1427:. This interaction depends on details of the crystal structure such as the bond length between magnetic ions and the angle formed by the bonds between magnetic and ligand ions. In magnetic insulators it usually is the main mechanism for magnetic ordering, and, depending on the orbital occupancies and bond angles, can lead to ferro- or antiferromagnetic interactions. As the strength of symmetric exchange depends on the relative position of the ions, it couples the spin orientations to the lattice structure. Coupling of spins to a collective distortion with a net electric dipole can occur if the magnetic order breaks inversion symmetry. Thus, symmetric exchange can provide a handle to control magnetic properties through an external electric field.
81:). The experimental confirmation came just a few months later when the effect was observed for the first time by D. Astrov. The general excitement which followed the measurement of the linear magnetoelectric effect lead to the organization of the series of Magnetoelectric Interaction Phenomena in Crystals (MEIPIC) conferences. Between the prediction of Dzyaloshinskii and the MEIPIC first edition (1973), more than 80 linear magnetoelectric compounds were found. Recently, technological and theoretical progress, driven in large part by the advent of multiferroic materials, triggered a renaissance of these studies and magnetoelectric effect is still heavily investigated.
1456:
component to another, realizing the magnetoelectric coupling. For an efficient coupling, a high-quality interface with optimal strain state is desired. In light of this interest, advanced deposition techniques have been applied to synthesize these types of thin film heterostructures. Molecular beam epitaxy has been demonstrated to be capable of depositing structures consisting of piezoelectric and magnetostrictive components. Materials systems studied included cobalt ferrite, magnetite, SrTiO3, BaTiO3, PMNT.
1003:
592:
998:{\displaystyle {\begin{aligned}F(E,H)&=F_{0}-P_{i}^{s}E_{i}-\mu _{0}M_{i}^{s}H_{i}-{\frac {1}{2}}\epsilon _{0}\chi _{ij}^{e}E_{i}E_{j}-{\frac {1}{2}}\mu _{0}\chi _{ij}^{v}H_{i}H_{j}\\&\qquad -\alpha _{ij}E_{i}H_{j}-{\frac {1}{2}}\beta _{ijk}E_{i}H_{j}H_{k}-{\frac {1}{2}}\gamma _{ijk}H_{i}E_{j}E_{k}+\ldots \end{aligned}}}
1410:
which determines preferential axes for the orientation of the spins (such as easy axes). An external electric field may change the local symmetry seen by magnetic ions and affect both the strength of the anisotropy and the direction of the easy axes. Thus, single-ion anisotropy can couple an external
1480:
Ferroelectricity developed from micromagnetic structure can appear in any magnetic material even in centrosymmetric one. Building of symmetry classification of domain walls leads to determination of the type of electric polarization rotation in volume of any magnetic domain wall. Existing symmetry
1455:
The overall effect is that the polarization of the ferroelectric substrate is manipulated by an application of a magnetic field, which is the desired magnetoelectric effect (the reverse is also possible). In this case, the interface plays an important role in mediating the responses from one
1451:
film. This process, called magnetostriction, will alter residual strain conditions in the magnetoelastic film, which can be transferred through the interface to the piezoelectric substrate. Consequently, a polarization is introduced in the substrate through the piezoelectric process.
1464:
Magnetically driven ferroelectricity is also caused by inhomogeneous magnetoelectric interaction. This effect appears due to the coupling between inhomogeneous order parameters. It was also called as flexomagnetoelectric effect. Usually it is describing using the
1366:. Therefore, the linear magnetoelectric effect may only occur if time-reversal symmetry is explicitly broken, for instance by the explicit motion in Röntgens' example, or by an intrinsic magnetic ordering in the material. In contrast, the tensor
1446:
and a piezoelectric component. This type of heterostructure is composed of an epitaxial magnetoelastic thin film grown on a piezoelectric substrate. For this system, application of a magnetic field will induce a change in the dimension of the
426:
298:
1149:
1439:/ferromagnetic materials), it is possible to couple magnetic and electric properties indirectly by creating composites of these materials that are tightly bonded so that strains transfer from one to the other.
2722:
Baryakhtar, V.G.; L'vov, V.A.; Yablonskiy, D.A. (1984). "Chapter 2 – Theory of electric polarization of domain boundaries in magnetically ordered crystals". In
Prokhorov, A.M.; Prokhorov, A.S. (eds.).
2015:
Newacheck, Scott; Webster, Taylor; Youssef, George (2018-10-22). "The effect of multidirectional bias magnetic fields on the converse magnetoelectric response of multiferroic concentric composite ring".
1069:
43:, who showed that a dielectric material moving through an electric field would become magnetized. The possibility of an intrinsic magnetoelectric effect in a (non-moving) material was conjectured by
2503:
Logginov, A.S.; Meshkov, G.A.; Nikolaev, A.V.; Nikolaeva, E.P.; Pyatakov, A.P.; Zvezdin, A.K. (2008). "Room temperature magnetoelectric control of micromagnetic structure in iron garnet films".
597:
1435:
Because materials exist that couple strain to electrical polarization (piezoelectrics, electrostrictives, and ferroelectrics) and that couple strain to magnetization (magnetostrictive/
1279:
describes the linear magnetoelectric effect, which corresponds to an electric polarization induced linearly by a magnetic field, and vice versa. The higher terms with coefficients
24:
in 1888, who found that a dielectric material moving through an electric field would become magnetized. A material where such a coupling is intrinsically present is called a
181:
1257:
1230:
151:
121:
1360:
1337:
1317:
1277:
505:
449:
1384:
1297:
2230:
Yang, J. J.; Zhao, Y.G.; et al. (2009). "Electric field manipulation of magnetization at room temperature in multiferroic CoFeO/Pb(MgNb)TiO heterostructures".
1203:
1176:
304:
153:
describe the electric and magnetic polarization responses to an electric, resp. a magnetic field, there is also the possibility of a magnetoelectric susceptibility
1481:
classification of magnetic domain walls was applied for predictions of electric polarization spatial distribution in their volumes. The predictions for almost all
1473:
crystal the four phenomenological constants approach is correct. The flexomagnetoelectric effect appears in spiral multiferroics or micromagnetic structures like
584:
564:
189:
2595:
1652:
2265:
Baryakhtar, V.G.; L'vov, V.A.; Yablonskiy, D.A. (1983). "Spin reversal in 180 domain walls of the spin-flop phase of ease-axis antiferromagnets".
1077:
522:
are another example of single-phase materials that can exhibit a general magnetoelectric effect if their magnetic and electric orders are coupled.
31:
Some promising applications of the ME effect are sensitive detection of magnetic fields, advanced logic devices and tunable microwave filters.
20:(ME) denotes any coupling between the magnetic and the electric properties of a material. The first example of such an effect was described by
538:
If the coupling between magnetic and electric properties is analytic, then the magnetoelectric effect can be described by an expansion of the
1442:
Thin film strategy enables achievement of interfacial multiferroic coupling through a mechanical channel in heterostructures consisting of a
530:
material. These two materials interact by strain, leading to a coupling between magnetic and electric properties of the compound material.
2152:
Xie, S.; Cheng, J.; et al. (2008). "Interfacial structure and chemistry of epitaxial CoFeO thin films on SrTiO and MgO substrates".
2067:
Srinivasan, G. (2002). "Magnetoelectric effects in bilayers and multilayers of magnetostrictive and piezoelectric perovskite oxides".
2837:
1954:
Delaney, Kris T.; Mostovoy, Maxim; Spaldin, Nicola A. (2009-04-17). "Superexchange-Driven
Magnetoelectricity in Magnetic Vortices".
2593:
Pyatakov, A.P.; Meshkov, G.A.; Zvezdin, A.K. (2012). "Electric polarization of magnetic textures: New horizons of micromagnetism".
510:
The first material where an intrinsic linear magnetoelectric effect was predicted theoretically and confirmed experimentally was Cr
1614:"Ueber die durch Bewegung eines im homogenen electrischen Felde befindlichen Dielectricums hervorgerufene electrodynamische Kraft"
1498:
1014:
2540:
Pyatakov, A.P.; Meshkov, G.A. (2010). "Electrically stabilized magnetic vortex and antivortex states in magnetic dielectrics".
1342:
The possible terms appearing in the expansion above are constrained by symmetries of the material. Most notably, the tensor
2693:
Baryakhtar, V.; L'vov, V.; Yablonsky, D. (1984). "Magnetic symmetry of the domain walls in magnetically ordered crystals".
1407:
484:
61:
1587:
1919:
Cardwell, M.J. (1969). "Measurements of the magnetic field dependent electric susceptibility of yttrium iron garnet".
2842:
539:
1805:
1691:
1403:
2505:
1474:
124:
94:
1363:
2738:
Tanygin, B.M. (2011). "Symmetry theory of the flexomagnetoelectric effect in the magnetic domain walls".
526:
are another way to realize magnetoelectrics. There, the idea is to combine, say a magnetostrictive and a
1501:
theory appears if energy terms with electrical polarization spatial derivatives are taken into account.
1448:
1443:
1436:
1009:
2806:
2757:
2704:
2659:
2614:
2559:
2514:
2461:
2416:
2355:
2302:
2275:
2239:
2196:
2161:
2118:
2103:
2076:
2025:
1973:
1928:
1820:
1667:
1625:
1553:
1520:
1470:
456:
183:
which describes a linear response of the electric polarization to a magnetic field, and vice versa:
70:
66:
1394:
There are several ways in which a magnetoelectric effect can arise microscopically in a material.
156:
2796:
2773:
2747:
2675:
2630:
2604:
2575:
2549:
2485:
2432:
2406:
2379:
2345:
2318:
2049:
1997:
1963:
1901:
1844:
1683:
1569:
523:
1653:"Multiferroic magnetoelectric composites: Historical perspective, status, and future directions"
40:
21:
1235:
1208:
129:
99:
2477:
2371:
2212:
2134:
2041:
1989:
1893:
1885:
421:{\displaystyle \mu _{0}M_{i}=\sum _{j}\mu _{0}\chi _{ij}^{v}H_{j}+\sum _{j}\alpha _{ij}E_{j},}
69:, using an elegant symmetry argument, derive the form of a linear magnetoelectric coupling in
1419:
The main interaction between spins of transition metal ions in solids is usually provided by
1345:
1322:
1302:
1262:
490:
434:
2814:
2810:
2793:
Inhomogeneous magnetoelectric effect on defect in multiferroic
Material: Symmetry prediction
2765:
2667:
2622:
2567:
2522:
2469:
2452:
2424:
2363:
2310:
2247:
2204:
2169:
2126:
2084:
2033:
1981:
1936:
1877:
1836:
1828:
1675:
1633:
1561:
1510:
1369:
1282:
2293:
Pyatakov, A.P.; Zvezdin, A.K. (2009). "Flexomagnetoelectric interaction in multiferroics".
1469:
invariant (i.e. single-constant coupling term). It was shown that in general case of cubic
1339:
describes a linear magnetoelectric effect which is, in turn, induced by an electric field.
1181:
1154:
2795:. Materials Science and Engineering. IOP Conference Series. Vol. 15. p. 012073.
2695:
293:{\displaystyle P_{i}=\sum _{j}\epsilon _{0}\chi _{ij}^{e}E_{j}+\sum _{j}\alpha _{ij}H_{j}}
56:
1590:. Laboratory for Multifunctional Ferroic Materials. Condensed matter research. ETH ZĂĽrich
51:
in 1926. A mathematical formulation of the linear magnetoelectric effect was included in
2818:
2761:
2708:
2663:
2618:
2563:
2518:
2465:
2420:
2359:
2306:
2279:
2243:
2200:
2165:
2122:
2080:
2029:
1977:
1932:
1824:
1671:
1629:
1557:
1865:
1490:
1482:
569:
549:
480:
476:
2831:
2777:
2679:
2650:
2634:
2579:
2489:
2436:
2397:
Tanygin, B.M. (2011). "On the free energy of the flexomagnetoelectric interactions".
2322:
2187:
Bibes, M.; Barthélémy, A. (2008). "Multiferroics: Towards a magnetoelectric memory".
2053:
1905:
1779:
1573:
1565:
1515:
1486:
1420:
1072:
527:
519:
464:
2383:
2001:
1868:; Fiebig, Manfred (2005-07-15). "The Renaissance of Magnetoelectric Multiferroics".
1848:
1687:
2671:
1985:
543:
44:
2450:
Kimura, T.; et al. (2003). "Magnetic control of ferroelectric polarization".
2367:
1750:
2314:
1651:
Nan, C.W.; Bichurin, M.I.; Dong, Shuxiang; Viehland, D.; Srinivasan, G. (2008).
1144:{\displaystyle M_{i}=-{\frac {1}{\mu _{0}}}{\frac {\partial F}{\partial H_{i}}}}
48:
2769:
2626:
2428:
2088:
1840:
2571:
2267:
1940:
52:
2045:
1889:
1637:
1881:
2481:
2375:
2216:
2138:
1993:
1897:
1205:
are the static polarization, resp. magnetization of the material, whereas
2350:
1466:
2473:
1494:
89:
Historically, the first and most studied example of this effect is the
39:
The first example of a magnetoelectric effect was discussed in 1888 by
2526:
2251:
2173:
2037:
1832:
1679:
2208:
2130:
1613:
1804:
Spaldin, Nicola A.; Cheong, Sang-Wook; Ramesh, Ramamoorthy (2010).
2801:
2752:
2609:
2554:
2411:
1968:
2648:
Dzyaloshinskii, I. (2008). "Magnetoelectricity in ferromagnets".
1259:
are the electric, resp. magnetic susceptibilities. The tensor
1544:
Fiebig, M. (2005). "Revival of the magnetoelectric effect".
2336:
Mostovoy, M. (2006). "Ferroelectricity in Spiral
Magnets".
1411:
electric field to spins of magnetically ordered compounds.
1386:
may be non-vanishing in time-reversal symmetric materials.
1064:{\displaystyle P_{i}=-{\frac {\partial F}{\partial E_{i}}}}
47:
in 1894, while the term "magnetoelectric" was coined by
1751:"On the Magneto-Electrical Effect in Antiferromagnets"
1431:
Strain driven magnetoelectric heterostructured effect
1372:
1348:
1325:
1319:
describe quadratic effects. For instance, the tensor
1305:
1285:
1265:
1238:
1211:
1184:
1157:
1080:
1017:
595:
572:
552:
493:
437:
307:
192:
159:
132:
102:
1744:
1742:
1008:Differentiating the free energy will then give the
1780:"The magnetoelectric effect in antiferromagnetics"
1485:conform with phenomenology in which inhomogeneous
1378:
1354:
1331:
1311:
1291:
1271:
1251:
1224:
1197:
1170:
1143:
1063:
997:
578:
558:
499:
443:
420:
292:
175:
145:
115:
1726:
1724:
1722:
1607:
1605:
2727:. Moscow, RU: Mir Publishers. pp. 56–80.
8:
1773:
1771:
1707:P. Curie J. Physique, 3ième série III (1894)
1539:
1537:
1535:
2740:Journal of Magnetism and Magnetic Materials
2596:Journal of Magnetism and Magnetic Materials
2399:Journal of Magnetism and Magnetic Materials
1806:"Multiferroics: Past, present, and future"
451:must be the same in both equations. Here,
2800:
2751:
2608:
2553:
2410:
2349:
1967:
1371:
1347:
1324:
1304:
1284:
1264:
1243:
1237:
1216:
1210:
1189:
1183:
1162:
1156:
1132:
1114:
1106:
1097:
1085:
1079:
1052:
1034:
1022:
1016:
979:
969:
959:
943:
929:
920:
910:
900:
884:
870:
861:
851:
838:
817:
807:
797:
789:
779:
765:
756:
746:
736:
728:
718:
704:
695:
685:
680:
670:
657:
647:
642:
629:
596:
594:
571:
551:
492:
436:
409:
396:
386:
373:
363:
355:
345:
335:
322:
312:
306:
284:
271:
261:
248:
238:
230:
220:
210:
197:
191:
164:
158:
137:
131:
107:
101:
1531:
2104:"Data storage: Multiferroic memories"
1860:
1858:
1546:Journal of Physics D: Applied Physics
7:
546:in the electric and magnetic fields
1733:Electrodynamics of Continuous Media
518:. This is a single-phase material.
2542:Moscow University Physics Bulletin
1125:
1117:
1045:
1037:
14:
1731:Landau, L.; Lifshitz, E. (1960).
1716:P. Debye, Z. Phys. 36, 300 (1926)
2725:Problems in solid-state physics
830:
507:has units of second per meter.
1986:10.1103/PhysRevLett.102.157203
1406:is responsible for single-ion
615:
603:
16:In its most general form, the
1:
2819:10.1088/1757-899x/15/1/012073
2368:10.1103/physrevlett.96.067601
1408:magnetocrystalline anisotropy
91:linear magnetoelectric effect
85:Linear magnetoelectric effect
62:Course of Theoretical Physics
1415:Symmetric Exchange striction
1362:must be antisymmetric under
176:{\displaystyle \alpha _{ij}}
93:. Mathematically, while the
1749:Dzyaloshinskii, I. (1960).
1460:Flexomagnetoelectric effect
2859:
2770:10.1016/j.jmmm.2010.10.028
2672:10.1209/0295-5075/83/67001
2627:10.1016/j.jmmm.2012.02.087
2429:10.1016/j.jmmm.2011.02.035
2315:10.1140/epjb/e2009-00281-5
2089:10.1103/physrevb.65.134402
1660:Journal of Applied Physics
1566:10.1088/0022-3727/38/8/R01
2572:10.3103/S0027134910040156
1941:10.1080/14786436908228077
1489:couples with homogeneous
1252:{\displaystyle \chi ^{v}}
1225:{\displaystyle \chi ^{e}}
146:{\displaystyle \chi ^{v}}
116:{\displaystyle \chi ^{e}}
2838:Condensed matter physics
1638:10.1002/andp.18882711003
1588:"Magnetoelectric Effect"
2811:2010MS&E...15a2073T
2506:Applied Physics Letters
2232:Applied Physics Letters
2018:Applied Physics Letters
1956:Physical Review Letters
1882:10.1126/science.1113357
1666:(3): 031101–031101–35.
1477:and magnetic vortexes.
1355:{\displaystyle \alpha }
1332:{\displaystyle \gamma }
1312:{\displaystyle \gamma }
1272:{\displaystyle \alpha }
500:{\displaystyle \alpha }
444:{\displaystyle \alpha }
125:magnetic susceptibility
95:electric susceptibility
2791:Tanygin, B.M. (2010).
1921:Philosophical Magazine
1612:Röntgen, W.C. (1888).
1380:
1379:{\displaystyle \beta }
1364:time-reversal symmetry
1356:
1333:
1313:
1293:
1292:{\displaystyle \beta }
1273:
1253:
1226:
1199:
1172:
1145:
1065:
999:
580:
560:
501:
445:
422:
294:
177:
147:
117:
18:magnetoelectric effect
2160:(18): 181901–181903.
1497:between symmetry and
1398:Single-ion anisotropy
1381:
1357:
1334:
1314:
1294:
1274:
1254:
1227:
1200:
1198:{\displaystyle M^{s}}
1173:
1171:{\displaystyle P^{s}}
1146:
1066:
1010:electric polarization
1000:
581:
561:
534:General phenomenology
502:
457:electric polarization
446:
423:
295:
178:
148:
118:
2102:Scott, J.F. (2007).
1521:Exchange interaction
1370:
1346:
1323:
1303:
1283:
1263:
1236:
1209:
1182:
1155:
1078:
1015:
593:
570:
550:
491:
435:
305:
190:
157:
130:
100:
2762:2011JMMM..323..616T
2709:1984JETP...60.1072B
2664:2008EL.....8367001D
2619:2012JMMM..324.3551P
2564:2010arXiv1001.0391P
2519:2008ApPhL..93r2510L
2474:10.1038/nature02018
2466:2003Natur.426...55K
2421:2011JMMM..323.1899T
2360:2006PhRvL..96f7601M
2307:2009EPJB...71..419P
2280:1983JETPL..37..673B
2244:2009ApPhL..94u2504Y
2201:2008NatMa...7..425B
2166:2008ApPhL..93r1901X
2123:2007NatMa...6..256S
2081:2002PhRvB..65m4402S
2030:2018ApPhL.113q2902N
1978:2009PhRvL.102o7203D
1933:1969PMag...20.1087C
1841:20.500.11850/190313
1825:2010PhT....63j..38S
1778:Astrov, D. (1960).
1758:Zh. Eksp. Teor. Fiz
1672:2008JAP...103c1101N
1630:1888AnP...271..264R
1558:2005JPhD...38R.123F
1404:spin–orbit coupling
802:
741:
690:
652:
524:Composite materials
368:
243:
71:chromium(III) oxide
67:Igor Dzyaloshinskii
65:. Only in 1959 did
1866:Spaldin, Nicola A.
1425:symmetric exchange
1390:Microscopic origin
1376:
1352:
1329:
1309:
1289:
1269:
1249:
1222:
1195:
1168:
1141:
1061:
995:
993:
785:
724:
676:
638:
576:
556:
497:
441:
418:
391:
351:
340:
290:
266:
226:
215:
173:
143:
113:
2843:Materials science
2603:(21): 3551–3554.
2527:10.1063/1.3013569
2405:(14): 1899–1902.
2252:10.1063/1.3143622
2174:10.1063/1.3006060
2038:10.1063/1.5050631
1876:(5733): 391–392.
1833:10.1063/1.3502547
1735:. Pergamon Press.
1680:10.1063/1.2836410
1139:
1112:
1059:
937:
878:
773:
712:
579:{\displaystyle H}
559:{\displaystyle E}
382:
331:
257:
206:
2850:
2823:
2822:
2804:
2788:
2782:
2781:
2755:
2735:
2729:
2728:
2719:
2713:
2712:
2703:(5): 1072–1080.
2690:
2684:
2683:
2645:
2639:
2638:
2612:
2590:
2584:
2583:
2557:
2537:
2531:
2530:
2500:
2494:
2493:
2447:
2441:
2440:
2414:
2394:
2388:
2387:
2353:
2351:cond-mat/0510692
2333:
2327:
2326:
2290:
2284:
2283:
2262:
2256:
2255:
2227:
2221:
2220:
2209:10.1038/nmat2189
2189:Nature Materials
2184:
2178:
2177:
2154:Appl. Phys. Lett
2149:
2143:
2142:
2131:10.1038/nmat1868
2111:Nature Materials
2108:
2099:
2093:
2092:
2064:
2058:
2057:
2012:
2006:
2005:
1971:
1951:
1945:
1944:
1916:
1910:
1909:
1862:
1853:
1852:
1810:
1801:
1795:
1794:
1784:
1775:
1766:
1765:
1755:
1746:
1737:
1736:
1728:
1717:
1714:
1708:
1705:
1699:
1698:
1696:
1690:. Archived from
1657:
1648:
1642:
1641:
1609:
1600:
1599:
1597:
1595:
1584:
1578:
1577:
1541:
1511:Piezoelectricity
1385:
1383:
1382:
1377:
1361:
1359:
1358:
1353:
1338:
1336:
1335:
1330:
1318:
1316:
1315:
1310:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1258:
1256:
1255:
1250:
1248:
1247:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1194:
1193:
1177:
1175:
1174:
1169:
1167:
1166:
1150:
1148:
1147:
1142:
1140:
1138:
1137:
1136:
1123:
1115:
1113:
1111:
1110:
1098:
1090:
1089:
1070:
1068:
1067:
1062:
1060:
1058:
1057:
1056:
1043:
1035:
1027:
1026:
1004:
1002:
1001:
996:
994:
984:
983:
974:
973:
964:
963:
954:
953:
938:
930:
925:
924:
915:
914:
905:
904:
895:
894:
879:
871:
866:
865:
856:
855:
846:
845:
826:
822:
821:
812:
811:
801:
796:
784:
783:
774:
766:
761:
760:
751:
750:
740:
735:
723:
722:
713:
705:
700:
699:
689:
684:
675:
674:
662:
661:
651:
646:
634:
633:
585:
583:
582:
577:
565:
563:
562:
557:
506:
504:
503:
498:
450:
448:
447:
442:
427:
425:
424:
419:
414:
413:
404:
403:
390:
378:
377:
367:
362:
350:
349:
339:
327:
326:
317:
316:
299:
297:
296:
291:
289:
288:
279:
278:
265:
253:
252:
242:
237:
225:
224:
214:
202:
201:
182:
180:
179:
174:
172:
171:
152:
150:
149:
144:
142:
141:
122:
120:
119:
114:
112:
111:
2858:
2857:
2853:
2852:
2851:
2849:
2848:
2847:
2828:
2827:
2826:
2790:
2789:
2785:
2737:
2736:
2732:
2721:
2720:
2716:
2696:Sov. Phys. JETP
2692:
2691:
2687:
2647:
2646:
2642:
2592:
2591:
2587:
2539:
2538:
2534:
2502:
2501:
2497:
2460:(6962): 55–58.
2449:
2448:
2444:
2396:
2395:
2391:
2338:Phys. Rev. Lett
2335:
2334:
2330:
2295:Eur. Phys. J. B
2292:
2291:
2287:
2274:(12): 673–675.
2264:
2263:
2259:
2229:
2228:
2224:
2186:
2185:
2181:
2151:
2150:
2146:
2106:
2101:
2100:
2096:
2066:
2065:
2061:
2014:
2013:
2009:
1953:
1952:
1948:
1918:
1917:
1913:
1864:
1863:
1856:
1808:
1803:
1802:
1798:
1787:Sov. Phys. JETP
1782:
1777:
1776:
1769:
1753:
1748:
1747:
1740:
1730:
1729:
1720:
1715:
1711:
1706:
1702:
1694:
1655:
1650:
1649:
1645:
1611:
1610:
1603:
1593:
1591:
1586:
1585:
1581:
1543:
1542:
1533:
1529:
1507:
1483:symmetry groups
1462:
1433:
1417:
1400:
1392:
1368:
1367:
1344:
1343:
1321:
1320:
1301:
1300:
1281:
1280:
1261:
1260:
1239:
1234:
1233:
1212:
1207:
1206:
1185:
1180:
1179:
1158:
1153:
1152:
1128:
1124:
1116:
1102:
1081:
1076:
1075:
1048:
1044:
1036:
1018:
1013:
1012:
992:
991:
975:
965:
955:
939:
916:
906:
896:
880:
857:
847:
834:
824:
823:
813:
803:
775:
752:
742:
714:
691:
666:
653:
625:
618:
591:
590:
568:
567:
548:
547:
536:
517:
513:
489:
488:
481:magnetic fields
433:
432:
405:
392:
369:
341:
318:
308:
303:
302:
280:
267:
244:
216:
193:
188:
187:
160:
155:
154:
133:
128:
127:
103:
98:
97:
87:
80:
76:
57:Evgeny Lifshitz
41:Wilhelm Röntgen
37:
26:magnetoelectric
22:Wilhelm Röntgen
12:
11:
5:
2856:
2854:
2846:
2845:
2840:
2830:
2829:
2825:
2824:
2783:
2746:(5): 616–619.
2730:
2714:
2685:
2640:
2585:
2548:(4): 329–331.
2532:
2513:(18): 182510.
2495:
2442:
2389:
2328:
2301:(3): 419–427.
2285:
2257:
2238:(21): 212504.
2222:
2195:(6): 425–426.
2179:
2144:
2117:(4): 256–257.
2094:
2075:(13): 134402.
2059:
2024:(17): 172902.
2007:
1962:(15): 157203.
1946:
1911:
1854:
1796:
1767:
1738:
1718:
1709:
1700:
1697:on 2020-04-12.
1643:
1601:
1579:
1530:
1528:
1525:
1524:
1523:
1518:
1513:
1506:
1503:
1461:
1458:
1449:magnetoelastic
1444:magnetoelastic
1437:magnetoelastic
1432:
1429:
1423:, also called
1416:
1413:
1399:
1396:
1391:
1388:
1375:
1351:
1328:
1308:
1288:
1268:
1246:
1242:
1219:
1215:
1192:
1188:
1165:
1161:
1135:
1131:
1127:
1122:
1119:
1109:
1105:
1101:
1096:
1093:
1088:
1084:
1055:
1051:
1047:
1042:
1039:
1033:
1030:
1025:
1021:
1006:
1005:
990:
987:
982:
978:
972:
968:
962:
958:
952:
949:
946:
942:
936:
933:
928:
923:
919:
913:
909:
903:
899:
893:
890:
887:
883:
877:
874:
869:
864:
860:
854:
850:
844:
841:
837:
833:
829:
827:
825:
820:
816:
810:
806:
800:
795:
792:
788:
782:
778:
772:
769:
764:
759:
755:
749:
745:
739:
734:
731:
727:
721:
717:
711:
708:
703:
698:
694:
688:
683:
679:
673:
669:
665:
660:
656:
650:
645:
641:
637:
632:
628:
624:
621:
619:
617:
614:
611:
608:
605:
602:
599:
598:
575:
555:
535:
532:
515:
511:
496:
440:
429:
428:
417:
412:
408:
402:
399:
395:
389:
385:
381:
376:
372:
366:
361:
358:
354:
348:
344:
338:
334:
330:
325:
321:
315:
311:
300:
287:
283:
277:
274:
270:
264:
260:
256:
251:
247:
241:
236:
233:
229:
223:
219:
213:
209:
205:
200:
196:
170:
167:
163:
140:
136:
110:
106:
86:
83:
78:
74:
36:
33:
13:
10:
9:
6:
4:
3:
2:
2855:
2844:
2841:
2839:
2836:
2835:
2833:
2820:
2816:
2812:
2808:
2803:
2798:
2794:
2787:
2784:
2779:
2775:
2771:
2767:
2763:
2759:
2754:
2749:
2745:
2741:
2734:
2731:
2726:
2718:
2715:
2710:
2706:
2702:
2698:
2697:
2689:
2686:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2652:
2644:
2641:
2636:
2632:
2628:
2624:
2620:
2616:
2611:
2606:
2602:
2598:
2597:
2589:
2586:
2581:
2577:
2573:
2569:
2565:
2561:
2556:
2551:
2547:
2543:
2536:
2533:
2528:
2524:
2520:
2516:
2512:
2508:
2507:
2499:
2496:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2454:
2446:
2443:
2438:
2434:
2430:
2426:
2422:
2418:
2413:
2408:
2404:
2400:
2393:
2390:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2352:
2347:
2344:(6): 067601.
2343:
2339:
2332:
2329:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2296:
2289:
2286:
2281:
2277:
2273:
2270:
2269:
2261:
2258:
2253:
2249:
2245:
2241:
2237:
2233:
2226:
2223:
2218:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2183:
2180:
2175:
2171:
2167:
2163:
2159:
2155:
2148:
2145:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2105:
2098:
2095:
2090:
2086:
2082:
2078:
2074:
2070:
2063:
2060:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2011:
2008:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1970:
1965:
1961:
1957:
1950:
1947:
1942:
1938:
1934:
1930:
1927:(167): 1087.
1926:
1922:
1915:
1912:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1861:
1859:
1855:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1813:Physics Today
1807:
1800:
1797:
1792:
1788:
1781:
1774:
1772:
1768:
1763:
1759:
1752:
1745:
1743:
1739:
1734:
1727:
1725:
1723:
1719:
1713:
1710:
1704:
1701:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1661:
1654:
1647:
1644:
1639:
1635:
1631:
1627:
1623:
1620:(in German).
1619:
1615:
1608:
1606:
1602:
1589:
1583:
1580:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1540:
1538:
1536:
1532:
1526:
1522:
1519:
1517:
1516:Multiferroics
1514:
1512:
1509:
1508:
1504:
1502:
1500:
1499:phenomenology
1496:
1492:
1488:
1487:magnetization
1484:
1478:
1476:
1472:
1471:hexoctahedral
1468:
1459:
1457:
1453:
1450:
1445:
1440:
1438:
1430:
1428:
1426:
1422:
1421:superexchange
1414:
1412:
1409:
1405:
1402:In crystals,
1397:
1395:
1389:
1387:
1373:
1365:
1349:
1340:
1326:
1306:
1286:
1266:
1244:
1240:
1217:
1213:
1190:
1186:
1163:
1159:
1133:
1129:
1120:
1107:
1103:
1099:
1094:
1091:
1086:
1082:
1074:
1073:magnetization
1053:
1049:
1040:
1031:
1028:
1023:
1019:
1011:
988:
985:
980:
976:
970:
966:
960:
956:
950:
947:
944:
940:
934:
931:
926:
921:
917:
911:
907:
901:
897:
891:
888:
885:
881:
875:
872:
867:
862:
858:
852:
848:
842:
839:
835:
831:
828:
818:
814:
808:
804:
798:
793:
790:
786:
780:
776:
770:
767:
762:
757:
753:
747:
743:
737:
732:
729:
725:
719:
715:
709:
706:
701:
696:
692:
686:
681:
677:
671:
667:
663:
658:
654:
648:
643:
639:
635:
630:
626:
622:
620:
612:
609:
606:
600:
589:
588:
587:
573:
553:
545:
541:
533:
531:
529:
528:piezoelectric
525:
521:
520:Multiferroics
508:
494:
486:
482:
478:
474:
470:
466:
465:magnetization
462:
458:
454:
438:
415:
410:
406:
400:
397:
393:
387:
383:
379:
374:
370:
364:
359:
356:
352:
346:
342:
336:
332:
328:
323:
319:
313:
309:
301:
285:
281:
275:
272:
268:
262:
258:
254:
249:
245:
239:
234:
231:
227:
221:
217:
211:
207:
203:
198:
194:
186:
185:
184:
168:
165:
161:
138:
134:
126:
108:
104:
96:
92:
84:
82:
72:
68:
64:
63:
58:
54:
50:
46:
42:
34:
32:
29:
27:
23:
19:
2792:
2786:
2743:
2739:
2733:
2724:
2717:
2700:
2694:
2688:
2658:(6): 67001.
2655:
2649:
2643:
2600:
2594:
2588:
2545:
2541:
2535:
2510:
2504:
2498:
2457:
2451:
2445:
2402:
2398:
2392:
2341:
2337:
2331:
2298:
2294:
2288:
2271:
2266:
2260:
2235:
2231:
2225:
2192:
2188:
2182:
2157:
2153:
2147:
2114:
2110:
2097:
2072:
2069:Phys. Rev. B
2068:
2062:
2021:
2017:
2010:
1959:
1955:
1949:
1924:
1920:
1914:
1873:
1869:
1816:
1812:
1799:
1790:
1786:
1761:
1757:
1732:
1712:
1703:
1692:the original
1663:
1659:
1646:
1621:
1617:
1592:. Retrieved
1582:
1549:
1545:
1493:. The total
1491:polarization
1479:
1475:domain walls
1463:
1454:
1441:
1434:
1424:
1418:
1401:
1393:
1341:
1007:
544:power series
537:
509:
472:
468:
460:
452:
430:
90:
88:
60:
45:Pierre Curie
38:
30:
25:
17:
15:
1624:(10): 264.
1552:(8): R123.
540:free energy
431:The tensor
49:Peter Debye
2832:Categories
2268:JETP Lett.
1819:(10): 38.
1618:Ann. Phys.
1527:References
53:Lev Landau
2802:1007.3531
2778:119111445
2753:1007.3524
2680:119672380
2635:118383876
2610:1211.2403
2580:122153369
2555:1001.0391
2490:205209892
2437:119225609
2412:1105.5300
2323:122234441
2054:125847351
2046:0003-6951
1969:0810.0552
1906:118513837
1890:0036-8075
1574:121588385
1374:β
1350:α
1327:γ
1307:γ
1287:β
1267:α
1241:χ
1214:χ
1126:∂
1118:∂
1104:μ
1095:−
1046:∂
1038:∂
1032:−
989:…
941:γ
927:−
882:β
868:−
836:α
832:−
787:χ
777:μ
763:−
726:χ
716:ϵ
702:−
668:μ
664:−
636:−
495:α
439:α
394:α
384:∑
353:χ
343:μ
333:∑
310:μ
269:α
259:∑
228:χ
218:ϵ
208:∑
162:α
135:χ
105:χ
2482:14603314
2384:36936649
2376:16606047
2217:18497843
2139:17351613
2002:27782114
1994:19518672
1898:16020720
1849:36755212
1688:51900508
1505:See also
1467:Lifshitz
1151:. Here,
1071:and the
485:SI units
477:electric
2807:Bibcode
2758:Bibcode
2705:Bibcode
2660:Bibcode
2615:Bibcode
2560:Bibcode
2515:Bibcode
2462:Bibcode
2417:Bibcode
2356:Bibcode
2303:Bibcode
2276:Bibcode
2240:Bibcode
2197:Bibcode
2162:Bibcode
2119:Bibcode
2077:Bibcode
2026:Bibcode
1974:Bibcode
1929:Bibcode
1870:Science
1821:Bibcode
1668:Bibcode
1626:Bibcode
1594:15 July
1554:Bibcode
1495:synergy
455:is the
35:History
2776:
2678:
2633:
2578:
2488:
2480:
2453:Nature
2435:
2382:
2374:
2321:
2215:
2137:
2052:
2044:
2000:
1992:
1904:
1896:
1888:
1847:
1793:: 708.
1764:: 881.
1686:
1572:
2797:arXiv
2774:S2CID
2748:arXiv
2676:S2CID
2631:S2CID
2605:arXiv
2576:S2CID
2550:arXiv
2486:S2CID
2433:S2CID
2407:arXiv
2380:S2CID
2346:arXiv
2319:S2CID
2107:(PDF)
2050:S2CID
1998:S2CID
1964:arXiv
1902:S2CID
1845:S2CID
1809:(PDF)
1783:(PDF)
1754:(PDF)
1695:(PDF)
1684:S2CID
1656:(PDF)
1570:S2CID
542:as a
483:. In
2478:PMID
2372:PMID
2213:PMID
2135:PMID
2042:ISSN
1990:PMID
1894:PMID
1886:ISSN
1596:2017
1299:and
1232:and
1178:and
566:and
479:and
475:the
471:and
463:the
123:and
55:and
2815:doi
2766:doi
2744:323
2668:doi
2651:EPL
2623:doi
2601:324
2568:doi
2523:doi
2470:doi
2458:426
2425:doi
2403:323
2364:doi
2311:doi
2248:doi
2205:doi
2170:doi
2127:doi
2085:doi
2034:doi
2022:113
1982:doi
1960:102
1937:doi
1878:doi
1874:309
1837:hdl
1829:doi
1676:doi
1664:103
1634:doi
1562:doi
73:(Cr
59:'s
2834::
2813:.
2805:.
2772:.
2764:.
2756:.
2742:.
2701:60
2699:.
2674:.
2666:.
2656:83
2654:.
2629:.
2621:.
2613:.
2599:.
2574:.
2566:.
2558:.
2546:65
2544:.
2521:.
2511:93
2509:.
2484:.
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