The University of New South Wales
Faculty of Science
School of Materials Science and Engineering
Theoretical investigation of novel spin-polarized materials for spintronic applications
A Thesis in Materials Science and Engineering
By Anh Pham
Submitted in Partial Fulfilment of the Requirement for the Degree of
Doctor of Philosophy
October 2014
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ORIGINALITY STATEMENT
‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’
Signed
Date
i ACKNOWLEDGEMENT
I would like to thank my supervisor Prof. Sean Li for all the supports that he has given me since I joined his group. Without his strong support I would not have completed this thesis. I also want to express my appreciation to Dr. Hussein Assadi for introducing me to the density functional theory technique as well as guiding me in the early stages of my study. I also acknowledge the financial support from the Australian Research Council during my research.
Finally I want to dedicate this thesis to my partner for always being there during my long scientific journey.
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ABSTRACT
Development of spin-polarized materials is important for the realization of spintronic devices. Two approaches are taken: (i) investigating magnetism in diluted magnetic semiconductors (DMSs), and (ii) developing novel 2D materials with intrinsic spin currents. The focus of this thesis is to study these two classes of materials using the density functional (DFT) theory. The first part focuses on ZnO based DMS. The systematic study of C/N doped ZnO with defects shows that ZnO:C exhibits ferromagnetism, but dopant complex C2 tends to inhibit this property. Most importantly, the computational method plays a key role in studying DMS. Due to the popularity of hydrogenated ZnO:Co, this system is tested using the different methods. The DFT+U method shows a very specific configuration of hydrogenated Co complex can generate ferromagnetism. The inadequacies of traditional method are further exposed when the dopant geometry is compared between the hybrid functional method and normal DFT. The hybrid method includes the Fock exchange which significantly improves the description of long-range inter-atomic forces in the geometry relaxation yielding a more accurate description. The second part focuses on the study of novel 2D materials with. The studies are devoted to increase the band gap of of novel 2D materials. In silicene and germanene, it is shown that the adsorption and substitution of Tl can increase the bandgaps to 0.29 eV at a small doping limit of ~2%. In addition, Tl can also tune the conductivity of silicene from n to p types depending on the doping sites, and preserving the high mobility of the undoped structure. Furthermore, the author also predicts several new wide bandgap quantum spin Hall insulator (QSHI). QSHI is a novel material with an insulating bulk state and spin-polarized metallic edges. By modifying the surfaces through hydrogenation, the orbital interactions of Bi, Pb, Cr, Mo, and W in the low energy level are confined on two dimensional resulting in novel QSHI with giant gap and strongly correlated property. These results represent important contribution to the study of spintronic materials. First, the understanding between the theoretical method and magnetic properties in semiconductors can help to accurately predict new spintronic materials. Most importantly, the study in orbital engineering of 2D materials with hydrogenation provides new insights into the nature topological materials which have practical applications for the next generation of electronic devices.
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List of Publications Included as Part of the thesis
1. Anh Pham, M. H. N. Assadi, Y. B. Zhang, A. B. Yu, and S. Li “Weak d0 magnetism in C and N doped ZnO” Journal of Applied Physics, 110 art. no. 123917 (2011). 2. Anh Pham, Y B Zhang, M H N Assadi, A B Yu and S Li “Ferromagnetism in ZnO:Co originating from a hydrogenated Co–O–Co complex” Journal of Physics: Condensed Matter 25, 116002 (2013). 3. Anh Pham, M H N Assadi, A B Yu and Sean Li “Critical role of Fock exchange in characterizing dopant geometry and magnetic interaction in magnetic semiconductors” Physical Review B 89, 155110 (2014). 4. Anh Pham, Carmen J Gil, and Sean Li “Orbital engineering of 2D materials with hydrogenation: a realization of giant gap and strongly correlated topological insulators”, Being Reviewed in Physical Review Letters (2014) 5. Anh Pham, Carmen J Gil, and Sean Li “Engineering silicene and germanene’s band gap and conductivity with diluted doping of heavy elements: a first principle study”, in preparation. 6. Carmen J Gil, A Pham, A Yu and Sean Li “An ab-initio study of transition metals doped
WSe2 for long-range room-temperature ferromagnetism in two-dimensional transition metal dichalcogenide” Journal of Physics: Condensed Matter 26, 306004 (2014).
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Statement of Contribution of Others
I, Anh Pham, was the person who mainly conducted the calculations including analyses as well as writing the manuscripts in all publication presented in this thesis.
Signature
I as a co-author endorse that this level of contribution by the candidate indicated above is appropriate. Dr. Mohammad H. N. Al Assadi Dr. Yuebin Zhang Prof. Aibing Yu Prof. Sean Li Ms. Carmen J Gil.
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List of Abbreviation
2D: Two-dimensional
3D: Three-dimensional
BMP: Bound Magnetic Polaron
CVD: Chemical Vapour Deposition
DFT: Density Functional Theory
DMS: Diluted Magnetic Semiconductor
GKA: Goodenough-Anderson-Kanokori
QH: Quantum Hall
QSH: Quantum Spin Hall
QSHI: Quantum Spin Hall Insulator
RKKY: Ruderman–Kittel–Kasuya–Yosida
TKKN: Thouless, Kohmoto, Nightingale and den Nijs
TM: Transition metal
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TABLE OF CONTENTS
Certificate of Originality……………………………………………………………………….i
Acknowledgement…………………………………………………………………………….ii
Abstract……………………………………………………………………………………….iii
List of Publications…………………………………………………………………………...iv
Statement of Contribution of Others…………………………………………………………..v
List of Abbreviation…………………………………………………………………………..vi
Table of Contents…………………………………………………………………………….vii
CHAPTER 1 Motivation and Outlines……………………………………………………….1
CHAPTER 2 Literature Review………………………………………………………………9
2.1 Diluted Magnetic Semiconductors…………………………………………...10
2.2 Novel Two Dimensional Materials…………………………………………..19
CHAPTER 3 Computational Method……………………………………………………….36
CHAPTER 4 Weak d0 Magnetism in C and N doped ZnO……………………………...... 45
CHAPTER 5 Ferromagnetism in ZnO:Co Originating from a Hydrogenated Co–O–Co
Complex……………………………………………………………………………………...53
CHAPTER 6 Critical Role of Fock Exchange in Characterizing Dopant Geometry and
Magnetic Interaction in Magnetic Semiconductors…………………………………………..62
CHAPTER 7 Engineering Silicene and Germanene’s Band Gap and Conductivity with
Diluted Doping of Heavy Elements: a First Principle Study………………………………...69
CHAPTER 8 Orbital Engineering of 2D Materials with Hydrogenation: A Realization of
Giant Gap and Strongly Correlated Topological Insulators………………………………….86
CHAPTER 9 Conclusions…………………………………………………………………105
APPENDEX An Ab-Initio Study of Ttransition Metals Doped with WSe2 for Long-Range Room Temperature Ferromagnetism in Two-Dimensional Transition Metal Dichalcogenide……………………………………………………………………………...109
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Chapter 1: Introduction
Spintronic materials represent a novel platform in which information can be controlled rapidly through the manipulation of the quantum mechanical properties of the electrons through an external electric or magnetic field. One of the earliest discoveries of such materials was the discovery of the colossal magneto resistance in metallic hetero- structures [1, 2]. Since then, the field of spintronics has expanded at a rapid rate with the discoveries of several exotic materials for practical applications [3-6]. These materials can be classified into two types based on different scientific approaches.
The first type of materials are known as diluted magnetic semiconductors (DMSs) in which the interplay between charge and spin properties are manipulated through the introduction of impurities. These impurities can be either magnetic/non-magnetic but their introductions in the host materials generate a ferromagnetic interaction between the dopants making the doped materials magnetic. Various host materials have been studied ranging from
II-VI semiconductors [7-9], III-V materials [10-12], IV-VI chalcogenides [13-15] based materials to wide band gap oxides [16-20] since the 1980s. Subsequently, the prediction of room temperature of ferromagnetism in oxide DMS materials by Dietl et al. [21] sparked off a new chapter in the field of DMS research with increasing focus in the II-VI oxide materials.
Several experiment studies were conducted in doping II-VI material like ZnO with different dopants to generate ferromagnetism [22-24]. However, these studies have generated several contradicting results which question the nature of magnetism in doped semiconductor [25-
27]. In addition, as it has been demonstrated in numerous studies the theoretical methods play an important in describing the magnetism in wide band gap materials like ZnO [28-30]. Thus, sophisticated simulation method is required to accurately understand the nature of ferromagnetism.
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Besides DMS based materials, another type of materials has been suggested for spintronic application due to their intrinsic novel properties without additional doping. These new materials have attracted great interests in the field of condensed matter physics and materials science. With the successful fabrication of a single layered carbon material known as graphene in 2004 [31], this represents a new route towards functional spintronic devices based on the exploitation of a new class of novel two dimensional (2D) materials. These 2D materials have ultra-high carrier mobility [32] and exhibits several novel properties such as the quantum spin Hall effect [33-35], spin-valley coupling [36], quantum anomalous Hall effect [37], valley polarized metallic phase [38], large spin-orbit interaction [39] and giant magneto-resistance [40]. Thus, extensive efforts have been devoted to investigate and manipulate the novel properties for eventual application in practical electronic devices. Two properties that gathered intense focuses by various scientific groups are the high-carrier mobility and the quantum spin Hall effect. In 2D materials like graphene, these properties essentially originate from the behaviours of the massless Dirac fermions which dictate the carriers’ properties. In addition, if an intrinsically large spin-orbit coupling exists in a graphene-like material a band gap can be opened up in the bulk structure accompanied with the helical edge states [41, 42]. Due to the time reversal symmetry, these edge states are robust against Anderson localization and back scattering in the presence of non-magnetic impurity. Thus, they are an attractive platform for nano-electronic devices because of their ability to conduct electricity without losses as well as acting as spin injection or spin filter materials.
The focus of this study is the applications of ab-initio calculations focusing in these main developments in the field of spintronic materials. In the first part, it is aimed to address the following questions: (i) the interplay between dopants and defects in influencing the magnetic properties of the doped structure, and (ii) how the theoretical methods can influence
2 the nature of ferromagnetism. In the second part, the effort is devoted to the theoretical study of engineering and predicting several novel 2D materials which have properties desirable for practical applications in nano-devices. As a result the thesis can be outlined as follow
Chapter 2: this chapter surveys the physics of magnetic exchange in diluted magnetic semiconductor in oxide materials, and the properties of novel properties of two dimensional with hexagonal symmetries.
Chapter 3: it outlines the general computational methodology known as Density
Functional Theory used to study the properties of magnetic oxides and novel two dimensional materials
Chapter 4: In this chapter, a comprehensive theoretical study of the interaction between non-magnetic dopants like C, N and defects like H and oxygen vacancy was investigated to understand the d0 ferromagnetism in ZnO based DMS materials. The theoretical results show a weak ferromagnetism existing in C substituting oxygen in ZnO.
However, this magnetic property is inhibited by the tendency of C to form a binary complex which is non-magnetic. In addition the results also show that both of hydrogen interstitial and oxygen vacancy does not enhance the ferromagnetic interaction between carbon impurity atoms.
Chapter 5: The magnetic properties of cobalt doped ZnO is investigated in the presence of hydrogen interstitial and oxygen vacancies. The motivation of this study is understand the roles of Co’s concentration and the specific geometrical configuration of Co and hydrogen complex in mediating the ferromagnetism. To correct the localization problem of d orbitals in the traditional density functional method (DFT), a Hubbard potential is applied on Zn’s d and Co’s d orbitals, which significantly improved the band gap of ZnO compared to normal DFT method. The results demonstrate that hydrogen can mediate the short range ferromagnetic interaction in ZnO:Co by forming a special hydrogenated complex
3 at low Co concentration (~ 5.6%), while oxygen tends to inhibit the ferromagnetism.
However, a combined effect of hydrogen and oxygen vacancy can enhance the ferromagnetism at particular defect geometry.
Chapter 6: Since the theoretical method can influence the description of magnetism in ZnO based DMS material, this chapter focuses on how the description of Hellmann-
Feynman forces, which is crucial in most DFT calculations, can be significantly different depending on the choices of the functional. Specifically, the geometry of the hydrogenated
Co complex is investigated since hydrogen is a mobile defect thus it can form multiple stable geometry in ZnO. The calculations were performed hybrid functional which includes the
Fock-exchange to correct the shortcomings of the local density approximation and the generalized gradient approximation. The results from the hybrid method demonstrate a different result compared to previous study hydrogenated Co in ZnO. This is due to the improved description of the Hellmann-Feynman forces which can accurately describe long- range Coulomb forces with the inclusion of the Fock exchange.
Chapter 7: Next the focus of the study is shifted to the investigation of novel 2D materials. Novel materials like germanene and silicene, 2D equivalence of graphene, are two newly discovered materials with tremendous potential in electronic devices. Similar to graphene, even though these materials have ultra-high carrier mobility their narrow bandgap at the K valley limits their practical applications. As a result, the attention is devoted in this study to increase the band gap but also preserves the ultra-high carrier mobility in doped germanene and silicene. Several adsorbed and substitutional configurations of heavy elements like Pt, Au, Tl and Pb were investigated which demonstrate a relatively large gap opening ~ 0.29 eV in the diluted doping limit of 2%. In addition, the linear dispersion is shown to be preserved in several doped configurations indicating high carrier mobilities.
These dopants also modify the conductivity of the host silicene and germanene from n (Au)
4 to p type (Tl) depending on the doping configurations which is attractive for electronic devices like the field effect transistor.
Chapter 8: This chapter details the prediction of several new quantum spin Hall insulator with giant band gap and strongly correlate property. By chemically modifying the surfaces of 2D Pb, Bi and 4d and 5d materials, the first principle calculations show that a giant gap ≥ 1 eV can be induced. This is because the hydrogenation effect acts as an orbital
2 2 filtering which removes the pz orbital in Pb and Bi and the d3z -r in Cr, Mo, W and Re leaving a large on-site spin orbit coupling to lift the degeneracy in p and d orbitals. In addition, the value of the band gap in MoH, WH and ReH is dependent on a subtle interplay between the strongly correlated property and the spin orbit effect making these materials a new class of strongly correlated quantum spin Hall insulator.
Chapter 9: the summary of the previous chapters is provided and direction for future studies is detailed in this chapter.
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Chapter 2: Literature review
The key element of spintronic materials is their novel properties of spin-polarized carriers which can be utilized in various electronic devices. In order to be applied in commercial devices, the carriers in these spintronic materials need to be able to perform different functions such as injection, carrier and transport, and detection. Extensive effort has been devoted to manipulate the electronic properties of various semiconductor based materials to function as spintronic devices. The earliest advancement in the field of semiconductor based spintronics was the proposal of a spin field effect transistor (FET) by
Datta and Das [1]. In this device, ferromagnetic materials act as a source and drain which emit and collect the spin-polarized carriers. The spin-polarized current generated by the carriers is manipulated via a gate voltage which controls the spin precession by effecting the spin-orbit interaction thus creating an “On” and “Off” state. The “On” state is when the carriers freely travel from the source to the drain in the same spin alignment. However, as the gate voltage is applied the carriers’ spin precess alters their natural spin alignment, which inhibits them from reaching the “drain” size due to the difference in spin alignment. This process represents the “Off” state. Such “On” and “Off” operation requires a much smaller energy and performs at a much higher switching rate compared the traditional FET. As a result, spintronic devices were demonstrated to be the next stage of low-power and highly efficient electronic devices. This discovery sparks off a great enthusiasm in the field of semiconductor based spinstronic research.
Ferromagnetic material has been proposed to be a great candidate for spintronic devices. This is because these materials have a highly unequal spin-up and spin-down carriers in their density of states at the Fermi level, which can be exploited as the materials for the source and drain in spin based FET. In addition, magnetism in these materials can be
9 manipulated via an external magnetic field thus the spin-polarized currents can be easily turned “on” and “off”. Unfortunately, most semiconducting materials are intrinsically non- magnetic. Thus, great efforts have been devoted to magnetize various semiconductors via chemical method like introducing external dopants. This technique underlies the development of diluted magnetic semiconductors since their discoveries in the early 1980s. Alternatively, certain 2D materials with graphene-like properties have an intrinsically large spin-orbit coupling which can also generate spin-polarized current without the need for external doping.
Furthermore, the spin-orbit interaction can be tune via an external magnetic field through gate voltage to control the spin currents. The purpose of this chapter is summarizing the developments of these novel spintronic materials.
2.1. Diluted Magnetic Semiconductors
Diluted magnetic semiconductors are magnetic materials whose magnetism originated from the introduction of external dopants. These doping impurities can be either magnetic or non-magnetic and replace either the cations or anions. In the host lattice, the impurities interact ferromagnetically via free carriers (long range), or through the neighbouring atoms
(short range) to create global magnetism. These exchange mechanisms are known as indirect exchange interactions. In indirect exchange interactions, the magnetic interactions between the two magnetic ions are mediated via various medium such as free carriers or the ligand field in the host lattice. In addition to the indirect interaction, the wavefunctions of the magnetic ions can overlap directly to produce the global magnetic properties. This type of direct exchange does not play a prominent role in most semiconductor materials due to the large inter-atomic distance. As a result, the attention is focused in various indirect exchange mechanisms: superexchange, double exchange and RKKY interaction.
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2.1.1. Superexchange
Superexchange interaction is the magnetic exchange between the two neighbouring magnetic ions which is mediated by a neighbouring atom. In addition, in superexchange mechanism the magnetic ion does not need to have a finite density at the Fermi level since the interaction does not require free carrier to mediate the magnetism [2]. The only necessary condition for superexchange is the hybridization of the magnetic ions’ wavefunction with a ligand field. In the case of oxides, the hybridization usually occurs between the d orbital of the magnetic ions and the p orbital of oxides. The superexchange interaction was first introduced by Goodenough [3, 4] as a theory behind the antiferromagnetic ground state of oxides like LaMnO3. This theory was further expanded by Anderson [5, 6] and Kanamori [7] to include ferromagnetic interaction which is now known as the Goodenough-Anderson-
Kanokori rule. Based on the GKA theory, the magnetic interaction between two neighbouring magnetic ions is characterized by a fourth order perturbation of the Hubbard
Hamiltonian which involves the hopping between the p and d orbitals (tpd) [2]. The sign of the hopping integral tpd determines whether the interaction is ferromagnetic or anti- ferromagnetic. Empirically, the sign of tpd has been found to depend strongly on the bond angle between the transition metal (TM) ion and the anion. When the anion forms a 180° angle with the magnetic ions, the two magnetic ions couple to the same p orbital yielding an antiferromagnetic configuration. This is because the two electrons occupy the same p orbitals form a singlet state (↑↓) due to the Pauli Exclusion Principle. Due to the overlapping of 2 d orbitals to the same p orbital, the antiferromagnetic superexchange bonding between a magnetic ion and an anion has been known as “semi-covalent” bond [3]. As the bond angle of
M-O-M deviates from 180° to 90°, the ferromagnetic interaction becomes more favourable.
The magnetic ions now couple to the two orthogonal p orbitals in the anion making a parallel
11 spin-spin configuration possible for the electrons occupying different p orbitals. In this case, superexchange is mediated via the Coulomb exchange on the connecting oxygen.
2.1.2 Double exchange
In certain oxides such as doped manganate perovskites, the magnetic ions change from an antiferromagnetic configuration to the ferromagnetic interaction after doping. For example, LaMnO3 becomes ferromagnetic with Ca doping for concentration between 0.2 and
0.4 [8]. After doping, the Mn ions no longer have a 3+ valence state but now they have a mixture of 3+ and 4+ valence states. To understand the connection between the magnetic properties and conductivities in the doped manganate perovskite, Zener [9], Anderson and
Hasegawa [10], and de Gennes [11] independently developed a theory of carrier mediated magnetic interaction known as double exchange interaction. Their findings show an intimate connection between the high carrier concentration and the magnetism of these materials.
Similar to superexchange exchange interaction, the Mn ions with unequal valence states can interact magnetically via a bridging anion like oxygen. However, different from superexchange the double exchange mechanism is mediated via carriers. The double exchange can be defined as a process of double hopping, in which an electron hops from one transition metal ion to the anion at same time as the transfer of electron from the anion to the other TM ion. The spin of these electrons are parallel to each other because of the Hund’s coupling which yields an overall ferromagnetism.
2.1.3. RKKY interaction
Another type of carrier mediated magnetic interaction is the in mechanism developed by Ruderman–Kittel–Kasuya–Yosida (RKKY) [12-14]. In this type of interaction, the localized magnetic ions can exchange ferromagnetically or anti-ferromagnetically via free
12 carriers which are treated as electron gas. Thus, different from other magnetic interactions like double exchange or superexchange RKKY interaction does not require an interaction between the magnetic ions with the ligand field. This theory was originally proposed to describe interaction between nuclear spin via the conduction electrons, and have been proven to be successful in characterizing the magnetic interaction in metals and highly degenerate semiconductors. Based on RKKY theory, the spin-spin interaction between the two magnetic ions is mediated via band electrons which can be represented by an s-d Kondo Hamiltonian
[2]. The strength of the interaction between two ions at position i and j is character by a factor
(E )k 3 (N )2 J F F o F (2k R)as originally obtained by Rudermann and Kittel [12]. In this ij 2 3 F
equation, (EF ) is the density of states of the mobile carriers at the Fermi level and No is
xcos sin x the exchange integral of the s-d interaction. The function F3 takes the form of . x 4
Thus, at short distance the magnetic interaction is ferromagnetic, and as the distance between the magnetic ions becomes larger the interaction oscillates with decreasing strength. Due to the requirement of high carrier concentration, RKKY is usually not suitable to describe the ferromagnetism in most doped semiconductors and other theories like superexchange or double exchange are more suitable.
2.1.4 Bound Magnetic Polaron (BMP)
In most doped semiconductors, the impurity and carrier concentration can be significantly lower than in metals or in narrow band gap semiconductors, but strong ferromagnetism can still occur. To understand such behaviours, Caledron et al. [15] formulated a theory known as bound magnetic polaron (BMP) to describe the ferromagnetism in diluted magnetic semiconductors. As noted by Coey et al. [16] recently, this type of interaction can play an important role in oxides since defects can play a role of carrier donors
13 thus opening up a possibility to create n-type magnetic semiconductor. The basis of this theory is the polarization of free carriers surrounding the magnetic impurities. This happens due to a combination of the Coulomb and magnetic exchange between the impurities and the carriers in their orbits. At a certain distance, exchange between the carriers and the impurity is strong enough to flip the carriers’ spins. Thus carriers surrounding the impurities at a certain distance are also magnetized and they orbit the centre impurity like a hydrogen atom.
As these BMPs reach a certain percolation threshold, they start to overlap to form long range magnetism. In oxides, BMP can be formed based on exchange interaction between highly correlated narrow impurity bands and the atomic spin of the magnetic ions. The interaction between the carriers and impurities can be either ferromagnetic or antiferromagnetic. The parallel and anti-parrallel configuration is different in energy thus a non-zero spin flip energy is characteristic of the BMP model making it strongly temperature dependent. At low energy, ferromagnetism is favourable due to the lack of thermal energy so the parallel spin-spin alignment between carriers and impurities lowering the total energy of the system. However, as the temperature increases, the spin alignment of the impurities can fluctuate which weakens the ferromagnetism.
2.1.5 d0 Magnetism
A peculiar case in wide band gap semiconductor is the existence of ferromagnetism even in the absence of magnetic doping. Ferromagnetism has been in various undoped oxides such as MoO2 [17], TiO2 [18], HfO2 [19, 20], SnO2 [21], ZnO [22, 23], In2O3 [24], CaO [25,
26], MgAl2O4 [27] and non oxide materials like, GaN [28] and BN [29]. These materials do not contain unpaired electron thus they are classified as d0 materials. The ferromagnetism in d0 materials has been shown to be intrinsically linked to the growth process. In the absence of structural disorder, these materials are predominantly diamagnetic while samples
14 fabricated with non-equilibrium growth technique exhibit ferromagnetic behaviour. In addition, several experiments have also reported the existence of strong ferromagnetism in various doped oxides with non-magnetic dopants. Cation substitution with non-magnetic elements has been shown to develop magnetic properties, as in, Mg-doped SnO2 [30], Ca- doped GaN [31], Li-doped ZnO [32], K-doped SnO2 [33]. Anion substitution has also been able to produce magnetism, for instance, O doped MgN [34], N doped MgO [35], In2O3 [36],
TiO2 [37], SrO [38], and C doped TiO2 [39], ZnO [40]. In these materials, the doping process can produce defects which introduce levels in the energy gap. At a certain percolation threshold, these defect levels can overlap with the impurity bands originated from the dopants like N or C to produce long range carrier mediated ferromagnetism.
2.1.6 ZnO based DMS
ZnO is one of most widely studied material for DMS application due to its novel properties such as an intrinsically wide bandgap and large exciton binding energy making it suitable for a wide range of spin-based optoelectronic application [41]. Ferromagnetism in
ZnO was first predicted by Dietl et al. [42] through the introduction of transition metal like
Mn. Using a GaMnAs as an example, Dietl modified Zener’s double exchange theory to generalize the magnetic interaction in doped semiconductor as a strong p-d coupling between hole and magnetic impurities to produce long range ferromagnetism. In this theory, the dopant acts as a shallow acceptor in the host lattice as well as having a local magnetic moment, and ferromagnetic correlation between these shallow acceptors and the localized spins creating the overall ferromagnetism. The coupling between the localized spins and free holes are strong enough to overcome the superexchange interaction or spin-flip based on thermal fluctuation. Consequently, several wide band gap materials were predicted to produce strong ferromagnetism with the Curie temperature well exceed room temperature.
15
Subsequently, Sato et al. [43, 44] and Wang et al. [45] proposed a model of N co-doped with
Mn in ZnO to sufficiently produce room-temperature ferromagnetism. Their finding shows that with 10% of Mn and N can stabilize the ferromagnetism in the doped ZnO. However, recent theoretical studies cast doubt on the p type conductivity of N doped ZnO due to the deep acceptor nature of N in ZnO [46].
Since the initial discovery of Mn doped ZnO, various other transition metal doped
ZnO have also been predicted to exhibit strong ferromagnetism [47-49]. Among many dopant, Co received wide-spread attention due to high solubility in ZnO and it was experimentally reported to produce room-temperature ferromagnetism in ZnO:Co [50].
However, this result has been contradicted by several theoretical and experimental studies.
Spaldin [51] demonstrated that robust ferromagnetism cannot be obtained in ZnO:Co in the absence of the hole doping, and the magnetic interaction between the Co ion has been calculated to be anti-ferromagnetic [52, 53]. Experimentally, antiferromagnetic interaction has also been observed in ZnO:Co [54] and the ferromagnetism in doped ZnO is attributed to
Co precipitation [55] in ZnO:Co thin film. Other co dopant such as Li [56] and Ga [57] was suggested to stabilize the ferromagnetism through theoretical and experimental investigations. Theory of the magnetic interaction in ZnO:Co reveals a complex competition between superexchange ferromagnetism and anti-ferromagnetism which is intrinsically linked to the orbital splitting of Co’s 3d states [58]. The 3d orbitals of the TM dopant in ZnO split into a t2g and eg state, and spin coupling is dependent on the hopping of electrons between these states. As a result to accurately understand the physics of ferromagnetism in TM doped
ZnO, it is necessary to characterize these orbitals at the correct energy level. However, theories used to predict room-temperature FM are based on density functional theory (DFT) which has been known to systematically underestimate band gaps specifically by placing the
16 conduction band minimum too low and the valence band maximum too high [59], thereby resulting in contradicting results of ferromagnetism in TM doped ZnO.
The intrinsic nature of ferromagnetism has also come into question due to the formation of defects in various doped ZnO. One of the prominent candidates is oxygen vacancy (VO). It has been suggested that oxygen vacancy is the original source of polarons which are responsible for ferromagnetism in many systems of ZnO [16]. Several experiments have been conducted to study the effect of oxygen vacancy on the ferromagnetism. ZnO:Co sample was annealed in air and Ar/H2 to control the concentration of VO [60]. The result show that Co doped ZnO annealed in air show an enhancement of magnetization compared to sample annealed in air. These results seem to validate the theory of oxygen vacancy as a source of electron donors. However, the role of oxygen vacancy as a shallow donor has also been questioned from the more accurate theoretical study of VO in ZnO. Using the DFT+U
[61] and hybrid functional method [62], it was demonstrated that VO is a deep donor and not a shallow donor thus it cannot be responsible for donating free carriers in ZnO. In addition to oxygen vacancy, hydrogen is also another important defect since it can exist in abundance and has been suggested to be responsible for the n-type conductivity in ZnO [63]. Hydrogen has been suggested be responsible for the parallel spin-spin interaction in ZnO:Co and
ZnO:Mn [64]. The influence of hydrogen on ferromagnetic properties in ZnO:Co have been tested in several experimental investigation which collaborated on the enhancement of magnetization of hydrogenated ZnO:Co [65, 66].
17
2.2. Novel two dimensional materials
2.2.1. Graphene and graphene-like materials
A. Graphene
The discovery of graphene in 2004 [67] sparked a dramatic increase in the research of two dimensional (2D) materials due to their unique properties. One of the novel properties is the ultra high mobility exceeding 105 cm2V-1s-1 [68]. This enhancement in carrier mobility is significant for spintronic application, since it presents possibilities for the development of faster transistors with high conductance, which results in large gains. As the limitation of silicon transistors has been reached, 2D material is a viable solution for the next generation of electronics. The origin of such high carrier mobility can be attributed to the massless fermions in graphene which can be described by the relativistic Dirac equation. This massless character of carriers can be visualized in graphene by examining the band structure in the
Brillouin zone which shows a linear dispersion at the K symmetry point. As a result, carriers in this region have an effective mass close to zero corresponding to the high carrier mobility.
In addition to the enhancement in carrier mobility, the relativistic nature of the charge carriers also leads to other novel phenomena such as Klein tunnelling and the quantum Hall effect.
Klein tunnelling refers to the ability of electron passing through a potential barrier with a probability of 100%. This phenomenon has been experimentally verified using devices with p-n junctions [69, 70]. Another fundamental property of two dimensional materials like graphene is the quantum Hall effect which has been demonstrated experimentally even at room temperature [71-73]. In the presence of a perpendicular magnetic field, the Hall conductance’s is quantized in values that are half-integer multiples of 4e2/h which validates the relativistic behaviour of fermions in graphene.
18
Geometrically, graphene is a single layer of carbon atoms with hexagonal symmetry formed by the sp2 hybridization. In the bulk structure (graphite), the layered carbon sheets interact weakly via Van-der-Waals interaction. Because of the weak van-der-Waals force, layers of graphene with different stacking geometry can be peeled off to create new structures. In addition to the single layer structure, various other 2D structures have also been fabricated like bilayer [74- 78] and trilayer [79, 80] graphene. Graphene layer can also be rolled into a 1D structure to form carbon nanotube, or 0D structure called fullerenene [81]. In addition, an analogous material 2D single layer carbon material called graphane [82, 83] has also been fabricated. This material was first predicted in 2007 by Sofo et al. [84] who studied a new graphene structure in which hydrogen atoms are covalently bonded to carbon in a 1:1 ratio. The hydrogen atoms modify the planar configuration to a buckled configuration, i. e. the two carbon atoms no longer lie in the same z coordinate. As a result, the bonding in graphane is change from sp2 to sp3. Furthermore, the hydrogen atom also significantly modifies the band structure of graphane creating a new wide band semiconductor in contrast to the semi-metallic structure in graphene [85]. The band structure is graphane is composed of the px and py orbitals in the valence band and a hybridization of the H’s s orbital and C’s pz orbital in the conduction band. This is significantly different from the band structure of graphene in which the low energy bands are dominated by the C’s pz orbitals in the valence and conduction bands which form the π bond.
Because of the novel properties and the geometrical flexibility, numerous graphene- based applications have been proposed such as novel field effect transistors [86], photo detectors [87], bio-sensors [88], gas sensors [89], and energy storage devices [90]. A single layer graphene can be synthesized using different methods. The original method developed by Geim and Novoselov was mechanical exfoliation [67] which used an adhesive tape to remove the top layers of a highly oriented pyrolytic graphite. The graphite layers are then
19 transferred to a substrate, and the top layer will interact strongly to the substrate to form a single layer of graphene. Currently, new fabrication techniques have been developed to produce defect free graphene. These methods include the use of chemical synthesis to exfoliate the single structure [91] or growing a few layer of graphene epitaxially with the usage of chemical vapour deposition (CVD) [92].
B. Silicene and Germanene
Recently other elements like Si and Ge have also been predicted to have stable 2D structure similar to graphene [93 -95]. These new materials have been named silicene and germanene in analogous to graphene. Different from graphene, the atoms in silicene and germanene single layer structure bond together via sp3 hybridization. Due to the weaken π bond, silicene and germanene both adopt buckled configuration with the buckling parameter increases from Si to Ge. The electronic structure of silicene and germanene is revealed to consist of a linear dispersion at the Fermi level indicating a semi-metallic ground state similar to graphene. This indicates that charge carriers would also behave as massless fermions; leading to potentially high carrier mobility which is promising for next generation nanoelectronics due to their compatibility with the current Si and Ge based commercial devices. In addition, the band structure of silicene and germanene can also be modified trough surface modification through hydrogenation [96]. Both silicane and germanane are shown to be wide band gap insulator due to their saturation of the pz orbital by hydrogen.
The experimental realization of single layer free-standing silicene and germanene has been challenging due to: (i) single layer structure of Si and Ge is less stable than their bulk counterparts, and (ii) silicene and germanene form sp3 bonding rather than sp2 hybridization like graphene. Currently, free-standing structure of silicene and germanene has not been successfully realized experimentally and stable single layer structures have only been able to
20 form on metallic substrates. Silicene has been fabricated successfully on substrates like Ag
(111) [97], ZrB2 [98] and Ir (111) [99]. Recent report has suggested chemical vapour deposition is a possible to fabricate thin layer of silicon sheets (<2nm) [100]. However, the fabrication of isolated silicene using CVD still has not been reported. The experimental study of germanene has been limited since it is shown to be less stable compared to graphene and silicene. The only success of fabricating a single layer of germanene is by Li et al. [101] and
Bianco et al. [102] who report a stable germanene layer on Pt (111) substrate and hydrogenated germanene.
2.2.2. Quantum Spin Hall Insulator
Even though the experimental realization of silicene and germanene is still challenging, silicene and germanene have received great interests due to their large spin-orbit coupling in comparison to graphene. Thus, this makes them good candidates for a new novel class of novel materials known as quantum spin hall insulators [103]. These materials in 2D
(3D) form are characterized an insulating bulk state and helical edge (surface) states. The carriers in the edges (surfaces) behave like massless fermions characterized by a topological invariant number making them robust against Anderson localization and backscattering. As a result, they demonstrate great potential in various novel applications like topological quantum computing. The theory of the quantum spin Hall insulator first originated in a seminal paper by Haldane [104], and later developed by Kane and Mele [105] using a graphene-like lattice with strong spin-orbit coupling. In the following section, a brief theoretical review of the quantum spin hall insulator is outlined.
A. Quantum Hall effect
21
Most insulating materials are characterized by a gap separating the occupied valence band states from the empty conduction-band states. The model, which derived from Bloch theorem, has proven to be successful in explaining numerous materials. However, this is not a complete picture of the insulating states existing in nature. One of the simplest counter examples of this band-gap model is a phenomenon called the quantum Hall (QH) effect
[106]. This novel effect occurs when electrons confined in semiconductor hetero-structures, are placed in a strong magnetic field at low temperature. In the classical Hall effect, a sheet of metal with a current running in the clockwise direction is placed inside a magnetic field. The magnetic field then induces another current running in the counter direction, which creates a
Hall resistance depending linearly on the magnetic field. In the case of the quantum Hall effect, the Hall conductance σxy is quantized [107], and it can be described as an integer multiplication of the product of the elementary charge squared (e2) and the inverse of the
Planck constant (h), i.e. , independent of the materials. Moreover, the electrons in
the QH state travel only along the edges of the semiconductor, and the two counter-flows of electrons are spatially separated into different currents located at the sample’s top and bottom edges [108]. This separation of electrons travelling in opposite direction eliminates the scattering between the electrons which reduces the resistance of the material, and gives rise to a chiral current travelling around the edges. Furthermore, the chiral edge states are also robust against impurities and Anderson localization, hence leading to a dissipationless current.
The QH effect provides the first example of a quantum state which is topologically distinct from all states of matter known before. In mathematics, the concept of topological invariance is introduced to classify different geometrical objects into broad classes. Different objects can be classified into one topological class with a unique genus number [109]. As a result, objects with different genus number cannot be transformed smoothly into each other without destroying the object. This feature is essential in connecting the mathematical
22 concept of topology with the conductivity of material. In the quantum Hall effect, a smooth deformation in the form of impurity does not change the Hall conductance which reveals the topological origin the conductance in two dimensional electron gas. In a landmark paper,
Thouless, Kohmoto, Nightingale and den Nijs (TKKN) [110] reveal a mathematical relationship between a quantized Hall conductance and the phase coherence of the many- body wavefunctions, which brought to light a topological meaning to the quantum Hall effect. The TKKN theory shows that the integer multiplication n in the conductivity is a topological invariant known as a Chern number [111].
B. The Quantum Spin Hall effect
Even though the quantum Hall effect can create novel properties, its limitation is the requirement of an external influence like the strong magnetic field which limits its potential applications. The search for an internal mechanism existing in materials which can create edge states has been the focus of recent research. A potential candidate for such a mechanism is the spin Hall effect, which was first proposed theoretically in 1971 [112], in which impurities in a conducting material deflect the spin-up and spin-down electrons in opposite directions through Mott scattering. Recent studies [113-115] have shown that spin currents can arise in certain materials due to an intrinsic mechanism known as the spin-orbit coupling.
Accordingly, good candidates to exhibit this phenomenon are semiconductors with band structures that are significantly coupled to spin-orbit interactions [116]. Ultilizing this concept, Kane and Mele [117] and Bernevig and Zhang [118] independently developed models for a new class of material with intrinsic spin-orbit coupling known as the quantum spin Hall insulator. Unlike the quantum Hall effect, the quantum spin Hall (QSH) insulator does not require an external magnetic field. The intrinsic spin-orbit coupling now acts as an internal magnetic field which causes electron with opposite spins travelling in opposite
23 direction. Hence, the magnetic field is upward for spin-up electrons moving counter- clockwise, and points down for spin-down electrons moving clockwise, with the net field being zero. There is only a net spin current, but zero charge current. From this conceptual framework, it is possible to consider QSH system as consisting of two super-imposed non interacting QH systems with opposite Hall conductivities, one for the spin-up electrons and the other one is for spin-down [105].
In the QH state, the electronic edge states propagate along different directions on the top and bottom sides of the sample to form chiral edge states which are spatially separated from each other. In quantum spin Hall states, the channels of propagation are each split into two. The top portion of the sample contains right-moving spin-up electrons as well as left- moving spin-down electrons, while the bottom contains left-moving spin-up electrons along with right-moving spin-down electrons. While there are both left-moving and right-moving channels in each part of the sample, backscattering by non-magnetic impurities is still forbidden. The reason is that when a right-moving spin-up electron backscatters from an impurity along a clockwise path, it becomes a left-moving spin-down electron, picking up a phase of π in the process. Alternatively, a right-moving spin-up electron can also pick up a phase of -π, by travelling along a counter-clockwise path. The two reflected wavefunctions would then interfere destructively with each other, thereby allowing perfect transmission. The phase change of the wavefunction is due to a property of quantum mechanics called time reversal symmetry T leading to reversal of both spin and momentum [119]. This is similar to the behaviour of anti-reflective coating layer used in camera lenses or glasses. Reflected light from the top and the bottom surfaces interfere with each other destructively, leading to zero net reflection and thus perfect transmission [120].
A pair of backward and forward moving electrons on each sides of the sample is called a Kramers pair [119]. The numbers of Kramers pairs can take either even or odd value.
24
If this number is even, left-moving spin-down electron can scatter into the right-moving spin- up channel without picking up a phase difference, which allows it to interfere destructively, resulting in resistance around the edges. However, if the number of Kramers pair is odd, a dissipationless transport around the edges is preserved. This odd-even effect was pointed out by Qi and Zhang [121], which ties directly to a topological invariant number called Z2 number. As the quantum Hall effect is characterized by a topological invariant called the
Chern number, the quantum spin Hall phase is distinguished from the ordinary insulator phase by a Z2 topological invariant, which basically characterizes the number of Kramers pairs of edge states, and can take only two different values: even or odd [122, 123]. If Z2 is odd, there is an odd number of Kramers pairs of edge states and the system is in QSH phase, while if Z2 is even, the system is in the usual insulator phase, since a system with an even number of edge states can be transformed into a system with no edge states.
C. Two dimensional single-layer topological insulator materials
The search for a real material that displays the quantum spin Hall effect made a breakthrough in 2006 by Bernevig, Hughes, and Zhang [124]. They designed a system consisting of a HgTe thin film sandwiched between two CdTe layers to form a semiconductor heterostructure quantum well. By varying the thickness of the HgTe layer, the energy band structure of the quantum well can be engineered to affect the energy gap. They show that if the HgTe layer is thin enough, the structure is a normal insulator. When the thickness of
HgTe layer is more than a critical value, the spin-orbit effect is strong enough to invert the conduction and valence bands, which leads to a formation of a quantum spin Hall insulator.
This prediction was soon realized in experiments by the Molenkamp group [125, 126] on
CdTe/HgTe/CdTe quantum wells grown by molecular beam epitaxy. They measured the conductance of a two terminal device as a function of the gate voltage, which tuned the
25 chemical potential from the conduction band to the valence band. Inside the band gap, if it is an ordinary trivial insulator, there is a zero conductance. But if it is in the quantum spin Hall phase, there is one pair of edge states with a conductance of 2e2/h. Moreover, they varied the width of the sample. Supposing the bulk conducts, the measured conductance should scale with the sample width. The wider the sample is the higher the conductance will be. The results show that no matter how wide the sample is, the conductance is always a constant equal to 2e2/h. Hence it is convincing that the current can only conduct on the edge, not inside the bulk. Further investigations also show other quantum well structures also exhibit the quantum spin hall insulating properties [127, 128].
Recent research in the field of 2D topological insulator has focused in group IV elements such as Si, Ge [103] and Sn [129] since they share the similar structures to graphene but also have a large spin-orbit coupling similar to the Kane-Mele model. QSHI based on group III-Bi materials [130] have also been discovered with a band gap as large as 0.56 eV.
An interesting aspect of these materials is their relatively large band gap compared to the previous quantum well structures. This is due to the fact that the global bandgap can be enhanced through the use of chemical functionalization of the surfaces. As noted in the example of graphane, the band gap of graphene can be enhanced by filtering out the pz orbital. As a result, researchers in several group independently proposed various new large gap quantum spin hall insulator by modifying the surface states of Ge [131] and Sn [128] with group VII elements (F-I). For instance, the band gap of stanene can be tuned from 0.1 eV to 0.3 eV by attaching elements in group VII. After the surface is chemically modified, the pz orbital no longer dominates the low-energy band structure, and the conduction and valence edges are now composed of the px and py orbitals. The spin orbit coupling now affects these two orbital by lifting their degeneracy at the intersection of the Fermi level
26 creating a larger bandgap. This strategy reveals a new method to create larger gap quantum spin hall insulators for high-temperature application.
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34
Chapter 3: Computational method
3.1. Density functional theory
Since magnetic and two dimensional materials possess semiconducting and magnetic properties, they can be well-characterized within the field of quantum mechanics. Within quantum theory, the physics of these materials are described by interacting electrons which obey the Schrodinger equation. Within the Born-Oppenheimer approximation, the interactions of the nuclei are ignored, and the Schrodinger equation is treated as an equation of N electrons moving in an external potential and repelling each other due to Coulomb interaction [1].
(1)
th where ri is the position of the i electron, N is the total number of electrons in the system, Vext is the external field in which the electrons move, E is the total energy, m is the total energy and e is the charge of an electron. The solution of the Schrodinger equation can be used to calculate the electron density ρ(r) which is a physical observable parameter.
However, it is impossible to model the electronic properties for real materials using the Schrodinger equation since it involves solving equations containing 3N spatial variables and N number of electrons, and N is in the order of 1023. As a result, density functional theory has emerged as a powerful alternative. The foundation of DFT is based on the work done by
Hohenberg and Kohn which proved that the ground state electron density uniquely determines the external potential Vext(r) [2]. Another important point that was made
Hohenberg and Kohn is that there is a universal functional independent of the external potential that would lead to the ground state energy, but they provided no information to
35 obtain such a functional [3]. As a result, the information of the materials can now be obtained from the physical parameter charge density which only contains 3 spatial variables x, y and z.
Another leap forward in the field of DFT was made by Kohn and Sham who reformulated the problem of calculating the total energy E as a functional of electron density, thus simplifies the task to solving a set of single particle Schrodinger-like equations [4].
Within the Kohn-Sham formulism, the kinetic energy T and the energy Eext due to external potential Vext can be written as the functional of ground state charge density [1]
(3)
(4)
The electron-electron Coulomb repulsion Ec is defined in terms of density functional as
(5)
Finally, the total energy functional can be written as
(6)
where Exc is called the exchange-correlation energy functional containing all the interactions that are not described by other terms.
Now the many-body problem can be reduced to N single particle equations
(7)
The ground state of this non-interacting system is the N occupied lowest energy single particle states and the ground state density is obtained from the occupied states as [1]
(8)
The relationship between Veff and the other potentials can be described as following
(9) where Vext is the external potential, VC is the Coulomb potential and Vxc is the exchange correlation potential
36
The total energy can be now modified in terms of the energy of the individual single particle equation
(11)
To solve the Kohn-Sham equations, we apply a method called the self-consistent approach. For a given electron density, the potential Veff can be calculated, thereafter the single particle wave function are constructed using the N single particle equations described abobe, from which new charge density is estimated. If the input and output density are identical within the allowed tolerance, the self-consistent calculation terminates.
The wavefunctions obtained from the Kohn-Sham equations are expanded in some basis, and the Coulomb and exchange potential are evaluated using auxiliary basis function.
In reality, these basis and auxiliary functions can be varied depending on the DFT code.
Based on the Kohn-Sham model, the exchange correlated energy has to be approximated to solve the Kohn-Sham equations. One of the approximations proposed by Kohn and Sham is the local density approximation [5]. The LDA is derived from the homogeneous electron gas and can be written as
(12)
where is an exchange correlation function of density. The exchange Vx and correlation
Vc part of the functional can be analytically given by
(13)
for high density limit
for low density limit
where the Winger-Seitz radius rs is related to the density as
37
The LDA exchange-correlation functional have used with success, especially for metallic system because metals can be described by an effective free electron model.
Transition metals, on the other hand, need gradient corrections commonly known as generalized gradient approximation [6] for better electronic structure description because of the participation of d-orbitals which are correlated.
3.2. DFT + U method
Since both of LDA and GGA are built from homogeneous electron gas, the interactions described by the mean field approximation which is not accurate enough to properly describe the strongly correlated phenomenon. An improvement is needed to have better description of exchange correlation phenomenon. One of the methods proposed is through the addition of Hubbard-like Hamiltonian into normal DFT to describe the on-site interactions. The first expression of DFT + U was first formulated by Anisimov et al. [7, 8] and was later developed by several groups [9]. The rotationally invariant form, which is more popular in use because of its simplicity, was developed Liechtenstein et al. [10] and Dudarev et al [11]. The energy functional from Dudarev et al.’s approach can be explicitly described as
(14)
where U is the on-site correlation, J is the intra atomic exchange term, nσ is the charge
density with spin σ, and is the density matrix for correlated electron with i and i’ as the orbital on which U is added.
Within this approach, U and J are not treated individually and the effect of the Hubbard
potential U is replaced by an effective . The potential corresponding potential can be written as
38
(15)
From the above description it is clear that the occupation of Hubbard-orbital with value more than 0.5 leads to attractive potential, while the occupation below 0.5 leads to repulsive potential. Even though DFT + U is an improvement on DFT through the addition of the Hubbard potential U to improve the exchange correlated potential, there are two drawbacks in the formalism. Firstly, it neglects non-spherical contribution of U, which could be of the importance for anisotropic hybridization. Secondly, the exchange potential with
DFT + U is still expressed in relation to the homogeneous electron gas.
3.3. Hybrid functional
In order to have a better understanding of the electronic structure, we have to go beyond the electron gas approximation used to model electron-electron interactions. One way to introduce such a correction is combining the Hartree-Fock (HF) approach which describes the exact exchange and the density functional treatments of the exchange and correlation effect.
The general form of Hybrid functional [12] is
(16) in which EHF is the HF exchange term, and the coefficient determines the amount of exact exchange mixing and is fitted semi-empirically. Hybrid methods are based on the adiabatic formula
(17)
which connects the non-interacting KS system where μ = 0, to the fully interacting real systems [13].
39
There are many forms of hybrid functionals, but the two most commonly used are
PBE0 and HSE03. The PBE0 scheme was developed by J. P. Perdew, M. Errnzerhof and K.
Burke [14], who constructed the hybrid functional by mixing 25% Fock exchange with 75%
PBE exchange (one of the formulation for GGA) [6]. The electron correlation is represented by the correlation part of the PBE density functional. The resulting expression for the exchange-correlation energy thus takes the form as
(18)
A general expression can be written as
(19) where is the fraction of exact-exchange used in the calculation
The advantage of hybrid functionals is that it tends to significantly improve the quality of DFT predictions. The disadvantage is they are much more computationally expensive due to the fully non-local nature of the exchange. In order to speed up the exact exchange computation, Hyde, Scuseria and Ernzerhof [13-15] have exploited the fact that the range of the exchange interaction decays exponentially in insulators, and algebraically in metals. This new, approximate hybrid functional, called HSE, applies a screened Coulomb potential to the exchange interaction in order to screen the long-range part of the HF exchange:
(20)
where is the short-range Hartree-Fock exchange, and and are the short-range and long-range components of PBE exchange, and = 0:25 is the Hartree-Fock mixing parameter. The short-range and long-range parts are determined by splitting the
Coulomb operator into short-range and long-range parts
(21)
40 where the left term is short-range and the right term is long range, and erf and erfc and the error and complementary error functions, respectively, and is the screening parameter based on molecular basis tests.
3.4. VASP simulation software
The theoretical studies used in this thesis were based on the Vienna Ab-initio
Simulation Package (VASP). VASP utilizes the plane wave basis to calculate the ground state wave function. VASP is superior than other plane-waves based softwares like CASTEP due to the advanced implementation of different functionals and electronic properties which significantly improve the simulation results. In addition, the usage of plane wave basis also has several advantages compared to other methods based on localized orbitals basis: (i) the plan-wave functions are independent of the location of atoms, (ii) plane-waves can be easily convert from real space to reciprocal space simply using the Fast Fourier Transform, and (iii) the convergence of the basis set can be tuned by increasing the cut-off energy to expand the basis set.
The main method implemented in VASP software to solve the Kohn-Sham equation is the projector-augmented method. This approach is based on the basis that the behaviour of the plane-waves behaves differently in the vicinity of the core and at the valence states.
Electrons near the core have very high kinetic energy, thus to describe their behaviours requires a large number of plane-waves in the basis set since the core electrons oscillate rapidly, making it computational expensive. In addition, since most electronic properties are mostly affected by the intermolecular bonding formed by the valence states, it is reasonable to simplify the potential which treats the core states and preserve the real behaviour of the wavefunctions in the valence states. An approach in this framework is the pseudopotential method [16, 17] which is available in simulation codes like CASTEP. In this approximation,
41 the potential of the core and valence electrons are approximated by a pseudo-potential which behaves smoothly near the nuclei but acts as normal wavefunctions in valence states. A disadvantage of this method is all information about the real wavefunction close to the nuclei is lost, making it hard to compute properties which rely on the core region.
A more general approach is the Projector-Augmented-Wave method (PAW) introduced by Blöchl [18, 19]. This method is an extension of the norm-conserving pseudopotential method [20] and the linear augmented wave method [21] to solve the Kohn-
Sham equation. The basic idea behind the projector augmented-wave method is to linearly transform the all-electron Kohn-Sham wave function in, which gives computationally convenient pseudo wavefunctions. Consequently, the PAW method, implemented in VASP yields an accuracy approaching of the all-electrons methods but it is much less computationally demanding.
The VASP program consists of four input files: INCAR, KPOINTS, POTCAR and
POSCAR. The INCAR file contains the input command to perform the electronic calculation, the KPOINTS file contain the information of how densely the irreducible Brillouin zone is to be sampled to calculate the total energy, the POTCAR file contains the pseudopotentials for each atomic species, and the POSCAR has the geometric information of the elements in the material. In order to obtain the electronic properties of a particular system, the geometrical optimization process is first performed to obtain the optimal geometry of the materials, i.e. lattices and symmetry, by minimizing the forces acting on ions. This process is achieved by minimizing the total energy of the system based on the Hellmann-Feynman forces theorem
[22]. After the optimal geometry is achieved, the desired electronic properties can be calculated based on the WAVECAR and the CHARGCAR which contains the wavefunction solution and the charge density of the Kohn-Sham equation respectively.
42
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44
CHAPTER 4: Weak d0 Magnetism in C and N doped ZnO
Status: Published in Journal of Applied Physics
Authors: A. Pham, M. H. N. Assadi, Y. B. Zhang, A. B. Yu and Sean Li
Ref: Journal of Applied Physics, 110 art. no. 123917 (2011).
The chapter investigates the nature of magnetism in C and N doped ZnO to understand the origin of d0 magnetism in oxides. Various configurations of doped ZnO containing C and N as well as hydrogen interstitials and oxygen vacancies were considered to provide a comprehensive investigation.
All the calculations and writings were done by Anh Pham. The idea of the project was conceived by Dr. Assadi, and he assisted with the writings and visualization of the manuscript. Dr. Zhang, Prof. Yu and Prof. Li provided useful comments and supervised the projects.
The paper is reprinted with permission from the journal editor.
45
CHAPTER 5: Ferromagnetism in ZnO:Co
Originating from a Hydrogenated Co–O–Co
Complex
Status: Published in Journal of Physics Condensed Matter
Authors: Anh Pham, Y. B. Zhang, M. H. N. Assadi, A. B. Yu and S. Li
Ref: J. Phys.: Condens. Matter 25 116002 (2013).
The chapter investigates the effect of the theoretical method on the nature of ferromagnetism in hydrogenated Co doped ZnO. Various configurations of hydrogen, oxygen vacancy and Co dopants were considered. The results between the DFT and DFT+U method were compared to emphasize the role of the correlation effect in understanding magnetism in transition doped
ZnO.
All the writings and visualizations were done by A. Pham. Y. B. Zhang conceived the idea.
M. H. N. Assadi and Y. B. Zhang assisted with the writing of the manuscript. A. B. Yu and S.
Li supervised the project.
The paper is reprinted with permission from the journal editor.
53
CHAPTER 6: Critical Role of Fock Exchange in
Characterizing Dopant Geometry and Magnetic
Interaction in Magnetic Semiconductors
Status: Published in Physical Review B
Authors: Anh Pham, M. H. N. Assadi, A. B. Yu and Sean Li
Ref: Phys. Rev. B 89, 155110 (2014).
The role of the correlation effect is further investigated by studying the difference of the dopant geometry resulted from the DFT and hybrid functional method. The effect of the
Hellman-Feynman forces in the geometry optimization were carefully analysed to reveal into the discrepancies originated from difference exchange functional.
All the calculations and visualizations and the drafting of the manuscript were done by A.
Pham. Dr. Assadi conceived the idea and assisted with the writing of the manuscript. A. B.
Yu and S. Li supervised the project.
The paper is reprinted with permission from the journal editor.
62
Chapter 7: Engineering Silicene and Germanene’s
Band Gap and Conductivity with Diluted Doping of
Heavy Elements: a First Principle Study
Status: In preparation
Authors: A. Pham, C. J. Gil, and Sean Li
Two dimensional materials like silicene and germanene have attracted great attentions due to their novel properties. The purpose of this study is to engineer these materials for advanced spintronic applications by increasing their band gap and inducing n and p-type conductivity without affecting their intrinsically high mobilities. Various heavy dopants were considered for these purposes.
The idea of this manuscript was conceived by A. Pham. A. Pham did most of the calculations, the writings and the visualization and the interpretation of the data. C. J. Gil assisted with the theoretical calculations and S. Li supervised the project.
69
Engineering silicene and germanene’s band gap and conductivity with diluted doping of
heavy elements: a first principle study
A. Pham1, C. J. Gil1,2, and S. Li1*
1 School of Materials Science and Enigeering, University of New South Wales, Sydney NSW
Australia 2033
2School of Chemical Engineering, University of Florida, Gainesville Fl USA 32611
Abstract: Using first principle calculations, we demonstrate the ability of heavy elements like Pt, Au, Tl and Pb to tune the electronic structures of silicene and germanene through adsorption and subtitutional doping. With the introduction of these dopants, a global band gap as large as 0.29 eV is induced in the doped silicene and germanene at the diluted doping limit of 2%. In addition, Au Tl and Pb adsorption can produce n-type conductivity for silicenen and germanene while substitutional Tl exhibits p-type conductivity, and adsorbed Pt and substitutional Pb show intrinsic behaviours. Furthermore, dopant like Tl also preserves the high mobility comparable to the undoped silicene and germanene making it an attractive material for electronic application.
I. Introduction
Two dimensional materials like silicene and germanene have attracted great attention due to their novel properties. Geometrically, a single layer silicene/germanene is stabilized in a hexagonal lattice, which has been confirmed through numerous theoretical and experimental studies [1-10]. Similar to graphene, these materials exhibit a linear Dirac dispersion at the K valley with ultra-high mobility ~ 105 cm2 V-1 [11]. In addition, various novel effects such as the quantum spin Hall effect [12], valley polarized metallic phase [13], large spin-orbit interaction [14] and giant magneto-resistance [15] have made these materials an attractive platform for spintronic devices as well as fundamental investigation. However,
70 germanene and silicene both have a very narrow band gap which severely limits its application in practical devices.
Extensive efforts have been devoted to increase the band gap of silicene and germaene without degrading their high carrier mobilities. It has been predicted theoretically that the band gap of silicene and germanene can be tuned by applying an external electric field [16] or through mechanical deformation [17]. However, these techniques are only limited to tuning the position of the band-edges leaving the Fermi level unaffected thus the conductivity remains intrinsic. In order to fabricate silicenene and germanene based field effect transistor, a band gap opening needs to be accompanied with n and p type conductivity.
Alternative approach such as alkali adsorption has been proposed to induce n-type doping as well as a large gap opening in the order of 0.5 eV [18]. However this technique requires a large doping concentration which is impractical for device fabrication. Recently, both n and p type dopings for silicene have been suggested with the adsorption and absorption of materials like B, N, Al and P [19]. Nevertheless, similar to alkali materials the band gap opening at small concentration (3.12%) is only ~ 0.1 eV.
Motivated by a recent theoretical investigation of Au and Pt adsorbed on silicene for tunnelling field effect transistor application [20], we systematically investigate the effect of adsorbing and subtitutional heavy impurities (Pt, Au, Tl and Pb) on silicene and germanene at the diluted doping limit (~ 2%). Our results demonstrate that using these doping elements can produce a large gap opening ~ 0.29 eV and produce intrinsic, n and p type conductivity depending on the doping sites. Specifically, dopant like Tl demonstrates several interesting properties such as inducing a large band gap, creating both n and p-type conductivity, as well as retaining the high mobility comparable to the undoped silicene and germanene structures making it an attractive dopant for electronic devices.
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II. Computational Method
The first principle calculations were performed within the projector augmented wave method [21] with the Perdew-Burke-Ernzerhof (PBE) functional [22] as implemented in the
VASP code. To study the effect of dopants 5×5×1 supercells were utilized resulting in a 2% doping concentrations. The supercell structures were constructed from the fully relaxed lattice parameter of 3.87 Å (silicene) and 4.04 Å (germanene) which are similar to previous studies [16]. The plane wave basis was expanded using a cut-off energy of 500 eV and a
6×6×1 kpoints grid in the Monkhrast-Pact scheme [23]. The internal ions coordinates were relaxed with self-consistent criteria of 10-4 eV till the forces less than 0.01 eV/ Å. To avoid interactions emerged from the periodic boundary conditions a vacuum layer of 20 Å is included along with the dipole correction. Different spatial configurations of the adsorbing
(top, valley, bridge and hollow) [Fig.1a] and substituting impurities were considered in our calculations as shown in Fig. To investigate the thermodynamic stabilities of the impurities, the adsorption [19] and substitution energy [24] were calculated as follow
Eads = Esilicene + Eimp ─ Ecompound
N 1 Esub = Esilicene + Eimp ─ Ecompound N where Esilicene is the total energy of the pristine silicene, Eimp is energy of an isolated metal atom, and Ecompound is the energy of the doped configuration, and N is the number of Si/Ge atoms in the 5x5 silicene/germanene unit cell.
To calculate the electronic structure of the 5×5 supercells, a denser k-point mesh of
21×21× 1 were used. Even though silicene and germanene have non-trivial spin orbit interaction [12], in this calculation the spin orbit effect only yields a trivial spin splitting which narrow the band gap. This effect has also been observed in previous study of metal adsorption and absorption on silicene surface [19, 20]. As a result, only the effect of non-spin interaction is reported in this paper.
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III. Results and Discussion
1) Geometrical structure
Based on our adsorption energy calculation, the energy difference between the most stable configurations and the other adsorbing sites is summarized in Fig. 1b and 1c .
According to Fig. 1b, the hollow site is preferable for Pt, Au and Tl while the valley site is the most stable configuration for Pb. Previous study of metal (Mn, Fe, Co, Ti, Pd [25,26], Au and Pt [20]) adsorption on silicene also indicates the hollow site as the most stable configuration. The value of Eads for the most stable configurations of Pt, Au, Tl and Pb are calculated to be 5.83 eV, 2.37 eV, 2.11 eV and 2.66 eV respectively. The large binding energies suggest an exothermic process. Our calculated values for Pt and Au are consistent with previous study of Pt and Au doped silicene. Based on these results, the adsorption energy shows a trend of decreasing value as the atomic radius of the impurities increases from Pt to Tl indicating weakening bonds between the silicene sheet and large dopants.
However, Pb is shown to be able to form strong bonds with the Si atoms on the silicene sheet illustrated by the large adsorption energy. This is due to the similar electronic configurations between Si and Pb since they both belong to group IV materials. In addition, Fig. 1b also reveals the diffusion energy for different dopants to migrate from the most stable doping site to the neighbouring sites. Specifically, the energy barrier varies between 0.62 eV – 0.74 eV for Pt, for Au the barrier is in the range of 0.32 eV – 0.41 eV, for Tl the energy ranges from
0.13 eV – 0.33 eV, and for Pb the barrier is 0.001 eV – 0.28 eV. As a result, Pt and Au are meta-stable dopant while Pb is unstable and can easily to migrate to the other sites. Our energy difference for Pt is larger than recent work by Ni et al. [20], which can be attributed to the lower doping concentration. For germanene, similar adsorption behaviours are also observed when the heavy elements are doped on the surface of the monolayer. However, the adsorption energy is much lower for Pt, Au, Tl and Pb doping. These lower binding energies
73 is because Ge has larger electronegativity than Si which decreases the electronegative difference resulting in lower charge transferred from impurities to Ge. Thus the bonding between Pt, Au, Tl and Pb is much weaker in germanene than in silicene. In addition, our calculations also reveal the diffusion barrier for all the dopants now decreases significantly making both of Tl and Pb at the hollow and valley sites unstable, while Pt and Au still occupy the meta-stable hollow sites.
Fig. 1. a) Adsorption sites of impurity on silicene/germanene, b) and c) Energy difference between the most stable adsorption site and the other doping site of different impurities on silicene and germanene respectively.
Geometrically, the adsorption of the heavy elements modifies the spatial positions of the Si/Ge atoms covered by the adsorbants on the monolayer structure. Table 1 details the effects of the dopants on the geometry and electronic structure of silicene and germanene.
After the relaxation, all the impurities are shown to be stabilized above the monolayer in contrast to other dopants like B, N, Al, and P which are all immersed inside the single layer structure. At the hollow site, impurities like Pt, Au and Tl forms bonds with the six surrounding Si atoms which decreases the local buckled parameter of the silicene ring. The bond length between the dopants and the top most Si atom is calculated to be 2.42 Å, 2.56 Å,
3.18 Å for Si-Pt, Si-Au and Si-Tl bonds. For Pb, it occupies the valley site which pushes the underlying Si atom at the doping site and increases the local buckled parameter due to strong
74 distortion. The Si-Pb bond is calculated to be 2.86 Å. Within the germanene sheet, the bond length between Ge and the impurities increases due to weaker bonding indicated through the smaller absorption energies. The bond length for Ge-Pt, Ge-Au, Ge-Tl and Ge-Pb is now 2.47
Å, 2.63 Å, 3.18 Å and 2.91 Å respectively.
Next, we consider the substitutional configuration in which the impurity atoms replace a single Si/Ge atom. Based on our energy calculations, the substitutional configurations result in much lower for formation energy in both of germanene and silicene structures compared to the adsorption process. The substitutional energy decreases from Pt >
Pb > Au > Tl in both of germanene and silicene. This indicates that subtitutional doping is more favourable for elements like Pt and Pb, whereas it is less favourable for Au and Tl doping. However, the Esub value in is calculated to be larger in germanene than in silicene indicating a stronger binding between the impurities in germanene than in silicene in the absorption process. Examining the crystal structures, the dopants strongly distort the structure of silicene by moving above the monolayer while in germanene the dopants are shown to be fully immersed in the germenene sheet which minimizes the distortion. In addition, because of the bonding between the impurity atoms and the neighbouring Si/Ge atoms the dopants’ distance relative to the silicene/germanene plane is smaller than when they are adsorbed on the surface. As a result, the bond lengths between impurity and Si/Ge are shortened in the substitutional structure. The bonds between the impurity and Si are now calculated to be 2.34
Å, 2.35 Å, 2.55 Å, 2.68 Å for Pt-Si, Au-Si, Tl-Si, and Pb-Si. While for germanene, the bond lengths are 2.42 Å, 2.46 Å, 2.66 Å, 2.67 Å for Pt-Ge, Au-Ge, Tl-Ge, and Pb-Ge respectively.
The introduction of the impurity in the silicene/germanene sheet also alters the bond angle between the neighbouring Si/Ge atoms. For instance, the bond angle of Si-Pt-Si is 103o compared to a value of 116o in the pristine structure. However, this distortion remains locally and as the distance between the Si/Ge atom and the impurity increases the bonding geometry
75 between the Si/Ge atoms is restored to the condition in the pristine silicene/germanene. It is also worth noticing that the bonding of Pb substituting Si/Ge is similar to a Pb atom adsorbed on silicene/germanene sheet thus a potential way of producing a substituting structure is by inducing local Si/Ge vacancy which can capture the impurity atom at the vacancy site.
Table1. Changes in geometrical and electronic structure with the introduction of different dopants at different doping sites. The adsorption (Eads) and substitutional energy
(Esub) are calculated for the most stable adsorption and subtitutional configurations. dX-Si/Ge represents the bond length between the upmost Si/Ge and the impurity atom. The band gap new global band gap is presented as well the conductivity of the doped structure (p, n, intrinsic or metallic).
Eads/Esub (eV) dX-Si/Ge (Å) Band gap (eV) p/i/n/metallic
Ads Pt Si 5.83 2.42 0.29 i
Sub Pt Si 4.29 2.34 N/A metallic
Ads Pt Ge 5.00 2.47 0.26 i
Sub Pt Ge 4.52 2.42 0.23 i
Ads Au Si 2.37 2.56 0.29 n
Sub Au Si 1.49 2.35 N/A metallic
Ads Au Ge 2.14 2.63 0.25 n
Sub Au Ge 1.64 2.46 N/A metallic
Ads Tl Si 2.11 3.18 0.29 n
Sub Tl Si 0.34 2.55 0.06 p
Ads Tl Ge 2.12 3.18 0.27 n
Sub Tl Ge 0.76 2.66 0.26 p
Ads Pb Si 2.66 2.86 N/A metallic
Sub Pb Si 1.62 2.68 0.12 i
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Ads Pb Ge 2.58 2.91 N/A metallic
Sub Pb Ge 1.97 2.67 0.27 i
2) Electronic structure
The introduction of dopants also significantly modifies the electronic structure of silicene and germanene by opening a band gap of 0.29 eV even at a very small doping concentration. Fig. 2 shows the band structure when dopants are adsorbed on the surface of silicene and germanene. According to Fig. 2, the Dirac cone at the K valley can no longer be observed as heavy elements are introduced in silicenen and germanene even at diluted doping concentration. The Dirac points are now folded inside the conduction and valence band with a small gap opening at the K point in all doping configurations as highlighted in Figs. 2. A new global direct gap is now observed at the K/2 point which varies based on the dopant chemistry. As a result, the adsorption of heavy elements on the surface of silicene and germanene changes these materials to narrow gap semiconductors. Based on Table 1, the energy gaps of the doped silicene are 0.29 eV for Pt, Au and Tl doping. These energy gaps are significantly larger than previous study of Au and Pt doped silicene [20] even though our concentration is much smaller. For Pb doping, the doped silicene has a magnetic semimetallic ground state with a small magnetic moment of 0.11 µB originated from Pb’s p orbitals. In addition to the band gap opening, the Fermi level is also tuned to be inside the conduction band for Au, Tl and Pb doping indicating n-type doping while for Pt doping the Fermi level is localized inside the band gap consistent with an intrinsic conductivity behaviour. These results suggest a charge transfer process from the adsorbant to the silicene sheet which raises the chemical potential inside the conduction band as observed in n-type conductivity. Similar to silicene, a band gap opening also occurs at the K/2 symmetry point resulting in a narrow gap semiconductor. The direct gap for the doped germanene is calculated to be 0.29 eV and
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0. 27 eV for Pt, Au, and Tl doping respectively with n-type conductivity observed for Au, Tl and Pb and an intrinsic behaviour for Pt. Within our DFT calculations, we also observed a magnetic semi-metallic ground state for Pb doped germanene with a small magnetic moment of 0.12 µB from Pb’s p states.
Fig. 2 Top row: band structures of a) Pt, b) Au, c) Tl and d) Pb adsorbed on silicene. Bottom row: band structures of e) Pt, f) Au, g) Tl and h) Pb adsorbed on germanene. The band structures are calculated based on the most stable adsorption sites.
For substitutional doping [Figs. 3], the impurity atom alters the band structure of silicene and germanene significantly. Interestingly, the Dirac cone at the K valley are mostly shielded inside the conduction and valenece band except for Tl and Pb doping in silicene which preserves the band structure at K with a small gap opening. For Pt doping, the Fermi level remains between the band gap indicating an intrinsic conductivity similar to the adsorption process. However the valence and conduction band of silicene and germanene
78 differs significantly with Pt doping. In silicene, Pt doping pushes the Dirac cones inside the valence and conduction band creating a small indirect band gap while a direct band gap is observed in germanene. For Au doping, the impurity atom creates a metallic ground state for both of germanene and silicene. On the silicene lattice, Tl opens up a small gap of eV at K and also preserves the linear dispersion. In addition, the chemical potential is now shifted to be inside the valence band indicating a p-type conductivity in contrast to the n-type behaviour of Tl adsorbed on silicene. Similar p-type conductive behaviour also exists in Tl absorbed in germanene but the Dirac cones are shielded in the bulk conduction and valence band. Now a direct band gap is observed at the K/2 point with a larger energy gap compared to Tl doped silicene. With Pb, because of the similar chemistry between Si/Ge and Pb the introduction of
Pb changes silicene/germanene to a direct band gap semiconductor with intrinsic conductivity. However, the direct gap is observed at the K point in Pb doped silicene, while the valence band maximum and conduction band minimum are shifted to the K/2 point in Pb doped germanene. As a result, the introduction of impurity atoms in silicene and germanene results in the gap opening regardless of the method of doping. This is due to the charge transfer process as well as the breaking of inversion symmetry as suggested in earlier theoretical studies For dopants like Au and Tl, these dopants create a charge transfer between the monolayer to the impurity (p-type) or from the impurity to the single layer (n-type) which creates an internal electric field resulting in band gap opening. While for doping material like
Pb (substitutional) and Pt, these dopants do not provide/accept carriers to the silicene/germanene but the gap opening can still occur due to the existence of the dopants on on one side of the surfaces which break inversion symmetry similar to the ABA stacking pattern in bilayer graphene. However, it is interesting to note that this gap opening is different between germanene and silicene for Pb doping. This phenomenon can be understood due to the strength of the π bonding in group IV elements. The π bond in silicene is stronger than in
79 germanene which explains the high buckling parameter in germanene to enhance the overlapping between the π and ϭ bonds similar to material like stanene. Since Pb has the weakest π bond compared Si and Ge, its formation in a 2D structure will be expected to result in a high buckling parameter. As a result, its introduction in 2D silicene causes a large distortion which is demonstrated by its large displacement above the hexagonal plane. While in germanene, since it has higher buckling parameter than Si the Pb atom can easily intergrate to the 2D structure results in small distortion. Consequently, the hopping between the Bloch wavefunctions of Ge and Pb is larger than in Si and Pb. The gap opening in Pb doped Si can be originated mainly from the bond breaking symmetry, but for Ge the gap opening is contributed both by the breaking of inversion symmetry and the hopping between Ge and Pb atoms resulting in a larger band gap.
To further understand the effect of doping sites on the band structure the density of states of all the elements are investigated in Fig 4. Based on Figs. 4a and 4i. the Pt’s d orbitals are shown to be completely occupied inside the valence band of silicene/germanene which explains the intrinsic behaviour of Pt doping for adsorption process in silicene and germanene. However, doping Au introduces an impurity band [Figs. 4b and 4j] which hybridizes strongly with the pz orbitals in the conduction band. This effect is further enhanced for the substitutional configuration in which the delocalized Au’s s [Figs. 4f and 4j] and Pt’s d [Figs. 4e and 4m] orbitals bridges the gap between the conduction and valence band resulting in a metallic ground state for Au/Pt absorbed in silicene and germanene. As it has been noticed in Figs. 4c, 4k and Figs. 4g, 4o, Tl doped silicene and germanene shows opposite conductivity behaviours depending on the doping mechanism. In the adsorption process, Tl’s p orbitals are shown to strongly overlap with Si’s pz orbitals in the conduction band causing the n-type conductivity. Conversely, when Tl substitutes a Si/Ge atom Tl’s delocalized p orbitals are now strongly hybridized with Si/Ge’s pz orbital in the valence band
80 and spread across the Fermi level resulting in the p-type conductivity. It should also be noted for germanene, the px and py also strongly hybridizes with the impurity atom which explains the difference in the band structure of doped silicene and germanene. Similar changes in orbital hybridization between the subtitutional and adsorbing sites have also be found in other elements in group III like B and Al which also creates p-type doping in the substitutional doping.
Fig. 3. Band structures of the subsitutional doping of different impurities. Top row: band structures of a) Pt, b) Au, c) Tl and d) Pb absorbed in silicene. Bottom row: band structures of e) Pt, f) Au, g) Tl and h) Pb absorbed in germanene.
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Fig. 4. Density of states (DOS) of different doping configurations in silicene (top two rows) and germanene (bottom two rows). First row: DOS of adsorption sites in silicene with a) Pt, b) Au , c) Tl and d) Pb. Second row: DOS of substitutional sites in silicene with e) Pt, f) Au , g) Tl and h) Pb. Third row: DOS of adsorption sites in germanene with i) Pt, j) Au , k) Tl and l) Pb. Fourth row: DOS of substitutional sites in germanene with m) Pt, n) Au , o) Tl and p)
Pb
One of the most important features of germanene and silicene is the high mobility of the massless Dirac fermions originated from the Dirac dispersion at the K point. Thus, it is important to understand the effect of dopant on the nature carrier’s mobility. The introduction of dopant (substitution/adsorption) alters the band structure significant which causes the
Dirac cones to be shielded inside the valence and conduction band as shown in Figs. 2 and
Figs. 3. Since only substitutional Tl preserves the linear dispersion in silicene and also produces p-type conductivity, we first focus our analysis in Tl doped silicene. The Fermi
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1 E velocity is calculated using v which characterizes the high mobility in Dirac F k materials like silicene. This property is weakly anisotropic as the band structure moves away from the K valley due to the symmetry breaking of Si-Si bonding in the presence of impurity.
The Fermi velocity is slightly reduced from 7.7×105 m/s (undoped) to 3.47×105 m/s (Tl doped). In the case of Tl doped germanene, even though the global band gap is shifted away from the K point but a linear dispersion still exists at the K valley which indicates a high
Fermi velocity. The Fermi velocity in Tl doped germanene is calculated to 8.85×104 m/s, which is smaller than the value 6.3×105 m/s of the undoped structure. This larger decrease in
Tl doped germanene can be attributed to the fact that the Tl is fully immersed in the germanene structure, thus from a tight binding perspective the orbital hopping between Ge-Tl needs to included which breaks the symmetry of the Ge-Ge bonds. This is in contrast to Tl doped silicene in which the Tl atom is suspended above the silicenen sheet. This reduces the
Si-Tl hopping thus a global Dirac dispersion is maintained, yielding a much smaller reduction in the Fermi velocity. For the n-type conductivity in Tl adsorption, the Fermi velocity shows a small reduction in silicene (1.17 × 105 m/s), and in germanene (2.41×105 m/s). As a result,
Tl doping is shown to retain the high Fermi velocity in both germanene and silicene, and accompanied with the n and p-type conductivity and large band gap making Tl an attractive dopant for electronic devices like field effect transistors.
IV. Conclusions
In conclusions, we have demonstrated that by utilizing heavy elements the electronic of germanene and silicene can be modified for suitable electronic applications. Our results show that only a diluted doping concentration is required to open a band gap as large as 0.29 eV. In addition, the small doping concentration preserves the high mobility of the undoped silicene and germanene which manifests through the persistence of the Dirac cone in both of the adsorbed and absorbed structures. In all those dopants, Tl shows promising properties
83 which can induce both the n and p type conductivity while Pb and Pt demonstrates an insitrinsic conductivity behaviour and Au shows mainly n-type conductivity when it is adsorbed on the surfaces. These results open up a possibility for using heavy elements to tune different electronic properties of 2D materials like silicene and germanene.
Acknowledgement
The research was conducted with the financial support from the Australian Research Council through the Future Fellowship and the Discovery Project. The computational resource was supported by the National Computing Infrastructure and Intersect Ltd.
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CHAPTER 8: Orbital Engineering of 2D Materials with Hydrogenation: A Realization of Giant Gap and Strongly Correlated Topological Insulators
Status: Being Reviewed by Referees in Physical Review Latter (Manuscript #: LZ14416)
Authors: A. Pham, C. J. Gil, and Sean Li
Quantum Spin Hall insulator (QSHI) is a novel class of 2D materials with an insulating bulk state and metallic spin-polarized edge states. These materials have tremendous potential for spintronic applications. However, the band gap of the current QSHIs is too small for practical application. The purspose of this study is to investigate new QSHIs with large band gaps and exotic properties. The electronic properties of various hydrogenated 2D metallic materials were studied as potential novel QSHI.
A. Pham did most of the calculations, the writings and the visualization and the interpretation of the data. C. J. Gil assisted with the theoretical calculations and S. Li supervised the project.
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Orbital engineering of 2D materials with hydrogenation: a realization of giant gap and strongly correlated topological insulators
Anh Pham,1 Carmen J Gil,1,2 Sean Li1*
1School of Materials Science and Engineering, University of New South Wales, Sydney
Australia 2033
2School of Chemical Engineering, University of Florida, Gainesville Florida USA 32611
Two dimensional (2D) topological insulators known as quantum spin Hall insulators
(QSHIs) are a novel class of materials characterized by an insulating bulk state with metallic edge states1,2,3. Several 2D materials with quantum well structures4,5,6,7 have been reported to be QSHI but their band gaps are not large enough for a wide range of practical applications. Recently, it has been reported that the band gap value of 2D
QSHI could be increased through chemical functionalization of the materials surfaces8.
This surface modification alters the orbital contribution at the low energy level and confines the effect of the spin orbit coupling in two degenerate orbitals. Our first principle calculations demonstrate that hydrogenation can function as a filter for p and d orbitals, thus resulting in novel QSHIs with giant gaps ≥ 1eV. Most importantly, the results reveal that the orbital engineering with hydrogenation process in the 4d and 5d elements, like Mo, W and Re, leads to a novel class of strongly correlated 2D topological insulators with large gaps.
The novel two-dimensional topological insulating materials first predicted by Kane and Mele9 have sparked off extensive research to develop new QSHI with large gap. In the
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Kane-Mele model, the insulating bulk gap is determined primarily by the strength of the spin orbit coupling (SOC). Based on this approach, new 2D materials, which composed of heavy elements8,10,11,12, have been studied as the candidates for large gap QSHI. These materials all share a key feature: the bulk gap originates from either the next nearest neighbouring interaction in the pz orbitals like in Si, Ge, and Sn, or the hoppings between px, py and pz orbitals like in Bi-bilayer and III-Bi. Therefore, these materials are not truly 2D topological insulator since the spin orbit effect is not confined on the in-plane px and py orbitals.
Recent studies have highlighted the importance of orbitals selection in determining
8,13 the bulk energy gap . For instance, stanene has a small bulk gap of 0.1 eV when the pz orbital dominates the effective low energy band structure. However, through fluorination, the spin-orbit interaction can be confined on the in-plane px and py orbitals of stanene, thus enhancing its bulk gap to 0.3 eV. As a result, orbital filtering through surface modification is demonstrated to be an effective technique to produce large gap QSHI. This raises an interesting question: can the orbital filtering effect be applied to the heavier elements with p orbitals as well as metallic elements with 4d and 5d orbitals ?
In this letter, we address the aforementioned question by theoretically demonstrating that hydrogenating the surfaces of 2D honeycomb lattice of materials with p and d orbitals can produce novel QSHI with giant gap. In the materials with p orbitals like 2D Pb and Bi, hydrogen saturates the π bonding composed of the pz orbitals, causing the large SOC to affect the px and py orbitals. For the potential 4d and 5d metals, a half-filled d valence band is essential to enable the character of d electrons changed from itinerant to localized by hydrogenation, similar to actinide elements like Pu and Am14. Therefore, our study focuses on Group VI metals due to the nature of their electronic structures with 4 and 5 d electrons as well as their neighbouring element. In two dimensions, the hexagonal crystal field splits the
2 2 five d orbitals of the metal atoms into three groups: two degenerate dxy and dx -y orbitals,
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2 2 doubly degenerate dyz and dxz orbitals, and singly degenerate d3z -r orbital. Due to this characteristic, the selection of a particular set of degenerate d orbitals is important to form a single band crossing at the Fermi level. Such a selection can be realized through hydrogenation as described above. Most importantly, coupled with a strong SOC, this process produces a new class of strongly correlated QSHI with large gap.
To demonstrate our insights, we perform ab-initio calculations of PbH, BiH and the hydrogenated 4d and 5d metals. The optimized 2D geometry and band gap value of all the materials are summarized in Table 1S and Fig. 1. For p-based materials, we strategically focus on the electronic structure of Pb, which is a representative of the p-based materials to highlight the effect of hydrogenation. In unhydrogenated Pb (Figs. 2a and 2b), the band gap
opening happens at the K point between the odd pz states at the low energy level in the presence of SOC. After the hydrogenation (Figs. 2c and 2d), the chemical functionalization not only saturates the pz orbital but also alters the parity of the occupied and unoccupied states from odd to even near the Fermi level. The odd Bloch states are now located deep inside the valence band while two even Bloch states dominate the valence and conduction band minimum. With the SOC effect, a giant energy gap of 0.98 eV occurs between the two
even px,y states at the Γ point. To confirm the nontrivial topological property, the topological invariant ν of Pb and PbH is calculated by counting the parities of the occupied
Bloch states based on the Fu and Kane’s formula15. The calculated value of ν is equal to 1,
verifying that PbH is a Z2 topological insulator. Similar band opening between the px,y states also occurs in the case of BiH (Fig. 1S), which has a giant indirect gap of 1.09 eV due to the intrinsically large SOC of Bi. The large bulk gaps observed in BiH and PbH validate our initial hypothesis that the energy gap can be enhanced by selecting the appropriate orbitals (px and py) in two-dimensional materials.
89
Figure 1| Crystal structure and the band gap of the hydrogenated materials. a, Top view of a two-dimensional hydrogenate d material. b, the side view of PbH (top) and WH
(bottom). c, lattices and the corresponding band gap of the hydrogenated 2D materials.
Table 1S| Lattice parameters, buckling parameters d and energy gap of the structures investigated in our study. Here we omitted 2D Cr since the geometry optimization did not converge after numerous attempts indicating an unrealistic structure.
The band gap and lattice were calculated within the DFT method except for the results of
ReH which were obtained using DFT+U method with Ueff = 4 eV since without Ueff ReH is metallic.
Lattice (Å) d (Å) Energy gap (eV)
CrH 3.56 0.122 -0.129
Mo 4.06 0 0.318
MoH 4.09 0.097 0.19
W 4.15 0 -0.117
WH 4.18 0.048 0.263
Re 4.09 0 N/A
ReH 4.07 0.409 0.355
Pb 4.06 1.97 0.319
PbH 5.03 0.807 0.976
90
BiH 5.51 0.096 1.089
Figure 2| Band structure the projected p orbitals of Pb and hydrogenated Pb. a, the positions of the p orbitals on the band structure of Pb at different valleys. b, the band structure of Pb in the presence of the full strength SOC. c, the projected p orbitals on the band structure of PbH. d, the gap opening of PbH at Γ due to the SOC effect. The odd parity is denoted as ─ and even as +.
Figure 1S| Crystal structure and the band structure of BiH. a, Top view and side view of a two-dimensional hydrogenated Bi. b, Band structure of BiH without SOC (left) and with
SOC (right).
91
Table 2S| Parities of values of the wavefunctions at the high symmetry points of Pb,
PbH, WH and MoH. The first column is the symmetry points (M, Γ, K), the second column contains the parities of the occupied bands: odd (─) and even (+), the last column is the overall parity at a particular symmetry point. The topological invariant ν is presented in the last row.
Pb
2M ─ + ─ + + + ─ ─ + + + + + + + + + + + ─ + ─ + + + ─ + + + + ─ + +
Γ ─ + + + + + + + + + + + + + ─ ─ + + + + ─ ─ ─ ─ ─ + ─ + + ─ ─ ─ +
K + + + + + + + + + + ─ + + ─ + + ─ ─ ─ ─ ─ ─ + + + ─ ─ + ─ + ─ ─ ─
ν 1
PbH
3M + + + + + + + + ─ ─ ─ ─ ─ ─ ─ ─ ─ + + + + + + + + + + + ─ + + + + + + + + + ─ ─ +
Γ + + + ─ ─ ─ ─ ─ + + + + + + ─ ─ + + + + + + ─ ─ ─ ─ ─ ─ + + ─ ─ ─ ─ ─ ─ ─ ─ + + ─
ν 1
WH MoH
2M ─ ─ + + + + ─ ─ ─ ─ + + ─ ─ + 2M ─ ─ + + + + ─ ─ + + ─ ─ ─ ─ + +
Γ + + + + ─ ─ ─ ─ ─ ─ ─ ─ + + + Γ + + + + ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ + +
K + + + + ─ + ─ ─ ─ + + ─ + + ─ K + + ─ + + + ─ ─ ─ + + ─ + ─ ─ ─
ν 1 ν 1
ReH
3M ─ + ─ + + ─ + + + + + + + + + + ─
Γ + + + + ─ ─ ─ ─ ─ + ─ + + + ─ ─ +
ν 1
92
For the materials containing 4d and 5d metals, we focus on Group VI metals with tungsten (W) as an example. Fig. 3a shows the band structure of a graphene-like tungsten sheet. The conduction and valence bands at Γ point are degenerate with a small energy gap,
2 2 which mainly consists of the doubly degenerate dxy and dx -y . At the K point, the orbitals
2 2 overlapping between the singly degenerate d3z -r and doubly degenerate dyz and dxz states form four band crossings. With the inclusion of SOC (Fig. 3b), the degeneracy at the Γ point is lifted while a gap occurs between the doubly degenerate dyz and dxz. However, 2D W still
2 2 remains semi-metallic with a band gap of -0.117 eV due to the d3z -r being unaffected by the
2 2 SOC. Similarly, for 2D Mo, which d3z -r orbitals contribute less to the band crossing at the
Fermi level, the presence of SOC lead to a gap opening in Mo between the dxz,yz states, thus
2 2 resulting in an insulating state. These results clearly indicate that the d3z -r orbital prohibits the metal to insulator transition in the bulk structure.
2 2 In order to filter out the d3z -r orbitals, the surface of W is chemically modified with the hydrogen, similar to the modification applied on Pb and Bi to from PbH and BiH. As demonstrated in Figs. 3c and 3d, the band structure of WH reveals a single linear dispersion occurring at the K point, which is composed of the doubly degenerate dyz and dxz, while the
2 2 dxy and dx -y states at Γ point are unaffected. In this case, the effect of SOC causes a band gap opening with the topological invariant ν = 1, making WH a Z2 topological insulator.
Further examination of the band structure reveals nontrivial behaviours at the Γ and K valleys, which determines the global band gap. At the Γ valley, SOC creates a spin splitting in the conduction and valence band similar to that in MoS2. However, at the K point, SOC
creates a band gap between the even d yz,xz states, which are topological nontrivial because
of the band inversion mechanism between the odd s and even d yz id xz states. Such an anomalous behaviour at the two different valley points indicates a coexistence of the massles
93 and massive Dirac fermions in the same 2D material, demonstrating a potential to tune these novel properties via optoelectronic method.
Figure 3| The orbital filtering effect in group VI metals through hydrogenation demonstrated through band structure calculation. a, the band structure of the W which contains the five d orbitals at different symmetry points. b, the band structure of W in the
2 2 presence of the full strength SOC. c, the effect of hydrogen in filtering out the d3z -r at K. d, the gap opening of WH at K due to the SOC effect. The odd parity is denoted as ─ and even as +.
Since the energy gap between the valence and conduction bands of the hydrogenated
Group VI metals is mainly composed of the doubly degenerate d yz,xz states, this value is sensitive to the strongly correlated effect. To investigate the robustness of the topological property, the DFT+U method were utilized to investigate the effect of Hubbard potential Ueff on the band structure of CrH, MoH and WH [Fig. 4a-4d]. Without the SOC effect, the Ueff opens a small gap between the valence and conduction at the K point. However, with the
SOC effects, the global energy gap is significantly changed. The band gap value as a function of Ueff is presented in Fig. 4d. It shows that CrH remains in a semi-metallic ground state for all Ueff. For WH, although the energy gap increases with the effect SOC and Ueff at the beginning, the system becomes semi-metallic when Ueff is greater than 3 eV. For MoH, a
94 giant band gap (1.28 eV) is observed when Ueff = 5 eV. Such a large value for the energy gap is comparable to BiH even though Mo has a much smaller SOC strength due to its smaller atomic radius.
Subsequently, our further analysis focuses on MoH and WH due to the semi-metallic nature of CrH. The band structure as a function of the Coulomb repulsion reveals a nontrivial interplay between the SOC and the Ueff at the Γ and K valleys (Fig. 4e). According to Figs. 2S
and 3S, the Ueff potential localizes the d yz,xz states, resulting in a large energy gap at K with the effect of SOC. However, the spin and orbital interaction is significantly different at the Γ point for MoH and WH. Without Ueff, the SOC only creates the spin splitting while the orbital
d 2 2 degeneracy of xy,x y remains. As the electron-electron interaction becomes more prominent, both of the spin and orbital degeneracy are significantly affected depending on the coupling limit of the Coulomb effect and the strength of the spin orbit effect. For WH, the
d 2 2 d xy and x y strongly overlap in all the values of Ueff while the spin splitting increases for large Ueff (> 3 eV), resulting in a narrower band gap. However, a different orbital interaction occurs in MoH at the strong Coulomb interaction limit. The Ueff creates both an orbital and spin splitting, which removes the degeneracy of the doublet state and
. These orbitals are no longer overlapped with the effect of SOC and Ueff but become two distinct states. This orbital splitting is synonymous with the Jahn-Teller effect. Thus, the
and states are localized in the valence and conduction bands by the Jahn-Teller stability, resulting in a giant band gap. Such an effect is absent in WH because of the
delocalization of the doublet dxy and , thus causing the insulator-to-metal transition.
95
Figure 4| The effect of the strongly correlated property on the band gap of hydrogenated Cr, Mo and W. a, band structure of CrH without SOC (top) and with SOC
(bottom) calculated with Ueff =3 eV. b, the band structure of MoH without SOC (top) and with SOC (bottom) calculated with Ueff =3 eV. c, the band structure of WH without SOC
(top) and with SOC (bottom) calculated with Ueff =2 eV. d, the dependence of the global band gap of CrH, MoH and WH at different values of Ueff. e, a schematic diagram of the effect of the SOC and Hubbard potential on the band structure at different valleys. Here the K and Γ symmetry lines are chosen since they have nontrivial changes in the band structure. The
figure demonstrates the competition between the d xy,x2 y2 at Γ and d yz,xz at K. An insulating
state occurs when the d xy,x2 y2 states are localized either through the Jahn-Teller effect
(MoH) or in the weak electron-electron interaction limit (WH) and the band energy is determined explicitly by the spin splitting at the K point between the states. An
96 insulator-to-metal occurs when the doublet d xy,x2 y2 become delocalized in the strong
Coulomb interaction resuting in band crossing of the originally located in the conduction band
Figure 2S| Band structure of MoH at different values of Ueff. a, Ueff = 0. b, Ueff = 3 eV. c,
2 2 Ueff = 5 eV. The projected d orbitals (dxy and dx -y ) of Mo are also shown on the band structure.
Figure 3S| Band structure of WH at different values of Ueff. a, Ueff = 0. b, Ueff = 2 eV. c,
2 2 Ueff = 4 eV. The projected d orbitals (dxy and dx -y ) of W are also shown on the band structure with the same colour scheme as Figure 2S.
97
Since Mo and W have the electron configurations of 4d55s1 and 5d46s2, we also study the effect of hydrogenation in the neighbouring elements with similar number of d valence electrons. Our results show that only Re (5d56s2) exhibits a nontrivial topological property in
2D. Fig. 5 demonstrates a metal-insulator transition in the hydrogenated Re with a large direct bulk gap of 0.355 eV at Γ. The hydrogenation process filters out the d bands at the
d 2 2 Fermi level, resulting in the doublet states dxy and x y to dominate the low energy level. The hexagonal crystal field is modified by Ueff, which localizes the doublet states, and the SOC lifts the degeneracy to create an insulator. This band opening mechanism is similar to the actinide strongly correlated 3D topological insulators like PuTe and AmN14. As a result, the hydrogenation is demonstrated to be an effective technique to induce topological phase transition in 4d and 5d metals, resulting in strongly correlated QSHI.
Figure 5| The evolutions of Re’s band structure under the effect of hydrogenation. a, the band structure of 2D Re calculated with the spin-orbit effect. b, the band structure of ReH obtained with the SOC. c, the band structure of ReH calculated with Ueff and SOC (Ueff = 4 eV).
In addition, one of the main features of the 2D topologically nontrivial materials is the protected gapless edge states. To investigate these properties, the armchair edges of the unhydrogenated and hydrogenated Pb, and the hydrogenated W structures were constructed
98 with widths of 15.6 nm, 9.6 nm and 8.5 nm for study [Fig. 6], respectively. These values are chosen so that the separation between the top and bottom edges is large enough to overcome their hybridization. A gapless state is observed at the Fermi level for both of Pb and PbH at the Fermi energy, making them the perfect candidates for topological spintronic devices.
However, the edge states of WH are shielded deep inside the valence band. At such a deep energy level, the edge states of WH cannot be tuned with the conventional doping methodology, but it is possible to observe theses deep states via modern method such as the scan tunnelling microscope (STM). Interestingly, there is a qualitative difference between the low energy gapless states of the unhydrogenated and hydrogenated structure. For Pb, a linear dispersion is observed at the Γ point while for PbH a linear band crossing occurs at Γ along with an energy splitting in the valence band (24.7 meV) and conduction band (34.1 meV).
The band splitting originates from the broken inversion arising in the 1D structure.
Figure 6| The zizag edge states of Pb, PbH and WH. a, the edge state of Pb calculated with using the zizag naribbon configuration with a width of 15.6 nm. b, the band structure of Pb form the zizag naribbon configuration with a width of 9.6 nm. c, the zizag edge of WH form with a width of 8.5 nm.
Finally, we discuss possibilities to experimentally realize these novel two dimensional materials on substrates. One potential strategy is to utilize these materials with strong
16 coupling to Pb and Bi like SiH (111) , which can saturate the pz orbitals similar to a free standing PbH or BiH. However, the drawback of this method is the strong chemical bonding
99 between the substrate and the material, thus significantly reducing the band gap.
Alternatively, these materials can be fabricated on weak interacting substrates like boron nitride or MoS2 without altering the electronic structure. On these substrates, the materials are stabilized through the weak attractive Van der Waal forces, and the surfaces of the two- dimensional sheets can be exposed to hydrogenation to further enhance the band gap. In addition, the hydrogenated Group VI materials can potentially be fabricated through
17 removing sulphur atoms on MoS2 using STM technique leaving a single layer of metals exposed. This 2D metal sheet can be hydrogenated subsequently to create a topological transition.
Method
The calculations were performed using the projected augmented wave method18 with a cut- off energy of 500 eV used to expand the plane wave basis as implemented in the VASP package. A kpoint mesh of 21×21×1 was used within the Monkhrast Pact method19. To optimize the geometrical structures, the lattice parameters and the internal ions coordinates were relaxed with the self-consistent criteria of 10-6 eV until the forces were less than 0.01 eV/Å. The calculations with and without spin-orbit interaction were conducted using the PBE exchange correlation functional20. For the DFT+U calculations, the Dudarev scheme was
21 utilized with an effective Hubbard potential Ueff defined as the difference between the
Coulomb potential U and the on-site exchange interaction J. The value of J is set to 0.5 eV in all calculations. The single layer structures were constructed with a vacuum layer of 20 Å to avoid the interactions between the layers. For the nanoribbon configurations, we included a vacuum layer of 20 Å in the y and z directions to confine the structure in one dimension. The parities were calculated by extracting the coefficients and wavefunctions from the output
WAVECAR of occupied valence bands. The bands are chosen from -10 eV to 0 eV since the
100 lower energy bands will not affect the parities of the wavefunctions at the high symmetry points. The parity results for Pb, PbH, MoH, WH and ReH are shown in Table. 2S.
Acknowledgement
This research was supported by the Australian Research Council Discovery Projects and
Future Fellowship. A. P. acknowledges the support of Prof. Andrew Feenstra at Carnegie
Mellon University for providing suitable codes to extract information from the WAVECAR binary file for parity calculation. C. J. Gil acknowledges support for her contributions from the Buick Achievers Scholarship Program (General Motors Foundation), the American
Chemical Society (ACS) Scholars Scholarship (American Chemical Society), and NSF IRES award number OISE-1129412. The computational resources were supported by Intersect Aust
Ltd and the National Computing Infrastructure.
Author contributions
A. P. conceived the idea. A.P and C. J. Gil performed the first principle calculations. A.P calculated the parities of occupied valence bands. S. Li supervised the project. All the authors contributed to the writing of the manuscript.
Note: While writing this paper, we have discovered two recent published works22,23 which focus on studying large gap 2D topological insulator with px and py orbitals. However, their works focus exclusively in the traditional materials with p orbitals like Bi. In our case, we generalize the chemical functionalization method to other materials with d orbitals to realize new QSHI.
101
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Chapter 9: Conclusions
9.1 Summary of studies
This thesis has outlined several advancements in the study of spintronic materials using density functional theory technique. In the first section (chapter 4-6), these chapters detail the theoretical study of ferromagnetism in wide bandgap semiconductor oxide like ZnO with two important conclusions. First, the nature of that d0 ferromagnetism was investigated which shows confirms the effect of C as a possible doping candidate to produce ferromagnetism. However, the magnetic interaction was shown to be inhibited by the formation of complex like structure like C2, which can explain the difference in various experimental studies of d0 magnetism in doped ZnO. In addition, the interaction between the dopants (C and N) and their effects on the ferromagnetism was also thoroughly investigated.
The results show that the potential sources of carrier donors such as hydrogen interstitial and oxygen vacancy show no significant effect on C indicating C as the origin of d0 ferromagnetism. Secondly, the significance of the theoretical method is emphasized in understanding magnetism in wide bandgap material. In order to investigate the effects, the system of hydrogenated Co doped ZnO was studied since it has been widely attributed to produce long-range ferromagnetism. The studies were conducted using the DFT+U method and the hybrid functional which both reveal a new finding in regards to previous studies.
Within DFT+U method, the results demonstrate the ferromagnetism in ZnOCoH is possible even with a low doping concentration of Co (~ 5.6%). Further the results also reveal the origin of this ferromagnetism as a very specific geometrical arrangement of hydrogenated Co.
In order to further understand the role of dopant geometry in mediating ferromagnetism, the system of ZnOCoH was further studied using the hybrid functional method. The dopant geometry obtained from the hybrid method is significantly different from those obtained from
105 traditional DFT method. The discrepancy is attributed to the difference in characterizing the
Hellman-Feynman forces when using functional like LDA/GGA and hybrid functional. Since the conventional DFT method like LDA and GGA does not include the Coulomb interaction, the results of these studies can yield inaccurate nature of mobile dopant complex which can result in a mischaracterization of important electronic properties like ferromagnetism. In the case of hybrid functional, the exchange and correlation effect is corrected with a short range
Coulomb potential allowing for a much more precise description of long range forces like van-der-waals forces which can be vital in obtaining the correct understanding of dopants in magnetic materials.
Since diluted magnetic semiconductors require dopants to generate spin-polarized current, the doping effect can generate unwanted defects as demonstrated in the studies of
ZnO. The defects can then inhibit the ferromagnetic properties. Thus, the study of spintronic materials is shifted to focus on 2D materials since they have intrinsic spin-polarized properties without the need for external doping. Materials like silicene, germanene and 2D topological insulators are chosen because of their ultra-high carrier mobilities and novel properties making them perfect spintronic candidates. However, a major drawback of these materials is that they have very narrow band gap which limit their potential applications.
Thus, the focus of the studies is to increase their bulk band gaps for the practical application in the next generation of electronic devices. For materials like silicene and germanene, the theoretical calculation shows that their electronic properties can be tuned from a semi- metallic state to a semiconducting state by breaking the hexagonal symmetry through the doping of heavy elements (Pt, Au, Tl and Pb). Since the doping concentrations were small
(~2%), the high carrier mobility of doped silicene and germanene is preserved which is indicated through the persistence of Dirac cone in the band structure. In addition, dopants like
Tl, Au and Pt can also induce n or p type conductivity depending on their doping site
106
(adsorption or absorption), which also increase the band gap of the host materials to ~ 0.29 eV. Such a large band gap opening is comparable to other dopants but the required doping concentration is significantly smaller making heavy elements good candidates to engineer 2D materials. In addition to be effective dopant, heavy elements in 2D form with hexagonal symmetry can also create novel quantum spin Hall insulators (QSHIs) due to their large spin orbit interaction. Specifically, the theoretical simulations show that 2D hydrogenated Pb, Bi ,
Cr, Mo, W and Re are giant band gap QSHI. This is due to the orbital filtering effect of
2 2 hydrogen which saturates unwanted orbital like pz (Pb and Bi) and dz -r (Cr, Mo, W and Re).
As a result, the low energy physics of these 2D hydrogenated materials only contain two
2 2 degenerate orbitals like px/y (Bi and Pb) or dxy/dx -y (Re) or dyz/dxz (Mo and W). As the spin- orbit interaction becomes effective the degeneracy in the p and d orbitals is lifted resulting in a non-trivial topological phase transition which indicates through a gap opening in the bulk state. For materials like hydrogenated Mo, W and Re, a large bulk band gap can only occur in the presence of the Hubbard potential, making them candidates for a new class of 2D strongly correlated topological insulator. Specifically, a large bandgap ≥ 1 eV exists in MoH making it a giant bandgap Mott topological insulator. This novel phenomenon can be understood as a nontrivial interplay between the correlation effect between d electrons and the effect of the spin-orbit interaction. The Hubbard potential which characterizes the strongly correlated effect localizes the d electrons while the spin orbit coupling lifts the spin degeneracy at the specific symmetry which results in a large bulk gap. As a results the hydrogenated 2D topological insulators are excellent candidates for both practical device applications as well as fundamental study into new topological materials.
9.2. Future studies
Future research will focus in the further studies of these novel two dimensional materials. Specifically, it is important to study the parities of these novel materials with
107 external effect like doping or strain since they can provide platform for other novel phenomenon such 2D topological superconductivity. In addition, experimental realization of these materials is also important as a result further studies on the effects of different substrates will be carried out to decide suitable candidates which can maintain the large gap topological properties. Another aspect of the study of novel spintronic materials like topological insulator is to generalize the orbital filtering technique from 2D to 3D to predict potential candidates for large gap 3D topological materials. Instead of hydrogenation, other effects such as strain or building multi-layer/heterostructure interfaces can be viable techniques to filter unwated orbitals in 3D materials. These subjects will be the topic of future research to investigate more exotic topological materials in 3D crystals.
108
APPENDEX: An Ab-Initio Study of Transition
Metals Doped with WSe2 for Long-Range Room
Temperature Ferromagnetism in Two-Dimensional
Transition Metal Dichalcogenide.
Status: Published In Journal of Physics Condensed Matter
Authors: Carmen J Gil, Anh Pham, Aibing Yu and Sean Li
Ref: J. Phys.: Condens. Matter 26 306004 (2014)
This purpose of this paper is to investigate the ferromagnetic properties of WSe2 with transition metal dopants as a novel candidate for 2D diluted magnetic semiconductors.
The idea of this project was conceived by Anh Pham. C. J. Gil did most of the theoretical calculations and drafted the manuscript with the inputs from Anh Pham. S. Li and A. Yu supervised the project. Anh Pham was the corresponding author.
109