University of Kurdistan logoNanoelectronics: A New Perspective on TransportWeek 4 · Ballistic and diffusive · C2
University of Kurdistan logo
WEEK 4 · BALLISTIC AND DIFFUSIVE TRANSPORT · C2

Nanoelectronics:
A New Perspective on Transport

Transit times, the new Ohm's law, modes, and conductance from fluctuation
Prof. Mohammad RazaghiUniversity of Kurdistan · Graduate course in Electronics Engineering
Based on S. Datta, Lessons from Nanoelectronics, Part A: Basic Concepts, Chapters 4–5
Part 1 is the fixed lecture (Ch. 4–5). Part 2, after the divider, is circular session C2 on Weeks 3–4.

RecapWhere we are

Week 3 gave us

  • G(E) = q²D(E)/2t(E): states × ease of transport
  • I = (1/q)∫G(E)(f₁ − f₂)dE
  • The heat goes to the contacts

Today's question

What sets the transit time t, and how does it turn G = q²D/2t into Ohm's law for long wires?

Week 4Today's goals

  1. Explain t ∝ L (ballistic) and t ∝ L² (diffusive), and combine them: t = L/u + L²/2D = tB(1 + L/λ).
  2. Derive the new Ohm's law G = GBλ/(L + λ) for 1D, 2D and 3D conductors.
  3. Explain Datta's mean free path λ = vτ × {2, π/2, 4/3} from angular averaging.
  4. Say what the number of modes M means and why GB comes in steps.
  5. Get the conductance from equilibrium noise: G₀ = CI/2kT.

Physics refresherRefresher: random walks

  • A bullet covers a distance ∝ t.
  • A random walker covers only ∝ √t: ⟨x²⟩ = 2Dt, with diffusion constant D.
  • To cross a length L: ballistic t = L/u, diffusive t ≈ L²/2D.
Doubling L doubles the ballistic time but quadruples the diffusive time.

Datta Ch. 4From transit time to the new Ohm's law

t = L/u + L²/2DEq. 4.7, a "polynomial" in L
t = tB(1 + L/λ),   λ = 2D/uEq. 4.1
G = GBλ/(L + λ)Eq. 4.2, with GB = q²D/2tB

Same law in every dimension (Eq. 4.5)

G = σ/(L + λ) × {1, W, A} for 1D, 2D, 3D, with σ = GBλ × {1, 1/W, 1/A}. Ohm's law G = σA/L is the L ≫ λ limit.

Even a zero-length channel with perfect contacts has the resistance of a channel of length λ: that is RB.
Ballistic: D ∝ AL and t ∝ L, so G_B is independent of L. Diffusive: t ∝ L², so G ∝ 1/L.

InteractiveExplore: the transit-time race

The transit-time race: ballistic (t ∝ L) versus diffusive (t ∝ L²)  (Datta Sec. 4.1, Eqs. 4.1–4.7)

The gold "bullet" never scatters. The blue electrons are backscattered at random (mean free path λ) and stumble along. Each keeps a stopwatch until it reaches the drain. Datta: t = L/u + L²/2D = tB(1 + L/λ), and since G = q²D/2t, G = GBλ/(L + λ).

Try this

  1. L = 300 nm, λ = 60 nm. Predict the average time before watching (bullet: 3 ps).
  2. Short channel (L < λ): scattered electrons finish near the bullet.
  3. Long channel: the times spread out; the average follows L²/2D.
  4. Press Measure: the red points land on t = tB(1 + L/λ) and on G/GB = λ/(L + λ).
  5. Change λ and measure again: Ohm's law (λ/L) appears only for L ≫ λ.

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

Prediction answer: about 18 ps = 3 ps × (1 + 300/60). In this 1D model the polynomial t is exact.

Datta Sec. 4.1, Appendix BAngular averaging: why λ is not just vτ

u = ⟨|vz|⟩ = v{1, 2/π, 1/2}Eq. 4.8 · 1D, 2D, 3D
D = ⟨vz²τ⟩ = v²τ{1, 1/2, 1/3}Eq. 4.9
λ = 2D/u = vτ{2, π/2, 4/3}Eq. 4.10

Even in 1D the factor is 2: with isotropic scattering only half the collisions reverse the electron.

This is why the Week 1 3D widget used λ = (4/3)vτ, and why its measurement agreed.

Datta Sec. 4.2Channels for conduction: modes

GB = (q²/h) × MEq. 4.12 · q²/h ≈ 40 µS, M integer
M = floor[(hDv/2L){1, 2/π, 1/2}]Eq. 4.13
  • Large conductors: GB ∝ area (Sharvin resistance, metals, 1969).
  • Small conductors (1980s experiments): GB drops in integer steps.
  • M ≈ number of half de Broglie wavelengths across the width, like waveguide modes (Ch. 6).
Point-contact experiments show steps of 2q²/h because each spatial mode carries two spins.

Datta Sec. 5.1Conductance from fluctuation

  • At V = 0 the average current is zero, but electrons still cross in both directions.
  • The current correlation CI = ∫dτ⟨I(t₀+τ)I(t₀)⟩ is not zero.
G₀ = CI / 2kTFluctuation–dissipation theorem, Eq. 5.4
The low-bias conductance is an equilibrium property: you can "listen" to a resistor to know it (Kubo formula).

Datta Sec. 5.2One level, then many

I₀ = (q/t) f₀(1 − f₀)Eq. 5.7 · one-way equilibrium current
CI = 2(q²/t) f₀(1 − f₀)Eq. 5.8 · each pulse adds area q²/t
G₀ = q² ∫dE [f₀(1 − f₀)/kT] D/2tEq. 5.10 · many levels

Two routes, one answer

Chapters 2–3: f₁ − f₂ → −∂f₀/∂E. Noise: f₀(1 − f₀)/kT, with 1 − f₀ from the exclusion principle. They agree because

−∂f₀/∂E = f₀(1 − f₀)/kTEq. 5.5

CI = 2qI₀: equilibrium noise looks like the shot noise of two opposing currents I₀. For phonons the factor is 1 + n (Bose).

InteractiveExplore: listening to a resistor

Conductance from fluctuation: listening to a resistor at equilibrium  (Datta Ch. 5)

With no voltage, electrons still hop through a one-level resistor in both directions at random. Each crossing is a current pulse of height q/t lasting t (Fig. 5.3). The average current is zero, but its fluctuation CI is not, and the fluctuation-dissipation theorem says G₀ = CI/2kT (Eq. 5.4).

Average current ⟨I⟩–
Fluctuation CI = Var(Q)/Tw–
G₀ from noise, CI/2kT–
G₀ theory (q²/t) f₀(1−f₀)/kT–

Try this

  1. At V = 0, watch the ± pulses: the average is zero, the spread is not.
  2. Press Collect 3000 more windows: G₀ from noise matches the theory within about 1%.
  3. Move ε away from µ₀: noise and G₀ fall together by f₀(1 − f₀).
  4. Turn on 5 mV: ⟨I⟩/V gives the same G₀, but only after many windows.

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

Engineering form: Johnson–Nyquist noise, current spectral density 4kTG (one-sided) = 2C_I.

ApplicationJohnson–Nyquist noise in numbers

irms = √(4kTGΔf) at 300 K, bandwidth Δf = 1 MHz:

4.07 nAR = 1 kΩ
0.80 nAR = 25.8 kΩ (one mode)
0.13 nAR = 1 MΩ
The better the conductor, the larger its current noise. Noise sets the floor for every low-level measurement in nanoelectronics.

CheckConcept check

One minute with a neighbour, then click to reveal.

1. Why is GB independent of length while the diffusive G ∝ 1/L? (click)
G = q²D/2t with D ∝ L. Ballistic t ∝ L → G independent of L; diffusive t ∝ L² → G ∝ 1/L.
2. Why is Datta's 1D mean free path 2vτ? (click)
With isotropic scattering only half the collisions reverse the electron, so backscattering happens every 2τ.
3. Why do small point contacts show conductance steps? (click)
GB = (q²/h)M with M an integer number of modes; opening the contact adds modes one at a time.
4. A resistor sits in a drawer at room temperature. Does current flow through it? (click)
Yes, randomly in both directions; the average is zero, and the fluctuation CI = 2kTG₀.

Wrap-upFixed session summary

Take-home

  • t = L/u + L²/2D = tB(1 + L/λ) ⇒ G = GBλ/(L + λ)
  • λ = vτ{2, π/2, 4/3} from angular averaging
  • GB = (q²/h)M: modes, steps in small conductors
  • G₀ = CI/2kT: noise knows the conductance

Next: circular session C2

Refresher clinic → lab: measure λ and RB → problem clinic → derivation task. Open week04_ballistic_diffusive_C2.ipynb (Part 2).

PART 2

Circular session C2

Covers Weeks 3–4 · about 85 minutes

A · 15 min
Refresher clinic
B · 35 min
Lab: measure λ and RB
C · 25 min
Problem clinic
D · 10 min
Derivation + explanations
Timing follows the Week 4 notebook, Part 2.

Circular session C2Block A · Refresher clinic

In pairs: one explains, one sketches. Swap after each item.

1 · Flux

"Flux = population ÷ residence time" with a non-physics example.

2 · Diffusion

Why does the diffusion time grow as L²?

3 · Modes

What is a mode, and why does GB come in steps?

4 · Two limits

Degenerate vs non-degenerate: sketch G(E) and the window for each.

Circular session C2Block B · Lab: measure λ and RB like an experimentalist

Run Part 2 of the Week 4 notebook: it "measures" R at six lengths with the 3D scattering model (vτ = 50 nm, M = 10) and fits a line.

  1. Is the extracted λ closer to vτ or (4/3)vτ? Why?
  2. Which part of R can better contacts reduce, and which part never?
  3. Set vτ = 200 nm: what happens to intercept and slope?
Expect RB ≈ 2.7 kΩ and λ ≈ 69 nm: within a few percent of the truth.
This is the transfer-length method used for real contact-resistance measurements.

Circular session C2Block C · Problem clinic

P1. L = 100 nm, λ = 50 nm, u = 10⁵ m/s. Find tB, t and G/GB. (click)
tB = L/u = 1 ps; t = tB(1 + L/λ) = 3 ps; G/GB = λ/(L+λ) = 1/3.
P2. A 2D conductor: M = 20, λ = 200 nm, L = 1 µm. Find G and R. (click)
G = (q²/h)·M·λ/(L+λ) = 38.74 µS × 20 × 200/1200 ≈ 129 µS; R ≈ 7.74 kΩ.
P3. A 1 kΩ resistor at 300 K: find CI and irms in 1 MHz. (click)
CI = 2kT·G ≈ 8.3×10⁻²⁴ A²s; irms = √(4kTΔf/R) ≈ 4.1 nA.

Circular session C2Block D · Derivation task and two-minute explanations

Derive Datta's Eq. 1.6 (click each step)

Step 1: start (click)
G = q²D/2t, D = density of states of the whole channel.
Step 2: diffusive time (click)
t = L²/2Ddiff, so G = q²D·Ddiff/L².
Step 3: conductivity (click)
σ = GL/A = q²(D/AL)Ddiff: density of states per volume × diffusion constant.
Press the button for a two-minute topic.

Topics: flux and residence time · why heat is in the contacts · degenerate vs non-degenerate · ballistic vs diffusive time · what a mode is · why noise reveals G.

Wrap-upNext week: where D(E) and M(E) come from

Before next class

  • Finish the Week 4 notebook (both parts)
  • Work through the Week 4 study toolkit
  • Read Datta Part A, Ch. 6 (pp. 59–74)

Week 5 preview

The E(k) relation, counting states in k-space, density of states in 1D, 2D, 3D, number of modes, and conductivity versus electron density.

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