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Nonlinear ellipsometry of Si(111) by second harmonic generation
Accepted Manuscript Title: Nonlinear ellipsometry of Si(111) by second harmonic generation Author: Cornelia Reitb¨ock David Stifter Adalberto Alejo-Molina Hendradi Hardhienata Kurt Hingerl PII: DOI: Reference:
S0169-4332(16)32247-4 http://dx.doi.org/doi:10.1016/j.apsusc.2016.10.131 APSUSC 34225
To appear in:
APSUSC
Received date: Revised date: Accepted date:
31-7-2016 20-10-2016 20-10-2016
Please cite this article as: Cornelia Reitbddotock, David Stifter, Adalberto AlejoMolina, Hendradi Hardhienata, Kurt Hingerl, Nonlinear ellipsometry of Si(111) by second harmonic generation, (2016), http://dx.doi.org/10.1016/j.apsusc.2016.10.131 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ip t
Nonlinear ellipsometry of Si(111) by second harmonic generation Cornelia Reitb¨ ocka,b,∗, David Stiftera,b , Adalberto Alejo-Molinac , Hendradi Hardhienatad , Kurt Hingerlb a Christian
an
us
cr
Doppler Laboratory for Microscopic and Spectroscopic Material Characterization, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria b Center for Surface- and Nanoanalytics, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria c CONACYT Center for Research in Engineering and Applied Science (CIICAp), Institute for Research in Pure and Applied Science (IICBA), UAEM Cuernavaca, Mor. 62209, Mexico d Theoretical Physics Division, Department of Physics, Bogor Agricultural University, Jl. Meranti, Gedung Wing S, Kampus IPB Darmaga, Bogor 16680, Jawa Barat, Indonesia
M
Abstract
In this paper we present experimental and simulated data of the second harmonic response for arbitrarily oriented linear input polarization and output
d
polarization on Si(111) surfaces. In contrast to other nonlinear optical ellip-
te
sometry experiments (e.g. nonlinear optical null ellipsometry) we azimuthally rotate the sample for additional analysis of the crystal structure. In our study,
Ac ce p
we used the simplified bond hyperpolarizability model (SBHM) to simulate the observed angular shifts of the nonlinear peaks and the change of symmetry features related to a modification of the beam polarizations. Our findings help to identify the corresponding interface dipolar and bulk quadrupolar SHG sources. Keywords: Second harmonic generation, Si(111), nonlinear ellipsometry, simplified bond hyperpolarizability model
1. Introduction
Second harmonic generation (SHG) based techniques have been proven to be powerful tools for studying surface and interface properties on an atomic scale. ∗ Corresponding
author
Preprint submitted to Journal of LATEX Templates
October 20, 2016
Page 1 of 13
[1, 2, 3, 4, 5, 6, 7] In a typical SHG experiment a laser beam at frequency ω 5
and with a polarization perpendicular (s-polarized) or parallel (p-polarized) to
ip t
the plane of incidence impinges on a nonlinear medium. Simultaneously, two
photons of frequency ω are annihilated and a photon with the second harmonic
cr
frequency (2ω) is created. Commonly, in SHG experiments the sample is az-
imuthally rotated in order to study the rotational anisotropy dependence of the sample (RA-SHG).
us
10
In electric dipole approximation the induced second order polarization is ~ ~ given by P~ (2) (ω) = χ(2) E(ω) E(ω) where χ(2) is the second order nonlinear
an
susceptiblity, a tensor of rank 3 comprising material properties. For materials with a center of inversion (i.e. being centro-symmetric) such as Si χ(2) vanishes 15
in the bulk as long as dipole approximation holds [8]. However, at an interface
M
inversion symmetry is broken, thus allowing dipolar contributions. In 2002 Powell et al. introduced the simplified bond hyperpolarizability model (SBHM) [9] modelling the SHG signals in terms of a far-field intensity that is
ing along the bond axis. Applying this microscopic approach enabled fitting
te
20
d
calculated as the square of the superposition of fields radiated by charges mov-
of RA-SHG data using simpler mathematics than previous phenomenological
Ac ce p
models [10]. Even though already offering a better understanding of the underlying physics, the model was further developed taking into account additional nonlinear contributions like the electric quadrupole contribution [11].
25
Starting in the early 2000s the field of nonlinear ellipsometry has emerged as
a novel approach to evaluate sign and phase information between the different nonzero χ(2) tensor elements of a given sample [12, 13]. In its simplest form a polarized laser beam at frequency ω impinges on the sample and the polarized
reflected beam at 2ω is detected. In literature the term ”Second harmonic ellip-
30
sometry” exists [14] but contrary to RA-SHG the sample is at a fixed position neglecting additional information about the structure of the sample. In this work we present experimental and simulated SHG response of Si(111) surfaces using - besides typical polarization states of e.g. s-polarized input, ppolarized output - arbitrarily linear polarized input beams and investigating the 2
Page 2 of 13
35
output polarization by stepwise rotating the polarizer at the output. Additionally to this nonlinear ellipsometer configuration our sample is rotated. Thus,
ip t
our experiments are performed in a rotating polarizer, azimuthally rotating sam-
ple, rotating analyzer (PR SAR AR ) configuration. In contrast to other nonlinear
40
cr
optical ellipsometry experiments (e.g. nonlinear optical null ellipsometry [12])
the additional azimuthal rotation of the sample allows analysis of the crystal
us
structure.
2. Material and methods
an
Si(111) wafers with a native oxide and a doping level below the critical value for influencing the SHG signal [15] were chosen as samples. We performed the polarization sensitive RA-SHG measurements based on
M
45
a typical RA-SHG system in reflection geometry as described in Reitb¨ock et al. [16]. Succinctly, the fundamental beam is generated by a picosecond laser
d
operating at 50 Hz and the SHG signal is detected after spectral filtering by a monochromator using a photomultiplier tube. Recent modifications of the above mentioned setup allow changing the po-
te
50
larization state of the fundamental as well as the frequency doubled beam not
Ac ce p
only from s-polarization to p-polarization and vice versa (typical for RA-SHG experiments), but in a continuous manner. All experiments are performed using the instrumentation schematically de-
55
picted in fig. 1, in which the half-wave plate (HWP) and the analyzer are rotated for polarization sensitive measurements. The Glan-laser prism (P) prior to the HWP ensures distinct s-polarized light.
3. Theory
As outlined in the introduction, the SBHM is widely used for the simulation 60
of p- or s-polarized SHG data. To the best of our knowledge, we show for the first time how the SBHM can be used to simulate RA-SHG data with arbitrarily linear input and output polarization. 3
Page 3 of 13
ip t cr us an M
Figure 1: Top view of setup for polarization sensitive rotational second harmonic generation
d
experiments; tunable laser system, mirror (M), beamsplitter (BS), photodetector, filter (F), lens (L), Glan-laser prism (P), rotatable half-wave plate (HWP), rotatable analyzer (A),
te
monochromator, photomultiplier tube (PMT).
Ac ce p
If we overcome dipole approximation, the induced polarization of a cen-
trosymmetric non-magnetic material such as silicon comprises two dominant
65
SHG sources: dipolar SHG at the SiO2 -Si interface and a quadrupolar contribution from within the bulk. In SBHM the two contributions are modelled corresponding to eqn. 1 [9, 17, 18] and eqn. 2 [19, 17]. (2),interface
PD
(2),bulk
PQ
(2)
= χD · ·Ein Ein n P (2) ˆ ˆ ˆ = V1 αj b b b j j j · ·Ein Ein
(1)
j=1
(2)
= χQ · · · Ein ∇b Ein n (2) P ˆ ˆ ˆ ˆ = V1 αb bj bj bj bj · · · Ein ∇b Ein
(2)
j=1
(2)
where for the polarization produced by dipolar interfacial sources χD is the
4
Page 4 of 13
third rank susceptibility tensor, Ein is the incoming field, V is the volume, n is the total number of bonds inside the considered fundamental tetrahedral el(2)
ement, αj
ˆ j is the bond is the interface dipolar SHG hyperpolarizability and b
ip t
70
(2)
unit vector. For the polarization due to bulk quadrupolar sources χQ is the (2)
αb 75
cr
fourth rank susceptibility tensor, ∇b is the gradient of the incoming field and
is the bulk quadrupolar SHG hyperpolarizability. Regarding the hyper-
polarizabilities we assign in the interface simulation an up hyperpolarizability
us
ˆ 1 ) and a hyperpolabelled as αu for the bond reaching up to the surface (b ˆ2 , b ˆ3 , b ˆ 4 ) taking into account the larizability αd for the three down bonds (b
an
tetrahedral molecular geometry of Si(111) (see fig. 2). In the simulation for the bulk contribution we only use one bulk hyperpolarizability αb for all of the bulk bonds because of symmetry.
Ac ce p
te
d
M
80
Figure 2: Coordinate system and bond orientation of the effective Si (111) fundamental tetrahedral element.
Furthermore the far field is calculated for each contribution thoroughly tak-
ing care about the corresponding Fresnel factors and the correct projection of the fields. The corresponding optical constants of Si and SiO2 (obtained from ellipsometry measurements) are presented in tab. 1. In a last step the intensi-
85
ties of each contribution are calculated for comparing them qualitatively with the experiment.
5
Page 5 of 13
Wavelength [nm]
Energy [eV]
n
k
Si
800
1.55
3.68
0.004
400
3.10
5.63
0.28
890
1.39
3.63
0.002
445
2.78
4.75
0.08
800
1.55
1.46
400
3.10
1.48
890
1.39
1.46
445
2.78
1.47
an
4. Results and discussion
us
SiO2
cr
Material
ip t
Table 1: Optical constants of Si and SiO2 at different wavelengths.
Depending on the polarization of the beam, 3-fold (e.g. sp, blue in plot
90
M
3 a)) or 6-fold (e.g. ps, black in plot 3 d)) patterns are measured on Si(111) surfaces in RA-SHG experiments. In order to investigate the origin of this SHG response, we show, using the SBHM that the observed angular shifts of the
d
nonlinear peaks and the symmetry features help to identify the corresponding
te
interface dipolar, respectively bulk quadrupolar SHG sources. Experimental data are presented here for 16 different polarization states, 95
namely s-,30◦ -,60◦ - and p-polarization combinations. A polarization of e.g. s30
Ac ce p
indicates a s-polarized (half-wave plate at 0◦ position) input beam and a 30◦ polarized (analyzer is rotated 30◦ clockwise, looking into (or against) the beam
direction) SHG output beam. Figure 3 shows experimental (symbols) and simulated data (solid line) at a
100
SHG wavelength of 400 nm sorted in such a way, that the incoming polarization of all 4 graphs in a plot is the same. The data are normalized and vertical offsets are introduced between the curves for clarity. The simulation using simply the interface dipole source of the SBHM rep-
resents in good agreement, although not perfect for some polarizations, the 105
angular shift of the peaks that are related to changes in the polarization. Especially, the 3-fold and the 6-fold symmetries are displayed in the simulation reflecting the trend of the experiment. Remarkably, this means that SBHM can 6
Page 6 of 13
ip t cr us an M d
Figure 3: Experimental (symbols) and simulated (solid line) data for 16 different polarization
shifted for clarity.
te
states measured on a Si(111) surface at a SHG wavelength of 400 nm. Plots are vertically
Ac ce p
simulate experimental data requiring only two (in general complex) independent parameters namely αu and αd for a fundamental beam of 800 nm.
110
In the following, investigations performed at an input wavelength of 890 nm
are presented. Figure 4 shows experimental and simulated data at four different polarization combinations, namely pp (blue), 60p (green), 30p (red) and sp (black). The data are normalized and for better visualization the plots are shifted vertically.
115
In a first step, the same procedure as before (fundamental beam at 800 nm)
was applied in the simulation for the 890 nm experiment, i.e. using only a surface dipole contribution, see fig. 4a). Comparing the simulated data of fig. 4a) with the experimental data depicted in fig. 4c) (dotted) shows that the angular shift of a peak (e.g. large peak moves from 60◦ in pp to small peak at
7
Page 7 of 13
120
120◦ in sp) is conserved in both plots. But, in contrast to the investigations at 800 nm, the symmetry features of the simulation and the experiment are not
ip t
in agreement. This is demonstrated explicitly in the 30p (red) graph where the relation between the large and the smaller peak is even inverted. Additionally,
125
cr
the 30p is rather 6-fold in the simulation of fig. 4a), whereas the experiment displays a rather 3-fold symmetry.
us
In order to overcome this discrepancy between the simulation with solely surface dipole contributions and the experiment, a simulation based on pure bulk quadrupole contributions was considered (see fig. 4b)). Again the angular
130
an
shifts are present, but instead of shifting 60◦ , the peaks are shifted by -120◦ , e.g. the large peak in pp (blue) at 240◦ evolves into the small peak at 120◦ . Furthermore, the simulation with solely bulk quadrupole contributions cannot
M
reveal the 6-fold symmetry of the peaks that is measured in sp polarization combination.
In a next step we combined the surface dipole and the bulk quadrupole contribution by coherently summing up the far-field radiations and multiplying the
d
135
te
sum with its complex conjugate (see fig. 4c) (full line). It is apparent that only by taking such a combination of dipolar surface and quadrupolar bulk contri-
Ac ce p
bution the same high-to-low peak ratio as in the experiment can be obtained in the simulation. Remarkably, the SBHM (containing both contributions) reveals
140
an identical angular shift of the peaks and moreover, a correct intensity signature by using only four independent parameters, two for the surface (αu and
αd = αb ) and two for the bulk (the complex gradient ∇b ). A table containing the simulation parameters is shown in fig. 4.
5. Conclusion
145
We have successfully demonstrated that our RA-SHG experiment in rotating polarizer, azimuthally rotating sample, rotating analyzer configuration is able to distinguish dipolar surface and quadrupolar bulk contributions. The findings are based on simulations using the SBHM model which reliably repro-
8
Page 8 of 13
ip t cr us an M d
te
Figure 4: Surface dipole a) and bulk quadrupole b) contributions are coherently added to simulate (full line in c)) experimental data (dotted in c)) at a fundamental wavelength of 890
Ac ce p
nm. Four different polarization states are presented: pp (blue), 60p (green), 30p (red) and sp (black). Fitting parameters are shown in the table. Plots are vertically shifted for clarity.
duces RA-SHG data of a Si(111) facet. Experimental and simulated data are in
150
good agreement for arbitrarily linear polarized fundamental and SHG signals, as well as for measurements using different wavelengths (i.e. having different penetration depths). Our results are encouraging and further work will focus on investigations at different wavelengths.
6. Acknowledgements 155
The financial support by the Austrian Federal Ministry of Science, Research and Economy and the Austrian National Foundation for Research, Technology
9
Page 9 of 13
and Development is gratefully acknowledged. K.H. acknowledges support of the
ip t
European Commission under the H2020 grant TWINFUSYON (GA692034).
References
[1] Y. Shen, Optical second harmonic generation at interfaces, Annual re-
cr
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view of physical chemistry 40 (1) (1989) 327–350. doi:10.1146/annurev.
us
physchem.40.1.327.
[2] Y. R. Shen, Surface studies by optical second harmonic generation: An
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h-terminated (111)si by using the carrier concentration and the strain: A second-harmonic-generation investigation, Journal of the Korean Physical
[4] I. Kravetsky, L. Kulyuk, J. McGilp, M. Cavanagh, S. Chandola, J. Boness,
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G. Marowsky, F. Harbsmeier, A study of ion-implanted si(111) and si(111)/silicon oxide by optical second harmonic generation, Surface Sci-
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ence 402-404 (1998) 542–546. doi:10.1016/S0039-6028(97)00982-5.
[5] S. A. Mitchell, R. Boukherroub, S. Anderson, Second harmonic generation
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at chemically modified si(111) surfaces, The Journal of Physical Chemistry B 104 (32) (2000) 7668–7676. doi:10.1021/jp000450d.
[6] T. F. Heinz, C. K. Chen, D. Ricard, Y. R. Shen, Spectroscopy of molecular monolayers by resonant second-harmonic generation, Physical Review Letters 48 (7) (1982) 478–481. doi:10.1103/PhysRevLett.48.478.
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[7] C. K. Chen, A. R. B. de Castro, Y. R. Shen, Surface-enhanced secondharmonic generation, Physical Review Letters 46 (2) (1981) 145–148. doi: 10.1103/PhysRevLett.46.145.
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[8] Robert W. Boyd, Nonlinear Optics, third edition Edition, Academic Press,
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[9] G.
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Powell,
J.-F.
Wang,
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ip t
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hyperpolarizability model of second harmonic generation,
bond-
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cr
Review B 65 (2002) 205320. doi:10.1103/PhysRevB.65.205320.
[10] G. L¨ upke, D. J. Bottomley, H. M. van Driel, Second- and third-harmonic
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America B 11 (1) (1994) 33. doi:10.1364/JOSAB.11.000033.
[11] H. J. Peng, E. J. Adles, J.-F. T. Wang, D. E. Aspnes, Relative bulk and interface contributions to optical second-harmonic generation in silicon,
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M
Physical Review B 72 (2005) 205203. doi:10.1103/PhysRevB.72.205203. [12] R. M. Plocinik, G. J. Simpson, Polarization characterization in surface sec-
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ond harmonic generation by nonlinear optical null ellipsometry, Analytica
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Chimica Acta 496 (1-2) (2003) 133–142. doi:10.1016/S0003-2670(03)
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R. M. Everly, Direct determination of effective interfacial optical constants by nonlinear optical null ellipsometry of chiral films, Analytical chemistry 77 (1) (2005) 215–224. doi:10.1021/ac049203m.
[14] A. J. Timson, R. D. Spencer-Smith, A. K. Alexander, R. Greef, J. G. Frey, Second harmonic ellipsometry, Measurement Science and Technology 14 (4)
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[15] B. Gokce, D. B. Dougherty, K. Gundogdu, Effect of p-type doping on the oxidation of h–si(111) studied by second-harmonic generation, Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 30 (4) 210
(2012) 040603. doi:10.1116/1.4721329. 11
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[16] C. Reitb¨ ock, D. Stifter, A. Alejo-Molina, K. Hingerl, H. Hardhienata, Bulk quadrupole and interface dipole contribution for second harmonic
10.1088/2040-8978/18/3/035501.
[17] J. Kwon, M. C. Downer, B. S. Mendoza, Second-harmonic and reflectance-
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generation in si(111), Journal of Optics 18 (3) (2016) 035501.
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an
[18] J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, D. E. Aspnes, Simplified bond-hyperpolarizability model of second harmonic generation: Application to si-dielectric interfaces, Journal of Vacuum Science & Tech-
doi:10.1116/1.1493783.
M
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d
[19] S. A. Yang, X. Li, A. D. Bristow, J. E. Sipe, Second harmonic generation from tetragonal centrosymmetric crystals, Physical Review B 80 (2009)
Ac ce p
te
165306. doi:10.1103/PhysRevB.80.165306.
12
Page 12 of 13
ip t cr us an M ed ce pt
Second harmonic generation data of Si(111) in PRSARAR configuration is presented. Experiments are simulated via the simplified bond hyperpolarizability model using only 4 parameters. The model considers dipolar surface and quadrupolar bulk contributions. Simulation and experiment match which is shown for arbitrarily chosen in- and output polarization.
Ac
Page 13 of 13
S0169-4332(16)32247-4 http://dx.doi.org/doi:10.1016/j.apsusc.2016.10.131 APSUSC 34225
To appear in:
APSUSC
Received date: Revised date: Accepted date:
31-7-2016 20-10-2016 20-10-2016
Please cite this article as: Cornelia Reitbddotock, David Stifter, Adalberto AlejoMolina, Hendradi Hardhienata, Kurt Hingerl, Nonlinear ellipsometry of Si(111) by second harmonic generation, (2016), http://dx.doi.org/10.1016/j.apsusc.2016.10.131 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ip t
Nonlinear ellipsometry of Si(111) by second harmonic generation Cornelia Reitb¨ ocka,b,∗, David Stiftera,b , Adalberto Alejo-Molinac , Hendradi Hardhienatad , Kurt Hingerlb a Christian
an
us
cr
Doppler Laboratory for Microscopic and Spectroscopic Material Characterization, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria b Center for Surface- and Nanoanalytics, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria c CONACYT Center for Research in Engineering and Applied Science (CIICAp), Institute for Research in Pure and Applied Science (IICBA), UAEM Cuernavaca, Mor. 62209, Mexico d Theoretical Physics Division, Department of Physics, Bogor Agricultural University, Jl. Meranti, Gedung Wing S, Kampus IPB Darmaga, Bogor 16680, Jawa Barat, Indonesia
M
Abstract
In this paper we present experimental and simulated data of the second harmonic response for arbitrarily oriented linear input polarization and output
d
polarization on Si(111) surfaces. In contrast to other nonlinear optical ellip-
te
sometry experiments (e.g. nonlinear optical null ellipsometry) we azimuthally rotate the sample for additional analysis of the crystal structure. In our study,
Ac ce p
we used the simplified bond hyperpolarizability model (SBHM) to simulate the observed angular shifts of the nonlinear peaks and the change of symmetry features related to a modification of the beam polarizations. Our findings help to identify the corresponding interface dipolar and bulk quadrupolar SHG sources. Keywords: Second harmonic generation, Si(111), nonlinear ellipsometry, simplified bond hyperpolarizability model
1. Introduction
Second harmonic generation (SHG) based techniques have been proven to be powerful tools for studying surface and interface properties on an atomic scale. ∗ Corresponding
author
Preprint submitted to Journal of LATEX Templates
October 20, 2016
Page 1 of 13
[1, 2, 3, 4, 5, 6, 7] In a typical SHG experiment a laser beam at frequency ω 5
and with a polarization perpendicular (s-polarized) or parallel (p-polarized) to
ip t
the plane of incidence impinges on a nonlinear medium. Simultaneously, two
photons of frequency ω are annihilated and a photon with the second harmonic
cr
frequency (2ω) is created. Commonly, in SHG experiments the sample is az-
imuthally rotated in order to study the rotational anisotropy dependence of the sample (RA-SHG).
us
10
In electric dipole approximation the induced second order polarization is ~ ~ given by P~ (2) (ω) = χ(2) E(ω) E(ω) where χ(2) is the second order nonlinear
an
susceptiblity, a tensor of rank 3 comprising material properties. For materials with a center of inversion (i.e. being centro-symmetric) such as Si χ(2) vanishes 15
in the bulk as long as dipole approximation holds [8]. However, at an interface
M
inversion symmetry is broken, thus allowing dipolar contributions. In 2002 Powell et al. introduced the simplified bond hyperpolarizability model (SBHM) [9] modelling the SHG signals in terms of a far-field intensity that is
ing along the bond axis. Applying this microscopic approach enabled fitting
te
20
d
calculated as the square of the superposition of fields radiated by charges mov-
of RA-SHG data using simpler mathematics than previous phenomenological
Ac ce p
models [10]. Even though already offering a better understanding of the underlying physics, the model was further developed taking into account additional nonlinear contributions like the electric quadrupole contribution [11].
25
Starting in the early 2000s the field of nonlinear ellipsometry has emerged as
a novel approach to evaluate sign and phase information between the different nonzero χ(2) tensor elements of a given sample [12, 13]. In its simplest form a polarized laser beam at frequency ω impinges on the sample and the polarized
reflected beam at 2ω is detected. In literature the term ”Second harmonic ellip-
30
sometry” exists [14] but contrary to RA-SHG the sample is at a fixed position neglecting additional information about the structure of the sample. In this work we present experimental and simulated SHG response of Si(111) surfaces using - besides typical polarization states of e.g. s-polarized input, ppolarized output - arbitrarily linear polarized input beams and investigating the 2
Page 2 of 13
35
output polarization by stepwise rotating the polarizer at the output. Additionally to this nonlinear ellipsometer configuration our sample is rotated. Thus,
ip t
our experiments are performed in a rotating polarizer, azimuthally rotating sam-
ple, rotating analyzer (PR SAR AR ) configuration. In contrast to other nonlinear
40
cr
optical ellipsometry experiments (e.g. nonlinear optical null ellipsometry [12])
the additional azimuthal rotation of the sample allows analysis of the crystal
us
structure.
2. Material and methods
an
Si(111) wafers with a native oxide and a doping level below the critical value for influencing the SHG signal [15] were chosen as samples. We performed the polarization sensitive RA-SHG measurements based on
M
45
a typical RA-SHG system in reflection geometry as described in Reitb¨ock et al. [16]. Succinctly, the fundamental beam is generated by a picosecond laser
d
operating at 50 Hz and the SHG signal is detected after spectral filtering by a monochromator using a photomultiplier tube. Recent modifications of the above mentioned setup allow changing the po-
te
50
larization state of the fundamental as well as the frequency doubled beam not
Ac ce p
only from s-polarization to p-polarization and vice versa (typical for RA-SHG experiments), but in a continuous manner. All experiments are performed using the instrumentation schematically de-
55
picted in fig. 1, in which the half-wave plate (HWP) and the analyzer are rotated for polarization sensitive measurements. The Glan-laser prism (P) prior to the HWP ensures distinct s-polarized light.
3. Theory
As outlined in the introduction, the SBHM is widely used for the simulation 60
of p- or s-polarized SHG data. To the best of our knowledge, we show for the first time how the SBHM can be used to simulate RA-SHG data with arbitrarily linear input and output polarization. 3
Page 3 of 13
ip t cr us an M
Figure 1: Top view of setup for polarization sensitive rotational second harmonic generation
d
experiments; tunable laser system, mirror (M), beamsplitter (BS), photodetector, filter (F), lens (L), Glan-laser prism (P), rotatable half-wave plate (HWP), rotatable analyzer (A),
te
monochromator, photomultiplier tube (PMT).
Ac ce p
If we overcome dipole approximation, the induced polarization of a cen-
trosymmetric non-magnetic material such as silicon comprises two dominant
65
SHG sources: dipolar SHG at the SiO2 -Si interface and a quadrupolar contribution from within the bulk. In SBHM the two contributions are modelled corresponding to eqn. 1 [9, 17, 18] and eqn. 2 [19, 17]. (2),interface
PD
(2),bulk
PQ
(2)
= χD · ·Ein Ein n P (2) ˆ ˆ ˆ = V1 αj b b b j j j · ·Ein Ein
(1)
j=1
(2)
= χQ · · · Ein ∇b Ein n (2) P ˆ ˆ ˆ ˆ = V1 αb bj bj bj bj · · · Ein ∇b Ein
(2)
j=1
(2)
where for the polarization produced by dipolar interfacial sources χD is the
4
Page 4 of 13
third rank susceptibility tensor, Ein is the incoming field, V is the volume, n is the total number of bonds inside the considered fundamental tetrahedral el(2)
ement, αj
ˆ j is the bond is the interface dipolar SHG hyperpolarizability and b
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70
(2)
unit vector. For the polarization due to bulk quadrupolar sources χQ is the (2)
αb 75
cr
fourth rank susceptibility tensor, ∇b is the gradient of the incoming field and
is the bulk quadrupolar SHG hyperpolarizability. Regarding the hyper-
polarizabilities we assign in the interface simulation an up hyperpolarizability
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ˆ 1 ) and a hyperpolabelled as αu for the bond reaching up to the surface (b ˆ2 , b ˆ3 , b ˆ 4 ) taking into account the larizability αd for the three down bonds (b
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tetrahedral molecular geometry of Si(111) (see fig. 2). In the simulation for the bulk contribution we only use one bulk hyperpolarizability αb for all of the bulk bonds because of symmetry.
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d
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80
Figure 2: Coordinate system and bond orientation of the effective Si (111) fundamental tetrahedral element.
Furthermore the far field is calculated for each contribution thoroughly tak-
ing care about the corresponding Fresnel factors and the correct projection of the fields. The corresponding optical constants of Si and SiO2 (obtained from ellipsometry measurements) are presented in tab. 1. In a last step the intensi-
85
ties of each contribution are calculated for comparing them qualitatively with the experiment.
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Wavelength [nm]
Energy [eV]
n
k
Si
800
1.55
3.68
0.004
400
3.10
5.63
0.28
890
1.39
3.63
0.002
445
2.78
4.75
0.08
800
1.55
1.46
400
3.10
1.48
890
1.39
1.46
445
2.78
1.47
an
4. Results and discussion
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SiO2
cr
Material
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Table 1: Optical constants of Si and SiO2 at different wavelengths.
Depending on the polarization of the beam, 3-fold (e.g. sp, blue in plot
90
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3 a)) or 6-fold (e.g. ps, black in plot 3 d)) patterns are measured on Si(111) surfaces in RA-SHG experiments. In order to investigate the origin of this SHG response, we show, using the SBHM that the observed angular shifts of the
d
nonlinear peaks and the symmetry features help to identify the corresponding
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interface dipolar, respectively bulk quadrupolar SHG sources. Experimental data are presented here for 16 different polarization states, 95
namely s-,30◦ -,60◦ - and p-polarization combinations. A polarization of e.g. s30
Ac ce p
indicates a s-polarized (half-wave plate at 0◦ position) input beam and a 30◦ polarized (analyzer is rotated 30◦ clockwise, looking into (or against) the beam
direction) SHG output beam. Figure 3 shows experimental (symbols) and simulated data (solid line) at a
100
SHG wavelength of 400 nm sorted in such a way, that the incoming polarization of all 4 graphs in a plot is the same. The data are normalized and vertical offsets are introduced between the curves for clarity. The simulation using simply the interface dipole source of the SBHM rep-
resents in good agreement, although not perfect for some polarizations, the 105
angular shift of the peaks that are related to changes in the polarization. Especially, the 3-fold and the 6-fold symmetries are displayed in the simulation reflecting the trend of the experiment. Remarkably, this means that SBHM can 6
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Figure 3: Experimental (symbols) and simulated (solid line) data for 16 different polarization
shifted for clarity.
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states measured on a Si(111) surface at a SHG wavelength of 400 nm. Plots are vertically
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simulate experimental data requiring only two (in general complex) independent parameters namely αu and αd for a fundamental beam of 800 nm.
110
In the following, investigations performed at an input wavelength of 890 nm
are presented. Figure 4 shows experimental and simulated data at four different polarization combinations, namely pp (blue), 60p (green), 30p (red) and sp (black). The data are normalized and for better visualization the plots are shifted vertically.
115
In a first step, the same procedure as before (fundamental beam at 800 nm)
was applied in the simulation for the 890 nm experiment, i.e. using only a surface dipole contribution, see fig. 4a). Comparing the simulated data of fig. 4a) with the experimental data depicted in fig. 4c) (dotted) shows that the angular shift of a peak (e.g. large peak moves from 60◦ in pp to small peak at
7
Page 7 of 13
120
120◦ in sp) is conserved in both plots. But, in contrast to the investigations at 800 nm, the symmetry features of the simulation and the experiment are not
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in agreement. This is demonstrated explicitly in the 30p (red) graph where the relation between the large and the smaller peak is even inverted. Additionally,
125
cr
the 30p is rather 6-fold in the simulation of fig. 4a), whereas the experiment displays a rather 3-fold symmetry.
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In order to overcome this discrepancy between the simulation with solely surface dipole contributions and the experiment, a simulation based on pure bulk quadrupole contributions was considered (see fig. 4b)). Again the angular
130
an
shifts are present, but instead of shifting 60◦ , the peaks are shifted by -120◦ , e.g. the large peak in pp (blue) at 240◦ evolves into the small peak at 120◦ . Furthermore, the simulation with solely bulk quadrupole contributions cannot
M
reveal the 6-fold symmetry of the peaks that is measured in sp polarization combination.
In a next step we combined the surface dipole and the bulk quadrupole contribution by coherently summing up the far-field radiations and multiplying the
d
135
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sum with its complex conjugate (see fig. 4c) (full line). It is apparent that only by taking such a combination of dipolar surface and quadrupolar bulk contri-
Ac ce p
bution the same high-to-low peak ratio as in the experiment can be obtained in the simulation. Remarkably, the SBHM (containing both contributions) reveals
140
an identical angular shift of the peaks and moreover, a correct intensity signature by using only four independent parameters, two for the surface (αu and
αd = αb ) and two for the bulk (the complex gradient ∇b ). A table containing the simulation parameters is shown in fig. 4.
5. Conclusion
145
We have successfully demonstrated that our RA-SHG experiment in rotating polarizer, azimuthally rotating sample, rotating analyzer configuration is able to distinguish dipolar surface and quadrupolar bulk contributions. The findings are based on simulations using the SBHM model which reliably repro-
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Figure 4: Surface dipole a) and bulk quadrupole b) contributions are coherently added to simulate (full line in c)) experimental data (dotted in c)) at a fundamental wavelength of 890
Ac ce p
nm. Four different polarization states are presented: pp (blue), 60p (green), 30p (red) and sp (black). Fitting parameters are shown in the table. Plots are vertically shifted for clarity.
duces RA-SHG data of a Si(111) facet. Experimental and simulated data are in
150
good agreement for arbitrarily linear polarized fundamental and SHG signals, as well as for measurements using different wavelengths (i.e. having different penetration depths). Our results are encouraging and further work will focus on investigations at different wavelengths.
6. Acknowledgements 155
The financial support by the Austrian Federal Ministry of Science, Research and Economy and the Austrian National Foundation for Research, Technology
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and Development is gratefully acknowledged. K.H. acknowledges support of the
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European Commission under the H2020 grant TWINFUSYON (GA692034).
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Second harmonic generation data of Si(111) in PRSARAR configuration is presented. Experiments are simulated via the simplified bond hyperpolarizability model using only 4 parameters. The model considers dipolar surface and quadrupolar bulk contributions. Simulation and experiment match which is shown for arbitrarily chosen in- and output polarization.
Ac
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