Study Plan

A brief look at the physics and maths required for studying about phonons based on the research paper by Kiarash Gordiz, and Asegun Henry

Key Equations to Learn (for phonon transport at interfaces)

1) Equations of motion → eigenproblem

  • Newton’s law for lattice vibrations: \[ M\,\ddot{u}(t) = -K\,u(t) \]
  • Harmonic ansatz: \[ u(t) = U\,e^{-i\omega t} \]
  • Generalized eigenproblem (natural frequencies/modes): \[ (K - \omega^2 M)\,U = 0 \]
  • Mass-normalized form: \[ D = M^{-1/2} K M^{-1/2}, \qquad D\,U = \omega^2 U, \qquad U^\top U = I \]

Relevance to the paper: This is the starting point; the paper’s central claim is that the correct modes for interfacial transport are the eigenmodes of the entire interface system (from the above eigenproblem), not the bulk modes of the isolated materials.

2) Lead dispersion (useful for band edges in 1D)

  • Monatomic 1D chain with mass \(m\) and spring \(k\): \[ \omega(q) = 2\sqrt{\frac{k}{m}}\,\Big|\sin\!\frac{qa}{2}\Big| \]

Relevance to the paper: Knowing lead band limits clarifies where propagating vs evanescent modes live, which affects which interface modes can actually carry energy across the junction.

3) Surface Green’s function (1D monatomic lead, closed form)

Let \(z=\omega^2+i\eta\), \(h_0=\tfrac{2k}{m}\), \(h_1=-\tfrac{k}{m}\). The surface Green’s function \(g\) solves \[ h_1^2\,g^2 - (z - h_0)\,g + 1 = 0, \qquad \text{choose the root with } \Im g < 0. \]

Relevance to the paper: Interfaces are treated by embedding semi-infinite leads; the self-energies below use these \(g\)’s, which then feed the device Green’s function used for mode-resolved transmission.

4) Embedding the leads (self-energies & broadenings)

  • Self-energies: \[ \Sigma_{L}(\omega) = V_L\, g_L(\omega)\, V_L^\top, \qquad \Sigma_{R}(\omega) = V_R\, g_R(\omega)\, V_R^\top \]
  • Broadening (coupling) matrices: \[ \Gamma_{L} = i\!\left(\Sigma_L - \Sigma_L^\dagger\right), \qquad \Gamma_{R} = i\!\left(\Sigma_R - \Sigma_R^\dagger\right) \]

Relevance to the paper: These capture how the bulk leads couple into the interface region; the paper’s mode-correction is about what basis you use when interpreting the transmission that results.

5) Device Green’s function (frequency domain)

\[ G(\omega) = \Big[(\omega^2 + i\eta)I - D - \Sigma_L(\omega) - \Sigma_R(\omega)\Big]^{-1} \]

Relevance to the paper: \(G(\omega)\) is the engine of transport; the paper emphasizes that mode analysis should be done in the eigenbasis of the full interface system associated with \(D\), not projected onto bulk eigenmodes.

6) Transmission (Caroli formula)

\[ T(\omega) = \mathrm{Tr}\!\Big[\Gamma_L\, G\, \Gamma_R\, G^\dagger\Big] \]

Relevance to the paper: This is the scalar quantity everyone reports. The paper clarifies how to decompose \(T(\omega)\) correctly into contributions of the interface’s own modes, avoiding misinterpretation from bulk-mode projections.

7) Spectral function & density of states (diagnostics)

  • Spectral function: \[ A(\omega) = i\!\left(G - G^\dagger\right) \]
  • Local/total phonon DOS: \[ \rho(\omega) = \frac{1}{2\pi}\,\mathrm{Tr}\!\big[A(\omega)\big] \]

Relevance to the paper: Diagonalizing \(A(\omega)\) (or projecting responses onto system modes \(U\)) lets you build mode-resolved channels consistent with the “correct modes” definition at the interface.

8) Landauer heat current and thermal conductance (optional but useful)

  • Heat current: \[ J = \int_0^\infty \frac{\hbar \omega}{2\pi}\, T(\omega)\,\Big[n_B(\omega,T_L) - n_B(\omega,T_R)\Big]\, d\omega \] where \(n_B(\omega,T)=\frac{1}{e^{\hbar\omega/k_BT}-1}\).
  • Linear-response thermal conductance: \[ \kappa(T) = \int_0^\infty \frac{(\hbar\omega)^2}{2\pi k_B T^2}\, T(\omega)\, n_B(\omega,T)\!\left(1+n_B(\omega,T)\right)\, d\omega \]

Relevance to the paper: While the paper is about mode definitions, these relations show how improved, correctly mode-resolved \(T(\omega)\) directly impacts macroscopic thermal predictions.

9) “Correct modes” participation at contacts (simple diagnostic)

  • A handy contact-participation metric for system mode \(n\): \[ p_n = \sum_{i \in \text{ends}} |U_{in}|^2 \]

Relevance to the paper: A practical way to see which interface eigenmodes actually couple to the leads—aligns with the paper’s message to interpret transport in terms of interface/system modes rather than bulk ones.

📚 Reading Guide: Numerical Linear Algebra for our Phonon-NEGF notebook

This sequence maps each concept we used (eigenmodes, Green’s functions as linear solves, stability, block ops, sparsity, randomized methods) to fastai’s Numerical Linear Algebra notebooks — with a quick “why this matters for NEGF” under each.

1) Big-picture & accuracy/conditioning basics

  • 1. why are we here — overview of matrix computations, accuracy, vectorization
    Sets intuition for why small numerical tweaks (like \(i\eta\)) and stable algorithms matter when solving \((\omega^2 I - D - \Sigma_L - \Sigma_R)\).

2) Optional: faster big-matrix approximations

  • 4. Randomized SVD — when your \(D\) or per-frequency solves get big
    Useful to compress large subspaces (e.g., for model reduction or preconditioners) when sweeping many \(\omega\).

3) Block algebra & factorizations (for lead/device partitioning)

  • 5. LU Factorization — Gaussian elimination, pivoting, Block Matrix Multiplication
    Block reasoning underpins self-energies and Schur complements for embedding leads into the device.

4) Sparse thinking & broadcasting (scale up your NEGF)

5) Stable linear solves (don’t “invert the matrix”)

  • 8. How to Implement Linear Regression — Normal eqs vs Cholesky/QR/SVD, matrix inversion is unstable, conditioning & stability
    Direct inverses of the Green’s-function matrix are numerically brittle; prefer factorizations/solves and watch conditioning.

6) Eigenstuff for modes (power/QR/Arnoldi)

  • 9. PageRank with Eigen Decompositions — power method, QR, Arnoldi (intuition for mode finding at scale)
    The “correct modes” are eigenvectors of \(D\); scalable eigentools help analyze large interface systems.

7) QR details (for stable decompositions)

  • 10. Implementing QR Factorization — Gram–Schmidt, Householder, stability examples
    QR/Householder give orthogonal transforms that stabilize solves/eigendecompositions used for \(G(\omega)\) and mode analysis.