SEMINAR

Iron based superconductorsAuthor:Kristjan ANDERLE

Mentor:doc. dr. Denis ARCON

March, 2011

Abstract

In this seminar I will review iron-pnictides, which have been observed to have superconductingproperties [1]. These are new high-temperature superconductors (HTS), which are not based on copper.However there are some strinking simmilarites between cuprates and iron pnictides and also somepeculiar differences. Several different crystal structures of iron-pnictides have been discovered, I willreview some of them. Various different experiments have been conducted in order to find out differentphysical properties. I will show results of some of this experiments. While we still lack completetheoretical explanation of HTS, I will give summary of theoretical study for magnetic spinconfiguration in iron-pnictide.

Contents1 Introduction 2

1.1 Spin-density waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Types of pnictides 42.1 RFeAsO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 AFe2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 AFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 FeSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Crystallographic data for four types of pnictides . . . . . . . . . . . . . . . . . 7

3 Review of experiments conducted on pnictides 73.1 Structual measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 NMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Magnetic theoretical studies 11

5 Summary 13

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1 IntroductionIn 1986 HTS were discovered and superconductivity once again (after BCS theory) becameprime focus of condensed matter physics. Critical temperature (Tc) was pushed higher andhigher and soon it was above the temperature of liquid nitrogen (77K). This was extremelyimportant, because superconductor could now be used practically i.e. MRI in medicine. ThisHTS were all based on copper and hence name cuprates (latin for copper). The highest Tc wasachieved in HgBa2Ca2Cu3Ox - 135K (155K under pressure).

However after two decades of discovery of cuprates interest in HTS had begun to decline.We still have not found a theory, which would bring good explanation for HTS. There areseveral models, but none of them describes all physical properties. But in 2008 HTS againcame into spotlight. Hosono et al. discovered superconductor, which was based on iron-arseniccompound[1], now known as pnictides. That was very interesting, because iron is famous for itsferomagnetism. And it is believed that SC and feromagnetisem exclude each other. Howeverpnictides go trough structural phase transition around 150 K and become antiferomagnetic.Spins are arranged in spin-density wave (SDW), which I will explain later on. At first it seemedthat SDW vanishes before pnictides become superconducting. But in some materials, underright doping, they co-exist. This raises many questions, especially since spin fluxuations arethe best candidate for electron coupling - similar to phonnons in conventional superconductors,explained by BCS[2].

Transition temperatures (Tc) vary, depending on the type of pnictide. The highest tempera-ture so far is around 50 K, without pressure. The highest Tc predicted by BCS is around 30K.Therefore we know that this is unconventional SC. What is the binding mechanism of electronsin HTS, is yet to be explained. There are some theoretical models, however none of them haveexplained everything. At this point I can stress out, that for now the throne of HTS still belongsto cupprates, since they have much higher Tc. But with more and more research on HTS wehope that someday, we will comprehend the mechanism behind it and will be able to produceSC with even higher Tc.

Crystal structure is somehow similar to copper-oxide SC. We have 2D layers of FePn,where Pn represents a pnictogen atom, atom of nitrogen group in periodic table (N, As, P...) -word comes form pnigein, which is greek for ”‘choke”’. Most common (and with highest Tc)used is arsenic. This FePn layers are separated with layers of different insulators. The onlyexception so far is FeSe, which layers are not separated. More focus on atomic and crystalstructure of different pnictides will be in the following chapters.

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1.1 Spin-density wavesUsual antiferomagnets are describe with Heisenberg model. We have two crystal sublattices- one with spin up, the other with spin down. With SDW the picture is more complicated,and becomes many-body problem. Spins become spatially modulated. We can describe spindensity with [3]:

ρ↑(↓)(r) =12

ρ0(r)(1±σ0 cosQ · r) (1)

ρ↑(↓) is density of spin, so that ρ = ρ↑+ρ↓, σ0 is amplitude of modulation and Q is the wavevector of SDW. The wavelength λ = 2π

Q is usually not a multiple of lattice constant a. This typeof SDW is called incommensurate, and when λ is a multiple of a it’s called commensurate.

The reason, why the electron gas becomes unstable below TSDW and forms SDW is callednesting of Fermi surface. That happens, where Fermi surface (FS) include large parallel faces,spanned by a (nesting) vector q. When we introduce some new pericodicly in system (i.e.doping), there will be an instability in system and gaps will open in this faces. Hence systemwill undergo phase transition into ”‘super”’ structure, i.e. SDW.

Figure 1: Schematic illustration of 1D SDW. First line shows linear arranged atoms, secondantifeormagnetic structure by Heisenberg model. Third and fourth line shows SDW (commen-surate and incommensurate).

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2 Types of pnictidesAs mention in introduction we have different types of pnictides. Different types are discov-ered often, since many scientist are researching this. Fig. 1 shows most of them and theircrystal structure. We can clearly see similar layers of FeAs. Fe atoms are arranged in squarelattice, while FeAs form tetragonal or orthorhombic crystal structure (depending on type andtemperature).

Most of the iron-pnictides become superconductors only when doped with holes or elec-trons. Tc depends on doping concentration as I will show later on.

Figure 2: Six different crystal structures of superconducting iron-pnictides identified at present.Shaded blue rectangle shows similar layers of FeAs in each structure, shdaed red rectangleshow insulating layer. Below each iron pnictide is their highest Tc achieved so far. Copyright[4]

2.1 RFeAsOThe first discovered iron-pnictide superconductor was LaFeAsO[1]. Soon after this discovery,several others with similar crystal structure, were discovered. We usually refer to them as”1111” pnictides, due to ratio of elements in formula. La can be substituted for almost any otherrare-earth element and we superconductivity will still exist. That is because of the alternatinglayered structure - FeAs and RO sheets (R as rare-earth element). From theoretical studies wecan predict, that conductivity (and SC) mainly occurs in FeAs layer, while RO layer provide acharge reservoir.

These materials go through structural phase transition around 160K - from tetragonal toorthorhombic lattice structure. If we drop temperature even lower, materials become antifero-magnetic. Spins form a long-range SDW. We can suppress SDW with doping and that’s whensuperconductivity kick’s in. This shows and interesting relation between both types (SDW and

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SC) and is yet to be theoretically explained. We see this properties in Fig. 3, which shows µSRspecter of LaFeAsO1−xFx[5]. In low doped sample (x < 0.05) we have spontaneous muon spinprecession, which indicates long-range SDW. When we substitute Oxygen with Fluorine (wedope with electrons), antiferomagnetic structure is suppressed and sample becomes paramag-netic.

Figure 3: µSR experiment on LaFeAsO1−xFx. Sample is antiferomagnetic when x < 0.05. Withincrease doping, sample becomes paramagnetic. Copyright [5]

Tab. 2 shows Tc for different R in RFeAsO. The highest Tc is with Sm - 55K. This clearlyshows, that iron-pnictide SC are indeed HTS. Tc are not as high as in cuprate HTS, but re-searches hope to raise it even higher.

R La Ce Pr Nd Sm Gd Tb DyT max

c [K] 28 41 52 52 55 36 46 45x 0.11 0.16 0.11 0.11 0.1 0.17 0.1 0.1

Table 1: Maximum Tc for each RFeAsO1−xFx. x is the concentration of F at which highest Tc isachieved.

2.2 AFe2As2Next type is AFe2As2, where A is Alkaline earth (Ba, Sr or Ca). This structure is reffered as”122”. Superconductivity can be achieved with either hole or electron doping. Hole dopingis achieved with substituting A for monovalent B+ (B =Cs,K,Na). So we get A1−xBxFe2As2.On the other hand we can partly substitute Fe for Co (A(Fe1−xCox)2 As2) and we get electrondoped pnictide which is also superconducting.

Magnetic properties of this type of pnictides is similar to RFeAs. SDW ad structural tran-sition occur at Ts ≈ 140K. This is slightly different from 1111 structure, where these phasetransitions (structural and SDW) occur at different Ts.

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As mentioned, superconductivity appears with doping. Fig. 4 shows TN (SDW transitiontemperature) and TC (SC transition temperature) dependance of doping level. This is typicalphase diagram for all pnictides. We can use TN decreasing with higher concentration of dopant.Then at some xmin superconductivity emerges. From resistivity measurements, SDW and SCstate co-existed in a small window of x. A lot of attention is put to this matter, since it wasbelieved that SDW must be surpressed in order for SC to appear.

Figure 4: Phase diagrams of TN and TC obtained from resistivity mesurments. Magnetic stateis described as SDW, however the subject is still debated. Copyright [6]

Instead of breaking symmetry with doping, we can do that by applying pressure. Theconnection between magnetic state and SC is unclarified as well. Study [7] shows a suppressionof SDW with appearing of SC. On the other hand the phase diagram of BaFe2As2 shows TNand TC in the same window. There seems to be a big connection between SDW and SC in HTS,but we do not understand it fully.

2.3 AFeAsType AFeAs is called ”111” structure. A stands for alkali elements (Li or Na). Crystal structureconsist of tetraedra FeAs4 layers, seperated with double layer of A ions. Distance between Fe-Fe atoms in different layers in significant shorter than in ”‘1111”’ or ”‘122”’ structure. LiFeAshas Tc = 18K without extra doping. This is important difference from ”‘1111”’ and ”‘122”’structure. These types of pnictide both needed extra doping or applied pressure, to becomesuperconducting. On the other hand, NaFeAs is not SC in normal state. It undergoes intoSC phase only under optimum doping. This is rather peculiar phenomena, since NaFeAs andLiFeAs do not differ much, except in slight difference in atomic radii. Another difference isin Tc. Tc of LiFeAs linearly decreases with applied pressure, while in NaFeAs increases withpressure.

Not all researchers have discovered SC in LiFeAs, which suggests that SC is extremelysensitive to sample preparation and that further investigation is needed.

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2.4 FeSeThe simplest form of superconducting pnictides is FeSe, abbirevated ”11” structure. Crystalstructure is similar to those FeAs layers mentioned above, only that this pnictide does not have a”‘seperating”’ layer. TC for ”‘11”’ type pnictide is around 8K. It shows superconductivity onlywhen prepared with Se deficency or substituting Se for Te. With this substitution, we increaseTC ≈ 15K for FeSe0.5Te0.5. We can also increase TC with applying pressure. Superconductingphase transition has been found as high as 27K at 1.5GPa.

The FeSe is also much easier to synthsize, since it doesn’t include toxic arsenic.

2.5 Crystallographic data for four types of pnictides

LaFeAsO BeFe2As2 LiFeAs FeSe0.91Tm[K] 175 297 300 300a [A] 4.03007 3.96275 3.7914 3.77376c [A] 8.7368 13.0168 6.364 5.52482

dFe−Fe [A] 2.84969 2.802 2.6809 2.668dFe−As(Se) [A] 2.407 2.403 2.4204 2.38

Table 2: Crystallographic data for all four types of pnictides discussed above. Tm is tempera-ture at which data was measured.

3 Review of experiments conducted on pnictides

3.1 Structual measurementsFig. 5, which was measured by Cruz et al. [8], shows Bragg’s reflection on LaFeAsO nearstructural phase transition. One can see a structural transition occur at TS ≈ 150K. At thattemperature tetragonal symmetry changes from tetragonal to orthorhombic. On the other handif we substitute Oxygen for Flouirne (LaFeAsO1−xFx) structural phase transition does not occur.Pnictide crystal remains in tetragonal symmetry.

Another phase transition occurs with magnetic ordering. Approximately 18K below TSLaFeAsO undergoes a second-order phase transition, as seen from Fig. 6. We can see tem-perature dependence of the square of the order magnetic moments, measured at Bragg peak.Magnetic moment is µ = 0.36µB/Featom measured at 8K. This is much smaller than predictedvalue of µ ≈ 2µB/Fe atom. There are two main candidates used for explaining the reducedmagnetic momentum - Magnetic frustrations and/or spin fluctuations.

Similar phase transitions were observed in ”112” structure. Crystal undergoes from tetrag-onal to orthorhombic crystal symmetry and magnetic moments order in SDW. Main diffirencefrom ”1111” structure is in TN and TS, which coincide in ”112” structure. TS is approx. 143K.The ”112” structure has simmilar dopping dependance to ”1111”. Structual and phase transi-tion are supressed with doping. The phenomena has been partialy explained with theoreticalmodel, which I’ll present later on.

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Figure 5: Temperature dependence of the (2,2,0) nuclear reflection indicative of a structuralphase transition at 155 K in LaFeAsO. Peak intensities at the (2,2,0)T (tetrago- nal) reflection(open symbols, right-hand scale) and position of the (2,2,0)T , (−2,2,0)O (orthorombic) and(2,2,0)O peaks (solid symbols, left-hand scale) as a function of temperature on cool- ing. Theinset shows the (2,2,0)T reflection at 175 K and the (−2,2,0)M and (2,2,0)M reflections at 4K. Copyright [8]

Figure 6: Temperature dependence of the square of the magnetic momentum. Two differentspectrometers were used - BT-7 (blue circles) and HB-1A (green squares). Blue line is simplefit to mean field theory, which gives TN = 137K. Copyright [8]

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3.2 NMR experimentsFrom NMR experiments of 139La in LaFeAsO one can see relation between magnetic orderingand structural transition [9]. Fig. 7 shows relationship between latice-spin relaxation time T1and electrical resistivity ρ and its temperature derivative dρ/dT . Relaxation time T1 slowsdown below 160K, where crystal undergoes a structural transition. Then at approx. 142K 1/T1has a peak, which coincides with temperature derative of resistivity and occurs at TN . Below142K T1 becomes static.

Figure 7: Temperature dependance of 1/T1, ρ and dρ/dT of 139La in LaFeAsO. Copyright [9]

Figure 8: Temperature dependance of (T1T )−1 for 75As for different doping conctrations x inLaFeAsO1−xFx. The solid arrows indicate TC, broken arrow T ∗ (detail in text). The broken anddotted line are a fitting data for x=0.11 and x=0.14 respectively. Dotted arrows indicate TN .Copyright [10].

Nakai et al. has also measured temperature dependance of T1 for 75As with different dopingconcentrations. Fig. 8 shows temperature dependance of (T1T )−1 for different x in LaFeAsOxF1−x.For low doping concentrations, sample follows Curie-Weis law:

1T1T

=1

T +Θ(2)

between 200K and 142K (32K for x = 0.04), with Θ ≈ 10K [10]. This indicate antiferomag-netic ordering, SDW in peticular. One can see, that SDW ordering is quickly suppresed withdoping. TN moves from approx. 142K for x = 0 to approx. 31K for x = 0.04 and vanishes for

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even higher doping concentrations. As mentioned before, ”‘1111”’ pnictide becomes super-conductor under doping and TC increases with doping concetration x. For doping concentrationof x = 0.07 sample exhibits some peculiar behavior. T1T remains nearly constant down toT ∗ ≈ 40K then decreases rapidly. As seen from 75As NMR spectra T ∗ can not be ascribed tomagnetic ordering and superconductivity appears at lower temperatures (TC ≈ 22K).

One of the properties of conventional superconductor is energy gap - energy needed tobreak Cooper pair to free electrons. Simmilar gap was discovered in hole-doped Cuprates,called pseudogap. Simmilar pseudogap behavior is observed for x = 0.11 and x = 0.14. Energygap was obtained with fit function[10]:

(T1T )−1 = a+bexp−∆PG/kBT (3)

and was the same (within experimental uncertainty) for both concentrations: ∆PG ≈ 169K ·kB. The formula is obtained empiracly and we do not yet completly understand the backgroundbehind it. We belive it is connected to electrons forming a SDW and therefore energy gapemerges - simmilar to conventional superconductors, were electrons are connected with phon-nons.

3.3 Phase diagramsPhase diagrams of iron pnictides are the most popular representation of their properties sincethey show very cleary structual, magnetic and supercondutic phase transition. Phase diagramsare very interesting, because they clearly show connection between SDW and supercondutingphase. Fig. 3.3 shows three diferent phase diagrams for iron pnictides. Based on the first two,one would conclude, that antiferomagnetic and superconducting phase exclude each other. Butone can see the co-exitance of both phases in ”122” structure. Because there was no phasemixing in cuprates, the mystery behind iron pnictide phase diagrams is even bigger.

Figure 9: Phase diagram for three different iron pnictides. First one exibits smooth transitionform SDW to SC phase. Second also shows exclusion of SDW and SC phase, but the transitionis abrupt. On last figure one can see coexistance of SDW and SC phase. Copyright [6].

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4 Magnetic theoretical studiesIn this section I will present theoretical studies preformed by Yildrim in reference [11]. Hewas interested in structural phase transition occurring at approx. 140K and in reduced mag-netic order of 0.36µB per Fe atom. His calculations were done using full-potential linearizedaugmented plane wave method (FP-LAWP), within local density approximation using Perdew-WangCeperlye-Alder exchange correlation.

Figure 10: Lattice of FeAs layer. Figure shows relation between primitive and√

2×√

2 supercell used in calculations by Yildrim[11]. Dark and light shaded areas indicate As atoms belowand above Fe square lattice, respectively. Copyright [11].

In his calculations, he has used√

2×√

2 super cell, shown in Fig. 10. In order to determineground state, he has taken four possible states into account. First with no spin polarization(NM - nonmagnetic), second with ferromagnetic spin ordering (FM) and two different an-tiferomagnetic ordering (AFM1 and AFM2). First antiferomagnetic configurations is wherenearest-neighbor spins are anti parallel to each other (Fig. 11a). In the second antiferomag-netic configurations, spins of Fe atoms on diagonal of square lattice are arranged antiparralel(Fig. 11b). In this configuration we get two simple square AFM sub lattices, penetrating eachother (red and blue lattices in Fig. 11b). There is exactly one Fe ion in the middle of squareAFM lattice, which means the mean field at each spin site is zero. Therefore one sub latticecan be rotated freely with respect to another without costing any energy. That makes AFM2configuration fully frustrated.

Figure 11: Two antiferomagnetic configurations considered in study (see text). Left showsAFM1 configurations with nearest-neighbor always aligned anti parallel. Right panel showsAFM2 were next-nearest neighbor is always alligened anti parallel. Copyright [11].

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Figure 12: Energy dependance of magnetic moment for four different spin configurations.Copyright [11].

In order to determine, which of four spin configurations (NM, FM, AFM1 and AFM2) isground state, Yildrim calculated total energy, using FP-LAWP. He fixed the magnetic momen-tum (M) per Fe atom and than calculated energies for all configurations. Results are displayedin Fig. 12. It is clear, that AFM2 has lower energy than NM spin configuration (M=0), whileother two have higher energies. This indicates that AFM2 is ground state of LaFeAsO system.From Fig. 12 magnetic moment per Fe atom seems to be 1µB, which differs from experimentalvalue of M = 0.36µB.

However, he investigated further and he took magnetic frustration into account. He cal-culated energy dependace of angle γ , shown in Fig. 13. Evidently in AFM2 configurations,LaFeAsO system has lower system for higher γ . This indicate different cell structure - it trans-forms from tetragonal to orthorombic (i.e. cell length a and b are no longer equal), with γ = 91.Energy diffirence from tetragonal case is E ≈ 12meV , which gives T ≈ 140K, which is rel-atively close to experimental temperature of structural phase transition (T=150K). Calculatedmagnetic moment M = 0.48µB is also with agreement to experimental value.

Figure 13: Energy dependace of angle gamma for four different spin configurations. Copyright[11].

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5 SummaryIron-pnictide may give us better understanding of HTS. Before their discovery in 2008, most ofthe research of HTS was done on cuprates and we still haven’t managed to comprehend micro-scopic background of HTS. Now with iron-pnictide, with very interesting magnetic and crystalproperties a theoretical explanation might emerge. We have seen, that there are two compet-ing states in iron-pnictides: antiferomagnetic order and superconducting state. The magneticmomentum per Fe atom is significantly lower than predicted, which by some studies originatefrom magnetic spin frustrations. While magnetic structure differs from cuprates, crystal hassimmilarites. They both have layered structure with conducting (CuO and FeAs) and insulat-ing layer (charge reservoir). However there are even simpler iron-pnictide (i.e. FeSe) with onelayer structure, that are also superconductors. Pseudogap behavior, similar to cuprates, wasalso observed in iron-pnicitide using NMR technique.

Iron-pnictide are in focus of most HTS researchers, but the dominion of cuprates in practicalview is not yet disturbed, since TC differs for approx. 100K. But, as mentioned, their goal is toexplore as many properties as possible in order for theoretical explanation to fit these data.

References[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono: J. Am. Chem. Soc. 130 (2008)

3296.

[2] Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Phys. Rev. 108, (1957) 11751204 .

[3] http://www.pi1.uni-stuttgart.de/glossar/SDW e.php (2010)

[4] C. W. Chu, Nature Physics 5 (2009) 787-789.

[5] 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. Buchner Nature Materials, Vol. 8, No.4. (2009), 305-309.

[6] H. Chen, Y. Ren, Y. Qiu, W. Bao, R. H. Liu, G. Wu, T. Wu, Y. L. Xie, X. F.Wang, Q.Huang, and X. H. Chen: Europhys. Lett. 85 (2009) 17006.

[7] S. Takeshita and R. Kadono: arXiv:0812.2323v2 (2009).

[8] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcli II, J. L. Zarestky, H. A. Mook, G. F.Chen, J. L. Luo, N. L. Wang, and P. Dai, Nature 453, (2008) 899.

[9] Y.Nakai, K. Ishida, Y.Kamihara, M.Hirano, and H. Hosono: J. Phys. Soc. Jpn. 77 (2008)073701.

[10] Yusuke Nakai, Shunsaku Kitagawa, Kenji Ishida, Yoichi Kamihara, Masahiro Hirano, andHideo Hosono Phys. Rev. 79, (2009) 212506

[11] T. Yildirim: Phys. Rev. Lett. 101 (2008) 057010.

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