GOLLUM 1.0
Contents
1. A B C D E F G H l J K Mode 1 Y Y v v v v v v Mode 2 v v v v v v v Mode 3 Y Y Y v v v v v Mode 4 v v v v v v v v v KEY MV PMV OV Definition A Mode Y Mode B Leadp Y Lead s PL and TPL properties C atom v EM atom properties D ERange v Energy range and number of data points E TRange v Temperature range and number of data points F Bias v Current voltage range and number of data points G Biasaccuracy Y Energy grid refinement H EFshift v Fermi energy shift I scissor Y Scissor corrections J anderson Y Coulomb blockade and Kondo physics K Vgate Y EM gate voltage MV v Mandatory variable PMV v Partial mandatory variable OV v Optional variable NOTE Mode 3 functions with two leads only 27 8 1 3 MANDATORY VARIABLE LEADP This variable contains one row per branch or lead and sets the branch properties The format is name leadp type matrix rows 2 columns 3 leadp 1 leadp 2 leadp 3 For each row branch the columns are as follows leadp 1 Set the number of PL in the branch leadp 2 Set the terminating PL in the branch Notice that leadp 2 must be equal to or smaller than leadp 1 leadp 3 This variable sets the Hamiltonian used to compute the Lead Green s function GF Possible values are 0 The lead GF is computed using the Hamiltonian and overlap matrices of the TPL using the Hamiltonian of2. The input file is defined in SECTION 8 1 The Extended_Molecule file is defined in SECTION 8 2 The Lead 1 and Lead_2 files is defined in SECTION 8 3 The output data files are given for the various mode directories and are described in SECTION 9 13 5 3 STEP BY STEP EXAMPLE DFT BASED 5 3 1 1 AU LINEAR CHAIN Enter the directory gollum 1 0 examples DFT based 1 Au_linear_chain Gold linear chain Objectives Simulate a one dimensional 1D system with perfect transmission equal to the number of open channels or bands at a certain energy Check that the transmission exactly coincides with the number of open channels Transport direction Results es y Transmission and number of open channels as a function The transmission has a step like shape of energy Calculated with Mode 1 of Gollum which corresponds to the number of open channels In this case there is only one s orbital in the valence of gold the rest of states are included in the pseudopotential so the transmission is equal to 1 in the range of energies that comprises the gold band The current linearly increases as ax1o expected since it is equal to the integral 1x10 of a constant function until it saturates and slightly decreases when the bias window starts covering the edges of the band which are rounded by the effect 1 5 Transmission of open channels HA 1x10 2x10
3. type matrix rows 1 columns 3 TRange 1 TRange 2 TRange 3 TRange 1 defines the minimum temperature for the calculation Tmin measured in Kelvin degrees TRange 2 defines the maximum temperature for the calculation Tmax measured in Kelvin degrees TRange 3 defines the number of temperature points to be performed in the range Tmin Tmax 29 8 1 8 PARTIAL MANDATORY VARIABLE FOR MODE 4 VARIABLE BIAS This variable controls the calculation of current voltage curves and is only used in Mode 4 The variable contains one row per lead In this example for a four lead junction the first row describes how the voltage in lead 1 is to be ramped up The voltage for the remaining leads is fixed to a given value given by V2 V3 V4 The format of the variable is as follows name Bias type matrix rows 4 columns 3 Bias 1 Bias 2 Bias 3 v200 V300 V400 Bias 1 defines the minimum voltage to be applied to the lead Vmin measured in volts Bias 2 defines the maximum voltage to be applied to the lead Vmax measured in volts Bias 3 defines the number of voltage points nvolt to be used in the range Vmin Vmax 8 1 9 OPTIONAL VARIABLES 8 1 10 OPTIONAL VARIABLE FOR MODE 4 VARIABLE BIASACCURACY By default the energy grid for the computation of transmission and current voltage curves contains 300 energy points which is sufficient for many junctions However this grid might miss sharp resonances for weakly co
4. All changes will be vetted by the advisory board Jaime Ferrer ferrer uniovi es Colin J Lambert c lambert lancaster ac uk Victor Garcia Suarez vm garcia cinn es Hatef Sadeghi h sadeghi lancaster ac uk Steven Bailey s bailey lancaster ac uk Please report bugs to the GOLLUM mailing list http www physics lancs ac uk gollum mail 2 3 CITING GOLLUM A full description of the functionality and capabilities of the code are to be found in The New Journal of Physics J Ferrer et al 2014 New J Phys 16 093029 doi 10 1088 1367 2630 16 9 093029 2 4 GOLLUM UNITS 1 Energy E in eV 2 Current intensities I in Amps 3 Voltages V in volts 4 Temperatures in Kelvins 5 All energies are referred to the Fermi energy Ef of the EM region 2 5 DISTRIBUTED FOLDER STRUCTURE The distribution contains the following folders El gj gollum 1 0 6 E Ji examples E ji DFT based Ej Tight binding based de gollum linux Ji gollum mac de gollum windows di manual Ji matlab src E a tools de input generator Ji interface to siesta Depending upon your operating system you will find in folders gollum linux gollum mac and gollum windows the executable file plus a generated Readme The examples folder is more complex but contains all the information to run the tutorial examples Once the zip files are extracted you will find two folders DFT based and Tight binding based
5. each of which will expand into the MACOSX versions and the Windows Linux versions All the Tight binding based examples labelled 1 6 contain e mode 1 mode 2 mode 3 mode 4 output data folders e Guide pdf which gives details of the system and plots of the expected calculation result e input showing the GOLLUM input file e Extended Molecule input to GOLLUM e Lead 1 inputto GOLLUM e Lead 2 input to GOLLUM All the DFT based examples labelled 1 8 contain mode 1 mode 2 mode 3 mode 4 output data folders e Guide pdf which gives details of the system and plots of the expected calculation result e input showing the GOLLUM input file e The input files needed to run the associated SIESTA calculations are to be found in lead fdf and emol fdf We also include the pseudopotential files psf NOTE the fdf files have the flags needed to extract the Hamiltonians using the DIRECT SIESTA INTERFACE METHOD shown in section 6 1 e A input showing the GOLLUM input file e Gollum files with the Extended Molecule Lead 1 and Lead 2 GOLLUM input files The folder manual contains this documentation The folder tools contains the indirect siesta interface to construct GOLLUM compatible Hamiltonians using the gollum input generator with a handy GOLLUM input file generator in MATLAB and direct siesta interface with the Fortran routines to install in
6. Current as a function of voltage of the voltage Calculated with Mode 4 of Gollum 14 System description and parameters Leads Extended molecule EM Au leads made of only 2 gold atoms Perfect gold chain made of 18 atoms 9 unit cells Lattice vectors long enough along the transverse of the electrodes and with the same transverse directions to make sure the system is 1D lattice vectors of the leads The system is also periodic along the transport direction z to avoid finite size effects Gollum parameters Transmission coefficients calculated between 5 0 eV and 8 0 eV in 200 energy points 2 principal layers on each electrode The terminating principal layer is the second on each Electronic structure obtained from the leads calculation No SAINT correction or Fermi level shift Bias voltage between 3 2 V and 3 2 V calculated in 25 voltage points The bias shift is applied on the first terminating layer of the left and right electrodes only 4 column of the atom variable set to different from 0 This is an approximation since the voltage should not fall or at least fall very slowly on a perfect infinite system Ab initio Siesta parameters Basis set Single SZ only the s orbital is included in the Au valence the rest are hidden in the pseudopotential LDA exchange and correlation functional CA parametrization Mesh cut off 150 Ry 90 k points along z in the lead calculation 10 k point
7. by Ci SEP D a Jf ki c a 2 vig Jv a 14 The matrix elements of the scattering matrix block connecting leads i and j can then be written as S ik Di qD G yt 16 C kj 15 Here G is the off diagonal block of the surface Greens function defined in Eq 6 that connects leads i and j and V is also defined in Eq 6 With the above if the incoming channel k of lead i is occupied with probability fj E i e in Eq 10 oy 1 with probability Fr E then the number of electrons per unit time enetreing the scattering region from reservoir i along channel k with energy between E and E dE is dig E dE h fy E 16 23 And the number per unit time per unit energy leaving the scatterer and entering reservoir i along channel q with energy between E and E dE is dle E dE h js s 2 fi 17 In many cases the incoming and outgoing channels of each lead i can be grouped into channels possessing particular attributes i e quantum numbers labelled a D etc This occurs when all incoming channels of a particular type a in lead i possess the same occupation probability fi E For example all quasi particles of type in reservoir i may possess a common chemical potential 4 and f E may take the form f f E a where f E is the Fermi function In this case if the incoming and outgoing channels of type a belonging to lead i possess wave vectors kap Ga t
8. by fitting to a band structure However this does not solve the problem of choosing parameters to describe the interface between two materials Often this problem is finessed by choosing interface parameters to be a combination of pure material parameters such as an arithmetic or geometric mean but there is no fundamental justification for such approximations Therefore we describe here methods to generate K using a DFT program where the inclusion of branches as part of the extended scatterer occurs naturally FIG 7 3 1 shows an example of a junction where the electrodes are identical The system is composed of super cells formed from a central scatterer and PLs There are periodic boundary conditions in the longitudinal direction such that the TPL of one branch of a super cell is linked smoothly to the TPL of a neighboring super cell Running a DFT program for such a super cell then automatically generates KP Provided the super cells 21 contain sufficient PLs the Hamiltonians KIP and KT associated with the TPLs will be almost identical to those generated from a calculation involving an infinite periodic lead i e if the Hamiltonians KgandK associated with the PLs are generated from a calculation involving an infinite periodic lead then provided the super cells contain sufficient PLs these will be almost identical to Kg and KT respectively In this case then there will be minimal scattering caused by the junction between the TPL and the le
9. gollum exe in your path For example in Windows 7 follow Control Panel System and Security System Advanced System Settings Environment variables then scroll down system variables to find path click on path then edit and use the command prompt to run the code but for simplicity we outline the following procedure Copy from the examples folder supplied with your distribution an example of your choice into a new directory of your choice Place gollum exe in the folder and run the calculation by double clicking gollum exe Your results of the calculation should match those to be found in the example matching your choice 4 1 2 INSTALLING THE MCR AND RUNNING LINUX From your GOLLUM distribution navigate to your version dependent executable in the gollum linux directory to locate gollum and run_gollum sh For a LINUX machine verify that you have the MCR e g using the terminal command locate MATLAB_Compiler_Runtime If you do not have the correct MCR then navigate to http www mathworks co uk products compiler mcr in MathWorks and locate the correct version to install LINUX requires Linux R2012a 64 bit If you requested the 32 bit version of GOLLUM then you will need the release Linux R2012a 32 bit Install as root to place the application in your usr local directory Enter your gollum_linux directory to find gollum and run_gollum sh NOTE ensure that you have full permis
10. of a junction the DFT program produces the Hamiltonians and Fermi energy of the EM and leads in separate runs as the Hartree potential is defined up to a constant which is usually different for the EM and for each lead This usually means that the energy of the EM and of the corresponding lead PLs as well as their Fermi energies do not agree with each other so Eq 1 must be rewritten as follows dd x PaL ES 74 7 where we have referred the energy of each lead to its own Fermi energy To fix the Hamiltonian mismatch we define a realignment variable for each lead as follows A Hou u H5 u u 8 where p indicates a relevant orbital or group of orbitals Then the Hamiltonian of each lead is realigned with that of the EM Ha E ENS 41 9 It turns out that the renormalized Fermi energies of each lead do not match perfectly with each other if the number of PLs in the EM region is not sufficiently large This is the case when for efficiency reasons it is desirable to artificially minimize the size of the EM Sometimes it is advisable to choose the Fermi energy of one of the leads as the reference energy In this case the overall shift is given by At or the quantity 6 E EM EEM 22 7 5 SCATTERING MATRIX AND TRANSMISSION IN MULTI TERMINAL DEVICES We note that the most general scattering state in a given lead i at a given energy E can be written as a linear combination of open and closed channels as follows
11. the EM which is read from the file extended scattering region 1 The Hamiltonian and overlap matrices of ideal leads are appended at the end of the TPL and become the new TPL These are read from the file Lead_i where i is the Lead number The Lead GF is computed using this new TPL The sign indicates whether the lead incoming channels are directed along the positive 1 or negative 1 z axis As an example for a junction containing two leads lead 1 left lead should use leadp 3 1 and lead 2 right lead should use leadp 3 1 For DFT based calculations it is advisable to generate and use the files Lead1 and Lead2 setting leadp 3 1 8 1 4 MANDATORY VARIABLE ATOM This variable defines the properties of all the atoms in the extended scattering region There is therefore a row per atom For example for an EM region containing 26 atoms the variable would look like name atom type matrix rows 26 columns 4 atom 1 atom 2 atom 3 atom 4 For each row the columns are defined as follows atom 1 Atom number in the DFT simulation atom 2 Sets the EM region where the atom is placed Possible regions are Molecule Surface atom 2 0 or one of the branches E g For Lead i atom 2 i atom 3 If atom 2 i 0 then this variable sets the PL in Lead to which the atom belongs to However if the atom belongs to the Surface Molecule region atom 2 0 then this variable is used to discern atoms b
12. 1 Direct SIESTA interface Method oooocccconcccocccnonnnonnnononnnnnnnnnonnnnnnnn nor nnnnn cnn nnne en nennen en nnne tenen ini nnne 16 6 2 Indirect SIESTA interface Method cccesesscesseeeseeeeeeseeeeeeeeeeeeeeeecaeceaeceaeceaecaaesaeesaeeeaeseeeseeeeerseeteeaees 16 6 2 1 Files needed from the Siesta calculation ooooncnnnccionccnnoccnonnnonancnnnnnonancnnnncnnnnnnnnnc no nc cnc crac nanncccns 16 6 2 2 Hamiltonian Interface files and structure sss ener enne nns 17 6 2 3 Generation of the Gollum files oo eee eeeeeeeeeeeeeceeeceeceeeceaecaaecaeecaeeeaeeeeeeeeeaeeeseseaeceaeceaeeaesaeeaeeeaes 17 6 2 4 Extended Molecule inicia ii rta e ata R dp eria Ede e EQ raa rot ic aS 17 6 3 EXUGAS e 17 7 Theoretical ApDEOSCh ee Hue oe ete tette aaa 18 7 1 Nomenclature rte Hohner tee ee ehe d ferte uie re correa ee ea oda ve toes eras 18 7 2 The surface Green s function for the current carrying leads ccccconococoonnnncnonononnnnnnncnncnnonnnnnononnnns 20 7 3 The extended scattering region Hamiltonian escena nnne 21 7 4 Hamiltonian Assembly cccccssssssscccececssssneusecececsessnensececessensnaseesecessesssansssecesessessnenseceeeesensnanseseeess 22 7 5 Scattering matrix and transmission in multi terminal devices esses 23 7 5 1 GOLLUM delivers the following functionalities ococonoooo
13. 4 columns 3 2 3 2 50 000 000 000 The variable name is Bias and it is a matrix containing four rows and three columns with the last four lines assigning values to the variables 8 1 THE INPUT FILE 8 1 1 MANDATORY VARIABLE MODE As you have completed the Examples to test your compilation you will note that the results appear in results Mode j where j 1 4 The concept of a Mode is used in GOLLUM to define the range of calculations to be carried out by GOLLUM The basic calculation carried out by GOLLUM appears in Mode 1 Historically this was the first set of calculations that the transport code could perform and is the default value if Mode is not given a defining number 4 name Mode Htype scalar j Mode 1 j 1 In this mode GOLLUM computes the equilibrium transport properties of a junction Mode 2 j 2 In this mode GOLLUM computes the zero voltage zero energy spin dependent Scattering matrix and Transmission coefficients 26 Mode 3 j 3 In this mode GOLLUM computes the zero voltage thermal properties of the junction as a function of temperature Mode 4 j 4 In this mode GOLLUM determines the finite voltage transmission coefficients and current voltage V curves of multi lead junctions 8 1 2 ASUMMARY OF THE MODE FUNCTION TABLE 9 1 2 Before giving the details of the next input variables and their inclusion for each Mode the organisation of the input structure is summarised below
14. A next GOLLUM generation 1 0 simulation too j for electron thermal and User Manual spin transport June 2014 lala DO E Universidad de Oviedo tad La Universidad de Asturias HH oo Bete Do REE e n On 3 aa vs v4 see eh e du a lt esos E RRR ses etet ete 4 at 1 p M9 Bef ogee ELE ege n 6 2a q PR eso S 2026 O e HL ES ete EEES GOLLUM Authors Steven Bailey Jaime Ferrer Victor Garcia Suarez Colin J Lambert Hatef Sadeghi Lancaster University Universidad de Oviedo Universidad de Oviedo Lancaster University Lancaster University GOLLUM Team Laith Algharagholy lain Grace Kaitlin Guillemot David Z Manrique Laszlo Oroszlani Rub n Rodriguez Ferradas David Visontai Lancaster University Lancaster University Lancaster University Lancaster University E tovos University Universidad de Oviedo Lancaster University The current version of this package is GOLLUM 1 0 released in June 2014 Itis freely distributed under the terms of GOLLUM Academic License Version 1 0 to be found on http www physics lancs ac uk gollum 1 Eo NAS N ee T 0 2 MEFODUCHION siieu a Ac 3 2 1 Updates and new functionality 2 irri e eerte eiecit nei eben erbe e Ede bee EE ea noeud nV a e ase diia 3 2 2 The Advisory Bo atd oie pem i a Eng ei E Raus ia ERAS EL RE RENE EXER TER BEAR EAR PER E
15. ANCE TABLE 10 2 1 The file contains the DOS of the molecule with and without scissor corrections as a function of the energy 9 2 ADDITIONAL MODE GENERATED FILES Key Comment spin refers to paramagnetic spin up down refers to spin polarized i 1 N where N the number of leads Mode 2 G_spinsumi dat The spin summed conductance G Gp Mode 2 G_downi dat The spin summed conductance G G Mode 2 G_upi dat The spin summed conductance G Gp Mode 2 S_matrix dat The S matrix Mode 3 G_spinsumi dat The spin summed thermal conductance G T Mode 3 G_downi dat The spin summed thermal conductance G T Mode 3 G upi dat The spin summed thermal conductance G T Mode 3 Thermopoweri dat Spin summed thermopower S T Mode 3 Peltieri dat Spin summed Peltier coefficients I T Mode 3 Thermal conductancei dat Spin summed thermal conductance k T Mode 4 Charge Currenti dat Current voltage Mode 4 Spin Currenti dat Spin current 9 2 2 MODE 2 GENERATED FILES Pi 9 2 2 1 CONDUCTANCE Mode 2 G_spinsumi dat The spin summed conductance G Go Mode 2 G_downi dat The spin summed conductance G Go Mode 2 G_upi dat The spin summed conductance G Go For each lead i the file G_spinsumi dat containing N columns where N is the number of leads is written Each column j provides the spin summed zero voltage conductance G Go For spin polarized calculations additional files G_upi dat and G_downi dat cont
16. ERRARE ARR ERR RIS 3 2 3 agracstEsumeE M 3 2 4 GOLLUM Unitat ad OE ERRAT HERRERA 3 2 5 Distributed Folder Str cture irr nnm re eth tn Fe ERR re eR REL aa Arn sp RE REFER eg YR REPE ERR 4 3 Thescopeofthegoll m project uie RE lord deis 5 4 Installation and RUnnllig n NN 7 4 1 MATLAB VETO e 7 4 1 1 Installing the MCR and running in Windows eene nennen nennen nennen 7 4 1 2 Installing the MCR and running LINUX ooocococccnnocccononononccononononcnononononcnonn nono ncnnn nono nnnnnn rra nn entente nens 8 4 1 3 Installing the MCR and running MAC ceeccecsseceececseceeeeecaeeeeeeecsaeceeeecaaeeeeeecsaeeeeeeecaaeseeeeeaeenses 9 5 Test Examples and TUTORIALS rtr rore tei teer E eaaa A aA AAAA E a AR a AEE a aaiae ue 10 5 1 Summary of the examples TABLES Lisiera terere eere een ene nee FERE reped eroe Sa Ee reatus 10 5 2 Step by step example Tight binding based ccoooccccnoncccnononnnononanonononcconannoncnnnonnnconnnnnnnnn nn enne enne 12 SZL 2 Single atom EA tnre ein o sethn ERE ke ag Run Ens a d ae Eua e RR ann aS EER a aan RENTE ERA 12 5 3 Step by step example DFT based oooocooocccncccooncconcnononcnnnnanonnnnnnnnnonnnranon cnn n ran cnn cnn ran nn en nennen nnne nennen enne 14 Bide ES AAA TN 14 6 Harniltoniam generation ousocee tort nne ntum shoes cnetadseagutendnnsedgadaanoieustan sheds saietandaasathesnns ea vao ine DR ae Rn oi 16 6
17. IESTA CALCULATION In order to generate the files that are used by Gollum to calculate the transport properties it is necessary first to redirect the Siesta output to a file A Some of the information needed by Gollum is already printed in the output file of Siesta the file where the Siesta output is redirected e g Ni out It is also necessary to include some options in the Siesta Ni fdf file so that Siesta prints the necessary information firstly name the output files accordingly and B SaveHS T With this option in the Ni fdf file Siesta generates a HSX file e g Ni HSX where the Hamiltonian and overlap matrix elements are stored C WriteEigenvalues T With this option in the fdf file Siesta generates a EIG e g Ni EIG file where the Fermi energy and the eigenvalues are stored Siesta should also generate by default a KP file e g Ni KP where the k points information is stored 16 6 2 2 HAMILTONIAN INTERFACE FILES AND STRUCTURE The interface is included in the directory tools interface to siesta There are two Fortran files f90 and a Makefile A hsx m f90 Module that reads and processes the Hamiltonian and overlap from the HSX file B siesta2gollum f90 Main program which reads the rest of the information and creates the Extended Molecule or Lead i files C Makefile Makes the siesta2gollum executable Edit the Makefile to specify the compiler after the FC Fortra
18. Im kp lt 0 and therefore propagate decay towards the left of FIG 5 These are called negative open closed channels By using the dual vectors D k and D kp which satisfy D k C k 65 and D k C kq 6 4 which are found by inverting the NxN matrices Q Cki neil ee Cte Cu 20 D Axa RE Dy 9751 D By 071 the required transfer matrices are constructed T Yl Cel D 3b f gt en p 3c The transfer matrices allow us to build the coupling matrix V the self energies X and the surface Green s functions Gi o V KAa T T 4 KT 5 Gia Ko E 6 NOTE the procedure might fail if K is singular We adopt a decimation method to remove the offending degrees of freedom see the GOLLUM paper for details 1 7 3 THE EXTENDED SCATTERING REGION HAMILTONIAN 000 3000 UE e DOODDE DT DC C NL TOMO Unit Cell Unit cell _ gt FIG 7 3 1 The infinite system where the FIG 7 3 2 The super cell containing the unit cell is linked by periodic boundary EM with vacuum buffer regions to left condition and right The extended scattering region Hamiltonian K can be provided as a model Hamiltonian or generated by a DFT or other material specific program One of the strengths of GOLLUM is an ability to treat interfaces with high accuracy In a tight binding description tight binding parameters of a particular material are often chosen
19. Manrique D Visontai L Oroszlany R Rodriguez Ferrad s Grace S W D Bailey K Guillemot H Sadeghi L A Algharagholy New J Phys 16 093029 2 J M Soler E Artacho J D Gale A Garc a J Junquera P Ordej n and D S nchez Portal J Phys Condens Matter 14 2745 2002 3 J P Lewis P Jelnek J Ortega A A Demkov D G Trabada B Haycock H Wang G Adams J K Tomfohr E Abad H Wang and D A Drabold Phys Stat Solidi B 248 1989 2011 40 Cite GOLLUM paper euin nte rt nnns 0 3 conductance iie cni eee idee igit 42 43 Current amps GOLLUM units ssss 3 29 42 45 DFT based Example and tutorial i Direct SIESTA method seen EM extended scattering region 31 33 35 36 39 41 3 19 20 23 24 29 30 Energy eV GOLLUM units sees 3 29 33 Examples Example and tutorial 0 4 8 9 10 11 12 15 17 Extended molecule input method ssssssesseeeeeeeneenennn 1 18 F Fermi energy EF GV isis 3 18 23 24 29 33 35 36 39 ALOLE A T a NE RR RENS 5 18 G Generation of GOLLUM files seuees 0 18 Gold lead viii osa aa O 11 15 16 18 GOLLUM paper1 O 3 4 5 6 7 8 10 14 16 17 18 19 20 21 22 23 26 27 28 33 35 36 43 44 45 H Hamiltonian generation theory O 1 5 17 18 20 21 22 23 27 30 35 37 38 Indirect SIES
20. PL PL PL PL Hol Hol Ho Ho Ho Ho Hol Ho Ho Positive direction FIG 7 2 1 Illustrates the infinite system used to generate the leads Hamiltonians Ki and Ki where the positive direction is towards the scatterer For non orthogonal basis sets the overlap matrices Si 1 and S must have the same structure as the Hamiltonian matrices and 2d Ha BS ii 1 K Hi ES 2 By expanding the Bloch eigenstates w k Yng ein c o o Qn 11 of the infinite system in a localised basis set a NxN secular equation can be found Ko K e K_ e C k 0 which if by choosing the energy E and solving for the allowed wave vectors produces 2N solutions with either real or complex wave vectors ky p 1 2N which GOLLUM solves to give the eigenvalues and eigenvectors C kp G a elke mete 92 a B The group velocities of the states corresponding to the real wave vectors is then given by C kp Ke K_ e P c kp v kp a orse ras uen C kp i v kp has units of energy and is real and the fully dimensioned group velocity is given by v kp a h where a is the spacing between PL s in a given lead The 2N wave vectors are split into two sets the first denoted kp are real complex wave vectors that have v gt 0 Im kp gt 0 and therefore propagate decay towards the right of FIG 7 2 These are called positive open closed channels The second set are denoted kp are real complex wave vectors that have vp lt 0
21. Spin summed Peltier coefficients In a similar layout to the conductance files for each lead i GOLLUM writes a file called Peltieri dat which contains the spin summed temperature dependent Peltier coefficient between lead i and lead j jzi 39 ee eee eee er ee eC ee eee eee ee ee ee eee eee eee eee eC ee eer a eee Cree err j 9 2 3 4 THERMAL CONDUCTANCE Mode 3 Thermal conductancei dat Spin summed thermal conductance In a similar layout to the conductance files for each lead i GOLLUM writes a file called Thermal conductancei dat which contains the spin summed temperature dependent thermal conductance between lead i and lead j ji 9 2 4 MODE 4 GENERATED FILES 9 2 4 1 CURRENT VOLTAGE Fr Mode 4 Charge_Currenti dat Current voltage For each lead i GOLLUM writes a file called Charge_Currenti dat which contains the charge current as a function of voltage I V The file gives the total charge currents flowing between lead i and lead j j i as a function of voltage measured in Amps 9 2 4 2 SPIN CURRENT P Mode 4 Spin_Currenti dat Spin current If the calculation is spin polarized a file called Spin Currenti dat is also written for each lead i containing the spin current defined as If p as a function of voltage 1 GOLLUM a next generation simulation tool for electron thermal and spin transport J Ferrer CJ Lambert V M Garcia Suarez D Zs
22. TA method oooooccconcnccocnnccnonnncnonnnnnonnnnnos 0 17 Input file variable mandatory partial 41 optional in 1 27 installation eese 0 7 A A O 17 18 L e O 14 16 M ELDER 9 MCR for Matlab ooooccnocccccoccncnonnnnononcncnonanonannnncnnnos 0 7 8 9 Mod e 1 nui ai ri cheese teo phie 28 39 40 41 Mode 2 AEE SE 2 28 40 41 42 43 Mode 3 ossessi 2 28 29 30 40 42 43 44 Mode 4 essseeeeeeenn enne 2 28 29 40 42 45 Modes input output 1 2 27 28 29 30 31 32 33 34 38 39 40 41 42 43 44 45 O Output TA sand E E 2 38 P PL principle layer 20 21 27 29 30 31 38 R PUMA odes aten de 0 7 17 23 S Shot noise eee err ero ERE Enni 2 39 40 41 SIESTA O siranae neea odoras eene Eseia ESETE 45 Step by step tight binding and DFT ccceesesseesteeeseeee 0 12 15 Summary che me 0 10 T Temperature K GOLLUM units eet 29 34 Theoretical cie rot xr creer vete ten 1 19 Tight binding based Example and tutorial c cccsessseeeseesees 0 4 10 12 terminating principle layer 20 23 29 30 TPL 42
23. ability to construct conductance statistics relevant to break junction and STM measurements of single molecule conductances e Integration of a wide range of phenomena including thermoelectrics spintronics superconductivity Coulomb blockade quantum pumps Kondo physics and topological phases e The ability to describe multi probe structures e An interface between classical molecular dynamics which handles interactions with the environment The GOLLUM work flow shown in FIG 3 1 below illustrates the versatility of the package to allow the user to generate the required Hamiltonians either by hand from prescribed tight binding parameters or by adapting the output of DFT codes A suitable methodology follows the following route Once the atomic arrangements are generated these are fed into the second stage where the Hamiltonian matrix is generated This stage is in practice independent of the previous geometry construction and can berun separately taking only the output geometries of the first stage The junction Hamiltonian can be generated using a variety of tools some of which are listed in box Il in FIG 3 1 A popular approach is the use of DFT codes that are able to write the Hamiltonian in a tight binding language In this way model tight binding Hamiltonians can also be easily generated Other approaches involvethe use of Slater Koster or semi empirical methods In addition GOLLUM has the ability to modify these Hamiltonian matrices as
24. ad Clearly there is a trade off between accuracy and CPU time because inserting more PLs increases the size and cost of the calculation In practice the number of PLs retained in such a super cell is increased in stages until the results do not change significantly as the number of PLs is increased further There exist situations where the electrodes are dissimilar either chemically or because of their different crystalline structure or because their magnetic moments are not aligned In these cases there cannot be a smooth matching between TPLs of neighboring super cells in FIG 7 3 1 To address this situation we use a setup similar to that displayed in FIG 7 3 2 where additional PLs are appended to the branches in the EM region These additional PLs are terminated by artificial surfaces and surrounded by vacuum The TPLs are then chosen to be one of the PLs near the middle of each branch and should be surrounded by enough PLs both towards its artificial surface and towards the central scattering region Then the PLs placed between the TPL and the artificial surface can be discarded These sacrificial PLs ensure that the chosen TPL is unaffected by the presence of the artificial vacuum boundary Clearly calculations of this sort are more expensive in numerical terms than those performed with super cells generated as in FIG 7 3 1 because they contain many more atoms 7 4 HAMILTONIAN ASSEMBLY In an ab initio calculation of the transport properties
25. aining the spin resolved conductances G Go are also given 38 eee ee ee eee eC eee eee ee ee ee eee ee ee eee ee Te eee eee ee eee eee e Ce erry 9 2 2 2 S MATRIX Mode 2 S matrix dat The S matrix The S matrix dat file contains the scattering matrix coefficients 9 2 3 MODE 3 GENERATED FILES 9 2 3 1 CONDUCTANCE E Mode 3 G_spinsumi dat The spin summed thermal conductance G T Mode 3 G_downi dat The spin summed thermal conductance G T Mode 3 G_upi dat The spin summed thermal conductance G T For each lead i GOLLUM writes a file called G_spinsumi dat which contains the spin summed temperature dependent conductance G T G G between lead i and lead j jzi There is a row for each temperature K where the first column in each row provides the temperature and the remaining columns give G T The spin resolved temperature dependent conductances are given in G_downi dat and G_upi dat r STIIS SINAES SAA PEI E ANNA TIA NSPA N S AE AIAI A I II AEI O AARI O DAENT sede NII IPI SIAE PAI IEAA Desc css pe ed ee AEREI I AO IS APII IA IS TA AIEA j 9 2 3 2 THERMOPOWER Mode 3 Thermopoweri dat Spin summed thermopower In a similar layout to the conductance files for each lead i GOLLUM writes a file called Thermopoweri dat which contains the spin summed temperature dependent thermopower Q T between lead i and lead j j i j 9 2 3 3 PELTIER COEFFICIENTS Mode 3 Peltieri dat
26. annnnos 31 8 1 12 Optional variable for all Modes variable scissors enne 31 8 1 13 Optional variable for all Modes variable andersonN cccconococconcnnccnnononnnnnonccnnonononononccnnanenonnos 31 8 1 14 Optional variable for all Modes variable Vgate ccccccccccccsessssececececsessaesecececsesseasseeecsseeeeeaes 32 8 1 15 Example sin Put file et reti eremo a aee de tad 32 8 2 The Extended Molec le file 2 ort tre reete e ote cene eerte eed epa nario ep don 33 8 2 1 An Example Extended Molecule file oooooocccnccononoooonnnnccnoonononnnononncnnonnnnnnncnncnnononncnncnnconnns 33 82 2 Vatiable nsplhxs eret Ree e eoe rotae eda nea one eec emet e deer ee voee ae de A nd en eae deer e agn 33 8 2 3 E AAA Petra a po briser ied Decoder a noA BE rere a es Eidos 33 82 4 Variables IOrD zov rre etre A NN 34 10 11 832 5 Mariablez kpOlbits 5 t re Der aida 34 826 Variable Minnie e eheu orte reine vo tue ee va ve ora oda oct Da 34 8 3 Lead iinput file RN O 35 8 3 1 Example idea le iii rene eterne eoe A reve eene eene a erae ande ve Ege eer e ag 35 83 2 Variable kpoints lead RS 35 8 3 3 Vatiable HShe rre ida tv deter ida oe aree peers ya e ine dena e oet eras 35 Output file format and notation cccconocooncnncnnnononannnnnoncnnnonononnnoncnnnnnonnnncnnnnnenonnnnnnnnnnnennnnnnnnnnnnnnonnnnncnnnnnnns 36 9 1 The Universal Mil eerte eder ees dee eter e oy vene nre aree pee re
27. ates Iz using real spherical harmonics iorb 5 Indicates the z linumber GOLLUM only uses iorb 1 The rest are placed for information purposes 8 2 5 VARIABLE KPOINTS name kpoints EM H type matrix rows 1 columns 3 kpoints 1 kpoints 2 kpoints 3 This variable sets the transverse kpoints used in the DFT simulation of the Extended Molecule There are as many rows as kpoints kpoints 1 Gives the value of kx kpoints 2 Gives the value of ky kpoints 3 Gives the weight in the k summation 8 2 6 VARIABLE HSM name HSM type matrix rows 36 columns 7 HSM 1 HSM 2 HSM 3 HSM 4 HSM 5 HSM 6 HSM 7 HSM 8 HSM 9 This variables set the overlap S i j k and H i j k TL for a given transverse k point given orbital pairs i J and spin T1 The number of columns is 7 if the calculation is paramagnetic nspin 1 or 9 if the calculation is spin polarized nspin 2 There are as many rows as non zero Hamiltonian or overlap matrix elements If for given i j k 11 all matrix elements are zero then this row is skipped HSM 1 Defines the k point number as defined in variable kpoints HSM 2 Defines the orbital number i as defined in variable iorb HSM 3 Defines the orbital number j as defined in variable iorb 34 HSM 4 Provides Real S i j k HSM 5 Provides Imag S i j k HSM 6 Provides Real H i j k or Real H i j k 1 nspin 1 or 2 respectively HSM 7 Provide
28. c two terminal device where electrons are driven from the left to the right lead through the extended scattering region The leads are possibly kept at chemical potentials 41 2 e V1 2 where V V1 V2 is the applied bias Bottom Each lead is composed of an infinite chain of PLs with Hamiltonians HO which are coupled with each other via the coupling Hamiltonians H1 The Extended Moleculecomprisesthe actual Molecule theelectrodessurfaces andthe Left and Right Branches These branches contain several PLs upto the TPL The TPL linkthe EMtotheleads Both leadsareassumedtobe identicalinthisfigureforsimplicity FIG 7 1 2 shows a two terminal device in more detail and introduces further terminology The regions in light blue are called electrodes or leads and are described by perfect periodic Hamiltonians subject to chosen chemical potentials Each lead i is formed by a semi infinite series of identical layers of constant cross section which we refer to as principal layers PLs FIG 7 1 1 shows only two PLs per lead coloured white although an infinite number is implied Furthermore in the figure the leads are identical and therefore the lead index i has been dropped These PLs are described mathematically by intra layer Hamiltonians Hi PLs must be chosen so that they are coupled only to their nearest neighbors by the Hamiltonians Hi If each PL contains Nt orbitals then Hj and H are square Ni N matrices The extended scatterer EM in dark b
29. curves 8 Handles vdW and LDA U functionals 9 Scissors corrections scheme for strongly correlated systems 10 Computes band structure of the leads and density of states DOS in the scattering region 11 Covers large samples enabling to analise ballistic to diffusive regime Version 2 0 will come soon with Kondo amp Coulomb blockade physics Magnetic field and integer quantum Hall effect Info about Local charge spin current and spin current densities inside the scattering region Scripts to handle multiscale simulations Oo O00 0 25 Gollum can read the Hamiltonian of the extended scattering region which includes a terminal principle layer as shown in FIG 4 and extract the PL to build the leads Green s function It can also read stand alone lead Hamiltonians to obtain the same information Therefore GOLLUM requires as a minimum an input file and an Extended Molecule file with an optional instruction to read Lead i files The variables in the input file are designed to be clear and easy to interpret and apart from the Mode variable which must appear first the variables can be placed in arbitrary order and some variables are optional The Extended Molecule file instructs GOLLUM how to read the Hamiltonian overlap matrix orbitals spin degrees of freedom and the number of transverse k points for the extended scattering region All variables are mandatory An example of an input variable name Bias Htype matrix rows
30. ded molecule of an iridium cluster between silver surfaces has been performed Siesta calculation dumped to e g IrAg out do the following siesta2gollum IrAg out 0 This generates the Extended Molecule file that should be included in the directory where Gollum is going to be run 6 3 EXTRAS 1 In tools input generator there is a MATLAB script to help generate the INPUT lists and convert the format of the INPUT files for GOLLUM For large simulations the input files are tedious to write out by hand and the MATLAB script input generator m reads input generator in to automate the process 17 For a full description please refer to the GOLLUM paper 7 1 NOMENCLATURE GOLLUM describes open systems comprising an extended scattering region coloured dark blue in FIGS 7 1 1 and 7 1 2 connected to external crystalline leads coloured light blue in FIGS 7 1 1 and 7 1 2 Depending on the problem of interest and the language used to describe the system the material M of interest forming the central part of the scattering region could comprise a single molecule a quantum dot a mesoscopic cavity a carbon nanotube a two dimensional mono or multi layered material a magneto resistive element or a region containing one or more superconductors FIG 7 1 1 shows an example of a 4 lead system whose central scattering region labeled M is a molecule It is important to note that in an accurate ab initio description of such a structure the pr
31. directory to locate the folder gollum app which contains the executable and the files readme txt and run_gollum sh locate MATLAB_Compiler_Runtime If you do not have the correct MCR then navigate to http www mathworks co uk products compiler mcr in MathWorks and locate the correct version to install Mac requires release Mac R2014a intel 64bit Once the MCR is correctly installed you can make your first test Follow the instructions in the MATLAB readme file making sure that you place not only the run_gollum sh but the gollum app in your test directory and type run_gollum sh lt mcr_directory gt where mcr_directory locates the MCR version you have installed If you have a compatible MCR then you will see the following window DITE verston 1 0 Oviedo and Lancaster Universities http www physics lancs ac uk gollum tart of run 17 Aug 2014 11 33 04 rror using gollum line 36 Extended Molecule file is not found Please include Extended Molecule file try again sghs x1 s M Ignore the Error at this stage as it indicates that as yet you have not included the correct input files to run a calculation You can of course place gollum in your path using the Mac terminal in a similar way to the Linux procedure to run the code To run an example for simplicity we outline the following procedure Copy from the examples folder supplied with your distribution an example of your ch
32. discussed above For example the Hamiltonian matrix can be modified to include scissor corrections Coulomb blockade physics a gate or bias voltage a magnetic phase factor or a superconducting order parameter Finally stage III is the actual quantum transport calculation This takes the Hamiltonian matrix as an input and calculates the s matrix and associated physical quantities such the electrical or spin current the conductance or the thermopower Additional inputs HE Pseudopotential Ps Force field parameters fe a Initial geometry construction Geometry optimization or MD Hamiltonian generator Script based or GUI based Geometry DFT Classical codes Geometry DFT and TB codi Hamiltonian Avogadro SIESTA LAMMPS Arguslab NWCHEM DLPOLY FIREBALL AMBER SIESTA DFTB GOLLUM Transport code NWCHEM Semi empirical Matrix FIREBALL methods ASE Conductance DI Current Hamiltonian modifier DI Thermopower Addign magnetic field e Static gate I Scissor correction Transmission coefficient Environment Attaching electrodes Solvation shell Harina rr rra rr rr FIG 3 1 Typical GOLLUM workflow with various optional software tools GOLLUM is distributed as a standalone MATLAB executable A copy can be obtained by logging onto the web page at http www physics lancs ac uk gollum and completing the application form Upon request a user specific compatible version of GOLLUM will be pro
33. e strongly recommend that the Hamiltonians are generated through a suitable DFT code At present GOLLUM is directly interfaced with SIESTA and the various directly compatible versions can be found in direct siesta interface Nevertheless GOLLUM is not restricted to these SIESTA versions and a suitable Hamiltonian can be obtained from other versions by using the INDIRECT SIESTA INTERFACE METHOD shown in SECTION 6 2 below 6 1 DIRECT SIESTA INTERFACE METHOD The direct interface requires that SIESTA is recompiled with the GOLLUM interface subroutines In the folder direct siesta interface the files that need to be included in the SIESTA main source folder are located Using the included Makefile will then compile a GOLLUM compatible SIESTA executable Running SIESTA will then produce the Extended_Molecule and or the Lead_i files if the flag EMol or Lead is included in the SIESTA fdf input file Examples of the flag usage are to be found in examples DFT based DFT based Other interfaces with flavours of SIESTA are also included Interface siesta 3 0 rc2 Interface siesta LDA U Interface siesta spinorbit Interface siesta trunk 367 VdW 6 2 INDIRECT SIESTA INTERFACE METHOD GOLLUM is designed to be flexible It can use any suitable DFT Hamiltonian by constructing the GOLLUM compatible Hamiltonian At present we have developed a set of routines to convert SIESTA output files as follows 6 2 1 FILES NEEDED FROM THE S
34. elonging to the Molecule itself atom 3 0 or the surface atom 3 1 which is used to apply scissors corrections or include a Kondo U only in the molecule 28 atom 4 If atom 4 i the variable instructs the program to shift the on site matrix elements of the orbitals in the atom by the voltage of Lead i If atom 4 0 there is no shift Variable atom can have very many rows and a simple MATLAB utility in 6 4 Extras 2 automates the generation of the file 8 1 5 PARTIAL MANDATORY VARIABLE These variables are mandatory for specific Modes only 8 1 6 PARTIAL MANDATORY VARIABLE FOR MODE 1 VARIABLE ERANGE This variable sets the energy range and number of energy points used to compute The variable format is as follows name ERange type matrix rows 1 columns 3 ERange 1 ERange 2 ERange 3 ERange 1 Defines Emin measured in eV units ERange 2 Defines Emax measured in eV units ERange 3 Defines nE The variable is only read ifMode 1 because the program uses default values for the other three modes If Mode 2 Emin and Emax are set to 0 and nE to 1 If Mode 3 Emin and Emax are set to 0 0025 Tmax and nE is set to 500 If Mode 4 Emin and Emax are set to Vmax 2 and nE is 300 8 1 7 PARTIAL MANDATORY VARIABLE FOR MODE 3 VARIABLE TRANGE This variable controls the temperature range in the calculation of thermal properties It is not used in the other three modes Its format is as follows name TRange
35. ements Hy i j k TL for a given transverse k point given orbital pairs i j and spin Tl The number of columns is 11 if the calculation is paramagnetic nspin 1 or 15 if the calculation is spin polarized nspinz2 There are as many rows as non zero Hamiltonian or overlap matrix elements If for given i j k TL all matrix elements are zero then this row is skipped The ordering of the variable is For nspin 1 i j k Real S Imag So Real Ho Imag Ho Real S Imag S Real H Imag H For nspin 2 i j k Real So Imag So Real Ho 1 Imag Ho 1 Real Ho Imag Ho Real S Imag S Real H 1 Imag H Real Hi Imag Hi The output files are written for each lead i of a junction containing N leads Some files are universal to all Modes but some are specific to each Mode defined in the input file The output file for calculations with N leads will have N 1 columns where the first column is the energy 9 1 THE UNIVERSAL FILES 9 1 1 THE UNIVERSAL OUTPUT AT A GLANCE TABLE 10 1 1 This table shows the universal output files as seen in the Results folder in the Examples Key Comment spin refers to paramagnetic spin up down refers to spin polarized o 1or2 Mode 1 2 3 4 Open_channels_per_spino dat The number of open channels per lead Mode 1 2 3 4 Open_channels_upo dat The number of open channels per lead Mode 1 2 3 4 Open channels downo dat The number of open channels per lead M
36. he above equations fg E can be replaced by fg GE fg E f E where f E is an arbitrary function of energy which in practice is usually chosen to be the Fermi function evaluated at a convenient reference temperature and chemical potential When comparing theory with experiment we are usually interested in computing the flux of some quantity Q from a particular reservoir From Eq 18 if the amount of Q carried by quasi particles of type a is Qo E then the flux of Q from reservoir i is Ih dE R Ya s Qa E P f E e In the simplest case of a normal conductor choosing Qo e independent of q then the above equation yields the electrical current from lead i Within GOLLUM a may represent spin and in the presence of superconductivity may represent hole A or particle p degrees of freedom In the latter case the charge Qp e and Qr e 7 5 1 GOLLUM DELIVERS THE FOLLOWING FUNCTIONALITIES 1 Simulates multi terminal devices 2 Reads either Tight Binding or DFT Hamiltonians 3 Computes the full scattering matrix 4 Computes charge transport 1 Number of open scattering channels 2 Transmission and reflection coefficents 3 Shot noise 5 Computes heat transport 1 Thermal conductance 2 Thermopower Seebeck coefficent 3 Peltier coefficent 6 Computes spin transport 1 Collinear spins systems 2 Non collinear spins systems 3 Systems with strong spin orbit interaction 7 Computes zero voltage as well as I V
37. hen the number of quasi particles per unit time of type a leaving reservoir i with energy between E and E dE is dli E dE h Xo P f CE 18 ij da kg 19 where p BE Mo EJ Sa p s Laake and Mi E is the number of open incoming channels of type a energy E in lead i Note that in the above summation qq runs over all outgoing wave vectors of energy E and type a of lead i and kg runs over all incoming wave vectors of energy E and type fj in lead j If i and j are different leads then Saik is often called the transmission amplitude and denoted ta whilst if they are of the same lead then s y is called the reflection amplitude rz Similarily if i j it is common to define the transmission coefficient as 2 Lj sU wip Laake 5 da kp 20 For i j we define the reflection coefficient as Li ij Rip x Lake se x 21 So that dI E dE h f E p Mi Eag Reg f Ejui py Taro fp 22 Note the unitarity of the scattering matrix requires that 2 2 ij dajkg 1 23 Lido S Like pbj si dajkp Hence the sum of the elements of each row and column of matrix P is zero ij Ww Lj jg Pab Aldi Pab 0 24 Or equivalently Dp Rave Lisi Tan Mi 25 24 and jj ij _ mi a Raig joio T M ab Bj 26 From Eqs 23 and 24 if j E is independent of j and fj then dI E 0 for all i and aj For this reason in Bj J j aj i t
38. i IPIko Yo i o E Jik Ok Daas L Ix 10 Vki VG where k and q denote here open positive and negative channels and V k and w q are their normalized kets Here the contribution of all the closed channels in lead i is described by the ket Ix Consequently the number of electrons per unit time flowing between two adjacent PLs within the lead is j E x Xue Fadal 11 We pick in this section the convention that positive direction in the lead means flow towards the EM region and vice versa So positive negative open channels are also called incoming outgoing channels of lead i With this notation the wave function coefficients of the incoming open channels of a given lead are determined by the properties of the reservoir connected to the lead The wave function coefficients of the open outgoing channels of lead i are obtained from the amplitudes of all incoming channels by lem U oJ Og Lik Sg Ok 12 Where q k is an outgoing incoming dimensionless wave vector of lead i or lead j It is therefore convenient to assemble the wave functions of the M incoming outgoing open channels of a given lead i in the column vectors 0 0 and all the scattering matrix elements connecting leads i and j into the matrix block SU of dimension M x M Therefore the above equation can be written 01 gli 12 IP ot 02 _ g21 922 g2P 02 13 OZ By normalizing the Bloch eigenvectors and their duals to unit flux
39. it cell with just one atom One orbital per atom 5 atoms in the calculation Single atom chain with spin polarization The Same parameters as in the leads exchange splitting between the spin up and spin down states is 2 eV Gollum parameters Transmission coefficients calculated between 8 0 eV and 8 0 eV in 200 energy points 2 principal layers on each electrode The terminating principal layer is the second on each Electronic structure obtained from the leads calculation No SAINT correction or Fermi level shift Bias voltage between 4 0 V and 4 0 V calculated in 11 voltage points The bias shift is applied on the first terminating layer of the left and right electrodes only 4 column of the atom variable set to different from 0 This is an approximation since the voltage should not fall or at least fall very slowly on a perfect infinite system Tight binding parameters Orthogonal basis set overlap matrix identity matrix On site energies both leads and EM 1 0 eV spin up and 1 0 eV spin down Nearest neighbour coupling matrix elements both leads and EM 2 0 eV Transfer the files input Lead_ 1 Lead_2 and Extended_Molecule to a new test folder of your choice Execute GOLLUM as described in SECTION 4 INSTALLATION and RUNNING and check your results The Hamiltonians are generated by hand from these tight binding parameters and are shown in Leads_1 and 2 with the Extended_Molecule
40. lue is composed of a central scattering region M and branches Each branch contains several PLs These PLs have an identical atomic arrangement as the PLs in the leads However their Hamiltonians differ from Hj and H due to the presence of the central scattering region PL numbering at each branch starts at the PL beside the central scattering region The outermost PL at each branch of the EM region is called the terminating principal layer TPL and must be described by Hamiltonians He and p which are close enough to H and Hi to match smoothly with the corresponding lead Hamiltonian For this reason GOLLUM requires that the EM contain at the very least one PL The central scatterer M itself is described by an intra scatterer Hamiltonian Hf and coupling matrices to the closest PLs of the branches In the example in FIG 7 1 1 the central scattering region M comprises a molecule and the atoms forming the electrode surfaces The surfaces in GOLLUM include all atoms belonging to the electrodes whose atomic arrangements cannot be cast exactly as a PL due to surface reconstructions etc For simplicity FIG 7 1 1 shows the case of a symmetric system although no such symmetries are imposed by GOLLUM All Hamiltonians are spin dependent but again for notational simplicity the spin index o is not written explicitly 19 7 2 THE SURFACE GREEN S FUNCTION FOR THE CURRENT CARRYING LEADS Unit cell H Ah Ay Ay Ay Ay H H PL gt PL PL PL PL PL
41. n compiler variable e g for a ifort compiler FC ifort 6 2 3 GENERATION OF THE GOLLUM FILES Once the siesta2gollum executable is generated the files that are needed to run GOLLUM should be produced by specifying the out file and whether the calculation was based on a lead or extended molecule calculation notice that the siesta2gollum file should be either in the same directory as the rest of the Siesta files or linked to that directory A Leads siesta2gollum Siesta output file Integer value between 1 and 123456789 that represents the electrodes where the lead files are going to be used i For example if a calculation of a nickel lead has been done Siesta output dumped to e g the Ni out file and it is going to be used in two electrodes of a certain system do the following siesta2gollum Ni out 12 This generates the files Lead 1 and Lead 2 which should be included in the directory where Gollum is going to be run ii Another example If a gold lead simulation has been performed Siesta output dumped to e g Au lead out file and such lead is going to be used in electrodes 2 7 and 9 of a multiterminal calculation do the following siesta2gollum Au lead out 279 This generates the files Lead 2 Lead 7 and Lead 9 which should be included in the directory where Gollum is going to be run 6 2 4 EXTENDED MOLECULE A siesta2gollum Siesta output file 0 i For example if a simulation of the exten
42. n energy 2 Single Simulate a one dimensional 1D system with perfect atom_chain Non coll gt 000 transmission equal to the number of open channels or een eee bands at a certain energy and non collinear magnetic configuration 3 Two Simulate a two level system H2 molecule coupled to level_system SAINT metallic atomic chains Check that the transmission has two peaks corresponding to the positions of the levels of 0 00 the molecule Apply a gate voltage to change the on site energies and the SAINT correction to increase the separation between both molecular levels Simulate a one dimensional 1D system with different 12244 orbitals and perfect transmission equal to the number of oc open channels or bands at a given energy 4 Two level chain 5 Square lattice TN CN FN Simulate a two dimensional 2D system with perfect transmission equal to the number of open channels or oes bands at a given energy 6 Four probe system Simulate a four probe system with two types of electrodes Calculate the transmissions from one to other electrodes See the effect of including a bias voltage on one of the electrodes Calculate the currents DFT based Comment example 1 Au linear chain Simulate a one dimensional 1D system with perfect transmission equal to the number of open channels or bands at a certain energy 2 Ir zigzag chain Simulate a one dimensional 1D system with perfect transmissi
43. n polarized calculations The block of data is headed by a line stating the voltage and the first column is the energy E giving N 1 columns in total Mode 4 gives finite bias results T E V in the same format as the other Modes 9 1 3 THE NUMBER OF OPEN CHANNELS Mode 1 2 3 4 Open_channels_per_spino dat The number of open channels per lead Mode 1 2 3 4 Open_channels_upo dat The number of open channels per lead Mode 1 2 3 4 Open_channels_downo dat The number of open channels per lead These files give the number of open channels as a function of energy in the lead The number of open channels is a dimensionless quantity 9 1 4 SHOT NOISE Mode 1 2 3 4 Shot_noise_per_spino dat Shot noise coefficients Mode 1 2 3 4 Shot_noise_per_upo dat Shot noise coefficients Mode 1 2 3 4 Shot noise per downo dat Shot noise coefficients The file provides the Shot Noise coefficients SN E V 0 using the same format as the transmission files Shot noise coefficients are dimensionless 9 1 5 BAND STRUCTURE Mode 1 2 3 4 Bandso dat From the GF of lead i These files contain the inverse band structure K E computed from the GF of lead i Wave vectors are measured in units of the lattice constant in the direction of transport 37 9 1 6 DENSITY OF STATES DOS Optional file initiated by DOS dat scissor corrections flag The EM DOS 9 2 1 MODE GENERATED FILES AT A GL
44. nonncnonononoonnncnnnnnrannnnnncnncononannnnncnnnnnnns 25 Input files format and notatio seene E ERRE aak ER regn nass ss estate assa sss ita 26 8 1 O 26 8 1 1 Mandatory Variable Mode ccccccccccccscssssssecececeeseaeeeeecsceesesassesecsceeeeaeeeeeeeceesesaeeeseesceesesaeaeseesceeees 26 8 1 2 Asummary of the Mode function TABLE 9 1 Z oooooo cccccnnonocoonnnncnnnonononnnoncnnonnonnnnnnncnnonnononnnnncnnnnnnns 27 8 1 3 Mandatory Variable Lead occccccnonoooonnnncnnnononnnnnonccnnnnonnonnnncnnonnnnnnnnnncnnnnnennnnnnncnnnnnnnnnnnncnnnnnnns 28 amp 1 4 rMandatory Variables atOM iced eH EE ERR YR Ee ee Ce EORR Fe EA ERE E PLU BER el Ee FE FER Heo LU 28 8 1 5 Partial Mandatory variable ccccccccccccssssscecececeesesesseeeeccecseeeeesececeeseseesecceceeseaaeseseesceeesasaeseesceeees 29 8 1 6 Partial Mandatory variable for Mode 1 variable ERange ccccconccoonnncnocononannnononnnnnonnnncnncnnnonnns 29 8 1 7 Partial mandatory variable for Mode 3 variable TRange esses enn 29 8 1 8 Partial mandatory variable for Mode 4 variable Bias enne 30 8 19 Optional variables eror ore too euet ee pA terere eene eee ter ee Pega ee gen oo aee ce tenet eri 30 8 1 10 Optional variable for Mode 4 variable BiaSacCuracy ccccccocooooncnnnnonononnnnnoncconannonnnonononananannos 30 8 1 11 Optional variable for all Modes variable EFshift oocccccnncnoooonncncconononannnonccnnanononnnnnccncnn
45. ode 1 2 3 4 Shot noise per spinc dat Shot noise coefficients Mode 1 2 3 4 Shot noise per upo dat Shot noise coefficients Mode 1 2 3 4 Shot noise per downo dat Shot noise coefficients Mode 1 2 3 4 T per spinc dat Transmission coefficients Mode 1 2 3 4 Tupo dat Transmission coefficients Mode 1 2 3 4 Tdowno dat Transmission coefficients Mode 1 2 3 4 Bandso dat From the GF of lead i Optional file initiated by DOS dat The EM DOS scissor corrections NOTE 1 Scissor corrections shift the HOMO and LUMO levels upwards and downwards in energy This has the effect of reducing both the transmission and therefore the current across the junction around the Fermi energy 36 NOTE 2 The results for parametric or spin polarized calculations are determined by the input Hamiltonians which for example depend upon your flags in the SIESTA input file and the value of nspin 1 for spin unpolarized or nspin 2 polarized in the Extended_Molecule input file 9 1 2 TRANSMISSION REFLECTION COEFFICIENTS Mode 1 2 3 4 T_per_spino dat Transmission coefficients Mode 1 2 3 4 Tupo dat Transmission coefficients Mode 1 2 3 4 Tdowno dat Transmission coefficients Mode 1 Mode 2 and Mode 3 give zero bias calculations For example the transmission coefficients are given by T E 0 iG 1 N in the columns where i j and the reflection coefficients for i j Files are written to give parametric spin calculations and up down for spi
46. oice into a new directory Place gollum app and run_gollum sh into the folder Type run_gollum sh mcr directory Your results of the calculation should match those to be found in the example The tutorials are to be found as a set of calculations in gollum examples in your distribution There are Tight binding based 6 examples for MAC and Windows Linux MATLAB and DFT based 8 examples for MAC and Windows Linux MATLAB The Example files and related systems are summarised in TABLE 5 1 We will take the first and simplest example from both the Tight binding and DFT based sets and run through the steps to carry out the calculation and relate the steps to the details found in this manual The more advanced cases are then clearly explained from the Guide pdf accompanying each of the examples It is important to run each of the examples to ensure that you are happy with your compilation by then comparing the results to be found in the output data mode folders and the plots shown in the Guide pdf Following all the examples and reading this document together with the GOLLUM A paper will complete the tutorial 5 1 SUMMARY OF THE EXAMPLES TABLE 5 1 Guide based information Tight binding System Comment based example 1 Single Simulate a spin polarized one dimensional 1D system atom_chain Spin with perfect transmission equal to the number of open bbdd channels or bands at a give
47. on equal to the number of open channels or 10 bands at a certain energy various channels and a non collinear magnetic configuration 3 C_chain gap NDR Simulate a one dimensional 1D system with a vacuum gap to see the effect of including a bias voltage that falls in the middle 4 Ni 001 bulk Simulate a magnetic three dimensional 3D system transverse periodic boundary conditions and k points with perfect transmission equal to the number of open channels or bands at a certain energy and various channels 5 CNT 5 5 Simulate a quasi one dimensional q1D system with perfect transmission equal to the number of open channels or bands at a certain energy and various channels 6 Graphene Simulate a perfect two dimensional 2D system transverse periodic boundary conditions and k points along y with perfect transmission equal to the number of open channels or bands at a certain energy and various channels 7 Au_001 BDT SAINT 8 Au_111 OPE Calculate the transport properties of a BDT 1 4 benzenedithiolate molecule between gold electrodes Correct the transport properties by applying a spectral adjustment SAINT to the occupied and unoccupied levels and shifting the Fermi level towards the HOMO Calculate the transport properties of a OPE oligophenyleneethylene based molecule between gold electrodes To run these examples simply follow the same procedure set ou
48. operties of the leads closest to the molecule or more generally the central scattering material will be modified by the presence of the central scattering region M and by the fact that the leads terminate In what follows we refer to those affected portions of the leads closest to the central scatterer as branches and include them as part of the extended scatterer denoted EM Consequently within GOLLUM a typical structure consists of an extended scatterer EM formed from both the central scatterer M and the branches The extended scattering region is connected to crystalline current carrying leads of constant cross section shown in light blue in the FIGS 7 1 1 and 7 1 2 For an accurate description of a given system the branches are chosen to be long enough such that they join smoothly with the light blue crystalline leads Crucially the properties of this interface region between the central scatterer M and the leads are determined by their mutual interaction and are not properties of either M or the electrodes alone Extended scattering region EM Central scattering region M FIG 7 1 1 Schematic plot of a four terminal device which includes an extended scattering region and four leads 18 Central Left Branch scattering Right Branch Left Lead Extended scattering region EM Right Lead Ho H Hi Hi Ho HU H H H HH HUHH OH HOH HHH H tV PL TPL PL FIG 7 1 2 Top Schemati
49. s Imag H i j k or Imag H i j k T nspin 1 or 2 respectively HSM 8 Exists if nspin 2 It provides Real H i j k 1 HSM 9 Exists if nspin 2 It provides Imag H i j k V 8 3 LEAD I INPUT FILE 8 3 1 EXAMPLE LEAD FILE There can be one of these files for each lead The file will be read and used in the corresponding variable leadp 3 is set to 1 The following is an example of a Lead file name kpoints Lead H type matrix rows 1 columns 3 0 0000000000E 00 0 0000000000E 00 1 0000000000E 00 name HSL type matrix rows 4 columns 11 111 1 00 0 2 0 0 00 0 0 0 0 1 0 0 122 1 00 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 3 0 0 0 0 0 0 0 2 0 0 8 3 2 VARIABLE KPOINTS_LEAD name kpoints Lead type matrix rows 1 columns 3 kpoints 1 kpoints 2 kpoints 3 This variable sets the transverse kpoints used in the DFT simulation of the Lead There are as many rows as kpoints It is mandatory because the ordering of k points in the DFT simulations of the Leads and of the Extended Molecule need not coincide kpoints 1 Gives the value of kx kpoints 2 Gives the value of ky kpoints 3 Gives the weight in the k summation 8 3 3 VARIABLE HSL name HSL type matrix rows 4 columns 11 HSL 1 HSL 2 HSL 11 HSL 15 35 This variables set the intra and inter PL overlap So i j k and Hamiltonian matrix el
50. s in the EM calculation to make sure the electronic structure exactly matches that of the leads 1 k point 7 point along the perpendicular directions Files necessary tor run the leads calculation lead fdf Au psf Files necessary to run the EM calculation emol fdf Au psf In this case the definitions of the Hamiltonians are generated from ab initio DFT parameters using SIESTA and are shown in Gollum_files Leads_1 and 2 with the Extended Molecule These files are generated using the method explained in SECTION 6 HAMILTONIAN GENERATION and in SECTION 6 2 DIRECT SIESTA INTERFACE METHOD This makes use of the files lead fdf and emol fdf with Au psf provided within the distribution You will need to obtain a licenced SIESTA version http departments icmab es leem siesta and understand how to run the code from the documentation Again transfer the files input Lead 1 Lead 2 and Extended Molecule to a new test folder Execute GOLLUM as described in SECTION 4 INSTALLATION and RUNNING and check your results The input file is defined in SECTION 8 1 The Extended Molecule file is defined in SECTION 8 2 The Lead 1 and Lead 2 files is defined in SECTION 8 3 The output data files are given for the various mode directories and are described in SECTION 9 15 The Tight Binding Hamiltonians can be constructed by hand to generate the Lead_i and Extended_Molecule input files used by GOLLUM nevertheless w
51. scattering region It uses mandatory variables to define how GOLLUM reads the Fermi energy the orbital information the spin degrees of freedom and the number of transverse k points used in the simulation 8 2 1 AN EXAMPLE EXTENDED MOLECULE FILE name nspin H type scalar 1 name FermiE H type scalar 0 name iorb H type matrix rows 10 columns 5 16001 16101 26001 56101 name kpoints EM type matrix rows 1 columns 3 0 0000000000E 00 0 0000000000E 00 1 0000000000E 00 name HSM type matrix rows 36 columns 7 1111 00 0 2 0 0 0 133 1 00 0 2 0 0 0 155 1 00 0 2 0 0 0 1 10 9 0 0 0 0 0 3 0 0 8 2 2 VARIABLE NSPIN name nspin type scalar nspin This variable defines whether the calculation is spin unpolarized nspin 1 or polarized nspin 2 8 2 3 VARIABLE FERMIE 33 name FermiE type scalar 0 This variable defines the value of the Fermi energy at the extended molecule in eV which is taken as the reference energy 8 2 4 VARIABLE IORB name iorb H type matrix rows 10 columns 5 iorb 1 iorb 2 iorb 3 iorb 4 iorb 5 This variable contains information about the orbitals used in the simulation The variable has as many rows as orbitals exist in the EM region iorb 1 Indicates the atom to which the orbital belongs iorb 2 Indicates the principal quantum number n iorb 3 Indicates the orbital quantum number l iorb 4 Indic
52. set dnd eroe A 36 9 1 1 The Universal output at a glance TABLE 10 1 1 oooooocnncncconocoancnnnnnnonononnnncnnnnnonnnnnnncnnconononnnnncnnnnnons 36 9 1 2 Transmission reflection coefficients cccccccesscceessscecseseeeceeseeceessesecsesaececssssececsaeseceesaeceeseeseeees 37 9 1 3 The number of open channels ccccsesscccccecsessaseeccceceeseaeseccesceeseeasseeecsceesesaeseeeesceesesasaeeeeseeeees 37 9 1 4 ShOt nolse ion rere NT 37 9 1 5 Band str cture i idein eet ede HU e Fe e EE Ga e EFE ER REEL E ATA REDE ATE EL ERE ECKE Le eR o FEDERER RE ait 37 91 6 Density of states DOS iei er ntes ee pe lentes 38 9 2 Additional Mode generated files ooonccononoooonnncnononannnnnnnnononanononnnncnnnononnnnnnncnnonnonnnnnnnccnnnnanonnos 38 9 2 1 Mode generated files at a glance Table 10 2 1 oo oooccccnnonocuonnnncnnnonononnnoncnncnnonnnnnnncnnonnononnnnncnnnnnnns 38 92 2 Mode generated iles rire REEL ERE OH EC RNER ERE AREE RECANTE REC GET ellie 38 92 3 Mode 3 generated Tile cc oot ette ee reser engeren eee te ee ete CEU e aue gen HR ex Eee Dee uS Een 39 92 4 Mode A generated files tenere A EN eE EATUR RR ERERE ERE ETUR Feed 40 AORTA 40 NON 41 2 1 UPDATES AND NEW FUNCTIONALITY Updates to bug fixes together with code development will be posted on the web site http www physics lancs ac uk gollum Currently none of the GOLLUM versions is explicitly parallelized 2 2 THE ADVISORY BOARD
53. sions in place for both files To test your MCR locate your MCR libraries which should be in usr local MATLAB MATLAB_Compiler_Runtime v717 Then type run_gollum sh usr local MATLAB MATLAB_Compiler_Runtime v717 A successful compile is seen by the following window CAREER NEE Hn a WELCOME TO GOLLUM DLE Verston 1 0 Oviedo and Lancaster Untverstttes http www physics lancs ac uk gollum tart of run 17 Aug 2014 11 33 04 rror using gollum line 36 Extended Molecule file is not found Please include Extended Molecule file try again sghs x1 s Bl Ignore the Error at this stage as it indicates that as yet you have not included the correct input files to run a calculation The location of the MCR libraries and gollum can be placed in your PATH by editing your bashrc or cshrc files or by typing the following to set the path prior to each calculation In s path source run gollum sh run gollum sh In s path source gollum Gollum For simplicity we outline the following procedure Copy from the examples folder supplied with your distribution an example of your choice into a new directory of your choice Place gollum and run_gollum sh into the folder Type run_gollum sh usr local MATLAB MATLAB_Compiler_Runtime v717 Your results of the calculation should match those to be found in the example matching your choice 4 1 3 INSTALLING THE MCR AND RUNNING MAC Enter the gollum mac
54. t in section 4 INSTALLATION AND RUNNING above 11 5 2 STEP BY STEP EXAMPLE TIGHT BINDING BASED rc a a a a aaa 5 2 1 1 SINGLE ATOM_CHAIN SPIN Enter the directory gollum 1 0 examples Tight binding based 1 Single atom_chain Spin from your distribution Single atom chain Spin polarized Objectives Simulate a spin polarized one dimensional 1D system with perfect transmission equal to the number of open channels or bands at a certain energy Check that the transmission exactly coincides with the number of open channels Results The transmission has a step like shape which corresponds to the number of open channels The spin up and spin down channels and transmissions have an exchange splitting of 2 eV The current linearly increases as expected since it is equal to the integral of a constant function until it saturates at large voltages when the bias window starts covering the edges of the band which are rounded by the effect of the voltage Transport direction Transmission and number of open channels as a function of energy Calculated with Mode 1 of Gollum I A 12 down a 08 0 8 3 A 0 6 E 0 6 3 o 04 04 02 02 wears 0 5 5 0 pee E Eg E Epi 2x10 1x10 0 1x10 4 x10 4 2 0 2 4 v v Current as a function of voltage Calculated with Mode 4 of Gollum System description and parameters Leads Extended molecule EM Un
55. to the SIESTA package as an option to generate the GOLLUM compatible Hamiltonians For developers only the MATLAB source code is found in matlab src GOLLUM is a program that computes the electrical and thermal transport properties of multi terminal nanoscale scale systems The program can compute transport properties of either user defined systems described by a tight binding or H ckel Hamiltonian or more material specific properties of systems composed of real atoms described by DFT Hamiltonians The program has been designed to interface easily with any DFT code which uses a localized basis set and currently read information from all the latest public flavors of the codes SIESTA SIESTA 3 1 SIESTA VDW SIESTA LDA U and FIREBALL Plans to generate interfaces to other codes are underway GOLLUM is based on equilibrium transport theory which means that it consumes much less memory than non equilibrium Green s function codes The program has been designed for user friendliness and takes a considerable leap towards the realization of ab initio multi scale simulations of conventional and more sophisticated transport functionalities These include e The ability to use either model tight binding or DFT Hamiltonians including the optional use of scissor corrections e The ability to compute non equilibrium current voltage curves e Access to the full scattering matrix including scattering amplitudes phases and Wigner delay times e An
56. upled junctions In this case variable Bias accuracy can be used to increase the energy grid Available values are 1 and 2 whereby the energy grid is set to 1000 and 3000 energy points respectively Consequently the calculation is slowed down roughly by a factor of 3 or 10 respectively name Bias accuracy H type scalar 1 30 8 1 11 OPTIONAL VARIABLE FOR ALL MODES VARIABLE EFSHIFT GOLLUM uses the Fermi energy of the extended molecule to set the zero energy This reference energy can be shifted up and down by setting EF_shift to positive or negative energy values measured in eV name EF_shift type scalar 8 1 12 OPTIONAL VARIABLE FOR ALL MODES VARIABLE SCISSORS The atoms to which this correction is applied are set in variable atom If scissor corrections are applied then GOLLUM also prints the Molecule s Density of States DOS with and without scissor corrections where thebibliographyenergy origin is chosen to be the Fermi energy of the EM region This functionality is implemented only if the calculation is spin unpolarized and the number of transverse k points is one The variable format is as follows name scissors type matrix rows 1 columns 5 scissors 1 scissors 2 scissors 3 scissors 4 scissors 5 scissors 1 Compute Scissor corrections if 1 or not otherwise scissors 2 Number of electrons in the molecule scissors 3 Energy value for the shift downwards of the HOMO scissors 4 Energ
57. vided for example to run on 32 bit machines Requests for the developer version will be considered by the advisory board 4 1 MATLAB VERSION As GOLLUM is a standalone executable all you need to run the programme is a GOLLUM compatible MATLAB Compiler Runtime MCR 4 1 1 INSTALLING THE MCR AND RUNNING IN WINDOWS From your GOLLUM distribution navigate to your version dependent executable in the gollum windows directory to locate gollum exe For a windows machine to verify if you have the MCR installed simply double click on gollum exe or run gollum exe from the command prompt If you have a compatible MCR then you will see the following window Oviedo and Lancaster Universities http www physics Lancs ac uk gollun start of run 17 Aug 2014 11 33 04 Error using gollum line 36 l Extended Molecule file is not found Please include Extended Molecule file ind try again sis E Ignore the Error at this stage as it indicates that as yet you have not included the correct input files to run a calculation If you do not have the correct MCR then navigate to http www mathworks co uk products compiler mcr in MathWorks and locate the correct version to install Windows requires release Windows R2012a 64 bit If you requested the 32 bit version of GOLLUM then you will need the release Windows R2012a 32 bit Once the MCR is correctly installed you can make your first test You can of course place
58. y value for the shift upwards of the LUMO scissors 5 Defines the distance for the screening image charge measured in Bohr 8 1 13 OPTIONAL VARIABLE FOR ALL MODES VARIABLE ANDERSON This variable sets the parameters for the computation of Coulomb blockade and Kondo Physics in the spirit of Dynamical Mean Field Theory The variable format is as follows name anderson type matrix rows 1 columns 3 anderson 1 anderson 2 anderson 3 anderson 1 Compute Kondo Coulomb blockade effects if 1 or not otherwise anderson 2 On site Coulomb interaction U term in eV anderson 3 Temperature in Kelvin 31 8 1 14 OPTIONAL VARIABLE FOR ALL MODES VARIABLE VGATE This variable sets the parameter for applying a constant gate voltage to the ExtendedMolecule The variable format is as follows name Vgate type scalar Vgate VgateValue of the gate voltage in eV 8 1 15 EXAMPLE INPUT FILE The following is an input file for a two lead junction in Mode 4 name Mode type scalar 4 name Bias type matrix rows 1 columns 3 0 3 2 30 000 name leadp H type matrix rows 2 columns 3 420 420 name scissors type matrix rows 1 columns 5 0 42 0 5 0 5 10000 0 name atom type matrix rows 26 columns 4 1141 32 8 2 THE EXTENDED MOLECULE FILE The Extended Molecule file enables GOLLUM to read the Hamiltonian and overlap matrices of the extended
Download Pdf Manuals
Related Search