Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects
Abstract
:1. Introduction
1.1. General Issues
1.2. Cationic Substitutions
1.3. PbX-CdX (X = Te, Se, S) Solid Solution
1.4. Knowledge on Thermostructural and Elastic Properties for PbTe and Pb1−xCdxTe
1.5. Pb-Te and PbTe-CdTe System
1.6. Aim
2. Materials and Methods
3. Results: Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te
3.1. Effect of Cd Substitution on Temperature Variation of Unit Cell Size, Thermal Expansion Coefficient and Cationic and Anionic Mean Square Displacements
3.1.1. General Issues
3.1.2. Variation of Unit Cell Size of PbTe and Pb0.884Cd0.116Te with Temperature
3.1.3. Variation of Thermal Expansion Coefficient with Temperature
3.1.4. Variation of Mean Square Displacements with Temperature
- (1)
- The fitted <u2>stat(T) curves for PbTe and Pb0.884Cd0.116Te behave differently. Namely:
- (a)
- The MSDs at 0 K, <u2>(T = 0), increase significantly (by about 0.002–0.004 Å2) with x rising from 0 to 0.116. We attribute this increase to the appearance of the static disorder expressed by the nonzero <u2>stat term resulting from fitting Equation (4) (the values of <u2>stat are quoted in Table 9). This effect is graphically presented in the insets of Figure 4a,b, where the variation of fitted <u2>stat values is displayed. Appearance of marginally small negative fitted value for anionic site in PbTe (instead of zero that represents the lack of disorder) is attributed to be the effect of imperfections of fitted <u2>(T) data points. The quoted values (Table 9) show that the disorder in the anionic sublattice is considerably higher than that at the cationic site. Summarizing, an increase of the static disorder term, <u2>stat, in Equation (4), from approximately zero to a value of the order of 3 × 10−3 Å2 is observed for the mixed crystal in respect to PbTe crystal. Namely, the rise is from 0.38(4) × 10−3 Å2 to 2.03(6) × 10−3 Å2 for cations, and from −0.54(7) × 10−3 Å2 (a value marginally different from zero) to 3.4(1) × 10−3 Å2 for anions.
- (b)
- At higher temperatures, the cationic MSDs are nearly equal for the two crystals, whereas the anionic ones differ markedly in the whole temperature range.
- (c)
- The slope of the cationic <u2>(T) curve decreases with rising x, whereas the anionic one apparently increases. The property of Equation (4) is that the slope of <u2>(T) is governed at high temperatures by the Debye temperature (for high slope the Debye temperature is low and vice versa; the corresponding θD values are discussed in detail in Section 3.3 and Section 4).
- (2)
- The MSDs for the cationic and anionic sites behave differently for x = 0 than for x = 0.116.
- (3)
- Comparison of Figure 4a,b shows that the cationic and anionic MSDs of Pb0.884Cd0.116Te are of comparable values in a broad temperature range. As this effect must depend on x, we expect that for x < 0.116, the <u2> values of anions are lower than those of cations, whereas for x > 0.116 (if the structure is stabilized), the anionic ones are higher.
3.2. Effect of Substitution of Cd in the PbTe Lattice on Variation of Unit-Cell Size and of Bulk Modulus with Pressure
3.3. Effect of Cd Substitution on Values of Debye Temperature
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
APW | augmented plane-wave |
BMEOS | Birch-Murnaghan Equation of State |
CDM | crystal dynamics models |
CM | calorimetry |
DFPT | density functional perturbation theory |
DFT | density functional theory |
DM | dilatometry |
DPS | double parton scattering (nuclear inelastic scattering) |
EC | elastic constants |
FP | full potential |
FPBTF | first principles Boltzmann transport framework |
GGA | generalized gradient approximation |
GULP | computer program for the symmetry adapted simulation of solids, authored by Julian D. Gale |
HPM | heat-pulse method |
HS | hydrostatic conditions |
HSEsolSOC | revised Heyd-Scuseria-Ernzerhof functional + spin-orbit coupling |
LAPW | linearized augmented plane-wave |
LDA | local density approximation |
LDY | lattice dynamics calculations |
LEDPXRD | laboratory energy-dispersive X-ray diffraction |
LKF | Lin-Kleinman formalism |
LSCXRD | laboratory single-crystal X-ray diffraction |
MD | molecular dynamics |
MSD | mean square displacement |
n.a. | not available |
ND | neutron diffraction |
NNI | nearest-neighbor interaction model by Kagan and Maslow |
NPD | neutron powder diffraction |
NS | neutron scattering |
PAW | projector augmented wave method |
PBE | Perdew-Bucke-Ernzerhof exchange-correlation functional |
PBEsol | Perdew–Bucke–Ernzerhof exchange-correlation functional revised for solids |
PM | the paramagnetic resonance. θD estimated by the temperature-dependent hyperfine splitting constant A(T) |
PTW | plane temperature waves method |
QHA | quasiharmonic approximation |
QHS | quasi-hydrostatic conditions |
RT | room temperature |
SCXRD | single crystal X-ray diffraction |
SCXRDS | single crystal X-ray difraction at synchrotron |
SO | soft-constraint based online |
SME | slave mode expansion |
SP | spectroscopy |
SPXRD | synchrotron powder X-ray diffraction |
SV | sound velocity method |
TC | thermal conductivity |
TEC | thermal expansion coefficient |
THD | thermodynamic calculations |
UPE | ultrasonic pulse-echo method |
UIM | ultrasonic interferometry |
UWV | ultrasonic wave velocity |
UWVSC | ultrasonic wave velocity in single crystal |
XRD/ND/PDF | X-ray diffraction and neutron diffraction, analyzed with pair distribution function (PDF) method |
Appendix A
Appendix B
T [K] | PbTe | Pb0.884Cd0.116Te | ||||
---|---|---|---|---|---|---|
a [Å] | <uC2> [Å2] | <uA2> [Å2] | a [Å] | <uC2> [Å2] | <uA2> [Å2] | |
15 | 6.4298(4) | 0.0020(1) | 0.0016(2) | 6.3775(5) | 0.0041(2) | 0.0057(3) |
20 | 6.4299(3) | 0.0027(1) | 0.0018(2) | 6.3773(4) | 0.0041(2) | 0.0055(3) |
25 | 6.4301(3) | 0.0024(1) | 0.0018(2) | 6.3776(5) | 0.0047(2) | 0.0055(3) |
30 | 6.4301(4) | 0.0028(1) | 0.0016(2) | 6.3777(5) | 0.0046(2) | 0.0052(3) |
35 | 6.4305(4) | 0.0030(1) | 0.0018(2) | 6.3783(5) | 0.0047(2) | 0.0055(3) |
40 | 6.4310(4) | 0.0037(1) | 0.0010(2) | 6.3787(7) | 0.0046(2) | 0.0059(3) |
45 | 6.4312(3) | 0.0036(1) | 0.0019(2) | 6.3792(6) | 0.0053(2) | 0.0058(3) |
50 | 6.4316(3) | 0.0038(1) | 0.0021(2) | 6.3793(6) | 0.0054(2) | 0.0060(3) |
55 | 6.4323(3) | 0.0044(1) | 0.0027(2) | 6.3800(7) | 0.0053(2) | 0.0066(3) |
60 | 6.4327(3) | 0.0047(1) | 0.0021(2) | 6.3805(5) | 0.0047(2) | 0.0067(3) |
65 | 6.4333(3) | 0.0052(1) | 0.0024(2) | 6.3810(7) | 0.0054(2) | 0.0064(3) |
70 | 6.4338(3) | 0.0059(1) | 0.0026(2) | 6.3817(6) | 0.0056(2) | 0.0071(3) |
75 | 6.4342(5) | 0.0057(2) | 0.0029(2) | 6.3823(5) | 0.0072(2) | 0.0072(3) |
80 | 6.4347(5) | 0.0058(2) | 0.0025(2) | 6.3829(6) | 0.0071(2) | 0.0074(3) |
85 | 6.4350(3) | 0.0062(1) | 0.0034(2) | 6.3833(5) | 0.0069(2) | 0.0084(3) |
90 | 6.4358(4) | 0.0064(2) | 0.0034(2) | 6.3840(4) | 0.0078(2) | 0.0077(3) |
95 | 6.4364(5) | 0.0072(2) | 0.0035(3) | 6.3848(4) | 0.0075(2) | 0.0087(3) |
100 | 6.4371(4) | 0.0074(2) | 0.0026(3) | 6.3855(4) | 0.0079(2) | 0.0089(3) |
110 | 6.4381(3) | 0.0083(2) | 0.0034(2) | 6.3863(5) | 0.0085(2) | 0.0079(4) |
120 | 6.4394(3) | 0.0092(2) | 0.0043(3) | 6.3875(4) | 0.0094(2) | 0.0092(4) |
130 | 6.4408(4) | 0.0095(2) | 0.0044(3) | 6.3891(4) | 0.0095(2) | 0.0094(3) |
140 | 6.4418(4) | 0.0100(2) | 0.0060(3) | 6.3903(4) | 0.0099(2) | 0.0097(4) |
150 | 6.4429(3) | 0.0111(2) | 0.0057(3) | 6.3915(4) | 0.0102(2) | 0.0104(4) |
160 | 6.4440(3) | 0.0117(2) | 0.0059(3) | 6.3931(5) | 0.0115(2) | 0.0115(4) |
170 | 6.4451(3) | 0.0120(2) | 0.0069(3) | 6.3941(5) | 0.0108(2) | 0.0129(4) |
180 | 6.4466(3) | 0.0118(2) | 0.0064(3) | 6.3953(5) | 0.0117(2) | 0.0103(4) |
190 | 6.4477(3) | 0.0133(2) | 0.0073(3) | 6.3969(5) | 0.0121(2) | 0.0129(4) |
200 | 6.4491(4) | 0.0144(2) | 0.0072(3) | 6.3981(5) | 0.0127(2) | 0.0132(4) |
210 | 6.4503(3) | 0.0147(2) | 0.0070(3) | 6.3997(4) | 0.0132(2) | 0.0131(4) |
220 | 6.4514(4) | 0.0142(2) | 0.0097(3) | 6.4007(5) | 0.0137(3) | 0.0138(4) |
230 | 6.4527(3) | 0.0167(2) | 0.0085(3) | 6.4020(4) | 0.0156(3) | 0.0137(4) |
240 | 6.4539(3) | 0.0170(3) | 0.0088(3) | 6.4035(3) | 0.0170(3) | 0.0135(4) |
250 | 6.4552(3) | 0.0166(2) | 0.0096(3) | 6.4045(3) | 0.0173(3) | 0.0153(4) |
260 | 6.4564(3) | 0.0171(2) | 0.0094(3) | 6.4062(3) | 0.0160(3) | 0.0148(4) |
270 | 6.4578(3) | 0.0180(3) | 0.0109(4) | 6.4071(3) | 0.0192(3) | 0.0160(5) |
280 | 6.4588(3) | 0.0192(3) | 0.0111(4) | 6.4086(4) | 0.0171(3) | 0.0165(5) |
290 | 6.4602(3) | 0.0178(3) | 0.0123(4) | 6.4100(4) | 0.0188(3) | 0.0176(5) |
300 | 6.4616(3) | 0.0195(3) | 0.0124(4) | 6.4116(4) | 0.0200(3) | 0.0166(5) |
T [K] | PbTe | Pb0.884Cd0.116Te | ||||||
---|---|---|---|---|---|---|---|---|
a [Å] | α [MK−1] | <uC2> [Å2] | <uA2> [Å2] | a [Å] | α [MK−1] | <uC2> [Å2] | <uA2> [Å2] | |
0 | 6.42972(5) | 0 | 0.0021(1) | 0.0011(1) | 6.37725(6) | 0 | 0.0036(1) | 0.0052(1) |
10 | 6.42973(5) | 0.70(7) | 0.0022(1) | 0.0012(1) | 6.37726(6) | 0.9(1) | 0.0037(1) | 0.0053(1) |
20 | 6.42986(6) | 4.2(3) | 0.0025(1) | 0.0013(1) | 6.37742(7) | 5.0(4) | 0.0040(1) | 0.0054(1) |
30 | 6.43026(9) | 8.5(4) | 0.0030(1) | 0.0015(1) | 6.37788(11) | 9.8(6) | 0.0043(1) | 0.0056(1) |
40 | 6.43091(12) | 11.7(4) | 0.0035(1) | 0.0017(1) | 6.37861(15) | 13.2(6) | 0.0048(1) | 0.0059(1) |
50 | 6.43173(15) | 13.9(4) | 0.0041(1) | 0.0020(1) | 6.37952(18) | 15.4(6) | 0.0052(1) | 0.0062(1) |
60 | 6.43266(17) | 15.3(4) | 0.0047(1) | 0.0023(1) | 6.38054(22) | 16.9(5) | 0.0057(1) | 0.0066(1) |
70 | 6.43367(20) | 16.2(4) | 0.0053(1) | 0.0027(1) | 6.38165(25) | 17.8(5) | 0.0062(1) | 0.0070(2) |
80 | 6.43474(22) | 16.9(4) | 0.0059(1) | 0.0030(1) | 6.38281(28) | 18.5(5) | 0.0068(1) | 0.0074(2) |
90 | 6.43584(24) | 17.4(4) | 0.0066(1) | 0.0033(1) | 6.38400(31) | 19.0(5) | 0.0072(2) | 0.0079(2) |
100 | 6.43697(27) | 17.8(4) | 0.0072(1) | 0.0037(1) | 6.38523(35) | 19.3(5) | 0.0078(2) | 0.0083(2) |
110 | 6.43812(29) | 18.0(4) | 0.0079(1) | 0.0041(1) | 6.38647(38) | 19.6(5) | 0.0084(2) | 0.0087(2) |
120 | 6.43929(31) | 18.3(4) | 0.0085(1) | 0.0045(1) | 6.38773(41) | 19.8(5) | 0.0089(2) | 0.0091(2) |
130 | 6.44047(34) | 18.5(4) | 0.0092(1) | 0.0049(1) | 6.38900(44) | 20.0(5) | 0.0095(2) | 0.0096(2) |
140 | 6.44167(36) | 18.6(4) | 0.0098(1) | 0.0052(2) | 6.39028(47) | 20.1(5) | 0.0100(2) | 0.0100(2) |
150 | 6.44287(39) | 18.7(4) | 0.0105(1) | 0.0056(2) | 6.39157(51) | 20.2(5) | 0.0106(2) | 0.0105(2) |
160 | 6.44408(41) | 18.8(4) | 0.0111(1) | 0.0060(2) | 6.39287(54) | 20.3(6) | 0.0111(2) | 0.0109(2) |
170 | 6.44529(44) | 18.9(4) | 0.0118(1) | 0.0064(2) | 6.39417(58) | 20.4(6) | 0.0117(2) | 0.0113(2) |
180 | 6.44652(47) | 19.0(4) | 0.0125(1) | 0.0068(2) | 6.39547(62) | 20.4(6) | 0.0122(3) | 0.0118(2) |
190 | 6.44774(50) | 19.1(4) | 0.0131(1) | 0.0072(2) | 6.39678(66) | 20.5(6) | 0.0128(3) | 0.0122(2) |
200 | 6.44898(53) | 19.1(5) | 0.0138(1) | 0.0076(2) | 6.39809(70) | 20.5(6) | 0.0133(3) | 0.0127(3) |
210 | 6.45021(56) | 19.2(5) | 0.0144(1) | 0.0080(2) | 6.39941(74) | 20.6(6) | 0.0139(3) | 0.0131(3) |
220 | 6.45145(59) | 19.3(5) | 0.0151(1) | 0.0084(2) | 6.40073(78) | 20.6(7) | 0.0144(3) | 0.0136(3) |
230 | 6.45270(62) | 19.3(5) | 0.0158(1) | 0.0088(2) | 6.40205(82) | 20.6(7) | 0.0150(3) | 0.0140(3) |
240 | 6.45394(65) | 19.3(5) | 0.0164(1) | 0.0091(2) | 6.40337(87) | 20.7(7) | 0.0156(3) | 0.0145(3) |
250 | 6.45519(69) | 19.4(5) | 0.0171(1) | 0.0095(2) | 6.40469(91) | 20.7(7) | 0.0161(3) | 0.0149(3) |
260 | 6.45644(72) | 19.4(5) | 0.0177(1) | 0.0099(2) | 6.40602(96) | 20.7(7) | 0.0167(3) | 0.0154(3) |
270 | 6.45770(76) | 19.5(6) | 0.0184(1) | 0.0103(2) | 6.40734(100) | 20.7(8) | 0.0172(4) | 0.0158(3) |
280 | 6.45896(79) | 19.5(6) | 0.0191(1) | 0.0107(2) | 6.40867(105) | 20.7(8) | 0.0178(4) | 0.0163(3) |
290 | 6.46022(83) | 19.5(6) | 0.0197(2) | 0.0111(2) | 6.41000(110) | 20.7(8) | 0.0183(4) | 0.0167(3) |
300 | 6.46148(87) | 19.6(6) | 0.0204(2) | 0.0115(2) | 6.41133(116) | 20.7(8) | 0.0189(4) | 0.0172(3) |
PbTe | Pb0.884Cd0.116Te | ||
---|---|---|---|
p [GPa] | V [Å3] | p [GPa] | V [Å3] |
0.33(2) | 271.1(15) | 0.30(2) | 265.40(50) |
0.80(2) | 267.9(12) | 0.80(2) | 261.10(80) |
1.30(2) | 266.8(15) | 1.40(2) | 256.50(16) |
1.80(2) | 264.8(16) | 2.50(2) | 251.78(10) |
2.40(2) | 259.5(16) | 3.00(2) | 247.41(12) |
3.00(2) | 257.2(15) | 3.50(2) | 245.37(11) |
3.70(2) | 254.7(12) | 4.00(2) | 241.37(14) |
4.50(2) | 251.8(15) | 4.50(2) | 241.23(14) |
p [GPa] | PbTe | Pb0.884Cd0.116Te | ||
---|---|---|---|---|
V [Å3] | K [GPa] | V [Å3] | K [GPa] | |
0 | 273.25 | 45.6(2.5) | 273.25 | 33.5(2.8) |
0.5 | 270.33 | 47.6(2.5) | 270.33 | 35.5(2.8) |
1.0 | 267.57 | 49.6(2.5) | 267.57 | 37.5(2.8) |
1.5 | 264.93 | 51.5(2.5) | 264.93 | 39.4(2.8) |
2.0 | 262.42 | 53.5(2.5) | 262.42 | 41.3(2.8) |
2.5 | 260.02 | 55.4(2.5) | 260.02 | 43.2(2.8) |
3.0 | 257.72 | 57.3(2.5) | 257.72 | 45.1(2.8) |
3.5 | 255.52 | 59.1(2.5) | 255.52 | 47.0(2.8) |
4.0 | 253.41 | 61.1(2.5) | 253.41 | 48.8(2.8) |
4.5 | 251.37 | 62.9(2.5) | 251.37 | 50.7(2.8) |
5.0 | 249.41 | 64.8(2.5) | 249.41 | 52.5(2.8) |
Appendix C
Function | Fitted Equation | Parameters for PbTe | Parameters for Pb0.884Cd0.116Te |
---|---|---|---|
V(T) | Equation (1) | = 265.813(6) Å3, Q = 2.86(3) × 10−18, b = 1.3(6), θD = 135.2(3.9) K | = 259.358(7) Å3, Q = 2.63(4) × 10−18, b = 0.4(7), θD = 130.1(4.4) K |
<u2>(T), for cationic site | Equation (4) | θD = 102.8(3) K, <u2>stat = 0.00038(4) Å2 | θD = 114.5(5) K, <u2>stat = 0.00203(6) Å2 |
<u2>(T), for anionic site | Equation (4) | θD = 169.2(1.1) K, <u2>stat = −0.00054(7) Å2 | θD = 158.1(1.3) K, <u2>stat = 0.0034(1) Å2 |
V(p) | Equation (6) | V0 = 273.3(7) Å3 K0 = 45.6(2.5) GPa K′ = 4 (fixed) | V0 = 267.7(1.5) Å3 K0 = 33.5(2.8) GPa K′ = 4 (fixed) |
Appendix D
Compound | K0 [GPa] | K′ | Method | Ref. | Year |
---|---|---|---|---|---|
PbTe | 45 | n.a. | LDA | ||
48 | n.a. | LDA | (a) | 1983 | |
51.7 | 4.52 | LAPW LDA | (b) | 1997 | |
51.44 (0 K) 40.30 (0 K) 49.82 (0 K) 39.5 (0 K) | 5.50 4.27 5.76 3.92 | FP-LAPW LDA FP-LAPW GGA FP-LAPW LDA+SO FP-LAPW GGA+SO | (c) | 2000 | |
41.4 51.4 | 3.352 4.080 | GGA LDA | (d) | 2002 | |
37.5 (0 K) | n.a. | FP-APW PBE | (e) | 2007 | |
40.4 (0 K) 50.3 (0 K) 30.7 | n.a. n.a. n.a. | LDA/GGA “ “ | (f) | 2009 | |
39.05 | 4.32 | FP-LAPW | (g) | 2011 | |
46.0 46.1 | 4.27 4.53 | LDA LDA+SO | (h) | 2012 | |
41.0 | n.a. | MD (GULP) | (i) | 2012 | |
39.1 | n.a. | PAW PBE | (j) | 2013 | |
47 (0 K) | n.a. | FP-LAPW | (k) | 2014 | |
38.54 (300 K) ~45.5 (0 K) | n.a. n.a. | PBEsol “ | (l) | 2014 | |
34.04 (0 K) | n.a. | LDA, GGA | (m) | 2014 | |
46.61 (0 K) | n.a. | PBEsol | (n) | 2015 | |
44.1(&) | n.a. | HSEsolSOC | (o) | 2016 | |
41.1 | n.a. | FP-LAPW | (p) | 2017 | |
36.19 (100 K) 37.52 (300 K) | n.a. n.a. | LDY “ | (q) | 2019 | |
48.242 (0 K) | 5.576 (0 K) | LDA | (r) | 2020 | |
43.6 | 4.6 | GGA-PBE | (s) | 2020 | |
Pb0.969Cd0.031Te | 46.42 | n.a. | GGA | (n) | 2015 |
Pb0.884Cd0.116Te | 45.90 | n.a. | GGA | ($) | 2021 |
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Compound | a(x) [Å] or V(x) [Å3] | xmax | aex [Å] | Ref. | Year |
---|---|---|---|---|---|
Pb1−xCdxTe | 6.459–0.30x | 0.20 (at 1139 K) | 6.159 | (a) | 1964 |
6.459–0.40x | 0.144 | 6.059 | (b) | 1980 | |
6.466–0.414x | 0.75 | 6.037 (*) | (c) | 1989 | |
6.462–0.433(5)x | 0.114 | 6.029 | (d) | 2009 | |
6.38–0.434x (&) | - | - | (e) | 2012 | |
Pb1−xCdxSe | 6.127–0.42x | 0.26 (at 1213 K) | 5.707 | (f) | 1965 |
6.128–0.38x - | 0.03 (at 523 K) 0.18 (at 873 K) | - | (g) | 1968 | |
- | ~0.057 (at 673 K) | - | (h) | 1973 | |
6.1263–0.3025x | 0.04 | - | (i) | 2019 | |
Pb1−xCdxS | 203.151–0.4389x ($) | 0.016 | - | (j) | 1971 |
5.9386–0.4302x | 0.40 | 5.5084 | (k) | 2014 | |
5.412 (&) | (l) | 2019 | |||
5.435, 5.45, 5.72 | (m) | 2021 |
Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|
Experiment | PbTe | 0–400 ($) | n.a. | (a) | 1971 |
PbTe | 120–298 | SCXRD | (b) | 1987 | |
PbTe | 15–500 | XRD/ND/PDF | (c) | 2010 | |
PbTe | 105–1000 | SPXRD | (d) | 2013 | |
PbTe | 105–600 | SPXRD | (e) | 2016 | |
PbTe | 10–500 | ND | (f) | 2016 | |
PbTe | 125–293 | SCXRDS | (g) | 2018 | |
PbTe | 50–600 | NPD | (h) | 2021 | |
PbTe | 20–622 | SCXRDS | (i) | 2021 | |
PbTe | 15–300 | SPXRD | this work | 2021 | |
Pb0.987Cd0.013Te, | 300–~600, ~900–1073, | SPXRD | (j) | 2009 | |
Pb0.944Cd0.056Te, | 300–~430, ~970–1073, | “ | “ | “ | |
Pb0.904Cd0.096Te | 300–~350 | “ | “ | “ | |
Pb0.98Cd0.02Te | 15–300 | SPXRD | (k) | 2011 | |
Pb0.884Cd0.116Te | 15–300 | SPXRD | this work | 2021 | |
Theory | PbTe | 0–300 | LDA, GGA | (l) | 2009 |
PbTe | 4–550 | PBEsol | (m) | 2014 | |
PbTe | 0–300 | LDA, GGA | (n) | 2014 | |
PbTe | 100–800 | QHA | (o) | 2018 | |
PbTe | 300–800 (*) | MD | (p) | 2018 | |
PbTe | 0–800 (&) | DFPT/LDA | (q) | 2019 |
Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|
Experiment | PbTe | 30-340 | DM | (a) | 1963 |
PbTe | 4–297 | CM | (b) | 1968 | |
PbTe | 10–500 | ND | (c) | 2016 | |
PbTe | 50–600 | NPD | (d) | 2021 | |
PbTe | 15–300 | SPXRD | this work | 2021 | |
Pb0.884Cd0.116Te | 15–300 | SPXRD | this work | 2021 | |
Theory | PbTe | 70–300 | CDM | (e) | 1966 |
PbTe | 0–300 | LDA, GGA | (f) | 2009 | |
PbTe | 0–300 | PBEsol | (g) | 2014 | |
PbTe | 0–300 | LDA, GGA | (h) | 2014 | |
PbTe | 0–350 | GGA | (i) | 2015 | |
PbTe | 0–300 | FPBTF | (j) | 2017 | |
PbTe | 0–800 | DFPT/LDA | (k) | 2019 |
Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|
Experiment | PbTe | 78–400 | PXRD, SCXRD | (a) | 1973 |
PbTe | 100–300 | SCXRD | (b) | 1978 | |
PbTe | 120–298 | SCXRD | (c) | 1987 | |
PbTe | 15–500 | XRD/ND/PDF | (d) | 2010 | |
PbTe | 105–1000 | SPXRD | (e) | 2013 | |
PbTe | 8–500 | SPXRD | (f) | 2014 | |
PbTe | 105–600 | SPXRD | (g) | 2016 | |
PbTe | 10–500 | ND | (h) | 2016 | |
PbTe (*) | 100–450 | SCXRDS | (i) | 2018 | |
PbTe | 20–300 | SCXRDS | (j) | 2021 | |
PbTe | (40)–700 | NPD | (k) | 2021 | |
PbTe | 15–300 | SPXRD | this work | 2021 | |
Pb0.884Cd0.116Te | 15–300 | SPXRD | this work | 2021 | |
Theory | PbTe | 0–400 | LKF | (l) | 1968 |
PbTe | 0–700 | MD (SME) | (m) | 2014 | |
PbTe (*) | 100–450 | MD | (i) | 2018 | |
PbTe | 300–800 | MD | (n) | 2018 | |
PbTe | 0–800 | DFPT/LDA | (o) | 2019 |
Mode | Compound | Variables | Pressure Range [GPa] | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|---|---|
Experiment | PbTe | cp(T), cv(T) | - | 20–260 | CM | (a) | 1954 |
PbTe | γ(T) | - | 30–340 | CM+XRD | (b) | 1963 | |
PbTe | K(T), C(T) | - | 4–297 | CM | (c) | 1968 | |
PbTe | cp(T) | - | 300–700 | PTW | (d) | 1983 | |
PbTe | θD(p), γ(p) | amb.–15, amb.–10.5 | - | UIM | (e) | 2013 | |
PbTe | K(T), γ(T), cv(T) | - | 10–300/300/260 | ND | (f) | 2016 | |
PbTe | V(p), K(p), γ(T) | amb.–4.5 | - | SCXRD | this work | 2021 | |
Pb0.884Cd0.116Te | V(p), K(p), γ(T) | amb.–4.5 | - | SCXRD | this work | 2021 | |
Theory | PbTe | θD(T) | - | 0–200 | CDM | (g) | 1966 |
PbTe | K(T) | - | 0–300 | LDA, GGA | (h) | 2009 | |
PbTe | K(p), C(T) | 0–14 | - | LDA | (i) | 2012 | |
PbTe | a(p,T), α(p,T), K(p,T), θD(p,T), cv(T) | 0–10 | 0–300 | LDA, GGA | (j) | 2014 | |
PbTe | K(T) | - | 0–600 | PBEsol | (k) | 2014 | |
PbTe | cv(T) | - | 0–400 | GGA | (l) | 2015 | |
PbTe | K(T), C(T) | - | 100–500, 0–500 | LDY | (m) | 2019 | |
PbTe (*) | cp(T), cv(T) | - | ~20–1000 | THD | (n) | 2019 | |
PbTe | V(p) | - | - | LDA | (o) | 2020 | |
PbTe | cv(p) | 0–6 | - | PBEsol | (p) | 2021 | |
Pb1−xCdxTe (*) | cp(T), cv(T) | - | ~20–1000 | THD | (n) | 2019 |
Compound | T | a [Å] | Ref. | Year |
---|---|---|---|---|
PbTe | 1 | 6.42962 (*) | (a) | 2016 |
0 | 6.42972(5) (*) | this work | 2021 | |
15 | 6.42977(5) (*) | this work | 2021 | |
15 | 6.4298(4) | this work | 2021 | |
Pb0.98Cd0.02Te | 10 | 6.42114 (*) | (b) | 2011 |
Pb0.884Cd0.116Te | 0 | 6.37725(6) (*) | this work | 2021 |
Pb0.884Cd0.116Te | 15 | 6.37733(7) (*) | this work | 2021 |
Pb0.884Cd0.116Te | 15 | 6.3775(5) | this work | 2021 |
Compound | T [K] | a [Å] | Ref. | Year |
---|---|---|---|---|
PbTe | 300 | 6.46179(3), 6.46201(4) | (a) | 2013 |
300 | 6.46255 (*) | (b) | 2016 | |
300 | 6.46040(4), 6.46054(4) | (c) | 2016 | |
293 | 6.4626(1) | (d) | 2018 | |
300 | 6.4651(*) | (e) | 2021 | |
300 | 6.459–6.462 ($), <a> = 6.46148(15) | (f) | 2021 | |
300 | 6.4616(3) | this work | 2021 | |
300 | 6.46148(87) (*) | this work | 2021 | |
Pb0.987Cd0.013Te | 300 | 6.457(2) | (g) | 2009 |
Pb0.944Cd0.056Te | 300 | 6.437(2) | (g) | 2009 |
Pb0.884Cd0.116Te | 300 | 6.41133(116) (*) | this work | 2021 |
Pb0.884Cd0.116Te | 300 | 6.4116(4) | this work | 2021 |
Compound | T [K] | α [Å] | Ref. | Year |
---|---|---|---|---|
PbTe | 300 | 19.94 | (a) | 1964 |
300 | 19.91 | (b) | 2016 | |
300 | 19.36 (*) | (c) | 2019 | |
300 | 18.12 | (d) | 2021 | |
300 | 19.6(6) | this work | 2021 | |
Pb0.884Cd0.116Te | 300 | 20.7(8) | this work | 2021 |
Temperature | Compound | T [K] | <uC2>(T) [Å2] | <uA2>(T) [Å2] | <uC2>stat [Å2] | <uA2>stat [Å2] | Ref. | Year |
---|---|---|---|---|---|---|---|---|
low | PbTe | 15 | 0.0037 | --- | (a) | 2010 | ||
temperature | PbTe | 0 | 0.0018 (*) | 0.0018 (*) | (b) (#) | 2014 | ||
PbTe | 1 | 0.00200 | 0.00315 | (c) | 2016 | |||
PbTe | 0 | 0.0021(1) (*) | 0.0011(1) (*) | 0.00038(4) | −0.00054(7) | this work (#) | 2021 | |
Pb0.884Cd0.116Te | 0 | 0.0036(1) (*) | 0.0052(1) (*) | 0.00203(6) | 0.0034(1) | this work (#) | 2021 | |
room | PbTe | 300 | 0.0233(15) | 0.0209(14) | (d) | 1973 | ||
temperature | PbTe | 298 | 0.0204(3) | 0.0141(3) | (e) | 1987 | ||
PbTe | 300 | 0.0231 | --- | (a) | 2010 | |||
PbTe | 300 | 0.0098(2) | 0.01847(9) | small | small | (f) | 2013 | |
PbTe | 300 | 0.0238 | 0.0171 | (b) | 2014 | |||
PbTe | 300 | 0.0202 | 0.0136 | (g) | 2016 | |||
PbTe | 300 | 0.02155 | 0.01548 | 0.00031 | 0.00130 | (c) | 2016 | |
PbTe | 300 | 0.0260(2) | 0.0157(1) | (h) | 2018 | |||
PbTe | 300 | 0.0204(2) (*) | 0.0115(2) (*) | 0.00038(4) | −0.00054(7) | this work | 2021 | |
Pb0.884Cd0.116Te | 300 | 0.0189(4) (*) | 0.0172(3) (*) | 0.00203(6) | 0.0034(1) | this work | 2021 |
Type of Experiment | Compound | T [K] | BMEOS Parameters | Method and Remarks | Ref. | Year | ||
---|---|---|---|---|---|---|---|---|
V0 [Å3] | K0 [GPa] | K’ | ||||||
X-ray diffraction | PbTe | RT | n.a. | 38.9(1) | 5.4 | LEDPXRD | (a) | 1984 |
PbTe | RT | 269.6(4) | 44(1) | 4 (fixed) | SPXRD (QHS) | (b) | 2013 | |
PbTe | 296 | 273.3(7) | 45.6(2.5) | 4 (fixed) | LSCXRD (t) (HS) | this work | 2021 | |
Pb0.884Cd0.116Te | 296 | 267.7(1.5) | 33.5(2.8) | 4 (fixed) | LSCXRD (t) (HS) | this work | 2021 | |
other | PbTe | 0 | n.a. | 45.6(4) | - | UWVSC | (c) | 1968 |
PbTe | RT RT | n.a. n.a. | 39.76 38.39 | 5.171 4.891 | EC (s) UWV (t) | (d) | 1981 | |
PbTe | RT | n.a. | 40.5(7) | 3.8(2) | SV | (e) | 2013 | |
PbTe | 0 RT | n.a. n.a. | 44.89 41.26 | - - | UWVSC (*) | (f) | 2016 |
Compound | θDUC [K] | θDUA [K] | θDU [K] (*) | θDV [K] | Method | Ref. | Year |
---|---|---|---|---|---|---|---|
PbTe | 95.5(2.0) | 127(3) | 111.3(3.0) | - | PXRD/SCXRD | (a) | 1973 |
- | - | - | 107(2), 108(3) | LPXRD | (a) | 1973 | |
- | - | - | 111 | SCXRD | (b) | 1978 | |
- | - | 87(1) | - | XRD | (c) | 2013 | |
91(3) (&) | 175(5) (&) | - | 133(4) (&) | NPD+CM | (d) | 2016 | |
99.6(2) | 156.0(5) | 127.8(4) | - | NPD | (d) | 2016 | |
101.4 | 157.0 | 129.2 | - | SCXRD | (d) | 2016 | |
- | - | - | 129(2) | NPD | (e) | 2021 | |
102(1) | 163(2) | 132.5 | - | NPD | (e) | 2021 | |
102.0 | 161.4 | 131.7 | - | NPD/PDXRDS | ($) | 2021 | |
- | - | - | 135.2(3.8) | LPXRD | this work | 2021 | |
102.8(3) | 169(1) | 135.9(7) | - | LPXRD | this work | 2021 | |
Pb0.884Cd0.116Te | - | - | - | 130.1(4.4) | LPXRD | this work | 2021 |
Pb0.884Cd0.116Te | 115.1(5) | 158(1) | 136.6(8) | - | LPXRD | this work | 2021 |
Compound | θD [K] | Method | Ref. | Year |
---|---|---|---|---|
PbTe | 127 (at 20 K), 125 at 200 K | CM | (a) | 1954 |
PbTe | 176.7(5) (at 0 K) (*) | UWV | (b) | 1968 |
PbTe | 110 | PM | (c) | 1975 |
PbTe | 168 | HPM | (d) | 1976 |
PbTe | 140 | SP | (e) | 1979 |
PbTe | 136 | n.a. | (f) | 1998 |
PbTe | 105 | TC | (g) | 2006 |
PbTe | 163 | UPE | (h) | 2011 |
PbTe | 136 | UWV | (i) | 2012 |
PbTe | 170(5) | DPS | (j) | 2013 |
PbTe | 143 | SV | (k) | 2013 |
PbTe | 95 | NS | (l) | 2014 |
PbTe | 128(1) | CM + NPD | (m) | 2016 |
Compound | θD [K] | Method | Ref. | Year |
---|---|---|---|---|
PbTe | 167 at 0 K 131 at 300 K (&) | CDM | (a) | 1968 |
PbTe | 177(1) (&) | NNI | (b) | 1986 |
PbTe | 152 (&) | GGA | (c) | 2012 |
PbTe | 141.5 (&) ($) | LDA, GGA | (d) | 2014 |
PbTe | 187.8 (&) | GGA | (e) | 2015 |
Pb0.969Cd0.031Te | 185.4 | GGA | (e) | 2015 |
Pb0.884Cd0.116Te | 178.8 | GGA | (*) | 2021 |
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Minikayev, R.; Safari, F.; Katrusiak, A.; Szuszkiewicz, W.; Szczerbakow, A.; Bell, A.; Dynowska, E.; Paszkowicz, W. Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects. Crystals 2021, 11, 1063. https://doi.org/10.3390/cryst11091063
Minikayev R, Safari F, Katrusiak A, Szuszkiewicz W, Szczerbakow A, Bell A, Dynowska E, Paszkowicz W. Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects. Crystals. 2021; 11(9):1063. https://doi.org/10.3390/cryst11091063
Chicago/Turabian StyleMinikayev, Roman, Fatemeh Safari, Andrzej Katrusiak, Wojciech Szuszkiewicz, Andrzej Szczerbakow, Anthony Bell, Elżbieta Dynowska, and Wojciech Paszkowicz. 2021. "Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects" Crystals 11, no. 9: 1063. https://doi.org/10.3390/cryst11091063
APA StyleMinikayev, R., Safari, F., Katrusiak, A., Szuszkiewicz, W., Szczerbakow, A., Bell, A., Dynowska, E., & Paszkowicz, W. (2021). Thermostructural and Elastic Properties of PbTe and Pb0.884Cd0.116Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects. Crystals, 11(9), 1063. https://doi.org/10.3390/cryst11091063