1
Imaging of Quantum Array Structures with Coherent and Partially Coherent Diffraction
I.A. Vartanyants, I.K. Robinson
Department of Physics, UIUC, Urbana-Champaign,USA
J. Synch. Rad. (2003), submitted
2
X-ray scattering on nanostructuresGISAXS measurements
J.Stangl et al. Material Science & Engineering, C19 (2002) 349
Qx (Å-1)
Qy (Å
-
1)
-0.03 0.0 0.03
0.03
-0.03
High-resolution AFM images of PbSe dots on PbEuTe showing the pyramidal island shape
3
Imaging of Quantum Dots with Coherent Beams
Source
CCD
Sample
Electron density of periodical array of QD’s
)(p)(s)S()p( rrrr z
Scattered amplitude
n
nnz )S()(s)(Acoh hqhq
Diffracted intensity 2n
2
n
nz2
)S()(s)(A)(I cohcoh hqhqq cross terms
S(r) – shape of coherently illuminated areasz(r) –projection of shape of one island s(r,z)
n
n )()(p rrr
4
2D array of QD’s and it’s diffraction pattern
2D array of QD’s
Image of individual island
Diffraction pattern of 2D array
Diffraction pattern of individual island
5
Reconstructed image of 2D array of QD’s
Support used for reconstruction
Reconstructed image
Diffraction intensity of reconstructed image
Reconstructed image withsuperposition of twin images
6
Imaging of QD’s with partially coherent beams
Intensity distribution with partially coherent illumination
)'(iin e)',(J)'p()p(')I( rrqrrrrrrq dd p(r) - electron density distribution,
Jin(r) - mutual intensity function
Complex coherence factor
2cohinin
inin
l
)'(exp
)'(I)(I
)',(J)'(
2
2rr
rr
rrrr lcoh – transverse
coherence length
Intensity distribution )()(A)()(I)I( incohincoh qqqqq 2
Diffracted intensity from 2D array of QD’s:
)()S()(s)I( inz qhqhq
2n
2
n
n cross terms
7
Tests of Partial Coherence Effects in Imaging
2D array used for tests of partial coherence effects
Shape of individual island
Diffraction pattern of 2D array
Diffraction pattern of individual island
A. Szöke Acta Cryst. (2001) A57, 586
8
Diffraction intensity from QD’s with partial coherent illumination
Complex Coherence Factorin(r)
Diffraction intensity distribution
lcoh= 124 px
lcoh= 64 px
lcoh= 31 px
lcoh=
9
Equatorial cross section of the diffraction patterns
-150 -100 -50 0 50 100 1501E-24
1E-22
1E-20
1E-18
1E-16
Coherent Illumination
Single dot
Array of dots
Inte
nsity
(ar
b. u
nits
)
q (pixels)-150 -100 -50 0 50 100 150
1E-8
1E-6
1E-4
0.01
1
100
10000
Partial Coherent Illumination
lcoh
=31 px
lcoh
=64 px
lcoh
=128 px
lcoh
=15 px
Inte
nsity
(ar
b. u
nits
)
q (pixels)
10
Iterative phase retrieval algorithm
sk(x) Ak(q)
Reciprocal Space Constraints
A'k(q)s'k(x)
Real Space Constraints
FFT
FFT-1
Real space constraints:•finite support•real, positive
Reciprocal space constraint:
)(I)(A expk qq
R.W.Gerchberg & W.O. Saxton, Optic (1972) 35, 237J.R. Fienup, Appl Opt. (1982). 21, 2758R.P. Millane & W.J. Stroud, J. Opt. Soc. Am. (1997) A14, 568
11
Reconstructed images with reduced coherence
Reconstructed image of central island
Reconstructed image of the whole array
12
Ge islands on prepatterned Si substrates
AFM micrographs of 1D (a) and 2D ordered Ge islands (b) with corresponding line scans
Zhenyang Zhong, A. Halilovic, F. Schäffler, G. BauerInstitut für Halbleiter-und Festkörperphysik, Johannes Kepler Universität Linz
13
Diffraction pattern around Ge (202) peak
Experiment on ID34 APS (June 2003)
Qx2.025 1.965
Qz
1.89
1.95
14
Diffraction pattern from Ge islandsGISAXS measurements
i> c i< c
Qy Qy
f
i
c
15
Conclusions
• Diffraction pattern from 2D array of QD’s can be inverted to give the correct shape and orientation of individual island
• For successful reconstruction experimental diffraction pattern has to be measured up to Q~2/a (a - average size of an island)
• Test calculations for very small transverse coherence lengths lcoh of the incoming beam were made
• Correct shape of the individual island can be obtained when lcoh~a
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