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A primer on Smeagol
http://www.smeagol.tcd.ie
Víctor García Suárez
Outline
1) Introduction
2) Theory
3) How to run the code
4) Simple examples
5) Some calculations
1) Introduction
Introduction to Smeagol
S pin and
M olecular
E lectronics in an
A tomically
G enerated
O rbital
L andscape
Why Smeagol
Need for smaller electronic devices. Atomic limit
- Faster
- Cheaper
- More compact
- Similar features as today electronic elements (rectification, NDR, etc).
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
+
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: and k-point calculations
Calculations of more than 100 atoms
Parallelized
2) Theory
The philosophy behind a Smeagol calculation
Left electrode Right electrodeExtended molecule
Left bank (reservoir) Junction Right bank (reservoir)
V
Direction of electronic transport (z-axis)
Low resistance Low resistanceLarge resistance
Leads and extended molecule
2/0
][
02/
eVHH
HnHH
HeVH
H
RRM
MRMMML
LML
i
iiM εfrψ rn )()()(2
HL [nL] – e L
L – eV/2
No No Fermi
distribution
Left electrode Molecule Right electrode
Bulk left electrodeLeft reservoir
Extended moleculeNon-equilibrium
Bulk right electrodeRight reservoir
HR [nR] – e R
R = + eV/2
Procedure to compute nM when the distribution function is not Fermi
?][ MM nH
Non-equilibrium Green’s function formalism
)()( EfEDF
Calculation of the leads properties
0H0H0H 0H
1H 1H 1H
- 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
0H0H0H 0H
1H 1H 1H1H
Self-energies and 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
M L/R0RL/RL/R M
RL/R ]ˆˆ))[((ˆ]ˆˆ)[(ˆ HSiEEGHSiE
The matrix contains information on the coupling between the extended molecule and the leads It is important in the calculation of the transmission
)](ˆ)(ˆ[i)(ˆ RR/L
RR/LL/R EEE
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
iiikkk ESHKeKKeK
; 0][ i10
i1
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
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)
z0 z1z-1
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
Buffer layers
- 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
- The Hamiltonian, overlap and DM of the buffer layers is substituted by those of the leads
TEST: infinite system
Energy, charge and transmission
T(E)
E
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
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
Calculation of the DM
Siesta:
Solve eigenvalue problem. Order-N or diagonalization
n
nnn fcccSEcH *ˆˆˆˆˆ
Smeagol:
* Semi-infinite leads* Non-equilibrium charge distribution
)(ˆ d2
1ˆ)(ˆ EGE
iEG
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
)()](ˆ Im[d
1ˆ R
EfEGE
1RL
RR
R )](ˆ)(ˆˆˆ)i[()(ˆ EEHSEEG
The density matrix is obtained by integrating along an energy axis
Complex energy contour
The lesser Green’s is calculated on an energy contour in equlibrium
Three parts: imaginary circle, imaginary line and Fermi poles
Image from Atomistixwww.quantumwise.com
nz nzGTEGEEGE )](ˆIm[ik2)(ˆ d)(ˆ d R
C
Brandbyge et al. Phys. Rev. B 65, 165401 (2002)
Tnzn k)12(i Fermi function poles
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
)]()[](ˆˆˆ[ d2
1ˆ R/LL/R
RR/L
RV EffEGGE
VL/R ˆˆˆ
C L/R
RL/R )()](ˆ Im[d
1ˆ
EfEGE
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
VL
R
+eV/2
-eV/2
Inside the Siesta version of Smeagol
Initial guess
Calculate effective potential
Calculate the DM using NEGF
Compute electron densityNo
Output quantities
Transmission and current
YesSelf-consistent?
)(ˆ d2
1ˆ EGE
i
Smeagol
Smeagol
Smeagol
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 matrices
]ˆˆˆˆ[Tr
)](ˆˆˆˆ[Tr )(RRLR
RLRL
RR
RL
GG
EGGET
The current is calculated by integrating the transmission
))](()()[( d
h
eRL EEfEfETEI
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 (0) is renormalized by the real part of the self-energy (1)
The imaginary part of the self energy give the inverse of the lifetime (width of the Breit-Wigner resonance)
1
2i
2ˆ R
L/R
01
i
1)(
1
R
EEG
221
2
)()(
E
ET
3) How to run the code
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
Example of bulk file
SystemName AuSystemLabel AuNumberOfAtoms 2NumberOfSpecies 1%block ChemicalSpeciesLabel 1 79 Au%endblock ChemicalSpeciesLabel%block PAO.BasisAu 1 n=6 0 1 5.0%endblock PAO.Basis%block Ps.lmax Au 1%endblock Ps.lmaxLatticeConstant 1.00 Ang%block LatticeVectors 10.00 0.00 0.00 0.00 10.00 0.00 0.00 0.00 4.08%endblock LatticeVectorsAtomicCoordinatesFormat Ang%block AtomicCoordinatesAndAtomicSpecies 0.00 0.00 0.00 1 Au 1 0.00 0.00 2.04 1 Au 2%endblock AtomicCoordinatesAndAtomicSpecies
%block kgrid_Monkhorst_Pack 1 0 0 0.0 0 1 0 0.0 0 0 100 0.0%endblock kgrid_Monkhorst_Packxc.functional GGAxc.authors PBEMeshCutoff 200. RyMaxSCFIterations 10000DM.MixingWeight 0.1DM.NumberPulay 8DM.Tolerance 1.d-4SolutionMethod diagonElectronicTemperature 150 KSaveElectrostaticPotential TBandLinesScale pi/a%block BandLines 1 0.00 0.00 0.00 200 0.00 0.00 0.50%endblock BandlinesBulkTransport TBulkLead LRDM.UseSaveDM T
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 500 . Number of points along the real axis (out of equilibrium)
NEnergImCircle 90 . Number of points in the imaginary circle
NEnergImLine 30 . Number of points in the imaginary line
Npoles 10 . Number of poles of the Fermi function
Delta 1.0d-4 . Small imaginary part of the Green’s Function
EnergyLowestBound -10.0 Ry . Energy of the lowest bound of the EC
Nslices 1 . Number of slices that are substituted by the bulk H and S
Calculation of the extended molecule. Out of equilibrium
- Variables related to the bias voltage (out of equilibrium):
VInitial 0.0 . Initial value of the bias voltage
VFinal 2.0 . Final value of the bias voltage
NIVPoints 10 . Number of points where the bias is going to be applied
AtomLeftVcte 9 . Left position where the bias ramp starts
AtomRightVcte 36 . 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
Calculation of the extended molecule. Transmission and VH
- Variables related to the calculation of the transmission
NTransmPoints 800 . Number of energy points where the transmission is going to be calculated
InitTransmRange -8.0 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 2.040 Ang . Left position where the correction is applied
HartreeLeadsRight 10.200 Ang . Right position where the correction is applied
HartreeLeadsBottom -1.711 eV . Value of the Hartree potential at a certain point in the leads
Example of extended molecule fileSystemName Au.emSystemLabel Au.emNumberOfAtoms 6NumberOfSpecies 1%block ChemicalSpeciesLabel 1 79 Au%endblock ChemicalSpeciesLabel%block PAO.BasisAu 1 n=6 0 1 5.0%endblock PAO.Basis%block Ps.lmax Au 1%endblock Ps.lmaxLatticeConstant 1.00 Ang%block LatticeVectors 10.00 0.00 0.00 0.00 10.00 0.00 0.00 0.00 12.24%endblock LatticeVectorsAtomicCoordinatesFormat Ang%block AtomicCoordinatesAndAtomicSpecies 0.00 0.00 0.00 1 Au 1 0.00 0.00 2.04 1 Au 2 0.00 0.00 4.08 1 Au 3 0.00 0.00 6.12 1 Au 4 0.00 0.00 8.16 1 Au 5 0.00 0.00 10.20 1 Au 6%endblock AtomicCoordinatesAndAtomicSpecies
...EMTransport TNEnergReal 500NEnergImCircle 60NEnergImLine 30NPoles 10Delta 2.d-4EnergLowestBound -8.d0 RyNSlices 1VInitial 0.d0 eVVFinal 2.d0 eVNIVPoints 10AtomLeftVCte 2AtomRightVCte 7TrCoefficients TNTransmPoints 800InitTransmRange -14.0 eVFinalTransmRange 8.0 eVPeriodicTransp TUseLeadsGF FHartreeLeadsLeft 2.040 AngHartreeLeadsRight 10.200 AngHartreeLeadsBottom -1.711 eV#%block SaveBiasSteps# 0 1 2#%endblock SaveBiasStepsDM.UseSaveDM T
4) Simple examples
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
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
Diatomic molecule. Equilibrium I
Changing the coupling configuration
Effect of separating the atoms from the leads
Diatomic molecule. Equilibrium II
Effect of changing the intramolecular distance
Decrease the distance Increase the distance
Diatomic molecule. Out of equilibrium
Different behaviour of the bonding and antibonding orbitals
Bias-dependent transmission Negative differential resistance (NDR)
Bonding Antibonding
4) Some calculations
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
Metallocenes inside carbon nanotubes
Chains of metallocenes inside CNTs
Reduction of the magnetoresistance due to charge transfer from the metallocene
CoCpCoCp + CNT CNT Magnetoresistance effect
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
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
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
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
(a) (b)
S
N
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
Fin
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