{'fraction_non_alphanumeric': 0.04039510541058529, 'fraction_numerical': 0.01729814732910708, 'mean_word_length': 4.47484530535378, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Critical radius and temperature for buckling in graphene

In this work, we find an analytical flat-membrane solution to the saddle point equations, derived by Guinea et al. [Phys. Rev. B 89, 125428 (2014)], for the case of a suspended graphene membrane of circular shape. We also find how different buckled membrane solutions bifurcate from the flat membrane at critical temperatures and membrane radii. The saddle point equations take into account electron-phonon coupling and this coupling provides a residual stress even for a flat graphene layer. Below a critical temperature (which is exceedingly high for an infinite layer) or above a critical size that depend on boundary conditions, different buckling modes that may be the germ of rippling appear. Our results provide the opportunity to develop new feasible experiments dealing with buckling in small suspended graphene membranes that could verify them. These experiments may also be used to fit the phonon-electron coupling constant or the bending energy.


I. INTRODUCTION
The discovery of graphene has spawned one of the most fertile scientific fields of today.
Thermal and external stresses have been used to create large ripples (wrinkles) in a layer of graphene which is suspended over trenches [16]. These ripples have been explained using classical elasticity [16] and seem to be qualitatively different from the nanometer sized ripples observed in [3,17,18]. The latter tend to be considered inherent to suspended graphene.
Even when ripples in graphene are not fully understood, one of the most promising approaches to ripple formation is the quantum mechanical study of the membrane and the phonon-electron interaction as a mechanism for rippling [7,8]. Since the graphene Debye temperature is T D ∼ 1000 K, the quantum treatment of graphene seems justified at room temperature. Perturbation treatments point to the vanishing of the renormalized bending energy of the membrane as a possible mechanism for ripple formation [8,10]. The destabilizing effects of quantum fluctuations on the bending rigidity of crystalline membranes (without electron-phonon interaction) have been considered in [19]. Within equilibrium theory and without phonon-electron interaction, the effect of (Matsubara) frequency-dependent renormalization of the anharmonic coupling after elimination of in-plane phonons has been considered in [20]. the graphene layer is not buckled initially, a strong enough electrostatic force can produce irreversible buckling [28]. A spontaneous buckling effect ("mirror buckling") is also found by Xu et al. in [29]. This effect is systematically studied in [15], in a complementary approach to that in [27]. In [15], the authors study a suspended graphene monolayer using scanning tunnelling microscopy (STM). The microscope tip is located on the centre of an initially non-buckled layer. The STM keeps a constant current and a variable potential between the tip and the sample. Once the current is fixed at a sufficiently high value, the sample buckles when the potential increases, similarly to the experiments in [27,28].
These experimental efforts have their theoretical counterpart through different approaches to the buckling of two dimensional (2D) layers such as graphene. A broad overview of the effects of strain in graphene and its relation with its electrical properties can be found in [30]. The study of spontaneous buckling in graphene due to doping is carried out in [7]. In polymerized membranes, buckling occurs below a critical temperature [31]. Phenomenological models of crystalline membranes coupled to spins undergoing Glauber dynamics also indicate that there are critical temperatures below which the membranes buckle [12,13].
In [9], Guinea et al. have developed a model for suspended graphene based in the coupling between flexural phonons and electrons. They consider a 2D membrane in equilibrium that is embedded in a 2 + d dimension space, where there are d dimensional out-of-membrane displacements (d = 1 in the physical situation). Then they eliminate the in-plane phonons, ignore the frequency dependence of the resulting couplings, use the self-consistent screening approximation and perform a saddle-point analysis of the free energy in the limit as d → ∞.
The resulting saddle-point equations (SPEs) are time-independent and will be analyzed in the present paper. They consist of one von Kármán type plate equation for the physical out-of-plane displacement coupled to two equations for two auxiliary fields: one "scalar" stress associated to membrane curvature and a field associated to charge fluctuations. We expect these equations to describe buckling and ripples of the graphene membrane in a stable stationary (equilibrium) configuration after all possible transients have decayed. We find flat membrane solutions with constant non-zero auxiliary fields. The linearization of the stationary SPEs about these solutions accompanied by appropriate boundary conditions provide an eigenvalue problem that yields critical values of temperature and membrane size corresponding to the bifurcation of buckling states from the flat membrane. Why?
The stationary SPEs provide stationary solutions to not yet derived dynamic SPEs that should describe the graphene membrane out of equilibrium. Among them, we have found the stationary flat membrane solution. Other solutions may bifurcate from it at appropriate parameter values. To find them, we have to solve the eigenvalue problem that governs the linear stability of the stationary flat membrane solution to dynamic SPEs. The corresponding eigenvalues give the growth of disturbances about the stationary solution. We do not know the precise shape of the dynamic SPEs but we may surmise that stationary solutions bifurcating from the flat membrane appear when these eigenvalues are zero. But linearized dynamic SPEs with zero eigenvalues are the same as linearized stationary SPEs, which we know from [9]. These linearized stationary SPEs have nonzero solutions only for particular values of the constant auxiliary fields that correspond to critical temperature and membrane sizes. The eigenmodes solving these equations give the shape of the buckling states near bifurcation points. A combination of eigenmodes may characterize ripples in graphene.
The rest of the paper is as follows. In section II we briefly revise the derivation of the stationary saddle-point equations [9] and write them in real (not Fourier) space. In section III, we find the flat membrane solution of the SPEs for constant auxiliary fields. In Section  • The deformation energy of the graphene sheet is the sum of kinetic energy, H kin , and of curvature and elastic energy, H elas , with Here u i , i = 1, 2, are the in-plane phonon displacements, h a , a = 1, ..., d are the out-ofplane flexural phonon modes, u ij : , λ and µ are the Lamé constants, and κ is the bending energy.
• The energy of the free Dirac fermions (in units such that = 1), Here σ = (σ x , σ y ) are the Pauli matrices, v F is the Fermi velocity of Dirac electrons in graphene such that, in these units, v F a = 5 eV (a = 1.4Å is the side of a graphene hexagon).
• The electron-phonon coupling where δρ(x) is the the charge distribution on the graphene layer, ρ 0 is the equilibrium carrier density, and g 0 is in the range between 4 and 50 eV.
• Finally the Coulomb interaction has the form where V 0 (q) = 2πe 2 0 q is the Fourier transform of the electrostatic potential, V 0 (r) = e 2 /( 0 r), 0 is the dielectric constant of the environment and −e < 0 is the charge of the electron.
Once all interactions are defined, the resultant Hamiltonian is integrated over the in-plane phonons, leading to a coupled theory of flexural phonon modes and electrons. The frequency dependence of the resulting couplings is ignored, as it is important only for temperatures below 90 K [20], that we do not consider here. In this process, the Coulomb interaction The effective Hamiltonian, depending only on the electrons and flexural phonon modes, is used to build a Matsubara equilibrium partition function. For later convenience, this partition function is transformed using two fluctuating auxiliary fields, σ and α, defined through the relation, and table I gives the dimensions of parameters and variables. In (7) where With this definition, Φ(x) is related to the Gaussian curvature (K) through the relation Φ(q) = K(q)/q 2 . In (7), σ(x) is a local linear combination of Φ(x) and the charge disturbance δρ(x), whereas α(x) is a nonlocal linear combination, as it involves the potential energy of one electron in x with respect to The action corresponding to the Matsubara partition function is where µ is the chemical potential. We now decompose the vertical displacements into an average and a fluctuating part, h a = h a + δh a , and assume symmetry breaking in the  (7), (11) and (12).
To obtain the saddle-point equations, we vary (10) with respect to σ, α and h 1 , thereby where we have replaced h 1 = h and not yet made µ = 0 as in [9]. Eq. (11) provides the average fields σ 0 = σ(x) and α 0 = α in terms of h. The average auxiliary fields are related to the averages of the charge distribution (4) and of the field (8), by the linear equation (7), which also holds for the averages of the corresponding quantities.
(12), (13) and (14) are von Karman plate equations (they appear in Fourier transform form in [9]). The stress tensor σ ij is generated by the average auxiliary field σ 0 and it is automatically in equilibrium as ∂ x i P T ij = 0. P T ij is the transversal projector, and ω n = 2πn/β and ω n = 2π(n + 1 2 )/β are the bosonic and fermionic Matsubara frequencies. The matrix Green function appearing in (11) satisfies (16) and G(x , x ) can be computed from (16) as . (17) It is clear from (17) . Using this translation invariant Green's function, we can write (11) as Henceforth from this equation we suppress the subscript 0 in all average quantities.
Let us now set µ = 0 as in [9]. It is important to note that the flat membrane h(x) = 0 is always a solution of (12). However, the right side of (18) is not zero for h(x) = 0 and therefore the trivial solution (h = σ = α = 0) is not a valid solution of the SPEs. As we will see in the next section, h = 0 and non-zero σ and α is a valid solution. Nonzero auxiliary fields σ and α provide a residual stress that induces plate buckling and rippling below a certain critical temperature and over a critical plate size. In a mathematically related phenomenon, a growing bacterial biofilm can be modeled as a plate with a growth tensor that modifies the elastic part of strain, acts as residual stress and may trigger ripples (called wrinkles in that application) [32]. The critical growth term may then be sought by solving an appropriate eigenvalue problem for h(x) coming from the linearized plate equations [33].
The right side of (20) is computed separately in Appendix A with the result where Λ = 2π/a is an ultraviolet momentum cutoff (a is the side of a graphene hexagon).
The term Λ 2 /2π in (21) is proportional to the area of the Brillouin zone, and it will play a key role in the computation of the critical temperature and radius for the graphene layer.
We discuss in Appendix B other possible interpretations of (20) that do not involve setting the chemical potential µ = 0 and turn out to be unphysical. From (20) we obtain The left side of (23) is much smaller than the right side and it can be ignored, thereby producing the solution which gives σ by insertion in (22).

IV. LINEARIZED EQUATION AND EIGENVALUE PROBLEM
We want to ascertain the linear stability of the solution found in the previous section, h = 0 and constant auxiliary fields. For this we need to know the dynamics and to linearize the corresponding SPEs about this solution. Let us assume that the dynamic SPEs are so that are the stationary SPEs (12)- (14) and (18). Let u 0 (p) be the flat solution h = 0 and constant σ(p) and α(p) of (22) and (24). Here p is a bifurcation parameter, later to be identified as the temperature or the radius of a circular graphene membrane. Linear stability about u 0 follows from substituting u(t) = u 0 + e υt U in (25) and keeping terms of order U in the result. We obtain the eigenvalue problem where δF δu , δG δ∂u/∂t and δG δ∂ 2 u/∂t 2 are functionals acting upon U ≡ U(x). To solve the linearized stationary SPEs, we need appropriate boundary conditions. To solve the linearized plate equation (12), we need the value of σ ij in (13) for the constant σ and α of (22) and (24). Now u = ∆ −1 σ solves the equation ∆u = σ with appropriate boundary conditions. A solution is For a circular plate with zero data Dirichlet boundary conditions at r = R the solution is u = σ 4 (r 2 − R 2 ). Then, by (13) and (14), σ ij = σδ ij /2, and (12) becomes There are different eigenvalue problems associated to solving Equation (29) with different boundary conditions for a finite graphene membrane. Each of these problems yield its critical temperature and size and allows us to know the shape of different eigenmodes that appear as buckled solutions of the suspended layer of graphene. We consider a finite circular layer of graphene with a free border (natural boundary conditions) or a graphene sheet clamped to a circular hole. Suspended graphene sheets are clamped at their boundaries but the case of a free border is easier to treat mathematically, so we start by analyzing it.
The solution of ∆h = H that satisfies the other boundary condition is for m = 0 (normalized so that h n,0 (0) = 0), and for m > 0.
We are interested in computing the critical temperature and radii. To this end, we use (31) with equations (22) and (24), and find that ripples appear below the critical temperature Note that the critical temperature decreases as the size R decreases. As R/a → ∞, the critical temperature tends to 2 √ 3v F a ≈ 16.05 eV (about 186,000 K). For finite membrane size, (34) with γ n,m instead of γ 1,0 produces the following numerical estimate, Here T n,m is the temperature (measured in Kelvin) at which the mode h n,m appears and R is measured in nanometers. The values considered for the parameters are g = 3.9 eV (the lowest value considered in [9]) and κ = 1 eV. At room temperature, equation (35) gives us different critical radii above which the different modes h n,m (characterised by γ n,m ) appear, see table II and figure 1.

B. Clamped Membrane
In this more realistic case, the boundary conditions areh = 0 and ∂ rh = 0 at r = R instead of (30). We useh instead of h to distinguish the out-of-plane displacement in the clamped case. Following the same procedure as for equations (31) and (32), we get instead of (31), and in which k is not yet determined. Now the radial parth m (r) satisfies For m = 0, this equation is in which we have used the boundary conditionh (R) = 0. Integrating once and using h(R) = 0, we obtainh The vertical displacement is unbounded at r = 0 unless Thenh n,0 (r) = Clamped boundary conditions yield The condition thath n,m be bounded at r = 0 produces c 1 = 0. Thus J m+1 (kR) = 0, i.e., k = γ n,m+1 /R and therefore Equation (40) becomes which agrees with (39) for m = 0.
According to (43), the critical radii are given by (35) with γ n,m+1 instead of γ n,m . Then the critical radii of table III are greater than those in table II, and buckling of a clamped graphene membrane should be observable only for radii over 24 nm. Thus a clamped membrane remains flat for larger radii than in the case of natural free boundary conditions. This is according to our intuition that it is harder to buckle a clamped membrane than a membrane with a free border. The lowest possible value of σ in (31) corresponds to γ 1,1 = 3.8317.

V. CONCLUSIONS
In conclusion, the stationary saddle-point equations give the vertical displacement of a graphene membrane and auxiliary fields associated to curvature and to charge fluctuations, provided elasticity and phonon-electron interaction are considered [9]. We have solved these equations for a flat circular graphene sheet under constant auxiliary fields. We have then solved the SPEs linearized about the flat membrane solution and subject to either free or clamped boundary conditions. The latter problem has nonzero solutions only for discrete values of the auxiliary fields. These discrete values correspond to critical values of temperature and membrane radius at which buckling membrane solutions issue forth from the flat membrane. The flat membrane is stable only for temperatures larger than critical and radii smaller than critical. For an infinite membrane, the predicted critical temperature is extremely large and unphysical (186,000 K). However, critical radii at room temperature are in the nanometer range. Different modes of membrane buckling appear below specific critical temperatures and over specific critical radii, see tables II and III. The stability of these modes is to be analytically determined, but experiments describing rippling [3] and buckling [15] of graphene show that the flat solution is not stable in most cases: Flat graphene has only been observed when it stands over a substrate [36].
The calculated critical radii (tables II and III) suggest that the different modes appearing in figures 1 and 2 could be observed for a specific temperature and membrane size. This brings the opportunity to develop new experiments, as no buckling should appear below the minimum critical radius R c . Similar experimental procedures to the ones in [27][28][29], where suspended graphene layers were forced to buckle using an external electrical potential or a purpose, suspended graphene layers may be clamped to holes of different dimensions, under different temperatures. Checking when the layer may or not buckle, it should be possible to get a better estimation of g and κ by fitting the results of experiments with equation (34).
A different approach to study the effects due to the existence of non-flat modes would be experiments with graphene resonators [34,35]. In this setting, when the external driving force is sufficiently strong to deform the graphene layer close to one of the stable buckling modes (away from the flat configuration), the response of the resonator should be extremely The sum over Matsubara frequencies in the expression on the right side of equation (20) gives [37] 1 β We now split the last integral on the right hand side of (A2) as The result is because the other integrals are of order O(e −βα ) 1. In these computations q is upper bounded by Λ = 2π/a.

Appendix B: Selecting the chemical potential
Keeping a nonzero chemical potential, (20) and (21) yield These are two equations for three unknowns, σ, α and µ, so that we can give a value to µ and solve for σ and α. We have already seen the consequences of setting µ = 0. What do we get from other choices?
1. Assume that µ is selected so that µ + v F Λ = 0 and therefore the right hand side of (B1) (which represents the Fermionic propagator in real space for x = 0) becomes finite as Λ → ∞. Following the same steps as in previous sections, e.g. using equations (23) and (31), we get (the minus sign corresponds to g < 0) The right hand side of this equation contains the product R/β = RT . This produces a critical radius (at which the graphene sheet starts buckling) that decreases when the temperature increases. This is unphysical because increasing temperature would favor rippling, contrary to experimental evidence [16].
2. Let us set R = ∞ and α = 0 in (B1), thereby obtaining If α is no longer zero for finite radius, (23) gives Then either α = 0 and σ = 0, which contradicts (31), or In this case, the same arguments based on Equations (23) and (31) provide Once the numerical values of the constants are taken into account, we notice that the left hand side of (B6) is positive, whereas the second term in the right hand side of this equation (the minus sign corresponds to g < 0) is proportional to T R, and this again produces the same unphysical effect as before.
Thus we conclude that the physically meaningful choice of chemical potential is µ = 0 as in [9]. We do not renormalize the parameters of the model and leave the natural ultraviolet and infrared cutoffs Λ = 2π/a and q 0 = 2π/R, respectively, in terms of the lattice constant