1. Theory

Definition: An X-ray diffractometer measures intensity scattered by crystal planes as a function of angle; high-resolution XRD (HRXRD) uses a conditioned, near-monochromatic beam to resolve lattice-spacing differences of order $1 \times 1 0^{- 4}$ in epitaxial layers (Wormington; NCU).
Bragg condition: Constructive interference occurs at $2 d sin θ_{B} = nλ$ , equivalently when the scattering vector equals a reciprocal lattice vector (Wormington).
Strain and composition: A strained epitaxial layer has a tetragonally distorted cell, so its peak shifts from the substrate; the shift gives out-of-plane strain and, with elastic constants and a lattice-constant law, composition (Wormington; background knowledge, Bowen and Tanner).
Thickness and defects: Interference fringes give layer thickness, and peak broadening gives dislocation density and mosaicity (Wormington; Ryan et al.).

2. Working Principle

Source and optics: A Cu tube with a multi-crystal monochromator selects Cu K-alpha and limits divergence (NCU; background knowledge).
Scans: Rocking curves and omega-2theta scans around a symmetric reflection such as Si 004 give the out-of-plane spacing (NCU; Wormington).
Reciprocal space maps: Combined scans around an asymmetric reflection separate in-plane and out-of-plane lattice constants, giving relaxation (Wormington; NCU).
Modelling: Dynamical-diffraction simulation is fitted to the measured curve to extract thickness, composition and strain (Wormington; background knowledge).

3. Applications

SiGe, Ge and Si:C epitaxy: Composition, strain and thickness of source-drain and channel layers (Wormington; Ryan et al.).
Strain-relaxed buffers: Relaxation and threading dislocations in Ge and SiGe buffers (Ryan et al.; Zhou et al.).
Patterned fins: Pitch, sidewall angle and pitch walk from grating rods in FinFET structures (Wormington; Ryan et al.).
In-line monitoring: Small-spot 300 $mm$ tools for production wafers (Ryan et al.).

Diffraction Geometry

Name	Formula	Related Formulae	Variables/Constants
Bragg’s law	$2 d sin θ_{B} = nλ$	$d_{hk l} = \frac{a}{h ^{2} + k ^{2} + l ^{2}}, Q = \frac{4 π s i n θ}{λ} = \frac{2 π}{d}$	$d : l a tt i ce - pl an e s p a c in g$ $θ_{B} : B r a g g an g l e$ $n : or d er$ $λ : X - r a y w a v e l e n g t h$ $a : c u bi c l a tt i ce co n s t an t, hk l : M i l l er in d i ces$ $Q : sc a tt er in g v ec t or; W or min g t o n : B r a g g l a w, Q = 2 s in θ \cdot 2 π / λ, L a u e co n d i t i o n Q = G e q u i v a l e n t; N C U : C u w a v e l e n g t h 1.54 \overset{˚}{A}, tw o - cr y s t a l m o n oc h r o ma t or; C u K α_{1} = 1.5406 \overset{˚}{A}, a_{S} i = 5.431 \overset{˚}{A} (ba c k g r o u n d k n o w l e d g e, B o w e n an d T ann er)$ $(co m p u t e d : S i 004 g i v es θ_{B} = 34.56^{\circ}, 2 θ = 69.13^{\circ} an d Q = 4.63 \overset{˚}{A}^{- 1})$
Instrumental resolution	$\frac{Δ d}{d} = \frac{Δ λ}{λ} + \frac{Δ θ}{t a n θ _{B}}$	$4-bounce monochromator: Δ θ \sim 10 arcsec$	$Δ d / d : f r a c t i o na l s p a c in g r eso l u t i o n, Δ λ / λ : w a v e l e n g t h s p r e a d$ $Δ θ : an g u l a r d i v er g e n ce; W or min g t o n : r eso l u t i o n r e l a t i o n; m o n oc h r o ma t or f i g u r es (ba c k g r o u n d k n o w l e d g e, B o w e n an d T ann er)$ $(a ss u m e d : Δ λ / λ = 1.4 \times 1 0^{- 4} an d 12 arcsec; co m p u t e d : Δ d / d = 2.2 \times 1 0^{- 4} a t S i 004)$
Peak splitting and out-of-plane strain	$\frac{Δ d}{d} = - Δ ω cot θ_{B}$	$ε_{⊥} = \frac{a _{⊥} - a _{s}}{a _{s}}$	$Δ ω : l a y er - s u b s t r a t e p e ak se p a r a t i o n, ε_{⊥} : o u t - o f - pl an e mi s f i t, a_{⊥} : l a y er l a tt i ce co n s t an t n or ma l t o t h e s u r f a ce$ $a_{s} : s u b s t r a t e l a tt i ce co n s t an t; W or min g t o n : r e l a t i o n, 22.5 nm S i_{0.51} G e_{0.49} 004 e x am pl e; N C U : s t r ain f r o m Δ θ a r o u n d (004) ω - 2 θ sc an s$ $(co m p u t e d : S i_{0.81} G e_{0.19} p se u d o m or p hi c o n S i g i v es ε_{⊥} = 1.27 \times 1 0^{- 2} an d Δ ω = - 1806 arcsec w i t h D i s m u k es, - 1998 arcsec w i t h V e g a r d)$

Strain, Composition and Relaxation

Name	Formula	Related Formulae	Variables/Constants
Tetragonal distortion	$a_{⊥} = a_{s} + (a_{r} - a_{s}) (1 + \frac{2 C _{12}}{C _{11}})$	$a_{∥} = a_{s} (pseudomorphic)$	$a_{r} : r e l a x e d (c u bi c) l a y er l a tt i ce co n s t an t$ $C_{11}, C_{12} : e l a s t i c co n s t an t s, a_{∥} : in - pl an e l a tt i ce co n s t an t; W or min g t o n : a_{S i G e, ∥} = a_{S i} f or f u l l y s t r ain e d an d l a r g er f or r e l a x e d l a y er s; bia x ia l - s t r ain r e l a t i o n, S i C_{11} = 165.8 an d C_{12} = 63.9 GPa (ba c k g r o u n d k n o w l e d g e, B o w e n an d T ann er)$ $(co m p u t e d : 2 C_{12} / C_{11} = 0.77, so a_{⊥} s hi f t s 1.77 t im es t h e c u bi c mi s ma t c h)$
Composition of SiGe	$a_{r} = a_{S i} + 0.2 x + 0.027 x^{2} \overset{˚}{A} (Dismukes)$	$a_{r} = a_{S i} (1 - x) + a_{G e} x (Vegard)$	$x : G e f r a c t i o n, a_{S i} = 5.431 \overset{˚}{A}, a_{G e} = 5.658 \overset{˚}{A}; D i s m u k es an d V e g a r d f or m s (ba c k g r o u n d k n o w l e d g e, D i s m u k es e t a l .); W or min g t o n : x = 0.19, 0.49 an d 0.67 e x am pl es, N I S T S R M 2000 o f 23.7 nm S i o n 48.2 nm S i_{0.84} G e_{0.16}, S i_{1 - x} C_{x} w i t h x = 1.4 p er ce n t, G e S n; R y an e t a l . : co m p os i t i o n f r o m S R B p e ak s an d f in L - sc an s, S R B ab o u t 70 p er ce n t G e (see t h e Q / A)$ $(co m p u t e d : x = 0.19 g i v es 5.4700 \overset{˚}{A} (D i s m u k es) an d 5.4741 \overset{˚}{A} (V e g a r d))$
Degree of relaxation	$R = \frac{a _{∥} - a _{s}}{a _{r} - a _{s}}$	$\frac{Δ a _{⊥}}{a} = \frac{Δ L}{L}, \frac{Δ a _{∥}}{a} = \frac{Δ H}{H}$	$R : r e l a x a t i o n (0 p se u d o m or p hi c, 1 f u l l y r e l a x e d)$ $H, L : r ec i p r oc a l - s p a ce coor d ina t es f r o m an a sy mm e t r i c R S M; W or min g t o n : R S M r e l a t i o n s, S i_{0.33} G e_{0.67} w i t h R = 70 p er ce n t, r e l a x e d l a y er s ab o u t 50 p er ce n t l ess s t r ain e d; Z h o u e t a l . : G e o n S i R o f 103.9 t o 105 p er ce n t an d 105.9 p er ce n t (ab o u t 0.16 p er ce n t t e n s i l e), G e C T E 5.8 t o 8.1 an d S i 2.6 t o 4.1 ppm/K; R y an e t a l . : 100 p er ce n t r e l a x e d G e (see t h e Q / A)$ $(co m p u t e d : x = 0.67 a t R = 70 p er ce n t l e a v es - 0.79 p er ce n t in - pl an e s t r ain a g ain s t - 2.62 p er ce n t f u l l y s t r ain e d)$

Thickness, Defects and Patterns

Name	Formula	Related Formulae	Variables/Constants
Layer thickness from fringes	$t = \frac{λ}{2Δ ω _{f} c o s θ _{B}}$	$t \propto \frac{1}{Δ L _{f r in g es}}, Λ = \frac{λ}{2Δ ω _{S L} c o s θ _{B}}$	$t : l a y er t hi c k n ess$ $Δ ω_{f} : P e n d e l l oes u n g f r in g e p er i o d$ $Λ : s u p er l a tt i ce p er i o d$ $Δ ω_{S L} : s a t e l l i t e s p a c in g; W or min g t o n : b o t h r e l a t i o n s, 37.6 nm S i_{0.81} G e_{0.19}, [8.2 nm S i_{0.9} G e_{0.1} / 22.4 nm S i] b y 5 s u p er l a tt i ce; R y an e t a l . : ab o u t 30 nm G e f r o m ω - 2 θ f r in g es, S i G e f in t hi c k n ess f r o m L - sc an f r in g es$ $(co m p u t e d : 37.6 nm g i v es 513 arcsec f r in g es; t h e 30.6 nm s u p er l a tt i ce p er i o d g i v es 631 arcsec s a t e l l i t es)$
Dislocation density from peak width	$ρ \approx \frac{β ^{2}}{9 b ^{2}}$	$\frac{w ( H )}{H} \propto \frac{1}{ρ} (as summarized from Wormington)$	$ρ : t h r e a d in g d i s l oc a t i o n d e n s i t y$ $β : r oc k in g - c u r v e b r o a d e nin g in r a d ian s$ $b : B u r g er s v ec t or, w (H) : l a t er a l w i d t h; A y er s r e l a t i o n (ba c k g r o u n d k n o w l e d g e, A y er s); W or min g t o n : w i d t h r e l a t i o n, ab o u t 1 \times 1 0^{10} c m^{- 2} in I I I - V t r e n c h es; R y an e t a l . : G e o n S R B d e t ec t i o n l imi t ab o u t 1 \times 1 0^{5} c m^{- 2}, f i t w i d t h s g i v in g d e f ec t d e n s i t y (see t h e Q / A)$ $(a ss u m e d : b = 4.0 \overset{˚}{A} f or G e, β = 100 arcsec; co m p u t e d : ρ = 1.6 \times 1 0^{7} c m^{- 2})$
Patterned structures and in-line use	$P = \frac{2 π}{Δ Q _{x}}$	$grating rods: pitch, sidewall angle, pitch walk$	$P : g r a t in g p i t c h$ $Δ Q_{x} : g r a t in g - r o d s p a c in g; W or min g t o n : P = 42.2 + / - 0.5 nm, s i d e w a l l 9 + / - 1^{\circ}, p i t c h w a l k 5 + / - 0.5 nm, S i G e f in s x = 25 p er ce n t an d t = 39.4 nm; R y an e t a l . : 176 nm p i t c h f r o m a 113 R S M, J V X 7300 L S I f or 300 mm w a f er s w i t h b e am s b e l o w 0.2 mm an d 50 μ m, R S M s in min u t es r a t h er t han h o u r s$ $(co m p u t e d : a 42.2 nm p i t c h g i v es r o d s s p a ce d b y 1.49 \times 1 0^{- 2} \overset{˚}{A}^{- 1})$

Important Questions

Ques: How does HRXRD measure strain in an epitaxial layer?

From the angular separation between layer and substrate peaks. Wormington gives $Δ d / d = - Δ ω cot θ_{B}$ , and the NCU page describes strain from the angle change around the (004) omega-2theta scan. For Si 004 with Cu K-alpha-1, $θ_{B}$ = 34.56 $^{\circ}$ (computed). A pseudomorphic $S i_{0.81} G e_{0.19}$ layer keeps the silicon in-plane lattice constant, so its cell stretches vertically by $1 + 2 C_{12} / C_{11}$ = 1.77 times the cubic mismatch (background knowledge, Bowen and Tanner; computed). This gives $ε_{⊥}$ = $1.27 \times 1 0^{- 2}$ and a layer peak 1806 $arcsec$ below the substrate (computed). Resolution near $2.2 \times 1 0^{- 4}$ in $Δ d / d$ for an assumed 12 $arcsec$ beam makes this shift easy to measure (Wormington relation; computed). Reference: M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary); Condensed Matter Laboratory, National Central University, ‘HRXRD’ instrument page (fetch-tool summary); D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998; J. P. Dismukes, L. Ekstrom and R. J. Paff, J. Phys. Chem. 68, 3021 (1964); J. E. Ayers, J. Cryst. Growth 135, 71 (1994) (background knowledge; not opened).

Ques: Should SiGe composition use Vegard’s law or the Dismukes relation?

Vegard’s linear interpolation between silicon at 5.431 $\overset{˚}{A}$ and germanium at 5.658 $\overset{˚}{A}$ and the Dismukes relation $a = 5.431 + 0.2 x + 0.027 x^{2}$ $\overset{˚}{A}$ give different lattice constants (background knowledge, Dismukes et al.). For x = 0.19 they give 5.4741 and 5.4700 $\overset{˚}{A}$ , and the pseudomorphic 004 peak shifts of -1998 and -1806 $arcsec$ (computed). Reading a -1806 $arcsec$ peak with Vegard’s law would give x of about 0.17 instead of 0.19 (computed). The Dismukes relation is more likely correct because it is fitted to measured SiGe lattice constants, which deviate slightly from linearity (background knowledge). Wormington’s SiGe examples, including NIST SRM2000, are intended for this kind of calibration. Reference: D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998; J. P. Dismukes, L. Ekstrom and R. J. Paff, J. Phys. Chem. 68, 3021 (1964); J. E. Ayers, J. Cryst. Growth 135, 71 (1994) (background knowledge; not opened); M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary).

Ques: Does 70 per cent relaxation mean 70 per cent or 50 per cent less strain?

Wormington shows $S i_{0.33} G e_{0.67}$ with R = 70 per cent and also states that relaxed layers carry about 50 per cent less strain than pseudomorphic ones. By definition $R = (a_{∥} - a_{s}) / (a_{r} - a_{s})$ , so 70 per cent relaxation leaves 30 per cent of the misfit strain: -0.79 per cent in plane against -2.62 per cent fully strained (Dismukes constants; computed). The 70 per cent figure is therefore more likely the quantitative result for that sample, and the 50 per cent statement a general comparison (my reading). Relaxation is measured from an asymmetric reciprocal space map, using $Δ a_{∥} / a = Δ H / H$ and $Δ a_{⊥} / a = Δ L / L$ (Wormington). Ryan et al. report 100 per cent relaxed Ge from omega-2theta peak positions. Reference: M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary); P. Ryan, M. Wormington, J. Sun, A. Hikavyy, Y. Shimura, L. Witters, H. Tielens, A. Schulze and R. Loo, ‘High Resolution X-ray Diffraction for In-line Monitoring of Ge MOSFET Devices’, Jordan Valley Semiconductors, hosted by NIST (fetch-tool summary).

Ques: How can a Ge layer on Si show relaxation above 100 per cent?

Because it is slightly tensile after cooling. Zhou et al. report relaxation of 103.9 to 105 per cent, about 0.16 per cent tensile strain, in Ge grown on Si. Ge relaxes fully by misfit dislocations at growth temperature, but its thermal expansion of 5.8 to 8.1 $ppm/K$ exceeds silicon’s 2.6 to 4.1 $ppm/K$ (Zhou et al.), so on cooling the Si substrate prevents the Ge from contracting. Over 630 $K$ the mismatch gives 0.20 to 0.25 per cent (computed), somewhat above the measured 0.16 per cent, consistent with some relaxation during cooling (my reading). Interdiffusion also matters: Zhou et al. treated one sample as $G e_{0.98} S i_{0.02}$ , giving 105.9 per cent, so the assumed composition changes R. Reference: G. Zhou, K. H. Lee, D. H. Anjum, Q. Zhang, X. Zhang, C. S. Tan and G. Xia, ‘Impacts of Doping on Epitaxial Germanium Thin Film Quality and Si-Ge Interdiffusion’, arXiv:1712.05468 (fetch-tool summary).

Ques: How is layer thickness measured from diffraction fringes?

From the spacing of Pendelloesung fringes. Wormington gives $t = λ / (2Δ ω_{f} cos θ_{B})$ , so thinner layers give wider fringes: 37.6 $nm$ gives 513 $arcsec$ and 22.5 $nm$ gives 858 $arcsec$ at Si 004 (computed). In a reciprocal space map the same idea is $t \propto 1/Δ L$ (Wormington). For his [8.2 $nm$ $S i_{0.9} G e_{0.1}$ / 22.4 $nm$ Si] by 5 superlattice, the 30.6 $nm$ period gives satellites about 631 $arcsec$ apart (computed). Ryan et al. measured about 30 $nm$ Ge from omega-2theta fringes and fin thickness from L-scan fringes. Fringes require smooth, defect-free interfaces; their loss signals roughness or relaxation (Ryan et al.; background knowledge). Fitting the full curve by simulation is more precise than reading a single fringe period. Reference: M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary); P. Ryan, M. Wormington, J. Sun, A. Hikavyy, Y. Shimura, L. Witters, H. Tielens, A. Schulze and R. Loo, ‘High Resolution X-ray Diffraction for In-line Monitoring of Ge MOSFET Devices’, Jordan Valley Semiconductors, hosted by NIST (fetch-tool summary); D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998; J. P. Dismukes, L. Ekstrom and R. J. Paff, J. Phys. Chem. 68, 3021 (1964); J. E. Ayers, J. Cryst. Growth 135, 71 (1994) (background knowledge; not opened).

Ques: Does peak width rise or fall with dislocation density?

It rises. The Ayers relation, $ρ \approx β^{2} / (9 b^{2})$ , links rocking-curve broadening to threading dislocation density; for Ge with b = 4.0 $\overset{˚}{A}$ , a broadening of 100 $arcsec$ gives about $1.6 \times 1 0^{7}$ $c m^{- 2}$ (background knowledge, Ayers; computed). The summary of Wormington’s slides gives $w (H) / H \propto 1/ ρ$ , which appears to say the opposite; that quantity is more likely a lateral correlation length, which shortens as dislocations multiply, than an angular width (my reading). The Ayers form is therefore more likely for peak broadening. Wormington reports about $1 \times 1 0^{10}$ $c m^{- 2}$ in III-V trenches, while Ryan et al. put the HRXRD limit for Ge on SRB near $1 \times 1 0^{5}$ $c m^{- 2}$ , below which broadening is not resolved. Reference: D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis 1998; J. P. Dismukes, L. Ekstrom and R. J. Paff, J. Phys. Chem. 68, 3021 (1964); J. E. Ayers, J. Cryst. Growth 135, 71 (1994) (background knowledge; not opened); M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary); P. Ryan, M. Wormington, J. Sun, A. Hikavyy, Y. Shimura, L. Witters, H. Tielens, A. Schulze and R. Loo, ‘High Resolution X-ray Diffraction for In-line Monitoring of Ge MOSFET Devices’, Jordan Valley Semiconductors, hosted by NIST (fetch-tool summary).

Ques: What can HRXRD measure on patterned fins?

Pitch, sidewall angle, pitch walk, composition and thickness. A periodic array produces grating rods in reciprocal space spaced by $2 π / P$ ; Wormington reports P = 42.2 +/- 0.5 $nm$ , a sidewall angle of 9 +/- 1 $^{\circ}$ and pitch walk of 5 +/- 0.5 $nm$ , plus SiGe fins with x = 25 per cent and t = 39.4 $nm$ . A 42.2 $nm$ pitch gives rods $1.49 \times 1 0^{- 2}$ $\overset{˚}{A}^{- 1}$ apart (computed). Ryan et al. obtained fin composition from the peak along the zeroth-order rod and thickness from its fringes, and a 176 $nm$ pitch from a 113 map. Their 300 $mm$ tool uses beams below 0.2 $mm$ or 50 $μ m$ to fit test pads, giving maps in minutes rather than hours. Reference: M. Wormington, ‘High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures’, Frontiers of Characterization and Metrology for Nanoelectronics (2015), slides hosted by NIST (fetch-tool summary); P. Ryan, M. Wormington, J. Sun, A. Hikavyy, Y. Shimura, L. Witters, H. Tielens, A. Schulze and R. Loo, ‘High Resolution X-ray Diffraction for In-line Monitoring of Ge MOSFET Devices’, Jordan Valley Semiconductors, hosted by NIST (fetch-tool summary).

References

M. Wormington, “High-Resolution X-ray Diffraction of Epitaxial Thin-Films and Patterned Nanostructures,” presented at Frontiers of Characterization and Metrology for Nanoelectronics, 2015. Available: https://www.nist.gov/document/wormingtonpdf
Condensed Matter Laboratory, National Central University, “HRXRD.” Available: https://condensedlab.phy.ncu.edu.tw/HRXRD.html
D. K. Bowen and B. K. Tanner, High Resolution X-ray Diffractometry and Topography, Taylor and Francis, 1998; J. P. Dismukes, L. Ekstrom and R. J. Paff, “Lattice parameter and density in germanium-silicon alloys,” Journal of Physical Chemistry, vol. 68, p. 3021, 1964; J. E. Ayers, “The measurement of threading dislocation densities in semiconductor crystals by X-ray diffraction,” Journal of Crystal Growth, vol. 135, p. 71, 1994 (cited from background knowledge).
P. Ryan, M. Wormington, J. Sun, A. Hikavyy, Y. Shimura, L. Witters, H. Tielens, A. Schulze and R. Loo, “High Resolution X-ray Diffraction for In-line Monitoring of Ge MOSFET Devices,” Jordan Valley Semiconductors. Available: https://www.nist.gov/document/38ryanpdf
G. Zhou, K. H. Lee, D. H. Anjum, Q. Zhang, X. Zhang, C. S. Tan and G. Xia, “Impacts of Doping on Epitaxial Germanium Thin Film Quality and Si-Ge Interdiffusion,” preprint arXiv:1712.05468. Available: https://arxiv.org/pdf/1712.05468

Quartz 5

Explorer

X-Ray Diffractometer (XRD and HRXRD)