← Back to Business Simulations Epitaxial Metrology & X-Ray Diffraction

HRXRD & Epitaxial Strain Lab

Simulate High-Resolution X-Ray Diffraction (HRXRD), symmetrical (004) rocking curves, Pendellösung thickness fringes (Δω = λ / 2t·cosθB), asymmetrical (224) Reciprocal Space Mapping (RSM), Matthews-Blakeslee critical thickness, and 20-year fab metrology tool DCF economics.

Metrology Presets:
Tetragonal Lattice Constants
a⊥: 5.5120 Å | a∥: 5.4310 Å
In-Plane & Out-of-Plane Strain
ε∥: -1.02% | ε⊥: +0.79%
Bragg Peak Separation (Δω)
-1740" (Δθ = -0.483°)
Pendellösung Fringe Period
Δω = 420" (t = 28.0 nm)
Epilayer Relaxation State
0% (Pseudomorphic)
Dislocation Density & Tilt
1.2e5 cm⁻² (FWHM: 25")
Wafer Inspection Throughput
30 WPH (Inline SPC)
20-Year Tool DCF NPV
$19.5M

Epitaxial Heterostructure Parameters

Alloy Mole Fraction (x / y) 25%
Epilayer Physical Thickness (t) 28.0 nm
Degree of Strain Relaxation (R) 0%
Mosaic Tilt FWHM Broadening (βtilt) 25"

Fab In-Line Metrology Tool DCF

Tool Capital Expenditure (CapEx) $3.4M
Wafer Throughput (WPH) 30 WPH
Fab Yield Excursion Scrap Reduction 1.4%
Discount Rate (WACC) 9.0%

Diffraction & Reciprocal Space Visualizer

Internal Rate of Return
32.6%
Discounted Payback
2.5 Years
Return on Investment
415%

High-Resolution X-Ray Diffraction Engineering Curriculum

1. Dynamical Bragg Diffraction & Peak Splitting

High-Resolution X-Ray Diffraction (HRXRD) uses monochromatic Cu Kα1 radiation (λ = 1.54056 Å) to probe crystal lattice spacings. For symmetrical (004) reflection, Bragg's Law relates diffraction angle θB to interplanar spacing d004 = a⊥ / 4:

λ = 2 · d004 · sin(θB)

The angular separation Δω between substrate and epitaxial peaks directly tracks out-of-plane lattice parameter distortion:

Δω ≈ -tan(θB,sub) · (a⊥ - asub) / asub

When finite epilayers reflect wavefields, dynamical interference yields Pendellösung fringes with period Δωfringe = λ / (2t·cosθB), providing a direct measurement of thickness t.

2. Tetragonal Distortion & Poisson Ratio

When an epilayer grows pseudomorphically on a lattice-mismatched substrate, in-plane atoms are constrained to the substrate lattice constant (a∥ = asub). Elastic energy causes Poisson expansion or contraction in the perpendicular axis:

ε⊥ = -2 · [ν / (1 - ν)] · ε∥

where ν is Poisson's ratio (0.28 for Si/SiGe). Compressive in-plane strain (ε∥ < 0) elongates the vertical lattice (a⊥ > arelax), shifting the Bragg peak to lower diffraction angles.

The degree of relaxation R = (a∥ - asub) / (arelax - asub) tracks the transition from pseudomorphic strain (R = 0%) to complete plastic relaxation (R = 100%).

3. Reciprocal Space Mapping (RSM) Coordinates

A 1D rocking curve cannot distinguish between alloy composition change and strain relaxation. Asymmetrical (224) Reciprocal Space Mapping maps 2D scattering vectors in reciprocal space:

Qx = 2π · 2√2 / a∥,   Qz = 2π · 4 / a⊥

Pseudomorphic epilayers share the substrate's in-plane lattice and align on the vertical line Qx,epi = Qx,sub. As misfit dislocations nucleate, the peak relaxes along the diagonal trajectory toward the fully relaxed line, isolating composition from strain.

4. 20-Year Cleanroom HRXRD Metrology Tool Economics

Advanced logic foundries (3nm/2nm Gate-All-Around nanosheets) and compound semiconductor fabs rely on in-line automated HRXRD tools:

  • GAA Nanosheet Cavity Control: Ensures exact Ge concentration and zero misfit relaxation across Si/SiGe superlattice stacks to prevent nanosheet collapse.
  • Power GaN-on-Si / SiC Qualification: Measures wafer bow, epilayer tilt, and threading dislocation density across 200mm/300mm power wafers.
  • Rapid Scrap Avoidance: Early detection of epitaxial growth temperature excursions or precursor depletion saves over $2.0M annually in catastrophic wafer-lot scrap.

20-Year In-Line HRXRD Metrology Tool DCF Cash Flow Schedule

Year Wafer Volume Gross Scrap Savings Tool Maintenance & OpEx Net Free Cash Flow Discount Factor Discounted FCF Cumulative NPV

Formative Knowledge Assessment: HRXRD & Reciprocal Space Mapping

Question 1

Why does an in-plane compressively strained epilayer (like SiGe on Si) shift its symmetrical (004) Bragg peak to a lower angle?

Question 2

In an asymmetrical Reciprocal Space Map (such as 224), how is a fully pseudomorphic epilayer recognized?

Question 3

What causes Pendellösung interference fringes in high-resolution rocking curves?

Frequently Asked Questions: HRXRD & Epitaxial Strain

High-Resolution X-Ray Diffraction utilizes monochromatic, highly collimated X-rays (such as Cu K-alpha 1 conditioned through a 4-bounce Ge 220 monochromator) to probe atomic lattice plane spacings with sub-picometer accuracy. When constructive Bragg interference occurs (lambda = 2*d*sin(theta)), sharp diffraction peaks reveal epilayer composition, tetragonal strain, layer thickness via Pendellosung fringes, and crystalline perfection.

In a 1D rocking curve, both alloy composition and strain alter out-of-plane lattice parameter d(004), creating parameter ambiguity. A 2D Reciprocal Space Map of an asymmetrical reflection (such as 224) decouples in-plane (Qx) and out-of-plane (Qz) reciprocal lattice coordinates. Fully pseudomorphic films align vertically with the substrate at identical Qx, while plastic relaxation shifts the peak toward the diagonal fully relaxed vector, isolating composition from strain.

Pendellosung fringes are fine, periodic intensity oscillations flanking an epitaxial Bragg peak. They originate from dynamical optical interference between X-ray wavefields reflecting from the top epitaxial surface and the buried heterojunction interface. The angular fringe spacing Delta-omega is inversely proportional to layer thickness (Delta_omega = lambda / (2*t*cos(theta_B))), yielding non-destructive thickness measurement down to single-nanometer precision.

The Matthews-Blakeslee critical thickness (tc) defines the thermodynamic equilibrium boundary where the strain energy stored in an elastic pseudomorphic epilayer exceeds the energy required to nucleate a misfit dislocation line. Below tc, the epilayer remains elastically strained without defects. Above tc, plastic relaxation occurs via misfit and threading dislocations, which degrade carrier mobility and increase gate leakage.

An automated in-line 300mm wafer HRXRD tool represents an initial CapEx of 2.8 to 5.5 million dollars. Foundries deploy HRXRD for high-speed statistical process control of strained SiGe pMOS channels, GAA nanosheet stacks, and GaN power HEMTs. Detecting epitaxial composition drifts or relaxation excursions before full-flow processing prevents catastrophic wafer-lot scrap, delivering rapid financial payback within 2 to 3 years.

Related Semiconductor & Metrology Simulations

X-Ray Reflectivity (XRR) Lab

Model X-ray total external reflection, Kiessig fringes, film density, interface roughness, and fab DCF.

Spectroscopic Ellipsometry Lab

Model spectroscopic ellipsometry, psi and delta angles, Cauchy dispersion, Mueller matrix polarimetry, and fab DCF.

X-Ray Photoelectron Spectroscopy Lab

Model XPS core binding energies, ARXPS non-destructive depth profiling, inelastic mean free path, and fab metrology DCF.

TXRF Wafer Metrology Lab

Model Total Reflection X-Ray Fluorescence (TXRF), trace metal surface contamination, standing waves, and fab DCF.

Helium Ion Microscopy & GFIS Lab

Model Gas Field Ion Source physics, sub-0.5nm probe sizes, secondary electron yield, and EUV mask repair.

All Business Simulations & Worksheets →

Browse our comprehensive directory of over 890 engineering, physics, financial, and manufacturing simulation labs.