Effect of pressure on the electronic structure of hcp Titanium
The effect of pressure on the hexagonal close-packed structure of titanium is investigated. The lattice parameters of the equilibrium structure were determined in terms of the Gibbs free energy using the Epitaxial Bain Path method. When this process was repeated for several pressures, the effect of...
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irk-123456789-1199802017-06-11T03:04:02Z Effect of pressure on the electronic structure of hcp Titanium Jafari, M. Jahandoost, A. Vaezzadeh, M. Zarifi, N. The effect of pressure on the hexagonal close-packed structure of titanium is investigated. The lattice parameters of the equilibrium structure were determined in terms of the Gibbs free energy using the Epitaxial Bain Path method. When this process was repeated for several pressures, the effect of pressure on the lattice parameters was revealed. The calculated lattice parameters were in good agreement with the experimental and theoretical results. The effects of pressure on parameters depending on the electronic structure such as conductivity and resistivity in the ground state were also investigated up to 30 GPa using density functional theory. Дослiджено вплив тиску на гексагональну щiльно упаковану структуру титану. Параметри ґратки визначалися в термiнах вiльної енергiї Ґiббса, використовуючи метод епiтаксiї шляхом Бейна. Коли цей процес повторювався для декiлькох тискiв, було виявлено вплив тиску на параметри ґратки. Обчисленi параметри ґратки добре узгоджувалися iз теоретичними та експериментальними результатами. Вплив тиску на параметри, такi як провiднiсть та опiр в основному станi, в залежностi вiд електронної структури, також було дослiджено аж до 30 GPa, використовуючи теорiю функцiоналу густини. 2011 Article Effect of pressure on the electronic structure of hcp Titanium / M. Jafari, A. Jahandoost, M. Vaezzadeh, N. Zarifi // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23601:1-7. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 61.50.Ks, 05.70.Ce, 72.15.Eb, 71.15.Mb DOI:10.5488/CMP.14.23601 arXiv:1107.3948 http://dspace.nbuv.gov.ua/handle/123456789/119980 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The effect of pressure on the hexagonal close-packed structure of titanium is investigated. The lattice parameters of the equilibrium structure were determined in terms of the Gibbs free energy using the Epitaxial Bain Path method. When this process was repeated for several pressures, the effect of pressure on the lattice parameters was revealed. The calculated lattice parameters were in good agreement with the experimental and theoretical results. The effects of pressure on parameters depending on the electronic structure such as conductivity and resistivity in the ground state were also investigated up to 30 GPa using density functional theory. |
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Article |
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Jafari, M. Jahandoost, A. Vaezzadeh, M. Zarifi, N. |
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Jafari, M. Jahandoost, A. Vaezzadeh, M. Zarifi, N. Effect of pressure on the electronic structure of hcp Titanium Condensed Matter Physics |
| author_facet |
Jafari, M. Jahandoost, A. Vaezzadeh, M. Zarifi, N. |
| author_sort |
Jafari, M. |
| title |
Effect of pressure on the electronic structure of hcp Titanium |
| title_short |
Effect of pressure on the electronic structure of hcp Titanium |
| title_full |
Effect of pressure on the electronic structure of hcp Titanium |
| title_fullStr |
Effect of pressure on the electronic structure of hcp Titanium |
| title_full_unstemmed |
Effect of pressure on the electronic structure of hcp Titanium |
| title_sort |
effect of pressure on the electronic structure of hcp titanium |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/119980 |
| citation_txt |
Effect of pressure on the electronic structure of hcp Titanium / M. Jafari, A. Jahandoost, M. Vaezzadeh, N. Zarifi // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23601:1-7. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT jafarim effectofpressureontheelectronicstructureofhcptitanium AT jahandoosta effectofpressureontheelectronicstructureofhcptitanium AT vaezzadehm effectofpressureontheelectronicstructureofhcptitanium AT zarifin effectofpressureontheelectronicstructureofhcptitanium |
| first_indexed |
2025-07-08T17:00:59Z |
| last_indexed |
2025-07-08T17:00:59Z |
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1837098925006258176 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23601: 1–7
DOI: 10.5488/CMP.14.23601
http://www.icmp.lviv.ua/journal
Effect of pressure on the electronic structure of hcp
Titanium
M. Jafari∗, A. Jahandoost, Me. Vaezzadeh, N. Zarifi
Physics Department, K.N. Toosi University of Technology, Tehran, Iran
Received October 27, 2010, in final form March 11, 2011
The effect of pressure on the hexagonal close-packed structure of titanium is investigated. The lattice pa-
rameters of the equilibrium structure were determined in terms of the Gibbs free energy using the Epitaxial
Bain Path method. When this process was repeated for several pressures, the effect of pressure on the lattice
parameters was revealed. The calculated lattice parameters were in good agreement with the experimental
and theoretical results. The effects of pressure on parameters depending on the electronic structure such as
conductivity and resistivity in the ground state were also investigated up to 30 GPa using density functional
theory.
Key words: effect of pressure, hcp-Ti, Gibbs free energy, lattice parameters, EBP method
PACS: 61.50.Ks, 05.70.Ce, 72.15.Eb, 71.15.Mb
1. Introduction
At room temperature and ambient pressure, Ti has a hexagonal close-packed structure called
the α-phase. The lattice parameters of this structure are a = 2.957 Å and c = 4.685 Å [1] in
which the unit cell has two atoms at (1/3, 2/3, 1/4), (2/3, 1/3, 3/4) and the space group number
is 194 (P63/mmc) with the c/a ratio of ∼ 1.59 [2]. Experimental results at room temperature
indicate that lattice parameters decrease and the c/a ratio increases with pressure [3]. At ambient
temperature and high pressure, it changes to the ω-phase [4, 5]. The lattice parameters of this
structures are a = 4.598 Å and c = 2.822 Å [1, 6] with three atoms per unit cell at (0, 0, 0),
(1/3, 2/3, 1/2), (2/3, 1/3, 1/2) and the space group is P6/mmm with the c/a ratio of ∼ 0.61 [2].
The α → ω transition in Titanium is a representative example of martensitic transformation.
Recently, Trinkel et al. have proposed two pathways for this transformation which are called TAO-
1 (“Titanium alpha to omega”) and TAO-2. This mechanism is a direct mechanism in which six-
atom transformation proceeds without a meta-stable intermediate phase and has small shuffles and
strains [7].
The effect of pressure on lattice parameters was investigated using the Epitaxial Bain Path
(EBP) method [8, 9]. In this method, the equilibrium structure is determined by minimized Gibbs
free energy. As a hexagonal structure is defined by its lattice parameters, the Gibbs free energy
[G ≡ E(a, c) + PV (a, c)] should be minimized with respect to a and c. The EBP method is
summarized and explained here. At T = 0 K, the Gibbs free energy was G = E + PV , where P
is the pressure, V was the volume per atom and E is the internal energy per atom. At P 6= 0, the
lattice parameter of a = a1 was chosen and the value of c was varied until one found the values
c = c1 and E = E1 . The energy of E for the lattice parameters c = ci (i = 1, 2, . . . ) with constant a1
was calculated using Wien2k until c1 and E1 values satisfied (∂E/∂c)a = −
(
Pa21 sin γ
)
/2, where γ
is the angle between a and c. When a suitable value of c1 was determined, E1 , V1 and the Gibbs free
energy G1 , which is G1 = E1+PV 1 were found at pressure P . This process was repeated for several
ai values at the same pressure and Gi was determined. For the equilibrium structure at pressure
P , the value giving a minimum GP was chosen. Thus, by choosing different values of P , the lattice
parameters a and c, ratio c/a and the volume were determined directly as the functions of pressure.
∗E-mail: jafari@kntu.ac.ir
c© M. Jafari, A. Jahandoost, Me. Vaezzadeh, N. Zarifi, 2011 23601-1
http://dx.doi.org/10.5488/CMP.14.23601
http://www.icmp.lviv.ua/journal
M. Jafari et al.
The energy was calculated using Wien2k Package [10], which employs a self-consistent Full-
Potential Linearized Augmented Plane wave plus local orbital (FLAPW+LO) method, under the
generalized gradient approximation (GGA) with the Perdew-Burke-Emzerh of 96 exchange Corre-
lation functional [11]. Moreover, the following parameters were used: Muffin-tin-radius, RMT =
2.3 bohr, Largest vector in the charge-density Fourier expansion, Gmax = 12 bohr−1, K point =
4954, Plane-wave cutoff, RKmax = 9, cutoff energy = −6 Ryd and charge convergence = 1×10−4e
(the charge convergence was used in order to optimize parameters in the SCF cycle). Except for
K point = 5599 and RMT = 2.247 bohr, other parameters which were related to the effect of
pressure on the electronic structure did not change.
2. Results and discussion
The effect of pressure on the hexagonal close-packed structure of titanium in the ground state
was investigated. The Gibbs free energy (figure 1), ratio of c/a (figure 2), lattice parameters a and
c (figure 3) and volume (figure 4) were calculated as functions of pressure. The experimental data
were obtained using figure 5 in Errandonea et al. [3] via Gate data software. These results were in
good agreement with theoretical [8] and experimental studies [3].
Figure 1. The pressure dependence of the Free
energy G.
Figure 2. The pressure dependence of ratio
c/a.
Figure 3. The pressure dependence of (a) lattice parameter a and (b) lattice parameter c.
23601-2
Effect of pressure on the electronic structure of hcp Titanium
Figure 4. The pressure dependence of atomic
volume.
In order to make a comparison between the
results of this study and those of the refer-
ences [3], only three points of the pressure (0, 4
and 7 Gpa) were chosen which was due to the
limitation of the experimental results in which
the experimental points were only considered
up to 14.5 GPa. These comparisons are tabu-
lated in tables 1 to 3 and shown in figures 5
and 6. As shown in the aforementioned figures,
based on the expectations, the lattice param-
eters decreased and c/a ratio increased with
pressure.
In the present study at T = 0 K and
P = 0 GPa, α-phase had the c/a ratio of
1.586 which was in good agreement with the
experimental 1.584 [3] and theoretical results
1.611 [8], 1.584 [12] or 1.583 [13] smaller than
the ideal value of 1.633 for an hcp crystal. Furthermore, the experimental results showed that the
c/a ratio for α-Ti gradually increased from 1.584 at atmospheric pressure to 1.622 at 14.5 GPa [3].
Table 1. Lattice parameter of hcp Ti at zero pressure; estimated deviations with experimental
work [3] are indicated.
Lattice parameter This work Experimentala [3] Theoreticalb [8]
a (Å) 2.93693±0.007 2.9575 2.92455
c (Å) 4.65834±0.006 4.68548 4.71120
c/a 1.58613± 0.011 1.58427 1.61091
Table 2. Lattice parameter of hcp Ti at 4 GPa; estimated deviations with experimental work [3]
are indicated.
Lattice parameter This work Experimentala [3] Theoreticalb [8]
a (Å) 2.89777±0.005 2.9137 2.88898
c (Å) 4.61230±0.008 4.65088 4.65735
c/a 1.59167±0.003 1.59621 1.61211
Table 3. Lattice parameter of hcp Ti at 7 GPa; estimated deviations with experimental work [3]
are indicated.
Lattice parameter This work Experimentala [3] Theoreticalb [8]
a (Å) 2.86973±0.006 2.88717 2.86523
c (Å) 4.58955±0.009 4.63199 4.62227
c/a 1.59930±0.003 1.60433 1.61323
a Experimental values are obtained by Gate Data Software from figure 5 of the reference [3].
b Theoretical values are obtained by Gate Data Software from figure 2 of the reference [8].
The α-phase can transform to the ω-phase under pressure. However, it is believed that both
structures can coexist in the pressure range studied here [5]. The effect of pressure on lattice
23601-3
M. Jafari et al.
Figure 5. Lattice parameters vs pressure for
hcp Ti. Estimated deviations with experimen-
tal work [3] are indicated.
Figure 6. Ratio of c/a parameters vs pressure
for hcp Ti. Estimated deviations with experi-
mental work [3] are indicated.
parameters of α-phase in the range of 0–14.5 GPa was also investigated, both experimentally in [3]
and theoretically in [8]. These results confirm the possibility of coexistence of both structures in
the pressure range of 2–9 GPa and even more.
In fact, the coexistence of these phases has been also reported by experiments within the
temperature range from room temperature to around 923 K [14] and by the theory of reconstructive
phase transitions [15].
Table 4. Partial charges in the s, p and d bands.
P (GPa) s p d
0 0.31515 0.2767 0.9858
4 0.32471 0.2870 2.0104
7 0.33159 0.2953 2.0426
15 0.34840 0.3125 2.0999
20 0.35676 0.3199 2.1242
25 0.36704 0.3315 2.1744
30 0.37429 0.3371 2.2127
Table 5. Density of states at the Fermi level
as a function of pressure.
Table 6. The pressure dependence of the Fermi
energy.
P (GPa) n(EF) (States/ev atom)
0 0.8930
4 0.8868
7 0.8500
15 0.8625
20 0.8000
25 0.8158
30 0.7908
P (GPa) EF (Ryd)
0 0.56228
4 0.58999
7 0.60881
15 0.65693
20 0.67996
25 0.70925
30 0.73021
Using the lattice parameters and Wien2k, the number of electrons in s, p and d bands (table 4),
the Fermi energy (table 5) and the density of states at this energy n(εF) (table 6) were calculated
for different pressures.
23601-4
Effect of pressure on the electronic structure of hcp Titanium
Figure 7. 1/n (εf ) as a function of pressure for
hcp Ti.
Moreover, the electrical conductivity can be
expressed as σ = e2τFv
2
F
n(εF)/3 [16, 17], where
vF is velocity at the Fermi energy and τF is the
relaxation time, but n(εF) has a greater effect
than the latter two parameters. Figure 7 shows
1/n(εF) as a function of pressure. Because ρ ∝
1/n(εF), figure 7 can be taken as a measure of
the effect of pressure on electrical resistivity.
According to Matthiessen’s rule, total elec-
trical resistivity ρ, due to electron scattering by
different factors is given by sum of these factors
ρ = ρTh + ρD + ρI , where ρTh is thermal resis-
tivity, ρD and ρI are resistivity due to defects
and impurities, respectively. The other factor
that can scatter electrons is electron-electron
(e-e) interaction, which is negligible. At high
temperature, the effects of impurities and de-
fects are negligible; thus, ρ ≈ ρTh ; at low temperature, ρTh is less than ρD+ρI , so ρ ≈ ρD+ρI . At
T = 0 K, ions were frozen in fixed positions and electrons do not scatter with phonon (ρTh = 0).
In the present study, pure titanium was investigated. It is chemically and thermodynamically im-
possible to avoid impurities and defects, so the resistivity at T = 0 is not zero (ρ = ρD + ρI 6= 0
where ρ = ρr is called residual resistance).
Figure 7 shows that at T = 0 K an increase in pressure causes an increase in electrical resistivity,
which contradicts the results of P.S. Balog, who investigated the phenomenon at 50–700◦C, where
ρ was due to electron-phonon interaction. A pressure increase (at constant temperature) leads to a
decrease in inter-atomic spacing and atomic vibrational amplitude, causing a decrease in electrical
resistivity [18]. However, in the present study, the investigation was carried out at T = 0 K and
electrical resistivity increased with pressure. According to table 4, an increase in pressure leads to
an increase in the number of electrons per volume.
Furthermore, according to table 5, an increase in pressure leads to a decrease in the density of
state, thus causing an increase in the electrical resistivity.
According to table 6, an increase in pressure leads to an increase in the width of the valence
band. In the formation of molecules, several atoms are arranged beside each other, so atomic orbitals
are split and several molecular orbitals are created while the number of orbitals is proportional to
the number of atoms. However, if the inter-atomic space is small, atomic orbital splitting is larger.
According to the calculations and figure 2, a pressure increase causes a decrease in the lattice
constant and inter-atomic space, thus increasing the orbital splitting and width of the valence band.
Table 7. Partial charges in the px + py , pz ; dz2 , dxy + dx2
−y2 and dxz + dyz .
P (GPa) px + py pz dxy + dx2
−y2 dz2 dxz + dyz
0 0.18896 0.08777 0.84783 0.46065 0.67736
4 0.19654 0.09052 0.86045 0.46494 0.68507
7 0.20200 0.09334 0.87419 0.47196 0.69646
15 0.21359 0.09895 0.90120 0.48449 0.71430
20 0.21833 0.10159 0.91282 0.48769 0.72351
25 0.22668 0.10482 0.93479 0.50003 0.73965
30 0.22877 0.10840 0.95242 0.50818 0.75218
Table 7 lists the number of electrons in s, p and d orbitals and the deviation from spherical
symmetry is shown in table 8. This deviation for p and d orbitals is given by [19]:
∆nd =
(
ndxy
+ nd
x2
−y2
)
−
1
2
(
ndxz
+ ndyz
)
− nd
z2
,
23601-5
M. Jafari et al.
∆np =
1
2
(
npx
+ npy
)
− npz
.
Table 8. Deviation from spherical symmetry of the p and d states as a function of pressure.
P (GPa) ∆nd ∆np
0 0.048 2.1528
4 0.052 2.1712
7 0.054 2.1641
15 0.059 2.1585
20 0.063 2.1491
25 0.064 2.1625
30 0.068 2.1104
If ∆np and ∆nd are close to zero, deviation from spherical symmetry will be just slight. Ac-
cording to table 8, this deviation increases with pressure for d orbitals.
3. Conclusion
The aim of the present study was to investigate the pressure effect on lattice parameters of hcp
structure in titanium. The obtained results showed that the c/a ratio of hcp was nearly constant.
However, it is believed that both structures can coexist in the pressure range studied. The alpha
phase was the most stable phase at ambient conditions and its transformation to the omega phase
in the pressure range of 2–9 GPa. Moreover, theoretical and experimental results confirmed the
possibility of coexistence of both structures within the pressure range of 2–9 GPa and even more.
Furthermore, effects of pressure on parameters depending on the electronic structure, such as
conductivity, resistivity, the Fermi energy and n(εF) in the ground state were also investigated up
to 30 GPa using density functional theory. Moreover, an increase in pressure leads to a decrease in
the density of state, thus causing an increase in the electrical resistivity.
23601-6
Effect of pressure on the electronic structure of hcp Titanium
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Вплив тиску на електронну структуру hcp титану
М. Джафарi, А. Джагандуст, М. Ваеззаде, Н. Зарiфi
Технологiчний унiверситет iм. К.Н. Тусi, Тегеран, Iран
Дослiджено вплив тиску на гексагональну щiльно упаковану структуру титану. Параметри ґратки
визначалися в термiнах вiльної енергiї Ґiббса, використовуючи метод епiтаксiї шляхом Бейна. Ко-
ли цей процес повторювався для декiлькох тискiв, було виявлено вплив тиску на параметри ґратки.
Обчисленi параметри ґратки добре узгоджувалися iз теоретичними та експериментальними резуль-
татами. Вплив тиску на параметри, такi як провiднiсть та опiр в основному станi, в залежностi вiд
електронної структури, також було дослiджено аж до 30 GPa, використовуючи теорiю функцiоналу
густини.
Ключовi слова: вплив тиску, hcp-Ti, вiльна енергiя Ґiббса, параметри ґратки, метод EBP
23601-7
http://dx.doi.org/10.1103/PhysRevLett.86.3068
http://dx.doi.org/10.1103/PhysRevLett.49.673
http://dx.doi.org/10.1016/j.physb.2004.10.030
http://dx.doi.org/10.1126/science.140.3562.72
http://dx.doi.org/10.1016/0079-6425(82)90002-0
http://dx.doi.org/10.1103/PhysRevLett.91.025701
http://dx.doi.org/10.1002/pssb.200540110
http://dx.doi.org/10.1103/PhysRevB.66.064111
http://dx.doi.org/10.1103/PhysRevLett.77.3865
http://dx.doi.org/10.1016/j.ssc.2008.02.012
http://dx.doi.org/10.1063/1.3407560
http://dx.doi.org/10.1007/s11661-010-0393-1
http://dx.doi.org/10.1088/0953-8984/11/5/014
http://dx.doi.org/10.1103/PhysRevB.38.9368
Introduction
Results and discussion
Conclusion
|