An embedding green function approach for electron transport through interfaces [Elektronische Ressource] / vorgelegt von Daniel Wortmann

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• Physik

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• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2003
• Nombre de lectures: 4
• Langue: English
• Poids de l'ouvrage: 3 Mo

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An Embedding Green Function Approach for
Electron Transport through Interfaces
Von der Fakultät für Mathematik, Informatik und
Naturwissenschaften der Rheinisch-Westfälischen Technischen
Hochschule Aachen zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom Physiker
Daniel Wortmann
aus Münster
Berichter:
Universitätsprofessor Dr. Stefan Blügel
Universitätsprofessor Dr. Peter H. Dederichs
Tag der mündlichen Prüfung: 04.07.2003
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.An Embedded Green Function Approach for Electron
Transport through Interfaces
The theoretical description of electron transport properties through nanoscale sys-
tems is one of the major challenges of contemporary solid state physics. Emerging
new fields such as magnetoelectronics, spin-electronics or molecular electronics, ar-
eas with the potential to extend or replace the present microelectronics, are fueled
by recent experimental successes in building such nanoelectronic systems.
In this thesis a new method is established, based on the combination of the embed-
ding Green function method and the full-potential linearized augmented plane-wave
(FLAPW) method, to describe the coherent and the sequential electron transport.
By the use of the density functional theory realistic systems can be described on
the atomic scale. The chosen numerical scheme, the FLAPW method, is today the
most reliable and exact method available for first principle electronic structure cal-
culations. However, different to standard bulk setups, the description of electron
transport requires the treatment of the scattering problem which is particularly
tricky when applying this method. Thus, a key part of the present thesis describes
thedevelopment ofanewcomputationalscheme which isabletodealwithascatter-
ing region sandwiched between semi-infinite leads. Based on the ideas put forward
by J. Inglesfield the existing FLEUR code is modified to calculate the single-electron
Green function for the embedded scattering region. The semi-infinite leads are de-
scribed in terms of a transfer-matrix formalism which enables one to obtain the
so-called complex bandstructure of bulk materials
The electron transport is described using either the Landauer model or Bardeen’s
formalism of tunneling. These two formulas are discussed as two different limits
of single-particle transport and their reformulation in terms of quantities readily
available from the embedding method is presented.
Besides the presentation of the theory and the details of its implementation the
thesis describes how the method has been applied to several different test systems
to validate the implementation. The spin-dependent transport properties of the
Fe/MgO/Fe tunneljunction, the model system of tunnel-magnetoresistance (TMR),
were investigated. It is shown that the details of the Fe/MgO interface in this
junction is of crucial importance for the tunneling conductance. While the pure
relaxationofaFe/MgOinterfacealreadychangestheconductance,evenmoredrastic
modifications are found as soon as one FeO layer is inserted or if the interface is
modified by interchanging the Mg and O atoms.Contents
1 Introduction 1
2 Density Functional Theory 7
2.1 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . . . . . 8
2.2 Single Particle Formulation . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The Exchange-Correlation Potential . . . . . . . . . . . . . . . . . . . 12
3 Concepts of Electronic Transport 15
3.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The Landauer Approach . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Interpretation of the Landauer Formula . . . . . . . . . . . . . . . . . 21
3.4 The Bardeen Approach to Tunneling . . . . . . . . . . . . . . . . . . 22
3.5 Experimental Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5.1 Scanning Tunneling Microscope . . . . . . . . . . . . . . . . . 27
3.5.2 The Junction-Magneto-Resistance . . . . . . . . . . . . . . . . 29
4 Green Function Embedding 33
4.1 The Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 The Schrödinger Equation in the Embedding Region . . . . . . . . . 37
4.4 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 The Embedding Potential . . . . . . . . . . . . . . . . . . . . . . . . 40
4.6 The Embedded Green Function . . . . . . . . . . . . . . . . . . . . . 41
4.7 Self-consistent Embedding . . . . . . . . . . . . . . . . . . . . . . . . 43
iContents
5 Complex Bandstructure, Embedding and Transport 45
5.1 The Transfer-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 T-Matrix and Complex Bandstructure . . . . . . . . . . . . . . . . . 49
5.3 Complex Bandstructure and Embedding Potential . . . . . . . . . . . 51
5.4 Wavefunction-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 Landauer Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Green Function formulation of Bardeen’s Approach to Tunneling . . . 61
6 The FLAPW-Method 63
6.1 The FLAPW Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 The Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Two regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 Auxiliary volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.5 Construction of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . 74
6.6 Calculation of the Green Function . . . . . . . . . . . . . . . . . . . . 75
6.7 The Transfer-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.8 Embedding Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.9 Construction of the Charge Density . . . . . . . . . . . . . . . . . . . 80
6.9.1 Interstitial region . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.9.2 Muffin-tin spheres . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.10 Potential Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7 First results 87
7.1 Some Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Complex Bandstructures . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Basis set in CBS calculations . . . . . . . . . . . . . . . . . . . . . . 95
7.4 Transmission from the Green Function . . . . . . . . . . . . . . . . . 98
7.5 Landauer conductance versus Bardeen’s tunneling . . . . . . . . . . . 102
iiContents
8 The Fe/MgO/Fe Tunneljunction 107
8.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3 Ferromagnetic Fe/MgO/Fe junction . . . . . . . . . . . . . . . . . . . 113
8.4 Bardeen’s approach applied to the Fe/MgO. . . . . . . . . . . . . . . 117
8.5 LDA+U for MgO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.6 Additional Oxide Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.7 Artificial Interface Setup . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.8 Antiferromagnetic Alignment, TMR . . . . . . . . . . . . . . . . . . . 127
8.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
9 Summary and Outlook 129
Bibliography 132
iiiContents
ivChapter 1
Introduction
Increasingly smaller and faster semiconductor circuitry has fueled over the past
decades the information technology, producing ever cheaper and faster computing
devices thathavevirtuallyenteredeverycorner ofourmodernlifeandisturningour
societies into a global information society. This was made possible since semicon-
ductor electronics have seen a sustained exponential decrease in size and a similar
increase in the level of integration over the past 40 years. Known as Moore’s Law,
since the mid seventies, the number of transistors on a semiconductor chip doubled
every 18 months. The characteristic feature size will soon fall below the 100 nm
mark and will continue to shrink to approximately 20 nm by the year 2015. To all
what is known today, the 20 nm structure size provides a barrier for conventional
electronics, beyond which a straightforward down-scaling of the feature size keeping
theoperationprinciples oftoday’selectronics will notwork. Characteristic phenom-
ena for structures of a few nanometers in size are the very few charge carriers, the
relevance of single dopant atoms, their non-uniformity, the relevance of single atom
defects, the large surface-to-volume ratio, and the high electric fields across small
structures. In addition, quantum phenomena will increasingly start to modify and
dominate the overall behavior of such devices.
At this frontier nanoelectronics is emerging as a new field geared to continuously
alter or replace the present microelectronics. Nanoelectronics research encompasses
e.g. magnetoelectronics, spin-electronics or spintronics, and molecular electronics or
molectronics – areas of great potential for future technology. Recent experimental
successes like the discovery of the giant magneto resistance effect in magnetic mul-<