лекция 5 graphen

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Graphene

1. Обзор методов вычислительнои 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.

Tight binding approximaIon “The band theory of graphite” by Wallace Phys. Rev. LeT. 71, 622, 1947

“The electronic properIes of graphene” A. H. Castro Neto Rev. Mod. Phys. 81, 109 2009

trino” billiards !Berry and Modragon, 1987; Miao et al.,2007". It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots !Milton Pereira et al.,2007", leading to unusual Coulomb blockade effects!Geim and Novoselov, 2007" and perhaps to magneticphenomena such as the Kondo effect. The transportproperties of graphene allow for their use in a plethoraof applications ranging from single molecule detection!Schedin et al., 2007; Wehling et al., 2008" to spin injec-tion !Cho et al., 2007; Hill et al., 2007; Ohishi et al., 2007;Tombros et al., 2007".

Because of its unusual structural and electronic flex-ibility, graphene can be tailored chemically and/or struc-turally in many different ways: deposition of metal at-oms !Calandra and Mauri, 2007; Uchoa et al., 2008" ormolecules !Schedin et al., 2007; Leenaerts et al., 2008;Wehling et al., 2008" on top; intercalation #as done ingraphite intercalated compounds !Dresselhaus et al.,1983; Tanuma and Kamimura, 1985; Dresselhaus andDresselhaus, 2002"$; incorporation of nitrogen and/orboron in its structure !Martins et al., 2007; Peres,Klironomos, Tsai, et al., 2007" #in analogy with what hasbeen done in nanotubes !Stephan et al., 1994"$; and usingdifferent substrates that modify the electronic structure!Calizo et al., 2007; Giovannetti et al., 2007; Varchon etal., 2007; Zhou et al., 2007; Das et al., 2008; Faugeras etal., 2008". The control of graphene properties can beextended in new directions allowing for the creation ofgraphene-based systems with magnetic and supercon-ducting properties !Uchoa and Castro Neto, 2007" thatare unique in their 2D properties. Although thegraphene field is still in its infancy, the scientific andtechnological possibilities of this new material seem tobe unlimited. The understanding and control of this ma-terial’s properties can open doors for a new frontier inelectronics. As the current status of the experiment andpotential applications have recently been reviewed!Geim and Novoselov, 2007", in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.

II. ELEMENTARY ELECTRONIC PROPERTIES OFGRAPHENE

A. Single layer: Tight-binding approach

Graphene is made out of carbon atoms arranged inhexagonal structure, as shown in Fig. 2. The structurecan be seen as a triangular lattice with a basis of twoatoms per unit cell. The lattice vectors can be written as

a1 =a2

!3,%3", a2 =a2

!3,! %3" , !1"

where a&1.42 Å is the carbon-carbon distance. Thereciprocal-lattice vectors are given by

b1 =2!

3a!1,%3", b2 =

2!

3a!1,! %3" . !2"

Of particular importance for the physics of graphene arethe two points K and K! at the corners of the grapheneBrillouin zone !BZ". These are named Dirac points forreasons that will become clear later. Their positions inmomentum space are given by

K = '2!

3a,

2!

3%3a(, K! = '2!

3a,!

2!

3%3a( . !3"

The three nearest-neighbor vectors in real space aregiven by

!1 =a2

!1,%3" !2 =a2

!1,! %3" "3 = ! a!1,0" !4"

while the six second-nearest neighbors are located at"1!= ±a1, "2!= ±a2, "3!= ± !a2!a1".

The tight-binding Hamiltonian for electrons ingraphene considering that electrons can hop to bothnearest- and next-nearest-neighbor atoms has the form!we use units such that #=1"

H = ! t )*i,j+,$

!a$,i† b$,j + H.c."

! t! )**i,j++,$

!a$,i† a$,j + b$,i

† b$,j + H.c." , !5"

where ai,$ !ai,$† " annihilates !creates" an electron with

spin $ !$= ! , " " on site Ri on sublattice A !an equiva-lent definition is used for sublattice B", t!&2.8 eV" is thenearest-neighbor hopping energy !hopping between dif-ferent sublattices", and t! is the next nearest-neighborhopping energy1 !hopping in the same sublattice". Theenergy bands derived from this Hamiltonian have theform !Wallace, 1947"

E±!k" = ± t%3 + f!k" ! t!f!k" ,

1The value of t! is not well known but ab initio calculations!Reich et al., 2002" find 0.02t% t!%0.2t depending on the tight-binding parametrization. These calculations also include theeffect of a third-nearest-neighbors hopping, which has a valueof around 0.07 eV. A tight-binding fit to cyclotron resonanceexperiments !Deacon et al., 2007" finds t!&0.1 eV.

a

a

1

2

b

b

1

2

K!

k

k

x

y

1

2

3

M

" "

"

A B

K’

FIG. 2. !Color online" Honeycomb lattice and its Brillouinzone. Left: lattice structure of graphene, made out of two in-terpenetrating triangular lattices !a1 and a2 are the lattice unitvectors, and "i, i=1,2 ,3 are the nearest-neighbor vectors".Right: corresponding Brillouin zone. The Dirac cones are lo-cated at the K and K! points.

112 Castro Neto et al.: The electronic properties of graphene

Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

f!k" = 2 cos!#3kya" + 4 cos$#32

kya%cos$32

kxa% , !6"

where the plus sign applies to the upper !!*" and theminus sign the lower !!" band. It is clear from Eq. !6"that the spectrum is symmetric around zero energy if t!=0. For finite values of t!, the electron-hole symmetry isbroken and the ! and !* bands become asymmetric. InFig. 3, we show the full band structure of graphene withboth t and t!. In the same figure, we also show a zoom inof the band structure close to one of the Dirac points !atthe K or K! point in the BZ". This dispersion can beobtained by expanding the full band structure, Eq. !6",close to the K !or K!" vector, Eq. !3", as k=K+q, with&q & " &K& !Wallace, 1947",

E±!q" ' ± vF&q& + O(!q/K"2) , !7"

where q is the momentum measured relatively to theDirac points and vF is the Fermi velocity, given by vF=3ta /2, with a value vF*1#106 m/s. This result wasfirst obtained by Wallace !1947".

The most striking difference between this result andthe usual case, $!q"=q2 / !2m", where m is the electronmass, is that the Fermi velocity in Eq. !7" does not de-pend on the energy or momentum: in the usual case wehave v=k /m=#2E /m and hence the velocity changessubstantially with energy. The expansion of the spectrumaround the Dirac point including t! up to second orderin q /K is given by

E±!q" * 3t! ± vF&q& ! $9t!a2

4±

3ta2

8sin!3%q"%&q&2, !8"

where

%q = arctan$qx

qy% !9"

is the angle in momentum space. Hence, the presence oft! shifts in energy the position of the Dirac point andbreaks electron-hole symmetry. Note that up to order!q /K"2 the dispersion depends on the direction in mo-mentum space and has a threefold symmetry. This is theso-called trigonal warping of the electronic spectrum!Ando et al., 1998, Dresselhaus and Dresselhaus, 2002".

1. Cyclotron mass

The energy dispersion !7" resembles the energy of ul-trarelativistic particles; these particles are quantum me-chanically described by the massless Dirac equation !seeSec. II.B for more on this analogy". An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as itssquare root !Novoselov, Geim, Morozov, et al., 2005;Zhang et al., 2005". The cyclotron mass is defined, withinthe semiclassical approximation !Ashcroft and Mermin,1976", as

m* =1

2!+ !A!E"

!E,

E=EF

, !10"

with A!E" the area in k space enclosed by the orbit andgiven by

A!E" = !q!E"2 = !E2

vF2 . !11"

Using Eq. !11" in Eq. !10", one obtains

m* =EF

vF2 =

kF

vF. !12"

The electronic density n is related to the Fermi momen-tum kF as kF

2 /!=n !with contributions from the twoDirac points K and K! and spin included", which leads to

m* =#!

vF

#n . !13"

Fitting Eq. !13" to the experimental data !see Fig. 4"provides an estimation for the Fermi velocity and the

FIG. 3. !Color online" Electronic dispersion in the honeycomblattice. Left: energy spectrum !in units of t" for finite values oft and t!, with t=2.7 eV and t!=!0.2t. Right: zoom in of theenergy bands close to one of the Dirac points.

FIG. 4. !Color online" Cyclotron mass of charge carriers ingraphene as a function of their concentration n. Positive andnegative n correspond to electrons and holes, respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations; solid curves arethe best fit by Eq. !13". m0 is the free-electron mass. Adaptedfrom Novoselov, Geim, Morozov, et al., 2005.

113Castro Neto et al.: The electronic properties of graphene


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a1 =a2

!3,%3", a2 =a2

!3,! %3" , !1"


b1 =2!

3a!1,%3", b2 =

2!

3a!1,! %3" . !2"


K = '2!

3a,

2!

3%3a(, K! = '2!

3a,!

2!

3%3a( . !3"


!1 =a2

!1,%3" !2 =a2

!1,! %3" "3 = ! a!1,0" !4"



H = ! t )*i,j+,$

!a$,i† b$,j + H.c."

! t! )**i,j++,$

!a$,i† a$,j + b$,i

† b$,j + H.c." , !5"



E±!k" = ± t%3 + f!k" ! t!f!k" ,


a

a

1

2

b

b

1

2

K!

k

k

x

y

1

2

3

M

" "

"

A B

K’




f!k" = 2 cos!#3kya" + 4 cos$#32

kya%cos$32

kxa% , !6"


E±!q" ' ± vF&q& + O(!q/K"2) , !7"



E±!q" * 3t! ± vF&q& ! $9t!a2

4±

3ta2

8sin!3%q"%&q&2, !8"

where

%q = arctan$qx

qy% !9"


1. Cyclotron mass


m* =1

2!+ !A!E"

!E,

E=EF

, !10"


A!E" = !q!E"2 = !E2

vF2 . !11"


m* =EF

vF2 =

kF

vF. !12"



m* =#!

vF

#n . !13"






f!k" = 2 cos!#3kya" + 4 cos$#32

kya%cos$32

kxa% , !6"


E±!q" ' ± vF&q& + O(!q/K"2) , !7"



E±!q" * 3t! ± vF&q& ! $9t!a2

4±

3ta2

8sin!3%q"%&q&2, !8"

where

%q = arctan$qx

qy% !9"


1. Cyclotron mass


m* =1

2!+ !A!E"

!E,

E=EF

, !10"


A!E" = !q!E"2 = !E2

vF2 . !11"


m* =EF

vF2 =

kF

vF. !12"



m* =#!

vF

#n . !13"











a1 =a2

!3,%3", a2 =a2

!3,! %3" , !1"


b1 =2!

3a!1,%3", b2 =

2!

3a!1,! %3" . !2"


K = '2!

3a,

2!

3%3a(, K! = '2!

3a,!

2!

3%3a( . !3"


!1 =a2

!1,%3" !2 =a2

!1,! %3" "3 = ! a!1,0" !4"



H = ! t )*i,j+,$

!a$,i† b$,j + H.c."

! t! )**i,j++,$

!a$,i† a$,j + b$,i

† b$,j + H.c." , !5"



E±!k" = ± t%3 + f!k" ! t!f!k" ,


a

a

1

2

b

b

1

2

K!

k

k

x

y

1

2

3

M

" "

"

A B

K’




hopping parameter as vF!106 ms!1 and t!3 eV, respec-tively. Experimental observation of the "n dependenceon the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene #No-voselov, Geim, Morozov, et al., 2005; Zhang et al., 2005;Deacon et al., 2007; Jiang, Henriksen, Tung, et al.,2007$—the usual parabolic #Schrödinger$ dispersion im-plies a constant cyclotron mass.

2. Density of states

The density of states per unit cell, derived from Eq.#5$, is given in Fig. 5 for both t!=0 and t!!0, showing inboth cases semimetallic behavior #Wallace, 1947; Benaand Kivelson, 2005$. For t!=0, it is possible to derive ananalytical expression for the density of states per unitcell, which has the form #Hobson and Nierenberg, 1953$

!#E$ =4

"2

%E%t2

1"Z0

F&"

2,"Z1

Z0' ,

Z0 = (&1 + )Et)'2

!*#E/t$2 ! 1+2

4, ! t # E # t

4)Et) , ! 3t # E # ! t ! t # E # 3t ,,

Z1 = (4)Et) , ! t # E # t

&1 + )Et)'2

!*#E/t$2 ! 1+2

4, ! 3t # E # ! t ! t # E # 3t ,, #14$

where F#" /2 ,x$ is the complete elliptic integral of thefirst kind. Close to the Dirac point, the dispersion is ap-proximated by Eq. #7$ and the density of states per unitcell is given by #with a degeneracy of 4 included$

!#E$ =2Ac

"

%E%vF

2 , #15$

where Ac is the unit cell area given by Ac=3"3a2 /2. It isworth noting that the density of states for graphene isdifferent from the density of states of carbon nanotubes#Saito et al., 1992a, 1992b$. The latter shows 1/"E singu-larities due to the 1D nature of their electronic spec-trum, which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective, graphene nanoribbons, whichalso have momentum quantization perpendicular to theribbon length, have properties similar to carbon nano-tubes.

B. Dirac fermions

We consider the Hamiltonian #5$ with t!=0 and theFourier transform of the electron operators,

an =1

"Nc-k

e!ik·Rna#k$ , #16$

where Nc is the number of unit cells. Using this transfor-mation, we write the field an as a sum of two terms,coming from expanding the Fourier sum around K! andK. This produces an approximation for the representa-tion of the field an as a sum of two new fields, written as

an . e!iK·Rna1,n + e!iK!·Rna2,n,

bn . e!iK·Rnb1,n + e!iK!·Rnb2,n, #17$

-4 -2 0 20

1

2

3

4

5

!(")

t’=0.2t

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

-2 0 2" /t

0

0.2

0.4

0.6

0.8

1

!(")

t’=0

-0.8 -0.4 0 0.4 0.8" /t

0

0.1

0.2

0.3

0.4

FIG. 5. Density of states per unit cell as a function of energy#in units of t$ computed from the energy dispersion #5$, t!=0.2t #top$ and t!=0 #bottom$. Also shown is a zoom-in of thedensity of states close to the neutrality point of one electronper site. For the case t!=0, the electron-hole nature of thespectrum is apparent and the density of states close to theneutrality point can be approximated by !#$$% %$%.



hopping parameter as vF!106 ms!1 and t!3 eV, respec-tively. Experimental observation of the "n dependenceon the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene #No-voselov, Geim, Morozov, et al., 2005; Zhang et al., 2005;Deacon et al., 2007; Jiang, Henriksen, Tung, et al.,2007$—the usual parabolic #Schrödinger$ dispersion im-plies a constant cyclotron mass.

2. Density of states

The density of states per unit cell, derived from Eq.#5$, is given in Fig. 5 for both t!=0 and t!!0, showing inboth cases semimetallic behavior #Wallace, 1947; Benaand Kivelson, 2005$. For t!=0, it is possible to derive ananalytical expression for the density of states per unitcell, which has the form #Hobson and Nierenberg, 1953$

!#E$ =4

"2

%E%t2

1"Z0

F&"

2,"Z1

Z0' ,

Z0 = (&1 + )Et)'2

!*#E/t$2 ! 1+2

4, ! t # E # t

4)Et) , ! 3t # E # ! t ! t # E # 3t ,,

Z1 = (4)Et) , ! t # E # t

&1 + )Et)'2

!*#E/t$2 ! 1+2

4, ! 3t # E # ! t ! t # E # 3t ,, #14$

where F#" /2 ,x$ is the complete elliptic integral of thefirst kind. Close to the Dirac point, the dispersion is ap-proximated by Eq. #7$ and the density of states per unitcell is given by #with a degeneracy of 4 included$

!#E$ =2Ac

"

%E%vF

2 , #15$

where Ac is the unit cell area given by Ac=3"3a2 /2. It isworth noting that the density of states for graphene isdifferent from the density of states of carbon nanotubes#Saito et al., 1992a, 1992b$. The latter shows 1/"E singu-larities due to the 1D nature of their electronic spec-trum, which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective, graphene nanoribbons, whichalso have momentum quantization perpendicular to theribbon length, have properties similar to carbon nano-tubes.

B. Dirac fermions

We consider the Hamiltonian #5$ with t!=0 and theFourier transform of the electron operators,

an =1

"Nc-k

e!ik·Rna#k$ , #16$

where Nc is the number of unit cells. Using this transfor-mation, we write the field an as a sum of two terms,coming from expanding the Fourier sum around K! andK. This produces an approximation for the representa-tion of the field an as a sum of two new fields, written as

an . e!iK·Rna1,n + e!iK!·Rna2,n,

bn . e!iK·Rnb1,n + e!iK!·Rnb2,n, #17$

-4 -2 0 20

1

2

3

4

5

!(")

t’=0.2t

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

-2 0 2" /t

0

0.2

0.4

0.6

0.8

1

!(")

t’=0

-0.8 -0.4 0 0.4 0.8" /t

0

0.1

0.2

0.3

0.4

FIG. 5. Density of states per unit cell as a function of energy#in units of t$ computed from the energy dispersion #5$, t!=0.2t #top$ and t!=0 #bottom$. Also shown is a zoom-in of thedensity of states close to the neutrality point of one electronper site. For the case t!=0, the electron-hole nature of thespectrum is apparent and the density of states close to theneutrality point can be approximated by !#$$% %$%.



where the index i=1 !i=2" refers to the K !K!" point.These new fields, ai,n and bi,n, are assumed to varyslowly over the unit cell. The procedure for derivinga theory that is valid close to the Dirac point con-sists in using this representation in the tight-

binding Hamiltonian and expanding the opera-tors up to a linear order in !. In the derivation, oneuses the fact that #!e±iK·!=#!e±iK!·!=0. After somestraightforward algebra, we arrive at !Semenoff,1984"

H $ ! t% dxdy"1†!r"&' 0 3a!1 ! i(3"/4

! 3a!1 + i(3"/4 0)!x + ' 0 3a!! i ! (3"/4

! 3a!i ! (3"/4 0)!y*"1!r"

+ "2†!r"&' 0 3a!1 + i(3"/4

! 3a!1 ! i(3"/4 0)!x + ' 0 3a!i ! (3"/4

! 3a!! i ! (3"/4 0)!y*"2!r"

= ! ivF% dxdy+"1†!r"# · ""1!r" + "2

†!r""* · ""2!r", , !18"

with Pauli matrices "= !#x ,#y", "*= !#x ,!#y", and "i†

= !ai† ,bi

†" !i=1,2". It is clear that the effective Hamil-tonian !18" is made of two copies of the massless Dirac-like Hamiltonian, one holding for p around K and theother for p around K!. Note that, in first quantized lan-guage, the two-component electron wave function $!r",close to the K point, obeys the 2D Dirac equation,

! ivF" · "$!r" = E$!r" . !19"

The wave function, in momentum space, for the mo-mentum around K has the form

$±,K!k" =1(2

' e!i%k/2

±ei%k/2 ) !20"

for HK=vF" ·k, where the & signs correspond to theeigenenergies E= ±vFk, that is, for the '* and ' bands,respectively, and %k is given by Eq. !9". The wave func-tion for the momentum around K! has the form

$±,K!!k" =1(2

' ei%k/2

±e!i%k/2 ) !21"

for HK!=vF"* ·k. Note that the wave functions at K andK! are related by time-reversal symmetry: if we set theorigin of coordinates in momentum space in the M pointof the BZ !see Fig. 2", time reversal becomes equivalentto a reflection along the kx axis, that is, !kx ,ky"! !kx ,!ky". Also note that if the phase % is rotated by2', the wave function changes sign indicating a phase of' !in the literature this is commonly called a Berry’sphase". This change of phase by ' under rotation is char-acteristic of spinors. In fact, the wave function is a two-component spinor.

A relevant quantity used to characterize the eigen-functions is their helicity defined as the projection of themomentum operator along the !pseudo"spin direction.The quantum-mechanical operator for the helicity hasthe form

h =12

" ·p-p-

. !22"

It is clear from the definition of h that the states $K!r"and $K!!r" are also eigenstates of h,

h$K!r" = ± 12$K!r" , !23"

and an equivalent equation for $K!!r" with inverted sign.Therefore, electrons !holes" have a positive !negative"helicity. Equation !23" implies that " has its two eigen-values either in the direction of !!" or against !"" themomentum p. This property says that the states of thesystem close to the Dirac point have well defined chiral-ity or helicity. Note that chirality is not defined in regardto the real spin of the electron !that has not yet ap-peared in the problem" but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as longas the Hamiltonian !18" is valid. Therefore, the existenceof helicity quantum numbers holds only as anasymptotic property, which is well defined close to theDirac points K and K!. Either at larger energies or dueto the presence of a finite t!, the helicity stops being agood quantum number.

1. Chiral tunneling and Klein paradox

In this section, we address the scattering of chiral elec-trons in two dimensions by a square barrier !Katsnelsonet al., 2006; Katsnelson, 2007b". The one-dimensionalscattering of chiral electrons was discussed earlier in thecontext on nanotubes !Ando et al., 1998; McEuen et al.,1999".

We start by noting that by a gauge transformation thewave function !20" can be written as





H $ ! t% dxdy"1†!r"&' 0 3a!1 ! i(3"/4

! 3a!1 + i(3"/4 0)!x + ' 0 3a!! i ! (3"/4

! 3a!i ! (3"/4 0)!y*"1!r"

+ "2†!r"&' 0 3a!1 + i(3"/4

! 3a!1 ! i(3"/4 0)!x + ' 0 3a!i ! (3"/4

! 3a!! i ! (3"/4 0)!y*"2!r"

= ! ivF% dxdy+"1†!r"# · ""1!r" + "2

†!r""* · ""2!r", , !18"


= !ai† ,bi


! ivF" · "$!r" = E$!r" . !19"


$±,K!k" =1(2

' e!i%k/2

±ei%k/2 ) !20"


$±,K!!k" =1(2

' ei%k/2

±e!i%k/2 ) !21"



h =12

" ·p-p-

. !22"


h$K!r" = ± 12$K!r" , !23"









H $ ! t% dxdy"1†!r"&' 0 3a!1 ! i(3"/4

! 3a!1 + i(3"/4 0)!x + ' 0 3a!! i ! (3"/4

! 3a!i ! (3"/4 0)!y*"1!r"

+ "2†!r"&' 0 3a!1 + i(3"/4

! 3a!1 ! i(3"/4 0)!x + ' 0 3a!i ! (3"/4

! 3a!! i ! (3"/4 0)!y*"2!r"

= ! ivF% dxdy+"1†!r"# · ""1!r" + "2

†!r""* · ""2!r", , !18"


= !ai† ,bi


! ivF" · "$!r" = E$!r" . !19"


$±,K!k" =1(2

' e!i%k/2

±ei%k/2 ) !20"


$±,K!!k" =1(2

' ei%k/2

±e!i%k/2 ) !21"



h =12

" ·p-p-

. !22"


h$K!r" = ± 12$K!r" , !23"









H $ ! t% dxdy"1†!r"&' 0 3a!1 ! i(3"/4

! 3a!1 + i(3"/4 0)!x + ' 0 3a!! i ! (3"/4

! 3a!i ! (3"/4 0)!y*"1!r"

+ "2†!r"&' 0 3a!1 + i(3"/4

! 3a!1 ! i(3"/4 0)!x + ' 0 3a!i ! (3"/4

! 3a!! i ! (3"/4 0)!y*"2!r"

= ! ivF% dxdy+"1†!r"# · ""1!r" + "2

†!r""* · ""2!r", , !18"


= !ai† ,bi


! ivF" · "$!r" = E$!r" . !19"


$±,K!k" =1(2

' e!i%k/2

±ei%k/2 ) !20"


$±,K!!k" =1(2

' ei%k/2

±e!i%k/2 ) !21"



h =12

" ·p-p-

. !22"


h$K!r" = ± 12$K!r" , !23"









H $ ! t% dxdy"1†!r"&' 0 3a!1 ! i(3"/4

! 3a!1 + i(3"/4 0)!x + ' 0 3a!! i ! (3"/4

! 3a!i ! (3"/4 0)!y*"1!r"

+ "2†!r"&' 0 3a!1 + i(3"/4

! 3a!1 ! i(3"/4 0)!x + ' 0 3a!i ! (3"/4

! 3a!! i ! (3"/4 0)!y*"2!r"

= ! ivF% dxdy+"1†!r"# · ""1!r" + "2

†!r""* · ""2!r", , !18"


= !ai† ,bi


! ivF" · "$!r" = E$!r" . !19"


$±,K!k" =1(2

' e!i%k/2

±ei%k/2 ) !20"


$±,K!!k" =1(2

' ei%k/2

±e!i%k/2 ) !21"



h =12

" ·p-p-

. !22"


h$K!r" = ± 12$K!r" , !23"







absorbed water !Sabio et al., 2007". In fact, experimentsin ultrahigh vacuum conditions !Chen, Jang, Fuhrer, etal., 2008" display scattering features in the transport thatcan be associated with charge impurities. Screening ef-fects that affect the strength and range of the Coulombinteraction are rather nontrivial in graphene !Fogler,Novikov, and Shklovskii, 2007; Shklovskii, 2007" and,therefore, important for the interpretation of transportdata !Bardarson et al., 2007; Nomura et al., 2007; San-Jose et al., 2007; Lewenkopf et al., 2008".

Another type of disorder is the one that changes thedistance or angles between the pz orbitals. In this case,the hopping energies between different sites are modi-fied, leading to a new term to the original Hamiltonian!5",

Hod = #i,j

$!tij!ab"!ai

†bj + H.c." + !tij!aa"!ai

†aj + bi†bj"% ,

!144"

or in Fourier space,

Hod = #k,k!

ak†bk! #

i,!!ab

!ti!ab"ei!k!k!"·Ri!i!!aa·k! + H.c.

+ !ak†ak! + bk

†bk!" #i,!!aa

!ti!aa"ei!k!k!"·Ri!i!!ab·k!, !145"

where !tij!ab" !!tij

!aa"" is the change of the hopping energybetween orbitals on lattice sites Ri and Rj on the same!different" sublattices !we have written Rj=Ri+!! , where!!ab is the nearest-neighbor vector and !!aa is the next-nearest-neighbor vector". Following the procedure ofSec. II.B, we project out the Fourier components of theoperators close to the K and K! points of the BZ usingEq. !17". If we assume that !tij is smooth over the latticespacing scale, that is, it does not have a Fourier compo-nent with momentum K!K! !so the two Dirac cones arenot coupled by disorder", we can rewrite Eq. !145" inreal space as

Hod =& d2r$A!r"a1†!r"b1!r" + H.c.

+ "!r"'a1†!r"a1!r" + b1

†!r"b1!r"(% , !146"

with a similar expression for cone 2 but with A replacedby A*, where

A!r" = #!!ab

!t!ab"!r"e!i!!ab·K, !147"

"!r" = #!!aa

!t!aa"!r"e!i!!aa·K. !148"

Note that whereas "!r"="*!r", because of the inversionsymmetry of the two triangular sublattices that make upthe honeycomb lattice, A is complex because of a lack ofinversion symmetry for nearest-neighbor hopping.Hence,

A!r" = Ax!r" + iAy!r" . !149"

In terms of the Dirac Hamiltonian !18", we can rewriteEq. !146" as

Hod =& d2r'#1†!r"! · A! !r"#1!r" + "!r"#1

†!r"#1!r"( ,

!150"

where A! = !Ax ,Ay". This result shows that changes in thehopping amplitude lead to the appearance of vector A!and scalar $ potentials in the Dirac Hamiltonian. Thepresence of a vector potential in the problem indicatesthat an effective magnetic field B! = !c /evF"! %A! shouldalso be present, naively implying a broken time-reversalsymmetry, although the original problem was time-reversal invariant. This broken time-reversal symmetryis not real since Eq. !150" is the Hamiltonian aroundonly one of the Dirac cones. The second Dirac cone isrelated to the first by time-reversal symmetry, indicatingthat the effective magnetic field is reversed in the secondcone. Therefore, there is no global broken symmetry buta compensation between the two cones.

A. Ripples

Graphene is a one-atom-thick system, the extremecase of a soft membrane. Hence, like soft membranes, itis subject to distortions of its structure due to eitherthermal fluctuations !as we discussed in Sec. III" or in-teraction with a substrate, scaffold, and absorbands!Swain and Andelman, 1999". In the first case, the fluc-tuations are time dependent !although with time scalesmuch longer than the electronic ones", while in the sec-ond case the distortions act as quenched disorder. Inboth cases, disorder occurs because of the modificationof the distance and relative angle between the carbonatoms due to the bending of the graphene sheet. Thistype of off-diagonal disorder does not exist in ordinary3D solids, or even in quasi-1D or quasi-2D systems,where atomic chains and atomic planes, respectively, areembedded in a 3D crystalline structure. In fact,graphene is also different from other soft membranesbecause it is !semi"metallic, while previously studiedmembranes were insulators.

The problem of bending graphitic systems and its ef-fect on the hybridization of the & orbitals has been stud-ied in the context of classical minimal surfaces !Lenoskyet al., 1992" and applied to fullerenes and carbon nano-tubes !Tersoff, 1992; Kane and Mele, 1997; Zhong-can etal., 1997; Xin et al., 2000; Tu and Ou-Yang, 2002". Inorder to understand the effect of bending on graphene,consider the situation shown in Fig. 25. The bending ofthe graphene sheet has three main effects: the decreaseof the distance between carbon atoms, a rotation ofthe pZ orbitals 'compression or dilation of the latticeare energetically costly due to the large spring con-stant of graphene )57 eV/Å2 !Xin et al., 2000"(, and arehybridization between & and ' orbitals !Eun-Ah Kim





Hod = #i,j

$!tij!ab"!ai


†aj + bi†bj"% ,

!144"


Hod = #k,k!

ak†bk! #

i,!!ab


+ !ak†ak! + bk

†bk!" #i,!!aa




Hod =& d2r$A!r"a1†!r"b1!r" + H.c.

+ "!r"'a1†!r"a1!r" + b1

†!r"b1!r"(% , !146"


A!r" = #!!ab

!t!ab"!r"e!i!!ab·K, !147"

"!r" = #!!aa

!t!aa"!r"e!i!!aa·K. !148"


A!r" = Ax!r" + iAy!r" . !149"


Hod =& d2r'#1†!r"! · A! !r"#1!r" + "!r"#1

†!r"#1!r"( ,

!150"


A. Ripples







Hod = #i,j

$!tij!ab"!ai


†aj + bi†bj"% ,

!144"


Hod = #k,k!

ak†bk! #

i,!!ab


+ !ak†ak! + bk

†bk!" #i,!!aa




Hod =& d2r$A!r"a1†!r"b1!r" + H.c.

+ "!r"'a1†!r"a1!r" + b1

†!r"b1!r"(% , !146"


A!r" = #!!ab

!t!ab"!r"e!i!!ab·K, !147"

"!r" = #!!aa

!t!aa"!r"e!i!!aa·K. !148"


A!r" = Ax!r" + iAy!r" . !149"


Hod =& d2r'#1†!r"! · A! !r"#1!r" + "!r"#1

†!r"#1!r"( ,

!150"


A. Ripples





Two Dirac cones are not coupled by disorder



Hod = #i,j

$!tij!ab"!ai


†aj + bi†bj"% ,

!144"


Hod = #k,k!

ak†bk! #

i,!!ab


+ !ak†ak! + bk

†bk!" #i,!!aa




Hod =& d2r$A!r"a1†!r"b1!r" + H.c.

+ "!r"'a1†!r"a1!r" + b1

†!r"b1!r"(% , !146"


A!r" = #!!ab

!t!ab"!r"e!i!!ab·K, !147"

"!r" = #!!aa

!t!aa"!r"e!i!!aa·K. !148"


A!r" = Ax!r" + iAy!r" . !149"


Hod =& d2r'#1†!r"! · A! !r"#1!r" + "!r"#1

†!r"#1!r"( ,

!150"


A. Ripples





1.  Grain Boundary.

2.  Liquid environment enhancement on mobility in graphene.

3.  X-‐ray irradiaIon of graphene.

Defects in graphene

J. Lahiri et al., Nature Nanotech. (2010)

Experimental observaIon of defects in graphene

Extended defect = Metallic wire

Meyer, Kisielowski, Erni, Rossell, Crommie, ZeTl, Nano Le.. (2008)

Vacancy

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;

pris2ne Cl-‐5577 GB-‐558 GB-‐575 rim inside rim inside rim inside C 6.1 3.8 4.2 3.1 SV 7.9 5.6 3.0 7.1 6.4 6.9 4.2

All energies are in eV

LocaIon of Vacancies with lower formaIon energies

FormaIon Energies of Defects

InteracIon of vacancies in graphene

Banhart, Kotakoski, Krasheninnikov, ACS Nano 2010 Cretu, Krasheninnikov et al. PRL 2010 Lee et al. PRL, 2005;

Stability of 585 and 555777 defects

555777 is 1.2 eV more stable

Dynamical InteracIon of Vacancies

Strain up to 4%

T = 2000 K trecombinaIon ~ 80 ps

Vacancy interacIon with grain boundaries

~1 nm

Vacancy-‐GB interacIon

Vacancy merges with GB

160 ps 85 5 8775 5

Further GB reconstrucIon

40 ps

Defect Dynamics: Vacancy near GB

Strain up to 8%

1 nm T = 3000 K

Reconstrucoon relieves strain treconstruct ~ 200 ps

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

treconfiguraoon = 0.5 ns

T = 2000 K

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.

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

Ø Theory: behavior of impuriIes on graphene

o  Temperature and concentraIon dependence.

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


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)


Defect related D-‐peak

•  increases x-‐ray exposure •  decreases a�er temperature anneal

Ozone exposure

a)

b)

0

2000

4000

6000

8000

Annea l15 Mrad(S iO2)8 Mrad(S iO

2)

Integrated

intens

ity A

rea

10 -‐keV X -‐ray D os e

G -‐P eak

D -‐P eak

P re0

20

40

60

80

I D/I G

(10

0%)

Kine2c Monte-‐Carlo KMC

Density Func2onal Theory DFT

TheoreIcal Approach

O

O dimer

O migraIon O desorpIon

•  Defect formaoon energies •  Migraoon/desorpoon barriers

Defect dynamics •  Temperature •  Inioal concentraoon

Top

Bridge

1.3 eV

0.8 eV

0.5 eV

1.3 eV

Oxygen Removal and Vacancy GeneraIon

CO, CO2 1.1 eV O2 1.1 eV

Oxygen: clustering behavior Removal of oxygen •  Pairs O2 •  Triplets CO, CO2, VC Device degradaoon

Residual oxygen atom Vacancy

High-‐temperature Annealing

Concentraoon of vacancies exceeds concentraoon of residual O

T

High vs Low Temperature Anneal

T, oC

Temperature Anneal IniIal Defect ConcentraIon Dependence

Lo

Low O, High V concentraoon

High O concentraoon

vacancy

oxygen

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

surface coverage

Method to prevent defect forma2on during irradia2on/annealing?

T

iniIal O surface coverage

Oxygen and Hydrogen on Graphene: Binding energies, MigraIon and ReacIon Barriers

O-‐H is most likely to desorb from graphene surface

Leaves carbon network intact

H

O

Effect of Hydrogen On Oxygen Annealing

Oxygen/Hydrogen

Concentra2ons

Low High

Low 2% O, 10% H

High 15% O, 1% H

15% O, 10% H

@ T = 300 C

Final defect concentraIons?

Removal of residual Oxygen Causes formaoon of large amount of Vacancies

t ~ 0.001 s

t ~ 1 s

t ~ 0.0001 s

t ~ 1 s


Residual Hydrogen Forms clusters L ~ 0.5 nm No Vacancies are formed

Higher Hydrogen concentraoon Higher Oxygen concentraoon Hydrogen is removed Oxygen is removed

Hydrogen is removed first, Removal of residual Oxygen Causes formaoon of Vacancies

High O, High H concentraIons


1.  BoloIn, K. I. et al. Solid State Comm. 2008 2.  Castro, E. V. et al. Phys Rev LeT. 2010

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

ü  Vacuum

ü  Hexane C6H14

ü  Toluene C6H5CH3

Image from Meyer, J. C .

Electron scaTering due to flexural ripples

𝐸↓𝑞 =𝜅𝑞↑4 ⟨|ℎ↓𝑞 |↑2 ⟩/2 = 𝑘↓𝐵 𝑇 /2 

Fourier components of bending correlaIon funcIon

Harmonic approximaIon

h at 300K

hq2~ Tκq4

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

PotenIal perturbaIon due to ripples -‐ random sign-‐changing ‘magneIc field’

γ= γ↓0 +(𝜕γ/𝜕𝑢↓𝑖𝑗  )0𝑢↓𝑖𝑗 

Morozov S. V. et. al, Phys. Rev. LeT 2006 M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A, 2008 Castro, E. V. et. al Phys Rev LeT (2010)

2

1

3

1/𝜏 ≈2𝜋/ℎ 𝑁(𝐸↓𝐹 )⟨𝑉↓𝑞 𝑉↓−𝑞 ⟩↓𝑞≈𝑘↓𝐹  ~ ⟨|ℎ↓𝑞 |↑2 ⟩↑2 

Hopping integrals γ are modified

𝜌↓𝑟𝑖𝑝𝑝𝑙𝑒 ~ 1/𝜏 ~ ⟨|ℎ↓𝑞 |↑2 ⟩↑2 

𝑉↑(𝑥) = 1/2 (2γ↓1 − γ↓2 − γ↓3 ) 𝑉↑(𝑦) = 1/2 (γ↓2 − γ↓3 )

Effect of liquids ü  Hexane C6H14 ü  Toluene C6H5CH3

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

h

2 0.89vacuumh = Å2

T = 300 K

Suspended graphene in hexane

Hexane molecules envelopes graphene sheet

C chain aligned parallel to the plane Mean square displacement

2 0.39hexaneh = Å2

Suspended graphene in toluene

2 0.42tolueneh = Å2

Toluene molecules envelopes graphene sheet

Mean square displacement C ring aligned parallel to the plane

Preferred molecule posiIon: DFT calculaIon

ΔE = 0.21 eV

3 Å

ΔE = 0.37 eV

3 Å

Van der Waals interacIon

h

Ripple height analysis

2 0.89vacuumh = Å2

2 0.42tolueneh = Å2

2 0.39hexaneh = Å2

Bending sIffness of graphene in liquid

𝜌↓𝑟𝑖𝑝𝑝𝑙𝑒 ≈ℏ/4𝑒↑2  (𝑘↓𝐵 𝑇/𝜅𝑎 )↑2 𝛬/𝑛 

Bending S2ffness

ü  Vacuum μ ~10,000 cm2/V s

ü  Liquid μ ~ 200,000 cm2/V s

Out-‐of-‐plane flexural phonons limit at room T

2

40

qTNhA qκ

=r

Liquid suppresses flexural phonons

Conclusion •  Liquid dielectric environment suppresses flexural phonons

•  Phonon suppression affects mobility through bending sIffness

ρ r( ) = ρn r

− Rn

( )n=1

Natoms

∑ρn r− Rn

( ) = 1− qnQA"

#$$

%

&''ρ0

A r− Rn

( )

ρn r( ) =ηr2 cme

−γmr2

m=1

Mgauss

∑

ρn r− Rn

( )d 3r =QA − qn*Ωcell

∫

Электронная плотность

Разложение по функциям Гаусса

Перенос заряда

a)

Figure 2. Real space 72x72x72 grid. a) (100) and b) (110) planes c) [111] direcIon

Etotal = W r ρ r( )!

"#$%&ρ r( )

Ωvolume

∫ d 3r + Wq ρ q( )!

"#$%&ρ q( )

Ωvolume

∫ d 3q+ Eion−ion

W r ρ r( )!

"#$%&=T ρ r

( )!

"#$%&+Vex ρ r

( )!

"#$%&

Wq ρ q( )!

"#$%&=Vps q

( )+Vhartree q

( ) ( ) ( ) ( )pseudo

psV q S q w q=

Полная энергия

TWang−Teter ρ r( )"

#$%&'=45128

3π 2( )23 ρ

56 r( )∫∫ w1 r

− r'( )ρ 5

6 r'( )d 3rd 3r '−

−21250

3π 2( )23 ρ

53 r( )∫ d 3r − 1

2ρ12 r( )∇2ρ

12 r( )∫ d 3r

( )( )2

1 21

45 3 3 1 2, ln8 4 5 2 8 2

q qw w q q and wq q

−− −⎛ ⎞= − + = +⎜ ⎟ +⎝ ⎠

Теория линейного отклика

Кинетическая энергия

corr corrWang Teter LDA atomT T T T−= + +

TLDAcorr ρ r

( )!

"#$%&= cn

n=1

6

∑ Δρn2 ri( )

)*+

,+

-.+

/+i=1

Ngrid

∑

Tatomcorr k( ) = cn

n=1

6

∑ πξn

"

#$$

%

&''

32

exp −k 2

4ξn

"

#$$

%

&''

SA ki( ) = 1−

kα

Nα

"

#$$

%

&''

α∈A∑ exp −ikα

iRα

( )

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

G0W0

Ширина запрещенной зоны

GaN

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