NbN detector block in 3-D

What you are looking at

The solver is the 3-D one from the vortex-tube page (ψ, the vector potential and the scalar potential μ, the current's own field on the boundary) with tdgl3d's heat model added: the generalized TDGL u(∂_t + iμ)ψ = ∇²ψ + (1 − θ − |ψ|²)ψ with u = 5.79, and one temperature θ = (T − 4.2 K)/(T_c − 4.2 K) heated by the full TDGL dissipation. Heat leaves the bottom of the film through a boundary resistance into four 5 nm SiO₂ layers, and the bottom layer loses it to the silicon. The contacts are normal metal held at 4.2 K. The first 5 nodes next to each contact have a stronger condensate, so the current converting from normal to supercurrent there does not heat the ends.

A photon raises θ in a Gaussian of radius 9 nm through the whole thickness, holding 55 % of its energy (the rest leaves as athermal phonons). In a 39 nm block that is a small kick: a 1550 nm photon (0.8 eV) warms its spot by about 3 K and the block recovers, which is why real detectors use 5 nm films. Near the depairing current, a bigger hit (a few eV, or the thinner block) drives the spot past T_c, vortices nucleate, and the block can switch to a resistive, self-heated state. Lower the bias or press Reset to recover it. In tdgl3d at 0.8 I_dep and no field, θ = 0.5 and 2 recover and θ = 5 switches.

Shunt. The bias source feeds the block through a series inductance L with a resistor R_sh in parallel, so L dI_block/dt = R_sh(I_bias − I_block) − V_block. Once a hotspot's resistance exceeds R_sh, the current moves into the shunt with time constant L/(R_sh + R_hotspot), the Joule heat drops and the block can cool; it then returns with L/R_sh. Two rules from the detector literature set R_sh: it must be small next to the hotspot so the block current falls below the current that keeps a hotspot alive (Skocpol, Beasley and Tinkham, J. Appl. Phys. 45, 4054 (1974)), and L/R_sh must be longer than the thermal recovery, or the current comes back before the film is cold and it latches (Kerman et al., PRB 79, 100509 (2009); Annunziata et al., J. Appl. Phys. 108, 084507 (2010); for an external shunt on a wide strip, arXiv:2304.08762). This block cools in roughly 150 ps (20 nm) to 280 ps (39 nm), so the default 20 Ω and 2 nH (100 ps) sits near the edge: try 0.5 nH, where the current returns before the film is cold, and 5 nH. On the CPU reference (20 nm block, 0.8 I_dep, a θ = 5 photon that latches with no shunt): 20 Ω with 0.5 nH pulls the current away before a hotspot forms, and 20 Ω with 3 nH lets a normal region grow to about 150 Ω, diverts the block current to 0.3 I_dep, and retraps by about 70 ps while the current returns. The voltage for the circuit is read inside the contact banks, which carry about 0.9 mV of conversion voltage that is not hotspot.

Two simplifications to keep it live: λ is far shorter than in real NbN (κ of 1 to 3 instead of about 50), because the explicit step shrinks as 1/κ², and the oxide and silicon have the film's footprint, so heat cannot spread sideways past the edges. Grid spacing is h = ξ/2 = 4.9 nm.

In ramp mode the bias sweeps a bipolar triangle, 0 → +peak → 0 → −peak → 0, over and over, and the I–V curve is drawn from it, so the trace is a real dynamic sweep, not a set of settled points: a fast sweep overshoots the true switching current. Vortices crossing the block make the first voltage, then the dissipation heats it and it switches to the normal, self-heated branch; on the way down it stays resistive until the heat can leave, so it retraps well below where it switched, or not until the current is gone. That hysteresis is the same mechanism that latches a real detector. At each zero crossing the sweep waits until the film is below about 4.4 K and |ψ| has grown back, so each half starts from a cold, superconducting block; the switching and retrapping currents are given for both signs and match by symmetry. The dashed line is the normal-state resistance across the voltage probes: NbN's 900 Ω per square at 5 nm, scaled to the block's thickness. The simulated normal branch lands on it (about 114 Ω for the 39 nm block), which checks the voltage units.

Why it latches: once the block has switched, it is normal everywhere and its own Joule heat holds it far above T_c. That heat can only leave downwards, through the film's boundary, 20 nm of SiO₂ and into the silicon, and a thick block stores much more of it per area than a 5 nm wire, so it cools slowly: about 280 ps (1/e) for the 39 nm block and 150 ps for 20 nm, against about 50 ps for a 5 nm film. A sweep of tens of ps cannot retrap it, and even a slow sweep only retraps at a small fraction of the switching current. At zero current it always recovers, which is why the ramp waits at each zero crossing until the block is cold and superconducting again before sweeping the other way (the state readout says "cooling" meanwhile). The heat-escape slider speeds up both steps of that path to show how a better heat-sunk film retraps higher.

Vortices without latching: the Flux flow button puts the 20 nm block in 0.5 H_c2 (1.7 T) along z at a bias of 0.2 I_dep. The field fills the block with about ten vortices, the bias pushes them across from one edge to the other, and new ones enter as old ones leave, so there is a steady flux-flow voltage of about a tenth of the normal-state value. That dissipation is small and spread out, so the oxide can carry it away: in the CPU reference the film stays below about 5 K for 600 τ with the real SiO₂ cooling. It latches only when the dissipation outruns the cooling. Raise the bias to 0.3 I_dep, or the field to 0.55 H_c2, and the vortices merge into a nearly normal, self-heated state. The Joule heating switch turns the heat source off, which gives the isothermal TDGL of the older 2-D demos: vortices flow at any bias and the block never runs away.

The |ψ| and phase maps are painted on the film's faces straight from the solver's nodes. In the phase view each vortex is a point where all the colours meet, and the colours wind once around it. The smooth colour bands between vortices depend on the gauge the solver uses for the field; only the windings are physical. The vortex-tube surface is drawn where |ψ| drops to the chosen fraction of the bulk |ψ| in each layer, so a field that weakens the whole film does not paint false sheets along its faces.

The bipolar sweep on the CPU reference (20 nm block, fastest rate, no field): from a fresh block the positive and negative halves are mirror images, switching at 1.07 I_dep and latching. The positive half reaches zero with the film near 15 K, and the block then waits about 3,100 τ (270 ps) at zero to cool below 4.4 K with |ψ| fully back. The negative half then switches the same way and ends a little hotter (about 16 K), because the oxide still holds a trace of the first half's heat.

What the reference sweep shows (the page's own model run on the CPU, 39 nm block, default sweep): the block stays superconducting up to about the depairing current (only the small L dI/dt voltage shows), switches at about 1.0 I_dep straight onto the normal line, and heats to well above T_c. A 39 nm block makes so much Joule heat per area that it stays normal all the way down and only recovers while resting at zero: it latches. The 20 nm block makes half the heat per area and retraps sooner; a real 5 nm detector, less again.

Checking the GPU solver against the double-precision reference…