{"data":{"id":"10.5061/dryad.dr7sqvbdq","type":"dois","attributes":{"doi":"10.5061/dryad.dr7sqvbdq","prefix":"10.5061","suffix":"dryad.dr7sqvbdq","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Ju, Y. Sungtaek","nameType":"Personal","affiliation":["University of California, Los Angeles"],"nameIdentifiers":[{"schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0000-0003-1238-8674","nameIdentifierScheme":"ORCID"}]}],"titles":[{"title":"Code from: Full-scattering-matrix phonon BTE solver"}],"publisher":"Dryad","container":{},"publicationYear":2026,"subjects":[{"subject":"FOS: Physical sciences","schemeUri":"https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf","subjectScheme":"fos"},{"subject":"FOS: Engineering and technology","schemeUri":"https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf","subjectScheme":"fos"},{"subject":"FOS: Nano-technology","schemeUri":"https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf","subjectScheme":"fos"},{"subject":"Phonons","schemeUri":"https://github.com/PLOS/plos-thesaurus","subjectScheme":"PLOS Subject Area Thesaurus"},{"subject":"Thermal conductivity","schemeUri":"https://github.com/PLOS/plos-thesaurus","subjectScheme":"PLOS Subject Area Thesaurus"},{"subject":"Nanoelectronics","schemeUri":"https://github.com/PLOS/plos-thesaurus","subjectScheme":"PLOS Subject Area Thesaurus"}],"contributors":[],"dates":[{"date":"2026-06-23T21:49:08Z","dateType":"Created"},{"date":"2026-07-16T14:25:39Z","dateType":"Submitted"},{"date":"2026-07-22T00:00:00Z","dateType":"Issued"},{"date":"2026-07-22T00:00:00Z","dateType":"Available"}],"language":"en","types":{"ris":"DATA","bibtex":"misc","citeproc":"dataset","schemaOrg":"Dataset","resourceType":"dataset","resourceTypeGeneral":"Dataset"},"relatedIdentifiers":[{"relationType":"IsCitedBy","relatedIdentifier":"10.1038/s41598-026-62876-7","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":["178402 bytes"],"formats":[],"version":"5","rightsList":[{"rights":"Creative Commons Zero v1.0 Universal","rightsUri":"https://creativecommons.org/publicdomain/zero/1.0/legalcode","schemeUri":"https://spdx.org/licenses/","rightsIdentifier":"cc0-1.0","rightsIdentifierScheme":"SPDX"}],"descriptions":[{"description":"A deterministic solver for the phonon Boltzmann transport equation (BTE)\n that retains the full phonon–phonon scattering\n matrix W — rather than the diagonal relaxation-time\n approximation (RTA) — for thermal transport in geometries ranging from\n one-dimensional slabs to a three-dimensional device.","descriptionType":"Abstract"},{"description":"# TT-BTE: Deterministic full-scattering-matrix phonon Boltzmann transport\n solver A deterministic solver for the phonon Boltzmann transport equation\n (BTE) that retains the **full phonon–phonon scattering matrix** *W* —\n rather than the diagonal relaxation-time approximation (RTA) — for thermal\n transport in geometries ranging from one-dimensional slabs to a\n three-dimensional FinFET-like device. This repository contains the solver\n code accompanying the manuscript *\"Deterministic full-matrix phonon\n BTE solver for 3D device geometries\"* (Y. S. Ju). It reproduces the\n full-scattering-matrix construction, the hybrid mode-space solver, and the\n device geometries reported there. The interatomic model used throughout is\n the Stillinger–Weber (SW) potential for silicon. Material-specific\n transport magnitudes (e.g., the bulk conductivity and the device\n temperature-rise correction) are properties of this SW model and should\n not be read as quantitative predictions for crystalline silicon; the\n structural results of the paper are independent of this choice. --- ##\n What the solver does The phonon BTE under study is ``` v_λ · ∇ f_λ(r) =\n Σ_μ W_λμ f_μ(r) + Q_λ(r), ``` where `f_λ` is the deviational phonon\n distribution for mode `λ`, `v_λ` the group velocity, `W` the full\n scattering operator, and `Q` a heat source. The defining feature of this\n solver is that it keeps the complete off-diagonal `W`while remaining\n tractable in 3D, by a **hybrid mode-space** decomposition: * **Streaming**\n (`v · ∇`) is handled by dense, upwind spatial sweeps, which are trivially\n mode-local and need no compression. * **Scattering** (`W`) is applied as a\n compressed operator — either a truncated SVD of the in-scattering block\n (with the out-scattering diagonal carried exactly) or a tensor-train (TT)\n representation — with grid-independent rank. The scattering matrix is\n constructed to satisfy three physical constraints that are load-bearing\n for a stable device solve: 1. **Energy conservation**, `W c = 0`, where\n `c` are the modal heat capacities. 2. **Detailed balance**, `c_i W_ij =\n c_j W_ji`. 3. **Negative semi-definiteness** on the non-equilibrium\n subspace. Conservation is not optional: a non-conserving operator makes\n the device solver diverge. The construction enforces all three\n analytically and then projects out the residual, reaching machine\n precision on each (see \"Verifying the install \" below). --- ##\n Repository layout ``` __init__.py Sets the version and its description\n core/ Tensor-train algebra (physics-agnostic) tensor_train.py TensorTrain\n class, index bookkeeping tt_operators.py TTOperator (MPO),\n add/scale/compress tt_solvers.py AMEn linear solver physics/ Phonon model\n and scattering operator sw_potential.py Stillinger–Weber energy and forces\n si_physical_analysis_v2.py Force constants, phonons, and the full-W\n scattering-matrix construction (the core physical contribution)\n sw_phonon_bridge.py SWPhononComputation: end-to-end SW → (v, τ, C, W)\n phonon_data.py PhononBandData container and reference models mode_space.py\n ModeSpaceData: the mode-resolved model consumed by every solver; factory\n mode_space_from_sw() svd_scattering.py SVDScatteringOperator: low-rank W\n with exact γ tt_scattering.py TTScatteringOperator: tensor-train W\n solvers/ BTE solvers (hybrid mode-space) bte_slab_1d_hybrid.py 1D slab\n (validation against analytic reference) bte_slab_2d_hybrid.py 2D slab\n bte_monolithic_2d.py 2D monolithic domain dense_sweep_3d_hybrid.py 3D box\n dense-sweep reference bte_3d_finfet.py 3D FinFET-like device (fin + base,\n Schwarz coupling) — the headline device geometry dsa_operator.py\n Diffusion-synthetic acceleration (base) dsa_operator_1d.py DSA, 1D\n dsa_operator_2d.py DSA, 2D dsa_operator_3d.py DSA, 3D (Fourier-cosine) ```\n Diffusion-synthetic acceleration (DSA) is\n an **optional **convergence accelerator and is **off by\n default**; it is not part of the production methodology and is provided\n only for cross-checking. All the code files are\n in Dryad_tt_bte_solver.zip. --- ## Requirements * Python ≥ 3.10 * NumPy *\n SciPy That is the complete runtime requirement. The solver runs entirely\n on CPU/NumPy. JAX is an *optional* dependency, used only by a single\n GPU-resident DSA correction method (`DSAOperator3D.apply_correction_jax`).\n It is not needed to run any solver or to reproduce any result in the\n paper; the relevant import is lazy, so the package loads and runs without\n JAX installed. Install the requirements with: ``` pip install numpy scipy\n ``` --- ## Installing and importing This is a flat package: put the\n directory that *contains* `core/`,  `physics/`, and `solvers/` on your\n Python path and import the subpackages directly. The simplest approach is\n to run Python from inside this directory, or to add it to `PYTHONPATH`:\n ``` export PYTHONPATH=/path/to/this/directory:$PYTHONPATH ``` Then, in\n Python: ```python from physics import SWPhononComputation,\n SVDScatteringOperator, ModeSpaceData from physics.mode_space import\n mode_space_from_sw from solvers import BTESlab1D, DenseSweep3DHybrid from\n solvers.bte_3d_finfet import DenseSweep3DFinFET ``` --- ## Verifying the\n install The following builds the SW phonon model on a small grid,\n constructs the full scattering matrix, and confirms the conservation,\n detailed-balance, and semi-definiteness gates. It takes a couple of\n minutes (most of the time is the force-constant computation) and requires\n only NumPy and SciPy. ```python from physics import SWPhononComputation\n comp = SWPhononComputation(q_grid_size=3) # small 3x3x3 q-grid\n comp.compute(build_W=True, verbose=True) # builds v, tau, C, and W # After\n construction the solver prints the matrix gates, e.g.: # Energy\n conservation ||W c|| / ||c|| ~ 1e-16 (Wc = 0) # Detailed balance max|c_i\n W_ij - ...| ~ 1e-16 # Eigenvalues all \u0026lt;= 0 (semi-definite)\n print(\"modes:\", comp.W.shape[0]) ``` A clean run prints\n corrected conservation and detailed-balance residuals at the level of\n machine precision (`~1e-15` or smaller) and no positive eigenvalues. ---\n ## Typical workflow ### 1. Build the phonon model ```python from physics\n import SWPhononComputation from physics.mode_space import\n mode_space_from_sw comp = SWPhononComputation(q_grid_size=7) # production\n grids use larger N comp.compute(build_W=True) mode_data =\n mode_space_from_sw(comp) # ModeSpaceData for the solvers\n print(mode_data.summary()) ``` `ModeSpaceData` exposes the mode-resolved\n velocities, heat capacities, the full `W`, the RTA operator `W_rta`, the\n thermally active mask, and helpers such as `bulk_thermal_conductivity()`\n and `project_to_1d()`. ### 2. Compress the scattering operator (optional\n but recommended) ```python from physics import SVDScatteringOperator\n svd_op = SVDScatteringOperator.from_mode_data(mode_data, rank=50)\n svd_op.print_rank_summary() ``` The out-scattering diagonal `γ` is\n retained exactly; only the in-scattering block is approximated, so the\n transport-critical relaxation rates are preserved. ### 3. Solve a geometry\n 1D slab (used for validation against the analytic Fuchs–Sondheimer\n reference): ```python from solvers import BTESlab1D slab =\n BTESlab1D(N_x=100, L=200e-9, mode_data=mode_data.project_to_1d()) result =\n slab.solve(tol=1e-8, verbose=True) ``` 3D FinFET-like device (the headline\n geometry): ```python from solvers.bte_3d_finfet import DenseSweep3DFinFET\n # Construct with the fin/base spatial resolution and geometry, supply the\n # mode_data and (optionally) the SVD-compressed operator, then solve. #\n See the DenseSweep3DFinFET docstring for the full constructor signature #\n and boundary-condition options (isothermal sink, diffuse side walls, #\n y-uniform line heat source). ``` Each solver returns a result object\n carrying the converged temperature field,  the heat flux, and solver\n diagnostics. --- ## Reproducing the paper's headline quantities The\n device result is the full-`W` correction to the peak temperature rise,\n defined as a same-grid difference between the full-`W` and RTA solutions\n on the FinFET-like geometry with a y-uniform line heat source on the top\n of the fin. For Brillouin-zone grids,s `N ≥ 11` the correction is ```\n δT_fW / (T_RTA − T_0) = 11.7 ± 0.3 %, ``` a bounded, grid-robust\n correction. The RTA baseline used for this comparison carries the same\n isotope channel as the full operator, so the two solutions differ only in\n the off-diagonal scattering. To reproduce a single point: build the model\n at the desired grid. `N` (`q_grid_size=N`), construct both the full and\n RTA operators from the same `ModeSpaceData` solve the FinFET geometry with\n each, and take the difference of the peak temperatures. The bulk\n conductivity of the SW model on this grid is `k_bulk = 148.0 W m⁻¹ K⁻¹`\n (isotropic to better than 0.003%), which is a useful cross-check that the\n phonon model is built correctly. --- ## Numerical notes and known\n properties * **Γ-centred Monkhorst–Pack sampling.** On Γ-centred grids,\n the absolute device temperatures converge non-monotonically at small odd\n `N` (the grid does not close under `q → −q`); monotone behaviour sets in\n by `N ≈ 11–15`. The same-grid *correction* is far less sensitive and is\n reported as a bounded band. * **SW group-velocity anisotropy.** On a\n Γ-centred gri,d the SW group velocities carry a fixed `[111]` ellipticity,\n giving a non-negligible off-diagonal Cartesian conductivity (`κ_xy/κ_xx ≈\n 0.56`). The scalar bulk conductivity is unaffected (the tensor is\n traceless-clean). The y-uniform line source usedfor thee device results\n makes`∂T/∂y ≈ 0`, so this term is geometrically inactive. *\n **Scattering-matrix scaling.** The number of three-phonon channels, and\n hence the nonzeros of`W`, scales as the square of the active mode count,\n so the operator is genuinely dense. Compression of the *solution* (not the\n operator) is what makes the solver efficient. If you use this solver, we\n encourage you to cite the accompanying paper \"*Deterministic\n full-matrix phonon BTE solver for 3D device geometries*,\" by Y. S. Ju\n in *Scientific Reports* (2026).","descriptionType":"TechnicalInfo"}],"geoLocations":[],"fundingReferences":[],"xml":"<?xml version="1.0" encoding="UTF-8"?>
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  <creators>
    <creator>
      <creatorName nameType="Personal">Ju, Y. Sungtaek</creatorName>
      <nameIdentifier nameIdentifierScheme="ORCID" schemeURI="http://orcid.org/">0000-0003-1238-8674</nameIdentifier>
      <affiliation affiliationIdentifier="https://ror.org/046rm7j60" affiliationIdentifierScheme="ROR" schemeURI="https://ror.org">University of California, Los Angeles</affiliation>
    </creator>
  </creators>
  <titles>
    <title>Code from: Full-scattering-matrix phonon BTE solver</title>
  </titles>
  <publisher publisherIdentifier="https://ror.org/00x6h5n95" publisherIdentifierScheme="ROR" schemeURI="https://ror.org/">Dryad</publisher>
  <resourceType resourceTypeGeneral="Dataset">dataset</resourceType>
  <publicationYear>2026</publicationYear>
  <subjects>
    <subject subjectScheme="fos" schemeURI="https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf">FOS: Physical sciences</subject>
    <subject subjectScheme="fos" schemeURI="https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf">FOS: Engineering and technology</subject>
    <subject subjectScheme="fos" schemeURI="https://web-archive.oecd.org/2012-06-15/138575-38235147.pdf">FOS: Nano-technology</subject>
    <subject subjectScheme="PLOS Subject Area Thesaurus" schemeURI="https://github.com/PLOS/plos-thesaurus">Phonons</subject>
    <subject subjectScheme="PLOS Subject Area Thesaurus" schemeURI="https://github.com/PLOS/plos-thesaurus">Thermal conductivity</subject>
    <subject subjectScheme="PLOS Subject Area Thesaurus" schemeURI="https://github.com/PLOS/plos-thesaurus">Nanoelectronics</subject>
  </subjects>
  <dates>
    <date dateType="Created">2026-06-23T21:49:08Z</date>
    <date dateType="Submitted">2026-07-16T14:25:39Z</date>
    <date dateType="Issued">2026-07-22T00:00:00Z</date>
    <date dateType="Available">2026-07-22T00:00:00Z</date>
  </dates>
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  <descriptions>
    <description descriptionType="Abstract">
      A deterministic solver for the phonon Boltzmann transport equation (BTE)
      that retains the full phonon–phonon scattering
      matrix&amp;nbsp;W&amp;nbsp;— rather than the diagonal relaxation-time
      approximation (RTA) — for thermal transport in geometries ranging from
      one-dimensional slabs to a three-dimensional device.
    </description>
    <description descriptionType="TechnicalInfo">
      # TT-BTE: Deterministic full-scattering-matrix phonon Boltzmann transport
      solver A deterministic solver for the phonon Boltzmann transport equation
      (BTE) that retains the **full phonon–phonon scattering matrix** *W* —
      rather than the diagonal relaxation-time approximation (RTA) — for thermal
      transport in geometries ranging from one-dimensional slabs to a
      three-dimensional FinFET-like device. This repository contains the solver
      code accompanying the manuscript *"Deterministic full-matrix phonon
      BTE solver for 3D device geometries"* (Y. S. Ju). It reproduces the
      full-scattering-matrix construction, the hybrid mode-space solver, and the
      device geometries reported there. The interatomic model used throughout is
      the Stillinger–Weber (SW) potential for silicon. Material-specific
      transport magnitudes (e.g., the bulk conductivity and the device
      temperature-rise correction) are properties of this SW model and should
      not be read as quantitative predictions for crystalline silicon; the
      structural results of the paper are independent of this choice. --- ##
      What the solver does The phonon BTE under study is ``` v_λ · ∇ f_λ(r) =
      Σ_μ W_λμ f_μ(r) + Q_λ(r), ``` where `f_λ` is the deviational phonon
      distribution for mode `λ`, `v_λ` the group velocity, `W` the full
      scattering operator, and `Q` a heat source. The defining feature of this
      solver is that it keeps the complete off-diagonal `W`while remaining
      tractable in 3D, by a **hybrid mode-space** decomposition: * **Streaming**
      (`v · ∇`) is handled by dense, upwind spatial sweeps, which are trivially
      mode-local and need no compression. * **Scattering** (`W`) is applied as a
      compressed operator — either a truncated SVD of the in-scattering block
      (with the out-scattering diagonal carried exactly) or a tensor-train (TT)
      representation — with grid-independent rank. The scattering matrix is
      constructed to satisfy three physical constraints that are load-bearing
      for a stable device solve: 1. **Energy conservation**, `W c = 0`, where
      `c` are the modal heat capacities. 2. **Detailed balance**, `c_i W_ij =
      c_j W_ji`. 3. **Negative semi-definiteness** on the non-equilibrium
      subspace. Conservation is not optional: a non-conserving operator makes
      the device solver diverge. The construction enforces all three
      analytically and then projects out the residual, reaching machine
      precision on each (see "Verifying the install " below). --- ##
      Repository layout ``` __init__.py Sets the version and its description
      core/ Tensor-train algebra (physics-agnostic) tensor_train.py TensorTrain
      class, index bookkeeping tt_operators.py TTOperator (MPO),
      add/scale/compress tt_solvers.py AMEn linear solver physics/ Phonon model
      and scattering operator sw_potential.py Stillinger–Weber energy and forces
      si_physical_analysis_v2.py Force constants, phonons, and the full-W
      scattering-matrix construction (the core physical contribution)
      sw_phonon_bridge.py SWPhononComputation: end-to-end SW → (v, τ, C, W)
      phonon_data.py PhononBandData container and reference models mode_space.py
      ModeSpaceData: the mode-resolved model consumed by every solver; factory
      mode_space_from_sw() svd_scattering.py SVDScatteringOperator: low-rank W
      with exact γ tt_scattering.py TTScatteringOperator: tensor-train W
      solvers/ BTE solvers (hybrid mode-space) bte_slab_1d_hybrid.py 1D slab
      (validation against analytic reference) bte_slab_2d_hybrid.py 2D slab
      bte_monolithic_2d.py 2D monolithic domain dense_sweep_3d_hybrid.py 3D box
      dense-sweep reference bte_3d_finfet.py 3D FinFET-like device (fin + base,
      Schwarz coupling) — the headline device geometry dsa_operator.py
      Diffusion-synthetic acceleration (base) dsa_operator_1d.py DSA, 1D
      dsa_operator_2d.py DSA, 2D dsa_operator_3d.py DSA, 3D (Fourier-cosine) ```
      Diffusion-synthetic acceleration (DSA) is
      an **optional&amp;#xA0;**&amp;#x63;onvergence accelerator and is **off by
      default**; it is not part of the production methodology and is provided
      only for cross-checking. All the code files are
      in Dryad_tt_bte_solver.zip. --- ## Requirements * Python ≥ 3.10 * NumPy *
      SciPy That is the complete runtime requirement. The solver runs entirely
      on CPU/NumPy. JAX is an *optional* dependency, used only by a single
      GPU-resident DSA correction method (`DSAOperator3D.apply_correction_jax`).
      It is not needed to run any solver or to reproduce any result in the
      paper; the relevant import is lazy, so the package loads and runs without
      JAX installed. Install the requirements with: ``` pip install numpy scipy
      ``` --- ## Installing and importing This is a flat package: put the
      directory that *contains* `core/`,  `physics/`, and `solvers/` on your
      Python path and import the subpackages directly. The simplest approach is
      to run Python from inside this directory, or to add it to `PYTHONPATH`:
      ``` export PYTHONPATH=/path/to/this/directory:$PYTHONPATH ``` Then, in
      Python: ```python from physics import SWPhononComputation,
      SVDScatteringOperator, ModeSpaceData from physics.mode_space import
      mode_space_from_sw from solvers import BTESlab1D, DenseSweep3DHybrid from
      solvers.bte_3d_finfet import DenseSweep3DFinFET ``` --- ## Verifying the
      install The following builds the SW phonon model on a small grid,
      constructs the full scattering matrix, and confirms the conservation,
      detailed-balance, and semi-definiteness gates. It takes a couple of
      minutes (most of the time is the force-constant computation) and requires
      only NumPy and SciPy. ```python from physics import SWPhononComputation
      comp = SWPhononComputation(q_grid_size=3) # small 3x3x3 q-grid
      comp.compute(build_W=True, verbose=True) # builds v, tau, C, and W # After
      construction the solver prints the matrix gates, e.g.: # Energy
      conservation ||W c|| / ||c|| ~ 1e-16 (Wc = 0) # Detailed balance max|c_i
      W_ij - ...| ~ 1e-16 # Eigenvalues all &lt;= 0 (semi-definite)
      print("modes:", comp.W.shape[0]) ``` A clean run prints
      corrected conservation and detailed-balance residuals at the level of
      machine precision (`~1e-15` or smaller) and no positive eigenvalues. ---
      ## Typical workflow ### 1. Build the phonon model ```python from physics
      import SWPhononComputation from physics.mode_space import
      mode_space_from_sw comp = SWPhononComputation(q_grid_size=7) # production
      grids use larger N comp.compute(build_W=True) mode_data =
      mode_space_from_sw(comp) # ModeSpaceData for the solvers
      print(mode_data.summary()) ``` `ModeSpaceData` exposes the mode-resolved
      velocities, heat capacities, the full `W`, the RTA operator `W_rta`, the
      thermally active mask, and helpers such as `bulk_thermal_conductivity()`
      and `project_to_1d()`. ### 2. Compress the scattering operator (optional
      but recommended) ```python from physics import SVDScatteringOperator
      svd_op = SVDScatteringOperator.from_mode_data(mode_data, rank=50)
      svd_op.print_rank_summary() ``` The out-scattering diagonal `γ` is
      retained exactly; only the in-scattering block is approximated, so the
      transport-critical relaxation rates are preserved. ### 3. Solve a geometry
      1D slab (used for validation against the analytic Fuchs–Sondheimer
      reference): ```python from solvers import BTESlab1D slab =
      BTESlab1D(N_x=100, L=200e-9, mode_data=mode_data.project_to_1d()) result =
      slab.solve(tol=1e-8, verbose=True) ``` 3D FinFET-like device (the headline
      geometry): ```python from solvers.bte_3d_finfet import DenseSweep3DFinFET
      # Construct with the fin/base spatial resolution and geometry, supply the
      # mode_data and (optionally) the SVD-compressed operator, then solve. #
      See the DenseSweep3DFinFET docstring for the full constructor signature #
      and boundary-condition options (isothermal sink, diffuse side walls, #
      y-uniform line heat source). ``` Each solver returns a result object
      carrying the converged temperature field,  the heat flux, and solver
      diagnostics. --- ## Reproducing the paper's headline quantities The
      device result is the full-`W` correction to the peak temperature rise,
      defined as a same-grid difference between the full-`W` and RTA solutions
      on the FinFET-like geometry with a y-uniform line heat source on the top
      of the fin. For Brillouin-zone grids,s `N ≥ 11` the correction is ```
      δT_fW / (T_RTA − T_0) = 11.7 ± 0.3 %, ``` a bounded, grid-robust
      correction. The RTA baseline used for this comparison carries the same
      isotope channel as the full operator, so the two solutions differ only in
      the off-diagonal scattering. To reproduce a single point: build the model
      at the desired grid. `N` (`q_grid_size=N`), construct both the full and
      RTA operators from the same `ModeSpaceData` solve the FinFET geometry with
      each, and take the difference of the peak temperatures. The bulk
      conductivity of the SW model on this grid is `k_bulk = 148.0 W m⁻¹ K⁻¹`
      (isotropic to better than 0.003%), which is a useful cross-check that the
      phonon model is built correctly. --- ## Numerical notes and known
      properties * **Γ-centred Monkhorst–Pack sampling.** On Γ-centred grids,
      the absolute device temperatures converge non-monotonically at small odd
      `N` (the grid does not close under `q → −q`); monotone behaviour sets in
      by `N ≈ 11–15`. The same-grid *correction* is far less sensitive and is
      reported as a bounded band. * **SW group-velocity anisotropy.** On a
      Γ-centred gri,d the SW group velocities carry a fixed `[111]` ellipticity,
      giving a non-negligible off-diagonal Cartesian conductivity (`κ_xy/κ_xx ≈
      0.56`). The scalar bulk conductivity is unaffected (the tensor is
      traceless-clean). The y-uniform line source usedfor thee device results
      makes`∂T/∂y ≈ 0`, so this term is geometrically inactive. *
      **Scattering-matrix scaling.** The number of three-phonon channels, and
      hence the nonzeros of`W`, scales as the square of the active mode count,
      so the operator is genuinely dense. Compression of the *solution* (not the
      operator) is what makes the solver efficient. If you use this solver, we
      encourage you to cite the accompanying paper "*Deterministic
      full-matrix phonon BTE solver for 3D device geometries*," by Y. S. Ju
      in *Scientific Reports* (2026).
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