[Paper Review] Generalized Drude-Lorentz Model Complying with the Singularity Expansion Method

Isam Ben Soltane, Félice Dierick+2 more|arXiv (Cornell University)|Jan 11, 2024

Photonic and Optical Devices4 citations

TL;DR

This paper introduces a Generalized Drude-Lorentz (GDL) model derived from the Singularity Expansion Method (SEM), treating dielectric permittivity as a meromorphic function in the complex frequency plane. By leveraging auto-differentiation via PyTorch, the method accurately retrieves model parameters from experimental data across metals, dielectrics, and 2D materials, with poles and residues revealing microscopic material behavior with sub-eV precision.

ABSTRACT

Deriving analytical expressions of dielectric permittivities is required for numerical and physical modeling of optical systems and the soar of non-hermitian photonics motivates their prolongation in the complex plane. Analytical models are based on the association of microscopic models to describe macroscopic effects. However, the question is to know whether the resulting Debye Drude Lorentz models are not too restrictive. Here we show that the permittivity must be treated as a meromorphic transfer function that complies with the requirements of complex analysis. This function can be naturally expanded on a set of complex singularities. This singularity expansion of the dielectric permittivity allows us to derive a generalized expression of the Debye Drude Lorentz model that complies with the requirements of complex analysis and the constraints of physical systems. We show that the complex singularities and other parameters of this generalized expression can be retrieved from experimental data acquired along the real frequency axis. The accuracy of this expression is assessed for a wide range of materials including metals, 2D materials and dielectrics, and we show how the distribution of the retrieved poles helps in characterizing the materials.

Motivation & Objective

• To overcome the limitations of classical Drude-Lorentz models in describing permittivity across the complex frequency plane, especially for non-Hermitian photonics and time-domain simulations.
• To establish a rigorous framework where dielectric permittivity is treated as a meromorphic transfer function compliant with complex analysis and physical causality.
• To develop a parameter retrieval method that accurately reconstructs permittivity from real-axis experimental data using modern optimization tools.
• To demonstrate that the distribution of complex poles and residues in the GDL model encodes meaningful physical information about material response across different frequency regimes.

Proposed method

• Formalize the dielectric permittivity as a meromorphic transfer function in the complex frequency plane, enabling exact singularity expansion via the Singularity Expansion Method (SEM).
• Reformulate the SEM expansion into a generalized Drude-Lorentz (GDL) model that includes frequency-dependent imaginary residues, distinguishing it from classical Lorentz models.
• Introduce a generalized Lorentz term with complex residues that accounts for the first derivative of the electric field in the time domain, enabling accurate temporal-domain modeling.
• Implement an optimization-based parameter retrieval method using auto-differentiation via PyTorch, minimizing error between experimental data and GDL model predictions.
• Apply the method to 9 materials (metals, oxides, 2D materials), fitting the GDL model over a broad spectral window (UV to NIR) using a minimal number of poles.
• Reconstruct physical insights by analyzing the distribution of poles and residues, linking them to material regimes such as metallic, dielectric, or near-constant permittivity behavior.

Experimental results

Research questions

• RQ1 Can the singularity expansion of permittivity provide a more general and physically consistent framework than classical Drude-Lorentz models?
• RQ2 How can the complex poles and residues of the permittivity function be retrieved from real-frequency experimental data with high accuracy?
• RQ3 What is the physical significance of the imaginary parts of the residues in the generalized Lorentz terms, and how do they affect time-domain behavior?
• RQ4 To what extent can the distribution of poles in the complex plane characterize transitions between metallic, dielectric, and transparent regimes?
• RQ5 Can a minimal set of generalized poles accurately describe permittivity across wide spectral windows, including for complex materials like 2D materials and oxides?

Key findings

• The GDL model achieves sub-eV accuracy in fitting experimental permittivity data across metals (e.g., Au), dielectrics (e.g., SiO₂), and 2D materials (e.g., hBN) over a 1–10 eV spectral window.
• For gold (Au), the retrieved Drude poles are purely imaginary and located near the origin, consistent with free-electron behavior, while Lorentz poles at 2.7–13 eV align with known band gaps.
• The generalized Lorentz terms include complex residues that introduce a frequency-dependent imaginary contribution, enabling accurate time-domain modeling through the first derivative of the electric field.
• The distribution of poles reveals distinct physical regimes: Drude poles mark metallic behavior, Lorentz poles indicate resonant transitions, and absence of poles near the real axis corresponds to near-constant permittivity.
• The auto-differentiation-based retrieval method achieves low error with only 5–7 poles across all materials, demonstrating high efficiency and robustness.
• The method successfully identifies that the Debye-like terms in the GDL model are subsumed into the generalized Lorentz framework, with their residues carrying physical significance tied to relaxation dynamics.