Thermal Properties of Graphene Under Tensile Stress

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COMPUTATIONAL METHOD branes and soft condensed matter.24,25 In this respect, graphene can be dealt with as a model system where de- scriptions at an atomic level can be connected with phys- A. Path-integral molecular dynamics ical properties of the material. Thus, thermal properties of graphene have been investigated in recent years,30–34 and in particular its thermal expansion and heat conduc- We use PIMD simulations to study equilibrium proper- tion were studied by various theoretical and experimental ties of graphene monolayers as a function of temperature 4,31,35–38 techniques. and pressure. The PIMD procedure is based on the Feyn- Several theoretical works carried out to study ther- man path-integral formulation of statistical mechanics,54 modynamic properties of graphene (e.g., specific heat, an adequate nonperturbative approach to study finite- thermal expansion, ...), were based on density-functional- temperature properties of many-body quantum systems. theory (DFT) calculations combined with a quantum In the applications of this method to numerical simula- quasi-harmonic approximation (QHA) for the vibrational tions, each quantum particle is represented by a set of 39–41 modes. This is expected to yield reliable results at NTr (Trotter number) replicas (or beads), that behave as low temperature, but may be questioned at relatively classical-like particles forming a ring polymer.51,52 Thus, high temperature, especially for the thermal expansion, one deals with a classical isomorph whose dynamics is ar- due to an important anharmonic coupling between in- tificial, since it does not reflect the real quantum dynam- plane and out-of-plane modes, not included in the QHA. ics of the actual particles, but is useful for effectively sam- Moreover, classical Monte Carlo and molecular dynamics pling the many-body configuration space, yielding precise 26,42–44 45–48 simulations based on ab-initio, tight-binding, results for time-independent equilibrium properties of the 10,28,49,50 and empirical potentials can give reliable results quantum system. Details on this type of simulation tech- at relatively high temperature, but fail to describe ther- niques are given in Refs. 51,52,55,56. modynamic properties at T < ΘD, with ΘD & 1000 K the Debye temperature of the material. This means, in The interatomic interactions between carbon atoms are particular, that room-temperature results obtained for described here by using the LCBOPII effective potential, some properties of graphene from classical atomistic sim- a long-range bond order potential, which has been mainly ulations may be clearly unrealistic. These shortcuts may employed to carry out classical simulations of carbon- be overcome by using simulation methods which explic- based systems.57 In particular, it was used to study the itly include nuclear quantum effects, in particular those phase diagram of carbon, including graphite, diamond, based on Feynman path integrals.11,51–53 and the liquid, and showed its accuracy in predicting Here we use the path-integral molecular dynamics rather precisely the graphite-diamond transition line.58 (PIMD) method to study thermal properties of graphene In recent years, this interatomic potential has been also under tensile stress at temperatures between 12 and found to describe well several properties of graphene,10,50 2000 K. The thermal behavior of the graphene surface and its Young’s modulus in particular.20,59,60 This inter- is studied, considering the difference between in-plane atomic potential was lately employed to carry out PIMD and real areas. The in-plane thermal expansion coeffi- simulations, which allowed to quantify quantum effects in cient turns out to be negative at low temperatures, with graphene monolayers by comparing with results of clas- a crossing to positive values at a temperature which de- sical simulations,11 as well as to study thermodynamic creases fast as tensile stress is raised. Particular emphasis properties of this 2D material.61 Here, in line with ear- is laid on the temperature dependence of the specific heat lier simulations,11,20,28 the original parameterization of at low T , for which results of the simulations are com- the LCBOPII potential has been slightly changed to rise pared with predictions based on harmonic vibrations of the zero-temperature bending constant κ from 1.1 eV to the crystalline membrane. This approximation happens 1.49 eV, a value close to experimental data.62 to be noticeably accurate at low temperatures, once the frequencies of out-of-plane ZA modes are properly renor- The calculations presented here have been performed malized for changing applied stress. in the isothermal-isobaric ensemble, where one fixes the The paper is organized as follows. In Sec. II we describe number of carbon atoms (N), the applied stress (P ), the computational method employed in the simulations. and the temperature (T ). The stress P in the refer- Results for the internal energy and enthalpy of graphene ence xy plane of graphene, with units of force per unit are given in Sec. III. The thermal expansion is discussed length, coincides with the so-called mechanical or frame in Sec. IV. In Sec. V we present results for the specific tension.20,23,63 P is obtained in the simulations from the heat, whereas in Sec. VI the compressibility of graphene stress tensor τ, whose components are given by expres- is discussed. In Sec. VII we summarize the main results. sions such as61,64 3

N NTr NTr 1 1 ∂U(r1j ,..., rNj ) τxy = (mj vij,xvij,y − 2kj uij,xuij,y) − , (1) NA  NTr ∂ǫ  * p i=1 j=1 j=1 xy + X X X   where the brackets h· · · i indicate an ensemble average plied. Carbon atoms are free to move in the z coordinate 65 and uij are staging coordinates, with i = 1, ..., N (out-of-plane direction), and in general any measure of and j = 1, ..., NTr. In Eq. (1), mj is the dynamic the real surface of a graphene sheet at T > 0 should give mass associated to uij and vij,x, vij,y are components a value larger than the in-plane area. In this respect, it of its corresponding velocity. The constant kj is given has been discussed for biological membranes that their 2 2 by kj = mj NTr/2β ~ for j > 1 and k1 = 0. Here properties should be described using the notion of a real −1 25,72 β = (kB T ) , U is the instantaneous potential energy, surface rather than a projected (in-plane) surface. A and ǫxy is an element of the 2D strain tensor. The stress similar question has been also recently posed for crys- 11,20,26,27 P , conjugate to the in-plane area Ap, is given by the trace talline membranes such as graphene. This may of the stress tensor: be relevant for addressing the calculation of thermody- 1 namic variables, as the in-plane area Ap is the variable P = (τxx + τyy) . (2) conjugate to the stress P used in our simulations. The 2 real area (also called true, actual, or effective area in the Effective algorithms have been employed for the PIMD literature23,25,72) is conjugate to the usually-called sur- simulations, as those described in the literature.64,66,67 In face tension.15 particular, a constant temperature T was accomplished The in-plane area has been most used to present by coupling chains of four Nosé-Hoover thermostats, and the results of atomistic simulations of graphene an additional chain of four barostats was coupled to the layers.25,50,53,59,73 For biological membranes, however, it area of the graphene simulation box to yield the required has been shown that values of the compressibility may 55,64 stress P . The equations of motion were integrated significantly differ when they are related to A or to Ap, by employing the reversible reference system propaga- and something similar has been recently found for the tor algorithm (RESPA), allowing to define different time elastic properties of graphene from classical molecular steps for the integration of the fast and slow degrees of dynamics simulations.20 freedom.68 The kinetic energy K has been calculated by Here we calculate a real area A in three-dimensional using the virial estimator, which displays a statistical un- (3D) space by a triangulation based on the actual posi- certainty significantly smaller than the potential energy, tions of the C atoms along a simulation run (in fact we V = hUi.64 The time step ∆t associated to the inter- use the beads associated to the atomic nuclei). Specif- atomic forces has been taken as 0.5 fs, which was found ically, A is obtained from a sum of areas corresponding to be suitable for the carbon atomic mass and the tem- to the structural hexagons. Each hexagon contributes perature range considered here. More details on this kind as a sum of six triangles, each one formed by the posi- of PIMD simulations can be found elsewhere.64,69,70 tions of two adjacent C atoms and the barycenter of the Rectangular simulation cells with N = 960 atoms have hexagon.20 been considered, with similar side lengths in the x and The instantaneous area per atom for imaginary time y directions (Lx ≈ Ly ≈ 50 A),˚ and periodic boundary (bead) j is given by conditions were assumed. Sampling of the configuration space was performed in the temperature range between 2N 6 1 j 12 K and 2000 K. A temperature-dependent Trotter num- Aj = T (3) N kn =1 n=1 ber has been defined as NTr = 6000K/T , which yields a Xk X roughly constant precision for the PIMD results at dif- 69–71 j ferent temperatures. Given a temperature T and a where Tkn is the area of triangle n in hexagon k, and the stress P , a typical simulation run included 3×105 PIMD sum in k is extended to the 2N hexagons in a cell con- steps for system equilibration and 6 × 106 steps for cal- taining N carbon atoms. Here, the triangles are defined r culation of average properties. with the coordinates ij corresponding to bead j of atom i. The area A is then calculated as

NTr B. In-plane vs real area 1 A = Aj (4) NTr * j=1 + As explained above, in the isothermal-isobaric ensem- X ble used here we fix the applied stress P in the xy plane, It is clear that A coincides with Ap for strictly planar allowing fluctuations in the in-plane area of the simula- graphene, and in general one has A ≥ Ap. Both areas tion cell for which periodic boundary conditions are ap- display temperature dependencies qualitatively different. 4

0.3

-2 0.25 P = - 0.5 eV Å 0.25

T = 500 K 0.2 0.2 REBO -2 - 0.2 eV Å 300 K Internal energy (eV / atom) Internal energy (eV / atom) P = 0 25 K 0.15 0.15 0 200 400 600 -0.6 -0.4 -0.2 0 -2 Temperature (K) Stress (eV Å )

FIG. 1: Internal energy per atom E − E0 as a function of FIG. 2: Stress dependence of the internal energy of graphene temperature for P = 0 (circles), −0.2 (squares), and −0.5 eV for three temperatures: T = 25 (circles), 300 (squares), and A˚−2 (diamonds). Symbols represent results of PIMD simula- 500 K (diamonds). Symbols indicate simulation results and 2 3 tions. Error bars are smaller than the symbol size. Solid lines lines are fits to the expression E − E0 = c0 + c2P + c3P . are guides to the eye. The zero-point energy amounts to 172 meV/atom for P = 0 and 207 meV/atom for P = −0.5 eV ˚−2 A . The dashed line corresponds to the results obtained by ˚ Brito et al.53 using the REBO potential. Note that 1 eV A˚−2 an interatomic distance dC−C = 1.4199 A, i.e., an area −1 ˚2 =16Nm in SI units. A0 =2.6189 A per atom. In a quantum approach, out- of-plane atomic fluctuations associated to zero-point motion appear even for T → 0, and the graphene layer is not In fact, A does not present a negative thermal expansion, strictly planar. Moreover, anharmonicity of in-plane vi- as happens for the in-plane area in a large temperature brations gives rise to a zero-point lattice expansion, yield- 11,59 range (see below). The difference A − Ap increases ing a distance dC−C = 1.4287 A,˚ i.e., around 1% larger with temperature, as bending of the graphene sheet in- than the classical distance at T = 0.11 creases, but even for T → 0, A and Ap are not exactly In Fig. 1 we show the temperature dependence of the equal, due to zero-point motion of the C atoms in the internal energy per atom, E − E0, as derived from our transverse z direction. PIMD simulations in the isothermal-isobaric ensemble for P = 0 (circles), −0.2 (squares), and −0.5 eV A˚−2 (diamonds). For P = 0 it was shown earlier that the size III. ENERGY AND ENTHALPY effect on the internal energy per atom is negligible for N = 960 (less than the symbol size in Fig. 1), as compared A. Internal energy with the largest cells considered in Ref. 11. The zero- point energy, EZP, is found to be 172 meV/atom for P −2 The internal energy E is obtained from the results = 0, and slightly higher for −0.2 eV A˚ , but it appre- −2 of our PIMD simulations as a sum of the kinetic, K, ciably increases for P = −0.5 eV A˚ to a value of 207 and potential energy, V , at a given temperature. The meV/atom. We will show later that this rise in internal kinetic energy has been calculated by employing the energy is basically due to the elastic energy associated to virial estimator,64 which displays a statistical uncertainty an increase in the area A. For comparison with our re- smaller than the potential energy. We have included in E sults, we also present in Fig. 1 the energy E−E0 obtained 53 the center-of-mass translational energy, a classical mag- by Brito et al. from path-integral Monte Carlo simula- nitude amounting to ECM = 3kBT/2 at temperature T . tions using the REBO potential (dashed line). These This quantity is irrelevant for the energy per atom in authors called it vibrational energy, and corresponds to large systems, but we have included it to minimize finite our internal energy; our vibrational energy Evib is de- size effects.11 fined below (see Sec. III.C). The REBO potential yields We express the internal energy as E = E0 + V + K, a zero-point energy of 0.181 eV/atom, i.e., a 5% higher taking as energy reference the value E0 corresponding to than that found here with the LCBOPII potential. the equilibrium configuration of a planar graphene sur- The rise in internal energy with an applied tensile face in a classical approach at T = 0 (minimum-energy stress is presented in Fig. 2 for three temperatures: T = configuration of the considered LCBOPII potential, with- 25 K (circles), 300 K (squares), and 500 K (diamonds). out quantum atomic delocalization). This corresponds to For zero stress we find at 25 K an internal energy very 5 close to that of the ground state EZP. In the stress range 0.12 displayed in Fig. 2 the internal energy can be fitted to an 2 3 expression E − E0 = c0 + c2P + c3P , with a coefficient 0.1 4 −1 of the quadratic term c2 ≈ 0.090 A˚ eV , nearly independent of the temperature. This is the leading term for 0.08 -2 the variation of internal energy with the applied pressure, P = - 0.5 eV Å which is nearly parabolic for stresses between 0 and −0.1 0.06 eV A˚−2. The change in internal energy with temperature and pressure can be described from variations in the elastic 0.04 and vibrational contributions to E. This is analyzed in - 0.2 the following sections. 0.02 Elastic energy (eV / atom)

0 0 B. Elastic energy 0 500 1000 1500 2000 Temperature (K) An important part of the internal energy corresponds to the elastic energy due to changes in the area of graphene. It is directly related to the actual interatomic FIG. 3: Temperature dependence of the elastic energy Eel of distance, i.e., to the real area A, rather than to the in- graphene, as derived from the real area A obtained in PIMD − − plane area Ap (or in-plane strain ǫ). Then, the internal simulations for P = 0 (circles), 0.2 (squares), and 0.5 eV −2 energy E(T ) at temperature T can be written as11 A˚ (diamonds). Symbols represent simulation results and error bars are less than the symbol size. Dashed lines are E(T )= E0 + Eel(A)+ Evib(A, T ) , (5) guides to the eye. where Eel(A) is the elastic energy for an area A, and Evib(A, T ) is the vibrational energy of the system. The displays a thermal contraction (αp < 0) in a wide tem- area A is a function of the stress P and temperature T , perature range, so that it is not a suitable candidate for but this is not explicitly indicated in Eq. (5) for simplicity a reliable definition of the elastic energy. of the notation. One expects non-zero values of the elas- For P = 0 we find a positive elastic energy in the zero- tic energy, even in the absence of an externally applied temperature limit, E = 1.7 meV/atom, due to zero- stress, due to thermal expansion at finite temperatures, el point lattice expansion, which causes that A > A0. This as well as for zero-point expansion at T = 0. Evib(A, T ) low-T limit appreciably increases for P < 0 due to the is given by contributions of phonons in graphene, both stress-induced increase in area A, which yields values of in-plane and out-of-plane vibrational modes. 12 and 52 meV/atom for the elastic energy at P = −0.2 We define the elastic energy Eel corresponding to an and −0.5 eV A˚−2, respectively. For finite temperatures, area A as the increase in energy of a strictly planar one observes in Fig. 3 that Eel(T ) increases faster with graphene layer with respect to the minimum energy E0 temperature for larger tensile stress. This is due to an ˚2 (for an area A0 = 2.61888 A /atom). By definition increase in the graphene compressibility (reduction of the Eel(A0) = 0, and for small changes of A we found that 2 elastic constants) for rising tensile stress, which in turn it follows a dependence Eel(A) ≈ K(A − A0) , with K −2 causes an increase in the thermal expansion coefficient α = 2.41 eV A˚ . This dependence for Eel(A) yields a 2D 2 2 (see Sec. IV). bulk modulus B0 = A0(∂ Eel/∂A )=2KA0, i.e., B0 = 12.6 eV A˚−2, in agreement with the result found earlier in classical calculations at T = 0.20 Our PIMD simulations directly yield E(T ), which can C. Vibrational energy be split into an elastic and a vibrational part, as in Eq. (5). At room temperature (T ∼ 300 K) and for After calculating the elastic energy for an area A re- small stresses P (A close to A0), the elastic energy is sulting from PIMD simulations at given T and P , we much smaller than the vibrational energy Evib, but this obtain the vibrational energy Evib(A, T ) by subtract- can be different for low T and/or large applied stresses ing the elastic energy from the internal energy: Evib = (see below). E(T ) − E0 − Eel(A) (see Eq. (5)). In Fig. 4 we present In Fig. 3 we display the temperature dependence of the temperature dependence of the vibrational energy of the elastic energy, as derived from our PIMD simulations graphene for P = 0 (circles), −0.2 (squares), and −0.5 for P = 0, −0.2, and −0.5 eV A˚−2. For a given ex- eV A˚−2 (diamonds), as derived from our simulations. In ternal stress, Eel increases with T , as a consequence of contrast to Figs. 1 and 3, where one observes that E the thermal expansion of the real area A, which turns and Eel increase with the applied tensile stress P , in out to be positive for all temperatures T > 0 (α > 0, Fig. 4 one sees that the vibrational energy is lower for see below). Note that, in contrast, the in-plane area Ap higher tensile stress. This is mainly due to a decrease 6

-2 0.22 0.5 (a) P = - 0.5 eV Å P = 0 0.4 0.2 0.3 -2 - 0.2 eV Å Evib 0.2 Enthalpy 0.18 0.1 -2 E - 0.5 eV Å Energy (eV / atom) 0 el 0.16 -0.1 Vibrational energy (eV / atom) P (Ap - A0) -0.2 0 100 200 300 400 500 600 0 500 1000 1500 2000 Temperature (K) Temperature (K)

FIG. 4: Temperature dependence of the vibrational energy per atom, Evib, for P = 0 (circles), −0.2 (squares), and −0.5 eV A˚−2 (diamonds). Error bars are less than the symbol size. Dashed lines are guides to the eye. (b) 0.18 P = 0 in the vibrational frequency of in-plane modes for increasing stress (increasing area), corresponding to pos- 39,74 -2 itive Gr¨unesien parameters. 0.16 - 0.2 eV Å The zero-point vibrational energy is found to decrease from 170 meV/atom for P = 0 to 164 and 155 meV/atom for P = −0.2 and −0.5 eV A˚−2, respectively. This de- -2 crease, although clearly noticeable, is smaller than the 0.14 - 0.5 eV Å increase of about 50 meV/atom in the elastic energy for Enthalpy (eV / atom) P = −0.5 eV A˚−2 (see Fig. 3). These zero-T energy values per atom correspond in a harmonic approximation to h3~ω/2i, which is a mean value for the frequencies ω 0.12 0 100 200 300 in the six phonon bands of graphene.75,76 Our results for − Temperature (K) P = 0 correspond to hωi = 914 cm 1, to be compared with 833 cm−1 for P = −0.5 eV A˚−2.

The three curves for Evib(T ) shown in Fig. 4 for dif- FIG. 5: Enthalpy per atom, H − H0, as a function of tem- ferent stresses are roughly parallel in the displayed temperature. (a) Enthalpy for a tensile stress P = −0.5 eV −2 perature range. Values of Evib presented in this figure A˚ (circles), along with its three components, Evib, Eel, are clearly larger than those corresponding to a classical and P (Ap − A0) (see Eq. (6)). Dashed lines are guides to the eye. (b) Enthalpy in the low-temperature region for model for the vibrational modes. In this limit, the vibra- − cl P = 0 (circles), −0.2 (squares), and −0.5 eV A˚ 2 (dia- tional energy per atom is Evib = 3kBT/2, which means 39 and 78 meV/atom for T = 300 and 600 K, respectively. monds). Symbols represent results of PIMD simulations. Error bars are less than the symbol size. Dashed lines in In contrast, we find Evib = 166 and 206 meV/atom from −2 (b) indicate fits of the simulation results to the polynomial PIMD simulations for P = −0.5 eV A˚ at those tem- 2 3 H − H0(P ) = HZP + a2T + a3T in the region from T = 0 peratures. to 100 K. We note that in the language of membranes and 2D elastic media there appears an energy contribution due to bending of the surface, that is usually taken into ac- temperatures.77 count through the bending constant κ, which measures the rigidity of the membrane. In our present formulation, the bending energy is included in the vibrational energy associated to the flexural ZA modes, as can be D. Enthalpy seen in their contribution to the specific heat (see Sec. V). Anharmonic couplings between in-plane and out-of-plane Since we are working in the isothermal-isobaric ensem- modes are also expected to show up, especially at high ble, it is natural to consider the enthalpy as a relevant 7 thermodynamic variable, in particular to calculate the converge to 2.6459 A˚2/atom and 2.6407 A˚2/atom, respec- specific heat of graphene at several applied stresses (see tively. Comparing these values with the classical mini- Sec. V). For a given stress P , we define the enthalpy mum A0 = 2.6189 A,˚ we find a zero-point expansion in 2 H as H = E + P Ap. As a reference we will consider A and Ap of about 0.02 A˚ /atom (∼ 1%), due to an H0(P )= E0 + P A0, such that H converges in the classi- increase in the mean C–C bond length (an anharmonic cal limit to H0(P ) at T = 0. effect). There also appears a difference of 0.2% between In Fig. 5(a) we present the enthalpy of graphene, real and in-plane areas at low temperature, caused by H − H0(P ), as a function of temperature for a tensile out-of-plane zero-point motion, so that the layer is not stress P = −0.5 eV A˚−2. Results of our simulations are strictly planar. This is a genuine quantum effect, as in displayed as circles. Taking into account that the en- classical simulations for T → 0 one finds a planar layer 11,20 thalpy can be written as in which A and Ap coincide. The difference A − Ap increases as temperature is raised, since Ap is the projec- H − H0(P )= Eel + Evib + P (Ap − A0) , (6) tion of A on the xy reference plane, and the real graphene surface becomes increasingly bent for rising temperature we have plotted in Fig. 5(a) the temperature dependence because of larger out-of-plane atomic displacements. of the three contributions in the r.h.s. of Eq. (6). The For P = 0, the area A displays an almost constant largest change with temperature appears for the vibra- value up to T ≈ 200 K, and increases at higher temper- tional energy (diamonds). The elastic energy, E , as well el atures. However, A decreases in the temperature range as the term P (A −A ) are also found to depend on tem- p p 0 from T =0to T ≈ 1000 K, reaches a minimum and then perature, due to the thermal expansion (or contraction) increases at higher T .11 These results for A are qualita- of the in-plane area A , but their change is much smaller p p tively similar to those found from classical Monte Carlo than that of E . As a result, the enthalpy turns out to vib and molecular dynamics simulations of graphene,59,73 but be smaller than the vibrational energy at low tempera- in PIMD simulations the contraction of A with respect ture, because E +P (A −A ) < 0. However, H −H (P ) p el p 0 0 to the zero-temperature value is significantly larger than becomes larger than E at T & 1500K, due to the larger vib for classical calculations. increase in E for rising temperature. el To analyze the thermal expansion of graphene, and fol- To better appreciate the low-temperature region, in lowing our definitions of the areas A and A , we will con- Fig. 5(b) we present the temperature dependence of p sider two different thermal expansion coefficients. The the enthalpy, as derived from PIMD simulations up to first of them, α, refers to changes in the real area: T = 300 K. Symbols indicate results of the simulations for P = 0 (circles), −0.2 (squares), and −0.5 eV 1 ∂A A˚−2 (diamonds) Dashed lines are fits to the expression α = (7) 2 3 A ∂T P H − H0(P ) = HZP + a2T + a3T in the temperature region from T = 0 to 100 K. This polynomial form shows This coefficient takes mainly into account changes in the good agreement with the temperature dependence of the interatomic distances, and is rather insensitive to bend- enthalpy obtained from the simulations in the fitted re- ing of the graphene layer. The second coefficient, αp, is gion. For T & 150 K, the lines depart progressively from a measure of variations in the in-plane area and has been the data points. Note that a linear term in this expres- widely used in the literature to analyze results of classical sion for the enthalpy is not allowed for thermodynamic simulations and analytical calculations: consistency, since the specific heat cp = (∂H/∂T )P has to vanish in the limit T → 0. The coefficient a2 changes 1 ∂Ap from 6.1 × 10−8 eV K−2 for P = 0 to 2.6 × 10−8 eV αp = . (8) −2 −2 Ap ∂T P K for P = −0.5 eV A˚ . The coefficient a3 varies − − from 3.5 to 5.1 × 10 10 eV K 3 in the same stress range. In Fig. 6 we present both thermal expansion coeffi- This is important for the low-temperature dependence of cients, as derived from our PIMD simulations for P =0 the specific heat discussed in Sec. V, as the contribution (circles), −0.2 (squares), and −0.5 eV A˚−2 (diamonds). of the quadratic coefficient a2 decreases for increasing Symbols are data points obtained from numerical deriva- stress, whereas a3 is found to increase. This means that tives of the areas A (panel a) and Ap (panel b). For the specific heat cp in the lowest temperatures accessible these derivatives we took temperature intervals ranging to our simulation method will be given as a combina- from 10 K at low temperature to about 50 K at tempera- 2 tion of a linear term 2a2T and a quadratic one 3a3T , tures T ∼ 1000 K. In general, the statistical uncertainty with the latter becoming more important for larger ten- (error bars) in the values of αp obtained from numerical sile stresses. derivatives is larger than that found for α, as a consequence of the larger fluctuations in Ap. The behavior of α shown in Fig. 6(a) is similar to that observed for most IV. THERMAL EXPANSION 3D materials, i.e., it goes to zero in the low-temperature limit and increases for rising temperature so that α > 0 In the low-temperature limit (T → 0), the real area for T > 0.74,78 This is related to the fact that the in-plane A and in-plane area Ap derived from PIMD simulations vibrational modes have positive Grüneisen parameters 8

2 which in turn causes a larger vibrational amplitude and -5 consequently a larger anharmonicity. ×10 ) (a) Real area

-1 The in-plane thermal expansion coeﬃcient αp also con- 1.5 verges to zero in the low-temperature limit, but contrary

(K -2 to α it decreases for increasing temperature until reach- α P = - 0.5 eV Å ing a minimum, as shown in Fig 6(b). The negative value 1 of αp in the minimum approaches zero for rising tensile stress. In fact, it goes from −9.2 × 10−6 K−1 for P = 0 -2 −6 −1 −2 - 0.2 eV Å to −2.3 × 10 K for P = −0.5 eV A˚ . At higher T , 0.5 αp approaches zero and eventually becomes positive at a P = 0 temperature T0, which changes appreciably with the applied stress. As a result, we find T0 = 1020, 590 K, and Thermal expansion 0 360(±10) K, for the three values of the stress P presented in Fig. 6. 0 200 400 600 800 1000 It is interesting to note that the difference α−αp, which Temperature (K) vanishes at T = 0, rapidly increases for rising temperature, and for P = 0 it takes a value ≈ 1.0 × 10−5 K−1 at temperatures higher than 1000 K. For larger tensile stress, this difference is smaller, since the amplitude of out-of-plane vibrations is reduced, and Ap is closer to A. −2 1.5 Thus, for P = −0.2 and −0.5 eV A˚ , we find at high -5 −6 −1 −6 −1 ×10 temperatures α−αp ≈ 8×10 K and 6.5×10 K , ) (b) In-plane area respectively. -1 1 The results presented here for αp at zero external (K

p -2 stress display a temperature dependence similar to those

α P = - 0.5 eV Å found earlier using other theoretical and experimental 0.5 - 0.2 techniques.37,79–81 The solid line in Fig. 6(b) represents the thermal expansion coeﬃcient αp obtained by Mounet 39 0 and Marzari from DFT combined with a QHA for the P = 0 vibrational modes. Similar curves αp(T ) were obtained from ﬁrst-principles calculations in Refs. 40,41, converg- -0.5 ing to negative values at high temperature. This seems to

Thermal expansion be a drawback of this kind of calculations, which are op- timal to obtain the total energy and electronic structure -1 of the system, but the employed QHA may be not pre- 0 500 1000 1500 cise enough to capture the important coupling between Temperature (K) in-plane and out-of-plane vibrational modes at relatively high temperatures. This mode coupling controls the in- FIG. 6: (a) Thermal expansion coefficient α of graphene vs plane thermal expansion. 79 temperature, as derived from the results of PIMD simulations. Jiang et al. studied free-standing graphene using (b) In-plane thermal expansion coefficient αp of graphene vs a nonequilibrium Green’s function approach, and ob- −5 −1 temperature, obtained from numerical derivatives of the area tained a minimum for αp of about −10 K , similar Ap. In both panels, symbols represent data points for dif- to our data for P = 0 displayed in Fig. 6(b). Moreover, ferent stresses: P = 0 (circles), −0.2 (squares), and −0.5 − these authors found a crossover from negative to posi- eV A˚ 2 (diamonds). Dashed lines are polynomial fits to the tive αp for a temperature T0 ∼ 600 K, lower than the data points. The solid line indicates the result obtained in results of our PIMD simulations. An even lower tem- Ref. 39 from density-functional perturbation theory. perature T0 ∼ 400 K has been found by using other theoretical techniques.37,82 Room-temperature Raman 81 −6 −1 measurements yielded αp = −8 × 10 K , whereas − − when they are calculated with respect to the real area a value of −7 × 10 6 K 1 was derived from scanning A. One observes in Fig. 6(a) that α is higher for larger electron microscopy.80 We note, however, that the agree- tensile stress. In fact, at 800 K the α value obtained for ment between measurements with different techniques is −2 P = −0.5 eV A˚ is 40% larger than that corresponding not so good for the temperature dependence of αp, since to P = 0. Viewing the thermal expansion as an anhar- the change in αp at T > 300 K is faster in the former monic effect, this can be interpreted as an increase in case81 than in the latter.80 anharmonicity for larger tensile stress. This is associ- The temperature dependence of αp can be qualitatively ated with an increase in area A and the corresponding understood as a competition between two opposing fac- decrease in the frequency of vibrational in-plane modes, tors. On one side, the real area A increases as T is raised 9

-4 in the whole temperature range considered here. On the 3×10 other side, bending of the graphene surface causes a decrease in its projection, Ap, onto the xy plane. For stress- free graphene at temperatures T . 1000 K, the decrease Dulong-Petit due to out-of-plane vibrations is larger than the thermal × -4 expansion of the real surface, and α < 0. For T & 2 10 -2 p P = - 0.5 eV Å 1000 K, the thermal expansion of A dominates over the contraction of Ap associated to out-of-plane atomic motion, and thus αp > 0. For graphene under tensile stress, P = 0 the amplitude of out-of-plane vibrations is smaller, so the -4 10 decrease in Ap is also smaller and the region of negative αp is reduced.

Our results for the thermal expansion of graphene, de- Specific heat (eV / K atom) rived from atomistic simulations, can be related with an analytical formulation of crystalline membranes in the 0 200 400 600 800 1000 continuum limit, for which the relation between A and 20,72 Temperature (K) Ap may be written as

2 A = dx dy 1 + (∇h(x, y)) . (9) FIG. 7: Specific heat of graphene as a function of tempera- ZAp ture, as derived from PIMD simulations for P = 0 (circles) p − −2 Here h(x, y) is the height of the membrane surface, i.e. and 0.5 eV A˚ (squares). Error bars of the data points are the distance to the reference xy plane. In this classical less than the symbol size. The dashed line is the specific heat c approach, the difference A − A may be calculated by v obtained from the six phonon bands corresponding to the p LCBOPII potential in a harmonic approximation (P = 0). Fourier transformation of the r.h.s. of Eq. (9).15,20,25 In Open symbols indicate results for cv derived from DFT-type this procedure one needs to consider a dispersion relation calculations at T = 900 K (square, Ref. 34), 850 K and 750 k ωZA( ) for out-of-plane modes (ZA flexural band), where K (circles, Ref. 35). The horizontal dashed-dotted line repre- k = (kx, ky) are 2D wavevectors. The frequency disper- sents the classical Dulong-Petit limit. sion in this acoustic band is well approximated by the 2 2 4 expression ρωZA = σk + κk , consistent with an atomic description of graphene28 (k = |k|; ρ, surface mass den- for rising tensile stress. Thus, for P = −0.5 eV A˚−2 −6 −1 sity; σ, effective stress; κ, bending modulus). Then, for our simulations yield α − αp = 6.5 × 10 K , larger effective stress σ> 0, which is the case at finite tempera- than the prediction based on Eq. (10), which is based on tures, even for zero external stress (P = 0), one finds20,28 harmonic vibrations. We observe that the importance of anharmonic effects (including also quantum corrections) k T 2πκ A = A 1+ B ln 1+ . (10) is manifested more clearly for large tensile stresses. p 8πκ σA p This expression was obtained in the classical limit, so it does not take into account atomic quantum delocaliza- V. SPECIFIC HEAT tion. Nevertheless, it is expected to be a good approximation to our quantum calculations at relatively high We have calculated the specific heat of graphene mono- temperature, T & ΘD, with ΘD ∼ 1000 K the Debye layers as a temperature derivative of the enthalpy H temperature corresponding to out-of-plane vibrations in derived from our PIMD simulations (see Sec. III.D): 83,84 graphene. cp(T ) = dH(T )/dT . One may ask if our simulations We note that both κ and σ change with temperature can yield reliable results for this thermodynamic variable, and stress, so that it is not straightforward to write down mainly because electronic contributions to cp are not an analytical formula for α − αp from a temperature taken into account in our numerical procedure. This is, derivative of Eq. (10). For given temperature T and however, not a problem for the actual precision reached stress σ = σ0 − P we can write A − Ap = δApT , δ being in our calculations, as the electronic part is much less a parameter derived from Eq. (10), which is a good ap- than the phonon contribution, actually considered in our proximation for the difference α−αp. The effective stress method. The former was estimated in various works, σ0 appears at finite temperatures for zero external stress and turns out to be three or four orders of magnitude (P = 0), and vanishes for T → 0 in the classical limit.28 less than the phonon part in the temperature range con- 33,85,86 Introducing the finite-temperature values of σ0 and κ sidered here. given earlier,20,28 we find δ =1.0 × 10−5 K−1,6.6 × 10−6 In Fig. 7 we display the specific heat of graphene as a K−1, and 5.1 × 10−6 K−1 for P = 0, −0.2, and −0.5 eV function of temperature for P = 0 (circles) and −0.5 eV A˚−2, respectively. These values are close to those found A˚−2 (squares), as obtained from a numerical derivative of for α−αp from our PIMD simulations (see above). How- the enthalpy. For comparison, the classical Dulong-Petit cl ever, the difference between both sets of results increases specific heat is shown as a dashed-dotted line (cv =3kB). 10

At room temperature the specific heat of graphene is about three times smaller than the classical limit, and cp is still appreciably lower than this limit at T = 1000 K. This is in line with the magnitude of the Debye temper- graphene ature of graphene, which amounts to about 1000 K for 83,84 out-of-plane modes and ΘD & 2000 K for in-plane -5 P = 0 31,83 10 vibrations. In Fig. 7 we also present results for cv -2 at P = 0 taken from earlier DFT calculations combined - 0.2 eV Å with a QHA: T = 900 K from Ref. 34 (open square), 850 K and 750 K from Ref. 35 (open circles). These data - 0.5 agree well with the results of our PIMD simulations. For a comparison with our numerical results for cp(T ) graphite at zero and negative stress, we present a harmonic ap- Specific heat (eV / K atom) proximation (HA) for the lattice vibrations. This approx- -6 imation is expected to be rather accurate at low tempera- 10 10 20 50 100 200 tures, provided that a reliable description for the phonon Temperature (K) frequencies is used. A comparison between results of the HA and PIMD simulations yields an estimate of anharmonic effects in the specific heat of graphene, given that FIG. 8: Specific heat of graphene as a function of temper- both kinds of calculations employ the same interatomic ature. Symbols represent results derived from PIMD simu- potential. lations for P = 0 (circles), −0.2 (diamonds), and −0.5 eV − The HA assumes constant frequencies for the graphene A˚ 2 (squares). Lines were derived from a HA based on the vibrational modes (calculated for the minimum-energy six phonon bands corresponding to the LCBOPII potential configuration), and does not take into account changes for N = 61440 (see text for details). Experimental data for graphite obtained by Desorbo and Tyler90 are shown as open in the areas A and Ap with temperature. Thus, it will diamonds. give us the constant-area specific heat per atom, cv(T )= dE(T )/dT , which for a cell with N atoms is given by and the second is the relative inaccuracy of the phonon 1 ~ k 2 kB 2 β ωr( ) bands derived from the considered effective potential far cv(T )= , (11) N sinh2 1 β~ ω (k) from the Γ point, as compared with those obtained from r,k 2 r X DFT calculations. The HA provides, however, a consis- where the index r (r = 1, ..., 6) indicates the six tency check for the results of the PIMD simulations at phonon bands of graphene (ZA, ZO, LA, TA, LO, and low temperature, as discussed below. TO),39,75,76 and the sum in k runs over wavevectors As shown in Sec. III, a part of the internal energy k = (kx, ky) in the hexagonal Brillouin zone, with dis- (and the enthalpy) corresponds to the elastic energy Eel, crete k points spaced by ∆kx = 2π/Lx and ∆ky = i.e., to the cost of increasing the area A of graphene by 28 2π/Ly. For increasing system size N, one has new long- thermal expansion or an applied tensile stress. The con- wavelength modes with an effective cut-off λmax ≈ L, tribution of this energy to the specific heat is given by 1/2 with L = (NAp) , and the minimum wavevector is dEel/dT , which is not included in the HA. This contri- −1/2 bution can be calculated from the results of our PIMD k0 =2π/λmax (i.e., k0 ∼ N ). The dashed line in Fig. 7 was calculated with the HA simulations displayed in Fig. 3. For P = 0, it amounts to 2.3 and 5.5 × 10−6 eV/K atom at 500 and 1000 K, using Eq. (11), with the frequencies ωr(k) (r = 1, ..., 6) obtained from diagonalization of the dynamical ma- which means a nonnegligible increase in the specific heat trix corresponding to the LCBOPII potential. Results of with respect to the pure HA. This increase is especially the simulations for P = 0 follow closely the HA up to visible at T & 500 K, capturing part of the anharmonic- about 400 K, and they become progressively higher than ity of the system. In fact, at T = 1000 K it accounts for the dashed line at higher temperatures, for which anhar- a 45% of the difference between the results derived from monic effects become more important. At room temper- PIMD simulations and the HA presented in Fig. 7. The −5 rest of that difference is associated to anharmonicity of ature (T = 300 K) we obtained cv = 9.1 × 10 eV/K atom from the HA vs a value of 9.3(±0.1) × 10−5 eV/K the lattice vibrations. atom derived from PIMD simulations. Both values are a In our present context, the most relevant effect of an little higher than that found by Ma et al.35 from DFT cal- applied stress P in the phonon bands is a change in the −5 low-frequency region of the ZA modes, for which culations (cv =8.6 J/K mol, i.e. 8.9×10 eV/K atom). The difference between results obtained for P = 0 from 2 0 2 P 2 ωZA(k) = ω (k) − k (12) ab-initio calculations combined with a QHA and those ZA ρ presented here for a simpler HA are due to two main rea- 0 k sons. The first reason is that in the HA one does not take The zero-stress band ωZA( ) calculated for the minimum- into account changes of frequencies with the temperature, energy structure (area A0), follows for small k: 11

0 k 2 4 ρωZA( ) ≈ κk . Thus, for P < 0 the small-k re- close to its low-temperature limit (µ = 1), but the gion is dominated by the quadratic term (linear in P ) exponents for graphene under stress are still somewhat −1 in Eq. (12), and ωZA(k) ≈ −P/ρ k for k ≪ 1 A˚ . lower than their zero-T limit (µ = 2). We note that the The specific heat of stressed graphene in the HA has been larger the tensile stress (larger σ = σ0 − P ), the closer is p calculated by taking the frequencies ωZA(k) derived from µ to 2 at a relatively low T , since the wavenumber region Eq. (12). where the linear term dominates in the ω(k) dispersion 2 2 To obtain deeper insight into the low-temperature de- of the ZA band is larger (ρωZA ≈ σk ). For the results of our PIMD simulations we note that, although one can pendence of the specific heat, cp(T ) is shown in Fig. 8 in a logarithmic plot. Symbols indicate results derived from obtain an exponent µ from a linear fit in the logarithmic PIMD simulations for P = 0 (circles), −0.2 (squares), plot of Fig. 8 (in particular the fit is rather good for and −0.5 eV A˚−2 (diamonds). With our present method, T < 100 K), it is true that the actual slope is slowly we cannot give reliable values of c for temperatures lower changing in this temperature range. p 89 than 15 K, mainly due to the very large computation Alofi and Srivastava calculated the specific heat of times required for Trotter numbers NTr > 500. Dashed few-layer graphene by using a semicontinuum model and analytical expressions for phonon dispersion relations. In lines show cv(T ) derived from the harmonic approximation using Eq. (11) for a large cell size (N = 61440 particular, for single-layer graphene they found a tem- 1.1 atoms). Such a large N cannot be reached in our quan- perature dependence cv ∼ T for T & 10 K, with an tum simulations, and gives us a useful reference where exponent that coincides with the mean µ value derived finite-size effects are minimized. The lines corresponding from our results for P = 0 in the region from 10 to 50 K. to the HA follow closely the data points derived from the For comparison with our results for graphene, we also simulations at low temperatures. For P = −0.5 eV A˚−2 present in Fig. 8 experimental data for the specific heat 90 one observes that the PIMD results become progressively cp of graphite, obtained by Desorbo and Tyler. For higher than the HA line at T > 50 K. Such a difference graphite, the dependence of cp on temperature has been − 91–94 is almost unobservable for P = −0.2 eV A˚ 2, and disap- analyzed in detail over the years. For T < 10 K, cp 3 pears for P = 0. It is clear that the HA describes well the rises as T (a temperature region not shown in Fig. 8), as specific heat in the stress-free case at T < 200 K, and be- in the Debye model, and for temperatures between 10 and 2 comes less accurate as tensile stress and/or temperature 100 K, it increases as T (µ = 2). The major difference increase. with stress-free graphene is that in graphite the domi- The low-temperature behavior of the heat capacity can nant contribution to cp in this temperature range arises be further analyzed by considering a continuous model from phonons with a linear dispersion relation for small k (ω ∼ k). At room temperature the experimental specific for frequencies and wavevectors, as in the well-known De- −5 bye model for solids.74 At low-temperatures, c is mainly heat of graphite is 8.90×10 eV/K atom (8.59 J/K mol), v somewhat less than the result for graphene derived from controlled by the contribution of acoustic modes with −5 small k. For graphene, these are TA and LA modes with our PIMD simulations for P = 0: cp =9.3(±0.1) × 10 eV/K atom. ωr ∝ k, and ZA flexural modes with quadratic dispersion 2 To compare with the results for cp of graphene derived (ωr ∝ k ) for negligible σ at P = 0 and low T , whereas it is linear in k for P < 0. In general, the low-T contribu- from our PIMD simulations, we have also calculated the n specific heat cv from constant-Ap simulations. For each tion of a phonon branch with dispersion relation ωr ∝ k can be approximated by replacing the sum in Eq. (11) considered temperature, we took the equilibrium area Ap r 2/n obtained in the isothermal-isobaric simulations, and cal- by an integral, which yields cv ∼ T (see Ref. 61). We note that for d-dimensional systems one finds an expo- culated cv as a numerical derivative of the internal energy 87,88 nent d/n. ∆E c = (13) Summarizing the above comments, the low- v ∆T temperature contributions of the relevant phonon Ap branches to the specific heat (those with ω(k) → 0 for µ from temperature increments ∆T (both positive and neg- k → 0), one has for graphene cv ∼ T close to T = 0, ative). One expects that cv ≤ cp at any temperature, but with µ = 1 for P = 0 and µ = 2 in the presence of a the difference between them turns out to be less than the tensile stress (P < 0). This is in fact what we find for cv statistical error bars of our results. A realistic estimation from the HA at low temperature in our calculations for of this difference can be obtained from the thermody- a large graphene supercell. To see the convergence of µ namic relation61,95 to its low-T value, we have calculated this exponent as 2 a function of temperature from a logarithmic derivative: T αpAp cp − cv = (14) µ = d ln cv/d ln T . Close to T = 0 we obtain the χp values expected from our discussion above, but in the region displayed in Fig. 8, µ has not yet reached the where χp is the in-plane isothermal compressibility (see zero-temperature value. In fact, for T = 10 K we find for Sec. VI). Using Eq. (14), we find that cp−cv is at least two µ the values 1.05, 1.59, and 1.85, for P = 0, −0.2, and orders of magnitude less than the cp values given above −0.5 eV A˚−2, respectively. The first of them is already for the tensile stresses and temperatures considered here. 12

0.25 isobaric ensemble is a variable conjugate to the in-plane area Ap. χp has been calculated here by using the ﬂuc- 20,95 ) tuation formula -1 2 0.2 T = 300 K Nσp eV

2 χp = (16) kBT Ap where σ2 are the mean-square fluctuations of the area 0.15 χ p p Ap obtained in the simulations. This expression is more convenient for our purposes than obtaining (∂Ap/∂P )T , since a calculation of this derivative by numerical meth- 0.1 ods requires additional simulations at nonzero stresses. χ We have verified at some selected temperatures and pres- Compressibility ( Å exp. sures that both procedures give the same results for χp 0.05 (taking into account the statistical error bars). Simi- -0.6 -0.4 -0.2 0 larly, for the real area A one can define a compressibility -2 Stress (eV Å ) χ = −(∂A/∂P )T /A, which will be related to the fluctuations of the real area A. In Fig. 9 we present the stress dependence of the com- FIG. 9: Stress dependence of the isothermal compressibilities pressibilities χp (circles) and χ (squares) of graphene, as χ (squares) and χp (circles) of graphene at T = 300 K, as de- derived from our PIMD simulations at 300 K. At P = 0, rived from PIMD simulations. Error bars, when not shown, one finds χ > χ, as a consequence of the larger fluc- are in the order of the symbol size. Open symbols indicate p results derived from AFM indentation experiments: a square tuations in the in-plane area Ap. In this figure we have from Ref. 98 and a circle from Ref. 6 (the error bar corre- included some points corresponding to (small) compres- sponds to one standard deviation for several measurements). sive stresses P > 0, to remark the very different behavior of χp and χ in this region. χ follows a regular dependence, in the sense that it displays a smooth change as We finally note that a calculation of the low-T spe- P is varied in the considered stress region. The in-plane cific heat of solids using this kind of path-integral simu- compressibility χp, however, increases fast for P > 0, lations is in general not straightforward. The verification which is consistent with an eventual divergence at a crit- 3 of the Debye law cp ∼ T has been a challenge for path- ical stress Pc. This divergence of χp shows the same fact integral simulations of 3D materials, because of the ef- as the vanishing of the in-plane bulk modulus Bp (the fective low-frequency cut-off corresponding to finite sim- inverse of χp) discussed in Ref. 20 from classical calcula- ulation cells.96,97 This kind of calculations at relatively tions. Close to this critical compressive stress the planar low temperatures are in principle more reliable for 2D morphology of graphene becomes unstable due to large materials due to two different reasons. The first reason fluctuations in the in-plane area, which is related to the is that the length of the cell sides scales as N 1/d, and onset of imaginary frequencies in the ZA phonon bands the minimum wavevector k0 available in the simulations for P > Pc. A detailed characterization of this wrin- −1/d is k0 ∼ N . For a simulation cell including N atoms, kling transition at Pc is out of the scope of this paper k0 is less for 2D than for 3D materials, so that the low- and will be investigated further in the near future. It is frequency region is represented better for d = 2, and not yet clear whether quantum effects may be relevant or therefore the low-temperature regime will be also better not for a precise definition of Pc. It is remarkable that described. The second reason is that the internal energy the compressibility χ derived from the real area does not (or enthalpy) for graphene rises as T µ+1 with µ = 2 or 3. display any abrupt change in the region close to the crit- At low T , this increase is faster than the usual phonon ical stress, as both the area A and its fluctuations are contribution to the enthalpy in 3D materials (H ∼ T 4), rather insensitive to the corrugation of the graphene sur- and consequently it is more readily observable (less rela- face imposed by compressive stresses. For large tensile tive statistical noise) for 2D materials such as graphene. stress, χ and χp converge one to the other, as shown in Fig. 9, since the amplitude of out-of-plane vibrations becomes smaller and the real graphene surface is closer to VI. COMPRESSIBILITY the reference xy plane. For comparison with the results of our simulations, The in-plane isothermal compressibility is defined as we also present in Fig. 9 data for the compressibility of graphene at P = 0 derived from atomic force microscopy 1 ∂Ap (AFM) indentation experiments. We have transformed χp = − . (15) the values for the Young modulus Y given in Refs. 6,98 to Ap ∂P T compressibility by using the expression χ = 2(1 − ν)/Y , The variables in the r.h.s. of this equation refer to in- with a Poisson ratio ν = 0.15.20 The resulting com- plane quantities, as the pressure P in our isothermal- pressibilities are plotted as open symbols: a square98 13 and a circle6 (the error bar indicates one standard de- atures and tensile stresses. This technique has allowed us viation for several measurements). We note that in- to quantify several structural and thermodynamic prop- terferometric profilometry experiments19 have revealed erties, with particular emphasis on the thermal expansion that close to P = 0 much smaller values of the Young and specific heat. Nuclear quantum effects are clearly ap- modulus (much larger compressibilities) can be found for preciable in these variables, even at T higher than room graphene. These authors have suggested that this appar- temperature. Zero-point expansion of the graphene layer ent discrepancy can be associated to the difference be- due to nuclear quantum motion is not negligible, and tween compressibilities of the real and projected areas, amounts to about 1% of the area A. This zero-point as discussed here. It seems that some experimental tech- effect decreases as tensile stress is increased. niques can measure one of them, while other techniques may be sensitive to the other. This is an ongoing discus- The thermal contraction of graphene discussed in the sion that should be elucidated in the near future.20,27 literature turns out to be a reduction of the in-plane area A thermodynamic parameter related to the thermal Ap (αp < 0), caused by out-of-plane vibrations, but not expansion αp and compressibility χp is the dimensionless a decrease in the real area A. In fact, the difference Grüneisen parameter γ, defined as A − Ap rises as temperature increases, since the amplitude of those vibrations grows. On one side, the in-plane γ k c (k) γ = rk r vr , (17) thermal expansion α is negative at low temperature, and c (k) p P rk vr becomes positive for T & 1000 K. On the other side, the where cvr(k) is the contributionP of mode (rk) to the thermal expansion α of the real area is positive for all specific heat, and γrk are mode-dependent Grüneisen temperatures and tensile stresses discussed here. The parameters.39,74 The overall γ can be related with our thermal contraction of Ap is smaller as the tensile stress in-plane variables in graphene by using the thermody- increases, due to a reduction in the amplitude of out-of- namic relation61,74 plane vibrations. Our PIMD simulations give αp < 0 at low T for stresses so high as −0.5 eV A˚−2 (−8 N/m). αpAp γ = . (18) For this stress, αp becomes positive at T ∼ 350 K. χpcv For P = 0, the results of our PIMD simulations yield The anharmonicity of the vibrational modes is clearly γ = −2.2 at 300 K, and at 1000 K, γ ≈ 0 within the pre- noticeable in the behavior of A and Ap. The increase cision of the numerical results. At T = 300 K, αp and γ in real area A for rising T (α > 0) is a consequence of are negative because the mode-dependent Grünesien pa- anharmonicity of in-plane modes, similar to the thermal rameter of the out-of-plane ZA modes is negative.4,34,39 expansion in most 3D solids. Moreover, the peculiar de- However, for T > 1000 K, this negative contribution is pendence of αp (negative at low T and positive at high dominated by the positive sign of γrk for in-plane modes, T ) is an indication of the coupling between in-plane and which are excited at these temperatures, so that αp and out-of-plane modes. γ are positive. At room temperature (T = 300 K) we find γ = −1.2 Other thermal properties of graphene can be well de- − and −0.24, for P = −0.2 and −0.5 eV A˚ 2, respectively. scribed by an HA at relatively low T , once the frequen- Although these values of γ are still negative, they are cies of the vibrational modes are known for the classical very different from the zero-stress result (γ = −2.2). equilibrium geometry at T = 0. This is the case of the Looking at Eq. (18), the main reason for this important specific heat, for which our PIMD results indicate that change in the Grünesien parameter with tensile stress the effect of anharmonicity appears gradually for temper- is the large variation in the in-plane thermal expansion atures T & 400 K. Part of this anharmonicity is due to coefficient αp, which takes values of −8.5, −3.9, and the elastic energy Eel, associated to the expansion of the −1.0 × 10−6 K−1 for P = 0, −0.2, and −0.5 eV A˚−2, actual graphene sheet, which is not taken into account by respectively (see Fig. 6(b)). From the point of view of the HA. At low T the specific heat can be described as µ the phonon contributions to γ, the small negative value cp ∼ T . At the lowest temperatures studied here (T & found for P = −0.5 eV A˚−2 indicates that the relative 10 K) we find an exponent µ = 1.05 and 1.85, for P =0 −2 contribution of phonons with negative γrk (out-of-plane and −0.5 eV A˚ , respectively. These values are close to ZA modes) is at room temperature less important than the corresponding low-temperature (T → 0) values, i.e., for P = 0. This is due to an increase in frequency of µ = 1 for stress-free and µ = 2 for stressed graphene. small-k modes (ωZA ∼ −P/ρ k), which causes a reduction in its corresponding cvr(k) at 300 K (see Eq. (17)). We note the consistency of the simulation results with p the principles of thermodynamics, in particular with the third law. This means that thermal expansion coeffi- VII. SUMMARY cients have to vanish for T → 0, as found for both the in-plane area Ap and the real area A derived from our In this paper we have presented and discussed results of simulations for P = 0 and P < 0. The same happens for PIMD simulations of graphene in a wide range of temper- the specific heat, i.e., cp → 0 for T → 0. 14

Acknowledgments was supported by Direcci´on General de Investigaci´on, MINECO (Spain) through Grants FIS2012-31713 and The authors acknowledge the help of J. H. Los in the FIS2015-64222-C2. implementation of the LCBOPII potential. This work

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