Fluctuation spectroscopy in granular superconductors with application to boron-doped nanocrystalline diamond

At a Glance

Metadata	Details
Publication Date	2021-09-13
Journal	Physical review. B./Physical review. B
Authors	David T. S. Perkins, Georgina M. Klemencic, J. M. Fellows, Robert A. Smith
Institutions	Cardiff University, University of Birmingham
Citations	1
Analysis	Full AI Review Included

Technical Documentation & Analysis: Fluctuation Spectroscopy in Boron-Doped Nanocrystalline Diamond

6CCVD Material Analysis Reference: Perkins et al., “Fluctuation Spectroscopy in Granular Superconductors with Application to Boron-doped Nanocrystalline Diamond” (arXiv:2105.08973v2)

Executive Summary

This research provides fundamental insights into the quantum transport properties of granular superconductors, specifically utilizing Boron-Doped Nanocrystalline Diamond (BNCD) films. The findings directly validate the need for highly controlled, heavily doped diamond materials, a core offering of 6CCVD.

Core Achievement: Detailed calculation of fluctuation conductivity ($\sigma_{fl}$) in granular metals, identifying three distinct power-law regions based on the reduced temperature ($\eta = (T - T_{c})/T_{c}$).
Theoretical Advancement: The model successfully combined internal (intragrain) and external (intergrain) degrees of freedom (DOFs) to consistently describe the close-to-T_c, intermediate, and far-from-T_c regimes.
Material Requirement: The study relies on experimental data from BNCD films, demonstrating the critical role of heavily Boron-Doped Diamond (BDD) as a granular superconducting system (T_c up to 3.8 K).
Key Experimental Discrepancy: Experimental data showed an unexpected $\eta^{-3}$ power law in the intermediate region, attributed to the Aslamazov-Larkin (AL) contribution dominating.
Material Implication: Achieving the observed $\eta^{-3}$ behavior requires a significant phase breaking rate ($\tau_{\phi}^{-1} \sim T_{c}$), necessitating highly controlled granular structures and doping levels achievable via MPCVD.
6CCVD Value Proposition: 6CCVD specializes in custom Boron-Doped Polycrystalline Diamond (PCD/BDD) wafers, offering precise control over thickness, grain size, and doping concentration required to replicate and advance these complex quantum transport experiments.

Technical Specifications

The following parameters, derived from the experimental data on BNCD films used for numerical comparison, define the required material characteristics for this superconducting granular system:

Parameter	Value	Unit	Context
Observed Transition Temperature (T_c)	3.8	K	Superconducting transition temperature of BNCD film
Thouless Energy (E_Th)	1	K	Intragrain energy scale
Electron Tunneling Rate (Γ)	2.62 x 10^-2	K	Intergrain energy scale
Mean Level Spacing (δ)	5.6 x 10^-3	K	Used to confirm granular metallic limit ($\delta \ll \Gamma$)
Intragrain Diffusion Coefficient (D₀)	13.1	cm s^-1	Derived parameter for internal transport
Typical Grain Size (a)	10^-7	m	Assumed cubic grain size (100 nm)
Carrier Concentration (n)	10²⁷	m^-3	Corresponds to heavy Boron doping
Experimental Film Thickness	329	nm	BNCD film used in referenced experiments [13]
Required Phase Breaking Rate ($\tau_{\phi}^{-1}$)	$\sim T_{c}$	K	Necessary to suppress anomalous MT term and observe $\eta^{-3}$ power law

Key Methodologies

The theoretical framework relies on extending the granular diagrammatic formalism to accurately model fluctuation conductivity in the metallic regime of BNCD.

Extended Diagrammatic Formalism: The standard granular diagrammatic theory was extended to include both internal (intragrain) and external (intergrain) degrees of freedom (DOFs) simultaneously, crucial for defining the three temperature regimes.
Fluctuation Contributions: Calculation focused on the leading order corrections to conductivity: Aslamazov-Larkin (AL), Maki-Thompson (MT), and Density of States (DOS) diagrams.
Temperature Regime Definition: Three distinct regions of reduced temperature ($\eta$) were defined by the crossovers at $\Gamma/T_{c}$ and $E_{Th}/T_{c}$, corresponding to changes in the Cooper pair coherence length relative to the grain size ($a$).
- Close-to-T_c ($\eta \ll \Gamma/T_{c}$): System appears homogeneous ($\xi_{T} \ge a$).
- Intermediate ($\Gamma/T_{c} \ll \eta \ll E_{Th}/T_{c}$): System appears quasi-zero dimensional ($\xi_{T} \approx a$).
- Far-from-T_c ($E_{Th}/T_{c} \ll \eta \ll 1$): System appears homogeneous again ($\xi_{g} \ll a$).
Material Modeling: Boron-Doped Nanocrystalline Diamond (BNCD) was modeled as a cubic lattice of grains (side length $a$) with quantized internal momenta based on Neumann boundary conditions.
Phase Breaking Inclusion: A constant phase breaking rate ($\tau_{\phi}^{-1}$) was introduced to numerically suppress the anomalous MT term, demonstrating that this suppression is necessary to match the experimentally observed $\eta^{-3}$ power law in the intermediate region.

6CCVD Solutions & Capabilities

The study of fluctuation spectroscopy in BNCD requires highly specialized, heavily Boron-doped diamond films with precise control over microstructure and dimensions. 6CCVD is uniquely positioned to supply the materials necessary to replicate and extend this cutting-edge research.

Applicable Materials

To achieve the superconducting granular metallic limit ($n \approx 10^{27}$ m^-3) and the required granular structure, 6CCVD recommends:

Heavy Boron-Doped Polycrystalline Diamond (PCD/BDD): Our MPCVD process allows for heavy Boron incorporation, achieving the metallic regime necessary for superconductivity (T_c up to 4 K and higher). The inherent nanocrystalline structure of our PCD films provides the granular architecture (grain size $a \sim 100$ nm) essential for observing the intergrain and intragrain fluctuation effects studied in this paper.
Custom BDD Film Thickness: We provide films ranging from 0.1 µm (329 nm used in the experiment is easily achievable) up to 500 µm, allowing researchers to explore dimensional effects on fluctuation conductivity.

Customization Potential

The complexity of granular transport studies demands materials tailored to specific theoretical parameters (e.g., controlling $E_{Th}$ and $\Gamma$ via grain size and doping).

Research Requirement	6CCVD Custom Capability	Benefit to Researcher
Custom Dimensions	Plates/wafers up to 125 mm (PCD).	Enables large-scale device fabrication and high-throughput testing.
Thickness Control	SCD/PCD films from 0.1 µm to 500 µm.	Precise control over film thickness (e.g., 329 nm replication) for dimensional studies.
Doping Precision	Heavy Boron Doping (BDD) control.	Allows tuning of carrier concentration ($n$) to optimize $T_{c}$ and the granular metallic limit.
Surface Preparation	Polishing to Ra < 5 nm (Inch-size PCD).	Essential for high-quality lithography and reliable contact formation for transport measurements.
Metalization	In-house deposition of Au, Pt, Pd, Ti, W, Cu.	Facilitates immediate integration of electrical contacts required for fluctuation spectroscopy measurements.

Engineering Support

The theoretical analysis confirms that the observed power laws are highly sensitive to material parameters, particularly the phase breaking rate ($\tau_{\phi}^{-1}$) and the ratio of energy scales ($\Gamma/T_{c}$ and $E_{Th}/T_{c}$).

Material Selection Expertise: 6CCVD’s in-house PhD team specializes in the growth and characterization of BDD materials for quantum transport and superconducting applications. We can assist researchers in selecting the optimal MPCVD recipe (doping, growth temperature, pressure) to influence grain size and intergrain coupling, thereby controlling the critical parameters $a$, $\Gamma$, and $E_{Th}$ for similar Granular Superconductor Fluctuation Spectroscopy projects.
Recipe Optimization: We offer consultation to design diamond materials that maximize or minimize specific fluctuation contributions (AL, MT, DOS) by tailoring the granular structure.

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

View Original Abstract

We perform a detailed calculation of the various contributions to the fluctuation conductivity of a granular metal close to its superconducting transition. We find three distinct regions of power law behavior in reduced temperature, η=(T−Tc)/Tc, with crossovers at Γ/Tc and ETh/Tc, where Γ is the electron tunneling rate, and ETh is the Thouless energy of a grain. The calculation includes both intergrain and intragrain degrees of freedom. This complete theory of the fluctuation region in granular superconductors is then compared to experimental results from boron-doped nanocrystalline diamond, using the assumption of a constant phase breaking rate τ−1ϕ. We find a semiquantitative agreement between the theoretical and experimental results only in the case of large phase breaking. We argue that there may be a phase breaking mechanism in granular metals worthy of further experimental and theoretical investigation.