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.
Download
Report
Transcript 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.
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
outof 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 ix 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 T rC G T rC T rC
T ru 2T ru 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:
2
2
r
1
2
Dr
2
F u,h d x[ h uii 2uij ]
2
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 3K3d
K0
hK3d n n 3d
2
121 n 2
The height of the membrane: h 121 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)
The Green’s function of the h field:
hqhq IdC dC
With the effective bending rigidity R q
kB T
R qq4
r
d q
D
kB T
2z
~
L
D
4
q
q
2
R
q 0
The bending rigidity R q ~ q , 4 D 2z .
z – the roughness exponent:
h
2
L1
q 0
In principal, one can define an elasticity exponent: , ~ q u
From Ward identitiesu 4 D 2
, which means
that h fluctuations
0 z 1
are rdivergent. But normal-normal fluctuations…
2r
1
r
L
r
h
dq
q2
2
2
2z 2
n
h
n
qh
~
~
L
0
2
4
r 2
L
2 R qq
1 h
Thus, an asymptotically flat phase.
Experimentally,
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 2uij ]
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 relationu 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
3.5731
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
a K
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
gn diag1,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†kbk' i ab e
r r r
r r
k k ' R i iaa 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 iab 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 Ar y
2
r r
g2 2uxy
Ar
v f uxx uyy
g2 ~ 1 3 eV
NN effect,
Keeping T
invariance
r
3a 2 2 2
r Duii g3
h
4
g3 ~ 10 eV
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 Sand 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 Ar 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 Aiij 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
k
q
r
1
d
k
k
0
0
V q0 0;q
T
r
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
T
r
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 dxln2cosh
2
2
4
V
1
4 ln2
1
x 1 x
1
4
T 1 1
hv f q 15
1
300ÞK 100A
Exact
P
4 ln2
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
2
2
2
q
4hv f
r
2
2
ˆ
V
hv
A
q
f
r r
g2 2uxy
Ar
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 quii 2quij
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
d 2q
2
2
r
q h quii 2uij 2Duii q V q; Vq
2
4
F u,h D
2
2
r
dq
2
2
2
V q; uii q Vq
r
D q 0; uii qr 0 V
V
4Dln2
hv f
2
V
r r
d xuii x
2
r r
d xuii x
2
Chemical potential leads to stress!
August 2009
4Dln2
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
q4 h q2 h 0 q
h
0 q* 0
!
hv f
2
T 1/ 2 D 1/ 2 V 1/ 2
2
144A
300ÞK
20eV
100meV
4Dln2 V
August 2009
Doron Gazit - on crystalline membranes and graphene
45
Physical dopings
Chemical adsorption:
T 1/ 2 D 1/ 2 ad 1/ 2 nad 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