VOLUME 86, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 12 MARCH 2001 Real-Space Imaging of Atomic Structure with White X Rays P. Korecki and G. Materlik Hamburger Synchrotronstrahlungslabor HASYLAB am Deutschen Elektronen-Synchrotron DESY, 22603 Hamburg, Germany (Received 9 August 2000) The first real-space x-ray image of an atomic structure was obtained by illuminating a crystal with white synchrotron radiation. The internal photocurrent signal served as a probe of the x-ray interference field strength at the atomic sites and was accordingly measured as a function of illumination direction to record the two-dimensional image. This novel method of real-space imaging makes use of the fact that the interference field intensity is energy independent with respect to contributions from those scattering atoms which are brought via sample rotation into the forward scattering condition. In contrast, contributions from other atoms oscillate with energy and vanish for broadband illumination. DOI: 10.1103/PhysRevLett.86.2333 PACS numbers: 61.10.­i, 07.85.Tt, 42.40.­i, 61.14.­x Although x rays have a well suited wavelength for real- detectors. We show the first projection phase-contrast im- space atomic structure imaging, the experimental resolu- age of an atomic structure recorded with x rays. tion has so far never reached the wavelength limit. The When the sample is illuminated by polychromatic radi- small numerical apertures of x-ray lenses limit the resolu- ation, described by the wave number spectrum N k and tion to at best 10 nm, and thus the atomic resolution x-ray the illumination direction k, the x-ray field intensity at a microscope is not feasible [1]. A lensless approach such given atomic detector can be evaluated as an incoherent as a point-projection microscope [2] is limited by the finite superposition of wave fields each corresponding to a given R size of the x-ray source, which in order to accomplish high k vector value, i.e., as I k 0 N k jE k j2 dk. In the magnification has to be brought close to the object. X-ray first Born approximation, E k may be written as a sum of diffraction methods, on the other hand, can be used only the incident plane wave E0 k and the superposition Es k for crystals and, in addition, the so called phase problem of all singly scattered waves: makes the direct inversion of the diffracted intensities from Z reciprocal to real space impossible. I k ~ N k jE0 k 1 Es k j2 dk Internal source holography (ISH) [3] and internal de- 0 tector holography (IDH) [4] are alternative lensless ap- Z proaches which were realized experimentally quite recently. N k jE0 k j2 1 2 ReE 0 k Es k 1 . . . dk . 0 They avoid optical elements and thus provide images with (1) atomic resolution by using x rays [5] or g rays [6]. In ISH, atoms inside a sample serve as coherent point sources of The factor jE0 k j2 provides the isotropic background fluorescent radiation which is scattered from nearby atoms. whereas the terms containing the squared scattered waves This produces an interference pattern corresponding to the can be neglected in a weak scattering approximation. The local environment around the emitting atoms. Since x-ray background normalized signal at the atomic detector sites fluorescence radiation has a sufficient temporal coherence is x k I k 2 I0 k I0 k . It is related to the second length (lc 1 mm) to image atomic distances, such a pat- term on the right side of Eq. (1) and can be written as [9] tern is an in-line Gabor hologram [7]. The reconstruction Z Z 1 of the hologram to real space is performed numerically [8]. x k ~ 2r0 Re dk N k r r ei kr1kr dV , r In IDH, the interference of elastically scattered waves is 0 V monitored by measuring the absorption of particular atoms (2) serving as internal detectors. The detection signal for mea- where r r is the electron density at position r relative to suring the local x-ray field intensity is provided by x-ray the absorbing atom, r0 is the classical electron radius, and fluorescence or electron yield and is recorded with an ex- the minus sign reflects the phase change of the scattered ternal detector. Because of the weak scattering of x rays waves. Equation (2) neglects the attenuation of the scat- the holographic modulation is only in the order of 1024 tered waves. For monochromatic illumination, i.e., N k of the total signal. In addition, for high spatial resolution, d k 2 k0 , Eq. (2) becomes the formula used in ISH. holograms have to be recorded over wide angular and en- Consider, however, the case of perfect white illumina- ergy ranges. Therefore, high-brilliance sources and fast tion of the sample. For N k 1 the integration yields two-dimensional energy resolving detectors are needed to Z 1 make these techniques widely applicable. x k ~ 2r0 r r d r 1 kr dV . (3) In this Letter we demonstrate that the atomic structure V r can be viewed directly in real space by using incoherent The presence of the Dirac function with argument r 1 white x rays in connection with atoms acting as internal kr means that, for a given illumination direction, only the 0031-9007 01 86(11) 2333(4)$15.00 © 2001 The American Physical Society 2333 VOLUME 86, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 12 MARCH 2001 atoms along the line r 2 kr (i.e., along the illumina- direction. Therefore, Fig. 1 shows a transition from coher- tion direction) contribute to the signal modulations. There- ent imaging of a structure in reciprocal space to incoherent fore, when the sample is illuminated by a white beam a imaging in real space. Figure 1 also demonstrates that, real-space spherical projection of the electron density is even for relatively broadband illumination (Dk k0 10), obtained, since all diffraction patterns corresponding to a holographic approach may be applied to image the different energies add up constructively only in the for- nearest atoms as was done using Bremsstrahlung photons ward direction, where the scattered waves are all in phase. in Ref. [11]. The 1 r factor in Eq. (3) enhances the contribution from An incoherent image in real space was recorded at the the nearest scatterers. x k will thus have minimum val- Hamburg Synchrotron Radiation Laboratory HASYLAB ues for directions corresponding to the nearest neighbors, DESY using the beam line D4. The scheme of the experi- and hence the information about bond directions can be ob- mental setup is shown in Fig. 2(a). The direct white radia- tained directly without performing any integral transform. tion from the bending magnet passed through a 7.6 mm This direct real-space imaging works only when all detect- thick aluminum absorber that was used to shape a broad- ing atoms have similar neighborhoods. Then the scattering band energy spectrum. A Si photodiode served as a test geometry can be viewed as the time-reversed version of a sample. Photodiodes are efficient detectors not only for lensless projection x-ray microscope [10] with an internal visible light but also for hard x rays [12]. X rays produce atomic source of illumination. high-energy photoelectrons and Compton electrons which However, because of experimental limitations, it is nec- are "thermalized" into a distribution of electron-hole pairs essary to consider the intermediate case of a band-limited with energies around 3.6 eV for Si. Since the Si photodi- N k spectrum approximating the incident radiation spec- ode is a perfect crystal, it is well suited to serve as sample trum by a Gaussian curve centered at k0 and spread by Dk. and detector simultaneously. The photocurrent of the diode Figure 1 shows diffraction patterns (k0 20 Å21) of a measures simply the total absorption yield. The diode was point scatterer probed by an atomic detector at a distance an ion implanted 300 mm thick n-type wafer with a mis- of r 2.35 Å calculated for several values of Dk. As cut of 7± relative to the (111) surface [13]. The diode was the spectrum of incident radiation becomes broader, the operated without bias in the photovoltaic mode as a cur- diffracted wavelets localize around the forward scattering rent source and the photocurrent was amplified by typi- direction, and simultaneously their modulation amplitude cally 106 V A. In the experiment, the diode was rotated is damped. In other words, the higher-order interference around two axes. The inclination angle u, measured with fringes vanish and only the zeroth-order diffraction dip sur- respect to the surface normal, was changed in 1± steps. vives in the forward direction. The increase of Dk reduces The f rotation around the surface normal was performed the temporal coherence length lc l2 Dl 2p Dk, continuously and the measured signal was integrated in while the transverse coherence is preserved. For practical 0.5± intervals. The measuring time of a single pattern purposes, this means that by using a broadband spectrum 0± , u , 55±, 0± , f , 360± was three hours. Eight (k0 2 , Dk , k0 5), one is also able to determine the di- patterns were added together. About 109 photons s pixel rections of the nearest bonds, without any a priori knowl- were absorbed by the sample. The measured signal was edge because the scattering contribution from a single normalized using a similar photodiode placed directly in atom is localized around the corresponding interatomic front of the sample. (a) (a) Monitor- slits 2 incident (b) forward scattering Si photodide DORIS 0.1x0.1mm plane wave forward scattering Sample- lc=31.4Å k=k/100 0 Al absorber direction k^ Si(111) . units] 7.6mm photodiode scattering l =3.1 [arb c Å k=k/10 0 (b) atom =0° l k=k/5 =30° c=1.6Å 0 . units] =45° r l =0.6 c Å k=k/2 0 [arb absorbing atom- - /2 /2 internal detector 0 number of photons scattering angle [rad] 20 40 60 80 100 FIG. 1. Diffraction of a plane wave (k Energy [keV] 0 20 Å21) by a single point scatterer probed by an atomic detector at a distance of FIG. 2. White beam experiment: measurement of the total ab- r 2.35 Å. (a) Scattering geometry. (b) Diffraction patterns sorption yield as a function of sample illumination direction. calculated for several Dk values. Note the localization of the (a) Experimental setup. (b) Calculated energy spectrum seen by diffraction pattern around the interatomic direction as the wave- the average atomic detector inside the sample diode shown for length spread is increased. various inclination angles. 2334 VOLUME 86, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 12 MARCH 2001 An average absorbing sample atom sees a wave-number the calculation and from multiple-scattering effects, which spectrum that depends first on the spectrum of the beam were not taken into account. The single-scattering approxi- produced at the bending magnet, second, on the absorp- mation can be applied because the broadband illumination tion and scattering of all elements placed into the beam, reduces the effective number of scatterers from N3 atoms and finally on the integrated absorption in the sample. The in a crystal volume to N atoms in each atomic row along function N k also contains the different energy-current the illumination direction. In addition, for x rays the scat- conversion ratios for photoelectrons and Compton elec- tering amplitude of a single atom has a very small value trons. The energy spectrum N e , with e ¯hck, calcu- and thus the multiple-scattering contribution is very weak. lated for an aluminum absorber with a thickness of 7.6 mm Apart from the inverse contrast resulting from the nega- is presented in Fig. 2(b) for several inclination angles. The tive value of the photon scattering amplitude, the measured spectrum has its maximum around 35 keV with FWHM pattern is very similar to photoelectron forward focusing of ca. 25 keV. Its shape does not change essentially with patterns measured for the Si(111) surface [15]. Forward the inclination angle. The calculated N e was checked scattering patterns of low-energy electrons [16] and chan- by comparison with the similar spectrum measured with a neling patterns [17] of high-energy electrons are routinely crystal monochromator. The longitudinal coherence length used for simple and direct real-space determination of bond of such a beam is less than 1 Å. direction at surfaces, at interfaces, and in bulk crystals. De- Figure 3(a) shows the recorded pattern of the total ab- spite the different scattering mechanism, the x-ray pattern sorption measured as a function of sample orientation rela- may be analyzed in the same way to provide structural in- tive to the x-ray beam direction. A simple, slowly varying formation in real space [18]. two-dimensional background subtraction and a threefold To demonstrate the sensitivity of the measured pattern symmetry operation were applied to the data. The rela- to structural parameters we performed calculations sepa- tive signal modulation is of the order of 4 3 1024. Note rately for atoms in clusters around the two nonequivalent that the modulation has the same order of magnitude as atomic sites in Si. Patterns calculated for detecting atoms the forward scattering amplitude on a nearest neighbor Si in positions Si(1) and Si(2) are shown in Figs. 4(a) and atom: fSi r Zr0 r 1.6 3 1024. This shows that the 4(b), respectively. Those patterns are less similar to the largest contributions to the pattern come from the local structure. Figure 3(b) shows the absorption pattern simulated for experimental conditions according to Eq. (2) and with the (a) (b) energy spectrum shown in Fig. 2(b). The calculation was done for two nonequivalent crystal sites: Si(1) in position 000 and Si(2) in position 1 1 1 of the unit cell within a 4 4 4 120 Å diameter cluster. The agreement with the measured pattern is very good. The reliability factor has a value of R 0.075 [14]. Some discrepancies in the fine struc- tures can result from the finite size of the cluster used in (c) (d) (a) (b) 4.0 (e) (f) -4.0 x10-4 FIG. 3. (a) The experimental pattern of total absorption yield Si(1) Si(2) caused by white illumination of the Si(111) crystal measured as the function of sample orientation. The linear gray scale is FIG. 4. (a),(b) Patterns of total absorption simulated separately proportional to the sample current (dark: low absorption; bright: for clusters around atomic detectors in the two nonequivalent high absorption). The observed modulation of the total signal positions Si(1) and Si(2) in the unit cell. (c),(d) Low-pass filtered is about 4 3 1024. (b) Pattern simulated for an atomic cluster patterns from (a) and (b). (e) Geometrical projection of atoms with radius of 60 Å around two nonequivalent atomic sites in in the 5 Å neighborhood of Si(1) and Si(2) sites. (f) Low-pass Si. Both patterns are shown in spherical projection. The center filtered image of the experimental pattern from Fig. 3(a) showing corresponds to the 111 direction and the edge to 48± away superimposed images of atoms above Si(1) and Si(2) atomic from it. sites. 2335 VOLUME 86, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 12 MARCH 2001 experimental data than is their superposition. The corre- this information is obtained in real space. This means sponding R factors are 0.12 and 0.17 which means nearly that the signal does not need to be measured on the whole 100% change with respect to the summed pattern. This hemisphere. If only particular directions are of interest the indicates the possibility of conventional R factor analysis data can be measured as one-dimensional angular scans. based on simple kinematical calculations. By changing the energy spectrum it will be possible to Because of the weak absorption of x rays, the measured move from space to frequency localization and to achieve pattern contains features coming from distant atoms. This more high-resolution information about bond lengths. contribution could cover the signal from the nearest atoms. We thank Dr. Dimitri Novikov for help during the To demonstrate that real-space local information can be ex- experiment and Dr. Yoshinori Nishino for valuable tracted, a low-pass filter was applied to the calculated data. discussions. The low-pass filter is not ideally suited for this purpose since the signal from an individual atom is neither distri- buted over the whole hemisphere nor is it a simple sine wave. 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