Statistical model analysis of local structure of quaternary sphalerite crystals
At the 2004 Ural International Winter School, we introduced the statistical strained tetrahedron
 model and discussed ternary tetrahedron structured crystals. The model allows one to interpret
 x-ray absorption fine structure (EXAFS) data and extract quantitative information on ion s...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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| Цитувати: | Statistical model analysis of local structure of quaternary
 sphalerite crystals / B.V. Robouch, A. Kisiel, A. Marcelli, E.M. Sheregii, M. Cestelli Guidi, M. Piccinini, J. Polit, J. Cebulski, A. Mycielski, V.I. Ivanov-Omskii, E. Sciesinska, J. Sciesinski, and E. Burattini // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 291-303. — Бібліогр.: 40 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860228352268304384 |
|---|---|
| author | Robouch, B.V. Kisie, A. Marcelli, A. Sheregii, E.M. Cestelli Guidi, M. Piccinini, M. Polit, J. Cebulski, J. Mycielsk, A. Ivanov-Omski, V.I. Sciesinska, E. Sciesinski, J. Burattini, E. |
| author_facet | Robouch, B.V. Kisie, A. Marcelli, A. Sheregii, E.M. Cestelli Guidi, M. Piccinini, M. Polit, J. Cebulski, J. Mycielsk, A. Ivanov-Omski, V.I. Sciesinska, E. Sciesinski, J. Burattini, E. |
| citation_txt | Statistical model analysis of local structure of quaternary
 sphalerite crystals / B.V. Robouch, A. Kisiel, A. Marcelli, E.M. Sheregii, M. Cestelli Guidi, M. Piccinini, J. Polit, J. Cebulski, A. Mycielski, V.I. Ivanov-Omskii, E. Sciesinska, J. Sciesinski, and E. Burattini // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 291-303. — Бібліогр.: 40 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | At the 2004 Ural International Winter School, we introduced the statistical strained tetrahedron
model and discussed ternary tetrahedron structured crystals. The model allows one to interpret
x-ray absorption fine structure (EXAFS) data and extract quantitative information on ion site occupation
preferences and on the size and shape of each elemental constituent of the configuration tetrahedra.
Here we extend the model to cover quaternary sphalerite crystal structures. We discuss the
two topologically different quaternary sphalerite systems: the pseudo balanced A₁₋xBxYyZ₁₋y (2:2
cation:anion ratio), and the unbalanced AxBx 
C₁₋x₋x 
Z or AXyYy 
Z₁₋y₋y (3:1 or 1:3 cation:anion ratios)
truly quaternary alloy systems. These structural differences cause preference values in pseudo
quaternaries to vary with the relative contents, but to remain constant in truly quaternary compounds.
We give equations to determine preference coefficient values from EXAFS or phonon spectra
and to extract nearest-neighbour inter-ion distances by EXAFS spectroscopy. The procedure is illustrated
and tested on CdMnSeTe, GaInAsSb, and ZnCdHgTe quaternary alloys.
|
| first_indexed | 2025-12-07T18:20:31Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3, p. 291–303
Statistical model analysis of local structure of quaternary
sphalerite crystals
B.V. Robouch1, A. Kisiel2, A. Marcelli1, E.M. Sheregii3, M. Cestelli Guidi1, M. Pic-
cinini1,4, J. Polit3, J. Cebulski3, A. Mycielski5, V.I. Ivanov-Omskii6, E. Sciesi~nska7,
J. Sciesi~nski7, and E. Burattini8
1INFN-Laboratori Nazionali di Frascati, Via E. Fermi, 40, Frascati I-00044, Italy
E-mail: robouch@Lnf.infn.it
2Instytut Fizyki, Universytet Jagiellonski, Reymonta, 4, Krakow 30-059, Poland
3Institute of Physics, University of Rzeszow, Rejtana, 16A, Rzeszow 35-310, Poland
4Dipartimento Scienze Geologiche, Universita Roma Tre, L. go S.L. Murialdo, Rome 00146, Italy
5Instytut Fizyki PAN, Al. Lotnikow, Warszawa 32/46, Poland
6A.F. Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia
7Institute of Nuclear Physics, Krakow, Poland
8University of Verona, Department of Informatics, str. Le Grazie 15, Verona 37134, Italy
Received October 4, 2006
At the 2004 Ural International Winter School, we introduced the statistical strained tetrahe-
dron model and discussed ternary tetrahedron structured crystals. The model allows one to interpret
x-ray absorption fine structure (EXAFS) data and extract quantitative information on ion site occu-
pation preferences and on the size and shape of each elemental constituent of the configuration tetra-
hedra. Here we extend the model to cover quaternary sphalerite crystal structures. We discuss the
two topologically different quaternary sphalerite systems: the pseudo balanced A1–xBxYyZ1–y (2:2
cation:anion ratio), and the unbalanced AxBx′C1–x–x′Z or AXyYy′Z1–y–y′ (3:1 or 1:3 cation:anion ra-
tios) truly quaternary alloy systems. These structural differences cause preference values in pseudo
quaternaries to vary with the relative contents, but to remain constant in truly quaternary com-
pounds. We give equations to determine preference coefficient values from EXAFS or phonon spec-
tra and to extract nearest-neighbour inter-ion distances by EXAFS spectroscopy. The procedure is il-
lustrated and tested on CdMnSeTe, GaInAsSb, and ZnCdHgTe quaternary alloys.
PACS: 61.43. Dq Amorphous semiconductors; metals and alloys.
Keywords: sphalerite quaternary A1–xBxYyZ1–y, AxBx′C1–x–x′Z, AXyYy′Z1–y–y′ systems; local crystal
structure; statistical model.
1. Introduction
In recent years interest in multinary sphalerite
structured semiconductors has developed rapidly,
moving from binary to ternary, then to quaternary
systems, as an extra ion-component leads to an addi-
tional degree of freedom in controlling material pa-
rameters. Indeed, the substitution of one cation by an-
other in the cation sublattice, and the substitution of
one anion by another in the anion sublattice, induces a
reconstruction of the electronic structure and the
phonon spectra as the composition is varied, leading to
almost unique property variations. As observed, ion
substitution in sphalerite crystals exhibits site occupa-
tion preferences (SOPs) linked to the thermodynamic
properties of the creation of the quaternary system.
We recall here that the elemental structure of a
sphalerite (zincblende) binary AZ compound is a regu-
© B.V. Robouch, A. Kisiel, A. Marcelli, E.M. Sheregii, M. Cestelli Guidi, M. Piccinini, J. Polit, J.Cebulski, A. Mycielski, V.I. Iva-
nov-Omskii, E. Sciesi~nska, J. Sciesi~nski , and E. Burattini, 2007
lar tetrahedron with alternatively at the center an A
(or a Z) ion, and at its four vertices the other ion Z (or
A). Alloying two binary compounds such as AZ + BZ
(or AZ+AY) leads to the formation of different ter-
nary A1–xBxZ (or AYyZ1–y) systems with A ions being
progressively substituted by B ions (or Z by Y). Thus
one of the sublattices remains homogeneously mono-
ion, while the other is modulated by the two compet-
ing ions. At the same time the tetrahedron is distorted,
becoming strained (see Fig. 1 in [1]), whence the
name of strained tetrahedron model introduced to
treat these systems.
In the following, we use the same nomenclature in-
troduced in [1] and used since [2–6] and define cat-
ions by A, B, C, anions by X, Y, Z, and the relative
contents of site competing ions by x, x′ for cations,
and by y, y′ for anions. We thus write AZ for a binary
compound, A1–xBxZ or AYyZ1–y for a ternary system,
and either A1–xBxYyZ1–y (2:2 cation:anion ratios, re-
ferred to as Q22) or A1–x–x′Bx′CxZ or AXyYy′Z1–y–y′
(with respectively (3:1 or 1:3 cation:anion ratios) for
quaternary systems. We shall refer to them, respec-
tively, as Q31 and Q13 systems.
We immediately note that configuration-struc-
ture-wise, the quaternary systems are of two different
natures: the Q31 and Q13 are quaternaries of the «un-
balanced» truly-quaternary (truly-Q) type with re-
spectively a 3:1 or 1:3 cation:anion ratio. Here, of the
possible fifteen elemental tetrahedron configurations,
ion-occupation-wise, three are binary, nine ternary,
and three quaternary; indeed, the elementary tetrahe-
dron contains respectively two, three, and four differ-
ent ions. On the other hand, the Q22 type of quater-
nary is balanced, contains only 2:2 cation:anion
species, and can be considered pseudo-quaternary
(pseudo-Q). Ion-occupation-wise Q22 can have six-
teen tetrahedron configurations, of which four are
binary, and twelve ternary. None of its sphalerite tet-
rahedron configurations can canonically be «quater-
nary». Indeed, in a Q22 system, each cation (anion)
has only two anions (cations) of the opposite polarity
that can occupy the four vertex sites of the sublattice.
Implanting into a vertex position of an ion with the
same polarity as the central ion corresponds to intro-
ducing it into the opposite sublattice, i.e., creating an
antisite occupation point defect. Thus we can consider
the four ternary constituents (each with its tetrahe-
dron configurations) as dissolved in a quaternary me-
dium, with a A1–xBxYyZ1–y consisting of A1–xBxY or
AYyZ1–y, A1–xBxZ or BYyZ1–y.
The Q31 (Q13) and Q22 systems are usually chem-
ically identified as quaternary alloys. However, there
is a basic structural difference between the pseudo-Q
and the truly-Q systems. Indeed, while a truly-Q has
one of its sublattices with a homogeneously mono-
ionic population, the complementary sublattice is
modulated by the three ion species competing for each
site. This leads to a crystal structure with alterna-
tively a homogenous ion-shell, followed by a heteroge-
neous shell with site-competing ions. On the other
hand, in pseudo-Q the sublattices, and hence the suc-
cessive ion shells, are all heterogeneous, have succes-
sively anions or cations, intracompeting for site occu-
pation in that shell. While the denomination is a
consequence of the existing configurations, the occur-
rence of successive homogeneous\inhomogeneous
shell structures conditions basic differences in the
properties of the two classes of quaternaries. The topo-
logical differences between quaternary sphalerite Q31
(Q13) and Q22 structured systems impose different
statistical model approaches and thus the statistical
analysis of the pseudo and truly-Q compounds is not a
simple generalisation. This explains why, after the
Q22 compound analysis [4], an ad hoc statistical
strained tetrahedron model for quaternary sphalerite
Q31 and Q13 systems is developed in Ref. 7.
An understanding of the properties of quaternary
alloys requires accurate and systematic structural in-
vestigations using different techniques such as neu-
tron scattering, extended x-ray absorption fine struc-
ture (EXAFS) [8,9] or far infrared (FIR) [10,11]
vibrational spectroscopy. In particular the last two
methods yield respectively information on the real lo-
cal structure and on the collective internal vibrations
(phonons) of the investigated systems.
As generally reported, ternary tetrahedron coor-
dinated A1–xBxZ and AYyZ1–y systems ([12–15] for
FIR and [16–28] for EXAFS), as well as balanced
quaternaries of type A1–xBxYyZ1–y (such as GaInAsSb
[29] for EXAFS, CdMnSeTe [30] for EXAFS and
[6,31] for FIR, analyzed in [32], ZnCdSeTe [33]
for FIR), and unbalanced quaternary A1–x–x′Bx′ CxZ
(such as MnCdHgTe [34], ZnCdHgTe [35,36]), or
AXyYy′Z1–y–y′ system FIR spectra all exhibit SOPs;
such preferences are linked to the thermodynamic
properties of each system.
To extract quantitative information on the local
structure of tetrahedron configured crystals, we origi-
nally developed the statistical strained tetrahedron
model to return site preferences as well as sizes and
shapes of the elemental tetrahedron configurations
from sphalerite and some wurzite EXAFS experimen-
tal data for ternary semiconductor alloys [1,2]. Suc-
cessively, the model was expanded to treat ternary
M3(XX′)1 type intermetallide alloys [3] and then to
interpret FIR spectra of ternary sphalerite structure
[4,5]. To our knowledge no EXAFS data for truly-Q
systems are available in the literature.
292 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
For wurzite systems, the model returns a valid
analysis of EXAFS data provided the electron wave
backscattering from ions beyond the next nearest
neighbours (NNN) remains negligible with respect to
that from nearest neighbours (NN) and from NNN
ions; the same restriction applies to both balanced
pseudo-Q [6] and unbalanced truly-Q [7] sphalerite
structured systems.
The basic assumptions or axioms of the statistical
model are [1,2]:
1. The elemental tetrahedra are free to have differ-
ent sizes and shapes (Fig. 1 in [1,2]).
2. Bernoulli binomial polynomials with preference
weight coefficients describe ion-pair and configuration
populations.
3. All NNNs and further fills are determined by NN
preferences.
4. The total coordination number is conserved
when dilution varies (this defines the range of values
with physical meaning and imposes bounds on coeffi-
cient values).
5. The formation of each ternary or quaternary ele-
mental configuration proceeds, respecting mass conser-
vation, until complete exhaustion of one of the binary
ingredients, or of both simultaneously as in the random
case. This axiom is crucial to understand and interpret
SOP estimations returned from EXAFS and FIR analy-
sis. Then to interpret EXAFS distance data [1]:
6. All nineteen elemental inter-ion ternary dis-
tances are tetrahedron constrained.
7. Elemental volumes of the two sublattices, for
each of the three strictly ternary configurations, relax
pair-by-pair to common values (i.e., three volume re-
laxation constraints [VRC] on the distance parame-
ters). For FIR spectra, the lines issued from the differ-
ent configurations by a given ion ij-dipole have an
invariant [4,5].
6′. Line width shape ij � and
7′. Oscillator strength ij s.
We compare the statistical models of balanced and un-
balanced quaternary compounds and quantify preferences
using both EXAFS and FIR experimental data. We at-
tempt to extract NN inter-ion distances from EXAFS
data. Finally, we apply the model to balanced quaternary
alloys CdMnSeTe, GaInAsSb and the available unbal-
anced quaternary ZnCdHgTe phonon spectra [35,36].
To facilitate comprehension of the text, we confine
all the equations to the appendix, but refer to the pub-
lished articles where they are derived and presented in
extensio so that the reader can interpret EXAFS data
or FIR vibrational spectra and extract quantitative in-
formation on SOP coefficients, or for just EXAFS, the
elemental configuration distances between NN and
NNN ions.
2. General considerations
Sphalerite (zincblende) systems are face centered
cubic (fcc) structure with tetrahedron coordinated
configurations with an anion (cation) at the center
and a cation (anion) at each of the four vertices. De-
pending on whether we have at the vertices one, two,
or three types of ions (with the extra ion type at the
center), the systems are respectively binary, ternary,
or quaternary. In all cases, the cations form one of the
two crystal sublattices, while the anions constitute
the complementary sublattice shifted by a 1/4 of the
length of the main diagonal of the elemental cube.
A binary AZ compound consists of regular tetrahe-
dra T a0( )AZ , is defined by its lattice constant aAZ
with inter-ion distances AZ
AZ
AA4;d a / d/
0
1 2
03� �
� �ZAZ
AZd a / /
0
1 22 , and its first and second shells
are strict spheres, one of cations, the other of anions;
thus, determining aAZ by interferometer spectrome-
try, defines the distribution and elemental dimen-
sions.
A ternary A1–xBxZ (or AYyZ1–y) system is ob-
tained by a progressive substitution of A by B ions (or
Z by Y) in the configuration tetrahedra { } ,Tk k�0 4,
where k stands for the number of substituting ions B
(or Y) at the vertices. Still, one of the two sublattices
remains strictly homogeneous mono-ionic, while the
other is modulated by the two competing ions. Com-
petition is conditioned by the preference of the one
competing ion to the other; thus the ion distribution
departs from pure randomness. At the same time, since
competing ions have different radii, the tetrahedron
becomes distorted i.e., strained (Fig. 1 in [1]).
A quaternary sphalerite structured semiconductor
system results from a further substitution of yet ano-
ther ion into a ternary alloy. Such an intrusion may or
may not keep one of the two sublattices strictly
homogeneous mono-ionic. When AxBx′ C1–x–x′Z or
AXyYy′Z1–y–y′ are generated, the «unbalanced» Q31
and Q13 have one sublattice mono-ionic homogeneous,
while the second is modulated by the three competing
ions and hence are truly-Q. Their configuration tetra-
hedra are { }, , ; ,Tk j k j k� � �0 4 0 4 , where k and j stand for
the number of substituting ions A and B (or X and Y)
at the vertices. On the other hand, in a A1–xBxYyZ1–y
system with a balanced Q22 with 2:2 cation:anion ratio
competition occurs in both sublattices. Indeed, a Q22
system cannot (excluding antisite point defects) have a
canonical quaternary configuration and hence is a
pseudo-Q [6]. Thus a Q22 A1–xBxYyZ1–y is a mixture
of four ternary systems, each with a distinct c-ion
(A,B,Y, or Z) at the center; their configuration tetra-
hedra are { } ,, ;c k kT � �03 c A,B,Y,Z where k stands for the
number of substituting ions Y (or B) at the vertices.
Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 293
In the absence of SOPs, ion distribution through-
out the configuration sites is random. The ion distri-
bution in the successive shells with N sites (4 in the
NN shell, 12 in the NNN shell) around the central
c-ion are described by Bernoulli binomials.
In the case of two competing ions (occupation of N
sites by mA ions and hence ( )N m� B ions with rela-
tive contents x and 1 � x), by
{ ( ) ( ) }[ ]
,pm x C x xN
m
N m N m
m N� � �
�1 0 (1)
with C N / m N mm
N � �! [ !( )!] the binomial coeffi-
cient and
{ ( )}[ ]
,
pm xN
m N
�
�
� 1
0
; { ( )}[ ]
,
mp x Nxm
N
m N
�
�
�
0
.
In the case of three competing ions, as for truly-Q
systems (with occupation of N sites by kA ions, jB
ions and (N–k–j)C ions, with respective relative con-
tents x, x′, and (1–x–x′)) (see for instance [37]) by
{ ( , )[ ]p x x C x C xkj
N
k
N k
j
N k j� � � ��
� � � � � �
� � �( ) } , ; ,1 0 0x x N k j
k N j N k (2a)
a set of fifteen terms, one for each configuration, with
j kk
kjp x x
� ��
�� � �
0 40 4
1
,,
{ ( , )} while 0 1� � �( )x x .
(2b)
Whenever the system presents preferences, the distri-
bution is no longer random, and to describe probabili-
ties, SOP weight coefficients W are applied to each
Bernoulli binomial [1]. Binary weight coefficients are
evidently Wbinary � 1. Thus, a ternary with its three
truly ternary configurations and two binary requires
three { } ,Wk k�1 3 SOP coefficients [1], while the Q22,
mixture of four ternary systems, requires 12 (= 4
3)
{ } , ;c k k cW � �1 3 A,B,Y,Z coefficients (Table 1 and [6]).
The truly-Q with { } , ; ,Wkj k j k� � �0 4 0 4 coefficients, with
the binary W W W00 04 40 1� � � (Table 4), needs 12
coefficients. Since all probability coefficients are non-
negative, the SOP W coefficient values are restricted.
Because preferences imply unbalanced populations
(with respect to random, W � 1), the excess (or scar-
city) of one of the constituent ions competing for sites
with respect to stoichiometry determined populations.
Actually when the minor constituent is no more avail-
able, the excess remains as a binary. Respect to the ran-
dom case the resulting nonbinary configuration pro-
babilities are depressed each time anyW-coefficient � 1,
and the corresponding depression factor CW �1 appears,
with the ions in excess going to enhance the correspond-
ing binary populations, see Appendix 1.
Once the values of the SOP coefficients are known,
all coordination numbers (CN), see Appendix 2, and
configuration probabilities (Pternary: Pk(x); Ppseudo-Q:
cPk(x,y); Ptruly-Q: Pkj(x,y)) as functions of the respec-
tive ion relative contents (x, x′, y, y′ as appropriate for
each system) are fully determined, see Appendix 3.
From these quantities we get the expressions for the
evaluation of the elemental configuration inter ij-ion
distances ( )i
j d , derivable from EXAFS �
i
j CN and �
i
j d
(see Appendix 4), or from FIR spectra the preference
SOP coefficients of the multinary (see Appendix 5). In
the following for the sake of clarity, we will use the
above-mentioned mnemonic symbolism rather than the
traditional Rj and � j EXAFS parameters.
We now concentrate on the two quaternary systems
Q22 and Q31 or Q13 and outline the consequence on
SOPs: Pseudo-Q is a mixture of ternaries, so SOPs
drift with contents! On the other hand, truly-Q SOP
coefficient values remain invariant as the relative ion
contents vary!
3. Pseudo-quaternary A1–xBxYyZ1–y sphalerite
structure
Tetrahedron coordinated sphalerite structure qua-
ternary systems of type A1–xBxYyZ1–y consist exclu-
sively of binary and ternary elemental tetrahedra, four
of the first and four of the latter, each one with three
configurations, i.e., a total of sixteen elemental tetra-
hedron configurations (Table 1). Counting the di-
poles, we have 4 from binary + (4
3
2) from the ter-
nary configurations; i.e., potentially a total of up to
28 phonon lines in 4 basic bands (Fig. 1) of closely
overlapping lines. However, these configurations can-
not contain simultaneously all four constituent atoms
in the same elemental tetrahedron; as a consequence
we can consider each ternary tetrahedron composition
as being diluted in the quaternary compound. Thus,
A1–xBxYyZ1–y EXAFS data can be treated by using
the strained tetrahedron model which, originally de-
veloped to deal with ternary systems, has already ex-
hibited excellent agreement with numerous experi-
mental data [1,2].
Table 1. The 16 tetrahedron configurations of a quater-
nary A1–xBxYyZ1–y system: 4 binary (AY, AZ, BY, BZ) +
(4·3) ternary (A1–xBxY, A1–xBxZ, AYyZ1–y, BYyZ1–y) =16.
No pure quaternary configuration occurs.
j
B
= \ k
A
= 0 1 2 3 4
0 CZ 1A3C:Z 2A2C:Z 3A1C:Z AZ
1 1B3C:Z 1A1B2C:Z 2A1B1C:Z 3A1B:Z
2 2B2C:Z 1A2B1C:Z 2A2B:Z
3 3B1C:Z 1A3B:Z
4 BZ
294 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
To determine ion site occupation preferences of qua-
ternary systems, we applied this model to GaInAsSb
[29] data available in the literature and to our EXAFS
data for CdMnSeTe [30], and we compared the results
with those derived from ternary EXAFS data for
GaInAs [16] and CdMnTe [17,18]. In both sets, as the
ternary is diluted in the quaternary system, different
preference values respect to the pure ternary are ob-
served. The present analysis of the experimental reflec-
tivity FIR phonon spectra of quaternary CdMnSeTe
crystals [6] confirms the model predictions and leads to
a consistent interpretation of the experimental data for
A1–xBxYyZ1–y quaternary systems [6]. From Table 1,
the total number of dipole contributions is 4 from bi-
nary + (12
2) from the ternary configurations; i.e., we
expect up to 28 phonon lines in 4 basic bands (Fig. 1).
4. Truly-quaternary AxBx�
C1–x–x�
Z or
AXyYy�
Z1–y–y�
sphalerite structured systems
Tetrahedron coordinated sphalerite quaternary sys-
tems of the type AxBx�C1–x–x�Z or AXyYy�Z1–y–y� con-
sist of fifteen elemental tetrahedra, each containing at
the center one ion and at the vertices one, two, or
three competing ions, i.e., respectively binary, ter-
nary, and truly-Q in composition (Table 2). Counting
the dipoles, we have 3 from binary + (3
3
2) from the
ternary + (3
3) from the quaternary configurations;
i.e., up to 30 phonon lines in 3 basic bands (Fig. 2).
By tuning the x-ray energy to the K- (or L- or M-)
edge of one of the constituent ions «c» ( where c = {A,
B, C, or Z} of the AxBx�C1–x–x�Z quaternary) we probe
the average (over the sample and over all configura-
tions) local structure around the selected atom; one
thus may retreive: the relative number of ions of a
given type around it, i.e., ion pairs or coordination
numbers (CN), <center
vertex CN(x,x�)>; inter-ion dis-
tance center
vertex< d(x,x�)> between the selected c-ion
and the NN or NNN vertex-ions around it.
Table 2. The 15 tetrahedron configurations of the quater-
nary AxB �x C1–x– �x Z or AXyY �y Z1–y– �y system: 3 binary (AZ,
BZ, CZ) + (3·3) ternary (A1–xBxZ, C1–xBxZ, AY Zy y1– ,
BY Zy y1� ) + (1·3) quaternary (1A1B2C:Z, 2A1B1C:Z,
1A2B1C:Z ) = 15
j
B
= \ k
A
= 0 1 2 3 4
0 CZ 1A3C:Z 2A2C:Z 3A1C:Z AZ
1 1B3C:Z 1A1B2C:Z 2A1B1C:Z 3A1B:Z
2 2B2C:Z 1A2B1C:Z 2A2B:Z
3 3B1C:Z 1A3B:Z
4 BZ
The NN average coordination numbers � �
i
y x xCN( , )
are a «count» of ij-ion-pair numbers, explicitly the av-
erage number of «j» ions around a central «i» ion. For
the truly-Q structure with alternate homogeneous
mono ion shells around the «competing» ions A, B, C,
the NN shells have only Z-ions, i.e., � �
�i
Z x xCN( , ) 4,
while around the Z-ions the expressions of probabili-
ties of the «competing» A, B, C ions occupying the
four NN sites, are given in the Appendix 2. Using the
configuration probabilities { ( , )} , ; ,( )P x xkj k j k� � � �0 4 0 4
(Appendix 3) that duly account for stoichiometry, the
expressions for the average NN inter-ion distances
� �
z d x xvertex ( , ) are given in Appendix 4 [1].
To interpret far infrared reflection\transmission
spectra, the Kramers—Kronig transformation is ap-
Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 295
100 150 200 250
0
4
8
MnSe
MnTe
CdSe
Im30K.dat Sp.fit
f100.2 f140 f142.8
f145.5 f148.3 f151
f153.8 f156.6 f159.3
f162.1 f164.8 f167.6
f170.4 f173.1 f175.9
f178.6 f181.4 f184.1
f186.9 f189.7 f192.4
f195.2 f197.9 f200.7
f203.5 f206.2 f209
f211.7 f214.5 f220
0.80.20.10.9Cd Mn Se Te
�
�(
)
2
Wavenumber –1�, cm
CdTe
Fig. 1. Deconvolution of the � �2( , )T function obtained by
the KK-transform of the reflectivity spectrum of the
Cd0.9Mn0.1Se0.2Te0.8 sample measured at T � 30 K (closed
circles). Sp-fit is the obtained best fit curve; the 30 pseudo
Voigt-shaped lines are labelled «f» followed by their pho-
non frequency. Experimental �-resolution = 2 cm–1; �2
uncertainty �0.06; error bars are shown. The four main
phonon bands [6] are addresed.
50 100 150 200
0
40
80
Possible ( )
Hg vacancies
ZnTe
CdTe
HgTe
KKdata Fit
L1 C3
C4 C5
C6 C7
C8 C9
C10 L10
C12 C13
C14 C15
C16 C17
C18 L18
C
Wavenumber –1�, cm
�
(
)
�
2
0.750.130.12Zn Cd Hg Te
Fig. 2. Deconvolution of a Zn0.12Cd0.13Hg0.75Te spectrum
(using Lorentzian line shapes). Experimental �-resolution
= 2 cm–1; �2 uncertainty bars = �0.63; error bars are shown.
Note the three main bands, and possibly Hg-vacancy point
defects [7].
plied (carrefully as the experimental �-range is fi-
nite), to obtain the corresponding complex dielectric
constant � �( , , )x x� , whose general expression is [10,11]:
� � � � � � �( , , ) { [( ) ]}
,
x x S / ij j
j n
j j� � � � ��
�
� 2 2
1
2 �
� � � ��� � � � � �� � � �plasma
2 / i x x i x x[ ( )] ( ) ( , , )� 1 2
(3)
where � �� �1( )x x� , and � �� �2( )x x� are respectively, the
real and imaginary parts of the dielectric constant;
� �( , , )x x� . In the equation, the first term �� is the optic
dielectric constant; the second term represents contribu-
tions by lattice and defect vibration modes, with � j the
frequency of the phonon mode, �j the damping con-
stant (or line half-width), and Sj the (dimensionless)
oscillator strength (OS) of the spectrum component n
lines; the third, the Drude term is related to plasma
contribution with �plasma the plasma frequency, and
� � 1/� , where � is the scattering time of free carries,
apparent in semimetal alloys. The line profiles are
assumed to be either Lorentzian, ( ( )yL � �
� � s /L L L L� �2 2 2
0
2 2 2 2� � � �[( ) ] [10]), indicated at
low temperatures, or whenever � �L G�� or Gaussian
( ( ) exp { [( ) ] }y s /G G G G� � �� � � 0
2� whenever the
experimental line profile half width is greater than the
natural spectral half width � �G L�� ; or a linear com-
bination of both, referred to as pseudo-Voigt line
shaped, y y yG LVoigt ( ) ( )� � �� �1 whenever �L �
� �G ; with 0 1� �� , pseudo-Voigt covers both extreme
and intermediate line shapes, leading to better quality
best fits. A detailed comparative study of the spectral
fit by Gaussian, Lorentzian, or pseudo-Voigt line shapes
can be found in Ref. 40. Each line is associated to an
ij-dipole ( ,i j� �A,B, or C Z, Y, or X) of the different
configurations: one phonon from each of the binary
configurations, two phonons from each of the three ter-
nary configurations, and three phonons from each of the
quaternary (see Appendix 5). The total number of
phonon lines and the corresponding number of mode
bands is particular to each system. The oscillator
strength of each phonon is determined by the configura-
tion probabilities P xk( ) (taking into account the SOPs,
see Appendix 3) and the specific oscillator strengths of
the binary compounds involved. The explicit expression
of the equation for � �2( ) is given in Appendix 5.
5. Discussion and conclusions
Experimental data of EXAFS and FIR of quaternary
systems are rather scarce. We retrieved and processed
EXAFS data of pseudo-Q GaInAsSb [29] (Fig. 3 and
Table 3) and our previous data of CdMnSeTe [30]
296 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
Table 3. Ga1–xInxAsySb1–y [6]: comparison of NN values for GaInAs(as T) [1,17] with those of GaInAs(in Q).
GaInAs
y
Sb
1–y
cTk
c = As k = 1
AsWk
2 3 1
AsCk
2 3 k = 1
As
Gadk
2 3 1
As
In dk
2
[�]
3
GaInAs
(in Q)
y
As
= 0.05
x
Ga
= {0.2, 0.5, 0.65,
0.8, 0.95}
1.88 1.54 1.32
0.71 0.46 0.05
2.81 2.45 — 2.66 2.53 �
y
As
= 0.10
x
Ga
= {0.1, 0.5, 0.9}
1.93 1.59 1.33
0.69 0.41 0
2.84 2.34 — 2.65 2.49 �
both together 1.89 1.56 1.33 0.70 0.44 0 2.80 2.44 — 2.65 2.52 �
GaInAs
(as T)
1.05 0.25 0.58 0.85 0.25 0.58 2.49 2.42 2.48 2.59 2.60 2.61
0 0.5 1.0
0
4
N
N
<
A
sG
a C
N
>
<
A
sIn
C
N
>
x Ga
Fig. 3. Analysis of Ga1–xInxAsySb1–y EXAFS
�
�CN Ga As K edge( )x data [31]. Experimental points (trian-
gles up y � 01. , triangles down y � 02. ); derived curves
�
As
Ga
GaCN( )x (solid); �
As GaCNln ( )x (dash dotted). Random
limits (dotted).
(Fig. 4 and Table 4), a system for which also the FIR
reflectivity spectra are available [6]. From these the
imaginary part of the respective dielectric functions
� �2( ) were obtained (Fig. 1), and the data interpreted
using the strained tetrahedron model. To our knowl-
edge no truly-Q EXAFS data are available in the litera-
ture. ZnCdHgTe FIR spectra are available [35,36] and
the analysis [7], partially presented here (Fig. 2) is
still in progress. Note that using pseudo-Voigt line
shapes gives a better fit (Fig. 1), than using Lorentzian
(Fig. 2), as observed in Refs. [6,38].
Table 4. Cd1–xMnxSeyTe1–y: comparison of NN values for
CdMnTe(as T) [1,18,19] with those of CdMnTe (in Q) [6].
Ternary cTk
c k = 1
cWk
2 3 k = 1
cCk
2 3
MnSeTe
(in Q)
Mn 3.23
3.22
3.23 0
0
0.20
0.40 0.26 0
0
0.40
0.20
CdMnSe
(in Q)
Se 2.18 0 1.33 0.54 0 0
CdSeTe
(in Q)
Cd 2.38 0 1.33 0.54 0 0
CdMnTe
(in Q)
CdMnTe
(as T)
Te 3.23
0.68
0
1.33
0
0
0.26
0.68
0
0.67
0
0
Quaternary sphalerite crystals have two distinct
system structures — pseudo-Q and truly-Q — and re-
quire distinct ad hoc theoretical treatments. The topo-
logical structure of the ideal, canonical crystal (i.e.,
with no point defects or impurities) attributes respec-
tively 28 and 30 basic phonon frequencies. In a real
crystal, due to defects and impurities the number of
frequencies further increases (see, for instance,
[39,40]). We discussed EXAFS and phonon spectra of
pseudo-Q alloys (GaInAsSb [29] (Fig. 3 and Table 3)
and our data of CdMnSeTe [30] (Fig. 4 and Table 4))
and the phonon spectra of truly-Q ZnCdHgTe. Three
of the four Q22 CdMnSeTe samples investigated by
EXAFS, have been re-examined with FIR reflectivity
[6], and the main results are summarized below.
A pseudo-Q zincblende system responds as a linear
superposition of the contributions of binary and ternary
elemental tetrahedra of the four ternary components, as
shown by the analysis of both EXAFS and FIR phonon
spectra. Thus, the strained tetrahedron model, designed
to interpret ternary tetrahedron coordinated systems,
successfully interprets A1–xBxYyZ1–y systems. Most se-
miconductors exhibit SOPs, as experimentally con-
firmed yielding quantitative SOP coefficient values reli-
ably using different models. The strained tetrahedron
model gives a good agreement for both EXAFS data
[1,2] and FIR spectra [4,5] of sphalerite ternary
A1–xBxZ and AYyZ1–y, and now to the Q22
A1–xBxYyZ1–y systems [6]. It also successfully inter-
prets EXAFS data of intermetallic M3(XX�) [3] sys-
tems, and within certain limitations may do so for some
wurzite crystalline structures [2], returning SOP coeffi-
cient values. As a ternary is progressively diluted within
the Q22, the evolution of the SOP-coefficients is a valu-
able index of the thermodynamic evolution of the sys-
tem. Indeed, the heterogeneous presence of competing
ions in the NNN shell profoundly modifies the SOPs
with respect to the corresponding pure ternary (which
has a perfectly homogeneous NNN shell composition),
and the system is affected by the site occupation compe-
tition in both the cation and the anion sublattices, i.e.,
all shells, hence, SOPs conditioned by relative concen-
trations. Thus in contrast with SOPs of a pure ternary
Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 297
0 0.1 0.2 0.3 0.5 1.0
0
2
4
<
M
nS
e C
N
(y
S
e
)>
,
<
M
nTe
C
N
(y
S
e
)>
y
Se
Mn
Se
CN
Mn
TeCN
0 0.1 0.2 0.3 0.5 1.0
0
2
4
Se
Cd
CN
Se
Mn
CN
<
S
eM
n C
N
(x
M
n) >
,
<
S
eC
d
C
N
(x
M
n)>
x
Mn
Fig. 4. Analysis of Cd1–xMnxSeyTe1–y EXAFS �
CN data
[32]. Experimental points: triangles pointing (a) down:
Mn
Se CN, up: Mn
Te CN, (b) left: Se
CdCN, right: Se
MnCN; de-
rived �
CN curves, within the sphalerite range (solid),
(dotted) beyond the sphalerite range. Random limit (dash
dotted).
for GaInAs(as T) in which the NNN ions around any
cation are all As-anions, in GaxIn1–xAs0.05Sb0.95,
GaInAs(in Q) they are almost all Sb-anions. In Q22 and
in ternary systems, line shapes and intensities are deter-
mined, to a first approximation, by the cation-anion di-
pole pair, and are unaffected by the center-vertex posi-
tion of the dipole within the tetrahedron configuration.
Indeed, observing the normalized variance s2 values of
successive best-fit results (Table 3 [6]) confirms our as-
sumption that the two phonons emitted by a dipole ZA
from the ternary AxB1–xZ and by AZ from the comple-
mentary AYyZ1–y, are substantially equivalent.
In contrast with Q22, the Q31 and Q13 systems
have SOP coefficients invariant with relative contents
values. Indeed, a truly-Q system, Q31 or Q13, is char-
acterised by three ions competing for site occupation
in a shell bounded on both sides by shells with homo-
geneous mono-ions of the complementary sublattice.
Many semiconductors exhibit SOPs with extreme
values. Such a behavior indicates that the ternary fill-
ing of one (sometimes two) of the three elemental tet-
rahedron configuration(s) is almost negligible. This
may be explained by a thermodynamic affinity of cat-
ion–anion pair components in ternary or quaternary
systems. A wide group of semiconductors is character-
ized by missing elementary configurations, a behavior
that is still not understood.
Noteworthy, Q22 FIR spectra exhibit a trend ver-
sus temperature, that could be correlated to the
EXAFS analysis of these materials at different temper-
atures and in particular to the behavior of the
Debye-Waller parameters.
Our analysis shows that one of the major conse-
quences of the difference among SOPs in truly-Q and
pseudo-Q is due to their structural difference. In fact
as the relative ion contents vary, in a truly-Q the
12 SOP coefficient values remain constant, while in
a pseudo-Q system, i.e., a mixture of ternaries, the
12 SOP values are functions of the relative contents.
Work is in progress to determine from available
FIR-spectra [7] the values of the twelve SOP-coeffi-
cients for the truly-Q ZnCdHgTe similarly to the
work performed in the ternary GaAsP [4,5].
Part of this work was supported by EU TARI-pro-
ject contract HPRI-CT-1999-00088.
298 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
Appendix (vademecum of Sphalerite system equations)
We here reassume the equations based on the assumptions of the statistical strained tetrahedron model as de-
rived in extenso in manuscripts [1–6]. The corresponding parameters are indicated.
Appendix 1. SOP W- and C-coefficients [1, 6]
All probability coefficients being nonnegative, restricts SOP W-coefficients values.
Preferences affect configuration probability C-coefficients as follows
System
ternary [1]
3 {W
k
}
k=1.3
parameters with
{0 �W
k
� 4/k}
k=13.
0 � {C
k
= min [W
k
,1,(4 – kW
k
)/(4 – k)] }
k=1.3
� 1
pseudo quaternary [6]
12 {
c
W
k
}
k=1.3;c=A,B,Y,Z
parameters with
{0 �
c
W
k
�4/k}
k=1.3;c=A,B,Y,Z
0 � {
c
C
k
= min [
c
W
k
,1,(4 – k
c
W
k
)/(4 – k)] }
k=1.3;c=A,B,Y,Z
� 1
truly quaternary
and for pure Q – configurations
12 {W
kj
}
k=0.3;j=0.3–k
with
0 �{(k+ j)W
kj
}
k=0.3;j=0.3–k
� 4, 0 � {kW
k,(4–k)
}
k=1.3
� 4,
and 0�W
2.1
�4/3, 0 �W
1.2
� 4/3 , 0 � W
1.1
� 2
0 � C
kj
= min {W
kj
, 1, [4 – (k+ j)W
kj
]/(4 – k – j)} �1
Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 299
Appendix 2. EXAFS – coordination numbers
equations defining the average NN coordination numbers
Ternary A1–xBxZ (or AYyZ1–y) [1]
Parameters 3: {Wk}k=1.3
<
A
ZCN (x)> = <
B
ZCN (x)> = 4
<
Z
BCN(x)> = �
k=0.3
�
j=1.4–k
[( j W
kj
) p
kj
(x)]
<
Z
ACN(x)> = �
k=1.4
�
j=0.4–k
[( k W
kj
) p
kj
(x)] = 4 – <
Z
BCN(x)>
Pseudo quaternary A1–xBxYyZ1–y [6]
Parameters 12: {cWk}k=1.3;c=A,B,Y,Z
NN dipole <CN(x,y)>
Pseudo quaternary
A
1–x
B
x
Y
y
Z
1–y
AZ <
A
ZCN (y)> = �
k=0.3
[(4 – k
A
W
k
)p
k
[4](y)] = 4 – <
A
YCN(y)>
<
Z
ACN (x)> = �
k=0.3
[(4 – k
Z
W
k
)p
k
[4](x)] = 4 – <
Z
BCN(x)>
BZ <
B
ZCN (y)> = �
k=0.3
[ (4 – k
B
W
k
)p
k
[4](y)] = 4 – <
B
YCN(y)>
<
Z
BCN(x)> = �
k=1.4
[ k
Z
W
k
p
k
[4](x)] = 4 – <
Z
ACN(x)>
AY <
A
YCN(y)> = �
k=1.4
[ k
A
W
k
p
k
[4](y)] = 4 – <
A
ZCN(y)>
<
Y
ACN(x)> = �
k=0.3
[ (4 – k
Y
W
k
) p
k
[4](x)] = 4 – <
Y
BCN(x)>
BY <
B
YCN(y)> = �
k=1.4
[ k
B
W
k
p
k
[4](y)] = 4 – <
B
ZCN(y)>
<
Y
BCN (x)> = �
k=1.4
[ k
Y
W
k
p
k
[4](x)] = 4 – <
Y
ACN(x)>
Truly quaternary AxB �x C1–x– �x Z or AXyYy�Z1–y–y� [7]
Parameters 12: {Wkj}
<
A
ZCN (x,x�)> = <
B
ZCN (x,x�)> = <
C
ZCN (x,x�)> = 4
<
Z
ACN(x,x�)> = �
k=1.4
�
j=0.4–k
[( k W
kj
) p
kj
(x,x�)]
<
Z
BCN(x,x�)> = �
k=0.3
�
j=1.4–k
[( j W
kj
) p
kj
(x,x�)]
<
Z
CCN(x,x�)> = �
k=0.3
�
j=0.3–k
[ (4 – k – j)W
kj
p
kj
(x,x�)] =
= 4 – <
Z
ACN(x,x�)> – <
Z
BCN(x,x�)>
Appendix 3. Tetrahedron configuration probabilities
(Pternary : Pk(x); PpseudoQ : Pk(x,y); PtrulyQ : Pkj(x,y))
Ternary A1–xBxZ (or AYyZ1–y) [1]
Non binary configurations: 3{Tk}k=1.3.Parameters 3: {Wk}k=1.3
Stoichiometry preference shifts:
Wk < 1 enhances binary AZ populations, while Wk > 1 that of binary BZ, i.e.
{P
k
(x) = C
k
p
k
[4]}
k=1.3
of ternary T
k
P
0
(x) = { p
0
[4](x) + �
k=1.3
[max (0, 1 – W
k
) p
k
[4] (x) ] } of binary AZ configuration T
0
P
4
(x) = { p
4
[4](x) + �
k=1.3
[max (0, k(W
k
– 1)/(4 – k)) p
k
(x) ] } of binary BZ configuration T
4
300 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
Pseudo quaternary A1–xBxYyZ1–y [6]
Non binary configurations: 12 {cTk}c=A,B,Y,Z;k=1.3. Parameters 12: {cWk}c=A,B,Y,Z;k=1.3
Stoichiometry preference shifts to binary:
AZ for ZWk < 1 and for AWk < 1, BZ for ZWk > 1 and BWk < 1,
AY for YWk <1 and AWk >1, BY for BWk >1 and YWk >1.
Let {cv(x,y)}c=A,B,Y,Z: = x for c = Y or c = Z , and = y for c = A or c = B ,
and {�c(x,y)}c=A,B,Y,Z;k=1.3) = { (1 – x)/2, x/2, y/2, (1 – y)/2}, with �c=A,B,Y,Z �c(x,y) �1
{
c
P
k
(x,y) = �
c
(x,y)
c
C
k
p
k
[4][
c
v(x,y)] }
k=1.3
of ternary
c
T
k
c
P
0
(x,y) = �
c
(x,y) { p
0
[4](x) + �
k=1.3
[max (0, 1 –
c
W
k
) p
k
[4](x) ] } of binary AZ configuration
c
T
0
cP
4
(x,y) = �
c
(x,y) { p
4
[4](x) + �
k=1.3
[max (0, k(
c
W
k
– 1)/(4 – k)) p
k
[4](x) ] } of binary BZ configuration
c
T
4
In the random case, when {Wk�1}k=1.3,{cPk
[4](x,y) ���c(x,y)pk[cv(x,y)]}k=0.4
Truly quaternary AxB �x C1–x– �x Z or AXyYy�Z1–y–y� [7]
Non binary configurations: 12 {Tkj}. Parameters 12: {Wkj}
{ P
kj
(x,x�) = C
kj
p
kj
(x,x�)}
k=0.4;j=0,(4–k)
for non binary
P
40
(x,x�) = �
k=1.3
�
j=0.3–k
[max (0,(k+ j)(W
kj
– 1)/(4 – k – j)) p
kj
(x,x�)]
+ �
k=1.3
[max (0,k(W
k,(4–k)
– 1)/(4 – k)) p
k,(4–k)
(x,x�)] for binary AZ
P
04
(x,x�) = �
k=0.3
�
j=1.3–k
[max (0,(k+ j)(W
kj
– 1)/(4 – k – j)) p
kj
(x,x�)]
+ �
k=1.3
[max (0,1 – W
k,(4–k)
) p
k,(4–k)
(x,x�)] for binary BZ
P
00
(x,x�) = �
k=0.3
�
j=0.3–k
[max (0, 1 – W
kj
) p
kj
(x,x�)]
for binary CZ
(4)
with
�
k=0.4
�
j=0.4–k
{P
kj
(x,x�)}+ P
40
(x,x�)+ P
04
(x,x�)+ P
00
(x,x�) =
= �
k=0.4
�
j=0.4–k
{p
kj
(x,x�)}
1
(5)
�
k=0.4;j=0.4–k
k*p
kj
(x,x�)}
4x ,
�
k=0.4;j=0.4–k
j*p
kj
(x,x�)}
4x�,
�
k=0.4;j=0.4–k
k*p
kj
(x,x�)}
4(1 – x – x�)
(6)
In the random case, when {W
kj
1}
k=0.4;j=0.4–k
, {P
kj
(x,x�) �p
kj
(x,x�)}
k=0.4;j=0.4–k
Appendix 4. EXAFS – average NN inter – ion distances <Z
vertexd(x,x�)>
(the binary distance values are from the literature)
Ternary A1–xBxZ (or AYyZ1–y) [1]
NN equations: 2. Parameters 9: 6 distance { AZdk , BZdk}k=1.3 + 3 {Wk}k=1.3
<
Z
Bd(x)> =
� �
k=1.4
[ (k C
k
)BZd
k
+ 4 Max (0,W
k
– 1) BZd
4
] p[4]
k
(x) }
/� �
k=1.4
[ (k C
k
) + 4 Max (0,W
k
– 1) ] p[4]
k
(x) }
<
Z
Ad(x)> =
� �
k=0.3
[(4 – k C
k
) AZd
k
+ 4 Max (0,1 – W
k
) AZd
0
] p[4]
k
(x) }
/� �
k=0.3
[ (4 – k C
k
) + 4 Max (0,1 – W
k
) ] p[4]
k
(x) }
Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 301
Pseudo quaternary A1–xBxYyZ1–y [6]
NN equations: 4. Parameters 36: 24 distances { c
cvdk }k=1.3;c=A,B,Y,Z;v=v1,v2 + 12 {cWk}c=A,B,Y,Z;k=1.3
<
A
Zd (x,y)> = � (1 – x) �
k=0.3
{ A
Zdk (4 – k
A
C
k
) + 4 A
AZd0 max (0,1 –
A
w
k
) + 4 A
AZd4 max [0, (k
A
W
k
– 1)/(4 – k)]} p
k
[4](y) } /
/ � (1 – x) �
k=0.3
{(4 – k
A
C
k
) + 4 max (0,1 –
A
W
k
) + 4 max [0, (k
A
W
k
– 1)/(4 – k)]} p
k
[4](y) }
<
B
Zd (x,y)> = { x �
k=1.4
{ B
BZdk (4 – k
B
C
k
) + 4 B
BZd4max (0,1 –
B
W
k
) + 4 B
AZd4 max [0, (k
B
W
k
– 1)/(4 – k)] } p
k
[4](y) }/
/ { x �
k=1.4
{ (4 – k
B
C
k
) + 4 max (0,1 –
B
W
k
) + 4 max [0, (k
B
W
k
– 1)/(4 – k)] }p
k
[4](y) }
<
Y
Ad (x,y)> = � y �
k=0.3
{ Z
AYdk(4 – k
Z
C
k
) + 4 Z
AYd0 max (0,1 –
Z
w
k
) + 4 Z
AYd4 max [0, (k
Z
W
k
– 1)/(4 – k)] } p
k
[4](x) }
/ �y �
k=0.3
{(4 – k
Z
C
k
) + 4 max (0,1 –
Z
W
k
) + 4 max [0, (k
Z
W
k
– 1)/(4 – k)] }p
k
[4](x) }
<
Z
Bd (x,y)> = { y �
k=1.4
{ Z
BYdk k
Z
C
k
+ 4 Z
BYd4max (0,1 –
Z
W
k
) + 4 Z
BYd4 max [0, (k
Z
W
k
– 1)/(4 – k)] }p
k
[4](x) }
/ { y �
k=1.4
{ k
Z
C
k
+ 4 max (0,1 –
Z
W
k
) + 4 max [0, (k
Z
W
k
– 1)/(4 – k)] }p
k
[4](x) }
Truly quaternary AxB �x C1–x– �x Z or AXyYy�Z1–y–y� [7]
NN equations: 3. Parameters 21: 9 distance { vZdk }k=1.3;v=A,B,C+ 12 {Wkj}
<
Z
Ad(x,x�)> = {�
k=1.4
�
j=0.4–k
[
Z
Ad
kj
k P
kj
(x,x�) ] + 4
Z
A d
40
P
40
(x,x�) }/4x
<
Z
Bd(x,x�)> = {�
k=0.3
�
j=1.4–k
[
Z
Bd
kj
j P
kj
(x,x�) ] + 4
Z
B d
04
P
04
(x,x�) }/4x�
<
Z
Cd(x,x�)> = {�
k=0.3
�
j=0.3–k
[
Z
Cd
kj
(4 – k – j) P
kj
(x,x�) ] + 4
Z
C d
00
P
00
(x,x�) }/4(1 – x – x�)
Appendix 5. FIR
Recall that by axioms �6 ad �7 : AZ
�����
AZ
�k �
AZ
�kj��
BZ
�����
BZ
�k�
BZ
�kj��
CZ
�����
CZ
�k�
CZ
�kj},
and AZs = {AZsk = AZskj},
BZs = {BZsk = BZskj},
CZs = {CZsk = CZskj} for all k = 0.4 and j = 0.4
Ternary A1–xBxZ (or AYyZ1–y) [1,4,5]
Parameters 15: 8 { AZ
�k}k=0.3 , { BZ
�k}k=1.4} + 3 {Wk}k=1.3 + 2 {AZ
��
BZ
�} + 2 {AZs,BZs }
�
�
�,x) = {{4 AZs
0
AZ
�
0
2 AZ
�
0
� / [
�2
�
AZ
�
0
2)2 + AZ
�
0
2
�
2]} P
0
(x) binary AZ
� �
k=1.3
{k BZs
k
BZ
�
k
2 BZ
�
k
� / [
�2
�
BZ
�
k
2)2 + BZ
�
k
�
�
2]
+ (4 – k) AZs
k
AZ
�
k
2 AZ
�
k
� /[
�2
�
AZ
�
k
2)2+ AZ
�
k
�
�
2]}P
k
(x) ternary ABZ
��4 BZs
4
BZ
�
4
2 BZ
�
4
� / [(�2
�
BZ
�
4
2)2 + BZ
�
4
2
�
2]} P
4
(x) } binary BZ
Pseudo quaternary A1–xBxYyZ1–y [6,4,5]
Parameters 52: 32 { ij
c�k} + 12 {cWk}c=A,B,Y,Z;k=1.3 + 4 {AZ
��
BZ
��
AY
��
BY
�} + 4 {AZs,BYs,AZs,BYs}
�
�
�,x) = �
�
�
�,x) +
�
�
�
�,x) +
Y
�
�
�,x) +
�
�
�
�,x)
each taken as defined in the above table where
�
�
�,x) =
�
�
�
�,x) was Z-centred with the binary allowed to have distinct
frequencies (whence the extra 4 parameters).
302 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
B.V. Robouch et al.
Truly quaternary AxB �x C1–x– �xZ or AXyYy�Z1–y–y�
Parameters 48: 30 { vZ
�kj}+ 12 {Wkj} + 3 {AZ
��
BZ
��
CZ
�} + 3 {AZs,BZs,CZs}
�
�
��,x,x�) = { k
A,
k
B
+ {4 AZs
0
AZ�
0
�/ [��2 �AZ�
0
2)2 + AZ�
0
2 �2]} p
00
(x,x�) binary AZ 4,0
��4 BZs
4
BZ�
4
�/ [(�2 �BZ�
4
2)2 + BZ�
4
2 �2]} p
40
(x,x�) binary BZ 0.4
+ {4 CZs
0
CZ�
0
�/ [��2 �CZ�
0
2)2 + CZ�
0
2 �2]} p
04
(x,x�) binary CZ 0,0
�
k=1.3
{ k ABZC
k
BZs
k
BZ�
k
�/ [��2 �BZ�
k
2)2 + BZ�
k
� �2] ternary ABZ k,0
+ (4 – k ABZC
k
) AZs
k
AZ�
k
�/ [��2 �AZ�
k
2)2+ AZ�
k
� �2]
4 Max (0, ABZW
k
– 1) BZs
4
BZ�
4
�/ [(�2 �BZ�
4
2)2 + BZ�
4
2 �2] excess in binary BZ
4 Max (0, 1 – ABZW
k
) AZs
4
�/ [(�2 �AZ�
4
2)2 + AZ�
4
2 �2]} p
k0
(x,x�) excess in binary AZ
��
k=1.3
{ k BCZC
k
CZs
k
CZ�
k
�/ [��2 �CZ�
k
2)2 + CZ�
k
� �2]
+ (4 – k BCZC
k
) BZs
k
BZ�
k
�/ [��2 �BZ�
k
2)2+ BZ�
k
� �2]
ternary CBZ k,(4 – k)
4 Max (0, BCZW
k
– 1) CZs
4
CZ�
4
�/ [(�2 �CZ�
4
2)2 + CZ�
4
2 �2] excess to binary CZ
4 Max (0, 1 – BCZW
k
) BZs
4
�/ [(�2 �BZ�
4
2)2 + BZ�
4
2 �2]} p
k(4–k)
(x,x�) excess to binary BZ
�
k=1.3
{ k CAZC
k
AZs
k
AZ�
k
�/ [��2 �AZ�
k
2)2 + AZ�
k
� �2]
+ (4 – k CAZC
k
) CZs
k
CZ�
k
�/ [��2 �CZ�
k
2)2+ CZ�
k
� �2]
ternary ACZ 0,k
4 Max (0, CAZW
k
– 1) AZs
4
AZ�
4
�/ [(�2 �AZ�
4
2)2 + AZ�
4
2 �2] excess to binary AZ
4 Max (0, 1 – CAZW
k
) CZs
4
�/ [(�2 �CZ�
4
2)2 + CZ�
4
2 �2]} p
0k
(x,x�) excess to binary CZ
+ { 1 C
11
AZs
11
AZ�
11
�/[��2 �AZ�2
11
)2 + AZ�
11
� �2] quaternary A
1
B
1
C
2
Z 1,1
+ 1 C
11
BZs
11
BZ�
11
�/[��2 �BZ�2
11
)2 + BZ�
11
� �2]
+ 2 C
11
CZs
11
CZ�
11
�/[��2 �CZ�2
11
)2 + CZ�
11
� �2]
+ Max (0, 1 – W
11
) 4 CZs
11
CZ�
11
�/[��2 �CZ�2
11
)2 + CZ�
11
� �2] excess to binary CZ
+ Max (0, W
11
– 1){ 1 BZs
21
BZ�
21
�/[��2 �BZ�2
21
)2 + BZ�
21
� �2] excess to binary BZ
+ 1 AZs
21
AZ�
21
�/[��2 �AZ�2
21
)2 + AZ�
21
� �2] } }p
11
(x,x�) excess to binary AZ
+ { 1 C
12
AZs
12
AZ�
12
�/[��2 �AZ�2
12
)2 + AZ�
12
� �2] quaternary A
2
B
1
C
1
Z 1,2
+ 2 C
12
BZs
12
BZ�
12
�/[��2 �BZ�2
12
)2 + BZ�
12
� �2]
+ 1 C
12
CZs
12
CZ�
12
�/[��2 �CZ�2
12
)2 + CZ�
12
� �2]
+ Max (0, 1 – W
12
) 4 CZs
12
CZ�
12
�/[��2 �CZ�2
12
)2 + CZ�
12
� �2] excess to binary CZ
+ Max (0, W
12
– 1){ 1 BZs
21
BZ�
21
�/[��2 �BZ�2
21
)2 + BZ�
21
� �2] excess to binary BZ
+ 2 AZs
21
AZ�
21
�/[��2 �AZ�2
21
)2 + AZ�
21
� �2] } }p
12
(x,x�) excess to binary AZ
+ { 2 C
21
AZs
21
AZ�
21
�/[��2 �AZ�2
21
)2 + AZ�
21
� �2] quaternary A
1
B
2
C
1
Z 2,1
+ 1 C
21
BZs
21
BZ�
21
�/[��2 �BZ�2
21
)2 + BZ�
21
� �2]
+ 1 C
21
CZs
21
CZ�
21
�/[��2 �CZ�2
21
)2 + CZ�
21
� �2]
+ Max (0, 1 – W
21
) 4 CZs
21
CZ�
21
�/[��2 �CZ�2
21
)2 + CZ�
21
� �2] excess to binary CZ
+ Max (0, W
21
– 1){ 2 BZs
21
BZ�
21
�/[��2 �BZ�2
21
)2 + BZ�
21
� �2] excess to binary BZ
+ 1 AZs
21
AZ�
21
�/[��2 �AZ�2
21
)2 + AZ�
21
� �2] } }p
21
(x,x�) excess to binary AZ
}
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Statistical model analysis of local structure of quaternary sphalerite crystals
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 303
|
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| last_indexed | 2025-12-07T18:20:31Z |
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| record_format | dspace |
| spelling | Robouch, B.V. Kisie, A. Marcelli, A. Sheregii, E.M. Cestelli Guidi, M. Piccinini, M. Polit, J. Cebulski, J. Mycielsk, A. Ivanov-Omski, V.I. Sciesinska, E. Sciesinski, J. Burattini, E. 2017-12-27T11:01:11Z 2017-12-27T11:01:11Z 2007 Statistical model analysis of local structure of quaternary
 sphalerite crystals / B.V. Robouch, A. Kisiel, A. Marcelli, E.M. Sheregii, M. Cestelli Guidi, M. Piccinini, J. Polit, J. Cebulski, A. Mycielski, V.I. Ivanov-Omskii, E. Sciesinska, J. Sciesinski, and E. Burattini // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 291-303. — Бібліогр.: 40 назв. — англ. 0132-6414 PACS: 61.43. Dq https://nasplib.isofts.kiev.ua/handle/123456789/127732 At the 2004 Ural International Winter School, we introduced the statistical strained tetrahedron
 model and discussed ternary tetrahedron structured crystals. The model allows one to interpret
 x-ray absorption fine structure (EXAFS) data and extract quantitative information on ion site occupation
 preferences and on the size and shape of each elemental constituent of the configuration tetrahedra.
 Here we extend the model to cover quaternary sphalerite crystal structures. We discuss the
 two topologically different quaternary sphalerite systems: the pseudo balanced A₁₋xBxYyZ₁₋y (2:2
 cation:anion ratio), and the unbalanced AxBx 
 C₁₋x₋x 
 Z or AXyYy 
 Z₁₋y₋y (3:1 or 1:3 cation:anion ratios)
 truly quaternary alloy systems. These structural differences cause preference values in pseudo
 quaternaries to vary with the relative contents, but to remain constant in truly quaternary compounds.
 We give equations to determine preference coefficient values from EXAFS or phonon spectra
 and to extract nearest-neighbour inter-ion distances by EXAFS spectroscopy. The procedure is illustrated
 and tested on CdMnSeTe, GaInAsSb, and ZnCdHgTe quaternary alloys. Part of this work was supported by EU TARI-project
 contract HPRI-CT-1999-00088. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Структура и свойства полупроводников с переходными элементами Statistical model analysis of local structure of quaternary sphalerite crystals Article published earlier |
| spellingShingle | Statistical model analysis of local structure of quaternary sphalerite crystals Robouch, B.V. Kisie, A. Marcelli, A. Sheregii, E.M. Cestelli Guidi, M. Piccinini, M. Polit, J. Cebulski, J. Mycielsk, A. Ivanov-Omski, V.I. Sciesinska, E. Sciesinski, J. Burattini, E. Структура и свойства полупроводников с переходными элементами |
| title | Statistical model analysis of local structure of quaternary sphalerite crystals |
| title_full | Statistical model analysis of local structure of quaternary sphalerite crystals |
| title_fullStr | Statistical model analysis of local structure of quaternary sphalerite crystals |
| title_full_unstemmed | Statistical model analysis of local structure of quaternary sphalerite crystals |
| title_short | Statistical model analysis of local structure of quaternary sphalerite crystals |
| title_sort | statistical model analysis of local structure of quaternary sphalerite crystals |
| topic | Структура и свойства полупроводников с переходными элементами |
| topic_facet | Структура и свойства полупроводников с переходными элементами |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/127732 |
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