Investigation of artificial spin ice structures employing magneto-optical Kerr effect for susceptibility measurements Report 15 credit project course Materials Physics division, Department of Physics and Astronomy, Uppsala University Student: Agne Ciuciulkaite Supervisors: Vassilios Kapaklis Henry Stopfel 2015 Abstract Artificial spin ice structures are two-dimensional systems of lithographically fabricated lattices of elongated ferromagnetic islands, which interact via dipolar interaction. These systems have been shown to be a suitable playground to study the magnetic, monopole-like, excitations, similar to those 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 ice structures 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. In this project, square artificial spin ice arrays were investigated employing magneto-optical Kerr effect for the measurement of the magnetic susceptibility. The susceptibility dependence on temperature was measured at different frequencies of the applied AC magnetic field for arrays of the different island spacing and at two different incident light directions with the respect to the direction of the islands. A peak shift of the real part of susceptibility, χ0, with increasing frequency towards the higher temperatures was observed. Furthermore, a rough estimation of the relaxation times of the magnetic moments in the islands is given by the analysis of the susceptibility data. Contents 1 Theoretical part 2 1.1 Spin-ice structures . .2 1.1.1 Thermodynamics of artificial spin-ice structures . .3 1.1.2 Dynamics in artificial spin ice structures . .5 1.2 Susceptibility and magneto-optical Kerr effect . .6 1.2.1 Magneto - optical Kerr Effect (MOKE) . .7 2 Experimental part 9 2.1 Samples . .9 2.2 Experimental setup . 10 2.3 Experiment protocol . 12 3 Results and Discussion 14 3.1 Data analysis . 14 3.1.1 Susceptibility calculation . 14 3.1.2 Peak deconvolution . 16 3.2 Summary of susceptibility measurements of the samples with 380, 420 and 460 nm spacings . 17 3.3 Analysis of dynamics in the square ASI systems . 21 3.3.1 Limitation of analysis of dynamics . 21 3.3.2 Analysis of spin dynamics employing N´eel-Brown model . 21 3.3.3 Analysis of spin dynamics employing critical dynamics model . 22 4 Conclusions 24 5 Outlook 25 Acknowledgement . 26 List of figures . 30 List of tables . 30 Bibliography . 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-dimensional systems exhibiting intrinsic frustration [1], [2], [3]. One of the examples of naturally occurring frustration phenomenon is a water ice, in which frustration occurs due to unachievable ground state of two long and two short oxygen-hydrogen bonds for one oxygen atom. This arrangement of bonds is called the "ice rule". The magnetic analogues are the rare-earth pyrochlores, in which frustration is exhibited due to crystal geometry of this system. This is manifested by the frustration of Ising-like magnetic moments, which would like to follow the "ice rule" of two spins in and two spins out at the 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 systems mimicking rare-earth pyrochlores in order to investigate systems exhibiting frustrated behavior. In artificial spin ice structures magnetic moments of the single domain magnetic islands take the role of spins in rare-earth pyrochlores [1], [4], [5]. The difference between pyrochlores and ASI structures is that in the latter spins are oriented in plane since these artificial spin ice structures are fabricated out of the thin magnetic films. Frustration in such ASI structures arises from the geometric structure of the array and the dipolar interactions occurring between the magnetic moments. Each island is a single magnetic domain with a defined magnetic moment along the long axis of the island due to the shape anisotropy [1], [4], [5]. There can be sixteen different configurations of magnetic moments in square artificial spin ice structures. According to the energy these configurations can be assigned into four different types as illustrated in Fig. 1.2. 2 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 I state is the doubly degenerate lowest energy state, since there are two kinds of spin configurations leading to that energy state. The Type II possess a dipole magnetic moment due to asymmetric spin configuration 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 are pointing inwards, while the fourth is pointing outwards, or vice versa. This configuration is of higher energy than the first two types vertices and is eight fold degenerate. The excited state with the highest energy is the Type IV vertex state is doubly degenerate and in this state all moments are pointing 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 magnetic field results in the realignment of the magnetic moments parallel to the direction of the applied P magnetic 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 in the external magnetic field, are called ferromagnetic materials [6]. The Ising Hamiltonian for a two-dimensional ferromagnet in an external magnetic field can be expressed as [6]: X X H^ = − JijSi · Sj + gµB Sj · B; (1.1) ij j where J is the coupling strength between two nearest neighbors, g is the so called Land´eg-factor, µB is the Bohr magneton. The sum over ij means that the sum is evaluated for the pairs of the nearest neighbor spins. That is, the spin at a site (i; j) will have four nearest neighbors in the square arrangement of the lattice at the sites (i; j − 1); (i; j + 1); (i − 1; j) and (i + 1; j). The second term defines the Zeeman energy, which is the amount by which spin energy levels split in the magnetic field B. Ferromagnetic materials possess a characteristic parameter, called Curie temperature, Tc. This is 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 spins in ferromagnetic material tend to order parallel to each other. 3 Figure 1.3: Phase diagram for artificial spin ice lattices (black lines (solid and dashed) correspond to energies, 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 increasing the temperature, thermal fluctuations of the spins in the artificial spin ice system are activated and due to this, the reversal energy decreases with temperature until finally it becomes equal to zero at Curie temperature, TC (See Fig. 1.3). As illustrated in this figure, the susceptibility measured in 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. This temperature is the intrinsic ordering parameter of the islands. At temperatures between TC and TB in the ferromagnetic region of the material, spin flipping is random and independent of the flipping of neighboring spins. However, going down in temperature towards the TB, spin flipping becomes correlated with the flipping of neighbor spins. Cooling down even further to TB results in critical slowing down of the spin flipping rate, which results in a random but well-defined frozen ground 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 spin configuration, the actual state reached would highly depend on the process through which the system is brought to the "frozen' state. That is, different cooling rates and other differing parameters of experimental 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 still have a probability to flip. Overall, changing of the temperature leads to re-ordering of the magnetic moments since spins are now excited to the higher energy states. Therefore, different types of vertices can be accessed at different temperatures. Bringing the islands closer together results in the increase of the dipolar interaction between them. 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.