Abstract

Coherent X-ray diffraction imaging (CXDI) of the displacement field and strain distribution of nanostructures in kinematic far-field conditions requires solving a set of non-linear and non-local equations. One approach to solving these equations, which utilizes only the object’s geometry and the intensity distribution in the vicinity of a Bragg peak as a priori knowledge, is the HIO+ER-algorithm. Despite its success for a number of applications, reconstruction in the case of highly strained nanostructures is likely to fail. To overcome the algorithm’s current limitations, we propose the HIOORM+ERM-algorithm which allows taking advantage of additional a priori knowledge of the local scattering magnitude and remedies HIO+ER’s stagnation by incorporation of randomized overrelaxation at the same time. This approach achieves significant improvements in CXDI data analysis at high strains and greatly reduces sensitivity to the reconstruction’s initial guess. These benefits are demonstrated in a systematic numerical study for a periodic array of strained silicon nanowires. Finally, appropriate treatment of reciprocal space points below noise level is investigated.

© 2013 OSA

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2013 (4)

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr.46, 312–318 (2013).
[CrossRef] [PubMed]

Y. Chushkin and F. Zontone, “Upsampling speckle patterns for coherent x-ray diffraction imaging,” J. Appl. Crystallogr.46, 319–323 (2013).
[CrossRef]

R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input-output method of phase retrieval,” Appl. Opt.52, 3031–3037 (2013).
[CrossRef] [PubMed]

S. T. Haag, M.-I. Richard, S. Labat, M. Gailhanou, U. Welzel, E. J. Mittemeijer, and O. Thomas, “Anomalous coherent diffraction of core-shell nano-objects: a methodology for determination of composition and strain fields,” Phys. Rev. B87, 035408 (2013).
[CrossRef]

2012 (5)

2011 (4)

K. Busch, C. Blum, A. M. Graham, D. Hermann, M. Köhl, P. Mack, and C. Wolff, “The photonic Wannier function approach to photonic crystal simulations: status and perspectives,” J. Mod. Opt.58, 365–383 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 568 (2011).
[CrossRef] [PubMed]

2010 (2)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature467, 436 (2010).
[CrossRef] [PubMed]

A. Diaz, V. Chamard, C. Mocuta, R. Magalhães-Paniago, J. Stangl, D. Carbone, T. H. Metzger, and G. Bauer, “Imaging the displacement field within epitaxial nanostructures by coherent diffraction: a feasibility study,” New J. Phys.12, 035006 (2010).
[CrossRef]

2009 (4)

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat. Mater.8, 291–298 (2009).
[CrossRef] [PubMed]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett.102, 018101 (2009).
[CrossRef] [PubMed]

G. Renaud, R. Lazzari, and F. Leroy, “Probing surface and interface morphology with grazing incidence small angle x-ray scattering,” Surf. Sci. Rep.64, 255–380 (2009).
[CrossRef]

2008 (3)

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett.101, 090801 (2008).
[CrossRef] [PubMed]

M. Eberlein, S. Escoubas, M. Gailhanou, O. Thomas, P. Rohr, and R. Coppard, “Influence of crystallographic orientation on local strains in silicon: a combined high-resolution x-ray diffraction and finite element modelling investigation,” Thin Solid Films516, 8042–8048 (2008).
[CrossRef]

2007 (7)

M. Hanke, Y. I. Mazur, J. E. Marega, Z. Y. AbuWaar, G. J. Salamo, P. Sch¨afer, and M. Schmidbauer, “Shape transformation during overgrowth of InGaAs/GaAs(001) quantum rings,” Appl. Phys. Lett.91, 043103 (2007).
[CrossRef]

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep.444, 101 (2007).
[CrossRef]

M. Eberlein, S. Escoubas, M. Gailhanou, O. Thomas, J.-S. Micha, P. Rohr, and R. Coppard, “Investigation by high resolution x-ray diffraction of the local strains induced in Si by periodic arrays of oxide filled trenches,” Phys. Status Solidi A204, 2542–2547 (2007).
[CrossRef]

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging,” Opt. Express15, 9954–9962 (2007).
[CrossRef] [PubMed]

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum.78, 011301 (2007).
[CrossRef]

S. Marchesini, “Phase retrieval and saddle-point optimization,” J. Opt. Soc. Am. A24, 3289–3296 (2007).
[CrossRef]

2006 (2)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature442, 63–66 (2006).
[CrossRef] [PubMed]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr., Sect. A: Found. Crystallogr.62, 248–261 (2006).
[CrossRef]

2004 (4)

M. Hanke, D. Grigoriev, M. Schmidbauer, P. Schäfer, R. Köhler, U. Pohl, R. Sellin, D. Bimberg, N. Zakharov, and P. Werner, “Diffuse x-ray scattering of InGaAs/GaAs quantum dots,” Physica E21, 684–688 (2004). Proceedings of the Eleventh International Conference on Modulated Semiconductor Structures.
[CrossRef]

M. Hanke, D. Grigoriev, M. Schmidbauer, P. Sch¨afer, R. Köhler, R. L. Sellin, U. W. Pohl, and D. Bimberg, “Vertical composition gradient in InGaAs/GaAs alloy quantum dots as revealed by high-resolution x-ray diffraction,” Appl. Phys. Lett.85, 3062 (2004).
[CrossRef]

S. Cuenot, C. Frétigny, S. Demoustier-Champagne, and B. Nysten, “Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy,” Phys. Rev. B69, 165410 (2004).
[CrossRef]

N. Jalili and K. Laxminarayana, “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences,” Mechatronics14, 907–945 (2004).
[CrossRef]

2003 (2)

F. J. Giessibl, “Advances in atomic force microscopy,” Rev. Mod. Phys.75, 949–983 (2003).
[CrossRef]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter15, R1233 (2003).
[CrossRef]

2002 (1)

G. R. Liu and S. S. Q. Jerry, “A finite element study of the stress and strain fields of InAs quantum dots embedded in GaAs,” Semicond. Sci. Technol.17, 630–643 (2002).
[CrossRef]

2001 (1)

I. A. Vartanyants and I. K. Robinson, “Partial coherence effects on the imaging of small crystals using coherent x-ray diffraction,” J. Phys.: Condens. Matter13, 10593–10611 (2001).
[CrossRef]

1999 (2)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).
[CrossRef]

T. Benabbas, Y. Androussi, and A. Lefebvre, “A finite-element study of strain fields in vertically aligned InAs islands in GaAs,” J. Appl. Phys.86, 1945–1950 (1999).
[CrossRef]

1996 (1)

1990 (2)

1987 (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58, 2059–2062 (1987).
[CrossRef] [PubMed]

1986 (2)

1984 (1)

1982 (3)

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 - theory,” IEEE Trans. Med. Imaging1, 81–94 (1982).
[CrossRef]

J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982).
[CrossRef] [PubMed]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttgart)61, 247–262 (1982).

1969 (1)

S. Takagi, “A dynamical theory of diffraction for a distorted crystal,” J. Phys. Soc. Jpn.26, 1239–1253 (1969).
[CrossRef]

AbuWaar, Z. Y.

M. Hanke, Y. I. Mazur, J. E. Marega, Z. Y. AbuWaar, G. J. Salamo, P. Sch¨afer, and M. Schmidbauer, “Shape transformation during overgrowth of InGaAs/GaAs(001) quantum rings,” Appl. Phys. Lett.91, 043103 (2007).
[CrossRef]

Adams, D. E.

Allain, M.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 568 (2011).
[CrossRef] [PubMed]

Androussi, Y.

T. Benabbas, Y. Androussi, and A. Lefebvre, “A finite-element study of strain fields in vertically aligned InAs islands in GaAs,” J. Appl. Phys.86, 1945–1950 (1999).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttgart)61, 247–262 (1982).

Bauer, G.

A. Diaz, V. Chamard, C. Mocuta, R. Magalhães-Paniago, J. Stangl, D. Carbone, T. H. Metzger, and G. Bauer, “Imaging the displacement field within epitaxial nanostructures by coherent diffraction: a feasibility study,” New J. Phys.12, 035006 (2010).
[CrossRef]

Baumbach, T.

M. Köhl, A. A. Minkevich, and T. Baumbach, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express20, 17093–17106 (2012).
[CrossRef]

P. Schroth, T. Slobodskyy, D. Grigoriev, A. Minkevich, M. Riotte, S. Lazarev, E. Fohtung, D. Hu, D. Schaadt, and T. Baumbach, “Investigation of buried quantum dots using grazing incidence x-ray diffraction,” Mater. Sci. Eng., B177, 721–724 (2012).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-Ray Scattering from Thin Films to Lateral Nanostructures (Springer, New York, 2004).
[CrossRef]

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A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
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M. Hanke, D. Grigoriev, M. Schmidbauer, P. Sch¨afer, R. Köhler, R. L. Sellin, U. W. Pohl, and D. Bimberg, “Vertical composition gradient in InGaAs/GaAs alloy quantum dots as revealed by high-resolution x-ray diffraction,” Appl. Phys. Lett.85, 3062 (2004).
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P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr., Sect. A: Found. Crystallogr.62, 248–261 (2006).
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C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett.101, 090801 (2008).
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[CrossRef] [PubMed]

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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature467, 436 (2010).
[CrossRef] [PubMed]

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[CrossRef]

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S. T. Haag, M.-I. Richard, S. Labat, M. Gailhanou, U. Welzel, E. J. Mittemeijer, and O. Thomas, “Anomalous coherent diffraction of core-shell nano-objects: a methodology for determination of composition and strain fields,” Phys. Rev. B87, 035408 (2013).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematics of the HIO O R M-algorithm according to Eqs. (7a) and (11): In addition to QΓ;λΓ in the HIOOR-algorithm the operator MM is applied before the calculation of the next iterative candidate ρ eff ( i + 1 ) ( x ) is performed. In the limit ML,n → 0 and MH,n → ∞ for all n, the algorithm reduces to HIOOR because MM → 1.

Fig. 2
Fig. 2

Illustration of the physical system used for our investigation of the HIO O R M + ER M-algorithm. Figure (a) shows the geometry and composition of the upper region of a periodic unit of the periodic Si-nanowire system. The hatched domain will become important in next sections. Figures (b)–(d) depict the phase field QB · u(x) of effective electron density ρeff(x) for this system for different values of the maximum strain εM on the wires’ symmetry axis in the crystalline domain, if the Bragg peak QB = [004] is investigated. Figures (e)–(g) show the scattering signal in presence of the low signal cutoff ΓN (see Sec. 1.4). Only the central region of the scattering signal around the Bragg peak QB is shown (QB is located at the yellow dot). Data points below the noise level ΓN are masked in dark cyan.

Fig. 3
Fig. 3

Behavior of the HIO O R M + ER M-algorithm (including its limiting cases without randomization and/or without constraints on the local scattering magnitude, see grey boxes with black border at the beginning of each row of four subfigures) for ideal data: Depicted are two-dimensional cuts of the success rate s through the three dimensional parameter space for either fixed strain εM, number of iterations i or angle φMax. We count all random initial trails as success for which the angle φ(i) (defined in Eq. (13)) to the reference solution ρeff is below φMax in iteration (i) for that particular value of strain εM. The domain ΩSub corresponds to the hatched domain in Fig. 2(a).

Fig. 4
Fig. 4

Characteristics of the HIO O R M + ER M-algorithm in presence of a low cutoff ΓN. Figures (a) and (b) compare the success rate s of the models defined in Sec. 1.4 for fixed strain εM = {0.20%, 0.28%} after i = 500 iterations. Figures (c)–(f) contain two-dimensional plots of the success rate s for model (E) (analogue to Fig. 3). For Figs. (a), (c) and (d), no constraints on ζ have been applied. Figures (b), (e) and (f) illustrate the improvement if bounds on the local scattering magnitude are taken into account. Note the different range of the strain axes.

Fig. 5
Fig. 5

Summary of our numerical investigation: Starting from standard HIO+ER, the improvements which are achieved by randomized overrelaxation and the additional constraints M on the local scattering magnitude ζ are simplified to two values. The upper value in each rectangular box is the maximum strain εM for which an almost perfect solution could be reconstructed within i = 500 iterations and for almost all random initial trails. The lower value is the maximum strain εM for which the reconstruction of the effective electron density ρ eff ( 500 ) was successful if the requirement for success is relaxed: Any strain εM for which at least some non-negligible fraction of initial guesses managed to achieve a result close to the solution ρeff (φMax ≲ 10.0°) is classified as suitable for the respective approach and constraints.

Tables (2)

Tables Icon

Table 1 Definition of κq for five models (A)–(E) which are investigated for treating data points below a given low cutoff ΓN.

Tables Icon

Table 2 Characteristics of reciprocal space input data Γq in case of low signal cutoff ΓN.

Equations (19)

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𝔣 ( Q ) = FT Q x { ρ el ( x ) } = d d d x ρ el ( x ) e i Q x
ρ eff ( x , Q B ) e i Q B u ( x ) ζ Q B ( x ) Ω ( x ) IFT x q { 𝔣 Q B ( q ) } .
I Q B ( q ) | 𝔣 Q B ( q ) | 2 = | FT q x { ρ eff ( x ) } | 2 .
𝔣 Q B ( i + 1 ) = H ^ ER 𝔣 Q B ( i ) , H ^ ER = ( FT ) P Ω ( IFT ) P Γ .
P Ω ρ eff ( i ) ( x ) = { ρ eff ( i ) ( x ) if x Ω , 0 if x Ω .
P Γ 𝔣 ( i ) ( q ) = Γ q e i arg ( 𝔣 ( i ) ( q ) ) .
ρ eff ( i + 1 ) ( x ) = { M C ( λ Γ ) ρ eff ( i ) ( x ) if x Ω , ρ eff ( i ) ( x ) β M C ( λ Γ ) ρ eff ( i ) ( x ) if x Ω ,
M C ( λ Γ ) = ( IFT ) Q Γ ; λ Γ ( FT ) .
Q Γ ; λ Γ = 1 + λ Γ ( P Γ 1 ) .
𝔣 Q B ( 0 ) ( q ) = Γ q e i Φ q
M L , j ζ ¯ j ζ ( x ) M H , j ζ ¯ j x Ω j Ω ,
M M ( j ) ρ eff ( i ) ( x ) = { M j ( i ) ( x ) e i arg ( ρ eff ( i ) ( x ) ) if x Ω j , ρ eff ( i ) ( x ) if x Ω j ,
M j ( i ) ( x ) = min ( M H , j ζ ¯ j ( i ) ; max ( M L , j ζ ¯ j ( i ) ; | ρ eff ( i ) ( x ) | ) ) ,
ζ ¯ j ( i ) = P Ω j | ρ eff ( i ) | ; P Ω j | ρ eff ( i ) | / P Ω j ; P Ω j .
M C ( λ Γ ) = M M ( IFT ) Q Γ ; λ Γ ( FT ) , M M = j M M ( j ) .
H ^ ER , M ( OP ) = ( FT ) P Ω M M ( IFT ) P Γ
φ ( i ) = arccos [ | ρ eff ( i ) ; ρ eff | / ρ eff ( i ) ; ρ eff ( i ) ρ eff ; ρ eff ] .
χ ( i ) = arccos [ | ρ eff ( i 1 ) ; ρ eff ( i ) | / ρ eff ( i 1 ) ; ρ eff ( i 1 ) ρ eff ( i ) ; ρ eff ( i ) ] .
P Γ 𝔣 ( i ) ( q ) = κ q e i arg ( 𝔣 ( i ) ( q ) ) .

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