N.22 The mech­a­nism of fer­ro­mag­net­ism

It should be noted that in solids, not just spa­tial an­ti­sym­me­try, but also sym­me­try can give rise to spin align­ment. In par­tic­u­lar, in many fer­rites, there is an op­po­site spin cou­pling be­tween the iron atoms and the oxy­gen ones. If two iron atoms are op­po­site in spin to the same oxy­gen atom, it im­plies that they must have aligned spins even if their elec­trons do not in­ter­act di­rectly.

It comes as some­what a sur­prise to dis­cover that in this time of high-tem­per­a­ture su­per­con­duc­tors, the mech­a­nism of plain old fer­ro­mag­net­ism is still not un­der­stood that well if the mag­netic ma­te­r­ial is a con­duc­tor, such as a piece of iron.

For a con­duc­tor, the de­scrip­tion of the ex­clu­sion ef­fect should re­ally be at least partly in terms of band the­ory, rather than elec­trons lo­cal­ized at atoms. More specif­i­cally, Aha­roni [2, p. 48] notes “There is thus no doubt in any­body’s mind that nei­ther the itin­er­ant elec­tron the­ory nor the lo­cal­ized elec­tron one can be con­sid­ered to be a com­plete pic­ture of the phys­i­cal re­al­ity, and that they both should be com­bined into one the­ory.”

Sproull notes that in solid iron, most of the 4s elec­trons move to the 4d bands. That re­duces the mag­ne­ti­za­tion by re­duc­ing the num­ber of un­paired elec­trons.

While Sproull [41, p. 282] in 1956 de­scribes fer­ro­mag­net­ism as an in­ter­ac­tion be­tween elec­trons lo­cal­ized at neigh­bor­ing atoms, Feyn­man [22, p. 37-2] in 1965 notes that cal­cu­la­tions us­ing such a model pro­duce the wrong sign for the in­ter­ac­tion. Ac­cord­ing to Feyn­man, the in­ter­ac­tion is thought to oc­cur with [4s] con­duc­tion band elec­trons act­ing as in­ter­me­di­aries. More re­cently, Aha­roni [2, p. 44] notes: “It used to be stated [...] that no­body has been able to com­pute a pos­i­tive ex­change in­te­gral for Fe, and a neg­a­tive one for Cu [...]. More mod­ern com­pu­ta­tions [...] al­ready have the right sign, but the mag­ni­tude of the com­puted ex­change still dif­fers con­sid­er­ably from the ex­per­i­men­tal value. Im­prov­ing the tech­niques [...] keeps im­prov­ing the re­sults, but not suf­fi­ciently yet.”

Batista, Bonča, and Gu­ber­natis note that “Af­ter seven decades of in­tense ef­fort we still do not know what is the min­i­mal model of itin­er­ant fer­ro­mag­net­ism and, more im­por­tantly, the ba­sic mech­a­nism of or­der­ing.” (Phys Rev Let 88, 2002, 187203-1) and “Even though the tran­si­tion met­als are the most well stud­ied itin­er­ant fer­ro­mag­nets, the ul­ti­mate rea­son for the sta­bi­liza­tion of the FM phase is still un­known.” (Phys Rev B 68, 2003, 214430-11)