Charge transport in molecular devices

1. Charge transport in molecular devices Aldo Di Carlo, A. Pecchia, L. Latessa, M.Ghorghe* Dept. Electronic Eng. University of Rome “Tor Vergata”, (ITALY) P. Lugli TU-Munich (GERMANY) Collaborations T. Niehaus, T. Frauenheim University of Paderborn (GERMANY) G. Seifert TU-Dresden (GERMANY) R. Gutierrez, G. Cuniberti *University of Regensburg (GERMANY) European Commission Project

2. What about realistic nanostructured devices ? Inorganics 1D (quantum wells): 100-1000 atoms in the unit cell 2D (quantum wires): 1000-10’000 atoms in the unit cell 3D (quantum dots): 100’000-1’000’000 atoms in the unit cell Organics Molecules, Nanotubes, DNA: 100-1000 atoms (or more) Traditionally, nanostructures are studied via k · p approaches in the context of the envelope function approximation (EFA). In this case, only the envelope of the nanostructure wavefunction is considered, regardless of atomic details. Modern technology, however, pushes nanostructures to dimensions, geometries and systems where the EFA does not hold any more. Atomistic approachesare required for the modeling structural, electronic and optical properties of modern nanostructured devices.

3. Transport in nanostructures active region where symmetry is lost + contact regions (semi-infinite bulk) The transport problem is: activeregion contact contact contact Open-boundary conditions can be treated within several schemes: • Transfer matrix • LS scattering theory • Green Functions …. These schemes are well suited for localized orbital approach like TB

4. Atomistic approaches: The Tight-Binding method f= orbital Cia Y We attempt to solve the one electron Hamiltonian in terms of aLinear Combination of Atomic Orbitals (LCAO) The approach can be implemented “ab-initio” where the orbitals are the basis functions and Hia ,jbis evaluated numerically

5. Scalability of TB approaches Empirical Tight-Binding Hamiltonian matrix elements are obtained by comparison of calculated quantities with experiments or ab-initio results. Very efficient, poor transferability. Semi-Empirical Tight-Binding Density Functional Tight-Binding Density-functional based methods permit an accurate and theoretically well founded description of electronic properties for a wide range of materials.

6. Si/SiO2 tunneling:. empirical TB sp3d5s* SiO2 Poly-Si-gate p-Si Staedele, et al.J. Appl. Phys. 89 348 (2001 ) Sacconi et al. Solid State Elect. 48 575 (‘04) IEEE TED in press b-critobalite b-quartz tridymite Empirical parameterizationsare necessarydue to the band gap problem of ab-initio approaches

7. Tunneling Current: Comparison with experimental data b-cristobalite b-cristobalite tridimite b-quartz Good agreement between experimental and TB results for the b-cristobalite polimorph Slope of the current density is related to the microscopic structure of SiO2 non-par. EMA par. EMA

8. Toward ”ab-initio” approaches.Density Functional Tight-Binding • Many DFT tight-binding:SIESTA(Soler etc.), FIREBALL(Sankey),DMOL(Delley),DFTB(Seifert, Frauenheim etc.)….. • TheDFTBapproach[Elstner, et al. Phys. Rev. B 58 (1998) 7260] • provides transferable and accurate interaction potentials. The numerical efficiency of the method allows for molecular dynamics simulations in large super cells, containing several thousand of atoms. • DFTB is fully scalable (from empirical to DFT) • DFTB allows also for TD-DFT simulations • We have extended the DFTB to account for transport in organic/inorganic nanostructures by using Non Equilibrium Green Function approach self-consistently coupled with Poisson equation

9. DFTB II order-expansion of Kohn-Sham energy functional [Elstner, et al. Phys. Rev. B 58 (1998) 7260] Tight-binding expansion of the wave functions [Porezag, et al Phys. Rev. B 51 (1995) 12947] DFT calculation of the matrix elements, two-centers approx. Self-Consistency in the charge density (SCC-DFTB)

10. Non equilibrium systems The contact leads are two reservoirs in equilibrium at two different elettro-chemical potentials. f2 f1 How do we fill up the states ? How to compute current ?

11. How do we fill up states ? (Density matrix) . . . . . . The crucial point is to calculate the non-equilibrium density matrix when an external bias is applied to the molecular device Three possible solutions: • Ignore the variation of the density matrix (we keep H0)Suitable for situations very close to equilibrium (Most of the people do this !!!!) • The new-density matrix is calculated in the usual way by diagonalizing the Hamiltonian for the finite systemProblem with boundary conditions, larger systems • The new-density is obtained from the Non-Equilibrium Green’s Function theory [Keldysh ‘60] [Caroli et al. ’70] [Datta ’90]

12. DFTB + Green Functions Systems close to the equilibrium • Molecular vibrations and current (details: Poster 16)

13. The role of molecular vibrations T= 300 K An organic molecule is a rather floppy entity • We compare: • Time-average of the current computed at every step of a MD simulation (Classical vibrations) • Ensemble average of over the lattice fluctuations (quantum vibrations = phonons). A. Pecchia et al. Phys. Rev. B. 68, 235321 (2003).

14. Molecular Dynamics + current Hamiltonian matrix = Current calculation [ J(t) ] t=t+Dt Calculation of the forces Molecular dynamics Molecular dynamics + current Hamiltonian matrix Atomic position update t=t+Dt Calculation of the forces Atomic position update Di Carlo, Physica B, 314, 211 (2002) The dynamics of the a-th atom is given by The evolution of the system is performed on a time scale of ~ 0.01 fs

15. Molecular dynamics limitations The effect of vibrations on the current flowing in the molecuar device,via molecular dynamics calculations, has been obtained withoutconsidering the quantization effects of the vibrational field. The quantum nature of the vibrations (phonons) is not considered ! However, vibration quantization can be considered by performing ensamble averages of the current over phonon displacements H. Ness et al, PRB 63, 125422 How does it compare with MD calculations ?

16. The lowest modes of vibration

17. Phonons H. Ness et al, PRB 63, 125422 The hamiltonian is a superposition of the vibrational eigenmodes, k: The eigenmodes are one-dimesional harmonic oscillators with a gaussian distribution probability for qk coordinates:

18. The current calculation • The tunneling probability is computed as an ensemble average over the atomic positions (DFTB code + Green Fn.) • We average the log(T) because T is a statistically ill-defined quantity (is dominated by few events). MC integration • The current is computed as usual:

19. Transmission functions Quantum average MD Simulations

20. Comparison: MD, Quantum PH, Classical PH QPH = phonon treatement CPH = phonons treatement without zero point energy

21. Frequency analysis of MD results Ph-twist S-Au stretch CC stretch Mol. Dynamics Fourier Transf. A. Pecchia et al. Phys. Rev. B. 68, 235321 (2003).

22. I-V characteristics Molecular dynamics Quantum phonons Harmonic approximation failure produces incorrect results of the quantum phonontreatement of current flowing in the molecule

23. DFTB + Non-Equilibrium Green Functions • Full Self-Consistent results • Electron-Phonon scattering (details: Posters 34 and 37)

24. Self-consistent quantum transport BULK Surface SELF-CONS. DFTB WITH POISSON 3D MULTI-GRID Device Surface BULK Self-consistent loop: Density Matrix Mullikan charges Correction SC-loop Di Carlo et. al. Physica B, 314, 86 (2002)

25. Charge and Potential in two CNT tips Potential Profile Equilibrium charge density Charge density with 1V bias Charge neutrality of the systemis only achived in large systems Net charge density Negative charge Positive charge

26. Self-consistent charge in a molecular wire 0.5 V 1.0 V

27. CNT-MOS: Coaxially gated CNT y z x VG VD VS=0 5 nm 1.5 nm Semiconducting (10,0) CNT CNT contact Insulator (εr=3.9)

28. CNT-MOS 2.10-5 0 -4.10-5 -8.10-5 Potential Charge Isosurfaces of Hartree potential and contour plot of charge density transfer computed for an applied gate bias of 0.2 V and a source-drain bias 0f 0.0 V

29. Output characteristics Gate coupling (capacitance) is too low. A precise design is necessary (well tempered CNT-MOS)

30. Electron-phonon self-energy Directly from DFTB hamiltonian The el-ph interaction is included to first order (Born approximation) in the self-energy expansion. [A. Pecchia, A. Di Carlo Report Prog. in Physics (2004)] Born –approximation

31. Simple linear chain system q=17 meV, E0 = 0.06 eV emission resonance absorption incoherent coherent

32. Inelastic scattering: Current + phonons I(E)

33. IV Current + phonons T=150 K T=0 K Coherent Coherent Incoherent Incoherent No phonons

34. Conclusions The method • Density Functional Tight-Binding approach has been extended to account for current transport in molecular devicesby using Self-consistent non-equilibrium Green function (gDFTB). • DFTB is a good compromise between simplicity and reliability. • The use of a Multigrid Poisson solver allows for study very complicated device geometries • Force field and molecular dynamics can be easily accounted in the current calculations. • Electron-phonon coupling can be directly calculated via DFTB • Electron-phonon interaction has been included in the current calculations. For the gDFTB code visit: http://icode.eln.uniroma2.it

35. Conclusions Results • Anharmonicity of molecular vibrations can limit the use of phonon concepts • Concerning ballistic transport, temperature dependence of current is better described whit molecular dynamics than ensamble averages of phonon displacements • Screening length in CNT could be long. • Coaxially gated CNT presents saturation effects but gate control is critical. • Electron-phonon scattering is not negligible close to resonance conditions of molecular devices All the details in A. Pecchia, A. Di Carlo Report Prog. in Physics (2004) For the gDFTB code visit: http://icode.eln.uniroma2.it