Scanning SQUID Microscopy (SSM) & Flux Lab

1. DC-SQUID Physics & Flux Quantization

A Direct Current Superconducting Quantum Interference Device (DC-SQUID) consists of a superconducting ring interrupted by two parallel Josephson junctions. In the superconducting state, the phase difference $\Delta \theta$ of the macroscopic Cooper-pair wavefunction around the loop is constrained by fluxoid quantization:

When an external magnetic flux $\Phi$ threads the pickup loop, quantum interference between the two junction supercurrents modulates the critical current: $I_c(\Phi) = 2 I_{c0} |\cos(\pi \Phi / \Phi_0)|$. Biasing the SQUID with a constant current $I_b > 2 I_{c0}$ converts flux variations into an oscillating voltage $V(\Phi)$ with steep transfer coefficients $V_{\Phi} = (\partial V / \partial \Phi)_{I_b} \approx 100\text{ to }500\,\mu\text{V}/\Phi_0$, enabling sensitivities down to $10^{-6}\,\Phi_0/\sqrt{\text{Hz}}$.

2. Spatial Point-Spread Function & Standoff Deconvolution

The spatial resolution $\delta x$ of an SSM is fundamentally governed by the sensor pickup loop geometry and its distance from the current source:

• Loop Diameter ($d_{\text{loop}}$): Modern micro-SQUIDs integrated on cantilever tips achieve sub-micron pickup loop apertures ($0.3\text{ to }1.5\,\mu\text{m}$).
• Standoff Height ($z_0$): Includes tip-to-package spacing plus the physical burial depth of the conductor through silicon or mold compound.
• Effective Resolution: $\delta x \approx \sqrt{d_{\text{loop}}^2 + z_0^2}$. At $z_0 = 2.5\,\mu\text{m}$, sub-3µm spatial fault localization is routinely obtained.

3. 2D Biot-Savart Fourier Inversion Algorithm

Reconstructing planar current density vectors $\mathbf{J}(x, y) = [j_x, j_y]$ from a measured magnetic field map $B_z(x, y)$ is an ill-posed inverse problem resolved via 2D Fast Fourier Transform (FFT) filtering:

Invoking the 2D continuity equation $\nabla \cdot \mathbf{J} = 0 \implies k_x j_x + k_y j_y = 0$, the individual Fourier components of the current density are directly inverted:

A high-frequency Hanning or Butterworth window suppresses exponential noise amplification at $k > 1/z_0$, isolating microampere shorts across 3D interposers.

4. 3D Advanced Packaging & Quantum Processor Applications

Scanning SQUID Microscopy delivers non-invasive diagnostics across two critical high-value domains:

• 3D Heterogeneous Chiplet Packaging: Tracing short-circuit current paths through through-silicon vias (TSVs) and solder micro-bumps without destructive de-encapsulation.
• Superconducting Qubit Devices: Imaging trapped Abrikosov flux vortices and stray magnetic dipoles that cause dephasing in transmon qubits at millikelvin temperatures.