The gdftb tool for quantum transport calculations. Alessandro Pecchia CNR - ISMN. University of Roma Tor Vergata

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University of Roma Tor Vergata A Gagliardi, G. Romano, G.

1 The gdftb tool for quantum transport calculations Alessandro Pecchia CNR - ISMN Institute for Nanostractured Materials University of Roma Tor Vergata A Gagliardi, G. Romano, G. Penazzi, M. Auf der Maur, M. Povolotskyi, F. Sacconi, Aldo Di Carlo

2 Outline Introduce NEGF extensions in DFTB Overlook on applications Electron-phonon interactions and heating in molecules Multiscale device simulations in TiberCAD CECAM Approx QM Bremen, September 20-24,

3 The DFTB approach DFTB = DFT based Tight-Binding method Kohn-Sham equation: 0 Scc k Hµν + Hµν [ δn] Ek S µν cν = 0 ν ε µ H = µν 0 0 µ V[ n + n ] ν µ ν onsite atomic energy levels two-centre density superposition S µν = µν Scc 1 H = S ( γ + γ ) q 2 µν µν µσ νσ σ σ [Elstner, et al. Phys. Rev. B 58 (1998) 7260] CECAM Approx QM Bremen, September 20-24,

4 Self-consistent loop (gdftb) L Active Region R µ L µ R Poisson Density matrix 1 < = de G ( E ) 2π i ρ µν µν CECAM Approx QM Bremen, September 20-24,

5 Green s operators L Active Region R ( ) 1 r G ( E ) = ES H Σ Σ L R < r a r a G ( E ) = if ( E) G ( E) Γ ( E) G ( E) + if ( E ) G ( E ) Γ ( E) G ( E) L L R R L -incoming DOS R -incoming DOS Equilibrium limit: < r a G ( E ) = if ( E )[ G ( E ) G ( E )] CECAM Approx QM Bremen, September 20-24,

6 Contour integration G < 1 Im r = de G π C ρ µν µν ρ µ 2 1 < µν = de Gµν iπ µ 1 µ R µ L The largest part of the integration is performed away from the real axis The integral is performed numerically via gaussian quadrature Parallelized with MPI CECAM Approx QM Bremen, September 20-24,

Iterative scheme H L τ τ H L τ HS, = τ H L τ τ H L τ τ P H τ L τ τ H L = τ H τ L τ H L τ τ q µ = P µν S νµ ν PROFILING Charge density and potential: Density of States (350 points): Peak memory: 20 h

7 Iterative scheme H L τ τ H L τ HS, = τ H L τ τ H L τ τ P H τ L τ τ H L = τ H τ L τ H L τ τ q µ = P µν S νµ ν PROFILING Charge density and potential: Density of States (350 points): Peak memory: 20 h 6 h 876 MB Calculations on single PC Linux core Intel(R) Core(TM)2 CPU 2.40GHz 2.4nm 2.4nm 21.3nm 7500 atoms O(N m 3 ) [Penazzi, et al. New J. Phys. 10 (2008) ] CECAM Approx QM Bremen, September 20-24,

Sub-partitioning of each layer ( ) 1 G ( E ) = ES H H g H 11 11 11 12 22 21 ( ) 1 g ( E ) = ES H 22 22 G ( E ) 11 G ( E ) = g H G 21 22 21 11 2 O(m 3 )/8 + MM mult O(m 3 )/3 G ( E ) = G H g 12 11 12

8 Sub-partitioning of each layer ( ) 1 G ( E ) = ES H H g H ( ) 1 g ( E ) = ES H G ( E ) 11 G ( E ) = g H G O(m 3 )/8 + MM mult O(m 3 )/3 G ( E ) = G H g G ( E ) = g + g H G H g O(m 3 )/9 + MM mult O(m 3 )/6 LibNEGF - General Sparse Matrices (CSR) - Automatic partitioning (METIS) - Parallel computations (MPI/OpenMP) - GPU acceleration (?) work in progress CECAM Approx QM Bremen, September 20-24,

9 Poisson solver 2-terminals Discretize in real space ρ( r) = qn( r) µ µ µ gated (3-term.) coaxially-gated The Poisson equation is solved with a multi-grid algorithm (MUDPACK). This allows to solve complex boundary conditions (bias, gate) 4-terminals 2 V = 4π ρ CECAM Approx QM Bremen, September 20-24,

10 CNT devices V DS < 0 E F, D h E F,S Drain p i p Self- consistent barriers Band to band tunneling Negative Quantum Capacity L. Latessa et al., PRB 72, (2005) CECAM Approx QM Bremen, September 20-24,

11 SiNW MOSFETs 1.2 nm (2.4 nm) p doped region oxide Intrinsic region p doped region Source oxide Drain 7.7 nm CECAM Approx QM Bremen, September 20-24,

12 Molecular Electronics OPV based transistor 10 Vds = 0.1 V na symmetric molecule F. Pump, G. Cuniberti Heiko Weber CECAM Approx QM Bremen, September 20-24,

13 Power Dissipation CECAM Approx QM Bremen, September 20-24,

Molecular heating&cooling Thermal effects at the molecular scale represent an increasingly hot topic Theoretical and experimental challenges to measure nanoscale temperatures N.

14 Molecular heating&cooling Thermal effects at the molecular scale represent an increasingly hot topic Theoretical and experimental challenges to measure nanoscale temperatures N. Néel et al. PRL 98, (2007) Z. Huang et al. Nano Lett. 6, 1240 (2006) Z. Ioffe et al. Nature Nanotech., on-line doi: /nnano CECAM Approx QM Bremen, September 20-24,

15 electron-phonon scattering μ L Device region Left contact Right contact V Elastic hν/e Inelastic μ R r DFT r r r G ( E ) = [ ES H Σ Σ Σ ] < r < < < G ( E ) = G [ Σ +Σ +Σ ] G < Σ LR, = iflr, ΓLR, L R scatt 1 L R scatt a < i q < q < Σ ph ( E ) = de ' α G ( E E ') α Dq ( E ') 2π q Σ < ( E ) = N γ G < ( E ω ) γ + ( N + 1) γ G < ( E + ω ) γ q q q q q q q q q CECAM Approx QM Bremen, September 20-24,

16 IETS simulations di/dv hν/e V d 2 I/dV 2 hν/e M. Reed Solomon et al., J. Chem Phys 124, (2006) CECAM Approx QM Bremen, September 20-24,

17 Phonon population Set up a steady-state solution for the vibronic populations Rate equation: dn dt q = R J [ N n ( T )] = 0 q q q q eff R = ( N + 1) E N A q q q q q CECAM Approx QM Bremen, September 20-24,

18 Molecular Temperature C 60 on Cu(110) Definition of molecular temperature: U = ω N = ω n ( T ) q q q q q mol q Bath coupling: ωqw q Nq nq( T ) = 0 q CECAM Approx QM Bremen, September 20-24,

0,0 0,5 Z [Å] I Crack [A] 10-5 0,5 1,0 1,5 2,0 2,5 3,0 Bias [V] P = I crack V crack is not a constant

19 C 60 burning experiment Tip approaches at fixed V until C 60 cracks. Molecules can be selectively burned I [na] A B C D E ,0-3,5-3,0-2,5-2,0-1,5-1,0-0,5 0,0 0,5 Z [Å] I Crack [A] ,5 1,0 1,5 2,0 2,5 3,0 Bias [V] P = I crack V crack is not a constant but shows features and plateaux G. Shulze et al., Phys. Rev. Lett. 100, (2008) CECAM Approx QM Bremen, September 20-24,

20 Model vs Experiments Fix critical T for degradation Theory Experiment CECAM Approx QM Bremen, September 20-24,

21 Multiscale/multiphysics outlook CECAM Approx QM Bremen, September 20-24,

22 Micro/macro scale Device should be accessible from a macro scale Number of atoms cannot grow to much in simulations Micro/macro scale details are as important as nanoscale features CECAM Approx QM Bremen, September 20-24,

23 Introduction: Multiscale/multiphysics Engineering Physics Time scale ms µs ns ps fs Length and time scale hierarchy Emp. Potentials Atomistic Semi-Empirical Atomistic Ab-initio Finite Elements Coarse Grains Systems nm µm mm Length scale Atoms, molecules Lumped devices, circuits CECAM Approx QM Bremen, September 20-24,

Hierarchy of transport models http://www.tibercad.

24 Hierarchy of transport models Engineering Time scale ms Circuit Level QM regions Strain Physics µs ns ps fs Boltzmann QM corrections Quantum Transport Hydrodynamic Drift-Diffusion Poisson/Drift Diffusion nm µm mm Length scale Atoms, molecules Lumped devices, circuits CECAM Approx QM Bremen, September 20-24,

25 Tiume scale ms µs DFTB as intermediate method Quantum Device ns ps Atomistic Semi-Empirical Atomistic Ab-initio nm µm mm CECAM Approx QM Bremen, September 20-24,

26 FEM/atomistic approaches FEM ATOMISTIC Projection on atoms Potentials (electrostatic, piezo, ) Strain and deformations domain 1 Projection on FEM Electrostatic map Deformation map (Wavefunctions) Charge density Current o domain 2 DOMAIN 1 DOMAIN 2 Different simulation domains exchange data CECAM Approx QM Bremen, September 20-24,

27 Light emitting GaN/AlGaN QD sp 3 d 5 s* Empirical TB atoms CECAM Approx QM Bremen, September 20-24,

28 Modeling of STM junction Tungsten STM tip 50 nm Idealized model of the tip 200 µm Construction of FEM Atomistic Model of the STM junction CECAM Approx QM Bremen, September 20-24,

29 Atomistic-FEM coupling CECAM Approx QM Bremen, September 20-24,

30 Geometry setup 1. Atomistic Generator 2. Starting guess 3. Geometry relaxations (Gunnar Shulze PhD Thesis) CECAM Approx QM Bremen, September 20-24,

31 Molecular rising 1 nm distance 1.2 nm distance 1.4 nm distance T vs V at 1.0 nm d=1.0, 1.4 nm at 2.0 V CECAM Approx QM Bremen, September 20-24,

the tip reduced thermal conductance due to size effects Low

32 Tip Heating Tip Subs P mol P=I V P mol <<I V Most of the power is dissipated in the contacts Fourier dissipation Large heating of the tip reduced thermal conductance due to size effects Low heating of the substrate CECAM Approx QM Bremen, September 20-24,

33 What we learned from gdftb DFTB is a versitile intermediate method bridging between abinitio and empirical potentials for electronic calculatons, structural relaxations and transport (gdftb). - Consistent computational framework (geometries, electronics, ) - Relatively fast but should improve SCC convergence Multigrid Poisson solver allows to study complicated device geometries Electron-phonon and heating effects can be included Electron-electron interactions (GW) still in progress CECAM Approx QM Bremen, September 20-24,

34 outlooks 1. A forthcoming release of dftb+ will contain NEGF (+examples, documentation ) 2. We are working at a general libnegf to compute electronic densities and current in equilibrium and under bias (FEM / Atomistic) 3. dftb+ has been also included in TiberCAD and we would like to develop QM/MM schemes for nanodevices (interfaces, defects, ) 4. Electron-phonon interactions in dftb+ (via libnegf) 5. Applications to nanodevices 6. CECAM Approx QM Bremen, September 20-24,

35 Acknowledgements Prof. Aldo Di Carlo Matthias Auf Der Maur, Post Doc Fabio Sacconi, Post Doc Alessio Gagliardi, Post Doc Michail Povolotsky, Post Doc Giuseppe Romano, PhD Gabirele Penazzi, PhD Marco Pacini, Student Luca Salvucci, Student Thank you CECAM Approx QM Bremen, September 20-24,

36 e-e correlations CECAM Approx QM Bremen, September 20-24,

37 Level Alignment problem RSH RS RS Adsorbate Gold (111) Obs PW91 PW91 PW91 Bridge FCC PW91 Obs. Band-gap error 5.6 ev Band lineup error 3.4 ev Bilić, Reimers and Hush J. Chem. Phys. 122 (2005) CECAM Approx QM Bremen, September 20-24,

38 GW on DFTB T.A. Niehaus et al., Phys. Rev. A, (2005) Energy Gap Very Good agreement on π-π* Worse on σ orbitals DFT+GW DFT Efficient GW: Can compute more than 30 rings CECAM Approx QM Bremen, September 20-24,

Complex bandstructures DFTB+GW bands DFTB bands E F β=0.32-0.

39 Complex bandstructures DFTB+GW bands DFTB bands E F β= Å -1 β= Å -1 I e -βl CECAM Approx QM Bremen, September 20-24,

40 GW on BDT junction Cu-S-Benzene-S-Cu HOMO LUMO GW Σ = 0 GW Σ 0 CECAM Approx QM Bremen, September 20-24,

41 Current calculation Transmission HOMO µ 1 µ LUMO 2 2 Landauer Formula for coherent transport: + 2e I( V) = T( E, V)[ f( E µ 1) f( E µ 2)] d E h E(eV) CECAM Approx QM Bremen, September 20-24,

42 What this is NOT good at Coulomb Blockades U I Mean field V H = εn + εn + Un n CECAM Approx QM Bremen, September 20-24,

Multiscale-Multiphysics modeling of nanostructured devices: The TiberCAD project.

Multiscale-Multiphysics modeling of nanostructured devices: The TiberCAD project. M. Auf der Maur, M. Povolotskyi, F. Sacconi, G. Romano, G. Penazzi, A Gagliardi, A. Pecchia, Aldo Di Carlo, Department