38:
1733:(SPEELS), very high energy surface magnons can be excited. This technique allows one to probe the dispersion of magnons in the ultrathin ferromagnetic films. The first experiment was performed for a 5 ML Fe film. With momentum resolution, the magnon dispersion was explored for an 8 ML fcc Co film on Cu(001) and an 8 ML hcp Co on W(110), respectively. The maximum magnon energy at the border of the surface Brillouin zone was 240 meV.
161:
1327:
169:
1253:, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the
878:
1508:
1674:
the energy loss of a beam of neutrons that excite a magnon is measured, typically as a function of scattering vector (or equivalently momentum transfer), temperature and external magnetic field. Inelastic neutron scattering measurements can determine the dispersion curve for magnons just as they can
1529:
The first term on the right hand side of the equation describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping
1261:
has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite
342:
647:
1076:
1403:
213:
1521:
is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the
465:
1721:, incident on a magnetic material, by spin waves, typically as a function of angle, temperature and applied field. Ferromagnetic resonance is a convenient laboratory method for determining the effect of
1742:
When magnetoelectronic devices are operated at high frequencies, the generation of spin waves can be an important energy loss mechanism. Spin wave generation limits the linewidths and therefore the
1164:
1388:
968:
636:
1757:
devices. The reciprocal of the lowest frequency of the characteristic spin waves of a magnetic material gives a time scale for the switching of a device based on that material.
873:{\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}S_{i}^{z}S_{j}^{z}-g\mu _{\rm {B}}H\sum _{i}S_{i}^{z}-{\frac {1}{4}}J\sum _{i,j}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})}
1312:
577:
201:
1288:
553:
519:
1239:
1199:
1796:
Plihal, M.; Mills, D. L.; Kirschner, J. (1999). "Spin wave signature in the spin polarized electron energy loss spectrum in ultrathin Fe film: theory and experiment".
1730:
1169:
but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator
1710:(usually at a convenient visible wavelength) reflected from or transmitted through a magnetic material. Brillouin spectroscopy is similar to the more widely known
1262:
number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state
1726:
1700:
1503:{\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \mathbf {M} \times \mathbf {H} -{\frac {\lambda \mathbf {M} \times (\mathbf {M} \times \mathbf {H} )}{M^{2}}}}
979:
337:{\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}\mathbf {S} _{i}\cdot \mathbf {S} _{j}-g\mu _{\rm {B}}\sum _{i}\mathbf {H} \cdot \mathbf {S} _{i}}
1249:. The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a
1771:
1081:
where N is the total number of
Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed.
164:
An illustration of the precession of a spin wave with a wavelength that is eleven times the lattice constant about an applied magnetic field.
412:
59:
398:
is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in
1932:
1909:
1890:
1833:"Spin-Polarized Electron Energy Loss Spectroscopy of High Energy, Large Wave Vector Spin Waves in Ultrathin fcc Co Films on Cu(001)"
1659:
81:
172:
The projection of the magnetization of the same spin wave along the chain direction as a function of distance along the spin chain.
1523:
1330:
An excitation in the middle of a grid of spins propagates by exchanging torque (and thus angular momentum) with its neighbours.
177:
1951:
1722:
1696:
1315:
1094:
1714:, but probes a lower energy and has a superior energy resolution in order to be able to detect the meV energy of magnons.
1671:
1639:
1340:
913:
52:
46:
1989:
403:
479:
1692:
139:
134:
excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a
63:
95:
585:
1663:
1587:
form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the
1084:
One might guess that the first excited state of the
Hamiltonian has one randomly selected spin at position
1684:
1615:
1847:
1805:
1647:
1397:
is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:
1293:
558:
182:
1680:
1538:
579:
can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:
111:
1265:
530:
496:
1212:
1172:
1968:
1947:
1928:
1905:
1886:
1863:
1258:
483:
372:
204:
1855:
1813:
1776:
1750:
1711:
1651:
1611:
1831:
Vollmer, R.; Etzkorn, M.; Kumar, P. S. Anil; Ibach, H.; Kirschner, J. (29 September 2003).
1832:
1071:{\displaystyle {\mathcal {H}}|0\rangle =\left(-Js^{2}-g\mu _{\rm {B}}Hs\right)N|0\rangle }
478:
dimensions this equation admits several integrable and non-integrable extensions like the
364:
352:
1851:
1809:
1994:
1743:
1578:
360:
135:
1983:
1921:
389:
144:
115:
1973:
127:
1859:
1717:
Ferromagnetic (or antiferromagnetic) resonance instead measures the absorption of
887:
has been taken as the direction of the magnetic field. The spin-lowering operator
110:
material. These low-lying collective excitations occur in magnetic lattices with
1817:
1607:
1254:
149:
1610:. The underlying reason for the difference in dispersion relation is that the
1541:. The dispersion relation for phonons is to first order linear in wavevector
1766:
1754:
1718:
1526:, which replaces the final term by a more "simple looking" equivalent one.)
168:
1867:
17:
1688:
1563:
is the velocity of sound. Magnons have a parabolic dispersion relation:
160:
1679:. Important inelastic neutron scattering facilities are present at the
1326:
460:{\displaystyle \mathbf {S} _{t}=\mathbf {S} \times \mathbf {S} _{xx}.}
1707:
1676:
131:
119:
107:
903:
annihilates the ground state with maximum spin projection along the
1537:
One important difference between phonons and magnons lies in their
1655:
1643:
1325:
167:
123:
893:
annihilates the state with minimum projection of spin along the
176:
The simplest way of understanding spin waves is to consider the
103:
521:
is that in which all spins are aligned parallel with the field
1614:(magnetization) for the ground-state in ferromagnets violates
1290:
does not have the full rotational symmetry of the
Hamiltonian
31:
1927:(27. repr. ed.). New York: Holt, Rinehart and Winston.
1299:
985:
653:
564:
219:
188:
1706:
Brillouin scattering similarly measures the energy loss of
1638:
Spin waves are observed through four experimental methods:
1883:
Concepts in solids : lectures on the theory of solids
1658:
scattering), inelastic electron scattering (spin-resolved
1618:. Two adjacent spins in a solid with lattice constant
1406:
1343:
1296:
1268:
1215:
1209:
back to its low-energy orientation, but the operator
1175:
1097:
982:
916:
650:
588:
561:
533:
499:
415:
216:
185:
1159:{\displaystyle S_{i}^{z}|1\rangle =(s-1)|1\rangle ,}
1920:
1725:on the dispersion of spin waves. One group at the
1534:are in many cases dominated by the eddy currents.
1502:
1382:
1306:
1282:
1233:
1193:
1158:
1070:
962:
872:
630:
571:
547:
513:
459:
336:
195:
1946:(2nd ed.). Oxford: Oxford University Press.
27:Wave which propagates through a magnetic material
1731:spin polarized electron energy loss spectroscopy
1383:{\displaystyle M={\frac {N\mu _{\rm {B}}gs}{V}}}
142:. The energies of spin waves are typically only
1885:(Repr. ed.). Singapore: World Scientific.
963:{\displaystyle S_{i}^{z}|0\rangle =s|0\rangle }
1727:Max Planck Institute of Microstructure Physics
1701:National Institute of Standards and Technology
1976:performing Brillouin scattering measurements.
8:
1969:Spin waves - The Feynman Lectures on Physics
1919:Ashcroft, Neil W.; Mermin, N. David (1977).
1277:
1150:
1121:
1065:
998:
957:
940:
542:
508:
1622:that participate in a mode with wavevector
1902:Basic notions of condensed matter physics
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1107:
1102:
1096:
1057:
1036:
1035:
1019:
990:
984:
983:
981:
973:for the maximally aligned state, we find
949:
932:
926:
921:
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861:
856:
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828:
823:
813:
808:
789:
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82:Learn how and when to remove this message
1904:. Cambridge, Mass.: Perseus Publishing.
631:{\displaystyle S^{\pm }=S^{x}\pm iS^{y}}
493:and the ground state of the Hamiltonian
159:
45:This article includes a list of general
1788:
1729:in Halle, Germany proved that by using
897:-axis, while the spin-raising operator
118:point of view, spin waves are known as
7:
1626:have an angle between them equal to
1245:-projection of the spin at position
1205:-projection of the spin at position
1362:
1037:
735:
298:
51:it lacks sufficient corresponding
25:
1772:Holstein–Primakoff transformation
1660:electron energy loss spectroscopy
1530:forces described by the constant
1524:Landau-Lifshitz-Gilbert equation
1480:
1472:
1461:
1447:
1439:
1414:
1334:In this model the magnetization
441:
432:
418:
324:
315:
276:
261:
36:
1314:, a phenomenon which is called
152:at room temperature and below.
130:that correspond roughly to the
1699:in Tennessee, USA, and at the
1517:is the gyromagnetic ratio and
1484:
1468:
1307:{\displaystyle {\mathcal {H}}}
1270:
1143:
1139:
1127:
1114:
1058:
991:
950:
933:
867:
801:
572:{\displaystyle {\mathcal {H}}}
535:
501:
196:{\displaystyle {\mathcal {H}}}
1:
1860:10.1103/PhysRevLett.91.147201
1723:magnetocrystalline anisotropy
1697:Oak Ridge National Laboratory
1316:spontaneous symmetry breaking
486:and so on. For a ferromagnet
1900:Anderson, Philip W. (1997).
1881:Anderson, Philip W. (1997).
1672:inelastic neutron scattering
1662:), and spin-wave resonance (
1640:inelastic neutron scattering
1818:10.1103/PhysRevLett.82.2579
2011:
1942:Chikazumi, SĹŤshin (1997).
1283:{\displaystyle |0\rangle }
548:{\displaystyle |0\rangle }
514:{\displaystyle |0\rangle }
1944:Physics of ferromagnetism
1693:High Flux Isotope Reactor
1234:{\displaystyle S_{j}^{-}}
1194:{\displaystyle S_{i}^{+}}
140:spontaneous magnetization
1683:in Oxfordshire, UK, the
1634:Experimental observation
480:Landau-Lifshitz equation
148:in keeping with typical
96:condensed matter physics
1840:Physical Review Letters
1664:ferromagnetic resonance
104:propagating disturbance
66:more precise citations.
1738:Practical significance
1685:Institut Laue-Langevin
1616:time-reversal symmetry
1504:
1384:
1331:
1308:
1284:
1235:
1195:
1160:
1072:
964:
874:
632:
573:
549:
515:
461:
406:equation has the form
404:Heisenberg ferromagnet
338:
197:
173:
165:
114:. From the equivalent
1505:
1385:
1329:
1309:
1285:
1236:
1196:
1161:
1073:
965:
875:
633:
574:
550:
516:
462:
339:
198:
171:
163:
106:in the ordering of a
1648:Brillouin scattering
1573:where the parameter
1539:dispersion relations
1404:
1341:
1294:
1266:
1213:
1173:
1095:
980:
914:
648:
586:
559:
555:is an eigenstate of
531:
497:
413:
214:
183:
1923:Solid state physics
1852:2003PhRvL..91n7201V
1810:1999PhRvL..82.2579P
1753:components used in
1681:ISIS neutron source
1230:
1190:
1112:
931:
866:
851:
833:
818:
768:
722:
707:
112:continuous symmetry
1559:is frequency, and
1500:
1380:
1332:
1304:
1280:
1231:
1216:
1201:will increase the
1191:
1176:
1156:
1098:
1068:
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837:
819:
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193:
174:
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1990:Magnetic ordering
1804:(12): 2579–2582.
1703:in Maryland, USA.
1498:
1427:
1378:
1259:discrete symmetry
785:
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672:
484:Ishimori equation
304:
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92:
91:
84:
16:(Redirected from
2002:
1957:
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1926:
1915:
1896:
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1871:
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1821:
1793:
1777:Spin engineering
1712:Raman scattering
1652:Raman scattering
1629:
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1621:
1606:
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1544:
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1418:
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1408:
1396:
1389:
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1367:
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1313:
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1146:
1117:
1111:
1106:
1088:rotated so that
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1042:
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62:this article by
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1892:981-02-3231-4
1888:
1884:
1879:
1878:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1834:
1827:
1824:
1819:
1815:
1811:
1807:
1803:
1799:
1792:
1789:
1782:
1778:
1775:
1773:
1770:
1768:
1765:
1764:
1760:
1758:
1756:
1752:
1748:
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1732:
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1724:
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1375:
1371:
1368:
1357:
1353:
1347:
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1337:
1336:
1335:
1328:
1322:Magnetization
1321:
1319:
1317:
1274:
1260:
1256:
1252:
1226:
1221:
1217:
1186:
1181:
1177:
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1124:
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1108:
1103:
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907:-axis. Since
901:
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769:
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698:
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666:
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644:
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642:
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623:
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581:
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454:
449:
446:
436:
428:
423:
409:
408:
407:
405:
396:
391:
390:Bohr magneton
383:
378:
366:
362:
354:
329:
319:
309:
305:
293:
289:
286:
281:
271:
266:
254:
251:
248:
244:
240:
235:
232:
227:
224:
210:
209:
208:
207:ferromagnet:
206:
179:
170:
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155:
153:
151:
146:
141:
137:
133:
129:
126:modes of the
125:
121:
117:
116:quasiparticle
113:
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105:
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75:
72:December 2013
65:
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34:
33:
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19:
1974:List of labs
1943:
1922:
1901:
1882:
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1839:
1826:
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1797:
1791:
1746:
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1646:scattering (
1642:, inelastic
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1257:magnet with
1250:
1168:
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889:
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640:
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472:1 + 1, 2 + 1
469:
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381:
346:
175:
150:Curie points
128:spin lattice
122:, which are
99:
93:
78:
69:
50:
29:
1608:dot product
1255:Ising model
178:Hamiltonian
136:ferromagnet
64:introducing
1984:Categories
1953:0191569852
1783:References
1719:microwaves
205:Heisenberg
47:references
18:Spin waves
1767:Magnonics
1755:microwave
1545:, namely
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1466:×
1458:λ
1452:−
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1436:γ
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1010:−
999:⟩
958:⟩
941:⟩
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830:−
787:∑
770:−
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679:∑
662:−
613:±
595:±
543:⟩
509:⟩
437:×
320:⋅
306:∑
294:μ
287:−
272:⋅
245:∑
228:−
100:spin wave
1868:14611549
1761:See also
1689:Grenoble
1555:, where
367:points,
203:for the
108:magnetic
1848:Bibcode
1806:Bibcode
1751:ferrite
1708:photons
1677:phonons
1581:." The
527:. That
388:is the
377:-factor
371:is the
351:is the
124:bosonic
120:magnons
60:improve
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1513:where
1393:where
883:where
491:> 0
482:, the
373:Landé
347:where
156:Theory
132:phonon
49:, but
1995:Waves
1836:(PDF)
1656:X-ray
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476:3 + 1
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361:spins
102:is a
1948:ISBN
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1675:for
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474:and
392:and
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