Application of iterative phase-retrieval algorithms to ARPES orbital tomography
Pavel Kliuiev, Tatiana Latychevskaia, Juerg Osterwalder, Matthias Hengsberger, Luca Castiglioni
aa r X i v : . [ c ond - m a t . o t h e r] S e p Application of iterative phase-retrieval algorithmsto ARPES orbital tomography
P Kliuiev, T Latychevskaia, J Osterwalder, M Hengsbergerand L Castiglioni
Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich,SwitzerlandE-mail: [email protected], [email protected]
June 2016
Abstract.
Electronic wave functions of planar molecules can be reconstructed viainverse Fourier transform of angle-resolved photoelectron spectroscopy (ARPES) data,provided the phase of the electron wave in the detector plane is known. Since therecorded intensity is proportional to the absolute square of the Fourier transform ofthe initial state wave function, information about the phase distribution is lost inthe measurement. It was shown that the phase can be retrieved in some cases byiterative algorithms using a priori information about the object such as its size andsymmetry. We suggest a more generalized and robust approach for the reconstructionof molecular orbitals based on state-of-the-art phase-retrieval algorithms currently usedin coherent diffraction imaging. We draw an analogy between the phase problem inmolecular orbital imaging by ARPES and of that in optical coherent diffraction imagingby performing an optical analogue experiment on micrometer-sized structures. Wesuccessfully reconstruct amplitude and phase of both the micrometer-sized objects anda molecular orbital from the optical and photoelectron far-field intensity distributions,respectively, without any prior information about the shape of the objects.
Keywords : phase retrieval, ARPES, orbital tomography, molecular orbital
1. Introduction
Organic semiconductors play a key role in modern devices such as organic light-emittingdiodes and photovoltaic cells [1, 2]. More recently, organic molecules have been usedas catalysts in photolytic water splitting, a promising route towards production ofhydrogen as renewable energy source [3]. Tailoring the physical properties of molecularoptoelectronic devices [4, 5, 6] crucially depends on a deep understanding of the chargetransfer mechanisms at metal-organic interfaces. The time-resolved spatial visualizationof such processes would hence be highly desirable.The frontier orbitals, i. e. the highest occupied (HOMO) and lowest unoccupied(LUMO) molecular orbitals, largely determine the chemical reactivity and electronic pplication of iterative phase-retrieval algorithms to ARPES orbital tomography I ( k f k , E kin ) is derived from Fermi’s golden rule as I ( k f k , E kin ) ∝ X i |h ψ f ( k f k , E kin , r ) (cid:12)(cid:12)(cid:12) A · p (cid:12)(cid:12)(cid:12) ψ i ( k i k , r ) E(cid:12)(cid:12)(cid:12) × δ ( E kin + Φ + E i − ¯ hω ) × δ ( k f k − k i k − G k ) , (1)where ψ i and ψ f denote initial and final state wave functions with correspondingmomentum components k i k and k f k parallel to the surface, respectively. The deltafunctions in the second line comprising photon energy ¯ hω , sample work functionΦ and reciprocal lattice vector G k ensure energy and momentum conservation inthe photoemission process. The transition matrix element is given in the dipoleapproximation, where p and A denote the momentum operator and the vector potentialof the exciting light. The photocurrent I ( k f k , E kin ) is obtained by summation over alltransitions from occupied initial states ψ i to the final state ψ f characterised by thekinetic energy E kin and the parallel component of the final state momentum k f k of thephotoelectron. The photoemission final state ψ f can be approximated by a plane wave ∝ e i k f r provided the following conditions are fulfilled [7, 8, 13]: (i) photoelectrons areemitted from π -orbitals of large planar molecules, for which all the contributing orbitalsare of the same p z character; (ii) the molecules consist of mainly light atoms (H, C, N,O) and final state scattering effects can thus be neglected. Under these assumptions,the measured ARPES intensity becomes proportional to the squared modulus of theFourier transform of the initial state wave function ψ i weakly modulated by a slowlyvarying angle-dependent envelope function [7, 8]: I ( k f k , E kin ) ∝ | A · p | |F { ψ i ( k i k , r ) }| (2)The recorded intensity pattern, however, does not contain any information aboutthe phase of the complex-valued electron wave distribution in the detector plane, whichinhibits the direct reconstruction of the molecular wave function via computation of aninverse Fourier transform. In certain cases, phase information can be inferred from theparity of the wave function [7] or from dichroism measurements [11] and be imposed ontothe measured data. However, the reconstruction of the molecular wave functions in sucha way is not applicable to the most general type of problems when the phase distributioncannot be deduced from symmetry considerations. This issue was addressed by L¨uftneret al. [10] by suggesting an iterative phase retrieval procedure similar to the Fienupalgorithm [14]. In the suggested procedure, one iterates back and forth between realand reciprocal spaces by computing Fourier transforms and satisfying the constraints inboth domains. In real space, the wave function is confined to a rectangular box whichroughly corresponds to the van der Waals size of the molecule and thus represents thesupport of the object. The absolute value of the wave function is reduced to 10% outside pplication of iterative phase-retrieval algorithms to ARPES orbital tomography
2. Methods
The microstructures for the optical CDI experiments were patterned in a 105 nm-thickCr film deposited on a 1.7 mm-thick fused silica substrate, thus providing transparentobjects in a non-transparent medium. The individual microstructures had an identicalshape but different sizes and were separated from one another by several millimeters toavoid interference between the neighbouring objects. The size of the microstructureswas selected in such way that the ratio between microstructure length (e.g., 15 µ m) andemployed laser wavelength (0 . µ m) was comparable to the ratio between length ofpentacene molecule ( ≈ . ≈ .
17 nm)at the used photon energy (50 eV). The experimental setup for optical CDI is shown inFig. 1. The laser beam profile had a Gaussian distribution as shown in the inset. ForCDI experiments, the laser beam is usually spatially filtered and then expanded usingtwo lenses, which ensures that the intensity profile in the object plane is constant [20].In our experiment, we employed the laser beam without expansion because the light pplication of iterative phase-retrieval algorithms to ARPES orbital tomography
532 nm CW laser Neutraldensity filterSample Screen CCD
Figure 1.
Experimental setup of optical CDI. The distance between sample and screenwas set to 22.5 cm. The size of the imaged screen area comprised 40 ×
40 cm sampledwith 1000 × A well-ordered sub-monolayer of pentacene molecules adsorbed on Ag(110) served asmodel system for orbital tomography. Pentacene ARPES data has been acquired duringa beamtime of A. Sch¨oll and coworkers (University of W¨urzburg) at the NanoESCAbeamline at Elettra synchrotron (Trieste, Italy) and has been provided to us forvalidation of our phase retrieval algorithm [22]. The crystal was prepared accordingto standard procedures [23] and pentacene molecules [7] were deposited from a home-built Knudsen cell [11]. ARPES constant binding energy (CBE) momentum maps of thepentacene LUMO were recorded with the p-polarized light at a photon energy of 50 eVusing the photoemission electron microscope (PEEM) [24, 25]. The setup of the PEEMand the experimental geometry are shown in Fig. 2(a) and Fig. 2(b), respectively. Themicroscope was operated in the momentum mode and allowed for detection of electronswith the acceptance angle of α = ± corresponding to slightly less than ± − at50 eV photon energy without any sample rotation. The CBE map was integrated overa 200 meV energy window, which is of the order of the electron analyzer resolution andof the full-width at half-maximum of the pentacene LUMO at the binding energy of0 . pplication of iterative phase-retrieval algorithms to ARPES orbital tomography Figure 2. (a) Schematic of the PEEM setup. (b) Experimental geometry. The photonenergy was 50 eV and the light was p-polarized with an incidence angle of 65 . Thephotoemitted electrons were collected by the PEEM objective lens with an acceptanceangle of α = ± . k f k and k f ⊥ denote parallel and normal components of the finalstate momentum of the photoelectrons. Prior to reconstruction of the pentacene LUMO, we tested the performance of thealgorithms on the optical CDI data set, taking advantage of the high dynamic rangeof these data. We employed a combination of the phase-constrained [18] hybrid input-output [14] (PC-HIO) and error reduction [14] (ER) algorithms. The usage of bothalgorithms in an alternating scheme has been shown to eliminate stagnation problemsand to provide faster convergence [14, 18, 26]. The support of the object was found usingthe shrinkwrap algorithm [19]. Following the conventional procedure of this algorithm,the initial estimate of the object support was obtained from the autocorrelation ofthe object by computing the inverse Fourier transform of the experimental diffractionpattern I ( X, Y ), convolving it with a Gaussian function (width σ = 5 pixels) andapplying a threshold at 10% of its maximum. The pixel values below the threshold werezeroed. The reconstruction began with 40 iterations of the PC-HIO algorithm followedby 2 iterations of the ER algorithm. We found that this number of iterations is sufficientto yield a resonable estimate of the object shape and thus to perform the first updateof the object support by using the shrinkwrap procedure [19] described in detail below.The scheme of the iterative phase retrieval procedure is shown in Fig. 3, which includedthe following steps:(i) In the first iteration k = 1, the experimental amplitude | F ( X, Y ) | = q I ( X, Y ) wascombined with a random phase and the inverse Fourier transform supplied an initialinput object distribution g k ( x, y ), where ( X, Y ) and ( x, y ) denote the coordinatesin the detector and object planes, respectively. We assume the most general case ofa complex-valued object distribution and keep both its real and imaginary parts.(ii) By computing the Fourier transform of g k ( x, y ), we obtain the complex-valueddistribution G k ( X, Y ) =
F { g k ( x, y ) } . pplication of iterative phase-retrieval algorithms to ARPES orbital tomography Fourier domain constraint: k k k Initial estimate Hybrid Input-Output,Error Reduction constraints k k k+1 k+1 k+1 k k k k Amplitude Phase
Amplitude Phase
Amplitude Phase
Amplitude Phase (i)(ii) (iii) (iv)(v)
Figure 3.
Iterative phase retrieval scheme. (iii) By replacing the calculated amplitude | G k ( X, Y ) | with the experimental amplitude | F ( X, Y ) | , while keeping the calculated phase distribution, we obtain an updatedcomplex-valued field distribution in the detector plane G ′ k ( X, Y ).(iv) Inverse Fourier transform of G ′ k ( X, Y ) provides the output object distribution g ′ k ( x, y ).(v) In the PC-HIO algorithm [14, 18], the input object for the next iteration g k +1 ( x, y )is obtained as g k +1 ( x, y ) = ( g ′ k ( x, y ) , if ( x, y ) ∈ γ,g k ( x, y ) − βg ′ k ( x, y ) , if ( x, y ) / ∈ γ, (3)where β = 0 . γ corresponds to a set of points whichcomply with the object domain constraints (belong to the support region and havetheir phases within an expected range). In the ER algorithm [14], the objectdistribution g k +1 ( x, y ) is calculated as g k +1 ( x, y ) = ( g ′ k ( x, y ) , if ( x, y ) ∈ γ, , if ( x, y ) / ∈ γ, (4)where γ fulfills the same criteria as in the PC-HIO algorithm.The output object distribution g ′ k ( x, y ) obtained in the last iteration of the ERcycle was used to update the object support. This was done by convolving g ′ k ( x, y ) witha Gaussian function and setting a threshold at 12% of its maximum, as it is typicallydone in the shrinkwrap algorithm [19]. The width of the Gaussian was initially set pplication of iterative phase-retrieval algorithms to ARPES orbital tomography E = vuut P N − X,Y =0 || F ( X, Y ) | − | G it ( X, Y ) || P N − X,Y =0 | F ( X, Y ) | , (5)where | F ( X, Y ) | is the experimental amplitude, | G it ( X, Y ) | is the iteratively obtainedamplitude. The solution of the phase problem requires the fulfillment of the oversamplingcondition [16, 17]. Given an N × N pixel sampled amplitude | F ( X, Y ) | = | P N − X,Y =0 f ( x, y ) e − πi ( xX + yY ) /N | in reciprocal space, we obtain a set of N equations,which have to be solved in order to find both the amplitude and phase of f ( x, y ). Miaoet al. [16] defined the oversampling ratio as σ = N total N unknown , (6)where N total is the total number of pixels and N unknown is the number of pixels withunknown values. The set of equations is solved by dense sampling of the diffractionpattern so that the object distribution is surrounded by a zero-padded region with σ > N ∆ ra , (7)where N is the linear number of pixels, ∆ r is the size of the pixel in the objectdomain and a is the largest extent of the object. The oversampling requirement thencorresponds to Ø > √
3. Results and discussion
Fig. 4 shows the results of the reconstruction of the micrometer-sized structures. Inoptical CDI, we employed micrometer-sized structures of 30 × µ m (sample 1) and pplication of iterative phase-retrieval algorithms to ARPES orbital tomography . × µ m (sample 2). The scanning electron microscope (SEM) images of samples 1and 2 are shown in Fig. 4(a,b) next to the experimental diffraction patterns (Fig. 4(c,d)).The size of the diffraction patterns sampled with 1000 × ×
40 cm ineach case, thus giving the size of the pixel in the detector plane ∆ p = 400 µ m. Thesize of the pixel in the object plane ∆ r can be related to the distance z = 22 . λ = 532 nm [29].The linear oversampling ratio defined by Eq. 7 can be rewritten as [16]:Ø = zλa ∆ p (8)For the samples 1 and 2 with the lengths a = 30 µ m and a = 15 µ m, thelinear oversampling ratios fulfilled the oversampling condition and were Ø ≈
10 andØ ≈ .
2, respectively. μ m (e) ((cid:0)(cid:1)(cid:2)(cid:3)(cid:4) μ m (c) (d) A m p li t u(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) (b) Int. (log. scale)8.74.0Int. (log. scale) (g) μ m (h) π P ha s e (r ad ) π P ha s e (r ad ) A m p li t ude ( a . u . ) Sample 1Sample 2 (a) μ m5 μ m 10 μ m5 μ m Figure 4.
Reconstruction of the micrometer-sized objects. (a,b) SEM images.(c,d) Experimental diffraction pattern intensities shown on logarithmic scale. (e,g)Reconstructed amplitudes. (f,h) Reconstructed phases.
Prior to application of the phase retrieval algorithms, the experimental diffractionpatterns were pre-processed: First, each of the recorded 1000 × ×
500 pixels around their centers because of the low signal-to-noise ratio pplication of iterative phase-retrieval algorithms to ARPES orbital tomography . · counts (sample 1) and 6 · counts (sample 2) were defined as missing and their values were updated in the course ofthe reconstruction by using the corresponding pixel values of the calculated amplitudesin the detector plane [19, 31]. In each case, the square root of the resulting diffractionpattern was fed into the algorithm. We found that 10 alternating cycles of the PC-HIO and ER algorithms, each followed by an update of the support, were enough toachieve a stable reconstruction. Further increase in the number of the reconstructioncycles was not necessary since it did not improve the quality of the reconstructed objectdistribution. At the end of 10 cycles, each reconstruction was stabilized by 100 iterationsof the ER algorithm [31]. In total, we performed 1000 independent reconstructions byemploying a random phase distribution for each reconstruction run. Eventually, the50 reconstructions with the smallest error E as defined by Eq. 5 were selected andaveraged [31] and are shown in Fig. 4(e-h). The reconstructed amplitudes correctlyreproduce the shape and dimension of the microstructures. Furthermore, as it wasexpected for a purely transmitting object illuminated by a Gaussian beam with analmost planar wavefront at the object site, the phase distributions turned out to bealmost constant. The lower quality of the reconstructed amplitude of sample 2 (Fig. 4(g)) can be attributed to the low signal-to-noise ratio in the respective diffractionpattern. We then applied the same algorithm to the ARPES data. Fig. 5 shows the results ofthe reconstruction of the pentacene LUMO. The experimental CBE map is shown inFig. 5(a). Given the resolution in reciprocal space of ∆ k ≈ .
01 ˚A − and the length of thepentacene molecule a ≈
15 ˚A, the linear oversampling ratio in the ARPES experimentcan be calculated using Eq. 7. Taking the relation ∆ r ∆ k = πN between the pixel size inobject space, ∆ r , and reciprocal space, ∆ k , into account, the linear oversampling ratiocan be expressed asØ = 2 πa ∆ k . (9)The linear oversampling ratio was Ø ≈
42 and thus fulfilled the oversamplingcondition [16]. The experimental CBE map was pre-processed following similar stepsas those applied to the reconstruction of the micrometer-sized objects: First, the imagewas centered and the quasi-constant noise of the CCD camera (average count rate of50 counts) was subtracted from each pixel. To ensure a sufficient number of pixelsallocated per unit length of the molecule, we zero-padded the experimental CBE mapto 2000 × pplication of iterative phase-retrieval algorithms to ARPES orbital tomography g ( x, y ) were reconstructed together with their conjugate g ∗ ( − x, − y ) or twin images [28].The identification of the twin images could be automated by a procedure proposed byFienup [28], but here they were easily identified by visual inspection and discarded. Fromthe remaining reconstructions, 50 with the smallest error E as defined by Eq. 5 wereselected and averaged. The reconstructed amplitude and phase of the pentacene LUMOare shown in Fig. 5 (b-c) together with the overlayed carbon frame of the molecule forcomparison. (c)(b)(a)(d) (e) (cid:13)(cid:14)(cid:15) Phase (rad)Amplit (cid:16)(cid:17)(cid:18) (cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)m(cid:25)(cid:26) min (cid:27) S(cid:28)(cid:29)(cid:30) e t r i z ed i n t en s i(cid:31) U!" d i n t en s i t-
10 10
Figure 5.
Reconstruction of the pentacene LUMO. (a) CBE map recorded withPEEM from a sub-monolayer of pentacene on Ag(110) at 50 eV photon energy. (b)Reconstructed amplitude of the LUMO. (c) Reconstructed phase. Image transparencyis weighted with the corresponding amplitude values for illustration purposes. (d) Thesame CBE map as in (a), but symmetrized with respect to the center. Reconstructionsof (e) amplitude and (f) phase obtained from (d).
It should be noted that we did not perform a normalization of the ARPES intensityby the angle-dependent factor | A · p | nor did we enforce any symmetry constraints inthe course of the reconstruction onto the amplitude and phase shown in Fig. 5 (b-c).The object distribution was let to freely evolve until the stable solution was reached,which makes the utilized algorithm independent of any symmetry properties imposedonto the object under reconstruction. Furthermore, we note that the recorded CBEmap shown in Fig. 5(a) contains features coming from the Ag(110) substrate (mostlyat high momenta), but they do not seem to have a profound effect on the results ofthe reconstruction. By comparing our results with the literature, we find that thephase distribution weighed with the correspondent amplitude values as well as theshape of the orbital correctly reproduce the DFT calculations [10, 32] as well as thedata reconstructed by L¨uftner et. al [10]. pplication of iterative phase-retrieval algorithms to ARPES orbital tomography
4. Summary and Conclusion
In this work, we show that the state-of-the-art phase retrieval algorithms currentlyemployed in CDI can be successfully used for the reconstruction of complex-valued wavefunctions of molecules adsorbed on single-crystalline substrates. We tested and appliedthese algorithms in an optical analogue experiment and then successfully applied themto the reconstruction of the LUMO of pentacene adsorbed on Ag(110). The advantageof using modern CDI algorithms and in particular the shrinkwrap algorithm for thereconstruction of molecular orbitals is that they do not require any a priori informationabout the shape of the object. Instead, they smoothly converge to the correct shape ofthe object in the course of the reconstruction. In case of molecular wave functions, this ishighly important, since precise estimation of the object support is difficult and cannot beguaranteed in every case. This applies, for instance, if the orbital tomography techniqueaims at visualizing chemical reactions or following the dynamics of excited states, whereeffective electronic wave functions are unknown. The availability of a general and robustreconstruction algorithm is thus an important step for further advancement of orbitaltomography.
Acknowledgments
Financial support by the Swiss National Science Foundation through NCCR MUSTis greatefully acknowledged. We thank Achim Sch¨oll and co-workers (University ofW¨urzburg) for making the pentacene ARPES data available to us. The Center for pplication of iterative phase-retrieval algorithms to ARPES orbital tomography
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