I-V Characteristic

The current-voltage (I-V) characteristic is the "identity card" of an electronic device — it fully describes how the device responds to an applied voltage. In a classical resistor, I=V/R (Ohm's law) — a linear characteristic. In semiconductor devices, I(V) is strongly nonlinear: a p-n diode exponentially passes current in one direction and blocks it in the other; a tunnel diode exhibits a negative differential resistance (NDR) region due to quantum tunneling; a Schottky contact creates a barrier at the metal-semiconductor interface. At the nanoscale, physics changes once more — molecular junctions carry current through one or a few quantum levels, and nanowires exhibit ballistic transport. I(V) measurement is the foundation of nanoelectronics experiments.

The ideal p-n diode I(V) characteristic:

I_s — reverse saturation current, typically 10⁻¹²–10⁻⁹ A for silicon. n — ideality factor: n=1 (diffusion current, ideal), n=2 (recombination in depletion region). V_T = kT/e — thermal voltage: 25.85 mV at 300 K. Forward bias: I ∝ exp(V/V_T) — strong exponential growth. Reverse bias: I ≈ -I_s — nearly zero.

The Esaki tunnel diode (1958, Nobel 1973) — heavily doped p-n junction where the depletion region is narrow (~10 nm) and electrons quantum-mechanically tunnel through. Current has three components:

NDR — current decreases with increasing voltage — impossible classically, only quantum tunneling explains it. PVR (Peak-to-Valley Ratio) = I_peak/I_valley — figure of merit for NDR quality. Applications: high-speed oscillators, memory (SRAM), logic circuits.

At a metal-semiconductor interface, a barrier φ_b forms due to the difference in work functions. Current via thermionic emission:

A* — effective Richardson constant [A/cm²/K²]: Si ≈ 110, GaAs ≈ 4. φ_b — barrier height: Si/Au ≈ 0.8 eV, Si/Al ≈ 0.7 eV, GaN/Ni ≈ 0.9 eV. Schottky diode vs p-n: much faster (no minority carrier storage), lower forward voltage drop (~0.3 V vs ~0.7 V for Si p-n).

A single molecular orbital at energy E₀ between two metallic leads. Breit-Wigner transmission:

E₀ — HOMO or LUMO energy relative to Fermi level. Γ_L, Γ_R — molecule-electrode coupling strengths. Γ_L = Γ_R — symmetric contact (T_max = 1). Γ_L ≠ Γ_R — asymmetric, T_max < 1. I(V) is often S-shaped: plateau where E₀ enters the transport window. Experiment: STM break junction, MCBJ.

Landauer-Büttiker formalism extended to include scattering:

L — wire length. λ — mean free path: carbon nanotube ~1 μm, Si nanowire ~10–100 nm. L ≪ λ: ballistic regime — R = R_Q/N, independent of L! L ≫ λ: diffusive (Ohmic) — R ∝ L. R_Q = h/2e² = 12.906 kΩ — quantum of resistance, fundamental lower bound. Experimental confirmation: gold nanocontacts (1997, van Wees et al.) — G changes in steps of G₀ as atoms are pulled apart.

dI/dV contains more information than I(V): peaks = resonant levels; negative region = NDR; near-zero value = zero-bias conductance. In STM spectroscopy, dI/dV(V) is measured directly.