Week 8 Study Toolkit · Nanoelectronics

1This week's plan

Read
Datta Part A, Chapters 10–11 (pp. 121–149).

Run
Colab notebook week08_qfl_hall_C4.ipynb: Part 1 before class, Part 2 (review lab and practice midterm) in circular session C4.

Practise
Quiz (aim for 8/10), the practice problems, then the practice midterm below with the formula sheet.

By the end of the week you should be able to

Find µ⁺ and µ⁻ across a defect and derive both Landauer formulas.
Explain what a voltage probe measures and why biased probes can show a negative resistance.
Use Büttiker's formula and its sum rules, including non-reciprocal (edge-state) circuits.
Derive the Hall resistance, explain its sign for n and p, and Datta's Fermi-surface view.
Describe edge states and the quantized Hall resistance h/(q²i).

Optional video: Datta's nanoHUB-U course, Unit 3: L3.5–L3.7 (Landauer and Büttiker formulas, Hall effect).

2Key ideas in five sentences

A defect splits the potentials. With transmission T, µ⁺ − µ⁻ = T(µ₁ − µ₂) on both sides, so I = (q/h)MT(µ₁ − µ₂), and each potential drops by (1 − T)(µ₁ − µ₂) at the defect.
Two Landauer formulas. Two-terminal G = (q²/h)MT; four-terminal G = (q²/h)MT/(1 − T); they differ by the interface resistance h/q²M.
Probes average. A probe reads αµ⁺ + (1 − α)µ⁻; identical probes recover Eq. 10.1, unequal ones can even give negative resistance. Büttiker's formula uses only terminal quantities and agrees for non-invasive probes.
Hall effect. R_H = B/qn for a single band; really a Fermi-surface property; its sign follows the sign of p/v, not the sign of the carriers' charge.
Edge states. At high B forward and backward movers live on opposite edges; backscattering vanishes and R = h/(q²i) to extraordinary precision.

3Equation sheet (Week 8)

G_4t = (q²/h)M·T/(1 − T)Eq. 10.1

Landauer's four-terminal formula (non-invasive probes).

G_2t = (q²/h)M·TEq. 10.2

Two-terminal formula; 1/G_2t − 1/G_4t = h/q²M.

µ_probe = αµ⁺ + (1 − α)µ⁻Eq. 10.4

α = g⁺/(g⁺ + g⁻); ½ for an atomic-scale probe.

µ⁺(R) = Tµ₁ + (1 − T)µ₂, µ⁻(L) = (1 − T)µ₁ + Tµ₂Eqs. 10.5–10.6

Potentials on the two sides of a defect.

I_m = (1/q)Σ_nG_m,n(µ_m − µ_n)Eq. 10.3

Büttiker formula; G_m,m = Σ_nG_m,n = Σ_nG_n,m (Eq. 10.13).

G_n,m(+B) = G_m,n(−B)Eq. 10.26

Onsager reciprocity (reverse the magnetization too, Eq. 10.27).

R_H = B/(qN/LW)Eq. 11.6

Textbook Hall resistance.

R_H(E) = 2BLW/[qD(E)v(E)p(E)]Eqs. 11.7–11.8

Fermi-surface version; average 1/R_H over −∂f₀/∂E.

ω_c = qB/m, r_c = p/qBEq. 11.10

ω_cτ = mobility × B.

V_H/W = (2/π)(δµ/p)B, δµ = qVλ/(L + λ)Eqs. 11.13–11.14

Datta's replacement for V_H/W = v_dB.

R = h/(q²i)Eq. 11.22

Quantum Hall resistance, i integer.

4Physics refresher: magnetic forces

Lorentz force

F = −qv × B is perpendicular to v: the speed stays, the direction turns at ω_c = qB/m. In a p-type band m = p/v < 0, so the turning sense reverses.

Cyclotron numbers

	m/m₀	ħω_c at 1 T
free electron	1	0.116 meV
Si	0.19	0.61 meV
GaAs	0.067	1.73 meV

Why ω_cτ = mobility × B

Mobility µ̃ = qτ/m and ω_c = qB/m, so ω_cτ = (qτ/m)B. With µ̃ in m²/Vs and B in tesla the product is dimensionless: 1 m²/Vs × 1 T = 1.

Filling factor

ν = nh/(qB): electrons per flux quantum h/q. Plateaus of R_xy = h/(q²i) appear when ν is near the integer i.

5Glossary

Term	Meaning
Two-terminal resistance	V/I measured at the current-carrying contacts; includes the interface resistance.
Four-terminal resistance	Voltage between separate probes divided by the current; excludes the contact parts if probes are non-invasive.
Non-invasive probe	Weakly coupled, draws no current and does not disturb the conductor.
Sum rule	Σ_nG_m,n = Σ_nG_n,m: no current at equilibrium and currents add to zero.
Non-reciprocal	G_m,n ≠ G_n,m; possible with three or more terminals in a magnetic field.
Hall resistance	Transverse voltage over longitudinal current, R_H = B/qn for one band.
Skipping orbit	A cyclotron orbit that bounces along an edge.
Edge state	A one-way channel along an edge; the basis of the quantum Hall effect.

6Calculators

One defect: two- and four-terminal resistances

Transmission TModes MLeft probe α_LRight probe α_R

R_B = h/q²M

R_2t

R_4t (identical probes)

Probes read · measured R_4t

Classical Hall bar

n (10¹² cm⁻²)B (T)Current I (µA)

R_H = B/qn

V_H = R_HI

Filling factor ν = nh/qB

Cyclotron motion

Effective mass m/m₀B (T)Mobility (cm²/Vs)n (10¹² cm⁻², g = 2)

ω_c

ħω_c

ω_cτ

r_c at E_F

7Self-check quiz

Click an answer: you get instant feedback. Score: 0 / 0

8Practice problems (Week 8)

h/q² = 25.81 kΩ, q = 1.602 × 10⁻¹⁹ C, m₀ = 9.109 × 10⁻³¹ kg.

P1. Two Landauer formulasEASY

A defect with T = 0.25 in a wire with M = 2 modes. Find R_B, R_2t and R_4t.

Solution

R_B = 12.9 kΩ; R_2t = R_B/T = 51.6 kΩ; R_4t = R_B(1 − T)/T = 38.7 kΩ; difference = R_B.

P2. Fair probesEASY

T = 0.3. What do fair probes (α = ½) read on the two sides (µ₁ = 1, µ₂ = 0)? R_4t/R_B?

Solution

1 − T/2 = 0.85 and T/2 = 0.15; R_4t/R_B = 0.7/0.3 = 2.33.

P3. Biased probesMEDIUM

T = 0.8, left probe α = 0 (sees only µ⁻), right probe α = 1. Find the readings and R_4t/R_B.

Hint

Left µ⁻ = 1 − T, right µ⁺ = T.

Solution

Readings 0.2 and 0.8; drop = 1 − 2T = −0.6; R_4t/R_B = −0.6/0.8 = −0.75: negative.

P4. Hall measurementEASY

A 2D electron gas with n = 2 × 10¹¹ cm⁻² at B = 0.5 T carries 1 µA. Find R_H and V_H. What changes for a p-type sample with the same density?

Solution

R_H = B/qn = 1.56 kΩ; V_H = 1.56 mV. p-type: same magnitude, opposite sign.

P5. Cyclotron motionMEDIUM

GaAs (m = 0.067 m₀) at B = 2 T, mobility 10⁵ cm²/Vs. Find ω_c, ħω_c and ω_cτ.

Hint

ω_cτ = mobility × B.

Solution

ω_c = 5.25 × 10¹² s⁻¹, ħω_c = 3.46 meV, ω_cτ = 10 m²/Vs × 2 T = 20: full orbits between collisions.

P6. PlateausMEDIUM

n = 3 × 10¹¹ cm⁻². At what field is ν = 1? ν = 4? Which R_xy plateaus do these correspond to?

Solution

ν = nh/qB: ν = 1 at 12.4 T (R = 25.8 kΩ); ν = 4 at 3.1 T (R = 6.45 kΩ).

P7. Two defectsHARDER

Two incoherent defects with T₁ = 0.5 and T₂ = 0.8 in a single-mode wire (M = 1). Find the total T and R_2t.

Hint

Four-terminal resistances add: (1 − T)/T = Σ(1 − T_i)/T_i.

Solution

(1 − T)/T = 1 + 0.25 = 1.25 ⇒ T = 0.444; R_2t = 25.8 kΩ/0.444 = 58.1 kΩ.

P8. A circulatorHARDER

Three terminals with perfect one-way transmission 1 → 2 → 3 → 1 (M channels each). Terminal 1 at V, terminal 3 grounded, terminal 2 a floating probe. Find V₂ and the current. What if B is reversed?

Hint

Write I₂ = 0 with Eq. 10.3: only G₂₁ and G₂₃ matter for terminal 2.

Solution

G₂₁ = M(q²/h), G₂₃ = 0 ⇒ I₂ = G₂₁(V₂ − V₁) = 0 ⇒ V₂ = V. I₁ = G₁₃V = (q²/h)MV. Reversed B (1 → 3 → 2 → 1): V₂ = 0, same current: a non-reciprocal probe reading.

9Common mistakes

MISTAKE"Two- and four-terminal measurements of a defect must agree."

CORRECTThey differ by the interface resistance h/q²M (Fig. 10.3).

MISTAKE"A voltage probe measures the local electrochemical potential."

CORRECTIt measures a weighted average of µ⁺ and µ⁻; unequal probes can even give negative resistances.

MISTAKE"Resistance at a defect means heat at the defect."

CORRECTIn an elastic resistor the IR drop is at the defect, but the I²R heat is released in the contacts.

MISTAKE"p-type Hall voltage proves that positive particles move."

CORRECTElectrons move in both cases; the sign comes from v being antiparallel to p in a p-type band.

MISTAKE"The Hall effect counts all the electrons in the sample."

CORRECTIt is a Fermi-surface property; R_H = B/qn needs a single band.

MISTAKE"Conductance matrices are always symmetric."

CORRECTWith three or more terminals in a magnetic field G_m,n ≠ G_n,m is allowed; only the sum rules and Onsager relations must hold.

10Prepare for circular session C4 and the midterm

A · The story so far

Be ready to explain, in two minutes, how any two weeks connect (for example: Week 4's λ in Week 7's boundary conditions, or Week 5's D(E) in Week 6's C_Q).

B · Lab

Part 2 of the Week 8 notebook: µ⁺ and µ⁻ with two defects in series (do four-terminal resistances add?) and the Hall Monte Carlo at high field with and without scattering.

C · Practice midterm

Do the practice midterm below under exam conditions (60 min, formula sheet allowed) before C4.

D · Strategy

Regime first (ballistic/diffusive, degenerate/not, low/high bias), then the window, then G(E), M(E) or D(E); check units with q²/h and kT.

11Formula sheet: Weeks 1–8

Constants: q = 1.602 × 10⁻¹⁹ C, h = 6.626 × 10⁻³⁴ J s, q²/h = 38.74 µS, h/q² = 25.81 kΩ, kT = 25.85 meV at 300 K, m₀ = 9.109 × 10⁻³¹ kg.

R = R_B(1 + L/λ), G_B = (q²/h)MWeek 1 · Eqs. 1.8, 4.12

Interface plus channel; σ = (q²/h)(M/A)λ.

f(E) = 1/(1 + e^(E−µ)/kT), µ₁ − µ₂ = qVWeek 2 · Eqs. 2.1–2.2

Two contacts, two Fermi functions.

f₁ − f₂ ≈ (−∂f₀/∂E)qV, ∫(−∂f₀/∂E)dE = 1Week 2 · Eq. 2.10

Window about 4kT wide.

G(E) = q²D(E)/2t(E), I = (1/q)∫G(f₁ − f₂)dEWeek 3 · Eqs. 3.2–3.3

Elastic resistor; heat in the contacts.

G = ∫(−∂f₀/∂E)G(E)dEWeek 3 · Eq. 3.1

Degenerate: G ≈ G(µ₀); non-degenerate: Boltzmann tail.

t = L/u + L²/2D = t_B(1 + L/λ), G = G_Bλ/(L + λ)Week 4 · Eqs. 4.1–4.7

Ballistic to diffusive.

λ = vτ{2, π/2, 4/3}, M = (hDv/2L){1, 2/π, 1/2}Week 4 · Eqs. 4.10, 4.13

1D, 2D, 3D.

G₀ = C_I/2kT, i_rms = √(4kTGΔf)Week 4 · Eq. 5.4

Fluctuation–dissipation.

p = n·h/L, N ∝ p^d, D = dN/dE, D·v·p = N·dWeek 5 · Eqs. 6.9–6.17

Counting states; Drude σ = q²nτ/m with m = p/v.

M = floor{1, 2W/(h/p), πA/(h/p)²}Week 5 · Eq. 6.20

Wavelengths across; 2D DOS gm/2πħ² per area.

I = (1/q)∫G(E − U)(f₁ − f₂)dE, U = U_L + U₀(N − N₀)Week 6 · Eqs. 7.1–7.7

Point-channel model, self-consistent.

C_Q = q²D₀, dU/d(−qV_G) = 1/(1 + C_Q/C_E)Week 6 · Eqs. 7.12–7.14

60 mV/decade below threshold.

µ(0) = µ₁ − qIR_B/2, µ(L) = µ₂ + qIR_B/2Week 7 · Eq. 8.6

Diffusion equation for any L/λ.

I = (q/h)M(µ⁺ − µ⁻), δµ = (µ₁ − µ₂)/(1 + L/λ)Week 7 · Eqs. 8.8, 11.13

Two quasi-Fermi levels.

µ = (µ − U) + U; BTE ∂f/∂t + v∂f/∂z + F∂f/∂p = S_opfWeek 7 · Eqs. 9.1, 9.5

Equilibrium: constant µ.

G_2t = (q²/h)MT, G_4t = (q²/h)MT/(1 − T)Week 8 · Eqs. 10.1–10.2

Defect; probes; Büttiker Eq. 10.3.

R_H = B/qn, ω_c = qB/m, R = h/(q²i)Week 8 · Eqs. 11.6–11.22

Hall effect and edge states.

12Practice midterm (Weeks 1–8)

60 minutes, formula sheet allowed. Try each question before opening the solution.

M1. Resistance (Week 1)EASY

A conductor has M = 50 modes and λ = 20 nm. Find its resistance at L = 100 nm. What fraction of it is the interface resistance?

Solution

R_B = 25.8/50 = 0.516 kΩ; R = R_B(1 + 5) = 3.10 kΩ; interface share 1/6 = 17%.

M2. The window (Week 2)EASY

At 77 K, what is kT, how wide is the window −∂f₀/∂E (about 4kT), and what is f at E = µ + 2kT?

Solution

kT = 6.64 meV; window ≈ 26.5 meV; f = 1/(1 + e²) = 0.119.

M3. An elastic resistor I–V (Week 3)MEDIUM

G(E) = 50 µS for E > E_c and zero below, with E_c = µ₀ + 0.1 eV. At T ≈ 0 apply V = 0.3 V with µ₁,₂ = µ₀ ± qV/2. Find I.

Hint

Only the part of the window above E_c conducts.

Solution

The window runs from µ₀ − 0.15 to µ₀ + 0.15 eV; the part above E_c is 0.05 eV wide: I = 50 µS × 0.05 V = 2.5 µA.

M4. Transit time (Week 4)EASY

L = 200 nm, λ = 50 nm, u = 10⁵ m/s. Find t_B, t and G/G_B.

Solution

t_B = 2 ps; t = 2(1 + 4) = 10 ps; G/G_B = 50/250 = 0.2.

M5. Modes (Week 5)EASY

A 32 nm wide strip has λ_F = 12 nm. Find M per spin and G_B with g = 2.

Solution

floor(64/12) = floor(5.33) = 5; G_B = 2 × 38.74 × 5 = 387 µS.

M6. Subthreshold (Week 6)MEDIUM

How much gate swing reduces the off-current 10⁵-fold at 300 K with gate efficiency 1? With 0.9?

Solution

5 × 59.5 mV = 298 mV; with 0.9: 331 mV.

M7. Where the voltage drops (Week 7)MEDIUM

M = 20, λ = 40 nm, L = 160 nm, V = 1 mV. Find R, I and the fraction of V dropped at each interface.

Solution

R_B = 1.29 kΩ; R = R_B(1 + 4) = 6.45 kΩ; I = 0.155 µA; each interface (R_B/2)/R = 10%.

M8. Two potentials (Week 7)MEDIUM

Explain in three sentences why a conductor with a built-in field carries no current at equilibrium, using the words drift, diffusion and electrochemical potential.

Solution

At equilibrium µ is constant although U varies. The field pushes electrons one way (drift) and the density gradient pushes them the other (diffusion). Both terms carry the same σ₀ (Einstein relation), so their sum −σ₀d(µ/q)/dz is zero.

M9. Hall (Week 8)EASY

n = 2 × 10¹¹ cm⁻², B = 0.5 T, I = 1 µA. Find V_H. How would you tell n- from p-type?

Solution

V_H = BI/qn = 1.56 mV. The sign of V_H for a given current and field direction.

M10. Probes (Week 8)HARDER

A defect with T = 0.6. Two probes have α_L = 0.5 and α_R = 0.5, then α_L = 1, α_R = 0. Find R_4t/R_B in each case.

Hint

Left (µ⁺, µ⁻) = (1, 0.4), right = (0.6, 0); I ∝ T.

Solution

Fair: (0.7 − 0.3)/0.6 = 0.667 = (1 − T)/T. Biased: left reads µ⁺ = 1, right reads µ⁻ = 0: (1 − 0)/0.6 = 1.667 = R_2t/R_B: these probes see the full contact potentials.

13Explore

Files are in the course folder (course_materials).

Week 8 slides Widget: one defect, two probes Widget: Hall bar Colab notebook Week 7 toolkit Week 1 toolkit (start of review)

Mini-investigations (15 min each)

In the defect widget, find a pair (α_L, α_R) that makes R_4t = 0 for T = 0.5. Explain why it works.
In the Hall widget (ballistic), double ω_c: does the transmission halve, as (π/2)r_c/W predicts?
In the notebook, use buttiker() for a 4-terminal quantum Hall bar with i = 3 and check R_xy = h/(3q²).

14Can I…?

Tick what you can do without notes. Your ticks are saved in this browser.

find µ⁺, µ⁻ across a defectderive G_2t and G_4t and explain their differenceexplain what a voltage probe measuresuse Büttiker's formula with probes (I = 0)derive R_H = B/qn and explain its sign for n and pcompute ω_c, ω_cτ and the filling factorexplain edge states and h/(q²i)solve the practice midterm in 60 minutes

15Looking ahead: Week 9

Smart contacts (Datta Ch. 12): when one pair of µ⁺ and µ⁻ is not enough. A p-n junction needs separate quasi-Fermi levels for its two bands, and a spin valve for its two spins.

In an idealized p-n junction no energy channel connects both contacts. How can any current flow?
What would a magnetic contact measure in a channel where up and down spins have different µ?

Quasi-Fermi levels, the Hall effect, and the mid-course review

1This week's plan

By the end of the week you should be able to

2Key ideas in five sentences

3Equation sheet (Week 8)

4Physics refresher: magnetic forces

5Glossary

6Calculators

One defect: two- and four-terminal resistances

Classical Hall bar

Cyclotron motion

7Self-check quiz

8Practice problems (Week 8)

9Common mistakes

10Prepare for circular session C4 and the midterm

11Formula sheet: Weeks 1–8

12Practice midterm (Weeks 1–8)

13Explore

Mini-investigations (15 min each)

14Can I…?

15Looking ahead: Week 9