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
Contents
Abstract x
R´esum´e 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 AboutthisManual ...... 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 MatlabCode ...... 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-probe device...... 24
3.1 Geometry optimization of an Fe/MgO/Fe device ...... 28
3.2 Contour integration of density matrix ...... 33
3.3 K-pointintegration ...... 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 accessible to 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 i