2. TranSIESTA 1. SIESTA. DFT In a Nutshell. Introduction to SIESTA. Boundary Conditions: Open systems. Greens functions and charge density

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2 1. SIESTA DFT In a Nutshell Introduction to SIESTA Atomic Orbitals Capabilities Resources 2. TranSIESTA Transport in the Nanoscale - motivation Boundary Conditions: Open systems Greens functions and charge density Notions of quantum transport Some details (preview)

4 Ψ({ r i }) n( r ) 1. (Hohenberg-Kohn Theorems) n( r ) = occ n ψ n ( r ) 2 h ˆ ψ n ( r ) = ε n ψ n ( r ) ˆ h = ˆ V eff [n] ˆ V eff [n]=v ext ( r) + V ˆ H [ n] ( r) + V ˆ xc [ n] LDA GGA

5 ψ n ( r )= µ c nµ φ µ ( r ) SCF ρ( r ) = occ n ψ n ( r ) 2

Spanish Initiative for Electronic Simulations with Thousands of Atoms Soler, Artacho, Gale, García, Junquera, Ordejón and Sánchez-Portal J. Phys.: Cond.

6 Spanish Initiative for Electronic Simulations with Thousands of Atoms Soler, Artacho, Gale, García, Junquera, Ordejón and Sánchez-Portal J. Phys.: Cond. Matt 14, 2745 (2002) Valence electrons: Flexible bases of Numerical Pseudo-atomic Orbitals with Finite Range (Pseudopotentials to elliminate the core electrons) Implements O(N) methodology Atomic forces and stress MD simulations Very efficient: capable of trating large systems with modest computers Parallelized Freely available for the academic community

7 Atomic orbitals: LCAO: ψ n ( r )= µ c nµ φ µ ( r ) Radial part: Numerical Pseudo-atomic Orbitals with Finite Range Spherical harmonics s p d Size: Number of orbitals for a given lm n Range: Spatial extension of the orbitals Shape: of the radial part f

8 Used to compute ρ(r) in order to calculate: - XC potential (non linear function of ρ(r) ) - Solve Poisson equation to get Hartree potential - Calculate Matrix Elements of KS Hamiltonian <φ i (r-r i ) V(r) φ j (r-r j )> (Hartree potential, XC potential, local pseudopotential...) - Mesh cut-off: highest energy of PW that can be represented with such grid.

9 Relaxations - Atomic coordinates - Cell shape & size Phonons, elastic constants,... Band structures (k-point sampling) Population analysis Charge distributions Electrostatic Potentials Electric Polarization Molecular Dynamics: - E, V - T, V (Nose Thermostat) - P (Parrinello-Rahman) - T, P - Quenching, annealing... Density of States, COOP,... Spin distributions Non-collinear spin states STM image simulation... LSDA, GGA (including Non-collinear spin states). Van der Waals Functionans (Langreth-Lundqvist) Time Dependent DFT (time evolution) LDA+U, SIC Spin-Orbit Coupling

Miniaturization of Electronic Devices Eniac computer

Pentium processor (2000) Current technologies: 65 to

theory starts to fail) Lithographic top-down

11 Miniaturization of Electronic Devices Eniac computer (1947) Challenges: Leak currents at gate oxide Pentium processor (2000) Current technologies: 65 to 22 nm Quantum confinement effects (semi-classical theory starts to fail) Lithographic top-down printing of smaller features Atomic limit: doping becomes unreliable

12 New approach to electronics: Bottom-up design Huge experimental progress in fabrication and characterization R. Stadler, M. Forshaw and C. Joachim, Nanotechnology 14, 138 (2003) Atomic wires: quantized conductance Diodes (with single molecules) Negative differential resistance Molecular Transistors (e.g., with nanotubes) Single-electron Transistor - Coulomb blockade Inelastic effects (phonons IETS) Kondo resonances...

13 Strong need for theoretical methods for molecular electronics and nanoeletronics: Quantum Behavior (semiclassical models are not applicable) L d R - V +

14 Infinite but non-periodic systems:

15 Retarded Green s Function: it is related to the spectral density and the density matrix occ ρ(e) = n FD (E) c * iµ c iν δ(e E i ) n Can be used in practice to compute the density matrix and the charge density without computing the eigenvectors/eigenvalues Useful for solving systems with extra interactions (for instance, among different subsystems)

17 If the system can be divided into two sub-systems, we can take the couping V as the interaction in Dyson s equation (note that V needs not to be small!): We have: with Using Dyson s eq. it is easy to prove that: Self energy Σ B - Non-hermitian, energy dependent potential - Accounts exactly for the effect of region B on A

18 1. Split the system into: Central region (where we are interested) Embedding or Electrodes region 2. Solve the Green Function G B of the embedding region 3. Calclate the self-energy due to the embedding 4. Solve the Green s Function of the central region

19 E µ R µ L = µ+δµ µ L µ R = δµ = ev k I = - 2e bands i BZ d k f ( k ) v g ( k, i) QUANTUM OF CONDUCTANCE

T ε ( ) =Tr [t + t] ε ( ) transmission matrix: I ev Non-interacting electrons Scattering-free leads (perfect crystalline electrodes) Electrons incident from

20 T ε ( ) =Tr [t + t] ε ( ) transmission matrix: I ev Non-interacting electrons Scattering-free leads (perfect crystalline electrodes) Electrons incident from left/right are in thermal equilibriumwith left/right reservoirs. Complete thermalization of electrons upon entering reservoir No back-scattering at lead-reservoir interface

21 L C R Coupling the finite contact to infinite electrodes Greens Functions

22 L C R Contact: Contains the molecule, and part of the Right and Left electrodes Sufficiently large to include the screening C B L R B Solution in finite system: Σ (ε ) = Selfenergies. Can be obtained from the bulk Greens functions Lopez-Sancho et al. J. Phys. F 14, 1205 (1984)

23 Bulk Greens functions and self-energies (unit cell calculation) Hamiltonian of the Contact region: Solution of GF s equations ρ(r) SCF PBC Landauer conductance: transmission probability: T ε ( ) =Tr [t + t] ( ε)

24 L C R Keldysh-Kadanoff-Baym Green s Functions Same equations can be derived using scattering states

25 L C R

26 Poisson s Eq: 2 V (r) = ρ(r) Given ρ(r), V H (r) is determined except up to a linear term: φ( r): particular solution of Poisson s equation +V/2 Au 12 au -V/2 Au a and b: determined imposing BC: the shift V between electrodes φ (r) computed using FFT s Linear term:

27 Good statistics: 1 atom=1 conductance quantum Onishi et al., Nature 395, 780 (1998) Linear I/V curves Brandbyge et al., PRB 52, 8499 (1995) Hansen et al., APL 77, 708 (2000)

28 Calculation: Bandstructure of infinite chain Experiment: dzx,dyz Same result (T 1) obtained, irrespective of chain length, electrode surface, strain, etc., which explains the narrow peak at 1 G o in the histograms

29 Voltage drop through the contact Almost linear I/V curve e Density Change in Density

30 1. DFT Calculations through SIESTA (numerical, confined atomic orbitals) 2. Transport Calculations: Define your system!. Contact region; Electrodes Bulk calculation for the electrodes Calculation of open system as Contact region + Self-energies Use NEGF s to obtain the charge densty and potential (under zero or finite bias) Obtain the transport properties from the Greens Functions (transmission matrix, conductance,...)

Ab initio study of Mn doped BN nanosheets Tudor Luca Mitran

Ab initio study of Mn doped BN nanosheets Tudor Luca Mitran MDEO Research Center University of Bucharest, Faculty of Physics, Bucharest-Magurele, Romania Oldenburg 20.04.2012 Table of contents 1. Density