Oddelek za fiziko Seminar 1 b 1. letnik, II. stopnja Topological insulators Author: Žiga Kos Supervisor: prof. dr. Dragan Mihailović Ljubljana, June 24, 2013 Abstract In the seminar, the basic ideas behind the theory of topological insulators will be presented. At first, quantum Hall and quantum spin Hall states will be explained, which show important similarities to a 3-dimensional topological insulator. Symmetrical and topological properties of these states will be emphasised. Experimental realization of such states will be explained and the scope for future and active research will be mentioned.
Contents 1 Introduction 3 2 Quantum Hall effect 4 2.1 Hall effect.......................................... 4 2.2 Hall conductivity...................................... 4 2.3 Edge states......................................... 6 3 Quantum spin Hall effect 7 3.1 Edge states protected by time reversal invariance.................... 7 3.2 HgTe-CdTe quantum wells................................. 9 4 3D topological insulators 10 4.1 Experimental observations................................. 11 4.2 Applications of topological insulators........................... 12 5 Conclusion 13 2
1 Introduction Usually, states of matter are recognized by which symmetry they spontaneously break. For example, translational symmetry in crystals or rotational symmetry in liquid crystals. In 1980, a new state of matter, describing the quantum Hall effect was proposed. It does not spontaneously break any symmetries. Instead of geometrical order, the quantum Hall state is described in terms of topological order. Although quantum Hall effect is not the main topic of this seminar, it is included, because of a great resemblance to the quantum spin Hall effect. The generalization of quantum spin Hall effect in 3 dimensions is called a topological insulator. The main characteristic of these effects is that such materials are insulators in the bulk, but have surface conducting states. They can be described by topological quantum numbers. Topology is a branch of mathematics that can be used to describe such properties of objects that do not change under continuous transformations. The most commonly used example is the topological equivalency between a torus and a coffee mug (figure 1), since one can be continuously transformed into another without closing or opening any holes. Another example would be the topology of knots (figure 1), where two knots on a closed loop are equivalent if they can be transformed one into another without cutting a string. To describe such structures, topological invariants are defined. Figure 1: a) topological equivalence between a mug and a donut, since they both have a single hole. To transform them into an orange, the hole must be closed. b) the simplest non-trivial knot the trefoil knot compared to the trivial closed loop the unknot. [1],[2] At the contact of two materials, the contact regions can exist, one with trivial topology and one with non trivial, where the topological quantum numbers have to change between the two cases. Since topological invariants are integers, this change can not be gradual. This indicates interesting behavior on the surface. Two theories are generally used to describe the topological state. Topological band theory tries to evaluate topological invariants from the band structure. It is valid only for non-interacting systems. Topological field theory takes different approach and is also valid for interacting systems. Under certain conditions, it can be reduced to topological band theory. The determination of topological invariants is not the main topic of this seminar, but we comment on the influence that they have on physical systems. 3
2 Quantum Hall effect 2.1 Hall effect When a conducting material is put in the magnetic field, and there is an electric current through the conductor, a voltage difference which is transverse to the magnetic field and the electric current occurs (see figure 2). This is known as Hall effect. Figure 2: Hall effect measurement setup. [3] The effect occurs due to the Lorentz force that separates opposite charged moving charges on the opposite sides of the conducting plate. The value of the Hall voltage V H depends on the properties of the conducting material, such as the density of the carrier electrons or whether a material is a conductor or a semiconductor. The above explanation includes electrons as particles moving through a conductor. It provides good evaluation of Hall voltage at room temperatures, but is invalid at extremely low temperature and high magnetic field. It these conditions we speak of quantum Hall effect. For it s discovery, von Klitzing was awarded Nobel Prize in 1985. 2.2 Hall conductivity In classical image, free electrons, without any momentum in direction of an external magnetic field, move in circular orbits. In quantum limit, however, the energy of these orbits is quantized in Landau levels ([1]): ɛ m = ω c (m + 1 ), (1) 2 where ω c represents cyclotron frequency ω c = eb m. The Hall conductivity, defined as σ xy = also quantized: σ xy = N e2 h. (2) N in equation 2 corresponds to the number of filled Landau levels. The degeneracy of each Landau level grows proportionally with the magnetic field strength ([4]), and so does the energy of each level. If the strength of the magnetic field is increased, the energy of higher Landau levels may raise above Fermi energy. In that case, the electrons would occupy lower levels and Hall conductivity would decrease. This is shown on figure 3 in terms of the Hall resistance, which is the inverse value of σ xy. The way to describe this system topologically ([1]) is to introduce the Chern number. In order for lattice translations to commute with one other in the presence of external perpendicular I V H is 4
12 N=2 10 8 6 4 6 5 4 3 2 1 2 3 4 5 6 7 magnetic field (T) Figure 3: Hall resistance as the strength of the magnetic field gets increased. The numbers above plateaus show how many Landau levels are filled. [5],[6] magnetic field, unit area enclosing a flux quantum must be defined ([1]). Only than can be states labeled with 2D crystal momentum ([1]). When a wave vector k is transported around Brillouin zone, Bloch wave function u(k) acquires a Berry phase, which is equal to the line integral of Berry connection A m : A m (k) = i u m (k) k u m (k), (3) where k represents the gradient in k-space. Berry flux is defined as F m = A m. (4) The Chern invariant is the integral of the Berry flux over the whole Brillouin zone: n m = 1 d 2 kf m. (5) 2π The integration is represented graphically in figure 4. Since the surface integral of the Berry flux is a multiple of 2π, n m is an integer. Chern number is the sum of n m over all occupied bands ([1]), 1. BZ Figure 4: Berry flux over the Brillouin zone. which are formed from Landau levels modified by the periodic potential: n = n m. (6) 5
Chern number in fact notes the number of filled Landau levels. Thouless, Kohmoto, Nightingale and den Nijs showed, that Chern number n is identical to N in equation 2. In mathematics, Chern number is associated with vector bundles. In our case, we consider mapping from Brillouin zone to the Hamiltonian in a Hilbert space. Chern number describes equivalence classes of H(k) that can be smoothly transformed into one another ([1]). The topological background of quantum Hall effect helps to explain why σ xy is quantized so rigidly. Since the topological invariants do not change under small variation of the Hamiltonian, the measurements of Hall conductivity can be very precise, despite of the imperfections of the material. It is of great importance in metrology. The precision of the measurement of the fine structure constant using quantum Hall effect is just slightly below the precision of the experiment measuring the electron s anomalous magnetic moment ([6]). Fine structure constant is known to the precision of 3 10 12, whereas the measurement involving quantum Hall effect gives the precision of 2 10 11 ([6]). 2.3 Edge states Quantum Hall effect takes place in two dimensions. It exhibits interesting behavior at the edge of the sample. In a semiclassical approximation this can be explained by orbits of the electrons. In the middle of the sample, there are no restrictions for the orbits, but when electrons hit the edge, they bounce back (see figure 5). But even after collision the electron propagates in the same direction. Since there are no states available for backscattering ([1]) this process is dissipationless. More adequate explanation is provided by the use of topological quantum numbers. The quantum Hall state has non-zero Chern number, opposed to the trivial insulator with Chern number 0. A trivial insulator, without any magnetic field required, is an example of a material with Chern number 0. For the Chern number to change, the energy gap on the edge between two materials must vanish. In that case the edge states become conductive. This conductive edges are the reason for the perfectly quantized Hall conductivity. The exact shape of the edge is not important. Insulator n=0 E Conduction band EF B Quantum Hall state n=1 Valence band 0 k Figure 5: Electrons orbits in quantum Hall state. In the bulk, Fermi energy lies in the energy gap. Boundary states have no energy gap they are conductive. Energy bands are shown in dependence on the component of the wave vector, which is parallel to the edge. Quantum Hall effect is the first example, how the change in topology between two bulk materials causes special robust edge states. Another, more recent example is quantum spin Hall effect. 6
3 Quantum spin Hall effect Quantum Hall effect requires large magnetic fields, which can be unfavorable in experimental realization. It also breaks time reversal symmetry. In 2005, quantum spin Hall state was proposed, which is time reversal invariant. The basic idea of such state is, that it does not exhibit quantum Hall effect. The quantum Hall states can be referred to as chiral, since the electrons propagate only in one direction along the edge of the sample and are not identical to their mirror image. In the quantum spin Hall effect, this is not the case. Electrons with spin up move in one direction and electrons with spin down in the other ([7]). Such states, where spin and momentum are connected, are referred to as helical. The role of the magnetic field is then taken over by the spin-orbit interaction. In figure 6 the electron transport is shown for the quantum Hall effect and the quantum spin Hall effect. In 1D, an electron can move backwards or forwards. In 2D, the quantum Hall effect separates this movement into two lanes. This transport can be described as chiral. It is clearly not time reversal invariant. In the quantum spin Hall state, electron has four degrees of freedom. It can have spin up or spin down and can move in two directions. In this state chiral movement for one spin orientation and movement in the opposite direction for the other spin orientation are combined. Under time reversal, particle s spin is reversed as well as it s momentum. As seen in figure 6, the quantum spin Hall state is time reversal invariant. The fact that the spin of the electron dictates the direction of it s momentum strongly suggests that spin-orbit coupling must be present. After certain approximations, spin-orbit term in the Hamiltonian can be expressed as H so = λl S, where λ > 0. if we look closely at the figure 6, we can see that for each electron, it s spin has the opposite orientation as it s angular momentum. Given the Hamiltonian H so, this corresponds to the lowest energy. Quantum Hall Quantum spin Hall Figure 6: Electron transport in the edge states. 1D movement is separated into 2 (quantum Hall) or 4 (quantum spin Hall) channels. 3.1 Edge states protected by time reversal invariance Not only are the edge quantum spin Hall states time reversal invariant, time reversal symmetry even protects them from disorder. Let T be the time reversal operator. For electrons, which have spin 1/2, it is true that T 2 = 1 ([1]). According to Kramers theorem, every eigenstate of the time reversal invariant Hamiltonian is at least twofold degenerate ([1]). Scattering between a state and its degenerate Kramers state is forbidden ([8]). Two lanes of electrons on the edge of a quantum spin Hall sample represent such two states. That is the reason, why electron transport along the edge is dissipationless. This statement is not completely true, since electron-electron interactions and phonon excitation can cause inelastic scattering. The mean free path of an electron is believed 7
to be of the order of few microns at low temperatures ([9]). Transport experiments (Hall resistance measurements) on HgTe quantum wells are performed at temperature below 30 mk on distances of few µm ([10]). The absence of the elastic scattering can be explained by a semiclassical example as well. On the edge, there are only two states. Dissipation can occour only from one state to another. There are two possibilities of this, represented on the figure 7. On the left example, electron s spin rotates clockwise by the angle of π and on the right example by the angle of π. On rotation of 2π, which is exactly the difference between two cases here, an electron s wave function acquires additional factor of 1. Therefore, both states interfere destructively and backscattering is not allowed. This holds if the impurity around which an electron would backscatter is non-magnetic. Magnetic impurity would break time reversal symmetry and the electron transport would no longer be dissipationless. Figure 7: Two possibilities of scattering around impurity. If impurity is non-magnetic both interfere destructively with one another. [8] In time reversal invariant systems, the Chern number is always zero ([1]). Additional topological invariant must be defined to describe the topological states of such systems. A new invariant is Z 2 invariant, which means it can have only 0 or 1 values ([1]). Due to Kramers theorem, edge state dispersions must cross where k = k, that is at k = 0 and k = π/a ([1]) at the edge of the Brilloiun zone. The dispersion lines can cross Fermi energy even or odd number of times, which is represented in figure 8. An even number of crossing can be eliminated by slight changes of dispersion. On the other hand, an odd number of crossings can not be eliminated. That is the case of topologically nontrivial insulator. This even or odd difference is the reason for the Z 2 invariant. E Conduction band E Conduction band EF EF Valence band Valence band 0 k 0 k Figure 8: Dispersion of 1D edge states intersects Fermi energy even or odd number of times, which leads to topologically trivial (even times) or nontrivial (odd) states. k represents a projection of 2D Brillouin zone on the axis parallel to the edge. [9] 8
3.2 HgTe-CdTe quantum wells In physics it does not happen often that a state of matter is first theoretically predicted, and only later experimentally realized. Topological insulators are such an example. In 2006, a mechanism to find topological insulators was proposed. It was known that quantum spin Hall effect would occur in the case of band inversion due to spin orbit coupling. It was predicted to occur in HgTe-CdTe quantum wells ([7]) and only later experimentally realized. The idea behind the choice of HgTe is band inversion. In CdTe, close to the Γ (k = 0) point, the valence band is formed from the p-type Γ 8 band and the conduction band is formed from the s-type Γ 6 band ([9]). In HgTe, spin-orbit coupling is much stronger. It is a relativistic effect and gets stronger in heavier elements. In HgTe, the Γ 6 and Γ 8 bands are inverted (figure 9). The band gap changed from 1.6 ev in CdTe to 300 mev in HgTe ([9]). Figure 9: Band structure of HgTe and CdTe around Γ point and inside the quantum well. [9] Quantum well is a structure, where one semiconductor is sandwiched between another with larger band gap to form a potential well for particles and holes. Easy manipulation of discrete energy levels makes quantum wells useful for optical devices. In our case, a layer of HgTe is sandwiched between two layers of CdTe. CdTe is chosen, because it has similar lattice constant as HgTe, but smaller spin-orbit interaction. In a well, only a rotational symmetry in the plane remains ([11]). 6 bands from figure 9 form 3 subbands E1, H1 and L1, each with two different spin orientations. Only E1 and H1 bands are important for this model. In the basis of E1, m J = 1 2, H1, m J = 3 2, E1, m J = 1 2, H1, m J = 3 2, one can write the Hamiltonian as ([7]): M(k) A(k x + ik y ) 0 0 H(k) = ɛ(k)1 + A(k x ik y ) M(k) 0 0 0 0 M(k) A(k x ik y ) (7) 0 0 A(k x + ik y ) M(k) ɛ(k) = C + Dk 2, M(k) = M Bk 2 (8) ɛ(k) is unimportant bending of the bands and the rest has a form of a Dirac Hamiltonian. The crossing point in the dispersion of the edge states is therefore called Dirac point (as seen on figure 9
10) or Dirac cone and is reoccurring theme in 3D topological insulators as well. M represents the mass or the gap parameter between E1 and H1 states. The thickness of a quantum well determines the energy gap. For thin wells, CdTe will dominate and for thick wells, the band will be in inverted regime ([11]), as shown on figure 9. At critical thickness d c, the bands E1 and H1 must cross. In HgTe-CdTe quantum wells, this critical thickness is d c 6.3 nm. As the thickness is varied over d c, the gap parameter M changes sign. It can be shown that the solutions of such Hamiltonian are edge states in which opposite spins have opposite momentum ([9]), shown on figure 6. Figure 10: Calculated energy spectrum for the thick Hg-Te quantum well. In blue and red, edge states are visible. They intersect at the Dirac point. [9] 4 3D topological insulators In 2006, it was predicted that the quantum spin Hall state can be generalized in three dimensions ([1]), so that conductive states on the surface would exist. In 2008, first such topological insulator (Bi 1 x Sb x ) was experimentally realized. A year after, so called second generation of topological insulators was identified, the most important example of which is probably Bi 2 Se 3 ([1]). Instead of one, as in the quantum spin Hall effect, four Z 2 invariants (ν 0, ν 1, ν 2, ν 3 ) must be defined to characterise a topological insulator ([1]). For the surface states, a 2D crystal momentum can be defined. In a quadratic Brillouin zone, there are four time reversal invariant points (Γ 1, Γ 2, Γ 3, Γ 4 ), which are seen on figure 11(a). These states must be degenerate. Each of a degeneracies is represented by a Dirac point (figure 11(c)). The way these points are connected defines the nature of a topological insulator. If a surface band between two Γ points intersects the Fermi energy odd or even number of times, defines wheater the material is trivial or nontrivial toplogical insulator ([1]). This is analogous to 2D quantum spin Hall effect and to figure 8. One way to construct 3D topological insulator is to stack layers of 2D quantum spin Hall insulators ([1]). Such an example is characterized by ν 0 = 0 and is named a weak topological insulators. A possible Fermi surface is shown on figure 11(a). The other three invariants characterize the orientation of the layers. Such surface states are not protected by time reversal symmetry. ν 0 = 1 identifies strong topological insulators (figure 11(b)) and is defined as even or odd number of Γ points inside surface Fermi circle ([1]). Bi 1 x Sb x, the first realized topological insulator, has 10
Figure 11: Fermi energy in the surface Brilloiun zone of (a) an weak and (b) a strong topological insulator. On figure (c) is a Dirac point in Γ 1. [1] complicated surface states and small band gap. The second generation topological insulators are much better improvement. Bi 2 Se 3, for example, has only one Dirac cone on the surface and a band gap of 0.3 ev (k b 3600K), which makes it suitable for use at room temperature ([1]). 4.1 Experimental observations ARPES (angle resolved photoemission spectroscopy) is the most commonly used technique to identify topological insulators. A photon ejects an electron from the crystal and then the electron s momentum is determined ([1]). Photon s momentum is also varied. Since the dispersion of surface states happens only in two dimensions, these states can be clearly distinguished from the bulk states that disperse along all directions of the wave vector ([1]). ARPES can be used to determine spin of the electrons as well ([1]), which is important, since spin and momentum of the surface states are deeply connected. It s disadvantage is that this method can not measure band states above Fermi energy. k y E K k x K 0 0.2 0.4-0.2 0 0.2 Figure 12: Spin dependent ARPES of Bi 2 Se 3 (first picture) and ARPES of Bi 2 Te 3 (second picture). From the first picture, it is visible, that surface spins are perpendicular to the momentum. The second picture shows the linear disperssion of surface-state band (SSB) above the Dirac point. The bulk-valence band (BVB) below Dirac point is visible. [7] Another possibility is to measure Hall resistance. This method can be used on 2D topological insulators. Since there are 4 channels in quantum spin Hall effect, instead of 2 as in quantum Hall 11
effect, resistance is quantized with the base level of h/2e 2. Figure 13 shows the measurement of resistance with respect to gate voltage for different thicknesses of the HgTe quantum well. When the thickness of the HgTe layer is beyond d c, system behaves as an insulator (left picture). On the right picture are the measurements for d > d c and the width of the samples is varied. They all show the same quantization of h/2e 2 at small enough voltages. This proves edge conducting states. gate voltage gate voltage Figure 13: Measured Hall resistance on HgTe quantum wells for d < d c (left picture) and d > d c (right picture). Three different curves present measurements for three different sample widths. Since they all reach the plateau at h 2e, this proves edge states. [7] Besides already mentioned methods, the surface Hall conductivity can also be observed by optical methods or by measuring the magnetic fields, induced by the surface currents ([1]). Another possibility is the use of scanning tunneling microscopy, which can provide additional information about impurities or the edges on the surface ([9]). 4.2 Applications of topological insulators With the use of topological insulators, some interesting quasiparticles can be created, that are otherwise very elusive. Such example would be Marjorana fermions. They can be created by combining topological insulators with superconductors. Majorana fermions are particles which are also their antiparticles. They are of great importance for quantum computing. Seperated pair of Majorana bound states represents a qubit ([1]). Quantum information would be topologically protected. Exchanging Marjorana states creates non-abelian statistic, meaning that the state of the system depends on the order in which the exchange was performed ([12]). Because of these properties, Marjorana states would be an important building block of a topological quantum computer. 12
5 Conclusion Topological insulators are new and exiting field of physics, that has evolved in the last decade. They are a great achievement for the theoretical physics, since their existence was first predicted by theoretical approach. The classification of insulators into topological classes has proven to be an powerful approach and can be extended to topological superconductors. Besides topological superconductors, the main theoretical challenges for the future are believed to be electron-electron interactions, disorder effects and fractional topological insulators ([9]). For the experimental physics, the challenges include the production of pure enough materials, that are completely insulating in the bulk, to tune Fermi level close to the Dirac point and to detect exotic quasiparticles ([9]). As explained in the seminar, topological insulators are a playground to produce and to experiment with many quaiparticles, some of them have not been previously produced anywhere else. The proximity effect of superconductors can produce Majorana fermions. These could be the basis of the topological quantum computers. Dissipationless channels in quantum spin Hall states could be used for their low power consumption. Topological insulators have the potential for spintronics devices, infrared detectors and thermoelectric applications ([9]). The field has extended way above the initial expectations and certainly holds many new interesting developments in the future. 13
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