The structure of crystalline
membranes and graphene
Doron Gazit
Institute for Nuclear Theory,
University of Washington.
Based on DG, arXiv: 0810.1062 (PRB 79, 113411(2009)),
0903.5012, 0907.3718.
Introduction
 Crystalline membranes are solid-like structures with 2D
character.
 Very common in our world:
 Cytoskeleton of red blood cells – whose structure is vital for the
operation and stability of the cell – forms a triangular lattice.
 In soft condensed matter, one can create crystalline lattices by
polymerizing liquid interfaces.
 In condensed matter layered materials of tens to hundreds of
layers.
 However – the isolation of graphene, and then of other single
layers, has conquered the final frontier – with many
implications.
Nelson, Piran, Weinberg (eds.), “statistical mechanics of membranes and surfaces”, (2004).
August 2009
Doron Gazit - on crystalline membranes and graphene
2
Novoselov et al. Science, 306, 666 (2004) ; PNAS, 102, 10451 (2005).
Introduction (2)
 Thus:
 Only one atom thick, graphene should represent the ultimate
crystalline membrane, and can be used as a simple model,
where calculations are feasible.
 The fact that graphene can be used to construct nanometer-sized
electronic applications, has only enhanced the need of a
profound understanding of its structure, which is a critical
ingredient in the design and quality control of such applications.
 However,
 Experiments show that graphene possesses intrinsic ripples, of
sizes 100-300 Å.
 Theory predicts a scale invariant cascade of corrugations.
 What is the origin of this difference?
August 2009
Doron Gazit - on crystalline membranes and graphene
3
Outline
 Crystalline membranes.
 The structure of physical crystalline membranes within the
self-consistent screening approximation.
 Graphene: an electronic crystalline membrane!
 On the correlation between charge inhomogeneities and ripples
in graphene.
 Spontaneous buckling of hole doped graphene.
 Outlook.
August 2009
Doron Gazit - on crystalline membranes and graphene
4
Physical Crystalline Membranes
 Membranes are D dimensional entities, embedded in a d
dimensional world.
 Physical membranes: D=2, d=3.
 Crystalline membranes are built of a lattice with fixed
connectivity.
 The main question: does a flat phase exist? What is its
structure?
 Is the Mermin-Wagner theorem violated?
 Phase stability is a long wavelength question, thus continuum
theory.
August 2009
Doron Gazit - on crystalline membranes and graphene
5
The flat phase of a crystalline membrane
 We need to describe an almost flat phase in the continuum.
 We use the Monge representation:
 Describe a deviation from the flat phase by:
r
r  (u
,...,u
,h
,...,h
)

u
,h


1
D
1
d
D
14 2 43 1 4 2 43
 The metric is:
in plane
outof  plane
gij  ij  2uij ; i, j  1,...,D
the strain tensor:


1 r
r
1
1
uij  i u j   j ui  i h   j h + i uk j uk
2
2
2
 We expand around a flat surface, keeping leading orders in h
and u.
August 2009


Doron Gazit - on crystalline membranes and graphene
6
The flat phase of a crystalline membrane (2)
 In addition, bending the membrane costs energy.
r
r  r
 This can be expressed using the curvature tensor: Cij  n 
x ix j
 For small deformations, the energetics is a sum of bending energy
and elastic energy:
2
r
1
F u,h 
2




r
2
2
2
d D x  TrC  G TrC  Tr C


 Tru  2Tr u 2 
2
 Note:

Linear terms are not included as the membrane  ij  F ; M ij  F
uij
Cij
is assumed to be free.
Cross terms break the symmetry between the two sides of the
membrane.

The Gaussian curvature energy is invariant under small deformations
for D=2, thus not important for the structure.
August 2009

Doron Gazit - on crystalline membranes and graphene
7

The flat phase of a crystalline membrane (3)
 Thus we will focus on the following free energy:
r
1
F u,h 
2

2
2
r
2
2
d x[  h  uii   2uij  ]
D
 For a bulk 3-d material, small perturbations are given in terms of the
elastic part only, characterized by 3d, and 3d (or equivalently the
Young modulus K3d and the Poisson ratio n3d).
 For a physical membrane with a lateral size h:
4   
h 3K 3d
K0 
 hK 3d n  n 3d  
2  
121 n 2 
 The height of the membrane: h  121 n 2  
K0
 Thus, the height of graphene is about 1 Angstrom, i.e. less than the
lattice
 size!

 Graphene, in the continuum limit is truly the ultimate membrane!
August 2009
Doron Gazit - on crystalline membranes and graphene
8
The flat phase of a crystalline membrane (4)

kB T
h
q
h
q

I




The Green’s function of the h field:
dC dC
 R qq 4
With the effective bending rigidity  R q
r
d q
kB T
2
2z
 z – the roughness exponent: h  
~
L
D
4


q
q


2



R

q 0
 The bending rigidity  R q ~ q,   4  D 2z .
L1
D
q 0
 In principal, one can define an elasticity exponent: ,  ~ q u


 From Ward identities u  4  D 2


, which means
that h fluctuations
are rdivergent. But normal-normal fluctuations…
2r
2
1
r
L
r
h
d
q
q
2
2
2z 2

n


h
n

qh
~
~
L
0

 2 2  qq4

r 2
L 
  R
1 h
 Experimentally,
0  z 1
  
Thus, an asymptotically flat phase.
August 2009

Doron Gazit - on crystalline membranes and graphene
9
The flat phase of a crystalline membrane (5)
 Thus we will focus on the following free energy:
r
1
F u,h 
2

2
2
r
2
2
d x[  h  uii   2uij  ]
D
 The in-plane phonon fields enter quadratically – thus can be
integrated out:

 The effective four-leg interaction:
August 2009
Doron Gazit - on crystalline membranes and graphene
10
The flat phase of a crystalline membrane (5)
 What is the relevant perturbation scheme?
 Expand in the number of bubbles, implying that the elastic
interaction is small compared to the bending energy.
 By power counting this is true only for q 
kB TY
applicable for the structure issue.
2
, thus not
 A different scheme hides in the fact:

 Every interaction contributes a factor 1/dC=1/(d-D).
 Every h field propagator adds dC=d-D.
 Thus, a relevant perturbation scheme is in powers of 1/dC.
Topologically, diagrams contributing to order n in perturbation
theory fulfill the condition: n=NR-Lh, (NR- number of
interactions, Lh-number of loops).
August 2009
Doron Gazit - on crystalline membranes and graphene
11
Physical Crystalline Membranes
 The interaction is separable:
 Feynman diagrams:
h propagator
Interaction propagator
Interaction vertex
 Dyson equations:
August 2009
Doron Gazit - on crystalline membranes and graphene
12
Self Consistent Screening Approximation
 SCSA is an extension of a consistent perturbative expansion.
 One replaces each propagator with the dressed propagator.
 Cutting the expansion at a specific order.
 SCSA for a first order expansion:
 Le Doussal and Radzihovsky, PRL 69, 1209 (1992).
 SCSA for a second order expansion:
 DG, arxiv: 0907.3718 (2009).
August 2009
Doron Gazit - on crystalline membranes and graphene
13
SCSA – general results
 The scaling relationu
 4  D 2 holds to all orders.
 At any order the SCSA equations:

 and y are polynomials in z0y0-2.

 For D>2, the long-wavelength Poisson ratio is -1/3.
 The first order SCSA coincides with:

First order expansion in 4-D.

d=D

Large dC expansion.
August 2009
Doron Gazit - on crystalline membranes and graphene
14
Second order SCSA for physical crystalline membranes
 Second order - a naïve 2-loop expansion or a 1/dC2 expansion?
 No solution for a naïve 2-loop expansion!
 Thus, the 1/dC character of the expansion is essential.
 However, even though dC=1, the results are very close: z
changes only by 2%.
 One should try and check if this still coincides with the e2
expansion, as this will give an indication of the error.
August 2009
Doron Gazit - on crystalline membranes and graphene
15
Method

u
z
References
1st order SCSA
0.821…
0.358…
0.590…
PRL, 69, 1209 (1992)
2nd order SCSA
0.789…
0.421…
0.605…
This work
e Expansion [1]
0.96
0.08
0.52
PRL 60, 2634 (1990)
Large dC [2]
2/3
2/3
2/3
EPL 5, 709 (1988)
Exact RG [3]
0.849
PRE 79, 040101 (2009)
Simulations [4]
0.75-0.85
0.50(1)
0.64(2)
PRE 48, R651 (1993), J.
Phys 6, 3521 (1996),
arxiv:0903.3847
Experiment [5]
-
-
0.65(10)
Science, 259, 952 (1993)
 All methods are consistent with each other.
 None show any unusual finite q behavior.
 One of the numerical simulations is for graphene – thus graphene
would have been a great experimental device.
 The other universal parameter implies:

1 kB TK R q
 3.5731
q 0 q
 R2 q
lim
Only 6% away from the 1st order SCSA prediction.
August 2009
Doron Gazit - on crystalline membranes and graphene

16
Structure of Graphene
 Meyer et al. [Nature, 446, 60 (2007)], have characterized suspended
graphene sheets:

Showed stability. No defects were found even at strain > 10%.

Used TEM diffraction patterns to determine that there is a
characteristic ripple on the surface of 100-250 Å.
 Different groups have isolated suspended graphene far above SiO2
substrate.

Guinea, Horovitz, and Le Doussal [Sol. St. Com. 149, 1140 (2009)],
suggested a mechanism that results in ripples due to stress in the
production process.
 It is commonly believed that these are real inherent ripples.
 External effects were still not ruled out, though.
 I suggest an inherent mechanism for ripple creation due to charge
inhomogeneities, energetically favored.
August 2009
Doron Gazit - on crystalline membranes and graphene
17
Atomistic simulations of graphene
 Los et al. [arxiv: 0903.3847] have used a carbon-carbon
potential to calculate normal-normal correlation.
 They found a behavior consistent with the theory of physical
membranes. No sign of ripples.
August 2009
Doron Gazit - on crystalline membranes and graphene
18
How would ripples
look in the normalnormal correlation
function?
Los, Fasolino, Katsnelson, Nature
Materials 6, 858 [2007].
August 2009
Doron Gazit - on crystalline membranes and graphene
19
Other experiments
 Gerringer et al., PRL 102, 076102 (2009).
 The experiments in J. Folk’s group are not yet published, but
show L~8nm.
August 2009
Doron Gazit - on crystalline membranes and graphene
20
So…
 As graphene is a real membrane, this difference is rather
disturbing.
 I suggest that the origin of this different is the additional
degree of freedom: the -electrons.
 Considering the fact that these electrons are responsible for the
specific size of the lattice (determine the resonant bond), such
an effect is reasonably large.
 How do ripples affect the electronic structure?
August 2009
Doron Gazit - on crystalline membranes and graphene
21
Dirac picture
 An effective theory around the Dirac points.
 Low momenta excitations are possible only around the Dirac
point, thus:
ai,  e
bi,  e
r r
iK  R n
r r
iK  R n
 Defining:
 Hamiltonian:
 The effective
H  ihv f  

August 2009
r
K
i,
r
K
i,
a e
b e
r r
iK ' R n
r r
iK ' R n
r
K'
i,
r
K'
i,
a
b
r
aK 
r
ˆ K   i,r 

i,
 K 
bi, 

r K r
r K' r
K† r r
K'† r r *
ˆ
ˆ
ˆ
ˆ r 
dxdy  r   r    r   


Semenoff
Doron Gazit - on crystalline membranes and graphene

1984
22
An effective Dirac Action
 In Euclidean space:
2
S  h  0 d 

r
d xy   y
2
 1
1

kB T
K 
y   K' 
 
0
i

,


2

;





;

  i 1
  n  n
3
3
gn  diag1,v f ,v f 
August 2009
Doron Gazit - on crystalline membranes and graphene
23
Effects of corrugation on the electronic structure
 The Dirac picture is an effective picture, due to the tight-
binding Hamiltonian.
 Thus, though attractive conceptually, it is not really a massless
fermion in curved space (however...).
 Possible effects:
 Deformation energy.
 Pseudo-magnetic gauge fields.
 Electric gauge fields
August 2009
Doron Gazit - on crystalline membranes and graphene
24
Deformation Energy
 In the presence of corrugations, the surface area changes:
S ~ a uii
2
 This changes the ion density, and thus the electron density.
 As a result, the electron’s chemical potential is locally changed
 an effective induced electric field:

V r  Duii ; D  20  30eV

August 2009
Suzuura, Ando PRB, 65, 235412 (2002).
Doron Gazit - on crystalline membranes and graphene
25
Effective Gauge Field
 The hopping integral changes due to the change in angles
between normals and distances in the lattice.
 Due to corrugation and ripples:
  
   0  
u 
uij
 ij 

August 2009
Doron Gazit - on crystalline membranes and graphene
26
 Thus:
H   a†k bk'  i abe 

r r r
r r
 k  k '  R i iaa  k '
r r
k ,k '
 h.c
i
 bk†bk'  a†k ak'  i aae
 
r r r
r r
 k  k '  R i iab  k '
i
 Or…
 2

S  h  0 d
 1




Deformation
Energy +
NNN effects
 With
August 2009

r r
r
r
r r
0
d xy   0  i r  hv f   i 5 Ar  y
2
r r
g2  2uxy 
Ar   

v f uxx  uyy 
g2 ~ 1 3 eV
NN effect,
Keeping T
invariance
r
3a 2 2 2
 r   Duii  g3
 h

4
g ~ 10 eV
3
Doron Gazit - on crystalline membranes
and graphene
27
Intrinsic ripples in graphene
 The structure of graphene is determined by a mutual
minimization of the lattice free energy and the electronic one.
 In the absence of electron inhomogeneities – this results in an
elastic free energy.
 However, allowing inhomogeneities, keeping only the
deformation energy:
August 2009
Doron Gazit - on crystalline membranes and graphene
28
Intrinsic ripples in graphene
 We estimate the electron-electron interaction by:
 The effective screening is big, Kotov et al., PRB 78, 035119
(2008) showed that:
 E=3-4.
 But, they do it perturbatively and the series does not seem to
converge, as the fine structure is of order 1.
August 2009
Doron Gazit - on crystalline membranes and graphene
29
Intrinsic ripples in graphene
 Integrating out the in plane phonon fields and charge
fluctuations:
 For 3D materials, negative Young modulus means instability.
 For 2D materials, this means inherent competition between
bending and stretching.
August 2009
Doron Gazit - on crystalline membranes and graphene
30
Intrinsic ripples in graphene
 The Dyson equations in this case:
 Estimating Sand  in first order:
 Searching for maximum in the normal-normal correlation
function.
August 2009
Doron Gazit - on crystalline membranes and graphene
31
Size of ripples
August 2009
Doron Gazit - on crystalline membranes and graphene
32
What is the meaning of all this?
 The best way to get some insight, is to rewrite the effective
free-energy as:
 The Gaussian curvature:

 Let us assume that impurities contribute to the charge density
as well:
August 2009
Doron Gazit - on crystalline membranes and graphene
33
The meaning
 Naively, without a bending energy term, one expects a glass-
phase, whose ground states are solutions to the equation:
 This is a basic difference from what was done previously, since
they looked for a correlation in the mean curvature.
August 2009
Deshpande
al.graphene
PRB 79,
Doron Gazit - on crystalline
membranesetand
205411 (2009)
34
Electron Hole puddles
Ldisorder~10-30 nm
Martin et al., Nature Physics, 4, 144 (2008).
August 2009
Doron Gazit - on crystalline membranes and graphene
35
Results
 The electron-phonon coupling originating from the deformation
energy competes with the electron-electron interaction.
 Formation of ripples correlated with electron-hole puddles is
favored, both reproduce the experimental length scale.
 Indeed, Graphane is found to have less corrugation, consistent with
no corrugation.
Elias et al., Science, 323, 610 (2009).
 Additional work is required to specify the form of graphene in the
presence of few impurities.
 The dynamics of the electrons was neglected.
 What about finite chemical potential?
August 2009
Doron Gazit - on crystalline membranes and graphene
36
Doped graphene
 We neglect electron-electron interaction!
2
S  h  0 d 

 1


r r
r
r
r
0
d xy   0  i r  hv f  i 5 Ar  y
2

 Since vf>>vph we integrate out the electronic degrees of
freedom.
August 2009
Doron Gazit - on crystalline membranes and graphene
37
Feynman Diagrams for -Electrons
i
k
 Fermion propagator
 Vector pot./Fermion vertex

r
i 5
 Electro-chem. Pot/Fermion
vertex
August 2009

i 0
Doron Gazit - on crystalline membranes and graphene
38
Integrating out the -electrons
 The resulting Lagrangian is pure gauge.
1
1 V 2
A
L  Ai ij A j   V
2
2
L

S 
r
d xL  Feff
2
 The structure is frozen to a very good approximation, thus
polarizations can be calculated using zero frequency.

August 2009
Doron Gazit - on crystalline membranes and graphene
39
“Structure” Polarization operators
r
d k





k

q


r
1

k
 0
 0

V q0  0;q     
Tr

 
2
2
2
  1  f

2   k
k  q

r

2


2


k

q


r
1
d
k

k

i
j



 ijA q0  0;q     
Tr




5
5
2
2
2
  1  f

2   k
k  q

r
2

f 
n 1
k   f ,k
2

2

August 2009

2


Doron Gazit - on crystalline membranes and graphene
40
Electro-chemical “Structure” Polarization Operator


P  hv f  q   hv f q  0 dx ln 2cosh
2


2
4
V
1

4 ln 2


1
 

x 1 x  
 

  1

4
 T 1  1
hv f q  15
 
 1
300ÞK  100A 
Exact
P
4 ln 2
qiq j 
r hv f q 
 q  
ij  2 
4 
q 
A
ij
r
V q  
q
4hv f


4


August 2009

Doron Gazit - on crystalline membranes and graphene
41
 Electrons contribution to the free energy
F 

r
dq q
2
2  4hv f
2
r
2 
 2
ˆ
V

hv
A

q
f




r r
g2  2uxy 
Ar   

v f uxx  uyy 
V  Duii

August 2009
Doron Gazit - on crystalline membranes and graphene
42
Elastic Free-Energy
r
1
F u,h  
2

r 
hv f q r 2 
2
dq
2
2
4
q h  quii   2quij  
A  qˆ 
2 
4

2  
2
D2q
q   
4hv f

g22q
q   
8hv f

The effect on the shear modulus is negligible.
August 2009
Doron Gazit - on crystalline membranes and graphene
43
The effect of an external electro-chemical
potential
r
1
F u,h  
2

r
2
d 2q
2
2
4
V

q
h


q
u

2

u

2D
u







q; Vq

ii
ij 
ii q
2
2 

r
r
dq
V
F u,h   D 

q; uii q Vq
2
2 
V r
 D q  0; uii qr  0 V 
2

4Dln 2
 hv f  
2

V

r r
d xuii x   
2



r r
d xuii x 
2
 Chemical potential leads to stress!

August 2009

4Dln 2
 hv f  
2
V
Doron Gazit - on crystalline membranes and graphene
44
Buckling term in the elastic free energy
 Let’s assume a tensionless membrane in a negative chemical
potential (hole doping).
 The stress is negative  Buckling of graphene, as it has zero
thickness!
 The merit of the buckling wave length:
F h 
1
2

r 2
r 
2
d 2 x h   h 


 


*

F
0
 q 4 h  q 2 h    0 q  
 h
*
  0
q
0

!
 hv f  
 T 1/ 2  D 1/ 2  V 1/ 2
  2
 144A 
  
  

300ÞK 
20eV 
100meV 
4Dln 2  V
2

August 2009
Doron Gazit - on crystalline membranes and graphene
45
Physical dopings
 Chemical adsorption:
 T 1/ 2  D 1/ 2  ad 1/ 2  n ad 1/ 2
  150A  
  
  
 

300ÞK 
30eV 
200meV  20% 
 Gate voltage:

 In any case ~F1/2.
August 2009
Doron Gazit - on crystalline membranes and graphene
46
Conclusions
 Graphene is an example of a new class of materials: electronic
crystalline membranes, in which a strong interplay exists between the
structure and the free electrons in the membrane.
 The -electrons induce:

Ripples due to competition between electron-electron interaction and
electron-phonon interaction.

Additional stress in the presence of a chemical potential, which can
lead to buckling in the case of hole doping.
 Outlook:

Additional work is needed: incorporate electron-electron interactions
without neglecting dynamics, …

Phase space in the presence of impurities.

Effect on transport.
August 2009
Doron Gazit - on crystalline membranes and graphene
47