Quantum Chemistry of Nanotubes: Electronic Cylindrical Waves

224 Pages·2019·26.119 MB·\224

by  Pavel N. D`yachkov (Author)

Quantum Chemistry of Nanotubes
Electronic Cylindrical Waves
Pavel N. D’yachkov
Quantum Chemistry Laboratory
Kurnakov Institute of General and Inorganic Chemistry
Russian Academy of Sciences
Moscow, Russia
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Names: Liu, Jian (Chemical engineer), editor. | Jiang, San Ping, editor.
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Title: Mesoporous materials for advanced energy storage and conversion
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Faculty of Science and Engineering, Curtin University, Perth, WA,
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I was attracted to the idea that even in the smallest particles of
matter can be found mathematical formulas.
Werner Heisenberg
Preface
Quantum chemistry is the quantum mechanics of atoms, molecules, and
solids. Until recently, the quantum chemistry of crystals was regarded as
part of a solid-state physics and was a theoretical foundation of materials
sciences. Now in the time of nanotechnology, when individual molecules
have become the building block of electronic devices, molecular quantum
chemistry has become an important part of materials science. Nanotubes
lie halfway from the molecules to crystals and they are at a center progress
in nanoelectronics.
Quantum chemistry of nanotubes is being developed by the
efforts of the molecular and solid-state theorists. At first, the electronic
structure of carbon nanotubes was described using the simplest quantum
chemical Hückel molecular orbital theory elaborated for π-conjugated
organic compounds. The qualitatively correct predictions obtained of
the relationships between the geometry and band structure of carbon
nanotubes engendered a large number of experimental studies of their
electronic properties and design of nanoelectronics elements. There was
a need for more accurate calculations of nanotubes, and an alternative
and somewhat theoretically more substantiated and detailed description
of the nanotubes electron states came from the solid-state quantum
chemistry when Slater’s augmented plane wave method was taken as
a starting point for tubules studies. However, the cylindrical (tubular)
geometry of nanotubes determines the formation of cylindrical electron
waves in tubules, rather than plane waves as in crystals. This book
provides a detailed exposition of nonrelativistic and relativistic linearized
augmented cylindrical wave technique and demonstrates its applications
to the various carbon and non-carbon, achiral and chiral, pure, intercalated
and doped single-walled, double-walled, and embedded nanotubes and
iv Quantum Chemistry of Nanotubes
nanowires. We hope to convince the reader that the use of cylindrical
waves for nanotubes offers a great advantages in the studies of their
properties.
Pavel N. D’yachkov
Contents
Preface iii
Introduction vii
1. Augmented Cylindrical Waves for Nonchiral Nanotubes
and Wires 1
1.1. Geometry of Carbon Nanotubes 1
1.2. Theory 2
1.3. Applications 20
2. Augmented Cylindrical Waves for Embedded and Double-Walled
Nanotubes 62
2.1. Embedded Nanotubes 62
2.2. Double-Walled Carbon Nanotubes 72
3. Symmetrized Augmented Cylindrical Waves for Chiral
Nanotubes 91
3.1. Theory 93
3.2. Applications 103
4. Augmented Cylindrical Wave Green’s Function Technique for
Impurities 124
4.1. Defects in Nanotubes 124
4.2. Theory 128
4.3. Applications 138
vi Quantum Chemistry of Nanotubes
5. Relativistic Augmented Cylindrical Wave Method 151
5.1. Spin-Orbit Interaction in Nanotubes 151
5.2. Theory 153
5.3. Applications 154
Appendix: Darwin and Mass-velocity Corrections 186
References 190
Index 210
Introduction
Nanotubes are the nanometer-scale chemical compounds with cylindrical
geometry (Iijima 1991). All the single-walled carbon nanotubes can be
constructed by rolling up a single graphite sheet, and the structures of
tubules can be labeled by the pair of integers n and n (where n ≥ n ≥ 0),
1 2 1 2
which, together with C-C bond length d = 1.42 Å determine a geometry
C-C
of the tubule. They are currently the focus of intense multidisciplinary
studies because of their unique physical and chemical properties and their
prospects for practical applications in molecular electronics (Saito et al.
1998, De Volder et al. 2013, Laird et al. 2015, Dekker 2018, Peng 2018, Bai
et al. 2018, He et al. 2018, Gupta et al. 2018, Cao et al. 2017) that is perhaps
the most intriguing area of nanotechnology. Chemically, the carbon
nanotubes are the aromatic conjugated compounds. The theoretical
simulation of the nanotubes electronic structure have received much
attention since 1992, when the first calculations for the band structures of
all-carbon nanotubes were done using the Linear Combination of Atomic
Orbitals (LCAO) and π-electron tight-binding techniques (Mintmire et al.
1992, Saito et al. 1992a, Saito et al. 1992b, Hamada et al. 1992, White et al.
1993). Particularly, it was shown that the electronic structure of any tubule
in a region of the occupied and unoccupied π states can be obtained within
the tight-binding model by folding the graphene π bands along a certain
direction in the two-dimensional Brillouin zone.
The zone-folding technique has proven immensely successful in
providing physically relevant information on the nature of electronic
interactions in nanotubes. In this approach, it was predicted that a nanotube
is a metal if n − n is a multiple of 3, or a semiconductor otherwise.
1 2
Later on, the modified tight-binding π electron and all-valence orbital
band structure models were developed that account for the effects of the
sπ-hybridization, misalignment of carbon p -orbitals on the curved
π
graphene surface, chirality, and diameter dependence of nearest-neighbor
viii Quantum Chemistry of Nanotubes
hopping integral (Saito et al. 2001, Mintmire et al. 1993, Mintmire and White
1998, Rubio et al. 1999, Hagen and Hertel 2003, Vadapalli and Mintmire
2006, Bulusheva et al. 1998, Bussi et al. 2005). This leads particularly to
modifications of the gap energies, which reveals that the single-walled
carbon nanotubes with n − n = 3n (n = 1,2,…) can be very-small-gap
1 2
semiconductors, and that these effects are of great importance in the case
of the small- and moderate radius nanotubes.
These include terms causing the trigonal warping effects, similar to
the zone folding technique results for the highest occupied and lowest
unoccupied π states of the single-walled carbon nanotubes that were
obtained using an effective-mass k·p approximation (Ajiki and Ando 1993,
1996).
Going beyond the semi empirical tight-binding methods or k·p
approximation, one can perform the first-principles calculations for actual
curved-surfaced tubule. A version of LCAO pseudopotential method, in
which the core electrons are replaced by the nonlocal norm-conserving
pseudopotentials and the valence electrons are treated using a linear
combination of multiple-ζ and polarized atomic orbitals, was applied
in the calculation of the electronic dispersion energies and density of
states in the Fermi energy region for several small tubules (Christ and
Sadeghpour 2007, Reich et al. 2002). In the Fermi energy region, the
electronic structure of nanotubes was also studied using the plane-wave
basic and ab initio pseudopotential Local-Density-Functional theory (DFT)
(Blasé et al. 1994, Charlier et al. 1996 a,b, Machón et al. 2002, Marinopoulos
et al. 2003, Zólyomi and Kürti 2004, Cabria et al. 2003, Dubay et al. 2002,
Li et al. 2001, Liu and Chan 2002, Kürti et al. 2004, Miyake and Saito 2005).
One assumes that predictive power and accuracy of the tight-binding
calculations are less robust compared with the ab initio DFT methods
(Christ and Sadeghpour 2007). Using this ab initio method, it was possible
to study in more detail the curvature-induced π- and s-band mixing
and the deviations of chemical bonding from the sp2 hybridization. In
particular, these effects were found to significantly alter the electronic
structure of narrow nanotubes compared to the predictions of the tight-
binding model. In the case of small-diameter insulating nanotubes, where
the diameter is less than 1 nm, the strong π-s rehybridization modifies
the low-lying nondegenerate conduction band states (Marinopoulos et al.
2003, Zólyomi and Kürti 2004). The pseudopotential calculations on the
narrow tubes such as (5, 0) revealed total closure of the gap.
Since the 90s, we are developing a Linearized Augmented Cylindrical
Wave (LACW) method for the nanotubes band structure (D’yachkov et
al. 1998, D’yachkov et al. 1999, D’yachkov and Kirin 1999). The LACW
method is an extension of the one-dimensional multiatomic systems with
tubular structure of the Augmented Plane Wave (APW) theory suggested
for the bulk materials (Slater 1937, Slater 1974) and developed the latter
Introduction ix
in a form of more efficient linearized APW (LAPW) technique (Andersen
1975, Koelling and Arbman 1975). During the past decades, the techniques
for APW and LAPW calculations have reached the point at which, with
the aid of large computers, a solution of the band structure problem may
be obtained for particular crystals, including those with heavy metals
(Singh 1994, Singh and Nordstrom 2006). There is good reason to believe
that the LAPW method is one of the most accurate computational schemes
in the bulk solid-state electronic structure theory. These developments
have challenged an extension of the LAPW approach to low-dimensional
systems such as nanofilms (Jepsen et al. 1978, Krakauer et al. 1979),
nanotubes and nanowires (D’yachkov 1997, 2004, 2016, Mokrousov et
al. 2005, 2006) and fullerene-type spherical clusters (D’yachkov and
Kuznetsov 2004).
The LACW method, as applied to the nanotubes, has an advantage
over the conventional LCAO and plane-wave pseudopotential methods.
While low level LCAO calculations are relatively easy to do, improving
the accuracy quickly becomes both technically demanding and
computationally very expensive. The main concern with approaches of
the LCAO method is the transferability of the basic set. Moreover, it is well
known from the band structure of bulk materials that the LCAO basis is
adequate to achieve good results for the valence band, but sometimes not
for the conduction band, because this basis does not include the delocalized
conducting plane-wave-type functions. The plane-wave pseudopotential
calculations suffer from a slow convergence and an unfavorable scaling:
The number of basic functions and the time taken to perform such a
calculation on a computer increase asymptotically with the cube of the
number of atoms (Skylaris et al. 2005). This method is computationally
quite cumbersome for calculating the band structure of the chiral tubes
without rotational symmetry and very large translational unit cells; hence,
the availability of such results in the literature is very limited. The purely
delocalized nature of the plane-wave basic set puts obstacles in the way
of calculating the inner low-energy states of the valence band, e.g., the
carbon nanotubes p and s bands. The basis of the LACW method has
s
both localized and delocalized components. Finally, the main argument
for using cylindrical waves is to account for the cylindrical geometry of
the nanotubes in an explicit form that offers the obvious advantages.
The purpose of this book is two-fold. First of all, it provides the
physical basis and gives the detailed exposition of the LACW method, so
that newcomers to nanomaterial’s studies can quickly learn the technique,
and if desired construct a working code. Connections are made between
the standard LAPW method for solids and the LACW for nanotubes that
must make the LACW approach more understandable for the readers
experienced in the solid-state LAPW methods. The experimental detection
of spin-orbit gaps in the carbon nanotubes stimulated an interest to the