Topological insulators
|
|
- Jared Todd
- 6 years ago
- Views:
Transcription
1 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.
2 Contents 1 Introduction 3 2 Quantum Hall effect Hall effect Hall conductivity Edge states Quantum spin Hall effect Edge states protected by time reversal invariance HgTe-CdTe quantum wells D topological insulators Experimental observations Applications of topological insulators Conclusion 13 2
3 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
4 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 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
5 12 N= 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
6 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 , whereas the measurement involving quantum Hall effect gives the precision of ([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
7 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
8 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
9 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) 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 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
11 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 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
12 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
13 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
14 References [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys (2010) [2] J. E. Moore, Nature (2010) [3] effect, (available on May 7, 2013) [4] hall effect, (available on May 26, 2013) [5] M. A. Paalanen et al, Phys. Rev. B (1982) [6] J. E. Avron et al, Phys. Today (2003) [7] X. L. Qi and S. C. Zhang, Phys. Today (2010) [8] J. Maciejko et al, Annu. Rev. Condens. Matter Phys (2011) [9] X. L. Qi and S. C. Zhang, Rev. Mod. Phys (2011) [10] M. König et al, J. Phys. Soc. Jpn (2008) [11] B. A. Bernevik et al, Science (2006) [12] fermion, (available on May 22, 2013) [13] R. Li et al, Nature Phys (2010) 14
Notes on Topological Insulators and Quantum Spin Hall Effect. Jouko Nieminen Tampere University of Technology.
Notes on Topological Insulators and Quantum Spin Hall Effect Jouko Nieminen Tampere University of Technology. Not so much discussed concept in this session: topology. In math, topology discards small details
More informationTopological Insulators
Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect
More informationTopological insulators
http://www.physik.uni-regensburg.de/forschung/fabian Topological insulators Jaroslav Fabian Institute for Theoretical Physics University of Regensburg Stara Lesna, 21.8.212 DFG SFB 689 what are topological
More informationarxiv: v1 [cond-mat.other] 20 Apr 2010
Characterization of 3d topological insulators by 2d invariants Rahul Roy Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK arxiv:1004.3507v1 [cond-mat.other] 20 Apr 2010
More informationLes états de bord d un. isolant de Hall atomique
Les états de bord d un isolant de Hall atomique séminaire Atomes Froids 2/9/22 Nathan Goldman (ULB), Jérôme Beugnon and Fabrice Gerbier Outline Quantum Hall effect : bulk Landau levels and edge states
More informationEffective Field Theories of Topological Insulators
Effective Field Theories of Topological Insulators Eduardo Fradkin University of Illinois at Urbana-Champaign Workshop on Field Theoretic Computer Simulations for Particle Physics and Condensed Matter
More informationIntroduction to topological insulators. Jennifer Cano
Introduction to topological insulators Jennifer Cano Adapted from Charlie Kane s Windsor Lectures: http://www.physics.upenn.edu/~kane/ Review article: Hasan & Kane Rev. Mod. Phys. 2010 What is an insulator?
More information3D topological insulators and half- Heusler compounds
3D topological insulators and half- Heusler compounds Ram Seshadri Materials Department, and Department of Chemistry and Biochemistry Materials Research Laboratory University of California, Santa Barbara
More informationSymmetry, Topology and Phases of Matter
Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum
More informationIntroductory lecture on topological insulators. Reza Asgari
Introductory lecture on topological insulators Reza Asgari Workshop on graphene and topological insulators, IPM. 19-20 Oct. 2011 Outlines -Introduction New phases of materials, Insulators -Theory quantum
More informationTopological Defects inside a Topological Band Insulator
Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv: 0908.2691 Part 1: Outline A toy model of
More informationHartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany
Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany Part I and II Insulators and Topological Insulators HgTe crystal structure Part III quantum wells Two-Dimensional TI Quantum Spin
More informationTopological insulators. Pavel Buividovich (Regensburg)
Topological insulators Pavel Buividovich (Regensburg) Hall effect Classical treatment Dissipative motion for point-like particles (Drude theory) Steady motion Classical Hall effect Cyclotron frequency
More informationTopological Insulators and Superconductors
Topological Insulators and Superconductors Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Lecture #3: Topological Superconductors, Majorana Fermions an Topological
More informationWhat is a topological insulator? Ming-Che Chang Dept of Physics, NTNU
What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU A mini course on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G K 0 G 0 G>0 G=0 K 0 G=0 G
More informationTopological insulator (TI)
Topological insulator (TI) Haldane model: QHE without Landau level Quantized spin Hall effect: 2D topological insulators: Kane-Mele model for graphene HgTe quantum well InAs/GaSb quantum well 3D topological
More informationTopological Physics in Band Insulators II
Topological Physics in Band Insulators II Gene Mele University of Pennsylvania Topological Insulators in Two and Three Dimensions The canonical list of electric forms of matter is actually incomplete Conductor
More informationARPES experiments on 3D topological insulators. Inna Vishik Physics 250 (Special topics: spectroscopies of quantum materials) UC Davis, Fall 2016
ARPES experiments on 3D topological insulators Inna Vishik Physics 250 (Special topics: spectroscopies of quantum materials) UC Davis, Fall 2016 Outline Using ARPES to demonstrate that certain materials
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationTopological Kondo Insulator SmB 6. Tetsuya Takimoto
Topological Kondo Insulator SmB 6 J. Phys. Soc. Jpn. 80 123720, (2011). Tetsuya Takimoto Department of Physics, Hanyang University Collaborator: Ki-Hoon Lee (POSTECH) Content 1. Introduction of SmB 6 in-gap
More informationTopological Insulators in 3D and Bosonization
Topological Insulators in 3D and Bosonization Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter: bulk and edge Fermions and bosons on the (1+1)-dimensional
More informationTopological insulator with time-reversal symmetry
Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationField Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti)
Field Theory Description of Topological States of Matter Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Topological States of Matter System with bulk gap but non-trivial at energies below
More informationTopological Insulator Surface States and Electrical Transport. Alexander Pearce Intro to Topological Insulators: Week 11 February 2, / 21
Topological Insulator Surface States and Electrical Transport Alexander Pearce Intro to Topological Insulators: Week 11 February 2, 2017 1 / 21 This notes are predominately based on: J.K. Asbóth, L. Oroszlány
More informationQuantum Hall effect. Quantization of Hall resistance is incredibly precise: good to 1 part in I believe. WHY?? G xy = N e2 h.
Quantum Hall effect V1 V2 R L I I x = N e2 h V y V x =0 G xy = N e2 h n.b. h/e 2 = 25 kohms Quantization of Hall resistance is incredibly precise: good to 1 part in 10 10 I believe. WHY?? Robustness Why
More informationPOEM: Physics of Emergent Materials
POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Tutorials: May 24, 25 (2017) Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction
More informationTopological Insulators
Topological Insulators A new state of matter with three dimensional topological electronic order L. Andrew Wray Lawrence Berkeley National Lab Princeton University Surface States (Topological Order in
More informationTopological insulators and the quantum anomalous Hall state. David Vanderbilt Rutgers University
Topological insulators and the quantum anomalous Hall state David Vanderbilt Rutgers University Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z
More informationteam Hans Peter Büchler Nicolai Lang Mikhail Lukin Norman Yao Sebastian Huber
title 1 team 2 Hans Peter Büchler Nicolai Lang Mikhail Lukin Norman Yao Sebastian Huber motivation: topological states of matter 3 fermions non-interacting, filled band (single particle physics) topological
More informationOrganizing Principles for Understanding Matter
Organizing Principles for Understanding Matter Symmetry Conceptual simplification Conservation laws Distinguish phases of matter by pattern of broken symmetries Topology Properties insensitive to smooth
More informationElectronic transport in topological insulators
Electronic transport in topological insulators Reinhold Egger Institut für Theoretische Physik, Düsseldorf Alex Zazunov, Alfredo Levy Yeyati Trieste, November 011 To the memory of my dear friend Please
More informationMassive Dirac Fermion on the Surface of a magnetically doped Topological Insulator
SLAC-PUB-14357 Massive Dirac Fermion on the Surface of a magnetically doped Topological Insulator Y. L. Chen 1,2,3, J.-H. Chu 1,2, J. G. Analytis 1,2, Z. K. Liu 1,2, K. Igarashi 4, H.-H. Kuo 1,2, X. L.
More informationTime Reversal Invariant Ζ 2 Topological Insulator
Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary
More informationThe Quantum Hall Effect
The Quantum Hall Effect David Tong (And why these three guys won last week s Nobel prize) Trinity Mathematical Society, October 2016 Electron in a Magnetic Field B mẍ = eẋ B x = v cos!t! y = v sin!t!!
More informationVisualizing Electronic Structures of Quantum Materials By Angle Resolved Photoemission Spectroscopy (ARPES)
Visualizing Electronic Structures of Quantum Materials By Angle Resolved Photoemission Spectroscopy (ARPES) PART A: ARPES & Application Yulin Chen Oxford University / Tsinghua University www.arpes.org.uk
More informationWelcome to the Solid State
Max Planck Institut für Mathematik Bonn 19 October 2015 The What 1700s 1900s Since 2005 Electrical forms of matter: conductors & insulators superconductors (& semimetals & semiconductors) topological insulators...
More informationPhysics of Semiconductors
Physics of Semiconductors 13 th 2016.7.11 Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo Outline today Laughlin s justification Spintronics Two current
More informationarxiv: v1 [cond-mat.mes-hall] 29 Jul 2010
Discovery of several large families of Topological Insulator classes with backscattering-suppressed spin-polarized single-dirac-cone on the surface arxiv:1007.5111v1 [cond-mat.mes-hall] 29 Jul 2010 Su-Yang
More informationMinimal Update of Solid State Physics
Minimal Update of Solid State Physics It is expected that participants are acquainted with basics of solid state physics. Therefore here we will refresh only those aspects, which are absolutely necessary
More informationDisordered topological insulators with time-reversal symmetry: Z 2 invariants
Keio Topo. Science (2016/11/18) Disordered topological insulators with time-reversal symmetry: Z 2 invariants Hosho Katsura Department of Physics, UTokyo Collaborators: Yutaka Akagi (UTokyo) Tohru Koma
More information5 Topological insulator with time-reversal symmetry
Phys62.nb 63 5 Topological insulator with time-reversal symmetry It is impossible to have quantum Hall effect without breaking the time-reversal symmetry. xy xy. If we want xy to be invariant under, xy
More informationTopological Insulators and Ferromagnets: appearance of flat surface bands
Topological Insulators and Ferromagnets: appearance of flat surface bands Thomas Dahm University of Bielefeld T. Paananen and T. Dahm, PRB 87, 195447 (2013) T. Paananen et al, New J. Phys. 16, 033019 (2014)
More informationExperimental Reconstruction of the Berry Curvature in a Floquet Bloch Band
Experimental Reconstruction of the Berry Curvature in a Floquet Bloch Band Christof Weitenberg with: Nick Fläschner, Benno Rem, Matthias Tarnowski, Dominik Vogel, Dirk-Sören Lühmann, Klaus Sengstock Rice
More informationSUPPLEMENTARY INFORMATION
A Dirac point insulator with topologically non-trivial surface states D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, and M.Z. Hasan Topics: 1. Confirming the bulk nature of electronic bands by
More informationv. Tε n k =ε n k T r T = r, T v T = r, I v I = I r I = v. Iε n k =ε n k Berry curvature: Symmetry Consideration n k = n k
Berry curvature: Symmetry Consideration Time reversal (i.e. motion reversal) 1 1 T r T = r, T v T = v. Tε n k =ε n k n k = n k Inversion Symmetry: 1 1 I r I = r, I v I = v. Iε n k =ε n k n k = n k θ
More informationTopological Defects in the Topological Insulator
Topological Defects in the Topological Insulator Ashvin Vishwanath UC Berkeley arxiv:0810.5121 YING RAN Frank YI ZHANG Quantum Hall States Exotic Band Topology Topological band Insulators (quantum spin
More informationFloquet Topological Insulators and Majorana Modes
Floquet Topological Insulators and Majorana Modes Manisha Thakurathi Journal Club Centre for High Energy Physics IISc Bangalore January 17, 2013 References Floquet Topological Insulators by J. Cayssol
More informationKonstantin Y. Bliokh, Daria Smirnova, Franco Nori. Center for Emergent Matter Science, RIKEN, Japan. Science 348, 1448 (2015)
Konstantin Y. Bliokh, Daria Smirnova, Franco Nori Center for Emergent Matter Science, RIKEN, Japan Science 348, 1448 (2015) QSHE and topological insulators The quantum spin Hall effect means the presence
More information3D Weyl metallic states realized in the Bi 1-x Sb x alloy and BiTeI. Heon-Jung Kim Department of Physics, Daegu University, Korea
3D Weyl metallic states realized in the Bi 1-x Sb x alloy and BiTeI Heon-Jung Kim Department of Physics, Daegu University, Korea Content 3D Dirac metals Search for 3D generalization of graphene Bi 1-x
More informationQuantum anomalous Hall states on decorated magnetic surfaces
Quantum anomalous Hall states on decorated magnetic surfaces David Vanderbilt Rutgers University Kevin Garrity & D.V. Phys. Rev. Lett.110, 116802 (2013) Recently: Topological insulators (TR-invariant)
More informationSUPPLEMENTARY INFORMATION
DOI: 1.138/NMAT3449 Topological crystalline insulator states in Pb 1 x Sn x Se Content S1 Crystal growth, structural and chemical characterization. S2 Angle-resolved photoemission measurements at various
More informationSymmetric Surfaces of Topological Superconductor
Symmetric Surfaces of Topological Superconductor Sharmistha Sahoo Zhao Zhang Jeffrey Teo Outline Introduction Brief description of time reversal symmetric topological superconductor. Coupled wire model
More informationTopological insulator part I: Phenomena
Phys60.nb 5 Topological insulator part I: Phenomena (Part II and Part III discusses how to understand a topological insluator based band-structure theory and gauge theory) (Part IV discusses more complicated
More informationTwo Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models
Two Dimensional Chern Insulators, the Qi-Wu-Zhang and Haldane Models Matthew Brooks, Introduction to Topological Insulators Seminar, Universität Konstanz Contents QWZ Model of Chern Insulators Haldane
More informationSupplementary Figure S1: Number of Fermi surfaces. Electronic dispersion around Γ a = 0 and Γ b = π/a. In (a) the number of Fermi surfaces is even,
Supplementary Figure S1: Number of Fermi surfaces. Electronic dispersion around Γ a = 0 and Γ b = π/a. In (a) the number of Fermi surfaces is even, whereas in (b) it is odd. An odd number of non-degenerate
More informationTopological Kondo Insulators!
Topological Kondo Insulators! Maxim Dzero, University of Maryland Collaborators: Kai Sun, University of Maryland Victor Galitski, University of Maryland Piers Coleman, Rutgers University Main idea Kondo
More informationGraphite, graphene and relativistic electrons
Graphite, graphene and relativistic electrons Introduction Physics of E. graphene Y. Andrei Experiments Rutgers University Transport electric field effect Quantum Hall Effect chiral fermions STM Dirac
More informationSplitting of a Cooper pair by a pair of Majorana bound states
Chapter 7 Splitting of a Cooper pair by a pair of Majorana bound states 7.1 Introduction Majorana bound states are coherent superpositions of electron and hole excitations of zero energy, trapped in the
More informationTopological Properties of Quantum States of Condensed Matter: some recent surprises.
Topological Properties of Quantum States of Condensed Matter: some recent surprises. F. D. M. Haldane Princeton University and Instituut Lorentz 1. Berry phases, zero-field Hall effect, and one-way light
More informationSpin Hall and quantum spin Hall effects. Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST
YKIS2007 (Kyoto) Nov.16, 2007 Spin Hall and quantum spin Hall effects Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST Introduction Spin Hall effect spin Hall effect in
More informationDefects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014
Defects in topologically ordered states Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 References Maissam Barkeshli & XLQ, PRX, 2, 031013 (2012) Maissam Barkeshli, Chaoming Jian, XLQ,
More informationObservation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator
Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator Authors: Yang Xu 1,2, Ireneusz Miotkowski 1, Chang Liu 3,4, Jifa Tian 1,2, Hyoungdo
More informationCoupling of spin and orbital motion of electrons in carbon nanotubes
Coupling of spin and orbital motion of electrons in carbon nanotubes Kuemmeth, Ferdinand, et al. "Coupling of spin and orbital motion of electrons in carbon nanotubes." Nature 452.7186 (2008): 448. Ivan
More informationNanostructured Carbon Allotropes as Weyl-Like Semimetals
Nanostructured Carbon Allotropes as Weyl-Like Semimetals Shengbai Zhang Department of Physics, Applied Physics & Astronomy Rensselaer Polytechnic Institute symmetry In quantum mechanics, symmetry can be
More informationLCI -birthplace of liquid crystal display. May, protests. Fashion school is in top-3 in USA. Clinical Psychology program is Top-5 in USA
LCI -birthplace of liquid crystal display May, 4 1970 protests Fashion school is in top-3 in USA Clinical Psychology program is Top-5 in USA Topological insulators driven by electron spin Maxim Dzero Kent
More informationSpin Superfluidity and Graphene in a Strong Magnetic Field
Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)
More informationFrom graphene to Z2 topological insulator
From graphene to Z2 topological insulator single Dirac topological AL mass U U valley WL ordinary mass or ripples WL U WL AL AL U AL WL Rashba Ken-Ichiro Imura Condensed-Matter Theory / Tohoku Univ. Dirac
More informationUniversal phase transitions in Topological lattice models
Universal phase transitions in Topological lattice models F. J. Burnell Collaborators: J. Slingerland S. H. Simon September 2, 2010 Overview Matter: classified by orders Symmetry Breaking (Ferromagnet)
More informationThe many forms of carbon
The many forms of carbon Carbon is not only the basis of life, it also provides an enormous variety of structures for nanotechnology. This versatility is connected to the ability of carbon to form two
More informationThe Quantum Spin Hall Effect
The Quantum Spin Hall Effect Shou-Cheng Zhang Stanford University with Andrei Bernevig, Taylor Hughes Science, 314,1757 2006 Molenamp et al, Science, 318, 766 2007 XL Qi, T. Hughes, SCZ preprint The quantum
More informationComposite Dirac liquids
Composite Dirac liquids Composite Fermi liquid non-interacting 3D TI surface Interactions Composite Dirac liquid ~ Jason Alicea, Caltech David Mross, Andrew Essin, & JA, Physical Review X 5, 011011 (2015)
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationBerry s phase in Hall Effects and Topological Insulators
Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.
More informationTopology of electronic bands and Topological Order
Topology of electronic bands and Topological Order R. Shankar The Institute of Mathematical Sciences, Chennai TIFR, 26 th April, 2011 Outline IQHE and the Chern Invariant Topological insulators and the
More informationElectrons in a weak periodic potential
Electrons in a weak periodic potential Assumptions: 1. Static defect-free lattice perfectly periodic potential. 2. Weak potential perturbative effect on the free electron states. Perfect periodicity of
More informationHIGHER INVARIANTS: TOPOLOGICAL INSULATORS
HIGHER INVARIANTS: TOPOLOGICAL INSULATORS Sponsoring This material is based upon work supported by the National Science Foundation Grant No. DMS-1160962 Jean BELLISSARD Georgia Institute of Technology,
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS
2753 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2011 Wednesday, 22 June, 9.30 am 12.30
More informationUniversal transport at the edge: Disorder, interactions, and topological protection
Universal transport at the edge: Disorder, interactions, and topological protection Matthew S. Foster, Rice University March 31 st, 2016 Universal transport coefficients at the edges of 2D topological
More informationWiring Topological Phases
1 Wiring Topological Phases Quantum Condensed Matter Journal Club Adhip Agarwala Department of Physics Indian Institute of Science adhip@physics.iisc.ernet.in February 4, 2016 So you are interested in
More informationDirac and Weyl fermions in condensed matter systems: an introduction
Dirac and Weyl fermions in condensed matter systems: an introduction Fa Wang ( 王垡 ) ICQM, Peking University 第二届理论物理研讨会 Preamble: Dirac/Weyl fermions Dirac equation: reconciliation of special relativity
More informationSection 10 Metals: Electron Dynamics and Fermi Surfaces
Electron dynamics Section 10 Metals: Electron Dynamics and Fermi Surfaces The next important subject we address is electron dynamics in metals. Our consideration will be based on a semiclassical model.
More informationWeyl semi-metal: a New Topological State in Condensed Matter
Weyl semi-metal: a New Topological State in Condensed Matter Sergey Savrasov Department of Physics, University of California, Davis Xiangang Wan Nanjing University Ari Turner and Ashvin Vishwanath UC Berkeley
More informationWeyl fermions and the Anomalous Hall Effect
Weyl fermions and the Anomalous Hall Effect Anton Burkov CAP congress, Montreal, May 29, 2013 Outline Introduction: Weyl fermions in condensed matter, Weyl semimetals. Anomalous Hall Effect in ferromagnets
More informationDirac fermions in condensed matters
Dirac fermions in condensed matters Bohm Jung Yang Department of Physics and Astronomy, Seoul National University Outline 1. Dirac fermions in relativistic wave equations 2. How do Dirac fermions appear
More informationHartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany
Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany Outline Insulators and Topological Insulators HgTe quantum well structures Two-Dimensional TI Quantum Spin Hall Effect experimental
More informationInAs/GaSb A New Quantum Spin Hall Insulator
InAs/GaSb A New Quantum Spin Hall Insulator Rui-Rui Du Rice University 1. Old Material for New Physics 2. Quantized Edge Modes 3. Andreev Reflection 4. Summary KITP Workshop on Topological Insulator/Superconductor
More informationTopological Phases of Matter Out of Equilibrium
Topological Phases of Matter Out of Equilibrium Nigel Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge Solvay Workshop on Quantum Simulation ULB, Brussels, 18 February 2019 Max McGinley
More informationScanning Tunneling Microscopy Studies of Topological Insulators Grown by Molecular Beam Epitaxy
EPJ Web of Conferences 23, 00020 ( 2012) DOI: 10.1051/ epjconf/ 20122300020 C Owned by the authors, published by EDP Sciences, 2012 Scanning Tunneling Microscopy Studies of Topological Insulators Grown
More informationOut-of-equilibrium electron dynamics in photoexcited topological insulators studied by TR-ARPES
Cliquez et modifiez le titre Out-of-equilibrium electron dynamics in photoexcited topological insulators studied by TR-ARPES Laboratoire de Physique des Solides Orsay, France June 15, 2016 Workshop Condensed
More information3.15. Some symmetry properties of the Berry curvature and the Chern number.
50 Phys620.nb z M 3 at the K point z M 3 3 t ' sin 3 t ' sin (3.36) (3.362) Therefore, as long as M 3 3 t ' sin, the system is an topological insulator ( z flips sign). If M 3 3 t ' sin, z is always positive
More informationQuantum Hall Effect in Graphene p-n Junctions
Quantum Hall Effect in Graphene p-n Junctions Dima Abanin (MIT) Collaboration: Leonid Levitov, Patrick Lee, Harvard and Columbia groups UIUC January 14, 2008 Electron transport in graphene monolayer New
More informationarxiv: v1 [cond-mat.mes-hall] 26 Sep 2013
Berry phase and the unconventional quantum Hall effect in graphene Jiamin Xue Microelectronic Research Center, The University arxiv:1309.6714v1 [cond-mat.mes-hall] 26 Sep 2013 of Texas at Austin, Austin,
More informationarxiv: v2 [cond-mat.mes-hall] 15 Feb 2015
Topological Insulators, Topological Crystalline Insulators, and Topological Kondo Insulators (Review Article) M. Zahid Hasan, 1,2 Su-Yang Xu, 1 and Madhab Neupane 1 1 Joseph Henry Laboratories: Department
More informationClassification of topological quantum matter with reflection symmetries
Classification of topological quantum matter with reflection symmetries Andreas P. Schnyder Max Planck Institute for Solid State Research, Stuttgart June 14th, 2016 SPICE Workshop on New Paradigms in Dirac-Weyl
More informationPreface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
More informationSTM probe on the surface electronic states of spin-orbit coupled materials
STM probe on the surface electronic states of spin-orbit coupled materials Author: Wenwen Zhou Persistent link: http://hdl.handle.net/2345/bc-ir:103564 This work is posted on escholarship@bc, Boston College
More informationBasics of topological insulator
011/11/18 @ NTU Basics of topological insulator Ming-Che Chang Dept of Physics, NTNU A brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator
More informationSUPPLEMENTARY INFORMATION
doi:1.138/nature12186 S1. WANNIER DIAGRAM B 1 1 a φ/φ O 1/2 1/3 1/4 1/5 1 E φ/φ O n/n O 1 FIG. S1: Left is a cartoon image of an electron subjected to both a magnetic field, and a square periodic lattice.
More informationA Short Introduction to Topological Superconductors
A Short Introduction to Topological Superconductors --- A Glimpse of Topological Phases of Matter Jyong-Hao Chen Condensed Matter Theory, PSI & Institute for Theoretical Physics, ETHZ Dec. 09, 2015 @ Superconductivity
More informationTOPOLOGICAL BANDS IN GRAPHENE SUPERLATTICES
TOPOLOGICAL BANDS IN GRAPHENE SUPERLATTICES 1) Berry curvature in superlattice bands 2) Energy scales for Moire superlattices 3) Spin-Hall effect in graphene Leonid Levitov (MIT) @ ISSP U Tokyo MIT Manchester
More information