Optical Properties of Exfoliated
NbSe
2
THESIS
submitted in partial fulfillment of the requirements for the degree of
BACHELOR OF SCIENCE
in PHYSICS
Author : Zhiyuan Cheng
Student ID : s2673568
Supervisor : Michiel de Dood
2ndcorrector : Sense Jan van der Molen
Optical Properties of Exfoliated
NbSe
2
Zhiyuan Cheng
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
June 19, 2020
Abstract
Superconducting Single-Photon Detector (SSPD) has been a topic in quantum information. In our lab, we use superconducting exfoliated
NbSe2thin flakes to construct an SSPD. To find thin flakes with few
layers to monolayer, we adopted the idea of a Fresnel-law-based model.
We managed to determine the wavelength (∼500nm) and thickness of
SiO2(∼50nm or∼250nm) to best visualize thin flakes. Further
experiments performed on Si wafer with 300nm-thick SiO2also show
great agreement with our model, and more solid proofs are also given by AFM scanning. From the model, we can also determine number of layers of the flakes from a linear relationship between the contrast and number
of layers even without light filters. Transparent SiO2wafer can also
visualize thin flakes with much lower contrast (∼0.1) from transmitted light. This can save time and effort of transferring the flakes, but will cost
Contents
1 Introduction 1
2 Background 3
2.1 Superconductivity 3
2.2 Superconducting Single-Photon Detector (SSPD) 4
2.2.1 Superconducting Nanowire Single-Photon Detector
(SNSPD) 4
2.2.2 Superconducting tunnel junction (STJ)-based detector 5
2.2.3 Superconducting Transition Edge Sensor 6
2.3 van der Waals Materials 7
2.3.1 Graphene 7
2.3.2 Hexagonal Boron Nitride 8
2.3.3 Transition Metal Dichalcogenides (TMDCs) 9
2.4 Visibility of Layered Structures 9
3 Methods 13
3.1 Theoretical Calculations 13
3.2 Exfoliation 14
3.3 Graphics Analysis 14
3.4 Another Method 16
4 Results and Discussion 19
4.1 Simulations 19
4.2 Experiments 23
4.3 Another Method 27
6 Appendix 33
Chapter
1
Introduction
Since its discovery, superconductivity has drawn much attention from sci-entists. There have been many advantageous examples of applications in superconductivity based on its unique electromagnetic (EM) properties, one of which is Superconducting Single-Photon Detector (SSPD).
With the development of quantum techniques, quantum particles ap-peal to be a perfect intermedia for high confidentiality. Determining quan-tum state of a single particle is then one of the prerequisites for further experiments in field of quantum information. This need eventually gave birth to the idea of SSPD. The core of an SSPD is a small piece of supercon-ductor, once hit by a single photon, superconductivity is destroyed and thus cause a detectable change of electrical or thermal properties, such as electrical resistance, of the system. Typically, there are many types of SSPDs making use of 2D, instead of 3D, superconductivity. It is widely ac-cepted that once the size of a superconductor goes below 4.4 by its coher-ence length, 2D superconductivity is then achieved. However, fabricating such thin pieces of superconductor with relatively large area effectively is extremely hard to achieve even with most advanced chemical techniques, finding a suitable material with naturally grown thin layers becomes an-other solution an-other than waiting for emerging of a more advanced chem-ical techniques.
In our group, we aim to construct a (P-)SNSPD, but fabricating long nanowires can be troublesome. On considering the fact that we are us-ing 2D superconductivity, why not find a naturally grown layered crystal instead? In fact, there are indeed plenty of crystals having such feature, which are called van der Waals materials or 2D materials.
The existence of isolated 2D materials was denied by Landau, since thermal fluctuation under extremely low temperature would destroy the
”net”[1]. However, after the group from Manchester and Chernogolovka successfully isolated monolayer graphene from bulk graphite in 2004[1], exfoliated few to monolayered van der Waals materials give a hint to a new type of SSPD. As a member of transition metal dichalcogenides, which is a large family of van der Waals materials, exfoliated single-layered NbSe2
still exhibits superconductivity with TC ∼ 2K[2], and this makes NbSe2 a
very suitable material for building an SSPD.
Though thin flakes of exfoliated NbSe2can be nearly transparent,
mod-ern techniques still give us many ways to find these flakes. One of the most effective way is optical microscopy. By placing the flakes onto a sili-con wafer with a certain thickness of oxide layer, even monolayers are able
to be noticed. From a Fresnel-law described model[3], we can determine
the visibility of the flakes. It is also possible to use this model to determine the number of layers of the flakes. For extension, we can also make some change of the model to find another possible way to find the flakes.
In this thesis, we will illustrate some basic ideas of
superconductiv-ity and some typical types of SSPD in Chapter 2. Some introduction of
van der Waals materials as well as a Fresnel-law-based model to visualize atomic thin layers will also be presented in Chapter2. In Chapter3, exper-iments will be instructed, followed by results as well as some discussions in Chapter4. Chapter 5will summarize the results, detailed calculations will be shown in Chapter6.
Chapter
2
Background
In order to have a clearer view of the whole project, it is necessary and useful to have some basic knowledge before achieving SSPD.
2.1
Superconductivity
As a representative of phase transition, dimensionality of the
supercon-ductor is playing an important role in superconductivity[4]. Generally
speaking, lower dimensionality would bring lower carrier density, while interactions between carriers would also be restrained[4]. Thus achieving
2D superconductivity is usually much harder than 3D cases[4]. However,
due to the fact that 2D systems sometimes act in a counter-intuitive way, it is also exciting to see what ”odd” appearances are expected in 2D cases. In earlier stage of 2D superconductivity research, experiments are per-formed on thin film structures, which are a strongly disordered and
amor-phous systems[5]. Since crystals are randomly grown into the thin film
formations, properties acquired from such systems are usually spatially symmetric, leaving out much detailed information like spin polarization which relies on spatial orientations[5]. Until recent 20 years, after more ad-vanced fabrication techniques such as molecular beam epitaxy (MBE) and exfoliation were introduced to 2D superconductivity research, it is even-tually possible for scientists to manipulate spatial orientation of crystals, deeper factors of superconductivity are then ready to be found.
Since 2D superconductor was first realised, it has quickly become a popular subject of research in many fields. For example, critical temper-ature (TC) of FeSe is 8 K in 3D case[6], however, TC can reach up to a
crys-tals are grown on top of SrTiO3 substrate. Besides, 3D insulators can
even be transformed into superconductors by changing electronic system around the interfaces, one of the most representative examples is the het-erostructure of La2−xSrxCuO4/La2SrCuO4[4]. Recently, a breakthrough
was found in graphene superconductivity, a team from MIT found su-perconductivity existing in bilayer graphene with two layers twisted by a certain angle[7,8].
2.2
Superconducting Single-Photon Detector (SSPD)
To meet great needs for quantum information, apparatuses are assembled to detect single-particle quantum states, one of the most popular detectors is Single-Photon Detector (SPD). Since EM fields can tremendously affect or even destroy superconductivity, superconductors are natually suitable to detect single photons. The working principle behind SSPD is to identify the change in the superconductivity induced by the absorption of a single photon, there are still some different methods to detech such change. Fol-lowed are some typical examples of SSPDs divided into different types based on their ways of function.
2.2.1
Superconducting Nanowire Single-Photon Detector
(SNSPD)
The sensor of an SNSPD is a string of nanowire, to maximum its effective interaction interface with photons, the nanowire is in zigzag pattern to maximize detecting area[9].
The functioning procedures can be illustrated by Figure2.1. Before
de-tecting photons, the nanowire must be cooled down below TC, and
cur-rent is running at I which is just below IC. Once a single photon hits
the wire, local superconductivity is then destroyed[9] and electrical resis-tance in this area rises, causing local current density to drop. To main-tain the total current, current density around the transitioned area would increase, which push beyond its critical current density[9] and supercon-ducting area will be transformed into normal conductor. Such avalanche effect will eventually destroy superconductivity of the local system, and the whole transition process can be easily seen from I-V curve, which sug-gests a single photon just interacted with the system.
SNSPD is highly sensitive and effective, but it requires a relatively long nanowire (≈ 100nm)[9], which can be very hard for fabrication. Besides,
2.2 Superconducting Single-Photon Detector (SSPD) 5
Figure 2.1: Operation circle of SNSPD, τ1 is the working time and τ2 is cooling
time, I and IC are the running curent inside the superconductor and critical
cur-rent with respect[10]
it is not capable of distinguish number of incident photons[9], since tran-sition would always happen no matter where multi photons interact with the superconductor. A clever way to increase its photon number resolution is to use parallel superconducting nanowires (P-SNSPD). By introducing parallel structure to the system, number of photons interacting with dif-ferent nanowires can be told from the signal, time resolution will also be largely improved as a result of reduced τ2.
2.2.2
Superconducting tunnel junction (STJ)-based
detec-tor
Different from SNSPD, core part of STJ is two superconducting thin layers (Shown in Figure2.2). Since binding energy of Cooper pairs is only around several meV while photons usually carry energy of several eV, breaking Cooper pairs can be very easy for photons. With biased voltage, broken Cooper pairs in the top layer will tunnel through the junction and reform in the bottom. Transport of the carriers would form detectable current, which is proportional to the energy of incident photons[9].
STJ is an outstanding SSPD with broad bandwidth and high time res-olution. But it is notable that it is using Cooper pair as intermedia, which only exist in Type-I superconductors, so the working temperature is very low, making detection hard to achieve. Besides, it has relatively low
effi-ciency and is also greatly affected by electronic fluctuations.
Figure 2.2: Structure of STJ. The junction is made up of two superconducting thin layers on bottom and top with respect, as well as an extremely thin (≈1nm) insulator in the middle[9]. In-plane magnetic field and weak bias between two thin layers would effectively stop unbroken Cooper pairs from tunneling through the junction[9,11]
2.2.3
Superconducting Transition Edge Sensor
Similar to SNSPD, the superconducting transition edge sensor (Figure2.3) is also making use of phase transition. However, unlike SNSPD, the sensor is a bolometer detecting the temperature change caused by the injecting photon[9].
By absorbing energy from incident photons, the temperature of the su-perconductor rises up. Then transition happens, and triggers high elec-trical resistance in the circuit. To avoid long cooling time, a fixed bias is
applied on the superconductor[9]. The bias slightly increases
tempera-ture of electrons beyond the temperatempera-ture of the substrate[9]. Once hit by a photon, temperature is lifted, causing larger resistance and less current density, thus reduces Joule heating of the sensor[9].
2.3 van der Waals Materials 7
Figure 2.3: (a) Electrical circuit of superconducting transition edge sensor. The superconductor is shown as R(T, I), RLis the eternal resistance of the voltage
source, L is the inductance of the circuit plus that introduced by superconduct-ing quantum-interference device (SQUID). (b) Thermal circuit of the sensor. The sensor with heat capacity of C is coupled with the reservoir by thermal conductor G, T and TBrepresents temperature of the sensor and reservoir with respect, PJis
the energy flowing into the sensor and PB is the energy flowing from the sensor
to the reservoir[12].
Superconducting transition edge sensor is a very sensitive and effec-tive device to detect single photons[9]. It is also merely affected by fluc-tuations and background noise and capable of distinguish number of in-cident photons. However, to effectively change the temperature of the sensor, heat capacity of the material must be very small, which leaves very limited choices of superconducting materials. Besides, due to weak electron-phonon interaction, cooling time of the superconductor can take a relatively long time.
2.3
van der Waals Materials
2.3.1
Graphene
Existence of individual 2D materials had been believed to be impossible until Geim and Novoselov declared their successful attempt on fabricating graphene from bulk graphite[1].
Graphene exhibits many appealing features. First of all, graphene has extraordinary high thermal and electrical conductivity, making it the known finest conductor[14]. Besides, due to its special structure, graphene is even
Figure 2.4:Honeycomb structure of graphene[13]
weak spin-orbit interaction as well as nearly zero nuclear spin, it is also an ideal material for spintronics[16]. Graphene sheets are atomically thin and transparent to notice, simulations show that graphene cannot be stable un-til number of atoms reaches ∼6000[17]. Recent results from Cao’s group show that the electronic band structures of the twisted bi-layer graphene depend on the twisted angles, at magic angles, it exhibit similar properties as any other unconventional superconductors[7,8].
2.3.2
Hexagonal Boron Nitride
Hexagonal boron nitride (hBN) has a very similar structure to graphene (shown in Figure2.5). It is also called as white graphene as it has a white outlook[18].
Opposite to graphene, hBN is an electric insulator with a wide band-width ranging from 5 to 6 eV[20]. However it can also be altered to con-ductor by the presence of Stone-wales defects within the structure or doping[20]. Besides, it is also an outstanding thermal conductor with high thermal stability, making it suitable for high-temperature applications[18]. hBN is
2.4 Visibility of Layered Structures 9
Figure 2.5: Structure of hBN, there are two different possible crystalline structures[19].
also frequently used in 2D device fabrications for providing an atomically flat surface with insulating properties for the possibility of gating[2,18].
2.3.3
Transition Metal Dichalcogenides (TMDCs)
TMDCs also belong to the family of 2D materials. Their chemical com-pound type is MX2, where M is a transition metal atom and X is a
chalco-gen atom[21]. Unlike graphene, a TMDC monolayer is like a sandwich
layered structure, with M atoms sandwiched between two layers of X atoms[21] (shown in Figure2.6).
Properties of different TMDCs may differ from each other significantly.
Take NbSe2 for example. Bulk NbSe2 can exhibit superconductivity with
TC ∼ 7.2 K[2], even a single layer of NbSe2 can still be
superconduct-ing under TC ∼2 K[2]. Adhered through relatively weak van der Waals
force between layers, it is possible to exfoliate NbSe2 down to
monolay-ers. Unlike bulk crystal, there can be Ising paring in superconducting monolayers of NbSe2[23]. Besides, quantum matallic state and strong
en-hancement of charge density wave are also found to exist in few layers to monolayers[24].
2.4
Visibility of Layered Structures
Generally, NbSe2flake that can be obtained from exfoliation usually range
Figure 2.6:Structure of 2H-NbSe2, with Nb atoms sandwiched between two
lay-ers of Se atoms[2,22].
effectively find an ideal flake out of many other flakes. Limited by low throughput, it would be very ineffecient and unpractical to use AFM or
STM to run scanning on such a large surface[3], a more appropriate way
to do such job is to use optical microscope[3].
Thin layers, especially monolayers, are nearly transparent, making them challenging to spot without a proper design of the optical system (system structure is shown in2.7). In this optical system, thin flakes are transpar-ent enough to contribute to the optical path, while existance of SiO2 layer
can change the color of the interfaces and thus make thin layers visible[3]. In all, as we alter the thickness of SiO2 and wavelength of incident light,
we can get different feedback from the system.
Figure 2.7:Optical structure of the system. Incident light comes from the air and is perpendicular to the surface, Si is thought to fill the half-infinite space.
2.4 Visibility of Layered Structures 11
To describe this system in a mathematical way, a Fresnel-law-based model[3] is introduced. The use of this model can quantity the difference caused by wavelength and intensity of reflected light. In practical, the Fresnel law is a very powerful tool, and it provides the theoretical base for this project.
Chapter
3
Methods
3.1
Theoretical Calculations
Using optical structure described above (Figure2.7), by using Fresnel law, the reflected light intensity can be written as[3]:
I =|(r1ei(Φ1+Φ2)+r2e−i(Φ1−Φ2)+r3e−i(Φ1+Φ2) +r1r2r3ei(Φ1−Φ2)) × (ei(Φ1+Φ2)+r1r2e−i(Φ1−Φ2) +r1r3e−i(Φ1+Φ2)+r2r3ei(Φ1−Φ2)−1|2 (3.1) where r1 = n0−n1 n0+n1, (3.2) r2 = n1−n2 n1+n2 , (3.3) r3 =n2 −n3 n2+n3 (3.4)
are the relative indices of refraction. Φ1 =2πn1d1/λ,Φ2 =2πn2d2/λ
are phase shifts introduced by the monolayer and SiO2 respectlively. A
variable called contrast is introduced for easy comparison. The contrast is defined as relative intensity of reflected light from the surface with (n1 =
n0 =1) and without (n16= n0) the monolayer:
C= I(n1 =1) −I(n1)
From Equation 3.5, the contrast reflects the difference by comparing light intensity from areas covered with flakes and the background. Since light intensity is relevant with wavelength and layer thickness, the con-trast can also be changed if the two factors change. This can also help us
find proper wavelength and SiO2thickness, and also give us information
about thickness of the flakes in practice.
3.2
Exfoliation
Before exfoliation, silicon wafers must be cleaned thoroughly in clean room. The substrates are cleaned through ultrasonication in acetone, isopropanol and distilled water for around 15 min respectively. Surface contaminations of dirt or particles are expected to be removed during this process.
After cleaning process, the wafer is ready for exfoliation. Actually, be-fore we use polymer gel films to exfoliate, the material has already been exfoliated into thin flakes by a scorch tape for several times, this is to get thin flakes with only several layers as many as we can. Then thin flakes of NbSe2can be exfoliated onto the wafer by the two following steps:
(i) Press the polymer gel film against the NbSe2 sample and tear them
apart quickly;
(ii) Press the film with NbSe2 flakes against the wafer and tear them
apart quickly again.
In our lab, we have two different types of gels (Gel 0 and Gel 4, pro-duced by Gel-Pak). While Gel 0 is less adherent than Gel 4, flakes are usually small and with less area of thin layer after exfoliation.
During this process, van der Waals interaction can determine the suc-cess of exfoliation. On the one hand, van der Waals force between the layer and gel has to be stronger than that to another layer[25]. This can make flakes attached to the gel. On the other hand, the force between the bot-tom layer and gel should be weaker than that to the wafer[25]. Otherwise few flakes can be transferred to the wafer. A more complete procedure of exfoliation is described in Ref.[25].
3.3
Graphics Analysis
To extract information from a certain light wavelength, a light filter can be added to the microscope. But if the light filters are not available,
informa-3.3 Graphics Analysis 15
tion regarding the wavelengths can still be separated by data buried in the CCD camera of the microscope.
In our lab, the camera (Nikon Digital Sight DS-Fi1) has built-in RGB channels, which can work as red, green and blue light filters(Figure3.1).
Figure 3.1: Quantum effeciency/Response spectrum of a newer camera model (Nikon Digital Sight DS-Fi3)[26]. Since no reliable description of currently used camera is found, it is helpful to have a basic idea of how our camera is working by using manual of a newer model of the same series.
The RGB channels can also be played to show information of the orig-inal light. In general, RGB of each pixel represent the intensity of the red, green and blue light, with values ranging from 0 (darkest) to 255 (bright-est). RGB can also be narrated as three orthogonal vectors in a so-called color space, and the module of a vector in the color space is the overall light intensity:
I2 =R2+G2+B2 (3.6)
If we replace I(n1 = 1) and I(n1) in Equation 3.5 with R, G, B or I
values respectively, we can then get the contrast from the photos taken in the lab.
3.4
Another Method
PDMS, a transparent polymer gel, is commonly used in transferring 2D materials in device manufacture. Since optical properties of different pieces of PDMS may differ from each other due to their properties, such as length of molecular chains and orientation of chains etc., we decide to use SiO2
wafers for simulation because we have already known optical properties of SiO2 better and both two materials are transparent. By using SiO2, we
can expect a very qualitative picture of how this idea works.
Figure 3.2: Optical model based on the novel idea. The overall structure is sim-ilar to the original model, though we no longer have Si layer, and SiO2 layer is
assumed to be much thicker than before (∼5µm), which is the thinnest wafer we can accurately produce. Besides, down below the SiO2wafer is assumed to exist
a semi-infinate-thick air layer.
In this novel model, we can get information of the system not just from reflection, but also from transmission thanks to the transparency of every material making up this model. After calculation∗, we find that the (rela-tive) reflected/transmitted light intensity IR/IT can be written as:
IR =| b0 a0 |2, (3.7) IT =|a3 a0 |2 (3.8) where
3.4 Another Method 17 a0 = n0+n1 2n0 n1+n2 2n1 n2+n0 2n2 a3e−i(Φ1+Φ2)+ n0+n1 2n0 n1−n2 2n1 n2−n0 2n2 a3e−i(Φ1−Φ2)+ n0−n1 2n0 n1−n2 2n1 n2+n0 2n2 a3e i(Φ1−Φ2)+ n0−n1 2n0 n1+n2 2n1 n2+n0 2n2 a3ei(Φ1+Φ2) b0 =n0 −n1 2n0 n1+n2 2n1 n2+n0 2n2 a3e −i(Φ1+Φ2)+ n0−n1 2n0 n1−n2 2n1 n2−n0 2n2 a3e−i(Φ1−Φ2)+ n0+n1 2n0 n1−n2 2n1 n0+n2 2n2 a3ei(Φ1−Φ2)+ n0+n1 2n0 n1+n2 2n1 n2−n0 2n2 a3e i(Φ1+Φ2) (3.9)
Knowing intensities of reflected/transmitted light, we can then use Equation3.5to determine if this method is suitable to find thin flakes.
Chapter
4
Results and Discussion
4.1
Simulations
Simulations are performed based on theoretical background introduced
in Chapter 3.1. By using equations 3.1 and 3.5, we can know how
con-trast changes as a function of incident light wavelength at different SiO2
thicknesses (plotted in Figure4.1).
In Figure4.1, we can see clearly that contrast curves look very different due to the change of SiO2 thickness. Contrast is around 0.5 at most, and
usually a none-zero value albeit variety of d2. Besides, we can also
no-tice that contrast can be negative, i.e. reflected light intensity from surface with flakes can sometimes be stronger than that from none-flake wafer. The occurrence of negative values means that within a range of wave-length, flakes exhibits metallic properties and thus reflect more light than the wafer does.
To determine the optimum thickness of SiO2, a more intuitive way to see how wavelength and SiO2thickness play their roles in determining the
contrast is to combine these 3 variables together in one color plot (Figure
4.2).
For a certain d2, the overall contrast is vanishing as wavelength
in-creases. Besides, around the regions of ∼ 50nm, 175nm 6 d2 6 250nm
and 300nm 6 d2 6 400nm, we can have the greatest contrast, especially
around 50 nm, when the contrast are relatively strong in most wavelength spectrum of the incident light.
However, according to Ref.[27], same work was done and the bright
pillars appears to be more vertical-like (Figure 4.3(a)). It suggests 75 nm
and 250 nm of the best options for d2, and the maximum contrast we can
Figure 4.1: Contrast plots at SiO2 thickness of (a) 50 nm, (b) 250 nm, (c) 300 nm.
Blue curves are the contrast curves and red dashed horizontal lines represent where C=0.
the use of refractive index of NbSe2, n1(λ). According to Ref.[28], the real
part of refractive index only varies within a small range of values at dif-ferent wavelength, while the imaginary part is varying over a wide range (shown in Figure4.3(b)). According to Ref.[3], the imaginary part is play-ing a very significant role in visualization of flakes, change of imaginary part would even bring the model to failure. In our model, we are only using n1(λ) = n1(500nm) due to lack of data, this eventually causes
dis-agreement with the work from Ref.[27]. It is sufficient to have a basic un-derstanding of the optical model. For a more specific analysis at a certain wavelength, it is also doable to change the refractive index according to
4.1 Simulations 21
Figure 4.2: Color plot of the contrast (n1(λ) = 3.4−1i, d1 = 1.1nm). There are
three visible yellow pillars in the plot, and the more yellow the area shows, the greater contrast can be acquired.
(a) (b)
Figure 4.3: 4.3(a) Color plot from Ref.[27], 4.3(b) Refraction index of bulk NbSe2[28]. Blue area and dots represent refractive index or real part of the
re-fractive index in our notation, while red area and dots represent extinction index or imaginary part of the refractive index.
Figure4.3(b).
The value of contrast also varies with the number of layers N of a par-ticular flake. For λ = 500nm, we can get color plot Figure 4.4. In later
experiments, we are using 300-nm-thick SiO2 layer, thus we pay more
curve shown in Figure4.5.
Figure 4.4:Color plot of the contrast vs. number of layers (N) and SiO2thickness
(d2).
(a) Large range (b) Small range
Figure 4.5: Contrast vs. N curve at d2 = 300nm, 4.5(a) is a wide range of N,
which gives an overview of how contrast acts as N changes, while we are more interested in thin layers, so zoom4.5(a)in and get4.5(b)where 16 N610.
In Figure4.5(a), we can see that C is oscillating significantly in thinner
range, and C comes to convergence when N ∼ 250. This is because at
certain layer number, which is below 250, NbSe2sample can no longer be
considered as microscopic, instead it becomes a thick bulk material which exhibits macroscopic metallic properties.
4.2 Experiments 23
As can be clearly seen in Figure4.5(b), the contrast is in a nearly linear
corresponding relation with N. Since it is evident that C = 0 when there
is no flake on top of the wafer (N =0), we can assume that as N doubles, C also doubles. This conclusion will be a very powerful analytical tool in later experiments.
4.2
Experiments
In the lab, we use Si wafers with 300-nm-thick SiO2 layer. Several few to
monolayer flakes have been exfoliated to the substrate, the analysis was mainly performed on one specific flake (shown in Figure4.6).
Figure 4.6: NbSe2flakes. We are most interested in the middle flake since it only
contains very few layers and also has a very clear layered structure with sharp edges. Yellow arrow line is the pixel scanning path, which go through five di-vided areas, namely, I (Background), II (Monolayer), III (Bilayer), IV (Quadru-layer) and V (Tri(Quadru-layer). The path also passes through a noise dot (marked in red circle).
If we scan every pixel on the path marked by the yellow arrow and summarize RGB and Intensity values in a plot, we can expect a step-like curve corresponding to different layers.
Figure 4.7:RGBI plot along the scanning path in Figure4.6. Horizontal axis marks distance from the beginning point of the path (in unit of 0.5 pixel). Edges are marked in G and I channels with yellow arrows, while red arrow marks the noise point.
From Figure4.7, it is very clear to see a step-like curve in Green chan-nel, while it is harder to notice in Blue and Intensity channels, and nearly no step could be seen in Red channel except for the monolayer. If we sum-marize RGBI values on each area by averaging, variance and medians, we can have a more straight-forward understanding from statistical view (Ta-ble4.1).
From Table 4.1, one can clearly see that R is only distinguishable be-tween I, II and III, there is barely a difference bebe-tween III, IV,and V, thus
R can only be used to find monolayers. From Figure 4.7, GBI channels
are all exhibiting change of height corresponding to different locations. However, while B is exhibiting clear difference at different locations, the variance of B is too high comparing to three other channels, making it hard to tell the edges of different areas. Meanwhile, I channel has low variance, but does not have clear distinction between II and III. For G channel, it is relatively excellent for low variance and being distinguishable enough
4.2 Experiments 25
Area R G
Average Variance Median Average Variance Median
I 73.276 3.835 73 97.778 3.205 98 II 61.531 3.302 62 99.556 3.649 99 III 45.223 3.281 45 107.321 3.968 107 IV 43.052 3.139 43 118.464 3.182 118 V 42.770 2.251 43 112.831 2.546 113 Area B I
Average Variance Median Average Variance Median
I 131.653 6.904 132 94.335 2.930 94
II 135.038 5.375 135 92.236 2.683 92
III 141.418 4.042 141 92.629 3.109 93
IV 146.609 4.281 147 99.115 2.657 99
V 144.456 3.739 144 95.460 1.901 95
Table 4.1:Average, variance and median of RGB values on each area
from average or median values. Generally speaking, G channel would provide a better view of the layered structures whereas R channel would be extremely helpful on finding monolayers.
From previously summarized data, we can then calculate the actual contrast by using the averaged values (Table4.2).
Area I II III IV V
CR 0 0.160 0.383 0.413 0.416
CG 0 -0.018 -0.098 -0.212 -0.154
CB 0 -0.026 -0.074 -0.114 -0.097
CI 0 0.022 0.018 -0.051 -0.012
Table 4.2: Actual contrast of RGBI, as area I is the background, it should not be surprising to find all contrast in area I turn out to be 0.
Since I channel is a mixture of RGB channels, it cannot be analysed in theory since no corresponding wavelength at a certain value can be found. First, let us try to confirm that area II is a monolayer while area III is a bi-layer. From Figure3.1, one might as well assume that Red channel
corre-sponds to λ=600nm, Green channel corresponds to λ=500nm and Blue
channel corresponds to λ = 450nm, then plot theoretical curves and data
points in one figure (Figure4.8).
from Figure4.8, one can see that all data points are not in perfect match with the calculated curve. It seems that there is something wrong with the
Figure 4.8: Theoretical curves and data points. Blue curve in the upper plot is the theoretical curve when N = 1, while the blue curve in the bottom plot the theoretical curve when N=2, data points are plotted as *.
optical model. Later experiments performed on AFM show agreement with the assumption that area II is a monolayer and area III is a bilayer (shown in Figure4.9).
From figure4.9(b), it is clear to see that the height difference of area II and III is around 1.5 nm, which agrees with the preset value of thickness of monolayer. Though height difference of area I and II is almost 3.5 nm, one should not consider this as the height of area II since the pin tip of AFM is experiencing different force on area I (SiO2) and II (NbSe2). As is stated
before, C can be thought as a linear function of N, one can see clearly that CGat area III, IV and V is around 2∗ (−0.5), 4∗ (−0.5)and 3∗ (−0.5)with
respect. Moreover, CGat area II lies between 0 and 2∗ (−0.5), despite the
fact that it is far from 1∗ (−0.5). So we can safely come into the conclusion that area II, III, IV and V corresponds to monolayer, bilayer, quadrulayer and trilayer with respect. And the disagreement between theory and ex-periments may be due to the following reasons:
* The refraction index of NbSe2is not correct due to lack of data;
* RGB channels cannot be simply regarded as RGB light filters since
bandwidth of RGB channels on the CCD is too broad (Figure3.1);
* From the AFM measurement 4.9(a), we can see a very rough
4.3 Another Method 27
(a) AFM Photo (b) AFM height
(c) 3D image of the surface
Figure 4.9: 4.9(a) Monolayer and bilayer photos by AFM, the white line is the scanning path. 4.9(b)Height curve along the scanning path in4.9(a). There are plenty of peaks on the flake which make the surface uneven in4.9(c).
bubbles trapped between the flake and the wafer during exfoliation, which might change the intensity by changing the phase factors in equation3.1.
4.3
Another Method
Similar to Section4.1, we can draw color plots of reflected contrast (Figure
4.10(a)) and transmitted contrast (Figure4.10(b)).
(a) Reflection
(b) Transmission
Figure 4.10: 4.10(a)Color plot of the contrast caused by reflected light. 4.10(b)
4.3 Another Method 29
from reflected light since contrast is 0 for almost all sets of λ and d2. But
one should still notice that there are several bright lines in the plot, still this is not ideal for experiments since it is almost impossible to have such a nar-row bandwidth and exact SiO2. As a matter of fact, if we take d2to be 5µm
and plot the contrast curve vs. λ (Figure4.11), we may find those bright lines are actually showing contrast far less than 0. Such abnormal behav-ior may due to the fact that NbSe2flakes have metallic properties and can
reflect more light with some certain wavelengths, whereas SiO2crystal is
almost transparent and is not capable of reflecting much light. Besides, even if we could manage to meet all the experimental requirements, this is still not giving much information about thickness of the flakes since even monolayers could achieve extremely high contrast and one cannot
tell much difference between area with C =10000 and C=20000.
Figure 4.11: Contrast as a function of λ. There are several peaks showing the contrast greatly exceeds below 0 at some certain wavelengths.
If we take a look at Figure4.10(b), things are less crazy as one can get
the absolute contrast up to∼ 0.14. However, this is much less than what
we can get on Si wafers. So it is much harder to find thin flakes on top of a SiO2wafer from transmitted light.
Chapter
5
Conclusion
We used an optical system to effectively find NbSe2 flakes. By placing
flakes on top of a Si wafer with 300-nm-thick thermal oxidised SiO2layer,
we have successfully find few to monolayer flakes with optical micro-scope.
The concept of contrast was used to quantify the ”easiness” in identify-ing monolayer flakes. Based on a Fresnel-law-based model, we managed to determine contrast as a function of wavelength λ, SiO2thickness d2and
number of layers N. From the results of simulation, we found d2 ≈50nm
and 250nm 6 d2 6 300nm to be the most suitable SiO2thickness for high
contrast (up to 0.6) at wide selection range of λ. Meanwhile, it is also
rec-ommended to use green light (∼ 500nm) for experiments since it is most
suitable for eyes while contrast value remains considerably high (∼0.5). Besides, contrast shows linear relation with number of layers (16 N6
10) when d2 = 300nm and λ = 500nm. Along with AFM scanning, it is
safe to conclude that the flake contains monolayer, bilayer, trilayer and quadulayer.
We also came up with another idea of new optical model in hope to save time and effort of transferring flakes. However, it is not practical to use such model to find flakes since both reflection and transmission cannot provide very clear signal about layered structures of NbSe2flakes.
However, there are discrepancies between the experimental data and the theoretical calculations. We assumed that this disagreement might be due to lack of proper data, broad bandwidth of RGB channels on the CCD and uneven surface of the sample.
In conclusion, the optical model is in general agreeing with the exper-iments. It proves itself to be a powerful tool on finding thin flakes and layered structures. From this project, one can use contrast to determine
number of layers of the material. Using linear relation between the
con-trast and number of layers (16 N610), one can easily know the number
of layers from G channel, even without light filters or AFM scanning. For
experiments performed on top of a transparent material, such SiO2 and
PDMS, it is still possible to find thin layers from transmitted light, though contrast is not high enough and one have to find the flakes with much more effort.
Chapter
6
Appendix
6.1
Calculation on the New Optical Model
Calculations are based on the idea from Ref. [29].
Assume the incident light is perpendicular to the interfaces, and wave vector is orienting towards +z. The electric fields at the uppermost air layer, flake, SiO2wafer and bottom air layer can be written as:
E0 =a0eik0z+b0e−ik0z (6.1)
E1 =a1eik1(z−d1)+b1e−ik1(z−d1) (6.2)
E2 =a2eik2(z−d1−d2)+b2e−ik2(z−d1−d2) (6.3)
E3 =a3eik3(z−d1−d2) (6.4)
where a0,1,2,3 marks the electric field when the wave vector is positive,
b0,1,2 marks the electric field when the wave vector is negative. k0,1,2,3 is
the corresponding wave vector.
Use boundary conditions of electric fields: a0+b0= a1e−ik1d1+b1eik1d1 a1+b1= a2e−ik2d2+b1eik2d2 a2+b2= a3 (6.5) k0(a0−b0) =k1(a1e−ik1d1 −b1eik1d1) k1(a1−b1) =k2(a2e−ik2d2 −b2eik2d2) k2(a2−b2) =k3a3 (6.6)
Then the solution can be presented with a3: a0 =n0 +n1 2n0 n1+n2 2n1 n2+n0 2n2 a3e−i(Φ1+Φ2)+ n0+n1 2n0 n1−n2 2n1 n2−n0 2n2 a3e −i(Φ1−Φ2)+ n0−n1 2n0 n1−n2 2n1 n2+n0 2n2 a3ei(Φ1−Φ2)+ n0−n1 2n0 n1+n2 2n1 n2+n0 2n2 a3ei(Φ1+Φ2) b0 = n0−n1 2n0 n1+n2 2n1 n2+n0 2n2 a3e −i(Φ1+Φ2)+ n0−n1 2n0 n1−n2 2n1 n2−n0 2n2 a3e−i(Φ1−Φ2)+ n0+n1 2n0 n1−n2 2n1 n0+n2 2n2 a3e i(Φ1−Φ2)+ n0+n1 2n0 n1+n2 2n1 n2−n0 2n2 a3ei(Φ1+Φ2) (6.7)
whereΦ1 =2πn1d1/λ,Φ2 =2πn2d2/λ are phase shifts introduced by
the monolayer and SiO2with respect. The reflected and transmitted light
intensity can be then written as:
IR =|b0 a0 |2, (6.8) IT =| a3 a0 |2 (6.9)
6.1 Calculation on the New Optical Model 35
Acknowledgements
Special thank to Xi’an Jiaotong University (XJTU) and Leiden University for offering me this great experience of ”3+1+2” scheme. This project was carried out in Quantum Matter & Optics team, Leiden Institute of Physics. The whole work was accomplished with generous guidance by Prof. M.J.A. de Dood and Ph.D candidate Xingchen Chen. During this project, Yuxuan Zhang from XJTU and Mihai Ghidoveanu from Facultatea de Matematica si Informatica - Bucuresti have offered much technical sup-port on Python programming. Some brilliant ideas were also generously provided by Matthijs Rog, Tim Reisinger and Thom Boudewijn. The
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