1. Thermal expansion coefficient and specific heat of graphene
V.Yu. Kachorovskii
Ioffe Physico-Technical Institute, St.Petersburg, Russia (also Landau/KIT)
Co-authors:
I.S. Burmistrov (Landau/KIT)
I.V. Gornyi (KIT/Ioffe/Landau)
M.I. Katsnelson (Radboud Un)
A.D. Mirlin (KIT/PNPI/Landau)
Petersburg Nuclear
Physics Institute

2. Outline Introduction. Elastic properties of a crystalline membrane.
Flexural phonons .
Phase diagram of the crystalline membrane. Crumpling and
buckling transitions
Thermal fluctuations in the classical regime. RG description.
Negative temperature –independent thermal expansion
coefficient
Quantum fluctuations RG description . Suppression of the
thermal expansion coefficient . Specific heat and
Gruneisen parameter

3. Flexural phonons (FP) In-plane phonons out-of-plane flexural mode
bending rigidity
soft dispersion of FP
In-plane phonons
in-plane
elastic
coefficients

4. Global shrinking of membrane induced by FP
stretching
parameter
in-plane and
out-of-plane
fluctuations
global
deformation

5. Phase diagram of the crystalline membrane
Quantum
effects ???
Guitter, David, Leibler, Peliti,
PRL (1988)
FLAT, T
STRETCHED, non-zero tension, s > 0
BUCKLED
CRUMPLED,

6. Crumpling transition Crumpled phase Flat phase
Physics behind: anharmonic
coupling between FP and
in-plane modes
D. Nelson, T. Piran, S. Weinberg Statistical Mechanics of Membranes and Surfaces (1989).
critical behavior
of bending rigidity
F.David and E. Guitter,
Europhys. Lett. (1988)
P. Le Doussal , L. Radzihovsky,
PRL (1992)
h - critical index
( 0.7)

7. Buckling transition < 1 - critical index of buckling transition
anomalous Hooke’s law
at small(!!!) tension:
Guitter, David, Leibler, Peliti, PRL (1988);
Aronovitz, Colubovic, Lubensky J.Phys. France (1989)
Gornyi, Mirlin, Kachorovskii.,arXiv

8. Classical versus quantum regimes
classical coupling constant (for graphene g ≈30
even at T=300 K)
Gornyi, Kachorovskii, Mirlin’15
h - critical exponent ( 0.7)
quantum coupling
constant (for
graphene g0 ≈1/20 )
Kats, Lebedev‘14
Physics behind: anharmonic
coupling of FP and in-plane modes

9. ??? Arbitrary temperature Solution : generation of tension ?
Amorim, Roldan, Capelluti, Fasolino, Guinea, Katsnelson ,
“Thermodynamics of quantum crystalline membranes” PRB (2014)
???
Solution : generation of tension ?
ultraviolet
fluctuations
Kats, Lebedev, Comment on “Thermodynamics of …” PRB (2014):
in the absence of
external force

10. Theory of crumpling transition (classical regime)
Paczuski, Kardar, Nelson , PRL (1988)
physical membranes: d=3, D=2
This talk:
Mean field

11. Mean-field theory
stretching factor
for zero tension, s = 0
flat phase
crumpled phase

12. Energy of fluctuations
Mean
field
Fluctuations
around MF
in-plane and out-of-plane
fluctuations
Contribution of fluctuations to elastic energy:
strong
anharmonicity
strain tensor

13. Beyond mean field energy of fluctuations stretching energy
Fluctuations change stretching factor :
Mean field:
energy of fluctuations
stretching energy
coupling between stretching and fluctuations
Physics behind: transverse
fluctuations lead to decrease
of membrane size in x-direction

14. x 0, for certain value of L
Renormalization of x
minimization
of energy
classical limit, harmonic approximation, zero tension
logarithmic divergence  RG
x 0, for certain value of L
flat phase is destroyed
by thermal fluctuations

15. How to stabilize the flat phase?
1) Anharmonicity
2) External tension

16. Renormalization of the bending rigidity
in-plane modes
are integrated out
interaction between
out-of-plane modes:

17. Renormalization of bending rigidity by screened interaction
self-energy
Interaction is screened:
polarization
operator

18. bare coupling drops out !
Universal scaling
q << q*
bare coupling drops out !
ultraviolet cutoff
(Ginzburg scale) c c

19. Anharmonicity-induced increase of the bending rigidity
large-d expansion
h 
numerical simulations

20. Crumpling transition , s=0
critical temperature of CT
for fixed bending rigidity
coupling constant in
the classical regime

21. What happens with decreasing the temperature ?
Quantum effects:
ultraviolet cutoff
for classical RG
(Ginzburg scale)
coupling constant in
the quantum regime:
for graphene g0≈1/20 ≪1
Kats, Lebedev‘14
Classical RG
Quantum RG

22. Free energy Partition function Calculation:
Decouple ( ξ2 – … )2 by global auxiliary field χ
Integrate over {du dh}
Calculate integral over dχ by stationary phase method
stationary value
is imaginary
Stationary
phase
condition
External
tension

23. Thermal expansion coefficient

24. RG in the quantum regime
retardation effects can be neglected

25. RG in the quantum regime
Kats, Lebedev‘14

26. Solution of quantum RG equations
Classical RG
Quantum RG

27. Thermal expansion coefficient at low temperature
bending rigidity at q=qT :
thermal expansion coefficient
drops to zero at extremely low T

28. Thermodynamics of membrane at low T
buckling
transition

29. Effective elastic coefficient
Thermal extension coefficient for σ≠0
for s>s* out-of-plane
fluctuations are suppressed

30. Specific heat
Gruneisen parameter 30

31. Main results
Anharmonicity crucially effects elastic properties of graphene
Thermal expansion coefficient of graphene is negative and temperature-independent in a very wide range of temperatures. It drops down to zero at extremely low (exponentially small) temperature.
Thermodynamic properties of graphene are controlled by the parameter s/s*, where tension s*=mT/k corresponds to suppression of critical fluctuations

32. Cancellation of uv-divergent terms in the self-energy
Partition function
Calculation:
Decouple ( ξ2 – … )2 by global auxiliary field χ
Integrate over {du dh}
Calculate integral over dχ by stationary phase method
stationary value
is imaginary FP
in-plane phonons

33. ??? Finite T dynamical term Stationary phase condition: + uv-divergent
Stationary phase condition:
+ uv-divergent
terms
External tension:
uv-divergent
???

34. Ultraviolet correction to self-energy

35. Buckling
- projected area

36. Anomalous Hooke’s law

37. Critical indices for buckling transition (clean case)
STRETCHED
BUCKLED T
For h=0 , Guitter, David, Leibler, Peliti, PRL (1988);
Aronovitz, Colubovic, Lubensky J.Phys. France (1989)