atomistix toolkit tutorial and reference guide

ATOMISTIX TOOLKIT

Tutorial and Reference Guide

Version 2.0

Atomistix A/SJuliane Maries Vej 30DK–2100 CopenhagenDenmarkTel: +45 3532 0630Fax: +45 3532 0635Email: [email protected]

Atomistixr and Virtual NanoLabr are registered trademarks of Atomistix A/S.Atomistix ToolKitTM is a trademark of Atomistix A/S.

Copyright c 2003–2006 Atomistix A/S.All rights reserved.

Revision date: April 24, 2006 (2.0.3)

mailto:[email protected]

Preface

The strength of the Atomistix ToolKit (ATK) software lies in its flexibility to describe sys-tems with different symmetries. It can describe isolated systems (molecules), periodicsystems (crystals), and systems of the type bulk–nanodevice–bulk (two-probe systems).In this tutorial, we will first start with a quick tour, which outlines the capabilities ofthe software, and then focus on each of the different symmetries, with emphasis on thetwo-probe geometry.

The ATK software uses a novel design which eases the setup of two-probe calculations.However, these types of calculations add complexity to the electronic structure calcu-lations, and a range of input parameters are required. The purpose of this document isto give the user a hands-on introduction to the ATK package, and to provide a guideto the various commands and input parameters. A detailed summary of all input andoutput parameters and available commands can be found in the appendices.

The background material, describing the underlying physics and algorithms, is de-scribed in a number of research papers listed in the References section (page 125)of the manual. The electron transport part is mainly described in Refs. 1, 11, 13. In thistutorial we will not delve deep into the physics details, but focus on describing how touse the program.

The proper citation to ATK includes a reference to the most important methodologypapers. We recommend users to cite:

� ATK version 2.0, Atomistix A/S (www.atomistix.com).

� M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys. Rev. B 65,165401 (2002)

� J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, andD. Sánchez-Portal, J. Phys. Condens. Matter 14, 2745 (2002).

� J. Taylor, H. Guo, and J. Wang, Phys. Rev. B 63, 245407 (2001).

Remember to visit our website http://www.atomistix.com for updates to this manualand the software!

Atomistix, April 2006.

http://www.atomistix.com

Contents

1 Introduction 1

1.1 About This Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Quick Tour 5

2.1 Molecular Orbitals of an H2 Molecule . . . . . . . . . . . . . . . . . . . 6

2.2 Band Structure of a Lithium Chain . . . . . . . . . . . . . . . . . . . . . 7

2.3 Setting up the Two-Probe System . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Electrode Calculation . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Determining the Li–H Bond Length . . . . . . . . . . . . . . . . 12

2.3.3 Placing H2 Between the Lithium Chains . . . . . . . . . . . . . . 13

2.3.4 Self-Consistent Calculation at Zero Bias . . . . . . . . . . . . . . 14

2.4 Analyzing the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Transmission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 MPSH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Current–Voltage Characteristics . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Voltage Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Relaxation of a Two-Probe System . . . . . . . . . . . . . . . . . . . . . 19

2.8 Including a Gate Electrode . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Molecular Systems 23

3.1 Water Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 The Self-Consistent Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 24

iv Contents

3.2.1 How To Cure Convergence Problems . . . . . . . . . . . . . . . 25

3.3 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Changing the Accuracy of the Calculation . . . . . . . . . . . . . . . . . 27

3.5 Molecular Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Spin-Polarized Calculations . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Periodic Systems 35

4.1 Band Structure of a Si Crystal . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Convergence of an Al Wire . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Two-Probe Systems 41

5.1 Geometry of an Ideal Lithium BCC Lattice . . . . . . . . . . . . . . . . 41

5.1.1 Lithium Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.2 Two-Probe Geometry . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.3 Miswrapped Electrode Atoms . . . . . . . . . . . . . . . . . . . 45

5.1.4 Transmission Spectrum . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.5 Energy Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Benzene Ring Coupled to Lithium Electrodes . . . . . . . . . . . . . . . 46

5.2.1 Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.2 Central Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2.3 Finite Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Heterogeneous Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Spin-Polarization and k-point Sampling of Two-Probe Systems 59

6.1 Transmission Through a Fe–MgO–Fe Interface . . . . . . . . . . . . . . 59

6.1.1 Fe Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.2 Fe-MgO-Fe Geometry . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.3 Transmission Spectrum . . . . . . . . . . . . . . . . . . . . . . . 62

6.2 k-point Sampling in Two-Probe Systems . . . . . . . . . . . . . . . . . . 62

6.3 k-point Sampling of the Transmission Spectrum . . . . . . . . . . . . . . 65

6.3.1 Plotting the k-dependent Transmission . . . . . . . . . . . . . . 65

Contents v

Appendix A ATK vs. TranSIESTA-C 69

A.1 Changes to Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1.1 Keywords No Longer Available . . . . . . . . . . . . . . . . . . 71

A.1.2 Keywords With New Names . . . . . . . . . . . . . . . . . . . . 72

A.1.3 Keywords With New Names and Syntax . . . . . . . . . . . . . 72

Appendix B Basis Sets 75

B.1 Basis Set Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.1.1 Single Zeta Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.1.2 Double Zeta Orbitals . . . . . . . . . . . . . . . . . . . . . . . . 76

B.1.3 Polarization Orbitals . . . . . . . . . . . . . . . . . . . . . . . . 76

B.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Appendix C Command Line Options 79

C.1 Input and Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.2 Environment Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Appendix D Parameters and Keywords 83

D.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

D.1.1 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

D.1.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

D.2 Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D.2.1 Analysis:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

D.2.2 AtomList:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D.2.3 Bulk:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

D.2.4 Electrode:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D.2.5 Method:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

D.2.6 Molecule:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D.2.7 NumOrb:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D.2.8 Relaxation:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

D.2.9 SCF:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

vi Contents

D.2.10 Simulation:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.2.11 System:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.2.12 TwoProbe:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.2.13 UnitCell:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Appendix E Output file 105

E.1 Header and self-consistent cycle . . . . . . . . . . . . . . . . . . . . . . 105

E.2 Total energy and atom list . . . . . . . . . . . . . . . . . . . . . . . . . 107

E.3 Mulliken population . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

E.4 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

E.4.1 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 111

E.4.2 Transmission spectrum . . . . . . . . . . . . . . . . . . . . . . . 111

Appendix F Writing Scripts 113

F.1 Bash Scripts for Linux and Cygwin . . . . . . . . . . . . . . . . . . . . . 113

F.2 Batch Files for Windows . . . . . . . . . . . . . . . . . . . . . . . . . . 115

F.3 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Appendix G Atomic Data 121

References 125

Index 127

C H A P T E R

1INTRODUCTION

ATOMISTIX TOOLKIT is a library of atomic scale modeling techniques that can be used tocalculate a wide range of properties of nanoscale systems. New methods are constantlybeing added to the toolkit, and for each new release ATK will be able to calculate moreproperties using a wider range of different modeling techniques. The most uniquefeatures is the ability to calculate the electrical properties of nanoscale devices, whichconsist of a scattering region coupled to two macroscopic bulk systems or electrodes.

Atomistix provides two different interfaces for performing electronic transport calcula-tions: VIRTUAL NANOLAB (VNL), which is a graphical user interface (GUI), and ATOM-ISTIX TOOLKIT (ATK), described in this manual, which is a file-based or command lineinterface. Which of the interfaces to use, is sometimes a matter of taste, and sometimesdetermined by the task at hand. Most users will probably find that VNL is the most ap-pealing option, because of its intuitive ease of use. Some prefer ATK because it is moretransparent, gives more control, and calculations can be automated by using scripts.Moreover, ATK provides immediate access to the newest functionality, whereas VNLinstruments for accessing these new features are often developed in a later step.

A modeling study generally consists of three parts: setting up the calculation, runningthe calculation and finally analyzing the output of the calculation. The purpose of thistutorial is to help the user getting control of these steps using ATK, often in combinationwith VNL.

When setting up a calculation, the first thing to consider is the symmetry of the system.ATK provides methods for describing three different system types; molecular, bulk, andtwo-probe systems. A two-probe system consists of a nanoscale region coupled to twomacroscopic bulk systems, or electrodes. The nanoscale region can be a molecule,nanotube, cluster of atoms, a piece of a semiconductor or an interface between theelectrodes. In this tutorial, we will first discuss the modeling of molecular and periodic

1

2 Introduction

Figure 1.1: ATK is capable of modeling isolated, periodic, and open systems using both veryaccurate first principle modeling methods and very fast semi-empirical methods.

systems before introducing the more complex two-probe systems.

Once the system to be studied is defined, the calculational methodology must be spec-ified. The description of the electronic structure in ATK is based on density functionaltheory (DFT) [5] and an important ingredient in the calculations is the approximationused for the exchange–correlation functionals.

In the current version of ATK, the local density approximation (LDA) [8] and the gener-alized gradient approximation (GGA), in the form of the Perdew–Burke–Ernzerhof (PBE)exchange–correlation functional [7], are implemented. The analogue spin-dependentexchange–correlation functional are implemented as well, which make it possible toperform spin-polarized calculations. For most applications, the PBE functional will bethe more accurate option.

Other sources of errors arise from the use of pseudopotentials to describe the effectof the core electrons, the finite size of the basis set and the numerical integrationroutines. These errors can in most cases be systematically reduced at the expenseof the computational time. The default parameters for ATK are chosen such that theapproximation for the exchange–correlation potential usually is the largest source oferror. In this tutorial, we will describe the other sources of errors, and explain howthe default parameters can be changed in order to reduce the computational cost orincrease the accuracy.

1.1 About This Tutorial 3

Figure 1.2: A typical two-probe system, consisting of a carbon nanotube between two metalsurfaces. The unique capability of ATK is its ability to study the electron transport through sucha system.

Finally, the output of the calculation must be analyzed to gain new insight from thestreams of data produced by the program. ATK can be used to calculate many differentproperties of the system, including electronic current, voltage drop, transmission coef-ficients, electron density, etc. The results can then be exported to standard file formatsso that they can be visualized in two or three dimensions, e.g. in VIRTUAL NANOLAB.

1.1 ABOUT THIS TUTORIAL

The first part of this tutorial is a quick tour, outlining the capabilities of ATK. In thefollowing three sections, we will go into more details about the available commandsand parameters relevant for different system symmetries. If you are new to ATK, it isrecommended to read the tutorial in its entirety, since many basic concepts will beintroduced and explained in the quick tour and then used in the subsequent sections.

We shall assume that the software has been properly installed and the environmentvariables set (including the path) as described in the Installation Guide, supplied withthe program. The Installation Guide is also available from Atomistix’ web site.

We recommend that you follow the tutorial, type the input file and enter the commandsmanually to gain experience with the steps needed for an ATK calculation. It is alsorecommended to create a new directory where you will run the examples. However, forreference, all input files and scripts used in this tutorial are located in the subdirectoriesof the directory examples in the program installation directory.

In this document, we symbolize the command prompt by a dollar sign $. Thus, state-ments like

$ atk h2.atk h2.out

are commands to be entered at the command prompt. In Linux and similar envi-ronments a command can be broken into multiple lines by using the backslash (’\’)character, cf. the box on page 114. In Windows this is not possible, and it is necessaryto type the entire command line on a single line, remembering to omit the backslash.

4 Introduction

For more detailed information about command line and input file parameters, pleaseconsult the reference sections, Appendices C and D.

Note that commands in the input file are not case sensitive. The capitalizations usedthroughout this manual are only meant to ease the reading.

Note! When editing files, be sure to save them as plain text files,using ANSI or ASCII encoding.

C H A P T E R

2QUICK TOUR

In this first part of the tutorial, you will be guided through a few examples that illustratethe general features and capabilities of ATK. The focus will be on physical propertiesthat you can study with ATK, rather than the particulars of all commands used to cal-culate them. Nevertheless, we shall try to explain the role and influence of most of theparameters used in the calculations when they appear.

The aim of the calculations is to study the conductance of a two-probe system con-sisting of a hydrogen molecule coupled with two lithium chains, as illustrated in Fig-ure 2.1. Our motivation for considering this structure is a recent measurement of theconductance of a single hydrogen molecule [10]. In the experiment, the molecule wassituated between two platinum wires, but for understanding the system, it is possibleto use lithium chains instead, which reduces the computational requirements substan-tially. We will investigate, why this system has a nearly perfect conductance despitethe huge hydrogen HOMO–LUMO gap.

To get experience it is recommended to type in the input file for these calculations your-self, but they can also be found in the directory examples/quicktour in the programinstallation directory.

Figure 2.1: A hydrogen molecule between two semi-infinite one-dimensional lithium chains.

5

6 Quick Tour

2.1 MOLECULAR ORBITALS OF AN H2 MOLECULE

Create a text file h2.atk with the following content

System::Type Molecule

Simulation::Type Relaxation

AtomList::Format Angstrom

%block Molecule::AtomList

H 0.0 0.0 -0.35

H 0.0 0.0 0.35

%endblock Molecule::AtomList

and execute the command

$ atk h2.atk h2.out

Note that no output is displayed on the screen while the program is running (exceptfor two lines indicating that the program started and finished), since we have specifieda second file, h2.out, to which all output is redirected. If this second parameter isomitted, the corresponding output is instead displayed on the screen. Hence, you mayalternatively redirect or pipe the output to another file, system device, or program byuse of the standard operators ’>’ and ’|’.

The first line of the input file tells ATK that we wish to study a molecular system and thesecond line instructs the program to calculate the equilibrium positions of the hydrogenatoms (i.e. to relax the geometry). This is achieved through a number of self-consistenttotal energy and force calculations where the position of the hydrogen atoms are movedalong the force directions. The initial guess for the distance between the atoms is setto 0.7 Å, as seen from the Molecule::AtomList block. The coordinates are given inÅngström (10–10 m), as indicated by the keyword AtomList::Format.

Note! If you mistyped any of the parameters in the input file,the program will most likely fail to run, and will produce an er-ror message specifying the offending input. Correct the typos andtry again. Note that by default ATK will not overwrite old outputfiles, and will refuse to run unless you either delete the old outputfiles or specifically tell it to overwrite them by using the --force

option.

Two output files are generated from the calculation:

h2.out Contains a log of the calculation and important results such as total energiescan be extracted from this file.

2.2 Band Structure of a Lithium Chain 7

h2.nc NETCDF data file with the final state of the calculation and data for plottingwith e.g. Atomistix’ visualization program VIRTUAL NANOLAB (see below). Thefile contains a complete description of the final state of the calculation and canthus be used to restart a calculation or to generate further analysis data. For moreinformation about NETCDF files, see Appendix C.1.

We can now find the relaxed distance between the two hydrogen atoms by looking forthe word "Positions" in the output file h2.out. The obtained result is approximately0.77 Å, which is in acceptable agreement with the experimental value of 0.741 Å, inparticular in view of the fact that the hydrogen molecule is the smallest molecule of all,and therefore poorly described by the local density approximation (LDA), which is thedefault exchange–correlation functional in ATK.

ATK can also be used to extract further results from the calculation by specifying ad-ditional analysis options. Keywords starting with Analysis:: do not affect the actualelectronic structure calculation, but only produce analysis output that is stored in aNETCDF file. This information can then be visualized in VIRTUAL NANOLAB.

If we, for example, wish to inspect the orbitals of the HOMO and LUMO levels, wecreate an input file h2_analysis.atk

Analysis::MolecularOrbital::A::Index 1

Analysis::MolecularOrbital::A::Label Lumo

Analysis::MolecularOrbital::B::Index 0

Analysis::MolecularOrbital::B::Label Homo

and run ATK with the command

$ atk -a h2.nc h2_analysis.atk h2_analysis.out

The option -a tells the program to read the data file h2.nc where the results of the elec-tronic structure calculation are stored, and use the data from this file for the analysis.

The molecular orbitals are written to the file h2_analysis.nc, and this file can now beimported into VIRTUAL NANOLAB to plot the eigenfunctions. Such a plot is illustrated inFigure 2.2. Note that the electron density and electrostatic potential are always writtento the NETCDF file by default.

2.2 BAND STRUCTURE OF A LITHIUM CHAIN

In the previous section, we looked at the first ingredient we need for our final two-probe system, namely the hydrogen molecule that will form the central region. Nowwe shall consider the leads, which in our example are semi-infinite one-dimensionalchains of lithium atoms.

8 Quick Tour

Figure 2.2: An isosurface of the LUMO orbital of the hydrogen molecule, created using Atomistixvisualization program VIRTUAL NANOLAB.

The following input file (lichain.atk) defines a lithium chain.

System::Type Bulk

NumOrb::MeshCutoff 100 Ry

NumOrb::BasisSet::Size SZ

Bulk::NumKPoints::C 200

UnitCell::LatticeConstant 3.0 Angstrom

%block Bulk::UnitCell

3. 0. 0.

0. 3. 0.

0. 0. 1.

%endblock Bulk::UnitCell


%block Bulk::AtomList

Li 0.0 0.0 0.0

%endblock Bulk::AtomList

Analysis::BandLine::L::Start (0., 0., 0.0) FracRecip

Analysis::BandLine::L::End (0., 0., 0.5) FracRecip

Analysis::BandLine::L::NumPoints 1000

In this case, we have a bulk system as indicated by the first line in the input file. Thesingle Li atom is repeated periodically in three dimensions according to the lattice vec-tors defined by the unit cell. In ATK, the three directions defined by the lattice vectorsare referred to as the A, B and C directions, respectively. This notation is reflectedin a number of keywords, such as Bulk::NumKPoints::C used above to specify thenumber of points that should be used in the direction of the reciprocal lattice vectorcorresponding to the C direction.

In the input file above, the lattice vectors defining the unit cell are given in units ofthe lattice constant, defined by UnitCell::LatticeConstant and thus, the lithium

2.2 Band Structure of a Lithium Chain 9

−10.106

−10.105

−10.104

−10.103

−10.102

2.82 2.84 2.86 2.88 2.9 2.92

Tot

al E

nerg

y (e

V)

Lattice Constant (Å)

Figure 2.3: Total energy versus the lattice constant of the lithium chain.

atom is repeated by 9, 9, and 3 Å in the x, y, and z directions, respectively. Lithiumatoms separated by 9 Å are too distant to interact significantly (the lattice constant 3.0 Åroughly corresponds to the experimental value) and hence, we have defined an infinitechain of lithium atoms extending in the z direction (or rather the C direction, but thesecoincide in the present case).

The command

$ atk lichain.atk lichain.out

calculates the total energy of the lithium chain for a lattice constant of 3.0 Å.

Inspecting the output file, one notes that it contains substantially more parameters thanwe specified in the input file. Input parameters which are not specified explicitly in theinput file are set to default values specified in the file $ATK_DATA_DIR/atk-defaults.

atk. The default values are also listed in Appendix D.2.

For example, if the parameters NumOrb::MeshCutoff and NumOrb::BasisSet::Size

are removed from lichain.atk, ATK will use the default parameters. This will actuallyresult in more accurate results; however, the calculation will also take more time toexecute. In particular, the SZ basis set is too small to yield an even reasonably accuratedescription of the band structure of lithium. It is nevertheless often useful to start witha small basis set to get a general picture of the system, before launching a larger andmore accurate, but also more time-consuming calculation.

In the next step, we will use ATK to find the equilibrium lattice constant of the chain,i.e. the value of the lattice constant that minimizes the total energy. In this case, it isnot possible to use the relaxation method employed for the hydrogen molecule, sincethis approach only relaxes the atoms within the unit cell and not the size of the unit

10 Quick Tour

−2

−1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2

Ene

rgy

(eV

)

kz (1/Å)

2.84 Å2.90 Å

Figure 2.4: Band structure of a lithium chain with an interatomic distance of 2.84 Å and 2.90 Å,respectively. The Fermi level is at zero energy, and thus the Li chain is metallic.

cell itself.

Instead, we employ the fact that ATK takes a number of command line options, aswe have already seen examples of. Of particular interest for our task is the option--param "parameter". Using this when calling ATK is equivalent to including thesame statement "parameter" in the input file. In this way, it becomes possible to writesmall scripts that perform complex tasks.

The scripts can be created in any suitable scripting language, but here we will fo-cus on the ATK commands, rather than the implementation of the scripts. However,in Appendix F, a collection of example scripts are provided for Linux/Cygwin (bashshell scripts), Windows (DOS batch files) and platform-independent scripts written inPython, for each of the tasks discussed in the tutorial.

To calculate the total energy for a given lattice constant (2.84 Å in this case), we usethe command

$ atk lichain.atk lichain2.84.out -o lichain2.84.nc \

--param "UnitCell::LatticeConstant 2.84 Angstrom"

The command may be written on a single line (without the ’\’), it is just broken here tobe easier to read. We are already familiar with the first line, which contains the namesof the input and output files. The option -o is required to create NETCDF files withunique names for each lattice constant (the default name is the same as the input file),and finally we specify the value of the lattice constant using the --param option, asdiscussed above.

A script can be written (see Appendix F) that runs such a calculation for a sequence of

2.3 Setting up the Two-Probe System 11

lattice constants in the range from 2.82 to 2.92 Å. In this way, a plot like the one inFigure 2.3 can be made. The figure shows that the chain with lattice constant 2.90 Å hasthe lowest energy and thus, we will use this lattice constant for the lithium electrodes inthe two-probe system. Note, that the total energy in itself is meaningless, only relativetotal energies are useful.

The last three lines in the script lichain.atk computes the band structure of thelithium chain between the � point, defined by k=(0, 0, 0), and the X point, definedby k = (0, 0, �/a), where a is the lattice constant (or rather, in our case, the spacing ofthe atoms in the C direction). The keyword FracRecip means that the wave vectors,such as (0.0, 0.0, 0.5), are interpreted in units of the reciprocal lattice vectors. Otherunits are also possible, cf. Appendix D.1.2.

The band structure is shown in Figure 2.4 for two different lattice constants; note thatthe calculations only need to be carried out for positive wave numbers kz, since theband structure always is symmetric about the � point.

2.3 SETTING UP THE TWO-PROBE SYSTEM

Our goal is to place the hydrogen molecule between two lithium chains and calculatethe transmission through the system. Thus, the system consists of three parts, the leftelectrode, the central region, and the right electrode. The left and right electrodes aresemi-infinite lithium chains, while the central region consists of the hydrogen moleculewith parts of the lithium chains. The geometry is illustrated in Figure 2.5.

2.3.1 ELECTRODE CALCULATION

We will first consider the electrode atoms, i.e. the red atoms in Figure 2.5. For this weneed to do a calculation with three-dimensional boundary conditions of the electrodeatoms, similar to the lithium chain calculation in the previous section. However, foran electrode calculation, it is assumed that there are only interactions between atomsin neighboring unit cells, meaning that all other interactions are set identically to zero.If, on the other hand, the physical interactions in reality have a longer reach than

Figure 2.5: Geometry of the hydrogen molecule coupled with semi-infinite lithium chains. Theelectrode atoms are colored red.

12 Quick Tour

the lattice constant (this is the situation that would occur in most realistic cases), itis necessary to create a conventional cell, or supercell, such that all interactions areproperly accounted for.

In the present case, we may safely assume that lithium atoms separated by more than 4lattice constants are not interacting significantly. Therefore, we include 4 lithium atomsin the supercell (see the input file below), which then of course has a four times largerlattice constant in the C direction. In this way, 4 atoms in one unit cell may interactwith the 4 atoms in an adjacent cell, and we thus account for all interactions extendingup to at least 4 of the (original) lattice constants along the chain.

To define the electrodes, create a file lielec.atk with the content

System::Type Electrode



Electrode::NumKPoints::C 500


%block Electrode::UnitCell

3. 0. 0.

0. 3. 0.

0. 0. 4.

%endblock Electrode::UnitCell


%block Electrode::AtomList

Li 4.35 4.35 0.00

Li 4.35 4.35 2.90

Li 4.35 4.35 5.80

Li 4.35 4.35 8.70

%endblock Electrode::AtomList

and produce the electrode NETCDF file lielec.nc with the command

$ atk lielec.atk lielec.out

2.3.2 DETERMINING THE LI–H BOND LENGTH

The next step is to place the hydrogen molecule between the lithium chains, and wetherefore need to estimate the Li–H bond length. We do this by relaxing a Li4H2 cluster.For this purpose, create a file li4h2.atk with the content



Relaxation::ForceTolerance 5.e-2 eV/Ang

Relaxation::MaxSteps 200

2.3 Setting up the Two-Probe System 13


NumOrb::BasisSet::Size::Li SZ



Li 0.00 0.00 -4.955

Li 0.00 0.00 -1.905

H 0.00 0.00 -0.375

H 0.00 0.00 0.375

Li 0.00 0.00 1.905

Li 0.00 0.00 4.955


Here we introduced some new parameters related to relaxation. They determine thetolerance (in the forces) for deciding when the calculations are considered to haveconverged, and how many steps this at most may take. We also specified a different(smaller) basis set specifically for the lithium atoms by appending the element symbol,Li, to the keyword NumOrb::BasisSet::Size. In this way, we still use the larger defaultbasis set for the hydrogen atoms, which results in a more accurate calculation of therelaxed H–H bond length.

To perform the calculation, execute the command

$ atk li4h2.atk li4h2.out

The calculation takes a while to complete.

From the output file, we find that the relaxed Li–H distance is estimated to be 2.37 Å,and the H–H bond is slightly stretched to 0.80 Å. In the next section, we use the valuesto set up the central region of the two-probe system.

2.3.3 PLACING H2 BETWEEN THE LITHIUM CHAINS

We now place the hydrogen molecule between the lithium contacts. In the centralregion, we have H2 coupled with three lithium atoms to each side. For the Li–H–H–Li,part we will use the relaxed geometry from the Li4H2 cluster, which can be extractedfrom the output file li4h2.out. The remaining Li–Li distances are set to the lithiumchain lattice constant found earlier (2.90 Å). The file lih2-0.0.atk defines the inputto ATK for the complete two-probe system:

System::Type TwoProbe

Method::Type NumOrb

NumOrb::BasisSet::Size::Li SZ


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%block TwoProbe::CentralAtomList

Li 0.0 0.0 -8.568

Li 0.0 0.0 -5.668

Li 0.0 0.0 -2.768

H 0.0 0.0 -0.402

H 0.0 0.0 0.402

Li 0.0 0.0 2.768

Li 0.0 0.0 5.668

Li 0.0 0.0 8.568

%endblock TwoProbe::CentralAtomList

TwoProbe::LeftElectrode::NetCDFFile lielec.nc

TwoProbe::RightElectrode::NetCDFFile lielec.nc

TwoProbe::LeftSurface::NumberOfAtoms 3

TwoProbe::RightSurface::NumberOfAtoms 3

The unit cell is automatically generated by ATK from the electrode file lielec.nc andthere is therefore no need to specify the unit cell in the input file. The electrodes areautomatically aligned with the central region using the electrode lattice constant.

To check that the system is set up correctly we will generate a data file with the two-probe geometry. With the -c option ATK does not perform an actual calculation. Typethe command

$ atk -c lih2-0.0.atk --xyz

The --xyz option is used to output the geometry into lih2-0.0.xyz, which can bevisualized with the Atomistix program VIRTUAL NANOLAB (see Figure 2.5).

2.3.4 SELF-CONSISTENT CALCULATION AT ZERO BIAS

The structure looks correct and next we perform the self-consistent calculation:

$ atk lih2-0.0.atk lih2-0.0.out --force

Note that this calculation will take rather long time. Note that we have included the op-tion --force to instruct ATK to overwrite and replace any existing output and NETCDFfiles. This is necessary in this case, since the previous command (at the end of theprevious section) already produced an output NETCDF file with the same name as theone we will create here.

It is instructive to inspect the output file. The first part of the file contains informationabout the system setup and input parameters. Below this, there is information aboutthe self-consistent cycle. Two self-consistent cycles are performed. The first cycle is

2.4 Analyzing the Results 15

carried out with periodic boundary conditions, and the only purpose of this part is toobtain a good initial guess for the density matrix. The second cycle is carried out withthe proper open system boundary conditions.

2.4 ANALYZING THE RESULTS

As already mentioned above, ATK offers a range of analysis options, all specifiedby Analysis:: keywords, which can be used to post-process the results of the self-consistent energy structure calculation. The data generated can be used to draw con-clusions about the results, and also for plotting.

We shall analyze the transmission spectrum of the system, and for this we create a filetrans.atk with the contents

Analysis::TransmissionSpectrum::NumPoints 100

Analysis::TransmissionSpectrum::E0 -5. eV

Analysis::TransmissionSpectrum::E1 5. eV

Analysis::MPSH::A::Index 0

Analysis::MPSH::A::Label Mpsh0

Analysis::MPSH::B::Index 1

Analysis::MPSH::B::Label Mpsh1

Analysis::DOS::A::Energy 0 eV

Analysis::DOS::A::Label total

Analysis::DOS::A::Type total

and run the command

$ atk trans.atk -a lih2-0.0.nc trans.out

The results will be saved in trans.nc, and some results can also be inspected in thelog file trans.out. Note how the output file name for the NETCDF file is created fromthe input atk file, unless a specific name is specified by using the option -o.

2.5 TRANSMISSION SPECTRUM

The main quantity of interest in most two-probe calculations is the transmission spec-trum. This is controlled by the keywords in trans.atk beginning with Analysis::

TransmissionSpectrum. The resulting spectrum is plotted in Figure 2.6. We observe,as indicated earlier, that the transmission is very high (above 0.6) at the Fermi level.This is a great puzzle, since the HOMO–LUMO gap of a hydrogen molecule is hugecompared to typical electron excitation energies.

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0.0

0.2

0.4

0.6

0.8

1.0

−5 −4 −3 −2 −1 0 1 2 3 4 5

Tra

nsm

issi

on C

oeffi

cien

t

Energy (eV)

Figure 2.6: Transmission coefficients of an H2 molecule placed between two lithium chains. Thedashed line shows the Fermi level.

2.5.1 MPSH

We shall now try to use the other analysis options specified in trans.atk to explainthe observed unusually high conductance of the hydrogen molecule in the lithiumchain. A very useful concept is the molecular projected self-consistent Hamiltonian(MPSH). The molecular projection refers to the part of the two-probe system whichremains when the electrode and surface atoms are excluded. In the input file lih2-0.

0.atk, we defined three surface atoms on each side of the hydrogen atoms by using theoptions TwoProbe::LeftSurface::NumberOfAtoms and TwoProbe::RightSurface::

NumberOfAtoms. These two parameters specify the number of surface atoms in theTwoProbe::CentralAtomList. If their values are n and m, respectively, the first natoms in the central atom list will be treated as surface atoms of the left electrode andthe last m atoms in the list will be treated as surface atoms of the right electrode.

Thus in our case, all lithium atoms are either electrode or surface atoms and the molec-ular projection only refers to the hydrogen molecule. This means that the MPSH is the

Figure 2.7: The two small red and turquoise ellipsoids in the center of the image show anisosurface (the color denotes the phase) of the wave function of the MPSH state with energy2.36 eV. The phase difference clearly indicates that this state corresponds to the LUMO. Thelarge balls are not part of the isosurface, but illustrate the positions of the lithium atoms.

2.6 Current–Voltage Characteristics 17

self-consistent Hamiltonian of the hydrogen molecule in the presence of the lithiumchain. The new eigenvalues are –8.02 eV for the HOMO and 2.36 eV for the LUMO.

The reduction of the HOMO–LUMO gap is directly responsible for the large value ofthe transmission coefficient of the hydrogen molecule observed experimentally [10].

In the file trans.atk, we included the lines Analysis::MPSH::A::Index and Analysis::

MPSH::A::Label. These lines mean that the MPSH wavefunctions are written in theanalysis file trans.nc and can then be visualized with VIRTUAL NANOLAB. In Fig-ure 2.7, we show a plot of the MPSH state with energy 2.36 eV, the MPSH LUMOorbital.

2.5.2 DENSITY OF STATES

From the previous section, we conclude that the transmission is mainly due to theLUMO, since the position of this orbital is closest to the transmission maximum ataround 2.5 eV (cf. see Figure 2.6). Another piece of evidence to support the conclusionthat the transmission mainly takes place through the LUMO is obtained from the densityof states (DOS) at the Fermi level. The last three lines in trans.atk, starting withAnalysis::DOS, control whether or not the DOS is output to the file trans.nc. Thecorresponding data has been visualized, spatially resolved, with VIRTUAL NANOLAB inFigure 2.8 , and we see that there is a substantial density of states concentrated to thehydrogen atoms at the Fermi level.

2.6 CURRENT–VOLTAGE CHARACTERISTICS

The next task is to calculate the current–voltage characteristics of the system. This re-quires a self-consistent calculation for a series of different values of the applied bias,and for this purpose we may again create a script (Appendix F), with the central com-mand (assuming a bias of 0.8 V)

$ atk lih2-0.0.atk lih2-0.8.out -o lih2-0.8.nc -i lih2-0.6.nc \

--param "TwoProbe::LeftElectrode::Voltage 0.8 eV"

Figure 2.8: Volume plot of the density of states of the LiH2 system at the Fermi energy (0 eV).The small spheres indicate the positions of the atoms.

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0

10

20

30

40

0 0.2 0.4 0.6 0.8 1

Cur

rent

(µA

)

Bias (V)

Single ConfigurationRelaxation

Figure 2.9: I–V curve of a hydrogen molecule between two lithium chains, comparing thecurrent–voltage characteristics when the atomic positions are relaxed (cf. Section 2.7) or not.

Note carefully the -i option, which refers to a NETCDF file for a different bias (0.6 Vin this case), which we thus assume has already been calculated (cf. the scripts). In thisway, i.e. by using the -i option, we can employ a previously calculated density matrixas the initial guess for the new calculation to improve the calculation speed.

The command above adds an additional parameter to the lih2-0.0.atk parameterfile. The parameter TwoProbe::LeftElectrode::Voltage adds a bias voltage to theleft electrode. The resulting current can then be extracted from the output file. Lookfor the line containing the word "Current". A positive value of the current indicates thatthe current flows from the left electrode to the right; obviously the electron flow is inthe opposite direction.

Figure 2.10: Voltage drop through the Li–H–H–Li system for an applied bias of 1 V.

2.7 Relaxation of a Two-Probe System 19

0.810

0.815

0.820

0 0.2 0.4 0.6 0.8 1 2.34

2.36

2.38

H−

H B

ond

Leng

th (

Å)

H−

Li, L

i−H

Bon

d Le

ngth

(Å

)

Bias (V)

H−HH−LiLi−H

Figure 2.11: Change in the H–H and Li–H bond lengths as a function of the bias.

2.6.1 VOLTAGE DROP

Next, we calculate the voltage drop through the structure. The voltage drop is given bythe difference in the effective potential between the finite bias calculation and the zerobias calculation. The effective potential of the zero bias calculation was stored in thefile trans.nc.

We extract the effective potential from the 1.0 V calculation with the command

$ atk trans.atk trans+1.0.out -o trans+1.0.nc -a lih2-1.0.nc

In Figure 2.10, the difference between the effective potentials from the two calculationsare plotted. Note, that for this effectively one-dimensional system the central region(the number of surface lithium atoms) is not large enough to screen the applied electricfield, and therefore an abrupt change occurs in the voltage drop at the right electrode.

2.7 RELAXATION OF A TWO-PROBE SYSTEM

In this section, we relax the two-probe geometry both at zero and finite bias.

Make a new file lih2-0.0-relax.atk with the same content as lih2-0.0.atk, andadd the line


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−20

−15

−10

−5

0

5

−10 −8 −6 −4 −2 0

MP

SH

Eig

enva

lues

(eV

)

Gate Voltage (eV)

HOMO (Self−Consistent)LUMO (Self−Consistent)

HOMO (Non Self−Consistent)LUMO (Non Self−Consistent)

Figure 2.12: The curved lines represent the self-consistent change in the MPSH eigenstates asfunction of applied gate voltage, while the straight lines are linear extrapolations (i.e. the initialshift of the eigenstates by the gate voltage) of the MPSH eigenstates at zero gate voltage.

We can reuse the script we used to calculate the current–voltage relationship, and justreplace the input file with the one containing the relaxation command above. Thus,the generic ATK command becomes (for the bias 0.8 V, using the 0.6 V as initial guess)

$ atk lih2-0.0-relax.atk lih2-0.8-relax.out \

--force -o lih2-0.8-relax.nc -i lih2-0.6-relax.nc\

--param "TwoProbe::LeftElectrode::Voltage 0.8 eV"

These calculations take a few hours on a modern PC. If you check the output file,you will find that only the hydrogen atoms have moved (the results are summarizedin Figure 2.11). This is because the positions of the electrode and surface atoms arekept fixed. Remember that surface atoms are specified by the commands TwoProbe::LeftSurface::NumberOfAtoms and TwoProbe::RightSurface::NumberOfAtoms.

2.8 INCLUDING A GATE ELECTRODE

As the final step of the quick tour, we demonstrate how to simulate the electrostaticpotential induced by a third electrode (a gate electrode) on the two-probe system.The strength of the electrostatic potential due to the gate is controlled by the keywordTwoProbe::Molecule::GateVoltage.

First, create a copy of the input file lih2-0.0.nc named gate-0.0.nc. This file willbe used as the initial guess for the lowest gate voltage. We can now write a script

2.8 Including a Gate Electrode 21

(see Appendix F), which performs a self-consistent calculation of the electronic levelsfor our two-probe system for gate voltages ranging from –1 V to –10 V by using thecommand (here at the gate voltage –5 V)

$ atk lih2-0.0.atk gate-5.0.out -o gate-5.0.nc -i gate-4.0.nc \

--param "TwoProbe::Molecule::GateVoltage -5.0 eV"

The gate voltage for ATK should be specified as an electrostatic potential energy (herein electron volts). The conversion of the gate voltage used in a physical system to thispotential energy is not trivial. It requires a specification of the geometry used in thephysical system and a solution of the Poisson equation that relates the electrostaticpotential on the molecule to the applied gate voltage.

The gate electrode is not included as a physical electrode in ATK, and, consequently,a current cannot run from the source or drain electrode to the gate electrode. In thecurrent implementation, we simulate the electrostatic effect of the gate electrode bysimply shifting the MPSH part of the Hamiltonian with the gate voltage (i.e. the electro-static potential energy). This corresponds to assuming that the gate electrode inducesan external potential localized to the molecular region. For metallic electrodes this willusually be a reasonable approximation.

Figure 2.12 shows the self-consistent shift of the MPSH eigenstates due to the appliedgate potential. For reference is also plotted the initial (non-self-consistent) shift. We seethat the self-consistent change in the MPSH eigenvalues is significantly smaller thanthe initial shift by the gate potential.

22 Quick Tour

C H A P T E R

3MOLECULAR SYSTEMS

In this chapter, we go into more details about how to use ATK to study molecular sys-tems. The purpose of the different examples is to show the syntax of the input file andto demonstrate how different input parameters affect the accuracy of the calculations.We recommend that you type the input files yourself, but they can also be found in thedirectory examples/molecule in the program installation directory.

3.1 WATER MOLECULE

Make an input file h2o-1.atk for an H2O molecule with the following lines


Method::Type NumOrb



H 0.757 0.586 0.000

H -0.757 0.586 0.000

O 0.000 0.000 0.000


Since the input file does not explicitly list parameters related to the basis set or whichmethod to use for the calculation, the default values will be used. These correspondto using density functional theory (DFT) with the LDA exchange–correlation functionalof Perdew and Zunger [8], and a double zeta basis with polarization orbitals for all theelements. Run the file

23

24 Molecular Systems

$ atk h2o-1.atk h2o-1.out

and inspect the results in the output file.

3.2 THE SELF-CONSISTENT LOOP

An important part of the DFT calculation is to obtain the self-consistent density, and anumber of parameters are dedicated to controlling this part of the calculation. We willnow take a look at some of the parameters. Add the following two lines to the previousinput file

SCF::Coordinate DensityMatrix

SCF::DiagonalMixingParameter 0.5

and run ATK on the new input file, which we call h2o-2.atk


By inspecting the output file h2o-2.out, you will find that ATK now converged in fewersteps.

The parameter SCF::Coordinate controls which system variable is mixed during theself-consistent cycle. For closed systems with a fixed number of electrons, i.e. molec-ular or bulk geometries, usually density matrix mixing is most efficient. For open two-probe systems, where the central region does not have a fixed number of electrons,Hamiltonian mixing is, based on our experience, most efficient.

The algorithm used for mixing the system variable is controlled by the keyword (SCF::Algorithm). In ATK, there are two different mixing algorithms, Broyden [2] and Pulaymixing [9]. Both schemes use a number of previous iterations to make a guess on thenew density. The mixing ratio between the new guess and the old guess is controlled bySCF::DiagonalMixingParameter. If the value of this parameter is too large, the self-consistent loop will not converge. If it is too small the self-consistent loop will take avery long time. For isolating systems, large values of SCF::DiagonalMixingParametercan be used (up to 0.5), whereas for metallic systems the value should usually bearound 0.1.

All parameters which control the self-consistent cycle are summarized in Table 3.1.

3.3 Relaxations 25

SCF::Coordinate DensityMatrix k Hamiltonian k EffectivePotentialThis keyword determines which coordinate is used for mixing.

SCF::Algorithm Pulay k BroydenThis keyword controls the mixing algorithm.

SCF::Tolerance realWhen the relative change of the density and the Hamiltonian matrix is less than thisvalue between two iterations, the self-consistent cycle is considered converged.

SCF::DiagonalMixingParameter realThis parameter defines the fraction of the new value of the SCF::Coordinate whichis mixed with the old value.

SCF::HistorySteps (integer)Number of previous steps used in the mixing algorithm to generate the next state.

SCF::MaxSteps integerMaximum number of steps in the self-consistency loop.

Table 3.1: Parameters used to control the self-consistent cycle. For units and default values, referto the keyword reference section on pp. 86– 104.

3.2.1 HOW TO CURE CONVERGENCE PROBLEMS

Convergence problems with molecular systems can usually be related to a small orvanishing HOMO–LUMO gap (the HOMO is highest occupied molecular orbital, theLUMO (the lowest unoccupied molecular orbital). In such cases, a very small perturba-tion can shift a molecular state from occupied to unoccupied, making the convergencedifficult. To remove this instability one can increase the electron temperature. Usuallya value of

Molecule::ElectronTemperature 0.2 eV

will be sufficient to obtain convergence.

If an increased electron temperature does not cure an convergence problem, increasingthe number of history steps used in the Pulay mixing to e.g. 20 (default is 6), mightwork. The number of history steps is specified by the keyword SCF::HistorySteps.Moreover, lowering the value of SCF::DiagonalMixingParameter can also improvethe convergence properties of a system; the required value may be as low as 0.01.

3.3 RELAXATIONS

ATK calculates the forces on all atoms, and this information can be used to relax theionic coordinates, as we already demonstrated in the quick tour. The relaxation modulecan be used for molecules, bulk or two-probe geometries.


Create a file h2o-3.atk with the parameters of h2o-1.atk, and add the following lines:


Relaxation::ForceTolerance 5.e-2 eV/Ang

Relaxation::MaxSteps 200

Now the command


will produce a relaxed geometry.

It is possible to fix the position of some of the atoms during the relaxation using the key-word Simulation::ConstrainAtoms. The following command will fix the positionsof the two hydrogen atoms (atoms number 0 and 1) during the relaxation:

$ atk h2o-3.atk h2o-3c.out -o h20-3c.nc \

--param "Simulation::ConstrainAtoms 0:1"

Relaxation::ForceTolerance real ForceUnitThe relaxation is completed when the largest force component is less than this value. Thetotal force magnitude is also returned in the output file. This is the length of the forcevector on all atoms, i.e. sX

i;k

jfi;kj2

where fi;k is the force component k on atom i, and i = 0; :::; n� 1 where n is the numberof atoms and k = x; y; z.

Relaxation::MaxSteps integerThe maximum number of energy evaluations during the relaxation run.

Relaxation::MaxDisplacement real LengthUnitSets the maximal displacement for each step in the relaxation. If this is too large therelaxation might fail, while a too small step will slow the relaxation down.

Simulation::ConstrainAtoms int:int[,int:int]Fixes the positions of certain atoms during a relation. Example: a system consists of 12atoms numbered 0, 1, 2, : : :, 11 (the numbering of atoms always begins at zero). With thecommand

Simulation::ConstrainAtoms 0:2,7:10,4:5

we constrain atoms 0, 1, 2, 7, 8, 9, 10, 4, 5. Negative atom indices correspond to countingbackwards from zero, so the command

Simulation::ConstrainAtoms 0:2,-5:-2,-8,5

is equivalent to the one above.

Table 3.2: Parameters used to control the relaxation. For units and default values, refer to thekeyword reference section on pp. 86–104.

3.4 Changing the Accuracy of the Calculation 27

3.4 CHANGING THE ACCURACY OF THE CALCULATION

Usually the main approximation in an ATK calculation is related to the choice of theexchange–correlation functional. The exchange-correlation potential is chosen usingthe keyword NumOrb::XCFunctional. Currently four different exchange–correlationfunctionals are available.

LDA-PZ The local density approximation (LDA) with the Perdew–Zunger parametriza-tion [8] of the correlation energy of a non spin-polarized homogeneous electrongas calculated by Ceperly–Alder [4].

LSDA-PZ The local spin density approximation (LSDA) with the Perdew–Zunger parametriza-tion [8] of the correlation energy of a spin-polarized homogeneous electron gascalculated by Ceperly–Alder [4].

GGA-PBE The non spin-polarized generalized gradient approximation (GGA) in thePerdew–Burke–Ernzerhof approximation (PBE) [7].

SGGA-PBE The spin-polarized generalized gradient approximation (SGGA) in the Perdew–Burke–Ernzerhof (PBE) approximation [7].

In the previous section, we relaxed the H2O molecule with the LDA-PZ functional.Inspection of the relaxed geometry shows that the H2O molecule has a bonding an-gle of �HOH=103.3� and an OH bond length of dOH=0.97 Å, which are close to theexperimental values of �HOH=104.5� and dOH=0.96 Å. We will now test the GGA-PBE

functional (the spin-polarized functionals, LSDA-PZ and SGGA-PBE will be described inSection 3.6). To change the exchange–correlation functional add the following line

NumOrb::XCFunctional GGA-PBE

to the h2o-3.atk and save it in a new file h2o-4.atk


The new values �HOH=103.0� and dOH=0.97 Å are not much different from the previousones. In general, structural properties do not depend much on the exchange-correlationfunctional (i.e. LDA versus GGA, or LSDA versus SGGA). In contrast, energy changesare more sensitive to the choice of exchange-correlation functionals, and the GGA-PBEfunctional will usually perform better than the LDA functional.

The accuracy of the calculation is also affected by the choice of the pseudopotential,the basis set and the numerical integration routine, and it is important to check that theerrors due to these parameters are small. In this section, we will show how the size ofthe basis set and the accuracy of the numerical integration is controlled.


To change the basis set to a minimal basis set, add NumOrb::BasisSet::Size SZ tothe input file h2o-3.atk and save the file as h2o-5.atk. Relax the H2O moleculedefined by this file. By inspection of the output file (h2o-5.out) we find that thegeometry has changed. For instance, �HOH=99.1�, whereas it was �HOH=103.3� for theDZP basis set. Thus, the use of the SZ basis set decreases the accuracy of the calculationsubstantially, whereas DZP basis sets usually are comparable to well-converged plane-wave basis sets.

ATK has 5 different built-in basis sets:

SZ A single basis orbital for each valence orbital.

DZ Two basis orbitals for each valence orbital.

SZP SZ and one basis orbital for the first unoccupied shell.

DZP DZ and one basis orbital for the first unoccupied shell.

DZDP DZ and two basis orbitals for the first unoccupied shell.

It is possible to use different basis sets for different atoms. For instance, the followingkeywords (h2o-6.atk)

NumOrb::BasisSet::Size DZ

NumOrb::BasisSet::Size::O DZP

assign a DZP basis to the oxygen atoms and a DZ basis to all other atoms. It is possibleto control the basis sets in much more detail, in order to improve the results further;this is described in Appendix B.

The parameter NumOrb::MeshCutoff controls the density of the grid used for real spaceintegrals. The real space resolution, �x is related to the mesh cutoff, Ecut through therelation �x = �=

pEcut, where Ecut is in Rydberg and �x in bohr.

Now make the following calculation

$ atk h2o-1.atk h2o-1-50Ry.out -o h2o-1-50Ry.nc \

--param "NumOrb::MeshCutoff 50 Ry"

You will note that the calculation is much faster now, but the total energy has increased.The cutoff value of 50 Ry is far too low to give any meaningful results for the totalenergy. The default value of NumOrb::MeshCutoff is 150 Ry. Let us try to increase thevalue to 200 Ry:

$ atk h2o-1.atk h2o-1-200Ry.out -o h2o-1-200Ry.nc \

--param "NumOrb::MeshCutoff 200 Ry"

In Figure 3.1, we show the total energy and the force in the water molecule as afunction of the mesh cutoff. The force converges more slowly than the total energy. Formost materials the system default of 150 Ry will be sufficient.

3.5 Molecular Unit Cell 29

−465.40

−465.35

−465.30

−465.25

0 50 100 150 200 250 3000.95

1.00

1.05

Tot

al e

nerg

y (e

V)

Larg

est f

orce

com

pone

nt (

eV/Å

)

NumOrb::MeshCutoff (Ry)

Total EnergyForce

Figure 3.1: The total energy of H2O and the largest force component (which always is on theoxygen atom, in the z-direction) as a function of the mesh cutoff. The geometry has not beenrelaxed.

3.5 MOLECULAR UNIT CELL

In ATK, a molecule is embedded in a unit cell with periodic boundary conditions. Thismeans that even molecular systems are treated as periodic systems, and there is in factnot only a single molecule in the system, but an infinite number of molecules.

The size of the unit cell, and thus the distance between the repeated copies of themolecule, is controlled by the parameter Molecule::ElectrostaticPaddingFactor.When this parameter is set to zero, the unit cell has the minimal size where there areno matrix elements, i.e. overlap between wave functions of molecules in neighboringcells, which in turn means that they only interact electrostatically. The default value is0.1.

Try to increase the size of the unit cell with the command

$ atk h2o-1.atk h2o-1-pad0.7.out -o h2o-1-pad0.7.nc \

--param "Molecule::ElectrostaticPaddingFactor 0.7"

By inspecting the output file you will find that the total potential energy has onlychanged slightly, whereas the molecular eigenvalues are shifted by about 0.04 eV.In Figure 3.2, we show the dependence of the total energy and the HOMO energyas a function of the padding factor. The change in the computed values is due to anelectrostatic interaction between the repeated images of the molecules.

We see that the total energy is well converged with the default choice of padding factor.The H2O molecule has a relatively strong dipole moment, so the default parameter


−465.34

−465.33

−465.32

0 0.2 0.4 0.6 0.8 1 1.2−6.09

−6.08

−6.07

−6.06

−6.05

−6.04

−6.03

Tot

al E

nerg

y (e

V)

HO

MO

Ene

rgy

(eV

)

Molecule::ElectrostaticPaddingFactor

Total EnergyHOMO

Figure 3.2: The total energy of H2O as a function of the value of Molecule::

ElectroStaticPaddingFactor , and the energy of the HOMO. Note that the scale on the twoy-axes is the same.

should be sufficient for most systems. The HOMO energy shows a larger variation withthe padding factor. This is because the molecular eigenvalues are reported relative tothe vacuum level (the value of the effective potential far away from the molecule), andthe determination of the vacuum level is more sensitive to the electrostatic interactionbetween the cells.

Thus, for molecules with a large dipole moment the padding factor must be increasedto obtain converged molecular eigenvalues relative to the vacuum level.

3.6 SPIN-POLARIZED CALCULATIONS

In this section, we introduce calculations with spin dependent exchange-correlationfunctionals by considering the O2 molecule.

First, make the file o2.atk with the context



AtomList::Format Ang


O 0.0 0.0 -0.6

O 0.0 0.0 0.6


NumOrb::XCFunctional SGGA-PBE

3.6 Spin-Polarized Calculations 31


$ atk o2.atk o2.out

By inspecting the output file, we see that there are two sets of eigenvalues, one for spinup and one for spin down, and that there are two more electrons with spin up than spindown. Hence, the electronic ground state is a triplet in agreement with experimentalevidence (or, at least, the spin along the quantization axis is MS = 2� 1

2~).

In the spin dependent calculations, the initial density is generated by superpositionof atomic densities with maximum spin. The calculation then automatically relaxesthe spin to that of the ground state, although for complicated systems, it might betrapped in a local minimum. We can modify the behaviour by using the keywordsSCF::InitialSpinPolarization and Molecule::TotalSpin, cf. Appendix D.2 Theformer one specifies the initial spin, whereas the latter constrains the total spin to agiven value. Below, we constrain the total spin in order to study the O2 molecule inthe quintet state by copying o2.atk to o2_5.atk and adding the line

Molecule::TotalSpin 2 hbar

Note, that the total spin is given in units of ~ and that the spin of a single electron is 1

2~.

Hence, the above line indicates that there should be 4 electrons in excess with spin up,the majority spin, corresponding to a quintet state. Next, we execute the command

$ atk o2_5.atk o2_5.out

An inspection of the output file reveals that the molecule is stretched significantly com-pared to the previous calculation without a constraint on the total spin.

The bond lengths from the two calculations and the corresponding energies are listedin Table 3.3 together with the results from a calculation of the singlet state, obtainedby executing

$ atk o2.atk o2_1.out -o o2_1.nc --param "Molecule::TotalSpin 0 hbar"

and a calculation with the spin-independent GGA exchange-correlation functional(GGA-PBE), obtained by executing

$ atk o2.atk o2-ns.out -o o2-ns.nc --param "NumOrb::XCFunctional GGA-PBE"

As can be seen in Table 3.3, the triplet state is indeed the groundstate, whereas thequintet and singlet states are not stable. Moreover, computing the singlet state with aspin-dependent exchange-correlation functional really just corresponds to performinga spin-indepedent calculation as already indicated in Table 3.3. However, the spin-independent calculation can be made at lower cost.


State XC Distance (Å) Total Energy (eV)Singlet GGA-PBE 1.22 –871.50Singlet SGGA-PBE 1.22 –871.50Triplet SGGA-PBE 1.22 –872.71Quintet SGGA-PBE 1.91 –867.25

Table 3.3: Relaxed O2 molecule in different spin states and computed with spin-dependent orindependent exchange-correlation functionals.

For some of the Analysis keywords, it is now necessary to specify the spin of interest.For instance, if we want to compare the HOMO of the triplet structure with that of thequintet structure, we may write an input file (o2-homo3.atk)


Analysis::MolecularOrbital::A::Label HOMO

Analysis::MolecularOrbital::A::Spin Up


$ atk o2-homo3.atk o2-homo3.out -a o2.nc

in order to get the HOMO of the triplet structure. To get the HOMO of the quintetstructure, we instead use (o2-homo5.atk)


Analysis::MolecularOrbital::A::Label HOMO

Analysis::MolecularOrbital::A::Spin Up


$ atk o2-homo5.atk o2-homo5.out -a o2_5.nc

Surface plots of the two orbitals are plotted in Figure 3.3. It is seen that the HOMOof the triplet state is an anti-bonding �� orbital, whereas that of the quintet state is an

Figure 3.3: The HOMO of the triplet (left panel) and quintet (right panel) states of O2 illustratedwith VIRTUAL NANOLAB. Note that the O–O distance is 1.2 Å in the left panel, whereas it is1.9 Å in the right panel.

3.6 Spin-Polarized Calculations 33

anti-bonding �� orbital. In fact, by studying the eigenvalues and plotting the corre-sponding orbitals, it can be realized that in the quintet state a spin-down electron in adegenerated and bonding � orbital (the spin down eigenvalues with index 3 and 4) hasbeen transferred to an anti-bonding �� orbital (the HOMO in the quintet state). This inturn explains the increased inter-atomic distance in the quintet state.

In general, spin-polarized calculations come at a higher cost, but in many cases theyare necessary in order to get meaningful energies and structures as indicated by theabove example. Other examples are individual atoms, radicals, and some systems con-taining transition metals like iron. An example of the latter is the spin filter describedon pp. 59–67.

C H A P T E R

4PERIODIC SYSTEMS

In a periodic system, the eigenfunctions obey Bloch’s theorem and can be labeled bythe wavevector k. In this section, we will show how k-point sampling is implementedin ATK, and how the number of k-points can be increased to obtain well convergedresults.

4.1 BAND STRUCTURE OF A SI CRYSTAL

Type the following lines into the file si-1.atk (the input files for this chapter can befound in examples/bulk in the installation directory)

System::Type Bulk

Numorb::XCFunctional GGA-PBE

Numorb::basisset::Size SZ

# FCC LATTICE

UnitCell::LatticeConstant 5.43 Ang


0. 0.5 0.5

0.5 0. 0.5

0.5 0.5 0.


Bulk::NumKPoints::A 3

Bulk::NumKPoints::B 3


35

36 Periodic Systems

# TWO POINT BASIS

AtomList::Format Scaled

AtomList::Scale 5.43 Angstrom


Si 0.00 0.00 0.00

Si 0.25 0.25 0.25


This input file demonstrates an optional way of specifying the atomic coordinates inthe atom list. By using the command AtomList::Format Scaled, we express all coor-dinates in units of the length AtomList::Scale, which here is set to 5.43 Å. Also notethat the unit cell is defined in terms of the lattice constant.

Also note the lines starting with the symbol #. This is the comment symbol, and theline will not be read by the input parser.

Run the file with the command

$ atk si-1.atk si-1.out

The command performs a self-consistent calculation for the silicon crystal with 3 � 3 �3=27 k-points uniformly spaced in the first Brillouin zone (i.e. in the reciprocal lattice)following the Monkhorst–Pack scheme [6]. The three directions defined by the latticevectors, specified in the unit cell block in the input, file are known in ATK as the A, Band C directions, respectively. The present case is a good example showing that thesedirections do not always coincide with the Cartesian axes (which of course are used todefine the components of the lattice vectors). The three lines containing the keywordNumKPoints specify the density of the k-point sampling in the directions of the threereciprocal lattice vectors corresponding to the A, B, and C directions, respectively.

If a, b, and c are the lattice vectors, then the reciprocal lattice vectors ka, kb and kc aredefined as

ka = 2�(b� c)=V; kb = 2�(c� a)=V; kc = 2�(a� b)=V;

where V = |a � (b � c)| is the volume of the unit cell. In the case of Si, which has adiamond crystal structure corresponding to fcc (face-centered cubic) with two atoms inthe basis, the reciprocal unit cell is the same as the real space unit cell of bcc (body-centered cubic), and thus

ka = 2�a(�1; 1; 1); kb =

2�a(1;�1; 1); kc =

2�a(1; 1;�1);

where a is the Si lattice constant as given in the input file above.

We would now like to plot the band structure from the � point k = (0, 0, 0) to the X pointk = (2�/a, 0, 0). Clearly, the X point corresponds to k = (kb + kc)/2, a relation whichis independent of the actual value of the lattice constant. Therefore it is convenient touse the unit FracRecip to specify this point.

4.1 Band Structure of a Si Crystal 37

−20

−15

−10

−5

0

5

10

XΓ

Ene

rgy

(eV

)

k (Å)

SZ−GGADZP−GGA

Figure 4.1: The band structure of a Si crystal along the �–X direction calculated with GGA.

To calculate the band structure along the line �–X, create the file siband.atk with thecontents

System::Type Bulk

UnitCell::LatticeConstant 5.43 Ang


0. 0.5 0.5

0.5 0. 0.5

0.5 0.5 0.


Analysis::BandLine::A::Start (0., 0., 0.) FracRecip

Analysis::BandLine::A::End (0., 0.5, 0.5) FracRecip

Analysis::BandLine::A::NumPoints 100

and run the command

$ atk siband.atk si-1-band.dat -a si-1.nc -o si-1-band.nc

It is crucial to inform ATK about the unit cell when using the FracRecip unit, otherwisethe reciprocal lattice vectors cannot be calculated correctly. If the unit cell is omittedfrom the input file in analysis mode, the program will use a default unit cell corre-sponding to a simple cubic system with lattice constant 1 Å. However, the programwill not read the unit cell from the input file unless we also state that the system type isbulk (this is done by the first line in the input file). If one does not use the FracRecip

unit, but instead specifies the start and end points of the bandline explicitly in inverseBohr or Angstrom, then one may exclude the unit cell from the input file.

38 Periodic Systems

Figure 4.1 shows the obtained Si band structure, which indicates that Si would have anindirect band gap of 1.73 eV. Experimentally, however, Si is known to have an indirectband gap of 1.13 eV. The reason for the discrepancy is, in this case, that the SZ basisset is not versatile enough to describe a Si crystal well enough. The SZ basis set isspherically symmetric, but a Si crystal is held together by strongly directed covalentbonds. Therefore, the next thing to try is to use a DZP basis set, by the command:

$ atk si-1.atk si-2.out -o si-2.nc --param "NumOrb::BasisSet::Size DZP"

Then, extract the band structure by the command

$ atk siband.atk si-2-band.dat -a si-2.nc -o si-2-band.nc

With the more accurate basis set we find a band gap below 0.6 eV, i.e. the band gapis severely underestimated. This is a well known artifact of DFT, or at least LDA andGGA-PBE.

4.2 CONVERGENCE OF AN AL WIRE

In this section, we will calculate the total energy of a one-dimensional aluminium wireand illustrate how the k-point sampling can be increased to converge the total energy.

Create the following file (alchain-1.atk)

System::Type Bulk

Method::Type NumOrb


Bulk::ElectronTemperature 1.e-3 Ry

UnitCell::LatticeConstant 1. Ang


15. 0. 0.

0. 15. 0.

0. 0. 2.8




Al 7.5 7.5 1.4


and run the calculation

$ atk alchain-1.atk alchain-1.out

4.2 Convergence of an Al Wire 39

−140.30

−140.25

−140.20

−140.15

−140.10

−140.05

0.01 0.1

Tot

al E

nerg

y (e

V)

Bulk::ElectronTemperature (eV)

812204060

120

Figure 4.2: Total energy as function of the electronic temperature. For each electron temperaturethere are data for runs with 8, 12, 20, 40, 60, and 120 k-points.

We will now check if the total energy is converged with respect to the number of k-points in the chain direction (the C direction). Try to perform runs with 8, 12, 20, 40,60 and 120 k-points by changing the value of the keyword Bulk::NumKpoints::C.Use the command

$ atk alchain-1.atk alchain-2.out -o alchain-2.nc \

--param "Bulk::NumKPoints::C 20"

or write a script to perform the whole series of calculations (cf. Appendix F).

In Figure 4.2, we show the total energy as a function of the number of k-points andthe electron temperature, and we see that more than 20 k-points are needed to ob-tain a well converged result at the low electron temperature used in the calculation(alchain-1.atk, 10–3 Ry = 0.013 eV).

The problem with the Al chain is that it is a metal and the Fermi level is thereforecrossing a band. The region around the Fermi level needs to be described accurately,which requires a dense k-point sampling. By introducing a finite electron temperature,the Fermi level crossing is smoothed out and fewer k-points are needed.. The elec-tron temperature is changed with the command Bulk::ElectronTemperature. Thefollowing command generates a run with 10 k-points and an electron temperature of0.1 eV.

$ atk alchain-1.atk alchain-t0.1.out -o alchain-t0.1.nc \

--param "Bulk::ElectronTemperature 0.1 eV"

ATK accepts energies given as temperature-equivalents, and it is often convenient tospecify the electron temperature as e.g. 300 K.

40 Periodic Systems

0.00

0.01

0.10

0.01 0.10

Rel

ativ

e E

nerg

y (e

V)

Bulk::ElectronTemperature (eV)

812204060

120

Figure 4.3: Total energy per atom of an Al chain where the bond lengths are alternatingly 2.7 Åand 2.9 Å, relative (on an absolute scale) to the energy of an Al chain where all bond lengths areequal to 2.8 Å (cf. Figure 4.2). Results are shown for calculations using 8, 12, 20, 40, 60 and120 k-points (symbols as in Figure 4.2). Note the different energy scales in Figure 4.2 and in thisfigure.

In Figure 4.2, the total energy is plotted as function of the electron temperature fordifferent numbers of k-points. As can be seen, the total energy is a function of boththese quantities, and for a given temperature it is important to choose sufficiently manyk-points to get a converged result, otherwise, artifacts may arise. However, as indicatedby Figure 4.2, a higher electron temperature facilitates convergence of the total energywith respect to the k-points.

Relative energies converge faster with respect to number of k-points than total energies.Moreover, increasing the electron temperature will typically change a relative energyonly slightly, which in turn can be used to reduce the number of k-points further. Thisis illustrated in Figure 4.3, where it can be seen that the relative energy is less sensitiveto the choice of k-points than the total energy (compare the different energy scales ofFigures 4.2 and 4.3), and that the relative energy only depends slightly on the electrontemperature.

C H A P T E R

5TWO-PROBE SYSTEMS

The unique capability of ATK is the calculation of so-called two-probe systems, and weshall in this last part of the tutorial devote special attention to setting up and analyzingthese nanoscale systems.

A two-probe geometry consists of two semi-infinite systems coupled via a scatteringregion. In ATK, this geometry is broken down into three subsystems: the left electrode,the central region, and the right electrode. Figure 5.1 illustrates this division of thegeometry for a lithium bcc crystal in the (100) direction. The central region mustinclude the part of the Li (100) surfaces perturbed by the scattering region (indicatedin the figure by a dashed line). Beyond the dashed line, the perturbation from thescattering region is screened out by the metallic leads, and the effective potential retainsits bulk value. Thus, the effective potential of the complete system can be obtained fromtwo independent calculations of the electrode regions, and a composite two-probecalculation of the central region.

5.1 GEOMETRY OF AN IDEAL LITHIUM BCC LATTICE

In the following, we describe the setup of the system illustrated in Figure 5.1. The inputfiles are, for reference, located in examples/tutorial in the installation directory.

41

42 Two-Probe Systems

5.1.1 LITHIUM ELECTRODES

The Li(100) electrodes are defined by the following input file (li-100.atk)


Method::Type NumOrb






1.0 0.0 0.0

0.0 1.0 0.0

0.0 0.0 4.0


AtomList::Format Fractional


Li 0.0 0.0 0.0

Li 0.5 0.5 0.125

Li 0.0 0.0 0.25

Li 0.5 0.5 0.375

Li 0.0 0.0 0.5

Li 0.5 0.5 0.625

Li 0.0 0.0 0.75

Li 0.5 0.5 0.875


We here introduce yet another way of specifying the atomic coordinates in the atomlist; the Fractional keyword implies that all coordinates are expressed in terms of thelattice vectors, defined in the UnitCell block. As before, we use the simplest basis set(SZ) in order to speed up the calculations in these examples. For more accurate results,

Figure 5.1: Two-probe geometry of a Li (100) crystal. The left electrode, thecentral region, and the right electrode are combined by matching the hatchedatoms with the electrode lattice vectors. The hatched atoms are definedthrough the keywords TwoProbe::LeftElectrode::EquivalentElectrodeAtom andTwoProbe::LeftElectrode::EquivalentCentralAtom.

5.1 Geometry of an Ideal Lithium BCC Lattice 43

use the default value by excluding the line which specifies the basis set from the inputfile.

Lithium is a bcc metal, and in the (100) direction there are two inequivalent planes, Aand B. The electrode file sets up the layer structure (ABABABAB). There is one atom perlayer, with periodic boundary conditions.

In choosing the electrode cell, three important considerations must be taken.

� The cell must be long enough in the transport direction, such that matrix elementsonly extend to the first repeated cell in the z direction. This is because ATKassumes that all other matrix elements are zero; if they, in reality, are not, thecontribution from these overlaps will not be properly accounted for.

� The cell must be periodic in all three directions.

� The C direction must be perpendicular to the A and B directions.

These requirements, in particular that of a periodic electrode cell in the C direction inconjunction with the first point, may sometimes make it necessary to use a very longelectrode cell. However, this only increases the computational efforts of generatingthe electrode, while the two-probe calculation is not affected, since the two-probecalculation only includes orbitals in the electrode region that are interacting with thecentral region.

We next calculate the effective potential of the electrode region with the command:

$ atk li-100.atk li-100.out

Since the electrode region of the left and the right electrode is the same, we can usethe same electrode file for both.

5.1.2 TWO-PROBE GEOMETRY

We next set up the central region with the commands (li-ideal.atk)


Method::Type NumOrb

# ----------------------------------

# Ideal Li (100) system

# ----------------------------------


TwoProbe::LeftElectrode::NetCDFFile li-100.nc

TwoProbe::RightElectrode::NetCDFFile li-100.nc

TwoProbe::Contour::DeltaEMin 2.00000 Ry

AtomList::Format Ang



Li 0.00000 0.00000 0.00000

Li 1.75500 1.75500 1.75500

Li 0.00000 0.00000 3.51000

Li 1.75500 1.75500 5.26500

Li 0.00000 0.00000 7.02000

Li 1.75500 1.75500 8.77500


Note that no unit cell parameters are included in the two-probe setup file; this infor-mation, including the k-point sampling, is obtained from the electrode files.

The next step is to align the central region with the electrode atoms. ATK makesan automatic alignment as illustrated in Figure 5.1. The automatic alignment re-lies upon the specification of equivalent atoms in the electrode and in the centralregion. These atoms are defined with the keywords TwoProbe::LeftElectrode::

EquivalentCentralAtom and TwoProbe::LeftElectrode::EquivalentElectrodeAtom.The two equivalent atoms will be positioned one electrode lattice vector apart.

By default the first atom in the central region is positioned one lattice vector from thefirst atom in the left electrode cell, and the last atom in the central region is positionedone lattice vector from the last atom in the right electrode cell. Naturally, similarkeywords TwoProbe::RightElectrode::... exist for the right electrode.

To check that the alignment is made properly, the structure should be inspected visually.When ATK is called with the command line option -c, it will stop after generating thetwo-probe geometry. The xyz file can then be viewed using e.g. VIRTUAL NANOLAB.

$ atk li-ideal.atk -o li-ideal-check.nc -c --xyz --xyz-na 3 --xyz-nb 3

The --xyz-na and --xyz-nb options repeat the structure the specified numbers oftimes in the x and y directions, and in this way it can be verified that the computationalcell is setup properly. The geometry is output in the file li-ideal.xyz.

When the atoms in the electrodes are aligned relatively to the atoms in the centralregion, the two-probe cell vectors can be determined from the boundaries of the elec-trode cells.

In the current version of ATK, the electrostatic problem is by default calculated by anFFT routine, and the entire system must form a periodic structure. This means that thefirst layer in left electrode must be similar to the first layer in the right electrode. Itis however possible to have different electrodes. In that case, one must include thekeyword TwoProbe::UseMultigridForElectrostatics in the input file, and as thename indicates the Poisson equation is then solved by a multigrid method instead.

5.1 Geometry of an Ideal Lithium BCC Lattice 45

5.1.3 MISWRAPPED ELECTRODE ATOMS

A frequent setup mistake with the two-probe geometry is miswrapped atoms. Mis-wrapping happens when the electrode atoms are aligned such that some atoms endup outside the supercell. An example of miswrapping is illustrated in Figure 5.2. Theexample is generated by adding an additional atom

Li 0.00 0.0 10.53

to the atom list in li-ideal.

5.1.4 TRANSMISSION SPECTRUM

To perform a two-probe calculation of li-ideal.atk, type the command

$ atk li-ideal.atk li-ideal.out

After some waiting, the self-consistent solution will be obtained and stored in theNETCDF file li-ideal.nc. From this file various information about the system canbe obtained. To calculate the transmission spectrum generate the file trans.atk withthe content




and run the command

$ atk trans.atk -a li-ideal.nc trans.out

Inspection of the file trans.out shows ideal transmission of the lithium bands as ex-pected.

Note, that small negative numbers can appear in the transmission spectrum, due toround-off or other numerical errors in the integration in a complex plane (cf. also be-low). The absolute values of these negative numbers can be reduced by reducing thevalue of Analysis::TransmissionSpectrum::Eta, cf. page 90.

Figure 5.2: Miswrapped two-probe geometry of a Li (100) crystal. The right electrode is alignedsuch that some atoms are outside the computational cell.


Figure 5.3: The contour used for the complex energy integral.

5.1.5 ENERGY INTEGRALS

The two-probe calculations are based on non-equilibrium Green’s function (NEGF)techniques, where the electron density is resolved in energy space. To obtain the totaldensity from this spectral density, we need to perform an integral along the energy axis.Part of the integral is performed on a complex contour, as illustrated in Figure 5.3. Adetailed description of the contour can also be found in Ref. 1.

In the complex plane, the functions are smooth and only few points are needed forvery accurate integration. The default parameters are conservative and will usuallybe sufficient. However, if the value of TwoProbe::Contour::DeltaEMin is too low,there may be bound states below the contour and this will result in charge missingin the integration. This will typically manifest itself in a rapidly diverging value of "q"during the SCF cycle (as reported in the log file), and what generally happens is thatthe calculation converges (easily), but not to the correct result but rather to a situationwhere there are no electrons in the central region any more.

If this is the case, it is necessary to increase TwoProbe::Contour::DeltaEMin. For anestimate of the required value, one can perform a molecular calculation for the isolatedatoms to obtain the eigenenergies of the valence states. Subsequently, check that thecontour starts below these states by increasing the value of TwoProbe::Contour::DeltaEMin accordingly. If the new value is much larger than the default, it is generallya good idea to also increase TwoProbe::Contour::NumPoints in order not to loseaccuracy in the contour integration.

5.2 BENZENE RING COUPLED TO LITHIUM ELECTRODES

In this example, we will investigate the conductance of a benzene ring coupled withtwo lithium fcc (111) electrodes. This system is similar to that considered in Ref. 12,where gold electrodes were considered. That calculation takes much longer time, so

5.2 Benzene Ring Coupled to Lithium Electrodes 47

TwoProbe::Contour::DeltaEMin real EnergyUnitThe distance from the Fermi level to the energy where the complex contour starts.Recommended value is twice the binding energy of the lowest eigenstate. If there areelectronic states below the complex contour, charge will be missing in the integration.

TwoProbe::Contour::RealAxis::PointDensity real EnergyUnitThe spacing between the points in the close-to-real axis part of the contour in thevoltage bias window.

TwoProbe::Contour::RealAxis::Eta real EnergyUnitSmall complex number added to the denominator in the real axis integration. A corre-sponding parameter defined by the keyword (Analysis::TransmissionSpectrum::Eta) is used for calculating the transmission spectrum.

Table 5.1: Important parameters for controlling the complex contour integration. For units anddefault values, refer to the keyword reference section on pp. 86– 104.

for this example we choose lithium, which has only a single s band; the behavior ofthe system still closely resembles gold, but is computationally cheaper. Figure 5.4illustrates the system.

5.2.1 ELECTRODES

The lithium electrodes form a 3 � 3 structure, with 9 atoms in each layer. The leftelectrode contains 3 layers with the stacking (BCA), while the right electrode has thestacking (CBA). The stacking of the electrodes is chosen to match the central DTB (di-thiol benzene) region which has the structure (BC–DTB–CBA). The complete two-probestructure has the geometry (BCA)BC–DTB–CBA(CBA).

The complete two-probe structure is not perfectly periodic, but has a stacking fault atthe interface. This can be seen by repeating the structure (BCA)BC–DTB–CBA(CBA)|(BCA)BC–DTB–CBA(CBA), which has a stacking fault (BA|BC) at the interface (the ideal interfaceis BA|CB). Such small disturbances of the periodicity are allowed, and do not affect thecalculation.

In order to setup the electrode files, we will use the ATK keyword Electrode::NumRep.

Figure 5.4: A benzene molecule between two Li (111) surfaces. Note that some of the atoms inthe benzene ring have been periodically wrapped. Wrapping of atoms in the central region doesnot affect the calculation, since periodic boundary conditions apply in the transverse directions.


With this keyword, the file for the left electrode (li111-bca.atk) becomes:


Method::Type NumOrb



Electrode::NumRep::A 3

Electrode::NumRep::B 3

UnitCell::LatticeConstant 5.4537 Bohr


0.500000 -0.866025 0.0000000

0.500000 0.866025 0.0000000

0.000000 0.000000 2.4494897


AtomList::Format Fractional


Li 0.3333333 0.66666666 0.0000000

Li 0.6666666 0.33333333 0.3333333

Li 0.0000000 0.00000000 0.6666666


The Electrode::NumRep parameter constructs a super cell consisting of the specifiedunit cell repeated Electrode::NumRep times along the specified axes. Hence, in theabove example, we construct a supercell with 3 � 3 = 9 atoms per layer. The benefitof setting up the system, which will contain 27 atoms in the end, through the keywordsElectrode::NumRep instead of typing in the coordinates of the 27 atoms, is that thesupercell calculation can be performed with 3 atoms and 9 k-points in the A/B-plane,instead of with 27 atoms and 1 k-point. The results would be the same, as one canconvince oneself of by applying Bloch’s theorem and by folding the supercell Brillouinzone accordingly, but the 27-atom case would take much longer since the calculationtime scales with the number of atoms to the third power, but only linearly with thenumber of k-points. Moreover, when running in a parallel computing environment,the program is designed to parallelize the k-point sampling, and so in this case the27-atom calculation would, with access to 9 parallel nodes, take the same time as a3-atom calculation.

ATK will automatically adjust the k-point grid to account for the zone folding, sothe k-point sampling that is specified in the electrode input file should be the de-sired sampling for the complete system, i.e. ultimately the composite two-probe sys-tem. To be specific, in the present example the final two-probe calculation will beperformed with a single k-point in the A/B plane, since we use the default valuesElectrode::NumKPoints::A/B (note that the sampling in the C-direction is a com-pletely different matter), while the actual electrode calculation will be performed with9 points.

The use of the keywords Electrode::NumRep is limited to the electrode input files. In

5.2 Benzene Ring Coupled to Lithium Electrodes 49

the central region, it is necessary to explicitly list the coordinates of all atoms. Thus, ifwe for example wish to include 3 surface layers of Li in the central region, we need togive the coordinates of all 27 surface atoms in the two-probe input file.

In the right electrode file li111-cba.atk, the electrode layers are shifted to reflect thestacking sequence CBA, i.e. it contains the following atom list


Li 0.3333333 0.66666666 0.0000000

Li 0.6666666 0.33333333 0.3333333

Li 0.0000000 0.00000000 0.6666666


All other lines are identical to li111-bca.atk.

5.2.2 CENTRAL REGION

To glue together the central region with the electrodes, we need to specify the equiva-lent electrode and central atoms. By default, ATK will use the first and the last atom inthe central region atom list to match the first and last atom of the left and the right elec-trodes, respectively. In the following example, we ensure this by placing the matchingatoms as the first and last atom in the central atom list:


Method::Type NumOrb


TwoProbe::LeftElectrode::ATKFile li111-bca.atk

TwoProbe::RightElectrode::ATKFile li111-cba.atk

#(BCA)BC-DTB-CBA(CBA)



Li 1.44298600 0.83310800 5.89096100

...

Li 0.00000000 0.00000000 23.00722200





The central atom list is too long to include here; it can be found in the file li-dtb-li+

0V.atk in the examples/twoprobe directory.


0

0.2

0.4

0.6

0.8

1

1.2

−4 −2 0 2 4

Tra

nsm

issi

on C

oeffi

cien

t

Energy (eV)

0 V1 V

Figure 5.5: The transmission coefficient of the Li–DTB–Li system at 0 V and 1 V bias.

As opposed to the previous example, we do not make independent runs of the elec-trodes. Instead, we use the keywords TwoProbe::LeftElectrode::ATKfile and TwoProbe::

RightElectrode::ATKfile. In this way, the electrode calculations are made implicitlywhen performing the two-probe calculation.

Calculate the transmission spectrum with the command

$ atk li-dtb-li+0V.atk li-dtb-li+0V.out

The calculation will take some time. The transmission spectrum, extracted from theoutput file, is illustrated in Figure 5.5.

5.2.3 FINITE BIAS

To perform a calculation at finite bias, we simply shift the chemical potential of the leftelectrode relative to the right electrode. To do this, add the command

TwoProbe::LeftElectrode::Voltage 1 eV

and run the new file:

$ atk li-dtb-li+1V.atk li-dtb-li+1V.out -i li-dtb-li+0V.nc

With the -i option we use the zero bias density as a initial guess for the self-consistentcalculation to speed up the calculation.

5.3 Heterogeneous Electrodes 51

The result is shown in Figure 5.5. Due to the applied bias, the band edge of lithium hasmoved closer to the HOMO level of the DTB, and part of the transmission is cut off atlow energies.

5.3 HETEROGENEOUS ELECTRODES

A new and unique feature in ATK is the possibility to have different electrode materials.This can be used to study systems such as a metal–nanotube contact, or an interfacebetween two different metals.

There are a couple of separate aspects to keep in mind when setting up a heterogeneousproblem, compared to all other two-probe system so far considered in this tutorial,where the electrodes were always the same (except for some differences in the stackingsequences).

First of all, we mentioned earlier in this chapter that there is a requirement in ATKthat the electrodes must be periodic in all directions, including the transport or Cdirection. The underlying reason for this requirement is that the Poisson equation,which is an integral part of the self-consistent procedure, is most efficiently solved by aFourier-transform method. There is, however, another method for solving this equationwhich does allow for the electrodes to be different. This method is called the multigridmethod, and it has now been implemented in ATK. By setting the keyword

TwoProbe::UseMultigridForElectrostatics

to True (by default it is False), the requirement of a periodic electrode cell in the Cdirection is eliminated. The multigrid method is usually quite a bit slower than theFourier transform method, which is why it is not used by default for electrodes whichindeed are periodic.

The procedure of solving the Poisson equation is of course not restricted to the trans-port direction. In the directions perpendicular to the interface plane, the periodicityrequirement still holds, which is very important to keep in mind. It means, for instance,that if we want to consider an interface between two different metals, these must havethe same lattice constant. Since it is highly unlikely to find two such materials, one hasto resort to various tricks to circumvent this problem.

One way is it stretch one of the materials slightly (i.e. introduce strained materials),an option which is viable if the difference in the lattice constants is small (such as forinstance between Au and Al). Another option is to try to come closer to commensuratecells by making surface supercells, i.e. repeat the two surface unit cells a differentnumber of times, in the hope that, say, 3 cells of one material and 4 cells of the otherbecome almost the same size.


Figure 5.6: The geometry of the heterogeneous [111] bcc Li–C linear chain two-probe system,visualized in VIRTUAL NANOLAB. Note that because we use the keywords NumRep to set up theLi electrode, only the primitive electrode unit cell is shown. In the central region, however, wehave full layers of Li atoms.

Note that it is not sufficient to just come close; the two electrode cells must have exactlysame size in the A and B directions. Thus, usually one additionally has to include somestrain to get a perfect match.

One type of system where this latter requirement is trivial to fulfill is if one electrode(or both) is a quasi one-dimensional system, such as a chain of atoms or a nanotube. Inthis case the unit cell in the A and B directions can be chosen arbitrarily (as long as it islarge enough to avoid interactions between the repeated copies of the central region).

To demonstrate this, we shall here consider an example of a contact between a lithiumsurface and a atomic carbon wire. This is admittedly a somewhat artificial system;a more realistic example would instead be, say, a carbon nanotube and a gold orpalladium surface. The principles involved in setting up and calculating such a systemare however precisely the same as for our toy model. The only real difference is thatour model can easily be calculated on a standard PC within an hour, whereas a morerealistic system probably requires the use of a cluster (to take advantage of the parallelpower in ATK, in particular for k-point sampling), and uses much more memory. Still,carbon wires are actually a topic of current research, at least when embedded insidecarbon nanotubes, so the system perhaps has some relevance anyway!

First, let us consider the left electrode, which is a Li bcc 3�3 [111] surface. The properway to set such a system up is by using the keywords Electrode::NumRep; that way weonly need to specify the three atoms in the primitive surface unit cell (li111-3x3.atk):





Electrode::NumKPoints::A 4

Electrode::NumKPoints::B 4


Electrode::NumRep::A 3

Electrode::NumRep::B 3

UnitCell::LatticeConstant 1. Angstrom


4.9629 0 0

-2.48145 4.298 0

0 0 6.07829




Li 2.48145 1.43267 0.506524

Li 0 0 1.51957

Li 0 2.86533 2.53262

Li 2.48145 1.43267 3.54567

Li 0 0 4.55871

Li 0 2.86533 5.57176


Three layers of Li are required to have a periodic cell for the [111] cleaving, and theseatoms are also sufficient to have a long enough electrode cell. In addition to run-ning much faster, the NumRep calculations uses far less memory, which can be a realconcern on workstations. We use a small basis set, again to make the calculationsmanageable and fast, but try to make the k-point sampling a bit more realistic by notjust using the default 1�1 sampling in the interface plane.

The right electrode is an atomic carbon wire. Precisely as we did for the Li chain inSection 2.2 we must determine the spacing between the carbon atoms by minimizingthe total energy. We can basically use the same script as we did then (and for thatreason, this is not included in the scrips shipped with ATK); the result is that the minimalenergy is obtained with a distance of 1.34 Å between the C atoms (using a SZ basisset).

In order to have a long enough electrode cell it is sufficient to include two repetitions,at least to ensure that all first-order interactions are accounted for (cchain.atk):










14.8887 0. 0.

-7.44435 12.894 0.

0. 0. 2.68




C 2.48145 7.16337 0

C 2.48145 7.16337 1.34


Pay attention to the unit cells in the two electrodes file, and note that they are identicalin the A and B directions, once the 3�3 repetition is applied, as required.

Note also that we are forced to use the same k-point sampling in both electrodes,despite the fact that we could have settled for only a single k-point in the A and Bdirections for the carbon chain. This is because the sampling in the central regionis defined to be the same as in the electrodes (which is why there are no keywordsTwoProbe::NumKPoints), and hence the electrodes must have the same sampling.

Running the electrodes,

$ atk li111-3x3.atk li111-3x3.out

$ atk cchain.atk cchain.out

should be rather fast.

In the central region we put three layers of Li and 8 carbon atoms. This is most likelyenough for proper screening on both sides (we will soon verify this by inspecting theresults); linear chains are generally quite poor at screening surfaces (a pure geometricalconsequence) so one should expect to use a couple of extra layers on the chain side,which fortunately not is particularly costly as each layer contains only a single atom.We consistently use the smallest basis set (SZ) and a reduced mesh cut-off to speed upthe calculations.

Thus in total there are 35 atoms in the central region (li-cchain.atk):


NumOrb::MeshCutOff 100 Ry


TwoProbe::LeftElectrode::NetCDFFile li111-3x3.nc


TwoProbe::RightElectrode::NetCDFFile cchain.nc

TwoProbe::Contour::DeltaEMin 4. Ry

TwoProbe::Contour::Circle::NumPoints 50

SCF::TwoProbe::InitialBulkRun T

SCF::DiagonalMixingParameter 0.2

SCF::HistorySteps 20

TwoProbe::UseMultigridForElectrostatics T


Li 0.0000 0.0000 6.5848

Li -2.4815 -1.4327 7.5979

...

C 0 5.73067 16.26

C 0 5.73067 17.60

C 0 5.73067 18.94

C 0 5.73067 20.28


As for the other parameters used in the two-probe input file, an initial bulk run is of-ten also useful for heterogeneous systems. We use a lot of history steps, otherwise itwill be difficult to reach convergence, and for safety we have increase the values ofTwoProbe::Contour::DeltaEMin and to avoid the risk of losing charges in the com-plex contour integration.

We shall run the two-probe system using the --verbose option to get a bit more infor-mation about the convergence progress:

$ atk li-cchain.atk li-cchain.out --verbose

The run will take about an hour. Considering that the total amount of k-points is 400, itis clear that this system (both electrode and two-probe calculations) is extremely wellsuited to calculate in parallel, and the performance should scale almost linearly withthe number of processors.

Inspecting the output file, we observe that the Mulliken populations change relativelylittle during the second self-consistent cycle, indicating that the convergence is stable,and that the initial bulk run did indeed provide a good initial guess for the calculationwith open boundary conditions. This is the reason why we can afford such a largemixing parameter (we set SCF::DiagonalMixingParameter to 0.2 in the two-proberun). We also see that the population of the C atoms quickly drops back to 4 as wemove away from the surface, but still


Figure 5.7: The left part of the figure shows a contour plot (cutting through the C atoms) of theso-called electrostatic difference potential. This potential, which is defined as the solution tothe Poisson equation for an electron density corresponding to the difference between the atomicdensities and the self-consistent density, is relatively constant far away from the contact region,both deep in the Li surface and in the C chain. The right plot show the effective single-electronpotential. Obviously the carbon chain extends to the left. The scale is different in the two plots,but the colors work the same way.

Pay attention to the forces in the final result. The C atom closest to the surface expe-riences an almost vanishing force in the z-direction (this was, in fact, the criterion forchoosing its distance to the surface), but the other atoms in the C chain are subject to(large) forces that alternate in direction. Thus it is clear that in the presence of the Lisurface one must reconsider the position of all C atoms in the central region, and aproper relaxation should be carried out. (There are also forces on the Li atoms in thecentral region, but one can expect that the influence on the surface from the chain ismuch smaller than the opposite effect.)

Let us look further at the Mulliken populations (some irrelevant lines are cut out).

%block Results::MullikenPopulations

0 Li q = 1.03155 [ s: 1.032 ]

1 Li q = 1.03915 [ s: 1.039 ]

2 Li q = 0.85846 [ s: 0.858 ]

3 Li q = 1.03078 [ s: 1.031 ]

4 Li q = 1.03918 [ s: 1.039 ]

...

13 Li q = 1.03556 [ s: 1.036 ]

14 Li q = 0.63550 [ s: 0.636 ]

15 Li q = 1.03078 [ s: 1.031 ]

16 Li q = 1.03616 [ s: 1.036 ]

17 Li q = 0.85840 [ s: 0.858 ]

...


27 C q = 4.27496 [ s: 1.510, y: 0.811, z: 1.144, x: 0.810 ]

28 C q = 4.18817 [ s: 0.961, y: 1.079, z: 1.068, x: 1.080 ]

29 C q = 4.07750 [ s: 0.942, y: 1.023, z: 1.089, x: 1.024 ]

30 C q = 4.07890 [ s: 0.919, y: 1.042, z: 1.076, x: 1.042 ]

31 C q = 4.05772 [ s: 0.922, y: 1.027, z: 1.081, x: 1.028 ]

32 C q = 4.04744 [ s: 0.920, y: 1.023, z: 1.081, x: 1.024 ]

33 C q = 4.03796 [ s: 0.918, y: 1.019, z: 1.081, x: 1.020 ]

34 C q = 4.05127 [ s: 0.927, y: 1.016, z: 1.092, x: 1.016 ]

# ----------------------------------------------------------------

# Total charge = 59.0256

# ----------------------------------------------------------------

%endblock Results::MullikenPopulations

Note how the bonding between the surface and chain is manifested in charge transferfrom the Li atom number 14 (the contact point) into the C chain. An isolated Li atomhas a population of 1 and a C atom 4, but in the converged two-probe system thecontact Li atom only has a population of 0.63, while the two closest C atoms in thechain harbor about 4.2–4.3 electrons. One can also observe how the C atom closestto the surface has its p-shell polarized along the z-direction, while all other atoms aremore symmetrically populated. Finally, the total Mulliken population approaches thebulk value 4 as we move away from the surface, and the most distance Li atoms alsohave populations close to unity, showing that the screening is sufficient.

The charge transfer between the Li and the C chain is not particularly large, and thesystem has a very low conductance. One might imagine that the conductance wouldbe larger if a different contact point was used (such as an interstitial surface point,where the chain could penetrate deeper). By moving the chain around one couldmap out the conductance as a function of the contact point. The conductance wouldmost likely also change considerably if we were to relax the positions of the C atomsmore carefully, as mentioned above. All this is however well beyond the scope of thistutorial, and so we leave it up to the interested reader to carry out this analysis!

C H A P T E R

6SPIN-POLARIZATION AND k-POINT

SAMPLING OF TWO-PROBE

SYSTEMS

In this chapter, we focus on some of the more advanced features in the two-probecalculations. The examples will illustrate how to treat spin polarization for two-probesystems and the use of k-point sampling. Other issues that will be addressed is how toachieve convergence for difficult cases and examples of how to extract data from theNETCDF file.

The input files for these calculations are located in examples/twoprobe-spin in theinstallation directory.

6.1 TRANSMISSION THROUGH A FE–MGO–FE INTERFACE

The example in this chapter illustrates the transmission through a Fe–MgO–Fe interface.This is a system which currently recieves a lot of attention in the field of spintronics [3,14], because it turns out to be an efficient spin filter, i.e. it only allows the majority spinelectrons of a current to pass through the interface.

In this chapter, we investigate a realistic application of ATK, and the calculations aretherefore rather time consuming. It takes about 24 hours to complete the calculationsin this chapter on a modern PC. Users with the possibility to run ATK in parallel shouldhowever experience a substantial speedup on these examples.

59

60 Spin-Polarization and k-point Sampling of Two-Probe Systems

Figure 6.1: Geometry of the electrode and central region of the Fe-MgO-Fe system. Fe atoms areturquise, O atoms red, and Mg atoms violet.

The system we will investigate is illustrated in Figure 6.1. The system has previouslybeen studied numericaly by Wortmann et. al. [15].

6.1.1 FE ELECTRODES

The Fe (100) electrodes are defined by the following input file (fe-100.atk)


Electrode::ElectronTemperature 0.1 eV

Method::Type NumOrb

NumOrb::BasisSet::Size SZP

NumOrb::XCFunctional LSDA-PZ

SCF::Tolerance 1.e-5


UnitCell::LatticeConstant 1. Bohr


5.42 0.00 0.00

0.00 5.42 0.00

0.00 0.00 16.26


AtomList::Format Bohr


Fe 1.355 1.355 1.355

6.1 Transmission Through a Fe–MgO–Fe Interface 61

Fe 4.065 4.065 4.065

Fe 1.355 1.355 6.775

Fe 4.065 4.065 9.485

Fe 1.355 1.355 12.195

Fe 4.065 4.065 14.905


By setting the exchange-correlation potential to LSDA-PZ the system is modelled as aspin polarized system. The initial spin is by default chosen as the maximum polariza-tion of each atom, and with the majority spin pointing up. The total initial spin can alsobe specified with the keyword SCF::InitialSpinPolarization. For instance, setting

SCF::InitialSpinPolarization -0.1 fractional

reverses the sign of the majority spin.

Note, that we have increased the electron temperature in the electrode. This is acomputational trick that will help us in getting a converged result.

To calculate the effective potential of the electrode region we use the command.

$ atk fe-100.atk fe-100.out

Because the electrode region of the left and the right electrode is the same, we can usethe same electrode file for both.

6.1.2 FE-MGO-FE GEOMETRY

We next set up the central region with the commands (fe-mgo-fe.atk)


Method::Type NumOrb

NumOrb::BasisSet::Size SZP

NumOrb::XCFunctional LSDA-PZ


AtomList::Format Bohr


Fe 1.355 1.355 1.355

Fe 4.065 4.065 4.065

Fe 1.355 1.355 6.775

Fe 4.065 4.065 9.485

Fe 1.355 1.355 12.035

Mg 4.065 4.065 16.205


O 1.355 1.355 16.205

Mg 1.355 1.355 20.355

O 4.065 4.065 20.355

Mg 4.065 4.065 24.505

O 1.355 1.355 24.505

Fe 1.355 1.355 28.675

Fe 4.065 4.065 31.225

Fe 1.355 1.355 33.935

Fe 4.065 4.065 36.645


TwoProbe::LeftElectrode::NetCDFFile ./fe-100.nc

TwoProbe::RightElectrode::NetCDFFile ./fe-100.nc

Note the use of the keyword SCF::HistorySteps. With this keyword, we improves theperformance of the SCF routine, at the expense of slightly more memory usage sincethe algorithm now uses information from 20 previous steps to make a new step.

To perform the SCF calculation use the command

$ atk fe-mgo-fe.atk fe-mgo-fe.out

6.1.3 TRANSMISSION SPECTRUM

Upon inspection of the output file, fe-mgo-fe.out, we notice that there are two sec-tions with the transmission spectrum, corresponding to the spin-up and spin-downcomponent of the electronic system. The transmission coefficients of the spin up anddown electrons are shown in Figure 6.2. Notice how the transmission of spin downelectrons are strongly suppressed at the Fermi-level. Hence, the calculation demon-strates that the MgO interface indeed functions as a spin filter.

6.2 k-POINT SAMPLING IN TWO-PROBE SYSTEMS

The Fe–MgO–Fe interface system has periodic boundary conditions in the two direc-tions parallel to the interface (the A and B directions). In these directions, the wave-functions are Bloch waves and we must average over all the k-vectors when calculatingphysical quantities. In our two-probe calculation, two different quantities depend onthe number of k-points used in the A and B directions. The first quantity is the thedensity matrix, the other is the transmission coefficient. Usually, it will require morek-points to accurately calculate the transmission coefficient, compared to the densitymatrix, and in ATK, it is therefore possible to specify different numbers of k-points forthe two quantities.

6.2 k-point Sampling in Two-Probe Systems 63

10−6

10−5

10−4

10−3

10−2

10−1

100

−2 −1 0 1 2

Tra

nsm

issi

on C

oeffi

cien

t

Energy (eV)

Spin UpSpin Down

Figure 6.2: Transmission coefficients of the spin up and down electrons for the calculation de-fined in the file fe-mgo-fe.atk.

The number of k-points used for the density matrix is defined in the electrode inputfile. To perform a two-probe calculation with a 6 � 6 k-point mesh in the A and Bdirections, we must specify this mesh for the electrode file. We can do this by addingto the file (fe-100.atk) following lines




and save it in fe-100-6x6.atk and subsequently execute the command

$ atk fe-100-6x6.atk fe-100-6x6.out

The result of the electrode calculation can now be compared with the calculation with-out k-points. We see that all quantities, like the total energy, Mulliken populations (onlyshown when running with the {-}{-}verbose option), etc have changed significantly.This is a clear sign that an insufficient k-point sampling was used in the first calcula-tion. To have a well converged k-point sampling, the change in the total energy/atomshould as minimum be less than 10 meV, and preferably in the order of 1 meV/atom.If we test different k-point samplings for the electrode, we find that indeed the 6 � 6produces well converged results. By increasing the mesh to a 8 � 8 mesh the totalenergy only changes by 1 meV/atom. A calculation with a 4 � 4 mesh, on the otherhand, produces results that differ by 10 meV/atom.

Thus, the electrode calculations show that the 6 � 6 mesh is a good choice, and wenext include the 6 � 6 electrode files in the two-probe input file simply by copying thefile fe-mgo-fe.atk to fe-mgo-fe-6x6.atk and replace the lines


10−6

10−5

10−4

10−3

10−2

10−1

100

−2 −1 0 1 2

Tra

nsm

issi

on C

oeffi

cien

t

Energy (eV)

Spin UpSpin Down

Figure 6.3: Transmission coefficients at the � point for the two-probe system in Figure 6.1. Thedifferent curves are for calculations using 1 � 1, 2 � 2, 4 � 4, 6 � 6, and 8 � 8 k-points for thedensity matrix calculation. The bold lines show the result of the 6 � 6 calculation, as defined inthe input file fe-mgo-fe-6x6.atk

TwoProbe::LeftElectrode::NetCDFFile ./fe-100.nc

TwoProbe::RightElectrode::NetCDFFile ./fe-100.nc

with

TwoProbe::LeftElectrode::NetCDFFile ./fe-100-6x6.nc

TwoProbe::RightElectrode::NetCDFFile ./fe-100-6x6.nc

The two-probe calculation will use the k-point mesh of the electrodes and we performthe self-consistent two-probe calculation with the command

$ atk fe-mgo-fe-6x6.atk fe-mgo-fe-6x6.out

Note that we have not yet specified a k-point mesh for the transmission calculation,and the transmission calculation therefore uses the default k-point mesh, which is a1 � 1 mesh, i.e. the � point.

In Figure 6.3 we show how the transmission spectrum at the � point as function of thenumber of k-points used for converging the two-probe calculation. The transmissionsfor 6 � 6 and 8 � 8 are indistinguishable, indicating that the 6 � 6 mesh is sufficient.

6.3 k-point Sampling of the Transmission Spectrum 65

6.3 k-POINT SAMPLING OF THE TRANSMISSION SPECTRUM

We now have a converged density matrix and thereby a converged Hamiltonian ofthe system. The next step is to obtain a k-point converged transmission spectrum. Thecalculation of the transmission spectrum is a post processing step, and we will run atk inthe analysis mode using the converged Hamiltonian from the file fe-mgo-fe-6x6.nc.

The k-point sampling of the transmission coefficient is specified with the keywords

Analysis::TransmissionSpectrum::NumKPoints::A

Analysis::TransmissionSpectrum::NumKPoints::B

To calculate the transmission coefficient at the Fermi-level with a 20 � 20 k-pointmesh, we use the following input file (transmission.atk)

Analysis::TransmissionSpectrum::Calculate T

Analysis::TransmissionSpectrum::E0 0 eV

Analysis::TransmissionSpectrum::E1 +1 eV

Analysis::TransmissionSpectrum::Eta 1.e-5 eV


Analysis::TransmissionSpectrum::NumKPoints::A 20

Analysis::TransmissionSpectrum::NumKPoints::B 20

To perform the calculation, type the command

$ atk transmission.atk transmission.out -a fe-mgo-fe-6x6.nc

In the transmission.out file, we can read the k-point averaged transmission coeffi-cient at the Fermi-level E = 0. We see that it is significantly different from the transmis-sion coefficient at the � point (compare the numbers with Figure 6.2).

The density matrix was well converged for a 6 � 6 k-point sampling, however, even20 � 20 k-points are not enough to converge the transmission coefficient. This is illus-trated in Figure 6.4, which shows the spin-up and spin-down transmission coefficientcalculated with different numbers of k-point. To obtain a converged result, more than20 000 k-points are needed.

6.3.1 PLOTTING THE k-DEPENDENT TRANSMISSION

To understand why so many k-points are needed to converge the transmission coef-ficient, it is instructive to plot the k-dependence of the transmission coefficient. Allthe data used for calculating the k-point averaged transmission coefficient are storedin NETCDF data file, transmission.nc. In future versions of VIRTUAL NANOLAB, it


10−4

10−3

10−2

10−1

10 100 1000 10000 100000

Tra

nsm

issi

on C

oeffi

cien

t

Number of k−points

Spin UpSpin Down

Figure 6.4: Transmission coefficients of the Fe-MgO-Fe system as function of the number ofk-points used for the transmission calculation.

will be possible to manipulate such data from the graphical user interface, however,for now you will have to extract the data manually and plot them with your favoriteplotting program

Many programs like MatLab or Python has NETCDF support, and can be used ratherconveniently to extract data from the NETCDF file. However, in the following we willuse the command ncdump to extract the data. This is not an elegant solution but thecommand is available on most platforms. For more information, see Appendix C.1.

To obtain a list of all the variables contained in the NETCDF file, type

$ ncdump -h transmission.nc

The variables we want to extract from the NETCDF file are TransmissionSpectrumTransmission\_Up and TransmissionSpectrumTransmission\_Down. To extract the data use thecommand

$ ncdump -v TransmissionSpectrumTransmission_Up,TransmissionSpectrumTransmission_Down transmission.nc

After the header part of the output, the spin up and down transmission spectra areprinted. There are 400 data elements in each variable. To get the values of the corre-sponding k-points we extract the variable TransmissionSpectrumKpoints

$ ncdump -v TransmissionSpectrumKPoints transmission.nc

We get 400 values, corresponding to the kx and ky of 200 k-points. Note that a(20 � 20) k-point mesh has 400 points, but ATK only calculates the transmission

6.3 k-point Sampling of the Transmission Spectrum 67

Figure 6.5: k-dependent transmission coefficient for majority (spin up) and minority spin (spindown) at the Fermi energy for the Fe–MgO–Fe system. A (200 � 200) k-point grid covering theentire Brillouin zone was used for the calculation.

at 200 of the points, as the transmission at the missing points can be obtained fromT(�k) = T(k).

We may now pair the k-points with the transmission coefficients, and obtain the plotshown in Figure 6.5. We see that there is a strong relation between the k-vector and thetransmission coefficient, and the behavior for the two different spin directions are ratherdifferent. For the majority spin direction, the � point gives the highest transmission,whereas for the minority spin direction, only some special points in the Brillouin zonecan transmit the current.

Can such spin filtering be observed in real devices? To answer this question note thatwe model a perfect interface, whereas in reality there will be some surface roughness.The surface roughness destroys the conservation of the momentum parallel to the inter-face (kx, ky). To quantify the effect of surface roughness one must perform calculationsof systems that resemble the experimental situation. The effect of surface roughnesscan easily be investigated with ATK by increasing the unit cell, and adding some fluc-tuations in the interface. However, such calculations are beyond the scope of thistutorial.

A P P E N D I X

AATK VS. TRANSIESTA-C

ATK constitutes a major upgrade over its predecessor TRANSIESTA-C in terms of per-formance and features. Although there are some changes in the interface, these arerather limited, and the users will hopefully agree that the changes serve to simplify theinput files and make the software easier to use. Thus, we are confident that users ofTRANSIESTA-C will feel right at home with this new generation of Atomistix’ electronicstructure software.

Here is a short list of new or updated features:

� A general performance boost has been achieved due to a complete redesign ofthe program, plus improvements in the basic algorithms. The program also usesless memory now.

� It is now possible to use different electrode materials in two-probe systems, bysetting the keyword TwoProbe::UseMultiGridForElectrostatics to T (the de-fault is F, in which case the faster FFT method is used, as in TRANSIESTA-C.).

� Spin polarization has been implemented.

� Support for GGA has been added.

� The mixing algorithms have been improved by adding the Pulay mixing method,and the calculations now often converge faster and easier than before.

� The code is parallelized (by using MPI, more specifically MPICH), which gives agreat performance boost in particular for two-probe systems.

� There are no longer any archive (ark) files; instead, all the analysis results as wellas the data required to restart a calculation is saved in NETCDF files with a publicformat, giving a much more portable and transparent interface to VNL and otherprograms that can read NETCDF files (see Appendix C.1 for more information).

69

70 ATK vs. TranSIESTA-C

� The program no longer uses ion files, instead all basis sets are generated on thefly, which allows for much more flexibility for the users to optimize their basisset.

� There are a couple of important bug fixes. Earlier, negative transmission valuessometimes appeared due to the fact that the surface Green’s function did not con-verge, but this has been fixed. Constraining atoms in relaxations now works prop-erly, and the calculation of the partial density of states (PDOS) has been updated.Finally, k-point sampling for the self-consistent cycle in two-probe calculationsplus k-point sampling for the transmission spectrum now works properly.

� ATK ships with a completely new set of improved pseudopotentials that cover theentire periodic table up to element 103 (Lr).

� ATK uses the FLEXlm license manager, which allows proper license control withfloating licenses, running in parallel, etc. This has important consequences, sinceall users who have not yet upgraded their licenses must do so in order to run ATK2.0. Please see the Installation Guide for more detailed information.

It should be noted that ATK employs double precision floating point variables as op-posed to the single precision used in in TRANSIESTA-C. Although this may appear to bea minor modification, which should only effect the last decimals of the results, the sit-uation is not always so simple. The self-consistent loop is a highly complex non-linearproblem, and as such very sensitive to small changes in the parameters. Therefore, ATKmay produce slightly different results compared to TRANSIESTA-C, even if all other pa-rameters are identically the same (including, notably, the pseudopotential; see below).However, the discrepancy should lie roughly within the tolerance of the self-consistentcycle, meaning that the same differences in the final numbers could have been ob-tained by just using a slightly different tolerance or mixing parameter.

There are other sources of potential differences in the results for systems calculated inTRANSIESTA-C and ATK. For many elements, we have updated and improved the pseu-dopotentials; it is of course still possible to run the calculations with the old potentials(or, for that matter, any other pseudopotential which conforms to the format understoodby ATK). Finally, the default values of some keywords have been changed, so unlessyou specify these keywords explicitly in the input file, a different value compared toTRANSIESTA-C will be used. Most notably, the default mixing algorithm is now Pulayby default, which should give faster convergence in almost all cases.

An important point to note is that ATK is not fully backward compatible with oldTRANSIESTA-C input files. Certain keywords have been changed, others removed (and,of course, some have been added). Most of the deprecated keywords are related tofunctions that now are performed automatically in the program, but these keywordswere seldom used in input files anyway. If such keywords are encountered in the in-put files, the program will inform about it, and the offending keywords should just beremoved from the input files.

More importantly, there are keywords for which the syntax has changed, or the keyword

A.1 Changes to Keywords 71

itself has a new name. These changes have been made in order to improve the logicalstructure of the input files and to make it easier for new users to learn the atk language.If such outdated keywords are found in the input file, ATK will inform about the newsyntax/keyword name, to make it easier to migrate the users’ existing input files to thenew format. The two most commonly used keywords in this category are NumOrb::

BasisType which now should be NumOrb::BasisSet::Size and SCF::Beta whichhas been renamed to SCF::DiagonalMixingParameter. For a complete list of changesin the keyword syntax, see the sections further ahead in this chapter.

In addition to the changes in the input keywords, users will also notice that the outputfile has a slightly new format, but it should still be simple enough to read off the resultsof the calculation.

A.1 CHANGES TO KEYWORDS

Below is a list of all the keywords which have been deprecated, or for which the syntaxhas changed.

A.1.1 KEYWORDS NO LONGER AVAILABLE

The following keywords cannot be used anymore:

Analysis::3DGrids::SaveEffectivePotential Done by default.

Analysis::3DGrids::SaveElectronDensity Done by default.

Analysis::3DGrids::SaveTo Only NetCDF output format is supported.

Analysis::3DLine::... Not available.

NumOrb::Accelerate::... Accelerated runs are no longer available.

NumOrb::BasisSet::IonFile The program no longer uses external ion files.

NumOrb::IonFileDirectory The program no longer uses external ion files.

NumOrb::RadialSampling::MaxRange Now calculated automatically.

SCF::ToleranceEbs Not available, instead use SCF::Tolerance.

SCF::ToleranceH Not available, instead use SCF::Tolerance.

SCF::ToleranceRho Not available, instead use SCF::Tolerance.

SCF::UseTmpFile Not available.


Simulation::AccumulateHistory Not available.

Molecule::TotalCharge Not available.

TwoProbe::Contour::UseTmpFile Not available.

A.1.2 KEYWORDS WITH NEW NAMES

The following keywords have new names, but otherwise the same syntax.

Bulk::FermiTemperature :is now Bulk::ElectronTemperature.

Electrode::FermiTemperature :is now Electrode::ElectronTemperature.

Molecule::FermiTemperature :is now Molecule::ElectronTemperature.

NumOrb::BasisType :is now NumOrb::BasisSet::Size.

NumOrb::MoleculePaddingFactor :is now Molecule::ElectrostaticPaddingFactor.

NumOrb::PseudoFileDirectory :is now NumOrb::BasisSet::PseudoPotentialDirectory.

SCF::Beta :is now SCF::DiagonalMixingParameter.

SCF::NumStepsReset :is now SCF::HistorySteps.

A.1.3 KEYWORDS WITH NEW NAMES AND SYNTAX

The following keywords have new names, and a slightly modified syntax. Please referto Appendix D.2.

SCF::MixH :is now SCF::Coordinate with a new syntax.

TwoProbe::LeftElectrode::ArchiveFile :is now TwoProbe::LeftElectrode::NetCDFfile orTwoProbe::LeftElectrode::ATKfile.

A.1 Changes to Keywords 73

TwoProbe::RightElectrode::ArchiveFile :is now TwoProbe::RightElectrode::NetCDFfile orTwoProbe::RightElectrode::ATKfile.

Analysis::BandLine::... :has a new syntax.

A P P E N D I X

BBASIS SETS

B.1 BASIS SET PARAMETERS

ATK uses SIESTA type numerical orbital basis sets [11]. In Table B.1 we list the param-eters used to define the SZ, DZ, DZP, DZDP basis sets. It is possible to change theseparameters in the input file and in this way tailor the basis set to a specific application.In the following subsections we describe the meaning of these parameters and the basisorbitals generation procedure.

B.1.1 SINGLE ZETA ORBITALS

The basis functions are found by solving the radial Schrödinger equation for the atomwith a confinement potential. The confinement potential is defined by the parameters,V0, rinn and rc through the equation

Vconf(r) =

8>><>>:0 if r < rinn

V0 exp[�1=(r � rinn)]=(rc � r) if rinn < r < rc

1 if rc < r

Figure B.1 shows the confinement potential and the corresponding basis functions usedfor constructing the ATK standard basis set for hydrogen.

75

76 Basis Sets

First zeta parameters

NumOrb::Basisset::EnergyShift (real EnergyUnit)The energy shift defines the confinement radius, rc of the numerical orbital. Theconfinement radius is taken at the position of the first node of the wave function withenergy Enl+EnergyShift, where Enl is the energy of the atomic eigenstate.(Default value: 0.01 Ry)

NumOrb::Basisset::DeltaRinn (real)Parameter used to determine the inner radius, rinn, of the soft confinement potential.rinn is determined by rinn = DeltaRinn*rc.(Default value: 0.8)

NumOrb::Basisset::V0 (real EnergyUnit)Parameter for defining the softness of the confined potential. If V0 = 0, a hard wallpotential is used.(Default value: 40. Ry)

NumOrb::Basisset::Charge (real)Setting a net charge of the ion, when generating basis orbitals.(Default value: 0.)

Second zeta parameters

NumOrb::Basisset::SplitNorm (real)Defines the matching radius of an analytical orbital which splits the first zeta orbitalinto a double zeta basis. The matching radius is determined by requiring that thetotal norm of the split orbital should be a certain fraction of the norm of the first zetaorbital.(Default value: 0.15)

Table B.1: Parameters for controlling the basis set generation.

B.1.2 DOUBLE ZETA ORBITALS

The double zeta orbital is constructed through a procedure reminiscent of the splittingof a Gaussian basis set. The split orbital is obtained by constructing an analytical basisorbital that matches the first zeta orbital smoothly at the radius rsplit. The functionalform used for the split orbital is

�2�l (r) =

(rl(al � blr

2) if r < rsplitl

�1�l (r) if r � rsplitl

The radius rsplit is determined by specifying the norm of the split orbital through thekeyword NumOrb::Basisset::SplitNorm.

B.1.3 POLARIZATION ORBITALS

The polarization orbitals have higher angular momentum than the valence orbitals. InATK such orbitals are generated by perturbing the single zeta orbitals by an electric

B.2 Pseudopotentials 77

Figure B.1: The lower part of the plot shows the l = 0 effective potential for hydrogen (dashed)with the soft confinement potential (solid). The upper part shows the lowest occupied state of theconfined potential (solid line), and the atomic s-wavefunction is indicated by the dashed curve.The dotted curve shows the solution with energy shift 0.01 Ry. The position of the first node ofthis solution defines the position of rc.

field, and extracting the l + 1 component of the perturbed orbital.

B.2 PSEUDOPOTENTIALS

ATK uses norm-conserving pseudopotentials, and is shipped with a database of pseu-dopotentials for the entire periodic table (cf. Appendix G). This database is constantlyupdated and improved to provide accurate, general purpose pseudopotentials. How-ever, some users may have their own pseudopotential databases which they would liketo use with ATK.

NumOrb::BasisSet::PseudoPotentialDirectory (string)The string specifies the directory to look for pseudopotentials.(Default value: $ATK_DATA_DIR/pseudopotentials/atk2.0)

NumOrb::BasisSet::PseudoPotentialFile::XX (string)The string gives the name of the pseudopotential file for element Xx.(Default value: Empty)

Table B.2: Parameters for controlling the pseudopotentials.

This is possible, since ATK can read several different pseudopotential formats. It readsthe unified pseudopotential format (upf) defined by the PWSCF consortium. From thewebsite http://www.pwscf.org it is possible to download tools that can convert sev-

http://www.pwscf.org

78 Basis Sets

eral different pseudopotential formats into upf. ATK can also read the psf format usedby the SIESTA software.

In Table B.3 we list the pseudopotential formats compatible with ATK, and give refer-ences for finding more information and sample pseudopotentials.

Extension HTML Reference.cpmd http://www.cpmd.org/

.fhi http://www.fhi-berlin.mpg.de/th/fhi98md/fhi98PP/

.psf http://www.uam.es/departamentos/ciencias/fismateriac/siesta/

.uspp http://www.physics.rutgers.edu/~dhv/uspp/

.upf http://www.pwscf.org/pseudo.htm

Table B.3: List of pseudopotential formats that can be used by ATK either directly (upf, psf) orthrough conversion to the upf format. The web references point to the authors’ websites whereadditional information on the format and sample pseudopotentials can be found.

http://www.cpmd.org/

http://www.fhi-berlin.mpg.de/th/fhi98md/fhi98PP/

http://www.uam.es/departamentos/ciencias/fismateriac/siesta/

http://www.physics.rutgers.edu/~dhv/uspp/

http://www.pwscf.org/pseudo.htm

A P P E N D I X

CCOMMAND LINE OPTIONS

The general syntax of a ATK call is

$ atk inputfile [outputfile] [OPTIONS]

where OPTIONS can be any combination of the following

--version

Print version number and exit.

-h, --help

Print available commands and exit.

--param "command1; command2; ..."Equivalent to including the lines command1, command2, ... in the input file.

--force

Overwrite existing output files if any. Required if the files already exist, otherwisethe program will not proceed.

-o, --netcdf-file NetCDFfile.ncWrite data into file NetCDFfile.nc

-c, --check

Invoke checking mode. Do not perform simulation. Exit after checks are per-formed. Note that this option produces a NETCDF file with the same name as theinput file (unless top option -o is used, of course).

-a, --analyze NetCDFfile.ncLoad NetCDFfile.nc and perform analysis.

79

80 Command Line Options

-i, --init-from NetCDFfile.ncInitialize density matrix data from NetCDFfile.nc.

-v, --verbose

Output additional information about progress of simulation.

-q, --quiet

Output less information about progress of simulation.

--xyz

Output positions to .xyz file for visualization.

--xyz-file filenameOutput xyz data to filename.

--xyz-na integerNumber of repetitions of the system along the A unit cell vector when generatingthe .xyz file. The default value, used if this option is omitted, is 1.

--xyz-nb integerNumber of repetitions of the system along the B unit cell vector when generatingthe .xyz file. The default value is 1.

--xyz-nc integerNumber of repetitions of the system along the C unit cell vector when generatingthe .xyz file. The default value is 1.

C.1 INPUT AND OUTPUT FILES

The standard input and output files involved in a ATK run (the $ sign symbolizes thecommand prompt)

$ atk filename.atk filename.out

are

filename.atk The input file with parameters for the calculation.

filename.out A text file containing the progress information and results of the calcu-lation, along with all input data for the calculation, including parameters whichare set to default values. If the output file name is omitted, the output will bewritten on the terminal, and can thus be piped to a file or inspected for progressinformation. For more details about the contents of the output file, see Section E.

C.2 Environment Variables 81

filename.nc A binary file in the NETCDF format containing all data of the calcula-tion. It can be used for restarting the calculation, as well as for analyzing thesimulation. Different types of data, like the electrostatic potential or the electrondensity, are contained in this file and can be viewed with e.g. VIRTUAL NANOLAB.To redirect data to another filename, use the command line option -o.

Note that the default behavior can be changed by using the command line options, inorder for instance to change the name of the output files. The output files are by default(i.e. if the output file names are given without paths) in the directory from where theprogram is run.

The NETCDF format is a platform independent binary format and therefore the file canbe copied between computers using different operating systems. For more informa-tion about NETCDF files in general, it is recommended to have a look at the websitehttp://www.unidata.ucar.edu/software/netcdf/, where one also can download varioustools for managing NETCDF files, e.g. the program "ncdump" for extracting data (val-ues) from NETCDF files. Moreover, NETCDF files can be read by other programs, e.g.MATLAB and PYTHON, plus of course, as mentioned several times in this manual, VIR-TUAL NANOLAB.

C.2 ENVIRONMENT VARIABLES

ATK_DATA_DIR This environment variable is used to locate the file containing the de-fault ATK parameters, atk-defaults.atk. It is also used in the file atk-defaults.atk itself to setup the path to the basis and pseudopotential files.

PATH It is recommended to include the ATK binary directory in your path.

For more details about how to set these variables up for different systems, please referto the Installation Guide.

http://www.unidata.ucar.edu/software/netcdf/

82 Command Line Options

A P P E N D I X

DPARAMETERS AND KEYWORDS

In this chapter all the keywords and parameters that ATK understands will be describedin detail. For each keyword, we will specify the syntax, list possible units and parame-ters, and explain what these parameters mean.

The last section will clarify some points of the output (log) file which is generated duringa calculation.

D.1 SYNTAX

The parameters that control the setup of the system geometry and the calculationsperformed with ATK are specified in an input file. Optionally, there is a possibilityto add keywords via the command line option --param. In either case, the generalspecification of the parameters involves a keyword which ATK recognizes and relatesto a particular property, and a value of this property.

Some, but not all, values have units, and for those that do the unit is mandatory; thereare no default units (except for atomic coordinates, for which the unit is specified in adifferent way; see below).

Each line in the input file looks like

keyword value [unit]

which is also the format for the --param commands. There are also parameters whichcorrespond to a table of values. These are set by

%block keyword

line 1 line 2 ...

83

84 Parameters and Keywords

%endblock keyword

Lines starting with the symbol # will not be read by the input parser, and can thus beused for leaving out certain lines temporarily, or to introduce comments and documen-tation of the calculational setup.

We shall in this Appendix consider first the values and units, and then go through thekeywords in alphabetical order. Note that ATK is case insensitive and the use of capitalletters in the documentation is only used in order to increase the readability.

In what follows, keywords are typeset in monospace, whereas unit specifications aregiven in italic. When there are different options for specifying the value, these optionsare separated by k. For example, AtomList::Format may be specified either by thekeywords Scaled or Fractional, or a LengthUnit such as Angstrom. Therefore, thecorresponding entry in the list below is given as

AtomList::Format LengthUnit k Scaled k Fractional

D.1.1 VALUES

A value may be another keyword (such as in the case of Method::Type which may be,among other things, NumOrb) or a value may be a quantity of one of the following kinds:

real Floating point variable of the form 117.5 or 1.175e+2.

integer Integer value of the form 117

logical Logical of the form T or F (meaning true or false).

string A text string.

D.1.2 UNITS

All values describing a physical parameter must have specified a unit. In the currentversion of ATK the basic output units are Å, eV and their combinations, e.g. eV/Å forforces. In the output NETCDF file, however, the units are generally Bohr and Rydberg;in each case, both in the log file and NETCDF files, all quantities are always given withunits, to avoid misunderstandings.

Valid units in the input files are

LengthUnit m, meter, nanometer, Bohr, a0, Ang, AngstromUnit of length (several alternative spellings are accepted, like "nanometres"); in

D.1 Syntax 85

SI units the keywords correspond to the following values:Bohr, a0 0.529177�10–10 mnm, nanometer 10–9 mAng, Angstrom 10–10 m

InverseLengthUnit 1/m, 1/nanometer, 1/Bohr, 1/a0, 1/Ang, FracRecipUnit of inverse length (see LengthUnit above for a more complete list). If the key-word FracRecip is used, coordinates are given in units of the reciprocal latticevectors. In this case, the unit cell must be given in the input file, also for analysisruns! In addition, the keyword System::Bulk must be included.

EnergyUnit eV, meV, Ha, Hartree, J, Joule, K, Kelvin, Ry, RydbergUnit of energy (some alternative spelling are also accepted, like "millielectron-volts"); in SI units the keywords correspond to the following values:eV 1.60219�10–19 JoulemeV 1.602�10–22 JouleHa, Hartree 4.359863412�10–18 JouleRy, Rydberg 2.179931706�10–18 JouleK, Kelvin 1.3807�10–23 Joule

ForceUnit Ry/Bohr, Ha/Bohr, eVPerAng, eV/Ang, nN, nanoNewtonUnit of force (alternative spellings like "RyPerBohr" or "Hartree/Bohr" are alsoaccepted); in SI units the keywords correspond to the following values:Ry/Bohr 41.19475536 NewtonHa/Bohr 82.38951072 NewtoneV/Ang 1.60219�10–9 NewtonnN, nanoNewton 10–9 Newton

TimeUnit s, seconds, AUT, fsUnit of time; in SI units the keywords correspond to the following values:AUT 2.418865878�10–17 secondsfs 10–15 seconds

SpinUnit HBar, fractionalUnit of spin; we use atomic units (~ or Hbar) or measure the total electron spinin fractions of the total number of electrons (fractional). In atomic units the spinof one electron is 1

2 . The degeneracy of the state is (2S+1), where S is the totalspin. Thus,0.0 Hbar Singlet0.5 Hbar Doublet1.0 Hbar Triplet1.5 Hbar Quartet2.0 Hbar Quintet

In some cases, in particular in analysis mode, spin is gives as keywords Up/Downto distinguish between e.g. the spin up or down molecular eigenvalues.

ConductanceUnit Siemens, G0Unit of conductance; in SI units the keywords correspond to the following values:G0 7.74806917�10–5 Siemens


D.2 KEYWORDS

Keywords are divided into different groups or classes according to a hierarchial system,like in a menu system with subentries. Each class of keywords have the same base-name, and the basename signifies which part of the calculation the keyword controls.The different levels in the hierarchy are separated by double colons ::. For exam-ple, the keyword Bulk::NumKPoints::A belongs to the major class Bulk, which has asubclass NumKPoints.

The major keyword classes are:

Analysis:: Controls the analysis of the simulation.

AtomList:: Controls how the atomic coordinates are defined in the input file.

Bulk:: Parameters for systems with bulk symmetry.

Electrode:: Parameters of the electrode region of a two-probe system.

Method:: Select which method to use for the electronic structure calculation.

Molecule:: Parameters for systems with molecular symmetry.

NumOrb:: Parameters of the DFT numerical orbital basis set.

Relaxation:: Parameters for the relaxation of the atomic coordinates.

SCF:: Parameters for the self-consistency loop.

Simulation:: Determine what type of simulation to perform.

System:: Determines the overall symmetry of the system.

TwoProbe:: Parameters for a two-probe system.

UnitCell:: Defines how the unit cell of a periodic system is defined in the input file.

As pointed out earlier in this manual, parameters not explicitly listed in the input atkfile will be given values from the defaults file (global or user-defined; cf. page 81).There are furthermore certain required keywords for each system type, without whichit is not possible to perform the calculation. These keywords are rather few, andquite logical. Naturally, for any calculation it is necessary to define the System::Type

and to give an Atomlist. In addition, it is necessary to specify a UnitCell for bulkand electrodes, and finally TwoProbe::LeftElectrode::NetCDFfile or TwoProbe::LeftElectrode::atkfile (and correspondingly for the right electrode) are requiredfor two-probe systems.

Below all the keywords in each class will be described in detail in terms of syntax. Thekeywords are given in alphabetical order, and for each one is also listed the defaultvalue used if the keyword is not explicitly given in the input file.

D.2 Keywords 87

D.2.1 ANALYSIS::

This class of keywords controls the analysis of the simulation performed after the sim-ulation has been performed.

Analysis::BandLine::X::Start

Analysis::BandLine::X::End (real, real, real) InverseLengthUnitPrint the band structure along the line from Analysis::BandLine::X::Start toAnalysis::BandLine::X::End with the number of points defined by Analysis::BandLine::X::NumPoints. X can be any letter in the alphabet and is used toidentify the bandlines when several lines are included in the calculation.(Default value: Empty)

Analysis::BandLine::X::NumPoints integerNumber of points along the band structure line associated with the label X. If notspecified, then the value Analysis::BandLine::NumPoints is used for all bandlines.(Default value: Empty)

Analysis::BandLine::NumPoints integerNumber of points along the band structure lines.(Default value: 100)

Analysis::BlochState::X::Index integerSpecification of the band index of the Bloch state to be calculated. X can be anyletter in the alphabet and is used to identify the Bloch state when several statesare included in the calculation. The numbering of the bands starts from 0.(Default value: Empty)

Analysis::BlochState::X::KPoint (real, real, real) InverseLengthUnitSpecification of the wave vector, i.e. the k-point, of the Bloch state associatedwith the index X (see Analysis::BlochState::X::Index).(Default value: �-point)

Analysis::BlochState::X::Label stringString used to label the Bloch state associated with the index X (see Analysis::

BlochState::X::Index).(Default value: Empty)

Analysis::BlochState::X::Spin Up k DownSpecification of the spin of the Bloch state associated with the index X (seeAnalysis::BlochState::X::Index). Only relevant for spin-polarized calcula-tions.(Default value: Up)

Analysis::DOS::X::Energy real EnergyUnitSpecifies the energy, at which a spatially resolved density of states, n(E; r;k; �),


should be returned. This can in turn be used to make 3D plots. Currently, thisworks only for two-probe systems. The spatially resolved DOS is defined by:

n(E; r;k; �) =X�

j �;k;�(r)j �(E � "�;k;�);

where E is the energy, specified by the keyword Analysis::DOS::X::Energy;k is a point in reciprocal space, specified by the keyword Analysis::DOS::X::

KPoint; and � is the spin specified by Analysis::DOS::X::Spin. The latterkeyword should only be specified for spin-polarized calculations. �;k;�(r) isan eigenstate with eigenenergy "�;k;�. X can be any letter in the alphabet andis used to identify the spatially resolved DOS with this particular energy (and k-point and spin).(Default value: Empty)

Analysis::DOS::X::KPoint real, real, real InverseLengthUnitSpecifies a point in reciprocal space, where the spatially resolved DOS associatedwith index X should be determined (see Analysis::DOS::X::Energy).(Default value: �-point)

Analysis::DOS::X::Label stringThe string used to label the spatially resolved DOS associated with the index X(see Analysis::DOS::X::Energy).(Default value: Empty)

Analysis::DOS::X::Spin Up k DownSpecifies the spin of the spatially resolved DOS associated with index X (seeAnalysis::DOS::X::Energy. Only relevant for spin-polarized calculations.(Default value: Up)

Analysis::DOS::X::Type Total k Left k Right k NonEquilibrium

This keyword controls which part of the real space density of states (DOS) thatis saved. Total means the total DOS, Left and Right mean the DOS fromthe left and right electrodes, respectively, and NonEquilibrium means the DOSweighted with the Fermi occupation, i.e. the occupied eigenstates.(Default value: Empty)

Analysis::MolecularOrbital::X::Index integerSpecification of the state index of the molecular eigenstate to be printed in theanalysis output file. X can be any letter in the alphabet and is used to identifythe molecular state, when several states are included in the calculation. Thenumbering of the states starts from 0.(Default value: Empty)

Analysis::MolecularOrbital::X::Label stringString used to label the molecular state associated with the index X (see Analysis::MolecularOrbital::X::Index).(Default value: Empty)

D.2 Keywords 89

Analysis::MolecularOrbital::X::Spin Up k DownSpecifies whether the orbital associated with index X should be a spin up or downorbital. Only relevant for spin-polarized calculations.(Default value: Up)

Analysis::MPSH::Calculate logicalThe eigenvalues of the Molecular Projected Self-consistent Hamiltonian (MPSH)are calculated if this parameter is set to T. The MPSH is defined as the Hamilto-nian of a two-probe system excluding the electrode and surface atoms. The num-ber of surface atoms is determined by the two parameters TwoProbe::LeftSurface::NumberOfAtoms and TwoProbe::RightSurface::NumberOfAtoms.(Default value: T)

Analysis::MPSH::EnergyWindow real EnergyUnitIf this parameter is positive (i.e. > 0), only eigenvalues in an energy window ofthe specified size around the average electrode Fermi level are reported.(Default value: 0. eV)

Analysis::MPSH::X::Index integerSpecification of the state index of the MPSH eigenstate to be printed in the anal-ysis output file. X can be any letter in the alphabet and is used to identify themolecular state when several states are included in the calculation. The number-ing of the states starts from 0.(Default value: Empty)

Analysis::MPSH::X::Label stringString used to label the MPSH state associated with the index X (see Analysis::

MPSH::X::Index).(Default value: Empty)

Analysis::MPSH::X::Spin Up k DownSpecifies whether the MPSH eigenstate associated with index X should be a spinup or down state. Only relevant for spin-polarized calculations.(Default value: Up)

Analysis::PrintMolecularEigenvalues logicalControls if the molecular eigenvalues are printed in the log file or not.(Default value: T)

Analysis::PrintMullikenPopulation logicalPerform Mulliken population analysis. The results are only displayed if the com-mand line option --verbose is used.(Default value: T)

Analysis::TransmissionSpectrum::Calculate logicalCalculate the transmission spectrum.(Default value: T)


Analysis::TransmissionSpectrum::E0 real EnergyUnitValue of first energy point of the transmission spectrum.(Default value: -2. eV)

Analysis::TransmissionSpectrum::E1 real EnergyUnitValue of last energy point of the transmission spectrum.(Default value: +2. eV)

Analysis::TransmissionSpectrum::Eta real EnergyUnitSmall complex number added to the energy when calculating the transmissionspectrum.(Default value: 1.e-5 eV)

Analysis::TransmissionSpectrum::NumEigenChannels integerThe transmission coefficient can be decomposed into transmission eigenchannelsby diagonalizing the transmission matrix tyt. The number of channels reported arecontrolled by the keyword Analysis::TransmissionSpectrum::NumEigenChannels,and the eigenvalues are reported for the eigenchannels with the highest transmis-sion.(Default value: 3)

Analysis::TransmissionSpectrum::NumKPoints::A integerNumber of k-points used for the transmission spectrum in the A direction.(Default value: 1)

Analysis::TransmissionSpectrum::NumKPoints::B integerNumber of k-points used for the transmission spectrum in the B direction.(Default value: 1)

Analysis::TransmissionSpectrum::NumPoints integerNumber of energy points used for the transmission spectrum.(Default value: 100)

D.2.2 ATOMLIST::

This class controls how the atomic coordinates are read into ATK. The actual coordi-nates are entered in a list, where each line defines an atom and its position, using fourentries:

%block keyword

atomname x y z

...

atomname x y z

%endblock keyword

where atomname is the symbol of the relevant chemical element (such as Au for agold atom), and keyword should be replaced by the respective keyword for the rel-

D.2 Keywords 91

evant system type: Bulk::AtomList for periodic systems, Molecule::AtomList formolecules, Electrode::AtomList for the electrodes in a two-probe system, and fi-nally TwoProbe::CentralAtomList for the scattering region in a two-probe system.

How the three real numbers x, y, and z on each line are interpreted is controlled bythe following two keywords:

AtomList::Format LengthUnit k Scaled k FractionalIf a LengthUnit is specified, then the coordinates x, y, and z are assumed to begiven in the specified unit of length (such as Angstrom). If the keyword Scaled

is used, the coordinates are assumed to be given in units of AtomList::Scale.Finally, if the keyword Fractional is used, then the three real numbers x, y, andz are assumed to specify the atomic positions r through the relation r = xa1 +ya2 + za3, where a1, a2, a3 are the three lattice vectors, specified by the relevantUnitCell keyword.(Default value: Ang)

AtomList::Scale real LengthUnitUsed to scale the atomic coordinates if AtomList::Format is set to Scaled.(Default value: Empty)

D.2.3 BULK::

This class specifies parameters which are specific to systems with bulk symmetry.

Bulk::AtomList %block

The coordinates of the atoms in the basis, given in a list (c.f. AtomList onpage 90).(Default value: Empty)

Bulk::ElectronTemperature real EnergyUnitThe electronic temperature of the bulk system.(Default value: 300. K)

Bulk::NumKPoints::A integerNumber of evenly spaced k-points in the A direction. ATK uses the Monkhorst–Pack method for choosing the mesh points. [6](Default value: 1)

Bulk::NumKPoints::B integerNumber of evenly spaced k-points in the B direction.(Default value: 1)

Bulk::NumKPoints::C integerNumber of evenly spaced k-points in the C direction.(Default value: 1)


Bulk::UnitCell %block

The three lattice vectors or unit cell vectors a1, a2 and a3, expressed in unitsof UnitCell::LatticeConstant in Cartesian coordinates. For example, a bcclattice with 2 Å lattice constant is most conveniently specified by the followinginput:

UnitCell::LatticeConstant 2. Angstrom


-0.5 0.5 0.5

0.5 -0.5 0.5

0.5 0.5 -0.5


This way of doing it allows the lattice constant to be changed with a single mod-ification of the file, instead of having to recalculate all components of all unitvectors. It is of course still possible to give the same unit cell as


-1 1 1

1 -1 1

1 1 -1


since the default value of UnitCell::LatticeConstant is 1 Å.(Default value: Empty)

D.2.4 ELECTRODE::

This class defines the electrode region of a two-probe system. The effective potential inthe electrode region is obtained from a corresponding bulk system, and the electrodeparameters set up the connection with this corresponding system.

Electrode::AtomList %block

The coordinates of the atoms in the electrode, given in a list (c.f. AtomList onpage 90).(Default value: Empty)

Electrode::ElectronTemperature real EnergyUnitThe electronic temperature of the electrode.(Default value: 300. K)

Electrode::NumKPoints::A integerNumber of evenly spaced k-points in the A direction. ATK uses the Monkhorst-Pack method for choosing the mesh points [6]. The k-point sampling specifiedfor the electrode in the A and B direction is both used for the electrode calcula-tion and the two-probe calculation. Since the same k-point sampling is used in

D.2 Keywords 93

the two calculations, usually a converged calculation can be obtained with fewpoints. Typically, a more dense sampling is needed for calculating the transmis-sion function, see Analysis::TransmissionSpectrum::NumKPoints::A.(Default value: 1)

Electrode::NumKPoints::B integerNumber of evenly spaced k-points in the B direction.(Default value: 1)

Electrode::NumKPoints::C integerNumber of evenly spaced k-points in the C direction. This parameter is usedwhen calculating the self-consistent effective potential of the corresponding bulksystem. The two-probe system is infinite in the C-direction and for a consistentelectrode, it is important that its electronic structure accurately describes theinfinite system. For this reason a rather large value of Electrode::NumKPoints::C must generally be chosen.(Default value: 100)

Electrode::NumRep::A integerNumber of replicas of the corresponding bulk system in the A direction needed tomatch the A direction of the two-probe system. Note, that when using Electrode::NumRep::A>1, the k-point sampling specified with Electrode::NumKPoints::A

is the k-point sampling for the two-probe system, i.e. the repeated electrode cell.The corresponding k-point sampling of the non-repeated electrode cell is auto-matically calculated by ATK.(Default value: 1)

Electrode::NumRep::B integerNumber of replicas of the corresponding bulk system in the B direction neededto match the B direction of the two-probe system.(Default value: 1)

Electrode::UnitCell %block

Similar to Bulk::UnitCell.(Default value: Empty)

D.2.5 METHOD::

Parameters related to the method used to calculate the total energy and forces.

Method::Type NumOrb The energy method used for the calculation. NumOrb corre-sponds to using density functional theory (DFT) with the electronic wave func-tions expanded in a basis set of atom-based numerical orbitals.(Default value: NumOrb)


D.2.6 MOLECULE::

This class specifies parameters which are specific to systems with molecular symmetry.

Molecule::AtomList %block

The atomic coordinates for the molecule, given in a list (c.f. AtomList on page 90).(Default value: Empty)

Molecule::ElectrostaticPaddingFactor realA molecule is represented within a periodic cell with side lengths a, b and c.The numerical orbitals have finite radius, and a0, b0, and c0 are defined as thelengths where the molecule is not interacting with any of its periodic images ineach respective direction. The side length of the computational cell is defined bya = (1+Molecule::ElectrostaticPaddingFactor) a0,and equivalently for b and c.(Default value: 0.1)

Molecule::ElectronTemperature real EnergyUnitThe electronic temperature of the molecular system.(Default value: 300. K)

Molecule::TotalSpin real SpinUnitBy default the spin of the system is determined self-consistently. With the Molecule::TotalSpin parameter it is possible to fix the spin of a molecule to a specificvalue. The spin is given in atomic units (~), in which the spin of a single elec-tron is 0.5. Note that a spin-polarized method (see NumOrb::XCFunctional) isrequired when using non-zero spin values. We recommend to set the SCF::

InitialSpinPolarization to the same value as this parameter.(Default value: Empty)

D.2.7 NUMORB::

This class specifies the parameters of the DFT numerical orbital basis set used to solvethe electronic structure problem.

NumOrb::BasisSet::Size::Xx SZ k DZ k SZP k DZP k DZDPThis keyword describes the size of the numerical orbital basis for element Xx,where Xx is Li for lithium, for example. The corresponding keyword without the::Xx extension gives a global value for all elements.

SZ A single basis orbital for each valence orbital.

DZ Two basis orbitals for each valence orbital.

SZP SZ and one basis orbital for the first unoccupied shell.

D.2 Keywords 95

DZP DZ and one basis orbital for the first unoccupied shell.

DZDP DZ and two basis orbitals for the first unoccupied shell.

The basis set is generated according to the valence shells listed on pp. 121–124.If semicore states are included in the pseudopotential, then separate zeta basisorbitals are generated for the semicore states.(Default value: DZP)

NumOrb::BasisSet::Charge::Xx (real)Defines the net charge of element Xx, when generating basis orbitals. The key-word without the ::Xx extension gives a global value for all elements.(Default value: 0.)

NumOrb::Basisset::DeltaRinn::Xx (real)Determines the inner radius, rinn, of the soft confinement potential for elementXx. The keyword without the ::Xx extension gives a global value for all elements.rinn is determined by rinn = DeltaRinn*rc. Valid numbers are in the range[0.0,1.0].(Default value: 0.8)

NumOrb::Basisset::EnergyShift::Xx (real EnergyUnit)Determines the confinement radius rc of the numerical orbitals of element Xx.The keyword without the ::Xx extension gives a global value for all elements.The confinement radius is taken at the position of the first node of the wavefunction with energy Enl+EnergyShift, where Enl is the energy of the atomiceigenstate.(Default value: 0.01 Ry)

NumOrb::Basisset::SplitNorm::Xx (real)Defines the relative norm of the second zeta, relative to the first zeta for elementXx. The keyword without the ::Xx extension gives a global value for all ele-ments. The splitnorm is used to find the matching radius of an analytical orbitalwhich splits the first zeta orbital into a double zeta basis. The matching radiusis determined by specifying that the norm of the split orbital relative to the normof the first zeta orbital should have the value NumOrb::Basisset::SplitNorm.Valid numbers are in the range [0.0,1.0].(Default value: 0.15)

NumOrb::Basisset::V0::Xx (real EnergyUnit)Parameter for defining the softness of the confined potential for element Xx. Thekeyword without the ::Xx extension gives a global value for all elements. IfV0 = 0, a hard wall potential is used.(Default value: 40. Ry)

NumOrb::BasisSet::PseudoPotentialDirectory stringThe string describes the directory in which to look for pseudopotential files.(Default value: $ATK_DATA_DIR/pseudopotentials/atk2.0)


NumOrb::BasisSet::PseudoPotentialFile::XX (string)The string gives the name of the pseudopotential file for element Xx.(Default value: Empty)

NumOrb::MeshCutoff real EnergyUnitThis parameter controls the fineness of the grid used for evaluating real spaceintegrals and for solving the Poisson equation. The real space resolution �x isrelated to the mesh cutoff Ecut through the relation �x = � /

pEcut, if Ecut is in

Rydberg and �x in Bohr.(Default value: 150. Ry)

NumOrb::XCFunctional LDA-PZ k LSDA-PZ k GGA-PBE k SGGA-PBEThis parameter defines the approximation used for the exchange–correlation func-tional.

LDA-PZ The local density approximation (LDA) with the Perdew–Zunger parametriza-tion [8] of the correlation energy of a non spin-polarized homogenous elec-tron gas calculated by Ceperly–Alder [4].

LSDA-PZ The local spin density approximation (LSDA) with the Perdew–Zungerparametrization [8] of the correlation energy of a spin-polarized homoge-neous electron gas calculated by Ceperly–Alder [4]. .

GGA-PBE The non spin-polarized generalized gradient approximation (GGA) inthe Perdew–Burke–Enzerhofen approximation. [7].

SGGA-PBE The spin-polarized generalized gradient approximation (GGA) in thePerdew–Burke–Ernzerhof approximation. [7].

If the functional describes spin polarization, the spin of the system will be deter-mined self-consistently. For molecular systems it is possible to fix the spin usingMolecule::TotalSpin.(Default value: LDA-PZ)

NUMORB:: SPECIAL PARAMETERS

This set of parameters defines the radial representation of the numerical orbitals. Werecommend to use the default values of these parameters.

NumOrb::RadialSampling::DR real LengthUnitThe separation between the points used for the radial representation of the nu-merical orbitals.(Default value: 0.005 Bohr)

NumOrb::Tables::Cutoff real EnergyUnitTwo center integrals are done in Fourier space, and this parameter controls themaximum Fourier component used for the integral.(Default value: 2500. Ry)

D.2 Keywords 97

NumOrb::Tables::NumPoints integerControls the number of points used for the two-dimensional table, which containsthe two center integrals.(Default value: 1024)

D.2.8 RELAXATION::

These keywords control the relaxation of the ionic coordinates.

Relaxation::ForceTolerance real forceUnitThe relaxation is converged when the magnitude of the largest force componentis lower than this value.(Default value: 5.e-2 eV/Ang)

Relaxation::MaxSteps intThe relaxation makes at most MaxSteps iterations.(Default value: 200)

Relaxation::MaxDisplacement real LengthUnitSets the maximum allowed length of each step in the relaxation routine.(Default value: 0.1 Bohr)

Relaxation::Type (CG)Specifies the algorithm used in the minimization. At present, only the conjugategradient method is available.(Default value: CG)

D.2.9 SCF::

This class of keywords controls the self-consistency loop.

SCF::Coordinate DensityMatrix k Hamiltonian k EffectivePotentialThis keyword determines which physical quantity that is mixed between iterationsin the self-consistent cycle.(Default value: Hamiltonian)

SCF::Algorithm Pulay k BroydenThis keyword controls which mixing algorithm to use. In most cases, in particularfor two-probe systems, Pulay mixing is more stable and faster than Broyden.(Default value: Pulay)

SCF::Tolerance realWhen the relative change of the density and the Hamiltonian matrix is less than


this value between two iterations, the self-consistent cycle is considered con-verged.(Default value: 1.e-4)

SCF::Criterion Strict

The criterion for when self-consistency is reached. The keyword Strict indicatesthat all three convergence criteria (density matrix, Hamiltonian and energy of alloccupied states) must reach the required tolerance.(Default value: Strict)

SCF::DiagonalMixingParameter realThis parameter defines the fraction of the new value of the SCF::Coordinate

which is mixed with the old value.(Default value: 0.1)

SCF::HistorySteps integerNumber of previous steps used in the mixing algorithm to generate the new state.Linear mixing corresponds to setting SCF::HistorySteps to unity, but this ishighly unstable and the calculations usually never converge. In case of conver-gence problems this parameter can be increased to include more of the previoussteps to find the optimal value of the new state. This requires more memory, butincreasing fileSCF::HistorySteps up to 20 or even a bit more should be safe onmost systems.(Default value: 6)

SCF::InitialSpinPolarization real SpinUnitSets the initial total spin of the system to a specified value. The parameter onlyhas an effect for a spin polarized calculation (see NumOrb::XcFunctional).

If this parameter is not specified, the spin occupation of each orbital is set sothat the initial polarization is equal to the maximum atomic polarization of eachelement. In formulas, this means that

o";i = min(Oi; 1); (D.1)

o#;i = max(Oi � 1; 0): (D.2)

where o";i is the occupation of the "up" spin channel in orbital i, and o#;i is theoccupation of the "down" channel in orbital i, and Oi = o";i + o#;i is the atomicoccupation of the shell. Since Oi can only be 0, 1 or 2, the two occupations oican take the values 0 (unoccupied orbital) and 1 (occupied orbital).

To be more concrete, we can consider a specific example. In its ground state,the d-shell of Fe has the configuration 3d6["#]["]["]["]["], where the [: : :] boxessymbolize the occupancies of the 5 degenerate d-states. Thus, o";3d

�2= 1,

o#;3d�2

= 1, o";3d�1

= 1, o#;3d�1

= 0, and so on.

With the keyword SCF::InitialSpinPolarization it is possible to choose an-other initial polarization, which is applied to all atoms alike. The keyword defines

D.2 Keywords 99

the parameter �, which re-scales the polarized state. The value of � is calculatedfrom the initial spin-polarization,

� = (SCF :: InitialSpinPolarization)=Xi

(o";i � o#;i)

The initial spin polarization is now chosen as

~o";i = �o";i + (1� �)Oi=2 (D.3)

~o#;i = �o#;i + (1� �)Oi=2 (D.4)

Note that if � > 1 then we set � = 1, and if � < –1 then we set � = –1.

Again, an example illustrates the point. The initial spin polarization of the groundstate of Fe as given above is 4; the spin polarization is defined as

Xi

(o";i � o#;i:

If instead we would like to have an initial polarization of 2, we set the key-word SCF::InitialSpinPolarization to 1 Hbar. Note carefully that the spinof one electron is ~=2! This gives � = 1=2, and the orbital occupations becomeo";3d

�2= 1, o#;3d

�2= 1, and o";i = 3=4, o#;i = 1=4 for all other d-orbitals.

It may seem weird to have a fractional occupation of the orbitals, but this isnot a problem; the occupations only gives the probability of finding the electronsomewhere, and this can of course be different from an integer (cf. the situationwith a finite electron temperature, where the HOMO and LUMO occupationsmay be fractional as well).

SCF::MaxSteps integerMaximum number of steps in the self-consistency loop.(Default value: 100)

SCF::TwoProbe::InitialBulkRun logicalIf this parameter is set to true, a two-probe system will initially be treated as aperiodic (bulk) system, subject to periodic boundary conditions, and a full self-consistent run will be performed on this system. The purpose of this is to generatea better initial guess of the density matrix for the full two-probe SCF loop. If theinitial bulk run is left out, the initial density matrix is evaluated from the neutralatom density, unless of course the density matrix is initialized from a previouscalculation by the use of the -i command line parameter; in that case, thereis naturally no point to perform any initial bulk run either, and the parameterSCF::TwoProbe::InitialBulkRun will be ignored. Experience shows that theinitial bulk run almost always improves the convergence rate and the probabilityfor successful convergence for two-probe systems.(Default value: T)


D.2.10 SIMULATION::

These keywords control the simulation steps performed, i.e. geometry relaxation, etc.

Simulation::ConstrainAtoms integer:integer[,integer:integer]Fixes the positions of certain atoms during a relaxation. Example: a system con-sists of 12 atoms numbered 0, 1, 2, : : :, 11 (the numbering of atoms always beginsat zero). With the command

Simulation::ConstrainAtoms 0:2,7:10,4:5

we constrain atoms 0, 1, 2, 7, 8, 9, 10, 4, 5. Negative atom indices correspondto counting backwards from zero, so the command

Simulation::ConstrainAtoms 0:2,-5:-2,-8,5

is equivalent to the one above.(Default value: Empty)

Simulation::Type SingleConfiguration k RelaxationParameter defining the overall simulation. A SingleConfiguration simulationcalculates the self-consistent energy and potential, while the ions are moved inorder to minimize the total energy in a Relaxation simulation.(Default value: SingleConfiguration)

D.2.11 SYSTEM::

Keywords describing the overall symmetry of the system.

System::Type Molecule k Bulk k Electrode k TwoProbeThe symmetry of the system. A Molecule is an isolated system and a Bulk systemis a periodic system. An Electrode system is similar to a bulk system except thatadditional information is calculated such that an electrode can be used to definea two-probe system. Finally, a TwoProbe system an open system where a centralregion (the scattering region) is connected with two electrode regions.(Default value: Empty)

D.2.12 TWOPROBE::

This class specifies parameters which are specific to two-probe systems.

TwoProbe::CentralAtomList %block

The coordinates of the atoms in the central region, given in a list (cf. AtomList

D.2 Keywords 101

on page 90).(Default value: Empty)

TwoProbe::LeftElectrode::ATKfile stringName of the ATK file that defines a calculation of the left electrode. The keywordcan be used instead of the keyword TwoProbe::LeftElectrode::NetCDFfile

in order to avoid a separate calculation of the left electrode.(Default value: Empty)

TwoProbe::LeftElectrode::EquivalentCentralAtom integerThe left electrode is aligned with the central region by requiring that the atom cor-responding to this parameter is displaced by one lattice vector relative to the atomcorresponding to the parameter TwoProbe::LeftElectrode::EquivalentElectrodeAtom.If the atom number is negative, it is counted backwards from the end of the atomlist.(Default value: 0)

TwoProbe::LeftElectrode::EquivalentElectrodeAtom integerDefines the number of the atom in the left electrode which is aligned with theatom in the central region specified by TwoProbe::LeftElectrode::EquivalentCentralAtom.(Default value: 0)

TwoProbe::LeftElectrode::NetCDFfile stringName of the NetCDF file that defines the left electrode. The keyword can besubstituted by TwoProbe::LeftElectrode::NetCDFfile in order to avoid a sep-arate calculation of the left electrode.(Default value: Empty)

TwoProbe::LeftElectrode::Voltage real EnergyUnitElectrochemical potential of the left electrode.(Default value: 0. eV)

TwoProbe::LeftSurface::NumberOfAtoms integerDefines which atoms in the central region that are regarded as surface atoms ofthe left electrode. These atoms are fixed in relaxations.(Default value: 0)

TwoProbe::Molecule::GateVoltage real EnergyUnitSimulate the effect of a gate electrode by shifting the MPSH part of the two-probeHamiltonian by this amount. (Default value: 0. eV)

TwoProbe::RightElectrode::ATKfile stringName of the ATK file that defines a calculation of the right electrode. Thekeyword can be used instead of the keyword TwoProbe::RightElectrode::

NetCDFfile in order to avoid a separate calculation of the right electrode.(Default value: Empty)

TwoProbe::RightElectrode::EquivalentCentralAtom integerDefines the number of the atom in the central region which is used for aligningthe right electrode with the central region. (Default value: -1)


Figure D.1: The contour used for the complex energy integral.

TwoProbe::RightElectrode::EquivalentElectrodeAtom integerDefines the number of the atom in the right electrode which is aligned with theatom in the central region specified by TwoProbe::RightElectrode::EquivalentCentralAtom.(Default value: -1)

TwoProbe::RightElectrode::NetCDFfile stringName of the NetCDF file which defines the right electrode. The keyword canbe substituted by TwoProbe::RightElectrode::NetCDFfile in order to avoid aseparate calculation of the right electrode.(Default value: Empty)

TwoProbe::RightElectrode::Voltage real EnergyUnitElectrochemical potential of the right electrode.(Default value: 0. eV)

TwoProbe::RightSurface::NumberOfAtoms integerDefines which atoms in the central region that are regarded as surface atoms ofthe right electrode. These atoms are fixed in relaxations.(Default value: 0)

TwoProbe::UseMultigridForElectrostatics logicalWhen set to T, the Hartree potential is solved with a multigrid method, and thisallows for different left and right electrodes. In this case it is crucial that the sizeof the unit cells in the A and B directions are the same for both electrodes (i.e.not the same for A and B, but the same in the A direction in the left and rightelectrodes, and the same for the B direction). When this option is set to F, theHartree potential is solved using FFT and periodic boundary conditions, and inthis case the left and right electrodes must be the same.(Default value: F)

TWOPROBE::CONTOUR

This subclass of keywords is related to the contour integration.

D.2 Keywords 103

TwoProbe::Contour::Circle::NumPoints integerThe number of points along the arc part of the complex contour starting atTwoProbe::Contour::DeltaEMin and ending at the Fermi energy Ef.(Default value: 30)

TwoProbe::Contour::DeltaEMin real EnergyUnitThe distance from the Fermi energy Ef to the starting energy of the complex energycontour. In a ATK run this value should be larger than the binding energy of thelowest eigenstate, otherwise there will be poles outside the contour and chargewill be missing in the integration.(Default value: 3. Ry)

TwoProbe::Contour::FermiLine::NumPoints integerThis keyword determines the number of points from the Fermi energy Ef to1. TheFermi function is used as a weight function, and the important energy range has awidth of about 4kBT where kB is the Boltzmann constant and T the temperature.(Default value: 10)

TwoProbe::Contour::KTPoles::NumPoints integerThe complex contour always stays a distance from the real axis. This meansthat it encloses some poles of the Fermi functions. In ATK the number of polesis specified by this keyword, and this indirectly determines the distance of thecontour from the real axis.(Default value: 4)

TwoProbe::Contour::RealAxis::PointDensity real EnergyUnitThe spacing between points on the close-to-real axis part of the contour in thevoltage bias window.(Default value: 0.02 eV)

TwoProbe::Contour::RealAxis::Eta real EnergyUnitSmall complex number added to the denominator in the real axis integration.(Default value: 0.01 eV)

D.2.13 UNITCELL::

This class defines the unit cell of the calculation.

UnitCell::LatticeConstant real LengthUnitThe lattice constant of the unit cell.(Default value: 1. Ang)

UnitCell::X0 real LengthUnitCartesian x coordinate of the starting point for the cell vectors.(Default value: 0. Ang)


UnitCell::Y0 real LengthUnitCartesian y coordinate of the starting point for the cell vectors.(Default value: 0. Ang)

UnitCell::Z0 real LengthUnitCartesian z coordinate of the starting point for the cell vectors.(Default value: 0. Ang)

A P P E N D I X

EOUTPUT FILE

During a run with ATK, a log or output file is always generated. In addition to informingthe user of the progress of the calculations, it also contains many important calculatedquantities such as total energy or final positions of a relaxation run. The log file alsodocuments the parameters used in the calculation.

Most parts of the log file are self-documented and easy enough to understand just bylooking at it. Here we will just point out some details which perhaps are not so obvious,usually due to special notation.

E.1 HEADER AND SELF-CONSISTENT CYCLE

The first portion of the log file, after the header which tells when, where, how and bywhom ATK was launched, is simply a record of the parameters used in the calculations.Parameters not found in the input atk file are taken from the defaults file, but the log filedoes not become a copy of the defaults file, since only parameters which are actuallyrelevant for the particular kind of calculation desired are displayed.

Unless ATK was started in analysis mode (using the -a option), the next part of the logfile is almost always the progress information from the self-consistent cycle. Except forthe first step, this consists of a single line of the form (it is all a single line in the log file)

# sc 1 : FermiEnergy = -3.8255 eV Ebs = -126.9 eV

dRho = 2.2543E-2 dEbs = 1.43532E-4 dH = 6.34533E-01

Here, "Ebs" is the band structure energy, i.e. the sum of the energies of all occupiedstates, while "dRho", "dEbs" and "dH" are the relative changes in the density matrix,band structure energy, and Hamiltonian, respectively, compared to the previous step.

105

106 Output file

These are the quantities which are compared to the tolerance, set by SCF::Tolerance,which must be met for convergence.

For a two-probe run with open boundary conditions (there is also, if so requested, aninitial run with periodic boundary conditions, as explained above in relation to thekeyword SCF::TwoProbe::InitialBulkRun), the line is slightly different:

# sc 3 : q = 8.13734 e Ebs = -20.45441 eV

dRho = 4.0454E-02 dEbs = 3.3454E-03 dH = 4.3454E-02

Here "q" is the total charge in the central region, in units of the electron charge. Formolecular, bulk and electrode systems, the total charge is of course strictly conserved,but this is not the case in a two-probe system (in particular under bias), where we haveopen boundary conditions.

Note carefully that the band structure energy "Ebs" is defined in a different way in theinitial bulk run compared to a two-probe system with open boundary conditions, whichis why this quantity will exhibit a jump when going from one self-consistent cycle tothe other. In the first case, the electrodes are included, in the latter they are not.

When running with the command line option --verbose, there is additional progressinformation for each step in the self-consistent cycle. First of all, the Mulliken popula-tions are displayed for each step (this can often be a useful piece of information whentrying to cure convergence problems), and in addition there is a total energy summaryin each step.

Moreover, with the --verbose option, ATK also prints the k-point sampling grid usedfor the calculation to the output file. The output looks something like

%block BrillouinZoneSampling

# ----------------------------------------------------------------

# KX (1/Ang) KY (1/Ang) KZ (1/Ang) Weight

# ----------------------------------------------------------------

-0.38500 -0.38500 -0.38500 0.25000

-1.15500 0.38500 0.38500 0.25000

0.38500 -1.15500 0.38500 0.25000

-0.38500 -0.38500 1.15500 0.25000

# ----------------------------------------------------------------

%endblock BrillouinZoneSampling

for a 2�2�2 grid in fcc Au with lattice constant 4.08 Å. Note that 2�2�2=8, but 4 ofthese can be related to the other 4 by time-reversal symmetry, which is used by ATK toreduce the number of required calculations.

E.2 Total energy and atom list 107

E.2 TOTAL ENERGY AND ATOM LIST

The total energy information is always displayed after the self-consistent loop has com-pleted. It is a large block,

%block Results::TotalEnergy

.

.

.

%endblock Results::TotalEnergy

which consists of two main parts. First, there is a summary of all the different terms inthe total energy:

# Total Energy = -465.33332 eV

# Potential Energy = -465.33332 eV

# Ion Kinetic Energy = 0.00000 eV

# --------------------------------------

# NumOrb: Energy of system

# --------------------------------------

# Charge = 8.0000

# FermiEnergy = -0.7348 eV

# Band Structure Energy = -101.6476 eV

# --------------------------------------

# El. Kinetic Energy = 337.3125 eV

# NL Energy = -113.9817 eV

# dEna = -5.4525 eV

# dUscf = 1.0067 eV

# XC Energy = -116.0787 eV

# ENA = 69.3998 eV

# Eions = -637.5394 eV

# --------------------------------------

# Electron kinetic energy = 337.3125 eV

# Electrostatic energy = -686.5671 eV

# Exchange-corr. energy = -116.0787 eV

# --------------------------------------

# Total potential energy = -465.3333 eV

# --------------------------------------

The first line, the total energy, is one of the most important parameters in a DFT cal-culation. The total energy (of the ions, as it is) is the sum of the ionic potential energyand the kinetic energy. For the calculations that presently can be performed in ATK theions are stationary, and thus the total and potential energies are always equal.

108 Output file

Note carefully that since ATK uses pseudopotentials to represent the core electrons,the absolute energies reported in the output files have no real meaning. Only energydifferences have physical relevance, be they between different states in the same sys-tem, or the same system calculated for different conditions (such as when relaxing thegeometry).

The total energy block also reports the charge in the system, and the Fermi energy andband structure energy. As mentioned already, the band structure energy is the sum ofthe energies of all eigenstates, weighted with their occupation, and the total charge inthe system is conserved for molecular, bulk and electrode systems. Note that for two-probe systems, the charge listed in the total energy block includes the charges locatedin the two electrode unit cells as well, whereas the value of "q" as presented during theself-consistent cycle only counts the charges in the central region.

For bulk and molecule calculations, the Fermi energy is calculated by requiring thesystem to be charge neutral. More precisely, the Fermi level is the energy which makesthe Fermi occupation of the orbitals equal to the number of electrons in the system.Note that for molecules, if there is a HOMO–LUMO gap, the Fermi level can varydramatically as a function of the temperature and other parameters, and is thus notreally well-defined. It is then physically more interesting to look at the HOMO level.The Fermi energy is not presented for two-probe systems, since it is an equilibriumquantity which is not well-defined when open boundary conditions apply.

The total energy is made up of contributions from several terms, which are also pre-sented in this block. For a more detailed presentation of the definitions of these, pleaserefer to Ref. 11. Finally, the total potential energy (or more specifically the potentialenergy of the ions) is the sum of the ground state energy of the electronic system andthe ion–ion interaction, and can be subdivided into three terms: the electron kineticenergy (i.e. the kinetic energy of the non-interacting electron gas), the electrostaticenergy (the mean-field electrostatic energy of the electron gas and the ions), and theexchange–correlation energy of the electron gas.

The second part of the total energy block is a report on the atomic configuration, listingthe positions (i.e. the final positions in the case of a relaxation run), plus the velocitiesand forces on each atom:

# ----------------------------------------------------------------

# Number Of Atoms = 3

# ----------------------------------------------------------------

# Positions in Angstrom

# Velocities in Angstrom/fs

# Forces in eV/Ang

# ----------------------------------------------------------------

# X Y Z FX FY FZ

# ----------------------------------------------------------------

0.757 0.586 0.000 5.419E-01 5.080E-01 7.978E-14

-0.757 0.586 0.000 -5.419E-01 5.080E-01 1.153E-13

E.3 Mulliken population 109

0.000 0.000 0.000 -5.608E-14 -1.017E+00 5.326E-14

# ----------------------------------------------------------------

# Total Force (eV/Ang) = 0.0 -0.0014080 0.0

# Total Magnitude (eV/Ang) = 0.00140804

# Largest force component (eV/Ang) = 1.01733

# Root-mean-squared Force Magnitude (eV/Ang) = 0.844244

# ----------------------------------------------------------------

%endblock Results::TotalEnergy

Actually, the example above is a slightly modified version of a real output from ATK. Wehave removed the velocities, which naturally always zero are in the presently supportedcalculations, simply in order to fit the output.

Unless you are running a relaxation, it is recommended to always check in the outputthat the forces are not too large. If they are, it is an indication that the system setupdoes not correspond to a natural configuration, and the results may be questionable forthis reason. The forces calculated by ATK are analytical DFT forces [11].

During a relaxation, there will be one self-consistent cycle per iteration in the conjugentgradient minimization of the total energy. Each step will display the self-consistent totalenergy block, plus the atomic configuration in that step, so that one can follow theprogress of how the atoms are moved into their relaxed positions.

E.3 MULLIKEN POPULATION

If the keyword Analysis::PrintMullikenPopulation is set to true (this is the default),and if ATK is run with the command line option --verbose, the Mulliken populationswill be evaluated and printed to the log file for every step in the self-consistent cycle, aswell as at the end of the calculation. Unless one is having trouble with the convergenceof the self-consistent cycle, however, it is often not necessary to use the --verbose

option, since it bloats the output file quite a bit. In that case, it is possible to performthe Mulliken population analysis as a simple post-processing analysis step, by running

atk file.atk mulliken.out -a file.nc --verbose --force

where file.atk is the name of the original input file for the calculation (which natu-rally must not set the keyword Analysis::PrintMullikenPopulation to False!), andfile.nc is the NETCDF file produced by the original run. The --force option is re-quired to overwrite file.nc, but in fact the file will not be modified.

The Mulliken populations are presented in a block with, except for a report on the totalcharge, simply consists of one line per atom in the setup (for two-probes, for each atomin the central region). For each atom, the total Mulliken population, which is the same

110 Output file

` m Label Solid harmonic ` m Label Solid harmonic

0 0 s 1 3 –3 f(–3) y(3x2 � y2)3 –2 f(–2) xyz

1 –1 py y 3 –1 f(–1) y(4z2 � x2 � y2)1 0 pz z 3 0 f(0) z(2z2 � 3x2 � 3y2)1 +1 px x 3 +1 f(1) x(4z2 � x2 � y2)

3 +2 f(2) z(x2 � y2)2 –2 d(–2) xy 3 +3 f(3) x(x2 � 3y2)2 –1 d(–1) yz2 0 d(0) 2z2 � (x2 + y2)2 +1 d(1) xz2 +2 d(2) x2 � y2

Table E.1: ATK employs solid harmonics to expand the angular part of the basis orbitals. Notethat normalization factors are omitted from all expressions here.

as the total number of electrons on that atom, is listed as "q", and after that followsa breakdown of how these electrons are distributed among the basis orbitals. Theseorbitals are labelled as follows

An example of such output, where a DZP basis set was used for both H and Au, couldlook like this:

%block Results::MullikenPopulations

0 H q = 1.00000 [ s: 1.014, s: -0.014, y: 0.000, z: 0.000, x: 0.000 ]

1 Au q = 11.00000 [ s: -0.006, s: 2.006, d(-2): 1.840, d(-1): 1.840,

d(0): 1.331, d(1): 1.840, d(2): 1.331, d(-2): 0.160, d(-1): 0.160,

d(0): 0.169, d(1): 0.160, d(2): 0.169, y: 0.000, z: 0.000, x: 0.000 ]

# ----------------------------------------------------------------

# Total charge = 12

# ----------------------------------------------------------------

%endblock Results::MullikenPopulations

There are two "s" orbitals for H, which is the "DZ" part of the basis set, plus the "P"(polarization) "p" orbitals. For Au, we have two sets of "s" and "d" orbitals (again "DZ"),plus a single set of polarization "p" orbitals. Note that the valence configuration for Auis 5d106s2, and thus the next higher shell, i.e. the polarization shell, is 6p.

E.4 Spectra 111

E.4 SPECTRA

E.4.1 ENERGY SPECTRUM

For molecular systems, the last quantity reported in the output file of a regular calcula-tion is the molecular eigenspectrum (provided the keyword Analysis::PrintMolecularEigenvalues

is set to true, which is the default value). The energy zero-level is chosen at the vacuumlevel, i.e. the value of the effective potential far away from the molecule.

In addition, the energies of the HOMO and LUMO levels are presented, and the oc-cupation of each eigenstate is indicated. Note that the HOMO and LUMO are notentirely well-defined for spin-dependent systems.

For bulk systems, the vacuum level is not well defined, in this case the energy zerois arbitrary. Instead, the energy spectrum (i.e. the output obtained from specifyingAnalysis::BandLine keywords) is always presented relative the Fermi energy. Allwave vectors in the output file are reported in inverse Ångström.

Finally, for two-probe systems, the MPSH eigenstates are reported relative to the aver-age Fermi level of the electrodes. Thus, if an equal voltage (say, 1 V) is applied to bothelectrodes, the MPSH spectrum does not change.

E.4.2 TRANSMISSION SPECTRUM

The last part of a two-probe output file contains the energy-dependent transmissionspectrum, the details of which are controlled by the keywords Analysis::TransmissionSpectrum.As for the MPSH eigenstates, the energy scale for the transmission spectrum is reportedrelative the average Fermi level.

Note that the transmission coefficient can be larger than unity if multiple transmissionchannels are available at a certain energy. Also, when the transmission coefficient isless than unity, it does not mean that this state is not transmitted; it only means thatthe probability of the incoming wave function to be transmitted is smaller than one(quantum mechanics is inherently statistical).

If the keywords Analysis::TransmissionSpectrum::NumKPoints::A/B are differentfrom unity (and in general they should be, in order to obtain a converged transmissionspectrum; the most notable exception being that of effectively one-dimensional elec-trodes), the transmission is first given as a weighted average (i.e. the actual transmissionspectrum), and then the spectrum for each sampled k-point in the 2D surface Brillouinzone is reported.

The two last column of the (average) transmission spectrum are the total density of

112 Output file

states (DOS) of the entire two-probe system, and the partial density of states (PDOS)of the central region excluding surface atoms, defined by the parameters TwoProbe::LeftSurface::NumberOfAtoms and TwoProbe::RightSurface::NumberOfAtoms.

For the k-point resolved spectra, there are also additional columns giving the largesteigenvalues of the decomposed transmission eigenchannels, as controlled by Analysis::TransmissionSpectrum::NumEigenChannels.

Finally, the output file also contains the calculated current (at finite bias) and conduc-tance. The value reported in the output file is essentially the transmission at the Fermilevel (times the quantum conductance unit, 77.48 �S); to calculate the differential con-ductance, it is necessary to perform a numerical differentiation of the current–voltagefunction.

A P P E N D I X

FWRITING SCRIPTS

Because ATK takes a number of command line options, writing small scripts is a verypowerful and yet simple way to perform complex calculations, as described in thistutorial.

One may create these scripts in any suitable scripting language which is able to callother executables. In the following sections we provide examples of such scriptsfor Linux/Cygwin (bash shell scripts) and Windows (DOS batch files), plus platform-independent scripts written in Python. All examples are related to tasks considered inthe quick tour section of this tutorial, as referred to in each case below.

As with the input files it is recommended that you type and run the scripts yourself, butfor reference they can also be found in the examples/tutorial/quicktour directory.

F.1 BASH SCRIPTS FOR LINUX AND CYGWIN

In order to execute the scripts you write, it is necessary to make them executablethrough the command

$ chmod +x script.sh

where script.sh of course is the name of the script and the $ sign symbolizes thecommand prompt. To run the script, simply type

$ ./script.sh

at the command prompt.

113

114 Writing Scripts

Note! In Linux (and similar environments such as Cygwin) thecommand interpreter or shell interprets each line in a script file asa separate command. However, many of the commands we wishto run are rather long, and do not fit on a single line. For improvedreadability we have therefore in the script listings below used a "n"sign at the end of the line. This is a "continuation sign", whichtells the command shell that the text on the following line is to beconsidered as a continuation of the present one, and not as a newcommand. If you wish, you may of course omit the continuationsign and enter the entire command on a single line.

The following script (latenergy.sh) calculates the total energy for a series of latticeconstants for the lithium chain discussed on page 10:

#!/bin/sh

for lat in 2.82 2.84 2.86 2.88 2.90 2.92 ; do

atk lichain.atk lichain$lat.out \

--force -o lichain$lat.nc \

--param "UnitCell::LatticeConstant $lat Angstrom" $*

done

Next, we have a script ivrun.sh that calculates the current–voltage characteristics ofthe Li–H–H–Li two-probe system considered on page 17:

#!/bin/sh

oldbias=0.0

for bias in 0.2 0.4 0.6 0.8 1.0 ; do

atk lih2-0.0.atk lih2-$bias.out \

--force -o lih2-$bias.nc -i lih2-$oldbias.nc \

--param "TwoProbe::LeftElectrode::Voltage $bias eV" $*

oldbias=$bias

done

For the same system, we also made an I–V calculation when the geometry was relaxedon page 19, using the script ivrunrelax.sh:

#!/bin/sh

oldbias=0.0

bias=0.0

atk lih2-$bias-relax.atk lih2-$bias-relax.out \

-o lih2-$bias-relax.nc --force $*

F.2 Batch Files for Windows 115

for bias in 0.0 0.2 0.4 0.6 0.8 1.0 ; do

atk lih2-0.0-relax.atk lih2-$bias-relax.out \

-o lih2-$bias-relax.nc -i lih2-$oldbias-relax.nc \

--param "TwoProbe::LeftElectrode::Voltage $bias eV" \

--force $*

oldbias=$bias

done

Note how the calculation for the previous value of the bias is used as an initial guessfor the density matrix by using the command -i.

Again for the Li–H–H–Li, we also considered the influence of a gate electrode (page 20).The corresponding script gaterun.sh is:

#!/bin/sh

oldgate=0.0

cp lih2-0.0.nc gate-0.0.nc

for gate in 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 ; do

atk lih2-0.0.atk gate-$gate.out \

-o gate-$gate.nc -i gate-$oldgate.nc \

--param "TwoProbe::Molecule::GateVoltage -$gate eV" \

--force $*

oldgate=$gate

done

Note that we use a negative gate voltage, but in order to avoid strange filenames we in-clude the negative sign explicitly in the --param command and loop over the absolutevalue of the potential.

F.2 BATCH FILES FOR WINDOWS

In Windows, script files are usually referred to as batch files, and are given the exten-sion .bat. Such files can be executed directly by double-clicking them in WindowsExplorer.

116 Writing Scripts

Note! In Windows it is not possible to break long lines in a batchfile; the commands must be given on a single line. However, manyof the commands we wish to run are rather long, and do not fit on asingle line. For improved readability we have therefore in the scriptlistings below anyway broken them, by using a "n" sign at the endof the line. This character must be removed, and the correspondinglines joined together, if you wish to copy/paste the batch files fromthis manual. The batch files provided in the examples directory arehowever correctly formatted already.

The following script (latenergy.bat) calculates the total energy for a series of latticeconstants for the lithium chain discussed on page 10 (this must all be entered on asingle line, either in a batch file or on a Windows command line; c.f. the box onpage 116):

for %%A in (2.82 2.84 2.86 2.88 2.90 2.92) do \

atk lichain.atk lichain%%A.out -o lichain%%A.nc \

--param "UnitCell::LatticeConstant %%A Ang" --force

Next, we have a script ivrun.bat that calculates the current–voltage characteristics ofthe Li–H–H–Li two-probe system considered on page 17:

set oldbias=0.0

for %%V in (0.2 0.4 0.6 0.8 1.0) do call :Loop %%V

goto eof

:Loop

atk lih2-0.0.atk lih2-%1.out -o lih2-%1.nc -i lih2-%oldbias%.nc \

--param "TwoProbe::LeftElectrode::Voltage %1 eV" --force

set oldbias=%1

:eof


For the same system, we also made an I–V calculation when the geometry was relaxedon page 19, using the script ivrunrelax.bat:

set oldbias=0.0

atk lih2-0.0-relax.atk lih2-0.0-relax.out -o lih2-0.0-relax.nc --force

for %%V in (0.2 0.4 0.6 0.8 1.0) do call :Loop %%V

goto eof

F.3 Python Scripts 117

:Loop

atk lih2-0.0-relax.atk lih2-%1-relax.out -o lih2-%1-relax.nc \

-i lih2-%oldbias%-relax.nc --force \

--param "TwoProbe::LeftElectrode::Voltage %1 eV"

set oldbias=%1

:eof

Again for the Li–H–H–Li, we also considered the influence of a gate electrode (page 20).The corresponding script gaterun.bat is:

set oldgate=0.0

copy /Y lih2-0.0.nc gate-0.0.nc

for %%A in (1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0) do call :Loop %%A

goto eof

:Loop

atk lih2-0.0.atk gate-%1.out -o gate-%1.nc -i gate-%oldgate%.nc \

--param "TwoProbe::Molecule::GateVoltage -%1 eV" --force

oldgate=%1

:eof


F.3 PYTHON SCRIPTS

In order to use Python scripts, you must have the Python interpreter installed on yourcomputer. To test if you do, simply type the command

$ python -V

at the command prompt in either Windows or Linux. If Python is installed, this com-mand should display the Python version number. Be careful to type a capital V; if youtype python -v you will enter the interactive mode in Python, and you must pressCtrl+D to exit.

Python scripts are executed by simply giving the script name as the only argument toPython:

$ python script.py

118 Writing Scripts

where script.py is the name of the script.

The following script (latenergy.py) calculates the total energy for a series of latticeconstants for the lithium chain discussed on page 10:

import os

latconst = ["2.82", "2.84", "2.86", "2.88", "2.90", "2.92"]

for lat in latconst:

os.popen("atk lichain.atk lichain"+lat+".out \

--param \"UnitCell::LatticeConstant "+lat+" Ang\" \

-o lichain"+lat+".nc" --force)

Next, we have a script ivrun.py that calculates the current–voltage characteristics ofthe Li–H–H–Li two-probe system considered on page 17:

import os

bias = ["0.2", "0.4", "0.6", "0.8", "1.0"]

oldV = 0.0

for V in bias:

os.popen("atk lih2-0.0.atk lih2-"+V".out \

--param "TwoProbe::LeftElectrode::Voltage "+V" eV" \

--force -i lih2-"+oldV".nc -o lih2-"+V+".nc")

oldV = V

For the same system, we also made an I–V calculation when the geometry was relaxedon page 19, using the script ivrunrelax.py:

import os

bias = ["0.2", "0.4", "0.6", "0.8", "1.0"]

oldV = 0.0

V = 0.0

os.popen("atk lih2-"+V"-relax.atk lih2-"+V"-relax.out \

-o lih2-"+V"-relax.nc" --force)

for V in bias:

os.popen("atk lih2-0.0-relax.atk lih2-"+V"-relax.out \

--param "TwoProbe::LeftElectrode::Voltage "+V" eV" \

-i lih2-"+oldV"-relax.nc -o lih2-"+V+"-relax.nc" --force)

oldV = V


F.3 Python Scripts 119

Again for the Li–H–H–Li, we also considered the influence of a gate electrode (page 20).The corresponding script gaterun.py is:

import os

os.popen("cp lih2-0.0.nc gate-0.0.nc")

gatepotential = ["1.0", "2.0", "3.0", "4.0", "5.0", \

"6.0", "7.0", "8.0", "9.0", "10.0"]

oldVg = 0.0

for Vg in gatepotential:

os.popen("atk lih2-0.0.atk gate-"+Vg".out \

--param "TwoProbe::Molecule::GateVoltage -"+Vg" eV" \

--force -i gate-"+oldVg".nc -o gate-"+Vg+".nc")

oldVg = Vg


120 Writing Scripts

A P P E N D I X

GATOMIC DATA

This appendix list the pseudopotentials included in ATK 2.0, along with the corre-sponding atomic configurations used for constructing the basis sets. From the valencecharge of each pseudopotential, the valence configuration can be obtained. In the Sin-gle Zeta (SZ) basis set there is one basis function for each valence orbital. For instance,for a Au (Z=79) pseudopotential with ionic charge 11, the valence orbitals will consistof 5d106s1.

The polarization orbitals will have lowest angular momentum not represented in theSZ basis set. For instance, for the Au system, the polarization orbital will be a p orbital.

Z Symbol Electron configuration Valence configuration Valence charge

1 H 1s1 1s1 12 He 1s2 2s2 23 Li [He]2s1 2s1 14 Be [He]2s2 2s2 25 B [He]2s22p1 2s22p1 36 C [He]2s22p2 2s22p2 47 N [He]2s22p3 2s22p3 58 O [He]2s22p4 2s22p4 69 F [He]2s22p5 2s22p5 7

10 Ne [He]2s22p6 2s22p6 811 Na [Ne]3s1 3s1 112 Mg [Ne]3s2 3s2 2

continued on next page

121

122 Atomic Data


13 Al [Ne]3s23p1 3s23p1 314 Si [Ne]3s23p2 3s23p2 415 P [Ne]3s23p3 3s23p3 516 S [Ne]3s23p4 3s23p4 617 Cl [Ne]3s23p5 3s23p5 718 Ar [Ne]3s23p6 3s23p6 819 K [Ar]3d04s1 4s1 120 Ca [Ar]3d04s2 4s2 221 Sc [Ar]3d14s2 4s23d1 322 Ti [Ar]3d24s2 4s23d2 423 V [Ar]3d34s2 4s23d3 524 Cr [Ar]3d44s2 4s23d4 625 Mn [Ar]3d54s2 4s23d5 726 Fe [Ar]3d64s2 4s23d6 827 Co [Ar]3d74s2 4s23d7 928 Ni [Ar]3d84s2 4s23d8 1029 Cu [Ar]3d104s1 4s13d10 1130 Zn [Ar]3d104s2 4s23d10 1231 Ga [Ar]3d104s24p1 4s24p1 332 Ge [Ar]3d104s24p2 4s24p2 433 As [Ar]3d104s24p3 4s24p3 534 Se [Ar]3d104s24p4 4s24p4 635 Br [Ar]3d104s24p5 4s24p5 736 Kr [Ar]3d104s24p6 4s24p6 837 Rb [Kr]4d05s1 5s1 138 Sr [Kr]4d05s2 5s2 239 Y [Kr]4d15s2 5s24d1 340 Zr [Kr]4d25s2 5s24d2 441 Nb [Kr]4d45s1 5s24d3 542 Mo [Kr]4d55s1 5s14d5 643 Tc [Kr]4d55s2 5s24d5 744 Ru [Kr]4d75s1 5s24d6 845 Rh [Kr]4d85s1 5s14d8 946 Pd [Kr]4d105s0 4d10 1047 Ag [Kr]4d105s1 5s14d10 1148 Cd [Kr]4d105s2 5s24d10 1249 In [Kr]4d105s25p1 5s25p1 350 Sn [Kr]4d105s25p2 5s25p2 451 Sb [Kr]4d105s25p3 5s25p3 5


123


52 Te [Kr]4d105s25p4 5s25p4 653 I [Kr]4d105s25p5 5s25p5 754 Xe [Kr]4d105s25p6 5s25p6 855 Cs [Xe]5d06s1 6s15p6 756 Ba [Xe]5d06s2 6s25p6 857 La [Xe]5d16s2 6s25d1 358 Ce [Xe]4f25d06s2 6s24f2 459 Pr [Xe]4f35d06s2 6s24f3 560 Nd [Xe]4f45d06s2 6s24f4 661 Pm [Xe]4f55d06s2 6s24f5 762 Sm [Xe]4f65d06s2 6s24f6 863 Eu [Xe]4f75d06s2 6s24f7 964 Gd [Xe]4f75d16s2 6s25d14f7 1065 Tb [Xe]4f95d06s2 6s24f9 1166 Dy [Xe]4f105d06s2 6s24f10 1267 Ho [Xe]4f115d06s2 6s24f11 1368 Er [Xe]4f125d06s2 6s24f12 1469 Tm [Xe]4f135d06s2 6s24f13 1570 Yb [Xe]4f145d06s2 6s24f14 1671 Lu [Xe]4f145d16s2 6s25d14f14 1772 Hf [Xe]4f145d26s2 6s25d2 473 Ta [Xe]4f145d36s2 6s25d3 574 W [Xe]4f145d46s2 6s15d5 675 Re [Xe]4f145d56s2 6s25d5 776 Os [Xe]4f145d66s2 6s25d6 877 Ir [Xe]4f145d76s2 6s25d7 978 Pt [Xe]4f145d106s0 6s15d9 1079 Au [Xe]4f145d106s1 6s15d10 1180 Hg [Xe]4f145d106s2 6s25d10 1281 Tl [Xe]4f145d106s26p1 6s26p1 382 Pb [Xe]4f145d106s26p2 6s26p2 483 Bi [Xe]4f145d106s26p3 6s26p3 584 Po [Xe]4f145d106s26p4 6s26p4 685 At [Xe]4f145d106s26p5 6s26p5 786 Rn [Xe]4f145d106s26p6 6s26p6 887 Fr [Rn]7s1 7s16p6 788 Ra [Rn]7s2 7s26p6 889 Ac [Rn]6d17s25f0 7s26d1 390 Th [Rn]6d27s25f0 7s26d2 4


124 Atomic Data


91 Pa [Rn]6d17s25f2 7s26d15f2 592 U [Rn]6d17s25f3 7s26d15f3 693 Np [Rn]6d07s25f5 7s26d15f4 794 Pu [Rn]6d07s25f6 7s25f6 895 Am [Rn]6d07s25f7 7s25f7 996 Cm [Rn]6d17s25f7 6d17s25f7 1097 Bk [Rn]6d07s25f9 7s25f9 1198 Cf [Rn]6d07s25f10 7s25f10 1299 Es [Rn]6d07s25f11 7s25f11 13

100 Fm [Rn]6d07s25f12 7s25f12 14101 Md [Rn]6d07s25f13 7s25f13 15102 No [Rn]6d07s25f14 7s25f14 16103 Lr [Rn]6d17s25f14 7s26d15f14 17

References

[1] M. BRANDBYGE, J.-L. MOZOS, P. ORDEJÓN, J. TAYLOR, AND K. STOKBRO, Density-functional method for nonequilibrium electron transport, Phys. Rev. B 65, 165401(2002). i, 46

[2] C. G. BROYDEN, A class of methods for solving nonlinear simultaneous equations,Math. Comput. 19, 577–593 (1965). 24

[3] W. H. BUTLER, X.-G. ZHANG, T. C. SCHULTHESS, AND J. M. MACLAREN, Spin-dependent tunneling conductance of fe|mgo|fe sandwiches, Phys. Rev. B 63,54416 (2001). 59

[4] D. M. CEPERLEY AND B. J. ALDER, Ground state of the electron gas by a stochasticmethod, Phys. Rev. Lett. 45, 566 (1980). 27, 96

[5] P. HOHENBERG AND W. KOHN, Inhomogeneous electron gas, Phys. Rev. 136, 864(1964). 2

[6] H. J. MONKHORST AND J. D. PACK, Special points for Brillouin-zone integrations,Phys. Rev. B 13, 5188 (1976). 36, 91, 92

[7] J. P. PERDEW, K. BURKE, AND M. ERNZERHOF, Generalized gradient approximationmade simple, Phys. Rev. Lett. 77, 3865 (1996). 2, 27, 96

[8] J. P. PERDEW AND A. ZUNGER, Self-interaction correction to density-functionalapproximations for many-electron systems, Phys. Rev. B 23, 5048 (1981). 2, 23,27, 96

[9] P. PULAY, Chem. Phys. Lett. 73, 393 (1980). 24

[10] R. H. M. SMIT, Y. NOAT, C. UNTIEDT, N. D. LANG, M. C. VAN HEMERT, AND

J. M. VAN RUITENBEEK, Measurement of the conductance of a hydrogen molecule,Nature 419, 906 (2002). 5, 17

[11] J. M. SOLER, E. ARTACHO, J. D. GALE, A. GARCÍA, J. JUNQUERA, P. ORDEJÓN, AND

D. SÁNCHEZ-PORTAL, The SIESTA method for ab initio order-N materials simula-tion, J. Phys. Condens. Matter 14, 2745 (2002). i, 75, 108, 109

[12] K. STOKBRO, J. L. MOZOS, P. ORDEJÓN, M. BRANDBYGE, AND J. TAYLOR, Theoret-ical study of the nonlinear conductance of Di-thiol benzene coupled to Au(111)surfaces via thiol and thiolate bonds, Comp. Mat. Sci. 27, 151 (2003). 46

125

[13] J. TAYLOR, H. GUO, AND J. WANG, Ab inito modeling of quantum transport prop-erties of molecular electronic devices, Phys. Rev. B 63, 245407 (2001). i

[14] Y. TSERKOVNYAK, A. BRATAAS, G.E.W. BAUER, AND B. I. HALPERIN, Cond-mat/0409242v2 (2005). 59

[15] D. WORTMANN, G. BIHLMAYER, AND S. BLÜGEL, ab initio calculations of interfaceeffects in tunneling through MgO barriers on Fe(100), J. Phys. Cond. Mat. 16,S5819 (2004). 60

Index

.nc file, see NetCDF file

.out file, see output file# (comment symbol), 36, 84$ (command prompt), 3

A/B/C directions, 8, 36, 43accuracy, 2, 27–28

basis set, 28force tolerance, 26

analysis, 7, 15, 17, 19Analysis::, 7, 15

3DGrids::SaveEffectivePotential, 71SaveElectronDensity, 71SaveTo, 71

BandLine::NumPoints, 87BandLine::X::, 73

End, 37, 87NumPoints, 87Start, 37, 87

BlochState::X::Index, 87KPoint, 87Label, 87Spin, 87

DOS::X::, 17Energy, 87Kpoint, 88Label, 88Spin, 88Type, 88

MolecularOrbital::X::, 7Index, 88Label, 88Spin, 89

MPSH::, 17Calculate, 89

EnergyWindow, 89MPSH::X::, 17

Index, 89Label, 89Spin, 89

PrintMolecularEigenvalues, 89, 111PrintMullikenPopulation, 89, 109TransmissionSpectrum::, 15, 45, 49

Calculate, 89E0, 90E1, 90Eta, 47, 90NumEigenChannels, 90, 112NumKPoints, 90NumPoints, 90

approximations, 2ATK_DATA_DIR, 81AtomList::

Format, 6, 36, 42, 91Scale, 36, 91

band structure, 11, 36energy, 105

basis set, 9, 23, 28bias, 17, 19, 50boundary conditions, 11, 15, 29, 43Broyden mixing, 24bulk systems, 7–11Bulk::

AtomList, 36, 91ElectronTemperature, 39, 91FermiTemperature, 72NumKPoints, 8, 36, 39, 91UnitCell, 8, 92

central region, 43, 47, 49, 61checking the geometry, 14command line options, see options, 79

128 Index

command prompt, 3comments, 36, 84complex contour, 46conductance, 5, 16, 46, 57, 85, 112

differential, 112unit, 85

constrain atoms, 26continuation symbol, 114, 116contour integration, 46convergence, 24, 28, 38

problems, 25current–voltage characteristics, 17

default parameters, 9, 86–104defaults file, 105density functional theory (DFT), 2, 23, 38density of states (DOS), 17DZ basis set, 28, 94DZDP basis set, 28, 95DZP basis set, 23, 28, 95

effective potential, 19electrode, 11–12, 42, 47, 49, 60

alignment, 44heterogeneous, 51NetCDF file, 12, 14, 43, 61

Electrode::, 12, 42AtomList, 12, 42, 92ElectronTemperature, 61, 92FermiTemperature, 72NumKPoints, 12, 42, 48, 92, 93NumRep, 47, 52, 93UnitCell, 42, 93

electron temperaturebulk, 39molecule, 25

energyFermi, see Fermi leveltotal, see total energyunit, 85zero-level, see zero-level, energy

environment variables, 81equivalent atoms, 44, 49exchange–correlation functional, 2, 23, 27

Fermi level, 10, 39, 47, 103, 108, 111

force, 6, 13, 25, 28, 108tolerance, 26unit, 85

FracRecip, 11, 85fractional

length unit, 91reciprocal length unit, 11, 85spin unit, 85

gate electrode, 20GGA-PBE, 27, 38, 96

HBar (spin unit), 85heterogeneous electrodes, 51HOMO, 25, 111

I–V curve, 17initial guess, 18, 50input and output files, 80interfaces, 59InverseLengthUnit, 85

k-point sampling, 36, 39, 44, 54, 62keywords, 86–104

lattice constant, 8, 11lattice vectors, 8, 36LDA-PZ, 23, 27, 31, 38, 96LengthUnit, 84logical variables, 84LSDA-PZ, 27, 61, 96LUMO, 25, 111

mesh cutoff, 28Method::Type, 93miswrapped electrode atoms, 45mixing, 24, 25, 98molecular orbital, 7molecular systems, 6–7, 23–30molecular unit cell, 29Molecule::

AtomList, 6, 94ElectronTemperature, 25, 94ElectrostaticPaddingFactor, 29, 72, 94FermiTemperature, 72TotalSpin, 31, 94

Monkhorst–Pack, 36, 91, 92

Index 129

MPSH, 16, 21, 111Mulliken population, 63, 89, 106, 109multigrid method, 51

NEGF, 46NetCDF file, 7, 10, 14, 17NumOrb::, 93

BasisSet::Charge, 76, 95DeltaRinn, 76, 95EnergyShift, 76, 95PseudoPotentialDirectory, 72, 77, 95PseudoPotentialFile::XX, 77, 96Size, 9, 13, 23, 28, 38, 72, 94SplitNorm, 76, 95V0, 76, 95

BasisType, 72MeshCutoff, 9, 28, 96RadialSampling::DR, 96Tables::

Cutoff, 96NumPoints, 97

XCFunctional, 23, 31, 61, 96

options-a, 7, 15, 19, 79-c, 14, 44, 79-h, 79-i, 18, 50, 80-o, 14, 79-q, 80-v, 80--analyze, 7, 15, 19, 79--check, 14, 44, 79--force, 6, 14, 79--help, 79--init-from, 18, 50, 80--netcdf-file, 14, 79--param, 10, 28, 79--quiet, 80--verbose, 80--version, 79--xyz, 14, 44, 80

output file, 6, 9, 14, 105–112overwriting output files, 6

parallel calculations, 55, 59path, 81periodic systems, 7–11, 35–40Poisson equation, 51pseudopotential, 77, 108Pulay mixing, 24

reciprocal lattice vectors, 8, 11, 36Relaxation::, 25

ForceTolerance, 13, 26, 97MaxDisplacement, 26, 97MaxSteps, 13, 26, 97Type, 97

relaxing the geometry, 6, 13, 19, 25–26Results::, 105–112

BandLines, 11TransmissionSpectrum, 18

scaled coordinates, 36, 91scattering region, 41SCF::, 24

Algorithm, 25, 97Beta, 72Coordinate, 24, 25, 72, 97Criterion, 98DiagonalMixingParameter, 24, 25, 55,

72, 98HistorySteps, 25, 55, 62, 72, 98InitialBulkRun, 55InitialSpinPolarization, 31, 61, 98MaxSteps, 25, 99MixH, 72NumStepsReset, 72Tolerance, 25, 97, 106TwoProbe::InitialBulkRun, 99

screening, 41scripts, 10, 113–119self-consistency, 24

tolerance, 25, 98SGGA-PBE, 27, 31, 96Simulation::

ConstrainAtoms, 26, 100Type::, 100

Relaxation, 6, 13, 20, 26, 100SingleConfiguration, 100

130 Index

spatially resolved density of states, 17spin polarization, 30–33, 59spin unit, 85stacking, 47supercell, 48supercell, electrodes, 12surface atoms, 20System::Type, 100

Bulk, 8, 36, 38Electrode, 12, 42, 61Molecule, 6TwoProbe, 14, 44

SZ basis set, 9, 28, 94SZP basis set, 28, 94

TimeUnit, 85tolerance

force, 26self-consistency, 25, 98

total energy, 6, 9–11, 28, 29, 38, 40, 63,100, 107

transmission eigenchannels, 90, 112transmission spectrum, 15, 45, 62two-probe systems, 11–21, 41–57, 59–67TwoProbe::

CentralAtomList, 14, 16, 44, 100Contour::

Circle::NumPoints, 103DeltaEMin, 46, 47, 55, 103FermiLine::NumPoints, 103KTPoles::NumPoints, 103NumPoints, 46RealAxis::Eta, 47, 103RealAxis::PointDensity, 47, 103

LeftElectrode::ATKfile, 72, 101EquivalentCentralAtom, 44, 101EquivalentElectrodeAtom, 42, 44, 101NetCDFfile, 14, 44, 72, 101Voltage, 17, 20, 50, 101

LeftSurface::NumberOfAtoms, 14, 16,20, 101, 112

Molecule::GateVoltage, 20, 101RightElectrode::

ATKfile, 73, 101

EquivalentCentralAtom, 44, 101EquivalentElectrodeAtom, 44, 102NetCDFfile, 14, 44, 73, 102Voltage, 50, 102

RightSurface::NumberOfAtoms, 14, 16,20, 102, 112

UseMultigridForElectrostatics, 44, 51,102

unit cell, 8bulk, 8, 36electrode, 12, 42molecule, 29two-probe, 14, 44

UnitCell::LatticeConstant, 8, 10, 103X0, 103Y0, 104Z0, 104

vacuum level, 30, 111voltage drop, 19

xyz file, 14, 44

zero-level, energybulk, 111molecule, 30, 111two-probe, 111

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