1. The g DFTB method applied to transport in Si nanowires and carbon nanotubes 1 Dip. di Ingegneria Elettronica, Universita` di Roma Tor Vergata 2 Computational Material Science, Universitat Bremen Alessandro Pecchia 1 L. Latessa 1, Th. Frauenheim 2, A. Di Carlo 1

2. ACS - San Francisco 2006 Retarded (r) and advanced (a) Green functions are defined as follow Let us write H and G in a block form NEGF + DFTB = g DFTB Self-energies Device region Lead HDHD HLHL HRHR  LD  RD -- ++

3. ACS - San Francisco 2006 Non equilibrium density In order to compute V(r) we need the local  (r) Density Matrix We can build the n.e. density matrix LL RR

4. ACS - San Francisco 2006 Self-consistent loop Density matrix ρ Multigrid Poisson solver Self-consistent solutions Evaluation of Green’s function External potential Hartree term Exchange- correlation (LDA) + +

5. ACS - San Francisco 2006 Implementation details Is performed by countour integration and has been parallelized (MPI) All matrices stored in dense format Green’s functions computed by direct inversions Sparse storage Implementation of a block-iterative construction Old gDFTB (2003-2005) New gDFTB (2006-)

6. ACS - San Francisco 2006 Block-iterative algorithm The device G.F. are computed with an iterative algorithm 1) Computation of partial Green’s 2) Computation of equilibrium Green’s 3) Additional blocks needed for non-equilibrium H PL

7. ACS - San Francisco 2006 Profiling SWCNT(20,0): 2880 atoms, 36 Principal Layers 2,5 nm 14,7 nm 60,2 5,4 0 20 40 60 80 Tempo (s) Old gDFTB New gDFTB 1743 186 0 500 1000 1500 2000 Peak Memory (MB) Old gDFTB New gDFTB Single node (P4 - 3.2 GHz), Single energy point

8. ACS - San Francisco 2006 Self-consistent potential eV

9. ACS - San Francisco 2006 Poisson Equation The Poisson’s equation is solved with a 3D Multi-grid algorithm. Discretize in real space This allows to solve complex boundary conditions (bias, gate) 2-terminals gated coaxially-gated 4-terminals

10. ACS - San Francisco 2006 projection back in AO Now we need to project the solution into the local basis set Can be viewed as an approximation of the rigorous matrix elements of V(r). This is consistent with standard DFTB

11. ACS - San Francisco 2006 Summary g DFTB implementation  Construction of H and S directly in sparse format  Solution of Green’s functions via block-iterative methods  Current Bottle-neck: Poisson equation - Very efficient for 1D type systems - Memory scales linearly (depends on PL size) - Can be used for O(N) calculations even in equilibrium - Considerable speed-up and memory save - Dense matrices never allocated - Multigrid with uniform grid, dense storage! - Need to implement more efficient methods (finite elements with adaptive grids)

12. ACS - San Francisco 2006 Applications of g DFTB Molecular Electronics. Incoherent Transport and Inelastic Tunneling Spectroscopy A. Pecchia et al., Nano Lett. 4, 2109 (2004) G. Solomon et al., J. Chem Phys 124, 094704 (2006) A. Pecchia, A. Di Carlo, Introducing molecular electronics, Springer Series, (2005) A. Pecchia, A. Di Carlo, Molecular Electronics: Analysis design and simulations, Elsevier (2006)

13. ACS - San Francisco 2006 Applications to CNT and SiNW

14. ACS - San Francisco 2006 Coaxially gated CNT VDVD VS=0VS=0 VGVG Semiconduct ing CNT Insulator (ε r =3.9) 10 nm 1.5 nm x y z CNT contact (INFINEON - Düsberg)

15. ACS - San Francisco 2006 Atomic Forces V GATE = 5 V Ang. GATE Forces [Ang] Application of V G changes CNT diameter

16. ACS - San Francisco 2006 Screening problem Quantum correction to the induced charge V ext Distance VGVG CNT electron gas CNT is not able to accumulate the electronic charge to completely screen the gate bias (λ > electron gas extension) λ CNT completely screens the external field. Classical electrostatics: charge induced on the CNT is λ

17. ACS - San Francisco 2006 HOWEVER: In a CNT the DOS is not the only contribution. Many body correction should be considered (XC) Screening in CNT: DOS limit Why charge induction is limited? DOS [Latessa et al., Phys. Rev. B 72, 035455 (2005)] Pauli exclusion principle limits the induced electrons to the number allowed by filling the DOS

18. ACS - San Francisco 2006 Many-body corrections Compressibility of an interacting electron gas Compressibility of an non-interact electron gas  Compressibility Capacity

19. ACS - San Francisco 2006 Evidence of Negative K Eisenstein, Pfeiffer and West, PRL 68, 674 (1992) Eisenstein, Pfeiffer and West, PRB 50, 1760 (1994) N (10 11 cm -2 ) 0.0 0.1 -0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 NcNc Compressibility of 2D electron gas Thomas-Fermi screening In 1D systems things can be more complicated because of D(E) Including exchange

20. ACS - San Francisco 2006 Negative C Q in CNTs Overscreening, C Q <0 [Latessa et al., Phys. Rev. B 72, 035455 (2005)]

21. ACS - San Francisco 2006 Negative C Q in CNTs Critical charge density n TS TOTAL SCREENING High density limit : PARTIAL SCREENING C Q approaches e 2 DOS(E F ) gDFTB calculation C Q proportional to DOS Low density limit: OVER-SCREENING Fit to analytic model Chalmers, PRB 52, 10841 (‘95) Fogler, PRL 94, 056405 (2005)

22. ACS - San Francisco 2006 XC in DFTB Hubbard and e-e repulsion integrals T.A. Niehaus, PRA 71, 022508 (2004)

23. ACS - San Francisco 2006 Diameter Dependence CNT (13,0) CNT (10,0)CNT (7,0)

24. ACS - San Francisco 2006 Output characteristics h V DS < 0 p p i E F,S E F,D Drain Source “Electrostatic saturation” L. Latessaet al.: IEEE Trans. Nanotechnol., in press

25. ACS - San Francisco 2006 Small sub-threshold swing (theoretical limit for silicon MOSFET: 60 mV/dec) I on /I off ~ 10 8 Unipolar behavior Trans-characteristics

26. ACS - San Francisco 2006 Band-to-well tunneling Generation of confined states in a quantum well

27. ACS - San Francisco 2006 SiNW FET SiO 2 shells has been removed and silicon is terminated with H D. D. D. Ma. et al., Science, vol. 299, pp. 1874-1877, 2003 L. J. Lauhon, et al., Nature, vol. 420, pp. 57-61, 2002. Coaxially gated Si nanowire FET

28. ACS - San Francisco 2006 Geometry relaxations d<10 nm 10<d<20 nm d>30 nm 1.22 nm 0.87 nm

29. ACS - San Francisco 2006 Device geometry P doped region Intrinsic region oxide 1.2 nm (2.4 nm) 7.7 nm 3.6 nm 6 nm Drain Source

30. ACS - San Francisco 2006 CNT vs SiNW CNT-FETSiNW-FET 6 nm

31. ACS - San Francisco 2006 Differences in S Coax. gated (7,0) CNTFET SiNW – FET |V GS | (V) Current, I DS (A) S = 180 mV/dec S = 75 mV/dec

32. ACS - San Francisco 2006 Conclusions of part II Atomistic Density Functional approach can be extended to account for current transport in molecular devices by using self-consistent non-equilibrium Green function. We use g DFTB is a good compromise between simplicity and reliability but there is room for improvement. The use of a Multigrid Poisson solver allows for study very complicated device geometries CNT and Silicon Nanowire FET has been studied with g DFTB Quantum capacitance in CNT is governed by XC Gate control in SiNW FETs is more delicate

33. ACS - San Francisco 2006 Surface Green’s function The surface G.F. is computed by iteration (decimation technique)   H PL g 1 2 3 4 5 -> Converged surface Green’s function Lopez Sancho et al., J. Phys. F: Met. Phys. 14 1205 (1984); ibid., 15 851 (1985)