Polaron Formation in Colossal Magnetoresistive Oxides 

The magnetic properties of the lanthanum manganese oxide class of materials have attracted tremendous interest recently because of the dramatic increase in conductivity these systems exhibit when the magnetic moments order ferromagnetically, either by lowering the temperature or applying a magnetic field.  This huge increase in the carrier mobility, which has been given the name "Colossal MagnetoResistivity" (CMR), is both of scientific and technological interest.  In particular, it is anticipated that the "half-metallic" behavior these materials exhibit could provide fully spin polarized electrons for use in "spintronics" applications, for sensors in a variety of applications such as in the automotive industry, and may also provide the next generation of read/write heads for the magnetic data storage industry. 

The CMR originates from a magnetically driven insulator-metal transition, where the magnetic, electronic, and structural degrees of freedom are intimately intertwined.  Neutron scattering measurements have been used to discover that the transition from the low temperature ferromagnetic-metallic state to the paramagnetic-insulator state is caused by the formation of combined structural/magnetic polarons, which have a size of about one nanometer.  The formation of these nanoscale polarons truncates the ferromagnetic phase in the regime of optimal doping, and thus explains the first-order nature of the transition.  These polarons form a well defined thermodynamic glass phase above the ferromagnetic ordering temperature, which then melts into a polaron fluid at higher temperatures as shown in the figures below.

There is a strong similarity between these nanoscale polarons observed in the CMR materials, the polar nanoregions that cause the dramatic piezoelectric response of relaxor ferroelectrics, and the formation of stripes in the high temperature superconducting cuprates.  Recent progress in our understanding of these intrinsic nanoscale structures has enabled a deeper understanding of the fundamental properties and shared concepts of all these perovskite-based materials.

    

"Polaron Mountains" shown on the left were discovered in the colossal magnetoresistive oxide materials.  They "pop up" as we go from the metallic ferromagnetic state at low temperature to the paramagnetic polaron state at high temperatures, whereas a magnetic field suppresses the polarons.  Thus the development of ferromagnetic order, either by lowering the temperature or increasing the magnetic field, is detrimental to polaron formation.

To understand the structure and dynamics of this ferromagnetic-metallic to paramagnetic-insulating transition, as the paramagnetic state is entered the purely elastic component to the structural polaron scattering signals the development of the correlated polaron glass phase.  This elastic scattering is accompanied by dynamic correlations that also peak at the same wave vector.  The dynamic polaron correlation length in this phase is also around one nanometer.  The strength of the elastic scattering diminishes with increasing temperature until the static polarons disappear at a higher temperature T*. The correlations remain above this temperature, but are then purely dynamic in character. The statics and dynamics of this scattering bear a remarkable similarity to the magnetic fluctuation spectrum recently observed in underdoped cuprates, suggesting that the underlying behavior has a similar origin.

Left:  Energy dependence of the scattering at the wave vector position of the polaron scattering.  There are two components of the scattering evident, a purely elastic component, and inelastic scattering that peaks at E=0 (quasielastic scattering).  Right:  Schematic block phase diagram for La1-xCaxMnO3, where we have sketched  the proposed polaronic glass phase and how it evolves into the long-range ordered Jahn-Teller transition for the undoped compound.  Basic ferromagnetic-metallic to paramagnetic-insulating, ferromagnetic-insulating to paramagnetic insulating, and antiferromagnetic to paramagnetic phase transitions are from previous work of other authors and us.  Note that at lower x the transitions are also dependent on the oxygen concentration and hence the heat treatment of the samples.  The ferromagnetic-paramagnetic transitions are from our work as well as the work of other authors (see references given in Phys. Rev. B 76, 014437 (2007)), while above optimal doping the transition becomes second order.  The polaron glass phase is based on our recent work.  Further work is in progress.

 

 

Chemical disorder can be a key factor in phase stabilization, and recently A-site ordered and disordered La1/2Ba1/2MnO3 samples have been synthesized.  The Curie temperature for the ordered sample is much higher than for the disordered sample, while the exchange interaction as determined from the low temperature spin waves is almost the same.  The spin dynamical data shown at the left/top demonstrate that the transition in the ordered sample (Tc=348.7 K) is continuous, while there is two-phase coexistence (right/bottom) for the disordered sample (Tc=297 K).  The results indicate that the energy for polaron formation is lower in the chemically disordered sample, which truncates the ferromagnetic state.
 

See link to Multiferroics 

 
Selected Publications

Unconventional Ferromagnetic Transition In La1-xCaxMnO3, J. W. Lynn, R. W. Erwin, J. A. Borchers, Q. Huang, A. Santoro, J. L. Peng, and Z. Y. Li, Phys. Rev. Lett. 76, 4046 (1996).

Magnetic, Structural, and Spin Dynamical Properties of La1-xCaxMnO3 , J. W. Lynn, R. W. Erwin, J. A. Borchers, Q. Huang, A. Santoro, J. L. Peng and R. L. Greene, J. Appl. Phys. 81, 5488 (1997).

Structure and Magnetic Order in Undoped Lanthanum Manganite, Q. Huang, A. Santoro, J. W. Lynn, R. W. Erwin, J. A. Borchers, J. L. Peng, and R. L. Greene, Phys. Rev. B 55, 14987 (1997).

Neutron Scattering Investigation of the Structure and Spin Dynamics in La0.85Sr0.15MnO3 , L. Vasiliu-Doloc, J. W. Lynn, A. H. Moudden, A. M. De Leon-Guevara and A. Revcolevschi, J. Appl. Phys. 81, 5491 (1997).

Structure and Spin Dynamics of La0.85Sr0.15MnO3, L. Vasiliu-Doloc, J. W. Lynn, A. H. Moudden, A. M. de Leon-Guevara, and A. Revcolevschi, Phys. Rev. B58, 14913 (1998).

Structure and Magnetic Order of La1-xCaxMnO3, Q. Huang, A. Santoro, J. W. Lynn, R. W. Erwin, J. A. Borchers, J. L. Peng, K. Ghosh, and R. L. Greene, Phys. Rev. B 58, 2684 (1998).

Spin Dynamics of the Magnetoresistive Pyrochlore Tl2Mn2O7, J. W. Lynn, L. Vasiliu-Doloc, and M. A. Subramanian, Phys. Rev. Lett. 80, 4582 (1998).

Spin Dynamics of Strongly Doped La1-xSrxMnO3, L.Vasiliu-Doloc, J. W. Lynn, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, J. Appl. Phys. 83, 7342 (1998).

Magnetic Correlations in the Bilayer Manganite La1.2Sr1.8Mn2O7, S. Rosenkranz, R. Osborn, S. K. Sinha, K. E. Gray, J. F. Mitchell, L. Vasiliu-Doloc, J. W. Lynn, and D. Argyriou, J. Appl. Phys. 83, 7348 (1998).

Neutron Scattering Investigation of Magnetic Bilayer Correlations in La1.2Sr1.8Mn2O7, R. Osborn, S. Rosenkranz, D. N. Argyriou, L. Vasiliu-Doloc, J. W. Lynn, S. K. Sinha, J. F. Mitchell, K. E. Gray, and S. D. Bader, Phys. Rev. Lett. 81, 3964 (1998).

Charge Melting and Polaron Collapse in La1.2Sr1.8Mn2O7, L. Vasiliu-Doloc, S. Rosenkranz, R. Osborn, S. K. Sinha, J. W. Lynn, J. Mesot, O. Seeck, G. Preosti, A. J. Fedro, and J. F. Mitchell, Phys. Rev. Lett. 83, 4393 (1999)

Charge Ordering and Polaron Formation in La0.7Ca0.3MnO3, C. P. Adams, J. W. Lynn, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, Phys. Rev. Lett. 85, 3954 (2000).

Temperature Dependence of Low-Lying Electronic Excitations of LaMnO3, M. A. Quijada, J. R. Simpson, H. D. Drew, J. W. Lynn, L. Vasiliu-Doloc, Y. M. Mukovskii, and S. G. Karabashev, Phys. Rev. B 64, 224426 (2001).

Charge Correlations in La0.7Ca0.3MnO3, J. W. Lynn, C. P. Adams, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, J.  Appl. Phys. 89, 6846 (2001).

Glass Transition in the Polaron Dynamics in CMR Manganites, D. N. Argyriou, J. W. Lynn, R. Osborn, B. Campbell, J. F. Mitchell, U. Ruett, H. N. Bordallo, A. Wildes, and C. D. Ling, Phys. Rev. Lett. 89, 036401 (2002).

Structure of Nanoscale Polaron Correlations in a Colossal Magnetoresistive Manganite, B. J. Campbell, R. Osborn, D. N. Argyriou, L. Vasiliu-Doloc, J. F. Mitchell, S. K. Sinha, U. Ruett, C. D. Ling, Z. Islam, and J. W. Lynn, Phys. Rev. B 65, 014427 (2002).

First-order Transition in the Itinerant Ferromagnet CoS1.9Se0.1, T. J. Sato, J. W. Lynn, Y.-S. Hor, and S.-W. Cheong, Phys. Rev. B 68, 214411 (2003).

Inhomogeneous Magnetism in La-doped CaMnO3:  (I) Phase Separation due to Lattice-coupled Ferromagnetic Interactions, C. D. Ling, E. Granado, J. J. Neumeier, J. W. Lynn, and D. N. Argyriou, Phys. Rev. B 68, 134439 (2003).

Inhomogeneous Magnetism in La-doped CaMnO3:  (II) Nanometric-scale Spin Clusters and Long-range Spin Canting, E. Granado, C. D. Ling, J. J. Neumeier, J. W. Lynn, and D. N. Argyriou, Phys. Rev. B 68, 134440 (2003).

First-order Nature of the Ferromagnetic Phase Transition in (La-Ca)MnO3 near optimal doping, C. P. Adams, J. W. Lynn, V. N. Smolyaninova, A. Biswas, R. L. Greene, W. Ratcliff, II, S-W. Cheong, Y. M. Mukovski, and D. A. Shulyatev, Phys. Rev.B 70, 134414 (2004).

Correlated and Uncorrelated Nanoscale Lattice Distortions in the Paramagnetic Phase of Magnetoresistive Manganites, V. Kiryukhin, A. Borissov, J. S. Ahn, Q. Huang, J. W. Lynn, and S-W. Cheong, Phys. Rev. B 70, 214424 (2004).

Crystal Structures and Magnetic Order of La0.5+dA0.5-dMn0.5+eRu0.5-eO3 (A=Ca, Sr, Ba):  Possible Orbital Glass Ferromagnetic State, E. Granado, Q. Huang, J. W. Lynn, J. Gopalakrishnan, and K. Ramesha, Phys. Rev. B 70, 214416 (2004).

Disorder-induced Polaron Formation in the Magnetoresistive Perovskite La0.54Ba0.46MnO3, T. J. Sato, J. W. Lynn, and B. Dabrowski, Phys. Rev. Lett. 93, 267204 (2004).

Field-induced Avalanche to the Ferromagnetic State in the Phase-separated Ground State of Manganites, F. M. Woodward, J. W. Lynn, M. B. Stone, R. Mahendiran, P. Schiffer, J. F. Mitchell, D. N. Argyriou, and L. C. Chapon, Phys. Rev. B 70, 174433 (2004)

 Inter-granular Giant Magnetoresistance in a Spontaneously Phase Separated Perovskite Oxide, J. Wu, J. W. Lynn, C. J. Glinka, J. Burley, H. Zheng, J. F. Mitchell, and C. Leighton, Phys. Rev. Lett. 94, 037201 (2005).

Magnetic Order and Spin Dynamics in Ferroelectric HoMnO3, O. P. Vajk, M. Kenzelmann, J. W. Lynn, S. B. Kim and S.-W. Cheong, Phys. Rev. Lett. 94, 087601 (2005).

Magnetic Inversion Symmetry Breaking and Ferroelectricity in TbMnO3, M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm, J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong, O. P. Vajk and J. W. Lynn, Phys. Rev. Lett. 95, 087206 (2005).

Electronically Smectic Liquid-crystal Phase in a Nearly Half-doped Manganite, F. Ye, J. A. Fernandez-Baca, P, Dai, J. W. Lynn, H. Kawano-Furukawa, H. Yoshizawa, Y. Tomioka, and Y. Tokura, Phys. Rev. B 72, 212404 (2005).

Strong Orbital Correlations in an Exchange-Frustrated Fe-substituted Manganite, E. Granado, R. R. Urbano, C. A. Perez, C. Azimonte, J. W. Lynn, R. A. Souza, N. M. Souza-Neto, A. Y. Ramos, G. L. Bychkov, S. V. Shiryaev, and S. N. Barilo, Phys. Rev. B 72, 052406 (2005).

Mn Magnetic Ordering in Fully Oxygenated Lu1-xScxMnO3 (x=0, 0.5, and 1), P. Huang, S. Y. Wu, F. C. Tsao, C. C. Yang, M. K. Chung, W-H. Li, J. W. Lynn, and H. Ku, J. Appl. Phys.  97, 10H703 (2005).

Evolution of Spin-Wave Excitations in Ferromagnetic Metallic Manganites, F. Ye, P. Dai, J. A. Fernandez-Baca, H. Sha, J. W. Lynn, H. Kawano-Furukawa, Y. Tomioka, Y. Tokura, and J. Zhang, Phys. Rev. Lett. 96, 047204 (2006).

Structural Anomalies at the Magnetic and Ferroelectric Transition in RMn2O5, C. R. dela Cruz, F. Yen, B. Lorenz, M. M. Gospodinov, C. W. Chu, W. Ratcliff II, J. W. Lynn, S. Park, and S-W. Cheong, Phys Rev. B 73, 100406 Rapid Communications (2006).

Spontaneous Spin-lattice Coupling in the Geometrically Frustrated Triangular Lattice Antiferromagnet CuFeO2,  F. Ye, Y. Ren, Q. Huang, J. A. Fernandez-Baca, P. Dai, J. W. Lynn, and T. Kimura, Phys. Rev. B Rapid Communications 73, 220404 (2006).

Complex Magnetic Order in the Kagome Staircase Compound Co3V2O8, Y. Chen, J. W. Lynn, Q. Huang, F. M. Woodward, T. Yildirim, G. Lawes, A. P. Ramirez, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, and A. B. Harris, Phys. Rev. B 74, 014430 (2006).

Magnetic Instability and Oxygen Deficiency in Na-doped TbMnO3, C. C. Yang, M. K. Chung, W.-H. Li, T. S. Chan, R. S. Liu, Y. H. Lien, C. Y. Huang, Y. Y. Chan, Y. D. Yao, and J. W. Lynn, Phys. Rev. B 74, 094409 (2006).

Reentrant Orbital Order and the True Ground State of LaSr2Mn2O7, Qing'An Li, K. E. Gray, H. Zheng, H. Claus, S. Rosenkranz, S. Nyborg Ancona, R. Osborn and J. F. Mitchell, Y. Chen and J. W. Lynn, Phys. Rev. Lett. 98, 167201 (2007).

Effect of Antiferromagnetic Spin Correlations on Lattice Distortion and Charge Ordering in Pr0.5Ca1.5MnO4, S. Chi, F. Ye, P. Dai, J. A. Fernandez-Baca, Q. Huang, J. W. Lynn, E. W. Plummer, R. Mathieu, Y. Kaneko, and Y. Tokura, Proceedings of the National Academy of Sciences 104, 10796 (2007).

A "non-Griffiths-like" clustered phase above the Curie temperature of the doped perovskite cobaltite La1-xSrxCoO3, C. He, M.A. Torija, J. Wu, J.W. Lynn, H. Zheng, J.F. Mitchell and C. Leighton, Phys. Rev. B 76, 014401 (2007).

Neutron Diffraction Study of Multiferroic Tb0.85Na0.15MnO3−y, T.S. Chan, R.S. Liu, C.C. Yang, W.-H. Li, Y.H. Lien, C.Y. Huang and J.W. Lynn, J. Mag. Mag. Materials 310, 1151 (2007).

Incipient Orbital Order in Half-Metallic Ba2FeReO6, C. Azimonte, J. C. Cezar, E. Granado, Q. Huang, J. W. Lynn, J. C. P. Campoy, J. Gopalakrishana, and K. Ramesha, Phys. Rev. Lett. 98, 017204 (2007).

Magnetic ordering of Mn and Ru in (La0.52Ba0.48) (Mn0.51Ru0.49)O3, S. Y. Wu, 1, W.-H. Li, C. C. Yang, J. W. Lynn, and R. S. Liu, Physica Stat. Sol. B 244, 2233 (2007).

Influence of Oxygen Defects on the Crystal Structure and Magnetic Properties of the (Tb1-xNax)MnO3-y (0 x 0.3) System, T. S. Chan,R. S. Liu, C. C. Yang, W.-H. Li, Y. H. Lien, C.Y. Huang, Jeff. W. Lynn, J. M. Chen, and H.-S. Sheu, Inorganic Chemistry 46, 4575 (2007).

Order and Dynamics of Intrinsic Nanoscale Inhomogeneities in Manganites, J. W. Lynn, D. N. Argyriou, Y. Ren, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, Phys. Rev. B 76, 014437 (2007).

Signature of Magnetic Phase Separation in the Ground State of Pr1-xCaxMnO3, Hao Sha, F. Ye, P. Dai, J. A. Fernandez-Baca, D. Mesa, J. W. Lynn, Y. Tomioka, Y. Tokura, and Jiandi Zhang, Phys. Rev. B 78, 052410 (2008).

Spin-valve Effect and Magnetoresistivity in Single Crystalline Ca3Ru2O7, W. Bao, Z. Q. Mao, Z. Qu, and J. W. Lynn, Phys. Rev. Lett. 100, 247203 (2008).

The Pressure Effect on the Magnetic Commensurability and Ferroelectricity in Multiferroic HoMn2O5, C. R. dela Cruz, B. Lorenz, W. Ratcliff, J. Lynn, M. M. Gospodinov, and C. W. Chu, Physica B 403, 1359 (2008).

Magnetic Switching and Phase Competition in the Multiferroic Antiferromagnet Mn1-xFexWO4, F. Ye, Y. Ren, J. A. Fernandez-Baca, H. A. Mook, J. W. Lynn, R. P. Chaudhury, Y.-Q. Wang, B. Lorenz, and C. W. Chu, Phys. Rev. B 78, 193101 (2008).

Spin Dynamics in the Magnetoelectric Effect LiCoPO4 Compound, W. Tian, J. Li, J. W. Lynn, J. L. Zarestky, and D. Vaknin, Phys. Rev. B 78, 184409 (2008).

Polaron Formation in the Optimally Doped Ferromagnetic Manganites La0.7Sr0.3MnO3 and La0.7Ba0.3MnO3, Y. Chen, B. G. Ueland, J. W. Lynn, G. L. Bychkov, S. N. Barilo, and Y. M. Mukovskii, Phys. Rev. B 78, 212301 (2008).

Antiferromagnetism in the Magnetoelectric Effect Single Crystal LiMnPO4, Jiying Li, Wei Tian, Ying Chen, Jerel L. Zarestky, Jeffrey W. Lynn, and David Vaknin, Phys. Rev. B79, 144410 (2009).

Electronic Self-organization in the Single-layer Manganite Pr1-xCa1+xMnO4, F. Ye, S. Chi, J. A. Fernandez-Baca, A. Moreo, E. Dagotto, J. W. Lynn, R. Mathieu, Y. Kaneko, Y. Tokura, and P. Dai, Phys. Rev. Lett. 103, 167202 (2009).

Re-entrant Spiral Magnetic Order and Ferroelectricity in Mn1-xFexWO4 (x=0.035), R. P. Chaudhury, B. Lorenz, Y. Q. Wang, Y. Y. Sun, C. W. Chu, F. Ye, J. Fernandez-Baca, H. Mook, and J. Lynn, J. Appl. Phys. 105, 07D913 (2009).

Doping Fluctuation-driven Magnetoelectronic Phase Separation in La1-xSrxCoO3 Single Crystals, C. He, S. El-Khatib, J. Wu, J. W. Lynn, H. Zheng, J. F. Mitchell, and C. Leighton, Europhysics Lett. 87, 27006 (2009).

Coupled Structural/Magnetocrystalline Anisotropy Transitions in the Doped Perovskite Cobaltite Pr1-xSrxCoO3, C. Leighton, D.D. Stauffer, Q. Huang, Y. Ren, B. Toby, S. El-Khatib, M.A. Torija, J. Wu, J. W. Lynn, L. Wang, N.A. Frey, H. Srikanth, J.E. Davies, K. Liu and J.F. Mitchell, Phys. Rev. B 79, 214420 (2009).

Tweaking the Spin Wave Dispersion and Suppressing the Incommensurate Phase in LiNiPO4 by Iron Substitution, Jiying Li, Thomas B. S. Jensen, Niels. H. Andersen, Jerel L. Zarestky, R. William McCallum, J.-H. Chung, Jeffrey W. Lynn, and David Vaknin, Phys. Rev. B 79, 174435 (2009).

Transport Signatures of Percolation and Electronic Phase Homogeneity in La1-xSrxCoO3 Single Crystals, C. He, S. El-Khatib, S. Eisenberg, M. Manno, J. W. Lynn, H. Zheng, J. F. Mitchell, and C. Leighton, Appl. Phys. Lett. 95, 222511 (2009).

Neutron Scattering Studies of LiCoPO4 and LiMnPO4, Wei Tian, Jiying Li, Haifeng Li, Jeffrey W. Lynn, Jerel L. Zarestky, and David Vaknin, J. Physics: Conf. Series 150, (2009).

Origin of Electric Field Induced Magnetization in Multiferroic HoMnO3, B. G. Ueland, J. W. Lynn, M. Laver, Y. J. Choi, and S.-W. Cheong, Phys. Rev. Lett. 104, 147204 (2010).

Short range magnetic correlations induced by La-substitution in Ho1-xLaxMn2O5, Chin-Wei Wang, Chun-Ming Wu, Chi-Yen Li, Sunil K. Karna, Chien-Kang Hsu, Carissa H. C. Li, Wen-Hsien Li, Chun-Chen Yu, Chun-Pin Wu, Hsiung Chou, and Jeffrey W. Lynn, J. Phys: Cond. Mat. 22, 246002 (2010).

Spontaneous Formation of an Exchange Spring Composite via Magnetic Phase Separation in Pr1-xCaxCoO3, S. El-Khatib, S. Bose, C. He, J. Kuptic, M. Laver, J. A. Borchers, Q. Huang, J. W. Lynn, J. F. Mitchell, and C. Leighton, Phys. Rev. B 82, 100411(R) (2010).

Phonon Response to Charge and Orbital Order in LaSr2Mn2O7, F. Weber, S. Rosenkranz, J. P. Castellan, R. Osborn, D. Reznik, H. Zheng, J. F. Mitchell, Y. Chen, and J. W. Lynn, Phys. Rev. Lett. 107, 207202 (2011).

Neutron Diffraction Study of the Mn Spin Correlations in Bi0.46Ca0.54Mn0.95Cr0.05O3, Chi-Hung Lee, Chun-Ming Wu, Daniel Hsu, Chin-Wei Wang, Chih-Jen Wang, Wen-Hsien Li, Chun-Chuen Yang, Jirong Sun, and Jeffrey W. Lynn, J. Phys. Soc. Japan 80, SB013 (2011).

Paramagnetic Spin Correlations in Colossal Magnetoresistive La0:7Ca0:3MnO3, Joel S. Helton; , Matthew B. Stone, Dmitry A. Shulyatev, Yakov M. Mukovskii, and Jeffrey W. Lynn, Phys. Rev. B 85, 144401 (2012).

Two-Dimensional Magnetic Correlations and Partial Long-Range Order in Geometrically Frustrated Sr2YRuO6, E. Granado, J. W. Lynn, R. F. Jardim, and M. Torikachvili, Phys. Rev. Lett. 110, 017202 (2013).

Polaron-mediated Spin Correlations in Metallic and Insulating La1-xAxMnO3 (A=Ca, Sr, or Ba), Joel S. Helton, Daniel M. Pajerowski, Yiming Qiu, Yang Zhao, Dmitry A. Shulyatev, Yakov M. Mukovskii, Georgii L. Bychkov, Sergei N. Barilo, and Jeffrey W. Lynn, Phys Rev B 90, 214411 (2014).

Spin Dynamics and Two-dimensional Correlations in an FCC Antiferromagnetic Sr2YRuO6, S. M. Disseler, J. W. Lynn, R. F. Jardim, M. S. Torikachvili, and E. Granado, Phys. Rev. B 93, 140407(R) (2016). 

 

 [please see Recent Publications list for full details to these references as well as additional ones]

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