Impurity states in antiferromagnetic Iron Arsenides

We explore theoretically impurity states in the antiferromagnetic spin-density wave state of the iron arsenide. Two types of impurity models are employed: one has only the intraband scattering while the other has both the intraband and interband scattering with the equal strength. Interestingly, the impurity bound state is revealed around the impurity site in the energy gap for both models. However, the impurity state is doubly degenerate with respect to spin for the first case; while the single impurity state is observed in either the spin-up or spin-down channel for the second one. The impurity-induced variations of the local density of states are also examined.

The recent discovery of iron-based superconductors [1] has triggered intensive efforts to unveil the nature of and interplay between magnetism and superconductivity in this family of materials. Series of iron arsenide have been synthesized, which possess many similar features of the normal and superconducting states. Experimental measurements have reported that the undoped ReFeAsO (where Re= rare-earth metals) and AFe 2 As 2 (where A=divalent metals such as Ba, Ca, Sr) compounds exhibit a long-range antiferromagnetic spin-density-wave (SDW) order [2,3,4,5,6,7]. Upon electron/hole doping the SDW phase is suppressed and superconductivity emerges with T c up to above 50 K [8,9,10,11,12].
At present, there is likely certain controversy on the understanding of the SDW state of the undoped FeAs-based parent compounds. Two kinds of theories have been put forward: 1) the itinerant antiferromagnetism, which takes advantage of proper Fermi surface (FS) nesting (or strong scattering) between different FS sheets [13,14,15,16]; and 2) the frustrated Heisenberg exchange model of coupled magnetic moments of the localized d-orbital electrons around the Fe atoms [17,18,19,20]. As for the itinerant electronic behavior, first principle band structure calculations [21] based on the density functional theory (DFT) indicate up to five small Fermi pockets with three hole-like pockets centered around the Γ point and two electron-like ones centered around the M point of the folded Brillouin zone of the FeAs layers, which have partially supported by the angle-resolved photoemission spectroscopy (ARPES) from different groups [22,23,24,25,26]. Motivated by the DFT calculation and experimental measurements, in Refs. [15,16], the excitonic mechanism [27] of itinerant carriers are employed taking account of the FS nesting between electron and hole pockets and the SDW phase are associated with triplet excitonic state, which can be understood as condensate of triplet electron-hole pairs [27].
In this paper, we explore theoretically the effect of a single impurity on the local electronic structure of an Febased antiferromagnet in the triplet excitonic phase. It is shown that impurity bound states are formed inside the SDW gap, which may be observed experimentally by local probes. Before introducing the impurity, we first propose an effective model Hamiltonian to address the triplet excitonic state, where Γ = (0, 0), X = (π, 0). We use the index i to label different valence bands around Γ point. Around X and Y points, there are two conduction bands. d ikσ and c k+Xσ are the annihilation operators of electrons in the Γi and X bands. Theoretically X and Y are two equivalent nesting directions. Note that, the structural phase transition occurred just above/on the SDW transition breaks this equivalency. Without loss of generality, it is assumed that only conduction band around the X point couples with the valence bands around the Γ point, which is characterized by the mean-field order parameters ∆ iσσ ′ . For the triplet excitonic phase (SDW), we have real order parameters satisfying ∆ i↑↑ = −∆ i↓↓ and ∆ i↑↓ = ∆ i↓↑ = 0 [27].
ε Γi (k) and ε X (k) are used to denote the band dispersions of the nonmagnetic normal state. For k in the vicinity of the Γ point (therefore, k + X in the vicinity of the X point), the normal-state energy dispersions have ap-  proximately the 2D parabolic forms as schematically shown in Fig. 1. Here m Γi and m X are the corresponding effective masses. In describing the X band, the elliptic FS is approximated by the circular one for simplicity. ǫ Γi 0 (ǫ X 0 ) denotes the top (bottom) of the hole (electron) bands. According to the ARPES measurement [22], two hole-like Fermi pockets are revealed around the Γ point for undoped BaFe 2 As 2 . The band parameters extracted from the experimental data are as follows. m Γ1 ≈ 2.8m e , m Γ2 ≈ 7.4m e , and m X ≈ 6.5m e , where m e is the mass of bare electron. ε Γ1 0 ≈ 4 meV, ε Γ2 0 ≈ 16 meV, and ε X 0 ≈ 24 meV. These parameters indicate that the nesting between the Γ2 band and X band is much better than that of the Γ1 band. Therefore it is natural to assume a larger order parameter ∆ 2 and a vanishingly small ∆ 1 .
Here E G = −2E g denotes the indirect gap between the top of the Γ band and the bottom of the X bands. Therefore, E g > 0 describes a semimetal and E g < 0 a semiconductor. With the help of E g and µ 0 and a further assumption of m Γ2 = m X = m ≈ 7m e . we can re-express the energy dispersions as Note that for µ 0 = 0, the hole and electron bands are perfectly nested since ε Γ2 (k) = −ε X (k + X) and the system is unstable with respect to infinitesimal Coulomb interaction while for nonzero µ 0 finite strength of Coulomb repulsion is needed.
For the reason that the order parameter ∆ 1 is set to zero, there is no coupling between the Γ1 band and the X band. The Hamiltonian of Eq. (1) is reduced to a model of two bands with one valence band (Γ2 band) and one conduction band (X band). Introducing the twocomponent Nambu operator,ψ † kσ = (d † 2kσ , c † k+Xσ ), the model Hamiltonian can be simplified aŝ where an impurity term has been added with the form, whereÛ k,k ′ represents a 2 × 2 matrix of the scattering potential associated with non-magnetic impurities. Here we use ∆ σ to denote ∆ σσ for short. The Green's function method is applied to study the single impurity effect. The matrix Greens functions are defined aŝ From the Hamiltonian defined in Eq. (7) we can derive the bare Green's function whereω = ω +µ 0 .τ 0 is the 2×2 unit matrix, andτ 1,3 are the pauli matrices. The T-matrix approximation is employed to compute the Green's function in the presence of impurities. For a single impurity, the T-matrix exactly accounts for the multiple scattering off the impurity. The single-particle Green's functionĜ can be obtained from the following Dyson's equation, where the T matrix is given bŷ (15) For a point-like scattering potential interacting with itinerant carriers just on the impurity site, the scattering matrix is isotropic,Û k,k ′′ =Û. The above equation is greatly simplified whereĜ 0 σσ (ω) = kĜ 0 σσ (k, ω). After some derivation we obtain with E c denoting the high-energy cutoff and N 0 = ma 2 /(2πh 2 ) the density of states per band per spin. Note that α(ω) and γ(ω) are independent of the spin index σ. The first impurity model we study is the scatteringpotential matrix with only intraband scattering terms, i.e.Û = V impτ0 , which was adopted in Ref. [28] to study effect of many impurities. From Eq. (16), we obtain The energy of the impurity bound state is determined by the pole ofT σσ (ω), determined by det

we have the equation for the energy of impurity bound state
For the spin triplet excitonic phase ∆ ↑ = −∆ ↓ , the above equation gives rise to impurity states with the same bound energy, i.e. the impurity states are doubly degenerate. Generally, the above equation has to be solved numerically to obtain the bound energyΩ. However, we can get some analytic results under certain approximations. Under the wide-band approximation E c , E g ≫ |∆ σ |, α(Ω) ≈ 1 and γ(Ω) ≈ γ 0 = π −1 ln(E c /E g ), so we havẽ and furthermore if the system has approximately the particle-hole symmetry E c ≈ E g , then γ 0 ≈ 0 and Ω/|∆ σ | = sgn(c)(1−c 2 )/(1+c 2 ) from the above equation.
For the second impurity model, the four matrix elements ofÛ is assumed to be the same, i.e. the intraand inter-band scattering terms are the same [29] witĥ U = V imp (τ 0 +τ 1 )/2. Then the T-matrix according to Eq. (16) isT withτ ′ =τ 0 +τ 1 . The energy of the impurity bound state is again determined by the pole ofT σσ (ω). From det[2V −1 imp −τ ′Ĝ0 σσ (Ω)] = 0 we have the equation forΩ, From the above equation we find thatΩ is independent of the function of γ(ω) for this case, which reflects the particle-hole asymmetry. Before solving the above equation for the bound energy, we study the existence of the impurity state. BecauseΩ 2 < ∆ 2 σ ,Ω + ∆ σ has the same sign as that of ∆ σ . Therefore, the solution of Eq. (24) exists only if the sign of c is opposite to that of ∆ σ . For the SDW state, i.e. the triplet excitonic phase, we have ∆ ↑ = −∆ ↓ and so there is exactly one impurity bound state in either the spin-up or spin-down channel. We may assume ∆ ↑ = −∆ ↓ = ∆ > 0 as well, then for attractive scattering V imp < 0, the impurity bound state only exists in the spin-up channel and its energy is given bỹ Ω/∆ = −(1 − c 2 )/(1 + c 2 ) under the wide-band approximation. If V imp > 0, however, the impurity state will be in the spin-down channel, andΩ/∆ = (1 − c 2 )/(1 + c 2 ). In general, the impurity bound-state energy is given bỹ in the valid regime of the wide-band approximation.
To apply the theoretical results to the iron arsenide, we try to pin down the parameters of our model by extracting them from the available experimental data for BaFe 2 As 2 [22]. ǫ Γ 0 ≈ 16 meV and ǫ X 0 ≈ 24 meV so that E g ≈ 20 meV. m Γ ≈ m X ≈ 7.0 m e and therefore N 0 ≈ 1.2 eV −1 . ∆ ↑ = −∆ ↓ = ∆ ≈ 20 meV. The high-energy cutoff is set as E c = 500 meV, which is of the same order of magnitude as the band width. Note that E g extracted from experimental data is very small, which is in the same order of magnitude of the order parameter ∆. Therefore, neither the wide-band approximation nor the particle-hole symmetry can be applied to the present case. Eqs. (21) and (24) have to be numerically solved. Now we examine the local characteristics induced by the impurity by looking into the variation of the local density of states (LDOS), which can be probed by the scanning tunneling microscopy (STM). The LDOS is defined as whereĜ σσ (r, r ′ , ω) the Green's function in real space. Applying the T-matrix approximation we have, Substituting Eq. (27) into Eq. (26) we may single out the variation of LDOS due to the presence of the impurity potential, For the second impurity model[30], Fig. 2(a) shows the LDOS as a function of energyω on the impurity site, namely N (0,ω), while Fig. 2(b) the impurity-induced LDOS at the bound energy as a function of radial distance r off the impurity site, i.e. N imp (r,Ω). V imp has been set as -0.36, -0.6, and -4.0 eV, giving rise to the impurity bound states seen as the sharp peaks located respectively at the energiesΩ/∆ = 0, −0.5, and −0.99 in Fig. 2(a). The probability densities of these bound states exhibit a kind of exponential decay with the Friedel oscillation, as seen in Fig. 2(b). Introducing two length scales, ξ 1 and ξ 2 to characterize the oscillation and decay, we obtain the asymptotic behavior of N imp (r,Ω) for large r, N imp (r,Ω) ∝ r −1 cos 2 (πr/ξ 1 ) exp(−r/ξ 2 ).
ξ 1 /a = π/ 4πN 0 E g which is approximately 5.7 in consistence with the numerical results shown in Fig.2(b). ξ 2 /ξ 1 = E g /(π ∆ 2 −Ω 2 ) = 0.32, 0.37, and 2.3 for the three cases of impurity states. This explains why we see clear Friedel oscillation for impurity state with bound energy near the gap edge. This work was supported by the NSFC grand under Grants Nos. 10674179 and 10429401, the GRF grant of Hong Kong.