Interlayer Physics in Few Layer Graphenes (2011)

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(Luo et al., Nanoletters 10, 777 (2010)). III. Nanoparticle shape-selection on FLG's. (Somers et al. Phys. Rev. B 82, 115430 (2010)). IV. Interlayer physics in ...
Interlayer Physics in Few-Layer Graphenes Experiment:

Theory:

Yaping Dan Sujit Datta Charlie Johnson Nick Kybert Zhengtang Luo Thomas Ly Luke Somers Doug Strachan

Charlie Kane Seungchul Kim Gene Mele Andy Rappe Natalya Zimbovskaya

Topics for today: History: How this work began I.

Surface potentials and layer charge distributions for FLG/SiO2 (Datta et al. Nanoletters 9, 7 (2009))

II.

Nanoparticle size-selection for gold on FLG’s (Luo et al., Nanoletters 10, 777 (2010))

III. Nanoparticle shape-selection on FLG’s (Somers et al. Phys. Rev. B 82, 115430 (2010)) IV. Interlayer physics in epitaxial twisted FLG’s (GM, Phys. Rev. B 81, 161405 (2010))

Some History: Spring 2007: Sujit Datta (Penn junior) is analyzing his undergraduate project: exfoliating graphenes on SiO2 and measuring their surface potentials.

Fall 1978: Michael Rice and collaborators publish their theory of layer charge distributions graphite-acceptor intercalation compounds

Phys. Rev. Lett. 41, 763 (1978)

New physics in the FLG problem: N ~ 20 (closer to continuum limit) New boundary conditions from the free surface Surface is experimentally accessible (Datta) Size selection for condensed species (Luo) Shape selection (Somers)

How this work began I.

Surface potentials and layer charge distributions for FLG/SiO2 (Datta et al. Nanoletters 9, 7 (2009))

II.

Nanoparticle size-selection for gold on FLG’s (Luo et al., Nanoletters 10, 777 (2010))

III. Nanoparticle shape-selection on FLG’s (Somers et al. Phys. Rev. B 82, 115430 (2010))

Height and Surface Potential of Few Layer Graphene

Power Law Decay in Surface Potential

 2 2 Ec  2m* D 2

?

Surface Potential and Layer Charge Profile

2  Ki  vF  i3/ 2 3

Analytic solution in the limit of weak interlayer tunneling D

D

dz 3/ 2 2 2 dz dz      ( z )   ( z )  2 e z o ( z )    e   ( z ) z  z '  ( z ') a a a ' 0 0 Nonlinear stiffness

External field

Interaction

Minimization of  is constrained by a conservation law f   ( z )sgn( ( z )) then 2 d  1  df  2 3  f 0      dz  2  dz  3a 

Classical particle sliding on an “inverted cubic” hill

Useful mechanical analogy

tz x f   initial speed  charge transfer comes to stop  compensate donated charge

Useful mechanical analogy

final position rD  initial position determines shape of layer charge distribution

Solution for layer charge distributions weak coupling

rD  1

rD  0.1

strong coupling

rD depends on a single dimensionless parameter  1/ 3 2 3    0D    2  3 d  

2 2  e   vF

Main Result: Surface Potential for Strong Coupling

 D  3 d 0 



 3 d 0



1  rD

1/ 3



1/ 3



3 1/ 3 D

1  r  

3d  D 2

Interface-specific dipole layer with a universal thickness dependence determined by the layer compressibility

Comparison with experiment

How this work began I.

Surface potentials and layer charge distributions for FLG/SiO2 (Datta et al. Nanoletters 9, 7 (2009))

II.

Nanoparticle size-selection for gold on FLG’s (Luo et al., Nanoletters 10, 777 (2010))

III. Nanoparticle shape-selection on FLG’s (Somers et al. Phys. Rev. B 82, 115430 (2010))

Growth of epitaxial gold on graphene HRSEM thick (graphitic): faceted islands thin (FLG): size-limited droplets

mean radius depends on layer count

Images of epitaxial gold on FLG’s

from Yaping Dan (Johnson’s Lab)

and from Zhou et al. JACS 132, 945 (2010)

Droplet nucleation and growth

unhappy dipoles

growth limited by repulsive interaction increasing faster than N

Size-scaling of the dipolar energy areal dipole density,  dipole depth, s work function mismatch, 

  4 e 2 s

   f (r ) f ( r ') 2 2 Ud  d r d r'   3 2 2  32 e r r ' 2

Dipole depth s regularizes short distance 1/r3 singularity

Regularize the near-field interaction

Near-field cutoff is the FLG thickness: for a circular drop of radius R

2 Ud  8vF Area-squared scaling rule

4 1 R 7/2   6 de D

R4 Ud   for m-layer FLG m

Droplet nucleation and growth (amended)

2D: U  U area  U edge  U d R4  aR  bR   m 2

Gives a size dependence to the addition energy: 2 U 1 U  2  R   *   N 2 nRh R R m h

which suppresses small droplets & limits growth of large droplets

Key result: m1/3 scaling of the droplet radius

Fit to diameter = Am p Expt: A = 6.46 ± 0.68 nm & p = 0.33 ± 0.06 Theory: A = 5.9 nm & p =1/3

How this work began I.

Surface potentials and layer charge distributions for FLG/SiO2 (Datta et al. Nanoletters 9, 7 (2009))

II.

Nanoparticle size-selection for gold on FLG’s (Luo et al., Nanoletters 10, 777 (2010))

III. Nanoparticle shape-selection on FLG’s (Somers et al. Phys. Rev. B 82, 115430 (2010))

Island patterns for other metals (Somers, 2010) Gold: “droplets”

Ytterbium: “fibrils”

Ferrofluid patterns: 2D colloidal suspensions of magnetic particles in a perpendicular field

FF Theory: Sign reversal of line tension from dipolar repulsion S. Langer et al. Phys. Rev A 46, 4894 (1992) M. Iwamoto et al. PRL 93, 206101 (2004)

Interaction-renormalized SHAPE energy (L.A. Somers et al. PRB 82, 115430 (2010))

Partition U  U N  U shape where

U shape N

0

3  1 2    2 r 1    U edge  2 h  r10   m 01 rm     1    m   2 m 2 r 0  m  2  d      

size: q  0 potential shape: q  0 asymptotics

Growth and shape instability of a metal island

fibril width quenched at w = 2rc

 2  8 e2 h 2

1/ 3

rc (m)  2(m  1)    d   

Width distribution for Yb/Gr Yb/Gr:   2.2 eV

  320 erg/cm2 h  1 nm

  0.067 diam  2rc (m  2)  5 nm

expt: diam(m  2)  4.01  0.08 nm

Combined Size and Shape Scaling

Yb

Au

Summary I.

Surface Potential FLG/SiO2: nonlinear interlayer screening

II. Size Selection for Au/FLG: limited by long range electrostatics

III. Shape Instability for Yb/FLG: shape selection by dipolar energy

Collaborators:

Support

Yaping Dan, Sujit Datta, Charlie Johnson Charlie Kane, Seungchul Kim, Nick Kybert Zhengtang Luo, Thomas Ly, Andy Rappe, Luke Somers, Doug Strachan, Natalya Zimbovskaya

with a sharp crossover from weak to strong coupling nearly uniform screening charge

  z  4 rD  2 

1 (1  z / 2 D) 4

collapses to bounding layer