The density-matrix renormalization group method is used to study the three-band zigzag ladder model that simulates the electronic state of the t 2g-orbital system of the double string of the edge-shared CrO 6 octahedra in the half-metallic ferromagnet K2Cr 8O16. The saturated ferromagnetism caused by the double-exchange mechanism is thereby demonstrated. We also use the Lanczos exact-diagonalization technique on small clusters to study the two Hubbard chains coupled with the ferromagnetic exchange interaction, whereby we consider the nonquasiparticle states in the half-metallic ferromagnet that appear in the single-particle spectral function.Ministry of Education, Culture, Sports, Science and Technology of Japa

Nakano, K.

Ohta, Y.

Shirakawa, T.

Journal of Physics : Conference Series

Institutionelles Repositorium der Leibniz Universität Hannover

This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 194.95.157.184This content was downloaded on 26/04/2017 at 12:36Please note that terms and conditions apply.Nonquasiparticle states in the half-metallic ferromagnet simulated by the two-band HubbardmodelView the table of contents for this issue, or go to the journal homepage for more2010 J. Phys.: Conf. Ser. 200 012143(http://iopscience.iop.org/1742-6596/200/1/012143)Home Search Collections Journals About Contact us My IOPscienceYou may also be interested in:Exact wave functions and excitation spectra of the one-dimensional double-exchange model with onemobile electronK Nakano, R Eder and Y OhtaExact wave functions and excitation spectra of the one-dimensional double-exchange model with onemobile electronK Nakano, R Eder and Y OhtaDimerization transition of three-leg Heisenberg tubeSatoshi Nishimoto and Mitsuhiro ArikawaSuperconductivity and ferromagnetism in charge-ordered metallic phases of extended hubbard modelsS Morohoshi and Y FukumotoCharge-order in the quasi-cubic manganitesP SchlottmannAnomalous electronic states of hollandite-type transition-metal oxidesY Ohta, T Toriyama, M Sakamaki et al.Magnetic phases of the two orbital Hubbard model in dynamical mean field theoryR Peters and T PruschkeElectron Correlation and Spin Dynamics in Iron Pnictides and ChalcogenidesRong Yu, Qimiao Si, Pallab Goswami et al.Charge-spin-orbital dynamics of one-dimensional two-orbital Hubbard modelHiroaki OnishiNonquasiparticle states in the half-metallicferromagnet simulated by the two-band HubbardmodelK Nakano1, T Shirakawa2 and Y Ohta11Department of Physics, Graduate School of Science, Chiba University, Japan2Institut fu¨r Theoretische Physik, Leibnitz Universita¨t Hannover, GermanyE-mail: ohta@science.s.chiba-u.ac.jpAbstract. The density-matrix renormalization group method is used to study the three-bandzigzag ladder model that simulates the electronic state of the t2g-orbital system of the doublestring of the edge-shared CrO6 octahedra in the half-metallic ferromagnet K2Cr8O16. Thesaturated ferromagnetism caused by the double-exchange mechanism is thereby demonstrated.We also use the Lanczos exact-diagonalization technique on small clusters to study the twoHubbard chains coupled with the ferromagnetic exchange interaction, whereby we considerthe nonquasiparticle states in the half-metallic ferromagnet that appear in the single-particlespectral function.1. IntroductionHalf-metallic ferromagnets [1] offer a unique opportunity for studying the electronic states ofstrongly correlated electron systems. Here, only the majority-spin electrons form the Fermisurface with a gapped minority-spin band and can couple with excitations of the spin (and insome cases orbital) degrees of freedom of the system. The situation therefore should attractmuch interest, in particular, when the relevant electrons are strongly correlated, leading thesystem to double-exchange ferromagnetism [2] and metal-insulator transition (MIT).In this paper, motivated by a recent discovery [3, 4] that a chromium oxide K2Cr8O16 with thehollandite-type crystal structure [5] belongs to this class of materials, we study electronic statesof the three-band zigzag ladder model by the density-matrix renormalization group (DMRG)method. We also study the model of two Hubbard chains coupled with ferromagnetic exchangeinteraction by the Lanczos exact-diagonalization technique on small clusters. We therebyconsider the origin of saturated ferromagnetism caused by the double-exchange mechanism andindications of nonquasiparticle states [1, 6, 7] appeared in the half-metallic ferromagnet withinthe simplest models.2. Method of calculationThe crystal structure of K2Cr8O16 belongs to a group of hollandite-type phases where one-dimensional (1D) double strings of edge-shared CrO6 octahedra (see Fig. 1) forms a Cr8O16framework of a tunnel structure, wherein K ions reside [5]. Cr ions are in the mixed-valent stateof Cr4+ (d2) : Cr3+ (d3) = 3 : 1, and hence with 2.25 electrons per Cr ion.International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics: Conference Series 200 (2010) 012143 doi:10.1088/1742-6596/200/1/012143c© 2010 IOP Publishing Ltd 1zyxCrOFigure 1. Schematic representations of the double string of the edge-shared CrO6 octahedrain the hollandite structure.To simulate the electronic states of this system, we use a model for the double string of Crions, retaining only the t2g orbitals (see Fig. 2 for its schematic representation). Our Hamiltonianis therefore of the following form:H = −∑〈iα,jβ〉,σtiα,jβ(c†iασcjβσ +H.c.)− JH∑iσσ′,α6=βc†iασc†iβσ′ciβσciασ′+ U∑iαniα↑niα↓ + U′∑i,α6=βniαniβ (1)where c†iασ (ciασ) is the creation (annihilation) operator of an electron at site i, orbital α, and spinσ =↑, ↓. We define the number operators niασ = c†iασciασ, niα = niα↑ + niα↓, and ni =∑α niα.JH is the Hund’s rule coupling, and U and U′ are the intra- and inter-orbital on-site Coulombrepulsions, respectively. We assume the relation U ′ = U − 2JH throughout the paper, which isvalid in the atomic limit. tiα,jβ is the hopping parameter between the orbital α on site i andorbital β on site j where α, β ∈ {dxy, dyz, dzx} in the coordinate system shown in Fig. 1. Weretain only the direct electron hopping between the t2g orbitals for simplicity. We then have theindependent nearest-neighbor hopping parameters ta, tb, and tc as shown in Fig. 2.tatbtcU', JH dxydzx dyz U++++++++++++++++++++++++Figure 2. Schematic representation of the t2g orbitals on the 1D zigzag chain of K2Cr8O16.Two of the four lobes for each of the three t2g orbitals are drawn.We also use the two-chain Hubbard model with the ferromagnetic exchange interactionbetween the chains [8]. We calculate the single-particle spectral function defined asA(q, ω) = A−(q,−ω) +A+(q, ω) (2)with the electron removal spectrumA−(q, ω) = −1piℑ〈Ψ0|c†qσ1ω + iη − (H− E0)cqσ|Ψ0〉 (3)International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics: Conference Series 200 (2010) 012143 doi:10.1088/1742-6596/200/1/0121432and electro addition spectrumA+(q, ω) = −1piℑ〈Ψ0|cqσ1ω + iη − (H− E0)c†qσ|Ψ0〉, (4)where η → +0, which is replaced by a small positive number in the actual calculations to give anartificial broadening of the spectra. We use a cluster of 16 sites with open boundary conditionfor these calculations.00.20.40.60.811.21.40 2 4 6 8 10 12 14 16n(ri)ridxy dzx dyzFigure 3. Left panel: Ground-state charge distribution of the three-band model calculated bythe DMRG method. The result for the finite-size cluster of 16 sites (or 48 orbitals) with openboundary condition is shown, where we assume the values of the parameters ta = 1, tb = tc = 0,U = 30, U ′ = 10 and JH = 10. Right panel: Ground-state phase diagram in the parameterspace where we assume ta = 1 and tb = tc = 0 with the assumption U = U′ + 2JH. In bothcalculations, the cluster contains 36 electrons.3. Results of calculationCalculated results for the ground state of the three-band model are shown in Fig. 3. In thephase diagram, we find the fully spin-polarized region S = Smax for the large values of U . Inthis ferromagnetic region, we find that the electron distribution is characteristic of the fullyoccupied dyz orbitals coexisting with the partially occupied dxy and dzx orbitals. Thus, we havea dualistic situation where the localized dyz electrons interact with the itinerant dxy and dzxelectrons via the Hund’s rule coupling, resulting in the ferromagnetic spin polarization via thedouble-exchange mechanism [2]. One should note that the role of orbitals is different from thatobtained in the generalized gradient approximation within the density functional theory wherethe dxy electrons are essentially localized and the dyz and dzx electrons are highly itinerant [4].The difference may come from the coupling between the double strings that is not taken intoaccount in our simple 1D model calculations.We also calculate the ground state of the two-chain Hubbard ladder model coupled by theHeisenberg-type ferromagnetic exchange interaction. We find the fully spin-polarized groundstate in a parameter space (see Ref. [8]). In this parameter region, we calculate the single-particle spectral function. The result is shown in Fig. 4. We find that the metal-like spectraappear in the majority-spin spectral function (where a gap-like feature due to finite-size effect isexpected to vanish in the thermodynamic limit), whereas the gap opens clearly in the minority-spin spectral function, indicating the system to be half-metallic. The indication of the presence ofnonquasiparticle state is not clear although the state just above the Fermi level in the minority-spin spectral function may correspond to the state. Further studies are in progress.International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics: Conference Series 200 (2010) 012143 doi:10.1088/1742-6596/200/1/0121433-4 -2 0 2 40.40.20.00.20.4Majority spinMinority spin  Figure 4. Momentum-integrated single-particle spectral function ρ(ω) =∑qA(q, ω) calculatedfor the two-chain Hubbard ladder model. The vertical line indicates the Fermi level. We use a16-site cluster with 6 electrons with the parameter values t = 1, U = 30, U ′ = 10, and J = 10,where the ground-state is fully spin-polarized.4. SummaryWe have applied the DMRG method to the three-band zigzag ladder model of the t2g-orbital system and have studied the saturated ferromagnetism caused by the double-exchangemechanism, whereby we have simulated the electronic states of the half-metallic ferromagnetK2Cr8O16 recently found. We have shown that the fully spin-polarized region actually appearsin the ground-state phase diagram, where the fully occupied dyz orbital coexists with partiallyoccupied dxy and dzx orbitals, as is expected in the double-exchange ferromagnet. We havealso studied the nonquasiparticle states in the single-particle spectral function by means ofthe Lanczos exact-diagonalization technique on small clusters of the two-chain Hubbard model.We have thereby discussed possible indications of the nonquasiparticle states appeared in thehalf-metallic ferromagnet, of which further studies are in progress.AcknowledgmentsThis work was supported in part by a Grant-in-Aid for Scientific Research (No. 19014004)from the Ministry of Education, Culture, Sports, Science and Technology of Japan. A partof computations was carried out at the Research Center for Computational Science, OkazakiResearch Facilities, and the Institute for Solid State Physics, University of Tokyo.References[1] Katsnelson M I, Irkhin V Y, Chioncel L, Lichtenstein A I and de Groot R A 2008 Rev. Mod. Phys. 80 315[2] Zener C 1951 Phys. Rev. 82 403[3] Hasegawa K, Isobe M, Yamauchi T, Ueda H, Yamaura J, Goto H, Yagi T, Sato H and Ueda Y 2009 Phys.Rev. Lett. 103 146403[4] Sakamaki M, Konishi T and Ohta Y 2009 Phys. Rev. B 80 024416[5] Tamada O, Yamamoto N, Mori T and Endo T 1996 J. Solid State Chem. 126 1[6] Edwards D M and Hertz J A 1973 J. Phys. F: Met. Phys. 3 2191[7] Irkhin V Y and Katsnelson M I 2006 Phys. Rev. B 73 104429[8] Shirakawa T, Nishimoto S and Ohta Y 2008 Phys. Rev. B 77 224510International Conference on Magnetism (ICM 2009) IOP PublishingJournal of Physics: Conference Series 200 (2010) 012143 doi:10.1088/1742-6596/200/1/0121434

Nonquasiparticle states in the half-metallic ferromagnet simulated by the two-band Hubbard model

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