241:
498:
248:. The separation between LO and TO phonon frequencies near the Γ-point (small wave vectors) is described by the LST relation. Note this plot shows much higher wavevectors than considered below, and the scale cannot not show the hybridization of the TO branch with light (which would be confined extremely close to Γ).
403:
228:
The relation holds for systems with a single optical branch, such as cubic systems with two different atoms per unit cell. For systems with many phonon branches, the relation does not necessarily hold, as the permittivity for any pair of longitudinal and transverse modes will be altered by the other
935:
668:
In these modes, the electric field is perpendicular to the wavevector, producing transverse currents, which in turn generate magnetic fields. As light is also a transverse electromagnetic wave, the behaviour is described as a coupling of the transverse vibration modes with the
223:
811:
In the case of multiple, lossy
Lorentzian oscillators, there are generalized Lyddane–Sachs–Teller relations available. Most generally, the permittivity cannot be described as a combination of Lorentizan oscillators, and the longitudinal mode frequency can only be found as a
1281:
940:
the absolute value is necessary since the phonon frequencies are now complex, with an imaginary part that is equal to the finite lifetime of the phonon, and proportional to the anharmonic phonon damping (described by
Klemens' theory for optical phonons).
508:. Red curves are the uncoupled phonon and photon dispersion relations, black curves are the result of coupling (from top to bottom: upper polariton, LO phonon, lower polariton). The LST relation relates the frequencies of the horizontal red curve (
275:
834:
991:
796:
Since the
Lyddane–Sachs–Teller relation is derived from the lossless Lorentzian oscillator, it may break down in realistic materials where the permittivity function is more complicated for various reasons:
150:
265:(over ranges much longer than the inter-atom distances). The relation assumes an idealized polar ("infrared active") optical lattice vibration that gives a contribution to the frequency-dependent
791:
1208:
1116:
1149:
115:
142:
650:. This remains 'split off' from the bare oscillation frequency even at high wave vectors, because the importance of electric restoring forces does not diminish at high wavevectors.
435:
1353:
1326:
1063:
1036:
735:
701:
648:
611:
560:
533:
492:
88:
57:
465:
1176:
398:{\displaystyle \varepsilon (\omega )=\varepsilon (\infty )+(\varepsilon (\infty )-\varepsilon _{st}){\frac {\omega _{\text{TO}}^{2}}{\omega ^{2}-\omega _{\text{TO}}^{2}}},}
703:
because magnetic restoring forces can be neglected: the transverse currents produce a small magnetic field and the magnetically induced electric field is also very small.
620:
In this mode, the electric field is parallel to the wavevector and produces no transverse currents, hence it is purely electric (there is no associated magnetic field).
1383:
1196:
1359:. As previously mentioned, the phonon frequencies used in the LST relation are those corresponding to the TO and LO branches evaluated at the gamma-point (
1705:
494:
is the "natural" oscillation frequency of the lattice vibration taking into account only the short-ranged (microscopic) restoring forces.
930:{\displaystyle {\frac {|\omega _{\text{LO}}|^{2}}{|\omega _{\text{TO}}|^{2}}}={\frac {\varepsilon _{\text{st}}}{\varepsilon _{\infty }}}}
1680:
956:
218:{\displaystyle {\frac {\omega _{\text{LO}}^{2}}{\omega _{\text{TO}}^{2}}}={\frac {\varepsilon _{\text{st}}}{\varepsilon _{\infty }}}}
570:
to find the complete set of normal modes including all restoring forces (short-ranged and long-ranged), which are sometimes called
1798:
Chang, I. F.; Mitra, S. S.; Plendl, J. N.; Mansur, L. C. (1968). "Long-Wavelength
Longitudinal Phonons of Multi-Mode Crystals".
1669:
1614:
754:
1732:
Irmer, G.; Wenzel, M.; Monecke, J. (1996). "The temperature dependence of the LO(T) and TO(T) phonons in GaAs and InP".
1715:
1694:
1648:
1623:
1867:
1827:
Casella, L.; Zaccone, A. (2021). "Soft mode theory of ferroelectric phase transitions in the low-temperature phase".
1276:{\displaystyle \nu _{\text{LO}}={\sqrt {\varepsilon _{\text{st}}/\varepsilon _{\infty }}}\times \nu _{\text{TO}}=7.9}
807:
There may be other electrically active degrees of freedom (notably, mobile electrons) and non-Lorentzian oscillators.
1088:
741:
effects that make the transverse magnetic back-action behave identically to the longitudinal electric back-action.
1121:
93:
710:
mode is primarily vibrational and its frequency instead coincides with the longitudinal mode, with frequency
120:
411:
661:
modes appear (actually, four modes, in pairs with identical dispersion), with complex dispersion behavior.
1331:
1304:
1041:
1014:
713:
679:
626:
589:
538:
511:
470:
66:
35:
824:
The most general
Lyddane–Sachs–Teller relation applicable in crystals where the phonons are affected by
676:
At high wavevectors, the lower mode is primarily vibrational. This mode approaches the 'bare' frequency
1640:
623:
The longitudinal wave is basically dispersionless, and appears as a flat line in the plot at frequency
440:
1154:
240:
793:. Solving this for the Lorentzian resonance described above gives the Lyddane–Sachs–Teller relation.
257:
The
Lyddane–Sachs–Teller relation applies to optical lattice vibrations that have an associated net
229:
modes in the system. The
Lyddane–Sachs–Teller relation is named after the physicists R. H. Lyddane,
950:
17:
1686:
949:
A corollary of the LST relation is that for non-polar crystals, the LO and TO phonon modes are
567:
262:
804:
Materials may have multiple phonon resonances that add together to produce the permittivity.
1807:
1778:
1741:
258:
28:) determines the ratio of the natural frequency of longitudinal optic lattice vibrations (
8:
1406:
1362:
1181:
1006:
501:
1811:
1782:
1769:
Lyddane, R.; Sachs, R.; Teller, E. (1941). "On the Polar
Vibrations of Alkali Halides".
1745:
1836:
1356:
738:
90:) for long wavelengths (zero wavevector). The ratio is that of the static permittivity
1757:
1711:
1690:
1665:
1644:
1634:
1619:
583:
571:
505:
1389:. This is also the point where the photon-phonon coupling most often occurs for the
1846:
1815:
1786:
1749:
1659:
994:
658:
230:
1850:
1609:
1394:
1386:
1861:
1819:
1761:
825:
234:
60:
1753:
737:. This coincidence is required by symmetry considerations and occurs due to
1790:
1390:
1065:
there is 100% reflectivity. This range of frequencies (band) is called the
813:
748:
266:
575:
63:
to the natural frequency of the transverse optical lattice vibration (
1511:
1509:
497:
1841:
1581:
1506:
801:
Real phonons have losses (also known as damping or dissipation).
1448:
1446:
986:{\displaystyle \varepsilon _{\text{st}}=\varepsilon _{\infty }}
673:
inside the material (in the figure, shown as red dashed lines).
29:
1431:
670:
586:
mode occurs with an essentially flat dispersion at frequency
1494:
1443:
993:. This indeed holds for the purely covalent crystals of the
1198:
THz. Using the LST relation, we are able to calculate that
1082:
245:
1393:
measured in Raman. Hence two peaks will be present in the
1521:
747:
The longitudinal mode appears at the frequency where the
117:
to the permittivity for frequencies in the visible range
1707:
Physical
Principles of Far-Infrared Radiation, Volume 10
1397:, each corresponding to the TO and LO phonon frequency.
1419:
1797:
1569:
1545:
1515:
1482:
1081:
The static and high-frequency dielectric constants of
1365:
1334:
1307:
1211:
1184:
1157:
1124:
1091:
1044:
1017:
959:
837:
757:
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682:
629:
592:
541:
514:
473:
443:
414:
278:
153:
123:
96:
69:
38:
1470:
997:, such as for diamond (C), silicon, and germanium.
786:{\displaystyle \varepsilon (\omega _{\text{LO}})=0}
1768:
1731:
1587:
1557:
1533:
1437:
1377:
1347:
1320:
1275:
1190:
1170:
1143:
1110:
1057:
1030:
985:
929:
785:
729:
695:
642:
605:
574:. These modes are plotted in the figure. At every
554:
527:
486:
459:
429:
397:
217:
136:
109:
82:
51:
1458:
1859:
269:described by a lossless Lorentzian oscillator:
1826:
1632:
1527:
1500:
1452:
828:damping has been derived in Ref. and reads as
1633:Ashcroft, Neil W.; Mermin, N. David (1976).
1301:One of the ways to experimentally determine
1111:{\displaystyle \varepsilon _{\text{st}}=5.9}
1144:{\displaystyle \varepsilon _{\infty }=2.25}
1657:
1425:
1840:
437:is the permittivity at high frequencies,
252:
1703:
1488:
535:) and the black curve intercept at k=0 (
496:
239:
110:{\displaystyle \varepsilon _{\text{st}}}
1291:
566:The above equation can be plugged into
261:, so that they can produce long ranged
1860:
1608:
1476:
1000:
819:
137:{\displaystyle \varepsilon _{\infty }}
1296:
944:
430:{\displaystyle \varepsilon (\infty )}
1829:Journal of Physics: Condensed Matter
1076:
1678:
1615:Introduction to Solid State Physics
1575:
1563:
1551:
1539:
1464:
1348:{\displaystyle \omega _{\text{LO}}}
1321:{\displaystyle \omega _{\text{TO}}}
1069:. The name derives from the German
1058:{\displaystyle \omega _{\text{LO}}}
1031:{\displaystyle \omega _{\text{TO}}}
730:{\displaystyle \omega _{\text{LO}}}
696:{\displaystyle \omega _{\text{TO}}}
643:{\displaystyle \omega _{\text{LO}}}
606:{\displaystyle \omega _{\text{LO}}}
555:{\displaystyle \omega _{\text{LO}}}
528:{\displaystyle \omega _{\text{TO}}}
487:{\displaystyle \omega _{\text{TO}}}
467:is the static DC permittivity, and
83:{\displaystyle \omega _{\text{TO}}}
52:{\displaystyle \omega _{\text{LO}}}
13:
1247:
1130:
978:
920:
421:
318:
300:
208:
129:
14:
1879:
460:{\displaystyle \varepsilon _{st}}
1588:Irmer, Wenzel & Monecke 1996
1438:Lyddane, Sachs & Teller 1941
1171:{\displaystyle \nu _{\text{TO}}}
578:there are three distinct modes:
706:At zero, or low wavevector the
1151:, and the TO phonon frequency
887:
871:
858:
842:
816:in the permittivity function.
774:
761:
424:
418:
340:
321:
315:
309:
303:
297:
288:
282:
1:
1658:Klingshirn, Claus F. (2012).
1597:
22:Lyddane–Sachs–Teller relation
1682:Optical Properties of Solids
1602:
1412:
1073:which means "residual ray".
7:
1725:
1400:
1011:In the frequencies between
10:
1884:
1641:Holt, Rinehart and Winston
1528:Casella & Zaccone 2021
1501:Ashcroft & Mermin 1976
1453:Ashcroft & Mermin 1976
1004:
751:passes through zero, i.e.
739:electrodynamic retardation
244:Phonon band structure in
1868:Condensed matter physics
1851:10.1088/1361-648X/abdb68
1820:10.1002/pssb.19680280224
1710:(1 ed.). Elsevier.
1704:Robinson, L. C. (1973).
1664:(4 ed.). Springer.
504:of phonon polaritons in
18:condensed matter physics
1800:Physica Status Solidi B
1754:10.1002/pssb.2221950110
1734:Physica Status Solidi B
1687:Oxford University Press
1791:10.1103/PhysRev.59.673
1379:
1349:
1322:
1277:
1192:
1172:
1145:
1112:
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987:
931:
787:
731:
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644:
607:
563:
556:
529:
488:
461:
431:
399:
263:electromagnetic fields
253:Origin and limitations
249:
219:
138:
111:
84:
53:
1618:(8 ed.). Wiley.
1380:
1350:
1323:
1278:
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1113:
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988:
932:
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557:
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500:
489:
462:
432:
400:
243:
220:
139:
112:
85:
54:
1661:Semiconductor Optics
1363:
1332:
1305:
1292:Experimental methods
1209:
1182:
1155:
1122:
1089:
1042:
1015:
957:
835:
755:
714:
680:
627:
590:
539:
512:
471:
441:
412:
276:
259:polarization density
151:
121:
94:
67:
36:
1812:1968PSSBR..28..663C
1783:1941PhRv...59..673L
1746:1996PSSBR.195...85I
1636:Solid State Physics
1407:Reststrahlen effect
1378:{\displaystyle k=0}
1191:{\displaystyle 4.9}
1007:Reststrahlen effect
1001:Reststrahlen effect
820:Anharmonic crystals
568:Maxwell's equations
502:Dispersion relation
388:
359:
185:
170:
1679:Fox, Mark (2010).
1578:, p. 287-289.
1554:, p. 277-278.
1375:
1357:Raman spectroscopy
1345:
1318:
1297:Raman spectroscopy
1273:
1188:
1168:
1141:
1108:
1055:
1028:
983:
945:Non-polar crystals
927:
783:
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457:
427:
395:
374:
345:
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107:
80:
49:
1516:Chang et al. 1968
1342:
1315:
1264:
1252:
1234:
1219:
1165:
1099:
1077:Example with NaCl
1052:
1025:
995:group IV elements
967:
925:
912:
898:
882:
853:
771:
724:
690:
637:
600:
584:longitudinal wave
572:phonon polaritons
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1629:
1591:
1590:, p. 85-95.
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1858:
1857:
1771:Physical Review
1728:
1718:
1697:
1672:
1671:978-364228362-8
1651:
1626:
1610:Kittel, Charles
1605:
1600:
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1426:Klingshirn 2012
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1129:
1125:
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1086:
1079:
1067:Reststrahl band
1049:
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659:transverse wave
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231:Robert G. Sachs
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12:
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5:
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1871:
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1855:
1824:
1806:(2): 663–673.
1795:
1777:(8): 673–676.
1766:
1727:
1724:
1723:
1722:
1717:978-0080859880
1716:
1701:
1696:978-0199573370
1695:
1685:(2 ed.).
1676:
1670:
1655:
1650:978-0030839931
1649:
1639:(1 ed.).
1630:
1625:978-0471415268
1624:
1604:
1601:
1599:
1596:
1593:
1592:
1580:
1568:
1566:, p. 280.
1556:
1544:
1542:, p. 277.
1532:
1520:
1505:
1503:, p. 551.
1493:
1491:, p. 363.
1481:
1479:, p. 414.
1469:
1467:, p. 209.
1457:
1455:, p. 548.
1442:
1430:
1417:
1416:
1414:
1411:
1410:
1409:
1402:
1399:
1395:Raman spectrum
1387:Brillouin zone
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1021:
1005:Main article:
1002:
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73:
42:
9:
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1489:Robinson 1973
1485:
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1428:, p. 86.
1427:
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961:
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942:
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906:
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829:
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684:
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631:
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294:
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235:Edward Teller
232:
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194:
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172:
166:
157:
145:
125:
98:
71:
62:
61:ionic crystal
40:
31:
27:
23:
19:
1832:
1828:
1803:
1799:
1774:
1770:
1740:(1): 85–95.
1737:
1733:
1706:
1681:
1660:
1635:
1613:
1583:
1571:
1559:
1547:
1535:
1523:
1496:
1484:
1472:
1460:
1433:
1421:
1391:Stokes shift
1300:
1080:
1070:
1066:
1010:
948:
939:
831:
823:
814:complex zero
810:
795:
749:permittivity
746:
707:
565:
407:
267:permittivity
256:
227:
147:
26:LST relation
25:
21:
15:
1477:Kittel 2004
1355:is through
953:, and thus
1842:2103.10262
1835:: 165401.
1598:References
1071:reststrahl
951:degenerate
826:anharmonic
576:wavevector
1762:0370-1972
1603:Textbooks
1413:Citations
1385:) of the
1337:ω
1310:ω
1259:ν
1255:×
1248:∞
1244:ε
1229:ε
1214:ν
1160:ν
1131:∞
1127:ε
1094:ε
1047:ω
1020:ω
979:∞
975:ε
962:ε
921:∞
917:ε
907:ε
877:ω
848:ω
766:ω
759:ε
719:ω
685:ω
632:ω
595:ω
544:ω
517:ω
476:ω
446:ε
422:∞
416:ε
376:ω
372:−
363:ω
347:ω
329:ε
325:−
319:∞
313:ε
301:∞
295:ε
286:ω
280:ε
209:∞
205:ε
195:ε
173:ω
158:ω
130:∞
126:ε
99:ε
72:ω
41:ω
1862:Category
1726:Articles
1612:(2004).
1576:Fox 2010
1564:Fox 2010
1552:Fox 2010
1540:Fox 2010
1465:Fox 2010
1401:See also
59:) of an
1808:Bibcode
1779:Bibcode
1742:Bibcode
30:phonons
1760:
1714:
1693:
1668:
1647:
1622:
408:where
233:, and
20:, the
1837:arXiv
708:upper
671:light
1758:ISSN
1712:ISBN
1691:ISBN
1666:ISBN
1645:ISBN
1620:ISBN
1328:and
1139:2.25
1118:and
1085:are
1083:NaCl
1038:and
657:two
246:GaAs
24:(or
1847:doi
1816:doi
1787:doi
1750:doi
1738:195
1283:THz
1271:7.9
1186:4.9
1178:is
1106:5.9
562:).
506:GaP
32:) (
16:In
1864::
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1833:33
1831:.
1814:.
1804:28
1802:.
1785:.
1775:59
1773:.
1756:.
1748:.
1736:.
1689:.
1643:.
1508:^
1445:^
1341:LO
1314:TO
1263:TO
1233:st
1218:LO
1164:TO
1098:st
1051:LO
1024:TO
966:st
911:st
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852:LO
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599:LO
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351:TO
237:.
199:st
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76:TO
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1781::
1764:.
1752::
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1530:.
1518:.
1440:.
1373:0
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1103:=
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901:=
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872:|
864:2
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781:0
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762:(
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450:s
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419:(
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336:t
333:s
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316:(
310:(
307:+
304:)
298:(
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289:)
283:(
189:=
182:2
167:2
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