Phonons and related crystal properties from density-functionalperturbation theory

Stefano Baroni, Stefano de Gironcoli, and Andrea Dal CorsoSISSA–Scuola Internazionale Superiore di Studi Avanzati and INFM–Istituto Nazionale diFisica della Materia, I-34014 Trieste, Italy

Paolo Giannozzi*Chemistry Department and Princeton Materials Institute, Princeton University, Princeton,New Jersey 08544

(Published 6 July 2001)

This article reviews the current status of lattice-dynamical calculations in crystals, usingdensity-functional perturbation theory, with emphasis on the plane-wave pseudopotential method.Several specialized topics are treated, including the implementation for metals, the calculation of theresponse to macroscopic electric fields and their relevance to long-wavelength vibrations in polarmaterials, the response to strain deformations, and higher-order responses. The success of thismethodology is demonstrated with a number of applications existing in the literature.

CONTENTS

I. Introduction 516II. Density-Functional Perturbation Theory 516

A. Lattice dynamics from electronic-structuretheory 516

B. Density-functional theory 5171. The Kohn-Sham equations 5172. Local-density approximation and beyond 518

C. Linear response 5191. Monochromatic perturbations 5202. Homogeneous electric fields 5213. Relation to the variational principle 5234. Metals 524

D. Phonons 5261. Vibrational states in crystalline solids 5262. Long-wavelength vibrations in polar

materials 5273. Interatomic force constants 528

E. Homogeneous deformations 5281. Elastic properties 5282. Piezoelectric properties 529

F. Higher-order responses 5301. The 2n!1 theorem 5302. Nonlinear susceptibilities 531

III. Implementations 532A. Plane waves and pseudopotentials 532B. Ultrasoft pseudopotentials 533C. Localized basis sets, all-electron implementations 535

IV. Other Approaches 536A. The dielectric approach 536B. Frozen phonons 537C. Vibrational properties from molecular dynamics 538

V. Applications 539A. Phonons in bulk crystals 539

1. Simple semiconductors 539a. Elemental and III-V semiconductors 539

b. II-VI semiconductors 540c. Diamond and graphite 541d. Silicon carbide 541e. Nitrides 541f. Other semiconductors 541g. Second-order Raman spectra in simple

semiconductors 542h. Piezoelectricity in binary semiconductors 542

2. Simple metals and superconductors 5423. Oxides 543

a. Insulators 544b. Ferroelectrics 545c. High-Tc superconductors 545

4. Other materials 546B. Phonons in semiconductor alloys and

superlattices 5471. GaAs/AlAs superlattices 5472. GaAs/AlAs alloys 5473. Si/Ge superlattices and alloys 5484. AlGaN alloys 5485. GaP/InP alloys 5486. Localized vibrations at defects 548

C. Lattice vibrations at surfaces 549D. Soft phonons and pressure-induced lattice

transformations 5491. Ferroelectrics 5492. Pressure-induced phase transitions 550

a. Cesium halides 550b. Cesium hydride 551c. Silicon dioxide 551d. Semiconductors 552e. Solid hydrogen 552f. Metals 553

3. Other phase transitions 553E. Thermal properties of crystals and surfaces 553

1. Metals 5532. Semiconductors and insulators 5543. Surfaces 5544. Alloys 554

F. Anharmonic effects 555G. Isotopic broadening of Raman lines 555H. Vibrational broadening of electronic core levels 556

VI. Conclusions and Perspectives 556Acknowledgments 557

*On leave of absence from Scuola Normale Superiore, Pisa,Italy.

REVIEWS OF MODERN PHYSICS, VOLUME 73, APRIL 2001

0034-6861/2001/73(2)/515(48)/$29.60 ©2001 The American Physical Society515

Phonons and related crystal properties from density-functionalperturbation theory

Stefano Baroni, Stefano de Gironcoli, and Andrea Dal CorsoSISSA–Scuola Internazionale Superiore di Studi Avanzati and INFM–Istituto Nazionale diFisica della Materia, I-34014 Trieste, Italy

Paolo Giannozzi*Chemistry Department and Princeton Materials Institute, Princeton University, Princeton,New Jersey 08544

(Published 6 July 2001)

This article reviews the current status of lattice-dynamical calculations in crystals, usingdensity-functional perturbation theory, with emphasis on the plane-wave pseudopotential method.Several specialized topics are treated, including the implementation for metals, the calculation of theresponse to macroscopic electric fields and their relevance to long-wavelength vibrations in polarmaterials, the response to strain deformations, and higher-order responses. The success of thismethodology is demonstrated with a number of applications existing in the literature.

CONTENTS

I. Introduction 516II. Density-Functional Perturbation Theory 516

A. Lattice dynamics from electronic-structuretheory 516

B. Density-functional theory 5171. The Kohn-Sham equations 5172. Local-density approximation and beyond 518

C. Linear response 5191. Monochromatic perturbations 5202. Homogeneous electric fields 5213. Relation to the variational principle 5234. Metals 524

D. Phonons 5261. Vibrational states in crystalline solids 5262. Long-wavelength vibrations in polar

materials 5273. Interatomic force constants 528

E. Homogeneous deformations 5281. Elastic properties 5282. Piezoelectric properties 529

F. Higher-order responses 5301. The 2n!1 theorem 5302. Nonlinear susceptibilities 531

III. Implementations 532A. Plane waves and pseudopotentials 532B. Ultrasoft pseudopotentials 533C. Localized basis sets, all-electron implementations 535

IV. Other Approaches 536A. The dielectric approach 536B. Frozen phonons 537C. Vibrational properties from molecular dynamics 538

V. Applications 539A. Phonons in bulk crystals 539

1. Simple semiconductors 539a. Elemental and III-V semiconductors 539

b. II-VI semiconductors 540c. Diamond and graphite 541d. Silicon carbide 541e. Nitrides 541f. Other semiconductors 541g. Second-order Raman spectra in simple

semiconductors 542h. Piezoelectricity in binary semiconductors 542

2. Simple metals and superconductors 5423. Oxides 543

a. Insulators 544b. Ferroelectrics 545c. High-Tc superconductors 545

4. Other materials 546B. Phonons in semiconductor alloys and

superlattices 5471. GaAs/AlAs superlattices 5472. GaAs/AlAs alloys 5473. Si/Ge superlattices and alloys 5484. AlGaN alloys 5485. GaP/InP alloys 5486. Localized vibrations at defects 548

C. Lattice vibrations at surfaces 549D. Soft phonons and pressure-induced lattice

transformations 5491. Ferroelectrics 5492. Pressure-induced phase transitions 550

a. Cesium halides 550b. Cesium hydride 551c. Silicon dioxide 551d. Semiconductors 552e. Solid hydrogen 552f. Metals 553

3. Other phase transitions 553E. Thermal properties of crystals and surfaces 553

1. Metals 5532. Semiconductors and insulators 5543. Surfaces 5544. Alloys 554

F. Anharmonic effects 555G. Isotopic broadening of Raman lines 555H. Vibrational broadening of electronic core levels 556

VI. Conclusions and Perspectives 556Acknowledgments 557

*On leave of absence from Scuola Normale Superiore, Pisa,Italy.

REVIEWS OF MODERN PHYSICS, VOLUME 73, APRIL 2001

0034-6861/2001/73(2)/515(48)/$29.60 ©2001 The American Physical Society515

dispersions of Si were calculated as well by Savrasov(1996) as a test of the LMTO implementation of DFPT.Dispersions for InP appear in a paper devoted to the(110) surface phonons of InP (Fritsch, Pavone, andSchroder, 1995); dispersions for both GaP and InP werepublished in a study of phonons in GaInP2 alloys (Ozo-lins and Zunger, 1998). For all these materials, phononspectra and effective charges are in very good agree-ment with experiments, where available. For AlAs—forwhich experimental data are very scarce—these calcula-tions provide the only reliable prediction of the entirephonon dispersion curve. For Si, the calculated phonondisplacement patterns compare favorably to those ex-tracted from inelastic neutron-scattering experiments(Kulda et al., 1994).

In all these materials the interatomic force constantsturn out to be quite long ranged along the (110) direc-tion. This feature had already been observed in earlycalculations (Herman, 1959; Kane, 1985; Fleszar andResta, 1986) and is related to the peculiar topology ofdiamond and zinc-blende lattices, with bond chainspropagating along the (110) directions.

The force constants of GaAs and of AlAs are espe-cially interesting in view of their use in complex GaAlAssystems such as superlattices, disordered superlattices,and alloys. While the phonon dispersions in GaAs areexperimentally well characterized, bulk samples of AlAsof good quality are not available, and little experimentalinformation on its vibrational modes has been collected.For several years it has been assumed that the forceconstants of GaAs and those of AlAs are very similarand that one can obtain the dynamical properties ofAlAs using the force constants of GaAs and the massesof AlAs (the mass approximation; Meskini and Kunc,1978). The DFPT calculations provided convincing evi-dence that the mass approximation holds to a very goodextent between GaAs and AlAs (Giannozzi et al., 1991).This transferability of force constants makes it possibleto calculate rather easily and accurately the vibrationalspectra of complex GaAlAs systems (Baroni, Giannozzi,and Molinari, 1990; Molinari et al., 1992; Baroni, de Gi-roncoli, and Giannozzi, 1990; Rossi et al., 1993). Some-what surprisingly, the mass approximation does notseem to be valid when the interatomic force constantsfor a well-known and widely used model, the bond-charge model (BCM), are employed. A six-parameterbond-charge model for GaAs that gives dispersionscomparing favorably with experiments and ab initio cal-culations, yields, when used in the mass approximation,AlAs dispersions quite different from first-principles re-sults. This clearly shows that information on the vibra-tional frequencies alone is not sufficient to fully deter-mine the force constants, even when complete phonondispersions are experimentally available. In order to ob-tain more realistic dispersions for AlAs in the mass ap-proximation, one has to fit the bond-charge model forGaAs to both frequencies and at least a few selectedeigenvectors (Colombo and Giannozzi, 1995).

b. II-VI semiconductorsThe II-VI zinc-blende semiconductors ZnSe, ZnTe,

CdSe, and CdTe present some additional difficulties in aplane-wave-pseudopotential (PP-PW) framework withrespect to their III-V or group-IV counterparts. The cat-ion d states are close in energy to the s valence states sothat the d electrons should be included among the va-lence electrons. Phonon calculations performed severalyears ago, when the inclusion of localized d states in thepseudopotential was difficult, showed that the effects ofcation d electrons could also be accounted for by includ-ing the d states in the core and by using the nonlinearcore-correction approximation. The results showed anaccuracy comparable to that previously achieved for el-emental and III-V semiconductors (Dal Corso, Baroniet al., 1993). Similar calculations have been more re-cently performed for hexagonal (wurtzite structure) CdS(Debernardi et al., 1997; Zhang et al., 1996) and CdSe(Widulle, Kramp et al., 1999) and compared with the re-sults of inelastic neutron scattering experiments.

FIG. 1. Calculated phonon dispersions and densities of statesfor binary semiconductors GaAs, AlAs, GaSb, and AlSb: !,experimental data. From Giannozzi et al., 1991.

540 Baroni et al.: Phonons and related crystal properties from DFPT

Rev. Mod. Phys., Vol. 73, No. 2, April 2001

sions over the entire Brillouin zone allows the calcula-tion of the electron-phonon (Eliashberg) spectralfunction !2F(") and of the mass enhancement param-eter # that enters the MacMillan equation for the tran-sition temperature Tc to superconductivity. Other im-portant quantities that can be calculated are thetransport spectral function ! tr

2 F(") and the # tr coeffi-cients, which determine the electrical and thermal resis-tivity in the normal state. In simple metals, calculationsof # and !2F(") have been performed for Al, Pb, andLi (Liu and Quong, 1996); for Al, Cu, Mo, Nb, Pb, Pd,Ta, and V (Savrasov and Savrasov, 1996); and for Al,Au, Na, and Nb (Bauer et al., 1998). Transport spectralfunctions and coefficients (Savrasov and Savrasov, 1996;Bauer et al., 1998) and phonon linewidths due toelectron-phonon coupling (Bauer et al., 1998) have alsobeen calculated. Figure 5 shows the results of Savrasovand Savrasov, 1996 for !2F(").

The calculation of electron-phonon coefficients founda remarkable application, beyond simple metals, in thestudy of the behavior of molecular solids S, Se, and Teunder pressure. With increasing pressure, these trans-form first to a base-centered orthorhombic supercon-ducting structure, followed by a rhombohedral $-Pophase, and finally for Se and Te by a bcc phase.

At the phase transition between the $-Po and the bccphase, a jump is observed in Tc in Te. The origin of this

jump was clarified (Mauri et al., 1996) through the studyof phonon dispersions and of the electron-phonon inter-actions. The phonon contribution to the free energy wasshown to be responsible for the difference in the struc-tural transition pressure observed in low- and room-temperature experiments.

In S, the $-Po phase is predicted to be followed by asimple cubic phase that is stable over a wide range ofpressures (280 to 540 GPa), in contrast to what is ob-served in Se and Te. The calculated phonon spectrumand electron-phonon coupling strength (Rudin and Liu,1999) for the lower-pressure $-Po phase is consistentwith the measured superconducting transition tempera-ture of 17 K at 160 GPa. The transition temperature iscalculated to drop below 10 K upon transformation tothe predicted simple cubic phase.

3. Oxides

Oxides present a special interest and a special chal-lenge for anyone interested in phonon physics. On theone hand, many very interesting materials, such as fer-roelectrics and high-Tc superconductors, are oxides. Onthe other hand, good-quality calculations on oxides areusually nontrivial, both for technical and for more fun-damental reasons. In a straightforward PW-PP frame-work, the hard pseudopotential of oxygen makes calcu-lations expensive: the use of ultrasoft pseudopotentialsis generally advantageous. The LDA is known to be in-sufficiently accurate in many cases (and sometimes the

FIG. 3. Calculated phonon dispersions for fcc simple metal Aland Pb and for the bcc transition metal Nb: solid lines, 0.3 eVsmearing width; dashed line 0.7 eV, smearing width; !, experi-mental data. From de Gironcoli, 1995.

FIG. 4. Calculated phonon dispersions in magnetic transitionmetals. Upper panel, bcc Fe. Solid lines, calculated GGA pho-non dispersions; ", inelastic neutron scattering data; dottedlines, dispersions calculated within local spin- density approxi-mation (LSDA). Lower panel, Ni. Solid lines, calculated GGAphonon dispersions; ", inelastic neutron scattering data; dot-ted lines, calculated LSDA dispersions. From Dal Corso andde Gironcoli, 2000.

543Baroni et al.: Phonons and related crystal properties from DFPT

Rev. Mod. Phys., Vol. 73, No. 2, April 2001

3

00.0

0.5

1.0

1.5

2.0

2.5

3D

2D

1D

1 2 3 4

F(q

) /

F(0

)

q

FIG. 1: (Color online) Momentum dependence of the d-dimensional static response functions, F (q) = ��(q,! = 0),corresponding to zero energy transfer, ! = 0. Here, momentaare expressed in rescaled units, i.e. in units of the Fermi mo-mentum, kF .

or

~22m

F (q) = 2

Z

kkF

ddk

(2⇡)d

⇣ 1

q

2 + 2k · q +1

q

2 � 2k · q

⌘.

(4.3)Using rescaled coordinates, i.e. by letting k ! k/kF , weobtain

F (q) =2kdF✏F

Z

k1

ddk

(2⇡)d2q2

q

4 � 4(k · q)2 . (4.4)

We will discuss now the three-dimensional realizations ofthe static response function corresponding to d = 1, 2and 3. We have:

• d = 3 case: We have

F

3

(q) =k

3

F

⇡

2

✏F

Z1

0

k

2 dk

Z1

�1

dµ

q

2 � 4k2µ2

=N

3

✏F

3

2q

Z1

0

k dk log���q + 2k

q � 2k

��� ,

which gives

F

3

(q) =N

3

✏F

3

4

h1 +

1� ( 12

q)2

q

log���1 + 1

2

q

1� 1

2

q

���i. (4.5)

• d = 2 case: We have

F

2

(q) =k

2

F

⇡

2

✏F

Z1

0

k dk

Z2⇡

0

d�

q

2 � 4k2 cos2 �. (4.6)

For q � 2, the above gives

F

2

(q > 2) =N

2

✏F

4

q

Z1

0

k dkpq

2 � 4k2

=N

2

✏F

⇣1�

p1� (2/q)2

⌘, (4.7)

whereas for q < 2, we obtain

F

2

(q < 2) =N

2

✏F

4

q

Z 12 q

0

k dkpq

2 � 4k2=

N

2

✏F, (4.8)

Hence, we obtain:

F

2

(q) =N

2

✏F

h1 � ⇥(q � 2)

p1� (2/q)2

i. (4.9)

• d = 1 case: We have

F

1

(q) =2kF⇡✏F

Z1

�1

dk

q

2 � 4k2, (4.10)

which gives

F

1

(q) =N

1

✏F

1

2qlog

���1 + 1

2

q

1� 1

2

q

��� . (4.11)

In Fig. 1, we depict the momentum dependence of theabove static response functions, F

1

(q), F2

(q), and F

3

(q).

V. LINDHARD FUNCTIONS

To calculate the Lindhard function, i.e.

�(q,!) =2

~

Z

kkF

ddk

(2⇡)dn

k

(1� n

k+q

) (5.1)

⇥⇣ 1

! � !

kq

+ i⌘� 1

! + !

kq

� i⌘

⌘,

we use the formal identity

1

! ± i⌘= P 1

!

⌥ i⇡�(!) . (5.2)

Therefore, we have

�(q,!) = Re�(q,!) + i Im�(q,!) , (5.3)

with

Re�(q,!) =2

~ PZ

ddk

(2⇡)dn

k

(1� n

k+q

)2!

kq

!

2 � !

2

kq

,

(5.4)and

Im�(q,!) = �2⇡

~

Zddk

(2⇡)dn

k

(1� n

k+q

) (5.5)

⇥h�(! � !

kq

) + �(! + !

kq

)i,

where we introduce the notation

~!kq

= ✏

k+q

� ✏

k

=~22m

⇣q

2 + 2k · q⌘. (5.6)

3

00.0

0.5

1.0

1.5

2.0

2.5

3D

2D

1D

1 2 3 4

F(q

) /

F(0

)

q

FIG. 1: (Color online) Momentum dependence of the d-dimensional static response functions, F (q) = ��(q,! = 0),corresponding to zero energy transfer, ! = 0. Here, momentaare expressed in rescaled units, i.e. in units of the Fermi mo-mentum, kF .

or

~22m

F (q) = 2

Z

kkF

ddk

(2⇡)d

⇣ 1

q

2 + 2k · q +1

q

2 � 2k · q

⌘.

(4.3)Using rescaled coordinates, i.e. by letting k ! k/kF , weobtain

F (q) =2kdF✏F

Z

k1

ddk

(2⇡)d2q2

q

4 � 4(k · q)2 . (4.4)

We will discuss now the three-dimensional realizations ofthe static response function corresponding to d = 1, 2and 3. We have:

• d = 3 case: We have

F

3

(q) =k

3

F

⇡

2

✏F

Z1

0

k

2 dk

Z1

�1

dµ

q

2 � 4k2µ2

=N

3

✏F

3

2q

Z1

0

k dk log���q + 2k

q � 2k

��� ,

which gives

F

3

(q) =N

3

✏F

3

4

h1 +

1� ( 12

q)2

q

log���1 + 1

2

q

1� 1

2

q

���i. (4.5)

• d = 2 case: We have

F

2

(q) =k

2

F

⇡

2

✏F

Z1

0

k dk

Z2⇡

0

d�

q

2 � 4k2 cos2 �. (4.6)

For q � 2, the above gives

F

2

(q > 2) =N

2

✏F

4

q

Z1

0

k dkpq

2 � 4k2

=N

2

✏F

⇣1�

p1� (2/q)2

⌘, (4.7)

whereas for q < 2, we obtain

F

2

(q < 2) =N

2

✏F

4

q

Z 12 q

0

k dkpq

2 � 4k2=

N

2

✏F, (4.8)

Hence, we obtain:

F

2

(q) =N

2

✏F

h1 � ⇥(q � 2)

p1� (2/q)2

i. (4.9)

• d = 1 case: We have

F

1

(q) =2kF⇡✏F

Z1

�1

dk

q

2 � 4k2, (4.10)

which gives

F

1

(q) =N

1

✏F

1

2qlog

���1 + 1

2

q

1� 1

2

q

��� . (4.11)

In Fig. 1, we depict the momentum dependence of theabove static response functions, F

1

(q), F2

(q), and F

3

(q).

V. LINDHARD FUNCTIONS

To calculate the Lindhard function, i.e.

�(q,!) =2

~

Z

kkF

ddk

(2⇡)dn

k

(1� n

k+q

) (5.1)

⇥⇣ 1

! � !

kq

+ i⌘� 1

! + !

kq

� i⌘

⌘,

we use the formal identity

1

! ± i⌘= P 1

!

⌥ i⇡�(!) . (5.2)

Therefore, we have

�(q,!) = Re�(q,!) + i Im�(q,!) , (5.3)

with

Re�(q,!) =2

~ PZ

ddk

(2⇡)dn

k

(1� n

k+q

)2!

kq

!

2 � !

2

kq

,

(5.4)and

Im�(q,!) = �2⇡

~

Zddk

(2⇡)dn

k

(1� n

k+q

) (5.5)

⇥h�(! � !

kq

) + �(! + !

kq

)i,

where we introduce the notation

~!kq

= ✏

k+q

� ✏

k

=~22m

⇣q

2 + 2k · q⌘. (5.6)

Kohn anomalies in 1D systems

Lattice Dynamics and Electron-Phonon Interaction in (3,3) Carbon Nanotubes

K.-P. Bohnen,1 R. Heid,1 H. J. Liu,2,3 and C. T. Chan2

1Forschungszentrum Karlsruhe, Institut fur Festkorperphysik, P.O.B. 3640, D-76021 Karlsruhe, Germany2Department of Physics, University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

3Department of Physics, Wuhan University, Wuhan 430072, People’s Republic of China(Received 13 August 2004; published 6 December 2004; publisher error corrected 8 December 2004)

We present a detailed study of the lattice dynamics and electron-phonon coupling for a (3,3) carbonnanotube which belongs to the class of small diameter based nanotubes which have recently beenclaimed to be superconducting. We treat the electronic and phononic degrees of freedom completely bymodern ab initio methods without involving approximations beyond the local density approximation.Using density functional perturbation theory we find a mean-field Peierls transition temperature of! 240 K which is an order of magnitude larger than the calculated superconducting transitiontemperature. Thus in (3,3) tubes the Peierls transition might compete with superconductivity. ThePeierls instability is related to the special 2kF nesting feature of the Fermi surface. Because of thespecial topology of the (n; n) tubes we also find a phonon softening at the ! point.

DOI: 10.1103/PhysRevLett.93.245501 PACS numbers: 63.22.+m, 63.20.Dj, 63.20.Kr, 71.15.Mb

During the past ten years carbon nanotubes havegained a lot of attention [1]. This is due to their potentialfor applications (e.g., for molecular electronics) as well asto the fact that they allow the study of electronic systemsin one dimension. Thus, they offer the opportunity toinvestigate effects like Peierls transition, superconductiv-ity, electron-phonon interaction, and the interplay be-tween them in a low-dimensional system. Despite thegreat interest in these materials, progress has been hin-dered until recently due to the difficulty of producingcarbon nanotubes with well-defined radii as well as dueto the difficulty to determine in detail the structure ofnanotubes present in a given sample. Recently, however,very promising progress in identifying the structure ofnanotubes has been made by combining Raman and pho-toluminescence measurements [2]. Size selection hasbeen also achieved in certain cases. For example, growingnanotubes in zeolite crystals has made it possible toproduce tubes with a very narrow radii distribution [3]thus allowing for a detailed comparison with moderndensity functional theory (DFT) based calculations [4–6]. Recently, superconductivity has been reported fornanotubes with radii of 4 "A [7]. This immediately raisesthe question as to the origin of superconductivity, theimportance of Peierls distortions, and of electron-electron correlations. Dealing with electron-lattice andstrong electron-electron interaction in a materials spe-cific way from ab initio methods has not been possible sofar even for much simpler systems than nanotubes. Thus,in the past either one had to restrict oneself to modelstudies of the electron-electron aspect [8] or study theelectron-lattice interaction with correlations taken intoaccount only on the level of local density approximation(LDA) based functionals [9,10].

In the latter case, however, modern DFT-based meth-ods allow for the parameter free microscopic calculationof phonon modes. So far, for nanotubes this scheme has

been used nearly exclusively for the determination ofRaman active modes. The results are generally in goodagreement with available Raman data, thus emphasizingthe reliability of this approach. Calculation of the fullphonon dispersion for a given nanotube is a much moreelaborate task. Based on supercell calculations the phonondispersion for a small number of tubes could be deter-mined recently [6,10]. This approach, however, has acertain disadvantage if one is interested in phononanomalies and electron-phonon coupling. Anomaliesshow up in most cases at phonon wave vectors whichare not commensurable with the underlying lattice, thuseven for approximate treatments, huge supercells wouldbe needed to study, for example, a Peierls transition bythis method. Only in very favorable cases, these anoma-lies appear at high-symmetry points and can be dealtwith accurately by the supercell approach (as, e. g., forgraphite [10,11]). Also the calculation of superconductingproperties in the framework of the Eliashberg theoryrequires a detailed knowledge of the phonon dispersionover the whole Brillouin zone (BZ). Especially systemswhich show phonon anomalies require usually a verydense mesh of phonon wave vectors which can not beobtained with supercell methods. The calculations geteven more demanding if the phonon anomalies are relatedto special nesting features of the Fermi surface. Since inone-dimensional systems the Fermi surface consists onlyof isolated points, the most extreme case for nestingproperties is reached here. This requires also a very densek-point mesh for calculating the electronic band structureand wave functions. Already, for simple systems likegraphene and graphite, this effect can be seen easily. Itis reflected in the high sensitivity of certain phononmodes at the K point to sampling effects of the Fermisurface as reported recently [10,11].

An alternative method is offered by using density func-tional perturbation theory (DFPT) which allows for the

PRL 93, 245501 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending10 DECEMBER 2004

0031-9007=04=93(24)=245501(4)$22.50 245501-1 © 2004 The American Physical Society

calculation of phonon modes at arbitrary wave vectorswithout relying on large supercells [12]. Using this ap-proach we have studied in detail the complete latticedynamics and electron-phonon interaction for a (3,3)nanotube. This belongs to the class of small-radii tubeswhich have been reported to be superconducting. OurLetter will try to answer the question whether or notelectron-phonon coupling can explain superconductivityin this tube. A special complication arises from the Peierlsinstability which has to show up in all one-dimensionalmetallic systems. However, this has not received muchattention in the past since general belief was that themean-field Peierls transition temperature for nanotubesis always very low so that it is of no practical conse-quence. These arguments are based on information fromgraphite using the folding concept [1]. However, thisapproach breaks down for tubes with small diameterwhich is seen, e.g., in the prediction of the wrong groundstate for small-radii nanotubes [4,13]. Recently an ap-proximate treatment of the lattice dynamics for (5,0),(6,0), and (5,5) tubes has been presented which showedfor (5,0) tube, indeed, that the estimated transition tem-perature is of the order of 160 K and thus not negligible[14]. However this approach was based on a nonselfcon-sistent tight-binding scheme and did use only an approxi-mate treatment of the polarization eigenvectors; thusresults might be questioned.

In contrast, our DFT-based method allows for a con-sistent calculation of electronic states, phonon modes andelectron-phonon coupling without introducing approxi-mations beyond the LDA level. In this Letter we presentfully ab initio results for phonon dispersion and eigenvec-tors as well as for the electron-phonon coupling for the(3,3) nanotubes using a well-tested norm-conservingpseudopotential of Hamann-Schluter-Chiang type [15].We use DFPT in the mixed-basis pseudopotential formal-ism [16] which has been successfully applied to studyelectron-phonon-mediated superconductivity [17,18]. Asbasis functions localized 2s and 2p functions are usedtogether with plane waves up to an energy cutoff of 20 Ry.For integration over the BZ, a Gaussian broadeningscheme is employed. As test of the reliability of ourphonon approach we have calculated the phonon disper-sion for graphene. Comparison with results publishedrecently show excellent agreement among the differentcalculations except for the highest mode at the K pointwhich is very sensitive to k-point mesh and broadening asalready mentioned [10,11].With a very dense k-point meshof 5184 points in the BZ we found still a fluctuation of1.5 meV for the highest mode when going from a broad-ening of 0.05 to 0.2 eV. The authors of Ref. [10] tried toavoid the complications due to very dense k-point sam-pling by increasing the broadening; however, for studyinginstabilities due to Fermi surface nesting that is not apractical way since any instability will be broadenedsubstantially. Thus, one of the complication for the cal-

culations for the nanotube is the requirement of a verydense k-point mesh and a small broadening.

Our calculations for the (3,3) nanotubes were done in asupercell geometry so that all tubes are aligned on ahexagonal array with a closest distance between adjacenttubes of 10 !A. The tube-tube interactions are very small[4]. We used 129 k points in the irreducible part of the BZand a broadening of 0.2 eV. The structure was fullyrelaxed and the optimal geometry agreed very well withthose given in Refs. [4,5]. The phonon calculation wascarried out with the DFPT method. For calculation of theelectron-phonon matrix elements we even used up to1025 k points. In Fig. 1 we have plotted the band structureclose to the Fermi energy and for comparison also theband structure as obtained by using the folding method.Our results agree well with those obtained previously[4,5]. At k ! kF ! 0:284 (in units of 2!=a, where a isthe lattice constant of the graphene honeycomb lattice)we see that two bands are crossing "F. This specialfeature which holds for all (n; n) tubes is important forphonon anomalies seen at the " point, as will be empha-sized later. The dispersion differs most notably from thoseobtained by using the folding technique by a shift of thekF value and a change in slope of the two bands crossingat "F. This has of course drastic consequences for thephonon modes.

Phonon results are shown in Fig. 2. Again we haveplotted folding results together with the ab initio disper-sion curves. These ab initio results were obtained by

0 0.1 0.2 0.3 0.4 0.5-0.1-0.2-0.3-0.4-0.5k

-2

-1

0

1

2

ε-ε F (

e V)

(a)

0 0.1 0.2 0.3 0.4 0.5-0.1-0.2-0.3-0.4-0.5k

-2

-1

0

1

2

ε-ε F (

eV)

(b)

FIG. 1. Calculated band structure of the (3,3) nanotube.Compared are results obtained (a) from the calculated grapheneband structure using the folding method, and (b) for the truenanotube geometry.

PRL 93, 245501 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending10 DECEMBER 2004

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Fourier interpolation on a 16-point grid. The electronicwave functions were obtained from a calculation with abroadening width of 0.2 eV. In general, the ab initiophonon frequencies are softer than the folding results.Furthermore, near q ! 2kF one sees phonon anomalies incertain phonon branches as well as a certain softening atthe ! point which is not present in the dispersion obtainedby folding. The !-point softening has been seen andstudied already for the G band in Refs. [9,10]. However,this effect is also present in another mode at " 90 meVwith the same symmetry. It is related to the fact that these!-point phonons couple the two electronic bands crossingright at kF (Fig. 1), leading to a nesting vector q ! 0.Therefore, this softening also indicates an underlyingPeierls instability; however, the corresponding Peierlsdistortion does not change the lattice periodicity. Notethat in our DFPT approach, the calculations of phononsinclude all screening effects due to lattice distortions incontrast to other treatments like, e.g., tight-binding meth-ods (see Ref. [14]).

To study the anomalies near q ! 0 and q ! 2kF inmore detail we have increased the number of k pointsand reduced the broadening from 0.2 to 0.025 eV. This canbe interpreted as a variation of the electronic temperaturefrom 1096 to 137 K. With finer sampling and reduced

temperature one of the modes with the anomalous behav-ior at q ! 2kF gets unstable. Results for Tel ! 137 K areshown in Fig. 3. Here only those modes which are sensi-tive to the temperature variation are shown. The modeswhich show anomalous behavior at the ! point stay stablefor all temperatures studied but should eventually go softat low enough temperatures. The Peierls transition tem-perature TP is defined by !2kF #TP$ ! 0. This leads im-mediately to a lower limit of 137 K for TP which is ofsimilar magnitude as the value given for a (5,0) tube inRef. [14]. It is important to note that, contrary to earlyestimates of TP for nanotubes of the order of 1 K [19], thistemperature is of sizeable magnitude and, thus, the Peierlsinstability might compete with the superconductingtransition.

To investigate the possibility of phonon-inducedsuperconductivity, we have calculated the microscopicelectron-phonon coupling parameters, which are centralto the Eliashberg theory of strong-coupling superconduc-tivity. The coupling constant for a phonon mode q! isgiven by

!q! ! 2

"hN#"F$!q!

Xk##0

jgq!k%q#0;k#j2$#"k#$$#"k%q#0$; (1)

where N#"F$ is the density of states at the Fermi energyand all energies are measured with respect to "F. Theelectron-phonon coupling matrix element is given by a

gq!k%q#0;k# / h%k%q#0 j$Vq!eff j %k#i;

with $Vq!eff being the change of the effective crystal

potential due to a phonon q! and j %k#> being theelectronic wave functions . Since this quantity is verysensitive to k-point sampling we have used a grid of 1025points. Because of the self-consistent determination of$V, the matrix elements include all screening effects incontrast to tight-binding approaches. Because the Fermisurface consists only of the points k ! &kF, contri-butions to the total electron-phonon coupling constant

0 0.1 0.2 0.3 0.4 0.5q

0

50

100

150

200

Freq

uenc

y (m

eV) (a)

0 0.1 0.2 0.3 0.4 0.5q

0

50

100

150

200

Freq

uenc

y (m

eV) (b)

FIG. 2 (color online). Calculated phonon dispersion curves ofthe (3,3) nanotube as obtained (a) by the folding method fromthe graphene phonon dispersions and (b) for the true nanotubegeometry. The latter corresponds to a high effective tempera-ture of 1096 K. The arrow indicates 2kF (folded back to the firstBrillouin zone).

0 0.1 0.2 0.3 0.4 0.5q

0

50

100

150

200

Freq

uenc

y (m

eV)

FIG. 3 (color online). Phonon dispersion curves for the twosymmetry classes which are affected by electron-phonon cou-pling. Shown are results obtained on a fine q grid and for asmall effective temperature of 137 K.

PRL 93, 245501 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending10 DECEMBER 2004

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Kohn anomalies in quasi 2D systems: MgB2

Adiabatic and nonadiabatic phonon dispersion in a Wannier function approach

Matteo Calandra,1 Gianni Profeta,2 and Francesco Mauri11IMPMC, Université Paris 6, CNRS, 4 Pl. Jussieu, 75015 Paris, France

2Consiglio Nazionale delle Ricerche-Superconducting and Innovative Devices (CNR-SPIN), 67100 L’Aquila, Italy!Received 13 July 2010; revised manuscript received 7 September 2010; published 8 October 2010"

We develop a first-principles scheme to calculate adiabatic and nonadiabatic phonon frequencies in the fullBrillouin zone. The method relies on the stationary properties of a force-constant functional with respect to thefirst-order perturbation of the electronic charge density and on the localization of the deformation potential inthe Wannier function basis. This allows for calculation of phonon-dispersion curves free from convergenceissues related to Brillouin-zone sampling. In addition our approach justifies the use of the static screenedpotential in the calculation of the phonon linewidth due to decay in electron-hole pairs. We apply the methodto the calculation of the phonon dispersion and electron-phonon coupling in MgB2 and CaC6. In both com-pounds we demonstrate the occurrence of several Kohn anomalies, absent in previous calculations, that aremanifest only after careful electron- and phonon-momentum integration. In MgB2, the presence of Kohnanomalies on the E2g branches improves the agreement with measured phonon spectra and affects the positionof the main peak in the Eliashberg function. In CaC6 we show that the nonadiabatic effects on in-plane carbonvibrations are not localized at zone center but are sizable throughout the full Brillouin zone. Our method opensperspectives in large-scale first-principles calculations of dynamical properties and electron-phonon interaction.

DOI: 10.1103/PhysRevB.82.165111 PACS number!s": 63.20.dk, 71.15.!m, 63.20.kd

I. INTRODUCTION

Electron-phonon !EP" interaction is responsible of manyimportant phenomena in solids. As an example, the tempera-ture behavior of the electron relaxation time in metals is to agreat extent due to the scattering between carriers and atomicvibrations1 such that finite-temperature transport is largelyruled by the EP interaction. Similarly, the high-temperatureheat capacity in metals is enhanced by the increased elec-tronic mass due to the interaction with lattice vibrations.2 Inmetals, at low temperatures, EP coupling can generate a su-perconducting state in which electrons move with no electri-cal resistance.3 It also can increase the effective mass of thecarriers so much that the system is driven from a metallic toan insulating state, as it happens in case of polaronic orPeierls instabilities.4 Finally, the electron-phonon scatteringis often the largest source of phonon damping in phonon-mediated superconductors.5

First-principles theoretical determination of the electron-phonon coupling strength in solids requires the calculation ofthe electronic structure, the vibrational properties, and theelectron-phonon coupling matrix elements. In state-of-the-artelectronic-structure calculations these quantities are obtainedusing the adiabatic Born-Oppenheimer approximation6 anddensity-functional theory !DFT" in the linear-responseapproach.7–9

More specifically, within the Born-Oppenheimer approxi-mation, the determination of phonon frequencies, that arerelated to the real part of the phonon self-energy, requires thecalculation, in a self-consistent manner, of the variation inthe Kohn-Sham potential VSCF with respect to a static ionicdisplacement u,7 namely,

"VSCF!r""u . As the displacement of the

ions is static, the obtained"VSCF!r"

"u is real. Conversely, thephonon linewidth, related to the imaginary part of the pho-non self-energy, is obtained in a nonself-consistent procedureusing the electron charge density and the previously deter-

mined"VSCF!r"

"u . The advantage of using a nonself-consistentprocedure to study phonon linewidth is mainly related to theless expensive computational load with respect to a self-consistent one. In addition, as recently demonstrated, inter-polation schemes10,11 of the electron-phonon matrix elementscan be used to calculate the imaginary part of the phononself-energy on ultradense k-point grid.

It is however unclear to what extent this procedure ofcalculating self-consistently the real part of the phonon self-energy and nonself-consistently the imaginary part is actu-ally correct. Indeed, the proper way of treating phononsshould be to consider a monochromatic time-dependent dis-placement !at the phonon frequency #" and perform a time-dependent self-consistent linear-response scheme. The result-ing variation in the self-consistent potential

"VSCF!r,#""u would

then be a complex quantity. The real part of the resultingphonon self-energy would then determine the phonon fre-quencies while its imaginary part would lead to the phononlinewidth. In this way, nonadiabatic !NA" !in the sense ofRef. 12" dynamical phonon frequencies could be accessed.Although feasible, in principle, this procedure would requirea full rewriting of the linear-response code including timedependence and it would also be more expensive then a stan-dard static linear-response calculation. Moreover, in the pres-ence of Kohn anomalies and long-range force constants,where extremely accurate k-point sampling of the Fermi sur-face is needed to converge the phonon self-energy, the cal-culation would be unfeasible.

Thus, it would be desirable to have a nonself-consistentlinear-response formulation to obtain both the real andimaginary phonon self-energy, both within the adiabatic/static and nonadiabatic/dynamic approximation. In this workwe develop a scheme to calculate nonself-consistently boththe real and the imaginary part of the phonon self-energy byusing a functional13 that is stationary with respect to thevariation in the self-consistent charge density. Our method

PHYSICAL REVIEW B 82, 165111 !2010"

1098-0121/2010/82!16"/165111!16" ©2010 The American Physical Society165111-1

are clearly found along !A using our new method but arecompletely missed by standard Fourier interpolation.29 Themain differences with respect to previous calculations38,41,43

are: !1" a prominent softening !Kohn anomaly" of the twoE2g modes at q#0.25!K is seen. The Kohn anomaly is alsopresent along the !M high symmetry direction and in a cir-cular region around the zone center. Its origin is due to anin-plane nesting between the two " cylinders. This Kohnanomaly could be probably inferred from previouscalculations5,38,43 on smaller k-points grids although the soft-ening of the E2g modes in these works is weaker.

!2" A second Kohn anomaly, absent in previouscalculations,5,38,43 is present on the E2g and B1g modes alongthe kz direction at q#0.8!A. This Kohn anomaly is due tonesting between different cylinders along !A. As it can be

seen by the comparison with previously available experimen-tal data in Fig. 3, its presence is confirmed by inelastic x-rayscattering measurements of the phonon dispersion.5,44 Thedata at q#0.8!A are indeed in disagreement with previousphonon-dispersion calculation5,38,41,43,44 even if this disagree-ment was overlooked in all previous publications.

!3" Overall the Wannier-interpolated phonon dispersion isin better agreement against experiments with respect to thelinear-response calculated one. These differences point outthe need of having an ultradense k-point sampling of theBrillouin zone not only for the electron-phonon coupling butalso for the phonon dispersion.

3. Electron-phonon coupling

Having interpolated the electron-phonon matrix elementsand the phonon frequencies it is possible to calculate thephonon linewidth and the electron-phonon coupling with ahigh degree of precision, eliminating source of errors comingfrom insufficient k-point sampling.

In Fig. 4 the Wannier-interpolated Eliashberg function iscompared with previous calculations done with different ap-proaches. There are two main improvements due to the betterk-point sampling.

Γ K M Γ A0

50

100

LR (Nk=303, Tph=0.02)

Wannier (Nk=303, T0=0.02 Ryd)

Ene

rgy

(meV

)

Γ K M Γ A0

10

20

30

40

50

60

70

80

90

100

LR (Nk=162x12, Tph=0.025)

ExperimentWannier (Nk=45

3, Tph=0.01)

Ene

rgy

(meV

)

FIG. 3. !Color online" Top panel: Wannier-interpolated adiabaticphonon dispersion !red line" compared to standard linear-responsecalculations performed on selected points in the Brillouin zone!black dots". Bottom panel: comparison between state-of-the-artcalculated phonon dispersion using Fourier interpolation !blackline" and Wannier interpolation !red line". Both interpolationschemes start from the same linear-response calculation. Experi-mental data in the bottom panel are from Ref. 5 !blue circles" andfrom Ref. 44 !black circles". Electronic temperatures are in rydberg.The acoustic sum rule is not applied. Interpolated branches are ob-tained by connecting the points at which the interpolation has beencarried out in the order of increasing energy.

0 50 100

0

1

2

3

4

Kong et al. PRB 64, 020501Bohnen et al. PRL 86, 5771This Work

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

1.5

2Choi et al. PRB 66, 020513Eiguren et al. PRB 78, 045124This Work

ω (meV)

α2 F(ω

)α2 F(

ω)

λ(ω)

FIG. 4. !Color online" Comparison of our Wannier-interpolatedMgB2 Eliashberg function with previous calculations. Kong et al.!Ref. 38" used first-principles calculations and a particular treatmentfor #qE2g

for q along !A. Bohnen et al. !Ref. 43" used linear-response calculations while Choi et al. !Ref. 41" performed frozen-phonon calculations. Finally Eiguren et al. !Ref. 11" used a differentimplementation of Wannier interpolation on the electron-phononmatrix elements with 403 k-points grid for electron momentum and403 k-points grid for phonon momentum, but did not interpolatephonon frequencies. In this work we Wannier interpolate bothelectron-phonon coupling !using 803 k-points grid for electron mo-mentum and 203 k-points grid for phonon momentum" and phononfrequencies !using 303 k-points grid for electron momentum and203 k-point grid for phonon momentum". The integrated #!$" fromour work is also shown in the bottom panel.

ADIABATIC AND NONADIABATIC PHONON DISPERSION… PHYSICAL REVIEW B 82, 165111 !2010"

165111-11

VOLUME 86, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 14 MAY 2001

Γ M K Γ A L−15

−10

−5

0

5

10

Ene

rgy

(eV

)

Σ Λ ∆

FIG. 1 (color). Band structure of MgB2 with the B p character.The radii of the red (black) circles are proportional to the B pz(B px,y) character.

rather small MT spheres for Mg. For the LMTO calcula-tions we used an atomic sphere of nearly the size of thefree Mg atom (up to 3.13aB!, and obtained, as expected,a larger charge of 2.8 electrons. However less than 25%of the charge has s character. The remaining charge of p,d, and f character arises not from Mg electrons but ratherfrom the tails of the B p orbitals and contributions fromthe interstitials. In fact, one can say that Mg is fully ion-ized in this compound, however the electrons donated tothe system are not localized on the anion, but rather aredistributed over the whole crystal.

The resulting band structure can be easily understood interms of the boron sublattice. The character of the bandsis plotted in Fig. 1. We show only the B p character, sinceother contributions near the Fermi level are very small.Observe two B band systems: two bands are derived from

−15 −10 −5 0 5 10Energy (eV)

0

0.5

1

1.5

2

DO

S (

stat

es/e

V)

total DOSMg DOSB DOSinterstitial DOS

FIG. 2 (color). Total density of states (DOS) and partial DOSfor the MgB2 compound. The small Mg DOS is partially dueto the small rMT of 1.8aB used.

B pz states and four from B px,y . All these bands arehighly dispersive (light), the former being quite isotropicand the latter more two dimensional. Both pz bands crossthe Fermi level (in different parts of the Brillouin zone),but only two bonding px,y bands do so, and only near theG point. They form two small cylindrical Fermi surfacesaround the G-A line (Fig. 3). However, due to their 2Dcharacter, they contribute more than 30% to the total N"0!.

In contrast, the pz bands have 3D character, since thesmaller intraplane distance compensates for a smaller(ppp vs pps) hopping. In the nearest neighbor tightbinding (TB) model their dispersion is ´k ! ´0 1 2tpps 3cosckz 6 tppp

p3 1 2 cosa1k 1 2 cosa2k 1 2 cosa3k

where a1,2,3 are the smallest in-plane lattice vectors. Theon-site parameter ´0 can be found from the eigenvalue atthe K point and is #1.5 eV above the Fermi energy. Weestimated tpps and tppp from the LMTO calculations as#2.5 and #1.5 eV, respectively. This model gives a verygood description of the pz band structure near and belowthe Fermi level, although the antibonding band acquiressome additional dispersion by hybridizing with the Mg pband. The role of Mg in forming this band structure canbe elucidated by removing the Mg atoms from the latticeentirely and repeating the calculations in this hypotheticalstructure. The in-plane dispersion of both sets of bands atand below the Fermi level changes very little (ppp bandsare hardly changed, while the pps in-plane dispersionchanges by #10%). The kz dispersion of the pz bands isincreased in MgB2 as compared with the hypothetical emptyB2 lattice by about 30%, and these bands shift down withrespect to the px,y bands by approximately 1 eV. Thisshift, as well as the additional dispersion, comes mainlyfrom the hybridization with the empty Mg s band, whichis correspondingly pushed further up, increasing the ef-fective ionicity. Substantial kz dispersion of the pz bands

FIG. 3 (color). The Fermi surface of MgB2. Green and bluecylinders (holelike) come from the bonding px,y bands, the bluetubular network (holelike) from the bonding pz bands, and thered (electronlike) tubular network from the antibonding pz band.The last two surfaces touch at the K point.

4657

kF

2kF

2kF