Quantum Simulation of Si/SiGe Devices with Experimentally Calibrated Micromagnet Effects

AI summary

Overview

Research area: Silicon spin-qubit hardware, specifically device-level simulation (quantum TCAD) of Si/SiGe heterostructure quantum dot arrays with integrated micromagnets for electric dipole spin resonance (EDSR). Technical level: Advanced. The paper assumes familiarity with Poisson–Schrödinger device solving, finite element magnetics, and rotating-wave-approximation qubit dynamics. Scope: An experimentally calibrated end-to-end 3D simulation pipeline for a 6-dot Si/SiGe device with two charge sensors, validated against measured qubit resonant frequencies and microwave power chevrons.

What This Paper Is About

Standard computer-aided design simulations of Si/SiGe spin-qubit devices use idealized assumptions — Anderson's rule for band alignment, perfect material interfaces, and constant micromagnet magnetization — and therefore miss effects that dominate real devices: interface dipoles, parasitic charge accumulation in the SiGe buffer, and magnetic hysteresis of on-chip micromagnets. The authors build a 3D model of a real 6-dot device, systematically correct each of these mismatches, and show that the calibrated model reproduces measured qubit resonant frequencies for all six qubits and the experimental microwave power chevron.

Key Contributions

1. Corrected band alignment for Si/SiGe and Si/Al₂O₃ interfaces. The default Anderson's rule produces incorrect band gaps for Si against Si₇₀Ge₃₀ and against Al₂O₃; the authors shift the electron affinities of Si₇₀Ge₃₀ and Al₂O₃ to recover the correct gaps, since interface dipoles need not match those at a material–vacuum boundary.
2. A semi-empirical model of parasitic charge in the SiGe buffer. The buffer is forced to be an insulator (removing unphysical classical screening), and discrete artificial charge domains under plungers and tunnel barriers are added to reintroduce the screening actually observed in experiment.
3. Jiles–Atherton micromagnet modeling. Cobalt micromagnet hysteresis and unknown pre-magnetization are captured with the Jiles–Atherton model inside COMSOL Multiphysics, with parameters fitted to three experimentally characterized dots and then applied to the full array.
4. Coupling of static device profiles to time-dependent qubit dynamics. The calibrated electrostatic and magnetic results feed a rotating-wave-approximation Hamiltonian solved as a time-dependent Schrödinger equation, producing simulated power chevrons that are compared with experiment and used to reason about microwave line losses.

Main Findings

• Anderson's rule is insufficient: the default model predicts the wrong band gap between Si and Si₇₀Ge₃₀ and between Si and Al₂O₃ (Fig. 2d–f); shifting electron affinities fixes the alignment (Fig. 2g–i).
• Classical charge in the SiGe buffer causes unrealistic screening: if classical charges are allowed there, the Poisson solver fills the well until the conduction band edge equals the Fermi level, screening the gates; making the buffer an insulator removes this but also removes the screening seen in the real device (Fig. 2j–l), motivating artificial negative charge domains (Fig. 2m–o).
• Parasitic well occupancy estimates: with classical charge transport enabled in the SiGe buffer, the parasitic wells could hold 251 electrons under QP0, 300 electrons under QP1, and 3159 electrons under SAP; the model instead places a uniform charge density with the same ratio but fewer electrons, assuming partial filling.
• Experimental evidence for screening: bias voltages applied to tunnel barriers and plunger gates in experiment are 20% to 50% higher than predictions without screening, and C–V curves show hysteresis attributed to charge accumulation in the SiGe buffer near the Si-cap.
• Lever-arm suppression is reproduced qualitatively: Fig. 3b shows the plunger-gate lever-arm dropping significantly once charges accumulate in the SiGe buffer. The authors state the simulation captures the qualitative trend rather than precise absolute values, and they tune the lever-arm transfer function to the filling of the first electron in each dot.
• Sensor operates in multielectron mode: integration over the sensor quantum dot gives approximately 665 electrons.
• Reservoir approximation: accumulation gates are assumed already activated, emulated by reservoir domains (SAA0_R, SAA1_R, SBA0_R, SBA1_R) doped at approximately 10¹⁹ 1/m³.
• Micromagnet calibration matters: under a 50 mT external field calibration — effectively the absence of a pre-magnetization procedure — the experimentally derived magnetic field is substantially higher than theoretical predictions, attributed to unknown initial magnetization of the micromagnets. With calibration, simulated resonant frequencies agree excellently with experiment across all six qubits (Fig. 4b).
• Residual error is at the device-variation level: remaining deviation is comparable in magnitude to device-to-device variation, attributed to lithographic geometric variation, film thickness fluctuations, and deposition-process and cobalt-purity inconsistencies.
• Microwave loss accounting: the cryogenic line attenuates the signal by 9 dB, wire bonds account for approximately 5 dB, and the microwave line has losses of approximately 15 dB at 8 GHz; 1 mV of amplitude corresponds to −50 dBm for 50 Ω (≈ −49 dBm in the experimental setup) and 10 mV to −30 dBm (≈ −29 dBm).

Methodology in Plain English

The team built a convex 3D finite-element model of the device. The in-plane geometry came from the device layout and SEM images; the vertical stack came from fabrication specifications. The Si₇₀Ge₃₀ buffer is treated as at least 200 nm thick, and the roughly 1 nm Si-cap is omitted because meshing such a thin layer is impractical and its influence on the field in the dot is minor. Overlapping gate layers are replaced by gate boundary conditions on only their non-overlapping parts, since the removed regions are screened anyway.

Specific regions — the two sensor dots and the 6-dot array — are declared "dot regions" where classical charge transport is switched off, so quantum dot properties can be studied. Reservoir domains with weak n-doping emulate activated accumulation gates. The solver first computes the classical conduction band edge from the Poisson equation with gate and ohmic boundary conditions; the single-electron Schrödinger equation is then solved separately for each sensor and for the array, giving eigenenergies and wavefunctions.

Fine-tuning uses two knobs: the gate work function (bounded between the reported titanium value of approximately 4.3 eV and the palladium value of approximately 5.6 eV, since palladium diffusion into titanium is uncertain) and the amount of charge placed in the parasitic wells under the plungers and barriers.

For the magnets, the authors use COMSOL with the Jiles–Atherton model, which tracks magnetization rather than assuming a fixed value. Its parameters are saturation magnetization, domain wall density, inter-domain coupling, pinning energy (which sets coercivity), and reversibility. The external field is slowly ramped to 50 mT, 300 mT, or 500 mT and then ramped down to the target field; fits are made against three experimentally known dots. Measured fields are inferred from qubit frequencies using f = gμ_B B/h with g = 2 for electrons.

Finally, the calibrated electrostatic and magnetic profiles are combined into a rotating-wave-approximation Hamiltonian with a sinusoidal time-dependent drive, and the time-dependent Schrödinger equation is solved to obtain measurement probabilities and power chevrons.

Why This Matters

Research impact: The work shows that idealized TCAD inputs are quantitatively inadequate for Si/SiGe spin qubits, and that hysteresis, interface dipoles, and buffer charging must be modeled to match experiment. It provides a validated route from layout to predicted qubit frequency and Rabi behavior.

Real-world applications:

• Co-designing gate stacks and dot layouts for scalable silicon spin-qubit processors before committing to fabrication.
• Designing on-chip micromagnets and EDSR drive schemes, including the effect of magnetization history on qubit frequency.
• Budgeting cryogenic microwave delivery: estimating line attenuation and inferring the voltage that actually reaches the device from a given source power.
• Engineering charge sensors and accumulation-gate/reservoir schemes for multielectron sensor operation.

Industry relevance: The pipeline runs in commercial FEM tools (QTCAD® and COMSOL Multiphysics) and targets a CMOS-compatible Si/SiGe process, which suits foundry-style design cycles where simulation must substitute for repeated tape-outs. It also gives a practical way to attribute simulation–experiment mismatch to device variation rather than model error.

Future Directions

• Replace the semi-empirical parasitic-charge tuning with a proper multielectron model, since the real buffer contains tens of electrons tunneling between parasitic dots rather than the hundreds a classical solver would assign.
• Improve treatment of the Si-cap and the graded SiₓGe₁₋ₓ layer, both currently excluded, which the authors identify as governing the z-shape of the potential and the biasing differences between device halves and between devices.
• Close the loop between simulated voltage chevrons and measured power chevrons, so that microwave line losses can be predicted rather than inferred.
• Extend the calibrated framework to larger dot arrays and more complex micromagnet systems, and find a way to fix or control the initial magnetization state that currently limits predictive accuracy at low applied fields.

Target Audience

Device physicists and quantum-hardware engineers working on silicon spin qubits; TCAD and FEM practitioners modeling semiconductor quantum devices; experimentalists who need to explain discrepancies between electrostatic simulations and measured gate voltages or qubit frequencies; and graduate students or advanced readers seeking a concrete, end-to-end example of calibrating a quantum device model against experiment.

Authors’ abstract

Silicon-based spin qubits in Si/SiGe heterostructures are a leading platform for scalable quantum computing, yet bridging the gap between theoretical computer-aided design(CAD) models and experimental reality remains a significant challenge. Standard simulations often fail to capture critical physical phenomena, such as interface dipoles, parasitic charge accumulation, and the magnetic hysteresis of on-chip micromagnets. In this work, we present a comprehensive, experimentally calibrated 3D simulation pipeline for a 6-dot Si/SiGe device. We refine the semiconductor band alignment and introduce a semi-empirical classical charge screening model to accurately capture the formation of parasitic wells and their suppression of gate lever-arms. Furthermore, we apply the Jiles-Atherton model to account for the hysteresis and pre-magnetization of integrated cobalt micromagnets, successfully reproducing the experimental resonant frequencies across all six qubits. By coupling these calibrated electrostatic and magnetic profiles into a time-dependent rotating wave approximation (RWA) Hamiltonian, we reproduce experimental observables, including microwave power chevrons. This framework provides a robust foundation for predicting device behavior, evaluating microwave line losses, and optimizing future scalable spin qubit architectures prior to fabrication.