Week 7 Study Toolkit · Nanoelectronics

1This week's plan

Read
Datta Part A, Chapters 8–9 (pp. 99–119). Appendix E is optional (AC transport).

Run
Colab notebook week07_two_potentials.ipynb: both widgets, the diffusion-equation solver, the Monte Carlo of µ^±, and Exercise 1.

Practise
Quiz (aim for 8/10), the practice problems and one mini-investigation.

By the end of the week you should be able to

Solve the diffusion equation with the modified boundary conditions (Eq. 8.6) and get G = σ₀A/(L + λ).
Explain µ⁺ and µ⁻, solve Eq. 8.13 with µ⁺(0) = µ₁, µ⁻(L) = µ₂, and use I = (q/h)M(µ⁺ − µ⁻).
Write the Boltzmann equation, explain each term, and the relaxation-time approximation.
Distinguish the electrochemical µ and the electrostatic U, and explain why equilibrium means constant µ.
Explain kinetic inductance and quantum capacitance in AC transport.

Optional video: Datta's nanoHUB-U course, Unit 3: L3.2–L3.4 and L3.8 (diffusion equation, electrochemical potentials, Boltzmann equation).

2Key ideas in five sentences

The diffusion equation also works for ballistic channels if its boundary conditions include an interface drop qIR_B/2 at each end; then G = σ₀A/(L + λ).
Two streams, two potentials. Right-movers carry µ⁺, left-movers µ⁻; I = (q/h)M(µ⁺ − µ⁻), with µ⁺(0) = µ₁ and µ⁻(L) = µ₂.
Ballistic is the cleanest case: µ⁺ = µ₁ and µ⁻ = µ₂ everywhere, so the whole drop sits at the two ends.
The Boltzmann equation is Newton's law for a cloud of particles in phase space plus a scattering term that pushes f towards equilibrium.
Two potentials: µ = (µ − U) + U. Equilibrium needs constant µ, not constant U: drift and diffusion share one σ₀ (Einstein relation). For AC signals both matter: L_K, C_Q from transport, L_M, C_E from electromagnetics.

3Equation sheet

dI/dz = 0, I/A = −(σ₀/q)dµ/dzEqs. 8.1–8.2

Continuity and the diffusion (drift–diffusion) equation.

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

Modified boundary conditions; R_B = λ/σ₀A = h/q²M (Eq. 8.7).

I = σ₀AV/(L + λ)Eq. 8.5

New Ohm's law from the diffusion equation.

I = (q/h)M(µ⁺ − µ⁻)Eq. 8.8

Current from the quasi-Fermi levels (Eq. 8.23 with thermally averaged M).

I = −(σ₀A/q)dµ^±/dzEq. 8.13

With µ⁺(0) = µ₁, µ⁻(L) = µ₂ (Eqs. 8.11–8.12); µ = (µ⁺ + µ⁻)/2 (Eq. 8.14).

µ⁺ − µ⁻ = (µ₁ − µ₂)/(1 + L/λ)Eq. 11.13

Constant along the channel (also Eq. 12.1).

f⁺ = f₁, f⁻ = f₂Eqs. 8.15–8.16

Exact for a ballistic channel with good contacts.

µ = (µ − U) + UEq. 9.1

Electrochemical = chemical + electrostatic.

∂f/∂t + v_z∂f/∂z + F_z∂f/∂p_z = S_opfEq. 9.5

Boltzmann transport equation; S_opf ≈ −(f − f̄)/τ (Eq. 9.8).

∂µ⁺/∂z = ∂µ⁻/∂z = −(µ⁺ − µ⁻)/λ, λ = 2|v_z|τEq. 9.16

Linearised BTE: the field terms cancel.

L_K = (h/q²)/⟨2Mv_z⟩, C_Q = (q²/h)⟨2M/v_z⟩Eq. 9.21

Kinetic inductance and quantum capacitance per unit length.

4Physics refresher: phase space, Poisson, potentials

Phase space

A particle is a point (z, p). Newton: dz/dt = ∂E/∂p, dp/dt = −∂E/∂z. Without collisions, a cloud of particles keeps its density along each trajectory (Liouville): that is the collisionless BTE.

Poisson and built-in fields

d²U/dz² ∝ charge. Wherever two materials meet (junctions, heterostructures, contacts) charge layers create a built-in field at equilibrium.

Einstein relation in two lines

For non-degenerate electrons n ∝ e^{−(U−µ)/kT}, so dn/dz = −(n/kT)dU/dz. Zero net flux at constant µ: the drift flux n·µ̃·(−dU/dz)/q must equal the diffusion term D·dn/dz = −D·(n/kT)dU/dz, hence D = µ̃·kT/q.

µ as a measure of filling

For a 2D band with constant D: n = D·kT·ln(1 + e^(µ−U)/kT). Given n and U you can read off µ, which is how the BTE widget plots µ(z).

5Glossary

Term	Meaning
Quasi-Fermi level	An electrochemical potential for a subset of states out of equilibrium (here µ⁺ for right-movers, µ⁻ for left-movers).
Interface (Sharvin) resistance	R_B = h/q²M, split as R_B/2 at each end of the channel.
Electrochemical potential µ	How high the states are filled; equal everywhere at equilibrium.
Electrostatic potential U	Where the states are; can vary at equilibrium (built-in fields).
Chemical potential µ − U	Degree of filling, set by the density.
Boltzmann transport equation	Evolution of f(z, p, t): drift in phase space plus scattering.
Relaxation-time approximation	S_opf ≈ −(f − f̄)/τ.
Kinetic inductance L_K	Inductance from the electrons' inertia, (h/q²)/⟨2Mv_z⟩ per length.

6Calculators

Diffusion equation with modified boundary conditions

Modes MMean free path λ (nm)Length L (nm)Voltage V (mV)

R_B = h/q²M

R = R_B(1 + L/λ)

Current I

Drop at each end / in channel

Two potentials µ⁺, µ⁻ (µ₁ = 1, µ₂ = 0)

L/λ

δµ = µ⁺ − µ⁻

µ⁺(0), µ⁻(0)

µ⁺(L), µ⁻(L)

T = λ/(L + λ)

Kinetic inductance and quantum capacitance

Modes M (incl. spin)Velocity v (10⁵ m/s)

L_K

C_Q

1/√(L_KC_Q)

√(L_K/C_Q)

7Self-check quiz

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

8Practice problems

q²/h = 38.74 µS, h/q² = 25.81 kΩ, kT/q = 25.85 mV at 300 K.

P1. Where is the drop?EASY

M = 10 (R_B = 2.58 kΩ), λ = 100 nm. For L = 50, 100 and 500 nm find R and the fraction of V dropped at each interface.

Solution

R = R_B(1 + L/λ): 3.87, 5.16, 15.5 kΩ. Each interface drops (R_B/2)/R: 33%, 25%, 8.3%.

P2. Two potentialsEASY

Find µ⁺ − µ⁻ (in units of µ₁ − µ₂) for L/λ = 0.2, 1 and 5. What is µ at z = 0 in each case?

Hint

δµ = 1/(1 + L/λ); µ(0) = µ₁ − δµ/2.

Solution

δµ = 0.833, 0.5, 0.167; µ(0) = 0.583, 0.75, 0.917 (with µ₁ = 1, µ₂ = 0).

P3. Current from QFLsEASY

A conductor has M = 20 modes and µ⁺ − µ⁻ = 1 meV at some point. Find the current.

Solution

I = (q/h)M(µ⁺ − µ⁻) = (q²/h) × 20 × 1 mV = 0.775 µA.

P4. A ballistic probeMEDIUM

A ballistic channel connects µ₁ = 1 and µ₂ = 0. A tiny probe in the middle measures (µ⁺ + µ⁻)/2. What does it read, and where is the voltage dropped?

Solution

µ⁺ = 1, µ⁻ = 0 everywhere, so the probe reads 0.5 anywhere in the channel: the whole drop is at the two interfaces, half at each.

P5. A built-in stepMEDIUM

A heterojunction has a 60 meV step in its band edge at 300 K, no bias. Find the ratio of electron densities on the two sides (non-degenerate) and the net current.

Solution

n_high/n_low = e^−60/25.85 = 0.098. Net current zero: drift and diffusion cancel because µ is constant.

P6. Einstein relationMEDIUM

Electrons with mobility 1000 cm²/Vs at 300 K (non-degenerate). Find the diffusion coefficient.

Hint

D = mobility × kT/q.

Solution

D = 0.1 m²/Vs × 0.02585 V = 2.59 × 10⁻³ m²/s = 25.9 cm²/s.

P7. A nanotube transmission lineHARDER

Metallic nanotube, M = 4, v = 8 × 10⁵ m/s. Find L_K and C_Q per µm, the wave speed 1/√(L_KC_Q) and the impedance √(L_K/C_Q).

Hint

L_K = (h/q²)/(2Mv), C_Q = (q²/h)(2M/v).

Solution

L_K = 4.03 nH/µm, C_Q = 387 aF/µm, speed = v = 8 × 10⁵ m/s, impedance = h/(2Mq²) = 3.23 kΩ. With the electrostatic C_E ≈ 28 aF/µm in series the wave speed rises to ≈ 3 × 10⁶ m/s.

9Common mistakes

MISTAKE"The diffusion equation only describes diffusive transport."

CORRECTWith the boundary conditions of Eq. 8.6 it gives the right answer for any L/λ.

MISTAKE"In a ballistic channel there is no voltage drop."

CORRECTThe drop is at the two interfaces, qIR_B/2 each; inside, µ⁺ and µ⁻ are flat.

MISTAKE"A voltage probe measures the electrochemical potential of the electrons."

CORRECTIt measures a weighted average of µ⁺ and µ⁻ (next week, Eq. 10.4).

MISTAKE"Equilibrium means no electric field."

CORRECTEquilibrium means constant µ; built-in fields with U(z) varying are common.

MISTAKE"Current is driven by the electric field."

CORRECTLow-bias current follows d(µ/q)/dz; the field part and the density part always come together.

MISTAKE"Kinetic inductance is just magnetic inductance."

CORRECTL_K comes from electron inertia and is thousands of times larger than L_M in a nanotube.

10Explore

Files are in the course folder (course_materials).

Week 7 slides Widget: µ⁺ and µ⁻Widget: Boltzmann equation Colab notebook Week 6 toolkit

Mini-investigations (15 min each)

In the µ^± widget, measure δµ for L/λ = 0.3, 1, 3 and plot 1/δµ against L/λ. Is it a straight line with intercept 1?
In the Boltzmann widget, set ΔU = 3 kT and τ long. Estimate the fraction of left-injected electrons that pass the step and compare with e⁻³.
In the notebook, change the nanotube to a 2D strip with M = 100 modes. How does L_K compare with L_M now?

11Can I…?

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

write the diffusion equation with the modified boundary conditionsderive G = σ₀A/(L + λ) from themexplain µ⁺ and µ⁻ and their boundary conditionscompute δµ and the interface drops for given L/λwrite the BTE and explain each termexplain why equilibrium means constant µ, not constant Ustate the Einstein relation and why it followsexplain kinetic inductance and quantum capacitance

12Looking ahead: Week 8 and circular session C4

Quasi-Fermi levels and the Hall effect (Datta Ch. 10–11), then C4: mid-course review and the midterm on Weeks 1–8.

A single defect with transmission T sits in a ballistic wire. Where does µ⁺ jump, and by how much?
Why might a two-terminal and a four-terminal measurement of the same defect give different resistances?
A magnetic field bends electrons sideways. What voltage appears across the width, and how does its sign tell n-type from p-type?

What and where is the voltage drop? Diffusion equation, Boltzmann equation, two potentials

1This week's plan

By the end of the week you should be able to

2Key ideas in five sentences

3Equation sheet

4Physics refresher: phase space, Poisson, potentials

5Glossary

6Calculators

Diffusion equation with modified boundary conditions

Two potentials µ⁺, µ⁻ (µ₁ = 1, µ₂ = 0)

Kinetic inductance and quantum capacitance

7Self-check quiz

8Practice problems

9Common mistakes

10Explore

Mini-investigations (15 min each)

11Can I…?

12Looking ahead: Week 8 and circular session C4