Tight-binding molecular dynamics study of mechanical and electronic properties in twisted graphene nanoribbons

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Tight-binding molecular dynamics study of
mechanical and electronic properties in twisted
graphene nanoribbons

• Satofumi Souma, Shozo Kaino, and Matsuto Ogawa
• Department of Electrical and Electronic
  Engineering,
• Kobe University, Japan

SISPAD 2012 Denber, Colorado, USA, Sep. 6, 2012
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Outline

• Introduction
• Calculation method
• Numerical results
• Summary

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Introduction

• Recent rapid progress in nanostructure
  electronics based on
• carbon based nano-scale materials, such as CNT,
  Graphene.
• --- electronic property is sensitive to the
  geometrical structure.
• It is essentially important to know the relaxed
  geometrical structure for reliable simulations.

First principles calculation (accurate, but
time-consuming), Tight-binding molecular
dynamics method. Classical valence force field
method (Tersoff, Keating, etc..) (suitable for
large-scale simulation)
Efficient algorithm is required to perform the
structural relaxation calculation for large scale
system.
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Purpose of this study

• Graphene
• High electron mobility and thermal conductivity
• However No bandgap at Fermi energy

Essentially required to open the bandgap in
graphene for switching device application.
example
Graphene nanoribbon(GNR) (especially AGNR)
Applying electric field to bi-layer graphene
Our question
In-plane/out-of-plane strain
Can we further control the electronic state of
AGNR (especially bandgap) by applying various
types of strain? (Stretching/Twisting)

• Electronic structure is sensitive to the atomic
  size geometry
• Essentially important to have the relaxed
  geometry under the strain
• We use Tight-Binding Molecular Dynamics (TBMD) )
  C. H. Xu, et al, J. Phys. Condens. Matter . 4,
  28 (1992).

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Bandgap in AGNR
In this study, we choose 722, 922, 1122 AGNR
to analyze the effect of streatching/twisting
11x22
9x22
7x22
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Flowchart of structural relaxation calc.based on
TBMD
Setup the TB Hamiltonian for initial atomic
positions
Total energy (band structure energy repulsion
energy ) Etot(EbsErep)
Force Ftot(tn) acting on atoms
Atomic positions ri(tn1) are updated by using
velocity Verlet method.
Force Fi(tn1) and velocity vi(tn1) are updated
Velocity Verlet method
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Conventional diagonalization method
In TBMD method, bandstructure energy Ebs is a sum
of occupied energy eigenvalue for H
In finite temperature
In conventional method, ei iis calculated by
diagonalizing the Hamiltonian matrix H
We employ the Fermi operator expansion method for
the calculation of Ebs
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Fermi Operator Expansion method
In FOE method, the bandstructure energy Ebs is
calculated as
requires the matrix multiplication of H
(Hamiltonian) and Fermi operator
Fermi distribution func.
In FOE, the Fermi distribution function is
expressed by Chebyshev polynomial expansion p(H),
which enables us to obtain the matrix form of
Fermi func.
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Chebyshev polynomial
? related to the computational accuracy
In Chebyshev expansion, the polynomial p(H)
satisfies the following equation
Matrix multiprication HTj(H)
For 1 column of Tj1(H) nn, which has to be
repeated for n-rows n3
By using the sparse nature of H cut-off radius
for matrix element, n3 is reduced to n.
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Comparison between diagonalization and FOE
Typical example AGNR with 14 atoms in 1 unit
cell. Number of unit cell is increased.
Estimation of the time required to obtain the
relaxed structure.
Significant reduction of calculation time
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Result 1 Stretched AGNR
Fracture limit
SR dependence of total energy
SR dendence of the bandgap
Stretching ratio
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Stretched AGNR
?SR dependence of the bandgap in stretched AGNR
Energy dependence of the transmission for
various SR Transport gap and electrode bandgap
can be different
Electrode band structure
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Result 2 Twisted AGNR
Twisting angle ? is assumed to be controlled
while the ribbon length L is fixed.
L
Twisting causes the stretching especially near
the edge of ribbon
It is interesting to compare twisting and
stretching to understand the effect of twisting
on the property of AGNR.
Twisting angle ? ? Stretching ratio SR ?
L
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Fracture limit of twisted AGNR
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How does the twisted AGNR look like?
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ParametertW
?rad ? tW ? SR
Pekka Koskinen, 2011, April. Phys. Lett. 99,
013105
This relationship is determined so that the
strain energy is comparable in stretched and
twisted cases

1. 922 AGNR 900(tW 1.62), Tube-like
2. 722 AGNR 900(tW 1.21), Ribbon-like
3. 722 AGNR 1210(tW 1.62) , Tube-like

Twisted ribbons with the same tW look similarly
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Fracture limit of twisted AGNR
Fracture limit is roughly proportional to the
aspect ratio

• GNR can be twisted more when the length L is much
  longer than the width W (consistent with
  intuition).

Fracture limit (tW) is constant (2.6 rad) for
L/Wgt6 (Lgt6W)
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Twisting and Stretching 1
At tW1.551.70 twisted AGNR changes its relaxed
geometry from ribbon like to tube like, by
reducing the total energy.
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Twisting and Stretching 2
Twisting and twisting agree well until tW1.55 ,
and after that they start to deviate
Geometrical structure changes from ribbon-like to
tube-like, and then the large curvature
influences the electronic band structure.
722 AGNR
922 AGNR
1122 AGNR
???
stretching twisting
tW1.55
tW1.55
tW1.62
?????
tW1.62
tW1.62
tW1.70
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Summary
Summary

• We employed order-N FOE method for TBMD
  structural relaxation calculation along with the
  parallel calculation, and succeeded to reduce the
  computation time significantly.
• In Twisted AGNR, the fracture limit is determined
  by the aspect ratio.
• Bandgap of AGNR can be controlled by stretching
  and twisting.
• Twisted AGNR changes its relaxed structure from
  ribbon-like to tube-like abruptly at around a
  critical curvature, then the electronic property
  also changes.

Future plan

• More care study on the effect of
  ribbon-like-to-tube-like transition to the
  electronic property.
• Similar study for zigzag edged GNR.

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damping factor

• ?????????????????????????????????????
• ? Chebyshev??????Jackson damping
  factor?????????TBMD??????time step????????????

M75
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????-FOE???
C2???????????????a0????????? ??????Etot??? a01.41
3Å???????????Ebs???
???? FOE O(N)
-22.113 eV -22.111 eV
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Eband???
EF
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Etot???
????????????????Etot?????????
EtotEbsErep
Ebs ???????, ??????????????????????????????
Erep ??????????????????????????
Ebs, Erep??2???????????
C2?? ?????????
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TBMD????????
Etot before -6.910828 eV/atom Etot after
-7.079077 eV/atom
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??????????

• ??????????????N?3???????????
• Ebs?????????????N?3????????????????????????????

Ebs?????????
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????N?
????N??? ?????????????????????N?3??????????????
???N?1??????(????N)???????????????
?????????N???? Fermi operator expansion(FOE)??????
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????????????
e?H???????
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Step1. ?????????????
TB?????????H?????Scaling??s(rij)??????????????????
????????????
??????????0?????????(???)??????
????
?
1
3
?
1
5
????????????
??????????????????????????? ????????(noff n2,
noff ?????????????)
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Step2. cutoff??rloc???
???????????????cutoff????
???????a1-ngt rloc???(H?n?)(Tj?1?)??????????? ?Tj
1????nloc??????
nloc????????????????? ?????????????????????????nof
f nlocn?????N???