лекция 5 memristor

Tight binding approxima0on “The band theory of graphite” by Wallace Phys. Rev. Le<. 71, 622, 1947

“The electronic proper0es 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.

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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







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







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.  Bolo0n, K. I. et al. Solid State Comm. 2008 2.  Castro, E. V. et al. Phys Rev Le<. 2010

Sca<ering 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 sca<ering due to flexural ripples

Fourier components of bending correla0on func0on

Harmonic approxima0on

h at 300K

hq2~ Tκq4

the Coulomb centre (Berestetskii et al. 1971), vacuum polarization effectsdiminish the initial supercritical value of b down to a critical value of one-half(Shytov et al. 2007).Nevertheless, the concentration dependence of resistivity shouldagain remain qualitatively the same as for the unscreened Coulomb potential.

3. Scattering by a generic random potential: cross-checkwith perturbation theory

As shown above, in the case of a small concentration of scattering centres theircontribution to resistivity can be calculated without any assumptions about thestrength of the potential. In this section, we use the perturbation theory to checkthe applicability of the phase-scattering approach and justify it.

Let us consider a generic perturbation of the form

V kk 0 ZV !0"kk 0 CsVkk 0 ; !3:1"

where k, k0 are the electron wavevectors and the Pauli matrices act on thepseudospin (sublattice) indices. This equation takes into account both scalar(electrostatic) and vector (pseudomagnetic) potentials created by defects (seealso McCann et al. 2006). Assuming that potential V is small in comparison withthe bandwidth and repeating the standard derivation of the Boltzmann equationin the Born approximation (Shon & Ando 1998), we find

1

tZ

4p

N!EF"X

kk 0

d!EkKEF"d!Ek 0KEF"!cos fkKcos fk 0"2jWkk 0 j2; !3:2"

where fk is the polar angle of the wavevector k and EkZZvFk is the electronenergy and

Wkk 0 ZV !0"kk 0

1Cexp#i!fkKfk 0"$2

C1

2V !x"

kk 0K iV !y"kk 0

! "exp!ifk"C V !x"

kk 0 C iV !y"kk 0

! "exp Kifk 0! "

h i: !3:3"

Equation (3.2) corresponds to the solution of the Boltzmann equation by avariational principle (Ziman 2001) or, equivalently, using the Mori formula(1965) for resistivity, in which only intra-band matrix elements of the currentoperator are taken into account. For the case of a small concentration of eithershort-range or Coulomb scatterers, one can check by direct calculations thatequation (3.2) gives the same concentration dependence of resistivity as thephase-scattering approach described in §2.

4. Scattering by ripples

A local curvature of a graphene sheet changes interatomic distances and anglesbetween chemical bonds and can be described by the following nonlinear term inthe deformation tensor (Nelson et al. 2004):

!uij Z1

2

vuivxj

Cvujvxi

Cvh

vxi

vh

vxj

# $; !4:1"

199Electron scattering in graphene

Phil. Trans. R. Soc. A (2008)

on February 22, 2012rsta.royalsocietypublishing.orgDownloaded from

where ui are the components of in-plane atomic displacements and h is thedisplacements normal to a graphene sheet. This curvature modifies the hoppingintegrals g as

gZg0 Cvg

v!uij

! "

0

!uij : !4:2"

The change in nearest-neighbour hopping parameters is equivalent to theappearance of a gauge field (Morozov et al. 2006) described by a ‘vector potential’

V !x" Z1

2!2g1Kg2Kg3"; V !y" Z

1

2!g2Kg3"; !4:3"

where the indices 1, 2 and 3 label the nearest neighbours that correspond totranslational vectors !Ka=

###3

p; 0", !a=2

###3

p;Ka=2" and !a=2

###3

p; a=2", respectively.

Changes in the next-nearest-neighbour hopping also lead to an electrostaticpotential V (0) that fluctuates in a randomly rippled graphene sheet(Kim Neto & Castro 2007). However, as follows from equations (3.2) and(3.3) both vector and electrostatic potentials contribute to r in a similarmanner and, for brevity, we will further discuss only the effect of vectorpotential (4.3). This potential is equivalent to a random sign-changing‘magnetic field’ that was previously shown to cause additional resistivity inconventional two-dimensional electron systems (Geim et al. 1994) andsuppression of weak localization in graphene (Morozov et al. 2006).

For a rough estimate, equation (3.2) can be rewritten in the simplified form

1

tz

2p

ZN!EF"hVqVKqiqzkF : !4:4"

According to equations (4.1)–(4.3), the vector potential is proportional toin-plane deformations and, thus, quadratic in derivatives vh/vx, vh/vy. Thisleads to the following expression:

hVqVKqizZvFa

! "2 X

q1q2

hqKq1hq1

hKqCq2hKq2

$ %#!qKq1"$q1$#!qKq2"$q2$: !4:5"

To describe scattering on ripples for the most general case, let us assume thattheir height-correlation function grows with increasing distance r ash#h!r"K h!0"$2ifr2H , where the exponent H characterizes the fractal dimensionof ripples (Ishigami et al. 2007). This expression immediately gives us the scalingbehaviour of the Fourier transform correlation function hjhq j2ifqK2!HC1" as wellas the q dependence of hVqVKqi that is defined by the convolution of twofunctions qK2H. The latter result corresponds to the decoupling of the four-hcorrelation function in equation (4.5) using the Wick theorem that is rigorousif fluctuations are Gaussian but can also be used for a qualitative estimate inother cases.

As a result, for 2H!1, the correlation function has a finite limit at qZ0

hVqVKqiqZ0zZv F

a

! "2 z4

R2 ; !4:6"

where z and R are the characteristic height and radius of ripples, respectively.

M. I. Katsnelson and A. K. Geim200



Electron sca<ering due to flexural phonons

Poten0al perturba0on due to ripples -‐ random sign-‐changing ‘magne0c field’

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

2

1

3

Hopping integrals γ are modified

Effect of liquids ü  Hexane C6H14 ü  Toluene C6H5CH3

1τ≈2πhN EF( ) VqV−q ~ hq

2 2

ρripple ~1τ~ hq

2 2

where ui are the components of in-plane atomic displacements and h is thedisplacements normal to a graphene sheet. This curvature modifies the hoppingintegrals g as

gZg0 Cvg

v!uij

! "

0

!uij : !4:2"

The change in nearest-neighbour hopping parameters is equivalent to theappearance of a gauge field (Morozov et al. 2006) described by a ‘vector potential’

V !x" Z1

2!2g1Kg2Kg3"; V !y" Z

1

2!g2Kg3"; !4:3"

where the indices 1, 2 and 3 label the nearest neighbours that correspond totranslational vectors !Ka=

###3

p; 0", !a=2

###3

p;Ka=2" and !a=2

###3

p; a=2", respectively.

Changes in the next-nearest-neighbour hopping also lead to an electrostaticpotential V (0) that fluctuates in a randomly rippled graphene sheet(Kim Neto & Castro 2007). However, as follows from equations (3.2) and(3.3) both vector and electrostatic potentials contribute to r in a similarmanner and, for brevity, we will further discuss only the effect of vectorpotential (4.3). This potential is equivalent to a random sign-changing‘magnetic field’ that was previously shown to cause additional resistivity inconventional two-dimensional electron systems (Geim et al. 1994) andsuppression of weak localization in graphene (Morozov et al. 2006).

For a rough estimate, equation (3.2) can be rewritten in the simplified form

1

tz

2p

ZN!EF"hVqVKqiqzkF : !4:4"

According to equations (4.1)–(4.3), the vector potential is proportional toin-plane deformations and, thus, quadratic in derivatives vh/vx, vh/vy. Thisleads to the following expression:

hVqVKqizZvFa

! "2 X

q1q2

hqKq1hq1

hKqCq2hKq2

$ %#!qKq1"$q1$#!qKq2"$q2$: !4:5"

To describe scattering on ripples for the most general case, let us assume thattheir height-correlation function grows with increasing distance r ash#h!r"K h!0"$2ifr2H , where the exponent H characterizes the fractal dimensionof ripples (Ishigami et al. 2007). This expression immediately gives us the scalingbehaviour of the Fourier transform correlation function hjhq j2ifqK2!HC1" as wellas the q dependence of hVqVKqi that is defined by the convolution of twofunctions qK2H. The latter result corresponds to the decoupling of the four-hcorrelation function in equation (4.5) using the Wick theorem that is rigorousif fluctuations are Gaussian but can also be used for a qualitative estimate inother cases.

As a result, for 2H!1, the correlation function has a finite limit at qZ0

hVqVKqiqZ0zZv F

a

! "2 z4

R2 ; !4:6"

where z and R are the characteristic height and radius of ripples, respectively.

M. I. Katsnelson and A. K. Geim200



This leads to excess resistivity

drzh

4e2z4

R2a2: !4:7"

For 2HO1, the resistivity is proportional to n1K2H and, for 2HZ1, to ln2(kFa).Belowwe argue that rippleswith 2HZ2 can naturally occur in graphene owing to

the way it is prepared. Indeed, single-layer samples studied in transportexperiments are currently obtained by micromechanical cleavage, a process inwhich individual graphene sheets detach from bulk graphite and then attach to aSiO2 substrate (Novoselov et al. 2005a). It is sensible to assume that during thedeposition process there is a transient state in which a detached sheet behaves as afree-standing membrane and, accordingly, exhibits dynamic out-of-plane fluctu-ations induced by the room temperature environment (e.g. Abedpour et al. 2007;Fasolino et al. 2007). As graphene touches the substrate, van der Waals forcesinstantly pin the rippled configuration (at least, partially), and any furtherreconstruction should be suppressed because this would require local movementsalong the substrate that stronglybinds the atomically thin sheet (onemight imaginethe deposition process as a microscopic equivalent of placing a cling film on top of atable). Therefore, the corrugations originally induced by temperature are expectedto become static and, also, persist to lower temperatures without notable changes.

To estimate scattering on such ripples, we note that, in the simple harmonicapproximation, the average potential energy per individual bending modeEqZkq4hjhqj2i=2 is equal to kBT/2 (kz1 eV is the bending stiffness of graphene),which yields

hjhq j2iZk BT

kq4: !4:8"

Numerical simulations using a realistic atomic interaction potential in graphenedirectly confirm this result for the case of bending fluctuations with a typicallength scale smaller than l#z7–10 nm (Fasolino et al. 2007). The linear ndependence of conductivity is experimentally observed for doping levels suchthat kFl

#O1, which justifies the use of the harmonic approximation here. Wethen use equation (4.8) for the pair correlation function and find the rippleresistivity as

rr zh

4e2!k BT=ka"2

nL; !4:9"

where the factor L is of the order of unity for kFl#y1 and weakly, as ln2!kFl

#",depends on n for kFl

#[1. The above equation shows that thermally createdripples lead to charge-carrier mobility m practically independent of n, inagreement with experiments. Importantly, equation (4.9) also yields m of thesame order of magnitude as observed experimentally. One can interpret!k BTq=ka"2z1012 cmK2 as an effective concentration of static defects (ripples)induced at the quench temperature Tq of 300 K. We emphasize that for a weakdisorder, that is, in the Born approximation, the above formalism can be used todescribe electron scattering by both static (quenched) and dynamic ripples,assuming that, first, they are classical scatterers and, second, their energy atrelevant q is smaller than the energy of scattered Dirac fermions ZqvF (Ziman2001). The former condition means that kk2

Fa2/kBTq and is satisfied in the

existing experiments; the latter holds if q/ZvF/k and is even less restrictive.




Molecular dynamics with classical poten0als

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

•  Large 0me scale ~ns

•  Bond-‐order poten0als for C-‐H

•  Boundary condi0ons 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 posi0on: DFT calcula0on

ΔE = 0.21 eV

3 Å

ΔE = 0.37 eV

3 Å

Van der Waals interac0on

h

Ripple height analysis

2 0.89vacuumh = Å2

2 0.42tolueneh = Å2

2 0.39hexaneh = Å2

Bending s0ffness of graphene in liquid

Bending S)ffness

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

hq2=TNκA0q

4

Liquid suppresses flexural phonons

This leads to excess resistivity

drzh

4e2z4

R2a2: !4:7"

For 2HO1, the resistivity is proportional to n1K2H and, for 2HZ1, to ln2(kFa).Belowwe argue that rippleswith 2HZ2 can naturally occur in graphene owing to

the way it is prepared. Indeed, single-layer samples studied in transportexperiments are currently obtained by micromechanical cleavage, a process inwhich individual graphene sheets detach from bulk graphite and then attach to aSiO2 substrate (Novoselov et al. 2005a). It is sensible to assume that during thedeposition process there is a transient state in which a detached sheet behaves as afree-standing membrane and, accordingly, exhibits dynamic out-of-plane fluctu-ations induced by the room temperature environment (e.g. Abedpour et al. 2007;Fasolino et al. 2007). As graphene touches the substrate, van der Waals forcesinstantly pin the rippled configuration (at least, partially), and any furtherreconstruction should be suppressed because this would require local movementsalong the substrate that stronglybinds the atomically thin sheet (onemight imaginethe deposition process as a microscopic equivalent of placing a cling film on top of atable). Therefore, the corrugations originally induced by temperature are expectedto become static and, also, persist to lower temperatures without notable changes.

To estimate scattering on such ripples, we note that, in the simple harmonicapproximation, the average potential energy per individual bending modeEqZkq4hjhqj2i=2 is equal to kBT/2 (kz1 eV is the bending stiffness of graphene),which yields

hjhq j2iZk BT

kq4: !4:8"

Numerical simulations using a realistic atomic interaction potential in graphenedirectly confirm this result for the case of bending fluctuations with a typicallength scale smaller than l#z7–10 nm (Fasolino et al. 2007). The linear ndependence of conductivity is experimentally observed for doping levels suchthat kFl

#O1, which justifies the use of the harmonic approximation here. Wethen use equation (4.8) for the pair correlation function and find the rippleresistivity as

rr zh

4e2!k BT=ka"2

nL; !4:9"

where the factor L is of the order of unity for kFl#y1 and weakly, as ln2!kFl

#",depends on n for kFl

#[1. The above equation shows that thermally createdripples lead to charge-carrier mobility m practically independent of n, inagreement with experiments. Importantly, equation (4.9) also yields m of thesame order of magnitude as observed experimentally. One can interpret!k BTq=ka"2z1012 cmK2 as an effective concentration of static defects (ripples)induced at the quench temperature Tq of 300 K. We emphasize that for a weakdisorder, that is, in the Born approximation, the above formalism can be used todescribe electron scattering by both static (quenched) and dynamic ripples,assuming that, first, they are classical scatterers and, second, their energy atrelevant q is smaller than the energy of scattered Dirac fermions ZqvF (Ziman2001). The former condition means that kk2

Fa2/kBTq and is satisfied in the

existing experiments; the latter holds if q/ZvF/k and is even less restrictive.




Conclusion •  Liquid dielectric environment suppresses flexural phonons

•  Phonon suppression affects mobility through bending s0ffness

Четвертый основной компонент электрической цепи current for a resistor (blue), capacitor (red), inductor (green) and memristor (purple). The lower figures show the current-voltage characteristics for the four devices, with the characteristic pinched hysteresis loop of the memristor in the bottom right. It is nearly obvious by inspection that the memristor curve cannot be constructed by combining the others.

There are also arguments that there are far more than four fundamental electronic circuit elements. In fact, Chua has shown that there are essentially an infinite number of two-terminal circuit elements that can be defined via various integral and differential equations that relate voltage and current to each other [L. O. Chua, Nonlinear Circuit Foundations for Nanodevices, Part I: The Four-Element Torus. Proc. IEEE 91, 1830-1859 (2003) – this is an interesting tutorial for the beginner], to which the memcapacitor and meminductor belong. It comes down to whether one wants to think of all of these possible circuit elements as being on an equal footing or choose the four lowest order relations to be a fundamental set with a large number of higher order cousins. Similar considerations apply in other fields – do we consider electrons, protons and neutrons fundamental or quarks or what?

Who 'Discovered' the Memristor? The memristor as a mathematical model or entity was discovered and made rigorous by Leon Chua.

Independent of and even preceding his discovery, there were experimental observations of pinched hysteresis loops in two-terminal electrical measurements in a variety of material systems and subsequent development of devices based on those observations. We are not aware of any useful mathematical models presented in any of these previous works for predicting the behavior of these devices in an electronic circuit. We are not aware that any of these researchers cited Chua's papers after they appeared in print. In turn, Chua was not aware of these studies (except for one that he discussed in his 2003 paper cited above) – but this is not surprising, since he is an electronic circuit theorist and the experimental

3

The nonlinearity exists because of coupled electronic and ionic conducOon, the laPer being mediated by defects, typically vacancies or

intersOOals.

Pd/WO3/W TiO2

•  R(w(t))=RONw(t)/D+ROFF(1-‐w(t)/D)

WRITE SPEED VS. RETENTION WRITE SPEED VS. RETENTION

D.Strukov et al. Appl.Phys.A 94 515 (2009)

linear ionic transport linear ionic transport

nonnonlinearlinear effect due to temperature and/or electric field

TI

I

write

store ~)(

)0(

V

V

DV

Vv

v=

=

= µ

!

!

)(~ writeB

A

storeB

A

Twrite

store Tk

U

Tk

U

eeV

V"

!

!

e.g. temperature only:

strong nonlinearity in ionic transport required for high retention, even

M.Janousch et al. Adv.Mat 19 2232

(2007)

perform 3D coupled electro-thermal simulations

map temperature on I-V

2.0 1.0 0.0 1.0420

2468

1000 K

900 K

voltage, V

curr

ent,

mA

- -

-

-

Joule hJoule heatingeatingXRF map

infrared

oother sources of nonlinearity?ther sources of nonlinearity?

Dmitri Strukov, UCSBSRC/NSF/A*STAR forum

October 2009

R.Waser et al. Adv.Mat 212632 (2009)

A Bmobile interacting ions

ionic conductor

Electrochemical effects…

Buttler Volmer reactions

L=100 nm30 nm

10 nm3 nm

0.1 0.2 0.3 0.5

107

104

103

101

10-1

UB (eV)

! sto

re/!

wri

te

D.Strukov (TBP)

high retention, even more for half select

J.Borghettiet al.JAP

(2009)to appear

v

RC

RON

ON state

dONz

r

0wON

OFF state

wOFF

TiO2

( I I)

dC

electrode ( E E)

metallicchannel( C C)

dOUT

L

fit experimental data using equivalent circuit

perform 3D

1100 K

D.Strukov et al. MRS (2009)

290K140K3K

15

10

5

0

-5

I (m

A)

-1.0 -0.5 0.0 0.5 1.0V (V)

ONOFF

INTERMEDIATE

ON

SHORT

OFF

600

500

400

300Loca

l Tem

pera

ture

(K

)

3020100I (mA)

Domain fitted on dataExtrapolation

extract geometry from fitting

Однополярный Биполярный

Биполярный механизм поверхностного переключения

The most common type of insulators in the sandwich structures are metal oxides with high concentra0ons of oxygen vacancies, such as NiO, HfO2, ZnO,, Al2O3, WO3, and TiO2

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] direc0on

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

лекция 5 memristor

Technology

Transcript of лекция 5 memristor