Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids

At a Glance

Metadata	Details
Publication Date	2016-01-06
Journal	Physical review. B./Physical review. B
Authors	Tianli Feng, Xiulin Ruan
Institutions	Purdue University West Lafayette
Citations	336
Analysis	Full AI Review Included

Technical Documentation & Analysis: High-Order Phonon Scattering in CVD Diamond

This document analyzes the research on four-phonon scattering in diamond and semiconductors, translating the findings into actionable technical specifications and demonstrating how 6CCVD’s advanced MPCVD diamond materials directly support and enable this critical research area, particularly in high-temperature thermal management.

Executive Summary

Critical Thermal Physics: The research successfully developed an Anharmonic Lattice Dynamics (ALD) formalism to explicitly calculate intrinsic four-phonon scattering rates ($\tau_{4,\lambda}^{-1}$), demonstrating their necessity for accurate thermal conductivity ($\kappa$) prediction.
High-Temperature Dominance: Four-phonon scattering rates are comparable to three-phonon rates ($\tau_{3,\lambda}^{-1}$) at medium and high temperatures, increasing quadratically with temperature ($\sim T^{2}$) versus the linear increase ($\sim T$) of three-phonon rates.
Impact on Diamond/Si/Ge: Inclusion of four-phonon processes significantly reduces predicted lattice thermal conductivity ($\kappa$) at high temperatures, with reductions up to 36% in Germanium and 15% in Diamond at 1000 K.
Optical Phonon Importance: For optical phonons in diamond, silicon, and germanium, fourth-order and higher-order scattering processes are found to be dominant even at low temperatures.
Material Requirement: Accurate high-temperature $\kappa$ prediction requires high-purity, well-characterized materials like SCD, validating the need for 6CCVD’s specialized MPCVD diamond for intrinsic thermal studies.

Technical Specifications

The following hard data points were extracted, focusing on the thermal behavior and computational parameters relevant to advanced material characterization.

Parameter	Value	Unit	Context
Materials Investigated	Diamond, Si, Ge, Ar	N/A	Less anharmonic bulk solids
Si $\kappa$ Reduction (1000 K)	~25	%	Due to four-phonon scattering inclusion
Diamond $\kappa$ Reduction (1000 K)	15	%	Due to four-phonon scattering inclusion
Ge $\kappa$ Reduction (1000 K)	36	%	Due to four-phonon scattering inclusion
Ar $\kappa$ Reduction (80 K)	> 60	%	Due to four-phonon scattering inclusion
$\tau_{3,\lambda}^{-1}$ Temperature Dependence	~ T	N/A	High temperatures (T > 300 K)
$\tau_{4,\lambda}^{-1}$ Temperature Dependence	~ T²	N/A	High temperatures (T > 300 K)
k-Mesh Size (Diamond, Si, Ge)	16 x 16 x 16	k-points	Brillouin Zone calculation
Si Intrinsic 4-Phonon Events	4.6 x 10⁷	Events	TA mode at k* = (0.5, 0, 0)
Si Resonant 3-Phonon Events	2.7 x 10⁴	Events	TA mode at k* = (0.5, 0, 0)
Maximum Temperature Studied	1200	K	Ge comparison with NMA
Diamond Debye Temperature	~2200	K	Classical MD comparison limitation

Key Methodologies

The study employed advanced computational physics techniques to overcome the long-standing challenge of calculating high-order phonon scattering.

Formalism Derivation: Developed an Anharmonic Lattice Dynamics (ALD) formalism based on perturbation theory, extending the derivation of Maradudin et al. to rigorously calculate four-phonon scattering probability matrices.
Thermal Transport Calculation: Lattice thermal conductivity ($\kappa$) was calculated using the Boltzmann Transport Equation (BTE) in the Single Mode Relaxation Time Approximation (SMRTA).
Interatomic Potentials: Empirical interatomic potentials were used: the classical Tersoff potential for Diamond, Silicon, and Germanium, and the Lennard-Jones potential for Argon.
Force Constant Calculation: Second-, third-, and fourth-order Interatomic Force Constants (IFCs) were obtained using the central difference method based on finite differences of energy derivatives.
Computational Cost Mitigation: Computational efficiency was achieved by:
- Excluding atomic combinations that yield negligible IFCs.
- Reducing the dimensions of the scattering matrices by pre-excluding mode combinations that do not satisfy momentum and energy selection rules.
Validation: Results were benchmarked against Molecular Dynamics (MD) simulations and Normal Mode Analysis (NMA) results, showing strong agreement for $\kappa_{3+4}$ (three- and four-phonon scattering included) at high temperatures.

6CCVD Solutions & Capabilities

This research confirms that accurate thermal modeling of diamond, especially for high-power and high-temperature applications, requires accounting for intrinsic four-phonon scattering. 6CCVD provides the necessary high-purity, custom-engineered MPCVD diamond materials to validate and extend these theoretical findings into practical engineering solutions.

Applicable Materials

The study emphasizes the intrinsic thermal properties of diamond. To replicate or extend this research, materials with minimal extrinsic scattering are essential:

Optical Grade Single Crystal Diamond (SCD): Ideal for fundamental phonon physics studies. Our SCD offers the highest purity (low nitrogen/boron content) and lowest defect density, ensuring that measured thermal properties reflect the intrinsic $\kappa$ required for validating ALD models up to 1000 K and beyond.
High-Purity Polycrystalline Diamond (PCD): Suitable for large-area thermal management applications. 6CCVD offers PCD plates up to 125mm, providing high thermal conductivity across large footprints for industrial validation of $\kappa_{3+4}$ models.

Customization Potential

The complexity of thermal measurements often requires unique sample geometries and interfaces. 6CCVD’s capabilities ensure researchers have access to tailored materials:

Research Requirement	6CCVD Capability	Technical Specification
Large-Area Testing	Custom Dimensions	Plates/wafers up to 125mm (PCD)
Precise Thickness Control	SCD/PCD Thickness	0.1µm to 500µm (Wafers)
Substrate Integration	Substrate Thickness	Up to 10mm
Low-Loss Interfaces	Ultra-Smooth Polishing	Ra < 1nm (SCD), Ra < 5nm (PCD)
Thermal/Electrical Contacts	Custom Metalization	In-house deposition of Au, Pt, Pd, Ti, W, Cu

Engineering Support

The findings demonstrate that the existing practice of limiting ALD to three-phonon scattering is insufficient for high-temperature applications.

Expert Consultation: 6CCVD’s in-house PhD team specializes in CVD growth optimization and material physics. We offer consultation on material selection and specification for High-Temperature Thermal Management projects, ensuring the chosen diamond grade meets the required intrinsic thermal properties for accurate BTE/ALD validation.
Global Logistics: We support global research efforts with reliable shipping, offering DDU (default) and DDP options worldwide.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

Recently, first principle-based prediction of lattice thermal conductivity\n$\kappa$ from the perturbation theory has achieved significant success.\nHowever, it only includes three-phonon scattering due to the assumption that\nfour-phonon and higher-order processes are generally unimportant. Also,\ndirectly evaluating the scattering rates of four-phonon and higher-order\nprocesses has been a long-standing challenge. In this work, however, we have\ndeveloped a formalism to explicitly determine quantum mechanical scattering\nprobability matrices for four-phonon scattering in the full Brillouin Zone, and\nby mitigating the computational challenge we have directly calculated\nfour-phonon scattering rates. We find that four-phonon scattering rates are\ncomparable to three-phonon scattering rates at medium and high temperatures,\nand they increase quadratically with temperature. As a consequence, $\kappa$ of\nLennard-Jones argon is reduced by more than 60% at 80 K when four-phonon\nscattering is included. Also, in less anharmonic materials — diamond, silicon,\nand germanium, $\kappa$ is still reduced considerably at high temperature by\nfour-phonon scattering. Also, the thermal conductivity of optical phonons is\ndominated by the fourth and higher orders phonon scattering even at low\ntemperature.\n