TVE-F 17 017 juni

Examensarbete 15 hpJuni 2017

Theoretical understanding and calculation of the Edelstein effect

Gustav ErikssonHampus Nyström

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Theoretical understanding and calculation of theEdelstein effect

Gustav Eriksson, Hampus Nyström

The main topic of this project is the so called Edelstein effect. This recently discovered effect consists in the possibility of converting an electric field (a current) into a magnetization in materials that fulfill specific characteristics, more specifically materials where an effective Rashba spin-orbit coupling is present. The Edelstein effect is appealing to the scientific community from the fundamental physics point of view as well as from the technological point of view. In fact the possibility of efficiently converting an electric signal into a magnetic signal could revolutionize the current information storage technology.

In this project, after a study of basic concepts of solid state physics: crystal structure, Bloch's theorem, spin-orbit coupling; we addressed the study of the basics of a powerful numerical tool, called density functional theory (DFT), for predicting the electronic properties of solids. This tool provides us with all the needed quantities for numerically calculating any kind of linear response, which we show that the Edelstein effect is a specific form of. Using a specific implementation of DFT, called augmented spherical wave (ASW), we calculate the Edelstein effect in iron and copper (where no effect is expected) and manganese silicide (where the effect is expected to appear). We also perform a systematic study on how the Edelstein effect depends on the symmetry of the material and the magnitude of the spin-orbit coupling. The calculations showed promising results from which we concluded that the numerical methods used could clearly distinguish between the presence of the Edelstein effect or not in mentioned materials.

ISSN: 1401-5757, TVE-F 17 017 juniExaminator: Martin SjödinÄmnesgranskare: Martin SjödinHandledare: Marco Berritta, Peter M. Oppeneer

Populärvetenskaplig sammanfattningAnvändningsområdena för datorer har ökat konstant med ett kontinuerligt ökande krav påkomponenterna både när det kommer till prestanda men även platseffektivitet. I dagensdatorer används elektroners laddning som som informationsbärare, dessa kan represen-teras av ett binärt talsystem, ettor och nollor, och har länge fungerat med tillförlitligatekniska implementationer. I framtiden kommer dock nya tekniker behövas vilket förmed sig ett krav på nya teoretiska modeller. Ett förslag innebär att använda elektroner-nas spinnmagnetisering som informationsbärare. Om spinnmagnetiseringen i ett materialskulle användas i komponenter, som ersättning till dagens RAM minnen till exempel, såskulle mer information kunna lagras på mindre yta. Tekniker för att manipulera mag-netiseringen i dessa komponenter tyder också på att informationen kan ändras snabbare,vilket möjliggör mer effektiv hantering av information i datorer. En utav dessa teorier ärden så kallade Edelsteineffekten.

Edelsteineffekten omvandlar ett elektriskt fält, eller en strömtäthet, till en spinnpolar-isation i material med vissa specifika egenskaper. I det här projektet har vi teoretisktundersökt Edelsteineffekten, vad som krävs för att den ska uppstå, och med numeriskametoder försökt beräkna den för tre olika material. Målet var att få en teoretisk förståelseför effekten och dess härkomst men också att verifiera om programmet, som de numeriskametoderna var implementerade i, kan användas i fortsatta studier. Alltså undersökaom de resultat programmet ger oss stämmer överens med de förväntningar teorin ger.Beräkningarna genomfördes på UPPMAX superdatorer i Uppsala och visade sig ge godaresultat. Vi kunde verifiera att programmet identifierade Edelsteineffekten i materialdär den var förväntad och gav indikationer på obefintlig effekt i material där den inteförväntades. Med hjälp av resultaten kunde vi estimera det numeriska felet och dra slut-satser gällande dessa metoders användningsområden när det kommer till experimentellamätningar av Edelsteineffekten. Vi noterade att programmet, och därmed de numeriskametoderna, fungerar väl och kan, med relativt få modifikationer, utökas till att undersökafler material och därmed användas för att i framtiden identifiera material lämpliga förimplementationer i tekniska komponenter.

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Contents

1 Introduction 5

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Bloch states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.4 Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.5 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.6 Density functional theory . . . . . . . . . . . . . . . . . . . . . . 12

1.2.7 Kubo linear response theory . . . . . . . . . . . . . . . . . . . . . 13

1.2.8 Edelstein effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 Reciprocal space discretization . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Electron density and potential calculation . . . . . . . . . . . . . 16

1.3.3 Edelstein effect calculation . . . . . . . . . . . . . . . . . . . . . . 17

1.3.4 Refining results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Results 18

2.1 Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Manganese silicide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Spin-orbit coupling dependency . . . . . . . . . . . . . . . . . . . . . . . 24

3 Discussion 26

4 Conclusions 27

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1 Introduction

1.1 Background

Nowadays the random-access memories (RAM) are based on devices where information isstored in the form of electric currents and electric charges. An alternative to the currenttechnology is based on the so called spintronic devices, where the information units areencoded through the magnetic states of a magnetic material. Implementing such a devicerequires a way to read and write (or manipulate) the magnetic information units. Meth-ods have already been developed for reading the information on such devices and can,in the case of antiferromagnetic (AFM) states, be based on different magnetoresistanceeffects.1 These methods basically measure effects that describe correlations between mag-netic fields and the resistance of a material in those fields.

The problem that is still being heavily investigated is instead how to efficiently ma-nipulate the magnetization. This can be done by applying an external magnetic field butan electrical method is more desirable, of which there are two approaches of current in-terest. One is to utilize the spin-transfer torque which requires a spin-polarized current,in essence transferring spin-angular momentum from the spin-polarized current to thematerial. The other way is using spin-orbit torque which instead is caused by spin-orbitcoupling in the material. This requires a material with a broken inversion symmetry butthe current does not need to be spin-polarized. The Edelstein effect is one of the effectsthat lead to such a spin-orbit torque and is therefore of great interest. With an electricfield along a certain direction, this effect induces in certain materials a magnetizationperpendicular to the electric field itself.

A way to quantify the Edelstein effect in different materials is introduced by the Kubolinear response theory, which can be used to determine the relationship between the in-duced spin polarization, M, and an applied electrical field, E, such that M = χE. Whereχ is a material dependent tensor. In previous studies a DFT (density functional theory)based method, which accounts for the specifics of the material under consideration, hasbeen implemented in a program, called ’MOKE’, to calculate the optical conductivityand the magneto-optical Kerr effect (MOKE) in different materials.2 Similar methodscan be used for calculating the mentioned χ tensor caused by the Edelstein effect. In thisproject we will use a modified version of the MOKE program, called ’EDEFF’, to calcu-late certain elements of the χ tensor. In particular we will run the EDEFF program foriron, copper and manganese silicide and study the results with different expectations dueto the symmetry properties of the materials. Comparing the results of the calculationswith these expectations will then provide us with a general idea whether the programworks and if so how it can be used. Additionally, we will conduct a theoretical studyof the required basics of solid state physics to obtain an understanding of the origins ofthe Edelstein effect. This also includes the basics of the needed tools to formulate andnumerically calculate the χ tensor.

Since iron and copper have centrosymmetric crystal structures we expect the Edelsteineffect to be non-present, i.e. we expect the χ tensor calculated from the program to be

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zero. According to the theory the effect is expected in materials without inversion sym-metry, which manganese silicide is an example of, so consequently we expect a non-zero χtensor from those calculations. The general aim of this project is to validate the program,to study the results of the calculations done with different settings and attempt to drawconclusions whether the program yields credible results. Furthermore, we want to gain adeeper understanding of the Edelstein effect and the physics behind it.

1.2 Theory

This section introduces the basic theory needed to understand the Edelstein effect andthe calculation of its linear response (the χ tensor).

1.2.1 Crystal Structure

The structure of a crystal, how the individual atoms are ordered in a material, is a majorfactor that determines the material properties. This is true for how it conducts electricityand heat but also for its optical absorption and many other properties.3 The Edelsteineffect is not different from these and therefore a basic understanding of crystal structuresis needed to comprehend it.

What defines a crystal is a unit cell (a portion of space which contains a set of atoms)which periodically repeats itself in the whole space. It can be proven that there exist nomore than 14 different ways in which a 3D structure can periodically repeat itself (5 in2D), these are called Bravais lattices. We can then classify all the possible crystals in 3Daccording to the so called Bravais lattices. On top of these "translational symmetries" itis possible to have other transformations transforming the crystal in itself. For instanceif the lattice that is formed from infinitely many cubes like the one in figure 1, whereall the vertices are equivalent (for example when all vertices are formed by one atom), arotation by 90° around the middle of any face would be a symmetry operation.

Figure 1: Simple cubic unit cell.

The composition of the translational symmetries of the Bravais lattices with the so calledcrystallographic point groups (of transformations which can transform the crystal in it-self, for instance rotations, reflections on a specific plane etc.) allows us to define a totalof 230 3D crystal space groups which all crystals adhere to. These 230 crystal structuresalways has an underlying Bravais lattice from which their unit cell is derived.

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What is seen in figure 1 is known as a unit cell of the simple cubic Bravais lattice.This is one way of defining a unit cell and it is good for visualizing how it looks but it isfar from the only one. The only requirement of a unit cell is that it should be a space thatwhen stacked produces the entire lattice and because of this definition it can be createdin many different ways, one of which is the Wigner-Seitz cell. Unlike the one in figure 1it only contains one lattice point and is defined as the space that is closer to that latticepoint than any other. For the simple cubic Bravais lattice this just results in anothercubic cell with one lattice point in the middle instead but for other Bravais lattices thechanges are greater. For instance the body-centered cubic (BCC) Bravais lattice (figure2) has the Wigner-Seitz cell seen in figure 3. Although the simple unit cell seen in figure2 gives a clear image of how the lattice looks the Wigner-Seitz cell is more useful in manycalculations.

Figure 2: Body-centered cubic unit cell.

Figure 3: Body-centered cubic Wigner-Seitz cell.

There are three lattices that are of particular interest in this paper. The first is the bodycentered cubic (BCC) lattice which has already been visualized in figure 2 and 3. Theother two Bravais lattices are body centered tetragonal (BCT) and orthorhombic Bravaislattices. In the case of copper it is worth noting that the BCT lattice, seen in figure 4,is equal to the FCC (face centered cubic) lattice due to the relation c =

√2 a, where c

the height of the cell and a is the side length of the square base. In all other ways thecomplete BCT lattice is formed in the same way as for the BCC lattice.

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Figure 4: Body-centered tetragonal unit cell.

The orthorhombic unit cell in turn looks the same as the two previous ones but withouta point in the middle where all the sides are of different length. What also can be statedis that iron has a BCC structure and copper a BCT (or FCC) structure which bothhave inversion symmetry, meaning that mirroring at a point in the lattice is a symmetryoperator that brings the lattice into itself. Manganese silicide on the other hand onlyhas the translational symmetry of an orthorhombic unit cell but consequently has anatom structure that breaks the full symmetry, in particular the inversion symmetry ofthe lattice, which is or great relevance to the Edelstein effect.

1.2.2 Reciprocal space

The reciprocal space, or k-space,3 is the Fourier transform of the Bravais lattice. In otherwords it is a representation of the periodicity of the crystal structure. The formal defini-tion is given as follows:

The set of all wave vectors K that yield plane waves with the same periodicity as agiven Bravais lattice, is known as it’s reciprocal space.

This give rise to the following condition; K belongs to the reciprocal space of a Bra-vais lattice if

eiK·(r+R) = eiK·r (1)

holds for any arbitrary vector r and all points R in the Bravais lattice.

There are several important qualities of the reciprocal space which are not easily re-alized at first. The first of these is the correlation between planes in the Bravais latticeand vectors in the reciprocal space. In any Bravais lattice it is possible to identify latticeplanes which are planes that contain at least three lattice points. Each of these belongto a family of lattice planes which are parallel and spaced an equal distance apart whichwhen added together contain all the points of the Bravais lattice. One of the interestingproperties of these planes are that for every vector in the reciprocal lattice there is a planein direct space that is orthogonal to the vector. This is helpful due to the relevance thatplanes have in the diffraction of photons. In other words the reciprocal space simplifiesthe calculations but they can still be made in the direct space. This effect transfers tomany more calculations where the wave properties of particles are of interest.

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As in the direct lattice, we can also define a unit cell in the reciprocal space. In thereciprocal space the Wigner-Seitz cell is instead called the first Brillouin zone but is oth-erwise defined the exact same way as for the direct lattice. Due to this it obviously hasthe same benefits and these benefits are intimately related to what is known as Bloch’sTheorem.

1.2.3 Bloch states

Bloch’s theorem (theorem 1) is a powerful tool for studying quantum effects in periodicpotentials of any kind.3

Theorem 1 The eigenstates ψ of the one-electron Hamiltonian H = −h2

2m∇2 + U(r),

where U(r+R) = U(r) for all vectors R in the Bravais lattice, can be chosen to have theform of a plane wave times a function with the periodicity of the Bravais lattice:

ψnk(r) = eik·runk(r) (2)

whereunk(r + R) = unk(r) (3)

for all R in the Bravais lattice.

This can also be written in an alternative form:

ψnk(r + R) = eik·Rψnk(r) (4)

The wave vector k which is introduced in Bloch’s theorem is of particular interest due tothe fact that we never have to consider any effects outside of the first Brillouin zone (theWigner-Seitz cell of the reciprocal space). The reason for this is that any k not in thefirst Brillouin zone can be written as k = kb + K where K is a vector in the reciprocallattice and kb lies in the first Brillouin zone. Due to equation 1 the relation eiK·R = 1holds for all vectors in the reciprocal lattice, which proves that if equation 4 holds for kbit also holds for k.

This wave vector is also interesting in the regard that it plays a similar role as the wavevector in the free electron model. This is not to say that this wave vector is a measureof the momentum; this can be proven to be false as they do not commute. Instead, it isto say it has many of the properties for which momentum is so useful and it describesthe translational symmetry of a periodic potential the same way that momentum is acharacteristic of the translational symmetry of free space. Therefore, it is a useful way oflabeling the eigenstates of the electronic wave function in a solid. This is also one of thereasons why calculations in the reciprocal space are so useful.

The index n in equations 2 to 4 is the result of the periodicity of the system and thecomplexity of the unit cell. For every such n we can vary k continuously which insteadof constant energy levels gives us continuous energy functions εn(k + K) = εn(k) withthe same periodicity as the reciprocal lattice. These functions are what defines the bandstructure of the material. One of the reasons for this name is quite apparent. Due to the

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fact that all εn(k) are both periodic and continuous they create an upper and a lowerbound which all values of εn(k) lies within, creating a band. These bands are a directresult of the Pauli exclusion principle and also gives rise to what is known as the Fermisurface.

1.2.4 Fermi surface

The ground state of a number of Bloch atoms3 is not as straightforward as for the freeelectron model. In the case of the free electron model the ground state is given by fillingall the energy levels ε(k) = h2k2

2mwith energies less than the Fermi energy εF . The Fermi

energy in turn is defined by requiring that the number of energy levels less than εF is thesame as the total number of electrons in the system. We can then say that εF characterizesthe ground state of a system with a specific number of electrons. The trouble when weare observing Bloch electrons is that ε is no longer dependent only on k but on both thecrystal and the band index n. This gives rise to two different scenarios when looking atthe ground state of a number of Bloch electrons. The first is that there are a number offull energy bands and all others are empty. This is when we can define what is known asa band gap in the material. The band gap is the difference in energy between the highestoccupied energy band and the lowest unoccupied one and in this band gap we find theFermi energy. The value of this band gap is very important for how a solid behaves,especially regarding the electrical conduction of the material. The other scenario is thatthere are a number of partially filled bands. In this case we can define a surface in thek-space that separates the levels that are filled from the ones that are not. The set of allthese surfaces is what is known as the Fermi surface and is important for understandinghow the electrons in a material behaves, especially for the electrical conductivity.

Figure 5: Visualization of the band structure of an insulating material.

What can be seen in figure 5 is a visualization of the band structure of an unknownmaterial. From this diagram we can draw the conclusion that it is not a metal, as theFermi energy is not in the middle of one or more energy bands. Instead, it must be eitheran insulator or a semiconductor, depending on the magnitude of the band gap. This is asimple theoretical case, as the energy bands rarely are evenly spaced but it gives a goodvisualization of what the Fermi energy and Fermi surface says about a material. When an

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electron is excited to the conduction band this allows charge to flow in the material, boththrough the electron in the conduction band but also through the hole it left behind inthe valence band. What decides how easily a material can conduct electricity is thereforehow easily an electron can pass from the valence band to the conduction band. As waspreviously stated the Fermi energy cuts right through one or more bands when it comesto metal, which can be seen in figure 6, and hence this band contain both occupied andunoccupied states. This is the reason why they conduct charges so easily; there is noenergy difference between the highest occupied and the lowest unoccupied state.

Figure 6: Visualization of the band structure of a metal.

For us, studying the Edelstein effect, we assume that there will always be a Fermi surfaceas we are attempting to manipulate conducting magnetic materials. The fact that weare attempting to manipulate the magnetization with a current in conducting materialsallows us rewrite the expression for the Edelstein effect. Using the relation ji = σijEj,where σij is the conductivity of the material and jy the current density, gives us Mx =

χxyEy = χxy(jyσyy

) where Mx is the spin-polarization and χxy is the material dependentlinear response of the Edelstein effect. Thus, the spin-polarization can be expressed interms of the current density asMx = χxyρyyjy, where ρyy is the resistivity of the material.

1.2.5 Spin-orbit coupling

It has been proven that the Edelstein effect is intimately related to the so called spin-orbit coupling,4 which can be pictured as the interaction between the magnetic momentof the spin of an electron and the magnetic moment due to its orbital angular momentum.It can be shown that for an electron in a centrosymmetric potential we can model thespin-orbit coupling by adding a term in the Hamiltonian of the electron of the form

HSOC =2µB

hmeec2

1

r∂U(r)∂r

L · S (5)

where µB is the Bohr magneton, me the mass of the electron, e the electrons charge, cthe speed of light, U(r) is the spherically symmetric potential, L is the orbital angularmomentum of the electrons and S is its spin.

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Depending on the symmetry of the system the spin-orbit coupling can assume differ-ent forms, in particular the Edelstein effect is intimately related to the Rashba spin-orbitcoupling which has the form

HR−SOC = α(p× S) · z (6)

where α is a coupling constant, p is the momentum of the electrons, S their spin andz a direction determined by the symmetry of the system. This additional term in theHamiltonian is what causes the Edelstein effect, therefore we expect a larger effect if thisterm grows and a smaller effect, or non-present, if this term decreases towards zero. Whatcan also be shown is that in crystals where there exists inversion symmetry the net effectof this coupling is not present and is therefore the reason for why the non-zero Edelsteineffect does not arise in such materials.

1.2.6 Density functional theory

Even though the Born-Oppenheimer approximation simplifies the many-body Schrödingerequation, fixing the nuclei to a static external potential, the problem of solving it is stilltoo complicated to be done conveniently.5 The interaction between electrons in the sys-tem prevents the equation to be divided into multiple simpler ones. So, in order tonumerically solve the many-body Schrödinger equation efficiently, an approximation hasto be used. This is where density functional theory (DFT) is useful.6 The DFT allowsus to identify the ground-state properties of a system without having to deal with themany-electron state. Using the concept of a spatially dependent electron density thetheory maps the many-body problem, including the electron interaction, onto a one-bodyproblem. The Born-Oppenheimer Hamiltonian (in Gaussian units) is defined by

HBO = −∑i

h2

2m∇2i −

∑i,A

e2ZAriA

+∑i>j

e2 1

rij+∑B>A

e2ZAZBRAB

(7)

where m is the electron mass, e the electron charge, ZA the atomic numbers of theatoms, ri the coordinates of the electrons and RAB are the coordinates of the nuclei. Incomparison, the Kohn-Sham Hamiltonian is defined as

HKS = − h2

2m∇2 + Vext +

∫dr′

ρ(r′)|r− r′|

+ Vxc(r) (8)

and can be seen as a single electron Hamiltonian. In fact while HBO act on state functionsof the form ψ(r1, r2, r3...), where r1, r2, r3... are the coordinates of the various electrons,HKS depends only on r. This simplifies the calculations significantly. In the Kohn-ShamHamiltonian Vxc is called exchange potential and is a crucial quantity in DFT, we willnot discuss its details here however since it is out of the scope of this project. It is worthnoting here however that the Kohn-Sham Hamiltonian must have the same symmetryof the crystal of the material it should model. Despite it can be proven that HKS isexact only for calculating the ground state of a material it provides a powerful tool toreliably calculate the band structure of a wide range of materials. In practice we nowhave a tool for calculating HKS|nk〉 = εnk|nk〉 where εnk are the bands and |nk〉 the

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eigenstates of HKS. We can then calculate, at least in principle, the matrix elementsof any observable, in particular a spin-polarization. Using this we essentially convertthe interaction potentials of the electrons into an outer density potential, providing uswith a numerically solvable one-body Hamiltonian which in turn provides us with theground-state properties needed to calculate the Edelstein effect.

1.2.7 Kubo linear response theory

The Kubo linear response theory is an approximation to treat how a time dependentperturbation affects the expectation value of an observable.7 For the derivation of theKubo formula we start from the density matrix formalism which does not restrict us tothe treatment of pure quantum states.8 For a given generic observable O we can, oncethe density matrix ρ is defined, calculate its expectation value as

〈O〉 = Tr{ρ(t)O} =∑j

pjj〈ψj|Oj|ψj〉, T r{a} =∑i

aii (9)

ρ(t) =e−βH

Z, Z = Tr{e−βH} and β =

1

kβT (10)

where kβ is the Boltzmann constant and T the temperature. When describing the re-action of a time independent system to a small perturbation it is helpful to use theinteraction picture instead of the Schrödinger picture. In the Schrödinger picture thetime dependence is strictly "attached" to the states while the observables are alwaystime independent. There is also the Heisenberg picture in which the operators carry allthe time dependence and the states are time independent. The interaction picture is amix of the two. We consider a Hamiltonian on the form H = H0 + H1(t) where H0 is theKohn-Sham Hamiltonian and H1(t) a time dependent perturbation.

The time dependence of the density operator in the interaction picture is given by

dρi(t)

dt=

1

ih[H i

1(t), ρi(t)] (11)

where the index i stands for "interaction picture". After integrating we get

ρi(t)− ρi(−∞) =1

ih

∫ t

−∞dt′[H i

1(t′), ρi(t′)], ρi(−∞) = ρ0. (12)

Combining equation 9 with equation 12 we obtain

〈O〉 = Tr{ρ(t)O} = Tr{ρi(t)Oi(t)} = Tr{ρ0O}+1

ih

∫ t

−∞dt′Tr{[H i

1(t′), ρi(t′)]Oi} (13)

⇒ 〈O〉 = Tr{ρ0Oi}+

i

h

∫ t

−∞dt′Tr{ρi(t′)[H i

1(t′), Oi]}. (14)

Only considering the first order of the density operator gives the first order approximationof the expectation value

〈O〉 ≈ 〈O〉0 +i

h

∫ t

−∞dt′〈[H i

1(t′), Oi(t)]〉0. (15)

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The change in the expectation value from the time independent solution can be writtenas

δ〈O〉 ≈ i

h

∫ t

−∞dt′〈[H i

1(t), Oi(t)]〉0. (16)

If we then consider H i1 as two parts H i

1 = P (t)F (t), where F is an external field and Pis the coupling between that field and an observable of the electronic system, we get

δ〈O〉 ≈ i

h

∫ t

−∞dt′〈[P i(t′), Oi(t)]〉0F (t′) (17)

If we define a response as, assuming a perturbation that is switched on at t’,

χ = Tr{ ihρ0[P i, Oi]}Θ(t− t′) (18)

where Θ(t) is the Heaviside function, we get

δ〈O〉 ≈∫ t

0

dt′χ(t, t′)F (t′) (19)

χ = Tr{ ihρ0[P i, Oi]}Θ(t− t′) =

i

h

∑n

ρnn〈n|[P i, Oi]|n〉Θ(t− t′) (20)

=i

h

∑n

ρnn〈n|(P iOi − OiP i)|n〉Θ(t− t′) (21)

=i

h

∑n

ρnn〈n|(eiH0t

′h P e

iH0(t−t′)h Oe

−iH0th − e

iH0th Oe

iH0(t′−t)h P e

−iH0t′

h )|n〉Θ(t− t′) (22)

=i

h

∑nm

ρnn(eiEnt′

h 〈n|P |m〉eiEm(t−t′)

h 〈m|O|n〉e−iEnt

h − eiEnth 〈n|O|m〉e

iEm(t′−t)h

〈m|P |n〉e−iEnt′

h )Θ(t− t′)(23)

=i

h

∑nm

ρnn(e−iEn(t−t′)

h eiEm(t−t′)

h PnmOmn − eiEn(t−t′)

h eiEm(t′−t)

h OnmPmn)Θ(t− t′) (24)

=i

h

∑nm

(ρnnei(Em−En)(t−t′)

h PnmOmn − ρmmei(Em−En)(t−t′)

h PnmOmn)Θ(t− t′) (25)

=i

h

∑nm

(ei(Em−En)(t−t′)

h PnmOmn(ρnn − ρmm))Θ(t− t′) (26)

Where ρnn is the Fermi-Dirac function. A Fourier transform with respect to δt = t − t′and adding limδt→0 e

−ηδt in order for the integral to converge gives us

χ(ω) = limη→0+

i

h

∑nm

PnmOmn(ρnn − ρmm)

∫ ∞0

ei(ωmn+ω+iη)δtdδt, ωmn =Em − En

h(27)

→ χ(ω) = limη→0+

−1

h

∑nm

PnmOmn(ρnn − ρmm)

ωmn + ω + iη. (28)

Additionally, it can be proven that a finite value of η can account for the so called linebroadening (lifetime of a state).

14

1.2.8 Edelstein effect

To measure the Edelstein effect we want to determine the off-diagonal term of the tensorχ and in order to do that we have to define the operator P , the field F and the observableO that are involved. The Edelstein effect is an effect where the electrical field affects thespin polarization of a material. This means that F is the electric field, P is the couplingbetween the electric field and the electron density and the observable O is the spin of thematerial. This gives rise to the following equations that will be substituted in equation28.

Onm = Sxnm, Pmn = erymn, rymn = − ih

me

pymnEm − En

(29)

where Sxnm is the spin in the x-direction, e is the electron charge and rymn is the positionof the electron. The third relation in equation 29 is due to the commutation relationpi = me

ih[H, ri], where me is the mass of the electron, pymn is the momentum and Em and

En are the energies. This leads to the following expression:

χ(ω) =ie

hme

∑nm

(ρnn − ρmm)

ωnm

pymnSxnm

ω − ωmn + iη(30)

=ie

hme

{∑n>m

(ρnn − ρmm)

ωnm

pymnSxnm

ω − ωmn + iη+∑m>n

(ρnn − ρmm)

ωmn

pymnSxnm

ω − ωmn + iη

}(31)

=ie

hme

{∑n>m

(ρnn − ρmm)

ωnm

pymnSxnm

ω − ωmn + iη+

(ρmm − ρnn)

ωmn

pynmSxmn

ω + ωmn + iη

}(32)

⇒ χ(ω) =ie

hme

∑n>m

(ρnn − ρmm)

ωnm

( pymnSxnm

ω − ωmn + iη+

pynmSxmn

ω + ωmn + iη

)(33)

In all of this derivation a summation of k (crystal momentum) has been consideredimplicitly, i.e. the Em term correspond to the previously mentioned Bloch energies εnk.In order to actually calculate the response tensor we need to consider a summation overk in the first Brillouin zone. Note also that all quantities in the summation carry the kdependence of |nk〉. In equation 34 the k dependence is incorporated in the relation:

χ(ω) =ie

hme

∑k,n>m

(ρknn − ρkmm)

ωknm

( pykmnSxknm

ω − ωkmn + iη+

pyknmSxkmn

ω + ωkmn + iη

). (34)

Density functional theory is then used to derive the solid state Hamiltonian of the system.In this Hamiltonian the structure of the material comes into play and the symmetries areof particular importance. If the crystal has complete inversion symmetry there also arisesa symmetry in the Hamiltonian which completely cancels out the Edelstein effect. Instead,in order for the Edelstein effect to be present we need a material with either locally orglobally broken inversion symmetries. In this computation of the Hamiltonian, Bloch’stheorem is also of great importance as it simplifies the computation. The eigenstatesthat are produced by the Hamiltonian are then used to compute p, S, ρ and ωkmn.This of course means that the indexes in equation 34 are the same as the band indexin the Bloch states. Additionally, note that we do not account here for the intrabandcontribution. There are however some hints that these contributions might be zero,9 butfurther numerical and/or symmetry based investigations are needed in this direction.

15

1.3 Program

This section describes each part of the EDEFF program calculating the χ tensor (it canbe divided into four parts) and describes in a general way the role each part played inthe calculation. The calculations are done in Rydberg atomic units (R.a.u.) where theinduced spin polarization is measured in units of the Bohr magneton, µB, where one Bohrmagneton corresponds to a complete switch of the polarization, and the electric field ismeasured in units of ERy/a0e, where a0 is the Bohr radius, e the electron charge and ERythe energy in Rydberg atomic units. Consequently, the χ tensor is measured in unitsof µBa0e/ERy and calculated versus the photon energy of the electric field measured inelectron volts.

1.3.1 Reciprocal space discretization

As previously mentioned, to calculate the linear response (χ tensor) an integral over onlythe first Brillouin zone is required. This can be done numerically by creating a meshin this volume in k-space and approximating the integrals with a weighted sum on thesepoints. Since for symmetry reasons it is possible to restrict the Brillouin zone to a smallerportion of the k-space, that contains exactly the same information, we use a routine thatallows us to choose a specific part of the Brillouin zone where to perform the integration.After which we can use mentioned symmetries to enlarge the solution to the whole Bril-louin zone. In addition to choosing the integration volume the method includes a choiceof number of k-points to be used. Using the tetrahedron method,10 the chosen part of theBrillouin zone is divided into smaller volumes, tetrahedrons, which correspond linearlyto the number of k-points. The method includes a choice to divide the first tetrahedronsin thirds instead of the standard division by two. For example, a structure with 4-1subdivisions would represent a total of four divisions where the first division is done inthirds and the remaining three in twos. Whereas 4-0 subdivisions corresponds to fourdivisions in twos.

By varying the number of divisions, and thus the number of k-points, a convergencestudy was done for a given choice of integration volume. These choices for iron, copperand manganese silicide respectively was called BCC (1/48th of the Brillouin zone), BCT(1/16th of the Brillouin zone) and ORTH2 (1/4th of the Brillouin zone). The optionsfor iron and copper was complemented with BCCT and BCTT where the last T standsfor total, i.e. integration over the whole Brillouin zone. These use a larger part of theBrillouin zone than the standard BCC and BCT options for the calculations and thuswill require more computing power but might result in more accurate results.

1.3.2 Electron density and potential calculation

To approximate the electron density and potential of the system, which is used to formu-late the Hamiltonian, the augmented spherical wave (ASW) method is used. The methodis based on DFT, described in section 1.2.6, and provides the basis set ψnk that is usedby the linear response program.

16

1.3.3 Edelstein effect calculation

The state properties of the system are calculated by solving the single-body Hamiltonian,given by the ASW method, on the k-points created by the discretization routine. Theseproperties are in turn implemented in equation 34 and the off diagonal elements of theχ tensor, the interband contribution, versus the photon energy of the electric field iscalculated. This routine includes an option to rescale the effect of the spin-orbit couplingin the calculations. How the Edelstein effect depends on the magnitude of the spin-orbitcoupling might suggest whether the effect is contributing to the results.

1.3.4 Refining results

A method to rescale and interpolate the resulting calculations is used. The rescalingtakes into account the integration volume used in the discretization routine, enlargingthe result to the whole Brillouin zone, so that it is comparable for different structuresand materials. The interpolation of the data points allows for better handling of theresults, especially plotting.

1.4 Method

A modified version of the MOKE program,2 called EDEFF (the main difference is in theimplementation of the Kubo formalism), was ran on the Rackham cluster on UPPMAX.The settings in the program chosen by us included the integration volume, the discretiza-tion in this volume, the magnitude of the of spin-orbit coupling and the rescaling of theresult depending on the integration volume used. Initially, we ran the discretization rou-tine where the integration volume and number of subdivisions where chosen. The electrondensity and potential calculations was already done by our supervisors and included inthe material specific code given to us. The main code was run, solving the Hamiltonianand calculating the specific elements of the χ tensor. When the calculations finished therefining method was ran (with a rescaling factor modified corresponding to the integra-tion volume used).

To determine how the calculated χ values depend on the discretization a convergencestudy was done comparing the results of the same material and integration volume cal-culated with multiple different discretizations. For most calculations a maximum of 5-1subdivisions were used except for iron with the BCCT options were 6-0 and 6-1 subdi-visions were used as well. This was done because the convergence of these calculationsshowed to be of particular interest, which will be discussed later. It was also noted thatthe calculations with 5-0 and 5-1 subdivisions for manganese silicide exceeded the timelimit on the Rackham cluster, there were indications however that using 4-1 subdivisionswas sufficient. The study of spin-orbit coupling dependency was done for iron, copperand manganese silicide with the respective options BCCT, BCTT and ORTH2. For thesecalculations 5-1 subdivisions was used for iron and copper and for manganese silicide 4-1subdivisions were used. The χ tensor was compared for each material with the spin-orbitcoupling scaled to 0, 1 and 10 times the normal magnitude (non present, normal andgreatly enhanced spin-orbit coupling). This is of interest since the Edelstein effect is aconsequence of the spin-orbit coupling and expected to depend linearly on its magnitude.

17

2 ResultsIn this section the results of the calculations are presented and compared for different ma-terials, integration volumes, discretizations and spin-orbit coupling scaling. The graphsshow the real part of the interband contribution values of the χ tensor plotted versus thephoton energy of the electric field in electron volts.

2.1 Iron

The results for iron using the BCC option in the discretization routine with 5-1 subdivi-sions is shown in figure 7. This configuration corresponds to a total of 442368 k-points inthe integration volume. We see a response of magnitude around 10-3 (R.a.u.) except fora region between 2 and 3 eV where the value grows. The graph for iron with the BCCToptions with 6-1 subdivisions (84934656 k-points) has a similar shape but with two or-ders of magnitude lower value, as seen in figure 8. The assumption that the Edelsteineffect is zero for iron and this result suggests that using symmetries in the Brillouin zoneto calculate the effect is not synonymous with using the whole zone, as the calculationstaking the whole Brillouin zone into account perform better.

0 1 2 3 4 5 6 7

Photon energy [eV]

-14

-12

-10

-8

-6

-4

-2

0

2

4

[Ba

0e/E

Ry]

10-3

Figure 7: The calculated element of the χ tensor plotted versus photon energy for iron with theBCC option with 5-1 subdivisions.

18

0 1 2 3 4 5 6 7

Photon energy [eV]

-8

-6

-4

-2

0

2

4

[Ba

0e/E

Ry]

10-5

Figure 8: The calculated element of the χ tensor plotted versus photon energy for iron with theBCCT option with 6-1 subdivisions.

2.2 Copper

Similar calculations as the ones for iron were done for copper. In figure 9 the results forthe BCT option with 5-1 subdivisions corresponding to 1327104 k-points is seen. Com-paring this with the results for the calculations done in the whole Brillouin zone, whichcan be seen in figure 10, further enhances the notion that the use of symmetries in theBrillouin zone results in a worse solution. For copper with a BCTT option with 5-1subdivisions (21233664 k-points) we see an order of magnitude of 10-19 (R.a.u.) while forregular BCT the magnitude is around 10-13 (R.a.u.). Both are however very small andconsidered to be zero.

0 1 2 3 4 5 6 7

Photon energy [eV]

-4

-3

-2

-1

0

1

2

[Ba

0e/E

Ry]

10-13

Figure 9: The calculated element of the χ tensor plotted versus photon energy for copper withthe BCT option with 5-1 subdivisions.

19

0 1 2 3 4 5 6 7

Photon energy [eV]

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

[Ba

0e/E

Ry]

10-19

Figure 10: The calculated element of the χ tensor plotted versus photon energy for copper withthe BCTT option with 5-1 subdivisions.

2.3 Manganese silicide

In figure 11 we see the results for manganese silicide with the ORTH2 option with 4-1 sub-divisions (663552 k-points). For low photon energies we see a larger order of magnitude,around 10-3 (R.a.u.), compared to both the corresponding values for iron and copperwith the BCCT and BCTT options respectively, as expected if the program works asintended. In the result of the calculations of the individual atoms for manganese silicide,figure 12, we note that the values of each atom varies a lot but are of similar magnitudeas the results calculated with the whole unit cell. This indicates that the Edelstein ef-fect is present in all atoms and that they contribute with a similar amount in the unit cell.

0 2 4 6 8 10 12

Photon energy [eV]

-8

-6

-4

-2

0

2

4

6

8

10

[Ba

0e

/ER

y]

10-4

Figure 11: The calculated element of the χ tensor plotted versus photon energy for manganesesilicide with the ORTH2 option with 4-1 subdivisions.

20

0 2 4 6 8 10 12

Photon energy [eV]

-1.5

-1

-0.5

0

0.5

1

1.5

[Ba

0e/E

Ry]

10-3

Atom 1

Atom 2

Atom 3

Atom 4

Atom 5

Atom 6

Atom 7

Atom 8

Figure 12: The calculated element of the χ tensor plotted versus photon energy for each atom inthe unit cell of manganese silicide.

2.4 Convergence

In figures 13 and 14 the results of the convergence study are shown (the four smallestdiscretizations are plotted here) for iron with the BCC and BCCT options respectively.For BCC it is clear that the shape of the graph is preserved, the χ tensor convergestowards non-zero values, for increased discretization. This is not the case for BCCThowever, we note that the magnitude for these calculations is decreasing with more k-points, indicating that the correct solution is approached by increasing the discretization.

0 1 2 3 4 5 6 7

Photon energy [eV]

-14

-12

-10

-8

-6

-4

-2

0

2

4

[Ba

0e/E

Ry]

10-3

16384 k-points

55296 k-points

131072 k-points

442368 k-points

Figure 13: The calculated element of the χ tensor plotted versus photon energy for iron with theBCC option with varying subdivisions.

21

0 1 2 3 4 5 6 7

Photon energy [eV]

-1.5

-1

-0.5

0

0.5

1

1.5

2

[Ba

0e/E

Ry]

10-4

393216 k-points

1327104 k-points

3145728 k-points

10616832 k-points

Figure 14: The calculated element of the χ tensor plotted versus photon energy for iron with theBCCT option with varying subdivisions.

Once again similar results as the ones for iron are achieved for copper, which can beseen in figures 15 and 16. The results with the BCT option quickly converges towardsnon-zero values while the results with BCTT decreases with increased discretizations.Although, as mentioned earlier, the order of magnitude for copper is much smaller thanfor iron. The convergence study for both iron and copper indicates once again that theuse of symmetries in the Brillouin zone causes the solution to not converge to zero as itshould. While using the whole Brillouin zone in the calculations yields a more realisticsolution that converges towards zero.

0 1 2 3 4 5 6 7

Photon energy [eV]

-4

-3

-2

-1

0

1

2

[Ba

0e/E

Ry]

10-13

49152 k-points

165888 k-points

393216 k-points

1327104 k-points

Figure 15: The calculated element of the χ tensor plotted versus photon energy for copper withthe BCT option with varying subdivisions.

22

0 1 2 3 4 5 6 7

Photon energy [eV]

-1

-0.5

0

0.5

1

1.5

2

[Ba

0e/E

Ry]

10-18

786432 k-points

2654208 k-points

6291456 k-points

21233664 k-points

Figure 16: The calculated element of the χ tensor plotted versus photon energy for copper withthe BCTT option with varying subdivisions.

For manganese silicide the results of the convergence study are shown in figure 17. Unlikefor the BCCT and BCTT studies for iron and copper the results for manganese silicidewith the ORTH2 option converges towards a non-zero shape. Since these calculations aredone in a relatively large part of the Brillouin zone (1/4th), compared to the BCC andBCT options (1/48th and 1/16th), the error caused by an extensive use of symmetriesin the Brillouin zone is likely not present, to the same degree, in this result. Providing asign that the program is able to recognize the Edelstein effect in manganese silicide.

0 2 4 6 8 10 12

Photon energy [eV]

-8

-6

-4

-2

0

2

4

6

8

10

12

[Ba

0e

/ER

y]

10-4

24576 k-points

82944 k-points

196608 k-points

663552 k-points

Figure 17: The calculated element of the χ tensor plotted versus photon energy for manganesesilicide with the ORTH2 option with varying subdivisions.

23

2.5 Spin-orbit coupling dependency

As previously mentioned the spin-orbit coupling dependency was determined with theBCCT, BCTT and ORTH2 options. For iron and copper (BCCT and BCTT) 5-1 sub-divisions was used and for manganese silicide 4-1 subdivisions was used. The results foriron and copper can be seen in figures 18 and 19, where we note that the Edelstein effectwith 10x scaling does not seem to follow the shape of the graph with normal (1x) spin-orbit coupling. The calculated values do not depend on the magnitude of the spin-orbitcoupling in any obvious way. To be compared to the corresponding results for manganesesilicide, seen in figure 20, where we note that the effect with good approximation scales lin-early with the increased spin-orbit coupling magnitude. This reinforces that the programidentifies the non-zero Edelstein effect in the calculations for manganese silicide but notfor iron or copper. In addition to this we see that the program treats the Edelstein effectas a direct consequence of the spin-orbit coupling, as intended, since the χ tensor val-ues are zero when the magnitude of the spin-orbit coupling is set to zero, for all materials.

0 1 2 3 4 5 6 7

Photon energy [eV]

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

[Ba

0e/E

Ry]

10-5

0x SOC scaling

1x SOC scaling

10x SOC scaling

Figure 18: The calculated element of the χ tensor plotted versus photon energy for iron with theBCCT option with varying influence of spin-orbit coupling.

24

0 1 2 3 4 5 6 7

Photon energy [eV]

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

[Ba

0e/E

Ry]

10-18

0x SOC scaling

1x SOC scaling

10x SOC scaling

Figure 19: The calculated element of the χ tensor plotted versus photon energy for copper withthe BCTT option with varying influence of spin-orbit coupling.

0 2 4 6 8 10 12

Photon energy [eV]

-3

-2

-1

0

1

2

3

4

5

6

[Ba

0e

/ER

y]

10-3

0x SOC scaling

1x SOC scaling

10x SOC scaling

Figure 20: The calculated element of the χ tensor plotted versus photon energy for manganesesilicide with the ORTH2 option with varying influence of spin-orbit coupling.

25

3 DiscussionAs mentioned in section 1.2.8 we do not expect the Edelstein effect to appear for iron orcopper. Nonetheless, the χ tensor for those materials assumes non-zero values regardlessof which integration volume is used. For the iron BCCT and copper BCTT calculationsthis appears to be due to a numerical error that decreases with increased discretization.This is clear from observing figures 14 and 16 as they both seem to converge towardszero. The magnitudes of the values for copper are better in these convergence studiesbut only by a constant factor as their dependence on the discretization seems to be thesame. The results of the convergence studies for iron BCC and copper BCT in figures13 and 15 are significantly worse. It is hard from these graphs to determine how theydepend on the discretization but what we can say is that they do not converge towardszero as expected. This is due to the portion of the Brillouin zone taken into account notcontaining enough information for the expression we are calculating. The reason for thiswould be that the spin-orbit coupling breaks some symmetry, so when we apply thesesymmetries and enlarge the results to the entire zone we get another value that the pro-gram converges to. This argument is further reinforced by the fact that the BCCT andBCTT options do converge to the correct value. An obvious solution to this is to use aslarge part of the Brillouin zone as possible (preferable the whole zone) when doing theEdelstein effect calculations as they appear to give the most credible results.

In any regard, using the best performing settings (the largest possible integration volumewith smallest discretization) we see a clear tendency towards low values of the Edelsteineffect for iron and copper. In figure 11 the result for manganese silicide with 4-1 sub-divisions show that for low photon energies the Edelstein effect is significantly strongerthan for iron and copper with χ values peaking at 10-3 (R.a.u.). This is also the case forthe individual atoms in manganese silicide as can be seen in figure 12. The magnitude ofthe response tensor in the individual atoms is of a similar magnitude as the unit cell as awhole, especially for low photon energies. Confirming that all atoms in the unit cell con-tributes with a similar amount. Worth noting concerning the calculations for manganesesilicide is also how close the results with varying discretizations are to each other, as canbe seen in figure 17. Unlike for iron or copper the calculations for manganese silicidequickly converge towards specific non-zero values. Indicating that the Edelstein effect isnoticed by the program and contributes to the result for manganese silicide.

As mentioned in section 1.2.5, spin-orbit coupling plays a crucial part of what causesthe Edelstein effect. So consequently an interesting feature in the program is how thecalculated response tensor depends on the magnitude of the spin-orbit coupling, or theabsence of it. An obvious confirmation that the program treats the existence of thespin-orbit coupling as intended is seen in figures 18, 19 and 20 where the χ tensor foriron, copper and manganese silicide respectively is zero when the spin-orbit coupling isset to zero. The more interesting comparison is when its influence in the calculations isincreased, in our cased scaled to ten times the normal effect. For iron this causes theχ tensor to oscillate in general, but not increase in magnitude. Thus, the shape of thegraphs is not preserved with increased spin-orbit coupling as would be expected if thevalues were due to the Edelstein effect. The same is true for copper, as seen in figure

26

19, where the results with increased scaling is of a higher magnitude but once again theshape of the normal plot is not preserved. Comparing this to corresponding results formanganese silicide, seen in figure 20, we see that the increased spin-orbit coupling be-haves differently. The scaling of the χ tensor is not entirely proportional to the magnitudeof the spin-orbit coupling but still follows its general shape, especially for small photonenergies, suggesting once again that the Edelstein effect is present and contributes to theresults for manganese silicide.

One reason for this deviation from complete linearity might be that the crystal potentialis not recalculated with respect to the spin-orbit coupling, which means that the potentialfor normal spin-orbit coupling is used for all three calculations, where in reality a largerspin-orbit coupling would have additional effects in the system. Since this is not the casefor iron or copper it also suggests that the non-zero results previously discussed for thosematerials are due to a numerical error in the calculations, an error that does not in anyobvious way depend on the spin-orbit coupling.

4 ConclusionsIf a large part of the Brillouin zone are used for the integrations (in our case the wholezone for iron and copper and 1/4th for manganese silicide) the program appears to givecredible results. We see for these cases, as expected, that the values for iron and cop-per are lower than for manganese silicide for both the unit cell as a whole but also forthe individual atoms in the cell. This is reinforced by the convergence study where thecalculated elements of the χ tensor converges to zero with increased discretization forboth iron and copper (assuming usage of the best performing integration volume). Wealso conclude that the results with scaled spin-orbit coupling is promising. For iron andcopper it reinforces the notion that the non-zero values we get is caused by a numericalerror and we can in the case of manganese silicide see that the spin-orbit coupling isintrinsically responsible for the Edelstein effect. This confirms that the program givescredible results if an integration over a large volume is used, but we also conclude thatthe numerical error in the calculations for iron is quite large compared to the values weget for manganese silicide. Even though the convergence study exhibits a realistic patternthe results is most likely not to be seen as true material properties, using symmetry in theBrillouin zone clearly generates an error. How large this error is in the case of manganesesilicide is unknown to us but it is most likely present, though probably not as large asour error for iron. In any case to confirm with certainty how trustworthy the program is,and how its results relate to true material properties, more calculations have to be doneon different kinds of materials. In addition, the Edelstein effect should be tested andmeasured experimentally, in which case it is relevant that previous tests of the relatedspin Hall effect has had an uncertainty of about 2 ·10−5 (R.a.u.). Which would be enoughto compare with our results for manganese silicide. Nonetheless, with all of this in mindthe program still seem to show much promise in the calculation of the Edelstein effect.

An important goal of the research regarding the Edelstein effect is of course possibletechnical applications. If the results of these calculations for manganese silicide are as-

27

sumed to be correct it is still a long way to go until the applications discussed in theintroduction are possible. For practical applications in switching the electron spin theeffect must be a lot bigger than 1 · 10−3 (R.a.u.) which our calculations show for man-ganese silicide. This is due to the fact that the atomic unit for the electrical field isaround 5.1 · 1011 V/m, meaning we need an extremely large field to switch the spin ina reasonable time frame. We also assume in the calculations that the electric field isa small perturbation which might no longer be the case if we applied a field of thatmagnitude. Therefore, the calculations would no longer be dependable which makes itclear that other solutions must be found for practical applications. One possibility is tosimply reorient the spin with the Edelstein effect as the induced magnetic field appliesa torque on the electron spin, this would solve the problem of requiring a large electricfield. Another possibility is that there exists materials were the χ tensor is much larger,in which case further tests with more materials is all that is needed. A recent proposal11

for technological applications consist in using AFM materials where, even though it isnot possible to impart a net magnetization, it is possible to orient the AFM using theEdelstein effect. Nonetheless, it is clear that programs like this one can be used to studythe Edelstein effect and will likely be useful to predict what materials might suitable forfuture technical implementations.

We can conclude by saying that spintronic devices, even if they won’t appear for sometime, can in the future improve and possibly replace the devices used for todays RAMand that numerical methods most likely will contribute in developing such technologies.

28

References1 M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988).

2 P. M. Oppeneer, in Magneto-Optical Kerr Spectra, edited by K. H. J. Buschow, Hand-book of Magnetic Materials Vol. 13 (Elsevier, Amsterdam, 2001), pp. 229–422.

3 N.W. Ashcroft and N.D. Mermin, Solid state physics 5th edition, Thomson Brooks/Cole(1976).

4 V. Edelstein Phys. Rev. B 67, 033104 (1990).

5 U. von Barth, Physica Scripta. Vol. T109, 9-39 (2004).

6 W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965).

7 G. Rickayzen, Green’s function and condensed matter, Academic Press, London (1980).P.M. Oppeneer, lecture notes.

8 J.J. Sakurai, Modern Quantum Mechanics, Rev. ed., Addison-Wesley publishing com-pany (1994).

9 A direct calculation of the intraband contribution to the off-diagonal conductivity atzero frequency showed that it was zero. P. M. Oppeneer, unpublished result.

10 P.M. Oppeneer, A. Lodder, J. Phys. F 17, 1885 (1987).

11 P. Wadley et al., Science 351, 587 (2016).

29