IFM The Department of Physics, Chemistry and Biology Lab 70 in TFFM08 Curie & Ising NAME PERS. -NUMBER DATE APPROVED Rev Aug 09 Agne
1 Introduction Magnetic materials are all around us, and understanding their properties underlies much of today's engineering efforts. The range of applications in which they are centrally involved includes audio, video and computer technology, telecommunications, automotive sensors, electric motors at all scales, medical imaging, energy supply and transportation, as well as the design of stealthy airplanes [1]. When it s not possible to solve a problem analytically, one can resort to computers as an alternative way for finding solutions. The first alternative is to try to solve numerically the problem equations. The second is to use a computer to simulate the system under study, which can be seen as a numerical experiment considered as a substitute for a real lab experiment. But even for problems that can be solved analytically, the computer remains a good way to learn as much as possible about the system under study in order to build a realistic model about it. In this laboratory exercise you will use simulations and physical models rather than experimental manipulations as for the other labs in this course. You will have to study two dimensional systems of interacting and non interacting magnetic moments based on the knowledge acquired mostly during previous courses. The objective is to get a grasp of magnetic phenomena through models and computer simulations. But for a deeper insight into the magnetic theory the course Condensed Matter, Part II is strongly recommended. To introduce you to computer simulations and give you some basic knowledge in magnetism and statistical mechanics this exercise is divided in two main parts: In Part 1 you will simulate a physical system with non interacting magnetic moments which can be set in thermal contact with a lattice and a heat reservoir; In Part 2 you will focus on ferromagnetic and anti-ferromagnetic interactions and magnetic ordering. Figure 1: Different kinds of magnetism: a) paramagnetism, b) ferromagnetism, c) antiferromagnetism. 2
2 Part 1: Curie simulation model Curie simulation model is based on a physical system with non interacting magnetic moments which can be set in thermal contact with a lattice and a heat reservoir. In this part of laboratory exercise we will concentrate on magnetic phenomena in paramagnetic materials. 2.1 Paramagnetism in brief Para-magnetism is briefly treated in this introduction. The purpose is to provide a qualitative understanding of the basics necessary for the simulation exercises, not a detailed description of paramagnetic phenomena. Those basics are further developed and described in other courses like e.g. Atomic and Nuclear physics, Thermodynamics and Statistical mechanics and Quantum mechanics. There are two possible atomic origins of magnetism which lead to the magnetization orbital motion and spin of electrons. An atom which has a magnetic moment caused by spin or by orbital motion of electrons or by both is generally called a magnetic atom. The magnetic moments of the important magnetic atoms such as iron, cobalt and nickel are caused mostly by spin motion of the electrons. If we consider that the material consists of atoms of type i with magnetic moment µi with a number density N i per unit volume, the magnetization of the material can be written as M =! N i µ i i. In paramagnetism the magnetization is proportional to the magnetic field H. In most cases paramagnetic substances contain magnetic atoms or ions whose magnetic moments are isolated from their magnetic environment and can more or less freely change their directions. At finite temperatures the magnetic moments are thermally agitated and take random orientations. Upon application of a magnetic field, the average orientations of the magnetic moments are slightly changed so as to produce a weak induced magnetization parallel to the applied magnetic field. M M 0 T 1 T 2 T 1 < T 2 M 0 = N µ BgS H Figure 2: Magnetization as function of magnetic field at two different temperatures. How the magnetization dependence on the magnetic field, at two different temperatures, is shown in figure 2. For high enough magnetic field, the magnetic moments are all parallel to the field, i.e. they occupy the lowest energy level (saturation level M 0 ). If at a given field the temperature is increased, the magnetization will decrease. The reason for this is that thermal energy is added, which means that some of the magnetic moments will be excited to higher energy levels, where they are no longer parallel with the field. One can say that while the magnetic field brings order (line up the moments); the temperature creates disorder (causes fluctuations in the direction of the moments). 3
2.2 Thermal interaction The transport of thermal energy is in general a microscopic disordered (random) motion while work comes from an ordered macroscopic movement. We take a gas in a container as an example. If the walls of the container have higher temperature than the gas temperature, the heat energy will be transferred to the gas. The atoms in the walls have initially a higher average energy than the gas atoms average energy. Every time a gas atom collides with the walls it will acquire energy, this process (disordered transfer of energy) goes on until the walls temperature is equal to the gas temperature. For an ideal gas we have, ( 1 2)m v 2 = ( 3 2)k B T i.e. the temperature of the gas atoms is proportional to the average energy motion. For a solid (container walls) things are not so simple but we can roughly say that the walls temperature is related to the average vibration energy of the atoms. In this laboratory exercise you will study a solid material where every atom has a magnetic moment and can vibrate i.e. lattice vibrations are also possible. The lattice can be put in contact or be isolated from the surrounding. The surrounding is much bigger than the system itself and acts as a heat reservoir: it can influence the system without been influenced by it in terms of average energy, temperature, etc. The system under study is thus formed by three components. The heat reservoir (surrounding) with temperature T; The lattice, whose temperature T l is related to the average vibration energy of the atoms; The spin system (magnetic moment) formed by the contribution of the spins of every atom. If a magnetic field is applied it will give rise to a field-dependent energy spectrum leading to the definition of the so-called spin temperature, T S. The three components of the system can exchange thermal energy, illustrated in figure 3. The heat reservoir can be put in contact with the lattice while the latter is always in contact with the spin system. Spin -system T S Lattice T l Heat reservoir T Figure 3: Illustration of the three- component system and how the different parts can interact thermally. Note that the spin system and the lattice cannot be separated in reality In equilibrium and in the case the heat reservoir is not isolated from the lattice, all three temperatures are equal i.e. T = T l = T S. Note that even in equilibrium there is an exchange of thermal energy at the microscopic level but the macroscopic measurable quantities like average energy, magnetization, etc. remain constant in time. If for example the temperature of the heat reservoir increases and it is in contact with the lattice, energy will be given by the heat reservoir until T l = T. Similarly the spin system will (if a magnetic field is applied) absorb energy until T S = T l = T. To analytically take directly into account thermal interactions between two systems is not yet possible. But in equilibrium this can be done indirectly. According to statistical mechanics the probability of finding a system (under thermal interaction) in a certain energy level is proportional to e!"e. This is the so-called Boltzmann distribution. With this probability distribution we can calculate thermodynamic quantities like e.g. magnetization, internal energy, etc. In this laboratory exercise you will use a computer to simulate the thermal interactions between the different components of the system under study. This will give the same qualitative results as statistical mechanics. 4
2.3 Description of the Curie model system The model system under study is two-dimensional and is formed by 900 atoms organized in 30x30-grid (lattice). Every atom has a magnetic moment (spin) and all these spins are assumed not to interact with each other. In the classical model energy of the system could be defined as E =!! µ " H!!, where magnetic moment is expressed as µ =!µ B g n S! n. Since M S can take 2S + 1 different values, it means that one atomic level will split into 2S + 1 level in a magnetic field (figure 4). µ B gh M S =S M S =S-1 H M S =-(S-1) µ B gh M S =-S Figure 4: Splitting of an atomic level in a magnetic field. However in the Ising model, where the spin operators are treated like classical vectors, spins (vectors) can just be parallel or anti-anti-parallel with the magnetic field. The energy for one spin is thus given by, E ± = ±H The system has two energy levels according to figure 5. In the lower level the magnetic moment is parallel with the field while the spin is anti-parallel and vice versa in the upper level. The number of spins in the lower and upper levels is denoted by N! and N + respectively. µ = 1, S = +1, N + 2H H µ = +1, S = 1, N Figure 5: Energy levels for a spin system in presence of a magnetic field H. Atoms can also vibrate around their equilibrium positions. These so-called lattice vibrations are described with Einstein model according to which all the atoms vibrate with the same fre- 5
quency ν. Similarly to light waves, the lattice vibrations are also quantized. The lattice vibrations quanta are called phonons and the energy for one phonon is! = h" where h is the Plank constant (we ignore here the zero point energy h! / 2 ). The simulation is built on the following interaction rules: Spin-lattice: A spin is chosen randomly. If the spin is in the upper energy level, it will flip and give its energy to a lattice point. A certain number of phonons are thus created. Lattice-Spin: A lattice point is chosen randomly. If the energy within this point is high enough to overcome the energy difference between spin energy levels and the spin is in a low energy level, energy will be given by the lattice. The spin will then flip to a higher level. Lattice-Lattice: Two lattice points are chosen randomly. The energy for these two lattice points add up and is shared afterwards randomly between these points. Lattice-Heat reservoir: The heat reservoir and the lattice are in contact and the lattice temperature is T l. If T l is different from the heat reservoir T, energy is exchanged between the two systems until T l = T. This exchange is made in the form of many small energy packets distributed randomly among the lattice points. In computations we assume that the whole process is accomplished by fewer large energy packets. This assumption is without consequences for the final result as long as the system reaches equilibrium, i.e. T l = T. Only the relaxation process (way to equilibrium) is important. For this reason T l = T is set directly in the program and T is subject to variation. One must therefore wait a while for the large energy packets to spread through interactions. The simulation itself is performed in the following way. At the starting point the phonons are distributed according to Boltzmann distribution and the spins are oriented randomly. In every time step of the simulation an interaction rule is chosen randomly from the above. However lattice and heat reservoirs are connected only if the insulation between them is removed and the temperature T is changed. 2.4 Using the program The program runs under a Macintosh II-Cx system. Start the Mac and open the directory marked Ising/Curie then click on the sub-directory Curie. The following interface should show up on the screen. If a spin points in the opposite direction to the magnetic field (S = 1), then the magnetic moment is parallel to the field and this is represented by a white square in the Spin array, and by a black square in the other case. Concerning the Lattice array, the darker a lattice point is the more phonons are there. To remove the insulation with the heat reservoir click on it. To increase or decrease the external magnetic field click on + respectively on the scale marked H. Heat source temperature is changed in the similar way. The square marked Magnetization shows the evolution of the magnetization with time. The magnetization fluctuates continuously and to measure it one must take an average value over an enough long time. This can be done by clicking on the button marked Average. The time average computation starts when by clicking a first time and finishes by clicking a second time. Observe that for a real measurement of the magnetization in the lab we measure also a time average value of the microscopic fluctuating magnetization. It s this time average value that we calculate, in an intuitive way without involving time constant, in statistical mechanic. The square marked Lattice temperature shows how the temperature in the lattice changes with time. 6
To start the program, choose Go under the menu Run and to stop it choose Stop. To close the program, choose Quit under the menu File. If Compute under menu Statistics is chosen the phonon statistics is displayed which is the average phonon number as function of the energy in the square marked Lattice statistics. Avoid using the Help menu as the command does not always work and can cause some overflow in the program. 7
2.5 Lab tasks Task 1 Start the simulation and remove the insulation between the lattice and the heat reservoir. Examine the relation between the average magnetization, the temperature and the magnetic field. Allow enough time for the system to reach equilibrium after every change in the system parameters. Use the Average button to get the magnetization values. Plot the average magnetization as a function of H for 3 different values of T (low, medium and high). Discuss what happens to the system when temperature and magnetic field vary. How would you find the Curie constant? Task 2 Start the simulation and let the lattice isolated from the heat reservoir with H = 0. Then increase the magnetic field rapidly to the maximum value. What happens to the lattice temperature? Explain. Task 3 a) Set the temperature to be more than 0. Put the lattice in contact with the heat reservoir and increase the magnetic field to its maximum value. b) When the magnetization saturates, isolate the lattice from the heat reservoir. c) Decrease slowly the magnetic field and notice how the lattice temperature is decreasing. Explain what happens during the process steps a)-c). This process is the so-called adiabatic demagnetization for cooling and is used to reach temperatures close to the absolute zero point. This method is more effective when the temperature is already enough low i.e. lower than 4.2 K (liquid He temperature). Try to repeat steps a)-c) several times to decrease the magnetic field even more to reach the lowest possible temperature and comment on that. 8
3 Part II: Ising simulation model In part 1, we simulated a paramagnetic spin-system in contact with a lattice which in its turn can be put in contact or isolated from a heat reservoir. In this part we will perform a so-called Monte Carlo (MC) simulation of two-dimensional spin system, where spins interact between each other, to study magnetic ordering. In contrast to part 1 where the thermal interaction between spins and lattice were simulated, we assume here that the spin system is always in thermal equilibrium with the lattice at a temperature T. It is well known from statistical mechanics that the probability, P n, to find the spin system in a quantum state dependent on energy is following Boltzmann distribution, i.e. P n! e "#$E n. The objective of MC simulation here is to generate that Boltzmann distribution. We will see this more in detail after a short introduction on magnetic ordering. 3.1 Magnetic ordering in brief It is very difficult to predict in detail the behavior of the magnetic moment of an individual atom embedded in a solid material rather then isolated. Such discussion is beyond the scope of this introduction and will not be pursued any further. However we notice that only some atoms have a permanent magnetic moment as constituents of a solid material. As mentioned in part 1 this occurs in paramagnetic solid material, but in most cases the magnetic moments (spins) interact between each other. The situation is generally simpler in isolators than in metals where the free conduction electrons complicate things considerably. There exist different kinds of magnetic ordering, but we will discuss here only the two more known kinds namely ferro- and anti-ferromagnetism. In a ferromagnetic material the magnetic moments interact in such way that the most favorable energy state is where there are parallel and anti-parallel in an anti-ferromagnetic material. The ordering process is disturbed by thermal energy. If we increase the temperature, thermal energy will be provided to the spin system i.e. a number of spins will excite to higher energy levels where they are no longer parallel or anti-parallel with other spins. Higher temperature causes disorder. If we consider a ferromagnet at H = 0 the magnetic moments are completely disordered at enough high temperatures. The magnetization is almost zero. But when the temperature is lowered slowly the transmitted thermal energy decreases and interactions between spins will become more and more dominant. And suddenly and a certain temperature (and lower), called critical temperature T c, the magnetization will take a nonzero value. A spontaneous magnetization has occurred, which means that a predominant number of momenta have a component in a given direction. As the temperature drops further below T c all momenta will align to become completely parallel. The system has undergone a phase transition from a paramagnetic phase ( M = 0, T > T c ) to a ferromagnetic phase ( M! 0, T > T c ). Iron e.g. is a ferromagnet with T c = 1043 K. For the antiferromagnet the momenta are anti-parallel below T c, which leads to M = 0. One cannot detect the phase transition by measuring M. But there are other properties characterizing phase transition: the specific heat e.g. diverges at T = T c and for a ferromagnet even the susceptibility diverges while for a anti-ferromagnet it shows a kink just above T c and below the kink the susceptibility depends on the direction of the magnetic field. The reason why the susceptibility diverges at T = T c, for a ferromagnet is that the susceptibility is proportional to the magnetization fluctuations (show this as an exercise). At T T c, the magnetization fluctuations are very big (infinite for an infinite system). What reason can make the magnetic momenta interact with each other? One can think that the interaction arises from the momentum, through the magnetic field either directly by magnetic dipole-dipole interaction, or, less directly by spin-orbit coupling. These mechanisms constitute rarely the dominant type of magnetic interaction. The undoubtedly most important one is the conventional electrostatic (Coulomb) electron-electron interaction. It is remarkable that many theories in magnetism, as first approximation, have neglected dipole-dipole interaction and spin-orbit coupling and kept the Coulomb interaction only. 9
3.2 Description of the Ising model system and the simulation model In this part of the laboratory exercise we simulate a two-dimensional spin system constituted by 50x50 spins on a grid (lattice). The boundary conditions are periodic, so that the first column lies next to the last column and the first line next to the last (called sometimes warp over ). The model that we will use for the magnetic interaction is the Ising model derived from the Heisenberg Hamiltonian. In the Ising model the spin operators! S i are replaced by classical spins (i.e. no commutation rules) which means that they can only takes the values S i = ±1. Let! = ( S 1, S 2,..., S N ) denote an Ising spin configuration. For a certain configuration the system is said to be in a certain state (microstate actually). Because of thermal interaction the spins will flip according to time, i.e. ν is a function of time. As the time passes the system experiences its different states (configurations) and this determines a trajectory in the configuration space. To illustrate this we associate numbers to different states. For two Ising spins there are 4 possible sates v 1 = (1,1), v 2 = (1,!1), v 3 = (!1,1), v 4 = (!1,!1). A possible trajectory is shown in the figure below where the time axis has been divided into steps.! (t)! 4! 3! 2! 1 t Figure 6: Trajectory example for a system of 2 Ising spins, for 8 time steps. Let be the value taken by a physical observable, like magnetization, internal energy, etc. in the state ν. In any experiment we measure the time average of the observable and the average value over a trajectory formed by T steps is defined as G T = 1 T T " t=1 G! ( t) Currently it s generally impossible to determine which trajectory a system is following in order to calculate the time average value according to the equation above. From statistical mechanics we know however that a system in equilibrium follows such trajectory and the probability to find it in a state ν, corresponding to energy E ν, is proportional to e!"e. This is called the Boltzmann distribution. To evaluate the thermal equilibrium average value we need to generate trajectories that give the Boltzmann distribution. It s often such trajectories that are generated in the so-called Monte Carlo simulations. The name comes from the fact that in these kinds of simulations one uses a randomness generator, as should a perfect roulette do. There exist many different MC algorithms and the one used in this laboration is called the Metropolis algorithm as a reference to a paper of N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller (1953), where this method was presented for the first time. 10
One chooses a start configuration ν. After that a spin is selected randomly, this is done by the randomness generator of the computer. The computer generates a random number between 0 and 1. A way to do this, if the spins bear numbers S(I) 1 < i < N, is to identify the spin by choosing i as the integer closest to the random number multiplied by N. After selecting randomly a spin, we consider the new configuration ν, obtained by flipping the selected spin. As an example! "!# can corresponds to (..., 1, -1, 1, 1,...) (..., 1, 1, 1, 1,...) - randomly chosen spin. This leads to the energy variation:!e vv' = E v' " E v If!E ""# $ 0, the spin flip is accepted. If!E ""# > 0 a random number x is generated between 0 and 1, and the new configuration is accepted only if e!"e vv ' # x. We summarize this (with ν(t) = ν) as v(t + 1) = v ' if!e vv' < 0, and otherwise v(t + 1) = v ' if e!"e vv ' $ # x % v if e!"e vv ' &' < x This procedure is then repeated over and over again and it can be shown that it generates trajectories over which an average value is calculated. This average value is similar to the thermal average value calculated by using the Boltzmann distribution. 3.3 Using the program From the already opened folder Ising/Curie select and double click on the icon called Ising. The interface of the program is similar to the one used in the Part I of the exercise. To start the simulation one can choose between Coldstart and Warmstart under the menu Start. Coldstart means that all spins point in the same direction at start while Warmstart means that the spins are oriented randomly. The type of interaction is selected under the menu Magnet. The magnetic field and the temperature can be changed in the same way as in Curie program. The magnetic field can either be positive or negative, i.e. the direction of the field can be selected. The magnetization is shown in the square as a function of time and the button Average is used to measure it as before. If a spin is parallel to the magnetic field, it s represented by a black square and a white square if the spin is anti-parallel. Allow enough time for the system to reach equilibrium state after any parameter change. Note that most often waiting time should be longer comparing to the program used in Part I in the exercise as the system takes longer time to reach equilibrium state. To close the program, choose Quit under the menu File. 11
3.4 Lab tasks Task 1 Select Ferromagnet under the menu Magnet and let H = T = 0. Choose coldstart. You should plot the equilibrium magnetization as a function of T, for H = 0. Increase T from 0 and up, and measure for every step the magnetization with Average button. Do not forget to allow some time to reach equilibrium after any change, i.e. until the average value of the magnetization becomes relatively constant. Use a 0.4 unit steps in the beginning, but when the magnetization begins to come close to zero use 0.2 unit steps instead. Be very careful when the magnetization is coming closer to zero (i.e. T comes closer to T c ) since the fluctuations increase a lot. Describe how the black-white pattern of the squares representing spins is changing as the temperature is rising. What is the approximate value of T c? The magnetization as function of T is called a phase diagram and magnetization is playing the role of an ordering parameter, it s zero in the unordered phase and nonzero in the ordered one. Mark the paramagnetic phase and the ferromagnetic phase in the phase diagram with P and F accordingly. Task 2 Select maximal temperature and let the magnetic field equal to zero. Choose warmstart. Decrease then the temperature to T = 0 and observe what happens. It takes fairly long time before reaching the final result at T = 0. Repeat the procedure, but let the temperature come close to zero in different ways. If you re lucky (and patient) you should get two different final situations, either all the spins point to the same direction or a part of the spins points to one direction and the other part to the opposite direction. In the last case domains are formed, with clear limits between them. Explain why the model permits domains formation even if this is not the lowest energy configuration for the system at T = 0 K. Do you have any suggestion on how to correct this in the simulation algorithm? When all spins point to the same direction, there exists a spontaneous magnetization. This is also observed experimentally. According to the latter finding is the Ising model and statistical mechanics consistent with spontaneous magnetization at T = 0? Task 3 Let T = H = 0 and choose warmstart. Increase H to the maximal value. At equilibrium, decrease the magnetic field to zero then increase it in the negative direction to the maximum. Should the spins flip in the opposite direction if the magnetic field is increased further? With the magnetic field at its maximum (in the negative side), increase the temperature with one unit. What happens then and why? Task 4 Select Antiferromagnet under menu Magnet and let T = H = 0. Choose warmstart and observe how the spins order themselves so that the nearest neighbours are anti parallel. Some disturbances occur in the chess board pattern. Describe this and explain why it happens. Increase the temperature and observe how the spin directions become random for enough high temperature. Experiment also the influence of the magnetic field. 12