Unified ab initio quantum-electrodynamical density-functional theory for cavity-modified electron-phonon-photon coupling in solids

Abstract

Quantum-electrodynamical density-functional theory (QEDFT) provides a first-principles framework for describing materials coupled to quantized electromagnetic fields. While QEDFT has successfully captured cavity-induced modifications of electronic structures in atoms and molecules, a fully self-consistent and accurate framework to simulate and predict the structural, phonon-related, polarization and optical response of periodic solids in optical cavities has remained elusive. Here, we introduce a unified QEDFT approach that incorporates collective light-matter coupling in the electronic ground state, density functional perturbation theory for phonons, and real-time time-dependent QEDFT for optical excitations. This framework enables \textit{ab initio} calculations of cavity-modified electronic and phononic dispersions, Born effective charges, dielectric tensors, and both resonant and non-resonant optical absorption spectra. Using wurtzite \ac{GaN} in an optical cavity as a case study, we demonstrate that the quantized vacuum field reshapes electronic, phononic and polarization properties, producing experimentally accessible signatures in the transmission and absorption spectra. These results establish QEDFT as a general first-principles platform for predicting and exploring cavity-modified quantum materials.

Figures (8)

• Figure 1: Schematic illustration of the unified ab initio QEDFT framework for describing materials inside optical cavities. The central panel represents the QEDFT treatment of an extended material collectively coupled to quantized cavity modes. This framework enables consistent calculations of cavity-modified electronic structures (top left), phononic structures obtained from DFPT (top right), electronic polarization (bottom left), and optical responses under a weak external probe laser (bottom right).
• Figure 2: (a) The crystal structure of wurtzite GaN. (b) The modified electron density difference $\Delta \rho_e=\rho_{\rm {QEDFT}}-\rho_{\rm {DFT}}$ of GaN, where $\rho_{\rm {QEDFT}}$ is the electron density under cavity photon field with the collective light-matter coupling parameter $\lambda^\prime_{\alpha}=0.1\omega_{\alpha}$ and the photon frequency $\omega_{\alpha}=0.037\ \rm{Ha}\ (1\ \rm{eV})$, and $\rho_{\rm {DFT}}$ is the electron density without photon coupling. The $x+y$ cavity photon modes are polarized parallel to the gray plane shown in (a). (c) Similar results to (b), but with the photon mode polarized along the $z$ direction. (d) Electronic band structures of GaN, along with the PDOS for Ga $4s$ and N $2p_{x,y,z}$ orbitals. The black lines indicate the band structures in the absence of cavity fields. The pink solid lines and blue dashed lines represent the band structures under $x+y$ and $z$ modes, respectively. The BZ of GaN is shown in the upper inset. (e) The change in the direct band gap of GaN as a function of the number of unit cells $N_{\rm{cell}}$ under the $x+y$ and $z$ modes, induced by the collective light-matter coupling parameter $\lambda_\alpha^\prime$. The mode volume $\Omega_\alpha$ is $10^{12}$ Bohr$^3$, and the photon energy is 1 eV. The gray dashed line indicates the band gap (2.175 eV) outside the cavity. (f) Similar results as (e), but with the mode volume $\Omega_\alpha=10^{13}$ Bohr$^3$.
• Figure 3: (a) Phonon dispersions of GaN, together with the DOS and PDOS for Ga and N atoms. The black lines represent the phonon dispersions in the absence of cavity fields. The pink (blue) solid lines correspond to the phonon dispersions under cavity photon modes polarized along the $x+y$$(z)$ direction, computed using a collective light-matter coupling parameter $\lambda^\prime_{\alpha}=0.1\omega_{\alpha}$ and a photon frequency $\omega_{\alpha}=0.037\ \rm{Ha}\ (1\ \rm{eV})$. For comparison, phonon dispersions obtained via the FD method are shown as blue open circles. As highlighted by the black dashed boxes, the two insets in the upper and lower panels display representative branches around the $\Gamma$ point. The irreducible representations of the optical modes at the $\Gamma$ point are marked in red. (b) The vibration patterns of two selected optical phonon modes, extracted from the black-line regions at the $\Gamma$ point in the two insets of (a). The high-frequency mode in the upper panel is 21.23 Terahertz (THz) and the low-frequency mode in the lower panel is 4.18 THz outside the cavity.
• Figure 4: (a) The in-plane (out-of-plane) Born effective charge $Z^{*}_{\perp}$ ($Z_{\parallel}^{*}$) of the Ga atom as a function of the ratio between the collective light-matter coupling parameter and photon frequency, $\lambda^\prime_{\alpha}/\omega_{\alpha}$. The inset shows the Born effective charge $Z^{*}$ ($-Z^{*}$) for the Ga (N) atom. (b) The in-plane (out-of-plane) high-frequency dielectric function $\varepsilon^{\infty}_{\perp}$ ($\varepsilon^{\infty}_{\parallel}$) as a function of $\lambda^\prime_{\alpha}/\omega_{\alpha}$. (c) Similar results to (b), but for the static dielectric function $\varepsilon^{0}_{\perp}$ ($\varepsilon^{0}_{\parallel}$). (d) The out-of-plane polarization $P_\parallel$ under the $x+y$ and $z$ modes as a function of $\lambda^\prime_{\alpha}/\omega_{\alpha}$. The gray dashed line indicates the polarization outside the cavity. The horizontal axis uses logarithmic scaling to better present the data.
• Figure 5: (a) A thin film of GaN with a thickness of $1$$\mu m$ inside a DBR cavity. The incident electric field is normally aligned with respect to the cavity structure and polarized along the $y$ direction. The refractive index of silicon (air) is $3.42$ (1.0). (b) Calculated transmission spectra of the cavity structure containing the GaN thin film with various static dielectric functions $\varepsilon^{0}_{\parallel}$. The two zoomed-in panels display the local spectral shift more clearly. (c) Spatial distribution of the amplitude of the electric field $E_{y}$ inside the cavity at the frequency of $2.9368$ THz.