Multi-Spin Interactions in Magnetic Solids Gustav Johansson Engineering Physics and Electrical Engineering, masters level 2016 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Abstract Exchange models are of great interest when studying magnetic properties at non-zero temperatures and the most common one is the Heisenberg model. In this work, in order to better describe the magnetic properties at non-zero temperatures or in non-collinear magnets, we propose an extended Heisenberg model that includes higher-order spin interactions. To this end, the electronic structure Elk-code [1] (based on density functional theory) has been further developed to include our new model for calculating the exchange parameters. This new method is based on earlier work of A. Jacobsson [2] and should be seen as an improvement of the spin-cluster method developed by Singer, Dieterman and Fähnle [14], as it scales better with the size of the system, making it more suitable for realistic systems. It is found that including multiple-spin exchange parameters gives a better description of the system and that the largest contribution of the multi-spin parameters comes from the biquadratic exchange interactions confirming earlier results by Singer, Dieterman and Fähnle. Contents 1 Introduction 1 1.1 Background . .1 1.2 Earlier work . .1 1.3 Purpose . .2 2 Theory 3 2.1 Models of exchange . .3 2.1.1 The Heitler-London model . .3 2.1.2 The Heisenberg model . .4 2.1.3 The extended Heisenberg model . .4 2.2 Density functional theory . .6 2.2.1 The Schrödinger equation . .6 2.2.2 The Hohenberg-Kohn theorems . .6 2.2.3 The Kohn-Sham equations . .7 2.2.4 Spin Density Functional Theory . .7 2.2.5 The exchange-correlation functional . .8 2.2.6 The eigenvalue problem . .8 2.2.7 Augmented Plane Waves (APW) . .9 2.3 First-principles calculations of exchange interactions . 10 2.3.1 Spin-spirals . 10 2.3.2 Exchange parameters from constraining fields . 10 3 Method 13 3.1 Determine the constraining fields . 13 3.2 Identify equivalent interactions . 13 3.2.1 Bilinear interactions . 13 3.2.2 Multi-spin interactions . 14 3.3 Set up the system of equations . 14 3.4 Determine the unit cell . 14 4 Results 17 4.1 Energy difference in spin-spirals . 17 4.2 Exchange parameter calculations for different configurations . 18 4.3 RSS for different number of parameters . 19 4.4 Bilinear exchange parameters including biquadratic parameters . 20 4.5 Multi-spin parameters for an increasing number of bilinear pa- rameters . 21 5 Concluding remarks 23 5.1 Conclusions . 23 5.2 Future work . 23 6 Acknowledgements 25 A DFT-calculations of bcc-Fe 27 A.1 Convergence test . 27 A.2 Volume dependence of energy and magnetic moment . 28 A.3 Density of states (DOS) . 29 B Spin-spirals in the Heisenberg model 30 Chapter 1 Introduction 1.1 Background The mechanism behind magnetic ordered systems, such as ferromagnetic and antiferromagnetic systems, is the exchange interaction. This interaction is often depicted by the Heitler-London model for the Hydrogen molecule in which it is seen that, even though the spin doesn’t explicitly enter the enter the energy calculations, it forces restrictions on the spatial wave function due to the Pauli exclusion principle. As a consequence, the total energy depends on the spin con- figuration and the energy associated with this interaction is called the exchange energy. This energy is considered, and effectively computed for small systems, by density functional theory (DFT). However, for larger systems a simpler model is needed. The most commonly used model for describing large systems is the classical Heisenberg model. It is a parametrized model describing the energy due to interactions between pairs of spins located at different lattice sites and is often used to describe magnetic properties at non-zero temperatures. The parameters are called exchange parameters and are, for realistic systems, computed from DFT calculations. Even though the classical Heisenberg model was developed for localized spin systems, it works surprisingly well for many itinerant systems. A well known example for the classical Heisenberg model’s failure is the spin-spiral configu- rations in bcc-Fe. The energy difference between a collinear configuration and a spin-spiral configuration with an opening angle θ should be proportional to sin2 θ. This is shown not to be true for bcc-Fe, which implies that the classical Heisenberg model is insufficient. 1.2 Earlier work Additional angle-dependence can be included by extending the classical Heisen- berg model to also consider higher order interactions. This has been done suc- 1 cesfully by Singer, Dieterman and Fähnle [14] using a spin-cluster expansion (SCE). They show how the description drastically improves when including non-Heisenberg interactions systematically in the SCE. By extending the clas- sical Heisenberg model with biquadratic interactions, they show that bilinear and biquadratic nearest neighbor interaction exhibits the same accuracy as the pure Heisenberg model including 20 pair-interactions. 1.3 Purpose The purpose of this work is to develop a routine that extracts the higher order exchange parameters using a much simpler approach then the SCE. In contrast to Singer, Dieterman and Fähnle, this approach is restricted to biquadratic, three- and four-spin interactions. The method used to extract the exchange parameters is based on the con- straining field method by Grotheer et al. [6] where constraining fields are added to the potential. In this method the exchange parameters are extracted directly from the magnitude and directions of the constraining fields. In contrast to Grotheer, the parameters are not calculated in reciprocal space and the mo- ments rather than the fields are fixed. Extending this method, it is also possible to extract the higher order ex- change parameters and thus verify the results by Singer, Dieterman and Fähnle. Moreover, this extended method also scales better with the size of the system, making it more suitable for realistic systems. 2 Chapter 2 Theory 2.1 Models of exchange The strong interaction responsible for the magnetic order in solids is the ex- change interaction. Although this interaction is considered in DFT calculations, for large systems a modelling is required for calculating the interaction. 2.1.1 The Heitler-London model The exchange interaction is often depicted by the Heitler-London model for the Hydrogen molecule1. The model considers the Hamiltonian2 of the four particle system given by 1 1 1 1 H^ = − r2 − − r2 − 2 1 jr − R j 2 2 jr − R j 1 1 2 2 (2.1) 1 1 1 1 − − + + : jr1 − R2j jr2 − R1j jr1 − r2j jR1 − R2j The first row is just the sum of the independent Hydrogen Hamiltonians which can be seen as a unperturbed state and the last row can be seen as a pertur- bation such that H = H0 + λV . It follows that the unperturbed state can be expressed as a product of two single electron functions. With the condition that 2 2 jΨ(r1; r2)j = jΨ(r2; r1)j , two possible wave functions are possible ΨS(r1; r2) = 1(r1) 2(r2) + 2(r1) 1(r2); (2.2) ΨA(r1; r2) = 1(r1) 2(r2) − 2(r1) 1(r2): (2.3) Using these two wave functions and the Hamiltonian, the energy for the two states can be calculated. More interesting is the energy difference between the 1A more formal description of the Heitler-London model and the Heisenberg model can be found in most literature concerning magnetism e.g. [3, 11, 8] 2 Atomic units are used, where ¯h; me; a0 and e are all equal to one. 3 two states which can be expressed as X − SC E − E = 2 : (2.4) S A 1 − S2 S is called the overlap integral, C the Coulomb integral and X the exchange integral. These integrals are given by S = h 1(r1); 2(r2)j 1(r2); 2(r1)i; (2.5) C = h 1(r1); 2(r2)jV^ j 1(r1); 2(r2)i; (2.6) X = h 1(r1); 2(r2)jV^ j 1(r2); 2(r1)i: (2.7) For a zero overlap i.e. S = 0 the energy difference is determined by the exchange integral alone and this energy is also called the exchange energy. 2.1.2 The Heisenberg model Since the total wave function also includes the spin and the Pauli exclusion prin- ciple states that the total wave function has to be antisymmetric with respect to particle exchange, the symmetric wave function Eq. 2.2 has to be combined with an antisymmetric spin function and vice versa. The antisymmetric spin function is known as the singlet state and the symmetric spin functions are know as the triplet states. It follows that the spin configuration determines the energy. Thus, it is possible to express the energy difference explicitly in terms of the spin H = J12s1 · s2; (2.8) where J12 is called the exchange parameter or exchange constant. This result can be generalized for more than two atoms, giving the classical Heisenberg model describing the spin or magnetic moment dependence of the exchange energy X H = − Jijsi · sj: (2.9) ij 2.1.3 The extended Heisenberg model An extended Heisenberg model can be motivated from a perturbation expansion of the Hubbard model [15]. X y X H = −t (ciσcjσ + h.c.) + U ni"ni# (2.10) i;j,σ i y In the Hubbard model, the operator ciσ creates an electron with spin σ on atom i, and the operator ciσ annihilates an electron with spin σ on atom i. The y product of these two operators niσ = ciσciσ counts the number of electrons with spin σ on site i.