鐵基超導體的非彈性中子散射

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(1)國立臺灣師範大學物理學研究所碩士論文. 鐵基超導的非彈性中子散射. 研究生：昌文宗指導教授：吳文欽. 中華民國九十八年七月.

(2) Inelastic Neutron Scattering in Iron-based Superconductor － Applications of two-band Model. A Thesis Submitted for the Degree of Master. Student : Wen-Zong Chang Advisor : Wen-Chin Wu The Department of Physics, National Taiwan Normal University, Taiwan.

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(5) 致. 謝. 首先我要鄭重感謝中研院原分所的魏金明老師和師大物理系的吳文欽老師。魏老師成全了我從事自己有興趣的研究，並且給予我經濟上的資助，也教導了我不少做研究上的事。吳老師在碩二的時候收留了我，細心的教導我如期完成這一年來的研究與論文，甚至於平日生活、感情問題都像朋友與長輩般的給予建議與鼓勵。並感謝我的口試委員張明哲老師和胡崇德老師，對我的論文提出了不少建議。感謝研究大樓與 A201 研究室的同學、學長姐弟妹們這一年多來的陪伴，從修課、做研究、寫程式、慶生、吃喝玩樂等大大小小事情都受到你們的幫忙，跟你們在一起的時光絕對是我碩士班最值得懷念的回憶。短短兩年碩士班，我卻換了三次指導教授，其中受到不少人的委屈與鼓勵，感謝曾經給我壓力與痛苦的人，讓我在逆境中堅強尋求出路，感謝曾經給我關心與建議的人，讓我能感受到人情的溫暖。還有感謝系辦助教鈞萍小姐，這兩年來幫忙我們處理了大大小小事務，細心親切的態度是我們物理系的驕傲。謝謝父母在家裡經濟不穩定的情況下還支持我完成學業。兩年的碩士生涯轉眼就過去了，這段日子真的學到很多事，最後最後由衷感謝所有曾經幫助過我的人。.

(6) 摘. 要. 超導電性一直是凝態物理中的一個重要研究課題。在二十幾年前 (1986 年)，銅氧化物高溫超導體被發現，後續在理論和實驗上的研究都有了許多進展。然而卻還是有許多重要的問題尚未有明確的解答，如高溫超導相變的機制與能隙的配對對稱性。去年 2008 年，由日本的研究團隊發現以鐵為基礎導電層的新型高溫超導體，許多科學家相信有可能藉由這類新型的高溫超導體的研究從而解開高溫超導的謎底。在這論文中，我們利用格林函數方法對鐵基高溫超導體的配對對稱性做一理論上的研究。我們計算在三種目前被提出的配對對稱性下的自旋響應函數，關聯至非彈性中子散射實驗。在我們的結果中顯示出無論是在電子或電洞掺雜下，s± 波的能隙形式是與近期的中子散射實驗結果最為吻合，我們也在考慮 Hund’s 耦合的頂角修正項，在所有考慮的配對對稱性下，響應函數的共振行為都會隨著 Hund’s 耦合強度增強而有劇烈的改變。.

(7) Abstract Superconductivity is an important research subject in condensed-matter physics. In last 20s years, the copper-based high-temperature superconductors (HTSC), called the cuprates, have been discovered and many subsequent studies have been done. However, there remains some unsolved subtlety, such as the superconducting mechanism and the pairing symmetry. In January of last year, the iron-based HTSC were discovered, and many scientists believed that these new type of HTSC might hold the key to solve problems of HTSC as mentioned previously. In this thesis, we present a theoretical study of pairing symmetry of iron-based HTSC (e.g. LaFxO1-xFeAs) by the Green’s function method. There are several candidates of the pairing symmetry proposed, to which we calculated the spin response for three types of gap function (i.e. s±, extended-s, and dx2 - y2 wave). It is found that the s± wave is most favored in either electron or hole doping and it agrees well with recent inelastic neutron scattering experiments. Moreover, we have also studied the effect of Hund’s coupling through the random-phase approximation vertex correction..

(8) Contents 1. Introduction Refernces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 5. 2. Two-Band Model of Iron-Based Superconductors 6 2.1 The Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Bare Spin Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 3. Inelastic Neutron Scattering of the Iron-Based Superconductors 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 14 15. 3.3 Dynamical Nesting Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 RPA-Corrected Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4. Conclusion. 32.

(9) Chapter 1 Introduction In January of last year, a paper published from the group of Hideo Hosono in Japan showing the existence of superconductivity in layered iron arsenide material with a transition temperature (Tc) of 26K. [1] The parent compound, LaOFeAs [see Fig. 1.1(a)], was not superconducting (SC), but upon replacing some of the oxygen by fluorine, the material became SC. Even the crystal structure was reminiscent of the cuprates, with layers of FeAs separated by spacer layers of LaO where the fluorine dopants were introduced.[2] [see Fig. 1.1(b)]. (a). (b). Figure 1.1 (a) The crystal structure of parent compound LaOFeAs.[Source: Ref. 1] (b) Electron carriers generated by F-doping into oxygen sites are injected into FeAs metallic layers as a result of the large energy offset between these two layers. [Source: Ref. 2]. Following these initial observations, subsequent data seemed to strengthen the connection between the cuprates and these so-called “pnictides” (i.e., compound of the nitrogen group). In particular, the discovery that the spins on the iron atoms in the parent 1.

(10) compound order antiferromagnetically seemed to confirm this picture. [3] As in the cuprates and other unconventional superconductors, the material is an antiferromagnet at low doping level and increased doping destroys the antiferromagnetism [see Fig. 1.2], leading to superconductivity. As a result, many researchers conjectured that the mechanism of superconductivity would be related to that of cuprates. But, as further work has shown, the story is not so simple and there are important difference between the pnictides and cuprates. Although the parent compound in the cuprates is indeed an antiferromagnet, it is a special type—a Mott insulator—where band theory says the material should conduct but the charge carriers are localized because of the large Coulomb repulsion, U, between the electrons. This is in contrast to LaOFeAs, which is an antiferromagnetic “spin-density wave” metal (with the spins periodically modulated in space) where the electrons appear to be more delocalized. The magnetic phase appears to be associated with a distortion of the crystal lattice from a tetragonal to an orthorhombic structure. [5. 6]. (a). (b). Figure 1.2 (a) Phase diagram of fluorine doped CeO1-xFxFeAs as determined by neutron scattering, showing a second-order phase transition from antiferromagnetism (AFM) at low doping to superconductivity at larger dopings. TN is the magnetic transition, with the inset the value of the staggered magnetic moment. (b) Phase diagram of fluorine doped LaO1-x FxFeAs, showing a more abrupt (first-order phase transition) change from spin-density-wave (SDW) antiferromagnetism to superconductivity as a function of fluorine content. Ts is the structural transition. [Source: Ref. 4]. 2.

(11) Band calculations based on the local-density approximation to density-functional theory emphasize this difference between these types of antiferromagnetism. For example, band theory predicts that the undoped cuprates are metallic, in contradiction to experiment, implying that LDA underestimates the correlations between the electrons [7]. In contrast, for the pnictides, the antiferromagnetism is predicted to be stronger than what is actually observed, meaning LDA may overestimate the correlations [8]. Band theory predicts a metal with several (mostly cylindrical) Fermi surfaces [9] that are separated by a wave vector that is consistent with the period of the magnetic phase [Fig. 1.3(a)], leading to a “nesting” picture for the origin of the magnetism, as occurs in chromium. The predicted Fermi surface has been directly observed by angle-resolved photoemission spectroscopy [4] [Fig. 1.3(b)]. Angle-resolved photoemission studies have in turn been able to map out the anisotropy in momentum of the superconducting energy gap [4], which is important in identifying the nature of the order parameter associated with the superconducting condensate.. 3.

(12) Figure 1.3 (a) The FS from LDA calculation [Source: Ref. 9] (b) The Fermi surface and superconducting gap (Δ) of Ba0.6K0.4Fe2 As2 as determined from angle-resolved photoemission spectroscopy (ARPES). The 3D plot shows the gap at 15 K as a function of the x and y components of the momentum (G, M, and X label high symmetry points of the two-dimensional Brillouin zone), with the colors indicating the gap magnitude (the gap amplitude vs temperature is shown in the inset). The gap anisotropy in momentum space is weak, though the gap magnitude differs between the various Fermi surfaces (α, β, γ). The image at the bottom is the photoemission intensity near the Fermi energy. [Source: Ref. 4] 4.

(13) References [1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). [2] H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hirano, and H.Hosono, Nature 453, 376 (2008). [3] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff II, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, and P. Dai, Nature 453, 899 (2008). [4] Michael R. Norman, Physics 1, 21 (2008) [5] J. Dong, H. J. Zhang, G. Xu, Z. Li, G. Li, W. Z. Hu, D. Wu, G. F. Chen, X. Dai, J. L. Luo, Z. Fang, and N. L. Wang, Europhys. Lett. 83, 27006 (2008). [6] T. Nomura, S. W. Kim, Y. Kamihara, M. Hirano, P. V. Sushko, K. Kato, M. Takata, A. L. Shluger, and H. Hosono, arXiv:0804.3569. [7] W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989). [8] I. I. Mazin, M. D. Johannes, L. Boeri, K. Koepernik, and D. J. Singh, Phys. Rev. B 78, 085104 (2008). [9] D. J. Singh and M-H. Du, Phys. Rev. Lett. 100, 237003 (2008); D. J. Singh, arXiv:0807.2643.. 5.

(14) Chapter 2 Two-band model of iron-based superconductors The Density-functional theory (DFT) band structure calculations have shown that superconductivity in iron-based superconductors is associated with the Fe-pnictide layer, and that the dominant contributions to the Fermi energy came from the Fe 3d orbits. [1] In light of the two-band model [2] that has many virtues and can simply reproduce the same topology of the LDA Fermi surface (FS) and exhibit finite q spin-density wave (SDW) fluctuations, we use this model to study the magnetic response of the Fe-pnictide superconductors by the Green’s function method. In this chapter, we will introduce the two-band model with the basic Hamiltonian, go through a Bogoliubov-transform diagonalization to produce the two bands and FS, and then use it to calculate the normal-state static and dynamic spin susceptibility.. 2.1. The model Hamiltonian. The structure of the FeAs layer of LaFeAsO viewed along the c-axis is illustrated in Fig. 2.1(a). The Fe ions form a square lattice which is interlaced with a second square lattice of As ions. These As ions sit in the center of each square of Fe lattice and are displaced above and below the plane of the Fe ions. As shown by various band structure calculations, the main contribution to the density of states within several eV of the FS comes from the Fe 3d orbits which disperse only weakly in the z-direction. The Fe 3d orbits hybridize among themselves and through the As p orbits leading to a complex of bands. However, as noted in Ref. 1, the band structure near the Fermi level is relatively simple in the unfolded one Fe/cell Brillouin zone (BZ) where it primarily involves three Fe orbits, dxz, dyz, and dxy (or dx2-y2). Based upon this observation and by making the further approximation that the role of the dxy (dx2-y2) orbit can be replaced by a next-nearest-neighbor hopping between dxz, dyz orbits, in this sense, this model should be call the “two-orbital model” rather than “two-band model”, because that there are just two orbits in the model Hamiltonian. We consider a two-dimensional square lattice with two degenerate “dxz, dyz” orbits per site. The tight-binding parameters of the two-orbital model that we will study are illustrated in Fig. 2.1(b). 6.

(15) Figure 2.1 (a) The Fe ions form a square lattice and the crystallographic unit cell contains two Fe and two As ions. The As ions are located above (solid circles) or below (dashed circles) the planes of the Fe square lattice. (b) The schematic showing the hopping parameters of the two-orbital model on a square lattice. The projections of the dxz(dyz) orbits onto the xy plane are depicted in green(white) ellipse. Where t1 is a nearest-neighbor hopping (σ -bonds) and t2 is a nearest-neighbor hopping (π -bonds), and the next-nearest-neighbor hopping t3 between similar orbits and the second-neighbor hopping t4 between different orbits (orbital coupling). [Source:Ref..2]. Due to the nature of the two-orbital model, it is convenient to introduce a. . . . . † † † † † two-component field operator k  [ d x (k ), d y (k )] . Here d x ( k ) [ d y ( k )] creates a. dxz (dyz) electron with spin σ and momentum k. The tight-binding non-interacting Hamiltonian can thus be written as.    ˆ ˆ H0    [(  (k )   )1    (k ) 3   xy ( k )ˆ1 ] k ,  †  k. (2.1). k. where τi are the Pauli matrices and.     x (k )   y (k )   (k )  , 2.   x (k )  2t1 cos k x  2t 2 cos k y  4t3 cos k x cos k y ,   y (k )  2t2 cos k x  2t1 cos k y  4t3 cos k x cos k y ,   xy (k )  4t4 sin k x sin k y .. (2.2). 7.

(16) One can rewrite Eq. (2.1) in a matrix form. H0    k†  k.     x (k )    xy ( k )    k .     xy (k )  y (k )    . (2. 3). To diagonalize this Hamiltonian, we take the Bogoliubov transformation as following,.        .   uk    v    k. vk   d x  uk   d y.   , . (2. 4). where uk and vk are coherent factors, which satisfy the following conditions,. uk2  vk2  1,    1  ( k ) 2  uk  1     2 2  2   (k )   xy (k )   ,    1  ( k ) 2  vk  1     2 2 2   (k )   xy (k )  . (2. 5). ,. and   (   ) is the decoupled field operator on the upper (lower) band, as a consequence,   one can obtain the diagonalized Hamiltonian with two decoupled bands, E (k ) and E (k ) ,.   E ( k ) 0   †   H0       k , k   E (k )  k  0 where.     E (k )    (k )    2 (k )   xy 2 ( k )   .. (2. 6). (2. 7). One can calculate the one-particle Green’s function, which are obtained to be.   ˆ G (k , in )    ein Tˆ k ( ) k† (0) d 0    [in    ( k )  ]1ˆ    ( k )ˆ3   xy ( k )ˆ1    [in  E ( k )][in  E ( k )] .. (2. 8). In Fig. 2.2(a) and (b) we show the two decoupled band for a specific choice of hopping parameters t1 = -1, t2 = 1.3, t3 = t4 = -0.85, in unit of |t1|. The folded energy spectrum for the two Fe/cell zone case is shown in Fig. 2.2(c). 8.

(17) (a). (b). (c). (d). Figure 2.2 (a) The two-d band structure for k x , k y  [ ,  ] . A saddle point exists for each band. (b) The band structure of the two-band model with t1 = -1, t2 = 1.3, t3 = t4 = -0.85, in unit of |t1| and μ = 1.45, plotted along the path (0, 0)  ( , 0)  ( ,  )  (0, 0) . (c) The band structure folded to the small BZ, with the Γ, X, M defined in the small BZ as shown in Fig. 2.3(b). (d) The density of states of the two-band model, with two Van Hove singularities. The dashed line shows the Fermi level corresponding to the choice of μ = 1.45.. 9.

(18) Due to the saddle points in the two bands [see Fig. 2.2(a)], there are two Van Hove singularities in the density of states, which also qualitatively agrees with the LDA results. In Fig. 2.3 we show the FS for the same set of parameters. On the large BZ [Fig 2.3(a)] associated with our model of one Fe/unit cell, there are two hole Fermi pockets labeled 1  and  2 defined by E (k )  0 , and two electron Fermi pockets 1 and  2 defined by  E (k )  0 . To compare with band structure calculations, one must fold the large BZ into a. smaller one which is associated with the crystallographic unit cell containing two Fe atoms. The dashed square in Fig. 2.3(a) marks this smaller zone and in Fig. 2.3(b) we show what happens as the 1,2 and 1,2 bands of Fig. 2.3(a) are folded back into the two Fe/cell BZ.. (a) Figure 2.3 (a) The FS of the two-orbital model on the large one Fe/cell BZ. Here α1,2 surfaces are hole Fermi pockets given by E-(k) = 0 and β1,2 are electron Fermi pockets by E+(k) = 0. The dashed square indicates the BZ of the case of two Fe/cell. (b) The FS folded down into the two Fe/cell BZ consists of two α surfaces around Γ and two elliptically deformed β surfaces (b). around the M point. Here the parameters are the same as those in Fig. 2.2.. 10.

(19) 2.2. The bare spin susceptibility. Now we study the one-loop spin susceptibility for the two-orbital model. Due to the existence of two degenerate orbits in our model, the spin-spin correlation function is orbital dependence, and is defined by      st ( q , iΩ )   eiΩ Tˆ S s ( q , ) St ( q , 0) d 0. . Here s, t = 1, 2 label the orbital indices, and S s (q )  spin. operator. . by  s ( q , iΩ ) . for. . the. orbital. s.. The. ,. (2. 9).    1  †  ( k  q )   ( q ) is the  s   s  2 k. total. spin. susceptibility. is. given.   ( q , iΩ ) . The one-loop spin susceptibility is obtained to be s ,t st.  1  s (q , iΩ )   2N. .  k , , '.    k  q , k , '. nF ( E ,k  q )  nF ( E ',k ). 2. iΩ  E , k  q  E ', k. .. (2. 10). Here E , k , with ν= +1(-1) is the eigenvalue of the upper (lower) band [see Eq. (2.7)], and nF ( E )  1 (e  E  1) is the Fermi distribution function..    k  q , k , '. 2. is the transition. amplitude, which consists of coherent factors given by Eq. (2.5). Eq. (2.10) describes a thermal scattering process that annihilates a particle with momentum k+q and creates a particle with momentum k inter- or intra- the FS of two bands, which costs energy E , k  q  E ', k . Fig. 2.4 shows the plot of the static spin susceptibility.   s (q , 0) versus q to which one. can see the structure associated with various nesting points and density of states features.. . For our choice of parameters, maximum Re  s ( q , 0) occurs around q = (±π, 0) and (0, ±π), which suggests a transition to an antiferromagnetic (AFM) ordered phase at some critical interaction strength. This is also in agreement with the result of band structure calculations. [1] Such a peak in the spin susceptibility comes from the nesting between the electron and hole Fermi pockets, which can be seen from the chemical potential dependence of the spin susceptibility. In Fig. 2.5, the real-part spin susceptibility at q = (π, 0) , which corresponds to the SDW modulations of q wave vector, is enhanced significantly due to the nesting effect at the chemical potential μ >1.4|t1|.. 11.

(20) (a). (b). Figure 2.4 (a) Real part and (b) imaginary part of the bare spin susceptibility at ω= 0 versus q for the same parameters as those used in Fig. 2.2. with the chemical potential μ = 1.54|t1|.. μ = 1.2|t1|. μ = 1.3|t1|. μ = 1.4|t1|. μ = 1.6|t1|. Figure 2.5 The real part of static spin susceptibility with different chemical potentials. A peak at QSDW = (π, 0) start to appear at μ ≧ 1.4|t1|and it is stronger at higher chemical potentials. 12.

(21) References [1] I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett. 101, 057003 (2008) [2] S. Raghu, Xiao-Liang Qi, Chao-Xing Liu, D. J. Scalapino, and Shou-Cheng Zhang, Phys. Rev. B 77, 220503 (2008). 13.

(22) Chapter 3 Inelastic neutron scattering of the Iron-based superconductors Recently neutron-scattering experiments have shown that the iron-based parent compound LaOFeAs undergoes a structural phase transition below ~150 K, which is then followed at ~137 K by the onset of the long-range spin-density wave (SDW) stripe magnetic order with a wave vector q = (0.5, 0.5, 0.5) π/a. [1, 2] When it is doped with F atom, which replaces the O atom and the compound becomes LaO1-xFxFeAs, both the structural distortion and the magnetic order are suppressed and the system becomes superconducting (SC). [2] To understand the pairing mechanism of a superconductor, it is crucial to identify the symmetry of the order parameter. To do this, we present a theoretical study of the dynamical magnetic properties on iron-based superconductors with different types of the gap functions.. 3.1. Introduction. Magnetic fluctuations in the cuprate high-Tc superconductors are long thought to have intimate relation to their mechanism of superconductivity. Based on their great similarity, it is also believed that the magnetism may hold the key to the superconducting pairing mechanism in iron-based high temperature superconductors. Therefore a complete understanding of the spin fluctuation in iron-based superconductors is quite important. Magnetic properties can be probed by many experiments such as the Nuclear Magnetic Resonance (NMR), Nuclear Quadrupole Resonance (NQR), and Inelastic Neutron Scattering (INS) measurements. In this chapter, I will focus on INS which may provide us with indirect verification of the pairing symmetry. The INS spectrum is proportional to Im χ (q, ω) besides some Bose-Einstein distribution factor due to the elementary excitation of bosons. Here χ (q, ω) is the dynamical spin susceptibility. (See later). We shall present theoretical study of χ (q, ω) with different pairing symmetries, which have been proposed. At the end of this chapter, we will show briefly the effect of including the inter-orbital coupling interaction. 14.

(23) 3.2. Formalism. The neutron-scattering cross section is proportional to the dynamical structure factor S(q,ω) which is related to the imaginary part of the susceptibility,.   S ( q ,  )  [1  n( )]Im  (q ,  ) ,. (3.1). where n(ω) = 1/[exp(ω/T)-1] is the Bose-Einstein distribution function. In the BCS framework, the orbital-dependent bare dynamical spin susceptibility is given below (for two-orbital model)       (k ) ' (k  q)  (k) ' (k  q)  1    (q, iΩ)   [ 1 ] 4 k , , ' E (k )E ' (k  q)       1 n(E (k ))  n(E ' (k  q)) 1 n(E (k ))  n(E ' (k  q))        ] [ iΩ E (k )  E ' (k  q) iΩ E (k )  E ' (k  q)       (3.2)  (k ) ' (k  q)  (k ) ' (k  q)    [ 1 ] E (k )E ' (k  q)       n(E (k  q))  n(E ' (k )) n(E (k  q))  n(E ' (k ))        ] [ iΩ E (k  q)  E ' (k ) iΩ E (k  q)  E ' (k ). {. },.    where ν, ν’= +,- are the eigenvalues of the bands and E (k )   (k )2   (k )2 is the quasiparticle excitation spectrum with ε and Δ the band dispersion and superconducting gap of band ν respectively. Equation (3.2) describes two kinds of excitation, the pair-annihilating (creating) excitation that excites two particles (holes) from the SC condensate and costs energy ±( Ek + Ek+q), and the thermal one-particle excitation that excites a particle from k + q to k which costs energy Ek – Ek+q. The two-particle excitation vanishes in the normal state and the one-particle excitation vanishes at zero temperature in the SC state. For isotropic s wave, for example, the gap function Δ(k) is k independent and no node exist on the Fermi surface. There exists a minimum excitation energy 2Δ to create a quasiparticle. In contrast, for a state with nodes, there exist lines where the gap function Δ(k) is zero due to the symmetry. At the intersections of these lines and the Fermi surface the quasiparticle excitation energy is zero. Let these points be labeled by ki; then E(ki) = 0, Δ(ki) = 0, ki  kF. Clearly at low temperature the quasiparticles are concentrated around the vicinity of ki. Therefore the low-frequency scattering processes will be dominated by quasiparticles with these momenta. In Eq.(3.2) the effect of the pair-breaking scattering is incorporated by a finite scattering rate Γ(after we take the analytic continuation, i    i ). In principle, one should do 15.

(24) this self-consistently and include the correction of the self-energy into the single-particle Green function. However, our emphasis here is the q dependence of χ(q, ω) at fixed temperature. It had been shown that the resonance phenomenon is quite insensitive to the different scattering rates. [3] The most important correction to the bare susceptibility should take the Coulomb interaction or antiferromagnetic (AFM) correlation between the quasiparticles into account. Such correlations are believed to exist as a residual interaction between the renormalized particles, and mean field decoupling as a nontrivial step beyond bare theories conventionally treats them. Then the susceptibility is written into a simple random-phase approximation (RPA) form as.     RPA ( q ,  )  [I   0 ( q ,  )]1  0 ( q ,  ) ,. (3.3). where I is a 2×2 identity matrix and the vertex Γ = UI with the U is on-site intraband Coulomb interaction. The RPA-type correction is important because it is perhaps the only way to treat correlation effect analytically. Theoretically, there are several possible gap pairing symmetries which were proposed, based on the 4-fold C4 symmetry of the crystal structure. Both weak and strong coupling approaches suggest that an extend s-wave pairing symmetry is favored [4, 5]. Based on a weak-coupling approach, an s-wave, (so call s±) state [5] in which the sign of order parameters changes between hole and electron pockets, is argued to be favored for repulsive inter-band interactions. In the strong coupling approach, in a recent paper [4], it showed that the pairing symmetry is determined mainly by the next neighbor antiferromagnetic exchange coupling J2 [6, 7] and has an explicit form in momentum space, Δ0 coskx．cosky, which resembles the order parameter s±. This result is model-independent, as long as the dominating interaction is next-nearest neighbor J2 and the Fermi surfaces are located close to the Γ and M points in the Brillouin zone. The magnitude of SC gaps measured by angle-resolved photoemission spectroscopy (APRES) at different Fermi surfaces are in a good agreement with the simple |coskx．cosky| form [8]. To study the pairing symmetry, we have done explicit calculations on a two-dimensional square lattice with the two-band model. Three different symmetries of the pairing state were investigated： (1) s± wave, Δ(k) = Δ0 (coskx．cosky);. (3.4a). (2) extended-s wave, Δ(k) = Δ0 (coskx+ cosky)/2;. (3.4b) 16.

(25) (3) dx2-y2 wave, Δ(k) = Δ0 (coskx- cosky)/2; (3.4b) In principle, for the two-band model, there are two gaps, Δ1(k) and Δ2(k). for simplicity, however, we consider only the case Δ1(k) = Δ2(k). The dx2-y2 and extended-s pairing symmetries are nodal and s± is nodeless on the FS. Note that s± exhibits a sign change between the hole and the electron pockets, while extended s does not.. 3.3. Dynamical nesting effect. This section is devoted to the discussion of the properties of the magnetic excitation with three kinds of the pairing symmetries, which were mentioned in Sec. 3.2. The imaginary part of single-band SC state spin-spin correlation function at T = 0 is given by.            (k ) (k  q )  (k )(k  q )    Im    [ 1  ]   (   E ( k )  E ( k  q )) . (3.5) 4 k E (k ) E ( k  q ) The integrand consists of a coherence factor that reflects the non-time-reversal invariance nature of the magnetic measurement, and a Dirac δ-function that imposes the energy conservation rule. We neglected the δ(ω- E(k)- E(k+ q)) term, because it is out of the energy window that we considered. The role of the δ-function should be first discussed since it is the most dominant. It limits the contributing region of the phase space by energy conservation Ek + Ek+q = ω. At ω ~ 0, the only possibility is Ek and Ek+q both ~ 0. At ω > 0, to satisfy Ek + Ek+q = ω, q can be any momentum that connects points on the contours Ek = ω1 and Ek+q = ω2, which satisfy ω1 + ω2 = ω. A situation that can contribute significantly when ω1 = ω2 = ω/2 and all the two frequency dependent contours become Ek = Ek+q = ω/2. This is so called “dynamical nesting effect”, it spans the whole FS when ω → 2Δ. Figure 3.1 shows the frequency-dependent of the imaginary part of the RPA spin susceptibility with the s± gap in the q space. Here we have set U = 0.9|t1|, gap amplitude Δ0 = 0.3|t1|, chemical potential μ = 1.45|t1|, and the temperature T = 0.0001|t1|. Maier and Scalapino [9] first pointed out that, in the s±-wave SC states, there is a strong coherent peak in the dynamic spin susceptibility at an energy  below the two-gap value and at a wave vector QSDW = (π, 0) or (0, π). It can be easily seen that there are apparent peaks in the q space with the wave vectors q = (±π, 0) or (0, ±π) at each frequencies. This is due to a dynamical nesting effect. Fig. 3.2(a) illustrates the nesting effect at a frequency ω = Δ0. One can see, in Fig 3.2(a), that there are strongly coherent nesting effects with the wave vector q = (±π, 0) or (0, ±π) which drives the inter-band quasiparticles scattering. 17.

(26) Figure 3.1：The imaginary part of spin susceptibility χRPA (q, ω) with respect to different frequencies ω in s± state. The frequencies are from ω = 0.1|t1| to ω = 0.8|t1|. 18.

(27) Figure 3.2(a)：Energy contour Ek = ω in the BZ for ω = Δ0. There are strong coherent nesting effects with the wave vector q = (±π, 0) or (0, ±π) which drives the inter-band quasiparticles scattering.. Figure 3.2(b)：Energy contour Ek = ω in the BZ for ω = 0.65|t1|. At this high frequency, the nesting vectors change from commensurate into incommensurate q* = (π, 0.15π).. 19.

(28) As can be seen in Eq. (3.2), the BCS coherence factors that enter the spin-spin response function depend upon the sign of Δ(k+q)Δ(k). When this is negative, it become the maximum value and there can be a resonance response at ωc = min( | Δ(k+q) | + | Δ(k)| ). Obviously, this coherence factor becomes large because that Δ(k+QSDW) = -Δ(k), this condition is only met by s± pairing. Furthermore, at ω ~ ωc, almost all the k’s around the  electron and hole pockets can contribute to the spectra weight  (  E ( k )  E (k  q )) . At low frequencies, ω below the one-gap amplitude, the imaginary part of the susceptibility almost vanished due to the gap opening. When the frequency is increased, the Imχ experiences a discontinuous jump at ωc [see Fig. 3.6]. Besides, the commensurate (π, 0) peak of the susceptibility has shifted to an incommensurate wave vector q* = (π, 0.15π) [see Fig. 3.2(b)]. The nesting effect is quite sensitive to the geometry of the FS. Although the two-orbital model can reproduce the same-topology FS as found from the LDA, there are many fine structure of FS geometry, such as ellipticity, this model had no shown. It is not surprised, because this model had only taken two of the five most contributing orbits into account, and had no included any interactions to calculate the band structure. Nevertheless, this model has the virtue of simplicity while qualitatively capturing the shapes of the relevant band structure and reproduce FS. In light of the simplicity of this model, we could easily do explicit calculations of the Green function and response function just need to go through a 2 × 2 matrix in normal state, and a 4 × 4 matrix in SC state. To obtain more precise FS, there are many ways which beyond the two-orbital model, such as the three-orbital model [10], and five-orbital model [11]. S. –L. Yu et al. had used the fluctuation exchange (FLEX) approximation and three-orbital model to study the band renormalization, FS reconstruction [12]. They found that the inter-orbital spin fluctuations lead to the strong anisotropic band renormalization and the renormalization is orbital dependent. As a result, the topology of FS shows distinct variation with doping from the electron type to the hole type, which is consistent with the recent experiments. This shows that the Coulomb interactions will have a strong effect on the band renormalization and the topology of the electron Fermi pocket. To adequately describe the region of the FS of the Fe-pnictides one needs at least three orbits and the orbital-band matrix elements are known to play an essential role in determining the q dependence of the magnetic susceptibility [10]. T. A. Maier et al. had studied the neutron scattering resonance for a five-orbital within RPA-BCS approximation [11]. Their model consists of a five d-orbital tight-binding fit to the LDA band structure, and taken account of the orbital-band matrix elements. They found that, in the normal state, the commensurate peak q = QSDW of the undoped system has moved to an incommensurate wave vector q* = (π, 0.15π) for doping x = 0.125. This is due to the fine structure of the FS; the different geometry will generate distinct results. 20.

(29) In addition, we have also showed the frequency-dependent spin susceptibility with both extended-s wave [see Fig. 3.4] and dx2-y2 wave [see Fig. 3.5] in q space. Evidently, in the extended-s wave state, there is no resonance peak, such as in the s± wave state, at each of the frequencies. One can see that our results are qualitatively consistent with Maier and Scalapino [9]. Although the momentum transfer QAFM = (π, π) could make the gap function Δ(k+q) sign changing for extended-s wave and dx2-y2 wave, but due to the Fermi-surface geometry, there are no nesting vectors between the Femi surface. Based on the effective four-band model [13], Korshunov and Eremin had analyzed the spin response in the normal and SC states of Fe pnictides. Their model had neglected the orbital-band matrix elements, and also produced the same FS topology but has quite different fine structure with the FS, which we obtained. In their case, the AFM wave vector QAFM not only connected the electron and hole pockets but also matched the condition Δ(k) = -Δ(k+ QAFM) for the extended-s wave, thus it showed a resonance peak for extended-s wave. This quite different result with us is mainly also due to the different structure of FS. In the case of dx2-y2 wave the situation is more complicated. A single resonant pole will occur for all components of the RPA spin susceptibility at ω≦Ωc. At the same time even for this symmetry the resonance condition can be fulfilled due to the fact that Δ(k) = -Δ(k+ QAFM). However, because of the smallness of Ωc  Δ0 the total RPA susceptibility shows a moderate enhancement with respect to the normal state value as seen in Ch.2. Therefore, the resonance peak is pronounced only for the s± order parameter [see Fig. 3.3].. Figure 3.3：The imaginary part of the RPA spin susceptibility χRPA at QSDW = (π, 0) for various SC pairing symmetries. Here we set chemical potential μ = 1.45|t1|, U = 0.9|t1|, and gap amplitude Δ0 = 0.3|t1|. The blue line represents the result for s± state, the green line represents the extended-s state, and the red line represents the dx2-y2 state.. 21.

(30) Figure 3.4：The imaginary part of the spin susceptibility χRPA (q, ω) with the extended-s wave gap symmetry. The extended-s wave has the form Δ0(coskx + cosky)/2. 22.

(31) (a). (b) Figure 3.5：Energy contour Ek = ω of excitation with extended-s wave (a) and dx2-y2 wave (b) in the BZ. In Fig. 3.5(a), for ω = Δ0, there are just a few nesting vectors q = (0.7π±δ, 0.7π±δ) which drives intra-band quasiparticles scattering. In Fig. 3.5(b), while the frequency ω ＞ Δ0, there also are some nesting vectors q = (π±δ, π±δ).. 23.

(32) Figure 3.6：The imaginary part of the spin susceptibility χRPA (q, ω) with the dx2-y2 wave gap symmetry. The dx2-y2 wave has the form Δ0(coskx - cosky)/2. 24.

(33) Fig. 3.7 shows that the imaginary part of spin susceptibility for two different dopings. When the doping increase, the resonance peak was shifted to lower energy.. (a). (b) Figure 3.7：The doped imaginary part of the RPA spin susceptibility χRPA at QSDW = (π, 0) for various SC pairing symmetries. (a) hole-doped side (μ = 1.4|t1|) and (b) electron-doped side (μ = 1.6|t1|).. 25.

(34) 3.4. RPA-Corrected Excitation. In this section we will discuss the effect of RPA correction on the dynamical spin susceptibility. Now we consider the effect of electron-electron Coulomb interaction in this model. For the two d orbits we considered, the generic form of the interaction term can be written as. H int   (U  nis  nis   Vni1ni 2  JSi1  Si1 ) , i. (3.6). s. with U and V the intraband and interband Coulomb repulsion, and J the Hund’s coupling. For an isolated Fe atom, the intraband U and interband V are similar in magnitude (U = V), while J is one order smaller in magnitude [14]. Due to the two-band nature of the model we considered, the RPA correction is determined by the matrix equation Eq.(3.3), where Γ is the interaction vertex defined by.  U J 2  . J 2 U . (3.7). We note that the interband interaction V does not contribute to the RPA response when only the spin fluctuations are considered. We set J = 0, which makes the interaction vertex Γ in Eq.(3.7) proportional to the identity. The RPA enhancement of the spin fluctuations is determined by the real part of the  det I   (q ,  ) , which is greater or equal to zero. It makes a resonance occur when it is  approaching zero. In Fig.3.8, we show the real part of the det I   (q ,  ) . Evidently,. there is only a zero at interaction strength U = 0.9|t1|, this would be responsible for the resonance. We have also carried out the calculation for a finite Hund’s coupling J > 0, and find that the spin fluctuations are enhanced by increasing J, but the structure of χ(q) remains qualitatively the same. The resonant frequency were shifted to lower energy when magnitude of J increasing. Fig.3.9 shows the imaginary part of the spin susceptibility for s± pairing state with variation of the magnitudes J, the resonant peak has no obvious change when J < 0.2|t1| and appears a striking enhancement while the interaction strength J > 0.2|t1|. Furthermore, the resonant peak were strongly sharpened when J = U/3. Fig.3.9 (a) to (c) shows the spin fluctuation at different doping level, the energy of resonant peak is below an one-gap energy when the chemical potential μ < 1.45|t1|, and shifted to higher frequency when the chemical potential is increased. 26.

(35)  Figure 3.8：The real part of det I   (q ,  ) . The black line corresponding to the U =. 0.1|t1|, red line corresponding to the U = 0.3|t1|, green line corresponding to the U = 0.6|t1|, blue line corresponding to the U = 0.9|t1|. There is a resonance peak at frequency below two-gap energy when U = 0.9|t1|.. Although the strength of J is much small than U, we can still see that the effect of correction with Hund’s J is quite conspicuous and significant [see Fig. 3.9(d)]. The bare response is small and broadness compares with the RPA-corrected responses. Likewise, the extended-s and dx2-y2 states are also appearing similar effect. The Fig. 3.10 shows the imaginary part of the spin susceptibility for extended-s pairing state with variation of the magnitudes J, the resonant peak is also sharpened as well as the s± pairing state. When the chemical potential μ >1.6, this peak is no longer sharp but broad, the high-excitation frequency is extension to an energy window δω ~ 0.3Δ0. Fig. 3.11 shows the spin fluctuation for dx2-y2 state with variation of the magnitudes J. the spectrum has wide-interval enhancement within the two-gap energy when the chemical potential μ >1.45. Besides, the resonant peak near the one-gap energy has no shifted respect to the change of doping. 27.

(36) ` (a). (c). (b). (d). Figure 3.9：The imaginary part of the RPA spin susceptibility χRPA at QSDW = (π, 0) for different doping with the vertex includes inter-band interaction (J>0) for the s± pairing symmetry, each plot included the Coulomb interaction U = 0.9|t1|. (a) μ = 1.6|t1|, (b) μ = 1.45|t1| and (c) μ = 1.4|t1|. In (a), (b), and (c), the blue line represents the J = 0.3, the red line represents the J = 0.2, the green line represents the J = 0.1, and the black line represents the J = 0. In (d) red line represents the bare response, the green line represents the response with on-site interaction U, the blue line represents the response with Hund’s coupling.. 28.

(37) (a). (b). (c). (d). Figure 3.10：The imaginary part of the RPA spin susceptibility χRPA at QSDW = (π, 0) for different doping with the extended-s wave pairing symmetry, each plot included the Coulomb interaction U = 0.9|t1|. (a) μ = 1.6|t1|, (b) μ = 1.45|t1| and (c) μ = 1.4|t1|. In (d) red line represents the bare response, the green line represents the response with on-site interaction U, the blue line represents the response with Hund’s coupling.. 29.

(38) (a). (b). (c). (d). Figure 3.11：The imaginary part of the RPA spin susceptibility χRPA at QSDW = (π, 0) for different doping with the dx2-y2 pairing symmetry. (a) μ = 1.6|t1|, (b) μ = 1.45|t1| and (c) μ = 1.4|t1|. In (d) red line represents the bare response, the green line represents the response with on-site interaction U, the blue line represents the response with Hund’s coupling.. 30.

(39) References [1] J. Dong et al., Europhys. Lett. 83, 27006 (2008) [2] C. de la Cruz et al., Nature (London) 453, 899 (2008) [3] J. P. Lu, Phys. Rev. Lett 68, 125 (1992) [4] K. Seo, B. A. Bernevig, and J. Hu, Phys. Rev. Lett 101, 206404 (2008) [5] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett 101, 057003 (2008) [6] C. Fang, H. Yao, W. –F. Tsai, J. Hu, and S. A. Kivelson, Phys. Rev. B 77, 224509 (2008) [7] Q. Si and E. Abrahams, arXiv：0804.2480 (2008) [8] K. Nakayama, T. Sato, P. Richard, Y. –M. Xu, Y. Sekiba, S. Souma, G. F. Chen, J. L. Luo, N. L. Wang, H. Ding, and T. Takahashi, arXiv：0812.0663 (2008) [9] T. A. Maier and D. J. Scalapino, Phys. Rev. B 78, 020514 (2008) [10] P. A. Lee and X. -G. Wen, arXiv: 0804.1739 (2008). [11] T. A. Maier, S. Graser, D. J. Scalapino, and P. Hirschfeld, Phys. Rev. B 79, 134520 (2009) [12] S. –L Yu, J. Kang, J. -X Li, Phys. Rev. B 79, 064517 (2009) [13] M. M. Korshunov, I. Eremin, Phys. Rev. B 78, 140509 (2008) [14] I. Schnell et al.,Phys. Rev. B 68, 245102 (2003). 31.

(40) Chapter 4 Conclusion It should be fair to say that the two-band model has the virtue of simplicity to study iron-based superconductors analytically. However, it just can serve as a qualitative analysis, because there are too many important parts which were ignored, such as dxy (or dx2-y2) orbit, thus one can not obtain a precise FS to calculate quantitatively physical quantities. On the other hand, while the two-band model can not control doping accurately, it is quite important for studying iron-based high-temperature superconductors at the moment. With the two-band model, we have obtained results which are in good agreement with INS experiments and other theoretical studies. In our calculations, s± wave is most favored over the extended-s and d wave, and after the RPA correction with Hund’s rule coupling the resonance behaviors were changed significantly, this may hold an important key to verify that whether the pairing mechanism of iron-based superconductors is magnetic mediated. There is no definite conclusion at this time to say about the exact form of pairing symmetry of iron-based superconductors. Future experiments may pursue higher precision and more comprehensive data, and future theoretical works may aim to make more definite predictions among different scenarios.. 32.

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