Multilayer graphene sheets:Computational studies for

electron holography

C. Degli Esposti Boschi1, L. Ortolani1, J. Simoni2, V. Morandi1

1. CNR-IMM, Sezione di Bologna, I-40129, Italia2. Trinity College Dublin, Dublin 2, Ireland

References[1] T. Björkman et al., Phys. Rev. Lett. 108 (2012) 235502 [4] V. R. Cooper, Phys. Rev. B 81 (2010) 161104R[2] S. Grimme, J. Comp. Chem. 27 (2006) 1787 [5] J. M. B. Lopes dos Santos et al.,[3] K. Lee et al., Phys. Rev. B 82 (2010) 081101R Phys. Rev. Lett. 99 (2007) 256802

Many existing DFT functionals fail (or are successful by chance) for compounds held together by dispersion forces (aka van der Waals or non-bonding interactions). The issue is still of recent investigation (e.g. [1]), either

-Empirically by introducing classical effective forces depending only on the cartesian coordinates of the ions' center r

i (Grimme's approach [2])

-or fully at the DFT level with some suitable form of the exchange-correlation (XC) functional depending non-locally on n(r).

Electron Holography in Transmission Electron Microscopy can provide information that are related to the phase difference between one unscattered beam and one beam that has interacted with the sample

λ being the wavelength, U the accelerating potential and m0 the electron

mass corrected by the relativistic factor ε=eU/m0 c2 ≃0.2 @ 100 kV.

The average inner potential

is simply related to the sample's thickness in bulk materials; how does one compute it for nanometric objects? Density-functional theory (DFT) seems to be a promising tool since it is based on the direct evaluation of the electron density field n(r), with spatial resolution

With a cutoff energy of 100Ry one has already a resolution of 0.005nm, well below the current experimental values.

In a pseudopotential (PP) framework we need a good description of n(r) also inside core regions (~0.1nm) so we employ norm conserving forms (and generalized gradient approximation - GGA).

a. Motivation

Δϕ= πλ U

mm0

V (x , y)

U=U (1+ϵ/2) m=m0(1+ϵ)

V (x , y)=∫beam dir.dzV (x , y , z )

λmin∝1/√E cut

b. Modeling of dispersion forces

V ion({r i})→V ion({r i})+V Grimme({r i})

Rij=RiVDW+R j

VDW

(RCarbonVDW =0.145 nm)

r ij=∣r i−r j∣

Careful selection among the various available ones in the Quantum Espresso suite, in

order to avoid spurious effects on V(r) due to numerical noise in

vacuum regions.

Warning: PP choice

c. Preliminary validation

Once a good PP is found (C.pbe-mt_fhi.UPF in our case), we proceed to estimate the error on our calculations taking into account:

- (super)cell dimension along z (beam direction)- cutoff energy and accuracy due to the related real-space grid for integration

We choose 5nm for the former and 100Ry for the latter, and checked that the extrapolated data fall within a range of 0.035Vnm.

Both the Grimme scheme [2] and the “vdw-df2-c09” with Cooper's parametrization [3,4] yield good results for the interlayer distances as well as binding energies in AA or AB graphite. However the Grimme scheme does not include the density-dependent part of dispersive forces so it is not strictly suitable for our purpose.

δ V=

Equilibrium interlayer distances found for G multilayers

(Grimme scheme)

AB: 0.326 nm AA: 0.353 nmABA: [0.328,0.328] nm AAA: [0.352,0.351] nmABAB: [0.325,0.323,0.324] nm AAAA: [0.352,0.350,0.351] nmABABA: [0.325,0.323,0.324,0.325] nm AAAAA: [0.352,0.350,0.350,0.352] nm

ABABABAB: [0.324, ...0.323, …, 0.325] nm

ABABABABAB: [0.325,0.324,...0.323,...,0.324,0.323,0.325] nm

(vdw-df2-c09 scheme)

AB: 0.331 nmABA: [0.331,0.331] nmABAB: [0.330,0.328,0.330] nmABABA: [0.330,0.329,0.329,0.331] nm

Fit over 1-5 layers (volt nm)

u(volt nm)

AA (Grimme) -4.06(3) -2.887(7)

AB (Grimme) -4.11(13) -2.94(3)

AB (vdw-df2-c09)

Direct evaluation

-4.38(11)

-4.32 (singlegraphene)

-3.04(3)

3.2 (bulk graphite)

Turbo i=1 (Grimme) -4.06(4) -3.18(7)

Turbo i=1 (vdw-df2-c09) -4.32(4) -3.28(7)(the digit within parenthesis is the std error affecting the last digit reported outside)

Effect on

Layer decoupling... beyond ~ 1nm

e. Dependence on the number of layers

V=σ+u(N layers−1)

Pretty linear behaviour!

V

¡Connection

with TEM exp

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¡Connection

with TEM exp

eriments

in L. Ortola

ni's talk on

Friday!

σ and u depend weakly on the relative orientation

Additional orientation examined:Turbostratic index

i=1, angle = 21.79°, 28 atoms perunit cell [5] (with map of

integrated internal potential)

E bind=(ETOT−N layers E 1layer)

N atoms

Careful selection among the various available ones in the Quantum Espresso suite, in

order to avoid spurious effects on V(r) due to numerical noise in

vacuum regions.

d. Dependence on the layers distance

A

B

B

A