Nanoresistor Size-Effect Simulator

Wire Geometry

Wire diameter D 20 nm

Grain size L_g 30 nm

Scattering Parameters

Surface specularity p 0.50

Grain-boundary reflection R 0.30

Live Readouts

Effective λ

— nm

Resistivity ρ

— μΩ·cm

ρ / ρ_bulk (Cu)

—×

Bulk copper: λ₀ = 39 nm, ρ₀ = 1.68 μΩ·cm at 300 K.

How it works

In the Drude model, resistivity is inversely proportional to the electron mean free path: ρ = m_ev_F/(n e² λ). In bulk copper λ₀ ≈ 39 nm at room temperature. Once a wire's diameter or grain size approaches λ₀, electrons scatter off surfaces and grain boundaries almost as often as they scatter off phonons in the bulk — and every one of those extra collisions randomizes momentum and adds resistance. This is why copper interconnects on a chip get resistive far faster than their cross-section shrinks.

Surface term (Fuchs–Sondheimer):
  ρ_surf/ρ0 = 1 + (3/8)(1−p)(λ0/D)

Grain-boundary term (Mayadas–Shatzkes):
  α = (λ0/Lg)·R/(1−R)
  ρ_gb/ρ0 = 1 / [1 − 1.5α + 3α² − 3α³·ln(1+1/α)]

Combined (Matthiessen's rule):
  ρ/ρ0 = ρ_surf/ρ0 + ρ_gb/ρ0 − 1
  λ_eff = λ0 / (ρ/ρ0)

D — wire diameter. Smaller D means the wire wall is reached more often, so this term matters more.
p — fraction of surface collisions that are specular (mirror-like, no momentum loss along the wire) vs. diffuse (randomizing, resistive). Real evaporated copper surfaces sit closer to p ≈ 0.5.
L_g — average grain size. Smaller grains mean electrons cross more grain boundaries per unit length.
R — probability an electron reflects off a grain boundary instead of transmitting through it.

The particles in the scene are a qualitative Monte Carlo illustration of the same three mechanisms (bulk phonon scattering, diffuse/specular wall bounces, grain-boundary reflection) driven by the same sliders; the numeric readouts come from the closed-form Fuchs–Sondheimer / Mayadas–Shatzkes / Matthiessen expressions above, which are the standard model used to predict copper-interconnect resistivity scaling in chip design.

Electron (drifting) Diffuse surface scatter Grain-boundary reflection Bulk (phonon) scatter

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