Excitons into one-axis crystals of zinc phosphide (Zn₃P₂)
Theoretical study of excitons spectra is offered in this report as for Zn₃P₂ crystals. Spectra are got in the zero approach of the theory of perturbations with consideration of both the anisotropy of the dispersion law and the selection rules. The existence of two exciton series was found, which c...
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irk-123456789-1199832017-06-11T03:03:23Z Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) Stepanchikov, D.M. Chuiko, G.P. Theoretical study of excitons spectra is offered in this report as for Zn₃P₂ crystals. Spectra are got in the zero approach of the theory of perturbations with consideration of both the anisotropy of the dispersion law and the selection rules. The existence of two exciton series was found, which corresponds to two valence bands (hh, lh) and the conductivity band (c). It is noteworthy that anisotropy of the dispersion law plus the existence of crystalline packets (layers) normal to the main optical axis, both will permit the consideration of two-dimensional excitons too. The high temperature displaying of these 2D-exciton effects is not eliminated even into bulk crystals. The calculated values of the binding energies as well as the oscillator's strength for the optical transitions are given for a volume (3D) and for two-dimensional (2D) excitons. The model of energy exciton transitions and four-level scheme of stimulated exciton radiation for receiving laser effect are offered. Представлено теоретичне дослiдження екситонних спектрiв для кристалiв Zn₃P₂. Енергетичний спектр екситонiв отримано у нульовому наближеннi теорiї збурень з урахуванням анiзотропiї закону дисперсiї та правил вiдбору. Зафiксовано прояв двох серiй екситонних станiв, якi вiдповiдають двом валентним зонам (hh, lh) i зонi провiдностi (c). Гранична анiзотропiя закону дисперсiї та шарувата кристалiчна структура дозволяють очiкувати на квазiдвовимiрний характер екситонiв. Наслiдком цього може стати високотемпературний прояв екситонних ефектiв, навiть в об’ємних кристалах. Розраховано значення енергiї зв’язку та сили осцилятора оптичного переходу для об’ємних (3D) i двовимiрних (2D) екситонiв. Запропоновано модель екситонних переходiв i чотирирiвневу схему стимульованого екситонного випромiнювання для отримання лазерного ефекту. 2009 Article Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) / D.M. Stepanchikov, G.P. Chuiko // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 239-248. — Бібліогр.: 34 назв. — англ. 1607-324X PACS: 71.18.+y DOI:10.5488/CMP.12.2.239 http://dspace.nbuv.gov.ua/handle/123456789/119983 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Theoretical study of excitons spectra is offered in this report as for Zn₃P₂ crystals. Spectra are got in the
zero approach of the theory of perturbations with consideration of both the anisotropy of the dispersion law
and the selection rules. The existence of two exciton series was found, which corresponds to two valence
bands (hh, lh) and the conductivity band (c). It is noteworthy that anisotropy of the dispersion law plus the
existence of crystalline packets (layers) normal to the main optical axis, both will permit the consideration of
two-dimensional excitons too. The high temperature displaying of these 2D-exciton effects is not eliminated
even into bulk crystals. The calculated values of the binding energies as well as the oscillator's strength for the
optical transitions are given for a volume (3D) and for two-dimensional (2D) excitons. The model of energy
exciton transitions and four-level scheme of stimulated exciton radiation for receiving laser effect are offered. |
| format |
Article |
| author |
Stepanchikov, D.M. Chuiko, G.P. |
| spellingShingle |
Stepanchikov, D.M. Chuiko, G.P. Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) Condensed Matter Physics |
| author_facet |
Stepanchikov, D.M. Chuiko, G.P. |
| author_sort |
Stepanchikov, D.M. |
| title |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) |
| title_short |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) |
| title_full |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) |
| title_fullStr |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) |
| title_full_unstemmed |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) |
| title_sort |
excitons into one-axis crystals of zinc phosphide (zn₃p₂) |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119983 |
| citation_txt |
Excitons into one-axis crystals of zinc phosphide (Zn₃P₂) / D.M. Stepanchikov, G.P. Chuiko // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 239-248. — Бібліогр.: 34 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stepanchikovdm excitonsintooneaxiscrystalsofzincphosphidezn3p2 AT chuikogp excitonsintooneaxiscrystalsofzincphosphidezn3p2 |
| first_indexed |
2025-07-08T17:01:18Z |
| last_indexed |
2025-07-08T17:01:18Z |
| _version_ |
1837098949204246528 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 239–248
Excitons into one-axis crystals of zinc phosphide
(Zn3P2)
D.M.Stepanchikov∗, G.P.Chuiko
Kherson national technical university, department of general and applied physics, research laboratory of the
theory of solids, Berislavske shosse, 24, 73008, Kherson, Ukraine
Received May 18, 2008, in final form May 18, 2009
Theoretical study of excitons spectra is offered in this report as for Zn3P2 crystals. Spectra are got in the
zero approach of the theory of perturbations with consideration of both the anisotropy of the dispersion law
and the selection rules. The existence of two exciton series was found, which corresponds to two valence
bands (hh, lh) and the conductivity band (c). It is noteworthy that anisotropy of the dispersion law plus the
existence of crystalline packets (layers) normal to the main optical axis, both will permit the consideration of
two-dimensional excitons too. The high temperature displaying of these 2D-exciton effects is not eliminated
even into bulk crystals. The calculated values of the binding energies as well as the oscillator’s strength for the
optical transitions are given for a volume (3D) and for two-dimensional (2D) excitons. The model of energy
exciton transitions and four-level scheme of stimulated exciton radiation for receiving laser effect are offered.
Key words: excitons, zinc phosphide, one-axis crystals
PACS: 71.18.+y
1. Introduction
The electronic properties of highly anisotropic systems such as confined systems or layered
crystals have received much attention due, among other reasons, to the possibility of growing high-
quality nanostructures with prescribed configurations, allowing the control of physical properties
such as carrier densities, band gaps and bandwidths, and even dimensionality. On the other hand,
excitons are important excitations that strongly affect electronic and optical properties of bulk and
low-dimensional solids.
In this respect, the II–V compound semiconductors are of interest for several reasons. Firstly,
these are possible applications and secondly come the band structure parameters and layered crys-
talline structures [1–11]. Now II–V compounds have come to show strong promise of constituting
the next generation of electronic materials. Among these, zinc phosphide (Zn3P2), the constituent
elements of which are known to be present in abundant deposits near the surface of the earth,
is drawing particularly strong attention as a material which it is hoped will make it possible to
produce highly efficient solar cells, sensors, lasers and the like at low cost [12–15].
Due to the large excitonic radii of II–V materials, they are expected to exhibit pronounced
size quantization effects. The electrons in such a semiconductor will become confined in crystals
much larger than for the analogous II–VI or III–V semiconductors. However, compared with the
significant progress in bulk and low-dimensional II–VI and III–V semiconductors, research on
II–V semiconductors has been lingering far behind due to the lack of appropriate and generalized
synthetic methodologies. Now this problem of producing II–V compound semiconductors (including
nanoscales) has been substantially decided [16–20]. Therefore, theoretical studies of the features of
band structure of II–V materials once again become an urgent problem.
The main features and the quality of semiconductor devices are determined by the edge of
optical absorption of the basic material. Excitons may substantially affect the form of this edge,
particular through the appearance of discrete levels inside the band gap. Knowledge of the levels
∗E-mail: step 75@mail.ru
c© D.M.Stepanchikov, G.P.Chuiko 239
D.M.Stepanchikov, G.P.Chuiko
located within the energy gap is important both for the potential applications and for the qualita-
tive evaluation of a material. For the practical application of the II–V compound semiconductors,
however, it is necessary to study the problems posed by the fact that they are defect semiconductors
having several vacancies in their unit cells. As a result, these semiconductors include numerous
defect deep levels attributed to the deficiency of atoms [4,21,22]. The interaction between exci-
tons and such defect levels can be a reason to exciton recombination. In this case, the creation of
multiple-exciton complexes is very possible too. Reduction of system dimensionality sharply inten-
sifies these effects. However, the exciton states in II–V semiconductors have not been extensively
examined yet. For Zn3P2 crystals, the exciton spectrum calculations have not been described so far.
This paper presents a theoretical study with fresh manner of modelling the exciton spectra as
for Zn3P2 crystals. For the first time the exciton binding energy and oscillator strength of exciton
transitions are calculated. Besides that, the layered crystalline structure, anisotropy of dispersion
law, possible two-dimensional exciton spectrum and selection rules are taken into account. The
electron, hole and exciton effective masses are calculated in the framework of generalized Kildal
band model. The model of Wannier-Mott excitons is used for exciton spectrum calculations. The
four-level scheme of possible stimulated exciton radiation at low temperature is received.
2. Crystalline structure
Zn3P2 is one of four crystallographically similar semiconductors of the type AII
3 BV
2 . The others
are Cd3P2, Zn3As2 and Cd3As2. Materials with a common chemical formula AII
3 BV
2 differ by the
variety of crystalline forms. They have anion sublattices in close proximity to the standard FCC
packing. Their cations occupy only three quarters of tetrahedral emptiness among anions. Thus,
their cation sublattices contain exactly a quarter of vacancies, which may be recognized even
as stoichiometric. Different ways of allocation of these vacancies determine the key features of all
crystalline forms. Fully disordered high-temperature phosphide β-phases have, for instance, a cubic
symmetry. The partly ordered crystals (α′′-phases) demonstrate a tetragonal symmetry.
Figure 1. A general view of the crystal
structure of Zn3P2 (bigger spheres present
P, smaller spheres – Zn atoms).
Phase transition β → α′′ for Zn3P2 crystals oc-
curs at T = 1118 K [23,24]. A positional order ari-
ses up in this transition, and typical energy of the
positional ordering is 0.078 eV [24]. Herein below
we should like to examine the partly ordered low-
temperature tetragonal α′′-phases. Zinc vacancy lev-
els were suggested as the origin of the 0.19 eV accep-
tor (singly ionized) and the 0.29 eV acceptor (doubly
ionized) [21].
The crystal structure of the tetragonal modifica-
tion of Zn3P2 (see figure 1) is a primitive tetragonal
lattice with a unit cell containing 16 P atoms and
24 Zn atoms. The space group is P42/nmc
(
D15
4h
)
with the following parameters of a unit cell: a = b =
8.0889 Å and c = 11.4069 Å [1–6].
The presence of a quarter of vacant sites into
the cation sublattice is a “logo” of these crystalline
structures. Each cation atom (Zn) has the tetra-
hedral coordination with their nearest neighbors,
i.e. with four phosphorus atoms, whereas those sur-
rounded by cation atoms are located only at six of the eight corners of a coordinating cube. Two
vacant sites are located either at diagonal corners of a cubic face or at the opposite spots of a space
cubic diagonal [1–6,24].
An examination of figure 1 shows that there are two kinds of phosphorus layers and four of
cations. The four cation layers differ in the positions of cation vacancies [1]. Other authors consider
this structure as a simplest polytype consisting of two layered packets normal to the main axis.
240
Excitons into one-axis crystals
In the extended elementary cell (a
√
2 × a
√
2×c) the crystalline structure of the α′′-modification
can be presented as the sequence of two non-equivalent layered packets alternated along the main
axis – X, Y . Every packet consists of four atomic layers. The packet contains two different layers
of the close packed anions (α, β) alternated by two layers of cations (A, B), so that: X = αAβB.
Furthermore, the packet X transforms to the packet Y , and vice versa, by partial translations
(
a
√
2 + c
)
/2 in the coordinate system of the extended elementary cell. The atomic layers, as well
as the layered packets, are normal towards the main axis of crystal [24].
Such structures, more tightly filled with the metal atoms, should enhance the interaction be-
tween them and increase the metallic contribution to the chemical bond in comparison with III–V
and II–VI compounds. The average number of electrons per bond is 4/3, and it is palpable less
than standard digit of 2 electrons per bond as for conventional semiconductors with the tetrahedral
coordination. Such relative deficiency of bonding electrons realizes itself in a descent of equilibrium
bond lengths. Most of the bond lengths (56 from 96) remain close to the sum of covalent radii
in the tetrahedral coordination (2.41 Å), but the remaining 40 bond lengths displace themselves
below the sum of ionic radii (2.86 Å). The value of fractional ionicity is 0.17÷ 0.19. The chemical
bond for Zn3P2 is therefore a complex ionic-metallic-covalent bond [2,5–7,11]. It is also possible to
think that forces of bonds are different: they may be much stronger into the layered packets, than
between them.
3. Band model
The dispersion law for AII
3 BV
2 semiconductors within the framework of model [8,25] has such
a form nearby the point k = 0 and with spherical coordinates (k, θ, ϕ):
(Γ(E) − (Pk)2(f1(E) sin2(θ) + f2(E) cos2(θ))2 − (Pk)2f2
3 (E) sin2(θ) = 0, (1)
Γ(E) = E (E − Eg)
((
E +
2∆
3
)(
E + δ +
∆
3
)
− 2∆2
9η2
)
, (2)
f1(E) =
(
E +
∆
3
) (
E + δ +
∆
3
)
− ∆2
9η2
, (3)
f2(E) = E
(
E +
2∆
3
)
η−4, (4)
f3(E) =
2
3η
E∆d. (5)
Thereto: (Eg, ∆, P ) – are three well-known Kane’s parameters (the energy gap, the spin-
splitting parameter and the matrix element of the impulse); δ – is the known parameter of the
crystal field; d – is another parameter of the crystal field, which describes the absence of symmetry
center; η = c/(a
√
2) – is the scalar factor taking into account the deformation of the lattice. The
numerical values of band parameters are presented in table 1 as if deserving the trust. It may be
pointed out that a dispersion law similar to (1) has been also found in wurtzite-type bulk crystals
and in some two-dimensional systems (e.g. heterojunctions and inversion layers) [27].
Hamiltonian (1) describes a surface of rotation around the main crystalline axis. This Hamil-
tonian is obviously of the fourth order, that is for k and has quite evident decomposition into the
product of two factors P+ and P− [25]:
P+ (k, E, θ) P− (k, E, θ) = 0, (6)
where
P± (k, E, θ) = Γ(E) − (Pk)2(f1(E) sin2(θ) + f2(E) cos2(θ) ± Pkf3(E) sin(θ). (7)
On the other hand, it might be simplified to the two identical surfaces, both of second order,
under the additional condition: d = 0; f3(E) = 0 ⇒ P+ = P−. Physically, this condition means
the presence of the symmetry center into a crystal [26]. This is just the case for P42/nmc (D15
4h)
241
D.M.Stepanchikov, G.P.Chuiko
modifications of AII
3 BV
2 compounds. Each energy level should be twice degenerated according to
this. Obviously, we deal with the well-known Kramer’s degeneration at this rate. Therefore, by
using expressions (2-5) the equations (7) become as follows:
P± (k, E, θ) =
E
9η4
[
(E − Eg) η2
(
−2∆2 + (3 (E + δ) + ∆) (3E + 2∆) η2
)
+3k2P 2
(
− (3E + 2∆) cos2(θ) − (3 (E + δ) + 2∆) η4 sin2(θ)
)]
= 0. (8)
Dispersion equation (8) has four non-identical solutions describing the conductivity band (c),
the heavy holes band (hh), the light holes (lh) and spin-orbital split bands (so), respectively (see
figure 2). These direct solutions might be obtained even in radicals, although proper expressions
are quite cumbersome and unbelievably long. But in a special case (k0 = 0, Γ-point, band extrema)
we get simple roots
Ec
0 = Eg; E
hh
0 = 0; Elh,so
0 = −3η (δ + ∆) ∓
√
9δ2η2 − 6δ∆η2 + ∆2 (8 + η2)
6η
. (9)
In the last equation the sign “–” and “+” correspond to lh-band and so-band, respectively. The
top of the hh-band is selected as the zero energy coordinates. Two of the three valence bands (hh
and lh) are somewhat narrow within this model.
Figure 2. Energy band structure of
Zn3P2 at the Γ point.
The knowledge of selection rules is also necessary for
any interpretation of the optical band-to-band transitions
and for classification of the exciton states. The selection
rules for Zn3P2 depend on the direction of polarization
vector (ep) of the light. As it is shown in [5], at the Γ
point there were indicated such “allowed” transitions (see
figure 2): Γ±
i → Γ±
j for the light polarized both perpendi-
cular (ep ⊥c) and parallel (ep ‖c) to the main crystalline
c-axis, and Γ±
i → Γ∓
j for the light polarized perpendic-
ular to the c-axis only (ep ⊥c; i, j = 6.7). “Forbidden”
transitions (Γ±
i → Γ∓
j in (ep ‖c) polarization conditions)
are unallowable exactly in k0 = 0 only. Near the k0 = 0
point these transitions are allowed. Only s-excitons are
capable of being generated for the“allowed” transitions,
whereas just p-excitons are capable of being generated
for the “forbidden” transitions [28]. In (ep ⊥c) polari-
zation conditions, both longitudinal and transversal ex-
citons are generated. However, the longitudinal excitons
are not generated in (ep ‖c) polarization conditions [29].
As we know, the generation of the transversal excitons
is more probable in layered crystals. Therefore, below we
considered transversal excitons only.
Let us rewrite the simplified equation (8) with Cartesian coordinates and in accordance with
the above supposition:
(
k2
x + k2
y
)
P 2f1(E) + k2
zP 2f2(E) − Γ(E) = 0. (10)
Certainly, the equation (10) describes a surface of the second order in the k-space. Therefore,
effective masses associate themselves with two semi-axes of this surface.
m⊥ =
~
2Γ(E)
2(E − E0)P 2f1(E)
=
~
2η2 (E − Eg)
(
(3 (E + δ) + ∆) (3E + 2∆) η2 − 2∆2
)
6P 2 (E − E0) (3E + 2∆)
, (11)
m‖ =
~
2Γ(E)
2(E − E0)P 2f2(E)
=
~
2η2 (E − Eg)
(
(3 (E + δ) + ∆) (3E + 2∆) η2 − 2∆2
)
2P 2 (E − E0) ((3E + ∆) (3 (E + δ) + ∆) η2 − ∆2)
. (12)
242
Excitons into one-axis crystals
Therein E0 is the energy of the corresponding band extreme at the Γ point (see equations (9).
The band energy values E were taken close to extremes E0 for identical values of a wave-vector
k → k0 and two polar angles θ = π
2
; 0. It corresponds to equations of the transversal (11) and
longitudinal (12) effective masses respectively. The band parameters and the calculated values of
energies E0, E and the effective masses m⊥, m‖ for actual bands are shown in table 1.
Table 1. The main band parameters of Zn3P2.
Eg, eV [1] ∆, eV [5] P , eVm [9] δ, eV [5] η
1.60 0.11 4.7 · 10−10 0.03 0.9971
parameter c-band hh-band lh-band so-band
E0, eV 1.60 0 -1.78 · 10−2 -1.22 · 10−1
E (θ = π/2), eV 1.61 -6.75 · 10−4 -1.80 · 10−2 -1.223 · 10−1
E (θ = 0), eV 1.61 -3.18 · 10−4 -1.81 · 10−2 -1.230 · 10−1
m⊥/m0 0.28 -0.57 -1.18 -1.11
m‖/m0 0.29 25.95 -0.53 -0.63
4. Calculations
Let us consider the model of Wannier-Mott excitons. Participation of phonons in exciton gen-
eration is of little probability due to a direct band structure of Zn3P2. So, the exciton-phonon
interaction is not considered below. Moreover, the exchange interaction and polariton effects are
not considered herewith too. It is known that an exciton spectrum and the wave functions are de-
termined from the Schrödinger equation. With spherical coordinates, this equation may be written
down for one-axis crystals as [30]:
[
−∇2 − 2
r
1√
1 − α cos2 θ
− E
]
Ψ (r, θ, ϕ) = 0, (13)
where parameter of anisotropy α is calculated from the relation
α = 1 − ε⊥µ⊥
ε‖µ‖
. (14)
The physical sense has a module of the parameter of anisotropy. Two marginal cases are the
maximal anisotropy if | α |−→ 1 and the full isotropy if | α |−→ 0.
Equation (13) is written in the atomic units of length aB⊥ = 4πε0~
2√ε‖ε⊥/(µ⊥e2) (effective
Bohr radius), energy EB = µ⊥e4/
(
32~
2π2ε2
0ε⊥ε‖
)
(effective Rydberg), µ⊥ = me
⊥mh
⊥/(me
⊥ +
mh
⊥); µ‖ = me
‖m
h
‖/(me
‖ + mh
‖) are the transversal and longitudinal effective masses of the exciton
respectively; me
⊥; me
‖; m
h
⊥; mh
‖ are the transversal and longitudinal effective masses of electrons
(e) and holes (h) (see equations 11, 12); ε⊥; ε‖ are two dielectric constants normal to the main
crystalline c-axis and along that, respectively. In our calculation there was accepted a simple
supposition: ε‖ = ε‖∞ = n2
‖∞, ε⊥ = ε⊥∞ = n2
⊥∞ and ε⊥=15.13 [32]. From relation ∆n∞ =
n‖∞ − n⊥∞ = 0.02 [5] the longitudinal dielectric constant was found ε‖=15.28.
In a space of q dimension (qD) the Laplace operator is [31]
∇2 =
1
rq−1
∂
∂r
(
rq−1 ∂
∂r
)
+
1
r2
(
1
sinq−2 θ
∂
∂θ
sinq−2 θ +
1
sin2 θ
∂2
∂ϕ2
)
. (15)
243
D.M.Stepanchikov, G.P.Chuiko
It is possible to use the perturbation methods for solving the equation (13). This way we get
the exciton energy spectrum in the zero-order approximation.
EqD
nlm (α) = Eg − EqD
b = Eg −
EB
(
n + q−3
2
)2
Z2
lm (α) , (16)
where EqD
b is the binding energy of an exciton; n is the main quantum number; Zlm (α) is an
efficient charge depending on anisotropy.
We consider the s-type (thus l = 0; m = 0) and p-type (thus l = 1; m = 0, ±1) states of an
exciton. Under these conditions Zlm (α) may be written as [30]:
Z0.0 (α) = I1 (α) , (17)
Z1.0 (α) =
3
2
[
1
α
I1 (α) − I2 (α)
]
, (18)
Z1,±1 (α) =
3
2
[(
1 − 1
2α
)
I1 (α) +
1
2
I2 (α)
]
. (19)
Here the coefficients I1 (α), I2 (α) are determined by piecewise-continuous expressions:
I1 (α) =
1√
α
arcsin (α) , α > 0,
1, α = 1,
1
√
|α|
Arsh
(
√
|α|
)
, α < 0,
(20)
I2 (α) =
√
1 − α
α
. (21)
Free excitons are capable of generating themselves owing to the direct optical transitions be-
tween the valence and conduction bands. Therefore, we expect to observe strong optical absorption
lines at energies close to Enlm (α). These will appear in the optical spectra at energies below the
fundamental band gap. Absorption efficiency is defined by the oscillator strength (fcv). For an
“allowed” transition (s-exciton states only) the oscillator strength may be determined as [30,28]:
f3D
cv =
2Ω0
πa3
Bm0Ecv
1
n3
|〈c |ep| v〉|2 ; f2D
cv =
2a2
πa2
Bm0Ecv
1
(n − 1/2)
3
|〈c |ep| v〉|2 . (22)
In addition, for a “forbidden” transition (p-exciton states only), the oscillator strength may be
determined as [28]:
f3D
cv =
8Ω0β
2
0
3πa5
Bm0Ecv
n2 − 1
n5
|〈c |ep| v〉|2 ; f2D
cv =
8a2β2
0
3πa4
Bm0Ecv
(n − 1/2)
2 − 1
(n − 1/2)
5
|〈c |ep| v〉|2 , (23)
where Ω0 = a2c is the volume of unit cell, β0 = 3.0490Å [6] is the distance between two nearest
zinc atoms, Ecv = ~ωcv = Enlm − Evb is the resonance photon energy (the difference between
exciton Enlm and valence Evb energy levels), |〈c |ep| v〉| – the optical matrix element.
5. Results and discussion
In our exciton energy calculations, the spin-orbit interaction was neglected. These assumptions
allow us to consider the two exciton series that correspond to the heavy holes and light holes
valence bands (hh, lh) and conduction band (c). Table 2 presents the main parameters of excitons,
i.e. the relative effective mass (µ), the Bohr radius (aB) and the parameter of anisotropy (α). It is
noted that hh→c transition is characterized by maximal anisotropy (|α| → 1).
244
Excitons into one-axis crystals
It is noteworthy that both anisotropy of the dispersion law plus the existence of crystalline
packets (layers) normal to the main c-axis, will allow us to consider not only the volume 3D-
excitons but also two-dimensional 2D-excitons thereto [30,28]. The display of high temperature
phenomena caused by 2D-excitons is not eliminated in these crystals. The calculated values of the
binding energies (Eb) as well as the strength of oscillators for the optical transitions (fcv) are given
in tables 3, 4 both for a volume (3D) and for a plane (2D) excitons.
Figure 3. Schematic diagram of the localized level positions and transitions in Zn3P2.
Table 2. The main exciton parameters of Zn3P2.
Transition µ⊥/m0 µ‖/m0 aB⊥, Å aB‖, Å α
hh→c 0.58 0.28 13.9 28.7 -1.04
lh→c 0.37 0.62 21.5 12.9 0.41
Table 3. Exciton binding energy.
exciton states
hh→c-bands, meV lh→c-bands, meV
3D 2D 3D 2D
1s 26.2(ep ⊥c) 104.7(ep ⊥c) 25.8 103.3
2s 6.5(ep ⊥c) 11.6(ep ⊥c) 6.4 11.5
2p0 5.3(ep ‖c) 9.5(ep ‖c) — —
2p± 7.2(ep ‖c) 12.8(ep ‖c) — —
The data of tables 3, 4 show that the “allowed” transitions of an exciton with n=1 jointly
provide the greatest contribution to optical effects. For the main exciton state, the binding energy
is equal or exceeds the average heat motion energy (k0T ≈ 25.8 meV) at room temperature.
Therefore, the cutting peaks of the absorption or luminescence in the optical gap will be expected
even at room temperature. This statement is also true for 2D-exciton spectrum either in the bulk
crystals or in the thin films and even with some strengthening.
A model of localized level positions and transitions in Zn3P2 determined by using our calcula-
tions is presented in figure 3. The energy-transition values listed in table 5 can be considered to
be relatively well confirmed by data from different experiments.
245
D.M.Stepanchikov, G.P.Chuiko
Table 4. The optical transitions oscillator strength.
exciton states
hh→c-bands lh→c-bands
3D 2D 3D 2D
1s 3.6 · 10−2 3.7 · 10−1 2.4 · 10−2 3.8 · 10−1
2s 4.5 · 10−3 1.3 · 10−2 2.9 · 10−3 1.3 · 10−2
2p0 2.1 · 10−4 4.6 · 10−4 — —
2p± 2.1 · 10−4 4.6 · 10−4 — —
Table 5. Energy values (in eV) of transition in Zn3P2 band structure.
“allowed” “forbidden” all values from
transitions∗ transitions∗ experiments∗∗
t1 1.38; 1.31 1.40; 1.39 1.23 PL [22]; 1.28 A [33];
t2 1.40; 1.39 1.41; 1.40 1.31 P [21]; 1.32 A (ep ‖c) [10];
t3 1.28; 1.21 1.30; 1.29 1.34 P (ep ‖c) [10];
t4 1.30; 1.29 1.31; 1.30 1.36 A (ep ⊥c) [10];
A1 →c 1.41 — 1.37 P (ep ⊥c) [10]; 1.41 P [21];
A2 →c 1.31 — 1.44 PC [34]
t5 1.57; 1.49 1.59; 1.58 1.46 A [10]; 1.52 R (ep ‖c) [10];
t6 1.59; 1.58 1.595; 1.59 1.54 R (ep ⊥c) [10]; 1.55-1.60 A [11];
t7 1.59; 1.51 — 1.59 A (ep ⊥c) [5];
t8 1.61; 1.60 — 1.59 R (ep ⊥c) [5]; 1.60 PL [22];
hh→c 1.60 (ep ⊥c) 1.60 (ep ‖c) 1.60 PC (ep ⊥c) [5]; 1.60 P [21];
lh→c 1.62 — 1.61 PC [34];1.62 A [5];1.63 PC,R [5]
∗the first numeral is energy for 3D excitons; the second numeral is energy for 2D excitons.
∗∗A, P, R, PL, PC denotes absorption, photovoltage, reflectivity, photoluminescence and
photoconductive measurements, respectively.
6. Conclusions
Our calculations have shown a high binding energy of an exciton with n = 1. Moreover, it is
noteworthy that both anisotropy of the dispersion law plus the existence of crystalline packets (lay-
ers) normal to the main optical axis, will make possible the existence of two-dimensional excitons
too. So, the high temperature displaying of exciton effects is not eliminated even into bulk crystals.
Comparison of available experimental data with our theoretical results testifies to an acceptable
correlation between them. A series of optical experimental data from different sources may be sim-
ply and uniformly explained by using our calculations that are defined by 3D or 2D-excitons. This
provides some argumentation for the applied modelling approach as well as for all consequences
following from that.
The offered model of energy transitions makes it possible to provide a simple explanation of
many problem-solving situations, which were earlier explained by means of artificial suggestions
about indirect band structure of Zn3P2. Under low temperature conditions, this model can be
considered as a four-level scheme of the stimulated exciton radiation. The full process of exciton
generation and annihilation can occur according to the scheme: v-band→ c-band→ exciton level→
acceptor or phonon oscillator level. The acceptor or phonon energy levels are par excellence occupied
246
Excitons into one-axis crystals
at a room temperature. However, at a low temperature these levels are loose. The automatic
inversion of populating the upper and lower levels exists. Such a suggestion appears to be of
interest both from practical and theoretical viewpoints and requires additional studies.
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247
D.M.Stepanchikov, G.P.Chuiko
Екситони в одновiсних кристалах фосфiду цинку (Zn3P2)
Д.М.Степанчиков, Г.П.Чуйко
Херсонський нацiональний технiчний унiверситет, кафедра загальної та прикладної фiзики,
науково-дослiдна лабораторiя теорiї твердого тiла, Бериславське шосе, 24, 73008, Херсон, Україна
Отримано 18 травня 2008 р., в остаточному виглядi – 18 травня 2009 р.
Представлено теоретичне дослiдження екситонних спектрiв для кристалiв Zn3P2. Енергетичний
спектр екситонiв отримано у нульовому наближеннi теорiї збурень з урахуванням анiзотропiї закону
дисперсiї та правил вiдбору. Зафiксовано прояв двох серiй екситонних станiв, якi вiдповiдають двом
валентним зонам (hh, lh) i зонi провiдностi (c). Гранична анiзотропiя закону дисперсiї та шарува-
та кристалiчна структура дозволяють очiкувати на квазiдвовимiрний характер екситонiв. Наслiдком
цього може стати високотемпературний прояв екситонних ефектiв, навiть в об’ємних кристалах.
Розраховано значення енергiї зв’язку та сили осцилятора оптичного переходу для об’ємних (3D)
i двовимiрних (2D) екситонiв. Запропоновано модель екситонних переходiв i чотирирiвневу схему
стимульованого екситонного випромiнювання для отримання лазерного ефекту.
Ключовi слова: екситони, фосфiд цинку, одновiснi кристали
PACS: 71.18.+y
248
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