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Unexpected zero bias conductance peak on the topological semimetal Sb(111) with a single broken layer Yam, Yau Chuen 2017

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Unexpected Zero Bias ConductancePeak on the Topological SemimetalSb(111) with a Single Broken Layerby a scanning tunneling microscopy and densityfunctional theory studybyYau Chuen YamB.Sc., The Hong Kong University of Science and Technology, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2017c© Yau Chuen Yam 2017AbstractThe signature of the long-sought Majorana fermion from a heterostructureof a superconductor and a topological material is the zero bias conductancepeak (ZBCP). Topological semimetal Antimony is a good material in mak-ing such heterostructure. Since it is of a bilayer crystal structure, it isexpected to be cleaved between bilayers. However, we found that on itscleaved surface there can be steps with step heights corresponding to the in-trabilayer distance, indicating that there is a broken layer underneath. ThedI/dV spectrum observed using scanning tunneling microscope on these ab-normal steps are quite different from the usual Sb spectrum and there is apronounced ZBCP. Using quasiparticle interference imaging, Landau levelspectroscopy and density functional theory (DFT), we found that the ZBCPis originated from the changed band structure through van Hove singularity.This shows that when we try to probe the signature of Majorana fermion inthe heterostructure, we need to make sure the ZBCP is not from this trivialorigin due to the imperfectness of the topological material.iiPrefaceThe results of this thesis will be going to be submitted to arXiv and possiblyPhysical Review Letters or Physical Review B in Dec 2016. The scanningtunneling microscope (STM) experiment was carried out in Hoffman lab inHarvard University by the author, Y.-C. Yam, and Y. He and the data wasanalyzed by the author and P. Chen. The sample used in the experimentwas provided by D. Gardner and Y. Lee in MIT. The DFT calculationswere performed and interpretated by the author and S. Fang in HarvardUniversity. The Vienna ab initio simulation package (VASP) was used forthe DFT simulation.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Scanning Tunneling Microscopy . . . . . . . . . . . . . . . . . 32.1 Tunneling Current . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Measurement Modes . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Topography . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Density of states . . . . . . . . . . . . . . . . . . . . . 62.2.3 Linecut and DOS map . . . . . . . . . . . . . . . . . 73 Overview of Density Functional Theory . . . . . . . . . . . . 83.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . 93.2 Two Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . 93.2.1 The First Hohenberg-Kohn theorem . . . . . . . . . . 103.2.2 The Second Hohenberg-Kohn theorem . . . . . . . . . 113.3 The Kohn-Sham method . . . . . . . . . . . . . . . . . . . . 113.3.1 A practical scheme to find the ground state density . 113.3.2 Approximation of the exchange-correlation functional 143.4 Pseudopotential to simplify actual computation . . . . . . . 154 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 184.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Sb Structure in literature versus the topography in our ex-periment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18ivTable of Contents4.3 dI/dV Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3.1 Zero bias peak on abnormal surface . . . . . . . . . . 204.3.2 Wiggling in the spectrum . . . . . . . . . . . . . . . . 204.3.3 No superconductivity involved . . . . . . . . . . . . . 234.4 Momentum resolved spectroscopic information . . . . . . . . 244.4.1 Quasiparticle interference (QPI) . . . . . . . . . . . . 244.4.2 Landau levels . . . . . . . . . . . . . . . . . . . . . . 265 DFT Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Band structure of normal and abnormal terrace . . . . . . . 285.2.1 Normal surface . . . . . . . . . . . . . . . . . . . . . . 285.2.2 Abnormal surface . . . . . . . . . . . . . . . . . . . . 285.2.3 Further investigation for magnetic breakdown . . . . 306 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33AppendicesA Band Structures of different thickness and depth . . . . . 37B Justification of Validity of DFT calculation . . . . . . . . . 38vList of Figures2.1 Schematic representation of tip-sample tunneling . . . . . . . 43.1 Schematic representation of nucleus potential and pseudopo-tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1 Terraces Structures of Antimony. . . . . . . . . . . . . . . . . 194.2 dI/dV Spectra observed on different terraces. . . . . . . . . . 214.3 Wigglings in dI/dV spectra on normal and abnormal surfacesand the energy difference between peaks . . . . . . . . . . . . 224.4 STM Topography of the normal and abnormal steps and itsFourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Dispersing modes from Quasiparticle Interference and Lan-dau Quantization of surface states. . . . . . . . . . . . . . . . 255.1 Band Structure of normal and abnormal terraces. . . . . . . . 295.2 Extra emergent band on the abnormal terrace cutting acrossthe Dirac cone . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.1 Band structures from DFT with different thickness and depthof the single broken layer . . . . . . . . . . . . . . . . . . . . 37B.1 Comparison of Band Structure infered from Landau levels indI/dV spectrum and that from DFT . . . . . . . . . . . . . . 38viAcknowledgementsI would like to express my gratitude to my research advisor Professor Jen-nifer Hoffman for giving me a chance to conduct research in her group andbringing me to work in her lab in Harvard University where I have attendedsome great classes, met numerous great minds and started to develope myscientific thoughts. I also would like to thank Dr. Yang He and Dr. Zhi-HuaiZhu for their tough and almost sufficient guidance in mastering the controlof STM, which make me tougher in working with coworkers.I am particularly grateful to Shiang Fang in Kaxiras group who has notonly assisted me so much with the use of the VASP software, but also sharedcountless of his computational resources with me, which make many of thecalculation involving large slab possible, and demonstrated his research skillsto me.Finally, I have to thank all the others who has given me assitance duringmy Master program, including but not limited to Dr. Pengcheng Chen, whotaught me his tip treament skills in STM, Dr. Mohammad Hamidian, whohelped me to start using Matlab for data analysis and explained some STMcomponents to me, Dr. Jhih-Shih You in Demler Group and Adrian Poin Vishwanath Group, with whom I had some very useful and interestingdiscussions on topological insulators.viiChapter 1IntroductionRealizing Majorana fermion has become a very hot research topic recentyears because of its potential in making qubits for quantum computing [1–3]. It has been proposed that Majorana fermions can be created at theinterface between an s-wave superconductor and a topological insulator [4].The signature of the Majorana fermion [5, 6] would be a zero bias anomalyin the tunneling conductance at the interface. So zero bias peaks (ZBP)observed in the spectrum of a number of experiments [7–10] have drawnmuch of the attention. However, as pointed out by Liu et al. [11] and Choet al. [12], there are many other possible origins of the zero bias anomalies,like the effects of disorder and Kondo effect. We must be very careful ininterpreting and verifying the origins of the ZBP. In this thesis, we aregoing to show that there is an even more unexpected trivial origin of theZBP, which is just due to a rapid change of sample in the experiment.Topological semimetal Sb(111) is a very good candidate in making theinterface with the superconductor because it has a clean, stable surface andwell-defined surface states [13]. Also, due to its simple bilayer structure,it has a much lower critical thickness than other topological materials todecouple surface states on its opposite sides [14]. Moreover, it is less fragile[15] than tetradymite topological insulators, so a superconducting film couldbe grown on top of it without destroying it. In addition, its semiconductingbulk screens chemical potential variations [16, 17], leading to a more homo-geneous surface state, more suited for the superconductor proximity effectstudy.However, in our scanning tunneling microscope (STM) experiment, wediscovered that a robust ZBP can appear on the cleaved surface of Sb with-out proximity to superconductivity. Using quasiparticle interference (QPI)imaging, Landau level (LL) spectroscopy in STM and band structure calcu-lation by density functional theory (DFT), we demonstrate that the peakoriginates from a single layer of Sb, isolated within its ordinary bilayer struc-ture. The isolated layer introduces additional states at negative energypushing a saddle point up to the Fermi Energy and producing a van Hovesingularity at the Fermi level. We thus introduce a cautionary note in the1Chapter 1. Introductionsearch for Majorana fermions as ZBP at Sb-superconductor interfaces, andmore generally exemplify how easily a trivial state can robustly mimic thelong-sought Majorana.This thesis is organized as follows. Chapter 2 is devoted to provide areview of the principle of STM. Chapter 3 presents an overview of DFT.Our experimental findings, including the ZBP observed on the unusuallycleaved Sb surface as well as the QPI and LL measurements, are providedin Chapter 4. The DFT band structure calculations and the explanation tothe origin of the ZBP are addressed in Chapter 5. Finally, a summary andconclusion is included in Chapter 6.2Chapter 2Scanning TunnelingMicroscopyThe scanning tunneling microscopy (STM) comprises a metallic tip sittingat the end of a piezo tube scanner which can control the motion of thetip in the x, y, z direction with sub-Angstrom precision with respect to aflat conducting sample surface. The tip is pointing towards the sample atcertain height z. When the tip and the sample are in equilibrium, theirfermi levels are at the same energy. But when a negative bias voltage -Vis applied to the sample, as shown in Fig. 2.1, all the electronics states ofthe sample would be shifted upwards by (−e)(−V ) = +eV with respect tothe tip. The electron with higher energy in the sample can tunnel to theempty states of the tip aross the vaccum space, although it does not haveenough energy to overcome the vacuum barrier. Classically this is forbidden.But quantum tunneling allows this kind of electron transition, resulting ina current flowing in the tip. Knowing the applied voltage V and the (x,y)location of the tip from the piezo, we can record the current as function ofV and location on the sample, which can in turn reflect different propertiesof the sample.2.1 Tunneling CurrentAs addressed in the beginning, if the sample is biased by -V, the energy ofthe sample states would be raised by eV , as illustrated in Fig. 2.1, and atunneling current would be created when the electron flow out of the filledstates in the sample and into the empty states in the tip. This tunnelingcurrent can be calculated by the Fermi Golden Rule.Consider the current from the sample to the tip for states at energy with respect to the shifted fermi energy of the sample (i.e. regarding shifted32.1. Tunneling CurrenttipstatessamplestatesEnergy (ε)sampleDOStipDOSseperation(z)e-V0-eVVacuumbarrierϕsϕtFigure 2.1: Schematic representation of tip-sample tunneling. Onthe sample side, there is a general density of states (DOS) with the filledstates in blue. On the tip side, a typical flat density of state is shown withthe occupied states in blue. Electron is tunneling from the sample to thetip as indicated by the orange arrow. φs and φt are the work fucntions forthe sample and the tip respectively.fermi level of the sample as 0),Isample→tip() = (−2e)Γsample→tip()= (−2e)[2pih¯|T |2(#filled state in sample)(#empty state in tip)]=−4epih¯|T |2(ns()f())(nt(+ eV )(1− f(+ eV )) (2.1)where in the first line, Γsample→tip is the transition probability per unit timefrom the sample to the tip and the ‘2’ take cares of the spin of the electron.In the second line, Fermi Golden rule has been used to express Γsample→tip,with |T |2 being the transition matrix element between the initial filled stateand the final empty states. In the third line, ns() and nt() are the densityof states of the sample and the tip of energy  and f is the occupationnumber with f() = 11+e/kBT.To get a form of |T |2, we can regard the vacuum barrier as a a square42.1. Tunneling Currentbarrier and use WKB approximation to find that:|T |2 = e−2γ (2.2)withγ =∫ z0√2mφh¯2dx =zh¯√2mφ (2.3)where m is the electron mass, φ = 12(φs + φt) is the average work functionof the tip and the sample and z is the seperation between the tip and thesample.Please be noted that although the transition from sample to the tipwould be dominant, electron could also transit from the tip to the sampleat the same time. By symmetry,Itip→sample() =−4epih¯|T |2(ns()(1− f()))(nt(+ eV )(f(+ eV )) (2.4)and |T |2 of Itip→sample() would be the same as that in Itip→sample(). There-fore, the current for states of energy  isI() = Isample→tip()− Itip→sample()=−4epih¯|T |2ns()nt(+ eV )[f()(1− f(+ eV ))− (1− f())f(+ eV )](2.5)To find the total current, we have to sum up (or integrate) all the I() from=−∞ to +∞. But since our measurement was taken at liquid Heliumtemperature, which is around 4.2K, kBT ≈ 0.36 meV, which is sufficientlysmall compared with our interested energy range of a few hundreds meV.So f() ≈ 1 − θ() where θ is the Heavyside step function. Then f()(1 −f(+ eV ))− (1− f())f(+ eV ) ≈ θ(+ eV )− θ() = 1 if −eV <  < 0 and0 otherwise. Thus the only concerned energy range is −eV <  < 0.Therefore, after inserting |T |2 from (2.2) and (2.3) into (2.5) and inte-grating over energy, the total tunneling current isI =∫ ∞−∞I() d ≈ −4epih¯e−2zh¯√2mφ∫ 0−eVns()nt(+ eV ) d (2.6)Furthermore, the interested range of V in the Sb experiment was from-300 meV to 300 meV. The tip that we used is made of PtIr which has a flatdensity of states in this range of energy. So nt(+ eV ) can be constantlywritten as nt(0). Hence, finally, the tunneling current becomesI ≈ −4epih¯e−2zh¯√2mφnt(0)∫ 0−eVns() d (2.7)52.2. Measurement Modes2.2 Measurement Modes2.2.1 TopographyA ‘topographic image’ showing the structural surface corrugation is the mostcommon measruement that STM can do.In the topography mode, apart from the constant bias voltage V , thetunneling current will also be set to be fixed. As (2.7) demonstarted, thetunneling current I exponentially depend on the seperation z between thetip and the sample. When there is corrugation on the sample surface, the z-peizo has to extend or retract to maintain the constant seperation z in orderto keep the constant current. By monitoring the extension of the z-peizo atdifferent (x,y) location of the sample, the z position of the tip can be inferedand from which we can interpret the relative height of the sample surfaceand map out its ‘surface corrugation’.Clearly, the ‘surface corrugation map’ here is in fact the contour mapof constant current rather than a direct measurement of local height of thesurface. In intrepretating the constant current map as a surface corrugationmap, we have to make an important assumption that∫ 0−eV ns() d is more orless constant at different location ~r = (x, y). This may not be a too severesimplication generally as long as the entire sample is of the same crystalwithout abrupt changes in any region and the density of state ns() doesnot vary much at different location. But this assumption may be seriouslyweakened if there is emergent impurities or geometric feature on the surfacethat can potentially change ns() severely locally. So we have to pay specialattention if we are looking at the topography in these cases.2.2.2 Density of states(2.7) let us know that if the tip-sample sepeartion z is fixed, the tunnelingcurrent I is proportional to∫ 0−eV ns() d, the integrated density of states.To get the density of state, we just need to take its derivative with respectto V . From (2.7),I = I0∫ 0−eVns() d for some constant I0= I0∫ +∞−∞ns()(θ(+ eV )− θ()) d62.2. Measurement ModesTaking derivative with respect to V ,dIdV= I0∫ +∞−∞ns()(dθ(+ eV )dV− 0) d= I0∫ +∞−∞ns()(eδ(+ eV )) d= eI0ns(−eV ) ∝ ns(−eV ) (2.8)So theoretically, if the bias voltage is −V , taking derivative of the current Iwith respect to V can tell us the density of states ns(−eV ) at −eV . Or inother words, by scanning a bias voltage of V one can find density of statens(eV ) of energy eV .However, if we measure dIdV by taking a derivative numerically, very highenergy resolution would be required to prevent errors. So, instead, a stan-dard lockin technique is employed. For the steady bias voltage V applied tothe sample, we add an oscillaton ∆V = Vaccos(ft) to it and a change of ∆Iwill appear in the tunneling current. (In our experiment, f = 1.115kHz.)By measuring the current response ∆I at frequency f and dividing this∆I by ∆Vrms =Vac√2, the differential conductance ∆I∆V is obtained and it isproportional to the density of states of the sample as shown by (2.8).In experiment, ns() is in general depends on the spatial location ~r onthe sample, i.e. ns = ns(, ~r). When we move to a new position, we wouldfix the seperation z between the tip and the sample by a constant set-pointvoltage Vset and a set-up current Iset (‘the feedback loop’) as explained inthe topography mode. Then we will turn off the feedback to allow current Ito change with the bias voltage while keeping the other variable, seperations between tip and sample, constant. After that, we will sweep through arange of bias voltage V with the addition of the voltage oscillation ∆V andrecord the response of ∆I at each voltage V. In this way, a spectrum of ∆V∆Iat different bias V can be obtained. Lastly, the feedback would be turnedon again before moving to the new position.2.2.3 Linecut and DOS mapPreviously, the measurement of the differential conductance spectrum atone spatial point was explained. With the piezo tube scanner, the tip canbe moved in the x, y direction with sub-Angstrom precision. So we canacquire the spectra of the sample on a number of points along a line or in aregion on the sample surface to form a ’linecut’ or ’DOS map’ to gain spatialresolution for the density of states.7Chapter 3Overview of DensityFunctional TheoryIn characterizing the properties of Sb with different terminations, apartfrom the experimental efforts, we also did some theoretical simulations usingdensity functional theory (DFT), which is an ab initio method widely appliedto study different properties of tremendous amount of molecules and solid.So this chapter will be devoted to discuss some essential ideas of DFT.The target of an ab initio method is to solve the many-body problem fora solid, which has an enormous number of nuclei and electrons. In particular,we want to solve the Schro¨dinger equation HˆΨ = EΨ whereHˆ = − h¯22∑i∇2~RiM− h¯22∑i∇2~rim+18piε0∑i 6=je2∣∣∣⇀r i − ⇀r j∣∣∣+18piε0∑i 6=je2Z2∣∣∣∣⇀Ri − ⇀Rj∣∣∣∣ −14piε0∑i,je2Z∣∣∣∣⇀Ri − ⇀r j∣∣∣∣ (3.1)in which ~Ri and ~ri are the position of the nucleus of mass M and electronof mass m respectively and Z is the number of electron contributed by eachnucleus. The first and second term of (3.1) give the kinetic energy of thenuclei and the electron. The thrid and fourth term are the electron-electronand nucleus-nucleus repulsions while the last term is the Coulomb interactionbetween the electrons and the nuclei.If there are N nuclei (and thus NZ electrons), there will be 3(N+ZN)degrees of freedom. It is well known that there would be no analyticalsolution for Ψ if there is more than one particle, not to mention that N istypically in the order of 1023 in a condensed matter system. Approximationsmust be taken in order to make it possible to solve for such system.83.1. Born-Oppenheimer Approximation3.1 Born-Oppenheimer ApproximationSince the masses of the nuclei are much larger than that of the electrons,they would move much slower than the electrons. We can well assume thosenuclei are not moving at all and consider only the motions of electrons. Inother words, we can drop the first term of (3.1) which concerns the kineticenergy of the nuclei. Also, since the nucleus-nucleus repulsion term is justa constant in the system, it can be ignored as well. Now we are left withHˆ = − h¯22∑i∇2~rim+18piε0∑i 6=je2∣∣∣⇀r i − ⇀r j∣∣∣ −14piε0∑i,je2Z∣∣∣∣⇀Ri − ⇀r j∣∣∣∣ (3.2)which is in the form of Hˆ = Tˆ + Vˆee + Vˆext and the scenario of problembecomes a number of negatively charged electrons, with repulsion amongthemselves resulting in the repulsive potential Vˆee, moving in the static pos-tive external potential Vˆext created by the nuclei. In this way, we can greatlyreduce 3N degrees of freedom in the problem. This is known as the Born-Oppenheimer Approximation.3.2 Two Hohenberg-Kohn TheoremsAlthough the dimensions of the problems have been largely reduced after ap-plying the Born-Oppenherimer Approximation, the many-body Schro¨dingerequation remains to be unsolvable because its dimension is still as high as3NZ where N is of orders 1023. There are two ways to further simplify theproblem. One is to look for a simpler effective Hamiltonians which can cap-ture the essential physics without directly working with the full Hamiltonian(3.2). Examples of this approach is the Hatree-Fock theory. Another wayto tackle this problem is to reformulate the problem so that the full Hamil-tonain (3.2) can be retained. This is the approach that DFT has taken.Suppose the ground state of the system is Ψ0(~r1, . . . , ~rNZ) with 3NZ de-gree of freedom. The spirit of DFT is that instead of dealing directly withthis Ψ0(~r1, . . . , ~rNZ), we could look at the ground state electron densityn0(~r) with only 3 degree of freedom (corresponding to the spatial directionsx, y and z). Although intuitively the density seems to contain less informa-tion than the wave function does, it is not the case as justified by the twoHohanberg-Kohn Theorems [18].93.2. Two Hohenberg-Kohn Theorems3.2.1 The First Hohenberg-Kohn theoremThe first Hohenberg-Kohn Theorem stated that the ground state electrondensity n0 can uniquely determine the external potential Vext(~r) and thus theHamiltonian Hˆ. Since the Schro¨dinger equation with an unique Hamiltonianwould yield an unique ground state wave function, the density would con-tain as much information as the wave function does. The consequence of thiswould be that we can write down any observable Oˆ of the system as a func-tional of the ground state electron density n0(~r), i.e. 〈Ψ|Oˆ|Ψ〉 = O[n0(~r)].This is where the name ‘Density Funtional Theory’ from.This theorem can be proved by contradiction. Assume there are twodifferent external potentials, Vext(~r) and V′ext(~r) giving rise to two differentHamiltonain, H(~r) and H ′(~r), corresponding to the same ground state elec-tron density n0(~r). Let Ψ0 and Ψ′0 be the ground states of Hˆ and Hˆ′ andE0 and E′0 be the corresponding ground state energy, i.e. E0 = 〈Ψ0|Hˆ|Ψ0〉and E′0 = 〈Ψ′0|Hˆ ′|Ψ′0〉. By variatonal principle,E0 < 〈Ψ′0|Hˆ|Ψ′0〉 = 〈Ψ′0|(Hˆ − Vext(~r) + V ′ext(~r)) + Vext(~r)− V ′ext(~r)|Ψ′0〉= 〈Ψ′0|Hˆ ′ + Vext(~r)− V ′ext(~r)|Ψ′0〉= 〈Ψ′0|Hˆ ′|Ψ′0〉+ 〈Ψ′0|Vext(~r)− V ′ext(~r)|Ψ′0〉= E′0 + 〈Ψ′0|Vext(~r)− V ′ext(~r)|Ψ′0〉= E′0 +∫n0(~r)[Vext(~r)− V ′ext(~r)] dr (3.3)Similarly,E′0 < 〈Ψ0|Hˆ ′|Ψ0〉 = E0 +∫n0(~r)[V′ext(~r)− Vext(~r)] dr (3.4)Summing (3.3) and (3.4),E0 + E′0 < E′0 + E0 +∫n0(~r)[Vext(~r)− V ′ext(~r)] + n0(~r)[V ′ext(~r)− Vext(~r)] dr= E′0 + E0 +∫n0(~r)[Vext(~r)− V ′ext(~r) + V ′ext(~r)− Vext(~r)] dr= E′0 + E0 + 0= E0 + E′0 (3.5)The relation of E0 +E′0 < E0 +E′0 in (3.5) is certainly an absurb. Thereforethe assumption that different external potentials Vext can have the sameelectron density n0 is incorrect and we can conclude that there is a one-to-one correspondence between the ground state electron density n0 and theexternal potential Vext.103.3. The Kohn-Sham method3.2.2 The Second Hohenberg-Kohn theoremThe second Hohenberg-Kohn theorem reformulates the variational principlein terms of electron density.The conventional variational principle states in terms of wave functionthat 〈Ψt|Hˆ|Ψt〉 ≥ E0 for any trial state Ψt and a general Hamiltonian Hˆwith ground state energy E0. By the First Hohenberg-Kohn theorem, theground state density n0 can take the place of the wave function Ψ0 and fullydescribe the ground state. So even the wave function Ψ itself can be writtenin terms of the electron density n. In other words, there is a functionalΨ[n(~r)] mapping n(~r) to Ψ(~r). Similarly, the observable ground state energyE0 can be expressed as a functional of density such that E0 = E[n0]. In thisway, we can completely work with the electron density for the variationalprinciple. So instead of using the trial wavefunction Ψt, if we are looking atany trial density nt, the variational would become E[nt(~r)] ≥ E0.This theorem implies that in order to find the true ground state density,we can look for the possible trial density nt which can minimize the energyE[nt(~r)].3.3 The Kohn-Sham method3.3.1 A practical scheme to find the ground state densityAlthough the Hohenberg-Kohn theorems present a formalism of denistyfunctional theory in terms of electron density, it does not provide a practicalprocedure in finding a ground state density. The Kohn-Sham method[19]going to be discussed in this section would fill the gap and offer a recipeto find the ground state density in a self-consistent manner. Until now, theKohn-Sham method remains to be the only practical tool in implemetatingDFT and any DFT calculation would imply that the this method has beenused.The original full interacting system is very difficult to be solved becausethe electron-electron interaction couples the differential equations together.So the spirit of the Kohn-Sham method is that rather than dealing withthis full interacting system, we could design and work with a reference non-interacting system with the same ground electron density. Since this refer-ence system is non-interacting, it would be reduced to the Schro¨dinger-likesingle-particle equations which is much easier to be solved.So how do we find such non-interacting reference system? The bad newsis that we do not know the exact way to do this. All we can do is group all113.3. The Kohn-Sham methodunknown terms together into a term called exchange-correlation energy andmake sensible approximation to this term.In the original interacting system, the total energy functional E[n(~r)]of the electron density n(~r) can be written as a sum of the kinetic energyterm T [n(~r)], the electron-electron potential energy term Uee[n(~r)] and theexternal potential energy term Uext[n(~r)] =∫V (~r)n(~r) d~r, i.e.E[n(~r)] = T [n(~r)] + Uee[n(~r)] + Uext[n(~r)] (3.6)In the above equation, Uee[n(~r)] can actually be broken into 2 parts, namelyUee[n(~r)] = UH [n(~r)] + Uex[n(~r)] where UH [n(~r)] =12∫e24pi0n(~r)n(~r′)|~r−~r′| d~r d~r′is the classical part concerning the Coulomb repulsions between electrons,which is also known as the Hartree-Fock term, and Uex[n(~r)] accounts for the‘exchange energy’ arising from the many-body effect whose form is unknown.Also, T [n(~r)] can also be written as T [n(~r)] = Ts[n(~r)] + Tc[n(~r)] whereTs[n(~r)] is going to be the kinetic energy of the non-interacting referencesystem, which is the sum of single-particle kinetic energy of all electronsin the reference system, and Tc[n(~r)], called the ‘correlation energy’, is thedifference of kinetic energy between the non-interacting reference systemand true kinetic energy in the interacting system, which is also orginatedfrom the many-body effect and whose form is unknown. The two unknownfunctionals, Uex[n(~r)] and Tc[n(~r)], can be put together into the co-calledexchange-correlation energy functional Exc[n(~r)]. The above statemens canbe summarised as follows:E[n(~r)] = Ts[n(~r)] + Tc[n(~r)] + UH [n(~r)] + Uex[n(~r)] + Uext[n(~r)]= Ts[n(~r)] + UH [n(~r)] + Uext[n(~r)] + (Tc[n(~r)] + Uex[n(~r)])= Ts[n(~r)] +12∫e24pi0n(~r)n(~r′)|~r − ~r′| d~r d~r′ +∫V (~r)n(~r) d~r + Exc[n(~r)](3.7)All we have done above is just rearranging terms of the original systemand it is still not useful yet. But now all the unknown many-body partshas been group into a single term. All other terms can be easily decoupledinto a single-particle operator form by taking a functional derivative of withrespect to n(~r). So, we can write down the single-particle Hamiltonian forthe ith electron, after taking the functional derivative of (3.7),HˆKS = − h¯22m~∇i2 +∫e24pi0n(~r′)|~r − ~r′| d~r′ + V + Vxc[n(~r)] (3.8)123.3. The Kohn-Sham methodwhere the exchange-correlation potential Vxc[n(~r)] =δExc[n(~r)]δn(~r) . HˆKS de-fined in (3.8) is the reference system that we are looking for. It is theHamiltonian for a single non-interacting classical electron, subjected to twoexternal potentials V and Vxc. Assuming Vxc is known, we can easily solvethe well-studied Schro¨dinger-like single-particle equationHˆKSφi = Eiφi (3.9)for the N lowest-energy wavefunction φi, if there are N electrons in theoriginal system, and we can construct the ground state electron density byn(~r) =N∑n=1φ∗i (~r)φi(~r) (3.10)With this Kohn-Sham method, to find the ground state density, we cannot only get around the problem of dealing with orginal coupled interactingdifferential equations system, but we even do not have to even make use ofthe second Hohenberg-Kohn theorem and we just have to solve the familiarsingle-particle Schro¨dinger equation.However, HˆKS is actually depending on the electron density n(~r), whichin turn depends on the wavefunction φi that we are trying to solve for. Sowe need to solve (3.9) self-consistently with the following scheme:1. Make an initial guess of density ni(~r)2. Construct the HˆKS3. Solve for the N lowest state φi4. Calculate the corresponding density nf (~r) by n(~r) =∑Nn=1 φ∗i (~r)φi(~r)5. If nf (~r) 6= ni(~r), use nf (~r) as an input to construct HˆKS and repeat 2to 4 until the input and output density are equal (up to a threshold).In the above procedure, we have assumed that the nuclei are static. If weallow them to change position after we have found the self-consistent electrondensity in each static setting, we can look for the position of nuclei which canminimize the energy of the system. This is so called the crystal relaxation.The iterative process can be extended as follows for this purpose:1. Assume a fixed configuration for nuclei position2. Calculate the electron ground state with the above self-consistent it-eration scheme133.3. The Kohn-Sham method3. Calculate net forces on the nuclei by the Hellmann-Feynman theorem.4. Move the nuclei to the new position and repeat the above process untilthe forces on them are minimized3.3.2 Approximation of the exchange-correlation functionalThe above methods are looking good. However, please be reminded thatthey are based on the assumption that we know the form of the exchangecorrelation functional Vxc[n(~r)] in (3.8) such that we have a known HˆKSto solve for. So before we could apply the mentioned methods, we need tofirst approximate the form of Vxc[n(~r)] in justified ways. This leads to animportant remarks - researchers have been complaining that experimentalmeasurements can be quite different from what DFT calculated. In DFT weare actually calculating the ground state of the non-interacting Kohn Shamreference system, but not exactly the system of interacting electrons directly.Whether electronic structure and band structure from DFT is physical reallydepends on how good the approxmation is. There are a number of ways todo the approximation as illustrated in the following. In fact there is stillmuch ongoing efforts on obtaining better forms of Vxc[n(~r)].The most basic and widely used approximation is called the Local Den-sity Approximation (LDA). The form of Vxc[n(~r)] from LDA is:Vxc[n(~r)] =∫n(~r)xc(n(~r)) d~r (3.11)which only depends on the local electron density at the spatial points.xc(n(~r)) is the exchange correlation energy for an uniform electron gassystem which can be determined numerical methods[20]. This approxima-tion is reasonable because we can divide the whole material into many verysmall volumes which only has an uniform electron density within themselves.Because of this, we can foresee that LDA would work well for systems withspatially slowly varying density.An improvement on LDA would be making Vxc[n(~r)] depends not onlyon local electron density n(~r), but also depends on its relation with itsneighbor density, i.e. the local gradient of the electron density ∇n(~r), sothat systems with more rapidly varying electron density in space can betreated more accurately. The form of Vxc[n(~r)] would thus be:Vxc[n(~r)] =∫f(n(~r),∇n(~r)) d~r (3.12)143.4. Pseudopotential to simplify actual computationThis slightly more general method is called the Generalized Gradient Ap-proximation (GGA). Unlike LDA, which has a uniform electron gas systemto be a reference system in determining xc(n(~r)), there is no such referencesystem for GGA to uniquely determine f(n(~r),∇n(~r)). As a results, differ-ent versions of GGA exists. Most of them would contain free parametersthat need to be fixed by experimental data, making GGA is not strictly an abintio method. However, there is a few forms of GGA that do not have suchfree parameters, like Perdew-Burke-Ernzerhof (PBE)[21] functional that wehave employed in our DFT calculation on the Sb project.3.4 Pseudopotential to simplify actualcomputationNow we have every element for implementing DFT and we can carry cal-culation using the scheme suggested by the Kohn-Sham method outlinedbefore. In solving the periodic Hamiltonian system, the eigenfunction, bythe Bloch theorem, would be:Ψn~k(~r) = un~k(~r)ei~k·~r (3.13)where n is the band index, ~k is the reciprocal vector in the first Brillouinzone and un~k(~r) is a periodic function with the same periodicity as the lattice,i.e. un~k(~r) =∑~K cn,~k~Kei~K·~r with ~K being the crystal momentum. So,Ψn~k(~r) = (∑~Kcn,~k~Kei~K·~r)ei~k·~r=∑~Kcn,~k~Kei(~k+ ~K)·~r (3.14)Now Ψn~k(~r) is expressed in the basis of ei(~k+ ~K)·~r, the so-called ‘plane wavebasis’. This is an infinite basis set because there are infinitely many ~K.But practically, the more ~K included, the longer is the computation time.So a cut-off should be set at certain ~Kmax, i.e. we should consider only~K < ~Kmax, where ~Kmax can be seen to relate to a cut-off energy Ecut byEcut ∼ h¯2 ~K2max2m .Actually, the highly oscillating component of the wave function withhigh energy only come from the region very close to the nucleus where thepotential is very negative, as shown in the left panel of Fig. 3.1. Thereforethere are two simplications that we can make.153.4. Pseudopotential to simplify actual computationFigure 3.1: Schematic representation of nucleus potential and pseu-dopotential. Left panel is showing the valence electron wavefunction Ψin the nucleus potential V˜ nuc. Right panel is showing the valence elec-tron (pseudo-)wavefunction φ in the pseudopotential V˜ ion which matchesthe original nuclues potential outside the red region and is much less nega-tive inside that region. (Credit: Efthimios Kaxiras, Atomic and ElectronicStructure of Solids (Cambridge University Press, 2003))[22]1. Focus our attention only on the valence electron, which play a mainrole in physical properties of the crystal and live mostly far away fromthe core, in the calculation and ignore the core electrons which alwaysgather around the nucleus and is not important in determining thecrystal properties. We will take the effect of core electron into accountonly through their screening of the nucleus potential.2. As shown in the right panel of Fig. 3.1, replace the original nuclueuspotential V˜ nuc by the pseudopotential V˜ ion which matches V˜ nuc out-side certain distance from the core and is much less negative than V˜ nucinside that threshold distance.In this way, we need not consider the highly oscillating core electrons and thevalance electron is much less oscillating in the region close to the nuclues.The calculation can be speeded up greatly because there is less electronneeded to be concerned and we can set a much lower energy cutoff Ecut (orcrystal momentum cutoff ~Kmax) after getting rid of the highly oscillatingcomponent. These simplications are reasonable because chemistry happens163.4. Pseudopotential to simplify actual computationin the region far away from the nucleus core and it is the valence orbitalswhich determine the chemical reactivity of the crystal and how the crys-tal response to the applied electromagnetic fields in various spectroscopy.This method for simplifying the calculation is known as the pseudopoentialmethod.17Chapter 4Experimental Results4.1 MethodsWe performed scanning tunneling microscope (STM) measurement using ahome-built STM at liquid helium temperature (∼4-5K) to study Sb.The Sb sample used was prepared by Dillon Gardner and Yong Lee inMIT. It is from a high-purity antimony (99.999%, supplied by Alfa Aesar R©)in shot form (10.15 g, 6 mm) which was sealed in an evacuated quartztube, and heated in a box furnace to 700◦C for 24 hours. The furnace wasthen cooled slowly (0.1◦C/min) to 500◦C, and subsequently cooled to roomtemperature.Before being inserted into the STM, single crystals of Sb were cleavedin-situ in cryogenic ultrahigh-vacuum to expose the (111) face. In the STM,a mechanically cut PtIr tip, cleaned by field emission and characterized ongold, was used for measurements. As introduced in a previous chapter, weacquired spectroscopy data by the lock-in technique at frequency f = 1.115kHz and conductance maps by recording out-of-feedback dI/dV spectra ateach spatial location.4.2 Sb Structure in literature versus thetopography in our experimentThe structure of Sb(111) is shown in Fig. 4.1(a), from which we can see thatthe inter-bilayer distance [23, 24] is 2.2 A˚ and the intra-bilayer length is 1.5A˚. Since the bonding is much stronger within the bilayer than that betweenthe bilayer, when the sample is cleaved, it is typically cleaved between thebilayer. If there were any steps presented on the surface after the cleavage,the expected step height would be 2.2 A˚ + 1.5 A˚ = 3.7 A˚.In our STM experiment, we observed a topography of a cleaved Sb(111)surface like what Fig. 4.1(b) is showing. There are a number of steps onit and its height profile along the grey line is shown in Fig. 4.1(d). Thereare two types of step heights whose values are around 1.5 A˚ and 2.2 A˚.184.2. Sb Structure in literature versus the topography in our experimentDistance (nm)601.5 ÅHeight (Å)2.2 Å~1.4 Å25 50 75 1000241.5 Հ2.2 Հ6.5 Å0 ÅA125nmB3B1 A2 B2 A3(d)(a) (b)Distance (nm)601.5 ÅHeight (Å)2.2 Å~1.4 Å25 50 75 1000241.5 Å2.2 Å(c)Figure 4.1: Terraces Structures of Antimony. (a) Bilayer crystal struc-ture of Sb(111). Intrabilayer distance is 1.5 A˚ and interbilayer distance is2.2 A˚. (b) Topography of steps observed on Sb(111) by scanning tunnelingmicroscope (STM). (sample bias, V0=300mV; juction resistance, RJ=3 GΩ;resolution=70×1200 pixels) (c) A cartoon model showing how different stepheight observed can be formed using bilayers and a single broken layer ofSb. The blue terrace is called normal and the orange terrace with a singlebroken layer is called abnormal. (d) Height profile along the grey line in (b).From the step heights observed, it can be identified that B1, B2 and B3 in(b) are abnormal terraces.194.3. dI/dV SpectraThey are much closer to the length of the intra- and inter- bilayer distancerespectively than that of the usual expected step height (3.7 A˚). These kindsof unusual step heights can be produced by the cartoon model in Fig. 4.1(c),where there is a single broken layer beneath the surface of a terrace (theorange terrace). In the following, we will call this kind of unusual terraceas ‘abnormal terrace and the terrace without a single broken layer (the blueterrace) as a ‘normal terrace. When the abnormal terrace is sandwichedbetween two normal terrace as in the model of Fig. 4.1(c), the step heightsbetween the normal and abnormal terraces on the left and right ends wouldcorrespond to the intra- and inter- bilayer distances respectively. It shouldalso be noted that the unusual step heights can be obtained no matter thesingle broken layer is located on the first bilayer (i.e. on the surface) orseveral bilayers below the surface. From the height profile in Fig. 4.1(d),we can see that terrace B1 and B2 in Fig. 4.1(b) could be identified as theabnormal terraces. The existing of the single broken layer can be due to thecleavage in our experiment or the imperfect of the sample.4.3 dI/dV Spectra4.3.1 Zero bias peak on abnormal surfaceFig. 4.2(a) is showing the averaged spectra on the surfaces of the normaland abnormal terraces in Fig. 4.1(b). Compared with the spectrum on thenormal terrace, there is a characteristic peak at zero bias of width ∼13 meV(by a Lorentzian fit) on the abnormal terrace. A linecut along the grey linein Fig. 4.1(b) is obtained and the result is shown in Fig. 4.2(b), where therelative color is showing the relative intensity of dI/dV (thus the density ofstates). It can be seen that the zero bias peaks (ZBP) are quite robust onthe abnormal terrace. Apart from that, there is a series of euqally spacedwiggling energy peaks in the dI/dV spectrum in Fig. 4.2(a) from -100 to0 meV on both the normal and abnormal terrace. These wigglings will beexplained in the next subsection.4.3.2 Wiggling in the spectrumThe wigglings in spectra are also observed by Seo et al. [25] on their stepsof Sb surface. This kind of quantized resonance are caused by the scatteringof the surface states from the step edges, reflecting the reflection propertiesof those steps.204.3. dI/dV Spectra0T,Abnormal9T, Abnormal(a)(b)-200 -100 0 100 20012080-20040 10020-300-100010020030060Bias (meV)Distance (nm) (a.u.)Bias (meV)low0Figure 4.2: dI/dV Spectra observed on different terraces. (a) Aver-aged dI/dV spectra observed on the normal and abnormal terraces, featuringthe zero bias conductance peak on the abnormal terrace. (V0 = 300 mV; RJ= 300 MΩ; V rms = 5 mV) The blue curve is from the normal terrace andthe orange curve is from the abnormal terrace. They are averaged along thegrey line on the A2 normal terrace and B1 abnormal terrace in Fig. 4.1(b)respectively. (b) A linecut of dI/dV spectra (arbitrary units) along the greyline in Fig. 4.1(b), normalized with respect to the maximum intenisty alongthe entire inecut, showing quasiparticle interference pattern. The color isthe magnitude of the dI/dV spectrum. (V0 = 300 mV; juction resistance,RJ = 300 MΩ; V rms = 5 mV; obtained at 5.4K)214.3. dI/dV SpectraBias(meV)-100 -50 0 50dI/dV (a.u.)ΔEEnergy (meV)-100 -80 -60 -40 -20 0024681012141618A1B1A2B2(a) (b) -50 0 0.2Bias (meV)dI/dV (a.u.)Energy (meV)-100 -80 -60 -40 -20 0024681012141618A1B1A2B2Figure 4.3: Wigglings in dI/dV spectra on normal and abnormalsurfaces and the energy difference between peaks (a) AverageddI/dV measurement on different surfaces featuring quantized peaks in en-ergy ranging from -100 to 0 meV. ∆E is defined to be the energy differencebetween adjacent peaks. The speactra are offset vertically for clarity. (Ob-tained at 2 meV per energy point; V0 = 300 mV; RJ = 300 MΩ; V rms = 5mV)(b) Energy spacing between quantized peaks in (a). Dotted lines arethe avearges energy seperation.As pointed out by Seo et al. [25], if scattering resonance is really theorigin of the wigglings, the energy peaks should be equally spaced in thespectra. So we look more closely on the wigglings in Fig. 4.3(a) and plot outthe energy spacing (∆E) between different wiggling peaks on each spectrumin Fig. 4.3(b). Indeed, given the sampling rate of the dI/dV spectra was 2meV per energy point and the lock-in V rms amplitude is 5 meV, the energyspacing of the wiggling peaks shown in Fig. 4.3(b) is quite constant in everydI/dV spectra.An immediate question would be: is the ZBP also caused by quantumresonance? On the abnormal steps B1 and B2, the average energy seperationbetween those peaks are 12.3 and 15 meV respectively. The energy differencebetween the ZBP and its closest wiggling peak are 10 and 8 meV on B1 andB2 respectively, which tend not to be in the trend of the equally spaced seriesof peaks. Also, the amplitude of the ZBP is much larger than those wigglingspeak. Noticibly, there is also a wiggling peak at 0 meV on the normal surfaceA1. But the amplitude is similarly small as the other wiggling peaks on thatsurface. So it can be seen that the ZBP on the abnormal surface may notbe due to the quantum resonance of the steps.224.3. dI/dV Spectra(b)(c)(a) (d)Low High10 nm 1 nm1 nmA2 B2A2B2Figure 4.4: STM Topography of the normal and abnormal stepsand its Fourier transform (a) STM Topography of the step with thenormal surface A2 on the left side and abnormal surface B2 on the rightside. The total field of view is 40×40 nm. (sample bias, V0 = 300mV;juction resistance, RJ = 3 GΩ; resolution = 512×512 pixels) (b,c) Zoomed-in of (a) for the abnormal B2 (b) and normal A2 (c) steps showing atomicresolution. (d) Fourier transfrom of (a) from which we can see there is onlyone set of hexagonal Bragg peaks corresponding to the lattice constant ofthe surfaces. This reveals that there is no lattice constant change on theabnormal surface.4.3.3 No superconductivity involvedIn our experiment, only pristine Sb was used. There is no superconductivityproximity effect to induce the zero bias peak, although Wittig [26] pointedout that Sb can be superconducting with critical temperature 3.5 K at highpressure, because the temperature in our experiments were 4.5 K - 5.4 K.However, Reale [27] suggested that there is a metastate of Sb of a moredensely packed face centered cubic structure with a much shorter latticeconstant can have a critical temperature of as high as 7.5 K under hugestrain and pressure. Therefore, we tried to compare the topography of thenormal and abnormal surfaces in Fig. 4.4 and see if there is any change inthe lattice constant.In the metastate of Sb of FCC structure suggested by Reale [27], thedistance between atoms in hexagonal pattern on the (111) surface wouldbecome about 3.2 A˚, around 25% smaller than the usual distance, which is4.3 A˚[24]. This would give us a set of bragg peak in the fourier space with234.4. Momentum resolved spectroscopic information1/3 larger magntitude corresponding to that shortened atomic distance.So when we Fourier transform a topography having both the normal andabnormal surfaces, if the abnormal surface had a diffferent lattice constantas the normal surface, we would expect two sets of hexagonal Bragg peaksin the Fourier space, where one set corresponding to the altered latticeconstant on abnormal surface has magnitude about 1/3 larger than theother set corresponding to the original lattice constant on the normal surface.However, as apparent in Fig. 4.4(d), there is only one set of hexagonal Braggpeak. This shows that there is no crystal structure change on the abnormalsurface and thus there would not be potential for the abnormal surface to besuperconducting at our operating temperature. In this way, we can verifythat there is no superducting involved in the ZBP on the abnormal terrace.4.4 Momentum resolved spectroscopicinformationTo clarify the origin of the zero bias peak on the abnormal terrace, weneed to know more about the momentum-resolved spectroscopic informationon this terrace. There are two ways STM can obtain this information,namely through quasiparticle interference (QPI) and Landau quantization.It has been showed that these two pieces of information can be obtainedsimultaneously [13].4.4.1 Quasiparticle interference (QPI)The surface state quasiparticle of Sb is scattered by the step on each terrace.The scattered outgoing surface states interfered with the incoming states tocreate interference pattern on terraces as shown in Fig. 4.2(b). This is knownas QPI. The peaks in the Fourier transform of the interference patterns atdifferent energy are corresponding to the possible scattering modes at thatenergy, which in turn reflect the band structure of the terrace. The bandstructure of the normal terrace of Sb is well-studied [13] and from that weknow the surface state of Sb that can be probed by QPI is from about−250 meV to about 100 meV. So we Fourier transformed the interferencepattern of the line-cuts in Fig. 4.2(b) for the normal and abnormal terracesrespectively. Fig. 4.5(a) is showing the results for the normal terrace alongthe Γ-M direction. There are two noticeable linear dispersions starting fromaround −200 meV and −100 meV. It agrees with what [13] has shown andthey are corresponding to the two scattering modes in the double cone of244.4. Momentum resolved spectroscopic informationhighlow-300Momentum, q ((2πÅ)  ) 0-20 40Sample Bias (meV)0-100-200 100Sample Bias (meV)0-3000. 0.080- 209T, Normal9T, Abnormal0T, Abnormal9T, AbnormalBias (meV)Bias (meV)dI/dV (a.u.)dI/dV (a.u.)(a)(b)NormalAbnormalalong Γ-Malong Γ-M(c)(d)a LL peakFigure 4.5: Dispersing modes from Quasiparticle Interference andLandau Quantization of surface states. (a,b) Representative 1DFourier Transforms (arbitrary unit) of the quasiparticle interference in thedI/dV linecuts of normal (a) and abnormal (b) in Fig. 4.2(b), showing twosimilar prominent dispersing modes along the Γ-M direction. Each graphhas intensity relative to itself. (c) Comparison of dI/dV spectra at mag-netic field B=9T on the abnormal terrace (purple line, V0 = 300mV; juctionresistance, RJ = 300 MΩ; V rms = 1 mV; obtained at 4.5K) and normal ter-race (the yellowish brown line). Both curves are normalized with respect tothemselevs. The pale grey lines behind the curves are comparing the energyposition of different Landau level peaks on the two curves, demonstratingthose peaks are at the same energy values on the normal and abnormal ter-races. (d) Zoomed-in comparison of the zero bias conductance peaks onabnormal terrace in magnetic field 9T (the purple line, obatined at 4.5K)and out of field (the orange line, obatined at 5.4K). We can see that thezero bias peak is still quite robust in magnetic field.254.4. Momentum resolved spectroscopic informationthe surface states of Sb. Fig. 4.5(b) is showing dispersion of the QPI onthe abnormal terrace. Similar to the normal terrace, there are two lineardispersing modes starting from about −200 meV and −100 meV. This showsthat the main features of the surface states of the abnormal terrace shouldbe quite similar to that of the normal terrace.4.4.2 Landau levelsThe other phenomena that can reflect the band structure is Landau levels(LL) in the dI/dV spectrum. Applying strong magnetic field (>4 T) tothe Sb sample can cause conductance oscillation in the DOS spectrum andthey are known as the LL peaks. As in other topological materials, wecan interpret the LLs in the Dirac fermion picture [28–31]. The energyof the nth LL increases with√nB . The Bohr-Sommerfeld quantizationrelation suggests the momentum space radius for the nth LL orbit is alsoproportional to√nB [28]. So, by tracing the energy of the nth peak in theDOS spectrum, we can deduce the dispersion of the energy of surface statesversus momentum. In other words, the energy location of the LL peaksis reflecting the band structures. Fig. 4.5(c) is comparing the LL peakson the normal Sb from [13] in magnetic field of 9T (the yellowish browncurve) and the peaks on the abnormal terrace of Sb in our experiment inthe same strength of magnetic field (the purple curve). As indicated by thegrey lines in the background, the energy of the peaks in both curves matchquite well with each other. This also suggests that the band structure ofthe surface states on the abnormal terrace are quite similar to that on thenormal terrace. Fig. 4.5(d) is demonstrating that in the presence of magneticfield of 9T, the zero bias peak is still robust, showing no clear evidence ofpeak splitting, for its width remains to be about 10 meV by the Lorentzianfit. Thus the possibility for Kondo effect [12, 32] to play a role in inducingthe ZBCP is not apparent from the experiment26Chapter 5DFT ResultsTo gain theoretical insight into the origin of the ZBP, density functionaltheory (DFT) simulation was carried out to compare the calculated bandstructure of the abnormal terrace and the normal terrace.5.1 MethodsThe DFT calculation was performed using the Vienna ab initio simula-tion package (VASP) code[33–35], with the projector augmented plane wavemethod. The exchange-correlation energy is estimated by the Perdew-Burke-Ernzerhof (PBE) functional and the energy cutoff for the plane-wavebasis function is 300 eV. A Γ-ceneterd k-point mesh of (21,21,1) was usedin the crystal structural relaxation while a much finer k-point mesh of(200,200,1) was used in the band structure calculation.The band structures and surface states of the normal and abnormalterraces were simulated from a slab geometry. Ideally, the larger the slab(i.e. the more layers in the slab) is, the more reliable will be the resultsbecause the less influenzed by the finite size effect. So we included a slab aslarge as possible until we hit the limit of our calculating power. Eventually,a normal slab of 46 layers and an abnormal slab of 47 layers (with an extrasingle broken layer) were used.In the structural optimization of the abnormal terrace, we first put thesingle broken layer at the surface of the slab and allow it to relax. Afterrelaxation of the layers, the single broken layer sunk into the bulk of the slab.It indicates that in experiment, even if the single broken layer was createdat the surface accidentally when we cleaved the sample in situ, the crystalwould relax and other layers would rearrange themselves so that the singlebroken layer would appear below the surface in order to lower the energy. Sowe put the single broken layer at different depth below the surface, allowedit to relax and calculated the band structure. We found that the brokenlayer would continue to fall to the lower layer until we put it more than 4bilayers below the surface. So we will focus on band structure calculated275.2. Band structure of normal and abnormal terracefrom abnormal slab with a single broken layer more than 4 layers below thesurface.Also, noticing in STM experiments, electronic states on the surface aremainly concerned because they are what mostly the tip can probe. So thebands available at the surface of the slab would be the most of our interest.Furthermore, the outermost valence electrons in Sb are in s and p orbitals.Since the pz orbital is pointing out of the page and much more far reachingthan the s orbital to the tip, it would be the closest to the tip in STM anddominate the measurement in experiment. Hence, to recognize the maincontributor in experiment and to compare what happened on the surfacesof the normal and abnormal terrace, we projected the available electronicstates in pz orbital on the surface in the band structure. In Fig. 5.1 andFig. A.1 showing the calculated band structures, the greys lines would bethe unavailable states on the surface and the available states are coloredaccording to their normalized relative contribution to the DOS spectrum.5.2 Band structure of normal and abnormalterrace5.2.1 Normal surfaceThe calculated band structure of the normal terrace along the M-K-Γ-Mdirection is plotted in Fig. 5.1(a). As in literature [13], the surface statefor the normal surface of Sb is a double Dirac cone ranging from around−250 meV to 100 meV. There is a saddle point (s) at around −0.1 eV. Theenergy of the minimum of the cones (B) are a bit above −300 meV andthat of an extreme T is just above 0.2 eV. The shapes and main features ofthe calculated band structure for normal terrace agree quite well with theother publication [13]. The subtle difference in the exact value could be dueto the finite size of the slab.5.2.2 Abnormal surfaceFig. 5.1(b) is showing the typical result for the abnormal terrace, in which aslab of 47 layers with the single broken layer 6 bilayers (or 12 layers) belowthe surface had been used.As implied by [36], the contribution of states in momentum space to theDOS spectrum will decay away from the Γ point exponentially. So we couldfocus at the band structure around the Γ point as zoomed in Fig. 5.1(c) and285.2. Band structure of normal and abnormal terraceBias (meV)Bias (meV)-300-200-1000100200300400200-200-4000Momentum MomentumK ΓM MK ΓM MMomentumK/3 Γ M/3MomentumK/3 Γ M/3012345678910-3 -2 -1 0 1 2 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1low HighΓMK(c)(a)(d)(b)Normal AbnormalNormal AbnormalεsεBεTFigure 5.1: Band Structure of normal and abnormal terraces. (a,b)Band Structure of the normal (a) and abnormal (b) terrace from densityfunctional theory (DFT). The color intenisty is the projection of the relativeamount of available states of pz orbital on the surfaces. The grey lines are theunavailable states on the surfaces. (c,d) Zoomed in of the band structuresof the normal (c) and abnormal (d) surfaces around the Γ point. The saddlepoints are indicated by the arrows.295.2. Band structure of normal and abnormal terrace(d) for the normal and abnormal surfaces. From Fig. 5.1(d), the double-cone feature of the surface state is preserved on the abnormal terrace. Itis still lying in the similar range of energy as in the normal terrace caseif we compare Fig. 5.1(c) and (d). This explains why similar QPI andLandau quantization, which reflect the characteristics of the surface state,were observed on the abnormal terrace in our experiment.Now we may turn our attention to the saddle point causing the zero biasconductance peak. As mentioned above, on the normal terrace, there is asaddle point (s) around −0.1 eV on the outer cone along the Γ-K direction.By [13], this saddle point causes a peak at around −0.1 eV in the DOS spec-trum of the normal terrace. However, on the abnormal terrace (Fig. 5.1(e)),there are more states appearing below this saddle point, pushing it up to theFermi energy (0 eV). This will constitute a van Hove singularity leading to apeak in the zero bias of the DOS spectrum. This pused-up saddle point canbe seen in slabs with the single broken layer at different depths (as long asit can be stabilized). Other plots of band structure for the abnormal terracewith slab of different thickness and single broken layer at different depth canbe found in Appendix A. A justification of DFT results by comparing withexperimental data is also included in Appendix B. Our findings suggestedthat the origin of the zero bias peak observed in experiment is just due tothe abrupt change of the crystal and is trivial.Other characteristics in the DOS spectrum of the abnormal terrace fromFig. 4.2(a) included a larger bump at negative energies from −200 meV to0 meV, compared with that of the normal terrace. Comparing the bandstructures of the normal (Fig. 5.1(c)) and abnormal (Fig. 5.1(d)) terraces,there are more available states on the abnormal terrace around the Γ pointfrom around −200 meV to 0 meV. This naturally explains why the dI/dVconductance is enhanced on the abnormal terrace from −200 meV to 0 meV.5.2.3 Further investigation for magnetic breakdownThe increased number of states on the abnormal terrace in the concernedenergy range from −200 meV to 200 meV may have another implication.Energy of Landau level is determined by the area enclosed by the state inmomentum space. On the abnormal terrace, when there are more statesat the same energy, they collide and interfere with the original state in themomentum space with an example shown in Fig. 5.2(a-b). (The evolution ofthe crossing is shown more clearly in Fig.5.2(c-h).) Hybridization betweenthe newly emergent states and the original states would happen. The areaenclosed by the original state could be affected. In this way, the Landau305.2. Band structure of normal and abnormal terrace(a) (b)Bias (meV)-300-200-1000100200300Bias (meV)-180-160-140-120-100-80-60MomentumK/3 Γ M/3Momentum, k (Å  ) 0.005 0.02 0.035-1-0.0200.02-0.0200.02k y0 0.02-0.02kx (Å  )-10 0.02-0.02kx (Å  )-10 0.02-0.02kx (Å  )-1(Å  )-1k y(Å  )-1(c) (d) (e)(f) (g) (h)-118 meV-131 meV-144 meV-171 meV -158 meV -150 meVFigure 5.2: Extra emergent band on the abnormal terrace cuttingacross the Dirac cone (a,b) (a) Band structure on the abnormal surfacefrom DFT (data from thickness 47 (depth 6) is employed). The red curve isrepresenting the inner Dirac cone and the blue curve is the emergent bandcutting cross the Dirac cone. Other states not being concenred here are ingrey. The portion along the Γ-M direction enclosed in the green box in (a)is enlarged in (b). (c-h) Energy contour of the inner cone (red) and theemergent band (blue) from energies below to above the crossing.levels should be changed. However, from our experiment, the Landau levelswas not altered by that. This problem is known as ‘magnetic breakdown’[37]. It was proposed that electrons could jump across the hybridized orbitaland remain in the original way of motion. Further investigation can beconducted to establish a more satisfactory and complete explanation forthat.31Chapter 6ConclusionWe reported that zero bias conductance peak, which can be a signature ofMajorana fermion, would be possibly resulted from an abrupt change of atopological material, Sb, with a trivial origin.The ZBP happens on the abnormal terrace of Sb, with a single brokenlayer beneath the surface. Using STM, we found that the QPI and LLs onthe abnormal surface are similar to that on the normal surface. Then wesimulated the band structures of those two kinds of surfaces by DFT. Itreveals that both the normal and abnormal terraces have a similar doublecone surface states, which provide an explanation for the similar QPI andLLs observed by STM. However, on the abnormal terrace, there is a saddlepoint around the Γ point being pushed up to energy ∼ 0 meV which coulda zero bias peak in dI/dV spectrum due to van Hove singularity.The lesson learnt is that when we are trying to probe Majorana fermionin the interface of a topological insulator and a superconductor, even if weare not working with Sb, we should make sure the signal is not trivially fromthe crystal itself.In addition, more complete mechanism can be sought to describe whythere is magnetic breakdown when extra state is introduced to interfere withthe original states.32Bibliography[1] Sergey Bravyi and Alexei Kitaev. Fermionic quantum computation.[2] Sergey Bravyi. Universal quantum computation with the ν=52 frac-tional quantum Hall state. Phys. Rev. A, 73(4):042313, 2006.[3] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, and MatthewP. a. Fisher. Non-Abelian statistics and topological quantum informa-tion processing in 1D wire networks. Nat. 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Cambride UniversityPress, 1984.36Appendix ABand Structures of differentthickness and depthDepth 7(a)Bias (meV)-300-200-1000100200300(d)Bias (meV)-300-200-1000100200300MomentumK/3 Γ M/3MomentumK/3 Γ M/3MomentumK/3 Γ M/3(b) (c)MomentumK/3 Γ M/3Thickness 47Thickness 43Depth 5 Depth 6012345678910-3 -2 -1 0 1 2 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1low HighFigure A.1: Band structures from DFT with different thickness anddepth of the single broken layer (a-c) Band structures from DFT usingslabs with 47 total layers and single broken layer at depth 5 (a), 6 (b) and7 (c). They all have a saddle point (indicated by the black arrow) around0 meV. (d) Band structure from a slab with different thickness (43 layersin total) and single broken layer at depth 6. Ideally the larger the slab, thesmaller the finite size effect and the more reliable is the calculation. (And47 total layers is the limit of our computational resources.) But comparing(b) and (d), we can still identify similar main features of the band structuresof the abnormal surfaces, especially the saddle point around 0 meV.37Appendix BJustification of Validity ofDFT calculationLandau LevelsDFT(a) (b)Landau LevelsDFTBias (meV)200-300-4000-200-100100Bias (meV)200-300-4000-200-1001000, k (Å  )0 Momentum, k (Å  )-1Normal AbnormalFigure B.1: Comparison of Band Structure infered from Landaulevels in dI/dV spectrum and that from DFT (a,b) Using the semi-classical Bohr-Sommerfeld quantization relation [28], the inner cone in theband structure can be deduced from the energies of Landau level peak inthe dI/dV spectrum. Red curves in the above figures are showing the innercones calculated from Landau levels on the normal (a) and abnormal (b)terraces. The inner cone from DFT (black curves, data from thickness 47(depth 6) is employed) are also drawn in the figures. It can be seen thatDFT match quite well with the experimental data, especially from -100 meVto 0 meV in which the semiclassical approximation for Landau levels is par-ticularly reliable. This comparison can be regarded as a test for the validityof the DFT results.38


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