Two-band superconductivity in (AlMg)B2: Critical temperature and isotope exponent as a function of carrier density

Physica C 468 (2008) 2299–2304

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Two-band superconductivity in (AlMg)B2: Critical temperature and isotope exponent as a function of carrier density J.J. Rodríguez-Núñez a,*, A.A. Schmidt b, A. Bianconi c, A. Perali d a

Lab. SUPERCOMP, Departamento de Física–FACYT, University of Carabobo, Valencia, Venezuela Departamento de Matemática, UFSM, 97105-900 Santa Maria, RS, Brazil c  di Roma ‘‘La Sapienza”, I-00185 Roma, Italy Physics Department, Universita d  di Camerino, Via Madonna delle Carceri, I-62032, Camerino (MC), Italy Physics Department, Universita b

a r t i c l e

i n f o

Article history: Received 15 December 2007 Accepted 15 May 2008 Available online 23 May 2008 PACS: 74.20.Fg 74.70.Ad Keywords: Two-band superconductivity Topological transition (AlMg)B2 Inter-band pairing

a b s t r a c t We consider as a minimal model for superconductivity in (AlMg)B2 a two-band superconductor, having an anisotropic electronic structure described by two tight binding bands, considering a pairing scenario driven by an attractive interaction in which the interband pairing terms assume an important role. We solve the two-gap equations at the critical temperature T ¼ T c and calculate T c and the chemical potential l as a function of the number of carriers n for various values of pairing interaction, V, and cut-off energy, xc . Using a self-consistent approach developed in a previous paper by two of the present authors, we calculate the isotope exponent a as a function of l. We find that the isotope exponent shows a minimum in the energy range around a dimensional electronic topological transition (ETT) where the Fermi surface of one of the bands changes from 2D to 3D dimensionality. We have been able to fix the parameters of the theory, namely, the attractive interaction, V, and the cut-off frequency, xc , by imposing experimental constraints on a, T c and l for undoped MgB2, specifically, 0.3, 40 K and 1.8 eV, respectively. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction It is well known that the paramagnetic impurities in a single band superconductor do not break the time reversal symmetry [1]. In a multiband system the pairing is described by a matrix where the diagonal elements are the intraband pairing processes and the non-diagonal elements describe the exchange-like interband pairing process associated with the transfer of pairs between the bands [2–4]. Commonly impurities in a multiband metal act as pair breaking centers for the interband pairing process as it has been well supported by the fact that nearly all multiband metals are reduced to effective single band superconductors in the dirty limit [1]. However in some very special materials it is possible that the impurity scattering of single electrons from one band to another is suppressed so that the interband pairing is not quenched and the multiband superconductor could be in the clean limit. It has been proposed that the superconductivity in the clean limit can be realized in particular heterogeneous materials made of superlattices of metallic units at atomic limit [5–7]. In fact in these systems the electronic states at the Fermi level in different * Corresponding author. Tel.: +58 416 5475940; fax: +58 241 8422878. E-mail address: [email protected] (J.J. Rodríguez-Núñez). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.050

subbands (determined by the charge density and superlattice geometry) have different symmetry and different spatial locations so that the impurity scattering of single electrons from one band to the other (single electron impurity interband scattering, SEIIS) is strongly reduced. In the clean limit the non-diagonal elements due to the exchange-like interband pairing could provide the driving mechanism for raising the critical temperature. Light element (A = Al or Mg) diborides AB2 are artificial intermetallics: first synthesized in 1935 was AlB2 [8] followed by MgB2 in 1953 [9,10] with the AlB2-type crystal structure where the boron atoms form a superlattice of graphite-like sheets separated by hexagonal layers of Mg atoms, very similarly to intercalated graphite compounds. Therefore also their electronic structure is very similar to intercalated graphite [11]. While in AlB2 the Fermi level crosses only the p-bands since it is above the top of the first r-bands like in graphite. On the contrary in MgB2 the Fermi level is shift down below the top of the r subband and it crosses both the p-bands and the r-bands. The light element diboride AlB2 is not superconducting, while in MgB2 the T c reaches the record for intermetallics T c ¼ 40 K [12]. After a few years of huge experimental and theoretical research on MgB2 it is now well established that this is the first system for which multiband superconductivity has independently been verified by several experimental techniques (heat capacity, tunneling

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spectroscopy, Raman spectroscopy, penetration depth measurements, the analysis of critical fields and ARPES [13,14]) since it has been determined that the impurity interband single electron scattering is negligible. The Fermi surface of this material consists of four sheets: two cylindrical sheets which correspond to quasi two-dimensional r-bands and two tubular networks which are three-dimensional p-bands. The single electron intraband impurity scattering r to r-band and the other p to p-band mixes the two r(p)-bands with the same symmetry reducing the system to an effective two-band superconductor. On the contrary the single electron interband impurity scattering r to p-band and vice versa is exceptionally small [15], so that the system is a two-band superconductor in the clean limit as predicted [16]. The most direct experimental evidence of superconductivity in the clean limit is the fact that the superconducting phase has two gaps and a single T c so it is also called two-gap superconductivity. In fact the interband pairing process described by the non-diagonal elements of the pairing matrix gives a single T c and in the dirty limit it will show a single gap. Tsuda et al. [17] have found definite experimental evidence for twoband superconductivity in the clean limit in MgB2, using high-resolution angle-resolved photoemission spectroscopy (ARPES). They directly measure both gaps in the two dispersing bands and they show that both gaps close at the bulk critical transition temperature, providing definitive evidence for the two-band superconductivity with strong interband pairing interaction in this compound. Recently compelling experimental evidence has been reported by many groups using ARPES, tunneling and specific heat measurements, that in ternary intermetallic diborides obtained by Al for Mg or C for B substitutions in MgB2 [18–20] it is possible to study the band filling or Fermi level tuning effects while the superconducting phase remains in the clean limit. In fact the single electron interband (r to p-band) impurity scattering due to atomic substitutions remains suppressed, with the inversion of the r to p-gap hierarchy [18] providing a clear support for earlier theories and experiments [21–25]. Klie et al. [26] and Putti et al. [27] have carefully studied the effects of intraband single electron impurity scattering in the electronic and transport properties. Carrington et al. [28] have reported de Haas–van Alphen (dHvA) studies of the electronic structure of Al-doped crystals of MgB2. They conclude that the mass of the quasiparticles on the larger r orbit is lighter in the pure case indicating a reduction of the electron–phonon coupling constant k. Their observations are compared with band-structure calculations, and they are found to be in good agreement. Also there is growing interest on controlling the critical temperature in two-gap superconductor MgB2 by Al substitutions. For example, Szabó et al. [29] have addressed the two–band superconductivity in Al- and C-doped MgB2 by point-contact spectroscopy. According to these authors the evolution of the two gaps as a function of the critical temperature in the doped systems suggests the dominance of the band-filling effects, but for the increased Al doping, the enhanced interband scattering must also be considered. Giubileo et al. [30] have performed an extensive study of the temperature and magnetic field dependencies of local tunneling spectra measured by means of variable temperature scanning tunneling spectroscopy on high quality Mg1xAlxB2 single crystals. Their results strongly suggest that different interband scattering occurs for different gap amplitudes. Pissas et al. [31] report the vortex matter phase diagrams of aluminum doped Mg1x Alx B2 crystals, deduced from local Hall ac-susceptibility and bulk dcmagnetization measurements. According to their results all the experimental observations could be qualitatively explained within the clean two-band approximation. Di Capua et al. [32] have used a series of MgB2 thin films systematically disordered by neutron irradiation and studied by scanning tunneling spectroscopy. Their results are discussed in the framework of two-band

superconductivity. Sologubenko et al. [33] report data of the thermal conductivity jðT; HÞ along the basal plane of the hexagonal crystal structure of superconducting Mg1x Alx B2 with x ¼ 0:02 and x ¼ 0:07 for 0:5K 6 T 6 50 K and in external magnetic fields for 0 6 H 6 70 kOe. The analysis of the jðT; HÞ data implies that the Al impurities provoke an enhancement of the intraband scattering rate, almost equal in magnitude for both the r and the p bands of electronic excitations. They also checked the validity of the Wiedemann–Franz law. Di Castro et al. [34] have carried out Raman measurements on neutron-irradiated and Al-doped MgB2 samples. The Raman spectrum changes substantially in both cases and in a similar manner. They show that the modifications of the Raman spectrum in both irradiated and Al-doped samples are due to disorder-induced violations of the Raman selection rules, which is supported by theoretical calculations. Daghero et al. [35] present new results of point-contact Andreev-reflection (PCAR) spectroscopy in single-phase Mg1x Alx B2 single crystals with x up to x ¼ 0:32. They interpret the decrease of Dp for T Ac < 30 K, where T Ac is the local critical temperature of the junctions, as being governed by the onset of inhomogeneity and disorder in the Al distribution that partially mask the intrinsic effects of doping and is not taken into account in standard theoretical models. Following the previous considerations, in this paper we adopt the point of view that the interband scattering term is indeed very important to explain the response of the superconducting phase to the variation of the chemical potential. So, as a minimum model for the ternary diborides as Mgx Al1x B2 as a function of x we consider the extreme case where the interband pairing V is the dominant term using an approach reported in a previous paper [36]. Here we report the study of the variations of the critical temperature and isotope coefficient as a function of the charge density and chemical potential near a critical point in the electronic structure. For the calculation of the isotope effect we have considered the case of a phonon mediated interband mechanism. In fact, in the case of a mechanism mediated by electronic excitations we expect a zero isotope effect. This paper is organized as follows. In Section 2 we present the two self-consistent equations to be solved, namely, the equation for the superconducting critical temperature T c coupled to the equation of the chemical potential l at a given particle density n, for different values of the interaction parameters, V and xc . In Section 3 we present our numerical results and finally we conclude in Section 4. 2. The effective Hamiltonian and the BCS equations The model Hamiltonian we use is the following:

H  lN ¼

X ~ k;r



e2 ð~kÞa~yk;r a~k;r þ

Xh

X

e3 ð~kÞb~yk;r b~k;r  V

~ k;r

i a~y ay ~ bk~0 ;" bk~0 ;# þ h:c: ;

~ k;k~0

k;" k;#

ð1Þ

where

e2 ð~kÞ ¼ 2 ð~kÞ  l; e3 ð~kÞ ¼ 3 ð~kÞ  l

ð2Þ ð3Þ

are the three-dimensional tight-binding bands of the r-(e2 ð~ kÞÞ and p-(e3 ð~ kÞÞ bands. In order to study the generalities of the effect of the electronic topological transition on the isotope effect, we consider the simple case of a cubic lattice. The energy band dispersion are given by [37]

2 ð~kÞ ¼ 02  2t2 ½cosðkx Þ þ cosðky Þ þ s2 cosðkz Þ; 3 ð~kÞ ¼ 03  2t3 ½cosðkx Þ þ cosðky Þ þ s3 cosðkz Þ:

ð4Þ ð5Þ

J.J. Rodríguez-Núñez et al. / Physica C 468 (2008) 2299–2304

The Hamiltonian given in Eq. (1) is the same used in Ref. [36], with the difference that the center of the two bands are here at different energies. Going through the same technical steps followed in Ref. [36] we obtain the following self-consistent equations [38]

1 ¼ F 2 ðT c ; lÞ  F 3 ðT c ; lÞ; V2 1  n ¼ F~2 ðT c ; lÞ þ F~3 ðT c ; lÞ;

ð6Þ ð7Þ

where

" # v ð~kÞ ei ð~kÞ ; d~ k i tanh 2kB T c kÞ 2ei ð~ " # Z 1 ei ð~ kÞ F~i ðT c ; lÞ ¼ d~ ; k tanh 2 2kB T c F i ðT c ; lÞ ¼

Z

ð8Þ ð9Þ

where kB is the Boltzmann constant and vi ð~ kÞ ¼ 1 if jei ð~ kÞj 6 xc and zero otherwise, i ¼ 2; 3 and xc is an energy cut-off for the effective interaction. Details of the numerical solution of the self-consistent Eqs. (6)–(9) are given in Appendix A and the results are presented in the next section. 3. Numerical results Before we present our results, let us present in Fig. 1 the two density of states corresponding to the energy dispersions of Eqs.

Fig. 1. The upper panel shows the density of states N 2 ðEÞ and N 3 ðEÞ functions for the r- and p-bands, respectively (see Eq. (A3)); the middle panel shows the square root of the product of the two DOS; the lower panel shows the number particle density per unit cell per spin component as a function of the chemical potential.

2301

(2) and (3) (see Eq. (A3)), where the parameters of the band structures in Eqs. (4) and (5) are t2 ¼ 0:6 eV, s2 ¼ 0:167, 02 ¼ 0:0 eV, t3 ¼ 0:9 eV, s3 ¼ 1:00, 03 ¼ 2:9 eV [39]. This we do in the upper panel: we have the density of states for the r-band and the p-band (dotted one), respectively. As the effective density of states which pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi enters in the calculation of T c is given by N eff ðxÞ ¼ N r ðxÞN p ðxÞ, we present in the middle panel of this figure, N eff ðxÞ vs. x. We see that for x > 2:50 N eff ðxÞ is zero. In the lower panel of this figure, we present q vs. l, where q ¼ n=2. In order to fix the parameters of our model we assume that the superconductor compound under study, MgB2, corresponds to T c ¼ 40 K and l ¼ 1:80 eV. For these values, we plot V vs. xc and a vs. xc (see Fig. 2). From experiments we know that MgB2 has an isotope exponent given by a  0:30. Then, for this value of a we must require that xc ¼ 0:6733 eV and V ¼ 2:274 eV. To derive a reasonable parameter for xc we have to take into account the fact that due to the presence of two bands, we have two effective masses. This complicates our expression for the isotope exponent, a, in Ref. [36]. Based on the work of Samuely et al. [20], we adopt a factor of 2/3 for the effective isotope exponent. This produces the values of xc and V given above. We also point out that the obtained value of V ¼ 2:274 eV is not a high value since in BCS theory the important parameter is the adimensional effective coupling k  V  N eff ðlÞ. From Fig. 1, we see that N eff ðlÞ  0:15 or less. This implies that k 6 0:23. As a consequence, the mean field approximation here adopted is justified. We point out that we have chosen the minimum value of xc which satisfies the experimental value of a (see Fig. 2). The value of xc we have obtained from our theory by experimental constraints is almost six times larger than the average energy of optical phonons in MgB2 compounds. We can

Fig. 2. Plot of V vs. xc (upper panel) and a vs. xc (lower panel). From the conditions T c ¼ 40 K and l ¼ 1:80 eV we find that V ¼ 2:274 eV and xc ¼ 0:6733 eV. The curve of a vs. xc is doubly valued. We have taken the minimum value of the two possible solutions. See text for more details.

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argue that the inclusion in the BCS gap equations of the intra-band attractive pairing, neglected in this paper to isolate the physics of the inter-band pairing, will increase the total effective pairing determining the critical temperature T c and therefore a smaller value of xc , of the same order of the MgB2 phononic energies, would be required to get experimental quantities. As l ¼ 1:80 eV corresponds to MgB2, we consider here the doping of Mg by Al, which leads to the compound Mg1x Alx B. Doping with Al is equivalent to changing the chemical potential l. In Fig. 3 we plot T c vs. l and a vs. l. The lower panel of Fig. 1 shows the behavior of q vs. l, which allows to calculate q for any value of l. From Fig. 3, we see that with doping T c decreases to both sides of l ¼ 1:80 eV, while a vs. l has a more complex behavior. The decreasing of T c vs. l around l ¼ 1:80 eV is in agreement with experiments [22]. We propose that experimentalists measure a vs. l to test our predictions given in the lower panel of Fig. 3. In Fig. 4 we present T c vs. l (upper panel) and a vs. z (lower panel), z  ðl  2:2Þ=0:5, for several values of xc ðeVÞ, namely, xc ¼ 1:20; 1:00; 0:80; 0:60; 0:40 and 0:20 (eV), for V ¼ 2:274 eV. In this way we are looking for the dependence of T c;max vs. xc . From Fig. 4 (lower panel) we see that the a vs. z is almost monotonous to the left side of some critical values of z and then increases rapidly. 4. Conclusions

Fig. 3. T c vs. l (upper panel) and a vs. l (lower panel) for xc ¼ 0:6733 eV and V ¼ 2:274 eV. We see that T c decreases at both sides of l ¼ 1:80 eV, which corresponds to the maximum value of T c , namely, T c;max ¼ 43 eV at l ¼ 2:03 eV and a ¼ 0:23 for MgB2.

We have investigated the two-band superconductivity scenario for (AlMg)B2 superconducting compounds by an extended BCS mean field approach, considering the prominent role of the interband pairing near a critical point of the electronic structure: an electronic topological transition (ETT). From our results we have obtained that the maximum superconducting critical temperature occurs at the chemical potential l  2 eV, in the proximity of the ETT that occurs at 2.2 eV. Moreover, we have found that the change of dimensionality in the electronic band dispersion from 2D to 3D dimensions, is signaled by the isotope exponent that exhibits a minimum. The small value of the minimum of the isotope coefficient at the ETT is in agreement with previous results of Angilella et al. [39]. Therefore, experimental investigation of the variation of the isotope coefficient by tuning the Fermi level, e.g., by Al doping, will provide relevant information for the pairing process in diborides. Extension of the present work in order to compare our theory with experimental results will require the inclusion in the hamiltonian of the hexagonal band structure, the intra-band attractive pairing together with the inter-band pairing and of the long range Coulomb repulsion between electronic carriers. Acknowledgements

Fig. 4. Same as Fig. 3 for V ¼ 2:274 eV, for several values of xc . We see that the results we have obtained are stable under variations of the cut-off parameter. See text for details.

The authors wish to thank kindly the following supporting Venezuelan agencies: FONACIT (Project S1 2002000448), European Community SREP Project 517039 ‘‘Controlling Mesoscopic Phase Separation (COMEPHS)” and CDCH-UC (Project 2004-014) (JJRN), and the Brazilian agencies CNPq and FAPERGS (AAS). Numerical calculations were performed at LANA (UFSM, Brazil) and SUPERCOMP (UC, Venezuela). We thank M.D. García for reading the manuscript. Interesting discussions with C.I. Ventura, M. Acquarone and R. Citro are fully acknowledged. In particular, one of the authors (JJRN) has held very interesting discussions with Giuseppe G.N. Angilella, while attending the Majorana School in 2006. Finally, one of us (J.J.R.N.) thanks the hospitality of ICTP Abdus Salam, Trieste, Italy as Senior Associate for a very productive stay (August 2005 and August–September 2007), where this paper was partially written. J.J. Rodríguez-Núñez is a member of the Venezuelan Scientific Program (P.P.I.-IV).

J.J. Rodríguez-Núñez et al. / Physica C 468 (2008) 2299–2304

References

Appendix A The set of Eqs. (8) and (9) can be given more explicitly for the superconducting critical temperature T c and the chemical potential l as

  Ni ðxÞ tanh ðx þ 0i  lÞ=2T c ; 2 ðx þ 0i  lÞ ai   Z di x þ 0i  l Ni ðxÞ dx tanh F~i ðT c ; lÞ ¼ ; 2 2T c ci Z

F i ðT c ; lÞ ¼

bi

dx

[1] [2] [3] [4] [5]

ðA1Þ [6]

ðA2Þ

[7]

where kB  1 and [8] [9] [10] [11]

ai ¼ maxfl  xc  0i ; min½i ð~ kÞgci ¼ min½i ð~ kÞ; 0 ~ kÞ: bi ¼ minfl þ xc  i ; max½i ðkÞgdi ¼ max½i ð~ N i ðxÞ is the density of states for the bands:

Ni ðxÞ ¼

Z

1 2p3 t i

Li ðxÞ

F

li ðxÞ

hp 2

r (i ¼ 2) and p (i ¼ 3)

[12] [13]

i ; K i ðx; yÞ dy;

ðA3Þ

where F½p2 ; x is the elliptic integral of the first kind and

[14] [15] [16]

1

li ðxÞ ¼ cos fmin½þ1; ðwi ðxÞ þ 2Þ=si g;

[17]

Li ðxÞ ¼ cos1 fmax½1; ðwi ðxÞ  2Þ=si g; " #12 ½wi ðxÞ  si cosðyÞ2 ; K i ðx; yÞ ¼ 1  4 wi ðxÞ ¼

[18] [19]

x  0i : 2t i

[20] [21]

The self-consistent Eqs. (6) and (7) can be re-written considering iterations as

h i12 T cjþ1 ¼ ðV=4Þ F 2 ðT cj ; lj Þ  F 3 ðT cj ; lj Þ ;

ljþ1 ¼

4ðn  1ÞT cj þ ½F~2a ðT cj ; lj Þ þ F~3a ðT cj ; lj Þ ; F~ ðT cj ; l Þ þ F~ ðT cj ; l Þ

[22]

ðA4Þ ðA5Þ

[23] [24]

where F 2 , F 3 , F~2a;b and F~3a;b are obtained from Eqs. (A1) and (A2) as

[25]

j

2b

F i ðT c ; lÞ ¼ F~ia ðT c ; lÞ ¼ F~ib ðT c ; lÞ ¼

Z

bi

3b

j

[26]

Ni ðxÞfi ðx; l; T c Þ dx;

ðA6Þ

Ni ðxÞ½x þ 0i fi ðx; l; T c Þ dx;

ðA7Þ

ai

Z Z

di

ci di

[27] [28]

Ni ðxÞfi ðx; l; T c Þ dx;

ðA8Þ

ci

[29]

where

[30]

tanh½X i ðx; l; T c Þ fi ðx; l; T c Þ ¼ ; X i ðx; l; T c Þ x þ 0i  l : X i ðx; l; T c Þ ¼ 2T c The numerical solution of Eqs. (6) and (7) are obtained, for a given pair of fixed values ðV; nÞ, by iterating the system of Eqs. (A4) and (A5) from proper initial values ðT c0 ; l0 Þ until desired absolute and relative tolerances are achieved: 7

jT ciþ1  T ci j; jliþ1  li j 6 10 eV;     T ciþ1  T ci  liþ1  li  ;   6 105 :    l   T ci

2303

i

A set of solutions, i.e., a line in the T c  l or q  l planes is obtained by fixing a value for V and varying the value of n ðn ¼ 2qÞ by small steps and using the solution values ðT c ; lÞ obtained from a previous nj1 quantity as initial values for the next nj quantity.

[31] [32] [33] [34] [35] [36] [37]

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 1=2 E1;2 ð~ kÞ ¼ V 1 Að~ kÞ þ Bð~ kÞ þ 4V 2 cosðkx a=2Þ pffiffiffi  cosð 3ky a=2Þ þ 2V 3 cosðkz cÞ h i2 pffiffiffi pffiffiffi Að~ kÞ  cosðky a= 3Þ þ 2 cosðky a=ð2 3ÞÞ cosðkx a=2Þ h pffiffiffi pffiffiffi i2 Bð~ kÞ ¼ 2 sinðky a=ð2 3ÞÞ cosðkx a=2Þ  sinðky a= 3Þ ;

ðA9Þ

2304

J.J. Rodríguez-Núñez et al. / Physica C 468 (2008) 2299–2304

where V 1;2 is the hopping integral between atoms placed on the hexagonal lattice (second type of atoms placed inside the primitive cell). V 3 is the effective hopping integral along the c-axis (or z-axis). However, in this paper, we follow the bandstructure of Eqs. (4) and (5). The hexagonal lattice (Eq. (A9)) is left for a future calculation.

[38] In the general case, i.e., with intra- and inter-band pairing, we have to follow the steps of Ref. [2]. However, in this paper, for simplicity, we use intra-band pairing only. An extension to inter-band pairing is under way. [39] G.G.N. Angilella, A. Bianconi, R. Pucci, J. Supercond. 18 (2006) 619.