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

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Reconstructed images with reduced coherence

Reconstructed image of central island

Reconstructed image of the whole array

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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

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Diffraction pattern around Ge (202) peak

Experiment on ID34 APS (June 2003)

Qx2.025 1.965

Qz

1.89

1.95

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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