Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals

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Davide Ceresoli,¹ T. Thonhauser,² David Vanderbilt,² and R. Resta³
¹Scuola Internazionale Superiore di Studi Avanzati (SISSA/ISAS) and DEMOCRITOS, via Beirut 2-4, 34014 Trieste, Italy
²Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
³Dipartimento di Fisica Teorica, Università di Trieste and DEMOCRITOS, strada Costiera 11, 34014 Trieste, Italy

Received 7 December 2005; revised 6 March 2006; published 12 July 2006

Wederive a multi-band formulation of the orbital magnetization in anormal periodic insulator (i.e., one in which the Chern invariant,or in two dimensions (2D) the Chern number, vanishes). Followingthe approach used recently to develop the single-band formalism [Thonhauser,Ceresoli, Vanderbilt, and Resta, Phys. Rev. Lett. 95, 137205 (2005)],we work in the Wannier representation and find that themagnetization is comprised of two contributions, an obvious one associatedwith the internal circulation of bulklike Wannier functions in theinterior and an unexpected one arising from net currents carriedby Wannier functions near the surface. Unlike the single-band case,where each of these contributions is separately gauge invariant, inthe multi-band formulation only the sum of both terms isgauge invariant. Our final expression for the orbital magnetization canbe rewritten as a bulk property in terms of Blochfunctions, making it simple to implement in modern code packages.The reciprocal-space expression is evaluated for 2D model systems andthe results are verified by comparing to the magnetization computedfor finite samples cut from the bulk. Finally, while ourformal proof is limited to normal insulators, we also presenta heuristic extension to Chern insulators (having nonzero Chern invariant)and to metals. The validity of this extension is againtested by comparing to the magnetization of finite samples cutfrom the bulk for 2D model systems. We find excellentagreement, thus providing strong empirical evidence in favor of thevalidity of the heuristic formula.

URL:	http://link.aps.org/doi/10.1103/PhysRevB.74.024408
DOI:	10.1103/PhysRevB.74.024408
PACS:	75.10.Lp; 73.20.At; 73.43.-f 75.10.Lp Band and itinerant models (magnetism) 73.20.At Surface states, band structure, electron density of states 73.43.-f Quantum Hall effects YEAR: 2006
KEYWORDS:	magnetisation, Wannier functions, band model of magnetism

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Phys. Rev. B 74, 024408 (2006) [13 pages]

Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals

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A new free weekly publication from APS