A first-principles study of the electronic structure of Iron-Selenium; Implications for electron-phonon superconductivity
Alexander P. Koufos, Dimitrios A. Papaconstantopoulos, Michael J. Mehl
AA first-principles study of the electronic structure of Iron-Selenium; Implications forelectron-phonon superconductivity
Alexander P. Koufos and Dimitrios A. Papaconstantopoulos
School of Physics, Astronomy, and Computational Sciences,George Mason University, Fairfax, Virginia, 22030, USA
Michael J. Mehl
Center of Computational Material Sciences, Naval Research Laboratory, Washington, D.C., USA (Dated: January 10, 2014)We have performed density functional theory (DFT) calculations using the linearized augmentedplane wave method (LAPW) with the local density approximation (LDA) functional to study theelectronic structure of the iron-based superconductor Iron-Selenium (FeSe). In our study, we haveperformed a comprehensive set of calculations involving structural, atomic, and spin configura-tions. All calculations were executed using the tetragonal lead-oxide or P4/nmm structure, withvarious volumes, c/a ratios and internal parameters. Furthermore, we investigated the spin po-larization using the LDA functional to assess ferromagnetism in this material. The paramagneticLDA calculations find the equilibrium configuration of FeSe in the P4/nmm structure to have avolume of 472.5au with a c/a ratio of 1.50 and internal parameter of 0.255, with the ferromagnetichaving comparable results to the paramagnetic case. In addition, we calculated total energies forFeSe using a pseudopotential method, and found comparable results to the LAPW calculations.Superconductivity calculations were done using the Gaspari-Gyorffy and the McMillan formalismand found substantial electron-phonon coupling. Under pressure, our calculations show that thesuperconductivity critical temperature continues to rise, but underestimates the measured values. PACS numbers: 71.15.Mb, 71.20.-b, 74., 74.20.Pq, 74.70.XaKeywords: Iron-Selenium; Electronic Structure; Superconductivity; Density functional Theory; Iron-basedSuperconductors
INTRODUCTION
Iron-based superconductors are the newest additionto high-temperature superconductivity. Current ex-perimental findings have made many believe that thesuperconductivity may not be due to electron-phononinteraction . Spin-fluctuations and spin-density waveshave been suggested as mechanisms for the high-temperature superconductivity, but without quantitativeassessment. It is therefore important to fully study theelectronic structure of these materials, and its implica-tions on superconductivity.Iron-selenium (FeSe) has the simplest structure of thecurrent iron-based superconductors. As shown in Fig. 1,under ambient conditions it forms in the tetrago-nal PbO structure,
Strukturbericht
B10, , space groupP4/nmm-D d ( / z ).There have been several studies of iron-selenium bothcomputationally and experimentally . Most of thecomputational works use the experimental equilibriumresults as input without optimization of all parametersthrough first-principles. In this work we have performedcalculations using the experimental parameters as wellas calculations based on first-principles energy minimiza-tion. FIG. 1. (Color online) Ground state structure of FeSe,
Struk-turbereicht designation B10. The space group is P4/nmm-D d ( I. COMPUTATIONAL DETAILS
Most calculations performed in this paper used theLinearized Augmented Plane Wave (LAPW) , im-plementation of Density Functional Theory (DFT) . a r X i v : . [ c ond - m a t . s up r- c on ] J a n . . . . E ne r g y ( m R y ) Volume (a.u. )FeSe LDA, tetra-PbO V exp = 529.5au V LAPW = 472.5au V VASP = 465.7au B exp = 31.0GPaB LAPW = 32.2GPaB
VASP = 34.8GPa *c/a and internal parameter varied to optimize energy minimization
LAPW VASP
FIG. 2. (Color online) Total energy of FeSe using the LAPW with LDA functional and VASP code with LDA functional. Bothmethods performed energy minimization to obtain the optimal energy for each volume. Although the equilibrium volume isunderestimated, the bulk modulus value is in very good agreement with experiment.
LAPW wave functions were used for the valence band,further augmented by local orbitals for the semi-corestates, using a code developed by Krakauer, Wei, andSingh . Exchange and correlation effects were ap-proximated by the Hedin-Lundqvist parametrization ofthe Local Density Approximation (LDA) . The rigidmuffin-tin approximation (RMTA) code developed byPapaconstantopoulos and Klein was used to apply theGaspari-Gyorffy theory . A Γ-centered k-point mesh of75 points was used for total energy and density of states(DOS) calculations. A larger mesh of 904 k-points wasused for the calculations of energy bands. All calculationsused a basis set size of 6x6x6, 7 core states (equilibriumstate of argon) for iron and 9 core states (argon + 3dstates) for selenium resulting in 28 total valence electrons.Local orbitals were also used with energies of 0.308 Ry for3s and 3p iron and -0.548 Ry and 0.158 Ry for 3s and 3pselenium, respectively. All calculations used fixed muffin-tin radii of 2.0 bohr for Fe and Se atoms. Structural op-timization was executed via energy minimization withrespect to both the tetragonal lattice constants a and c and the internal Selenium parameter z . As a check onour LAPW results, we performed calculations on the B10structure using the Vienna Ab initio
Simulation Package(VASP) using the VASP implementation of theProjector Augmented-Wave (PAW) method . To en-sure convergence we used a plane-wave cutoff of 500eV,and used the same k-point mesh as in the LAPW calcu- lations. II. TOTAL ENERGY OF IRON-SELENIUM
Fig. 2 shows the optimized (variation of volume, c/aand internal parameter z) total energy calculations per-formed with our LAPW code, as well as our opti-mized VASP calculations with the LDA functional. BothLAPW and VASP methods with LDA functional under-estimate the measured lattice parameters by 4.5% and2.2% for the a and c parameters, respectively. Thiswas expected since the LDA functional usually underesti-mates the lattice parameters, although for simple materi-als the difference is usually smaller. Bulk modulus resultswere overestimated by 3.9% using the LAPW method.The calculated and experimental structural results andbulk moduli of FeSe are given in Table I. Structural re-sults from another optimized study of FeSe using PAWand LDA functional completed by Winiarski, et. al. agree with our calculations. They found lattice param-eters of a = 6 . c = 10 . au and z = 0 . a and c parameters, respectively.Comparing our structural results under pressure withthe Winiarski paper, we find comparable results for allpressures.All calculations presented in this paper are for param-agnetic FeSe. We also performed ferromagnetic calcula-tions which yield nearly the same results for total energyto the paramagnetic calculations. Furthermore, the cal-culated equilibrium parameters are nearly equivalent tothe paramagnetic case, and therefore show no significantdifference between the two cases. Further calculations us-ing the ferromagnetic, antiferromagnetic, and other mag-netic orders should be considered for further study, butare beyond the scope of this paper. TABLE I. Volume, c/a ratio, internal parameter ( z ) and bulkmoduli for FeSe at ambient pressure. Calculated results arefrom optimized calculations of the corresponding method andLDA functional. Experimental results are taken from Kumaret. al. and Ksenofontov, et. at. .volume a c/a z Bulk Mod.(au ) (au) (au) (GPa) LAPW . .
804 1 .
50 0 .
255 32 . VASP . .
771 1 .
50 0 .
256 34 . Experiment . . . . . III. ELECTRONIC STRUCTURE
Density of States (DOS) and energy band calculationswere performed using the LAPW results from the opti-mized LDA total energy calculations. These calculationsare performed down to 76% V exp , where V exp = 529 . from Table I. This corresponds to pressures as high as8GPa. The DOS results are then used to calculate su-perconductivity properties.Fig. 3 shows the total, Se-p, and Fe-d decomposed DOSfor FeSe at ambient pressure, i.e. V exp . The d-componentof the Fe DOS is the largest contributor to the total DOSnear and at the Fermi level, E F . Around E F , these Fe-dstates are localized and do not extend into the interstitialregion, while the Se-p and the Fe-d tails have significantcontribution in the interstitial space as shown in Fig. 3.The Se-s semi-core states, not shown in Fig. 3, are foundapproximately 1Ry below E F . Fig. 4 displays the energybands of FeSe at ambient pressure. These two figuresare in agreement with calculations of the DOS and en-ergy bands performed by other groups . Table II givesa comparison of the DOS between our results for theexperimental lattice parameters with those from Subediet. al. , and Bazhirov and Cohen . The specific latticeparameters used for the DOS calculations presented inTable II are a=7.114a.u., c=10.1154a.u. and z=0.2343for both Subedi et. al. and us. However, it is unclearwhat structural parameters were used in reference . DOS S t a t e s / R y / C e ll Energy(Ry) ε F Total DOSDOS Se-pDOS Fe-d
FIG. 3. (Color online) Total, Se-p, and Fe-d density of statesof FeSe at ambient pressure. Notice that the Fe-d componentis the largest contributor to the total DOS around the Fermilevel. Dashed vertical (blue) line represents the Fermi level. calculations for various a, c, and z leads to different val-ues at the Fermi surface. These differences in structuralparameters are influencing the calculation of supercon-ducting properties and will be discussed in more detailin the following section. The differences in N ( E F ) shownin Table II are probably due to the details of the methodused to calculate the DOS and possibly the number ofk-points. -0.4-0.3-0.2-0.1 0 0.1 Γ ∆
X Y M
Σ Γ Λ Z E - E F e r m i ( R y ) FIG. 4. (Color online) Energy bands of FeSe at ambient pres-sure. Solid horizontal (black) line represents the Fermi level.
IV. SUPERCONDUCTIVITY
As mentioned, the DOS calculations are used to cal-culate superconductivity parameters. For each atomtype, we calculate the electron-ion matrix element known
TABLE II. Comparison of the total DOS at the ambient pres-sure. Our DOS calculations, as well as those of Subedi et.al. , are for the experimental structural parameters with theinternal parameter z=0.2343. It is not entirely certain thestructural parameters used to calculate the values given byBazhirov and Cohen . N ( E F ) (states/Ry/cell)This paper 32 . . . as the Hopfield parameter, η i , using the followingformula , η i = N ( E F ) < I > i (1)where N ( E F ) is the total DOS per spin at E F and < I > i is the electron-ion matrix element for each atomtype, which is calculated by the Gaspari and Gyorffy the-ory (GG) . The electron-ion matrix element is given bythe following equation, < I > i = E F π N ( E F ) (cid:88) l =0 l + 1) sin ( δ il +1 − δ il ) N il +1 N il N (1) ,il +1 N (1) ,il (2)where N il are the per spin angular momenta ( l ) compo-nents of the DOS at E F for atom type i , N (1) ,il are theso-called free-scatterer DOS for atom type i , and δ il arescattering phase shifts calculated at the muffin-tin radiusand at E F for atom type i . Free-scatterer DOS are cal-culated by N (1) l = √ E F π (2 l + 1) (cid:90) R s r u l ( r, E F ) dr (3)and scattering phase shifts are calculated bytan δ l ( R S , E F ) = j (cid:48) l ( kR S ) − j l ( kR S ) L l ( R S , E F ) n (cid:48) l ( kR S ) − n l ( kR S ) L l ( R S , E F ) (4)where R S is the muffin-tin radius, j l are spherical Besselfunctions, n l are spherical Newnaun functions, L l = u (cid:48) l /u l is the logarithmic derivative of the radial wavefunc-tion, u l , evaluated at R S for different energies, k = √ E F and u l is computed by solving the radial wave equation ateach k point in the Brillouin zone. The Hopfield parame-ter is then used to calculate the electron-phonon couplingconstant, which is obtained using the following equationfrom McMillan’s strong-coupling theory λ = (cid:88) i =1 η i M i < ω > (5)where M i is the atomic mass of atom type i , and < ω > is the average of the squared phonon frequency takenfrom the experimentally calculated Debye temperature of Ksenofontov, et. at. . The Debye temperature isrelated to the phonon frequency by < ω > = 12 Θ D (6)where Θ D is found to be 240K . The critical su-perconductivity temperature is given by the McMillanequation , T C = Θ D .
45 exp (cid:20) − . λ ) λ − µ ∗ (1 + 0 . λ ) (cid:21) (7)where µ ∗ is the Coulomb pseudopotential, given by theBennemann-Garland equation µ ∗ = 0 . N ( E F )1 + N ( E F ) (8)In equation 8, N ( E F ) is expressed in eV and given in aper cell basis. The prefactor 0.13 was chosen such that µ ∗ = 0 .
10 at the experimental volume. Θ D was calcu-lated as a function of volume, V, by the following formulaΘ D = C ( V − V ) + Θ D (9)where C is given as the slope between experimental De-bye temperature of Ksenofontov, et. at. at ambientand 6.9GPa pressures at their corresponding volumes.We have C = − . V =530.3au .In Table III, we show the total DOS at the Fermilevel, Hopfield parameters, electron-phonon couplingconstants, µ ∗ and critical superconductivity temperaturefor FeSe at various volumes, using our LAPW results.We have included constant c/a and z calculations in Ta-ble III, where c/a is set at the experimental value. TABLE III. Total DOS at E F , N ( E F ), Hopfield parameters, η , electron-phonon coupling constants, λ , µ ∗ and critical su-perconductivity temperature, T c fo FeSe at various volumes,corresponding to pressures as high a 8GPa. Experimentalvalues are calculated using a fixed c/a and z taken from ex-perimental parameters (c/a=1.4656 and z=0.260/z=0.2343)V N ( E F ) η Fe η Se λ Fe λ Se λ µ ∗ T c (au ) ( statesRy/cell ) ( eV / ˚ A ) (K) Experimental c/a; optimized z (c/a=1.4656; z=0.2343)530 32 .
28 1 .
47 0 .
46 0 .
500 0 .
114 0 .
614 0 .
10 4 . Experimental parameters (c/a=1.4656; z=0.260)530 44 .
82 1 .
50 0 .
36 0 .
513 0 .
090 0 .
603 0 .
10 4 . .
05 2 .
15 0 .
47 0 .
558 0 .
088 0 .
646 0 .
095 6 . .
23 2 .
58 0 .
53 0 .
600 0 .
089 0 .
689 0 .
093 8 . .
44 2 .
75 0 .
55 0 .
618 0 .
088 0 .
707 0 . . First we note that at the experimental volume we ob-tain T c ≈
5K reasonably close to the measured value of8K. We also note that if the c/a ratio and z are held con-stant, specifically at the experimental values, throughoutthe DOS calculations for the given volumes, an increasein all T c values are found for increased pressure (decreas-ing volume), as seen in Table III. Although, the total N ( E F ) is found to monotonically decrease during the in-crease in pressure, the parameter η undergoes a rapidincrease. This decrease in total N ( E F ) also contributesto a subtle reduction in the µ ∗ at larger pressures. Itis also of interest that the change of internal parameterdoes not seem to influence the overall superconductiv-ity properties. We continue to see the increase of η dueto the complexity of equation 2, that gives the electron-ion matrix elements, < I > . This shows that the totalDOS at the Fermi level is not the only, nor the major,influence in calculating superconductivity properties. Itis important to note from Table III that λ F e is approxi-mately 6 to 7 times larger than λ Se . This is not surpris-ing since the Fe states dominate near Ef, but also justifiesour use of the Debye temperature in estimating the aver-age phonon frequency. Varying the c/a ratio for a givenvolume does, however, cause a significant change in thesuperconductivity properties. This makes the search ofabsolute optimized parameters quite hard and time con-suming. We show the trend of increasing superconduc-tivity critical temperature for increasing pressure usingthe experimental structural parameters, with a value ofabout 9K, which is too small to account for the measuredvalue of 30K.At ambient conditions, our electron-phonon couplingconstant calculation, λ = 0 . , λ = 0 .
65, byinverting the McMillan equation using the measured θ D .Other computational papers find values of approx-imately λ =0.15 using linear response theory. It is notclear to us what is the source of discrepancy between ourcalculations based on the Gaspari-Gyorffy theory and thelinear response theory-based calculations. It is possiblethat our use of the relationship between average phononfrequency and the Debye temperature is an oversimplifi-cation or on the other hand the Brillouin-zone samplingsperformed in the linear response codes are not sufficientlyconverged. It would be helpful for resolving this issue tohave separate calculations of < I > by the linear re-sponse method. In any case, our results presented in thefollowing paragraphs regarding our agreement with thesmall λ = 0 .
23 we obtained for LaFeAsO needs to beunderstood.In this other iron-based superconducting material,LaFeAsO, electron-phonon coupling constant of approx-imately 0.2 have been reported. We have also per-formed superconductivity calculations of LaFeAsO usingthe experimental lattice constants . Our calculationsof the electron-phonon coupling constant for LaFeAsOis consistent with the results of these groups. LaFeAsOis known to be on the verge of magnetic instability .This suggests that spin-fluctuations are important in this material. TABLE IV. Calculated DOS and superconductivity relatedresults of LaFeAsO using LAPW with LDA functional. Ex-perimental critical temperature result is taken from Takahashiet. al LaFeAsO N ( E F ) (states/Ry/cell)53 . N s ( E F ) N p ( E F ) N d ( E F ) N f ( E F ) η ( eV / ˚ A ) λ Fe 0 .
029 0 .
562 44 .
052 0 .
014 0 .
967 0 . .
006 0 .
826 0 .
439 0 .
119 0 .
186 0 . .
006 0 .
126 0 .
330 0 .
278 0 .
034 0 . .
009 0 .
264 0 .
042 0 .
005 0 .
008 0 . D (K) λ Total µ ∗ T c T c (exp)319 0 .
23 0 .
13 0 . . Table IV shows our calculated LAPW DOS and super-conductivity results using the LDA functional and experi-mental parameters for LaFeAsO. The Hopfield parameterof the Fe atom is seen to be quite low for the LaFeAsOmaterial, in contrast to what we find in FeSe. Similarly,the electron-phonon coupling is much larger for FeSe.
V. CONCLUSIONS
In this paper we present calculations of the band struc-ture of FeSe which are in good agreement with otherworks regarding mechanical and electronic properties ofthis material. We have also presented calculations of theparameters entering the McMillan equation for the su-perconducting critical temperature. Our view is that anelectron-phonon mechanism can explain superconductiv-ity in FeSe at zero pressure, but it does not give enough ofan enhancement to the value of T c under pressure. Com-paring with the multicomponent compound LaFeAsO wesee the following picture emerging: In LaFeAsO, a com-bination of the small value of the calculated parameter η and a large value of the measured θ D invalidates theelectron-phonon coupling. On the other hand, in the caseof FeSe, the combination of large η and small θ D supportselectron-phonon coupling, at least at zero pressure. ACKNOWLEDGMENTS
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