Nonequilibrium Photocarrier and Phonon Dynamics from First Principles: a Unified Treatment of Carrier-Carrier, Carrier-Phonon, and Phonon-Phonon Scattering

Technical Summary: Nonequilibrium Photocarrier and Phonon Dynamics from First Principles

Problem Statement
Recent advances in ultrafast pump-probe experiments have enabled the monitoring of photocarrier and phonon dynamics in semiconductors with high accuracy across femtosecond to picosecond timescales. These experiments reveal complex phenomena such as nonthermal melting, coherent phonon dynamics, and light-induced phase transitions. However, existing theoretical frameworks lack a unified, predictive, and computationally efficient approach to describe these coupled nonequilibrium dynamics. Previous methods, including Non-Equilibrium Green's Functions (NEGF), Semiconductor Bloch equations, and semiclassical Boltzmann approaches, often neglect one or more of the three critical scattering mechanisms: carrier-carrier, carrier-phonon, and phonon-phonon scattering. Furthermore, many approaches fail to treat light-matter coupling explicitly, ignore time-dependent quasiparticle renormalizations, or rely on phenomenological parameters rather than ab initio inputs. The absence of a framework that simultaneously treats all three scattering channels on an equal footing, while accounting for coherent atomic motion and time-dependent screening, limits the ability to quantitatively interpret experimental data.

Methodology
The authors develop a first-principles many-body framework to describe the real-time dynamics of photocarriers and phonons in semiconductors. The methodology is built upon the following pillars:

1. Hamiltonian and Basis: The approach starts from the full quantum Hamiltonian of electrons and nuclei under the Born-Oppenheimer approximation. The electronic dynamics are treated in a maximally localized Wannier function (MLWF) basis, derived from Density Functional Theory (DFT) and Density Functional Perturbation Theory (DFPT) calculations. This choice ensures gauge-consistent scattering integrals and allows for dense momentum sampling via direct interpolation schemes.
2. Equations of Motion (EOMs): The time evolution is governed by quantum kinetic equations for electronic occupations ($f_{kn}$), microscopic polarizations ($p_{knm}$), and phonon occupations ($n_{q\nu}$). These EOMs are derived from the density matrix formalism, incorporating:
   • Light-Matter Coupling: Explicit inclusion of a time-dependent electric field via the minimal-coupling substitution in the dipole approximation.
   • Quasiparticle Renormalization: Time-dependent corrections to electronic eigenvalues and Rabi frequencies, and phonon frequencies, arising from the real part of the self-energy. This includes Hartree, COHSEX (Coulomb-hole plus screened-exchange), Fan-Migdal, and Debye-Waller terms.
   • Collision Integrals: Scattering terms derived from the imaginary part of the self-energy using the Generalized Kadanoff-Baym Ansatz (GKBA) and the Markov approximation. These include:
     • Carrier-Carrier: Screened Coulomb interaction (second-order Born approximation).
     • Carrier-Phonon: Fan-Migdal approximation with statically screened electron-phonon vertices.
     • Phonon-Phonon: Anharmonic scattering derived from third-order force constants (Fermi's golden rule).
3. Coherent Atomic Motion: The framework includes a displacive excitation mechanism where photoexcited carriers generate non-equilibrium forces on ions. These forces are calculated using a linear-response DFPT approach that accounts for the screening of the electron-phonon vertex by the evolving electronic distribution, going beyond standard Ehrenfest dynamics where orbitals are fixed.
4. Computational Implementation: The method is implemented in the open-source epiq suite, interfaced with Quantum ESPRESSO and Yambo. It utilizes adaptive time-stepping (Runge-Kutta Dormand-Prince) and parallelization (MPI) to handle the computational cost of evaluating collision integrals over dense Brillouin zones.

Key Contributions
The paper presents a unified theoretical and computational framework that achieves the following:

• Unified Scattering Treatment: It is the first ab initio framework to simultaneously treat carrier-carrier, carrier-phonon, and phonon-phonon scattering on an equal footing in real-time dynamics.
• Dynamic Screening and Renormalization: It explicitly captures time-dependent quasiparticle energy renormalizations (bandgap shrinking) and phonon frequency shifts driven by the evolving photocarrier distribution and screening.
• Gauge-Consistent Interpolation: By utilizing a Wannier representation with a direct-interpolation scheme, the method enables ultra-dense momentum sampling necessary for accurate scattering integrals without the prohibitive cost of direct ab initio calculations at every time step.
• Coherent Phonon Dynamics: It provides a first-principles calculation of coherent atomic motion and damping, deriving forces from the screened electron-phonon coupling rather than fixed orbitals.
• Scalability: The implementation is scalable and computationally efficient, allowing for parameter-free comparisons with experimental pump-probe conditions at realistic fluences and temperatures.

Results
The framework is applied to monolayer MoS$_2$ and hexagonal boron nitride (h-BN):

• MoS$_2$ (Ultrafast Dynamics):

  • Carrier Relaxation: The inclusion of carrier-carrier scattering is shown to be essential for realistic ultrafast carrier equilibration (tens of femtoseconds). Neglecting it leads to relaxation times that are an order of magnitude too long.
  • Hot Phonons and Thermalization: Carrier-phonon scattering initially populates specific "hot" phonon modes (e.g., $A_{1g}$). However, neglecting phonon-phonon scattering results in incorrect long-time lattice thermalization and overestimates the coherent $A_{1g}$ phonon damping time by a factor of two.
  • Renormalization: The method captures significant photoinduced bandgap reduction (from ~2.6 eV to ~1.8 eV) due to screening changes, consistent with time-resolved ARPES data.
  • Coherent Motion: The calculated displacive forces and $A_{1g}$ mode damping times agree well with experimental transient absorption data, validating the treatment of screened forces.

• h-BN (Quasi-Equilibrium Properties):

  • Screening and Band Structure: In the quasi-equilibrium state following carrier thermalization, the framework quantifies the fluence-dependent enhancement of screening. This leads to a significant reduction of the single-particle bandgap.
  • Exciton Melting: The method demonstrates the melting of excitonic features (1s and 2s peaks) in the optical absorption spectrum as photocarrier density increases, with the 1s exciton persisting up to ~$3.8 \times 10^{13}$ cm$^{-2}$.
  • Phonon Renormalization: Calculated phonon frequency shifts due to excited carriers agree closely with constrained DFPT (cDFPT) benchmarks, validating the self-energy approach for predicting lattice responses in photoexcited materials.

Significance and Claims
The authors claim that this work establishes a versatile and quantitatively predictive computational platform for interpreting ultrafast experiments in semiconductors. By treating all three scattering mechanisms and time-dependent renormalizations on an equal footing, the framework resolves discrepancies found in simplified models (e.g., neglecting phonon-phonon scattering or carrier-carrier interactions). The ability to directly compare with experimental observables (reflectivity, absorption, phonon lifetimes) without adjustable parameters represents a significant step forward in the theoretical description of light-matter interactions. The paper positions this approach as a tool to guide future experiments and investigate nonequilibrium phenomena, including light-induced structural phase transitions, by coupling with constrained DFT for longer time scales. The authors explicitly state that their results highlight the regimes where simplified descriptions fail qualitatively, emphasizing the necessity of their unified treatment for accurate modeling.