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Two-band soliton-like polaron model for MgB2 superconductivity
Physics Letters A 317 (2003) 483–488 www.elsevier.com/locate/pla
Two-band soliton-like polaron model for Mg B2 superconductivity Yi Wang a,∗ , Francesco Ancilotto b , Flavio Toigo b , Kelin Wang a a Structure Research Laboratory and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, PR China b Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica G. Galilei, Università di Padova,
via Marzolo 8, 35131 Padova, Italy Received 27 June 2003; accepted 27 August 2003 Communicated by J. Flouquet
Abstract Using a two-band soliton-like polaron model of a deformable continuum, the critical temperature, energy gap and other parameters in Mg B2 superconductor are studied in the framework of finite temperature Green’s function technique. Two group of equations for the two energy gaps and two transition temperatures are derived by taking into account the different dimensionality of the relevant bands involved. Based on the ratio between the quasi-2D σ -band energy gap ∆1 and the quasiparticle ground-state energy |Eg1 |, we find, in agreement with recent findings, that the conventional ratio 2∆1 /kB Tc1 for the 2D σ -band can take a value ∼ 4.4. Based on the ratio of the quasi-particle kinetic energy and quasi-3D energy gap ∆2 , we find instead for the smaller π-band ratio 2∆2 /kB Tc2 the value ∼ 2.3. Both these estimates are in quantitative agreement with experimental measurements. The calculated transition temperatures, energy gaps, coherence length and Fermi velocity, are also in agreement with recent experimental data and ab initio calculations. 2003 Elsevier B.V. All rights reserved. PACS: 74.25.Jb; 63.20.Kr; 11.10.Lm; 74.70.Ad
1. Introduction The recent remarkable discovery of superconductivity at Tc ≈ 39 K in magnesium diboride, Mg B2 , has arouse great interest in the scientific community [1–3]. Experiments indicate that the superconducting state of Mg B2 is a conventional s-wave paring state mediated by phonon exchange [1]. At variance with other superconductor materials of the same type, the Mg B2 band structure is characterized by two energy gaps, the larger gap ∆1 being associated with the nearly cylin* Corresponding author.
E-mail address: [email protected] (Y. Wang). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.08.052
drical σ quasi-two-dimensional (quasi-2D) sheets of the Fermi surface, while the smaller gap ∆2 is associated with the three-dimensional (3D) π sheets [4–6]. In order to gain a better understanding of the nature of superconductivity of Mg B2 , intense effort has been directed towards studying the various properties of this material. Researches on isotope effect [7,8], magnetoresistivity and upper critical field [2,9], transport [10], tunneling [11–13], thermodynamic [14], lattice properties [15–17] and thin film effect [18] have recently been reported. The very recent experimental study of the anisotropic superconductor Mg B2 using a combination of scanning tunneling microscopy and spectroscopy reveals two distinct energy gaps at
484
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
∆1 = 7.1 meV and ∆2 = 2.3 meV at 4.2 K [19]. The π -band superconducting gap, ∆2 = 2.2 meV, was also recently observed by scanning tunneling spectroscopy [20]. Many results show that conventional phonon-mediated electron pairing theory can explain superconductivity in Mg B2 when both the anisotropy of the nonlinear electron–phonon interaction and the anharmonicity of the phonons are properly taken into account [17,21–24]. In this Letter we have focused our investigation on the strong coupling of carriers with optical phonons in the B-layers of Mg B2 [25] and anharmonic phonons induced by the deformation of the continuum. By employing a two-band soliton-like polaron model, two critical transition temperatures, two energy gaps and other properties in Mg B2 superconductor are examined in the framework of finite temperature Green’s function technique. The validity of two-bands models for Mg B2 has been confirmed by recent transport experiments where it has been shown that the impurity scattering between the σ - and π -bands is exceptionally small, and thus the σ - and π -bands behave as two separate conduction channels in parallel [26].
2. Soliton-like polaron model The system Hamiltonian that we consider contains a Fröhlich type electron–phonon interaction plus a deformation potential term: H = H0 + Hint h¯ 2 2 + hωa + ¯ q aq 2m q Vq aq eiq·r + Vq∗ aq+ e−iq·r , +
H0 = −
(1)
q
where Vq is electron–phonon coupling coefficient, h¯ 1/2 1/2 Vq = h¯ ωs 2πλ for two-dimensional sysSq 2mωs tem (i.e., the strongly anharmonic in-plane B–B stret h¯ 1/21/2 ching modes in Mg B2 and Vq = h¯ ω V4πλ q2 2mω for three-dimensional system, respectively. Here S is the total surface area, V is the volume of the system, ωs and ω are two-dimensional (2D) and threedimensional (3D) longitudinal optical (LO) phonon frequency, respectively, and λ is the dimensionless electron–phonon coupling constant. Hint represents the potential energy of electrons subject to a lattice de-
formation potential [27], Hint(r) = − dr1 Z(r, r1 )∆(r1 ),
(2)
where ∆(r) is the dimensionless deformation variable of the continuum at r, Z(r1 , r) gives the dependence of the electronic potential energy at r1 on the deformation of the continuum at r [27]. As given in our previous article [28] that the wave function (solution) of such a deformable polaron model satisfy a nonlinear Schrödinger equation: i h¯
h¯ 2 2 dΨ (r, t) =− Ψ (r, t) + AΨ (r, t) dt 2m r 2 − B Ψ (r, t) Ψ (r, t),
(3)
where A, which is the polaron ground-state energy within second-order perturbation theory, is equal to A = −λh¯ ω for 3D system, and A = − π2 λh¯ ωs for 2D system. B = dg 2 /k, where d is the interatomic separation, g is the coupling coefficient of the deformable potential energy Z(r1 , r) and k is the Hooke’s law stiffness constant. Assuming that the medium is uniform along the directions y and z, the general 3D wave function describing the quasi-particle can be written as Ψ (r, t) = Φ(x, t) exp(ip · r/h¯ ), where p = (0, py , pz ) and Φ(x, t) represents a soliton-like solution [28]: Φ(x, t)
1/2 η = 2 exp i h¯ −1 mvx x − A + 12 mvx2 − × cosh[η(x − vx t)]
h¯ 2 η2 2m t
. (4)
Here η = mB/2h¯ 2 and vx is the soliton velocity. When vx = 0, the ground state energy of the quasiparticle described by such state is found to be [28] 2 Eg = A − mB2 . Note that, the ground state energy is 8h¯ always negative, representing a true bound state of the system even when the soliton velocity is zero. By employing the finite temperature Green’s function technique [29], the superconductivity of Mg B2 was investigated according to the BCS-like treatment. We define the single quasi-particle Green’s function as G(r, r ) = − Tτ Ψ (r)Ψ (r ) , where · · · denotes the
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
average of the grand canonical ensemble, τ is imaginary time and r = (r, τ ). By using the equation of motion:
∂ ∂Ψ (r) − G(r, r ) = δ(r − r ) + Tτ Ψ (r ) , (5) ∂τ ∂τ and the evolution equation (3), one gets −
h2
∂ ¯ G(r, r ) = δ(r − r ) + 2 G(r, r ) ∂τ 2m − B Tτ Ψ (r)Ψ + (r)Ψ (r )Ψ + (r ) − µG(r, r )
(6)
(note that in the above equation the quantity A has been incorporated in the definition of the chemical potential µ). Based on Gorkov’s assumption, the fouroperator term of above equation can be treated as Tτ Ψ (x)Ψ + (x)Ψ (x )Ψ + (x ) = Tτ Ψ (x)Ψ (x ) Tτ Ψ + (x)Ψ + (x ) , (7) where F (x, x ) = Tτ [Ψ (x)Ψ (x )], F + (x, x ) = Tτ [Ψ + (x)Ψ + (x )] are defined as the “anomalous” Green’s functions. Such that, the equation of motion of the quasi-particle can be rewritten as −
∂ h¯ 2 2 G(x, x ) = δ(x − x ) + G(x, x ) ∂τ 2m − BF + (x, x )F (x, x ) − µG(x, x ). (8)
If the system is translationally invariant, F (x, x ) = F (x − x ), thus yields F (x, x) = F (0+ ). Define BFαβ (0+ ) = ∆αβ , the two equations satisfied by G(x, x ) and F (x, x ) are derived separately to be h¯ 2 2 ∂ − + µ Gαβ (x − x ) − ∂τ 2m + ∆αγ Fγ+β (x − x ) = δ(x − x )δαβ ,
(9)
h¯ 2 2 ∂ + − + µ Fαβ (x − x ) − ∆∗αγ Gγ+β (x − x ) ∂τ 2m = 0. (10)
The Fourier transform of above equations with respect to frequency yield (−iωn − ξk )G(k, ωn ) + ∆F + (k, ωn ) = 1, +
∗
(iωn − ξk )F (k, ωn ) − ∆ G(k, ωn ) = 0,
(11) (12)
485
which can be solved immediately to yield iωn + ξk , + ξk2 + |∆|2 ∆∗ F + (k, ωn ) = , 2 ωn + ξk2 + |∆|2
G(k, ωn ) = −
(13)
ωn2
(14)
where ξk = h¯ 2 k2 /2m − µ. Considering ∆ = ∆∗ and the energy dispersion spectrum of the quasi-particle,
ε(k) = ξk2 + ∆2 , the following energy gap equation can thus be derived 1 − 1 = BT e−iωn 0 2 + ε 2 (k) ω n n k 1 βc ε(k) tanh , =B (15) 2ε(k) 2 k
where βc ≡ 1/(kB Tc ). We would like to stress here that, although the gap equation looks like the usual gap equation of the microscopic BCS theory, in the present context additional important informations, as shown in the following, can be obtained due to the different dimensionality of the relevant bands involved in Eq. (15). We next analyze the content of the previous gap equation by treating it separately according to the different topology of the bands involved in Mg B2 , i.e., the quasi-2D σ -bands and the quasi-3D π -bands.
3. Two-band energy gaps and discussions For the boron-derived σ -band structure, at the transition temperature Tc1 the energy gap ∆1 is equal to zero, ∆1 = 0. By turning the summation into an integral one gets from Eq. (15): m 1 = B1 4π h¯ 2
Emax1
d ln ξ(k) tanh −|Eg1 |
βc1 ξ(k) , 2
(16)
where B1 , as mentioned before, is a strain coefficient determined by the coupling coefficient g of the deformation potential, the Hooke’s law stiffness constant k and the interatomic separation d in the σ -band sheets (see Eq. (3)). By performing explicitly the integration one finds 1/2 − κ 2 2γ e Nσ (0)g d , Emax1 |Eg1 | kB Tc1 ≈ (17) π
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where ln γ = C ≈ 0.577 is the Euler’s constant, 2γ π ≈ 1.134. βc1 ≡ 1/kB Tc1 , and Nσ (0) is the density of the state at the Fermi level for 2D-bands. Eg1 is the ground state energy of the 2D quasi-particles, and, as stated previously, can achieve negative values being determined not only by electron–phonon coupling, but also by the deformation of the continuum. Emax1 may be regarded as the σ -band Fermi energy in the boron layers. At zero temperature, β → ∞ and tanh β → 1, and assuming ∆1 (0) = γ1 |Eg1 |, we get from Eq. (15) 1/2 − κ 2 e Nσ (0)g d . ∆1 = 2 1 + γ12 + 2 Emax1 |Eg1 | (18) We can thus express, by using Eqs. (17) and (18), the ratio between the energy gap and the transition temperature in terms solely of γ1 as follows: 8 1 + γ12 + 8 2∆1 R1 ≡ (19) . = KB Tc1 1.134 The above relation can be thought as the generalization of R for different types of superconductors, the value of R for a given system being determined by the competition between energy gap and quasi-particle ground-state energy. We see that, for very low values of γ1 , the ratio R approaches to the conventional BCS weak-coupling limit, R = 3.53. Low values of γ1 do not imply however that the energy gap is strictly zero, but rather ∆1 (0) |Eg1 |. For most materials characterized by strong electron–phonon coupling, the ground state energy is large [30]. A higher value of γ , on the other hand, does not only means that the system has a lower ground state energy, but rather a large energy gap. Assuming that Eg1 is about one half of the energy gap, |Eg1 | 12 ∆1 , we find R1 ≈ 4.4. This gap ratio is close to the recent experiments and theoretical calculations, R1 = 4.2, 4.3, 4.4 and 4.18, respectively [19,25, 31–33], showing that the assumption made is reasonable. By using the above value for the ratio R1 , and taking for the transition temperature the experimental value, Tc1 = 38 K, the energy gap is found to be ∆1 = 7.2 meV. This result is in agreement with the experimental values [4,5,12,25]. In particular, it compares very well with the most recent experimental determination, ∆1 = 7.1 ± 0.2 meV [19].
Once the relevant physical parameters are given, Tc and ∆ value can be evaluated according to Eqs. (17) and (18). In the simplest model we may assume that g is a simple function of lattice constants a, c, namely, assuming g = g0 (c/a)1/2 . By defining C1 ≡ (Emax1 |Eg1 |)1/2 , C2 ≡ κ/[g02 dNσ ], we have 1.134C1 −C2 a/c e , kB ∆1 = 2.54404C1e−C2 a/c . Tc1 =
(20)
By employing two groups of data given in Refs. [8, 9,21,25], which refer to measurements on two different boron isotopes Mg10 B2 and Mg11 B2 : (i) a = 3.0860 Å, c = 3.5204 Å, Tc1 = 38.9 K; (ii) a = 3.1432 ± 0.0315 Å, c = 3.5193 ± 0.0323 Å, Tc1 = 40.2 K, and putting them into the first formula in (20), 1 we get 1.13C kB = 220.543 K and C2 = −1.98906. As mentioned above, Emax1 ≈ EF 1 (the σ -band Fermi energy) and |Eg1 | ≈ 12 ∆1 , so the Fermi energy is found approximately, by using the second of Eq. (12) with the calculated values of C1 and C2 , EF 1 ∼ 78 meV. The Fermi velocity vF 1 can then be
7 ∗ F1 calculated vF 1 = 2E m∗ ∼ 3.3 × 10 cm/s. Here m ∗ is the quasi-particle effective mass, m ≈ 0.25me [34, 35]. One can see that the calculated Fermi velocity is in reasonable agreement with the value for the average Fermi velocity of vF 1 = 4.7 × 107 cm/s given in Refs. [10,36]. The Pippard coherence length is also an important parameter for superconductors, and it is given ¯ F1 approximately by ξ0 ≈ hv 2π∆ . By using the values quoted above for vF 1 , we obtain ξ0 ≈ 4.8 nm, a value which is agreement with the recent experimental estimates ξ0 ≈ 4.3 nm and 4.9 nm [2,37,38]. It has been shown [14] that the Ginzburg–Landau parameter for Mg B2 is κ ≈ 26, so the in-plane London penetration depth λH = κξ0 is approximately 125 nm at T = 0, which is close to the experimental results of 112 nm and 140 nm, respectively [2,14]. According to formula Hc2 = φ0 2 , where φ0 = 2.07 × 10−7 Gcm2 is the flux 2πξ0
quantum, the upper critical magnetic field Hc2 can be estimated to be 14 T. Again this value is close to the experimental predictions Hc2 (0) = 14, 16.4 and 14 T, respectively [2,9,38]. By following the same procedure leading to Eqs. (17) and (18), but considering now the quasi-3D π -band of Mg B2 , we can obtain the following pair of
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
equations for the smaller energy gap ∆2 and for the critical transition temperature Tc2 :
1
2/3
3 1 + γ22 2/3 3/2 × B2 Nπ (0) Emax 2 + |Eg2 |3/2 , (21) 2/3 2/3 3/2 1 1 B2 Nπ (0) Emax 2 + |Eg2|3/2 , kB Tc2 = 2 3 (22) ∆2 =
where B2 is the strain coefficient in the π -band sheets. Nπ (0) is the density of state at the Fermi level for π -bands, Emax 2 can be regarded as the π -band Fermi energy, and Eg2 is the quasi-particle ground-state energy. We thus have for the ratio between the energy gap and the transition temperature: R2 ≡
2∆2 4 = . KB Tc2 (1 + γ22 )1/3
(23)
Here γ2 = ξk /∆2 is the ratio of the kinetic energy of the quasiparticles to the energy gap. Considering ξk as an average value of the kinetic energy, i.e., the energy of conducting carriers which can overcome the barrier of the energy gap, it can be simply estimated as 52 kB Tc2 ≈ 5 meV. Based on the above estimate, we take ξk ≈ 2∆2 and then find R2 ≈ 2.34. This value is very close to recent experimental results, as reported in Refs. [5,19,20,31], where R values ranging from 1.0 to 2.8 are reported. We notice that possible values for R2 , which are determined by the kinetic energy of the quasiparticles relative to the smaller energy gap ∆2 , range from ∼ 3.2 to 0.86 for γ2 between 1 and 10. Many evidence show that the smaller π -band gap-temperature ratio 2∆2 /kB Tc2 is located in this region [4,5,19,20,31–33,36]. Taking ∆2 ≈ 2.2 meV [5,19,20,39], the transition temperature of smaller band is found to be approximatively Tc2 ≈ 21.8 K. This value is also close to the recently calculated and experimental values 22, 20, 19 and 25.4 K, respectively [4,19,21,32]. In summary, by using finite temperature Green’s function technique within a simple two-band approach, superconductivity in magnesium diboride (Mg B2 ) is investigated using a soliton-like polaron model based on phonon-mediated interaction and localized deformation of the continuum [27,28]. Our
487
theoretical results for the transition temperature Tc , energy gap ∆, coherence length ξ0 , Fermi velocity vF and the upper critical magnetic field Hc2 are found to be in reasonable agreement with the experimental data, confirming that both the electron–phonon interaction and the anharmonicity of the phonons are key ingredients in the superconducting behavior of Mg B2 .
Acknowledgements Y.W. acknowledges the financial support from project “Accordo di Cooperazione Scientifica CNR/CAS” of the Consiglio Nazionale delle Ricerche. This work was also supported by National Natural Science Fundation of China (grant No. 50272063).
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Two-band soliton-like polaron model for Mg B2 superconductivity Yi Wang a,∗ , Francesco Ancilotto b , Flavio Toigo b , Kelin Wang a a Structure Research Laboratory and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, PR China b Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica G. Galilei, Università di Padova,
via Marzolo 8, 35131 Padova, Italy Received 27 June 2003; accepted 27 August 2003 Communicated by J. Flouquet
Abstract Using a two-band soliton-like polaron model of a deformable continuum, the critical temperature, energy gap and other parameters in Mg B2 superconductor are studied in the framework of finite temperature Green’s function technique. Two group of equations for the two energy gaps and two transition temperatures are derived by taking into account the different dimensionality of the relevant bands involved. Based on the ratio between the quasi-2D σ -band energy gap ∆1 and the quasiparticle ground-state energy |Eg1 |, we find, in agreement with recent findings, that the conventional ratio 2∆1 /kB Tc1 for the 2D σ -band can take a value ∼ 4.4. Based on the ratio of the quasi-particle kinetic energy and quasi-3D energy gap ∆2 , we find instead for the smaller π-band ratio 2∆2 /kB Tc2 the value ∼ 2.3. Both these estimates are in quantitative agreement with experimental measurements. The calculated transition temperatures, energy gaps, coherence length and Fermi velocity, are also in agreement with recent experimental data and ab initio calculations. 2003 Elsevier B.V. All rights reserved. PACS: 74.25.Jb; 63.20.Kr; 11.10.Lm; 74.70.Ad
1. Introduction The recent remarkable discovery of superconductivity at Tc ≈ 39 K in magnesium diboride, Mg B2 , has arouse great interest in the scientific community [1–3]. Experiments indicate that the superconducting state of Mg B2 is a conventional s-wave paring state mediated by phonon exchange [1]. At variance with other superconductor materials of the same type, the Mg B2 band structure is characterized by two energy gaps, the larger gap ∆1 being associated with the nearly cylin* Corresponding author.
E-mail address: [email protected] (Y. Wang). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.08.052
drical σ quasi-two-dimensional (quasi-2D) sheets of the Fermi surface, while the smaller gap ∆2 is associated with the three-dimensional (3D) π sheets [4–6]. In order to gain a better understanding of the nature of superconductivity of Mg B2 , intense effort has been directed towards studying the various properties of this material. Researches on isotope effect [7,8], magnetoresistivity and upper critical field [2,9], transport [10], tunneling [11–13], thermodynamic [14], lattice properties [15–17] and thin film effect [18] have recently been reported. The very recent experimental study of the anisotropic superconductor Mg B2 using a combination of scanning tunneling microscopy and spectroscopy reveals two distinct energy gaps at
484
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
∆1 = 7.1 meV and ∆2 = 2.3 meV at 4.2 K [19]. The π -band superconducting gap, ∆2 = 2.2 meV, was also recently observed by scanning tunneling spectroscopy [20]. Many results show that conventional phonon-mediated electron pairing theory can explain superconductivity in Mg B2 when both the anisotropy of the nonlinear electron–phonon interaction and the anharmonicity of the phonons are properly taken into account [17,21–24]. In this Letter we have focused our investigation on the strong coupling of carriers with optical phonons in the B-layers of Mg B2 [25] and anharmonic phonons induced by the deformation of the continuum. By employing a two-band soliton-like polaron model, two critical transition temperatures, two energy gaps and other properties in Mg B2 superconductor are examined in the framework of finite temperature Green’s function technique. The validity of two-bands models for Mg B2 has been confirmed by recent transport experiments where it has been shown that the impurity scattering between the σ - and π -bands is exceptionally small, and thus the σ - and π -bands behave as two separate conduction channels in parallel [26].
2. Soliton-like polaron model The system Hamiltonian that we consider contains a Fröhlich type electron–phonon interaction plus a deformation potential term: H = H0 + Hint h¯ 2 2 + hωa + ¯ q aq 2m q Vq aq eiq·r + Vq∗ aq+ e−iq·r , +
H0 = −
(1)
q
where Vq is electron–phonon coupling coefficient, h¯ 1/2 1/2 Vq = h¯ ωs 2πλ for two-dimensional sysSq 2mωs tem (i.e., the strongly anharmonic in-plane B–B stret h¯ 1/21/2 ching modes in Mg B2 and Vq = h¯ ω V4πλ q2 2mω for three-dimensional system, respectively. Here S is the total surface area, V is the volume of the system, ωs and ω are two-dimensional (2D) and threedimensional (3D) longitudinal optical (LO) phonon frequency, respectively, and λ is the dimensionless electron–phonon coupling constant. Hint represents the potential energy of electrons subject to a lattice de-
formation potential [27], Hint(r) = − dr1 Z(r, r1 )∆(r1 ),
(2)
where ∆(r) is the dimensionless deformation variable of the continuum at r, Z(r1 , r) gives the dependence of the electronic potential energy at r1 on the deformation of the continuum at r [27]. As given in our previous article [28] that the wave function (solution) of such a deformable polaron model satisfy a nonlinear Schrödinger equation: i h¯
h¯ 2 2 dΨ (r, t) =− Ψ (r, t) + AΨ (r, t) dt 2m r 2 − B Ψ (r, t) Ψ (r, t),
(3)
where A, which is the polaron ground-state energy within second-order perturbation theory, is equal to A = −λh¯ ω for 3D system, and A = − π2 λh¯ ωs for 2D system. B = dg 2 /k, where d is the interatomic separation, g is the coupling coefficient of the deformable potential energy Z(r1 , r) and k is the Hooke’s law stiffness constant. Assuming that the medium is uniform along the directions y and z, the general 3D wave function describing the quasi-particle can be written as Ψ (r, t) = Φ(x, t) exp(ip · r/h¯ ), where p = (0, py , pz ) and Φ(x, t) represents a soliton-like solution [28]: Φ(x, t)
1/2 η = 2 exp i h¯ −1 mvx x − A + 12 mvx2 − × cosh[η(x − vx t)]
h¯ 2 η2 2m t
. (4)
Here η = mB/2h¯ 2 and vx is the soliton velocity. When vx = 0, the ground state energy of the quasiparticle described by such state is found to be [28] 2 Eg = A − mB2 . Note that, the ground state energy is 8h¯ always negative, representing a true bound state of the system even when the soliton velocity is zero. By employing the finite temperature Green’s function technique [29], the superconductivity of Mg B2 was investigated according to the BCS-like treatment. We define the single quasi-particle Green’s function as G(r, r ) = − Tτ Ψ (r)Ψ (r ) , where · · · denotes the
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
average of the grand canonical ensemble, τ is imaginary time and r = (r, τ ). By using the equation of motion:
∂ ∂Ψ (r) − G(r, r ) = δ(r − r ) + Tτ Ψ (r ) , (5) ∂τ ∂τ and the evolution equation (3), one gets −
h2
∂ ¯ G(r, r ) = δ(r − r ) + 2 G(r, r ) ∂τ 2m − B Tτ Ψ (r)Ψ + (r)Ψ (r )Ψ + (r ) − µG(r, r )
(6)
(note that in the above equation the quantity A has been incorporated in the definition of the chemical potential µ). Based on Gorkov’s assumption, the fouroperator term of above equation can be treated as Tτ Ψ (x)Ψ + (x)Ψ (x )Ψ + (x ) = Tτ Ψ (x)Ψ (x ) Tτ Ψ + (x)Ψ + (x ) , (7) where F (x, x ) = Tτ [Ψ (x)Ψ (x )], F + (x, x ) = Tτ [Ψ + (x)Ψ + (x )] are defined as the “anomalous” Green’s functions. Such that, the equation of motion of the quasi-particle can be rewritten as −
∂ h¯ 2 2 G(x, x ) = δ(x − x ) + G(x, x ) ∂τ 2m − BF + (x, x )F (x, x ) − µG(x, x ). (8)
If the system is translationally invariant, F (x, x ) = F (x − x ), thus yields F (x, x) = F (0+ ). Define BFαβ (0+ ) = ∆αβ , the two equations satisfied by G(x, x ) and F (x, x ) are derived separately to be h¯ 2 2 ∂ − + µ Gαβ (x − x ) − ∂τ 2m + ∆αγ Fγ+β (x − x ) = δ(x − x )δαβ ,
(9)
h¯ 2 2 ∂ + − + µ Fαβ (x − x ) − ∆∗αγ Gγ+β (x − x ) ∂τ 2m = 0. (10)
The Fourier transform of above equations with respect to frequency yield (−iωn − ξk )G(k, ωn ) + ∆F + (k, ωn ) = 1, +
∗
(iωn − ξk )F (k, ωn ) − ∆ G(k, ωn ) = 0,
(11) (12)
485
which can be solved immediately to yield iωn + ξk , + ξk2 + |∆|2 ∆∗ F + (k, ωn ) = , 2 ωn + ξk2 + |∆|2
G(k, ωn ) = −
(13)
ωn2
(14)
where ξk = h¯ 2 k2 /2m − µ. Considering ∆ = ∆∗ and the energy dispersion spectrum of the quasi-particle,
ε(k) = ξk2 + ∆2 , the following energy gap equation can thus be derived 1 − 1 = BT e−iωn 0 2 + ε 2 (k) ω n n k 1 βc ε(k) tanh , =B (15) 2ε(k) 2 k
where βc ≡ 1/(kB Tc ). We would like to stress here that, although the gap equation looks like the usual gap equation of the microscopic BCS theory, in the present context additional important informations, as shown in the following, can be obtained due to the different dimensionality of the relevant bands involved in Eq. (15). We next analyze the content of the previous gap equation by treating it separately according to the different topology of the bands involved in Mg B2 , i.e., the quasi-2D σ -bands and the quasi-3D π -bands.
3. Two-band energy gaps and discussions For the boron-derived σ -band structure, at the transition temperature Tc1 the energy gap ∆1 is equal to zero, ∆1 = 0. By turning the summation into an integral one gets from Eq. (15): m 1 = B1 4π h¯ 2
Emax1
d ln ξ(k) tanh −|Eg1 |
βc1 ξ(k) , 2
(16)
where B1 , as mentioned before, is a strain coefficient determined by the coupling coefficient g of the deformation potential, the Hooke’s law stiffness constant k and the interatomic separation d in the σ -band sheets (see Eq. (3)). By performing explicitly the integration one finds 1/2 − κ 2 2γ e Nσ (0)g d , Emax1 |Eg1 | kB Tc1 ≈ (17) π
486
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
where ln γ = C ≈ 0.577 is the Euler’s constant, 2γ π ≈ 1.134. βc1 ≡ 1/kB Tc1 , and Nσ (0) is the density of the state at the Fermi level for 2D-bands. Eg1 is the ground state energy of the 2D quasi-particles, and, as stated previously, can achieve negative values being determined not only by electron–phonon coupling, but also by the deformation of the continuum. Emax1 may be regarded as the σ -band Fermi energy in the boron layers. At zero temperature, β → ∞ and tanh β → 1, and assuming ∆1 (0) = γ1 |Eg1 |, we get from Eq. (15) 1/2 − κ 2 e Nσ (0)g d . ∆1 = 2 1 + γ12 + 2 Emax1 |Eg1 | (18) We can thus express, by using Eqs. (17) and (18), the ratio between the energy gap and the transition temperature in terms solely of γ1 as follows: 8 1 + γ12 + 8 2∆1 R1 ≡ (19) . = KB Tc1 1.134 The above relation can be thought as the generalization of R for different types of superconductors, the value of R for a given system being determined by the competition between energy gap and quasi-particle ground-state energy. We see that, for very low values of γ1 , the ratio R approaches to the conventional BCS weak-coupling limit, R = 3.53. Low values of γ1 do not imply however that the energy gap is strictly zero, but rather ∆1 (0) |Eg1 |. For most materials characterized by strong electron–phonon coupling, the ground state energy is large [30]. A higher value of γ , on the other hand, does not only means that the system has a lower ground state energy, but rather a large energy gap. Assuming that Eg1 is about one half of the energy gap, |Eg1 | 12 ∆1 , we find R1 ≈ 4.4. This gap ratio is close to the recent experiments and theoretical calculations, R1 = 4.2, 4.3, 4.4 and 4.18, respectively [19,25, 31–33], showing that the assumption made is reasonable. By using the above value for the ratio R1 , and taking for the transition temperature the experimental value, Tc1 = 38 K, the energy gap is found to be ∆1 = 7.2 meV. This result is in agreement with the experimental values [4,5,12,25]. In particular, it compares very well with the most recent experimental determination, ∆1 = 7.1 ± 0.2 meV [19].
Once the relevant physical parameters are given, Tc and ∆ value can be evaluated according to Eqs. (17) and (18). In the simplest model we may assume that g is a simple function of lattice constants a, c, namely, assuming g = g0 (c/a)1/2 . By defining C1 ≡ (Emax1 |Eg1 |)1/2 , C2 ≡ κ/[g02 dNσ ], we have 1.134C1 −C2 a/c e , kB ∆1 = 2.54404C1e−C2 a/c . Tc1 =
(20)
By employing two groups of data given in Refs. [8, 9,21,25], which refer to measurements on two different boron isotopes Mg10 B2 and Mg11 B2 : (i) a = 3.0860 Å, c = 3.5204 Å, Tc1 = 38.9 K; (ii) a = 3.1432 ± 0.0315 Å, c = 3.5193 ± 0.0323 Å, Tc1 = 40.2 K, and putting them into the first formula in (20), 1 we get 1.13C kB = 220.543 K and C2 = −1.98906. As mentioned above, Emax1 ≈ EF 1 (the σ -band Fermi energy) and |Eg1 | ≈ 12 ∆1 , so the Fermi energy is found approximately, by using the second of Eq. (12) with the calculated values of C1 and C2 , EF 1 ∼ 78 meV. The Fermi velocity vF 1 can then be
7 ∗ F1 calculated vF 1 = 2E m∗ ∼ 3.3 × 10 cm/s. Here m ∗ is the quasi-particle effective mass, m ≈ 0.25me [34, 35]. One can see that the calculated Fermi velocity is in reasonable agreement with the value for the average Fermi velocity of vF 1 = 4.7 × 107 cm/s given in Refs. [10,36]. The Pippard coherence length is also an important parameter for superconductors, and it is given ¯ F1 approximately by ξ0 ≈ hv 2π∆ . By using the values quoted above for vF 1 , we obtain ξ0 ≈ 4.8 nm, a value which is agreement with the recent experimental estimates ξ0 ≈ 4.3 nm and 4.9 nm [2,37,38]. It has been shown [14] that the Ginzburg–Landau parameter for Mg B2 is κ ≈ 26, so the in-plane London penetration depth λH = κξ0 is approximately 125 nm at T = 0, which is close to the experimental results of 112 nm and 140 nm, respectively [2,14]. According to formula Hc2 = φ0 2 , where φ0 = 2.07 × 10−7 Gcm2 is the flux 2πξ0
quantum, the upper critical magnetic field Hc2 can be estimated to be 14 T. Again this value is close to the experimental predictions Hc2 (0) = 14, 16.4 and 14 T, respectively [2,9,38]. By following the same procedure leading to Eqs. (17) and (18), but considering now the quasi-3D π -band of Mg B2 , we can obtain the following pair of
Y. Wang et al. / Physics Letters A 317 (2003) 483–488
equations for the smaller energy gap ∆2 and for the critical transition temperature Tc2 :
1
2/3
3 1 + γ22 2/3 3/2 × B2 Nπ (0) Emax 2 + |Eg2 |3/2 , (21) 2/3 2/3 3/2 1 1 B2 Nπ (0) Emax 2 + |Eg2|3/2 , kB Tc2 = 2 3 (22) ∆2 =
where B2 is the strain coefficient in the π -band sheets. Nπ (0) is the density of state at the Fermi level for π -bands, Emax 2 can be regarded as the π -band Fermi energy, and Eg2 is the quasi-particle ground-state energy. We thus have for the ratio between the energy gap and the transition temperature: R2 ≡
2∆2 4 = . KB Tc2 (1 + γ22 )1/3
(23)
Here γ2 = ξk /∆2 is the ratio of the kinetic energy of the quasiparticles to the energy gap. Considering ξk as an average value of the kinetic energy, i.e., the energy of conducting carriers which can overcome the barrier of the energy gap, it can be simply estimated as 52 kB Tc2 ≈ 5 meV. Based on the above estimate, we take ξk ≈ 2∆2 and then find R2 ≈ 2.34. This value is very close to recent experimental results, as reported in Refs. [5,19,20,31], where R values ranging from 1.0 to 2.8 are reported. We notice that possible values for R2 , which are determined by the kinetic energy of the quasiparticles relative to the smaller energy gap ∆2 , range from ∼ 3.2 to 0.86 for γ2 between 1 and 10. Many evidence show that the smaller π -band gap-temperature ratio 2∆2 /kB Tc2 is located in this region [4,5,19,20,31–33,36]. Taking ∆2 ≈ 2.2 meV [5,19,20,39], the transition temperature of smaller band is found to be approximatively Tc2 ≈ 21.8 K. This value is also close to the recently calculated and experimental values 22, 20, 19 and 25.4 K, respectively [4,19,21,32]. In summary, by using finite temperature Green’s function technique within a simple two-band approach, superconductivity in magnesium diboride (Mg B2 ) is investigated using a soliton-like polaron model based on phonon-mediated interaction and localized deformation of the continuum [27,28]. Our
487
theoretical results for the transition temperature Tc , energy gap ∆, coherence length ξ0 , Fermi velocity vF and the upper critical magnetic field Hc2 are found to be in reasonable agreement with the experimental data, confirming that both the electron–phonon interaction and the anharmonicity of the phonons are key ingredients in the superconducting behavior of Mg B2 .
Acknowledgements Y.W. acknowledges the financial support from project “Accordo di Cooperazione Scientifica CNR/CAS” of the Consiglio Nazionale delle Ricerche. This work was also supported by National Natural Science Fundation of China (grant No. 50272063).
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