A gate-all-around nanowire confines electrons in a cylinder of radius R. Treating the channel as an infinite circular well gives quantized radial subbands from the zeros of the Bessel function Jl:
E_l,n = ħ² α_l,n² / (2 m* R²)
α_01 = 2.405 (1st subband, l=0)
α_11 = 3.832 (2nd subband, l=±1)
α_02 = 5.520 (3rd subband, l=0)
Because E ∝ 1/R², shrinking the wire pushes the lowest subband up in energy — the gate has to work harder to pull it down to the source Fermi level before current can flow. That extra gate voltage is modeled here as a threshold-voltage shift:
V_th(D) = V_th0 + E₁(D) / q
I_D ∝ exp[(V_G − V_th) / (n·kT/q)] below threshold (subthreshold leakage)
I_D ∝ (V_G − V_th)² above threshold (square-law MOSFET)
- Diameter — smaller D raises every subband energy and Vth; watch the accumulation ring in the cross-section thin out and dim at a fixed gate voltage as you shrink the wire.
- Material — a lighter effective mass (InAs) confines much more weakly than Si for the same diameter, since E₁ ∝ 1/m*.
- Gate voltage — sweeps the channel from depleted (barrier, only leakage current) through threshold into strong inversion (full conduction); the band diagram's dip under the gate deepens as VG rises.
- Temperature — sets the thermal energy kT that both smears the subthreshold turn-on and thermally populates higher subbands.
- Cross-section / Band diagram — switch the view: the cross-section (drag to pan, scroll to zoom) shows the wire and gate-all-around shell face-on; the band diagram shows the conduction-band edge bending along the channel length and electrons flowing under bias.
Real-world relevance: this radial-confinement threshold shift is exactly what limits how far gate-all-around silicon nanowire and nanosheet transistors — the leading device architecture beyond FinFETs at sub-3 nm process nodes — can be scaled down before Vth control and drive current collapse.