Ab initio simulations of topological phase transitions in ...lnu.diva-portal.org/smash/get/diva2:1290892/FULLTEXT01.pdf · In condensed matter physics the classi cation of the phases

Universita degli studi dell’Insubria

DIPARTIMENTO DI SCIENZA E ALTA TECNOLOGIA

Corso di Laurea Magistrale in Fisica

Tesi di laurea magistrale

Ab initio simulations of topological phase transitions in Diracsemimetal Cd3As2 doped with Zn and Mn impurities.

Candidate:

Andrea RancatiMatricola 731470

Examiner:

Prof. Carlo Canali

Supervisors:

Prof. Alberto DebernardiProf. Fhokrul IslamProf. Alberto Parola

Anno Accademico 2017/18

Acknowledgments

I would rst like to thank my thesis supervisors, Professors Alberto De-bernardi and Fhokrul Islam, and my examiner Professor Carlo Canali, fortheir support and devotion of their time, and for all the valuable experiencesthey allow me to live. I will always remember our weekly meetings, whichgave me the opportunity to partcipate to genuine scientic discussions thatcontribuited to my progress as a physics student.

I also thank my parents, Lina and Roberto, for providing me with continuoussupport throughout all my years of study. This accomplishment would nothave been possible without them.

Finally, I owe great thanks to Camilla, whose unfailing encouragement ac-companied me through this journey, and in my life.

Contents

1 Theoretical background 1

1.1 The Bloch's theorem . . . . . . . . . . . . . . . . . . . . . . . 11.2 General introduction to topology . . . . . . . . . . . . . . . . 21.3 3D Topological insulators . . . . . . . . . . . . . . . . . . . . . 41.4 Topological semimetals . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Weyl semimetals . . . . . . . . . . . . . . . . . . . . . 111.4.2 Dirac semimetals . . . . . . . . . . . . . . . . . . . . . 161.4.3 From Dirac to Weyl and further: topological phase

transitions . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Computational background 23

2.1 The density functional theory . . . . . . . . . . . . . . . . . . 232.1.1 The Hohenberg-Kohn theorems and Kohn-Sham ansatz 242.1.2 Determination of the electronic structure and accu-

racy: pseudopotential vs all-electron scheme . . . . . . 282.2 Wannier functions and tight-binding models . . . . . . . . . . 29

3 Methodology and results for the undoped Cd3As2 system 35

3.1 The cadmium-arsenide system Cd3As2 . . . . . . . . . . . . . 373.2 Results and discussion of the Cd3As2 system . . . . . . . . . . 38

4 Results and discussion of the non-magnetic doped systems 51

4.1 Breaking of the rotational C4 symmetry . . . . . . . . . . . . 514.1.1 DFT results in the presence of SOC . . . . . . . . . . . 524.1.2 Topological properties . . . . . . . . . . . . . . . . . . 55

4.2 Breaking of both inversion I and rotational C4 symmetry . . . 574.2.1 DFT results in the presence of SOC . . . . . . . . . . . 594.2.2 Topological properties . . . . . . . . . . . . . . . . . . 62

4.3 Breaking of the I symmetry . . . . . . . . . . . . . . . . . . . 664.3.1 Results for the unrelaxed structure in the presence of

SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

i

ii CONTENTS

4.3.2 The relaxed structures . . . . . . . . . . . . . . . . . . 744.3.2.1 Results for the total relaxed structure in the

presence of SOC . . . . . . . . . . . . . . . . 744.3.2.2 Eect of external strains on the system . . . . 77

5 Results and discussion of the magnetic-doped systems 87

5.1 Results for the 40 atoms cell in the presence of SOC . . . . . . 885.1.1 Results for the 80 atom supercell . . . . . . . . . . . . 94

A Convergence tests 103

B Wannier90 numerical errors 107

Conclusion 102

Introduction

In condensed matter physics the classication of the phases of matter has al-ways played an important role since the Landau's theories, but in the recentyears the study of new phenomena, such as the quantum Hall eect [1], hasled to a dierent classication paradigm based on the notion of topologicalorder.The rst realization of a topological material in solid state physics has beenthe topological insulator (TI) [2], a particular insulating systems in whichthe spin-orbit interaction provides a non-trivial topological order resulting insome unique properties, such as the presence of metallic surface states, whichare robust against disorder and other perturbations. The two dimensionalFermi contour that can be dened on a surface form closed loops in momen-tum space and, at some special points called Dirac points, the surface statesbecome doubly degenerate. Furthermore, away from this degenerate point,the low-energy dispersion becomes linear and form a conical intersection.Recently another class of topological materials has attracted large interestbecause of their unique properties: the topological Dirac semimetals (DSM)and Weyl semimetals (WSM) [3, 4, 5]. The non-trivial topology, as in caseof TI, comes out because of the spin-orbit interaction but, in contrast withthe latter, DSM and WSM present a linear-dispersed low-energy excitationsand conical intersection at some special momentum points of the bulk, calledDirac or Weyl points. The two dimensional Fermi contour present on thesurface now form open lines in momentum space, called Fermi arcs, whoseend points are projections of two Dirac or Weyl fermion nodes on the surfaceBrillouin zone.The main dierences between DSM and a WSM are the follows: a Diracmaterial, close to the Dirac node, has 2-fold degenerate excitations and gen-erally requires three symmetries (time reversal, inversion and an extra crystalsymemtry) to be stable. On the other hand, in a Weyl material the degener-acy is lifted by the breaking of either T or I, and the nodes always come inpairs since they can be viewed as a Dirac node split in the momentum spaceby the symmetry breaking. The two nodes are characterized by having an

iii

iv INTRODUCTION

opposite chirality (orientation of the Berry curvature), and since they do notrequire any such symmetry-dependent protection, their topological proper-ties are more robust than the Dirac's ones. Both DSM and WSM exhibitnovel quantum phenomena that are not only of interest for their fundamen-tal physics but also may hold potential for technological applications such asquantum computing or low-energy spintronics devices [6].

In this work we exploit the unique characteristics of a Dirac semimetal ma-terial to be symmetry-protected, to investigate dierent topological phasetransitions provided by chemical dopings, focusing in particular on the elec-tronic, magnetic and topological properties of the doped systems, studiedby the mean of rst-principles methods based on density functional theory(DFT) approach [7]. In particular these doped systems, besides being of in-terest for investigating the role of topology in solid state physics, could have agreat potential for practical application since the dierent topological phasesthat come along with the chemical dopings allow one to exploit the uniqueproperties of topological materials. The starting point for our study will bethe material called cadmium-arsenide (Cd3As2), an example of a topologicalDirac semimetal, which is chemically stable at ambient conditions.Chapter I presents a general introduction to topology, especially in condensedmatter physics, and to the main physical properties of the topological mate-rials we mentioned. Then, in chapter II, we briey present the methods andthe computational tools that we used for our study. In chapter III a moredetailed introduction to our work is given, along with a schemetic view ofthe path we followed, together with the results that we obtained for pristineCd3As2, which we use as bench mark for our computational methods. Fi-nally, in chapter IV and V, the results for the doped systems are presentedand discussed, respectevely for the non-magnetic (IV) and magnetic (V) dop-ings.Our study has enabled us to discern how doping can give rise to see dierenttopological phase transitions. Specically our work shows that dierent re-alizations of non-magnetic doping gives rise to dierent topological phases:the topological Weyl semimetal phase, which is of great interest since it cansupport a robust quantum spin Hall eect, and a very special mixed Dirac +Weyl phase, where surprisingly both a Dirac and a Weyl phase can coexist inthe same system. Furthermore, magnetically doped systems show the emer-gence of a magnetic Weyl phase, which can support a quantum anomalousHall eect. Our work can be the starting point for future studies, both the-oretical and experimental, in which the unique physical properties we foundin the doped Cd3As2 systems can be further investigated, in order to exploitthem for practical applications.

vi INTRODUCTION

Chapter 1

Theoretical background

The rst two chpaters are intended to introduce the reader to the physicsconcepts, theories and methods used in this work, mostly related to the elec-tronic structure and topology in condensed matter physics.

1.1 The Bloch's theorem

Understanding the behavior of electrons in materials is an essential key tostudy the dierent large-scale properties of matter that can be exploited forseveral practical uses; for instance their excitations in solids or moleculesallow us to study electronic, magnetic and optical properties of materials.Taking account of the many variables involved it's not always easy to studytheir behaviors in complex systems and it's clear that accurate theoretical ap-proaches and computational methods are needed. Among them the Bloch'stheory helps to understand the behavior of electrons in crystals, namely sys-tems having atoms arranged in a regular and periodic pattern. This approachis based on the Bloch's theorem [7] which states that

The eigenstates ψ of a generic one-electron Hamiltonian H = −~252

2m+ V (~r)

with a periodic potential V (~r), with the same periodicity of the crystal lat-tice, can be chosen to be a plane wave times a function which have the sameperiodicity of the potential.

Therefore the wavefunction for an electron in a crystal can be written as

ψn,~k(~r) = ei~k·~run,~k(~r), (1.1)

1

2 CHAPTER 1. THEORETICAL BACKGROUND

where ψn,~k(~r) is the wavefunction, ~k is the so called crystal momentum de-ned in the Brillouin zone, and un is a periodic function, with the sameperiodicity of the potential V (~r), which modules the plane wave in the realspace.Although in general the number of crystal momenta is discrete for a systemconsisting of N atoms, in the limit of large crystals the spacings betweenthe ~ki go to zero and the crystal momentum can be considered a continuousvariable. The eigenstates of the Hamiltonian H can be found separately foreach momentum in the primitive cell, and for each ~k there is a discrete set ofn eigenstates: choosing a particular value of n one obtains continuous func-tions of ~k (the bands) or eigenvalues εn(~k), separated by energy gaps.The bloch theorem is essential for this work since the Bloch states and thebands of solids will allow us to characterize topology in condensed matterphysics, as showed in the next sections.

1.2 General introduction to topology

Topology is a branch of mathematics that deals with a special classes ofobjects, for example surfaces in the 3D space, of which some particular prop-erties are invariant under continuous smooth transformations (homotopies).The most straightforward example is the smoothly transformation of a muginto a doughnut (a torus) as shown in gure 1.1 which can be done withoutclosing nor opening any holes: here the number of handles, counted by theso called genus g, is precisely the property that doesn't change during thetransformation, namely the topological invariant, which makes the two ob-jects topologically equivalent. For instance a topologically dierent objectis the sphere, which has no holes and cannot be smoothly transformed in atorus without opening new holes.A more rigorous way to dene the topological invariant for orientable sur-faces is given by the Gauss-Bonnet theorem [8] which allows one to dene aninteger topological invariant, called Euler characteristic

χ =1

2π

∫s

KdA = 2− 2g, (1.2)

that is the surface integral of the Gaussian curvature K [8]. For instance asphere of radius r has gaussian curvatureKs = 1

r2and g=0 (since the integral

is equal to 2) whereas for a at plane, a cylinder or a torus the integral of

1.2. GENERAL INTRODUCTION TO TOPOLOGY 3

Figure 1.1: A smooth tranformation of a mug into a dough-nut, topologically equivalent objects. Image downloaded fromhttps://www.cems.riken.jp/en/laboratory/qmtrt.

the Gaussian curvature gives zero and therefore we have g=1.The Gauss-Bonnet theorem can be generalized to include more abstract vec-tor spaces such as the band structures of a condensed matter system, in whichthe general concepts will return in a very similar way. In solid state physicstopology enters the game when one wants to characterize phases of matters;it's well known that some phases, such as magnets or superconductors, canbe understood in terms of continuous symmetries which they spontaneouslybreak [9]. However some particular phases have a more subtle kind of orderthat can be understood using the topology. The more famous example isthe quantum Hall eect [1]: the quantum hall state doesn't break any con-tinuous symmetry but it has been shown that its properties are insensitiveto smooth changes of the material parameters. It's clear that this fact canbe understood as a consequence of the topological structure of the quantumstate.

Dealing with crystals we can use the band structures and the Bloch's statespresented in the previous section in order to classify dierent topologicalphases: the shape of the bands and the way they are knotted characterizethe phases whereas topological invariants can be presented as the propertiesthat don't change upon adiabatic transformations of the system [7]. Materi-als realising non-trivial topological order are called topological materials.The rst historical example of a topological material is the class of topologi-cal insulators [2], widely studied in the past years for their unique propertyto be an insulator with a metallic topologically protected surface. Recentlyanother class of topological materials, the so called topological semimetals[3], has become of great interest: they are characterized by having particularconducting states in the bulk as well as unique metallic surface states.In the next sections we are going to present the main aspects of both topo-


logical insulators and semimetals, focusing in particular on how topology canbe formally characterized in these systems.

1.3 3D Topological insulators

In solid state physics insulators are usually presented as materials having anite energy gap between the valence and the conduction bands, everywherein the whole Brilluoin zone. Here the denition of topological equivalency isquite trivial: if one passes from one insulator state to another one using anadiabatic transformation, namely slowly changing the Hamiltonian parame-ters such as the system always remains in the ground state, and an energygap never closes along all the path, then it's said that the two states aretopologically equivalent.An evidence of topology can be found looking at interfaces between two in-sulating materials that belong to dierent topological phases: along a pathpassing through the interface the gap of the bands must closes somewhereotherwise, from the previous denition, the two materials would belong tothe same topological phase. This fact is known as the bulk-boundary corre-spondence and explains the presence of metallic states on the surface despitethe insulating bulk. This particular condition is found to be topologicallyprotected from external perturbations [2].In order to characterize these particular materials we need to dene topolog-ical invariants, and particular crystal symmetries provide us with a way todo this.

Time Reversal and Inversion symmetries

Time reversal symmetry (T ) is the key ingredient to dene topological in-variants in topological insulators. It can be viewed as a quantum operatorwhich results in reversal momentum and ipping of spins

T = iσyK, (1.3)

where σy is the complex Pauli's matrix and K is the complex-conjugationoperator; note that the particular form of the operator presented in eq. (1.3)is valid for system of half-integer spins equal to s = 1

2, as the ones considered

in this work. Furthermore this operator is antiunitary, namely T 2 = −1.

1.3. 3D TOPOLOGICAL INSULATORS 5

In crystals, where Bloch states are well dened, the operator T acts on thehamiltonian H and its eigenvalues as

H(−~k) = TH(~k)T−1 (1.4a)

En,↑(~k) = En,↓(−~k) (1.4b)

We can notice that at some particular crystal momenta of the Brillouin zonecalled time reversal invariant momenta (TRIMs), in which ~k = −~k + ~G, thebands are at least double degenerate: this is a consequence of the Kramer'stheorem which states that in systems of half-integer spins the two states ψ(k)and Tψ(k) (Kramer's pair) are orthogonal and share the same energy, makingthem (at least) double degenerate. Furthermore if the unitary inversionsymmetry I (I2 = +1), acting as

H(−~k) = IH(~k)I−1 (1.5a)

En,σ(~k) = En,σ(−~k), (1.5b)

is also present in the system the double degeneracy holds for each k-point inthe whole Brillouin zone, as one can see from equations (1.4b) and (1.5b),leading to double degenerate bands everywhere.

Topological invariants

Using the time reversal symmetry and the denition of the TRIMs Fu andKane dened a topological invariant for bidimensional systems [10], general-ized to the 3D case by Moore and Balents [11]. For a 2D system the startingpoint is the denition of a unitary matrix wmn(~k) =< u−~k,m|u~k,n >, anti-symmetric at the 4 TRIMs (Γi) present in the system. This antisymmetricproperty allows one to calculate the Pfaan of the matrix w(Γi)

Pf [w(Γi)] = w12(Γi)w34(Γi)...w2N−1,2N(Γi) (1.6)

(Pf [w(Γi)])2 = det[w(Γi)]

and subsequently a compact expression for a factor of the topological index


can be written as

δ(Γi) =Pf [w(Γi)]√det[w(Γi)]

. (1.7)

Provided |um(k) > is chosen continuously throughout the Brillouin zone thebranch of the square root can be specied gloablly and a Z2 invariant can bedened as follow

(−1)ν =4∏i=1

δ(Γi), (1.8)

where ν is the Z2 inviariant which can just take the values ν = 0, 1.Moore and Balents showed that four independent Z2 invariants, which in-volves a total of eight TRIMs, can be dened in the 3D case as follow

(−1)ν0 =∏

ni=0,π

δ(Γn1,n2,n3), (1.9)

(−1)νl =∏

ni=π,nj 6=i=0,π

δ(Γn1,n2,n3). (1.10)

where l = 1, 2, 3.The four indices are often written as (ν0;ν1,ν2,ν3) and, as in the 2D systems,each can just take the two values 0, 1.The index ν0, the so called strong index, is unique to a 3D material andcharacterize a special kind of topological insulators called strong topologicalinsulators [12]. The other three indeces νi are called weak indeces since theycharacterize the so called weak topological insulators [12].Note that if both T and I symmetries are present, the expressions for thetopological indeces simplify and they can be calculated as the product of theparity eigenvalues P of the parity opertor of the occupied Kramers pairs atthe TRIMs [13]

δ(Γi) =N∏i=1

P (Γi). (1.11)

Surface states, weak and strong TI

As already announced in the previous sections one main feature of topolog-ical insulator is the bulk-boundary correspondence which shows that bands

1.3. 3D TOPOLOGICAL INSULATORS 7

closings must happen on a boundary separating two dierent topological ma-terials and for instance can explain the presence of the surface states. Closeto the TRIMs, where the band touchings happen, the dispersion become lin-ear and the typical Dirac cone is shaped, as shown in 1.3c (we will see thismany times in this work).It can be showed that the surface states of a 3D topological insulator areprotected if the time reversal symmetry is present. A pair of TRIMs statesbelonging to the same surface can only be linked in two dierent ways inthe bidimensional momentum space, as showed in gure 1.2, by means of thesurface states: if the number of crossings between the surface states and the

Figure 1.2: Electronic dispersion between two TRIMs Γa and Γb. In (a) thenumber of edge states crossing Ef is even, whwereas in (b) is odd. An odd numberof crossings leads to topologically protected metallic boundary states. From [2]

Fermi energy Ef is odd, as in the case of topological materials, then there isno way to avoid the metallic state, not even changing the value of Ef , andthe topological phase is protected.

It has been shown [2] that dierent shapes of surface states can arise, butin any case they form closed lines, a loop, which represent the 2D corre-sponding of the Fermi surface in 3D. By the mean of this Fermi line wecan distinguish between weak or strong topological systems, looking at howmany Dirac points the surface state encloses (see gure 1.3): if the numberof enclosed points is even then the material is a weak topological insulatorwhereas if it's odd then the system is a strong topological insulator.The main physical dierence is that a strong topological insulator is topo-logically protected from the disorder localization whereas a weak topologicalinsulator is not [14].


Figure 1.3: Fermi loops in surface Brillouin zone for (a) weak and (b) strongtopological insulators. (c) is a single Dirac cone enclosed by the Fermi loop of thestrong topologial insulator.

The importance of the spin-orbit coupling

Spin orbit coupling (SOC) is a well known relativistic eect between thespin angular momenta and the orbital angular momenta of particles whichcontribution to the Hamiltonian is given by

Hsoc = λ~L · ~σ, (1.12)

where L is the orbital angular momentum, σ is the vector of the Pauli ma-trices and λ is a term that includes all the dependencies coming from theproperties of the system such as the potential, the atomic number Z (it isproportional to ∼ Z4) and so on.In crystalline solids the impact of this interaction could be signicant: itcauses the mixing of dierent spin and orbital momenta leading to level re-pulsions whose strength depends on λ and, most important, it could alsomove the bands in such a way to invert their order, a key mechanism fornon-trivial topology. In gure 1.4is shown an example of band inversion mechanism provided by SOC whichshift the order of the bands close to the Fermi enegy (+ and - represents theparities of the bands); the gap opening in in gure 1.4b is due to the mixingof states at the crossing points.

It should be noticed that in some special cases the crossings in the bulkBrillouin zone could be protected from an opening of a gap, resulting in aconducting bulk and a new topological phase (see gure 1.4c). In the nextsection we will present two example of topological materials which are char-acterized by having these bands crossing in the bulk, the so called Dirac andWeyl semimetals; in particular we will see that the Dirac semimetal phasecould be achieved by using the band inversion mechanism.

1.4. TOPOLOGICAL SEMIMETALS 9

Figure 1.4: Example of band inversion mechanism provided by the spin-orbit cou-pling, in which the parities ± of the k = 0 state are inverted. In (a) SOC is notpresent, in (b) the SOC reverses the parities resulting in an insulating system (thestates at the crossings are mixed) whereas in (c) the crossings are protected by thecrystal symmetries, resultic in a semimetal. From [3]

1.4 Topological semimetals

In the previous section we showed that in a 3D topological insulator thebands closing happens on the boundary as a direct consequence of a non-trivial topology. It can be said that 3D topological semimetals are the gaplesscousins of topological insulators in that besides the gapless surface states thebands closings can also occur in the bulk of the Brillouin zone.The name semimetal comes from the presence of a small overlap between thebottom of the bulk conduction band and the top of the bulk valence band atthe fermi energy, in this particular case in the form of points (see for examplethe band structure in gure 1.4c).As already mentioned, topological insulators and semimetals belong to dif-ferent topological phases but they are not completely unrelated; rst of allin both cases the non-trivial topology is strictly related to the presence ofSOC in the system, with the dierence that here some special k-points areprevented from band openings. Furthermore these semimetals phases canbe considered as particular intermediate topological phases in the transitionbetween two dierent insulating phases with fully gapped bands in the bulksince in order to break (and change) the topology a band closing in the bulkit's necessary. These bulk band touchings occurs (principally) linearly, re-


sembling the Dirac cone shape already presented in gure 1.3c.There are two dierent types of topological semimetal, depending on whichsymmetries are present in the crystal: Dirac and Weyl semimetals [3, 5, 4, 15].These two names remind us the eld of particle physics where Dirac and Weylname two dierent types of relativistic fermions which obey, respectevely, theDirac and the Weyl equations [16]

(i~γµ∂µ −mc)ψ = 0, (1.13)

i~∂0ψR/L = ±c(~σ · ~p)ψR/L, (1.14)

where γmu are the 4x4 Dirac matrices, µ is the index of the Einstein sum-mation convention, ψ is a four component spinor whereas the ψR/L are twocomponent spinors, ~σ is the vector of 2x2 Pauli matrices, which representthe spin variable, and ~p is the momentum operator.The two 2x2 Weyl equations come directly from the decoupled 4x4 Diracequation in the ultrarelativistic regime, when the mass term m goes to zero.In this situation, the energy dispersion of the Dirac equation E2 = c2p2+m2c4

becomes linear and the o-diagonal terms of the Dirac equation disappear

i~∂0(ψRψL

)= c(+~σ · ~p 0

0 −~σ · ~p)(ψRψL

)(1.15)

In this case the 4x4 Dirac equation can be viewed as a merging of two 2x2equations with opposite sign, sharing the same momentum ~p, which can beseparated forming two dierent Weyl equations. In analogy, when the ultra-relativistic regime is taken in account, a Dirac fermion can be viewed as astable merging of two dierent Weyl fermions, described by the same equa-tion but with opposite sign; the presence of two dierent signs are associatedwith a special property called chirality, which in the world of particle physicstells us that a Weyl fermion can just move parallel or antiparallel to its spindirection.

The linear energy dispersion of a Weyl fermion recovers the same behaviorone has at the band touchings of a semimetal system: close to the touch-ing points of the bulk bands structure it can be shown that the eectivelow-energy Hamiltonian describing the excitations of these systems resem-bles the relativistic equations for the massless fermions (see next sectionsfor details). Furthermore we'll see that the property of chirality can be also


found in topological semimetals [17]. One of the most important eect com-ing from the presence of these chiral electrons in materials is the so calledChiral anomaly [15] describing the adiabatic electron pumping caused by thejoint eect of the external magnetic and electric elds between Weyl pointswith opposite chirality: this generates a current along the direction of themagnetic eld leading to negative magneto resistance when the magnetic andelectric elds are parallel to each other.These solid-state realizations of Dirac and Weyl fermions oer a platformwhere prediction made by relativistic theories can be tested adding at thesame time entirely new properties coming from the context of condensedmatter such as the Fermi arcs, the surface states of semimetals. Furthermoretheir unique properties make them very interesting for practical applicationsamong which we cite the enhancement of the catalytic activity [18], quantumspin Hall eect [19] and anomalous Hall eect at room temperature [20], spinlters transistors [21], IR detectors [22] and room temperature + topologicalsuperconductors [23].

1.4.1 Weyl semimetals

In solid-state physics Weyl fermions are observed in Weyl semimetals in theform of low-energy excitations around the bulk bands touchings, called Weylnodes (or Weyl points), in which conduction and valence bands touch at aneven number of momenta in the BZ close to the Fermi energy. The crossingsoccur generically at low-symmetry momenta, have a linear dispersion arounda Weyl node and usually at these points the bands are 2-fold degenerate:in 3D systems this require either T or I (or both) to be broken, otherwisebands would be double degenerate everywhere in the bulk resulting in a 4-fold degenerate touching point, characteristic of a Dirac material.Excitations in Weyl semimetals can be described by the eective low-energyHamiltonian expanded around the Weyl nodes which in the most simple casetake the form

H(~k) = ±vf~k · ~σ, (1.16)

where vf is a constant that plays the same role the speed of light c plays inthe Weyl equation (1.14) (this is not the speed of light in the materials), ~k isthe crystal momentum and ~σ is the vector of Pauli matrices which representnot the real spin variable but a pseudo-spin which gives rise to the bandtouchings in Weyl materials [3, 5].


First of all we can see that the property of chirality arises also in solids:in particle physics it tells us that a Weyl fermion can propagate parallelor antiparallel to its spin whereas in solid-state physics, as we'll see later,chirality is related to the so called charge of the Weyl point which characterizethe topology of the system.Furthermore from equation (1.16) we notice that a Weyl node is protectedfrom a generic external perturbation, which expanded around the Weyl nodetakes the general form

Hp = δ0I2x2 + ~δ · σ, (1.17)

where δ may be expressed as Taylor expansions in the crystal momentum ~kand I2x2 is the 2x2 identyty matrix. It's clear that the perturbation can justmove the band touching around the BZ without opening a gap: a Weyl nodeis a topologically stable object and its stability is provided by the topology.The only way one can destroy a Weyl point is annihilating it with anotherWeyl point of opposite chirality, namely merging two nodes with oppositechiralities at the same crystal momentum ~k0.The topology characterization in Weyl semimetals is furnished by the Berrycurvature [8], presented in the next section, which gives an explicit way tocalculate the chirality and the topological invariant. The non-trivial topol-ogy of Weyl semimetals is signalled by the presence of peculiar surface statescalled Fermi arcs represented by open lines which connect the projection oftwo Weyl nodes on a surface.

Topology in Weyl semimetals: Berry curvature and Chern number

A key role in the topological band theory is played by the so called Berryphase, which arise because of the intrinsic phase ambiguity of quantum me-chanical wavefunctions. This argument is well presented in the rst chapterof the Kane's book on topological insulators [8], and here we are going tofollow his description.In crystalline systems it's easy to see that a generic Bloch state |u(~k) > isinvariant under the transformation

|u(~k) >→ eiφ(~k)|u(~k) >, (1.18)

which reminds us of electromagnetic gauge transformations. An object calledthe Berry connection, similar to a magnetic vector potential in momentum


Figure 1.5: Berry curvature generated by the presence of two Weyl nodes withopposite chiralities (orange positive and blue negative). The integral of the ux ofthe curvature through the green surface, surrounding a single Weyl point, can tellone the chirality of the node. From [5]

space, can be dened taking into account the periodic part of the Bloch states

~A(~k) =N∑n=1

−i < un(~k)| 5~k |un(~k) >, (1.19)

where N is the number of the occupied bands.The connection is not Gauge invariant since it transforms like ~A(~k)→ ~A(~k)+

∆~kφ(~k). On the other hand the analog of the magnetic ux in this situationis Gauge invariant and for a closed loop l in the momentum space we candene an invariant phase, called the Berry phase, as follow

γl =

∮l

~A(~k) · d~k =

∫S

~Ω(~k)dS. (1.20)

Here we used the Stoke's theorem and the surface integral is done using theso called Berry curvature ~Ω(~k) = ∆× ~A(~k) which is a sort of magnetic elddened in the crystal momentum space.Similarly to the Gauss's law of magnetism the ux of the Berry curvaturein the entire Brillouin zone is equal to zero, but we can always select closedsurfaces inside the BZ for which the ux may not vanish; for instance it hasbeen shown that Weyl points in the Brillouin zone act like source or drain ofthe Berry curvature, as they were magnetic monopoles in momentum space(see gure 1.5), and exactly the ux through a surface enclosing each ofthese points furnish the topological invariant (the topological charge) weneed, called the Chern number C

C =1

2π

∫S

~Ω(~k)dS, (1.21)


This number takes only integer values, dening a Z topological invariant, anddepending on the sign it denes also the chirality of the node. Furthermore itcan be shown [24] that a Chern number |C|= 1, calculated for a single node,is related to the linear dispersion of the bands whereas a greater number, forexample |C| = 2, lead to a parabolic or higher dispersions.As already mentioned the ux of the curvature through the whole Brillouinzone boundary must vanish (in order to avoid divergences of the curvature)and therefore the number of Weyl points in the system can only be even.Furthermore depending on which symmetries (I or T ) are present in thesystem the minumum number of Weyl nodes can be 2 or 4: table 1.1 showsthe action of the two symmetry operators on the Berry curvature and we cansee that if only I is maintained a node at ~k0 is copied in −~k0 with oppositechirality and the presence of just 2 Weyl points in the system is possible.For the same reasoning when only T is maintained we need at least 4 Weylpoints in order to have a zero total chirality in the whole Brillouin zone.

Table 1.1: Action of the operators T and I on the Berry curvature.

Operator Operation ResultT T ~Ω(~k) −~Ω(−~k)

I I ~Ω(~k) ~Ω(−~k)

Fermi arcs

As in 3D topological insulators the bulk-bundary correspondence holds alsoin Weyl semimetals and surface states must be present as a physical con-sequence of topology. The dierence with the insulator phase is that herethe surface states are present in the form of arcs, namely open lines whichconnects the projections of two Weyl nodes on a boundary surface. On theother hand the merger of two arcs present on opposite surfaces, linked by thebulk nodes, generate a closed loop resembling a closed Fermi line as in caseof topological insulators.In order to explain the presence of these arcs we can use the argument pre-sented in [25]. First of all consider a system in which 2 Weyl points, ofopposite chiralities and located at dierent momenta, are present in the bulkBrillouin zone, and right now focus on one of them, for instance the pointwith a positive chirality. Take a tubular surface (in the momentum space),with xed radius, enclosing the Weyl point as shown in gure 1.6a, and de-


Figure 1.6: Illustration of surface states arising from bulk Weyl points. (a) Bulkstates as a function of (kx,ky), with an arbitrary kz, which show one Weyl point;a cylinder enclosing the node, whose base denes a 1-D circular BZ (kλ), is alsodrawn. (b) The cylinder unrolled on a plane gives the spectrum of the 2-D systemH(kλ, kz), with kz xed: a chiral edge states appears at the boundary of the systemdue to the non-zero Chern number. (c) Meaning of the surface states back in 3-D. The chiral state appear as a surface connecting two Weyl points with oppositechiralities, and the intersection of the plane with the Fermi level gives the Fermiarc. From [25].

ne two periodic parameters kλ and kz characterizing the momentum on thisbidimensional system, which due to the periodic boundary conditions can beviewed as a torus, a closed surface. Since just one Weyl point is enclosedby the torus, the ux of the Berry curvature through this surface, calculatedusing equation (1.21), leads to a non-trivial Chern number C=+1, namelythis toroidal system has a non-trivial topology; in particular a bidimensionalsystem, such as our torus, in which a non-zero Chern number can be denedis called a Chern insulator.Since the Weyl nodes are the only band touchings present in the bulk Bril-louin zone, this particular 2D system is fully gapped everywhere; if we nowconsider a nite system, in which boundaries appears, the presence of a non-trivial topology on the torus must be accompanied with the presence of edgestates at the boundary, as in case of topoloigcal insulators. Furthermore a2D Chern insulator with a Chern number C = |1| has one single chiral edgestate [3], and in our system it is shown in gure 1.6b, namely the line whichcrosses the Fermi energy once.Now it's easy to see that such a crossing can be obtained for each chosenradius of the torus, and the junction of these points on the boundary surfaceof the 3D system creates a line; since another Weyl point with opposite chi-rality C = −1 is present in the system, for which holds the same argumentpresented before, it's clear that the same surface line must link the projec-


tions of the Weyl points on the surface, forming an open line called Fermiarc. In gure 1.6c is shown an example of a Fermi arc formed between twoWeyl nodes of opposite chiralities.Generally dierent systems with dierent number of Weyl nodes can showvarious and more complex surface state patterns, but a Weyl Fermi arc it'seasily recognisable since we know it will always resemble an open line, linkingthe projection on the surface of two nodes with opposite chiralities.

Anomalous Hall eect

The simplest manifestation of the Weyl physics is the so called anomalousHall eect [3], closed related to the well known Hall eect with the dierencethat here no external magnetic eld is needed. On the other T must be brokenalso in this case in order to achieve the Hall state, but this can be easily doneusing magnetic atoms. In order to explain this eect we can consider a simplesystem with 2 Weyl nodes of opposite charges at ~k0 and −~k0, separated inthe momentum space by d = 2k0 along the z direction. Being respectevelya source and a drain of the Berry curvature we expect a non zero ux onthe planes kz = a, with −k0 < a < k0, and therefore a non trivial Chernnumber: each plane denes a Chern insulator whose conducting helical edgestate at the Fermi energy has a quantized Hall conductance σ = C e2

h, where

C is the Chern number. Now if the Weyl points of our 3D system are ator very close to the Fermi energy it's clear that each plane between the twonodes contribute to a total Hall conductance, given by the formula

σxy =e2

2πhd, (1.22)

where d = 2k0 is the distance (in the momentum space) between the twonodes, and therefore to an anomalous Hall eect.Note that the term helical referred to the edge state stands for the char-acteristic of the edge current to ow in a particular direction, clockwise oranti-clockwise, depending on the sign of the Chern number.

1.4.2 Dirac semimetals

As discussed in the previous section Weyl nodes, namely a 2-fold degenerateband touching, occur in 3D materials when either time reversal or inversionsymmetries are broken, otherwise the bands would become doubly degenerate


everywhere and 2-fold degenerate band touchings were no longer possible. Inthe latter case a 4-fold degenerate band touching could occurs, for instanceif 2 Weyl points of opposite charges are located at the same crystal mo-mentum, but we know that, in general, this procedure annihilate the points,resulting in a nite gap; nevertheless some crystal symmetries can stabilizethe simultaneous presence of two Weyl nodes with opposite chiralities atthe same crystal momentum, producing a composite 4-fold degenerate pointcalled Dirac node (or Dirac point). Therefore, in analogy with the particlephysics, a Dirac node can be viewed as the stable merging of two Weyl nodes,with opposite chiralities, at the same crystal momentum.It has to be noticed that, in general, this point is not topologically protectedsince its total Chern number is zero and projecting the states into a degen-erate subspace can mix them resulting in an opening of a gap; anyway insome special cases this mixing can be forbidden by space group symmetries,giving origin to a stable Dirac node.As in case of Weyl points it shows a linear dispersion in its surrounding, andthe simplest eective low energy Hamiltonian expanded around the Diracnode can be described by the following 4x4 equation (note the similaritieswith equation 1.15)

H(~k) =(vf~σ · ~k 0

0 −vf~σ · ~k

). (1.23)

In this case we can see that if o-diagonal terms are present in equation(1.23), for example when external perturbations are taken in account, thedispersion becomes quadratic and a band gap is opened at the Dirac point;therefore the degeneracy of the point must be protected, and some specialcrystal symmetries are responsable for that protection.Although a Dirac point has a trivial Chern number it has been shown thatDirac semimetals present some non-trivial topological properties, such as sur-face states, and topological invariants can be dened. However it has to benoticed that topology is fragile, just protected by the same crystal symmetrywhich makes possible the merger of two Weyl nodes in the bulk.The Dirac semimetal phase can occur in normal-topological insulator transi-tions tuned by composition or strain, but it can also appear accidentally inmaterials, along some high symmetry lines, when the band inversion mecha-nism is present. We are going to deepen the latter mechanism since it leadsto the Dirac semimetal phase in the cadmium arsenide system Cd3As2, thematerial studied in this work.


Dirac semimetals from band inversion mechanism

As already mentioned before, the band inversion mechanism due to SOCcan lead to a topological phase transition between insulating phases, but insome special cases this gap opening can be avoided by the mean of crystalsymmetries. Band crossings are generically prevented at low-symmetry mo-menta, but along some special lines called high symmetry lines, on which ismore likely to have symmetry elements that copy a point onto itself, one cansymmetry-enforce a band touching if two crossing bands belong to dierentirreducible representations of the group.In general this mechanism generates a pair of Dirac points along the linewhich, as in case of Weyl points, can be moved around the Brillouin zoneand possibly annihilated, producing a fully gapped system.The two most notable examples of Dirac semimetals which underlie thismechanism are the sodium bismuth Na3Bi [26] and Cd3As2 [19] systems.Although their crystals structures belong to dierent space groups (hexago-nal and tetragonal, respectively) both present 2 Dirac points in the primitivecell, located along a crystallographic axis. In both cases the extra crystalsymmetry which provides the stabilty of the nodes is the rotational symme-try Cn, more precisely the C3 in Na3Bi and the C4 in the Cd3As2 system.

Topological invariants and Fermi arcs

Despite having a gapless bulk band structure Dirac semimetals support quan-tized topological invariants similar to the insulators' ones. From the bandinversion mechanism we saw that a pair of Dirac nodes is generally created at~k0 and −~k0, along an high symmetry line, due to the simultaneous presenceof T , I and the protecting crystal symmetries, for instance the Cn symmetry.Now suppose that the nodes lie along the line passing through the Γ point(~k = 0) along the z direction, as shown in gure 1.7: the kz = 0 plane is afully gapped time reversal invariant plane (TRIP) on which a Z2 invariantν2D can be dened as in the case of topological insulators. Furthermore if n(in Cn) is an even number grater than 2 this particular plane is also a mirrorplane and another invariant, the so called mirror Chern number [27], can bedened.As in Weyl semimetals one of the consequences of topology is the presenceof surface states called Fermi arcs, but here they are more subtle since thebulk Dirac point carries a zero Chern number. On the other hand it can beviewed as the stable merging of two Weyl points with opposite chirality at


Figure 1.7: Representation of the Fermi Arcs in a Dirac system, with a pair ofDirac points along the kz axis. The plane kz = 0 is a TRIP on which a topologicalinvariant Z2 can be dened, non-trivial in this case. From [5].

the same momentum ~k0 and in this sense the unique surface states of a Diracsemimetal can be explained as the simultaneous projection of the two arcscoming from the Weyl nodes on the same surface momentum, creating theso called double Fermi arcs shown in gure 1.7.Moreover a non-trivial Z2 topological invariant of the TRIP kz = 0 is relatedto the presence of 2 edge states, which can be viewed as the crossing betweenthis plane and the Fermi arcs present on the perpendicular boundary surface.Being related to the Weyl arcs, we know that each edge state is an helicalstate associated to a non-trivial Chern number but since C = 0 in the wholecell it's clear that the two states must have opposite helicity. In particulareach state is related to a dierent spin channel (up and down), leading to aso called quantum spin hall insulator(QSHI) [8, 19]: in contrast to anoma-lous quantum Hall insulators this particular bidimensional system supportsno electric current on the edge although two dierent spin currents, owingin opposite directions depending on the spin channels, are present. A QSHIallows one to exploit the spin currents more than electric ones making thismaterial a fundamental tool for spintronic devices.

1.4.3 From Dirac toWeyl and further: topological phase

transitions

An example of a topological phase transition between two insulating stateshas already been shown in previous sections. Of course many other kindsof transitions are possible, but here we will use the Dirac semimetal phase


as a starting point taking advantage of the fact that the Dirac nodes is justprotected by some crystal symmetries. In particular we are interested tosee if it's possible to achieve dierent topological phases using principallychemical dopings, namely substituting some of the atoms of the original cellwith dierent non-magnetic or magnetic impurities, in order to break variouscrystal symmetries.

Particularly interesting will be the transition between a Dirac and Weylsemimetal: since a Dirac node can be regarded as the stable merger of twoWeyl nodes at the same crystal momentum, protected by some crystal sym-metries, we expect that breaking these protecting symmetries leads to a stableseparation of these Weyl points in the Brillouin zone.It's clear that depending on which symmetries we decide to break, in partic-ular T, I or Cn since they play the main roles, we can distinguish dierentkind of transitions:

1. If we only break the Cn symmetry, the one who protects the Diracnode, we expect the Dirac point to be broken. The presence of Weylnodes is forbidden by symmetries since T and I make the bands doubledegenerates. In this case the nal state could be a topological insulator(just like a trivial one) or a metal, depending on the position of theFermi energy.

2. Breaking both the Cn and the I the separation of the Dirac point intwo Weyl nodes coud be possible. Furthermore looking at the eectof the T symmetry on the Berry curvature, presented in table 1.1, weexpect at least 2 Weyl nodes (or a multiple) with opposite chiralities.

3. The simultaneous breaking of Cn and T could also lead to a Weylsemimetal phase, but in this case the minimum number of nodes mustbe 4 (or a multiple).

4. Breaking either T or I (or both), maintaining the Cn, could lead to aninteresing situation in which, in general, the presence of Weyl points isnot forbidden although they cannot directly arise from the Dirac nodesince it is still present in the system, protected by the crystal symmetry.

5. If all the 3 symmetries are broken a Weyl semimetal phase it's in prin-ciple possible and no restrictions on the number of Weyl nodes are


present.

Note that the breakings of the symmetries can be equivalently done usingexternal parameters, such as strains or magnetic elds. In the third chap-ter is presented a particular system on which, beside the chemical doping,an external strain is considered: in this case the external parameter doesn'tbreak any particular symmetry, but allows us to study an interesting phasetransition.


Chapter 2

Computational background

Most of the arguments present in this chapter, in particular the ones presentin the rst two sections and in the Wannier function's one, are adapted fromthe Martin's book Electronic structure [7].

2.1 The density functional theory

Our present understanding of the electronic structure of materials comesmostly from quantum mechanics and the starting point is the Hamiltonianoperator for the assembly of electrons and nuclei

Htot = −∑i

~252~ri

2me

−∑I

~252~RI

2MI

+

+e2

4πε0(∑j>i

1

|~ri − ~rj|+∑J>I

ZIZJ

|~RI − ~RJ |−∑i,J

ZJ

|~ri − ~RJ |),

(2.1)

where electrons are denoted by lower case subscripts and nuclei, with chargeZ and mass M, are denoted by upper case subscripts (all the eects com-ing from relativity, magnetic elds and quanutm electrodynamics are notincluded here). The rst two terms of equation (2.1) represent the kineticenergies of electrons and nuclei whereas the last three terms represent themutual interactions electrons-electrons, nuclei-nuclei and electrons-nuclei.The term which includes the fraction 1

MIcan be neglected (nuclei are way

more heavy than electrons) introducing the so called the Born-Oppenheimerapproximation. This is particularly useful since it allows us to separate theelectronic degrees of freedom from the nuclear ones: the electrons act as if

23

24 CHAPTER 2. COMPUTATIONAL BACKGROUND

they are moving in an external potential Vext created by the xed nuclei. Fur-thermore the mutual interaction of the nuclei contribute just as an additiveterm. Therefore the Born-Oppenheimer approximation leads to the elec-tronic Hamiltonian which can be used to solve the well known Schrödingerequation

i~dΨ(~ri, t)

dt= HΨ(~ri, t) (2.2)

where Ψ is the many-body wavefunction for the electrons.There are many sophisticated methods to solve the eigenvalue problem, onefor all the self-consistent Hartree-Fock method, but the huge computationaleort required makes them very inecient for big and complex systems suchas molecules or solids. Here we got help from the density functional the-ory (DFT) which provides a more versatile and feasible way to solve themany-body problem: the fundamental tenet of DFT is that any property ofa system of many interacting particles can be viewed as a functional of theground state density n0(~r), a scalar function of position, instead of a functionof position itself.Hohenberg and Kohn proved the existence of such functionals but they pro-vide no guiding for constructing them. On the other hand the ansatz madeby Kohn and Sham provides a way to construct approximated ground statefunctionals for real system of many electrons: they replaced the interactingmany-particle problem with an auxiliary independent-particle one in whichall the many-body eects are included in a functional called the exchange-correlation functional.

2.1.1 The Hohenberg-Kohn theorems and Kohn-Sham

ansatz

The Hohenberg and Kohn approach is valid for any system of interactingparticles in an external potential Vext, such as the one described by the many-body Hamiltonian (2.1) present in the previous chapter, for which the twofollowing theorems hold:

• For any system of interacting particles in an external potential Vext(~r)the latter is uniquely determined, except for a constant, by the groundstate particle density n0(~r) (and vice versa, namely n0 ↔ Vext).

2.1. THE DENSITY FUNCTIONAL THEORY 25

• A universal functional for the energy E[n] in terms of the density n(~r)can be dened, valid for any external potential Vext(~r). For any partic-ular Vext(~r) the exact ground state energy of the system is the globalminumum value of this functional, and the density n(~r) that minimizesthe functional is the exact ground-state density n0(~r).

The consequence of these two theorems is that, in principle, the wavefunc-tion of any state can be determined by solving the eigenvalue problem andamong all the solutions consistent with a given density the unique ground-state wavefunction is the one with the lower energy. Therefore they provethe existence of the universal functional and the dependency of the groundstate on the ground state electron density, but it's clear that no prescriptionhas been given to solve the original problem since we still need to solve themany-body problem in presence of an external potential Vext(~r).

Kohn and Sham (KS) provided a more feasible way to solve the problem:the idea is to replace the dicult many-body interacting system with anindependent-particle auxiliary system in which all the complicated many-body terms are incorporeted into an exchange-correlation functional of thedensity. Solving the KS equations one nds the ground-state density and theenergy of the original interacting problem with an accuracy limited to theapproximations of the exchange-correlation functional.The single particle KS Hamiltonian is given by

HKS = −~52

2me

+ VKS(~r), (2.3)

where VKS is the Kohn-Sham potential and the kinetic term is the non-interacting particle kinetic energy. The density can be calculated startingfrom the wavefunctions as follow

n(~r) =N∑i=1

|φi(~r)|2, (2.4)

and therefore we can write the ground state energy functional in terms ofthe density

EKS = Ts[n] +

∫Vext(~r)n(~r)d~r + Eh[n] + EII + Exc[n], (2.5)

where Vext is the external potential generated by the nuclei plus other external


elds (if present), Ts is the sum of the independent particle kinetic energies,Eh[n] is the Hartree energy functional dening the classical Coulomb inter-action

Eh =1

2

∫n(~r)n(~r′)

|~r − ~r′|d~rd~r′, (2.6)

and Exc[n] is the exchange-correlation energy, which includes either the non-classical part of the electron-electron interactions and the kinetic energiesdierence between the interacting and the non-interacting systems

Exc[n] =< T > −Ts[n]+ < Vint > −Eh[n]. (2.7)

The variation of the energy functional with respect to the density n(~r) givesthe the expression of the Kohn-Sham potential

VKS(~r) = Vext(~r) +δEhδn(~r)

+δExc

δn(~r). (2.8)

Once a suitable form of the exchange-correlation Exc is chosen (see the nextsection for details) the corresponding potential can be calculated analiticallyand the KS equations can be solved self-consistentely following this recipe:

1. Choose an initial (trial) wavefunction and the corresponding density n0

using (2.4).

2. Calculate VKS using equation (2.8).

3. Solve the KS equation (2.3).

4. Calculate the new electronic density n′.

5. Check the convergence (energy, forces, ...) and, if it's not convergent,restart the procedure using the new density n′.

Note that in all the previous formula the spin variable is not taken into ac-count, but its addition is trivial since it's sucient to separate the calculation

2.1. THE DENSITY FUNCTIONAL THEORY 27

in two parallel pieces in which the total density n is separated in n↑ and n↓(n = n↑+n↓), each one describing a dierent spin channel; this will be usefulin magnetic systems, in which one has to take into account the spin polar-ization.Furthermore extra interactions such as the spin-orbit coupling can be in-cluded in the system simply adding the corresponding term to the totalHamiltonian (for instance adding the term HSOC present in equation (1.12)). Then the eigenvalue problem is usually solved using a second variationstep [28] in which the potential and the density coming from a rst con-vergent calculation without SOC are used as a starting point for a secondself-consistent calculation in which the spin-orbit coupling term is added tothe Hamiltonian and a scalar-relativistic eigenfunctions basis is chosen (aDirac Hamiltonian is considered).

Exchange-correlation functional

Looking at the Khon-Sham equations it's easy to realize that if the termExc were known the exact ground state energy and density could be found.In practice it's necessary to make approximations for the functional, amongwhich we cite the two most famous and used: the Local (spin) density approx-imation (L(S)DA) [29] and the generalized gradient approximation (GGA)[29], widely used in this work. The L(S)DA approximation treats the densitylocally in the space and it assumes that the exchange-correlation functionalat each point can be considered the same as that of an uniform electron gas

ELDAxc =

∫ρ(~r)εxc(ρ)d~r, (2.9)

where εxc(ρ) is the exchange-correlation energy per particle.The problem is that L(S)DA fails in cases of rapid changes of the density,such as in molecules, but the GGA approximation helps to overcome this issuesince, as its name suggests, it takes into account the gradient of the electrondensity in the exchange-correlation functional, related to the variation of thedensity in space

Exc = Exc[ρ(~r),5ρ(~r)]. (2.10)


2.1.2 Determination of the electronic structure and ac-

curacy: pseudopotential vs all-electron scheme

From the previous sections it's clear that computationally treating the wave-functions as accurately as possible is important as having an accurate exchange-correlation functional; on the other hand this is not always possible since thehighly oscillating behavior of the wavefunctions close to the nuclei make thisjob very hard.

Since we are interested in calcuating properties that usually depend mostlyon the valence electrons one can usually approximate the highly oscillatingbehavior of the wavefunctions in the core region maintaining the behaviorin the valence region as accurate as possible. In particular this prevalentelydone using two dierent methods, namely the all-electrons schemes and thepseudopotentials methods. In the rst case the main idea is to choose a par-ticular basis set on which expand the wavefunction, the so called linear aug-mented planewave (LAPW) [28], maintaining the full form of the potential:in spheres centered on the nuclei, with a chosen radius called the mun-tinradius, the core electron's wavefunctions are expanded to reproduce atom-like orbitals whereas in the interstitial region the expantion is given by themeans of plane-waves. The constrain for this particular basis set is that thewavefunctions must match at the boundaries of the two regions. One of themost famous package based on this approach is Wien2k [28], widely used inthis work.The second approach simplify the complicated behavior of the core region in adierent way, using the so called pseudopotentials: here the idea is to replacethe full-potential with an eective potential in order to obtain a smoothercore electron's wavefunctions which require fewer Fourier modes and makethe plane-waves basis set practical to use. The new wavefunctions and thepseudopotential should coincide with the original ones beyond a certain cut-o radius rs. One of the packages based on this method, the one used in thiswork, is Quantum Espresso [30].In both cases the accuracy of a calculation is strictly related to a chosencuto values, related to the number of basis functions involved: in Wien2kthis is given by the R ∗ Kmax parameter, where R is the smallest atomicsphere radius in the unit cell (each type of atom has a dierent radius) andKmax is the largest K vector used in the linear combination of (L)APWs thatgives the total wavefunction. In Quantum Espresso the cuto values, relatedto the number of planewaves involved, are given by Ewfc (the cuto for thewavefunctions) and Erho (the cuto for the charge density and potential), thelatter just used when ultrasoft pseudopotentials are considered. The higher

2.2. WANNIER FUNCTIONS AND TIGHT-BINDING MODELS 29

the cutos the more accurate are the results but of course they will requiremore computational power and time.Another important aspect that has to be taken into account for the accu-racy of the results, which is valid for all the packages, is the computationaltreatment of the crystal momentum: although in general this can be viewedas a continuous variable numerically the momentum is discretized (~k → ~ki)in equidistant steps along all the directions, creating a uniform mesh. Apractical choice for the k-mesh, for instance used by Quantum espresso, isthe so called Monkhorst-Pack [31] mesh that allows one to choose a groupof k-points homogeneously distributed in the entire Brillouin zone; a similarhomogeneus k-mesh, used by Wien2k, is the tetrahedron mesh [28].As in the case of cuto energies the higher the number of k-points the higherthe accuracy, but note that also the computational power and time demandincrease with the number of k-points.

2.2 Wannier functions and tight-binding mod-

els

Wannier functions are widely used as they provide an easy and practical toolfor calculating the electronic structures [7]. Their intrinsical non-uniquenessrelated to the dependence upon the choice of a gauge may appears as anobstacle for practical uses but the construction of the maximally localizedWannier function (MLFW) [32, 33] brings out their usefulness, for instanceproviding an elegant connection to the Berry's phase formulation to polar-ization [7].The Wannier functions can be viewed as the Fourier transforms of the Bloch'sstates: for the nth band the function associated with the cell labeled by thelattice point ~R is

wn(~r − ~R) =Vcell

(2π)3

∫d~kei

~k·(~r−~R)un,~k(~r) (2.11)

where Vcell is the real-space primitive cell volume, un,~k(~r) is periodic part ofthe bloch function and the integral is performed in the whole BZ.In general these functions are dened as a linear combination of Bloch func-tions of dierent bands and therefore the periodic term in equation (2.11)can written as


un,~k =∑m

U~kmnum,~k, (2.12)

where U~kmn is a ~k-dependent unitary transormation.The Wannier functions form an orthogonal set but they are not unique, incontrast to the Bloch states. We already know that the overall phase of aBloch function is arbitrary and a Gauge transformation of this states leaveunchanged all the physically meaningful properties (see equation 1.18). Nowif we apply the same transformation to a Wannier function it's clear fromequations (2.11) and (2.12) that the changing is not trivial since simultane-ously change the relative phases and amplitudes of the Bloch functions atdierent ~k and bands n.In order to overcome this obstacle one can x a Gauge obtaining a localizedWannier function in real space, in which the grade of localization can bemeasured with the spread functional

Ω =

Nbands∑i=1

[< r2 >i − < ~r >2i ], (2.13)

where < ... >i means the expectation values over the ith wannier functionin the unit cell, with ~R = 0. Finally, minimizing the spread functional, onenally obtains the MLWF.The package which allows us to computationally use this procedure is Wan-nier90 [34]: the starting point are the outputs of a self-consistent ground-state calculation, for instance coming from Wien2k or Quantum Espresso,from which the matrix M~k,~b

mn =< um,~k|un, ~k+b > of the overlaps of the periodicparts of the Bloch states is calculated (m,n are the indices of the matrixwhereas ~b is the vector linking to the k-neighbors). This matrix is computa-tionally useful since the position operator present in equation (2.13) can bewritten in terms of the matrix Mmn as

~r = − 1

N

∑~k,~b

wb ~b Im[lnM (~k,~b)mn ], (2.14)

where wb is an appropriate weight.In order to work properly Wannier90 also needs a set of initial trial functionsused as a starting Wannier functions, which in general are taken projectingthe Bloch functions in the real space, onto atomic or hybridized orbitals.Note that during the procedure the orbitals can be mixed.In general the Wannierization can be done choosing a group of bands which


are well separated by a nite gap from all the other bands, since in this casean orthogonal set of states is always possible. On the other hand when thischoice is not possible, namely when the chosen bands are linked to otherbands (for instance if one chooses to considering all the valence bands of ametal), a further calculation is necessary, the so called disentanglement pro-cedure [32]: this allows us to separate the chosen group of bands from theother bands and create an orthogonal set of states to be used in the Wanniercalculation.

Finally MLWF can be used to construct tight-binding (TB) models [7] thatprovide an easy way to calculate the topological properties of a system. Inparticular the TB model matrix, with the so called hopping terms tmn, canbe constructed via the formula

tmn(0− ~R) =< w0,m|H|wR,n > (2.15)

where the w are the dierent MLWF. Fourier transforming we nally obtainthe tight binding matrix

Hmn(~k) =1

N

∑~R

ei~R·~ktmn(0− ~R). (2.16)

Then Wannier90 provides the code to construct the tight-binding matrixwhose le can be used by the WannierTools [35] package to study the topo-logical properties of the system. First of all it allows us to nd all the nodes(gapless points) present in the system, using a script called nd nodes, whichbasically nd the local minima of the energy gap function in the whole 3DBZ.Furthermore this package allows one to calculate the Z2 in the TRIPs andthe Chern topological invariants by looking at the evolution of the so calledwannier charge centers (WCC) [36, 37]; this method is particularly usefulsince it also works when inversion symmetry or time reversal symmetry arelacking (see next section for details).Another important features of Wanniertools are the capability of calculatingthe Berry curvature on lines or planes [35], very useful if one wants to seedivergences related to the presence of Weyl points, and the possiblity of sim-ulating a surface calculation from an original bulk one by the mean of theiterative Green's function methods [38], which allow to calculate the surfacestate spectra and consequentely to study the surface states of a 3D topolog-ical material, for instance the Fermi arcs in case of Weyl or Dirac semimetals.


Wannier charge centers and Z2 topological invariant

As mentioned in the previous section the WannierTools package is very usefulfor our purpose since it has the capability of calculating the Z2 invariantson the TRIPs, looking at the evolution of particular objects calels Wanniercharge centers in the plane. In this section we are going to briey introducethe WCCs (for more details look at [36, 37]), but most important we willfurnish a method which allows one to deduce the topology of a non-magneticsystem looking at the evolution of these WCCs on the TRIPs.We know that at time reversal invariant momenta the Bloch states are 2-folddegenerate due to the presence of T , but if we link two dierent TRIMs wewill see that at the intermediate points the degeneracy is generally brokenand the energy levels are splitted. Starting from the Bloch states one canconstruct the WCCs from hybrid Wannier functions (HWFs), in which weFourier transform the state along one particualar direction maintaining theother two unchanged

|n; lx, ky, kz >=ax2π

∫ πax

− πax

eikxlxax|ψn,~k > dkx (2.17)

where lx is an integer layer index, ax is the lattice constant along x. Thisparticular choice of HWF is localised along x, therefore its position can bedened as xn(ky, kz) =< n; 0, ky, kz|rx|n; 0, ky, kz >, translated in the homeunit cell (R = 0). Note that WannierTools automatically furnishes all thenecessary WCCs coming from all the occupied states.Tracking the evolution of the WCCs between two TRIMs located on the bidi-mensional TRIP (an example is shown in gure 2.1) can provide informationabout the topology of the plane: it has been shown [37] that this particularplot and the corresponding bands plot of the bidimensional system are closelyrelated, especially regarding the topological and the symmetry properties ofthe system, and therefore one can relate the pairs of branches we see in thegure 2.1 (the ones which touch at ky = 0) with the edge states we tipicallysee in the band structure plot, similar to the case presented in gure 1.2.As for the edge states, in which we counted the number of crossings withthe Fermi energy in order to understand the topology of the system, herewe can consider a generic horizontal line which goes from ky = 0 to ky = 2π

b

and count the number of crossings between the line and the branches, whichequivalently tells us the topology: if the number of crossings is odd, than theplane has a non-trivial topology (Z2 = 1), otherwise the topology is trivial.


Figure 2.1: Evolution of the occupied WCCs between two TIRIMs ky = 0 2πay

and

ky = 0.5 2πay

located on the kz = 0 TRIP, in the Dirac system Cd3As2. A generichorizontal line linking the two TRIMS, here the blue line, cuts the WCCs branchesonce, reecting the fact that it has a non-trivial topology (Z2 = 1).


Chapter 3

Methodology and results for the

undoped Cd3As2 system

In the introductory part we briey presented our work, intended to study thetopological phase transitions in topological semimetals induced by chemicaldoping using rst-principles methods, focusing in particular on the electronicand magnetic properties of the systems and on the characterization of theirtopology. In order to do this, we are going to use the computational meth-ods and tools presented in the previous chapter, which provide us with thegeneral path we can follow during our study. The procedure can be resumedin 5 steps:

1. Relax the structure, either nding the cell parameters or the atompositions (or both) which minimize the stresses acting on the cell andthe forces acting on the atoms. This is done using Quantum espresso,within the DFT approach.

2. Do a self-consistent calculation following the DFT scheme, adding thespin-orbit coupling in a second variation approach, as explained in theprevious chapter. This is done using principally Wien2k.

3. Calculate and plot the density of states (DOS) and the band structure:other than being useful to understand the physical properties of thesystem, they give us a hint about the construction of the tight-bindingmodel. Wien2k has been used for these calculations.

4. According to the previous point choose the initial projections of theBloch states and construct the maximally localized Wannier functions(MLWF) using theWannier90 package. The tight-binding model Hamil-tonian is given as an output.

35

36 CHAPTER 3. THE PURE CD3AS2 SYSTEM

5. Use the tight-binding Hamiltonian along with WannierTools in orderto calculate the various topological properties of the system.

In literature similar theoretical works are present, but usually the construc-tion of the low-energy eective model Hamiltonians starts from a non-spin-orbit DFT calculation, using pre-built parametric models (to be tted) withthe addition of the SOC term, considered as an extra parmeter. For instancein the Wang work [19], which studied the Cd3As2 system, the low-energyeective model in the presence of SOC is based on a modied eight-bandsKane model in which an external parameter, related to the strength of thespin-orbit interaction, is tuned to recover the DFT results.In our work we used a dierent approach: the spin-orbit coupling is includedat the self-consistent level (step 2 ), and the resulting band structure andDOS are used to construct the tight-binding model in the Wannier functionsbasis (step 4 ). Since we didn't use pre-built models, generally we expect toobtain more accurate tight-binding models and therefore more realistic bansstructures; for instance in the case of pure Cd3As2 system we can compareour results with the theoretical works present in literature, in order to see ifany substantial dierences arise.

For our study we chose to use a Dirac semimetal phase as a starting pointsince its peculiar toplogy, protected by the crytal symmetries, makes it theperfect candidate for our goal: taking advantage of its fragility we will try tochange the topological phase of the initial system breaking dierent crystalsymmetries, mostly by means of chemical dopings, analizing time to time theoutcomes of these operations. In particular we will see that Weyl semimetalphases arise, particularly interesting for their peculiar properties, as well astopological and trivial metal phases.

The material we are considering is cadmium-arsenide (Cd3As2), an exam-ple of a topological Dirac semimetal arising from the band inversion mech-anism, which provides the pair of Dirac points present in the system [19].In the theoretical chapter we saw that three specic symmetries are fun-damental in a Dirac semimetal, namely the time reversal symmetry T, theinversion symmetry I and the rotational symmetry C4, and in our study weare going to break them in dierent ways, for instance by substituting someCadium atoms with dierent elements, in dierent percentages: the role ofnon-magnetic dopants will be played by the Zinc (Zn), whereas the Man-ganese (Mn) will be used as a magnetic impurity. These choices were madetaking in account the electronic congurations (they all have d and s valence

3.1. THE CADMIUM-ARSENIDE SYSTEM CD3AS2 37

electrons) just like the chemical valence, since they are supposed to act assubstituted impurities of Cadmium.The dierent types of chemical dopings and the dierent impurity concen-trations that we are considering are the following:

1. 4 Zn atoms in the 40-atoms unit cell, with 24 Cd atoms (∼ 17% ofdopants), placed in order to maintain I while breaking the C4.

2. 1 Zn atom (40-atoms cell, ∼ 4% of dopants), in order to break bothinversion I and C4.

3. 12 Zn atoms (40-atoms cell, 50% of dopants), which simulates δ dopings(dopants placed on planes perpendicular to some axis), in order to breakI.

4. 1 Mn atom (∼4% in the 40 atoms cell and ∼2% in the 80-atoms su-percell with 48 Cd atoms), in order to break all the three symmetries,including the T .

Before studying the doped systems a calculation on the pure Cd3As2 systemis done; this can be considered as a test for our method since, as alreadymentioned, we can compare our results with the theoretical works presentin literature, but also it allows us to enlight some important aspects of thetopological Dirac semimetal phase, for instance the importance of the spin-orbit coupling in such a systems.

3.1 The cadmium-arsenide system Cd3As2

The cadmium arsenide system attracted great interest after the predictionof the topological Dirac phase in the sodium bismuth Na3Bi system [26].Unfortunately Na3Bi is not stable at ambient conditions, which requiredto nd materials with similar properties but more suitable for experimentalstudies and practical uses. The two systems share the property of having apair of Dirac points, generated by the band inversion mechanism, althoughthe nodes are protected by dierent symmetries, the C3 in the Na3Bi systemand the C4 in the Cd3As2 system.In addition to be the ideal alternative of Na3Bi for its chemical stability, thepeculiar characteristic of Cd3As2 system of having an high carrier mobility(up to 1, 5m2

V sat room temperature) and the suggestion of its quantum well


structure to naturally support quantum spin hall insulators [19] make it avery promising candidate for future experimental studies and practical ap-plications.

The Cd3As2 system belongs to the II3V2 types narrow gap semiconductors,among which it stands out for having an inverted band structure, a funda-mental property for a non-trivial topology.The crystal structure at the atmospheric pressure can be related to a tetrag-onally distorted antiuorite structure with oredered 1/4 Cd site vacanciesleading to a primitive tetragonal structure D15

4h (P42/nmc), with 40 atoms inthe unit cell [39], or to a body-centered tetragonal structure C12

4v (I41/acd)with 80 atoms [40], depending on which temperature range we are consid-ering (room temperature for the 40 atoms cell, ∼ 100 K for the 80 atoms cell).

Altohugh the two structures belong to dierent crystal groups, they sharethe properties of having all the three desired symmetries T , I and C4, whichis fundamental for the topological semimetal phase. In both cases the pairof Dirac points are located along the tetragonal axis. Since we expect thetopology to be the same in the two systems, we decided to focus on the 40atoms because of its feasibility (its simulation requires a reasonable eortin term of computer time and memory). Furthermore the large number ofatoms present in the unit cell allows us to easily modify the percentage ofdopants, in particular reaching quite diluite regimes of dopings (until 4% inthe case of one single impurity).

3.2 Results and discussion of the Cd3As2 sys-

tem

Relaxation of the structure

Following the scheme presented at the beginning of this chapter the rst thingto do is relaxing the 40-atoms structure, initially choosing the experimentalparameters taken from the crystallographic table [39]: the cell parametersare a′ = b′ = 16.887347 Bohr and c′ = 23.962859 Bohr ( c

′

a′= 1.418983),

whereas the positions of the 6 inequivalent atoms, in terms of crystal latticeparameters, are presented in table 3.1, along with their Wycko position1

1A point belonging to a set of points for which site symmetry groups are conjugatesubgroups of the space group.

3.2. RESULTS AND DISCUSSION OF THE CD3AS2 SYSTEM 39

Table 3.1: Experimental values of the positions of the 6 inequivalent atoms in termsof the experimental lattice parameters a′, b′ and c′. The Wycko positions are alsoshown.

Atom W.p. X (a) Y (b) Z (c)Cd1 8g 0.250000 0.537300 0.850300Cd2 8g 0.250000 0.028600 0.129100Cd3 8g 0.250000 0.005800 0.390800As1 4c 0.750000 0.250000 0.000600As2 4d 0.250000 0.250000 0.009400As3 8f 0.496600 0.503400 0.250000

The 6 inequivalent atoms, 3 Cd and 3 As atoms, provide the positions of allthe other atoms by means of the crystal symmetries present in the P42/nmcgroup (16 in total). Note that after the relaxation the nal cell parametersand atom positions will maintain all the same crystal symmetries.The relaxation has been made with Quantum Espresso, using a non-spin-

Figure 3.1: Crystal structure of the relaxed unit cell of Cd3As2.

polarized calculation (we are studying a non-magnetic system). The cuto-energies are chosen to be Ewfc = 80 Ry for the wavefunction and Erho = 600Ry for the charge density and potential; a uniform Monkhorst−Pack meshof 7x7x5 k-points has been used, reduced to a total of 30 k-points by thesymmetries, namely in the irreducible Brillouin zone. The thresholds for the


calculations are 10−6 Ry for the energy, 1 mRyau

for the forces, acting on eachatom, and 0.5 Kbar for the stresses, acting on the cell along each direction.Note that in this calculation no smearing2 has been taken into account.The particular choices of the cuto energies and the k-mesh are based on aconvergence test, previously made on the Cd pseudopotential, the toughestamong all the considered pseudopotentials, and we will use these parametersfor all the relaxation calculations present in our work (Appendix A).

During the relaxation of this particular system two calculations has beenperformed, either including or not the spin-orbit coupling, although we no-ticed that the eect of SOC on the relaxed parameters is not relevant. Thespin-orbit coupling could only be taken in account during the relaxation cal-culation in Quantum Espresso. Since in Wien2k the calculation of the forcesin the presence of SOC is not implemented yet, the relaxation calculationsof all the structures in this work are performed using Quantum Espresso.

At the end of the calculation the new cell paramters are a = b = 17.273283Bohr and c = 24.272846 Bohr ( c

a= 1.405225), whereas the inequivalent

atomic positions, in terms of the new crystal lattice parameters, are pre-sented in table 3.2. The resulting cell is shown in gure 3.1.Such a variation of the atomic positions and cell parameters will not change

Table 3.2: Relaxed values of the positions of the 6 inequivalent atoms in terms ofthe relaxed lattice parameters a, b and c. The Wycko positions are also shown.

Atom W.P. X Y ZCd1 8g 0.250000 0.531213 0.855053Cd2 8g 0.250000 0.034167 0.132560Cd3 8g 0.250000 0.994409 0.396994As1 4c 0.750000 0.250000 0.005427As2 4d 0.250000 0.250000 0.992112As3 8f 0.494033 0.505967 0.250000

the topological phase of the system, and we will see later that the relaxedsstructure is still a topological Dirac semimetal.

2Usually this parameter helps a lot when a metallic (or semi-metallic) system is con-sidered since its sharply drops of the occupation at the Fermi energy could make the self-consistent convergency very dicult to achieve: in practice the smearing has no physicalvalue, but it allows one to simulate a fake temperature that spreads out the occupationsat the Fermi level creating a smoothly decay, such as in insulators, easier to handle.


Band structure and DOS without SOC

In order to enlight the eect of the spin-orbit interaction in the topologi-cal Dirac semimetal we rst present a self-consistent calculation, along withthe resulting outcomes, in which SOC is not taken in account: in this way acomparison with the spin-orbit case is possible, allowing us to study its eecton the system. Since the spin-orbit case is more relevant for the topologicalproperties, we are going to present the results for a non-spin-orbit calculationonly for the Cd3As2 system; furthermore we expect the main eects of SOCon the other system to be very similar.

For all these calculations we used the Wien2k package along with a non-spin-polarized calculation, since we are considering a non-magnetic system.For the self-consistent cycle a cuto parameter R ∗Kmax = 9 is used, alongwith a uniform k-mesh of 7x7x5 k-points, reduced to a total of 30 k-points inthe irreducible zone. The mun tin radii are chosen to be RMT = 2.5 Bohrin the case of cadmium and RMT = 2.41 Bohr in the case of arsenic. Allthe scalar relativistic eect of the core electrons have been considered andwe also added a little Fermi smearing, choosing a value of TEMP = 0.002Ry, since it helps us to easily nd a convergent result. Finally a thresholdof 10−5 Ry is chosen for the convergence of the energy, along with a chargethreshold of 10−3 e. Note that, as in Quantum Espresso, the choice of thecuto parameter and the k-mesh is dictated by a convergence test we made:in the next sections, if not specied, we will use exactly the same value ofR ∗ Kmax and the same k-mesh for all the Wien2k calculations. (AppendixA)Once the self-consistent cycle has converged and the ground state of thesystem has been found we can plot the density of states, showed in gure3.2a, using a uniform k-mesh 7x7x5: this plot, which represents the totalDOS, tells us the total occupancies of the states at dierent energies, addingall the contributions coming from all the dierent atomic orbitals. In thiscase we notice that, close to the Fermi level, the occupancy drops almostto zero, reecting the character of a semimetal; furthermore the dropping isparabolic, at least on the valence side, which could be an hint of the lineardispersion of the bands.In order to get a better idea of the occupancies we also plotted the projectedDOS, showed in 3.2b, which at dierent energies shows portions of the to-tal occupancy coming from specic orbitals of specic atoms, which givesan hint on which atoms and orbitals are more involved close to the Fermienergy; in the plot we took into account all the inequivalent atoms, since thecontribution coming from the ohter atoms is the same, and we can see that


Figure 3.2: Density of states, in absence of SOC, for the relaxed Cd3As2 system.(a) shows the total DOS whereas (b) the projected one, in which we took into accountthe s, d orbitals of all the cadmium atoms and the s, p orbitals of all the arsenicatoms present in the unit cell.


the valence electrons have a prevalent p character, coming from the 4p ofthe arsenic atoms, slightly hybridized with the 5d orbitals of the cadmiumatoms, whereas in the conduction region we have an hybridized sp character,in which the p part comes from the arsenic orbitals whereas the s part comesfrom either the 4s of the arsenic and the 5s of the cadmium atoms.Along with the DOS we plotted the band structure, shown in 3.3a, which

(a)

Figure 3.3: Projected band structure of Cd3As2 along the Z−Γ−X−M −Γ path,when SOC is not taken in account. (a) shows a general view of the bands, alongwith their dierent characters coming from S (red) and P (blue) orbitals (lightpurple represent hybrid SP bands). (b) shows a zoom of the band structure alongZ − Γ, in which the band inversion between sp-conduction and p-valence bands isevident; furthermore it also shows a 6-fold degenerate crossing between the bands.The energy scale is in eV.

provides us with further informations. This calculation has been made alongthe high symmetry path Z − Γ − X −M − Γ, presented in 3.4, using 250k-points; the colours just represent the contribution of dierent orbitals onthe bands, which help us to visualize the character of each band.Although not evident it should be noted that all the bands are at least doubledegenerate, due presence of T and I (the Kramer's degeneracy holds every-where). Furthermore we can see in action the band inversion mechanism,which shifts the bands around Γ, as shown in gure 3.3b: in this region wecan see that the hybrid conduction sp band (the light purple one, with aprevalent s character hybridized with pz) switches with the valence band,


Figure 3.4: The brillouin zone of the Cd3As2 system, along with the path used forthe band structures plot.

the one with a pure p character (the blue one, consiting of pz and px,y or-bitals). Note that the presence of the inversion at this level tells us that theband inversion mechanism, in this particular case, is provided by the crystalstructure itself, not by the spin-orbit coupling.Related to the inversion of the bands we can see that some band crossingshappen close to the Fermi energy, among which we now focus the 6-fold de-generate point along the Z − Γ: the degeneracy of the touching point is dueto the fact that the crossing is between the 2-fold degenerate sp band and the4-fold degenerate p band (pz and px,y states are degenerate on this line); thisspecial crossing, along with the other one present on the tetragonal axis, atopposite momentum, are related to the 4-fold degenerate Dirac points whicharise in the presence of SOC, whereas all the other crossings, for instance theones along Γ−X, will be gapped.

The eect of SOC on the results

As already mention in the second chapter, the addition of the spin-orbitcoupling has been made with a second variation approach (see the compu-tational section for details), using the same Wien2k parameters presented inthe non-spin-orbit calculation.Moving to the results we rst present the density of states in the presenceof SOC, either total and projected, showed in gure 3.5; we can see that thegeneral behavior is quite similar to the previous case, in that the projectedDOS shows that the same orbitals are involved close to the Fermi level, witha prevalence As-p orbitals in the velence region and an hybrid sp character


in the conduction region, whereas the total DOS shows that the occupancygoes almost to zero parabolically at the Fermi energy, resulting in a typicalsemimetallic behavior.On the other hand the band structure close to the Fermi energy, shown in

Figure 3.5: Density of states in presence of SOC for the Cd3As2 system. (a) showsthe total DOS whereas (b) the projected one, in which we took in account the s, dorbital of all the cadmium atoms and the s, p orbitals of all the arsenic atoms.

gure 3.6, plotted along the same Z − Γ − X −M − Γ using 250 k-point,changes considerably in presence of SOC. First of all we notice that all thebands are still double degenerate (SOC does not break the Kramer's degen-eracy), and the bands around Γ are still inverted. Furthermore we see thatthe previous 4-fold degenerate p-bands along the Z − Γ line is now splittedin two separated 2-fold degenerate bands: this happens because the SOC


mixes the px,y/z states (l and m are no longer good quantum numbers), andthe resulting p 1

2p 3

2states are no longer degenerate. Related to this fact wee

see that the band touching point along Z − Γ, the one close to the Fermienergy located at ~kD1 = (0, 0, k0) = (0, 0, 0.165) (in terms of reciprocal lat-tice vector), is now 4-fold degenerate whereas all the other crossings are nowgapped.This special degenerate crossing involves two bands belonging to dierentrepresentations of the crystal group, having dierent characters (sp and p 3

2),

which avoid the presence of a gap: the dierent representations prohibitshybridizations between them, resulting in a protected band touching. Inparticular the action of the C4 symmetry on the tetragonal axis provides thedegeneracy, and therefore we can say that this symmetry protects the bandtouching point. We will see in the next section that the topological analysiswill characterize this special point as a Dirac point, as well as the other bandtouching located at ~kD2 = (0, 0,−k0), due to the presence of I and T .

With the exception of the Dirac points, the main eect of SOC on the

E-Ef

Figure 3.6: Projected band structure of Cd3As2 along the Z−Γ−X−M −Γ path,when SOC is taken into account. (a) shows a general view of the bands, along withtheir dierent characters (red for s, blue for p, purple for hybrid sp). (b) shows azoom along Z−Γ, in which the inverted bands cross, resulting in a 4-fold degenerateband touching close to Ef , namely the Dirac point, circled in red. The energy scaleis in eV .


bands is the opening of a gap at almost every crossing points related to theinverted bands. This can be seen, for instance, looking at the anti-crossingspresent along Γ−X or Γ−M , resulting in a formation of a nite gap due tothe mixing of the states. Away from the tetragonal axis there are no crystalsymmetries which protect these crossings: the C4 just protects the crossingsalong the tetragonal axis, namely the two Dirac points.The last thing we have to mention is that the two Dirac nodes are locatedat the same negative energy ED = −0.0078 eV (respect to Ef ), as shown ingure 3.6. On the other hand we can see that along the Γ−X high symmetryline a small portion of the highest valence band exceed the energy of eitherthe Dirac points and the Fermi level , namely the peak shown in gure 3.6:although this is not a big issue here, this will aect the future study of thesurface states since the projections of these bulk states on the surface willcover, in part, the Fermi arcs.

Topological properties

From the previous calculations we obtained a lot of informations about thesystem, in particular about the atoms and the characters mostly involvedclose to the Fermi energy, very useful for the construction of the tight-bindingmodel. In order to do this we are going to use the Wannier90 package, whichcalculates the tight-binding Hamiltonian matrix used by WannierTools tocalculate the topological properties of the system.We initially chose to project the Bloch states over all the s orbitals of the24 Cadmium atoms and over the s and p orbitals of the 16 Arsenic atoms,in order to obtain the starting Wannier functions. Furthermore, taking intoaccount the spin-orbit coupling, we have to consider the Wannier functions asspinors, namely two component states, and therefore a total of 176 Wannierfunctions have been chosen (88 initial projections, considered twice). Notethat the number of Wannier functions are quite large for constructing a low-energy eective model, but this choice was necessary rst of all because wehad to take into account all the main characters involved close to the Fermienergy, namely the ones considered, and also because during our tests wenoticed that involving more orbitals ususally leads to a more accurate result(Appendix B).Furthermore since the number of the bands must coincide with the numberof Wannier functions, in this calculation we took in account a large numberof conduction bands, although in general this is not necessary. For the samereason we mentioned before, this choice helps us to get accetable accuraciesand therefore accetable results.


Then we calculated the overlaps matrix M using a full 10x10x7 mesh, big-ger respect to the one used during the self-consistent DFT calculation: werealized that in this system such a choice allows us to reduce the numericalerror coming from the Wannier procedure (Appendix A). Then a disentangle-ment procedure was performed, using the frozen energy window [−0.4, 0.5]eV since we expect the most relevant eects to happen in this region. Notethat the same k-mesh and frozen window will be used in the next calcula-tions, if not specied dierently.At the end of the calculation, when the Maximally localized Wannier func-tions have been found, we plotted the bands structure directly from thetight-binding model, shown in gure 3.7, comparing the results with the out-come of the Wien2k calculations. It's clear that within the frozen window

Figure 3.7: Two comparisons of the band structures coming from the DFT cal-culations (red lines) and from the tight-binding model (green lines). On the left ageneral view is presented whereas on the right a zoom around the Γ point is shown.The sparse red dots in the plots represent the small dierencies between the nalresults obtained from the two computational methods.

the tight-binding bands are in good agreement with the DFT ones, althoughsome small numerical errors (not visible on this scale) appear during the Wan-nierization procedure: for instance the Kramer's degeneracy of the bands areslightly broken and a very little split is present, in the order of few decimalsof meV , but it can be considered negligible.The reliability of the model was also checked by looking at the spreads ofeach Wannier function, which should not exceed the value of the smallest cellparameter (17.273283 bohr). Eventually, considering the comparison of thebands and the values of the spreads, we decide to consider our tight-bindingmodel accetable.The tight-binding Hamiltonian obtained from Wannier90 is then used in


Wanniertools to calculate the topological properties of the system. First ofall we used the nding nodes algorithm, which in the whole Brillouin zonejust found the two already known Dirac points, as expected.Then we calculated the Z2 topological invariant in the fully gapped timereversal invariant planes (TRIPs), namely on the kz = 0, π and kx,y = πplanes (the other TRIPs present band touchings and the Z2 number cannotbe calculated): the evolution of the WCCs, presented in gure 3.8, suggest anon-trivial Z2 invariant only in the kz = 0 plane, as expected from the pres-ence of the two Dirac points along the tetragonal axis. Just to be sure abouttheir topological nature, we also checked the chirality of each Dirac point,calculating the ux of the Berry curvature on small spheres surrounding thesepoints, and the results was in both cases trivial (C = 0), as expected.The study of the surface states, nally, complete the picture. Having a non-

Figure 3.8: WCCs plot on dierent TRIPs, respectevely (from the left to right) theKz = 0, kz = π, kx = π, ky = π. A non-trivial topology (Z2 = 1) is found onlyin the rst case, in the kz = 0 plane, whereas all the other TRIPs are trivial 2Dsystems.

trivial Z2 invariant in the kz = 0 plane we expect to see two Fermi arcs on allthe surfaces perpendicular to that plane, for instance on the (100) surface,


showed in gure 3.9a (the plot is done at E = −0.02 eV ): we can glimpsethe two arcs but, as already anticipated, they are in part covered by theprejection of the bulk states on the surface which doesn't allow us to see thejunction point of the arcs. Finally a comparison with gure 3.9b, which justshows the contribution of the surface spectra coming from the bulk states,makes easier to distinguish the real surface states, namely the Fermi arcs.

Figure 3.9: Surface states spectra for the Cd3As2 system in the (100) surface(k1 = kx and k2 = kz). On the left the complete spectra is shown, in which we canspot the two Fermi arcs, partly covered from the projections of the bulk states. Onthe right we isolated the part of the spectra coming from the bulk. Note that thescale of the density of states (the colored one) is logarithmic, and it is dierent inthe two plots.

Although partly covered, the presence of such a surface states conrm thenon-trivial topology of the system, which along with all the other results (4-fold degenerate band touhcings, Z2 invariants, Dirac cone) allow us to char-acterize the pure cadmium arsenide system as a topological Dirac semimetal,the same conclusion present in the Weng's article [19]. A more detailed com-parison with this work shows that the numerical results are slightly dierent,due to the dierent methods and packages used, but the main electronicproperties of the system, including the non-trivial topology, are the same.Therefore we can conclude that our method were able to reproduce the al-ready known results of the Cd3As2 system and then it can be safely used forour study.

Chapter 4

Results and discussion of the

non-magnetic doped systems

As already mentioned in the previous chapter, in non-magnetic chemical dop-ings we are substituting some cadmium atoms with zinc atoms in the pureCd3As2 system. This choice is mostly related to the electronic congurationof the zinc (3d104s2), very similar to the Cd one (4d105s2) since they belongto the same group of the chemical table of elements: we expect the Zn tointeract with the As in a very similar way, at least maintaining the chemicalstability of the original structure.The substitutions follow dierent ways, breaking dierent symmetries caseby case, in order to see if topological phase transitions, starting from theoriginal topological Dirac semimetal phase of the pure Cd3As2 system, oc-cur. Furthermore we used dierent percentage of dopants in order to breakspecic symmetries, and this allows us to enlight the eect of dierent con-centration of dopants on the system.Finally, in this chapter we present the results for systems in which the spin-orbit coupling is taken into account, since only this case is the most relevantfor studying the topological aspects of the system.

4.1 Breaking of the rotational C4 symmetry

The rst example of non-magnetic chemical doping is the (Cd(1−x)Znx)3As2(x = 0.167) system, in which 4 Cd atoms are substituted with 4 Zn atomsin the relaxed Cd3As2 structure, presented in table 3.2. The substitution isdone in order to maintain the inversion I and the time reversal T symme-tries, while breaking the rotational C4 symmetry: this is only possible with

51

52 CHAPTER 4. NON-MAGNETIC DOPINGS

an even number of dopants, since the inversion symmetry always link a pairof atoms in the unit cell, and therefore we chose to use 4 Zn atoms, as shownin gure 4.1. Of course we could have chosen to substitute just 2 Zn atoms,but we push it further mostly because we wanted to emphasise the eectof the dopants on the system. Along with the breaking of the C4 we alsobroke other crystal symemtries, not relevant for the topological phase, andnow the system just has the inversion and the time reversal symmetries. Thenew doped system belong to a new crystal group, namely the P − 1 group.Before presenting the results, it should be noted that this particular doping

Figure 4.1: Crystal structure of the doped (4 Zn) unit cell of Cd3As2.

is very dicult to realize in practice, since in this case we are substitutingspecic atoms in very specic positions, necessary to maintain the inversion.On the other hand this calculation has an academic interest since help usto understand the importance of the rotational C4 symmetry in the Diracsystem (we remind that this is the crystal symmetry that protects the Diracnodes from an opening of the gap when SOC is considered).

4.1.1 DFT results in the presence of SOC

As in the case of pure Cd3As2 we are going to plot both the density of statesand the bands structure in the presence of SOC, which in addition to providea way to study the electronic properties of the system, provide also a hint for

4.1. BREAKING OF THE ROTATIONAL C4 SYMMETRY 53

the construction of the tight-binding model. Since the potential for futurepractical uses is limited, in this case we chose not to relax the doped systemsince we expect the relaxation to slightly modify the bands and the DOS,without changing the topological properties of the system.For all the DFT calculation we used the Wien2k package, along with thesame parameters (R ∗Kmax, k-mesh and smearing) we used in the pure sys-tem calculation, since the structure of the doped system is quite similar; themun tin radii are RMT = 2.5 bohr for the Cd and the Zn atoms, andRMT = 2.41 bohr for the Arsenic atoms.Moving on to the results, we rst present the density of states, shown in

Figure 4.2: Density of states for the doped (4 Zn) Cd3As2 system. (a) shows thetotal DOS whereas (b) the projected one, in which we considered the s orbital ofone single Zn impurity along with its closest As and Cd nighbor, for which we tookinto account the p and the s orbitals respectevely.


gure 4.2, both total and projected. The total DOS we can see that thegeneral behavior of the occupancy is quite similar to the one of the pureCd3As2 system, presented in gure g 3.2, although the occupancy at theFermi energy slightly increases. Since the electronic conguration of the Znis very similar to the one of the Cd, we expect a very similar chemical be-havior, especially in the interaction with its surrounding atoms in the bulk,namely 4 As atoms. In order to see this we can project the DOS on the sorbital of one Zn atom, on the p orbitals of its nearest As neighbor and alsoon the s orbital of its nearest Cd neighbor, as shown in gure 4.2. Note thatthe results for the other 3 As atoms which sourround the impurity and forthe other Zn atoms, not shown here, are very similar. It is clear that theorbitals involved close to Fermi level are the same we found in the previouscalculation, namely the valence region presents a prevalence of p electronscoming from the the 4p orbitals of the As atoms, and the conduction statespresent a hybrid sp character in which the p part comes from the As atomswhereas the s one come from both Cd and Zn atoms.So far we didn't nd relevant dierencies with the pure system, but the bandsstructure, shown in gure 4.3, presents some substantial modications. Firstof all note that all the bands are still double degenerate, due to the pres-ence of I and T (Kramer's degeneracy). Furthermore we can see that theinversion mechanism is still present around the Γ point, at which the valencep 3

2band (the blue one) and the hybrid conduction sp band (the light purple

one) switch, leading to an inverted order of the bands. The rst dierence wenote, respect to the Cd3As2 case, is that the old band crossing related to theinversion of the bands, the one shown in gure 1.1 along the high symmetryline Z−Γ, is now gapped: the original Dirac point is now broken and a gap ofδE = 12 meV is opened at k0 = (0, 0,±0.197) (in terms of reciprocal latticeparameter), namely the presence of the dopants introduced a mass term inthe Dirac equation (1.23), which describes the low-energy excitation aroundthe Dirac point. In particular we can relate this opening to the lacking ofthe C4 symmetry, which allow the states at k0 to mix since now they don'tbelong to dierent representations of the new crystal group.In general we cannot exclude the possibility that the Dirac point moved insome other k-point away from the tetragonal axis, but a further analysis(presented later) shows that, close to the Fermi energy, no other band touch-ings occur in the Brillouin zone. Therefore the breaking of the rotational C4

symmetry leads to the breaking of the Dirac points, no longer protected, andthe topological Dirac phase is no longer present. To characterize whetherthis new phase is topologically trivial or non-trivial, we have investigated itstopology, which we present in the next section.

4.1. BREAKING OF THE ROTATIONAL C4 SYMMETRY 55

Figure 4.3: Projected band structure of the doped (4 Zn) Cd3As2 system, alongthe Z − Γ − X −M − Γ path. A general view of the bands is shown, along withthe dierent characters coming from the S (red) and P (blue) orbitals of all theatoms (light purple bands represent hybrid SP bands). All the bands are doublydegenerate. The energy scale is in eV.

4.1.2 Topological properties

Looking at the results obtained from the DFT it is clear that the constructionof the tight-binding model can be done with the same parameters used inthe Cd3As2 system, since the main characteristics of the cell and the orbitalsinvolved in the region close to the Fermi energy are the same, although littledierences arise from the presence of the dopant. Therefore we chose to ini-tially project the Bloch states on the s orbital of Cadmium and Zinc atomsand on the s,p orbitals of Arsenic atoms, leading to a total of 176 Wannierfunctions in which the SOC is taken into account.


As in the previous calculations, after the minimization and the construc-

Figure 4.4: A comparisons of the band structure coming from DFT calculations(red lines) and from the tight-binding model (green lines), around the Γ point. Thesparse red dots in the bands represent the small dierencies which arise between thenal results obtained from the two computational methods.

tion of the MLWF (and the tight-binding model), we have conrmed thereliability of our results by rst verifying that the spreads of all the Wannierfunctions are less than the smallest lattice constant of the unit cell, and thenby comparing the band structure from DFT and tight-binding calculations,as shown in gure 4.4, which tells us that our model is quite good. As inthe case of pure Cd3As2 system, here the original double degenerate bandspresent a little splitting, but the chosen parameters allow us to keep thesenumerical errors low (order of decimals of meV ).First of all we look for other bands touchings in the Brillouin zone, but noone has been found. Then we can study the topology of the system lookingat the Z2 invariants of all the 6 TRIPs, since now all these planes are fullygapped. Note that in this particular case we don't have a proper insulatorbut a semimetal system, since we saw that the states at the Fermi energyare occupied, creating little hole and electrons pockets (see gure 4.2). Theconduction and the valence bands are well separated at each k-point of Bril-louin zone, and then the calculation of the topological invariants is still validand the topological properties of the system can be studied.Figure 4.5 shows the evolution of the WCC on the kz = 0 TRIP (the resultsfor the Kx,y = 0 are the same, whereas kx,y,z = π have trivial invariant),which tell us that this particular system has the same indices of a strongtopological insulator, namely (1; 1, 1, 1). Because of its strong topologicalnature, we expect to see non-trivial surface states on all the surfaces. Fig-ure 4.6 shows an example of surface spectra plotted in the (010) surface, atE = −0.0135 eV : unfortunately the projection of the bulk states on the

4.2. BREAKINGOF BOTH INVERSION I ANDROTATIONAL C4 SYMMETRY57

Figure 4.5: WCCs plot on the kz = 0 TRIP, which shows a non-trivial topologyon this plane.

surface is quite overwhelming, but in the two holes we can see some specialrounded states which we expect to be related to the non-trivial topology ofthe system, similarly to the case of surface states in strong topological insu-lators.Finally the Z2 topological invariants calculation, the bands structure andthe presence of non-trivial surface states tell us that the (Cd(1−x)Znx)3As2(x = 0.167) is a topological semimetal with the indeces of a strong topo-logical insulator, namely characterized by a non-trivial topology similar tothe strong insulators one. Therefore in this particular doped system we sawthe rst example of a topological phase transition provided by non-magneticchemical dopings. These results also conrm the crucial role C4 symmetryplays to protect the topological Dirac phase.

4.2 Breaking of both inversion I and rotational

C4 symmetry

In the previous section we saw that the breaking of the rotational C4 sym-metry in the (Cd(1−x)Znx)3As2 (x = 0.042) system leads to a breaking of theDirac topological phase, resulting in a phase transition which led to strongtopological phase with the character of a semimetal.In this section we are going to break not only the rotational C4 symmetry but


Figure 4.6: Surface spectra of the (010) surface (k1 = ky, k2 = kz), mainly occupiedby the projection of the bulk states. Inside the two holes, the two regions free fromthe projection of bulk states, two rounded non-trivial surface states can be spotted.The scale for the density of state is logarithmic.

also the inversion I symmetry, substituting one Cd atom with one Zn atom,leading to the (Cd(1−x)Znx)3As2 (x = 0.042) system. Since here we are notconstrained by the presence of the inversion, and furthermore we are notinterested in the other crystal symmetries, we can substitute any Cd atom inthe relaxed Cd3As2 cell, obtaining the structure showed in gure 4.7 that wehave used as starting point for the relaxation procedure. The crystal groupof the new cell is now the Pm one, which just cointain one crystal symmetrybesides the time reversal one.

Since the inversion symmetry is broken, a non-magnetic Weyl semimetalphase could arises, with at least 4 Weyl points in the Brillouin zone (remem-ber the action of the operator T , present in table 1.1): here we expect oneoriginal Dirac point to separate into two distinct Weyl points, with oppositechirality, located at dierent crystal momenta. Another important aspect toaddress is if the lacking of the C4 rotational symmetry is derimental for thepresence of Weyl points, as it does in the case of topological Dirac semimetal.Later we will see that this particular symmetry is not relevant for the cre-ation of the Weyl semimetal phase.

This type of doping could be particular interesting for practical uses, sinceit simulates a sort of random chemical doping in the system. Therefore we

4.2. BREAKING OF I AND C4 59

decide to relax the system before proceeding with further calculations, inorder to obtain a more reliable results. The relaxation was performed usingQuantum Espresso, along with the same parameters we used in the case ofthe pure cadmium arsenide system and no SOC is taken in account, sincewe saw that the presence of this interaction does not modify the relaxedparameters signicantly. Note that we just performed a relaxation of the


atomic positions, since we saw that the stresses acting on the cell are reason-ably small (∼ 1 Kbar along each direction). The cell parameters are keepedxed to the bulk values of the undoped Cd3As2. Figure 4.7 shows the nalstructure, the starting point for the DFT calculations and the tight-bindingmodel construction.

4.2.1 DFT results in the presence of SOC

As in the previous cases here we present the nal results, in which the spin-orbit coupling is taken in account, since this is the most relevant case regard-ing the topological properties of the system.The self-consistent cycle, along with all the calculations presented in thissection, have been made using the Wien2k package. The mun tin radii arenow RMT = 2.5 Bohr for the Cd atoms, RMT = 2.27 Bohr for the Arsenic


atoms and RMT = 2.39 Bohr for the Zinc atoms.First of all, following the path of the other sections, we are going to presentthe total density of state, shown in gure 4.8, plotted using a 7x7x5 k-mesh.Note that we decided not to show the projected DOS is in this case since,from the 4 Zn doping case, we already know the main eects of the Zn im-purity on its surrounding (see gure 4.2), and therefore the orbitals moreinvolved close to the Fermi energy. Furthermore the relaxation did not movethe atoms signicantly fro their starting position, then we do not expect con-sistent changings.Due to the low percentage of dopants, also the total DOS does not presentmain changings respect to the the pure Cd3As2, plotted in gure 3.5, andthe general behavior is quite the same, for instance we still have a semimetalsystem with a parabolic dispersion close to Ef .More interesting is the plot of the projected bands structure, shown in g-

Figure 4.8: Total density of states for the doped (1 Zn) Cd3As2 system.

ure 4.9, plotted using 250 k-points along the Z − Γ−X −M − Γ path (seegure 3.3b). First of all we can still see the inversion of the conduction andvalence bands around Γ (dierent colors correspond to dierent character),a typical sign of a non-trivial topological system. Furthermore, although notso evident since the eect of the impurity is small, the double degeneracyof the bands is generally broken, due to the lacking of the inversion symme-try, but at TRIMs the bands are still degenerate (Kramer's), for instance atk = Z or k = Γ. Note that the breaking of the degeneracy is positive for ourpurpose since it could pave the way to the creation of 2-fold degenerate bandtouchings close to the Fermi energy (obviously beside the TRIMs), namely


the Weyl nodes.Finally we can see that original Dirac point located along the Z − Γ line is

XΓZ ΓM

Figure 4.9: Projected band structure of the doped (1 Zn) Cd3As2 system, alongthe Z −Γ−X −M −Γ path. A general view of the bands is shown, along with thedierent characters coming from the S (red) and P (blue) orbitals. The bands areno longer degenerate due to the lacking of I. The energy scale is in eV.

now broken (a gap is opened), similar to the 4 Zn doping case, but in this casethe presence of 2-fold degenerate band touchings, possibly Weyl points, arenot forbidden because the bands are no longer degenerate everywhere. Sincewe did not see band crossings in the previous band structure, along the highsymmetry lines, we expect them to be located at low-symmetry momenta andtherefore we will use the tight-binding model along with the WannierToolspackage, which also allows us to study the topolgical properties of the system.


4.2.2 Topological properties

Based on the results coming from the DFT calculations, we constructed thetight-binding model using the Wannier90 package. In particular, since themain characteristics of the system are not so dierent from the pure Cd3As2system, we decided to use the same parameters and project on the sameorbitals, plus the s coming from the Zn impurity, to perform the Wanniercalculations (we still have 176Wannier functions, taking in account the SOC).Eventually, after the disentanglement procedure and the minimization of thespreads, we obtained the tight-binding model Hamiltonian, used by Wan-nierTool.As in the previous cases the reliability of the model has been tested lookingat the spreads of the Wannier functions, checked to be lower than the valuesof the cell parameters, and the tight-binding bands structure, plotted in g-ure 4.10 compared with the DFT one, which shows a good agreement, withinan error of a fraction of meV . Since now the bands are non degenerate, wedon't have the problem of the non physical splittings of the bands and thenumerical errors have been estimated looking at the dierence with the DFTbands, which is very small also in this case.As already anticipated we expect to see some Weyl points in low-symmetry

Figure 4.10: A comparisons of the band structure coming from DFT calculations(red lines) and from the tight-binding model (green lines), zoomed around Γ. Thesparse red dots in the bands represent the small dierencies between the nal resultsobtained from the two computational methods.

momenta, generated from the original Dirac points; therefore we can rstrun the nd nodes algorithm in the whole Brillouin zone, and see if newband touchings appear. Eventually the script found a total of 4 nodes in the


cell, shown in gure 4.11 from dierent views of the Brillouin zone of thesystem: as expected the new nodes are located at low-symmetry momenta,as shown in table 4.1, which also gives us information about their energies.Furthermore the number of the nodes we found is consistent with a non-

Table 4.1: Crystal positions, energy and chiralities of the 4 Weyl points.

W. points Kx (2πa) Ky (2π

b) Kz (2π

c) E (eV) Chir.

Wp1 +0.097 +0.036 -0.087 -0.0028 +1Wp2 +0.097 -0.036 -0.087 -0.0025 -1Wp3 -0.097 +0.036 +0.087 -0.0028 -1Wp4 -0.097 -0.036 +0.087 -0.0025 +1

magnetic Weyl system, in which the inversion symmetry is broken but thetime reversal is still present (see chapter one for details).In order to study the topoology of the system, namely to check if the new

Figure 4.11: Weyl points in the momentum space, seen from dierent views of theBrillouin zone (from above and from the sides). The red dots represent Weyl nodeswith a positive chirality, whereas the blue ones the nodes with a negative chirality.

nodes are really Weyl points, we can rst look at the Berry curvature, sincewe know that a Weyl point acts as a source or a drain of the Berry curvature


in the momentum space. In gure 4.12 has been shown an example of a Berrycurvature plot, drawn in the xz plane with xed ky = 0.036, in which twonodes are present. Here the presence of two divergent points (red and bluedots), respectevely source and drain of the Berry curvature, it's evident andit can be related to the presence of two Weyl points of opposite chiralities.We can check the chiralities of all the 4 nodes using WannierTools, which

Figure 4.12: Berry curvature in the kx−ky plane, at xed ky = 0.036 2πb . Two Weyl

points with opposite chiralities are shown (blue for negative, brown for positive), atwhich the Berry curvature diverges.

calculates how the Berry phase changes along a path which surrounds thenode, and eventually can tell us if this changing is not-trivial: the nal re-sults tell us that the 4 band touchings have non-trivial chiralities, namelytwo points have C = +1 whereas the other two have C = −1, as shown intable 4.1 and in gure 4.11, in which the red dots stand for positive chirali-ties whereas the blue ones for negative chiralities. Note that the total Chernnumber in the cell is zero, as it should be.

Further evidences of the Weyl phase can be obtained plotting the surfacestates spectra, for instance on the (010) surface, on which the projections ofthe bulk states are quite small. In gure 4.14 we see the results for the surfacespectra calculation, at constant E = −0.007 eV , from which we can clearlysee two Fermi arcs, coming from two dierent Weyl points. The presence ofa double Fermi arcs plot, similar to the Dirac case, comes from the fact thatpairs of Weyl points are projected at the same points in the surface, namely


Figure 4.13: Surface states spectra in the (010) surface (k1 = ky, k2 = kz). TwoFermi arcs are present, coming from the projection of 4 dierent Weyl points on thesame surface, projected in pairs on the same 2D-surface momentum at the junctionpoints of the arcs.

at the junction points of the two arcs, but actually they come from dierentpairs of Weyl points and therefore the arcs have to be considered separately.The shapes of the Fermi arcs on the (010) surface suggests that, similar toa Dirac semimetal, the plane that cuts the Fermi arcs in the middle, namelythe fully gapped kz = 0 TRIP, is characterized by a non-trivial Z2 invariant:this happens because either above and below the plane we have 2 weyl pointswith opposite chiralities, for a total Chern number of C = 0, but similarlyto a topological Dirac system we expect a non-trivial topology in the TRIPslocated between these 4 Weyl points.From the calculation of the Z2 invariant we found a value of Z2 = 1 on thekz = 0 TRIP, suggesting the possible presence of a spin hall eect in theplane. Furthermore, since the topological Weyl phase is more robust than aDirac semimetal phase (the rst one does not need crytal protecting symme-tries), this type of quasi-random doped system could be very interesting forpractical applications.

All the previous results lead to the conclusion that the presence of a singleZn dopant in the 40 atoms Cd3As2 system lead to a new topological phase,a Weyl semimetal phase, in which the 2 original Dirac points are separatedin 4 dierent Weyl points, located at low-symmetry momenta. Furthermorewe found that this non-magnetic Weyl phase could support a SHE, at least


in the kz = 0 TRIP, but since the same argument could be extended to allthe gapped kx,y=0 TRIPs, we expect a SHE on all these planes.This type of chemical doping, which in this case simulates a random dop-ing, provides the topological phase transition we were looking for, between atopological Dirac semimetal and a Weyl semimetal phase. At the end we alsoestablished that the C4 rotational symmetry is not relevant for the creationof the Weyl points.

4.3 Breaking of the I symmetry

In this section we are presenting the last example of non-magnetic doping,potentially the most interesting. In this particular system we are going tosimulate a so called delta doping, namely a particular type of doping in whichthe dopants are disposed on planes perpendicular to one direction: as shownin gure 4.14 in our structure the Zn atoms, the dopants, substitute 12 Cad-mium atoms in two dierent parallel planes perpendicular to the tetragonalaxis (the z direction), resulting in a new crystal group, namely the P42/mc.Although the precentage of dopants is quite high (50 %), we need such a


4.3. BREAKING OF THE I SYMMETRY 67

doping in order to achieve our aim in this particular system 1 , namely tomaintain the rotational C4 symmetry while discarding the inversion symme-try.The doped (Cd(1−x)Znx)3As2 (x = 0.5) system could be very interestingsince, rst of all, delta dopings are experimetally feasible, for instance byusing the molecular beam epitaxial (MBE) [41] technique, which allows oneto control the construction of the material layer by layer. Furthermore weare particularly interested in this system since it could show new topologicalphases, dierent from the simple Dirac or Weyl semimetal ones: as we knowwe broke the inversion symemtry I, but since we maintained the rotationalC4 we expect to nd again the two 4-fold degenerate Dirac points along thetetragonal axis, protected by the action of the latter symmetry along thisparticular axis. Furthermore the breaking of I makes possible the presenceof 2-fold degenerate band touchings, possibly Weyl points, away from thetetragonal axis. These nodes cannot be directly generated from the originalDirac points, as they did in the case of a single Zn doping, because in thiscase they are still present in the system.

Therefore, in principle, a coexisting Weyl+Dirac phase could arise, but wewill see that just chemical doping the system is not enough to obtain sucha situation since relaxing the cell leads to a trivial material in this system:an external parameter, in our case the strain, has to be taken into accountin order to add an extra degree of freedom to play with. Obviously othertopological phases can be obtained as an outcome of the doping, for instancea Dirac semimetal phase or a trivial phase, but notice that in all the cases the4-fold band touchings along the tetragonal axis are always present, protectedby the C4.

The path we follow in this section will be slightly dierent with respect to theprevious cases, in that we are going to present dierent situations regardingthe same doped system (in presence of SOC), either considering unrelaxedand relaxed structures or simulating external strains acting on the cell, tunedas an external parameter. After a brief introduction to the computationaldetails, we rst present the results for the unrelaxed structure, which showsthe particular Weyl+Dirac phase, the total relaxed structure, a trivial mate-rial, and the strained systems, grouped in one section, in order to study theevolution of the electronic and topological properties of the system in pres-

1The two matrices representing the rotational C4 symmetry in the original structureare translated by some crystal lattices in the unit cell, making impossible to consider just4 atoms for the substitution.


ence of dierent external strains, which passes from a trivial (relaxed cell)to a topological Dirac phase (strained cell) hopefully through the particularDirac+Weyl phase.

Computational details

We grouped all the computational details in one section because all the stud-ied systems are very similar, at least at the computational level, and thereforewe can use exactly the same parameters in all the calculations.First of all the relaxation calculation of the atomic poistions has been madeusing Quantum espresso with the following parameters and thresholds: anenergy cuto of Ewfc = 80 Ry, a charge-density cuto energy of Erho = 600Ry, a uniform Monkhorst − Pack k-mesh of 7x7x5, reduced to 30 k-pointsin the irreducible zone. The calculation thresholds are 10−5 Ry for the en-ergy, 1 mRy/au for the forces and 0.5 Kbar for the stresses. For this type ofcalculation the spin-orbit-coupling has not been taken in account, since theparameters are not expected to change signicantly.The self-consistent cycles are done using Wien2k along with the followingparameters and thresholds: R ∗ Kmax = 9, a uniform k-mesh of 7x7x5 k-points, reduced to a total of 30 k-points as in the previous case, and a littleFermi smearing of TEMP = 0.002 Ry. The chosen thresholds are 10−5 Ryfor the energy and 10−3 e for the charges. Note that we chose the RMTscase by case, since the relaxation procedure changes the relative positions ofthe atoms, possibly leading to overlapping spheres and therefore to dierentvalues of the radii.Finally, during the Wannier calculation of the tight binding model, we choseto project the Bloch states over the s orbitals of the Cd, Zn and As atoms andover the p of the Arsenic atoms, for a total of 176 Wannier functions, takingin account the SOC: this choice is based on the DOS calculation we presentin the next section. The overlap matrix was calculated using a k-mesh of10x10x7, in order to minimize the numerical errors, and a disentanglementprocedure was carried out using the frozen window [−0.4 : 0.5] eV .

4.3.1 Results for the unrelaxed structure in the pres-

ence of SOC

In this section we present the rst calculation we did, in which we simplytook the relaxed Cd3As2 cell and we substituted 12 Cd atoms with Zn atoms,simulating delta dopings, as shown in gure 4.14 (the cell parameters of the


original structure are a = b = 17.273283 Bohr and c = 24.272846 Bohr). Itshould be noted that the unrelaxed doped structure may not be chemically avery stable system, but we wanted to show the results we obtained, namelythe particular Dirac+Weyl topological phase, bacause it could explain theintermediate phase we expect to see in the evolution of the strained systems,arising in the middle between the topological Dirac phase and the trivialphase (see next sections for more details).First of all, as in the other cases, we are going to show the DOS, calculated

Figure 4.15: Density of states for the doped (12 Zn) Cd3As2 system, unrelaxed.(a) shows the total DOS whereas (b) the projected one, in which we considered thes orbitals coming from all the atoms present in the unit cell and the p ones comingfrom all the As atoms.

using a mesh of 7x7x5 k-points, and the bands structure, calculated alongthe Z − Γ−X −M − Γ path using 250 k-points. From a comparison of the


DOSs obtained for the doped cell, presented in gure 4.15, with the resultsof the pure Cd3As2 system, shown in gure 3.2, it is clear that the signicantZn doping does not aect the general behavior of the occupancy close to theFermi energy, which always shows a semimetallic behavior with a quadraticdispersion. Furthermore from the projected DOS, in which we are consid-ering the orbitals of all the inequivalent atoms, we can see that the orbitalsclose to the Fermi energy remain the same, namely the p-As in the valenceregion and both s,p in the conduction region, in which the s character comesprevalentely from the Zn atoms.

The bands structure, plotted in gure 4.16, shows some dierences with thepure case: rst of all now we do not have degenerate bands everywhere, dueto the lacking of the inversion, but along the high symmetry line Z − Γ thebands are still double degenerate because of the action of the rotational C4

symmetry on this particular axis.Furthermore we can see that the substitution with the Zinc atoms lowersboth the hybrid sp bands (light purple) coming from the conduction region:the lowest one now intersects, along the Z − Γ line, either the p 3

2valence

band, forming a crossing point, and the p 12band, forming an anti-crossing.

Therefore the only 4-fold band touching, involving the hybrid sp1 conductionband and the p 3

2valence band, is the original Dirac point, which survives in

the doped system. Note that the other signicant k-points related to the in-version of these two bands, namely the ones along Γ−X andM−Γ, presentsanti-crossings with a small gap.Note that the Dirac point now presents a peculiar characteristic in that oncewe move away from the Z −Γ line, namely along kx or ky, the lacking of theinversion symmetry results in a splitting of the bands and the Dirac cone isno longer doubly degenerate.

The bands plot also shows that the second hybrid sp2 conduction band,higher (in energy) respect to the one already cited, switches with the p 3

2

band cose to the Fermi energy, in the conduction region. Although not soclear from the plot, an anti-crossings with a small gap is formed, and there-fore we found no other band touchings so far.

In order to look for the Weyl points, since we expect they to be located atlow-symmetry momenta, we need to use the nd nodes script (implementedin WannierTools) acting on the whole BZ, and therefore we need to constructthe tight-binding model. Afetr we ran the script we found two band touch-ings along the tetragonal axis, namely the Dirac points, and 8 new 2-folddegenerate points in the kz = 0 plane, shown in gure 4.17a, which also


Figure 4.16: Projected band structure of the doped (12 Zn) Cd3As2 system (unre-laxed), along the Z − Γ−X −M − Γ path. A general view of the bands is shown,along with the dierent characters coming from the S (red) and P (blue) orbitalsof all the atoms (light purple bands represent hybrid SP bands). The bands aregenerally not degenerate (I is broken), but along Z−Γ the rotational symmetry C4

protects the 2-fold degeneracy of the bands, and the 4-fold degenerate Dirac pointsare still present. The energy scale is in eV.


Figure 4.17: In the upper gure the positions of the Weyl nodes in the kz = 0 planeis shown, along with their chiralities (blue for negative, red for positive); kx and kyare in unit of reciprocal lattice parameters.In the lower gure the plot of the Berry curvature in the kz = 0 plane is shown,zoomed around two Weyl nodes of opposite chiralities (red and blue dots), fromwhich we can see the divergencies of the curvature.


shows their chiralities (blue for C = −1 and red for C = +1), which suggestsus that these 8 band touchings are actually Weyl points.Another evidence of the presence of Weyl points is furnished by the plot ofthe Berry curvature on the kz = 0 plane, zoomed around two nodes as shownin gure 4.17b, in which we can spot the 2 divergences caused by the presenceof the Weyl points.In order to complete the topological picture we should also check the topol-ogy related to the Dirac points, for instance looking at the Z2 invariants onthe kz = 0 TRIP, but in this case it is not possible since this particular planepresent band touchings, namely the 8 Weyl nodes, and a Z2 invariant cannotbe dened (it needs a fully gapped system). We could also check the surfacestates on surfaces peprendicular to kz = 0, but unfortunately we found thatthe projection of the bulk states is overwhelminf and doesn't allow us tostudy in detalis the spectra.

To summarise we found that this unrelaxed (Cd(1−x)Znx)3As2 system is aparticular topological system, in which the original 4-fold degenerate Diracpoints of the pure Cadmium-Arsenide system, protected by the C4, and thenew 8 Weyl points cohexist, resulting in a new topological phase. Recentlya similar phase has been studied in [42], in which a coexistence of Diracand Weyl fermions has been found in polar hexagonal ABC crystals, suchas HgPb. In this particular system 2 Dirac points are present along thekz axis, similar to our case, but since they are protected by the rotationalC6 symmetry, instead of the C4 present in our system, a dierent numberof Weyl points, namely 12, has been found on the kz = 0 plane. Further-more, changing the buckling distance of HgPb layer, they noticed that atopological phase transition occurs in the system, which passes from a trivialto a topological Dirac semimetal phase with an intermedite phase, namelythe particular Dirac+Weyl one. We also studied the possibility of coexistingDirac + Weyl phase in a externally strained system, which will be discussedlater.

The structures presented in the next sections will not present substantialmodication of the physical structure other than the relative distances ofthe atoms, and therefore we expect to nd a similar characteristic, at leastfor the DFT results, and possibly the same topological phase we found herein some case; in particular we will see that when external strain is taken inaccount, this particular phase could arise as an intermediate phase betweenthe topological Dirac phase and the trivial one, since the topology has tochange through bulk gap closings.


4.3.2 The relaxed structures

Since we just saw an interesting new topological phase in the unrelaxed struc-ture, we need to understand if it could be reproduced in a more physical sys-tem. The rst thing we can do, in order to obtain a more physical system,is relaxing both the atomic positions and cell parameters. After that, sincethe total relaxed structure will present a trivial phase, we can move furtherand using its relaxed parameters as a starting point for the strained cell.

4.3.2.1 Results for the total relaxed structure in the presence of

SOC

As already anticipated we rst present the results for the total relaxed cell,in which both the atomic position and the cell parameters are completelyrelaxed: at the end we saw that the As atoms in contact with the Zn planesmoved closer to these planes (and far from the Cd atoms), whereas the cellparameters reduced to a = b = 16.567031 Bohr and c = 23.487859 Bohr( ca

= 1.417747), resulting in a smaller unit cell.We then performed the DFT calculations: rst of all we see no substantialchangings in the total and projected density of states, presented in gure4.18, compared with the unrelaxed result (just a little bit higher occupancyof p orbitals at Ef along with a slightly lower occupancies in the conductionregion close to this level).From the band structure, presented in gure 4.19 and plotted along Z −Γ−X −M − Γ using 250 k-points, we can see that generally the bands move,in particular the two hybrid sp conduction bands which go up in energy, butessentially we have the same main characterisics of the unrelaxed case, exceptthat the lowest hybrid sp band now is higher than the p 1

2valence band; the

4-fold band touching point along Z − Γ is still present, slightly moved alongthe line.

Since the situtation is not so dierent we wonder what happend to the8 Weyl points we found in the unrelaxed case. In order to nd them we rstconstruct the tight-binding model, as usual, and then we used WannierToolsto study the topology of the system. In this case the nd-nodes algorithm,in contrast with the unrelaxed system, found just 2 ungapped points (theoriginal Dirac nodes) along the tetragonal axis, whereas the Weyl points inthe kz = 0 plane disappeared.We also checked the topology of the system calculating the Z2 invariants inthe now gapped TRIPs, especially on the kz = 0 plane, whose evolution is


Figure 4.18: Density of states for the 12 Zn doped Cd3As2 system, totally relaxed.(a) shows the total DOS whereas (b) the projected one, in which we considered the sorbitals coming from all the atoms present in the unit cell and the p orbitals comingfrom all the As atoms.

plotted in gure 4.20. We found that the Z2 invariant is trivial on everygapped plane, but we need to consider with more details the kz = 0 case,the one shown in gure. We can see that here a generic horizontal line cutsthe WCC branches an even number (two) or zero times, in contrast withthe original topological Dirac semimetal in which there is a single cross: thismeans that although a non-trivial topological aspects could be found, forinstance the non-trivial surface states, they are not protected by the timereversal symmetry as they were in real topological materials, and generallythey disappear when some external perturbation is added to the system, re-sulting in a trivial material.


E-Ef

p1/2

sp1

S P

Figure 4.19: Projected band structure of the doped (12 Zn) Cd3As2 system (totallyrelaxed), along the Z −Γ−X −M −Γ path. A general view of the bands is shown,along with the dierent characters coming from the S (red) and P (blue) orbitals ofall the atoms (light purple bands represent hybrid SP bands). Note that the 4-folddegenerate Dirac point is still present along Z − Γ. The energy scale is in eV.


Figure 4.20: On the left: WCCs plot in the kz = 0 TRIP, which shows that ageneric horizontal line cuts the WCCs branches an even number (2) or zero times.On the right: zoomed-in around C=0.85 of the left gure to highlight the evolutionof WCCs. At the TRIM ky = 0, each of the pair is degenerate. Along the ky > 0path, they split into 4 branches but recombine at TRIM ky = 0.5 with the time-reversed partner, conrming that the system is topologically trivial.

Therefore in this particular case the non-trivial topology related to the Diracphase is broken, although the 4-fold degenerate nodes are still present in thesystem protected by the rotational C4 symmetry, and no other topologicalaspects are found in the system, resulting in a trivial semimetal.

4.3.2.2 Eect of external strains on the system

In the previous cases we found the (Cd(1−x)Znx)3As2 (x = 0.5) system intwo dierent topological phases, namely a trivial phase for the relaxed struc-ture and the particular Dirac+Weyl phase in the unrelaxed case, althoughthe latter may not be considered as a stable physical system. Since we areinterested to reproduce the Dirac+Weyl phase in a more physical system,in which the atomic positions are relaxed, and since we saw that the trivialrelaxed system present smaller cell parameters respect to the unrelaxed one,we decided to introduce an external parameter, namely an external stain,which allows us to enlarge the cell and hopefully, at some point, to repro-duce the wanted Dirac+Weyl phase. In our calculations we simulated anexternal strain acting on the cell modifying the values of the total relaxedcell parameters at our will, relaxing the atomic positions time by time.

In this section all the most relevant results of all the strained system are pre-sented and compared, in order to see how they modify using dierent strains.In particular we are interested to see how the band structures change, in par-


ticular looking at the gaps around the Fermi energy which, at some points,must close in order to change the topology. In fact we will see that tak-ing increasingly larger values of the strain, namely enlarging the cell moreand more, the trivial phase of the relaxed structure slowly changes into atopological Dirac phase; since the 4-fold band touchings along the tetragonalaxis, the original Dirac points, are always present in all the strained systems,the changing of the topology must occur through an intermediate phase andsome gaps have to be closed somewhere else in the Brillouin zone. We expectthis phase to be very similar to the particular Weyl+Dirac phase we foundin the total unrelaxed system, and therefore we hope to see the Weyl pointsat particular values of the strain.

The strained systems

Here we are going to introduce the strained systems we used, explaining themethod we followed and focusing on the values of the cell parameters weused. First of all we decided to maintain constant the cell parameters ratioca

= k = 1.417747 of the relaxed system, changing case by case the valueof the rst cell parameter a (b = a, c = ka), considering increasingly largervalues.We studied a total of 5 strained structures, grouped in the following list alongwith the values of the strains, acting on x,y and z.

1. a = 17.273283 Bohr, the same parameter of the unrelaxed structure,but here we relaxed the atom positions. A nal strain of −44.90 Kbarsalong x,y is found, whereas along z we have a strain of −43.43 Kbars.

2. a = 17.300000 Bohr, with a strains of −45.0 Kbars along x,y and−42.65 Kbars along z.

3. a = 17.342376 Bohr, with a strain of −47.09 Kbars along x,y, whereasalong z we have a strain of −44.47 Kbars.

4. a = 17.446016 Bohr, with a strain of −51.19 Kbars along x,y −47.97Kbars along z.

5. a = 17.618749 Bohr, with a strain of −55.36 Kbars along x,y, whereasalong z we have a strain of −50.23 Kbars.


Since the changing the cell parameters are quite small, we expect the generalbehavior of DOS and bands to be very similar, but we will see that the littledierences arising in the bands structure lead to topological phase transi-tions; in particular, since the initial system 1 and the nal system 5 willshow dierent topology, we expect to see an intermediate phase with somenew band closings somwhere in the bulk, namely new nodes, very similar tothe ones arising in the unrelaxed cell.

Results

As already mentioned before we expect similar outcomes for the various den-sity of states and therefore we present, once for all, the results for the rststructure, which can be considered valid also for the other cases; gure 4.21

Figure 4.21: Total density of states for the strained doped (12 Zn) Cd3As2 system.

shows the total DOS of the rst system, and a comparison with the old re-sults shows no substantial changes.More interesting properties come out from the band structures, in which wecan clearly see how the strain modify the electronic properties of the sys-tem, especially close to the Fermi energy. In gure 4.22 we show the bandsstructure of the rst strained system, along the Z − Γ − X −M − Γ path,from which we can rst see that, away from the Fermi level, we have thegeneral same behavior we had in the total relaxed system (4.19), just like inthe other strained cases not shown here.


Close to the Fermi energy we can see that, beside the usual 4-fold band

S P

Figure 4.22: Projected band structure of the strained 12 Zn doped Cd3As2 system,along the Z − Γ −X −M − Γ path. A general view of the bands is shown, alongwith the dierent characters coming from the S (red) and P (blue) orbitals of allthe atoms (light purple bands represent hybrid SP bands). The Dirac point is stillpresent along Z − Γ. The energy scale is in eV.

touching along Z − Γ, some k-points along Γ − X and M − Γ, namely inthe kz = 0 plane, present what seems to be band touchings; anyway a moredetailed analysis tells us that these are actually gapped points, with a tinygap of ∆E =∼ 0.0005 eV alongM−Γ, as shown in gure 4.22b, even smallerthan the one present in the total relaxed system. Furthermore the topolog-ical analysis of this strained system shows no new nodes in the Brillouinzone, and the Z2 calculations on the kz = 0 TRIP, whose WCC evolutionplot is shown in gure 4.23, tells us that the system has the same topologyof the total relaxed one (see gure 4.20), namely some topological aspectscould arise but they are not protected by the time reversal symmetry and,in general, the material can be considered trivial.


Figure 4.23: WCCs plot in the kz = 0 TRIP for the rst strained system, whichshows that a generic horizontal line cuts the WCCs branches an even (2) or zeronumber of times, resulting in a trivial system.

The analysis of the rst strained cell gived us hint on how we can look forthe wanted Weyl nodes: since we saw the gaps along Γ−X and Γ−M linesbecome smaller with a larger cell, we expect on an even larger cell that someclosings of the bands will happen close to these high symmetry lines, namelyon the kz = 0 plane, resembling the unrelaxed doped system. In order to seehow these gaps modify with dierent strains we plotted in gures 4.24 and4.25 the evolution of the bands, zoomed around the gaps, along the Γ − Xand Γ−M , in all the strained systems.First of all we can see that the gaps are very small until the third case, afterwhich they become larger and larger signaling that, if there are some bandclosings, they must happen with an intermediate value of the strain, betweenthe rst and the last system. Furthermore, from the topological analysis ofthe last strained structure, in which we expect a dierent topology, we foundno other nodes beside the Dirac ones along Z − Γ and the Z2 invariant cal-culation on the the kz = 0 TRIP, whose WCC plot is shown in gure 4.26,found a non-trivial topology on this plane; this results, along with the trivialtopology found on the other TRIPs, tells us that the system is a topologicalDirac semimetal, similar to the case of pure Cd3As2.Since in all the strained structures the two Dirac points are always present,and since the topology changes between the rst and the last strained system,we know that somewhere in between a dierent topological phase appears,namely an intermediate phase between the trivial and the Dirac ones, and


Figure 4.24: Evolution of the bands along Γ − X in a region close to Γ, whichshows how the gap between the conduction and the valence bands modies with thestrain. The order of the gures respect the order in which we presented the strainedstructures in the previous section.

some new band closings must occur somewhere in the bulk BZ in order tobreak the topology. In particular we saw that the smallest gaps are locatedalong Γ −X and Γ −M , and therefore we expect to nd 2-fold degeneratenodes in the kz = 0 plane, similarly to the unrelaxed case, since the bandsaway from the tetragonal line are generally split by the SOC.Therefore we checked the topology for the other strained system, but unfor-tunately we found no new nodes in all the cases. Furthermore the Z2 resultson the kz = 0 TRIP for all the intermediate values of the strain (second,third and fourth cases) are exactly the same as that of the last strained sys-tem (gure 4.26), namely we found a topological Dirac semimetal. Note thatthe second strained system, the one with the smallest gap (see the secondgures of 4.24 and 4.25), present very little gap between the conduction andthe valence bands, in the order of ∆E =∼ 0.0004 eV , sign that we are veryclose to the transition, apparently quite dicult to achieve.

Therefore, although we considered dierent values of the strains, very closeto each other, eventually we couldn't nd the particular Dirac+Weyl phasewe hoped to see, but we expect that, in a very small range of strains, between


Figure 4.25: Evolution of the bands along M − Γ in a region close to Γ, whichshow how the gap between the conduction and the valence bands modies with thestrain.

the rst and the second strained system, this phase must appears since thetopology changes with the strain. On the other hand this particular phaseseems to be very dicult to achieve, and probably our idea to enlarge thecell uniformly is not the correct path to follow. For instance we think thatstressing the cell in dierent ways along x,y and z (in a more consistent waywith respect to the values we took) could enable us to nd this particularDirac+Weyl phase. Another approach could be to consider a dierent typeof (heavier) doping, since we saw that the relaxation of the atomic positions,which moved the As atoms towards the Zn planes (Zn are lighter than Cd),broke the topology of the unrelaxed system.

Summarizing the results for the (Cd(1−x)Znx)3As2 (x = 0.5) doped systemwe saw two dierent types of topological phase transitions, between the orig-inal topological Dirac phase of the pure system to the trivial phase of thetotal relaxed cell and the topological Dirac+Weyl phase of the unrelaxed cell.Furthermore, if strain is taken into account, we saw that below the value −45Kbars the system remains in a topological Dirac semimetal phase whereasfor greater values it passes to a trivial semimetal phase, and an intermediatephase, similar to the unrelaxed case, is expected to appear for a very spe-


Figure 4.26: WCCs plot in the kz = 0 TRIP for the last strained system, whichshows that a generic horizontal line cuts the WCCs branches an odd number of times(1), resulting in a non-trivial topological material, in this case a Dirac semimetal.

cic small range of values of the strain. A phase diagram of the topologicalevolution of the system is shown in gure 4.27, in terms of the ratio r = a′

a,

which tells us how much the rst lattice parameters of a strained cell (a′) isenlarged respect to the one of the totally relaxed system (a). In particular,for the kind of strain and procedure we used (see previous sections), thisratio can be directly related to the value of the strain and therefore the dia-gram tells us how the topological phase of the system modies with the strain.


Figure 4.27: Phase diagram of the topological evolution of the system, in terms ofthe ratio r = a′

a . a and a′ are the rst lattice constants of the totally relaxed andof a strained system, respectevely.


Chapter 5

Results and discussion of the

magnetic-doped systems

In this last chapter we present an example of magnetic doping, introduc-ing one Manganese atom in the relaxed pure Cd3As2 system. The magneticdopings of the topological Dirac semimetal could be very interesting for prac-tical uses since a possible magnetic Weyl phase could arises, and we knowthat such a system supports the anomalous quantum Hall eect (see the rstchapter for details). Note that magnetism could be also taken in accountusing an external magneic eld, quite easy to tune, but following the path ofthis work we decided instead to use chemical dopings: in this way if a Weyltopological phase is achieved, the related system may become very interest-ing for practical uses and devices.Since we expect a consistent eect of the Manganese impurity on the elec-tronic properties of the system, which will make quite dicult the study ofthe doped Cd3As2Mn system, especially the band structure, we chose, at thebeginning, to substitute just one Cd atom, corresponding to the lowest possi-ble percentage of dopant (∼ 4 % in the 40 atoms cell); furthermore, in orderto consider a smaller and more isotropic eect of the impurity, a quasi-cubicsupercell made of 80 atoms is constructed from the original Cd3As2 system(see the last section for details), in which 1 Cd is substituted, obtaining a ∼2 % of doping.The method of the supercells is very useful for our purpose since it allows usto consider exactly the same structure while taking a larger unit cell, so thatit is possible to further decrease the percentage of the doping and lower theeect of the impurity. Furthermore the way we constructed the cell, whichis still tetragonal but almost cubic ( c

a= 0.993644), allows one to consider

a more isotropic eect of the Manganese impurity, since now the distancebetween two Mn atoms, present in two adjacent unit cells, is the same in all

87

88 CHAPTER 5. MAGNETIC DOPINGS

directions.

In either 40/80 atoms cells we are going to break all the important sym-metries we talked about in the previous chapters, namely the time-reversalT , the inversion I and rotational C4 symmetry; furthermore, just as in thecase of the 1 Zn doping, this type of magnetic doping reproduces a kind ofrandom doping, since we didn't really choose a particular Cd atom to sub-stitute, and this could be very interesting for practical applications.In this chapater we are going to present rst the results for the 40 atomscell, but we will see that in this case the Mn impurity modify a lot the elec-tronic properties of the original system, making dicult a detailed study ofthe topology, the most interesting part. On the other hand, moving to the80 atoms cell, we obtain a slightly better results for the electronic proper-ties, especially for the band structure, but unfortunately we were unable tocomplete the study of the topological aspects for this case as the cell as thecell and the number of bands involved are too big, making not feasible thecomputational calculation.

5.1 Results for the 40 atoms cell in the presence

of SOC

Electronic properties

After the subtituion of the Cadmium atom with one Manganese atom in theoriginal 40 atoms cell, we performed a spin-polarized (magnetism is takenin account) relaxation of the atomic positions using Quantum espresso, re-sulting in the structure showed in gure 5.1 which is now part of the Pmcrystal group; the lattice parameters used during this calculation are thesame we used for the original cell, but we added a small smearing to thesystem (degauss = 0.004). SOC is not taken in account at this level.Then we took the relaxed cell and we calculated the electronic properties ofthe system within the DFT approach, namely the DOS and the band struc-ture in presence of SOC, using Wien2k and a spin polarized calcualtion. Inthis case we used a dierent cuto value, namely R ∗ Kmax = 7.8, and weconsidered a larger value for the Fermi smearing (we expect a more metallicsystem), which is now TEMP = 0.005 Ry.

Focus on the results we can rst plot the total DOS, shown in gure 5.2,separating the results for the two dierent spin channels (up and down, re-

5.1. RESULTS FOR THE MAGNETIC DOPINGS 89

Figure 5.1: Crystal structure of the doped (1 Mn) unit cell of Cd3As2, made of 40atoms.

spectevely majority and minority states). The dierent eect of magnetismon the two channels it is evident, and we can see that now the two DOS donot coincide anymore. The occupancy at the Fermi energy is small, althoughit increased a little bit in the doped system. Furthermore we can see thatthe two dierent channels present extended regions of low occupancies closeto the Fermi energy, which probably are real gaps without the smearing (re-member that it spreads out the states close to the Fermi energy). Therefore ifthe low occupancy of the minority states in the valence region was a real gap,this could lead to an half-metallic system in which, close to Ef , the minoritystates present a gap whereas the majority states have a metallic behavior.This could be very interesting for spintronic applications, since allows one toexploit separately the dierent spin channels.Another important properties of the system are the magnetic moments of theentire cell and of the single atoms, calculated by Quantum Espresso duringthe relaxation procedure 1 , in which we obtained a total magnetic momentof µtot = 4.94 µb, mainly coming from the Mn impurity and its surroundingAs atoms. The total value is quite similar to the magnetic moment of anisolated Manganese atom (µMn = 5 µb, coming from the 5 electrons in the3d orbital), which in this particular system lowers to µ = 3.97 µb, as if one

1In this way we are not limitated just to consider the magnetic moment inside themun-tin spheres, as Wien2k does.


Figure 5.2: Total density of states for the doped (1 Mn) Cd3As2 system made of40 atoms, separated in the two spin channels up (red) and down (blue).

electron was lost in the cell. Since the Mn atom is surrounded by 4 As atoms,its closest neighbors, we expect it to share the other 4 electrons with them,creating some pd bonding resulting in an eective magnetic interaction pro-vided by a p-d exchange.To further clarify this aspect we plotted the projected DOS, shown in g-ure 5.3, taking in account the d orbital of the Mn and the p orbital of itsclosest As atom, separating the two spin channels (we expect for the otheratoms very similar results): from the shape of the DOS we can see thatin the valence region, close to Ef , the majority d states is hybridized withthe majority p states of the As and the same happens to the the minoritystates (ferromagnetic coupling). Furthermore the minority states present alow-occupancy zone just below the Fermi energy, as in the plot of the totalDOS.From the previous results, and from the calculation of the magnetic moments,we expect an eective ferromagnetic interaction provided by a p-d exchange,similar to the conclusion present in [43], in which they study the eect ofdierent transition metal dopings on the Cd3As2, inculding the manganese,although they focus on chromium dopants, for which they found very similarmagnetic properties.

Now we can move to the band structure, shown in gure 5.4, plotted using250 k-points along Z − Γ−X −M − Γ. The rst thing we can notice froma comparison with the pure Cadmium arsenide (3.6) is that now the system


Figure 5.3: Projected density of states for the doped (1 Mn) Cd3As2 system, inwhich we considered the d orbital of the Mn impurity along with the p orbitals ofits closest As atom. The two spin channels are considered separately.

has a more metallic character, but the plot is quite messy, and we can bearlyrecognize the original bands; in particular we will see that a complete anal-ysis of the topological properties, usually based on a clear understanding ofelectronic properties, is quite dicult since the result is not so clear.The main problem with the bands is that the presence of magnetism furthersplits all the bands close to the Fermi energy, already splitted by the actionof the SOC in absence of the inversion symmetry, and now all the bands arenon-degenerate: in this case not even the Kramer's degeneracy holds, andalso the old degenerate TRIMs now become non-degenerate. Furthermorethe lacking of C4 breaks also the degeneracy along the tetragonal axis: theoriginal 4-fold degenerate Dirac point along Γ − Z disappeared and it's noteven recognizible in the plot.The valence band closest to the Fermi energy, the old p 3

2bands in the pure

Cd3As2 system (see gure 3.6), now present an hybrid pd(s) characters (thecontribution of the s states is low), mainly coming from the p-majority statesof arsenic atoms hybridized with the d majority states of manganese. Onthe other hand the bands coming from the conduction region, which switchwith the valence band around Γ (band inversion is still present), has stilla prevalent sp character, but sligthly hybridized with the d orbitals of themanganese, in which both majority and minority spins are involved (notethat the coupling is ferromagnetic). Therefore the large modication of thebands close to the Fermi energy is explained, since we expect the magnetism


Efp-d

sp

Figure 5.4: Total band structure of the doped (1 Mn) Cd3As2 system. The dierentcolors represent dierent bands, and the main characters of the two bands closestto the Fermi energy are enlighted.

to aect mainly the p/d orbitals in this system, the ones more involved inthis region.

Topology

Although the large splitting of the bands make dicult the topological analy-sis, the previous result help us in the construction of the tight-binding model,at least in the choice of the initial projections of the Bloch states: in partic-ular we decided to take in account the s orbitals of all the dierent type ofatoms, in addition to the p orbitals of As atoms and the d orbitals of themanganese impurity, for a total of 186 Wannier functions (with SOC), andwe chose to use the same parameters we took in the pure cadmium arsenidesystem for the Wannier procedure.At the end of the minimization of the spreads we rst checked the spreads of


the WFs and we compared the results for the band structure, shown in gure5.5, from which we can see that the two band structures, the DFT and thetight-binding ones, are in good agreement.Then we rst look for nodes in the whole BZ, in an energy values which

Figure 5.5: Comparison of the bands, the ones coming from the DFT calculations(red) and the Wannier90 calculations (green), around the Γ point. The red dotsenlight the dierencies between the two plots.

extends from E1 = −0.1 eV to E2 = 0.05 eV , since we expect the Weylpoints to be close to Ef . We found that many nodes are present, at dierentenergies, but here we only present the ones with non-trivial chiralities: intable 5.1 are listed the 10 nodes we found, along with their energies and chi-ralities. This number of nodes is quite strange compared with the two Diracpoints of the original structure, but probably the bands are so splitted by thepresence of the Manganese impurity that new nodes, completely unrelatedfrom the old Dirac points, could appear.Furthermore we checked the topology of the system looking at the Chern

number in the kz = 0 plane (calculated by the mean of WCCs, similar to theZ2 invariants), since from the previous table we can see that we have 5 nodes(Ctot1 = −1) above and 5 nodes below (Ctot2 = +1) this plane and thereforewe expect to nd a non-trivial topological number (note that Z2 cannot bedened when T is broken, since not even TRIP are present). Eventually wefound a non-trivial Chern number C = |1| on the plane, sign that this planecan be viewed as a 2D Chern insulator, and since we are breaking the timereversal symmetry using magnetic dopings, we expect this particular systemto support an anomalous quantum Hall eect, as the one presented in therst chapter.


Table 5.1: Crystal positions of the 10 Weyl points, along with their energies and

chiralities.

WP Kx (2πa) Ky (2π

b) Kz (2π

c) E-Ef (eV) Chir.

Wp1 -0.033 -0.167 -0.050 -0.002 +1Wp2 -0.033 -0.012 0.178 -0.058 +1Wp3 0.039 0.114 -0.103 -0.044 +1Wp4 0.029 0.009 -0.180 -0.057 -1Wp5 -0.039 0.109 0.098 -0.056 -1Wp6 0.105 0.007 0.092 -0.095 +1Wp7 -0.107 -0.001 -0.088 -0.097 -1Wp8 0.036 -0.160 0.032 -0.005 -1Wp9 -0.046 -0.129 0.043 -0.037 -1Wp10 0.046 -0.135 -0.032 -0.032 +1

At the end we also tried to plot the surface spectra, but in this system theprojection of the bulk did not allow one to plot the surface states (they areall covered), and we couldn't see the Fermi arcs which could have given afurther evidence of the non trivial topology.At the end we found that the topology of the Mn doped Cd3As2 system seemsto have a non-trivial topology, related to the presence of 10 nodes with non-trivial chirality in the BZ and to a non-trivial chern number C = |1| denedon the kz = 0 plane, possible sign of the presence of a 2D Chern insulatorsystem in which an anomalous quantum Hall eect can be supperted. There-fore, although the bands are largely splitted by the presence of the impurity,making dicult a complete topological analysis, we could nd reasonableresults which drive us to say that the original Dirac system has passes aphase transition, and the doped system is now a topological magnetic-Weylsemimetal.

5.1.1 Results for the 80 atom supercell

Since in the previous section we saw that the presence of just one manganeseatom in the 40 atoms cell is quite overwhelming and its large eect on thesystem make it dicult for a detailed analysis of the band structure, andto some extent also of topology, we decided to lower the eect of the impu-rity by diluiting the doping, namely considering one manganese dopant in alarger cell, an almost-cubic 80 atoms supercell, constructed starting from therelaxed doped 40 atom structure. Figure 5.6 shows a general view of the nal


relaxed cell (we just relaxed the atomic positions), along with a particularview in which one can spot the old 40 atoms structure: the height of thenew cell are the same of the old structure (c′ = c = Bohr), whereas alongx,y we took a larger values for the cell parameters (a′ = b′ =

√2a = Bohr,

ca

= 0.993644).This particular choice for the supercell allows one, rst of all, to reduce the

Figure 5.6: Crystal structure of the doped (1 Mn) super cell of Cd3As2, made of80 atoms. The upper gure shows a general view of the cell, the bottom/left a sideview and the bottom/right a view from above. The red lines show how the original40 atoms unit cell can be recognized in the supercell.

eect of the impurity on the system, since now we are considering a ∼ 2% ofdoping, and furthermore, since the cell is now almost cubic, the eect of the


Mn can be considered isotropic and the Manganese atoms are now (almost)equidistant in all the directions.Now we present the DFT results for the electronic structure of this largersystem, for which we used a cuto of R ∗Kmax = 7 (larger values were notfeasible) and a k-mesh of 5-5-5 k-points.First we present the total DOS plot, shown in gure 5.7, since we expect

Figure 5.7: Total density of states for the doped (1 Mn) 80 atoms supercell of theCd3As2, separated in the two spin channels up (red) and down (blue).

the eect of Mn on the surrounding atoms to be the same; futhermore thetotal DOS is a general property of the system and we can compare the resultwith the 40 atoms case, looking at the eect of the dierent percentage ofdopants. First of all we can see that the splitting between the majority andthe minority states is less evident with respect to the 40 atom case. At thefermi energy the occupancy is now smaller, and furthermore the minoritystates present, just below the Fermi level, a region of almost zero occupancy:as already anticipated in the previous chapter if this region presents a realgap (probably without smearing this would be a real gap) we would see thetypical behavior of an half-metal, which could be very interesting for practi-cal spintronic applications.

The magnetic moment of the untit cell is now 4.99 µb, whereas for the Mnatom we have 3.97 µb, therefore the argument presented in the 40 atomsystem holds also in this case and we expect the magnetic iteraction to bemediated by an eective p-d exchange.


We can now move to the band structure, presented in gure 5.8, from

Ef

Figure 5.8: Total band structure of the doped 80 atoms supercell of Cd3As2. Thedierent colors just represent dierent bands.

which we can see that, although the bands are still splitted in the same waythey were before, the general eect is now less evident and a sligthly moreclear results came out. For instance we can spot the old Dirac point alongZ−Γ, now gapped, and futhermore we can see that the valence and the con-duction bands, with dierent characters, intersect close to the Fermi energy,although no band touchings are present along these high symemtry lines.Since they are very close, we expect that some band touchings occur at lowsymmetry momenta, hopefully Weyl points, but we should rst construct thetight-binding model to nd them and to check the topolgy of the system.Unfortunately, using our method, the construction of the tight-binding modeland the topological analysis was not possible in this case since the large celltogether with the large number of Wannier function we had to take intoaccount made the Wannier calculation not feasible, both in time and power


demandings. Therefore, although the results for the 80 atoms cell are promis-ing, we were not able to study the topology of the system and therefore andanother (more feasible) methods have to be taken into account.

Conclusions and outlooks

In this work we used rst-principles methods based on density functionaltheory to study the electronic, magnetic and topological properties of twonovel classes of topological materials, known as topological Dirac and Weylsemimetals (DSM andWSM). In particular we focused on the Dirac semimetalcadmium-arsenide (Cd3As2). The main purpose of the work was to inves-tigate whether it is possible to trigger topological phase transitions in thissystem, for example the WSM phase, by chemical doping. Since three sym-metries are important for the non-trivial topological Dirac phase of Cd3As2,namely I, T and C4, we broke them in dierent ways, adding some impuritiesto the original system, and case by case we studied the physical propertiesof the doped material. In particular we chose to use two types of impurities,substituting some Cadmium atoms with Zinc atoms (non-magnetic doping)or Manganese atoms (magnetic doping).

For each system we considered we rst perform a relaxation calculation,when needed, and then we calculated the electronic ground state using aself-consistent scheme within the DFT approach, which allows one to studythe electronic and magnetic properties of the material. Based on the DFTresults, we then constructed the tight-binding model Hamiltonian, used tocalculate the topological properties. In recent years various theoretical stud-ies have been done on Dirac and Weyl materials, but in our work we useda dierent computational method which doesn't involve pre-built eectivemodels: the tight-binding models we obtained come directly from our nu-merical calculations, in which we directly took in account the spin-orbit cou-pling, fundamental in topological materials, and therefore we expect a moreaccurate results.

After a brief introduction to the theory and the computational tools (chapterI and II), in chapter III we tested our method on the pure Cd3As2 Dirac sys-tem, in order to see if we were able to reproduce the electronic and topologicalproperties of this system already published in other theoretical works: even-

99

100 CHAPTER 5. CONCLUSIONS

tually we found a good agreement on the electronic structure and the sametopological properties, namely a Dirac phase with two Dirac points along thetetragonal axis, along with the non-trivial Z2 invariant on the kz = 0 tmie-reversal invariant plane (TRIP), which in principle can support a quantmspin Hall eect.

In chapter IV we analyzed the rst doped system, in which the C4 symmetryis broken. Although this system cannot be easily realized experimentally, itserved as a benchmark to test the eect of breaking the C4 on the topologicalDirac phase while maintaining the inversion and the time reversal symme-try. We found that for this doping the original Dirac semi-metal topologicalphase is replaced by another one, characterized by indices similar to the onesdisplayed by topological insulators, despite the system remains a semimetal.

In the second doped system we used one Zn impurity, simulating a randomdoping in which we broke both I and C4, while maintaining the T symme-try. In this case we found a new topological phase, namely a transition to anon-magnetic Weyl phase, in which the two original Dirac points separatedinto 4 dierent Weyl points, two with positive chirality C = +1 and two withnegative chirality C = −1. As for the original pristine Cd3As2 system, thisdoped phase is also characterized by a non-trivial topological index Z2 on thekz = 0 TRIP, possibly leading to a quantum spin Hall eect. However, incontrast to the Dirac topological semi-metal phase, this phase is more robustsince it is topologically protected and does not require any extra-symmetryto be stable.

In the third non-magnetic doped system we substituted twelve Cd atomswith twelve Zn atoms in two parallel planes, in order to maintain the ro-tational C4 symmetry while breaking the inversion one, simulating a sortof delta doping. In this case we studied dierent cases, namely the unre-laxed, the totally relaxed and some strained systems, in which we introducethe strain as an external parameter. We saw that the unrelaxed system,although it is not a real physical system, presents a particular topologicalphase in which the two orginial Dirac points are still present, protected bythe C4 symmetry, but eight new Weyl points appear in the kz = 0 plane,resulting in the cohexistence of both Dirac and Weyl phases. This result isparticularly signicant and unexpected, and highlights the subtle role playedby the dierent symmetries in these quantum materials. Unfortunately therelaxation procedure breaks this particular phase, leading to a trivial topo-logical phase. We then explored the possibility to inducing such mixed Dirac+ Weyl topological phase in a physically system by means of an external

101

strain. We found that, using dierent values of the strain strength, the sys-tem goes from a trivial phase (for the relaxed system) to a topological Diracphase (for a system under a strong enough strain). It is our conviction thatthis transition must occur via an intermediate phase which we expect to beprecisely the mixed Dirac + Weyl phase found for the un-relaxed system.For the kind of strain procedure enforced here we were unable to identify theprecise point of the transition. Nevertheless this is an area that is certainlyworth investigating further more by using other strain procedure and dier-ent types of dopants.

Finally, we consider a doped system where all the symmetries, includingthe time-reversal, are broken, by introducing random magnetic impurities.Specically, we consider a system where one Cd atom in the supercell is re-placed by a Mn atom, which typically makes the ground state of the systemferromagnetic. We rst doped the 40 atoms cell: from the DFT results wenoted that the eect of the magnetic impurity on the system is quite large,especially on the bands, making dicult a detailed analysis of the physicalproperties of the system.Nevertheless, despite the complexity of the bands caused by the magneticorder, we were able to construct the corresponding tight-binding model andcarry out the topological analysis. We found the emergence of a magneticWeyl phase with a total of 10 Weyl pointsclose to the Fermi energy. Fur-thermore we found that the plane kz = 0 has a non-trivial C = |1| Chernnumber, and therefore we expect it to support an anomalous quantum Halleect.We also considered a more diluite magnetic doping, using 1 Mn atom in aquasi-cubic 80 atoms supercell: in this case the DFT results are more trans-parent than the ones of the fourty-atom supercell, since for this more diluitingdoping the eect of magnetism is reduced. Unfortunately a larger supercellalso leads to a smaller Brillouin zone with many more bands, making the ex-traction of tight-binding model and the ensuing topologically analysis com-putationally unfeasible. The study of magnetic doping is left for future works.

102 CHAPTER 5. CONCLUSIONS

Appendix A

Convergence tests

In this appendix we are going to present the convergence tests we did inQuantum Espresso and Wien2k, an essential preliminary step for a numer-ical simulation such as the ones presented in this work; more precisely thisprocedure can tells one to what extent the computational results are accurate,and if they can be considered valid for a comparison with the correspondingphysical systems.We know that all the computational methods bring numerical errors, de-pending of the choices of some parameters, but it's clear that more accurateresults require more power and time demandings, and therefore we need tobalance these two aspects.In particular, either in Quantum Espresso and Wien2k, we need to under-stand which cuto values and k-mesh are needed in order to get accetablephysical results: we can do this looking at the values of the total energy ofthe system, and see how they change with the dierent choices of the pa-rameters. Furthermore we can also look at the electronic properties of thesystem, namely the DOS and the bands structure, and see how they changewith the cutos.

We rst present the test done in Wien2k for the choice of the k-mesh, whichwe also maintain in Quantum Espresso, considering the pure Cd3As2 system.In order to do this we looked at the values of the total energies related to twodierent choices of the mesh, namely 7x7x5 and 10x10x7, choosing the cutovalue R ∗ Kmax = 9 and considering the spin-orbit coupling: the dierencebetween the two total energies is E10 −E7 = 0.00008 Ry, therefore we choseto use the 7x7x5 since it accetable for our purpose.In Wien2k we also did a convergence test for the cuto parameters R∗Kmax,using a xed k-mesh of 7x7x5 and without considering the spin-orbit in-teraction: this chioce is mainly due to the high computational eorts the

103

104 APPENDIX A. CONVERGENCE TESTS

calculation required for high values of cutos (note that the developer ofWien2k usually suggests a maximum value of R ∗Kmax = 9), which in pres-ence of SOC increase exponentially. Therefore we considered three values,namely R ∗ Kmax = 9/10/11, altohugh the last one took a lot of time andmemory. The total energies, shown in table A.1, shows that dierence be-tween the last two values is E11 − E10 = 0.0066 Ry, quite big, but sincewe are mostly interested in the nal results, such as the band structure orDOS, wa can look at how they modify with the dierent cutos. In gure

Table A.1: Position of atoms.

R ∗Kmax (Ry) Energy (Ry)9 -340964.533217910 -340964.562288911 -340964.5689949

A.1 we plotted a comparison of the bands structure and the total DOS whenthe three dierent cutos, from which we can see that the eect on the nalresults is relatively small, and therefore we chose to use R ∗ Kmax = 9 Ry,since a value of 10 would be too large and many computationial eorts wouldbe required.

Regarding the Quantum espresso package we need to choose the correct en-ergy cuto values Ew and Ep, respectevely the energy cutos for the wave-function and for the charge-density/potential. Since we are simulating asemimetallic system with ultra-soft preudopotentials, the cuto value Ep canbe generally calculated from Ew from the formula Ep ∼ 8Ew, and thereforewe just need to check the value of Ew. In order to do this we tested thetoughest pseudopotential, namely the Cadmium one, in the bulk system ofpure cadmium in the presence of spin-orbit coupling, since we expect thecuto energies to be the same when we consider the more complex systemCd3As2. From table A.2 we can see that the dierence between the values

Table A.2: Total energies for the bulk Cd system, for dierent Ew values.

Ew (Ry) Energy (Ry)60 -233.666662670 -233.666697980 -233.666791390 -233.6668020

Ew = 80 Ry and Ew = 90 Ry is ∆E = 0.00001 Ry, therefore we chose to use

105

Figure A.1: In the upper gure is shown how the bands plot close to Γ modify withthe value of the cuto R ∗Kmax. In the bottom gure is shown hoe the total DOSmodify. .

106 APPENDIX A. CONVERGENCE TESTS

Ew = 80 eV in our calculations, along with Ep = 7.5 ∗ Ew = 600 eV .

Appendix B

Wannier90 numerical errors

As in the case of computational tools based on DFT, also the constructionof the tight-binding model provided by the Wannierization procedure bringssome numerical errors related to the choice of the parameters. AlthoughWannier90 allows one to tune a lot of dierent parameters, the two whichmostly aect the results are the choice of the initial projections and the choiceof the k-mesh used to calculate the overlap matrix M.Since the goal of Wannier90 is to construct an accetable tight-binding modelable to reproduce the DFT results, for instance the band structure and thecharacter of the states, it is clear that we need to take into account all theorbitals most involved in the bondings, and ususally we can base our choiceon the nal results of previous DFT calculations (note that this choice isquite obliged).On the other hand a correct choice of the k-mesh is equally essential, but inthis case we have to test this parameter: in particular we saw that it has asignicant eect on the the band structure, and therefore from a comparisonwith the 'correct' DFT results we can choose the most appropriate k-meshfor the Wannier90 calculations.

Here we are considering the pure Cd3As2 system, using the same initialprojections, testing two dierent k-mesh (7x7x5 and 10x10x7) and compar-ing their respective outcome.First of all we show a general view of the bands around Γ, shown in gureB.1, from which we can see that the two band structures (red and green)related to the dierent k-meshes are in good agreement with the DFT result(the blue line), altohugh the smaller k-mesh present more dierencies. Onthe other hand if we consider a smaller region, the one enlighted by the redrectangle shown in gure B.1, and we plot the band structure for the two k-meshes, as shown in gure B.2, we can see that the original degenerate DFT

107

108 APPENDIX B. WANNIER90 NUMERICAL ERRORS

Figure B.1: Comparison of the bands around Γ, in which we plot the two bandstructures coming from the W90 calculations, red and green, which stand for dif-ferent k-mesh choices, and the band structure coming from the DFT calculations,in blue. The red rectangle shows the region in which we plot the bands presented inthe next gure.

Figure B.2: A zoom of the W90 bands around the conduction valley, in the regionshown in the previous gure. Note that the energy scale is dierent, and thereforethe splitting. .

109

band (2-fold degenerate due to I and T ) are now splitted because, in general,the Wannierization procedure minimize the spreads of the Wannier functionswithout taking into account the symmetries of the system and some numer-ical errors of this type may occur. Anyway this unphysical numerical errorcan be used as a test for our models, since we can see which choice of themesh make the split smaller: we noted that this splitting can be decreasedto an accetable value by considering a larger k-mesh, as shown in gure B.2.A numerical error of this type, in this case ∼ 0.1meV , can be consideredaccetable for our purpose and therefore we chose tu use the 10x10x7 k-mesh.

110 APPENDIX B. WANNIER90 NUMERICAL ERRORS

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