1. Introduction
The recent discovery of superconductivity (SC) with rather high critical temperature in the family of doped iron pnictide compounds [1, 2], has motivated a great interest to these materials (see the reviews [3, 4]). Unlike the extensively studied cuprate family [5], that present insulating properties in their initial undoped state, the undoped LaOFeAs compound is a semimetal. As was established by the previous physical and chemical studies (see, e.g., [6, 7]), this material has a layered structure, where the SC state is supported by the FeAs layer with a 2D square lattice of Fe atoms and with As atoms located out of plane, above or below the centers of square cells (Fig. 1). Its electronic structure, relevant for constructing microscopic SC models, have been explored with high-resolution angle-resolved photoemission spectroscopy (ARPES) techniques [8, 9]. Their results indicate the multiple connected structure of Fermi surface, consisting of electron and hole pockets and absence of nodes in both electron and whole spectrum gaps [8], suggesting these systems to display the so-called extended
To study the band structure, the first principles numeric calculations are commonly used, outlining the importance of Fe atomic
Having established the SC state parameters, important effects of disorder, in particular by impurities, on the system electronic properties, have been studied for doped iron pnictides. Alike the situation in doped perovskite cuprates, impurity centers can result here either from dopants, necessary to form the very SC state, or from foreign atoms and other local defects. Within the minimal coupling model, an interesting possibility for localized impurity levels within SC gaps in doped LaOFeAs was indicated, even for the simplest, so-called isotopic (or non-magnetic) type of impurity perturbation [24, 25]. This finding marks an essential difference from the traditional SC systems with \emph{s}-wave gap on a single-connected Fermi surface, were such perturbations are known not to produce localized impurity states and thus to have no sizeable effect on SC order, accordingly to the Anderson theorem [26].
In presence of localized quasiparticle states by isolated impurity centers, the next important issue is the possibility for collective behavior of such states at finite (but low enough) impurity concentration. They are expected to give rise to some resonance effects like those well studied in semiconductors at low doping concentrations [27]. This possibility was studied long ago for electronic quasiparticles in doped semiconducting systems [28] and also for other types of quasiparticles in pnononic, magnonic, excitonic, etc. spectra under impurities [29], establishing conditions for collective (including coherent) behavior of impurity excitations. Thus, indirect interactions between impurity centers of certain type (the so-called deep levels at high enough concentrations) in doped semiconductors can lead to formation of collective band-like states [28, 30]. This corresponds to the Anderson transition in a general disordered system [31], and the emerging new band of quasiparticles in the spectrum can essentially change thermodynamics and transport in the doped material [32]. In this course, the fundamental distinction between two possible types of states is done on the basis of general Ioffe-Regel-Mott (IRM) criterion that a given excitation has a long enough lifetime compared to its oscillation period [32, 33].
Analogous effects in superconductors were theoretically predicted and experimentally discovered for magnetic impurities, both in BCS systems [34-36] and in the two-band MgB2 system [37, 38]. In all those cases, the breakdown of the Anderson's theorem is only due to the breakdown of the spin-singlet symmetry of an
Therefore the main physical interest in SC iron pnictides from the point of view of disorder in general is the possibility for pair-breaking even on non-magnetic impurity [41-43] and for related localized in-gap states [21, 44-46]. This theoretical prediction was confirmed by the observations of various effects from localized impurity states, for instance, in the superfluid density (observed through the London penetration length) [47, 48], transition critical temperature [49, 50] and electronic specific heat [51], all mainly due to an emerging spike of electronic density of states against its zero value in the initial band gap. An intriguing possibility for banding of impurity levels within the SC gap [38, 52], similar to that in the above mentioned normal systems, was recently discussed for doped iron pnictides [53]. Here a more detailed analysis of the band-like impurity states is also focused on their observable effects that cannot be produced by localized impurity states.
We apply the Green function (GF) techniques, similar to those for doped cuprate SC systems [54], using the minimal coupling model by two orbitals for host electronic structure and the simplest isotopic type for impurity perturbation. The energy spectrum near in-gap impurity levels at finite impurity concentrations, emergence of specific branches of collective excitations in this range, and expected observable effects of such spectrum restructuring are discussed. Then specific GFs for SC quasiparticles are used in the general Kubo-Greenwood formalism [55, 56] to obtain the temperature and frequency dependences of optical conductivity. These results are compared with available experimental data and some suggestions are done on possible practical applications.
2. Model Hamiltonian and Green functions
For the minimal coupling model of Fig. 1, the hopping Hamiltonian
where
Here the 2×2 matrix
includes the Pauli matrices
with
(
Here
correspond to the two subbands in the normal state spectrum that respectively define electron and hole pockets of the Fermi surface. There are two segments of each type, defined by the equations
The adequate basis for constructing the SC state is generated by the operators of electron and hole subbands:
giving rise to the "multiband-Nambu" 4-spinors
where the 4×4 matrix
includes the Pauli matrices
The electronic dynamics of this system is determined by the (Fourier transformed) GF 4×4 matrices [58, 29, 54]:
whose energy argument
the explicit GF for the unperturbed SC system with the Hamiltonian
where the denominators
3. Impurity perturbation and self-energy
We pass to the impurity problem where local perturbation terms due to non-magnetic impurities [24] on random sites
are added to the Hamiltonian
through the 4×4 scattering matrix
Following Refs. [29, 53], the solution for Eq. (10) with the perturbed Hamiltonian
by specific routines of its iterating for the "scattered" GF's
where the self-energy matrix
Here
The next term to the unity in the brackets in Eq. (16):
describes the effects of indirect interactions in pairs of impurities, separated by vector
An alternative iteration routine applies Eq. (14) to
The NRF self-energy admits a GE:
At the first step, we restrict GE to the common T-matrix level to find the possibilities for localized quasiparticle states and related in-gap energy levels by single impurities [21]. Next, at finite impurity concentrations, formation of (narrow) energy bands of specific collective states near these levels is studied. Finally, the criteria for such collective states to really exist in the disordered SC system follow from the analysis of non-trivial GE terms. We notice that RF for GF's
This diagonal form (restricted only to the "intraband" matrix elements) follows from the aforementioned cancellation of integrals with cos
The functions
Here
it presents two symmetric poles within the gap, at
where
First of all, there are modified initial bands
whose main difference from the unperturbed SC bands with dispersion
the position of mobility edge
As follows from Eq. (25), the
All these spectrum bands would contribute to the overall density of states (DOS) by the related quasiparticles:
at
More peculiar contributions to DOS come from the
at
Formation of the
4. Group expansion and Ioffe-Regel-Mott criteria
Let us now study the crossover from band-like to localized states near the limits of
To simplify calculation of the scalar function Γ
both proportional to the matrix
where the length scales both for the monotonous decay:
and for the sine factor:
Now calculation of Γ
(since the
At next stage, summation of these results in
Numerical calculation of the latter integral results in:
where the function
giving an (
Its comparison with the full extension of this band,
For typical values of
However, the r. h. s. of Eq. (33) vanishes at
Under the condition of Eq. (36), this vicinity is yet narrower than Γ0 by Eq. (35), defining the true, even wider, range of extended states within the impurity band.
Otherwise, for
with
defining its broadening due to inter-impurity interactions. Within this range, the DOS function for localized states can be only estimated by the order of magnitude, but outside it is given by:
Notably, the total number of states near the impurity level is
alike that of extended states in the impurity band by Eq. (26). The summary of evolution of this area of quasiparticle spectrum in function of impurity concentration is shown in Fig. 7.
5. Impurity effects on superconducting characteristics
The above results on the quasiparticle spectrum in the disordered SC system can be immediately used for calculation of impurity effects on its observable characteristics. Thus the fundamental SC order parameter ∆ is estimated from the modified gap equation:
where
For finite
For
and leads to the modified gap equation as:
Its approximate solution for
and for the values of
To study another important dependence, that of the SC transition temperature
It is of interest to compare the present results with the known Abrikosov-Gor'kov solution for BCS SC with paramagnetic impurities in the Born approximation [59, 60]. In that approximation, the only perturbation parameter is the (constant) quasiparticle lifetime
Also, a notable impurity effect is expected on the London penetration depth
When compared to its unperturbed value in the pure SC system
the effect by the last term in Eq. (45) produces a considerable slowing down of the low-temperature decay of the difference
Finally, a similar analysis can be applied for the impurity effect on the electronic specific heat in the SC state, whose dependence on inverse temperature
and naturally divided in two characteristic contributions,
The resulting function
The same approach can be then used for other observable characteristics for SC under impurity effect, such as, e.g., heat conductivity, differential conductivity for scanning tunneling spectroscopy or absorption coefficient for far infrared radiation that is done in the next section.
6. Kubo-Greenwood formalism for multiband superconductor
The relevant kinetic coefficients for electronic processes in the considered disordered superconductor follow from the general Kubo-Greenwood formulation [55, 56], adapted here to the specific multiband structure of Green function matrices. Thus, one of the basic transport characteristics, the (frequency and temperature dependent) electrical conductivity is expressed in this approach as:
for
This function is defined in the whole
Other relevant quantities are the static (but temperature dependent) transport coefficients, as the heat conductivity:
and the thermoelectric coefficients associated with the static electrical conductivity
and the Seebeck coefficient
It is worth to recall that the above formulae are only contributed by the band-like states, that is the energy arguments
Next, we consider the particular calculation algorithms for the expressions, Eqs. (47, 49, 50), for the more involved case of dynamical conductivity, Eq. (47), that can be then reduced to simpler static quantities, Eqs. (49, 50).
7. Optical conductivity
The integral in Eq. (47) is dominated by the contributions from
For practical calculation of each contribution, the relevant matrix Im
is a smooth enough function while the above referred peaks result from zeros of Re
where
Here
but within the
Within the
Since four peaks in Eq. (47) are well separated, the
for
with characteristic scale
vs the threshold frequency
with the Dawson function
Calculation of the
and the energy integration limits:
at 0 <
Extrapolation of these asymptotics to the center of impurity band gives an estimate for the maximum of
To emphasize, the considered impurity features in optical conductivity cannot be simply treated as optical transitions between localized impurity states (or between these and main bands) since localized states can not contribute to currents. Such effects only appear at high enough impurity concentrations,
8. Concluding remarks
Resuming, the GF analysis of quasiparticle spectra in SC iron pnictides with impurities of simplest (local and non-magnetic) perturbation type permits to describe formation of impurity localized levels within SC gap and, with growing impurity concentration, their evolution to specific bands of extended quasiparticle states, approximately described by quasimomentum but mainly supported by the impurity centers. Explicit dispersion laws and densities of states are obtained for the modified main bands and impurity bands. Further specification of the nature of all the states in different energy ranges within the SC gap is obtained through analysis of different types of GEs for self-energy matrix, revealing a complex oscillatory structure of indirect interactions between impurity centers and, after their proper summation, resulting in criteria for crossovers between localized and extended states. The found spectral characteristics are applied for prediction of several observable impurity effects.
Besides the thermodynamical effects, expected to appear at all impurity concentrations, that is either due to localized or band-like impurity states, a special interest is seen in the impurity effects on electronic transport in such systems, only affected by the impurity band-like states. It is shown that the latter effects can be very strongly pronounced, either for high-frequency transport and for static transport processes. In the first case, the strongest impurity effect is expected in a narrow peak of optical conductance near the edge of conductance band in non-perturbed crystal, resembling the known resonance enhancement of impurity absorption (or emission) near the edge of quasiparticle band in normal systems. The static transport coefficients at overcritical impurity concentrations are also expected to be strongly enhanced compared to those in a non-perturbed system, including the thermoelectric Peltier and Seebeck coefficients.
The proposed treatment can be adapted for more involved impurity perturbations in SC iron pnictides, including magnetic and non-local perturbations, and for more realistic multiorbital structures of the initial iron pnictide system. Despite some quantitative modifications of the results, their main qualitative features as possibility for new narrow in-gap quasiparticle bands and related sharp resonant peaks in transport coefficients should be still present. The experimental verifications of such predictions would be of evident interest, also for important practical applications, e.g., in narrow-band microwave devices or advanced low-temperature sensors, though this would impose rather hard requirements on quality and composition of the samples, to be extremely pure aside the extremely low (by common standards) and well controlled contents of specially chosen and uniformly distributed impurity centers. This can be compared to the requirements on doped semiconductor devices and hopefully should not be a real problem for modern lab technologies.
Acknowledgments
Y.G.P. and M.C.S. acknowledge the support of this work through the Portuguese FCT project PTDC/FIS/101126/2008. V.M.L. is grateful to the Special Program of Fundumental Research of NAS of Ukraine.
Parts of the chapter are reproduced from the authors' previous publication [53]
References
- 1.
Kamihara Y, Hiramatsu H, Hirano M, Kawamura R, Yanagi H, Kamiya T, Hosono H. Iron-Based Layered Superconductor: LaOFeP. J. Am. Chem. Soc. 2006;128:10012–10013. DOI: 10.1021/ja063355c - 2.
Kamihara Y, Watanabe T, Hirano M, Hosono H. Iron-based Layered Superconductor La[O1-xFx]FeAs (x=0.05-0.12) with Tc = 26K. J. Am. Chem. Soc. 2008;130:3296-3297. DOI: 10.1021/ja800073m - 3.
Sadovskii MV. High-temperature superconductivity in iron-based layered iron compounds. Phys. Uspekhi. 2008;178:1201-1227. DOI: 10.1070/PU2008v051n12ABEH006820 - 4.
Izyumov YA, Kurmaev EZ. Materials with strong electron correlations. Phys. Uspekhi. 2008;51:23-56. DOI: 10.1070/PU2008v051n01ABEH006388 - 5.
Ginsberg DM, editor. Physical Properties of High Temperature Superconductors I. World Scientific;1989. 509 p. ISBN: 9971-50-683-1 9971-50-894-X(pbk) - 6.
Takahachi H, Igawa K, Arii K, Kamihara Y, Hirano M, Hosono H. Superconductivity at 43K in an iron-based layered compound LaO1-xFxFeAs. Nature 2008;453:376-378. DOI: 10.1038/nature06972 - 7.
Norman MR. High-temperature superconductivity in the iron-pnictides. Physics 2008;1: 21-26. DOI: 10.1103/Physics.1.21.6 - 8.
Ding H, Richard P, Nakayama K, Sugawara K, Arakane T, Sekiba Y, Takayama A, Souma T, Takahashi T, Wang Z, Dai X, Fang Z, Chen GF, Luo JL, Wang NL. Observation of Fermi-surface-dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2. Europhysics Lett. 2008;83:47001. DOI: 10.1209/0295-5075/83/47001 - 9.
Kondo T, Santander-Syro AF, Copie O, Liu C, Tillman ME, Mun ED, Schmalian J, Bud’ko SL, Tanatar MA, Canfield PC, Kaminski A. Momentum Dependence of the Superconducting Gap in NdFeAsO0.9F0.1. Phys. Rev. Lett. 2008;101:147003. DOI: http://dx.doi.org/10.1103/PhysRevLett.101.147003 - 10.
Mazin II, Singh DJ, Johannes MD, Du MH. Unconventional Superconductivity with a Sign Reversal in the Order Parameter of LaFeAsO1-xFx. Phys. Rev. Lett. 2008;101:057003. DOI: http://dx.doi.org/10.1103/PhysRevLett101.057003 - 11.
Luetkens H, Klauss HH, Kraken M, Litterst FJ, Dellmann T, Klingeler R, Hess C, Khasanov R, Amato A, Baines C, Kosmala M, Schumann OJ, Braden M, Hamann-Borrero J, Leps N, Kondrat A, Behr G, Werner J, Buechner B. The electronic phase diagram of the LaO1-xFxFeAs superconductor. Nature Materials Lett. 2008; DOI: 10.1038/nmat2397 - 12.
Singh DJ, Du MH. Density Functional Study of LaFeAsO1-xFx: A Low Carrier Density Superconductor Near Itinerant Magnetism. Phys. Rev. Lett. 2008;100:237003. DOI: http://dx.doi.org/10.1103/PhysRevLett100.237003 - 13.
Haule K, Shim JH, Kotliar G, Correlated Electronic Structure of LaO1-xFxFeAs. Phys. Rev. Lett. 2008;100:226402. DOI: http://dx.doi.org/10.1103/ PhysRevLett100.226402 - 14.
Xu G, Ming W, Yao Y, Dai X, Zhang SC, Fang Z. Doping-dependent phase diagram of LaOMAs (M=V-Cu) and electron-type superconductivity near ferromagnetic instability. et al , Europhys Lett. 2008;82 :67002. DOI : 10.1209/0295-5075/82/67002 - 15.
Raghu S, Qi XL, Liu CX, Scalapino DJ, Zhang SC. Minimal two-band model of superconducting iron oxypnictides. Phys. Rev. B 2008;77:220503. DOI: http://dx.doi.org/10.1103/PhysRevB.77.220503 - 16.
Kuroki K, Onari S, Arita R, Usui H, Tanaka Y, Kontani H, Aoki H. Unconventionaol Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsO1-xFx. Phys. Rev. Lett. 2008;101:087004. DOI: http://dx.doi.org/10.1103/PhysRevLett.101.087004 - 17.
Boeri L, Dolgov OV, Golubov AA. Is LaFeAsO1-xFx an Electron-Phonon Superconductor ? Phys. Rev. Lett. 2008;101:026403. DOI: http://dx.doi.org/10.1103/PhysRevLett.101.026403 - 18.
Gao Y. Interorbital Pairing and its Physical Consequences in Iron Pnictide Superconductors. Phys. Rev. B 2010;81:104504. DOI: 10.1103/PhysRevB.81.104504 - 19.
Si Q, Abrahams E. Strong correlations and magnetic frustration in the high Tc iron pnictides. Phys. Rev. Lett. 2008;101:076401. DOI: http://dx.doi.org/10.1103/PhysRevLett.101.076401 - 20.
Daghofer M, Moreo A, Riera JA, Arrigoni E, Scalapino DJ, Dagotto E. Model for the Magnetic Order and Pairing Channels in Fe Pnictide Superconductors. Phys. Rev. Lett. 2008;101:237004. DOI: http://dx.doi.org/10.1103/PhysRevLett.101.237004 - 21.
Tsai WF, Zhang YY, Fang C, Hu JP. Impurity-induced bound states in iron-based superconductors with s-wave cos kx cos ky pairing symmetry. Phys. Rev. B 2009;80:064513. DOI: http://dx.doi.org/10.1103/PhysRevB.80.064513 - 22.
Graser S, Maier TA, Hirschfeld PJ, Scalapino DJ. Near-degeneracy of several pairin g channels in multiorbital models for the Fe pnictides. New J. Phys. 2009 ;11:025016. DOI: 10.1088/1367-2630/11/2/025016 - 23.
Maier TA, Graser S, Scalapino DJ, Hirschfeld PJ. Origin of gap anisotropy in spin fluctuation models of the iron pnictides. Phys. Rev. B 2009;79:224510. DOI: http://dx.doi.org/10.1103/PhysRevB.79.224510 - 24.
Zhang D. Nonmagnetic Impurity Resonances as a Signature of Sign-Reversal Pairing in FeAs-Based Superconductors. Phys. Rev. Lett. 2009;103:186402. DOI: http://dx.doi.org/10.1103/PhysRevLett.103. 186402 - 25.
Zhang YY, Fang C, Zhou X, Seo K, Tsai WF, Bernevig BA, Hu J. Quasiparticle scattering interference in superconducting iron pnictides. Phys. Rev. B 2009;80:094528. DOI: http://dx.doi.org/10.1103/PhysRevB.80. 094528 - 26.
Anderson PW. Theory of dirty superconductors. J. Phys. Chem. Solids 1959;11:26-30. DOI: 10.1016/0022-3697(59)90036-8 - 27.
Shklovskii BI, Efros AL. Electronic properties of doped semiconductors, Springer-Verlag, 1984, 387 p. DOI: 10.1007/978-3-662-02403-4 - 28.
Ivanov MA, Pogorelov YG, Electron Properties of Two-Parameter Long-Range Impurity States. Sov. Phys. JETP 1985;61:1033-1039. DOI: 10.1134/0038-5646/85/051033-07 - 29.
Ivanov MA, Loktev VM, Pogorelov YG, Long-range impurity states in magnetic crystals. Phys. Reports 1987;153:209-330. DOI: 10.1016/0370-1573(87)90103-7 - 30.
Yamamoto H, Fang ZQ, Look DC. Nonalloyed ohmic contacts on low-temperature molecular beam epitaxial GaAs: Influence of deep donor band. Appl. Phys. Lett. 1990;57:1537-1539. DOI: http://dx.doi.org/10.1063/1.103345 - 31.
Anderson PW. Absence of Diffusion in Certain Random Lattices. Phys. Rev. 1958;109:1492-1505. DOI: http://dx.doi.org/10.1103/PhysRev.109.1492 - 32.
Mott NF. Electrons in disordered structures. Adv. Phys. 1967;16:49-144. DOI: 10.1080/00018736700101265 - 33.
Ioffe AF, Regel AR. Non-crystalline, amorphous and liquid electronic semiconductors. Prog. Semicond. 1960;4:237-291. - 34.
Shiba H. Classical Spins in Superconductors. Prog. Theor. Phys. 1968;40:435-451. DOI: 10.1143/PTP.40.435 - 35.
Rusinov AI. On the Theory of Gapless Superconductivity in Alloys Containing Paramagnetic Impurities. Sov. Phys. JETP 1969;29:1101-1106. DOI: 10.1134/0038-5646/69/061101-06 - 36.
Maki K. Anomalous Scattering by Magnetic Impurities in Superconductors. Phys. Rev. 1967;153:428-434. DOI: http://dx.doi.org/10.1103/PhysRev.153.428 - 37.
Gonnelli RS, Daghero D, Ummarino, Calzolari A, Tortello M, Stepanov VA, Zhigadlo ND, Rogacki K, Karpinski J, Bernardini F, Massidda S. Effect of Magnetic Impurities in a Two-Band Superconductor : A Point-Contact Study of Mn-Substituted MgB2 Single Crystals. Phys. Rev. Lett. 2006;97:037001. DOI: http://dx.doi.org/10.1103/PhysRevLett.97. 037001 - 38.
Moca CP, Demler E, Janko B, Zarand G. Spin-resolved spectra of Shiba multiplets from Mn impurities in MgB2. Phys. Rev. B 2008;77:174516. DOI: http://dx.doi.org/10.1103/PhysRevB.77.174516 - 39.
Balatsky AV, Salkola MI, Rosengren A. Impurity-induced virtual bound states in d-wave superconductors. Phys. Rev. B 1995;51:15547. DOI: http://dx.doi.org/10.1103/PhysRevB.51.15547 - 40.
Pogorelov YG, Ground state symmetry and impurity effects in superconductors. Sol. St. Commun. 1995;95:245-249. DOI: 10.1016/0038-1098(95)00260-X - 41.
Onari S, Kontani H. Violation of Anderson’s Theorem for the Sign-Reversing s-Wave State in Iron-Pnictide Superconductors. Phys. Rev. Lett. 2009;103:177001. DOI: http://dx.doi.org/10.1103/PhysRevLett.103.177001 - 42.
Kontani H, Onari S. Orbital-Fluctuation-Mediated Superconductivity in Iron Pnictides: Analysis of the Five-Orbital Hubbard-Holstein Model. Phys. Rev. Lett. 2010;104157001. DOI: http://dx.doi.org/10.1103/PhysRevLett.104.157001 - 43.
Efremov DV, Korshunov MM, Dolgov OV, Golubov AA, Hirschfeld PJ. Disorder-induced transition between s± and s++ states in two-band superconductors. Phys. Rev. B 2011;84:180512. DOI: http://dx.doi.org/10.1103/PhysRevB.84.180512 - 44.
Senga Y, Kontani H. Impurity Effects in Sign Reversing Fully-Gapped Superconductors: Analysis of FeAs Superconductors. Journ. Phys. Soc. Japan 2008;77:113710. DOI: 10.1143/JPSJ.77.113710 - 45.
Kariado T, Ogata M. Single-Impurity Problem in Iron-Pnictide Superconductors. Journ. Phys. Soc. Japan 2010;79:083704. DOI: http://dx.doi.org/10.1143/JPSJ.79.083704 - 46.
Beaird R, Vekhter I, Zhu JX, Impurity states in multiband s-wave superconductors: Analysis of iron pnictides. Phys. Rev. B 2012;86:140507. DOI: http://dx.doi.org/10.1103/PhysRevB.86.140507 - 47.
Gordon RT, Kim H, Tanatar MA, Prozorov R, Kogan VG. London penetration depth and strong pair breaking in iron-based superconductors. Phys. Rev. B 2010;81:180501. DOI: http://dx.doi.org/10.1103/PhysRevB.81.180501 - 48.
Kitagawa S, Nakai Y, Iye T, Ishida K, Guo YF, Shi YG, Yamaura K, Takayama-Muromachi E. Nonmagnetic pair-breaking effect in La(Fe1-xZnx)AsO0.85 studied by 75As and 139La NMR and NQR. Phys. Rev. B 2011;83:180501(R). DOI: http://dx.doi.org/10.1103/PhysRevB.83.180501 - 49.
Guo YF, Shi YG, Yu S, Belik AA, Matsushita Y, Tanaka M, Katsuya Y, Kobayashi K, Nowik I, Felner I, Awana VPS, Yamaura K, Takayama-Muromachi E. Large decrease in the critical temperature of superconducting LaFeAsO0.85 compounds doped with 3% atomic weight of nonmagnetic Zn impurities. Phys. Rev. B 2010;82:054506. DOI: http://dx.doi.org/10.1103/PhysRevB.82. 054506 - 50.
Li Y, Tong J, Tao Q, Feng C, Cao G, Chen W, Zhang F, Xu Z. Effect of a Zn impurity on Tc and its implications for pairing symmetry in LaFeAsO1-xFx. New J. Phys. 2010 ;12 :083008. DOI : 10.1088/1367-2630/12/8/083008 - 51.
Hardy F, Burger P, Wolf T, Fisher RA, Schweiss P, Adelmann P, Heid R, Fromknecht R, Eder R, Ernst D, Löhnheysen Hv, Meingast C. Doping evolution of superconducting gaps and electronic densities of states in Ba(Fe1-xCox)2As2 iron pnictides. Europhys. Lett. 2010;91:47008. DOI: 10.1209/0295-5075/91/47008 - 52.
Loktev VM, Pogorelov YG. Formation of d-wave superconducting order in a randomly doped lattice. Low Temp. Phys. 2001;27:767-776. DOI: http://dx.doi.org/10.1063/1.1401186 - 53.
Pogorelov YG, Santos MC, Loktev VM. Specifics of impurity effects in ferropnictide superconductors. Phys. Rev. B 2011;84:144510. DOI: http://dx.doi.org /10.1103/ PhysRevB.84.144510 - 54.
Pogorelov YG, Santos MC, Loktev VM. Green function study of impurity effects in high-Tc superconductors. In: Carmelo JMP, Lopes dos Santos JMB, Rocha Vieira V, Sacramento PD, editors. Strongly Correlated Systems, Coherence and Entanglement, World Scientific;2007. p. 443-494. ISBN 978-981-270-572-3. cap. 17 - 55.
Kubo R. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Jpn. 1957 ;12 :570-586. DOI: http://dx.doi.org/10.1143/JPSJ.12.570 - 56.
Greenwood DA. The Boltzmann Equation in the Theory of Electrical Conduction in Metals. Proc. Phys. Soc. 1958;71:585-596. DOI: 10.1088/0370-1328/71/4/306 - 57.
Economou EN. Green's Functions in Quantum Physics. Springer, Berlin;2006. 477 p. DOI: 10.1007/3-540-28841-4 - 58.
Mazin II, Schmalian J. Pairing symmetry and pairing state in ferropnictides: Theoretical overview. Physica C 2009;469:614-627. DOI:10.1016/j.physc.2009.03.019 - 59.
DeWeert MJ. Proximity-effect bilayers with magnetic impurities: The Abrikosov-Gor’kov limit. Phys. Rev. B 1988;38:732-734. DOI: http://dx.doi.org/10.1103/PhysRevB.38.732 - 60.
Srivastava RVA, Teizer W. Analytical density of states in the Abrikosov-Gorkov theory. Solid State Commun. 2008;145:512-513. DOI: 10.1016/j.ssc.2007.11.030 - 61.
Note that this static limit of Eq. (47) only defines the conductivity by normal quasiparticles, seen e.g. in normal resistivity by the magnetic flux flow in the mixed state, but otherwise short circuited by the infinite static conductivity due to supercurrents. - 62.
Dolgov OV, Efremov DV, Korshunov MM, Charnukha A, Boris AV, Golubov AA. Multiband Description of Optical Conductivity in Ferropnictide Superconductors. J. Supercond. Nov. Magn. 2013;26:2637-2640. DOI: 10.1007/s10948-013-2150-3 - 63.
Abramowitz MVL, Stegun IA, editors. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Natl. Bureau of Standards;1964. 1046 p. ISBN 0-486-61272-4 DOI: 10.2307/2008636 - 64.
Broude, Prikhot'ko AF, Rashba EI. Some problems of crystal luminescence. Sov. Phys. Uspekhi 1959;2:38-49. DOI: http://dx.doi.org/10.1070/PU1959v002n01ABEH003107