Statistical strained-tetrahedron model of local ternary zinc blende crystal structures

The statistical strained-tetrahedron model was developed to overcome two common assumptions of previous models: 1) rigid undistorted ion sublattice of regular tetrahedra throughout all five configurations and 2) random ion distribution. These simplifying assumptions restrict the range of applicab...

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Datum:	2004
Hauptverfasser:	Robouch, B.V., Sheregii, E.M., Kisiel, A.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2004
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spelling	irk-123456789-1203422017-06-12T03:05:21Z Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Robouch, B.V. Sheregii, E.M. Kisiel, A. Фазовые переходы и структура полупроводниковых соединений The statistical strained-tetrahedron model was developed to overcome two common assumptions of previous models: 1) rigid undistorted ion sublattice of regular tetrahedra throughout all five configurations and 2) random ion distribution. These simplifying assumptions restrict the range of applicability of the models to a narrow subset of ternary alloys for which the constituent binaries have their lattice constants and standard molar enthalpies of formation (∆fH₀) equal or quasi-equal. Beyond these limits predictions of such models become unreliable, in particular, when the ternary exhibits site occupation preferences. The strained-tetrahedron model, free from rigidity and stochastic limitations, was developed to better describe and understand the local structure of ternary zinc blende crystals, and interpret experimental EXAFS and far-IR spectra. It considers five tetrahedron configurations with the shape and size distortions characteristic of ternary zinc blende alloys, allows nonrandom distributions and, hence, site occupation preferences, conserves coordination numbers, respects stoichiometry, and assumes that next-neighbor values determine preferences beyond next-neighbor. The configuration probabilities have three degrees of freedom. The nineteen inter-ion crystal distances are constrained by tetrahedron structures; to avoid destructive stresses, we assume that the average tetrahedron volumes of both sublattices relax to equal values. The number of distance free-parameters ≤ 7. Model estimates, compared to published EXAFS results, validate the model. Knowing the configuration probabilities, one writes the dielectric function for far-infrared absorption or reflection spectra. Constraining assumptions restrict the number of degrees of freedom. Deconvolution of the experimental spectra yields site-occupation- preference coefficient values and/or specific oscillator strengths. Validation again confirms the model. 2004 Article Statistical strained-tetrahedron model of local ternary zinc blende crystal structures / B.V. Robouch, E.M. Sheregii, A. Kisiel // Физика низких температур. — 2004. — Т. 30, № 11. — С. 1225–1234. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 78.30.Er http://dspace.nbuv.gov.ua/handle/123456789/120342 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Фазовые переходы и структура полупроводниковых соединений Фазовые переходы и структура полупроводниковых соединений
spellingShingle	Фазовые переходы и структура полупроводниковых соединений Фазовые переходы и структура полупроводниковых соединений Robouch, B.V. Sheregii, E.M. Kisiel, A. Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Физика низких температур
description	The statistical strained-tetrahedron model was developed to overcome two common assumptions of previous models: 1) rigid undistorted ion sublattice of regular tetrahedra throughout all five configurations and 2) random ion distribution. These simplifying assumptions restrict the range of applicability of the models to a narrow subset of ternary alloys for which the constituent binaries have their lattice constants and standard molar enthalpies of formation (∆fH₀) equal or quasi-equal. Beyond these limits predictions of such models become unreliable, in particular, when the ternary exhibits site occupation preferences. The strained-tetrahedron model, free from rigidity and stochastic limitations, was developed to better describe and understand the local structure of ternary zinc blende crystals, and interpret experimental EXAFS and far-IR spectra. It considers five tetrahedron configurations with the shape and size distortions characteristic of ternary zinc blende alloys, allows nonrandom distributions and, hence, site occupation preferences, conserves coordination numbers, respects stoichiometry, and assumes that next-neighbor values determine preferences beyond next-neighbor. The configuration probabilities have three degrees of freedom. The nineteen inter-ion crystal distances are constrained by tetrahedron structures; to avoid destructive stresses, we assume that the average tetrahedron volumes of both sublattices relax to equal values. The number of distance free-parameters ≤ 7. Model estimates, compared to published EXAFS results, validate the model. Knowing the configuration probabilities, one writes the dielectric function for far-infrared absorption or reflection spectra. Constraining assumptions restrict the number of degrees of freedom. Deconvolution of the experimental spectra yields site-occupation- preference coefficient values and/or specific oscillator strengths. Validation again confirms the model.
format	Article
author	Robouch, B.V. Sheregii, E.M. Kisiel, A.
author_facet	Robouch, B.V. Sheregii, E.M. Kisiel, A.
author_sort	Robouch, B.V.
title	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures
title_short	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures
title_full	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures
title_fullStr	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures
title_full_unstemmed	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures
title_sort	statistical strained-tetrahedron model of local ternary zinc blende crystal structures
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2004
topic_facet	Фазовые переходы и структура полупроводниковых соединений
url	http://dspace.nbuv.gov.ua/handle/123456789/120342
citation_txt	Statistical strained-tetrahedron model of local ternary zinc blende crystal structures / B.V. Robouch, E.M. Sheregii, A. Kisiel // Физика низких температур. — 2004. — Т. 30, № 11. — С. 1225–1234. — Бібліогр.: 18 назв. — англ.
series	Физика низких температур
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first_indexed	2025-07-08T17:41:48Z
last_indexed	2025-07-08T17:41:48Z
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fulltext	Fizika Nizkikh Temperatur, 2004, v. 30, No. 11, p. 1225–1234 Statistical strained-tetrahedron model of local ternary zinc blende crystal structures B.V. Robouch1, E.M. Sheregii2, and A. Kisiel3 1Laboratori Nazionali di Frascati INFN, DAFNE-L, C.P.:13, 00044 Frascati (RM), Italy E-mail: Robouch@lnf.infn.it 2Institute of Physics, Rzeszow University, Rejtana 16A, 35-310 Rzeszow, Poland 3Instytut Fizyki, Universytet Jagiellonski, Reymonta 4, 30-059 Krakow, Poland Received June 1, 2004 The statistical strained-tetrahedron model was developed to overcome two common assump- tions of previous models: 1) rigid undistorted ion sublattice of regular tetrahedra throughout all five configurations and 2) random ion distribution. These simplifying assumptions restrict the range of applicability of the models to a narrow subset of ternary alloys for which the constituent binaries have their lattice constants and standard molar enthalpies of formation (�fH0) equal or quasi-equal. Beyond these limits predictions of such models become unreliable, in particular, when the ternary exhibits site occupation preferences. The strained-tetrahedron model, free from rigid- ity and stochastic limitations, was developed to better describe and understand the local structure of ternary zinc blende crystals, and interpret experimental EXAFS and far-IR spectra. It considers five tetrahedron configurations with the shape and size distortions characteristic of ternary zinc blende alloys, allows nonrandom distributions and, hence, site occupation preferences, conserves coordination numbers, respects stoichiometry, and assumes that next-neighbor values determine preferences beyond next-neighbor. The configuration probabilities have three degrees of freedom. The nineteen inter-ion crystal distances are constrained by tetrahedron structures; to avoid de- structive stresses, we assume that the average tetrahedron volumes of both sublattices relax to equal values. The number of distance free-parameters � 7. Model estimates, compared to published EXAFS results, validate the model. Knowing the configuration probabilities, one writes the di- electric function for far-infrared absorption or reflection spectra. Constraining assumptions re- strict the number of degrees of freedom. Deconvolution of the experimental spectra yields site-oc- cupation-preference coefficient values and/or specific oscillator strengths. Validation again confirms the model. PACS: 78.30.Er 1. Introduction The abundance of articles in the literature devoted to sphalerite (zinc blende) ternary semiconductors is ample evidence of the interest paid to them. In the hope of better understanding their local structure, we considered the interpretation of extended x-ray ab- sorption fine structure (EXAFS) (see theoretical con- siderations [1,2] since 1981) and vibrational spectra observed in the far-infrared region (FIR spectra). EXAFS was applied soon after [3,4] with, alas, no re- view paper covering the abundant literature devoted to it. For literature on the FIR spectra, see, for in- stance, the review articles [5–7] and book [8]. With that aim we developed the statistical strained-tetrahe- dron model, validating it on published EXAFS zinc blende data [9,10] and, after an adaptation, on inter- metallide materials [11]. The model was then ex- tended to describe and interpret FIR spectra [12]. We propose to briefly recall here the model de- veloped and its validation and then to dwel more on FIR-spectrum interpretation, applicability, and limits. 2. The statistical strained-tetrahedron model To create any model, one has to 1) describe as closely as possible the object under study, using © B.V. Robouch, E.M. Sheregii, and A. Kisiel, 2004 proper parameters for that; 2) reduce the set of these parameters through motivated constraints to deter- mine the minimum number of degrees of freedom/pa- rameters; 3) check the model-predicted values against experimental ones; 4) consequently, discard the model or retain it as valid depending on the reproducibility thus obtained. With this in mind we recall our mo- deling. 2.1. The object under study Zinc blende fcc structures are tetrahedrally coordi- nated, characterized by a central ion surrounded by four nearest-neighbor (NN) ions (first shell) defining the four vertices of a tetrahedron, and 12 next near- est-neighbor (NNN) ions (second shell). Binary com- pounds AZ (we use A, B, … for cations, and Z, Y, … for anions), have their successive shells alternate- ly fully filled by A then Z ions. All tetrahedra are symmetric, regular, and identical; thus, by simple trigonometry, equal interbond angles �(A:Z:A) = = �(B:Z:A) = 109.47�, and the inter-ion distances (ijd) are defined in terms of the lattice constant a (known from x-ray diffraction analysis): AZd = 31/2a/4, AAd = ZZd = a/21/2. For ternary A1–xBxZ (or AYyZ1–y), in the binary compound AZ, cations A are partially substituted by B ions. This, leads to five different elemental tetrahe- dra {Tk}k=0,4 where the subscript k indicates the num- ber of B ions at the vertices of the tetrahedron, with (4–k) A-ions [T0(Z:4A), T1(Z:3A+1B), T2(Z:2A+2B), T3(Z:1A+3B), T4(Z:4B)]. Prior to ours, simulations had considered the five {Tk}k=0,4 tetrahedra as exter- nally rigid with the central ion free to be displaced. The ion distribution fillings (k B-ions into a shell with N sites, from relative contents x and 1–x) were as- sumed stochastic and defined by the random Bernoulli binomial polynomials p k [N](x)=N!/[k!(N–k)!]xk(1–x)N–k with k = 0, ..., N. Thus around a central Z ion the first shell four A/B ions are described by pk [4](x), while the second shell contains twelve Z ions! On the other hand, around an A or B ions, the first shell contains four Z ions, while the second shell distribution of the twelve ions A/B is described by pk [12](x). This allows ap- proximate evaluations, avoiding analytical difficul- ties. However, to assume a stochastic filling with ions A and B around Z means that the Z-ion preference for ei- ther is the same. Thermodynamically this implies that the enthalpies of formation of AZ and BZ pairs are identical. But we are aware that in nature equality is the exception that confirms the rule of inequality. In- deed, the standard molar enthalpies of formation of bi- naries, �fH0, kJ/mol, are generally different. That is why the stochastic approach is unable to describe the site occupation preferences (SOPs) reported experi- mentally! 2.2. Statistical strained-tetrahedron model assumptions We build our model discarding both restrictions: 1) deviating from stochastic filling of ions, and 2) freeing the tetrahedra of the unnatural constraint of rigidity. The price for such a more general model is the num- ber of parameters needed to describe the crystal struc- ture. But as we shall demonstrate, realistic assump- tions (checked at the end) reduce the degrees of freedom to an acceptable value. To quantify results departing from stochastic dis- tribution, we attribute to each Bernoulli binomial a SOP weight coefficient. This leads to five NN terms Wk pk [4](x), thirteen NNN terms Awk pk [12](x) (for central A ions), and thirteen Bwk pk [12](x) terms (for central B ions), a total of 31 parameters! Fortunately it is the Z ion that determines the choices, and we claim that ALL higher shell fillings are determined by linear expressions of the five NN SOP coefficients {Wk}k=0,4 . But binary tetrahedron configurations T0, T4 have NO preferences. Thus W0 =W4 �1 (!) and we are left with only {W1,W2,W3}. The probabilities of finding B and A ions in a Tk configuration are proportional to {kWk}k=0,4 and to {4–kWk}k=0,4, respectively (conservation of coordi- nation numbers). As probabilities cannot be negative, we have 0 � Wk � 4/k. There are thus only three bounded free parameters {W1,W2,W3}. Expressions {Awk}k=0,12 , {Bwk}k=0,12 for the NNN shell, are determined by combinatorial probabilities in terms of those around the Z ions of the NN shell. This hypothesis leads for the zinc blende structure, to the linear expressions of the NNN SOP coefficients Awk and Bwk as functions of the three Wk’s given in Table 1,a [9]. To illustrate that the assumption is gen- eral and applies to other crystal structures also, the expressions for intermetalides M3(X1–x �X x)1 are given in Table 1,b [11]. A random integer ion distribution (k and 4–k) fully respects stoichiometry. With SOP coefficients Wk � 1 the situation departures from stochastic equilibrium, with a consequent attenuation of the ternary configu- ration populations caused by the observed scarcity of one of the two ion populations {P k [4](x) = C k p k [4](x) } k=1,3 for ternary T k with {Ck(Wk)}k=1,3 , corrective weight factors im- posed by stoichiometry 1226 Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 B.V. Robouch, E.M. Sheregii, and A. Kisiel 0 � {C k =min [W k ,1,(4 – kW k )/(4–k)]} k=1,3 � 1 Wk < 1 enhances the binary AZ populations and Wk > 1 that of binary BZ, i.e. P0 [4](x) = p0 [4](x) + {max ( , – ) ( )}[ ] , 0 1 4 13 W p xk k k� for binary AZ configuration T0, P4 [4](x) = p4 [4](x) + � � � {max( , ( ) ( )) ( )} , 0 1 4 13 k W / k p xk k k for binary BZ configuration T4. (1) In the random case, when {Wk � 1}k=1,3, {Pk(x) � � pk(x)}k=0,4. Note however, that even if the SOPs enhance the two binary populations with respect to corresponding populations of the random case, it by no means leads to clustering, since the spatial distribution remains perfectly stochastic. Ion-pair and configuration populations are NOW determined with due account of the SOPs. This allows us to interpret local crystal structures. We have five tetrahedra freed from any constraint, with two of which (T0 and T4) are binary, regular, different sized, and well defined (as remarked above), and three (T1, T2, and T3) are ternary and distorted (strained tetrahedra), with nineteen unequal inter- ionic ijd distance parameters and, consequently, al- tered interbond angles (see Fig. 1 [9]). The geometri- cal symmetry of each Tk configuration yields three Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 1227 Table 1,a NNN SOP-coefficients in terms of NN SOPs: All possible NNN distributions and resulting SOP-coefficients for ternary. 0 � {Wk}k=1,3 � 4/k, while W0=W4 = 1. Zinc blende A1–xBxZ with a B or A as central ion [9]. k All 12 possible NNN B-ion fills of the 4 tetrahedra Resulting B-weights Resulting A-weights Bw [12] k = j=0,4 �BM j,k W[4] j } Aw [12] k = j=0,4 �AM j,k W[4] j } 0 0 0 0 0 W 1 W 0 = 1 1 0 0 0 1 (3W 1 +W 2 )/4 (3W 0 +W 1 )/4 2 0 0 0 2/0 0 1 1 (5W 1 + 2W 2 +W 3 )/8 (5W 0 + 2W 1 +W 2 )/8 3 0 0 0 3/0 0 1 2/0 1 1 1 (6W 1 +4W 2 +W 3 +W 4 )/12 (6W 0 +4W 1 +W 2 +W 3 )/12 4 0 0 1 3/0 0 2 2/0 1 1 2/1 1 1 1 (5W 1 +7W 2 +3W 3 +W 4 )/16 (5W 0 +7W 1 +3W 2 +W 3 )/16 5 0 0 2 3/0 1 1 3/0 1 2 2 (4W 1 +3W 2 +3W 3 +2W 4 )/12 (4W 0 +3W 1 +3W 2 +2W 3 )/12 6 0 0 3 3/0 1 2 3/1 1 1 3/0 2 2 2/1 1 2 2 (4W 1 +6W 2 +6W 3 +4W 4 )/20 (4W 0 +6W 1 +6W 2 +4W 3 )/20 7 0 1 3 3/0 2 2 3/1 1 2 3 (2W 1 +3W 2 +3W 3 +4W 4 )/12 (2W 0 +3W 1 +3W 2 +4W 3 )/12 8 0 2 3 3/1 2 2 3/1 1 3 3/2 2 2 2 (W 1 +3W 2 +7W 3 +5W 4 )/16 (W 0 +3W 1 +7W 2 +5W 3 )/16 9 0 3 3 3/1 2 3 3/2 2 2 3 (W 1 +W 2 +4W 3 +6W 4 )/12 (W 0 +W 1 +4W 2 +6W 3 )/12 10 1 3 3 3/2 2 3 3 (W 2 +2W 3 +5W 4 )/8 (W 1 +2W 2 +5W 3 )/8 11 2 3 3 3 (W 3 +3W 4 )/4 (W 2 +3W 3 )/4 12 3 3 3 3 W 4 = 1 W 3 Table 1,b Intermetallides M3(X �X)1 around X or �X [11]. {Xw k = j=0,4 XM kj W j } k=0,6 {X’w k = j=0,4 X’M kj W j } k=0,6 Xw 0 = 1 3 W 1 +2 3 W 2 X’w 0 = 1 3 W 0 +2 3 W 1 Xw 1 = 10 36 W 1 +23 36 W 2 +3 36 W 3 X’w 1 = 10 36 W 0 +23 36 W 1 +3 36 W 2 Xw 2 = 16 72 W 1 +41 72 W 2 +13 72 W 3 +2 72 W 4 X’w 2 = 16 72 W 0 +41 72 W 1 +13 72 W 2 +2 72 W 3 Xw 3 = 6 36 W 1 +15 36 W 2 +11 36 W 3 +4 36 W 4 X’w 3 = 6 36 W 0 +15 36 W 1 +11 36 W 2 +4 36 W 3 Xw 4 = 2 72 W 1 +13 72 W 2 +41 72 W 3 +16 72 W 4 X’w 4 = 2 72 W 0 +13 72 W 1 +41 72 W 2 +16 72 W 3 Xw 5 = 3 36 W 2 +23 36 W 3 +10 36 W 4 X’w 5 = 3 36 W 1 +23 36 W 2 +10 36 W 3 Xw 6 = +2 3 W 3 + 1 3 W 4 X’w 6 = +2 3 W 2 + 1 3 W 3 constraints each, which reduces the number of inde- pendent distance parameters from nineteen to ten. Material strength considerations lead us, to avoid destructive intercrystal stresses, to impose the condi- tion that the average tetrahedron volume of the four vertex tetrahedra be equal to the central one (one con- straint per configuration), leaving us with only (10–3) = 7 distance parameters, while for SOP ex- treme values, configurations disappear, and their distances become virtual, i.e., � 7. Indeed �, as for extreme SOP values configurations disappear, and their distances become virtual! On the basis of the above probabilities, expressions for the average pair coordination <i:jCN(x)> and dis- tances <i:jd(x)> as functions of x, for any two-ion pair i:j={AZ, BZ, BB, BA, AA, ZAZ, ZBZ} of zinc blende ternary alloys A1–xBxZ are given in Table 2 [9]. On the basis of these, deconvolving a given set of EXAFS data such as GaAsyP1–y [13], one obtains the dimensions of all the elemental tetrahedra involved: the inter-ion distances and angles (see Table 3). Having defined a 31+19 parameter model and iden- tified the relative constraint relations, we have re- duced the problem to 3+7 independent parameters. The model is ready for confrontation of its estimations with experimental data. 1228 Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 B.V. Robouch, E.M. Sheregii, and A. Kisiel Table 2 Expressions of average pair coordinations <i:jCN(x)> and distances <i:jd(x)>, as a function of x, for any two-ion pair i:j = = {AZ, BZ, BB, BA, AA, ZAZ, ZBZ} of zinc blende A1–xBxZ ternary [9]. Average Coordination numbers for NN ion pairs AZ, BZ <BZCN(x)> = k=0,4 � kW k p[4] k (x)� <AZCN(x)> = k=0,4 �(4 – kW k )p[4] k (x)� = 4– <BZCN(x)> For NNN ion pairs BB, BA, AA <BBCN(x)> = k=0,12 {k Bw k p[12] k (x)} <ABCN(x)> = k=0,12 {k Aw k p[12] k (x)} <AACN(x)> = �� <ABCN(x)> = k=0,12 {(12 – k Aw k ) p[12] k (x)} <BACN(x)> = �� <BBCN(x)> = k=0,12 {(12 – k Bw k ) p[12] k (x)} <ZBZCN(x)> = k=0,4 {3 k W k p[4] k (x)} = 3 <BZCN(x)> <ZAZCN(x)> = k=0,4 {3 (4 – kW k ) p[4] k (x)} = 3 <AZCN(x)> Average Distances for NN ion pairs AZ, BZ <BZd(x)> = �� k=1,4 { k C k BZd k + 4 Max [0, k(W k – 1)/(4 – k)] BZd 4 } p[4] k (x)} /�� k=1,4 { k C k + 4 Max [0, k(W k – 1)/(4 – k)] } p[4] k (x)} <AZd(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)} For NNN ion pairs Z:A:Z or Z:B:Z <ZBZd(x)> = �� k=1,4 { k C k ZBZd k + 4 Max [0, k(W k – 1)/(4 – k)] ZBZd 4 } p[4] k (x)} /�� k=1,4 { k C k + 4 Max [0, k(W k – 1)/(4 – k)] } p[4] k (x)} <ZAZd(x)> = �� k=0,3 { (4–k C k ) ZAZd k + 4 Max [0,(1–W k )] ZAZd 0 } p[4] k (x)} /�� k=0,3 { (4–k C k ) + 4 Max [0,(1–W k )] } p[4] k (x)} T0 T1 T2 T3 T4 BinaryBinary regularregular Ternary distored Fig. 1. Aspect of the five elemental tetrahedron configura- tions {Tk}k=0,4 of A1–xBxZ (or AYyZ1–y) ternary alloys [9]. Small open circles indicate the would-be ion-positions as per rigid tetrahedron hypothesis. 2.3. Model verification To confirm the validity of the model and its as- sumptions, we checked the quality of the model with its restricted free parameters. 1. Comparing the experimental distance-EXAFS points and error bars reported in the literature with model fit curves (see Fig. 2 (36 points with 10 free pa- rameters) and the curves reported in [9–11]). 2. Comparing the «coordination number» curve predictions on the basis of SOP values obtained from distance-EXAFS measurement analysis, against inde- pendently measured coordination number values (see Fig. 3 and also [9–11]). 3. Checking for a correlation between the thermo- dynamic standard molar enthalpies of formation, �fH 0, kJ/mol, of materials (Table 4 [10]) and the corresponding values obtained for the SOP coeffi- cient. The validity of the model with its restrictive as- sumptions is thus confirmed. 3. FIR dielectric function �(�,x) for ternary zinc blende alloys The dielectric function � � � � � � �( ) { [( ) ] , � � � � S / ij j j j j n 2 2 2 1 � of phonon spectra of solids can be extracted from ex- perimental reflectivity or transmission coefficients of a crystal, fitting the measured spectra via the Kra- mers–Kronig (KK) analysis. The KK output Im [�(�)] directly yields the maxima for each oscillator line, as- sumed Lorentzian, with its three parameters {�j, �j, and Sj}, respectively, the frequency, the line half- Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 1229 Table 3 GaAsxP1–x complete set: determined SOP coefficients, distances, angles and volumes for all five elemental configuration tetrahedra. Eleven fit parameters (bold) (3-SOP + 8-distance (PGaP NNN data not reported) to check VRC. 37 available experimental points from a set of 16 measurements [9]. GaAsxP1–x Configurations T 0 T 1 T 2 T 3 T 4 k 0 1 2 3 4 W k 1 0.93 1.15 1.07 1 BZd [Å] — 2.42 2.43 2.44 2.450 AZd 2.359 2.37 2.37 2.38 — BZBd — — 3.90 3.90 4.001 AZAd 3.852 3.88 3.98 — — BZAd — 3.91 3.90 3.99 — ZBZd — 3.95 3.97 3.98 4.001 ZAZd 3.852 3.87 3.87 3.89 — �(B:Z:B) [deg] — — 106.7 106.9 109.47 �(A:Z:A) 109.47 109.7 114.3 — — �(B:Z:A) — 109.3 107.36 112.0 — <Vol B.centred > [Å 3] — 7.28 7.37 7.41 7.55 <Vol A.centred > 6.74 6.84 6.86 6.91 — <Vol Z.sublattice > 6.74 6.95 7.12 7.29 7.55 <Vol Z.centred > 6.74 6.95 7.07 7.28 7.55 �Difference � [%] 0.0 0.1 0.1 0.2 0.0 3.9 4.0 GaAsP 3.8 3.9 4.0 0 00 0.5 0.50.5 1.0 1.01.0 2.35 2.40 2.45 As AsAs G aA sG a G aP G a P G a A sG a P P A sP A sA s A ve ra g e in te r- io n d is ta n ce ,Å a b c Fig. 2. Average inter-ion distances, Å, as a function of relative content x for GaAsxP1–x [13]: comparison of model best fit curves versus reported experimental data. Points are (circles) for As-related (top curves), (triangles) for P-related (bottom curves), and (diamond) for mixed AsP ion distances. Linear combination of weighted average distances (LCWAD) curves (thin dashed lines) and corresponding reference Vegard law lines (thin dotted) are all reported [9]. 1230 Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 B.V. Robouch, E.M. Sheregii, and A. Kisiel 0 0.5 1.0 2 4 6 8 10 12 AsGa As AsGa P PGa P PGa As P G a P P P As 0.5 1.00 2 4 6 8 10 12 GaAs Ga GaP Ga P As 0.5 1.00 1 2 3 4 As Ga P Ga As P a G a G a G a G a G a G a G a G a G a A s A s A s A s A s < N N N co o rd in at io n n u m b er > < N N N co o rd in at io n n u m b er > < N N co o rd in at io n n u m b e r> A s 0.5 1.00 1 2 3 4 Ni 3(Al 1-xFex ) 1 NiFe NiAl x (Fe relative content) C o o rd in at io n n u m b e rs c 0.5 0.5 0.5 1.0 1.0 1.0 0 0 0 2 4 6 8 10 12 Mn Mn 2 4 6 8 10 12 SeMn Se SeZn Se Mn 1 2 3 4 Mn Zn < N N N co o rd in at io n n u m b er > < N N N co o rd in at io n n u m b er > < N N co o rd in at io n n u m b e r> M n M n M n M n M n S e S e S e S e S e S e S e S e S e S e Z n Z n Z n M n Z n Z n Z n b Fig. 3. Average coordination numbers as a function of relative content x: comparison of model best fit curves using SOP values deduced from distance measurements [{0.98,1.07,1.03} GaAsxP1–x [9] (a); {0.62,1.67,0} ZnMnxSe1–x [10] (b); {1.01,0.86,1.33} Ni3(Al1–xFex)1 [11] (c)] versus independently measured coordination number data. width, and the oscillator strength (OS). Note that while {iZ�k and iZ�k} are prime values, {iZSk} are sums over all the specific OSs {iZsk} multiplied by the rela- tive ion-pair populations, taking into account ion-pair multiplicities (Eq. (4)), and by three SOP parame- ters {W1,W2,W3}, which express the thermodynamics of the considered alloy. The introduction of SOPs links them to the OS of each ternary line. The sum Im[ ( , )] ( ), � � � � � � � x Sj j j j jj n � � � 2 2 2 2 2 1 � �� (2) describes the total activity of all the oscillators over the frequency range considered. In zinc blende ter- nary A1–xBxZ (or AYyZ1–y) compounds, each vibrat- ing ion dipole pair AZ and BZ from each of the five elemental tetrahedron configurations {Tk}k=0,4 con- tributes a phonon line to the spectrum (this idea was first presented by Verleur and Backer [14], who pro- posed a pioneering single-parameter model; the limits of the model were later discussed by us [15]). Thus Im[��,x)] of the A1–xBxZ spectra can be written as Im[ ( , )] ( ) ( )� � � � � � �� x s P x� � 4 0 0 2 0 2 0 2 0 2 2 0 AZ AZ AZ AZ AZ � � binary AZ � � � � �� k sk k k k kk BZ BZ BZ BZ BZ � � � � � 2 2 2 2 2 2 13 4� �( ) ( , k s P x k k k k k k ) ( ) ( ) AZ AZ AZ AZ AZ � � � � � 2 2 2 2 2 2 � �� ternary ABZ � 4 4 4 4 2 4 2 2 4 2 2 4 BZ BZ BZ BZ BZ S P x � � � � � � � �( ) ( ) binary BZ (3) with the Pk(x)’s defined in Eqs. (1). Thus, the OS iZSk of each mode can be expressed by BZS k (x) = BZs k k P k (x) and AZS k (x) = AZs k (4–k) P k (x), (4) with the specific OS AZs0 = AZs and BZs4 = BZs proper to the two binary constituents, and to {AZsk}k=0,3 {BZsk}k=1,4 of the three ternary configurations. If all four specific OSs for a given iZ pair are equal and independent of x, i.e., {BZs k } k=1,4 = BZs and {AZs k } k=0,3 = AZs for a random distribution of A and B ions, the total OS of the respective modes AZ and BZ of A1–xBxZ al- loys is reduced identically to two linear functions of x, Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 1231 Table 4 Standard molar enthalpies of formation, �fH 0, kJ/mol, of materials and corresponding SOP coefficients. Column (§) gives origin and comments for reported SOP coefficient values derived from: c — coordination number data, d — distance data, 0* — assumed W3 = 0 value [10] Material components � f H0, kJ/mol SOP (§) ABZ/ AYZ AZ+BZ/ AZ+AY AZ BZ/ AY W 0 W 1 W 2 W 3 W 4 ZnMnSe ZnSe +MnSe –163.0 a –106.7 b 1 0.67 1.67 0.04 1 c 1 0.62 1.70 0 1 d GaInAs GaAs +InAs –71.0 a –58.6 a,c 1 0.58 0.25 1.05 1 d GaAsP GaP +GaAs –88 a –71.0 a,c 1 0.93 1.15 1.07 1 d ZnMnS ZnS +MnS –206.0 a, –205.98 c ! –214.2 a,c, –207.0 b 1 1.78 0 0.01 1 d ZnMnTe ZnTe +MnTe –92.7 a,c , –120.5 b –94.7 b 1 0.25 2.0 0.01 1 d CdMnTe CdTe +MnTe –102.5 b –94.7 b 1 0.68 1.33 0* 1 d+0* BZ BZS x x sk k ( ) ,� � 0 4 4 and AZ AZS x x sk k ( ) ( – ) ,� � 0 4 4 1 , (5) often referred to as the linear dependence on x. To treat FIR spectra we make two FIR assumptions. 1. Specific OSs relative to a given ion pair is the same for all configurations, {BZs k } k=1,4 = BZs , {AZs k } k=0,3 = AZs . (6) 2. Analogously, we assume that for each of the two constituent ion pair populations, the line widths � of any given composition spectrum are invariant: {AZ� k } k=0,3= AZ� and {BZ� k } k=1,4= BZ�. (7) Thus to deconvolve a ternary spectrum with its 8 lines/spectrum, we have THREE SOP coefficients and TWO OS coefficients! As was shown, the true tetrahedron populations in crystal lattices are determined by the alloy composi- tion «x» (or «y» for AYyZ1–y compounds) and the three SOP coefficients {W1, W2, W3}. To assess the credibility of the model FIR assump- tions, a best-fit test is carried out to «derive» the two binary OS {AZs, BZs} values from the GaAsyP1–y spec- tra (Fig. 4) [14] that have a rich documentation in EXAFS [13], yielding SOP values. 1232 Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 B.V. Robouch, E.M. Sheregii, and A. Kisiel 200 250 300 350 400 450 0 50 100 Frequency, cm –1 Frequency, cm –1 200 250 300 350 400 450 0 50 100 150 GaAs yP 1–y R e fle ct iv ity ,% Im [ ( ,y )] sp e ct ra � � a b Fig. 4. GaAsyP1–y: reflectivity [%] spectra [14] (a); corre- sponding normalized Im [��,y)] Kramers–Kronig derived spectra (b). Curves: y = 0.01 (solid), 0.15 (dashed), 0.44 (dash-dotted), 0.72 (dash-dot-dotted), 0.94 (dotted). Table 5 Individual deconvolution of GaAsyP1–y spectra for each spectrum parameter {iGa�k, iGa�}i=As,P;k=0,4 and {GaAss, GaPs} taken as free. {W1=0.975, W2=1.0715, W3=1.034, as per Wu et al. data}: table of best fit parameter values for dipoles GaAs k = 1,4 and for GaP k = 0,3. Amplitudes are given to two decimals. Phonon dipole GaAs GaP configuration = 4As0P k = 4 3As1P 3 2As2P 2 1As3P 1 0As4P 0 1As3P 1 2As2P 2 3As1P 3 y As � [cm–1] 1% 257.8 261.9 264.3 268.4 363.5 365.5 377.0 379.0 15% 259.6 261.6 263.6 271.2 362.9 364.9 366.9 381.0 44% 265.1 269.2 271.2 273.2 350.7 352.7 363.6 373.9 72% 269.7 271.7 273.7 279.6 342.8 344.8 350.9 360.6 94% 269.5 271.5 278.5 280.5 345.8 347.8 349.8 351.8 � [cm–1] 1% 8.64 4.06 15% 10.04 6.58 44% 10.74 11.52 72% 6.00 9.24 94% 3.99 11.50 A 1% 0.00 0.00 0.00 0.04 1.84 0.05 0.00 0.00 (given 15% 0.02 0.01 0.08 0.17 1.05 0.55 0.09 0.01 to two 44% 0.13 0.21 0.28 0.12 0.26 0.57 0.42 0.11 decimals) 72% 0.52 0.46 0.18 0.02 0.02 0.13 0.31 0.26 94% 1.63 0.28 0.02 0.00 0.00 0.00 0.03 0.17 4. Summary and conclusions The spectrum of any pure canonical, zinc blende ter- nary ABZ (or AYZ) material with its 5 tetrahedron con- figurations {Tk} exhibits 8 phonon lines (4AZ + 4BZ). The number of lines can be less than 8 when in pure defect-free materials extreme preferences prevent the formation of some configuration; this is observed with a transient element in B = {Mn, Fe,...} [10,11]: ZnMnSe lacks one, ZnMnS lacks two; however, GaAlN also lacks one with another nearly evanescent [16]. Thus only 6, 4, 4 intense + 2 weak phonon lines, respectively. More than 8 lines are observed when point defects occur (antisites, vacancies, …), respon- sible for the extra lines (as reported for HgCdTe [17]). Thus FIR admits the detection of defects: va- cancies, intersites, antisites, H-loading deformations, quantifying amount of impurity ions [18]. The statistical model of the optical dielectric func- tion is applied to five GaAsyP1–y (of type AYxZ1–x) FIR spectra [14] (see Fig. 4). In spite of the restricted number of parameters, the results show a good fit of the spectra (Fig. 5, Table 5); but most important, the best fit yields for GaAss and GaPs values that overlap with published values within the uncertainty bars (Table 6). Such a satisfactory reproduction validates the mo- del assumptions and gives confidence the model is use- ful in giving a deeper understanding of the FIR re- sults. Equations, tables and figures taken from our previ- ous publications (as referenced) are documented in greater detail. Part of the work was supported by the EU TARI- project contract HPRI-CT-1999-00088. Statistical strained-tetrahedron model of local ternary zinc blende crystal structures Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 1233 250 300 350 400 0 50 100 150 Experimental data Spectrum fit 4As0P 0As1P 2 As 2 P 1 As 3 P 0As1P 2As2P 1 As 3 P 0 As 4 P 250 300 350 400 0 50 100 150 250 300 350 400 0 50 100 150 GaAs GaP Legend 250 300 350 400 0 50 100 150 250 300 350 400 0 50 100 150 y = 0.01 y = 0.15 y = 0.44 y = 0.72 y = 0.94 Frequency, cm –1 Im [ ( ,y )] � � Im [ ( ,y )] � � Im [ ( ,y )] � � Im [ ( ,y )] � � Im [ ( ,y )] � � { { Fig. 5. Model best fit unfolding of GaAsyP1–y Im [��,y)] spectrum for SOP coefficients {W1=0.98, W2=1.07, W3 = 1.03}. The four line bands of {GaAs�k}k=1,4 and of {GaP�k}k=0,3 are distinctly seen below and above � = 300 cm–1, respectively; experimental points (circles), best fit (solid lines), deconvolved lines (various discontin- uous lines); the frequencies and intensities obtained are given in Table 5. Table 6 Individual unfolding of GaAsyP1–y spectra, {W1=0.975, W2=1.0715, W3=1.034}: table of best fit values {GaAssy, GaPsy}, their average values, and comparison to values in literature. An asterisk indicates unreliable low-signal values. y As GaAs GaP 1% * 1.97 15% 2.18 1.98 44% 1.68 2.49 72% 1.58 2.77 94% 2.00 * All 5 together (global values) 1.75 1.98 Aver. experimental 1.84±0.11 2.24±0.17 Aver. literature 1.815±0.21 2.06±0.16 ratio 0.90±0.14 1.15±0.17 1. P.A. Lee, P.H. Citrin, P. Eisenberger, and B.M. Kin- caid, Rev. Mod. Phys. 93, 769 (1981). 2. B.K. Teo, EXAFS: Basic Principles and Data Analy- sis, Springer Verlag (1986). 3. J.C. Mikkelsen and J.B. Boyce, Phys. Rev. B28, 7130 (1983). 4. A. Balzarotti, M.T. Czyzyk, A. Kisiel, N. Motta, M. Podgorny, and M. Zimnal-Starnawska, Phys. Rev. B30, 2295 (1984), ibid. 31, 7526 (1985). 5. A.S.Barker and J. Sievers, Rev. Mod. Phys. 47, 51 (1975). 6. D.W. Taylor, in: Dynamical Properties of Solids, vol. 2, G.H. Horton and A.A. Maradudin (eds.), North-Holland, N.-Y. (1975). 7. D.W Taylor, in: Optical Properties of Mixed Cry- stals, R.J. Elliot and I.P. Ipatova (eds.), Elsevier Science Publishers B.V. (1988), p. 35. 8. Sadao Adachi, Optical Properties of Crystalline Solids and Amorphous Semiconductors. Materials and Fundamental Principles, Kluver Academic Publishers, Boston (1999). 9. B.V. Robouch, A. Kisiel, and J. Konior, J. Alloys Compounds 339, 1 (2002). 10. B.V. Robouch, A. Kisiel, and J. Konior, J. Alloys Compounds 340, 13 (2002). 11. B.V. Robouch, E. Burattini, A. Kisiel, A.L. Suvorov, and A.G. Zaluzhnyi, J. Alloys Compounds 359, 73 (2003). 12. B.V. Robouch, E.M. Sheregii, and A. Kisiel, Phys. Status Solidi (in print) 13. Z. Wu, K. Lu, Y. Wang, J. Dong, H. Li, Ch. Li, and Zh. Fang, Phys. Rev. B48, 8694 (1993). 14. H.W. Verleur and A.S. Barker, Phys. Rev. 149, 715 (1966). 15. B.V. Robouch, A. Kisiel, and E.M. Sheregii, Phys. Rev. B64, 73204 (2001). 16. B.V. Robouch, A. Kisiel, I. Kutcherenko, and L.K. Vodopyanov, Appl. Phys. Lett. (submitted). 17. S.P. Kosyrev, L.K. Vodopyanov, R. Triboulet, Phys. Rev. B58, 1374 (1998). 18. A. Kisiel, B.V. Robouch, E. Burattini, A. Marcelli, M. Piccinini, M. Cestelli Guidi, P. Calvani, A. Nuca- ra, E.M. Sheregii, J. Polit, and J. Cebulski, 5-th In- ternational Ural Seminar Radiation Damage Physics of Metals and Alloys, Book of abstracts, Snezhinsk, Russia (2003). 1234 Fizika Nizkikh Temperatur, 2004, v. 30, No. 11 B.V. Robouch, E.M. Sheregii, and A. Kisiel

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