Electron-phonon interaction in Fe-based superconductors: Coupling of magnetic moments with phonons in LaFeAsO 1−x F x
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Electron-phonon interaction in Fe-based superconductors: Coupling of magneticmoments with phonons in LaFeAsO − x F x Felix Yndurain ∗ Departamento de F´ısica de la Materia Condensada and Instituto de Ciencia de Materiales”Nicol´as Cabrera”. Universidad Aut´onoma de Madrid. Cantoblanco. 28049 Madrid. Spain. (Dated: December 27, 2018)The coupling of Fe magnetic moments in LaFeAsO − x F x with the As A g phonon is calculated.We present first principles calculations of the atomic and electronic structure of LaFeAsO as a func-tion of electron doping. We perform calculations using the virtual crystal approximation as well assupercell calculations with F substitutional impurity atoms. The results validate the virtual crystalapproximation for the electronic structure near the Fermi level. Its is found that the electronicdensity of states at the Fermi level is maximum for x=0.125, enhancing the electron-phonon interac-tion. An additional increase of the electron-phonon parameter λ is obtained if the coupling betweenthe A g phonon and the Fe magnetic moment is included. It is found that the electron-phononinteraction can be one order of magnitude larger than its value if no spin resolution is included inthe calculation. The implications of these results on the superconducting transition are discussed PACS numbers: 71.20.-b, 71.38.-k, 74.20.-z, 74.70.-b
Since the discovery of the Fe-based superconductors afew years ago [1, 2], several properties common to mostof the compounds have been well established and generalagreement has been reached. However, there are veryimportant aspects that have not been settled [3]. Themost important one, is the mechanism responsible fortheir superconducting behavior. Early theoretical work[4, 5] ruled out any phonon’s role. More recent calcula-tions [6] reveal that, in the non-magnetic phase, phononscoupled with magnetism contribute to superconductivityalthough not enough to explain the high critical tem-perature. Another aspect to be clarified involves the Femagnetic moment. The low temperature stripped an-tiferromagnetic order in the parent compound is gener-ally accepted, although the magnitude of the calculatedmagnetic moment is much larger than the observed one[7]. Fluctuations mechanisms as well as orbital magneticordering [8, 9] have been proposed to explain this dis-crepancy. The coupling of the Fe magnetic momentswith phonons has been clearly established at least inCaFe As , in the undoped orthorombic phase [10], and inSmFeAsO − x F x [11] . Also the possibility of a non-zeroFe magnetic moment in the superconducting phase hasnot been ruled out [12].Here we propose that the magnetic moment in Featoms does not disappear in the superconducting phase.Instead, what disappears is the long-range collinear an-tiferromagnetic order while keeping the As-mediated an-tiferromagnetic local coupling between the Fe atoms. Tothis end, we have calculated in the manner describedin [13] how the energy barrier between two equivalentcollinear antiferromagnetic configurations varies with theextra electron concentration in LaFeAsO − x F x . We as-sume that a non-collinear configuration takes place be-tween the two minima as discussed in [13]. The resultsof the calculation are shown in Figure 1. The results indicate that the barrier between two equivalent antifer-romagnetic arrangement is very small beyond x=0.1 upto the disappearance of the magnetic moment aroundx=0.3. This low barrier can be responsible for the disap-pearance of the antiferromagnetic long-range order sincethe system may fluctuate between the two equivalentmagnetic arrangements. E non - c o ll - E an t i f e rr o ( m e V ) x LaFeAsO F x FIG. 1: (Color online) Energy barrier for the transition be-tween two equivalent antiferromagnetic stripped collinear ar-rangements (left inset) through an intermediate non collinearstate (right inset) in LaFeAsO − x F x as a function of x. Theenergy difference per formula unit is shown. To study how the electronic structure, and eventually,the electron-phonon interaction varies with doping we re-strict ourself to the antiferromagnetic collinear arrange-ment since the non-collinear one has a similar electronicstructure (see [13]) and its calculation is much more com-putational demanding. To obtain the electronic structurewe have performed density functional [14, 15] calculationsusing the SIESTA code [16, 17] which uses localized or-bitals as basis functions [18]. In our calculation we usea double ζ polarized basis set, non-local norm conserv-ing pseudopotentials and a local density approximation(LDA) for exchange and correlation. The calculations areperformed with stringent criteria in the electronic struc-ture convergence (down to 10 − in the density matrix),Brillouin zone sampling (up to 18000 k -points), real spacegrid (energy cut-off of 500 Ryd) and equilibrium geom-etry (residual forces lower than 10 − eV/˚A). Due to therapid variation of the density of states at the Fermi level,we used a polynomial smearing method [19]. To simulatethe effect of doping we use the virtual crystal approxima-tion (VCA) [20].The calculated total densities of states for various fluo-rine contents are drawn in Figure 2. We first observe theantiferromagnetic pseudo gap at the Fermi level for x=0.As the excess of electrons increases with the fluorine con-centration the size of the gap decreases and eventuallydisappears between x=0.25 and x=0.30. It is interestingto notice that the antiferromagnetic gap lies fully bellowthe Fermi level for x ≥ .
15. Moreover, the peak in thedensity of states at the antiferromagnetic gap upper edgecrosses the Fermi level between x=0.10 and x=0.15. -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6010203040506070 D en s i t y o f S t a t e s ( a r b . un i t s ) E-E F (eV) LaFeAsO F X X=0.30X=0.25X=0.20X=0.15X=0.10X=0.05
X=0.00
FIG. 2: (Color online) Variation of the total density of statesof LaFeAsO − x F x with the F content x. The calculation isfor the virtual crystal approximation. A small imaginary con-tribution has been added to the energy to smooth the curves. Before we go into the discussion of the electron-phononinteraction parameter we analyze to what extent the vir-tual crystal approximation is a good starting point tostudy the electronic structure of these materials. To thisend, we have calculated the equilibrium atomic structureand the electronic density of states for various super-cells size substituting an oxygen atom by a fluorine one.In this manner we have performed ”exact” calculationsfor x=0.25, 0.125 and 0.0625. The calculated densityof states after full relaxation of the atomic positions areshown in Figure 3 which shows an excellent agreementwith the VCA results. The VCA density of states repro-duces the main features of the super-cell calculations, inparticular, the peak in the density of sates crossing theFermi level at around x=0.125. From these results it isclear that the details of how impurities are incorporatedin the oxygen layer does not affect much the electronicstructure around the Fermi level which is mainly dom-inated by the Fe electrons. The main effect of dopingimpurities at the oxygen layer is to pump electrons intothe As-Fe-As layer, as anticipated [21] .
FIG. 3: Calculated LaFeAsO − x F x electronic densities ofstates using super-cells (”exact”) (panels (a), (b) and (c))and using the virtual crystal approximation (panels (d),(e)and (f)) for various values of x. A small imaginary contribu-tion has been added to the energy to smooth the curves. We have also calculated the influence of an F impurityon the electronic charges and magnetic moments of theFe atoms. In Table I we show the results for a supercellFe As La O F which corresponds to x=0.0625. Wenotice that the extra charge is distributed rather evenlythrough the Fe atoms and does not distinguish betweenthose closer to the impurity and the other Fe atoms. Onthe other hand, although the magnetic moment of the Featoms closer to the F impurity display a magnetic mo-ment larger than the others, the difference is rather smalland we can rule out any relevant polaron formation. Theantiferromagnetic coupling between successive Fe layeragrees with the experimental magnetic order of this ma-terial. The above results stress the appropriateness ofthe VCA approximation. Also, there is no significant in-fluence of the presence of F impurities in the geometryof the Fe-As layers next to it. We find that the Fe-Asdistance depends on the extra charge at the As-Fe-Aslayers, rather than on the details concerning the oxygenlayer and the substitutional impurities. TABLE I: Electronic charge and magnetic moment (in Bohrmagnetons) at the Fe atoms of Fe As La O F in thesuper-cell calculation. The corresponding VCA results areindicated. Fe Layer Q Q Q Q ∗ ∗ µ µ µ µ ∗ ∗ V CA : 8.301; µ V CA : 1.247* Fe atoms closest to the F impurity
To address the electron-phonon interaction in thissystem and the superconducting λ parameter, we haveconsidered the symmetric out-of-plane As A g mode at k || = 0, in which the Fe atoms remain fixed whereasthe As atoms move perpendicularly to the FeAs layers,expanding and compressing the Fe-As bonds. For thisphonon mode the diagonal electron-phonon matrix ele-ment can be written [22]: < −→ k , n | H e − ph ( ν ) | −→ k , n > = ( ~ / M a ω ν ) / ǫ aν ·−→ D ( −→ k , n )where ǫ aν is the polarization vector, ω ν is the frequencyof the vibrational mode involved and −→ D ( −→ k , n ) is the de-formation potential for the state ( −→ k , n )The λ parameter associated to a vibrational mode canbe written as a sum of the contributions of the electronicbands crossing the Fermi level in the form: λ A g = M As ω A g P n N n ( E F ) D n , -0.5-0.4-0.3-0.2-0.10.00.10.20.30.4-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 E ne r g y r e l a t i v e t o E F ( e V ) (cid:17)(cid:19) (cid:22) (cid:22)(cid:21)(cid:30)(cid:26)(cid:28)(cid:1) (cid:18) Γ (cid:15) (cid:1) (cid:2)(cid:20)(cid:3) (cid:31)(cid:11)(cid:1) (cid:7)(cid:6)(cid:7)(cid:7)(cid:1) !(cid:31)(cid:11)(cid:4)(cid:7)(cid:6)(cid:7)(cid:9)(cid:1) !(cid:1) (cid:31)(cid:11)(cid:5)(cid:7)(cid:6)(cid:7)(cid:9)(cid:1) ! (cid:14)(cid:19)(cid:13)(cid:22)(cid:12)(cid:29)(cid:16)(cid:7)(cid:6)(cid:10)(cid:13)(cid:7)(cid:6)(cid:8)(cid:1) (cid:1) (cid:1) (cid:1) (cid:12)(cid:29)(cid:1) (cid:12)(cid:8)(cid:23)(cid:1) (cid:27)(cid:24)(cid:26)(cid:25)(cid:26)(cid:25) (cid:2)(cid:19)(cid:3) (cid:1) FIG. 4: (Color online) Band structure with frozen A g phonon(vibration of As atoms perpendicularly to the Fe plane) fora doping of 0.10 electrons per formula unit. The results arefor the virtual crystal approximation. The dashed bands cor-respond to zero phonon amplitude and the red (blue) bandscorrespond to an expansion (compression) of the Fa-As planesdistance u of 0.02 ˚A. (a) Spin independent calculation. (b)Stripped antiferromagnetic magnetic order. where N n ( E F ) is the nth-band density of states at theFermi level, ω A g the phonon frequency, M As the mass ofthe arsenic atoms and D n is the variation of the energyband E n at the Fermi energy the phonon with the phononamplitude: D n = √ dE n ( E F ) du Before we calculate the λ parameter it is worth look-ing at the deformation potential associated to the A g phonon. Although a supercell approximation could bemore precise [23], the calculations are performed withinthe VCA. In Figure 4 we show, for the fluorine concen-tration of 10%, how the electronic band structure is per-turbed by the presence of the phonon. In the case of anon magnetic calculation (Figure 4 (a)) the effect of thephonon in the bands at the Fermi level is small in all thebands but one ( d xy ). On the contrary, in the case of thecalculation for the antiferromagnetic configuration (Fig-ure 4 (b)), all the bands are perturbed by the presenceof the phonon, and, in addition, there is a flattening ofthe bands due to the antiferromagnetic pseudo-gap. Theeffect of the A g phonon is to modulate the size of theFe magnetic moment with the vibration amplitude andtherefore inducing an additional shift to the bands. Asimilar behavior was previously discussed [13]. FIG. 5: (Color on line) Variations of the density of statesat the Fermi level and parameter λ of LaFeAsO − x F x withthe fluorine concentration x. (a) Density of states for theantiferromagnetic collinear order (black squares) and non-magnetic (red circles) calculations. (b) Calculated parame-ter λ for the antiferromagnetic collinear order (black squares)and non-magnetic (red circles) . We have calculated the λ parameter associated to thisphonon in the approximation discussed above and withinthe VCA. The results, along with the variation of the den-sity of states at the Fermi level, are shown in Figure 5,where a phonon energy of 25 meV is considered. We ob-serve that the antiferromagnetic pseudo-gap induces anincrease of the density of states at the Fermi energy withrespect to the non-magnetic calculation (see Figure 2).This increase would enhance the λ parameter by a factorof two at the most. However, if the deformation potentialis included, the λ parameter can be one order of magni-tude larger than its value for the non-magnetic calcula-tion (Figure 5 (b)). In addition, if we consider the renor-malization of the parameter λ due to the renormalizationof the phonon mode ω = ω λ then λ = λ − λ which forthe maximum value at x=0.125 becomes λ = 0 .
16. Thisis indeed a substantial increase with respect to the non- magnetic calculation but not large enough to explain byitself the superconducting critical temperature. All thevibrational modes should be included in the calculationto obtain the total λ parameter. The connection betweenthe Fe-As bond length and the magnetic moments in Featoms is important (as discussed by Yildirim [24]) and isrelevant in the electron-phonon interaction.In summary, we propose that in the superconduct-ing phase of LaFeAsO − x F x , the Fe atoms have a finitemagnetic moment that fluctuates between two equiva-lent collinear antiferromagnetic configurations. The pres-ence of this moment varies substantially the density ofstates near the Fermi level and enhances dramaticallythe electron-phonon interaction at least fort the As A g phonon mode. The maximum of the calculated λ param-eter takes place at x=0.125. These results suggest thatelectron-phonon interaction, coupled with the Fe mag-netic moments, must be carefully revised, before rulingout its connection with Fe-based compounds supercon-ductivity.I am indebted to J.M. Soler, for very enlightening dis-cussions and for a critical reading of the manuscript.I also thank E. Bascones, M. J. Calderon, M.L.Cohen,G. Gomez-Santos, S.G. Louie, D. Sanchez-Portal and B.Valenzuela for very helpful comments about this work.This work was supported by the Spanish Ministry of Sci-ence and Innovation through grants FIS2009-12712 andCSD2007-00050. ∗ Electronic address: [email protected].[1] Y. Kamihara, H. Hirmatsu, M. Hirano, R. Kawamura,H. Yanagi, T. Kamiya, and H. Hosono, J. Am. Chem.Soc. , 10012 (2006).[2] Y. Kamihara, T. Watanable, M. Hirano, and H. Hosono,J. Am. Chem. Soc. , 3296 (2008).[3] M. R. Norman, Physics , 21 (2008).[4] I. I.Mazin, M. D.Johannes, L. Boeri, K. Koepernik, andD. J. Singh, Phys. Rev. B , 085104 (2008).[5] L. Boeri, O. V. Dolgov, and A. A. Golubov, Phys. Rev.Lett. , 026403 (2008).[6] L. Boeri, M. Calandra, I. I.Mazin, O. V.Dolgov, andF. Mauri, Phys. Rev. B , 020506 (2010).[7] M. A. Korotin, S. V. Streltsov, A. O. Shorikov, andV. I. Anisimov, Journal of Experimental and Theoret-ical Physics , 649 (2008).[8] E. Bascones, M. J. Calderon, and B. Valenzuela, Phys.Rev. Lett. , 227201 (2010).[9] C.-C. Lee, W.-G. Yin, and W. Ku, Phys. Rev. Lett. ,267001 (2009).[10] S. Hahn, Y. Lee, N. Ni, P. Canfield, A. I. Goldman,R. McQueeney, B. Harmon, A. A, B.M-Leu, E. Alp, et al.,Phys. Rev. B , 220511(R) (2009).[11] M. Le Tacon, T. Forrest, C. Ruegg, A. Bosak, A. Wal-ters, R. Mittal, H. M. Ronnow, N. Zhigadlo, S. Katrych,J. Karpinski, et al., Phys. Rev. B , 220504(R) (2009).[12] D. Reznik, K. Lokshin, D. C. Mitchell, D. Parshall, W. Dmowski, D. Lamago, R. Heid, K.-P. Bohnen, A. S.Sefat, M. A. McGuire, et al., Phys. Rev. B , 214534(2009).[13] F. Yndurain and J. M. Soler, Phy. Rev. B , 134506(2009).[14] P. Hohenberg and W. Kohn, Phys. Rev. , B864(1964).[15] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965).[16] P. Ordejon, E. Artacho, and J. M. Soler, Phys. Rev. B , R10441 (1996).[17] J. Soler, E. Artacho, J. Gale, A. Garcia, J. Junquera,P. Ordejon, and D. Sanchez-Portal, J. Phys.: Condens.Matter , 2745 (2002). [18] O. F. Sankey and D. J. Niklewski, Phys. Rev. B , 3979(1989).[19] M. Methfessel and A. T. Paxton, Phys. Rev. B , 3616(1989).[20] L. Bellaiche and D. Vanderbilt, Phys. Rev. B , 7877(2000).[21] H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hi-rano, and H. Hosono, Nature , 376 (2008).[22] F. S. Khan and P. B. Allen, Phys. Rev. B , 3341 (1984).[23] J. Noffsinger, F. Giustino, S. G. Louie, and M. L. Cohen,Phys. Rev. Lett. , 147003 (2009).[24] T. Yildirim, Phys. Rev. Lett.102