Magnetism in 2D

This exercise investigates magnetic order in two dimensions. While a collinear spin-density functional theory calculation might reveal that it is energetically favorable for a given 2D material to order magnetically, two dimensional magnetic order at finite temperatures also requires presence of magnetic anisotropy, usually arising from the spin-orbit coupling.

This exercise will teach you how to extract magnetic exchange and anisotropy parameters for a localized spin model based on first principles calculations. It will also touch upon the Mermin-Wagner theorem and show why anisotropy is crucial in order to sustain magnetic order in two dimensions.

In the first part of the project, you will calculate the Curie temperature of a CrI3 monolayer. Afterwards, you will investigate the magnetic order in VI2, which has antiferromagnetic coupling and noncollinear order. Finally, you will finish the project by performing a search for new magnetic 2D materials with high critical temperatures based on a database of hypothetical monolayers.

Part 1: Critical temperature of CrI3

Following the instructions in the Jupyter notebook magnetism1.ipynb you will in this first part of the project set up and relax a monolayer of CrI3 after which you will calculate its critical temperature using a prerelaxed structure file (CrI3.xyz).

The procedure will be as follows:

  1. Set up the atomic structure and optimize the geometry of CrI3

  2. Calculate the nearest neighbour Heisenberg exchange coupling based on a total energy mapping analysis

  3. Show that the magnetic ground state is thermodynamically unstable when anisotropy is neglected (The Mermin-Wagner theorem)

  4. Calculate the single-ion magnetic anisotropy and estimate the critical temperature

Part 2: Noncollinear magnetism in VI2

In materials where the dominant magnetic exchange coupling is antiferromagnetic, or in cases where different exchange couplings compete, the ground state may have a complicated noncollinear magnetic order. Completing the notebook magnetism2.ipynb, you will examine a prototypical monolayer with a noncollinear ground state, namely VI2. Starting from the structure file VI2.xyz, you will:

  1. Relax the atomic structure using LDA

  2. Compare a collinear antiferromagnetic structure with the ferromagnetic state

  3. Obtain the noncollinear ground state

  4. Calculate the magnetic anisotropy and discuss whether or not the material will exhibit magnetic order at low temperatures

Part 3: Find new ferromagnetic monolayers with high critical temperatures

In this last part of the project, you will try to find new ferromagnetic monolayers that can preserve their magnetic ordering at elevated temperatures. Using the notebook magnetism3.ipynb, you will:

  1. Search through a database of monolayers to pick a material you might expect to have a high critical temperature

  2. Carry out a total energy mapping analysis to obtain exchange coupling and anisotropy parameters

  3. Calculate a first principles estimate of the critical temperature

You are welcome to repeat this procedure for as many monolayers as you like.