First-principles electron-phonon scattering in real-time TDDFT

Technical Summary: First-Principles Electron-Phonon Scattering in Real-Time TDDFT

Problem Statement
Real-time time-dependent density functional theory (rt-TDDFT) is a established first-principles tool for simulating coherent electron dynamics in laser-driven solids. However, its standard formulation relies on a unitary evolution of the Kohn-Sham orbitals, which describes a closed quantum system. Consequently, conventional rt-TDDFT cannot capture irreversible processes such as carrier relaxation, thermalization, and decoherence arising from the coupling of excited electrons to their environment. In real materials, electron-phonon (e-ph) interactions are the dominant mechanism for these dissipative processes, yet integrating them into a first-principles real-time framework while retaining access to spectroscopic observables has remained a central challenge. Existing approaches often rely on phenomenological models or semiclassical Boltzmann equations that track only carrier populations, failing to describe the decay of quantum coherences essential for ultrafast dynamics.

Methodology
The authors develop a dissipative rt-TDDFT framework that treats the electronic subsystem as an open quantum system weakly coupled to a thermal phonon bath. The methodology integrates three key components:

1. Open-System Density Matrix Formulation: Starting from first-principles e-ph matrix elements, the authors derive a collision integral for the reduced one-body density matrix within the Born–Markov approximation. The equation of motion for the reduced density matrix $\hat{\rho}^{(1)}$ combines the coherent Liouville–von Neumann evolution driven by the time-dependent Kohn-Sham Hamiltonian ($\hat{H}_{KS}$) with a dissipative term $I_{e-ph}$:
   $$\frac{\partial \hat{\rho}^{(1)}(t)}{\partial t} = -i [\hat{H}_{KS}(t), \hat{\rho}^{(1)}(t)] + I_{e-ph}[\hat{\rho}^{(1)}(t)]$$
   The collision integral $I_{e-ph}$ is constructed to simultaneously describe phonon-assisted population redistribution (gain-loss terms) and the decay of interband coherences (off-diagonal damping), ensuring the evolution remains completely positive and trace-preserving (CPTP).

2. Mixed Real-Space/Active-Space Propagation: To maintain computational feasibility while preserving the full capabilities of real-space rt-TDDFT, the authors employ a hybrid representation.

   • Coherent Dynamics: The unitary propagation of Kohn-Sham orbitals is performed in real space using the Octopus code, capturing nonlinear light-matter interactions and continuum effects.
   • Dissipative Updates: The collision integral is applied in a selected "active space" defined by a time-dependent adiabatic basis (instantaneous eigenstates of $\hat{H}_{KS}$). The density matrix is projected onto this basis, updated via the collision integral, and then mapped back to the real-space orbital representation.
   • Basis Construction: To avoid norm loss and gauge ambiguities, the Hilbert space is partitioned into an active space ($P$) and a residual space ($Q$). A gauge-fixing procedure based on the orthogonal Procrustes algorithm ensures the continuity of orbitals between time steps.

3. Coarse-Grained Transition Rates: Since first-principles e-ph scattering requires ultra-fine momentum grids to resolve meV-scale phonon energies (prohibitive for real-time propagation), the authors implement a coarse-graining scheme. Microscopic transition rates calculated on a dense grid (using the EPW code) are averaged over momentum patches to define effective rates for the coarser grid used in the rt-TDDFT simulation.

Key Results
The framework was validated through two benchmark applications:

• Ultrafast Carrier Relaxation in Bulk Silicon:

  • Simulating a 2.5 eV laser pulse, the method captured the redistribution of photoexcited carriers.
  • Energy vs. Momentum Relaxation: The results revealed a separation of timescales: momentum redistribution (scattering out of valleys) occurred on a 8–14 fs timescale, while energy relaxation (cooling) was slower (40–106 fs). This supports the formation of a quasi-equilibrium hot-electron ensemble before macroscopic cooling.
  • Quantitative Agreement: Extracted energy lifetimes (40 fs at 2.6 eV, 106 fs at 1.6 eV) showed quantitative agreement with experimental measurements, validating the dissipative approach.

• Valley Dynamics in Monolayer WS$_2$:

  • The study modeled intervalley scattering following circularly polarized (CP) and linearly polarized (LP) excitation.
  • Valley Depolarization: Under CP light, the initial valley polarization (K valley) decayed as carriers scattered to K', Q, and Q' valleys. The extracted intervalley scattering time was 20 fs (17 fs for LP), consistent with recent experimental values (16 fs).
  • tr-ARPES Connection: By coupling the dissipative dynamics with the tSURFF photoemission framework, the authors generated fully ab initio time-resolved ARPES (tr-ARPES) spectra. These simulations successfully reproduced the valley-selective excitation and subsequent redistribution, linking microscopic scattering rates to measurable photoemission signals.

Significance and Claims
The paper claims to provide a practical, first-principles route for simulating dissipative quantum dynamics in realistic crystalline materials without resorting to phenomenological parameters. Its primary contributions are:

1. Unified Description of Relaxation and Decoherence: Unlike semiclassical Boltzmann approaches that track only populations, this framework retains the full single-particle density matrix, allowing for the simultaneous description of population relaxation and coherence decay driven by the same microscopic scattering channels.
2. Integration with Spectroscopy: By embedding the dissipative update within the real-space rt-TDDFT framework of the Octopus code, the method preserves access to the full suite of observables, including nonlinear optical responses and time-resolved spectroscopic signatures (e.g., tr-ARPES).
3. Scalability: The active-space formulation limits the computational cost of the collision integral to the relevant subset of states, making the approach viable for realistic materials while maintaining the rigorous connection to first-principles e-ph theory.

The authors position this work as a controlled realization of a general open-system rt-TDDFT framework, capable of addressing ultrafast phenomena where coherent driving and dissipative scattering compete, such as in high-harmonic generation and pump-probe spectroscopy. They note that while the current implementation assumes a thermal phonon bath and Born–Markov approximations, the strategy is extensible to other environmental channels (e.g., photons, magnons) and regimes involving strong non-Markovian effects.