2551:
1656:
240:
20:
3456:
1780:
3447:. Many of these techniques only measure either the direction or strength of the interaction and make assumptions on the symmetry or coupling of the spin interaction. A recent advancement in broadband electron spin resonance coupled with optical detection (OD-ESR) allows for characterization of the DMI vector for rare-earth ion materials with no assumptions and across a large spectrum of magnetic field strength.
1372:
2546:{\displaystyle {\begin{aligned}\delta E&=2\lambda \sum \limits _{m}{\frac {J(nn'mn')}{E_{n}-E_{m}}}\langle n|\mathbf {L_{i}} |m\rangle \cdot \\&+2\lambda \sum _{m'}{\frac {J(nn'nm')}{E_{n'}-E_{m'}}}\langle n'|\mathbf {L_{j}} |m'\rangle \cdot \\&=2i\lambda \sum \limits _{m,m'}\left\cdot \\&=\mathbf {D} _{ij}\cdot .\end{aligned}}}
290:
as the microscopic mechanism of the antisymmetric exchange interaction. Moriya referred to this phenomenon specifically as the "antisymmetric part of the anisotropic superexchange interaction." The simplified naming of this phenomenon occurred in 1962, when D. Treves and S. Alexander of Bell
3657:
Most multiferroic materials are transition metal oxides due to the magnetization potential of the 3d electrons. Many can also be classified as perovskites and contain the Fe ion alongside a lanthanide ion. Below is an abbreviated table of common multiferroic compounds. For more examples and
3601:
above this temperature undergoes a structural change where its total spin vector no longer points along the crystal axis but at a slight angle along the basal (111) plane. This is what causes the iron-containing compound to display an instantaneous ferromagnetic moment above 250K, while the
3647:, because the systems tend to enhance the magnetic interaction energy at the cost of lattice energy. This mechanism is called "inverse Dzyaloshinskii–Moriya effect". In certain magnetic structures, all ligand ions are shifted into the same direction, leading to a net electric polarization.
3602:
chromium-containing compound shows no change. It is thus the combination of the distribution of ion spins, the misalignment of the total spin vector, and the resulting antisymmetry of the unit cell that gives rise to the antisymmetric exchange phenomenon seen in these crystal structures.
1651:{\displaystyle J(nn'mm')=\int \int \phi _{n}^{*}(\mathbf {r_{1}} -\mathbf {R} )\phi _{n'}^{*}(\mathbf {r_{2}} -\mathbf {R'} ){\frac {e^{2}}{r_{12}}}\phi _{m}(\mathbf {r_{2}} -\mathbf {R} )\phi _{m'}(\mathbf {r_{1}} -\mathbf {R'} )\mathrm {d} \mathbf {r_{1}} \mathrm {d} \mathbf {r_{2}} }
3350:
1341:
3430:
The
Dzyaloshinskii–Moriya interaction has proven difficult to experimentally measure directly due to its typically weak effects and similarity to other magnetoelectric effects in bulk materials. Attempts to quantify the DMI vector have utilized
209:
2863:
4105:
Cyril
Laplane; Emmanuel Zambrini Cruzeiro; Florian Frowis; Phillipe Goldner; Mikael Afzelius (2016). "High-precision measurement of the Dzyaloshinskii-Moriya interaction between two rare-earth ions in a solid".
349:
3491:
The image on the right displays a coordinated heavy metal-oxide complex that can display ferromagnetic or antiferromagnetic behavior depending on the metal ion. The structure shown is referred to as the
3085:
3224:
is constrained by symmetry, as discussed already in Moriya’s original publication. Considering the case that the magnetic interaction between two neighboring ions is transferred via a single third ion (
2971:
3650:
Because of their magneto electric coupling, multiferroic materials are of interest in applications where there is a need to control magnetism through applied electric fields. Such applications include
1705:
1785:
474:
291:
Telephone
Laboratories simply referred to the interaction as antisymmetric exchange. Because of their seminal contributions to the field, antisymmetric exchange is sometimes referred to as the
3587:
structure, the spins of the first and third metal ion are positive while the second and fourth are negative. Both compounds are antiferromagnetic at cold temperatures (<250K), however
3267:
461:
417:
3187:
3420:
3382:
3262:
3222:
2591:
469:
99:
70:
2726:
1749:
1727:
111:
1772:
3133:
260:
The discovery of antisymmetric exchange originated in the early 20th century from the controversial observation of weak ferromagnetism in typically antiferromagnetic
3654:(TMR) sensors, spin valves with electric field tunable functions, high-sensitivity alternating magnetic field sensors, and electrically tunable microwave devices.
375:
3156:
2996:
2751:
2654:
3107:
3016:
2905:
2885:
2771:
2698:
2674:
2631:
2611:
1364:
2776:
3627:
configurations that are stabilized by the
Dzyaloshinskii-Moriya interaction. Skyrmions are topological in nature, making them promising candidates for future
4580:
4041:
4255:
Mihailova, B.; Gospodinov, M. M.; Guttler, G.; Yen, F.; Litvinchuk, A. P.; Iliev, M. N. (2005). "Temperature-dependent Raman spectra of HoMn
3639:
Antisymmetric exchange is of importance for the understanding of magnetism induced electric polarization in a recently discovered class of
306:
4435:
Kundys, Bohdan; Simon, Charles; Martin, Christine (2008). "Effect of magnetic field and temperature on the ferroelectric loop in MnWO
303:
The functional form of the DMI can be obtained through a second-order perturbative analysis of the spin-orbit coupling interaction,
4212:
Nan, C. W.; et al. (2008). "Multiferroic magnetoelectric composites: Historical perspective, status, and future directions".
3565:
space group symmetry. The upper half unit cell displayed shows four M ions along the space diagonal of the rhombohedron. In the Fe
3021:
2910:
275:
102:
1664:
278:
provided evidence that the interaction was due to the relativistic spin lattice and magnetic dipole interactions based on
219:
of otherwise parallel or antiparallel aligned magnetic moments and thus, is a source of weak ferromagnetic behavior in an
3345:{\displaystyle \mathbf {D} _{ij}\propto \mathbf {r} _{i}\times \mathbf {r} _{j}=\mathbf {r} _{ij}\times \mathbf {x} }
3651:
428:
384:
4364:
3847:
3161:
3967:
D. Treves; S. Alexander (1962). "Observation of antisymmetric exchange interaction in
Yttrium Orthoferrite".
3573:
structure, the spins of the first and last metal ion are positive while the center two are negative. In the
3522:
3440:
3387:
1336:{\displaystyle {\begin{aligned}\delta E=\sum _{m}&{\Biggl }\\+\sum _{m'}&{\Biggl }\end{aligned}}}
3355:
3235:
3195:
2564:
2561:
In an actual crystal, symmetries of neighboring ions dictate the magnitude and direction of the vector
4074:
75:
46:
4585:
4540:
4458:
4401:
4363:
Chaudhury, R. P.; Yen, F.; Dela Cruz, C. R.; Lorenz, B.; Wang, Y. Q.; Sun, Y. Y.; Chu, C. W. (2007).
4329:
4280:
4221:
4178:
4125:
4053:
4011:
3976:
3941:
3903:
3842:
3436:
2703:
204:{\displaystyle H_{i,j}^{\rm {(DM)}}=\mathbf {D} _{ij}\cdot (\mathbf {S} _{i}\times \mathbf {S} _{j})}
40:
4492:
Jana R.; et al. (2015). "Direct
Observation of Re-entrant Multiferroic CuO at High Pressures".
3932:
I. Dzyaloshinskii (1958). "A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics".
287:
1732:
1710:
4556:
4530:
4493:
4474:
4448:
4417:
4391:
4345:
4319:
4237:
4149:
4115:
4087:
3444:
224:
239:
4194:
4141:
4079:
3644:
3616:
3432:
1754:
19:
3112:
4548:
4466:
4409:
4337:
4288:
4229:
4186:
4169:
4133:
4069:
4061:
4019:
3984:
3949:
3911:
3799:
3455:
23:
Determination of the orientation of the
Dzyaloshinskii–Moriya vector from the local geometry
4513:
Zajdel P.; et al. (2017). "Structure and
Magnetism in the Bond Frustrated Spinel, ZnCr
3704:
220:
4002:
F. Keffer (1962). "Moriya
Interaction and the Problem of the Spin Arrangements in βMnS".
2858:{\displaystyle \mathbf {D} \parallel \mathrm {mirror\ plane\ or} \ \mathbf {D} \perp AB.}
354:
4544:
4462:
4405:
4333:
4284:
4225:
4182:
4129:
4057:
4015:
3980:
3945:
3907:
3138:
2978:
2733:
2636:
3497:
3092:
3001:
2890:
2870:
2756:
2683:
2659:
2616:
2596:
1349:
4574:
4560:
4478:
4421:
4349:
4153:
3953:
3867:
3857:
3852:
3659:
3640:
3229:
378:
283:
228:
4306:
Rovillain P.; et al. (2010). "Magnetoelectric excitations in multiferroic TbMnO
4241:
4091:
4137:
3894:
T. Moriya (1960). "Anisotropic
Superexchange Interaction and Weak Ferromagnetism".
216:
4552:
4470:
4413:
4341:
4292:
4167:
Gajek, M.; et al. (2007). "Tunnel junctions with multiferroic barriers".
3628:
3384:
is oriented perpendicular to the triangle spanned by the involved three ions.
279:
4023:
3862:
3619:
is a magnetic texture that occurs in the magnetization field. They exist in
4198:
4145:
4083:
3915:
1729:, etc. If the ground state is non-degenerate, then the matrix elements of
4396:
3493:
3533:
trigonal space group. The structure also contains the same unit cell as
227:
and explains the magnetoelectric effects in a class of materials termed
4233:
3988:
4190:
4065:
3225:
256:
pictured as Haematite, the main source of iron for the steel industry
4535:
4498:
4120:
4453:
4324:
3454:
238:
18:
344:{\displaystyle {\hat {\mathbf {L} }}\cdot {\hat {\mathbf {S} }}}
419:
is a 3-dimensional vector of angular momentum operators on ion
3080:{\displaystyle \mathrm {D} \perp \mathrm {two-fold\ axis} .}
3459:
Corundum crystal structure displaying the crystal forms of
2966:{\displaystyle \mathbf {D} \perp \mathrm {mirror\ plane} .}
3643:. Here, small shifts of the ligand ions can be induced by
2593:. Considering the coupling of ions 1 and 2 at locations
4365:"Pressure-temperature phase diagram of multiferroic Ni
4042:"Multiferroics: a magnetic twist for ferroelectricity"
223:. The interaction is fundamental to the production of
3390:
3358:
3270:
3238:
3198:
3164:
3141:
3115:
3095:
3024:
3004:
2981:
2913:
2893:
2873:
2779:
2759:
2736:
2706:
2686:
2662:
2639:
2619:
2599:
2567:
1783:
1757:
1735:
1713:
1667:
1375:
1352:
472:
431:
387:
357:
309:
114:
78:
49:
1700:{\displaystyle \phi _{n}(\mathbf {r} -\mathbf {R} )}
3496:crystal structure, named after the primary form of
463:is a 3-dimensional spin operator of the same form:
3414:
3376:
3344:
3256:
3216:
3181:
3150:
3127:
3101:
3079:
3010:
2990:
2965:
2899:
2879:
2857:
2765:
2745:
2720:
2692:
2668:
2648:
2625:
2605:
2585:
2545:
1766:
1743:
1721:
1699:
1650:
1358:
1335:
455:
411:
369:
343:
203:
93:
64:
1324:
912:
879:
502:
2975:When a two-fold rotation axis perpendicular to
1707:the ground orbital wavefunction of the ion at
284:theory of phase transitions of the second kind
215:In magnetically ordered systems, it favors a
16:Contribution to magnetic exchange interaction
8:
4035:
4033:
3889:
3887:
3885:
3883:
2431:
2390:
2312:
2281:
2114:
2073:
1910:
1879:
1281:
1204:
992:
920:
846:
779:
577:
510:
3232:mechanism (see Figure), the orientation of
39:), is a contribution to the total magnetic
4075:11370/f0777dfc-d0d7-4358-8337-c63e7ad007e7
3934:Journal of Physics and Chemistry of Solids
3927:
3925:
3664:
381:formalism. Note the notation used implies
4534:
4497:
4452:
4395:
4323:
4119:
4073:
3397:
3392:
3389:
3365:
3360:
3357:
3337:
3325:
3320:
3310:
3305:
3295:
3290:
3277:
3272:
3269:
3245:
3240:
3237:
3205:
3200:
3197:
3165:
3163:
3140:
3114:
3094:
3033:
3025:
3023:
3003:
2980:
2922:
2914:
2912:
2892:
2872:
2838:
2788:
2780:
2778:
2758:
2735:
2707:
2705:
2685:
2680:When a center of inversion is located at
2661:
2638:
2618:
2598:
2574:
2569:
2566:
2527:
2522:
2512:
2507:
2491:
2486:
2465:
2460:
2450:
2445:
2418:
2411:
2406:
2401:
2376:
2358:
2318:
2304:
2297:
2292:
2287:
2272:
2259:
2219:
2197:
2161:
2156:
2146:
2141:
2128:
2123:
2101:
2094:
2089:
2084:
2059:
2041:
2001:
1990:
1957:
1952:
1942:
1937:
1924:
1919:
1902:
1895:
1890:
1885:
1870:
1857:
1817:
1811:
1784:
1782:
1756:
1736:
1734:
1714:
1712:
1689:
1681:
1672:
1666:
1641:
1636:
1631:
1624:
1619:
1614:
1601:
1591:
1586:
1572:
1560:
1550:
1545:
1536:
1524:
1514:
1508:
1495:
1485:
1480:
1471:
1461:
1449:
1439:
1434:
1425:
1420:
1374:
1351:
1323:
1322:
1308:
1290:
1268:
1262:
1251:
1249:
1248:
1238:
1227:
1225:
1224:
1215:
1198:
1187:
1185:
1184:
1174:
1163:
1161:
1160:
1122:
1098:
1080:
1068:
1057:
1055:
1054:
1044:
1033:
1031:
1030:
984:
978:
967:
965:
964:
954:
943:
941:
940:
931:
917:
911:
910:
895:
878:
877:
868:
855:
838:
832:
821:
819:
818:
808:
797:
795:
794:
785:
773:
762:
760:
759:
749:
738:
736:
735:
697:
678:
665:
653:
642:
640:
639:
629:
618:
616:
615:
569:
563:
552:
550:
549:
539:
528:
526:
525:
516:
507:
501:
500:
490:
473:
471:
456:{\displaystyle {\hat {\mathbf {S} }}_{i}}
447:
436:
434:
433:
430:
412:{\displaystyle {\hat {\mathbf {L} }}_{i}}
403:
392:
390:
389:
386:
356:
330:
328:
327:
313:
311:
310:
308:
192:
187:
177:
172:
156:
151:
131:
130:
119:
113:
85:
80:
77:
56:
51:
48:
3487:(Metal ions in red, oxygen ions in blue)
3182:{\displaystyle \mathbf {D} \parallel AB}
43:between two neighboring magnetic spins,
3879:
2867:When there is a mirror plane including
2676:, The following rules may be obtained:
1751:are purely imaginary, and we can write
286:. In 1960, Toru Moriya identified the
101:. Quantitatively, it is a term in the
4040:S.-W. Cheong and M. Mostovoy (2007).
2730:When a mirror plane perpendicular to
7:
3415:{\displaystyle \mathbf {D} _{ij}=0}
3264:is obtained by the simple relation
2194:
1808:
3070:
3067:
3064:
3061:
3055:
3052:
3049:
3046:
3040:
3037:
3034:
3026:
2956:
2953:
2950:
2947:
2944:
2938:
2935:
2932:
2929:
2926:
2923:
2831:
2828:
2822:
2819:
2816:
2813:
2810:
2804:
2801:
2798:
2795:
2792:
2789:
1632:
1615:
138:
135:
14:
3377:{\displaystyle \mathbf {D} _{ij}}
3257:{\displaystyle \mathbf {D} _{ij}}
3217:{\displaystyle \mathbf {D} _{ij}}
2586:{\displaystyle \mathbf {D} _{ij}}
293:Dzyaloshinskii–Moriya interaction
33:Dzyaloshinskii–Moriya interaction
3393:
3361:
3338:
3321:
3306:
3291:
3273:
3241:
3201:
3166:
2915:
2839:
2781:
2708:
2570:
2523:
2508:
2487:
2466:
2462:
2451:
2447:
2412:
2408:
2298:
2294:
2162:
2158:
2147:
2143:
2129:
2125:
2095:
2091:
1958:
1954:
1943:
1939:
1925:
1921:
1896:
1892:
1737:
1715:
1690:
1682:
1642:
1638:
1625:
1621:
1603:
1592:
1588:
1561:
1551:
1547:
1497:
1486:
1482:
1450:
1440:
1436:
1252:
1228:
1188:
1164:
1058:
1034:
968:
944:
822:
798:
763:
739:
643:
619:
553:
529:
437:
393:
331:
314:
188:
173:
152:
94:{\displaystyle \mathbf {S} _{j}}
81:
65:{\displaystyle \mathbf {S} _{i}}
52:
3422:if the three ions are in line.
2721:{\displaystyle \mathbf {D} =0.}
4581:Magnetic exchange interactions
4138:10.1103/PhysRevLett.117.037203
3666:Common multiferroic materials
3192:The orientation of the vector
2533:
2503:
2472:
2442:
2419:
2402:
2349:
2324:
2305:
2288:
2250:
2225:
2171:
2168:
2138:
2120:
2102:
2085:
2032:
2007:
1967:
1964:
1934:
1916:
1903:
1886:
1848:
1823:
1694:
1678:
1611:
1583:
1565:
1542:
1505:
1477:
1454:
1431:
1404:
1379:
1269:
1256:
1232:
1216:
1192:
1168:
1156:
1131:
1062:
1038:
1026:
1001:
985:
972:
948:
932:
839:
826:
802:
786:
767:
743:
731:
706:
647:
623:
611:
586:
570:
557:
533:
517:
441:
397:
335:
318:
198:
168:
141:
132:
1:
3954:10.1016/0022-3697(58)90076-3
1744:{\displaystyle \mathbf {L} }
1722:{\displaystyle \mathbf {R} }
2633:, with the point bisecting
2557:Effects of crystal symmetry
4602:
4553:10.1103/PhysRevB.95.134401
4471:10.1103/PhysRevB.77.172402
4414:10.1103/PhysRevB.75.012407
4342:10.1103/PhysRevB.81.054428
4293:10.1103/PhysRevB.71.172301
3969:Journal of Applied Physics
1366:is the exchange integral,
3699:Type of ferroelectricity
105:which can be written as
3652:tunnel magnetoresistance
1767:{\displaystyle \delta E}
4220:(3): 031101–031101–35.
4108:Physical Review Letters
4024:10.1103/PhysRev.126.896
3441:electron spin resonance
3128:{\displaystyle n\geq 2}
4310:by Raman scattering".
3916:10.1103/PhysRev.120.91
3658:applications see also
3521:), which displays the
3488:
3416:
3378:
3346:
3258:
3218:
3183:
3152:
3129:
3103:
3081:
3012:
2992:
2967:
2901:
2881:
2859:
2767:
2747:
2722:
2694:
2670:
2650:
2627:
2607:
2587:
2547:
1768:
1745:
1723:
1701:
1652:
1360:
1337:
457:
413:
371:
345:
257:
205:
95:
66:
29:antisymmetric exchange
24:
3458:
3417:
3379:
3347:
3259:
3219:
3184:
3153:
3130:
3104:
3082:
3013:
2993:
2968:
2902:
2882:
2860:
2768:
2748:
2723:
2695:
2671:
2651:
2628:
2608:
2588:
2548:
1769:
1746:
1724:
1702:
1653:
1361:
1338:
458:
414:
372:
346:
242:
206:
96:
67:
22:
3843:Exchange interaction
3810:magnetically driven
3794:magnetically driven
3755:magnetically driven
3738:magnetically driven
3437:Brillouin scattering
3388:
3356:
3352:. This implies that
3268:
3236:
3196:
3162:
3139:
3113:
3093:
3022:
3002:
2979:
2911:
2891:
2871:
2777:
2757:
2734:
2704:
2684:
2660:
2637:
2617:
2597:
2565:
1781:
1755:
1733:
1711:
1665:
1373:
1350:
470:
429:
385:
355:
307:
112:
76:
47:
41:exchange interaction
31:, also known as the
4545:2017PhRvB..95m4401Z
4463:2008PhRvB..77q2402K
4406:2007PhRvB..75a2407C
4334:2010PhRvB..81e4428R
4285:2005PhRvB..71q2301M
4226:2008JAP...103c1101N
4183:2007NatMa...6..296G
4130:2016PhRvL.117c7203L
4058:2007NatMa...6...13C
4016:1962PhRv..126..896K
3981:1962JAP....33.1133T
3946:1958JPCS....4..241D
3908:1960PhRv..120...91M
3848:Spin–orbit coupling
3667:
1476:
1430:
370:{\displaystyle i,j}
288:spin-orbit coupling
276:Igor Dzyaloshinskii
274:crystals. In 1958,
146:
3665:
3611:Magnetic skyrmions
3489:
3445:neutron scattering
3412:
3374:
3342:
3254:
3214:
3179:
3151:{\displaystyle AB}
3148:
3125:
3099:
3077:
3008:
2991:{\displaystyle AB}
2988:
2963:
2897:
2877:
2855:
2763:
2746:{\displaystyle AB}
2743:
2718:
2690:
2666:
2649:{\displaystyle AB}
2646:
2623:
2603:
2583:
2543:
2541:
2213:
2000:
1816:
1764:
1741:
1719:
1697:
1648:
1457:
1416:
1356:
1333:
1331:
905:
495:
453:
409:
367:
341:
258:
225:magnetic skyrmions
201:
115:
91:
62:
25:
4441:Physical Review B
4234:10.1063/1.2836410
3989:10.1063/1.1728631
3834:
3833:
3645:magnetic ordering
3617:magnetic skyrmion
3451:Material examples
3433:X-ray diffraction
3102:{\displaystyle n}
3089:When there is an
3060:
3011:{\displaystyle C}
2943:
2900:{\displaystyle B}
2880:{\displaystyle A}
2837:
2827:
2809:
2766:{\displaystyle C}
2693:{\displaystyle C}
2669:{\displaystyle C}
2626:{\displaystyle B}
2606:{\displaystyle A}
2388:
2279:
2193:
2071:
1986:
1877:
1807:
1530:
1359:{\displaystyle J}
1320:
1259:
1235:
1195:
1171:
1110:
1065:
1041:
975:
951:
891:
875:
829:
805:
770:
746:
685:
650:
626:
560:
536:
486:
444:
400:
338:
321:
4593:
4565:
4564:
4538:
4510:
4504:
4503:
4501:
4489:
4483:
4482:
4456:
4432:
4426:
4425:
4399:
4397:cond-mat/0701576
4381:
4360:
4354:
4353:
4327:
4303:
4297:
4296:
4252:
4246:
4245:
4209:
4203:
4202:
4191:10.1038/nmat1860
4170:Nature Materials
4164:
4158:
4157:
4123:
4102:
4096:
4095:
4077:
4066:10.1038/nmat1804
4046:Nature Materials
4037:
4028:
4027:
3999:
3993:
3992:
3975:(3): 1133–1134.
3964:
3958:
3957:
3929:
3920:
3919:
3891:
3668:
3592:
3578:
3552:
3538:
3528:
3520:
3519:
3518:
3510:
3509:
3478:
3464:
3421:
3419:
3418:
3413:
3405:
3404:
3396:
3383:
3381:
3380:
3375:
3373:
3372:
3364:
3351:
3349:
3348:
3343:
3341:
3333:
3332:
3324:
3315:
3314:
3309:
3300:
3299:
3294:
3285:
3284:
3276:
3263:
3261:
3260:
3255:
3253:
3252:
3244:
3223:
3221:
3220:
3215:
3213:
3212:
3204:
3188:
3186:
3185:
3180:
3169:
3157:
3155:
3154:
3149:
3134:
3132:
3131:
3126:
3108:
3106:
3105:
3100:
3086:
3084:
3083:
3078:
3073:
3058:
3029:
3017:
3015:
3014:
3009:
2997:
2995:
2994:
2989:
2972:
2970:
2969:
2964:
2959:
2941:
2918:
2906:
2904:
2903:
2898:
2886:
2884:
2883:
2878:
2864:
2862:
2861:
2856:
2842:
2835:
2834:
2825:
2807:
2784:
2772:
2770:
2769:
2764:
2752:
2750:
2749:
2744:
2727:
2725:
2724:
2719:
2711:
2699:
2697:
2696:
2691:
2675:
2673:
2672:
2667:
2655:
2653:
2652:
2647:
2632:
2630:
2629:
2624:
2612:
2610:
2609:
2604:
2592:
2590:
2589:
2584:
2582:
2581:
2573:
2552:
2550:
2549:
2544:
2542:
2532:
2531:
2526:
2517:
2516:
2511:
2499:
2498:
2490:
2478:
2471:
2470:
2469:
2456:
2455:
2454:
2438:
2434:
2430:
2422:
2417:
2416:
2415:
2405:
2400:
2389:
2387:
2386:
2385:
2384:
2368:
2367:
2366:
2352:
2348:
2337:
2319:
2308:
2303:
2302:
2301:
2291:
2280:
2278:
2277:
2276:
2264:
2263:
2253:
2249:
2238:
2220:
2212:
2211:
2177:
2167:
2166:
2165:
2152:
2151:
2150:
2134:
2133:
2132:
2113:
2105:
2100:
2099:
2098:
2088:
2083:
2072:
2070:
2069:
2068:
2067:
2051:
2050:
2049:
2035:
2031:
2020:
2002:
1999:
1998:
1973:
1963:
1962:
1961:
1948:
1947:
1946:
1930:
1929:
1928:
1906:
1901:
1900:
1899:
1889:
1878:
1876:
1875:
1874:
1862:
1861:
1851:
1847:
1836:
1818:
1815:
1773:
1771:
1770:
1765:
1750:
1748:
1747:
1742:
1740:
1728:
1726:
1725:
1720:
1718:
1706:
1704:
1703:
1698:
1693:
1685:
1677:
1676:
1657:
1655:
1654:
1649:
1647:
1646:
1645:
1635:
1630:
1629:
1628:
1618:
1610:
1609:
1597:
1596:
1595:
1582:
1581:
1580:
1564:
1556:
1555:
1554:
1541:
1540:
1531:
1529:
1528:
1519:
1518:
1509:
1504:
1503:
1491:
1490:
1489:
1475:
1470:
1469:
1453:
1445:
1444:
1443:
1429:
1424:
1403:
1392:
1365:
1363:
1362:
1357:
1342:
1340:
1339:
1334:
1332:
1328:
1327:
1321:
1319:
1318:
1317:
1316:
1300:
1299:
1298:
1284:
1280:
1272:
1267:
1266:
1261:
1260:
1255:
1250:
1243:
1242:
1237:
1236:
1231:
1226:
1219:
1214:
1203:
1202:
1197:
1196:
1191:
1186:
1179:
1178:
1173:
1172:
1167:
1162:
1152:
1141:
1123:
1115:
1111:
1109:
1108:
1107:
1106:
1090:
1089:
1088:
1074:
1073:
1072:
1067:
1066:
1061:
1056:
1049:
1048:
1043:
1042:
1037:
1032:
1022:
1011:
988:
983:
982:
977:
976:
971:
966:
959:
958:
953:
952:
947:
942:
935:
930:
918:
916:
915:
904:
903:
883:
882:
876:
874:
873:
872:
860:
859:
849:
842:
837:
836:
831:
830:
825:
820:
813:
812:
807:
806:
801:
796:
789:
778:
777:
772:
771:
766:
761:
754:
753:
748:
747:
742:
737:
730:
719:
698:
690:
686:
684:
683:
682:
670:
669:
659:
658:
657:
652:
651:
646:
641:
634:
633:
628:
627:
622:
617:
610:
599:
573:
568:
567:
562:
561:
556:
551:
544:
543:
538:
537:
532:
527:
520:
508:
506:
505:
494:
462:
460:
459:
454:
452:
451:
446:
445:
440:
435:
424:
418:
416:
415:
410:
408:
407:
402:
401:
396:
391:
376:
374:
373:
368:
350:
348:
347:
342:
340:
339:
334:
329:
323:
322:
317:
312:
265:
247:
210:
208:
207:
202:
197:
196:
191:
182:
181:
176:
164:
163:
155:
145:
144:
129:
100:
98:
97:
92:
90:
89:
84:
71:
69:
68:
63:
61:
60:
55:
4601:
4600:
4596:
4595:
4594:
4592:
4591:
4590:
4571:
4570:
4569:
4568:
4520:
4516:
4512:
4511:
4507:
4491:
4490:
4486:
4438:
4434:
4433:
4429:
4379:
4376:
4372:
4368:
4362:
4361:
4357:
4309:
4305:
4304:
4300:
4270:
4266:
4262:
4258:
4254:
4253:
4249:
4211:
4210:
4206:
4166:
4165:
4161:
4104:
4103:
4099:
4039:
4038:
4031:
4004:Physical Review
4001:
4000:
3996:
3966:
3965:
3961:
3931:
3930:
3923:
3896:Physical Review
3893:
3892:
3881:
3876:
3839:
3822:
3818:
3786:
3771:
3767:
3763:
3746:
3730:
3726:
3708:
3696:
3689:
3680:
3637:
3613:
3608:
3600:
3596:
3588:
3586:
3582:
3574:
3572:
3568:
3564:
3561:which possess D
3560:
3556:
3548:
3546:
3542:
3534:
3526:
3517:
3514:
3513:
3512:
3508:
3505:
3504:
3503:
3501:
3486:
3482:
3474:
3472:
3468:
3460:
3453:
3428:
3391:
3386:
3385:
3359:
3354:
3353:
3319:
3304:
3289:
3271:
3266:
3265:
3239:
3234:
3233:
3199:
3194:
3193:
3160:
3159:
3137:
3136:
3111:
3110:
3091:
3090:
3020:
3019:
3000:
2999:
2998:passes through
2977:
2976:
2909:
2908:
2889:
2888:
2869:
2868:
2775:
2774:
2755:
2754:
2753:passes through
2732:
2731:
2702:
2701:
2682:
2681:
2658:
2657:
2635:
2634:
2615:
2614:
2595:
2594:
2568:
2563:
2562:
2559:
2540:
2539:
2521:
2506:
2485:
2476:
2475:
2461:
2446:
2423:
2407:
2393:
2377:
2372:
2359:
2354:
2353:
2341:
2330:
2320:
2293:
2268:
2255:
2254:
2242:
2231:
2221:
2218:
2214:
2204:
2175:
2174:
2157:
2142:
2124:
2106:
2090:
2076:
2060:
2055:
2042:
2037:
2036:
2024:
2013:
2003:
1991:
1971:
1970:
1953:
1938:
1920:
1891:
1866:
1853:
1852:
1840:
1829:
1819:
1794:
1779:
1778:
1753:
1752:
1731:
1730:
1709:
1708:
1668:
1663:
1662:
1637:
1620:
1602:
1587:
1573:
1568:
1546:
1532:
1520:
1510:
1496:
1481:
1462:
1435:
1396:
1385:
1371:
1370:
1348:
1347:
1330:
1329:
1309:
1304:
1291:
1286:
1285:
1273:
1247:
1223:
1207:
1183:
1159:
1145:
1134:
1124:
1113:
1112:
1099:
1094:
1081:
1076:
1075:
1053:
1029:
1015:
1004:
963:
939:
923:
919:
906:
896:
885:
884:
864:
851:
850:
817:
793:
758:
734:
723:
712:
699:
688:
687:
674:
661:
660:
638:
614:
603:
592:
548:
524:
509:
496:
468:
467:
432:
427:
426:
420:
388:
383:
382:
353:
352:
305:
304:
301:
273:
269:
261:
255:
251:
243:
237:
221:antiferromagnet
186:
171:
150:
110:
109:
79:
74:
73:
50:
45:
44:
17:
12:
11:
5:
4599:
4597:
4589:
4588:
4583:
4573:
4572:
4567:
4566:
4529:(13): 134401.
4518:
4514:
4505:
4484:
4447:(17): 172402.
4436:
4427:
4374:
4370:
4366:
4355:
4307:
4298:
4279:(17): 172301.
4268:
4264:
4260:
4256:
4247:
4204:
4177:(4): 296–302.
4159:
4097:
4029:
3994:
3959:
3921:
3878:
3877:
3875:
3872:
3871:
3870:
3865:
3860:
3855:
3850:
3845:
3838:
3835:
3832:
3831:
3829:
3826:
3823:
3820:
3816:
3812:
3811:
3808:
3805:
3802:
3796:
3795:
3792:
3790:
3787:
3784:
3780:
3779:
3777:
3775:
3772:
3769:
3765:
3761:
3757:
3756:
3753:
3750:
3747:
3744:
3740:
3739:
3736:
3734:
3731:
3728:
3724:
3720:
3719:
3716:
3713:
3710:
3706:
3701:
3700:
3697:
3694:
3687:
3681:
3678:
3674:Ferroelectric
3672:
3636:
3633:
3612:
3609:
3607:
3604:
3598:
3594:
3584:
3580:
3570:
3566:
3562:
3558:
3554:
3544:
3540:
3515:
3506:
3498:Aluminum oxide
3484:
3480:
3470:
3466:
3452:
3449:
3435:interference,
3427:
3424:
3411:
3408:
3403:
3400:
3395:
3371:
3368:
3363:
3340:
3336:
3331:
3328:
3323:
3318:
3313:
3308:
3303:
3298:
3293:
3288:
3283:
3280:
3275:
3251:
3248:
3243:
3211:
3208:
3203:
3190:
3189:
3178:
3175:
3172:
3168:
3147:
3144:
3124:
3121:
3118:
3098:
3087:
3076:
3072:
3069:
3066:
3063:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3032:
3028:
3007:
2987:
2984:
2973:
2962:
2958:
2955:
2952:
2949:
2946:
2940:
2937:
2934:
2931:
2928:
2925:
2921:
2917:
2896:
2876:
2865:
2854:
2851:
2848:
2845:
2841:
2833:
2830:
2824:
2821:
2818:
2815:
2812:
2806:
2803:
2800:
2797:
2794:
2791:
2787:
2783:
2762:
2742:
2739:
2728:
2717:
2714:
2710:
2689:
2665:
2645:
2642:
2622:
2602:
2580:
2577:
2572:
2558:
2555:
2554:
2553:
2538:
2535:
2530:
2525:
2520:
2515:
2510:
2505:
2502:
2497:
2494:
2489:
2484:
2481:
2479:
2477:
2474:
2468:
2464:
2459:
2453:
2449:
2444:
2441:
2437:
2433:
2429:
2426:
2421:
2414:
2410:
2404:
2399:
2396:
2392:
2383:
2380:
2375:
2371:
2365:
2362:
2357:
2351:
2347:
2344:
2340:
2336:
2333:
2329:
2326:
2323:
2317:
2314:
2311:
2307:
2300:
2296:
2290:
2286:
2283:
2275:
2271:
2267:
2262:
2258:
2252:
2248:
2245:
2241:
2237:
2234:
2230:
2227:
2224:
2217:
2210:
2207:
2203:
2200:
2196:
2192:
2189:
2186:
2183:
2180:
2178:
2176:
2173:
2170:
2164:
2160:
2155:
2149:
2145:
2140:
2137:
2131:
2127:
2122:
2119:
2116:
2112:
2109:
2104:
2097:
2093:
2087:
2082:
2079:
2075:
2066:
2063:
2058:
2054:
2048:
2045:
2040:
2034:
2030:
2027:
2023:
2019:
2016:
2012:
2009:
2006:
1997:
1994:
1989:
1985:
1982:
1979:
1976:
1974:
1972:
1969:
1966:
1960:
1956:
1951:
1945:
1941:
1936:
1933:
1927:
1923:
1918:
1915:
1912:
1909:
1905:
1898:
1894:
1888:
1884:
1881:
1873:
1869:
1865:
1860:
1856:
1850:
1846:
1843:
1839:
1835:
1832:
1828:
1825:
1822:
1814:
1810:
1806:
1803:
1800:
1797:
1795:
1793:
1790:
1787:
1786:
1763:
1760:
1739:
1717:
1696:
1692:
1688:
1684:
1680:
1675:
1671:
1659:
1658:
1644:
1640:
1634:
1627:
1623:
1617:
1613:
1608:
1605:
1600:
1594:
1590:
1585:
1579:
1576:
1571:
1567:
1563:
1559:
1553:
1549:
1544:
1539:
1535:
1527:
1523:
1517:
1513:
1507:
1502:
1499:
1494:
1488:
1484:
1479:
1474:
1468:
1465:
1460:
1456:
1452:
1448:
1442:
1438:
1433:
1428:
1423:
1419:
1415:
1412:
1409:
1406:
1402:
1399:
1395:
1391:
1388:
1384:
1381:
1378:
1355:
1344:
1343:
1326:
1315:
1312:
1307:
1303:
1297:
1294:
1289:
1283:
1279:
1276:
1271:
1265:
1258:
1254:
1246:
1241:
1234:
1230:
1222:
1218:
1213:
1210:
1206:
1201:
1194:
1190:
1182:
1177:
1170:
1166:
1158:
1155:
1151:
1148:
1144:
1140:
1137:
1133:
1130:
1127:
1121:
1118:
1116:
1114:
1105:
1102:
1097:
1093:
1087:
1084:
1079:
1071:
1064:
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3868:Multiferroics
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3858:Landau theory
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3853:Superexchange
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229:multiferroics
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4384:Phys. Rev. B
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4052:(1): 13–20.
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3606:Applications
3589:
3575:
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3191:
3109:-fold axis (
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262:
259:
244:
217:spin canting
214:
36:
32:
28:
27:In Physics,
26:
4586:Spintronics
3426:Measurement
103:Hamiltonian
4575:Categories
4536:1701.08227
4499:1508.02874
4121:1605.08444
4010:(3): 896.
3940:(4): 241.
3874:References
3718:lone pair
3629:spintronic
299:Derivation
280:Lev Landau
4561:119502126
4479:119271548
4454:0806.0117
4422:117752707
4350:118430304
4325:0908.0061
4154:206278388
3902:(1): 91.
3863:Skyrmions
3683:Magnetic
3631:devices.
3335:×
3302:×
3287:∝
3228:) by the
3171:∥
3120:≥
3044:−
3031:⊥
2920:⊥
2844:⊥
2786:∥
2519:×
2501:⋅
2458:×
2440:⋅
2432:⟩
2391:⟨
2370:−
2316:−
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2282:⟨
2266:−
2195:∑
2191:λ
2154:⋅
2118:⋅
2115:⟩
2074:⟨
2053:−
1988:∑
1984:λ
1950:⋅
1914:⋅
1911:⟩
1880:⟨
1864:−
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1789:δ
1759:δ
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1670:ϕ
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1473:∗
1459:ϕ
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1427:∗
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1411:∫
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1257:^
1245:⋅
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1169:^
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756:⋅
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522:λ
511:⟨
488:∑
478:δ
442:^
398:^
336:^
325:⋅
319:^
184:×
166:⋅
4263:and TbMn
4242:51900508
4199:17351615
4146:27472133
4092:23304200
4084:17199121
3837:See also
3671:Material
3625:hedgehog
3494:corundum
3135:) along
2656:denoted
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4541:Bibcode
4459:Bibcode
4402:Bibcode
4330:Bibcode
4281:Bibcode
4222:Bibcode
4179:Bibcode
4126:Bibcode
4054:Bibcode
4012:Bibcode
3977:Bibcode
3942:Bibcode
3904:Bibcode
1774:out as
235:History
4559:
4477:
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4348:
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3621:spiral
3443:, and
3226:ligand
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2836:
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2808:
1346:where
425:, and
4557:S2CID
4531:arXiv
4494:arXiv
4475:S2CID
4449:arXiv
4418:S2CID
4392:arXiv
4380:(PDF)
4346:S2CID
4320:arXiv
4238:S2CID
4150:S2CID
4116:arXiv
4088:S2CID
3789:13.5
3743:TbMnO
3712:1100
3705:BiFeO
1661:with
4195:PMID
4142:PMID
4080:PMID
3825:110
3815:ZnCr
3807:230
3804:230
3783:MnWO
3774:6.5
3723:HoMn
3715:653
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