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TWO-DIMENSIONAL STRONG SPIN-ORBIT COUPLING MATERIALS A first-principle study WU DI A0083153L DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2015
TWO-DIMENSIONAL STRONGSPIN-ORBIT COUPLING MATERIALS
A first-principle study
WU DI
A0083153L
DEPARTMENT OF PHYSICSNATIONAL UNIVERSITY OF SINGAPORE
2015
Acknowledgement
I would like to thank my supervisor Assistant Professor Lin Hsin for his consistentguidance and support throughout this project. His contagious enthusiasm and encour-agement are precious inspirations for me, and I truly value his constructive advice everystep of the way.
I also wish to pay special acknowledgement to Dr.Chang Tay-Rong and Dr.Zeng Ming-gang for their frequent discussions and technical help on VASP and Matlab codes. Theyhad been extremely supportive and encouraging at later stage of this project. I reallyappreciate their unwavering support.
I am grateful to all the members from Professor Lin’s group: Dr.Tanmoy Das, ChangGupqing, Liu Yu-Tzu, Huang Shin-Ming, Wang Bokai, Le Quy Duong, and all the mem-bers from Professor Feng Yuan Ping’s group: Dr.Yang Ming, Dr.Wu Qingyun, Dr.ChintalapatiSandhya, Luo Yongzheng, Ling Hu Jiajun, Zhou Jun, Xu Lei and Deng Jiawen for theirvaluable discussion and generous help. Moreover, I want to thank our experimental col-laborator - Assistant Professor Liu Zheng from School of Materials Science & Engineering,Nanyang Technological University for his contribution on the experimental results usedas a reference.
Finally, I would like to acknowledge Department of Physics and Centre for Advanced2D Materials of National University of Singapore for this research opportunity and finan-cial support.
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Contents
1 Introduction 81.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.1 Metal-Based Spintronics . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Silicon-Based Spintronics . . . . . . . . . . . . . . . . . . . . . . . . 91.1.3 Graphene-Based Spintronics . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Two-Dimensional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Transition Metal Dichalcogenides . . . . . . . . . . . . . . . . . . . 13
1.3 Concept of Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Time Reversal Symmtery and Spatial Inversion Symmetry . . . . . 171.3.3 Effect of SOC in Solids . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Objectives of the Current Project . . . . . . . . . . . . . . . . . . . . . . . 20
2 Computational Methods 212.1 ab initio Total Energy Calculations . . . . . . . . . . . . . . . . . . . . . . 212.2 Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 222.2.2 Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . 232.2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 242.2.4 Exchange-Correlation Functional Approximation . . . . . . . . . . . 25
3 Results and Discussion 283.1 SnSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Effect of SOC and Layer Dependence Study . . . . . . . . . . . . . 283.1.2 HSE Band Structure Correction . . . . . . . . . . . . . . . . . . . . 32
3.2 PbI2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Effect of SOC and Layer Dependence Study . . . . . . . . . . . . . 343.2.2 Effect of Substrates on Monolayer PbI2 . . . . . . . . . . . . . . . . 38
3.3 NbSe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.1 Effect of SOC and Layer Dependence Study . . . . . . . . . . . . . 443.3.2 Spin Texture of Monolayer NbSe2 . . . . . . . . . . . . . . . . . . . 50
4 Conclusion 52
Appendices 54.1 Effect of SOC and Layer Dependence Study . . . . . . . . . . . . . . . . . 54.2 PbI2 on Graphene and MoS2 Substrates . . . . . . . . . . . . . . . . . . . 61
References 68
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Abstract
Spin-based electronics, or spintronics, represent a new paradigm for future electronics.Its fundamental objective is to achieve active manipulation of the spins in the materi-als, in addition to the charges and the ultimate goal is to build spin-based transistorsthat would replace the conventional transistors in the integrated logic circuits and mem-ory devices. Spintronics promise to be smaller, faster and more versatile, but the currentchallenge still remains on the search for suitable materials. Since the discovery of grapheneand other 2-dimensional ver-der-Waals structured materials, or transition metal dichalco-genides (TMDs), extensive research efforts have been concentrated on the investigation ofelectric, magnetic, optical and chemical properties of such type of materials. With simplemodifications, certain properties can be changed significantly, thus, materials as such arevery promising for the development of next-generation electronic products. In the currentproject, we used first-principle calculations based on density functional theory (DFT) tostudy the effect of spin-orbit interaction and number of layers on several TMDs–SnSe2,PbI2, NbSe2, PtS2, PtSe2 and PdSe2. We realized that the effect of spin-orbit couplingis closely related to the structural symmetry and it also affects the band structures of amaterial to a certain extent. In the layer dependence study, we discovered that the bandgaps could be engineered through controlling the number of layers for some material, suchas SnSe2, but not so much for other materials such as PbI2. During the project, we havealso learned to perform some crucial analyzing techniques in first-principle calculations(i.e. use Heyd-Scuseria-Ernzerhof (HSE) hybrid functional to make energy band gap cor-rections; examine the effect of substrates on a material for different positions; and toevaluate the spin texture of a system). These techniques were readily applied to somematerials studied throughout this project that aims to provide theoretical predictions onthe physical properties, which if well understood would be valuable to the applications ofreal electronic devices in the future.
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1 Introduction
1.1 Spintronics
The famous Moore’s Law states that the number of transistors on an affordable CUPwould double every two years. Despite some inaccuracy, it is rather a simple reflectionon the rapid progress in the development of semiconductor electronic devices. However,as the size of the devices is getting smaller than 100 nanometers, physicists and engineerswill inevitably be facing the intimidating problem associated with quantum mechanics –that the behaviour of the particles is governed by probabilistic wavelike properties. Al-though physicists still have doubts about this theory, many believe that quantum physicsoffers us an unprecedented opportunity to define a drastically new device that would haveunique advantages over the existing information technology.
One of the unique characteristics in quantum mechanics is the spin of the electrons,which is closely related to magnetism. The current semiconductors ignored this propertyand are solely based on charge of the electrons. Hence, the new devices we are interestedwill rely on the spin degree of freedom to perform their functions. This is the founda-tion of spintronics, which is short for spin-based electronics or spin transport electronics.It is not the electron charge but the electron spin that carries the information. Hencespintronics is anticipated to be the devices that combine the standard microelectronicswith the spin-dependent effects arising from the interaction between spin and magneticproperties of the materials.
Nowadays, information processing and communication devices (such as integrated cir-cuits) are based on semiconductors whereas information storage devices (such as computerhard drives) rely on magnetic materials. The controls of these two types of devices areestablished on manipulation of the motion of electric charges and re-orientation of mag-netic domains, respectively. Spintronics are therefore expected to perform the all theseoperations, logic, communication and storage within one device. In fact, surprisingly,adding spin degree of freedom to conventional charge-based electronics will substantiallyintroduce more capability and performance to the electronic products. The advantagesinclude non-volatility, increased data processing speed, decreased electric power consump-tion and increased integration densities. However, the theoretical and technical challengesremained to be resolved are also overwhelming – the optimization of electron spin lifetime,transport of spin polarized current across interfaces, detection of spin coherence in thestructures, the realization of room temperature operating devices and the manipulationof both electron and spin sufficiently fast[1]. Despite all these challenges, many have pro-posed a number of practical applications of spin-based devices, such as spin-field effecttransistor, spin-light emitting diode, spin resonant tunneling device and quantum bitsfor quantum computation. Nevertheless, all of these promising features will ultimatelydepend on the better understanding of the fundamental physics behind the profound spin
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interactions in solid-state materials.
1.1.1 Metal-Based Spintronics
The discovery of giant magnetoresistive effect (GMR)[2][3] in 1988 marked the beginningof spin-based electronics. GMR was observed in a thin-film structure, which consistsof alternating ferromagnetic and nonmagnetic metal layers. Depending on the relativeorientation of the magnetizations in the magnetic layers, the resistance of the materialchanges from lowest when the magnetic moments in ferromagnetic layers are aligned, tohighest when they are anti-aligned. The structure will exhibit substantial change in themagnetoresistance when subjected to a relatively small magnetic field. Followed by thisdiscovery were a number of GMR-based devices, such as spin valve[4], magnetic tun-nel junction (TMJ)[5], galvanic isolators and magnetoresistive random access memory(MRAM)[6].
1.1.2 Silicon-Based Spintronics
Besides this metal-based spintronics, attentions have also been directed to other types ofmaterials suitable for spintronic applications and two of those are silicon and graphene.The development of silicon spintronics is guided by three steps – creation, manipulationand detection[7]. The creation of spin polarization in silicon is necessary because siliconis non-magnetic in equilibrium; the manipulation of the spin polarization in response toan external stimulus provides the gate operation; detection of the state of the spin po-larization is required to obtain an output signal. Over the past few decades, tremendousprogress has been made in silicon spintronic devices. In 2007, Appelbaum et al. firstdemonstrated the electrical injection, transport and detection of spins in the undopedsilicon at low temperature[8] as well as the coherent spin precession over long distance[9];Jonker et al. reported the electrical spin injection into silicon from a ferromagnetic Fecontact across an Al2O3 tunnel barrier at low temperature[10]; and later that year, van’tErve et al. reported electrical injection, transport and detection of spin in silicon at 10Kwith ferromagnetic tunnel contacts[11]. The low temperature problem was resolved in2009, when Dash et al. demonstrated the electrical creation, detection and manipulationof spin polarization in silicon at 300K[12]. The electric field control of spin in silicon quan-tum well at low temperature was first reported in 2010 by Jansen et al [13]. In addition tothe advance in experimental results, theory of spin transport and spin-dependent opticaltransitions in silicon also took a leap forward by introducing the spin-orbital couplingand phonons into the picture[14][15][16]. In spite of some engineering problems, all thesemajor achievements for the implementation of spin in silicon indicate the great potentialand promise of this material to be readily applied in the short future.
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1.1.3 Graphene-Based Spintronics
Graphene, a single-layer graphite, is also a favorable spin-based device not only due tomany successful development both theoretical and experimental in spin related field, butalso for its own unique properties such as high electrical mobility and gate-tunable carrierconcentration. Here, I shall just list a few major accomplishments in graphene spintronicdevices. In 2007, van Wees et al.[17] first reported the non-local electrical spin transportand precession in graphene at room temperature shortly after the isolation of monolayergraphite. High spin injection efficiency was achieved in 2010 by Han et al.[18], who used asmooth MgO thin film as the tunnel barrier to alleviate the conductance mismatch prob-lem. The spin injection efficiency was improved from 10% to approximately 30%; and laterstudies demonstrated a highest efficiency of about 60%[19]. Subsequently, spin injectionand transport in large-area graphene and high-mobility graphene have been investigated.Spin transport in large-area graphene fabricated by chemical vapour deposition has beenachieved in 2011, which marks a key step towards graphene spintronics at wafer scale[20].Moreover, many have focused their attention on spin transport in high-mobility grapheneby using suspended graphene[21]. Despite the relatively low quality in the experiments,long spin diffusion length over several micrometers[22], longer spin-relaxation length andspin lifetime[23] are all being observed over the past few years. Although graphene doesnot contain any d or f orbitals, making graphene magnetic is still highly tempting due tothe possibility of giving rise to room temperature diluted magnetic material for the appli-cation of information storage. Fortunately, many theoretical and experimental efforts havecome to fruition for the observation of magnetic moments in graphene as a result of va-cancy defects[24][25][26], adatoms[27][28] and coupling to ferromagnetic substrates[29][30].The room temperature physical characteristics will unavoidably make graphene one of themost favourable candidates for spintronics in the spin logic and data storage applications.
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1.2 Two-Dimensional Materials
1.2.1 Graphene
Graphene, a flat monolayer of carbon atom arranged in a two-dimensional (2D) hexag-onal lattice, is the basic building block for graphitic materials of higher dimensions[31].Graphene has been studied theoretically as long ago as 1947, and the name graphenewas first mentioned in 1987 by S. Mouras and the co-workers. It was presumed not toexist in the free state, being described as an academic material and was believed to beunstable with respect to the formation of curved structures[31]. However, the theoreticalmodel suddenly turned into reality, when free-standing graphene was unexpectedly foundin 2004 by Andre Geim and Konstantin Novoselov from University of Manchester, andespecially when the follow-up experiments confirmed that its charge carriers were indeedmassless Dirac fermions. This was when the golden age of graphene has begun.
The most important reason for the rapid development in graphene is probably dueto its extraordinary properties. Many graphene characteristics measured in experimentshave exceeded those obtained in any other material: room-temperature electron mobil-ity of 2.5×105cm2(Vs)−1[32] (the corresponding resistivity would be 10−6 cm, the lowestresistivity known at room temperature); a Young’s modulus of 1.0 TPa and intrinsicstrength of 130 GPa[33]; and very high thermal conductivity of above 3,000WmK−1[34].Single-layer graphene has a unique electronic structure with band overlap at K point inthe Brillouin zone. Its charge carriers, known as massless Dirac fermions, are electronswithout rest mass[35][36]. At room temperature, single-layer graphene exhibits anoma-lous (half-integer) quantum Hall effect (QHE)[37] and its charge carriers can be alternatedbetween electrons and holes depending on the gate voltage[38]. There are a few impor-tant applications based on these properties, one of which is the gas sensing capability.Single-layer graphene-based gas sensor can be used to detect adsorption and desorptionof single molecules of gases, such as CO, NH3 and NO2[39]. Being one the strongest mate-rials with Young’s modulus to be around 1.0TPa, Chen et al.[40] has even engineered thesingle-layer graphene into graphene papers, which is believed to be biocompatible. Unlikesingle-layer graphene, bilayer graphene has massive Dirac fermions and it is considered asa gapless semiconductor. Few-layer graphene has no band gap and the structure becomesmetallic as with increasing number of layers[41]. Moreover, the study of graphene edge-state attracted rising attention due to the appearance of magnetic properties, includingferromagnetism in the structure[42]. With proper chemical modifications to generate mag-netism in graphene, we are looking at very promising applications in data storage devices.So far, the applications of graphene-based materials range from electronics, sensor, andstorage devices to biotechnology. With the current amount of resources being divertedin the field of graphene research, I believe the near future application based on graphenewill be limitless.
The unsuccessful attempts of synthesizing monolayer graphite had deterred the real-world application of carbon-based materials, until the experimental discovery of graphene
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by Novoselov and Geim in 2004, which was credited with the Nobel Prize in 2010. Thetechniques have been followed and improved since then, along with efforts to develop newmethods to efficiently synthesize graphene on a large scale. Here I shall briefly introducea few commonly used synthesize methods. The first is exfoliation and cleavage. Sincegraphite has strong covalent bonding with each layer and weak van der Waals force inbetween layers of graphene sheets, it is intuitively attempting to separate the layers bybreaking the van der Waals bonds. Exfoliation and cleavage use mechanical or chemicalenergy to break these bonds. In 2003, Viculis et al. made the first shot on this method. Heused potassium metal to intercalate a graphite sheet and then exfoliated it with ethanol toform dispersion of carbon sheets. However, only 40±15 layers of graphene were producedin each sheet[43]. In the successful attempt by Novoselov et al.[38], a commercially avail-able highly oriented pyrolytic graphite (HOPG) sheet of 1 mm thickness was subjectedto dry etching in oxygen plasma to make many 5 µm deep mesas. This was then puton a photoresist and baked, to stick the mesas to the photoresist. Then, a scotch tapewas used to peel off layers from the graphite sheet. Thin flakes, attached to the pho-toresist, released in acetone and transferred to a Si substrate, were found to have single-to few-layer graphene sheets[44]. A similar approach of exfoliation is the liquid phaseexfoliation. Hernandez et al. proposed to cleave graphite in N-methyl-pyrrolidone andthe monolayer yield can be improved from 1 wt% to 12 wt% with further processing[45].The process of exfoliation, both mechanical and aqueous, has shown great potential inlarge-scale industrial production. The second method is thermal chemical vapour depo-sition (CVD) technique. The first report on few planar few layer graphene synthesizedby CVD was by Somani et al. in 2006[46]. They used a natural, low cost precursor,camphor, to synthesize graphene on Ni foils. Camphor was first evaporated at 180◦C andthen pyrolyzed, in another chamber of the CVD furnace, at 700 to 850◦C, using argonas the carrier gas. Upon cooling to room temperature, few-layer graphene sheets wereobserved on Ni foils[44]. Through thermal CVD, Yu et al. reported three to four layergraphene formation on polycrystalline Ni foils in 2008[47]. A precursor gas mixture ofCH4, H2 and Ar (0.15:1:2 ratio) was adopted for the synthesis process. And Raman spec-troscopic analysis later confirmed the growth of graphene on Ni under moderate coolingrates. Tremendous progress has been made to grow graphene on a large-scale by ther-mal CVD through the understanding of the mechanical process. The achievements ingraphene growth through thermal CVD method have demonstrated reproducibility andgood quality on a centimeter scale substrate undoubtly proved its promising potential.The last but not least method is thermal decomposition of silicon carbide (SiC). Thismeans of growing graphene requires thermal decomposition of Si on the (0001) surface ofsingle crystal 6H-SiC[48]. As the surface of 6H-SiC was heated to approximately 1300◦Cfor a short period of time, usually about 10 minutes, graphene sheets would be formed.The graphene grown by this method typically has 1 to 3 layers (the number of layersdependent on the decomposition temperature) and has fairly high quality since the crys-tallites tend to approach hundreds of micrometers in size[49]. However, the expensivecost of the SiC wafers and the high temperatures limit the usage and application in theindustries. In a further development in this method, a relatively lower temperature ofaround 750◦C[50] is required for the growth of continuous films of graphene synthesized
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on a Ni thin film coated SiC substrate. The advance of this improvement is the growthof continuous graphene film over the entire Ni-coated surface, which paves the way tothe large area production of graphene in an industrial scale. Nevertheless, there is stillno one single method, mature enough for industrial level production of graphene sheetscurrently. Future research needs to focus on the understanding of the growth mechanism,the effect of interface between substrates and graphene, and how to apply the knowledgein manufacturing practical devices.
1.2.2 Transition Metal Dichalcogenides
Many 2D materials exist in bulk form as stacks of strongly bonded layers with weakinterlayer attraction, allowing exfoliation into individual, atomically thin layers. Theform receiving the most attention today is graphene. Other well-known 2D materials aretransition metal oxides including titania- and perovskite-based oxides[51], and grapheneanalogues such as boron nitride (BN). In particular, transition metal dichalcogenides(TMDs) show a wide range of electronic, optical, mechanical, chemical, and thermalproperties[52][53][54] for the use in photovoltaic devices, lithium ion batteries, transistorsphoto-detectors, DNA detection and memory devices[55]. Currently there is a resurgenceof scientific and engineering interest in TMDCs in their atomically thin 2D forms becauseof recent advances in sample preparation, optical detection, transfer and manipulation of2D materials, and physical understanding of 2D materials learned from graphene[56].
Transition metal dichalcogenides have a sandwich structure with a general chemicalformula MX2, where M is the transition metal from group IV (Ti, Zr, Hf etc.), group V(V, Nb, Ta etc.) or group VI (Mo, W, etc.), and X is the chalcogen (S, Se or Te). Thesematerials form layered structure of the arrangement X-M-X, with the chalcogen atoms intwo hexagonal planes separated by a plane of transition metal atoms. Like graphene, theadjacent layers are weakly held together by van der Waals forces, which make exfoliationinto single- or few-layer structure possible by physical or chemical means. The bulk crys-tal can have a variety of polytypes according to the stacking order and transition metalatoms coordination. The overall symmetry of TMDs is hexagonal or rhombohedral, andthe metal atoms have octahedral (1T) or trigonal prismatic (2H) coordination[56]. Dueto quantum confinement and surface effect at lower dimensions, monolayer and few-layerTMDs exhibit a range of interesting properties absent in their bulk counterparts. For ex-ample, bulk semiconducting trigonal prismatic TMDs possess an indirect band gap, buttheir monolayer structures exhibit direct band gaps. This change can be utilized to fieldeffect transistors[57], logic circuits[58] and enhanced photoluminescence applications[59].
The synthesis of 2D TMDs can be generally classified into two categories: top-down(mechanical or chemical exfoliation from bulk materials) and bottom-up (chemical vapourdeposition) approaches. Because these methods are very closely related the synthesis ofgraphene, in fact, many are directly modified from growth of graphene, here I shall onlybriefly discuss these two types of methods. Mechanical exfoliation simply means to peel
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thin films of 2D TMDs from their bulk counterparts by adhesive tapes, like what Novoselovet al. did to obtain graphene. This simple method produces single-crystal flakes up tohigh purity, but it does not allow systematic control of the flakes size and thickness, thusmaking it impossible for large-scale production. To obtain large quantities of exfoliated2D TMD sheets, physicists and engineers use liquid-phase exfoliation method instead.Bulk TMDs are cleaved in liquid by ionic species[60]. The process usually starts by sub-merging bulk TMD powders in a solution containing lithium compound for more than aday to allow lithium ions to intercalate, followed by reacting the intercalated material withwater to release hydrogen gas in the process, so as to effectively separates the layers[61].Such chemical exfoliation method can produce thin films of TMDs up to micrometer insize and the whole process can be better controlled, despite at a relatively high cost. Asdiscussed in the previous section for the growth of graphene, the development of wafer-scale synthesis through chemical vapour deposition (CVD) and epitaxial growth on SiCsubstrate has achieved large-scale production. From these successes, CVD methods arealso adopted for the synthesis of 2D TMDs. For the synthesis of thin film MoS2, there area couple of CVD methods been reported: a solid precursor, such as sulphur powder, beenvaporized and co-deposited onto a substrate[62]; a thin layer of Mo metal deposited ontoa wafer heated with solid sulphur[63]; substrates dip-coated in a solution of (NH4)2MoS2
and heated in the presence of sulphur gas[64]. Although the precise control of the numberof layers over a large area has not been achieved nor the processes are well understood,these CVD methods still demonstrate promising results for the future development.
One of the most promising near-term applications of semiconducting 2D TMDs isin high-performance flexible electronics. Mechanical measurements for single-layer MoS2
showed that it was 30 times as strong as steel and can be deformed up to 11% beforebreaking[65]. Semiconducting 2D TMDs have unique features that make them attractiveas a channel material in field-effect transistor (FET), such as the lack of dangling bonds,structural stability and carrier mobility comparable to silicon[66]. One of the earliestuses of TMDs in FETs was reported in 2004[67], when WSe2 crystals exhibited mobilitycomparable to the best single-crystal silicon FETs, ambipolar and a 104 on/off ratio at atemperature of 60K. However, the room temperature mobility was fairly low (at least 3orders of magnitude lower than that of graphene) in early years of work[68]. It was onlywhen single-layer MoS2 FET was reported with a moderate mobility of 60-70 cm2(Vs)−1
and a large on/off ratio of about 108 at room temperature in 2011 that brought thisfield much attention and excitement. With the initial success of high performance 2DTMD FETs, many research groups have started to consider the possibility of more com-plicated devices based on FETs, such as inverters. An inverter is a device comprising twoFETs in series operated to convert logic 0 with low input voltage to logic 1 with highoutput voltage, and vice versa. After the first invertor based on monolayer MoS2 was re-ported in 2011[58] by Radisavljevic et al., many other groups that followed demonstrateda wide range of possible semiconducting TMDs to be applied as inverters[69][70]. Withits own unique properties, 2D TMDs can also be readily combined with other materi-als to open up new opportunities for improvement in certain technology. For example,Lee et al. has demonstrated a top-gate nonvolatile memory FETs with single- to triple-
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layer MoS2 nanosheets adopting a ferroelectric polymer[71]. In addition, the direct bandgaps, the large excitation binding energies and strong photoluminescence effect observedin 2D TMDs unsurprisingly make them favorable candidates for optoelectronic appli-cations, such as photodetectors[72], photovoltaic cells[73] and light-emitting devices[74].Nonetheless, despite all these possible revolutionary applications on 2D TMDs (as well asgraphene) being presented so far, most of the development is still in the early theoreticalstage. Much more efforts and resources are expectedly to be streamed in this excitingfield to truly make these novel materials commercial products in the imminent future.
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1.3 Concept of Spin-Orbit Coupling
1.3.1 Spin-Orbit Interaction
Spin-orbit coupling (SOC) is the interaction between the spin of the electron and itsangular motion. The effect of SOC causes a lift in the degeneracy of the one-electronenergy levels due to the electromagnetic interaction between the spin and the magneticfield generated by the orbital motion of the electron. The simplest form of Schrodingerequation in solid state physics neglect the spin-dependent term in the Hamiltonian, so theenergies obtained are doubly degenerate spin-up and spin-down states. However, we caninclude SOC term by considering relativistic correction to the Schrodinger equation. Here,we shall discuss briefly the origin of this correction following the approach introduced byJ. Kessler in his book Polarized Electrons [75]. As the starting point, let us derive Diracequation, which includes the electron spin and relativistic behaviour, from the relativisticenergy equation: [76]
H2 = c2p2 +m2c4. (1)
This equation can be re-expressed, by including electric and magnetic potentials φ andA, and substituting p by p− ( ε
c)A and H by H − εφ, where the electric charge e = −ε.
Equation (1) becomes(H − eφ)2 = (cp− εA)2 +m2c4. (2)
Note that p and H are quantum operators with
p = −ih∇ and H = ih∂
∂t. (3)
Without external forces, the wave equation, as follows from Eq.(2), becomes
(H2 − c2∑µ
p2µ −m2c4)ψ = 0 (4)
where pµ is the components of momentum operator. Writing Eq.(4) in the form of
(H − c∑µ
αµpµ − βmc2)(H + c∑µ
αµpµ − βmc2)ψ = 0 (5)
where the coefficients α and β satisfy
{αµ, αµ′} = 2δµµ′
{αµ, β} = 0
β2 = 1
One would realize that the solution to the first part of Eq.(5) is the solution to the wholeequation and it is referred to as Dirac Equation:
(H − c∑µ
αµpµ − βmc2)ψ = 0 (6)
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with the four-component vector ψ.
Comparing Schrodinger equation with Dirac equation in the non-linearized form of
[H − εφ− cα · (p− ε
cA)− βmc2][H − εφ+ cα · (p− ε
cA) + βmc2]ψ = 0 (7)
and consider the approximation that the kinetic and potential energies are small comparedto mc2, two components of the spin function can be neglected, and Eq.(7) takes the form
[1
2m(p− ε
cA)2 + εφ− εh
2mcσ ·B + i
εh
4m2c2E · p− εh
4m2c2σ · (E× p)]ψ = Wψ (8)
where W + mc2 is the total energy. The first two terms in the bracket are equivalentto the terms in Schrodinger equation with the presence of external electric and magneticfields. The third term corresponds to the interaction energy (-µ · B) of the magneticdipole, where the magnetic moment µ = εh
2mcσ = ε
mcs.The fourth term is the relativistic
correction to the energy and the last term describes the spin-orbit interaction of the sys-tem.
If the electric field is spherically symmetric, it can be expressed in the following form:
E = −1
ε
r
r
dV
dr.
Therefore, the spin-orbit interaction can be written as
− εh
4m2c2σ · (E× p) =
ε
2m2c2s · (−1
ε
r
r
dV
dr× p) =
1
2m2c2
1
r
dV
dr(s · l) (9)
where s = h2σ is the spin, and l is the angular momentum.
1.3.2 Time Reversal Symmtery and Spatial Inversion Symmetry
As mentioned earlier, the effect of SOC is to lift the degeneracy of one-electron energylevels by splitting the orbital into spin parallel and spin anti-parallel states. In a freeatom, SOC will always separate the energy levels with the same orbital wave function butwith opposite spins. However, such simple splitting of energy levels may not happen insolids due to the presence of crystal symmetries[76].
Time reversal symmetry preserves the Kramer’s degeneracy, which states that a wavefunction ψ(r,s) only differs from its complex conjugate ψ∗(r,s) by a reversal of wave vectorand electron spin. From here, one arrives at the condition for time reversal symmetry inthe Brillouin zone:
E(k, ↑) = E(−k, ↓) (10)
which simply says that the energy of a state with wave vector k and spin-up at a pointin Brillouin zone is the same as the state with wave vector -k and spin-down at the same
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point.
On the other hand, if the crystal lattice has spatial inversion symmetry (i.e. crystallattice does not change under the operation r 7→-r), the energy of the bands will satisfy
E(k, ↑) = E(−k, ↑) and E(k, ↓) = E(−k, ↓). (11)
If the crystal has both time reversal and spatial inversion symmetry, the band structureof the crystal would satisfy both Eq.(10) and Eq.(11):
E(k, ↑) = E(k, ↓). (12)
Hence, the energy level splitting will not only depend on SOC but also on the sym-metries present in the solids. In other words, the energy bands will stay spin degeneratewhen there is any crystal symmetry even if the spin orbital interaction is considered inthe calculation.
1.3.3 Effect of SOC in Solids
From the previous section, we should understand that the importance of crystal symmetryon the band structures. SOC will only have significant effect on the band structures atcertain special points in the Brillouin zone. Now, let us consider an example of a materialwith cubic lattice structure[76] at the centre of the Brillouin zone – Γ point. For a freeatom without the presence of SOC in a tight binding model, there will be three degeneratep bands at Γ point (see Fig. 1(a)), namely px, py and pz. Each of these p bands can takeup to 2 electrons with the same wave vector but different directions of spin, according toPauli exclusion principle. Therefore, each of them is doubly degenerate and altogetherthere is a six-fold degeneracy at Γ point. When we include SOC, the p bands will split intotwo sub-bands with four-fold and two-fold degeneracies respectively. Now, the behaviourof each p band will differ when we move away from Γ point, depending on the crystalsymmetry. If the structure has spatial inversion symmetry, or if it has an inversion centre,the bands will have to satisfy Eq.(11), so each p band will preserve its spin degeneracy,as illustrated in Figure 1(b). If the structure does not have spatial inversion symmetry,then each p band will further split into two bands, effectively lifting spin degeneracy withthe existence of SOC when we move away from Γ point, as shown in Figure 1(c), and thisis the Rashba splitting.
18
Figure 1: Effect of spin-orbit coupling on the p orbital at Γ point in the Brillouin zone: (a) six-fold degeneracy is observed
at Γ point without SOC; (b) SOC leads to four-fold and two-fold degeneracy at Γ point when the crystal has spatial
inversion symmetry; (c) SOC leads to complete removal of spin degeneracy when the crystal does not pocess spatial
inversion symmetry, except at Γ point.
Notice that the bands are always spin degenerate at Γ point itself no matter if thestructure has spatial inversion symmetry or not. This is because time reversal symmetryis always present at Γ point. Thus, the bands passing through Γ point should alwayssatisfy Equation (10), where energy of the bands of spin-up and spin-down is the same.It is only when we leave Γ point the spin degeneracy is removed by SOC and the absenceof spatial inversion symmetry. Therefore, as we consider a real solid-state material, weneed to take into account of all these factors when we analyze its band structure.
19
1.4 Objectives of the Current Project
Motivated by the fabrication of graphene and other 2D materials as well as the extraordi-nary physical properties for the exciting applications of spintronic devices in the imminentfuture, we are going to investigate the electronic properties of this type of materials underthe effect of spin-orbit coupling and layer dependence in the current project. We aim tounderstand how these two factors influence the behaviour of the materials by comparingthe calculations with and without the presence of SOC and for different number of layers;and hope to discover some interesting features or phenomena that would help to advancethe research in this field eventually.
20
2 Computational Methods
2.1 ab initio Total Energy Calculations
It is known that most physical properties of a system are related to the total energy or tothe difference between the total energies. For example, the equilibrium lattice constantof a crystal and the surfaces or defects of solids in the structure are all determined bythe minimization of the total energies. If the total energy can be calculated, many phys-ical properties, such as bulk modulus, equilibrium lattice constants and phase transitionpressure and temperature, can be found computationally[77][78]. Various methods havebeen developed for calculating a wide range of physical properties of materials. Thesemethods, which require only the specification of the ions/nuclei present in the structure,are usually referred to as ab initio methods[79].
While the numerical solution of Schrodinger Equation is difficult to found, the exactsolution is impossible even for the smallest and simplest systems. Approximations maythen be introduced to reduce the equations to a form that can be solved in polynomialterms, but at the cost of losing some degree of accuracy and predictive power. The treat-ment of electron-electron interactions is the primary source of difficulty – most physicaland chemical properties of a system depend essentially on the strong interactions of theelectrons with each other and with the nuclei. These interactions cannot easily be singledout or treated without approximations. Nevertheless, physicist and mathematicians havedeveloped a series of approximation methods that substantially improve the accuracy andefficiency of the solution to realistic problems. Popular methods are density-functionaltheory (DFT)[80][81] for the electron-electron interaction, pseudopotential theory[82] forthe electron-ion interaction, supercell for the system with periodic geometries, and itera-tive minimization technique[83] for the relaxation of electron coordination.
21
2.2 Approximation Methods
Many physical properties of atoms, molecules and crystals may be understood by solvingthe time-independent Schrodinger equation:
HΨ = EΨ, (13)
where the non-relativistic many-body Hamiltonian can be represented by five items:
H = Tn(rα) + Te(ri) + Ven(ri, rα) + Vnn(rα) + Vee(ri), (14)
corresponding to the kinetic energy of the nuclei, kinetic energy of the electrons, inter-action energy between the electrons and the nuclei, interaction energy among the nucleiand interaction energy among the electrons, respectively. However, solving Eq.(14) for amany-particle system is impossible. Hence, we need to adopt some forms of approxima-tion in the calculations to make them practical problems.
2.2.1 Born-Oppenheimer Approximation
Because nuclei are much more massive than electron, their characteristic velocity is muchslower. The electrons can generally react so quickly to the motion of nuclei so that wecan effectively consider nuclei as stationary particles. Thus, the interaction energy amongnuclei Vnn(rα) will be irrelevant when considering the electron motion. Hence, nuclei canbe treated adiabatically, leading to a separation of electronic and nuclear coordination inthe wave function[84]:
Ψ(ri, rα) = ψ(ri, rα)Φ(rα), (15)
where ψ(ri, rα) and Φ(rα) are wave functions of electrons and nuclei, satisfying Schrodingerequations of the following form:
[− h2
2m
∑i
∇2i −
∑i,α
Zαe2
4πε0|ri − rα|+
∑i,j<i
e2
4πε0|ri − rj|]ψ(ri, rα) = E(rα)ψ(ri, rα) (16)
[−∑β
h2
2mβ
∇2β −
∑β,γ<β
ZβZγe2
4πε0|rβ − rγ|+ E(rα)]Φ(rα) = εΦ(rα), (17)
where indices i, j run over electrons and α, β run over the nuclei. Z is the nuclear number,mβ is the mass of the nuclei, and ε is the eigenvalue of the nuclei wave function.
In this approximation, the kinetic energy of the nuclei is neglected and the interactionenergy among the nuclei can be handled classically. Thus, the many-body problem is re-duced to one regarding a system of interacting electrons moving in an external potentialformed by the frozen-in nuclei configuration. However, due to the electron-electron inter-action, we still cannot solve the equations easily. Further approximations have to be made.
22
2.2.2 Hartree-Fock Approximation
The approximation proposed by Hartree and later improved by Fock was based on thefundamental concept of variational method. Hartree suggested a variational calculation inwhich the wave function is approximated by the product of one-electron wave functions,and the energy of the form < Ψ|H|Ψ > / < Ψ|Ψ > is minimized. This variationalprocedure leads directly to the Hartree equation from which the one-electron functionsthat minimize the energy can be determined:[85]
[− h2
2m∇2 + V (r) +
∑j 6=i
e2∫ ψ?j (r
′)ψj(r′)drdr′
|r− r′|]ψi(r) = εiψi(r). (18)
The summation term represents the potential due to the other electrons in which ith
electron moves around (screening effect) and εi is the energy contributed from the ith
electron. The total energy is then found to be
< Ψ|H|Ψ >
< Ψ|Ψ >=
∑i
εi −1
2
∑i,j
e2∫ ψ?j (r
′)ψj(r′)ψ?i (r)ψi(r)drdr′
|r− r′|. (19)
We can solve these equations self-consistently by iteration. Assuming a particular setof approximated eigenstate, we can compute the potential and using this potential tore-calculate the eigenstate. The new eigenstate can be substituted back to obtain a newpotential, which can then be used to calculate another set of eigenstate. This process willrepeat itself until the convergence is reached with a satisfactory level of accuracy.
Nonetheless, the Hartree method contains one serious fault. The wave function of amany-electron system must be anti-symmetric under the exchange of any two electrons.This anti-symmetrization of the wave function produces a spatial separation betweenelectrons with the same spin, thus reducing the Coulomb interaction of the system. Thisdecrease in energy due to anti-symmetric wave function is called the exchange energy.Therefore, the Hartree-Fock approximation is generally referred to as the calculation oftotal energy including this exchange interaction. This approach starts with a determi-nantal wave function satisfying Pauli exclusion principle:
ΨHF =1√N !det[ψ1(r1, σ1)ψ2(r2, σ2)...ψN(rN , σN)], (20)
where (r,σ) denote the position and spin coordinates. Performing variational method onEq.(20) leads to the Hartree-Fock equation of optimized one-electron functions:[86]
[− h2
2m∇2+V (r)+
∑j 6=i
e2∫ ψ?j (r
′)ψj(r′)dσ′dr′
|r− r′|]ψi(r)−
∑j 6=i
e2ψi(r)∫ ψ?j (r
′)ψi(r′)dσ′dr′
|r− r′|= εiψi(r)
(21)The extra term in this equation is the exchange interaction as distinguished from thedirect interaction is the Hartree equation and the difference between the energy found
23
in a many-body system and the energy calculated from Hartree-Fock method is calledcorrelation energy[87].
Despite being a powerful tool applied in self-consistent calculation, the energy calcu-lated by solving Hartree-Fock equation does not agree with that obtained from solvingmany-body Schrodinger equation under Born-Oppenheimer approximation. It turns outthat the correlation energy of a complex system is very challenging to compute. Thus,overcoming these difficulties to some extent is the driving force to develop alternativemethods to describe the electron-electron interaction more precisely, so as to solve theproblem in a more practical way.
2.2.3 Density Functional Theory
Density functional theory (DFT) takes a radically different approach than the methodsdiscussed previously. It is both a profound theory for interacting electrons and a practicalprescription of calculation in terms of single-electron equation. Its contribution in boththese respects received the highest recognition with the award of Nobel Prize for chem-istry in 1998 to Walter Kohn and John Pople. Since its formulation in the mid 1960sand early 1970s, DFT has been used extensively in condensed matter physics in almostall band structure and electronic structure calculations. It is based on the following totheorems in terms of the electron density function ρ(r):[80][81]
Theorem I.If the number of electrons in the system is conserved, the external potential V(r) uniquelydetermines the ground state density ρ0(r).
Theorem II.There exists a universal functional of ρ, E[ρ], which is minimized by the ground statedensity ρ0(r).
Kohn and Sham carried these theorems further and obtained a single-particle Schrodingerequation– Kohn-Sham equation as follow:
[− h2
2m∇2 + e2
∫ ρ(r′)d3r′
|r− r′|+δExc[ρ(r)]
δρ(r)+ Vext]Ψi(r) = εiΨi(r), (22)
where the four terms on the left hand side of the equation correspond to the kinetic energy,Hartree-Fock potential, exchange-correlation potential and the external ionic potential,respectively. Kohn-Sham equation maps a many-particle interacting system into a systemof non-interacting electrons moving in an effective potential formed by the nuclei and otherelectrons. If the exchange-correlation energy functional were known exactly, taking thefunctional derivative with respect to the density would produce an exchange-correlationpotential that includes the effects of both exchange and correlation. The standard proce-dure for solving Kohn-Sham equation is through iteration until self-consistency is attained.
24
Starting from an assumed density ρ(r), the Coulomb and exchange-correlation potentialsare calculated; then we can solve Kohn-Sham equation to obtain the wave functions. Withthe new wave functions, a new density can be constructed by
ρ(r) =∑i
|Ψi(r)|2, (23)
wher i goes over all the occupied states. This procedure is repeated until the consis-tent input and output density is achieved. DFT provides a strong foundation for simplemodels of the many-body effects that dominate the computation of wave function-basedelectronic structures.
2.2.4 Exchange-Correlation Functional Approximation
Comparing with the many-body Schrodinger equation, solving Kohn-Sham equation ismuch easier for a real system. In spite of all the items are very well treated in Kohn-Sham equation, the exchange-correlation Exc [ρ(r)] is unknown. The search for an accurateExc [ρ(r)] has proven to be extremely difficult, thus various approximations have to bemade accordingly. Among the approximations, local density approximation (LDA)[81] isa simple and powerful tool to simplify the exchange-correlation functional. It is based onthe assumption that for a system with slowly varying density, the electron density in asmall region near point r can be treated as if it is homogeneous. Therefore, under LDA,Exc [ρ(r)] is given by
Exc[ρ(r)] ≈ ELDAxc [ρ(r)] =
∫ρ(r)εxc(r)dr (24)
where εxc(r) = εhomoxc [ρ(r)] is the exchange-correlation energy per particle of a homoge-neous system of density ρ(x).
Despite its simplicity, the LDA is remarkably successful in predicting the structuraland electronic properties of real materials. On the other hand, there are also several prob-lems associated with LDA. For example, LDA does not provide a good description on theexcited states, thus underestimating the band gap of semiconductors and insulators; it isalso not a suitable approximation to be used in structures with van der Waals interactions.
An improvement for LDA is the generalized gradient approximation (GGA)[88], inwhich the exchange-correlation functional is treated not just by the functional of theelectron density but also by the gradient of the electron density in the following form
EGGAxc [ρ(r)] =
∫ρ(r)εxc(ρ,∇ρ)dr. (25)
GGA can correct some problems associated with LDA, such as overbinding tendency andthe wrong prediction of non-magnetic ground state of Fe. However, the strong correlatedsystems as well as the van der Waals interactions still cannot be handled well by GGA.
25
Nonetheless, both LDA and GGA provide fairly good approximations to be effectivelyand efficiently applied in diverse calculations.
LDA and GGA are semi-local functionals, whose exchange-correlation energy densityat r depends only on the density and Kohn Sham orbitals in an infinitesimal region aroundr. Simple semi-local density functionals can accurately model many ground state prop-erties including lattice parameters and bulk moduli[89]. However, they have an intrinsicself-interaction error[90], of which the semi-local exchange-correlation functionals allowelectrons to interact with themselves. This serious error would inevitably lead to inaccu-racies in the prediction of energies of many-body system. Hence, more advanced function-als were meant to be developed; and one of the most popular choices today is the hybridexchange-correlation functionals, which incorporates a fraction of the exact Hartree-Fockexchange correlation in addition to the original LDA or GGA functionals. To further ex-tend the success of this idea, Heyd, Scuseria and Ernzerhof proposed the screened hybridfunctional that avoids the problematic effects of long-range Hartree-Fock exchange; andthis is know as Heyd-Scuseria-Ernzerhof (HSE) functional[91]. HSE is based on the PBEhglobal hybrid[92] of the Perdew-Burke-Ernzerhof GGA and it includes 25% short-rangeHartree-Fock exchange and no long-range Hartree-Fock exchange[93]. This has dramati-cally improved the many molecular properties relative to semi-local functionals. Perhapsthe most significant achievement of HSE correction is the precise treatment of semicon-ductor band gaps. An extensive study of 40 systems (35 semiconductors and 5 insulators)was performed to compare the experimental band gaps with the calculated values fromHSE and semi-local functionals[94]. And the results indicate the HSE is much more ac-curate among all the semi-local functionals. Besides the correction to band gap, otherstudies revealed that HSE also provides the improved lattice constants[95] and more ac-curate treatment on semiconductor defects[96]. But the computational issues inherent tothe range-separated hybrids remain unresolved. Despite tremendous efforts towards thecomputationally efficient implementations, HSE is still much more computationally ex-pensive than the simpler semi-local functionals. Nevertheless, it is still a rather accurateand powerful tool for theoretical calculations.
26
In this project, all the electronic properties of the materials obtained are based on firstprinciple calculations within density functional theory (DFT) as implemented in VASP[97](Vienna ab initio simulation package) codes. The ion-electron interaction was modeledby projector augmented plane wave (PAW)[98] and the electron exchange correlation wastreated by Perdew-Burke-Ernzerhof (PBE) parameterization of the generalized gradientapproximation. The plane wave basis set with kinetic energy cutoff of 500 eV was usedand a vacuum layer of 15A was adopted to avoid interactions between the neighboringsurfaces for 2D structures. The Brillouin zone was sampled by a 8×8×3 Γ-centred k-pointmesh for bulk and a 8×8×1 mesh for monolayer and few-layer structures.
27
3 Results and Discussion
3.1 SnSe2
3.1.1 Effect of SOC and Layer Dependence Study
Due to its brilliant optical and electrical properties, tin diselenide (SnSe2) has been widelystudied in recent years in the hope to achieve applications in lithium ion batteries[99],photovoltaic devices[100], and solar cells[101]etc. Therefore, many groups have also at-tempted to synthesize SnSe2 in the laboratory by organic solution method[102], directvapour transport technique[103], photoelectrochemical deposition method[104] and chem-ical bath deposition method[105]. SnSe2, as a n-type semiconductor, has a hexagonal crys-tal structure of the type CdI2 and a symmetry group of D3
3d[106]. One layer of tin atom issandwiched between two layers of selenide atoms such that each tin atom is octahedrallybounded by six selenium atoms (1T polytype with P3m1 space-group), forming a 2D Se-Sn-Se layered structure. Within the layers are the strong covalent bonds and in betweenthe layers are the weak van der Waals forces, holding the structure just like graphene.There are three atoms per unit cell, which has lattice constants of a = b = 3.811A andc = 6.141A[107]. The bond length between Sn and Se atom is 2.680A and between Sn andSn atom is 3.785A(See Fig.2). Moreover, the inversion centre, located at the geometricalcentre of the unit cell, indicated by the red dot in Fig.2 (b), guarantees the energy bandssatisfy Equation (11) due to spatial inversion symmetry.
28
Z
3.173Å
(a) (b)
(c) (d)
Figure 2: Structure information of SnSe2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of SnSe2 with the dark
blue atom being tin and the yellow atom being selenium; the red dot at the centre of the unit cell indicates the inversion
centre; (c) top-view of SnSe2 supercell; (d) side-view of SnSe2 supercell with the separation of two layers being 3.173A in
the z-axis.
There has been a contradiction concerning the fundamental energy band gap when theoptical property of SnSe2 was studied. The most frequently reported band gap in the lit-eratures are approximately 0.95eV for indirect forbidden transitions[108][109], while somegroups predicted a slightly lower value of about 0.81eV using empirical pseudopotentialmethod[110]. Direct forbidden transitions were also observed with band gap values tobe 1.59-1.78eV[110][111]. From our first-principle calculation, under PBE form of gener-alized gradient approximation, both bulk and monolayer SnSe2 have indirect band gapsof 0.736eV and 0.971eV, respectively. As depicted in Figure 3 (a), the conduction bandminimum is located in between M and L points and valence band maximum is in betweenK and Γ points for bulk SnSe2 band structure; while in Figure 3 (c), the conduction bandminimum is at M point, and valence band maximum is in between Γ and M points formonolayer SnSe2 band structure. Due to the presence of spatial inversion symmetry inboth bulk and monolayer SnSe2, all the bands are found to be spin degenerate. WhenSOC is turned on, we can see a significant splitting at the top of valence bands for bothcases. In Figure 3 (b), we can observe a strong splitting for the top two valence bandsat high-symmetry points of A and H with splitting energy of about 272meV and 90meV,respectively. Similarly, the same strong splitting is also seen in monolayer band structure(Fig.3 (d)) at Γ and K points with splitting energy of 270meV and 71meV. However, all
29
these bands are still spin degenerate as a result of crystal symmetries in the structure.Comparing the results with and without SOC in SnSe2, it is realized the effect of SOCis mainly concentrated at the top valence band but not at the bottom conduction bandsnear Fermi level, which is indicated by the dotted line at 0eV; and it does not change theshape of the bands nor the band gaps of the material very much.
1 Γ M K Γ
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC
Eg = 0.971eV
Eg = 0.710eV Eg = 0.736eV
Eg = 0.969eV
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a) (b)
(c) (d)
Ene
rgy
(eV
)
Ene
rgy
(eV
)
Ene
rgy
(eV
) Γ A H K Γ M L H A Γ
Γ M K Γ
Figure 3: Band structures of SnSe2: (a) band structure of bulk SnSe2 excluding SOC with an indirect band gap of
0.736eV; (b) band structure of bulk SnSe2 including SOC with an indirect band gap of 0.710eV; (c) band structure of
monolayer SnSe2 excluding SOC with an indirect band gap of 0.971eV; (d) band structure of monolayer SnSe2 including
SOC with an indirect band gap of 0.969eV.
To further analyze the band structures, we have also studied the atomic contributionsto the bands for bulk (Fig.4 (a) and (b)) and monolayer (Fig.4 (c) and (d)) SnSe2 withSOC. As shown in Figure 4, the blue and magenta coloured lines indicate the atomiccontribution from Sn and Se atom, respectively. The density of the colour signifies thelevel of contribution from a particular atom. For example, the magenta coloured lines aredominant at the top few valence bands in both bulk and single-layer cases, which implies
30
that these bands have more atomic contribution from selenium. The minimum conduc-tion bands are more or less equally contributed by both atoms as the colour densities areapproximately equivalent.
1
Se Sn
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
Figure 4: Atomic contribution of the band structures of SnSe2 including the effect of SOC with the blue and magenta
coloured lines correspond to the contribution from Sn and Se atoms, respectively: (a) Sn contribution to bulk SnSe2; (b)
Se contribution to bulk SnSe2; (c) Sn contribution to monolayer SnSe2; (d) Se contribution to monolayer SnSe2.
Following the idea of manipulating the band structures of such 2D materials and theexperimental success to synthesize not only monolayer structure but also few-layer struc-tures of these TMDs, we then investigated the effect of number of layers on the bandstructure of 2D SnSe2. From the experimental lattice parameters used to obtain mono-layer SnSe2, without structure relaxation, we have created bi-layer, 3-layer and 4-layermodels in MaterialStudio for our calculation. The results are shown in Figure 5. Asmentioned above, SnSe2 possesses an inversion centre in monolayer structure, and thisinversion symmetry is also present in the structures of higher number of layers as wellas in bulk. Hence, all the bands are doubly spin degenerate. From Figure 5, when the
31
number of layer is increased, the number of bands increases accordingly with each bandcomprised of a linear combination of states from all the layers. Larger the separationamong the bands indicates stronger interaction at a particular point. Thus, clearly theinteraction at K is much stronger than the interaction at M point shown by the relativeseparations in between the bands. Moreover, although the shape of the band structuresdoes not change much, the band gaps are becoming smaller and smaller. Excluding otherpossible factors and considering only the layer dependence alone, this provides us with asimple practical tool to engineer the band gap of 2D SnSe2.
−3
−2
−1
0
1
2
3
Γ M K Γ
Ene
rgy
(eV
)�
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
Γ M K Γ−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�Γ M K Γ
(a) (b) (c)
Figure 5: Band structure of 2-dimensional SnSe2: (a) band structure of bi-layer SnSe2; (b) band structure of 3-layer
SnSe2; (c) band structure of 4-layer SnSe2.
3.1.2 HSE Band Structure Correction
In addition, taking into account of the experimental evidence of a larger band gap valuefor bulk SnSe2 and the limitations of our theoretical calculation, we have also made fur-ther corrections to our previous calculation for the band structure. As briefly discussedin the Computational Methods section, semi-local density based approximations for theexchange-correlation functionals often underestimate the band gaps of semiconductorsand insulators. Thus, the optical, electrical and even magnetic properties predicted fromthe band structures are less accurate compared with the experimental results. Therefore,it is imperative that we improve our results by taking extra steps in the calculation, andHSE correction is one of the popular choices because of its high accuracy and relativelylow computational cost. Figure 6 demonstrates the result of bulk SnSe2 band structure,with and without SOC, after HSE correction. Evidently, the indirect band gap of thestructure has been increased significantly to 1.457eV and 1.419eV shown in Figure 6 (a)and (b). Comparing the diagrams, we can see the same effect of SOC to split bands atcertain high symmetry points as before. Our results have again proved the power of HSE
32
method to improve band structure of a semiconductor.
Γ A H K Γ M L H A Γ�−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
Γ A H K Γ M L H A Γ�E
nerg
y (e
V)�
Γ A H K Γ M L H A Γ�
(a) (b)
Eg = 1.457eV Eg = 1.419eV
Figure 6: Band structure of bulk SnSe2 with HSE correction: (a) band structure of bulk SnSe2 with HSE correction
excluding SOC; (b) band structure of bulk SnSe2 with HSE correction including SOC.
33
3.2 PbI2
3.2.1 Effect of SOC and Layer Dependence Study
Organic-inorganic hybrid perovskite materials have been given more and more attentionsin recent years due to its potential application in solid-state chemistry, planar-structuredevices and optoelectronic devices[112], such as high power light emitting diodes[113] andsolar cells[112]. The advantages of this type of materials include high efficiency, rela-tively low cost of production and easier to fabricate[114][115]. Orgnic-lead iodide (PbI2)perovskite such as CH3NH3PbI3 has been applied in solid-state mesoscopic solar cellswhere they were found to act not only as a light harvester, but also as an electron andhole conductor[116]. Although many research interests have been concentrated on thephysical properties of low-dimensional lead iodide compounds, its inorganic counterpartPbI2 has yet been given enough attention. It is known that PbI2 is a semiconductorwith one atomic layer of Pb sandwiched by two layers of I atoms, forming a I-Pb-I TMDstructure. Just like SnSe2 studied previously, the interlayer interaction is the weak vander Waals forces while the intralayer interaction is the strong covalent forces. Because ofthe weak interaction between the atomic sheets, we can form a variety of PbI2 structuresaccording to different ways of stacking orders. The most common structure of PbI2 isthe 2H polytype, where one lead atom is trigonal-prismatically coordinated by six io-dide atoms with a P6m2 space group[117]. The lattice constants are a = b = 4.648A,c = 7.119A and the bond length of Pb-I is 3.293A. (See Fig.7) Early theoretical[118]and experimental[119][120] studied have revealed that bulk PbI2 has a direct band gapwith conduction band minimum and valence band maximum occurring at A point inthe Brillouin zone. Applications of inorganic PbI2 comprise of room-temperature pho-tocells, X-ray and γ-ray detectors[121] and it can be used as a colloidal nanoparticlesuspension[122] and for making quantum dots in solution[123] etc.
34
(a)
(c) (d)
(b)
Z
3.303Å
Figure 7: Structure information of PbI2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of PbI2 with the red
atom being lead and the dark blue atom being iodine; the red dot at the centre of the unit cell indicates the inversion
centre; (c) top-view of PbI2 supercell; (d) side-view of PbI2 supercell with the separation of two layers being 3.303A in the
z-axis.
From our calculation, it is realized that bulk PbI2 is a semiconductor with a directband gap of 2.234eV at A point. Monolayer PbI2 is also a semiconductor with an indirectband gap of 2.555eV for the conduction band minimum is at Γ point and valence bandmaximum is in between Γ and K points, shown in Figure 8(a) and (c). Because of thespatial inversion symmetry (inversion centre is indicated by the red dot in Fig. 7(b))present in both even number of layered and odd number of layered PbI2, all the bandsare doubly spin degenerate. Again when we turned on SOC, we can observe a strongsplitting in the band structures. From Figure 8(b), the splitting energy for the bottomtwo conduction bands at A is 0.469eV for bulk PbI2. The same situation is also witnessedin monolayer PbI2 (Fig. 8(d)) with strong splitting energy of 0.488eV at Γ and 0.899eVat K point for the bottom two conduction bands. Unlike the case in SnSe2, the significanteffect of SOC in PbI2 is mainly focused on the bottom conduction bands and the bandgaps are considerably changed by 32.8% for bulk and 27.4% for monolayer. Hence, fromthis initial analysis, we can see that spin-orbit interaction is surprisingly an effective wayfor band gap engineering in PbI2.
35
1
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC Eg = 2.234eV
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
Eg = 1.501eV
(b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
Γ M K Γ
Eg = 2.555eV
(c)
Ene
rgy
(eV
)
Eg = 1.855eV
(d)
Ene
rgy
(eV
)
Γ M K Γ
Figure 8: Band structures of PbI2: (a) band structure of bulk PbI2 excluding SOC with a direct band gap of 2.234V; (b)
band structure of bulk PbI2 including SOC with a direct band gap of 1.501eV; (c) band structure of monolayer PbI2
excluding SOC with an indirect band gap of 2.555eV; (d) band structure of monolayer PbI2 including SOC with an
indirect band gap of 1.855eV.
Figure 9 presents the results for atomic contribution investigation of bulk (Fig.9(a) and(b)) and monolayer (Fig.9(c) and (d)) PbI2 with SOC. The blue and magenta colouredlines indicate the atomic contribution from Pb and I atom, respectively. Clearly from thegraphs, we can see the bottom few conduction bands are predominantly contributed bylead atom and the top few valence bands by iodine atom for both bulk and monolayerPbI2, implied from the density of the coloured lines.
36
1
I Pb −3
−2
−1
0
1
2
3
4
5E
nerg
y (e
V)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3
4
5(b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
−3
−2
−1
0
1
2
3
4
5
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3
4
5(d)
Ene
rgy
(eV
)
Γ M K Γ
Figure 9: Atomic contribution of the band structures of PbI2 including the effect of SOC with the blue and magenta
coloured lines correspond to the contribution from Pb and I atoms, respectively: (a) Pb contribution to bulk PbI2; (b) I
contribution to bulk PbI2; (c) Pb contribution to monolayer PbI2; (d) I contribution to monolayer PbI2.
Subsequently, we studied the layer dependence of the band structure. Similar to theresult of SnSe2, from the band diagrams of bi-layer, 3-layer and 4-layer PbI2 in Figure10, the number of bands increases linearly with increasing number of layers. However,the band gaps seem to have a weak relation with the number of layers, as the band gapof 4-layer PbI2 is 1.794eV, only 3.3% different from the band gap of monolayer PbI2. Inaddition, due to inversion centres are found in both odd number of layered and even num-ber of layered PbI2, by changing the number of layers will not remove spatial inversionsymmetry, thus all the bands are still doubly spin degenerate.
37
−3
−2
−1
0
1
2
3
4
5
Γ M K Γ
Ene
rgy
(eV
)�
(a)
−3
−2
−1
0
1
2
3
4
5
Ene
rgy
(eV
)�Γ M K Γ
(b)
−3
−2
−1
0
1
2
3
4
5
Ene
rgy
(eV
)�
Γ M K Γ
(c)
Figure 10: Band structure of 2-dimensional PbI2: (a) band structure of bi-layer PbI2; (b) band structure of 3-layer PbI2;
(c) band structure of 4-layer PbI2.
3.2.2 Effect of Substrates on Monolayer PbI2
Due the potential in high-performance optoelectronic applications, the synthesis of organometalperovskite materials on a large scale has always been a key focus in this field. By far,most of the van der Waals solids were fabricated using inorganic 2D crystals, such asgraphene and BN, as building blocks in popular synthesis methods. In this section, weprovide a preliminary theoretical calculation on the effect of 2D PbI2 on three favourablevan der Waals solids–BN, Graphene and MoS2–as substrates. In any theoretical studyon the possibility of experimental growth of a material on a substrate, one has to resolvethe problem related to the lattice mismatch between the material and the substrate first.Here, taking BN as an example, whose lattice constants are a=b=2.510A and c=6.690A(Fig.11). The lattice mismatch is calculated by the following formula:
|m(aPbI2)− n(aBN))
aPbI2| × 100%, (26)
where m and n are real numbers.
38
(b) (a)
Figure 11: Structure information of PbI2 and BN: (a) unit cell of PbI2 with the red atom being lead and the dark blue
atom being iodine; (b) unit cell of BN with the light blue atom being boron and yellow atom being nitrogen.
Thus, to minimize the lattice mismatch is equivalent to find the most suitable pairof m and n values. After comparing the lattice constants of BN and PbI2, we found theminimum lattice mismatch possible is 4.2% with the choice of m=1 and n=
√3. This is
equivalently stretching BN lattice by√
3 times. Nevertheless, from Figure 12, the bandstructure before and after scaling does not indicate noticeable changes but only with adecrease of direct band gap energy at K by 6.6%.
Γ A H K Γ M L H A Γ�
Ene
rgy
(eV
)�
(a)
Eg = 4.681eV
Ene
rgy
(eV
)�
(b)
Eg = 4.371eV
Γ M K!!!!!!!!!!!!!!!!!!!!!!Γ Γ M K!!!!!!!!!!!!!!!!!!!!!!Γ
Figure 12: Band structures of BN: (a) band structure of normal BN with a direct band gap of 4.618eV; (b) band structure
of stretched BN by 4.2% with a direct band gap of 4.371eV.
Following the stretching of BN lattice constants to match that of PbI2, one needs todecide how to position PbI2 on the substrate BN. The four possible ways – Pb on B site, Pbon N site, bridge site and hollow site – are displayed in Figure 13 below. It is recognized,
39
from the band structure of the composite system (see Fig.14), that all the structures havedirect band gaps of around 2.5eV at Γ and the most stable site is bridge site, of whichPb atom is located in between B and N atoms. Setting the structure with the highestenergy as 0eV, the bridge site has a total energy of -1.1meV, making it theoretically themost stable structure out of the four possibilities.
40
Figure 13: Different positions of PbI2 on substrate BN: (a) Pb on B atom top view; (b) Pb on B atom side view; (c) Pb
on N atom top view; (d) Pb on N atom side view; (e) bridge site (B-Pb-N) top view; (f) bridge site (B-Pb-N) side view;
(g) hollow site (Pb in the centre of BN hexagonal ring) top view; (h) hollow site (Pb in the centre of BN hexagonal ring)
side view.
1
Hollow Bridge
Pb-B Pb-N
Eg = 2.547eV E = -0.6meV
Ene
rgy
(eV
)
(a) Eg = 2.548eV E = 0.0meV
(b)
Ene
rgy
(eV
)
Γ M K Γ
Eg = 2.547eV E = -1.1meV
(c)
Ene
rgy
(eV
)
Eg = 2.549eV E = -0.9meV
(d) E
nerg
y (e
V)
Γ M K Γ
Γ M K Γ Γ M K Γ
Figure 14: Band structures of 2-dimensional PbI2 on monolayer BN with different positions: (a) band structure for site
Pb on B atom with a direct band gap at Γ of 2.547eV and total energy 0.6meV less than the larges energy; (b) band
structure for site Pb on N atom with a direct band gap at Γ of 2.548eV and total energy equal to the larges energy, which
is set to be 0.0eV; (c) band structure for bridge site with a direct band gap at Γ of 2.547eV and total energy 1.1meV less
than the larges energy; (d) band structure for hollow site with a direct band gap at Γ of 2.549eV and total energy 0.9meV
less than the larges energy.
In addition, we have also investigated the effect of SOC in the bridge-site structure.It is noted that the SOC introduced a very significant splitting in the bottom few con-duction bands (see Fig.15), effectively reducing the band gap by 24.8%. This impliesthat the structure is very susceptible to the effect of spin-orbit interaction. Moreover,the atomic contribution for the bottom few conduction bands and top few valence bandsmainly comes from Pb atom and I atom, respectively indicated by the red and dark blue
41
coloured lines in Figure 16. Although all these results are theoretical within DFT, theystill can be used as valuable reference data for the real synthesis of PbI2 on van-der-Waals-solid substrates in the experiment.
The results for the calculations of PbI2 on graphene and MoS2 substrates are shownin Appendices section II.
−3
−2
−1
0
1
2
3
4
Ene
rgy
(eV
)�
(a)
Eg = 2.547eV
−3
−2
−1
0
1
2
3
4
Ene
rgy
(eV
)�
(b)
Eg = 1.916eV
Γ !M K Γ Γ !M K Γ
Figure 15: Band structure for bridge site without SOC and with SOC: (a) band structure for bridge site excluding SOC
with a direct band gap at Γ of 2.547eV; (b) band structure for bridge site including SOC with a direct band gap at Γ of
1.916eV.
42
1
N B
Ene
rgy
(eV
) (a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
) I Pb
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
Γ M K Γ Γ M K Γ
Figure 16: Atomic contribution to the band structures of PbI2 on BN substrate with the green, yellow, red and dark blue
coloured lines correspond to the contribution from B, N, Pb and I atoms, respectively: (a) B contribution to the
composite structure of PbI2 on BN; (b) N contribution to the composite structure of PbI2 on BN; (c) Pb contribution to
the composite structure of PbI2 on BN; (d) I contribution to the composite structure of PbI2 on BN.
43
3.3 NbSe2
3.3.1 Effect of SOC and Layer Dependence Study
Although TMDs are categorized as one group of material based on their structural simi-larities, they can be further divided into more classes depending on their various electronicnatures. For example, titanium selenide (TiSe) and hafnium telluride (HfTe) are semi-conducting; vanadium disulfide (VS2) and tantalum diselenide (TaSe2) are semimetalsor narrow band metals; and platinum ditelluride (PtTe2) is metallic.[124] Among them,niobium dichalcogenides attracted much attention. Apart from being metallic at roomtemperature, they also exhibit superconducting behaviour at low temperature, discov-ered by Maaren et al.[125] in the 1960s. As the temperature goes below a critical valueof around 7.2K, niobium diselenide (NbSe2) of 2H polytype would become a type IIsuperconductor[126]. Moreover, high quality 2H-NbSe2 samples can even have supercon-ductivity coexists with a charge density wave, although the nature of the competitionbetween these two states has not been well understood yet[127]. The superconductivityin correlation to other physical properties of NbSe2 was studied in the past few decadeswith the hope to integrate this exciting feature into room-temperature functioning de-vices. Frindt et al.[128] mechanically exfoliated a few molecular layers of 2H-NbSe2 flakesin the laboratory and observed a decrease in the critical temperature with decreasing flakethickness. Novoselov et al.[68] reported measurements of field effect transistors producedfrom single molecular layer of NbSe2 and observed a semi-metallic behaviour at 300K.Later, Staley et al.[129] studied thin NbSe2 flakes in a two-terminal field effect transistorsfabricated by mechanical exfoliation onto Si/SiO2 substrates and they realized supercon-ducting features in the devices, which were about two molecular layers thick with a criticaltemperature as high as 2.5K. In addition to the studies on superconductivity, many othergroups focused on the modification of 2H-NbSe2 properties through a variety of ways.Particularly popular is chemical doping: V[130], Cr[131], Ti[132], Fe[133], Cu[134] andGe[135] doped 2H-NbSe2 were reported with enhanced electronic, magnetic, lubricationand transport properties.
Each niobium atom is trigonal-prismatically coordinated with six selenium atoms,forming a hexagonally packed Se-Nb-Se layer with strong covalent bonds. (see Fig.17)The lattice constants are a = b = 3.440A and c = 12.482A[136] and the bond lengthbetween Nb and Se atom is 2.619A. There is no inversion centre for single layer NbSe2,thus the odd number of layered NbSe2 does not need to obey Equation (11) for spatialinversion symmetry. However, spatial inversion symmetry will be restored for bi-layer andthus even number of layered NbSe2 with the introduction of inversion centres. (Bi-layerinversion centre is shown in Fig.17 (b) by the orange dot.) This unique structure, differentfrom the previously presented materials, has a distinct feature that will be discussed inthe following section.
44
(a) (b)
(c)
Z
2.964Å
(d)
Figure 17: Structure information of NbSe2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of NbSe2 with the
blue atom being niobium and the dark red atom being selenium; the orange dot at the centre of the unit cell indicates the
inversion centre; (c) top-view of NbSe2 supercell; (d) side-view of NbSe2 supercell with the separation of two layers being
2.964A in the z-axis.
Our investigation on the band structures (see Fig.18) unambiguously proves the metal-lic nature of NbSe2, indicated by the several bands (Fig.18 (a)) for bulk and one band(Fig.18 (c)) for monolayer crossing Fermi level, which is calibrated to 0eV. This metal-lic character is also revealed in its density of states (DOS) plot (Fig.19), which showsa highly localized state cutting through Fermi level. Partial DOS analysis reveals thatthis electronic state at Fermi level primarily comes from Nb 5d and Se 4p orbitals, whichare hybridized nearly in the entire energy range, indicating a strong covalency betweenthese two atoms. Now when SOC is introduced, we can observe a distinct and robustspin splitting at K point (Fig.18 (d)) for monolayer NbSe2. Due to both the breaking ofspatial inversion symmetry in odd number of layered structures and the presence of SOC,spin degeneracy at K point is lifted with a large splitting energy about 140meV. FromFigure 18 (d), it is seen that the spin splitting only occurs at K point, but not at M and Γpoints. This is because M and Γ are also time reversal invariant points in hexagonal latticestructure; thus, Equation (10) prevents the energy splitting at these two points. However,K is not a time reversal invariant point; by introducing SOC all the degeneracies imposedby the structural symmetries are lifted at this point. As 2-dimensional NbSe2 turns intoa 3-dimensional structure, spatial inversion symmetry would be restored. Therefore, allthe bands in bulk band structures (Fig.18 (a) and (b)) are doubly spin degenerate.
45
1
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
Γ A H K Γ M L H A Γ
Figure 18: Band structures of NbSe2: (a) band structure of bulk NbSe2 excluding SOC; (b) band structure of bulk NbSe2
including SOC; (c) band structure of monolayer NbSe2 excluding SOC; (d) band structure of monolayer NbSe2 including
SOC.
46
- 8
- 4
0
4
8
- 4 - 3 - 2 - 1 0 1 2 3 4
- 2
0
2
- 2
0
2
Densi
ty of
States
(Stat
e/eV)
S e p
E n e r g y ( e V )
t o t a l
N b d
Figure 19: Density of States for 2D NbSe2: the plots from the top are total DOS of NbSe2 (black colour line), d orbital
PDOS of Nb atom (red colour line) and p orbital PDOS of Se atom (green colour line), respectively. Fermi level is aligned
to 0eV, indicated by the blue dashed line.
Figure 20 presents the results for atomic contribution of bulk (Fig.20 (a) and (b)) andmonolayer (Fig.20 (c) and (d)) NbSe2 with SOC. The blue and magenta coloured linesindicate the atomic contribution from Nb and Se atom, respectively. Clearly, the bandscrossing Fermi level are more significantly contributed by Nb atom implied from the den-sity of the coloured lines. This again confirms the result of PDOS plot, that niobiumatom has a larger peak at Fermi level.
47
1
Se Nb −3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
Γ A H K Γ M L H A Γ
Figure 20: Atomic contribution of the band structures of NbSe2 including the effect of SOC with the blue and magenta
coloured line correspond to the contribution from Nb and Se atoms, respectively: (a) Nb contribution to bulk NbSe2; (b)
Se contribution to bulk NbSe2; (c) Nb contribution to monolayer NbSe2; (d) Se contribution to monolayer NbSe2.
In 2011, Kuc and Heine et al.[137] have investigated the quantum confinement effecton the electronic structure of NbS2 and found that the metallic character is always presentin the material irrespective of the number of layers. In 2013, Kumar et al.[138] discoveredthat this property also occurs in NbSe2, that there are always one or a few bands cutacross the Fermi level. From our calculation with the effect of SOC for bi-layer, 3-layerand 4-layer NbSe2 (Fig.21), the metallic nature remains intact in all the systems. As thenumber of layer is increased, the number of band crossing Fermi level increases accordingly.For bi-layer NbSe2, spatial inversion symmetry is restored, all the bands return to spindegenerate states. The spin splitting energy due to SOC is approximately the same asthat of monolayer at K point. The similar situation also occurs in 4-layer NbSe2. From theenlarged diagram in Figure 22 (a), there are two doubly degenerate bands at K point witha separation of about 20meV, indicating the strength of interlayer coupling. However,spatial inversion symmetry is broken in 3-layer NbSe2 due to the absence of inversioncentre. Hence, the three distinctive bands in Figure 22 (b) are non-spin degenerate at Kpoint.
48
1"
−3
−2
−1
0
1
2
3
Γ M K Γ
Ene
rgy
(eV
)�
(a)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�Γ M K Γ
(b)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
Γ M K Γ
(c)
Figure 21:Band structure of 2-dimensional NbSe2: (a) band structure of bi-layer NbSe2; (b) band structure of 3-layer
NbSe2; (c) band structure of 4-layer NbSe2.
Figure 22: Band structures of 4-layer and 3-layer NbSe2 with SOC. The enlarged diagram on the right is for the area
encircled by the red box: (a) band structure of 4-layer NbSe2 with the two spin degenerate bands; (b) band structure of
3-layer NbSe2 with the three spin non-degenerate bands.
49
3.3.2 Spin Texture of Monolayer NbSe2
Inspired by the observation[139] and control[140] of valley polarization in pristine mono-layer MoS2 through optical pumping with circularly polarized light as well as the strikingsimilarities in the structures between NbSe2 and MoS2, one would naturally inquire ifsuch phenomenon can also be observed in monolayer NbSe2. Following the investiga-tion on the effect of spin-orbit interaction previously, here we analyze the spin textureof monolayer NbSe2. To begin with, a rectangular mesh-grid with 21 points along bothkx and ky directions is generated to cover the entire Brillouin zone in reciprocal space.There are in total 32 energy bands in our system, but we are only interested in the twonon-degenerate bands (number 25 and 26) that cut through Fermi level at K point. Theresult is presented in Figure 23 (Fig. 23 (a) and (c) are the spin texture plots for 25th
and 26th band; Fig. 23 (b) and (d) are the monolayer band structures with the corre-sponding band highlighted in red). The centre of the plot on Figure 23 (a) and (c) is theΓ point in the Brillouin zone and the black arrows surrounding it represent the directionof the in-plane spins. Value of the spin at each point on the mesh-grid is denoted by thelength of the arrow and clearly the only observable in-plane spins are concentrated nearΓ point. The background colour in the plots indicates the out-of-plane spins with theintensities signify the value of these out-of-plane spins. It is obvious from the results thatthe spins at K point are out-of-plane polarized in the opposite direction with the blueand red colour represents spin-down and spin-up in band 25 and 26, respectively. Tak-ing the idea of optically pumping electrons from top valence band to the spin polarizedbottom conduction bands at K point in MoS2, we propose a similar process to populatethe spin polarized bands (band 25 and 26) in NbSe2 from the band immediately belowthem, shown in Figure 23 (b) and (d) by the red arrow. More interestingly, if the Fermilevel could be tuned, such that only one non-degenerate band is above it; after populatingthis band one would effectively generate a spin polarized current in the material. Justlike MoS2[141], with almost 100% out-of-plane polarization at K point, monolayer NbSe2
demonstrates very promising characteristics for valleytronic devices in the near future.
50
Figure 23: Spin texture analysis on monolayer NbSe2 with SOC: (a) spin texture of band 25 with the red and blue colour
representing spin-up and spin-down, respectively; (b) band structure of monolayer NbSe2 with the band in red indicates
band 25; (c) spin texture of band 26; (d) band structure of monolayer NbSe2 with the band in red indicates band 26.
51
4 Conclusion
In summary, we have systematically investigated the electronic properties of several TMDsin the hope to find suitable materials in the application of spintronic devices. First-principle electronic calculations were performed to analyze the effect of spin-orbit couplingand number of layers on the ver-der-Waals structured materials. From the study, it is real-ized that the effect of SOC is closely related to the inherent crystal symmetry. SnSe2 andPbI2 have spatial inversion symmetry both in odd numbered and even numbered layers,thus, all the energy bands remain spin degenerate with the inclusion of SOC in the calcu-lations. However, NbSe2 breaks this spatial inversion symmetry in odd numbered layers,a rather significant splitting of 140meV at K point due to SOC is observed, consequentlylifting the spin degeneracy. When spatial inversion symmetry is present, SOC also playsan important role in altering the electronic and optical properties of the material. Asshown previously, the top few valence bands of SnSe2 and bottom few conduction bandsof PbI2 are considerably affected by SOC, which introduced noticeable splitting in the en-ergy bands, effectively changing the band gaps of the materials. In the layer dependencestudy, we examined the band structures of bi-layer, 3-layer and 4-layer structures. Besidesthe case for NbSe2 of which the number of layers determines the presence or absence ofspatial inversion symmetry that would affect the degeneracy of the bands, the nature (i.e.direct or indirect) and value of band gaps would also be modified. However, from ourpartial investigation, it is still difficult to predict how the band structures would changewith the presence of SOC or with different number of layers, because these two factorsdoes not show a consistent effect on different TMDs. In addition, we have also learnedseveral useful techniques in the analysis of the electronic property of this type of materials.For example, HSE hybrid functional was applied to make corrections to the PBE formedGGA approximation for SnSe2 band structure and the results obtained were much morecomparable to that from the experiments; the growth of PbI2 on ver-der-Waals structuredsubstrates (BN, graphene, MoS2) was modeled to provide theoretical calculations for themost stable configuration of layer stacking; moreover, spin texture of 2D NbSe2 was alsoexamined with the result of spins to be approximately 100% polarized in the out-of-planedirection at K point. Although there are limitations on theoretical DFT calculations, ourresults can still provide some insights on the electronic properties of the materials andviabilities of spintronic applications in the future.
52
Appendices
.1 Effect of SOC and Layer Dependence Study
Effect of SOC and Layer dependence study on PtS2:
Z
2.834Å
(a) (b)
(c) (d)
Figure A: Structure information of PtS2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of PtS2 with the dark
blue atom being platinum and the pink atom being sulfur; the red dot at the centre of the unit cell indicates the inversion
centre; (c) top-view of PtS2 supercell; (d) side-view of PtS2 supercell with the separation of two layers being 2.834Ain the
z-axis.
54
1
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC E
nerg
y (e
V)
Γ A H K Γ M L H A Γ
(a) (b)
Ene
rgy
(eV
)
Γ M K Γ
(c)
Ene
rgy
(eV
)
(d)
Ene
rgy
(eV
)
Γ M K Γ
Eg = 1.056eV Eg = 1.030eV
Eg = 1.791eV Eg = 1.769eV
Γ A H K Γ M L H A Γ
Figure B: Band structures of PtS2: (a) band structure of bulk PtS2 excluding SOC; (b) band structure of bulk PtS2
including SOC; (c) band structure of monolayer PtS2 excluding SOC; (d) band structure of monolayer PtS2 including
SOC.
1
S Pt
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
Figure C: Atomic contribution of the band structures of PtS2 including the effect of SOC with the blue and magenta
coloured line correspond to the contribution from Pt and S atoms, respectively: (a) Pt contribution to bulk PtS2; (b) S
contribution to bulk PtS2; (c) Pt contribution to monolayer PtS2; (d) S contribution to monolayer PtS2.
55
1"
−3
−2
−1
0
1
2
3
Γ M K Γ
Ene
rgy
(eV
)�
(a)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�Γ M K Γ
(b)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
Γ M K Γ
(c)
Figure D: Band structure of 2-dimensional PtS2:PtS2: (a) band structure of bi-layer PtS2; (b) band structure of 3-layer
PtS2; (c) band structure of 4-layer PtS2.
Effect of SOC and Layer dependence study on PtSe2:
(a) (b)
(c)
Z
2.591Å
(d)
Figure E: Structure information of PtSe2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of PtSe2 with the
yellow atom being platinum and the pink atom being selenium; the red dot at the centre of the unit cell indicates the
inversion centre; (c) top-view of PtSe2 supercell; (d) side-view of PtSe2 supercell with the separation of two layers being
2.591Ain the z-axis.
56
1
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC E
nerg
y (e
V)
Γ A H K Γ M L H A Γ
(a) (b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
Γ M K Γ
(c)
Ene
rgy
(eV
) Eg = 1.393eV
(d)
Ene
rgy
(eV
)
Γ M K Γ
Eg = 1.269eV
Figure F: Band structures of PtSe2: (a) band structure of bulk PtSe2 excluding SOC; (b) band structure of bulk PtSe2
including SOC; (c) band structure of monolayer PtSe2 excluding SOC; (d) band structure of monolayer PtSe2 including
SOC.
1
Se Pt −3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
Γ A H K Γ M L H A Γ
−3
−2
−1
0
1
2
3
Γ M K Γ
(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Γ M K Γ
Figure G: Atomic contribution of the band structures of PtSe2 including the effect of SOC with the blue and magenta
coloured line correspond to the contribution from Pt and Se atoms, respectively: (a) Pt contribution to bulk PtSe2; (b) Se
contribution to bulk PtSe2; (c) Pt contribution to monolayer PtSe2; (d) Se contribution to monolayer PtSe2.
57
1"
−3
−2
−1
0
1
2
3
Γ M K Γ
Ene
rgy
(eV
)�
(a)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�Γ M K Γ
(b)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
Γ M K Γ
(c)
Figure H: Band structure of 2-dimensional PtSe2: (a) band structure of bi-layer PtSe2; (b) band structure of 3-layer
PtSe2; (c) band structure of 4-layer PtSe2.
Effect of SOC and Layer dependence study on PdSe2:
(a)
(c) (d)
(b)
Z
2.444Å
Figure I: Structure information of PdSe2: (a) 3-dimensional hexagonal Brillouin zone; (b) unit cell of PdSe2 with the
jungle green atom being palladium and the dark yellow atom being selenium; (c) top-view of PdSe2 supercell; (d)
side-view of PdSe2 supercell with the separation of two layers being 2.444Ain the z-axis.
58
1
1-Layer GGA+SOC 1-Layer GGA
Bulk GGA Bulk GGA+SOC E
nerg
y (e
V)
(a) (b)
Ene
rgy
(eV
)
Z Γ Y S X U R T Z Z Γ Y S X U R T Z
−3
−2
−1
0
1
2
3(c)
Ene
rgy
(eV
)
Eg = 1.578eV
Ene
rgy
(eV
)
Eg = 1.585eV
(d)
Γ Y T Z Γ Γ Y T Z Γ
Figure J: Band structures of PdSe2: (a) band structure of bulk PdSe2 excluding SOC; (b) band structure of bulk PdSe2
including SOC; (c) band structure of monolayer PdSe2 excluding SOC; (d) band structure of monolayer PdSe2 including
SOC.
1
Se Pd −3
−2
−1
0
1
2
3
Ene
rgy
(eV
)
(a)
−3
−2
−1
0
1
2
3(b)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(c)
Ene
rgy
(eV
)
−3
−2
−1
0
1
2
3(d)
Ene
rgy
(eV
)
Z Γ Y S X U R T Z Z Γ Y S X U R T Z
Γ Y T Z Γ Γ Y T Z Γ
Figure K: Atomic contribution of the band structures of PdSe2 including the effect of SOC with the blue and magenta
coloured line correspond to the contribution from Pd and Se atoms, respectively: (a) Pd contribution to bulk PdSe2; (b)
Se contribution to bulk PdSe2; (c) Pd contribution to monolayer PdSe2; (d) Se contribution to monolayer PdSe2.
59
1"
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
(a)
Γ Y T Z Γ−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
(b)
−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
(c)
Γ Y T Z Γ Γ Y T Z Γ
Figure L: Band structure of 2-dimensional PdSe2: (a) band structure of bi-layer PdSe2; (b) band structure of 3-layer
PdSe2; (c) band structure of 4-layer PdSe2.
60
.2 PbI2 on Graphene and MoS2 Substrates
PbI2 on Graphene Substrate:
PbI2 Unit Cell� Graphite Unit Cell�
Lattice Constants: a = 4.648 Å b = 4.648 Å c = 7.119 Å
Lattice Constants: a = 2.470 Å b = 2.470 Å c = 6.790 Å
(a) (b)
Figure AA: Structure information of PbI2 and graphene.
Γ A H K Γ M L H A Γ�
Ene
rgy
(eV
)�
(a)
Γ M K!!!!!!!!!!!!!!!!!!!!!!!Γ
Ene
rgy
(eV
)�
(b)
Γ M K!!!!!!!!!!!!!!!!!!!!!!!Γ
Normal Stretched 5.8%
Figure BB: Band structure of normal graphene and band structure of graphene stretched by 5.8%.
61
Figure CC: Different positions of PbI2 on substrate graphene.
Ene
rgy
(eV
)
Ene
rgy
(eV
)
Ene
rgy
(eV
)
E = 0.0meV E = 0.0meV
E = -0.8meV
Γ M K Γ Γ M K Γ
Γ M K Γ
(a) (b)
(c)
Pb on C Bridge
Hollow
Figure DD: Band structures of 2-dimensional PbI2 on graphene with different positions.
62
−3
−2
−1
0
1
2
3E
nerg
y (e
V)�
(a)
Γ !M K Γ−3
−2
−1
0
1
2
3
Ene
rgy
(eV
)�
(b)
Γ !M K Γ
Hollow Site Hollow Site + SOC
Figure EE: Band structures for hollow site (the most stable configuration) without SOC and with SOC.
Ene
rgy
(eV
)�
C I
Ene
rgy
(eV
)�
Ene
rgy
(eV
)�
Pb
Γ M K Γ Γ M K Γ Γ M K Γ
(a) (c) (b)
Figure FF: Atomic contribution to the band structures of PbI2 on graphene substrate with the green, dark blue and red
coloured lines correspond to the contribution from C, I and Pb atoms, respectively.
63
PbI2 on MoS2 Substrate:
PbI2 Unit Cell� MoS2Unit Cell�
Lattice Constants: a = 4.648 Å b = 4.648 Å c = 7.119 Å
(a) (b)
Lattice Constants: a = 3.140 Å b = 3.140 Å c = 12.530 Å
Figure GG: Structure information of PbI2 and MoS2.
(a) (b) Normal Stretched 2.5%
Ene
rgy
(eV
)�
Ene
rgy
(eV
)�
Γ M K!!!!!!!!!!!!!!!!!!!!!Γ Γ M K!!!!!!!!!!!!!!!!!!!!!Γ
Eg = 1.848eV
Eg = 0.933eV
Figure HH: Band structure of normal MoS2 and band structure of MoS2 stretched by 2.5%.
64
Figure II: Different positions of PbI2 on substrate MoS2.
65
Pb on Mo
Γ M K Γ Γ M K ΓE
nerg
y (e
V)
I on S
Γ M K Γ
Ene
rgy
(eV
)
Pb on S
Γ M K Γ
Ene
rgy
(eV
)
I on Mo
Γ M K Γ
Hollow Bridge
Γ M K Γ
Ene
rgy
(eV
)
Ene
rgy
(eV
)
(a)
(d) (f) (e)
(c) (b) E
nerg
y (e
V) Eg = 1.828eV
E = -6.4meV
Eg = 1.826eV E = -1.9meV
Eg = 1.825eV E = -1.1meV
Eg = 1.826eV E = 0.0meV
Eg = 1.828eV E = -1.8meV
Eg = 1.827eV E = -6.5meV
Figure JJ: Band structures of 2-dimensional PbI2 on MoS2 with different positions.
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Ene
rgy
(eV
)�
(a)
Γ !M K Γ
Bridge Site
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Ene
rgy
(eV
)�
(b)
Γ !M K Γ
Bridge Site + SOC
Eg = 1.827eV Eg = 1.240eV
Figure KK: Band structures for bridge site (the most stable configuration) without SOC and with SOC.
66
1
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Ene
rgy
(eV
)
(a)
Γ M K Γ−2
−1.5
−1
−0.5
0
0.5
1
1.5
2(b)
Ene
rgy
(eV
)
Γ M K Γ
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Γ M K Γ
(c)
Ene
rgy
(eV
)
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2(d)
Ene
rgy
(eV
)
Γ M K Γ
Mo S
Pb I
Figure LL: Atomic contribution to the band structures of PbI2 on MoS2 substrate with the cyan, yellow, red and dark
blue coloured lines correspond to the contribution from Mo, S, Pb and I atoms, respectively.
67
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