Studying Nanometer Scale Substrate
Effects on Optical Properties of M oS2
using Near-field Optical Microscopy and
Spectroscopy
Masters Thesis
Alen Kamalov
Freie Universität Berlin
Department of Physics
Reviewer:
Prof. Dr. Alexei Erko
Supervisor:
Prof. Dr. Stephanie Reich
24.05.2022
ii
Freie Universität Berlin
fu-berlin.de
Contents
List of Figures
iii
List of Tables
iv
List of Abbreviations
iv
1. Introduction
1
2. Theoretical Foundations
3
2.1. Transition Metal Dichalcogenites . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2. Basic Principles of Atomic Force Microscopy
. . . . . . . . . . . . . . . . . . . . . .
7
2.3. Scanning Near-Field Optical Microscope (s-SNOM) . . . . . . . . . . . . . . . . . . .
8
2.3.1. Near-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2. Observables in a SNOM Measurement . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3. Demodulation by Cantilever Tapping . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4. Interferometric Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4. Quantitative Measurement of Dielectric Function with s-SNOM . . . . . . . . . . . . 18
3. Experimental details
20
3.1. Details of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Bibliography
A.
I
First Appendix Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI
Affidavit
VII
Prototype Video Publication Agreement
VIII
i
List of Figures
2.1. Schematic representation of M oS2 inner structure [23] . . . . . . . . . . . . . . . . .
3
2.2. Bulk M oS2 on a piece of scotch tape (a). Monolayer M oS2 on a hBN flake (b). . . .
4
2.3. Calculated band structure of M oS2 bulk (a), quadri- (b), bi- (c), and monolayer
(d). Authors apply density functional theory with generalized gradient approximation using the PWscf package. The dashed line shows the Fermi level. Black arrow
indicates the most probable transition from the valence band (blue line) to the conduction band(red line). The transition becomes direct in a monolayer film [32]. . . .
5
2.4. Photoluminescence intensity comparison for different thickness of M oS2 layers. As
the thickness increases, the PL intensity lowers and the peak position red shifts
(shown in the inset) [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.5. PL and differential reflectance spectra indicating positions of A, B and C excitons in
different TMDs (a). Grey lines show reflectance spectra. Colored lines represent PL
spectra. Corresponding energy transitions calculated using density functional theory
(b) [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.6. Schematic representation of a standard AFM setup [22]. The cantilever with a tip
is oscillated by a shaker piezo with an amplitude of several tens of nanometers near
the sample surface. Position of the cantilever is controlled via a reflection of laser
light from the cantilever onto a position sensitive diod. . . . . . . . . . . . . . . . . .
7
2.7. Scanning electron microscopy picture of a cantilever with a tip [22]. Long white slab
is the cantilever with a small probe underneath. The right picture demonstrates the
tip apex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.8. Schematic of an aperture SNOM setup (a) and a scatterer-type SNOM setup (b) [18]
9
2.9. Numerical simulation of electric field enhancement based on the shape of the scatterer
[18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.10. Numerical simulation of electric field enhancement based on the curvature radius of
the tip apex [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.11. Schematic representation of an experiment proving the existence of near-field. Green
arrows indicate fluorescent radiation with wavelength different from the incident light
in the prism [24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.12. Elongated tip is treated as a spherical dipole which is in turn considered to be a
point dipole outside of the tip [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
ii
List of Figures
2.13. Different types of distorted SNOM images. Contrast jumps (a) occur due to different
inhomogeneity of experimental conditions (external vibrations or temperature fluctuation). Residual background due to insufficient demodulation (b) is represented
by the series of parallel interference stripes. Noise-dominated images (c) and (d) are
obtained when the laser spot is not fully focused onto the tip or the tip itself is not
sharp enough. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.14. Schematic picture of an interferometric demodulation setup with a reference beam.
The reference beam is modulated at the frequency M << Ω (a). Resulting frequency
spectrum after PSHet demodulation (b) [27] . . . . . . . . . . . . . . . . . . . . . . . 16
3.1. Schematic representation of the experimental setup. The laser light emitted by the
C-Wave laser passes through an optical density filter, further through a series of
mirrors the beam is guided towards a beam expander, passing trough which, hits the
probe apex. After scattering from the tip and the sample, the light is guided to a
detector. Picture provided by Gabriela Luna Amador. . . . . . . . . . . . . . . . . . 20
3.2. Amplitude (a) and Phase (b) SNOM images of a 4x4 µm area demodulated at 4-th
harmonic taken at 605 nm excitation wavelength. M oS2 monolayer is placed on top
of a hBN layer on a Si substrate. The blue dashed line represents the edge of the
M oS2 flake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
iii
List of Tables
iv
1. Introduction
With the discovery of graphene in 2004 [25], the study of two-dimensional materials has gained
great popularity and interest. With the introduction of a reliable yet simple method of acquiring
monoatomic layers of graphite by only using scotch tape, it became widely possible to investigate
two-dimensional (2D) structures. To produce small quantities of atomically thin films a mechanical
exfoliation method is used which utilizes the weak van-der-Waals interaction between different
monolayers of material. By performing several iterations of sticking a bulk material to the tape
there is a probability that only one atom thick layer will be left on the substrate.
Numerous scientific groups around the world have stepped into this field, making it a fast
growing area of research. Soon many other materials were characterised forming 2-D structures,
i.e. transition metal dichalcogenites (TMDs) [3], hexagonal Boron Nitride [4], phosphorene [15] and
many others.
Due to a lack of inversion symmetry, 2D materials have properties that are completely different
from those in bulk state. The above-mentioned graphene, for example, has a huge electrical [30] and
thermal conductivity [5], incomparable strength [28], and has already found multiple applications
in nanoelectronics [10, 31].
In this work we will focus on molybdenum disulfide (M oS2 ), which is an example of a transition
metal dichalcogenite. TMDs are semiconductors with indirect bandgap in their bulk form. On
passing to a two-dimensional configuration, such substances become direct-gap semiconductors
once the bandstructure changes [19]. Due to the 2D nature of the material the electronic screening
is much less thereby enabling excitons to exist with high binding energies at room temperature.
Because of the small Coulomb screening, excitons determine optical properties of the material.
Due to their size, there is a limited set of methods and experimental approaches to study
the various properties of 2D materials. These methods include spectroscopic methods, atomic
force microscopy, electron microscopy, etc. To characterize optical properties of the material one
can measure its dielectric function. This quantity describes the wavelength dependent electric
response of a material to an incident radiation. The most widely used method for determining
the dielectric function of thin films is spectroscopic ellipsometry [41, 11]. In this work, we show
that a scanning near-field optical microscope (s-SNOM) can be used to measure the dielectric
function of MoS2 in the visible wavelength range. The technique is based on comparing the SNOM
contrast of an unknown material with that of a substance whose dielectric function is previously
determined. We will also demonstrate the subwavelength sensitivity of this method in measuring
local dielectric function of a sample. As an additional challenge, dielectric function of a thin sample
can be significantly affected by substrate, since the thickness of the monolayer sample is, of course,
incomparable with the size of the substrate underneath. As a result, most of the information
obtained from the measurement can mainly originate from the influence of the substrate.
The aim of this thesis is to investigate the excitonic influence on optical properties of M oS2 as
1
well as to analyze the effect of the substrate on excitonic response of the sample. We investigate
SNOM contrast of images of monolayers of M oS2 lying on Si and hBN substrates by raster scanning
of the sample’s surface using AFM based s-SNOM in the visible range between 1.77 and 2.07 eV.
2
2. Theoretical Foundations
2.1. Transition Metal Dichalcogenites
Transition metal dichalcogenites (TMDs) is a class of materials exhibiting unusual optical
properties while being of monolayer thickness. A TMD molecule consists of one transition metal
atom and two achalcogen atoms placed in a Chalcogen-Metal-Chalcogen order. In the past, TMDs
were widely used as a lubricant for different mechanical applications thanks to their low friction
due to relatively weak Van-der-Vaals interaction between different layers. This exact property
has opened an easy way to repeatedly fabricate monolayer TMDs. Using mechanical exfoliation
technique it is possible to split off very thin layers of TMDs using a scotch tape. Monolayers of
TMDs exhibit direct band gap transition in contrast to bulk samples being indirect semiconductors
[19]. Thin films Monolayer structures represent two-dimensional objects, In thin films electrostatic
screening is suppressed in out-of-plane direction because of only one or few atoms stacked in this
direction. As a result the Coulomb interaction becomes dominant, which in turn allows creation
of stable bound electron-hole pairs named excitons [36]. Here we investigate the properties of one
specific TMD semiconductor - M oS2 . The inner structure of the compound can be visualized as a
series of stacked crystal layers which consist of three atomic layers in a S − M o − S order. Atoms
within one crystal layer are covalently bound in a trigonal prismatic configuration. Schematic
illustration of M oS2 inner structure can be seen in Fig. 2.1. Real M oS2 bulk and monolayer
pictures are shown in Fig. 2.2. Fig. 2.2 (a) shows a bulk M oS2 crystal on a piece of scotch tape,
(b) is an optical microscope picture of a monolayer M oS2 flake on top of a hBN flake. The red
contour shows the position of the M oS2 sample.
Figure 2.1.: Schematic representation of M oS2 inner structure [23]
3
2.1. Transition Metal Dichalcogenites
Figure 2.2.: Bulk M oS2 on a piece of scotch tape (a). Monolayer M oS2 on a hBN flake (b).
In Fig. 2.3 one can observe a transition from indirect (a) bulk M oS2 to direct semiconductor
(d) energy transition based on calculations. As a consequence of this emergent direct transition
one can observe a strong enhancement in Photoluminescence (PL) signal from monolayer M oS2
compared to multiple layers (Fig. 2.4) [32]. The brightest PL comes from a monolayer sample
decreasing in intensity with increasing thickness. The center of the peak also red shifts with
growing flake thickness. Bulk M oS2 provides much weaker PL signal because of the additional
momentum required for the indirect transition. The explanation of this effect is the following:
when the bandgap increases in thin MoS2 samples, the intraband relaxation rate originating from
the excitonic states lowers and the PL signal is enhanced.
4
2.1. Transition Metal Dichalcogenites
Figure 2.3.: Calculated band structure of M oS2 bulk (a), quadri- (b), bi- (c), and monolayer (d).
Authors apply density functional theory with generalized gradient approximation using
the PWscf package. The dashed line shows the Fermi level. Black arrow indicates the
most probable transition from the valence band (blue line) to the conduction band(red
line). The transition becomes direct in a monolayer film [32].
Figure 2.4.: Photoluminescence intensity comparison for different thickness of M oS2 layers. As the
thickness increases, the PL intensity lowers and the peak position red shifts (shown in
the inset) [12].
Absorbing of light by M oS2 and all TMDs in general leads to a formation of excitons - bound
electron-hole pairs. Excitons are electrically neutral charge particles which are used to describe
5
2.1. Transition Metal Dichalcogenites
a transfer of energy without a charge transfer [38]. The binding energy of these excitons in 2-D
M oS2 (0.2 eV [43]) is substantially higher than those for semiconductor quantum wells ( 0.01 eV
[2]). Such prominent binding energy values of TMDs can be explained by strong in-plane charge
confinement and low dielectric screening arising from 2-D structure. Additionally, it has been shown
in [29, 14] that at the K point electrons and holes exhibit large effective masses which also favors
the formation of excitons. Due to such large excitonic binding energies the optical properties of
M oS2 are mostly determined by excitonic effects even at ambient conditions. Presence of excitons
can easily be detected using differential reflectance spectroscopy - the spectra exhibit maxima at
three different energy values corresponding to A, B and C exciton respectively (Fig. 2.5(a)). In this
particular study we have investigated optical properties of A (1.88eV) and B (2.04 eV) excitons
[16]. Fig. 2.5 (b) demonstrates where PL peaks A and B come from from the point of view of
energy levels.
Due to Molybdenum having relatively large atomic number (42), Spin-Orbit Coupling, which
describes an interaction of an electron with a positively charged nucleus, can create a remarkable
splitting in the Valence Band (Splitting is also present in the Conduction Band but is much lower in
magnitude) [17]. Therefore, existence of two excitonic resonances is due to two different electronic
transitions occurring from two split valence band states. Since Spin-Orbit coupling strength is
proportional to atomic number, the separation between two excitons is larger for WS2 compared
to MoS2 because atomic number of tungsten is 74 (Fig. 2.5(a)).
Figure 2.5.: PL and differential reflectance spectra indicating positions of A, B and C excitons in
different TMDs (a). Grey lines show reflectance spectra. Colored lines represent PL
spectra. Corresponding energy transitions calculated using density functional theory
(b) [16].
6
2.2. Basic Principles of Atomic Force Microscopy
2.2. Basic Principles of Atomic Force Microscopy
Atomic Force Microscopy (AFM) is a powerful high resolution technique for determining sample’s
topography by raster scanning of the sample surface. For this scanning a small cantilever is used
which is a peace of metal or dielectric forming a long slab or cantilever with a very sharp tip at
its end. The curvature of the tip’s apex is usually in the order of tens of nanometers (Fig. 2.7).
When a tip approaches a surface it experiences local microscopic forces (typically van der Waals
forces), which results in the bending of the tip. By knowing the tip’s spring constant, one can easily
measure these molecular forces, thus acquiring information about tip-sample distance. By keeping
the force of the tip bending constant throughout the whole sample area one can create a pixel-bypixel topography of the investigated sample by scanning the whole area moving the tip over it. This
method is called ”Contact mode”. Besides that there exists a variety of other approaches. One of
them is named ”Non-contact mode” (NC). As an example of NC mode realization one can force the
cantilever to oscillate at its characteristic frequency with a certain amplitude. Oscillation amplitude
change may only occur because of alteration of van der Waals forces affecting the tip, therefore
using special feedback mechanisms one can keep this tapping amplitude constant by lowering and
ascending the tip from the sample. A typical AFM configuration is shown in Fig. 2.6. Upper part
of the tip is coated with a highly reflective metal and is illuminated by a laser source. Reflected
light is focused onto a special light sensitive diode which keeps track of lateral and vertical tip
movement. The cantilever with the tip underneath is connected to a vibrating device that provides
tapping of the cantilever. When an inhomogeneity is encountered by the tip, tapping amplitude
gets affected which gets recorded on the light sensitive diode straight away.
Figure 2.6.: Schematic representation of a standard AFM setup [22]. The cantilever with a tip is
oscillated by a shaker piezo with an amplitude of several tens of nanometers near the
sample surface. Position of the cantilever is controlled via a reflection of laser light
from the cantilever onto a position sensitive diod.
7
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
Figure 2.7.: Scanning electron microscopy picture of a cantilever with a tip [22]. Long white slab is
the cantilever with a small probe underneath. The right picture demonstrates the tip
apex.
Some materials, while being subjected to a mechanical strain, create electric charges in the
material. The opposite process is also observed: piezoelectric materials deform when an external
electric field is applied to them [40]. The last mentioned effect is utilised for a feedback loop in
an AFM. In Tapping mode, tapping amplitude is chosen to be the feedback parameter. To keep
it constant the z-piezo drive regulates the sample position down to nanometer precision. Based on
these movements of the z-piezo drive one can obtain a precise topography of the sample’s surface
[37].
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
The optical microscope was one of the first powerful tools for investigating the micro-world.
The first iteration was invented already in the beginning of 17th century [39]. But the usual
optical microscope resolution has its natural limit - the diffraction limit. Due to the light being
an electromagnetic wave, only objects which are larger than half of the light’s wavelength are
clearly resolved in the image. In spite of an optical microscope always relying on visible light with
wavelength λ ≈ 500nm, only objects larger than several hundreds of nanometers can be resolved.
With the growing interest of researchers in the study of ever smaller objects, the diffraction limit
8
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
has become a significant obstacle, the overcoming of which was necessary for the further study of
the microworld.
The first idea of a possible solution for overcoming the diffraction limit dates back to 1928 [34].
It is based on exploiting the evanescent modes of an electric field. In the first SNOM configurations
to create evanescent radiation small apertures were used. After the planar light wave hits an
opaque screen with a microscopic aperture, the light is scattered on this aperture (Fig. 2.8(a)).
This configuration is called aperture SNOM. In the very vicinity of the scatterer so-called near
field appears. Near Field is also sometimes referred to as evanescent waves. Through the use
of evanescent modes the diffraction limit can be overcome due to the fact that evanescent modes
don’t obey Abby’s diffraction limit. Near field modes don’t propagate but decay exponentially with
the distance from scatterer. Therefore, in order to acquire well resolved optical pictures with the
SNOM, a sample-aperture distance has to be maintained in the subwavelength region.
Figure 2.8.: Schematic of an aperture SNOM setup (a) and a scatterer-type SNOM setup (b) [18]
Instead of using the aperture to act as a light scatterer one can also utilize tip scatterers which
size can be lowered down to several nanometer at the apex (Fig. 2.8(b)). Such configuration is
called scattering-type SNOM (s-SNOM) and is the main technique used in this research. Scattering
of light from this very sharp tip apex also creates evanescent waves. Resolution of scattering-type
configuration can reach up to 1 nanometer [42]. To maintain a constant tip-sample distance an
atomic force microscope (AFM) is used as a feedback system (Chapter 2.2) [1]. Simultaneously the
AFM is also utilized for obtaining the topography of the sample.
Electric field in the close proximity to the scatterer is enhanced and confined due to the
lightning rod effect [26]. The degree of confinement and enhancement depends on size and shape of
a scattering tip. This can be shown by numerical calculations by solving the electromagnetic wave
equation under electrostatic approximation (Fig. 2.9,2.10).
9
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
Figure 2.9.: Numerical simulation of electric field enhancement based on the shape of the scatterer
[18]
Figure 2.10.: Numerical simulation of electric field enhancement based on the curvature radius of
the tip apex [18]
2.3.1. Near-Field
The first experimental observations of near field were performed using the Total Internal
Reflection effect causing light to be fully reflected without penetrating into the reflective medium.
The experiment setup performed by Mandelstam in the beginning of the 20th century was based
on a glass triangular prism which bottom part was immersed into water mixed with a fluorescent
solution. The light was directed onto a prism under such an angle so it would experience total
reflection from the bottom side of the prism, which is immersed in liquid (Fig. 2.11). Due to the
Total Internal Reflection effect the light would be reflected onto the different prism wall and leave
the prism. Beneath the prism two crossed optical filters would be placed with one of them only
10
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
letting the initial light’s wavelength through and the second one would be set to only transmit
fluorescent light’s wavelength. As a result, one can observe fluorescent radiation below the prism
which would not be possible if the light did not penetrate past the prism’s wall.
Figure 2.11.: Schematic representation of an experiment proving the existence of near-field. Green
arrows indicate fluorescent radiation with wavelength different from the incident light
in the prism [24]
Near field is often referred to as evanescent waves, which describes its rapid exponential decay.
Following the Fresnel equations describing the border conditions for the interface between the prism
and water in our example we conclude that z-component of near-field’s wavevector is imaginary,
⃗ =
therefore if we assume traditional harmonic representation of evanescent electric field as E
⃗ 0 ei(⃗k⃗r−ωt) , then we will come up with a term e−kz z which describes a decaying wave. In general
E
near-field is defined as an electric field with an at least one imaginary component of the wavevector.
Evanescent waves due to the imaginary part of the wavevector can contain higher frequencies,
which can provide us the information about very small features of our object of interest. In s-SNOM
setup a nanoscopic tip scatterer is used as a source of the near-field. In the vicinity of a very sharp
metallic tip near the sample locally confines and enhances the near-field creating a so-called hot
spot - an area of strongly enhanced local electric field. By keeping the tip close to the sample area
at a distance much less than the wavelength of the light, it becomes possible to optically observe
nanoscopic details of the sample’s topography. NF modified with the presence of the sample, can be
transferred back into the far field and detected therefore revealing a detailed picture of the sample
with nanoscopic resolution.
2.3.2. Observables in a SNOM Measurement
For a better understanding of what one actually observes in a SNOM measurement and what
quantities can affect the results, we will introduce a model treatment of the metallic tip which
serves as the main near-field source while scattering the light. Upon being illuminated by light,
surface plasmons are formed at the surface of the tip. These plasmonic modes can be polarized by
the incoming light. Within the point-dipole model the tip is considered to be a small polarizable
sphere and the tip’s shaft is not taken into consideration. From solving Maxwell equations in the
electrostatic approximation one can obtain the polarizability of a sphere (eq. 2.1). Derivation of
11
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
this quantity can be found for example in [35].
α = 4πa3 (
ϵt − 1
)
ϵt + 2
(2.1)
Here ϵt stands for dielectric function of the spherical tip, a is the radius of a sphere. The
near-field is formed between the tip and the surface of the sample. To simplify the geometry one
can consider a mirror dipole to be present under the sample’s surface (Fig. 2.12). This mirror
dipole polarization is parallel to the real tip polarization. The polarizability of the mirroring tip is
given by αβ where β is the quantity which only depends on the dielectric function of the sample
[13].
β=
ϵs − 1
ϵs + 1
(2.2)
Two electrostatically interacting dipoles form a coupled system with the effective polarizability
αef f given by eq. 2.3.
αef f =
α(1 + β)
αβ
1 − ( 16π(α+z)
3)
(2.3)
z is the tip-sample distance.
Effective polarizability fully describes influence of the sample on the near-field signal. Let us
describe this statement with more details.
Figure 2.12.: Elongated tip is treated as a spherical dipole which is in turn considered to be a point
dipole outside of the tip [13]
⃗ i . Then electric field
Let us denote the incident electric field generated by the laser source as E
⃗ i with rs describing the fraction of the incident
present at the tip can be expressed as (1 + rs )E
field reflected by the sample. The incident light excites surface plasmon polaritons in the tip shaft
12
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
⃗ i . The scattered electric field will thus be
creating an effective dipole moment p⃗ = αef f (1 + rs )E
⃗ s = (1+rs )⃗
given by E
p in analogy with the incident light scattered light comes directly from the tip
and from the sample surface after reflection. Further, we introduce the scattering coefficient σ =
Es
Ei
or σ = αef f (1 + rs )2 [7]. Scattering coefficient σ is indirectly obtained in a SNOM measurement
which will be discussed more thoroughly in the corresponding chapter. We should note that values
described in this chapter such as α, β or σ are complex quantities.
From the point dipole moment model it has been concluded that the scattered electric field
is proportional to polarizability of the point dipole. In a SNOM experiment one can indirectly
obtain information about the scattering coefficient. αef f , in turn, contains optical properties of the
investigated sample via the dielectric function ϵs . Therefore, a SNOM can be used as a powerful
tool to determine local dielectric function with high precision [6]. Signal that reaches the detector
is proportional to the scattered electric field corrected by a response function of the detector. This
signal is then demodulated using higher harmonics of the oscillation frequency of the tip. Tipsample distance and therefore the detector signal is time-dependent, thus a Fourier transform can
be applied to it using eq. 2.4. H(t) is time-dependent tip position, nΩ are higher harmonics of tip
oscillation frequency Ω.
Z
σn =
σ(H(t))einΩt dt = F̂n [σ(H(t))]
(2.4)
Since every detector brings a correction into the signal, one does not directly measure σn . To
eliminate the influence of the detector, SNOM contrast should always be normalized to a reference
scattering coefficient σn,ref and a total relative SNOM contrast is thus given by a complex value
2.5 [7].
ηn =
σn
σn,ref
=
Sn
Sn,ref
eiϕn −iϕn,ref
(2.5)
2.3.3. Demodulation by Cantilever Tapping
The total electric field reaching the detector does not only contain the one scattered by the
tip and thus is not totally informative, as it is mostly comprised of background noise, which has
to be avoided in the experiment. Multiple interference effects can occur between tip scattered and
background scattered radiation resulting in parallel interference stripes or fringes on the image [33]
(Fig.2.13 (b)). In general, a SNOM image can be distorted in many other ways such as: contrast
jumps (Fig. 2.13 (a)), dominant noise (Fig. 2.13(d)), imprecise laser alignment (Fig. 2.13(c)).
All of the above-mentioned artifacts except (b) arise from external perturbations like mechanical
instabilities of the setup or external vibrations, as well as the tip quality. The artifact depicted
in Fig. 2.13(b), however, is an internal problem overcoming of which is described in this and the
following chapter.
13
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
Figure 2.13.: Different types of distorted SNOM images. Contrast jumps (a) occur due to different
inhomogeneity of experimental conditions (external vibrations or temperature fluctuation). Residual background due to insufficient demodulation (b) is represented
by the series of parallel interference stripes. Noise-dominated images (c) and (d) are
obtained when the laser spot is not fully focused onto the tip or the tip itself is not
sharp enough.
A typical SNOM signal filtration procedure consists of several steps. The first one is lock-in
amplification. Since the scattering tip is oscillated at a certain frequency (hundreds of kHz) due to
AFM tapping regime the near-field part of the signal acquires this modulation. The background
component of the detector signal arises from sample scattering, tip shaft scattering etc. due to a
focused laser spot having a diameter on the µm scale whereas the tip’s apex size does not exceed
several tens of nanometers. The background contribution is not modulated and therefore can be
filtered out using lock-in amplifier. However, it is not possible to entirely eliminate the background
contribution by only using a lock-in technique.
Intensity at the detector is given by:
Idet ∼ Isc = |Esc |2 = |Enf + Ebg |2 = (Enf + Ebg )(Enf + Ebg )∗
(2.6)
The near-field contribution is proportional to the incident electric field via the near-field scattering coefficient defined in eq. (2.4): Enf = σnf Einc . The scattering coefficient is time dependent
due to modulation and can now be expressed by the Fourier expansion:
σsc =
∞
X
Esc
= σnf + σbg =
(σnf,n + σbg,n )einΩt
Einc
n=−∞
(2.7)
Modulation has a different effect on background and near-field parts of the electric field.
The tapping amplitude is a negligibly low quantity for background far-field radiation (the tapping
14
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
amplitude is much less than a wavelength). Far-field radiation can be considered almost independent
on the tapping amplitude. In contrast, for the near-field, several tens of nanometers is already
enough to observe a drastic change in the signal, which exhibits a non-linear behavior with respect
to the tapping amplitude. For describing a small varying quantity with a Fourier transform only
first few coefficients in the expansion matter, while for a non-linear near-field Fourier expansion
more coefficients σnf,n are non-vanishing, therefore allowing the filtration of the signal at higher
harmonics (usually n>3 for the visible region) [8, 27]. At the same time the harmonic at which the
background impact is negligible depends on the illumination wavelength and tapping amplitude.
λ n
Each subsequent harmonic is reported to increase signal to noise ratio by a factor of ( 2πa
) where
a is the probe apex radius [9]. Therefore, it is necessary to keep the tapping amplitude as low as
possible.
Considering all these simplifications by combining (2.6) and (2.7) the detector intensity is given
by:
∗
∗
Idet,n ∝ σbg,0 σnf,n
+ σbg,0
σnf,n
(2.8)
Here background scattering coefficients of order n=1 and higher were neglected due to their relatively small impact compared to σnf,n . Although, the non-vanishing coefficients of the order 0 are
still present in the signal at the detector. Such residual background effect is sometimes referred to
as multiplicative background [20].
2.3.4. Interferometric Demodulation
As it has been shown in the previous chapter, it is impossible to completely eliminate the noise
contribution in the SNOM signal by only demodulating at the tip’s tapping frequency. Multiplicative background is still present after demodulation. To avoid the interference between near-field
and background signals, a reference beam is introduced which is modulated by oscillating the reference mirror at much lower lower frequencies M compared to the tapping frequency Ω (M is in the
order of hundreds of Hz, whereas Ω lies in hundreds of kHz range). The setup of this experimental
technique simply represents the use of a Michelson interferometer (Fig. 2.14(a)) adapted for SNOM
experiments. A monochromatic laser beam is directed onto a beam-splitter. A fraction of the beam
reflected by the beam-splitter is directed towards an oscillating (with frequency M ) reference mirror
generating the reference beam. Laser light which passes through the beam-splitter, hits the tip and
creates the near-field signal. Both beams are then focused on the detector, where the interference
of these two rays can be disentangled yielding SNOM amplitude and phase signals. This approach
is often referred to as pseudo-heterodyne demodulation or PSHet.
15
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
Figure 2.14.: Schematic picture of an interferometric demodulation setup with a reference beam.
The reference beam is modulated at the frequency M << Ω (a). Resulting frequency
spectrum after PSHet demodulation (b) [27]
To further understand the PSHet demodulation one can consider the complex scattered electric
field:
Esc =
X
τn einΩt
(2.9)
n
where τn = σnf,n + σbg,n . This electric field creates a voltage at the detector (depending on the
intensity of the laser light), which is proportional to a square of (2.9). The voltage itself is therefore
also time-dependent and can be expressed via Fourier transform with coefficients given by:
Un = k
X
∗
τj−n τj∗ + τj−n
τj ≈ k(τ0 τn∗ + τn τ0∗ )
(2.10)
j≥n
Complex coefficient k describes the detector sensitivity. Additionally, in (2.10) it is assumed
that τ0 >> τn for n>0 because of initial tapping demodulation. Equation (2.10) can be rewritten
in terms of σ, which yields:
∗
∗
Un = k[σbg,0 (σnf,n
+ σbg,n ) + σbg,0
(σnf,n + σbg,n )]
(2.11)
Due to the tapping demodulation higher harmonics of background vanish, resulting in eq.2.12
∗
∗
Un ≈ k(σbg,0 σnf,n
+ σbg,0
σnf,n ) = 2kSbg,0 Snf,n cos(ϕbg,0 − ϕnf,n )
Scattering coefficients in (2.12) are re-expressed in correspondence with (2.5).
16
(2.12)
The reference
2.3. Scanning Near-Field Optical Microscope (s-SNOM)
beam after reflection from oscillating mirror can be expressed as a phasor via 2.13
ER = ρeiγsinM t+iψR
(2.13)
with γ being the modulation depth and ψR is the phase difference accumulated because of optical
path difference. Equation (2.13) can be also expanded in a Fourier series:
ER =
X
ρm eimM t
(2.14)
m
ρm = ρJm (γ)eiψR +imπ/2
(2.15)
Jm (γ) are Bessel functions of the 1-st kind.
When two beams, the reference and the scattered beam from the tip, reach the detector,
they interfere creating sidebands of the modulated harmonics at frequencies nΩ + mM , basically
demodulating previously demodulated harmonics, which is shown in Fig. 2.14(b). The background
signal is only present in the laser beam passing through the sample and not in the reference beam,
thus it is only contained in the frequencies nΩ but not in nΩ + mM .
Complete suppression of background noise is not the only advantage provided by interferometric
demodulation. Additionally, it grants access to real and imaginary components of the scattering
coefficient at the same time. Voltage at the detector (2.12) with the use of (2.14) and (2.15) is
described as follows:
Un,m = 2kρJm (γ)Sm,n cos(ϕm,n − ψR − mπ/2)
(2.16)
For even and odd values of m the expression (2.16) becomes proportional to real or imaginary components of the scattering coefficient respectively. The complex coefficient of the Fourier
expansion τn can be obtained from 2.16 via
Un,l
Un,j
τn = κ
+i
Ji (γ)
Jl (γ)
(2.17)
iψ
where κ = ( e2kρR ). The demodulation depth is a variable parameter and can be chosen to be equal
to 2.63 for convenience. Then for l=1 and j=2 the resulting expression for τn is following
τn = 2.16κ(Un,2 + iUn,1 )
(2.18)
The value of κ is in principle unknown but in a usual SNOM measurement not the absolute contrast
is obtained, but rather the relative contrast given by the equation (2.5) thus simplifying κ factor
[27].
17
2.4. Quantitative Measurement of Dielectric Function with s-SNOM
2.4. Quantitative Measurement of Dielectric Function with s-SNOM
A standard SNOM picture usually shows provides only qualitative information. We can see
which regions reflect more or less light in the amplitude image and which areas absorb more or less
light in the phase image. Quantitative measurements are also possible by comparing the contrast
of a sample with an unknown dielectric function to a reference sample which dielectric function is
previously determined. For a comparison to be valid, both these areas must be present in a single
SNOM picture. Reference material should have a rather stable dielectric function in a region of
interest for a reasonable comparison.
Dielectric function is an important material property which can be studied to characterise
light-matter coupling in the material. Complex dielectric function is affected by many intrinsic
material properties such as charge transfer effects, excitons, doping, mechanical strain or surface
inhomogeneity. Additionally, the dielectric function provides the information about a material’s
absorption and transmission [6].
We assume a point dipole model for a tip apex. The effective polarizability of the tip is given
by the eq. (2.3), which can be rewritten as:
αef f =
with f (H) =
α0
16π(a+H)3
α0
1 − f (H)β(ϵ)
(2.19)
and β(ϵ) defined in (2.2) The scattering coefficient σ and αef f are related
via:
σ = αef f (1 + rs )2
where rs is the reflection coefficient of the sample.
(2.20)
In a real SNOM experiment instead of
measuring σ directly, we, in fact, measure its Fourier harmonics σn (2.4). As our initial goal is to
obtain the value of ϵ, we encounter a challenge since the relation between the measured σn and
desired ϵ is non-algebraic.
One of the possible solutions of this problem is to expand αef f in a Taylor series (2.21) with
α(j)
as expansion coefficients
αef f (f β) =
∞
X
α(j) (f β)(j)
(2.21)
j=0
The expansion is valid when the product f β<1 which is true for the most of SNOM experiments
[7]. σn is then expressed as:
σn = (1 + rs )2
∞
X
F̂n [α(j) f (j) ]β (j)
(2.22)
j=1
Equation 2.22 can be truncated at a desired order, thus representing only a polynomial relation
between σn and β. The equation can be then inverted to determine β which is in simple relation
18
2.4. Quantitative Measurement of Dielectric Function with s-SNOM
with ϵ (eq. (2.2)).
To take into account the response function of the detector, the measured contrast should be
divided to a reference value as described in (2.5). The value of σn,ref can be numerically calculated
as the dielectric value of the reference in known [7].
19
3. Experimental details
3.1. Details of the experiment
Monolayer M oS2 samples on Silicon and hBN substrates were created using mechanical exfoliation on a PDMS (Polydimethylsiloxan) polymer and stamped onto a desired substrate. The
samples were precharacterised by the means of PL spectroscopy. The PL maps were recorded with
a Horiba Jobin-Yvon XploRA microRaman spectrometer as a function of laser position on the
sample with a 0.90NA (NA: numerical aperture) 100x objective. For excitation we used a laser
with 532nm wavelength and 1 mW power on the sample, with an acquisition time of 1 s and a 600
grooves per mm grating.
SNOM pictures were obtained using a commercial s-SNOM (NeaSNOM from Neaspec GmbH,
Germany) at 5 different excitation wavelengths. For excitation a tunable wavelength laser (Hübner
C-Wave) was used with a possible photon energy range of 1.91-2.76 eV. The laser power for all
s-SNOM image acquisition was 1 mW at the tip with an integration time of 16 ms. Pt-Ir coated
AFM tips were used as near-field probes (NanoWorld) with an apex radius less than 30 nm. The
tip oscillated at ∼250 kHz frequency with a tapping amplitude ∼55 nm. The setup scematics is
shown in Fig. 3.1. The light emitted by the C-Wave laser passes through an optical density filter,
further through a series of mirrors the beam is guided towards a beam expander, passing trough
which, the light is focused onto the probe apex. After scattering from the tip and the sample, the
light is guided to a detector.
Figure 3.1.: Schematic representation of the experimental setup. The laser light emitted by the CWave laser passes through an optical density filter, further through a series of mirrors
the beam is guided towards a beam expander, passing trough which, hits the probe
apex. After scattering from the tip and the sample, the light is guided to a detector.
Picture provided by Gabriela Luna Amador.
20
3.2. Results and Discussion
3.2. Results and Discussion
SNOM images were obtained at 5 different wavelengths: 605, 617, 629, 637, 690 nm. Both
Si and hBN demonstrate almost constant dielectric function in this region, therefore can be both
used as a reference. The result for 605 nm can be seen in the Fig. 3.2. 5×5 µm area was scanned
containing 4 different regions: silicon substrate, M oS2 on Si, hBN on Si and M oS2 on hBN. The
near-field signal was demodulated at the 4-th harmonic of the tapping frequency and normalized
to the 3-rd harmonic to artifacts arising from interference at the edges of flakes [21]. Images clearly
demonstrate subwavelength resolution as we are able to resolve nanoscopic bubbles on top of M oS2
for example. We also observe a clear correlation between the amplitude and the phase picture.
Figure 3.2.: Amplitude (a) and Phase (b) SNOM images of a 4x4 µm area demodulated at 4-th
harmonic taken at 605 nm excitation wavelength. M oS2 monolayer is placed on top of
a hBN layer on a Si substrate. The blue dashed line represents the edge of the M oS2
flake.
First, we have qualitatively estimated the behavior of the amplitude and phase contrasts
with changing wavelength. For that we have plotted a normalized M oS2 contrast dependence on
wavelength. Each SNOM image was first processed in the Gwyddion software to filter measurement
artifacts. Then an area of ∼ 1 × 1µm2 was taken from the M oS2 flake and Si positions to further
average the data yielding a single contrast value. The M oS2 contrast value was then divided by
the similarly obtained Si contrast for both amplitude and phase images.
21
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V
Appendix
A. First Appendix Section
...
VI
Affidavit
Ich versichere hiermit wahrheitsgemäß, die Arbeit selbstständig verfasst und keine anderen als die
angegebenen Quellen und Hilfsmittel benutzt, die wörtlich oder inhaltlich übernommenen Stellen
als solche kenntlich gemacht und die Satzung des Karlsruher Instituts für Technologie (KIT) zur
Sicherung guter wissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet zu haben.
Karlsruhe, 24.05.2022
Alen Kamalov
VII
Prototype Video Publication Agreement
I hereby agree that the prototype video submitted by me may be published on the Internet.
Karlsruhe, 24.05.2022
Alen Kamalov
VIII