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