Importance of Itinerancy and Quantum Fluctuations for the Magnetism in Iron Pnictides
Yu-Zhong Zhang, Hunpyo Lee, Ingo Opahle, Harald O. Jeschke, Roser Valenti
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Importance of Itinerancy and Quantum Fluctuations for the Magnetism in Iron Pnictides
Yu-Zhong Zhang ∗ ,a,b , Hunpyo Lee a , Ingo Opahle a , Harald O. Jeschke a , Roser Valent´ı a a Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany b Department of Physics, Tongji University, Shanghai, 200092 P. R. China
Abstract
By applying density functional theory, we find strong evidence for an itinerant nature of magnetism in two families of iron pnictides.Furthermore, by employing dynamical mean field theory with continuous time quantum Monte Carlo as an impurity solver, weobserve that the antiferromagnetic metal with small magnetic moment naturally arises out of coupling between unfrustrated andfrustrated bands. Our results point to a possible scenario for magnetism in iron pnictides where magnetism originates from a stronginstability at the momentum vector ( π , π , π ) while it is reduced by quantum fluctuations due to the coupling between weakly andstrongly frustrated bands. Key words:
Iron pnictides, itinerant magnetism, density functional theory, dynamical mean field theory, multi-band Hubbard model.
PACS: + h
1. Introduction
Since the discovery of high-T c superconductivity inLaOFeAs [1], great e ff ort has been devoted to pursuing highersuperconducting transition temperatures T c [2] as well as to theunderstanding of the various phase transitions in these materi-als as a function of temperature, pressure and doping [3, 4, 5].Density functional theory (DFT) calculations [6, 7] show thatelectron-phonon coupling – though non-negligible – is notstrong enough to explain the high T c observed in these systems.Instead, magnetically mediated pairing has been proposed toaccount for the superconducting state due to its proximity to astripe-type antiferromagnetic (AF) metallic state [8, 9].However, the origin of the stripe-type AF metal, i.e., whetherit comes from the itinerant nature of the Fermi surface (FS)nesting [9, 10, 11, 12, 13] or a localized picture of exchange in-teractions between local spins [14, 15, 16], is still under debate.The scenario of FS nesting was severely challenged by recentDFT calculations [17], which found a disconnection betweenFe moment and FS nesting. However, a recent revision of thenature of magnetism in the iron pnictides – also within DFT, butwith more precise optimized structures – has reestablished theclose connection between itinerancy and magnetism [13]. Inthis work, we will present further clear evidence of the itinerantnature of magnetism by performing DFT calculations for a fewfamilies of iron pnictides.On the other hand, the disagreement [18, 19] of the mag-nitude of the Fe moment in the stripe-type AF metal betweenexperimental observations and DFT calculations based on ex-perimental structures is still controversely discussed. Vari-ous mechanisms have been proposed from DFT calculations, ∗ Corresponding author.
Email address: [email protected] (Yu-Zhong Zhang) for example, negative e ff ective on-site electronic interaction U [20, 21], a low moment solution within GGA + U stabilizedby the formation of magnetic multipoles [22, 23] or frustrationbetween local spins [14, 15, 24, 25]. Recently, also mean-fieldcalculations based on a five-band Hubbard model with positive U obtained a small magnetic moment comparable to experi-mental results [26, 27, 28]. However, the existence of varioussets of DFT-derived hopping parameters [29, 30, 31] casts doubton the reliability of the model parameters used in the model cal-culations.From DFT calculations it is well known [8, 18, 19] that thephysical properties of the iron pnictides are highly sensitive todetails of the structure as well as to details of the exchange andcorrelation (XC) functionals. For instance, structural optimiza-tion within the local spin density approximation (LSDA) leadsto almost perfect agreement [18] with the most recent experi-mental value of the Fe moment [32]. This indicates, that quan-tum fluctuations, which are only insu ffi ciently incorporated inthe common approximations to DFT, could strongly improvethe agreement with experiment.To further explore this, we employ dynamical mean-fieldtheory (DMFT) [33] combined with continuous time quantumMonte Carlo (CTQMC) [34] simulations. A feature that is com-mon to di ff erent sets of DFT-derived hopping parameters is thepresence of some strongly and some weakly frustrated Fe 3 d bands [29, 30, 31]. In the present work we consider a minimal two-band Hubbard model which should capture this feature byconsidering one band with frustration and the second one with-out frustration and we investigate the e ff ect of the coupling ofthese two bands on the magnetism of the system. We would liketo remark that while a realistic description of the Fe systemsneeds at least a five-band model, the present two-band modelshould be su ffi cient for the proposed analysis. We find that an Preprint submitted to Journal of Physics and Chemistry of Solids March 2, 2018
F metallic state is present in a wide range of the interactionparameter U when one band is highly frustrated and the secondone unfrustrated, while the state is absent when both bands areequally frustrated [35].Our results from DFT and DMFT suggest that a strong insta-bility at a momentum vector ( π , π , π ) is the promising mech-anism for the itinerant magnetism observed in iron pnictideswhile quantum fluctuations originating from the coupling be-tween weakly and strongly frustrated bands reduce the mag-netic moment and make it more comparable to the experimentalobservations.
2. Method and Model
In order to quantify the FS nesting and hence the itinerantnature of the magnetism, we calculate (i) the q -dependent Paulisusceptibility at ω = χ ( q ) = − X k αβ f ( ε k α ) − f (cid:16) ε k + q β (cid:17) ε k α − ε k + q β + i δ (1)where α and β are band indexes and q and k are momentum vec-tors in the Brillouin zone, and (ii) the k z dispersion of the FS.These calculations were performed with the full potential lin-earized augmented plane wave method as implemented in theWIEN2k code [36] with RK max =
7. While 40000 k points areused in calculating the k z dispersion of the FS, a three dimen-sional grid of 128 × × k and q points are employed forthe susceptibility. All calculations were performed in the scalarrelativistic approximation.In order to perform the DFT analysis, (i) we use the avail-able experimental structures for LaOFeAs [37], BaFe As [38]and (ii) we obtain fully optimized structures for BaFe − x Co x As within the virtual crystal approximation and for a few hy-pothetical compounds like LaOFeSb and BaFe Sb using theCar-Parrinello [39] projector-augmented wave [40] method.Our optimized structures compare well with the experimentalones [13, 41]. Part of our results are double-checked by the fullpotential local orbital (FPLO) method [42]. Results are consis-tent among these methods. Throughout the paper, the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) toDFT has been used.For our model calculations we consider the following two-band Hubbard model H = − X h i j i m σ t m c † im σ c jm σ − X h i j ′ i m σ t ′ m c † im σ c j ′ m σ + U X im n im ↑ n im ↓ + X i σσ ′ (cid:0) U ′ − δ σσ ′ J z (cid:1) n i σ n i σ ′ , (2)where t m ( t ′ m ) is the intra-band hopping integral betweennearest-neighbor (next nearest-neighbor) sites with band in-dices m = ,
2. For simplification, we neglect inter-bandhybridizations. U , U ′ and J z are the intra-band, inter-bandCoulomb interaction and Ising-type Hund’s coupling, respec-tively. In our calculations we set U ′ = U and J z = U which fulfills the rotational invariance condition of U = U ′ + J andignore the spin-flip and pair-hopping processes. The operatorsare written in the standard notation of the multi-band Hubbardmodel [35]. In order to solve this model we employ the two-sublattice DMFT method [33] which includes the local quan-tum fluctuation e ff ects and can account for the AF state, com-bined with CTQMC simulations [34]. Existing DMFT studiescombining with DFT calculations are focused on paramagneticstate [43, 44, 45, 46, 47]. Our calculations are performed on theBethe lattice.
3. Results and Discussion (b)
BaFe As BaFe Sb ( S , S , S )(0,0, S ) F ( q ) / F , A s ( q ) q (a) LaOFeAs LaOFeSb ( S , S , S )(0,0, S ) F ( q ) / F , A s ( q ) q Figure 1: (Color online) Comparison of normalized q -dependent Pauli suscep-tibilities calculated within GGA at fixed q z = π along the [110] direction be-tween arsenide and antimonide of (a) 1111 compounds and (b) 122 compounds.The normalization factors are the susceptibilities of the corresponding arsenidesystems for each type of compound at q = (0 , , π ). Please note that the peakposition is not exactly at q π = ( π, π, π ) since the electron and hole FSs are nearlynested rather than perfectly nested.Table 1: Magnetic moment calculated within spin-polarized GGA based onexperimental structures of LaOFeAs and BaFe As and optimized structuresof the hypothetical compounds LaOFeSb and BaFe Sb . LaOFeAs LaOFeSb BaFe As BaFe Sb m ( µ B ) 1.8 2.2 2.0 2.5In Fig. 1, we present the comparison of q -dependent Paulisusceptibilities between arsenide and antimonide 1111 and122 compounds normalized by the susceptibilities of the cor-responding arsenide systems for each type of compound at q = (0 , , π ). Here we only consider q z = π since in bothLaOFeAs and BaFe As the Fe spins are AF ordered along c asexperimentally observed and we show the results for q x = q y .We observe that in all these four compounds a strong peakaround q π = ( π, π, π ) is found, which supports the presence ofstripe-type magnetically ordered states as observed experimen-tally. Most importantly, in contrast to an earlier DFT study [17]where the magnetic moment increases as As is replaced by Sbin both LaOFeAs and BaFe As while the susceptibilities at q π decrease in BaFe Sb compared to BaFe As , which is inter-preted to be evidence of disconnection between FS nesting andmagnetism, our results show that in both compounds, the sus-ceptibilities at q π increase as the replacement takes place andsimultaneously the magnetic moment is also enhanced as dis-played in Table. 1. This discrepancy is attributed to the im-proved precision of optimized lattice structures in our DFT cal-2ulation [13]. Therefore, the close connection between itiner-ancy and magnetism remains. (a) LaOFeAs BaFe As ( S , S , S )( S , S ,0) F ( q ) / F ( q ' ) q (b) BaFe Co x As x=0 x=0.2 x=0.5 ( S , S , S )(0,0, S ) F ( q ) / F ( q ) q Figure 2: (Color online) (a) Normalized q -dependent Pauli susceptibilities ofLaOFeAs and BaFe As at fixed q x = q y = π along the [001] direction cal-culated within GGA. The normalization factors are the susceptibilities of eachcompound at q ′ = ( π, π, q -dependent Pauli susceptibilities of BaFe − x Co x As at fixed q z = π alongthe [110] direction calculated within GGA. Here x = , . , .
5. The normal-ization factors are the susceptibilities of each compound at q = (0 , , π ). In Fig. 2 (a), we show the q -dependent Pauli susceptibilitiesof LaOFeAs and BaFe As at fixed q x = π and q y = π , normal-ized by the susceptibilities of each compound at q ′ = ( π, π, q ′ to q π in LaOFeAs, indicating a dispersionless FS along c and therefore nearly perfect two dimensional physical prop-erties, a stronger enhancement is seen in BaFe As suggestinga three dimensional FS topology as shown in Fig. 3 (a), (b),even though it is a layered compound. Such a three dimension-ality of the FS has been proposed to be the mechanism for anearly isotropic critical field in (Ba,K)Fe As [48]. The com-mon feature in the susceptibilities for LaOFeAs and BaFe As is that the values at q π are larger than those at q ′ which canaccount for the AF arrangement of Fe spins along c , althoughthe interlayer interaction is believed to be small. Such a con-sistency again implies a close relation between itinerancy andmagnetism. Figure 3: (Color online) k z dispersion of FS calculated within GGA as a func-tion of Co-doping in BaFe − x Co x As around Γ at x = x = . x = . X at x = x = . x = . It has already been shown that changes in the FS topologycan explain the di ff erent nature of structural and magnetic phasetransitions in the 122 compounds under pressure [11, 49]. In thefollowing we will further demonstrate that the phase transitionsin BaFe As under Co-doping are also related to the changeof FS topology. It is known from experiments [50] that forBaFe − x Co x As at x =
0, the system shows a stripe-type AF metal. At x = . x = .
5, both magnetization and superconductivitydisappear. In Fig. 3 we show the k z dispersion of the FS forthese three cases around Γ (left panels) and X (right panels).We find that the FS along c becomes more dispersive with Codoping. At x = q π as shownin Fig. 2 (b) solid curve, supporting the AF state. At x = . q π instability (see Fig. 2 (b) dashed curve) and therefore of themagnetization. However, the superconducting state which, ac-cording to the spin fluctuation theory [51], is related to the in-stabilities around q π may in such a situation be more favourablethan magnetization. At x = .
5, the FS around Γ shrinks to aFermi pocket and is even more distorted around X , suggestingthat no obvious instability in the susceptibility will be present(see Fig. 2 (b) dotted curve) and leading to the disappearanceof both magnetic ordering and superconductivity.After having presented various evidence for the itinerant na-ture of magnetism in iron pnictides, we will investigate inwhat follows a possible mechanism for the reduced magneticmoment observed experimentally compared to DFT calcula-tions [11, 18, 19]. As mentioned in Section 1, we would liketo extract essential physics from a simplified model as intro-duced in Section 2. In this model, local quantum fluctuationsare included in the calculations by employing DMFT. Fig. 4shows the magnetization as a function of U at two tempera-tures. Combining these results with the analysis of density ofstates in Ref. [35], we conclude that when two bands are equallyhighly frustrated (see Fig. 4 (a)), a first-order phase transitionfrom a paramagnetic metal to an AF insulator state is observedand an AF metallic state is absent, while, if one band is unfrus-trated and the second one highly frustrated (see Fig. 4 (b)), sev-eral continuous phase transitions appear separately in these twobands, and an AF metal with small magnetic moment appears.This indicates that an AF metallic state with small magneticmoment emerges out of the coupling between highly frustratedand unfrustrated bands, which is the case in iron pnictides asmentioned in Section 1, rather than out of a pure frustratione ff ect. (b) m agne t i z a t i on / band U/t t band, T/t=1/16 t,t' band. t band, T/t=1/32 t,t' band.t'/t=0.65 (a) m agne t i z a t i on / band U/t
T/t=1/16 T/t=1/32t'/t=0.58
Figure 4: (Color online) Magnetization per band of a two-band Hubbard modelat two temperatures calculated by DMFT(CTQMC) as a function of interactionstrength U with (a) two bands equally highly frustrated and (b) one band highlyfrustrated, the other unfrustrated. . Conclusion In summary, we presented various evidence for the close re-lation between itinerancy and the magnetism observed experi-mentally. Our results support the itinerant nature of magnetismin iron pnictides and suggest that a strong instability at the nest-ing vector q π is responsible for the magnetism in iron pnictides.We also propose that the reason why the reduced magnetic mo-ment in the iron pnictides cannot be reproduced by LSDA orGGA calculations employing the experimental lattice structureis the insu ffi cient incorporation of quantum fluctuations in suchcalculations. By applying the DMFT approach to a simpli-fied two-orbital Hubbard model, we propose that coupling ofstrongly frustrated and unfrustrated bands may be the mecha-nism for the reduction of the magnetic moment, rather than apure frustration e ff ect.We gratefully acknowledge useful discussions and commentsfrom L. Nordstr¨om and the Deutsche Forschungsgemeinschaft(DFG) for financial support through SFB / TRR 49, SPP 1458,Emmy Noether programs and the Helmholtz Association forsupport through HA216 / EMMI.
References [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. , 3296 (2008).[2] Z. A. Ren, W. Lu, J. Yang, W. Yi, X. L. Shen, Z. C. Li, G. C. Che, X.L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao, Chin. Phys. Lett. , 2215(2008).[3] H. Luetkens, H.-H. Klauss, M. Kraken, F. J. Litterst, T. Dellmann, R.Klingeler, C. Hess, R. Khasanov, A. Amato, C. Baines, M. Kosmala, O.J. Schumann, M. Braden, J. Hamann-Borrero, N. Leps, A. Kondrat, G.Behr, J. Werner, B. B¨uchner, Nature Materials , 305 (2009).[4] S. A. J. Kimber, A. Kreyssig, Y.-Z. Zhang, H. O. Jeschke, R. Valent´ı,F. Yokaichiya, E. Colombier, J. Yan, T. C. Hansen, T. Chatterji, R. J.McQueeney, P. C. Canfield, A. I. Goldman and D. N. Argyriou, NatureMaterials , 471 (2009).[5] S. Medvedev, T.M. McQueen, I. Trojan, T. Palasyuk, M.I. Eremets, R.J.Cava, S. Naghavi, F. Casper, V. Ksenofontov, G. Wortmann, C. Felser,Nature Materials , 630 (2009).[6] L. Boeri, O. V. Dolgov, A. A. Golubov, Phys. Rev. Lett. , 026403(2008).[7] L. Boeri, M. Calandra, I. I. Mazin, O. V. Dolgov, F. Mauri,arXiv:1004.1943.[8] D. J. Singh and M.-H. Du, Phys. Rev. Lett. , 237003 (2008).[9] I. I. Mazin, D. J. Singh, M. D. Johannes and M. H. Du, Phys. Rev. Lett. , 057003 (2008).[10] V. Cvetkovic and Z. Tesanovic, Europhys. Lett. , 37002 (2009).[11] Y.-Z. Zhang, H. C. Kandpal, I. Opahle, H. O. Jeschke, and R. Valent´ı,Phys. Rev. B , 094530 (2009).[12] A. N. Yaresko, G.-Q. Liu, V. N. Antonov, O.K. Andersen, Phys. Rev. B , 144421 (2009).[13] Y.-Z. Zhang, I. Opahle, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B ,094505 (2010).[14] Q. Si and E. Abrahams, Phys. Rev. Lett. , 076401 (2008).[15] T. Yildirim, Phys. Rev. Lett. , 057010 (2008).[16] B. Schmidt, M. Siahatgar, P. Thalmeier, Phys. Rev. B , 165101 (2010).[17] C.-Y. Moon, S. Y. Park, and H. J. Choi, Phys. Rev. B , 054522 (2009).[18] I. Opahle, H. C. Kandpal, Y. Zhang, C. Gros, and R. Valent´ı, Phys. Rev.B , 024509 (2009).[19] I. I. Mazin, M. D. Johannes, L. Boeri, K. Koepernik, and D. J. Singh,Phys. Rev. B , 085104 (2008).[20] H. Nakamura, N. Hayashi, N. Nakai, M. Machida, arXiv:0806.4804.[21] J. Ferber, Y. Z. Zhang, H. O. Jeschke, R. Valent´ı, arXiv:1005.1374.[22] F. Cricchio, O. Grån¨as, L. Nordstr¨om, Phys. Rev. B , 140403(R) (2010). [23] The stabilized solution with small magnetic moment was found inGGA + U with a positive U and employing the double counting schemeof around mean field. This solution ceases to be favorable when the dou-ble counting scheme of fully localized limit is considered.[24] F. Ma, Z.-Y. Lu, T. Xiang, Phys. Rev. B , 224517 (2008).[25] M. J. Han, Q. Yin, W. E. Pickett, and S. Y. Savrasov, Phys. Rev. Lett. ,107003 (2009).[26] E. Kaneshita, T. Morinari, T. Tohyama, Phys. Rev. Lett. , 247202(2009).[27] E. Bascones, M. J. Calder´øn, and B. Valenzuela, Phys. Rev. Lett. ,227201 (2010).[28] P. M. R. Brydon, Maria Daghofer, and Carsten Timm,arXiv:1007.1949v1.[29] T. Miyake, K. Nakamura, R. Arita, M. Imada, J. Phys. Soc. Jpn. ,044705 (2010).[30] S. Graser, T. A. Maier, P. J. Hirschfeld, D. J. Scalapino, New J. Phys. ,025016 (2009).[31] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H.Aoki, Phys. Rev. Lett. , 087004 (2008).[32] N. Qureshi, Y. Drees, J. Werner, S. Wurmehl, C. Hess, R. Klingeler, B.B¨uchner, M. T. Fern´andez-D´ıaz, and M. Braden, arXiv:1002.4326.[33] A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, Rev. Mod. Phys. ,13 (1996).[34] A. N. Rubtsov, V. V. Savkin, A. I. Lichtenstein, Phys. Rev. B , 035122(2005).[35] H. Lee, Y.-Z. Zhang, H. O. Jeschke, and R. Valent´ı, Phys. Rev. B ,220506(R) (2010).[36] P. Blaha, K. Schwarz, G. Madsen, D. Kvaniscka, and J. Luitz, WIEN2K,An Augmented Plane Wave + Local Orbitals Program for CalculatingCrystal, edited by K. Schwarz (Techn. University,Vienna, Austria, 2001).[37] T. Nomura, S. W. Kim, Y. Kamihara, M. Hirano, P. V. Sushko, K. Kato,M. Takata, A. L. Shluger, H. Hosono, Supercond. Sci. Technol. ,125028 (2008).[38] Q. Huang, Y. Qiu, Wei Bao, M. A. Green, J. W. Lynn, Y. C. Gasparovic,T. Wu, G. Wu, and X. H. Chen, Phys. Rev. Lett. , 257003 (2008).[39] R. Car, M. Parrinello, Phys. Rev. Lett. , 2471 (1985).[40] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[41] S. Thirupathaiah, S. de Jong, R. Ovsyannikov, H. A. D¨urr, A. Varykhalov,R. Follath, Y. Huang, R. Huisman, M. S. Golden, Yu-Zhong Zhang, H. O.Jeschke, R. Valent´ı, A. Erb, A. Gloskovskii, and J. Fink, Phys. Rev. B ,104512 (2009).[42] K. Koepernik and H. Eschrig, Phys. Rev. B , 1743 (1999).http: // , 226402 (2008).[44] L. Craco, M. S. Laad, S. Leoni, H. Rosner, Phys. Rev. B , 134511(2008)[45] S. L. Skornyakov, A. V. Efremov, N. A. Skorikov, M. A. Korotin, Yu. A.Izyumov, V. I. Anisimov, A. V. Kozhevnikov, D. Vollhardt, Phys. Rev. B , 092501 (2009)[46] H. Ishida, A. Liebsch, Phys. Rev. B , 054513 (2010)[47] M. Aichhorn, L. Pourovskii, V. Vildosola, M. Ferrero, O. Parcollet, T.Miyake, A. Georges, S. Biermann, Phys. Rev. B , 085101 (2009).[48] H. Q. Yuan, J. Singleton, F. F. Balakirev, S. A. Baily, G. F. Chen, J. L.Luo, N. L. Wang, Nature , 565 (2009).[49] Y.-Z. Zhang, I. Opahle, H. O. Jeschke, and R. Valent´ı, J. Phys.: Con-densed Matter , 164208 (2010).[50] J.-H. Chu, J. G. Analytis, C. Kucharczyk, and I. R. Fisher, Phys. Rev. B , 014506 (2009).[51] T. Moriya, K. Ueda, Rep. Prog. Phys. , 1299 (2003)., 1299 (2003).