Nanowire

Transverse quantization, ballistic transport and conductance quantization in nanowires

Parameters
Subband Dispersion E(k)
Transverse Energy Levels
1. What is a Nanowire?

A nanowire is an ultra-thin conducting or semiconducting rod — 1–100 nm in diameter and thousands of nanometers long. In such a structure, electrons are quantum-confined in two transverse directions, while along the wire axis they move freely. This creates a quasi-one-dimensional (1D) system — one of the most important objects in nanophysics.

GaAsm* = 0.067 m₀, mean free path ~500 nm
InAsm* = 0.026 m₀, mean free path ~800 nm
Sim* = 0.26 m₀, mean free path ~100 nm
2. Transverse Quantization

In the transverse cross-section, the electron "feels" the boundaries — similar to a quantum well, but now in two directions. As a result, energy is quantized — subbands appear.

Rectangular nanowire (W × H):

$$ E_{mn} = \frac{\hbar^2\pi^2}{2m^*}\left[\left(\frac{m}{W}\right)^2 + \left(\frac{n}{H}\right)^2\right], \quad m,n = 1,2,3,\ldots $$

Cylindrical nanowire (radius R):

$$ E_{mn} = \frac{\hbar^2 x_{mn}^2}{2m^* R^2} $$

where x_mn is the m-th zero of Bessel function J_n. Total energy along the wire axis with momentum k_z:

$$ E(k_z) = E_{mn} + \frac{\hbar^2 k_z^2}{2m^*} $$

Each (m,n) pair creates a separate 1D subband — parabolic dispersion from threshold energy E_mn.

3. Ballistic Transport and Landauer Formula

If the wire length L is less than the mean free path (L ≪ l_mfp), electrons traverse the wire without scattering — ballistic transport. In this case, conductance follows the Landauer formula:

$$ G = G_0 \cdot M(E_F) = \frac{2e^2}{h} \cdot M(E_F) $$

where M(E_F) is the number of occupied subbands (modes) at Fermi energy, G₀ = 2e²/h ≈ 77.48 μS — conductance quantum (factor 2 for spin degeneracy). When E_F crosses a new subband threshold, conductance increases step-by-step in multiples of G₀ — conductance quantization.

$$ G_0 = \frac{2e^2}{h} \approx 77.48\,\mu\text{S}, \qquad R_0 = \frac{1}{G_0} \approx 12.9\,\text{k}\Omega $$

This quantization was first observed in 1988 in GaAs/AlGaAs quantum point contacts (van Wees et al., Wharam et al.).

4. Diffusive vs Ballistic Transport

The nature of electron transport depends on the wire length relative to the mean free path (l_mfp):

Regime Condition Conductance
Ballistic L ≪ l_mfp G = G₀·M (L-დამოუკიდებელი)
Diffusive L ≫ l_mfp G = G₀·M·(l_mfp/L) ∝ 1/L
$$ \frac{1}{G_{total}} = \frac{1}{G_{ball}} + \frac{1}{G_{diff}} = \frac{h}{2e^2 M}\left(1 + \frac{L}{l_{mfp}}\right) $$

In the diffusive regime, Ohm's law applies — R ∝ L. Ballistic regime is observed even at room temperature in metallic nanowires (Au, Ag). As temperature increases, l_mfp decreases (phonon scattering) and the system transitions from ballistic to diffusive regime.

5. Quantities Reference
G₀ = 2e²/hConductance quantum ≈ 77.48 μS
R₀ = h/2e²Resistance quantum ≈ 12.9 kΩ
M(E_F)Number of occupied subbands (modes)
l_mfpMean free path — distance between scatterings
E_mnTransverse subband threshold energy
x_mnm-th zero of Bessel function J_n
m*Electron effective mass in material [units of m₀]
6. Applications
Nano-FETInAs, GaAs nanowire — transistor channel
Quantum sensorConductance change upon single molecule approach
Quantum computingMajorana fermions in InAs/Al nanowire
PhotodetectorInP, GaAs nanowire — single photon detection
ThermoelectricsSi nanowire — Seebeck coeff. 10× larger than bulk Si
Beginner Guide

Step 1 — Transverse Levels
GaAs, rectangular W=20, H=20 nm. After calculation, note the first subband threshold E₁₁ ≈ 0.028 eV. This means: an electron needs at least 0.028 eV to traverse the wire.

Step 2 — Ballistic Transport
Switch to "Ballistic Transport", T=4K. Notice the step-by-step conductance — at each threshold G increases by G₀ (77.48 μS). At T=300K the steps are smeared by thermal broadening.

Step 3 — Diffusive vs Ballistic
Switch to "Diffusive vs Ballistic". Note: in a short wire (L < l_mfp) G = const (ballistic), in a long wire G ∝ 1/L (Ohm's law). Crossover occurs near l_mfp.

Advanced Guide

InAs vs GaAs
InAs has m* = 0.026 m₀ (2.5× smaller than GaAs). Therefore subbands in InAs are much more widely spaced — first threshold is higher than GaAs. Also l_mfp = 800 nm — ballistic transport is more easily observable.

Rectangular vs Cylindrical
In rectangular wire (m,n) levels — two quantum numbers, (1,2) and (2,1) — are degenerate. In cylindrical wire Bessel zeros differ — asymmetric thresholds, n>0 levels are doubly degenerate (±n).

Nanowire — Quiz

1. The conductance quantum G₀ = 2e²/h ≈ 77.48 μS. The factor 2 means:

2. The condition for ballistic transport is:

3. When nanowire width W is doubled (W → 2W), the first subband threshold E₁₁:

4. In diffusive regime, nanowire resistance R vs length L: