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

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

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

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

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

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)

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.,arXiv1603.00398

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

??? 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

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

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

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

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

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

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

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

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

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

Anharmonicity-induced increase of the bending rigidity large-d expansion h  0.7-0.8 numerical simulations

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

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

Free energy Partition function Calculation: Decouple ( ξ2 – 1 + … )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

Thermal expansion coefficient

RG in the quantum regime retardation effects can be neglected

RG in the quantum regime Kats, Lebedev‘14

Solution of quantum RG equations Classical RG Quantum RG

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

Thermodynamics of membrane at low T buckling transition

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

Specific heat Gruneisen parameter 30

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

Cancellation of uv-divergent terms in the self-energy Partition function Calculation: Decouple ( ξ2 – 1 + … )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

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

Ultraviolet correction to self-energy

Buckling - projected area

Anomalous Hooke’s law

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)