Eyzo Stouten (10024301) Physics Faculty of Science

Supervised by: dr. V. Gritsev Universiteit van Amsterdam dr. B. Nienhuis Universiteit van Amsterdam Amount of ECTs: 12 July 24, 2014

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Abstract In this thesis the integer Quantum Hall (IQH) effect and Quantum spin Hall (QSH) effect will be introduced with the use of Haldanes model of graphene and the HgTe model of Bernhevig Huges and Zhang. The topological interpetation in the band theory of topological insulators (TIs) will be described and a way to induce these propreties on quantum systems using electro magnetic radiation inducing the Floquet topological state. Starting from Blochs formalism and the Berry phase the Chern insulator is introduced followed by time reversal symmetry and the Z2 TI. Lastly Floquet topological insulators (FTI) will be introduced with as goal to give reader some insight in the reasons why TIs are "topological" without the necessaty of a very deep understanding in the field of topology.

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Brief popular scientific discription in Dutch Topologische isolatoren zijn een nieuwe klasse van vaste stoffen die hele speciale eigenschappen vertonen, ze kunnen zowel geleiden als isoleren op het zelfde moment! Het geleiden gebeurt in topologische isolatoren niet op de manier waarmee iedereen bekend is, geleiden doen ze alleen via hun oppervlak en isoleren aan de binnenkant. Bovendien geleiden topologische isolatoren netto geen stroom doordat de stroom een rondje om de isolerende binnenkant heen loopt, dus over tegenovergestelde randen van het materiaal staan de stromen in tegenovergestelde richting. Het lijkt misschien niet zo spectaculair als er netto geen stroom geleid wordt maar in 2005 werd er door Charles Kane en Eugene Mele een theoretisch voorstel gedaan voor een speciale klasse topologische isolatoren die zogenaamde spinstromen in tegenovergestelde rondjes over de rand geleiden (zie figuur 1b). Materialen die spinstromen geleiden zijn een integraal onderdeel voor het ontwikkelen van spintronics een techniek die nog in de kinderschoenen staat maar als een van de kandidaten wordt gezien om normale transistor technieken te vervangen. Deze spinstromen kunnen gebruikt worden om de wet van Moore in leven te houden en zijn een belangrijke eigenschap voor onderdelen van de kwantumcomputer. In dit artikel wordt een korte beschrijving gegeven van hoe topolgische isolatoren werken en wat ze temaken hebben met topologie, een tak van de wiskunde die zich bezig houd met de behouden invarianten van een ruimte onder vervorming. Om topologische isolatoren te onderscheiden van normale isolatoren wordt een korte introductie in banden structuren gegeven, in figuur 1a zijn twee bekende bandenstructuren weergegeven. Een banden structuur is een weergave van de verdeling aan energieën die elektronen (1a weergegeven in grijs) in een vaste stof kunnen hebben. Een elektron in een vaste stof kan als gevolg van kwantummechanica niet elke energie aannemen, toegestane energieën worden weergegeven met banden, de zwarte lijnen in 1a. Doordat niet alle energieën toegestaan zijn ontstaan er gaten tussen de zwarte lijnen genaamd bandkloven. In vaste stoffen wordt er onderscheid gemaakt tussen de geleidingsband en de valentieband, dit zijn de banden die respectievelijk net boven en onder het Fermi niveau liggen (het Fermi niveau is de maximale energie die elektronen bij een temperatuur van T = 0K hebben). Elektronen in de geleidingsband kunnen zich vrij door het materiaal bewegen en die in de valentie band zitten vast aan de atomen die de het materiaal opbouwen (naar valentie elektronen). Om van een niet geleidende naar een geleidende toestand over te gaan moeten de elektronen over springen van de ene naar de andere band, hiervoor is genoeg energie nodig om de band kloof over te steken. In geleidende materialen is de bandkloof klein genoeg zodat er altijd elektronen vrij zijn om stroomt te geleiden en zich dus in de geleidingsband bevinden echter in isolatoren is dit niet het geval. De banden structuur van een topologische isolator ziet er drastisch anders uit dan die van een normale isolator, er zijn namelijk banden die de valentie band met de geleidingsband verbinden 1b. Omdat er geen bandkloof is voor de elektronen die de twee banden verbinden zijn deze elektronen vrij om te bewegen, deze banden beschrijven de stromen op het oppervlak van de topologische isolator. Zonder de eerder genoemde banden ziet de structuur eruit als die van een gewone isolator, deze banden zijn een direct effect van de topologie die veranderd aan de randen van de isolator. Zoals genoemd in de inleiding is topologie een tak van de wiskunde die zich bezig houd met de behouden invarianten van een ruimte onder vervorming, onder vervorming wordt verstaan dat je alles met een object mag doen behalve een gat erin prikken. Een voorbeeld van topologie in de geometrie is dat een bol gelijk is aan een kubus omdat je de bol kan omvormen tot een kubus zonder een gat erin te prikken. Echter is een bol niet gelijk aan een torus (doughnut vorm) doordat er eerst een gat gemaakt moet worden om hem daarin om te vormen. Het aantal gaten in een geometrische vorm wordt weergegeven met de genus, deze is g = 0 voor de bol ook

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Figure 1: Bandenstructuren

(a) De bandenstructuur van een isolator en een geleider met op verticale as energie E op de horizontale as de impuls k en het Fermi niveau µ aangegeven met een gestippelde lijn, in donkergrijs de elektronen[10]

(b) In de linker figuur zijn twee spinstromen te zien die in tegengestelde richting langs de rand van de topologische isolator bewegen en rechts de banden structuur van dit systeem [16]

wel de triviale oplossing genoemd en g = 1 voor de torus. De topologie van een banden structuur kan op de volgende manier gezien worden: een gewone isolator heeft een triviale banden structuur en kan vervormd worden tot een geleider zonder de bandkloof dicht te maken. Bij een topologische isolator lopen er banden door de bandkloof, deze bandenstructuur kan niet uit die van de triviale bandenstructuur gemaakt worden zonder de bandkloof te dichten. Het dichten van de bandkloof is equivalent aan dichten van het gat in een torus om een bol te maken, dit veraderen van de topologie is wat de geleiding op de randen van de topologische isolatoren veroorzaakt. Als een topologische isolator naast een triviale isolator geplaatst wordt moet de topologische invariant veranderen aan de rand waar de twee isolatoren elkaar raken en wordt het gat in de banden structuur gedicht. De reden dat de topologische isolator allen geleid aan het oppervlak komt doordat alleen daar de invariant niet goed bepaalt is (du g = 0 v g 6= 0), omdat het vacuüm ook isoleert gebeurt dit ook zonder dat de topologische isolator in fysiek contact staat met een ander materiaal. Natuurlijk zijn er nog veel meer moeilijke concepten nodig om precies te beschrijven wat er gebeurt in een topologische isolator zoals de exacte kwantummechanische beschrijving van topologische isolatoren. Het is al gelukt om topologische isolatoren te maken met verschillende stoffen echter werken deze materialen nog niet goed genoeg, er zijn echter manieren om ze beter te laten werken zonder een helemaal nieuwe chemisch samengestelde topologische isolator te maken. Deze manieren om topologische isolatoren te beïnvloed zijn al bedacht bedacht en er wordt aan geëxperimenteerd, bijvoorbeeld met gepulseerde lasers (ook wel bekend als Floquet Topologische Isolatoren) of door er druk op te zetten. Het is belangrijk voor de vaste stof fysica dat deze technieken verder ontwikkeld en onderzocht worden zodat de toepassingen zoals spintronics in de toekomst misschien gerealiseerd kunnen worden.

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Brief Popular scientific description in English Topological insulators are a phase of solid-state matter which exhibit very interesting properties, they conduct as well as insulate at the same time! Conducting in topological insulators does not happen in the same way everyone is familiar with; they conduct via the surface while their bulk is an insulator. Most notably is that there is no net current in topological insulators because the currents encircle the isolating bulk cancelling the currents on opposite sides of the sample thus cancelling each other. The previous might seem like a rather underwhelming property but in 2005 Charles Kane and Eugene Mele did a theoretical proposal for a special kind of topological insulator which conduct so called spin sorted currents in opposite directions across along the edge see figure 2b. Materials conducting spin currents are of great interest because of their applications in spintronics which is said to be one of the candidates to replace regular transistor techniques. Spintronics can be used to keep up with Moores law and play an important role in quantum computing. In this article a short description on the theory behind topological insulators and their connection to topology (a branch of mathematics) will be given. To extinguish topological insulators from regular insulators a short introduction to band theory will be given, in figure 2a some common band structures are depicted. A band structure is a representation of the dispersion of the energies electrons have in a solid (electrons are represented in grey in 2a). Due to quantum mechanics an electron can not possess a continuous energy dispersion, the allowed energies are depicted with bands (black in figure 2a). There are also "forbidden" energies that cannot be occupied by any electrons, as a result not all energies in the band structure are covered leaving a so-called band gap between the black lines. Two bands will be of special interest: the valence and the conduction band, these bands lie just above and below the Fermi energy respectively (the Fermi energy is the maximum energy at temperature T = 0K). Electrons in the valence band are free to move trough the material and electrons in the valence band are confined to their respective atoms (as in valence electrons). To change from an insulating to a conducting state the electrons have to cross the band gap which requires an amount of energy, this is related to the size of the band gap and differs for each material. In conducting materials the band gap is sufficiently small such that conduction is always possible, for insulators the gap is too large thus no conduction occurs. The band structure of topological insulators has a drastically different structure than that of a regular insulator, in figure 2b it can be seen that they posses bands crossing the gap. Because these materials exhibit no band gap the electrons in the bands crossing the gap are free to move, these bands describe the surface states. Without these bands the band structure is equivalent to that of an ordinary insulator, the gap crossing bands are a direct effect of the changing of a topological invariant across the surface. Topology is a branch of mathematics concerned with conserved quantities of a system under continuous deformation, under continuous deformation any change is allowed but cutting and poking holes. An example of topology in geometry is that a sphere is considered topologically equivalent to a cube because these shapes can be related under continuous deformation. A sphere however is not equivalent to a torus (a doughnut shape), which has one hole and thus it is necessary to change the topology of the system by cutting a hole in the sphere. The amount of holes is related to the topological invariant called the genus, g = 0 is called the non-trivial solution for the sphere and g = 1 for the torus. The topology of a band structure can be interpreted in the following manner: a regular insulator and conductor has a trivial band structure because the can be deformed into one another without cutting the bands. The bandstructure of the topological has bands crossing the gap thus a trivial insulator cannot be continuously deformed to the non-trival case without closing the gap. Closing the gap can be interpreted as similar to closing the hole of the torus to

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Figure 2: Bandenstructuren

(a) The bandstructure of an isolator and a conductor with on the vertical axis energy E and on the horizontal axis the momentum k. The Fermi energy mu is depicted with the dotted line and shaded in dark gray are the electrons [10]

(b) In the figure on the left two spincurrents are depicted traversing in opposite directions along the edge of the topological insulator. On the right the band structure describing this state is depicted [16]

make a sphere. If a topologically trivial and nontrivial material are placed next to each other the topological invariant has to change across the interface; as a result the band gap is closed and edge states are observed at the surface. From this behaviour it can be seen that a topological insulator has a different topology than a regular insulator, because the vacuum also isolates it is not necessary to place the topological insulator in physical contact with an insulating material to display this behaviour. Naturally more an understanding of more complicated processes is mandatory to exactly describe the behaviour displayed by topological insulators. Although physicist have succeeded in engineering an array of topological insulators, they have not yet succeeded in making them operate optimally. There have however been numerous propositions to alter the behaviour of these materials without physically changing their chemical composition, which is an important feature for the sake of research. A couple of these techniques involve altering the system by studying the influence of pulsating lasers (also known as Floquet topological insulators) or applying strain tot the sample. It is important for the field of solid state physics that these new techniques are studied and developed such that in the future everyone may be able to benefit of these materials through for example spintronics or quantum computing.

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Contents 1 Introduction 2 Theory 2.1 Periodic systems 2.2 Berry phase . . . 2.3 Chern integer . . 2.4 Two state system

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3 Integer Quantum Hall systems 15 3.1 Dirac points in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Quantum Spin Hall systems 19 4.1 Kramers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Finding the Z2 topological invariant . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Two state spin Hamiltonian and models . . . . . . . . . . . . . . . . . . . . . . . 22 5 Floquet Topological insulators 25 5.1 Floquet Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Implications and models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Conclusion and outlook 27 6.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 Appendices 28 7.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.1.1 Time inversion symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.1.2 Parity symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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1

Introduction

The aim of this thesis will be to give an introduction on the subject of topological insulators (TIs), what their link is to topology and how these states of matter come to exist. The band theory of solids is an important tool to understand the basic idea of a TI, it is used to describe the energy eigenvalues of infinite solids as a function of their crystal momentum. The properties of materials can be inherently connected to their bandstructure, the gap between the filled states (valence-) and unfilled states (conduction-band) distinguishes whether a system is a conductor or an insulator (see figure 3). The difference with topological insulators is that these systems can

Figure 3: Finite gap bandstructures [10] give rise to both these effects at the same time. An infinite TI has a bandstructure equivalent to a trivial insulator, their non-trivial topological nature becomes apparent when the TI is adjacent to a trivial insulator and bulk boundary correspondence is taken into account. At the boundary between these two phases of matter a topological phase transition has to occur giving rise to currents running along the edge of the system. These currents will appear in the bandstructure as bands crossing from the conduction to the valence band and are very robust due to their insensitivity to backscattering. Topology is a branch of geometry that studies how systems (that is mathematical spaces, knots, bandstructures, etc.) are connected, this is done by stuying whether they can to be continuously transformed into one another. An often used example is that a torus can not continuously be deformed into a sphere without cutting it open, here the sphere is the trivial system identified by its genus g = 0 with no holes and the torus g = 1 has one. The bands of topological insulators possess an equivalent property, because of the band crossing the gap the bandstructure can not be continuously deformed into the trivial case. The concept of topological classes is not new to band theory, the first realization that a system could have non trivial topology was discovered by Thouless, Khomoto, Nightendale and den Nijs in 1982 [27]. They found that the Hall conductance of the Integer Quantum Hall (IQH) effect is quantized by the topological Chern invariant. This invariant was induced by breaking time reversal symmetry T with a magnetic field, just like in the regular Hall effect. This class of topological insulators is the Chern class and only identifies 2D effects, at the boundary of these systems chiral edge states can be observed. These edge states are chiral in a sense that they only propagate in one direction along the edge of the sample and it is the quantization of these edge states that cause the IQH effect. Until 2005 it was thought that topological insulators where only possible under the breaking

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T but in this year Kane and Mele [17] found that this was not the case. They studied the effect of spin orbit coupling (SOC) in graphene on with spin1 /2 states and discovered the Z2 topological class which quantizes the Quantum Spin Hall (QSH) effect. This topological class was first realized in the HgTe/CdTe system and has no net edge current but in stead spin filtered edge states traversing in opposite directions [6]. These states dubbed helical edge states can also extist in three dimensional systems because unlike Chern insulators the Z2 class also generalizes to higher dimensions. The discovery and realization of the Z2 insulator led to a big impulse in the field of condensed matter physics because of their exotic behaviour. QSH systems can give Majorana bound states and can be used for spintronic technology. Both these properties are of great interest because of their applications for Quantum computing and other alternatives for future computing.[24] Controlling the bandstructure of TIs is important to acquire a topogical phase, this can be done by changing the interaction terms but this is difficult to do without physically changing the crystal structure. To gain control over the bandstructure it is more convenient alter the bandstructure by using perturbations such as putting the crystal under strain or perturbing the system wit time dependent electric or magnetic fields. In this theses perturbing the system with electro magnetic radiation will be considered, this perturbation is described by Floquet theory and gives rise to a new topological class called Floquet Topological Insulators (FTI). This thesis will be structured in the following way first a short theoretical introduction will be given in 2, here a basic introduction to Bloch will be given, the notion of the Berry phase and the Chern integer will be introduced. In section 3 Chern insulators will be described and in section 4 Z2 TIs. The last section 5 will be dedicated to Floquet Topological insulators.

2 2.1

Theory Periodic systems

Throughout this theses two kinds of periodical systems will be studied; crystalline solids which are described by the band theory of solids and systems periodic in time which are described by Floquet theory. In this section the aforementioned theories will be described. Band theory is the study of eigenstates and energies of crystalline solids as a function of the crystal momentum p = k~, here k is the crystal wave vector which becomes periodic due to the spatial periodicity of the system. In band theory the periodicity of a crystal is exploited to define a periodic Hamiltonian such that H(r) = H(r + Rn )

(1)

V(r) = V(r + Rn )

(2)

This can be done because a crystalline lattice can be described as a finite set of atoms each with a potential V(r) when translated over the primitive lattice vector Rn = n1 a1 + n2 a2 + n3 a3 construct the whole crystal lattice. The minimum volume spanned by Rn defines the unit cell. When the unit cell is translated by the subset of vectors Rn it fills up all space completely without overlap and displays the full symmetry of the system. [28] Introducing discrete symmetry implies that any eigenstate of (1) can be translated by a lattice spacing Rn without changing the system, this enables one to define a translational operator that commutes with the periodic Hamiltonian. Tˆ = exp[iRn · r] [H, Tˆ] = 0

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(3) (4)

Figure 4: Brillouin Zone represented in toroid and plane shape [13] The previous conditions imply that the complete set of eigenfunctions must be eigenfunctions of H as well as Tˆ, Bloch’s theorem states that these states can be written as. ψn,k (r) = exp[ik · r]un,k (r)

(5)

Here un,k (r) is a periodic function with the same periodicity as the lattice such that un,k (r + R) = un,k (r), the exponent is the plane wave and n the band index. For ψn,k (r) to also remain unchanged over a translation Rn it is required that k · Rn = 2πn for all n. Describing periodicity through Rn is not unique, by introducing the Bloch wave vector the periodicity of the system can also be induced by the crystal moment momentum. Let Gm = m1 b1 + m2 b2 + m3 b3 with the following properties k0 = k + Gm , {Gm · Rn = 2πn : ∀n ∈ Z}. The vector Gn defines the Brillouin zone (BZ) analogous to the primitive cell in real space: Let Bragg planes be the planes that perpendicularly bisect all the vectors Gm then the first BZ is the set of points in k-space that can be reached from the origin without crossing any Bragg plane. The position space and momentum space interpretation are linked by the Fourier transform. X f˜(k) = f (r) exp[−ik · r] (6) r∈Rn

ˆ

dn kf˜(k) exp[ik · r]

f (r) =

(7)

BZ

These transformations can also be used to transform a state ψn or coefficient cn , it is also convenient to rewrite the Hamiltonian in the crystal momentum basis H(r) → H(k) also known as the Bloch Hamiltonian. [3] An n dimensional periodical system is defined on an n dimensional torus Tn , this is no different for the BZ an example can be seen in figure 4. In figure 4 the sides of a two dimensional BZ can be identified with each other because the eigenstates are periodic and can be connected continuously throughout the BZ. In chapter 5 another periodic system will be introduced, the Floquet formalism concerns systems which are periodic in time. In this thesis the temporal periodicity is induced by perturbing the system with electromagnetic radiation. Analogous to the spatially periodic system a perturbation of period T is added to the time independent Hamiltonian. H(k, t) = H0 (k) + H0 (t)

(8)

0

0

(9)

0

0

(10)

H (t) = H (t + T ) V (t) = V (t + T )

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Here H0 is the unperturbed Hamiltonian with a complete set of eigenstates {φn (k)} and eigenvalues {En (k)}. The full Hamiltonian H(k, t) should obey the time dependent Schrödinger equation [H(k, t) − i1∂t ]Ψµ (k, t) = 0 (11) According to the Floquet theorem the solutions of (8) are of the form Ψµ (k, t) = exp[−iµ t/~]Φµ (k, t)

(12)

here Φµ (k, t) is the Floquet mode obeying Φµ (k, t) = Φµ (k, t + T ) and µ = ~ω = 2π~n/T is the quasi energy analogous to the crystal momentum k. Like with the crystal momentum the quasi energy can also be used to quantize the system n,µ = µ + n~ω, {n,µ · T = 2πn : ∀n ∈ Z}. Again an operator can be defined which can be used to evolve Ψµ over time ˆk (t, t0 )Ψµ (k, t0 ) Ψµ (k, t) = U ˆ t 0 ˆ Uk (t, t0 ) = Tt exp −i dt H(k, t)

(13) (14)

t0

where Tt is the time ordering operator. From the previous expression the Floquet Hamiltonian HF can be defined ˆk (T + t0 , t0 ) exp[−iHF (k)T ] = U (15) this Hamiltonian is effectively a static Hamiltonian that describes the system after each periodic evolution T . [9][23]

2.2

Berry phase

The Berry phase is a special case of the geometric phase crucial in understanding the Chern integer which is the topological invariant characterizing integer quantum Hall (IQH) and Floquet topological systems. Although the regular phase factor of a quantum state does not carry any intrinsic meaning due to phase freedom, the difference in Berry phase across the BZ can accumulate to a finite change of the eigenstate. Derivation of the Berry phase starts with the adiabatic approximation for a time dependent Hamiltonian. The adiabatic theorem states that a quantum state remains in eigenstate under an adiabatic deformation of the Hamiltonian. This can be proven with first order perturbation theory. From the most general solution of a time dependent Hamiltonian the eigenstates pick up a phase factor: H(R)ψn (R) = En (R)ψn (R),

Ψn (R, t) = ψn (R) exp [−iEn (R)t/~]

(16)

Here R is some set of parameters independent of time, when the Hamiltonian H(R(t)) is changing with respect to time the eigenstates will do so too.[5] H(R(t))ψn (R(t)) = En (R(t))ψn (R(t)) ˆ i t 0 Ψn (x, R(t)) = ψn (R(t)) exp − En (R(t))dt exp[iγn (R(t))] ~ 0

(17) (18)

Here γn (R(t)) is the geometric phase, the integral in the exponent is the dynamic phase which does not affect the geometric phase and is equivalent to the phase in (16). The expression of the geometric phase is obtained by plugging Ψn (R(t)) into the time dependent Schrödinger equation to find. ˆ t ˙ 0 ) · hψn (R(t0 ))|∇R |ψn (R(t0 ))i γn (t) = i dt0 R(t (19) 0

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The Berry phase is a special case of the geometric phase, when R(t) describes a closed path in parameter space such that R(T ) = R(0). Berry’s insight in his 1982 paper was that in the case for the previous path the geometric phase could be rewritten into a closed path integral around C, around this path the geometric phase is not necessarily single valued γn (T ) 6= γn (0). Calculating the Berry phase is thus equivalent to finding the change in geometrical phase around the closed loop in parameter space C which is an invariant. ˆ T ˙ 0 ) · hψn (R(t0 ))|∇R |ψn (R(t0 ))i dt0 R(t (20) γn (C) ≡ i 0 ˛ (21) = i dR · hψn (R)|∇R |ψn (R)i C ˛ dR · An (R) (22) = C ˆ dS · Bn (R) (23) = S

To simplify these expressions the Berry connection is defined as. An (R) = ihψn (R)|∇R |ψn (R)i

(24)

Using Stokes theorem (22) can be rewritten as the surface integral dS over the parameter space encircled by C. Bn (R) = ∇R × An (R) (25) The curl of the Berry connection is defined as the Berry curvature [1] or Berry flux [16]. The Berry curvature gives the difference in Berry phase after one adiabatic transformation over a period T therefor enclosing one loop in parameter space R(t). These quantities can be interpreted as an effective vector potential An and magnetic field Bn in the parameter space of R. Naturally if the Berry connection (24) is single valued across dS the total Berry flux results in a trivial phase change γn (C) = 0. When An is not single valued across dS then by gauge freedom it is generally possible to define (for example two) separate regions S1 and S2 where An is single valued. The two regions should together cover S and An on these domains can be defined by a gauge transformation. Changing the gauge also effects the phase of |ψn (R)i. |ψn (R)i → |ψn0 (R)i exp[−iχ(R)] An (R) →

A0n (R)

+ i∇R χ(R)

(26) (27)

Now let A0n (R) and An (R) be the Berry connections in region S1 and S2 , then the total Berry curvature can be found by connecting the two surfaces resulting in an integral over dS. ˆ ˆ γn (C) = dS1 · [∇R × An (R)] + dS2 · [∇R × A0n (R)] (28) S1 S2 ˆ = dS · [∇R × An (R) − ∇R × A0n (R)] (29) S ˆ = dR · [An (R) − A0n (R)] (30) ˆC = dR · ∇R χ(R) (31) C

In the previous expression it can be seen that the change in Berry phase indeed is equivalent to the change in phase between the eigenstates |ψn (R)i and |ψn0 (R)i. [5][20] 12

Figure 5: Closed curve [30]

2.3

Chern integer

The Chern number is a topological invariant identifying the Chern classes, these classes are associated with the theory of fiber bundles. Although studying topological concepts like the aforementioned is necessary to understand the deeper structure of topological field theory or its generalization to other dimensions (Zhang [29]) this is not necessary for the models studied in this thesis. Therefor in this chapter the description of the Chern class will be omitted and rather a more applicable description of its topological nature will be described. In the previous chapter it is shown that a change in Berry curvature across the closed surface of parameter space leads to a change in Berry phase. For the phase change in (28) to give rise to topological effects periodicity of the system is a necessity (either in space: Bloch or time: Floquet). In Fig. 5 an example for a 2D periodic system is given, to induce a non-trivial solution let An (ky ) such that {kx , ky } span the periodic parameter space. This restrains total Berry flux be quantized because the systems periodic eigenstates |un (k)i have to be defined continuously after encircling the Brillouin torus. The total Berry curvature over T2 is thus given by γn = 2πm where m is the Chern integer. ˆ 1 Cn = d2 k · Bn . (32) 2π Where n is the band index, the total PNChern number of is a topological invariant given by the sum over all occupied bands C = n Cn . Note that this is integral is only defined for a two dimensional surface, therefor the Chern number only characterizes 2D systems. It is possible to have 3D structures which expressing the characteristics of a Chern system but these can be viewed as stacked 2D systems. [16][10] In the previous section the notion of the gauge freedom of An was introduced as a result of the phase freedom of |ψn i. In the case of the 2D periodic system the change in phase implies a locally single valued basis for |un (R)i can not be defined thus the two domains S1 and S2 are not simply connected or has a genus of g = 1. Because of the non-contractible base space T2 this singularity can not be erased without drastically altering the Hamiltonian or non adiabatic transformations. Any adiabatic change in the Hamiltonian of the system thus keeps it in the same topological state it is this property which gives topological insulators and their edge states their robust nature. These edge states arise because the topological invariant has to change across the boundary where a non-trivial system is in contact with a trivial state (as will be 13

shown in chapter 3.2). Naturally the Chern number can take any integer value when the domain is multiply connected witn g > 1, this also gives rise to more Chiral edge states. [10][20]

2.4

Two state system

Another way to find the Chern integer is by inspecting the Hamiltonian of a system directly, a good model for this is the two state system. The two state system is a reoccurring subject when when studying TIs because the topological nature manifests itself between the valance and conduction band of the studied system. In the two state interpretation the two bands can be represented by a mapping from the two dimensional Hilbert space H2 ' C2 onto S2 and give a figurative way of interpreting the topology of the system. [10] In this section the general two state Hamiltonian will be considered h0 + hz hx − ihy H= (33) hx + ihy h0 − hz with basis states {|φ1 i, |φ2 i}. The Hamiltonian can be simplified by using the Pauli spin matrices σµ≥1 . H = hµ σµ = h0 1 + ~h · ~σ (34) The ~h terms are interaction terms with for example a magnetic field and nearest neighbor or spin orbit coupling (SOC) interactions. The eigenvalues E± of this system can be obtained by solving the characteristic equation. [5] E± = h0 ± ||~h||

(35)

As can be seen from the previous expression for ~h 6= 0 the eigenstates are symmetric in ~h. The eigenenergy is degenerate when all three terms of ~h = 0, these variables (~h) are usually a function of k can be used to parameterize the Hamiltonian on the surface of S2 . Because the energy shift h0 and the norm h = ||~h|| do not affect the topological properties of the system the ˆ defined as mapping can be simplified by only considering a vector h ~ ˆ = h = (sin θ cos φ, sin θ sin φ, cos θ) h ||~h||

(36)

spanning S2 . The eigenstates of this system are the regular solution to the two state system where φi are the two states from (34). |ψ± i = α± |φ1 i + β± |φ2 i

(37)

With coefficients. α+ = exp[−iφ] cos α− = − sin

θ 2

β+ = sin

θ 2

θ 2

β− = exp[iφ] cos

(38) θ 2

(39)

ˆ runs over the surface of sphere with polar angle θ and the azimuthal angle φ as This vector h depicted in figure 6. [7] [10] The two state system is a known example for the Berry phase equivalent to a spin-1 /2 system which is rotated around the sphere along a magnetic field in the direction of H(k) and has known

14

ˆ (36) enclosing a path on S2 [31] Figure 6: h solutions. Each state of the system acquires a change in Berry phase equal to half the solid angle swept out by the vector H on the surface of the sphere. [14] 1 1 γ± (C) = ∓ dΩ = ∓ sin θdθdφ 2 2

(40)

Due to the coefficients of (37) when θ → 0 the eigenstates can not have a well defined phase over the whole BZ. The phase convention can be changed by changing the phase or multiplying the states with exp[iφ] but this only moves the ill defined phase to θ → π. Because the eigenstates have to be continuous on T1 there are only a view solutions for Ω possible, Ω = 0 if the H remains in the upper hemisphere and does not enclose any area (trivial case). If the H does cover the sphere it encloses an area multiple of 4π else if the Hamiltonian is real Ω = ±2π and is confined on a circle.[10] The Chern invariant is again given by (32) which for the two state system can be rewritten ˆ as a function of h. ˆ 1 ˆ × ∂k h) ˆ ·h ˆ Cn = d2 k(∂kx h (41) y 4π This is a convenient representation which counts the amount of times H wraps S2 . Of course it is possible for this vector to enclose the surface of S2 multiple times resulting in a higher Chern number. [29][10]

3

Integer Quantum Hall systems

As discussed in the introduction a trivial insulator is a material where a band gap separates the filled bands from the empty conduction bands. These trivial insulators are insensitive to boundary conditions because their Hamiltonians can be changed adiabatically from a conducting state to an insulating state without closing the gap. Therefor any adjacent trivial insulator does not break symmetry and no phase transition will occur. Differend behaviour arises when one of 15

them is topologically non-trivial in this case the topological invariant has to change across the boundary giving rise to edges states.[13] The first notion of a non-trivial topological system was found by Thouless, Kohmoto, Nightengale and de Nijs [27] who found that the Integer Quantum Hall (IQH) system was quantized by the Chern invariant. By breaking the time reversal (T ) symmetry of the system with a magnetic field the Chern invariant of the system was altered, they found that it was this integer 2 which quantizes the Hall conductance σxy = Cn eh . This Hall conductance is a direct effect of the Chiral edge states encirling the insulator, these will be studied in section 3.2. [10][27]

3.1

Dirac points in graphene

An example capturing most of the important concepts from Chern topological insulators which also arise in quantum spin Hall (QSH) insulators is the quantum Hall (QH) effect in the bandstructure of graphene. In this chapter the concept of Dirac cones, the effect of Time reversal symmetry breaking and its effects on the Chern number will be studied.

Figure 7: Crystal lattice of Graphene [?] In his 1988 paper Haldane studied graphene and considered the effects of symmetry breaking across a bipartite lattice (for more on symmetry see appendix 7.1). He considered the crystal lattice to consist of inter penetrating triangular lattices A and B as in figure 7. By breaking time reversal symmetry T with a magnetic field that has zero flux trough the (A, B) unit cell but is non zero in individual cells A and B the topology of the system can be changed. [16] To observe the effect of the magnetic field on the topology of graphene the methods in chapter 2.4 will be exploited, to do so the Hamiltonian needs to be rewritten in the form of (34). The Hamiltonian of this system can simply be found through the tight binding approximation describing the pz orbitals therefor it is a two by two matrix with on the off diagonal elements the neighbor interactions (first and second). Because the Hamiltonian is of the 2 × 2 form the general two state Hamiltonian can be used in the k basis and σ representation.[16] H(k) = hµ (k)σµ

(42)

The exact coefficients of hµ (k) can be found in [15] but for the purpose of this example an exact formulation is not necessary. By studying the symmetries of the system the effect of the magnetic field on the system can be found. Without a magnetic field present time reversal symmetry (T ) is conserved implying T hz (k) = hz (−k). Also inversion symmetry P has to be preserved because all atoms in the unit cell are equivalent, this leads to the constraint Phz (k) = −hz (−k). Together these constraints require hz = 0 and thus (42) gives the regular graphene bandstructure as depicted in figure 8. [16][10]

16

Figure 8: Reciprocal lattice and bandstructure of Graphene [32] A special feature of graphene is that the conduction and valence band touch each other in points K and K0 forming Dirac points. At these points all coefficients of ~h = 0 thus ~h can be linearly expanded around q ≡ k − K for small k resulting in ~h(q) = ~vF q. Substituting the previous expression into (42) H results in a Hamiltonian equivalent to the massless Dirac Hamiltonian [10][16] H(q) = ~vF q · ~σ (43) were vF is the Fermi velocity. This Hamiltonian was found by Dirac to describe relativistic particles quantum mechanically, the massles Dirac Hamiltonian thus describes a massles relativistic particle which has a linear energy dispersion whith respect to k. Haldanes insight was that by breaking P or T the Dirac point opens up due to hz (K) 6= 0 this is possible because the symmetry constraints no longer have to be satisfied both. To break the symmetry two cases where considered: changing an atom in the lattice or introducing the magnetic field for both cases when hz is small the Dirac Hamiltonian is no longer massless. H(q) = ~vF q · ~σ + mσz (44) p Here ~σ = (σx , σy ), hence the eigenvalues are E(q) = ± |~vF q|2 + m2 . There are two cases to consider; when P symmetry is broken, then the Dirac point gains a mass m = hz (K) and T symmetry requires m0 = hz (K0 ) thus both Dirac points have mass m. By inspecting H = ~h(k)/|~h(k)| using the techniques from chapter 2.4 it can be seen that when the two masses are equal throughout the BZ, H remains confined to the upper hemisphere of S2 therefor no phase is acquired resulting in a trivial bandstructure. When T symmetry is broken P symmetry requires m = hz (K) and −m0 = −hz (K0 ) thus 0 m = −m. In this case H changes from the north to the south pole across the BZ covering S2 and the subtended solid angle Ω = 4π resulting in a Chern number of Cn = 1. Here it is also possible for the Chern number to be greater then one, for this to happen higher order interactions (third nearest neighbor and up) must be taken into account. Like in the IQH system the non-trivial topology results in chiral edge states propagating at the boundary where the topologically nontrivial system cones into contact with a trivial system, these states can be solved exactly in the following chapter. [16][10]

3.2

Edge states

In the previous chapter only the bulk properties of graphene have been described. The bandstructure for an infinite TI looks similar to that of a trivial insulator, only when the boundary 17

effects are taken into account the distinct crossing of an energy band from the conduction to the valence band is observed (figure 9b). Band crossing is a consequence of the topological invariant changing at the boundary of the TI and an adjacent trivial insulator. This band describes the chiral edge state propagating along the boundary of the TI, which are chiral in a sense that they only propagate in one direction. The edge states are robust because there are no states available for backscattering. [16]

Figure 9: a) A chiral edge state and b) its energy band crossing the gap [16] In this chapter the band crossing of the edge states will be derived following Fruchard and Carpentier [10] by inspecting the boundary of graphene adjacent to a trivial insulator. At the boundary the mass of the Dirac Hamiltonian has to change for the Chern invariant to become zero. The non-trivial case has negative Dirac mass at one side of the BZ and postive on the other whereas in the trivial case both masse should be equal as discussed in the previous section. When the boundary lays along the x direction this leads to n(y < 0) = 1 → n(y > 0) = 0 to obtain this result the negative mass has to change at the boundary. Consider both masses of both Dirac points m0 and m are now functions of y resulting in. m0 (y) = m;

∀y

m(y) < 0;

∀y < 0

m(y) > 0;

∀y > 0

Everywhere TI Insulator

(45) (46) (47)

To find eigenstates of this system the massive Dirac Hamiltonian (44) must be put in the position basis. m(y) −i∂x − ∂y H(y) = −i∇ · ~σ + m(y)σz = (48) −i∂x + ∂y −m(y) This form can be rewritten to a separable differential equation by means of a unitary operator, 1 1 1 √ U= (49) 2 1 −1 resulting in. U HU

−1

α −i∂x ∂y + m(y) α α = =E β −∂y + m(y) i∂x β β

(50)

This set of differential equations has an exact solution as found by Jackiw and Rebbi (1976). ˆ y 1 0 0 0 ψkx (x, y) ∝ exp[ikx x] exp − dy m(y )dy vF (51) 1 0 18

with eigenvalue E(kx ) = EF + ~vF kx . This eigenvalue is a linear function in kx thus gives a linear band across the gap with velocity dE/dqx = ~vF . In the previous example a right moving edge state was discussed naturally if a system has multiple edge states propagating in opposite directions the left and right moving edge states (NR , NL ) cancel so the net Chern integer is equivalent to the difference in right and left moving states. ∆Cn = NR − NL

(52)

However it is possible to have multiple edge states crossing the gap if Cn > 1 by having Berry flux > 2π across the BZ. [10][16]

4

Quantum Spin Hall systems

In this section the Z2 topological invariant will be studied, this differen topological class was discovered by Kane and Mele (2005 [17]) this topological invariant describes the Quantum Spin Hall (QSH) state. Unlike in Chern insulators the QSH state preserves T symmetry and examines its effect on spin 1 /2 systems. Presreving T symmetry is done by using spin orbit coupling (SOC) insted of a magnetic field as the interaction term.

Figure 10: QSH edge state [16] Like Chern insulators Z2 TIs have bands crossing the gap and edge states, the difference is that for the QSH state the SOC separates the spin up and spin down states of the conductuion band. The separate bands give rise to two symmetric bands crossing the gap and due to T resulting in two spin filtered edge states running in opposite direction. Because the bands of the QSH state are symmetric the Chern number is always zero across the gap therefor the Z2 invariant is necessary to describe this topological phase (figure 10). [10][16]

4.1

Kramers theorem

The time riversal symmetry of a spin system is rooted within the Kramers theorem. To implement T symmetry to a spin 1 /2 system a time reversal operator is introduced, this is an antiunitairy operator (see chapter 7.1). ˆ = exp[iπSy /~]K Θ

19

(53)

where K is the complex conjugation, Sy is the spin operator and π is a rotation in spin space. For spin 1 /2 particles T symmetry restricts spin up and down states to be two fold degenerate. ΘH(k)Θ−1 = H(−k)

(54)

Kramers theorem states that at any T reversal invariant point in the Brillouin zone the spin up and spin down states should have equal eigenvalues, as a result the spin up and spin down bands coincide forming Kramers points as can be seen in figure 11. Bands which share a Kramers point are called Kramers pairs (solid lines in figure 11 left), if T symmetry is conserved everywhere in the lattice the trivial bandstructure is retrieved and the spin up and down bands are everywhere degenerate (dotted lines in figure 11). [16][21]

Figure 11: Symmetric bandstructure and TRIM in the 2d BZ [10][13] In TIs T symmetry is only conserved at special points in the Brillouin zone called time reversal invariant momenta (TRM), the TRM will be denoted by k = Γi . There are four of these points in two dimensions and eight in three dimensions in the first BZ satisfying −Γi = Γi + G, here G is the reciprocal lattice vector. In points other than the TRM spin orbit coupling splits the spin up and dow bands (Zeeman splitting) as can be seen in figure 11.[10][16]

4.2

Finding the Z2 topological invariant

The invariant of the Z2 class can be defined in two as well three dimensions. In two dimensions the Z2 invariant takes the two values ν = 0 (trivial insulator) or ν = 1 (non-trivial QSH insulator). In three dimensions there are four independent invariants (ν0 ; ν1 ν2 ν3 ) the first distinguishes between strong ν0 = 1 and weak ν0 = 0 topological insulators. It is thus possible to define a distinct Z2 insulator in three dimensions that is not a stacked version of 2D TIs but for the sake of simplicity only the two dimensional case will be considered in this thesis. [16] The simplest way to acquire ν is to inspect the bandstructure of the TI, the two bandstructures with edge states are shown in figure 12. Depending on the Hamiltonian there might be states bound inside the gap, if this is the case than Kramers theorem requires them to be degenerate at the Γi points. There are two ways the Kramers partners can connect either by 20

Figure 12: The edge states for ν = 0 (left) and ν = 1 right plotted between Γa = 0 and Γb = π/2, only half the BZ is shown because T requires it to be symmetric around k = 0. [16] crossing the Ferimi surface an even or odd amount of times in the half BZ. For the even case the the Hamiltonian can be changed adiabatically such that there are no longer any bands crossing the Fermi surface and thus this state is trivial. However for the odd case this is not possible and thus the edge states can not be eliminated and are thus topologically protected. The Z2 invariant can therfor be defined by NK = ∆ν

mod 2

(55)

Here NK is the number of Kramers bands crossing the Fermi energy which is equal to the change in ν across the surface.[16] There are many different ways to obtain ν and it is not always possible to obtain it by studying the the change ∆ν by inspecting the bandstructure. A more general method was discovered by Fu and Kane [12]. They introduced the sewing matrix which relates the phase change between two states forming a Kramers pair. wmn (k) = hu−k,m |Θ|uk,n i

(56)

Here |uk,n i are the Bloch functions of the occupied bands. This sewing matrix is anti symmetric for k = Γi . The determinant of an anti symmetric matrix can be found by its Pfaffian which has the following property Pf[w(Γi )]2 = det[w(Γi )]. The Pfaffian can be used to express the phase change between two states forming a Kramers pair between two TRIM in the BZ. " # X Pf[w(Γb )] = exp χΓb ,n − χΓa ,n (57) Pf[w(Γa )] n For for k = Γi these states have to be Kramers degenerate and |uk,n (Γi )i can either connect up even or odd at the edge of the BZ. The connection can result in either a trivial or non-trivial phase change, this connection in each TRIM can be expressed by. p det[w(Γi )] δi = = ±1 (58) Pf[w(Γi )] The Z2 invariant is given by the product of the δi ’s. Y (−1)ν = δi i

21

(59)

The previous equation can be interpreted as such; For the trivial case all δi = 1 thus the phase change of |uk,n i is only trivial therefor no boundary effects are observed. If there is an odd phase change in an even amount of TRIM then (δi = −1) is found two times and ν = 0. For the even case the spin sorted bands cross the Fermi surface an even amount of times as can be seen in figure 12 left. Else if the phase change is odd over the whole BZ, |uk,n i can not be defined continuously at the edges resulting in the non-trivial case (fig 12 right) or the product of δi is −1. [16][17] It was found bay Fu and Kane that when inversion symmetry is present finding ν is easyer, by exploiting the parity of |un i at the TRIM. δi =

N Y

ξ2m (Γi )

(60)

m=1

Here, ξ2n (Γi ) = ±1 is the parity eigenvalue of the 2nth band, these values are tabulated in band theory literature. Note that ξ2n = ξ2n−1 at the TRM because the odd bands are Kramers degenerate with the even bands and have the same parity. [13] When Sz is conserved throughout the system finding the Z2 integer is simplified even more. For this case the Chern integers for the separate spin bands can be evaluated as n↑ and n↓ . Then the Z2 invariant is found by ν=

(n↑ − n↓ ) 2

mod 2

(61)

In the following section a few models will be studied showing the application of the methods described in this section. [16]

4.3

Two state spin Hamiltonian and models

To show how the concepts discussed in the previous section can be applied two models will be considered; first the Kane-Mele QSH system in graphene [17] and second the Bernevig-HugesZang model for HgTe quantum wells [4]. For both models a general four level Bloch Hamiltonian can be defined, 5 X H(k) = d0 (k)1 + di (k)Γi (62) i=1

with eigenvalues corresponding tot the Kramers degenerate pairs and eigen energys. v u 5 uX E± (k) = d0 (k) ± t d2i (k)

(63)

i=1

This Hamiltonian is a 4 × 4 matrix with structure similar to (34), the difference here is the eigenstates are in the "sub-lattice tensor spin" basis for the graphene case or in the "orbital tensor spin" basis for the HgTe case. (A, B) ⊗ (↑, ↓) = (A ↑, A ↓, B ↑, B ↓)

or

(s, p) ⊗ (↑, ↓)

= (s ↑, s ↓, p ↑, p ↓)

(64)

Therefor the Hamiltonian is also represented in this basis, to represent this it is rewrittem function of the Dirac matrices Γi (to not confuse these matrices with the TRIM a lower index will be used instead of Γi ). These matrices are Hermitian and obey the Clifford algebra {Γa , Γb } = 2δ a,b they can be used to put restrictions on di (k). Because these elements of the Hamiltonian determine the eigenstates |ui (k)i similar arguments as in section 2.4 can be used to determine 22

the constraints on di (k) by using the fact that one can not define a global basis where Kramers pairs (65) can be formed over T2 . Θ|u1 (k)i = |u2 (−k)i

(65)

To do this an exact formulation for the Dirac matrices must be chosen, note that these matrices are not unique and can be chosen to contain more symmetries of the studied system. For both models the following time reversal operator will be considered Θ = i(1 ⊗ sy )K

(66)

which conserves orbital quantum numbers but flips spin, K is the complex conjugation.[10][13] ˆ In the graphene model of Kane-Mele the Dirac matrices are chosen to be even under Pˆ Θ, Γ1,2,3,4,5 = (σx ⊗ 1, σy ⊗ 1, σz ⊗ sx , σz ⊗ sy , σz ⊗ sz ) where the Γ1 = Pˆ resulting in the following properties. +Γi i ˆ −1 i ˆ −1 ˆ ˆ ΘΓ Θ = P Γ P = −Γi

for i = 1 for i = 6 1

(67)

(68)

For the Hamiltonian to be consistent with both constraints necessarily all di≥2 (k) are odd and d1 (k) even, at the TRIM k = Γi however this requires for di≥2 (Γi ) = 0. Now (60) can be used to find the Z2 integer by exploiting the parity E(Γi )|ui = d1 (Γi )Γ1 |ui = d1 (Γi )ξ(Γi )|ui

(69)

sign[ξ(Γi )] = −sign[d1 (Γi )]

(70)

thus. This method can be used provided that there is an energy gap throughout the BZ, this is always the case for graphene due to the SOC. For ν 6= 0 it the product of δi must be equal to -1 which can only be obtained when multiple di are non zero. The minimal model where ν 6= 0 considered by Kane en Mele takes d1,2,5 6= 0 which conserves sz (see 67). The eigenstates for this system are 0 d5 − ||d|| 1 1 0 −d5 − ||d|| |u1 i = |u i = (71) 2 0 N1 N2 d1 + id2 d1 + id2 0 p Here N1,2 are normalization constants and ||d|| = d21 + d22 + d25 . For the limit that d1 , d2 → 0 these states can be rewritten using d1 + id2 = t exp[iθ]. p 0 d5 − |d5 | 1 + (t/d5 )2 p 2 1 1 0 −d5 − |d5 | 1 + (t/d5 ) |u1 i = |u2 i = (72) 0 t exp[iθ] N1 N2 t exp[iθ] 0 When d1 , d2 → 0 then t → 0 and the phase θ can not be well defined leading to a non-trivial Chern number for both bands. Using (61) it can be seen that this gives the non-trivial solution ν = 1 resulting in a QSH state. [10][13] The second model is another 2D QSH insulator proposed by Bernevig-Hughes-Zhang (2006) which was the first QSH system to be actually realized. Unlike the graphene model where a 23

Figure 13: bandstructure of HgTe/CdTe quantum well [23] Dirac point is opened here the QSH phase is induced by a band inversion. The system studied is a layer of HgTe sandwiched between layers of CdTe, these two lattices are arranged in a diamond structure formed by two interpenetrating face-centered-cubic lattices with a different atom at each sublattice. The interesting properties of this system are due to the strong spin orbit coupling in HgTe, this element has an inverted bandstructure near the Fermi level that is s-type conduction and p-type valence band. The bandstructure of CdTe has a p-type conduction band and an s-type valence band. By growing the layers of CdTe and HgTe in a different thickness the gap in the bandstructure can be changed to the point where the bands invert and HgTe terms dominant (figure 13). [4][16] The band inverted system gives rise to a QSH state, to show this the non inverted system will be examined and then it will be shown that for a small band gap non-trivial solutions for this system are possible. The non inverted system can be described by following effective Hamiltonian H(k) 0 (73) Hef f (k) = 0 H(−k)∗ where.

~ H(k) = d0 (k)1 + d(k) · ~σ

(74)

These block matrices are spanned by the states describing the s and p bands, mj = (±1/2, ±3/2) with a plus sign for the upper block and the lower block its time reversed partner. To not make this example overcomplicated the following parameters are taken from the literature. ~ d(k) = (Akx , −Aky , M (k)) 2

M (k) = M − B(k )

(75) (76)

The parameters A and M are dependent on the band gap geometry. The mass of the gap around k = 0 is given by M (k unlike the Dirac points in graphene this effective mass is not a relativistic dispersion and besides the Dirac mass term M also a classical mass term B shall be taken in to account (which is quadratic in k).[10][16]

24

For this system to have non-trivial topology one can study the Chern number generated by the separate bands as in (61). To find the band Chern number the methods from chapter 2.4 can be used. ˆ 1 ˆ · [∂k d(k) ˆ × ∂k d(k)] ˆ d2 kd(k) (77) C± = ± x y 4π From the previous equation can be seen that the system only displays non-trivial topological behavior if the M − Bk2 has a relative positive sign near k = 0 and negative sign near the edges of the BZ. [4][23] The edge states for QSH systems can be found in a similar fashion to chapter 3.2 by using the block matrix interpetation as seen with the HgTe/CdTe example. Both the block matrices can be transformd into a position dependent basis and can be solved to find the two spin filtered edge states. An equivalent block matrix form can be used to describe the graphene QSH system, this shall be used in the next chapter. [10]

5

Floquet Topological insulators

The goal of this chapter is to describe the effects of time periodic perturbation on the Chern and Z2 insulators. A perturbation will extend the periodicity of the Bloch Hamiltonian to the time domain. This makes it possible to change the topological properties of the system by other means then just statical perturbations such as magnetic fields or SOC and thus grant a higher degree of tunabillity of the system. Perturbations by electro magnetic radiation will be considered which add a periodic time dependent term to the many body Hamiltonian the periodicity of this system can be described using Floquet theory (see chapter 2.1). [6][23]

5.1

Floquet Chern number

The unperturbed system considered in this section can either be as a 4 × 4 Hamiltonian for simplicity only we consider these Hamiltonians to be comprised of two 2 × 2 block Hamiltonians. The time independent 2 × 2 Hamiltonian is of the form as described in section 4.3. H0 (k) = h0 (k)1 + ~h(k) · ~σ

(78)

Perturbing these Hamiltonians with electro magnetic radiation a time dependent driving potential is added to H0 V(k, t) = V(k) · ~σ cos(ωt) (79) The previous potential describes a linearly polarized monochromatic wave but different perturbations (such as circularly polarized light) can be considered leading to different effects on the system. Using the processes described in section 2.1 a Floquet Hamiltonian can be defined of the form HF (k) = n0 (k)1 + ~n(k) · ~σ (80) Important to note is that the Floquet approach can only be used if T is the shortest time scale of the system, this can be more easily obtained for low temperature systems and high frequency driving potentials. [6][23] The form of (80) is equivalent to (78) with the methods described in chapter 2.4 a Floquet Chern number can be defined ˆ 1 CF = d2 k(∂kx n ˆ × ∂ky n ˆ) · n ˆ (81) 4π here n ˆ (k) = ~n(k)/|~n(k)|. The Floquet Chern number can be non zero while the regular Chern number of the time independent system is not, this implies that a system can have chiral edge states while the unperturbed system is a trivial Chern or Z2 insulator.[6] 25

5.2

Implications and models

In Haldanes graphene model a magnetic field is used to open up a gap at the Dirac points changing Dirac mass across the BZ, creating a non-trivial Chern insulator with a QH state. With polarized light it is also possible to open up a gap at the Dirac points called a photo induced gap. The time dependent Hamiltonian for such a system near the Dirac points is of the following form H(k, t) = vF (~σ q + eA(t)) (82) where e is the electron charge and as in section 3.1 the momentum near the Dirac points is linearaly expanded such that q = k ± K. The difference here is a vector potential A(t) is added which is the coupling term with the electro magnetic radiation defined as. A = (−A0 sin(ωt), −A0 sin(ωt − φ))

(83)

Here φ is the is the polarization angle of the wave and A0 ω = E0 the field amplitude, the wave is taken to be perpendicular incident to the plane of the graphene lattice. For small laser power eA0 vF