arXiv:cond-mat/0412643v1 [cond-mat.mtrl-sci] 22 Dec 2004 sterpeetadptnilapiain nsc di- such in applications nanoelectronics potential as fields and verse present their as fdaodadistemlpoete aebe stud- been have experiments properties with thermal extensively its ied spectrum and phonon the of Although calculations. diamond, first- dy- from principles graphite of rhombohedral structural, and properties the graphite, thermodynamical of and determination namical, accurate verged, thermal heat. the specific as or such cal- expansion to quantities needed dis- thermodynamical is can that phonon culate excitations information the the and electronic all materials, provide thus, bands these persions in massless 8); neglected two Ref. be the also where e.g. point (see K cross the van- at gap the only but temperatures. ishes semi-metals, high are to graphene and up Graphite properties thermal for account iso h uk ned imn en ieband wide a being diamond Indeed, (E proper- material bulk. a gap thermodynamic the play the of properties determining ties vibrational in respect, role citations this and crucial 3,4,5,6,7 broad In Refs. a characteri- e.g. therein). under (see complete stability conditions of a thermodynamic a range lack their of (i.e. still zation graphene layer) and graphite ex- graphite Even single sin diamond, derivatives, elements. crystalline their the gle and all nanotubes, among fullerenes, place cluding special a them h hnnsetu fgaht ssiludrac- under still is graphite of investigation spectrum tive phonon the eonzdt enegative been be long has to expansion recognized thermal in-plane Graphite ties. h xrodnr ait fcro lorps swell as allotropes, carbon of variety extraordinary The h i fti ae st rvd con- a provide to is paper this of aim The ihacrc rtpicpe eemnto ftestruc the of determination first-principles High-accuracy ASnmes 32.j 54.b 11.b 81.05.Uw 71.15.Mb, 65.40.-b, 63.20.Dj, e numbers: membrane PACS mo the acoustic of ZA manifestation cases, direct both ago. in a years large; in as contraction, experiments. times the with three agreement be good to found very quas in the temperature, t via c room thermal-expansion calculated the negative been in-plane expansion, have distinctive thermal a heat shows The specific the calculations. and moduli the for chosen is exc the very A with Overall, dispersions, den level. phonon GGA-PBE and and the properties calculations at structural dynamics total-energy comput lattice (DFT) are (DFPT) theory graphite) rhombohedral functional (graphene, derivatives h soitdeatccntnsadpoo iprin.B dispersions. phonon and constants elastic associated the hroyaia rpriso imn,gaht,adderi and graphite, diamond, of properties thermodynamical h tutrl yaia,adtemdnmclpropertie thermodynamical and dynamical, structural, The hnndsesosaebogtt ls gemn ihavai with agreement close to brought are dispersions phonon g .INTRODUCTION I. . V,eetoi xiain onot do excitations electronic eV), 5.5 = 12,13 swl sistemlproper- thermal its as well as , 14,15 ascuet nttt fTcnlg,Cmrde A USA MA, Cambridge, Technology, of Institute Massachusetts 1 n thsee been even has it and , rbioengineering or 9,10 eateto aeil cec n Engineering, and Science Materials of Department n calculations and ioa Mounet Nicolas Dtd eray2 2008) 2, February (Dated: 2 gives , 11 ∗ - , n ioaMarzari Nicola and ilygraphite cially n pcfi heat. expansion specific thermal and as such properties lattice temperature hseprmna nu srqie ic F,i its in DFT, since required is input experimental data. experimental experimental the with of This agreement use the the improves that greatly rhombohedral argue and experi- graphite any we of require graphite case not the For do input. and calcu- mental ab-initio graphene, and fully diamond are of lations first- case from the graphene For or principles. graphite of properties thermal the diamond on studies early the of PBE- most the adopted We GGA transferability. and performance h irtoa reeeg scluae ntequasi- the in calculated (QHA) is supercells. approximation of energy harmonic use free the vibrational to wave- resort The arbitrary to any having at without in frequencies vector, phonon appearing the graphite compute dia- for DFPT data of (GGA-PBE). (GGA-PBE, 13,27,28,29 some Refs. cases graphene with and the 12,13), 21) Refs. for Ref. mostly (GGA-PBE, cal- mond appeared GGA have (LDA). approximation culations density local the using to respect with pseudo-potentials convergence ultrasoft full and which reach size, sets, basis to basis plane-wave to allow with es- tool easily paired accurate properties, when and linear-response efficient pecially and very ground-state a obtain is DFT culations. est-ucinlprubto hoy(DFPT) used to theory and we perturbation and (DFT) materials, density-functional theory questions, these density-functional open for ab-initio the picture extensive coherent of a some provide resolve To effect). eae otelreepnini the in expansion large the to related suggested otebs forkolde hsi h rtsuyon study first the is this knowledge, our of best the To ecetta ece h iiu around minimum the reaches that oefficient 19 t h C the oth pino the of eption xhnecreainfntoa,a ainewith variance at functional, exchange-correlation meauedpnec fteelastic the of dependence emperature duigacmiaino density- of combination a using ed 7,15 al aaoc h experimental the once data lable -amncapoiain Graphite approximation. i-harmonic etpeitdb isizoe fifty over Lifshitz by predicted ffect iyfntoa etraintheory perturbation sity-functional fdaod rpieadlayered and graphite diamond, of s hra otato ngahn is graphene in contraction Thermal e r hw ob epnil for responsible be to shown are des odareeti on o the for found is agreement good htti a edet h nenlstresses internal the to due be may this that 13,22,23,24,25,26 33 † lsi osatadteΓto Γ the and constant elastic c/a ua,vbainland vibrational tural, ai ngaht and graphite in ratio hc aebe performed been have which , 11,30 vatives c opeitfinite- predict to , 16,17 ieto (Poisson direction 11,20,21 c/a ste sdto used then is 18 o optimal for n espe- and 16,17 cal- c/a 2 current state of development, yields poor predictions for of the total crystal energy versus displacements of the the interlayer interactions, dominated by Van Der Waals ions: dispersion forces not well described by local or semi-local exchange correlation functionals (see Refs. 31 and 32 for details; the agreement between LDA predictions and ex- ∂2E perimental results for the c/a ratio is fortuitous). It is Cαi, βj(R − R′) = ∂u (R)∂u (R′) αi βj equil found that the weak interlayer bonding has a small influ- ence on most of the properties studied and that forcing ion elec = C (R − R′)+ C (R − R′) the experimental c/a corrects almost all the remaining αi, βj αi, βj ones. This allowed us to obtain results for all the materi- (2) als considered that are in very good agreement with the available experimental data. Here R (R’) is a vector, i (j) indicates th th The article is structured as follows. We give a brief the i (j ) atom of the unit cell, and α(β) repre- ion summary of our approach and definitions and introduce sents the cartesian components. Cαi, βj are the second DFPT and the QHA in Section II. Our ground-state, derivatives17 of Ewald sums corresponding to the ion- zero-temperature results for diamond, graphite, graphene ion repulsion potential, while the electronic contributions elec and rhombohedral graphite are presented in Section III: Cαi, βj are the second derivatives of the electron-electron Lattice parameters and elastic constants from the equa- and electron-ion terms in the ground state energy. From tions of state in subsection III A, phonon frequencies and the Hellmann-Feynman17 theorem one obtains: vibrational density of states in subsection III B, and first- principles, linear-response interatomic force constants in subsection III C. The lattice thermal properties, such elec ∂n(r) ∂Vion(r) as thermal expansion, mode Gr¨uneisen parameters, and Cαi, βj (R − R′)= Z ∂uαi(R) ∂uβj(R′) specific heat as obtained from the vibrational free energy 2 ∂ Vion(r) 3 are presented in section IV. Section V contains our final + n0(r) d r remarks. ∂uαi(R)∂uβj(R′) (3)

II. THEORETICAL FRAMEWORK (where the dependence of both n(r) and Vion(r) on the displacements has been omitted for clarity, and Vion(r) A. Basics of Density-Functional Perturbation is considered local). Theory It is seen that the electronic contribution can be ob- tained from the knowledge of the linear response of the In density-functional theory33,34 the ground state elec- system to a displacement. The key assumption is then tronic density and wavefunctions of a crystal are found by the Born-Oppenheimer approximation which views a lat- solving self-consistently a set of one-electron equations. tice vibration as a static perturbation on the electrons. In atomic units (used throughout the article), these are This is equivalent to say that the response time of the electrons is much shorter than that of ions, that is, each time ions are slightly displaced by a phonon, electrons 1 2 (− ∇ + VSCF (r))|ψii = εi|ψii, (1a) instantaneously rearrange themselves in the state of min- 2 imum energy of the new ionic configuration. Therefore, static linear response theory can be applied to describe the behavior of electrons upon a vibrational excitation. n(r′) 3 δExc VSCF (r)= d r′ + + Vion(r), (1b) For phonon calculations, we consider a periodic pertur- Z |r − r′| δ(n(r)) bation ∆Vion of wave-vector q, which modifies the self- consistent potential VSCF by an amount ∆VSCF . The linear response in the charge density ∆n(r) can be found 2 n(r)= |ψi(r)| f(εF − εi), (1c) using first-order perturbation theory. If we consider its Xi Fourier transform ∆n(q + G), and calling ψo,k the one- particle wavefunction of an electron in the occupied band where f(εF −εi) is the occupation function, εF the Fermi “o” at the point k of the Brillouin zone (and εo,k the cor- energy, Exc the exchange-correlation functional (approx- responding eigenvalue), one can get a self-consistent set imated by GGA-PBE in our case), n(r) the electronic- of linear equations similar to Eqs. (1a,1b,1c)35: density, and Vion(r) the ionic core potential (actually a sum over an array of pseudo-potentials). Once the unperturbed ground state is determined, 1 (ε + ∇2 − V (r))∆ψ = Pˆk+q∆V q ψ phonon frequencies can be obtained from the interatomic o,k 2 SCF o,k+q e SCF o,k force constants, i.e. the second derivatives at equilibrium (4a) 3

of independent harmonic oscillators. In a straightforward manner, it can be shown37 that: 4 i(q+G) r k+q ∆n(q + G)= hψ k|e− · Pˆ |∆ψ k qi V e, e o, + Xk,o (4b) F ({ai},T ) = E({ai})+ Fvib(T )

∆n(r′) 3 d δExc ¯hωq,j ∆VSCF (r)= d r′ + ∆n(r) = E({ai})+ Z |r − r | dn δ(n(r)) 2 ′ n0(r) Xq,j

¯hωq,j +∆Vion(r) (4c) + k T ln 1 − exp − (7) B   k T  Xq,j B ˆk+q Pe refers to the projector on the empty-state manifold at k + q, V to the total crystal volume, and G to any where E({ai}) is the ground state energy and the sums reciprocal lattice vector. Note that the linear response run over all the Brillouin zone wave-vectors and the band contains only Fourier components of wave vector q + G, index j of the phonon dispersion. The second term in the q so we added a superscript q to ∆VSCF . We have implic- right hand side of Eq.(7) is the zero-point motion. itly assumed for simplicity that the crystal has a band If anharmonic effects are neglected, the phonon fre- gap and that pseudo-potentials are local, but the general- quencies do not depend on lattice parameters, therefore ization to metals36 and to non-local pseudo-potentials17 the free energy dependence on structure is entirely con- are all well established (see Ref. 16 for a detailed and tained in the ground state equation of state E({ai}). complete review of DFPT). Consequently the structure does not depend on temper- Linear-response theory allows us to calculate the re- ature in a harmonic crystal. sponse to any periodic perturbation; i.e. it allows direct Thermal expansion is recovered by introducing in access to the dynamical matrix related to the interatomic Eq.(7) the dependence of the phonon frequencies on the force constants via a Fourier transform: structural parameters {ai}; direct minimization of the free energy

1 iq R F ({ai},T ) = E({ai})+ Fvib(ωq,j ({ai}),T ) D˜αi, βj(q)= Cαi, βj (R) e− · (5) MiMj XR p ¯hωq,j({ai}) th = E({ai})+ (where Mi is the mass of the i atom). 2 Phonon frequencies at any q are the solutions of the Xq,j eigenvalue problem: ¯hωq,j ({ai}) + k T ln 1 − exp − B   k T  Xq,j B 2 ω (q)uαi(q)= uβj(q)D˜ αi, βj (q) (6) (8) Xβj provides the equilibrium structure at any temperature T. In practice, one calculates the dynamical matrix on a rel- This approach goes under the name quasi-harmonic ap- atively coarse grid in the Brillouin zone (say, a 8 × 8 × 8 proximation (QHA) and has been applied successfully to 11,38,39 grid for diamond), and obtains the corresponding inter- many bulk systems . The linear thermal expansion atomic force constants by inverse Fourier transform (in coefficients of the cell dimensions of a lattice are then this example it would correspond to a 8 × 8 × 8 super- cell in real space). Finally, the dynamical matrix (and 1 ∂ai phonon frequencies) at any q point can be obtained by αi = (9) ai ∂T Fourier interpolation of the real-space interatomic force constants. The Gr¨uneisen formalism40 assumes a linear dependence of the phonon frequencies on the three orthogonal cell dimensions {ai}; developing the ground state energy up B. Thermodynamical properties to second order, (thanks to the equation of state at T = ∂F 0K), one can get from the condition ∂a = 0 the  i T When no external pressure is applied to a crystal, the alternative expression equilibrium structure at any temperature T can be found by minimizing the Helmholtz free energy F ({ai},T ) = U −TS with respect to all its geometrical degrees of free- Sik −a0,k ∂ωq,j αi = cv(q, j) (10) dom {a }. If now the crystal is supposed to be perfectly V0  ω0,q,j ∂ak  i Xq,j Xk 0 harmonic, F is the sum of the ground state total energy and the vibrational free energy coming from the parti- We follow here the formalism of Ref. 41: cv(q, j) is the tion function (in the canonical ensemble) of a collection contribution to the specific heat from the mode (q, j), 4

Sik is the elastic compliance matrix, and the subscript C. Computational details “0” indicates a quantity taken at the ground state lattice parameter. The Gr¨uneisen parameter of the mode (q, j) All the calculations that follow were performed us- is by definition ing the ESPRESSO42 package, which is a full ab- initio DFT and DFPT code available under the GNU Public License43. We used a plane-wave ba- −a ∂ω 18 γ (q, j)= 0,k q,j (11) sis set, ultrasoft pseudo-potentials from the standard k ω ∂a 44 45 0,q,j k 0 distribution (generated using a modified RRKJ ap-

proach), and the generalized gradient approximation For a structure which depends only on one lattice param- (GGA) for the exchange-correlation functional in its PBE eter a (e.g. diamond or graphene) one then gets for the parameterization19. We also used the local density ap- linear thermal expansion coefficient proximation (LDA) in order to compare some results be- tween the two functionals. In this case the parameteriza- 46 1 −a0 ∂ωq,j tion used was the one proposed by Perdew and Zunger . q α = 2 cv( , j) (12) d B0V0 ω0 q ∂a For the semi-metallic graphite and graphene cases, we Xq,j , ,j 0 used 0.03 Ryd of cold smearing47. We carefully and ex-

2 tensively checked the convergence in the energy differ- ∂ E where B0 is defined by B0 = V0 ∂V 2 (V represents the ences between different configurations and the phonon volume of a three-dimensional crystal such as diamond frequencies with respect to the wavefunction cutoff, or the surface of a two-dimensional one like graphene), d the dual (i.e. the ratio between charge density cut- is the number of dimensions (d = 3 for diamond, d = 2 off and wavefunction cutoff), the k-point sampling of for graphene), and V0 is the volume (or the surface) at the Brillouin zone, and the interlayer vacuum spacing equilibrium. for graphene. Energy differences were converged within In the case of graphite there are two lattice parameters: 5 meV/atom or better, and phonon frequencies within 1 a in the basal plane and c perpendicular to the basal 1 − 2 cm− . In the case of graphite and graphene phonon plane, so that one gets frequencies were converged with respect to the k-point sampling after having set the smearing parameter at 0.03 Ryd. Besides, values of the smearing between 0.02 Ryd 1 −a0 ∂ωq,j and 0.04 Ryd did not change the frequencies by more αa = cv(q, j) (S11 + S12) 1 V0  2ω0 q ∂a than 1 − 2 cm− . Xq,j , ,j 0 In a solid, translational invariance guaranties that

−c ∂ωq three phonon frequencies at Γ will go to zero. In our + S 0 ,j (13a) 13 ω ∂c  GGA-PBE DFPT formalism this condition is exactly sat- 0,q,j 0 isfied only in the limit of infinite k-point sampling and full convergence with the plane-wave cutoff. For the case of graphene and graphite we found in particular that an 1 −a0 ∂ωq,j αc = cv(q, j) S13 exceedingly large cutoff (100 Ryd) and dual (28) would be V0  ω0 q ∂a Xq,j , ,j 0 needed to recover phonon dispersions (especially around

Γ and the Γ − A branch) with the tolerances mentioned; −c ∂ω + S 0 q,j (13b) on the other hand, application of the acoustic sum rule 33 ω ∂c  0,q,j 0 (i.e. forcing the translational symmetry on the inter-

atomic force constants) allows us to recover these highly The mode Gr¨uneisen parameters provide useful insight converged calculations above with a more reasonable cut- to the thermal expansion mechanisms. They are usu- off and dual. ally positive, since phonon frequencies decrease when the Finally, the cutoffs we used were 40 Ryd for the wave- solid expands, although some negative mode Gr¨uneisen functions in all the carbon materials presented, with du- parameters for low-frequency acoustic modes can arise als of 8 for diamond and 12 for graphite and graphene. and sometimes compete with the positive ones, giving a We used a 8×8×8 Monkhorst-Pack k-mesh for diamond, negative thermal expansion at low temperatures, when 16 × 16 × 8 for graphite, 16 × 16 × 4 for rhombohedral only the lowest acoustic modes can be excited. graphite and 16 × 16 × 1 for graphene. All these meshes Finally, the heat capacity of the unit cell at constant were unshifted (i.e. they do include Γ). Dynamical ma- 2 ∂ Fvib 37 trices were initially calculated on a 8 × 8 × 8 q-points volume can be obtained from Cv = −T ∂T 2 :  V mesh for diamond, 8 × 8 × 4 for graphite, 8 × 8 × 2 for rhombohedral graphite and 16 × 16 × 1 for graphene. 2 ¯hωq,j 1 Finally, integrations over the Brillouin zone for the vi- Cv = cv(q, j)= kB brational free energy or the heat capacity were done us- 2k T  2 hω¯ q,j q,j q,j B sinh X X 2kB T ing phonon frequencies that were Fourier interpolated on   (14) much finer meshes. The phonon frequencies were usually 5

TABLE I: Equilibrium lattice parameter a0 and bulk mod- ulus B0 of diamond at the ground state (GS) and at 300 K 4.5 (see Section IV), compared to experimental values. 4 Present calculation Experiment (300 K) a Lattice constant a0 6.743 (GS) 6.740 (a.u.) 6.769 (300 K) 3.5 b c/a Bulk modulus B0 432 (GS) 442 ± 2 (GPa) 422 (300 K) 3

aRef. 49 2.5 bRef. 50 2 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 computed at several lattice parameters and the results a (Bohr) interpolated to get their dependence on lattice constants. A final remark is that we were careful to use the same parameters (cutoffs, k-points sampling, smearing, etc.) FIG. 1: Contour plot of the ground state energy of graphite in the determination of the ground state equation of state as a function of a and c/a (isoenergy contours are not equidis- and that of the phonon frequencies, since these two terms tant). need to be added in the free energy expression. 0.1

III. ZERO-TEMPERATURE RESULTS 0.08

A. Structural and elastic properties 0.06

0.04 We performed ground state total-energy calculations c/a = 2.725 c/a = 3.45 on diamond, graphite, and graphene over a broad range

E (eV / atom) 0.02 of lattice parameters. The potential energy surface can then be fitted by an appropriate equation of state. The 0 minimum gives the ground state equilibrium lattice pa- rameter(s). The second derivatives at that minimum are -0.02 related to the bulk modulus or elastic constants. 2 2.5 3 3.5 4 4.5 For the case of diamond we chose the Birch equation c/a of state48 (up to the fourth order) to fit the total energy FIG. 2: Ground state energy of graphite as a function of vs. the lattice constant a: c/a at fixed a = 4.65 a.u.. The theoretical (PBE) and the experimental c/a are shown. The zero of energy has been set to the PBE minimum. 2 2 2 3 9 a0 a0 E(a)= −E0 + B0V0 − 1 + A − 1 8  a    a   2 4 2 5 th a0 a0 equation of state was fitted by a two-dimensional 4 or- +B − 1 + O − 1 (15)  a    a   der polynomial of variables a and c. To illustrate the very small dependence of the ground state energy with where B0 is the bulk modulus, V0 the primitive cell vol- the c/a ratio, we have plotted the results of our calcu- 3 a lations over a broad range of lattice constants in Figs. 1 ume (V0 = 4 here) and A and B are fit parameters. The Murnaghan equation of state or even a polynomial and 2. would fit equally well the calculations around the min- A few elastic constants can be obtained from the sec- imum of the curve. A best fit of this equation on our ond derivatives of this energy22: data gives us both the equilibrium lattice parameter and the bulk modulus; our results are summarized in Table I. The agreement with the experimental values is very good, 2 1 ∂ E even after the zero-point motion and thermal expansion C11 + C12 = 2 √3c0 ∂a are added to our theoretical predictions (see Section IV).   2 The ground state equation of state of graphene was fit-  2c0 ∂ E  C = 2 2 th Stiffness coefficients  33 √3a ∂c (16a) ted by a 4 order polynomial, and the minimum found  0 for a =4.654 a.u., which is very close to the experimen- 2 1 ∂ E tal in-plane lattice parameter of graphite. The graphite  C13 =  √3a0 ∂a∂c   6

t 1 data. Tetragonal shear modulus C = [(C11 + C12) 6 In Table III and IV we summarize our results at high- symmetry points and compare them with experimental +2C33 − 4C13] (16b) data. In diamond, GGA produces softer modes than LDA11 on the whole (as expected), particularly at Γ (op- C (C + C ) − 2C2 tical mode) and in the optical Γ-X branches. For these, Bulk modulus B = 33 11 12 13 0 6Ct the agreement is somehow better in LDA; on the other (16c) hand the whole Γ-L dispersion is overestimated by LDA. The results on graphite require some comments. In We summarize all our LDA and GGA results in Ta- Table IV and Figs. 4, 5, 6 and 7, modes are classified ble II: For LDA, both the lattice parameter a0 and the as follow: L stands for longitudinal polarization, T for c0/a0 ratio are very close to experimental data. Elastic in-plane transversal polarization and Z for out-of-plane constants were calculated fully from first-principles, in transversal polarization. For graphite, a prime (as in the sense that the second derivatives of the energy were LO’) indicates an optical mode where the two atoms in taken at the theoretical LDA a0 and c0, and that only each layer of the unit cell oscillate together and in phase these theoretical values were used in Eqs. (16a). Elastic opposition to the two atoms of the other layer. A non- constants are found in good agreement with experiments, primed optical mode is instead a mode where atoms in- except for the case of C13 which comes out as negative side the same layer are “optical” with respect to each (meaning that the Poisson’s coefficient would be nega- other. Of course “primed” optical modes do not exist for tive). graphene, since there is only one layer (two atoms) per Fully theoretical GGA results (second column of Ta- unit cell. ble II) compare poorly to experimental data except for We observe that stacking has a negligible effect on all 1 the a0 lattice constant, in very good agreement with the frequencies above 400 cm− , since both rhombohedral experiments. Using the experimental value for c0 in graphite and hexagonal graphite show nearly the same Eqs. (16) improves only the value of C11 + C12 (third dispersions except for the Γ-A branch and the in-plane column of Table II). Most of the remaining disagree- dispersions near Γ. The in-plane part of the dispersions is ment is related to the poor value obtained for c/a; if the also very similar to that of graphene, except of course for 1 second derivatives in Eqs. (16a) are taken at the exper- the low optical branches (below 400 cm− ) that appear imental value for c/a all elastic constants are accurately in graphite and are not present in graphene. recovered except for C13 (fourth column of Table II). For graphite as well as diamond GGA tends to make In both LDA and GGA, errors arise from the fact that the high optical modes weaker while LDA makes them Van Der Waals interactions between graphitic layers are stronger than experimental values. The opposite hap- poorly described. These issues can still be addressed pens for the low optical modes, and for the Γ-A branch within the framework of DFT (as shown by Langreth and of graphite; the acoustic modes show marginal differences collaborators, Ref. 31) at the cost of having a non-local and are in very good agreement with experiments. Over- exchange-correlation potential. all, the agreement of both LDA and GGA calculations Zero-point motion and finite-temperature effects will with experiments is very good and comparable to that be discussed in detail in Section IV. between different measurements. Some characteristic features of both diamond and graphite are well reproduced by our ab-initio results, such B. Phonon dispersion curves as the LO branch overbending and the associated shift of the highest frequencies away from Γ. Also, in the We have calculated the phonon dispersion relations case of graphite, rhombohedral graphite and graphene, for diamond, graphite, rhombohedral graphite and the quadratic dispersion of the in-plane ZA branch in graphene. For diamond and graphene, we used the the- the vicinity of Γ is observed; this is a characteristic fea- oretical lattice parameter. For graphite, we either used ture of the phonon dispersions of layered crystals60,61, the theoretical c/a or the experimental one (c/a =2.725). observed experimentally e.g. with neutron scattering58. We will comment extensively in the following on the role Nevertheless, some discrepancies are found in graphite. of c/a on our calculated properties. The most obvious one is along the Γ-M TA branch, where Finally we also calculated the phonon dispersions for EELS55 data show much higher frequencies than calcula- rhombohedral graphite, which differs from graphite only tions. Additionally several EELS experiments56,57 report in the stacking of the parallel layers: in graphite the a gap between the ZA and ZO branches at K while these stacking is ABABAB while it is ABCABC in rhombohe- cross each other in all the calculations. In these cases the dral graphite, and the latter unit cell contains six atoms disagreement could come either from a failure of DFT instead of four. We therefore used the same in-plane lat- within the approximations used or from imperfections in tice parameter and same interlayer distance as in graphite the crystals used in the experiments. c 3 (that is, a a ratio multiplied by 2 ). Results are presented There are also discrepancies between experimental in Figs. 3, 4, 5, 6 and 7, together with the experimental data, in particular in graphite for the LA branch around 7

TABLE II: Structural and elastic properties of graphite according to LDA, GGA, and experiments

LDAfully GGAfully GGAusing GGAwith Experiment nd theoretical theoretical exp. c0 2 derivatives (300K) inEqs.(16a) takenatexp. c0/a0 a Lattice constant a0(a.u.) 4.61 4.65 4.65 4.65(fixed) 4.65 0.003 c0 ± a a0 ratio 2.74 3.45 3.45 2.725(fixed) 2.725 0.001 ± b C11 + C12 (GPa) 1283 976 1235 1230 1240 40 ± b C33 (GPa) 29 2.4 1.9 45 36.5 1 ± b C13 (GPa) -2.8 -0.46 -0.46 -4.6 15 5 ± c B0 (GPa) 27.8 2.4 1.9 41.2 35.8 Ct (GPa) 225 164 207 223 208.8 c

aRefs. 51,52,53, as reported by Ref. 22. bRef. 6 cRef. 54, as reported by Ref. 22 1500 ) -1 1000

500 Frequency (cm 0 Γ K X Γ LXWL VDOS

FIG. 3: GGA ab-initio phonon dispersions (solid lines) and vibrational density of states (VDOS) for diamond. Experimental neutron scattering data from Ref. 9 are shown for comparison (circles).

Table IV show results of GGA calculations performed at TABLE III: Phonon frequencies of diamond at the high- 1 1 the theoretical c/a. Low frequencies (below 150 cm ) symmetry points Γ, X and L, in cm− . − between Γ and A are strongly underestimated, as are ΓO XT A XTO XLO LT A LLA LTO LLO the ZO’ modes between Γ and M, while the remaining LDAa 1324 800 1094 1228 561 1080 1231 1275 branches are barely affected. b GGA 1289 783 1057 1192 548 1040 1193 1246 The high-frequency optical modes are instead strongly Exp.c 1332 807 1072 1184 550 1029 1206 1234 dependent on the in-plane lattice constant. The differ- aRef. 11 ence between the values of a in LDA and GGA explains bPresent calculation much of the discrepancy between the LDA optical modes c Ref. 9 and the GGA ones. Indeed, a LDA calculation performed at a = 4.65 a.u. and c/a = 2.725 (not shown here) brings the phonon frequencies of these modes very close K: EELS data from Ref. 56 agree with our ab-initio re- to the GGA ones obtained with the same parameters, 1 sults while those from Ref. 57 deviate from them. while lower-energy modes (below 1000 cm− ) are hardly affected. Finally, we should stress again the dependence of the graphite phonon frequencies on the in-plane lattice pa- Our final choice to use the theoretical in-plane lattice rameter and c/a ratio. The results we have analyzed parameter and the experimental c/a seems to strike a so far were obtained using the theoretical in-plane lat- balance between the need of theoretical consistency and tice parameter a and the experimental c/a ratio for both that of accuracy. Therefore, the remaining of this section GGA and LDA. Since the LDA theoretical c/a is very is based on calculations performed using the parameters close to the experimental one (2.74 vs. 2.725) and the discussed above (a = 4.61 for LDA, a = 4.65 for GGA interlayer bonding is very weak, these differences do no and c/a =2.725 in each case). matter. However this is not the case for GGA, as the the- Elastic constants can be extracted from the data on oretical c/a ratio is very different from the experimental sound velocities. Indeed, the latters are the slopes of one (3.45 vs. 2.725). Fig. 7 and the second column of the dispersion curves in the vicinity of Γ and can be ex- 8 1800 LO 1600 LO TO 150 1400 LO' TO

) 100

-1 LA 1200 TO' 50 LA TA 1000 0 A Γ ZO 800 ZO

600 LA Frequency (cm TA 400 TA

200 ZO' ZO' ZA ZA 0 A Γ M K Γ VDOS

FIG. 4: GGA (solid lines) and LDA (dashed line) ab-initio phonon dispersions for graphite, together with the GGA vibrational density of states (VDOS). The inset shows an enlargement of the low-frequency Γ-A region. The experimental data are EELS (Electron Energy Loss Spectroscopy) from Refs.55, 56, 57 (respectively squares, , and filled circles), neutron scattering from Ref. 58 (open circles), and x-ray scattering from Ref. 12 (triangles). Data for Refs. 55 and 57 were taken from Ref. 13.

1800 1800 LO LO LO 1600 1600 LO TO 150 TO 1400 TO TO 1400 ) ) 100 -1 -1 1200 1200 50 LA 1000 1000 0 ZO ZO A Γ ZO 800 800 ZO

600 LA LA 600 LA Frequency (cm Frequency (cm TA TA 400 TA TA 400

200 ZA ZA 200 ZO' ZO' ZA ZA 0 0 Γ M K Γ VDOS A Γ M K Γ VDOS

FIG. 5: GGA ab-initio phonon dispersions for graphene (solid FIG. 6: GGA ab-initio phonon dispersions for rhombohe- lines). Experimental data for graphite are also shown, as in dral graphite. The inset shows an enlargement of the low- Fig. 4. frequency Γ-A region.

pressed as the square root of linear combinations of elas- Table V. tic constants (depending on the branch considered) over The overall agreement with experiment is good to very the density (see Ref. 62 for details). We note in pass- good. LDA leads to larger elastic constants, as expected ing that we computed the density consistently with the from the general tendency to “overbind”, but still agrees geometry used in the calculations (see Table IV for de- well with experiment. For diamond, the agreement is tails, first column for LDA and third one for GGA), and particularly good. As for C13 in graphite, it is quite not the experimental density. Our results are shown in difficult to obtain it from the dispersion curves since it 9

TABLE IV: Phonon frequencies of graphite and derivatives at the high-symmetry points A, Γ, M and K, in cm−1. The lattice constants used in the calculations are also shown. Graphite Rhombo.graphite Graphene Graphite Functional LDA GGA GGA GGA GGA Experiment In-plane lattice ct. a0 4.61 a.u. 4.65 a.u. 4.65 a.u. 4.65 a.u. 4.65 a.u. 4.65 a.u. Interlayer distance/a0 1.36 1.725 1.36 1.36 15 1.36 ′ a AT A/T O 31 6 29 35 ′ a ALA/LO 80 20 96 89 ALO 897 880 878 AT O 1598 1561 1564 ′ a ΓLO 44 8 41 35 49 ′ b a ΓZO 113 28 135 117 95 , 126 b ΓZO 899 881 879 879 881 861 b f ΓLO/T O 1593 1561 1559 1559 1554 1590 , 1575 1604 1561 1567 a b d MZA 478 471 477 479 471 471 , 465 , 451 d MT A 630 626 626 626 626 630 b MZO 637 634 634 635 635 670 c MLA 1349 1331 1330 1330 1328 1290 c MLO 1368 1346 1342 1344 1340 1321 c b MT O 1430 1397 1394 1394 1390 1388 , 1389 d d e KZA 540 534 540 535 535 482 , 517 , 530 d e KZO 544 534 542 539 535 588 , 627 KT A 1009 999 998 998 997 c c KLA/LO 1239 1218 1216 1216 1213 1184 , 1202 g g d e KT O 1359 1308 1319 1319 1288 1313 , 1291 aRef. 58 bRef. 55 cRef. 12 dRef. 57 eRef. 56 fRef. 59 gNote that a direct calculation of this mode with DFPT (instead of the Fourier interpolation result given here) leads to a significantly lower value in the case of graphite — 1297 cm−1 instead of 1319 cm−1. This explains much of the discrepancy between the graphite and graphene result, since in the latter we used a denser q-points mesh. This effect is due to the Kohn anomaly occurring at K29.

1800 LO LO TABLE V: Elastic constants of diamond and graphite as 1600 calculated from the phonon dispersions, in GPa. TO 1400 TO 50 Diamond Graphite )

-1 LO' 1200 25 Functional GGA Exp. LDA GGA Exp. LA b a TO' LA C11 1060 1076.4 ± 0.2 1118 1079 1060 ± 20 1000 0 b a A Γ ZO C12 125 125.2 ± 2.3 235 217 180 ± 20 b a 800 ZO C44 562 577.4 ± 1.4 4.5 3.9 4.5 ± 0.5 a C33 - - 29.5 42.2 36.5±1 600 LA Frequency (cm TA TA aRef. 6 400 bRef. 50 200 ZA ZO' ZA ZO' 0 A Γ M K Γ VDOS and thus several thermodynamical quantities. Before ex- ploring this in Section IV, we want to discuss the nature FIG. 7: GGA ab-initio phonon dispersions for graphite at and decay of the interatomic force constants in carbon the theoretical c/a. The inset shows an enlargement of the based materials. low-frequency Γ-A region.

C. Interatomic force constants enters the sound velocities only in a linear combination involving other elastic constants, for which the error is As explained in Section II A, the interatomic force con- almost comparable to the magnitude of C13 itself. stants Ci, j (R−R′) are obtained in our calculations from An accurate description of the phonon dispersions al- the Fourier transform of the dynamical matrix D˜ i, j (q) low us to predict the low-energy structural excitations calculated on a regular mesh inside the Brillouin zone 10

0 0

-5 -5 Graphene Graphene ln(C(r)) ln(C(r))

-10 Diamond -10 Graphite

-15 -15 0 10 20 30 40 50 0 10 20 30 40 50 r (Bohr) r (Bohr)

FIG. 8: Decay of the norm of the interatomic force constants FIG. 9: Decay of the norm of the interatomic force constants as a function of distance for diamond (thin solid line) and as a function of distance for graphite (thin solid line) and graphene (thick solid line), in a semi-logarithmic scale. The graphene (thick solid line). dotted and dashed lines show the decay for diamond along the (100) and (110) directions.

(8×8×8 for diamond, 8×8×4 for graphite and 16×16×1 that a simple truncation is not comparable to the VFF for graphene). This procedure is exactly equivalent (but or 4NNFC models, where effective interatomic force con- much more efficient) than calculating the interatomic stants would be renormalized. force constants with frozen phonons (up to 47 neighbors Figs. 10 and 11 show the change in frequency for se- in diamond and 74 in graphene). At a given R, C (R) i, j lected modes in diamond and graphene as a function of is actually a 2nd order tensor, and the decay of its norm the truncation range. The modes we chose are those most (defined as the square root of the sum of the squares of strongly affected by the number of neighbors included. all the matrix elements) with distance is a good measure to know the effect of distant neighbors. In Fig. 8 we For diamond, our whole supercell contains up to 47 have plotted the natural logarithm of such a norm with neighbors, and the graph shows only the region up to respect to the distance from a given atom, for diamond 20 neighbors included, since the selected modes do not and graphene. The norm has been averaged on all the 1 vary by more than 1 cm− after that. With 5 neigh- neighbors located at the same distance before taking the bors, phonon frequencies are already near their converged logarithm. value, being off by at worst 4% off; very good accuracy The force-constants decay in graphene is slower than 1 (5 cm− ) is obtained with 13 neighbors. in diamond, and it depends much less on direction. In diamond decay along (110) is much slower than in other For graphene, our 16×16×1 supercell contains up to 74 directions due to long-range elastic effects along the cova- neighbors, but after the 30th no relevant changes occur. lent bonds. This long-range decay is also responsible for At least 4 neighbors are needed for the optical modes to the flattening of the phonon dispersions in zincblende and be converged within 5-8%. Some acoustic modes require diamond along the K-X line (see Fig. 3 more neighbors, as also pointed out in Ref. 24. As can be and Ref. 17, for instance). seen in Fig. 11, the frequency of some ZA modes in the In Fig. 9 we show the decay plot for graphite and Γ-M branch (at about one fourth of the branch) oscillates graphene, averaged over all directions. The graphite in- strongly with the number of neighbors included, and can teratomic force constants include values corresponding to even become imaginary when less than 13 are used, re- graphene (in-plane nearest neighbors) and smaller values sulting into an instability of the crystal. This behavior corresponding to the weak interlayer interactions. does not appear in diamond. Also, the KT O mode keeps It is interesting to assess the effects of the truncation varying in going down from 20 to 30 neighbors, though 1 of these interatomic force constants on the phonon dis- this effect remains small (8 − 9 cm− ). This drift could persion curves. This can be done by replacing the force signal the presence of a Kohn anomaly64. Indeed, at the constants corresponding to distant neighbors by zero. In K point of the Brillouin zone the electronic this way the relevance of short-range and long-range con- vanishes in graphene, so that a singularity arises in the tributions can be examined. The former are relevant for highest optical phonon mode. Therefore a finer q-point short-range force-constant models such as the VFF (Va- mesh is needed around this point, and longer-ranged in- lence Force Field)8 or the 4NNFC (4th Nearest-Neighbor teratomic force constants. This effect is discussed in de- Force Constant)63 used e.g. in graphene. Note however tail in Ref. 29. 11

1500 1290 6.84

1275 Frequency (cm Frequency ) 1260

-1 1000 6.82

1035

1020 500 6.8

1005 -1 ) Frequency (cm 560

0 6.78 550 Lattice parameter (Bohr) 0 5 10 15 20 10 15 20 Number of neighbors included before truncation 6.76 0 500 1000 1500 2000 2500 FIG. 10: Phonon frequencies of diamond as a function of the Temperature (K) number of neighbors included in the interatomic force con- stants: ΓO (solid line), XTO (dotted line), and LT A (dashed FIG. 12: Lattice parameter of diamond as a function of tem- line). perature

1565 1560 expansion). In particular, our “corrected” equation of 1500 1555 state is obtained by fitting with a fourth order polyno- ) Frequency (cm Frequency mial the true equation of state around the experimental -1 1300 1000 a and c/a, and then dropping from this polynomial the 1290 linear order terms. Since the second derivatives of the 636 polynomial are unchanged, this is to say we keep the

500 635 elastic constants unchanged. The only input from exper- -1 Frequency (cm 634 ) iments remains the c/a ratio. We have also checked the 45 effect of imposing to C13 its experimental value (C13 is 0 40 the elastic constant that is less accurately predicted), but 35 0 10 20 30 15 20 25 30 the changes were small. Number of neighbors included before truncation The dependence of the phonon frequencies on the lat- tice parameters was determined by calculating the whole FIG. 11: Phonon frequencies of graphene as a function of phonon dispersions at several values and interpolating the number of neighbors included in the interatomic force these in between. For diamond and graphene we used constants: ΓLO/TO (solid line), KTO (dot-dashed), MZO four different values of a (from 6.76 to 6.85 a.u. for di- (dashed), and for the dotted line a phonon mode in the ZA amond, and from 4.654 to 4.668 a.u. for graphene) and branch one-fourth along the Γ to M line. interpolated them with a cubic polynomial. For graphite, since the minimization space is two dimensional, we re- stricted ourselves to a linear interpolation and calculated IV. THERMODYNAMICAL PROPERTIES the phonon dispersions at three different combinations of the lattice constants : (a,c/a)=(4.659,2.725), (4.659,2.9) We present in this final section our results on the and (4.667,2.725). thermodynamical properties of diamond, graphite and Before considering thermal expansion, we examine the graphene using the quasi-harmonic approximation and zero-point motion. Indeed, lattice parameters at 0 K phonon dispersions at the GGA level. As outlined in Sec- are different from their ground state values. The effects tion II B we first perform a direct minimization over the of the thermal expansion (or contraction) up to about lattice parameter(s) {ai} of the vibrational free energy 1000 K are small compared to the zero-point expansion F ({ai},T ) (Eq. 8). This gives us, for any temperature T, of the lattice parameters. In diamond, a expands from the equilibrium lattice parameter(s), shown in Figs. 12, 6.743 a.u. (ground state value) to 6.768 a.u., a difference 13 and 14. For diamond and graphene, we used in Eq. 8 of 0.4%. For graphene, a is 4.654 a.u. at the ground the equations of state obtained from the ground state state and 4.668 a.u. with zero-point motion corrections calculations presented in Section III A. For graphite this (+0.3%); for graphite a increases from 4.65 to 4.664 a.u. choice would not be useful or accurate, since the theoret- (+0.3%) and c from 12.671 to 12.711 (+0.3%). The in- ical c/a is much larger than the experimental one. So we crease is similar in each case, and comparable to the dis- forced the equation of state to be a minimum for a=4.65 crepancy between experiments and GGA or LDA ground c states. a.u. and a =2.725 (fixing only c/a and relaxing a would give a=4.66 a.u., with negligible effects on the thermal The coefficients of linear thermal expansion at any T 12

7

4.67 6 Graphite

) 5 −1

Κ 4 4.66 −6

a (Bohr) 3

α(Τ) (10 2 Graphene 4.65 1

0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 3000 Temperature (K) Temperature (K)

FIG. 13: In-plane lattice parameter of graphite (solid line) FIG. 15: Coefficient of linear thermal expansion for diamond and graphene (dashed line) as a function of temperature as a function of temperature. We compare our QHA-GGA ab- initio calculations (solid line) to experiments (Ref. 10, filled circles), a path integral Monte-Carlo study using a Tersoff 13.4 empirical potential (Ref. 65, open squares) and the QHA- LDA study by Pavone et al 11 (dashed line). The QHA-GGA thermal expansion calculated using the Gr¨uneisen equation 13.2 (Eq. 12) is also shown (dotted line).

2 13

c (Bohr) 1 Graphite

12.8 ) -1 0 K -6 12.6 -1 0 500 1000 1500 2000 2500 (T) (10 Temperature (K) a -2

α Graphene

FIG. 14: Out-of-plane lattice parameter of graphite as a func- -3 tion of temperature -4 0 500 1000 1500 2000 2500 Temperature (K) are obtained by numerical differentiation of the previous data. Results are shown in Figs. 15, 16 and 17. FIG. 16: In-plane coefficient of linear thermal expansion as a For the case of diamond, we have also plotted the lin- function of temperature for graphite (solid line) and graphene ear thermal expansion coefficient calculated using the (dashed line) from our QHA-GGA ab-initio study. The ex- Gr¨uneisen formalism (Eq. 12) instead of directly mini- perimental results for graphite are from Ref. 14 (filled circles) mizing the free energy. While at low temperature the and Ref. 7 (open diamonds). two curves agree, a discrepancy becomes notable above 1000 K, and direct minimization should be performed. This difference between Gr¨uneisen theory and direct min- expansion is underestimated by about 30% at 1000 K. imization seems to explain much of the discrepancy be- In-plane, the coefficient of linear thermal expansion is tween the calculations of Ref. 11 and our results. Fi- confirmed to be negative from 0 to about 600 K. This nally a Monte-Carlo path integral study by Herrero and feature, absent in diamond, is much more apparent in Ram´irez65, which does not use the QHA, gives very sim- graphene, where the coefficient of linear thermal expan- ilar results. sion keeps being negative up to 2300 K. This thermal con- For graphite, the in-plane coefficient of linear thermal traction will likely appear also in single-walled nanotubes expansion slightly overestimates the experimental values, (one graphene sheet rolled on itself)66. Some molecular but overall the agreement remains excellent, even at high dynamics calculations41,67 have already pointed out this temperatures. Out-of-plane, the agreement holds well up characteristic of SWNT. to 150 K, after which the coefficient of linear thermal To further analyze thermal contraction, we plotted 13

40 3

35

30 0 ) -1

K 25 -6 (q)

20 j -3

γ ZO' ZO'

(T) (10 15 c

α 10 -6 ZA ZA 5

0 -9 0 500 1000 1500 2000 2500 A Γ M K Γ Temperature (K) FIG. 19: Ab-initio in-plane mode Gr¨uneisen parameters for FIG. 17: Out-of-plane coefficient of linear thermal expansion graphite. as a function of temperature for graphite from our QHA-GGA ab-initio study (solid line). The experimental results are from 4 Ref. 14 (filled circles), and Ref. 7 (open diamonds). 2 2 0

1.5 -2 (q) j γ -4 (q)

j 1

γ ZA -6 ZA

-8 0.5 Γ M K Γ

FIG. 20: Ab-initio mode Gr¨uneisen parameters for graphene. 0 Γ K X Γ LXWL

FIG. 18: Ab-initio mode Gr¨uneisen parameters for diamond. tive Gr¨uneisen parameters are still not excited) the con- tribution from the negative Gr¨uneisen parameters will be predominant and thermal expansion (from Eq. 12) nega- in Figs. 18, 19, 20 and 21 the mode Gr¨uneisen pa- tive. rameters (see Section II B) of diamond, graphene and The negative Gr¨uneisen parameters correspond to the graphite. These have been obtained from an interpo- lowest transversal acoustic (ZA) modes, and in the case lation of the phonon frequencies by a quadratic (or lin- of graphite to the (ZO’) modes, which can be described ear, for graphite) polynomial of the lattice constants, and as “acoustic” inside the layer and optical out-of-plane computed at the ground state lattice parameter. (see Section III B). Indeed, the phonon frequencies for The diamond Gr¨uneisen parameters have been already such modes increase when the in-plane lattice parameter calculated with LDA (see Refs. 11,20); our GGA re- is increased, contrary to the usual behavior, because the sults agree very well with these. In particular, all layer is more “stretched” when a is increased, and atoms the Gr¨uneisen parameters are shown to be positive (at in that layer will be less free to move in the z direction odds with other group IV semiconductors such as Si (just like a rope that is stretched will have vibrations of or Ge). The situation is very different in graphite and smaller amplitude, and higher frequency). In graphite graphene, where some bands display large and nega- these parameters are less negative because of the inter- tive Gr¨uneisen parameters (we have used the definition action between layers: atoms are less free to move in the γ (q)= − a dωj (q) ). j 2ωj (q) da z-direction than in the case of graphene. While not visible in the figure, the Gr¨uneisen parame- This effect, known as the “membrane effect”, was pre- ter for the lowest acoustic branch of graphite becomes as dicted by Lifshitz61 in 1952, when he pointed out the low as -40, and as low as -80 for graphene. Therefore, at role of these ZA modes (also called “bending modes”) low temperatures (where most optical modes with posi- in layered materials. In particular, several recent stud- 14

6 1.002

5 (298 K) 0 1 (T)/B

4 0 0.998 B 0 100 200 300 400 3 Temperature (K) (q) j

γ 440 2 420 400 380

1 (T) (GPa) 0

B 360 340 0 0 500 1000 1500 2000 2500 3000 A Γ M K Γ Temperature (K)

FIG. 21: Ab-initio out-of-plane mode Gr¨uneisen parameters FIG. 22: Lower panel: Bulk modulus B0(T ) of diamond as a for graphite. function of temperature. The filled circle indicates the value of the bulk modulus (as in Table I) before accounting for zero- point motion. Upper panel: theoretical (solid line) and ex- ies have highlighted the relevance of these modes to the perimental values (Ref. 72, open circles) for the ratio between thermal properties of layered crystals such as graphite, B0(T ) and B0(298K) in the low temperature regime. boron nitride and gallium sulfide68,69,70. The knowledge of the equilibrium lattice constant(s) 1200 at any temperature allows us also to calculate the de- 1100 C +C pendence of elastic constants on temperature. To do so 11 12 we calculated the second derivatives of the free energy 1000 (Eq. 8) vs. lattice constant(s) at the finite-temperature 40 C equilibrium lattice parameter(s). We checked that this 33 30 B0 was equivalent to a best fit of the free energy at T around the equilibrium lattice parameter(s). 20 Results are shown in Figs. 22 and 23 (diamond and 3 Elastic constants (GPa) graphite respectively). Again, the zero-point motion has 0 C13 a significant impact on the elastic constants; the agree- -3 ment with experimental data for the temperature depen- 0 500 1000 1500 2000 2500 dence of the ratio of the bulk modulus of diamond to its Temperature (K) 298 K value is excellent (upper panel of Fig. 22). We note that the temperature dependence of the bulk FIG. 23: Elastic constants of graphite (C11 + C12, C13, C33) modulus of diamond has already been obtained by Karch and bulk modulus (B0) as a function of temperature. The et al 71 using LDA calculations. filled circles (at 0 K) indicate their ground state values (as in As final thermodynamic quantities, we present results Table II) before accounting for zero-point motion. on the heat capacities for all the systems considered, at constant volume (Cv) and constant pressure (Cp). Cv has been computed using Eq. 14, in which we used at V. CONCLUSION each temperature T the interpolated phonon frequencies calculated at the lattice constant(s) that minimize the We have presented a full ab-initio study of the struc- respective free energy. To calculate Cp, we added to Cv tural, vibrational and thermodynamical properties of di- 2 the additional term Cp − Cv = T V0B0αV where V0 is the amond, graphite and graphene, at the DFT-GGA level unit cell volume, αV the volumetric thermal expansion and using the quasi-harmonic approximation to derive and B0 the bulk modulus. All these quantities were taken thermodynamic quantities. All our results are in very from our ab-initio results and evaluated at each of the good agreement with experimental data: the phonon dis- temperatures considered. The difference between Cp and persions are well-reproduced, as well as most of the elastic Cv is very small, at most about 2% of the value of Cv constants. In graphite, the C33 elastic constant and the for graphite and 5% for diamond. Note that Cp and Cv Γ to A phonon dispersions (calculated here with GGA for shown on the figures are normalized by dividing by the the first time) were found to be in good agreement with unit cell mass. experimental results provided the calculations were per- The heat capacity of diamond, graphite and graphene formed at the experimental c/a. Only the C13 constant are almost identical except at very low temperatures. remains in poor agreement with experimental data. Agreement with experimental data is very good. The decay of the long-ranged interatomic force con- 15

2500 stants was analyzed in detail. It was shown that interac- tions in the (110) direction in diamond are longer-ranged 2000 than these in other directions, as is characteristic of the zincblende and diamond structures. For graphene and )

-1 graphite, in-plane interactions are even longer-ranged 1500 .kg and phonon frequencies sensitive to the truncation of the -1 interatomic force constants. 1000 Cp (J.K

500 Thermodynamical properties such as the thermal ex- pansion, temperature dependence of elastic moduli and 0 0 500 1000 1500 2000 2500 3000 specific heat were calculated in the quasi-harmonic ap- Temperature (K) proximation. These quantities were all found to be in close agreement with experiments, except for the out- FIG. 24: Constant pressure heat capacity for diamond (solid of-plane thermal expansion of graphite at temperatures line). Experimental results are from Refs. 49 and 73 (circles), higher than 150 K. Graphite shows a distinctive in- as reported by Ref. 65. plane negative thermal-expansion coefficient that reaches 2500 the minimum around room temperature, in very good agreement with experiments. This effect is found to be three times as large in graphene. In both cases, the 2000 mode Gr¨uneisen parameters show that the ZA “bending”

) acoustic modes are responsible for the contraction, in a -1 1500 direct manifestation of the membrane effect predicted by .kg 61 -1 Lifshitz in 1952. These distinctive features will likely affect the thermodynamical properties of single-walled 1000 and multiwall carbon nanotubes41,66,67. Cp (J.K

500

0 0 500 1000 1500 2000 2500 Temperature (K)

FIG. 25: Constant pressure heat capacity for graphite (solid line). Experimental results are from Ref. 74 (squares), as reported by Ref. 75.

2500

2000 ) -1 1500 500 .kg ) -1 400 -1

.kg 300 1000 -1 Acknowledgments 200 Cv (J.K

Cv (J.K 100

500 0 0 50 100 150 200 T (K)

0 0 500 1000 1500 2000 2500 3000 The authors gratefully acknowledge support from Temperature (K) NSF-NIRT DMR-0304019 and the Interconnect Focus Center MARCO-DARPA 2003-IT-674. Nicolas Mounet FIG. 26: Constant volume heat capacity for graphite (solid personally thanks the Ecole Polytechnique of Palaiseau line), graphene (dashed line) and diamond (dotted line). The (France) and the Fondation de l’Ecole Polytechnique for inset shows an enlargement of the low temperature region. their help and support. 16

∗ Electronic address: [email protected] P. Hyldgaard, S. I. Simak, D. C. Langreth, and B. I. † URL: http://nnn.mit.edu/ Lundqvist, Phys. Rev. Lett. 91, 126402 (2003). 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, 32 W. Kohn, Y. Meir, and D. E. Makarov, Phys. Rev. Lett. Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. 80, 4153 (1998). Firsov, Science 306, 666 (2004). 33 W. Kohn and L. J. Sham, Phys. Rev. A 140, A1133 (1965). 2 R. J. Chen, S. Bangsaruntip, K. A. Drouvalakis, N. W. S. 34 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). Kam, M. Shim, Y. Li, W. Kim, P. J. Utz, and H. Dai, 35 S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. Proc. Natl. Acad. Sci. U.S.A. 100, 4984 (2003). 58, 1861 (1987). 3 G. Galli, R. M. Martin, R. Car, and M. Parrinello, Phys. 36 S. de Gironcoli, Phys. Rev. B 51, R6773 (1995). Rev. Lett. 63, 988 (1989). 37 A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory 4 G. Galli, R. M. Martin, R. Car, and M. Parrinello, Phys. of lattice dynamics in the harmonic approximation (Aca- Rev. Lett. 62, 555 (1989). demic Press, New York, 1963), vol. 3 of Solid-state physics, 5 A. De Vita, G. Galli, A. Canning, and R. Car, Nature 379, pp. 45–46. 523 (1996). 38 A. A. Quong and A. Y. Liu, Phys. Rev. B 56, 7767 (1997). 6 P. Delhaes, ed., Graphite and Precursors (Gordon and 39 S. Narasimhan and S. de Gironcoli, Phys. Rev. B 65, Breach, Australia, 2001), chap. 6. 064302 (2002). 7 H. O. Pierson, Handbook of carbon, graphite, diamond, and 40 T. H. K. Barron, J. G. Collins, and G. K. White, Adv. fullerenes : properties, processing, and applications (Noyes Phys. 29, 609 (1980). Publications, Park Ridge, New Jersey, 1993), pp. 59–60. 41 P. K. Schelling and P. Keblinski, Phys. Rev. B 68, 035425 8 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical (2003). Properties of Carbon Nanotubes (Imperial College Press, 42 S. Baroni, A. Dal Corso, S. de Gironcoli, P. Gi- London, 1998). annozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, 9 J. L. Warren, J. L. Yarnell, G. Dolling, and R. A. Cowley, G. Chiarotti, P. Focher, A. Pasquarello, et al., URL Phys. Rev. 158, 805 (1967). http://www.pwscf.org. 10 G. A. Slack and S. F. Bartram, J. Appl. Phys. 46, 89 43 URL http://www.gnu.org/copyleft/gpl.html. (1975). 44 A. Dal Corso, C.pbe-rrkjus.upf, URL 11 P. Pavone, K. Karch, O. Sch¨utt, W. Windl, D. Strauch, http://www.pwscf.org/pseudo/1.3/UPF/C.pbe-rrkjus.UPF. P. Giannozzi, and S. Baroni, Phys. Rev. B 48, 3156 (1993). 45 A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. 12 J. Maultzsch, S. Reich, C. Thomsen, H. Requardt, and Joannopoulos, Phys. Rev. B 41, R1227 (1990). P. Ordej´on, Phys. Rev. Lett. 92, 075501 (2004). 46 J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 13 L. Wirtz and A. Rubio, Solid State Commun. 131, 141 47 N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, (2004). Phys. Rev. Lett. 82, 3296 (1999). 14 A. C. Bailey and B. Yates, J. Appl. Phys. 41, 5088 (1970). 48 E. Ziambaras and E. Schr¨oder, Phys. Rev. B 68, 064112 15 J. B. Nelson and D. P. Riley, Proc. Phys. Soc., London 57, (2003). 477 (1945). 49 O. Madelung, ed., Physics of Group IV and III-V com- 16 S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, pounds (Springer-Verlag, Berlin, 1982), vol. 17a of Landolt- Rev. Mod. Phys. 73, 515 (2001). B¨ornstein, New Series, Group III, p. 107. 17 P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, 50 M. H. Grimsditch and A. K. Ramdas, Phys. Rev. B 11, Phys. Rev. B 43, 7231 (1991). 3139 (1975). 18 D. Vanderbilt, Phys. Rev. B 41, R7892 (1990). 51 Y. X. Zhao and I. L. Spain, Phys. Rev. B 40, 993 (1989). 19 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. 52 M. Hanfland, H. Beister, and K. Syassen, Phys. Rev. B 39, Lett. 77, 3865 (1996). 12598 (1989). 20 J. Xie, S. P. Chen, J. S. Tse, S. de Gironcoli, and S. Baroni, 53 J. Donohue, The Structures of the Elements (Kreiger, Mal- Phys. Rev. B 60, 9444 (1999). abar, 1982), p. 256. 21 F. Favot and A. Dal Corso, Phys. Rev. B 60, 11427 (1999). 54 O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, 22 J. C. Boettger, Phys. Rev. B 55, 11202 (1997). and T. Weng, J. Appl. Phys. 41, 3373 (1970). 23 M. C. Schabel and J. L. Martins, Phys. Rev. B 46, 7185 55 C. Oshima, T. Aizawa, R. Souda, Y. Ishizawa, and (1992). Y. Sumiyoshi, Solid State Commun. 65, 1601 (1988). 24 O. Dubay and G. Kresse, Phys. Rev. B 67, 035401 (2003). 56 S. Siebentritt, R. Pues, K.-H. Rieder, and A. M. Shikin, 25 P. Pavone, R. Bauer, K. Karch, O. Schutt, S. Vent, Phys. Rev. B 55, 7927 (1997). W. Windl, D. Strauch, S. Baroni, and S. de Gironcoli, 57 H. Yanagisawa, T. Tanaka, Y. Ishida, M. Matsue, Physica B 219/220, 439 (1996). E. Rokuta, S. Otani, and C. Oshima (2004), submitted 26 L. H. Ye, B. G. Liu, D. S. Wang, and R. Han, Phys. Rev. to SIA. B 69, 235409 (2004). 58 R. Nicklow, N. Wakabayashi, and H. G. Smith, Phys. Rev. 27 I. H. Lee and R. M. Martin, Phys. Rev. B 56, 7197 (1997). B 5, 4951 (1972). 28 K. R. Kganyago and P. E. Ngoepe, Mol. Simul. 22, 39 59 F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126 (1999). (1970). 29 S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and 60 H. Zabel, J. Phys. Condens. Matter 13, 7679 (2001). J. Robertson, Phys. Rev. Lett. 93, 185503 (2004). 61 I. M. Lifshitz, Zh. Eksp. Teor. Fiz. 22, 475 (1952). 30 P. Pavone, Ph.D. thesis, SISSA, Trieste (Italy) (1991). 62 C. Kittel, Introduction to Solid State Physics (Wiley, New 31 H. Rydberg, M. Dion, N. Jacobson, E. Schroder, York, 1976), chap. 3, 5th ed. 17

63 T. Aizawa, R. Souda, S. Otani, Y. Ishizawa, and C. Os- L. N. Alieva, Phys. Solid State 44, 1859 (2002). hima, Phys. Rev. B 42, 11469 (1990). 71 K. Karch, T. Dietrich, W. Windl, P. Pavone, A. P. Mayer, 64 W. Kohn, Phys. Rev. Lett. 2, 393 (1959). and D. Strauch, Phys. Rev. B 53, 7259 (1996). 72 65 C. P. Herrero and R. Ram´irez, Phys. Rev. B 63, 024103 H. J. McSkimin and P. Andreatch, J. Appl. Phys. 43, 2944 (2000). (1972). 73 66 N. Mounet and N. Marzari, in preparation. A. C. Victor, J. Chem. Phys. 36, 1903 (1962). 74 67 Y. K. Kwon, S. Berber, and D. Tom´anek, Phys. Rev. Lett. R. R. Hultgren, Selected Values of the Thermodynamic 92, 015901 (2004). Properties of the Elements (American Society for Metals, 68 G. L. Belenkii, R. A. Suleimanov, N. A. Abdullaev, and Metals Park, Ohio, 1973). 75 V. Y. Stenshraiber, Sov. Phys. Solid State 26, 2142 (1984). L. E. Fried and W. M. Howard, Phys. Rev. B 61, 8734 69 N. A. Abdullaev, Phys. Solid State 43, 727 (2001). (2000). 70 N. A. Abdullaev, R. A. Suleimanov, M. A. Aldzhanov, and