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Transcript
Spin-resolved spectroscopic studies of topologically ordered materials David Hsieh A THESIS PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS Advisor: M. Zahid Hasan June 2009 c Copyright by David Hsieh, 2009. All rights reserved. ° Abstract It has recently been proposed that band insulators with strong spin-orbit coupling can support a new phase of quantum matter called a ‘topological insulator’. This exotic phase of matter is a subject of intense research because it is predicted to give rise to dissipationless spin currents, axion electrodynamic phenomena and non-Abelian quasi-particles. However, it is experimentally challenging to identify a topological insulator because unlike ordinary phases of matter such as magnets, liquid crystals or superconductors, topological insulators are not described by a local order parameter associated with a spontaneously broken symmetry but rather by a quantum entanglement of its wavefunction, dubbed ‘topological order’. Because conventional experimental probes are designed to be sensitive to local order parameters, methods of measuring topological order are relatively unknown. Topologically ordered phases of matter are extremely rare, the most well known example being the quantum Hall phase, which is realized in a cold two-dimensional electron system subject to a large external magnetic field. Its topological order is identified by measuring a quantized magneto-transport, which is carried by robust conducting states localized along the one-dimensional edges of the sample. In topological insulators, intrinsic spin-orbit coupling simulates the effect of a spin-dependent external magnetic field, leading to quantum Hall-like physics without any external magnetic field. However, unlike quantum Hall phases, topological insulators exhibit no quantized transport response, therefore its topological order cannot be detected iii by means of a transport measurement. In this thesis, we show that topological insulators exhibit robust conducting states that are localized on the two-dimensional surfaces of the sample. These surface states have an unusual spin-polarized band structure that cannot be realized on the surfaces of ordinary insulators, nor in purely two-dimensional electron systems. By measuring the spin-polarized band dispersion of surface states using a combination of synchrotron based spin- and angle-resolved photoemission spectroscopy (ARPES), we show that their topological order can be detected. In Chapter 1, we describe the relationship between the topological orders found in quantum Hall systems and in topological insulators. In Chapter 2, we provide a description of the experimental technique of spin-resolved ARPES and a general procedure for mapping spinpolarized surface state band structures. In Chapter 3, we apply this method to study the Bi1−x Sbx alloy series and demonstrate that a topological insulator is realized in a specific composition range. The work presented in this thesis constitutes the first experimental evidence of a topological insulator in nature. iv Acknowledgements First and foremost I would like to thank my advisor Zahid Hasan for his mentorship and support. He has been a constant source of encouragement and advice throughout my graduate career, and his enthusiasm and leadership are the driving forces of our lab. I am also indebted to my colleagues for their help and friendship. Yinwan Li introduced me to the theory and operation of neutron scattering instruments when I first arrived at Princeton. Dong Qian taught me everything I know about the ARPES technique, and has been an extremely generous and patient science teacher to me. I thank Lewis Wray for his constant willingness to help, insatiable curiosity and unique ability to keep me entertained during over-night experiments. I also thank Matthew Xia for his team-first attitude and many insightful discussions. Throughout the course of my Ph.D., I have had the unique opportunity of working with several different experimental techniques that has taken me to a number of national facilities around the world. Over the course of this work, I benefitted greatly from interactions with local scientists. I thank Jeff Lynn, Qing Huang and William Ratcliffe from NIST for spending a great deal of time teaching me about neutron scattering. I thank Alexei Fedorov, Sung Kwan Mo and Kiyohisa Tanaka from the ALS for going above and beyond to ensure that our experiments always went smoothly. I thank Donghui Lu and Rob Moore from the SSRL for their fantastic beamline support. I thank Sebastian Janowski and Hartmut Hochst from the SRC for their always helpful support. Finally I thank Hugo Dil, Fabian Meier and v Jurg Osterwalder from the SLS for teaching me everything I know about SR-ARPES and for generously hosting me during visits. I have also been very fortunate to have been able to learn from world renowned scientists at Princeton. I thank Bob Cava and members of his team including Yew San Hor, Tyrel McQueen, Satoshi Watauchi, Garret Lau, Katie Holman and Emilia Morosan for their generosity in providing pristine and innovative samples and teaching me about solid state chemistry. I thank Princeton theorists David Huse, Shivaji Sondhi, Duncan Haldane and Phillip Anderson for giving me incredible insight into the problems I have worked on. In particular, I thank Andrei Bernevig for spending countless hours discussing experimental results and introducing me to the most current theories. I have also benefitted greatly from discussions with Princeton experimentalists including members of Ali Yazdani’s group and Phuan Ong’s group, who are always willing to help with either sample characterization or manuscript reading. I am grateful to Phuan Ong for agreeing to be second reader on my thesis. Finally I thank Michael Romalis and members of his group from whom I learned a great deal during my experimental project. Of course, none of the work presented in this thesis would have been possible without the architects of topological band theory, with whom I have had great discussions with over the years. I am especially thankful to Liang Fu, Charlie Kane, Ashvin Vishwanath and Joel Moore for teaching me many of the theoretical aspects of this work. Pursuing a Ph.D. in physics can sometimes be a mentally trying task and it is my friends who keep me sane. For this I owe them big time. I thank all the great friends I’ve made at Princeton, my Stanford buddies, and of course my lifelong CIS playmates who have all had to endure hours upon grueling hours of listening to me ramble on about physics. I feel your pain and I truly appreciate it. vi Finally, I am so grateful to Jessica and my parents Jeff and Janet for always keeping me on the right track in life. They are my steady source of inspiration and happiness. vii Contents Abstract iii Acknowledgements iv 1 Introduction 1 1.1 TKNN topological order and the quantum Hall effect . . . . . . . . . 1 1.2 Z2 topological order and the quantum spin Hall effect . . . . . . . . . 4 2 Spin- and angle-resolved photoemission spectroscopy 2.1 2.2 10 ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 Sample preparation for ARPES . . . . . . . . . . . . . . . . . 16 SR-ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Basic principles of Mott polarimetry . . . . . . . . . . . . . . 18 2.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Fitting routine for vectorial spin analysis . . . . . . . . . . . . 23 3 SR-ARPES on Bi1−x Sbx 26 3.1 3D strong topological insulators and their surface states . . . . . . . . 26 3.2 Predicted bulk and surface electronic structure of Bi1−x Sbx . . . . . . 31 viii 3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Topological surface states in insulating Bi1−x Sbx . . . . . . . . 38 3.3.2 Topological surface states in metallic Sb . . . . . . . . . . . . 57 3.3.3 Evolution of surface state spectrum from Bi to Sb . . . . . . . 71 4 Conclusions Bibliography 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 78 Chapter 1 Introduction 1.1 TKNN topological order and the quantum Hall effect The insulating state is one of the most elementary quantum phases of matter [1]. It is characterized by a completely filled set of electronic bands that are separated from a completely empty set of bands by an energy gap, which, according to the semiclassical Boltzmann approach to electron transport [1], makes it electrically inert at low energies. Naively, one might therefore expect the low energy physics of insulators (or gapped systems in general) to be trivial, and expect interesting properties to occur only in gapless systems where there are low energy degrees of freedom. The discoveries of the integer [2] and fractional [3] quantum Hall effects in twodimensional electron systems in the early 1980s revealed a phenomenon that appeared to violate this picture, namely a gapped system that exhibits highly non-trivial transport properties. In particular, transverse magneto-conductance σxy is quantized in extremely precise (to nearly one part in a billion) rational multiples of e2 /h despite an energy gap for electronic excitations. The solution to this paradox comes from 1 1.1. TKNN topological order and the quantum Hall effect (a) (b) Trivial 2 Non-trivial a k2 k2 k1 k1 Figure 1.1: (a) Illustration of parallel transport of a vector along a closed path on a sphere. (b) Schematic diagram of the phase of a wavefunction in the magnetic Brillouin zone with and without the introduction of a topological defect. the fact that in addition to their energy dispersion, the wavefunctions of electronic states may also play a role in charge transport. As was first pointed out by Laughlin [4], the quantum Hall current is contingent upon phase coherence of the many body wavefunction. An explicit expression for how the phase is related to σxy was later derived by Thouless, Kohmoto, Nightingale and den Nijs (TKNN) [5, 6], which brought to light a topological meaning to the quantum Hall effect. A useful measure of the topology of some manifold is the phase acquired by a vectorial object after being parallel transported through a closed path in that manifold [7]. An intuitive example is the parallel transport of a vector n̂(t)e−iα(t) along a closed loop C (parametrized by t) on a sphere (Figure 1.1(a)), where the total phase mismatch α(C) is given by I α(C) = dα (1.1) dα = −in̂∗ · dn̂ (1.2) C with 1.1. TKNN topological order and the quantum Hall effect 3 The quantity α(C) is a topological invariant in the sense that it does not change under continuous deformations of the sphere. Only through some drastic deformation that changes its topology, such as the puncturing of a hole in the sphere, will α(C) change. Because electronic states are represented by vectors in Hilbert space, Berry proposed that a similar geometric phase (coined a “Berry phase”) is accumulated when a state is adiabatically transported through a closed path in parameter space [8]. By choosing this parameter to be the crystal wave vector k, TKNN showed that a topological invariant n characterizes the Berry phase mismatch in momentum space. The quantum Hall conductivity can be expressed as σxy = ne2 /h, where [5] 1 n= 2π Z [∇k × A(kx , ky )]z d2 k (1.3) BZ with A = −ihuk |∇k |uk i (1.4) where the integral is taken over the area of the magnetic Brillouin zone [5, 6] and uk is the periodic part of the Bloch wavefunction [1]. Since the Berry curvature B(k) = ∇k × A(k) is odd under time-reversal, for inversion symmetric systems, σxy must vanish in the absence of an external magnetic field. However, because σxy is proportional to a topological (TKNN) invariant, it cannot smoothly evolve from zero as a function of magnetic field. Rather, it must jump from one discrete value to another once the magnetic field is strong enough to change the topology of the Hilbert space. Formally, a change in topology takes place through the introduction of a topological defect, a magnetic monopole, in k-space (Figure 1.1(b)). The integral nature of these local defects and their stability explain the precision with which σxy is quantized and its experimental robustness. Unlike an Ohmic current, which comes from the deviation of some distribution function from equilibrium and is carried only by states at the Fermi level [1], the 1.2. Z2 topological order and the quantum spin Hall effect 4 quantum Hall current is a topological current that is carried by all occupied states below the Fermi level [4]. Because there are no states below the Fermi level in which to scatter, the bulk topological current is inherently dissipationless, which has made such materials attractive for energy efficient electronics technologies. In practical devices however, phase coherence of the bulk wavefunction is destroyed at low temperatures by Anderson localization [9] due to the inevitable presence of random impurities. Therefore no current carrying states exist in the sample bulk. Instead, building upon Laughlin’s arguments [4], Halperin showed that for a system with Hall conductivity σxy = ne2 /h, there must exist exactly n branches of gapless one-dimensional edge states [10]. These edge states form a chiral Luttinger liquid, which means that they cannot be localized by weak impurity disorder, and are the states responsible for carrying the Hall current. As later shown by Hatsugai [11, 12], these edge states have a topological invariant of their own, which he showed to be equivalent to the bulk TKNN invariant. A von-Klitzing type measurement of the edge state conductance thus represents a direct method of probing the bulk topological invariant. 1.2 Z2 topological order and the quantum spin Hall effect The spin Hall effect is the generation in a material of a spin current at right angles to an applied charge current. The spin Hall effect was first proposed theoretically in 1971 based on an “extrinsic mechanism” [13], in which impurities in a conducting material deflect the spin-up and spin-down electrons in opposite directions through Mott scattering. Interest in the early 2000’s shifted towards the possibility of an “intrinsic mechanism” where spin currents arise from the spin-orbit field inherent to the band structure of the material, independent of impurities. Like the quantum Hall current, 1.2. Z2 topological order and the quantum spin Hall effect 5 the spin current in the “intrinsic case” is generated by the Berry curvature of the Bloch states. Although the charge current is still dissipative, the spin current is dissipationless. This effect was predicted to occur in a number of both three-dimensional [14] and two-dimensional doped semiconductors [15], and has been indirectly observed through the detection of edge spin accumulation by optical methods [16]. In contrast to doped semiconductors, Murakami, Nagaosa and Zhang proposed in 2004 that certain band insulators could exhibit an intrinsic spin-Hall conductivity SH σxy without any charge conductivity, making these systems completely dissipationless [17]. Such “spin-Hall insulators” are realized in materials whose energy gap arises due to spin-orbit coupling, and its mechanism is put briefly as follows. Spin-orbit coupling gives rise to a splitting of bands into multiplets of the total angular momentum J = L + S. Because all occupied bands contribute to a topological current, if the Fermi level lies between two bands within the same J multiplet, the spin currents in the totally occupied bands do not cancel. The presence of a bulk energy gap in spin-Hall insulators raised the question of whether a topological invariant that characterizes its many body wavefunction exists, SH which acts to quantize σxy . To understand how this might arise, Bernevig and Zhang, building upon earlier work by Haldane [18], presented the following model for a quantum spin Hall effect in two-dimensions [19]. Landau level quantization in the quantum Hall effect arises from a velocity dependent term in the Hamiltonian A · p, where A is the vector potential of the applied magnetic field and p is the particle momentum. By choosing the symmetric gauge A = B (−y, x, 0), 2 the velocity dependent term is proportional to B(xpy − ypx ). In the absence of a magnetic field, a spin-orbit term (p × E) · σ, where σ is the Pauli spin matrix and the electric field E is chosen such that E ∼ E(x, y, 0), can give rise to a similar velocity dependent term Eσz (xpy − ypx ). Such a system behaves as if a spin-up electron experiences an 1.2. Z2 topological order and the quantum spin Hall effect 6 Figure 1.2: Energy bands of a one-dimensional strip of graphene. Closely spaced lines show the discrete bulk energy levels projected onto the edge Brillouin zone. The green and red lines show the spin-filtered bands localized on either edge of the strip. (a) Spectrum of the Z2 = 1 and (b) the Z2 = 0 phase achieved by tuning the parameter λv (inest). [Figure adapted from [20]] effective upward pointing orbital magnetic field while a spin-down electron experiences a downward pointing one. This way, the spin-up electrons are described by a TKNN invariant n↑ while spin-down electrons are described by n↓ = −n↑ , leading to an overall vanishing charge Hall conductivity conductivity e (n↑ −n↓ ). 4π e2 (n↑ h + n↓ ) but a non-vanishing spin Hall It was proposed that a measurement of the spin-filtered edge state (in the sense that states with opposite spin propagate in opposite directions) in such systems could reveal its time-reversal symmetric topological invariants. Unfortunately, as Kane and Mele pointed out in 2005 [20], a topological characterization based on the spin TKNN invariant (n↑ − n↓ ) only holds true if σz is conserved, which inevitably breaks down in real systems due, for example, to inter-band mixing, disorder or interactions. Rather, they postulated that the topological properties of the quantum spin Hall state are encoded in a new stable Z2 topological invariant that is particular to time-reversal symmetric systems. Like the TKNN invariant, the Z2 1.2. Z2 topological order and the quantum spin Hall effect 7 invariant can be formulated as a defect in momentum space and becomes equivalent to the parity of the spin TKNN invariant when σz is conserved [21, 22]. Its value, either 0 or 1, determines whether the system is an ordinary or quantum spin Hall insulator respectively. Like Halperin’s edge state theory of the quantum Hall effect, spin filtered edge states play a crucial role for spin Hall transport in the theory of Kane and Mele. To illustrate the different behavior of the edge states when Z2 = 0 or 1, they constructed a model of graphene with spin-orbit coupling that could be switched between Z2 = 0 and 1 by tuning the strength of the staggered sublattice potential λv [23]. In bulk graphene, all bands must be doubly spin degenerate owing to a combination of inversion symmetry [E(k, ↑) = E(−k, ↑)] and time-reversal symmetry [E(k, ↑) = E(−k, ↓)]. Because inversion symmetry is broken at a terminated edge, the spin degeneracy of the one-dimensional edge states is generally lifted. However, by Kramers theorem [24], they must remain doubly degenerate at the center (k = 0) and boundary (k = π) of the edge Brillouin zone, since those momenta are invariant under time-reversal. The topological distinction between Z2 = 0 and Z2 = 1 insulators is manifest in the number of times, even and odd respectively, that their edge state Fermi surface encloses these time-reversal invariant momenta (TRIM). The edge state spectrum for the Z2 = 1 case (Figure 1.2(a)) shows a single pair of bands that cross the Fermi level, resulting in a Fermi surface that encloses an odd number of TRIM, namely once around k = π. In contrast, the ordinary insulating phase (Figure 1.2(b)) shows a Fermi surface that encloses an even number of TRIM, namely none at all. The gapless nature of the edge states in the quantum spin Hall insulator is guaranteed by time-reversal symmetry. Moreover, because there is no counter-propagating channel in which a spin polarized electron at the edge can backscatter, weak dis- 1.2. Z2 topological order and the quantum spin Hall effect 8 order will not lead to localization and transport is dissipationless [20, 23]. Due to the presence of these robust metallic edge states, dubbed a helical liquid [25], the spectrum of a quantum spin Hall insulator cannot be adiabatically deformed into an ordinary insulator with no edge states. Unlike the quantum Hall insulator, the topological properties of the ground state wavefunction of the quantum spin Hall insulator are manifest in its energy dispersion. This provides a great experimental advantage for measuring topological invariants because the relevant temperatures scales are no longer set by phase coherence lengths but rather by spin-orbit gap energies. The first realistic proposal for a quantum spin Hall insulator was made in 2006 by Bernevig, Hughes and Zhang for the HgTe/(Hg,Cd)Te quantum well system [26]. However, there are several practical limitations to a direct imaging of the edge state energy dispersion. First, the edge states lie at a buried interface between two semiconductor films, which makes them difficult to access with scattering probes. Second, the spatial extent of the edge state is typically small because they are localized to within a few atomic layers near the sample edge, which is far smaller than the focused spot sizes of most scattering probes. In 2007, Konig et al. gave the first indirect evidence for helical edge states through a remarkable set of charge conductance measurements in HgTe/(Hg,Cd)Te quantum wells [27]. However, the Z2 invariant was not directly measured because the spin degeneracy and the energy dispersion of the edge states were not imaged. The prospect for a direct measurement of Z2 topological invariants arose between 2006 and 2007 following a series of theoretical predictions [22, 28, 29, 30] showing that three-dimensional insulators can also be characterized by Z2 invariants. Rather than a single Z2 invariant, three-dimensional time-reversal symmetric band structures are described by four Z2 invariants ν0 ; (ν1 ν2 ν3 ), of which only ν0 is robust in the presence of disorder. While the ν0 = 0 phase describes ordinary three-dimensional 1.2. Z2 topological order and the quantum spin Hall effect 9 insulators, the ν0 = 1 phase describes an intrinsically three-dimensional topological phase of matter called the “strong topological insulator” [22, 28], which exhibits unusual spin-filtered surface states. This is akin to the predicted, but never observed, three-dimensional quantum Hall insulator [31]. In this thesis, I will detail an experimental method of directly measuring the Z2 invariant ν0 in strong topological insulators. Chapter 2 will explain the basic principles of spin and angle-resolved photoemission spectroscopy that can be used to measure the spin-polarization and energy dispersion of surface states. Chapter 3 will first describe how to identify candidate strong topological insulator materials, and then discuss the photoemission based experimental method we used to measure the Z2 invariant in the Bi1−x Sbx alloys. Chapter 2 Spin- and angle-resolved photoemission spectroscopy 2.1 2.1.1 ARPES Basic principles Angle-resolved photoemission spectroscopy is a method of studying the electronic structure of solids by using the photoelectric effect [32, 33]. The basic principles of an ARPES experiment are as follows. A monochromatic beam of light with energy hν, typically from a synchrotron radiation source, impinges on a sample and photoexcites electrons into the vacuum. These photoelectrons are then collected in an electrostatic analyzer that measures their kinetic energy (Ekin ) as a function of emission angles (ϑ,ϕ) relative to the sample surface. This way, the wave vector (K = p/h̄) of the photoelectrons in vacuum is completely determined via 10 2.1. ARPES 11 (a) E kin (b) Spectrum EF z hn E hn e- J Sample Evac EF y N(Ekin ) W j EB E0 hn x Core level N(E) Figure 2.1: (a) Relation between energy levels in a solid and the electron energy spectrum produced by photons of energy hν. The electron energy distribution inside the solid is expressed in terms of the binding energy (EB ), which is referenced to the Fermi level (EF ), whereas the photoelectron kinetic energy is referenced to the vacuum level (Evac ). (b) Geometry of an ARPES experiment. The emission direction of the photoelectrons is specified by the polar (ϑ) and azimuthal (ϕ) angles. 2.1. ARPES 12 1p 2mEkin sinϑcosϕ h̄ 1p 2mEkin sinϑsinϕ Ky = h̄ 1p 2mEkin cosϑ Kz = h̄ Kx = (2.1) By exploiting energy and momentum conservation, it is straightforward to relate the measured kinetic energy of the photoelectron to its binding energy (EB ) while inside the sample Ekin = hν − W − |EB | (2.2) where W is the work function, which is typically 4-5 eV for metals. The photon momentum has been neglected since it is much smaller than a typical Brillouin zone dimension at VUV photon energies. Because of translational symmetry in the plane of the sample surface, the parallel component of momentum is conserved in a photoemission process and thus Kk = (Kx , Ky , 0) can be related to the parallel component of the electron crystal momentum h̄kk via |kk | = |Kk | = 1p 2mEkin sinϑ h̄ (2.3) The perpendicular component of momentum h̄k⊥ , on the other hand, is not conserved due to the surface potential barrier. Although there exist several experimental methods to measure this quantity absolutely, under certain circumstances a simpler alternative is to invoke a three-step model of the photoemission process and make some a priori assumption about the dispersion of the electron final states. The three-step process consists of 1) an optical excitation of an electron in the solid from a low to high energy Bloch state, followed by 2) transport of the electron to the surface, and finally 3) transmission of the electron from a high energy Bloch state beneath the 2.1. ARPES 13 (a) E optical transport excitation to surface escape to vacuum (b) E kin Ef 1 2 3 G Ef hn hn E vac EF Ei E0 Ei W K V0 -p a 0 p a z 0 Figure 2.2: (a) Illustration of the three-step model of the photoemission process. (b) Kinematics of the photoemission process within the three-step nearly-free-electron final state model. surface into a free electron state in vacuum (Figure 2.2(a)). If the final Bloch state is assumed to be free electron like, then electron escape will take place at the surface via transmission between parabolic bands inside and outside the sample that are offset by some energy V0 known as the inner potential. For the band structure of a nearly free electron model (Figure 2.2(b)), the final state energy is then given by h̄2 (k2k + k2⊥ ) − |E0 | Ef (k) = 2m (2.4) where Ef and E0 are both referenced to the Fermi level. Substituting Ef = Ekin + W and Equation 2.3 into Equation 2.4, one arrives at the expression |k⊥ | = 1p 2m(Ekin cos2 ϑ + V0 ) h̄ (2.5) where V0 = |E0 | + W , which relates |k⊥ | to the measured values Ekin and ϑ once V0 is known. In this work, two independent methods of determining V0 are used: (i) 2.1. ARPES 14 agreement between experimental and theoretical band structures is optimized and (ii) the periodicity of the dispersion E(k⊥ ) measured at normal emission (ϑ = 0◦ ) is used to extract V0 . Although the free electron final state approximation is most accurate for materials with weak crystal potentials or at high excitation energies where the kinetic energy of the electron far exceeds the lattice potential, it has proven successful for mapping bands in even in high Z materials such as bismuth [34] and using VUV photon energies. The ability to map band dispersions perpendicular to the surface plane provides an elegant way to distinguish surface from bulk electronic states in three-dimensional systems. This is especially useful for systems whose bulk states disperse strongly with k⊥ and have small c-axis lattice parameters because the electron mean free path (the average distance that an excited state electron travels in the solid with no change in energy and momentum) is limited to about 5Å for VUV photon energies [32, 33]. 2.1.2 Experimental setup Our non spin-resolved ARPES measurements were performed using linearly polarized synchrotron radiation and hemispherical electron analyzers from Scienta. Experiments took place at beamlines 10.0.1 and 12.0.1 of the Advanced Light Source in Berkeley, California, at beamline 5-4 of the Stanford Synchrotron Radiation Laboratory in Stanford, California, and at beamlines PGM(A) and U1 NIM at the Synchrotron Radiation Center in Stoughton, Wisconsin. The spectrometer setup at all of these end stations is shown in Figure 2.3. Due to the surface sensitivity of ARPES, the samples are cleaved and maintained in ultra high vacuum at pressures less than 5 × 10−11 torr to minimize the adsorption of atoms to its surface. Electrons are photoemitted from the sample in situ and are decelerated and focused onto a w by l sized entrance slit of the hemispherical analyzer 2.1. ARPES 15 V2 R1 V1 R2 w lens z J hn sample x E 2D detector y Figure 2.3: Schematic of a hemispherical electron analyzer. The path of photoelectrons within some narrow energy range ∆Ekin centered about Epass is marked in gray. via an electrostatic input lens. The hemispherical analyzer consists of two concentric hemispheres of radius R1 and R2 maintained at a potential difference of ∆V = V2 V1 . Therefore only those electrons reaching the entrance slit within a narrow kinetic energy range ∆Ekin centered at the value Epass = e∆V /(R1 /R2 − R2 /R1 ) will be able to travel around the analyzer and onto a multichannel 2D detector. Because the electrons are spread apart along the detector y axis as a function of their kinetic energy, a ∆Ekin (typically around ±0.1Epass ) slice of phase space can be imaged simultaneously. At the same time, photoelectrons emitted along a finite angular range (either 9◦ , 14◦ or 38◦ depending on the analyzer model) defined by the entrance slit length l are spread apart along the detector x axis. Therefore a 2D snap shot of phase space of dimension ∆Ekin by ∆ϑ is captured at once. To obtain an ARPES spectrum near EF , the first step is to determine the kinetic energy that corresponds to EF by taking an energy distribution scan of polycrystalline 2.1. ARPES 16 gold, and fitting this to a convolution of a Gaussian resolution function with a FermiDirac distribution. A detailed scan near EF can then be obtained by setting the input lens voltage so as to decelerate photoelectrons from this particular kinetic energy to Epass , and sweeping the input lens voltage in incremental energy steps in case energy ranges greater than ±0.1Epass need to be covered. The majority of our experiments were taken with an Epass of either 5 eV or 10 eV. At these values of Epass , the total energy resolution, which accounts for the resolving power of both the beamline monochromator and electron analyzer, is around 15 meV. 2.1.3 Sample preparation for ARPES The samples we use for the work presented in this thesis are single crystals of both doped and undoped Bi1−x Sbx . These were cleaved using a razor blade from a boule grown from a stoichiometric mixture of high-purity elements, resulting in shiny flat silver surfaces. The boule was cooled from 650 ◦ C to 270 ◦ C over a period of five days and was annealed for seven days at 270 ◦ C. X-ray diffraction patterns of the cleaved single crystals, collected on a Bruker D8 diffractometer using Cu Kα radiation (λ = 1.54 Å), exhibit only the (333), (666), and (999) peaks, showing that the cleaved surface is the (111) plane. Powder X-ray diffraction measurements were also taken to check that the samples were single phase (rhombohedral A7 crystal structure, point group R3̄m). The following steps were taken to prepare the samples for an ARPES measurement (Figure 2.4). (1) Single crystals were typically cut to around 2 mm × 2 mm × 0.5 mm in size, and were mounted onto a copper sample post using silver epoxy. Because of the relatively poor electrical conductivity of the samples (Chapter 3), silver epoxy was used to ensure good electrical contact with the copper post so as to prevent sample charging from photon exposure. (2) The Laue back-reflection technique was used 2.1. ARPES 17 (b) (a) ceramic post epoxy sample silver epoxy (c) copper post Figure 2.4: . (a) Sample mounting schematic. (b) Example of a Laue back-reflection image of bismuth, adapted from [35]. (c) Example of a LEED image of bismuth. to determine the in-plane orientation of the crystal axes. A three-fold rotationally symmetric diffraction pattern is typically observed, consistent with the three-fold rotational symmetry of the A7 structure around the [111] axis. (3) A ceramic toppost is mounted onto the exposed face of the sample using an epoxy resin and the entire object is then covered with a layer of silver paint for improved electrical contact and then a layer of colloidal graphite for masking purposes. The copper post is finally affixed to the inside of the ARPES chamber, where it is cooled and pumped down, after which the sample is cleaved by knocking the ceramic top-post in situ. After the experiment, the crystal structure of the cleaved surfaces was examined using low energy electron diffraction (LEED) to ensure a good quality. Detailed analyses of LEED [36] and photoelectron diffraction measurements [37] of the surfaces of bismuth and antimony show negligible surface structural reconstruction or relaxation effects, which provides assurance that the surface state spectra measured using ARPES are representative of states arising from the terminated bulk crystal. 2.2. SR-ARPES 18 P ki NL k f Ù nL Ù nR NR kf q q Figure 2.5: Schematic of a Mott scattering geometry. Incident electrons (red dot) with a polarization P on a high Z nucleus (yellow dot) are backscattered to the left and right with a probability that is dependent on P. 2.2 2.2.1 SR-ARPES Basic principles of Mott polarimetry The interaction Hamiltonian between a photon and spin 1/2 electron can be described by the Dirac equation Hint = e ´2 eh̄ eh̄ 1 ³ ieh̄ p − A +eΦ− σ ·(∇×A)+ E·p− σ ·(E×p) (2.6) 2m c 2mc 4m2 c2 4m2 c2 where p is the electron momentum, A is the photon vector potential, Φ is the scalar potential, E is the electric field and σ is the electron spin. However by using linearly polarized photons in the UV to soft x-ray regime, it has been shown [38] that the spin dependent terms in Equation 2.6 are greatly suppressed, and the photon electron interaction Hamiltonian can be well approximated by the Schrodinger model Hint = −(e/mc)A · p, which conserves spin. Provided the photoemission process is spin-conserving, the spin of the initial state of an electron in a solid can be determined by measuring its spin after it has been photoemitted. Mott electron polarimetry [39] is a method of separating electrons of 2.2. SR-ARPES 19 different spin from such a photoemitted beam based on the use of spin-orbit (Mott) scattering of electrons from nuclei. The physical principle of Mott scattering can be understood from the classical picture of a moving electron scattering off of a stationary bare nucleus of charge Ze. At low incident energies, the electron interacts with the nucleus predominantly via its charge, and scattering is described by the Rutherford cross section σR (θ), where the scattering angle θ is typically small. At high incident energies and in cases where Z is large, the velocity v of the electron in the electric field E of the nucleus can result in a considerable magnetic field B in its rest frame given by 1 B=− v×E c (2.7) which, using E = (Ze/r3 )r, can be written as B= Ze Ze r×v= L 3 cr mcr3 (2.8) where L = mr × v is the electron orbital angular momentum. The interaction of this magnetic field with the electron spin S creates a spin-orbit (L · S) term in the scattering potential and introduces a spin dependent correction to the Rutherford cross section σ(θ) = σR (θ)[1 + S(θ)P · n̂] (2.9) where S(θ) is the asymmetry or Sherman function, P is the polarization (2/h̄)(hSx i, hSy i, hSz i), and n̂ is the unit normal to the scattering plane defined by n̂ ≡ ki × kf |ki × kf | (2.10) where ki and kf are the initial and final wave vectors of the electron respectively. The direction n̂ depends on whether scattering to the left or right is being considered. 2.2. SR-ARPES 20 This spin-orbit scattering relation allows for the measurement of the component of spin polarization perpendicular to the scattering plane in the following way. Consider a beam of N incident electrons with N↑ of them polarized along +ẑ and N↓ of them polarized along −ẑ, which leads to a net polarization Pz = (N↑ − N↓ )/(N↑ + N↓ ). When the scattering of this beam from a nucleus takes place in the xy plane, there results a left-right scattering asymmetry Az (θ) defined as Az (θ) = NL − N R NL + NR (2.11) where NL and NR are the number of electrons scattered to the left and right respectively through an angle θ. Substituting the relations NL ∝ N↑ [1 + S(θ)] + N↓ [1 − S(θ)] and NR ∝ N↑ [1 − S(θ)] + N↓ [1 + S(θ)] derived from Equation (2.9) into Equation (2.11) yields Pz = Az (θ) S(θ) (2.12) which shows that given the Sherman function, measurement of Az (θ) yields Pz . In a single Mott polarimeter, it is therefore possible to measure two orthogonal spin components of an electron beam by arranging four detectors in two orthogonal scattering planes in front of a target. 2.2.2 Experimental setup Our spin-resolved ARPES measurements were performed using the COmplete PHoto Emission Experiment (COPHEE) spectrometer [41] with linearly polarized VUV synchrotron photons generated at the Swiss Light Source in Villigen, Switzerland. In the COPHEE spectrometer, energy and momentum analysis of the photoelectrons again takes place using a hemispherical electrostatic analyzer (Figure 2.6). Electrons at some selected energy and momentum are then accelerated to high energy (typically 2.2. SR-ARPES (a) 21 (b) +J y +j z x x’ y’ z’ Polarimeter 1 y’ x’ z’ Polarimeter 2 Figure 2.6: (a) Schematic of the spin-resolved ARPES spectrometer COPHEE adapted from [40]. Photoelectrons are energy and momentum analyzed using a hemispherical electrostatic analyzer and are alternately deflected at a frequency of 1 Hz into two orthogonally mounted Mott polarimeters. The dual polarimeter system is shown rotated by 90◦ for clarity. (b) The relationship between the sample and Mott coordinate systems. When ϑ and ϕ are both zero, the sample coordinates can be transformed into the Mott coordinates via a 45◦ rotation about their common z axis. The Mott axes marked red denote the spin components that the polarimeter is sensitive to. 2.2. SR-ARPES 22 around 40 keV) and are alternately deflected into two Mott polarimeters that are mounted perpendicular to one another so that a total of four (three independent) components of spin are measured. Each Mott polarimeter consists of a gold foil target with silicon diode detectors positioned to its left, right, top and bottom. To account for unequal sensitivities between a detector pair, we applied a small multiplicative factor to the intensity from one detector to ensure that the unpolarized background intensity yields zero polarization. The two Mott polarimeters share a common coordinate (primed) frame, with polarimeter 1 sensitive to the y 0 and z 0 components of spin and polarimeter 2 sensitive to the x0 and z 0 components. The sample coordinate frame (unprimed) is positioned such that when ϑ and ϕ are both equal to zero, it is related to the Mott coordinate frame by a 45◦ rotation about z 0 (Figure 2.6(b)). A spin polarization vector Px0 measured in the Mott coordinate frame can then be expressed in sample coordinates Px via a matrix transformation T T = cosϑcosϕ+sinϕ √ 2 −cosϑcosϕ+sinϕ √ 2 −cosϑsinϕ+cosϕ √ 2 cosϑsinϕ+cosϕ √ 2 sinϑ √ 2 −sinϑ √ 2 −sinϑcosϕ sinϑsinϕ cosϑ where Px = T Px0 . A typical in-plane Fermi wave vector (kF ∼ 0.1 Å−1 ) can be accessed at a tilt angle ϑ ∼ 2◦ using 30 eV photons. At such small tilt angles, the effect of ϑ contributes only a very small correction to P (∼ 1%) and thus setting ϑ to zero is often a good approximation. The accuracy of the spin polarization measurement depends critically on knowledge of the Sherman functions of both Mott polarimeters. While the Sherman function for electron scattering from a single atomic nucleus can be calculated, scattering from a solid state target is complicated by multiple scattering and inelastic scattering events that reduce the analyzing power. In practice, the Sherman function of both polarimeters are experimentally calibrated using a source of electrons with known po- 2.2. SR-ARPES 23 larization, from a magnetized sample for instance [40], to yield an effective Sherman function (Sef f ), which can be cross checked by measuring the spin polarization along the common (z 0 ) axis shared between them. Typical values of Sef f in our experiments lie in the range 0.07 to 0.085. An additional source of inefficiency comes from the fact that only a fraction N/N0 of the total number of incoming electrons are backscattered into the detectors. The overall efficiency of a Mott polarimeter is thus usually quantified by the figure of merit 2 ² = (N/N0 )Sef f (2.13) which typically ranges between 10−3 to 10−4 . The statistical error of a polarization measurement is given by 1 δAα0 1 δPα0 = =√ Pα0 Sef f Aα0 ²N0 (2.14) Due to the low efficiency of detection, spin-resolved ARPES measurements are usually taken with increased photon beam sizes and analyzer slit widths w to increase the photoelectron flux to the Mott polarimeters, which compromise the energy and momentum resolution of the measurement. Nevertheless, typical hours long counting times are still required to achieve a high signal to noise ratio scan, which is the reason gold is used as the scattering target because it presents a compromise between high atomic number and material inertness. Typical electron counts on the detector reach 5 × 105 , which places an error bar of approximately ±0.01 for each point on our measured polarization curves. 2.2.3 Fitting routine for vectorial spin analysis In a typical SR-ARPES experiment, the electron beam polarization at some fixed energy is measured as a function of the manipulator angle ϑ to yield a continuous trace 2.2. SR-ARPES 24 P(ϑ). However because of the overlap of several bands due to intrinsic broadening or resolution effects and the presence of a background, the polarization data do not directly reveal the spin polarization vector of each band individually. To overcome this issue, we employ the following fitting routine developed by Meier et al. [42] to perform quantitative spin analysis. The spin-averaged intensity Itot (ϑ), defined as the sum of the intensities measured by each left (L) right (R) pair of Mott detectors over all three spin components ¡ ¢ P α0 = x0 , y 0 , z 0 Itot = α0 IαL0 + IαR0 , is first fit to a sum of Lorentzians (I i ), one for each band, and a background B, Itot = n X Ii + B (2.15) i where n is the number of bands traversed by the ϑ scan. Each band is then assigned a polarization vector Pi = (Pxi , Pyi , Pzi ) = ci (cosθi cosφi , cosθi sinφi , sinθi ) (2.16) where 0 ≤ ci ≤ 1 is the magnitude of the spin polarization vector of band i and θi and φi are the polar and azimuthal angles measured relative to the sample coordinate frame. The polarization is then transformed into the Mott coordinate frame via T −1 , which can be well approximated as ϑ independent for small ϑ. This is used to define a spin-resolved spectrum for each peak i Iαi;↑,↓ = I i (1 ± Pαi 0 )/6 , α0 = x0 , y 0 , z 0 0 (2.17) where the + and - correspond to ↑ and ↓ respectively, which mean spin parallel or antiparallel to the α0 direction. The entire spin-resolved spectrum for some component α0 can then be constructed from spin-resolved spectra of the individual bands via 2.2. SR-ARPES 25 Iα↑,↓ 0 = n X Iαi;↑,↓ + B/6 0 (2.18) i=1 where the background is divided equally between the different spatial directions. The spin polarization for each spatial component α0 can be expressed as Pα0 = Iα↑0 − Iα↓0 Iα↑0 + Iα↓0 which is used to fit the measured polarization spectra Pα0 = Aα0 /Sef f . (2.19) Chapter 3 SR-ARPES on Bi1−xSbx 3.1 3D strong topological insulators and their surface states When an infinite three-dimensional crystal is terminated at a two-dimensional surface, new states that are localized near the surface can emerge, which are formed by matching evanescent Bloch wave states into the crystal to evanescent plane waves into the vacuum [1]. In the presence of spin-orbit coupling, these two-dimensional dispersing surface states are generally spin-split. Because the spin-splitting is induced entirely electrically, the surface states of strongly spin-orbit coupled materials have been heavily investigated for their potential in spintronics technologies [43] such as the proposed Das-Datta spin transistor [44]. Recent theoretical works [22, 29, 30] have shown that on the surfaces of spin-orbit coupled insulators, topologically distinct varieties of surface states can be formed, which are related to the topological properties of the bulk band structure. In general, time-reversal symmetric band structures in three dimensions are characterized by four topological Z2 invariants ν0 ;(ν1 ν2 ν3 ) [22, 29, 30], which gives rise to 16 distinct bulk 26 3.1. 3D strong topological insulators and their surface states 27 kz (0,p) (0,0) (p,p) (p,0) (0,0,p) (0,p,0) ky (p,0,0) kx Figure 3.1: Bulk Brillouin zone of a three-dimensional cubic lattice and its projected (001) surface Brillouin zone. The eight bulk TRIM are located at the corners of the bold cube and the four surface TRIM are located at the corners of the bold square. 3.1. 3D strong topological insulators and their surface states 28 topological classes. However, of these four topological invariants, only ν0 is robust in the presence of disorder. Therefore only two classes, the ν0 = 0 ordinary insulator and the ν0 = 1 strong topological insulator (STI) are expected to be observable in nature. The relationship between the topology of the surface states on an arbitrary crystal face and the bulk Z2 invariant was first established by Fu, Kane and Mele in 2007 as follows [28, 45]. In a general three-dimensional Brillouin zone, there are eight independent time-reversal invariant momenta (TRIM) ki that satisfy ki = −ki modulo G, where G is any reciprocal lattice vector. For a surface perpendicular to G, the surface Brillouin zone has four independent TRIM Λ̄a . These are located at the projections of pairs ki=a1 , ki=a2 , that differ by G/2, onto the plane perpendicular to G (Figure 3.1). The connectivity of the spin polarized surface state bands between Λ̄a and Λ̄b is determined by the quantities δ(ki ) = ±1, which in turn are related to ν0 via (−1)ν0 = 8 Y δ(ki ) (3.1) i=1 In particular, whether a surface band crosses EF an even or odd number of times between Λ̄a and Λ̄b depends on whether the product of their surface fermion parities π(Λ̄a )π(Λ̄b ) equals +1 or -1 respectively, where the surface fermion parities are given by π(Λ̄a ) = δ(ka1 )δ(ka2 ) and π(Λ̄b ) = δ(kb1 )δ(kb2 ). It follows that the surface state Fermi surface forms a boundary in the surface Brillouin zone separating two regions, one containing the surface TRIM with surface fermion parity +1 and the other containing the surface TRIM with surface fermion parity -1. It was proven in general [28] that when ν0 = 0, each region must contain an even number of TRIM and when ν0 = 1, each region must contain an odd number of TRIM (Figure 3.2). 3.1. 3D strong topological insulators and their surface states (a) (c) n0 = 0 n0 = 1 kz kz + + + + - + + + ky ky kx kx (b) (d) E E EF EF La 29 ky Lb La ky Lb Figure 3.2: (a) Schematic of the surface state Fermi surface for ν0 = 0 and (b) ν0 = 1 cases. The surface fermion parities at the surface TRIM are labeled by + and − signs. (c) Schematic of the surface band dispersion for the ν0 = 0 case along the direction marked by the red arrow in (a). The shaded grey regions represent the projection of the bulk bands onto the (001) surface. (d) Analogous schematic for the ν0 = 1 case illustrating the partner switching. 3.1. 3D strong topological insulators and their surface states 30 The topological distinction between the ν0 = 0 and ν0 = 1 surface states is similar to that between the edge states of Z2 = 0 and Z2 = 1 planar insulators [23]. Because the surface bands must be doubly spin degenerate at both Λ̄a and Λ̄b by Kramers theorem, a pair of spin-polarized bands that intersect at Λ̄a must either intersect with one another again at Λ̄b , which is the case for ν0 = 0, or must separately intersect with different spin-polarized bands at Λ̄b , which is allowed only when ν0 = 1 (Figure 3.2). The latter phenomenon is dubbed “partner switching” [28]. The prediction of a two-dimensional surface state with a Fermi surface enclosing an odd number of TRIM represents a fundamentally new and exotic metallic system. Firstly, its gapless electrons are protected against time-reversal symmetric perturbations. Secondly, because the wave function of a single electron spin acquires a geometric phase factor of π [24] as it evolves by 360◦ in momentum space along a Fermi contour enclosing a TRIM, an odd number of Fermi pockets enclosing TRIM in total implies a π Berry’s phase, which prevents these gapless electrons from being localized by disorder [28]. Such an electronic spectrum is well known to not be realizable in purely two-dimensional electronic systems with Rashba or Dresselhaus spin-orbit coupling because of the fermion doubling theorem [46]. To predict a real material that exhibits these surface states, it is essential to be able to calculate the δ(ki ) from its bulk many body wavefunction. This can in principle be done by calculating Brillouin zone integrals of Berry phase terms or finding zeroes of Pfaffian functions [22, 28], however these are very computationally intensive. In 2007, Fu and Kane found that for inversion symmetric systems [22], the expression for δ(ki ) is greatly simplified, and depends only on the parity eigenvalues ξ2m (~ki ) = ±1 of the occupied bulk states at the eight bulk TRIM via the relation δ(ki ) = N Y m=1 ξ2m (~ki ) (3.2) 3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx 31 where N is the number of bulk Kramers degenerate states below EF . By applying this formalism to the calculated band structures of a number of inversion symmetric spinorbit insulators, they identified the compounds Bi1−x Sbx , α-Sn, HgTe, Pb1−x Snx Te as candidates for being an STI [28]. With the exception of Bi1−x Sbx however, all of these compounds require the application of an external strain to realize the STI phase, which is the reason we focus on Bi1−x Sbx . 3.2 Predicted bulk and surface electronic structure of Bi1−xSbx The crystal structure of bismuth and antimony has a rhombohedral A7 symmetry (space group R3̄m) typical of the group V semimetals, which is inversion symmetric and consists of two interpenetrating trigonally distorted FCC lattices with two atoms per unit cell (Figure 3.3(a)). Using a rhombohedral Bravais lattice, the three primitive translation vectors of the lattice are µ 1 1 1 √ a, − a, c a1 = 2 3 2 3 ¶ µ 1 1 1 √ a, a, c a2 = 2 3 2 3 ¶ µ 1 1 a3 = − √ a, 0, c 3 3 ¶ (3.3) with the relative position of the two basis atoms given by d = (0, 0, 2µ)c. Measured values of the lattice parameters for (Bi, Sb) are a = (4.5332 Å, 4.3007 Å), c = (11.7967 Å, 11.2221 Å) and µ = (0.2341, 0.2336). The structure can be viewed as a set of puckered triangular lattice bilayers stacked along the rhombohedral [111] direction, with 3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx z (a) 32 (b) 1st layer 2nd layer 3rd layer 4th layer y a3 a1 x a2 (c) 1.59 Å 2.35 Å y z -x x Figure 3.3: (a) Crystal structure of group V semimetals such as bismuth and antimony. Volume enclosed by the shaded planes denotes the rhombohedral unit cell. (b) Top view of the first three atomic layers after cleavage along the (111) plane. Each layer consists of a triangular lattice. (c) Side view of the first four atomic layers along the mirror (yz) plane, showing the first and second neighboring layers of each triangular lattice plane. The distances shown are for bismuth. each atom having three nearest-neighbor atoms within the bilayer and three nextnearest-neighbor atoms in the adjacent bilayer (Figure 3.3(b) and (c)). Because of a much weaker van der Waals type inter-bilayer bonding, the natural cleavage plane is along the (111) plane. 3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx 33 The reciprocal lattice vectors are given by ¶ 1 1 1 b1 = 2π √ , − , 3a a c ¶ µ 1 1 1 b2 = 2π √ , , 3a a c ¶ µ 1 2 b3 = 2π − √ , 0, c 3a µ (3.4) and throughout our work, the (hkl) plane denotes the surface that is perpendicular to the reciprocal lattice vector hb1 + kb2 + lb3 . The bulk Brillouin zone is shown in Figure 3.4(a), and consists of eight TRIM, Γ, T , 3 × L and 3 × X. The bulk band structures of Bi and Sb, derived from a tight binding model [47], are shown in Figure 3.4(b) and (c). Eight doubly spin degenerate bands are shown, which originate from the atomic s, px , py and pz orbitals located on the two atoms in the unit cell. Because of the large atomic weight of the Bi and Sb atoms, a large atomic spin-orbit coupling mixes the p orbitals such that the six bands nearest to the Fermi energy have strongly mixed px , py and pz character. In bismuth, the valence band crosses EF near the T point, forming a small hole pocket, while the conduction band crosses EF at the three L points, forming small electron pockets. In this tight binding calculation, the bottom of the conduction band at L is composed of a symmetric linear combination of p-orbitals (Ls symmetry) and is separated by a small energy gap from the top of the valence band at L, which is composed of an anti-symmetric linear combination of p-orbitals (La symmetry) (Figure 3.4(e)). In antimony, on the other hand, the hole pocket is near the H point instead of near the T point. Although the electron pocket is still near the L point, the symmetry of the valence and conduction bands at this momentum are now switched relative to those in bismuth (Figure 3.4(g)). 3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx (a) (b) Bismuth (c) Antimony (f) 0.07 < x < 0.22 34 (d) T H L G X (e) Pure Bismuth EF La Ls La H Pure Antimony La Ls T (g) L T H L Ls T H L Figure 3.4: (a) Bulk 3D Brillouin zone of Bi1−x Sbx showing the eight bulk TRIM (T , Γ, 3×L, 3×X). (b) Theoretical bulk band structure of Bi and (c) Sb based on the tight binding model of Liu & Allen [47]. (d) Table of symmetry labels of the five occupied valence bands of Bi and Sb, the right column displays the product δ(ki ). Tables adapted from [22]. (e)-(g) Schematic evolution of the near EF band structure from Bi to Sb. 3.2. Predicted bulk and surface electronic structure of Bi1−x Sbx T (K) r (mW-cm) ´ Bi 1-x Sb x r (mW-cm) r (mW-cm) Bi 1-x Sb x 35 T (K) T (K) Figure 3.5: Resistivity versus temperature curves of undoped and doped Bi1−x Sbx . The resistivity curve of the x=0 sample in the left panel has been multiplied by a factor of 80. Although both Bi and Sb are semimetals, because there is a finite direct energy gap separating the valence and conduction band at every point in k-space, they can both be adiabatically connected to an insulating state by pulling the conduction band above EF and pushing the valence band below. Therefore Equation 3.2 can be applied to calculate their Z2 invariant. Such a calculation was carried out in [22] and the δ(ki ) are displayed in the rightmost columns of Figure 3.4(d). The only difference between Bi and Sb is in their value of δ(L), which arises from the La to Ls symmetry inversion. By substituting these values of δ(ki ) into Equation 3.1 , Fu and Kane concluded that Bi has trivial topological properties (ν0 = 0) while Sb is adiabatically connected to an STI (ν0 = 1). A true STI phase was postulated to exist in a Bi1−x Sbx alloy [22] provided there is a composition range x where the material is both in an insulating state and in the inverted band regime where E(La ) > E(Ls ). Based on extensive quantum oscillation, magneto-optic and transport data previously collected on the Bi1−x Sbx series [48], it is widely accepted that the band inversion transition takes place at x ∼ 0.04, 3.3. Experimental results 36 and that the alloy exhibits an insulating trend between 0.07 < x < 0.22. The highly insulating trend of samples near x = 0.1 is supported by our transport data (Figure 3.5), although we note that the resistivity of these samples saturates at low temperatures suggestive of either conduction in an impurity band or through surface states. Therefore the latter regime represents a true STI, which is the main subject of this thesis. Based on the calculated values of δ(ki ), the topology of the surface state Fermi surface of insulating compositions of Bi1−x Sbx can be predicted. In the (111) surface Brillouin zone, the four TRIM are located at Γ̄ and 3×M̄, which are the projections of the (Γ,T ) and 3×(L,X) bulk TRIM pairs onto the (111) surface (Figure 3.6(a)). It can be checked that each pair of bulk TRIM are indeed separated by G[111] /2. The surface fermion parities at each of these four surface TRIM can be calculated from the table in Figure 3.6(b), which show that for Bi1−x Sbx , π(Γ̄) = +1 and π(M̄) = −1. The Fermi surface shown in Figure 3.6(b), which encloses an odd number of TRIM (7), is consistent with these surface fermion parities (Figure 3.6(d)). 3.3 Experimental results In this section, we will show that a measurement of ν0 for spin-orbit insulators (and in some instances metals) can be made using a combination of two techniques: 1) Incident photon energy modulated angle-resolved photoemission is first used to map and distinguish between the bulk and surface state band structures of a material; 2) Having isolated the surface state Fermi surface, spin-resolved ARPES is then employed to measure the spin degeneracy and chirality of each piece of the Fermi surface in order to determine the surface fermion parities and in turn the bulk ν0 number. 3.3. Experimental results 37 (111) (a) (b) K G M M M M (L,X) T L X G 1 L 2 G (G,T) M (L,X) 3 L X X (d) (c) BiSb and Sb d(G) d(L) d(T) d(X) n0 Path p(La)p(Lb) Fermi level crossings Bi -1 -1 -1 -1 0 1 -1 odd Sb -1 +1 -1 -1 1 2 +1 even BiSb -1 +1 -1 -1 1 3 -1 odd Figure 3.6: (a) Bulk 3D Brillouin zone of Bi1−x Sbx showing the projection of the eight bulk TRIM (T, Γ, 3×L, 3×X) onto the four surface TRIM (Γ̄, 3×M̄) on the (111) surface Brillouin zone. (b) Schematic of a Fermi surface formed by the surface states of a Z2 topological insulator showing an odd number of Fermi level crossings along Γ̄M̄ and an even number along M̄-M̄. (c) Bulk parity invariants of the Bi1−x Sbx system and their corresponding Z2 topological number ν0 . (d) Product of surface fermion parities between two surface TRIM on a Z2 topological insulator. 3.3. Experimental results 3.3.1 38 Topological surface states in insulating Bi1−x Sbx The Bi1−x Sbx alloy series has been investigated intensively since the 1960s for its thermoelectric properties [48]. The high thermoelectric figure-of-merit has been theoretically attributed to the highly non-parabolic electronic structure of pure bismuth near the L point, which is described by the massive (3+1)-dimensional relativistic Dirac p equation [49]. The resulting dispersion relation, E(k) = ± (v · k)2 + ∆2 ≈ v · k, is highly linear owing to the combination of an unusually large band velocity v and a small inter-band gap ∆ (such that |∆/v| ≈ 5 × 10−3 Å−1 ), which has successfully explained various peculiar properties of bismuth [50, 51, 52]. Substituting bismuth with antimony is believed to change the critical energies of the band structure according to Figure 3.8(e). At an Sb concentration of x ≈ 4%, the gap ∆ between La and Ls closes and a massless three-dimensional Dirac point is realized. As x is further increased this gap re-opens with inverted symmetry ordering, which leads to a change in sign of ∆ at each of the three equivalent L points in the Brillouin zone. For concentrations greater than x ≈ 7% there is no overlap between the valence band at T and the conduction band at L, and the material becomes an inverted-band insulator. Once the band at T drops below the valence band at L, at x ≈ 7 − 8%, the system evolves into a direct-gap insulator whose low energy physics is dominated by the spin-orbit coupled Dirac particles at L [22, 48]. Although this Dirac electronic structure has been experimentally inferred via transport, quantum oscillation and magneto-optical measurements [48], no direct observation has ever been made. We used incident photon energy modulated ARPES to measure the electronic band dispersion along various momentum space trajectories in the 3D Brillouin zone of single crystals of Bi0.9 Sb0.1 . Examples of such trajectories are shown in Figure 3.7, which are constructed by invoking the free-electron-final state approximation with an inner potential V0 = -10 eV [34]. ARPES spectra taken along two orthogonal cuts 3.3. Experimental results 39 k z (Å-1) 3 29 eV 2 18 eV 1 X L 0 -3 -2 -1 0 1 2 3 -1 k x (Å ) Figure 3.7: Location of the L (black circles) and X (red circles) points in the bulk Brillouin zone in the kx − kz plane together with the constant energy contours that can be accessed by changing the angle ϑ. The two contours correspond to hν = 29 eV and 18 eV and span an angular range of ±80◦ 3.3. Experimental results 40 through the L point in the third bulk Brillouin zone using 29 eV photons are shown in Figures 3.8(a) and (c). A Λ-shaped dispersion which intersects within 50 meV below the Fermi energy (EF ) can be seen along both directions. Additional features originating from surface states (SS) that do not disperse with incident photon energy are also seen. Owing to the finite intensity between the bulk and surface states, the exact binding energy (EB ) where the tip of the Λ-shaped dispersion lies is unresolved. The linearity of the bulk Λ-shaped bands is observed by locating the peak positions at higher EB in the momentum distribution curves (MDCs), and the energy at which these peaks merge is obtained by extrapolating linear fits to the MDCs. Therefore 50 meV represents a lower bound on the energy gap ∆ between La and Ls . The extracted band velocities along the kx and ky directions are 7.9 ± 0.5 × 104 ms−1 and 10.0 ± 0.5 × 105 ms−1 respectively, which are similar to the tight binding values 7.6×104 ms−1 and 9.1×105 ms−1 calculated for the La band of bismuth [47]. Our data is consistent with the extremely small effective mass of 0.002me observed in x = 0.11 samples by magneto-reflection measurements [53]. The Dirac point in graphene, coincidentally, has a comparable band velocity (|vF | ≈ 106 ms−1 ) [54] to what we observe for Bi0.9 Sb0.1 , but the spin-orbit coupling in graphene is several orders of magnitude weaker [23] and the only known method of inducing a gap in the Dirac spectrum of graphene is by coupling to an external chemical substrate [55]. The Bi1−x Sbx series thus provides a rare opportunity to study relativistic Dirac Hamiltonian physics in a 3D condensed matter system where the intrinsic (rest) mass gap can be easily tuned. Studying the band dispersion perpendicular to the sample surface provides a way to differentiate bulk states from surface states in a 3D material. To visualize the near-EF dispersion along the 3D L-X cut (X is a point that is displaced from L by a kz distance of 3π/c, where c is the lattice constant), in Figure 3.9(a) we plot energy distribution curves (EDCs), taken such that electrons at EF have fixed in- 3.3. Experimental results 41 ® k = (0.8, ky, 2.9) = L ± dky 0.1 0.1 ky 0.0 0.0 EB (eV) (c) (b) (a) Low 0.1 2 ky L -0.1 -0.1 kx L -0.1 -0.05 -0.3 -0.3 -0.3 -0.4 -0.4 -0.4 -0.2 0.0 -1 kdeg y (Å ) SS 1 kx SS SS -0.10 -0.15 -0.5 -0.5 -0.5 SS 0.00 kx -0.2 -0.2 -0.2 ky 0.05 L SS 0.0 SS ® k = (kx ,0, 2.9) = L ± dkx High -0.20 L-0.1 0.2 L L+0.1 0.4 0.6 0.8 1.0 1.2 -1 k xdeg (Å ) -1 k (Å ) kz M (e) (d) G T 2 1 T (f) ky kx E K X L L X LS La Bi -1 k y (Å ) 4% 7% 8% x -1 k x (Å ) Figure 3.8: Selected ARPES intensity maps of Bi0.9 Sb0.1 are shown along three ~kspace cuts through the L point of the bulk 3D Brillouin zone (BZ). The presented data are taken in the 3rd BZ with Lz = 2.9 Å−1 with a photon energy of 29 eV. The cuts are along (a), the ky direction, (b), a direction rotated by approximately 10◦ from the ky direction, and (c), the kx direction. Each cut shows a Λ-shaped bulk band whose tip lies below the Fermi level signalling a bulk gap. The surface states are denoted SS. (d), Momentum distribution curves (MDCs) corresponding to the intensity map in (a). (f), Log scale plot of the MDCs corresponding to the intensity map in (c). The red lines are guides to the eye for the bulk features in the MDCs. (e) Schematic evolution of bulk band energies as a function of x is shown. The composition we study here (for which x = 0.1) is indicated by the green arrow. 3.3. Experimental results 42 plane momentum (kx , ky ) = (Lx , Ly ) = (0.8 Å−1 , 0.0 Å−1 ), as a function of photon energy (hν). There are three prominent features in the EDCs: a non-dispersing, kz independent, peak centered just below EF at about −0.02 eV; a broad non-dispersing hump centered near −0.3 eV; and a strongly dispersing hump that coincides with the latter near hν = 29 eV. To understand which bands these features originate from, we show ARPES intensity maps along an in-plane cut K̄M̄K̄ (parallel to the ky direction) taken using hν values of 22 eV, 29 eV and 35 eV, which correspond to approximate kz values of Lz − 0.3 Å−1 , Lz , and Lz + 0.3 Å−1 respectively (Figure 3.9(b-d)). At hν = 29 eV, the low energy ARPES spectral weight reveals a clear Λ-shaped band close to EF . As the photon energy is either increased or decreased from 29 eV, this intensity shifts to higher binding energies as the spectral weight evolves from the Λ-shaped into a ∪-shaped band. Therefore the dispersive peak in Figure 3.9(a) comes from the bulk valence band, and for hν = 29 eV the high symmetry point L = (0.8 Å−1 , 0 Å−1 , 2.9 Å−1 ) appears in the third bulk BZ. In the maps of Figure 3.9(b) and (d) with respective hν values of 22 eV and 35 eV, overall weak features near EF that vary in intensity remain even as the bulk valence band moves far below EF . The survival of these weak features over a large photon energy range (17 to 55 eV) supports their surface origin. The non-dispersing feature centered near −0.3 eV in Figure 3.9(a) comes from the higher binding energy (valence band) part of the full spectrum of surface states, and the weak non-dispersing peak at −0.02 eV reflects the low energy part of the surface states that cross EF away from the M̄ point and forms the surface Fermi surface. As an additional check that we have indeed correctly identified the bulk bands of Bi0.9 Sb0.1 in Figures 3.8 and 3.9, we also measured the dispersion of the deeper lying bands well below the Fermi level and compared them to tight binding theoretical calculations of the bulk bands of pure bismuth following the model of Liu and 3.3. Experimental results 43 ® k = L ± dk z Bulk state (b) G Intensity (arb. units) M EB(eV) (a) hn = 22eV 00 High -1-1 -2-2 -3-3 -4-4 Low -0.1 0.0 0.1 0.2 -0.10 0.00 0.10 0.20 deg (c) EB(eV) SS hn = 29eV 0 0 -1-1 -2-2 -3-3 -4-4 -0.10 0.00 0.10 0.20 -0.1 0.0 0.1 0.2 deg (d) kZ EB(eV) M G K M K ky kX T X 3 0 -1 -2 -3 -4 L L X hn = 35eV -1.0 -0.5 0.0 -0.10 0.00 0.10 0.20 -0.1 0.0 0.2 0.1 deg -1 EB(eV) -1 k y (Å ) ; kx = 0.8 Å Figure 3.9: (a) Energy distribution curves (EDCs) of Bi0.9 Sb0.1 with electrons at the Fermi level (EF ) maintained at a fixed in-plane momentum of (kx = 0.8 Å−1 , ky = 0 Å−1 ) are obtained as a function of incident photon energy to identify states that exhibit no dispersion perpendicular to the (111)-plane. Selected EDC data sets with photon energies of 28 eV to 32 eV in steps of 0.5 eV are shown for clarity. The energy non-dispersive (kz independent) peaks near EF are the surface states (SS). (b-d) ARPES intensity maps along cuts parallel to ky taken with electrons at EF fixed at kx = 0.8 Å−1 and with photon energies of hν = 22 eV, 29 eV and 35 eV are shown. The faint Λ-shaped band at hν = 22 eV and hν = 35 eV shows some overlap with the bulk valence band at L (hν = 29 eV), suggesting that it is a resonant surface state which is degenerate with the bulk state in some limited k-range near EF . The flat band of intensity centered about -2 eV in the hν = 22 eV scan originates from Bi 5d core level emission from second order light. 3.3. Experimental results 44 Allen [47]. A tight-binding approach is known to be valid since Bi0.9 Sb0.1 is not a strongly correlated electron system. As Bi0.9 Sb0.1 is a random alloy (Sb does not form a superlattice [48]) with a relatively small Sb concentration (∼0.2 Sb atoms per rhombohedral unit cell), the deeper lying band structure of Bi0.9 Sb0.1 is expected to follow that of pure Bi because the deeper lying (localized wave function) bands of Bi0.9 Sb0.1 are not greatly affected by the substitutional disorder, and no additional back folded bands are expected to arise. Since these deeper lying bands are predicted to change dramatically with kz , they help us to finely determine the experimentally probed kz values. Figure 3.10(f) shows the ARPES second derivative image (SDI) of a cut parallel to K̄M̄K̄ that passes through the L point of the 3D Brillouin zone, and Figure 3.10(h) shows a parallel cut that passes through the 0.3 XL point (Figure 3.7), which were achieved as follows. By adjusting ϑ such that the in-plane momentum kx is fixed at approximately 0.8 Å−1 (the surface M̄ point), at a photon energy hν = 29 eV, electrons at the Fermi energy (EB =0 eV) have a kz that corresponds to the L point in the third bulk BZ. By adjusting ϑ such that the in-plane momentum kx is fixed at approximately -0.8 Å−1 , at a photon energy hν = 20 eV, electrons at a binding energy of -2 eV have a kz near 0.3 XL. There is a clear kz dependence of the dispersion of measured bands A, B and C, pointing to their bulk nature. The bulk origin of bands A, B and C is confirmed by their good agreement with tight binding calculations (bands 3, 4 and 5 in Figures 3.10(g) and (i)), which include a strong spin-orbit coupling constant of 1.5 eV derived from bismuth [47]. The bands labeled 3 to 6 are derived from 6p-orbitals and their dispersion is thus strongly influenced by spin-orbit coupling. The fact that there is a close match of the bulk band dispersion between the data and calculations further confirms the presence of strong spin-orbit coupling. The slight differences between the experimentally measured band energies and the calculated band energies at ky 3.3. Experimental results 45 kz (a) (c) (b) (d) T X L L ky kx X (e) kz = L E B(eV) (f) (g) kz = L kz = 0.3XL (h) 6 00 (i) 0 0.0 EB(eV) -1-1 -2-2 -0.5 -3-3 -1.0 -1 5 C B 4 -2 A 3 -3 -4-4 -0.15 -0.15 0.0 0.15 0.15 deg -0.4 C 5 B 4 -4 -0.2 -0.2 0.0 0.0 deg 0.2 0.2 0.4 -0.2 0.0 0.2 -1 ky(Å ) -0.4 -0.2 -0.2 0.0 0.0 0.2 0.2 0.4 -0.2 0.0 0.2 deg Figure 3.10: (a), Energy distribution curves (EDCs) along a k-space cut given by the upper yellow line shown in schematic (c) which goes through the bulk L point in the 3rd BZ (hν = 29 eV). The corresponding ARPES intensity in the vicinity of L is shown in (e). (b), EDCs along the lower yellow line of (c) which goes through the point a fraction 0.3 of the k-distance from X to L (hν = 20 eV). (This cut was taken at a kx value equal in magnitude but opposite in sign to that in (a). (f,h) The ARPES second derivative images (SDI) of the raw data shown in (a) and (b) to reveal the band dispersions. The flat band of intensity at EF is an artifact of taking SDI. (g,i) Tight binding band calculations of bismuth including spin-orbit coupling, using Liu and Allen model [47], along the corresponding experimental cut directions shown in (f) and (h). Band 3 drops below −5 eV at the 0.3 XL point. The inter-band gap between bands 5 and 6 is barely visible on the scale of (g). The circled curves mark the surface state dispersion, which is present at all measured photon energies (no kz dispersion). (d) Tight binding valence band (5) dispersion of bismuth in the ky -kz momentum plane showing linearity along both directions. 3.3. Experimental results 46 = 0 Å−1 shown in Figure 3.10(f-i) are due to the fact that the ARPES data were taken in constant ϑ mode. This means that electrons detected at different binding energies will have slightly different values of kz , whereas the presented tight binding calculations show all bands at a single kz . We checked that the magnitude of these band energy differences is indeed accounted for by this explanation. Even though the La and Ls bands in Bi0.9 Sb0.1 are inverted relative to those of pure Bi, calculations show that near EF , apart from an insulating gap, they are “mirror” bands in terms of k dispersion (see bands 5 and 6 in Figure 3.10(g)). Therefore an overall close match to calculations, which also predict a linear dispersion along the kz cut near EF (Figure 3.10(d)), provides strong support that the dispersion of band C, near EF , is in fact linear along kz . Focusing on the Λ-shaped valence band at L, the EDCs (Figure 3.10(a)) show a single peak out to ky ≈ ±0.15 Å−1 demonstrating that it is composed of a single band feature. Outside this range however, an additional feature develops on the low binding energy side of the main peak in the EDCs, which shows up as two well separated bands in the SDI image (Figure 3.10(f)) and signals a splitting of the band into bulk representative and surface representative components (Figure 3.10(a),(f)). Unlike the main peak that disperses strongly with incident photon energy, this shoulderlike feature is present and retains the same Λ-shaped dispersion near this k-region (open circles in Figures 3.10(g) and (i)) for all photon energies used, supporting its 2D surface character. This behaviour is quite unlike bulk band C, which attains the Λ-shaped dispersion only near 29 eV (Figure 3.9). Having established the existence of an energy gap in the bulk state of Bi0.9 Sb0.1 near L (Figures 3.8 and 3.9) and observed linearly dispersive bulk bands uniquely consistent with strong spin-orbit coupling model calculations [50, 51, 52, 47], we now discuss the topological character of its surface states, which are found to be gapless 3.3. Experimental results 47 (a) (b) G M SS of Bi1-x Sbx -0.2 -0.2 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 0.0 2 M1 -0.1 0.6 0.8 1.0 1.2 k x (Å-1 ) 0.6 0.8 1.0 -k x (Å-1 ) (d) Topological Hall insulator 1 M2 0.0 1.2 1.0 k x (Å-1 ) (c) 0.1 0.04 -0.04 0.0 0.0 EB(eV) -1 kDeg y (Å ) 0.2 0.2 -0.2 -0.2 EB(eV) k y (Å-1 ) high M’ low G M1 3 4,5 (e) 0.0 G M2 M -0.1 0.0 0.2 0.4 0.6 -1 G - k X (Å ) 0.8 M 1.0 -0.2 -0.1 0.0 EB(eV) Figure 3.11: (a) Spin-integrated ARPES intensity map of the SS of Bi0.91 Sb0.09 at EF . (b) High resolution ARPES intensity map of the SS at EF that enclose the M̄1 and M̄2 points. Corresponding band dispersion second derivative images (SDI) are shown below. The left right asymmetry of the band dispersions are due to the slight offset of the alignment from the Γ̄-M̄1 (M̄2 ) direction. (c) The surface band dispersion SDI of Bi0.9 Sb0.1 along Γ̄-M̄. The shaded white area shows the projection of the bulk bands based on ARPES data, as well as a rigid shift of the tight binding bands to sketch the unoccupied bands above the Fermi level. The Fermi crossings of the surface state are denoted by yellow circles, with the band near −kx ≈ 0.5 Å−1 counted twice owing to double degeneracy. The red lines are guides to the eye. The EDCs along Γ̄-M̄ are shown to the right (d). (e) Schematic of the surface Fermi surface observed in our experiments, which is consistent with a ν0 = 1 topology. 3.3. Experimental results 48 (Figure 3.9). On the (111) surface of Bi0.9 Sb0.1 , the four TRIM are located at Γ̄ and three M̄ points that are rotated by 60◦ relative to one another. Owing to the three-fold rotational symmetry of the bulk crystal (A7 crystal structure) and the timereversal symmetry of the two-dimensional surface states, the surface state dispersion around these three M̄ points are equivalent, and we henceforth refer to them as a single point, M̄. This is also experimentally verified in Figure 3.11(b). The surface state Fermi surface of Bi0.9 Sb0.1 (111) was obtained by collecting ARPES intensity in a narrow energy window about EF as a function of kx and ky , which is displayed in Figure 3.11(a). It shows a hexagonal Fermi surface enclosing Γ, a petal shaped Fermi surface along the Γ-M̄ line that does not enclose any TRIM, and a dumbbell shaped Fermi surface that encloses M̄. Due to the narrowness of the Fermi surface near M̄, it is necessary to study its band dispersion below EF to deduce the topology. The surface state band dispersion along a path connecting Γ̄ to M̄ is shown in Figure 3.11(c). Like in pure bismuth, two surface bands emerge from the bulk band continuum near Γ̄ to form a central electron pocket enclosing Γ̄ and an adjacent hole lobe [56, 57, 58]. It has been established that these two bands result from the spin-splitting of a Kramers pair and are thus singly spin degenerate [59, 58]. On the other hand, the surface band that crosses EF at −kx ≈ 0.5 Å−1 , and forms the narrow electron pocket around M̄, is clearly doubly degenerate, as far as we can determine within our experimental resolution. This is indicated by its splitting below EF between −kx ≈ 0.55 Å−1 and M̄, as well as the fact that this splitting goes to zero at M̄ in accordance with Kramers theorem. In semimetallic single crystal bismuth, only a single surface band is observed to form the electron pocket around M̄ [60, 61]. Moreover, this surface state overlaps, hence becomes degenerate with, the bulk conduction band at L owing to the semimetallic character of Bi. In Bi0.9 Sb0.1 on the other hand, the states near M̄ fall completely inside the bulk energy gap 3.3. Experimental results 49 (a) x = 0.1 k y (Å-1) 0.06 (b) 2 4,5 0.00 1 M -0.06 0.6 (c) 0.8 1.0 EB(eV) 0.0 -0.1 -0.2 0.04 1.0 0.8 2 EB(eV) 0.0 -0.04 -0.08 0.6 -0.2 M -0.1 0.0 Intensity (arb. units) 0.6 (e) (d) Intensity (arb. units) 1 1.0 0.8 -1 - k x(Å ) -0.08 (f) -0.04 0.0 EB(eV) Figure 3.12: (a) ARPES intensity integrated within ±10 meV of EF originating solely from the surface state crossings. The image was plotted by stacking along the negative kx direction a series of scans taken parallel to the ky direction. (b) Outline of Bi0.9 Sb0.1 surface state ARPES intensity near EF measured in (a). White lines show scan directions “1” and “2”. (c) Surface band dispersion along direction “1” taken with hν = 28 eV and the corresponding EDCs (d). The surface Kramers degenerate point, critical in determining the topological Z2 class of a band insulator, is clearly seen at M̄, approximately 15 ± 5 meV below EF . (We note that the scans are taken along the negative kx direction, away from the bulk L point.) (e), Surface band dispersion along direction “2” taken with hν = 28 eV and the corresponding EDCs (f). This scan no longer passes through the M̄-point, and the observation of two well separated bands indicates the absence of Kramers degeneracy as expected. 3.3. Experimental results 50 preserving their purely surface character at M̄. The surface Kramers doublet point can thus be defined in the bulk insulator (unlike in Bi [59, 56, 57, 58, 60, 61]) and is experimentally located in Bi0.9 Sb0.1 samples to lie approximately 15 ± 5 meV below EF at k = M̄ (Figure 3.11(c)). These results show that the surface band crosses EF five times between Γ̄ and M̄, which is consistent with a ν0 = 1 topology where π(Γ̄)π(M̄) = −1. The dumbbell shaped Fermi surface segment enclosing M̄ must therefore be regarded as two Fermi surfaces, which leads to an overall Fermi surface (Figure 3.11(e)) that encloses the four surface TRIM a total of 7 times. The imaging of the Kramers point at M̄ is an important demonstration that our alignment is strictly along the Γ̄-M̄ line. To see the effects of slight misalignment, we contrast high resolution ARPES measurements taken along the Γ̄-M̄ line with those that are slightly offset from it. Figure 3.12 shows that with ky offset from the Γ̄-M̄ line by less than 0.02 Å−1 , the Kramers degeneracy is lifted and the top branch of the Kramers pair crosses EF to form part of the bow-shaped electron distribution (Figure 3.12(a) and (b)). In order to confirm that the surface states of insulating Bi1−x Sbx belong to the ν0 = 1 class, the spin-polarization of the surface bands must be measured. Below we present results of the spin-resolved ARPES analysis on bulk insulating Bi0.91 Sb0.09 . For this experiment, we used incident photons in the VUV regime and analyzed the photoelectron spins using a single Mott polarimeter. By using VUV photons, spin conserving photoemission processes (where the electric field of light only acts on the orbital degree of freedom of the electron inside a solid) dominate over spin non-conserving processes (which arise from coupling to the magnetic field of light) [38]. Therefore the spin polarization of a photoemitted electron is representative of its spin polarization inside the crystal. 3.3. Experimental results (e) (c) Bi 0.91 Sb 0.09 l3 l2 l1 r1 EB(eV) 0.0 0.0 -0.1 -0.1 -0.2 -0.2 0.1 0.0 Pz’ Py’ -0.1 -M -0.2 -0.4 -0.6 Intensity (arb. units) l4,5 Polarization 0.1 0.1 -0.6 G -0.4 k x (Å ) Intensity (arb. units) 5 4 r2 r1 l1 l2 l3 l1 l2 r1 l3 l4,5 2 1 -0.6 -0.4 0.0 -0.2 k x (Å-1 ) (f) 0.5% Te doped BiSb 1 1 -M l4,5 M G l3 l2 l1 0.0 0.0 I tot Py 0 0 Pz 2 -1 -1 1 -1 0 -0.4 3 0.1 0.1 EB = -25 meV 3 -0.6 Spin up Spin down 4 0.0 (d) 6 5 k x (Å-1 ) -1 (b) -0.2 -0.2 0.0 k x (Å-1 ) 0 Px 1 0 1 Pin plane EB(eV) (a) 51 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.2 0.0 0.2 0.4 k x (Å-1 ) Figure 3.13: (a) The surface band dispersion ARPES second derivative image (SDI) along the Γ̄ to -M̄ direction of bulk insulating Bi0.91 Sb0.09 . Dashed white lines are guides to the eye. The intensity of bands l4, 5 is scaled up for clarity. (b) MDC of the spin averaged spectrum at EB = -25 meV [green line below EF in (a)] using a photon energy hν = 22 eV, together with the Lorentzian peaks of the fit. (c) Measured spin polarization curves (symbols) for the y 0 and z 0 (Mott coordinate) components together with the fitted lines. (d) The in-plane and out-of-plane spin polarization components in the sample coordinate frame obtained from the spin polarization fit. The symbols refer to those in (b). The fitted parameters are consistent with 100% polarized spins. (e) Spin resolved spectra for the y component based on the fitted spin polarization curves shown in (c). Spin up (down) refers to the spin direction being approximately parallel to the +(-)ŷ direction. (f) The surface band dispersion SDI centered about Γ̄ of (Bi0.925 Sb0.075 )0.995 Te0.005 . Electron doping through Te reveals that bands l2 and l3 are connected above EF . 3.3. Experimental results 52 Figure 3.13(b) shows a spin-averaged momentum distribution curve (MDC) along the Γ̄ to -M̄ direction taken at EB = -25 meV, which intersects the surface bands at the k positions shown in Figure 3.13(a). This MDC was obtained by averaging the signal from the four electron detectors surrounding the Mott polarimeter. A sum of Lorentzian lineshapes I i and a non-polarized background B are fitted to this MDC, which are used as inputs to the two-step fitting routine developed by Meier et al. [42] as described in Chapter 2. Because a single Mott polarimeter is being used, only two spin components are measured. Therefore an additional constraint that the electron spins are fully polarized with magnitude h̄/2 (i.e. |P| = 1) had to be imposed in the fit. It is known that spin polarization of surface bands can become less than 1 as a band approaches the bulk band continuum [62]. Because the surface band intersections with the EB = -25 meV line (Figure 3.13(a)) are located far from the bulk continuum, this imposed constraint is realistic. Moreover, common strong spin-orbit coupled materials such as gold and bismuth based surface alloys have been experimentally shown to exhibit 100% spin polarized surface states [63, 42]. To review the fitting procedure briefly, a spin polarization veci tor P~M = (Pxi0 , Pyi0 , Pzi0 ) = (cos ϑi cos ϕi , cos ϑi sin ϕi , sin ϑi ) is assigned to each band, where ϑi and ϕi are referenced to the primed Mott coordinate frame. A spin-resolved spectrum is then defined for each peak i using the relation Iαi;↑,↓ = I i (1 ± Pαi )/6, where α = x0 , y 0 , z 0 , and + and − correspond to the spin direction being parallel (↑) or antiparallel (↓) to α. The full spin-resolved spectrum is then given by P Iα↑,↓ = i Iαi;↑,↓ + B/6, from which the spin polarization of each spatial component can be obtained as Pα = (Iα↑ − Iα↓ )/(Iα↑ + Iα↓ ). This latter expression is a function of ϑi and ϕi , which are the parameters that are varied to fit the experimental spin polarization data. 3.3. Experimental results 53 (a) EB (b) EB spin down La EF G H kx La EF Ls spin up T spin down L M Ls spin up T G H kx L M Bulk conduction band Bulk valence band Figure 3.14: Two possible band connection scenarios in Bi1−x Sbx that are generated from (a) first principles calculations and (b) tight-binding calculations. The spin polarization data for the y 0 and z 0 components (i.e. Py0 and Pz0 ) are obtained by taking the difference between the intensities of the left-right (or top-bottom) electron detectors over their sum, normalized by the Sherman function. These data are shown as symbols in Figure 3.13(c), and the solid lines are the fits based on the fitting procedure described above. The best fit parameters (Pxi0 , Pyi0 , Pzi0 ) are transformed into the sample coordinate frame and are displayed in Figure 3.13(d). Although the measured polarization curves only reaches a magnitude of around ±0.1, which is similarly seen in studies of Bi thin films [59], this is not a true measure of the actual spin polarization but is rather an artifact of the non-polarized background and overlap of adjacent peaks with different spin polarization. A true measure of the spin polarization can only be extracted through fitting, which yield polarization values consistent with 1 (Figure 3.13(d)). 3.3. Experimental results 54 The fitted spin polarization vectors of the surface bands suggest that all spins are nearly aligned parallel to the in-plane ±ŷ direction along the Γ̄-M̄ cut, and Figure 3.13(e) shows the spin-resolved spectrum calculated from Py . This suggests that the electric field in the Rashba spin-orbit Hamiltonian (p × E) · σ points predominantly normal to the sample surface. The fact that the fitted spin polarizations are not strictly along the ±ŷ direction is likely due to slight misalignments of the scan away from the Γ̄-M̄ cut, which is difficult to eliminate given the very narrow Fermi surface features. Bands l1 and r1 having nearly opposite spin because time-reversal symmetry demands spins directions to be reversed on either side of Γ̄. Bands l1 and l2 also have opposite spin, which shows that they form a Kramers pair. These observations confirm that the surface states of bulk insulating Bi1−x Sbx are representative of the ν0 = 1 class. Because of a dramatic intrinsic weakening of signal intensity near crossings l4 and l5, and the small energy and momentum splittings of bands l4 and l5 lying at the resolution limit of modern spin-resolved ARPES spectrometers, we are unable to extract the spin polarizations of these two bands. However, whether bands l4 and l5 are both singly or doubly spin degenerate does not change the fact that an odd number of spin-polarized Fermi surfaces enclose the surface TRIM. Recent theoretical work [64] has uncovered a discrepancy between Bi(111) surface state calculations using the tight-binding parameters given by Liu and Allen [47] and using first principles methods [65]. In particular, they disagree as to whether bands l1 and l2 cross each other above the Fermi level as shown in Figure 3.14. Our observation that bands l2 and l3 have the same spin suggests that they originate from the same band, and thus the band dispersion above the Fermi level should follow that shown in Figure 3.14(a). A more direct way to show that bands l2 and l3 connect above EF is to map their energy dispersion above EF . While inverse photoemission [33] can be used for this purpose, these results can often be difficult to interpret. Instead, 3.3. Experimental results 55 Te-doped BiSb 0.1 (a) 0.1 E B (eV) Deg -1 k y (Å ) 0.2 0.2 0.0 0.0 1 -0.2 -0.2 -0.2 (b) 0.0 3 2 0.2 0.4 0.6 0.8 1.0 -0.1 -0.1 -0.1 -0.2 -0.2 -0.1 E B (eV) -1 Deg -0.2 0.0 0.2 0.1 undoped BiSb -0.2 0.0 0.2 0.4 0.6 0.8 0.2 1.0 1.2 (g) -1 Sn-doped BiSb Deg -0.2 -0.2 -0.2 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 -1 k x (Å ) 0.8 0.8 1.0 1.0 1.2 1.2 (h) -0.1 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.1 0.0 0.1 -0.1 0.0 0.1 (j) -0.1 (k) 0.0 0.0 0.0 -0.1 -0.1 -0.1 -0.1 -0.2 -0.2 -0.1 0.0 0.1 0.0 0.1 0.0 0.1 (l) 0.0 -0.2 0.2 0.1 0.1 3 0.0 (i) 0.0 M E B (eV) k y (Å ) -0.2 -0.2 0.1 0.0 K 0.0 0.2 0.0 0.1 0.0 0.0 0.0 0.2 G -0.2 0.1 2 -0.2 (c) (f) 0.0 0.1 -0.2 -0.2 (e) 0.0 1.2 0.0 0.0 Sn-doped BiSb 0.1 0.0 -0.2 0.2 0.2 k y (Å ) 1 (d) 0.0 -0.2 Te-doped BiSb undoped BiSb 0.1 -0.1 0.0 0.1 -0.1 -1 k x (Å ) Figure 3.15: The (111) surface state Fermi surfaces of (a) 1% Te doped, (b) undoped and (c) 1% Sn doped Bi1−x Sbx . (d) to (l) show ARPES dispersion maps along the constant kx cut directions marked by the white arrows in (a) for all three samples. 3.3. Experimental results 56 0.1 (a) Sn-doped BiSb 0.1 E B (eV) 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.3 E B (eV) 0.10.0 (b) undoped BiSb 0.0 -0.1 -0.1 -0.2 -0.2 0.1 (c) Te-doped BiSb E B (eV) 0.10.0 -0.1 0.0 -0.2 -0.1 -0.3 -0.2 0.0 -1 k x (Å ) 0.8 Figure 3.16: The ARPES intensity map along the Γ̄-M̄ cut of (a) 1% Te doped, (b) undoped and (c) 1% Sn doped Bi1−x Sbx together with their corresponding energy distribution curves. we mapped the surface band dispersion of Te doped Bi1−x Sbx , where Te acts as an electron donor [66]. Figure 3.13(f) shows that the hole band formed by crossings l2 and l3 in insulating Bi1−x Sbx has sunk completely below EF with 0.5% Te doping, and are in fact part of the same band. More generally, we show that the doping level of the surface Fermi surfaces of Bi1−x Sbx can be controlled by either hole doping the bulk with Sn or electron doping it with Te. Doping also changes the Fermi level in the sample bulk causing them to become metallic (Figure 3.5). Figures 3.15(a)-(c) show the (111) surface state Fermi surfaces of a 1% Te doped, undoped and 1% Sn doped sample of Bi0.93 Sb0.07 . The effect of Te doping is clearly to enlarge the sizes of the central hexagonal Fermi surface and the dumbbell shaped Fermi surface around M̄, while shrinking the size 3.3. Experimental results 57 of the petal shaped Fermi surfaces. Because the former two Fermi surface segments are electron like while the latter is hole like, the effect of Te doping is to raise the chemical potential of the surface states. The effect of Sn doping on the other hand is just the opposite. Figure 3.16 shows that in a range of 1% hole or electron doping, the surface state dispersion can be tuned from having the electron like bands around M̄ completely pulled above EF to having the hole like bands pushed completely below EF respectively. This study shows that small amounts of bulk doping can be used to finely control the carrier density of the topological surface metals. 3.3.2 Topological surface states in metallic Sb The ν0 = 1 topology of bulk insulating Bi1−x Sbx is predicted to be inherited from the bulk wavefunctions of pure Sb (Figure 3.4(d)). Because there is a finite direct energy gap between the bulk valence and conduction bands at each k-point in Sb, ν0 is well defined for Sb and its surface states should therefore also exhibit a non-trivial energy dispersion that is adiabatically connected to that of Bi1−x Sbx . In this section, we will apply the previous experimental procedures to study metals such Sb. The low lying electronic states of single crystal Sb(111) were studied first using incident photon energy modulated ARPES, which we employ to isolate the surface from bulk-like electronic bands over the entire BZ. Figure 3.17(b) shows momentum distribution curves of electrons emitted at EF as a function of kx (k Γ̄-M̄) for Sb(111). The out-of-plane component of the momentum kz was calculated for different incident photon energies (hν) using the free electron final state approximation with an experimentally determined inner potential of 14.5 eV [68, 69]. There are four peaks in the MDCs centered about Γ̄ that show no dispersion along kz and have narrow widths of ∆kx ≈ 0.03 Å−1 . These are attributed to surface states and are similar to those that appear in Sb(111) thin films [68]. As hν is increased beyond 20 eV, a 3.3. Experimental results 58 kz (a) M K ky kx EB = 0 eV X L G hn = 26 eV L H U U X T Intensity (arb. units) (c) (b) (111) G 3.2 L k z (Å-1 ) 3.0 hn = 26 eV H 2.8 18 eV 2.6 U 14 eV -1.0 -0.8 -0.6 -0.4 -0.2 -1 k x (Å ) T 0.0 hn = 14 eV 0.2 -0.4 -0.2 0.0 0.2 0.4 -1 k x (Å ) Figure 3.17: (a) Schematic of the bulk BZ of Sb and its (111) surface BZ. The shaded region denotes the momentum plane in which the following ARPES spectra were measured. (b) Select MDCs at EF taken with photon energies from 14 eV to 26 eV in steps of 2 eV, taken in the T XLU momentum plane. Peak positions in the MDCs were determined by fitting to Lorentzians (green curves). (d) Experimental 3D bulk Fermi surface near H (red circles) and 2D surface Fermi surface near Γ̄ (open circles) projected onto the kx -kz plane, constructed from the peak positions found in (c). The kz values are determined using calculated constant hν contours (black curves). The shaded gray region is the theoretical hole Fermi surface calculated in [67]. 3.3. Experimental results (a) hn = 24 eV 59 0 Intensity (arb. units) 20 0.0 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.4 -0.5 -0.5 EB(eV) (b) hn = 20 eV 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.4 -0.5 -0.5 (c) hn = 18 eV 0.0 -0.1 -0.1 -0.2 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.4 -0.5 -0.5 0.4 -1 k x (Å ) 1 (f) hn = 18 eV 0.0 0.0 0.0 0 (e) hn = 20 eV 0.0 -0.4 (d) hn = 24 eV G 1.0 0.4 M -1 k x (Å ) Figure 3.18: ARPES intensity maps of Sb(111) as a function of kx near Γ̄ (a)-(c) and M̄ (d)-(f) and their corresponding energy distribution curves, taken using hν = 24 eV, 20 eV and 18 eV photons. The intensity scale of (d)-(f) is a factor of about twenty smaller than that of (a)-(c) due to the intrinsic weakness of the ARPES signal near M̄. broad peak appears at kx ≈ -0.2 Å−1 , outside the k range of the surface states near Γ̄, and eventually splits into two peaks. Such a strong kz dispersion, together with a broadened linewidth (∆kx ≈ 0.12 Å−1 ), is indicative of bulk band behavior, and indeed these MDC peaks trace out a Fermi surface (Figure 3.17(c)) that is similar in shape to the hole pocket calculated for bulk Sb near H [67]. Therefore by choosing an appropriate photon energy (e.g. ≤ 20 eV), the ARPES spectrum at EF along Γ̄-M̄ will have contributions from only the surface states. The small bulk electron pocket centered at L is not accessed using the photon energy range we employed. 3.3. Experimental results 60 Having distinguished the bulk from surface contributions to the ARPES signal at EF , we proceed to map their band dispersions below EF . ARPES spectra along Γ̄-M̄ taken at three different photon energies are shown in Figure 3.18. Near Γ̄ there are two rather linearly dispersive electron like bands that meet exactly at Γ̄ at a binding energy EB ∼ -0.2 eV. This behavior is consistent with a pair of spin-split surface bands that become degenerate at the TRIM Γ̄ due to Kramers degeneracy. The surface origin of this pair of bands is established by their lack of dependence on hν (Figure 3.18(a)-(c)). A strongly photon energy dispersive hole like band is clearly seen on the negative kx side of the surface Kramers pair, which crosses EF for hν = 24 eV and gives rise to the bulk hole Fermi surface near H (Figure 3.17(c)). For hν ≤ 20 eV, this band shows clear back folding near EB ≈ -0.2 eV indicating that it has completely sunk below EF . Interestingly, at photon energies such as 18 eV where the bulk bands are far below EF , there remains a uniform envelope of weak spectral intensity near EF in the shape of the bulk hole pocket seen with hν = 24 eV photons, which is symmetric about Γ̄. This envelope does not change shape with hν suggesting that it is of surface origin. Due to its weak intensity relative to states at higher binding energy, these features cannot be easily seen in the energy distribution curves in Figure 3.18(a)-(c), but can be clearly observed in the MDCs shown in Figure 3.17(b) especially on the positive kx side. Centered about the M̄ point, we also observe a crescent shaped envelope of weak intensity that does not disperse with kz (Figure 3.18(d)-(f)), pointing to its surface origin. Unlike the sharp surface states near Γ̄, the peaks in the EDCs of the feature near M̄ are much broader (∆E ∼80 meV) than the spectrometer resolution (15 meV). The origin of this diffuse ARPES signal is not due to surface structural disorder because if that were the case, electrons at Γ̄ should be even more severely scattered from defects than those at M̄. In fact, the occurrence of both sharp and diffuse surface states originates from a k 3.3. Experimental results 61 (a) EB n 0 = 1 topology (Sb) K La EF Ls H T G (b) EB kx M G L M n0 = 0 topology (Au-like) K EF Ls La H T G kx M G L M Figure 3.19: Schematic of the bulk band structure (shaded areas) and surface band structure (red and blue lines) of Sb near EF for a (a) topologically non-trivial and (b) topological trivial (gold-like) case, together with their corresponding surface Fermi surfaces are shown. dependent coupling to the bulk. As will be discussed in Figure 3.20(b), the spin-split Kramers pair near Γ̄ lie completely within the gap of the projected bulk bands near EF attesting to their purely surface character. In contrast, the weak diffuse hole like band centered near kx = 0.3 Å−1 and electron like band centered near kx = 0.8 Å−1 lie completely within the projected bulk valence and conduction bands respectively. Therefore they are hybrid states of the bulk and surface states, dubbed “resonance states” [33], and their ARPES spectra exhibit the expected lifetime broadening due to an increase in elastic decay channels into the underlying bulk continuum [70]. The topological properties of Sb depend on whether the surface Kramers pair observed near Γ̄ switch partners between Γ̄ and M̄, which can be deduced from the shape 3.3. Experimental results 62 of the surface Fermi surface. In topologically trivial spin-orbit metals such as gold, a free-electron like surface state is split into two parabolic spin-polarized sub-bands that are shifted in k-space relative to each other [63], which do not switch partners between TRIM (Figure 3.19(b)). As a result, two concentric spin-polarized Fermi surfaces are created, one having an opposite sense of in-plane spin rotation from the other, that enclose Γ̄. Such a Fermi surface, like the schematic shown in Figure 3.19(b), does not support a non-zero Berry’s phase because the TRIM are enclosed an even number of times and their geometrical phases cancel. In a topologically non-trivial metal on the other hand, a partner switching behavior must occur through the spin up and spin down bands emerging from Γ̄ separately merging into the bulk valence and conduction bands respectively between Γ̄ and M̄ (Figure 3.19(a)). This way, the surface bands that formed the outer ring-shaped Fermi surface in gold now instead form petal shaped Fermi surfaces that do not enclose Γ̄, leaving behind a single Fermi surface enclosing Γ̄. Figure 3.20 shows a spin-integrated ARPES intensity spectrum of Sb(111) from Γ̄ to M̄. The previosuly discussed systematic incident photon energy dependence study of such spectra, previously unavailable with helium lamp sources [71], made it possible to identify two V-shaped surface states centered at Γ̄, a bulk state located near kx = −0.25 Å−1 and resonance states centered about kx = 0.25 Å−1 and M̄. An examination of the ARPES intensity map of the Sb(111) surface and resonance states at EF (Figure 3.20(c)) reveals that the central surface FS enclosing Γ̄ is formed by the inner V-shaped SS only. The outer V-shaped SS on the other hand forms part of a tear-drop shaped FS that does not enclose Γ̄, unlike the case in gold. This tear-drop shaped FS is formed partly by the outer V-shaped SS and partly by the hole-like resonance state. The electron-like resonance state FS enclosing M̄ does not affect the determination of ν0 because it must be doubly spin degenerate as will be 3.3. Experimental results 63 shown later. Such a FS geometry (Figure 3.20(e)) suggests that the V-shaped SS pair may undergo a partner switching behavior expected in Figure 3.19(a). This behavior is most clearly seen in a cut taken along the Γ̄-K̄ direction since the top of the bulk valence band is well below EF (Figure 3.20(d)) showing only the inner V-shaped SS crossing EF while the outer V-shaped SS bends back towards the bulk valence band near kx = 0.1 Å−1 before reaching EF . The additional support for this partner switching band dispersion behavior comes from tight binding surface calculations on Sb (Figure 3.20(b)), which closely match with experimental data below EF . Our observation of a single surface band forming a FS enclosing Γ̄ suggests that pure Sb is likely described by ν0 = 1, and that its surface may support a π Berry’s phase. Detailed ARPES intensity maps of Sb(111) along the -K̄−Γ̄−K̄ direction are shown in Figure 3.21(c), which shows that the inner V-shaped band that was observed along the -M̄−Γ̄−M̄ direction retains its V-shape along the -K̄−Γ̄−K̄ direction and continues to cross the Fermi level, which is expected since it forms the central hexagonal Fermi surface. On the other hand, the outer V-shaped band that was observed along the -M̄−Γ̄−M̄ direction no longer crosses the Fermi level along the -K̄−Γ̄−K̄ direction, instead folding back below the Fermi level around ky = 0.1 Å−1 and merging with the bulk valence band (shaded regions in Figure 3.21(c)). This confirms that it is the Σ1(2) band starting from Γ̄ that connects to the bulk valence (conduction) band, in agreement with the tight binding calculations. Here we give a detailed explanation of why the surface Fermi contours of Sb(111) that overlap with the projected bulk Fermi surfaces can be neglected when determining the ν0 class of the material. Although the Fermi surface formed by the surface resonance near M̄ encloses M̄, we will show that this Fermi surface will only contribute an even number of enclosures and thus not alter the overall evenness or oddness of TRIM enclosures. Consider some time reversal symmetric perturbation that lifts the 3.3. Experimental results (a) SS M (b) e RS 0.4 0.4 0.4 EB(eV) EB(eV) h RS { BS { G 0.0 0.0 high low Sb(111) 0.2 0.2 64 -0.2 -0.2 0.000 EF S1 -0.2 -0.2 n M = -1 -0.4 -0.4 -0.4 -0.6 -0.6 -0.5 0.0 0.5 -1 1.0 kx (Å ) (c) G -1 M kx (Å ) (d) -K ¬ G ® K (e) M 0.0 0.0 0.3 0.2 0.2 EB(eV) -1 S2 0.2 0.2 -0.4 -0.4 Deg ky (Å ) Band structure calculation e RS -0.1 -0.1 0.1 0.0 0.0 -0.1 -0.2 -0.2 -0.3 -0.5 -0.5 0.0 0.0 0.5 0.5 -1 kx (Å ) -0.2 -0.2 SS -0.3 -0.3 G h RS M 1.0 1.0 -0.2 -0.1 0.0 0.1 0.2 k y (Å-1) Sb (111) topology Figure 3.20: (a) Spin-integrated ARPES spectrum of Sb(111) along the Γ̄-M̄ direction. The surface states are denoted by SS, bulk states by BS, and the hole-like resonance states and electron-like resonance states by h RS and e− RS respectively. (b) Calculated surface state band structure of Sb(111) based on the methods in [47, 64]. The continuum bulk energy bands are represented with pink shaded regions, and the lines show the discrete bands of a 100 layer slab. The red and blue single bands, denoted Σ1 and Σ2 , are the surface states bands with spin polarization hP~ i ∝ +ŷ and hP~ i ∝ −ŷ respectively. (c) ARPES intensity map of Sb(111) at EF in the kx -ky plane. (d) ARPES spectrum of Sb(111) along the Γ̄-K̄ direction shows that the outer V-shaped SS band merges with the bulk band. (e) Schematic of the surface FS of Sb(111) showing the pockets formed by the surface states (unfilled) and the resonant states (blue and purple). 3.3. Experimental results 65 bulk conduction La band completely above EF so that there is a direct excitation gap at L. Since this perturbation preserves the energy ordering of the La and Ls states, it does not change the ν0 class. At the same time, the weakly surface bound electrons at M̄ can evolve in one of two ways. In one case, this surface band can also be pushed up in energy by the perturbation such that it remains completely inside the projected bulk conduction band (Figure 3.21(a)). In this case there is no more density of states at EF around M̄. Alternatively the surface band can remain below EF so as to form a pure surface state residing in the projected bulk gap. However by Kramers theorem, this SS must be doubly spin degenerate at M̄ and its FS must therefore enclose M̄ twice (Figure 3.21(b)). In determining ν0 for semi-metallic Sb(111), one can therefore neglect all segments of the FS that lie within the projected areas of the bulk FS (Figure 3.20(e)) because they can only contribute an even number of FS enclosures, which does not change the modulo 2 sum of TRIM enclosures. As was the case in Bi1−x Sbx , confirmation of a surface π Berry’s phase in Sb rests critically on a measurement of the relative spin orientations (up or down) of the SS bands near Γ̄. Spin detection of the photoelectrons was again measured using a single Mott polarimeter that measures two orthogonal spin components, which are along the y 0 and z 0 directions of the Mott coordinate frame and lie predominantly in and out of the sample (111) plane respectively (Figure 3.22(a)). Spin-resolved momentum distribution curve data sets of the SS bands along the −M̄-Γ̄-M̄ cut at EB = −30 meV (Figure 3.22(b)) are shown for maximal intensity. Figure 3.22(d) displays both y 0 and z 0 polarization components along this cut, showing clear evidence that the bands are spin polarized, with spins pointing largely in the (111) plane. In order to extract the 3D spin polarization vectors from a two component measurement, we carried out the two-step fitting routine [42] with the spin polarization vectors again constrained to have length one. Our fitted polarization vectors are displayed in the sample (x, y, z) 3.3. Experimental results (a) 66 (c) Scenario 1 Sb La EF Ls Sb (111) 0.0 0.0 -0.1 -0.1 M G (c) -K ¬ G ® K -0.2 -0.2 Scenario 2 La -0.3 -0.3 EF Ls G M -0.2 -0.1 0.0 0.1 0.2 -1 k y (Å ) Figure 3.21: (a) Schematic of the surface band structure of Sb(111) under a time reversal symmetric perturbation that lifts the bulk conduction (La ) band above the Fermi level (EF ). Here the surface bands near M̄ are also lifted completed above EF . (b) Alternatively the surface band near M̄ can remain below EF in which case it must be doubly spin degenerate at M̄. (c) ARPES intensity plot of the surface states along the -K̄−Γ̄−K̄ direction. The shaded green regions denote the theoretical projection of the bulk valence bands, calculated using the full potential linearized augmented plane wave method using the local density approximation including the spin-orbit interaction (method described in [65]). Along this direction, it is clear that the outer V-shaped surface band that was observed along the -M̄−Γ̄−M̄ now merges with the bulk valence band. 3.3. Experimental results 67 (a) 40 kV e beam accelerating optics q sample (b) E B (eV) 0.0 l2 z’ z y’ x’ x y ¬ -M Au foil G l1 r2 spin || y spin ||- y S2 -0.1 S1 l1 -0.2 -0.2 -0.1 0.0 10 0.1 G r1 0.2 -1 k x (Å ) 4 Momentum distribution of spin (d) 0.4 0.2 0.0 -0.2 Py ’ Pz ’ -0.4 0 -0.1 0.0 0.2 0.1 0.3 k x (Å-1) (e) M® r1 r2 r1 l1 l2 -0.2 Intensity (arb. units) e 20 EB = -30 meV Spin polarization Mott spin detector hn Intensity (arb. units) (c) -0.2 0.0 0.1 (f) l2 r1 l1 r2 1 1 2 Py 0 0 Pz -1 -1 -1 0 -0.2 -0.1 0.0 0.1 0.2 0.2 k x (Å-1) EB = -30 meV Iy Iy -0.1 0 Px 1 0 1 Pin plane -1 k x (Å ) Figure 3.22: (a) Experimental geometry of the spin-resolved ARPES study. (b) Spin-integrated ARPES spectrum of Sb(111) along the −M̄-Γ̄-M̄ direction. The momentum splitting between the band minima is indicated by the black bar and is approximately 0.03 Å−1 . A schematic of the spin chirality of the central FS based on the spin-resolved ARPES results is shown on the right. (c) Momentum distribution curve of the spin averaged spectrum at EB = −30 meV (shown in (b) by white line), together with the Lorentzian peaks of the fit. (d) Measured spin polarization curves (symbols) for the detector y 0 and z 0 components together with the fitted lines using the two-step fitting routine [42]. (e) Spin-resolved spectra for the sample y component based on the fitted spin polarization curves shown in (d). Up (down) triangles represent a spin direction along the +(-)ŷ direction. (f) The in-plane and out-of-plane spin polarization components in the sample coordinate frame obtained from the spin polarization fit. Overall spin-resolved data and the fact that the surface band that forms the central electron pocket has hP~ i ∝ −ŷ along the +kx direction, as in (e), suggest a left-handed chirality. 3.3. Experimental results 68 coordinate frame (Figure 3.22(f)), from which we derive the spin-resolved momentum distribution curves for the spin components parallel (Iy↑ ) and anti-parallel (Iy↓ ) to the y direction as shown in Figure 3.22(e). There is a clear difference in Iy↑ and Iy↓ at each of the four momentum distribution curve peaks indicating that the surface state bands are spin polarized. Each of the pairs l2/l1 and r1/r2 have opposite spin, consistent with the behavior of a spin split Kramers pair, and the spin polarization of these bands are reversed on either side of Γ̄ in accordance with the system being time reversal symmetric. This measured spin texture of the Sb(111) surface states together with the connectivity of the surface bands (Figure 3.20), uniquely determines its belonging to the ν0 = 1 class. Therefore pure Sb can be regarded as the parent metal of the Bi1−x Sbx topological insulator class. In other words, the topological order originates from the Sb wavefunctions as predicted by Fu and Kane [22]. In addition to a Z2 topological invariant ν0 , Teo, Fu and Kane predicted in 2008 that spin polarized ARPES measurements can uncover a new type of topological quantum number nM that provides information about the spin chirality of the surface Fermi surface [64]. Electronic states in the mirror plane (ky = 0) (Figure 3.23(a)) are eigenstates of the mirror operator M (ŷ) with eigenvalues ±i. M (ŷ) is closely related to, but not exactly the same as the spin operator Sy . It may be written as M (ŷ) = P C2 (ŷ): the product of the parity operator P : (x, y, z) → (−x, −y, −z) and a twofold rotation operator C2 (ŷ): (x, y, z) → (−x, y, −z). For a free spin, P does not affect the pseudovector spin, and C2 (ŷ) simply rotates the spin. Thus, M (ŷ) = exp[−iπSy /h̄]. For spin eigenstates Sy = ±h̄/2, this gives M (ŷ) = ∓i. In a crystal with spin-orbit interaction on the other hand, Sy is no longer a good quantum number, but M (ŷ) still is. The energy bands near the Fermi energy in Bi1−x Sbx are derived from states with even orbital mirror symmetry and satisfy M (ŷ) ∝ −i sgn(hSy i). Unlike the bulk 3.3. Experimental results 69 (a) (d) kz (111) G M Insulating Bi Sb bulk conduction band E K ky kx spin up L spin down EF L L Mirror plane bulk valence band M (b) (c) nM = 1 G bulk conduction band E E E M Pure Sb (e) n M = -1 G kx G spin up S1 spin down S2 EF EF EF S2 S1 bulk valence band G kx G kx M G kx M Figure 3.23: Implications of k-space mirror symmetry on the surface spin states. (a) 3D bulk Brillouin zone and the mirror plane in reciprocal space. (b) Schematic spin polarized surface state band structure for a mirror Chern number (nM ) of +1 and (c) -1. Spin up and down mean parallel and anti-parallel to ŷ respectively. The upper (lower) shaded gray region corresponds to the projected bulk conduction (valence) band. The hexagons are schematic spin polarized surface Fermi surfaces for different nM , with yellow lines denoting the mirror planes. (d) Schematic representation of surface state band structure of insulating Bi1−x Sbx and (e) semi metallic Sb both showing a nM = −1 topology. Yellow circles indicate where the spin down band (bold) connects the bulk valence and conduction bands. 3.3. Experimental results 70 states which are doubly spin degenerate, the surface state spin degeneracy is lifted due to the loss of crystal inversion symmetry at the surface, giving rise to the typical Dirac like dispersion relations near time reversal invariant momenta (Figure 3.23(b) and (c)). For surface states in the mirror plane ky = 0 with M (ŷ) = ±i, the spin split dispersion near kx = 0 has the form E = ±h̄vkx . Assuming no other band crossings occur, the sign of the velocity v is determined by the topological mirror Chern number (nM ) describing the bulk band structure. When nM = 1, the situation in Figure 3.23(b) is realized where it is the spin up (hSy i k ŷ) band that connects the bulk valence to conduction band going in the positive kx direction (i.e. the spin up band has a velocity in the positive x direction). For nM = −1 the opposite holds true (Figure 3.23(c)). Because the central electron-like FS enclosing Γ̄ intersects six mirror invariant points (Figure 3.23(b) and (c)), the sign of nM distinguishes two distinct types of handedness for this spin polarized FS. For both Bi1−x Sbx and Sb, a single surface band with a positive velocity at the Dirac point, which switches partners at M̄, connects the bulk valence band to the bulk conduction band, so |nM | = 1. From our spin-resolved ARPES data on both insulating Bi1−x Sbx and pure Sb, this surface band has hP~ i ∝ −ŷ along the kx direction, suggesting a left-handed rotation sense for the spins around this central FS thus nM = −1. Therefore in both Bi1−x Sbx and Sb, the bulk electron wavefunctions exhibit the anomalous value nM = −1 predicted in [64], which is not realizable in free electron systems with full rotational symmetry. There is an intimate physical connection between a 2D quantum spin Hall insulator and the 2D k-space mirror plane of a 3D strong topological insulator. In the former case, the occupied energy bands for each spin eigenvalue will be associated with an ordinary Chern integer n↑,↓ , from which a non-zero spin-Chern number can be defined ns = (n↑ − n↓ )/2. In the latter case, it is the mirror eigenvalue of the occupied energy 3.3. Experimental results (a) l = 1.28eV l = 1.1eV (b) 71 (d) (e) (f) (g) (c) Figure 3.24: (a) The phase diagram of the Bi1−x Sbx system as a function of a dimerization parameter ∆d and a spin-orbit coupling parameter λ. (b) Surface Fermi surface near M̄ for and λ = 1.1 eV (c) λ = 1.28 eV. (d),(e) Surface state band dispersion along the Γ̄-M̄ direction for λ = 1.1 eV and (f),(g) for λ = 1.28 eV. All panels are taken from [72]. bands that have associated with them Chern integers n+i,−i , from which a non-zero mirror Chern number can be defined nM = (n+i − n−i )/2. 3.3.3 Evolution of surface state spectrum from Bi to Sb Having established that the topological invariant ν0 of both insulating Bi1−x Sbx and metallic Sb is equal to 1, we investigate the topological properties of pure bismuth, which is predicted to be characterized by ν0 = 0 (Figure 3.4). The topologically trivial (ν0 = 0) properties of bismuth were deduced from the relative energy ordering of its La and Ls bands, which were calculated using tight-binding methods [47, 64], first principles methods [65, 72] as well as more direct numerical methods based on evaluating Chern invariants [73]. 3.3. Experimental results 0.00.0 72 x=0 0.0 -0.1-0.1 E B (eV) -0.2-0.2 0.0 0.0 -0.1 -0.2 0.0 x = 0.02 0.5 deg 1.0 0.0 -0.1 -0.1 0.5 1.0 0.0 0.5 1.0 0.5 0.5 1.0 1.0 -0.2 0.0 x = 0.04 0.5 deg 1.0 0.0 -0.1 -0.1 -0.2 -0.2 0.0 x = 0.075 deg -0.1 -0.2 -0.2 0.00.0 x = 0.09 x = 0.065 deg -0.1 -0.2 0.0 0.0 0.5 0.5 deg 1.0 1.0 G M 0.0 0.0 -1 G M deg k G M (Å ) Figure 3.25: Surface state band dispersion along the Γ̄-M̄ direction for x=0, x=0.02, x=0.04, x=0.065, x=0.075 and x=0.09. Each panel includes energy distribution curves (right) and the corresponding second derivative image plot (left). The energy distribution curves from kx =0.25Å−1 to kx =1.25Å−1 are scaled up and are shown separately from the energy distribution curves ranging from kx =-0.15Å−1 to kx =0.25Å−1 in order to enhance the weak features near M̄. In the first principles study of Zhang et al. [72], the phase diagram of the Bi1−x Sbx system is parameterized by a dimerization parameter ∆d of two Bi layers and a spinorbit coupling parameter λ. It is seen in Figure 3.24(a) that at a fixed ∆d, the system can cross-over from the ν0 = 1 to ν0 = 0 regime by increasing λ. In these two regimes, the surface states near M̄ show a subtle but qualitative difference. In Sb, the dumbbell shaped Fermi surface around M̄ does not enclose M̄ (Figure 3.24(b)) whereas in Bi it does (Figure 3.24(c)). This is because in Sb, the surface bands between Γ̄ and M̄ switch partners whereas in Bi they do not (Figure 3.24(d) and (e)). In Bi, the spin-split band pair emerging from the valence band near Γ̄ reconnects again to the valence band near M̄. This is consistent with the scenario where a band inversion transition at L takes place near x ∼ 0.04. Figure 3.25 shows the evolution of the surface band dispersion between Γ̄ and M̄ measured by ARPES. Like the dispersions shown in Figures 3.24(e) and (g), as x 3.3. Experimental results 73 x = 0.03 0.0 M ¬ G ® M (b) K ¬ G ® K (a) G ¬ M ® G (c) 0.0 0.0 K ¬ M ® K (d) 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.1 -0.1 -0.1 EB(eV) -0.2 -0.2 -0.2 x = 0.09 0.0 Low (e) High (f) 0.0 0.0 0.6 0.8 1.0 (g) -0.1 -0.1 -0.2 -0.2 -0.1 0.0 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 -0.1 0.0 0.1 -1 k (Å ) -0.1 0.0 0.1 k x (Å-1) (h) 0.0 0.0 -0.2 0.6 0.8 1.0 k x (Å-1) -0.1 0.0 0.1 -1 k (Å ) Figure 3.26: High resolution ARPES intensity maps of x=0.03 and x=0.09 along the (a),(e) K̄-Γ̄-K̄, (b),(f) M̄-Γ̄-M̄, (c),(g) Γ̄-M̄-Γ̄ and (d),(h) Γ̄-K̄-Γ̄ directions. The cut directions in the surface Brillouin zone are shown in the cartoons above. is increased from zero, the band close to the Fermi level around M̄ goes from being relatively flat to acquiring a w-shaped dispersion. At the same time, the Kramers degeneracy near Γ̄ sinks lower in binding energy consistent with a lowering of the hole-like band at T . However, exactly how the surface bands connect to the bulk valence and conduction bands near M̄ could not be directly measured because the surface bands become very weak in intensity near M̄ and any intensity coming from the bulk conduction band overlaps the intensity of the surface state bands near M̄. High resolution ARPES data near M̄ show that the appearance of the w-shaped feature at higher x comes from a new band that emerges near EF , which was shown in Figure 3.11 to be its Kramers pair. The presence of such a double band is confirmed by the two-peak feature in the EDC (Figure 3.27(c)). On the other hand, a one- 3.3. Experimental results 74 peak feature is observed for lower x. By performing a series of such fits for pure Bi and Bi0.9 Sb0.1 , we extract a dispersion relation between Γ̄ and M̄ that is shown in Figure 3.27(a) and (b). By superposing the calculated positions of the projected bulk bands, it is seen that a partner switching behavior is exhibited by both Bi and Bi0.9 Sb0.1 , which would suggest that Bi0.9 Sb0.1 is also topologically non-trivial. However, further studies of possible surface state band bending effects [1], which may alter the energy positions of the surface states relative to the bulk states, are required to clarify this issue. As of the writing of this thesis, there is no direct evidence for a topological phase transition at x ∼ 0.04. 3.3. Experimental results 75 (a) (c) k x = 0.6 Å x = 0.0 x = 0.0 Intensity (arb. units) 0.0 EB(eV) -0.1 -0.2 (b) x = 0.1 0.0 x = 0.1 -0.1 -0.2 -0.2 0.0 0.2 0.4 0.6 k x (Å-1) 0.8 1.0 1.2 -0.1 0.0 EB(eV) Figure 3.27: The surface state band dispersion along Γ̄-M̄ obtained by fitting to the high resolution ARPES data is shown for (a) x=0.03 and (b) x=0.09. The shaded regions show the projected bulk band structure onto the (111) plane based on a rigid shift of the tight-binding bands according to the model of Golin [74]. (c) Energy distributions curves taken at a constant electron emission angle such that electrons at the Fermi level have kx =0.6Å−1 and ky =0Å−1 are shown for x=0 and x=0.1. The contrast between the single band and double band behavior is clearly seen. Chapter 4 Conclusions The work presented in this thesis constitutes the first experimental evidence of a strong topological insulator (STI) phase of matter in nature that is realized in bulk insulating Bi1−x Sbx alloys [75, 76, 77]. Recently, our work has been reproduced by Nishide et al. [78]. More generally, this work demonstrates that a combination of spinand angle-resolved photoemission spectroscopy is a new tool with which to measure Z2 topological invariants and identify new exciting topological phases of matter that do not require a large external magnetic field. Large parts of this work can be found in two publications [79, 80]. Following our discovery, there have been great advances in terms of new materials candidates for STIs, as well as new theoretical proposals for novel macroscopic phenomena that are associated with the STI phase. On the materials front, an ARPES study performed by our group in 2008 [81] showed that the surface states of Bi2 Se3 are topologically non-trivial and are much simpler than those in Bi1−x Sbx because there is only one as opposed to five Fermi level crossings. This coincided with theoretical work by Zhang et al. [82] predicting that Bi2 Se3 , Bi2 Te3 and Sb2 Te3 are all STIs that are realized without the need for external pressure. However, as-grown Bi2 Se3 is slightly n-doped in the bulk, which means that a true single surface band STI is not 76 77 realized in this system. Rather, two recent spin-ARPES [83] and ARPES [84] works have established that Bi2 Te3 may be a true single surface band STI with a large gap insulating bulk. 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