the internal picture.
Two terminals with a B field (click)
The sum rule forces G₁₁ = G₁₂ = G₂₁: no non-reciprocity in linear response.
Three or more terminals (click)
G₁₂ ≠ G₂₁ is allowed: edge states in a magnetic field act like a circulator (Fig. 10.11).

Onsager (Sec. 10.3.4)

Gn,m(+B) = Gm,n(−B) (Eq. 10.26), and with magnets Gn,m(+B, +M) = Gm,n(−B, −M) (Eq. 10.27). It follows from the multi-terminal Kubo formula and time reversal of trajectories.

Notebook: a 6-terminal quantum Hall bar gives Rxy = h/(q²i) and Rxx = 0 from Eq. 10.3 alone.

Datta Sec. 11.1–11.2The Hall effect, Datta's way

dθ/dt = qvB/p,   ωc = qB/m, m = p/vEqs. 11.9–11.10

Same force, same dp/dt; but in a p-type band v is antiparallel to p, so trajectories curve the other way. Only electrons move.

RH(E) = 2BLW/[qD(E)v(E)p(E)]Eq. 11.7 · averaged as 1/RH over −∂f₀/∂E (Eq. 11.8)

A Fermi-surface property

With D·v·p = 2N (Eq. 6.9 in 2D) Eq. 11.7 becomes B/(qN/LW): Eq. 11.6, but only for a single band. Amorphous materials without an E(p) often give strange Hall results yet sensible thermopower.

Equilibrium currents in a B field involve all electrons, but they carry no transport: eliminate them.

Datta Sec. 11.3The µ profile across the width

δµ = µ⁺ − µ⁻ = qVλ/(L + λ)Eq. 11.13 · plays the role of the drift velocity
µ(z, θ, x) = µ̄(z) + (2/π)δµ cosθ
− (2/π)(δµ/p)qBxEq. 11.21 · solves the BTE with F·∇pµ
VH/W = (2/π)(δµ/p)BEq. 11.14

InteractiveExplore: the Hall bar

Hall effect in a 2D bar: from a small Hall voltage to edge states  (Datta Ch. 11, Figs. 11.1–11.6)

Linear response: the source injects extra electrons (μ₁ = 1, μ₂ = 0). They move at the Fermi speed, curve in the magnetic field (dp/dt = −qv×B, Eq. 11.1), scatter elastically and reflect at the side walls. The density across the width at mid-bar is the transverse profile of μ; its top-to-bottom difference is the Hall voltage qVH.

Carriers:
qVH/(μ₁ − μ₂) from two edge probes–
Datta Eq. 11.14 (valid for ωcτ ≪ 1)–
Transmission to the drain–
Cyclotron radius rc/W–

Try this

  1. Low field, diffusive: compare the edge-probe VH with Eq. 11.14.
  2. Switch to p-type: same force, opposite sign of VH.
  3. Raise ωcτ above 1: the linear formula fails.
  4. Press High field, ballistic: skipping orbits, one edge at µ₁, the other at µ₂; VH ≈ V.

Scroll inside the panel for all controls. Swipe or use the arrow buttons outside it to change slide.

Wall-hitting electrons stand for non-invasive edge probes. Transmission (pi/2) r_c/W is the classical count of edge channels.

Datta Sec. 11.4Edge states: a divided highway

  • High B: source electrons hug one edge, drain electrons the other (skipping orbits).
  • One edge at µ₁, the other at µ₂: the Hall voltage equals the applied voltage.
  • Backscattering needs a jump across the whole width: almost never, even over hundreds of µm.
R = h/(q²i)Eq. 11.22 · i integer; why integer needs Part B

ApplicationThe quantum Hall effect in numbers

25.813 kΩh/q², i = 1
12.906 kΩi = 2
8.604 kΩi = 3
6.453 kΩi = 4

GaAs, n = 3 × 10¹¹ cm⁻²: filling factor ν = nh/qB = 2 at 6.2 T; classical slope B/qn = 2.08 kΩ/T.

von Klitzing 1980, Nobel prize 1985; used as the resistance standard.

CheckConcept check

One minute with a neighbour, then click to reveal.

1. Why is 1/G2t − 1/G4t exactly h/q²M? (click)
The two-terminal measurement includes the interface resistance of the contacts; non-invasive probes inside the ballistic wire exclude it.
2. How can a four-terminal resistance be negative? (click)
Probes that weight µ⁺ and µ⁻ differently (Eq. 10.4), e.g. α = 0 left, 1 right, T > ½.
3. Electrons move in both n- and p-type. Why opposite Hall signs? (click)
v = dE/dp is antiparallel to p in a p-type band: same force, opposite curvature.
4. Why is the quantum Hall resistance so precise? (click)
Forward and backward movers sit on opposite edges: no backscattering; only the interface resistance h/(q²i) is left.

Wrap-upFixed session summary

Take-home

  • Defect: µ⁺ − µ⁻ = T(µ₁ − µ₂); G2t = (q²/h)MT, G4t = (q²/h)MT/(1 − T)
  • Probes read weighted averages; Büttiker uses terminal quantities
  • Hall: RH = B/qn (one band), sign from p/v
  • Edge states: R = h/(q²i)

Next: circular session C4

Mid-course review of Weeks 1–8, a lab on quasi-Fermi-level profiles and Hall bars, and a practice midterm. Open week08_qfl_hall_C4.ipynb (Part 2) and the Week 8 study toolkit.

PART 2

Circular session C4

Mid-course review of Weeks 1–8 · midterm preparation

A · 15 min
The story so far
B · 25 min
Lab: QFL profiles, Hall bar
C · 35 min
Practice midterm
D · 10 min
Exam strategy
The midterm itself (25% of the grade) uses the format of Block C: concepts and short calculations on Weeks 1–8.

Circular session C4Block A · The story so far

WeekBig ideaKey equation
1Interface + channel resistanceR = RB(1 + L/λ), GB = (q²/h)M
2Two Fermi functions, the windowf₁ − f₂ ≈ (−∂f₀/∂E)qV
3Elastic resistorG(E) = q²D/2t, I = (1/q)∫G(f₁ − f₂)dE
4Ballistic to diffusive; noiset = tB(1 + L/λ), λ = vτ{2, π/2, 4/3}, G₀ = CI/2kT
5Counting states and modesD·v·p = N·d, M = floor(2W/λ)
6NanotransistorU = UL + U₀(N − N₀), CQ = q²D₀, 60 mV/dec
7Where the voltage dropsµ(0) = µ₁ − qIRB/2, I = (q/h)M(µ⁺ − µ⁻)
8Probes, Hall effectG4t = (q²/h)MT/(1 − T), RH = B/qn, h/(q²i)

In pairs: pick two rows and explain how the later builds on the earlier.

Circular session C4Block B · Lab: quasi-Fermi-level profiles and Hall bars

B1. Two defects in series: count µ⁺ and µ⁻ with the Monte Carlo and check that four-terminal resistances add:

(1 − T)/T = Σi(1 − Ti)/TiT₁ = T₂ = ½ ⇒ T = ⅓ (notebook: 0.336)

B2. Hall Monte Carlo at ωc = 20 v/W: ballistic gives top/bottom probes 0.95/0.00 and T = 0.075 ≈ (π/2)rc/W; with τ = 0.3 bulk scattering cuts T to 0.034.

Coherent (quantum) defects do not add this way: Part B, Chapter 19.

Circular session C4Block C · Practice midterm (1/2)

Q1. A ballistic conductor has M = 4 (spin included). RB? Then L = 3λ: R? (click)
RB = 25.8/4 = 6.45 kΩ; R = RB(1 + 3) = 25.8 kΩ.
Q2. Why is f₁ − f₂ ≈ (−∂f₀/∂E)qV for qV = 2 meV at 300 K, and how wide is it? (click)
qV ≪ kT (linear response, Eq. 2.10); width ≈ 4kT ≈ 100 meV around µ₀.
Q3. Where is the heat of an elastic resistor dissipated, and how much per electron? (click)
In the contacts: µ₁ − ε (source) + ε − µ₂ (drain) = qV per electron.
Q4. vτ = 30 nm. Datta's λ in 3D and in 2D? (click)
(4/3) × 30 = 40 nm; (π/2) × 30 = 47.1 nm.

Circular session C4Block C · Practice midterm (2/2)

Q5. 2D gas, m = 0.2 m₀, g = 2, EF − Ec = 30 meV: n and modes per spin in 1 µm? (click)
n = 8.35×10¹³ × 0.03 = 2.5×10¹² cm⁻²; λF = 15.8 nm, M = 126.
Q6. CQ = 13.4, CE = 1.77 µF/cm²: gate efficiency above threshold? Subthreshold limit? (click)
1/(1 + 7.6) ≈ 0.12; kT ln10 ≈ 60 mV/decade at 300 K.
Q7. L = λ: δµ = µ⁺ − µ⁻ and the drop at each interface? (click)
δµ = 0.5(µ₁ − µ₂); each interface drops 0.25(µ₁ − µ₂).
Q8. Defect T = 0.25 in a wire with M = 2: R2t and R4t? (click)
RB = 12.9 kΩ; R2t = 51.6 kΩ; R4t = 38.7 kΩ.

Circular session C4Block D · Exam strategy

Every problem

  1. Which regime? Ballistic or diffusive, degenerate or not, low or high bias.
  2. Write the window (−∂f₀/∂E or f₁ − f₂) first.
  3. Then the channel quantity: G(E), M(E) or D(E).
  4. Check units: q²/h = 38.7 µS, kT = 25.9 meV.

The midterm

Weeks 1–8 (Datta Part A, Ch. 1–11): short concept questions and short calculations like Block C. The Week 8 study toolkit has a longer practice midterm with worked solutions and a formula sheet.

Wrap-upAfter the midterm

Before the midterm

  • Finish the Week 8 notebook (both parts)
  • Do the practice midterm in the Week 8 toolkit
  • Re-run one widget per week and explain it aloud

Week 9 preview

Smart contacts (Datta Ch. 12): separate quasi-Fermi levels for conduction and valence bands in a p-n junction, and for up and down spins in a spin valve.

Prof. Mohammad Razaghi · University of Kurdistan · after S. Datta, Lessons from Nanoelectronics