Theory of Non-equilibrium Vertex Correction Youqi Ke 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 Youqi Ke, 2010 Contents Abstract xi R´esum´e xiii Statement of Originality xv Acknowledgments xvii 1 Introduction 1 2 Density Functional Theory 10 2.1 Basic theorems . 10 2.2 Kohn-Sham equations . 14 2.3 Exchange-Correlation energy EXC .................... 17 2.4 Using KS-DFT as a mean field theory . 22 3 Tight-Binding Linear Muffin Tin Orbital Method 24 3.1 Solving the Kohn-Sham equations . 24 3.2 Muffin Tin approximation for crystals . 26 3.3 The bare envelop function . 28 3.4 Screened envelope function . 30 3.5 Augmentation to Energy Dependent MTO . 32 3.6 Energy linearization . 34 3.7 Atomic sphere approximation: Hamiltonian and overlap matrices . 36 3.8 Transformation to the nearly orthogonal basis . 38 3.9 Summary ................................. 40 iii iv Contents 4 The non-equilibrium Green's function formalism 41 4.1 Basic definitions . 41 4.2 Non-equilibrium Green's Function . 45 4.3 The self-energies Σ< and Σ . 51 4.4 Electric current . 52 4.5 Implementation with local orbital basis set . 55 4.6 The recursive Green's function technique . 62 4.7 Green's function method in TB-LMTO-ASA . 65 4.8 Summary ................................. 71 5 Theory of non-equilibrium vertex correction 72 5.1 Basic definitions . 74 5.2 Coherent potential approximation . 78 5.3 Non-equilibrium vertex correction . 83 α,Q α,<,Q 5.4 Conditionally averaged site-diagonal quantities: gRR and gRR . 89 5.5 Testing NVC . 92 5.6 Charge density . 94 5.7 Electronic potential . 97 5.8 Transmission coefficient . 99 5.9 Summary . 101 6 Effects of interface roughness in magnetic tunnel junctions 102 6.1 Calculation overview . 105 6.2 Perfect junctions . 106 6.3 Rough interface junctions . 110 6.4 Summary . 116 Contents v 7 Role of oxygen vacancy in Fe/MgO/Fe MTJ 117 7.1 Calculation overview . 119 7.2 Coherent tunneling in ideal junction . 121 7.3 Disorder effects of oxygen vacancy . 122 7.4 Summary . 131 8 Surface roughness scattering in Copper interconnects 132 8.1 Calculation overview . 134 8.2 Surface roughness scattering . 136 8.3 Searching for coating material . 141 8.4 Summary . 145 9 Conclusion 146 References 149 List of Figures 4.1 Schetch of a two probe device. Here, Hc, Hl and Hr describe the + central device region, the left and right electrodes respectively; τl, τl , + τr and τr correspond to the interactions between the electrodes and the central device region. The electrodes extend to z = ±∞. 46 4.2 Schetch of two probe device in principle layers. A portion of the elec- trode is included in the central device region for screening, the central device region contains N principle layers. 61 5.1 Diagrammatical representation of the average auxiliary NEGF of Eq.(5.51) 88 5.2 Results for spin-up (a) and spin-down (b) electron in the test structure at equilibrium plotted against the iterative steps in solving Eq.(5.54). σ Q,σ Black Square: the deviation ∆ΩNVC ; The deviation ∆gRR ( red circles for the Cu atom, green up-triangles for the Co atom). All quantities are iterated to zero as dictated by the fluctuation-dissipation theorem. 93 5.3 Flowchart of the NEGF-DFT-NVC self-consistent calculation. 98 5.4 Diagrammatical representation of the transmission coefficient of Eq.(5.84) 100 5.5 Schematic plot for illustrating specular transport (a) and diffusive transport due to random disorder scattering (b). 101 6.1 Schetch of magnetic configuration in magnetic tunnel junction . 103 6.2 Schematic of atomic structure of the Fe/VA/Fe magnetic tunnel junc- tion. The two Fe/VA interfaces have roughness disorder. Fe: yellow spheres; vacuum: white spheres. 106 6.3 Conductance versus thickness of the vacuum spacer for spin-up and " spin-down channels in PC and APC of ideal MTJs. Red circles: GPC ; # σ " # blue up-triangles: GPC ; black squares: GAP C . Note GAP C = GAP C for the symmetric junctions we calculated. Inset: TMR versus thickness of vacuum spacer for the perfect junctions. 107 vi List of Figures vii 6.4 kk resolved transmission coefficient T = T (EF ; kx; ky) in 2D BZ for the spin-up and spin-down channels in PC and APC of perfect junctions having 3,5,6 ML vacuum spacer, shown in logarithmic scale . (a)(d)(h): APC spin-up channel. The APC spin-down channel is the same as the spin-up channel; (b)(e)(i): PC spin-up channel; (c)(f)(j):PC spin-down channel. .................................. 109 6.5 (a) Spin polarized current versus bias voltage for spin-up and spin-down channels in PC and APC of a junction having 5 ML vacuum. Black " # # squares: IPC ; red circles: IPC ; up-triangles: IAP C ; down-triangles: " IAP C . (b) TMR value as a function of bias voltage. 110 6.6 (a) total conductance G";# versus disorder x at equilibrium. Red circles: # " # GPC ; black squares: GPC in PC. Blue down-triangles: GAP C in APC; " green up-triangles: GAP C in APC. (b) TMR value versus x. 111 6.7 Conductance versus disorder x at equilibrium. Black squares: specular part, the first term of Eq.(5.84). Red circles: vertex correction part, the second term of Eq.(5.84). Blue up-triangles: total conductance. PC PC AP C AP C (a): G" . (b): G# . (c): G" . (d): G# . 112 6.8 (a) Comparison of I-V curves with disorder x = 0:05. Solid lines (green): current for PC (up-triangles) and APC (down-triangles) with- out using NVC in the density matrix self-consistent iteration. Dashed lines (red): current for PC (circles) and APC (squares) using the full NVC formalism.(b) Bias induced electrostatic potential change versus atomic layer of the disordered junctions at Vb = 0:544V . Up-triangle: x=0.8; Red Circle: x=0.5; Black Square: x=0.2. the marked layers are the two interfacial disordered atomic layers. 113 6.9 Total current for PC and APC and TMR versus bias voltage for differ- ent value of x. (a)(b) for x=0.05; (c)(d) for x=0.3, and (e)(f) for x=0.5. in (a)(c)(e), black squares: total current for PC IPC ; red circles: total current for APC IAP C . 114 6.10 (a) Spin currents versus disorder x at bias Vb = 0:544V, for PC and APC. Red circles and black squares: spin currents for spin-up and - down in PC; green up-triangles and blue down-triangles: spin currents for spin-up and -down in APC. (b) TMR versus x at the same Vb. Inset of (b): TMR versus Vb for a device where left and right interfaces have different values of x, on the left interface x = 0:3, on the right x = 0:05. 115 7.1 Atomic structure of the Fe/MgO/Fe MTJ with 13 ML MgO. Blue sphere: Fe; red: O; Green: Mg; White: Oxygen Vacancy. the junction is periodically extended in the transverse x,y directions. the numbers label the MgO layers from left to right. 120 viii List of Figures 7.2 Conductance versus thickness of MgO barirer for spin up and spin " down channels in PC and APC for perfect MTJ. Black square: GPC ; # σ " # red circle: GPC ; blue up-triangle: GAP C . In APC, GAP C = GAP C for the symmetric perfect junctions. Inset: TMR versus thickness of MgO barrier. 122 7.3 kk resolved transmission coefficient T = T (EF ; kx; ky) in 2D BZ for the spin-up and -down channels in PC and APC of perfect junctions with 3 and 7 MLs thick MgO barrier, shown on logarithmic scale . (a)(d): PC spin-up channel, (b)(e): PC spin-down channel; (c)(f): APC spin-up and -down channels. Due to the symmetry in perfect junctions, the spin-down channel transmission is the same as that of spin-up channel in APC. 123 7.4 Conductance versus interfacial OV concentration x at equilibrium for spin-up (a)(c) and -down(b)(d) channels in PC (a)(b) and APC (c)(d). Layer-1 of the MgO is fixed with 3% OV; Layer-13 with x%. Blue up-triangle: total conductance; Red Circle: Vertex Correction; Black Square: Coherent part. 123 7.5 TMR versus disorder x for three types of MTJ. Red Square: for sym- metrical junctions with x% OV on both MgO layers of 1 and 13; Black Circles: for asymmetric junctions with 3% OV fixed on layer layer-1 and x% on layer 13; Blue star: same disorder distribution as the black circles but for a junction with 7 ML thick MgO barrier. 125 7.6 Conductance versus interfacial OV concentration x at equilibrium for spin-up (a)(c) and -down(b)(d) channels in PC (a)(b) and APC (c)(d). Layer-2 of the MgO is fixed with 3% OV; Layer-12 with x%. Blue up-triangle: total conductance; Red Circle: Vertex Correction; Black Square: Coherent part. 125 7.7 Conductance versus OV concentration x for spin-up and -down chan- nels in PC and single spin channel in APC. 3% OV is fixed on interfacial MgO layer of 1 and 13, x% OV is put on the layer-7. Black Square: " # σ GPC ; Red Circle: GPC ; Blue up-triangle: GAP C .