Coherent electronic transport in nanostructures and beyond INFM Natl. Res. Center S3 University of Modena and Reggio Emilia Andrea Ferretti
Acknowledgments People @S3: External Collaborators: Andrea Ferretti Arrigo Calzolari Carlo Cavazzoni Rosa Di Felice Franca Manghi Elisa Molinari Marco Buongiorno Nardelli (NCSU, US) Nicola Marzari (MIT, US) Marilia J. Caldas (USP, Brazil) National Research Center on nanoStructures and bioSystems at Surfaces via Campi 213/A, 41100 Modena, Italy
Outline • Motivations • The method Ab initio electronic transport from max. loc. Wannier Functions: • Development • Implementation (WanT code) • Application to nanoscale systems • Inclusion of correlation in transport Application to short range e-e interaction regime
H. Ohnishi et Al.,Nature, 395, 780 (1998) Motivations • Novel systems for electronic devices (nanotubes, atomic chains, molecular systems,…) M. Reed et al., Science (1997) C. Dekker et Al., Nature 429, 389 (2004) D.Porath et al., Nature (2000) • Semiclassical transport theory breaks down • Full quantum mechanical approach Landauer Theory
Landauer Formula • Transmittance from real-space Green’s functions techniques Fisher & Lee formulation • Need for a localized basis set Ballistic transport Conductance as transmission through a nano-constriction • Ballistic transport:exclusion of non-coherent effects (e.g. dissipative scattering or e-e correlation). • Quantum conductance: depends on the local properties of the conductor (transmission - scattering) and the distribution function of the reservoirs
The “WanT” approach • Create a connection between • ab initio description of the electronic structure by means of state-of-the-art DFT- plane wave calculations • Real space Green’s function techniques for the calculation of quantum conductance. • Idea:Unitary transformation of delocalized Bloch-states intoMaximally localized Wannier functions WanT method A. Calzolari, PRB 69, 035108 (2004).
Single band transformation: Generalized transformation: Non-uniqueness of WFs under gauge transformation U(k)mn Goal: Calculation of WFs with the narrowest spatial distribution Maximally localized Wannier functions* Wannier functions (WFs): definition * N. Marzari, and D. Vanderbilt, PRB 56, 12847 (1997).
Wannier functions: localization Spread functional Maximal localization given by the minimization of the spread wrt U: WF advantages: WF disadvantages: orthonormality completness minimal basis set adaptability direct link to phys. prop. no analytical form computational cost algorithm stability
Flow diagram DFT Conductor(supercell) Leads (principal layer) WFs GFs All quantities onWannierbasis Zero bias Linear response QC Quantum conductance
WanT Code www.wannier-transport.org Features: • Input fromPW-PP, DFTcodes. • Maximally localizedWannier Functions computation. • Transport propertieswithin amatrix GF’sLandauer approach. • GNU-GPL distributed
Si Nanotubes: Si defect Zigzag (5,0) carbon nanotube with a substitutional Si defect • Si polarizes the WF’s in its vicinity affecting the electronic and transport properties of the system • General reduction of conductance due to the backscattering at the defective site • Characteristic features (dips) of conductance of nanotubes with defects A. Calzolari et al., PRB 69, 035108 (2004).
Evidences of strong e-e correlation effects: • Kondo effect • Coulomb blockade J. Park et. al., Nature 2002 T.W. Odom et al., Science290, 1549 (2000) J. Park et al., Nature417, 722 (2002) W. Liang et al., Nature417, 725 (2002) Correlated transport in nanojunctions • Goal:Ab initio description of electronic transport in the presence of strong electron-electron coupling, from atomistic point of view.* • Landauer formalism breaks down: Need for a novel theoretical treatment * A. Ferretti et al., PRL 94, 116802 (2005)
Effective transmittance Formalism Re-formulate the theory from more general conditions: Meir-Wingreen approach Generalized Landauer-like formula A. Ferretti et al., PRL 94, 116802 (2005) Correlated transport Landauer + e-e correction to the Green’s functions • Conductor GF’s are interacting • Lambda is also given by:
Implementation Correlation effects: • Three Body Scattering(3BS) method* • Describes thestrong short rangeelectron-electron interaction • Based on a configuration interaction scheme up to 3 interacting bodies (1particle + 1e-h pair) of the generalized Hubbard Hamiltonian • Hubbard U is an adjustable parameter * F. Manghi, V. Bellini and K. Arcangeli, PRB 56, 7159 (1997).
Flow diagram DFT Conductor(supercell) Leads (principal layer) Atomicpdos 3BS WFs Correlation Mean field Σnn’k(ω) GFs All quantities onWannierbasis QC Quantum conductance
C I T Correlated Pt chain Pt Pt Pt 3 correlated atoms NC Transport components coherent comp. incoherent comp. total C I T quasi-particle finite lifetimes
Nanotubes: Co impurity • EXP:Cobalt impurities adsorbed on metallic CNT T.W. Odom et al., Science290, 1549 (2000) • Transition metal(TM) often present as catalyzers • Interplay between CNT and TM physics • changes on electronic and transport properties • THEO:Co @ 5,0 CTN Work in progress
Conclusions and outlook • Development of the freely availableWanT (Wannier-Transport) code. • Inclusion of electron correlation(incoherent, non-dissipative model), in the strong short-range regime (by 3BS method). • Application to Pt chains: • renormalization effects on quantum transmittance and conductance • Importance of finite QP lifetimes
Computational DFT calc. Transport calc. Wannier functions basis PW basis WFs determination Stability issues withhundreds of WFs Strategies: • Existing code optimization • PAW / USPP implementation • Variational functional redefinition, minimization procedure
Molecular nanostructures J. Park et al.,Nature 417, 722 (2002) Free molecule • Co coordination complex • Prototype forcorrelation effectsin molecular electronics • Computationally challenging Device configuration Work in progress
Correlation within LDA+U Pt Pt Pt 3 correlated atoms Static coherent description
Transmittance Pt@Au chain Au Pt Au 3 correlated atoms Au chain Mean field Pt@Au Correlated Pt@Au Interface effects do not suppress correlation
C R L Transport: problem definition L= left lead C= conductor R= right lead Hypotheses: Leads arenon-interacting The problem isstationary Definitions: Operators in block matrix form Using a localized basis set:
C R L Coupling to the leads Leads self-energies • From the block inversion of the hamiltonian • Allows to treat the coupling to the leads • Computational interest Retarted, Advanced SE Coupling functions
Expression for the current • Exact expression from Meir & Wingreen* In the interacting case Ng-ansatz for G>,< Which results in N. Sergueev et al., PRB 65, 165303 (2002)
Simple analitycal structure and spectral analisys Retarded, Advanced Correlation functions Direct access to observable expectation values Equilibrium Green Functions • Various definitions: Allows perturbation theory (Wick’s theorem) Time ordered
Im ω Im ω Im ω Re ω xxxxxxx Re ω xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx Re ω Analitycal properties Retarded GF Time ordered GF Advanced GF Fermi Energy = 0.0
Equilibrium Green Functions Gr, Ga, G<, G>are enough to evaluate all the GF’s and are connected by physical relations General identity Fluctuation-dissipation th. Spectral function • Just one indipendent GF
Non-Equilibrium GF’s • Electric fields (TD laser pulses) • Coupling to contacts at different chemical potentials • Contour-ordered perturbation theory: Gr, Ga, G<, G> are all involved in the PT 2 of them are indipendent Ordering contour Only the identity holds(no FD theorem)
Non-Equilibrium GF’s In the time-indipendent limit • Two Equations of Motion Dyson Equation Keldysh Equation Gr, G< coupled via the self-energies Computing the (coupled) Gr, G< allows for the evaluation of transport properties