X-Ray Reflectometer (XRR)

1. Theory

• Definition: An X-ray reflectometer measures the specular reflectivity of a flat sample at grazing angles, typically 0 to a few degrees, and extracts thickness, density and roughness of thin films and multilayers (Colorado State deck; HZDR; Schreiber and Gerlach).
• Refractive index: For X-rays $n=1-\delta -i\beta$ with $\delta$ near $1\times 10^{-5}$, so total external reflection occurs below a critical angle ${\alpha }_c=\sqrt{2\delta }$ that depends on electron density (Schreiber and Gerlach; HZDR).
• Interference: Waves reflected from the film surface and interfaces interfere, producing Kiessig fringes whose period gives thickness (Colorado State deck; Schreiber and Gerlach).
• Roughness: Interface roughness damps reflectivity and fringes through a Debye-Waller or Nevot-Croce factor (HZDR; Schreiber and Gerlach).

2. Working Principle

• Beam: A Cu tube with a parabolic multilayer (Goebel) mirror gives a parallel, monochromatic beam (HZDR).
• Geometry: Theta-2theta specular scans from below the critical angle upward, over five to six decades of intensity (Colorado State deck; HZDR).
• Alignment: The sample height and tilt are set at an angle below ${\alpha }_c$, such as 0.2 ${}^{\circ }$ (Colorado State deck).
• Analysis: The Parratt recursion with roughness factors is fitted to the measured curve (Schreiber and Gerlach; background knowledge, Parratt).

3. Applications

• Gate and capacitor dielectrics: Thickness, density and roughness of SiO2, high-k oxides and nitrides (background knowledge; Colorado State deck).
• Metal and barrier films: Ta, Ti and TaN stacks and their oxidation (HZDR; Colorado State deck).
• Multilayers: Superlattices and mirrors (HZDR; Schreiber and Gerlach).
• Organic and soft films: Neutron and X-ray reflectivity of molecular layers (Schreiber and Gerlach).

Refraction and Critical Angle

Name	Formula	Related Formulae	Variables/Constants
X-ray refractive index	$n=1-\delta -i\beta$	$\delta =\frac{{\lambda }^2{r}_e{\rho }_e}{2\pi },\beta =\frac{\mu \lambda }{4\pi }$	$n:complexrefractiveindex$ $\delta :dispersiondecrement$ $\beta :absorptionindex$ $\lambda :wavelength$ $r_e:classicalelectronradius2.818\times 10^{-15}\mathrm{m}$ ${\rho }_e:electrondensity$ $\begin{array}{r}\mu :linearabsorptioncoefficient;Colorado\\ StatedeckandSchreiberandGerlach:\\ theserelations;HZDR:\delta about1\times 10^{-5}and\beta \\ about1\times 10^{-6}\end{array}$ $\begin{array}{r}(computed:Siat2.33g/c{\mathrm{m}}^{3}has{\rho }_e=\\ 6.99\times 10^{29}{\mathrm{m}}^{-3}and\delta =7.4\times 10^{-6}at1.5406\mathring{\mathrm{A}})\end{array}$
Critical angle	${\alpha }_c=\sqrt{2\delta }$	$Q_c=\frac{4\pi \sin {\alpha }_c}{\lambda },{\alpha }_c\text{at}R=R_{max}/2$	$\begin{array}{r}{\alpha }_c:criticalanglefortotalexternal\\ reflection\end{array}$ $\begin{array}{r}{Q}_c:criticalmomentumtransfer;Schreiber\\ andGerlach:{\alpha }_c=\sqrt{2\delta },typical0.1to0.5\\ {}^{\circ };HZDR:0.2to1{}^{\circ }forabout0.1\mathrm{nm};\\ ColoradoStatedeck:{\alpha }_{c}readathalf\\ maximumreflectivity,alignmentat0.2{}^{\circ }\\ below{\alpha }_c(seetheQ/A)\end{array}$ $\begin{array}{r}(computedwithCuK\alpha :Si0.221{}^{\circ },SiO2at\\ 2.20g/c{\mathrm{m}}^{3}0.215{}^{\circ },HfO2at9.68g/c{\mathrm{m}}^3\\ 0.413{}^{\circ },Ta0.532{}^{\circ },Ir0.62{}^{\circ })\end{array}$
Momentum transfer	$Q_z=\frac{4\pi \sin {\alpha }_i}{\lambda }$	$k_{z,j}=k_0\sqrt{n_j^2-{\cos }^2{\alpha }_i}$	$Q_z:momentumtransfernormaltothesurface$ ${\alpha }_i:grazingincidenceangle$ $\begin{array}{r}{k}_{z,j}:normalwavevectorinlayerj,k_0=\\ 2\pi /\lambda ;SchreiberandGerlach:Q_{z}relation;\\ ColoradoStatedeck:Snell-Descartes\\ \cos {\alpha }_I/\cos {\alpha }_T=n_0/n;normal-wavevectorform\\ (backgroundknowledge,Als-Nielsenand\\ McMorrow)\end{array}$ $(computed:Sigives{Q}_c=0.0315{\mathring{\mathrm{A}}}^{-1})$

Reflectivity and Fringes

Name	Formula	Related Formulae	Variables/Constants
Fresnel reflectivity	$r_{j,j+1}=\frac{k_{z,j}-k_{z,j+1}}{k_{z,j}+k_{z,j+1}}$	$R_F\approx {\left(\frac{{\alpha }_c}{2{\alpha }_i}\right)}^4({\alpha }_i\gg {\alpha }_c)$	$r_{j,j+1}:Fresnelamplitudecoefficient$ $\begin{array}{r}{R}_F:reflectivityofanidealsurface;\\ SchreiberandGerlach:Fresnelcoefficient\\ and{R}_{F}proportionalto{\alpha }_i^{-4};ColoradoState\\ deck:{\alpha }_I^{-4}law,dataover5to6ordersof\\ magnitude;asymptoticform(background\\ knowledge,Als-NielsenandMcMorrow)\end{array}$ $\begin{array}{r}(computed:for{\alpha }_c=0.22{}^{\circ },R_{F}is1.5\times 10^{-4}at\\ 1{}^{\circ }and9.2\times 10^{-6}at2{}^{\circ })\end{array}$
Kiessig fringes and thickness	${\sin }^2{\alpha }_m-{\sin }^2{\alpha }_c={\left(\frac{m\lambda }{2t}\right)}^2$	$\Delta \alpha \approx \frac{\lambda }{2t},\Delta Q_z=\frac{2\pi }{t}$	${\alpha }_m:angleoffringemaximumm$ $m:fringeorder$ $t:filmthickness$ $\begin{array}{r}\Delta \alpha :fringespacingwellabove{\alpha }_c;Colorado\\ Statedeck:fringeequation,thickness\\ fromtheslopeof{\alpha }_m^{2}against{m}^{2}and{\alpha }_{c}from\\ theintercept,range5\mathring{\mathrm{A}}to400\mathrm{nm};\\ SchreiberandGerlach:periodicity2\pi /d\end{array}$ $\begin{array}{r}(computed:10\mathrm{nm}gives0.441{}^{\circ },50\mathrm{nm}\\ 0.088{}^{\circ }and400\mathrm{nm}0.011{}^{\circ }fringes)\end{array}$
Parratt recursion for multilayers	$X_j=\frac{r_{j,j+1}+X_{j+1}{e}^{2i{k}_{z,j+1}{z}_j}}{1+r_{j,j+1}{X}_{j+1}{e}^{2i{k}_{z,j+1}{z}_j}}{e}^{-2i{k}_{z,j}{z}_j}$	$X_{N+1}=0,R={\left|X_1\right|}^2$	$\begin{array}{r}{X}_j:ratioofreflectedtotransmitted\\ amplitudeatthetopoflayerj\end{array}$ $z_j:interfaceposition$ $\begin{array}{r}N:numberoflayers;Schreiberand\\ Gerlach:recursionstartingfrom{T}_1=1and\\ R_{N+1}=0;explicitform(background\\ knowledge,Parratt)\end{array}$ $\begin{array}{r}(myreading:eachlayeraddstwofitted\\ parametersbeyondthickness,densityand\\ roughness,sofitsneedseveralfringe\\ orders)\end{array}$

Roughness, Density and Precision

Name	Formula	Related Formulae	Variables/Constants
Interface roughness	$R_{rough}=R_F\exp \left(-Q_z^2{\sigma }^2\right)$	$r_{j,j+1}^{rough}=r_{j,j+1}\exp \left(-2k_{z,j}{k}_{z,j+1}{\sigma }_j^2\right)$	$\begin{array}{r}\sigma :rmsroughness;HZDR:Debye-Waller\\ factor{R}_F\exp (-K_z^2{\sigma }_{rms}^2);SchreiberandGerlach:\\ amplitudedampingprintedas{e}^{-2k_z{k}_z\sigma }(read\\ as{\sigma }^2)andexp(-Q_z^2{\sigma }^2)fordiffuse\\ scattering,Nevot-Croceforminthe\\ Parrattscheme;ColoradoStatedeck:\\ surfaceroughnesslowersthewholecurve,\\ interfaceroughnessdampsfringes,0to20\\ \mathring{\mathrm{A}}(backgroundknowledge,NevotandCroce)\end{array}$ $\begin{array}{r}(computed:\sigma =3\mathring{\mathrm{A}}at{Q}_z=0.2{\mathring{\mathrm{A}}}^{-1}leaves70\\ percent;\sigma =5\mathring{\mathrm{A}}at0.4{\mathring{\mathrm{A}}}^{-1}leaves1.8per\\ cent)\end{array}$
Density from the critical angle	${\rho }_m=\frac{2\pi \delta }{{\lambda }^2{r}_e}\frac{A}{N_{A}Z}$	$\frac{\delta \rho }{\rho }\approx 2\frac{\delta {\alpha }_I}{{\alpha }_c}$	${\rho }_m:massdensity$ $A:molarmass$ $Z:electronsperformulaunit$ $N_A:Avogadroconstant$ $\begin{array}{r}\delta {\alpha }_I:goniometerstep;ColoradoStatedeck:\\ densityaccuracywithfactor2,densities\\ Si2.33,SiO22.65,Ti4.51,Ta16.65and\\ Ir22.65g/c{\mathrm{m}}^3;inversionofthe\delta relation\\ (backgroundknowledge,Als-Nielsenand\\ McMorrow;seetheQ/A)\end{array}$ $\begin{array}{r}(computed:a0.002{}^{\circ }stepat{\alpha }_c=0.22{}^{\circ }\\ gives1.8percentdensityaccuracy)\end{array}$
Thickness precision and dynamic range	$\frac{\delta t}{t}\approx \frac{1}{m_{max}}\frac{\delta {\alpha }_I}{{\alpha }_c}$	$R_{min}\sim 10^{-6}(\text{laboratory source})$	$\begin{array}{r}{m}_{max}:highestfringeorderresolvedtohalf\\ aperiod;ColoradoStatedeck:precision\\ relation,5to6decades;Schreiberand\\ Gerlach:laboratorysourcestoabout\\ 1\times 10^{-6},synchrotronslargerrangeand\\ diffusescattering;HZDR:D5000\\ theta-thetadiffractometerwithGoebel\\ mirrorandslit,0.154and0.1033\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:m_{m}ax=20witha0.002{}^{\circ }step\\ gives4.5\times 10^{-4},about0.005\mathrm{nm}on10\mathrm{nm})\end{array}$

Important Questions

Ques: Why does XRR show total reflection below a critical angle?

Because X-ray refractive indices are slightly below one. With $n=1-\delta -i\beta$ and $\delta ={\lambda }^2{r}_e{\rho }_e/(2\pi )$ (Schreiber and Gerlach; Colorado State deck), X-rays entering a denser medium bend away from the normal, and at grazing angles below ${\alpha }_c=\sqrt{2\delta }$ no transmitted wave exists. For silicon at 2.33 $g/c{\mathrm{m}}^3$ the electron density is $6.99\times 10^{29}$ ${\mathrm{m}}^{-3}$, $\delta$ is $7.4\times 10^{-6}$ and ${\alpha }_c$ is 0.221 ${}^{\circ }$ at Cu K-alpha (computed). The Colorado State deck locates ${\alpha }_c$ where reflectivity has fallen to half its maximum. Above ${\alpha }_c$ the reflectivity falls as ${\alpha }_i^{-4}$, to $9.2\times 10^{-6}$ at 2 ${}^{\circ }$ (computed), which is why five to six decades of dynamic range are needed. Reference: F. Schreiber and A. Gerlach, ‘X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial’, Oxford University, hosted by ORNL (fetch-tool summary); ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled).

Ques: Is the critical angle 0.1 to 0.5 degrees or 0.2 to 1 degree?

Schreiber and Gerlach give 0.1 to 0.5 ${}^{\circ }$ for typical $\delta$ of $1\times 10^{-6}$ to $1\times 10^{-5}$, whereas HZDR quotes 0.2 to 1 ${}^{\circ }$ for wavelengths near 0.1 $\mathrm{nm}$. At Cu K-alpha the computed values are 0.215 ${}^{\circ }$ for SiO2, 0.221 ${}^{\circ }$ for Si, 0.413 ${}^{\circ }$ for HfO2, 0.532 ${}^{\circ }$ for Ta and about 0.62 ${}^{\circ }$ for Ir, the densest element in the Colorado State list (computed). The 0.1 to 0.5 ${}^{\circ }$ range is therefore more likely for semiconductor films, including low-density organics near 0.1 ${}^{\circ }$, while 1 ${}^{\circ }$ is an upper bound rarely reached even for heavy metals (my reading). Because ${\alpha }_c$ scales with $\lambda$, shorter wavelengths shrink all these values (background knowledge, Als-Nielsen and McMorrow). Reference: F. Schreiber and A. Gerlach, ‘X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial’, Oxford University, hosted by ORNL (fetch-tool summary); Helmholtz-Zentrum Dresden-Rossendorf, ‘X-ray reflectometry (XRR)’ page (fetch-tool summary); ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011; L. G. Parratt, Phys. Rev. 95, 359 (1954); L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980) (background knowledge; not opened).

Ques: How is film thickness obtained from Kiessig fringes?

From fringe positions corrected for refraction. The Colorado State deck gives ${\sin }^2{\alpha }_m-{\sin }^2{\alpha }_c=(m\lambda /2t{)}^2$, so plotting ${\alpha }_m^2$ against $m^2$ gives thickness from the slope and ${\alpha }_c$ from the intercept. Well above ${\alpha }_c$ the spacing approaches $\lambda /(2t)$: 0.441 ${}^{\circ }$ for 10 $\mathrm{nm}$, 0.088 ${}^{\circ }$ for 50 $\mathrm{nm}$ and 0.011 ${}^{\circ }$ for 400 $\mathrm{nm}$ (computed), which explains the deck’s range of 5 $\mathring{\mathrm{A}}$ to 400 $\mathrm{nm}$, since thicker films need finer angular steps. Its precision estimate $\delta t/t\approx (1/m_{max})(\delta {\alpha }_I/{\alpha }_c)$ gives $4.5\times 10^{-4}$ for 20 fringes and a 0.002 ${}^{\circ }$ step at 0.22 ${}^{\circ }$ (computed). Schreiber and Gerlach express the same periodicity as $2\pi /d$ in $Q_z$, which is independent of wavelength. Reference: ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled); F. Schreiber and A. Gerlach, ‘X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial’, Oxford University, hosted by ORNL (fetch-tool summary).

Ques: Is the density of SiO2 2.65 or about 2.2 grams per cubic centimetre?

The Colorado State deck lists SiO2 at 2.65 $g/c{\mathrm{m}}^3$ among its reference densities, a value that matches crystalline quartz, whereas amorphous thermal or deposited oxide on silicon is typically 2.2 to 2.3 $g/c{\mathrm{m}}^3$ (background knowledge, Als-Nielsen and McMorrow). The difference moves ${\alpha }_c$ from 0.215 to 0.236 ${}^{\circ }$ (computed), easily resolved with a 0.002 ${}^{\circ }$ step, whose density accuracy is about 1.8 per cent by the deck’s own relation (computed). For thin films on wafers, 2.2 to 2.3 $g/c{\mathrm{m}}^3$ is more likely, and fits forced to 2.65 would distort thickness and roughness (my reading). XRR density is in fact a sensitive check of oxide quality, since porous or nitrided films shift it. Reference: ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011; L. G. Parratt, Phys. Rev. 95, 359 (1954); L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980) (background knowledge; not opened).

Ques: How does roughness enter the reflectivity?

As a Gaussian damping in momentum transfer. HZDR writes the surface term as $R_F\exp (-K_z^2{\sigma }_{rms}^2)$, and Schreiber and Gerlach apply $\exp (-2k_{z,j}{k}_{z,j+1}{\sigma }^2)$ to each interface in the Parratt scheme, the Nevot-Croce form (background knowledge, Nevot and Croce); their printed amplitude factor omits the square on $\sigma$, read here as a typesetting slip (my reading). A 3 $\mathring{\mathrm{A}}$ surface keeps 70 per cent of the intensity at 0.2 ${\mathring{\mathrm{A}}}^{-1}$, while 5 $\mathring{\mathrm{A}}$ at 0.4 ${\mathring{\mathrm{A}}}^{-1}$ keeps only 1.8 per cent (computed). The Colorado State deck notes that surface roughness lowers the whole curve while interface roughness damps the fringes, and gives 0 to 20 $\mathring{\mathrm{A}}$ as the measurable range. Reference: Helmholtz-Zentrum Dresden-Rossendorf, ‘X-ray reflectometry (XRR)’ page (fetch-tool summary); F. Schreiber and A. Gerlach, ‘X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial’, Oxford University, hosted by ORNL (fetch-tool summary); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011; L. G. Parratt, Phys. Rev. 95, 359 (1954); L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980) (background knowledge; not opened); ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled).

Ques: How are multilayer stacks fitted?

With Parratt’s recursion from the substrate upward. Schreiber and Gerlach define Fresnel coefficients $r_{j,j+1}=(k_{z,j}-k_{z,j+1})/(k_{z,j}+k_{z,j+1})$ and build the reflected amplitude layer by layer, starting from no reflection below the substrate (background knowledge, Parratt). Each layer contributes thickness, density and roughness, so a three-layer gate stack already has about ten parameters (my reading). Fits are constrained by the critical angle, which fixes the top density, and by fringe envelopes, which reveal individual layer thicknesses. HZDR highlights multilayers as a special strength and shows an oxidized Ta layer on silicon. Data over five to six decades, as the Colorado State deck requires, keep the fit from settling on correlated thickness-roughness solutions. Reference: F. Schreiber and A. Gerlach, ‘X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial’, Oxford University, hosted by ORNL (fetch-tool summary); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011; L. G. Parratt, Phys. Rev. 95, 359 (1954); L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980) (background knowledge; not opened); Helmholtz-Zentrum Dresden-Rossendorf, ‘X-ray reflectometry (XRR)’ page (fetch-tool summary); ‘Specular X-ray Reflectivity’ slide deck (Colorado State University; author and year not shown), hosted by the University of Cincinnati (fetch-tool summary; symbols partly garbled).

References

1. F. Schreiber and A. Gerlach, “X-Ray and Neutron Reflectivity for the Investigation of Thin Films - A Short Tutorial,” Physical and Theoretical Chemistry Laboratory, Oxford University. Available: https://neutrons.ornl.gov/sites/default/files/beamline_04A_xray_neutron_reflectivity_tutorial.pdf
2. “Specular X-ray Reflectivity,” slide deck (Colorado State University), hosted by the University of Cincinnati. Available: https://www.eng.uc.edu/%7Ebeaucag/Classes/Characterization/ReflectivityLab/Colorado%20State%20Univ%20XRR.pdf
3. Helmholtz-Zentrum Dresden-Rossendorf, “X-ray reflectometry (XRR).” Available: https://www.hzdr.de/db/Cms?pOid=11657
4. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley, 2011; L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Physical Review, vol. 95, p. 359, 1954; L. Nevot and P. Croce, Revue de Physique Appliquee, vol. 15, p. 761, 1980 (cited from background knowledge).

Related pages in this vault: Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS), X-Ray Diffractometer (XRD and HRXRD), X-Ray Fluorescence Spectrometer (XRF and TXRF), Rutherford Backscattering Spectrometer (RBS), X-Ray Photoelectron Spectrometer (XPS), Spectroscopic Ellipsometer, Spectral Reflectometer, MOS Capacitor, Picosecond Ultrasonic Film Metrology, Quartz Crystal Microbalance (QCM), Atomic Layer Deposition (ALD) Reactor.