An efficient full-potential linearized augmented-plane-wave electronic structure method for charge and spin transport through realistic nanoferronic junctions [Elektronische Ressource] / Frank Freimuth

F.Freimuth

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2011
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 11 Mo

Extrait

An Efficient Full-Potential Linearized
Augmented-Plane-Wave Electronic Structure Method
for Charge and Spin Transport through realistic
Nanoferronic Junctions
Von der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaften
der Rheinisch-Westf¨alischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften
genehmigte Dissertation
vorgelegt von
Diplom-Physiker
Frank Freimuth
aus Ellern
Berichter: Universit¨atsprofessor Dr. S. Blu¨gel
Universit¨atsprofessor Dr. P. H. Dederichs
Tag der mu¨ndlichen Pru¨fung: 02.02.2011
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfu¨gbar.Abstract
Thefull-potentiallinearizedaugmented-plane-wave(FLAPW)electronicstructuremethod,
aprecise relativisticall-electronmethodenablingthedescriptionoftheelectronicstructure
and ground-state properties of multi-component complex bulk solids and low-dimensional
systems from the first principles of quantum mechanics, has been extended along two
research lines:
(A) An efficient order-N implementation of the FLAPW method was developed to per-
form ab initio calculations of electronic transport properties in magnetic tunnel junctions
(MTJs) and all-metallic spin-valves within the spin-density and vector spin-density formu-
lations of the local density approximation (LDA) and the generalized gradient approxima-
tion (GGA) to the density functional theory (DFT), whose computational effort scales not
3proportionallytotheconventionalcubicpower(N ),butlinearlywiththenumberoflayers
N of the system. The method is based on the Green-function embedding technique, which
allows to treat open systems and to solve the scattering problem of electronic transport.
In order to achieve the order-N scaling behavior, the system is partitioned into layers,
which are calculated separately. Due to its order-N scaling, the implementation allows to
profit from the high precision of the FLAPW method at a low computational cost. Com-
putational efficiency is an important aspect in calculations of electronic transport as the
scattering region often exceeds 2-5 nm in thickness and may contain several hundreds of
atoms per unit cell. The applicability of the order-N FLAPW electronic structure method
is much more general and allows the efficient investigation of quantities based on the self-
consistent solution of the electron charge and vector magnetization densities such as the
electronic structure, the total energy and other ground state properties. The method is
ideally suited to heterostructures and has been extended to treat surfaces and thin films.
Aparticularasset is thepossibility toinvestigate quantities slowly varyinginspace such as
a charge or spin-density wave. The implementation was validated for the electronic, mag-
netic and structural properties of films and surfaces as well as their electronic transport
properties.
The method was applied to the investigation of the spin-transfer torque (STT), an
alternative to the conventionally used Oersted field to switch the magnetization in spin-
valvesandtunneljunctions,whosestrengthscalesfavorablywiththeincreasingintegration
density of the actual devices. Within the order-N Green function embedding method, two
formulations of the spin torque were implemented: The spin torque is calculated (i) from
the variation of the spin current and (ii) directly as the torque, which the exchange fieldexerts on the non-equilibrium spin density. The second formulation allows the calculation
of the spin-torque in the presence of spin-orbit coupling. Calculations of the spin-torque
were performed for Co/Cu/Co and Fe/Ag/Fe spin-valves and for the Fe/MgO/Fe MTJ.
For the all-metallic spin-valves the asymmetries of the torque per current were determined
and found to be in good agreement with the theory of Slonczewski and in the case of the
Fe/Ag/Fespin-valvealsoingoodagreementwithexperiments. IncontrasttoSlonczewski’s
model,theout-of-planetorqueisavailablefromtheabinitiocalculationsofthespin-torque.
It is found to be negligible for thick free layers. In the case of the Fe/MgO/Fe MTJ good
quantitative agreement with experiments is found.
(B) For the description of many properties in condensed matter physics, the Wannier
function is superior over the Bloch function, which is the underlying concept of most elec-
tronic structure methods describing periodic solids, including the FLAPW method. The
Wannierfunctionapproachtoelectronicstructureprovidesareal-spacedescriptionofsolid-
state properties and enables for example a very intuitive picture on bonding properties. In
the context of electronic transport, Wannier functions have advantages in several respects:
(i) They may be used to set up realistic single particle or many-body model Hamiltonians
with parameters as determined from ab initiocalculations. (ii)They areintimately related
to the Berry phase, a quantity entering the modern theory of ferroelectric polarization,
orbital magnetism and the Hall conductivities. (iii) They provide an efficient basis set
of localized functions, which is optimal for the study of local correlation effects on elec-
tronic transport. Within the FLAPW formalism, maximally localized Wannier functions
were implemented. The implementation was validated for bulk, films and one-dimensional
systems, with and without spin-orbit coupling. The ferroelectric polarization was com-
puted from the Wannier functions for several ferroelectric and multiferroic materials (e.g.
HoMnO ) and found to be in good agreement with experimental data, where available.3Contents
1 Introduction 1
2 Density Functional Theory and Beyond 9
2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 The Hohenberg-Kohn Theorems . . . . . . . . . . . . . . . . . . . . 10
2.1.2 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Spin Density Functional Theory . . . . . . . . . . . . . . . . . . . . 12
2.1.4 The Local Spin Density Approximation . . . . . . . . . . . . . . . . 13
2.1.5 The full-potential linearized augmented-plane-wave method . . . . . 13
2.1.6 The APW+lo method . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.7 Non-Collinear Magnetism . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Dynamical Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Perturbation Theory in Infinite Dimensions . . . . . . . . . . . . . 19
2.2.2 Mapping onto the Anderson Impurity Model . . . . . . . . . . . . . 20
2.2.3 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Ballistic Transport and Beyond 27
3.1 Ballistic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 The Conductance Quantum . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Landauer Formulation of Ballistic Transport . . . . . . . . . . . . . 29
3.1.3 Calculation of electronic transport within ground-state DFT . . . . 30
3.2 NEGF Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 The Meir-Wingreen formula . . . . . . . . . . . . . . . . . . . . . . 33
4 The embedding method 35
4.1 Derivation of the Embedding Method . . . . . . . . . . . . . . . . . . . . . 37
4.1.1 The embedding potential . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.2 Variational principle for the embedded region . . . . . . . . . . . . 40
4.2 Embedding within the FLAPW method . . . . . . . . . . . . . . . . . . . 43
4.3 The surface projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Scattering States and Conductance . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Transfer Matrix and Embedding Potential . . . . . . . . . . . . . . . . . . 47
4.6 The embedding-based order-N concept . . . . . . . . . . . . . . . . . . . . 49
iii Contents
4.7 The surface projector for a curvy surface . . . . . . . . . . . . . . . . . . . 51
4.8 An alternative expression for the curvy surface projector . . . . . . . . . . 54
4.9 Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.10 Transformation of the Potential and the Charge Density . . . . . . . . . . 57
4.11 Computation of Surfaces within the embedding method . . . . . . . . . . . 59
4.12 Flow chart: Non-Self-Consistent Embedding . . . . . . . . . . . . . . . . . 59
5 Self-Consistent Embedding 61
5.1 Generation of the charge density . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Construction of the Coulomb Potential . . . . . . . . . . . . . . . . . . . . 63
5.2.1 The Pseudocharge Method . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.2 Construction of the Coulomb Potential in the Interstitial . . . . . . 64
5.2.3 Construction of the Coulomb potential (Surface Calculations) . . . 66
5.3 Self-Consistency scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Evaluation of total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5 Evaluation of atomic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5.1 Atomic Forces within standard FLAPW . . . . . . . . . . . . . . . 68
5.5.2 Atomic Forces within the Embedding Method . . . . . . . . . . . . 71
6 Efficient Embedding 73
6.1 Solution of a linear system of equations . . . . . . . . . . . . . . . . . . . 73
6.2 Spectral representation of the Green function. . . . . . . . .