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Scuola di Dottorato “Vito Volterra” Dottorato di Ricerca in Fisica– XXIII ciclo Exotic superconductors: an infrared spectroscopy study Thesis submitted to obtain the degree of Doctor of Philosophy (“Dottore di Ricerca”) in Physics October 2010 by Chiara Mirri Program Coordinator Thesis Advisors Prof. Vincenzo Marinari Prof. Paolo Calvani Prof. Stefano Lupi

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  • Scuola di Dottorato “Vito Volterra”

    Dottorato di Ricerca in Fisica– XXIII ciclo

    Exotic superconductors:

    an infrared spectroscopy study

    Thesis submitted to obtain the degree of

    Doctor of Philosophy (“Dottore di Ricerca”) in Physics

    October 2010

    by

    Chiara Mirri

    Program Coordinator Thesis Advisors

    Prof. Vincenzo Marinari Prof. Paolo CalvaniProf. Stefano Lupi

  • ii

  • To Elena

  • Contents

    Introduction vii

    1 Theory 11.1 Broken symmetry states of metals . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Transition probabilities and coherence effects . . . . . . . . . 21.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2.1 The London model . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . 9

    1.3 The spin density wave instability . . . . . . . . . . . . . . . . . . . . 101.4 Optical properties of matter . . . . . . . . . . . . . . . . . . . . . . . 10

    1.4.1 Kramers-Kronig transformations . . . . . . . . . . . . . . . . 121.4.2 The Drude-Lorentz model . . . . . . . . . . . . . . . . . . . . 121.4.3 Extended Drude model . . . . . . . . . . . . . . . . . . . . . 151.4.4 Optical spectral weight and Sum Rules . . . . . . . . . . . . . 15

    1.5 The optical response of a superconductor . . . . . . . . . . . . . . . 171.5.1 Models for the superconducting state . . . . . . . . . . . . . . 191.5.2 Reflectivity ratios . . . . . . . . . . . . . . . . . . . . . . . . . 20

    1.6 The optical response of SDW . . . . . . . . . . . . . . . . . . . . . . 22

    2 Physical properties of CaAlSi 272.1 Lattice properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    3 Physical properties of the Iron-based superconductors 373.1 Lattice properties and phase diagrams . . . . . . . . . . . . . . . . . 37

    3.1.1 ReO1−xFxFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.2 AFe2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.3 FeQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.2 Electronic properties and Fermi surfaces . . . . . . . . . . . . . . . . 433.2.1 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . 43

    v

  • 3.2.2 Fermiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Spin Density Waves and antiferromagnetism in the undoped compounds 503.4 Superconductivity in the doped compounds . . . . . . . . . . . . . . 523.5 Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    3.5.1 Transport properties of ReO1−xFxFeAs . . . . . . . . . . . . 553.5.2 Transport properties of different-doped AFe2As2 . . . . . . . 583.5.3 Transport properties of FeQ . . . . . . . . . . . . . . . . . . . 61

    3.6 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    4 Experimental technique 734.1 Fourier-Transform IR Spectroscopy . . . . . . . . . . . . . . . . . . . 734.2 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    4.2.1 Incoherent Synchrotron Radiation . . . . . . . . . . . . . . . 754.2.2 Coherent Synchrotron Radiation at Bessy . . . . . . . . . . . 76

    4.3 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Reflectivity measurements . . . . . . . . . . . . . . . . . . . . 784.3.2 BOMEM DA-3 Interferometer . . . . . . . . . . . . . . . . . 794.3.3 Closed-cycle cryostat and heating system . . . . . . . . . . . 834.3.4 Infrared experimental stations at BESSY and Elettra . . . . 844.3.5 Liquid-helium cryostat and pumped bolometer . . . . . . . . 85

    4.4 Samples growth and characterization . . . . . . . . . . . . . . . . . . 874.4.1 CaAlSi single crystal . . . . . . . . . . . . . . . . . . . . . . . 874.4.2 SmOFeAs and SmO0.93F0.07FeAs polycrystalline samples . . . 894.4.3 FeTe0.91 and FeTe0.7Se0.3 single crystals . . . . . . . . . . . . 894.4.4 BaFe2As2 single crystal . . . . . . . . . . . . . . . . . . . . . 90

    5 Infrared properties of CaAlSi 935.1 Optical properties of the ab plane . . . . . . . . . . . . . . . . . . . . 93

    5.1.1 Optical spectral weight . . . . . . . . . . . . . . . . . . . . . 965.1.2 Spectra of the superconducting ab plane . . . . . . . . . . . . 98

    5.2 Optical properties of the c axis . . . . . . . . . . . . . . . . . . . . . 1025.2.1 Superconductivity along the c axis . . . . . . . . . . . . . . . 104

    6 Infrared properties of Pnictides 1116.1 SmO1−xFxFeAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

    6.1.1 Optical properties of the undoped compound . . . . . . . . . 1116.1.2 Optical properties of the doped compound . . . . . . . . . . . 1146.1.3 Comparison between the undoped and the doped sample . . . 117

    6.2 Optical properties of FeTe0.91 . . . . . . . . . . . . . . . . . . . . . . 1186.2.1 The analysis of the optical conductivity . . . . . . . . . . . . 1206.2.2 The optical spectral weight . . . . . . . . . . . . . . . . . . . 124

    6.3 Optical properties of single-crystal BaFe2As2 . . . . . . . . . . . . . 1276.3.1 The analysis of the optical conductivity . . . . . . . . . . . . 129

    vi

  • 6.3.2 The optical spectral weight . . . . . . . . . . . . . . . . . . . 1326.4 Optical properties of FeTe0.7Se0.3 . . . . . . . . . . . . . . . . . . . . 134

    6.4.1 The analysis of the optical conductivity . . . . . . . . . . . . 1376.4.2 The optical spectral weight . . . . . . . . . . . . . . . . . . . 141

    6.5 Comparison between the undoped and the doped FeTe1−xSex . . . . 142

    Conclusions 147

    Acknowledgments 151

    vii

  • Introduction

    Since the discovery of high critical temperature (Tc) superconductivity in layeredcopper-based oxides (cuprates) [1], extensive efforts have been devoted to the explo-ration of new ceramic systems, in a hope to reach higher Tc. In fact, it is believed thatthe high-Tc values of copper oxides are related to the strong electronic correlationassociated with the transition metal. The standard Bardeen, Cooper and Schieffer(BCS) theory [2] well describes the superconductivity (SC) in conventional metals,where the electronic correlation is not too strong. In this theory the formation of elec-tron pairs (Cooper pairs) leads to an instability of the Fermi surface and therefore tothe superconducting state. The basic interaction involved is the electron-phonon one(e-ph) which gives low Tc, experimentally observed in many systems, and a reducedgap 2∆/kBTc ∼= 3.52 which can increase for strong e-ph interaction. The maximumvalue of 39 K, observed in MgB2, cannot be explained through an e-ph interactionapproach and should be due to a different pairing mechanism in the superconductingstate in a BCS framework. In the cuprates Tc is even higher and their physicalproperties cannot be explained by this theory. Therefore the electronic correlationand other effects have to be taken into account. The discovery of superconductivityup to 26-43 K in LaO1−xFxFeAs [3] has disclosed a new family of superconductors(oxypnictides), with rather high Tc and a structure where Fe-As planes replace theCu-O planes of the cuprates. Higher critical temperatures, up to about 50 K, havebeen obtained by substituting La with Sm, Ce, Pr and Nd [4]. Soon, oxygen-freecompounds were also found to exhibit superconductivity: AFe2As2 (A = Ba, Sr,Eu) [5, 6], LiFeAs [7] and FeQ (Q = Se, Te)[8]. Their structures contain the samestack of tetrahedra in which Fe atoms mainly contribute to the electrical conduction.The parent compounds are poor metals with a magnetic phase characterized bya spin density wave (SDW) which forms below a transition temperature TSDW .Superconductivity appears when the antiferromagnetic ordering in the Fe-layer issuppressed by doping the system with electrons or holes. As the e-ph interaction isnot strong enough to explain the rather high Tc of the pnictides, one is led to lookfor other mechanisms which may establish a SC state in these compounds.

    This work is aimed at investigating the optical properties of different super-conducting materials and of their parent compounds. We shall start by describingthe optical properties of a novel superconductor, CaAlSi, which however can stillbe explained by the BCS theory. Then, we describe and discuss the infrared and

    ix

  • optical spectra of different pnictides, with and without oxygen, in order to increasethe experimental information which is being collected by a variety of techniques tounderstand the superconducting phase in these compounds. It will be clear from thisimplicit comparison how the difference between conventional and unconventionalsuperconductors also reflects into their optical properties.

    The technique here used is Fourier-Transform Infrared (IR) Spectroscopy in awide range of frequencies which span from the Terahertz (THz) to the visible. Itallows one to measure the optical conductivity, a quantity which can also calculatedfrom the Hamiltonian and is then particularly suitable for checking models andtheories. One can thus study the electrodynamics of a system at low energy, wheremost excitations relevant to the onset of superconductivity can be observed: phonons,polarons, free-carrier excitations and the gap opening below Tc. Moreover, the effectof electronic correlation can be studied by extending the measurements up to theVisible and Ultraviolet range and by looking for possible violations of the opticalsum rules.

    The optical conductivity can be obtained, via the Kramers-Kronig transforma-tions, from normal-incidence reflectivity measurements. They have been carried outin a wide range of temperatures and frequencies, depending on the sample, in threedifferent laboratories: the IRS Laboratory of the University of Rome "La Sapienza",the IRIS beamline of the Synchrotron BESSY II in Berlin and the SISSI beamlineof the Synchrotron Elettra in Trieste.

    In the present thesis we first present an optical study of CaAlSi, which has thesame hexagonal structure of MgB2 and a critical temperature Tc = 6.7 K. As it isstrongly anisotropic, we have performed reflectivity measurements with the radiationpolarized along both the ab plane and the c axis. On the ab plane above Tc, wherethe conduction takes place, the effect of the electronic correlation is studied withthe analysis of the optical spectral weight, comparing the results with those oneof a conventional metal like gold, a weak correlated material (Sr3Ru2O7) and astrong correlated one (La1.88Sr0.12CuO4). We will show that the correlations effectsin CaAlSi are much weaker. Across Tc, reflectivity-ratio measurements have beenperformed both on the ab plane and the c axis in order to study the anisotropy ofthe superconducting gap and the possible multi-gap nature of this compound, inanalogy with the isostructural MgB2. From the gap values and other parameters wehave found that CaAlSi is well described by conventional BCS theory.

    The second part of this thesis is devoted to the optical study of three differentpnictides: polycristalline SmO1−xFxFeAs (with x = 0, 0.07) and single crystals ofBaFe2As2, FeTeSe (FeTe0.91, and FeTe0.7Se0.3). Absolute reflectivity measurementshave been performed on all those samples in a wide range of temperatures andfrequencies.

    SmO1−xFxFeAs present optical signatures of their polycrystalline nature, butthey clearly show the physical and the optical properties expected for the SDWin the x=0 sample and for the superconducting state in the x=0.07 sample. Inthe latter case, reflectivity ratios show the effect of the superconducting transition,

    x

  • even if it is not possible to evaluate the gap values along different crystal directions.BaFe2As2 and FeTe0.91 show optical signatures of the SDW transition. An estimateof the SDW gap is given. Neither sample shows the optical spectral-weight (SW)behavior that one expects for conventional or strong correlated metals. However, SWtransfer occurs between different spectral ranges. In metals the SW can be describedthrough one-band tight-binding model and has a T 2 dependence obtained throughthe Sommerfeld expansion [9]. On the contrary, pnictides do not present a metallicresponse, as revealed by their physical properties, for example the semiconductingbehavior of the resistivity, which provides that the Fermi-liquid theory is not correctto describe these systems. In FeTe0.7Se0.3, optical measurements, according to thephase diagram in Ref. [10], suggest that both the SDW and the SC are present.

    The thesis is organized as follows. The first Chapter is devoted to an overviewof the theory involved in the materials studied in this work: the broken symmetrystates of metals (SC and SDW), their optical response, and the models needed toanalyze the experimental data.

    The second and the third chapters are focused on the physical properties ofCaAlSi and of the Pnictides, respectively. As, regarding the Pnictides, many aspectsof the mechanisms driving both the SDW and the SC transitions are still unclear, adescription of the models proposed up to now is provided.

    In Chapter 4 the Fourier-Transform Infrared Spectroscopy is introduced, togetherwith a description of the experimental apparatus and of the sample growth andcharacterization.

    Chapter 5 is entirely devoted to the discussion of the experimental resultsobtained on both the ab plane and the c axis of CaAlSi and to their analysis.

    Finally, in Chapter 6 the experimental results obtained on the three classes ofpnictides and their analysis are presented.

    xi

  • Bibliography

    [1] J. G. Bednorz, K. A. Müller, Z. Phys. B: Condens. Matter 64, 189 (1986).

    [2] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

    [3] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono J. Am. Chem. Soc., 130(11), 3296-3297 (2008).

    [4] Chen Gen-Fu, Li Zheng, Wu Dan, Dong Jing, Li Gang, Hu Wan-Zheng, ZhengPing, Luo Jian Lin, Wang Nan-Lin, Chin. Phys. Lett., 25, No. 6, 2235 (2008).

    [5] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes and R. Pöttgen,Phys. Rev. B 78, 020503R (2008).

    [6] M. Rotter, M. Tegel and D. Johrendt, Phys. Rev. Lett. 101, 107006 (2008).

    [7] X. C. Wang, Q. Q. Liu, Y. X. Lv, W. B. Gao, L. X. Yang, R. C. Yu, F. Y. Li,and C. Q. Jin, Solid State Communications 148, 538 (2008).

    [8] B. C. Sales, A. S. Sefat, M. A. McGuire, R. Y. Jin, D. Mandrus and Y.Mozharivskyj, Phys. Rev. B 79, 094521 (2009).

    [9] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Cornell University,Thomson Learning, 2002.

    [10] R. Khasanov, M. Bendele, A. Amato, P. Babkevich, A. T. Boothroyd, A.Cervellino, K. Conder, S. N. Gvasaliya, H. Keller, H.-H. Klauss, H. Luetkens,V. Pomjakushin, E. Pomjakushina, and B. Roessli, Phys. Rev. B 80, 140511(R)(2009).

    xiii

  • Chapter 1

    Theory

    1.1 Broken symmetry states of metals

    The interactions involving electrons may lead to broken symmetry ground stateswhich make the Fermi surface unstable and cause the formation of electron-electron(e-e) or electron-hole (e-h) pairs.

    The pairing Hamiltonian is in the form

    H =�

    k,σEka

    +k,σak,σ +

    k,k�,l,l’,σ,σ�Vk,k�,l,l’a

    +k,σa

    +k�,σ�al,σal’,σ� (1.1)

    where a+k,σ and ak,σ are the creation and annihilation operators of an electron ofmomentum k and spin σ. The first term of the sum is for free electron gas and thesecond term is due to interactions.

    In the case of a one-dimensional metal, where the Fermi surface (FS) consists oftwo points at kF and −kF , the four conditions shown in table 1.1 can occur. In thiswork both singlet-superconducting (SC) and spin density wave (SDW) systems havebeen studied. In particular these two states of matter are competing in the physicsof pnictides.

    pairing Total spin Total momentumSinglet superconductor electron-electron S = 0 q = 0

    e,σ; e,−σ k’=-kTriplet superconductor electron-electron S = 1 q = 0

    e,σ; e,σ k’=-kSpin density wave electron-hole S = 1 q = 2kF

    e,σ; h,−σ k’=k - 2kFCharge density wave electron-hole S = 0 q = 2kF

    e,σ; h,σ k’=k - 2kFTable 1.1. Different pair formation in the case of broken symmetry ground states ofone-dimensional metals: e = electron, h = hole.

    1

  • 2 1. Theory

    1.1.1 Transition probabilities and coherence effects

    Generally the effect of an external perturbation on the electrons in a metal can bewritten as

    Hint =�

    kσk�σ�

    �k�,σ� |H|k,σ

    �a

    +k�σ�akσ (1.2)

    with the same operators a+k,σ and ak,σ introduced in the section before. They can bereplaced by Bogoliubov operators γk,0 which describe the creation or the destructionof pairs of quasi-particles and are a linear combination, with coefficient uk and vk,of the single particle operators. The transition probabilities are determined by theso-called coherence factors (u∗k�uk + ηv

    ∗k�vk�) where η = ±1. The sign depends on

    the type of interaction involving electrons:

    • Case I: electron-phonon interaction, as in the case of ultrasonic attenuation;the interaction depends only on the momentum change and not on the sign ofk or σ, and the two terms add coherently.

    • Case II: interaction of the electrons with the electromagnetic field.

    For a superconductor one can find the coherence factors

    F (E , E �) ≈�

    1 if η = -1 case 10 if η = +1 case 2 (1.3)

    which multiply the matrix element of transitions in the calculation of energy-absorption rate.

    In the case of SDW one can find that the case 1 and case 2 coherence fac-tors, with the interactions involved, are the opposite to those which apply for thesuperconducting ground state.

    In order to determine an expression for the absorption rate W we can start fromFermi’s golden rule. The Fermi distribution function is

    f(E) = 11 + e EkT

    (1.4)

    where E is the energy difference respect to EF .Given the density of state Ds(E) and the energy of the external probe �ω, the

    transition rate WEE � of absorbed energy in a transition from an occupied to anunoccupied state is proportional to the number of occupied quasi-particle statesf(Es)Ds(Es) and of unoccupied quasi-particle states [1− f(Es + �ω)]Ds(Es + �ω).The rate of absorbed energy can be written as

    Ws ∝

    �|pEE � |

    2F (E , E + �ω)Ds(E)Ds(E + �ω) [f(E)− f(E + �ω] dE (1.5)

    where |pEE � |2 is the matrix element of the transition from the state with energy E tothe state with energy (E + �ω).

  • 1.2 Superconductivity 3

    By normalizing(1.5) to the value in the normal state, we obtain the absorptionrates Ws/Wn from equation (1.6) which is reported in figure 1.1 for SC and SDWas a function of ω at T � 0 (fig. 1.1a) and as a function of T at �ω � 0.1∆ (fig.1.1b). The quantity ∆ is the gap that opens in the Fermi surface below the criticaltemperature of transition (see Sections 1.2 and 1.3). Note that, as seen before, caseI and case II apply to the electromagnetic absorption in the SDW and SC groundstate respectively.

    Ws

    Wn= 1�ω

    � +∞

    −∞

    ��E(E + �ω) + η∆2�� [f(E)− f(E + �ω)]

    (E2 −∆2)1/2 [(E + �ω)2 −∆2]1/2dE (1.6)

    Figure 1.1. Absorption rate evaluated from equation (1.6). (a) Frequency dependence atT=0; (b) temperature dependence at �ω � 0.1∆[1].

    By looking at figure 1.1a we can see that, at T = 0, there are no thermallyexcited quasi-particles and the absorption rate is zero for energies below the gapvalue. For �ω > 2∆ and case 2 the transition probabilities cancel the singularityof the density of states and the absorption rate smoothly increase to 1. For SDWthere is a peak in the absorption rate due to the coherence factor 1.

    For T > 0 the scattering of thermally excited single-particle states becomespossible. For case 1 the transition rate is suppressed when compared with the normalstate transition. For case 2 the divergence at Tc, which is present at ω = 0, isremoved, but a maximum near Tc is observed.

    1.2 Superconductivity

    Superconductivity was discovered by K. Onnes in 1911 when he observed the zeroresistivity of mercury below a temperature Tc = 4.18 [2]. Another observable effectof the superconductivity is the perfect diamagnetism below Tc: if a magnetic field isapplied to a superconductor it does not enter the sample. If it is already presentinside the material it is expelled once the sample is cooled down to T < Tc (Meissner

  • 4 1. Theory

    effect, [3]). A magnetic field can destroy the superconducting state if is strongerthan a critical field Hc, which depends on temperature and goes to zero at T = Tc[4].

    1.2.1 The London model

    As we will see deeply in Section 1.4, if an electric field E is applied on electrons, withcharge e and mass m, they undergo scattering processes with impurities, phononsand other electrons. We can think that in a superconductor some electrons, withdensity ns, give contributions to the conduction without scattering processes. Thenwe can write

    dvdt

    = em

    E (1.7)

    where v is the drift velocity.Through the Ohm’s law for superconducting density current Js = nsev and the

    second Maxwell equation we obtain the two London equations [4].

    dJsdt

    = nse2

    mE = c

    2

    4πλ2LE (1.8)

    H = −4πc∇× (λ2LJs) (1.9)

    where λL is the London penetration depth

    λ2L =

    mc2

    4πnse2. (1.10)

    From the Maxwell equation ∇×H = 4πc

    Js we find

    ∇2H = H

    λ2(1.11)

    For a magnetic field Hy(x), with the only y-component parallel to the superconduc-tor’s surface, equation (1.11) turns into (1.12) where the field follows an exponentiallaw inside the superconductor with a typical penetration depth λ which is empiricallyexpressed by equation (1.13) where λ(0) = λL.

    H(x) = H(0)e−xλ (1.12)

    λ(T ) = λ(0)�

    1−�T

    Tc

    �4�−1/2(1.13)

    However, the penetration depth is always greater than λL: this is due to nonlocal effects for which we can introduce the length ξ0. In a superconductor, with anelectric field E applied, the current at r depends on the field at any point r’ inside avolume of size ξ0 centered in r. From indetermination principles and by consideringthat only electrons with energy ≈ kTc around EF give their contribution to pairformation we can write

    ξ0 ∝�vFkTc

    (1.14)

  • 1.2 Superconductivity 5

    By including the scattering mechanism, namely a mean free path � in whichelectrons are not scattered, we can define a coerence length ξ by the equation

    = 1ξ0

    + 1�

    (1.15)

    and the penetration depthλ ∝ λL(

    ξ

    �)

    12 (1.16)

    1.2.2 The BCS theory

    The basic idea of the BCS theory, formulated in 1957 by Bardeen, Cooper andSchrieffer [5], is that in the presence of an arbitrary small mutual attractive interactionV the electrons are unstable with respect to the formation of fermionic bound statesof opposite momentum and spin, called Cooper pairs (see table 1.1 in section 1.1).Pairs have an energy lower than the Fermi energy EF by a quantity ∆ calledsuperconducting gap.

    We can qualitatively derive the form of the attractive interaction in the case ofCooper pairs by adding to a Fermi gas two electrons with opposite momentum k intwo points r1 and r2 with a mutual interaction V (r1− r2). The wave function of thesystem of the two electrons ψ(r1 − r2) follows the Schrödinger equation (1.17) witheigenvalue E + 2EF . E is the energy difference between the total energy of the twoelectrons and the Fermi energy: for E0) the total energy is lower (greater)than the Fermi energy.

    −�22m(∇

    21 +∇22) + V (r1 − r2)

    ψ(r1 − r2) = (E + 2EF )ψ(r1 − r2) (1.17)

    In the Fourier expansion (1.18) of ψ(r1 − r2) we can introduce the function g(k)that is the probability that the two added electrons have momenta k and -k. Forthe Pauli principle this probability is zero for |k| < kF , because all the states belowthe Fermi energy are occupied.

    ψ(r1 − r2) =�

    kg(k)eik·(r1−r2) (1.18)

    The Schroedinger equation for k, by considering also the matrix elements Vkk�of the interaction V in equation (1.20), is

    �2k2mg(k) +

    k�Vkk�g(k�) =

    E + �2k

    2F

    m

    g(k) (1.19)

    Vkk� =� ∞

    0V (r)ei(k−k�)·rdr (1.20)

    If V is an attractive interaction (Vkk� < 0) values |k| > kF may exist, such thatthe total energy of the pairs is lower than the Fermi energy: bound states exist witha energy difference of ∆ with respect to the Fermi energy EF .

  • 6 1. Theory

    The attractive interaction can be provided by phonons. Indeed, if an electronslightly distorts the positive ion lattice, a second electron will be perturbed by thedistortion and an effective attractive interaction results between the two electrons.

    By using the jellium model, where a solid system is sketched as a fluid of electronsand ions, we can consider different contributions to the interaction V (q,ω), withthe total momentum q = k-k’:

    • Coulomb interaction V (r) = e2r =⇒ V (q) =4πe2q2

    • screening by conduction electrons with screening length 1ks→ V (q) = 4πe2q2+k2s

    • electron-phonon scattering with a phonon frequency ωq → V ∝ 1ω2−ω2q

    The total interaction can be written as

    V (q,ω) = 4πe2

    q2 + k2s+ 4πe

    2

    q2 + k2sω

    2q

    ω2 − ω2q(1.21)

    where the first term is the Coulomb screened interaction and the second term is thephonon-mediated interaction.

    We can consider an attractive interaction (1.22) for electrons interacting withphonons of energy �ωc, where ωc is the Debye cutoff frequency.

    Vkk� =

    −V

    Ek = �2k2m > EF

    E�k < �ωc

    0 otherwise

    (1.22)

    We want to calculate the superconducting gap ∆, that is the energy difference Ewith respect to the Fermi level. Switching from discrete to continue formalism, thatis replacing sums with integrals weighted with the density of states N(E), we canexpect that near the Fermi level (E = 0) the density of states is constant and equalto N(0). Moreover we consider just a weak interaction which means N(0)V

  • 1.2 Superconductivity 7

    ∆(T ) = ∆(0) tanh�

    1.785�Tc

    T− 1�1/2�

    (1.24)

    ∆(T ) = ∆(0)(1− TTc

    )12 (1.25)

    2∆(0) = 3.52kBTc (1.26)

    Figure 1.2. Temperature dependence of the gap described by equation (1.24) [1].

    By combining equations (1.23) and (1.26) and taking into account that in aharmonic approximation the phonon frequency depends on ion mass M through thespring constant ke (Eq. (1.27), the isotope effect in the equation (1.28) is obtainedfor Tc. This means that if in a superconductor, with critical temperature Tc1, wereplace the ions of mass M1 with isotopes of mass M2, Tc2 will be related to Tc1through the square root of M1/M2.

    ωc1 ∼

    �ke

    M1=⇒ ωc2 ∼

    �M1M2ωc1 (1.27)

    kBTc ∝ �ωce−2/(N(0)V ) =⇒Tc2

    Tc1∝

    �M1M2

    (1.28)

    For a BCS superconductor the reduced gap 2∆(0)/kBTc defines the strength of thecoupling:

    2∆(0)kTc

    � 3.52 weak − coupling

    > 3.52 strong − coupling(1.29)

    In section 1.2.1 we have introduced two fundamental lengths to describe thebehavior of a superconductor: the penetration depth λL and the coherence length ξ0.Comparing these quantities with the mean free path � we can distinguish differenttypes of superconductors.

  • 8 1. Theory

    • Type I SC : ξ0 >>λ L

    • Type II SC: ξ0 ξ0

    • Dirty limit : � >λ L the non-local electrodynamics becomesimportant and the q dependence has to be considered resembling the anomalousskin effect in the metallic normal state. At low temperatures, when the skin depthδ0, which is the distance over which the field is attenuated by the metal, is lessthan the mean free path the anomalous skin effect occurs: the electrons movingperpendicular to the surface crosses the skin depth layer without being scatteredand only the electrons travelling parallel to the surface are absorbed. In type I SCthe same argument can be used with ξ and λ playing the same role of � and δ0,respectively.

    Figure 1.3. Diagram showing different superconducting limits as a function of the penetra-tion depth λ, the coherence length ξ and the mean free path � [1].

  • 1.2 Superconductivity 9

    1.2.3 Electron-phonon interaction

    We saw that in the BCS theory phonons play an important role to describe thesuperconducting state. In the Eliashberg theory the electron phonon couplingconstant λe−ph allows one to write the effective mass m∗ = m(1+λe−ph) [1]. λe−ph isdue to phonon contributions, each weighted with the spectral function (Eliashberg’sfunction) α2F (ω), as one can see in equation (1.30). F (ω) is the phonon density ofstates and α2 is the matrix describing the electron-phonon interaction.

    λe−ph = 2�dω

    ωα

    2F (ω) (1.30)

    Let us consider, for strong electron-phonon interaction, only the electrons nearthe Fermi surface and a phonon with wave vector q and frequency ν. The interactioncan be written through an average, on the Fermi surface, of the interactions of eachelectron with the phonons. The matrix elements of the electron-phonon interactionare included in

    γqν = 2πωqν

    kjj�|g

    qνk−qjikj |

    2δ(εkj − εF )δ(εk−qj − εF ) (1.31)

    and can be written as gqνk−qjikj . The spectral function is a density of phonon statesweighted with the coupling of each phonon with the electrons on the Fermi surfaceand can be written as the sum of each term γqν weighted with the phonon frequency.

    α2F (ω) = 12πN(εF )

    γqν

    ωqνδ(ω − ωqν) (1.32)

    The strong coupling with a generic phonon of frequency ωph gives rise to a peak inthe spectral function at ωph. Then, we can write the total electron-phonon couplingconstant λe−ph by considering all the contribution λqν of each phonon:

    λqν =γqν

    πN(εF )ω2qν=⇒ λe−ph =

    λqν (1.33)

    Each coupling λqν gives a contribution to define the Tc, as shown by the McMillan-Dynes formula in equation (1.35) where the free parameter µ∗ takes into account thescreening of the Coulomb interaction. The frequency ωln is a logarithmic average ofthe phonon frequencies defined in equation (1.34). These calculations allow one toobtain an expected Tc to be compared with experimental results.

    ωln = exp��

    qν λqν lnωqν�qν λqν

    (1.34)

    Tc =ωln

    1.2 exp�

    −(1 + λe−ph)1.04

    λe−ph − µ∗(1 + 0.62λe−ph)

    (1.35)

  • 10 1. Theory

    1.3 The spin density wave instability

    The pairing Hamiltonian discussed in section 1.1 can give rise to pair formationin the spin density wave (SDW) channel through the following conditions in theHamiltonian (1.1): k’=k - 2kF l’=l - 2kF and σ� = −σ. The SDW formation isa typical consequence of the electron-electron interaction in a system displayinggood nesting properties of the FS with a nesting vector q at which the susceptibilitydiverges. It develops as a consequence of the divergence of the fluctuations at q =2kF and one can observe a periodic spatial variation of the spin density

    ∆S = S1 cos(2kF · r + φ) (1.36)

    We can distinguish commensurate and incommensurate density waves, where theformer ones have a period λDW = π/ |Q| multiple of the lattice translation vectorR. Let us consider a one dimensional chain of atoms along the x coordinate. If weapply a magnetic field to the hamiltonian 1.1 the interaction with the field generatesa magnetization which, in absence of Coulomb interaction, can be written as

    �Mq� = µB(�nq,↑� − �nq,↓�) = χ0(q)Heffq (1.37)

    Here nq,↓ (nq,↑) indicates electrons with spin down (up), Heffq is the effective magneticfield seen by the electrons and χ0(q) is the magnetic susceptibility.

    We already saw that for SDW a discussion similar to that of SC can be done,with some exception. For example the gap equation and the form of the density ofstates remain the same, but the energy scales involved are different as the characterof the ground states. In case of SDW no retarded electron-phonon interactionare involved and the cutoff energy is the bandwidth of the metallic state. Thisis usually significantly larger than the phonon energies, therefore the transitiontemperatures TSDW and the SDW gaps for density waves are, in general, larger thanthe superconducting ones. The equation of motion of the phase condensate is

    d2φ

    dt2− v

    2Fm

    m∗d

    dr2= em∗

    kF ·E(q,ω) (1.38)

    where m∗ is the mass ascribed to the dynamic response of the density wave conden-sates. The interaction responsible for the formation of SDW is the electron-electroninteraction, so m∗ is simply the free-electron mass or the band mass in the metallicstate.

    1.4 Optical properties of matter

    The optical properties of a solid system can be described by frequency dependentlinear response functions. In this work reflectivity measurements have been performed.The measured reflectivity R(ω) is related to the complex reflectance r̃(ω) throughthe relation

  • 1.4 Optical properties of matter 11

    lnr̃(ω) = ln�R(ω) + iφr(ω) (1.39)

    From r̃(ω) one can extract, using Fresnel relations, the complex refractive indexñ(ω) = n(ω) + ik(ω). At near normal incidence one can write

    r̃(ω) = ñ(ω)− 1ñ(ω) + 1 ⇒ R(ω) =

    (n− 1)2 + k2(n+ 1)2 + k2 (1.40)

    In case of a thin sample, analogous relations can be written for the transmissioncoefficient t̃ to obtain the transmittance T (ω) =

    ��t̃��2 of a single surface.

    T (ω) = 4n(ω)(n(ω) + 1)2 + k2(ω) (1.41)

    For a bi-layered system (as a film on a substrate) with thickness of the layers smallerthan the skin depth, the multireflections can be calculated by Fresnel relations[1]. These lead to the equations (1.42) and (1.43) for reflection and trasmissioncoefficients, respectively, where 1,4 are the external media, 2 and 3 are the two layersof the system.

    r̃bi =r̃12 + r̃23e2iδ2 + r̃34e2i(δ2+δ3) + r̃12r̃23r̃34e2iδ31 + r̃12r̃23e2iδ2 + r̃23r̃34e2iδ3 + r̃12r̃34e2i(δ2+δ3)

    (1.42)

    t̃bi =t̃12t̃23t̃34ei(δ2+δ3)

    1 + r̃12r̃23e2iδ2 + r̃23r̃34e2iδ3 + r̃12r̃34e2i(δ2+δ3)(1.43)

    The complex angle δj is defined for every layer with thickness dj as

    δj = βj + iαjdj/2 = 2πdj(nj + ikj)/λ0 (1.44)

    where λ0 is the wavelength in the vacuum.By using Maxwell equations for the electromagnetic field we can write the real

    and imaginary part of the dielectric function ε̃(ω) = ñ(ω)2 = ε1(ω) + iε2(ω)

    ε̃1(ω) = n2(ω)− k2(ω) (1.45)ε̃2(ω) = 2n(ω)k(ω) (1.46)

    By using the relation ε̃ = 1 + i4πσ̃/ω one can derive the complex conductivityσ̃(ω) = σ1(ω) + iσ2(ω):

    σ1(ω) =ω

    4πε2(ω) (1.47)

    σ2(ω) =ω

    4π [1− ε1(ω)] (1.48)

  • 12 1. Theory

    1.4.1 Kramers-Kronig transformations

    The real and the imaginary part of a complex response function are related to eachother by the so called Kramers-Kronig (KK) transformations. These mathematicrelations were first introduced by Kramers [6] and Kronig [7] and can be derivedfrom general considerations involving causality [1, 8]. They are of great practicalimportance for the evaluation of the components of the complex dielectric functionor conductivity when only one optical parameter, such as the reflected or absorbedpower, is measured.

    For a general linear response function χ(ω) = χ�(ω)+ iχ��(ω) the Kramers-Kronigtransformation are:

    χ�(ω) = 2

    πP

    � ∞

    0

    ω�χ��(ω�)

    ω�2 − ω2dω� (1.49)

    χ��(ω) = −2ω

    πP

    � ∞

    0

    χ�(ω�)

    ω�2 − ω2dω� (1.50)

    where P is the principal part of the integral. For reflectivity measurements χ(ω) =lnr̃(ω) (See equation (1.39)).

    By applying equation (1.50) one can obtain

    φr(ω) = −2ωπP

    � ∞

    0

    ln�R(ω�)

    ω�2 − ω2dω� (1.51)

    Therefore, from experimental values of R(ω) one can calculate n(ω) and k(ω)through equations (1.51) and (1.40) and then obtain optical conductivity σ1(ω).But, in order to perform the integral in (1.51), it is necessary to have R(ω) downto zero frequency and up to a maximum frequency. For this reason a model hasto be used to fit experimental data and extrapolate the low-frequency behavior ofR(ω), below the minimum measured frequency. At high frequencies a ω−4 behavioris applied above the maximum measured frequency.

    1.4.2 The Drude-Lorentz model

    The linear response functions of a system, either a metal or an insulator, can bedecomposed into a sum of contributions related to different charge and latticeexcitations.

    The Drude-Lorentz model describes the response to electromagnetic radiation ofthe electron gas and other excitations, like phonons, in terms of a sum of Lorentzoscillators. In this model, when an electric field E is applied on a system, the electronsare subject to both an elastic and a viscous force, the latter being proportional totheir velocity. The equation of motion for an electron with mass m and charge e is

    d2rdt2

    = − em

    Eloc − γdrdt− ω

    20r (1.52)

    where Eloc is the local field felt by each electron.

  • 1.4 Optical properties of matter 13

    From the solution of equation (1.52) one can find the one-electron dielectricfunction, which can be written through equation (1.53). The real and imaginaryparts of ε̃(ω) is shown in figure 1.4.

    ε̃(ω) = 1 + 4πNe2

    m

    20 − ω

    2 − iγω(1.53)

    Figure 1.4. Real and imaginary part of the dielectric function for a single oscillator obtainedfrom equation (1.53).

    If Nj is the number of electrons with the same ω0j and γj we find

    ε̃(ω) = 1 +�

    j

    4πe2Njm

    20j − ω

    2 − iγjω(1.54)

    If N electrons are completely free (electron gas) the equation of motion for anelectron can be written as the (1.52) without the elastic term (Drude model). Theiroscillating frequency is ω0j = 0 and they contribute through the Drude term, whichis tipical for a metal, characterized by a plasma frequency ωp (ω2p = 4πNe2τ/m) andΓ = 1/τ . The relaxation time τ is the average time between the scattering processesof free electrons with phonons, impurities and other electrons and can be describedby the Matthiessen rule

    = 1τe−ph

    + 1τe−e

    + 1τe−imp

    (1.55)

    where τe−ph, τe−e e τe−imp are the relaxation time related to scattering processes ofelectrons with phonons, other electrons and impurities respectively.

    We can introduce the constant ε∞ which includes all the contributions to thedielectric function coming from the frequencies greater than the maximum measuredone. Then, the total dielectric function in the Drude-Lorentz model is

    ε̃(ω) = ε∞ −ω

    2p

    ω2 + iωΓ +�

    j

    I2j

    ω20j − ω

    2 − iγjω(1.56)

  • 14 1. Theory

    where I2j is the strenght of the oscillation j. The complex conductivity can be writtenas

    σ̃(ω) = iω4π

    1− ε∞ +ω

    2p

    ω2 + iωΓ −�

    j

    I2j

    ω20j − ω

    2 − iγjω

    (1.57)

    If we consider only the Drude term in equations (1.56) and (1.57) we can obtaininformation about the response of a conventional metal. At the plasma frequency ε2is almost zero and ε1 crosses the zero line becoming positive (see also figure 1.4).The frequency of the zero of ε1 is related to ωp, ε∞ and Γ. The complex conductivityof the metal within the Drude model is

    σ̃(ω) = Ne2τ

    m

    11− iωτ =

    ω2pτ

    4π(1− iωτ) =σdc

    1− iωτ (1.58)

    Figure 1.5. Reflectivity and optical conductivity calculated with the Drude-Lorentz modelfor: (a),(b) a metal with ωp = 15000 cm−1, Γ = 200 cm−1); (c),(d) an insulator with aphonon (I21 = 1000cm−2, γ1 = 20 cm−1 and ω1 = 250 cm−1) and a optical electron transitionin the visible range (I22 = 100000 cm−2, γ2 = 15000 cm−1 and ω2 = 25000 cm−1).

    In figure 1.5a and b the frequency-dependent reflectivity and the optical conduc-tivity are shown as calculated through the Drude model for a good metal with ωp= 15000 cm−1 and Γ = 200 cm−1. The reflectivity presents a drop at the so calledplasma edge frequency and the conductivity shows the Drude peak at ω = 0 with ahalf-height width Γ.

    For an insulator there is no contribution of free carriers, so in equation 1.57 theDrude term is not present. In figure 1.5c and d the results of the model are shownfor a system with a IR excitation, for example a phonon (I21 = 1000 cm−2, γ1 = 20

  • 1.4 Optical properties of matter 15

    cm−1 and ω1 = 250 cm−1), and an optical electron transition in the visible range(I22 = 100000 cm−2, γ2 = 15000 cm−1 and ω2 = 25000 cm−1).

    1.4.3 Extended Drude model

    The Drude-Lorentz model may not be sufficient to describe the electrodynamics of ametal, in particular in case of bands of d and f electrons where the electron-electroninteraction becomes strong. Generally, neither the other interactions involvingelectrons can be taken into account simply through the relation (1.55). By extendingthe model reported in the preceding section we can assume a complex frequency-dependent relaxation rate Γ̃(ω) = Γ1(ω)+ iΓ2(ω) and write the complex conductivityas

    σ̃(ω) =ω

    2p

    4π [1 + λ(ω)]1

    Γ1(ω)1+λ(ω) − iω

    (1.59)

    where λ(ω) = −Γ2(ω)/ω. This expression reminds the simple Drude term (1.58)with a renormalized frequency-dependent scattering rate

    1τ∗

    = Γ1(ω)1 + λ(ω) (1.60)

    which approaches the constant 1/τ as λ→ 0.The interactions "dress" the electrons with an effective mass m∗ = (1 + λ(ω))mb

    where mb is the band mass. Then the conductivity is

    σ̃(ω) =ω

    2p/4π

    Γ1(ω)− iω [m∗(ω)/mb]. (1.61)

    Therefore in the extended Drude model one can write

    Γ1(ω) =ω

    2p

    4πσ1(ω)

    σ21(ω)− σ22(ω)

    (1.62)

    m∗(ω)mb

    2p

    4πσ2ω/ω

    σ21(ω)− σ22(ω)

    . (1.63)

    Then, these two frequency-dependent quantities can be calculated by complexconductivity obtained from experimental measurements of reflectivity R(ω).

    1.4.4 Optical spectral weight and Sum Rules

    The optical functions follow sum rules that can be obtained from the KK transforma-tions (1.49) and (1.50). With the Drude-Lorentz model we can consider one oscillatorwith frequency ω0 and width γ and its dielectric function 1.53. For frequenciesω >> ω0 > γ we can make the approximation

    ε1(ω) = 1−4πNe2mω2

    = 1−ω

    2p

    ω2(1.64)

  • 16 1. Theory

    By applying (1.49) and considering the relation (1.47) between ε2(ω) and theoptical conductivity we obtain

    π

    2ω2p = P

    � ∞

    0ωε2(ω)dω (1.65)

    � ∞

    0σ1(ω)dω =

    ω2p

    8 =πNe

    2

    2m (1.66)

    Therefore, the area below σ1(ω) is temperature independent: by changing T , atransfer of spectral weight, which conserves the total one, can be observed betweendifferent frequency ranges. For a solid system we have to consider all the contributionsto σ1(ω), namely both electronic and ionic transitions. As phonons do not influenceappreciably the optical spectral weight, we can neglect their contribution.

    If we limit the integral in equation (1.66) to a specific frequency Ω, the tem-perature independence is not guaranteed. The integral (1.67) is the number of thecarriers, per unit formula, which participate to optical processes at all the frequenciesup to Ω and at the temperature T .

    W (Ω, T ) = mm∗neff (Ω, T ) =

    2mVπe2

    � Ω

    0σ1(ω, T )dω (1.67)

    In a conventional metal the reflectivity has a cutoff (plasma edge) at ωp (see fig.1.5a) which separates low (intraband) to high (interband) charge excitations. Wecan expect that W (Ω, T ) is conserved at ωp. From the Sommerfeld expansion ([9])of electron density one can obtain

    W (Ω, T ) �W (Ω, 0)−B(Ω)T 2 (1.68)

    where W (Ω, 0) is temperature independent and B(Ω) depends on the material[10, 11]. The effect, expressed by the equation (1.68), is due to the correlationbetween the carriers which in a conventional metal is absent and gives rise toa restricted sum rule at the plasma frequency Ωp, which defines the conductionband, W (ωp, T ) ∼= W (ωp, 0), that is B(Ωp) ∼= 0. This means that W (ωp, T ) innon correlated metals counts only the carriers in the conduction band and B(Ωp),which depends on the density of states at Fermi energy, is related to the electroniccorrelation in the system. This suggests to study the frequency dependence ofB(Ωp) in order to evaluate the strength of the correlation in unconventional metals.To compare different materials it is necessary to study the normalized quantity ofequation (1.69) and in particular the normalized coefficient b(Ω) = B(Ω)/W (Ω, 0)[12]. This gives information about the per cent variation with temperature due tothe electronic correlation.

    W (Ω, T )W (Ω, 0) = 1− b(Ω)T

    2 (1.69)

    By fitting the spectral weight in the square of T , one can extract b(Ω) and itsbehavior with respect to frequencies normalized with ωp. In figure 1.6 a comparison

  • 1.5 The optical response of a superconductor 17

    between b(Ω) of gold (conventional metal), the ruthenate Sr3Ru2O7 (weakly corre-lated, [12]) and the superconducting cuprate La1.88Sr0.12CuO4 (strongly correlated,[10]) is shown. In gold, due to the restricted sum rule, b(ωp) � 0, while in the othertwo unconventional metals b(ωp) is as greater as more correlated the material is.

    Figure 1.6. Comparison of coefficients b(Ω) for different correlated materials. b(ωp) is asgreater as more correlated the material is [12]. The frequency Ω is normalized with respectto the plasma frequency of each metal.

    1.5 The optical response of a superconductor

    We introduced in Section 1.2.2 different types of superconductors depending on thevalues of ξ, λ and �. By considering � = vF τ we can characterized the clean anddirty limit of superconductors through the comparison between the gap ∆ and thescattering rate Γ.

    ξ ∝vF

    � ≈vF

    Γ

    =⇒��

    ξ∝

    ∆Γ

    • Clean limit � >> ξ =⇒ Γ ∆

    A superconductor in the dirty limit is a perfect reflector of electromagneticradiation at frequencies ω < 2∆/�: its reflectivity is equal to 1 (at T = 0) at thesefrequencies. If instead we illuminate it by radiation of frequency ω > ∆/�, someCooper pairs are broken, and the absorption processes occur again. Therefore, the

  • 18 1. Theory

    Figure 1.7. Bulk reflectivity R(ω) of a superconducting metal as a function of frequency[1].

    reflectivity of the superconductor turns to that of a normal metal. In figure 1.7 thereflectivity of a superconductor is reported at different temperatures below Tc.

    From London equations (1.8) and (1.9) with E ∼ eiωt one can obtain

    σ2(ω) =nse

    2

    mω(1.70)

    By applying the first Kramers-Kronig relation (1.50) and the sum rule (1.66) we find

    σ1(ω) =πnse

    2

    2m δ{ω = 0} =c

    2

    8λ2Lδ{ω = 0} (1.71)

    � +∞

    0(σn1 − σs1) dω =

    πnse2

    2m = A (1.72)

    where A is the missing spectral weight. It can be calculated as the difference betweenthe area below the optical conductivity of the normal (σn1 ) and the superconducting(σs1) state and is related to the penetration depth λ which can be written, fromequation (1.71), as

    λ = 12π√

    8A(1.73)

    where A is in cm−2.Figure 1.8a and b shows the optical conductivity of a superconductor, both in the

    normal and the superconducting state, for the clean and the dirty limit, respectively.In the clean limit the only contribution to σ1(ω) is given by (1.71): all the spectral

    weight collapses in the collective mode at zero frequency and the penetration isgoverned by the London penetration depth (1.10).

    In the dirty limit, for ω > 2∆� , σs1 recovers the normal behavior and the penetration

    depth isλ(�) ≈ λL(

    ξ0�

    )12 (1.74)

  • 1.5 The optical response of a superconductor 19

    Figure 1.8. Real part of optical conductivity of a superconductor in: (a) clean limit; (b)dirty limit

    Then, by calculating the missing optical spectral weight we can obtain theseinformation about the superconducting state

    • penetration depth λ → Eqn. (1.73)

    • the density of the condensate ns → Eqn. (1.72)

    1.5.1 Models for the superconducting state

    The Mattis-Bardeen model for the optical conductivity takes into account the non-local effects for superconductors in the condition 2∆

  • 20 1. Theory

    Figure 1.9. Behavior of σ1/σn (a) and σ1/σn (b)predicted by the Mattis-Bardeen modelat temperatures below Tc.

    In equation (1.76) the lower extreme of integration is −∆ for �ω > 2∆ and theresulting behavior of σ2(ω)/σn(ω) is shown in figure 1.9b.

    Another model has been formulated by Zimmermann et al. considering ahomogeneous isotropic BCS superconductor and displays its power in the infraredspectroscopy of thin films [13].

    1.5.2 Reflectivity ratios

    We saw in figure 1.7 that a superconductor in dirty limit has a reflectivity almost equalto 1 for frequencies ω < 2∆ (R(ω) = 1 at T = 0). For a BCS s-wave superconductorwe can expect bulk reflectivity ratios RT (ω)/RTn(ω) between the reflectivity at T <Tc and at Tn ≥ Tc, peaked at frequencies ω = 2∆. This situation is shown in figure1.10a that reports reflectivity ratio calculated through the Zimmerman model for abulk system with the parameters T = 6 K, Tn = 22.5 K, ∆(0) = 25 cm−1, ωp = 8000cm−1, Γ = 2000 cm−1. From zero frequency and below 2∆ the ratios increas for thedecreasing of the reflectivity in the normal state; above 2∆, with the breaking ofCooper pairs, the reflectivity in the superconducting state becomes more and moresimilar to the normal one for increasing frequency.

    The theoretical models discussed up to here can be used to fit the data obtainedby infrared spectroscopy on a variety of superconductors. In this work we appliedthe Mattis-Bardeen model in order to estimate the superconducting gap of CaAlSi.

    In figure 1.11 the experimental reflectivity-ratios of B-doped diamond are shown[14]. Here a BCS model can be applied in order to evaluate the superconductinggap ∆. The resulting fit are reported with the experimental values.

    In a superconductor with two gaps ∆1 and ∆2 (∆1 < ∆2), reflectivity ratiosallow one to estimate the gap with the lower value. In fact for 2∆1 < ω < 2∆2 thematerial is still a superconductor and the reflectivity ratios show a peak at 2∆1.The larger gap ∆2 can affect the reflectivity ratio changing its shape. Therefore amodel with two Mattis-Bardeen contributions is needed to fit the data in order toextract the values of the gap(s).

  • 1.5 The optical response of a superconductor 21

    Figure 1.10. Reflectivity ratios calculated through the Zimmermann model [13] for atypical BCS superconductor with T = 6 K, Tn = 22.5 K, ∆(0) = 25 cm−1, ωp = 8000 cm−1,Γ = 2000 cm−1

    Figure 1.11. Reflectivity ratios of B-doped diamond [14] at different temperatures T

  • 22 1. Theory

    1.6 The optical response of SDW

    In Section 1.3 we introduced the SDW ground state. In general, non-local effects arenot important in the SDW state and the q dependence can be neglected. Therefore,the real and imaginary part of the conductivity can be written as

    σcoll1 (ω) =

    πNse2

    2m∗ δω = 0 (1.77)

    σcoll2 (ω) = −

    2ωπ

    � ∞

    −∞

    σcoll1 (ω)ω2 − ω2

    dω� = Nse

    2

    m∗ω(1.78)

    These equations are the same derived for the superconducting state (see equations(1.71) and (1.70)). However, the electrodynamics of the spin density wave groundstate is different from the electrodynamics of the superconducting case. We alreadysaw that phonons are not involved here and also case 1 coherence factors apply,then the transition probabilities are different from those of a superconductor. Theelectrodynamics of a SDW is due to the translational motion of the whole densitywave (collective mode). The absorption due to the quasi-particles ideally occurs,in the absence of lattice imperfections, at zero frequency. At zero temperature,the onset of the quasi-particle absorption is set at the BCS gap 2∆. The SDWproduces a periodic modulation of the spin density which is incommensurate with theunderlying lattice in one direction only; in the other directions the lattice periodicityplays an important role in pinning the condensate to the underlying lattice. If thisis the case, the collective mode contribution to the conductivity is absent due to thelarge restoring force exerted by the lattice [1].

    Let us now see what can be extracted from optical spectroscopy. The theoreticalreflectivity at different temperatures below Tc is shown in figure 1.12. A suppressionof R(ω) is observed at low frequencies indicating the opening of the SDW gap at theFS.

    In the case of finite scattering effects, �/πξ = 0.1, the real and imaginary part ofthe conductivity σ̃(ω) can be described by the equations (1.79) and (1.80) in thecase of dirty limit.

    σs1(ω, T )σn

    = 2�ω

    � ∞

    −∞

    f([E − f(E + �ω)] (E2 −∆2 + �ωE)(E2 −∆2)1/2 [(E + �ω)2 −∆2]

    dE (1.79)

    σs2(ω, T )σn

    = 1�ω

    � ∆

    ∆−�ω,−∆

    [1− 2f(E + �ω)] (E2 −∆2 + �ωE)(∆2 − E2)1/2 [(E + �ω)2 −∆2]

    dE (1.80)

    The resulting frequency dependence of σ1(ω) and σ2(ω) derived from these equationsare reported in figure 1.13a and b respectively.

    Below the single particle gap 2∆, σ1(ω) is lower than the normal one due tothe gapping of the FS and the consequent reducing of the number of carriers. Thisspectral weight is transferred above 2∆ where an enhancement of σ1(ω) is observed,similar to that in a one-dimensional semiconductor and due the case I coherence

  • 1.6 The optical response of SDW 23

    Figure 1.12. Bulk reflectivity R(ω) of a metal in the spin density wave ground state as afunction of frequency for different temperatures [1].

    factor (see section 1.1.1). On the other hand σ2(ω) displays a minimum at thesingle-particle gap.

    If we now calculate the contribution to the optical spectral weight of the collectivemode and of the single particle excitations of a generic density wave state we obtainrelations (1.81) and (1.82) respectively.

    Acoll =

    � ∞

    0σcoll1 (ω)dω =

    πNe2

    2m∗ (1.81)

    Asp =

    � ∞

    0σsp1 (ω)dω =

    � ωg

    0σn1 (ω)− σcoll1 (ω)dω =

    πNe2

    2mb

    �1− mbm∗

    �(1.82)

    For SDW m∗/mb = 1 and all the spectral weight is associated with the collectivemode, while single-particle excitations do not contribute to the optical conductivity.This is valid for the clean limit (ξ0 � ) thespectral weight of the collective mode is expressed by equation (1.83) with therelation (1.84) between the optical spectral weight in the case of clean (Acoll0 ) anddirty (Acoll) limit.

    A �

    � ωg

    0σn1 (ω)dω �

    ω2p

    4πωgτ �ω

    2p

    2π2��

    ξ0

    �1/2(1.83)

    Acoll0Acoll

    = (1 + ξα�

    )1/2 (1.84)

  • 24 1. Theory

    Figure 1.13. Frequency dependence of the real and imaginary parts of the optical conduc-tivity of a spin density wave (case 1) at different temperatures as evaluated from equations(1.79) and (1.80) with �/πξ = 0.1 [1].

  • Bibliography

    [1] M. Dressel and G. Grüner, Electrodynamics of solids, (Cambridge UniversityPress, 2002).

    [2] H. K. Onnes, Comm. Phys. Lab. Univ Leiden, Nos. 119, 120, 122 (1911).

    [3] W. Meissner and R. Ochsenfeld, Naturwiss. 21, 787 (1933).

    [4] M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1975).

    [5] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

    [6] H. A. Kramers, Nature (London) 117, 775 (1926); H. A. Kramers, in Estrattodagli Atti del Congresso Internazionale di Fisica, Vol. 2 (Zanichelli, Bologna,1927), p. 545; Collected Scientific Papers (North-Holland, Amsterdam, 1956)

    [7] R. de L. Kronig, Journal of the Optical Society of America 12, 547 (1926); Ned.Tjidschr. Natuurk. 9, 402 (1942).

    [8] F. Wooten, Optical properties of solids (Academic Press, New York, 1972).

    [9] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Cornell University,Thomson Learning, 2002.

    [10] M. Ortolani, P. Calvani and S. Lupi, Phys. Rev. Lett. 94, 067002 (2005).

    [11] A. Toschi, M. Capone, M. Ortolani, P. Calvani,S. Lupi and C. Castellani, Phys.Rev. Lett 95, 097002 (2005).

    [12] C. Mirri, L. Baldassarre, S. Lupi, M. Ortolani, R. Fittipaldi, A. Vecchione, andP. Calvani, Phys. rev. B 78, 155132 (2008).

    [13] W. Zimmermann, E.H. Brandt, M. Bauer, E. Seider and L. Genzel, Physica C183, 99-104 (1991).

    [14] M. Ortolani, S. Lupi, L. Baldassarre, U. Schade, P. Calvani, Y. Takano, M.Nagao, T. Takenouchi and H. Kawarada, Phy. Rev. Lett. 97, 097002 (2006).

    25

  • Chapter 2

    Physical properties of CaAlSi

    2.1 Lattice properties

    CaAlSi is a superconductor with an hexagonal-layered structure of (Al,Si) and Caplanes, similar to the one of the two-band superconductor MgB2 (Tc ≈ 39 K, [1]).It belongs to the P6mmm group symmetry and its structure is shown in figure 2.1with lattice constants a = 4.19Åand c = 4.39Å.

    Figure 2.1. hexagonal crystal structure of MgB2 (on the left) and CaAlSi (on the right)

    In some samples other crystallographic orders have been observed due to sub-structures in the (Al,Si) planes characterized by different types of chemical bonds.The possible two substructures A and B are rotated with respect to each other andare shown in figure 2.2.

    Figure 2.2. Possible sub-structures of (Al,Si) planes between Ca layers.

    27

  • 28 2. Physical properties of CaAlSi

    Due to the distorted bonds, each layer, A- or B-type, of Al and Si atoms can beplain or pleated. The repetition of different sub-structure types generates two kindsof superstructures belonging to different symmetry groups [2].

    • 5-fold superstructure (figure 2.3a)trigonal symmetry group → P3

    – 5 substructures → cs � 5c

    – 3 B-type and 2 A-type

    – 1 plain Al-Si layer

    • 6-fold superstructure (figure 2.3b)hexagonal symmetry group → P63

    – 6 substructures → cs � 6c

    – 3 B-type and 3 A-type

    – 2 plain Al-Si layers between two pleated layers with the same substructure

    Figure 2.3. Superstructure of CaAlSi: (a) 5-fold (b) 6 fold[2].

    In figure 2.4 the phonon dispersion calculated in [3] is shown with the indicationof the four optical phonons, of which two are double degenerate at the Γ point:

  • 2.2 Electronic properties 29

    • E2g =⇒ stretching Al ←→ Si in plane (Raman active)

    • A2u =⇒ Ca vs (Al,Si) out of plane (IR active)

    • E1u =⇒ Ca vs. (Al,Si) in-plane (IR active)

    • B1g =⇒ Al vs Si out of plane (Raman active)

    Figure 2.4. Phonon dispersion calculated in [3]

    2.2 Electronic properties

    The superstructures have not been taken into account for the band structure calcula-tions of CaAlSi [3, 4, 5]. The hexagonal structure in the reciprocal space of CaAlSiis shown in figure 2.5.

    Figure 2.5. Structure of CaAlSi in the reciprocal space

    The px, py and s orbitals of Al and Si atoms hybridize to generate three σ bandsfull and bi-dimensional due to strong bonds in the plane. Those playing a role atthe Fermi level are the p orbitals of Al and Si atoms and the d orbitals of Ca atoms.The π∗ band obtained from Al and Si pz orbitals crosses the Fermi level as does thed-band due to the mixing between Ca-d and (Al,Si)-p orbitals. The resulting band

  • 30 2. Physical properties of CaAlSi

    structure shown in figure 2.6 [3] is compared with the one of MgB2[6]. In the lattercase two σ bands are not full and this affects the transport properties of MgB2 aswe will see in section 2.3.

    Figure 2.6. Comparison between band structure of (a) CaAlSi and (b) MgB2

    ARPES experiments agree with theoretical calculations as reported in figure 2.7where we can see the agreement between theory and the experimental k-dispersion[7].

    Figure 2.7. Band dispersion of CaAlSi: comparison between theoretical and experimentalresults [7].

  • 2.3 Transport properties 31

    2.3 Transport properties

    The anisotropy of CaAlSi can be observed in resistivity data collected along the abplane and the c axis as shown in figure 2.8: the ab plane is more metallic than the caxis and both present a conventional (decreasing) resistivity vs. T [8].

    Figure 2.8. Resistivity of CaAlSi along the ab plane and the c axis [8]

    The resistivity of MgB2 is shown for comparison in figure 2.9 [9]: its values arelower than the ones of CaAlSi.

    Figure 2.9. Resistivity of MgB2 along the ab plane and the c axis [9].

    The Seebeck coefficient shown in figure 2.10a is negative, indicating that thecarriers in this compound are electrons [10]. This coefficient relates the currentflowing between two metals with their temperature difference. The sign of thecoefficient is determined by the direction of the current with respect to the gradientof temperature and provides the sign of the charge of the carriers. On the otherhand, in MgB2 the carriers are holes, as shown in figure 2.10b where RH is positive[11].

  • 32 2. Physical properties of CaAlSi

    Figure 2.10. Seebeck coefficient of (a) CaAlSi [10] and (b) MgB2 [11]

    2.4 Superconductivity

    Up to now, CaAlSi is known as a BCS superconductor of second type (λ >> ξ) inthe dirty limit.

    The critical temperature varies between 6-8 K, probably depending on thepresence or the absence of superstructures [12].

    The spectral function α2F (ω) calculated for CaAlSi has a strong contributionfrom the Raman-active mode B1g. This is shown in the right panel of figure 2.11where, in the left panel, the phonon dispersion of CaAlSi is also shown. In α2F (ω),we can also see a contribution from E1u mode [3].

    Figure 2.11. Spectral function α2F (ω) of CaAlSi calculated in [3] shown in the right panel(red curve). A strong contribution from the B1g mode is observed.

    Penetration depth measurements give the behavior of λab(T) shown in figure 2.12where λab vs. T/Tc is reported with a BCS fit [13]. In this case the reduced-gap value2∆ab/kBTc is comparable with the BCS estimation. Samples with different criticaltemperatures Tc have the same behavior of λab versus T/Tc [14]. The BCS theoryfits these data only at low temperatures as one can see in figure 2.13a. Moreoveralong the c axis the reduced-gap values of the two samples with different Tc are

    • 2∆c/kTc = 4.21 sample with Tc = 6.2 K

  • 2.4 Superconductivity 33

    • 2∆c/kTc = 4.03 sample with Tc = 7.3 K

    They are greater than the one predicted by the BCS theory in the weak couplingindicating a strong electron-phonon interaction. The difference in 2∆/kBTc ratioalong the c axis with respect to the ab plane suggests a relatively strong anisotropyin the superconducting gap.

    Figure 2.12. Penetration depth λab fitted through BCS theory. [13].

    Figure 2.13. Penetration depth along (a) the ab plane and (b) the c axis of CaAlSi withBCS fit for two samples with different Tc [14].

    Under the assumption of s-wave superconductivity with an elliptical gap (eccen-tricity ε) we can obtain the anisotropic gap, normalized to the value ∆BCS = ∆ab,shown in figure 2.14: as Tc increases the gap anisotropy decreases.

    The superconducting parameters of CaAlSi depend on the presence of a su-perstructure. In table 2.1 the values of Tc, ξ and ∆(0) are reported for the threetypes of structure: 1-fold (absence of superstructure), 5-fold and 6-fold. The Tcvariation is due to electron-phonon coupling that changes for the effect of new phonon

  • 34 2. Physical properties of CaAlSi

    modes raising from the superstructure. This enhancement of the electron-phononinteraction also results in a greater reduced-gap value.

    A part of this work will be dedicated to the optical properties of CaAlSi. Reflec-tivity measurements, performed with radiation polarized along ab plane and c axis,allowed us to study the anisotropy in the crystal. For the superconducting state thereflectivity ratios seen in 1.5.2 have been performed in order to estimate the gap ∆and give an evaluation of the possible multi-gap nature of CaAlSi.

    Figure 2.14. Left: anisotropic gap of CaAlSi normalized to ∆ab. Right: comparison withan isotropic BCS gap (dotted curve) [14].

    Table 2.1. Superconducting parameters of CaAlSi with different structures: 6-fold (6H-CAS), 5-fold (5H-CAS) e 1-fold (1H-CAS) [12].

  • Bibliography

    [1] J. Nagamatsu, N. Nakagaua, T. Muranaka, Y. Zenitani and J. Akimitsu, Nature(London) 410, 63 (2001).

    [2] H. Sagayama, Y. Wakabayashi, H. Sawa, T. Kamiyama, A. Hoshikawa, S. Harjo,K. Uozato, A. K. Ghosh, M. Tokunaga and T. Tamegai, J. Phys. Soc. Jpn. 75,043713 (2006).

    [3] Matteo Giantomassi, Lilia Boeri and Giovanni B. Bachelet, Phys. Rev. B 72,224512 (2005)

    [4] G. Q. Huang, L. F. Chen, M. Liu and D. Y. Xing, Phys. Rev. B 69, 064509(2004).

    [5] I.I. Mazin and D.A. Papaconstantopoulos., Phys. Rev. B 69, 1805 (2004).

    [6] G. Satta, G. Profeta, F. Bernardini, A. Continenza and S. Massidda, Phys. Rev.B, 64, 104507 (2001).

    [7] S. Tsuda, T. Yokoya, S. Shin, M. Imai, and I. Hase, Phys. Rev. B 69, 100506(R)(2004).

    [8] T. Tamegai, K. Uozato, A. K. Ghosh and M. Tokunaga, Int. J. Of modern Phys.B 19, 369 (2004).

    [9] Yu. Eltsev, K. Nakao, S. Lee, T. Masui, N. Chikumoto, S. Tajima, N. Koshizuka,and M. Murakami, Phys. Rev. B 66, 180504(R) (2002).

    [10] B. Lorenz, J. Lenzi, J. Cmaidalka, R. L. Meng, Y. Y. Sun, Y. Y. Xue, C. W.Chu, cond-mat/0208341 (2002).

    [11] M.Putti, E.Galleani d’Agliano, D.Marrè, F.Napoli, M.Tassisto, P.Manfrinettiand A.Palenzona, cond-mat/0109174 (2001).

    [12] S. Kuroiwa, H. Sagayama, T. Kakiuchi, H. Sawa, Y. Noda, and J. Akimitsu,Phys. Rev. B 74, 014517 (2006).

    [13] A. K. Ghosh, Y. Hiraoka, M. Tokunaga, and T. Tamegai, Phys. Rev. B 68,134503 (2003).

    35

  • 36 Bibliography

    [14] R. Prozorov, T. A. Olheiser, R. W. Giannetta, K. Uozato and T. Tamegai,Phys. Rev. B 73, 184523 (2006).

  • Chapter 3

    Physical properties of the

    Iron-based superconductors

    In this Chapter a description of the physical properties of the Iron-based supercon-ductors will be given. In the last few years several studies have been performedon this family, discovered in 2008, in order to understand the physics involved inthese compounds. Many aspects regarding the mechanism driving both the SDWand the SC transitions in these materials are still unclear. Therefore, a generaldescription of different hypothesis formulated on these compounds will be providedin this Chapter.

    3.1 Lattice properties and phase diagrams

    Superconducting pnictides are layered compounds containing a transition metal (Fe,Co, Ni) bound with a group V element (Pn = As, P) giving rise to a layered structurewith Fe2Pn2. Since the discovery of superconductivity in LaFeAsO1−xFx [1] extensiveefforts have been devoted to the exploration of new systems containing Fe atoms,aimed at reaching higher transition-temperatures Tc. Therefore, several materialshave been synthesized exhibiting superconductivity at different Tc depending ontheir structure: the so called 1111 systems ReFeOPn (oxypnictides), with rare earth(Re =La, Sm, Nd, Ce, Pr), the 122 systems AFe2As2 (A = Ba, Sr, Eu), the 111systems like LiFeAs and the 11 systems FeQ (Q = Se, Te). We will discuss here thephysical properties of the 1111, the 122 and the 11 systems which have been studiedin this work.

    3.1.1 ReO1−xFxFeAs

    The oxypnictides have a layered crystal structure belonging to the tetragonal P4/nmmspace group with an alternate stack of Re3+O2− and Fe2+Pn3− layers as shown forLaOFeAs in Figure 3.1. The conduction takes place along the FePn layer wherethe carriers are two-dimensionally confined, causing strong interactions among the

    37

  • 38 3. Physical properties of the Iron-based superconductors

    electrons. This two-dimensional character of ReOFePn involves different types ofchemical bonding, which is strongly ionic in the ReO layers and rather covalentin the FePn layers. Below a characteristic temperature TSDW around 150 K thetransition to a SDW phase occurs and a concomitant structural distortion changesthe symmetry from tetragonal (P4/nmm) to monoclinic (space group P112/n), [2], ororthorhombic (space group Cmma), [3]. By applying external pressure to the parent

    Figure 3.1. Crystal structure of the 1111-system LaOFeAs [1]

    compound, superconductivity takes place by suppressing the SDW transition. Thesame effect can be obtained through hole doping, by substitution of O2− with F−which provides an extra positive charge and a negative charge in the insulating andin the conduction layer, respectively. This gives rise to a superconducting transitionbelow a Tc which reaches 55 K in SmFeAsO1−xFx [4]. The evolution of TSDW andTc with doping is shown in figure 3.2a and b for ReOFeAs with Re = Ce and Sm,respectively.

    Figure 3.2. Phase diagram of two oxypnictides: (a) CeOFeAs [5], (b) SmOFeAs [6]

    Regarding the superconductivity, an isotope effect on Fe atom obeying the BCSrelation has been reported [7]. This result implies that phonons should play animportant role in the superconductivity mechanism in these materials. Figure 3.3

  • 3.1 Lattice properties and phase diagrams 39

    shows the phonon dispersion of ReOFeAs with Re = La and the phonon densityof states (PDOS, right panel) as calculated through a Density Functional Theory(DFT) method, [8]. Of course the higher-frequency contributions are due mainlyto vibrations of O atoms, while the lower ones are modes with a mainly metalliccharacter. As it will be shown, the optical modes related to Fe vibrations showalmost similar features in the three types of pnictides described in this Introduction.

    Figure 3.3. Phonon dispersion and, on the right panel, phonon density of states of LaOFeAs.The mode originated from Fe vibrations is drawn with thick curves [8].

    Let us now discuss the main effects of the doping on these systems. The effect ofthe F-doping is a charge redistribution which can be observed in figure 3.4, wherethe integrated difference in the charge density between undoped LaOFeAs andSmFeAsO1−xFx , with x = 0.125, is reported [1]. Most of the charge is redistributedfrom the Fe atoms in the interstitial region and near the F dopants, concomitantwith a depletion of electrons on the iron layer.

    Figure 3.4. Integrated difference in the charge density between undoped and F-doped(with x = 0.125) LaOFeAs [9].

    A significant phonon softening with doping has been observed through inelasticx-ray scattering [10] which might imply a strong electron-phonon interaction for some

  • 40 3. Physical properties of the Iron-based superconductors

    specific modes. Figure 3.5 reports in panel (a) the comparison of the experimentaldifference, obtained by inelastic x-ray scattering, between the PDOS of undopedand F-doped NdFeAsO (with x = 0.15) and in panel (b) the calculated differencebetween the PDOS of undoped and F-doped LaOFeAs ([10, 9]). The phonon spectraof these two undoped compounds are expected to be quite the same because of theirsame structure, the low difference in their lattice constants and the mass of rareearth which is nearly the same.

    Figure 3.5. (a) Experimental difference, obtained by inelastic x-ray scattering, betweenthe PDOS of undoped and F-doped NdOFeAs with x = 0.15 [10]; (b) calculated differencebetween the PDOS of the same compounds [9].

    Experimental results on F-doped NdOFeAs show a softening around 170 cm−1(≈ 21 meV) ([10]). Also the calculation for LaOFeAs reveals a similar softeningassociated with vibrations involving predominantly the La and As motion along thec axis leading to this Raman active mode [9]. This softening can be assigned to themodified bonding arrangement in the La-(O/F) layers and, analogously, most of theobserved changes in the PDOS can be explained in terms of lattice deformation upondoping. Therefore this feature must be assigned to the structural deformations dueto the substitutional replacement of O atoms by F atoms and not to electron-phononcoupling effects.

    3.1.2 AFe2As2

    The ternary iron arsenides AFe2As2 belong to the tetragonal I4/mmm space groupand contain layers of FeAs tetrahedra very similar to those of oxypnictides, but theyare separated by A atoms instead of ReO sheets (Fig. 3.6a). In these compoundsthere is an excess of charge in the FeAs layer due to the transfer of an electron toFeAs according to (A2+0.5FeAs−). At TSDW the space group symmetry changes fromtetragonal to orthorhombic (Fmmm). The structural and magnetic transition occursimultaneously for the parent compound, at different T ’s for the doped systems.

    Superconductivity appears upon pressure and/or with doping by substituting,

  • 3.1 Lattice properties and phase diagrams 41

    Figure 3.6. (a) Crystal structure of a 122-system AFe2As2 [11]; (b) Phase diagram of thebilayered Ba1−xKxFe2As2 [12]. Here TS has the same meaning as TSDW in the text.

    for example, A2+ with K+, Fe2+ with Co+ or As3− with P3−. The phase diagramin Figure 3.6b reports transition temperatures as a function of K-doping level xshowing a coexistence of SDW and SC phases for some x values. The optimal dopingfor superconductivity is greater than that in ReOFeAs [12].

    Figure 3.7 shows the phonon dispersion of BaFe2As2 and its PDOS in the rightpanel [8]. By a comparison with figure 3.3 it can be seen that higher frequencyphonons are absent because they are related to vibrations involving O atoms, whichare not present in the 122 systems, while the modes originated from Fe vibrationshave almost the same features dispersion in both types of pnictides.

    Figure 3.7. Phonon dispersion and, on the right panel, phonon DOS of BaFe2As2. Themode originated from Fe vibrations is drawn with thick curves [8].

  • 42 3. Physical properties of the Iron-based superconductors

    3.1.3 FeQ

    The α-FeQ (Q = non-metal ion) compounds crystallize at room temperature withthe tetragonal PbO-type structure with stacks of FeQ4 tetrahedra similar to thoseseen in the iron oxyarsenides (Figure 3.8a, [13]). These materials are prototypesamples in the pnictides family because of their simple structure, even if theirproperties are different from those of the previous families. Below a TSDW ≈ 80K the crystal structure becomes metrically orthorhombic (space group Cmma)displaying a distortion of the FeQ slabs identical to that observed for the FePnlayers in the iron oxypnictides family and this effect is accompanied by a magneticordering transition.

    In FeTe systems doped with atoms, like Se, a superconducting transition occursat a temperature Tc of the order of 10 K. FeSe shows no static magnetism and ittransforms to a hexagonal phase under pressure [14]. In figure 3.8b a schematicphase diagram of the Se-doped FeTe is shown, with a TN=TSDW ≈ 70 K for theparent compound [15].

    Figure 3.8. (a) Crystal structure of FeTe [13]; (b) Phase diagram of FeSexTe1−x [15]

    We have seen for the other members of the pnictide family that the externalpressure affects the superconductivity by changing Tc. Usually, the phonon frequencyincreases with increasing applied pressure, and Tc is expected to rise up if thesuperconductivity arises from the phonons. An enhancement of Tc has been observedin FeSe under high pressure [16]: it is 13.5 K at 0 GPa and 27 K at 1.48 GPa.Therefore, one must understand the correlation between the pressure and the phononfrequencies. In Figure 3.9 the phonon dispersion of FeSe and the PDOS for twoexternal pressures (0 GPa dotted curves, 1.48 GPa solid curves) is shown. Thedispersion due to Fe-atoms contribution is similar to that seen in the previoussystems. Between 6 and 10 THz (198-330 cm−1) the dispersion, which mainly comesfrom Fe atoms, shifts upwards with increasing the pressure. This hardening can bealso seen in the PDOS. Generally, the existence of the high frequency phonon iscrucial for observing a high Tc. Then, this result may support the importance of

  • 3.2 Electronic properties and Fermi surfaces 43

    phonons, but it is necessary to examine the electron-phonon coupling to understandthe role of phonons in the superconductivity mechanism.

    Figure 3.9. Phonon dispersion and, on the right panel, phonon DOS of FeSe. The solidand dotted curves correspond to the external pressures 1.48 GPa and 0 GPa, respectively[8].

    3.2 Electronic properties and Fermi surfaces

    A general discussion can be done about the electronic properties of different pnictidesby considering the local density approximation (LDA), LDA + DMFT (dynamicalmean field theory) calculations and angle resolved photoemission spectroscopy(ARPES) experiments.

    3.2.1 Electronic properties

    As shown in the previous section a common feature in the pnictides family is thelayered structure wherein the planes containing Fe atoms play a fundamental role inthe electronic properties of these compounds. The 3d-Fe orbitals have six valenceelectrons and if we consider a simple tetrahedral crystal field the five d orbitalssplit into low-lying two-fold eg (d3z2−r2 and dx2−y2) states and up-lying three-foldt2g states (figure 3.10). The distortion of the tetrahedra from their normal shapefurther splits the eg and t2g manifolds, making the final orbital distributions rathercomplicated.

    In figure 3.11, electronic spectra of LaOFeAs and PrOFeAs, obtained withinthe LDA approach, are reported [17]. No evident difference can be noticed bythe replacement of the rare-earth, in particular around the Fermi energy, which isrelevant to superconductivity.

    This can also be seen by looking at the partial-electronic DOS reported in figure3.11a, which are almost the same around the Fermi level [17]. This partial DOSshows that the relevant electronic states at the Fermi level are due to Fe-d states

  • 44 3. Physical properties of the Iron-based superconductors

    Figure 3.10. d-orbitals splitting due to a simply tetrahedral crystal field.

    Figure 3.11. Electronic spectra of LaOFeAs and PrOFeAs calculated through a LDAapproach [17].

    hybridized with As-p states [18]. The only noticeable difference in the spectra of thesystems with different rare-earth ions is in the tetrahedral splitting due to latticecompression, which appears at energies of the order of -1.5 eV for d states of Fe, and-3 eV for p states of As. The insensitivity of electronic spectra to the type of the Reion is due to the fact that the electronic states of ReO layers are far from the Fermilevel, and p states of O only weakly overlap with the d states of Fe and p states ofAs in FeAs layers. This suggests again a quasi-two-dimensional character as seenfrom the layered crystal structure shown in the previous section. If we comparethe total and partial electron DOS in the LaOFeAs and BaFe2As2 samples, bothreported in figure 3.12b, we can observe similar features around Fermi energy.

    The LDA calculation does not take into account the dynamical correlations amongelectrons, as the LDA+DMFT does. Then we can write the total Hamiltonian byadding an interaction term to the free electron one H0 =

    �k,a,σ �a(k)c∗k,a,σck,a,σ

    where the index a refers to each d orbital. The total Hamiltonian is

    H =�

    k,a,σ�a(k)c∗k,a,σck,a,σ + U

    i,a

    nia↑nia↓ + U ��

    i,a �=bnianib − JH

    i,a,b

    Sia · Sib (3.1)

    where JH is the Hund constant which couples the spin S of electrons belonging tothe different orbitals a and b.

  • 3.2 Electronic properties and Fermi surfaces 45

    Figure 3.12. Total (in the upper panels) and partial DOS of LaOFeAs in comparison withthat of (a) PrOFeAs and (b) BaFe2As2 [17]

    Figure 3.13. Comparison between the orbital-resolved DOS obtained by LDA andLDA+DMFT calculations. The insets show the imaginary part of the correspondingself-energies, showing a clear evidence of non-FL behavior.

    With a total band filling n = 6 and values U = 4 eV, U � = 2.6 eV and JH=0.7eV, LDA and LDA+DMFT calculations for Fe-d orbitals give the result shown infigure 3.13 [19]. We can observe a strong modification of the spectral function and alarge-scale spectral-weight redistribution due to the electronic correlation. In the

  • 46 3. Physical properties of the Iron-based superconductors

    inset the imaginary part of the orbital-resolved self-energy is reported, showing a nonFermi liquid state. In fact, the interorbital electron-electron correlation produces alarge scattering between different orbitals which are the more localized d3z2−r2,x2−y2 ,the intermediately localized dxz,yz, and the itinerant dxy electronic states. Therefore,a large damping at EF is observed and an incoherent, pseudogapped, bad-metallicstate is realized.

    3.2.2 Fermiology

    The Fe-d and As-p hybridization leads to a complex Fermi surface (FS) in the Bril-louin zone (shown in figure 3.14a [20]). Several calculations and ARPES experimentsdescribe the FS as consisting of two or more electron and hole cylinders around theM and the Γ point, respectively. An example is shown in figure 3.14b which reportsthe FS calculated for a 10% F-doped LaOFeAs [18].

    Figure 3.14. (a) Brillouin zone of a pnictide system [20]; (b) FS calculated for a 10%F-doped LaOFeAs [18].

    The above FS can be evaluated through a tight-binding approximation byconsidering the schematic structure of a single FeAs layer displayed in figure 3.15a[21]. In this scheme, Fe ions form a square lattice and are surrounded by layers ofAs ions which also form a square lattice. As ions are at the centers of squares of Feions and are displaced upwards or downwards with respect to the Fe lattice.

    The complex electronic structure due to Fe-d orbitals can be treated througha simplified model which primarily takes into account three orbitals of Fe: dxz,dyz and dxy (or dx2−y2z). By introducing transfer integrals on the second nearestneighbors between dxz and dyz orbitals one can calculate the electronic spectrumby the two-orbital model of figure 3.15b. Here t1 is the transfer integral betweennearest σ-orbitals and t2 is the transfer integral between nearest π-orbitals. t3 andt4 are the transfer integrals between identical and different orbitals, respectively, onthe second nearest neighbors. The electronic spectrum gives rise to the FS shown infigure 3.16a, corresponding to one Fe ion per elementary cell. The real cell consistsof two Fe ions and the Brillouin zone is twice as small. Therefore, the spectrum mustbe folded down into this new zone and the resulting FS is reported in figure 3.16b.

  • 3.2 Electronic properties and Fermi surfaces 47

    Figure 3.15. (a) Structure of a FeAs layer and (b) transfer integrals used in the two-orbitalsmodel [21]

    It shows two hole-like FS, α1 and α2, around the Γ point and two electron-like FS,β1 and β2, around the M point.

    Figure 3.16. Fermi surface in the two-orbital model in the Brillouin zone corresponding toone (a) and two (b) Fe ions per elementary cell [21]

    Experimental results confirm this shape of FS for either undoped and dopedmembers of the pnictides family. In figure 3.17 the ARPES-intensity maps andthe FS of NdO0.9F0.1FeAs (Tc = 53 K) are reported, showing agreement with LDAcalculations, though only one hole-like cylinder is resolved around the Γ point [22].

    A similar FS is observed in 122 systems. The calculated FS of the parentcompound BaFe2As2 is shown in figure 3.18 [23]. ARPES measurements, carried outalong different scanning directions and around the Γ point, show three hole pocketsaround the Fermi energy. Comparison between experimental data and calculationshows that these bands are related mainly to Fe-dx2−y2 , Fe-dyz and Fe-dxz stateswhich are almost degenerate at Γ [20].

    ARPES measurements performed in doped 122 systems show also different results.For example, in Ba1−xKxFe2As2 one can find FS electron-like sheets around the X

  • 48 3. Physical properties of the Iron-based superconductors

    Figure 3.17. (a,c) ARPES intensity maps and (b,d) FS determined for NdO0.9F0.1FeAs attwo different photon energies at T = 70 K

    Figure 3.18. FS calculated through LDA approach for the parent compound BaFe2As2[23]

    point and a pocket at X, while other structures surround X, but they are hole-like[24]. This is revealed by the momentum dependence of the photoemission intensityselected at constant energies below and above the Fermi energy, reported in figure3.19b. In fact, if one looks at the change of the size of the structures with the bindingenergy, the hole-like pockets increase their size with increasing binding energy, whilethe electron-like ones have the opposite behavior. Figure 3.19 reports a comparisonbetween ARPES data of the parent compound ([20]) and the K-doped system ([24])as momentum dependence of the photoemission intensity at constant energy cuts.As said before, the type of pockets at Γ and M points are clearly identified as hole-and electron-like, respectively. The electron doped 122 compounds (BaFe2−xCoxAs2)show similar features, with the hole pocket at Γ being small with respect to theelectron-like one at the M point [25].

    Also in the 11 systems the FS is characterized by electron and hole pockets, as

  • 3.2 Electronic properties and Fermi surfaces 49

    Figure 3.19. Momentum dependence of the photoemission intensity at constant energycuts obtained for (a) BaFe2As2 [20] and (b) Ba1−xKxFe2As2 [24]. The size of the holepockets decrease with the increasing of the bonding energy, while the electron pockets showthe opposite behavior.

    shown for FeTe in figure 3.20, where the momentum dependence of the photoemissionintensity at various binding energies are reported [26].