лекция 5 graphen

Графен,-‐
материал
будущего
или

поиск
ниши
для
применения

Graphene

1.
Обзор
методов
вычислительной
физики

Много-‐масштабное
моделирование:
от
дефектов
к

ошибкам
в
приборах

2.

Локальная
структура
металлических
сплавов:

диффузионное
рассеяние
и
атомные
смещения.

3.
Дефекты
в
полупроводниках
и
поведение
приборов:
GaN,

SiC
и
AlSb.

4.
Проблемы
функциональности
материалов
для

мемристора
TiO2
и
ZnO.

5.
Графен,-‐
материал
будущего
или
поиск
ниши
для

применения.

Saito, R., M. lower ͑ Dresselhaus, is M. S. from Eq.
minus sign the Fujita, G. ␲͒ band. Itand clear Dresselhaus, ͑6͒
that Tight
bspectrum is Lett. 60, 2204. around zero energy if tЈ
lectronic propertiespproximaIon

the inding
a of graphene
1992a, Appl. Phys. symmetric
Saito,finite values graphite”
by
Wallace
Phys.
Rev.
LeT.
71,
622,
1947
is
= 0. For

“The
band
theory
of

R., M. Fujita, of tЈ, the electron-hole symmetry

G. Dresselhaus, and M. S. Dresselhaus,
broken and theRev.and ␲1804.
, 1992b, Phys.
␲ B 46, * bands become 1asymmetric. In
San-Jose, P., E. Prada, and D. Golubev,k2007,bPhys. Rev. B 76,
-
Fig. 3, we show theAfullBband structure of graphene with y
195445.
,
both t and S.,. 2007, Phys. Rev.figure, we also show a zoom in
Saremi, tЈ δ 3 the1 same B 76, 184430.
In δ K
,
of the band D., E. H. Hwang, and W. K.ofΓ the Dirac points ͑at
Sarma, S. structure close to one Tse, 2007, Phys. Rev. B
s
the K or KЈ a 1
75, 121406. pointδ 2in the BZ͒. This dispersion can be M kx
c Schakel, A. M.2J., 1991, the full band structure, Eq. ͑6͒,
obtained by expanding Phys. Rev. D 43, 1428.
a
K’
t
close to the K ͑orGeim, vector, Eq. ͑3͒, as kE. H. + q, P.
Schedin, F., A. K. KЈ͒ S. V. Morozov, D. Jiang, = K Hill, with
b2
a Blake, and K. S. Novoselov, 2007, Nature Mater. 6, 652.
͉q ͉ Ӷ ͉K͉ ͑Wallace, 1947͒,
n Schomerus, H., 2007, Phys. Rev. B 76, 045433.
- Schroeder,͑ColorM. + O͓͑q/K͒2͔, and A. Javan, 1968, Phys. ͑7͒
FIG. 2. online͒ Honeycomb lattice and its Brillouin
E±͑q͒ Ϸ P. R., ͉q͉ S. Dresselhaus,
± vF
; zone.Lett. 20, 1292.structure of graphene, made out of two in-
Rev. Left: lattice
Semenoff, G. momentumRev. ͑a1 53, a2 are
where q is theW.,triangular latticesLett. and 2449. the latticethe
terpenetrating 1984, Phys. measured relatively to unit
- Sengupta, and ␦i G. 1 , 2the are the nearest-neighbor by vF
vectors, and
Dirac pointsK., and, viF=Baskaran, 2008, Phys. Rev. B given vectors͒.
is , 3 Fermi velocity, 77, 045417.
Seoanez, C., a value vandf
1raphene”
The This result are lo-
“The
electronic
properIes
o A. H. Castro Neto, 2007, Phys.
F. Guinea, Ӎ g ϫ 106 m / s. Dirac cones was
Right: corresponding Brillouin zone.
= 3ta / 2,.
Castro
Neto
Rev.
Mod.
Phys.
81,
109
2009

- A.
H with 125427. KЈ F
catedB 76, K and
Rev. at the points.
first obtained by Wallace ͑1947͒.
-

form Left: energy spectrum ͑in units of t͒ for finite values of
lattice. ͑Wallace, 1947͒
t and tЈ, with t = 2.7 eV and tЈ = −0.2t. Right: zoom in of the
E±͑k͒ = ± tͱ3 + f͑k͒ − tЈf͑k͒,
energy bands close to one of the Dirac points.

1
f͑k͒ = 2 of tЈ is not 4 cos ͩ ͪ ͩ ͪ
ͱ3
ky cos kxa
3
The valuecos͑ͱ3kya͒ + well knownabut ab initio , calcula
͑6͒
͑Reich et al., 2002͒ find 0.02t Շ tЈ2 0.2t depending on the t
Շ 2
binding parametrization. These the upperet͑al.:alsoelectronic pro
where the plus sign applies to calculations␲*Theand the
Castro Neto
͒ include
effect of a third-nearest-neighbors hopping, which has a v
minus sign the lower ͑␲͒ band. It is clear from Eq. ͑6͒
of around 0.07 eV. A tight-binding fit to cyclotron reson
that the spectrum is symmetric around zero energy if tЈ
experiments ͑Deacon et al., 2007͒ finds tЈ Ϸ 0.1 eV.
= 0. For finite values of tЈ, the electron-hole symmetry is
broken and the ␲ and ␲* bands become asymmetric. In
Fig. 3, we show the full band structure of graphene with
both t and tЈ. In the same figure, we also show a zoom in
of the band structure close to one of the Dirac points ͑at
the K or KЈ point in the BZ͒. This dispersion can be

anប = 1͒ ͚ e−ik·Rna͑k͒,
͑we use units such that = ͱN k
͑15͒
c

H=−t ͚is
by A = 3ͱ3a / 2.͗i,j͘,␴
2
It
†
͑a␴,ib␴,j + H.c.͒
where Nc is the number of unit cells.
c
tates for graphene is mation, we write the field an as a
. Dirac fermions
͚ † †
coming+ b␴,ib␴,j + H.c.͒, Fourier s
of carbon nanotubes ͑a␴,ia␴,j
− tЈ from expanding the
er shows 1 / ͱE singu- ␴ K. This produces an Јapproximation
͗͗i,j͘͘,
We consider the Hamiltonian ͑5͒ withas a sum of two ne
tion of the field an t = 0 and the
their electronic spec- the electron operators,
ourier transform of ͒ annihilates ͑creates͒ an electron
where the ͑ai,␴ †
antization of ai,␴ mo-
ular spin ␴ tube axis. ͒ on site Ӎ i −iK·RsublatticeЈ·Rna ͑an equ
to the ͑␴ = ↑ , ↓ an e
R on na + e−iK A ,
1,n 2,n
nanoribbons, whichis used for sublattice B͒, t͑Ϸ2.8 eV͒ i
lent definition
1
= ͚ e−ik·Rna͑k͒,
anearest-neighbor hopping energy ͑hopping between
ͱN c k
perpendicular to the
n ͑16͒
milar ferent sublattices͒, and t is the next −iKЈ·Rn
to carbon nano-
bn Ӎ e−iK·Rnb1,n + e nearest-neig
Ј b2,n ,
hopping energy1 ͑hopping in the same sublattice͒.
where Nc is the number of unit cells. Using this transfor-
energy bands derived from this Hamiltonian have
mation, we write the field an as a sum of two terms,
2009
form ͑Wallace, 1947͒

͒/4 †
ͪ ͵ͩ
͑ai ,‫ץ‬b†͒
+
xi
ͫͩ
0 − ͱ3͒/4
3a͑− i guage, the two-component
͑i = 1 , 2͒. It is clear that ‫ץ‬around K ͱhas the fo
mentum y ⌿1͑r͒
ˆ
the effective Ham
ͪ
−ˆ3a͑i − ͱ3͒/4 3a͑1 − iͱclose to the K3a͑− i − 3͒/4 obeys
point, ˆ
ͩ
ͪͬ ͪͬ
ͩͪ ͬ ͪ
0 0 ‫+ץ‬
3͒/4 0
oniant ͑18͒ is madeͱ3͒/4 two copies3a͑i − ͱthe1massless Di
of 0 of 3͒/4
ͬ
dxdy⌿†͑r͒

ͩ ͪ ͪͩ
HӍ− ‫ ץ‬⌿ ͑r͒
0 −i␪k/2
e
1 x y 1
− 3a͑1 + i −
ke Hamiltonian, 3a͑1− iͱ3͒/4 ‫␺ + ץ‬forF␴ 3a͑i −ͱ 3͒/4 ‫= ץ‬⌿␪k/2 and
0
ˆ ͑r͒ †
ͫͩ
0 one holding−0iv ͑k͒ ·=ٌ␺͑r͒ iÊ␺͑r͒.
3a͑i + ͱ 3͒/4
ther fori −−ͱaround K0Ј. Note The wave function, in m
+⌿
− 3a͑− p 3a͑1 − i
2
3͒/4 ͱ3͒/4
0
ˆ ±,K p around K
y ⌿ ͑r͒
‫3 −ץ‬a͑− i2− ͱ3͒/4
x
ͱ
2 ±e ͑r͒
that, in first quantized
0
y 2

= − i ͵ dxdy͓⌿ ͑r͒␴ · ٌ⌿ ͑r͒ + ⌿ ͑r͒␴ · ٌ⌿ for H =
uage,v the two-componentmentum vFwave where the ␺
ˆ ˆ † ˆ
electron ␴ · k, function
ˆ ͑r͔͒,
†
K around K has the
F 1 * 2
1 2

r͔͒, ͑18͒

ͩ ͪ
lose to the K point, obeyseigenenergies E = ± vF␪k, that
the 2D Dirac equation,
respectively, and 1 ␪k −i kgiven
e is /2
͑k͒ momentum/2arou
␺±,Kthe =
− iv ␴ · ٌ␺͑r͒ = E␺͑r͒. tion for
F
with Pauli matrices ␴ = ͑␴x , ␴y͒, ␴* = ͑␴x , −␴y͒, and ˆ
⌿† 2 ±e
i␪k
ˆ 1
͑
p ͱ
ͩ ͪ
i h= ␴· .
= ͑a† , b†͒ ͑i = 1 , 2͒. It is clear that the effective Hamil-
i i 2 ͉p͉
The wave function, in momentum ␴ · k, wherethe m
onian ͑18͒ is made of two copies of the massless Dirac- space, for/2 the
1 e i␪k
for HK =͑k͒ = definition of h that the
ike Hamiltonian, one holding for p around K and the␺±,KЈ vF the
ͱ
It is clear from ˆ
mentum around K has theeigenenergies E =±evof␪k, tha
⌿†ˆ p form ␺ Ј͑r͒ are also eigenstates−i hk,/2
other for p around KЈ. Note that, in first quantized lan-
2 ± Fˆ
ˆ = 1 ␴ · 2D Dirac equation,
and

ͩ ͪ
guage, the two-component electron wave function ␺͑r͒,
i K
h . ͑22͒
lose to the K point, obeys the
mil- 2 ͉p͉−i␪k/2 respectively, ␺ ͑r͒, ␪k is given
and
− iv ␴ · ٌ␺͑r͒ = E␺͑r͒. 1 e for
͑19͒ H KЈ
h␺= v= ␴ * · k. Note that t
ˆ ͑r͒ ±
F K
1
K
tion for the momentum aro
2
ac- ␺ ͑k͒ =
F
͑
ͱ2 the definitionTherefore,that the statesЈ a/2positive
mentum around K has the form ±e
he
It is clear from
K and related by time-rever
The wave function, in momentum space, /2 the mo-Ј arean equivalent equation for ␺ ͑r͒ with in
±,K i␪k for
ˆ electrons ͑holes͒ i␪ ␺ ͑r͒
of hcoordinateshavekmom
K

an- ␪ −i k/2
origin of Equation ͑23͒ impliesein ␴ has its
helicity. 1 that
K
1 e

he hopping energies between different sites are m
͑5͒,
ed, leading to a new term to the original Hamilto
5͒, H = ͕␦t͑ab͒͑a†b + H.c.͒ + ␦t͑aa͒͑a†a + b†b ͖͒,
od ͚ ij i j
i,j
ij i j i j

Hod = ͚ ͑ab͒ †
͕␦tij ͑ai bj + H.c.͒ + ͑aa͒ †
␦tij ͑ai aj + †
bi bj͖͒,
͑144
i,j
or in Fourier space, ͑1

͚ ͚
† ជ
͑ab͒ i͑k−kЈ͒·Ri−i␦aa·kЈ
r Hod =
in Fourier space,
a kb kЈ ␦ti e + H.c.
k,kЈ ជ
i,␦ab

͚ ͚
† ជ
͑ab͒ i͑k−kЈ͒·Ri−i␦aa·kЈ ជ
Hod = a†kbkЈ † ␦ti e ͑aa͒ i͑k−kЈ͒·Ri−i␦ab·kЈ + H.c.
+ ͑akakЈ + ជ kbkЈ͒
k,kЈ
b
i,␦ab ជ
͚
␦ti e , ͑145
i,␦aa

͚
† ͑aa͒ † ជ
͑aa͒ i͑k−kЈ͒·Ri−i␦ab·kЈ
͑ab͒͑a a k + b b k ͒ , ͑
where ␦tij+ ͑␦tij Ј ͒ is k Ј changei ofethe hopping energ
k the ␦t
ជ

real space as

͵
A͑r͒ = Ax͑r͒ + iAy͑r͒.
d2r͕A͑r͒a†͑r͒b ͑r͒ + y
d can
= Dirac Hamiltonian1͑18͒, we isorder

In terms of D
Two

Eq. ͑146͒ as
1
͑149͒
Hodthe irac
cones
are
not
coupled
bH.c. rewrite

+ ␾͑r͓͒a†͑r͒a1͑r͒ + b†͑r͒b1͑r͔͖͒,
͵similar expression for cone 2 but with A replace
1 1 ͑14
Hod = ˆ ជ ˆ ˆ ˆ
d2r͓⌿†͑r͒␴ · A͑r͒⌿1͑r͒ + ␾͑r͒⌿†͑r͒⌿1͑r͔͒,
1 1
with a
by A*, where ͑150͒
ជ
where A = ͑Ax , Ay͒. This result shows that changes in the
͚ ͑ab͒ ជ
−i␦ab·K
A͑r͒ = ␦t ͑r͒e , ជ ͑14
hopping amplitude lead to the appearance of vector A
␦abជ
and scalar ⌽ potentials in the Dirac Hamiltonian. The
presence of a vector potential in the problem indicates

also be present, ជ
͚ ͑aa͒
␾͑r͒ = ␦t ͑r͒e −i ជ B =
ជ
.
ជ
that an effective magnetic ﬁeld␦aa·K͑c / evF͒ ٌ ϫ A should
naively implying a broken time-reversal
͑14
␦aa
symmetry, although the original problem was time-
reversal invariant. This broken time-reversal symmetry
is not real since Eq. ͑150͒ ͑r͒ the *͑r͒, because of the inversio
Note that whereas ␾ is = ␾ Hamiltonian around

Defects
in
graphene

1.  Grain
Boundary.

2.  Liquid
environment
enhancement
on
mobility
in
graphene.

3.  X-‐ray
irradiaIon
of
graphene.

Experimental
observaIon
of

defects
in
graphene

Vacancy
Extended
defect
=
Metallic
wire

Meyer,
Kisielowski,
Erni,
Rossell,
Crommie,
ZeTl,

Nano
Le..
(2008)

J.
Lahiri
et
al.,
Nature
Nanotech.

(2010)

Grain
Boundary
and

Point
Defects

What
is
Ime
scale
and
range
of
interacIon
between
defects
and
GB?

~1
nm

ComputaIonal
Method

•  Quantum
Molecular
Dynamics

Ø  Density
FuncIonal
Theory
(DFT)

Ø  Natoms
~
300

•  Classical
Molecular
Dynamics
(CMD)

Ø  adapIve
intermolecular
reacIve
bond-‐order
(AIREBO)
potenIal.

Ø  Isothermal-‐isobaric
(NPT)
ensemble;
Natoms
~1000;

FormaIon
Energies
of
Defects

pris2ne

Cl-‐5577

GB-‐558

GB-‐575

All
energies

rim

inside

rim

inside

rim

inside

are
in
eV

C

6.1

3.8

4.2

3.1

SV

7.9

5.6

3.0

7.1

6.4

6.9

4.2

LocaIon
of
Vacancies
with

lower
formaIon
energies

InteracIon
of
vacancies
in
graphene

Stability
of
585
and
555777
defects

Banhart,
Kotakoski,
Krasheninnikov,
ACS
Nano
2010
555777
is
1.2
eV
more
stable

Cretu,
Krasheninnikov
et
al.
PRL
2010

Lee
et
al.
PRL,
2005;

Dynamical
InteracIon
of
Vacancies

Strain
up
to
4%

T
=
2000
K

trecombinaIon
~
80
ps

Vacancy

interacIon
with
grain
boundaries

Vacancy-‐GB
interacIon

~1
nm

Vacancy
merges
with
GB

Further
GB
reconstrucIon

8
160
ps
5
5

8
40
ps
7
7
5
5

Defect
Dynamics:
Vacancy
near
GB

1
nm
T
=
3000
K

Strain
up
to
8%

treconstruct
~
200
ps
Reconstrucoon
relieves
strain

Vacancy
and
adatom
recombinaIon
near
GB

Vacancy
and
adatom
recombinaIon
near
GB

•  Adatoms
are
very
mobile
–
low
diﬀusion
barrier

•  Stretched
C-‐C
at
the
heptagon
accumulate
adatoms

T
=
2000
K

treconﬁguraoon
=
0.5
ns

B. Wang, Y. Puzyrev, S. T. Pantelides, Carbon (2011)

Conclusion

•  Vacancies
interact
and
recombine
t
~
10
ns

•  Point
defects
interact
with
grain
boundaries
d
~
2
nm

•  Grain
boundaries
act
as
sinks
for
vacancies
and
adatoms

Enhanced
defect
reacovity
at
grain
boundaries

Graphene
device
degradaIon

•  Graphene
fabricated
by
mechanical

exfoliaIon
from
Kish
graphite

•  Sweep
VG
with
VDS=5mV

MoIvaIon
and
Outline

Ø Experiment
[1]

o  Graphene’s
resisIvity
response
to
x-‐ray
radiaIon,

ozone
exposure,
annealing.

o  Defect
related
Raman
D-‐peak
appears
a…er

§  x-‐ray
irradiaIon
in
air

§  ozone
exposure,
decreases
a…er
annealing.

Ø Theory:
behavior
of
impuriIes
on
graphene

o  Temperature
and
concentraIon
dependence.

o  Need
to
remove
oxygen
without
vacancy
formaIon
(would
H
help?)

[1]
E.-‐X.
Zhang
et
al,
IEEE
Trans.
Nucl.
Sci.
58,
2961
(2011)

Graphene
device
degradaIon

Two-‐probe
resistances
measured
on

•  10
keV
irradiated
graphene

•  prisIne
graphene

•  ozone
exposed
graphene
(1
min)

•  annealed

(300C
for
2
hrs
in
200
sccm
Ar)

Graphene
device
degradaIon

Ozone
exposure

a)
80

8000

G -‐P e a k 60
Inte g ra te d
inte ns ity
A re a

6000

I D /I G
(1 0 0 % )

40 Defect
related
D-‐peak

4000

20 •  increases
x-‐ray
exposure

2000
D -‐P e a k •  decreases
a…er
temperature
anneal

0 0
P re 8
Mra d(S iO 2 ) 15
Mra d(S iO 2 ) A nne a l
1 0 -‐ke V
X -‐ra y
D os e
b)

TheoreIcal
Approach

O
O
desorpIon
Density
Func2onal
Theory

O
migraIon
DFT

•  Defect
formaoon
energies

•  Migraoon/desorpoon
barriers

O
dimer

Kine2c
Monte-‐Carlo

KMC

Defect
dynamics

•  Temperature

•  Inioal
concentraoon

Oxygen
Removal
and
Vacancy
GeneraIon

1.3
eV
Oxygen:
clustering
behavior

0.5
eV

0.8
eV
Removal
of
oxygen

Bridge 1.3
eV
•  Pairs

O2

•  Triplets

CO,
CO2,
VC

Top

Device
degradaoon

1.1
eV
CO,
CO2
1.1
eV
O2

High-‐temperature
Annealing

Vacancy

Concentraoon
of
vacancies
exceeds

Residual
oxygen
atom
concentraoon
of
residual
O

High
vs
Low
Temperature
Anneal

T,
oC

T

Temperature
Anneal

IniIal
Defect
ConcentraIon
Dependence

High
O
concentraoon

Lo

vacancy

surface
coverage

Low
O,
High
V
concentraoon

oxygen

T

iniIal
O
surface
coverage

High
T:
Removal
of
oxygen
>
0.05
iniIal
surface
coverage
leads
to
vacancy
formaIon

Low
T:
Oxygen
stays
on
the
surface
and
forms
clusters

Decrease
of
D-‐peak,
Increase
in
resisovity

Method
to
prevent
defect
forma2on
during
irradia2on/annealing?

Oxygen
and
Hydrogen
on
Graphene:

Binding
energies,
MigraIon
and
ReacIon
Barriers

O-‐H
is
most
likely
to
desorb

O
from
graphene
surface

H

Leaves
carbon
network
intact

Eﬀect
of
Hydrogen
On

Oxygen
Annealing

Oxygen/ Low
High

Hydrogen

Concentra2ons

Low
2%
O,
10%
H

@
T
=
300
C

Final
defect

concentraIons?

High
15%
O,
1%
H
15%
O,
10%
H

Eﬀect
of
Hydrogen
On
Oxygen
Annealing

Higher
Oxygen
concentraoon
Higher
Hydrogen
concentraoon

Hydrogen
is
removed
t
~
0.001
s

Oxygen
is
removed
t
~
0.0001
s

t
~
1
s

t
~
1
s

Removal
of
residual
Oxygen

Residual
Hydrogen

Causes
formaoon
of
large
Forms
clusters

L
~
0.5
nm

amount
of
Vacancies
No
Vacancies
are
formed

High
O,
High
H
concentraIons

Hydrogen
is
removed
ﬁrst,

Removal
of
residual
Oxygen

Causes
formaoon
of
Vacancies

Eﬀect
of
Hydrogen
On
Oxygen
Annealing

ScaTering
mechanisms
in
graphene

•  Suspended
graphene
at
4K
μ
~200,000

cm2/V

[1]

•  Suspended
graphene
at
300K

μ
~10,000

cm2/V
s

ü  Out-‐of-‐plane
flexural
phonons
limit
[2]

•  Suspended
graphene
in
non-‐polar
liquid

μ
~60,000

cm2/V
s

•  Effect
of
liquids
on
the
flexural
phonons
Image
from
Meyer,
J.
C
.

ü  Vacuum

ü  Hexane
C6H14

ü  Toluene
C6H5CH3

1.  BoloIn,
K.
I.
et
al.
Solid
State
Comm.

2008

2.  Castro,
E.
V.
et
al.
Phys
Rev
LeT.
2010

Electron
scaTering
due
to
ﬂexural
ripples

𝐸↓𝑞 = 𝜅 𝑞↑4 ⟨|ℎ↓𝑞 |↑2 ⟩/2 = 𝑘↓𝐵 𝑇 /2 

Harmonic
approximaIon

Fourier
components
of

2 T
hq
 ~
bending
correlaIon
funcIon
κ q4

h
at
300K

Deformaoon
tensor

𝑢 ↓𝑖𝑗 =1/2 ( 𝜕 𝑢↓𝑖 /𝜕 𝑥↓𝑗  + 𝜕 𝑢↓𝑗 /𝜕 𝑥↓𝑖  + 𝜕ℎ/𝜕 𝑥↓𝑖   𝜕ℎ/𝜕 𝑥↓𝑗  )

Electron
scaTering
due
to
flexural
phonons

Hopping
integrals
γ
are
modified

2

γ=
γ↓0 +( 𝜕γ/𝜕 𝑢↓𝑖𝑗  ) 𝑢↓𝑖𝑗 

0
1

PotenIal
perturbaIon
due
to
ripples

-‐

random
sign-‐changing
‘magneIc
field’
3

1/𝜏 ≈2 𝜋/ℎ 𝑁( 𝐸↓𝐹 )〈 𝑉↓𝑞  𝑉↓− 𝑞 〉↓𝑞≈ 𝑘↓𝐹  
~⟨|ℎ↓𝑞 |↑2 ⟩↑2 
𝑉↑( 𝑥) =1/2 (2γ↓1 −γ↓2 −γ↓3 )
𝑉↑( 𝑦) =1/2 (γ↓2 −γ↓3 )

𝜌↓𝑟𝑖𝑝𝑝𝑙𝑒 ~1/𝜏 ~⟨|ℎ↓𝑞 |↑2 ⟩
↑2 

Effect
of
liquids

ü  Hexane
C6H14
Morozov
S.
V.
et.
al,
Phys.
Rev.
LeT
2006

ü  Toluene
C6H5CH3
M.
I.
Katsnelson
and
A.
K.
Geim,
Phil.
Trans.
R.
Soc.
A,

2008

Castro,
E.
V.
et.
al
Phys
Rev
LeT
(2010)

Molecular
dynamics
with
classical
potenIals

•  Large
system
10,000-‐50,000
atoms
L
~10nm

•  Large
Ime
scale
~ns

•  Bond-‐order
potenIals

for
C-‐H

•  Boundary
condiIons

ü  NPT
–
constant
pressure

ü  NVT
–
constant
volume,
corresponding
to
P~0

Strain-‐free
suspended
graphene

T
=
300
K

h

h2 = 0.89 Å2

vacuum

Suspended
graphene
in
hexane

Hexane
molecules
envelopes
graphene
sheet

C
chain
aligned
parallel
to
the
plane
Mean
square
displacement

h2 = 0.39 Å2

hexane

Suspended
graphene
in
toluene

Toluene
molecules
envelopes
graphene
sheet

C
ring
aligned
parallel
to
the
plane
Mean
square
displacement

h2 = 0.42 Å2

toluene

Preferred
molecule
posiIon:
DFT
calculaIon

ΔE
=
0.21
eV
ΔE
=
0.37
eV

3
Å

3
Å
Van
der
Waals
interacIon

Ripple
height
analysis

h

h2 = 0.89 Å2

vacuum

h2 = 0.42 Å2

toluene

h2 = 0.39 Å2

hexane

Bending
sIffness
of
graphene
in
liquid

2 TN 𝜌↓𝑟𝑖𝑝𝑝𝑙𝑒 ≈ℏ/4 𝑒↑2  ( 𝑘↓𝐵 𝑇/
hq
r = 𝜅𝑎 )↑2  𝛬/𝑛 

κ A0 q 4

Liquid
suppresses

Bending
S2ffness

flexural
phonons

Out-‐of-‐plane
flexural

phonons
limit
at
room
T

ü  Vacuum
μ
~10,000

cm2/V
s

ü  Liquid
μ
~
200,000

cm2/V
s

Conclusion

•  Liquid
dielectric
environment
suppresses
flexural
phonons

•  Phonon
suppression
affects
mobility
through
bending
sIffness

Электронная
плотность

Разложение
по
функциям
Гаусса

  " q %
   
   
ρ r =()
N atoms

∑ (
ρn r − Rn ) ( ) $ Q '
# A&
(
ρn r − Rn = $1− n ' ρ0A r − Rn )
n=1

Перенос
заряда

 M gauss
−γ m r 2
()
2
ρn r = ηr ∑ cme
m=1

  
∫
Ωcell
( )
ρn r − Rn d 3r = QA − qn
*

Полная
энергия

   
! $
Etotal = ∫ () ()
W r #ρ r & ρ r d 3r + ∫ W q !ρ q $ ρ q d 3q + Eion−ion
# & () ()
Ω
" % Ω
" %
volume volume

  
! $ ! $ ! $
W r #ρ r
" () &
%
= T #ρ r
" () &
% " ()
+Vex #ρ r &
%

  
! $
() ()
W q #ρ q & =V ps q +Vhartree q
" % () Vps ( q ) = S ( q ) w pseudo ( q )

Кинетическая энергия
corr corr
T = TWang −Teter + TLDA + Tatom

    5 
" % 45 2 5
()
TWang−Teter $ρ r ' =
# & 128 ( )
3π 2 3
∫∫ () ( ) ( )
ρ 6 r w1 r − r ' ρ 6 r ' d 3rd 3r '−

21 2 5  1 1  1 
−
250
( )
3π 2 3
∫ () () ()
ρ 3 r d r − ∫ ρ 2 r ∇ ρ 2 r d 3r
3

2
2

Теория линейного отклика

1 (q − 4)
2
5 ⎛ −1 3 2 3 ⎞ 2−q
w1 = ⎜ w ( q ) − q + , and w = +
⎟ ln
8 ⎝ 4 5 ⎠ 2 8q 2+q

 N grid
)6
+ n  -
+
corr ! $
"()
TLDA #ρ r &=
% ∑ *∑ cnΔρ ri .
i=1 + n=1
,
2
+
/
()
3
corr
6 "π % 2 " k2 %
()
Tatom k = ∑ cn $ ' exp $ −
$ξ '
# n&
$ 4ξ '
#
'
n=1 n&


 " k %  

()
S A ki = ∑ $1− α ' exp −ikα iRα
$ N '
α ∈A # α &
( )

λ=1 upper limit von Weizsäcker
λ=1/9 gradient expansion second order
λ=1/5 computational Hartree-Fock

1. Phase Diagram

2. Elastic Properties

3. Defect Formation Energies

Ширина
запрещенной
зоны

G0W0
GaN

лекция 5 graphen

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