2 Two-Band Model of Iron-Based Superconductors
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 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|.
(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.
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)
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 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.
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,
( , ) [1 ( )] Im ( , )
S q n 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) 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, kikF.
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, ii). In principle, one should do
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
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)
(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
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.
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|.
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).
As can be seen in Eq. (3.2), the BCS coherence factors that enter the spin-spin response 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.
In addition, we have also showed the frequency-dependent spin susceptibility with both 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.
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.
(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).
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.
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|).
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
int is is i1 i2 i1 i1
i s
H ( U n n
Vn n JS S )
, (3.6) 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 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 ( , )
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.
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.
`
(a) (b)
(c) (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.
(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.
(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.
References
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[5] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett 101, 057003 (2008)
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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.