Investigating thermally assisted control of magnetization using plasmons
Master project
By: Louiza Chettouh
Supervisors: Vassilios Kapaklis
Richard Rowan-Robinson Merlin Pohlit
June 2019
Contents
1. Introduction ……… 1
2. Theory ………. 3
2.1 Optical properties of metals………. 3
2.1.1 Relative permittivity and refractive index……….
3
2.1.2 Plasma frequency ……….
4
2.1.3 Localized surface plasmons ………..
6
2.2 Magnetism ……….. 9
2.2.1 Ferromagnetism / Paramagnetism/ Diamagnetism………….
9
2.2.2 Compensated ferrimagnets TbCo ……….
10
2.3 Electrical properties……… 11
2.3.1 Drude model……….
11
2.3.2 Hall effect in metals ………
13
2.3.3 Anomalous Hall effect ………
14
3. Method………. 15
3.1 Spike……… 15
3.2 Transport set-up ………..… 16
4. Results and discussion……….. 20
4.1 Sample A ……….. 20
4.2 Sample B ……….. 24
5. Conclusion ………. 35
1. Introduction
Surface plasmons have been found more than 100 years ago, and in the last two decades, the researches put a lot of interest on it. Many books detail the operating principle of surface plasmon and their use. This area proved to be one of the most dynamic areas of electromagnetic research. This interest is due to the fact that the surface plasmon has the ability to firmly confine the electric field in a region that is on the order of the subwavelength and this is called localized surface plasmon. Surface plasmon is the resonant interaction obtained under certain conditions between electromagnetic radiation (especially light) and free electrons. The localized surface plasmon is the result of the confinement of the surface plasmon in a nanoparticle of a size which is comparable or smaller than the wavelength of light used to excite the plasmon.
The modern electronics device that are used in phones, hard disks, microscopes, and others mechanisms tend to be more and more tiny, engineers desire to ultraminiaturize this technology, and this is how the localized surface plasmon intervenes since it is the size of a subwavelength. To facilitate researches and the measurements with the aim of producing tiny electronic components, the electronical readout out offers automatic output display to make the reading of the data easier. Furthermore, it ensures accuracy and avoid errors that the human eyes can do. The electronics readout has also auto polarity functions which can read negative values, so there is no risk to place the probes into the opposite polarity. As localized surface plasmon and electrical readout have significant advantages, the association of these two methods is a considerable way to have modern and revolutionary devices by controlling the magnetism of material by using light. Although, it not an easy thing to accomplish, as the electrical readout needs conductive surface to work but not the plasmons. A compromise might be found to combine these two technical by playing on conductive surfaces to use.
To control magnetism with light, the use of plasmonics is one of the phenomena that can successfully master this area. The kind of plasmonics that is used is the localized surface plasmon and the key property is that this phenomenon deals with infinitely small parts as mentioned above: the size of the subwavelength namely the nanoparticles. Thus, in this case, the possibility of heating a nanometric region is not an issue. Additionally to that, in these regions the electromagnetic field is enhanced and the resulting electromagnetic field in the vicinity of the nanoparticle can exceed the excitation field of several orders of magnitude.
Furthermore, to characterize the magnetic state of materials, one of the methods that can be used is the electronics readout. Therefore, the plasmonics systems must be developed in such a way that they are fully compatible with these electronics readout gear. Thus, to be done, the device used here is the measurement of the anomalous Hall effect also called extraordinary Hall effect which is a fundamental transport process in solids and it was first detected in ferromagnetic metal. This tool has several advantages, the signal coming out is very big and of high quality. The advantage of this method is also its way of adapting the scale to the sample measured in the device, so it can measure specific region of the size of the
nanometer. However, measuring the anomalous Hall effect is done on a conductor material and the localized surface plasmon occur on an isolated surface, combining these two phenomena causes problems so it is complex to evaluate the extraordinary Hall effect for localized plasmon surface which are on a conductive material. A compromise may be possible to combine these two things, consequently the aim of this project is to find what is this compromise and how to do it and finish at the end with a cross of a conductive material with a plasmonic antenna on it and finally control the magnetism with light.
2. Theory
2.1 Optical properties of metals
The optical properties of metals can be explained by a plasma model, where a gas of free electrons of number density n moves against a fixed background of positive ion cores 1. In this section, we discuss how the optical properties of metals, beginning with the relative permittivity, arise from the electronic properties of the free-electron gas.
2.1.1 Relative permittivity and refractive index
Permittivity is a constant of proportionality that relates the electric field E in a material to the electric displacement D in that material. It characterizes the tendency of the atomic charge in an insulating material to distort in the presence of an electric field. The larger the tendency for charge distortion (also called electric polarization), the larger the value of the permittivity2. Its symbol is ε. The permittivity of a medium is expressed as the product of the relative permittivity constant and the free space permittivity:3
ε= ε0 εr. (1.1) with Dielectric constant = relative permittivity = εr
Absolute permittivity = ε
Free space permittivity = Vacuum permittivity = ε0
The term ε0 is the permittivity of vacuum (no atoms present), and εr is the dielectric constant which is always greater than or equal to 1. This means the E-field is always reduced relative to what the E-field would be in free space. The permittivity is a measure of this reduction. The permittivity is the multiplier that relates the electrical excitation and the electric field
D = εE = ε0 εr E (1.2) Since the response of a medium, to an applied electric field depends on the fields frequency (omega) often the dielectric constant is split into a real part and an imaginary part:
εr(ω) = ε1(ω)+iε2(ω). (1.3) There is a link between the polarization P, the electrical field E and the displacement field D which is:
D = ε0 E + P. (1.4)
The refractive index, also called index of refraction, is a measure of the bending of a ray of light when passing from one medium into another4. In an isotropic medium, the index of refraction (n) at a point is given by the following formula5 : n=c0/c, where c0 is the speed of light in the vacuum and c the speed of light in the medium. There is a link between the
dielectric function and the refractive index ñ(ω) = √εr 1. At optical frequencies, εr can be experimentally determined for example via reflectivity studies and the determination of the complex refractive index ñ(ω) = n(ω) + iκ(ω) of the medium1.
The two parts of the dielectric function have distinct signification. When |ε1| ≫ |ε2|, ε1 determine the real part n of the refractive index, quantifying the lowering of the phase velocity of the propagating waves due to polarization of the material. The imaginary part ε2 of the dielectric function determines the amount of absorption inside the medium.1
2.1.2 Plasma frequency
(This part is following the description of Stefan Maier book plasmonics: fundamentals and applications)
The plasma frequency is the collective longitudinal oscillation of the conduction electron gas versus the fixed positive background of the ion cores in a plasma slab and it is recognized as the natural frequency of a free oscillation of the electron sea. A collective displacement of the electron cloud by a distance x, leads to a surface charge density σ = ±neex (with ne the charge density and e the charge of the electron) at the slab boundaries and the quanta of these charge oscillations are called volume plasmons.
The behaviour of the conduction electron gas has been studied for many years. The first model was the Drude model, which was proposed in 1900. In this model, one assumes that the electrons are free to move in the material only interacting by scattering with the fixed lattice of positive ions. The equation that describes the motion of the electron oscillations which are subjected to an external electric field is x(t) = x0e-i t. The complex amplitude x0
incorporates any phase shifts between driving field and response via:
𝒙(𝑡) =𝑚(𝑤2𝑒+𝑖𝛾𝑤) E(t). (2.1) with γ = 1 τ⁄ characteristic collision frequency and τ is known as the relaxation time of the
free electron gas.
e the charge of the electron m mass of the electron
The displaced electrons contribute to the macroscopic polarization P = −neex, explicitly given by:
P = − m(wne2+iγw)2 E(t). (2.2)
Inserting this equation in (1.3) gives:
Here wp2 = ne2⁄ε0m is the plasma frequency of the free electron gas. Therefore, by comparing eq. 2.3. with eq. 1.2, the dielectric function of the free electron gas is found:
εr(ω) = 1 − m(ww2+iγw)p2 . (2.4) The real and the imaginary parts of this dielectric function are written like this:
ε1(ω) = 1 − 1+ wwp2τ22τ2 (2.5) ε2(ω) = w (1+ wwp2τ22τ2) (2.6) where γ = 1 τ⁄ .
The dielectric function is studied for two regimes which are: w < wp and w > wp. First, one begins with the regime of low frequencies where w < wp. In this region, metals retain their metallic character and they are mainly absorbing and reflecting. At low frequencies, the absorption coefficient is given by,
α = (2wcp22τw)
1⁄2
. (2.7)
For low frequencies, the electric fields fall off rapidly inside the metal as 𝑒−𝑧⁄𝛿, with 𝛿 the skin depth given by
𝛿 = 2𝛼 . (2.8)
For frequencies approaching wp, the electrons are less capable of screening the electric field.
The imaginary part of the dielectric function becomes negligible while on the other side, the real part increases. In this regime, the losses are very low, since the frequency is higher than the electron scattering rate. This means that the light penetrates deeper into the metal and scattering is reduced. Although, in real materials, the interband transitions intervene, the imaginary part increases again and the metals recover its ability to absorb light. To describe the interband transitions which occur at these high frequencies, one can add corrections, such as 𝜀∞ constant, to the expression of the dielectric function 6:
εr(ω) = 𝜀∞− (w2𝑤+iγw)𝑝2 (2.9) Figures 2.1 shows the real and imaginary parts of silver with the interband transition1: Plasmonic excitations are strongest in the frequency range where ε is small, before interband contributions begin to dominate. In the next section, we discuss the phenomena of localised surface plasmons.
Figure 2.1: Dielectric function of the silver. At low energy, the imaginary part of the dielectric function is low so there are less losses and the medium is less absorbing. At high energy, the dielectric function starts to increase the interband transition occurs so there are more losses and the medium is more absorbing 1.
2.1.3 Localized surface plasmons
(This part is following the description of Xabier Inchausti master thesis).
The localized surface plasmons (LSP) are non-propagating excitations of the conduction electrons in metallic nanostructures excited by light. One describes the localized surface plasmons by a point-dipole and assumes a small spherical homogeneous and isotropic nanoparticle of radius a with dielectric constant εr embedded in a dielectric medium with εd. When incident light comes and interacts with the particle, an electric dipole moment is induced on the center of the sphere. The response of the nanoparticle is an induced point- dipole moment which is expressed as:
𝑝 = 4𝜋𝜀0𝜀𝑑𝑎3 𝜀𝜀𝑟−𝜀𝑑
𝑟+2𝜀𝑑 𝐸0. (3.1)
This equation, combined with the general definition for the dipole moment given by p = ε0εd αE0 and the volume of the sphere 𝑉 = 43𝜋𝑎3, defines the polarizability of the sphere
as:
𝛼 = 3𝑉 𝜀𝜀𝑟−𝜀𝑑
𝑟+2𝜀𝑑 (3.2)
The LSP is the resonant dipole-moment, where the polarizability is maximized, thus the equation 2.10 shows that the resonance occurs when the denominator εr + 2εd reaches a minimum.
The real part of the corrected dielectric function (equation 2.9) is very important and plays a crucial role. Indeed, using the expression for the polarizability (2.11), it can be deduced that the plasmon resonance occurs when,
This equation is known as the Fröhlich condition.
One of the most attractive aspects of the LSP is the sensitivity of its resonance to subtle changes in parameters such as the nature of the metal, the geometry and the size. The size of the nanoparticles has a considerable effect on the position and the intensity of the surface plasmon resonance peak. In the case of small nanoparticles (R~ 50 nm), the size mainly affects the intensity and the width of the resonance peak while its effects on the position of the peak are small. For large nanoparticles (R > 50 nm), the effect of changing size is pronounced in both of the intensity and position of the resonance peak7.
Figure 2.2: Representation of the distribution of the charges in spherical nanoparticles under the effect of an electromagnetic field. (a) Case of a nanoparticle of very small size in comparison to the wavelength. (b) Case of a nanoparticle of a size comparable to the wavelength. 7
Figure 2.3: The maximum in the extinction spectrum for a Gold sphere is around 525 nm, then when the shape changes, one sees that the peaks are shifted. When a2 is smaller than before, the peak is shifted to the left at around 510nm. When a2 is smaller but a3 is bigger, the peak is shifted to the right at around 550nm. So, one can see that the shape plays a role in the shift of the peaks of intensity. Thus a slight morphological deviation from the spherical shape has a considerable impact on the resonance. 6
The environment of the nanoparticles influences the LSP resonance in a significant way. Figure 2.4 shows the extinction spectra of a spherical gold nanoparticle in different materials with a different dielectric constant. A shift of the LSP resonance toward the red range is accompanied by an increase in amplitude. The spectral width is narrowed which means that the lifetime of the surface plasmon increases when the dielectric constant increases and that corresponds to the Fröhlich condition in (3.3) 8.
Figure 2.4: Extinction spectra for a gold disk with a diameter of 10 nm on different substrates with different dielectric constant7.
In this project, the environment the plasmonic nanostructure are mounted on a conductive medium, rather than a dielectric substrate. The conductive surface screens the electric field so the energy penetrates with difficulty. The dielectric surface redshift the resonance so it needs less energy than in the conductive surface. One sees what will happen and try to find if there is a compromise to make everything works correctly.
2.2 Magnetism
2.2.1 Ferromagnetism / Paramagnetism/
Diamagnetism
Ferromagnetic materials have a large, positive susceptibility to an external magnetic field.
They exhibit a strong attraction to magnetic fields and are able to retain their magnetic properties after the external field has been removed. Ferromagnetic materials have some unpaired electrons so their atoms have a net magnetic moment.
Paramagnetic materials have a small, positive susceptibility. These materials are slightly attracted by a magnetic field and the material does not retain the magnetic properties when the external field is removed. Paramagnetic properties are due to the presence of some unpaired electrons, and from the realignment of the electron spins caused by the external magnetic field.
Diamagnetic materials have a weak, negative susceptibility. Diamagnetic materials are slightly repelled by a magnetic field and the material does not retain the magnetic properties when the external field is removed. In diamagnetic materials, all the electrons are paired so there is no permanent net magnetic moment per atom.
Figure 2.5: representation of a diamagnetic, paramagnetic and ferromagnetic materials with and without a magnetic field.
2.2.2 Compensated ferrimagnets TbCo
A little located at the intermediary between ferromagnetism (all the electrons are oriented in the same direction) and antiferromagnetism (the electrons are oriented in two opposite directions), we find the ferrimagnetism. Here, the magnetic moments of the electrons are well opposed two by two, but those being in the sense the external magnetic field are stronger than those in the opposite direction. The total magnetic moment in one direction is therefore greater than that in the other direction and the total magnetic moment of the sample is therefore not zero.
Tb and Co are ferromagnet material but the interaction between them is antiferromagnetic. TbCo is very sensitive to the temperature. At high temperature Co dominates the Tb and it is in the same direction of the magnetic field and at low temperature Tb dominates and it is aligned with the magnetic field. Thus, TbCo is called ferrimagnet because one magnetic momentum is bigger than the other and aligned with B when the temperature change.
Diamagnetic Paramagnetic Ferromagnetic
B = 0 B B = 0 B B = 0 B
Figure 2.6: representation of a ferrimagnetic material.
2.3 Electrical properties
2.3.1 Drude model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals)8. Electrons are seen as "classical" material points of mass m, with a velocity v, subject to completely elastic collisions, considered as instantaneous, which cause their speed to change abruptly. For an electron between two collisions, in the presence of a constant and uniform external electric field E, the following fundamental relationship applies9 :
𝑑𝑣
𝑑𝑡 = 𝑚𝑞 𝐸 (4.1)
𝑣 = 𝑞𝐸𝑚 𝑡 + 𝑣𝑜 (4.2)
By averaging on all the electrons and on all the collisions, we get the average speed9:
< 𝑣 > = 𝑞𝐸𝑚 < 𝑡 > + < 𝑣𝑜> (4.3) Because the initial speed at which the electron emerges after collision has a random direction the average < 𝑣𝑜 > is equal to zero9.
The average of t is none other than the time of relaxations or in other words the average free time that was previously mentioned: 𝜏. Thus, the expression of the average speed (3.3) is9:
< 𝑣 > = 𝑞𝐸𝑚 𝜏 = 𝑣𝑑 (4.4) with 𝑣𝑑 drift velocity due to the electric field E.
Ferrimagnetic
If n is the density of electrons per unit of volume, the volume density of current is9:
𝑗 = 𝑛𝑞𝑣𝑑 (4.5)
and by inserting the expression of the drift velocity (3.4), one gets9:
𝑗 =𝑛𝑞𝑚2𝜏 𝐸 (4.6)
As a result, the Drude formula for conductivity is9:
𝜎 =𝑛𝑞𝑚2𝜏 (4.7)
The resistivity 𝜌 is reciprocal to the conductivity: 𝜌 = 𝜎1 . This means that as conductivity increases, resistivity decreases. Likewise, as conductivity decreases, resistivity increases.
Resistance is an electrical quantity that measures how the device or material reduces the electric current flow through it. The resistance is measured in units of ohms (Ω). The electrical resistance of a circuit component or device is defined as the ratio of the voltage applied to the electric current which flows through it:
𝑅 = 𝑉𝐼 (4.8)
The electrical resistance of a wire would be expected to be greater for a longer wire, less for a wire of larger cross-sectional area, and would be expected to depend upon the material out of which the wire is made. Experimentally, the dependence upon these properties can be determined for a wide range of conditions, and the resistance of a wire can be expressed as10:
𝑅 = 𝜌𝑙𝐴 (4.9)
Resistivity is determined by the scattering of electrons. The greater the scattering, the higher the resistance 11.
Sheet resistance (also known as surface resistance or surface resistivity) is a common electrical property used to characterise thin films of conducting and semiconducting materials. It is a measure of the lateral resistance through a thin square of material, i.e. the resistance between opposite sides of a square. The key advantage of sheet resistance over other resistance measurements is that it is independent of the size of the square - enabling an easy comparison between different samples. Another advantage is that it can be measured directly using a four-point probe. Sheet resistance (RS) is commonly defined as the resistivity (ρ) of a material divided by its thickness (t) and its unit is ohms (Ω)12:
𝑅𝑠 = 𝜌𝑡 (4.10)
2.3.2 Hall effect in metals
The Hall effect was discovered in 1879 by the U.S physicist Edwin Herbert Hall13. When a material is traversed by a current and subjected to a perpendicular magnetic field, there appears a so-called Hall electromotive force in a direction perpendicular to the current and the magnetic field. This effect is due to the deflection of the flow of electrons/charge carriers participating in the current by the field: it is the strong accumulation of electrons on one side of the solid, thus of "holes" on the opposite face, which creates the potential difference called
" Hall-Voltage". This voltage has a stable value because the concentration of electrons (and/or
"holes") creates an electrostatic field that has an opposite effect on the magnetic field. Charge carriers which pass through the magnetic field undergo the Lorentz force14:
𝑭𝒎 = 𝑞𝒗 ∧ 𝑩 = 𝑞𝑣𝐵 (5.1)
Due to the concentration of electrons, there is an electrostatic field E which will act on the electron and will oppose the magnetic force according to14:
𝑭𝒆 = 𝑞𝑬 (5.2)
When the two forces are balanced, the charge carriers are no longer deflected and the voltage does not increase any more. At equilibrium:15
𝑞𝑣𝑩 = 𝑞𝑬 = 𝑞𝑉𝐷𝐻 (5.3)
𝑉𝐻= 𝑣𝐵 (5.4)
with 𝑉𝐻 the Hall voltage and one can define the Hall constant which is:
𝑅𝐻 = 𝐸 𝐵𝑗⁄ = 1 𝑛𝑞⁄ (5.5) where j the current density and n is the density of the charge carriers.
This figure is taken from15:
Figure 2.7: representation of the classical Hall effect.
with: d Thickness of the plate D Height of the plate L Length of the plate q Charge carriers
B Magnetic field perpendicular to the plate v Charge velocity
𝑉𝐻 Hall voltage
E Electric field due to Hall voltage: E = 𝑉𝐻/ D
2.3.3 Anomalous Hall effect
Hall effect measurements in magnetic materials do not follow the classical Hall effect law, mainly because of the presence of a spontaneous magnetization which modifies the transport. Often, in Hall resistivity measurements performed on magnetic compounds it possible to identify two distinct contributions. The first contribution describes the classical Hall effect while the second term explicitly depends on the magnetic nature of the compound.
The empirical law obtained in ferromagnetic materials has the following form 16:
𝜌 = 𝑅𝐻𝐵 + 𝑅𝑠𝑀 (6.1)
with 𝑅𝑠 the anomalous Hall constant.
3. Method
3.1 Spike
The optical characterisation of the sample was done in reflexion configuration because the substrate is made with silicone and it does not transmit light. The light source was a lamp that swiped different wavelengths. For this experiment, the wavelength was varied from 400 nm to 800 nm. The configuration of this experiment was such that the light came out of the lamp, then it passed by a lens then by a chopper to have a periodic beam then by another lens. To have a well-defined spot, the beam of light passes through a diaphragm. After the diaphragm, the light goes through a filter which serves to kill the harmonics. To configure the polarization, the light goes through a polarizer. A small glass is placed after the polarizer to split the beam into two part one goes in the monitor and the other on the sample. Finally, the light hits the sample which is placed on a holder which can move so as to have the spot of light on the part that one wishes on the sample. A camera is placed near to the sample which is used to make the motion of the spot easier. The monitor was placed on the left side if we look at the set-up behind the lamp, and the detector right in front of the light source. The beam which was going in the detector was the reflective one. The monitor and the detector were protected by black boxes to avoid that they detect the parasitic lights of the room. The whole set-up was protected by a black curtain.
Figure 3.1: Sketch of the optical bench.
Lamp
Lens s
Lens Chopper
Diaphragm
Polarizer Filter
Beam splitter
Sample
Detector
Monitor 4 deg
The software collects the intensity of light from the monitor and the intensity of the reflective beam from the detector. First, one has to take a reference curve where the sample does not have patches for example for the sample B it is on the top so we have the ratio of:
𝐼𝑑𝑒𝑡,𝑟𝑒𝑓
𝐼𝑚𝑜𝑛 = 𝜅(𝜆)
with 𝐼𝑑𝑒𝑡,𝑟𝑒𝑓 is the intensity in the detector of the reference film 𝐼𝑚𝑜𝑛 is the intensity in the monitor
𝜅(𝜆) is a constant of the ratio of the intensity So, one deduces that 𝐼𝑑𝑒𝑡,𝑟𝑒𝑓 = 𝜅(𝜆) ∗ 𝐼𝑚𝑜𝑛 .
Afterwards, the sample can be studied by hitting it with the light spot so:
𝐼𝑑𝑒𝑡,𝑛𝑎𝑛𝑜
𝐼𝑑𝑒𝑡,𝑟𝑒𝑓 = % 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦
with 𝐼𝑑𝑒𝑡,𝑛𝑎𝑛𝑜 is the intensity in the detector when the light is on the nanostructure so, one deduces that:
% 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 𝐼𝑑𝑒𝑡,𝑛𝑎𝑛𝑜
𝐼𝑑𝑒𝑡,𝑟𝑒𝑓 = 𝐼𝑑𝑒𝑡,𝑛𝑎𝑛𝑜 𝜅(𝜆) ∗ 𝐼𝑚𝑜𝑛
This is what the software does and collect the data and plot the reflectivity against the wavelength. The last step is to observe the curves to see if there is a minimum and where is it knowing that this minimum refers to the plasmon resonance.
3.2 Transport set-up
The transport experiment is used to determine the resistance of the films using two methods: the row probes and the square probes. A second experiment was done and it was the measure of the anomalous Hall effect. In this part, one uses the reference film and not the sample with the gold ellipses. Let focus first on measuring the resistance.
For the line contacts, one uses the row probes to measure the resistance which is:
Figure 3.2: Sketch of the row probes.
The current was going through the 1 and the 4 probes. There are two voltages in this set-up:
the V2W which is measured in 1 and 4 probes and the V4W which is in the 2 and 3 probes. The V2W voltage is higher than the V4W because V2W consider the wire resistance, the contacts resistance and the sample resistance. Contrariwise, the V4W consider only the resistance of the sample so this value is more important than the V2W.
The four contacts measure, one uses the squares probes and it is like:
Figure 3.3: Sketch of the square probes.
1 2 3 4
V
2WV
4WI
1
2 3
V4w 4
I V2W
The current was going through the 2 and 3 probes. In this method, there is also the V4W and the V2W voltage and have the same meaning as in the row probes. The V2W was measured between the 2 and 3 probes and the V4W was measured between the 1 and 4 probes.
In order to measure the resistance, it was necessary to change the value of the current by hand while waiting a few seconds. The current was going from -50 uA until 50 uA. The software drew a current curve as a function of time. Then, one took the values of the two voltages and drew them according to the current I. At the end, one had two linear curves that passed through 0, one corresponded to the V4W and the other to the V2W. By doing a linear fit, one had access to the value of the resistance which was the value of the slope since V = RI.
After knowing the value of the resistance, one computes the sheet resistance of the film by using this formula for the row probes:
𝑅𝑆 = 𝜋 ln (2)
∆𝑉 𝐼
Whilst the above equation for sheet resistance is independent of sample geometry, this only applies when the sample is significantly larger (typically having dimensions 40 times greater) than the spacing of the probes, and if the sample is thinner than 40% of the probe spacing. If this is not the case (and in this project, we are not), the possible current paths between the probes are limited by the proximity to the edges of the sample, resulting in an overestimation of the sheet resistance. To account for this difference, a correction factor based upon the geometry of the sample is required.
The correction factors are on a guide, in this project I used the one which is on ossila.com12.
For the square probes, one uses the figure 12 of the review: The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems17.
The second part of this transport measurement is the anomalous Hall effect. The holder which was used is the square holder:
Figure 3.4: Sketch of the holder for the anomalous Hall effect.
The current was going through the 1 and 3 probes and the Hall voltage was measured in the 2 and 4 probes. The reference film was between the bobbins and one applies the magnetic field. The value of B was set in the software and it was going from -500 mT until 500 mT. The software shows in these measurements a loop and it means when the magnetic field is strong enough to change the direction of the film magnetization. It is a phenomenon which happens instantaneously.
1
2 3
4
VHALL I
4. Results and discussion
4.1 Sample A
In this chapter, we look at the results obtained from the experiences. First, we will focus on the sample A describing the results of the two experiences: optical and transport measurements. Secondly, we will study the sample B and then finish with a comparison of the two samples.
In this section, we will focus on the sample A. We will describe and discuss the results of the optical and transport measurements. This sample has different patches with different size of the ellipses and with different doses as explained in the method section. The optical measurements were done on the patches with a dose of 11 for X and Y polarization. The transports measures were done on the reference film.
Optical measurements:
In the figure 4.1 we do not see any resonance and we see that everything is shifted in both cases: x and y polarization. The shift is probably caused by the absorption of the gold because the imaginary part of the dielectric function of the gold showed that in this range the interband transition occurs so the metal character is strong there. We assume that the LSP is blue-shifted because of the conductive surface so that is why we built the sample B which has bigger ellipses. By doing that, we wanted to red-shift the LSP.
In the y polarization, the patch with the ellipses with the length of 130nm is missing because the measurements were not good, it is because of the reference curve thus we just removed it.
Figure 4.1: Optical measurements for the sample A. The measurements were done for all ellipses. (a) X polarization along the long axis of the ellipse. (b) Y polarization along the short
axis of the ellipse.
Transport measurements:
This measurement gives the value of the resistance by computing the slope. In the figure 4.2 and 4.3 we see here that the value of the resistance in the row holder is bigger than in the square holder, it is because of the correction factor.
Calculation of the sheet resistance:
➔ For the row probes:
First, we have to calculate this formula: 𝑅𝑆 = ln (2)𝜋 ∆𝑉𝐼 with ∆𝑉𝐼 the value of the resistance.
Then we have to find the correction factor in the table in ossila.com12 by finding the ratio if 𝑤 𝑙 with 𝑙 is the length of the sample and 𝑤 is the width of the sample. Then we have to find another ratio which is 𝑤𝑠 with 𝑠 is the spacing between the probes.
So: 𝑤 𝑙 = 10 𝑚𝑚 10 𝑚𝑚 = 1 and 𝑤𝑠 =10𝑚𝑚2𝑚𝑚 = 5 Then the correction factor is equal to: C = 0.7744
290nm 320nm
Xpol 290nm
320nm
Ypol
400 500 600 700 800
0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l150
˚l165
˚l180
˚
˚
Relative˚reflectivity
Wavelength˚(nm)
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0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l130
˚l150
˚l165
˚l180
˚
˚
Relative˚reflectivity
Wavelength˚(nm)
a b
The sheet resistance for the sample A with the row probes is:
𝑅𝑆 = ln (2)𝜋 ∆𝑉𝐼 ∗ 𝐶 = 4.53236 ∗ 12.8524 ∗ 0.7744 = 45.1 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
➔ For the square probes:
According to the review17, we just multiply the value of the resistance by the correction factor.
The correction factor is deduced like this:
We have to find first the ratio 𝑠𝑙 and here it is equal to: 4.9 𝑚𝑚 10 𝑚𝑚 = 0.49 then we have to check on the figure 12 of the review. So, the correction factor for the square probes is: C = 5.2 The sheet resistance for the sample A with the square probes:
𝑅𝑠 = 0.00881 ∗ 103∗ 5.2 = 45.8 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
So, after computing the value if the sheet resistance for the sample A for the row and square probes, we see that they are almost the same value.
Figure 4.3: Resistance measurements for the sample A with the square probes.
The loop in the figure 4.5 is clearly defined the measurements is not noisy. The Hall voltage is normalized.
4.2 Sample B
In this section, we will focus on the sample B. We will describe and discuss the results of the optical and transport measurements. This sample has a row of ellipses with different sizes.
Optical measurements
In the figure 4.6, we see that everything is shifted in the beginning as in the sample A it is also due the absorption of the gold. In the X polarization which is along the 750nm periodicity, we see a minimum at 600nm for the biggest ellipses. We assume that it is the LSP because according to the imaginary part of the dielectric function of the gold, this is the range were the resonance appears. And if we compare to the sample A, we see that with bigger ellipses the LSP is no longer in the blue range but it moves toward the red. Therefore, we do not see relevant results for the other patches.
In the Y polarization which is along the short axis, we do not see any minimum for the three patches.
Figure 4.6: Optical measurements for the sample B. The measurements were done for all ellipses. (c) X polarization along the long axis of the ellipse. (d) Y polarization along the short
axis of the ellipse.
810 nm 750 nm
X pol
810 nm 750 nm
Y pol
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0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l200nm
˚l250˚nm˚
˚l300˚nm
˚
˚
Relative˚reflectivity
Wavelength˚(nm)
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0.6 0.7 0.8 0.9 1.0
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˚ref
˚l200
˚l250
˚l300
˚
˚
Relative˚reflectivity
Wavelength˚(nm)
c d
Here we rotated the sample in the figure 4.7. We still see the absorption of the gold at the beginning. For the X polarization, along the short axis nothing important happen. Therefore, for the Y polarization along the long axis we see that a small deep appears in the l250nm at 550nm. A deeper minimum arises in the l300nm and it is shifted a little at 600nm. We still assume that is the LSP but we do not know why there is this double deep. The small feature at 750nm is due to the periodicity of the lattice.
With this rotation, we deduce that the relevant thing happens along the long axis of the biggest ellipse. So, more ellipses are big more we see promising things.
Figure 4.7: Optical measurements for the sample B. The measurements were done on the rotated sample. (e) X polarization along the short axis of the ellipse. (f) Y polarization along
the long axis of the ellipse.
Transport measurements:
In the figure 4.8 and 4.9, we see here like in the sample A that the value of the resistance in the row holder is bigger than in the square holder.
We see that the value in the sample A is higher than in the sample B because the layer of platinum is thicker in the sample B.
Calculation of the sheet resistance:
We follow the same steps as in the sample A, the value of the resistance only will change because it is not the same sample but they have the same size and shape.
Ypol 810 nm
750 nm Xpol
810 nm 750 nm
400 500 600 700 800
0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l200
˚l250
˚l300
˚
˚
Relative˚reflectivity
Wavelength˚(nm)
400 500 600 700 800
0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l200
˚l250
˚l300
˚
˚
Relative˚reflectivity
Wavelengh˚(nm)
e f
➔ Row probes:
𝑅𝑆 = 𝜋 ln (2)
∆𝑉
𝐼 ∗ 𝐶 = 4.53236 ∗ 8.806 ∗ 0.7744 = 31 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
➔ Square probes:
𝑅𝑠 = 0.00607 ∗ 103∗ 5.2 = 31.6 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
Figure 4.8: Resistance measurements for the sample B with the row probes.
Figure 4.9: Resistance measurements for the sample B with the square probes.
The loop is clearly defined in the figure 4.10. The Hall voltage is normalized.
Figure 4.10: Anomalous Hall effect of the sample B.
After seeing these results, we decide to mill the sample B because the optical measurements showed that it was more promising than the sample A. We took this decision to reduce the layer of platinum which was on top the TbCo to improve the LSP. Thus, the results of the sample B are:
Optical measurements
In the figure 4.11 we see the same minimum at 600nm that before milling on the X polarization along the long axis of ellipses for the biggest one. As before the others patches with different dimension of the ellipses do not show any resonance. Along the short axis there is nothing relevant just like before.
Figure 4.11: Optical measurements for the sample B after milling. (g) X polarization along the long axis of the ellipse. (h) Y polarization along the short axis of the ellipse.
Transport measurements
In the figure 4.12 and 4.13, we noticed that the resistance in the sample B before milling is smaller than after milling because the platinum was milled but we do not know how much.
Thus, a part of the conductive layer was removed after millings so that is why the resistance
400 500 600 700 800
0.6 0.7 0.8 0.9 1.0
1.1400 500 600 700 800
0.6 0.7 0.8 0.9 1.0 1.1
˚ref
˚l200
˚l250
˚l300
˚
˚
Reflectivity
Wavelength˚(nm) 810 nm
750 nm
X pol 810 nm
750 nm
Y pol
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1.1400 500 600 700 800
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˚ref
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˚l250
˚l300
˚
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Reflectivity
Wavelength˚(nm)
g h
Calculation of the sheet resistance:
➔ Row probes:
𝑅𝑆 = 𝜋 ln (2)
∆𝑉
𝐼 ∗ 𝐶 = 4.53236 ∗ 29.18 ∗ 0.7744 = 102.4 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
➔ Square probes:
𝑅𝑠 = 0.02161 ∗ 103∗ 5.2 = 112.4 𝑜ℎ𝑚/𝑠𝑞𝑢𝑎𝑟𝑒
Figure 4.12: Resistance measurements for the sample B after milling with the row probes.
Figure 4.13: Resistance measurements for the sample B after milling with the square probes.
Now, with these results we can compare between the sample A and the sample B, between the sample B before and after milling. In this part, we removed the error bar from the plot because something wrong happens with the software.
In this figure 4.14, we compared the biggest ellipses for the sample A and B. The measurements were done in both cases on the X polarization. We see that there is no minimum in the sample A, no resonance appears. In the sample B, we see that there is a deep at 600nm and we assume that is the LSP since the dielectric function showed that the plasmon resonance it is in this range for the gold. This is the reason why we choose to study and improve the sample B instead of the sample A.
Figure 4.14: Comparison of the biggest ellipses of the sample A and B. X polarization along the long axis.
Here in the figure 4.15, we compared the sample B before and after milling for the biggest ellipses. We clearly see that before milling was better. The minimum is deeper before milling.
The reason is may be that when we milled the sample we milled the gold ellipses too, so the plasmonic effect was reduced.
Figure 4.15: Comparison between before and after milling for the biggest ellipse of the sample B
Comparison of the Anomalous Hall effect:
Figure 4.16: Comparison of the anomalous Hall effect between the sample A and the sample B.
The loop of the anomalous Hall effect in the figure 4.16 is the same in both samples. The mount of terbium and cobalt is the same in the sample A and the sample B. So, that is why the loop is exactly the same. The Hall voltage is normalized.
Figure 4.17: Comparison of the anomalous Hall effect between the sample B before milling and the sample B after milling.
The loop of the anomalous Hall effect in the figure 4.17 after milling is smaller than before milling. Maybe during the milling, a part of terbium cobalt was milled too. Thus, after milling the sample has less magnetic matter.
Some value important:
Value of the resistance:
Rrow (ohm) Errorrow Rsquare (ohm) Errorsquare
Sample A 12.3524 3.61527E-10 8.81 4.85598E-10
Sample B 8.808 7.6013E-10 6.07 2.65835E-9
Sample B after
milling 29.1827 5.98642E-10 21.61 1.03855E-9
Value of the sheet resistance:
Rsheet row (ohm/square) Rsheet square (ohm/square)
Sample A 45.1 45.8
Sample B 31 31.6
Sample B after milling 102.4 112.4
5. Conclusion
This project is a part of a big project which aim to control magnetism with light, in a cross of TbCo and pillars of gold on a part of it. To succeed this work, the phenomena which was chosen here is the localized surface plasmon because of its several advantages. Thus, to characterize the magnetization of the cross, the electrical readout which was used is the anomalous Hall effect. The issue of this project is the fact that the localized surface plasmon occurs on a dielectric surface like glass but the anomalous Hall effect needs a conductive surface to work. The goal of all that we did in this research project is to find a compromise between these two phenomena. So, we designed two sample A and B made of plasmonic medium which was on a conductive surface. We assumed that the sample A had small ellipses and that the resonance was blue shifted and that is why we decided to design the sample B with larger ellipses to try to redshift the resonance. According to the results obtained from the experiments made, the sample B was promising and we notice a minimum along the long axis of the biggest ellipse and nothing along the short axis. Therefore, we deduced that more the ellipses are big more we have chances to see a resonance.
We learned a lot in this project and we know what are the problematic things and how to correct it. So for the future, there is several things that can be suggest. First, one can replace the silicon by a dielectric medium which conduct well thermally. Secondly, build bigger ellipses to push the LSP toward the near infrared but one has to be careful to not build too big ellipses because after we are no longer in the subwavelength range. Finally, instead of using gold as a plasmonic surface one can use silver or aluminum because the interband transition occurs at the short wavelength so we could see a LSP in the visible range. That are my suggestion for the further study.
I would like to thank Mr. Vasilios Kapaklis who accepted me as part of his team and allowed me to work on a project of such importance and very interesting. I would also like to thank Richard Rowan-Robinson and Merlin Pohlit who helped me a lot during this research project.
They had the patience and kindness to explain to me the details of the work to be done. They also managed to transmit me the information necessary for the success of this project.
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