A theoretical study of longitudinal and transverse spin fluctuations in disordered Fe64Ni36 alloys

Linköpings universitet SE–581 83 Linköping

Linköping University | Department of Physics, Chemistry and Biology

Master’s thesis, 30 ECTS | Applied Physics and Electrical Engineering - Theory, Modelling,

Computer calculations

2020 | LITH-IFM-A-EX–20/3820–SE

A theoretical study of

longitudinal and transverse

spin fluctuations in disordered

Fe

64

Ni

36

alloys

Amanda Ehn

Supervisor : Björn Alling and Davide Gambino Examiner : Rickard Armiento

Datum

Date

2020-08-24

Division, Department

Department of Physics, Chemistry and Biology

Linköping University

URL för elektronisk version

ISBN

ISRN: LITH-IFM-A-EX--20/3820--SE

_________________________________________________________________ Serietitel och serienummer ISSN

Title of series, numbering ______________________________

Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________ Titel Title

A theoretical study of longitudinal and transverse spin fluctuations in disordered Fe64Ni36 alloys

Författare Author

Amanda Ehn

That certain iron and nickel alloys exhibit an anomalously low thermal expansion of a wide temperature range has been observed since late 1800s, and this effect is known as the Invar effect. Since then, many theories have been proposed to explain the phenomenon. While it is generally agreed that the effect is related to magnetism, a full explanation of the effect has yet to be found. One recent theory connected the effect to spin-flips in the iron atoms' magnetic moment and that the probability for a spin-flip to occur depends on the atom's local chemical environment.

The aim of this thesis is to perform a theoretical investigation into the magnetic energy landscapes for atomic magnetic moments in different local chemical environments in disordered Fe64Ni36 alloys, and the change in pressure upon populating different parts of the magnetic energy landscape. Constrained calculations are performed to obtain the energy landscapes for both iron and nickel atoms in ferromagnetic Fe64Ni36. The calculated nickel atoms all show one global minimum between 0.64 to 0.72μB. The calculated iron atoms all exhibit two local minima: one where the magnetic moment's direction is the same as the ferromagnetic background's direction and has a size between 2 to 3μB, one where the magnetic moment is flipped and has a reversed direction in regards to the ferromagnetic background with a size between -2.5 to -1.9μB. A weak trend is seen for the energy difference between the two local minima: for atoms with rich local environments the energy difference is smaller than for iron-atoms with nickel-rich local environments.

The energy landscapes for a moment rotated with respect to the background show that it is energetically favored to rotate the moment from the spin-up local minimum to the spin-flipped local minimum, rather than shrink in size and then increase in size in the opposite direction. This indicates that the negative local minimum might not be a local minimum, but further calculations are needed to determine if the spin-flipped state is a local minimum or just a saddle point in the complete size- and angle magnetic energy landscape.

It is observed that the pressure varies little for different magnetic moment sizes for a nickel atom, but shows a larger variation for different magnetic moment sizes for an iron atom. The pressure difference between the magnetic local minima is about 6-9 kbar, and from thermodynamical

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Abstract

That certain iron and nickel alloys exhibit an anomalously low thermal expansion of a wide temperature range has been observed since late 1800s, and this effect is known as the Invar effect. Since then, many theories have been proposed to explain the phenomenon. While it is generally agreed that the effect is related to magnetism, a full explanation of the effect has yet to be found. One recent theory connected the effect to spin-flips in the iron atoms’ magnetic moment and that the probability for a spin-flip to occur depends on the atom’s local chemical environment.

The aim of this thesis is to perform a theoretical investigation into the magnetic energy landscapes for atomic magnetic moments in different local chemical environments in dis-ordered Fe64Ni36alloys, and the change in pressure upon populating different parts of the

magnetic energy landscape.

Constrained calculations are performed to obtain the energy landscapes for both iron and nickel atoms in ferromagnetic Fe64Ni36. The calculated nickel atoms all show one

global minimum between 0.64 to 0.72 µB. The calculated iron atoms all exhibit two

lo-cal minima: one where the magnetic moment’s direction is the same as the ferromagnetic background’s direction and has a size between 2 to 3 µB, one where the magnetic moment

is flipped and has a reversed direction in regards to the ferromagnetic background with a size between -2.5 to -1.9 µB. A weak trend is seen for the energy difference between the

two local minima: for iron-atoms with iron-rich local environments the energy difference is smaller than for iron-atoms with nickel-rich local environments.

The energy landscapes for a moment rotated with respect to the background show that it is energetically favored to rotate the moment from the up local minimum to the spin-flipped local minimum, rather than shrink in size and then increase in size in the opposite direction. This indicates that the negative local minimum might not be a local minimum, but further calculations are needed to determine if the spin-flipped state is a local minimum or just a saddle point in the complete size- and angle magnetic energy landscape.

It is observed that the pressure varies little for different magnetic moment sizes for a nickel atom, but shows a larger variation for different magnetic moment sizes for an iron atom. The pressure difference between the magnetic local minima is about 6-9 kbar, and

Acknowledgments

First of all, I would like to thank my supervisor Björn Alling for the opportunity to do this project and for invaluable guidance and encouragement throughout the work on this thesis. I would also like to thank my second supervisor Davide Gambino for sharing his knowledge with me.

Finally, to my family and friends: without your love and support I would never have made it this far. Thank you.

Contents

Abstract vi

Acknowledgments ix

Contents xi

1 Introduction 1

1.1 The INVAR effect . . . 1

1.2 Thesis outline . . . 2

2 Theoretical Background 3 2.1 Density Functional Theory . . . 3

2.2 Bloch’s theorem . . . 5

2.3 Special quasirandom structures . . . 6

2.4 Magnetism in solids . . . 6

2.5 Longitudinal Spin Fluctuations and Thermodynamical simulations . . . 6

3 Computational Details 9 3.1 Lattice parameter . . . 9

3.2 Convergence tests . . . 10

3.3 Spin flips in collinear calculations . . . 11

3.4 Energy landscapes . . . 12

4 Results and Discussion 13 4.1 Energy landscapes . . . 13

4.2 Thermodynamical simulations . . . 20

4.3 Pressure . . . 26

5 Conclusions and Future work 33

1

Introduction

1.1

The INVAR effect

In 1897 C.-É. Guillaume discovered that face-centered cubic (fcc) alloys of iron and nickel with a concentration of 35 at. % nickel exhibits an anomalously low thermal expansion over a wide range of temperature [1]. The origin of the name comes from Guillaume himself who considered the expansion of these alloys to be invariable. The phenomenon has since then became known as the Invar effect. Guillaume understood that the effect was somehow related to magnetism and in his Nobel lecture described the magnetic properties of the alloy [2]. His results showed that the linear thermal expansion coefficient, α, for Fe65Ni35at 300 K was approximately 1.2 ¨ 10´6K´1. This value for α is thus an order of magnitude smaller than for the pure components of iron and nickel. The effect is not limited to ferromagnetic (FM) fcc FeNi alloys. The effect has been observed in other alloy systems, but it is noteworthy that these system are rich in at least one 3d-transition element, which points toward magnetism as an explanation for the effect [3]. Although it has been a century since Guillaume’s Nobel lecture and years of study of the effect, a full understanding of the effect has not yet been found.

Two main approaches have been used to understand the magnetism in solids. One is based on the localized electron picture (Heisenberg model). Here the atom has its own per-manent and temperature independent moment that can rotate in direction. The second ap-proach is based on the itinerant electron picture (Stoner model) [3].

One of the early attempts at a theoretical description of the Invar effect was Weiss’ so-called 2γ-state model [4], which belongs to the group of localized electron picture models. In this model the (fcc) γ-Fe atoms in the alloy can at finite temperatures occupy two states: an anti-ferromagnetic low-volume state and a ferromagnetic high-volume state. Weiss argued that the usual thermal lattice expansion was compensated by a transition from the high spin, high volume state to the low spin, low volume state. Weiss also assumed that the energy difference between the two states depended on the Ni concentration. This model has been questioned due to inconsistencies between theoretical predictions and experimental data.

Later explanations for the Invar effect include one developed by Kakehashi [5, 6]. In this model the local moments of Fe-atoms are reduced by a temperature induced magnetic dis-order. Another explanation, proposed by van Schilfgaarde et al. suggests that the anomalies in Invar properties arise from continuous variations of the spin alignment of the Fe-atoms

1.2. Thesis outline

and slightly reduced local moments when the volume is decreased [7]. Recently A.V. Ruban has developed a model for describing the thermal expansion in Invar Fe65Ni35 that uses the spin-wave method. The model includes consideration of the magnetic short-range order ef-fects at temperatures both below and above the Curie temperature. It also considers magnetic entropy related to temperature induced spin fluctuations [8]. The model is in semiquantitive agreement with experimental data that found uncorrelated longitudinal spin-fluctuations co-existing with transverse spin fluctuations [9].

Another theory on the origin of the Invar effect has been proposed by Liot and coauthors [10, 11]. In their calculations they show that the lattice constant is reduced upon an increased number of neighboring Fe-Fe spin-pairs where one of them is in a spin-flipped state, i.e. the magnetic moments point in opposite directions. This, they argue, is the physical mechanism of the Invar effect as they show that the average separation of nearest-neighbour Fe-Fe pairs is larger for pairs with moments that have the same direction. With their theory and calculations they have been able to reproduce some of the original results from Guillaume.

F. Liot et al. shows that the local chemical environment of an iron atom is an important aspect in regards to the possibility of a spin-flip of an atom’s magnetic moment, and how this couples to the local atomic volume [10, 11]. In their supercell calculations they used free floating collinear magnetic moments and thus only local and global magnetic energy minima in the collinear picture could be explored. It needs to be considered that temperature might induce fluctuations on the magnetic energy landscape in ways that have gone undetected by previous work. Thus the topology of this energy landscape with regards to size and direction of the local moments in different local chemical environments is still not fully explored.

This work aims to reduce this gap in knowledge by investigating the energy landscapes for atomic magnetic moments in different local chemical environments in Fe64Ni36, taking into account both longitudinal and transverse fluctuations. The following questions will be addressed:

• What are the relative energetics of the spin-flipped local minima, the energy landscape between the local minima in terms of longitudinal changes, and the effects of rotating the magnetic moments?

• How do the energy landscapes depend on the local chemical environment?

• How does the pressure change when populating different parts of the magnetic energy landscape, and how does it change with temperature?

1.2

Thesis outline

Following is a brief description of the contents of this thesis.

Chapter 2: Theoretical backgroundAn explanation of the basic concepts of Density Func-tional Theory (DFT) and techniques used to solve the Kohn-Sham equations, Bloch’s theorem, basic thermodynamic concepts, and theory for longitudinal spin fluctuations (LSF).

Chapter 3: Computational details Descriptions of the calculations carried out in the project.

Chapter 4: Results and DiscussionThe results are presented and discussed. Chapter 5: Conclusions and Future work

2

Theoretical Background

2.1

Density Functional Theory

This section is dedicated to introduce density functional theory (DFT).

The many-body problem

In 1926, Erwin Schrödinger published the now famous equation ˆ

HΨ=EΨ, (2.1)

which is the time-independent Schrödinger equation. ˆH is the Hamiltonian operator, and when acting on the wave functionΨ gives the energy E. This equation determines the prop-erties of a material. The equation can be solved analytically for system with a low number of electrons but when the number of particles increases, finding an analytical solution to the equation becomes very difficult due to the many degrees of freedom. This is known as the many-body problem.

For the many-body problem, the Hamiltonian can be divided into separate components: the kinetic and potential energies of the particles in the system. The Hamiltonian is thus, for a system of n electrons and N nuclei

ˆ H=´1 2 n ÿ i=1 ¯h2 me ∇2_i ´1 2 N ÿ I=1 ¯h2 MI ∇2_I´ n,N ÿ i,I ZIe2 |ri´RI| + +1 2 ÿ i‰j e2 |ri´rj| +1 2 ÿ I‰J ZIZJe2 |RI´RJ|, (2.2)

where meand MIare the electron and nucleus mass, respectively. riand RIare the positions of the electron and nuclei, e is the negative charge of the electron. The first and second term represent the electronic and nuclei kinetic energy, respectively. The third term represents the electron-nucleus attraction. The fourth and fifth terms stand for the electron-electron and nucleus-nucleus repulsion, respectively.

2.1. Density Functional Theory

The Hamiltonian can, in the Born-Oppenheimer (BO) approximation, be simplified [12]. The idea is to take advantage of the fact that the nuclei are massive as compared to the elec-trons and move slower than the elecelec-trons. Thus the nuclei can be interpreted as static, i.e. the nuclei are fixed in space. The second and fifth term in Eq. (2.2) can be neglected in the calculation for the electron.

Even with the BO approximation it is not possible to solve the Schrödinger equation for many-particle system. Further approximations need to be made.

In 1964 Kohn and Hohenberg [13] proposed two theorems:

• For any system with interacting particles in an external potential Vext(r), this potential is uniquely determined by the ground state density, n0(r).

• The electron density in the ground state minimizes the total energy functional, E[n0(r)]. For a system with many electrons the wave function becomes very complex and difficult to solve even with computational efforts, but the number of coordinates for electron density of a system does not depend on the number of electrons. Thus these theorems would simplify the calculation for larger systems. However, the theorems proposed by Hohenberg and Kohn do not say what the energy functional is or how to find it.

Kohn-Sham equations

In 1965 Kohn and Sham [14] continued on the work of Hohenberg-Kohn, proposing that a fictitious system of non-interacting electrons could be constructed with the same density as the real interacting system. These electrons move under the influence of an effective field, Ve f f, different from the real external potential. Since each electron behaves independently the Hamiltonian is a sum of one-electron operators, and the single-particle Schrödinger equation

looks as follows _# ´¯h 2 2m∇ 2_+V e f f + ψi(r) =eiψi(r), (2.3) in which the effective potential is

Ve f f(r) =Vext(r) + ż n(r1₎ |r ´ r1_|dr 1_+V xc. (2.4)

For the system of N electrons the single-particle wavefunctions correlate to the electronic density n as n(r) = n ÿ i=1 |ψi(r)|2. (2.5)

Equations (2.3)-(2.5) are commonly referred to as the Kohn-Sham equations. Vxcin Eq. (2.3) is the exchange-correlation term and is the only approximated quantity.

The total energy of the system is given by E[n(r)] = ż Vext(r)n(r)dr+1 2 ij n(r)n(r1₎ |r ´ r1_| drdr 1_{+ T} s[n(r)] + Exc[n(r)], (2.6) where the first term is the potential energy from the charge of the nuclei, the second term is the Coulomb interaction, the third term is the kinetic energy of the non-interacting particles, and the fourth term is the exchange-correlation energy, which exact form is unknown.

The equations depend on the electron density, n(r), which in turn depends on the single-particle wavefunctions. The solutions may thus be found self-consistently. An initial value for n(r)is assumed and this density is used to obtain single particle wavefunctions that solve the Kohn-Sham equations. From the solution of the Kohn-Sham equations a new electron density is obtained. Either convergence is found and thus the ground state energy, or the scheme reiterates with the new electron density.

2.2. Bloch’s theorem

Exchange correlation approximations

The Local Density Approximation (LDA) is a frequently used functional for the exchange-correlation energy, and depends only on the the electronic density. It is assumed that the density can be treated locally as a uniform electron gas. The exchange-correlation energy is at each point in the system the same as for a uniform electron gas

E_xcLDA[n(r)] = ż

exc[n(r)]n(r)dr, (2.7) where excis the exchange-correlation energy per particle of a uniform electron gas [14]. By taking into account the electron spin this approximation can be improved [15] and written as

E_xcLSDA[(nÒ(r), nÓ(r))] = ż

excnÒ(r), nÓ(r) n(r)dr, (2.8) where exc(nÒ(r), nÓ(r))is the uniform gas of spin densities nÒ(r)and nÓ(r).

In the Generalized Gradient Approximation (GGA) [15], the exchange-correlation energy is expanded to include a dependence on the gradient of the density, and is written as

E_xcGGA[(nÒ(r), nÓ(r))] = ż

excnÒ(r), nÓ(r),∇nÒ(r),∇nÓ(r) n(r)dr. (2.9) Using either LDA or GGA to describe the exchange-correlation functional can yield dif-ferent results. LDA performs well when applied to materials with a slowly varying electron density, but performs poorly when the electron density is less uniform and shows large fluc-tuations [16]. GGA on the other hand has produced results in better agreement with exper-iments for finite systems, such as atoms and molecules [17]. A limitation of LDA is that it tends to overbind solids, i.e. lattice parameters are typically underestimated. Other features, for example cohesive energies, are overestimated when using LDA. In the case of GGA, the lattice parameters are typically overestimated for solids [18]. Further, there are several im-plementations of GGA, which can be used depending on what system is under investigation [17].

2.2

Bloch’s theorem

Bloch’s theorem states that the wavefunction of a particle in a periodic potential can be ex-pressed as

ψk=uk(r)eikr, (2.10)

where uk(r)is a periodic function with the periodicity of the potential, i.e. uk(r+T) =uk(r), where T is a translation vector of the considered lattice. r is the position, and k is the crystal wave vector confined to the first Brillouin zone. Due to the periodicity of u(r), u(r)may be expressed as a Fourier series

u(r) =ÿ

G

˜

ukGeiGr, (2.11)

where G is a reciprocal lattice vector. Combining Eqs. (2.10) and (2.11) yields

ψ(r) =uk(r)eikr= ÿ

G

ckei(k+G)r, (2.12)

where ck is the Fourier coefficient. Now the wave function needs only to be solved within the first Brillouin zone, albeit for an infinite number of k. However, since the wavefunctions will be nearly identical for k-values that are sufficiently close, the wavefunctions can be rep-resented by a single k-point in a region around k. Therefore it is sufficient to consider the electronic states at a finite number of k-points [19].

2.3. Special quasirandom structures

2.3

Special quasirandom structures

Special quasirandom structures (SQS) is a method that can be used to generate supercells with chemical disorder for random alloys. In this method the atoms of the alloy are placed in the lattice so that the correlation functions in the supercell mimic the behavior of the average correlation function for an infinite lattice for the first few correlation shells. Different degrees of short range order can be included, depending on the size of the supercell [20].

2.4

Magnetism in solids

Two models have been used to describe magnetic materials. One is based on a localized electron picture (Heisenberg model) and the second is based on an itinerant electron picture (Stoner model).

In the Heisenberg model the interaction between magnetic moments is usually described with a Heisenberg Hamiltonian

H=´ÿ i,j‰i

Jijeiej, (2.13)

where Jij is the exchange interaction between moments i and j, and ei is the direction of moment i. Each magnetic atom is assumed to have a magnetic moment that does not change in size with temperature [21].

Equation (2.13) can be rewritten so that the exchange interactions Jij are independent of the size of the magnetic moments

H=´ÿ i‰j Jijeiej=´ ÿ i‰j r Jijmi¨mj, (2.14) where rJij=Jij/(|mi||mj|).

A model related to the Heisenberg model is the Ising model in which each magnetic mo-ment is represented by a spin. The spins can be in one of two states: either +1 (’up’) or -1 (’down’). In this model, the Hamiltonian is

HI=´ ÿ i‰j

JijSiSj, (2.15)

where Jijis the exchange interaction between moments and Siis the spin state [22].

The itinerant model is based on the band theory of electrons and describes ferromag-netism. In this model the electronic bands are split into spin-up and spin-down bands and an imbalance between the number of spin up and spin down electrons results in a ferromagnetic material [22].

2.5

Longitudinal Spin Fluctuations and Thermodynamical simulations

2.5. Longitudinal Spin Fluctuations and Thermodynamical simulations

Figure 2.1: The left side of the figure illustrates longitudinal spin fluctuations. The right side of the figure illustrates transversal spin fluctuations.

The Heisenberg Hamiltonian allows only transversal spin fluctuations. The Heisenberg Hamiltonian in Eq. (2.13) fails to describe system with some degree of itineracy; however, including a term representing longitudinal spin fluctuations (LSFs) makes it possible to in-vestigate more itinerant systems within this framework. This modification can be done by adding an on-site term that depends on the local environment [23]

HLSF=´ ÿ i‰j r Jijmi¨mj+ ÿ i E(mi). (2.16)

The second term in Eq. (2.16) is the term associated with LSFs and represents the energy of moment mias a function of its own magnitude. Inspired by Ginzburg-Landau theory of magnetism, this term has the general form of

Ei(mi) = 8 ÿ n=0

anm2ni «am2i +bm4i. (2.17)

A 4th order polynomial is needed to approximate a moment landscape, but higher order polynomials might also be required to obtain an accurate fit [24].

The Hamiltonian of a system can be used to calculate thermodynamical quantities such as entropy, average magnetic moment size, and more. To calculate these quantities first the partition function needs to be calculated. The partition function is defined as

Z= ż

dm1dm2...dmNe ´ H

kbT_{, (2.18)}

where kBis the Boltzmann constant, T the temperature, and the integration includes all con-figurations of the moments mi. In the paramagnetic limit the partition function can be written as a product of N partition functions Z=ś Zi, one partition function for each moment, since the first term in Eq. (2.16) is zero sincemi¨mj

=0. The single-moment partition function Ziis here defined as Zi= ż dmie ´Ei(mi) kbT ₌ ż8 0 dmiPSM e ´Ei(mi) kbT _{, (2.19)}

where PSM is the phase space measure. For the thermodynamical simulations in this thesis, the phase space measure is simply 1.

From the partition function thermodynamical quantities can be calculated by the relation

hAi(T)i = 1 Zi ż8 0 dmiAie ´Ei(mi) kbT _{, (2.20)}

2.5. Longitudinal Spin Fluctuations and Thermodynamical simulations

3

Computational Details

All calculations in this project are carried out with the Vienna ab initio simulation package (VASP) [26] employing projector-augmented wave (PAW) potentials [27, 28]. A 64 atoms supercell consisting of 41 Fe atoms and 23 Ni atom is used. The supercell was generated by using a Special quasirandom structure approach. By using this approach the supercell imitates a random alloy and thus each atom has its own unique local environment [29, 20]. In all calculations the ionic positions are allowed to relax under a conjugated gradient algorithm while the cell shape and volume are kept fixed.

3.1

Lattice parameter

The first calculations are aimed to determine the equilibrium lattice parameter. This is done by collinear calculations in which the size of the lattice parameter varies. The result is pre-sented in Figure 3.1. A k-mesh of 2x2x2 and a cutoff energy of 400 eV is used. The calculations are performed with ferromagnetic moments. The resulting equilibrium lattice parameter is 3.598 Å and is used in later calculations.

3.2. Convergence tests

Figure 3.1: Interpolated curve for energy per atom as a function of the lattice parameter.

3.2

Convergence tests

Two convergence tests are performed: one for the number of k-points and one for the cutoff energy, and the results are presented in Figures 3.2 and 3.3. Both tests are carried out with collinear calculations. From these tests, taking into consideration the available computational resources, a grid of 3x3x3 k-points distributed according to the Monkhorst-Pack method [30] and a cutoff energy of 500 eV are decided to be used when calculating the energy landscapes for the atoms in the 64 atom SQS cell.

3.3. Spin flips in collinear calculations

Figure 3.2: Energy convergence of the system with a k-grid of n ˆ n ˆ n where n=2, 3, 4, 5, 6. For example, 3x3x3 gives 14 k-points. 6x6x6 give 112 k-points.

Figure 3.3: Energy convergence of the system with regards to the cutoff energy.

3.3

Spin flips in collinear calculations

Collinear calculations in which an increasing number of Fe-atoms are in a spin-flipped state are carried out, starting from a fully ferromagnetic state to a state where nearly all Fe-atoms’ magnetic moment are flipped in the opposite direction. These calculations are made with the same values for lattice parameter, cutoff energy, and k-mesh as described in previous sections.

3.4. Energy landscapes

3.4

Energy landscapes

The magnetic configuration of the material is found by collinear calculations in which the magnetic moment sizes are unconstrained. A calculation of this type is initiated with all mo-ments having the same direction. The resulting magnetic configuration of this calculation is later referred to as a ’FM-background’, and in this configuration both the size and direction of the magnetic moments are defined. Identical calculations with the exception of one atom’s magnetic moment is in a spin-flipped state are also carried out to find new magnetic con-figurations for when a ’spin-flip’ occurred. These concon-figurations will later be referred to as ’SF-backgrounds’. Seven ’SF-backgrounds’ are calculated for seven atoms, but only two of these ’SF-backgrounds’ are used to calculate two new energy landscapes.

The energy landscapes are calculated with a cutoff energy of 500 eV, and the first Brillouin zone is sampled with a 3x3x3 k-mesh. Both direction and size of the moments are constrained to the magnetic moments found by the unconstrained calculations, with the constraining pa-rameter λ = 25. Worth noting is that the size of the input moments needs to be smaller than the desired size of the output moments in the constrained calculations for technical reasons. The iron input moments are between 73% and 75% smaller than the output moments. The input moments for the nickel atoms are between 83% and 89% smaller than the output mo-ments. In the constrained calculations the value for RWIGS has to be specified. The RWIGS specifies the radius for the sphere over which the density is integrated to obtain the magnetic moment size. In RWIGS values for iron and nickel in the above described calculations are 1.302 and 1.286 Å, respectively.

For two atoms, number 9 and 22, the energy landscapes are calculated twice; once with the FM-background and once with the SF-background. Additionally, for atom number 9 another three landscapes are calculated in which the atom’s magnetic moment is rotated in a plane, and then scaled, for three different angles: 45˝_{, 90}˝_{, and 135}˝_{. All calculations where the} magnetic moments are constrained are noncollinar calculations.

4

Results and Discussion

4.1

Energy landscapes

Iron atoms

Energy landscapes for for 19 Fe-atoms are calculated. The iron atoms are chosen based on their local chemical environment. An iron atom in the supercell has at most 11 and at least 5, out of the total 12, nearest neighbouring atoms that are iron. The chemical environment of each calculated iron atom is presented in Table 4.1.

4.1. Energy landscapes

Table 4.1: The chemical environments for the 19 calculated iron atoms. The first column is the atom number in the SQS supercell. The second column presents how many of the twelve nearest neighbouring atoms are iron atoms. The third column presents how many of the twelve nearest neighbouring atoms are nickel atoms.

Atom # # of Fe-NN # of Ni-NN

1 9 3 2 8 4 3 5 7 4 8 4 5 7 5 6 8 4 7 6 6 8 9 3 9 10 2 10 5 7 11 6 6 12 9 3 13 8 4 14 9 3 15 5 7 21 7 5 22 11 1 28 6 6 39 10 2

The energy landscapes for the 19 iron atoms are presented in Figure 4.1 and the difference in energy between the local minima is presented in Table 4.2. All landscapes, with the excep-tion of atom 13, have a clear energy barrier between the two local energy minima. For atom 13, the energies at magnetic moments -2.106 and -1.441 µBdo have an energy difference of 2.21 meV, where the energy at -2.106 µB is lower than the energy at -1.441 µB. There is also reason to believe that the energies for the negative magnetic moments near the local minima are an upper limit and might be smaller. This topic is discussed further in a later section.

4.1. Energy landscapes

Figure 4.1: Energy landscapes for 19 iron atoms calculated in a ferromagnetic background.

Table 4.2: The energy difference between the two local minima, sorted from smallest to largest energy difference. The first column gives the atom number. The second column presents how many of the twelve nearest neighbouring atoms are iron atoms. The third column presents how many of the twelve nearest neighbouring atoms are nickel atoms. The fourth column gives the energy difference between the two local minima.

Atom # # of Fe-NN # of Ni-NN Energy difference (eV)

9 10 2 0.404 2 8 4 0.453 12 9 3 0.477 1 9 3 0.488 22 11 1 0.491 4 8 4 0.494 8 9 3 0.505 39 10 2 0.525 6 8 4 0.544 21 7 5 0.546 28 6 6 0.552 14 9 3 0.554 5 7 5 0.617 3 5 7 0.640 7 6 6 0.653 10 5 7 0.655 13 8 4 0.668 15 5 7 0.714 11 6 6 0.746

4.1. Energy landscapes

Nickel atoms

Energy landscapes for 3 nickel atoms are calculated and are presented in Figure 4.2. As for the iron atoms, the nickel atoms are chosen based on their chemical environment. The chemical environment of each nickel atom is presented in Table 4.3. The curves of the three nickel landscapes are similar, with global energy minima between «0.6-0.8µB. There is no observed local minimum for negative moments for Ni atoms.

Table 4.3: The chemical environments for the three calculated nickel atoms. The first columns is the atom number. The second column presents how many of the twelve nearest neigh-bouring atoms are iron atoms. The third column presents how many of the twelve nearest neighbouring atoms are nickel atoms.

Atom # # of Fe-NN # of Ni-NN

45 11 1

50 5 7

57 8 4

Figure 4.2: Energy landscapes for three nickel atoms calculated in a ferromagnetic back-ground.

Combined landscapes

As described in Section 3.4 unconstrained calculations are carried out to find a magnetic con-figuration starting point for the atoms in the supercell. For each of seven iron atoms a spin-flip magnetic configuration is produced by unconstrained calculations. From this, two local minima for each atom and corresponding magnetic moments are obtained, one for when the material is ferromagnetic and one for when the magnetic moment of one atom is flipped.

4.1. Energy landscapes

A comparison of the local minima from the unconstrained calculations and the interpolated minima from the energy landscapes is presented in Table 4.4.

Table 4.4: A comparison of the iron atoms’ magnetic moments at the local energy minima found in the unconstrained calculations and the energy landscapes for 7 iron atoms. Column one is the atom number. Column two presents the atom types of the 12 nearest neighbour-ing atoms. The first number is the number of iron atoms and the second is the number of nickel atoms. In columns three to six, C and UC stands for constrained and unconstrained, respectively, referring to the type of calculation. NLM refers to the negative local minimum and PLM refers to the positive local minimum. Thus, the third column presents the negative magnetic moments where the local minimum is found in the energy landscapes. The fourth column presents the negative magnetic moments where the local energy minimum in the un-constrained calculations is found. The fifth column presents the negative magnetic moments where the local energy minimum is found in the energy landscape. The sixth column presents the positive magnetic moments where the local energy minimum in the unconstrained calcu-lations is found.

Atom # Fe/Ni NN C, NLM UC, NLM C, PLM UC, PLM (µB) (µB) (µB) (µB) 2 8/4 -2.02 -0.67 2.56 2.57 9 10/2 -2.13 -0.46 2.59 2.55 10 5/7 -2.38 -0.53 2.71 2.71 11 6/6 -2.31 -0.94 2.68 2.68 12 9/3 -2.13 -0.62 2.58 2.57 22 11/1 -2.19 -2.29 2.59 2.59 39 10/2 -2.19 -0.85 2.58 2.58

In general, the local minima at positive magnetic moments are in very good agreement between the two types of calculations. However, a large discrepancy (ą 1.3µB) for the local minima at negative magnetic moments can be seen. One exception can be noted for atom number 22, for which the magnetic moments are in relative good agreement between the two types of calculations for both local minima. It appears as if the unconstrained calculations get ’stuck’ in a configuration due to some feature in the computational method, thus unable to reach the local minima as found in the energy landscapes. It is also observed in the new unconstrained calculations that the magnetic moments for all atoms are smaller in size when a spin-flip occurred. In particular the nearest neighbouring iron atoms’ magnetic moments are smaller, generally by around 0.1 µB.

Another possibility is that the local minima found in the unconstrained calculations are, in fact, the correct local minima. As can be seen in Figure 4.1 some landscapes have a small energy barrier and relatively flat curvatures in the region between -2 and 0 µB, for example atoms 2 and 13. It is plausible that, in the unconstrained calculations starting from a fully ferromagnetic state, the local minima are unable to be reached due to the surrounding atoms’ magnetic moments sizes being too large. Due to this, new energy landscapes for two atoms are calculated with the magnetic configuration obtained in the unconstrained calculations where the atom’s magnetic moment is flipped. If the local energy minimums obtained in the unconstrained calculations are indeed the ’correct’ minimums, it is expected that they should show in the new energy landscapes, since the surrounding magnetic moments are constrained to smaller magnetic moment sizes.

The two atoms chosen for these new landscapes are atoms 22 and 9. Figures 4.3 and 4.4 present the energy landscapes for the original landscapes (i.e. the same ones as in Figure 4.1) and the landscapes calculated with the new magnetic configuration. The original landscape is presented as ’FM-background’ in the figures and the new landscape as ’SF-background’. In general it is seen that the original landscapes have higher energies around the local minima

4.1. Energy landscapes

at negative moments and that the new landscapes show higher energies for the local minima at positive moments.

Further, in Figures 4.3 and 4.4 one more landscape (black circles) is considered, namely a combination of the two landscapes. These landscapes are the results of interpolating the positive magnetic moments from the FM-backgrounds and the negative magnetic moments from the SF-background.

First it is worth noting that no local minimum is found near the magnetic moments pre-dicted by the unconstrained calculations. Second, the energies for the negative local minima are smaller in the combined landscapes than in the original landscapes. It is thus worth con-sidering that other atoms’ energies at the negative local minima are, too, smaller, and that the energy barriers might be higher than shown in Figure 4.1. However, atom 9 and 22 are both surrounded by iron-rich environments and more combined energy landscapes for atoms with less iron-rich environments could confirm if this shift in energy is consistent for all local chemical environments.

Figure 4.3: Three energy landscapes for atom 22, which has 11 iron nearest neighbours. The ’Combined landscape’ is the result of combining the data points from the two other landscapes. For magnetic moments below -1 µB it considers the data points from the ’SF-background’ and for magnetic moments above 1 µB it takes into account data points from the ’FM-background’. It takes into account the data point near 0 µBfrom both the FM- and SF-background.

4.1. Energy landscapes

Figure 4.4: Three energy landscapes for atom 9, which has 10 iron nearest neighbours. The ’Combined landscape’ is the result of combining the data points from the two other landscapes. For magnetic moments below -1 µB it considers the data points from the ’SF-background’ and for magnetic moments above 1 µB it takes into account data points from the ’FM-background’. It takes into account the data point near 0 µB from the FM- and SF-background.

LSF energy landscapes for rotated moments

For atom 9 another three energy landscapes are investigated. The atom’s magnetic moment is rotated in a plane, and then scaled to produce three new energy landscapes. The rota-tional angles are 45˝_{, 90}˝_{, and 135}˝_{, compared to the direction of the magnetic moment in the} original energy landscape. Except for the magnetic moment of atom 9, the other magnetic moments have the original, ferromagnetic, direction. The resulting landscapes are presented in Figure 4.5. The 0˝_{landscape is the original landscape (same as in Figure 4.1). While it has} not been calculated, the 180˝ _{rotation is here represented by mirroring the landscape of the} original landscape, i.e. the 0˝_landscape.

Considering the theories of Weiss and Liot, the least costly way for the magnetic moment to go from one local minimum to the other is for the magnetic moment to rotate, according to the energy landscapes in Figure 4.5. For example, consider the ferromagnetic case (filled blue circles). To go from the local minimum at around 2.5 µBto the local minimum around -2.3

µB by taking the smallest energy steps would be to rotate from 45˝ Ñ90˝ Ñ135˝ Ñ180˝. This rotation energetics indicate that the spin-flipped state might not even be a local minima, although more detailed calculations for more angles of rotations are needed to see if there is an energy barrier.

4.2. Thermodynamical simulations

Figure 4.5: Energy landscapes for one atom which magnetic moment is rotated in a plane by five angles. The rotational angles are 0˝_{, 45}˝_{, 90}˝_{, 135}˝_{and 180}˝_.

4.2

Thermodynamical simulations

The energy landscapes from the previous chapter can be used to directly gain qualitative un-derstanding of the magnetic couplings and behaviour in Fe64Ni36, but they can also be used as input to thermodynamical simulations of properties. Section 2.5 describes the method that is used to make the thermodynamical simulations in this work. The method described is here applied to a ferromagnetic material instead of a paramagnetic material, as the method is in-tended for. This should be considered when evaluating the results in the following sections. It should be noted that these simulations only probes a subset of the allowed magnetic excita-tions. It includes individual longitudinal magnetic fluctuations, but not individual rotations or collective magnetic excitations. For this reason, the temperature dependencies observed should be seen as illustrations of how different energy landscapes is transferred into temper-ature dependent effects, not as exact quantitative material properties.

Iron atoms

From simulations based on the original energy landscapes the average magnetic moment (Figure 4.6) and average absolute moment (Figure 4.7) both show a decrease in magnetic moments at increased temperature. Taking into account that the energy landscapes might exhibit a higher energy barrier and shifts in energy near the local minimums (see Figures 4.3-4.4), similar thermodynamical properties are calculated for the combined energy landscapes and are presented in a later section. These calculations show that the average absolute mag-netic moment are very close to the original energy landscapes, but that the average magmag-netic moment show a steeper decrease after „ 700 K.

4.2. Thermodynamical simulations

Figure 4.6: The simulated average absolute magnetic moment versus temperature, for the 19 iron atoms. The average of the 19 atoms is represented by the black curve (’All atoms’).

Figure 4.7: The simulated average magnetic moment versus temperature, for the 19 iron atoms. The average of the 19 atoms is represented by the black curve (’All atoms’).

Nickel atoms

The average absolute magnetic moment (Figure 4.8) decreases with increased temperature until it reaches a temperature at which no more decrease is found. This temperature wary somewhat for the three nickel atoms. The average magnetic moment (Figure 4.9) decreases nearly linearly across all temperatures.

4.2. Thermodynamical simulations

Figure 4.8: The simulated average absolute magnetic moment versus temperature, for the three nickel atoms. The average of the three atoms is represented by the black curve (’All atoms’).

Figure 4.9: The simulated average magnetic moment versus temperature, for the three nickel atoms. The average of the three atoms is represented by the black curve (’All atoms’).

Combined landscapes

For both atom 9 and 22 the average absolute magnetic moments based on the original energy landscapes and the combined landscapes are nearly identical, while the average absolute magnetic moments calculated from the spin-flipped landscapes have similar curvature but a downward shift in moment size, see Figures 4.10 and 4.13.

The average magnetic moment calculated from the combined energy landscapes closely follows the original landscapes curve for temperatures 100-700 K, at which point it has a 22

4.2. Thermodynamical simulations

steeper decrease in average magnetic moment. As for the average absolute magnetic moment, the spin-flipped landscape curve has similar curvature to the other two curves but has a downward shift in moment size, see Figures 4.11 and 4.14.

Figures 4.12 and 4.15 show the probability for the atom’s magnetic moment being a cer-tain size at various temperatures from 100-1900 K. The probability for both atoms across all temperatures is highest around the 2.5 µBmagnetic moment.

Figure 4.10: The simulated average absolute magnetic moment versus temperature, based on the three energy landscapes for atom 9.

Figure 4.11: The simulated average magnetic moment versus temperature, based on the three energy landscapes for atom 9.

4.2. Thermodynamical simulations

Figure 4.12: The probability of the magnetic moment for atom 9 being a certain size at various temperatures.

Figure 4.13: The simulated average absolute magnetic moment versus temperature, based on the three energy landscapes for atom 22.

4.2. Thermodynamical simulations

Figure 4.14: The simulated average magnetic moment versus temperature, based on the three energy landscapes for atom 22.

Figure 4.15: The probability of the magnetic moment for atom 22 being a certain size at vari-ous temperatures.

4.3. Pressure

4.3

Pressure

Along with energy and magnetic moment size, the pressure for the supercell is obtained from both the constrained and unconstrained calculations. In this section the pressure for the calculated points in the energy landscapes are presented. One section presents the results from collinear unconstrained calculations where the number of Fe-atoms that exhibited a spin-flip is increased.

Iron atoms

The pressure on the supercell when varying one atom’s magnetic moment size is presented in Figure 4.16, one curve for each atom which magnetic moment size varied. Most atoms have a minimum between -0.5 and 0.5 µB, but some have their minimum outside of this range, for example atom 10. As for the original landscapes in Figure 4.1, there is reason to believe the curves shown in 4.16 are not wholly representative. When carrying out unconstrained calculations in which one atom has an opposite direction of its magnetic moment in regards to the other atoms in the cell, the pressure is consistently lower the curves in Figure 4.16. This matter will be addressed later.

Figure 4.16: Pressure versus magnetic moment for the calculated atoms (see Table 4.1. The curves are fitted by a 4th order polynomial.

Nickel atoms

The pressure curves for the three Ni-atoms have a smaller variation in pressure across the calculated magnetic moment sizes compared to the Fe-atoms, especially near the local mini-mums in the energy landscapes (0.6-0.8 µB).

4.3. Pressure

Figure 4.17: Pressure versus magnetic moment curves for atoms 45, 50, and 57.

Combined landscapes

Pressure curves for the calculations to produce two energy landscapes for atom 9 and 22 also result in two pressure curves for each atom. As with the energy landscapes, a pressure curve that combines the two is also generated here, by the same method as the combined energy landscapes, i.e. taking the data points for positive moments from the FM-background calculation and the data points for negative moments from the SF-background calculation. The pressure curves for atom 9 are presented in Figure 4.18 and pressure curves for atom 22 are presented in Figure 4.20. The pressure curves from calculations with a SF-background are shifted downward across all magnetic moment sizes for both atoms. For atom 9 the shift is about 6 kbar and for atom 22 the shift is about 7 kbar. Thus, as with the energy landscapes, a dependence on the magnetic background configuration is observed.

By combining these pressure curves with the thermodynamical simulations based on the corresponding energy landscapes, the pressure as a function of temperature is generated. This is done for atoms 9 and 22 and the results are presented in Figures 4.19 and 4.21, re-spectively. In the case of atom 9 and the combined PT-curve, it is close to linear for lower temperatures, 100-500 K. The decrease in pressure is more pronounced in temperatures be-tween 600 and 1200 K. For temperatures above 1200 K, the curve is again close to linear, but still with a steeper decrease than for the low temperature range. In the same case for atom 22 (Figure 4.21), the curve is close to linear for lower temperatures, 100-700 K. The curve takes on a steeper decrease for temperatures above 700 K but this decrease abates slightly for temperatures above 1400 K.

4.3. Pressure

Figure 4.18: Pressure-magnetic moment curves for the three energy landscapes for atom 9. The ’Combined landscape’ curve is a combination of the two other curves. The curves are fitted by a 5th order polynomial.

Figure 4.19: P-T curves based on the three energy landscapes for atom 9.

4.3. Pressure

Figure 4.20: Pressure-magnetic moment curves for the three energy landscapes for atom 22. The ’Combined landscape’ curve is a combination of the two other curves. The curves are fitted by a 5th order polynomial.

4.3. Pressure

LSF energy landscapes for rotated moments

Pressure curves in the case of when the atom’s magnetic moment is rotated in a plane is pre-sented in Figure 4.22. As shown in the section regarding the combined energy landscapes, both energy at the local minima and the pressure depend on the magnetic background con-figuration. The energy landscapes for rotated moments are calculated in the FM-background. For the 0˝ _{rotation a pressure difference of approximately 6 kbar for negative magnetic} mo-ments is observed when the calculation is carried out with a SF-background. Thus the pres-sures at negative values are not representative for these moments for the 0˝ _{rotation. Based} on the calculations carried out in this project it is unclear if the curves for the other rotations are representative or how they might change if they are calculated with a different magnetic background configuration.

Figure 4.22: Pressure curves for when one atom’s magnetic moment is rotated in a plane, and scaled. The rotational angles are 0˝_{, 45}˝_{, 90}˝_{, and 135}˝_{. The curves are fitted by a 4th order} polynomial.

Flipping of moments

Previous sections showed the change in pressure over various magnetic moment sizes but only for one atom at a time. A separate set of calculations are carried out to investigate how the pressure changes when more than one atom’s magnetic moment is in a spin-flipped state. Figure 4.23 presents the change in pressure as the number of Fe-atoms with a spin-flipped moment increases. For comparison, flipping one Fe-atom’s magnetic moment in the com-bined pressure curves (Figures 4.20 and 4.18) resulted in a pressure change of about 6-9 kbar. From Figure 4.23 it can be seen that the maximum change in pressure occurs when roughly half of the Fe-atoms moments have flipped, and the change from the fully ferromagnetic configuration is roughly 55 kbar. Figure 4.24 shows the change in energy as the number of Fe-atoms with a spin-flipped moment increases.

4.3. Pressure

Figure 4.23: The change in pressure when the number of Fe-atoms with a spin-flipped mo-ment increases, in collinear calculations. The supercell has 41 Fe-atoms and 23 Ni-atoms.

Figure 4.24: The change in free energy when the number of Fe-atoms with a spin-flipped moment increases. The supercell has 41 Fe-atoms and 23 Ni-atoms.

4.3. Pressure

5

Conclusions and Future work

A theoretical investigation of the magnetic properties of Fe64Ni36 random alloys has been carried out. The energy landscapes for changing the magnetic moment size of a number of iron atoms with different local chemical environments have been derived, as well as a few energy landscapes for nickel atoms with different local chemical environments. The nickel energy landscapes all exhibit a single global minima with magnetic moment size between 0.64-0.72 µB, but in all energy landscapes for the iron atoms two local minima are observed, with an energy barrier between them with a peak near the zero magnetic moment.

A weak trend between the local chemical environment and the size of the energy differ-ence between the two local energy minima can be discerned. Atoms with a more iron-rich environment have a smaller energy difference while atoms with a more nickel-rich local en-vironment have a larger energy difference.

For at least one atom in the supercell, it is shown that to move between the two local minima it is energetically favorable to rotate the moment from the spin-up local minimum to the spin-flipped local minimum, rather than shrink in size and then increase in size in the opposite direction. The rotation energetics indicate the spin-flipped state might not even be a local minimum. Further and more detailed calculations with more rotational angles are needed to determine if the spin-flipped state is a local minimum or just a saddle point in the complete size- and angle magnetic energy landscape.

It is observed that the pressure varies little upon populating different parts of the magnetic energy landscape for nickel atoms, but varies more for iron atoms. The difference in pressure between the two local minima, thus for flipping one atom’s magnetic moment, is about 6-9 kbar. The maximum change in pressure upon flipping more and more of the atoms’ magnetic moments is about 55 kbar. Lastly, from thermodynamical simulations it is observed that a small, nonlinear, decline in pressure with increased temperature.

One aspect that has not been investigated in this work but should be considered for fu-ture work, is whether vibrational effects at the relevant temperafu-tures are of the same order of magnitude and with opposite sign of the here observed magnetic effect on the pressure, so that they together could explain the Invar effect. Possible future work also includes cal-culating the energy landscapes but using LDA or other exchange-correlation functionals, to see if and how it would affect the results. It should also, if possible, be good to determine the cause of the discrepancy for the energy minima between the unconstrained and constrained calculations.

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