Week 5 Study Toolkit · Nanoelectronics

1This week's plan

Read
Datta Part A, Chapter 6 (pp. 59–74), Sections 6.1–6.6.

Run
Colab notebook week05_band_model.ipynb: three widgets, the counting and identity checks, and Exercises 1–2.

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

By the end of the week you should be able to

Count states in momentum space (p = integer × h/L) and write N(p) and D(E) in 1D, 2D and 3D.
State and use D·v·p = N·d, and show how Datta's σ reduces to Drude's q²nτ/m for one band.
Compute the number of modes M = floor(2W/λ) and the ballistic conductance g(q²/h)M.
Say what "n" means for n-type, p-type and graphene, and why conduction is a Fermi-surface property.
Explain G_B ∝ n^(d−1)/d and the √n of ballistic graphene.

Optional video: Datta's nanoHUB-U course, Unit 2: L2.2–L2.7 (E(p), counting states, modes, electron density, conductivity vs n).

2Key ideas in five sentences

States are evenly spaced in momentum. Fitting an integer number of wavelengths into a box gives p = integer × h/L, so N(p) ∝ p^d and D(E) = dN/dE.
One identity links everything. For any single E(p) band, D(E)v(E)p(E) = N(E)·d; with it Datta's σ = q²(D/V)𝒟 becomes Drude's q²nτ/m with m = p/v.
Modes are wavelengths across the conductor. M = floor(2W/λ) in 2D; G_B = g(q²/h)M, with steps visible only when kT is smaller than the step spacing.
Conduction is a Fermi-surface property. Only states within a few kT of µ₀ matter; in a p-type band the window counts holes, in graphene electrons above and holes below the touching point.
Scaling with density. M ∝ p^d−1 and n ∝ p^d, so G_B ∝ n^(d−1)/d: √n in 2D.

3Equation sheet

σ = q²nτ/m, µ̃ = qτ/mEqs. 6.1–6.3

Drude formula and mobility (single-band special case).

σ(E) = q²(D/{L, WL, AL})𝒟(E)Eq. 6.5

Datta's general form (degenerate Einstein relation), averaged over −∂f₀/∂E (Eq. 6.6).

E = E_c + p²/2m | E = E_c + v₀pEqs. 6.12–6.13

Parabolic and linear dispersion; p = ħk.

v = dE/dp, m = p/vEqs. 6.14, 6.11

Velocity is the slope; Datta's mass, energy dependent for a linear band.

p = integer × h/LEq. 6.15

Allowed momenta; one state per h/L in each direction.

N(p) = {2L/(h/p), πLW/(h/p)², (4π/3)LA/(h/p)³}Eq. 6.17

States with momentum below p, per spin, in 1D, 2D, 3D.

D·v·p = N·dEq. 6.9

Holds for any single E(p) band.

M = floor{1, 2W/(h/p), πA/(h/p)²}Eqs. 6.19–6.20

Number of modes = wavelengths that fit in the cross-section.

R_B = (h/q²)/(gM)Sec. 6.4.1

g = 2 (spin), g = 4 (spin × valley: nanotubes, graphene).

n = N(µ₀)/{L, WL, AL}Eq. 6.21

Window average of N(E) = electrons (n-type) or holes (p-type).

σ = (q²/h)Mλ{1, 1/W, 1/A}Eq. 6.22

Conductivity from modes and mean free path.

G_B ∝ g^1/dn^(d−1)/dEqs. 6.24–6.25

√n in 2D (ballistic graphene).

4Physics refresher: waves, wavelengths and bands

Waves in a box

A string of length L fixed at both ends vibrates only when L = n·λ/2. For electrons, Datta uses a ring (periodic boundary): L = n·λ, so p = h/λ = n·h/L.

de Broglie

λ = h/p, k = 2π/λ, p = ħk. For a parabolic band p = √(2m(E − E_c)). Typical numbers: Si-like 0.2 m₀ at 25 meV, λ ≈ 17 nm; GaAs 0.067 m₀ at 50 meV, λ ≈ 21 nm; copper at E_F = 7 eV, λ ≈ 0.46 nm.

Effective mass in one paragraph

An electron in a periodic crystal is a Bloch wave. Near the bottom of a band its energy is approximately E_c + p²/2m with an effective mass m that encodes the effect of the ions. Light masses (GaAs 0.067 m₀) mean long wavelengths and few states; heavy masses mean many states per unit energy.

Why D(E) is constant in 2D

N ∝ p² ∝ (E − E_c), so dN/dE is constant. Per unit area and with g = 2: D = gm/(2πħ²) ≈ 8.35 × 10¹³ eV⁻¹cm⁻² for m = 0.2 m₀.

5Glossary

Term	Meaning
Dispersion relation E(p)	Energy versus momentum of a band; parabolic near most band edges, linear in graphene.
Effective mass	m = p/v (Datta); constant only for a parabolic band.
Periodic boundary conditions	Treating the conductor as a ring so that an integer number of wavelengths fits; convenient, exact only for large or ring-shaped conductors.
N(E)	Number of states with energy below E (above E for a p-type band).
Degeneracy factor g	Number of equivalent channels per mode: spins × valleys.
Fermi-surface property	A quantity that depends only on states within a few kT of µ₀ (conductivity, Hall coefficient).
Hole	An empty state in a downward band; "n" in the Drude formula for p-type conductors.
Double-ended D(E)	A density of states extending above and below a touching point, as in graphene (Fig. 6.3).

6Calculators

Wavelength and number of modes in a strip

E − E_c (meV)Effective mass m/m₀Width W (nm)

λ = h/p

2W/λ

M per spin

G_B (g = 2)

A 2D electron gas from its density

n (10¹² cm⁻²)Effective mass m/m₀Degeneracy g

D per area

E_F − E_c

λ_F

G_B per µm

Graphene from its density

n (10¹² cm⁻²)v₀ (10⁶ m/s)

E_F

λ_F

m = p/v

G_B per µm (g = 4)

7Self-check quiz

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

8Practice problems

h = 6.626 × 10⁻³⁴ J s, m₀ = 9.109 × 10⁻³¹ kg, q²/h = 38.74 µS.

P1. A wavelengthEASY

Find the de Broglie wavelength of an electron 50 meV above the band edge in GaAs (m = 0.067 m₀).

Solution

p = √(2mE) = 3.13 × 10⁻²⁶ kg m/s, λ = h/p = 21.2 nm.

P2. A 2D density of statesEASY

For m = 0.2 m₀ and g = 2, find D per unit area, and the density when E_F − E_c = 30 meV at low temperature.

Hint

D = gm/(2πħ²).

Solution

D = 8.35 × 10¹³ eV⁻¹cm⁻²; n = D × 0.030 eV = 2.5 × 10¹² cm⁻².

P3. Modes in a stripEASY

A 50 nm wide strip has λ_F = 12 nm. How many modes per spin, and what is G_B with g = 2?

Solution

2W/λ = 8.33 ⇒ M = 8; G_B = 2 × 38.74 µS × 8 = 620 µS (1.61 kΩ).

P4. A 2D channelMEDIUM

n = 5 × 10¹² cm⁻², m = 0.2 m₀, g = 2, width 1 µm. Find E_F − E_c, λ_F, M and G_B.

Hint

E_F − E_c = n/D.

Solution

59.8 meV, 11.2 nm, M = floor(2000/11.21) = 178, G_B = 13.8 mS (R_B ≈ 72 Ω).

P5. GrapheneMEDIUM

Graphene (g = 4, v₀ = 10⁶ m/s) with n = 10¹² cm⁻². Find p_F, E_F, λ_F and Datta's mass p/v.

Hint

Per spin and valley, n/g = πp²/h².

Solution

p_F = h√(n/4π) = 1.87 × 10⁻²⁶ kg m/s; E_F = v₀p_F = 117 meV; λ_F = 35.4 nm; m = p/v₀ = 0.021 m₀, and it grows as √n.

P6. Drude is a special caseMEDIUM

2D parabolic band, m = 0.2 m₀, E − E_c = 50 meV, τ = 0.1 ps. Compute σ = (q²/h)(2p/h)(π/2)vτ and q²nτ/m with n = πp²/h² (one spin).

Hint

Both reduce to q²πp²τ/(h²m).

Solution

Both give 294 µS per square (sheet resistance ≈ 3.4 kΩ).

P7. Scaling with densityHARDER

By what factor does G_B change if the carrier density is multiplied by 4 in a 2D conductor? By 8 in a 3D conductor? By 2 in a 1D wire with one subband occupied?

Hint

G_B ∝ n^(d−1)/d.

Solution

2D: 4^1/2 = 2. 3D: 8^2/3 = 4. 1D: no change (M = 1 as long as one subband is occupied).

9Common mistakes

MISTAKE"Insulators have few electrons."

CORRECTThey have about as many as metals; they lack states (modes) near µ₀.

MISTAKE"Conductivity is proportional to the total electron density."

CORRECTIt depends on states within a few kT of µ₀; Drude's n is meaningful only for a single band, and means holes in p-type.

MISTAKE"Allowed energies of a box are evenly spaced."

CORRECTAllowed momenta are evenly spaced (h/L); energies are not, which is why D(E) depends on E and dimension.

MISTAKE"Effective mass is always a constant."

CORRECTm = p/v is constant only for a parabolic band; for graphene it grows with energy.

MISTAKE"M is just a continuous number like 201.59."

CORRECTIt is an integer; the fractional value only appears after averaging over the thermal window.

MISTAKE"In p-type material the current is carried by positive particles."

CORRECTElectrons move in both cases; for a downward band the window simply counts empty states.

10Explore

Files are in the course folder (course_materials).

Week 5 slides Widget: counting states Widget: number of modes Widget: who conducts?Colab notebook Week 4 toolkit

Mini-investigations (15 min each)

In the modes widget, find the largest width W for which at most one mode is open at E − E_c = 20 meV (m = 0.2 m₀). Compare with λ/2 from the calculator.
In the "who conducts" widget (semiconductor), lower T from 600 K to 200 K with µ₀ at mid-gap. By how many orders of magnitude does G_B change? Compare with e^−E_g/2kT.
In the notebook, extend the G_B-versus-n plot to 3D (M = πA/(h/p)², n = (4π/3)(p/h)³) and check the slope 2/3.

11Can I…?

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

explain why allowed momenta are evenly spaced (h/L)write N(p) and D(E) in 1D, 2D and 3Dstate D·v·p = N·d and use it to get the Drude formulacompute λ and the number of modes for a stripexplain why conductance steps disappear at room temperatureexplain what n means in n-type, p-type and grapheneexplain why conduction is a Fermi-surface propertyexplain G_B ∝ n^(d−1)/d

12Looking ahead: Week 6 and circular session C3

The nanotransistor (Datta Ch. 7): what a gate does to D(E), why the current saturates, the role of charging and quantum capacitance. C3 is a Colab lab on DOS and modes in 1D–3D and a self-consistent nanotransistor.

A positive gate voltage lowers the channel states. Does that increase or decrease the current of an n-type channel?
Once µ₂ falls below the band edge, why should the current stop growing with drain voltage?
Two capacitors in series, C_E and C_Q: which one dominates when one of them is much smaller?

Energy band model: where D(E) and M(E) come from

1This week's plan

By the end of the week you should be able to

2Key ideas in five sentences

3Equation sheet

4Physics refresher: waves, wavelengths and bands

5Glossary

6Calculators

Wavelength and number of modes in a strip

A 2D electron gas from its density

Graphene from its density

7Self-check quiz

8Practice problems

9Common mistakes

10Explore

Mini-investigations (15 min each)

11Can I…?

12Looking ahead: Week 6 and circular session C3