Supplementary material for “Si-compatible candidates for high-K dielectrics with the Pbnm perovskite structure” Sinisa Coh,1, ∗ Tassilo Heeg,2 J. H. Haeni,3 M. D. Biegalski,4 J. Lettieri,3, † L. F. Edge,3 K. E. O’Brien,3 M. Bernhagen,5 P. Reiche,5 R. Uecker,5 S. Trolier-McKinstry,3 Darrell G. Schlom,2 and David Vanderbilt1 1
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019, USA Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, USA 3 Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA 16802, USA Center for Nanophase Materials Science, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA 5 Leibniz Institute for Crystal Growth, Max-Born-Straße 2, D-12489 Berlin (Adlershof ), Germany (Dated: April 26, 2010) 2
4
In this supplemental material we present the results of our calculations of the zone-center phonon frequencies, as well as the infrared activities for those modes that are infrared-active, for the compounds considered in the main article. PACS numbers: 77.22.-d,77.55.df,85.50.-n
I.
SUPPLEMENT: PHONON PROPERTIES
As stated in the main text, in the frequency range of the lattice vibrations, the dielectric tensor ǫαβ can be decomposed as ion ǫαβ (ω) = ǫel αβ + ǫαβ (ω),
(1)
where ǫel αβ is the purely electronic contribution (assumed to be independent of frequency in this range) and ǫion αβ (ω) is the ionic (or lattice) contribution. The values of the electronic contributions are given in the main text, and also repeated in this Supplement in Table I for completeness. The ionic contribution can be reconstructed from the information given in Tables II, III and IV using the relation 2 ǫion αβ (ω) = Ω0
X Zα⋆n Zβ⋆n , ωn2 − ω 2 n
(2)
where Zα⋆n =
X iβ
1 n √ Zi,αβ ζiβ Mi
is the dimensionless mode dynamical charge and s 4πe2 Ω0 = V m0
(3)
(4)
has units of a plasma frequency. Here ωn is the phonon frequency, Zi,αβ is the atomic Born effective-charge tenn sor, ζiβ is the dimensionless eigenvector of the dynamical matrix normalized to unity, Mi is the mass of i-th ion in atomic mass units, e is the electron charge, V is the unit cell volume, and m0 is the atomic mass unit. The values of Ω0 are reported in Table II. The results for the frequencies ωn and for the activities of the infrared-active modes are presented in Tables III and
IV for simple perovskites and double perovskites respectively. For the infrared-active modes, note that only the non-zero elements of the Zα⋆n tensor are reported.
2 TABLE I: Electronic part of the dielectric tensors for the compounds considered.
LaScO3 PrScO3 NdScO3 SmScO3 GdScO3 TbScO3 DyScO3 LaYO3 CaZrO3 SrZrO3 SrHfO3 LaHoO3 LaErO3
ǫel xx 5.0 5.0 5.0 4.9 4.9 4.8 4.8 4.7 4.6 4.6 4.3 4.7 4.7
Electronic part ǫel yy 5.0 5.0 4.9 4.9 4.8 4.8 4.8 4.6 4.7 4.6 4.3 4.6 4.6
ǫel zz 4.8 4.8 4.7 4.7 4.6 4.6 4.5 4.3 4.6 4.6 4.3 4.3 4.3
LaTmO3 LaYbO3 LaLuO3 CeTmO3 CeYbO3 CeLuO3 PrYbO3 PrLuO3 NdLuO3 La2 MgZrO6 La2 MgHfO6 La2 CaZrO6 La2 CaHfO6
ǫel xx 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.8 4.6 4.7 4.6
Electronic ǫel yy 4.6 4.5 4.5 4.6 4.6 4.5 4.5 4.5 4.5 4.6 4.5 4.6 4.5
part ǫel zz 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.5 4.4 4.4 4.3
ǫel xz
−0.06 −0.05 −0.06 −0.04
TABLE II: Values of Ω0 (in cm−1 ) for all compounds considered (see Eq. 4). LaScO3 425.80
PrScO3 429.94
NdScO3 431.89
SmScO3 436.40
GdScO3 440.32
TbScO3 442.13
DyScO3 443.82
LaYO3 398.31
CaZrO3 434.61
SrZrO3 419.11
SrHfO3 426.55
LaHoO3 397.40
LaErO3 399.3
LaTmO3 401.19
LaYbO3 402.93
LaLuO3 404.35
CeTmO3 403.17
CeYbO3 404.94
CeLuO3 406.36
PrYbO3 406.97
PrLuO3 408.41
NdLuO3 410.28
La2 MgZrO6 427.70
La2 MgHfO6 431.74
La2 CaZrO6 407.17
La2 CaHfO6 410.54
∗
Electronic address:
[email protected]
†
Deceased.
3 TABLE III: Γ-point phonon frequencies for P bnm perovskites we considered. Phonon frequencies are given in cm−1 . For infrared-active modes, the non-zero values of the dimensionless mode charges Zα⋆ are given (see Eq. 3). B3u modes are active along x-axis, B2u along y, and B1u along z. B1u
LaScO3
Zz⋆ 1.16 0.23 1.66 1.17 0.79 0.03 0.97
ω 145.6 182.3 279.9 304.1 349.4 440.6 457.0
B1u
PrScO3
Zz⋆ 1.11 0.14 1.55 1.15 1.01 0.02 0.95
ω 144.2 183.3 281.1 316.4 352.1 442.0 459.4
B1u
NdScO3
ω 142.1 183.3 280.8 321.0 354.3 442.9 459.8
Zz⋆ 1.09 0.10 1.52 1.09 1.12 0.02 0.93
Infrared active B2u ω Zy⋆ 104.1 0.20 175.1 1.27 222.2 0.40 266.9 1.27 308.9 1.56 320.0 0.67 403.2 0.56 455.1 0.99 510.0 0.20 Infrared active B2u ω Zy⋆ 102.5 0.18 187.1 1.13 227.8 0.55 271.8 1.23 317.7 1.70 335.6 0.52 407.6 0.75 461.0 0.87 517.5 0.25 Infrared active B2u ω Zy⋆ 101.2 0.17 190.5 1.06 229.6 0.64 274.3 1.16 321.1 1.75 341.7 0.52 410.2 0.84 464.7 0.81 520.9 0.27
B3u Zx⋆ 0.17 1.37 0.66 1.27 1.17 1.42 0.44 0.10 0.77
ω 107.5 159.9 255.9 267.9 314.7 348.4 371.4 427.8 484.7 B3u
Zx⋆ 0.17 1.31 1.38 0.49 0.83 1.53 0.69 0.13 0.73
ω 109.9 163.8 266.7 277.2 316.6 357.3 382.5 446.0 492.0 B3u ω 109.9 165.1 268.4 282.9 317.3 360.8 387.8 454.4 496.3
Zx⋆ 0.16 1.28 1.44 0.32 0.72 1.52 0.81 0.15 0.72
Ag ω 101.6 121.6 204.7 280.2 339.8 407.6 462.5
Raman active B1g B2g ω ω 111.2 124.7 144.8 283.7 226.0 402.9 322.1 450.6 401.3 648.7 475.8 631.5
Ag ω 108.7 124.2 220.5 295.9 365.6 419.9 467.1
Raman active B1g B2g ω ω 113.5 121.9 151.5 286.7 253.7 416.1 329.3 452.1 421.1 647.1 488.8 630.8
Ag ω 110.0 124.3 226.1 301.7 377.0 425.3 470.5
Raman active B1g B2g ω ω 113.4 119.3 152.8 287.5 264.4 421.6 332.2 452.4 430.0 646.6 494.9 631.3
B3g ω 144.1 174.4 334.3 456.6 588.4
B3g ω 137.9 194.0 346.0 458.4 590.2
B3g ω 133.6 200.6 350.5 459.9 590.2
Inactive Au ω 90.0 162.9 219.7 241.4 299.8 348.0 424.6 461.4 Inactive Au ω 88.8 161.7 219.9 241.8 309.0 352.2 426.0 466.0 Inactive Au ω 86.9 160.9 219.7 241.4 312.5 354.9 425.6 468.9
4
B1u
SmScO3
Zz⋆ 1.08 0.05 1.48 1.00 1.24 0.01 0.87
ω 133.3 183.5 278.6 326.6 360.1 445.2 458.7
B1u
GdScO3
Zz⋆ 1.08 0.004 1.46 0.92 1.32 0.11 0.78
ω 124.1 183.7 275.3 330.5 366.3 448.3 457.4
B1u
TbScO3
Zz⋆ 1.08 0.02 1.45 0.89 1.34 0.20 0.71
ω 119.3 183.7 273.2 332.4 369.6 449.6 457.0
B1u
DyScO3
ω 113.6 183.6 271.1 333.6 372.8 450.5 457.2
Zz⋆ 1.09 0.05 1.44 0.87 1.37 0.31 0.62
Infrared active B2u ω Zy⋆ 98.1 0.18 191.7 1.02 230.7 0.69 276.8 1.03 326.2 1.76 349.2 0.59 415.2 0.98 471.2 0.76 528.3 0.31 Infrared active B2u ω Zy⋆ 95.8 0.18 192.1 0.98 231.8 0.73 278.4 0.92 330.1 1.73 355.0 0.66 420.4 1.11 478.4 0.73 535.3 0.33 Infrared active B2u ω Zy⋆ 95.2 0.19 192.3 0.97 231.9 0.74 279.0 0.87 331.5 1.71 357.7 0.69 422.9 1.17 481.9 0.72 538.7 0.34 Infrared active B2u ω Zy⋆ 94.0 0.20 192.0 0.96 232.0 0.74 279.5 0.83 332.6 1.68 359.9 0.72 425.3 1.23 485.3 0.71 542.0 0.36
B3u Zx⋆ 0.19 1.25 1.43 0.26 0.59 1.49 0.96 0.10 0.73
ω 109.0 165.0 271.4 288.8 319.9 364.0 399.3 467.8 505.7 B3u
Zx⋆ 0.21 1.21 1.41 0.26 0.49 1.46 1.05 0.05 0.75
ω 107.9 165.7 274.6 290.9 324.2 366.1 410.7 479.1 515.3 B3u
Zx⋆ 0.22 1.20 1.39 0.29 0.44 1.45 1.08 0.02 0.76
ω 107.8 166.5 276.3 291.1 326.7 366.9 416.4 484.2 520.2 B3u ω 107.3 166.8 277.7 290.7 329.2 367.6 421.8 489.0 524.8
Zx⋆ 0.23 1.18 1.38 0.32 0.40 1.44 1.10 0.01 0.77
Ag ω 110.2 123.3 233.7 308.0 396.0 432.7 477.9
Raman active B1g B2g ω ω 112.6 112.3 153.5 287.1 280.2 428.6 335.7 452.1 441.6 646.0 506.8 634.4
Ag ω 109.2 122.2 239.0 312.0 411.9 440.0 486.0
Raman active B1g B2g ω ω 111.3 105.7 153.3 286.6 292.5 434.0 339.0 451.8 451.5 646.5 517.1 638.8
Ag ω 108.8 122.2 241.4 313.4 419.3 443.4 490.1
Raman active B1g B2g ω ω 111.2 103.0 153.7 286.4 297.8 436.0 340.8 451.9 456.0 647.0 522.0 641.4
Ag ω 107.7 121.5 242.9 314.6 425.9 446.9 493.6
Raman active B1g B2g ω ω 110.5 99.6 153.2 285.9 302.2 437.3 342.4 452.3 460.3 647.6 526.5 643.9
B3g ω 125.2 208.0 356.4 464.0 589.0
B3g ω 118.4 211.6 361.3 469.1 586.9
B3g ω 115.7 212.6 363.8 471.7 585.7
B3g ω 112.3 212.7 366.2 474.2 584.5
Inactive Au ω 83.5 157.4 218.9 240.1 317.0 361.2 422.5 475.4 Inactive Au ω 79.7 154.3 217.4 239.6 320.6 368.1 418.4 482.8 Inactive Au ω 78.5 153.0 215.8 239.6 322.2 371.7 416.0 486.7 Inactive Au ω 76.8 151.2 214.1 239.6 323.6 375.1 413.7 490.5
5
B1u
LaYO3
Zz⋆ 0.96 0.29 0.97 0.25 1.75 0.57 0.19
ω 79.6 138.0 194.4 245.6 306.3 391.1 411.5
B1u
CaZrO3
Zz⋆ 1.18 0.59 1.61 0.45 1.37 0.29 1.44
ω 85.6 133.3 196.8 267.4 328.8 447.2 472.6
B1u
SrZrO3
Zz⋆ 0.94 0.60 1.92 0.63 0.98 0.16 1.61
ω 109.0 128.9 197.0 226.9 312.5 454.9 477.8
B1u
SrHfO3
ω 94.5 123.8 187.2 214.1 259.3 490.7 509.9
Zz⋆ 0.23 0.77 1.60 0.62 1.27 0.08 1.47
Infrared active B2u ω Zy⋆ 76.2 0.37 150.4 0.96 171.3 0.22 221.9 0.56 239.7 1.59 276.9 0.17 347.9 0.94 421.3 1.17 467.3 0.56 Infrared active B2u ω Zy⋆ 118.4 0.58 184.5 1.19 204.7 0.56 232.0 1.14 260.0 1.70 308.0 0.38 399.2 0.36 469.6 1.62 475.9 0.25 Infrared active B2u ω Zy⋆ 99.9 0.52 136.0 1.05 187.2 1.27 203.3 1.23 239.9 1.17 262.0 0.55 373.5 0.04 447.8 0.30 471.2 1.65 Infrared active B2u ω Zy⋆ 97.8 0.43 123.5 0.79 165.5 0.43 200.8 1.68 223.0 0.14 230.9 1.09 352.8 0.12 426.8 0.21 505.7 1.52
B3u Zx⋆ 0.39 1.19 0.28 1.10 0.51 0.72 1.69 0.39 0.65
ω 91.6 132.9 197.2 216.9 253.2 299.9 335.5 426.1 466.5 B3u
Zx⋆ 0.57 1.23 1.50 0.06 0.00 1.69 0.33 0.20 1.35
ω 125.5 174.2 201.1 260.7 274.0 335.6 369.3 396.1 495.9 B3u
Zx⋆ 0.35 0.88 1.92 0.58 0.61 1.08 0.60 0.26 1.51
ω 91.6 146.7 192.4 205.2 253.9 308.3 315.7 356.7 488.4 B3u ω 87.9 139.3 179.3 192.2 230.7 277.6 293.8 332.5 521.0
Zx⋆ 0.24 0.49 0.84 1.32 1.13 1.18 0.46 0.07 1.41
Ag ω 99.1 106.8 208.1 257.8 365.9 398.6 461.1
Raman active B1g B2g ω ω 110.4 92.3 145.2 244.5 247.9 378.4 305.1 417.5 379.3 592.7 473.4 593.5
Ag ω 134.0 178.3 252.6 281.5 354.6 420.5 532.1
Raman active B1g B2g ω ω 185.0 153.3 204.3 297.0 231.2 413.9 337.9 517.1 409.1 747.5 454.6 728.2
Ag ω 94.7 105.9 170.2 250.3 277.2 398.5 525.3
Raman active B1g B2g ω ω 116.2 124.4 132.4 280.8 144.6 390.1 315.7 518.5 378.3 762.4 425.8 735.6
Ag ω 83.7 105.9 162.6 234.5 278.6 414.6 564.1
Raman active B1g B2g ω ω 110.4 128.3 114.9 296.3 137.4 405.4 325.5 556.9 393.9 793.4 438.9 766.2
B3g ω 98.3 164.6 336.1 454.1 529.3
B3g ω 169.7 220.6 342.2 524.2 685.5
B3g ω 140.0 153.0 306.6 517.5 694.7
B3g ω 123.2 161.4 314.8 556.8 721.8
Inactive Au ω 75.2 120.3 145.9 175.1 229.4 304.2 363.9 433.7 Inactive Au ω 99.6 136.7 165.0 182.5 242.9 327.6 433.0 476.8 Inactive Au ω 93.1 122.7 138.9 192.9 235.7 307.3 443.8 471.4 Inactive Au ω 86.1 108.5 109.8 202.8 216.3 264.8 476.7 504.6
6
B1u
LaHoO3
Zz⋆ 0.85 0.44 0.82 0.58 1.71 0.59 0.14
ω 68.9 109.6 155.1 193.4 305.3 396.7 416.0
B1u
LaErO3
Zz⋆ 0.80 0.48 0.83 0.63 1.68 0.62 0.15
ω 75.1 110.9 157.8 194.0 306.0 404.1 421.7
B1u
LaTmO3
Zz⋆ 0.75 0.51 0.83 0.68 1.65 0.65 0.17
ω 79.7 112.1 160.6 195.0 307.0 412.1 427.8
B1u
LaYbO3
ω 83.2 112.9 162.3 195.3 308.1 419.7 433.6
Zz⋆ 0.69 0.55 0.81 0.75 1.63 0.67 0.19
Infrared active B2u ω Zy⋆ 67.9 0.33 133.3 0.72 139.8 0.01 197.4 0.69 221.0 1.63 244.3 0.18 342.1 0.90 426.4 1.14 456.9 0.63 Infrared active B2u ω Zy⋆ 70.8 0.31 134.7 0.71 139.9 0.00 200.1 0.67 223.4 1.66 245.0 0.20 344.3 0.88 431.9 1.12 457.2 0.61 Infrared active B2u ω Zy⋆ 73.1 0.29 136.0 0.70 140.3 0.04 202.7 0.63 225.8 1.68 246.0 0.23 346.3 0.85 437.7 1.11 457.6 0.59 Infrared active B2u ω Zy⋆ 75.1 0.27 136.6 0.68 140.0 0.08 204.3 0.60 227.8 1.71 246.5 0.26 348.2 0.83 443.3 1.10 458.0 0.56
B3u Zx⋆ 0.32 0.99 0.35 1.15 0.72 0.31 1.75 0.34 0.64
ω 83.0 121.3 149.7 189.8 218.2 290.3 329.8 428.9 469.9 B3u
Zx⋆ 0.29 0.97 0.36 1.16 0.74 0.27 1.74 0.28 0.66
ω 84.4 122.9 150.6 192.4 218.2 291.2 330.6 430.8 472.2 B3u
Zx⋆ 0.26 0.95 0.36 1.16 0.76 0.24 1.73 0.22 0.68
ω 85.8 124.2 152.0 194.9 218.4 292.1 331.5 432.4 475.4 B3u ω 86.7 125.0 152.2 196.7 217.9 292.7 332.2 433.5 478.9
Zx⋆ 0.23 0.93 0.36 1.15 0.80 0.20 1.73 0.17 0.69
Ag ω 98.2 107.6 209.3 259.5 368.5 402.6 480.8
Raman active B1g B2g ω ω 110.5 91.3 145.4 244.4 248.0 382.8 309.9 436.5 380.9 601.2 475.2 602.2
Ag ω 100.5 107.6 209.4 261.4 369.1 401.6 483.8
Raman active B1g B2g ω ω 110.9 94.6 145.7 247.2 248.2 384.4 311.3 442.3 382.8 605.8 475.0 605.9
Ag ω 101.8 107.6 209.7 263.3 369.4 400.2 487.0
Raman active B1g B2g ω ω 111.1 97.7 146.0 249.9 248.1 386.0 312.8 448.5 384.5 610.5 474.9 609.6
Ag ω 103.5 107.8 209.8 265.0 368.8 399.8 489.8
Raman active B1g B2g ω ω 111.3 100.3 146.2 252.6 247.8 387.7 314.0 454.2 386.2 614.9 474.9 613.0
B3g ω 97.7 164.5 345.9 474.6 537.3
B3g ω 101.3 166.3 346.4 478.0 542.5
B3g ω 104.9 168.1 347.2 481.5 547.9
B3g ω 108.0 169.6 347.8 484.7 553.2
Inactive Au ω 65.6 103.5 119.0 146.4 173.7 301.5 369.3 439.1 Inactive Au ω 68.2 104.2 119.5 151.1 174.8 302.3 377.4 444.3 Inactive Au ω 70.4 105.2 120.0 155.4 176.4 303.5 386.1 449.9 Inactive Au ω 71.9 105.5 119.7 158.6 177.2 304.5 394.3 455.2
7
B1u
LaLuO3
Zz⋆ 0.65 0.58 0.79 0.80 1.61 0.69 0.21
ω 85.7 113.9 164.2 196.4 309.3 427.1 439.8
B1u
CeTmO3
Zz⋆ 0.78 0.46 0.82 0.66 1.67 0.59 0.15
ω 76.5 111.5 159.0 196.2 316.4 411.9 430.8
B1u
CeYbO3
Zz⋆ 0.72 0.50 0.81 0.72 1.64 0.62 0.16
ω 80.4 112.4 160.8 196.3 317.5 419.6 436.3
B1u
CeLuO3
ω 83.1 113.3 162.9 197.2 318.9 427.0 442.4
Zz⋆ 0.68 0.53 0.80 0.77 1.62 0.64 0.18
Infrared active B2u ω Zy⋆ 76.9 0.26 137.2 0.66 140.4 0.17 206.0 0.56 229.8 1.73 247.5 0.28 350.2 0.81 449.2 1.10 458.7 0.52 Infrared active B2u ω Zy⋆ 72.5 0.29 137.4 0.66 140.7 0.01 203.4 0.66 230.6 1.64 250.1 0.29 356.6 0.95 442.4 1.06 463.8 0.65 Infrared active B2u ω Zy⋆ 74.5 0.27 138.0 0.64 140.3 0.03 205.1 0.63 232.7 1.66 251.0 0.35 358.6 0.93 448.1 1.06 464.3 0.62 Infrared active B2u ω Zy⋆ 76.3 0.26 138.8 0.62 140.7 0.14 206.9 0.60 234.7 1.68 252.3 0.39 360.4 0.91 453.8 1.05 464.8 0.60
B3u Zx⋆ 0.22 0.91 0.37 1.13 0.84 0.18 1.72 0.13 0.70
ω 87.7 125.8 153.2 198.6 218.2 293.6 333.2 434.6 483.2 B3u
Zx⋆ 0.25 0.92 0.33 1.18 0.75 0.32 1.73 0.21 0.66
ω 86.6 125.9 151.1 195.7 222.2 298.3 342.3 440.8 482.9 B3u
Zx⋆ 0.23 0.90 0.34 1.18 0.78 0.29 1.73 0.16 0.67
ω 87.4 126.7 151.4 197.6 221.7 299.2 343.0 442.3 486.0 B3u ω 88.5 127.5 152.6 199.6 221.9 300.1 344.0 443.8 489.8
Zx⋆ 0.21 0.88 0.33 1.17 0.80 0.27 1.72 0.12 0.68
Ag ω 104.6 108.1 209.9 266.5 368.5 400.0 492.8
Raman active B1g B2g ω ω 111.4 102.2 146.3 254.6 247.3 389.2 315.2 459.5 387.7 619.1 475.0 616.4
Ag ω 102.3 109.7 214.9 270.0 379.5 410.6 491.6
Raman active B1g B2g ω ω 111.8 95.9 148.8 250.9 256.6 394.4 317.3 447.7 395.0 612.1 482.6 612.6
Ag ω 103.9 110.0 215.2 272.1 379.7 410.3 494.2
Raman active B1g B2g ω ω 112.0 98.4 149.0 253.5 256.7 396.1 318.8 453.6 396.8 616.4 482.6 615.9
Ag ω 105.1 110.4 215.5 273.7 379.5 410.4 497.0
Raman active B1g B2g ω ω 112.2 100.4 149.2 255.6 256.5 397.6 320.1 458.9 398.3 620.3 482.6 619.1
B3g ω 110.6 170.7 348.4 487.9 558.0
B3g ω 101.7 171.4 354.0 485.1 547.4
B3g ω 104.7 173.2 354.7 488.2 552.7
B3g ω 107.2 174.7 355.4 491.2 557.4
Inactive Au ω 72.9 106.1 119.8 161.3 178.9 305.9 402.4 461.0 Inactive Au ω 67.2 104.8 121.6 153.4 176.6 313.0 384.6 455.6 Inactive Au ω 68.8 105.1 121.3 156.7 177.1 314.0 393.0 460.7 Inactive Au ω 70.0 105.8 121.4 159.7 178.6 315.4 401.2 466.2
8
B1u
PrYbO3
Zz⋆ 0.77 0.45 0.81 0.68 1.66 0.56 0.14
ω 76.3 111.5 159.0 197.2 324.3 418.7 438.8
B1u
PrLuO3
Zz⋆ 0.72 0.48 0.80 0.73 1.64 0.59 0.15
ω 79.6 112.4 161.3 198.0 325.8 426.2 444.6
B1u
NdLuO3
ω 75.8 111.4 159.5 198.7 332.0 425.0 447.3
Zz⋆ 0.76 0.43 0.81 0.70 1.65 0.53 0.12
Infrared active B2u ω Zy⋆ 73.5 0.28 138.7 0.62 140.6 0.01 205.4 0.66 235.5 1.62 254.2 0.40 366.0 1.00 451.9 1.03 470.2 0.67 Infrared active B2u ω Zy⋆ 75.3 0.27 139.6 0.60 141.0 0.10 207.3 0.63 237.6 1.63 255.7 0.44 368.0 0.98 457.4 1.02 470.7 0.65 Infrared active B2u ω Zy⋆ 74.3 0.27 139.9 0.52 140.8 0.27 207.2 0.64 240.1 1.58 258.7 0.52 375.4 1.04 461.5 0.98 476.1 0.70
B3u Zx⋆ 0.24 0.87 0.31 1.18 0.78 0.35 1.71 0.17 0.66
ω 87.9 127.8 150.4 198.3 224.6 304.5 351.5 449.9 492.7 B3u
Zx⋆ 0.22 0.86 0.32 1.17 0.80 0.33 1.71 0.13 0.67
ω 88.8 128.5 151.6 200.2 224.8 305.5 352.5 451.6 496.0 B3u ω 89.0 129.3 150.3 200.9 226.7 310.7 360.9 458.6 502.5
Zx⋆ 0.23 0.82 0.29 1.17 0.83 0.39 1.69 0.13 0.66
Ag ω 103.5 111.6 219.4 276.2 388.1 418.5 498.4
Raman active B1g B2g ω ω 112.6 95.9 151.1 254.3 263.4 402.4 322.4 451.9 405.0 617.6 489.6 618.8
Ag ω 104.7 112.2 219.8 278.1 388.4 418.3 500.9
Raman active B1g B2g ω ω 112.8 97.9 151.3 256.5 263.5 403.9 323.6 457.5 406.5 621.4 489.7 621.7
Ag ω 103.6 112.8 223.0 282.1 396.5 426.3 505.4
Raman active B1g B2g ω ω 112.4 95.1 152.1 256.8 269.9 409.8 327.1 455.6 414.7 622.9 496.0 624.8
B3g ω 101.7 175.6 359.9 491.3 551.8
B3g ω 104.3 177.4 360.7 494.2 556.6
B3g ω 101.0 178.8 365.3 497.6 555.2
Inactive Au ω 65.9 104.5 122.5 154.5 177.0 321.4 390.7 465.6 Inactive Au ω 67.4 105.3 122.5 157.6 178.2 322.8 399.0 470.8 Inactive Au ω 64.6 104.3 123.1 156.0 178.1 329.9 396.3 475.9
9 TABLE IV: Γ-point phonon frequencies for the La2 BB′ O6 perovskites considered. Phonon frequencies are given in cm−1 . For infrared-active modes, non-zero components of mode charges Zα⋆ are given (see Eq. 3). Bu modes are active in x-z plane and Au modes along y-axis. Infrared active Au
La2 MgZrO6
Zy⋆ 0.08 0.08 0.69 0.95 0.25 0.59 0.12 1.15 0.75 0.03 0.88 0.82 0.63 0.23 0.85 0.19 0.19
ω 85.6 99.5 160.7 173.1 189.1 236.9 246.7 265.7 294.2 311.9 314.9 337.4 385.6 430.8 490.8 503.4 540.1
ω 99.2 146.8 159.2 174.9 252.1 260.7 274.8 309.4 318.1 331.4 336.5 355.8 425.5 482.0 494.6 519.7
Bu Zx⋆ 0.11 0.20 1.26 0.18 −0.35 1.18 0.002 −0.09 0.50 0.37 1.54 0.61 0.05 0.22 0.39 0.69
Zz⋆ 0.02 1.10 −0.22 0.18 0.56 −0.003 1.19 1.22 −0.18 0.72 −0.15 0.39 0.19 0.30 0.71 −0.42
Infrared active Au
La2 MgHfO6
ω 81.7 97.9 140.0 151.4 165.6 229.0 249.6 254.5 288.4 305.8 320.6 340.4 387.2 448.2 499.0 506.8 530.5
Zy⋆ 0.08 0.03 0.41 0.65 0.44 1.08 0.31 1.20 0.33 0.15 0.56 1.00 0.55 0.20 0.72 0.26 0.19
ω 92.0 142.8 146.8 155.4 237.5 243.0 254.8 308.7 318.4 330.9 339.1 355.7 425.6 488.5 502.5 527.0
Bu Zx⋆ 0.06 0.45 0.52 0.70 1.41 0.49 0.21 −0.03 0.08 −0.37 1.51 0.58 0.10 0.18 0.33 0.59
Zz⋆ −0.03 0.82 −0.31 −0.35 −0.53 1.26 0.18 1.03 0.35 0.95 0.10 0.39 0.15 0.12 0.67 −0.34
Raman active Ag Bg ω ω 98.9 110.0 122.0 139.8 126.8 152.9 203.0 172.0 269.0 219.7 296.4 304.4 328.0 350.0 399.9 402.6 406.6 455.1 466.7 492.3 477.8 580.1 657.7 648.7
Raman active Ag Bg ω ω 96.3 110.2 124.8 141.5 130.1 157.2 200.9 167.3 268.2 210.6 305.5 308.2 326.1 363.4 410.8 410.1 417.7 468.1 486.3 504.8 496.4 594.3 675.0 665.3
10 Infrared active Au
La2 CaZrO6
Zy⋆ 0.23 0.15 0.27 0.81 0.73 0.47 0.17 0.69 0.63 1.20 0.20 0.70 0.98 0.04 0.87 0.82 0.27
ω 71.4 91.3 131.9 155.5 167.4 191.0 211.5 243.2 272.4 273.9 295.8 316.8 363.5 402.2 431.8 460.2 504.2
ω 94.6 101.5 144.1 157.4 229.8 236.2 239.7 278.0 282.4 313.6 318.3 343.9 413.3 424.6 455.7 481.9
Bu Zx⋆ 0.39 0.23 1.21 −0.23 0.30 0.50 0.93 0.10 0.13 1.00 0.51 1.49 0.16 0.40 −0.08 0.87
Zz⋆ 0.47 0.87 −0.14 0.31 0.68 0.38 −0.88 0.002 −0.02 −1.00 1.25 0.35 0.04 0.81 0.28 −0.32
Infrared active Au
La2 CaHfO6
ω 69.1 93.5 121.8 135.4 152.2 187.0 209.8 244.8 251.0 275.3 288.2 311.0 356.5 417.7 441.4 461.7 496.2
Zy⋆ 0.16 0.13 0.14 0.39 0.66 0.76 0.43 0.19 1.12 1.14 0.21 0.45 0.97 0.06 0.63 0.86 0.25
ω 89.9 105.8 135.8 140.9 213.8 219.8 233.9 264.2 275.0 307.9 316.7 338.2 412.6 437.7 463.9 486.5
Bu Zx⋆ −0.08 0.44 0.34 0.91 0.97 −0.60 −0.65 0.43 −0.11 0.78 0.10 1.56 0.18 0.36 −0.04 0.73
Zz⋆ 0.21 0.82 0.19 −0.31 0.65 0.67 0.84 −0.12 0.19 −0.77 1.30 0.41 −0.07 0.71 0.28 −0.26
Raman active Ag Bg ω ω 94.0 96.7 106.9 120.1 110.2 148.8 207.5 173.1 235.6 247.0 287.8 283.1 353.8 359.0 385.5 394.3 399.1 449.5 446.7 495.2 475.5 538.3 635.4 638.0
Raman active Ag Bg ω ω 97.3 99.9 108.7 124.4 113.7 149.7 208.4 175.0 237.0 245.9 297.1 284.7 354.7 371.4 394.2 402.9 407.6 458.9 464.9 505.0 490.5 550.7 649.6 651.5
