Good Diffraction Practice Webinar Series

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$Good Diffraction Practice Webinar Series High Resolution$

1 Good Diffraction Practice Webinar Series High Resolution X-ray Diffractometry (1) Mar 24,

Martin Zimmermann Applications Scientist, XRD Bruker AXS GmbH

2 Welcome Heiko Ress Global Marketing Manager Bruker AXS Inc. Madison, Wisconsin, USA Dr. Martin Zimmermann Applications Scientist, XRD Bruker AXS GmbH Karlsruhe, Germany March 24,

3 Overview Introduction Basics of HRXRD The experimental setup for HRXRD Preparing an HRXRD measurement Data analysis March 24,

4 What is High Resolution X-Ray Diffraction? (HRXRD) A X-ray scattering technique Non-destructive method Wavelength probes on atomic scale Diffraction Scattering from a crystal lattice High resolution Highly parallel and monochromatic beam High spatial resolution in q-space March 24,

5 What is High Resolution X-Ray Diffraction? (HRXRD) A X-ray scattering technique Non-destructive method Wavelength probes on atomic scale Diffraction Scattering from a crystal lattice High resolution Highly parallel and monochromatic beam High spatial resolution in q-space What information can HRXRD extract from a sample? March 24,

6 Analytical tasks Layer thickness Chemical composition Mismatch & relaxation Lattice parameters Defects & crystal size Lateral structure March 24,

7 Audience Poll Please use your mouse to answer the question to the right of your screen: Which structural properties are you mainly interested in? (Check all that apply.) Lattice parameters Layer thickness Concentration Strain & relaxation Defects Others March 24,

8 Introduction The general scattering geometry k r i k r f 2θ r Q = r r k f k i Probed quantity r 2 r rr r F( Q) ρ( )exp( iqr ) d V 2 Probed length scale d = λ 2sin θ March 24,

9 Scattering from a crystal: The concept of reciprocal space Real space q-space a r 2 Crystal lattice a r 1 Fourier transform r a r = 2πδ i b k r r exp = ik ( i G R ) 1 Reciprocal lattice b r 1 b r 2 r R = n r r r 1 a1 + n2 a2 + n3 a3 r G = r hb r r 1 + k b2 + l b3 March 24,

10 Accessible region in reciprocal space Experimental contraints of a single atom cubic crystal 5 4 l [001] h [100] March 24,

11 Accessible region in reciprocal space Limitation due to wavelength Wavevector k = 2π / λ 5 4 Q > 2k l [001] 3 2 Wavelength λ [ nm ] = 1.24 E [ kev ] h [100] The range of accessible reflections can be increased by using X-rays of a higher energy. March 24,

12 Accessible region in reciprocal space Geometrical contraints 5 4 Q > 2k l [001] transmission θ < 0 < 0 i transmission θ f h [100] Reflections very close to the half-spheres have grazing incidence or grazing exit geometry surface sensitivity March 24,

13 The crystal in reciprocal space Lattice Unit cell Crystal + = March 24,

The crystal in reciprocal space Lattice Unit cell Crystal + = r F( Q) Lattice 64748 r4r exp( iqr ) = n n Structure factor of the unit cell 6444

14 The crystal in reciprocal space Lattice Unit cell Crystal + = r F( Q) Lattice r4r exp( iqr ) = n n Structure factor of the unit cell 6444 r74448 rr f ( Q)exp( iqr ) f m : Atomic scattering factor of the atom at r m m m m exp r r = F(Q) r ( i G R ) 1 non-zero for Q r = G r March 24,

15 A silicon cystal in reciprocal space FCC-lattice with Si atoms at (0, 0, 0) and (¼, ¼, ¼ ) l [001] h [110] March 24,

16 Ewald construction for the Si(115+) reflection March 24,

17 Bandgap vs. Lattice Constant Poll Results March 24,

Epitaxial Layers Lattice Parameters and Strain Status Lattice structure is defined by these related parameters: lattice parameters of bulk materials a, b, c, α, β, γ

18 Epitaxial Layers Lattice Parameters and Strain Status Lattice structure is defined by these related parameters: lattice parameters of bulk materials a, b, c, α, β, γ concentration c x ( and c y ) for solid solutions mismatch in vertical direction Δc/c mismatch in lateral direction Δa/a lattice relaxation degree R March 24,

19 Layer Lattice Misfit relaxed layer lattice mismatch: Δa a rel = a L a a S S a L Si a S Ge Si 1-x Ge x Δa a rel > 0 March 24,

20 Layer Lattice Misfit relaxed layer lattice mismatch: Δa a rel = a L a a S S a L a L Si a S a S Ge Si 1-x Ge x Si 1-x C x C Δa a rel > 0 Δa a rel < 0 March 24,

21 Relation between relaxed and strained lattice parameters Measurement of strained lattice parameters Relaxed lattice parameters March 24,

22 Relation between relaxed and strained lattice parameters Measurement of strained lattice parameters Lateral lattice mismatch Vertical lattice mismatch Δd d = d L a a S Δd d S S = d L a a S Relaxed lattice parameters March 24,

= d L a a S Relaxed lattice mismatch Relaxed lattice parameters Δa a rel = P Δd d Theory

23 Relation between relaxed and strained lattice parameters Measurement of strained lattice parameters Lateral lattice mismatch Vertical lattice mismatch Δd d = d L a a S Δd d S S = d L a a S Relaxed lattice mismatch Relaxed lattice parameters Δa a rel = P Δd d Theory of elasticity: Δd d + Δd d P 0,5 1,0 ( Depending on the surface normal ) March 24,

24 Relaxation Degree R Relaxation degree R = Δd Δa d a rel pseudomorphic layer R = 0 Δd d = 0 d L = a S March 24,

25 Relaxation Degree R Relaxation degree R = Δd Δa d a rel pseudomorphic layer partially relaxed layer R = 0 Δd d = 0 d L = a S Δd d d L a S 0 March 24,

26 Relaxation Degree R Relaxation degree R = Δd Δa d a rel pseudomorphic layer partially relaxed layer completely relaxed layer R = 0 R = 1 Δd d = 0 Δd d 0 Δd d = Δd d = Δa a rel d L = a S d L a S d = d = L L a L March 24,

27 Epitaxial Layers in Reciprocal Space Pseudomorphic Layer substrate fully strained layer March 24,

28 Epitaxial Layers in Reciprocal Space Pseudomorphic Layer Completely Relaxed Layer substrate fully strained layer substrate fully relaxed layer March 24,

29 The relaxation line The reflection of a fully strained layer is located on a perpendicular line. A reflection of a fully relaxed layer is on a line through the substrate reflection and the (000). Reflection of partly relaxed layers are on the relaxation line. tanα = tanτ / D Theory of elasticity: D 0.5, K, 1 March 24,

30 Reciprocal lattice point displacement (2) Tilted layer on the substrate substrate tilted layer March 24,

tilt of a layer relative to the substrate causes a rotation

31 Reciprocal lattice point displacement (2) Tilted layer on the substrate substrate tilted layer substrate tilted layer The tilt of a layer relative to the substrate causes a rotation of the layer s reciprocal lattice around (000). March 24,

32 Deformation of reciprocal lattice points (1) Lattice constant variation Δc/c variation Δa/a variation Depends on (hkl) March 24,

33 Deformation of reciprocal lattice points (1) Lattice constant variation Finite size effects Δc/c variation Δa/a variation Finite layer thickness Lateral granularity Depends on (hkl) Independent of (hkl) March 24,

34 Deformation of reciprocal lattice points (2) Mosaicity layer substrate March 24,

35 Deformation of reciprocal lattice points (2) Mosaicity layer substrate Mosaicity causes a smearing of the reflection on a circle around (000). substrate relaxed layer with high mosaicity March 24,

36 The different structural properties can be distinguished and seperated by measuring at multiple reflections. March 24,

37 The experimental setup for HRXRD Double-axis geometry Triple-axis geometry March 24,

38 The experimental setup for HRXRD Double-axis geometry Triple-axis geometry March 24,

$A diffractometer with double-axis geometry The incident beam is conditioned by a monochromator.$

39 A diffractometer with double-axis geometry The incident beam is conditioned by a monochromator. The detector is opened and integrates over 2θ. Variable slits on the detector side can reduce the noise. March 24,

$A diffractometer with double-axis geometry ω is the scan axis.$

40 A diffractometer with double-axis geometry ω is the scan axis. χ ϕ and - drives can be used to bring a reflection into the scattering plane. March 24,

41 Parameter extraction from a symmetric reflection good quality layers exhibit thickness fringes around their Bragg reflection ω S Film thickness: λ t = 2cos ω Δω Vertical mismatch: ω L Δс c = с L c c S S = sinωs sinω L 1. Assuming Δa/a = 0, the concentration of Ge can be evaluated. March 24,

42 Parameter extraction from rocking curves on asymmetric reflections Rocking Si(224-) Rocking Si(224+) Δω = Δ ω + = March 24,

43 Parameter extraction from rocking curves on asymmetric reflections From Δω - and Δω +, the unknown Δa/a and Δc/c of the layer can be calculated and the concentration and relaxation can be obtained. The layer thickness can be obtained from λ sin( θb + τ ) t = Δω sin(2θ ) B If the layer is tilted with respect to the substrate, a rocking curve at a third reflection is required to determine the tilt angle. March 24,

44 A perfect sample acts as an analyzer crystal Rocking curve on SiGe/Si RSM on SiGe/Si(004) March 24,

45 Samples with high mosaicity require an analyzer crystal RSM on GaN(002) Rocking curve on GaN(002) March 24,

46 The experimental setup for HRXRD Double-axis geometry Triple-axis geometry March 24,

$A diffractometer with triple-axis geometry An additional$ Measurements with high angular resolution in 2θ and ω.

47 A diffractometer with triple-axis geometry An additional analyzer crystal provides resolution on the 2θ axis. Measurements with high angular resolution in 2θ and ω. For sample alignment the two-axis mode is required. March 24,

48 A perfect sample acts as an analyzer crystal March 24,

49 Samples with high mosaicity require an analyzer crystal March 24,

50 Remarks on the triple-axis geometry Pros Provides local resolution in reciprocal space (2θ and ω). Better resolution allows thickness determination of thicker films. The analyzer crystal removes the diffuse scattering and fluorescence. Cons Loss in intensity compared to 2-axis geometry. Tilted layers can be easily missed. March 24,

51 Audience Poll Please use your mouse to answer the question to the right of your screen: What material class(es) are you mainly working on? (Check all that apply.) IV IV semiconductors ( Si, SiGe, SiC, ) III V semiconductors ( GaAs, InP, ) Nitrides ( GaN ) Oxides ( STO, BTO, ) Others March 24,

52 Preparing the HRXRD experiment Choosing the appropriate optics Mounting the sample March 24,

53 Different monochromators to obtain better results Different monochromators to balance resolution and intensity 2 x Ge(220a) 2 x Ge(004a) 4 x Ge(220a) 4 x Ge(220s) 2-bounce 4-bounce March 24,

54 Monochromators: Angular resolution vs. intensity Resolution Intensity 4-bounce monochromators have a better overall resolution for the cost of intensity. The type of the 2-bounce monochromator should be adapted to the sample. Poll Results March 24,

Automated change of the resolution on the

55 Automated change of the resolution on the detector side Motorized slit Motorized and softwarecontrolled switch between Double-axis geometry with motorized slit Triple-axis geometry with PATHFINDER optics 1xGe(220) or 3xGe(220) crystals March 24,

$Documentation of the experimental setup The resolution of the diffractometer is given by Monochromators Analyzer crystals Slits A$

56 Documentation of the experimental setup The resolution of the diffractometer is given by Monochromators Analyzer crystals Slits A detailed documentation of the experimental setup is mandatory for proper data analysis and should be stored in every data file. March 24,

57 Preparing the HRXRD experiment Choosing the appropriate optics Mounting the sample March 24,

The sample mounting should not bend or deform the substrate.

58 Mounting the sample (1) A proper mounting of the sample is mandatory for obtaining correct results. The sample mounting should not bend or deform the substrate. Samples with a flat subtrate can be directly mounted on a vacuum chuck. A porous ceramic chuck applies less force on the substrate. March 24,

The glass plate can then be mounted on a vacuum chuck.

59 Mounting the sample (2) Small samples can be glued on a flat glass plate. Use rubber glue, e.g. FixoGum from Marabu. The glass plate can then be mounted on a vacuum chuck. Large samples, like wafers, should be held by a 3-point vacuum suckers. Vacuum suckers March 24,

60 Analysis of HRXRD measurements Fitting of HRXRD data Examples March 24,

Evaluation of the sample - Fitting Procedure Sample Model parameterized by {p 1, p N } HRXRD Simulation in Dynamical theory Comparison with Experiment, χ2 cost

61 Evaluation of the sample - Fitting Procedure Sample Model parameterized by {p 1, p N } HRXRD Simulation in Dynamical theory Comparison with Experiment, χ2 cost function Minimization of χ 2 using Genetic Algorithm, Levenberg-Marquardt, Simplex, Simulated Annealing, etc. in view of {p 1..p N } Tolerance March 24,

62 Defining the materials properly Space group Unit cell Lattice constants Elastic properties Definition of solid solutions March 24,

63 Analytical calculation of the resolution function from the used optics A proper simulation requires a precise knowledge of the resolution function March 24,

64 Analysis of HRXRD measurements Fitting of HRXRD data Examples March 24,

65 HRXRD from SiGe films on Si substrates 65nm Si 0.9 Ge 0.1 substrate Si 25nm Si 50nm Si 0.85 Ge 0.15 substrate Si March 24,

66 HRXRD from SiGe films on Si substrates March 24,

67 Analysis of Super-Lattice structures (SL) 100nm GaAs 3nm In 0.15 Ga 0.85 As 26nm GaAs x 15 substrate GaAs March 24,

68 Analysis of Super-Lattice structures (SL) 100nm GaAs Position of zero-th order SL peak: < Δc / c > 15 x 3nm In 0.15 Ga 0.85 As 26nm GaAs substrate GaAs d SL Distance between SL-peaks: d SL 2 λ cos θ Δθ March 24,

69 Analysis of Super-Lattice structures (SL) Linking parameters 100nm GaAs 3nm In 0.15 Ga 0.85 As 26nm GaAs substrate GaAs x 15 First determine the thickness of the bi-layer d 1 + d nm. Fit d 1 using the constraint d nm - d 1 March 24,

70 Analysis of Super-Lattice structures (SL) Linking parameters 100nm GaAs 3nm In 0.15 Ga 0.85 As 26nm GaAs substrate GaAs x 15 First determine the thickness of the bi-layer d 1 + d 2 28,6nm. Fit d 1 with the constraint d 2 28,6nm - d 1 March 24,

Analysing measurements taken at different (hkl) reflections The structure factors of the materials in the sample can differ significantly at different reflections

71 Analysing measurements taken at different (hkl) reflections The structure factors of the materials in the sample can differ significantly at different reflections Structure factor for Zincblende structure F004 = 4 f III + 4 f V F002 = 4 f III 4 f V Atomic scattering factor f(0)=z F 004 F 002 GaAs AlAs March 24,

72 Analyzing measurements taken at different (hkl) reflections (1) (004) small contrast (002) huge contrast 1500nm GaAs 100nm AlAs substrate GaAs March 24,

73 Analysing measurements taken at different (hkl) reflections (1) (004) small contrast (002) huge contrast 1500nm GaAs 100nm AlAs substrate GaAs March 24,

74 Analysing measurements taken at different (hkl) reflections (2) 10X 5nm GaAs 100nm Al 0.8 Ga 0.2 As 100nm Al 0.2 Ga 0.8 As substrate GaAs (004) (002) March 24,

75 Analysing measurements taken at different (hkl) reflections (2) 10X 5nm GaAs 100nm Al 0.8 Ga 0.2 As 100nm Al 0.2 Ga 0.8 As substrate GaAs (004) (002) March 24,

76 Summary Adapt the experimental setup to your sample type and the parameters you want to extract. Mount the sample stable. Make sure that the substrate is not deformed by external forces. Choose the reflections depending on the sample structure and the structural properties you want to extract. Try to keep your sample model simple. March 24,

77 Summary Adapt the experimental setup to your sample type and the parameters you want to extract. Mount the sample stable. Make sure that the substrate is not deformed by external forces. Choose the reflections depending on the sample structure and the structural properties you want to extract. Try to keep your sample model simple. Thank you for your attention... March 24,

78 Any Questions? Please type any questions you may have in the Q&A panel and then click Send. 78 March 24,

79 March 24,

Quantum Condensed Matter Physics Lecture 5

Quantum Condensed Matter Physics Lecture 5 detector sample X-ray source monochromator David Ritchie http://www.sp.phy.cam.ac.uk/drp2/home QCMP Lent/Easter 2019 5.1 Quantum Condensed Matter Physics 1. Classical