Disordered local moment picture

File:Disordered Local Moment (DLM) Picture.pdf.]]

The disordered local moment (DLM) picture is a method, in theoretical solid state physics, for describing the electronic structure of a magnetic material at a finite temperature, where a probability distribution of sizes and orientations of atomic magnetic moments must be considered.{{Cite journal |last1=Pindor |first1=A J |last2=Staunton |first2=J |last3=Stocks |first3=G M |last4=Winter |first4=H |title=Disordered local moment state of magnetic transition metals: a self-consistent KKR CPA calculation |url=https://iopscience.iop.org/article/10.1088/0305-4608/13/5/012 |journal=Journal of Physics F: Metal Physics |publication-date=1983 |volume=13 |issue=5 |pages=979–989 |doi=10.1088/0305-4608/13/5/012 |bibcode=1983JPhF...13..979P |issn=0305-4608|url-access=subscription }}{{Cite journal |last1=Staunton |first1=J. |last2=Gyorffy |first2=B. L. |last3=Pindor |first3=A. J. |last4=Stocks |first4=G. M. |last5=Winter |first5=H. |date=1984 |title=The "disordered local moment" picture of itinerant magnetism at finite temperatures |url=https://linkinghub.elsevier.com/retrieve/pii/0304885384903676 |journal=Journal of Magnetism and Magnetic Materials |volume=45 |issue=1 |pages=15–22 |doi=10.1016/0304-8853(84)90367-6 |bibcode=1984JMMM...45...15S |issn=0304-8853|url-access=subscription }}{{Cite journal |last1=Staunton |first1=J |last2=Gyorffy |first2=B L |last3=Pindor |first3=A J |last4=Stocks |first4=G M |last5=Winter |first5=H |title=Electronic structure of metallic ferromagnets above the Curie temperature |url=https://iopscience.iop.org/article/10.1088/0305-4608/15/6/019 |journal=Journal of Physics F: Metal Physics |publication-date=1985 |volume=15 |issue=6 |pages=1387–1404 |doi=10.1088/0305-4608/15/6/019 |bibcode=1985JPhF...15.1387S |issn=0305-4608|url-access=subscription }}{{Cite journal |last1=Gyorffy |first1=B L |last2=Pindor |first2=A J |last3=Staunton |first3=J |last4=Stocks |first4=G M |last5=Winter |first5=H |title=A first-principles theory of ferromagnetic phase transitions in metals |url=https://iopscience.iop.org/article/10.1088/0305-4608/15/6/018 |journal=Journal of Physics F: Metal Physics |publication-date=1985 |volume=15 |issue=6 |pages=1337–1386 |doi=10.1088/0305-4608/15/6/018 |bibcode=1985JPhF...15.1337G |issn=0305-4608|url-access=subscription }} It was pioneered, among others, by Balázs Győrffy, Julie Staunton, Malcolm Stocks, and co-workers.

The underlying assumption of the DLM picture is similar to the Born-Oppenheimer approximation for the separation of solution of the ionic and electronic problems in a material. In the disordered local moment picture, it is assumed that 'local' magnetic moments which form around atoms are sufficiently long-lived that the electronic problem can be solved for an assumed, fixed distribution of magnetic moments.{{Citation |last=Mendive Tapia |first=Eduardo |title=Disordered Local Moment Theory and Fast Electronic Responses |date=2020 |work=Ab initio Theory of Magnetic Ordering: Electronic Origin of Pair- and Multi-Spin Interactions |series=Springer Theses |pages=29–54 |editor-last=Mendive Tapia |editor-first=Eduardo |url=https://link.springer.com/chapter/10.1007/978-3-030-37238-5_3 |access-date=2024-09-25 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-37238-5_3 |isbn=978-3-030-37238-5|url-access=subscription }} Many such distributions can then be averaged over, appropriately weighted by their probabilities, and a description of the paramagnetic state obtained. (A paramagnetic state is one where the magnetic order parameter, $\backslash mathbf\; m$, is equal to the zero vector.)

The picture is typically based on density functional theory (DFT) calculations of the electronic structure of materials. Most frequently, DLM calculations employ either the Korringa–Kohn–Rostoker (KKR){{Cite book |last1=Faulkner |first1=J. S. |url=https://iopscience.iop.org/book/mono/978-0-7503-1490-9 |title=Multiple Scattering Theory: Electronic structure of solids |last2=Stocks |first2=G. Malcolm |last3=Wang |first3=Yang |date=2018-12-01 |publisher=IOP Publishing |isbn=978-0-7503-1490-9 |language=en |doi=10.1088/2053-2563/aae7d8|bibcode=2018mste.book.....F }} (sometimes referred to as multiple scattering theory) or linearised muffin-tin orbital (LMTO) formulations of DFT, where the coherent potential approximation (CPA) can be used to average over multiple orientations of magnetic moment. However, the picture has also been applied in the context of supercells containing appropriate distributions of magnetic moment orientations.{{Cite journal |last1=Mendive-Tapia |first1=Eduardo |last2=Neugebauer |first2=Jörg |last3=Hickel |first3=Tilmann |date=2022-02-17 |title=Ab initio calculation of the magnetic Gibbs free energy of materials using magnetically constrained supercells |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.105.064425 |journal=Physical Review B |volume=105 |issue=6 |pages=064425 |doi=10.1103/PhysRevB.105.064425|arxiv=2202.11492 |bibcode=2022PhRvB.105f4425M }}

Within the context of the KKR method, and in the absence of spin-orbit coupling, the CPA condition describing the paramagnetic state (where the net magnetisation is zero) can be shown to be equivalent to the CPA condition for an Ising 'alloy' of 'up' and 'down' magnetic moments. Once the effects of spin-orbit coupling are included, and magnetic moments are coupled to the crystal axes, it is formally necessary to perform a full ingtegral over all possible magnetisation directions, in practice by sampling an angular mesh of possible magnetisation directions.{{Cite journal |last=Staunton |first=Julie B. |date=2007 |title=Relativistic Effects and Disordered Local Moments in Magnets |url=https://psi-k.net/download/highlights/Highlight_82.pdf |journal=Psi-k Scientific Highlight of the Month |volume=82}}

Though originally developed as a means by which to describe the electronic structure of a magnetic material above its magnetic critical temperature (Curie temperature), the disordered local moment picture has since been applied in a number of other contexts. This includes precise calculation of Curie temperatures and magnetic correlation functions for transition metals,{{Cite journal |last1=Pinski |first1=F. J. |last2=Staunton |first2=J. |last3=Gyorffy |first3=B. L. |last4=Johnson |first4=D. D. |last5=Stocks |first5=G. M. |date=1986-05-12 |title=Ferromagnetism versus Antiferromagnetism in Face-Centered-Cubic Iron |url=https://link.aps.org/doi/10.1103/PhysRevLett.56.2096 |journal=Physical Review Letters |language=en |volume=56 |issue=19 |pages=2096–2099 |doi=10.1103/PhysRevLett.56.2096 |pmid=10032856 |bibcode=1986PhRvL..56.2096P |issn=0031-9007|url-access=subscription }} rare-earth elements,{{Cite journal |last1=Hughes |first1=I. D. |last2=Däne |first2=M. |last3=Ernst |first3=A. |last4=Hergert |first4=W. |last5=Lüders |first5=M. |last6=Poulter |first6=J. |last7=Staunton |first7=J. B. |last8=Svane |first8=A. |last9=Szotek |first9=Z. |last10=Temmerman |first10=W. M. |date=2007 |title=Lanthanide contraction and magnetism in the heavy rare earth elements |url=https://www.nature.com/articles/nature05668 |journal=Nature |language=en |volume=446 |issue=7136 |pages=650–653 |doi=10.1038/nature05668 |pmid=17410171 |bibcode=2007Natur.446..650H |issn=1476-4687|url-access=subscription }}{{Cite journal |last1=Mendive-Tapia |first1=Eduardo |last2=Staunton |first2=Julie B. |date=2017-05-11 |title=Theory of Magnetic Ordering in the Heavy Rare Earths: Ab Initio Electronic Origin of Pair- and Four-Spin Interactions |url=http://link.aps.org/doi/10.1103/PhysRevLett.118.197202 |journal=Physical Review Letters |language=en |volume=118 |issue=19 |page=197202 |doi=10.1103/PhysRevLett.118.197202 |pmid=28548504 |issn=0031-9007|arxiv=1610.08304 |bibcode=2017PhRvL.118s7202M }} and transition metal oxides;{{Cite journal |last1=Hughes |first1=I D |last2=Däne |first2=M |last3=Ernst |first3=A |last4=Hergert |first4=W |last5=Lüders |first5=M |last6=Staunton |first6=J B |last7=Szotek |first7=Z |last8=Temmerman |first8=W M |date=2008-06-06 |title=Onset of magnetic order in strongly-correlated systems from ab initio electronic structure calculations: application to transition metal oxides |url=https://iopscience.iop.org/article/10.1088/1367-2630/10/6/063010 |journal=New Journal of Physics |volume=10 |issue=6 |pages=063010 |doi=10.1088/1367-2630/10/6/063010 |issn=1367-2630|arxiv=0802.3660 |bibcode=2008NJPh...10f3010H }} as well as a description of the temperature dependance of magnetocrystalline anisotropy.{{Cite journal |last1=Staunton |first1=J. B. |last2=Ostanin |first2=S. |last3=Razee |first3=S. S. A. |last4=Gyorffy |first4=B. L. |last5=Szunyogh |first5=L. |last6=Ginatempo |first6=B. |last7=Bruno |first7=Ezio |date=2004-12-14 |title=Temperature Dependent Magnetic Anisotropy in Metallic Magnets from an Ab Initio Electronic Structure Theory: L 1 0 -Ordered FePt |url=https://link.aps.org/doi/10.1103/PhysRevLett.93.257204 |journal=Physical Review Letters |language=en |volume=93 |issue=25 |page=257204 |doi=10.1103/PhysRevLett.93.257204 |pmid=15697934 |issn=0031-9007|arxiv=cond-mat/0407774 }}{{Cite journal |last1=Staunton |first1=J. B. |last2=Szunyogh |first2=L. |last3=Buruzs |first3=A. |last4=Gyorffy |first4=B. L. |last5=Ostanin |first5=S. |last6=Udvardi |first6=L. |date=2006-10-17 |title=Temperature dependence of magnetic anisotropy: An ab initio approach |url=https://link.aps.org/doi/10.1103/PhysRevB.74.144411 |journal=Physical Review B |language=en |volume=74 |issue=14 |page=144411 |doi=10.1103/PhysRevB.74.144411 |bibcode=2006PhRvB..74n4411S |issn=1098-0121|url-access=subscription }} The approach has found particular success in describing the temperature-dependence of magnetic quantities of interest in rare earth–transition metal permanent magnets, such as SmCo₅{{Cite journal |last1=Patrick |first1=Christopher E. |last2=Kumar |first2=Santosh |last3=Balakrishnan |first3=Geetha |last4=Edwards |first4=Rachel S. |last5=Lees |first5=Martin R. |last6=Petit |first6=Leon |last7=Staunton |first7=Julie B. |date=2018-02-28 |title=Calculating the Magnetic Anisotropy of Rare-Earth–Transition-Metal Ferrimagnets |url=https://link.aps.org/doi/10.1103/PhysRevLett.120.097202 |journal=Physical Review Letters |language=en |volume=120 |issue=9 |page=097202 |doi=10.1103/PhysRevLett.120.097202 |pmid=29547338 |issn=0031-9007|arxiv=1803.00235 |bibcode=2018PhRvL.120i7202P }} and Nd₂Fe₁₄B,{{Cite journal |last1=Bouaziz |first1=Juba |last2=Patrick |first2=Christopher E. |last3=Staunton |first3=Julie B. |date=2023-01-05 |title=Crucial role of Fe in determining the hard magnetic properties of Nd 2 Fe 14 B |url=https://link.aps.org/doi/10.1103/PhysRevB.107.L020401 |journal=Physical Review B |language=en |volume=107 |issue=2 |doi=10.1103/PhysRevB.107.L020401 |issn=2469-9950|arxiv=2301.02868 }} which are of interest for a range of energy generation and conversion technologies.