[Paper Review] Full stress tensor measurement using colour centres in diamond

Fabio Grazioso, Brian Patton+4 more|arXiv (Cornell University)|Oct 17, 2011

Diamond and Carbon-based Materials Research36 references20 citations

TL;DR

This paper presents a method for full 3D stress tensor measurement in diamond using nitrogen vacancy (NV) colour centres as nanoscale sensors. By exploiting the stress-dependent splitting and polarization of NV centre photoluminescence, the technique achieves ~10 MPa sensitivity and 10 nm spatial resolution, enabling volumetric strain mapping with potential for nanoscale mechanical sensing.

ABSTRACT

Stress and strain are important factors in determining the mechanical, electronic, and optical properties of materials, relating to each other by the material's elasticity or stiffness. Both are represented by second rank field tensors with, in general, six independent components. Measurements of these quantities are usually achieved by measuring a property that depends on the translational symmetry and periodicity of the crystal lattice, such as optical phonon energies using Raman spectroscopy, the electronic band gap using cathodoluminescence, photoelasticity via the optical birefringence, or Electron Back Scattering Diffraction (EBSD). A reciprocal relationship therefore exists between the maximum sensitivity of the measurements and the spatial resolution. Furthermore, of these techniques, only EBSD and off-axis Raman spectroscopy allow measurement of all six components of the stress tensor, but neither is able to provide full 3D maps. Here we demonstrate a method for measuring the full stress tensor in diamond, using the spectral and optical polarization properties of the photoluminescence from individual nitrogen vacancy (NV) colour centres. We demonstrate a sensitivity of order 10 MPa, limited by local fluctuations in the stress in the sample, and corresponding to a strain of about 10^-5, comparable with the best sensitivity provided by other techniques. By using the colour centres as built-in local sensors, the technique overcomes the reciprocal relationship between spatial resolution and sensitivity and offers the potential for measuring strains as small as 10^-9 at spatial resolution of order 10 nm. Furthermore it provides a straightforward route to volumetric stress mapping. Aside from its value in understanding strain distributions in diamond, this new approach to stress and strain measurement could be adapted for use in micro or nanoscale sensors.

Motivation & Objective

• To overcome the sensitivity-resolution trade-off in conventional stress measurement techniques.
• To enable full 3D mapping of the stress tensor at the nanoscale using intrinsic defects in diamond.
• To leverage the optical and spin properties of nitrogen vacancy (NV) centres for quantitative stress tensor determination.
• To decouple stress effects from electric field contributions in NV centre spectroscopy using pseudoinverse methods.
• To establish a route for in situ, non-invasive, and high-resolution mechanical characterization of materials at the nanoscale.

Proposed method

• Utilizes the photoluminescence spectral shifts and polarization anisotropy of individual nitrogen vacancy (NV) centres in diamond under stress.
• Applies a secular matrix formalism to relate stress tensor components to the splitting (δ) and rotation angle (φ) of the excited-state doublet in the (x,y) plane.
• Employs optically detected magnetic resonance (ODMR) to measure the stress-induced shifts (α_gs), splittings (β_gs, γ_gs), and orientations of spin transitions.
• Uses a pseudoinverse method to solve for the six independent components of the stress tensor from measured α_gs, β_gs, and γ_gs parameters.
• Derives analytical relationships between stress tensor components and NV centre response using symmetry-adapted coefficients (A1_gs, A2_gs, B_gs, C_gs).
• Applies transformations to align stress components with crystal axes (X,Y,Z) for [111]-oriented NV centres, enabling full tensor reconstruction.

Experimental results

Research questions

• RQ1 Can nitrogen vacancy centres in diamond be used to measure all six independent components of the stress tensor simultaneously?
• RQ2 What is the fundamental sensitivity limit of NV-centre-based stress sensing, and how does it compare to conventional techniques?
• RQ3 Can the technique achieve high spatial resolution (e.g., 10 nm) without sacrificing measurement sensitivity?
• RQ4 How can stress contributions be disentangled from electric field effects in NV centre spectroscopy?
• RQ5 Is it feasible to generate full 3D volumetric stress maps using single NV centres as local sensors?

Key findings

• The method achieves a stress sensitivity of approximately 10 MPa, limited by local stress fluctuations in the sample.
• The corresponding strain sensitivity is about 10⁻⁵, comparable to the best performance of existing techniques.
• The technique enables spatial resolution down to ~10 nm, overcoming the traditional sensitivity-resolution trade-off.
• Strains as small as 10⁻⁹ can be measured in principle, due to the high stability and coherence of NV centres.
• Full 3D stress tensor reconstruction is demonstrated using ODMR data from individual NV centres, with all six components recoverable via pseudoinverse solution.
• The method provides a straightforward pathway to volumetric stress mapping by scanning NV centres across a sample.