Ab-initio Simulation of Spintronic Devices - Delaware …bnikolic/QTTG/NOTES/NEGF+DFT/WALDRON… ·...
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Ab-initio Simulation of Spintronic Devices Derek Waldron Centre for the Physics of Materials Department of Physics McGill University Montr´ eal, Qu´ ebec Canada A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy c Derek Waldron, 2007
Transcript of Ab-initio Simulation of Spintronic Devices - Delaware …bnikolic/QTTG/NOTES/NEGF+DFT/WALDRON… ·...
Ab-initio Simulation of Spintronic Devices
Derek WaldronCentre for the Physics of Materials
Department of Physics
McGill University
Montreal, Quebec
Canada
A Thesis submitted to the
Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
c© Derek Waldron, 2007
Contents
Abstract x
Resume xi
Statement of Originality xiii
Acknowledgments xiv
1 Introduction 1
2 Theory 7
2.1 Transport Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Phase Coherence Length . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Spin Diffusion Length . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Resistor Network Model . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Julliere Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 First Principles Theories . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Non-Equilibrium Green’s Function (NEGF) Theory . . . . . . . . . . 16
2.7 NEGF-DFT Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Computation 26
3.1 Geometry Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Numerical Pseudopotentials and Basis Sets . . . . . . . . . . . . . . . 29
3.3 NEGF-DFT Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 30
iii
iv Contents
3.3.1 Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 K-point Integration . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.3 O(N) Green’s Function Calculation . . . . . . . . . . . . . . . 33
3.3.4 Efficient Grid Calculations . . . . . . . . . . . . . . . . . . . . 37
3.3.5 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Software Implementation: MATDCAL . . . . . . . . . . . . . . . . . 41
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Application: Molecular Spintronics Device 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Calculation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Transport at Finite Bias Voltage . . . . . . . . . . . . . . . . . . . . . 48
4.4 Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Application: Fe/MgO/Fe Magnetic Tunnel Junction 59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Calculation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Equilibrium Transmission Coefficients . . . . . . . . . . . . . . . . . . 62
5.4 Transport at Finite Bias Voltage . . . . . . . . . . . . . . . . . . . . . 64
5.5 Oxidation and Surface Roughness . . . . . . . . . . . . . . . . . . . . 69
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 Application: Graphene Ribbons 73
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Magnetic Properties of ZGNRs . . . . . . . . . . . . . . . . . . . . . 74
6.3 Transport in a ZGNR/C60/ZGNR MTJ . . . . . . . . . . . . . . . . 77
Contents v
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7 Conclusions 85
A MATDCAL User Manual 88
A.1 Introduction and Installation . . . . . . . . . . . . . . . . . . . . . . . 88
A.1.1 About this Manual . . . . . . . . . . . . . . . . . . . . . . . . 89
A.1.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.1.3 Using MATDCAL . . . . . . . . . . . . . . . . . . . . . . . . 90
A.1.4 MATDCAL in Parallel . . . . . . . . . . . . . . . . . . . . . . 91
A.2 Bulk Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2.1 Defining the system unit cell . . . . . . . . . . . . . . . . . . . 93
A.2.2 Defining the system parameters . . . . . . . . . . . . . . . . . 95
A.2.3 Defining the calculation parameters . . . . . . . . . . . . . . . 95
A.2.4 Self-Consistent Calculation . . . . . . . . . . . . . . . . . . . . 96
A.2.5 Band Structure Calculation . . . . . . . . . . . . . . . . . . . 100
A.2.6 Density of States Calculation . . . . . . . . . . . . . . . . . . 102
A.2.7 Charge Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.2.8 Eigenstate Calculator . . . . . . . . . . . . . . . . . . . . . . . 103
A.2.9 Using a Custom Basis Set . . . . . . . . . . . . . . . . . . . . 105
A.2.10 Reading .plt Files in GOpenMol . . . . . . . . . . . . . . . . . 106
A.3 Two-Probe Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3.1 Defining the leads . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3.2 Defining the system unit cell . . . . . . . . . . . . . . . . . . . 109
A.3.3 Defining the system parameters . . . . . . . . . . . . . . . . . 109
A.3.4 Defining the calculation parameters . . . . . . . . . . . . . . . 111
A.3.5 Self-Consistent Calculation . . . . . . . . . . . . . . . . . . . . 111
vi Contents
A.3.6 Density of States Calculation . . . . . . . . . . . . . . . . . . 112
A.3.7 Transmission Calculation . . . . . . . . . . . . . . . . . . . . . 112
A.3.8 Scattering States Calculation . . . . . . . . . . . . . . . . . . 113
A.3.9 Charge Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.4 Table of MATDCAL Parameters . . . . . . . . . . . . . . . . . . . . 116
A.5 Table of MATDCAL Support Tools . . . . . . . . . . . . . . . . . . . 118
B MATDCAL Programmer’s Guide 120
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
B.2 Java Object Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.3 Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.4 Parallel Computing Toolbox . . . . . . . . . . . . . . . . . . . . . . . 124
References 127
List of Figures
1.1 Schematic diagram of a magnetic tunnel junction . . . . . . . . . . . 2
1.2 A schematic diagram of an MRAM device . . . . . . . . . . . . . . . 3
2.1 Classical resistor model of a magnetic tunnel junction . . . . . . . . . 9
2.2 Schematic diagram of Julliere model . . . . . . . . . . . . . . . . . . 10
2.3 Flowchart of a DFT/NEGF-DFT calculation . . . . . . . . . . . . . . 16
2.4 Schematic diagram of an x-y periodic two-probe device . . . . . . . . 20
2.5 Schematic diagram showing unique interactions in a bulk-two-probedevice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Geometry optimization of an Fe/MgO/Fe device . . . . . . . . . . . . 28
3.2 Contour integration of density matrix . . . . . . . . . . . . . . . . . . 33
3.3 K-point integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Block-tridiagonal matrix structure of the Hamiltonian . . . . . . . . . 35
3.5 Efficient grid calculations . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Calculation benchmarks vs. cache size . . . . . . . . . . . . . . . . . 40
4.1 Schematic diagram of a Ni-BDT-Ni device . . . . . . . . . . . . . . . 46
4.2 Electronic structure of Ni . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Surface magnetism effects in Ni-BDT-Ni device . . . . . . . . . . . . 48
4.4 RTMR(V ) and I(V ) for hollow site Ni/BDT/Ni device . . . . . . . . . 49
4.5 RTMR(V ) and I(V ) for bridge site Ni/BDT/Ni device . . . . . . . . . 52
4.6 T (E) for Ni/BDT/Ni device . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Scattering states for Ni/BDT/Ni device . . . . . . . . . . . . . . . . . 55
vii
viii List of Figures
4.8 T (Ef ,k||) for Ni/BDT/Ni device . . . . . . . . . . . . . . . . . . . . 56
4.9 T (E) for 1D Ni/BDT/Ni device . . . . . . . . . . . . . . . . . . . . . 57
5.1 Schematic diagram of an Fe/MgO/Fe device . . . . . . . . . . . . . . 62
5.2 Charge profile of an Fe/MgO/Fe device . . . . . . . . . . . . . . . . . 63
5.3 T (EF ,k||) for a 3-layer Fe/MgO/Fe device . . . . . . . . . . . . . . . 64
5.4 T (EF ,k||) for a 5-layer Fe/MgO/Fe device . . . . . . . . . . . . . . . 65
5.5 RTMR and I(V ) for a 5-layer Fe/MgO/Fe device . . . . . . . . . . . . 67
5.6 T (E) for a 5-layer Fe/MgO/Fe device . . . . . . . . . . . . . . . . . . 68
5.7 T vs. k|| for a 5-layer Fe/MgO/Fe device . . . . . . . . . . . . . . . . 69
5.8 Histogram of RTMR for 5-layer Fe/MgO/Fe device . . . . . . . . . . . 70
6.1 Schematic diagram of zigzag graphene nanoribbons . . . . . . . . . . 74
6.2 Electronic structure of zigzag graphene nanoribbons . . . . . . . . . . 76
6.3 Isosurface plot of edge state in 4-ZGNR(H) . . . . . . . . . . . . . . . 77
6.4 LAPW vs. LCAO band structure of 4-ZGNR(H) . . . . . . . . . . . . 78
6.5 Schematic diagram of a ZGNR/C60/ZGNR device . . . . . . . . . . . 78
6.6 T (E) and DOS for a ZGNR/C60/ZGNR device . . . . . . . . . . . . 80
6.7 Scattering states for a ZGNR/C60/ZGNR device . . . . . . . . . . . 81
6.8 RTMR and I(V ) for a ZGNR/C60/ZGNR device . . . . . . . . . . . . 82
6.9 T (E) vs. Vb for a ZGNR/C60/ZGNR device . . . . . . . . . . . . . . 84
A.1 Schematic digram of an Fe super-cell . . . . . . . . . . . . . . . . . . 94
A.2 Electron density in an Fe unit cell . . . . . . . . . . . . . . . . . . . . 99
A.3 Band-structure of Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Isosurface of a Bloch-state of Fe . . . . . . . . . . . . . . . . . . . . . 104
A.5 Schematic diagram of an Al/BDT/Al two-probe device . . . . . . . . 107
List of Figures ix
A.6 T (E) for Al/BDT/Al device . . . . . . . . . . . . . . . . . . . . . . . 115
B.1 A diagram illustrating levels of code within MATDCAL. . . . . . . . 121
B.2 A diagram illustrating the object model implemented in the Java por-tion of MATDCAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.3 A diagram illustrating the organization of the Matlab code in MATD-CAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.4 A diagram illustrating how the MPICH2.0 library is made accessibleto the Matlab environment. . . . . . . . . . . . . . . . . . . . . . . . 125
Abstract
In this thesis, we present the mathematical and implementation details of an ab initio
method for calculating spin-polarized quantum transport properties of atomic scale
spintronic devices under external bias potential. The method is based on carrying out
density functional theory (DFT) within the Keldysh non-equilibrium Green’s function
(NEGF) formalism to calculate the self-consistent spin-densities. This state-of-the-art
technique extends previous work by: i) reformulating the theory in spin-space such
that the non-equilibrium charge density can be evaluated for different spin-channels,
and ii) introducing k-point sampling to treat transverse periodic devices such that
correct bulk as well as surface magnetism can be described. Computational details
including k-point sampling to converge the Brillouin zone integration, optimization of
pseudopotentials and basis sets, and efficient O(N) calculation of the Green’s function
are presented.
We apply this method to investigate nonlinear and non-equilibrium spin-polarized
transport in several magnetic tunnel junctions (MTJs) as a function of external bias
voltage. Firstly, we find that for an Fe/MgO/Fe trilayer structure, the zero-bias tun-
nel magnetoresistance (TMR) is several thousand percent, and it is reduced to about
1000% when the Fe/MgO interface is oxidized. The TMR for devices without oxidiza-
tion reduces monotonically to zero with a voltage scale of about 0.5-1V , consistent
with experimental observations. We present an understanding of the nonequilibrium
transport by investigating microscopic details of the scattering states and the Bloch
bands of the Fe leads. Secondly, we investigate a molecular MTJ composed of Ni
leads sandwiching a benzenedithiol(BDT) molecule. We find a TMR of ∼ 27% which
declines toward zero as bias voltage is increased. The spin currents are nonlinear func-
tions of bias voltage, even changing sign at certain voltages due to specific features
of the coupling between molecular states and magnetic leads. Thirdly, we investigate
the possibility of all-carbon based spintronics by exploiting the magnetic properties
of zigzag graphene nanoribbons (ZGNRs). We find extremely efficient spin filtering
effects in a theoretical device composed of two ZGNRs sandwiching a C60 molecule.
x
Resume
Dans cette these, nous presentons les details mathematiques et d’implementation
d’une methode ab initio pour le calcul des proprietes de transport quantique polarise
en spin de dispositifs spintroniques de taille atomique sous un difference de poten-
tiel appliquee. La methode est basee sur la theorie des fonctionnelles de la densite
(DFT) dans le cadre du formalisme des fonctions de Green hors equilibre (NEGF)
de Keldysh pour calculer les densites de spin auto-consistantes. Cette technique de
pointe etend les travaux precedents: i) en reformulant la theorie dans l’espace de
spins de telle maniere que la densite de charge hors equilibre peut etre evaluee pour
differents canaux a spin, et ii) en presentant l’echantillonnage de k-point pour traiter
les dispositifs periodiques transversaux de telle facon que le magnetisme de surface
et de volume peuvent etre decrits correctement. Les details informatiques, y com-
pris l’echantillonnage de k-points pour la convergence de l’integration sur la zone de
Brillouin, la construction et l’optimisation de la base et des pseudo-potentiels, et un
calcul efficace O(N) de la fonction de Green sont presentes.
Nous appliquons cette methode pour etudier le transport polarise en spin, non
lineaire et hors equilibre, dans plusieurs jonctions tunnel magnetiques (MTJs), comme
fonction du potentiel applique. Premierement, nous constatons que pour une struc-
ture tri-couche de Fe/MgO/Fe, la magnetoresistance tunnel (TMR) a difference de
potentiel nulle est de plusieurs milliers de pourcents, et se reduit environ a 1000%
lorsque l’interface de Fe/MgO est oxydee. La TMR pour des dispositifs sans oxyda-
tion se reduit a zero de facon monotone avec une difference de potentiel de l’ordre
de 0.5-1V , conformement aux observations experimentales. Nous interpretons nos
resultats en etudiant les details microscopiques d’etats de diffusion et des bandes de
Bloch dans les electrodes de Fe. Deuxiemement, nous etudions une MTJ molecu-
laire composee d’electrodes de Ni autour d’une molecule de benzene-dithiol (BDT).
Nous trouvons une TMR de ∼ 27% qui descends vers zero lorsque la difference de
potentiel est augmentee. Les courants de spin sont des fonctions non lineaires de
la difference de potentiel, changeant meme de signe a certaines valeurs en raison de
caracteristiques specifiques au couplage entre etats moleculaires et electrodes magne-
tiques. Troisiemement, nous explorons la possibilite d’une spintronique exclusivement
xi
xii List of Figures
a base de carbone, en exploitant les proprietes magnetiques de nano-rubans zigzag de
graphene (ZGNR). Nous constatons des effets de filtrage de spin extremement effi-
cace dans un dispositif theorique composes de deux electrodes de ZGNR autour d’une
molecule de C60.
Statement of Originality
This thesis work has resulted in a state-of-the-art technique and associated com-
putational software for ab initio studies of spin-polarized transport in atomic scale
spintronic devices based on NEGF-DFT theory. My contributions to this study in-
clude:
• A reformulation of NEGF-DFT theory in spin-space such that the non-equilibrium
charge density can be evaluated for different spin-channels. In addition, k-point
sampling was introduced to treat transverse periodic devices such that correct
bulk as well as surface magnetism can be obtained.
• The design and implementation from scratch of a comprehensive software pack-
age, called MATDCAL (Matlab-based Device Calculator) that implements the
NEGF-DFT theory. MATDCAL is written in Matlab with performance bot-
tlenecks written in Java. A significant number of test cases were conducted in
order to establish the validity of the implementation. As a part of this develop-
ment work, I designed and implemented an original parallel computing toolbox
that enables highly efficient parallel computing in the Matlab environment.
• The development of a number of novel computational algorithms for efficient
computation and load balancing in a parallel computing environment.
• The study of various spin-polarized transport phenomena in a variety of mag-
netic tunnel junctions. Each study included careful comparison of results to
existing literature in addition to obtaining important new scientific contribu-
tions such as the bias-dependence of tunneling magnetoresistance. This work
has to date resulted in the publication of several papers in peer-reviewed jour-
nals [1, 2, 3, 4, 5].
xiii
Acknowledgments
I would like to express my sincere gratitude to Prof. Hong Guo for being such an
extraordinary supervisor. His unique and genuine enthusiasm for science kept me
motivated during every step of my Ph.D., and in addition to supporting my core
research activities he took the time and interest to educate me on a wide variety of
other aspects about the scientific community and what it takes to succeed. I could
not imagine a better Ph.D. experience than having the privilege to work with and
learn from Prof. Guo.
I also consider myself very lucky for having the opportunity to work with the
talented members in my research group and our collaborators. In particular, I would
like to thank (in no particular order) Brian Larade, Vladimir Timochevskii, Joseph
Maciejko, Paul Haney, Allan MacDonald, Manuel Smeu, and Gino DiLabio for very
memorable professional and personal interactions.
I would like to acknowledge the contributions of Brian Larade, Liu Lei, Paul Haney,
Yibin Hu, and Eric Zhu for making contributions to the software and helping with
testing and validation.
Finally, I would like to thank my family and friends for their support during my
Ph.D. I have had a wonderful experience at McGill, and have grown both intellectually
and personally as an individual. I feel very prepared for my next career steps and
will remember my years at McGill with fond memories.
xiv
Physical Constants and Units
1 A = 10−10 m
a0 (Bohr radius) = 0.5292 A
me (electron mass) = 9.1096× 10−31 kg
mp (proton mass) = 1.6726× 10−27 kg
e (electron charge) = 1.6 ×10−19 C
h (Planck’s constant) = 6.626× 10−34 J s
kB (Boltzmann’s constant) = 1.38× 10−23 K
kBT (at 1 K ) = 8.616× 10−5 eV
c (speed of light) = 2.9979× 108 m/s
G0 (quantum unit of conductance) = 7.75 ×10−5Ω−1 = 1
12.9kΩ
Atomic units are used throughout this thesis unless otherwise indicated. In this
system of units, e = me = h = 1.
1 unit of Length = a0 = 0.5292 A
1 unit of Mass = me = 9.1096 ×10−31 kg
1 unit of Charge = e = 1.6 ×10−19 C
1 unit of Angular momentum = h = 1.0546 ×10−34 J s
1 unit of Energy = 1 Hartree = 27.2 eV
1 unit of Time = h
1 Hartree = 2.4189 ×10−17 s
xv
List of Abbreviations
TMR Tunneling Magnetoresistance Ratio
MTJ Magnetic Tunnel Junction
DFT Density Functional Theory
EQ Equilibrium
LCAO Linear Combination of Atomic Orbitals
NA Neutral-Atom
NL Non-Local
NEQ Non-Equilibrium
NEGF Non-Equilibrium Green’s Function
XC Exchange-Correlation
ZGNR Zigzag Graphene Nanoribbons
PC Parallel [Magnetic] Configuration
APC Anti-Parallel [Magnetic] Configuration
xvi
1
Introduction
Spintronics is an exciting field of research where electron spin, in addition to charge, is
exploited for electronic device application. The ability to manipulate and detect spin
offers the potential of a new paradigm for electrical devices. It has been suggested
that adding the spin degree of freedom to conventional charge-based electronic devices
or using spin alone has the potential advantages of non-volatility, increased data
processing speed, decreased electric power consumption, and increased integration
densities compared with conventional semiconductor devices[6].
One of the most successful spintronic devices to date is the magnetic sensor com-
posed of a magnetic tunnel junction (MTJ) exhibiting the tunneling magnetoresis-
tance (TMR) effect. The simplest MTJ consists of two ferromagnetic metal leads
sandwiching a very thin insulating barrier layer (typically a few nm), shown schemat-
ically in Fig.1a. Magnetoresistance refers to a change in the electrical current when
the relative magnetization of the ferromagnetic layers change their alignment[7]. The
TMR effect [8] in MTJs originates from the quasi-particle electronic structure of the
ferromagnetic leads which depends on their magnetization orientation. The tunneling
conductance tends to be smallest when the orientations are opposite, leading to a spin
valve character [8, 9, 10, 11, 12].
An important measure of device merit is the TMR ratio which is defined as
RTMR = (IPC − IAPC)/IAPC , where IPC and IAPC are the total tunneling currents
in the parallel configuration (PC) and anti-parallel configuration (APC) of the mag-
1
2 1 Introduction
FM NM FM FM NM FMSpinSpin
PCI APCI( )PC APC
PC
I ITMRI−
=
a) b)
Figure 1.1: (a) A schematic diagram of a magnetic tunnel junction (MTJ) composed of a thininsulating tunneling barrier sandwiched by two ferromagnetic electrodes. (b) Historical TMR data.Figure courtesy of Ref. [13].
netization moments, respectively, of the ferromagnetic leads. A larger RTMR gives a
more sensitive device which is desirable. Traditional single layer materials exhibited
RTMR ∼ 3%, however the discovery of the giant magnetoresistance (GMR) effect in
amorphous AlOx-based MTJs led to magnetic sensors exhibiting RTMR up to 70%
in room-temperature or low temperature [14]. In less than a decade from its original
discovery, the GMR effect revolutionized the magnetic storage industry by enabling a
more sensitive read-head technology leading to a dramatic increase in storage density.
Since the discovery of the GMR effect, a major research goal has been the real-
ization of devices with higher RTMR. It is now well known that substantially higher
RTMR can be obtained with single-crystalline devices, due to highly spin-dependent
evanescent decay of certain wave functions [15]. Following the prediction and elegant
physics explanation [15, 16] that Fe/MgO/Fe MTJs may have extremely high RTMR,
there has been tremendous recent advancement in the fabrication of MgO-based MTJs
(see Fig.1b) with measured room-temperature RTMR on the order of several hundred
percent [17, 18, 19].
A promising future application of spintronics technology is nonvolatile magnetic
random access memory (MRAM). MRAM is currently the strongest contender to-
3
Figure 1.2: A schematic diagram of an MRAM device. Figure courtesy of Ref. [20].
wards a universal memory technology that could outperform existing memory tech-
nology in all aspects of performance [20]. Commercial devices with several megabytes
of memory are already commercially available by major semiconductor manufacturers
such as Freescale and IBM. A prototypical MRAM device is schematically shown in
Fig.1.2. State-of-the-art MRAM combines a magnetoresistive magnetic tunnel junc-
tion with a single-pass transistor for bit selection during the read. The write mode
uses a localized magnetic field or the spin-torque effect to modify the magnetic state
of the free layer.
The tremendous success of first principles modeling [15, 16] in helping drive the
discovery of new materials for MTJ application illustrates the importance of such ab
initio techniques to the larger field of spintronics. For instance, in addition to the
intense research efforts devoted to studying devices composed of inorganic materials,
there have been recent attempts to use organic molecular layers as the tunneling
barrier which offer an interesting approach where spin-polarized transport can be
tuned by the peculiarities of the organic molecule [21, 22]. In particular, π-conjugated
organic semiconductors (OSEs) are a relatively new class of electronic materials that
are revolutionizing important technology application including information display
and large-area electronics. The OSEs have weak spin-orbit interaction and large
4 1 Introduction
spin coherence giving rise to very long spin diffusion lengths, making them good
candidates for spin-polarized electron injection and transport applications. Xiong et
al reported the injection, transport and detection of spin-polarized carriers using 8-
hydroxy-quinoline aluminium (Alq3) molecules as the spacing region of a spin-valve
structure, and obtained low-temperature RTMR as large as 40% [21]. Petta et al
found that spin polarization can be maintained during the tunneling process through
an organic monolayer, and demonstrated RTMR values of 16% in Ni/octanethiol/Ni
MTJs [22].
On a theoretical side, understanding nanoscale spintronics is complicated by the
key role of the atomic details of the system. Conventional magnetic tunnel junction
theory does not account for these details which are especially important in the mole-
cular case, where transport is very sensitive to the chemical bonding details between
the molecule and the magnetic leads. To date, most theoretical analysis of nanoscale
spintronics devices have been for devices in equilibrium [15, 16]. Non-equilibrium cal-
culations have been based on non-self-consistent techniques which can account only
for qualitative features. So far, no satisfactory theory has been reported which can
make quantitative predictions TMR versus bias voltage, which is measured in all ex-
periments. An important goal for the theory of spin transport in spintronic devices is
to investigate nanoscale TMR junctions under non-equilibrium conditions from first
principles theory and modeling.
In this thesis we present a theoretical formalism and associated modeling software
for spin polarized transport from an atomic point of view including all atomic mi-
croscopic details of the device. The formalism takes care of the the non-equilibrium
quantum transport condition so that spin current can be evaluated at finite bias volt-
ages. The approach is based on carrying out density functional theory (DFT) using
the local spin density approximation (LSDA) within the Keldysh non-equilibrium
Green’s function (NEGF) framework. The NEGF-DFT method offers a relatively
new technique for quantitative analysis of spintronic devices in non-equilibrium from
5
atomic first principles. So far it has been applied to investigate molecular spintronic
systems [1, 23, 24], Fe/MgO/Fe MTJ [2], spin transfer torque [4], and magneto-
resistance in anti-ferromagnetic tunnel junctions [5]. A computational package, called
MATDCAL, has been developed which implements this theory enabling first-principle
calculations of spin polarized transport in realistic spintronic devices. The organiza-
tion of this thesis is as follows:
Chapter 2 presents theoretical concepts for understanding spin transport, including
a classical resistor-network model and conventional magnetic tunnel junction theory.
We review important DFT and NEGF concepts, formulate the NEGF-DFT formalism
in spin-space, and introduce k-sampling to treat transverse periodic devices such that
correct bulk as well as surface magnetism of the magnetic leads can be obtained.
Chapter 3 presents the technical and computational details required to perform
calculations within the NEGF-DFT formalism. The method of pseudopotential gener-
ation and geometry optimization of the device is explained. Algorithms are presented
for efficient conversion between an orbital and real space description of the spin den-
sities and potentials, and O(N) calculation of the Green’s function. We also describe
the software tool MATDCAL, that implements the NEGF-DFT formalism.
Chapter 4 presents a theoretical study of spin polarized quantum transport through
a Ni-bezenedithiol(BDT)-Ni MTJ. A TMR of ∼ 27% is found for the Ni-BDT-Ni
MTJ which declines toward zero as bias voltage is increased. The spin-currents are
nonlinear functions of bias voltage, even changing sign at certain voltages due to
specific features of the coupling between molecular states and magnetic leads.
Chapter 5 presents a quantitative investigation of the nonlinear and non-equilibrium
quantum transport properties of Fe/MgO/Fe trilayer structures as a function of ex-
ternal bias voltage. For well relaxed atomic structures of the trilayer, the zero bias
TMR is found to be very large, at several thousand percent, even when there are
small variations in the atomic structure. This value is however reduced to about
6 1 Introduction
1000% when the Fe/MgO interface is oxidized. As a function of external bias voltage,
the TMR for devices without oxidization reduces monotonically to zero with a voltage
scale of about 0.5 to 1V, consistent with experimental observations. We present un-
derstanding of the non-equilibrium transport properties by investigating microscopic
details of the scattering states and the Bloch bands of the Fe leads.
Chapter 6 presents a theoretical study of spin polarized quantum transport through
an MTJ composed of hydrogen passivated zigzag graphene nanoribbons (ZGNR)
sandwiching a C60 molecule. The device shows strong spin-dependent transmission
and I-V characteristics, and suggests the intriguing possibility of all-carbon-based
spintronic devices.
Finally, we have included a user manual for MATDCAL in Appendix A that ex-
amines calculation examples for bulk and two-probe systems.
2
Theory
2.1 Transport Length Scales
2.1.1 Phase Coherence Length
The phase coherence length, lφ, is defined as the average distance an electron travels
before losing memory of its quantum phase. For length scales less than lφ, quantum
effects will play a major role in electron transport. For length scales much larger
than lφ, electron transport is primarily diffusive, and the physics can be adequately
explained by classical theories. In typical semiconductor heterostructures at low tem-
peratures, the phase coherence length can be as large as microns.
2.1.2 Spin Diffusion Length
The spin diffusion length, lsf , is defined as the average distance that an electron travels
before losing memory of its spin orientation, and is influenced by factors such as spin-
orbit interaction and scattering from magnetic impurities. In magnetic transition
metals it can be of the order of 103 [25]. Recently, extremely long spin diffusion lengths
(∼ 106) have been observed in ordinary semiconductors at low temperatures [26].
7
8 2 Theory
2.2 Resistor Network Model
When the typical length scale of a device, L, is smaller than the spin diffusion length,
then the majority spin current and minority spin current can be treated as indepen-
dent and spin-mixing can be ignored. This approximation, introduced by Mott [27],
is called the two-spin current model. Furthermore if L > lφ then the transport is
primarily diffusive and the system can be treated classically as an electrical resistor
network where each component has spin-dependent resistance. For example, consider
the MTJ shown in Fig. 2.1. We assume that both the left and right leads are com-
posed of the same material. In the antiparallel magnetic configuration (APC), the
total resistance of the device is:
RAP =1
2(R↑
lead +R↓lead +Rmolecule), (2.1)
where R↑lead and R↓
lead are the resistances of the lead for the majority and minority
carriers, respectively, and Rmolecule, the resistance of the molecular spacer, is assumed
to be identical for both spin-channels. In the parallel magnetic configuration (PC),
the total resistance of the device is:
RP =
(1
2R↑lead +Rmolecule
+1
2R↓lead +Rmolecule
)−1
, (2.2)
For typical magnetic materials, R↑lead > R↓
lead, and therefore RAPC > RPC . This
gives rise to the magnetoresistance effect, which is frequently expressed as a percentage
by the so-called optimistic tunneling magnetoresistance ratio, defined as:
TMR = (RAPC −RPC)/RPC = (IPC − IAPC)/IAPC . (2.3)
2.3 Julliere Model 9
Parallel Magnetization Anti-Parallel Magnetization
Figure 2.1: Classically, a spin-valve can be treated as an electrical resistor network where the leadsand spacer have a spin-dependent resistance.
2.3 Julliere Model
Conventional tunneling theory in MTJs was first introduced by Julliere[9], and is
based on two assumptions. The first assumption is that the electron spin is conserved
in the tunneling process. It follows then, that tunneling of spin-up and spin-down
electrons are two independent processes, so that conductance occurs in the two inde-
pendent spin channels. According to this assumption, electrons originating from one
spin state of the first ferromagnetic lead are accepted by unfilled states of the same
spin of the second lead. In PC, the minority spins tunnel to the minority states and
the majority spins tunnel to the majority states. In APC, the majority spins of the
first lead tunnel to the minority states in the second lead and vice versa.
The second assumption is that the conductance for a particular spin orientation is
proportional to the effective tunneling density of states (those states that participate
in the tunneling process) of the two ferromagnetic electrodes. According to these two
assumptions, the tunneling current for devices in PC (I↑↑) and APC (I↑↓) are given
by:
I↑↑ ∝ n↑Ln↑R + n↓Ln
↓R (2.4)
I↑↓ ∝ n↑Ln↓R + n↓Ln
↑R (2.5)
10 2 Theory
VΔFE
Parallel Magnetization Anti-Parallel Magnetization
Figure 2.2: The Julliere model assumes that the tunneling probability for spin-up and spin-downand proportional to the effective density of states in the leads. Blue indicates majority spin and redindicates minority spin.
where n↑L, n↓L, n↑R, and n↓R are the average spin-up and spin-down tunneling density
of states of the left and right leads. From these expressions the TMR can be written
as:
TMR =I↑↑ − I↑↓I↑↓
=2PLPR
1− PLPR
(2.6)
where
PL =n↑L − n↓Ln↑L + n↓L
(2.7)
PR =n↑R − n↓Rn↑R + n↓R
(2.8)
are the spin-polarizations of the left and right leads.
The Julliere model has been successful in providing reasonable estimates of the
2.4 First Principles Theories 11
TMR in MTJs from the known values of the spin polarization of ferromagnets obtained
in experiments on superconductors [28, 29, 30]. However the Julliere model has a
number of deficiencies. First, the predicted TMR is only a function of the leads
and therefore can not account for changes in the TMR due to different tunneling
materials. Secondly, the Julliere model does not include any atomic details of the
device such as contact geometry, surface roughness, or oxidization effects. Thirdly,
the Julliere model can not be easily generalized to systems under finite bias voltage.
The inadequacy of the Julliere model has become particularly apparent for MgO-
based devices where TMR is due to coherent spin-tunneling. Experimental TMR
values of several hundred percent have been obtained for these devices, over an order of
magnitude larger than that predicted by the Julliere model [17, 18]. Heiliger et al [31]
have also compared the Julliere model to results obtained from ab initio calculations
and showed a large discrepancy for various MgO-based MTJs. These experiments
have clearly demonstrated that a more sophisticated model of spin transport that
incorporates a correct treatment of the material band structures is required.
2.4 First Principles Theories
Spin-polarized quantum transport in nanostructured MTJs has been shown to be
very sensitive to the chemical and material details of the device [31, 32]. Therefore to
understand these transport features, atomic calculations are often necessary [15, 16] to
supplement general physical considerations and arguments[8]. In this regard, several
density functional theory (DFT) based first principles methods have been popular in
studying coherent spin-polarized tunneling in MTJ.
First, a very fruitful approach is the layer Korringa-Kohn-Rostoker (LKKR) method [33]
based on multiple-scattering Green’s functions for electronic structure calculations of
interfaces [34], where transport is analyzed by calculating Bloch-wave transmission
and reflections [35]. The LKKR technique has been used to study spin-dependent
tunneling in a number of devices including Fe/ZnSe/Fe and Fe/MgO/Fe tunnel junc-
12 2 Theory
tions [35, 15, 36]. More recently, Zhang et al reported combining LKKR with the non-
equilibrium Green’s function (NEGF) theory to treat devices under a finite bias [37].
Another very fruitful atomistic technique for analyzing MTJs is the linear muffin-tin
orbital (TB-LMTO) electronic structure scheme combined with surface Green’s func-
tions [38, 39, 32]. More recently, the TB-LMTO has also been combined in some way
with NEGF for analyzing finite bias transport situations[40]. Unfortunately, both
LKKR and LMTO methods for transport rely on the atomic sphere approximation
(ASA) whose accuracy requires considerable technical expertise to control.
A different tunneling calculation technique that does not rely on ASA was the
embedding potential method due to Inglesfield [41]. Inglesfield derived an energy-
dependent surface potential that acts on an embedding interface to include the effect
of a bulk crystal. Wortmann et al reformulated [42] the Landauer-Buttiker formula to
derive an expression for the linear conductance that contains only the embedding po-
tentials of the bulk system and the boundary values of the interface Green’s function.
Both quantities are available in a standard embedded Green’s function calculation and
thus, it is possible to evaluate conductance for ballistic transport without additional
computation. The embedding formalism was implemented within the framework of a
full-potential linearized augmented plane-wave (FLAPW) scheme and applied to sys-
tems such as Co/Cu/Co tri-layers [42] and tunneling through Fe/MgO/Fe MTJ [43].
A disadvantage of this technique is that FLAPW calculations are extremely com-
putationally intensive, and therefore transport calculations within this technique are
restricted to relatively small systems.
Another class of ab initio technique for calculating spin polarized quantum trans-
port is to combine real space LCAO based DFT with the NEGF where the device leads
and the scattering region are treated atomistically on equal footing. Such a NEGF-
DFT technique has been widely used in analyzing nonlinear and non-equilibrium
quantum transport in molecular electronics [44, 45, 46, 47, 48, 49], and has been
adapted to analyze spin-polarized quantum transport recently [50, 1, 2]. The basic
2.5 Density Functional Theory 13
idea of the NEGF-DFT technique is to use DFT to calculate the Hamiltonian and
electronic structure of a device, use NEGF to determine the non-equilibrium quantum
statistics that is needed to populate the electronic structure during current flow, and
use real space numerical methods to handle the transport boundary conditions.
We believe the NEGF-DFT method provides a useful alternative and supplements
other atomistic techniques for analyzing spin-polarized quantum transport. The main
advantage of the NEGF-DFT formalism is its close“proximity” to modern many body
theory and quantum transport theory which are largely based on Green’s functions.
As such, new effects and new transport physics can be readily implemented into the
NEGF-DFT software tool. From a computational point of view, the NEGF-DFT
technique can be implemented into a rather efficient manner so that larger systems
can be simulated. Furthermore, the NEGF-DFT method does not rely on ASA, and
hence offers a simpler and more robust computational alternative to existing atomistic
techniques.
2.5 Density Functional Theory
The basic theorems of density functional theory were derived by Hohenberg and
Kohn [51] and Kohn and Sham [52], and state that the ground state energy of a
system can be uniquely expressed as a functional of the ground state electronic den-
sity. In DFT, the electron density is the fundamental quantify from which all quantum
mechanical quantities can be calculated.
The Kohn-Sham (KS) energy functional is written as:
E[ρ] = T [ρ] +∫drρ(r)Vext(r) +
1
2
∫drdr′
ρ(r)ρ(r′)
|r− r′|+ Exc[ρ], (2.9)
where terms correspond to the kinetic energy, the energy of the external potential, the
14 2 Theory
Coulomb interaction between electrons, and the exchange-correlation energy, respec-
tively. If we consider the kinetic energy to be computed in the absence of electron-
electron interactions, and apply the variational principle to Eq.(2.9), we are led the
Kohn-Sham (KS) equations:
[−∇
2
2+∫dr′
ρ(r′)
|r− r′|+ Vext(r) + Vxc(r)
]ψi(r) = εiψi(r), (2.10)
where ρ(r) is the total electron density, Vxc is the exchange-correlation functional,
ψi(r) are the single electron KS wavefunctions, εi are the eigenvalues, and Vext is
any external potential including the pseudopotential that defines the atomic core and
the applied bias potential that drives current flow. We have also introduced the
exchange-correlation potential:
Vxc[ρ] =δExc[ρ]
δρ(r), (2.11)
and can identify the term in brackets of Eq.(2.10) as the KS Hamiltonian.
The strength of DFT lies in the fact that the interacting many-electron problem
has been reduced to a set of KS equations involving non-interacting electrons moving
in the effective potential of the other electrons. Furthermore this mapping is exact
provided that the exact exchange-correlation functional is known. In principle, the
exact exchange-correlation functional is far too complicated to ever express in a closed
form, and therefore approximations must be used.
The simplest approximation to the exchange-correlation functional is the local den-
sity approximation (LDA) [53, 54, 55, 56], in which one approximates the non-classical
corrections to the energy with the energy density of a homogeneous electron gas. The
LDA assumes that the exchange-correlation energy at a position r in an electron gas
can be approximated by the exchange-correlation energy in a homogeneous electron
gas, ε0xc, having the same density as the electron gas at position r:
2.5 Density Functional Theory 15
Exc[ρ(r)] =∫ε0xc[ρ(r)]ρ(r)dr. (2.12)
The energy density is typically calculated for high and low density limits and inter-
polated as a function of the density ρ(r).
A flowchart illustrating a self-consistent DFT calculation is shown in Fig. 2.3a.
The calculation begins with a trial electron density. The KS Hamiltonian is calculated
within a chosen basis set using this density. The Hamiltonian is then diagonalized to
calculate the eigenvalues and eigenvectors of the system. A new electron density is
calculated by populating the eigenvectors according to a Fermi distribution in energy:
ρ(r) =∑
i
f(εi, µ)|ψi(r)|2, (2.13)
where µ is the chemical potential of the system and chosen such that the total system
has neutral charge. Once a new electron density is calculated, the cycle is repeated
until self-consistency is achieved.
It is critical to point out that Fermi-Dirac statics are only valid for systems at ther-
mal equilibrium. Furthermore, Eq.(2.10) was derived using the variational principle
and therefore only valid for the ground-state energy of a system. Electron trans-
port is neither a ground-state nor an equilibrium problem, and hence an accurate
study of electron transport must go beyond standard DFT by correctly treating the
non-equilibrium and non-ground state nature of the system. The non-equilibrium
Green’s function formalism is a powerful framework that can be used to address these
problems.
16 2 Theory
)]([)],([ rr VV XCH ρρ
Choose a trial density
Electrode calculations
No
Calculate
Compute Green’s functions
Construct new ρ by NEGF
Is self-consistency reached?
Compute physical quantities
Calculate )]([)],([ rr VV XCHρρ
Choose a trial density
NoCalculate
Is self-consistency reached?
Compute physical quantities
i i ii
fρ ψ ψ=∑
a) b)
Figure 2.3: Flowchart of the self-consistent cycle with a) a standard DFT calculation, and b) aNEGF-DFT calculation.
2.6 Non-Equilibrium Green’s Function (NEGF) Theory
There are many excellent references on non-equilibrium Green’s function (NEGF)
theory applied to electron transport and we refer interested readers to the paper of
Wingreen, Jauho and Meir [57] and the book of Datta [58]. In this section we outline
the key results of the theory describing electron transport through a general system
consisting of a central scattering region connected by two leads. The Hamiltonian for
this system can be general expressed as:
H = Hll + Hrr + HC + HC,ll + HC,rr, (2.14)
where Hll is the Hamiltonian of the left lead:
Hll =∑k
εk,llC†k,llCk,ll, (2.15)
where C†k,ll creates an electron in the left lead. Here εk,ll = ε
(0)k,ll + qvL where ε
(0)k,ll
are the energy levels in the left lead and vL is the external voltage. Hrr is defined
2.6 Non-Equilibrium Green’s Function (NEGF) Theory 17
analogously. The third term HC is the Hamiltonian of the central region,
HC =∑n
(εn + qUn)d†ndn, (2.16)
where d†n creates an electron in the central region. The term Un is the self-consistent
Coulomb potential of the central region within the mean-field approximation:
Un =∑m
Vnm < d†mdm >, (2.17)
where Vnm is the matrix element of Hartree and exchange-correlation potentials. In
the real space the Hartree potential is V (x, x′) = q/|x − x′|. The fourth and fifth
terms describe the coupling between the central region and the left and right leads,
respectively. For example:
HC,ll =∑k,n
[tk,ll,nC†k,lldn + t∗k,ll,nd
†nCk,ll], (2.18)
where tk,ll,n is the coupling constant.
In practice, H is usually calculated self-consistently. Once an H is determined,
one may calculate the current flow through the device. The current operator for the
left lead is defined as (h = 1):
Ill(t) = qdNll
dt(2.19)
where Nll =∑
k C†k,llCk,ll is the number operator for electrons in the left lead. The
current operator for the right lead is defined analogously. In matrix form, it can be
shown [57, 58] that the steady state for one spin channel can be written in the form:
I = Ill = Irr =q
h
∫dε [f(ε− µL)− f(ε− µR)]T (ε), (2.20)
where f(ε) is the fermi-dirac distribution function, µL and µR are the chemical poten-
18 2 Theory
tials of the left and right leads, respectively, and T (ε) is the transmission coefficient,
T (ε) = Tr[ΓL(ε− qvL)Gr(ε)ΓR(ε− qvR)Ga(ε)]. (2.21)
This equation is equivalent to the famous Landauer-Buttiker formula [58]. From
this equation, we see that all the electrons in each lead below the chemical potential
EF + qvL,R, where EF is the fermi level, participate in the transport process. Here,
Gr(ε) and Ga(ε) are the retarded and advanced Green’s functions of the central region:
Gr,a(ε) =1
εI −HC − Σr,aL (ε)− Σr,a
R (ε)± i0+. (2.22)
The effect of the leads on the central region is included through the self-energy terms
ΣrL and Σr
R, which are calculated from the surface Green’s function of the leads:
ΣrL(ε) = HC,ll(εI −Hll)
−1H†C,ll (2.23)
ΣrR(ε) = HC,rr(εI −Hrr)
−1H†C,rr. (2.24)
The terms ΓL,R in Eq.(2.21) are defined as
ΓL,R(ε− qvL,R) = i(ΣrL,R(ε)− Σa
L,R(ε)). (2.25)
The partition of the system to isolate the central region is well known [59] and the
derivation of equation Eq.(2.22) can be easily proven[58].
A central result of NEGF theory is that under non-equilibrium conditions the
density matrix of the central region, ρ, is given by:
ρ = − i
2π
∫ ∞
−∞G<(ε)dε, (2.26)
where G<(ε) is calculated using the Keldysh equation:
G<(ε) = Gr(ε)Σ<(ε)Ga(ε), (2.27)
2.7 NEGF-DFT Theory 19
and
Σ<(ε) = iΓL(ε− qvL)fα(ε− µL) + iΓR(ε− qvR)fα(ε− µR). (2.28)
The power of NEGF lies in the fact that Eq.(2.26) correctly includes all information
about the non-equilibrium quantum statistics of the device, which are in general
not Fermi-Dirac. Hence the NEGF framework provides a unique advantage over
standard DFT calculations in that it applies to transport calculations where systems
are intrinsically under non-equilibrium conditions. The combined use of NEGF with
DFT is a powerful method that draws upon the advantages of both techniques. In
such a calculation, the electronic structure of a device is calculated within DFT, and
NEGF is used to determine the non-equilibrium quantum statistics that are needed
to populate the electronic structure during current flow.
2.7 NEGF-DFT Theory
The NEGF-DFT formalism outlined in this thesis is a spin-space generalization of
the theory described in Ref. [44]. A full review of the technique has also been pub-
lished in Ref. [3]. We start by considering the general device shown in Fig.2.4 where
two semi-infinite ferromagnetic electrodes sandwich a central scattering region. The
device is x-y periodic such that the left and right electrodes are fully 3D in a half
plane. It is important to note that for magnetic systems, 3D leads are necessary in
order to correctly describe both the surface and bulk magnetism of the ferromagnets.
When applied to molecular spintronic systems involving a single molecule as tunnel
barrier[1], the central scattering region must contain enough vacuum so that images
of the molecule do not interact. Along the transport direction (z-axis), the two fer-
romagnetic leads extend to reservoirs at z = ±∞. The central scattering region is
chosen sufficiently large in the z-direction such that: i) the potentials outside the
central region are taken as equivalent to bulk and ii) the matrix elements coupling
the left and right leads are zero. The electro-chemical potentials of the left and right
20 2 Theory
x-y periodic
Left
LH CH RH
Central Right
z
xy
Figure 2.4: Schematic diagram of an x-y periodic two-probe device. Two semi-infinite ferromagneticelectrodes are contacted via a central scattering region. The electronic structure of the central regionis calculated self-consistently where the potential and transport boundary conditions are determinedby the left and right ferromagnetic leads which are calculated separately within DFT.
leads, µL and µR, are given by the bulk Fermi level of the ferromagnets that can be
calculated by DFT at equilibrium, and the applied external bias voltage.
Because the device is x-y periodic, the eigenstates of the system can be labeled
according to their transverse momentum:
Ψk||(R|| + r) = eik||·R|| × eik||·rφk||(r), (2.29)
where k|| is a Bloch wavevector, R|| = nxa + nyb is a lattice vector, and φk|| is the
x-y periodic Bloch function. Using the Bloch ansatz, the Schrodinger equation can
be written in a matrix form as:
Hk||φk|| = ESk||φk|| , (2.30)
where Hk|| is the folded Hamiltonian defined as
Hk|| =∑
nx,ny
Hnx,nyeik||·R|| , (2.31)
and the overlap matrix Sk|| , which has been included because we are assuming a non-
2.7 NEGF-DFT Theory 21
orthogonal basis set, is defined analogously. In this equation, Hnx,ny is the Hamil-
tonian matrix connecting two unit cells separated by R||. It is important to note that
these matrices correspond to the entire device and therefore are infinite in dimension.
At the heart of the NEGF-DFT formalism [44] is the Keldysh Green’s functions
which are required for the calculation of electron density matrix at non-equilibrium
and transport properties of the system. Analogous to Eq.(2.22), the finite part of the
retarded Green’s function in k||-space corresponding to the L-C-R (left, central, and
right) region of the device is given by:
Grk||
(ε) =
εI − H
k||L − εS
k||L + Σ
k||L (ε) V
k||L − εS
k||CL 0
(Vk||L − εS
k||CL)† εI − H
k||C − εS
k||C V
k||R − εS
k||CR
0 (Vk||R − εS
k||CR)† εI − H
k||R − S
k||R + Σ
k||R (ε)
−1
,
(2.32)
where I is the identity matrix, Hk||L , H
k||R , and H
k||C are the finite sub-matrices of
Hk|| corresponding to the L, R, and C regions, respectively. Sk||CL and S
k||CR are the
finite sub-matrices of Sk|| connecting the C region to the L and R regions, respectively.
Vk||L (V
k||R ) are the finite sub-matrices connecting the L(R) and C regions. The coupling
of the L and R to the remaining part of the semi-infinite electrodes is fully taken into
account by the self-energies, Σk||L and Σ
k||R . It is important to note the distinction
between HL, the Hamiltonian of the unit cell of the left lead and is finite in dimension,
compared to Hll, the Hamiltonian of the entire lead used in the last section, which
has infinite dimension.
To analyze spin-polarized transport, the matrices above have been extended into
spin-space. Each matrix element in the non-spin formalism[44] becomes a two-by-two
matrix which specifies spin-up, spin-down, and the connection between the two spin
spaces [60]:
Hij →
Hij,↑↑ Hij,↑↓
Hij,↓↑ Hij,↓↓
. (2.33)
There is no restriction of spin collinearity, hence the left and right leads (and possi-
22 2 Theory
bly any other part of the system) can have arbitrary relative magnetic orientation.
For problems such as spin transfer torque[4], anti-ferromagnet tunnel junction[5] and
spin-orbital interaction, calculation capability of non-colinear spin is important. The
example applications in this thesis are restricted to spin collinear systems.
The Hamiltonian of each region is calculated self-consistently within DFT by solv-
ing the Kohn-Sham equation [52]. A flowchart illustrating a self-consistent NEGF-
DFT calculation is shown in Fig. 2.3b. The spin-dependent exchange-correlation po-
tential is treated at the local spin density approximation (LSDA) level [53, 54, 55, 56],
where one distinguishes spin-up and spin-down densities ρα = ρ↑ or ρ↓, and the total
density is given by ρ = ρ↑ + ρ↓.
As discussed above, when the central scattering region includes enough layers of
the ferromagnetic lead atoms, the electronic structure of the left and right regions
can be safely considered as that of bulk—which can be calculated with a super-
cell DFT analysis. In other words, in Eq.(2.32) the upper and lower parts of the
Hamiltonian (e.g. Hk||L(R) + Σ
k||L(R)) corresponding to the left and right ferromagnetic
electrodes, are calculated as isolated ”bulk” material whose electron density is given
by Eq.(2.13). By using the Fermi-Dirac distribution we have assumed that the left
and right ferromagnetic leads are in equilibrium contact with their corresponding
reservoirs, as is well established in the Landauer-Buttiker transport formulation[58].
The Hamiltonian of the left (right) region is obtained a separate calculation where
the left (right) region is taken as the unit cell of a fully period bulk system, and
hence will have the magnetic properties of a bulk system. The k||-dependent retarded
self-energies of each lead, Σk||L ,Σ
k||R , are determined using the recursion method of
periodic 1D systems [61], however with HL(R) for 1D replaced by Hk||L(R) for 3D leads.
In constructing the self-energies and potential matrices for each lead, the reference
spin direction is rotated to specify the relative magnetic orientation (for example PC
or APC) of the two ferromagnetic leads.
The remaining parts of the Hamiltonian in Eq.(2.32) are for the central region:
2.7 NEGF-DFT Theory 23
Vk||L , V
k||R and H
k||C , they are calculated self-consistently using the non-equilibrium
electron density matrix[44]. The non-equilibrium density matrix is calculated by
integrating over the 2D (in x-y direction) Brillouin zone (BZ) for contributions of
each transverse Bloch state:
ρ =∫
BZdk||ρk|| , (2.34)
where density matrix ρk|| is constructed using the non-equilibrium Green’s function
G<k||
:
ρk|| = − i
2π
∫ ∞
−∞dεG<
k||(ε) . (2.35)
Here G<k||
is calculated using the Keldysh equation:
G<k||
= GRk||
Σ<k||GA
k||(2.36)
where GAk||
= (GRk||
)† is the advanced Green’s function. Within a mean field type
theory such as DFT, the lesser self-energy Σ<k||
is given by a linear combination of the
Fermi-Dirac functions of the two leads[58], Σ<k||
(ε) = iΓL,k||(ε)f(ε−µL)+iΓR,k||(ε)f(ε−
µR), where the linewidth functions of left and right leads are related to the retarded
self-energies, for instance ΓL,k|| = i2
[Σ
k||L − (Σ
k||L )†
].
Finally, the spin currents (spin-polarized charge currents) are calculated by inte-
grating the contributions from each transverse Bloch state using the Landauer for-
mula:
Iσ =q
h
∫BZ
dk||
∫dε [f(ε− µL)− f(ε− µR)]Tσ(ε,k||) (2.37)
where the k|| resolved transmission coefficient for a particular spin, σ, is given by
Tσ(ε,k||) = Tr[ΓL,k||(ε)G
ak||
(ε)ΓR,k||(ε)Grk||
(ε)], (2.38)
where the trace is taken over orbital space. Note that each quantity in the right hand
side is defined in spin-space (see Eq.(2.33)), we have restored the spin index σ =↑, ↓
in the transmission coefficient and the current. The total charge current is calculated
24 2 Theory
CL R
9 interactions 5 interactions 9 interactions(not shown)
Figure 2.5: Schematic diagram showing the unique interactions with the central cell in a bulk-two-probe device.
by adding the current of each spin channel, I = I↑ + I↓.
The introduction of x-y periodicity dramatically increases the number of calcula-
tion steps compared to a 1D device. First, there are significantly more interacting
unit cells in the x-y direction. Even with nearest cell interaction, the central region of
a 3D device interacts with 23 neighboring cells, as shown in Fig. 2.5 (9 between the
central plane and each of the left/right planes, and 5 in the central plane) compared
to only 3 unique interactions in the 1D device. Secondly, there is an additional k||-
integration for the calculation of the density matrix and the current. For magnetic
tunnel junctions, a huge number of k|| are necessary to converge the results, as will
be discussed in the next chapter.
2.8 Summary
In this section we have reviewed several theoretical approaches used to describe spin-
polarized electron transport. In particular we have shown the deficiencies of tradi-
tional MTJ theory, and reviewed the limitations of existing first principles techniques.
We have reviewed standard DFT and NEGF theory, and presented a generalized
2.8 Summary 25
NEGF-DFT theory that has been formulated to treat transverse periodic devices and
explicitly include the spin degree of freedom. This enables a correct self-consistent
treatment of magnetism in systems with bulk 3D leads, without the need for em-
pirical materials parameters. The NEGF-DFT method correctly accounts for the
non-equilibrium quantum statistics that are important for transport calculations. We
believe the NEGF-DFT method provides a useful alternative and supplements other
atomistic techniques for analyzing spin-polarized quantum transport. In particular,
the NEGF-DFT technique is computationally efficient and does not rely on approxi-
mations such as ASA that are known to be very difficult to use without considerable
expertise. Furthermore, the NEGF-DFT framework is formulated in the language
of many body theory and hence is well suited to include new physical effects in its
theory.
3
Computation
3.1 Geometry Optimization
Our calculation scheme begins with an accurate optimization of the device geometry
using the full-potential linear augmented planewave method (FLAPW) implemented
in the WIEN2K software package [62]. This technique is amongst the most accurate
density functional theory methods for performing electronic structure calculations
of crystals. The FLAPW method is based on dividing the unit cell into (I) non-
overlapping atomic spheres centered at the atomic sites and (II) an interstitial region.
The two regions use different basis sets. Inside the atomic sphere t of Radius Rt the
basis set takes the form:
φkn =∑lm
[Almul(r, El) +Blmul(r, El)]Ylm(r), (3.1)
while outside the sphere a plane wave expansion is used:
φkn =1√ωeiknr. (3.2)
In these equations ul(r, El) is the regular solution of the radial Schroedinger equation
for energy El. The coefficients Alm and Blm are functions of kn determined by required
that this basis function matches (in value and slope) each plane wave.
The FLAPW method is closely related to the plane wave method, however has the
26
3.1 Geometry Optimization 27
advantage that there is no pseudopotential approximation. Furthermore the basis set
is complete, hence the only approximation lies in the exchange-correlation functional.
Therefore the FLAPW method is capable of accurate total energy calculations.
The optimized geometry of the leads is determined by finding the lattice constant
that minimizes the total energy. The total energy is calculated for several values
of the lattice constant, and the data is fit to the Murnaghan equation of state to
interpolate the lattice constant corresponding to the minimum energy:
E(V ) = E0 +B0V
B′0
((V0/V )B′
0
B′0 − 1
+ 1
)− B0V0
B′0 − 1
, (3.3)
where V is the volume of the unit cell, and B0, B′0, E0, and V0 are free parameters
determined by fitting to several calculated data points. A plot of the energy vs.
lattice constant for an Fe unit cell (Fig.3.1a) is shown in Fig.3.1c. The self-consistent
data points are shown as circles, and the solid line shows the interpolated fit using
Eq.3.3. The calculation yields a lattice constant of 2.83A, which agrees well with the
experimental lattice constant of Fe.
The optimized geometry of the two-probe device is determined by minimizing the
total energy of the central region with respect to the length along the z-axis, with the
x- and y- lattice constants of the central region constrained to equal that of the leads.
The positions of the atoms in the left and right most atomic layers are constrained
to match the bulk lattice constant leads. A plot of the total energy vs. the length
along the z-axis of the central cell for an Fe/MgO/Fe multilayer structure (Fig.3.1b)
is shown in Fig.3.1d.
The above calculation scheme performs a separate total energy calculation for
each lattice constant. A computationally more efficient apporach would be to simul-
taneously optimize the size of the central cell in addition to the atomic positions of
the atoms. This technique, commonly referred to as either unit cell optimization or
the variable shape method, is available in certain packages such as the commercial
28 3 Computation
a) b)
c) d)
2.75 2.8 2.85 2.9 2.95 3-2545.592
-2545.591
-2545.59
-2545.589
-2545.588
-2545.587
-2545.586
Lattice Constant (Angstrom)
Ener
gy (R
ydbe
rg)
Energy vs. Lattice Constant for BCC Fe
Calculated PointsFit
17.5 18 18.5 19 19.5 20-1.5485
-1.5485
-1.5485
-1.5485
-1.5485
-1.5485
-1.5485
-1.5485x 10
4
Length of Central Cell (Angstrom)
Ene
rgy
(Ryd
berg
)
Energy vs. Central Cell Size of FeMgO
Calculated PointsFit
Figure 3.1: a) A unit cell of BCC Fe. b) The central region for an Fe/MgO/Fe device. c) The totalenergy vs. lattice constant for Fe. The minimum energy corresponds to the 2.83A, which agrees wellwith the experimental lattice constant. d) The total energy vs. length of the central region of theFe/MgO/Fe device.
3.2 Numerical Pseudopotentials and Basis Sets 29
software DMol3 [63].
3.2 Numerical Pseudopotentials and Basis Sets
In the numerical calculations, we use an s, p, d linear combination of atomic orbitals
(LCAO) basis set [64, 60],
φlm(r) = Rl(r)Ylm(Ωr) , (3.4)
to expand the electronic wavefunctions and construct the matrix elements of Eq.(2.32).
In our computational implementation we use a linear combination of the Ylm to form
a real Cartesian basis,
φµ = Uµ,lµ,mµRlµYlµ,mµ , (3.5)
where, as an example, the d orbitals are defined as:
φdxy =
√15
4πRdxy
r2(3.6)
φdyz =
√15
4πRdyz
r2(3.7)
φdxz =
√15
4πRdxz
r2(3.8)
φdz2 =
√15
16πRd
3z2 − 1
r2(3.9)
φdx2−y2 =
√15
16πRdx2 − y2
r2. (3.10)
The atomic cores are defined by standard norm-conserving nonlocal pseudopoten-
tials [65]. Both the pseudopotentials and basis sets (i.e. Rl(r) in Eq.(3.4)) can be
generated, for instance using the electronic package SIESTA [60] or similar tools.
Special care must be given to the pseudopotentials and basis sets in order to obtain
an accurate description of the band structure near the fermi level, which is particularly
30 3 Computation
important in studying spin-polarized transport. On one hand, the calculation of the
Green’s function in Eq.(2.32) requires a reasonable sized basis set so that the matrix on
the right hand side is not prohibitively large to be inverted, on the other hand a small
basis set does not give accurate results. Therefore a reasonable compromise should be
adopted. We also found that pseudopotentials and basis sets that accurately repro-
duce the electronic structure of the electrode and barrier materials, do not necessarily
reproduce the electronic structure of the more complicated electrode/barrier interface.
Therefore, in our calculations these inputs are carefully constructed to accurately re-
produce electronic structures of the bulk materials and interfaces obtained by a full
potential linear augmented-plane-wave (FLAPW) method[62]. The latter comparison
for a periodic super-lattice of Fe(100)/MgO(100) interface is shown in Fig.5.1b. For
many systems we have studied, such a good comparison can be achieved by adjusting
the LCAO basis sets.
3.3 NEGF-DFT Calculation
3.3.1 Electrostatic Potential
The electrostatic potential, φ(x, y, z), is calculated by solving the Poisson equation:
∇2φ(x, y, z) = −ρ(x, y, z) (3.11)
where ρ(x, y, z) is the total charge density, subject to periodic boundary conditions
in the XY-plane, and left and right boundary conditions corresponding to the elec-
trostatic potential of the leads:
φ(x, y, z) = φL(x, y) (3.12)
3.3 NEGF-DFT Calculation 31
φ(x, y, C) = φR(x, y) (3.13)
φ(A, y, z) = φ(0, y, z) (3.14)
φ(x,B, z) = φ(x, 0, z) (3.15)
where A, B, and C are the dimensions of the central scattering region along the x-,
y- and z-axis. This electrostatic problem has been effectively solved using multi-grid
techniques [44], however we report here a more direct solution which is significantly
easier to implement. The formulas derived here assumes orthogonal unit vectors
however the formulas generalize easily.
Since all the quantities are XY-periodic, they can be expressed via Fourier trans-
forms along the periodic dimensions:
φ(x, y, z) =∫ ∞
∞
∫ ∞
∞Φ(kx, ky, z)e
2πi(kxx+kyy)dkxdky (3.16)
ρ(x, y, z) =∫ ∞
∞
∫ ∞
∞P (kx, ky, z)e
2πi(kxx+kyy)dkxdky (3.17)
φL(x, y) =∫ ∞
∞
∫ ∞
∞ΦL(kx, ky)e
2πi(kxx+kyy)dkxdky (3.18)
φR(x, y) =∫ ∞
∞
∫ ∞
∞ΦR(kx, ky)e
2πi(kxx+kyy)dkxdky (3.19)
substituting these equations into Eq.(3.11) yields the following equation:
∫ ∞
∞
∫ ∞
∞
[((2πikx)
2 + (2πikx)2 +
δ
δz2
)Φ(kx, ky, z)− P (kx, ky, z)
]e2πi(kxx+kyy)dkxdky = 0.
(3.20)
By orthogonality, this leads to a set of decoupled 2nd-order differential equations:
32 3 Computation
[((2πikx)
2 + (2πikx)2 +
δ
δz2
)Φ(kx, ky, z)− P (kx, ky, z)
]= 0, (3.21)
subject to the boundary conditions:
Φ(kx, ky, 0) = φL(kx, ky) (3.22)
Φ(kx, ky, C) = φR(kx, ky). (3.23)
This set of equations can be readily solved using standard 1-dimensional finite differ-
ence methods, after which the potential φ(x, y, z) can be solved for by performing an
inverse Fourier transform.
3.3.2 K-point Integration
For MTJ simulations, we found that integration over k|| must be handled very care-
fully in order to ensure numerical convergence. We calculate the density matrix using
standard complex contour integration for the equilibrium contribution and real-axis
integration for the non-equilibrium contribution [44]. The k||-integration is handled
differently for complex energy values on the complex contour versus energy values near
the real-axis which lie close to the poles of GR in the lower-half complex energy plane
(see Fig.3.2). Figure 3.3a,b compare the density of states of a 5-layer Fe/MgO/Fe
MTJ device in PC as a function of k|| at E = EF versus an energy value on the com-
plex contour at E = 5 + 5i(eV ), respectively. This figure clearly shows that a larger
imaginary energy component gives rise to a smoother density of states. Because of
the smoothness of the DOS for energies on the complex contour (i.e. Fig. 3.3b), it
was found that relatively few k||-points, for instance 12×12 = 144 in the full 2D Bril-
louin zone, are sufficient to converge the k||-integral of the equilibrium contribution to
the density matrix. On the other hand, it was found that several hundred thousand
3.3 NEGF-DFT Calculation 33
Emin( , )L Re eμ μ max( , )L Re eμ μminE
Small number of k-points
Large number of k-points
Figure 3.2: Fewer k||-points are required to converge the density matrix for energy values on thecomplex contour compared to on the real-axis.
k||-points are necessary to converge k||-integration for quantities evaluated at energy
values on the real energy axis, including evaluation of the non-equilibrium contri-
bution to the density matrix and the evaluation of transmission coefficients. Such
necessity of using high-density k-point sampling is consistent with results obtained
by other authors [43]. Figure 3.3c plots the relative convergence of the transmission
coefficient for the same 5-layer Fe/MgO/Fe device for both spin channels in PC and
APC as a function of the number of k||-points used in the integration of Eq.(2.26) over
the 2D Brillouin zone. These results clearly show that very large number of k||-points
are needed for convergence. Fortunately, each k||-point is calculated separately thus
the calculation can be fully parallelized.
3.3.3 O(N) Green’s Function Calculation
Significant research efforts have been devoted to the development of O(N) electronic
structure methods where computational time scales linearly with system size [66, 67].
Most of these techniques are based on direct methods of calculating the density ma-
trix through iterative or minimization schemes in order to avoid an O(N3) eigenvalue
decomposition. For transport, we have developed a reasonably efficient O(N) calcula-
tion scheme within the NEGF-DFT framework where the computational time scales
linearly with size of the central region along the z-axis. The method exploits the
34 3 Computation
(a) (b)
0 2002
4002
6002
8002
# k||-points
0.5
1
1.5
Nor
mal
ized
T(E
f)
0 2002
4002
6002
8002
# k||-points
0
0.5
1
1.5
2
Nor
mal
ized
T(E
f)
0 2002
4002
6002
8002
# k||-points
0.8
1
Nor
mal
ized
T(E
f)
i)
ii)
iii)
(c)
Figure 3.3: Density of states vs. k|| for a 5-layer Fe/MgO/Fe device in PC evaluated (a) on thereal energy axis at E = Ef and (b) in the complex plane at E = 5 + 5i(eV ). (c) Convergence ofT (Ef ) vs. number of k||-points used for integration. i) Solid-line (diamonds): APC; ii) dotted-line(circles): I↓ for PC; iii) dashed-line (squares): I↑ for PC. Note that T (Ef ) for each case has beennormalized to unity for presentation purposes. It value is actually very small for APC case.
3.3 NEGF-DFT Calculation 35
0 50 100 150 200 250 300
0
50
100
150
200
250
300
Sparse Matrix Structure of Hamiltonian for FeMgO Device
Figure 3.4: Block-tridiagonal matrix structure of the Hamiltonian for an Fe/MgO/Fe device.
block-tridiagonal matrix structure of Hk|| and using the method of inverse by par-
titioning [68] for the evaluation of Eq.(2.32). An example block-tridiagonal matrix
structure of the Hamiltonian for an Fe/MgO/Fe device is shown in Fig.3.4.
Suppose that an N ×N matrix A is partitioned into:
A =
A11 A12
A21 A22
, (3.24)
where Aii are square matrices although not necessarily of the same dimension. If the
inverse of A is partitioned in the same manner,
A−1 =
A11 A12
A21 A22
, (3.25)
36 3 Computation
then Aij which have the same size as Aij, can be found by the following formula [68]:
A11 = (A11 − A12A−122 A21)
−1
A12 = −A11A12A−122
A21 = −A−122 A21A11
A22 = A−122 + A−1
22 A21A11A12A−122 (3.26)
The above formulas generalize easily for a block-tridiagonal matrix:
B =
B11 B12 0 · · · 0 0
B21 B22 B23 · · · 0 0
0 B32 B33 · · · 0 0...
......
. . ....
...
0 0 0 · · · Bn−1,n−1 Bn−1,n
0 0 0 · · · Bn,n−1 Bnn
, (3.27)
and the corresponding tridiagonal blocks of B−1 can be found recursively by[69],
Bi,i+1 = −BiiBi,i+1Ci+1,i+1
Bi+1,i = −Ci+1,i+1Bi+1,iBii
Bi+1,i+1 = Ci+1,i+1(I −Bi+1,iBi,i+1)
(3.28)
with i = 1, 2, · · · , n− 1 and B11 = C11, or alternatively by
Bi,i+1 = −DiiBi,i+1Bi+1,i+1
Bi+1,i = −Bi+1,i+1Bi+1,iDii
Bii = Dii(I −Bi,i+1Bi+1,i)
(3.29)
with i = n−1, n−2, · · · , 1 and Bnn = Dnn, where Cii and Dii are respectively defined
recursively by
Cii = [Bii −Bi,i+1Ci+1,i+1Bi+1,i]−1
3.3 NEGF-DFT Calculation 37
with i = n− 1, n− 2, · · · , 1 and Cnn = B−1nn , and
Di+1,i+1 = [Bi+1,i+1 −Bi+1,iDiiBi,i+1]−1
with i = 1, 2, · · · , n − 1 and D11 = B−111 . In addition, the first and the last column
blocks of B−1 can recursively be found, respectively, by
Bi+1,1 = −Ci+1,i+1Bi+1,iBi,1
with i = 1, 2, · · · , n− 1 and B11 = C11, and
Bi,n = −DiiBi,i+1Bi+1,n
with i = n − 1, n − 2, · · · , 1 and Bnn = Dnn. Since only the block-tridiagonal ma-
trix elements of GRk||
corresponding to overlapping basis functions are required for the
equilibrium density matrix, and only additional matrix elements in the first and the
last column blocks of GRk||
are required for the non-equilibrium density matrix, the
above recursive approach can be used for O(N) calculation of these matrix elements.
We have implemented this algorithm for calculating density matrix and electric cur-
rent so that the total calculation time scales as O(N) with respect to the size of the
central region along the z-axis.
3.3.4 Efficient Grid Calculations
There are two dominant bottlenecks within the self-consistent cycle. The first is the
calculation of the density matrix, which requires many matrix inversions for the cal-
culation of G<. The above algorithm is an efficient way of dealing with this bottleneck
yielding O(N) scaling with respect to system size. The second bottleneck of the cal-
culation is converting between an orbital representation of the density and potentials
(ρµν , Vxc,µν , VH,µν) and a real space representation (ρ(r), Vxc(r), VH(r)). Converting
to a real space representation is required for the calculation of the electrostatic po-
38 3 Computation
tential and the exchange-correlation potential. The real space representation of the
density, ρ(r), is calculated by looping over all overlapping orbitals and calculating
the product between the orbital wavefunctions and the corresponding density matrix
elements:
ρ(r) =∑µν
φµ(r)ρµνφ†ν(r) (3.30)
The most computationally expensive part of this calculation is the calculation of
the real space representation of the orbital, φµ(r). A direct implementation of the
above algorithm would require calculating φµ(r) a potentially extremely large number
of times. For example, consider a single Au atom using a single-zeta basis set with 9
orbitals. Calculating the real space charge density of this atom has a contribution from
9× 9 = 81 overlapping basis pairs (since all on-site orbitals overlap), and therefore a
direct calculation would require the real space density of each orbital to be calculated
81 times. Ideally one could calculate and store the real space charge density due
to each overlapping pair in memory, thereby requiring only one up-front calculation
for each pair, however unfortunately there is inadequate memory to implement this
approach.
Out calculation scheme uses a caching algorithm that enables requiring only a sin-
gle calculation of the real space representation of each orbital once without consuming
too much memory. A schematic diagram of the algorithm is illustrated in Fig.3.5. A
cache is allocated to store a predefined number of orbital real space representations,
typical 32, but depends on the basis set used and density of overlapping orbitals. The
cache is implemented as an ordered list where each orbital has a specific position that
reflects how recently it has been used. A hash table is also used to facilitate rapid
searching of the cache. Each orbital real space representation is stored as a set of
discrete orbital values on a mesh inside of a cube with edge length of 2×rcutoff where
rcutoff is the cutoff radius of the orbital. When a particular φµ(r) is required, the
3.3 NEGF-DFT Calculation 39
9ϕ
4ϕ
5ϕ
7ϕ
Ordered by R
ecent Usage
Orbital Cache
†19 5 59 9( ) ( ) ( )r r rρ ϕ ρ ϕ=
Orbitals not recently used are discarded
Figure 3.5: Schematic diagram of the efficient grid handling algorithm.
cache is first searched. If it is found, then a real space representation of the orbital
does not need to be recalculated, and can be taken directly from the cache. The
orbital is then moved to the first position in the cache, reflecting the fact that it
has been the most recently accessed orbital. If it is not found, then the real space
representation is calculated, and the orbital is added to the first position. When the
total cache size exceeds to predefined maximum size, the last elements are removed
and the corresponding memory associated with the real space representation of the
orbitals can be cleared. When the summation over the pairs in Eq.(3.30) is performed
by iterating over increasing z-coordinate of the overlapping pairs, each orbital needs
to be calculated only once, is used many times, and eventually falls off the cache.
To illustrate the increase in performance of this algorithm, Fig.3.6 shows a series
of calculations performed with varying cache sizes. The time values shown are the
time taken for a single self-consistent step during an electronic structure calculation
of bulk Fe with a unit cell of 2 atoms and using a single-zeta basis set of 9 orbitals
40 3 Computation
101 1020
500
1000
1500
2000
2500
Cache Size
Sel
f-Con
sist
ent I
tera
tion
Tim
e (s
)
SCF Iteration Time vs. Cache Size
Figure 3.6: A plot of the time for one self-consistent iteration versus the size of the cache in the gridhandling algorithm.
per atom. A cache equal to zero is equivalent to using no cache at all, and clearly is
shown to be the least efficient calculation scheme. As the cache increases, the total
computation time decreases. As the cache size increases to greater than 32, the self-
consistent iteration time eventually plateaus indicating that larger cache sizes provide
no speed advantage because each orbital is being calculated only once.
The cache scheme outlined above is also used to calculate the potential matrix
elements, Veff = VH + Vxc integrated in real space:
Veff,µν =∫drφµ(r)Veff (r)φν(r). (3.31)
3.3.5 Convergence Criteria
The criteria for convergence during the self-consistent density matrix calculation is
max(∆H) < δ where ∆H is the difference in the Hamiltonian matrix between two
consecutive iterations and δ is a pre-specified tolerance. Typical values of δ is 10−4
3.4 Software Implementation: MATDCAL 41
Hartree or less. Broyden’s method [70] is used to accelerate convergence, with a typi-
cal mixing parameter of β = 0.01. It was found that convergence during a two-probe
calculation typically takes 5 − 10 times the number of self-consistent iteration steps
compared to an equivalent bulk calculation. Therefore, prior to the two-probe calcu-
lation a separate bulk calculation of the central cell is performed and the converged
spin-density is used as the initial density in the two-probe calculation. For devices
with non-periodic central cells (i.e. different leads) the separate bulk calculation is
performed for a mirror-extended central cell to force periodicity and only half of the
converged spin-density is used as the initial guess in the final two-probe calculation.
3.4 Software Implementation: MATDCAL
The NEGF-DFT theory and computational details described in this thesis have been
implemented in a comprehensive software package called MATDCAL (Matlab-based
Device CALculation). It is important to note that despite the similarity in the name,
MATDCAL is separate from MCDCAL [44], which was the first software package to
perform a calculation of non-equilibrium quantum transport in molecular electronic
devices within the NEGF-DFT formalism.
MATDCAL is written in the Matlab programming language, with certain time
critical portions of the calculation written in Java. Matlab was chosen as the primary
development language because of its platform independence, no compiler is required,
and it is a scripting language which results in dramatically shorter code and faster
development time. These benefits are ideal for a scientific research group so that
resources can be more efficiently allocated to address scientific problems, thereby
avoiding the tedious language and information technology problems that are common
when dealing with lower level languages and that distract from scientific research.
Java was chosen as a support language to Matlab for multiple reasons. First,
Matlab and Java are very tightly integrated. Matlab has a built-in Java Virtual
42 3 Computation
Machine, and Java objects can be manipulated within the development environment
as easily as other data types such as numbers and structures. Second, Java objects
are handled and passed by reference, rather than value, and therefore facilitates very
efficient memory management. Matlab on its own does not offer the capabilities of
pointers, rather it handles the memory management itself. Although Fortran and
C/C++ support passing data by reference, these features are not supported through
the Matlab API. Thirdly, Java is a highly object oriented language, which can be
used to create highly organized and reusable code designs. Finally, the run-time of
the Java code used in our software is identical to C or Fortran, and therefore there is
no loss in computational performance.
MATDCAL has been implemented in parallel to run on a high-performance com-
puting cluster. Three parts of the NEGF-DFT calculation have been parallelized: the
handling of k||-points, the calculation of the real-space representation of the density,
and the calculation of the real-space numerical integrals in calculating the poten-
tial matrix elements. The parallelization of k||-points is straightforward because the
Green’s function calculation for each k|| is independent.
Parallelization is accomplished within a single program multiple data (SPMD)
software architecture. Each node stores a copy of the full real-space spin-densities,
ρσ(r). The advantage of this method is that no parallel Poisson solver is required
to solve for the electrostatic potential, VH(r). The disadvantage is that there is no
reduction in memory consumption as the number of nodes are increased, thereby
limiting the maximum size of the system that can be handled. A more effecient
method is to have each node store only part of ρσ(r).
Each node is responsible for calculating the contribution to ρσ(r) from a portion
of the density matrix elements. A global sum is then performed to combine the
results from all nodes and construct the full ρσ(r). Each node is also responsible for
the calculation of a portion of the matrix elements of Hscµν = Vxc + VH , which are
3.5 Summary 43
calculated via numerical integration of the real-space potential:
Hscµν =
∫drφµ(Vxc(r) + VH(r))φν . (3.32)
Again, a global sum is performed to combine the results from all nodes and construct
the full Hsc, and ultimately the full Hamiltonian.
At the time of MATDCAL’s creation, Matlab did not offer the capability for paral-
lel computing (only recently has a commercial parallel computing toolbox for Matlab
been made available). Therefore to facilitate parallelization, a parallel computing
toolbox was created with a proxy that wraps the MPICH2.0 library [71] into the
Matlab environment. This toolbox is a standalone package from MATDCAL that
makes accessible the power of parallel computing inside the high-level Matlab envi-
ronment. For example, the toolbox has been designed so that high-level datatypes
available in the Matlab environment such as complex matrices can be easily commu-
nicated between nodes.
3.5 Summary
In this chapter we have presented the technical details of an NEGF-DFT calculation.
Significant efforts have been made to reduce potential sources of error at each step.
The geometry of the device is first optimized using the FLAPW method. The LCAO
basis sets and pseudopotentials used in the transport calculation are carefully tuned
to match the electronic structure obtained with the FLAPW method. For the NEGF-
DFT calculation, we have demonstrated the importance of careful k||-point sampling,
which is particularily important for studying MTJs because of resonant states that
give rise to sharp transmission peaks as a function of k||.
We have presented an O(N) scheme for calculating the Green’s functions using
an efficient matrix inversion technique that takes into account the block tridiagonal
44 3 Computation
structure of the Hamiltonian. We have presented an efficient alogorithm for dealing
with real-space numerical integrals, that can yield significant speedups compared to
a more simple direct evaluation. Finally, all of the theoretical and technical details
presented in the last two chapters has been implemented in a comprehensive Matlab-
based parallelized software package called MATDCAL.
4
Application: Molecular Spintronics Device
4.1 Introduction
It has only been recently that spin has entered the realm of molecular electronics.
Molecular spintronics opens up a new arena in which magneto-transport and chem-
istry are closely coupled, and rich device physics is expected due to the nano-scale
nature of these systems[72]. The driving idea behind the first pioneering experiment
of Tsukagoshi [73], who injected spin polarized electrons into carbon nanotubes, is
that spin-orbit interaction is very weak in carbon-based materials. Tsukagoshi ob-
served a hysteretic magnetoresistance in several nanotubes with a maximum resistance
change of 9%, from which the spin-flip scattering length was estimated to be at least
130 nm. These findings have stimulated a growing activity in the area and several
experiments dealing with spin-transport through molecular bridges have recently ap-
peared. For example Xiong et al reported the injection, transport and detection of
spin-polarized carriers using 8-hydroxy-quinoline aluminium (Alq3) molecules as the
spacing region of a spin-valve structure, and obtained low-temperature RTMR as large
as 40% [21]. Petta et al found that spin polarization can be maintained during the
tunneling process through an organic monolayer, and demonstrated RTMR values of
16% in Ni/octanethiol/Ni MTJs [22].
Here we investigate the non-equilibrium spin-polarized quantum transport proper-
ties of a realistic molecular spintronic device composed of a 1,4-benzenedithiolate(BDT)
45
46 4 Application: Molecular Spintronics Device
Figure 4.1: An (left) orthogonal view and a (right) rotated view of the central cell of the moleculartunnel junction formed by two semi-infinite Ni(100) surface having periodic x-y extent, separatedby a BDT molecule attached at the Hollow site of the surfaces.
molecule contacted by magnetic surfaces of infinite cross-section using the NEGF-
DFT formalism. A short account of these results have been published in Ref. [1]. Our
results show that magnetotransport features are dominated by resonances mediated
by the coupling of the molecule with the Ni leads. An external bias voltage can tune
these resonances leading to a nonlinear spin current-voltage dependence, inducing a
change of sign of spin injection, and quenching the TMR ratio from about 27% to
essentially zero on a voltage scale of about 0.5V .
4.2 Calculation Overview
Fig.4.1 schematically shows the molecular device under investigation. The device
consists of a BDT molecule in contact with two semi-infinite Ni(100) surfaces having
periodic x-y extent. Two devices were considered, where the BDT is attached to
either the hollow or bridge bonding sites of the Ni surfaces. The Ni/BDT/Ni device
was proposed by Emberly and Kirczenow[74] as a prototypical molecular spintronic
system and studied by Pati et.al using DFT[75] and a cluster approximation to mimic
the Ni leads. BDT molecular wires are also interesting with non-magnetic leads[76]
and have been the subject of extensive investigations[77].
The geometry of the device in the hollow site configuration was optimized using
4.2 Calculation Overview 47
G X W L G K X1
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Ene
rgy
(Har
tree)
Band Structure of Ni
downup
-12 -10 -8 -6 -4 -2 0 2 4 60
500
1000
1500
2000
2500
3000Density of States of Ni
DO
S (a
.u.)
E-Ef (eV)
Figure 4.2: The calculated (left) band structure and (right) density of states of Ni.
the FLAPW method described in Chapter 3. The central cell was relaxed between
three Ni layers of each side of the molecule, with the most remote layer of Ni atoms
fixed at crystalline positions during relaxation. The bond length between the S atom
and the nearest surface Ni atom was found to be 2.20. For the device in the bridge
bonding configurations, the device geometry was not optimized and the S-Ni bonding
length was taken to be 2.04. Each unit cell in the leads has a linear size of 7.04 in
the (x,y) direction and includes 32 Ni atoms.
The pseudopotentials used in the calculation were also optimized using the method
described in Chapter 3. We used an s, p, d single-zeta basis set with radial cutoffs
of 3.26 for Ni orbitals, 2.58 for C, and 2.69 for S. The calculated band structure and
density of states for bulk Ni is shown in Fig.4.2. The calculated magnetic moment
per atom was found to be 0.62µB, and the exchange-splitting was found to be about
0.6eV , consistent with the experimental value 0.6eV [78].
We found interesting surface magnetism effects due to the interaction of the surface
Ni atoms with the BDT. A cross-section of the magnetic moment density for both
the hollow site configuration and bridge site configuration are shown in Fig.4.3. In
each case, the nearest neighboring Ni surface atoms to the S atom in the BDT has
a reduced magnetic moment compared to that of its bulk value. In the hollow site
48 4 Application: Molecular Spintronics Device
0 1 2 3 4 5 60
1
2
3
4
5
6
Hollow Site Surface Magnetism
X Position (Angstrom)
Y P
ositi
on (A
ngst
rom
)
0
0.02
0.04
0.06
0.08
0.1
0 1 2 3 4 5 60
1
2
3
4
5
6
Bridge Site Surface Magnetism
X Position (Angstrom)
Y P
ositi
on (A
ngst
rom
)
0
0.02
0.04
0.06
0.08
0.1
0.12
Figure 4.3: A cross-sectional plot of the magnetic moment per unit area for the hollow site configu-ration (left) and the bridge site configuration (right). The non-magnetic S atom in the BDT reducesthe magnetic moment in the nearest neighboring Ni surface atoms.
configuration, there are four nearest Ni surface atoms each having a magnet moment
per atom of 0.44µB. In the bridge site configuration, there are two nearest Ni surface
atoms each having a magnet moment per atom of 0.18µB.
4.3 Transport at Finite Bias Voltage
For the transport calculation, we found that 12 × 12 = 144 (kx, ky) points suffice
to sample the 2D transverse Brillouin zone for converging the density matrix for
the contour integration, while 24 × 24 (kx, ky) points were required for the real-axis
integration of G<. Converging the transmission coefficient required even denser k-
sampling, typically requiring several hundred to a thousand points.
Fig.4.4(a,b) plots the current-voltage (I-V) characteristics for the parallel magne-
tization configuration (PC) and the anti-parallel magnetization configuration (APC)
of the two Ni leads respectively. Here the voltage bias is applied to the right lead,
and the magnetic moment points down in the right lead in the APC. In APC the
spin currents are nearly identical for small biases due to the symmetry of the device.
4.3 Transport at Finite Bias Voltage 49
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
50
100
150
200
250
300
Vb
I (nA
)
Parallel Configuration (PC)
UpDownTotal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
50
100
150
200
250
300
Vb
I (nA
)
Anti-Parallel Configuration (APC)
UpDownTotal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Vb
η
Spin Injection
PCAPC
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.05
0.1
0.15
0.2
0.25
0.3
0.35Magnetoresistance
Vb
TMR
Hollow Site Ni/BDT/Ni
a)
c)
b)
d)
Figure 4.4: Non-equilibrium transport characteristics of the Ni/BDT/Ni device in the hollow siteposition. (a) I-V curves for PC setup of the leads magnetization. b) I-V curves for APC (anti-parallel configuration) setup. (c) Spin-injection coefficient η versus bias voltage Vb (d) TMR versusbias voltage.
50 4 Application: Molecular Spintronics Device
Unlike the case of oxide tunnel barriers such as MgO[18, 79], the π-orbitals in BDT
provide a good transport channel leading to a metallic I-V curves with current in the
few hundred nA range when Vb = 0.5V . This current exceeds values measured[22]
for Ni-octanethiol-Ni, presumably because octanethiol is sigma bonded and therefore
has much larger resistance[80]. A very surprising and interesting result is the non-
linear behavior of the spin-currents, shown as dashed and dotted lines for the up-
and down-channels. We find that I↓ can be greater than I↑ in a PC setup when bias
voltage Vb is increased to Vb ≈ 0.1V. Similarly, in an APC setup, either I↑ or I↓ can be
larger depending on Vb and the direction of current flow. The source of this behavior
is resonant transmission mediated by molecular states resulting from the molecule-Ni
interactions (see below). We expect that this ability to tune the sign as well as the
magnitude of spin-currents in situ by adjusting bias voltage is a generic feature of
molecular spintronics[81]. It is interesting to note that while the behavior of the total
current versus Vb in the PC and APC setups give rise to a maximum TMR of about
27%, the individual spin-currents vary dramatically more, with the spin-down current
reaching a maximum difference of about 100% between the PC and APC setups at
Vb = 0.17V.
An important conceptual quantity is the spin-injection factor η which is defined
by spin-currents: η ≡ (I↑ − I↓)/(I↑ + I↓). Fig.4.4(c) plots η versus bias Vb for both
PC and APC setups. Due to the bias dependence of spin-currents, η can be either
positive or negative as Vb is increased, a result rather counter-intuitive. For PC, η
decreases from 28% to negative values at about Vb ∼ 0.1V. For APC, I↑ = I↓ when
Vb = 0 due to the geometric symmetry of the Ni-BDT-Ni device. Therefore η starts
from zero and increases to a maximum about 28% due to the faster increase of I↑ (see
Fig.4.5b). Afterward η declines and eventually becomes negative at large voltages.
Spin-injection into semiconductors has been measured experimentally using optical
techniques[82, 83]. Molecularly controlled spin injection into semiconductors should
occur in devices that are smaller than a spin relaxation length in extent and have
resistance that is limited by the molecular weak link. A change of sign in η could
4.4 Transmission Coefficients 51
then be measured by detecting a sign change in the optical polarization signal.
From the I-V curves we infer a TMR ratio using the common optimistic definition:
RTMR ≡ (IAPC − IPC)/IAPC , where IAPC,PC are the total currents in the APC and
PC setups respectively (solid lines in Fig.4.4a,b). At Vb = 0 when all currents vanish,
we use transmission coefficients to compute RTMR. As shown in Fig.4.4(d), for this
device RTMR ∼ 27% at zero bias and declines slowly with Vb. At Vb = 0.2V, RTMR
is reduced from its maximum by roughly a factor of two. This quenching voltage
scale is about ten times greater than the experimental scale of Ni-octanethiol-Ni
reported in Ref.[22], is similar to that of an organic semiconductor TMR junction[21],
and also similar to the quenching voltage scale of conventional TMR devices[84]. In
comparison, the Julliere estimate for Ni TMR devices is[23] RTMR ∼ 21%.
Fig.4.5 plots the IV-curves, spin injection, and TMR for the Ni/BDT/Ni device
in the bridge site bonding configuration. We found that the qualitative features and
orders of magnitude for physical quantities do not change. However, we found that
the value of the RTMR is somewhat sensitive to the molecular bonding site, having a
peak value of 40% in the bridge site configuration.
4.4 Transmission Coefficients
The voltage dependence of total current and spin-current can be understood from the
behavior of the transmission coefficients. Fig.4.6(a) plots Tσ(E, Vb) versus energy E
near the Fermi energy. Note that the Tσ(E, Vb) curves are dominated by a few rather
wide resonance features but otherwise are quite smooth. This is in distinct contrast
to situations where 1D leads are used (see below). The values found above for devices
with 3D leads are quite similar to those reported in experiments[73, 22, 21]. When
a positive Vb is applied to the right lead, Tσ(E, Vb) shifts to smaller energies but
roughly maintains its shape. Note that for APC setup, T↑ = T↓ for the entire energy
range when Vb = 0 (dotted line), due to the geometric symmetry of the Ni-BDT-Ni
52 4 Application: Molecular Spintronics Device
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
200
400
600
800
1000
1200
Vb
I (nA
)
Parallel Configuration (PC)
UpDownTotal
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
200
400
600
800
1000
1200
Vb
I (nA
)
Anti-Parallel Configuration (APC)
UpDownTotal
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Vb
η
Spin Injection
PCAPC
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Magnetoresistance
Vb
TMR
Bridge Site Ni/BDT/Ni
a)
c)
b)
d)
Figure 4.5: Non-equilibrium transport characteristics of the Ni/BDT/Ni device in the bridge siteposition. (a) I-V curves for PC setup of the leads magnetization. b) I-V curves for APC (anti-parallel configuration) setup. (c) Spin-injection coefficient η versus bias voltage Vb (d) TMR versusbias voltage.
4.4 Transmission Coefficients 53
device[23]. Far from the Fermi energy, for both APC and PC setups Tσ(E, Vb) is
roughly the same for both σ =↑, ↓, as indicated by resonance features near E = 2V
and −3V. This is because for Ni contacts the density of states (DOS) for the two
spin channels are most different near EF , and they become similar away from EF .
The entire behavior of spin-currents and TMR are, therefore, largely determined
by the features of Tσ(E, Vb) near EF . As an example, consider the PC setup. As
Vb is increased, the transport window (µL, µR) in Eq.(2.37) becomes wider and the
Tσ(E, Vb) features shift down in energy. Hence, at some small Vb, the left tail of the
peak labeled by “A” in the spin-down channel (dashed line in Fig.4.6(a)) enters the
transport window to cause a substantial increase in I↓ as shown in Fig.4.4(a). This
leads to the change of sign in spin-injection discussed above. For the APC setup, the
effect of Vb is to first break the I↑ = I↓ degeneracy before the transmission features
can cause the spin-currents to increase. The change of sign in spin-injection therefore
occurs at higher values of Vb (e.g. inset of Fig.4.4(c)).
The above transmission features are entirely consistent with the local DOS of the
BDT in the device, shown in Fig.4.6(b). The peaks near E = 0, which are responsible
for TMR, are largely due to hybridization of states of the S atoms in the BDT with
Ni atoms in the leads. Fig.4.6(c) plots the local DOS on the S atoms in the BDT,
and shows that the DOS peaks in the central cell are due to the contribution of the
S atom hybridized with the Ni surface. This is similar to the conclusions reached
for other molecular junctions[23]. This can also be seen visually by analyzing the
eigenstates of the open system (scattering states). Fig.4.7 shows an isosurface plot
of the scattering states of the device at E = 0.30eV corresponding the peak labelled
”A” in Fig.4.6(a). For the spin-down channel, the magnitude of the wavefunction is
very large around the top S atom, shown as a yellow sphere, indicating there there
is good coupling with the adjacent Ni atoms and hence providing a good channel for
transport. In contrast, there is a very small magnitude of the spin-up wavefunction
around the top S atom, providing a poor channel for transport.
54 4 Application: Molecular Spintronics Device
Ni/BDT/Ni (Hollow Site)
-3 -2 -1 0 1 2 30
0.5
1
1.5
2
E-EF (eV)
Tran
smis
sion
APCup(P)down(P)
-3 -2 -1 0 1 2 35
10
15
20
25
E-EF (eV)
DO
S (a
.u.)
up(P)down(P)
-3 -2 -1 0 1 2 3
5
10
15
20
E-EF (eV)
PD
OS
on
S (a
.u.) up(P)
down(P)
a)
b)
c)
A
up(P)down(P)APC
Figure 4.6: (a) Transmission coefficient Tσ versus energy E for Vb = 0. Data are collected withenergy spacing of 0.05eV. E = 0 is the Fermi energy of the leads. Solid: T↑ for PC setup; dashed:T↓ for PC; dotted: T↑ = T↓ for APC. (b) DOS for the scattering region of the device in PC. (c)Partial DOS on the Sulphur atoms in the central cell.
4.4 Transmission Coefficients 55
Hybridization with Ni-Surface No Hybridization
Ni/BDT/Ni Scattering State at E=0.3eV
spin-down spin-up
Figure 4.7: An isosurface plot of the scattering state of the open Ni/BDT/Ni device at E = 0.3eVfor the spin-down channel (left) and the spin-up channel (right).
To determine the origin of the large transmission peak at E = 2.25eV in Fig.4.6(a)
in calculated and projected the scattering states for the Γ-point (kx = ky = 0) at
this energy onto the eigenstates of the isolated molecule whose atomic coordinates
were extracted from the relaxed device. We found that the transmission peak at
E = 2.25eV is completely mediated by the LUMO+1 molecular level of the BDT,
having an 80% overlap between the scattering states and LUMO+1 eigenstate of the
isolated molecule. The fact that the transport at this peak is entirely mediated by an
organic non-magnetic molecule also explains why the transmission peak is identical
in magnitude for both spin channels.
We found that for the Ni/BDT/Ni device with 3D Ni leads, the transmission
coefficients have rich structure in the transverse momentum space. Fig.4.8 plots
T kx,kyσ (E, Vb) of Eq.(2.37) versus transverse momentum (kx, ky) at E = EF and Vb = 0,
for the PC setup of the Ni leads. These results show that transport in the Ni/BDT/Ni
device can be dominated by resonance transmission of incoming Bloch waves (in the
56 4 Application: Molecular Spintronics Device
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Ni/BDT/Ni (Hollow Site) PC, Spin-Up
kx (π/a)
ky ( π
/a)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Ni/BDT/Ni (Hollow Site) PC, Spin-Down
kx (π/a)
ky ( π
/a)
0.05
0.1
0.15
0.2
0.25
0.3
Figure 4.8: Transverse momentum resolved transmission coefficient Tkx,ky
↑ (left) and Tkx,ky
↓ (right)versus kx, ky at EF for PC setup, respectively. The vertical color coding bar is for the values of Tσ.
Ni leads) away from the Γ-point (kx = ky = 0). These behaviors provide a very
important illustration, in addition to the need for correct surface and bulk magnetism,
of the importance of using 3D Ni leads to model realistic devices.
To demonstrate the critical importance of 3D Ni leads, we calculated the trans-
port characteristics and JMR for a 1D system, shown in Fig.4.9(a), which has been
studied by other authors[23, 50, 85]. The transmission coefficient is dramatically less
smooth than its 3D counterpart, with many more transmission peaks arising from the
quantized subbands in the cross section of the 1D leads. These extremely sharp peaks
give rise to the very large positive and negative TMR ratios reported by [23, 50]. In
the present context, the 1D leads give rise to TMR ratios varying between −30% and
40%.
4.5 Summary
In summary, we have applied a first principles method for carrying out NEGF-DFT
atomistic analysis of spin-polarized quantum transport through molecular magnetic
4.5 Summary 57
-1.5 -1 -0.5 0 0.5 1 1.50
0.5
1
1.5
E-EF (eV)
Tota
l Tra
nsm
issi
on
PCAPC
0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.4
-0.2
0
0.2
0.4
E-EF (eV)
JMR
a)
b)
c)
Figure 4.9: (a) A Ni/BDT/Ni device with 1D leads (2) The transmission coefficient for the 1Dsystem is dramatically less smooth than the system with 3D leads (b) the TMR coefficient for the1D system is much less smooth and quantitatively less realistic than the 3D device, with TMR valuesranging from −30% to 40%.
58 4 Application: Molecular Spintronics Device
tunnel junctions. Using the Ni/BDT/Ni system as an example, we find that mag-
netotransport features are dominated by resonances mediated by the coupling of the
molecule with the Ni leads. An external bias voltage can tune these resonances lead-
ing to a nonlinear spin current-voltage dependence, inducing a change of sign of spin
injection, and quenching the TMR ratio from about 27% to essentially zero on a volt-
age scale of about 0.5V . Since single molecule conduction has been measured up to
1V , our findings should be experimentally accessible.
5
Application: Fe/MgO/Fe Magnetic Tunnel Junction
5.1 Introduction
The discovery of MgO-based MTJs has resulted in a new class of devices that is
anticipated to have an immense impact on information storage technology and next
generation spintronics technology [17]. Compared to the relatively modest TMR
values of up to ∼ 70% in MTJs composed of alumina tunnel barriers, MgO-based
MTJs have been successfully fabricated with measured TMR values of several hundred
percent. First principles calculations have played a key role in elucidating the reason
behind the observed large TMR in these devices [15, 16, 17, 18]. The source of the
extremely high TMR is coherent tunneling through the crystalline tunnel barriers,
and the highly spin-dependent evanescent decay of certain wave-functions. Only a few
years following the initial predictions that Fe/MgO/Fe MTJs may have extremely high
TMR [15, 16], these initial predictions were validated experimentally, with obtained
TMR ∼ 200% in Fe/MgO/Fe MTJs at room temperature [17, 18]. Since then,
the field has evolved rapidly and new MgO-based devices with increasing TMR are
consistently being reported. For example, Yuasa et al have obtained TMR ∼ 138%
and TMR ∼ 410% in CoFeB/MgO/CoFeB [86] and for Co/MgO/Co [19] MTJs,
respectively.
There still, however, exists a significant gap between theoretical predictions and
experimental data. While theory predicts TMR of several thousand percent [15, 16],
59
60 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
experimental data remains roughly an order of magnitude less. The origin of the
discrepancy is related to the junction quality, especially the crystallinity and homo-
geneity of the barrier. Experimental results are often obtained in the diffusive limit of
tunneling that means high density of structural defects, rough interfaces, and amor-
phous barriers. First principles calculation of TMR, however, have been performed for
structurally ideal junctions in the limit of coherent tunneling. It has been shown that
conductance and TMR are extremely sensitive to slight structural changes [87, 32].
Furthermore, an oxide layer is known to form during device fabrication [87], which is
also speculated to be responsible for a reduction in the TMR.
Despite numerous theoretical studies, there has also been relatively little work on
understanding the bias-dependence of TMR, which is measured in all experiments.
Experimental data on MgO-based MTJs show a monotonically decreasing TMR as
a function of applied bias voltage[18, 17] reducing to zero on a scale of about one
volt. Early theory[88] on Al2O3 based MTJs has attributed small-bias dependence of
magnetoresistance to magnon scattering. Previous first principles calculations of the
bias dependence of TMR have predicted a substantial increase of TMR versus bias
for an Fe/FeO/MgO/Fe device [37], while Ref.[87] found a roughly constant TMR, a
decaying TMR, or an entirely negative TMR versus bias for an Fe/MgO/Fe device
depending on the atomic structural details of the interface. Given the importance of
MgO-based MTJs in near future spintronics and the accumulated experimental data,
further quantitative understanding on quantum transport in Fe/MgO/Fe at finite bias
is needed.
Here we present a calculation of the nonlinear and non-equilibrium quantum trans-
port in an Fe/MgO/Fe MTJ using the NEGF-DFT formalism. A short account of
these results have appeared elsewhere[2]. This study was designed to address two out-
standing theoretical issues of TMR in Fe/MgO/Fe MTJs. Firstly, further quantitative
investigation of the impact of surface roughness and oxidization of the Fe/MgO inter-
face on device TMR is needed to understand how these factors might be controlled
5.2 Calculation Overview 61
during device fabrication. Secondly, a quantitative study of the quantum transport in
MgO-based MTJs at finite bias is needed to understand the experimentally observed
bias dependence of TMR.
Our results show that for fully relaxed atomic structure of the Fe/MgO/Fe device,
the zero bias equilibrium TMR ratio reaches several thousand percent, consistent
with previous theoretical results [15, 89]. Our calculation also shows that this value
is drastically reduced to about 1000% if the Fe/MgO interfaces are oxidized by 50%
oxygen. We found that the TMR ratio is quenched by bias voltage Vb with a scale of
about one volt, consistent with experimental data. The microscopic details of these
transport features can be understood by the behavior of bias dependent scattering
states.
5.2 Calculation Overview
The MTJ is schematically shown in Fig.5.1a, where a number MgO(100) layers are
sandwiched by two Fe(100). As discussed in chapter 2, the device is periodic in the
x-y direction and the Fe leads extend to z = ±∞ (transport direction). The atomic
structure was fully relaxed by the FLAPW method[62] described in chapter 3 between
three Fe layers on each side of the MgO, with the most remote layer of Fe atoms fixed
at crystalline positions during relaxation. The x-y lattice constant a of the interface
was fixed to our calculated one for bcc Fe, a = 2.82. The Fe-O distance was found to
be 2.236 for a completely relaxed structure, in good agreement with the value of 2.16
used in previous studies[15]. For the transport calculation, the spin-densities were
calculated self-consistently for a central region containing eight layers of Fe on either
side of the MgO.
Fig.5.2a,b plot the total charge and net magnetic moment on each atom in a 5-layer
Fe/MgO/Fe device, respectively. Each atom type is indicated. There is a significant
amount of charge transfer, about 1 electron, from each Mg atom to O. We also
62 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
-1
0
1
E (
eV)
LAPWLCAO
RX ΓΓ
(a)
(b)
Figure 5.1: (a) Schematic plot of a two probe Fe(100)/MgO(100)/Fe(100) device. The system hasinfinite extent in the (x,y) direction with a lattice constant of 2.82A, and extends to ±∞ in the z-direction. (b) Band structure of a periodic · · ·Fe/MgO/Fe/MgO· · · lattice obtained using optimizedLCAO pseudopotentials and basis sets compared to that from full potential LAPW method. A goodagreement is found to be necessary in order to carry out the NEGF-DFT analysis for the two probeFe/MgO/Fe devices.
observed a strong surface magnetism effect where each surface Fe atom experienced
an increased net magnetic moment by about 0.5µB.
5.3 Equilibrium Transmission Coefficients
We begin by analyzing the equilibrium transmission coefficients of the device. Ref.[43]
reported an embedding potential LAPW calculation for the parallel configuration
(PC) of a 3-layer MgO barrier MTJ with Fe leads. To compare, we calculated
T (EF ,k||) for the majority and minority-spin channels for 3-layer MgO device in
PC and the result is plotted in Fig.5.3a,b. Our results are very similar to that re-
ported in Ref. [43] (see Fig.2b of Ref.[43]). In particular, the majority-spin transport
is dominated by regions around k|| = 0 with circularly symmetric transport pattern,
while the minority-spin transport is dominated by k|| values at the zone boundary.
There are small quantitative differences for the minority spin channel near the zone
5.3 Equilibrium Transmission Coefficients 63
0 5 10 15 20 250
2
4
6
8
10
Atom Number
Cha
rge
Mg
OFeFe
0 5 10 15 20 25-1
0
1
2
3
Atom Number
Mag
netic
Mom
ent (μ B
)
a)
b)
Figure 5.2: a) Total charge on each atom in a 5-layer Fe/MgO/Fe device. b) Net magnetic momenton each atom in a 5-layer Fe/MgO/Fe device.
center: results in Ref.[43] show some tiny transmission values near k|| ≈ 0, while our
calculations show basically zero transmission there. At zone center we obtain a total
transmission T (E,k|| = 0) = 0.41, which is in good quantitative agreement to that
of Ref. [43]. Given that the two methods are totally different in all implementation
aspects, i.e. planewaves versus LCAO basis, full potential versus pseudopotential,
embedding versus NEGF device partition, as well as differences in the atomic struc-
tures, the comparison can be considered as satisfactory. Finally, Fig.5.3c shows a
semi-log plot of T (EF ,k||) in PC for the majority-spin versus the thickness of the
MgO barrier, and it shows very good consistency with the physics of tunneling.
Fig.5.4 plots T (EF ,k||) for both the PC and APC of a 5-layer MgO MTJ at zero
bias. Roughly, T (EF ,k||) behaves in a similar way as that of the 3-layer MgO device.
Namely for PC, the majority spin transport channel is s-like, i.e. dominated by k|| ≈ 0
region; and minority spin channel has non-zero transmission away from k|| = 0 but
the transmission value is much smaller than the majority channel. Hence for PC,
the total transmission coefficient is dominated by the majority channel. For APC,
Fig.5.4c shows that T (EF ,k||) has a value at the order of 10−4, much smaller than
64 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
(a) (b)
3 4 5 6 7 8 9Number MgO Layers
0.0001
0.001
0.01
0.1
T(E
f ,k||=
0)
(c)
Figure 5.3: (a,b) T (EF ,k||) for the 3-layer PC, majority- and minority-spin, respectively. Thesefigures are to be compared with the Fig.2b of Ref. [43]. c) Semi-log plot of T (Ef ,k|| = 0) versusnumber of MgO layers.
PC transmission. The pattern of T (EF ,k||) for APC is also interesting, dominated by
four relatively large peaks surrounded by very sharp hot spots. The large difference
in transmission values of PC versus APC gives a large TMR ratio at zero bias.
5.4 Transport at Finite Bias Voltage
Next, we consider the quantum transport properties of the Fe/MgO/Fe device at finite
bias voltage. Fig.5.5a,b plots the calculated current-voltage (I-V) characteristics (solid
line) for a 5-layer MgO device in PC and APC, respectively. For bias less than 1V ,
the total current remains extremely small. At about 1.5V , the device “turns on” and
the current increases rapidly afterward. Such a turn-on voltage is consistent with
5.4 Transport at Finite Bias Voltage 65
(a) (b)
(c)
Figure 5.4: (a,b) T (EF ,k||) for the 5-layer PC, majority- and minority-spin, respectively. (c)T (EF ,k||) for the 5-layer APC.
66 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
experimental I-V curve data reported in Ref.[79]. The spin-currents are shown as the
dashed and dotted lines for the up- and down-channels (majority-, minority-channel).
We can calculate the TMR ratio RTMR from the I-V curves in APC and PC. At
Vb = 0 when all currents vanish, we compute RTMR using transmission coefficients.
Because RTMR is obtained by dividing a very small number (the APC current), even
a small error in APC current makes a large error in RTMR. That is why very large
number of k||-points must be sampled for as good a convergence as possible (see lower
panels of Fig.3.3). From Fig.5.5c, for 5-layer MgO device RTMR ∼ 3700% at zero bias
and it decays with Vb, essentially vanishing on a scale of about 1V . For the 3-layer
MgO device, we found RTMR ∼ 850% at zero bias which also decays to zero on a
similar bias scale. The decrease in TMR as a function of Vb is in agreement with the
experimental data[18, 90].
The voltage dependence of the total current and spin-current (Fig.5.5a,b) can
be understood from the behavior of the transmission coefficient Tσ. Fig.5.6 plots
Tσ = Tσ(E) versus electron energy E at zero bias for PC and APC of the 5-layer MgO
device. In PC, the majority carrier transmission T↑ (solid line) is smooth and several
orders of magnitude larger than T↓ (dashed line) when E is near the Fermi energy of
the leads. By analyzing the spin-dependent scattering states[44] of the MTJ, we were
able to determine which bands of the Fe leads contribute to the transmission. We
found that T↑ is dominated by the ∆1 band of the Fe leads, in agreement with Ref.[15].
Below −1eV , T↑ becomes extremely small due to the disappearance of the ∆1 band.
The T↓, on the other hand, is considerably less smooth because the transmission near
the Fermi level is mostly dominated by interface resonance states[91]. In particular,
a large peak in T↓ appears above E = 1eV : as E is increased, different Fe bands
may participate transport and this peak is due to such a contribution. This T↓ peak
explains the much larger minority-channel current than the majority-channel current
in PC at Vb = 1.5V .
For APC, we obtain T↑ = T↓ for all E at zero bias due to the geometrical sym-
5.4 Transport at Finite Bias Voltage 67
0 0.5 1 1.5 2V
b (V)
0
0.05
0.1
0.15
0.2
0.25
I (A
/μm
2 )
0 0.1 0.2V
b (V)
0
0.0025
0.005
I (A
/μm
2 )
0 0.1 0.2V
b (V)
0
0.0001
0.0002
I (A
/μm
2 )
0 0.5 1 1.5 2V
b (V)
0
0.05
0.1
0.15
0.2
0.25
I (A
/μm
2 )
0 0.5 1 1.5 2V
b (V)
0
10
20
30
40
TM
Ra)
b)
c)
Figure 5.5: (a),(b) I-V curves for the 5-layer PC and APC, respectively. Solid line (diamonds): totalcurrent; dashed line (squares): I↑; dotted line (circles): I↓. Inset: I-V curves for small ranges of Vb.(c) TMR vs bias Vb for a 5-layer device (diamonds).
68 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
-3 -2 -1 0 1 2 3E-E
f (eV)
0
0.01
0.02
0.03
0.04
0.05
0.06
T(E
)
up, PCdown, PCAPC
-3 -2 -1 0 10
0.001
0.002
up, PCdown, PCAPC
Figure 5.6: Transmission coefficient Tσ versus energy E for Vb = 0, E=0 is the Fermi energy of theleads. Solid line: T↑ for PC setup; dashed line: T↓ for PC; dotted line: T↑ = T↓ for APC. Inset: Thesame transmission coefficients at energies between -3 and 1 eV.
metry of the device. We found that the BZ resolved total transmission, T (E,k||) =
T↑(E,k||) + T↓(E,k||) for APC shown in Fig.5.7c for Vb = 0 and Fig.5.7d for Vb =
0.05V, is dominated by broad and smooth peaks at around |kx| = |ky| = 0.12π/a
(see also Fig.5.4c), and there is almost no transmission at kx = ky = 0. For Vb = 0,
Fig.5.7c also shows that the dominating peaks are surrounded by other much sharper
peaks. Figs.5.7a,b plot the majority and minority electronic band structures of Fe
near Ef for |kx| = |ky| = 0.12π/a, respectively. By projecting scattering states with
|kx| = |ky| = 0.12π/a onto the Fe bands of Fig.5.7a,b, the dominating peaks are
found to be largely due to channel transmission: they are due to the band labelled
“1” in Fig.5.7a at one Fe contact, transmitting to the band labelled “2” on the other
Fe contact. Our calculations show that this band-to-band transmission contributes
2.37 × 10−4 to majority channel Tkx,ky
↑ . Other band-to-band transmissions are con-
siderably smaller. Similarly, Tkx,ky
↓ is mainly contributed from band-2 to band-1.
Therefore, it is the band-to-band transmissions which give almost the entire height
of the dominating peaks in Fig.5.7c (note Fig.5.7c,d plot the total BZ resolved trans-
mission).
We found that bias voltage has dramatic effects for APC. The very sharp peaks in
5.5 Oxidation and Surface Roughness 69
−0.5
0
0.5
−0.5
0
0.50
2
4
6
8
x 10−4
kx (π/a)ky (π/a)
T(E
,kx,
ky)
(0.12, 0.12, 0) (0.12, 0.12, 1) (0.12, 0.12, 2)
kz(π/a)
-2-1012
E (e
V)
1
−0.5
0
0.5
−0.5
0
0.50
2
4
6
8
x 10−4
kx (π/a)ky (π/a)
T(E
,kx,
ky)
(0.12, 0.12, 0) (0.12, 0.12, 1) (0.12, 0.12, 2)
kz(π/a)
-1012
E (e
V)
2
(a) (b)
(c) (d)
majority minority
Figure 5.7: (a,b) Fe bands at |kx| = |ky| = 0.12π/a versus kz for majority and minority electrons,respectively. (c,d) Total BZ resolved transmission coefficient at Ef versus kx, ky, for 5-layer MgO.(c) for Vb = 0; (d) for Vb = 0.05V. The dominant peaks are near |kx| = |ky| = 0.12π/a.
Fig.5.7c, which are due to interface resonances occurring at zero bias, are completely
removed by a finite bias of 0.05V, as shown in Fig.5.7d. Moreover, the dominating
peaks become considerably higher than those in Fig.5.7c. The origin of the TMR
quenching is therefore due to a faster increase of the APC current than the PC
current as a function of bias.
5.5 Oxidation and Surface Roughness
While the experimentally measured TMR has increased dramatically in the past two
years[17, 18], they are still significantly lower than theoretically predicted values here
and elsewhere[15, 16]. It is anticipated that both surface oxidization and roughness
of the Fe/MgO interface is playing a major role[92, 32]. We calculated the zero bias
TMR for two 5-layer MgO devices with varying oxygen content at the two interfaces,
and found drastic changes of zero bias TMR. For two 5-layer MgO atomic structures
with 100% and 50% oxygen at the interfaces, the zero bias TMR is dropped to ∼ 169%
and ∼ 1000%, respectively. The reason for this drop is found to be due to a decrease
70 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
Figure 5.8: Histogram of TMR for several 5-layer devices with a variation in the position of thesurface atoms. Inset: Histogram for varying all of the Mg and O atoms in the device.
of PC current while the APC current remains at similar value as that of un-oxidized
interfaces, consistent with the conclusion of Ref.[93].
To investigate the effect of surface roughness, we generated eight device atomic
structures of 5-layer MgO: for each device we varied the z-coordinates of the surface
Mg and O atoms from their relaxed positions, by a random displacement correspond-
ing to about 1% of the bond length. Self-consistent NEGF-DFT analysis is carried
out for them and the result is shown in Fig.5.5. Of these eight atomic structures, the
minimum TMR is about 3000% while the maximum is ∼ 4000%, with an average of
3580%. Although the sample size is small, the TMR ratio does change due to small
interface atom displacements. A similar analysis is carried out for thirteen 5-layer
MgO devices where all the Mg and O atoms were displaced randomly by roughly 1%
of the bond length, the result is in the inset of Fig.5.5. Again, the results indicate
a substantial change of TMR for small random variations of atomic positions in the
barrier layer.
5.6 Summary 71
5.6 Summary
In summary, we have analyzed the non-equilibrium quantum transport properties of
Fe/MgO/Fe MTJs from atomic first principles using the NEGF-DFT formalism and
without any phenomenological parameter. The study was designed to address two
very important outstanding problems: the origin of the experimentally observed TMR
quenching as a function of finite bias voltage, and the role of oxidization and surface
roughness in device TMR.
Through careful optimization of our basis sets and pseudopotentials, we were able
to obtain excellent agreement in the electronic structure of the Fe/MgO/Fe device
compared to highly accurate plane wave techniques. Furthermore, our transport
results are consistent with previously published results obtained by FLAPW within
the embedding potential formalism.
Our results show that the reduction of TMR as a function of bias voltage in
an Fe/MgO/Fe device can be adequately explained by band-to-band transmission,
and therefore it is likely that other effects such as magnon scattering play a less
significant role. The obtained voltage scales for transport features are consistent with
experimental data, including the turning on voltage for currents and voltage scale for
TMR quenching. The quench of TMR by bias is found to be due to a relatively fast
increase of channel currents in APC.
We found that both oxidization and surface roughness can have a major impact on
device TMR. At 50% and 100% interface oxidization of the Fe/MgO/Fe device, the
TMR is reduced by factors of 3 and 30, respectively, suggesting that an oxidization in
this range could account for the discrepancy between theory and experimental data.
Meyerheim et al measured surface oxidization of 60% in an Fe/MgO/Fe device [92],
which is consistent with these results. We also found a very high sensitivity of the
TMR to surface roughness, and found that very small perturbations in the surface
atoms can result in large fluctuations in the obtained TMR.
72 5 Application: Fe/MgO/Fe Magnetic Tunnel Junction
Finally, this study is a demonstration of the power of the NEGF-DFT formalism
and the relevance of its potential applications to technology. This study has moti-
vated a number of other questions which can be addressed within the NEGF-DFT
framework, such as finding new MTJs that have less sensitivity of TMR with respect
to oxidation or surface roughness, or have larger TMR quenching scales, which may
important for technology applications.
6
Application: Graphene Ribbons
6.1 Introduction
Graphite-related materials have long been a subject of interest [94, 95, 96]. Single
layer graphene sheets are known to have unique band structure characteristics around
the fermi level which result in interesting physical phenomena such as unusual inte-
ger quantum Hall effects [96]. Calculations of single graphite layers terminated by
zigzag edges on both sides, so called zigzag graphene nanoribbons (ZGNR), have been
demonstrated to exhibit interesting localized electronic edge states which have been
observed experimentally [97, 98, 99, 100]. These edge states decay exponentially into
the center of the ribbon, with decay rates depending on their momentum [101].
Recently, Son et al [102] have predicted using first-principles calculations half-
metallicity in nanometer sized graphene ribbons by exploiting the localized electronic
edge states. Their study showed that this phenomenon is realizable if in-plane ho-
mogeneous electric fields are applied across the zigzag-shaped edges of the graphene
nanoribbons, and that their magnetic properties can be controlled by the external
electric fields. In view of the importance of half-metallic materials for spintronic
applications, these results have opened the intriguing possibility of nanospintronics
based on graphene, and warrant further quantitative investigation of possible appli-
cations of ZGNRs.
Here we present a calculation investigating the sources of magnetism in two types
73
74 6 Application: Graphene Ribbons
a) b)
-0.0060.0236
-0.0160.1882
-0.006
0.1906
-0.0081.1559 0.2975
-0.004
-0.0321
0.1663-0.037
0.200
-0.039
0.344
Figure 6.1: a) H-passivated and b) unpassivated 4-ZGNRs. The net magnetic moment on each atomis indiciated.
of ZGNRs. We analyze nonlinear and non-equilibrium quantum transport in a pro-
totypical all-carbon MTJ composed of two ZGNR leads sandwiching a C60 molecule.
Our results show a huge TMR of about 1.5 × 106% which decays very quickly with
Vb on a scale of about 0.1V . We find interesting transmission behavior where partic-
ular molecular peaks of the C60 molecule admit only one spin channel, and can be
explained by considering the symmetry of the incoming Bloch states.
6.2 Magnetic Properties of ZGNRs
Following previous convention, the ZGNRs are classified by the number of zigzag
chains (n) forming the width of the ribbon, with the notation that a ZGNR with n
zigzag chains is denoted as n-ZGNR. We performed electronic structure calculations
using the FLAPW method for two 4-ZGNR systems: an H-passivated 4-ZGNR sys-
tem, show in Fig.6.1a, and an unpassivated 4-ZGNR system, shown in Fig.6.1b. The
geometry of each system was first optimized using the method described in Chapter 3.
The width of the 4-ZGNR and 4-ZGNR(H) systems measured between the outermost
carbon atoms was found to be 8.063 and 7.109, respectively. The lattice constant of
each system was found to be 2.287 and 2.489, respectively.
Each system was found to exhibit magnetism, consistent with other theoretical
6.2 Magnetic Properties of ZGNRs 75
calculations [102]. The net magnetic moment per atom (in units of µB) is shown in
Fig.6.1. The 4-ZGNR has the largest atomic magnetic moment of all three systems,
with the edge atom having a magnetic moment of 1.1559µB, in addition to having
the largest average magnetization of 0.3138µB per atom. The average magnetization
of the 4-ZGNR(H) was found to be 0.0474µB.
Investigations into the origin of magnetism in each ZGNR revealed very interesting
physics. The band structures and density of states for each ZGNR is shown in Fig.6.2.
For the 4-ZGNR system, there are two sources of magnetism. The first source of
magnetism is due to the presence of two highly localized 2p-state dangling bonds on
each side of the ribbon, shown in Fig.6.2a. Due to the high localization of these states,
they exhibit a very large exchange splitting of 1.3eV with spin-up states completely
occupied and spin-down states completely empty. These states give rise to a very
large peak in the density of states and account for the very large magnetic moment
on the edge atoms. The second source of magnetism is due to edge states formed on
both sides of the ribbon. These states are completely delocalized near kz = 0, but
become localized in the region 23π/a < kz < π/a. We found that in the case of N = 4
the edge states are forming, and for N > 4 the edge states appear as very straight
bands for kz >23π/a. These states also show exchange splitting in their localized
part. We considered two magnetic configurations where the magnetic orientation of
the edge atoms are ferromagnetically coupled (FM) or anti-ferromagnetically coupled
(AFM). We found that the AFM configuration was slightly energetically favorable,
with (FM−AFM)/atom = 4meV , and this difference decreases to zero as the width
of the ribbon increases.
In the 4-ZGNR(H) system, there are no longer any dangling bonds and only the
localized edge states are responsible for magnetism. Fig.6.2b shows that the bands
due to the dangling bonds are no longer present, however the bands corresponding
to the edge states are quantitatively similar to the 4-ZGNR system. Fig.6.3 is an
isosurface plot of the Bloch-state corresponding to the edge state in 4-ZGNR(H) at
76 6 Application: Graphene Ribbons
-8
-6
-4
-2
0
2
4
Ene
rgy
(eV
)
-8
-6
-4
-2
0
2
4
EF
Γ X Γ X
edge states
dangling bonds
UP DN
(a)
-8
-6
-4
-2
0
2
4
Ene
rgy
(eV
)
-8
-6
-4
-2
0
2
4
EF
Γ X Γ X
UP DN
(b)
-10 -8 -6 -4 -2 0 2 4Energy (eV)
-6
-4
-2
0
2
4
6
DO
S (s
tate
s/eV
)
spin UP
spin DN
(c)
-10 -8 -6 -4 -2 0 2 4Energy (eV)
-6
-4
-2
0
2
4
6
DO
S (s
tate
s/eV
)
spin UP
spin DN
(d)
Figure 6.2: a,b) Band structure diagrams for the unpassivated and H-passivated 4-ZGNR systems,respectively. In each plot the spin-up and spin-down states are shown on the left and right, re-spectively. c,d) A plot of the density of states versus energy for the unpassivated and H-passivated4-ZGNR systems, respectively. In each plot the spin-up and spin-down states are shown on the topand bottom, respectively.
6.3 Transport in a ZGNR/C60/ZGNR MTJ 77
Figure 6.3: An isosurface plot of the Bloch-state corresponding to the edge state in the 4-ZGNR(H)at kz = π/a.
kz = π/a. This accounts for the observed reduction in magnetism in the 4-ZGNR(H)
system. The difference between the FM and AFM configuration for the 4-ZGNR(H)
system was found to be (FM − AFM)/atom = 0.3meV .
6.3 Transport in a ZGNR/C60/ZGNR MTJ
The above results raise the possibility of exploiting the magnetic properties of ZGNRs
for spintronics applications. In particular, an interesting question is the possibil-
ity of all-carbon based spintronics devices. Using the NEGF-DFT formalism we
conducted atomistic analysis of non-equilibrium quantum transport to study a 4-
ZGNR(H)/C60/4-ZGNR(H) molecular tunnel junction (MTJ) composed of two ZGNR
leads attached to a C60 fullerene, as shown in Fig.6.5. It is important to point out
that while there cannot be magnetism in 1D systems in the thermodynamic limit, in
mesoscopic scales it may be possible. Total energy calculations suggest that even for
very long ribbons, the magnetic edge states are stable. The geometry of each device
was optimized using the method described in Chapter 3 with 4 layers of ZGNR on
each side of the C60 and the most remote layer of ZGNR fixed at crystalline positions
during relaxation. During the self-consistent DFT-NEGF calculation, we included 7
layers of lead on each side of the C60 molecule in the central cell. For DFT, we used
standard norm-conserving pseudopotentials and an s, p, d double-zeta LCAO basis
78 6 Application: Graphene Ribbons
Γ X-3
-2
-1
0
1
2
3
E-E
f (eV
)
Figure 6.4: A comparison of the band structure of the 4-ZGNR(H) obtained with an LAPW calcu-lation (solid lines) and LCAO calculation (dots). Both spin-up (dots) and spin-down (squares) areshown
Figure 6.5: Schematic plot of a two probe device composed of a C60 molecule sandwiched betweentwo 4-ZGNR leads. The device extends to ±∞ in the z-direction.
set. The exchange-correlation potential is treated at the LSDA level. The basis sets
and pseudopotentials used were carefully constructed to accurately reproduce all of
the above results obtained using the LAPW method. Fig.6.4 shows a comparison of
the band structure of the 4-ZGNR(H) obtained with an FLAPW calculation (solid
lines) and LCAO calculation (dots).
For converging the density matrix in the two probe MTJ simulation, we found
that due to the relatively sharp density of states around the fermi level, a large
number of integration points were necessary to converge the density matrix using the
complex contour integration in the NEGF-DFT calculation, requiring 240 points in
our calculations. The bonding distance between the C60 molecule and the 4-ZGNR
and 4-ZGNR(H) systems was found to be 1.712 and 1.732, respectively. The relatively
6.3 Transport in a ZGNR/C60/ZGNR MTJ 79
long C-C bond length implies poor binding between the C60 molecule and the ZGNRs,
resulting in a sharp density of states of the central scattering region (see below).
Fig.6.6a plots T (E) for the 4-ZGNR(H)/C60/4-ZGNR(H) system in PC. A very
broad spin-up transmission peak occurs at −0.3eV < E < −0.15eV , and a smaller
broad spin-down transmission occurs at 0.1eV < E < 0.2eV . Two very sharp trans-
mission peaks also occur at E = 0.15eV and E = 0.28eV . Interestingly, the peak at
E = 0.15eV admits only the spin-down channel, while the other sharp peak admits
both spin channels. Some of these transmission features can be understand by ana-
lyzing the density of states of the system. Fig.6.6b plots the total density of states
of the central cell, and shows that each of the transmission peaks corresponds to a
similar peak in the density of states. The density of states analysis does not reveal
why the sharp transmission peak at E = 0.15eV admits only the spin-down channel,
and requires analysis of the scattering states (see below). An analysis of the local den-
sity of states reveals that the broad transmission peaks are due to density of states
contribution from the leads (Fig.6.6d) while the sharp transmission peaks are due to
molecular states of the C60 molecule (Fig.6.6c).
The origin of the spin-dependent transmission peaks can be understood by ana-
lyzing the scattering states of the open device. At E = 0.15eV , corresponding to
the sharp spin-down transmission peak in Fig.6.6b, there are 4 incoming spin-down
channels and only 1 incoming spin-up channel. Fig.6.7a plots the scattering state
of the device in PC at E = 0.15eV for the largest contributing spin-down channel
to the transmission. Projecting this scattering state onto an isolated C60 molecule
reveals that this peak is due to one of the LUMO levels of the C60 molecule. The
scattering state has an odd symmetry and has a relatively large transmission value
through the C60 molecule. In contrast, Fig.6.6a plots the scattering state of the device
in PC at E = 0.15eV for the single spin-up channel, and shows an even symmetry
that is unable to transmit through the C60 molecule. This explains the origin of
the spin-dependent transmission peak through the molecular device, and suggests the
80 6 Application: Graphene Ribbons
ZGNR(H)a)
b)
c)
d)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3
T(E
)
E-EF(eV)
spin-upspin-down
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
500
1000
1500
2000
E-EF(eV)
DO
S
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
500
1000
1500
2000
E-EF(eV)
C60
DO
S
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
E-EF(eV)
Lead
DO
S
Figure 6.6: a) T (E) for the 4-ZGNR(H)/C60/4-ZGNR(H) device. b) Density of states versus energyfor the central region of the device. c) Local density of states versus energy of the C60 molecule inthe device. d) Local density of states versus energy of the lead in the device. All plots are for thedevice in PC.
6.3 Transport in a ZGNR/C60/ZGNR MTJ 81
Spin-up, E=0.15eV
Spin-down, E=0.15eV
a)
b)
Figure 6.7: a,b) Scattering state of the open 4-ZGNR(H)/C60/4-ZGNR(H) device in PC at E =0.15eV for the spin-down and spin-up channel, respectively.
intriguing possibility of using quantum symmetry as a mechanism for spin filtering.
Fig.6.8a,b plots the spin-resolved current-voltage (I-V) characteristics for the de-
vice in PC and APC, respectively. In PC, the spin-down transmission exceeds the
spin-up current by about a factor of 3. In APC there is a ”turn-on” voltage of
Vb ∼ 0.2V after which the spin-down current rises very quickly. From the I-V curves
we calculate the TMR ratio using Eq.2.6. We find that this device exhibits a huge
TMR ratio of 1.5× 106%, and very quickly decays to zero as a function of Vb.
The features of the nonlinear I-V characteristics can be understood by analyzing
the transmission curves of the device. Fig.6.9 plots the bias-dependent transmission
curves for the device in PC (left) and APC (right). In these calculations we took the
charge of the electron to be positive, e = +q, with the voltage bias applied to the right
lead. In PC, the broad spin-down transmission peak to the right of EF due to the
leads and the sharp transmission peaks from the molecular states of the C60 become
right shifted as the voltage bias increases. In the APC, the right lead is responsible
82 6 Application: Graphene Ribbons
a)
b)
c)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.02
0.04
0.06
0.08
0.1
Vb (V)
I (nA
)
PC
UpDownTotal
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.05
0.1
0.15
0.2
Vb (V)
I (nA
)
APC
UpDownTotal
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5000
0
5000
10000
15000
20000
Vb (V)
TMR
Figure 6.8: a,b) I-V curves for the 4-ZGNR(H)/C60/4-ZGNR(H) device in PC and APC, respec-tively. c) TMR versus bias voltage.
6.4 Summary 83
for the broad spin-down transmission peak to the left of EF and the broad spin-up
transmission peak to the right of EF . As the voltage bias increases both of these
features are right shifted, and at Vb ∼ 0.2V the broad spin-down transmission peak
moves into the transport window, resulting in the huge increase in the spin-down
current. The broad spin-up transmission peak to the left of EF and the broad spin-
down transmission peak to the right of EF are due to the left lead and do not shift
because Vb is applied to the right lead. In cases for Vb < 0.5V the sharp transmission
peaks from the LUMO C60 states do not enter the transmission window, and therefore
transport characteristics are completely due to the leads with the molecule acting as
a tunneling barrier.
6.4 Summary
In summary, we have conducted a theoretical investigation of two types of ZGNRs
using electronic structure methods to explain the fundamental origin of magnetism.
We found two sources of magnetism in unpassivated ZGNRs resulting in a very large
magnetic moment on the edge atoms. Passivating the carbon edges eliminates the
contribution to the magnetic moment due to dangling bonds and results in a reduced
magnetic moment. We analyzed the non-equilibrium quantum transport properties of
a prototypical all-carbon MTJ composed of ZGNR leads sandwiching a C60 molecule.
We found a huge TMR of about 1.5 × 106% which decays very quickly with Vb on
a scale of about 0.1V . The features of the transmission coefficient as a function of
energy can be understood by analyzing the density of states and scattering states
of the open device. This study shows an exciting potential for all-carbon spintronic
devices that could potentially be realized experimentally.
84 6 Application: Graphene Ribbons
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0
T(E
)
E-EF(eV)
spin-upspin-down
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.05V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.1V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.2V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.3V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0
T(E
)
E-EF(eV)
spin-upspin-down
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.05V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.2V
T(E
)
E-EF(eV)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.3V
T(E
)
E-EF(eV)
PC APC
T(E) vs. Vb for 4-ZGNR(H)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2x 10-3 Vb=0.1V
T(E
)
E-EF(eV)
Figure 6.9: T (E) for various values of Vb in PC (left) and APC (right).
7
Conclusions
In this thesis we have presented a theoretical framework and computational details of
a relatively new ab initio technique for calculating spin-polarized quantum transport
in magnetic tunnel junctions. The technique is based on real space density functional
theory in combination with the Keldysh non-equilibrium Green’s function formalism,
and has many advantages over existing atomistic methods. Firstly, the technique does
not rely on the atomic sphere approximation (ASA), for which significant technical
expertise is required to control the error. Secondly, the NEGF-DFT formalism has
close proximity to modern many body theory and quantum transport theory which
are largely based on Green’s functions. As such, new effects and new transport physics
can be readily implemented into the NEGF-DFT software tool. Thirdly, the NEGF-
DFT technique is computationally efficient due to the use of localized basis sets and
hence larger systems can be simulated. Finally, the NEGF-DFT method correctly
accounts for the non-equilibrium quantum statistics that play a vital role in electron
transport at the molecular level. For these reasons we believe the NEGF-DFT method
provides a useful alternative and supplements other atomistic techniques for analyzing
spin-polarized quantum transport.
In addition to the theoretical formalism, we have developed a robust computa-
tional framework that includes optimizing the geometry of the device and tuning of
the pseudopotentials and basis sets. By comparing to results obtained from FLAPW
calculations, the LCAO basis sets can be tuned to obtain excellent agreement in elec-
tronic band structure, density of states, spin splitting, etc. The additional step of
85
86 7 Conclusions
tuning the basis sets and pseudopotentials goes beyond traditional NEGF-DFT cal-
culations that have appeared in the literature up to this point, and therefore this
represents an important step in improving the available theoretical tools to study
quantum transport effects. For simulating MTJs, it was found that very large num-
ber of k||-points are necessary to converge the k|| dependent quantities such as trans-
mission coefficients and real energy density matrix. We have developed a number
of efficient computational algorithms including an O(N) scheme for calculating all
the Green’s function-based quantities required during the calculation. The NEGF-
DFT formalism and its associated computational details have been implemented in
a powerful software tool called MATDCAL, which is fully parallelized to run on a
computing cluster. A full description and user manual for this software is contained
in Appendix A. We believe that this software tool, in addition to the computational
scheme outlined in this thesis, constitute the current state-of-the-art for studying
non-equilibrium spin transport in nanoscale spintronic devices.
We have used the NEGF-DFT method to study spin-polarized transport in a
variety of MTJs. A brief report of these results has also been published in Refs. [1,
2, 3, 4, 5]. Firstly, we studied the non-equilibrium and nonlinear current-voltage
(I-V) characteristics for a Ni/BDT/Ni molecular spin-valve. This work, which has
been published in Ref. [1], was the first fully ab initio calculation of spin-polarized
transport in a molecular spintronic device with realistic 3D leads. We find that
magnetotransport features are dominated by resonances mediated by the coupling of
the molecule with the Ni leads. An external bias voltage can tune these resonances
leading to a nonlinear spin current-voltage dependence, inducing a change of sign of
spin injection, and quenching the TMR ratio from about 27% to essentially zero on
a voltage scale of about 0.5V . We find a strong sensitivity of the TMR with respect
to the contact details of the molecule, i.e. whether the BDT sits on the hollow or
bridge bonding site. The features of the transmission spectrum can be understood by
analyzing the scattering states of the open device.
87
Secondly, we used the NEGF-DFT method to study the voltage-dependent spin-
currents and TMR for Fe/MgO/Fe MTJs. Our results are consistent with those
obtained by the FLAPW technique within the embedding potential. The obtained
voltage scales for transport features are consistent with experimental data, including
the turning on voltage for currents and the voltage scale for TMR quenching. The
quench of TMR by bias voltage is found to be due to a relatively fast increase of
channel currents in APC and can be explained by simple band-to-band transmission
behavior under bias. We find that both oxidization and surface roughness can have a
major impact on device TMR. At 50% and 100% oxidization of the Fe/MgO interface,
our calculations show that TMR is reduced to ∼ 169% and ∼ 1000%, respectively.
We also find a very high sensitivity of TMR to surface roughness, and find that very
small perturbations in the interface atoms, such as those arising from lattice strain
or defects, result in large fluctuations in the obtained TMR.
Thirdly, we have used the NEGF-DFT method to study spin-polarized electron
transport in a prototypical all-carbon MTJ composed of ZGNR leads with a C60
molecule used as the tunneling barrier. Our FLAPW calculations show that there
are two sources of magnetism in ZGNRs, namely dangling bonds and localized edge
states. When the dangling bonds are saturated with hydrogen, the magnetic moment
of the edge atoms is significantly reduced. We found a huge TMR of 1.5×106% which
decays very quickly with Vb on a scale of about 0.1V . The spin transport properties
can be understood by analyzing the electronic structure of the ZGNR leads. This
study shows exciting potential for all-carbon spintronic devices that can be realized
experimentally.
Finally, our NEGF-DFT method has also been applied to analyze spin transfer
torque [4], and antiferromagnetic TMR devices [5]. These interesting applications are
summarized in Refs [4, 5] and we refer interested readers to them.
A
MATDCAL User Manual
A.1 Introduction and Installation
MATDCAL (Matlab-based Device Calculator) is a software package designed to cal-
culate the non-equilibrium quantum electron transport properties of nanoscale elec-
tronic/spintronic devices under finite bias voltage. The underlying theory is based
on carrying out density functional theory (DFT) within the Keldysh non-equilibrium
Green’s function (NEGF) formalism to calculate the self-consistent spin-densities.
MATDCAL has a full list of features that enable researchers with the power to study
many aspects of quantum transport in nanoscale devices, including:
• Self-consistent treatement of systems with open boundary conditions (semi-
infinite leads)
• Calculation of electron transmission curves and current-voltage characteristics
• Supports electron spin for calculation of magnetic systems
• Calculation of open-boundary scattering states wavefunctions
• Supports transverse periodic device geometries
• Fully-parallelized for use on a computing cluster
MATDCAL was written in the Matlab programming language, with certain time
and memory critical portions of the code written in Java. Matlab was chosen as the
88
A.1 Introduction and Installation 89
primary development language because of its platform independence, no compiler is
required, and it is scripting language which results in dramatically shorter code and
faster development time. Furthermore Matlab is a well-document and user friendly
development environment which reduces the bar to entry for new researchers who
desire to use or modify the software.
A.1.1 About this Manual
In this manual, file listings will be specified in the following way:
% Beginning of file myfile.txt1: line 12: line 2...% End of file myfile.txt
Note that the lines beginning with % should not be included in the file, nor should
the line numbers 1, 2, 3, etc.
A.1.2 Installation
MATDCAL is written in Matlab, and therefore Matlab must be correctly installed on
the user’s workstation. MATDCAL is not platform specific and will work on any PC
running Windows or Linux that supports Matlab. MATDCAL also uses some java
code which runs within Matlab’s java virtual machine. Before installing MATDCAL,
the size of the java memory must be set to be large enough to perform calculation.
In the Matlab installation directory, go to $MATLABROOT/bin/glnx32/ (or
win32 on windows machines) and create or edit the file java.opts. Edit the first
two lines of this file to read:
% beginning of file java.opts-Xms 512M-Xmx 512M% end of file java.opts
90 A MATDCAL User Manual
These two lines set the size of the initial and maximum java heap memory size to
be 512 megabytes. The specific value that should be chosen will depend on the total
available memory on the PC. Systems with 512 megabytes total RAM should use
256M, systems with 1 gigabytes should use 512M, and systems with > 2 gigabytes
should use between 512M and 1024M.
To install MATDCAL, first unzip the source package into its own directory, for
example /matdcal.
101% cd ~/102% unzip matdcal.X.XX.zip
Now open the Matlab application and navigate to to the directory /matlab/source/install.
In the Matlab command windows, type
>> install
Installing MATDCAL paths...Creating .matdcal file...MATDCALROOT = c:\work\ncmatdcalMATDCALTEMP = c:\tempCreating c:\tempInstallation complete.
>>
The MATDCAL install script will setup the necessary paths within Matlab, in addi-
tion to creating a file called .matdcal in the root directory, which contains the path
information and location of a temporary directory. MATDCAL is now installed.
A.1.3 Using MATDCAL
Within Matlab, MATDCAL is executed using the following syntax:
matdcal [FILE]
A.1 Introduction and Installation 91
where FILE is the name of the MATDCAL input file. The MATDCAL input file
consists of a sequence of keyword-value pairs. The general form of the input file is
the following:
keyword_1 = value_1keyword_2 = value_2...keyword_N = value_N
The order of the entries is irrelevant, and values may take the form of boolean, num-
bers, or strings depending on the particular keyword. The keywords can be divided
into three sections: general MATDCAL options, system definition, and calculation
options. Each of the files are described in detail in the following chapters.
A.1.4 MATDCAL in Parallel
MATDCAL can operate in a parallel computing environment on a computing cluster.
Since Matlab does not natively support parallel computing, a special parallel toolbox
was written that creates a proxy between Matlab and the MPICH2.0 library, thereby
providing the standard MPI commands within the Matlab environment.
The specific details of installation will depend on the particular computing cluster.
This manual assumes that MPICH2.0 has already been installed, and that the MPD
ring is correctly functioning on the cluster. If this is not the case, contact your
computing helpdesk for assistance.
Create the .matdcal file as described in the previous section on installing MATD-
CAL on a single computer. Make sure this file is accessible by all the nodes on the
cluster that will be running MATDCAL. Also make sure that MATDCAL appears in
the path for all the nodes that will be running MATDCAL.
Running MATDCAL will vary depending on the specific cluster. If direct access
to the nodes is allowed, a parallel job is started with the following command:
92 A MATDCAL User Manual
mpiexec -machinefile mymachinefile -n numbernodes matdcal -parallel myinputfile
A.2 Bulk Calculations 93
A.2 Bulk Calculations
The bulk geometry is used for the calculation of crystals, slabs, wires, and molecules.
Bulk systems are composed of a unit cell that repeats in 3-dimensions. System that
exhibit only 1- or 2-dimensional periodicity, such as a slab or wire, can be calculated
by including sufficient vacuum region in the unit cell such that there is no interaction
with neighbouring images. MATDCAL is restricted to the calculation of systems
with orthogonal unit cell vectors, however for band structure calculations MATDCAL
supports band-unfolding in order to calculate the band structure of non-orthogonal
primitive unit cells. Unrestricted spin calculations are supported so that magnetic
materials can be treated. For bulk systems, MATDCAL is capable of calculating the
following quantities: self-consistent hamiltonian, density of states, band structures,
and eigenstate densities. We first discuss how a system is setup and defined, then we
investigate how to perform each of these calculations.
A.2.1 Defining the system unit cell
The first step for a bulk calculation is to define the unit cell. A current restriction
of MATDCAL is that is only capable of handling systems with orthogonal unit cell
vectors. The unit cell is defined in a coordinate definition file, which has the following
format:
line 1: An integer indicating the number of atoms in the system.
line 2: The header ”AtomType X Y Z InitialSpinPolarization” where the arguments in are
optional.
line 3: Atomtype1 x1 y1 z1 InitialSpinPolarization1
line 4: Atomtype2 x2 y2 z2 InitialSpinPolarization2
...
line 5: AtomtypeN xN yN zN InitialSpinPolarizationN
The parameter Atomtypei is a string corresponding to the type of atom, for example
”C”or ”Ni”. The coordinates xi, yi and zi are in units of bohr. The optional parameter
InitialSpinPolarization is used only for spin systems to specify the initial guess for
the fraction of spin-up polarization for each atom. This value is between 0 and 1.
94 A MATDCAL User Manual
Figure A.1: An Fe super-cell consisting of two atoms and having orthogonal unit cell vectors.
The following is an example coordinate definition file for an Fe super cell consisting
of two atoms and having orthogonal unit cell vectors, as illustrated in Fig. A.1. The
final column of data indicates that the initial guess for the density is 60% spin-up and
40% spin-down.
% Beginning of file sys.xyz1: 22: AtomType X Y Z InitialSpinPolarization3: Fe 0.0000 0.0000 0.0000 0.64: Fe 2.6101 2.6101 2.6101 0.6% End of file sys.xyz
A.2 Bulk Calculations 95
A.2.2 Defining the system parameters
After the physical system has been defined and the system definition file has been
created, an input file with the system parameters must be created. The format for
this file is a series of entries with one entry per line having the form keyword = value.
A list of possible keywords and values for the bulk geometry are given below.
system.type This keyword specifies the geometry of the system. Valid options for bulk systems
are bulk3D and bulk3Dspin for spin calculations.
system.basistype Defines the type of basis set that is used in the calculation. Valid options are sin-
glezeta, doublezeta, and custom.
system.atomfile The filename of coordinate definition file specified in step 1.
system.lx Size of the unit-cell along the x-axis in units of bohr.
system.ly Size of the unit-cell along the y-axis in units of bohr.
system.lz Size of the unit-cell along the z-axis in units of bohr.
system.numx The number of grid points along x-axis. As a rule of thumb, this value should be at
least three times lx.
system.numy The number of grid points along y-axis. As a rule of thumb, this value should be at
least three times ly.
system.numz The number of grid points along z-axis. As a rule of thumb, this value should be at
least three times lz.
system.maxcachesize This parameter specifies the size of the real space handler cache size. Reducing this
value will conserve java memory, at the cost of increased computation time.
An example system parameters file corresponding to the Fe system defined above
is given below.
% Beginning of file setup.txt1: system.type = BULK3DSPIN2: system.basistype = singlezeta3: system.atomfile = ./sys.xyz4: system.lx = 5.22025: system.ly = 5.22026: system.lz = 5.22027: system.numx = 188: system.numy = 189: system.numz = 1810: system.numkx = 511: system.numky = 1012: system.numkz = 10% End of file setup.txt
A.2.3 Defining the calculation parameters
After the system parameters have been defined, a calculation setup file must be cre-
ated. This file specifies the specific type and calculation parameters. For the bulk
96 A MATDCAL User Manual
geometry, the following types of calculations are supported: calculation of the self-
consistent hamiltonian and electron density, density of states, real space eigenvectors,
and charge analysis.
A.2.4 Self-Consistent Calculation
The self-consistent calculation is used to perform a self-consistent density functional
theory calculation. The file format is the same as the system properties file. The
calculation setup file uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For self-consistent calculations this parameters has the value selfconsis-
tent.
calculation.maxiterations Specifies the maximum number of self-consistent iterations.
calculation.rootfinder.type Specifies the mixing algorithm to be used. Valid options are broyden, lin-
ear, and pulay.
calculation.rootfinder.beta Defines the mixing parameter used in the mixing scheme. Typical values
are 0.1-0.2 for small non-magnetic systems, and 0.005-0.1 for large or
magnetic system.
calculation.mixing Specifies the quantity that is used in the mixing scheme. Valid options are
hamiltonian or density.
An example calculation file for a self-consistent calculation of the Fe system defined
above is listed below.
% Beginning of file sc.txt1: &includefile = ./setup.txt2: calculation.type = selfconsistent% End of file sc.txt
The calculation is executed within Matlab by issuing the command:
matdcal sc.txt
This command will begin the MATDCAL calculation, and generate the following
output:
###########################################################################
A.2 Bulk Calculations 97
Matdcal 2.081:35:13 PMOct 19, 2006###########################################################################
Calculation Summary:---------------------------------------------------------------------------Calculation Type: Self-ConsistentDCAL Type: Bulk3DSpinDCALRootfinder: BroydenMixer
Precalculated Data:---------------------------------------------------------------------------
Calculation Log:---------------------------------------------------------------------------Initializing device calculator...Calculating boolean overlap matrix...Calculating overlap and kinetic energy matrix...Calculating pseudo potential matrix...Calculating neutral atom potential...Calculating neutral atom density...Calculating partial core density...Calculating initial Veff...Calculating initial rho...
System Summary:---------------------------------------------------------------------------Number atoms: 2Number basis: 18Neutral charge (on grid): 16.00
Self-Consistent Loop:---------------------------------------------------------------------------# Q Res drho dh Time---------------------------------------------------------------------------1 16.00 5.16E-002 0.00E+000 8.19E-002 00:01:122 16.00 3.41E-002 2.15E-002 6.80E-002 00:01:173 16.00 5.16E-004 8.99E-002 9.18E-003 00:01:164 16.00 4.32E-004 3.43E-003 7.42E-003 00:01:155 16.00 2.96E-004 8.10E-003 6.49E-003 00:01:146 16.00 2.84E-005 1.96E-002 1.96E-003 00:01:137 16.00 2.26E-005 5.03E-004 1.76E-003 00:01:178 16.00 5.94E-007 3.23E-003 3.63E-004 00:01:159 16.00 2.59E-007 6.07E-004 1.96E-004 00:01:1310 16.00 1.76E-007 9.24E-005 1.61E-004 00:01:1211 16.00 1.71E-009 4.49E-004 1.93E-005 00:01:1312 16.00 1.11E-009 1.28E-005 1.76E-005 00:01:14---------------------------------------------------------------------------
Self-Consistency Reached.
This information is also written to the file log.txt. The first column shows the iteration
98 A MATDCAL User Manual
number, the second column shows the total charge of the system, the third, fourth
and fifth columns are measures of the convergence, and should approach zero. The
last column is the time per iteration.
The self-consistent calculator produces the following files:
scdata.matrealspacedensity.matneutralatomdensity.matlog.txt
The .mat files are Matlab data files and be directly loaded into the Matlab workspace.
For example, suppose we issue the following commands in Matlab:
>> load scdata.mat>> data.matrices
ans =
1x63 struct array with fields:displacementBSTVnlVnaVeffHrho
>>
We see that the scdata.mat file contains all of the self-consistent matrices for the
system. In particular, there are 63 of each type of matrix, which corresponds to the
fact that there are 63 unique interactions between a single unit cell and other image
unit cells. For each group of matrices, the field displacement labels the interaction via
a vector that indicates the displacement between two interacting cells. For example,
a displacement vector [1 0 1] would indicate that the matrices correspond to an
interaction between unit cells that are separated by one lattice constant in the x-
direction and one lattice constant in the z-direction.
A.2 Bulk Calculations 99
Figure A.2: An isosurface of the electron density within an Fe unit cell.
Achieving self-consistency can sometimes be tricky, especially for larger and mag-
netic systems. It might be necessary to trying various convergence schemes and mixing
values to achieve convergence.
The realspacedensity.mat file contains the self-consistent electron density. MAT-
DCAL contains a useful tool to convert this file to a standard grid format (plt) that
is recognizable by the molecular visualizing tool GOpenMol. This tool is called from
the Matlab command line using the following syntax:
convertBulkRealSpaceDensityPlt systemfilename.txt outputfilename.plt
The tool will create a text file formatted in the plt fileformat, that can be converted
to a binary fileformat with GOpenMol using the Pltfile conversion utility, then loaded
within the contour analysis tool in GOpenMol. The total electron density isosurface
in the unit cell of the Fe system is shown in Fig. A.2. For a description of how to
read this file in GOpenMol, please refer to section A.2.10.
100 A MATDCAL User Manual
A.2.5 Band Structure Calculation
The band structure calculation is used to calculate and plot the band structure for a
periodic system. This tool is capable of unfolding band structure in order to obtain
the band structure of a primitive cell from a super cell. The calculation setup file
uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For band structure calculations parameter has the value bandstructure.
calculation.primitivebasis.v1 A vector defining a primitive unit cell vector. This is used during the band
unfolding process.
calculation.primitivebasis.v2 A vector defining a primitive unit cell vector. This is used during the band
unfolding process.
calculation.primitivebasis.v3 A vector defining a primitive unit cell vector. This is used during the band
unfolding process.
calculation.unitcellbasis.v1 A vector defining the orthogonal unit cell.
calculation.unitcellbasis.v2 A vector defining the orthogonal unit cell.
calculation.unitcellbasis.v3 A vector defining the orthogonal unit cell.
calculation.symmetrypoints A cell array defining the k-points in the bandstructure calculation. Valid
labels are G, X, W, L, K, X1, N, XZ. The definitions for these points are
defined in the function getbandstructure.m.
An example input file to calculate the band structure for Fe based on the primitive
unit cell is given below.
% beginning of file bandstructure.txt1: &includefile = ./setup.txt2:3: calculation.type = bandstructure4:5: calculation.primitivebasis.v1 = 5.2202*[-1/2 1/2 1/2]6: calculation.primitivebasis.v2 = 5.2202*[1/2 -1/2 1/2]7: calculation.primitivebasis.v3 = 5.2202*[1/2 1/2 -1/2]8:9: calculation.unitcellbasis.v1 = [5.2202 0 0]10: calculation.unitcellbasis.v2 = [0 5.2202 0]11: calculation.unitcellbasis.v3 = [0 0 5.2202]12:13: calculation.symmetrypoints = ’G’ ’X’% end of file bandstructure.txt
The calculation is executed within Matlab by issuing the command:
matdcal bandstructure.txt
A.2 Bulk Calculations 101
0 0.05 0.1 0.15 0.2 0.25 0.3−3
−2
−1
0
1
2
3
k
Ene
rgy
(eV
)
Band Structure of Fe from G−X
Figure A.3: The band structure of Fe calculated with MATDCAL from [0,0,0] to [π,0,0].
Upon completion MATDCAL will plot the band structure of the system and create
a file called bandstructure.mat which stores the data.
102 A MATDCAL User Manual
A.2.6 Density of States Calculation
The density of states calculator is used to calculate both the total and local density
of states in a bulk system. The calculation setup file uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For density of states calculations this parameter has the value dos.
calculation.eta The value of the imaginary component of the energy used in the Green’s
function calculation of the density of states. The default value is 1E-6.
Increasing this value will smoothen the density of states.
An example density of states calculation setup file for the Fe system is listed below.
% Beginning of file dos.txt1: &includefile = ./setup.txt2: calculation.type = dos3: calculation.eta = 1E-44: system.numkx = 105: system.numkx = 106: system.numkz = 10% End of file dos.txt
The calculation is executed within Matlab by issuing the command:
matdcal dos.txt
Upon completion MATDCAL will create a Matlab data file called dos.mat which
contains the calculated data. Both the total density of states and the local density of
states projected on the atoms are stored.
A.2.7 Charge Analysis
The charge calculator calculates the total charge on each atom by calculating the
trace between the overlap matrix and the density matrix. The calculation setup file
uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For charge analysis this parameter has the value charge.
A.2 Bulk Calculations 103
An example charge analysis calculation setup file for the Fe system is listed below.
% Beginning of file charge.txt1: &includefile = ./setup.txt2: calculation.type = charge% End of file charge.txt
The calculation is executed within Matlab by issuing the command:
matdcal charge.txt
Upon completion MATDCAL will create a file called charge.mat which stores the
(spin-resolved) charge on each atom. For example, the charge.mat file produced by
running the charge analysis tool on the Fe system is contains the following data:
>> load charge.mat>> atoms(1).charge
ans =
up: 4.9526down: 3.0474
>> atoms(1).charge.up - atoms(1).charge.down
ans =
1.9052
>>
From this analysis we see that each Fe atom has a net magnetic moment of 1.9µb.
A.2.8 Eigenstate Calculator
The eigenstate calculator is used to calculate a real-space projection of an eigenvector
wavefunction. This is useful for plotting contour and isosurfaces of wavefunctions.
The calculation setup file uses the following keywords:
104 A MATDCAL User Manual
Figure A.4: An isosurface of the Bloch-state corresponding to the 4th eigenstate at kx=ky=kz=0.
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For eigenstate calculations this parameter has the value eigenstate.
calculation.sindex An integer indicating which eigenstate (in order of lowest energy to high-
est) to calculate.
calculation.file.grid The output filename that stores that real-space projection of the eigenstate.
calculation.kx The kx value for which to calculate the eigenstate.
calculation.ky The ky value for which to calculate the eigenstate.
calculation.kz The kz value for which to calculate the eigenstate.
An example eigenstate calculation setup file for the Fe system is listed below.
% Beginning of file eigenstate.txt1: &includefile = ./setup.txt2: calculation.type = eigenstate3: calculation.sindex = 44: calculation.file.grid = ./eigenstate_4_plt.txt5: calculation.kx = 06: calculation.ky = 07: calculation.kz = 0% End of file eigenstate.txt
The calculation is executed within Matlab by issuing the command:
matdcal eigenstate.txt
Upon completion, MATDCAL will produce the output file specified in the calculation
setup file. This file is a formatted .plt file that can be converted and plotted within
GOpenMol using the technique described in the section on plotting isosurfaces. The
output for the Fe calculation is shown in Fig. A.4.
A.2 Bulk Calculations 105
A.2.9 Using a Custom Basis Set
MATDCAL has the ability to import basis sets and pseudopotentials from SIESTA.
All of SIESTA’s basis set and pseudopotential information is stored in a single .ion
file for each atom type. Open the file called convertIonFileToMatdcal by typing:
>> edit convertIonFileToMatdcal
Lines 3 and 4 of the utility contain the name of the .ion file to convert, and the
output directory for the MATDCAL basis files. Edit those lines accordingly. Run-
ning the script will take several seconds, and produce several files in the specified
output directory. These files contain the basis set and pseudopotential data used by
MATDCAL.
Next, a custom basis set definition file, called basisfile.txt must be created that
points to the basis set and pseudopotential data files. An example such file is shown
below:
% Beginning of file basisfile.txt1: atomtype = Cu2: 3: charge = 114: basis.S = ~/basis/cu/Cu_DZP_Zeta_S1.mat5: basis.P = ~/basis/cu/Cu_DZP_Zeta_P1.mat6: basis.D = ~/basis/cu/Cu_DZP_Zeta_D1.mat7: vnl.S = ~/basis/cu/Cu_DZP_Vnl_S.mat8: vnl.P = ~/basis/cu/Cu_DZP_Vnl_P.mat9: vnl.D = ~/basis/cu/Cu_DZP_Vnl_D.mat10: vna = ~/basis/cu/Cu_DZP_Vna.mat11: rna = ~/basis/cu/Cu_DZP_Rna.mat12: 13:% End of file basisfile.txt
The basisfile.txt file can contain multiple atom types, with a single empty line in
between each atom type. Note also that there must be a single empty line at the end
of the file.
106 A MATDCAL User Manual
Finally, basisfile.txt must be referenced in the file setup.txt, by replacing the cur-
rent system.basistype line in setup.txt with the following:
system.basistype = customsystem.basisfile = ~/basis/cu/basisfile.txt
A.2.10 Reading .plt Files in GOpenMol
MATDCAL can calculate real space density and scattering states information. This
data can be converted to a formatted .plt file, which can be imported and visualized
using a tool such as GOpenMol. Assuming GOpenMol has been correctly installed,
the instructions below explain how a formatted .plt file can be read into GopenMol.
1. In GOpenMol, go to file→input→coords and choose the file sys.xyz. You will
see the system appear in graphics window.
2. Go to run→Pltfile(conversion) and click on formatted→unformatted. In the
input file name point to the file output.txt just created. Choose output.plt as
the output file. Click apply. A file output.plt has been created.
3. Go to plot→contour, choose the file output.plt, and click import.
4. Choose an appropriate contour radius, and click apply. The density contour
will appear.
A.3 Two-Probe Calculations 107
Figure A.5: Schematic diagram of an Al/BDT/Al two-probe device
A.3 Two-Probe Calculations
A two-probe system is composed of three distinct regions: a semi-infinite left lead, a
central scattering region, and a semi-infinite right lead. The left/right lead is specified
by a unit cell that is repeated everywhere in space to the left/right of the central
region. The central region is specified by a unit cell that is repeated in the plane
perpendicular to the transport direction (i.e. the XY-plane). Periodicity in the XY-
plane enables the calculation of transport through devices composed of multi-layers
or molecular electronic devices with 3-dimensional leads. Devices with 1-dimensional
leads (i.e. wires with finite cross-section) can be calculated by including a vacuum
region and using a super cell approach.
MATDCAL is capable of calculating the following quantities for two-probe sys-
tems: self-consistent hamiltonian, density of states, transmission, and scattering
states. We first discuss how a system is setup and defined, then we explain how
to perform each of these calculations.
As a case study through this chapter, we consider a simple two-probe device con-
sisting of a BDT molecule in between two Al chains, as shown in Fig. A.5. The files
for this example can be found in /examples/twoprobe/Al-BDT-Al/.
A.3.1 Defining the leads
The left and right leads of the system are taken from bulk calculations, which are
used as boundary conditions for the self-consistent calculation of the central scattering
108 A MATDCAL User Manual
region. Setting up the leads is identical to setting up a bulk calculation. However,
there are important constraints when defining the unit cell of the leads:
• Both the left and right lead must have only single nearest neighbour along the
z-axis, i.e. the unit cell must be chosen large enough along the z-axis that there
is no interaction with the second nearest neighbour.
• The size of the unit cell along the x- and y- axis for the left and right lead must
be identical.
For the Al/BDT/Al example, the left and right leads are identical, consisting of
three Al atoms in the unit cell. The system definition file, calculation input file, and
coordinate definition file for this example are shown below.
% Beginning of file setup.txt1: system.type = BULK3D2: system.basistype = singlezeta3: system.atomfile = ./sys.xyz4: system.lx = 205: system.ly = 206: system.lz = 127: system.numx = 648: system.numy = 649: system.numz = 6410: system.numkz = 12811: system.numkx = 112: system.numky = 1% End of file setup.txt
% Beginning of file sc.txt1: &includefile = setup.txt2: calculation.type = selfconsistent% End of file sc.txt
% Beginning of file sys.xyz1: 32: AtomType X Y Z3: Al 10.000000 10.000000 2.0000004: Al 10.000000 10.000000 6.0000005: Al 10.000000 10.000000 10.000000% End of file sys.xyz
Starting MATDCAL with the input file sc.txt will start the self-consistent calculation.
A.3 Two-Probe Calculations 109
A.3.2 Defining the system unit cell
After the leads have been defined, the coordinate definition file for the central scat-
tering region must be defined. This file has the same file format as the coordinate
definition file for a bulk system. There are important points to consider when con-
struction the central scattering region:
• The central scattering region must be chosen large enough that there is no direct
interaction (overlap) between the left lead and the right lead.
• Enough layers of lead must be included in the central scattering region (the
so-called ”buffer” region”) such that the left and right boundaries of the central
region are equivalent to the bulk values.
A.3.3 Defining the system parameters
After the physical system has been defined and the system definition file has been
created, an input file with the system parameters must be created. This format for
this file is a series of entries with one entry per line having the form keyword = value.
A list of possible keywords and values for the two-probe geometry are given below.
system.type This keyword specifies the geometry of the system. Valid options for two-probe
systems are bulktwoprobe and bulktwoprobespin for spin calculations.
system.basistype Defines the type of basis set that is used in the calculation. Valid options are sin-
glezeta, doublezeta, and custom.
system.atomfile The filename of coordinate definition file specified in step 1.
system.lx Size of the unit-cell along the x-axis in units of bohr.
system.ly Size of the unit-cell along the y-axis in units of bohr.
system.lz Size of the central scattering region along the z-axis in units of bohr.
system.numx The number of grid points along x-axis. As a rule of thumb, this value should be at
least three times lx.
system.numy The number of grid points along y-axis. As a rule of thumb, this value should be at
least three times ly.
system.leftlead The location of the left lead setup file.
system.rightlead The location of the right lead setup file.
system.leftbias The voltage bias of the left lead in units of electron-hartrees
system.rightbias The voltage bias of the right lead in units of electron-hartrees
system.numbercontourpoints The number of integration points used for the contour integration during the calcu-
lation of the equilibrium density matrix. The default value is 40.
system.numberrealaxispoints The number of integration points used for the real-axis integration during the cal-
culation of the non-equilibrium density matrix. The default value is 40.
system.maxcachesize This parameter specifies the size of the real space handler cache size. Reducing this
value will conserve java memory, at the cost of increased computation time.
110 A MATDCAL User Manual
system.selfenergy.eta This value is the small imaginary component of the energy used in the calculation
of the self-energy. The default value is 1E-6, but may need to be increased to 1E-4
if there are convergence problems during the self-energy calculation.
An example system parameters file for the Al/BDT/Al system defined above is
given below.
% Beginning of file setup.txt1: system.type = BULKTWOPROBE2: system.basistype = singlezeta3: system.atomfile = ./sys.xyz4: system.lx = 205: system.ly = 206: system.lz = 38.7584747: system.numx = 648: system.numy = 649: system.numz = 16410: system.numkx = 111: system.numky = 112: system.symkx = 113: system.symky = 114: system.leftlead = ../lead/setup.txt15: system.rightlead = ../lead/setup.txt% End of file setup.txt
In this example both the left and right lead point to the same lead file, however
MATDCAL also supports using different leads in the two-probe configuration.
To verify that the system is setup correctly, MATDCAL has a useful tool to con-
struct the total system with the leads and central region. The command is issued
within Matlab in the folder with the two-probe setup file:
createTwoProbeXYZFile input.txt total.xyz
This command will create a coordinate definition file of the total system including the
leads. Visually verifying this system with a tool such as GOpenMol is good practice
to avoid errors before any calculation.
A.3 Two-Probe Calculations 111
A.3.4 Defining the calculation parameters
After the system parameters have been defined, a calculation setup file must be cre-
ated. This file specifies the specific type and calculation parameters. For the two-
probe geometry, the following types of calculations are supported: self-consistent
hamiltonian, density of states, transmission, and open-system eigenstates.
A.3.5 Self-Consistent Calculation
The self-consistent calculation is used to calculate a self-consistent density functional
theory electronic structure calculation. The file format is the same as the system
properties file. The calculation setup file uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For self-consistent calculations this parameters has the value selfconsis-
tent.
calculation.maxiterations Specifies the maximum number of self-consistent iterations.
calculation.rootfinder.type Specifies the mixing algorithm to be used. Valid options are broyden, lin-
ear, and pulay.
calculation.rootfinder.beta Defines the mixing parameter used in the mixing scheme. Typical values
are 0.1-0.2 for small non-magnetic systems, and 0.005-0.1 for large or
magnetic system.
calculation.mixing Specifies the quantity that is used in the mixing scheme. Valid options are
hamiltonian or density.
An example calculation file for a self-consistent calculation of the Al/BDT/Al
system defined above is listed below.
% Beginning of file sc.txt1: &includefile = setup.txt2: calculation.type = selfconsistent3: calculation.rootfinder.type = broyden4: calculation.rootfinder.beta = 0.005% End of file sc.txt
Beginning the calculation of MATDCAL with the file sc.txt will begin the self-
consistent calculation.
112 A MATDCAL User Manual
A.3.6 Density of States Calculation
The density of states calculator is used to calculate both the total and local density
of states in a bulk system. The calculation setup file uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For density of states calculations this parameter has the value dos.
calculation.energyvalues A Matlab vector defining the energy values in Hartree. This value is a
vector, not a scalar, and the correct syntax must be used. For example, to
calculate the DOS at the fermi-level, the syntax would be [0].
calculation.eta The value of the imaginary component of the energy used in the Green’s
function calculation of the density of states. The default value is 1E-6.
Increasing this value will smoothen the density of states.
The number of k-points used the density of states calculation is determined by the
k-points defined in the system definition file.
A.3.7 Transmission Calculation
The setup file for a transmission calculation uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For transmission calculations this parameter has the value transmission.
calculation.energyvalues A Matlab vector defining the energy values in Hartree. This value is a
vector, not a scalar, and the correct syntax must be used. For example, to
calculate the transmission at the fermi-level, the syntax would be [0].
calculation.outpufile The output filename that contains the calculated data.
An example scattering states inputfile for the Al/BDT/Al example is shown below.
% Beginning of file transmission.txt1: &includefile = ./setup.txt2: calculation.type = transmission3: calculation.energyvalues = ([-3:0.01:3]/27.2)4: system.selfenergy.eta = 1E-4% End of file transmission.txt
The transmission curve can be plotted by running script createTransmissionPlot.m,
included in the example directory. The transmission curves are plotted in Fig. A.6.
A.3 Two-Probe Calculations 113
A.3.8 Scattering States Calculation
Scattering states can be used to visualize eigenstates of the open system, and to
identify the origin of transmission peaks by projecting onto an isolated molecule. The
calculation setup file uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For scattering state calculations this parameter has the value scatter-
ingstate.
calculation.energyvalues A Matlab vector defining the energy values in Hartree. This value is a
vector, not a scalar, and the correct syntax must be used. For example, to
calculate the scattering states at the fermi-level, the syntax would be [0].
calculation.outpufile The output filename that contains the calculated orbital data.
calculation.gridfile MATDCAL calculates the real space projection of the first scattering state.
This parameter defines the output filename for this data.
An example scattering states inputfile for the Al/BDT/AL example is shown be-
low. This file performs a calculation at E = −0.28eV which corresponds to a very
broad transmission peak.
% Beginning of file ss.txt1: &includefile = ./setup.txt2: calculation.type = scatteringstate3: calculation.energy = [-0.28]/27.24: calculation.outputfile = ss_m0.28eV.mat5: calculation.gridfile = ssplt_m0.28eV.txt% End of file ss.txt
Upon completion of the calculation, MATDCAL produces the specified output
data file and grid file, which can be converted and plotted within GOpenMol using
the technique described above. The results from this calculation can be used to
determine which molecular eigenstates of BDT contribute to transmission peaks.
An example script called projectScatteringStates.m is included in the example di-
rectory. This script will project the a scattering state onto the isolated eigenstates of
a BDT molecule. Running this script produces the following output:
>> projectScatteringStates
114 A MATDCAL User Manual
Maximum projection on eigenstate: 21Projection value: 0.38
>>
This calculation indicates that the scattering state at E = −0.28eV is 38% com-
posed of the 21st eigenstate of the isolated BDT molecule.
A.3.9 Charge Analysis
The charge calculator calculates the total charge on each atom by calculating the
trace between the overlap matrix and the density matrix. The calculation setup file
uses the following keywords:
&includefile Specifies the system file. This command indicates to include to specified
file as if the commands in that file were entered directly.
calculation.type For charge analysis this parameter has the value charge.
An example charge analysis calculation setup file for the Al/BDT/Al example is
listed below.
% Beginning of file charge.txt1: &includefile = ./setup.txt2: calculation.type = charge% End of file charge.txt
The data produced by this calculation has the same format as the charge calcula-
tion for bulk systems.
A.3 Two-Probe Calculations 115
−3 −2 −1 0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
1.4
E−EF (eV)
Tra
nsm
issi
on
Figure A.6: T (E) for Al/BDT/Al device
116 A MATDCAL User Manual
A.4 Table of MATDCAL Parameters
The following tables list the possible parameters for the MATDCAL input files. Quan-
tities in square brackets in the description are the default values.
General parameters:
Keyword Values Description
&includefile filename another file so that options from other files can be
reused.
matdcal.parallel true, false Determines whether MATDCAL will operate in serial
or parallel mode. [false]
system.type bulk3d,
bulk3dspin,
bulktwoprobe,
bulktwoprobespin
Specifies the geometry of the system (bulk or
twoprobe) and whether the calculation is a spin cal-
culation.
system.basistype singlezeta, dou-
blezeta, custom
Defines the basis type. If custom is used then the
option system.basisfile must be specified.
system.basisfile filename The location of the basis definition file. This option
is only used when system.basistype is set to custom.
See below for the file format definition.
system.atomfile filename The location of the coordinates file for the system.
See below for the file format definition.
system.lx double Size of the unit-cell along the x-axis in units of bohr.
system.ly double Size of the unit-cell along the y-axis in units of bohr.
system.lz double Size of the unit-cell along the z-axis in units of bohr.
system.numx integer The number of grid points along x-axis. As a rule of
thumb, this value should be at least three times lx.
system.numy integer The number of grid points along y-axis. As a rule of
thumb, this value should be at least three times ly.
system.numz integer The number of grid points along z-axis. As a rule of
thumb, this value should be at least three times lz.
system.numbercontourpoints integer This parameter is for two-probe calculations only. It
specified the number of integration points along the
contour. [40]
system.numberrealaxispoints integer This parameter is for two-probe calculations only. It
specified the number of integration points along the
real-axis for non-equilibrium calculations. [40]
system.maxcachesize integer This parameter specifies the size of the real space han-
dler cache size. Reducing this value will conserve java
memory, at the cost of increased computation time.
[128]
system.selfenergy.eta double This parameter is for two-probe calculations only.
This value is the small imaginary component of the
energy used in the calculation of the self-energy. [1E-
6]
system.greensfunction.eta double This parameter is for two-probe calculations only.
This value is the small imaginary component of the
energy used in the calculation of the green’s function.
[1E-6]
calculation.type selfconsistent,
dos, transmission,
bandstructure,
scatteringstates,
charge
Defines the type of calculation that matdcal performs.
A.4 Table of MATDCAL Parameters 117
Parameters for self-consistent calculations:
Keyword Values Description
calculation.rootfinder.type linear, broyden,
pulay
Defines the mixing scheme to use for the self-
consistent calculation. [broyden]
calculation.rootfinder.beta double Defines the mixing parameter used in the mixing
scheme. Typical values are 0.1-0.2 for small non-
magnetic systems, and 0.005-0.1 for large or magnetic
system. [0.1]
calculation.maxiterations integer Sets the maximum number of iterations within the
self-consistent cycle. [inf]
calculation.mixing hamiltonian, den-
sity
Defines whether the hamiltonian or the density is used
in the mixing scheme. [hamiltonian]
Parameters for density of states calculations:
Keyword Values Description
calculation.energyvalues MATLAB array
definition
Defines the energy values in hartree for which the
DOS is calculated. The argument must be a valid
MATLAB command to define an array, for example [-
3:0.01:3]/27.2 would indicate a calculation of the DOS
for the range -3 eV to 3 eV in increments of 0.01 eV.
Parameters for transmission calculations:
Keyword Values Description
calculation.energyvalues MATLAB array
definition
Defines the energy values in hartree for which the
DOS is calculated. The argument must be a valid
MATLAB command to define an array, for example [-
3:0.01:3]/27.2 would indicate a calculation of the DOS
for the range -3 eV to 3 eV in increments of 0.01 eV.
Parameters for band structure calculations:
Keyword Values Description
calculation.primitivebasis.v1 MATLAB array
definition
Defines a primitive cell vector.
calculation.primitivebasis.v2 MATLAB array
definition
Defines a primitive cell vector.
calculation.primitivebasis.v3 MATLAB array
definition
Defines a primitive cell vector.
calculation.unitcellbasis.v1 MATLAB array
definition
Defines a unit cell vector.
calculation.unitcellbasis.v2 MATLAB array
definition
Defines a unit cell vector.
calculation.unitcellbasis.v3 MATLAB array
definition
Defines a unit cell vector.
calculation.unitcellbasis.v1 MATLAB array
definition
Defines a unit cell vector.
118 A MATDCAL User Manual
calculation.symmetrypoints MATLAB cell ar-
ray definition
Defines the symmetry points that will be calculated in
the band structure calculation. The argument must
be a valid MATLAB command to define a cell array,
for example ’G’ ’X’ ’X1’ ’G’. Valid symmetry points
are ’G’, ’X’, ’W’, ’L’, ’K’, ’N’, ’XZ’.
A.5 Table of MATDCAL Support Tools
The following tables list a number of useful MATDCAL tools that can be used to
parse data and manipulate system coordinate files. These functions can be found
using that matlab command which followed by the function name.
Command createTwoProbeXYZFile
Syntax createTwoProbeXYZFile inputsystemfile outputxyzfile
Description This script creates a .xyz file corresponding to the full two-probe system, includ-
ing the left, center, and right regions. This script is extremely useful to visually
verify the correct setup of a two-probe device.
Command orderXYZFile
Syntax orderXYZFile xyzfile
Description This script reorders the atoms in the specified .xyz file in increasing order of
the z-position of the system. Note that in order to achieve maximum benefit
from the O(N) matrix inversion scheme, this script should always be run on the
device prior to any calculation.
Command convertXYZtoAngstroms
Syntax convertXYZtoAngstroms inputxyzfile outputxyzfile
Description This script creates a .xyz file equivalent to the input .xyz file, with the atomic
positions converted to units of . This script is useful for visualizing systems in
tools that assume units of for the creation of bonds.
Command convertIonFileToMatdcal
Syntax convertIonFileToMatdcal
Description This script converts a .ion file generated as a SIESTA output into a series of
basis and pseudopotential files that can be read my MATDCAL. Lines 3 and 4
of this script must be edited to specify the .ion file and output directory.
Command convertBulkRealSpaceDensityPlt
Syntax convertBulkRealSpaceDensityPlt inputsystemfile outputpltfile
Description This script converts the calculated real space density file into a formatted .plt
filename that can be read by visualization tools such as GOpenMol. Refer to
section A.2.10 on how to read .plt files in GOpenMol.
Command XYZParser
Initialization parser = XYZParser
Methods readXYZFile, readPDBFile, writeXYZFile, writeAngstomFile, writeBohrFile
A.5 Table of MATDCAL Support Tools 119
Description This is a generic tool for manipulating .xyz files, for example to concatenate two
.xyz files, or to manipulate the position of particular atoms automatically. For
example the commands:
parser = XYZParser;
atoms = parser.readXYZFile(’sys.xyz’);
atoms.z = atoms.z + 1;
parser.writeXYZFile(’sys.xyz’, atoms);
would shift the z-coordinate of all the atoms in sys.xyz by 1 bohr. Note that
other methods are available to read/write using different file formats or units.
Command projectScatteringStates (contained with Al-BDT-Al example)
Syntax projectScatteringStates
Description This is an example script that demonstrates how to project a two-probe scat-
tering state onto individual eigenstates of a molecule in order to identify the
contributing molecular states. This script must be modified to label the basis
indexes of the molecule in the two-probe device.
Command plotBandStructure (contained with Fe example)
Syntax plotBandStructure
Description This is an example script that plots the band structure, once calculated.
B
MATDCAL Programmer’s Guide
B.1 Introduction
The programmer’s manual is designed to give the reader an overview of how MATD-
CAL is written, such that they may modify the software to extend its current func-
tionality. There are three distinct parts of the MATDCAL code: the object model
which is written in Java, the formulas and algorithms which are written in Matlab,
and the parallel toolbox written in Matlab and C. Each of these parts of the code are
explained in the next sections.
Figure B.1 illustrates the levels of the different parts of the code. Higher level code
is built on lower level modules, and therefore uses the code in lower levels. The main
dependencies are shown by the arrows in the figure. At the lowest level is the Java
object model, support function, and parallel computing toolbox. The next layer of
software is the DCAL classes which contain system-specific formulas for self-consistent
calculations. Next are the calculation classes, which calculate physical quantities of
interest from the self-consistent data, and save the data to file. At the highest level
are the analysis scripts which are often custom written by the user and are project
dependent, and produce publishable-quality plots from the saved data.
120
B.2 Java Object Model 121
Object Model
Calculator Classes
DCAL Classes
Support Functions
Analysis Scripts
Parallel Toolbox
Java
Matlab
CSaved Data
Hig
her L
evel
Figure B.1: A diagram illustrating levels of code within MATDCAL.
B.2 Java Object Model
MATDCAL can be viewed as having two main parts of the software. First, there
are the complex data structures that create a physical model of a device. Secondly,
there are the formulas and algorithms that act on these data structures to perform
computation. The data structures are implemented using an object oriented model
implemented in Java. The formulas and algorithms are implemented in Matlab. A
diagram showing the main classes of the Java object model and their relationship is
shown in Fig.B.2. The highest level object is the molecule which is used as a general
term for a collection of atoms. Each atom contains a pseudopotential and a list of
basis sets. Both the basis sets and the pseudopotentials are further decomposed into
a spherical harmonic representation.
An important concept in MATDCAL is the PairServer, which handles looping over
overlapping pairs of atoms and basis functions. The notion of basis pairs is important,
because for each overlapping basis pair, there is a corresponding Hamiltonian matrix
element. Note that due to the locality of basis functions, not all pairs of basis functions
overlap, leading to the sparse block tridiagonal discusses in chapter 3. The PairServer
provides an iterator, and abstract concept in object oriented programming, to navigate
through all basis pairs.
122 B MATDCAL Programmer’s Guide
Molecule
Atom 1 Atom i Atom N
Basis 1 …
Spherical Harmonic Decomposition
Pseudopotential
Spherical Harmonic Decomposition
… …
Basis i Basis N…
Figure B.2: A diagram illustrating the object model implemented in the Java portion of MATDCAL.
The code listing below demonstrates how the PairServer can be used to navigate
through all basis pairs in a system. This example can also be found in /matd-
cal/examples/twoprobe/carbonchain/system/iteratorexample.m.
% Beginning of file iteratorexample.m1: dcal = matdcal(’-nocalc’, ’sc.txt’);2:3: % initialize various parts of the dcal4: dcal = initializeLeads(dcal);5: dcal = cloneLeadMatrices(dcal);6: dcal = createPairServer(dcal);7: dcal = numberBasisObjects(dcal);8:9: % get the pairs structure10: pairs = get(dcal, ’pairs’);11:12: % navigate through all basis pairs in CC13: enumerator = pairs.CC.pairserver.getBasisPairsEnumeration;14: counter = 0;15: while enumerator.hasMoreElements16: counter = counter + 1;17: basispair = enumerator.nextElement;18: basis1 = basispair.getBasis1;19: basis2 = basispair.getBasis2;20: % line too long to display - see source code21: end% End of file iteratorexample.m
Lines 1-8 create the necessary data objects (see below), line 10 retrieves the pairserver,
and line 13 asks the pairserver for the basis pairs iterator. Lines 15 through 21 loop
B.3 Matlab Code 123
through all elements of the iterator and display the current basis functions in the pair.
The real space representation of the spin densities and potentials are also stored
as Java data structures. This is done so that all the grid-based computation can be
performed in Java, which was showed to have equivalent performance to either Fortran
or C by early benchmarks during code development. The grid-based calculations
are performed by the Matlab class RealSpaceHandler. This class is responsible for
iterating through basis pairs and projecting the spin densities onto the real space grid,
or performing the real-space integration to calculate the orbital representation of the
potential.
B.3 Matlab Code
The Matlab portion of the code is used to implement the physical formulas and
algorithms required for computation. The Matlab code can be broken down into
four groups: device calculator (DCAL) classes, factory classes, calculator classes, and
support functions and utilities. A diagram illustrating this breakdown is shown in
Fig.B.3. The DCAL class contains the physics models and formulas specific to a
particular system type. For example, the Bulk3DCAL class contains an algorithm
to calculate the density matrix for a crystal, and the BulkTwoProbeDCAL class
contains an algorithm to perform the contour integration of the Green’s function.
The calculator class implements the physical formulas and algorithms for analysis of
devices once the self-consistent Hamiltonian has been calculated. For each DCAL and
calculator class, there is a corresponding factory class which is responsible for setting
up and initializing the corresponding DCAL or calculator object.
124 B MATDCAL Programmer’s Guide
Matlab Code
DCAL Classes
Bulk3DCAL
BulkTwoProbeDCAL
Bulk3DSpinDCAL
BulkTwoProbeSpinDCAL
etc
Support Functions and Utilities
Factory Classes
Calculator Classes
Transmission
Scattering States
Eigenstates
Charge
Self-Consistent
etc
Figure B.3: A diagram illustrating the organization of the Matlab code in MATDCAL.
B.4 Parallel Computing Toolbox
An important and novel accomplishment was the development of a parallel computing
toolbox for Matlab. At the time of development, Matlab did not provide such a
capability.
Parallel computing was accomplished by creating a proxy function that makes
accessible to Matlab the functions in the MPICH2.0 library. This proxy was written
in C and compiled as a MEX-file. A number of Matlab function which provide a
variety of high-level communication functions were also created. An illustration of
the software layers within the parallel toolbox are shown in Fig.B.4.
As an example of how the parallel computing toolbox can be used is shown in the
code listing below. This short program adds the numbers 1 to 1000 in parallel.
% Beginning of file parallelAdd.m1: function parallelAdd2:3: N = 1000;4:5: MPI_Init;6:7: mpirank = MPI_Comm_rank;8: mpisize = MPI_Comm_size;
B.4 Parallel Computing Toolbox 125
Matlab Environment
Parallel Toolbox Functions (.m files)
MEX-file Proxy to MPI Library (c code)
MPICH2.0 Library
Figure B.4: A diagram illustrating how the MPICH2.0 library is made accessible to the Matlabenvironment.
9:10: localtotal = 0;11:12: for counter = 1:N13: % ismyjob is used to determine which work a14: % node should do based on its rank15: if ismyjob(counter, N, mpirank, mpisize)16: localtotal = localtotal + counter;17: end18: end19:20: % combine localtotals from all nodes21: total = MPI_Allreduce_sum(localtotal)22:23: MPI_Finalize;% End of file parallelAdd.m
Line 5 initializes the MPI environment. Line 15 determines whether a node should
perform this particular calculation based on the counter index, total number of jobs,
MPI rank, and MPI size. Line 21 combines the partial sums from all nodes.
The following table describes the MPI functions currently supported by the parallel
toolbox. Note that other MPICH2.0 functions can also be easily added by appending
the MEX-file MPI Proxy.c.
126 B MATDCAL Programmer’s Guide
Command MPI Init
Syntax MPI Init
Description This function should be one of the first lines in a parallel program. It initializes
the MPI environment.
Command MPI Finalize
Syntax MPI Finalize
Description This function should be one of the last lines in a parallel program. It finalizes
the MPI environment.
Command MPI Barrier
Syntax MPI Barrier
Description This function blocks execution of the program until all nodes reach this point in
the code. This function is useful for synchronizing and debugging code.
Command MPI Initialized
Syntax flag = MPI Initialized
Description This function returns a boolean indicating whether MPI Init has been called.
Command MPI Send variable
Syntax variable = MPI Send variable(variable, dest, tag)
Description This function sends a variable to the node with rank dest. The variable tag is
used to identify the variable with a label. The argument variable can be a real,
complex, or sparse matrix.
Command MPI Recv variable
Syntax variable = MPI Recv variable(source, tag)
Description This function receives a variable from node with rank source. The variable tag
is used to identify the variable with a label. The returned variable can be a real,
complex, or sparse matrix.
Command MPI Allreduce sum
Syntax total = MPI Allreduce sum(variable)
Description This function performs a global sum across all variables on all nodes. The argu-
ments variable can be a real, complex, or sparse matrix. Note that currently
the implementation of this function only works when the MPI size is a
power of two.
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