Week 4 Study Toolkit · Nanoelectronics

1This week's plan

Read
Datta Part A, Chapters 4–5 (pp. 43–56) and Appendix B (angular averaging).

Run
Colab notebook week04_ballistic_diffusive_C2.ipynb: Part 1 before class, Part 2 (lab) in circular session C2.

Practise
Quiz (aim for 8/10), practice problems, and the C2 preparation list.

By the end of the week you should be able to

Explain t ∝ L (ballistic) and t ∝ L² (diffusive), and use t = L/u + L²/2D = t_B(1 + L/λ).
Derive G = G_Bλ/(L + λ) and the conductivity in 1D, 2D and 3D.
Explain λ = vτ{2, π/2, 4/3} from angular averaging.
Explain modes and conductance quantization, G_B = (q²/h)M.
Use the fluctuation–dissipation theorem G₀ = C_I/2kT and the Johnson–Nyquist formula.

Optional video: Datta's nanoHUB-U course, Unit 1: L1.5–L1.8 (ballistic, diffusive, angular averaging).

2Key ideas in five sentences

Two time scales. A bullet crosses in t_B = L/u; a random walker needs ≈ L²/2D; Datta joins them as t = t_B(1 + L/λ).
The new Ohm's law. With G = q²D/2t and D ∝ L, G = G_Bλ/(L + λ): length-independent when L ≪ λ, Ohm's law when L ≫ λ.
λ includes an angular factor. λ = 2D/u = vτ × {2, π/2, 4/3} in 1D, 2D, 3D, because collisions in 2D/3D usually deflect rather than reverse.
Modes. G_B = (q²/h)M: M parallel channels, which makes small conductors' conductance rise in steps.
Noise knows the conductance. At equilibrium, random current pulses give C_I = 2kTG₀ (fluctuation–dissipation); in engineering form i_rms = √(4kTGΔf).

3Equation sheet

t_B = L/uEq. 4.6

Ballistic transit time; u = ⟨|v_z|⟩.

t = L/u + L²/2DEq. 4.7

Ballistic + diffusive "polynomial"; D = ⟨v_z²τ⟩.

t = t_B(1 + L/λ), λ = 2D/uEq. 4.1

Same thing, defining Datta's mean free path.

G = G_Bλ/(L + λ)Eq. 4.2

G_B = q²D/2t_B (Eq. 4.3).

G = σ/(L + λ) × {1, W, A}Eq. 4.5

σ = G_Bλ × {1, 1/W, 1/A}; Ohm's law for L ≫ λ.

λ = vτ{2, π/2, 4/3}Eq. 4.10

From u = v{1, 2/π, 1/2} and D = v²τ{1, 1/2, 1/3} (Eqs. 4.8–4.9).

G_B = (q²/h)MEq. 4.12

M modes (integer); M = floor[(hDv/2L){1, 2/π, 1/2}] (Eq. 4.13).

G₀ = C_I/2kTEq. 5.4

C_I = ∫dτ⟨I(t₀+τ)I(t₀)⟩ at equilibrium.

C_I = 2(q²/t)f₀(1 − f₀)Eq. 5.8

One-level resistor; I₀ = (q/t)f₀(1 − f₀) (Eq. 5.7).

−∂f₀/∂E = f₀(1 − f₀)/kTEq. 5.5

Links the two-contact picture to the noise picture.

i_rms = √(4kTGΔf)Johnson–Nyquist

One-sided current noise in bandwidth Δf (= 2C_IΔf).

4Physics refresher: random walks and averages

Random walk

Each step in a random direction: the mean displacement stays zero but ⟨x²⟩ = 2Dt grows linearly in time, so the typical distance grows as √t. Crossing L takes ≈ L²/2D.

Angular averages

	1D	2D	3D
⟨\|cosθ\|⟩	1	2/π	1/2
⟨cos²θ⟩	1	1/2	1/3
λ/vτ	2	π/2	4/3

Why 2 in 1D?

With isotropic scattering in 1D, after a collision the electron goes forward or backward with equal chance, so only half the collisions reverse it: the backscattering time is 2τ and λ = 2vτ.

Spectral density vs correlation

Datta's C_I is the zero-frequency current correlation. The familiar one-sided noise spectral density is S_I = 2C_I = 4kTG, so i_rms² = 4kTGΔf.

5Glossary

Term	Meaning
Ballistic	L ≪ λ: electrons cross without backscattering; t ∝ L, G independent of L.
Diffusive	L ≫ λ: random walk; t ∝ L², G ∝ 1/L (Ohm's law).
Diffusion constant D	⟨v_z²τ⟩; ⟨x²⟩ = 2Dt.
Mode (channel)	An independent transverse state (like a waveguide mode) that carries q²/h of conductance.
Sharvin resistance	Ballistic resistance of a wide contact, inversely proportional to its area.
Conductance quantization	Steps of 2q²/h in the conductance of narrow point contacts (two spins per spatial mode).
Fluctuation–dissipation theorem	Links the low-bias conductance to equilibrium current fluctuations: G₀ = C_I/2kT (Kubo formula).
Johnson–Nyquist noise	Thermal current noise of a resistor, S_I = 4kTG.

6Calculators

Transit time and the new Ohm's law

Length L (nm)Mean free path λ (nm)Average velocity u (10⁵ m/s)Modes M

Ballistic t_B

Transit time t

G / G_B

Resistance R

From vτ to Datta's λ

Mean free flight vτ (nm)

λ in 1D

λ in 2D

λ in 3D

Johnson–Nyquist noise

Resistance R (Ω)Temperature T (K)Bandwidth Δf (Hz)

C_I = 2kT/R

S_I = 4kT/R

i_rms

7Self-check quiz

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

8Practice problems

q²/h = 38.74 µS, k = 1.381 × 10⁻²³ J/K.

P1. Transit timesEASY

L = 100 nm, λ = 50 nm, u = 10⁵ m/s. Find t_B, t and G/G_B.

Solution

t_B = 1 ps; t = 1 × (1 + 2) = 3 ps; G/G_B = 50/150 = 1/3.

P2. A 2D conductorEASY

M = 20, λ = 200 nm, L = 1 µm. Find G and R.

Solution

G = 38.74 µS × 20 × 200/1200 = 129 µS; R = 7.74 kΩ.

P3. Angular factorsEASY

For vτ = 30 nm, find Datta's λ in 2D and in 3D.

Solution

2D: (π/2) × 30 = 47.1 nm; 3D: (4/3) × 30 = 40 nm.

P4. From v and τ to tMEDIUM

3D conductor: v = 10⁵ m/s, τ = 0.1 ps. Find D, u, λ and the transit time through L = 1 µm.

Hint

D = v²τ/3, u = v/2.

Solution

D = 3.33 × 10⁻⁴ m²/s (3.33 cm²/s), u = 5 × 10⁴ m/s, λ = 2D/u = 13.3 nm = (4/3)vτ. t = L/u + L²/2D = 20 ps + 1.50 ns = 1.52 ns: deep in the diffusive regime.

P5. A quantum point contactMEDIUM

A narrow constriction in a 2D electron gas with Fermi wavelength 40 nm is 100 nm wide. Estimate the number of spatial modes and the conductance, counting two spins.

Hint

Modes ≈ floor(2W/λ_F).

Solution

floor(200/40) = 5 spatial modes; G = 5 × 2q²/h = 387 µS, R = 2.58 kΩ.

P6. Noise of one levelMEDIUM

A one-level resistor with ε = µ₀, transit time t = 1 ps, at 300 K. Find the one-way current I₀ and G₀ from the noise.

Hint

f₀(1 − f₀) = 1/4 at ε = µ₀.

Solution

I₀ = (q/t)/4 = 40 nA. G₀ = (q²/t)f₀(1 − f₀)/kT = 1.55 µS (as in the noise widget).

P7. Johnson noiseHARDER

Find C_I and i_rms in 1 MHz at 300 K for R = 1 kΩ, 25.8 kΩ and 1 MΩ.

Solution

1 kΩ: C_I = 8.3 × 10⁻²⁴ A²s, i_rms = 4.07 nA; 25.8 kΩ: 0.80 nA; 1 MΩ: 0.13 nA. Better conductors are noisier in current.

9Common mistakes

MISTAKE"Datta's λ is the average distance between collisions, vτ."

CORRECTλ = vτ × {2, π/2, 4/3}: it is a backscattering length, longer than vτ.

MISTAKE"Diffusive transport just takes a bit longer than ballistic."

CORRECTThe diffusive time grows as L²: a 10× longer wire takes 100× longer.

MISTAKE"The ballistic conductance always grows smoothly with width."

CORRECTIn narrow conductors it rises in steps of q²/h per mode (2q²/h per spatial mode).

MISTAKE"No voltage, no current."

CORRECTAt equilibrium current fluctuates in both directions; its noise C_I = 2kTG₀.

MISTAKE"A good conductor is a quiet conductor."

CORRECTCurrent noise 4kTG grows with G: lower resistance means larger current noise.

MISTAKE"Ohm's law is the general rule and ballistic transport an exception."

CORRECTBoth are limits of one law, G = G_Bλ/(L + λ).

10Prepare for circular session C2

A · Refresher clinic (pairs)

Flux = population ÷ residence time, with a non-physics example
Why diffusion time grows as L²
What a mode is, and why G_B comes in steps
Degenerate vs non-degenerate: sketch both

B · Lab: measure λ and R_B

Run Part 2 of the Week 4 notebook. Bring answers: Is λ closer to vτ or (4/3)vτ? Which part of R can better contacts reduce? What changes when vτ = 200 nm?

C · Problem clinic

Problems like P1, P2 and P7 above.

D · Derivation + two-minute explanation

Derive σ = q²(D/AL)D_diff from G = q²D/2t and t = L²/2D_diff. Be ready to explain any Week 3–4 idea for two minutes without equations.

11Explore

Files are in the course folder (course_materials).

Week 4 slides Widget: transit-time race Widget: conductance from noise Widget: 3D nanowire (λ = 4/3 vτ)Colab notebook Week 3 toolkit

Mini-investigations (15 min each)

In the transit-time widget, measure t for λ = 30 nm and λ = 120 nm. Plot t/t_B against L/λ for both: do they fall on one line?
In the noise widget, find the level position ε at which G₀ falls to 10% of its peak (300 K). Compare with the f₀(1 − f₀) prediction.
In the 3D nanowire widget, use the quick measurement at L = 100, 200, 400 nm and extract λ from 1/T = 1 + L/λ.

12Can I…?

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

explain t ∝ L (ballistic) and t ∝ L² (diffusive)use t = t_B(1 + L/λ) to get G = G_Bλ/(L + λ)write the conductivity for 1D, 2D and 3D conductorsexplain λ = vτ{2, π/2, 4/3}explain modes and conductance stepsstate G₀ = C_I/2kT and explain why noise reveals Gcompute Johnson noise for a given resistorextract R_B and λ from R versus L data

13Looking ahead: Week 5

Energy band model (Datta Ch. 6): where D(E) and M(E) come from.

In a box of length L, which wave vectors k are allowed for an electron wave?
If allowed k values are evenly spaced, are allowed energies E = ħ²k²/2m evenly spaced too?
How does the number of states with energy below E grow with E in 1D, 2D and 3D?

Ballistic and diffusive transport, modes, and conductance from fluctuation

1This week's plan

By the end of the week you should be able to

2Key ideas in five sentences

3Equation sheet

4Physics refresher: random walks and averages

5Glossary

6Calculators

Transit time and the new Ohm's law

From vτ to Datta's λ

Johnson–Nyquist noise

7Self-check quiz

8Practice problems

9Common mistakes

10Prepare for circular session C2

11Explore

Mini-investigations (15 min each)

12Can I…?

13Looking ahead: Week 5