Warwick, June 015 Multiscale modelling callenges for transport problems Neopytos Neopytou Scool of Engineering, University of Warwick, Coventry, U.K.
Empirical metods WCPM page
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
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. 011 1 nm 5 nm ~ μm, million atoms 4
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
Hamiltonian construction towards nanostructures Te Hamiltonian is directly been built from te geometry and bonding info 0 1 3 4... N 1 3 4... 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
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
MVFF: Low-dimensional ponon spectrum optical quasiacoustic acoustic 1D nanowire D ultra-tin layer 8
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
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
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
Force constant metod 1
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, 1998 13
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 0 0 0 0 0 0 ( N ) to cosij sinij 0 sin cos 0 0 0 1 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, 1998 14
Valence force field metod 15
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
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
Tigt-Binding 18
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.. 0 1 3 4... N 1 3 4... 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
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