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- Herausgeber: Wiley-VCH GmbH
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- Sprache: Englisch
In contrast to the existing books, where conventional simulations for traditional materials are usually presented, the developed methods for novel materials are comprehensively reviewed in this book.
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Seitenzahl: 502
Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Preface
1 An Introduction to Density Functional Theory (DFT) and Derivatives
1.1 The Problem of a N‐electron System
1.2 The Thomas–Fermi Theory for Electron Density
1.3 The First Hohenberg–Kohn Theorem
1.4 The Second Hohenberg–Kohn Theorem
1.5 The Kohn–Sham Equations
1.6 The Local Density Approximation (LDA)
1.7 The Generalized Gradient Approximation (GGA)
1.8 The LDA+U Method
1.9 The Heyd–Scuseria–Ernzerhof Density Functional
1.10 Introduction to
k
⋅
p
Perturbation Theory
References
2 New Physical Effects Based on Band Structure
2.1 Valley Physics
2.2 Rashba Effects
References
3 Ferromagnetic Order in Two‐ and One‐Dimensional Materials
3.1 Intrinsic Ferromagnetic Order in 2D Materials
3.2 Intrinsic Ferromagnetic Order in 1D Molecular Nanowires
References
4 Two‐Dimensional Topological States
4.1 Topological Insulators
4.2 Topological Crystalline Insulators
4.3 Quantum Anomalous Hall Effect
4.4 Antiferromagnetic Topological Insulators
4.5 Mixed Topological Semimetals
References
5 Calculation of Excited‐State Properties
5.1 Green's Function Many‐Body Perturbation Theory
5.2 Excitonic Effects and Band Gap Renormalization in Two‐Dimensional Materials
5.3 Electron–Phonon Effects on the Excited‐state Properties
5.4 Nonlinear Optical Response
5.5 Optical Properties of van der Waals Heterostructures of Two‐Dimensional Materials
References
6 Charge Carrier Dynamics from Simulations
6.1 Time‐Dependent Density Functional Theory and Nonadiabatic Molecular Dynamics
6.2 Applications of TDDFT and NAMD in Two‐Dimensional Materials
References
7 Simulations for Photocatalytic Materials
7.1 Photocatalysis and Photocatalytic Reactions
7.2 Photoresponsivity and Photocurrent from Simulations
7.3 Simulation for Localized Surface Plasmon Resonance
References
8 Simulations for Electrochemical Reactions
8.1 Single‐atom Catalysts
8.2 Stability of Catalyst
8.3 Electrochemical Reactions
References
Index
End User License Agreement
List of Tables
Chapter 8
Table 8.1 Entropies for common molecules under standard condition
p
0
= 0.1 ...
Table 8.2 Binding energy (
E
b
) for transition metal atoms of consideration a...
List of Illustrations
Chapter 2
Figure 2.1 Schematic diagram of the valley filter based on graphene. Top pan...
Figure 2.2 Crystal structure and valley physics in monolayer TMDs. (a) 2D he...
Figure 2.3 Electronic band structures calculated for the MX
2
monolayer syste...
Figure 2.4 The variation of degree of circular polarization P with temperatu...
Figure 2.5 (a) Top and side views of the crystal structure of monolayer VClB...
Figure 2.6 (a) Side and top views of the crystal structure of monolayer LaBr
Figure 2.7 The band structure of monolayer VSi
2
N
4
with considering both spin...
Figure 2.8 (a) Band structures of pure monolayer MoS
2
with considering SOC. ...
Figure 2.9 Band structures of monolayer MoS
2
adsorbed by Sc (a), Ti (b), V (...
Figure 2.10 Spin‐polarized band structures of TM atom doped monolayer WS
2
wi...
Figure 2.11 (a and b) Orbital resolved band structures of WS
2
/VN heterostruc...
Figure 2.12 Band structure of monolayer AgBiP
2
Se
2
(a) without and (b) with c...
Figure 2.13 Band structures of monolayer SrFBiS
2
(a) without and (b) with co...
Figure 2.14 Band structures of monolayer BiTeX. The upper panels show the ba...
Figure 2.15 Band structures of monolayer SbTeI with considering SOC. (b) Enl...
Figure 2.16 Band structures of monolayer MoSeTe with considering SOC under v...
Figure 2.17 Charges on W, Se, and Te atoms for monolayer WSeTe with doping (...
Figure 2.18 Band structures of monolayer NX with considering SOC under an ex...
Figure 2.19 Band structures of monolayer (a) SeInGaTe and (b) SInGaSe withou...
Figure 2.20 Band structures of monolayer MSiGeN
4
without and with considerin...
Chapter 3
Figure 3.1 Spin‐polarized band structures of monolayer (a) MnF
3
and (c) MnI
3
Figure 3.2 Band structures of monolayer CrOCl with (a) majority and (c) mino...
Figure 3.3 (a) Crystal structure of VNp SWN. (b) Band structure of VNp SWN. ...
Chapter 4
Figure 4.1 Band structure for one‐dimensional nanoribbon of graphene. The ba...
Figure 4.2 (a) Top and (b) side view of optimized structures for H‐decorated...
Figure 4.3 Orbitally resolved band structures for (a), (b) H‐decorated Bi (1...
Figure 4.4 (a) The layered crystal structure of 3D bulk ZrTe
5
in space group...
Figure 4.5 (a) Schematic diagram of band evolution in single‐layer Bi
4
Br
4
. (...
Figure 4.6 (a) Top view of the crystal structure of (001) Sn/PbTe monolayers...
Figure 4.7 Orbitally resolved band structures for the monolayers of (a and b...
Figure 4.8 The Berry curvature distribution associated with (a) −
i
and (b) +
Figure 4.9 Variation of the energy gaps at
X
and
Y
points for TlSe monolayer...
Figure 4.10 Band structures of 1D nanoribbon edged with Tl atoms for TlSe in...
Figure 4.11 (a) Crystal structure of bulk Na
3
Bi with
P6
3
/mmc
symmetry. (b) T...
Figure 4.12 (a) Orbitally resolved band structures for Na
3
Bi monolayer (a) w...
Figure 4.13 Localization‐resolved edge states of the Na
3
Bi monolayer (a) wit...
Figure 4.14 (a) Sketch of a 2D square lattice, in which the mirror symmetrie...
Figure 4.15 Band structure of (a) majority and (b) minority states of NpSb m...
Figure 4.16 Evolution of Wannier charge centers (WCCs) for NpSb monolayer wi...
Figure 4.17 (a) Total density of states (DOS) and partial DOS of the unhydro...
Figure 4.18 Band structures of EuCd
2
As
2
QLs (a), (b) without SOC and (c) wit...
Figure 4.19 Schematics of the band inversion (band gap opening) for (a) QAH ...
Figure 4.20 (a) Sketch of the tight‐binding model for a 2D antiferromagnet o...
Figure 4.21 Orbitally resolved band structures for SrMnPb QLs (a) without an...
Figure 4.22 Band structures of (a) AFM and (d) FM configurations in honeycom...
Figure 4.23 The orbitally resolved band structures of EuCd
2
Bi
2
QLs under (a)...
Figure 4.24 (a) The magnetization direction
of a 2D magnet encloses the an...
Figure 4.25 Phase diagrams of (a) TlSe, (b) Na
3
Bi, and (c) GaBi with respect...
Figure 4.26 (a) Spin‐resolved band structure and (b) energy dispersion of a ...
Figure 4.27 Realization of mixed topological semimetals in two‐dimensional f...
Chapter 5
Figure 5.1 Exciton, i.e. the Coulomb bounded electron–hole pair, which can b...
Figure 5.2 Typical excitonic absorption in low‐dimensional materials; absorp...
Figure 5.3 Poles of the Green's function, with
I
being the ionization energy...
Figure 5.4 (Top) Concept of the quasi‐particle;
v
is the bare Coulomb intera...
Figure 5.5 Effective interaction between quasi‐particles.
W
(
r
1
,
r
2
)
means th...
Figure 5.6 The self‐energy Σ and one‐particle Green's function
G
can be dete...
Figure 5.7 Flow chart for the
GW
+BSE calculations starting from the density ...
Figure 5.8 Definition of the exciton binding energy (
E
b
),
E
g
is the quasi‐pa...
Figure 5.9 Relationship between the energy gap and exciton: for a system wit...
Figure 5.10 Momentum‐forbidden dark excitons consist of electrons and holes ...
Figure 5.11 Short and intense laser pulse induced different processes in a s...
Chapter 6
Figure 6.1 Photoinduced processes competing with electron transfer between a...
Figure 6.2 Absorption of a photon (1) by either electron donor or electron a...
Figure 6.3 Scheme of the photoinduced electron transfer mechanism. Adiabatic...
Figure 6.4 The vdW‐coupled multilayers for photocatalytic and photovoltaic a...
Figure 6.5 In experiments, in‐plane heterostructures of two‐dimensional TMDC...
Chapter 7
Figure 7.1 Type‐II band alignment in heterogeneous catalysts; arrows indicat...
Figure 7.2 Two‐dimensional
g
‐C
3
N
4
monolayer.
Figure 7.3 Band energy positions are shown for
g
‐C
3
N
4
and corresponding redo...
Figure 7.4 Photoelectric events in photocatalysts. Excited electrons and hol...
Figure 7.5 Schematic of a two‐probe device model, with
μ
L
and
μ
R
b...
Figure 7.6 Schematics of the device structure for intuitive.
Figure 7.7 Photocurrent maps (
I
ph
), and output power density (
P
out
) with var...
Figure 7.8 Schematic of plasmon resonance absorption of metal nanoparticles....
Figure 7.9 Near‐field enhancement of Au@TiO
2
for incident light with the int...
Figure 7.10 Schematic showing the Au core–multilayer MoS
2
shell structure, a...
Chapter 8
Figure 8.1 Electronic structure of single‐atom catalysts supported on two‐di...
Figure 8.2 Transtion metal–N
4
active center in graphene.
Figure 8.3 Models for obtaining the total energies of bulk metal
(
E
bulk
)
and...
Figure 8.4 In a typical HER process, H
2
gas generation pathway differs depen...
Figure 8.5 Schematic for a complete HER process. (a) Heyrovsky mechanism; (b...
Figure 8.6 The donation–backdonation mechanism for N
2
electrochemical activa...
Figure 8.7 Orbitals of N
2
molecule by calculating projected density of state...
Figure 8.8 Schematic depiction of distal, alternating, enzymatic, and mixed ...
Figure 8.9 The proposed “Five‐Step” strategy for screening NRR candidate cat...
Figure 8.10 Structure of H4,4,4‐graphyne from top view; bond lengths are den...
Figure 8.11 Orbital‐projected band structure for H4,4,4‐graphyne with two Di...
Figure 8.12 Transition metal (the bigger spheres) adsorption on H4,4,4‐graph...
Figure 8.13 (a) Gibbs free energy change of hydrogen adsorption for transiti...
Figure 8.14 Schematic of the four‐step elementary reactions for the OER proc...
Figure 8.15 Gibbs free energy diagram of transition metal atoms on H4,4,4‐gr...
Figure 8.16 (a) Scaling relationship for the Gibbs free energy change of ads...
Figure 8.17 (a) Scaling relationships between Gibbs free energy change of ea...
Figure 8.18 (a) Optimized structure of TMB
2
, with the rectangle denoting the...
Figure 8.19 Twenty‐nine transition metal atoms used to construct TMB
2
struct...
Figure 8.20 (a) Calculated cohesive energy of 13 TMB
2
monolayers. (b) In‐pla...
Figure 8.21 Fukui function of (a)
F
+
(
r
) and (b)
F
−
(
r
) of represent...
Figure 8.22 (a) Gibbs free energy change of N
2
adsorption on TMB
2
in end‐on ...
Figure 8.23 Crystal orbital Hamilton population (COHP), integrated COHP (ICO...
Figure 8.24 Correlation between excess electrons on *N
2
and ICOHP and N
2
ads...
Figure 8.25 Gibbs free energy diagram for NRR on ReB
2
at zero (blue line) an...
Figure 8.26 (a) Gibbs free energy diagram and (b) intermediates for NRR on R...
Figure 8.27 Kinetic barriers for N
2
conversion to NH
3
on ReB
2
; insets show t...
Guide
Cover Page
Table of Contents
Title Page
Copyright
Preface
Begin Reading
Index
End User License Agreement
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Calculations and Simulations of Low‐Dimensional Materials
Tailoring Properties for Applications
Ying DaiWei WeiYandong MaChengwang Niu
Authors
Prof. Ying Dai
Shandong University
School of Physics
Shandananlu 27
250100 Jinan
China
Prof. Wei Wei
Shandong University
School of Physics
Shandananlu 27
250100 Jinan
China
Prof. Yandong Ma
Shandong University
School of Physics
Shandananlu 27
250100 Jinan
China
Prof. Chengwang Niu
Shandong University
School of Physics
Shandananlu 27
250100 Jinan
China
Cover Image: © Yuichiro Chino/Getty Images
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details, or other items may inadvertently be inaccurate.
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© 2023 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
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Print ISBN: 978‐3‐527‐34909‐8
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Preface
In modern society, theoretical calculation and simulation are advancing the development of physics, chemistry, and materials science, etc., along with the great successes in computer technology. After the successful exfoliation of graphene from graphite by Sir Andre Geim and Konstantin Novoselov, then various two‐dimensional materials are predicted, synthesized, transferred, and characterized, and we are witnessing the coming of the ear of two‐dimensional materials. In comparison to the corresponding bulk counterparts, materials in low dimension exhibit unprecedented novel chemical and physical properties. Meanwhile, many novel phenomena as well as promising potential devices are proposed based on low‐dimensional materials.
Among these intriguing physics in two‐dimensional lattices, for example, the valley physics, excitonic effects, and Rashba effect attract special interest from the standpoints of both fundamental and applied level. Thus, there is an impressing progress in the research on these exciting topics. For the magnetic property, it was extensively studied in conventional bulk systems, as it had long been considered to survive in low‐dimensional systems due to the thermal fluctuations according to the Mermin–Wagner theorem. In recent years, with the discovery of long‐range magnetic order in two‐dimensional lattice, magnetic materials in two dimensions spurred extensive attention. Currently, there are many works devoted to investigate the magnetic properties in two‐dimensional materials. Also, the past decades have witnessed the evolution of topological phases and topological materials which reshape our understanding of physics and materials. First‐principles calculations based on the density functional theory provide effective descriptions of topological phases and play important roles in predicting realistic topological materials, from insulators to semimetals, and from nonmagnetic systems to magnetic ones. In fields of photochemistry and photoelectrochemistry, the theoretical calculations and simulations are so powerful in designing, screening, and characterizing the photocatalysts and electrocatalysts of high stability, selectivity, and activity. Indeed, calculation and simulation are guiding, leading, and moving forward these fields.
The book is divided into eight chapters. In Chapter 1, density functional theory and its derivatives (e.g., the tight‐binding method) are briefly introduced, and in Chapter 2, the recent development of valley physics and Rashba effect in two‐dimensional materials from the first‐principles are reviewed. In Chapter 3, calculations and simulation results for low‐dimensional ferromagnetic materials are discussed. Chapter 4 provides a gentle introduction of topological phases and topological materials especially in two dimensions. The concepts of different topological states are discussed using first‐principles band structures and topological invariants. In Chapter 5, ingredients in the many‐body Green's function perturbation theory for calculating the excited‐state properties of low‐dimensional materials are summarized, and in Chapter 6, time‐dependent density functional theory and nonadiabatic molecular dynamics are introduced for those who are studying photoexcited charge carrier dynamics. Process of how to calculate and simulate photocatalytic reaction is presented in Chapter 7, and electrochemical reactions from calculations and simulations are reviewed in Chapter 8.
This book would not have been possible without help from many people.
Jinan, China
Summer, 2022
Ying Dai
Wei Wei
Yandong Ma
Chengwang Niu
1An Introduction to Density Functional Theory (DFT) and Derivatives
1.1 The Problem of a N‐electron System
The density functional theory (DFT) is first resulted from the work by Hohenberg and Kohn [1], wherein the complicated individual electron orbitals are substituted by the electron density. Namely, the DFT is entirely expressed in terms of the functional of electron density, rather than the many‐electron wave functions. In this case, DFT significantly reduces the calculations of the ground state properties of materials. That is why DFT is useful for calculating electronic structures, especially with many electrons. As the foundation of DFT, two theorems are proposed by Hohenberg and Kohn [1]. The first theorem presents that the ground state energy is a functional of electron density. The second theorem shows that the ground state energy can be achieved by minimizing system energy on the basis of electron density.
It should be noted that, although Hohenberg and Kohn point out there are relations between properties and electron density functional, they do not present the exact relationship. But fortunately, soon after the work of Hohenberg and Kohn, Kohn and Sham simplified the many‐electron problems into a model of individual electrons in an effective potential [2]. Such a potential contains the external potential and exchange‐correlation interactions. For exchange‐correlation potential, it is a challenge to describe it rigorously.
The simplest approximation for treating the exchange‐correlation interaction is the local density approximation (LDA) [3], wherein the exchange and correlation energies are obtained by the uniform electron gas model and fitting to the uniform electron gas, respectively. LDA can provide a realistic description of the atomic structure, elastic, and vibrational properties for a wide range of systems. Yet, because LDA treats the energy of the true density using the energy of a local constant density, it cannot describe the situations where the density features rapid changes such as in molecules [4, 5]. To address this problem, the generalized gradient approximation (GGA) is proposed [6–8], which depends on both the local density and the spatial variation of the density. And in principle, GGA is as simple to use as LDA. Currently, in the vast majority of DFT calculations for solids, these two approximations are adopted.
By considering the Born–Oppenheimer and non‐relativistic approximations, the effective Hamiltonian of a N‐electron system in the position representation can be given by,
The first term is kinetic energy operator. The second term is an external potential operator. In systems of interest to us, the external potential is simply the Coulomb interaction of electrons with atomic nuclei:
where the ri is the coordinate of electron i and the charge on the nucleus at Rα is Zα. The third term of Eq. (1.1) is the electron‐electron operator. The electronic state can be obtained by the Schrödinger equation:
Here, Ψ(r1, r2, rN) is a wave function in terms of space‐spin coordinates. Apparently, the wave function is antisymmetric under exchanging the coordinates. Under Dirac notation, the Eq. (1.1) can be expressed in representation‐independent formalism:
In principle, the ground state energy E0 of the N‐electron system can be found based on the variational theorem, which is obtained by the minimization:
Here, the search is over all antisymmetric wave functions Ψ. In this regard, better approximations for Ψ can readily result in the ground state energy E0 of the N‐electron system, but the computational cost would be very high. Therefore, the direct solution is not feasible. To address this issue, DFT is developed, which is based on a reformulation of the variational theorem in terms of electron density.
We know that |Ψ|2 = Ψ*Ψ represents the probability density of measuring the first electron at r1, the second electron at r2, … and the Nth electron at rN. By integrating |Ψ|2 over the first N − 1 electrons, the probability density of the Nth electron at rN is determined. Then the probability electron density that defines any of the N electrons at the position r is given by:
And the electron density is normalized to the electron number:
The energy of the system is expressed as:
Here,
1.2 The Thomas–Fermi Theory for Electron Density
Before discussing the Hohenberg–Kohn theorems, we first introduce the Thomas–Fermi theory. The Thomas–Fermi theory is important as it gives the relation between external potential and the density distribution for interacting electrons moving in an external potential:
Here,
and μ is the r independent chemical potential. The second term is Eq. (1.12) is the classical electrostatic potential raised by the density ρ(r). Based on the Thomas–Fermi theory, Hohenberg and Kohn build up the connection between electron density and the Schrödinger equation. And in the following, we will introduce the two Hohenberg–Kohn theorems, which lie at the heart of DFT.
1.3 The First Hohenberg–Kohn Theorem
By replacing the external potential vne(r) with an arbitrary external local potential v(r), the corresponding ground state wave function Ψ can be found by solving the Schrödinger equation. Based on the obtained wave function, the ground state density ρ(r) can be computed. And obviously, two different local potentials would give two different wave functions and thus two different electron densities. This gives the map:
Based on the Thomas–Fermi theory, Hohenberg and Kohn demonstrated that the preceding mapping can be inverted, namely, the ground state electron densityρ(r)of a bound system of interacting electrons in some external potential v(r) determines the potential uniquely:
This is known as the first Hohenberg–Kohn theorem.
To demonstrate this theorem, we consider two different local potentials v1(r) and v2(r), which differ by more than the constant. These two potentials yield two different ground state wave functions Ψ and Ψ′, respectively. And apparently, these two ground state wave functions are different. Assume v1(r) and v2(r) correspond to the same ground state wave function, then
By subtracting Eq. (1.17) from Eq. (1.16), we can obtain:
which can be expressed in position representation,
This suggests that
thus in contradiction with the assumption that v1(r) and v2(r) differ by more than a constant. Accordingly, two different local potentials that differ by more than the constant cannot share the same ground state wave function, which demonstrate the map:
Then, we demonstrate the map:
Let Ψ and Ψ′ be the ground state wave functions corresponding to v1(r) and v2(r), respectively. Assuming that Ψ and Ψ′ exhibit the same ground state electron density ρ(r), then the variational theorem gives the ground state energy as:
By subtracting Eq. (1.23) from Eq. (1.24), we can obtain:
This makes no sense. This finally leads to the conclusion that there cannot exist two local potentials differing by more than an additive constant that has the same ground state density.
1.4 The Second Hohenberg–Kohn Theorem
According to the first Hohenberg–Kohn theorem, the ground state density ρ(r) determines the local potential v(r), and in turn determines the Hamiltonian. Therefore, for a given ground state density ρ0(r) that is generated by a local potential, it is possible to compute the corresponding ground state wave function Ψ0. That is to say, Ψ0 is also a unique functional of ρ0(r):
According to Eq. (1.26), the ground state energy E0 is also a functional of ρ0(r):
Hohenberg and Kohn define the universal density functional:
Here, Ψ[ρ] is any ground state wave function corresponding to the ground state density ρ(r). By combining Eq. (1.10), the total energy functional can be defined as:
From the Ritz principle, we have:
This is known as the second Hohenberg–Kohn theorem.
1.5 The Kohn–Sham Equations
For a system with noninteracting electrons, the effective Hamiltonian n can be given by:
The corresponding Schrödinger equation is:
Then the density is given by:
Here, the single particle orbital ψi(r) is constructed based on the effective potential vs(r).
The total energy can be expressed as:
The first term is the kinetic energy of the noninteracting electrons, which is given by:
The second term is the effective potential, which is given by:
Accordingly, the total energy can be given by:
By combining Eqs. (1.36) and (1.37), the kinetic energy can be expressed as:
Using the method of Lagrange multipliers, we can obtain the following equation:
Solution of Eq. (1.39) yields the density ρs(r).
The classical electrostatic interaction energy is given by:
And
Then the following equation is obtained:
The third term is exchange and correlation energy functional:
The exchange‐correlation potential is defined as:
Using the method of Lagrange multipliers, we can obtain the following equation:
And solution to Eq. (1.45) is ρ(r).
Therefore, given the relation:
We have:
We then arrive at the Kohn–Sham equation:
Solving this equation gives the orbital and then the density of the original interacting system.
The exchange and correlation functional can be written as:
Here, εxc(r′; ρ) is the exchange‐correlation energy density. And the exchange‐potential is defined as:
Then the total energy can be given by:
From preceding equation, we can see that, except for the exchange‐correlation functional, all the aforementioned expressions are exact. In practice, we have to use approximations for exchange‐correlation potential, as the exact form is unknown.
1.6 The Local Density Approximation (LDA)
LDA is one of the most widely used and simplest approximations for Exc. In LDA, the exchange‐correlation functional is approximated as:
Here, is the exchange‐correlation energy per electron in homogeneous electron gas at density ρ. LDA works well for homogeneous electron gas, and thus is valid for systems where electron density does not change rapidly.
The exchange‐correlation energy density can be broken down into two parts:
The first term is the exchange term, which is given by:
While for the second term of Eq. (1.53), it is the correlation density, which does not have an analytic formula. However, the correlation energies can be obtained numerically from quantum Monte Carlo (QMC) calculations by Ceperley and Alder [9].
And for spin‐polarized systems, the spin‐up and spin‐down densities are taken as two independent densities in the exchange‐correlation energy. And in this case, the Eq. (1.52) can be expressed as:
For calculating the electronic structure, LDA approach is estimated to be successful. However, for some systems, it does not work. As a result, many efforts are devoted to improve it. One of them is to include the gradient of the density in the exchange correlation functional, as we will show next.
1.7 The Generalized Gradient Approximation (GGA)
To introduce the gradient of the density in the exchange correlation functional, the gradient expansion approximation (GEA) is first proposed. Starting from the uniform electron gas, a slowly varying external potential v(r) is introduced. And then the exchange‐correlation energy is expanded in terms of the gradients of the density:
Here, Cxc(ρ) is the sum of the exchange and correlation coefficients of the gradient expansion. As the reduced density gradient is small, GEA approach should be superior to LDA approach. However, because the reduced density gradient can be large in some region of space for real systems, GEA is shown to be worse than LDA.
To overcome the shortcomings of GEA, the GGA is developed. In GGA, the exchange‐correlation functional is approximated as:
Here, f is some function. Many GGA functionals have been proposed, including B88 [10], Lee‐Yang‐Parr (LYP) [11], PW91 [12], and Perdew–Burke–Ernzerhof (PBE) [13] exchange‐correlation functionals.
1.8 The LDA+U Method
While LDA and GGA are estimated to be able to deal with the many systems and phenomena, they do not work well for the systems with rare‐earth and late‐transition metal elements. This is because the effective single‐particle methods are applicable for highly delocalized band states but not for strongly localized states. For the d and f electrons of rare‐earth and late‐transition metal elements, they essentially retain their atomic character in solids. As a consequence, standard DFT functionals such as LDA, local spin density approximation (LSDA), and GGA itinerant d states and metallic ground state for many transition metal oxides, for which semiconducting behavior is demonstrated experimentally. For improving these issues, LDA+U method is developed [14–16]. Here, if not specified elsewhere, +U indicates a Hubbard, and LDA indicates the standard DFT functionals, i.e. LDA, LSDA, and GGA. The idea of LDA+U method is on the basis that the strongly correlated electronic states (i.e. d and f sates) are treated by the Hubbard model, and the rest of the valence electrons are described by the standard DFT functionals. Therefore, the total energy within LDA+U method can be given by:
Here, the first term is the standard DFT total energy functional being corrected, the second term represents the Hubbard Hamiltonian to model correlated states, and the third is the double‐counting term. The LDA+U method can well describe the electronic properties of the Mott insulators and increase the band gaps in the Kohn–Sham spectrum.
1.9 The Heyd–Scuseria–Ernzerhof Density Functional
In standard DFT, the Fock exchange energy is computed based on a local energy density and its derivatives. However, the exact form for the Fock exchange energy is known as nonlocal from the Hartree–Fock theory. To improve the accuracy, the PBE exchange energy should be mixed with a fraction of the exact nonlocal Fock exchange energy, giving rise to the hybrid functionals, such as the Heyd–Scuseria–Ernzerhof (HSE) functional [17]. The HSE exchange‐correlation energy is given by:
Here, and , respectively, represent the exact Fock exchange energy and the PBE exchange energy. is the PBE correlation energy. μ is the range‐separation parameter. α is the mixing parameter. From Eq. (1.59), it can be seen that the HSE functional is split into short‐ and long‐range terms. In this case, it can improve the accuracy, while avoiding the computational cost.
1.9.1 Introduction to Tight‐Binding Approximation
Consider in a single atom there are multiple atomic orbitals ϕm(r) with m being the orbital indices. Here, ϕm(r) must be eigenfunctions of the Hamiltonian of that single atom Hatom. When we place it in a crystal with plenty of atoms, the wave function of different atoms overlap each other to form a different wave function. Due to that ϕm(r) is not a real eigenfunction for a Hamiltonian of crystal and we need to find out what the true eigenfunctions are. If the overlap of one atom on another is small enough, we can still assume that electrons are tightly bound to the corresponding atoms, which is exactly the reason why we call it as tight‐binding approximation. The approximate Hamiltonian is H(r) = Hatom(r) + ΔU(r), whose Bloch wave function can be taken as a combination of all the isolated orbitals: [18, 19]
where Rn denotes all lattice points and bm(Rn) is just a coefficient number for orbital m. In the presence of translation symmetry, coefficient numbers can be replaced by a Bloch form, which gives:
1.9.2 Matrix Elements of Tight‐Binding Hamiltonian
To get Hamiltonian in the momentum space, we shall do a basis transformation such as: [20]
When Ri is equal to Rj, we will find the onsite energy represented for the atomic energy shift due to the overlap of other atoms, which can be given as:
If Ri is not equal to Rj, hopping energy between different lattice sites can be defined as:
1.9.3 Matrix Elements with the Help of Wannier Function
Usually, Bloch wave functions are not orthogonal, which may result in some problems. To resolve that we should define the orthogonal Wannier function as:
By using a bra‐ket notation, the Hamiltonian in the real space takes the form of: [21]
where βi and tij denote the onsite energy and hopping energy, respectively. To get the energy of Hamiltonian, we shall do a basis transformation into the momentum space, which is implemented by a Fourier transform:
1.9.4 Example for a Graphene Model
By using graphene lattice [22] as an example, we give a detailed description about how to use tight‐binding method. Here, we consider two atoms with s orbital in a single unit cell of unit lattice constant, whose lattice vector can be written as:
Each atom is connected with three nearest‐neighbor atoms with a distance of:
In the momentum space, we consider the onsite energy for two s orbitals is ε1. The nearest‐neighbor hopping strength is t, which leaves a Hamiltonian matrix as follows:
We can get the energy dispersion with an implementation of diagonalization, which results in:
1.10 Introduction to k ⋅ p Perturbation Theory
A single electron in a periodic potential V(r) obeys a Schrödinger equation, and such a form can be written as [23, 24]:
where the eigenvalues and eigenfunctions can be written as En(k) and ψn(k, r) = eik ⋅ run(k, r). The periodic part that is called as a Bloch function satisfies such an equation as:
where the Hamiltonian located at momentum point of k0 can be given as: . If the eigenvalues and eigenfunctions are assumed to be solved for point as k0, we can get the solutions of nearby points such as k = k0 + δk through the following equation:
Then, the perturbation Hamiltonian gains a form of . With the help of perturbation theory, we can solve the energy of nearby points under two different situations, which is illustrated in the following chapter.
1.10.1 Solution for Non‐degenerate Bands
If the bands are not degenerate for k0, the perturbated eigenvalues of the points k = k0 + δk can be given by [25, 26]:
where the first term and second term of right‐hand side are zero order approximation, and the third term and fourth term serve as the first and second order approximation. Here, we give an explicit example to describe how to get a Hamiltonian based on the following equation. We assume the eigenvalues and eigenfunctions of k0 = 0 are known by first‐principles calculations or experiment. Consider a cubic lattice with a point group of Oh and two bands |u1k(r)〉 and |u2k(r)〉 that transform as and representation, and the symmetry representation of H′ is the vector representation as .
For the first order approximation, the direct product changes the representation into , resulting in a vanishing matrix. While for the second order approximation, the direct product limits the to the representation, which is just the band representation of antibonding p bands. There are three basis x, y, and z for representation; only three terms can exist as a cross term of , , and according to the selection rules. Finally, we can get the total eigenvalues as:
1.10.2 Solution for Degenerate Bands
If the bands are degenerate for k0, the eigenfunction must be a linear combination of degenerated bands. Here, we assume band i is degenerate with band j, and the first order perturbation equation can be written as: [27]
2New Physical Effects Based on Band Structure
2.1 Valley Physics
Valley refers to the local minimum in the conduction band or the local maximum in the valence band in the momentum space. Given the systems with two or more valleys, if the valleys can be polarized and detected, the valley can form a new degree of freedom of carriers in addition to charge and spin. Analogy to charge for electronics and spin for spintronics, the possibility of utilizing valley degree of freedom as information carrier gives rise to the concept of valleytronics [1–3]. The idea of using valley degree of freedom dates back to the studies in the late 1970s, which investigated the valley behavior and inter‐valley coupling in silicon inversion layers [4–6]. Moreover, by tuning the intravalley exchange and correlation, the electrons would preferentially occupy the valleys, thereby generating the valley polarization. Subsequently after the works on silicon inversion layer, several other systems like the AlAs quantum well, silicon heterostructures, bismuth, and diamond are also shown to feature the valley feature [7–11]. Despite these extensive efforts, there lacks an intrinsic physical property that correlated with the valley occupancy. This is in sharp contrast with the case of spin, where the magnetic moment, the spin optical selection rule, and the spin‐orbit coupling (SOC) can be employed as the intrinsic properties associating with the spin. Therefore, the utilization of the valley index is severely limited as compared with the spintronics, and the intrinsic property associated with the valley occupancy is under exploration.
The recent rise of 2D materials [12–14] provides an unprecedented opportunity to address the aforementioned issues. The first 2D system harboring the valley physics is proposed in graphene (Figure 2.1) [1]. In graphene with broken inversion symmetry, the band edges of the band structure lie at the +K and −K points of the 2D Brillouin zone, forming two degenerate by nonequivalent valleys. Importantly, the carriers in the +K and −K valleys are subjected to opposite Berry curvatures and orbital magnetic moment. These two quantities can be used to distinguish the valleys. The Berry curvature can behave like effective magnetic fields, which would result in an anomalous velocity perpendicular to an electric field. This is a Hall effect, which is referred to as the valley Hall effect [15]. While for the orbital magnetic moment, it results from self‐rotating, so an energy shift can be expected under the magnetic field. It should be noted that, under the inversion symmetry, the +K and −K valleys can be transformed into each other, making it not possible for distinguishing the valleys. Therefore, breaking the inversion symmetry is a necessary condition for utilizing Berry curvature and orbital magnetic moment to distinguish the valley occupancy. Currently, several approaches are proposed to break the inversion symmetry of graphene, including the creation of edge modes [16] and defect lines [17]. However, these approaches are challenging in experiments, limiting their practical applications.
Figure 2.1 Schematic diagram of the valley filter based on graphene. Top panel: Dispersion relation in the wide and narrow regions. An electron in the first valley (modes n = 0, 1, 2, …) is transmitted (filled circle), whereas an electron in the second valley (modes n = −1, −2, …) is reflected (open circle). Middle panel: Honeycomb lattice of carbon atoms in a strip containing a constriction with zigzag edges. Bottom panel: Variation of the electrostatic potential along the strip for the two cases of an abrupt and smooth potential barrier (solid and dashed lines). The polarity of the valley filter switches when the potential height, U0, in the constriction crosses the Fermi energy, EF.
Source: Rycerz et al. [1]/with permission of Springer Nature.
After discovering the valley behaviors in graphene, the monolayer group‐VI transition metal dichalcogenides (TMDs), i.e. MoS2, WS2, MoSe2, and WSe2, are identified as the most promising platform to study the novel valley‐contrasting physics [18–21]. In monolayer TMDs, the M atomic layer is sandwiched between two X atomic layers, which breaks the inversion symmetry naturally (Figure 2.2a). Moreover, there are two degenerate but nonequivalent valleys locating at the +K and −K points, generating two valleys. Therefore, under an in‐plane electric field, the carriers in +K and −K valleys would be accumulated at the opposite edges of the sample, which is related to the opposite Berry curvatures at the two valleys (Figure 2.2c). More importantly, as the valleys are dominated by the d orbitals of the M atoms, the valleys experience a valley spin splitting due to the strong SOC strength within M‐d orbitals [22]. And due to the in‐plane character of the d orbitals for the valleys in the valence bands, they have a very large valley spin splitting. While for the valleys in the conduction bands, because of the out‐of‐plane character, the valley spin splitting is rather small (Figure 2.3). Under such splitting, we can manipulate the carrier with one spin in one valley. This indicates novel spin and valley physics that are absent in graphene. Considering these merits, monolayer TMDs are considered as one of the most promising 2D valleytronic materials.
Figure 2.2 Crystal structure and valley physics in monolayer TMDs. (a) 2D hexagonal lattice. (b) Valley contrasting optical selection rules in a 2D hexagonal lattice with broken inversion symmetry. The interband transition in valley K (−K) couples to σ+ (σ−) circularly polarized light only (circular arrows). (c) Diagram of the valley Hall effect in monolayer TMDs. Arrows suggest the pseudo‐vector quantities, i.e. Berry curvature or orbital magnetic moment, of the carriers.
Source: Xu et al. [18]/with permission of Springer Nature.
Figure 2.3 Electronic band structures calculated for the MX2 monolayer systems with (solid line) and without (dotted line) inclusion of the spin‐orbit interaction.
Source: Zhu et al. [22]/with permission of American Physical Society.
Besides graphene and monolayer TMDs, many other 2D materials presenting the valley‐contrasting physics are proposed. For example, using first‐principles calculations, we reported that monolayer H‐Tl2O is a compelling 2D valleytronic material with spin‐valley coupling [23]. Soon after the work on monolayer H‐Tl2O, we discovered another class of 2D valleytronic materials in monolayer MN2X2 (M = Mo, W; X = F, H), which are shown to be dynamically and thermally stable. Monolayer MN2X2 is an indirect gap semiconductor with the valence band maximum locating at the +K and −K points, forming two degenerate but nonequivalent valleys in the valence band [24].
Recently, Lu et al. proposed that hexagonal monolayer MoSi2N4 and MoSi2As4 exhibit a pair of valleys at the +K and −K points [25]. The circularly polarized photons can be adopted to distinguish the +K and −K valleys. Arising from the valley‐contrasting Berry curvature, the intriguing valley Hall and spin Hall effects can be realized. In addition to the traditional two level valleys, the valleys in monolayer MoSi2As4 are shown to be multiple folded. This suggests an additional intrinsic degree of freedom for monolayer MoSi2N4 and MoSi2As4. Moreover, monolayer MoSi2N4 and MoSi2As4 have several advantages over monolayer TMDs. For example, while monolayer TMDs suffer from the limited size of growth [26] and low electron/hole mobility [27], which restricts them for further quantum‐transport applications, monolayer MoSi2N4 and MoSi2As4 were successfully synthesized with large size up to 15 mm × 15 mm [28] and is predicted to be with a large electron/hole mobility of four to six times larger than that of monolayer MoS2[29]. Therefore, monolayer MoSi2N4 and MoSi2As4 are believed to be promising candidates for the potential of valleys to be applied in multiple information processing.
After discovering these promising valleytronic materials, the next step is to explore how to use these materials for applications in valleytronic devices. To make a valleytronic device, the crucial step is to lift the degeneracy between the +K and −K valleys, thereby producing the valley polarization [15]. In 2D valleytronic materials, the valley‐contrasting orbital magnetic moment is accompanied by a valley‐contrasting optical selection rule. In detail, the interband transitions in the vicinity of the +K and −K valleys correlate with the right‐ and left‐handed circularly polarized, see Figure 2.2b [18]. To this end, the method of optical pumping is proposed to achieve the valley polarization in monolayer TMDs. Cui et al. found that the photoluminescence in monolayer MoS2 has the same helicity as the circularly polarized component of the excitation laser [30]. This is a strong signature of the valley polarization induced by optical pumping. Below temperature 90 K, a high photoluminescence circular polarization is obtained, and it decays with temperature, see Figure 2.4. To confirm that the polarized photoluminescence is related to the valley instead of spin, the Hanle effect is investigated. They showed that the persistent photoluminescence polarization is observed in monolayer MoS2 when traversing the magnetic field. This confirms that the polarized photoluminescence is attributed to the valley polarization as a transverse magnetic field causes the spin to precess. Furthermore, for bilayer MoS2 with inversion symmetry, the photoluminescence is found to be unpolarized by applying the same excitation condition, which again confirms the optically pumped valley polarization in monolayer MoS2. Similar to this work, Mak et al. also identified the viability of optical control of valleys and the valleytronics in monolayer MoS2[20]. Besides these two experimental works, there are also many other experimental breakthroughs in optical pumped valley polarization. For more detail, please see the review of Ref. [18].
Figure 2.4 The variation of degree of circular polarization P with temperature. For fitting the relationship, assuming an inter‐valley scattering proportional to the phonon population is assumed.
Source: Zeng et al. [30]/with permission of Springer Nature.
Although valley polarization is successfully achieved through optical pumping, it is subjected to the quite short lifetime of carriers. Considering this point, optical pumping is unable to robustly tune the valley, making it not applicable for developing practical information devices. An alternative approach for realizing the valley polarization is to break time‐reversal symmetry. That is because the time‐reversal symmetry requires the spins at the +K and −K valley to be energetically degenerate, but opposite, forming the valley‐spin locking relationship. By breaking the time‐reversal symmetry, the valley degeneracy would be lifted. In fact, lifting the valley degeneracy in monolayer TMDs has been achieved by applying an external magnetic field in some experimental works [31, 32]. However, this approach is shown to be rather modest. For instance, the valley polarization is estimated to be 0.1–0.2 meV T−1 [31, 32]. This is suitable for the development of valleytronic devices where the large valley polarization is required, analogous to large spin polarization for spintronics. Recently, the research efforts for breaking the time‐reversal symmetry in this field have been devoted to introducing foreigner atom doping and proximity‐induced magnetic interaction, and many breakthroughs have been made. In addition, with the recent discovery of 2D magnetic materials, the intrinsic valley polarization is also identified, which attracts more and more attention. In the following, we will discuss the recent theoretical developments on the spontaneous valley polarization, valley polarization induced by foreign atom doping, and valley polarization in van der Waals heterostructures.
2.1.1 Spontaneous Valley Polarization
Spontaneous valley polarization was first proposed in 2013 [33]. Using the tight‐binding model, Feng et al. proposed the theory of spin and valley physics of an antiferromagnetic honeycomb lattice. In the antiferromagnetic honeycomb lattice, there is an emergent electronic degree of freedom of carriers, which is characterized by the product of spin and valley indices (s⋅τ). This gives rise to the s⋅τ‐dependent optical selection rule. When the spin‐valley coupling is weak, namely the valley degeneracy is considered to be kept, the system would exhibit an s⋅τ‐selective circular dichroism (CD). Illuminated by the left‐polarized light, the spin‐up electrons at the +K valley and spin‐down electron at the −K valley would be excited to the conduction band. And by illustrating the right‐polarized light, the spin‐down electrons at the +K valley and spin‐up electron at the −K valley would be excited to the conduction band. For the case with strong spin‐valley coupling, the valley degeneracy is lifted, and thus the gaps at the +K and −K points are different. In this regard, by tuning the frequency of the polarized light, one spin from one valley can be excited. Besides the optical selection rule, the Berry curvature is also s⋅τ‐dependent. Charge carrier with opposite s⋅τ indices would exhibit opposite transversal anomalous velocities. After establishing the theory of spin and valley physics in an antiferromagnetic honeycomb lattice, Feng et al. suggested one actual material of monolayer manganese chalcogenophosphates, MnPX3, X = S, Se [33]. By calculating their band structure, they showed that the gaps at +K and −K differ by 43 meV, indicating the spontaneous valley polarization and the s⋅τ‐dependent physics.
After discovering the spontaneous valley polarization in the antiferromagnetic honeycomb lattice, the spontaneous valley polarization in the ferromagnetic lattice is also proposed. Using the two‐band k·p model, Duan et al. also proposed that the coexistence of SOC and intrinsic exchange interaction can result in that the valley polarization occurs spontaneously [34]. They further predicted one such real material of monolayer 2H‐VSe2.
Similar to the anomalous Hall effect, Duan et al. defined the valley Hall effect in ferromagnetic material as anomalous valley Hall effect [34]. Note that the charge Hall current is easier to be detected experimentally, and the anomalous valley Hall effect would provide a promising way to realize the valley‐based information storage. On the basis of the anomalous valley Hall effect, Duan et al. proposed the possible electrically reading and magnetically writing memory devices [34]. This significantly advances the practical applications of valleytronics.
Based on the preceding two works [33, 34], the spontaneous valley polarization is well established. However, for monolayer MnPX3, they show the s⋅τ‐selective, rather than the τ‐selective, CD, and Berry curvature. And valley is not an independent degree of freedom of carriers any more. While for monolayer 2H‐VSe2, this monolayer phase is not stable in experiments. Therefore, the candidate materials with spontaneous valley polarization are still under exploration. In the following, we will discuss the systems that are recently identified with spontaneous valley polarization.
Using first‐principles calculations, Lu et al. investigated the valley physics in monolayer VAgP2Se6[35]. They found that monolayer VAgP2Se6 can be easily exfoliated from its bulk counterpart. Monolayer VAgP2Se6 is intrinsically spin polarized. Monolayer VAgP2Se6 is a ferromagnetic semiconductor. The easy magnetization axis is along the out‐of‐plane, with a magnetocrystalline energy of 1.5 meV per unit cell, enabling the possible stable ferromagnetism. The spin‐polarized band structure of monolayer VAgP2Se6
