MSE 560 Project Presentation

MSE 560 Project PresentationSachin V MuleyVivek SaraswatModelling of 2D Ising system using Monte Carlo2D Ising modelUsed to model solids that exhibit a phase transition resembling ferromagnetism Each site on the lattice is assigned a spin-up or spin-down state2D Ising model can be used to model simple systemsOften MC is used to simulate the evolution of 2D Ising system

Metropolis Algorithm Used to draw sample configurations of 2D Ising systems in thermal equilibrium at a given temperature We start with a NxN grid with randomly assigned spins Next 10% of spins are flipped and energy of resulting configuration is calculatedIf E < 0, the new configuration is accepted. Otherwise, the new configuration is accepted with a probability of exp(-E/kT), where E > 0The system is then evolved for 1000 stepsThe above steps are repeated for 500 randomly chosen temperaturesTypical simulation run

Black = spin-down (-1)White = spin-up (+1)Tnorm is expressed in the units of J/kBT, where J is the interaction parameter and T is the actual temperature

ResultsCurie Temperature ~ 2.5A. Magnetization per site vs Temperature for a ferromagnetic materialMeta-stable statesn = grid sizeu = number of temperature values consideredl = number of simulations at each temperatureGround StatesGround StatesResultsB. Energy per site vs Temperature for a ferromagnetic material

At lower temperatures, lower energy configurations dominate

ResultsC. Magnetization per site vs Energy per site for a ferromagnetic materialHigh temperature phaseLow temperature ground stateLow temperature ground stateLow temperature meta-stable stateResultsD. Magnetic Susceptibility vs Temperature for a ferromagnetic material

Paramagnetic like behavior above curie temperature

For T > TcCurie Weiss LawResultsE. Heat Capacity vs Temperature for a ferromagnetic material

ResultsF. Number of up-spin domains vs Temperature for a ferromagnetic material

At higher temperatures, number of up-spin domains increaseResultsF. Hysterisis Loop for a ferromagnetic material, T = 0.2

=1=2=5=PermeabilityCoercivity (Hc) decreases on increasing permeability

G. Hysterisis Loop for a ferromagnetic material, T = 1Results

Coercivity (Hc) decreases on increasing permeability=1=2=5H. Hysterisis Loop for a ferromagnetic material, T = 5At high temperatures, Coercivity (Hc) tends to zero. Moreover, at low values of , the effect of magnetic field disappears at high temperatures=0.2=1=5Results

ResultsI. Magnetization per site vs Temperature for anti-ferromagnet

At low temperatures, neighbouring spins are perfectly aligned anti-parallel to each otherAt high temperatures, random configurations tend to appear, Ms 0ResultsJ. Staggered magnetization per site vs Temperature for anti-ferromagnet

High ordering at low temperaturesOrdering destroyed at high temperaturesNeel TemperatureResults

Paramagnetic like behavior above Neel temperatureK. Magnetic susceptibility vs Temperature for anti-ferromagnetResultsL. Heat Capacity vs Temperature for anti-ferromagnet

Neel TemperatureM. Hysterisis loop for anti-ferromagnetResults

=1=5ConclusionsMonte-carlo is a great technique to simulate simple 2D ising systemsUsing 2D ising model, a number of magnetic systems can be studied and their physical properties calculatedHowever, 2D ising model can only be used for simple systems and most magnetic materials cannot be accurately simulated this way. For instance in real materials, interaction between second and third nearest neighbours plays an important roleMonte-carlo technique must be run for longer times to gain accurate results, which requires more computational power

Questions?