T H E S E A R C H F O R 2 D D I R A C F E R M I O N S I N A 3 D T O P O L O G I C A L S Y S T E M An experimental and theoretical study of SrMnSb2
ja n s h e n k e
Master of Science (MSc) Institute of Physics
Faculteit der Natuurwetenschappen en Informatica University of Amsterdam
Jans Henke: The search for 2D Dirac fermions in a 3D topological system, An exper-imental and theoretical study of SrMnSb2, Master of Science (MSc), © August
2017
s u p e r v i s o r s: Mark Golden Jasper van Wezel
l o c at i o n:
Amsterdam, the Netherlands t i m e f r a m e:
A B S T R A C T
In recent decades, a central goal of condensed matter physics has been to un-derstand and characterise so-called topological phases of matter, which fall outside Landau’s symmetry-breaking description of phase transitions. One such phase is a two-dimensional (2D) Dirac semimetal, hosting 2D Dirac fermi-ons, which behave according to the relativistic Dirac equation and provide us with an easily accessible platform for studying the basic principles of relativity. Such a material would also be a fantastic conductor, hosting (nearly) massless charge carriers with a high mobility. The one-atom thick material graphene came to fame as being such a material in 2005, although these properties were later shown not to be robust against spin-orbit coupling (SOC) and easily af-fected by the substrate and adsorbants on the surface. In 2015, Young and Kane proposed that the combination of two specific crystal symmetries and time-reversal symmetry will result in a 2D Dirac semimetal phase within a bulk crystal that is protected from any effects that do not break those symmet-ries, including SOC. In this thesis, we study one specific material − SrMnSb2
−to determine whether it is indeed the 2D Dirac semimetal we’ve been look-ing for. We study its bulk electronic properties uslook-ing transport measurements, and study its band structure using angle-resolved photoemission spectroscopy (ARPES) as well as a tight-binding model combined with density-functional theory calculations. We show it to be a quasi-2D material hosting low-mass, high-mobility charge carriers, but despite these hopeful signs, the Dirac fermi-ons we were hoping to find were not those involved in conduction. Our study proves the value of the combination of thorough experimental and theoretical research, and may open new avenues of research into the role of magnetism in (otherwise) Young and Kane-like systems.
A C K N O W L E D G M E N T S
This project was a collaborative effort of a group of great experimental and theoretical physicists, which spanned beyond my own work. With regards to the work described in this thesis, there are many people that deserve some credit for either guiding me through the work that I did myself, or contributing new insight into some of the data central to the story of SrMnSb2.
To start with, I’d like to give a word of thanks to Mark, who really shaped the project and helped us all see the bigger picture. His great out-of-the-blue humour was especially appreciated during the beamtimes. I am truly indebted to Jasper, who guided me through how to set up my tight-binding model, and always made time to answer my endless questions in a way that everything would make sense again. With ARPES (and LEED, and STM), Shyama was the real workhorse, and she deserves the credit for analysing and making sense of all the key ARPES data while the rest of us mostly just helped with these measurements on a more practical level. The transport measurements, and the analysis thereof, were only possible with the guidance of the transport master: Anne, whose endless pool of knowledge on this front was invaluable. This en-tire project would not have been possible without the crystals themselves, so thank you Yingkai, you are ridiculously good at what you do. The DFT calcu-lations were all performed by Zhicheng, who was able to give us the pieces needed to complete the puzzle, almost entirely via email. Erik, aside from your hard work on the beamtimes, you made sure we actually drew some concrete conclusions, so thank you for the critical discussions these last few months. Of course I have to thank my lab buddy Rajah, who is the cool cucumber that made my experimental struggles a whole lot more enjoyable; and Huib, who took us both under his arms and got me knowing my own way around the lab and its endless cupboards. (One day, our strontium evaporation project will be useful for something, I know it!) Hans and the rest of the Technology Centre staff, without you nothing in the lab would work, and cutting my own cleav-age posts was oddly rewarding. Also thanks to all the beamline scientists at Diamond Light Facility, and (although I didn’t go there myself) those of the 13 ARPES end station at BESSY. And, last but not least, I have to thank those
who contributed to this project before I even started my own work: Yu Pan, Stephan Bron, and Adriaan Schiphorst.
I don’t know if I would have completed this thesis without those around me − aside from those working on the project itself − who made this last year enjoyable. Alona, Steef, Milo, Kris, Artem, George, Huaqian and Dona, our many coffee breaks and extended group lunches definitely helped keep me motivated to work (see quote below). Speaking of coffee, to the ladies of the secretariat: you were always friendly and super helpful, and I already look forward to future IOP Christmas parties and summer BBQs! Lia, you’re the coolest flatmate, thanks for the motivational comments and the chill-out sessions on the couch. To my family and my Amsterfam, you are awesome
and I love you all. And don’t worry, I don’t expect any of you to actually read my thesis, if you’ve gotten this far then you’ve read enough ;)
It is more fun to talk to someone who doesn’t use long, difficult words,
but rather short, easy words like "what about lunch?" — Winnie the Pooh
C O N T E N T S
i b a s i c p r i n c i p l e s 1 1 i n t r o d u c t i o n 3
2 t h e o r e t i c a l p r i n c i p l e s 7 2.1 Atomic orbitals 7
2.2 Crystals and symmetries 9 2.3 Modelling crystalline materials 12
2.3.1 Tight-binding model 12 2.3.2 Density functional theory 17 2.4 The Berry phase 19
2.5 Young and Kane and protected crossings 21 3 t h e c a n d i d at e: SrMnSb2 25
3.1 Crystal structure 26
3.2 Experimentally determined parameters 28 ii e l e c t r i c a l t r a n s p o r t 31
4 e l e c t r i c a l t r a n s p o r t: method and theory 33 4.1 Experimental setup 33
4.2 The Hall effect 34
4.3 Shubnikov-de Haas oscillations 36 5 t r a n s p o r t m e a s u r e m e n t s 41
5.1 Resistivity with temperature 41 5.2 Hall resistance 42
5.2.1 Hall charge carrier density and mobility at 2K 43 5.2.2 Temperature dependence of Hall charge carrier density
and mobility 45 5.3 Quantum oscillations 46
5.3.1 General features of the quantum oscillations 46 5.3.2 Nature of the Fermi surface 49
5.3.3 Temperature dependence 52 5.3.4 Berry phase determination 53 iii b a n d s t r u c t u r e o f SrMnSb2 61 6 p h o t o e m i s s i o n s p e c t r o s c o p y 63
6.1 Photoemission theory 63
6.2 The hemispherical electron analyser 68
6.3 Experimental requirements and sample preparation 70 6.4 Measurement set-ups 70
7 a r p e s m e a s u r e m e n t s 75 7.1 Constant energy cuts 76 7.2 The peak aroundΓ 78 7.3 The peak around Y 79
7.4 Electron-doping via surface deposition 81
viii c o n t e n t s
8 t i g h t-binding model set-up 85 8.1 Spin-independent terms 86 8.2 Spin-orbit coupling 89 8.3 Magnetism 92
8.4 Adding strontium atoms to the model 93 9 t i g h t-binding model results 95
9.1 Young and Kane protected crossings 95 9.2 Tight-binding model fitting 98
9.2.1 Fitting to DFT 98
9.2.2 Comparing to ARPES measurements 102 9.3 Determining the Berry phase of the Y cone 105 iv d i s c u s s i o n a n d c o n c l u s i o n 111
10 discussion 113
a t i g h t-binding model code 117 b i b l i o g r a p h y 119
L I S T O F F I G U R E S
Figure 1.1 A schematic diagram of the band structures of trivial and topological insulators, and a Dirac semimetal. 4 Figure 2.1 The boundary surfaces of the s and 2p orbitals 8 Figure 2.2 Types of orbital overlaps possible for s and p
orbit-als. 9
Figure 2.3 Examples of symmetries in a crinkled two-dimensional square lattice. 10
Figure 2.4 A one-dimensional solid. 13
Figure 2.5 An example of the band structure of a simple one-dimensional solid, calculated with a tight-binding model. 15
Figure 2.6 Protection of the band crossings in systems with a non-symmorphic symmetry, inversion, and time-reversal sym-metry. 22
Figure 2.7 Tight-binding calculations for lattices a,b and c consid-ering only s orbitals. 24
Figure 3.1 The structure of SrMnSb2 26
Figure 3.2 The Sr-Sb-Sr layer in SrMnSb2 27
Figure 3.3 The Brillouin zone of SrMnSb2. 27
Figure 3.4 Types of anti-ferromagnetic ordering. 29
Figure 4.1 The measurement pucks used for transport measure-ments in the Physical Property Measurement System® (PPMS). 34
Figure 4.2 The two contact configurations used for transport meas-urements. 34
Figure 4.3 The PPMS DynaCool™ cryostat. 35
Figure 4.4 Landau tubes for a spherical Fermi surface. 38 Figure 4.5 Quantum oscillations modelled using Lifshitz-Kosevich
theory. 39
Figure 5.1 In-plane and out-of-plane resistivity of SrMnSb2[40]. 41
Figure 5.2 Longitudinal resistivity measured on different samples of SrMnSb2. 42
Figure 5.3 Two Hall resistance measurements on the same sample of SrMnSb2, batch 20151022. 44
Figure 5.4 The temperature dependence of the Hall charge carrier density of SrMnSb2. 45
Figure 5.5 The temperature dependence of the Hall mobility, as measured on a sample from batch 20151022 and by Liu et al. [40]. 46
Figure 5.6 Magnetoresistance measurements showing quantum os-cillations. 47
Figure 5.7 A comparison of the quantum oscillations from the same sample, in separate measurements. 48
x List of Figures
Figure 5.8 Fast Fourier transform of quantum oscillations meas-ured in SrMnSb2at 2K. 49
Figure 5.9 The angular dependence of the main frequencies found in the quantum oscillations in SrMnSb2. 51
Figure 5.10 Quantum oscillations measured in SrMnSb2 at different
temperatures. 52
Figure 5.11 Frequencies of quantum oscillations measured in SrMnSb2
at different temperatures. 53
Figure 5.12 Lifshitz-Kosevich (LK) fit to the maximum amplitude of quantum oscillations measured in SrMnSb2 at different
temperatures. 54
Figure 5.13 The quantum oscillations (QO) in ρxxand ρxymeasured
on a sample from batch 20151022, and the oscillations in the conductivity σxx. 55
Figure 5.14 Fast Fourier transform of quantum oscillations meas-ured in SrMnSb2(batch 20151022) at 2K. 56
Figure 5.15 Landau level (LL) index fan diagrams for QO in the conductivity of SrMnSb2. 57
Figure 5.16 A LL index fan diagram for QO in ρxxof SrMnSb2. 59
Figure 6.1 Schematic of the three-step and one-step models de-scribing the photoemission process. 64
Figure 6.2 The energetics of the photoemission process. 65 Figure 6.3 The kinematics of the photoemission process within the
three-step model. 66
Figure 6.4 The geometry of a photoemission spectroscopy experi-ment. 67
Figure 6.5 A schematic of the hemispherical electron analyser used in photoemission measurements. 69
Figure 6.6 A photo of the Omicron sample plate used in angle-resolved photoemission spectroscopy (ARPES) measure-ments. 70
Figure 6.7 A bird’s eye view of the Diamond Light Source syn-chrotron. 71
Figure 6.8 A schematic of an undulator. 72
Figure 6.9 A schematic of the beamline layout of One-Cubed at the BESSY II synchrotron. 72
Figure 6.10 The beamline layout at Diamond Light Source I05. 74 Figure 7.1 The band structure of SrMnSb2, measured by ARPES. 75
Figure 7.2 Two Fermi surface maps of SrMnSb2 measured with
ARPES. 76
Figure 7.3 Constant energy cuts through the ARPES data shown in Figure 7.1. 78
Figure 7.4 Energy dispersion of the bands along Γ−M and Γ−Y, measured with ARPES. 79
Figure 7.5 Energy dispersion along X−Y measured with ARPES and a linear fit of the Y peak. 80
Figure 7.6 A kz-map of the energy bands along Γ−Y at the Fermi
level. 81
Figure 7.7 The energy dispersion alongΓ−Y before and after evap-orating potassium onto the surface. 82
Figure 7.8 Constant energy maps and a fit of the Y peak, indicat-ing the Fermi level shift due to deposition of potassium onto the surface of SrMnSb2. 82
Figure 8.1 The lattice structures used in the tight-binding model presented here. 86
Figure 8.2 A visual representation of the structure of the Hamilto-nian matrix. 86
Figure 8.3 A schematic of the px and py orbitals in the antimony
plane. 87
Figure 8.4 A schematic of the origin of intrinsic spin-orbit coup-ling. 90
Figure 8.5 A schematic of the origin of extrinsic spin-orbit coup-ling. 92
Figure 8.6 A schematic of how magnetism is incorporated in the tight-binding model. 92
Figure 9.1 Comparing tight-binding calculations for lattices a,b and cwith s and p orbitals. 96
Figure 9.2 The effects of intrinsic spin-orbit coupling and canted anti-ferromagnetism on the tight-binding band struc-ture. 97
Figure 9.3 The effects of anti-ferromagnetism (AFM) and ferromagnetism (FM) fields on the modelled band structure. 98
Figure 9.4 A comparison of two density-functional theory calcula-tions of the band structure of SrMnSb2. 99
Figure 9.5 The best found fit of the tight-binding model band struc-ture to a density-functional theory calculation. 101 Figure 9.6 The best found fits of the tight-binding model band
structure with Sr atoms included to a density-functional theory calculation. 102
Figure 9.7 Comparison of the density-functional theory (DFT) cal-culated band structure to the best tight-binding model (TBM) fits to the DFT and to ARPES data. 103 Figure 9.8 A comparison of constant energy contours measured
with ARPES and calculated with the TBM. 104 Figure 9.9 The fitted TBM bands partially overlaid on energy
dis-persions measured with ARPES. 105
Figure 9.10 Calculated constant energy contours about the Y point for several energies. 107
Figure 9.11 The Berry vector potential field for the Y pocket. 108 Figure 9.12 The Berry vector potential field around the Y point with
the points used to determine the Berry phase. 109
xii List of Tables
L I S T O F TA B L E S
Table 3.1 Atomic parameters of SrMnSb2 28
Table 9.1 Atomic parameters of SrMnSb2 used in the DFT
A C R O N Y M S
AFM anti-ferromagnetism
ARPES angle-resolved photoemission spectroscopy BZ Brillouin zone
DFT density-functional theory DOS density of states
EDC energy distribution curve EDM energy dispersion map FFT Fast Fourier transform FM ferromagnetism FS Fermi surface
GGA Generalised Gradient Approximation LDA Local Density Approximation
LK Lifshitz-Kosevich LL Landau level
MDC momentum distribution curve PES photoemission spectroscopy
PPMS Physical Property Measurement System®
QO quantum oscillations QSH quantum spin Hall RRR residual resistance ratio SdH Shubnikov-de Haas SOC spin-orbit coupling TBM tight-binding model
TB-MBJ Tran-Blaha Modified Becke-Johnson UHV ultra-high vacuum
XPS X-ray photoemission spectroscopy 2D two-dimensional
3D three-dimensional
Part I
1
I N T R O D U C T I O NA central goal of condensed matter physics is to characterise phases of matter. Many phases can be described by Lev Landau’s theory of phase transitions, in which phases are characterised by the symmetries they break [1–3]. However, in recent decades it has become apparent that there are phases that cannot be distinguished by symmetry breaking, and so require a different form of classification. The study of materials hosting different quantum Hall phases led to the notion of topological order [3, 4]. Topology is the mathematical study of the (qualitative) properties of objects and spaces which remain constant under continuous deformations. Materials with a non-trivial topological order have certain fundamental properties − such as the quantisation of the Hall conductance and the number of gapless boundary modes − that are robust under any change in the Hamiltonian that preserves all symmetries and does not open or close any band gaps [3]. This is interesting conceptually, but it can also lead to properties that may have useful future applications, such as in spintronics and quantum computation [3, 5, 6].
Perhaps the most famous example of a topological system is that of graphene, a single layer of carbon atoms arranged in a honeycomb lattice. This material came to fame in 2005 with a break-through experimental study by Novoselov et al. in which they show that its conduction electrons behave according to the Dirac equation [7]. Graphene presented itself as an easily accessible plat-form on which to test basic principles of relativity, because its electrons mimic relativistic massless particles with an effective "speed of light" c∗ ≈ 106 m/s [7]. The interesting linearly dispersing electronic band crossings (so-called two-dimensional (2D) Dirac cones) of this material were already noted by P.R. Wal-lace in a theoretical study in 1947 [8], but it was long thought that such a 2D material would not be stable, making its isolation by K. Novoselov and A. Geim all the more remarkable. Although initially hailed as a 2D Dirac semi-metal, spin-orbit coupling (SOC) actually leads to the formation of a gap at the Dirac node, formally making it a quantum spin Hall (QSH) insulator [9]. The SOC acts as an effective "magnetic field" that is opposite for spin up and spin down electrons, resulting in spin-polarised currents running in opposite dir-ections [10]. This effect is similar to the normal quantum Hall effect that may occur in the presence of large (actual) magnetic fields, where the transverse conductance of a material is quantised. The link to topology arises from the fact that this QSH phase is topological in nature, characterised by a topological invariant [11, 12].
The discovery of the topological nature of graphene was soon followed up by the discovery and subsequent experimental confirmation that SOC in other materials can lead to what is called a topological insulator. This insulating phase is characterised by the switching of the parity of the valence and con-duction bands (band inversion) at some points in the bulk band structure, as
4 i n t r o d u c t i o n
is shown on the right-hand side of Figure 1.1. At the edge or surface of such a system, where it meets the topologically trivial (insulating) vacuum lacking this band inversion, all bands must match up according to their parity. This necessarily leads to a band crossing and thus enforces conducting edge and surface modes, protected by topology [3]. More generally, a topological phase transition is always accompanied by the opening or closing of a band gap.
Figure 1.1: A schematic diagram of the band structures of trivial and topological insu-lators, and that of a Dirac semimetal. The band colours and labels indicate their parity. Image: [13]
The natural question arose whether it is possible for such symmetry-protected band crossings to exist within the bulk of a single crystal. This would be what we call a topological Dirac semimetal, hosting Dirac cones as were thought to be present in graphene [13]. (Note that to first order − so long as a first-order term is allowed by the symmetry of the Hamiltonian−any crossing of bands is linear.) In 2013 and 2014, Cd3As2 and Na3Bi were discovered to host
three-dimensional (3D) Dirac cones in their bulk band structure [13–16]. Un-like 2D Dirac cones, which do not disperse in kz, 3D Dirac cones have a node
in all three dimensions (kx, ky, kz). Having shown these to exist, the question
remained whether there exist systems that host 2D Dirac cones that, unlike in graphene, are protected from spin-orbit gapping. Such systems would exhibit behaviour distinct from both graphene and 3D Dirac semimetals, making them fundamentally interesting to investigate [17].
In 2015, S.M. Young and C.L. Kane proposed that systems will host precisely such 2D Dirac crossings in the presence of a three particular types of symmetry [17]. What these symmetries are and why they lead to protected crossings will be explained in detail in Chapter 2. The primary aim of this project was to test this prediction in a specific compound: SrMnSb2. Experimentally, we use
trans-port measurements and angle-resolved photoemission spectroscopy (ARPES) to determine whether SrMnSb2 could be the 2D Dirac semimetal we’ve been
looking for. This is accompanied by a theoretical investigation using a tight-binding model, which is qualitatively compared to the model presented by Young and Kane, and fitted to density functional theory (DFT) calculations and ARPES data.
i n t r o d u c t i o n 5
Before delving into the research done in this project, the remaining two chapters in Part I introduce the relevant theoretical principles and the candid-ate 2D Dirac semimetal mcandid-aterial studied. Part II discusses the electrical trans-port measurements performed in this material. Part III then presents work done on understanding the band structure of SrMnSb2, via ARPES and the
tight-binding model. Finally, Part IV will conclude with an overarching discus-sion and suggestions for future research.
2
T H E O R E T I C A L P R I N C I P L E SThis chapter introduces the theoretical principles necessary for understand-ing our study of the material SrMnSb2. Section 2.1 will start from the very
beginning of the story, with the quantum mechanics of electrons bound to an atomic core. Bringing many atoms together can lead to the formation of a crystal, in which atomic (electron) energy levels will become energy bands. Crystal structures, and how to use symmetries of a crystal system to under-stand its electronic behaviour, are explained in Section 2.2. It is useful to be able to model these energy bands, for which two methods are described in Section 2.3. Section 2.4 will then introduce the concept of a Berry phase in the context of (topological) band structures. And, last but not least, Section 2.5 will use the concepts introduced in the other sections to explain Young and Kane’s reasoning behind their proposed symmetry protected 2D Dirac points, which was the starting point of this project.
2.1 at o m i c o r b i ta l s
To determine whether SrMnSb2 hosts 2D Dirac fermions, we need to
invest-igate its electronic properties. The conduction of any crystal is dominated by the valence electrons of the atoms in the lattice, i.e. the electrons least strongly bound to the atomic cores. To be able to understand conduction, it is important to first understand how electrons behave when bound to an atomic nucleus, and then how bringing atoms close together can allow for electrons to move from one atom to the next.
In an isolated atom, we describe the wavefunctions of bound electrons by what we call atomic orbitals: the solutions of the Schrödinger equation for the (hydrogen) atom, in which the atomic nucleus acts as a (Coulomb) potential well. These wavefunctions can be described mathematically (in spherical co-ordinates) by:
|ψnlm(r, φ, θ)i = |Rnl(r)i|Ylm(φ, θ)i. (2.1)
Here |Rnl(r)iis the radial distribution function, which determines the size of
the atomic orbital, and|Ylm(φ, θ)iis the spherical harmonic, which determines
the shape of the orbital. n is the principal quantum number, l is the azimuthal quantum number, and m the magnetic quantum number, which are all integer-valued. By convention, atomic orbitals are labelled by s for l=0, p for l =1, d for l =2 and f for l=3. For each value of l, there are 2l+1 possible values of m. As n increases,|Rnligets an increasing number of radial nodes (non-zero
values of r for which|Rnli =0), but the shape (|Ylmi) stays the same.
When explicitly writing out the expressions for the p orbitals
|ψn11(r, φ, θ)iand|ψn1−1(r, φ, θ)i, it becomes clear that these are not real-valued,
due to the dependence of |Y1±1i on e±iφ [18]. Fortunately, |ψn11iand |ψn1−1i
8 t h e o r e t i c a l p r i n c i p l e s
are both solutions to the Schrödinger equation with the same energy En, such
that we can take any linear combination of these and still have a solution with the same energy. To ensure that we obtain real orbitals, we define the following:
|ψnpxi = 1 √ 2(|ψn1−1i − |ψn11i), |ψnpyi = i √ 2(|ψn1−1i + |ψn11i), |ψnpzi = |ψn10i. (2.2)
When atoms are brought close together, their atomic orbitals can overlap. Valence electrons are relatively loosely bound to the atomic nuclei, and can thus transfer from one atom to the next. The larger the overlap between neigh-bouring atomic orbitals, the lower the energy cost of doing so. Orbitals are not all isotropic, so their ’shape’ is relevant for understanding how these electrons can move within a lattice. s orbitals are spherical (and thus isotropic), whereas p orbitals are aligned along one of three orthogonal axes (x, y, z) with a nodal plane perpendicular to the axis along which the orbital is oriented (centred at the origin (0, 0, 0)). For larger values of n, more radial nodes appear. To visual-ise what these atomic orbitals look like, we can plot the boundary surface, i.e. a surface of constant probability density (|ψnlm|2) (see Figure 2.1). For SrMnSb2,
it is relevant to note that the valence electrons of Sr reside in the 5s orbital, and those of Sb in the 5p orbitals.
Figure 2.1: The surface of constant probability density (|ψnlm|2) of the s and 2p orbitals.
The colours represent the sign of the phase. Image made using [19]. When neighbouring atomic orbitals overlap, they form a chemical bond. s orbitals only form one type of bond, called a σ bond. However, p orbitals can form two types of bonding states: σ and π. The former describes bonding along the orbital axis, and the latter describes an overlap parallel to the or-bital axis (see Figure 2.2). The sign of the overlapping wavefunctions affects the type of bond formed between them: if the signs are the same, a bonding state is formed; if they’re different, an anti-bonding state forms, in which the total wavefunction of the state has a node between the two atoms. Terms in a Hamiltonian describing such bonding states acquire a negative or positive sign, respectively. The orientation and ’shape’ of the atomic orbitals present thus play an important role in the bonding states formed.
2.2 crystals and symmetries 9
Figure 2.2: Types of orbital overlaps possible for s and p orbitals. The colours indicate the sign of the phase of the wavefunction.
2.2 c r y s ta l s a n d s y m m e t r i e s
Although so far we have discussed only bonds formed between pairs of atoms, the same physics holds for larger groups of atoms brought together. The (valence) atomic orbitals can form chemical bonds, which can lead to the form-ation of a periodic array of atoms: a crystal. (The types of orbitals involved affect the type of crystal structure formed, especially regarding its rotational symmetry.) In a crystal lattice, valence electron wavefunctions are no longer localised around specific atomic cores, but the system as a whole acquires col-lective energy eigenstates, forming what is called a band structure. At this point, it becomes more effective to describe the system on a more macroscopic level than that of individual atoms and electrons. In the current project, we study a crystalline compound, so it is important to understand how the crys-tal structure affects electronic behaviour, and what the role of symmetries is in this context.
Mathematically, we can describe a crystal structure in terms of a Bravais lattice: an (infinite) array of discrete points with position vectors R, where (in three dimensions)
R=n1a1+n2a2+n3a3 (2.3)
Here ni are integers and aiare the primitive lattice vectors [18]. At the location
of each point there can be a single atom, groups of atoms, molecules, ions, and so forth. Bravais lattices can be characterised by the set of operations that take the lattice onto itself. These are called symmetry operations, and a set of these form a space group. If a lattice has a symmetry, any Hamiltonian describing the energy of the system must commute with the symmetry operator. There are 230 distinct space groups, and 14 distinct Bravais lattices, which we can use to classify a crystal structure.
The space group of a crystal consists of three types of symmetries:
1. Translation symmetry t: shifting the entire crystal lattice by one (or mul-tiple) unit cells in any direction.
2. Symmorphic symmetries g, also known as point group symmetries: symmet-ries that leave at least one point in a fixed location. This includes rota-tions, reflecrota-tions, and inversions.
10 t h e o r e t i c a l p r i n c i p l e s
3. Non-symmorphic symmetries {g|t}: combinations of point group symmet-ries and a fractional translation.
Figure 2.3 shows examples of (a) symmorphic and (b) non-symmorphic sym-metries present in a simple square lattice, buckled in the out-of-plane direction such that it is made up of two identical sublattices (black and grey), one lying in a plane slightly above the other.
(a) Symmorphic symmetries (b) Non-symmorphic symmetries
Figure 2.3: Examples of symmetries in a crinkled square lattice. The circles represent atoms, of which the grey ones lie in a plane slightly above the black atomic plane. Each unit cell (yellow) contains one grey and one black atom. Sym-metries shown are: rotation (blue); reflection (red); inversion (green); and screw axes{C2 ˆx|120}(light green) and{C2 ˆy|012}(orange).
For the arguments presented by Young and Kane, explained in Section 2.5, two types of non-symmorphic symmetries are relevant. The first is a screw axis, an example of which is (in two dimensions) {C2 ˆx|120}. Here g = C2 ˆx
represents a two-fold rotation about the x-axis, and t = (12 0) a translation of half a unit cell in the x-direction, and no translation along the y-direction. When, as depicted in Figure 2.3b, screw axes exist along both x and y, there will also exist a glide mirror plane:{Mˆz|1212}, where g= Mˆzrepresents a reflection
in the xy plane, in this case halfway between the black and the grey atomic planes.
Just as in real space the crystal lattice can be described by Equation 2.3, by taking the Fourier transform one can construct an equivalent lattice in recip-rocal space (also known as k-space):
G= q1b1+q2b2+q3b3. (2.4)
Here qi are integers and bi the primitive reciprocal lattice vectors. The
primit-ive reciprocal lattice vectors bi are defined such that
bi·aj =2πδij, where δijis the Kronecker delta. G is the reciprocal lattice vector,
defined such that the plane wave eiG·r has the same periodicity as the lattice, where r is the position vector. This condition requires that eiG·(R+r) =eiG·r, or
eiG·R=1. (2.5)
While a crystal lattice in real space is periodic with R −which defines a unit cell−in k-space it is thus periodic in G. One repeating unit in reciprocal space is called a Brillouin zone (BZ). By convention, the so-called first BZ spans from
−G
2.2 crystals and symmetries 11
According to Bloch’s theorem, energy eigenstates for an electron in a lattice can be described by Bloch waves:
|ψnk(r)i =eik·r|unk(r)i, (2.6)
where |unk(r)iis a periodic function with the same periodicity as the lattice,
n is the band index, and k is the crystal momentum [3, 20]. An equivalent formulation of Bloch’s theorem is
|ψnk(r+R)i =eik·r|ψnk(r)i. (2.7)
The Bloch states|ψnk(r)iare eigenstates of the Bloch HamiltonianH(k). (Bloch
Hamiltonians are essentially Hamiltonians with translation symmetry.) The corresponding eigenvalues define n energy bands (En(k)) that collectively
form the band structure [3].
In contrast to the real space lattice, symmetries in reciprocal space are not global: the operation of each symmetry of the lattice will leave only certain points or lines in the BZ invariant. As an example, let us take a system with a mirror symmetry in the yz-plane (g = Mˆx), as shown by the red line in
Figure 2.3a. This operator puts x→ −x, or in reciprocal space kx → −kx. This
operation leaves the system invariant only where kx =0, or at the BZ boundary
kx= ±aπ1 (the points G2 and−G2 are equivalent). This particular symmetry thus
has invariant lines in the BZ at these values of kx, but the symmetry does not
hold for other points in k-space. In this respect, the Γ point (k= 0) is unique because it is invariant under any point group symmetry operation.
Another relevant symmetry − one which falls outside of the space group classification of lattice structures−is time-reversal symmetry (T), which reverses the arrow of time. In general, T can be represented as
T=UK, (2.8)
where U is a unitary matrix− i.e. its Hermitian conjugate (U† = (U∗)T) is its inverse, UU† =U†U= I where I is the identity matrix−and K is the operator
of complex conjugation. Letting T act on a state twice must give us the state we started with, up to a phase factor. This gives us the following:
T2 =UKUK=UU∗ =φ, (2.9)
with φ being a diagonal matrix of phases. The transpose of a diagonal matrix is the matrix itself, such that φT = φ. Using this in combination with the
properties of a unitary matrix, we can manipulate Equation 2.9 to obtain [21]
U= φUT, UT =Uφ, (2.10)
which gives us
U= φUT = φUφ. (2.11)
This can only be true for φ = ±1, such that T2 = ±1. Spinless particles have φ=1, such that T2=1, and for spin-12 particles T2 = −1 [21].
12 t h e o r e t i c a l p r i n c i p l e s
Since the position operator ( ˆx) is independent of time, it commutes with T. However, the momentum operator, which is proportional to ∂tˆx, flips its sign
when acted upon by T. This tells us
TˆxT−1= ˆx, TˆpT−1 = −ˆp. (2.12) As already mentioned, T is proportional to complex conjugation, which can also be deduced from the following:
T[ˆx, ˆp]T−1 =Ti¯hT−1 = −i¯h= −[ˆx, ˆp]. (2.13) Relevant for crystal lattices is the effect of T on crystal momentum k and electron spin σ:
T : k→ −k σ→ −σ
An example of a T-invariant point in k-space is thus at k = 0, because there
k = −k. Importantly, time-reversal invariant systems containing 12-integer spin particles (such as electrons) obey Kramer’s theorem: all eigenstates of the Hamiltonian are at least two-fold degenerate at T-invariant points in reciprocal space [21]. In terms of the band structure, Kramer’s theorem thus enforces band degeneracies at, for instance k = 0, while bands away from this point are split.
2.3 m o d e l l i n g c r y s ta l l i n e m at e r i a l s
To be able to understand the electronic behaviour of a material, it is useful to be able to model its electronic band structure. Two common methods for doing this are a tight-binding model (TBM) and a numerical density-functional theory (DFT) calculation, which are both used in this research project. The first is based on the tight-binding approximation, which assumes that the overlap of atomic orbitals is enough for electrons to be able to ’hop’ from one atom to the next, but not so much that the atomic description becomes irrelevant [18]. The latter assumes the electrons are delocalised, the atomic nuclei being mostly shielded by the core electrons and so leaving only a weak periodic potential, and numerically minimises the energy as a functional of the electron density n. In the following two subsections I describe these two methods in more detail. 2.3.1 Tight-binding model
Originally put forward by Bloch [20], the tight-binding approximation assumes the wavefunctions of electrons in a lattice are essentially localised in orbitals around the atomic nuclei by the Coulomb potential. The movement of electrons throughout the lattice is possible through the overlapping of the orbitals with those of neighbouring atoms. To describe this, we make a linear combination of atomic orbitals located at the atomic positions, with coefficients being given by the plane wave eik·R at the Bravais lattice points R [22].
As an example, let us take an infinitely long one dimensional chain of N identical atoms, each having a valence s orbital (see Figure 2.4). A Bravais lattice in one dimension is simply R= ma, where m is an integer and a is the
2.3 modelling crystalline materials 13
Figure 2.4: A one-dimensional solid with a lattice parameter a. The yellow circles show the surface constant probability of ’finding’ an electron for valence s or-bitals, which overlap slightly. We describe the behaviour of an electron initially localised at position x=R0.
atomic spacing. For an atom whose electrons are so tightly bound to the core they do not interact with electrons of other atoms,
Hat(R)|φn(x−R)i =Eat,n|φn(x−R)i, (2.14)
where |φniis the n-th atomic orbital and En the associated energy [18]. Since
we only consider the valence s orbitals of the atoms in our one-dimensional solid, we can drop the index n for now. The Hamiltonian of the full crystal lattice can − in the vicinity of each lattice point − be approximated by the atomic HamiltonianHat. This holds for all|φ(x−R)iso long as x−R is small
compared to a. Further away from the lattice positions, we must account for the possible overlapping of |φ(x−R)iwith the neighbouring atomic orbitals,
as well as the influence of the Coulomb potential from the neighbouring atoms on |φ(x−R)i. We can view the periodic lattice potential as being made up of
the sum of the potential from the atom of the site we are at (Vat(x−R0)) and
the potential from all other atoms (∆V(x−R) =∑R6=R0Vat(x−R)). Filling this
in to the Schrödinger equation, we obtain:
H|ψ(x)i = (Hat(R0) +∆V(x−R)) |ψ(x)i = E(k)|ψ(x)i. (2.15)
The wavefunction for an electron in this system (|ψ(x)i) should satisfy Bloch’s
condition (Equation 2.6), but close to each atom it should again resemble the atomic wavefunction |φ(x−R)i. These conditions are satisfied if we take
|ψk(x)i =
1
√
N
∑
ReikR|φ(x−R)i. (2.16)
Our end goal is to calculate the energy dispersions E(k). (Since k-space is discrete, this is technically not a (continuous) function of momentum k, but we will treat it as such nonetheless.) To do so, we first multiply Schrödinger’s equation (Equation 2.15) on the left hand side byhφ(x−R0)|(terms rearranged
for simplicity):
(E(k) −Eat)hφ(x−R0)|ψk(x)i = hφ(x−R0)|∆V(x−R0)|ψk(x)i, (2.17)
where the terms are rearranged for simplicity. We can now evaluate the two integrals separately. On the left-hand side, the inner product will be dominated by one term in the expression for|ψk(x)i:
hφ(x−R0)|ψk(x)i ≈ √1
Ne
14 t h e o r e t i c a l p r i n c i p l e s
On the right-hand side, the integral can be divided into two terms of compar-able magnitude: hφ(x−R0)|∆V(x−R0)|ψk(x)i = 1 √ Ne ikR0hφ(x−R 0)|∆V(x−R0)|φ(x−R0)i +
∑
R6=R0 1 √ Ne ikR0h φ(x−R0)|∆V(x−R0)|φ(x−R)i. (2.19)Substituting in x0 = x−R0, Equation 2.17 can now be rewritten as
E(k) =Eat+ hφ(x0)|∆V(x0)|φ(x0)i
+
∑
R6=0
hφ(x0)|∆V(x0)|φ(x0−R)i. (2.20)
The two integrals in this expression each have their own physical meaning. The first,
∆V ≡ −hφ(x)|∆V(x−a)|φ(x)i, (2.21)
is the energy correction to Eatfor atoms at each lattice site from the Coulomb
potential of its neighbouring atoms. The second, t≡
∑
R6=0
hφ(x)|∆V(x)|φ(x−R)i, (2.22)
is the so-called hopping amplitude, taken by convention to be a positive num-ber. The size of the hopping amplitude is determined by the size of the overlaps of neighbouring orbitals, and so it parameterises the likelihood of hopping oc-curring between orbitals |φ(x−a)i and |φ(x)i. Note that the phases of the
atomic orbitals involved determine whether the resulting there is a+ or −in front of t: if two overlapping orbitals have the same phase, they form a bond-ing state, associated with−t; if the two phases have opposite signs, the overlap forms an anti-bonding state, which gives+t. Essentially, the sign in front of t determines whether the electron wavefunction|ψk(x)iis even or odd.
We can simplify what we have so far by making the approximation that the atomic orbitals at each lattice site will only be affected by its nearest neigh-bours (R= ±a). This assumes that the orbital overlap and the influence of the Coulomb potentials of atoms further away are so small that they are negligible. The more neighbours that are included, the more accurate a TBM becomes, al-though usually including only (next-)nearest neighbours is sufficiently accur-ate. Considering only nearest neighbour hopping, we can simplify the above expression to
E(k) =Eat−∆V−t(eika+e−ika)
=Eat−∆V−2t cos(ka).
(2.23)
Plotting our expression for E(k) (Equation 2.23), having provided values for the chemical potential µ = Eat−∆V and t, we are plotting a band structure.
2.3 modelling crystalline materials 15
Figure 2.5: An example of three bands modelled for a simple one-dimensional solid. The yellow lines indicate the Brillouin zone (BZ).
above is shown in Figure 2.5, including three bands. (This entails repeating the same procedure as described above for the n atomic orbitals considered at each lattice site.)
The description above for a very simple one-dimensional solid can be gen-eralised to higher-dimensional structures by taking R → R and k → k, and considering all possible hopping amplitudes: each type of hop will have a dif-ferent associated amplitude, dependent on the two orbitals involved and the real-space vector connecting the atomic sites they are centred at. The atomic orbitals will then be described by φn(r−Ri)for an atom at the vector position
Ri, and Equation 2.16 will be replaced by (here not normalised) [22]:
|ψk(r)i =
∑
Rieik·Ri|φ
n(r−Ri)i, (2.24)
where the sum extends over the atoms at equivalent positions in all unit cells. These so-called Bloch sums can be determined for each atomic orbital of an atom, for each distinct atom in a unit cell. Note that these Bloch sums will not necessarily be orthogonal to each other, because the atomic orbitals |φni
located on different atoms are not orthogonal. This difficulty can be avoided by defining new atomic orbitals |ϕnifrom linear combinations of the original
|φniwhich are orthogonal [22]. When this is done, Equation 2.24 is normalised
by a factor √1
N where N is the number of unit cells along Ri which defines the
periodic boundary conditions.
The tight-binding Hamiltonian H is a matrix of hopping terms from all the types of atom and their relevant orbitals to all other possible orbitals, so the simpler the starting lattice, the easier it is to compute the eigenvalues. If one includes spin, there are twice as many possible electronic states, so the Hamiltonian matrix grows even larger. When setting up a TBM, a finite set of relevant atomic orbitals are chosen to be included. The matrix components between two Bloch sums can be denoted
∑
Rj
eik·(Rj−Ri)hϕ
16 t h e o r e t i c a l p r i n c i p l e s
where Rj−Ri is the vector displacement from the atom with orbital |ϕni to
the atom with orbital |ϕmi. Remember that each |ϕni can be made up of a
linear combination of |φni. The integral in the equation above becomes a
lin-ear combination of integralshφn(r−Ri)|∆V(Rk)|φm(r−Rj)i, which are called
three-centre integrals because their three components are each centred at a different position [22]. The evaluation of such an integral can be extremely difficult.
There are of course ’two-centre’ integral terms considering the Coulomb po-tential at the same lattice site as one of the atomic wavefunctions (e.g. where
Rk = Ri), which are much easier to compute. The three-centre integrals are
smaller than the two-centre integrals because they are interactions between three spatially separated functions rather than just two, so we choose to neg-lect them. This allows us to evaluate the possible integrals and to get general expressions for the atomic orbital overlaps for different types of orbitals, de-pendent on the normalised vector components of Rj−Ri. This was computed
and presented by Slater and Koster for all possible two-centre integrals in [22]. The hopping amplitudes tij calculated for all possible orbital overlaps in
a crystal can thus be rewritten in terms of these energy integrals, which in combination with symmetries in the crystal can greatly reduce the number of hopping parameters in the TBM.
In addition to the hopping terms in a tight-binding Hamiltonian, effects such as spin-orbit coupling (SOC) and magnetism can also be included, simply by determining their contributing matrix elements and adding these separately. These concepts and how they are included in our own model will be explained in detail in Chapter 8.
As a final note, the model described so far was a single-particle model, scribing the behaviour of a single electron. A many-particle system can be de-scribed by the same tight-binding Hamiltonian if it is rewritten in the ’second quantisation’ form using the creation(annihilation) operators c†(c) to effectively
create(annihilate) electron states in the system. In a one-dimensional atomic chain as described above, the Hamiltonian describing the electron hopping is then given by
H = −t
∑
R
c†(R)c(R+a) +c†(R+a)c(R). (2.26) This expression can be Fourier transformed using the convention
c†(x) → √1 N
∑
k e −ikxc† k, c(x) → √1 N∑
k e +ikxc k, (2.27) giving H = − t N∑
R∑
kk0 e−i(k−k0)Rheik0a+e−ikaic†kck0 . (2.28) Using N1 ∑Re−i(k−k 0)R=δkk0, this can be simplified to H = −t
∑
k h eika+e−ikaic†kck = −2t∑
k cos(ka)c†kck. (2.29)2.3 modelling crystalline materials 17
In this formalism, a general tight-binding Hamiltonian can be denoted by
H = −
∑
i,j tijfij c†icj+h.c. −∑
i µici†ci+..., (2.30)where i and j denote atomic orbitals at different atomic sites, tij is the hopping
amplitude between these states, fij is a hopping prefactor (such as the cosine
in Equation 2.29), and µi is the chemical potential of filling electron state i.
2.3.2 Density functional theory
In contrast to TBMs− which simplify the modelled system such that one can extract exactly how each modelling parameter and symmetry present in the system affects the band structure − DFT calculations minimise the energy of the entire system, such that the output includes bands from all atomic orbitals present in the system rather than just a select few. Because they have fewer free parameters, DFT calculations are also usually able to provide relatively reliable magnitudes of the energy dispersions. In this section, we explain the basic principles behind this modelling method.
As a starting point, DFT calculations treat the electrons in a lattice as a nearly-free electron gas in a periodic potential. The time-independent Schrödinger equation for an N-electron stationary state in a lattice reads
Hψ=Tˆ +Vˆ +Uˆψ = " N
∑
i − ¯h 2 2mi ∇2i ! + N∑
i V(ri) + N∑
i<j U(ri, ri) # =Eψ, (2.31)with a normalised wavefunction ψ(r, r2, ..., rN)[23]. Here ˆT is the kinetic energy,
ˆ
V is an external potential energy (in this case the periodic Coulomb potential from the atomic nuclei), and ˆU is the electron-electron interaction energy. Of these three, ˆT and ˆU are universal, whereas ˆV depends on the lattice. The interaction term ˆU makes this expression not separable into a linear combina-tion of single-particle equacombina-tions. To solve this problem, density-funccombina-tional the-ory (DFT) calculations map the many-body problem with ˆU onto a single-body problem without ˆU. The key to doing this, as was first described by Hohenberg and Kohn in 1964 [23], is to look at the electron density n(r):
n(r) =N Z
d3r2...
Z
d3rNψ∗(r, r2, ..., rN)ψ(r, r2, ..., rN). (2.32)
Using this expression, it is possible to−for a given ground-state electron dens-ity n0(r) −calculate the corresponding ground-state wavefunction ψ0(r, r2, ..., rN).
This entails that ψ is a unique functional of n0 [23]:
ψ0= ψ[n0]. (2.33)
This, in turn, means that any ground-state observable is a functional of n0,
which includes the ground-state energy E0:
18 t h e o r e t i c a l p r i n c i p l e s
The contribution of the external potential potentialhψ0|Vˆ|ψ0ican explicitly be
written in terms of n0, or more generally, the contribution ofhψ|Vˆ|ψiis given
by
V[n] = Z
V(r)n(r)d3r. (2.35)
For a given external potential ˆV, the problem then reduces to minimising the energy
E[n] =T[n] +U[n] + Z
V(r)n(r)d3r. (2.36) In 1965, Kohn and Sham derived a method for minimising this energy func-tional using the Lagrangian method of undetermined multipliers, applicable to systems with a slowly varying or high charge carrier density n [24]. Put simply, they define a system without the electron-electron interaction term U[n], such that it reduces to a solvable single-electron system which has the same energy density as the original many-body system:
Es[n] = hψs[n]|Tˆ +Vˆs|ψs[n]i, (2.37)
where ˆT is again the kinetic energy of the electrons and ˆVs is the effective
external potential in which the electrons are moving such that ns(r) ≡n(r). To
now find the many-body electron density, one needs to solve the equation " − ¯h 2 2mi ∇2+Vs(r) # φ∗i(r) =eiφ∗(r) (2.38)
for the effective orbitals φi∗, where [24] n(r) ≡ns(r) =
∑
i
|φ∗(r)|2. (2.39)
Vs includes the electron-electron Coulomb repulsion as well as so-called
exchange-correlation potential, which include all many-particle correlation ef-fects [24]. There are different variations of the Kohn-Sham DFT, which depend on the way that Vs is approximated. Different approximations give different
results in a DFT calculation, so it is relevant to note which method was used. The simplest approximation for Vsis the so-called Local Density
Approxim-ation (LDA). It approximates the exchange-correlApproxim-ation potential as ExcLDA =
Z
n(r)exc(n↑(r), n↓(r))d3r (2.40)
where exc(n↑+n↓)is the exchange-correlation energy per particle of an
elec-tron gas with uniform spin densities n↑ and n↓, and n = n↑+n↓ [25]. This is
valid for systems where the spin densities vary slowly with r, which is a con-dition not satisfied in most atoms, molecules and crystal structures. A more accurate approximation is the Generalised Gradient Approximation (GGA), which considers the gradient of the electron density, i.e.
ExcGGA = Z
2.4 the berry phase 19
There are several variations of GGA, which differ in their parameterisations of the functional. A commonly used version of it is the Perdew, Burke and Ernzerhof (PBE) functional, in which all parameters are fundamental constants [25, 26]. One weakness of GGA is that it is not reliable in correctly estimating band gaps. To improve DFT’s predictions in this respect, Tran and Blaha [26] adapted a potential presented earlier by Becke and Johnson [27], resulting in the so-called Tran-Blaha Modified Becke-Johnson (TB-MBJ) potential.
To conclude, both DFT calculations and TBMs have advantages and disad-vantages. On the one hand, a TBM can provide good qualitative insight into why bands behave the way they do. However, its inherent parameter freedom means that wildly different band structures can be calculated from a single model, and there are no built-in reality checks. On top of this, it quickly be-comes tedious to model more than just a few bands, because modelling n bands requires finding the eigenvalues of a Hamiltonian that effectively forms an n×n matrix. On the other hand, DFT calculations are able to model all bands present in the system. For systems where electron correlation effects are weak, such that the approximation for the exchange correlation potential is good enough, the modelled bands are generally good quantitative agreement with the experimentally measured band structure. A downside of DFT calcu-lations is that they are a ’black box’ in the sense that they are not able to give specific insight to why the band structure looks as it does. These two methods are most powerful when they are combined: quantitative information can then be derived from the DFT calculation, while the TBM can be fitted to this and provide extra qualitative insight.
2.4 t h e b e r r y p h a s e
As was introduced in Chapter 1, condensed matter physicists are currently interested in understanding and characterising topological phases of matter. In this research project, we wish to determine whether the material we studied
− SrMnSb2 − is a topological 2D Dirac semimetal or not, as this would be a
realisation of a new topological phase of matter with unique and potentially useful properties. To determine whether a band structure is topological at all, we can use the concept of the geometrical Berry phase, presented by M.V. Berry in 1984 [28]. This phase was not initially described in the context of Bloch elec-trons, but more generally: particles travelling adiabatically could in principle pick up an extra phase factor on top of the dynamical phase. Joshua Zak later applied this concept to Bloch-periodic systems, where the Bloch momenta (k) are varied in closed loops (i.e. bands or Fermi surfaces) [21]. The following derivation is based on [21].
Consider a time-varying HamiltonianH(R), which depends on time through several parameters labelled by the vector R(t). Since we are interested in an adiabatically evolving system, the parameters R(t)are varied slowly. We can then introduce an orthonormal basis of instantaneous eigenstates |n(R)i of
H(R)by diagonalising the Hamiltonian at each point R:
20 t h e o r e t i c a l p r i n c i p l e s
We assume the basis function|n(R)iis normalised. This equation determines
|n(R)i up to a phase γ, which may depend on R. We can fix this phase by choosing a gauge. For convenience, we can pick a gauge where γ is smooth and single valued along the chosen closed path C in parameter space. This is not always possible for a full pathC, although it is then possible to divide the path into several slightly overlapping segments in which the phase is defined smoothly.
We want to analyse the phase of the wavefunction of a system prepared in an initial pure state |n(R(0))i as we slowly move R(t) along path C. A system starting in a pure eigenstate will evolve with H(R) and hence stay an eigenstate of the Hamiltonian. Throughout the adiabatic evolution of the system with the Hamiltonian H(R), the phase γ(t) of the state |ψ(t)i =
e−iγ(t)|n(R(t))i need not be zero. The time evolution of the system is given by
H(R(t))|ψ(t)i =i¯hd
dt|ψ(t)i. (2.43)
This can be translated to the differential equation En(R(t))|n(R(t))i =¯h d dtγ(t) |n(R(t))i +i¯hd dt|n(R(t))i. (2.44) If we then take the scalar product of this expression withhn(R(t))|, and move the far-right term to the left-hand side of the equation, we have
En(R(t)) −i¯hhn(R(t))| d dt|n(R(t))i = ¯h d dtγ(t) . (2.45)
Solving this for γ(t):
γ(t) = 1 ¯h t Z 0 En(R(t0))dt0−i t Z 0 hn(R(t))|d dt|n(R(t))idt 0. (2.46)
The first term is the dynamical phase, which is a phase factor dependent on the eigenenergies of the wavefunction, i.e. it is dependent on the Hamiltonian of which the wavefunctions are eigenfunctions. This factor is called dynamical because it depends on how (quickly) the chosen path is traversed. The negative of the second term is called the geometrical Berry phase (γB), which arises
from the fact that the states at t and t+dt are not the same:
γB(t) =i t Z 0 hn(R(t))|d dt|n(R(t))idt 0 =iZ C hn(R)|∇R|n(R)idR, (2.47) where in the last step the time dependence is contained in R(t). In contrast to the dynamical phase, the Berry phase is geometrical in the sense that it is not dependent on the dynamics of the path (i.e. how it is traversed), but only on the path itself.
Despite the imaginary sign in front of the Berry phase, it is real-valued. This is becausehn(R)|∇R|n(R)iis itself imaginary. To understand this, note that the
2.5 young and kane and protected crossings 21
wavefunctions of the Hamiltonian are normalised such that hn(R)|n(R)i = 1. If we now take the gradient of both sides of this inner product and simplify the outcome, we obtain
hn(R)|∇R|n(R)i = −hn(R)|∇R|n(R)i∗ ∈C. (2.48) The Berry phase can thus also be written as
γB(t) = Im
Z
C
hn(R)|∇R|n(R)idR, (2.49) In analogy with electron transport in an electromagnetic field, we can also define a vector function called the Berry connection, or Berry vector potential:
AB(R) =ihn(R)|∇R|n(R)i, γB =
Z
C
dR·AB(R). (2.50)
2.5 y o u n g a n d k a n e a n d p r o t e c t e d c r o s s i n g s
In 2015, Steve Young and Charles Kane published a paper in which they ar-gued that 2D materials that have a non-symmorphic symmetry, inversion, and time-reversal symmetry will have Dirac nodes that are protected from SOC-induced gapping, making such a material an ideal candidate for being a to-pological 2D Dirac semimetal [17]. As was already mentioned in Chapter 1, graphene was originally hailed for being such a material, and some of the properties associated with 2D Dirac semimetals were detected in graphene: electrons inside it can be described relativistically, with a Fermi velocity of around 106 m/s acting as the ’speed of light’, accompanied by an extremely high electron mobility and low effective electron mass [7]. However, SOC ac-tually introduces a small gap at the Dirac node in graphene, such that it is formally a quantum spin Hall insulator [12]. Aside from this, the fact that graphene is only one atom thick means its electronic properties are very sens-itive to external factors such as the substrate on which it lies and adsorbants on its surface. Young and Kane’s recipe for robust 2D Dirac semimetallic phases within 3D bulk crystals was therefore heartily welcomed by the condensed matter physics community.
It is easiest to understand how Young and Kane’s protection works for a 2D lattice where the unit cell of a symmorphic crystal is doubled, i.e. with a non-symmorphic symmetry {g|t} where t is a half-translation. The general band structure for such a system is depicted in Figure 2.6. In this figure, the x-axes represent a direction in k-space along which gk= kholds; An example of this, in a system with a screw axis along ˆx ({C2 ˆx|120}), would be along a line where
ky = 0, because C2 ˆx(kx, ky) = (kx,−ky). The y-axes show the energy. G is the
reciprocal lattice vector, so a cut through one BZ is shown.
Figure 2.6a depicts the band structure of such a system without any other symmetries present. The doubling of the unit cell by the half-translation means the energy bands must fold back, and the back-folded bands necessarily cross an odd number of times as they cross the BZ. More mathematically robust,
22 t h e o r e t i c a l p r i n c i p l e s
Figure 2.6: The band structure of a 2D system with a non-symmorphic symmetry {g|t} where t is a half-translation. (a) The half-translation in {g|t} causes the bands to fold back, such that pairs of bands cross an odd number of times as they cross the BZ. + and - denote the eigenvalue of {g|t} (±λ).
(b) Adding time-reversal symmetry, but without spin-orbit coupling (SOC) such that all bands are spin-degenerate, the crossing occurs at G/2 because the bands must be symmetric about k=0. (c) Including SOC, the bands are spin-split, and bands at momentum k with spin σ have a partner at −k
with spin−σ. (d) Also including inversion symmetry, all bands are again
spin-degenerate and a four-fold degeneracy occurs at G/2. Image: [17]
the Bloch states can be chosen to be eigenstates of the non-symmorphic sym-metry:{g|t}|u±nki = ±λeik·t|u±nki. Crossing the BZ, such that k→k+G, with
eiG·t = −1 due to t being a half-translation, requires that the eigenvalues (±λ,
represented by red/green) must switch places. This enforces that an odd num-ber of band crossings occur. Going across two BZ (k→ k+2G), bands must end up where they started.
Imposing time-reversal symmetry on the system adds more constraints. Ig-noring spin-orbit interactions, effectively T2 = 1 and all bands are spin-degenerate. Figure 2.6b depicts what the band structure would look like under these constraints. Without spin playing a role, applying the point group symmetry twice will return the system to its initial state (g2 = 1), which in combination with T2=1 means that the eigenvalue λ = ±1 at k= jG (j∈Z).
Time-reversal symmetry implies that if there is a state at k, there must also exist a state at−k, while the non-symmorphic symmetry still enforces that the band eigenvalues switch places as the bands cross the BZ. k = G2 is a time-reversal invariant momentum, where T interchanges the eigenstates of {g|t}
with eigenvalues ±i. This can also be understood as the Bloch Hamiltonian commuting with ˜T = {g|t}T, satisfying ˜T2 = −1. This is the equivalent of
having time-reversal symmetry with spin-12 particles, this enforces a Kramer’s degeneracy at the ˜T-invariant momentum k= G2 (see Section 2.2).
2.5 young and kane and protected crossings 23
Adding spin-orbit interactions into the equation (i.e. T2 = −1), the bands will no longer be spin degenerate although there must still be a pair of Kramer’s degeneracies at k= G2 following the same arguments as above. Time-reversal symmetry implies that for every eigenstate at k with spin σ = ±12 there must be an eigenstate at−kwith σ= ∓12 (i.e. opposite spin). This enforces Kramer’s degeneracies at k = 0 and k = G. Moreover, the non-symmorphic symmetry still enforces that eigenvalues of the bands switch sign as they cross the BZ, such that the Kramer’s partners at k = 0 and k = Ghave opposite eigenval-ues, while Kramer’s partners at k = G2 have the same eigenvalues. This leads to the band structure shown in Figure 2.6c.
The third, and final, symmetry that is required in this recipe for band cross-ings protected from SOC is inversion symmetry, which does the following:
P : k→ −k σ→σ
This symmetry operator thus acts very much like time-reversal symmetry, al-though it doesn’t flip the spin. This means that the bands must also be spin-degenerate, or in other words they must be Kramers degenerate for all k, which results in a band structure as shown in Figure 2.6d. This means there must be a four-fold band degeneracy at k = G2. This band crossing forms a cone-like structure that disperses in kx and ky, where each state can host both
a spin up and a spin down electron, which can be described by the (2D) Dirac equation. This thus constitutes a 2D Dirac cone.
In their paper ([17]), Young and Kane present a simple tight-binding model for four types of 2D lattices made up of atoms with valence s orbitals. Of these, the following three are relevant:
a. A simple square lattice, taking a unit cell with two atoms (a
√
2×√2 su-percell) such that the calculated band structure can be directly compared to the following lattices.
b. The same square lattice but now split into two square sublattices A and B, where one lies in a plane slightly above the plane of the other. They call this ’crinkling’ the lattice. This breaks {E|t}(where E is the identity operator and t a half-translation) but leaves non-symmorphic symmetries
{C2 ˆx|120},{C2 ˆy|012}, and{Mˆz|1212}intact.
c. The same lattice as in b but now with sublattice A shifted slightly along the ˆy direction with respect to B. This breaks the screw axis{C2 ˆy|012}and
the glide mirror plane{Mˆz|1212}. Only the screw axis along ˆx remains.
The Hamiltonian used to represent electrons hopping within these lattices is
H =2tτxcos kx 2 cos ky 2 +t2 cos kx+cos ky +tSOτz σysin kx−σxsin ky . (2.51)
τ and σ are Pauli matrices describing the lattice (A/B) and spin degrees of
freedom. t is the nearest-neighbour hopping parameter, t2 the next-nearest
24 t h e o r e t i c a l p r i n c i p l e s
effective next-nearest neighbour spin-orbit coupling (SOC) effect, also called the extrinsic SOC. The latter will be explained in more detail in Chapter 8. Finding the eigenvalues of this Hamiltonian, the resulting band structure for
Figure 2.7: Tight-binding calculations for lattices a,b and c considering only s orbitals. The top left shows the real-space lattice structure, the top right shows the locations of line and point nodes in the reduced BZ, and the lower images show the band structures. E/t is the energy normalised by the hopping parameter t. Image adapted from [17].
the three types of lattices are shown in Figure 2.7. The reduced Brillouin zone is shown in the upper right corners, with the band degeneracies marked in blue. These same degeneracies can be seen in the band structure plotted along different high-symmetry directions in the lower images. For lattice a, the bands are degenerate all along M−X and Y−M. Crinkling the lattice (lattice b) breaks the inversion symmetry centre between two next-nearest neighbours, which allows for a non-zero extrinsic SOC effect (the tso term in the Hamiltonian).
This splits the bands everywhere apart from at the high-symmetry points M, X and Y. Adding the shift in the y direction (lattice c)−such that the screw axis
{C2 ˆy|012}is broken−gaps the degeneracy at Y, but maintains the crossings at
