# Twist angle: moiré geometry, bandwidth and sample variation

Connect twist angle to moiré filling and interlayer coupling while preserving device-specific transition criteria, pressure history and stiffness-model limits.

## Scope

A twist angle controls the relative orientation of two layers, changing the moiré pattern and the coupled electronic Hamiltonian. It does not independently specify carrier filling, interlayer tunneling, disorder, pairing interaction or superconducting topology. The original continuum model explains narrow bands and suppressed Dirac velocity at particular parameter values; observing a moiré pattern alone does not demonstrate a flat band or superconductivity. [R1](#ref-source-28797be9898b68b5)

Magic-angle graphene supplies experimentally accessible correlated phases, but an adjacent insulating state does not uniquely identify a one-band Mott mechanism or a common pairing mediator. The 2018 correlated-insulator and superconductivity papers are distinct sources, with different questions and evidence. Pressure-dependent superconductivity further shows why nominal angle cannot be treated as the only tuning parameter. [R2](#ref-source-be5e45f3ed46d90b) [R3](#ref-source-90c67b111a6e919c) [R4](#ref-source-e80d36514bb4144d)

## Mathematical assumptions

For two matched, unstrained hexagonal lattices, define $a=0.246$ nm as graphene's lattice constant, **not** its carbon–carbon bond length. Then $L_m=a/[2\sin(\theta/2)]$, $A_m=\sqrt3L_m^2/2$ and $L_m\approx a/\theta$ for a small angle expressed in radians. Bistritzer–MacDonald's manuscript instead calls the bond length $a$, so its numerator is $\sqrt3a$: these conventions are equivalent. Using the bond length in the lattice-constant formula would underestimate the period by $\sqrt3$ and overestimate the inferred full-filling density by a factor of three. [R1](#ref-source-28797be9898b68b5) [R4](#ref-source-e80d36514bb4144d)

With four spin–valley states per moiré band, define the full-band density $n_s=4/A_m$ and filling $\nu=nA_m=4n/n_s$, measured relative to charge neutrality. At 1.10°, the ideal geometry gives $L_m\approx12.81$ nm and $n_s\approx2.81\times10^{12}$ cm⁻². This is a calculated reference, not a measured device value. The small-angle sensitivity is $\delta n_s/n_s\approx2\delta\theta/\theta$; the exact logarithmic derivative is $d\ln n_s/d\theta=\cot(\theta/2)$. A contact-averaged filling estimate cannot resolve a local twist/strain map, and broken-symmetry states need not retain fourfold degeneracy. [R1](#ref-source-28797be9898b68b5) [R4](#ref-source-e80d36514bb4144d)

Let $k_\theta=2K\sin(\theta/2)$, $K=4\pi/(3a)$, and $\alpha=w/(\hbar v_Fk_\theta)$, using wavevector units for $k_\theta$. In the first-shell approximation with the small layer-rotation correction neglected, $v_*/v_F=(1-3\alpha^2)/(1+6\alpha^2)$. Its zero at $\alpha=1/\sqrt3$ illustrates velocity cancellation, not an exact experimental magic angle. Tunneling, lattice relaxation and strain affect a realistic calculation. Our local check covers reciprocal geometry, density conversion and the stated algebraic zero; it does not diagonalize the complete continuum Hamiltonian or calculate a superconducting transition. [R1](#ref-source-28797be9898b68b5) [R4](#ref-source-e80d36514bb4144d)

For the Tanaka author-version circuit, distinguish the parallel termination inductance from the bulk MATBG branch: $L_{\parallel}^{-1}=L_{\rm prx}^{-1}+L_{\rm TBG}^{-1}$. This is not the inductance of the entire resonator. With active length $\ell$ and width $W$, the manuscript's $D_s=\ell/(W L_{\rm TBG})$ has units of inverse henries. In the phase-energy convention $F/A=J|\nabla\phi|^2/2$, the local conversion is $J=\hbar^2D_s/(4e^2)$, with $J$ an energy. Thus $D_s$ must not be compared numerically with an energy stiffness without converting units. A synthetic example with $L_{\rm TBG}=2.64$ nH and $L_{\rm prx}=1.32$ nH gives $L_{\parallel}=0.88$ nH; attributing this whole parallel response to the bulk would overestimate its stiffness threefold. These are an illustrative calculation and a source-based circuit parameter, not a measured bulk inductance. Active aspect ratio and branch separation remain model assumptions; an incompatible subtraction does not establish a negative physical stiffness. [R5](#ref-source-be7282623ca96056)

## Experimental criteria

Cao's published superconductivity paper labels M1 at 1.16° and M2 at 1.05°. Figure 2 assigns maximum half-normal-resistance transition temperatures of about 0.5 K to M1 and 1.7 K to M2; the figure and caption resolve the device assignment. Both reached the reported zero-resistance state at the 70 mK measurement floor. A separate tentative cubic current–voltage fit for M2 at $n=-1.44\times10^{12}$ cm⁻² gave $T_{\rm BKT}\approx1.0$ K. The midpoint, inferred BKT scale and measurement floor must remain separate records. [R3](#ref-source-90c67b111a6e919c)

Yankowitz's nominal 1.27° device D2 developed superconducting transport under hydrostatic pressure. The published pressure plot defines $T_c$ by the intersection of low- and high-temperature linear fits, and the reported resistance falls to a roughly 10-ohm noise floor. The author supplement lists approximately 3.1 K at 1.33 GPa and 1.7 K at 2.21 GPa; the 300 mK base temperature is separate. These fitted transition temperatures should not be relabeled as exact zero-resistance or half-resistance thresholds. The same supplement reports removal, solvent cleaning and reloading between these pressures, which adds a possible disorder change to the pressure comparison. [R4](#ref-source-e80d36514bb4144d)

The same study's ambient-pressure D1, at approximately 1.14°, had fitted transition scales near 0.4 K for holes and 0.25 K for electrons, with a 10 mK base temperature. Different contacts showed different full-filling densities; the final text reports moiré unit-cell-area variations up to roughly 30% within a device. The authors used full-filling features and quantum-oscillation extrapolations to estimate angle. They did not establish a uniform angle at every point merely by quoting two decimal places. [R4](#ref-source-e80d36514bb4144d)

Tanaka and colleagues' 2025 study combines microwave response and transport. The checked author v2 calibrates a proximitized-edge branch using an insulating-state resistance assumption; strong normal-metal damping limits resonance extraction. Its conventional-stiffness comparison uses a Fermi velocity inferred from nonlinear transport, not an independent band-velocity measurement. Refrigerator base (20 mK), effective device temperature (about 40 mK), and transition temperatures are different quantities. Supplementary Figure 12 defines onset by intersecting two fitted lines; the half-resistance criterion crosses half the normal-line intercept at zero temperature, and the “zero” criterion extrapolates the falling line to zero. The approximately 1 K half-resistance scale is therefore not a universal measured-zero threshold. The comparison with base-temperature stiffness does not directly measure the renormalized BKT jump. Geometric and gap interpretations remain model-dependent; neither a unique pairing mediator nor a model-independent geometric fraction follows. [R5](#ref-source-be7282623ca96056)

| Comparison | Observable or definition | Interpretation limit |
| --- | --- | --- |
| Nominal angle and filling | Matched-lattice geometry; contact-dependent full-filling density | A device-average angle is not a local uniformity map. [R1](#ref-source-28797be9898b68b5) [R4](#ref-source-e80d36514bb4144d) |
| Cao M1 and M2 | Midpoint transition, tentative nonlinear BKT fit, and measurement floor | Keep the device, filling and criterion attached to every temperature. [R3](#ref-source-90c67b111a6e919c) |
| Yankowitz D2 pressure sequence | Fit-intersection temperature; sample removed and reloaded between pressures | Pressure, disorder history and detection floor cannot be collapsed into one tuning variable. [R4](#ref-source-e80d36514bb4144d) |
| Tanaka microwave circuit | Bulk and proximity branches; assumed active-region aspect ratio | Sheet inverse inductance and energy stiffness have different units; a full resonator is not a single bulk inductor. [R5](#ref-source-be7282623ca96056) |
| Tanaka temperature criteria | Two-line onset, intercept-based half resistance and extrapolated zero | Fitted transitions, refrigerator base and effective device temperature are separate records. [R5](#ref-source-be7282623ca96056) |

## Limits and disputes

Compare local geometry, filling calibration, contacts and thermal history before interpreting differences between devices. Pressure-dependent transition curves, a nearby insulator or a broadened dome are not unique fingerprints of a pairing mechanism. Distinguish measured resistance, nonlinear response and stiffness from a proposed microscopic explanation. Likewise, normal-state band topology does not alone fix the topology of the superconducting quasiparticles. Transferring graphene conclusions to a lattice-mismatched semiconductor stack requires a new model and platform-specific evidence. [R1](#ref-source-28797be9898b68b5) [R2](#ref-source-be5e45f3ed46d90b) [R3](#ref-source-90c67b111a6e919c) [R4](#ref-source-e80d36514bb4144d) [R5](#ref-source-be7282623ca96056)

This review checks selected BM author equations, Cao final figures, Yankowitz final pressure passages plus selected author supplementary methods, and Tanaka author-v2 circuit/temperature definitions, including the visually checked supplementary fit diagram. The inherited source attributed the 2025 stiffness paper to Randeria; the publisher identifies Tanaka and colleagues. The complete BM and Tanaka final-version comparisons and complete supplementary/raw-data audits remain open. The correlated-insulator paper is used at publisher-abstract scope here, not as a completed microscopic-mechanism audit. [R1](#ref-source-28797be9898b68b5) [R2](#ref-source-be5e45f3ed46d90b) [R3](#ref-source-90c67b111a6e919c) [R4](#ref-source-e80d36514bb4144d) [R5](#ref-source-be7282623ca96056)

## Discovery implications

Optimize angle together with tunneling, strain, filling and phase stiffness. Store each transition under its device, carrier branch, pressure and criterion; keep measurement floors separate. Use local imaging and contact-resolved transport to test uniformity. A useful design can combine favorable pairing and adequate phase coherence, but narrow bandwidth alone is not a sufficient target or a universal guarantee of higher $T_c$. [R3](#ref-source-90c67b111a6e919c) [R4](#ref-source-e80d36514bb4144d) [R5](#ref-source-be7282623ca96056)

## Related topics

Compare [flat bands](/scwiki/flat-band-pairing), [quantum geometry](/scwiki/geometric-superfluid-weight), [pressure](/scwiki/pressure-tuning), [electrostatic gating](/scwiki/electrostatic-gating) and [dimensionality](/scwiki/dimensionality).

## Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.

| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |
|---|---|---|---|---|---|---|
| C (aliases: MATBG, twisted bilayer graphene, Cao M1); Cao M1, nominal1.16deg | 0.5 K; Maximum Figure2 transition at50% normal resistance. | transition; resistive-midpoint | Four-probe resistance | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-90c67b111a6e919c) |
| C (aliases: MATBG, twisted bilayer graphene, Cao M2); Cao M2, nominal1.05deg | 1.7 K; Maximum Figure2 transition at50% normal resistance. | transition; resistive-midpoint | Four-probe resistance | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-90c67b111a6e919c) |
| C (aliases: MATBG, twisted bilayer graphene, Cao M2); Cao M2 atn=-1.44e12cm^-2 | 1 K; Tentative cubic-power fit; not the1.7K midpoint. | transition; bkt-current-voltage-fit | Nonlinear current–voltage | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-90c67b111a6e919c) |
| C (aliases: MATBG, twisted bilayer graphene, Cao M1, Cao M2); Cao M1/M2 measurement floor | 0.07 K; 70mK measurement floor at reported zero resistance; not Tc. | measurement; measurement-floor | Four-probe transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-90c67b111a6e919c) |
| C (aliases: MATBG, twisted bilayer graphene, Yankowitz D2); Yankowitz D2 hole branch, nominal1.27deg | 3.1 K; Approximate author TableS1 value; final Fig2 uses low/high-temperature fit intersection, not zero resistance. | transition; resistive-fit-intersection | Four-terminal resistance | high; 1.33 to 1.33 GPa | equilibrium | experiment; [source](#ref-source-e80d36514bb4144d) |
| C (aliases: MATBG, twisted bilayer graphene, Yankowitz D2); Yankowitz D2 hole branch, nominal1.27deg | 1.7 K; Approximate author TableS1 value; sample removed/cleaned/reloaded between pressures. | transition; resistive-fit-intersection | Four-terminal resistance | high; 2.21 to 2.21 GPa | equilibrium | experiment; [source](#ref-source-e80d36514bb4144d) |
| C (aliases: MATBG, twisted bilayer graphene, Yankowitz D1 holes); Yankowitz D1 hole branch, nominal1.14deg | 0.4 K; Low/high-temperature fit crossover on logarithmic resistance plot. | transition; resistive-fit-intersection | Four-terminal resistance | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-e80d36514bb4144d) |
| C (aliases: MATBG, twisted bilayer graphene, Yankowitz D1 electrons); Yankowitz D1 electron branch, nominal1.14deg | 0.25 K; Carrier branch separate from hole branch. | transition; resistive-fit-intersection | Four-terminal resistance | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-e80d36514bb4144d) |
| C (aliases: MATBG, twisted bilayer graphene, Tanaka device); Tanaka wired device | 0.04 K; Approximately40mK effective device temperature; refrigerator base20mK and near1K half-resistance scale are distinct. | measurement; effective-device-temperature | Microwave circuit and transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-be7282623ca96056) |
| C (aliases: MATBG, twisted bilayer graphene, Tanaka refrigerator); Tanaka refrigerator base | 0.02 K; 20 mK refrigerator base; not the approximately40 mK effective device temperature or a transition temperature. | measurement; refrigerator-base-temperature | Cryostat temperature setting | ambient; unknown GPa | unknown | experiment; [source](#ref-source-be7282623ca96056) |

## Sources

### Ref source-28797be9898b68b5

[R1] Moiré bands in twisted double-layer graphene (2011). [DOI 10.1073/pnas.1108174108](https://doi.org/10.1073/pnas.1108174108). Publication: journal-article. Check: selected-author-geometry-and-first-shell-equations. Checked: 2026-10-18. Author v1 pp1–3 selected; p2 visually checked including bond-length convention and first-shell Eq4. Final journal July2011, author submission September2010. Full continuum diagonalization, supplementary derivation and final comparison open.
### Ref source-be5e45f3ed46d90b

[R2] Correlated insulator behaviour at half-filling in magic-angle graphene superlattices (2018). [DOI 10.1038/nature26154](https://doi.org/10.1038/nature26154). Publication: journal-article. Check: publisher-abstract-and-bibliography. Checked: 2026-10-18. Publisher original abstract and citation checked: online5March2018, Nature556,80–84. Correlated half-filling states consistent with Mott-like interpretation; no complete microscopic mechanism audit.
### Ref source-90c67b111a6e919c

[R3] Unconventional superconductivity in magic-angle graphene superlattices (2018). [DOI 10.1038/nature26160](https://doi.org/10.1038/nature26160). Publication: journal-article. Check: selected-final-device-and-criterion-figures. Checked: 2026-10-18. Final pp44–45 selected text and captions; p45 Fig2 visually checked. M1 1.16deg0.5K versus M2 1.05deg1.7K at50% criterion; M2 tentative1K BKT fit and70mK measurement floor separate. Full methods/supplement/rawdata review open.
### Ref source-e80d36514bb4144d

[R4] Tuning superconductivity in twisted bilayer graphene (2019). [DOI 10.1126/science.aav1910](https://doi.org/10.1126/science.aav1910). Publication: journal-article. Check: selected-final-pressure-text-and-author-supplement. Checked: 2026-10-18. Final pp1–3 selected text/Fig2 caption checked against authorv1 pp1–6; author supplement pp11–13 methods/TableS1/reloading read, p12 table visually checked. D2 fitted3.1K at1.33GPa and1.7K at2.21GPa remain author-table values. Published24January2019, issue8March2019. Author PDF displays a later compilation date; version identity follows arXiv1808.07865v1 record. Full final supplement and pressure-medium calibration not audited. Final p3 Figure2 was also visually checked.
### Ref source-be7282623ca96056

[R5] Superfluid stiffness of magic-angle twisted bilayer graphene (2025). [DOI 10.1038/s41586-024-08494-7](https://doi.org/10.1038/s41586-024-08494-7). Publication: journal-article. Check: selected-author-v2-circuit-and-supplementary-fit-definition-review. Checked: 2026-10-20. Published Tanaka et al identification retained. Author v2 pp5–7 and31–32 revisited; pp6,7,31,32 visually inspected. Supplement Fig12 fixes the half-normal reference at the zero-temperature intercept of the normal-state fit; the zero criterion is a line extrapolation. Branch calibration and active aspect ratio are model assumptions. Final-version comparison and complete supplements/raw-data audit remain open.

## Claim evidence and conditions

### scwiki-e9-claim-scope-1

A twist angle controls the relative orientation of two layers, changing the moiré pattern and the coupled electronic Hamiltonian. It does not independently specify carrier filling, interlayer tunneling, disorder, pairing interaction or superconducting topology. The original continuum model explains narrow bands and suppressed Dirac velocity at particular parameter values; observing a moiré pattern alone does not demonstrate a flat band or superconductivity.

Evidence: experiment. Sources: source-28797be9898b68b5.

Source relationship: context-for-editorial-analysis. Editorial comparison of geometry, device identity, observable and inference boundaries; no single source is credited with the entire synthesis.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-scope-2

Magic-angle graphene supplies experimentally accessible correlated phases, but an adjacent insulating state does not uniquely identify a one-band Mott mechanism or a common pairing mediator. The 2018 correlated-insulator and superconductivity papers are distinct sources, with different questions and evidence. Pressure-dependent superconductivity further shows why nominal angle cannot be treated as the only tuning parameter.

Evidence: experiment. Sources: source-be5e45f3ed46d90b, source-90c67b111a6e919c, source-e80d36514bb4144d.

Source relationship: context-for-editorial-analysis. Editorial comparison of geometry, device identity, observable and inference boundaries; no single source is credited with the entire synthesis.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-mathematical-assumptions-1

For two matched, unstrained hexagonal lattices, define $a=0.246$ nm as graphene's lattice constant, **not** its carbon–carbon bond length. Then $L_m=a/[2\sin(\theta/2)]$, $A_m=\sqrt3L_m^2/2$ and $L_m\approx a/\theta$ for a small angle expressed in radians. Bistritzer–MacDonald's manuscript instead calls the bond length $a$, so its numerator is $\sqrt3a$: these conventions are equivalent. Using the bond length in the lattice-constant formula would underestimate the period by $\sqrt3$ and overestimate the inferred full-filling density by a factor of three.

Evidence: theory. Sources: source-28797be9898b68b5, source-e80d36514bb4144d.

Source relationship: mixed-source-and-local-derivation. Source equations supply the model and convention; unit conversion, sensitivity and synthetic calculations are explicitly local, not experimental values or a full continuum/microwave reconstruction.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-mathematical-assumptions-2

With four spin–valley states per moiré band, define the full-band density $n_s=4/A_m$ and filling $\nu=nA_m=4n/n_s$, measured relative to charge neutrality. At 1.10°, the ideal geometry gives $L_m\approx12.81$ nm and $n_s\approx2.81\times10^{12}$ cm⁻². This is a calculated reference, not a measured device value. The small-angle sensitivity is $\delta n_s/n_s\approx2\delta\theta/\theta$; the exact logarithmic derivative is $d\ln n_s/d\theta=\cot(\theta/2)$. A contact-averaged filling estimate cannot resolve a local twist/strain map, and broken-symmetry states need not retain fourfold degeneracy.

Evidence: theory. Sources: source-28797be9898b68b5, source-e80d36514bb4144d.

Source relationship: mixed-source-and-local-derivation. Source equations supply the model and convention; unit conversion, sensitivity and synthetic calculations are explicitly local, not experimental values or a full continuum/microwave reconstruction.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-mathematical-assumptions-3

Let $k_\theta=2K\sin(\theta/2)$, $K=4\pi/(3a)$, and $\alpha=w/(\hbar v_Fk_\theta)$, using wavevector units for $k_\theta$. In the first-shell approximation with the small layer-rotation correction neglected, $v_*/v_F=(1-3\alpha^2)/(1+6\alpha^2)$. Its zero at $\alpha=1/\sqrt3$ illustrates velocity cancellation, not an exact experimental magic angle. Tunneling, lattice relaxation and strain affect a realistic calculation. Our local check covers reciprocal geometry, density conversion and the stated algebraic zero; it does not diagonalize the complete continuum Hamiltonian or calculate a superconducting transition.

Evidence: theory. Sources: source-28797be9898b68b5, source-e80d36514bb4144d.

Source relationship: mixed-source-and-local-derivation. Source equations supply the model and convention; unit conversion, sensitivity and synthetic calculations are explicitly local, not experimental values or a full continuum/microwave reconstruction.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-mathematical-assumptions-4

For the Tanaka author-version circuit, distinguish the parallel termination inductance from the bulk MATBG branch: $L_{\parallel}^{-1}=L_{\rm prx}^{-1}+L_{\rm TBG}^{-1}$. This is not the inductance of the entire resonator. With active length $\ell$ and width $W$, the manuscript's $D_s=\ell/(W L_{\rm TBG})$ has units of inverse henries. In the phase-energy convention $F/A=J|\nabla\phi|^2/2$, the local conversion is $J=\hbar^2D_s/(4e^2)$, with $J$ an energy. Thus $D_s$ must not be compared numerically with an energy stiffness without converting units. A synthetic example with $L_{\rm TBG}=2.64$ nH and $L_{\rm prx}=1.32$ nH gives $L_{\parallel}=0.88$ nH; attributing this whole parallel response to the bulk would overestimate its stiffness threefold. These are an illustrative calculation and a source-based circuit parameter, not a measured bulk inductance. Active aspect ratio and branch separation remain model assumptions; an incompatible subtraction does not establish a negative physical stiffness.

Evidence: theory. Sources: source-be7282623ca96056.

Source relationship: mixed-source-and-local-derivation. Source equations supply the model and convention; unit conversion, sensitivity and synthetic calculations are explicitly local, not experimental values or a full continuum/microwave reconstruction.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-experimental-criteria-1

Cao's published superconductivity paper labels M1 at 1.16° and M2 at 1.05°. Figure 2 assigns maximum half-normal-resistance transition temperatures of about 0.5 K to M1 and 1.7 K to M2; the figure and caption resolve the device assignment. Both reached the reported zero-resistance state at the 70 mK measurement floor. A separate tentative cubic current–voltage fit for M2 at $n=-1.44\times10^{12}$ cm⁻² gave $T_{\rm BKT}\approx1.0$ K. The midpoint, inferred BKT scale and measurement floor must remain separate records.

Evidence: experiment. Sources: source-90c67b111a6e919c.

Source relationship: direct-support. Source-specific devices, fit definitions and reported measurements are separated from the authors' physical interpretation; author-v2 evidence is not represented as a final-version comparison.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-experimental-criteria-2

Yankowitz's nominal 1.27° device D2 developed superconducting transport under hydrostatic pressure. The published pressure plot defines $T_c$ by the intersection of low- and high-temperature linear fits, and the reported resistance falls to a roughly 10-ohm noise floor. The author supplement lists approximately 3.1 K at 1.33 GPa and 1.7 K at 2.21 GPa; the 300 mK base temperature is separate. These fitted transition temperatures should not be relabeled as exact zero-resistance or half-resistance thresholds. The same supplement reports removal, solvent cleaning and reloading between these pressures, which adds a possible disorder change to the pressure comparison.

Evidence: experiment. Sources: source-e80d36514bb4144d.

Source relationship: direct-support. Source-specific devices, fit definitions and reported measurements are separated from the authors' physical interpretation; author-v2 evidence is not represented as a final-version comparison.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-experimental-criteria-3

The same study's ambient-pressure D1, at approximately 1.14°, had fitted transition scales near 0.4 K for holes and 0.25 K for electrons, with a 10 mK base temperature. Different contacts showed different full-filling densities; the final text reports moiré unit-cell-area variations up to roughly 30% within a device. The authors used full-filling features and quantum-oscillation extrapolations to estimate angle. They did not establish a uniform angle at every point merely by quoting two decimal places.

Evidence: experiment. Sources: source-e80d36514bb4144d.

Source relationship: direct-support. Source-specific devices, fit definitions and reported measurements are separated from the authors' physical interpretation; author-v2 evidence is not represented as a final-version comparison.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-experimental-criteria-4

Tanaka and colleagues' 2025 study combines microwave response and transport. The checked author v2 calibrates a proximitized-edge branch using an insulating-state resistance assumption; strong normal-metal damping limits resonance extraction. Its conventional-stiffness comparison uses a Fermi velocity inferred from nonlinear transport, not an independent band-velocity measurement. Refrigerator base (20 mK), effective device temperature (about 40 mK), and transition temperatures are different quantities. Supplementary Figure 12 defines onset by intersecting two fitted lines; the half-resistance criterion crosses half the normal-line intercept at zero temperature, and the “zero” criterion extrapolates the falling line to zero. The approximately 1 K half-resistance scale is therefore not a universal measured-zero threshold. The comparison with base-temperature stiffness does not directly measure the renormalized BKT jump. Geometric and gap interpretations remain model-dependent; neither a unique pairing mediator nor a model-independent geometric fraction follows.

Evidence: experiment. Sources: source-be7282623ca96056.

Source relationship: direct-support. Source-specific devices, fit definitions and reported measurements are separated from the authors' physical interpretation; author-v2 evidence is not represented as a final-version comparison.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-limits-and-disputes-1

Compare local geometry, filling calibration, contacts and thermal history before interpreting differences between devices. Pressure-dependent transition curves, a nearby insulator or a broadened dome are not unique fingerprints of a pairing mechanism. Distinguish measured resistance, nonlinear response and stiffness from a proposed microscopic explanation. Likewise, normal-state band topology does not alone fix the topology of the superconducting quasiparticles. Transferring graphene conclusions to a lattice-mismatched semiconductor stack requires a new model and platform-specific evidence.

Evidence: experiment. Sources: source-28797be9898b68b5, source-be5e45f3ed46d90b, source-90c67b111a6e919c, source-e80d36514bb4144d, source-be7282623ca96056.

Source relationship: context-for-editorial-analysis. Editorial comparison of geometry, device identity, observable and inference boundaries; no single source is credited with the entire synthesis.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-limits-and-disputes-2

This review checks selected BM author equations, Cao final figures, Yankowitz final pressure passages plus selected author supplementary methods, and Tanaka author-v2 circuit/temperature definitions, including the visually checked supplementary fit diagram. The inherited source attributed the 2025 stiffness paper to Randeria; the publisher identifies Tanaka and colleagues. The complete BM and Tanaka final-version comparisons and complete supplementary/raw-data audits remain open. The correlated-insulator paper is used at publisher-abstract scope here, not as a completed microscopic-mechanism audit.

Evidence: experiment. Sources: source-28797be9898b68b5, source-be5e45f3ed46d90b, source-90c67b111a6e919c, source-e80d36514bb4144d, source-be7282623ca96056.

Source relationship: context-for-editorial-analysis. Editorial comparison of geometry, device identity, observable and inference boundaries; no single source is credited with the entire synthesis.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

### scwiki-e9-claim-discovery-implications-1

Optimize angle together with tunneling, strain, filling and phase stiffness. Store each transition under its device, carrier branch, pressure and criterion; keep measurement floors separate. Use local imaging and contact-resolved transport to test uniformity. A useful design can combine favorable pairing and adequate phase coherence, but narrow bandwidth alone is not a sufficient target or a universal guarantee of higher $T_c$.

Evidence: theory. Sources: source-90c67b111a6e919c, source-e80d36514bb4144d, source-be7282623ca96056.

Source relationship: context-for-editorial-analysis. Editorial comparison of geometry, device identity, observable and inference boundaries; no single source is credited with the entire synthesis.

Conditions: Keep device, carrier branch, pressure, local geometry, circuit branch, stiffness units and resistance-fit definitions distinct; measurement temperatures are not transition temperatures.

Verification scope: Selected original source passages retained; Tanaka author-v2 pp5–7 and31–32 revisited with visual checks of pp6,7,31,32; bounded local circuit/unit calculations. Full independent acceptance remains open.

## Revision and review

Article ID: scwiki-e9. Revision: 1272dba0ba465d9d9e46. Updated: 2026-10-20.

Review scope: Tanaka author-v2 circuit, aspect ratio, nonlinear-transport velocity estimate and supplementary fit definitions; local branch/unit calculations. Prior selected BM/Cao/Yankowitz reading retained.

- Complete scientific acceptance, raw-data and full supplementary reviews remain open.
- BM and Tanaka final-version comparisons incomplete; Cao correlated-insulator source used at publisher-abstract scope.
- Yankowitz exact pressure Tc values/reloading history taken from selected author supplement; final supplement and pressure-medium calibration remain unchecked.
- Geometry checks do not compute a full continuum band structure or establish a pairing mechanism.
- The circuit check is a synthetic parallel-inductor and energy-convention calculation, not full resonator simulation, raw-data fitting, a measured BKT jump or a unique quantum-geometric decomposition.

Source: [E_tuning_parameters/E9_twist_angle.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/E_tuning_parameters/E9_twist_angle.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 ab35c4154f39717a9a6d10219fab53b3706bebb01b338e7eb679e8703d052c03.