Appendix A Acoustic sum rules for the interatomic force constants

104 Pages·2006·4.31 MB·English

Structural, vibrational and thermodynamic
properties of carbon allotropes from
first-principles: diamond, graphite, and nanotubes
by
Nicolas Mounet
Submitted to the Department of Materials Science and Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Materials Science and Engineer iMSS
OFTECHNOLOGY
at the
JUL 2 2 2005
MASSACHUSETTS INSTITUTE OF TECHNOLO
LIBRARIES
June 2005
) Massachusetts Institute of Technology 2005. All rights reserved.
Autho..r.. ................... .............. ..........
Department of Materials Science and Engineering
February 23, 2005
Certified by .............................. . ......
Nicola Marzari
AMAX Assistant Professor in Computational Materials Science
Thesis Supervisor
Accepted by .....................
' -~~~~~~C&b adeder
R. P. Simmons Professor of Materials Science and Engineering
Chairman, Department Committee on Graduate Students
ARCHIVES
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Structural, vibrational and thermodynamic
properties of carbon allotropes from
first-principles: diamond, graphite, and nanotubes
by
Nicolas Mounet
Submitted to the Department of Materials Science and Engineering
on February 23, 2005, in partial fulfillment of the
requirements for the degree of
Master of Science in Materials Science and Engineering
Abstract
The structural, dynamical, and thermodynamic properties of different carbon al-
lotropes are computed using a combination of ab-initio methods: density-functional
theory for total-energy calculations and density-functional perturbation theory for
lattice dynamics. For diamond, graphite, graphene, and armchair or zigzag single-
walled nanotubes we first calculate the ground-state properties: lattice parameters,
elastic constants and phonon dispersions and density of states. Very good agree-
ment with available experimental data is found for all these, with the exception of
the c/a ratio in graphite and the associated elastic constants and phonon disper-
sions. Agree:ment with experiments is recovered once the experimental c/a is chosen
for the calculations. Results for carbon nanotubes confirm and expand available,
but scarce, experimental data. The vibrational free energy and the thermal expan-
sion, the temperature dependence of the elastic moduli and the specific heat are
calculated using the quasi-harmonic approximation. Graphite shows a distinctive
in-plane negative thermal-expansion coefficient that reaches its lowest value around
room temperature, in very good agreement with experiments. The predicted value
for the thermal-contraction coefficient of narrow single-walled nanotubes is half that
of graphite, while for graphene it is found to be three times as large. In the case of
graphene and graphite, the ZA bending acoustic modes are shown to be responsible
for the contraction, in a direct manifestation of the membrane effect predicted by
I. M. Lifshitz over fifty years ago. Stacking directly hinders the ZA modes, explaining
the large numerical difference between the thermal-contraction coefficients in graphite
and graphene, notwithstanding their common physical origin. For the narrow nan-
otubes studied, both the TA bending and the "pinch" modes play a dominant role.
For larger single-walled nanotubes, it is postulated that the radial breathing mode
will have the! most significant effect on the thermal contraction, ultimately reaching
the graphene limit as the diameter is increased.
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Thesis Supervisor: Nicola Marzari
Title: AMAX Assistant Professor in Computational Materials Science
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Acknowledgments
The work that I am going to present in the following pages was performed at MIT
from January 2004 to February 2005. During that period, many things would not
have been possible if a lot of people had not helped me in various ways. I would like
to thank all of them, and I apologize in advance for any omission.
First and foremost, I am very grateful to my thesis supervisor, Prof. Nicola
Marzari, for his exceptional kindness and availability, for his attention on all the
issues, scientific or not, that I met, and for his strong support. He fully inspired
and motivated the work I am presenting here. I also greatly enjoyed being part of
his research group, both because of the great competence of all of its members and
the very friendly atmosphere that was always present. They helped me on countless
occasions with patience and care, and I personally thank all of them. In alphabetical
order, they are Mayeul D'Avezac, Dr. Matteo Cococcioni, Ismaila Dabo, Dr. Cody
Friesen, Boris Kozinsky, Heather Kulik, Young-Su Lee, Nicholas Miller, Dr. Damian
Scherlis, Patrick Sit, Dr. Paolo Umari, and Brandon Wood.
I thank Dr. Paolo Giannozzi and Dr. Stefano de Gironcoli who were always
very helpful in answering my questions. I also thank all the people developing the
v-Espresso code ( http://www.pwscf.org/ ) for the truly exceptional work they are
doing on this freely available ab initio code.
I gratefully acknowledge financial support from NSF-NIRT DMR-0304019 and the
Interconnect Focus Center MARCO-DARPA 2003-IT-674.
I would also like to thank the Ecole Polytechnique of Palaiseau (France) and the
Fondation de l'Ecole Polytechnique for making my studies in the USA possible.
I give many thanks to my parents and brothers for all their useful advice and their
constant support.
Finally, I give my very special thanks to my wife Irina, who made my life so much
easier. She has always been the strongest supporter of my work and study, whatever
the cost was for her. My gratitude goes far beyond what I could express with words,
and I am dedicating this thesis to her.
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Contents
1 Introduction 15
2 Theoretical framework 19
2.1 Crystalline structures studied.
19
2.1.1 Diamond . 19
2.1.2 Graphene, graphite and rhombohedral graphite 20
2.1.3 Achiral nanotubes . 20
2.2 Density-Functional Perturbation Theory. 21
2.3 Thermodynamic properties . 27
2.4 Comnputational details. 29
3 Zero-temperature results 33
3.1 Structural and elastic properties .... . . . . . . . . . . . . . . 33
3.1. 1 Diamond . . . . . . . . . . . . . . . 33
3.1.2 Graphene and graphite ..... . . . . . . . . . . . . . . 34
3.1.3 Single-walled nanotubes . . . . . . . . . . . . . . . 38
3.2 Phonon dispersion curves ........ . . . . . . . . . . . . . . .42
3.2.1 Diamond and graphite ..... . . . . . . . . . . . . . . .42
3.2.2 Armchair and zigzag nanotubes . . . . . . . . . . . . . . 49
3.3 Interatomic force constants .
. . . . . . . . . . . . . . .52
4 Thermodynamic properties
59
5 Conclusions 77
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A Acoustic sum rules for the interatomic force constants 79
A.1 Preliminary definitions. 80
. . . .symmetry
A.2 Properties of the IFCs .................... 82
. . . .
A.3 A new approach to apply the acoustic sum rules and index symmetry
constraints . 87
. . . . . .
A.4 Complexity of the algorithm ................. 90
A.4.1 Memory requirements. 90
. . . . . .
A.4.2 Computational time. . . . . . . 91
A.5 Conclusion. .......................... . . . . . . 93
Bibliography 95
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List of Figures
... . 19
2-1 Crystal structure of diamond ................
... . 20
2-2 Crystal structure of graphene ...............
... . 21
2-3 Crystal structure of graphite and rhombohedral graphite
... . 22
2-4 Chiral vectors for armchair and zigzag SWNTs ......
... . 22
2-5 Structure of an armchair (5,5) SWNT ...........
... . 23
2-6 Structure of a zigzag (8,0) SWNT .............
... . 23
2-7 Axial view of an armchair (5,5) SWNT ..........
... . 24
2-8 Axial view of a zigzag (8,0) SWNT ............
3-1 Ground state energy of diamond vs. lattice parameter ......... 34
3-2 Ground state energy of graphene vs. lattice parameter ........ 35
3-3 Contour plot of the ground state energy of graphite vs. lattice param-
eters a and c/a .............................. 36
3-4 Ground state energy of graphite vs. c/a at fixed a = 4.65 a.u..... 37
3-5 Contour plot of the ground state energy of an armchair SWNT vs. r
and I .................................... 40
3-6 Ground state energy of a relaxed armchair SWNT vs. 1 ........ 41
3-7 Phonon dispersions of diamond ..................... 43
3-8 Phonon dispersions of graphite (at the experimental c/a)....... 43
3-9 Phonon dispersions of graphene ..................... 44
3-10 Phonon dispersions of rhombohedral graphite ............. 44
3-11 Phonon dispersions of graphite (at the theoretical c/a) ........ 45
3-12 Phonon dispersions of an armchair (5,5) SWNT ............ 50
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3-13 Phonon dispersions of a zigzag (8,0) SWNT .............. 51
3-14 Decay of the interatomic force constants vs. distance for diamond and
graphene .................................. 53
3-15 Decay of the interatomic force constants vs. distance for graphite and
graphene .................................. 54
3-16 Decay of the interatomic force constants vs. distance for graphene,
armchair (5,5) and zigzag (8,0) SWNTs ................. 55
3-17 Phonon frequencies of diamond as a function of the number of neigh-
bors included in the interatomic force constants ............ 56
3-18 Phonon frequencies of graphene as a function of the number of neigh-
bors included in the interatomic force constants ............ 57
4-1 Lattice parameter of diamond vs. temperature ............ 60
4-2 In-plane lattice parameter of graphite and graphene vs. temperature . 61
4-3 Out-of-plane lattice parameter of graphite vs. temperature ...... 61
4-4 Axial lattice parameter of an armchair (5,5) SWNT vs. temperature . 62
4-5 Axial lattice parameter of a zigzag (8,0) SWNT vs. temperature . .. 62
4-6 Coefficient of linear thermal expansion for diamond .......... 63
4-7 In-plane coefficient of linear thermal expansion for graphite and graphene 64
4-8 Out-of-plane coefficient of linear thermal expansion for graphite . .. 65
4-9 Coefficient of linear thermal expansion along the axis for armchair (5,5)
and zigzag (8,0) SWNTs ......................... 66
4-10 Mode Griineisen parameters for diamond ................ 67
4-11 In-plane mode Griineisen parameters for graphite .......... . 68
4-12 Mode Griineisen parameters for graphene ................ 68
4-13 Out-of-plane mode Griineisen parameters for graphite ......... 69
4-14 Mode Griineisen parameters along the axis for zigzag (8,0) SWNTs 69
4-15 Bending mode of a zigzag (8,0) SWNT ................. 71
4-16 "Pinch" mode of a zigzag (8,0) SWNT ................. 71
4-17 Radial breathing mode of a zigzag (8,0) SWNT ............ 72
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