Quantum and classical ripples in graphene

Hasik, Juraj; Tosatti, Erio; Martoňák, Roman

doi:10.1103/physrevb.97.140301

Cited by 27 publications

(16 citation statements)

References 34 publications

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“…For physical membranes, the scaling exponent was calculated at leading and next-to-leading order, giving η 0.821 and η 0.789 respectively [18][19][20]. These results are compatible with predictions obtained by numerical simulations [13,19,[21][22][23][24][25][26][27][28][29]. Recently, the statistical mechanics of crystalline membranes has been revisited by non-perturbative renormalization group (NPRG) techniques [26,[30][31][32][33][34].…”

Section: Introductionsupporting

confidence: 77%

Scaling behavior of crystalline membranes: an $ε$-expansion approach

Mauri,

Katsnelson

2020

Preprint

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We study the scaling behavior of two-dimensional (2D) crystalline membranes in the flat phase by a renormalization group (RG) method and an ε-expansion. Generalization of the problem to non-integer dimensions, necessary to control the ε-expansion, is achieved by dimensional continuation of a well-known effective theory describing out-of-plane fluctuations coupled to phonon-mediated interactions via a scalar composite field, equivalent for small deformations to the local Gaussian curvature. The effective theory, which will be referred to as Gaussian curvature interaction (GCI) model, is equivalent to theories of elastic D-dimensional manifolds fluctuating in a (D + d c )-dimensional embedding space in the physical case D = 2 for arbitrary d c . For D = 2, instead, the GCI model is not equivalent to a direct dimensional continuation of elastic membrane theory and it defines an alternative generalization to generic internal dimension D. After decoupling interactions through a Hubbard-Stratonovich transformation, we study the GCI model by perturbative field-theoretic RG within the framework of an expansion in ε = (4 − D). We calculate explicitly RG functions at two-loop order and determine the exponent η characterizing the long-wavelength scaling of correlation functions to order ε 2 . The value of η at this order is shown to be insensitive to Feynman diagrams involving vertex corrections. As a consequence, the self-consistent screening approximation for the GCI model is shown to be exact to O(ε 2 ). In the physical case of a single out-of-plane displacement field, d c = 1, the O(ε 2 ) correction is suppressed by a small numerical prefactor. As a result, despite the large value of ε = 2, extrapolation of the first and second order results to D = 2 leads to very close numbers, η = 0.8 and η 0.795. The calculated exponent values are close to earlier reference results obtained by nonperturbative renormalization group, the self-consistent screening approximation and numerical simulations. These indications suggest that a perturbative analysis of the GCI model could provide an useful framework for accurate quantitative predictions of the scaling exponent even at D = 2.

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Section: Introductionsupporting

confidence: 77%

Scaling behavior of crystalline membranes: an $ε$-expansion approach

Mauri,

Katsnelson

2020

Preprint

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“…We would like to reinforce that even neglecting out-of-plane vibrational modes, our results for the flat membrane are a good reference for first estimates because they resemble the physics of a free-standing graphene monolayer in agreement with experiments. Very recently, PIMC simulation results [48] in agreement with [49] have shown that the crossover between classical and quantum contribution for the out-of-plane vibrational modes should occur around 50K. For lower temperatures quantum fluctuations should leave the membrane flat on the large scale.…”

Section: Numerical Results and Discussionsupporting

confidence: 52%

Strong anharmonicity in pristine graphene

Rabelo

Cândido

2018

J. Phys. Commun.

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The thermodynamic coefficients of a free standing infinite graphene monolayer are calculated using the quasi-classical unsymmetrized self-consistent field method (USF). The basic nonlinear integral equations of this theory are solved numerically in the strong anharmonic approximation. The isothermal and adiabatic elastic bulk moduli, the isochoric and isobaric heat capacities, the thermal expansion, thermal pressure coefficient, and the macroscopic Gruneisen parameter are calculated in terms of the derivatives of a specifically chosen interatomic potential function for different values of stress and for temperatures ranging from below room temperature up to the point of loss of thermodynamic stability. The nearest-neighbor distances vary from 1.4 Å to 1.8 Å for zero stress. Under stress, these distances decrease. At room temperature the molar heat capacities are ∼5.0Jmol −1 K −1 . The elasticity moduli vary from 15.0eV 2 -Å up to zero at the temperature of loss of stability and are increased by stress. The thermal expansion coefficient has a strong dependence on the temperature and is negative for temperatures lower than ∼340K. For high temperatures it monotonically increases and decreases with stress. The macroscopic Gruneisen parameter has a strong nonlinear dependence with temperature and is estimated in about 3.0 3.7;at ∼340K its value decreases to ∼1.0K and for even lower temperature it shows a peak and deep structure similar to what has been earlier reported for fullerene C 60 .The thermodynamic properties of crystals are related to atomic vibrations around their equilibrium positions. The relative displacements are small in comparison with the interatomic distance. Even in quantum crystals, where the zero point vibrations are large, the mean square deviations from the sites are about 30% of interatomic distances while for most of the solids they do not exceed about 10% up to the melting temperatures. This allows for the expansion of the potential energy in a power series of the nuclei displacements in the Born-Oppenheimer approximation [22]. For crystals with strong anharmonicity, perturbative techniques based on harmonic or quasiharmonic approximations do not work. The pseudoharmonic approximation [19][20][21] reduces the study of strong anharmonicity to the analysis of weak interacting phonons and because of that it is more suited to the investigation of the phonon spectrum and related phenomena.The USF has been applied over the years to several types of crystals but mostly with Bravais lattices, and for non Bravais ionic crystals [23,24]. It has been also applied to noncrystalline solids [25]. Recently [26], we have reported on a generalization of the USF to the case of layered structured crystals. In that work though, the thermodynamics of a free standing graphene monolayer was investigated in the weak anharmonic approximation. The anharmonicity has a close relation to the thermal properties of graphene and on-going debates about the contribution of different phonon modes to heat conduction. The...

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“…In contrast to our MD results for σ 0 , previous Monte Carlo (MC) works claim that such term should not be present. 20,52 Both MD and MC methods should provide identical results. The origin of this disagreement is not clear, but it might be due to inaccuracies in the sampling of the sluggish long-wavelength modes.…”

Section: Surface Tensionmentioning

confidence: 99%

Thermal control of graphene morphology: A signature of its intrinsic surface tension

Ramı́rez

Herrero

2018

Phys. Rev. B

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The surface tension σ of free-standing graphene is studied by path-integral simulations as a function of the temperature and the in-plane stress. Even if the applied stress vanishes, the membrane displays a finite surface tension σ due to the coupling between the bending oscillations and the real area of the membrane. Zero-point effects for σ are significant below 100 K. Thermal cooling drives the membrane from a planar to a wrinkled morphology. Upon heating the change is reversible and shows hysteresis, in agreement to recent experiments performed on supported graphene.

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Quantum and classical ripples in graphene

Cited by 27 publications

References 34 publications

Scaling behavior of crystalline membranes: an $ε$-expansion approach

Scaling behavior of crystalline membranes: an $ε$-expansion approach

Strong anharmonicity in pristine graphene

Thermal control of graphene morphology: A signature of its intrinsic surface tension

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