UNIVERSITY OF CALIFORNIA

RIVERSIDE

Magnetic Properties of Ferromagnetic and Antiferromagnetic Materials and

Low-Dimensional Materials

A Dissertation submitted in partial satisfaction

of the requirements for the degree of

Doctor of Philosophy

in

Electrical Engineering

by

Md. Rakibul Karim Akanda

September 2021

Dissertation Committee:

Dr. Roger K. Lake, ChairpersonDr. Alexander BalandinDr. Ran Cheng

Copyright byMd. Rakibul Karim Akanda

2021

The Dissertation of Md. Rakibul Karim Akanda is approved:

Committee Chairperson

University of California, Riverside

Acknowledgments

I would like to express my hear-felt gratitude and appreciation to my supervisor and mentor

Roger K. Lake for his guidance, support and encouragement during the past ve years. I

have learned a lot from him about magnetism, micromagnetic simulation, condensed mat-

ter physics, material science, nonequilibrium Greens functions, programming languages,

manuscript writing, proposal writing, and data science. When I was struggling with my

project, he encouraged and guided me. He encouraged me to learn new and diverse topics.

He also encouraged and helped me to learn dierent softwares and programming languages.

Moreover, before being a PhD candidate, I was not good at writing manuscripts. He guided

me and helped me to learn how to write a good manuscript. He often suggested reading

several manuscripts, which helped me to learn new information and to learn about writing a

manuscript. It goes without saying that devoid of his constant support and guidance, none

of the works appearing in the thesis would have ever been possible.

I want to thank my committee member Prof. Alexander Balandin, as I have learned

much about the experimental issues from Dr. Balandin and his group. I also want to thank

my committee member Prof. Ran Cheng, as I have learned much from his lecture classes on

magnetics and magnetic properties of materials.

Special thanks to my current lab mates in LATTE, Sohee Kwon, Yuhang Liu,

Topojit Denbath, and Hector Gomez, as well as my former colleagues Dr. Mahesh Neupane,

Dr. In Jun Park, Dr. Tonmoy Kumar Bhowmick, Dr. Bishwajit Debnath, Dr. Protik

Das, Dr. Shanshan Su, Dr. Nima Djavid, and Dr. Yizhou Liu, for their help and support

with many ideas, collaborative projects and enjoyable discussions. A special thanks to Prof.

iv

Shane Cybart for his funding support during several semesters of my PhD.

On the personal side, I would like rst to acknowledge my parents for their guidance

and support for my higher education. Lastly, I would like to thank to my dear wife Azmeri

Ferdous for her love and great support.

The text of this dissertation, in part or in full, is a reprint of the material as it

appears in the following journal and/or proceedings:

• Physical Review B [1]. Reprinted with permission from [1]. © [2020] APS

• Applied Physics Letters [2]. Reprinted with permission from [2]. © [2021] AIP

The co-author Roger K. Lake, listed in the above publications directed and super-

vised the research which forms the basis for this dissertation. The remaining co-authors

listed provided technical expertise and support as collaborators.

This work was supported in part by the Spins and Heat in Nanoscale Electronic

Systems (SHINES) an Energy Frontier Research Center funded by the U.S. Department of

Energy, Oce of Science, Basic Energy Sciences under Award #DE-SC0012670. This work

used the Extreme Science and Engineering Discovery Environment (XSEDE) [3] which is

supported by National Science Foundation Grant No. ACI-1548562 and allocation ID TG-

DMR130081. Used resources include Stampede-2 and Comet.

v

To my parents and dear wife Azmeri Ferdous, love you forever.

vi

ABSTRACT OF THE DISSERTATION

Magnetic Properties of Ferromagnetic and Antiferromagnetic Materials andLow-Dimensional Materials

by

Md. Rakibul Karim Akanda

Doctor of Philosophy, Graduate Program in Electrical EngineeringUniversity of California, Riverside, September 2021

Dr. Roger K. Lake, Chairperson

Magnetic skyrmions oer various advantages compared to domain wall based mem-

ory applications. A skyrmion based shift register is analyzed that uses either ferromagnetic

or synthetic anti-ferromagnetic skyrmions. The energy requirement to shift a ferromagnetic

(FM) skyrmion is less than that required to create or annihilate a skyrmion in conventional

race track memory. Two dierent types of skyrmion shift register structures are investigated.

They dier in the geometry used to conne a skyrmion into an individual cell. One struc-

ture uses a rectangular channel combined with a modication of the gate metal above the

channel to dene individual cells. In the the second structure, the channel width is modied

into an hourglass type shape, and the narrow regions of the channel serve to conne the

skyrmion to an individual cell. The eects of the shape of the structures, the thickness of the

top heavy metal layer, the gap between the gate regions, and the presence of disorder and

nite temperature are analysed using micromagnetic simulations. Synthetic antiferromag-

netic (SAFM) skyrmions provide higher velocity and reduced switching energy compared

to ferromagnetic skyrmions, and they can be stable above room temperature. All of the

vii

necessary phase diagrams for obtaining stable skyrmions are presented, which can help to

select the correct ferromagnetic material and heavy metal combinations to obtain skyrmions

with desired diameters.

To stabilize skyrmions, the interfacial Dzyaloshinskii-Moriya interaction (iDMI)

plays a vital role. The interface between a ferromagnet (FM) or antiferromagnet (AFM)

and a heavy metal (HM) results in an antisymmetric exchange interaction known as the

interfacial Dzyaloshinskii-Moriya interaction (iDMI) which favors non-collinear spin cong-

urations. The iDMI is responsible for stabilizing noncollinear spin textures such as skyrmions

in materials with bulk inversion symmetry. Interfacial DMI values have been previously de-

termined theoretically and experimentally for FM/HM interfaces, and, in this work, they

are calculated for the metallic AFM MnPt and the insulating AFM NiO. The heavy metals

considered are W, Re, and Au. Values for the iDMI, exchange, and anisotropy constants

are determined for dierent AFM and HM thicknesses. The iDMI values of the MnPt het-

erolayers are comparable to those of the common FM materials, and those of NiO are lower.

In few-layer lms of (001) MnPt, the high spin orbit coupling of the Pt layers can give rise

to a small DMI in the absence of a HM layer.

Magnetic properties have also been determined for a new family of two-dimensional

ferromagnetic materials. The recent demonstration of MoSi2N4 and its exceptional stability

to air, water, acid, and heat has generated intense interest in this family of two-dimensional

(2D) materials. Among these materials, monolayers of NbSi2N4, VSi2N4, and VSi2P4 are

semiconducting, easy-plane ferromagnets with negligible in-plane magnetic anisotropy. They

thus satisfy a necessary condition for exhibiting a dissipationless spin superuid mode. The

viii

Curie temperatures of monolayer VSi2P4 and VSi2N4 are determined to be above room

temperature based on Monte Carlo and density functional theory calculations. The magnetic

moments of VSi2N4 can be switched from in-plane to out-of-plane by applying tensile biaxial

strain or electron doping.

Besides magnetic materials, topological materials also oer various prospective ap-

plications. There have been also a tremendous interests in nding 1D and 2D materials. To

facilitate identifying materials of interest, 27 one dimensional and 300 two dimensional topo-

logical materials are identied using several databases. 1D and 2D topological insulators and

semimetals are identied. 1D and 2D topological materials which have been experimentally

demonstrated, are also identied.

ix

Contents

List of Figures xii

List of Tables xvi

1 Rationale 1

1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Above Room Temperature Low Power FM and SAFM Skyrmion Shift

Register in Presence of Disorder 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Device Structure and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Interfacial Dzyaloshinskii-Moriya Interaction of Antiferromagnetic Mate-

rials 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Magnetic Properties of NbSi2N4, VSi2N4, and VSi2P4 Monolayers 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6 Curie temperatures of other 2D Materials . . . . . . . . . . . . . . . . . . . . 60

x

4.7 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 1D and 2D Topological Insulators and Semimetals 62

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Summary and Outlook 94

7 Appendix 97

Bibliography 107

xi

List of Figures

2.1 (a) 3D view of FM skyrmion shift register where presence or absence of askyrmion is used as a memory bit instead of creation and annihilation of theskyrmion. The skyrmion will be detected under only one gate between twoconsecutive gates and the remaining gate will be used to aid in shifting theskyrmion back and forth. (b) Shifting the skyrmion by applying current alongthe Y-axis in the top metal gate region. (c) Cross section of the FM rectan-gular gate structure showing a 5nm wide gap region between two consecutivegates where the gap region acts as a barrier for skyrmion motion and thedummy gates to the left and right are used to create a symmetric potentialwithin each cell to bring the skyrmion to rest near the center of a cell. (d)Cross section of SAFM rectangular gate structure where current is applied inthe top gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Micromagnetic simulation of two dierent types of FM structures (165 nm× 64 nm × 0.8 nm) showing top view at 0 K: (a) Shifting operation of therectangular gate structure having a 5 nm gap where the skyrmion moves backand forth due to positive and negative current along the width direction and(b) Hourglass structure with single metal gate on top of the FM showing ashifting operation where the notch diameter acts as a barrier for skyrmionmotion and separates two cells. Only the FM rectangular and hourglassstructures at 0K are shown, however both FM and SAFM having these twotypes of structures are simulated at 0K and at 300K. . . . . . . . . . . . . . . 10

2.3 Change of current for dierent iDMI strengths in the gap region betweentwo gates. The FM structure is simulated keeping the iDMI strength of 1.93mJ/m2 constant under the gate regions and reducing the iDMI strength inthe gap region from 1.93 mJ/m2: (a) FM structure having width of 64nm at0 K, (b) FM structure having width of 128nm at 300 K. The SAFM structureis simulated keeping the iDMI strength of 3.5 mJ/m2 constant under the gateregion and reducing the iDMI strength in the gap region from 3.5 mJ/m2: (c)SAFM structure having width of 55nm at 0 K and (d) SAFM structure havingwidth of 86nm at 300 K. Current requirement increases with the increase ofiDMI variation in the gap region. Variation of iDMI in the gap region can beachieved by reducing the Pt thickness or by using HfO2. . . . . . . . . . . . 14

xii

2.4 Switching time versus current for the rectangular gate structure where thereduction of the iDMI strength in the gap region with respect to the gateregion is 0.25 mJ/m2. (a) FM structure having a width of 64nm at 0 K, (b)FM structure having a width of 128nm at 300 K, (c) SAFM structure havinga width of 55nm at 0 K, and (d) SAFM structure having width of 86nm at300 K. The switching time reduces with increasing injected current, and theswitching time of the SAFM skyrmion is less than that of the FM skyrmion. . 16

2.5 Switching time versus current in the hourglass structure for three dierentnotch diameters: (a) FM structure having width of 64nm at 0 K, (b) FMstructure having width of 128nm at 300 K, (c) SAFM structure having widthof 64nm at 0 K and (d) SAFM structure having width of 128nm at 300 K. . . 19

2.6 Velocity of SAFM skyrmion with current owing only in the top gate, andfor current in both the top and bottom gates. In our simulations, current isonly applied in the top gate, since such types of structures are easier to buildexperimentally. But if current is injected in both the top and bottom gatesin opposite directions, then velocity of the SAFM skyrmion becomes almostdouble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 The rectangular gate structure used as a racetrack memory. Double skyrmionsare simulated to show the repulsion between skyrmions: (a) Rectangular gateswith two SAFM skyrmions undergoing a shifting operation at T = 0 K. Theskyrmion and anti-skyrmion in the top and bottom layers are individuallyshown. Current ows only in the second from the left gate, labeled 2ndGate, and it moves both skyrmions due to the repulsion force between them.(b) The same as (a) but at T = 300 K. . . . . . . . . . . . . . . . . . . . . . . 21

2.8 Distance between two FM skyrmions and two SAFM skyrmions versus theskyrmion diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.9 Phase diagrams of a FM skyrmion in a 40nm wide region. The color shows theskyrmion diameter as indicated by the color bars. Parameters that are notvaried are given by the values listed in column 2 of Table 2.1. (a) InterfacialDMI strength versus Heisenberg exchange energy stiness. (b) iDMI strengthversus anisotropy energy. (c) iDMI strength versus external magnetic eld.(d) iDMI strength versus saturation magnetization. . . . . . . . . . . . . . . 24

3.1 (a) Unit cell of MnPt having stripe-type AFM spin congurations. (b) Unitcell of NiO having G-type AFM spin congurations. Spin directions are shownin 3d orbital magnetic materials (Mn or Ni) and atoms without spin are Pt(grey) and oxygen (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Spiral spin congurations used in the DMI calculations for (a,b,e) MnPt and(c,d) NiO. A 4× 1× 1 unit cell supercell of MnPt is constructed with a spinspiral in the (a) clockwise and (b) counterclockwise directions. A 2 × 2 × 1unit cell supercell of NiO is constructed with a spin spiral in the (c) clockwiseand (d) counterclockwise directions. (e) Clockwise rotated spin congurationin one layer of MnPt with labels used in Eq. (3.2). . . . . . . . . . . . . . . . 29

3.3 (a) MnPt on top of W. Mn atoms are purple. (b) NiO on top of Au. Oxygenatoms are red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xiii

3.4 (a) Dtot and (b) Dµ of MnPt-W, MnPt-Re, and NiO-Au as a function of thethickness of the AFM. The thickness of the HM is xed at 3 layers. PositiveDMI indicates that the CCW spin spiral is the lower energy state. . . . . . . 36

3.5 iDMI of monolayer AFM for two dierent HM thicknesses: (a) Dtot with onemagnetic layer and (b) Dµ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Magnetic moments and saturation magnetization of (a) MnPt-W, (b) MnPt-Re and (c) NiO-Au as a function of the thickness of the AFM. The thicknessof the HM is xed at 3 layers. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Local density of states of Mn atom in 3 layers of MnPt on top of 3 layers ofRe: (a) middle Mn atom which is far from Re, (b) bottom Mn atom which isclose to Re and is aected by proximity induced magnetic moment. . . . . . 39

3.8 (a) Magnetic anisotropy constant of MnPt with 3 layers of heavy metals. (b)Exchange stiness of MnPt with 3 layers of heavy metals. . . . . . . . . . . . 40

4.1 Structures and magnetic orientations of α1-NbSi2N4, α1-VSi2N4, and α1-VSi2P4. Blue atoms are Si, white atoms are N, and pink atoms are P. Thedirections of the magnetic moments are shown on the transition metals Nband V. All materials are easy-plane, semiconducting ferromagnets. . . . . . . 44

4.2 Spin resolved energy bands of VSi2N4 and VSi2P4 calculated with (a,b)HSE06 and (c,d) PBE+U. Spin up bands are red and spin down bands areblue. The two spin-split, narrow bands on either side of the Fermi level ared-orbital bands centered on the transition metal. . . . . . . . . . . . . . . . . 49

4.3 Monte Carlo calculations of the normalized magnetization as a function oftemperature for (a) VSi2N4 and (b) VSi2P4. The solid lines show the bestts to the analytical expression given by Eq. (4.2). . . . . . . . . . . . . . . . 52

4.4 MAE as a function of uniaxial and biaxial strain calculated with PBE(SOC)+U.MAE versus uniaxial strain: (a) VSi2N4 and (b) VSi2P4. MAE versus biaxialstrain: (c) VSi2N4 and (d) VSi2P4. Values are shown for the strain coecientsfor dierent regions discussed in the text. The orientation of the magnetiza-tion of VSi2N4 rotates from in-plane to out-of-plane with 3.3% biaxial strain.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 MAE as a function of band lling calculated with PBE(SOC)+U. MAE as afunction of excess electrons per unit cell for (a) VSi2N4 and (b) VSi2P4. Valuesare shown for the lling coecients discussed in the text. The orientationof the magnetization of VSi2N4 rotates from in-plane to out-of-plane withelectron doping of 2.0× 1014 cm−2. . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Orbital resolved PBE band structure of VSi2N4 at equilibrium: (a) s-orbitalcontribution, (b) p-orbital contribution, and (c) d-orbital contribution. Theweight is given by the color bars at right. The two isolated, narrow bands nearthe Fermi level are d-orbital bands centered on the transition metal atomsvanadium (V). The higher valence bands are primarily p-orbital bands whichcome from the N atoms. The lower conduction bands are primarily d-orbitalbands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xiv

4.7 d-orbital resolved PBE band structure of VSi2N4: (a) spin up bands, (c) spindown bands. d-orbital resolved band structure of VSi2P4: (b) spin up bands,(d) spin down bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.8 Spin resolved PBE band structure: (a) NbSi2N4, (b) VSi2N4, and (c) VSi2P4.Spin resolved PBE+U band structure: (d) NbSi2N4, (e) VSi2N4, and (f)VSi2P4. Spin resolved HSE06 band structure: (g) NbSi2N4, (h) VSi2N4, and(i) VSi2P4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.9 MC calculations of the normalized magnetization as a function of temper-ature with exchange constants extracted from PBE DFT for (a) NbSi2N4,(b) VSi2N4, and (c) VSi2P4. Curie temperatures are calculated from MonteCarlo simulation using the VAMPIRE software. . . . . . . . . . . . . . . . . . 59

5.1 Flowchart of identifying 1D and 2D topological materials from several database.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

xv

List of Tables

2.1 LLG parameters used for simulating the FM and SAFM structures. . . . . . . 112.2 Comparison of energy, switching time, charge current density, and width

among two dierent types of FM and SAFM structures at 0K and at 300K.The gap is 5nm in the rectangular gate structures. In the hourglass struc-tures, the notch diameter is 10nm for FM, 15nm for SAFM at 0K and 45nmfor SAFM at 300K. SAFM generally requires less switching energy comparedto FM. The current requirement increases at 300K to overcome the ther-mal eld. A wider simulation region is needed at 300K as the skyrmion islarger compared to 0K. SAFM rectangular structure can be made narrowercompared to FM as skyrmion moves in a straight line in the SAFM struc-ture. Hourglass structures generally need less current and energy comparedto rectangular structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Interlayer distances in the system of three layers of MnPt on top of threelayers of W or Re and in the system of three layers of NiO on top of threelayers of Au. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 Magnetic moment, exchange energy per link, equilibrium magnetic anisotropyenergy (MAE), and Curie temperature, calculated using dierent functionalsfor NbSi2N4, VSi2N4, and VSi2P4. ∗TC is calculated using the HSE06 ex-change energy and the PBE(SOC)+U MAE. . . . . . . . . . . . . . . . . . . . 51

5.1 2D topological insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 2D topological semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 1D topological insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 1D topological semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.5 2D topological insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.6 2D topological semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.7 2D experimental topological insulators . . . . . . . . . . . . . . . . . . . . . . 925.8 2D experimental topological semimetals . . . . . . . . . . . . . . . . . . . . . 93

xvi

Chapter 1

Rationale

1.1 Objectives

Magnetic based memory is one of the leading approaches for non-volatile memory.

Skyrmions are being considered as an alternative to other magnetic based memory devices,

such as the magnetic tunnel junction (MTJ), since skyrmion based memory can require lower

power to operate. [4] Previous works on ferromagnetic (FM) and synthetic antiferromagnetic

(SAFM) skyrmion shifting or moving did not have any mechanism to x the skyrmion in

a certain region, since the skyrmion always tries to move where it feels least repulsion

force. [510] In racetrack type memories, this property is a feature, since skyrmions have the

ability to move around defects. [11] Here, two dierent types of structures are investigated,

which can force skyrmion to stay at a specic location even in presence of thermal energy.

These two types of structures are compared for FM and SAFM skyrmions in terms of energy

and switching time at room temperature in presence of disorder. SAFM skyrmions, where

two ferromagnetic layers are antiferromagnetically exchange coupled by a spacer layer, oers

1

higher velocity compared to FM skyrmions. [12]

Antiferromagnetic (AFM) materials oer various promising spintronic applications

exploiting skyrmions or domain walls, which can be used as non-volatile memory and race-

track memory [1320]. A bilayer consisting of a heavy metal (HM), which oers high spin or-

bit coupling (SOC), and a magnetic material can be created that breaks inversion symmetry

at the interface. At a HM/FM or HM/AFM interface, the interfacial Dzyaloshinskii-Moriya

interaction (iDMI), which is an anti-symmetric exchange interaction, can stabilize Néel type

domain walls and skyrmions [2131]. There have been prior theoretical and experimental

studies of the iDMI in engineered or synthetic antiferromagnetic systems [3235]. This the-

sis describes the results of ab initio calculations of the iDMI for three dierent material

combinations of HMs and AFMs.

Two dimensional (2D) layered materials with transition metals have been of in-

terest for many decades due to the correlated phenomena and multiple polymorphs and

phases that they can exhibit such as charge density waves, [36] superconductivity, [37] and

magnetism. [38] The most recent addition to the family of 2D materials are the transition

metal silicon nitrides, phosphides, and arsenides with the chemical formulas MA2Z4, where

M is the transition metal, A ∈ Si, Ge, and Z ∈ N, P, As. [39] High quality multilayers

and monolayers of MoSi2N4 were grown using chemical vapor deposition, and what was

particulary notable was the stability of the material. In this thesis, we theoretically inves-

tigate the magnetic properties of the three Si based magnetic materials: VSi2N4, VSi2P4,

NbSi2N4 which are from MA2Z4 family. The exchange constants are determined, the Curie

temperatures are calculated, and the Curie temperature of VSi2N4 is found to be at room

2

temperature. The magnetic anisotropy energies of VSi2N4, VSi2P4, and NbSi2N4 are calcu-

lated as a function of uniaxial and biaxial strain and electron and hole doping.

1.2 Organization

The rest of the dissertation is organized as follow: Chapter 2 presents skyrmion

based ferromagnetic and synthetic antiferromagnetic memory devices, which can operate

near room temperature. Chapter 3 presents the interfacial Dzyaloshinskii-Moriya interac-

tion of three antiferromagnetic Materials which are needed to stabilize skyrmions. Chap-

ter 4 presents magnetic properties such as the Curie temperatures, exchange energies and

magnetic anisotropy energies of ferromagnetic NbSi2N4, VSi2N4, and VSi2P4 Monolayers.

Chapter 5 describes the identicaiton of 1D and 2D topological materials from several ma-

terial databases. Chapter 6 is a summary of all the interesting ndings and an outlook for

future work. In the Appendix, the codes for topological materials, skyrmions, Monte Carlo

simulation, band structures plotting are documented.

3

Chapter 2

Above Room Temperature Low

Power FM and SAFM Skyrmion Shift

Register in Presence of Disorder

2.1 Introduction

Magnetic based memory oers certain advantages over traditional charge based

memory, since it is non-volatile and can require less power to operate. Skyrmion based

memory is being considered as an alternative to other spintronic based memory devices such

as the magnetic tunnel junction (MTJ) since it can require less power to operate. [4] A

skyrmion is a topologically protected swirling spin texture that relies on the presence of

the Dzyaloshinskii-Moriya interaction (DMI) for its stabilization. The DMI requires broken

inversion symmetry in a magnetic material. The B20 compounds are bulk magnetic materials

4

that satisfy this criterion, but they are not very common. However, inversion symmetry

is always broken at an interface. Bilayers consisting of a heavy metal (HM) with high

spin orbit coupling and a ferromagnetic material have an interfacial DMI (iDMI) that can

support an interfacial skyrmion [21]. These engineered structures have made stabilization

of skyrmions near room temperature possible [2224]. There are several ways to control

the iDMI: changing the FM or heavy metal (HM) or oxide layer material [40, 41], changing

the FM or HM or oxide layer thickness [4247], increasing the number of layers [48], using

congurations such as HM/FM, HM1/FM/HM2, HM/FM/Oxide, HM1/HM2/FM/Oxide,

HM/FM/Metal/Oxide [6, 6, 47, 4953], and by changing their relative positions. Similar to

controlling the iDMI, the sign and amplitude of the spin Hall angle (θSH) of the HM layer

can be adjusted with the appropriate choice of heavy metal and its relative position. The

eective spin Hall angle can be increased by adding top and bottom HM layers with additive

iDMI [5456].

5

Figure 2.1: (a) 3D view of FM skyrmion shift register where presence or absence of a

skyrmion is used as a memory bit instead of creation and annihilation of the skyrmion.

The skyrmion will be detected under only one gate between two consecutive gates and the

remaining gate will be used to aid in shifting the skyrmion back and forth. (b) Shifting

the skyrmion by applying current along the Y-axis in the top metal gate region. (c) Cross

section of the FM rectangular gate structure showing a 5nm wide gap region between two

consecutive gates where the gap region acts as a barrier for skyrmion motion and the dummy

gates to the left and right are used to create a symmetric potential within each cell to bring

the skyrmion to rest near the center of a cell. (d) Cross section of SAFM rectangular gate

structure where current is applied in the top gate.

Racetrack memory based on domain wall motion is a low energy memory [57].

Skyrmion based racetrack memory requires even less energy, and it has the added advantage

that skyrmions move around obstacles such as defects. In skyrmion based racetrack memory,

skyrmion creation and annihilation are needed each time to represent 1/0 bit. Instead of

6

creating and annihilating a skyrmion each time, the presence or absence of the skyrmion can

be considered as a memory bit. The skyrmion can move along the X-direction by applying

current in Y-direction in the adjacent heavy metal layer using the property of the skyrmion

Hall angle and the edge repulsion force. [58] This shifting or sliding switch memory (Fig. 2.1a

and Fig. 2.1b) requires lower energy compared to the switching energy of a magnetic tunnel

junction (MTJ). [59]

Previous works on FM and SAFM skyrmion shifting or moving do not include

a mechanism to x the skyrmion in a certain region, since the skyrmion always tries to

move where it feels least repulsion force. [510] Here, two dierent types of structures are

considered which can force the skyrmion to stay at a specic location even in presence of

thermal energy. These two types of structures are compared for FM and SAFM skyrmions

in terms of energy and switching time at room temperature in the presence of disorder.

SAFM skyrmions, where two FM layers are antiferromagnetically exchange coupled by a

spacer layer, oer higher velocity compared to FM skyrmions [12]. The proposed structures

can work both as a shift memory where a single skyrmion is shuttled back and forth between

two cells and as racetrack memory where several skyrmions will be clocked along a gated

track. All of the necessary phase diagrams are calculated showing the relation among DMI

strength, exchange stiness, anisotropy energy, magnetic eld and saturation magnetization

considering the diameter variation of the skyrmion which in turn determines the device

dimension.

7

2.2 Device Structure and Operation

Two dierent types of structures are made using FM and SAFM skyrmions. The

FM device structure, shown in Fig. 2.1c, has three layers. (i) The bottom layer or back gate

or substrate can be made of a HM, HM1/HM2, oxide, or metal/oxide. The choice will aect

the iDMI. This back gate or bottom layer provides iDMI even when a top HM layer is not

present. This layer is considered in the simulation by adding the appropriate iDMI. (ii) The

middle FM layer consists of a 0.8 nm thick CoFeB lm. The Heisenberg exchange energy

stiness, anisotropy energy, damping constant, and saturation magnetization are used to

represent its properties. (iii) The top Pt heavy metal gate increases the iDMI, and the

charge current is injected in this layer.

The SAFM device structure, shown in Fig. 2.1d, is made of two FM layers antifer-

romagnetically coupled by a spacer layer The SAFM device has ve layers. (i) The bottom

layer is Pt which increases iDMI. (ii) The bottom cobalt layer supports a skyrmion. (iii) The

spacer layer antiferromagnetically couples the top and bottom Co layers and also provides

iDMI. (iv) The top cobalt layer supports an anti-skyrmion with opposite magnetization to

that in the bottom Co layer. (v) The top Pt layer increases the iDMI and carries charge

current that injects spin through the spin Hall eect.

Two dierent structure geometries are simulated for both FM and SAFM skyrmions:

a rectangular gate structure (Fig. 2.2a) and an hourglass structure (Fig. 2.2b). In the rect-

angular gate structure, in the gap between the two gates, the iDMI is lower, and the reduced

iDMI serves as a barrier to conne the skyrmion within an individual cell after the drive

current is turned o. The gap region of the rectangular gate structure can be made by

8

reducing the Pt thickness or by replacing the Pt with an insulator. Pt resistivity is strongly

dependent on its thickness. The resistivity increases to a high value for 1 nm thick Pt,

which then acts as an insulator in the gap region of the rectangular gate structure. [60] The

thinner Pt also reduces the iDMI in the gap region, so that the gap region acts as a barrier

to conne skyrmions within an individual cell after the drive current is switched o. An

insulator like HfO2 can also be used in the gap region, and it also provides reduced iDMI

to the magnetic layers. [61,62] DMI can be increased by reducing the thickness of FM layer

(CoFeB or Cobalt). [63] DMI can also be increased by increasing the heavy metal layer (Pt)

thickness. [64] Dummy gates at two ends help to create a symmetric potential in each cell so

that the skyrmion comes to rest at the center of a cell. Centering the skyrmion maximizes

the magnetoresistive readout performed with an MTJ.

The Hourglass shape provides control of the skyrmion with the use of a single gate

over the entire FM region. The narrow region of the hourglass shape acts as a barrier for

localizing the skyrmion when there is no current.

9

Figure 2.2: Micromagnetic simulation of two dierent types of FM structures (165 nm ×

64 nm × 0.8 nm) showing top view at 0 K: (a) Shifting operation of the rectangular gate

structure having a 5 nm gap where the skyrmion moves back and forth due to positive

and negative current along the width direction and (b) Hourglass structure with single

metal gate on top of the FM showing a shifting operation where the notch diameter acts

as a barrier for skyrmion motion and separates two cells. Only the FM rectangular and

hourglass structures at 0K are shown, however both FM and SAFM having these two types

of structures are simulated at 0K and at 300K.

A skyrmion can be moved back and forth by reversing the Y-direction of current in

the top heavy metal layer (Fig. 2.1a and Fig. 2.1b). Larger width structures show skyrmion

velocities with both X and Y components as a result of the skyrmion Hall eect. The

Y-component vanishes once the skyrmion is pushed against the channel edge boundary.

For narrow width channels, the velocity is always along the X-direction, since the motion

along Y-direction is nullied due to the repulsion from the nearby edge. A skyrmion can

move back and forth in this structure by applying positive and negative current along the

10

Parameters FM (CoFeB) [6, 65] SAFM (Cobalt) [7

10,66,67]

Msat 0.9 MA/m 0.58 MA/m

Intralayer Aex 14 pJ/m 15 pJ/m

Interlayer Aex 0 −1 pJ/m

Ku 0.779 MJ/m3 0.8 MJ/m3

α 0.5 0.3

Bext 12.5 mT 0

Dind 1.93 mJ/m2 3.5 mJ/m2

Dind in gap region 1.68 mJ/m2 3.25 mJ/m2

Table 2.1: LLG parameters used for simulating the FM and SAFM structures.

width (Y) direction of the HM gate layer. At smaller currents, the channel edge serves as a

repulsive boundary that connes the skyrmion. At higher drive currents, the channel edge

will annihilate the skyrmion as it is driven towards the edge.

2.3 Theory

The spin dynamics of skyrmions in these two types of structures can be described

by the Landau-Lifshitz-Gilbert (LLG) equation, [68]

∂mi

∂t= −γmi ×Heff + αmi ×

∂mi

∂t, (2.1)

11

where i denotes lattice site, α is the Gilbert damping, and the eective eldHeff is computed

as Heff = − 1|µi|

∂H∂mi

.

The Slonczewski torque due to the ow of current is transformed to the Landau-

Lifshitz formalism and is added in the LLG equation which is dened in MuMax3 [69].

Generally, the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation takes the form which

contains only the anti-damping torque due to the spin Hall eect. Later it was found that

there is an additional eld-like (FL) spin orbit torque (SOT) which had been previously

neglected [70].

The total Hamiltonian incudes the classical Heisenberg ferromagnetic exchange

interaction (represented by J), the Dzyaloshinskii-Moriya interaction (Dij), the uniaxial

anisotropy (Ku) along the z axis, and the magnetic eld (h) applied to the system in the

positive z direction. Therefore, the total Hamiltonian of the system is given by [68]

H =− J∑<i,j>

mi ·mj +∑<i,j>

Dij · [mi ×mj ]

−∑i

Ku(ez ·mi)2 −

∑i

|µi|h ·mi, (2.2)

where Dij is the DMI vector and mi is the unit vector of a magnetic moment µi = −ℏγS

with S being the atomic spin and γ the gyromagnetic ratio. With this Hamiltonian, the

dipole interaction energy and thermal energy are also included. The dipole interaction

or demagnetizing eld is included using a nite dierence discretization, which allows the

demagnetizing eld to be evaluated as a (discrete) convolution of the magnetization with

a demagnetizing kernel as implemented in MuMax3. Finite temperature is included by

calculating the thermal eld using a time step and a random vector from a standard normal

distribution whose value is changed after every time step [71]. The eective eld, Heff in

12

the torque of Eq. (2.1) is computed from the total Hamiltonian as Heff = − 1|µi|

∂H∂mi

.

The velocity of skyrmion motion can be found from Thiele's equation:

G× v − α←→D · v + FSH = 0 (2.3)

where FSH is the force due to the spin current, α the Gilbert damping, and←→D the dissipative

tensor. Here, the gyrotropic vector is represented as G =(0, 0,−4πQ

), where Q is the

skyrmion topological number, and the velocity has X and Y components, v =(vx, vy

). Both

components of the velocity depend on the magnitude and direction of the current, and the

velocity perpendicular to the current also depends on the skyrmion number Q.

13

Figure 2.3: Change of current for dierent iDMI strengths in the gap region between two

gates. The FM structure is simulated keeping the iDMI strength of 1.93 mJ/m2 constant

under the gate regions and reducing the iDMI strength in the gap region from 1.93 mJ/m2:

(a) FM structure having width of 64nm at 0 K, (b) FM structure having width of 128nm at

300 K. The SAFM structure is simulated keeping the iDMI strength of 3.5 mJ/m2 constant

under the gate region and reducing the iDMI strength in the gap region from 3.5 mJ/m2:

(c) SAFM structure having width of 55nm at 0 K and (d) SAFM structure having width of

86nm at 300 K. Current requirement increases with the increase of iDMI variation in the gap

region. Variation of iDMI in the gap region can be achieved by reducing the Pt thickness or

by using HfO2.

14

2.4 Simulation Details

LLG parameters used here are for room temperature interfacial skyrmion hosting

materials CoFeB and Cobalt. The saturation magnetization, exchange stiness, uniax-

ial anisotropy constant in the Z-direction, Landau-Lifshitz damping constant, interfacial

Dzyaloshinskii-Moriya strength are used to simulate the CoFeB or Co layers. An external

eld is applied only for the FM, and an antiferromagnetic exchange stiness is used to sim-

ulate the spacer layer between the top and bottom Co layers in the SAFM (Table 2.1). For

the Pt metal gate on top of the FM layer, the spin Hall angle is almost −0.1. In MuMax3,

the Slonczewski spin torque equation consists of polarization term which is set as 0.5 here.

The spin torque equation includes the spin Hall angle, and these two forms of equations are

compared to nd the actual charge current needed in the top Pt metal layer. The eect of

disorder has been simulated by using 10% variation of magnitude and direction of uniaxial

anisotropy. The eect of temperature is included by using Heun's method which oers 2nd

order Runge-Kutta method for convergence and a 1st order method for error estimation.

15

2.5 Results

Figure 2.4: Switching time versus current for the rectangular gate structure where the

reduction of the iDMI strength in the gap region with respect to the gate region is 0.25

mJ/m2. (a) FM structure having a width of 64nm at 0 K, (b) FM structure having a width

of 128nm at 300 K, (c) SAFM structure having a width of 55nm at 0 K, and (d) SAFM

structure having width of 86nm at 300 K. The switching time reduces with increasing injected

current, and the switching time of the SAFM skyrmion is less than that of the FM skyrmion.

The iDMI in the gap region can be varied by changing the Pt thickness in the gap

region or by lling with an insulator material like HfO2. As the iDMI in the gap is reduced,

the current required to push a skyrmion through the gap region from one cell to another

increases as shown in Fig. 2.3. The required current also increases for wider gap regions.

To determine the eect of current on the switching time, the iDMI in the gap region is kept

16

Device Energy Time JC (A/cm2) Width

(nm)

FM Rectangular (0 K) 38.3 fJ 3.4 ns 3× 108 64

FM Hourglass (0 K) 5.36 fJ 4.2 ns 0.7× 108 64

SAFM Rectangular (0 K) 18.8 fJ 0.7 ns 5× 108 55

SAFM Hourglass (0 K) 7.82 fJ 3 ns 1× 108 64

FM Rectangular (300 K) 121 fJ 1.6 ns 5.5× 108 128

FM Hourglass (300 K) 233 fJ 2.8 ns 4× 108 128

SAFM Rectangular (300

K)

104 fJ 2.05 ns 5.5× 108 86

SAFM Hourglass (300 K) 45.8 fJ 2.2 ns 2× 108 128

Table 2.2: Comparison of energy, switching time, charge current density, and width among

two dierent types of FM and SAFM structures at 0K and at 300K. The gap is 5nm in

the rectangular gate structures. In the hourglass structures, the notch diameter is 10nm for

FM, 15nm for SAFM at 0K and 45nm for SAFM at 300K. SAFM generally requires less

switching energy compared to FM. The current requirement increases at 300K to overcome

the thermal eld. A wider simulation region is needed at 300K as the skyrmion is larger

compared to 0K. SAFM rectangular structure can be made narrower compared to FM as

skyrmion moves in a straight line in the SAFM structure. Hourglass structures generally

need less current and energy compared to rectangular structures.

17

xed (0.25 mJ/m2 lower compared to gate region) and the switching time is plotted as a

funciont of the drive current in Fig. 2.4. To determine the switching time, A large number

of images are generated from the micromagnetic simulation as a funciton of time to track

the position of the skyrmion with time. The switching time is calculated by nding the time

for a skyrmion to move from its initial relaxed position in one gate/cell, to the nal relaxed

position in another gate/cell. The switching time of the SAFM skyrmion is less than that

of the FM skyrmion, since its velocity is higher and it travels in a straight path whereas FM

skyrmion moves in a curved path due to the skyrmion Hall eect. In presence of temperature

(300K) and disorder, the size of the skyrmion increases, but it can still move perpendicular

to the channel in the wider simulation regions. For the FM structure, the required width

of the FM region increases from 64 nm at 0 K to 128 nm at 300K. The SAFM requires less

width (55 nm at 0 K and 86 nm at 300 K) since the SAFM skyrmion moves in a straight

path whereas the FM skyrmion follows a curved path.

18

Figure 2.5: Switching time versus current in the hourglass structure for three dierent notch

diameters: (a) FM structure having width of 64nm at 0 K, (b) FM structure having width of

128nm at 300 K, (c) SAFM structure having width of 64nm at 0 K and (d) SAFM structure

having width of 128nm at 300 K.

In the hourglass shape structure shown in Fig. 2.2b the skyrmion can pass the

narrow region if the current is sucient. The switching times as a function of current are

shown in Fig. 2.5. The FM skyrmion moves in a curved way, so the notch diameter in the

FM structure is generally smaller compared to that in the SAFM structure. We have also

varied the diameter of the notches to restrict the passageway of the skyrmions. As the notch

diameter increases, it makes the skyrmion passageway narrower, and the current requirement

increases. A comparison of current, switching time, energy and width are shown in Table

2.2 for the rectangular gate structure and the hourglass structure made of FM and SAFM

(at 0K and at 300K). The switching energy is calculated from the resistance of a 3 nm thick

19

Pt metal gate and the charge current owing through the metal. It is the I2R power times

the switching time.

Figure 2.6: Velocity of SAFM skyrmion with current owing only in the top gate, and for

current in both the top and bottom gates. In our simulations, current is only applied in the

top gate, since such types of structures are easier to build experimentally. But if current is

injected in both the top and bottom gates in opposite directions, then velocity of the SAFM

skyrmion becomes almost double.

The SAFM skyrmion can move if current is injected in both the top and bottom

gate, or if it is injected only in the top gate. The skyrmion and antiskyrmion pairs in the

top and bottom FM layers of the synthetic antiferromagnet require opposite directions of

the top and bottom currents to drive them in the same direction, since their topological

charges are opposite. In all of the previous simulations, current is injected only in the top

gate, since such a structure is easier to build experimentally. If the current is applied in

both gates as shown in Fig. 2.6, the velocity almost doubles.

20

Figure 2.7: The rectangular gate structure used as a racetrack memory. Double skyrmions

are simulated to show the repulsion between skyrmions: (a) Rectangular gates with two

SAFM skyrmions undergoing a shifting operation at T = 0 K. The skyrmion and anti-

skyrmion in the top and bottom layers are individually shown. Current ows only in the

second from the left gate, labeled 2nd Gate, and it moves both skyrmions due to the

repulsion force between them. (b) The same as (a) but at T = 300 K.

Apart from the application as a switching memory, the proposed structure can

also be used as racetrack memory where multiple skyrmions can be clocked along the track.

Multiple skyrmions can be controlled individually, and due to the presence of multiple

skyrmions, the skyrmion-skyrmion repulsion force needs to be considered. Two synthetic

AFM skyrmions are simulated to address the issue of repulsion, and, as shown in Fig. 2.7,

they always stay some distance from each other due to the repulsion force between skyrmions.

A skyrmion moving under one gate with current ow, can move the skyrmion under another

gate even though there is no current owing in the other gate. We must determine the gate

width required to keep only one skyrmion under one gate to avoid errors during the read

operation using a magnetic tunnel junction (MTJ). The distance between two skyrmions

21

depends on their diameters. The variation of the distance with FM and SAFM skyrmion

diameter (by changing iDMI) is shown in Fig. 2.8. The distance is calculated as the center-

to-center distance of the skyrmions where image processing has been used to determine

the the center of the skyrmions. The center-to-center distance is always greater than the

diameter, and the FM skyrmions stay farther apart compared to the SAFM skyrmions.

Figure 2.8: Distance between two FM skyrmions and two SAFM skyrmions versus the

skyrmion diameter.

To understand the eect of the LLG parameters on the skyrmion diameter, which in

turn will determine the dimension of the device, all necessary phase diagrams are included

here for the FM skyrmion. Field-dependent size and shape variations of single magnetic

skyrmions are available in the literature [72]. The Ku-Dind phase diagram for a small range

is available, but it does not contain information of the skyrmion size. [68] Here, the Aex-Dind

phase diagram of the skyrmion diameter is determined. For this purpose, a 128 nm × 40

nm × 0.8 nm ferromagnetic material is simulated for a large array of Aex and Dind values.

Skyrmion formation depends on a specic range of ratios between Aex and Dind, as shown in

the phase diagram Fig. 2.9(a). The skyrmion diameter stays well below the width of the FM

22

region (40nm) due to the repulsion force from the boundary. For small Dind, the exchange

energy dominates and the system is ferromagnetic with all spins aligned. At larger ratios

of Dind/Aex, the iDMI dominates and the system is helical. In between, there is a balance

between the exchange energy favoring parallel alignment of the spins and the iDMI favoring

perpendicular alignment, with the result being the skyrmion phase.

Fig. 2.9b shows the phase diagram of the anisotropy constant (Ku) and the iDMI

strength (Dind). A larger Ku favors out-of-plane spin alignment and a larger Dind favors

nearest-neighbor spins to be perpendicular to each other resulting in in-plane components

of the spin. Thus, for larger Ku, a larger Dind is required to support a skyrmion with in-

plane components to the spins. Fig. 2.9b also shows that to stabilize skyrmion, a minimum

amount of DMI and anisotropy is required to support a skyrmion. The phase diagram of

the external magnetic eld Bext versus the iDMI strength (Dind) (Fig. 2.9c) shows behavior

similar to that of the Ku-Dind phase diagram. This is not surprising since the z-directed

magnetic eld also favors out-of-plane spin alignment. A larger magnetic eld requires a

larger iDMI to stabilize the skyrmion, and larger magnetic eld makes the skyrmion diameter

smaller for a xed iDMI [72]. The phase diagram of the saturation magnetization (Msat)

versus the iDMI strength (Dind), shown in Fig. 2.9d, shows a pattern opposite to that

of the Ku-Dind phase diagram, because increasing Msat decreases the eective anisotropy

(Keff = Ku−2πM2sat) [73]. A higherMsat reduces the eective anisotropy, and the skyrmion

can be stabilized with a smaller iDMI [73]. Since the iDMI, Heisenberg exchange, anisotropy,

and Zeeman energy do not depend on the damping constant (α), the skyrmion stability does

not depend on α. However, the damping constant does eect the velocity of skyrmion, so that

23

the switching time increases with an increase in the damping constant of the ferromagnetic

material. [70]

Figure 2.9: Phase diagrams of a FM skyrmion in a 40nm wide region. The color shows the

skyrmion diameter as indicated by the color bars. Parameters that are not varied are given

by the values listed in column 2 of Table 2.1. (a) Interfacial DMI strength versus Heisenberg

exchange energy stiness. (b) iDMI strength versus anisotropy energy. (c) iDMI strength

versus external magnetic eld. (d) iDMI strength versus saturation magnetization.

2.6 Conclusions

The energy requirement for SAFM skyrmion switching is much less compared to

that for FM skyrmion switching, and the switching time is less. The SAFM skyrmions have

higher velocity and do not experience the skyrmion Hall eect. For prospective applications,

these phase diagrams can be useful for selecting the right combinations of materials. The

successful simulation of two types of shift registers will pave the way for low power memory

24

design with minimized area.

2.7 Acknowledgements

This work was supported as part of Spins and Heat in Nanoscale Electronic Systems

(SHINES) an Energy Frontier Research Center funded by the U.S. Department of Energy,

Oce of Science, Basic Energy Sciences under Award #DE-SC0012670.

25

Chapter 3

Interfacial Dzyaloshinskii-Moriya

Interaction of Antiferromagnetic

Materials

3.1 Introduction

Antiferromagnetic (AFM) materials are of current interest due to their insensitivity

to external magnetic elds, absence of demagnetizing elds, scalability to small dimensions,

widespread availability in nature with high Néel temperatures, and operation in the THz

frequency range [7481]. Antiferromagnetic materials exhibit various interesting phenomena

such as a large anomalous Hall eect, spin Seebeck eect, spin Hall magnetoresistance and

spin galvanic eects [8294]. They oer various promising spintronic applications exploiting

skyrmions or domain walls which can be used as non-volatile memory and racetrack memory

26

Figure 3.1: (a) Unit cell of MnPt having stripe-type AFM spin congurations. (b) Unit

cell of NiO having G-type AFM spin congurations. Spin directions are shown in 3d orbital

magnetic materials (Mn or Ni) and atoms without spin are Pt (grey) and oxygen (red).

[1320].

Noncollinear spin congurations such as skyrmions and chiral helices are found

in non-centrosymmetric materials with broken inversion symmetry [9597]. Since non-

centrosymmetric magnetic materials are not common in nature, heterolayers consisting of

a heavy metal (HM), which oers high spin orbit coupling (SOC), and a magnetic mate-

rial can be created that breaks inversion symmetry at the interface. At an HM/FM or

HM/AFM interface, the interfacial Dzyaloshinskii-Moriya interaction (iDMI), which is an

anti-symmetric exchange interaction, can stabilize Néel type domain walls and skyrmions

[2131]. The interfacial DMI of FM materials is generally measured using Brillouin light

scattering (BLS) [98104]. Experimentally AFM spin textures are measured using spin po-

larized scanning tunneling microscopy (STM), and the bulk DMI in non-centrosymmetric

materials is measured using inelastic neutron scattering [105,106].

There have been prior theoretical and experimental studies of iDMI in engineered

27

or synthetic antiferromagnetic systems [3235]. In [34,35], spin-polarized scanning tunneling

microscopy was used to study heterostructures of monolayer Mn on W(110) and Re(0001),

and the iDMI was calculated using density functional theory (DFT). The same theoretical

approach was used to study the iDMI of Fe/Ir bilayers on Rh(001) [32]. In [33], the two-

dimensional AFM-Rashba model was introduced to calculate the iDMI in 2D Rashba AFMs.

This paper describes the results of ab initio calculations of the iDMI for three

dierent material combinations of HMs and AFMs. Since the iDMI helps to stabilize Néel

type skyrmions in which spins rotate from out-of-plane to in-plane, the AFM materials

MnPt and NiO are chosen with Néel vectors oriented out-of-plane [107111]. Also, both

MnPt and NiO have high Néel temperatures (975K and 530K, respectively) [112114], which

are required for on-chip applications. MnPt has a stripe-type AFM spin texture, and NiO

has a G-type AFM spin texture. Creating heterolayers for density functional theory (DFT)

calculations requires lattice matching to create a periodic structure. This generally requires

choosing materials with lattice constants that are relatively close to minimize strain. For

this reason, the two HMs paired with MnPt are tungsten (W) and rhenium (Re), and the

HM paired with NiO is gold (Au). MnPt-W has lattice mismatch of 1.49%, MnPt-Re has a

mismatch of 1.25%, and NiO-Au has a mismatch of 0.7%. The eects of thickness variation

of the AFM layer and the heavy metal layer on the iDMI are discussed. The magnetic

moments, magnetizations, exchange constants, and magnetic anisotropy constants are also

calculated as functions of layer thicknesses. These values provide required input parameters

for micromagnetic modeling of AFM spin textures.

28

Figure 3.2: Spiral spin congurations used in the DMI calculations for (a,b,e) MnPt and

(c,d) NiO. A 4 × 1 × 1 unit cell supercell of MnPt is constructed with a spin spiral in the

(a) clockwise and (b) counterclockwise directions. A 2 × 2 × 1 unit cell supercell of NiO is

constructed with a spin spiral in the (c) clockwise and (d) counterclockwise directions. (e)

Clockwise rotated spin conguration in one layer of MnPt with labels used in Eq. (3.2).

29

Interlayer Distance (Å)

Mn-Pt 1.85

Mn-Re 1.84

Mn-W 1.81

Ni-Ni 2.09

Ni-Au 2.59

Table 3.1: Interlayer distances in the system of three layers of MnPt on top of three layers

of W or Re and in the system of three layers of NiO on top of three layers of Au.

3.2 Method

The interfacial DMI is evaluated using rst principles calculations based on the

Vienna Ab initio Simulation Package (VASP) [115]. The electron-core interactions are de-

scribed by the projected augmented wave (PAW) potentials [116], and the exchange correla-

tion energy is included with the generalized gradient approximation (GGA) parameterized

by Perdew-Burke-Ernzerhof (PBE) [117]. The cuto energy for the plane wave basis is 550

eV in all calculations. The Monkhorst-Pack scheme is used with Γ-centered 10 × 10 × 1

k-point grids to make sure the total energy is converged within 10-5 eV per unit cell.

For calculations of the iDMI, the unit cells in Fig. 3.1 are repeated as shown in

Fig. 3.2, so that spin orientations can be rotated in clockwise (CW) and counter-clockwise

(CCW) directions [118120]. MnPt and W (or Re) unit cells are repeated four times to

make a supercell (4× 1× 1) in which spins of the Mn atoms are rotated one period over the

30

length of the supercell as shown in Fig. 3.3(a). A vacuum layer of 15 Å is added along the

out-of-plane direction. NiO and Au unit cells are repeated to make a supercell (2×2×1) in

which the spins of the Ni atoms are rotated once over the length of the supercell as shown

in Fig. 3.3(b).

Calculations are performed in three steps to nd the interfacial Dzyaloshinskii-

Moriya interaction [118120]. The supercell structure is relaxed until the forces are smaller

than 0.01 eV/Å to determine the most stable interfacial geometries. Table 3.1 shows the

relaxed interlayer distances between the interfacial atoms of the magnetic layer and the

heavy metal layer. Next, the KohnSham equations are solved without spin orbit coupling

(SOC) to determine the charge distribution of the system's ground state. Finally, spin orbit

coupling (SOC) is included and the total energy of the system is determined for the CW

and CCW spin spirals having 90 degree rotation. The total DMI strength, Dtot, is found

from the energy dierence between the CW and CCW spin congurations [118].

The iDMI is calculated using the approach of Yang et al. [118] generalized for an

AFM system. The energy due to the Dzyaloshinskii-Moriya interaction can be written as

EDMI =∑⟨i,j⟩

dij · [Si × Sj ] (3.1)

where dij is the DMI vector and Si is the unit vector of a magnetic moment. dij is found

from d(z × uij), where uij is the unit vector between sites i and j, and z is the direction

normal to the lm which is oriented from the heavy metal to the magnetic lm [118].

To numerically determine the value of d, AFM Nèel-vector spirals are created in

a supercell. Fig. 3.2(e) illustrates such a spiral along the x-direction. The spins that form

the anti-aligned spin pairs of the AFM primitive cells are numbered 1,1′, 2,2′, 3,3′, and

31

4,4′. Each pair is successively rotated 90 with respect to the pair on the left. The local

magnetization of each primitive cell remains zero. Writing out the sum in Eq. (3.1) over

the four nearest neighbours of atom 2, the energy of atom 2 is

E2 =1

2[d22′ · (S2 × S2′) + d22′′ · (S2 × S2′′)

+ d23′ · (S2 × S3′) + d23′′ · (S2 × S3′′)]

+ Eother (3.2)

where Eother is the spin independent, anisotropy and symmetric exchange energy contribu-

tions. The rst two terms, 2,2′ and 2,2′′, on the right side of Eq. (3.2) are zero due to

parallel or antiparallel magnetic moments in the cross products. The 2,2′ term is an intra

primitive cell term, and atom 2′′ is equivalent to atom 2′ but in the next primitive cell in

the y-direction. The non-zero terms come from the two nearest neighbor pairs 2,3′ and 2,3′′

resulting in

E2 =1

2[1√2Dtot +

1√2Dtot] + Eother (3.3)

where the factors of 1√2are due to the 45 angle between d23′ and S2 × S3′ . Dtot is the

total DMI strength which is considered as the DMI strength concentrated in a single atomic

layer [118]. The energies of atom 2 resulting from CW and CCW spirals are

E2,CW =1√2Dtot + Eother (3.4)

E2,CCW = − 1√2Dtot + Eother (3.5)

A single Mn layer of a supercell that contains eight magnetic atoms is shown in Fig. 3.2(e).

Note that the last line of atoms along the x and y directions are repeated and belong to the

32

next supercell, and are therefore not counted. For the Mn layer of Fig. 3.2(e),

∆EDMI = (ECW − ECCW ) = 8√2Dtot, (3.6)

so that Dtot is

Dtot = (ECW − ECCW )/m, (3.7)

where m = 8√2 for both MnPt and NiO. From Dtot, we determine the micromagnetic DMI,

Dµ, which is the parameter used in micromagnetic simulations. Considering four nearest

neighbours, Dµ is found from [118],

Dµ =4Dtot

NLa2(3.8)

where a is the lattice constant, and NL is the number of magnetic layers.

This approach for determining iDMI values has been used in several dierent stud-

ies and applied to dierent heterostructure systems [118120]. The approach assumes that

the iDMI can be described in a nearest neighbor approximation. It results in a minimal

length periodic spin spiral and minimal size supercell as shown in Fig. 3.2. There are

other approaches. Longer spin spirals with smaller angles can be used, however the su-

percells quickly become extremely large. For example, with 30 between nearest neighbor

spins, a 3 layer MnPt / 3 layer HM structure results in a supercell containing 216 atoms.

Furthermore, in this approach, several dierent spin spirals with dierent nearest neighbor

angles must be simulated and the resulting energies tted to a parabolic curve to extract

the iDMI [32,34,35,121,122]. Beyond the above two methods of creating spin spirals in su-

percells, there are several other approaches such as (i) the Korringa-Kohn-Rostoker (KKR)

Green function method [123], (ii) the coned full potential linearized augmented plane wave

33

generalized Bloch theorem (FLAPW-gBT) method [124, 125], (iii) the at FLAPW-gBT

method [124, 125], and (iv) the coned supercell spin spiral method [126]. Variations of the

DMI values calculated from the dierent methods have been studied [126]. Compared to

the approach used in this work, the KKR method varied by 6.7%, the coned FLAPW-gBT

method varied by 23%, the at FLAPW-gBT method varied by 21%, and the coned super-

cell method varied by 6.8% for monolayer Co on Pt [126]. Several papers containing both

experimental and theoretical results showed that the method used in this paper provides

DMI values which are comparable to experimental results for dierent systems [120,127].

The magnetic moment and magnetization are calculated from the collinear AFM

spin congurations of the three dierent combinations of materials where the magnetization

is the magnetic moment per unit volume. The magnetic anisotropy energy is calculated

from the energy dierence between the in plane and out of plane spin congurations of

MnPt [127].

The exchange energy (J) is calculated from the total energy dierence between

collinear antiferromagnetic (EAFM ) and ferromagnetic (EFM ) congurations [126,128130].

With four nearest neighbours, the nearest-neighbor exchange energy is found from VASP

[129,130],

J =EAFM − EFM

8. (3.9)

The micromagnetic exchange stiness (A) is related to J by [126,128]

A =1

2VΩ

∑j

J0j(R0j)2 (3.10)

where VΩ is the volume of the primitive AFM unit cell, R0j = a√2is the distance between

nearest-neighbor magnetic atoms, and a is the lattice constant of the primitive AFM unit

34

Figure 3.3: (a) MnPt on top of W. Mn atoms are purple. (b) NiO on top of Au. Oxygen

atoms are red.

cell. For four nearest neighbours, the micromagnetic parameter is [126],

A =a2J

NLVΩ. (3.11)

3.3 Results

Calculated values ofDtot andDµ for MnPt-Re, MnPt-W, and NiO-Au as a function

of AFM layer number are shown in Fig. 3.4. The structures consist of the AFM layers on

top of three layers of HM. The trends and quantitative values are similar to those in HM/FM

interfaces [131]. For a monolayer of MnPt, the strength of the iDMI, in terms of Dtot, is

larger with W than with Re, and this is consistent with the fact that the SOC of W is

larger than that of Re. Both MnPt-W and MnPt-Re show larger iDMI compared to that

of NiO-Au, and this is consistent with the fact that the magnetic moment of Mn is larger

than that of Ni. The trend that higher magnetic moments provide larger interfacial DMI

following Hund's rule [131] is also observed in HM/FM interfaces.

35

Figure 3.4: (a) Dtot and (b) Dµ of MnPt-W, MnPt-Re, and NiO-Au as a function of the

thickness of the AFM. The thickness of the HM is xed at 3 layers. Positive DMI indicates

that the CCW spin spiral is the lower energy state.

The magnitude of the volume averaged interfacial DMI (Dµ), shown in Fig. 3.4(b),

decreases with increasing thickness of the AFM material, and it will eventually approach

zero for thick magnetic material [63, 118]. However, this does not necessarily hold true

for the rst two to three layers in terms of the total DMI strength Dtot, as shown in Fig.

3.4(a). In MnPt, each layer of magnetic atoms (Mn) is sandwiched between layers of HM

atoms (Pt). Thus, Dtot depends not only on the spin orbit coupling (SOC) arising from

the interfacial heavy metal W (or Re), but also on the SOC from the inner heavy metal Pt,

and the resulting total DMI strength initially varies non-monotonically with layer number

as shown in Fig. 3.4(a). In contrast, NiO contains no HM atoms, and the magnitude of Dtot

decreases monotonically with thickness.

The iDMI values of antiferromagnetic materials are comparable to those of fer-

romagnetic materials. The iDMI of monolayer MnPt on W or Re is comparable to that

of monolayer Fe sandwiched between Pd and Ir, which is well known for hosting nm scale

36

Figure 3.5: iDMI of monolayer AFM for two dierent HM thicknesses: (a) Dtot with one

magnetic layer and (b) Dµ.

skyrmions [132134], and also to monolayer Co on Pt [133]. Three layers of MnPt on W has

Dµ = 3.8 mJ/m2, which is comparable to that of three layers of Co on Pt [118,119]. Three

layers of Ni on graphene has Dµ ≈ 0.5 mJ/m2 [120], and three layers of AFM NiO on Au

has an Dµ = 0.4 mJ/m2.

A thicker heavy metal layer increases the iDMI, since increasing the thickness of

the heavy metal layer initially increases the proximity SOC in the AFM. The increase in the

iDMI with thickness of the HM layer is shown in Fig. 3.5. Since the HM provides proximity

SOC, increasing the HM thickness from one layer to several layers increases the iDMI before

it saturates after 3 or 4 layers [135].

Since the magnitude of the iDMI depends on the local magnetic moments, com-

parisons of the magnetic moments and saturation magnetizations of MnPt and NiO with

three layers of heavy metal are shown in Fig. 3.6 as a function of the AFM thickness. The

highest magnetic moment is shown. NiO has smaller magnetic moments than MnPt, which

contributes to its lower iDMI compared to MnPt-W and MnPt-Re. Overall, the values are

smaller than those of the bulk. The magnetic moments increase with increasing thicknesses

37

Figure 3.6: Magnetic moments and saturation magnetization of (a) MnPt-W, (b) MnPt-Re

and (c) NiO-Au as a function of the thickness of the AFM. The thickness of the HM is xed

at 3 layers.

of the AFM thin-lms, and they approach the bulk values of MnPt and NiO [136138].

There are two reasons for smaller magnetic moments: (i) reduced thickness of the

magnetic materials [139,140] and (ii) orbital hybridization at the AFM-HM interface [131].

The hybridization of the 3d orbitals of the AFM with the 5d orbitals of the HM reduces

the magnetic moments of the AFM [40,41], and it gives rise to non-zero magnetic moments

in the HM layer. In FM/HM systems, a higher proximity induced magnetic moment in the

heavy metal reduces the iDMI [118], and this is consistent with what we observe with the

AFM/HM system. One layer of MnPt on three layers of Re induces a proximity magnetic

moment of 0.211 µB on the rst layer of Re atoms, and one layer of MnPt on three layers

of W induces a proximity magnetic moment of 0.206 µB on the rst layer of W atoms. The

38

Figure 3.7: Local density of states of Mn atom in 3 layers of MnPt on top of 3 layers of Re:

(a) middle Mn atom which is far from Re, (b) bottom Mn atom which is close to Re and is

aected by proximity induced magnetic moment.

enhanced magnetic moment in the HM is accompanied by a reduced magnetic moment in

the interfacial magnetic atoms. For a system with three layers of MnPt on top of three

layers of Re, the magnetic moment of a Mn atom at the HM interface is 2.18 µB, and the

magnetic moment of a Mn atom in the middle layer is 3.13 µB. This eect is apparent

in the spin-resolved local density of states (LDOS) of the Mn atoms. Fig. 3.7 shows the

spin-resolved LDOS of Mn atoms in the (a) middle layer and (b) interfacial layer from the

heterostructure of three layers of MnPt on three layers of heavy metal (Re). The upper (red)

39

Figure 3.8: (a) Magnetic anisotropy constant of MnPt with 3 layers of heavy metals. (b)

Exchange stiness of MnPt with 3 layers of heavy metals.

and lower (blue) curves show the spin-up D↑ and spin down D↓ LDOS, respectively. The

magnetic moment is proportional to the integration of D↑ −D↓ over the occupied energies

E ≤ EF . The LDOS of the minority down spin is larger in the interfacial layer than in the

middle layer, with the result that the Mn atoms at the interface are less polarized.

MnPt consists of alternating layers of magnetic Mn atoms and heavy metal Pt

atoms with high SOC. A nite (001) slab of MnPt, consisting of an integer number of unit

cells, breaks inversion symmetry since one face is a Mn layer and the opposing face is a Pt

layer. The combination of high SOC and broken inversion symmetry gives rise to a small

amount of DMI even without a proximity HM layer. For a three layer slab of MnPt with

the Mn layer on the bottom and the Pt layer on the top, Dtot = −1.02 meV/3d atom, and

Dµ = −1.37 mJ/m2. In contrast, NiO does not contain any heavy metal atoms, 3 layers

remain geometrically inversion symmetric, and it exhibits zero DMI in the absence of an

adjacent heavy metal layer.

Calculated values for the magneto-crystalline anisotropy constants and exchange

40

stinesses are shown in Fig. 3.8 for dierent numbers of magnetic layers. The magnetic

anisotropy constant decreases with thickness, and the exchange stiness slightly increases

with the number of layers, which is consistent with a previous study [127].

3.4 Conclusions

The interfacial Dzyaloshinskii-Moriya interaction of MnPt/W, MnPt/Re, and NiO/Au

are calculated for dierent thicknesses of both the AFM and the HM. Values of iDMI, mag-

netic moments, exchange and magnetic anisotropy constants required for the micromagnetic

simulations are determined. The values of the iDMI of the MnPt/HM heterolayers are com-

parable to those of the common ferromagnetic materials such as Fe, Co or CoFeB. The iDMI

of the NiO/Au system is approximately a factor of 7 lower. In general, iDMI is maximized by

choosing AFM materials with larger magnetic moments, heavy metals with high spin orbit

coupling, thinner AFM layers from 1 - 3 monolayers, and HM thicknesses of at least 3 to 4

layers. Few layer (001) MnPt which contains both HM atoms and magnetic atoms exhibits

an intrinsic DMI, albeit smaller than the iDMI induced by a proximity HM layer. These

results and the quantitative values provided will help guide experimental realization and

provide needed parameters for micromagnetic simulations of AFM materials with domain

walls and skyrmions supported by iDMI.

3.5 Acknowledgement

This work was supported as part of Spins and Heat in Nanoscale Electronic Systems

(SHINES) an Energy Frontier Research Center funded by the U.S. Department of Energy,

41

Oce of Science, Basic Energy Sciences under Award #DE-SC0012670. This work used the

Extreme Science and Engineering Discovery Environment (XSEDE) [3], which is supported

by National Science Foundation Grant No. ACI-1548562 and allocation ID TG-DMR130081.

42

Chapter 4

Magnetic Properties of NbSi2N4,

VSi2N4, and VSi2P4 Monolayers

4.1 Introduction

Two dimensional (2D) layered materials with transition metals have been of interest

for many decades due to the correlated phenomena and multiple polymorphs and phases that

they can exhibit such as charge density waves, [36] superconductivity, [37] and magnetism.

[38] The ability to exfoliate or grow single monolayers renewed the interest in these materials

for possible electronic and optoelectronic applications [141,142] by both traditional transistor

type devices [143,144] and by exploiting external control of their phase transitions. [145] The

relatively recent demonstration of magnetism in single monolayer of CrI3 [146] and bilayers

of Cr2Ge2Te6 [147] has spurred intense experimental and theoretical activity to nd other

2D magnetic materials with higher transition temperatures. [148155]

43

Figure 4.1: Structures and magnetic orientations of α1-NbSi2N4, α1-VSi2N4, and α1-VSi2P4.

Blue atoms are Si, white atoms are N, and pink atoms are P. The directions of the magnetic

moments are shown on the transition metals Nb and V. All materials are easy-plane, semi-

conducting ferromagnets.

The most recent addition to the family of 2D materials are the transition metal

silicon nitrides, phosphides, and arsenides with the chemical formulas MA2Z4, where M is

the transition metal, A ∈ Si, Ge, and Z ∈ N, P, As. [39] High quality multilayers and

monolayers of MoSi2N4 were grown using chemical vapor deposition, and what was particu-

larly notable was their stability to air, water, acid, and heat that was unprecedented among

transition metal 2D materials. [39] This rather mundane property is highly desirable for

manufacturing applications. While BN encapsulation is an eective solution for stabilizing

reactive 2D materials for laboratory experiments, [156] it is less than ideal for manufactur-

ing. Only MoSi2N4 was experimentally characterized in detail, WSi2N4 was also grown, and

12 materials were simulated with density functional theory (DFT) and found to be stable.

Among these 12, two of the nitrides, VSi2N4 and NbSi2N4, were identied as magnetic.

44

This work motivated immediate follow-on theoretical investigations of this ma-

terial family both determining properties of the materials and extending the list of stable

materials. [157165] The most extensive theoretical survey found 32 thermodynamically and

dynamically stable compounds of the form MA2Z4. [160] Of these, 6 were magnetic: VSi2N4,

VSi2P4, NbSi2N4, VGe2N4, VGe2P4, and TaGe2N4. Based on the formation enthalpies, a Si

based compound MSi2Z4 is approximately 3 times more stable than its equivalent Ge based

compound MGe2Z4, and a silicon nitride compound MSi2N4 is also approximately 3 times

more stable than its equivalent silicon phosphide compound MSi2P4.

Other recent works investigated MSi2Z4 bilayers (M = Ti, Cr, Mo; Z = N, P) for

their sensitivity to vertical strain [163, 164], and MoSi2N4 and WSi2N4 for their sensitivity

to biaxial strain and vertical applied electric eld. [163] It was found that vertical strain can

cause an insulator to metal transition; [164] biaxial strain can lead to an indirect to direct

bandgap transition; [163] and that an applied electric eld can result in an insulator to

metal transition. [163] A theoretical investigation of 2D-2D contacts to monolayer MoSi2N4

using NbS2 and graphene found ultralow p-type Schottky barriers with NbS2 contacts and

approximately equal n-type and p-type Schottky barriers with graphene contacts. [158] The

graphene Schottly barriers were shown to be tunable with an applied vertical electric eld.

In this letter, we theoretically investigate the magnetic properties of the three Si

based magnetic materials: VSi2N4, VSi2P4, NbSi2N4. Values of the exchange constants and

magnetic anisotropy energies are determined, and the Curie temperatures are calculated.

The Curie temperatures of VSi2N4 and VSi2P4 are near or at room temperature and above,

depending on the model used, whereas the Curie temperature of NbSi2N4 is low. Therefore,

45

the primarily focus will be on the two vanadium compounds. The magnetic anisotropy

energies of VSi2N4 and VSi2P4 are calculated as a function of uniaxial and biaxial strain

and electron and hole doping.

These materials can exist in a variety of hexagonal phases. Ref. [ [160]] calculated

the formation energies of 30 dierent phases and found that the lowest energy phase of

NbSi2N4 and VSi2N4 was the α1 phase, which is the same phase as the lowest energy phase

of MoSi2N4. The formation energy of the next higher energy phase (δ4) of NbSi2N4 was

13 meV per atom higher than the α1 phase, and the formation energy of the next higher

energy phase (β2) of VSi2N4 was 6 meV per atom higher than the α1 phase. For VSi2P4,

the lowest energy phase was δ4 with a formation energy 0.3 meV below that of α1. Since

the formation energy of the α1 phase of VSi2P4 is so close to that of the δ4 phase, and

since the δ4 phase has been previously analyzed [160], we will consider the α1 phase of all

3 materials: NbSi2N4, VSi2N4, and VSi2P4. We note that the piezoelectric and magnetic

properties of the α1 phase of VSi2P4 have recently been investigated with DFT using the

generalized gradient approximation (GGA). [157]

4.2 Method

DFT combined with Monte Carlo (MC) calculations are applied to determine the

electronic and magnetic properties of these materials. The magnetic anisotropy energy

(MAE) and exchange energy are evaluated from DFT calculations implemented in the Vi-

enna ab initio simulation package (VASP). [115] The electron-core interactions are described

by the projected augmented wave (PAW) potentials. [116] Electronic structure is calculated

46

using three dierent functionals: GGA as parameterized by Perdew-Burke-Ernzerhof (PBE),

PBE plus the Hubbard U correction (PBE+U), and the Heyd-Scuseria-Ernzerhof hybrid

functional (HSE06). [117, 166168] The PBE+U calculation includes a Hubbard U correc-

tion term Ueff = 1 eV for the V atom where Ueff = U − J . [169, 170] For calculations of the

MAE, spin orbit coupling (SOC) must also be included. All plots shown for the MAE are

the results from PBE(SOC)+U calculations. The cuto energies for expanding the plane

wave basis are 600 eV. Integration over the Brillouin zone uses a MonkhorstPack scheme

with a Γ-centered 16× 16× 1 k-point grid, an energy broadening parameter of 50 meV, and

the total energy is converged to 10−6 eV. Structures are relaxed until the forces are less than

0.001 eV/Å. A vacuum spacing of 20 Å is used in the direction normal to the 2D mono-

layer to eliminate the interactions from periodic images. The calculated lattice constants

for NbSi2N4, VSi2N4, and VSi2P4 are 2.96 Å, 2.88 Å, and 3.47 Å, respectively, and they are

very close to previous reported results. [39, 157]

Uniaxial strain is applied along the x axis corresponding to lattice vector a1. The

applied strain is evaluated using ϵ = (a − a0)/a0 × 100%, where a and a0 are the lattice

parameters of the strained and unstrained monolayer. Biaxial strain is applied by uniformly

varying both in-plane lattice constants. For each strain, the atomic positions are relaxed

using the DFT parameters described above. Spin-polarized self consistent calculations are

performed with the relaxed structure for each strain to obtain the charge density. Using the

charge densities, total energies are calculated in the presence of spin orbit coupling (SOC) for

in-plane (E||) and out-of-plane (E⊥) magnetization to nd the magnetic anisotropy energy

(MAE). The MAE is dened as EMAE = E⊥ − E||. Positive magnetic anisotropy indicates

47

that in-plane magnetization is favored. The values provided in meV are per magnetic atom,

which for these 3 materials under consideration, are also per unit cell.

To investigate the transition temperatures, a nearest neighbor Heisenberg type

Hamiltonian with magnetic anisotropy and long range dipole-dipole interactions is con-

structed,

H = −12J

∑⟨i,j⟩

Si · Sj + ku∑i

(Szi )

2 +Hdd, (4.1)

where Si is the spin for magnetic atom i, ⟨i, j⟩ are the indices of nearest neighbor magnetic

atoms, ku > 0 is the easy-plane magnetic anisotropy energy per magnetic atom, and Hdd is

the dipole-dipole interaction. The exchange energies (J) of these materials are calculated

from the total energy dierences between the antiferromagnetic (EAFM ) and ferromagnetic

(EFM ) states. [126,128130] With six nearest neighbours in the monolayer of 2D hexagonal

lattice, the nearest-neighbor exchange energy is given by, [1,129,130] J = (EAFM−EFM)/12.

In these monolayer materials all of the magnetic atoms are in the same plane. The second

neighbor exchange energies were previously determined for VSi2P4 and found to be negligi-

ble compared to the nearest neighbor energies, [160] and, therefore, they are ignored. The

Curie temperatures are determined from Monte Carlo (MC) calculations as implemented

in the VAMPIRE software package using a 10 × 10 supercell. [171, 172] The MC calcula-

tions incorporate the magnetic anisotropy energies, the magnetic moments, the exchange

energies, lattice constants, and atomic positions determined from the DFT calculations, and

they also include the long range dipolar interactions. This MC approach has been used to

determine the Curie temperatures in other monolayer 2D materials such as VSi2P4, VI3,

CrSBr, and CrSeBr. [160, 173, 174] We note that the Curie temperatures in other mono-

48

Figure 4.2: Spin resolved energy bands of VSi2N4 and VSi2P4 calculated with (a,b) HSE06

and (c,d) PBE+U. Spin up bands are red and spin down bands are blue. The two spin-split,

narrow bands on either side of the Fermi level are d-orbital bands centered on the transition

metal.

layer 2D materials determined from the MC approach have been shown to compare well to

temperatures determined from renormalized spin wave theory (RSWT). For example, the

MC/RSWT predicted Curie temperatures for monolayer CrSBr and CrSeBr were 168/150

K and 150/152 K, respectively. [174]

4.3 Results

The structures and the ground state orientations of the magnetic moments of

NbSi2N4, VSi2N4, and VSi2P4 are shown in Fig. 4.1, and the spin-resolved band struc-

ture for VSi2N4 and VSi2P4, calculated with the HSE06 and PBE+U functionals, are shown

49

in Fig. 4.2. The band structure for all three materials calculated with PBE, PBE+U, and

HSE06 are shown in the Fig. S3 of the Supplement, and the d-orbital resolved bands of

VSi2N4 and VSi2P4 are shown in Fig. S2. For all 3 materials at Γ, the two isolated narrow

bands near the Fermi level are spin-polarized, dz2 − orbital bands centered on the transi-

tion metal atoms. Near K, these bands transition to dx2−y2 . The next higher conduction

band is primarily dx2−y2 at Γ, and it transitions to dz2 at K. The electronic structures from

the HSE06 calculations show the following properties. In VSi2N4 and VSi2P4, the higher

spin-up conduction band at Γ crosses the lower spin-down band near K where the orbital

composition of the two bands switch, dz2 ⇐⇒ dx2−y2 . VSi2N4 is direct gap (0.78 eV) at K

with a dx2−y2 spin-up valence band and a dz2 spin-up conduction band. VSi2P4 is indirect

gap (0.84 eV) with the dz2 spin-up valence band edge at Γ and a mixed dz2 + dx2−y2 spin-

down conduction band edge at M (the spin-up band at K is 40 meV higher) . NbSi2N4 is

indirect gap (0.54 eV) with the spin-up dz2 valence band at Γ and a mixed orbital dz2 +

dx2−y2 spin-down conduction band near M. The lower valence bands are primarily p-orbital

bands which come from the N and P atoms. At the PBE level of theory, all 3 materials

are semi-metals. Adding the Hubbard U correction creates a small gap at the Fermi level

for VSi2N4 and VSi2P4. Adding a percentage of exact exchange with the HSE06 functional

increases the gap substantially. We note that the HSE06 calculations match those reported

in the Supplement of Ref. [ [160]].

The equilibrium NbSi2N4, VSi2N4, and VSi2P4 monolayers are easy-plane, semi-

conducting ferromagnets. The calculated equilibrium magnetic moments, exchange energies,

magnetic anisotropy energies, and Curie temperatures are shown in Table 4.1 for dierent

50

Materials (Theory) Magnetic Mo-

ment (µB)

Exchange

energy

(×10−21J)

MAE (MJm3 ) TC (K)

NbSi2N4 (PBE) 0.32 0.064 0.30 12

VSi2N4 (PBE) 0.93 1.50 0.24 301

VSi2P4 (PBE) 0.96 1.11 0.14 235

VSi2N4 (PBE+U) 1.05 2.53 0.25 452

VSi2P4 (PBE+U) 1.04 1.77 0.11 350

VSi2N4 (HSE06) 1.19 2.80 −− ∗506

Table 4.1: Magnetic moment, exchange energy per link, equilibrium magnetic anisotropy

energy (MAE), and Curie temperature, calculated using dierent functionals for NbSi2N4,

VSi2N4, and VSi2P4. ∗TC is calculated using the HSE06 exchange energy and the

PBE(SOC)+U MAE.

levels of theory. The magnetic moments are comparable to those from prior studies. [39,157]

The positive MAE values indicate in-plane alignment of the magnetic moments. Within

the energy resolution of our calculations (1 µeV), the total energy is independent of the

angle of the magnetic moment within the plane of the monolayer. Since the monolayers

are insulating, easy-plane FMs with extremely weak in-plane anisotropy, they satisfy the

necessary conditions for exhibiting a dissipationless spin superuid mode. [175]

The normalized magnetizations, determined from Monte Carlo calculations, are

plotted as a function of temperature for VSi2N4 and VSi2P4 in Fig. 4.3. The solid lines

51

Figure 4.3: Monte Carlo calculations of the normalized magnetization as a function of

temperature for (a) VSi2N4 and (b) VSi2P4. The solid lines show the best ts to the

analytical expression given by Eq. (4.2).

show the best ts to the analytic expression,

m(T ) = (1− T/TC)β. (4.2)

The tted values of TC and β are shown on the plots. At the PBE+U level of theory, both

monolayer VSi2P4 and VSi2N4 have Curie temperatures above room temperature, 350 K

and 452 K, respectively. The eect of the increasing levels of theory, PBE, PBE+U, and

HSE06, is to successively increase the bandgap between the two spin-polarized dz2 bands,

so that at the HSE06 level, the two bands are completely gapped which maximizes the spin

polarization, magnetic moment, and the exchange constant. The predicted TC increases

with the increasing gap, giving a maximum value of 506 K for VSi2N4 using the exchange

constant determined from the HSE06 calculation. Since VSi2N4 has the same structure,

surface chemistry, and formation energy [160] as the experimentally characterized MoSi2N4,

we expect VSi2N4 to also be an air/water-stable material with Tc > 100 C, which is a

criterion for operation in a modern integrated circuit environment.

52

For context, we compare to several other 2D materials with similar predicted Curie

or Néel temperatures. A more comprehensive comparison is provided in the supplementary

material. The calculated Curie-temperatures for monolayers of the transition metal dichalco-

genides (TMDs) VS2, VSe2, and VTe2 are 292 K, 472 K, and 553 K, respectively. [176] The

predicted monolayer Curie temperatures for the ternary 2D materials CrSeCl, VSeTe, and

CrSeI are 320 K, 350 K, and 360 K, respectively. [153,174,177] Monolayers of ferromagnetic

semiconductors TcSiTe3 have a predicted Curie temperature of 538 K. [178] We note that

Tc is radioactive, so that Tc compounds are unlikely to see magnetic applications. Mono-

layers of RuI3, MnN, Co2S2, and 3R-MoN2 have Curie temperatures of 360 K, 368 K, 404

K, and 420 K, respectively. [179182] We note that synthesis of layered 3R-MoN2 requires

high pressure, [183] which is not amenable to thin lm growth techniques, and the proposed

graphitization synthesis route to achieve MnN monolayers [184] has not yet been demon-

strated. Within the list above, VSi2N4 stands out for its combination of air stability and

relatively high Curie temperature.

The equilibrium values of the MAE are listed in Table 4.1. The equilibrium values

of VSi2N4 and VSi2P4 range from 0.11 to 0.25 meV / magnetic atom. For comparison,

monolayers of CrCl3, CrBr3, and CrI3 have equilibrium MAEs of 0.02 meV, 0.16 meV,

and 0.8 meV, respectively. [185] Two-dimensional FeCl2, NiI2, Fe3P, and Fe3GeTe2 have

equilibrium MAE values of 0.07 meV, 0.11 meV, 0.72 meV, and 1 meV, respectively in their

monolayer limit. [186189] In general, the MAE values of VSi2N4, and VSi2P4 are on the

lower end of values for other 2D magnetic materials.

The calculated values of the MAE of VSi2N4 and VSi2P4 as a function of uniaxial

53

ε μeV

ε μeV

ε μeV

ε μeV

ε μeV

ε μeV

Figure 4.4: MAE as a function of uniaxial and biaxial strain calculated with PBE(SOC)+U.

MAE versus uniaxial strain: (a) VSi2N4 and (b) VSi2P4. MAE versus biaxial strain: (c)

VSi2N4 and (d) VSi2P4. Values are shown for the strain coecients for dierent regions

discussed in the text. The orientation of the magnetization of VSi2N4 rotates from in-plane

to out-of-plane with 3.3% biaxial strain.

54

n = -0.29 meV n = 0.31 meV

Figure 4.5: MAE as a function of band lling calculated with PBE(SOC)+U. MAE as a

function of excess electrons per unit cell for (a) VSi2N4 and (b) VSi2P4. Values are shown

for the lling coecients discussed in the text. The orientation of the magnetization of

VSi2N4 rotates from in-plane to out-of-plane with electron doping of 2.0× 1014 cm−2.

strain and biaxial strain are shown in Fig. 4.4. We quantify the sensitivity by dening a

strain coecient as αϵ = dEMAE/dϵ. For small strain of both types in both materials, the

values for αϵ are well below the value of 32 µeV/%strain recently calculated for a 1.1 nm

slab of CrSb. [190] For compressive uniaxial and biaxial strain in VSi2N4, αϵ = −10 and −6

µeV/%strain, respectively. For VSi2P4, αϵ = −6 µeV/%strain for uniaxial tensile strain, -10

µeV/%strain for biaxial tensile strain, and 17 µeV/%strain for biaxial compressive strain.

For large (3-4%) biaxial strain in VSi2N4, the magnitude of the sensitivity increases. At a

strain of 3.3%, the MAE changes sign, and the spins rotate from in-plane to out-of-plane.

This magnitude of strain should be experimentally accessible, since 2D materials can sustain

large strain. [191]

The eects of band lling on the MAE are shown in Fig. 4.5. The MAE is reduced

in VSi2N4 and VSi2P4 for both electron and hole doing. To quantify the sensitivity of the

55

MAE to charge lling we dene the parameter [190] αn = dEMAE/dn evaluated at zero lling.

For electron lling in VSi2N4, αn = −0.29 meV, and for hole lling in VSi2P4, αn = 0.31

meV. What is most interesting is that the sign of the MAE can be reversed in VSi2N4 at an

electron lling of 0.28 electrons per unit cell corresponding to a sheet carrier concentration

of ns = 2.0 × 1014 cm−2. At this doping density, the orientation of the magnetization of

VSi2N4 rotates from in-plane to out-of-plane. These densities are high, however since the

thickness of the monolayer is under 1 nm, such densities could be experimentally accessible

using electrolytic gating.

4.4 Conclusions

In conclusion, NbSi2N4, VSi2N4, and VSi2P4 are easy-plane, semiconducting FMs

with negligible in-plane anisotropy. Exchange constants extracted from PBE+U DFT used

in MC calculations predict Curie temperatures for monolayer VSi2P4 and VSi2N4 of 350 K

and 452 K, respectively. The eect of the increasing levels of theory, PBE, PBE+U, and

HSE06, is to successively increase the energy gap between the two spin-polarized dz2 bands,

so that at the HSE06 level, the two bands are completely gapped which maximizes the spin

polarization, magnetic moment, and the exchange constant. The predicted TC increases

with the increasing gap, giving a maximum value of 506 K for VSi2N4 using the exchange

constant determined from the HSE06 calculation. Since VSi2N4 has the same structure,

surface chemistry, and formation enthalpy as MoSi2N4, it is expected to also be air and

water stable. The magnetic anisotropy energies of VSi2N4 and VSi2P4, determined by the

dierence in the total energies resulting from out-of-plane versus in-plane alignment of the

56

magnetic moments, are small, ranging from 0.11 to 0.25 meV / magnetic atom. Tensile

biaxial strain of 3.3% in VSi2N4 causes the MAE to change sign such that the magnetic

moments rotate from in-plane to out-of-plane. Band lling of 0.28 electrons per unit cell

(2.0 × 1014 cm−2) also causes VSi2N4 to switch from in-plane to perpendicular magnetic

anisotropy.

4.5 Supplementary Material

The supplementary material contains plots of orbital projected band diagrams,

comparisons of electronic bandstructures calculated with PBE, PBE+U, and HSE06 func-

tionals, and MC calculations of the normalized magnetization as a function of temperature

with exchange constants extracted from PBE DFT. It also contains an extended comparison

of Curie temperatures of 2D materials. Fig. 4.6 shows the orbital resolved band structure of

VSi2N4. Fig. 4.7 shows the d-orbital resolved band structures of VSi2N4 and VSi2P4. Spin

resolved PBE, PBE+U, and HSE06 band structures are shown in Fig. 4.8. Monte Carlo

calculations using exchange constants extracted from PBE DFT are shown in Fig. 4.9.

57

Figure 4.6: Orbital resolved PBE band structure of VSi2N4 at equilibrium: (a) s-orbital

contribution, (b) p-orbital contribution, and (c) d-orbital contribution. The weight is given

by the color bars at right. The two isolated, narrow bands near the Fermi level are d-orbital

bands centered on the transition metal atoms vanadium (V). The higher valence bands are

primarily p-orbital bands which come from the N atoms. The lower conduction bands are

primarily d-orbital bands.

Figure 4.7: d-orbital resolved PBE band structure of VSi2N4: (a) spin up bands, (c) spin

down bands. d-orbital resolved band structure of VSi2P4: (b) spin up bands, (d) spin down

bands.

58

Figure 4.8: Spin resolved PBE band structure: (a) NbSi2N4, (b) VSi2N4, and (c) VSi2P4.

Spin resolved PBE+U band structure: (d) NbSi2N4, (e) VSi2N4, and (f) VSi2P4. Spin

resolved HSE06 band structure: (g) NbSi2N4, (h) VSi2N4, and (i) VSi2P4.

Figure 4.9: MC calculations of the normalized magnetization as a function of temperature

with exchange constants extracted from PBE DFT for (a) NbSi2N4, (b) VSi2N4, and (c)

VSi2P4. Curie temperatures are calculated from Monte Carlo simulation using the VAM-

PIRE software.

59

4.6 Curie temperatures of other 2D Materials

For context, we provide comparisons to some of the more heavily studied 2D mag-

netic materials and other predictions in the literature. CrI3 has Curie temperatures of

45 K in monolayers and 61 K in bulk. [146] Cr2Ge2Te6 (CGT) has Curie temperatures of

30 K for bilayers and 66 K for bulk. [147] Magnesium chloride has Curie temperature of

250 K. [192] The ternary compound Fe3GeTe2 (FGT) has Curie temperatures of 130 K for

monolayers [193] and 205 K for bulk [148]. MnBi2Te4, a ferromagnetic insulator, has a Curie

temperature of about 20 K for monolayers. [194] MnPS3 and FePS3 have antiferromagnetic

ordering temperatures of 78 K and 118 K, respectively. [195] Ta3FeS6 is reported to be air

stable with a Curie temperature of 80 K. [196] There are many theoretical predictions in

the literature. The calculated Curie temperatures of the Dirac half-metal monolayers of

VCl3 and VI3 are 80 K and 98 K, respectively. [197] The predicted Curie-temperatures of

the transition metal dichalcogenide monolayers tend to be high, but these materials suer

from air/water instability. The calculated Curie-temperatures for monolayers of VS2, VSe2,

and VTe2 are 292 K, 472 K, and 553 K, respectively. [176] The predicted monolayer Curie

temperatures for the ternary 2D materials VSeTe, CrSBr, CrSeBr, CrSeCl and CrSeI 350

K, 168 K, 150 K, 320 K and 360 K, respectively. [153, 174, 177] Monolayers of ferromag-

netic semiconductors TcSiTe3, TcGeSe3, and TcGeTe3 have predicted Curie temperatures

of 538 K, 212 K, and 187 K, respectively. [178] We note that Tc is radioactive, so that

the Tc compounds are unlikely to see magnetic applications. Monolayers of Co2S2, ScCl,

RuI3, 3R-MoN2 and MnN have Curie temperatures of 404 K, 185 K, 360 K 420 K, and

368 K, respectively. [179182,198] We note that synthesis of layered 3R-MoN2 requires high

60

pressure, [183] which is not amenable to thin lm growth techniques, and the proposed

graphitization synthesis route to achieve MnN monolayers [184] has not yet been demon-

strated.

4.7 Acknowledgement

This work used the Extreme Science and Engineering Discovery Environment

(XSEDE), [3] which is supported by National Science Foundation Grant No. ACI-1548562

and allocation ID TG-DMR130081. MRKA thanks A. Bafekry from University of Guilan,

and San-Dong Guo from Xi'an University of Posts and Telecommunications for providing

initial crystal structures.

61

Chapter 5

1D and 2D Topological Insulators and

Semimetals

5.1 Introduction

One dimensional (1D) and two dimensional (2D) materials provide promising ap-

plications in nanoelectronics and optoelectronics due to their ultimately scaled sizes and

thicknesses [199201]. The availability of a large number of 1D and 2D candidate materi-

als will accelerate progress in this area. Previously, data mining of the Materials Project

database identied 1173 two dimensional materials and 487 one dimensional materials [202].

Approximately 6000 two dimensional materials have been identied and generated in a sep-

arate work [203]. At rst, a geometry-based algorithm was used to classify 2D materials

within existing material databases [202], and then the number of 2D materials was enlarged

by chemical substitution of elements in known 2D materials by others from the same group

62

in the periodic table, followed by DFT calculations [203]. 1,036 easily exfoliable 2D materials

that have been experimentally demonstrated have also been identied [204].

Topological insulators and semimetals are also gaining interest due to their var-

ious prospective applications, and they can oer various functionalities similar to transis-

tors [205210]. Using rst-principles calculations, 8,056 topological materials have been

identied considering spinorbit coupling [211]. A total of 10,007 topological semimetals,

4321 topological insulators, and 550 magnetic topological materials have been found found

from ICSD materials database [212214]. From several databases of 1D, 2D, and topological

materials, we identied a large number of 1D and 2D topological materials.

5.2 Results

A summary of the results is shown in Fig. 5.1. Topological insulators and semimet-

als from the topological materials database [211] and 2D materials from 2D matpedia [203]

are used to identify 150 two dimensional topological insulators and 98 two dimensional

topological semimetals listed in Tables 5.1 and 5.2, respectively. Topological insulators and

semimetals from the topological materials database [211] and 1D materials from database

of Evan Reed et al. [202] are used to identify the 12 one dimensional topological insulators

and 15 one dimensional topological semimetals listed in Tables 5.3 and 5.4, respectively.

Topological insulators and semimetals from the topological materials database [211] and

2D materials from the database of Evan Reed et al. [202] are used to identify the 93 two

dimensional topological insulators and 46 two dimensional topological semimetals listed in

Tables 5.5 and 5.6), respectively. Topological insulators and semimetals from the topological

63

materials database [211] and 2D materials from the experimental database [204] are used to

identify the 27 experimentally demonstrated two dimensional topological insulators and 8

two dimensional topological semimetals listed in Tables 5.7 and 5.8, respectively.

Figure 5.1: Flowchart of identifying 1D and 2D topological materials from several database.

MPID Material Formula

mp-570122 Cr3Te4

mp-2430 ZrGe2

mp-20738 FeSi2

mp-504594 ZrBr

mp-568746 Bi2Pd

mp-570197 Bi2Pd

Continued on next page

64

Table 5.1 continued from previous page

MPID Material Formula

mp-1168 HfTe5

mp-1334 Y2C

mp-20311 FeSe

mp-7655 BaSi2

mp-30428 BaCu

mp-1201 Sb2Te3

mp-583234 Bi4I

mp-541837 Bi2Se3

mp-7459 Te2Mo

mp-2089 ZrTe3

mp-2330 Nb2Se3

mp-692 Tb2C

mp-542634 Ta2Te3

mp-571033 NiSe

mp-605 ZrTe5

mp-399 Te2Pt

mp-570451 Nb2Te3

Continued on next page

65

Table 5.1 continued from previous page

MPID Material Formula

mp-29652 TaSe3

mp-27440 ZrCl

mp-9897 Te3As2

mp-1371 Sn3Pd

mp-30500 TiCd

mp-21296 PtPb4

mp-8094 Ba2Hg

mp-984703 CeGe5

mp-567510 BaSn2

mp-570286 BiSe

mp-628781 GeTe

mp-21042 Te4Pt3

mp-22562 YS

mp-615366 FeS2

mp-554055 HoS

mp-1879 SmS

mp-1056059 O2

Continued on next page

66

Table 5.1 continued from previous page

MPID Material Formula

mp-12574 Dy2C

mp-1078737 HoCoGe2

mp-130 P

mp-1008498 Ca

mp-1080587 CdNCl3

mp-1078753 LiTl

mp-1009757 Se

mp-13962 HfSiSe

mp-1078904 Sb2Te3

mp-1091417 AgN

mp-1025084 HoBC

mp-13963 HfSiTe

mp-1070135 Er2Mg2Ru

mp-1080184 LaIO3

mp-1080466 Nb2PdS6

mp-11 As

mp-1078645 YCoGe2

Continued on next page

67

Table 5.1 continued from previous page

MPID Material Formula

mp-104 Sb

mp-1079936 TbCoGe2

mp-10283 SmSi3Ni

mp-13683 HfGe2

mp-157 P

mp-19810 TaNiTe2

mp-1065583 LaSe

mp-2578 NiTe2

mp-19917 ZrSiTe

mp-158 As

mp-23224 BiTe

mp-1080067 ErCoGe2

mp-27910 Bi2Te2S

mp-27947 Sn(SbTe2)2

mp-1006619 Sn3PO6

mp-22777 NaN3

mp-15605 ScSi3Ni2

Continued on next page

68

Table 5.1 continued from previous page

MPID Material Formula

mp-2755 SiTe2

mp-24242 CuHOF

mp-27848 TiIN

mp-23896 LuHCl

mp-20757 MnAlGe

mp-24081 ScHCl

mp-23152 Bi

mp-27948 Ge(BiTe2)2

mp-24051 ErHCl

mp-27902 BiSe

mp-545706 Ca3Cu2(BrO2)2

mp-20506 NbNiTe2

mp-31507 Sb2Te4Pb

mp-4384 Nb2CS2

mp-7716 SbTe

mp-571550 Sb2TeSe2

mp-8677 ZrTiTe4

Continued on next page

69

Table 5.1 continued from previous page

MPID Material Formula

mp-48 C

mp-19607 K(MoO3)3

mp-3525 Sb2Te2Se

mp-79 Zn

mp-94 Cd

mp-569831 Zr3(Al2C3)2

mp-980004 SmGa2Co

mp-570930 Bi2Pb2Se5

mp-30485 Ca3Zn

mp-8211 GaGeTe

mp-7574 AlBMo

mp-567970 Hf3(Al2C3)2

mp-569044 Bi2Te5Pb2

mp-557597 K2Ca(NO2)4

mp-977592 Bi2Rh3S2

mp-34202 Bi2Te3

mp-642803 Zr2HBr2

Continued on next page

70

Table 5.1 continued from previous page

MPID Material Formula

mp-8742 DyCoSi2

mp-542640 GePt

mp-31406 Bi2TeSe2

mp-3534 Ge2Sb2Te5

mp-568050 Ca3SiBr2

mp-569358 Bi

mp-9936 DyBC

mp-644389 Li2H2Pd

mp-568730 Ge4Te7As2

mp-3128 ErCoSi2

mp-9379 Sr(SnAs)2

mp-8806 Sr2Cu2O3

mp-557619 Bi14Te13S8

mp-29666 Bi2Te2Se

mp-972406 YbSi2Ni

mp-4628 ZrSiSe

mp-571471 NbCoTe2

Continued on next page

71

Table 5.1 continued from previous page

MPID Material Formula

mp-864790 Nd2BC

mp-541449 CrGeTe3

mp-9983 Ta2Se3

mp-634817 CaZn2(PO)2

mp-639496 EuIn2Au

mp-624889 C

mp-31025 Sc5NCl8

mp-8436 Ta2PdSe6

mp-644136 Li2H2Pt

mp-541070 Ta2NiSe5

mp-782 Te2Pd

mp-643743 CuHClO

mp-558696 CuTeO3

mp-7814 Ta2CS2

mp-3208 ZrGeTe

mp-676250 Bi2Te4Pb

mp-1072085 CeSi5

Continued on next page

72

Table 5.1 continued from previous page

MPID Material Formula

mp-7263 CaGaN

mp-973938 LuCoGe2

mp-24522 CuH2SO5

mp-557946 Cu3Se2(ClO3)2

mp-28487 Ge5(Te4As)2

mp-27548 CuTe2O5

mp-683904 Yb2Te5O13

mp-1078720 Pt(NCl)4

Table 5.1: 2D topological insulators

MPID Material Formula

mp-20826 CuTe

mp-23313 InBi

mp-2129 NdGe2

mp-740 NdTe3

Continued on next page

73

Table 5.2 continued from previous page

MPID Material Formula

mp-580226 CuSe

mp-1080586 Sn4Ir

mp-9399 SmTe3

mp-570356 Ca5Ir

mp-604914 VSe

mp-12351 PrTe3

mp-7542 GeP3

mp-525 NbSe3

mp-998972 TiAu

mp-28149 Bi9I2

mp-29941 Sc2C

mp-30527 TaS3

mp-945077 YTe3

mp-7541 SnP3

mp-1229 CuF2

mp-12750 SmGe2

mp-2294 Al3Ir

Continued on next page

74

Table 5.2 continued from previous page

MPID Material Formula

mp-13152 TiSe

mp-1861 Cu2Te

mp-1001825 LiBe

mp-569701 ErTe3

mp-29663 Si3Ir

mp-1008600 ZrN2

mp-1008682 TiS2

mp-10021 Ga

mp-1079923 H3W

mp-1087512 TmTe3

mp-1084847 TbTe3

mp-1077588 SiNiSb

mp-1018658 CaCoSi

mp-10033 NbS2

mp-1077298 TbMgNi

mp-1078651 ErNiGe2

mp-1086673 DyTe3

Continued on next page

75

Table 5.2 continued from previous page

MPID Material Formula

mp-1086674 YbCoGe2

mp-1079251 W2N3

mp-1095661 YbCuPbSe3

mp-1372 Cd3As2

mp-10231 LiMnSe2

mp-1094089 NiS2

mp-1018654 BaMgSn

mp-1018653 BaMgPb

mp-1072320 HgSBr

mp-1077470 CaNiGe

mp-1080689 LuTe3

mp-1018808 MgCoGe

mp-1078069 Al2ZnS4

mp-1078637 Bi

mp-1080714 Li2AgSn2

mp-20698 LiCoAs

mp-28616 NbTe5Pd

Continued on next page

76

Table 5.2 continued from previous page

MPID Material Formula

mp-28754 Nb3GeTe6

mp-22992 TiClO

mp-22965 BiTeI

mp-23002 TiBrO

mp-17801 Hf2CuGe4

mp-22152 NaFeAs

mp-23095 Ca3Cu2(ClO2)2

mp-24164 CuH2(SeO3)2

mp-2194 TiSe2

mp-18279 Zr2CuGe4

mp-505137 Nb3SiTe6

mp-554027 Y7C3I6O

mp-541937 Zn(InS2)2

mp-8768 K2CoSe2

mp-29772 YGe3

mp-622009 TaNi2TeSe

mp-971806 TmFeGe2

Continued on next page

77

Table 5.2 continued from previous page

MPID Material Formula

mp-500 TaSe2

mp-570141 Sm2(PPt2)3

mp-3421 CuCN2

mp-4437 NdTe2Se

mp-30125 PrTe2Se

mp-601693 YbBrO

mp-8999 NbNiTe5

mp-8998 TaNiTe5

mp-3250 Ni2SbTe2

mp-567826 Sr

mp-608254 SmClF

mp-973954 LuFeGe2

mp-8848 TaFeTe3

mp-505042 Cu(BiO2)2

mp-7754 Si2Ru

mp-630773 CdInGaS4

mp-20049 LiFeP

Continued on next page

78

Table 5.2 continued from previous page

MPID Material Formula

mp-9391 TaNi2Te3

mp-28667 Ta(NiTe)2

mp-1078991 ErTe3

mp-1018798 MgCuGe

mp-505206 Ta3SiTe6

mp-14815 TaTe5Pt

mp-1078612 LaTe3

mp-4034 LaTe2Se

mp-11879 BaNiSn3

Table 5.2: 2D topological semimetals

MPID Material Formula

mp-637982 As5Pb3S10

mp-583234 Bi4I

mp-568758 BiBr

Continued on next page

79

Table 5.3 continued from previous page

MPID Material Formula

mp-568388 BiI

mp-7302 CrP4

mp-29279 CuClF10

mp-11508 MoP4

mp-558408 Nb4Te9I4O

mp-654051 Nb6SI9

mp-29579 Sc7NCl12

mp-684021 Sn4Sb6S13

mp-680836 Ta4Te9I4O

Table 5.3: 1D topological insulators

MPID Material Formula

mp-28661 Ba2Cu3P4

mp-21653 BaNiN

mp-28149 Bi9I2

Continued on next page

80

Table 5.4 continued from previous page

MPID Material Formula

mp-3199 CuSe2O5

mp-568002 HfI3

mp-17867 Mn2NbP12

mp-754514 N2

mp-568616 Nb2Br5

mp-14 Se

mp-19 Te

mp-571143 TiCl3

mp-8251 VP4

mp-9847 YbP5

mp-23247 ZrBr3

mp-22871 ZrCl3

Table 5.4: 1D topological semimetals

81

MPID Material Formula

mp-11 As

mp-21032 BaIn2Pt

mp-23152 Bi

mp-557619 Bi14Te13S8

mp-570930 Bi2Pb2Se5

mp-570197 Bi2Pd

mp-541837 Bi2Se3

mp-29666 Bi2Te2Se

mp-34202 Bi2Te3

mp-676250 Bi2Te4Pb

mp-569044 Bi2Te5Pb2

mp-23435 Bi2TeI

mp-31406 Bi2TeSe2

mp-581990 Bi3Rh

mp-27607 Bi4Se3

mp-23005 Bi4Te7Pb

mp-680214 Bi8Se7

mp-27902 BiSe

Continued on next page

82

Table 5.5 continued from previous page

MPID Material Formula

mp-23224 BiTe

mp-568286 C

mp-570122 Cr3Te4

mp-541449 CrGeTe3

mp-559094 Cu2H3ClO3

mp-643743 CuHClO

mp-24242 CuHOF

mp-12574 Dy2C

mp-30196 ErCuP2

mp-24051 ErHCl

mp-20311 FeSe

mp-8211 GaGeTe

mp-27948 Ge(BiTe2)2

mp-3534 Ge2Sb2Te5

mp-568730 Ge4Te7As2

mp-28487 Ge5(Te4As)2

mp-29644 GeBi4Te7

Continued on next page

83

Table 5.5 continued from previous page

MPID Material Formula

mp-29641 GeSb4Te7

mp-8645 GeTe7As4

mp-13962 HfSiSe

mp-13963 HfSiTe

mp-1168 HfTe5

mp-1640 Ho2C

mp-568574 LaGeI

mp-22716 LiCu2P

mp-23896 LuHCl

mp-570849 Mg5Tl2

mp-4384 Nb2CS2

mp-571471 NbCoTe2

mp-20506 NbNiTe2

mp-864790 Nd2BC

mp-571033 NiSe

mp-2578 NiTe2

mp-104 Sb

Continued on next page

84

Table 5.5 continued from previous page

MPID Material Formula

mp-640862 Sb16Te3

mp-6997 Sb2Te

mp-3525 Sb2Te2Se

mp-1201 Sb2Te3

mp-31507 Sb2Te4Pb

mp-12826 Sb8Te3

mp-7716 SbTe

mp-31025 Sc5NCl8

mp-24081 ScHCl

mp-2755 SiTe2

mp-27947 Sn(SbTe2)2

mp-7814 Ta2CS2

mp-541070 Ta2NiSe5

mp-8436 Ta2PdSe6

mp-542634 Ta2Te3

mp-510426 Ta4Te16Pd3

mp-19810 TaNiTe2

Continued on next page

85

Table 5.5 continued from previous page

MPID Material Formula

mp-29652 TaSe3

mp-692 Tb2C

mp-782 Te2Pd

mp-399 Te2Pt

mp-9897 Te3As2

mp-541180 Te3Pt2

mp-8628 Te4Pt3

mp-11693 TePt

mp-27848 TiIN

mp-579614 Y16B4C8I19

mp-1334 Y2C

mp-568046 Y5Mg13Cu5

mp-683904 Yb2Te5O13

mp-680388 ZnCuTeCl2O3

mp-642803 Zr2HBr2

mp-504594 ZrBr

mp-27440 ZrCl

Continued on next page

86

Table 5.5 continued from previous page

MPID Material Formula

mp-3208 ZrGeTe

mp-4628 ZrSiSe

mp-19917 ZrSiTe

mp-2089 ZrTe3

mp-605 ZrTe5

mp-570062 ZrTiSe4

mp-8677 ZrTiTe4

Table 5.5: 2D topological insulators

MPID Material Formula

mp-2284 AgF2

mp-11078 BaIn2Rh

mp-28229 Bi4Te3

mp-22965 BiTeI

mp-630773 CdInGaS4

Continued on next page

87

Table 5.6 continued from previous page

MPID Material Formula

mp-721072 Cu3As2H2PbO10

mp-24164 CuH2(SeO3)2

mp-580226 CuSe

mp-20826 CuTe

mp-10021 Ga

mp-23313 InBi

mp-4034 LaTe2Se

mp-28628 Li3FeN2

mp-30739 Mg3Ir

mp-645592 Nb3(BiO3)5

mp-28754 Nb3GeTe6

mp-505137 Nb3SiTe6

mp-8999 NbNiTe5

mp-10033 NbS2

mp-2207 NbSe2

mp-525 NbSe3

mp-28616 NbTe5Pd

Continued on next page

88

Table 5.6 continued from previous page

MPID Material Formula

mp-4437 NdTe2Se

mp-740 NdTe3

mp-3250 Ni2SbTe2

mp-10264 NiTe

mp-30125 PrTe2Se

mp-12351 PrTe3

mp-29941 Sc2C

mp-9399 SmTe3

mp-28667 Ta(NiTe)2

mp-505206 Ta3SiTe6

mp-505132 Ta3Te14Pd3

mp-8848 TaFeTe3

mp-9391 TaNi2Te3

mp-622009 TaNi2TeSe

mp-8998 TaNiTe5

mp-30527 TaS3

mp-500 TaSe2

Continued on next page

89

Table 5.6 continued from previous page

MPID Material Formula

mp-14815 TaTe5Pt

mp-23002 TiBrO

mp-22992 TiClO

mp-2194 TiSe2

mp-604914 VSe

mp-554027 Y7C3I6O

mp-561015 Y9C4I8O

Table 5.6: 2D topological semimetals

MPID Material Formula

mp-23152 Bi

mp-541837 Bi2Se3

mp-29666 Bi2Te2S

mp-29666 Bi2Te2Se

mp-34202 Bi2Te3

Continued on next page

90

Table 5.7 continued from previous page

MPID Material Formula

mp-31406 Bi2TeSe2

mp-48 C

mp-24051 ErHCl

mp-20311 FeSe

mp-8211 GaGeTe

mp-13963 HfSiTe

mp-568574 LaGeI

mp-23896 LuHCl

mp-558544 MoS2

mp-5745 Nb2CS2

mp-104 Sb

mp-3525 Sb2Te2Se

mp-1201 Sb2Te3

mp-571550 Sb2TeSe2

mp-24081 ScHCl

mp-2755 SiTe2

mp-504594 ZrBr

Continued on next page

91

Table 5.7 continued from previous page

MPID Material Formula

mp-27440 ZrCl

mp-19917 ZrSiTe

mp-1018107 ZrTe2

mp-570062 ZrTiSe4

mp-8677 ZrTiTe4

Table 5.7: 2D experimental topological insulators

MPID Material Formula

mp-2284 AgF2

mp-568390 Bi2Te3

mp-1861 Cu2Te

mp-623012 CuTe

mp-2207 NbSe2

mp-500 TaSe2

mp-2194 TiSe2

Continued on next page

92

Table 5.8 continued from previous page

MPID Material Formula

mp-1079251 W2N3

Table 5.8: 2D experimental topological semimetals

93

Chapter 6

Summary and Outlook

In conclusion, the energy requirement for switching with SAFM skyrmions is much

less compared to that for FM skyrmions, and the switching time is less. The phase diagrams

of skyrmion diameters as a function of dierent magnetic parameters can be used for selecting

the right combinations of materials. The simulations of rectangular and hourglass skyrmion

shift registers show an approach for low power memory design using SAFM skyrmions that

reduces chip area, reduces power, and increases switching speed.

To stabilize skyrmions, the interfacial Dzyaloshinskii-Moriya interaction plays a vi-

tal role. The interfacial Dzyaloshinskii-Moriya interaction of antiferromagnetic/heavy metal

systems MnPt/W, MnPt/Re, and NiO/Au are calculated for dierent thicknesses of both the

AFM and the HM. Values of iDMI, magnetic moments, exchange and magnetic anisotropy

constants for these bilayer structures required for the micromagnetic simulations are deter-

mined. The values of the iDMI of the MnPt/HM heterolayers are comparable to those of

the common ferromagnetic materials such as Fe, Co or CoFeB. The iDMI of the NiO/Au

94

system is approximately a factor of 7 lower. In general, the iDMI is maximized by choosing

AFM materials with larger magnetic moments, heavy metals with high spin orbit coupling,

thinner AFM layers from 1 - 3 monolayers, and HM thicknesses of at least 3 to 4 layers.

Few layer (001) MnPt which contains both HM atoms and magnetic atoms exhibits an in-

trinsic DMI, albeit smaller than the iDMI induced by a proximity HM layer. These results

and the quantitative values provided will help guide experimental realization and provide

needed parameters for micromagnetic simulations of AFM materials with domain walls and

skyrmions supported by iDMI.

Besides antiferromagnetic systems, ferromagnetic systems are also analyzed. The

Curie temperatures, exchange energies, and magnetic anisotropy energies as a function of

strain and band lling have been calculated for a new class of transition metal 2D mono-

layers of ferromagnetic NbSi2N4, VSi2N4, and VSi2P4. NbSi2N4, VSi2N4, and VSi2P4 are

easy-plane, semiconducting FMs with negligible in-plane anisotropy. Exchange constants

extracted from PBE+U DFT used in MC calculations predict Curie temperatures for mono-

layer VSi2P4 and VSi2N4 of 350 K and 452 K, respectively. The eect of the increasing levels

of theory, PBE, PBE+U, and HSE06, is to successively increase the energy gap between the

two spin-polarized dz2 bands, so that at the HSE06 level, the two bands are completely

gapped which maximizes the spin polarization, magnetic moment, and the exchange con-

stant. The predicted TC increases with the increasing gap, giving a maximum value of 506

K for VSi2N4 using the exchange constant determined from the HSE06 calculation. Since

VSi2N4 has the same structure, surface chemistry, and formation enthalpy as MoSi2N4, it

is expected to also be air and water stable. The magnetic anisotropy energies of VSi2N4

95

and VSi2P4, determined by the dierence in the total energies resulting from out-of-plane

versus in-plane alignment of the magnetic moments, are small, ranging from 0.11 to 0.25

meV / magnetic atom. Tensile biaxial strain of 3.3% in VSi2N4 causes the MAE to change

sign such that the magnetic moments rotate from in-plane to out-of-plane. Band lling of

0.28 electrons per unit cell (2.0× 1014 cm−2) also causes VSi2N4 to switch from in-plane to

perpendicular magnetic anisotropy.

To facilitate identifying materials of interest, 1D and 2D topological materials are

identied using several databases. 1D and 2D topological insulators and semimetals are

identied. 1D and 2D topological materials which are experimentally feasible, have also

been identied.

Going forward, the list of identied materials can serve as a starting point for

selecting and investigating materials with desired and improved characteristics. The ex-

tracted magnetic properties of ferromagnetic and antiferromagnetic materials can facilitate

experimental realization of improved memory devices at room temperature.

96

Chapter 7

Appendix

Code for Topological Materials

from pymatgen import MPRester , Composition

from pymatgen . e l e c t r on i c_s t r u c t u r e . p l o t t e r import BSPlotter

import pymatgen . a n a l y s i s . f ind_dimension

import pymatgen . i o . c i f as p c i f

import pandas as pd

import pymatgen . a n a l y s i s . f ind_dimension as pfd

from tqdm import tqdm

df1 = pd . read_excel ( "500 Experimental 2d mate r i a l s c loud . x l sx " ,0 )

df2 = pd . read_excel ( " space group Al l TI and TCI Chinese Database3000 New. x l sx "

,0 )

p r i n t ( df1 . columns )

p r i n t ( df2 . columns )

TI_id = df1 [ ' Formula ' ] . t o l i s t ( )

TI_symmetry = df1 [ ' Spacegroup ' ] . t o l i s t ( )

97

Reference_All_id = df2 [ ' formula ' ] . t o l i s t ( )

Reference_symmetry = df2 [ ' spacegroup ' ] . t o l i s t ( )

Unknown_dict = d i c t ( z ip (TI_id , TI_symmetry ) )

p r i n t (Unknown_dict )

Ref_dict = d i c t ( z ip ( Reference_All_id , Reference_symmetry ) )

p r i n t ( Ref_dict )

import re

f i l ename = ' seq_match_CompareChineseTIandExperimental2d_with_material . csv '

f = open ( f i l ename , 'w ' )

headers = 'TI_id , TI_symmetry , Reference_All_id , Reference_symmetry\n '

f . wr i t e ( headers )

f o r TI_id , TI_symmetry in Unknown_dict . i tems ( ) :

f o r Reference_All_id , Reference_symmetry in Ref_dict . i tems ( ) :

m = re . fu l lmatch ( s t r (TI_id ) , s t r ( Reference_All_id ) )

i f m:

f . wr i t e ( s t r (TI_id ) + ' , ' + s t r (TI_symmetry ) + ' , ' + s t r (

Reference_All_id ) + ' , ' + s t r ( Reference_symmetry ) + ' \n ' )

f . c l o s e ( )

Code for Generating Skyrmion

// s t r u c tu r e

s e t g r i d s i z e (128 , 128 , 4)

s e t c e l l s i z e (1 e=9, 1e=9, 0 . 2 e=9)

//magntic parameters

Msat = 900 e3

Aex = 14e=12

98

anisU = vector (0 , 0 , 1)

Ku1 = 0.779 e6

alpha = 0 .5

Dind = 1.68 e=3

// ex t e rna l f i e l d

B_ext = vector (0 , 0 , 12 .5E=3)

// skyrmion

m = BlochSkyrmion (1 , =1) . s c a l e (3 , 3 , 1)

r e l a x ( )

// cur rent

j = vec to r (0 , 2e11 , 0)

po l = 0 .5

// Schedule output

run (10 e=9)

Monte Carlo Simulation Code

# Creat ion a t t r i b u t e s :

c r e a t e : c r y s t a l=s t r u c tu r e=hcp

c r ea t e : p e r i od i c=boundaries=x

c r ea t e : p e r i od i c=boundaries=y

# System Dimensions :

dimensions : unit=c e l l=s i z e=x = 3.47 !A

dimensions : unit=c e l l=s i z e=y = 3.47 !A

dimensions : unit=c e l l=s i z e=z = 9.24 !A

dimensions : system=s i z e=x = 4 .0 !nm

dimensions : system=s i z e=y = 4 .0 !nm

99

dimensions : system=s i z e=z = 9.24 !A

# Mater ia l F i l e s :

mate r i a l : f i l e=mate r i a l . mat

# Simulat ion a t t r i b u t e s :

sim : temperature=300

sim :minimum=temperature=0

sim :maximum=temperature=400

sim : temperature=increment=5

sim : time=s teps=increment=1

sim : e qu i l i b r a t i o n=time=s t ep s=2000

sim : loop=time=s t ep s=2000

sim : enable=dipo le= f i e l d s

# Program and i n t e g r a t o r d e t a i l s

sim : program=cur i e=temperature

sim : i n t e g r a t o r=monte=c a r l o

# data output

output : r ea l=time

output : temperature

output : magnet i sat ion

sc r e en : magnet isat ion=l ength

sc r e en :mean=magnet isat ion=l ength

PyProcar Code for Band Structures and Density of States [215]

import pyprocar

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=2 ,2] ,mode=' p l a i n ' , c o l o r='

blue ' , code=' vasp ' )

100

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=2 ,2] ,mode=' p l a i n ' , c o l o r='

blue ' , k p o i n t s f i l e='KPOINTS ' )

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=2 ,2] ,mode=' p l a i n ' , c o l o r='

blue ' , k t i c k s =[0 ,39 ,79 ,119 ,159 ] , knames=[ 'G' , 'X ' , 'M' , 'G' , 'R ' ] , code=' vasp ' )

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=5 ,5] , k t i c k s

=[0 ,39 ,79 ,119 ,159 ] , knames=[ 'G' , 'X ' , 'M' , 'G' , 'R ' ] , cmap=' j e t ' ,mode='

parametr ic ' , sp in=1)

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=5 ,5] , k t i c k s

=[0 ,39 ,79 ,119 ,159 ] , knames=[ 'G' , 'X ' , 'M' , 'G' , 'R ' ] , cmap=' j e t ' , mode='

parametr ic ' , atoms=[1 ] )

pyprocar . bandsplot ( 'PROCAR' , outcar='OUTCAR' , e l im i t =[=5 ,5] , k t i c k s

=[0 ,39 ,79 ,119 ,159 ] , knames=[ 'G' , 'X ' , 'M' , 'G' , 'R ' ] , cmap=' j e t ' ,mode='

parametr ic ' , o r b i t a l s = [4 , 5 , 6 , 7 , 8 ] )

pyprocar . dosp lo t ( f i l ename=' vasprun . xml ' ,

mode=' p l a i n ' ,

e l im i t =[=4, 4 ] ,

o r i e n t a t i o n=' ho r i z on t a l ' ,

l a b e l s =[ r ' $\uparrow$ ' , r ' $\downarrow$ ' ] ,

t i t l e=r ' Total Density o f Sta t e s SrVO$_3$ ' )

pyprocar . dosp lo t ( f i l ename=' vasprun . xml ' ,

mode=' p l a i n ' ,

e l im i t =[=4, 4 ] ,

s p in s =[0 ] ,

o r i e n t a t i o n=' v e r t i c a l ' ,

l a b e l s =[ r ' $\uparrow$ ' , r ' $\downarrow$ ' ] ,

t i t l e=r ' Total Density o f Sta t e s SrVO$_3$ ' ,

s a v e f i g=' dos_plain_ver . png ' )

101

pyprocar . dosp lo t ( f i l ename=' vasprun . xml ' ,

mode=' parametr ic ' ,

o r b i t a l s =[1 , 2 , 3 ] ,

atoms=[2 , 3 , 4 ] ,

e l im i t =[=4, 4 ] ,

o r i e n t a t i o n=' ho r i z on t a l ' ,

p l o t_tota l=True ,

t i t l e=r ' Pro jec ted DOS o f p o r b i t a l s o f Oxygen in SrVO$_3$ ' )

pyprocar . bandsdosplot ( f i l ename=' vasprun . xml ' ,

mode=' parametr i c_l ine ' ,

o r b i t a l s =[1 , 2 , 3 ] ,

atoms=[2 , 3 , 4 ] ,

e l im i t =[=4, 4 ] ,

l a b e l s =[ 'O=up ' , 'O=down ' ] ,

p l o t_tota l=True )

pyprocar . dosp lo t ( f i l ename=' vasprun . xml ' ,

mode=' s tack ' ,

c o l o r s =[ ' lawngreen ' , ' orangered ' , ' r oya lb lu e ' ] ,

i tems=d i c t ( Sr=[4 , 5 , 6 , 7 , 8 ] , O=[1 , 2 , 3 ] , V=[0 , 1 , 2 , 3 ] ) ,

o r i e n t a t i o n=' ho r i z on t a l ' ,

e l im i t =[=4, 4 ] ,

p l o t_tota l=True )

pyprocar . dosp lo t ( f i l ename=vasprun ,

mode=' s tack_spec i e s ' ,

o r i e n t a t i o n=' ho r i z on t a l ' ,

e l im i t =[=4, 4 ] ,

p l o t_tota l=True )

102

pyprocar . dosp lo t ( f i l ename=vasprun ,

mode=' s t a ck_orb i t a l s ' ,

atoms=[2 , 3 , 4 ] ,

o r i e n t a t i o n=' ho r i z on t a l ' ,

e l im i t =[=4, 4 ] ,

p l o t_tota l=True )

Code for HSE Band Structures Using Files from VASPKIT

[216]

c l e a r a l l

c l c

c l f

[ k , down1 , down2 , down3 , down4 , down5 , down6 , down7 , down8 , down9 , down10 , down11 , down12 ,

down13 , down14 , down15 , down16 , down17 , down18 , down19 , down20 , down21 , down22 ,

down23 , down24 , down25 , down26 , down27 , down28 , down29 , down30 , down31 , down32 ] =

text read ( 'REFORMATTED_BAND_DW. dat ' , '%f %f %f %f %f %f %f %f %f %f %f %f %f

%f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f ' ) ;

f i g u r e (1 )

p l o t (k , down1 , 'b ' )

hold on

p l o t (k , down2 , 'b ' )

p l o t (k , down3 , 'b ' )

p l o t (k , down4 , 'b ' )

p l o t (k , down5 , 'b ' )

p l o t (k , down6 , 'b ' )

103

p lo t (k , down7 , 'b ' )

p l o t (k , down8 , 'b ' )

p l o t (k , down9 , 'b ' )

p l o t (k , down10 , 'b ' )

p l o t (k , down11 , 'b ' )

p l o t (k , down12 , 'b ' )

p l o t (k , down13 , 'b ' )

p l o t (k , down14 , 'b ' )

p l o t (k , down15 , 'b ' )

p l o t (k , down16 , 'b ' )

p l o t (k , down17 , 'b ' )

p l o t (k , down18 , 'b ' )

p l o t (k , down19 , 'b ' )

p l o t (k , down20 , 'b ' )

p l o t (k , down21 , 'b ' )

p l o t (k , down22 , 'b ' )

p l o t (k , down23 , 'b ' )

p l o t (k , down24 , 'b ' )

p l o t (k , down25 , 'b ' )

p l o t (k , down26 , 'b ' )

p l o t (k , down27 , 'b ' )

p l o t (k , down28 , 'b ' )

p l o t (k , down29 , 'b ' )

p l o t (k , down30 , 'b ' )

p l o t (k , down31 , 'b ' )

p l o t (k , down32 , 'b ' )

[ k , up1 , up2 , up3 , up4 , up5 , up6 , up7 , up8 , up9 , up10 , up11 , up12 , up13 , up14 , up15 , up16 , up17

104

, up18 , up19 , up20 , up21 , up22 , up23 , up24 , up25 , up26 , up27 , up28 , up29 , up30 , up31 ,

up32 ] = text read ( 'REFORMATTED_BAND_UP. dat ' , '%f %f %f %f %f %f %f %f %f %f

%f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f %f ' ) ;

p l o t (k , up1 , ' r ' )

p l o t (k , up2 , ' r ' )

p l o t (k , up3 , ' r ' )

p l o t (k , up4 , ' r ' )

p l o t (k , up5 , ' r ' )

p l o t (k , up6 , ' r ' )

p l o t (k , up7 , ' r ' )

p l o t (k , up8 , ' r ' )

p l o t (k , up9 , ' r ' )

p l o t (k , up10 , ' r ' )

p l o t (k , up11 , ' r ' )

p l o t (k , up12 , ' r ' )

p l o t (k , up13 , ' r ' )

p l o t (k , up14 , ' r ' )

p l o t (k , up15 , ' r ' )

p l o t (k , up16 , ' r ' )

p l o t (k , up17 , ' r ' )

p l o t (k , up18 , ' r ' )

p l o t (k , up19 , ' r ' )

p l o t (k , up20 , ' r ' )

p l o t (k , up21 , ' r ' )

p l o t (k , up22 , ' r ' )

p l o t (k , up23 , ' r ' )

p l o t (k , up24 , ' r ' )

105

p lo t (k , up25 , ' r ' )

p l o t (k , up26 , ' r ' )

p l o t (k , up27 , ' r ' )

p l o t (k , up28 , ' r ' )

p l o t (k , up29 , ' r ' )

p l o t (k , up30 , ' r ' )

p l o t (k , up31 , ' r ' )

p l o t (k , up32 , ' r ' )

p l o t ( [ 0 , 3 . 1 2 0 ] , [ 0 , 0 ] , ' k== ' )

hold o f f

ax = gca ;

ax . FontSize = 12 ;

s e t ( gca , ' fontname ' , ' t imes ' )

y l ab e l ( [ ' \ f o n t s i z e 12E=E_f [ eV ] ' ] )

xl im ( [ 0 3 . 2 1 4 ] )

yl im ([=4 4 ] )

x=[0.000 0 .841 2 .102 3 . 2 1 4 ] ;

s e t ( gca , 'XTick ' , x )

s e t ( gca , ' XTickLabel ' , str2mat ( 'K' , ' \Gamma ' , 'M' , 'K ' ) , ' f o n t s i z e ' , 12)

106

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