The band energy structure of RbKSO₄ crystals
The energy band structure of mechanically free and compressed RbKSO₄ single crystals is investigated. It is established that the top of the valence band is located at the D point of the Brillouin zone [k = (0.5, 0.5, 0)], the bottom of the conduction band lies at the Γ point, and the minimum direct...
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Інститут фізики конденсованих систем НАН України
2007
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| Цитувати: | The band energy structure of RbKSO₄ crystals / O.V. Bovgyra, V.Yo. Stadnyk, A.V. Franiv, M.R. Tuzyak, O.Z. Chyzh // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 39-44. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1180572017-05-29T03:05:20Z The band energy structure of RbKSO₄ crystals Bovgyra, O.V. Stadnyk, V.Yo. Franiv, A.V. Tuzyak, M.R. Chyzh, O.Z. The energy band structure of mechanically free and compressed RbKSO₄ single crystals is investigated. It is established that the top of the valence band is located at the D point of the Brillouin zone [k = (0.5, 0.5, 0)], the bottom of the conduction band lies at the Γ point, and the minimum direct band gap Eg is equal to 5.80 eV. The bottom of the conduction band is predominantly formed by the K s, Li p, Rb s, and Rb p states hybridized with the S p and O p antibonding states. The pressure coefficients of the energy position of the valence and conduction band states and the band gap Eg are determined. Дослiджено енергетичну структуру механiчно вiльних i затиснених одновiсним тиском монокристалiв RbKSO₄. Встановлено, що вершина валентної зони локалiзована в точцi D зони Брiллюена [k = (0.5, 0.5, 0)], дно зони провiдностi знаходиться в точцi Γ, а найменша ширина прямої забороненої щiлини Eg рiвна 5,80 еВ. Дно зони провiдностi сформовано в основному K s, Li p, Rb s i Rb p станами, гiбридизованими з антизв’язуючими S p i O p станами. Визначенi баричнi коефiцiєнти енергетичної змiни валентної зони, зони провiдностi та ширини забороненої зони Eg. 2007 Article The band energy structure of RbKSO₄ crystals / O.V. Bovgyra, V.Yo. Stadnyk, A.V. Franiv, M.R. Tuzyak, O.Z. Chyzh // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 39-44. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 71.20.Ps, 78.20.Ci DOI:10.5488/CMP.10.1.39 http://dspace.nbuv.gov.ua/handle/123456789/118057 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The energy band structure of mechanically free and compressed RbKSO₄ single crystals is investigated. It is established that the top of the valence band is located at the D point of the Brillouin zone [k = (0.5, 0.5, 0)], the bottom of the conduction band lies at the Γ point, and the minimum direct band gap Eg is equal to 5.80 eV. The bottom of the conduction band is predominantly formed by the K s, Li p, Rb s, and Rb p states hybridized with the S p and O p antibonding states. The pressure coefficients of the energy position of the valence and conduction band states and the band gap Eg are determined. |
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Article |
| author |
Bovgyra, O.V. Stadnyk, V.Yo. Franiv, A.V. Tuzyak, M.R. Chyzh, O.Z. |
| spellingShingle |
Bovgyra, O.V. Stadnyk, V.Yo. Franiv, A.V. Tuzyak, M.R. Chyzh, O.Z. The band energy structure of RbKSO₄ crystals Condensed Matter Physics |
| author_facet |
Bovgyra, O.V. Stadnyk, V.Yo. Franiv, A.V. Tuzyak, M.R. Chyzh, O.Z. |
| author_sort |
Bovgyra, O.V. |
| title |
The band energy structure of RbKSO₄ crystals |
| title_short |
The band energy structure of RbKSO₄ crystals |
| title_full |
The band energy structure of RbKSO₄ crystals |
| title_fullStr |
The band energy structure of RbKSO₄ crystals |
| title_full_unstemmed |
The band energy structure of RbKSO₄ crystals |
| title_sort |
band energy structure of rbkso₄ crystals |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118057 |
| citation_txt |
The band energy structure of RbKSO₄ crystals / O.V. Bovgyra, V.Yo. Stadnyk, A.V. Franiv, M.R. Tuzyak, O.Z. Chyzh // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 39-44. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-08T13:17:35Z |
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2025-07-08T13:17:35Z |
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| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 39–43
The band energy structure of RbKSO4 crystals
O.V.Bovgyra, V.Yo.Stadnyk, A.V.Franiv, M.R.Tuzyak, O.Z.Chyzh
Franko Lviv National University, 8 Kyrylo and Mefodij Str., Lviv, 79005 Ukraine
Received September 4, 2006
The energy band structure of mechanically free and compressed RbKSO4 single crystals is investigated. It is
established that the top of the valence band is located at the D point of the Brillouin zone [k = (0.5, 0.5, 0)],
the bottom of the conduction band lies at the Γ point, and the minimum direct band gap Eg is equal to 5.80 eV.
The bottom of the conduction band is predominantly formed by the K s, Li p, Rb s, and Rb p states hybridized
with the S p and O p antibonding states. The pressure coefficients of the energy position of the valence and
conduction band states and the band gap Eg are determined.
Key words: uniaxial pressure, energy band, structure, valence band, conduction band
PACS: 71.20.Ps, 78.20.Ci
1. Introduction
Crystals of potassium rubidium sulfate RbKSO4 have a pseudohexagonal lattice structure simi-
lar to the structure of ammonium lithium sulfate LiRbSO4 [1–3]. At room temperature they possess
the mmm point symmetry with the unit cell parameters a = 7.552, b = 5.805 and c = 10.116 Å
[4]. The first order phase transition (PT) at 116 K was observed. It is accompanied by jump-like
changes of the refractive indices ni and birefringence ∆ni. It was found that an application of
uniaxial mechanical stresses along the principal crystallographic axes results in PT displacements
towards both lower and higher temperatures [4].
Investigations of phase transitions in crystals by optical and luminescence spectroscopy, as
well as the analysis and interpretation of the optical and luminescence spectra of the compounds
studied, are impossible without a clear understanding of the energy band structure of the crystals.
Band-structure calculations of RbKSO4 crystals have not been performed so far.
This paper reports on the results of ab initio theoretical calculations of the energy band struc-
ture and the imaginary part ε2 of the complex permittivity for mechanically free single crystals of
the RbKSO4 compound at room temperature.
2. Experiments
RbKSO4 crystals were grown by a slow evaporation of an aqueous stoichiometric solution of
K2SO4 and Rb2SO4 substances at stable temperature. The crystals grown were of a good optical
quality and were of a pseudo-hexagonal shape. They comprised three well-defined blocks, the
pseudo-orthorombic axes a of which were oriented at 120◦ one to another.
The first-principles calculations of the energy band structure of RbKSO4 single crystals were
performed using the nonlocal norm-conserving pseudopotential method. The computational tech-
nique was described in detail in [5,6]. The theoretical details and basic approximations can be
summarized as follows.
1. The total electron energies of the single crystals were self-consistently calculated in the local-
density functional approximation with the use of a plane-wave basis set.
2. The electron energies and electron densities of states were determined from the Kohn-Sham
equations.
c© O.V.Bovgyra, V.Yo.Stadnyk, A.V.Franiv, M.R.Tuzyak, O.Z.Chyzh 39
O.V.Bovgyra, et al.
3. The ion potentials were represented by the Bachelet-Hamann-Schlüter norm-conserving pseudo-
potentials [7]. The correlation potential was determined according to the Ceperley-Alder re-
lationship for rs = (3/4πρ)1/3 and the Gell-Mann-Brueckner relationship in the high-density
limit. The charge density distribution was calculated by the special-point method [8,9] with
the charge damping algorithm.
4. Relaxation of ion positions was simulated by calculating the atomic forces and by determining
the total stress of the unit cell for each crystal structure.
The calculations were carried out in a basis set consisting of 9196 plane waves (the cutoff
kinetic energy was Ecut = 1
2
G2
max = 450 eV). The self-consistent potential was determined using
12 iterative cycles.
3. Results and discussion
Figure 1 presents the results of the calculations of the energy band structure along the high-
symmetry lines of the Brillouin zone. It can be seen in this figure that, on the whole, the two
bands forming the band gap are characterized by a relatively weak dispersion in the k space. An
exception is provided by the lines at the center of the Brillouin zone in the vicinity of the Γ point.
Figure 1. Energy band structure of RbKSO4 crystals at room temperature.
The top of the valence band is located at the D point [k = (0.5, 0.5, 0)], which is taken as zero
energy (0 eV). The bottom of the conduction band is located at the Γ point (E = 5.16 eV). The
minimum direct band gap (at the Γ point) is equal to 5.80 eV. However, it should be remembered
that the given value of the band gap is underestimated. This situation is typical of the majority
of the calculations performed within the framework of the local-density formalism.
The spectral dependences of the total density of states and the partial densities of states for
atoms with the appropriate orbital moments are shown in figure 2. The valence band of the RbKSO4
single crystal consists of narrow energy bands separated by band gaps. The density of states in the
range of binding energies close to −27 eV is determined by the K 2s bound states. The valence
bands in the energy range between −22.5 and −24.5 eV are predominantly formed by the Rb 5s
and O 2p states with an admixture of the S 2p states. The bands of the O 2p states are separated
into three wide subbands in the energy ranges from −1.5 to −3.5 eV, from −5 to −8 eV, and from
−18 to −19 eV. In other ranges, the S 2p states are admixed to a large extent. The Rb p orbitals
make the main contribution to the density of states in the range of −8.5 eV.
The top of the valence band is formed by the S p bounding orbitals. The bottom of the
conduction band is predominantly formed by the K s, K p, Rb s, and Rb p states hybridized with
the S p and O p antibonding states. In other words, the fundamental optical absorption is primarily
associated with the intra-anion transitions.
40
The band energy structure of RbKSO4
(a) (b)
Figure 2. Partial density of states of RbKSO4 crystals.
The relation between the interband transitions and their optical response can be best analysed
in terms of the spectral dependences of the imaginary part of the permittivity ε2. However, it should
be noted that optical spectra have often been identified using the calculated reduced densities of
states.
Ab initio calculations of the electronic and structural properties of simple solids permit one
to obtain information on the crystal structure, lattice parameters, elastic constants (such as the
bulk elastic modulus and the shear modulus), binding energies, phonon spectra, etc. Moreover, the
derivatives of the direct and indirect band gaps with respect to the pressure, as well as the x-ray
structure factors, can also be obtained by analysing the electronic properties. All these quantities
can be determined from the total energy of the system under investigation.
The properties of the ground state of a crystal can be calculated from the total energy with
the use of the equation of state for p = 0 and T = 0; that is,
(
∂E
∂V
)
S
= 0, (1)
where S is the entropy. Our calculations were performed with the most frequently employed equa-
tion of state, namely, the third-order Birch-Murnaghan equation of state [9,10]
E(V ) = E0 +
9
8
B0V0
(
(
V0
V
)
2
3
− 1
)2
×
(
1 +
(
4 − B′
2
)(
1 −
(
V0
V
)
2
3
)
)
, (2)
where E0 is the minimum total energy and V0 is the equilibrium volume. The equilibrium volume
V0 (the unit cell volume for which the total energy is minimum), the bulk modulus B0, and the
derivative of the bulk modulus B′
0 with respect to the pressure were determined by least-squares
fitting of the equation of state (2) to the calculated total energies.
When optimizing the structural parameters, the equilibrium unit cell volume was determined
for fixed experimental values of the ratios a/b and c/b [4]. Then, the theoretically obtained unit
cell volume was used to optimize the ratios of the lattice parameters a/b and c/b.
Table 2 presents the first and second pressure derivatives of the difference between the energies
of the states of the upper (valence) and lower (conduction) bands, which are responsible for the
optical transitions in the vicinity of the fundamental absorption edge.
As a result, we obtained the following equilibrium parameters of the RbKSO4 single crystal: the
minimum total energy E0 = −11 513.58 eV, the equilibrium unit cell volume V0 = 442.69Å3, the
hydrostatic bulk modulus B0 = 98.6 GPa, and the first derivative of the hydrostatic bulk modulus
B′
0 = 4.45.
41
O.V.Bovgyra, et al.
Table 1. Eqilibrium structural parameters of RbKSO4 single crystal (the unit cell parameters
a, b, and c are given in Å, the unit cell volume, Å3).
a b c a/b c/b V
Theory 7.549 5.801 10.109 1.301 1.742 442.69
Experiment 7.552 5.805 10.113 1.301 1.742 443.35
Table 2. Pressure coefficients of the energy position E of the valence (subscript ν) and conduction
(subscipt c) band states with respect to the top of the valence band of RbKSO4 crystals.
Γ Γ Yν Yc Bν Bc Dc
E,eV –0.09 5.0 –0.14 5.86 –0.08 5.83 6.86
dE/dp, meV/GPa –58.2 –206.2 –49.9 –121.5 –96.9 –211.2 –146.1
The energy band structure of the RbKSO4 single crystal was calculated from the theoretically
obtained structural parameters. The energy band structure of the RbKSO4 single crystal under
compression was also calculated by simulating the relaxation of the ion positions.
The pressure coefficient of the minimum direct band gap between the states located at the Γ
point in the Brillouin zone is determined to be dEg/dp ∼ −1.48 · 10−5 eV/bar. This indicates
that the direct band gap in the structure of the RbKSO4 single crystal decreases with increasing
pressure.
The equilibrium unit cell volume was determined for the fixed experiment values of ratios a/b
and c/d by optimizing the structural parameters. Then, the unit cell volume obtained theoretically
was used to optimize the ratios of lattice parameters a/b and c/d (table 1).
4. Conclusions
Thus, the energy band structure of RbKSO4 single crystals was determined by the pseudopo-
tential method with the use of the Bachelet-Hamann-Schlüter pseudo-potentials. It was revealed
that the top of the valence band is located at the D point [k = (0.5, 0.5, 0)] and the bottom
of the conduction band lies at the Γ point (E = 5.16 eV) of the Brillouin zone. The minimum
direct band gap is 5.80 eV. The top of the valence band is formed by the S p bounding orbitals.
The bottom of the conduction band is predominantly formed by the K s, K p, Rb s, and Rb p
states hybridized with the S p and O p antibonding states. The pressure coefficients of the energy
position E of the valence and conduction band states were determined. The pressure coefficient of
the minimum direct band gap between the states located at the Γ point of the Brillouin zone is
equal to ∼ 1.48 · 10−5 eV/bar.
42
The band energy structure of RbKSO4
References
1. Aleksandrov K.S., Beznosikov B.V. Structural Phase Transitions in Crystals (A Family of Potassium-
Sulfate). Nauka, Novosibirsk, 1993 (in Russian).
2. Steurer W., Wittmann H., Jagodzinski H. Acta Crystallogr., Sect. B: Struct. Sci., 1986, 42, 11.
3. Kunishge H., Mashiyama H.J. J. Phys. Soc. Jpn.,1987, 56 (9), 3189.
4. Stadnyk V.Yo., Romanyuk N.A., Brezvin R.S. Kristallografiya, 2005, 50 (6), 1038 (in Russian).
5. Hasebe K., Asahi T. Phys. Rev. B: Condens. Matter, 1990, 41 (10), 6794.
6. Kolinko M.I. J. Phys.: Condens. Matter, 1994, 6 (1), 167.
7. Bachelet B., Hamann D.R., Schlüter M. Phys. Rev. B, 1982, 26 (8), 4199.
8. Chadi D.J., Cohen M.L. Phys. Rev. B: Solid State, 1973, 8 (5), 5747.
9. Cohen R.E., Gulseren O., Hemley R.J. Am. Mineral, 2000, 85, 338.
10. Ravindran P., Subramoniam G., Asokamani R. Phys. Rev. B, 1996, 53 (3), 1129.
Зонно-енергетична структура кристалiв RbKSO4
О.В.Бовгира, В.Й.Стадник, А.В.Франiв, М.Р.Тузяк, О.З.Чиж
Львiвський нацiональний унiверситет iм.I.Франка, фiзичний факультет,
79005 Львiв, вул.Кирила i Мефодiя 8
Отримано 4 вересня 2006 р.
Дослiджено енергетичну структуру механiчно вiльних i затиснених одновiсним тиском монокри-
сталiв RbKSO4. Встановлено, що вершина валентної зони локалiзована в точцi D зони Брiллюена
[k = (0.5, 0.5, 0)], дно зони провiдностi знаходиться в точцi Γ, а найменша ширина прямої заборо-
неної щiлини Eg рiвна 5,80 еВ. Дно зони провiдностi сформовано в основному K s, Li p, Rb s i Rb
p станами, гiбридизованими з антизв’язуючими S p i O p станами. Визначенi баричнi коефiцiєнти
енергетичної змiни валентної зони, зони провiдностi та ширини забороненої зони Eg.
Ключовi слова: одновiсний тиск, енергетичнi зони, структура, валентна зона, зона провiдностi
PACS: 71.20.Ps, 78.20.Ci
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