Multiscale modelling challenges for transport problems

Transcription

1 Warwick, June 015 Multiscale modelling callenges for transport problems Neopytos Neopytou Scool of Engineering, University of Warwick, Coventry, U.K.

2 Empirical metods WCPM page

3 Multiscale features at te nanoscale quantum wells CNT nanowires nanocomposites Source Top Gate Insulator Drain nanotubes Ucida et al., IEDM 03 Substrate Trivedi, Nano Lett. 011 Delft group S Gate Insulator: 4nm HfO (k=16) D Gate.5 nm Doping: N D = 10 9 /m 5 nm Intrinsic.5 nm Doping: N D = 10 9 /m How electrons/ponons flow in multi-scale disordered systems? 3

4 Nanomaterials Modeling callenges Atomistic to continuum details 1) Atoms are countable ) Material variations 3) Interfaces 4) Dimensionality issues Ucida et al., IEDM 03 Trivedi et al., Nano Lett nm 5 nm ~ μm, million atoms 4

5 Empirical metods to obtain band structure Tigt binding Valence Force Fields Force constants NN sp 3 d 5 s*-so 1 st Electronic Brillouin Zone of Si Ponon bandstructure of bulk Ponon bandstructure of grapene Map parameters to BULK data: Genetic algoritm (TB), optical data, etc 5

6 Hamiltonian construction towards nanostructures Te Hamiltonian is directly been built from te geometry and bonding info N N E aa i, j i, j s, p, p, p, s*, x y z d, d, d, x y, z r xy yz zx B ac gi, j V ac i, j 1. Connecting atoms. Passivated 3. Periodic BCs 6

7 Low-dimensional bandstructure examples (100), (110), (11) (100), (110), (111) <100>, <110> Source Cannel Drain [100] [110] [111] [100] [110] [111] NN sp 3 d 5 s*-so Conduction band Valence band 1 st Brillouin Zone 7

8 MVFF: Low-dimensional ponon spectrum optical quasiacoustic acoustic 1D nanowire D ultra-tin layer 8

9 Approac transport NEGF for ponons G( E) E I D 1 1 Landauer D Dl exp iq. R ( q) q ( q) l E Tp ( ) Trace 1G G T p ( ) M ( ) p 1 n Kl Tp d T 0 9

10 Linearized Boltzmann transport v v k g k k k v 0 k ( ) f0 0 kt E B R q d 0 R S kb 0 R q R 1 0 At all κ-point, subbands: velocity density of states v g n n 1D E E E k 1 n x 1 1 v E n e k T q R 1 B R 0 0 R 10

11 Were it migt fail? Problems: We can scale te computation, still atomistically to large sizes, but.. 1) How transferable are te parameters at te nanoscale? ) Do te parameters apply at te edges/interfaces? ) Wat appens if you ave material variations? 3) Varying strain fields? 4) Dimensionality mixing? 5) Amorpous regions? Typical problems tat reviewers raise all te time.. Opportunities ere: Can we modify tese parameters to accommodate better description? 11

12 Force constant metod 1

13 Force constant metod Pononic structure: Fourt nearest-neigbor force constant metod LO LA, TA ZO LA, TA, ZA R. Saito et al., Pysical Properties of Carbon Nanotubes,

14 Approac Force constant metod Pononic structure: Fourt nearest-neigbor force constant metod ij U Dmn, i, j N A and m, n[ x, y, z] i j r r m n D D D ij ij ij xx xy xz ij ij ij ij yx yy yz ij ij ij Dzx Dzy Dzz D D D D D U K U 1 ( ij) ij m 0 m K ( N ) r ( ij) ( N ) 0 ti ( N ) to cosij sinij 0 sin cos U m ij ij D 1 D i j D i j ij ( ij) [ D33 ] MM il i j li R. Saito et al., Pysical Properties of Carbon Nanotubes,

15 Valence force field metod 15

16 Modified Valence Force Field Metod (MVFF) Keating Modified U ij bs 3 8 r ij dij d ij bond-stretcing U jik bsbs 3 8 rij dij rik dik dd ij ik cross bond stretcing U jik bb jik 3 8 dd ij ik bond-bending U jik bsbb 3 8 rij dij jik dd ij ik cross bond stretcing/ bending U jikl bbbb 3 8 jik ikl d d d ij ik kl coplanar bond bending 16

17 Modified Valence Force Field Metod (MVFF) Keating Modified jk 1 U U U U U U j k l ij jik jik jik jikl bs bb bsbs bsbb bbbb in A jnni j, knni j, k, lcop i D ij ij ij Dxx Dxy D xz U ij ij ij Dij yx yy yz D D D ij ij ij Dzx Dzy D zz ij ij mn mn i j rm rn l D D exp iq. R q I 0 l l 17

18 Tigt-Binding 18

19 Tigt binding multiscale opportunities Eac of te sp 3 d 5 s * -SO orbital, as an overlap wit te oter orbitals Need a large set of fitting parameters - ~0-30 Usually we average te parameters wen we create an alloy Good Si parameters are available, for III-Vs, D materials, so and so.. People use DFT to extract parameters nowadays N N E aa i, j i, j B ac gi, j V ac i, j s, p, p, p, s*, x y z d, d, d, x y, z r xy yz zx 1. Connecting atoms. Passivated 3. Periodic BCs 19

20 Conclusions Electrons: sp 3 d 5 s * bandstructure model: Ponons: Modified Valence Force Field metod (MVFF) Force constant metod Several possible opportunities for multiscale/multipysics improvements 0