Impact of electronic correlations on the equation of state and transport in ε -Fe
L. V. Pourovskii, J. Mravlje, M. Ferrero, O. Parcollet, I. A. Abrikosov
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Impact of electronic correlations on the equation of state and transport in ǫ -Fe L. V. Pourovskii,
1, 2
J. Mravlje, M. Ferrero, O. Parcollet, and I. A. Abrikosov Centre de Physique Th´eorique, CNRS, ´Ecole Polytechnique, 91128 Palaiseau, France Swedish e-science Research Centre (SeRC), Department of Physics,Chemistry and Biology (IFM), Link¨oping University, Link¨oping, Sweden Josef Stefan Institute, SI-1000, Ljubljana, Slovenija Institut de Physique Th´eorique (IPhT), CEA, CNRS, URA 2306, F-91191 Gif-sur-Yvette, France Department of Physics, Chemistry and Biology (IFM), Link¨oping University, Link¨oping, Sweden
We have obtained the equilibrium volumes, bulk moduli, equations of state of the ferromagneticcubic α and paramagnetic hexagonal ǫ phases of iron in close agreement with experiment using an ab initio dynamical mean-field theory approach. The local dynamical correlations are shown to becrucial for a successful description of the ground-state properties of paramagnetic ǫ -Fe. Moreover,they enhance the effective mass of the quasiparticles and reduce their lifetimes across the α → ǫ transition leading to a step-wise increase of the resistivity, as observed in experiment. The calculatedmagnitude of the jump is significantly underestimated, which points to non-local correlations. Theimplications of our results for the superconductivity and non-Fermi-liquid behavior of ǫ -Fe arediscussed. PACS numbers:
Understanding properly pressurized iron is importantfor the geophysics of the inner Earth core [1] as well asfor technological applications of this metal. In the range12-16 GPa [2, 3] a martensitic transition from the body-centered-cubic (bcc) ferromagnetic phase ( α -Fe) to thehexagonal close-packed (hcp) phase ( ǫ -Fe) takes place.This ǫ -Fe phase, discovered in 1956, [4] exhibits surpris-ing magnetic and electronic properties, including super-conductivity in the range of pressures from 13 to 31 GPawith a maximum transition temperature of about 2 K[5]as well as a non-Fermi-liquid normal state observed inthe same pressure range [6].Perhaps the most puzzling aspect of ǫ − Fe is its mag-netism, or rather an unexpected absence of it. Indeed,density-functional-theory (DFT) ab initio calculationswithin the generalized-gradient approximation (GGA)predict an antiferromagnetic ground state for the ǫ phase,with either collinear [7–9] or non-collinear [10] order,which has up to date eluded experimental detection.While an anomalous Raman splitting observed in ǫ -Fe[11] has been related to a possible antiferromagnetic or-der [12], no magnetic splitting has been detected in thisphase by M¨ossbauer spectroscopy down to temperaturesof several Kelvins [13, 14]. A collapse of the orderedmagnetism across the α − ǫ transition has also been ob-served in the x-ray magnetic circular dichroism and in thex-ray absorption spectroscopy measurements [2]. A spin-fluctuation paring mechanism that has been proposed forthe superconducting phase [15] seems as well incompat-ible with the large-moment antiferromagnetism. If thenon-magnetic ground state is imposed, the DFT-GGAtotal energy calculations predict an equation of state thatdrastically disagrees with experiment. The bulk modulusis overestimated by more than 50% and the equilibriumvolume is underestimated by 10% compared to the exper-imental values [7]. Therefore, the ground state properties of the observed non-magnetic ǫ -Fe remain theoreticallyunexplained.Another puzzling experimental observation is a largeenhancement in the resistivity across the α - ǫ transition,with the room temperature total resistivity of ǫ -Fe beingtwice as large as that of the α phase [16]. The electron-phonon-scattering contribution to resistivity calculatedwithin GGA is in excellent agreement with the exper-imental total resistivity for the α phase [17], however,these calculations predict virtually no change in the re-sistivity across the transition to antiferromagnetic hcp-Fe. The enhancement of resistivity in ǫ -Fe seems likelyto be caused by the ferromagnetic spin-fluctuations asthe resistivity ρ follows ρ ∝ T / [6, 18]. This again is atodds with the antiferromagnetism suggested by the GGAcalculations.All this points to the possible importance of the elec-tronic correlations that are not correctly incorporatedin the local or semi-local DFT. In this paper, we showthat including local dynamical many-body effects sig-nificantly improves the description of iron. Within alocal-density-approximation+dynamical mean-field the-ory (LDA+DMFT) framework we obtain the groundstate properties and the equation of states (EOS) of bothferromagnetic α and paramagnetic ǫ phases of iron as wellas the α − ǫ transition pressure and volume change in goodagreement with experiment. The strength of the elec-tronic correlations is significantly enhanced at the α → ǫ transition. This leads to a reduced binding, which ex-plains the relatively low value of the measured bulk mod-ulus in ǫ − Fe. The calculated resistivity has a jump atthe transition but the magnitude of the jump is severelyunderestimated compared with the experimental value,which points to additional scattering not present in ourlocal approach.Previously, the local correlations have been found toimprove the description of the high-temperature param-agnetic α -Fe [19, 20]. Moreover, the recently discoveredelectronic topological transition in ǫ -Fe has been success-fully explained by LDA+DMFT but not captured withinLDA and GGA [21]. However, no attempts to studyground state and transport properties of ǫ -Fe within anLDA+DMFT approach has been reported up to date.We have employed a fully self-consistent method[22, 23] combining a full-potential band structure tech-nique [24] and LDA for the exchange/correlations withthe dynamical mean-field theory (DMFT) [25] treat-ment of the onsite Coulomb repulsion between Fe 3 d states. The Wannier orbitals representing Fe 3 d stateswere constructed from the Kohn-Sham states within theenergy range from -6.8 to 5.5 eV. The DMFT quan-tum impurity problem was solved with the numericallyexact hybridization-expansion continuous-time quantumMonte-Carlo (CT-QMC) method [26] using an implemen-tation based on the TRIQS libraries [27] package. Thelocal Coulomb interaction in the spherically-symmetricform was parametrized by the Slater integral F = U =4 . J = 1 . α -Fe of 2.2 µ B at its experimental volume. These values aresomewhat larger than F = U = 3 . J = 0 . α -Fe. The 30% increasewith respect to the static cRPA values accounts for thefrequency dependence of the Coulomb vertex [29]. Weused the around-mean-field form of the double countingcorrection [30] in this work. We verified that the lowest-energy collinear anti-ferromagnetic order ”AFM-II” [7]collapses at the largest experimental volume of ǫ -Fe inLDA+DMFT-CTQMC calculations with a rotationally-invariant local Coulomb repulsion [31]. However, non-density-density terms in the Coulomb vertex increasedramatically the computational cost of CT-QMC andpreclude reaching the high accuracy required to extractan equation of state. Hence, we adopted the paramag-netic phase for ǫ -Fe and employed the density-density ap-proximation for the local Coulomb interaction through-out. This allowed us to use the fast ”segment picture”algorithm of the CT-QMC [26] reaching an accuracy ofabout 0.1 meV/atom in the total energy computed in ac-cordance with Ref. [23]. All our calculations were donefor a relatively low temperature T = 290 K. Thus thephonon and entropic contributions were neglected in thephase stability calculations. The conductivity in DMFTis ρ − = 2 πe ~ R dω P k − ( ∂f /∂ω ) v k A k ( ω ) v k A k ( ω ) withimplicit summation over band indices [32]. We calcu-lated the band velocities v k using the Wien2k opticspackage[24, 33] and we constructed the spectral functions A k ( ω ) from the highly precise DMFT self-energies (com-puted using 10 CT-QMC moves) which we analyticallycontinued to the real axis using Pad´e approximants.The obtained LDA+DMFT total energies vs. volume
TABLE I: Equilibrium atomic volume V (a.u. /atom) andbulk modulus B (GPa) of bcc and hcp Fe computed by dif-ferent ab initio approaches. The F M , P M , and NM sub-scripts indicate ferromagnetic, paramagnetic, non-magneticstate, respectively, AF M − II is the lowest energy collinearmagnetic structure of ǫ -Fe in accordance with GGA calcula-tions of Ref. [7]. . bcc LDA+DMFT
F M
GGA
F M expt.
F M
V 78.4 76.5 a , 77.2 b , 77.9 c a , 174 b , 186 c hcp LDA+DMFT PM GGA NM GGA
AF M − II expt.V 73.4 68.9 a , 69 d d e B 191 288 a , 292 d d e LDA+DMFT values are from this work. GGA and exp.values are from a). this work b). [34], c). [9], d). [7], e). [35] in the α and ǫ phases are plotted in Fig 1a. Our cal-culations predict the ferromagnetic bcc α phase to bethe ground state. The transition to a paramagnetic ǫ -Fe(the common tangent shown in Fig 1a) is predicted tooccur at a pressure P c of 10 GPa. The paramagnetic α phase is about 10 mRy or 1500 K higher in energy, ingood correspondence to the Curie temperature of α -Fe.In Table I we list the resulting LDA+DMFT equilibriumatomic volumes and bulk moduli obtained by the fittingof calculated energy-volume data with Birch-MurnaghanEOS [36]. Also shown are GGA lattice parameters andbulk moduli obtained by us and in previous works, as wellas corresponding experimental values. The LDA+DMFTdramatically improves agreement with the experiment forparamagnetic ǫ -Fe for both the volume and bulk modu-lus, thus correcting the large overbinding error of GGA.The paramagnetic LDA+DMFT results are still closerto experimental values than the AFM GGA ones. Hence,even by adopting a magnetic state, which is not observedin experiment, one can only partually account for the in-fluence of electronic correlations within the DFT-GGAframework. For α -Fe we obtain a relatively small correc-tion to GGA, which already reproduces the experimentalvalues quite well.In Fig 1b we compare the LDA+DMFT and GGAEOS with the one measured from Ref. [37]. Again,for both phases the LDA+DMFT approach successfullycorrects the GGA overbinding error, which is relativelysmall in α -Fe and very significant in ǫ -Fe. Conse-quently, LDA+DMFT also reproduces correctly the vol-ume change at the α − ǫ transition (about 5%), which isgrossly overestimated in GGA. The c/a ratio in the hcp ǫ phase is also affected by the electronic correlations. Asshown in Fig 1c, the GGA calculations predict a reduc-tion of the c/a ratio with increasing volume, from 1.59 atV=58 a.u. /atom to 1.58 at V=70 a.u. /atom. WithinLDA+DMFT the c/a ratio remains almost constant andis close to the value of 1.60, in good agreement with ex- FIG. 1: (Color online). a). LDA+DMFT total energy vs. volume per atom for bcc (ferromagnetic, solid blue line, andparamagnetic, dot-dashed black line) and hcp (dashed red line) Fe. The error bars are the CT-QMC method stochastic error.The orange long dash-dotted straight line indicates the common tangent construction for the α − ǫ transition. b). Equations ofstates (EOS) for ferromagnetic bcc (low pressure) and paramagnetic hcp (high pressure) Fe. Theoretical results are obtainedby fitting the LDA+DMFT (thick line) and GGA (thin line) total energies, respectively, using the Birch-Murnaghan EOS [36].The experimental EOS of iron shown by green filled squares is from Dewaelle et al., Ref. [37]. c). The ratio of lattice parametersc/a of ǫ -Fe vs. volume per atom obtained in LDA+DMFT (blue circles, dashed line) and GGA (red squares, solid line). Theexperimental data are from Dewaelle et al. [37] (diamonds) and Glazyrin et al. [21] (pink circles).FIG. 2: (Color online) The ratio of the average inverse quasi-particle lifetime h Γ i to temperature (the left axis) and theaverage mass enhancement h m ∗ i /m (the right axis) vs. vol-ume per atom. The solid lines (filled symbols) and dashedlines (hatched symbols) are h Γ i /T and h m ∗ i /m , respectively.The values for bcc and hcp phases are shown by blue squaresand red circles, respectively. The black stars indicate the bccand hcp atomic volumes at the transition point, respectively. perimental measurements [47].One may see that the ground-state properties (bulkmodulus, equilibrium volume, etc.) of the ǫ phaseare significantly modified within the LDA+DMFT ap-proach as compared to GGA. In contrast, for ferro-magnetic α -Fe those modifications are much weaker.In order to understand the origin of this differencewe have evaluated the strength of the correlation ef- fects in both phases from the low-frequency behav-ior of the local DMFT self-energy Σ( iω ) on the Mat-subara grid. Namely, we computed the average massenhancement h m ∗ i /m as P s m ∗ s N s ( E F ) / P s N s ( E F ),where the s index designates combined spin and or-bital quantum numbers { σ, m } , Σ s ( iω ) and N s ( E F ) arethe imaginary-frequency DMFT self-energy and partialdensity of states at the Fermi level for orbital s , re-spectively, m ∗ s = 1 − [ d ℑ Σ s ( iω ) /dω | ω → ] is the corre-sponding orbitally-resolved mass enhancement. We havealso evaluated the average inverse quasiparticle lifetime h Γ i = − m h m ∗ i P s N s ( E F ) ∗ℑ Σ s ( ω =0) P s N s ( E F ) . The resulting averagemass enhancement and inverse quasiparticle lifetime areplotted in Fig. 2. In ferromagnetic α -Fe both quantitiesslowly decay with decreasing volume and then they ex-hibit a large enhancement across the α − ǫ transition,indicating a more correlated nature of ǫ -Fe. The latter ischaracterized by heavier quasiparticles, a larger electron-electron scattering and a stronger volume dependence ofthe correlation strength as compared to the bcc phase.This analysis clearly demonstrates that dynamical many-body effects are enhanced in the ǫ phase.Why are the electronic correlations in ǫ -Fe strongerand why does the DFT fail there? The crucial differenceis the magnetism. In α -Fe, the physics is governed by thelarge static exchange splitting which easily polarizes theparamagnetic state characterized by a peak in the densityof states (DOS) close to the Fermi energy. Therefore, thespin-polarized DFT-GGA calculations, which are ableto capture this static exchange splitting, reproduce theground state properties of ferromagnetic α -Fe rather well.In antiferromagnetic ǫ -Fe obtained within DFT-GGA the Pressure (GPa) ρ ( µ Ω * c m ) ρ bcc theory ρ hcp theory ρ exp /10 bcc hcp FIG. 3: (Color online) The electron-electron contribution tothe resistivity in bcc α -Fe (black circles, ) and hcp ǫ -Fe (bluesquares) computed within LDA+DMFT for T=294 K. Theexperimental room-temperature resistivity [16] (red triangles,dashed line) is shown divided by 10 . The vertical dot-dashedline indicate the theoretical transition pressure. static exchange splitting also reduces the bonding leadingto an improuved agreement with the experimental equa-tion of state. In contrast, dynamical many-body effectsmust be included to describe the experimental paramag-netic state of ǫ -Fe properly. In this respect we note thatthe many-body corrections to the total energy and spec-tral properties were shown to be important [20, 38, 39]for the high-temperature nonmagnetic state of α -Fe aswell. If one suppresses magnetism, α -Fe is actually evenmore correlated than ǫ − F e , which is a consequence of itslarge DOS close to the Fermi energy[40]. This larger DOSimplies slower quasiparticles which are influenced by theinteractions, especially the Hund’s rule coupling [41].We now turn to transport. The drop in the quasi-particle lifetime across the α − ǫ transition affects theelectron-electron-scattering (EES) contribution to the re-sistivity ρ el . − el . . We have calculated the evolution of theroom-temperature ρ el . − el . versus pressure in both phases,as displayed in Fig. 3. One may see that the behavior of ρ el . − el . under pressure reflects that of the inverse quasi-particle lifetime Γ. The resistivity decays with pressureexcept at the transition point where a step-wise increaseis found. A rapid enhancement of ρ el . − el . in the ǫ phase atpressures below P c =10 GPa is in agreement with very re-cent measurements [42], in which a large hysteresis in the α − ǫ transition was obtained with the transition shiftedto 7 GPa at the depressurization, and a very similar rapidincrease in the total resistivity upon the decrease of pres-sure was observed in ǫ -Fe for pressures below the usualexperimental P c of 12-15 GPa.Now we discuss a very interesting point: despite thegood qualitative agreement, the magnitude of the resis- tivity enhancement through the transition to ǫ − Fe in ourcalculations underestimates the values found in the ex-periment [16], by a factor of 10, see Fig. 3 [48]. The cal-culations of the electron-phonon-scattering (EPS) con-tribution to resistivity in Ref. [17] reproduce the resis-tivity of the bcc-Fe well but exhibit almost no changeacross the transition. Therefore, the corresponding jumpof the measured total resistivity has to be attributed toelectron-electron scattering. If the experimental issues,such as the sample thinning can be excluded, then themissing scattering that we find in comparison with ex-periments has to be associated with non-local long-rangeeffects, which are not dealt with in our calculations. In-terestingly, except for the magnitude, the pressure depen-dence of the resistivity is accounted well by our results.This might be understood by recognizing that the localcorrelations that suppress the coherence scale (the ki-netic energy) also make the electronic degrees of freedommore prone to the effects of the coupling to the long-rangespin-fluctuations.It is interesting to compare ǫ -Fe and Sr RuO , awidely investigated unconventional superconductor withsimilar transition temperature [43], to make a furtherlink with spin fluctuations. Both materials display low-temperature unconventional superconductivity [43] andseveral mechanisms have been discussed to be at its ori-gin [44]. In Sr RuO superconductivity emerges froma well-established Fermi liquid with T FL = 25K. Lo-cal approaches, like the one used in this work, yieldmuch shorter lifetimes [45] and a resistivity which agreeswith experiments at low temperatures within 30%. [46]The picture is different for ǫ -Fe, which displays a non-Fermi-liquid T / temperature dependence of its low-temperature resistivity, [18] extending up to a temper-ature T ∗ which reaches the peak T ∗ max ≈
35K at a pres-sure where superconductivity reaches its maximum [42].Spin fluctuations, which are believed to be responsiblefor this behavior [15] have, hence, a very strong effect in ǫ -Fe. Because their non-local nature cannot be capturedwithin our framework, we believe that they are at theorigin of the discrepancy between the experimental andour calculated resistivities.In summary, including local correlations crucially im-proves the theoretical picture of ǫ − Fe by correctly ac-counting for a set of experimental observations withinthe paramagnetic state. This solves the long-standingcontroversy between theory and experiment for this ma-terial. The successful description of Fe within the para-magnetic state, together with an underestimation of theresistivity found in our local approach, hints at the im-portance of spin fluctuations, which supports scenariosrelating them to the origin of superconductivity.Acknowledgment: We are grateful to A. Georges andD. Jaccard for useful discussions. The input of X. Dengin the development of the transport code is gratefullyacknowledged. L. P. acknowledges the travel supportprovided by PHD DALEN Project 26228RM. J. M. ac-knowledges the support of College de France where apart of this work was done and the Slovenian researchagency program P1-0044. O. P. acknowledges supportby ERC under grant 278472 - MottMetals. The com-putations were performed on resources provided by theSwedish National Infrastructure for Computing (SNIC)at the National Supercomputer Centre(NSC) and PDCCenter for High Performance Computing. [1] S. Tateno, K. Hirose, Y. Ohishi, and Y. Tatsumi, Science , 359 (2010).[2] O. Mathon, F. Baudelet, J. P. Iti´e, A. Polian,M. d’Astuto, J. C. Chervin, and S. Pascarelli, Phys. Rev.Lett. , 255503 (2004).[3] A. Monza, A. Meffre, F. Baudelet, J.-P. Rueff,M. d’Astuto, P. Munsch, S. Huotari, S. Lachaize,B. Chaudret, and A. Shukla, Phys. Rev. Lett. ,247201 (2011).[4] D. Bancroft, E. L. 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