- Email: [email protected]
Effect of quasiparticle delocalization on μSR measurements of a multiband superconductor in the vortex state
ARTICLE IN PRESS
Physica B 374–375 (2006) 195–198 www.elsevier.com/locate/physb
Effect of quasiparticle delocalization on mSR measurements of a multiband superconductor in the vortex state F.D. Callaghan, M. Laulajainen, C.V. Kaiser, J.E. Sonier Department of Physics, Simon Fraser University, Burnaby, Canada BC V5A 1S6
Abstract We present mSR measurements of the low temperature magnetic field dependence of the vortex core size in the multiband superconductor NbSe2 . The spatially extended bound core states associated with the smaller energy gap rapidly delocalize with increasing magnetic field, resulting in a shrinking of the vortex core size. The more tightly bound core states associated with the larger energy gap, on the other hand, lead to a field-independent core size for fields greater than 4 kOe. The field dependence of the extracted magnetic penetration depth is explained in terms of the effect of delocalized quasiparticles on the spatial field variation around the vortex cores. r 2005 Elsevier B.V. All rights reserved. Keywords: Superconductivity; Multi-band; Vortex core size; Penetration depth
1. Introduction Interest in multi-band superconductivity (MBSC) has been reignited by its discovery in MgB2 , an s-wave type-II superconductor with an unusually high transition temperature T c [1]. MBSC implies that below T c more than one superconducting energy gap opens up, with each one existing on a distinct sheet of the Fermi surface. Both experiment [2–4] and band structure calculations [5] indicate that NbSe2 is a multiband superconductor. The currently accepted picture is that an energy gap of 1 meV exists on two bands derived from the Nb 4d orbitals, and that a smaller gap exists on a band derived from the Se 4p orbitals. Because of its large value of T c (39 K) and its potential for technological applications, MgB2 has received a huge amount of attention in recent years. However, mSR studies of the vortex state in MgB2 have been performed only on polycrystalline samples [6–10], as large single crystals are not yet available. On the other hand, high quality single crystals of NbSe2 provide an opportunity to study the
Corresponding author. Tel.: +1 604 291 5506; fax: +1 604 291 3592.
E-mail address: [email protected] (F.D. Callaghan). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.053
behaviour of the vortex core size in a multiband superconductor by mSR. 2. Experiment Transverse-field mSR measurements were performed on a single crystal of NbSe2 (T c ¼ 7:0 K and upper critical field H c2 ¼ 45 kOe) on the M15 beamline at TRIUMF. In order to isolate the effects of magnetic field from those of temperature, the sample was cooled to T ¼ 20 mK in a dilution refrigerator. The asymmetry spectra were fit by assuming that the local magnetic field due to an ideal hexagonal vortex lattice is described by the following analytical Ginzburg–Landau (GL) model [11]: BðrÞ ¼ B0 ð1 b4 Þ
X eiGr uK 1 ðuÞ G
l2ab G2
,
(1)
where b ¼ B=Bc2 , B0 is the average internal field, G are the reciprocal lattice vectors, K 1 ðuÞ is a modified Bessel function, u2 ¼ 2x2ab G 2 ð1 þ b4 Þ½1 2bð1 bÞ2 , xab is the GL coherence length, and lab is the magnetic penetration depth. Vortex lattice disorder and nuclear dipolar fields are accounted for by convoluting the theoretical field distribution with a Gaussian distribution of fields. xab is a measure
ARTICLE IN PRESS F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
of the vortex core size, which is particularly sensitive to changes in the core size due to changes in the vortex core electronic structure [12]. Although GL theory is strictly valid only at T ! T c , previous analysis using Eq. (1) at ToT c has resulted in values of lab and xab which are consistent with expectation [13]. Fig. 1 shows fast Fourier transforms (FFTs) of both the data and the fit for H ¼ 4 kOe and T ¼ 20 mK. All of the data at HX1 kOe were analysed in a reference frame rotating at a frequency 1 MHz with respect to the average muon spin precession frequency. For Ho1 kOe, however, much better fits were obtained by analysing the data in the laboratory reference frame. For example, a fit to the H ¼ 0:5 kOe data at a rotating reference frame (RRF) frequency of 5.9 MHz resulted in w2 =NDF ¼ 2:65 (NDF number of degrees of freedom), whereas the same data analysed in the laboratory frame resulted in w2 =NDF ¼ 1:40 (see Fig. 2). A previous mSR study on NbSe2 also found that data acquired at Ho1 kOe was best analysed in the laboratory frame [14]. The fact that reasonable fits can be obtained by working in the laboratory frame of reference suggests that BðrÞ is still well described by Eq. (1) in this low-field range (we note that Eq. (1) is expected to be valid in the field range b51 [11]). The problem lies in the numerical conversion of the data from its raw single-histogram form to the corrected asymmetry spectrum. Numerical determination of the normalization constant and uncorrelated background is most accurate when the oscillations are fast enough to be averaged over in a time much shorter than the muon lifetime [15]. Otherwise, particularly in the case of a fast
χ2/NDF = 1.40 Corrected Asymmetry (a.u.)
196
χ2/NDF = 2.65
0
2
4 Time (µs)
6
8
Fig. 2. Time dependence of the muon spin polarization in NbSe2 at H ¼ 500 Oe and T ¼ 20 mK (circles), and the fit to Eq. (1) (solid lines) in the laboratory frame (upper trace) and in a rotating reference frame (lower trace).
relaxing signal, the numerical reduction produces a slight distortion. Any such slowly varying distortions will be transformed to oscillations in the RRF, which may not be averaged out by rebinning after the RRF transformation. The conclusion is that great care should be taken in choosing a reference frame for low field data and that, if possible, such data is best analysed in the laboratory frame.
3. Results and discussion 6
Asymmetry
5
3.1. Field dependence of the vortex core size T = 20 mk H = 4 kOe
Real Amplitude
4
3 Time (µs) 2
1 0 53
54
55 Frequency (MHz)
56
57
Fig. 1. Fast Fourier transforms of both the time dependence of the muon spin polarization in NbSe2 at T ¼ 20 mK and H ¼ 4 kOe (open circles) and the fit (solid curve) to Eq. (1). The secondary peak at 54:1 MHz is due to the fraction ð 30%Þ of muons which did not stop in the sample. Inset: muon spin precession signal and fit to Eq. (1) (in a rotating reference frame).
Fig. 3 shows the field dependence of xab in NbSe2 determined from our analysis. Also shown is the electronic thermal conductivity ke ðHÞ in NbSe2 from Ref. [3]. It can be seen that xab shrinks rapidly at low fields and saturates at H 4 kOe. The rapid shrinkage of the vortex cores is accompanied by a sharp increase in ke , and the saturation of xab ðHÞ is accompanied by a change in slope of ke ðHÞ at approximately the same field. Measurements of xab ðHÞ may be understood by considering the bound quasiparticles (QPs) which occupy discrete energy levels in the vortex cores of s-wave type-II superconductors [16]. As H is increased, the wavefunctions of localized QPs in neighbouring vortices begin to overlap, hence becoming delocalized and causing an increase in ke . Similar behaviour has been observed in the single-band superconductor V3 Si, but in fields above 7 kOe [13]. However, in order to fully understand the field dependence of xab in NbSe2 , it is necessary to take into account the effect of MBSC on the vortex core structure. Calculations show that the electronic core states associated with the smaller energy gap in a two-band
ARTICLE IN PRESS F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
197
λab (Å)
2400
2000 1600 1400
1600
1200 0.0
0.5
1.0
1200 0 Fig. 3. Field dependence of the vortex core size xab in NbSe2 at T ¼ 20 mK (squares) and the electronic thermal conductivity ke normalised to its normal-state value kn (circles). The dashed lines are guides for the eye.
superconductor are spatially more extended than those associated with the larger energy gap [17]. The wavefunctions of these loosely bound states begin to overlap at relatively low fields leading to rapid delocalization of the bound core states and a corresponding shrinking of the core size. The data presented in Fig. 3 confirm this picture. We note that the observed saturation of xab above H 4 kOe is consistent with STM measurements which show that the low energy core states are more spatially confined at 10 kOe than at 1 kOe [18]. The overlap of these confined states will be much weaker leading to the observed fieldindependent xab above 4 kOe. In other words, the vortex core size is basically determined by the spatial extent of the QP states that remain localized in the core. 3.2. Field dependence of lab As shown in Fig. 4, the field dependence of lab is nonlinear, with a rapid increase at low fields and a weaker, though significant, dependence at higher fields. The field dependence of lab in s-wave type-II superconductors has previously been explained in terms of the supercurrentinduced shift in the QP energy spectrum [19]. However, this effect is only significant if the thermal energy kB T is comparable to the size of the smaller energy gap DS . At T ¼ 20 mK; kB T103 meV, while DS 101 meV. Therefore, we cannot attribute the observed field dependence of lab to this effect. Calculations have shown that the presence of extended QP states significantly modifies the current density, and hence BðrÞ, around the vortex core [20]. With increasing magnetic field these modified regions begin to overlap. Since Eq. (1) does not account for the changing electronic structure, the effect on BðrÞ shows up in our measurements as a field dependent lab . We note that the field dependence of lab is stronger at low field due to the more rapid delocalization of QPs from the vortex cores.
5
10 H (KOe)
15
Fig. 4. Field dependence of lab in NbSe2 at T ¼ 20 mK. Inset: linear fit to low field data.
The fact that lab is an effective decay length, and not a direct measure of the superfluid density, should be taken into account when interpreting mSR measurements. We note that the extrapolated value lab ðT ! 0; H ! 0Þ is expected to be a reliable estimate of the ‘‘true’’ magnetic penetration depth, because at low field and low temperature the QPs are highly localized in the vortex cores. Accurate determination of lab ð0; 0Þ requires measurement of lab ðTÞ at several fields followed by extrapolation of the zero temperature values of lab ðHÞ to H ¼ 0. As the data in Fig. 4 were acquired at T ¼ 20 mK, they can be considered to be measurements of lab ð0; HÞ. A linear fit to our three lowest field measurements of lab (see inset, Fig. 4), yields lab ð0; 0Þ ¼ 1256 29 A˚. This value is in excellent agreement with the value of 1240 A˚ obtained from Meissner state magnetization measurements [21]. We note that this method has also resulted in values of lab ð0; 0Þ for the high temperature superconductor YBa2 Cu3 O6þx which agree with those obtained from electron spin resonance experiments [22]. Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Canadian Institute for Advanced Research (CIAR). The authors thank the TRIUMF support staff for technical assistance, and Jess Brewer and Roger Miller for useful discussions and assistance with data acquisition. References [1] [2] [3] [4] [5] [6]
J. Nagamatsu, et al., Nature 410 (2001) 63. T. Yokoya, et al., Science 294 (2001) 2518. E. Boaknin, et al., Phys. Rev. Lett. 90 (2003) 117003. J.G. Rodrigo, S. Vieira, Physica C 404 (2004) 306. R. Corcoran, et al., J. Phys.: Condens. Matter 6 (1994) 4479. C. Panagopoulos, et al., Phys. Rev. B 64 (2001) 094514.
ARTICLE IN PRESS 198 [7] [8] [9] [10] [11]
F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
Ch. Niedermayer, et al., Phys. Rev. B 65 (2002) 094512. K. Ohishi, et al., J. Phys. Soc. Japan 72 (2003) 29. S. Serventi, et al., Phys. Rev. Lett. 93 (2004) 217003. M. Angst, et al., Phys. Rev. B 70 (2004) 224513. A. Yaouanc, P. Dalmas de Re´otier, E.H. Brandt, Phys. Rev. B 55 (1997) 11107. [12] J.E. Sonier, J. Phys.: Condens. Matter 16 (2004) S4499. [13] J.E. Sonier, et al., Phys. Rev. Lett. 93 (2004) 017002. [14] R.I. Miller, et al., Physica B 326 (2003) 296.
[15] J.E. Sonier, J.H. Brewer, R.F. Kiefl, Rev. Mod. Phys. 72 (2000) 769. [16] C. Caroli, P.G. de Gennes, J. Matricon, Phys. Lett. 9 (1964) 307. [17] M. Ichioka, et al., Phys. Rev. B 70 (2004) 144508. [18] H.F. Hess, et al., J. Vac. Sci. Technol. A 8 (1990) 450. [19] R. Kadono, J. Phys.: Condens. Matter 16 (2004) S4421. [20] F. Gygi, M. Schlu¨ter, Phys. Rev. B 43 (1991) 7609. [21] J.J. Finley, B.S. Deaver Jr., Solid State Commun. 36 (1980) 493. [22] T. Pereg-Barnea, et al., Phys. Rev. B 69 (2004) 184513.
Physica B 374–375 (2006) 195–198 www.elsevier.com/locate/physb
Effect of quasiparticle delocalization on mSR measurements of a multiband superconductor in the vortex state F.D. Callaghan, M. Laulajainen, C.V. Kaiser, J.E. Sonier Department of Physics, Simon Fraser University, Burnaby, Canada BC V5A 1S6
Abstract We present mSR measurements of the low temperature magnetic field dependence of the vortex core size in the multiband superconductor NbSe2 . The spatially extended bound core states associated with the smaller energy gap rapidly delocalize with increasing magnetic field, resulting in a shrinking of the vortex core size. The more tightly bound core states associated with the larger energy gap, on the other hand, lead to a field-independent core size for fields greater than 4 kOe. The field dependence of the extracted magnetic penetration depth is explained in terms of the effect of delocalized quasiparticles on the spatial field variation around the vortex cores. r 2005 Elsevier B.V. All rights reserved. Keywords: Superconductivity; Multi-band; Vortex core size; Penetration depth
1. Introduction Interest in multi-band superconductivity (MBSC) has been reignited by its discovery in MgB2 , an s-wave type-II superconductor with an unusually high transition temperature T c [1]. MBSC implies that below T c more than one superconducting energy gap opens up, with each one existing on a distinct sheet of the Fermi surface. Both experiment [2–4] and band structure calculations [5] indicate that NbSe2 is a multiband superconductor. The currently accepted picture is that an energy gap of 1 meV exists on two bands derived from the Nb 4d orbitals, and that a smaller gap exists on a band derived from the Se 4p orbitals. Because of its large value of T c (39 K) and its potential for technological applications, MgB2 has received a huge amount of attention in recent years. However, mSR studies of the vortex state in MgB2 have been performed only on polycrystalline samples [6–10], as large single crystals are not yet available. On the other hand, high quality single crystals of NbSe2 provide an opportunity to study the
Corresponding author. Tel.: +1 604 291 5506; fax: +1 604 291 3592.
E-mail address: [email protected] (F.D. Callaghan). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.11.053
behaviour of the vortex core size in a multiband superconductor by mSR. 2. Experiment Transverse-field mSR measurements were performed on a single crystal of NbSe2 (T c ¼ 7:0 K and upper critical field H c2 ¼ 45 kOe) on the M15 beamline at TRIUMF. In order to isolate the effects of magnetic field from those of temperature, the sample was cooled to T ¼ 20 mK in a dilution refrigerator. The asymmetry spectra were fit by assuming that the local magnetic field due to an ideal hexagonal vortex lattice is described by the following analytical Ginzburg–Landau (GL) model [11]: BðrÞ ¼ B0 ð1 b4 Þ
X eiGr uK 1 ðuÞ G
l2ab G2
,
(1)
where b ¼ B=Bc2 , B0 is the average internal field, G are the reciprocal lattice vectors, K 1 ðuÞ is a modified Bessel function, u2 ¼ 2x2ab G 2 ð1 þ b4 Þ½1 2bð1 bÞ2 , xab is the GL coherence length, and lab is the magnetic penetration depth. Vortex lattice disorder and nuclear dipolar fields are accounted for by convoluting the theoretical field distribution with a Gaussian distribution of fields. xab is a measure
ARTICLE IN PRESS F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
of the vortex core size, which is particularly sensitive to changes in the core size due to changes in the vortex core electronic structure [12]. Although GL theory is strictly valid only at T ! T c , previous analysis using Eq. (1) at ToT c has resulted in values of lab and xab which are consistent with expectation [13]. Fig. 1 shows fast Fourier transforms (FFTs) of both the data and the fit for H ¼ 4 kOe and T ¼ 20 mK. All of the data at HX1 kOe were analysed in a reference frame rotating at a frequency 1 MHz with respect to the average muon spin precession frequency. For Ho1 kOe, however, much better fits were obtained by analysing the data in the laboratory reference frame. For example, a fit to the H ¼ 0:5 kOe data at a rotating reference frame (RRF) frequency of 5.9 MHz resulted in w2 =NDF ¼ 2:65 (NDF number of degrees of freedom), whereas the same data analysed in the laboratory frame resulted in w2 =NDF ¼ 1:40 (see Fig. 2). A previous mSR study on NbSe2 also found that data acquired at Ho1 kOe was best analysed in the laboratory frame [14]. The fact that reasonable fits can be obtained by working in the laboratory frame of reference suggests that BðrÞ is still well described by Eq. (1) in this low-field range (we note that Eq. (1) is expected to be valid in the field range b51 [11]). The problem lies in the numerical conversion of the data from its raw single-histogram form to the corrected asymmetry spectrum. Numerical determination of the normalization constant and uncorrelated background is most accurate when the oscillations are fast enough to be averaged over in a time much shorter than the muon lifetime [15]. Otherwise, particularly in the case of a fast
χ2/NDF = 1.40 Corrected Asymmetry (a.u.)
196
χ2/NDF = 2.65
0
2
4 Time (µs)
6
8
Fig. 2. Time dependence of the muon spin polarization in NbSe2 at H ¼ 500 Oe and T ¼ 20 mK (circles), and the fit to Eq. (1) (solid lines) in the laboratory frame (upper trace) and in a rotating reference frame (lower trace).
relaxing signal, the numerical reduction produces a slight distortion. Any such slowly varying distortions will be transformed to oscillations in the RRF, which may not be averaged out by rebinning after the RRF transformation. The conclusion is that great care should be taken in choosing a reference frame for low field data and that, if possible, such data is best analysed in the laboratory frame.
3. Results and discussion 6
Asymmetry
5
3.1. Field dependence of the vortex core size T = 20 mk H = 4 kOe
Real Amplitude
4
3 Time (µs) 2
1 0 53
54
55 Frequency (MHz)
56
57
Fig. 1. Fast Fourier transforms of both the time dependence of the muon spin polarization in NbSe2 at T ¼ 20 mK and H ¼ 4 kOe (open circles) and the fit (solid curve) to Eq. (1). The secondary peak at 54:1 MHz is due to the fraction ð 30%Þ of muons which did not stop in the sample. Inset: muon spin precession signal and fit to Eq. (1) (in a rotating reference frame).
Fig. 3 shows the field dependence of xab in NbSe2 determined from our analysis. Also shown is the electronic thermal conductivity ke ðHÞ in NbSe2 from Ref. [3]. It can be seen that xab shrinks rapidly at low fields and saturates at H 4 kOe. The rapid shrinkage of the vortex cores is accompanied by a sharp increase in ke , and the saturation of xab ðHÞ is accompanied by a change in slope of ke ðHÞ at approximately the same field. Measurements of xab ðHÞ may be understood by considering the bound quasiparticles (QPs) which occupy discrete energy levels in the vortex cores of s-wave type-II superconductors [16]. As H is increased, the wavefunctions of localized QPs in neighbouring vortices begin to overlap, hence becoming delocalized and causing an increase in ke . Similar behaviour has been observed in the single-band superconductor V3 Si, but in fields above 7 kOe [13]. However, in order to fully understand the field dependence of xab in NbSe2 , it is necessary to take into account the effect of MBSC on the vortex core structure. Calculations show that the electronic core states associated with the smaller energy gap in a two-band
ARTICLE IN PRESS F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
197
λab (Å)
2400
2000 1600 1400
1600
1200 0.0
0.5
1.0
1200 0 Fig. 3. Field dependence of the vortex core size xab in NbSe2 at T ¼ 20 mK (squares) and the electronic thermal conductivity ke normalised to its normal-state value kn (circles). The dashed lines are guides for the eye.
superconductor are spatially more extended than those associated with the larger energy gap [17]. The wavefunctions of these loosely bound states begin to overlap at relatively low fields leading to rapid delocalization of the bound core states and a corresponding shrinking of the core size. The data presented in Fig. 3 confirm this picture. We note that the observed saturation of xab above H 4 kOe is consistent with STM measurements which show that the low energy core states are more spatially confined at 10 kOe than at 1 kOe [18]. The overlap of these confined states will be much weaker leading to the observed fieldindependent xab above 4 kOe. In other words, the vortex core size is basically determined by the spatial extent of the QP states that remain localized in the core. 3.2. Field dependence of lab As shown in Fig. 4, the field dependence of lab is nonlinear, with a rapid increase at low fields and a weaker, though significant, dependence at higher fields. The field dependence of lab in s-wave type-II superconductors has previously been explained in terms of the supercurrentinduced shift in the QP energy spectrum [19]. However, this effect is only significant if the thermal energy kB T is comparable to the size of the smaller energy gap DS . At T ¼ 20 mK; kB T103 meV, while DS 101 meV. Therefore, we cannot attribute the observed field dependence of lab to this effect. Calculations have shown that the presence of extended QP states significantly modifies the current density, and hence BðrÞ, around the vortex core [20]. With increasing magnetic field these modified regions begin to overlap. Since Eq. (1) does not account for the changing electronic structure, the effect on BðrÞ shows up in our measurements as a field dependent lab . We note that the field dependence of lab is stronger at low field due to the more rapid delocalization of QPs from the vortex cores.
5
10 H (KOe)
15
Fig. 4. Field dependence of lab in NbSe2 at T ¼ 20 mK. Inset: linear fit to low field data.
The fact that lab is an effective decay length, and not a direct measure of the superfluid density, should be taken into account when interpreting mSR measurements. We note that the extrapolated value lab ðT ! 0; H ! 0Þ is expected to be a reliable estimate of the ‘‘true’’ magnetic penetration depth, because at low field and low temperature the QPs are highly localized in the vortex cores. Accurate determination of lab ð0; 0Þ requires measurement of lab ðTÞ at several fields followed by extrapolation of the zero temperature values of lab ðHÞ to H ¼ 0. As the data in Fig. 4 were acquired at T ¼ 20 mK, they can be considered to be measurements of lab ð0; HÞ. A linear fit to our three lowest field measurements of lab (see inset, Fig. 4), yields lab ð0; 0Þ ¼ 1256 29 A˚. This value is in excellent agreement with the value of 1240 A˚ obtained from Meissner state magnetization measurements [21]. We note that this method has also resulted in values of lab ð0; 0Þ for the high temperature superconductor YBa2 Cu3 O6þx which agree with those obtained from electron spin resonance experiments [22]. Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Canadian Institute for Advanced Research (CIAR). The authors thank the TRIUMF support staff for technical assistance, and Jess Brewer and Roger Miller for useful discussions and assistance with data acquisition. References [1] [2] [3] [4] [5] [6]
J. Nagamatsu, et al., Nature 410 (2001) 63. T. Yokoya, et al., Science 294 (2001) 2518. E. Boaknin, et al., Phys. Rev. Lett. 90 (2003) 117003. J.G. Rodrigo, S. Vieira, Physica C 404 (2004) 306. R. Corcoran, et al., J. Phys.: Condens. Matter 6 (1994) 4479. C. Panagopoulos, et al., Phys. Rev. B 64 (2001) 094514.
ARTICLE IN PRESS 198 [7] [8] [9] [10] [11]
F.D. Callaghan et al. / Physica B 374–375 (2006) 195–198
Ch. Niedermayer, et al., Phys. Rev. B 65 (2002) 094512. K. Ohishi, et al., J. Phys. Soc. Japan 72 (2003) 29. S. Serventi, et al., Phys. Rev. Lett. 93 (2004) 217003. M. Angst, et al., Phys. Rev. B 70 (2004) 224513. A. Yaouanc, P. Dalmas de Re´otier, E.H. Brandt, Phys. Rev. B 55 (1997) 11107. [12] J.E. Sonier, J. Phys.: Condens. Matter 16 (2004) S4499. [13] J.E. Sonier, et al., Phys. Rev. Lett. 93 (2004) 017002. [14] R.I. Miller, et al., Physica B 326 (2003) 296.
[15] J.E. Sonier, J.H. Brewer, R.F. Kiefl, Rev. Mod. Phys. 72 (2000) 769. [16] C. Caroli, P.G. de Gennes, J. Matricon, Phys. Lett. 9 (1964) 307. [17] M. Ichioka, et al., Phys. Rev. B 70 (2004) 144508. [18] H.F. Hess, et al., J. Vac. Sci. Technol. A 8 (1990) 450. [19] R. Kadono, J. Phys.: Condens. Matter 16 (2004) S4421. [20] F. Gygi, M. Schlu¨ter, Phys. Rev. B 43 (1991) 7609. [21] J.J. Finley, B.S. Deaver Jr., Solid State Commun. 36 (1980) 493. [22] T. Pereg-Barnea, et al., Phys. Rev. B 69 (2004) 184513.