Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite

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Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite

Michele Lazzeri,¹Claudio Attaccalite,²Ludger Wirtz,³ and Francesco Mauri¹

1IMPMC, Universités Paris 6 et 7, IPGP, CNRS, 140 rue de Lourmel, 75015 Paris, France

2Unidad de Fisica de Materiales, European Theoretical Spectroscopy Facility (ETSF), Universidad del Pais Vasco, 20018 San Sebastian, Spain

3Institute for Electronics, Microelectronics, and Nanotechnology, CNRS, 59652 Villeneuve d’Ascq, France 共Received 29 May 2008; published 26 August 2008兲

We compute the electron-phonon coupling共EPC兲of selected phonon modes in graphene and graphite using variousab initio methods. The inclusion of nonlocal exchange-correlation effects within theGW approach strongly renormalizes the square EPC of the A₁⬘ ^K mode by almost 80% with respect to density-functional theory in the LDA and GGA approximations. Within GW, the phonon slope of the A₁⬘^K mode is almost two times larger than in GGA and LDA, in agreement with phonon dispersions from inelastic x-ray scattering and Raman spectroscopy. The hybrid B3LYP functional overestimates the EPC at K by about 30%. Within the Hartree-Fock approximation, the graphene structure displays an instability under a distortion following the A₁⬘ phonon atK.

DOI:10.1103/PhysRevB.78.081406 PACS number共s兲: 78.30.Na, 71.15.Mb, 63.20.kd, 81.05.Uw

The electron-phonon coupling共EPC兲is one of the funda- mental quantities in condensed matter. It determines phonon dispersions and Kohn anomalies, phonon-mediated super- conductivity, electrical resistivity, Jahn-Teller distortions, etc.

Nowadays, density-functional theory共DFT兲within local and semilocal approximations is considered the “standard model”

to compute ab initio the electron-phonon interaction and phonon dispersions.¹Thus, a failure of DFT would have major consequences in a broad context. In GGA and LDA approximations,²the electron exchange-correlation energy is a local functional of the charge density, and the long-range character of the electron-electron interaction is neglected.

These effects are taken into account by Green’s-function ap- proaches based on the screened electron-electron interaction W such as the GW method.³ GW is considered the most preciseab initioapproach to determine electronic bands, but so far it has never been used to compute EPCs nor phonon dispersions. The semiempirical B3LYP functional² partially includes long-range Hartree-Fock 共HF兲 exchange. B3LYP has been used to compute phonon frequencies but, so far, not the EPC.

The EPC is a key quantity for graphene, graphite, and carbon nanotubes. It determines the Raman spectrum, which is the most common characterization technique for graphene and nanotubes^4,5 and the high-bias electron transport in nanotubes.⁶Graphene and graphite are quite unique systems in which the actual value of the EPC for some phonons can be obtained almost directly from measurements. In particular, the square of the EPC of the highest optical-phonon branch共HOB兲at the symmetryKpoint is proportional to the HOB slope nearK.⁷The HOBKslope can be measured by inelastic x-ray scattering共IXS兲^8,9or by the dispersion of the D and 2D lines as a function of the excitation energy in a Raman experiment.^5,10–13 A careful look at the most recent data suggests that the experimental phonon slopes共and thus the EPC兲 are underestimated by DFT.⁵ The ability of DFT 共LDA and GGA兲in describing the EPC of graphene was also questioned by a recent theoretical work.¹⁴

Here, we show that:共i兲theGWapproach, which provides the most accurateab initiotreatment of electron correlation, can be used to compute the electron-phonon interaction and the phonon dispersion; 共ii兲 in graphite and graphene, DFT 共LDA and GGA兲underestimates, by a factor of 2, the slope

ξ=0 0.25 0.5 0.75 1/0 0.25 0.5 q=2π/a(ξ,0,0) q=2π/a(1,ξ,0) 400

800 1200 1600

Phononfrequency(cm-1)

Γ K M Γ

1200 1400 1600

Phononfrequency(cm-1)

E_2g A₁^’

Γ K M M

FIG. 1. Upper panel: Phonon dispersion of graphite. Lines are DFT calculations, dots and triangles are IXS measurements from Refs. 8 and 9, respectively. Lower panel: Phonon dispersion of graphene from DFT calculations. Dashed lines are obtained by subtracting, from the dynamical matrix, the phonon self-energy between the␲bands共␻qin the text兲.

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of the phonon dispersion of the highest optical branch at the zone boundary and the square of its EPC by almost 80%;共iii兲 GWreproduces the experimental phonon dispersion nearK, the value of the EPC, and the electronic band dispersion;共iv兲 the B3LYP hybrid functional² gives phonons close to GW but overestimates the EPC atKby about 30%; and共v兲within HF the graphite structure is unstable.

In Fig. 1, we show the phonon dispersion of graphite computed with DFT_GGA.¹⁵In spite of the general good agreement with IXS data, the situation is not clear for the HOB near K. In fact, despite the scattering among experimental data, the theoretical HOB is always higher in energy with respect to measurements, and the theoretical phonon slope 共for the HOB near K兲is underestimating the measured one.

It is also remarkable that while the DFT K frequency is

⬃1300 cm⁻¹, the highest measured is much lower at

⬃1200 cm⁻¹.

The dispersion of the HOB near K can also be obtained by Raman measurements of the graphene and graphite D line 共⬃1350 cm⁻¹兲.¹² The D-line frequency ␻Ddepends on the energy of the exciting laser ⑀L. According to the double- resonance model,^12,13⑀Lactivates a phonon of the HOB with momentum q=K+ 2⌬q along the K-M line⁵ and energy ប␻D. ⌬q is determined by ⑀K−⌬q,␲^ⴱ−⑀K−⌬q,␲=⑀L−ប␻D/2 where⑀_k,␲/␲ⴱ is the energy of the␲/␲^ⴱelectronic state with momentumk. Thus, by measuring␻Dvs⑀Land considering the electronic ␲bands dispersion from DFT, one can obtain the phonon dispersion␻Dvsq.¹²The phonon dispersion thus obtained is very similar to the one from IXS data and its slope is clearly underestimated by DFT共Fig.2, upper panel兲.

The same conclusion is reached by comparing the D-line dispersion ␻Dvs ⑀L共directly obtained from measurements兲 with calculations共Fig.2, lower panel兲. Note that the dispersions of the Raman 2D line⁵is consistent with that of the D line and thus in disagreement with DFT共LDA and GGA兲as well.

The steep slope of the HOB nearKis due to the presence of a Kohn anomaly for this phonon.⁷In particular, in Ref.7, it was shown that the HOB slope is entirely determined by the contribution of the phonon self-energy between␲ ^bands 共Pq兲to the dynamical matrixDq.␻q=

冑

Dq/mis the phonon pulsation, where m is the mass. For a given phonon with momentum q,

Dq=Bq+Pq, Pq= 4

Nk

兺

_k ^兩D_⑀_k,␲^共^k+q₋^兲␲_⑀_k+q,␲^ⴱ^,k^␲^兩²_ⴱ^, ^共1兲

where the sum is performed onNkwave vectors all over the Brillouin zone;D_共_k+q_兲_i,kj=具k+q,i兩⌬Vq兩k,j典is the EPC;⌬Vq

is the derivative of the Kohn-Sham potential with respect to the phonon mode; and 兩k,i典 is the Bloch eigenstate with momentumk, band indexi, and energy⑀k,i.␲共␲^ⴱ兲identifies the occupied共empty兲␲band. In Fig.1 we show a fictitious phonon dispersion ␻q obtained by subtracting Pq from the dynamical matrix 共␻q=

冑

Bq/m兲 for each phonon. The HOB is the branch which is mostly affected and, for the HOB,␻q

becomes almost flat nearK. Thus, DFT共LDA or GGA兲fails in describing the HOB slope nearK, the slope which is determined byPq.Pqis given by the square of EPC divided by

␲-band energies. Thus, the DFT failure can be attributed to a poor description of the EPC or of the ␲-band dispersion.

In graphene and graphite, it is known that standard DFT provides an underestimation of the␲^{- and}␲^ⴱ-band slopes of

⬃10– 20%.^16,17 A very precise description of the bands, in better agreement with measurements, is obtained using GW.^16,17 We thus computed the ␲ bands with DFT 共both LDA and GGA兲¹⁸andGW共Ref.19兲and compared with HF 共Ref. 20兲and B3LYP.²⁰ Details are in Ref.21. The different methods provide band dispersions whose overall behavior can be described by a scaling of the ␲ ^energies.¹⁶ ^{The dif-} ferent scaling factors can be obtained by comparing⌬⑀g, the energy difference between the ␲^ⴱ ^and␲ bands at the symmetry point M共L兲 for graphene共graphite兲.⌬⑀g is larger in

K

ξ= 0.4 0.5 0.6 0.7 0.8 q=2π/a(ξ,0,0) 1200

1300 1400

Phonon frequency (cm

-1

)

Measurements:

IXSRaman

Calculations:

DFTGW

1.0 2.0 3.0

Excitation energy ε

_L

(eV)

1300 1350 1400

D-line frequency ω

D

(c m

-1

)

Calculations:

DFTGW

FIG. 2. 共Color online兲. Upper panel: Dispersion of the highest optical phonon in graphite near K. Calculations are from DFT or corrected to includeGWrenormalization of the EPC. Here, the DFT dispersion is vertically shifted by −40 cm⁻¹to fit measurements.

Dots and triangles are IXS data from Refs. 8and 9, respectively.

Squares, plus, and diamonds are obtained from Raman data of Refs.

10–12, respectively, using the double-resonance model 共Refs. 12 and13兲. Lower panel: Dispersion of the Raman D line.

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GW than in DFT共Table I兲. Thus, inclusion of theGW cor- rection to the electronic bands alone results in a larger de- nominator in Eq.共1兲, providing a smaller phonon slope and a worse agreement with experiments. The underestimation of the Kphonon slope in DFT is thus due to the EPC.

The EPC can be computed with linear response as, e.g., in Ref.7but, at present, the use of this technique withinGWis not feasible. Alternatively, the EPC associated with a phonon mode can be determined by the variation of the electronic band energies by displacing the atoms according to the considered mode. In graphene, atK, there are doubly degenerate

␲electronic states at the Fermi level. The HOB corresponds to the E_2g phonon at⌫ and to the A₁⬘ ^atK. As an example, we consider the EPC associated with the⌫-E_2g phonon and we displace the atoms according to its phonon pattern 共see Fig.3兲. Following symmetry arguments,²²one can show that, in an arbitrary base of the two-dimensional space of the ␲ bands atK, the Hamiltonian is the 2⫻2 matrix

H= 2

冑

具D_⌫²典F

冉

^b^a^ⴱ ⁻^b^a

冊

^d⁺^O共d²^兲, ^共2兲

whereeach atom is displaced by d,兩a兩²+兩b兩²= 1, and具D_⌫²典F

=兺_i,j^␲,␲^ⴱ兩D_Ki,Kj兩²/4, where the sum is performed on the two degenerate ␲ bands. Diagonalizing Eq. 共2兲, we see that an atomic displacement following the⌫-E2gphonon induces the splitting⌬E_⌫=⑀K,␲^ⴱ−⑀K,␲and

具D_⌫²典F= lim

d→0

1

16

冉

^⌬E^d^⌫

冊

²^. ^共3兲

In analogous way, we define 具DK

2典F=兺_i,j^␲,␲^ⴱ兩D_共2K_兲i,Kj兩²/4 for the A₁⬘ ^{phonon at} K. Let us consider a

冑

3⫻

冑

3 graphene supercell. Such a cell can be used to displace the atoms following the K-A₁⬘ ^phonon 共Fig. 3兲 since the K point is refolded in ⌫. Let us call ⌬EK the splitting of the⑀K,␲ bands induced by this displacement共sinceKis refolded in⌫, here

⑀_K,␲ denotes the energies of the ⌫ band of the supercell corresponding to the ␲ ^{band at} ^Kin the unit cell兲. Consid- ering the atomic distortion of Fig. 3 and displacing each atom byd, one can show that

具DK 2典F= lim

d→0

1

8

冉

^⌬^E^d^K

冊

²^. ^共⁴^兲

In practice, by calculating band energies in the distorted structures of Fig. 3 and using Eqs. 共3兲and共4兲, one obtains the EPCs of the⌫-E2gandK-A₁⬘phonons between␲ ^states.

Similar equations can be used for graphite.²³ Results are in TableItogether with the computed phonon frequencies. The EPCs from DFT_GGAare in agreement with those from linear response.⁷ We also remark that, within the present “frozen- phonon” approach, the Coulomb vertex corrections are im- plicitly included within GW.

To study the effect of the different computational methods on the phonon slope共which is determined by Pq兲 we recall thatPqis the ratio of the square EPC and band energies关Eq.

共1兲兴. Thus, we have to compare ␣q=具Dq

2典F/⌬⑀g. As an example, assuming that the change in Pqfrom DFT toGW is constant for qnearK,

P_q^GW P_q^DFT⯝ ␣K

GW

␣K

DFT=r^GW 共5兲

and r^GW provides the change in theK phonon slope going from DFT toGW. To understand the results, we recall that in standard DFT the exchange-correlation depends only on the local electron density. In contrast, the exchange interaction in HF and GW is nonlocal. Furthermore, in GW, correlation effects are nonlocal since they are described through a dy- namically screened Coulomb interaction. The hybrid functional B3LYP gives results intermediate between DFT and HF.

Both ␣_⌫ ^and ␣K are heavily overestimated by HF, the K-EPC being so huge that graphene is no more stable 共the KA₁⬘ phonon frequency is not real兲. Indeed, the HF equilibrium geometry is a

冑

3⫻

冑

3 reconstruction with alternating double and single bonds of 1.40 and 1.43 Å lengths as in Fig.3共with a gain of 0.9 meV/atom兲. These results not only demonstrate the major effect of the long-range character of the exchange for the K-EPC 共Ref. 14兲 but also the impor- tance of the proper inclusion of the screening 共included in GWbut neglected in HF兲. Notice also that␣K

GWof graphite is smaller with respect to graphene by⬃10%. This is explained by the larger screening of the exchange in graphite 共due to the presence of adjacent layers兲 than in graphene. On the contrary, within GGA and LDA, the graphite phonon frequencies and EPCs are very similar to those of graphene TABLE I. EPC of the⌫-E_2gandK-A₁⬘phonons computed with

various approximations. ⌬⑀g 共eV兲, 具D_q²典F 共eV²/Å²兲, and ␣q

共eV/Å²兲are defined in the text.␻_⌫共␻K兲is the phonon frequency of the E_2gA₁⬘^mode共cm⁻¹兲. TheGW␻Kfor graphite共in parenthesis兲 is not computed directly共see the text兲.i=

冑

−1 is the imaginary unit.

Graphene

⌬⑀g 具D_⌫²典F ␣_⌫ ␻_⌫ 具D_K²典F ␣K ␻K

DFT_LDA 4.03 44.4 11.0 1568 89.9 22.3 1275 DFT_GGA 4.08 45.4 11.1 1583 92.0 22.5 1303

GW 4.89 62.8 12.8 – 193 39.5 –

B3LYP 6.14 82.3 13.4 1588 256 41.7 1172

HF 12.1 321 26.6 1705 6020 498 960⫻i

Graphite

⌬⑀g 具D_⌫²典F ␣_⌫ ␻_⌫ 具D_K²典F ␣K ␻K

DFT_LDA 4.06 43.6 10.7 1568 88.9 21.8 1299 DFT_GGA 4.07 44.9 11.0 1581 91.5 22.5 1319

GW 4.57 58.6 12.8 – 164.2 35.9 共1192兲

c) b) K-A’₁

a) Γ-E_2g

FIG. 3. 共a兲and共b兲Patterns of the⌫-E_2gandK-A₁⬘^{phonons of} graphene. Dotted and dashed lines are the Wigner-Seitz cells of the unit cell and of the

冑

3⫻

冑

3 supercell.共c兲HF equilibrium structure.

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since these functionals do not take into account the electron- electron interaction screening.

Concerning the phonon slope, ␣_⌫^GW is only 15% larger than ␣_⌫^DFT. Indeed, DFT reproduces with this precision the phonon frequency and dispersion of the HOB at ⌫. On the contrary, ␣K

GW is 60% larger than ␣K

DFT for graphite. This large increase with respect to DFT could explain the disagreement between DFT and the measured A₁⬘ phonon dispersion near K. To test this, we need to determine the GW phonon dispersion that, using Eq. 共5兲, becomes ␻q

GW

⯝

冑

共Bq

GW+r^GWP_q^DFT兲/m, where r^GW= 1.6. Moreover, we can assumeB_q^GW⯝B_K^GWsince theBqcomponent of the dynamical matrix关Eq.共1兲兴is not expected to have an important depen- dence onq共Fig.1兲. The value ofB_K^GWis obtained as a fit to the measurements of Fig. 2.²⁴ The resulting K A₁⬘ ^phonon frequency is 1192 cm⁻¹, which is our best estimation and is almost 100 cm⁻¹ smaller than in DFT. The phonon dispersion thus obtained and the corresponding D-line dispersion are both in better agreement with measurements共Fig.2兲.

The partial inclusion of long-range exchange within the semiempirical B3LYP functional leads to a strong increase in the EPC atKas compared to the LDA and GGA functionals.

However, comparing to the GWvalue, the EPC is overesti-

mated by 30% and the corresponding frequency for theK-A₁⬘ mode at 1172 cm⁻¹falls well below the degenerateKmode, which is around 1200 cm⁻¹in the experiment^8,9共Fig.1兲and at 1228 cm⁻¹ in our phonon calculation with B3LYP. We have checked that tuning the percentage of HF exchange in the hybrid functional allows to match the EPC value of the GW approach 共in which case, the K-A₁⬘ mode remains the highest mode兲. This may be a good way to calculate the full phonon dispersion of graphite/graphene within DFT, yet with an accuracy close to the one of the GWapproach.

Concluding,GWis a general approach to compute accu- rate EPC where DFT functionals fail. Such a failure in graphite/graphene is due to the interplay between the two- dimensional Dirac-type band structure and the long-range character of the Coulomb interaction.¹⁴However,GWcan be also used in cases 共in which the EPC is badly described by DFT兲where the electron exchange and correlation are more short ranged.²⁵

Calculations were done at IDRIS 共Projects No. 081202 and No. 081827兲. C.A. and L.W. acknowledge French ANR Project No. PJC05_6741. We thank D. M. Basko, A. Rubio, J. Schamps, and C. Brouder for the discussions and A.

Marini for the codeYAMBO.

1S. Baroniet al., Rev. Mod. Phys. 73, 515共2001兲.

2LDA, GGA and B3LYP refer, respectively, to D. M. Ceperleyet al., Phys. Rev. Lett. 45, 566共1980兲; J. P. Perdewet al.,ibid. 77, 3865共1996兲; A. D. Becke, J. Chem. Phys. 98, 5648共1993兲.

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Aulburet al., Solid State Phys. 54, 1共2000兲; G. Onidaet al., Rev. Mod. Phys. 74, 601共2002兲.

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5A. C. Ferrariet al., Phys. Rev. Lett. 97, 187401共2006兲; D. Graf et al., Nano Lett. 7, 238共2007兲.

6Z. Yaoet al., Phys. Rev. Lett. 84, 2941共2000兲.

7S. Piscanecet al., Phys. Rev. Lett. 93, 185503共2004兲.

8J. Maultzschet al., Phys. Rev. Lett. 92, 075501共2004兲.

9M. Mohret al., Phys. Rev. B 76, 035439共2007兲.

10I. Pócsiket al., J. Non-Cryst. Solids 227, 1083共1998兲.

11P. H. Tanet al., Phys. Rev. B 66, 245410共2002兲.

12J. Maultzschet al., Phys. Rev. B 70, 155403共2004兲.

13C. Thomsenet al., Phys. Rev. Lett. 85, 5214共2000兲.

14D. M. Baskoet al., Phys. Rev. B 77, 041409共R兲共2008兲.

15Calculations were done as in Ref.1. Technical details and thus phonon dispersions are the same as in Ref.7.

16S. Y. Zhou et al., Phys. Rev. B 71, 161403共R兲共2005兲; S. G.

Louie, inTopics in Computational Materials Science, editor by C. Y. Fong共World Scientific, Singapore, 1997兲, p. 96.

17A. Grüneiset al., Phys. Rev. Lett. 100, 037601共2008兲.

18LDA and GGA calculations were done with the code PWSCF

共http://www.quantum-espresso.org兲, with pseudopotentials of the type; N. Troullieret al., Phys. Rev. B 43, 1993共1991兲.

19GW calculations were done with the codeYAMBO 共the YAMBO

project, http://www.yambo-code.org/兲, within the nonself- consistent G₀W₀ approximation starting from DFT-LDA wave functions and using a plasmon-pole model for the screening,

following; M. S. Hybertsenet al., Phys. Rev. B 34, 5390共1986兲.

20HF and B3LYP calculations were done with the codeCRYSTAL

共V. R. Saunderset al.,CRYSTAL03 Users Manual共University of Torino, Torino, 2003兲, using the TZ basis by Dunning共without the diffuse P function兲.

21For graphite we used the experimental lattice parameters 共a

= 2.46 Å , c= 6.708 Å兲. For graphene we useda= 2.46 Å and a vacuum layer of 20 a.u. EPCs were calculated on a structure distorted byd= 0.01 a . u. For graphene, the electronic integra- tion on the 1⫻1 cell was done with a 18⫻18⫻1 grid for LDA/

GGA, 36⫻36⫻1 for GW, and 66⫻66⫻1 for B3LYP/HF. For graphite it was 18⫻18⫻6. For the

冑

3⫻

冑

3 cell we used the nearest equivalentkgrid. Plane waves are expanded up to 60-Ry cutoff. We used a Fermi-Dirac smearing with 0.002-Ry width for B3LYP/HF/GW and a Gaussian smearing with 0.02-Ry width for LDA/GGA.

22J. C. Slonczewski et al., Phys. Rev. 109, 272共1958兲; See also Suppl. information to S. Pisanaet al., Nat. Mater. 6, 198共2007兲.

23In graphite, at the high-symmetryHpoint the four␲bands are degenerate two by two, ⌬⑀0 being the energy difference. By displacing the atoms according to the⌫E_2gphonon, these bands remain degenerate and the energy difference is⌬⑀. In analogy to Eqs.共3兲 and 共4兲 we define 具D_⌫²典F=共⌬⑀²−⌬⑀02兲/共16d²兲. By displacing the atoms according to theKA₁⬘phonon, the four bands are no longer degenerate, being␲^ⴱ共␲兲the two bands which are up 共down兲 shifted and ⌬⑀=⑀␲^ⴱ−⑀␲. We define 具D_K²典F=共⌬⑀²

−⌬⑀0

2兲/共8d²兲, where⌬⑀²indicates the average between the four possible␲^ⴱ−␲couples.

24In principle, a direct calculation of␻K

GW共and thus ofB_K^GW兲could be obtained, e.g., by finite differences from a prohibitively ex- pensiveGWtotal energy calculation.

25P. Zhanget al., Phys. Rev. Lett. 98, 067005共2007兲.

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