Ayers Model: Dislocation Density Formula from XRD FWHM

The Ayers Model Formula and Required Inputs

Use several XRD rocking-curve widths to separate rotational and strain broadening, estimate threading dislocation density from each, and check whether the two estimates agree. Do not substitute an arbitrary measured FWHM directly into a density formula.

For the Gaussian broadening model in Ayers (1994), the rotational component gives

$$D_\alpha=\frac{\beta_\alpha^2}{2\pi\ln(2)b^2}=\frac{K_\alpha}{2\pi\ln(2)b^2}.$$

Here $\beta_\alpha$ is the separated rotational width, in radians, rather than the entire measured width. The denominator coefficient is $2\pi\ln(2)\approx4.355$.

For the simplified multi-reflection method, subtract the intrinsic specimen width and the monochromator/reference-crystal contribution:

$$\beta_{\mathrm{adj},i}^2=\beta_{m,i}^2-\beta_{0,i}^2-\beta_{d,i}^2 =K_\alpha+K_\varepsilon\tan^2\theta_i.$$

Conditions: approximately Gaussian contributions; appropriate nondispersive optics; negligible finite-size and curvature contributions; and reflection geometry consistent with the chosen dislocation model. Fit at least three reflections at different Bragg angles. The intercept is $K_\alpha$ and the slope is $K_\varepsilon$. More points help assess residuals and uncertainty. If size or curvature matters, include those terms before interpreting this straight-line fit.

The independent strain estimate is obtained by solving

$$K_\varepsilon=c_\varepsilon b^2D_\varepsilon\left|\ln\!\left[(2\times10^{-7}\,\mathrm{cm})\sqrt{D_\varepsilon}\right]\right|.$$

For the original (001) zinc-blende treatment with symmetric reflections, $c_\varepsilon=0.160$ for the assumed 60-degree dislocations and $0.090$ for screw dislocations. These coefficients belong to those particular assumptions; they are not interchangeable constants. Use $b$ in cm and $D_\varepsilon$ in $\mathrm{cm}^{-2}$ in this expression. The logarithm is natural and its argument is dimensionless. The original derivation assumes a Poisson ratio of 1/3, a core cutoff of $10^{-7}\,\mathrm{cm}$ and an outer strain-field radius of half the mean dislocation spacing. Changing those assumptions requires reconsidering the coefficients and logarithmic term. Because density also appears inside the logarithm, the strain equation requires a numerical solution on a physically justified range.

These relationships are from Ayers (1994), equations 7–8, 13–14 and 18–20; the coefficients are also stated in Healey et al. (1995), equations 5–6. Ayers (1994); Healey et al. (1995).

All angular widths in the formulas above are in radians; the example labels its arcsecond inputs explicitly. Use one consistent correction convention. If an instrument response already includes the monochromator contribution, subtracting both would double-count it. Negative corrected squared widths indicate an unresolved contribution or an inconsistent correction; they do not define a negative density.

Why Rocking-Curve Broadening Relates to Dislocation Density

A dislocation distorts the surrounding lattice. Different local orientations satisfy diffraction at different rocking angles, while changes in lattice spacing shift the local Bragg condition. The first produces rotational broadening; the second produces strain broadening.

The angle dependence of strain broadening follows directly by differentiating Bragg's law at fixed wavelength:

$$2d\sin\theta=\lambda,\qquad \Delta\theta\approx-\frac{\Delta d}{d}\tan\theta.$$

Thus a distribution of small strains gives an angular variance proportional to $\tan^2\theta$. This explains why a multi-reflection plot can separate a constant rotational term from an angle-dependent strain term. It does not, by itself, derive the numerical dislocation coefficients; those require the statistical defect assumptions in the cited papers.

Similarly, the typical spacing of a random dislocation network scales as $D^{-1/2}$, and its characteristic misorientation scales as $b\sqrt D$. Squaring therefore gives the dependence $\beta_\alpha^2\propto b^2D$. The Gaussian mosaic model supplies the specific coefficient used above.

Instrument resolution, finite coherent size and specimen curvature also affect peak width. Moreover, wavelength dispersion can produce the same $\tan^2\theta$ dependence as strain. A straight line alone therefore does not prove that dislocations caused every contribution. Ayers (1994) discusses these conditions; Healey et al. (1995) treats the instrument contributions explicitly.

Worked Example: From Measured FWHM to a Density Estimate

Teaching example — simulated inputs, not a measured specimen or a reproduction of the published data. Consider a (001) GaAs layer measured using nondispersive, Cu K-alpha rocking curves. Assume approximately Gaussian profiles, the original 60-degree dislocation model, $b=0.400\,\mathrm{nm}=4.00\times10^{-8}\,\mathrm{cm}$, and independently negligible size and curvature contributions. The Bragg angles are rounded teaching inputs. Take the effective monochromator contribution as 12 arcsec for each reflection in this example only.

The chosen symmetric reflections match the coefficient assumptions above. They provide three distinct angles for the demonstration; an experimental reflection set should also give adequate angle range and measurable intensity.

1. Convert and correct every width

$$1\,\mathrm{arcsec}=\frac{\pi}{180\times3600}\,\mathrm{rad}\approx4.84814\times10^{-6}\,\mathrm{rad}.$$

For (004), for example,

$$\beta_{\mathrm{adj}}^2=(180.889^2-8.7^2-12.0^2)(4.84814\times10^{-6})^2 \approx7.63921\times10^{-7}\,\mathrm{rad}^2.$$

Apply the same calculation to the other two reflections:

2. Fit the multi-reflection relationship

A least-squares fit of $y=K_\alpha+K_\varepsilon x$ gives, within the rounding of these inputs,

$$K_\alpha\approx6.96825\times10^{-7}\,\mathrm{rad}^2,\qquad K_\varepsilon\approx1.59094\times10^{-7}\,\mathrm{rad}^2.$$

3. Calculate the rotational estimate

$$D_\alpha=\frac{6.96825\times10^{-7}}{2\pi\ln(2)(4.00\times10^{-8})^2} \approx1.00\times10^8\,\mathrm{cm}^{-2}.$$

4. Solve the strain equation independently

Use $c_\varepsilon=0.160$, as specified for this example. Solve the strain equation on an illustrative interval $10^6\le D_\varepsilon\le10^9\,\mathrm{cm}^{-2}$ using bisection or another bracketed root solver. This yields

$$D_\varepsilon\approx1.00\times10^8\,\mathrm{cm}^{-2}.$$

Substitution checks the result:

$$0.160(4.00\times10^{-8})^2(1.00\times10^8) \left|\ln(2\times10^{-7}\sqrt{1.00\times10^8})\right| \approx1.59094\times10^{-7}.$$

5. Interpret agreement and uncertainty

The two routes agree because these synthetic widths were constructed from the model. This verifies the calculation, rather than demonstrating experimental accuracy. Do not add $D_\alpha$ and $D_\varepsilon$: they are alternative estimates of the same density under the assumptions, not separate defect populations. Here $1.00\times10^8\,\mathrm{cm}^{-2}=1.00\times10^{12}\,\mathrm{m}^{-2}$.

For an experimental rotational estimate, independent small errors give approximately

$$\left(\frac{\sigma_D}{D}\right)^2\approx \left(\frac{\sigma_{K_\alpha}}{K_\alpha}\right)^2+ 4\left(\frac{\sigma_b}{b}\right)^2.$$

For example, a 10% intercept uncertainty and 1% Burgers-vector uncertainty would give about 10.2% propagated uncertainty. This excludes model error. Obtain slope/intercept uncertainties and covariance from the fit; propagate the strain estimate numerically. No measured uncertainty can be inferred from the simulated table.

Assumptions, Limitations, and Common Errors

For a concrete boundary case, Pelzmann et al. (1996) applied an Ayers-style analysis to GaN and found differences between reflection groups and TEM estimates. It illustrates why adapting the equations requires testing their assumptions. Kaganer et al. (2005) shows how full-profile analysis can include dislocation correlations and non-Gaussian tails. Pelzmann et al. (1996); Kaganer et al. (2005).

Common errors have specific consequences:

• Using degrees or arcseconds in the density formula: the angular conversion is squared, so the error is much larger than a simple unit-label error. Convert before calculating; fitted arcsec$^2$ coefficients need the square of the conversion factor.
• Using $2\theta$ as $\theta$: this changes the horizontal fit coordinate and corrupts the slope.
• Mixing coefficients or Burgers vectors: selecting 0.090 when the declared strain model requires 0.160 changes the inferred density. State the model with the result.
• Calling Scherrer size a dislocation measurement: coherent-domain size is a length. Its reciprocal square has density units, but dimensions alone do not establish a dislocation model. An arbitrary rocking-curve width is also not a substitute for the size contribution in a coupled diffraction profile.

Ayers Model vs. Other XRD Dislocation-Density Methods

Ayers (1994) studied (001) zinc-blende semiconductors, including GaAs on Si(001). Silicon in the substrate name does not make the worked example a silicon-film calibration. For silicon or SiC data, first identify the crystal structure or polytype, orientation, defect population and matching model. The same numerical conversion cannot be assumed from the words “XRD FWHM” alone.

For GaN, Kaganer et al. (2005) analyzed complete diffraction profiles to obtain edge and screw densities and correlation ranges. Its Gaussian peak centers and power-law tails provide information discarded by a single FWHM. Use such a method when the profile or defect structure makes a width-only interpretation inadequate. Cross-check XRD with TEM or etch-pit measurements where suitable, and investigate disagreements instead of choosing the most favorable number.

Frequently Asked Questions

Can I estimate dislocation density from a single FWHM value?

Only conditionally, if a justified material-specific model assigns that width to the relevant defect component. One width cannot separate both $K_\alpha$ and $K_\varepsilon$ or perform the Ayers consistency check. See the formula section and the multi-reflection example.

Which angular units should I use?

Use radians in the width-to-density equations. The strain formula written here uses cm for $b$ and cm$^{-2}$ for density. Convert degree-valued Bragg angles when evaluating trigonometric functions in radian mode.

Is rocking-curve FWHM the same as theta–2theta peak width?

No. An omega scan rocks the specimen through diffraction; a coupled scan varies specimen and detector angles together. Their different scan directions prevent treating the widths as interchangeable inputs.

Can the same formula be used for GaAs, GaN, and SiC?

Do not assume so. Crystal structure, orientation, Burgers vector, defect character and reflection sensitivity must match the model. See the method comparison; a similar-looking coefficient does not establish identical applicability.

Why can XRD estimates differ from TEM or etch-pit measurements?

XRD infers defects through a model over the diffracting volume, whereas imaging or etching counts visible features in selected regions. Sampling, depth variation, defect visibility and unmodeled broadening can cause differences. Check these conditions and the two XRD calculation routes before interpreting disagreement.