Transcript Document 7242574

HUJI condensed matter seminar
A Spontaneous Buckling of
Hole Doped Graphene?
Based on arXiv: 0810.1062
Doron Gazit
Institute for Nuclear Theory
University of Washington
Doron Gazit
- INT
Background
picture
by J. Meyer, Max Planck Institute for Solid State Research, Germany.
Outline
Introduction.
Crystalline membranes.
Electronic structure.
Electron interaction with deformations in Graphene.
p electrons contributions to the elastic free energy.
Applications:
– Sound attenuation and directional softening.
– Spontaneous buckling of electrically and chemically
hole doped Graphene.
• Possible experimental signatures for the effects.
• Conclusions and outlook.
•
•
•
•
•
•
Doron Gazit - INT
HUJI condensed matter seminar
2
Carbon Allotropes
Diamond, Graphite
Graphene –
The origin of all allotropes
Carbon
nanotubes
Doron Gazit - INT
Fullerenes
HUJI condensed matter seminar
3
Discovery – at the tip of a pencil
Novoselov, Geim et al., Science 2004.
Doron Gazit - INT
HUJI condensed matter seminar
4
Why 2D materials shouldn’t be
• According to the Mermin-Wagner theorem:
long range order cannot exist for d<3.
• However, Graphene seems to be stable, even
when it is suspended.
• How come?
– Graphene flakes are small, it might be that large
flakes are not stable.
– This is not JUST a 2D material, but a 2D
membrane embedded in a 3D world!
Doron Gazit - INT
HUJI condensed matter seminar
5
Statistical mechanics of crystalline
membranes
• Membranes are d=2 dimensional entities,
embedded in a D=3 dimensional world.
• Crystalline membranes contain a lattice, with
small fluctuations of atoms around the
equilibrium lattice distance.
• Examples from soft-matter to solid state.
• The main question: does a flat phase exist? Is
the Mermin-Wagner theorem violated?
• Phase stability is a long wavelength question,
thus continuum theory.
condensed
matter seminar
Doron
Gazit
- INT (editors),HUJI
6
Nelson,
Piran,
Weinberg
“statistical
mechanics
of membranes and surfaces”, (2004).
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
)
1
d
1
Dd
1 2 3 1 4 2 43
in plane
outof  plane
– The metric is also almost flat:gij  ij  2uij ;i, j  1,...,d
the strain tensor, to leading orders in u and h:

Doron Gazit - INT
uij 
1
1

u


u

i h j h

i j
j i
2
2

HUJI condensed matter seminar
7

The flat phase of a crystalline
membrane
• The leading order free-energy is the elastic
free energy:
2
2
r
1
2
2r
2
F u,h   d x k  h  luii   2muij 


2
– k – the deformation energy = 1.1 eV.
– m– shear modulus
= 9 eV/Å2.
– lm – bulk modulus
= 11 eV/Å2.
• Gaussian theory in terms of the u field
1
r
1
F h   d x[k  h   K

]
;


P  h h
{
2
2
2
eff
2
2
2
0
T
ij i
j
4 m  m  l
2m  l
Doron Gazit - INT
HUJI condensed matter seminar
8
Elastic screening
K(q)
kR(q)
Doron Gazit - INT
HUJI condensed matter seminar
9
Self Consistent Screening
approximation
4
r

r
ˆ  qˆ
2
k
K
k
K 0 kB T
d k R
k R q

1

r r
2
2
 k
k
2p  K 0 k R q  k r r 4


qk

k

r  r 1
ˆ  qˆ 4
K q 1
2
k
k
k
 R    1 1 K 0 k2B T  d k 2  R 
r r


2 k
k
 K 0 
2
p
k
q
  
 R  k qr  kr 4


k


 



 

k R q 
~q
k
K R q
~ q u
K0
  0.81...
u  2  2

• As a result, the RG flow leads
to:
1
Feff h 
2

 h

2
~
r
d 2q
4
k
q
q
hq hq


2 R
2p 
L
r
d 2q
1
2 
2
~
L


 0.59
2
4
2
2p  k R qq
Doron Gazit - INT
Radzihovski, Le Doussal PRL 1991
HUJI condensed matter seminar
10
So…
r
r
h
n
r
1 h
 
n
2
2
r
 h
  nq

2
~

L
r
dq
q2
2 2
~
L
0
2

4
L 
2p  k R qq
2
• In other words: a stable asymptotically flat
 phase…
• This was verified in many systems.
Doron Gazit - INT
HUJI condensed matter seminar
11
Suspended Graphene
• Experimental and numerical
investigations show that indeed
Graphene is crumpled.
• However, they also show a preferred
wavelength, of the order of 50-150 Å.
• Also – a perfect lattice!
• Is it the carbon bond? other
peculiarities? Two-loop corrections?
Meyer et al., Nature 446, 60 (2007)
HUJI condensed matter seminar
Doron Gazit - INT
Los, Fasolino, Katsnelson, Nat. Mat. (2007)
12
Possible Explanations for the
Buckling
Monte-Carlo
• Fasolino et al. – carbon
bond.
• Guinea et al. – a term
proportional to uii is still
allowed by symmetry
considerations, and thus
will be produced by RG
flow.
Doron Gazit - INT
Experimental
• Some esoteric electronic
effect?
• Thompson-Flagg et al. –
Adsorption of molecules?
HUJI condensed matter seminar
13
Electronic structure
Wallace, 1947
Doron Gazit - INT
HUJI condensed matter seminar
14
Dirac picture
• An effective theory around the Dirac points.
• Low momenta excitations are possible only
around the Dirac point, thus:a  e a  e a
r r
iK  R n
i,
a
ˆ  


a
r
K
i,
r
K
i,
r
K'
i,




bi,  e
r r
iK  R n
r
K
i,
r
K
i,
b e
r r
iK ' R n
r r
iK ' R n
r
K'
i,
r
K'
i,
b
• Defining:

• The effective Hamiltonian:
H  ihv f 
 

Doron Gazit - INT
r K r
r K' r
K† r r
K'† r r *
ˆ
ˆ
ˆ
ˆ r 
dxdy  r   r    r   


HUJI condensed matter seminar

Semenoff 1984
15
An effective Dirac Action
• In Euclidean space:
2
S  h  0 d 

r
d x  m m
2
 1
1

kB T
K 
   K' 
 
0
i

,


2

;





;

  i 1
 m   m
3
3
gm  diag1,v f ,v f 
• Pseudo-spin eigenstates of massless relativistic
Fermions: Klein paradox, zitterbewegung, etc.
Doron Gazit - INT
HUJI condensed matter seminar
16
Effects of corrugation on the
electronic structure
• The Dirac picture is an effective picture, due
to the tight-binding Hamiltonian.
• Thus, though an attractive conceptually, it is
not really a massless fermion in curved
space (however...).
• Possible effects:
– Deformation energy.
– Pseudo-magnetic gauge fields.
Doron Gazit - INT
HUJI condensed matter seminar
17
Deformation Energy
• In the presence of corrugations, the surface
area changes: S ~ a 2 uii
• As screening is not complete, 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  10  30eV
Ando PRB, 2006
Doron Gazit - INT
HUJI condensed matter seminar
18
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 
Doron Gazit - INT
HUJI condensed matter seminar
19
• 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
Doron Gazit - INT

r r
r
r
r
0
d x   0  i r  hv f  i 5 Ar  
2
r r
g2  2uxy 
Ar   

v f uxx  uyy 
NN effect,
Keeping T
invariance
g2 ~ 1 3 eV
HUJI condensed matter seminar
20
p electrons effect on the surface
structure
• The surface structure is determined by the
elastic free energy:
– A low q expansion of the free energy.
– The contribution of the  electrons.
• At low q, the p electrons are well described
by the Dirac picture.
• All we have to do is to evaluate the
contribution of the Dirac particles to the
free-energy.
Doron Gazit - INT
HUJI condensed matter seminar
21

p Electrons contribution to the free
energy
• The acoustic velocity is much smaller than
the Fermi velocity: v  2 10 cm  v 10 cm
sec
sec
• Thus – one can integrate out the degrees of
freedom of the electrons.
6
8
ph
Z

r 
DuDhDexp S 
f

r
DuDh exp Seff 
• Seff should be found by means of functional
integration.
– Exact in 1D via bosonization.
– Perturbatively, via Feynman diagrams.
Doron Gazit - INT
HUJI condensed matter seminar
22
Feynman Diagrams for p-Electrons
i
• Fermion propagator
mkm
• No Vector/Electro-chemical potential
propagator.
• Vector pot./Fermion vertex 
r
i 5

• Electro-chem. Pot/Fermion
vertex
Doron Gazit - INT
HUJI condensed matter seminar
i 0
23
Integrating out the p-electrons
• The resulting Lagrangian is pure gauge.
1
1 V 2
A
L  Ai ij A j   V
2
2
L
S 


0
d 
r
2r
d xL   d xL  Feff
2
• The structure is frozen to a very good
approximation, thus polarizations can be
 calculated using zero frequency.
Doron Gazit - INT
HUJI condensed matter seminar
24
“Structure” Polarization operators
r
d k





k

q


r
1

k
 0
 0

V q0  0;q     
Tr

 
2
2
2
  1  f

2p   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

2p   k
k  q

r
2p
m
f 
n 1
k   f ,k
2

2

Doron Gazit - INT

2


HUJI condensed matter seminar
25
Electro-chemical “Structure”
Polarization Operator


P  hv f  q   hv f q  0 dx ln 2cosh
2
p

2
4
V
1

4 ln 2


1
 
p
x 1 x  
 

  1

4
 T 1 l 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
p

4


Doron Gazit - INT

HUJI condensed matter seminar
26
p Electrons contribution to the free
energy
Fp 

r
dq q
2
2p  4hv f
2
r
2 
 2
ˆ
V

hv
A

q
f




r r
g2  2uxy 
Ar   

v f uxx  uyy 
V  Duii

Doron Gazit - INT
HUJI condensed matter seminar
27
Elastic Free-Energy
r
1
F u,h  
2

r 
hv f q r 2 
2
dq
2
2
4
kq h  lquii   2mquij  
A  qˆ 
2 
4

2p  
2
D2q
lq  l 
4hv f

g22q
mq  m 
8hv f

• Attenuation of acoustic waves.
• Preferred directions.
Doron Gazit - INT
HUJI condensed matter seminar
28
Sound Attenuation – Figure of Merit
• For waves of the order of 100Å:
 D 2  l 1 eV
D2q
lq  l 
 2  2.5
 

30eV  100A  A2
4hv f

 g2 2  l 1 eV
g22q
mq  m 
 9  0.03
 

3eV  100A  A2
8hv f
• The bulk modulus is affected substantially.

Doron Gazit - INT
HUJI condensed matter seminar
29
Preferred Direction
• The last term reduces the free energy:
r
r 2qx qy 
A  qˆ ~ u  
q 2  q 2 

 x
y 
• Favors strains which follow the lattice
directions.

• Vanishes for q0.
• The size of the effect is 3 meV for ripples of
100Å.
Doron Gazit - INT
HUJI condensed matter seminar
30
So…
• The prominent effect is a substantial change
in the bulk modulus.
– Due to the electro-chemical potential.
• Should be observed experimentally.
• Are the preferred directions noticeable?
What else is hiding there?
Doron Gazit - INT
HUJI condensed matter seminar
31
The effect of an external electrochemical potential
• The deformation energy has the effect, and
symmetry,
of an electro-chemical potential.
2
S  h  0 d 



r r
r
r
r
0
d x   0  iV r  hv f  i 5 Ar  
2
 1

• Let’s add a constant, external, chemical
potential:
V  Duii  V
1
Fp 
2


r
dq V r
 q; Vq2
2p 
2
r
Vq  Duii q  2p  V q 
2

Doron Gazit - INT
HUJI condensed matter seminar
32
The effect of an external electrochemical potential
• Neglecting quadratic terms in the external
potential:
r
r
1
dq
F u,h   
kq h  lqu   2mu   2Du   q; V 

2
2p
2
2
4
2
2
 
ii
2
r
r
dq
V
F u,h   D 

q; uii q Vq
2
2p 
V r
 D q  0; uii qr  0 V 
2

4Dln 2
hv  
2

V

V
ii q
ij
q

r r
r
2r
d xuii x     d xuii x 
2
f
Doron Gazit - INT
HUJI condensed matter seminar
33
Buckling terms in the elastic free
energy
• For simplicity – let’s use vanishing in-plane
strains, and leading order in the out-of-plane
r 
1
r 
F
h

d
x
k
h



deformations:   2      h 
• Minimize the free-energy:
2
2
2


*
! F

0
 kq 4 h  q 2 h    0 q  k
h
  0
q*  0


• This has been verified in MC simulations,
even for non-vanishing strains.

Doron Gazit - INT
HUJI condensed matter seminar
34
Spontaneous Buckling of Graphene
Under Negative External Potential
• For V<0 we get:
4Dln 2

hv  
2
V  0
f
• The resulting buckling wave length:
p hv  k
 T 
 D 

  2p
 144A 
  
  
2
1/ 2
1/ 2
f

4Dln 2  V
300ÞK 
20eV 
1/ 2

100meV 
V
• However, calling V an external potential is

a bit misleading…
Doron Gazit - INT
HUJI condensed matter seminar
35
What is the external potential?
• An external electro-chemical potential in the
single electron picture, is the single electron
chemical potential.
• Sources of chemical potential are numerous:
– Chemical doping by adsorbents.
– Doping by charge impurities.
– Electric doping using gate voltage.
Doron Gazit - INT
HUJI condensed matter seminar
36
Chemical Doping
• The sign of V relates to endothermic and
exothermic reactions.
• NO2 molecules behave like holes, of energy
~100meV, per molecule.
• Water vapors are absorbed, behave like
holes, and have chemical potential of ~200
meV.
• NH3 behaves like an electron doping.
• An ideal place to check the effect.
Doron Gazit - INT
HUJI condensed matter seminar
37
Ripples due to chemical adsorption
• This can be the source of ripples found by
Meyer et al (Nature 2007), due to the
adsorption of water vapors.
 T 1/ 2  D 1/ 2  mad 1/ 2  n ad 1/ 2
  150A  
  
  
 

300ÞK 
30eV 
200meV  20% 
• The adsorption parameters have temperature
 dependence.
• More work is needed.
Doron Gazit - INT
HUJI condensed matter seminar
38
Electric Doping – Electric Field
Effect in Graphene
Novoselov, Geim, et al., Science, 2004
• Using the electric field effect, one can use
gate voltage to dope.
• The gate induces charge density,
via its capacitance, on the membrane.
n
0Vg
 n ~ 1012 cm2  Vg ~ 100V
te
Vg  gate voltage
  substrate dielectric constan t
Bolotin et al.,
PRL 2008
t  substrate thickness
• Very attractive property!
Doron Gazit - INT
HUJI condensed matter seminar
39
Buckling in electrically hole doped
Graphene
• The relation between the single electron
chemical potential and the doping:
 f ~ hv f pn
• Thus:


Doron Gazit - INT
1/ 4
 V 
g


 ~ 270A

10V


HUJI condensed matter seminar
40
Circumstantial evidence for
buckling - resistivity
• Ripples provide new source of scattering for
charge carriers.
• Should increase the resistivity.
• Many measurements show asymmetry
between hole- and electron-doped Graphene.
• The leading effect is electron-phonon
scattering (deformation energy).
pD2 kB T
  2
4e hmv 2f v 2ph
Doron Gazit - INT
Hwang, Da-Sarma, PRB 2008
Stauber et al,PRB 2006 41
HUJI condensed matter seminar
Bolotin et al. 2008
Doron Gazit - INT
HUJI condensed matter seminar
42
Temperature and density dependence of the resistivity
Bolotin et al.,
PRL 2008
Doron Gazit - INT
HUJI condensed matter seminar
43
Morozov et al., PRL 2008
Is this another circumstantial
evidence?
Can the difference between
measurements be explained
by chemical adsorbents,
causing ripples?
Vg ,T  L Vg  S T 

AT+BT5
Doron Gazit - INT
HUJI condensed matter seminar
44
Ripples induced resistivity
Katsnelson, Geim, Phil. Trans. R. Soc. A 2008
• The only work about the subject.
• Assumes thermal phonon distribution, up to
the buckling wavelength, which is assumed
quenched.
• Clearly, an improvement is needed.
2


h T 
  2   
e k a 
Doron Gazit - INT

HUJI condensed matter seminar
45
Spontaneous Buckling and
Resistivity
Adsorbent molecules
• No gate voltage
dependence.
1/ 2
 T 

300ÞK 
  150A 
Electrical doping
• Only for negative gate
voltage.
n ad T 1/ 2
 

20%


2
 ~ T 2 2 ~ T  mn1/
ad T 
• Morozov et al ?!
Doron Gazit - INT

 V 1/ 4  T 1/ 2
g

 ~ 270A

10V  

 300ÞK 
T
~T  ~
n
2 2
• Bolotin et al ?!

HUJI condensed matter seminar
46
Conclusions
• A novel method of including p-electrons
effect on the structure of Graphene was
presented.
• Possible effects are sound attenuation, and
preferred directions of ripples.
• The most interesting byproduct is a
spontaneous buckling of hole doped
Graphene.
Doron Gazit - INT
HUJI condensed matter seminar
47
Conclusions (2)
• Hole doped Graphene can be electrically and
chemically produced.
• Some circumstantial evidence was presented.
• However, a microscopic investigation is highly
called for.
• With the “proliferation” of Graphene based
electronics, structure effects are important.
• Finally, another esoteric property of Graphene.
Doron Gazit - INT
HUJI condensed matter seminar
48
Outlook
• Quantitative calculation of ripples using the
free energy.
• Effect of the ripples on the resistivity.
• Lattice simulations (w/ J. Drut).
• What is the source of the buckling in MC
simulations?
Doron Gazit - INT
HUJI condensed matter seminar
49