Intraband vs. interband scattering rate effects in neutron irradiated MgB₂

M. Putti¹, P. Brotto¹, M. Monni¹, E. Galleani d'Agliano¹, A. Sanna² and S. Massidda²

¹ CNR-INFM-LAMIA and Dipartimento di Fisica, Università di Genova - Via Dodecaneso 33, 16146 Genova, Italy
² CNR-INFM-SLACS and Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, I-09124 Monserrato (CA), Italy

Received 27 November 2006; accepted in final form 10 January 2007
Published 20 February 2007

After the observation of two superconducting energy gaps in MgB₂ which open in different sheets of Fermi surface [1, 2] many investigations have been carried out to understand features and consequences of this peculiar phenomenon. Back in 1959 Matthias, Suhl and Walker [3] pointed out that, within the BCS theory, a higher transition temperature, T_c, should subsist if more variational degrees of freedom are provided, e.g. by allowing different order parameters in different bands. This is expected only for very clean samples because interband scattering (IBS) with impurities should suppress superconductivity in almost the same way, as scattering with magnetic ones does in a one-band superconductor [4]. When IBS rates become comparable with the relevant phonon frequency a complete isotropization over the whole Fermi surface is expected; the two gaps merge into one and T_c drops to the isotropic value of about 20–25 K [5–7]. However, in MgB₂, the IBS rate is quite small even in rather dirty samples, due to the different symmetry of σ and π electronic states [8], and it is believed that two distinct gaps are observable because of this fortunate coincidence. To verify these predictions several efforts have been done to introduce defects systematically by substitutions and by irradiation. Unfortunately, substitutional defects introduce charge doping, which complicates the understanding of the role of IBS itself. In this respect, irradiation with neutrons [9–13] and alpha-particles [14, 15] is very appealing since it produces homogeneous defect structures, without introducing charge doping. With increasing irradiation resistivity increases monotonously and T_c decreases correspondingly; in heavily irradiated samples superconductivity is completely suppressed. By plotting T_c, as a function of the residual resistivity, ρ₀, a linear relation is found [13, 14] without any sign of T_c saturation around 20 K. A similar linear behaviour was observed in conventional superconductors like amorphous transition metals and damaged A15 superconductors [16], which suggests that a common approach could be exploited.

The role of IBS has been mainly investigated by studying the evolution of the energy gaps, Δ_σ and Δ_π, with disorder (ref. [17] and references therein). While Δ_σ decreases linearly with T_c, Δ_π in weakly disordered samples increases in agreement with theoretical predictions [18]; however, with a further increase in disorder, Δ_π decreases and the merging of the gaps has finally been observed in neutron irradiated polycrystals at a critical temperature (11 K) much lower than the one predicted for isotropic MgB₂. These results prove that the IBS is able to drive from two- to single-gap superconductivity, but suggest that in irradiated samples other mechanisms cooperate to the suppression of T_c.

In this letter we show that a full understanding of the T_c behavior in irradiated MgB₂ can be provided by introducing intraband electron lifetime effects in analogy with A15 superconductors. For these materials, the degradation of superconductivity induced by disorder was explained through the smearing of the density of states (DOS) produced by disorder. The model proposed by Testardi and Mattheiss [19] assumed that the smearing is due to the mixing of electron states in an energy region comparable to the inverse electron lifetime. In this letter we discuss the experimental behaviour of T_c in irradiated MgB₂ samples within a generalization of Testardi-Mattheiss model to the multi-band case. The suppression of T_c will be estimated in an anisotropic and an isotropic case and compared to experimental results.

The electron lifetime model proposed in ref. [19] assumes that the defects just broaden the DOS via the electron relaxation rate Γ, which increases with disorder. If N(E, Γ = 0) is the DOS of a perfectly ordered material, disorder effects can be taken into account with the convolution:

where S(E, E', Γ) is a broadening function that depends on the electron relaxation rate. Since N(E, Γ) changes with increasing Γ, the Fermi energy in the disordered system, E_F(Γ), is determined with the constraint that the total number of states remains constant [20]. The exact form of the broadening function is not crucial when the broadening is large compared to any fine structure in N(E). Within a semiclassical approach the proper weighting function should be a Lorentzian; however, to avoid complication from the broad Lorentzian wings, Testardi and Mattheiss suggested a thermal broadening with S = –∂ f/∂ E, where f is the Fermi-Dirac function with T replaced by T_B = Γ/k_B, where Γ is simply related to the residual resistivity ρ₀ by

In the Testardi-Mattheiss approach Ω_p(Γ) is the plasma frequency of the disordered material obtained by applying to Ω_p(E, Γ = 0) the same convolution of eq. (1) [19, 20]. In this way the effects due to lifetime broadening are approximately accounted for and, with some additional assumptions, T_c can be calculated as a function of ρ₀. In A15 superconductors [19, 20] the electron-phonon coupling constant, λ, was assumed to be proportional to the DOS of disordered material, λ (Γ) N(Γ) and, using the McMillan equation with fixed values of the pseudopotential μ* and of the average phonon frequency, the degradation of T_c was reproduced without farther free parameters, as a function of the measured ρ₀. In this work we generalize this model and use it to rationalize the behaviour of irradiated MgB₂.

In a two-band superconductor the effect of broadening has to be considered separately on the partial DOS (PDOS), N_σ and N_π. In fact, as long as the IBS rates (Γ_σπ, Γ_{π σ} = Γ_σπN_σ/N_π) are negligible in comparison with intraband scattering rates (Γ_σ, Γ_π), Γ_σ and Γ_π should produce the broadening of N_σ and N_π, respectively. The hypothesis Γ_σπ, Γ_{π σ}  Γ_σ, Γ_π, well verified in pure MgB₂ due to the different parity of σ and π electronic states, is plausible in irradiated samples as long as defects do not disrupt significantly the sp² bonding pattern of the crystal.

The PDOS^Note1 and the plasma frequencies of the clean MgB₂ have been computed within the local density approximation using the FLAPW method and the broadening was taken into account by the Fermi-Dirac function. Ω_p(Γ) comes out to be practically unaffected by disorder, varying at most by 1%, while N_σ(Γ) and N_π(Γ) present significant deviations.

Figure 1 shows N(Γ)/N(0) as a function of Γ, for the σ- and π-band of MgB₂; here and N(0) = N(E_F, 0) is the DOS at the Fermi level of the clean material. For comparison we plot N(Γ)/N(0) calculated in ref. [20] with the same approach for V₃Si and Nb₃Sn. With increasing Γ up to 0.2 eV, N_σ(Γ) and N_π(Γ) remain quite constant and then decrease; N_σ(Γ), which is more reduced than N_π(Γ), is about 80% of N_σ(0) for Γ = 0.7 eV. It is worth noting that a comparable reduction of DOS (0.75%) was estimated from NMR measurements in a strongly irradiated MgB₂ sample (T_c = 7 K) [21].

The behaviour of A15 is very different: N(Γ) sharply decreases at low Γ and then tends to saturate, being nearly one half of N(0) for Γ = 0.4 eV. These differences can be easily understood by looking at the DOS of the clean materials. A15 superconductors present a peak close to E_F [22] which quickly reduces by mixing the states. On the other hand, the DOS of MgB₂ is rather flat around E_F [23], however, N_σ becomes zero at E = 0.7 eV and when the smearing involves states at such a distance from E_F, N_σ(Γ) decreases.

In the presence of multiple gaps, T_c depends on the IBS rates. For negligible Γ_σπ, T_c≡T_c⁽⁰⁾ is given, in the two-square-well model, by the multiband analog of the Allen-Mitrovic equation [24, 25]

where ψ (z) is the digamma function and represents a characteristic cutoff phonon energy. Λ⁽⁰⁾ is the largest eigenvalue of the matrix Here, λ_ij is the asymmetric pairing interaction matrix, λ_ij = V_ijN_j, where V_ij = V_ji is the symmetric pairing potential matrix and μ_ij* is the Coulomb pseudopotential matrix. We notice that for 600–700 K and T_c⁽⁰⁾ = 39 K we have /2π T_c⁽⁰⁾ + 1/2 3, so that we cannot take the usual asymptotic expression ψ (z)~ln z.

In the opposite limit, the IBS rate is larger than the relevant phonon frequency (Γ_σπ ≥ ω_c) and a full isotropization of all Fermi surfaces occurs. In this case only one gap is present and extending the analysis previously done in the pure BCS case we obtain the following expression for T_c≡T_c^(∞):

where

is the mass renormalized PDOS [26, 27], with

To treat the general case, between T_c⁽⁰⁾ and T_c^∞, we have followed a path very similar to that of ref. [4] including in addition the renormalization of the PDOS due to electron-phonon interaction. We have derived in this way the generalisation of eq. (3) which includes IBS. This equation takes the form

where, now, Λ is the smallest eigenvalue of the matrix

with

and

Here is proportional to

with

In order to take into account the effects of intraband scattering, the dependence of the matrices λ_ij and μ_ij* on disorder has to be considered. From the relation λ_ij = V_ijN_j, and assuming V_ij independent of Γ we have

where λ_{σ σ} (0) = 1.017, λ_σπ (0) = 0.213, λ_{π π} (0) = 0.448 and λ_{π σ} (0) = 0.155. We also assume a scaling law of the pseudopotential matrix μ_ij* with the PDOS; following ref. [28] we write

To check the reliability of our calculation, we calculate the Sommerfeld coefficient γ as a function of Γ and compare it with experimental values. γ (Γ) is given by

With N_σ(0) = 0.302 states/eV cell and N_π(0) = 0.406 states/eV cell from eq. (8) we estimate γ (0) = 3.1 mJ/mole K² to be compared with 3.0 mJ/mole K² and 2.5 mJ/mole K² [29] estimated from specific heat measurements. In fig. 2, γ (Γ)/γ (0) is plotted as a function of Γ for MgB₂ and V₃Si: for V₃Si we assume λ (Γ) = λ (0)N(Γ)/N(0) with λ (0) = 1.12 and N(Γ) from ref. [20]. Reflecting the behaviour of the DOS, in MgB₂ γ (Γ)/γ (0) remains nearly constant for values of Γ less than 0.2 eV, and then slowly decreases. In V₃Si it sharply decreases at low Γ and then tends to saturate.

Experimental data can be plotted in fig. 2, provided that the relaxation rate is known. For a single band metal, Γ can be simply estimated by the measured ρ₀ (see eq. (2), while in the case of MgB₂Γ_σ and Γ_π cannot be both evaluated. For the sake of simplicity in the following we assume equal intraband scattering rates in each band (Γ_σ~Γ_π ≡ Γ). This assumption, which is crude for substituted samples, is realistic when high degree of disorder is introduced homogeneously by irradiation.

Experimental γ values estimated from specific heat in irradiated MgB₂ samples [17, 29] and in V₃Si [30] are shown in fig. 2 for comparison. The Γ values have been estimated for each sample from eq. (2) by keeping for (this value theoretically computed, has been recently confirmed by optical measurements [31] and for V₃SiΩ_p(Γ) ≈ Ω_p(0) = 4.0 eV.

As shown by fig. 2, both in MgB₂ and V₃Si theoretical calculations reproduce quite well the experimental values, which indicates that the Testardi-Mattheiss model is capable of taking into account lifetime effects in materials which present very differently shaped DOS.

Now we are ready to discuss the T_c vs. Γ behaviour. Experimental data obtained in neutron irradiated polycrystalline samples are shown in fig. 3; on this set of samples the two-gap feature is present above 21 K (empty symbols) and the merging of the gaps was observed below 11 K (full symbols). Other comparable T_c vs. ρ₀ data reported in the literature[14, 15] are not shown in fig. 3 for the sake of clarity.

Continuous lines in fig. 3 are the critical temperatures, T_c⁽⁰⁾ and T_c^(∞), given by eqs. (3) and (4) obtained by choosing ω_c = 53 meV ( = 615 K) and μ₀ = 0.050 (which implies μ_{σ σ}* = 0.26, μ_σπ* = 0.12, μ_{π π}* = 0.22 and μ_{π σ}* = 0.088) in good agreement with first principles calculations. With the above choices of ω_c and μ₀ we obtain T_c⁽⁰⁾(Γ = 0) = 39.4 K and T_c^(∞)(Γ = 0) = 21.3, in agreement with other predictions, [5–7, 32].

Starting from Γ = 0, T_c⁽⁰⁾ remains nearly constant for Γ < 0.1 eV; then it decreases, reaching a value of about 18 K, for Γ = 0.7 eV(ρ₀~100 μΩ cm). In this case, the IBS being completely neglected, T_c⁽⁰⁾ decreases solely as a consequence of the reduction of the PDOSs, with increasing of intraband scattering. Also in the isotropic limit T_c^(∞)is nearly constant for Γ < 0.1 eV values and it decreases as Γ increases because of the PDOS reduction, reaching a value of about 8 K for Γ = 0.7 eV. Remarkably, the T_c of the samples that present single-gap superconductivity (full symbols in fig. 3) falls on T_c^(∞). On the other hand, the T_c values of the samples that present two gaps (empty symbols) decrease roughly linearly with Γ laying in between the two limiting curves for T_c⁽⁰⁾and T_c^(∞). In the range of weakly disordered samples (Γ ≤ 0.1 eV), where the PDOSs are constant, this behaviour indicates that IBS is the main mechanism which suppresses T_c. When Γ increases, also Γ_σπ increases, but the two-gap feature survives, since the limit of strong IBS (Γ_σπ < ω _c) is not yet reached.

The dotted line in fig. 3 has been calculated by solving eq. (5) which takes into account both the effects of intraband scattering (PDOS reduction) and IBS. Assuming Γ_σπ = α Γ with α = 0.059, it reproduces pretty well the experimental values up to Γ~0.3 eV, while, for a further increase in disorder, the experimental data fall rather on T_c^(∞). Therefore, at low level of disorder the experimental data are well reproduced with Γ_σπ increasing proportionally with Γ, the condition Γ_σπ /Γ  1 being fulfilled; on the other hand, above a certain level of disorder a faster increase of Γ_σπ is expected because the π and σ states are loosing their symmetry. We point out that the samples showing a single gap must have Γ_σπ ≥ ω_c~53 meV with Γ = 0.5-0.6 eV, so that Γ_σπ /Γ < 1 remains true also at this level of disorder.

The details of our results depend on the assumption we made about the intraband scattering rates (Γ_σ~Γ_π). In weakly irradiated thin films the ratio Γ_σ/Γ_π was estimated to be 1.5–2 [33]. By varying Γ_σ/Γ_π from 1 to 2, the values of ω_c and μ₀, necessary to describe the experiments vary at most 10% (ω_c from 53 to 48 meV and μ₀ from 0.050 to 0.046). The most affected parameter is α that varies from 0.059 to 0.035.

However, our main results are still confirmed: we rationalize the decrease of T_c in irradiated samples, as well as the observation of the gap merging at 11 K rather than at the temperature predicted by IBS only (20–25 K). The latter point, which has been considered a puzzling problem so far, is naturally explained by considering that when the irradiation brings the system to the limit of strong IBS, the intraband scattering rate Γ has obviously increased as well, reducing significantly the λ_ij. This mechanism could play a role also in doped MgB₂ samples, mainly when substitutions affect mostly intraband scattering of σ-bands, as it should occur in C-doped samples.

The analysis performed up to now, which provides a satisfactory explanation of the decreasing of T_c for ρ₀ values less than 100 μΩ cm, cannot be simply extended to the regime of extreme disorder. First of all, in this regime experimental data are quite scattered because irradiation progressively reduces the connectivity between grains [13–15] and resistivity is hardly estimated; as a consequence, the resistivity at which T_c is completely suppressed is not well defined. On the other hand, at such level of disorder some of our assumptions could fail. For instance, the electron mean free path of samples with ρ₀~100 μΩ cm is of the order of the unit cell, and under these conditions one cannot neglect Anderson localization effects, which increase the effective Coulomb repulsion [34].

We also point out that in our calculations we neglected changes in the phonon modes. Experimentally, the normal state specific heat increases weakly with irradiation and in the most damaged sample a reduction of the Debye temperature by 15% can be estimated, accompanied by an increase of the cell volume by about 1.7%. Raman spectroscopy of irradiated samples [35] shows that disorder causes the appearance of high-frequency spectral structures, similar to those observed in doped MgB₂ samples that were considered as an evidence of violations of the selection rules rather than as a stiffening of the E_2g mode. These results did not emphasize strong modifications of the phonon spectrum, but they cannot be ruled out, especially at high level of disorder. A detailed description of these effects, which will certainly affect to some extent the quantitative results, is beyond the purpose of the present investigation.

In conclusion, the comparison with A15 materials has suggested that also in irradiated MgB₂ electronic lifetime effects reduce the PDOS, mainly for the σ bands, producing a reduction of T_c both in an anisotropic and an isotropic limit. The comparison with experiments suggests that at low level of disorder the pair breaking due to IBS plays the main role. However, when disorder increases and IBS progressively cancels the two-gap structure, the intraband scattering increases as well, reducing the PDOSs. As a result, the merging of the gaps is found at a critical temperature nearly one half of the value previously predicted, in excellent agreement with observation.

Acknowledgments

We acknowledge financial support by MIUR under the projects PRIN2004022024, PRIN2006021741 and PON-CyberSar.

References

[1]

Bouquet F., Fisher R. A., Phillips N. E., Hinks D. G. and Jorgensen J. D. 2001 Phys. Rev. Lett. 87 047001

[2]

Giubileo F. et al 2001 Phys. Rev. Lett. 87 177008

[3]

Suhl H., Matthias B. T. and Walker L. R. 1959 Phys. Rev. Lett. 3 552

[4]

Golubov A. A. and Mazin I. I. 1997 Phys. Rev. B 55 15146

[5]

Liu A. Y., Mazin I. I. and Kortus J. 2001 Phys. Rev. Lett. 87 087005

[6]

Choi H. J. et al 2002 Nature 418 758

[7]

Floris A. et al 2005 Phys. Rev. Lett. 94 037004

[8]

Mazin I. I. et al 2002 Phys. Rev. Lett. 89 107002

[9]

Kar'kin A. E., Voronin V. I., D'yachkova T. V., Kadyrova N. I., Tyutyunik A. P., Zubkov V. G., Zainulin Yu. G., Sadovski M. V. and Goshchitskii B. N. 2001 JEPT Lett. 73 570

[10]

Eisterer M., Zehetmayer M., Tonies S., Weber H. W., Kambara M., Babu N. H., Cardwell D. A. and Greenwood L. R. 2002 Supercond. Sci. Technol. 15 L9

[11]

Putti M., Braccini V., Ferdeghini C., Gatti F., Manfrinetti P., Marré D., Palenzona A., Pallecchi I., Tarantini C., Sheikin I., Aebersold H. U. and Lehmann E. 2005 Appl. Phys. Lett. 86 112503

[12]

Wilke R. H. T., Budko S. L., Canfield P. C., Farmer J. and Hannahs S. T. 2006 Phys. Rev., B 73 134512

[13]

Tarantini C., Aebersold H. U., Braccini V., Celentano G., Ferdeghini C., Ferrando V., Gambardella U., Gatti F., Lehmann E., Manfrinetti P., Marré D., Palenzona A., Pallecchi I., Sheikin I., Siri A. S. and Putti M. 2006 Phys. Rev. B 73 134518

[14]

Gandikota R., Singh R. K., Kim J., Wilkens B., Newmann N., Rowell J., Pogrebnyakov A. V., Xi X. X., Redwing J. M., Xu S. Y. and Li Qi 2005 Appl. Phys. Lett. 86 012508

[15]

Gandikota R., Singh R. K., Wilkens B., Newmann N., Rowell J. M., Pogrebnyakov A. V., Xi X. X., Redwing J. M., Xu S. Y., Li Q. and Moeckly B. H. 2005 Appl. Phys. Lett. 87 072507

[16]

Sweedler A. R., Cox D. E. and Moehlecke 1978 J. Nucl. Mater 72 50
Rowell J. M. and Dynes R. C. 1978 unpublished

[17]

Putti M., Affronte M., Ferdeghini C., Manfrinetti P., Tarantini C. and Lehmann E. 2006 Phys. Rev. Lett. 96 077003

[18]

Di Capua R., Aebersold H. U., Ferdeghini C., Ferrando V., Orgiani P., Putti M., Xi X. X. and Vaglio R. 2007 Phys. Rev. B 75 014515 [cond-mat/0606606]

[19]

Testardi L. R. and Mattheiss L. F. 1978 Phys. Rev. Lett. 41 1612
Mattheiss L. F. and Testardi L. R. 1979 Phys. Rev. B 20 2196

[20]

Soukoulis C. M. and Papaconstantopoulos D. A. 1982 Phys. Rev. B 26 3673

[21]

Gerashenko A. P., Mikhalev K. N., Verkhovskii S. V., Karkin A. E. and Goshchitskii B. N. 2002 Phys. Rev. B 65 132506

[22]

Pickett W. E. 1982 Phys. Rev. B 26 1186 and references therein

[23]

Mazin I. I. and Antropov V. P. 2003 Physica C 385 49

[24]

Allen P. B. and Mitrovic B. 1982 Solid State Physics Vol. 37, ed Ehrenreich H., Seitz F. and Turnbull D. (New York: Academic) p. 57

[25]

Carbotte J. P. 1990 Rev. Mod. Phys. 62 1027

[26]

Golubov A. A., Kortus J., Dolgov O. V., Jepsen O., Kong Y., Andersen O. K., Gibson B. J., Ahn K. and Kremer R. K. 2002 J. Phys.: Condens. Matter 14 1353

[27]

Dolgov O. V., Kremer R. K., Kortus J., Golubov A. A. and Shulga S. D. 2005 Phys. Rev. B 72 024504

[28]

Ummarino G. A., Daghero D., Gonnelli R. S. and Moudden A. H. 2005 Phys. Rev. B 71 134511

[29]

Wang Y., Bouquet F., Sheikin I., Toulemonde P., Revaz B., Eisterer M., Weber H. W., Hinderer J. and Junod A. 2003 J. Phys. Condens. Matter 15 883

[30]

Viswanathan R. and Caton R. 1978 Phys. Rev. B 18 15

[31]

Kakeshita , Lee S. and Tajima S. 2006 Phys. Rev. Lett. 97 037002

[32]

Braccini V., Gurevich A., Giencke J. E., Jewell M. C., Eom C. B., Larbalestier D. C., Pogrebnyakov A., Cui Y., Liu B. T., Hu Y. F., Redwing J. M., Li Qi, Xi X. X., Singh R. K., Gandikota R., Kim J., Wilkens B., Newman N., Rowell J., Moeckly B., Ferrando V., Tarantini C., Marré D., Putti M., Ferdeghini C., Vaglio R. and Haanappel E. 2005 Phys. Rev. B 71 012504

[33]

Pallecchi I., Ferrando V., Galleani D'Agliano E., Marré D., Monni M., Putti M., Tarantini C., Gatti F., Aebersold H. U., Lehmann E., Xi X. X., Haanappel E. G. and Ferdeghini C. 2005 Phys. Rev. B 72 184512

[34]

Anderson P. W., Muttalib K. A. and Ramakrishnan T. V. 1983 Phys. Rev. B 28 117

[35]

Di Castro D., Cappelluti E., Lavagnini M., Sacchetti A., Palenzona A., Putti M. and Postorino P. 2006 Phys. Rev. B 74 100505(R)

Notes

Note1 The determination of the PDOS in a wide energy range requires some explanations. In fact, while close to the E_F we can safely say that only σ and π states contribute to the total DOS, away from E_F we also have different contributions (e.g., boron s). We estimated the PDOS by projecting the electron wave functions over σ and π components, as customarily done in electronic structure calculations. However, procedure depends on the arbitrarily chosen atomic spheres. To limit this problem, we normalize these PDOS so as to give, close to E_F, the PDOS obtained by band and Brillouin zone partition, well known in the literature.