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

2. U Tor Vergata What about realistic nanostructured devices ? 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) Inorganics 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 approaches are required for the modeling structural, electronic and optical properties of modern nanostructured devices.

3. U Tor Vergata Transport in nanostructures The transport problem is: active region contact active region where symmetry is lost + contact regions (semi-infinite bulk) 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. U Tor Vergata Atomistic approaches: The Tight-Binding method The approach can be implemented “ab-initio” where the orbitals are the basis functions and H i ,j  is evaluated numerically We attempt to solve the one electron Hamiltonian in terms of a Linear Combination of Atomic Orbitals (LCAO)  CiCi  orbital

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

6. U Tor Vergata SiO 2 p-Si Poly-Si-gate Si/SiO 2 tunneling:. empirical TB sp 3 d 5 s* Empirical parameterizations are necessary due to the band gap problem of ab-initio approaches  -critobalite  -quartz tridymite Staedele, et al. J. Appl. Phys. 89 348 (2001 ) Sacconi et al. Solid State Elect. 48 575 (‘04) IEEE TED in press

7. U Tor Vergata Tunneling Current: Comparison with experimental data  -cristobalite  -quartz tridimite  -cristobalite Good agreement between experimental and TB results for the  -cristobalite polimorph Slope of the current density is related to the microscopic structure of SiO 2 non-par. EMA par. EMA

8. U Tor Vergata Toward ”ab-initio” approaches. Density Functional Tight-Binding Many DFT tight-binding: SIESTA (Soler etc.), FIREBALL (Sankey), DMOL (Delley), DFTB (Seifert, Frauenheim etc.) ….. The DFTB approach [ 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. U Tor Vergata DFTB 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) II order-expansion of Kohn-Sham energy functional [Elstner, et al. Phys. Rev. B 58 (1998) 7260]

10. U Tor Vergata Non equilibrium systems The contact leads are two reservoirs in equilibrium at two different elettro-chemical potentials. How do we fill up the states ? f2f2 f1f1 How to compute current ?

11. U Tor Vergata 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: 1.Ignore the variation of the density matrix (we keep H 0 ) Suitable for situations very close to equilibrium (Most of the people do this !!!!) 2.The new-density matrix is calculated in the usual way by diagonalizing the Hamiltonian for the finite system Problem with boundary conditions, larger systems 3.The new-density is obtained from the Non-Equilibrium Green’s Function theory [Keldysh ‘60] [Caroli et al. ’70] [Datta ’90]...

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

13. U Tor Vergata The role of molecular vibrations An organic molecule is a rather floppy entity T= 300 K 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. U Tor Vergata Molecular Dynamics + current The dynamics of the  -th atom is given by The evolution of the system is performed on a time scale of ~ 0.01 fs = Hamiltonian matrix Calculation of the forces Atomic position update t=t+  t Molecular dynamics Hamiltonian matrix Calculation of the forces Atomic position update Current calculation [ J(t) ] t=t+  t Molecular dynamics + current Di Carlo, Physica B, 314, 211 (2002)

15. U Tor Vergata Molecular dynamics limitations The effect of vibrations on the current flowing in the molecuar device, via molecular dynamics calculations, has been obtained without considering 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. U Tor Vergata The lowest modes of vibration

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

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

19. U Tor Vergata Transmission functions MD Simulations Quantum average

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

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

22. U Tor Vergata I-V characteristics Molecular dynamics Quantum phonons Harmonic approximation failure produces incorrect results of the quantum phonon treatement of current flowing in the molecule

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

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

25. U Tor Vergata Equilibrium charge density Charge density with 1V bias Net charge density Negative chargePositive charge Charge and Potential in two CNT tips Potential Profile Charge neutrality of the system is only achived in large systems

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

27. U Tor Vergata CNT-MOS: Coaxially gated CNT VDVD VS=0VS=0 VGVG Semiconducting (10,0) CNT Insulator (ε r =3.9) 5 nm 1.5 nm x y z CNT contact

28. U Tor Vergata CNT-MOS 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 Potential Charge 2. 10 -5 -8. 10 -5 0 -4. 10 -5

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

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

31. U Tor Vergata Simple linear chain system  q =17 meV, E 0 = 0.06 eV absorption emission resonance incoherent coherent

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

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

34. U Tor Vergata Conclusions  Density Functional Tight-Binding approach has been extended to account for current transport in molecular devices by 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 The method

35. U Tor Vergata Conclusions  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) Results For the gDFTB code visit: http://icode.eln.uniroma2.it