Investigation of artificial spin ice structures employingmagneto-optical Kerr effect for susceptibility measurements

Report15 credit project course

Materials Physics division, Department of Physics and Astronomy, UppsalaUniversity

Student: Agne Ciuciulkaite

Supervisors: Vassilios KapaklisHenry Stopfel

2015

Abstract

Artificial spin ice structures are two-dimensional systems of lithographically fabricated lattices ofelongated ferromagnetic islands, which interact via dipolar interaction. These systems have beenshown to be a suitable playground to study the magnetic, monopole-like, excitations, similar tothose in three-dimensional rare-earth pyrochlores. Therefore, such artificial structures can be po-tential materials for investigations of magnetricity [1]. The investigations of these artificial spin icestructures stretches from the direct imaging of the magnetic configurations among the islands to indi-rect investigation methods allowing to determine the phase transitions occurring in such systems. Inthis project, square artificial spin ice arrays were investigated employing magneto-optical Kerr effectfor the measurement of the magnetic susceptibility. The susceptibility dependence on temperaturewas measured at different frequencies of the applied AC magnetic field for arrays of the differentisland spacing and at two different incident light directions with the respect to the direction of theislands. A peak shift of the real part of susceptibility, χ′, with increasing frequency towards thehigher temperatures was observed. Furthermore, a rough estimation of the relaxation times of themagnetic moments in the islands is given by the analysis of the susceptibility data.

Contents

1 Theoretical part 21.1 Spin-ice structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Thermodynamics of artificial spin-ice structures . . . . . . . . . . . . . . . . . 31.1.2 Dynamics in artificial spin ice structures . . . . . . . . . . . . . . . . . . . . . 5

1.2 Susceptibility and magneto-optical Kerr effect . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Magneto - optical Kerr Effect (MOKE) . . . . . . . . . . . . . . . . . . . . . . 7

2 Experimental part 92.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Experiment protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Results and Discussion 143.1 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Susceptibility calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Peak deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Summary of susceptibility measurements of the samples with 380, 420 and 460 nmspacings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Analysis of dynamics in the square ASI systems . . . . . . . . . . . . . . . . . . . . . 213.3.1 Limitation of analysis of dynamics . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Analysis of spin dynamics employing Neel-Brown model . . . . . . . . . . . . 213.3.3 Analysis of spin dynamics employing critical dynamics model . . . . . . . . . . 22

4 Conclusions 24

5 Outlook 25Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1

Chapter 1

Theoretical part

1.1 Spin-ice structures

Artificial square spin ice structures are the two dimensional analogues of the real three-dimensionalsystems exhibiting intrinsic frustration [1], [2], [3]. One of the examples of naturally occurringfrustration phenomenon is a water ice, in which frustration occurs due to unachievable ground stateof two long and two short oxygen-hydrogen bonds for one oxygen atom. This arrangement of bondsis called the ”ice rule”. The magnetic analogues are the rare-earth pyrochlores, in which frustrationis exhibited due to crystal geometry of this system. This is manifested by the frustration of Ising-likemagnetic moments, which would like to follow the ”ice rule” of two spins in and two spins out atthe corners of tetrahedron as shown in Fig. 1.1.

Figure 1.1: Illustration of the spin ice structure of rear-earth pyrochlores from J. Snyder et al. [2]

Artificial spin ice structures were fabricated as the two dimensional (2D) frustrated model systemsmimicking rare-earth pyrochlores in order to investigate systems exhibiting frustrated behavior. Inartificial spin ice structures magnetic moments of the single domain magnetic islands take the roleof spins in rare-earth pyrochlores [1], [4], [5]. The difference between pyrochlores and ASI structuresis that in the latter spins are oriented in plane since these artificial spin ice structures are fabricatedout of the thin magnetic films. Frustration in such ASI structures arises from the geometric structureof the array and the dipolar interactions occurring between the magnetic moments. Each island isa single magnetic domain with a defined magnetic moment along the long axis of the island due tothe shape anisotropy [1], [4], [5].

There can be sixteen different configurations of magnetic moments in square artificial spin icestructures. According to the energy these configurations can be assigned into four different types asillustrated in Fig. 1.2.

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Figure 1.2: Illustration of vertex types in square artificial ice lattices taken from Kapaklis et al. [4]

Type I and Type II vertices are the states in which the ”ice rule” is fulfilled. The Type Istate is the doubly degenerate lowest energy state, since there are two kinds of spin configurationsleading to that energy state. The Type II possess a dipole magnetic moment due to asymmetric spinconfiguration and therefore its energy is higher and this state is four fold degenerate. In contrast,in Type III and Type IV vertices the ”ice rule” is broken. In Type III vertices three moments arepointing inwards, while the fourth is pointing outwards, or vice versa. This configuration is of higherenergy than the first two types vertices and is eight fold degenerate. The excited state with thehighest energy is the Type IV vertex state is doubly degenerate and in this state all moments arepointing either outwards or inwards in the vertex. [2], [4], [5].

1.1.1 Thermodynamics of artificial spin-ice structures

Placing a paramagnetic material, in which all the magnetic moments are random, in a magneticfield results in the realignment of the magnetic moments parallel to the direction of the appliedmagnetic field. Therefore, the sum of all spin magnetic moments,

∑j Sj = ms, leads to a finite

magnetization. Materials, which exhibit spontaneous magnetization even without being placed inthe external magnetic field, are called ferromagnetic materials [6]. The Ising Hamiltonian for atwo-dimensional ferromagnet in an external magnetic field can be expressed as [6]:

H = −∑ij

JijSi · Sj + gµB

∑j

Sj ·B, (1.1)

where J is the coupling strength between two nearest neighbors, g is the so called Lande g-factor,µB is the Bohr magneton. The sum over ij means that the sum is evaluated for the pairs of thenearest neighbor spins. That is, the spin at a site (i, j) will have four nearest neighbors in the squarearrangement of the lattice at the sites (i, j − 1), (i, j + 1), (i − 1, j) and (i + 1, j). The second termdefines the Zeeman energy, which is the amount by which spin energy levels split in the magneticfield B.

Ferromagnetic materials possess a characteristic parameter, called Curie temperature, Tc. Thisis the temperature at which the transition from the ferromagnetic to the paramagnetic state occurs.At temperatures below the Tc, the coupling strength J is stronger than the thermal energy so spinsin ferromagnetic material tend to order parallel to each other.

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Figure 1.3: Phase diagram for artificial spin ice lattices (black lines (solid and dashed) correspond toenergies, while blue curve is the corresponding magnetic susceptibility dependence on temperature)

Lets consider the artificial spin ice structure, consisting of certain magnetic domain island lattice.Reversal energy, the energy, required to reverse the magnetization, Er, is given as product KV ,where K is a uniaxial anisotropy constant, determined by the shape of the nanometer sized islands,and V is the volume of those islands [1]. At 0 K temperature, the Er is the highest. With increasingthe temperature, thermal fluctuations of the spins in the artificial spin ice system are activated anddue to this, the reversal energy decreases with temperature until finally it becomes equal to zeroat Curie temperature, TC (See Fig. 1.3). As illustrated in this figure, the susceptibility measuredin such experiment would show a peak at this temperature. Drawing the line of thermal energy,kBT , its intersection with the reversal energy, Er, line would give a point on temperature axis,TB, which defines the critical temperature of the islands, called the blocking temperature. Thistemperature is the intrinsic ordering parameter of the islands. At temperatures between TC andTB in the ferromagnetic region of the material, spin flipping is random and independent of theflipping of neighboring spins. However, going down in temperature towards the TB, spin flippingbecomes correlated with the flipping of neighbor spins. Cooling down even further to TB results incritical slowing down of the spin flipping rate, which results in a random but well-defined frozenground state of the spins, which now are completely aware of and correlated with their neighbors[1], [7], [5]. Although spins should ”fall” into the lowest energy ground state, that is, Type I spinconfiguration, the actual state reached would highly depend on the process through which the systemis brought to the ”frozen’ state. That is, different cooling rates and other differing parameters ofexperimental protocol would determine what kind of state will be reached in the end [1], [4], [5].Below the TB islands are completely frozen for the time window of the measurement, but they stillhave a probability to flip. Overall, changing of the temperature leads to re-ordering of the magneticmoments since spins are now excited to the higher energy states. Therefore, different types of verticescan be accessed at different temperatures.

Bringing the islands closer together results in the increase of the dipolar interaction betweenthem. This modifies the reversal energy, Er, that is, the total energy increases by the dipolar energy,Edipolar and the energy barrier of the spin flipping is increased. This way the blocking temperatureof the island is shifted towards higher temperatures and this means that the dipolar energy stabilizesthe magnetic state.

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1.1.2 Dynamics in artificial spin ice structures

Neel-Brown model

Commonly used model to investigate magnetic systems in which particles or domains are non-interatcting, and finding critical parameters is Arrhenius-Neel law [8] :

τ =1

ω=

1

2πf= τ0exp

(KV

kbT

), (1.2)

where f is the magnetic field frequency (Hz), ω is the angular inverse attempt frequency (rad/s),τ0 is the angular inverse attempt frequency, which defines the individual island flipping rate (s),KV is the anisotropy energy (J), with K as anisotropy constant (J/m3) and V – the volume of theparticle (m3), kBT is the thermal energy with kB– Boltzmann constant (1.380650·10−23 J/K) and T– temperature (T). Taking the natural logarithm of the both sides linearizes the Arrhenius-Neel law:

ln1

2πf= lnτ0 +

KV

kbT, (1.3)

where lnτ0 is the intersection of the line with the x-axis and obviously provides information on τ0.

Critical dynamics

Alternative method to analyze the behavior of artificial spin ice systems is assuming criticaldynamics of such systems, which account for the dipolar interactions between magnetic moments inthe system. Magnetic phase transition in the system is revealed the best by the divergence of thecorrelation length, ξ, which diverges as a · ε−ν , where a is the average distance between interactingspins, ε is the reduced temperature, ε = T/Tg − 1, where Tg is the phase transition temperature, andν is the critical exponent, which defines the divergence rate of the correlation length, when T → T+

g .The correlation time, τc, with the microscopic relaxation time of the individual magnetic moment

due to correlated dynamics, τ ∗, is related by the dynamic critical exponent, z, via relation τc ∝τ ∗(ξ/a)z [8], [9]. This means that the data can be fitted to the power law [7], [8], [10]:

τ =1

ω=

1

2πf= τ ∗

(TmTg− 1

)−zν, (1.4)

where Tm is peak position of χ′, zν is the dynamic critical exponent, Tg is the static glass temper-ature of spin glass systems. In the case of artificial spin ice systems, it is the static island blockingtemperature, TB. When the TB is approached from higher temperatures, the transition from super-paramagnetic to ferromagnetic state occurs. At this temperature the magnetic moment fluctuationsin the islands are completely blocked. Tg can be obtained from the static susceptibility measurementas a position of the χ′ peak, that is, when the magnetic field frequency is 0 Hz.

Time scales

The important parameter when talking about the dynamics in artificial spin ice structures, orstructures exhibiting magnetic excitations as a whole, is the time scale of those fluctuations. Ob-serving the peak shifts of the real and the imaginary parts of susceptibility, χ′ and χ”, respectively,towards higher temperatures with increasing frequency one can obtain the information about relax-ation dynamics of the system by analyzing χ′ peak position Tm at respective attempt frequencies,f . There are a couple of methods to determine the spin relaxation times from Tm variation with fin this kind of magnetic systems. First one is by using the model considering Neel-Brown behav-ior of non-interacting nanoparticles (see section 1.1.2). The typical time scales reported for various

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systems of non-interacting particles are from 2 ·10−12 to 3 ·10−11 s for non-interacting Fe-C nanopar-ticles in 0.06% volume concentration sample, reported by Hansen et al. in [8] and 10−23 s for CoFenanoparticles in Co80 Fe20 /Al2O3 multilayers reported Sahoo et al. in [10]. Second one is the modelof interacting magnetic moments described by the power law, describing critical dynamics in thesystem (See section 1.1.2). Typically obtained values for τ ∗ is in the range of 10−8.3±1–10−8±1 s forinteracting magnetic Fe-C (5 and 17 vol.% samples) and CoFe nanoparticle systemsas as reportedby Hansen et al. in [8] and Sahoo et al. in [10], respectively.

1.2 Susceptibility and magneto-optical Kerr effect

One of the ways to investigate magnetism in solid state is the susceptibility measurements. Thisquantity possess a singularity at the transition from paramagnet to ferromagnet, i.e., at the transitiontemperature (for ferromagnetic materials – Curie temperature and Neel temperature for antiferromag-netic materials), the susceptibility diverges to infinity. Study of the magnetic susceptibility allowsanalyzing the critical behavior of the magnetic material at the phase transition. The informationwhich can be obtained from the susceptibility measurements can be used to evaluate such parametersas the critical exponents, the magnetic moment and the critical temperature of the specimen. Inorder to experimentally measure the susceptibility, it is required to carry out the experiment at lowfrequencies below 10 kHz and the low magnetic fields (below mili Tesla’s to below couple of Tesla’sdepending on the material). In this way the magnetization of the sample does not reach the satura-tion value [6], [11], [12]. From the definition, susceptibility is the change in magnetization obtaineddue to the change in applied magnetic field:

χ =dM

dH, (1.5)

where M is magnetization and H is the applied magnetic field. This definition is generic andapplicable to both AC and DC susceptibility. The ac susceptibility is a complex quantity, defined in[11] as

χ = χ′ + iχ′′, (1.6)

where χ′ is the real part of the susceptibility, which is related to reversible magnetization process, andstays in-phase with the oscillating field; χ′′ is the imaginary part of the susceptibility, which identifiesthe losses in the system that arises from magnetic inhomogeneities, geometry and morphology of thespecimen as well as to the magnetic anisotropy [11], [13].

The susceptibility is usually measured at the low magnetic fields and the minor loops of themagnetization are recorded. From the slope of the minor loop the susceptibility value can be obtainedas in equation 1.5. The maxima of the real part of the susceptibility defines the critical temperature,Tc. Below the Tc, material is in the ferromagnetic state and the recorded minor loop possesses almostnegligible area. The slope of the loop is very small as well and therefore χ′ goes to 0. However, whenT approaches Tc, remanent magnetization and coercivity go to zero, which results in the significantincrease in the magnitude of the applied magnetic field H with respect to the coercive field. In thisway, the measured χ′ increases markedly as well, since at Tc the phase transition from ferromagneticto paramagnetic region takes place. In the paramagnetic region no hysteresis is present since themagnetic moments cannot be aligned by the small driving magnetic field and thermal fluctuationsovercomes their inclination to order ferromagnetically at high temperatures.

Imaginary part of susceptibility, χ′′, corresponds to losses in the system. It is shifted to lowertemperatures and has a peak in the ferromagnetic region, at temperature lower than that of the χ′

peak. At temperatures below the Tc, the coercive field is much larger than the applied magnetic

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field, and therefore, the energy required to overcome magnetization reversal barrier is not accessible.Therefore, spins in the islands cannot be flipped and since no absorption occurs, the χ′′ is approxi-mately zero. Furthermore, in the paramagnetic region the remanent magnetization and coercive fieldare zero, so no energy loss is present in the system since temperatures are high and spins can flipwithout any constrains, so the imaginary part of susceptibility is zero [11].

Magnetic measurements can be carried out at AC or DC magnetic fields depending on the prop-erties of interest. Its roots are in the investigation of thin films. In AC measurements, AC field isapplied and therefore the resulting AC moment, induced in the sample, is measured, providing theinformation about the magnetization dynamics, since the induced moment is time-dependent. Incontrast, during DC magnetometry measurements, the sample is magnetized by a constant appliedmagnetic field and therefore yields information about the equilibrium magnetization in a sample [14].

Magnetic properties of magnetic samples usually are measured employing vibrating sample mag-netometry (VSM) [15], superconducting quantum interference device (SQUID) [16], magneto-opticKerr effect (MOKE), magnetic circular X-ray dichroism (MCXD) [11], [12]. The MOKE techniquewas the one employed in this project and is described in the following section.

1.2.1 Magneto - optical Kerr Effect (MOKE)

Origin of MOKE

The magneto-optical such as Kerr effect and Faraday effect are employed in the investigationsof magnetic materials. The magneto-optical Kerr effect (MOKE) takes place when the polarizationplane of the light is rotated upon reflection from the magnetic surface, while Faraday effect occurswhen the light is transmitted through the transparent magnetic surface [12], [17]. The general prin-ciples of these two effects are described further. reflective magnetic surface magneto-optic Kerr effecttakes place, while for transparent magnetic surfaces it is the Faraday effect. As the polarized lightinteracts with the magnetic material, its electrical field induces motions of the electrons containedin that medium [17]. In the absence of the external magnetic field the electrons would follow theelectric field of the polarized light, that is, if the light were left-circularly polarized (LCP), the elec-trons would move in left-circular manner, and vice versa for the right-circularly polarized (RCP)light. Upon application of external magnetic field along the propagation direction of the light, therewill be an additional Lorentz force acting upon each electron in medium [17]. This will change theradius of the electron’s circular motion and result in a change of dielectric constants. This is the socalled Faraday effect which originates from the Lorentz force of the applied magnetic field.

The reason why the Faraday effect occurs is the spin-orbit interaction due to which electron spin iscoupled to its motion and thus gives a substantial Faraday rotation in a ferromagnetic material [17].As the electron moves through the electric field of the light, it ”feels” the magnetic field of the light ina way that its spin interacts with the magnetic component of the electromagnetic radiation. In thisway, magnetic moment of the electron is coupled with its motion. This effect is mostly pronouncedin ferromagnetic materials due to the unequal number of spin-up and spin-down electrons, whereasin nonmagnetic materials this effect is still present but insignificant due to balance between electronsof opposite spins [12].

Experimental facilitation of MOKE

Magneto-optic Kerr effect is exploited in studies of magnetic materials. Depending on the opticaland magnetic configurations, that is, the direction of incident light and the direction of the magneticfield in the system, MOKE experiments can be carried out in three different modes, that is, polar,transverse or longitudinal MOKE modes [12], [17], [18] :

• In polar MOKE (PMOKE ) magnetization is measured at normal incidence, with magneticfield vector being perpendicular to the sample surface;

7

• In transverse MOKE (TMOKE ) magnetization is measured at oblique angles with magneticfield vector in plane of sample surface and parallel to the optical plane of incidence;

• In longitudinal MOKE (LMOKE ) magnetization is also measured at oblique angles with themagnetic field vector in plane of sample surface and perpendicular to the optical plane ofincidence.

This is illustrated in the Figure 1.4 below.

Figure 1.4: Different configurations of MOKE measurement: ES – s–polarized light, E ′p – additionalp–polarized light component due to the reflection from a magnetic sample, HT – transverse MOKE,HL — longitudinal MOKE, HP – polar MOKE

Depending on the chosen MOKE geometry the result of the reflection from the magnetic surfaceis different for linearly polarized light: in PMOKE and LMOKE the reflected light is polarizedorthogonally compared to the incident light; in TMOKE only the amplitude of the incident light isaffected due to reflection [12]. Usually, for investigation of ASI structures LMOKE or TMOKE isemployed, since these structures consists of single domain islands in which magnetic moment lies inplane along the long axis due to shape anisotropy.

8

Chapter 2

Experimental part

2.1 Samples

Samples, investigated in this project were fabricated on MgO substrate by sputtering techniqueand the resultant composition of the samples was the following: Pd(20A)/Fe(2.2ML)/Pd(400A)/V(15A)/MgO substrate (See Fig. 2.1). In such material composition, iron is the ferromagnetic material (intemperature region below T c), which polarizes the palladium film below and above. The polarizedPd layer smoothen the magnetic profile of the sample.

Figure 2.1: Schematics of an island structure

The square artificial spin ice structures were fabricated by an electron beam lithography (EBL)on the described δ-doped Pd films. Fabricated islands are of 300 nm in length and 100 nm in width(See Fig. 2.2) and make up three kinds of arrays different inter-island spacing parameters of 380,420 and 460 nm (See Table 2.1 and Fig. 2.3) .

Figure 2.2: Island dimensions (Arrow illustrates the color code of the magnetic moment orientationof the island)

Sample Width [nm] Length [nm] Spacing [nm]Large1

100 300380

Large2 420Large3 460

Table 2.1: Sample characteristics

9

Figure 2.3: Illustration of arrays with different island spacings: 380, 420 and 460 nm

From earlier measurements and calculations the energies of vertex states were calculated by thesoftware MuMax3 (see Table 2.2).

Demagnetizing energies ·10−19 [J]Type I Type II Type III Type IV

Samples, pitch [nm]380 1.085 1.114 1.177 1.398420 1.131 1.161 1.196 1.339460 1.155 1.180 1.202 1.297

Table 2.2: Demagnetizing energies for the 4 vertex types and the 3 investigated samples

These energies can be related to the energy required to change one vertex type to another byreversing the magnetization direction of a certain island. For example, changing the magnetizationof an island in a Type I vertex, would lead to a Type III vertex, and the energy required to do sowould be equal to the difference between the demagnetizing energies of Type III and Type I vertices(See Fig. 1.2)

2.2 Experimental setup

The custom made experimental setup employed for MOKE measurements, named HOMER, dur-ing this project consisted of a laser, polarizer and analyzer, irises, Helmholtz coils isolated in theMu-metal tube (isolating magnetic fields), sample chamber with cryostat and sensors to measuretemperature, detector, lock-in amplifier (See Fig. 2.5).

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Figure 2.4: Photograph of LMOKE setup HOMER

Figure 2.5: Scematics of LMOKE setup HOMER. L – laser, F – lens, P – polarizer, S – sample, M– Helmholtz coils, I – iris, A – analizer, D – detector.

The working principle of the employed MOKE setup is following: a light, from a laser withwavelength of 660 nm and output power of 60 mW, is p–polarized by passing through the polarizer.Then the polarized light enters the Mu-metal cylinder (through a hole in it) that shields the samplefrom the Earth magnetic field. This is needed since the magnetic susceptibility measurements arevery sensitive to the external magnetic fields. The cylinder also contains Helmholtz coils, whichproduce the magnetic field, modifying the magnetization in the sample, which is placed in thecryostat. The polarized light interacts with the sample and is reflected out of the tube into thesecond polarizer, which works as analyzer, and thus lets polarized light with rotated polarizationplane (due to interaction with the sample) to pass through into the photodetector. If the sample ismagnetic, the reflected beam consists of the main p–polarized as well as of an additional s–polarizedlight component. The ratio of component of s–polarized light (Es) to component of p–polarizedlight (Ep) is the Kerr rotation Es/Ep (in radians). The measurement of the additional polarizedlight component is the goal of the MOKE measurements of the magnetic materials since it reflectsthe change in the polarization of the light. Experimentally this is implemented by placing a linearpolarizer in front of the detector at a small angle, δ, out from the s–axis.

Two quantities, obtained from the AC susceptibility measurements, are used for calculations ofits real and imaginary components: the magnitude of the susceptibility, R, which can be expressed

11

as in Eq. 1.6 and the phase shift between the real and imaginary parts of susceptibility, φ. The realpart of the susceptibility, χ′, corresponds to the in-phase component, while imaginary part, χ′′, tothe out-of-phase component. These two components are related by equations given in [12], [14]:

χ′ = R · cosφ (2.1)

and

χ′′ = R · sinφ, (2.2)

The AC susceptibility measurement is sensitive to the changes in the M(H), therefore, smallshifts can be detected in the magnetization of the sample. This allows determination of the criticalbehavior of the magnetic system. For instance, measurement of χ vs. T allows determination of thecritical temperature of the material, at which it undergoes the ferromagnet-paramagnet transition[11], [14].

2.3 Experiment protocol

Samples were measured in HOMER LMOKE setup in magnetic fields of 0.05, 0.1 and 0.2 mT atdifferent frequencies of 0.45, 4.5, 45 and 450 Hz. The amplitudes used for the field generation in theHelmholtz coils to obtain the desired magnetic field at each frequency are provided in a Table 2.3below.

Field [mT] Frequency [Hz] Amplitude [mV]0.05 450 1600.1 45 35

450 3150.2 0.45 30

4.5 3045 70450 630

Table 2.3: Used field amplitudes to generate the magnetic fields at different frequencies.

Furthermore, each of the samples was measured in two directions, [10] and [11]. This allows toinvestigate the magnetic response of two or four islands, respectively (See fig. 2.6).

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Figure 2.6: Schematics of the measurement setup and illustration of measurements in [10] and [11]direction.

Measuring samples along two different directions (Fig. 2.6) allows accessing two kinds of ex-citations along those directions. Measurement along [10] direction allows accessing excitations ofspin flipping in the vertex along the islands with long axis parallel to the direction of light, whereasexcitations of islands with long axis perpendicular to the direction of incident light are not probedin this configuration. On contrary, when the light is along [11] direction, spin flipping is probed fromall four islands, since each magnetic moment of those islands has a projection along the direction ofincident light.

The experiment protocol was the following: firstly, the sample was heated up to 380 K degrees in azero magnetic field and afterwards it was zero-field cooled (ZFC) to 100±1 K at the rate of 1 K/min.The susceptibility measurement was started at 100±1 K temperature after allowing the system toequilibrate for 10 min and then turning on the desired magnetic field in the coils by setting the fieldamplitude and frequency. Afterwards temperature was raised at the constant heating rate of 0.2, 0.5or 2 K/min during different measurements. The lock-in time constant during the measurements was30 s. That is, in order to obtain one point, the amplitude R and φ (among the other parameters)are measured for 30 s and measurement is stopped for 90 s after which the second point is againmeasured for 30 s, and so on. After the measurement was finished the field was turned off and thesample was ZFC to 100 K again.

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Chapter 3

Results and Discussion

3.1 Data analysis

3.1.1 Susceptibility calculation

The output data of the MOKE measurement, employed for the susceptibility evaluation, are thesample temperature, T, amplitude of the signal, R and phase, θ. Employing the following equations:

χ′ = R · cos(θ − θ0) (3.1)

andχ′′ = R · sin(θ − θ0), (3.2)

where θ0 is the phase shift between the applied field and the measured response (this phase shifthas different origins such as lock-in cables), the real and imaginary parts of the susceptibility areobtained.

To begin with, the obtained data was used to calculate real and imaginary parts of susceptibility,assuming that θ0 = 0. Recalling the fact that at high temperatures, in paramagnetic region, behav-ior of magnetic moments is fully determined by thermal fluctuations, and spins are allowed to flipwithout energy losses, the χ′′ was set to zero in high temperature region, T &330 K. It gives thatθ0 = θ in the high temperature region, and therefore, the θ was averaged and taken as θ0 in thattemperature range. Afterwards, the obtained θ0 value was plugged in into the previous equations andthe real and imaginary parts of the susceptibility were recalculated. The described data treatmentis illustrated in figures 3.1 and 3.2.

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Figure 3.1: χ′ and χ′′ raw data for sample of 460nm spacing at 4.5 Hz and 0.2 mT magnetic field

Figure 3.2: θ and estimated θ0 for sample of 460nm spacing at 4.5 Hz and 0.2 mT magnetic field

From the Fig. 3.2 it can be seen that the θ0 line corresponds to averaging of the θ values in thenoisy, high temperature region, where spin fluctuations are purely thermal, therefore, this region canbe called the superparamagnetic region. This sets the imaginary part of the susceptibility to be zeroin that superparamagnetic region (see Fig. 3.3).

Figure 3.3: Recalculated χ′ and χ′′ with θ0 taken into calculations for sample of 460 nm spacing at4.5 Hz and a 0.2 mT field

However, the θ dependence on temperature was not this straightforward to interpret for thehighest frequency of 450 Hz. As can be seen in the Fig. 3.4 below, above island blocking temperature,TB, θ exhibits a sudden jump to the opposite phase. At the moment the reason behind this abruptθ phase change is not clear, but it can only be suspected that this is related to the material phasetransition at the Curie temperature from the superparamagnetic to paramagnetic (See Fig. 1.3).Therefore the θ0 was obtained by averaging θ0 in the region 270K . T . 290K.

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Figure 3.4: θ and estimated θ0 for sample of 460 nm spacing at 450 Hz 0.2 mT field

3.1.2 Peak deconvolution

After the recalculation of the susceptibility peaks, all of them were deconvoluted. In all of the χ′

vs. T curves two peaks can be observed, while in some of them even up to four peaks can be spotted(See Fig. 3.5 and 3.6).

Figure 3.5: Deconvolution of χ′ into two Gaus-sian functions for χ′ vs. T of sample of 380 nmspacing at 4.5 Hz and 0.2 mT field

Figure 3.6: Deconvolution of χ′ into four Gaus-sian functions for χ′ vs. T of sample of 380 nmspacing at 4.5 Hz 0.2 mT field

However, due to unknown origin of those features and in order to maintain consistent data analy-sis, peaks were deconvoluted using two Gauss functions: one as the most intense peak and the secondpeak at around 300 K present in all of the the χ′ vs. T curves. After deconvolution, the Peak 1 (seenotation in Fig. 3.5) position on temperature axis, denoted as Tm, was employed for further dataanalysis.

16

3.2 Summary of susceptibility measurements of the samples

with 380, 420 and 460 nm spacings

In this section results of the susceptibility measurements of the samples with different latticespacings, 380, 420 and 460 nm, measured in the field of 0.2 mT at different frequencies, i.e. 0.45,4.5, 45 and 450 Hz by a 0.5 K/min heating rate are provided.

In Figures 3.7 – 3.12 it can be seen that for all samples measured at both directions ([10] and[11]) the χ′ and χ′′ peak positions shift towards higher temperatures with increasing frequency of theapplied field. This can be explained by the following. At low frequencies of the magnetic field thespins in the islands have a longer time to change their direction and when the temperature is reachedat which their flipping rate matches the magnetic field frequency, or in other words, when spinsbecome susceptible to the magnetic field oscillations, the susceptibility peak is obtained. However,when the magnetic field frequency is increased, spins are forced to flip at the higher rate and theyhave a shorter time to accommodate to the field frequency. Therefore, the increasing temperaturemakes spins to flip faster and more susceptible to the magnetic field oscillation frequency. In thisway, the increase in temperature matches the spin flipping rate with the magnetic field rate, andthus the susceptibility peak is obtained at higher temperature for increased frequency.

The intensity of χ peaks for samples of 380 and 460 nm lattice spacings appears to be higherfor peaks measured at [11] direction than for [10] direction. This could be explained by looking atfigure 2.6: when the light is incident onto a sample at [11] directions, spin excitations induced in allfour islands have a projection along the direction of incident light, and the intensity of fluctuationcoming from each of the four islands is proportional to cos(45◦). On the other hand, when thelight is incident along [10] direction, only two islands have projections along direction of light, andthe intensity of fluctuations measured of both of these islands theoretically should be maximum.Simultaneously, the other two islands, perpendicular to the direction of the incident light, should notbe probed. Therefore, the overall fluctuation intensity detected should be smaller than that detectedwhen measuring along [11] direction. Of course, increased number of islands being probed results inincreased noise in the signal, which results in a smaller signal-to noise-ratio for the χ′ and χ” peaks,as is evidenced in right graphs of the figures 3.7 – 3.12. However, this observation about the intensitybeing higher when measured along the [11] direction than along the [10] direction is conditional, sincein practice the measured intensity is highly dependent on the system alignment, and this statementcan be false, as is for the sample of 420 nm lattice spacing.

Figure 3.7: χ′ dependence on temperature at 0.2 mT field for sample with 380 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph)

17

Figure 3.8: χ′′ dependence on temperature at 0.2 mT field for sample with 380 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph)

Figure 3.9: χ′ dependence on temperature at 0.2 mT field for sample with 420 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph)

Figure 3.10: χ′′ dependence on temperature at 0.2 mT field for sample with 420 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph)

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Figure 3.11: χ′ dependence on temperature at 0.2 mT field for sample with 460 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph)

Figure 3.12: χ′′ dependence on temperature at 0.2 mT field for sample with 460 nm spacing measuredat [10] direction (left graph) and [11] direction (right graph).

After peak deconvolution into Gaussian functions, the obtained Peak 1 positions are summarizedin a Table 3.1

19

Tm [K]Spacing [nm] Direction [10]

0.45 Hz 4.5 Hz 45 Hz 450 Hz380 250.78±2.62 253.94±0.81 264.06±0.68 272.70±0.06420 233.86±1.05 242.85±0.42 250.46±0.08 258.99±0.06460 222.36±0.32 229±0.21 238.86±0.29 250.21±0.04

Tm [K]Spacing [nm] Direction [11]

0.45 Hz 4.5 Hz 45 Hz 450 Hz380 257.34±1.83 261.67±1.17 265.95±0.24 274.78±0.09420 237.21±2.31 244.55±1.42 252.98±0.14 261.25±0.07460 224.58±1.11 234.50±0.46 240.66±0.11 248.80±0.06

Table 3.1: Tm positions obtained at the different magnetic field frequencies at two measurementdirections for different island spacings.

Figure 3.13 serves as a visual summary of the Peak 1 (See Fig. 3.5) position tracking overdifferent island spacing parameters for χ′ measured at four decades of frequencies at two differentdirections: [10] and [11]. Figure 3.13 shows that increasing the lattice spacing results in shiftingof the Tm position to lower temperatures. This is expected from the thermodynamics of artificialspin ice structures (see Fig. 1.3). Bringing islands closer to each other results in stronger dipolarinteractions between them, which adds up to reversal energy and therefore increases the effectivereversal energy. This results into TB shifted to higher temperatures since the intersection point ofEr + Edipolar curve with kBT is shifted further. Furthermore, Tm increases with increasing appliedmagnetic field frequency. This Tm dependence is further used to explore the dynamics in the squareartificial spin ice structures (see section 3.3).

Figure 3.13: Tm position dependence on island spacing at different frequencies and different samplemeasurement directions

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3.3 Analysis of dynamics in the square ASI systems

3.3.1 Limitation of analysis of dynamics

Before starting the analysis of dynamics of square ASI systems, it is important to stress outthat the measurements carried out in this project were done as a basis for further investigations ofthis type of square ASI systems. The amount of data points measured for each sample at each oftwo directions is four, since we measured at four different frequencies, corresponding to four decades(0.45, 4.5, 45 and 450 Hz). The expected linear behavior in the logarithmic plot of the measurementscan be identified. However, the fitting to either the Arrhenius equation (Eq. 1.2) or the power law(Eq. 1.4) is just a rough estimation to get an idea about the dynamical behavior of such ASI systems.This is due to the fact that two parameters have to be obtained from fitting the data to Arrheniusequation and three parameters – from fitting to the power law. Therefore, the obtained parametersfrom the fits have the standard deviation of 2σ from the mean parameter value.

3.3.2 Analysis of spin dynamics employing Neel-Brown model

The susceptibility measurement of the spin ice structures reveals more information about suchsystems than just a characteristic shift towards higher temperatures of the susceptibility peak withthe increasing frequency of the magnetic field. As was seen before in section, the obtained suscepti-bility peaks contain two features at different temperatures. The observed features can be correlatedto the different vertex states of ASI systems. Each vertex type has a specific energy which can betranslated into a temperature in the susceptibility curve. The identification of possible transitionsand calculation of energies from the susceptibility peaks can be related to the demagnetizing energiesof the vertices (See Table 2.2) since the interaction strength is changed with increasing temperature:at temperatures below the critical temperature the anisotropy energy is stronger than thermal energyand magnetic moments incline to stay ordered ferromagnetically. However, at the critical temper-ature thermal energy begins to compete with the anisotropy energy and phase transition from theferromagnetic to paramagnetic state occurs. At even higher temperatures the thermal energy is muchhigher than the anisotropy energy and magnetic moments are completely random. Here the previ-ously described Arrhenius-Neel model (eq. 1.2) is employed to investigate the square ASI systems.The parameters that can be obtained from the fit of lnτ and T to eq.1.3 are τ0 and anisotropy energyKV . A summary of the obtained parameters is provided in the Table 3.2 and a presentation of thefitting is provided in Fig. 3.14. The obtained parameters have a 2σ standard deviation from themean parameter value.

Spacing[nm]

Direction τ0 [s] KV [J]

38010 4.32·10−36 2.76·10−19

11 2.28·10−48 3.84·10−19

42010 2.98·10−32 2.31·10−19

11 1.33·10−33 2.44·10−19

46010 7.63·10−28 1.88·10−19

11 2.87·10−32 2.23·10−19

Table 3.2: Fitting parameters from Arrhenius-Neel law

21

Figure 3.14: Arrhenius-Neel law fitting of data obtained for sample of 460 nm spacing measured at10 direction

Obtained anisotropy energy values are of the order of 10−19 J. These values, although with thelarge standard deviation of 2σ, are comparable to the demagnetizing energies provided in Table 2.2.However, this model does not take into account the interaction between the islands, however, theislands, comprising the square ASI samples do interact. Therefore, it is not unexpected that thefitting to Arrhenius - Neel law gives physically meaningless results, that is, τ0 is in the order of10−38±10 s.

3.3.3 Analysis of spin dynamics employing critical dynamics model

The frequency, f, and peak 1 position, Tm, was fitted to the power-law function given in eq. 1.4in order to extract the following parameters: Tg, zν and τ ∗. Tg is extracted as the peak positionof the susceptibility curve obtained upon zero field cooling. The fitting parameters were obtainedwith big error bars due to the small data set (4 data points) in comparison to the amount of fittingparameters (3 fitting parameters).

Presenting the fitting in the log-log plot shows that the data points are not laying perfectly onthe fitted curve as can be seen in Fig. 3.15. This is due to the large error bars given by the 2σstandard deviation from the mean value of fitted parameter.

Figure 3.15: Power law fitting on log-log scale for the sample of 460 nm lattice spacing measured at10 direction

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Nonetheless, the fitted parameters are given in the Table 3.3 below. Values obtained for therelaxation time τ ∗, are in the nano- to microsecond range and are comparable with the resultsreported for the Fe-C nanoparticle samples of 5 and 17% volume concentrations in [8] and CoFemagnetic nanoparticles in Co80Fe20/Al2O3 multilayers in [10].

Spacing[nm]

Direction τ ∗ [s] Tg [K] zν

38010 4.74·10−9 245.3 -4.76711 5.31·10−8 251.7 -4.145

42010 5.49·10−6 224.7 -3.45811 1.91·10−7 224.8 -4.988

46010 1.29·10−6 214.5 -3.79211 4.90·10−6 214.0 -3.72

Table 3.3: Fitting parameters

23

Chapter 4

Conclusions

In this project, square artificial spin ice structures with different lattice spacing parameters wereinvestigated. The investigation was conducted employing magneto-optical Kerr effect to measure thesusceptibility dependence on temperature at series of different frequencies.

The χ′ peak position shifts towards higher temperatures with decreasing island spacing. This isdue to the increasing dipolar interaction energy, which increases the effective magnetization reversalenergy. Therefore, the higher temperature is required to overcome the magnetization reversal barrier,that is, to flip the spin.

Furthermore, the χ′ peaks shift towards higher temperatures with the increasing frequency. Atthe low magnetic field frequencies spins have a longer time to flip than at the high frequencies.Furthermore, temperature affects the spin flipping rate. The increasing temperatures lead to thedecreasing magnetization reversal barrier. That is, the higher is the temperature, the faster is thespin flipping. Therefore, the susceptibility peak position is at the temperature which induces thespin flipping of the same rate as the magnetic field oscillations. Thus, at the low frequencies spinsare susceptible to magnetic field oscillations at lower temperatures, and therefore susceptibility peakmaximum is at the lower temperatures. Whereas at the higher frequencies, the higher temperatureis needed to flip the magnetization direction of the island at the same rate as the magnetic fieldfrequency.

The position of the χ′ peak was obtained from each measurement, carried out at the differentmagnetic field frequencies. This data was employed for the analysis of the dynamics in the squareASI systems employing different methods. One of them is the Neel-Brown model, which does nottake the particle interactions into account. As can be expected, this model is not suitable for theinvestigated square artificial ice structures since islands comprising these structures interact viadipolar interactions. And indeed, fitting the data to the Arrhenius-Neel law gave unphysical valuesof island flipping rate, τ0, that is 10−38±10 s. The second model employed in this project for the dataanalysis was considering critical dynamics in artificial spin ice structures. The data was fitted to thepower law and the obtained values of τ ∗ are in the order of nano- to microseconds. However, theobtained values of observables are not completely trustworthy due to the small set of frequencies usedfor measurements which results in 2σ standard deviation from the parameter value. Nevertheless,this project serves as a guideline for further investigations and data analysis.

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Chapter 5

Outlook

Further measurements should be carried out in order to better understand the nature of thesquare artificial spin ice structures:

• Measurements at more frequencies between 0.45 and 450 Hz should be carried out in order toincrease the accuracy of the data fitting;

• Measure the susceptibility after field-cooling (FC) to observe the effect of the magnetic field,which tends to align the spins and induce a certain vertex configuration. This effect canbe observed when measuring the susceptibility after FC, since there some more pronounced ordiminished features in the susceptibility peak could be expected, compared to the measurementafter the zero-field-cooling;

• Carry out measurements of the susceptibility during cooling so that the probing of spin fluctua-tions would be started from the same state, in which spins are allowed to fluctuate independentlyof their neighbors, and not from ground state, which is random;

• Measure the susceptibility over wider temperature range to obtain the peak at Curie tempera-ture or measure susceptibility of the structures with lower TC to observe two phase transitions,that is, the material phase transition from the paramagnetic to ferromagnetic material and theisland transition from superparamagnetic to ferromagnetic regime. The measurement could aswell allow to explain th abrupt change in the phase angle θ at temperatures above TB observedin measurements at 450 Hz frequency.

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Acknowledgement

I would like to express sincere gratitude to my supervisor Dr. Vassilios Kapaklis for providingan opportunity to participate in the research of the interesting phenomena of the artificial spin ice,for sharing his knowledge and experience, and for the invaluable help, guidance and patience duringthis project.

Moreover, I would like to acknowledge my second supervisor PhD student Henry Stopfel forteaching about the experimental part of the project, for helping to obtain the data, for sharing hisknowledge and insights, for the help to prepare the presentation and the feedback on this report.

Furthermore, I would like to thank Dr. Spyridon Pappas for the guidance in the laboratory, forthe useful scientific discussions which lead to a better understanding of the problems and methodsof solving them.

Last but not the least, I would like to acknowledge PhD student Erik Ostman for his technicalhelp throughout the project, for sharing his knowledge about theoretical and experimental aspectsof the topic, for providing the demagnetizing energies of the vertices for the investigated samples andfor the feedback on this report.

List of Figures

1.1 Illustration of the spin ice structure of rear-earth pyrochlores from J. Snyder et al. [2] 21.2 Illustration of vertex types in square artificial ice lattices taken from Kapaklis et al. [4] 31.3 Phase diagram for artificial spin ice lattices (black lines (solid and dashed) correspond

to energies, while blue curve is the corresponding magnetic susceptibility dependenceon temperature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Different configurations of MOKE measurement: ES – s–polarized light, E ′p – addi-tional p–polarized light component due to the reflection from a magnetic sample, HT

– transverse MOKE, HL — longitudinal MOKE, HP – polar MOKE . . . . . . . . . . 8

2.1 Schematics of an island structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Island dimensions (Arrow illustrates the color code of the magnetic moment orientation

of the island) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Illustration of arrays with different island spacings: 380, 420 and 460 nm . . . . . . . 102.4 Photograph of LMOKE setup HOMER . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Scematics of LMOKE setup HOMER. L – laser, F – lens, P – polarizer, S – sample,

M – Helmholtz coils, I – iris, A – analizer, D – detector. . . . . . . . . . . . . . . . . 112.6 Schematics of the measurement setup and illustration of measurements in [10] and [11]

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 χ′ and χ′′ raw data for sample of 460 nm spacing at 4.5 Hz and 0.2 mT magnetic field 153.2 θ and estimated θ0 for sample of 460 nm spacing at 4.5 Hz and 0.2 mT magnetic field 153.3 Recalculated χ′ and χ′′ with θ0 taken into calculations for sample of 460 nm spacing

at 4.5 Hz and a 0.2 mT field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 θ and estimated θ0 for sample of 460 nm spacing at 450 Hz 0.2 mT field . . . . . . . . 163.5 Deconvolution of χ′ into two Gaussian functions for χ′ vs. T of sample of 380 nm

spacing at 4.5 Hz and 0.2 mT field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 Deconvolution of χ′ into four Gaussian functions for χ′ vs. T of sample of 380 nm

spacing at 4.5 Hz 0.2 mT field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 χ′ dependence on temperature at 0.2 mT field for sample with 380 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph) . . . . . . . . . . . 173.8 χ′′ dependence on temperature at 0.2 mT field for sample with 380 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph) . . . . . . . . . . 183.9 χ′ dependence on temperature at 0.2 mT field for sample with 420 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph) . . . . . . . . . . 183.10 χ′′ dependence on temperature at 0.2 mT field for sample with 420 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph) . . . . . . . . . . 183.11 χ′ dependence on temperature at 0.2 mT field for sample with 460 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph) . . . . . . . . . . 193.12 χ′′ dependence on temperature at 0.2 mT field for sample with 460 nm spacing mea-

sured at [10] direction (left graph) and [11] direction (right graph). . . . . . . . . . . 19

27

3.13 Tm position dependence on island spacing at different frequencies and different samplemeasurement directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.14 Arrhenius-Neel law fitting of data obtained for sample of 460 nm spacing measured at10 direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.15 Power law fitting on log-log scale for the sample of 460 nm lattice spacing measuredat 10 direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

28

List of Tables

2.1 Sample characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Demagnetizing energies for the 4 vertex types and the 3 investigated samples . . . . . 102.3 Used field amplitudes to generate the magnetic fields at different frequencies. . . . . . 12

3.1 Tm positions obtained at the different magnetic field frequencies at two measurementdirections for different island spacings. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Fitting parameters from Arrhenius-Neel law . . . . . . . . . . . . . . . . . . . . . . . 213.3 Fitting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

29

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