1. A primer on Smeagol
Víctor García Suárez

2. Outline 1) Introduction 2) Theory 3) How to run the code
4) Simple examples
5) Some calculations

3. 1) Introduction

4. S pin and M olecular E lectronics in an A tomically G enerated
Introduction to Smeagol
S pin and
M olecular
E lectronics in an
A tomically
G enerated
O rbital
L andscape

5. Why Smeagol Need for smaller electronic devices. Atomic limit - Faster
- Cheaper
- More compact
- Similar features as today electronic elements (rectification, NDR, etc). 5

6. + General features of Smeagol - Density functional theory (DFT)
First-principles code based on localized functions: Siesta, Fireball, etc +
- Non-equilibrium Green’s functions
Calculation of the density matrix, transmission and current under finite biases 6

7. Smeagol Characteristics and Capabilities
Smeagol and spintronics: exploit the spin degree of freedom
- Spin polarized
- Non collinear
- Spin orbit
Calculations of both extended and isolated systems: G and k-point calculations
Calculations of more than 100 atoms
Parallelized 7

8. 2) Theory

9. Direction of electronic transport (z-axis) Right bank (reservoir)
The philosophy behind a Smeagol calculation V
Large resistance
Low resistance
Low resistance
Direction of electronic transport (z-axis)
Left bank (reservoir)
Junction
Right bank (reservoir)
Left electrode
Extended molecule
Right electrode 9

10. Leads and extended molecule
Left electrode
Molecule
Right electrode
No m
No Fermi
distribution
HL [nL] – e mL
mL = m – eV/2
HR [nR] – e mR
mR = m + eV/2
Bulk left electrode
Left reservoir
Extended molecule
Non-equilibrium
Bulk right electrode
Right reservoir
Procedure to compute nM when
the distribution function is not Fermi
Non-equilibrium
Green’s function formalism 10

11. Calculation of the leads properties
- Unit cell that is repeated along the transport direction (z)
- Use of k-points. Necessary to converge the density of states
- At the end of the calculation:
* Hamiltonian and overlap matrices (H0, S0 and H1, S1) → Surface GF
* Density matrix
* Fermi energy 11

12. Self-energies and G matrices
The self-energies are calculated with the couplings and the surface GF
G0R is the retarded Green’s function of the leads, calculated using a semi-analytic formula
The G matrix contains information on the coupling between the extended molecule and the leads
It is important in the calculation of the transmission 12

13. Calculation of the surface GF
Since the GF depends on energy it is necessary to calculate k(E) from the block vectors instead of E(k) → Solve the inverse secular equation
This involves obtaininig the inverse of the K1 matrix, which can not always be inverted → Identify the singularities (GSVD) and get rid of them (decimation)
Once the bulk GF is constructed the surface GF is obtained by applying the appropriate boundary conditions
The surface GF should vanish at z1 → Add a wave function to the bulk GF
z-1 z0 z1 13

14. Calculation of the extended molecule
- Includes molecule or central scattering region + part of the leads
- The first and last part of the EM must coincide with the unit cells of each of the leads (buffer layers). This implies that:
* The same general parameters as in the bulk calculation have to be used in the unit cell: temperature, mesh, perpendicular k-points, spin-polarization, etc
* The same particular parameters for the leads have to be used in the buffer layers and rest of the leads in the EM: basis set, atomic coordinates, etc.
Buffer layers 14

15. Buffer layers
- The Hamiltonian, overlap and DM of the buffer layers is substituted by those of the leads
- The buffer layers ensure that:
* The electronic structure at the beginning and end of the EM is that of the leads
* (forced) Convergence between the electronic strucrure of the leads near the scattering region and deep into the leads
* Absence of spurious effects in case of surfaces
TEST: infinite system
Energy, charge and
transmission
T(E) E 15

16. Equal leads
- In case of equal leads it is possible to make the calculation of the EM periodic to avoid the presence of surfaces. This does not mean that the system is periodic
This does not mean that the system is periodic
No k-points are necessary 16

17. Energy mismatch between the bulk and EM calculations
In general the energy origin in a calculation of an infinite system is arbitrary
It is necessary to determine such mismatch and correct it. Otherwise the Fermi energy can be incorrectly defined and the system can win or loose charge
The correction is made at certain positions of the bulk slices
It is necessary to calculate the bulk Hartree potential X X 17

18. Calculation of the DM Siesta:
Solve eigenvalue problem. Order-N or diagonalization
Smeagol:
* Semi-infinite leads
* Non-equilibrium charge distribution 18

19. Density matrix in equilibrium
In equilibrium it is only necessary to know the retarded Green’s function
Built with the Hamiltonian, overlap and self-energies
The density matrix is obtained by integrating along an energy axis 19

20. Complex energy contour
The lesser Green’s is calculated on an energy contour in equlibrium
Three parts: imaginary circle, imaginary line and Fermi poles
Fermi function poles
Image from Atomistix
Brandbyge et al. Phys. Rev. B 65, 165401 (2002) 20

21. Density matrix out of equilibrium
Out of equilibrium it the GF is not analytic inside the contour
The calculation of the DM is divided in two parts 21

22. Out of equilibrium voltage profile
The Hartree potential is defined up to a constant and a linear ramp (solution of the Poisson equation)
A linear ramp related to the bias voltage is added to help the convergence out of equilibrium
+eV/2 mL V mR
-eV/2 22

23. Smeagol Smeagol Inside the Siesta version of Smeagol Smeagol
Initial guess
Calculate effective potential No
Smeagol
Calculate the DM using NEGF
Smeagol
Smeagol
Compute electron density
Self-consistent?
Output quantities
Transmission and current
Yes 23

24. Calculation of the transmission and current
The transmission is calculated at the end of the self-consistent cycle
It is possible to simplify it due to the small size of the G matrices
The current is calculated by integrating the transmission 24

25. Single level coupled to wide band leads
Wide band leads have a constant density of states at the Fermi level
After coupling the level the onsite energy (e0) is renormalized by the real part of the self-energy (e1)
The imaginary part of the self energy give the inverse of the lifetime (width of the Breit-Wigner resonance) e1 25

26. 3) How to run the code

27. Calculation of the leads
First run before calculating the transport properties of the extended molecule
- Include two variables in the input file:
BulkTransport T . It specifies wehter or nor the bulk parameters are written
BulkLead LR . Left (L), right (R) or both (LR) leads
* At the end of the calculation three or four files are generated:
- bulklft.DAT and bulkrgt.DAT: contain the label of the system and basic information such as the Fermi energy, temperature, etc.
- SystemLabel.HST: contains the Hamiltonian and overlap matrices
- SystemLabel.DM: contains the density matrix 27

28. Example of bulk file SystemName Au SystemLabel Au NumberOfAtoms 2
NumberOfSpecies
%block ChemicalSpeciesLabel Au
%endblock ChemicalSpeciesLabel
%block PAO.Basis
Au 1 n=
5.0
%endblock PAO.Basis
%block Ps.lmax
%endblock Ps.lmax
LatticeConstant Ang
%block LatticeVectors
%endblock LatticeVectors
AtomicCoordinatesFormat Ang
%block AtomicCoordinatesAndAtomicSpecies
Au 1
Au 2
%endblock AtomicCoordinatesAndAtomicSpecies
%block kgrid_Monkhorst_Pack
%endblock kgrid_Monkhorst_Pack
xc.functional GGA
xc.authors PBE
MeshCutoff Ry
MaxSCFIterations
DM.MixingWeight 0.1
DM.NumberPulay 8
DM.Tolerance 1.d-4
SolutionMethod diagon
ElectronicTemperature 150 K
SaveElectrostaticPotential T
BandLinesScale pi/a
%block BandLines
%endblock Bandlines
BulkTransport T
BulkLead LR
DM.UseSaveDM T 28

29. Calculation of the extended molecule. General
Run the extended molecule file in the same directory that contains the bulk files
- Variable to define the transport calculation:
EMTransport T . Performs the transport calculation
- Variables related to the energy contour:
NEnergReal Number of points along the real axis (out of equilibrium)
NEnergImCircle Number of points in the imaginary circle
NEnergImLine Number of points in the imaginary line
Npoles Number of poles of the Fermi function
Delta d-4 . Small imaginary part of the Green’s Function
EnergyLowestBound Ry . Energy of the lowest bound of the EC
Nslices Number of slices that are substituted by the bulk H and S 29

30. Calculation of the extended molecule. Out of equilibrium
- Variables related to the bias voltage (out of equilibrium):
VInitial Initial value of the bias voltage
VFinal Final value of the bias voltage
NIVPoints Number of points where the bias is going to be applied
AtomLeftVcte Left position where the bias ramp starts
AtomRightVcte Right position where the bias ramp ends
At each bias point the electronic structure is converged. Optionally, it is also possible to relax the atomic coordinates
When the electronic structure converges the transmission and current at that bias point are calculated 30

31. Calculation of the extended molecule. Transmission and VH
- Variables related to the calculation of the transmission
NTransmPoints Number of energy points where the transmission is going to be calculated
InitTransmRange eV . Initial value of the transmission range
FinalTransmRange 2.0 eV . Final value of the transmission range
- Variables related to the energy mismatch between bulk and EM
HartreeLeadsLeft Ang . Left position where the correction is applied
HartreeLeadsRight Ang . Right position where the correction is applied
HartreeLeadsBottom eV . Value of the Hartree potential at a certain point in the leads 31

32. ... Example of extended molecule file SystemName Au.em
SystemLabel Au.em
NumberOfAtoms
NumberOfSpecies
%block ChemicalSpeciesLabel Au
%endblock ChemicalSpeciesLabel
%block PAO.Basis
Au 1 n=
5.0
%endblock PAO.Basis
%block Ps.lmax
%endblock Ps.lmax
LatticeConstant Ang
%block LatticeVectors
%endblock LatticeVectors
AtomicCoordinatesFormat Ang
%block AtomicCoordinatesAndAtomicSpecies
Au 1
Au 2
Au 3
Au 4
Au 5
Au 6
%endblock AtomicCoordinatesAndAtomicSpecies
...
EMTransport T
NEnergReal
NEnergImCircle
NEnergImLine
NPoles
Delta d-4
EnergLowestBound -8.d0 Ry
NSlices
VInitial d0 eV
VFinal d0 eV
NIVPoints
AtomLeftVCte
AtomRightVCte
TrCoefficients T
NTransmPoints
InitTransmRange eV
FinalTransmRange eV
PeriodicTransp T
UseLeadsGF F
HartreeLeadsLeft Ang
HartreeLeadsRight Ang
HartreeLeadsBottom eV
#%block SaveBiasSteps
# 0 1 2
#%endblock SaveBiasSteps
DM.UseSaveDM T 32

33. 4) Simple examples

34. Infinite atomic chain. Equilibrium
Band structure and transmission
Two atoms in the unit cell → two crossing bands in the Brillouin zone
One single channel at every energy
Perfect squared transmission 34

35. Infinite atomic chain. Out of equilibrium
Out of equiilibrium transmission
Starts to disappear at the edges, where the bands of both leads mismatch
Current
Ohmic regime 35

36. Diatomic molecule. Equilibrium I
Effect of separating the atoms from the leads
Changing the coupling configuration 36

37. Diatomic molecule. Equilibrium II
Effect of changing the intramolecular distance
Decrease the distance
Increase the distance 37

38. Diatomic molecule. Out of equilibrium
Bonding
Antibonding
Different behaviour of the bonding and antibonding orbitals
Bias-dependent transmission
Negative differential resistance (NDR) 38

39. 4) Some calculations

40. Magnetoresistance effects in nickel chains
Properties of this system:
- Highest magnetic moments in the middle of the chain
- The spins invert close to the leads
- Abrupt change (collinear) of the magnetization
Spin-polarized
Non-collinear
Symmetric, parallel
Symmetric, antiparallel
Aymmetric, antiparallel 40

41. Metallocenes inside carbon nanotubes
Chains of metallocenes inside CNTs
Reduction of the magnetoresistance due to charge transfer from the metallocene
CoCp + CNT
CoCp
CNT
Magnetoresistance effect 41

42. Even-odd effect in monoatomic chains
Different conductance depending on the number of atoms in the chain
The oscillations depend on the type of contact configuration
Gold
Sodium 42

43. Conductance of H2 molecules couples to Pt or Pd leads
Two possible configurations: parallel or perpendicular to the transport direction
In case of Pd the H2 molecule can go inside the bulk and contacts
Continuous line: Pt
Dashed line: Pd 43

44. Oscillations in Pt chains
Chains with a number of atoms between 1 and 5
Structural oscillations due to changes in the levels at the Fermi level as the chain is stretched from a zigzag to a linear configuration
2 atoms
3 atoms 44

45. I-V calculations of polyynes between gold leads
Two types of molecules with different coupling atoms
Two types of NDR due to a different evolution of the resonances with bias S N
(a)
(b) 45

46. Porphyrins between gold leads
Very large calculations with more than 500 atoms
Evolution of the conductance with the number of porphyrin units and the angle between them 46

47. Fin