Journal of Superconductivity

, Volume 18, Issue 5, pp 625–636

Feshbach Shape Resonance in Multiband Superconductivity in Heterostructures


  • Antonio Bianconi
    • Department of PhysicsUniversity of Rome

DOI: 10.1007/s10948-005-0047-5

Cite this article as:
Bianconi, A. J Supercond (2005) 18: 625. doi:10.1007/s10948-005-0047-5


We report experimental data showing the Feshbach shape resonance in the electron doped MgB2 where the chemical potential is tuned by Al, Sc, and C substitutions. The scaling of the critical temperature Tc as a function of the Lifshitz parameter z = EΓEF, where EF is the chemical potential and EΓ is the energy of the Γ critical point where the σ Fermi surface changes from the 3D to a 2D topology, is reported. The resonant amplification of Tc(z) driven by the interband pairing is assigned to a Feshbach shape resonance characterized by quantum superposition of pairs in states corresponding to different spatial location and different parity. It is centered at z = 0 where the chemical potential is tuned to a Van Hove-Lifshits feature for the change of Fermi surface dimensionality in the electronic energy spectrum in one of the subbands. In this heterostructure at atomic limit the multiband superconductivity is in the clean limit because the disparity and negligible overlap between electron wavefunctions in different subbands suppresses the single electron interband impurity scattering rate. The emerging scenario from these experimental data suggests that the Feshbach shape resonance could be the mechanism for high Tc in particular nanostructured architectures.


feshbach resonanceshape resonancedoped diborides


The macroscopic quantum coherence in a superconducting condensate appears below a critical temperature Tc because it is suppressed by decoherence effects at high temperature. Understanding high Tc superconductivity [1] means to shed light on the mechanism that allows the quantum condensate to resist to the decoherence attacks of temperature. Advances in this topic may be useful, first, to design new room temperature superconducting wires and devices; second, to design a functional quantum computer where one has to face how to suppress decoherence effects to perform some useful information processing; third, to understand the unknown mechanism behind the macroscopic biological coherence in living matter. In fact quantum phase coherence could be behind the unknown fundamental physical properties of the living cell where it is possible that the evolution has refined its nanostructures and nanomachines to acquire their properties by deploying quantum tricks. In this scenario the special nanoscale architectures of biosystems could bestow decoherence-evading qualities on their components.

Following the failure of the single band theoretical models for a homogeneous 2D superconducting plane in cuprates the research is now shifting toward particular features of nanoscale heterogeneity. As so often in physics, the key for solving this problem could be in subtle details. Looking for details in the theory of superconductivity one has to go beyond the standard approximations [2,3] for a generic homogeneous systems (i) the high Fermi energy: the Fermi energy is assumed at an infinite distance from the top or the bottom of the conduction band, (ii) the isotropic approximation: the pairing mechanism is not electronic state dependent. The BCS wave-function of the superconducting ground state has been constructed by configuration interaction of all electron pairs (+k with spin up, and −k with spin down) on the Fermi surface in an energy window that is the energy cut off of the interaction,
$$\left| {\Psi _0 } \right\rangle = \prod\limits_k {(u_k + v_k c_{k \uparrow }^ + c_{ - k \downarrow }^ + )\left| 0 \right\rangle }$$
where \(\left| 0 \right\rangle\) is the vacuum state, \(c_{k \uparrow }^ +\) is the creation operator for an electron with momentum k and spin up. The construction of the BCS wavefunction has been inspired by the configuration interaction theory developed in nuclear [46 ] and atomic [7] physics.

In anisotropic superconductivity one has to consider configuration interaction between pairs, in an energy window ΔE around the Fermi level, in different locations of the k-space with a different pairing strength, and in the particular case of inhomogeneous systems also in different spatial locations, that gives a k-space dependent superfluid order parameter, i.e., a k-dependent superconducting gap. A particular case of anisotropic superconductivity is multiband superconductivity, where the order parameter is mainly different in different bands, that was developed on the basis of the Bogolyubov transformations [8,9]. The theory of multiband superconductivity has been proposed for standard metals [1042], for doped cuprate perovskites [43114], for magnesium diboride [115184] and for few other materials as Nb doped SrTiO3 [185], Sr2RuO4 [186188] YNi2B2C, LuNi2B2C [189], and NbSe2 [190]. We provide a nearly complete reference list on this subject since it is not available elsewhere.

The multiband superconductivity shows up only in the “clean limit,” where the single electron mean free path for the interband impurity scattering satisfies the condition \(l > hv_{\rm F} /\Delta _{{\rm av}}\) where vF is the Fermi velocity and \(\Delta _{{\rm av}}\) is the average superconducting gap [24,28,35]. This is a very strict condition therefore most of the metals are in the “dirty limit” where the interband impurity scattering mixes the electron wave functions of electrons on different spots on Fermi surfaces and it reduces the system to an effective single Fermi surface for the pairing interaction.

In the BCS theory an attractive interaction is required for the formation of Cooper pairs. The “interchannel pairing” or “interband pairing”, that transfers a pair from the “a”-band to the “b”-band and viceversa, is a non-BCS pairing process in fact here the interaction may be repulsive as it was first noticed by Kondo [14]. This term in the multiband pairing theory is expressed by
$$\sum\limits_{k,k'} {J(k,k')(a_{k\, \uparrow }^* a_{ - k\, \downarrow }^* b_{ - k\, \downarrow } b_{k \,\uparrow } )}$$
where a* and b* are creation operators of electrons in the “a” and “b” band, respectively and J(k, k′) is an exchange-like integral. It has been shown that in the case of repulsive interband interaction the order parameter shows the sign reversal [26,136]. The non-BCS nature of this pairing process is indicated also by the fact that the isotope effect vanishes when it is dominant even if the intra-band attractive interaction in each band is due the electron–phonon coupling. Moreover, the effective repulsive Coulomb pseudopotential in the Migdal Eliasberg theory goes to zero (so the effective coupling strength increases) where the interband pairing is dominant. A particular case of multiband superconductivity is where a Van Hove-Lifshits feature [191] in the electronic energy spectrum within the energy window of the pairing interaction is taken into account [32,62] and it is similar to consider in anisotropic superconductivity the Van Hove singularity in a hot spot in the k-space as it occurs at antinodal points in cuprate superconductors.


The “shape resonances” have been described by Feshbach in elastic scattering processes for neutron capture and nuclear fission [5,6] in the cloudy crystal ball model of nuclear reactions, where the theory is dealing with configuration interaction between multi-channel processes. Therefore, these resonances can be called also Feshbach shape resonance. The Feshbach resonance is a quantum phenomena that appears in many fields of physics and chemistry [191]. The Feshabach resonance is related with the Fano configuration interaction theory in atomic spectroscopy between two photo-ionization channels giving two different final states (1) a discrete quasi-bound state of two excited electrons and (2) a continuum [7] so they are also called Fano-Feshbach resonances. The Feshbach resonances and shape resonances are also common in the molecular association and dissociation processes.

The process for increasing Tc by “shape resonance” was first proposed by Blatt and Thompson [15,16,19] in 1963 for a superconducting film and extended in 1993 to heterostructures at atomic limit made of superlattices of superconducting units intercalated by a different material [75,76,80,81,8790,9397,102,109,110] shown in Fig. 1. In the physics of ultracold atomic gasses it has been proposed for the manipulation of the inter-atomic interaction to tune the chemical potential of the atomic gas around the energy of a discrete level of a biatomic molecule controlled by an external magnetic field [193]. This quantum phenomenon has been used by Ketterle to achieve the Bose-Einstein condensation (BEC) in dilute bosonic gases of alkali atoms [194]. Feshbach resonances have recently been used to get a BCS-like condensate in fermionic ultra-cold gases with large values of Tc/TF [195].
Fig. 1.

The metal heterostructures at the atomic limit: (from bottom to top) (a) superlattices of stripes in the CuO2 layers intercalated by X layers in cuprates; (b) superlattices of C60 spheres, intercalated by X ions as in fullerides; (c) superlattices of metallic layers (B2) intercalated by layers of a X material according with the claims of the patent “High-temperature superconductors made by metal heterostructures at the atomic limit” [69,70].

The shape resonance described by Blatt occurs in a superconducting thin film of thickness L where the chemical potential crosses the bottom En of the nth subband of the film, a quantum well, characterized by \(k_z = n\pi /L\); with n > 1 [15,16,19]. Therefore, it occurs where the chemical potential EF is tuned near the critical energy EF = En for a 2.5 Lifshitz electronic topological transition (ETT) [191] of type (I) as shown in Fig. 2. At this ETT a small Fermi surface of a second subband disappears while the large 2D Fermi surface of a first subband shows minor variations. In the “clean limit” the single electrons cannot be scattered from one to the other band but configuration interaction between pairs in different bands is possible in an energy window around EF = En. Therefore, the Feshbach shape resonance occurs by tuning the Lifshitz parameter z = EFEn around z = 0 where in the Blatt proposal En is tuned by changing the film thickness. The prediction of Blatt and Thompson of the oscillatory behavior of Tc as a function of film thickness L has been recently confirmed experimentally for a superconducting film [196] although phase fluctuations due to the low 2D dimension suppresses the critical temperature.
Fig. 2.

The different types of 2.5 Lifshitz electronic topological transition (ETT): The upper panel shows the type (I) ETT where the chemical potential EF is tuned to a Van Hove singularity (vHs) at the bottom (or at the top) of a second band with the appearance (or disappearance) of a new detached Fermi surface region. The lower panel shows the type (II) ETT with the disruption (or formation) of a “neck” in a second Fermi surface where the chemical potential EF is tuned at a vHs associated with the gradual transformation of the second Fermi surface from a two-dimensional (2D) corrugated cylinder to a closed surface with three-dimensional (3D) topology characteristics of a superlattice of metallic layers.

Later we have proposed to increase Tc via a Feshbach shape resonance in a different class of systems: superlattices of metallic units [75,76,80,81,8790,9397,102,109,110]. The idea is that high Tc superconductivity is possible in particular heterogeneous architectures formed by a superlattice of metallic superconducting units intercalated by a different material. The superconducting units can be dots, spheres, wires, tubes, layers, films, or quantum wells, and some practical realizations of these architectures are shown in Fig. 1: (i) superlattices of fullerene (quantum spheres), nanotubes (quantum wires), or graphene layers (quantum wells) using carbon atoms intercalated by different atoms can be a practical realization of heterostructures at atomic limit; (ii) superlattices of lead layers intercalated by germanium or silicon layers; (iii) superlattices of bcc metallic layers intercalated by fcc rocksalt layers of a different metal rotated by 45°; (iv) Superlattices of stripes in the CuO2 planes of doped cuprate perovskites; (v) Superlattices of boron graphene monolayers intercalated by hcp metallic monolayers forming the composite AlB2 crystallographic structure.
Fig. 3.

The type (III) ETT for a 2D superlattice of quantum stripes with the ETT transition from the one-dimensional (left panel) to two-dimensional (right panel) topology of the Fermi surface of a subband for a superlattice. Going from the left panel to the right panel the chemical potential EF crosses a vHs singularity at Ec associated with the change of the Fermi topology going from EF > Ec to EF < Ec, while the first Fermi surface retains its one-dimensional (1D) character. A relevant inter-band pairing process with the transfer of a pair from the first to the second subband and viceversa is shown.

The Feshbach shape resonances in superconductivity are characterized by the configuration interaction between (1) first pairs in a first subband where the particle group velocity goes to zero at a critical energy and (2) second pairs in a second wide band with high Fermi velocity where these two set of states correspond to different spatial locations [102,192].

The generic feature of the electronic structure of the superlattices is the presence of different subbands where the charge density associated with each subband is non homogenously distributed in the real space therefore they realize the quantum tricks for high Tc. First, the disparity and negligible overlap between electron wavefunctions of different subbands suppresses the impurity scattering rate that allows multiband superconductivity in the clean limit. Second, tuning the chemical potential in superlattices it is possible to realize different types of shape resonances at different ETT that are shown in Fig. 2. The type (I) ETT occurs by tuning the chemical potential through the bottom of a subband where a new Fermi surface appears or disappears. The type (II) ETT is characterized by the opening or closing of a neck in the one of the Fermi surfaces as shown in Fig. 2 where the chemical potential is tuned in the region where one Fermi surface changes from a the three-dimensional (3D), for EF > Ec to a two-dimensional (2D) topology, for EF < Ec and it is characteristic of superlattices of metallic layers.

The type (III) ETT, characteristic of a superlattice of quantum wires (or stripes or nanotubes), is shown in Fig. 3. It is characterized by a first one-dimensional (1D) Fermi surface that coexists with a second Fermi surface that changes from a two-dimensional (2D) to one-dimensional (1D) topology at the Lifshitz critical point EF = Ec. The calculation of Feshbach shape resonance in the interband pairing strength is difficult since it involves the quantum interference effects for the overlap of the four wave functions of the two electron pairs as in all Fano Feshbach resonances [102]. These calculations have been carried out for a superlattice of quantum stripes of width L, shown in Fig. 1, [9396,102] where the chemical potential is tuned near a type (III) ETT. The electronic structure of this heterostructure at atomic limit is determined by a periodic potential barrier V(x, y), with period λp, amplitude Vb and width W along the y direction and constant in the x direction expressed for x = constant as
$$V(y) = - V_{\rm b} \theta \left( {\frac{L}{2} - \tilde y} \right)\\ {\rm where} \ \tilde y = y - q\lambda _p - \frac{{\lambda _p }}{2}$$
and q is the integer part of \(\frac{y}{{\lambda _{\rm p} }}\), The solution of the Schrödinger equation for this system, \( - \frac{{\hbar ^2 }}{{2m}}\nabla ^2 \psi\;\, (x,y) + V(x,y)\psi\;\, (x,y) = E\psi\;\, (x,y)\) is given as
$$\psi\;\, _{n,k_x ,k_y } (x,y) = e^{ik_x x} \cdot e^{ik_y q\lambda _p } \psi\;\, _{n,k_y } (y)$$
where in the stripe
$$\psi\;\, _{n,k_y } (y) = \alpha e^{ik_w \tilde y} + \beta e^{ - ik_w \tilde y} \quad {\rm for}\quad \left| {\tilde y} \right| < L/2\\ k_w = \sqrt {2m_w (E_n (k_y ) + + V_b )/\hbar ^2 }$$
and in the barrier
$$\psi\;\, _{n,k_y } (y) = \gamma e^{ik_b \tilde y} + \delta e^{ - ik_b \tilde y} \quad {\rm for}\quad \left| {\tilde y} \right| \ge L/2\\ k_b = \sqrt {2m_b E_n (k_y )/\hbar ^2 }$$
The coefficients α, β, γ, and δ are obtained by imposing the Bloch conditions with periodicity λp, the continuity conditions of the wave function and its derivative at L/2, and finally by normalization in the surface unit. The solution of the eigenvalue equation for E gives the electronic energy dispersion for the n subbands with energy \(\varepsilon _n (k_x ,k_y ) = \varepsilon (k_x ) + E_n (k_y )\) where \(\varepsilon (k_x ) = (\hbar ^2 /2m)k_x^2\) is the free electron energy dispersion in the x direction and \(E_n (k_y )\) is the dispersion in the y direction. There are Nb solutions for \(E_n (k_y )\), with 1 ≤ n ≤ Nb, for each ky in the Brillouin zone of the superlattice giving a dispersion in the y direction of the Nb subbands with kx = 0. The superlattice with its characteristic wavevector q = 2π/λp induces a relevant k dependent interband pairing interaction \(V_{n,n\prime } (k,k\prime )\). This is the non BCS interband effective pairing interaction (of any repulsive or attractive nature [14,26]) with a generic cutoff energy \(\hbar \omega _{\rm o}\). The interband interaction is controlled by the details of the quantum superposition of states corresponding to different spatial locations, i.e., between the wave functions of the pairing electrons in the different subbands of the superlattice
$$V_{n,n'} (k,k') = V_{n,k_y;\, n',\, k'_y }^o \theta (\hbar \omega _0 - |\varepsilon _n (k) - \mu |)\theta (\hbar \omega _0\\ - |\varepsilon _{n'} (k') - \mu |)$$
where \(k = (k_x ,k_y )\) and
$$V_{n,k_y ;n',k'_y }^{\rm o} = - J\int\limits_S {dxdy\psi\;\, _{n, - k} (x,y)\psi\;\, _{n', - k'} (x,y)}\\ \psi\;\, _{n,k} (x,y)\psi\;\, _{n',k'} (x,y) \\ = - J\int\limits_S {dxdy|\psi\;\, _{n,k} (x,y)|^2 |\psi\;\, _{n',k'} (x,y)|^2 }$$
n and n′ are the subband indexes. kx (kx′) is the component of the wavevector in the stripe direction (or longitudinal direction) and ky, (ky′) is the superlattice wavevector (in the transverse direction) of the initial (final) state in the pairing process, and μ the chemical potential. In the separable kernel approximation, the gap parameter has the same energy cut off \(\hbar \omega _{\rm o}\) as the interaction. Therefore, it takes the values Δn (ky) around the Fermi surface in a range \(\hbar \omega _{\rm o}\) depending from the subband index and the superlattice wave vector ky. The self consistent equation, for the ground state energy gap Δn (ky) is
$$\Delta _n (\mu ,k_y )\\ \quad = - \frac{1}{{2N}}\sum\limits_{n'k'_y k'_x } {\frac{{V_{n,n'} (k,k')\Delta _{n'} (k'_y )}}{{\sqrt {(E_{n'} (k'_y ) + \varepsilon _{k'_x } - \mu )^2 + \Delta _{n'}^2 (k'_y )} }}}\qquad $$
where N is the total number of wavevectors. Solving iteratively this equation gives the anisotropic gaps dependent on the subband index and weakly dependent on the superlattice wavevector ky. The structure in the interaction gives different values for the gaps Δn giving a system with an anisotropic gaps in the different segments of the Fermi surface. The critical temperature Tc of the superconducting transition can be calculated by iterative method
$$\Delta _n (k) = - \frac{1}{N}\sum\limits_{n'k'} {V_{nn'} (k,k')\frac{{tgh\left( {\frac{{\xi _{n'} (k')}}{{2T_{\rm c} }}} \right)}}{{2\xi _{n'} (k')}}} \Delta _{n'} (k')\qquad $$
where ξn(k) = ɛn(k)−μ. These calculations show that the interband pairing enhances Tc [93,95] by tuning the chemical potential in an energy window around the Van Hove singularities associated with a change of the topology of the Fermi surface from 1D to 2D (or 2D to 3D) of one of the subbands of the superlattice in the clean limit.


MgB2 provides the simplest high Tc superconductor therefore it could play a key role for understanding high Tc superconductivity as atomic hydrogen for quantum mechanics. There is now growing evidence that MgB2 is a practical realization of the proposed Tc amplification process driven by Feshbach shape resonances in interband pairing [116,119,120,124,125,132,143,162,176,180].

The light-metal diborides AB2 (A: Mg, Al) are not common natural compounds. They have been discovered as residuals in the chemical processing [197,198] for the reduction of boron oxide with electropositive metals to obtain elemental boron. Following the synthesis and characterization of aluminum diboride (AlB2) in 1935 [199] magnesium diboride (MgB2) was synthesized in 1953 [200,201] when the chemical interest in borides was driven by nuclear power industry for control rods and neutron shields. For many years MgB2 was not considered as a possible superconductor by the scientific community on the basis of conventional theories or material science rules for search of high Tc superconductors. The AlB2 crystalline structure is a heterostructure at the atomic limit made of superconducting layers (boron monolayers) intercalated by different layers (Al or Mg hcp monolayers) shown in Fig. 2.
Fig. 4.

Two different types of 2.5 Lifshitz electronic topological transition (ETT) in electron doped MgB2 associated with changes of the Fermi surface topology. The panel (a) shows the type (I) ETT: with the appearance of the closed 3D σ Fermi surface (at the point indicated by a arrow) by tuning the Fermi energy at the critical point EF = EA where EA is the energy of the A point in the band structure calculated by I.I. Mazin. The panel (b) shows the type (II) ETT with the disruption of a “neck” at the point indicated by an arrow, in the σ Fermi surface by tuning the chemical potential EF at the critical point in the band structure EF = EΓ where the σ Fermi surface changes from a 2D corrugated tube for EΓ > EF to a closed 3D Fermi surface for EΓ < EF while the large Fermi surface π band retains its 3D topology.
Fig. 5.

The Lifshitz parameter z = EΓEF as a function of the number of holes in the σ subband in Al doped MgB2. The quantum uncertainty in the z value is indicated by the error bars that are given by ΔE = D\(\sqrt {\langle \sigma _{x,y}^2 \rangle }\) where D is the deformation potential and \(\sqrt {\langle \sigma _{x,y}^2 \rangle }\) is the mean square boron displacement at T = 0 K associated with the E2g mode measured by neutron diffraction [204]. The curve EΓEA indicates the position of the bottom of the σ band. The Tc amplification by Feshbach shape resonance occurs in the σ hole density range shown by the double arrow indicating where the 2D/3D ETT sweeps through the Fermi level because of zero point lattice motion, i.e., where the error bars intersect the z = 0 line.

The σ band is usually fully occupied in non superconducting diborides where there are no holes in the σ band, so these diborides, as AlB2, have a Fermi surface of the type shown by the case EF > EA shown in Fig. 4. In MgB2 the σ band is partially unoccupied due to the electron transfer from the boron layers to the magnesium layers. In fact the chemical potential EF in MgB2 is at about 750 meV below the energy EA of the top of the σ band. Moreover, the chemical potential EF in MgB2 is also at about 350 meV below the energy of the Γ point in the band structure (EΓ > EF). Therefore, the σ Fermi surface of MgB2 where EF < EΓ < EA has the corrugated tubular shape with a two-dimensional topology of the type shown in Fig. 4 for the case EF < EΓ. Going from below (EF < EΓ < EA) to above the energy of Γ point the σ Fermi surface becomes a closed FS with 3D topology like in AlMgB4 that belongs to the Fermi surface type shown in Fig. 4 for the case EΓ < EF < EA. Therefore, by tuning the chemical potential EF by electron doping the σ subband it is possible to reach the point where EF is tuned at the 2D/3D Van Hove singularity. This is a type (II) 2.5 Lifshitz electronic topological transition (ETT) with the disruption of a “neck” in the σ Fermi surface with the critical point at EF = EΓ, therefore, this transition will be studied here as a function of the Lifshitz parameter z = EΓEF. The influence of the proximity to a type (II) electronic topological transition on the anomalous electronic and lattice properties of MgB2 is shown by the anomalous pressure dependence of the E2g phonon mode and Tc [202,203]. The response of the superconducting properties of diborides to Fermi level tuning has been studied by electron doping using atomic substitutions, in fact it is possible to reduce the number of holes in the σ band from about 0.15 holes per unit cell in MgB2 up to reach zero σ holes at the top of the σ band where EF = EA. We report in Fig. 5 the variation of the Lifshitz parameter z = EΓEF as a function of the number of holes in the σ subband for the case of Al doped Mg1−x AlxB2. In order to understand the the physics of superconductivity ion diborides we have to consider that the Lifshitz parameter z has an intrinsic uncertainty due to quantum fluctuations. The error bars in Fig. 5 indicate the quantum uncertainty in the z value induced by zero point lattice fluctuations. In fact the in-plane boron phonon mode with E2g symmetry near the zone-center, split the partially occupied σ bands. The energy splitting of the two degenerate σ bands is given by ΔE = D\(\smash{\sqrt {\langle \sigma _{x,y}^2 \rangle }}\) where D is the deformation potential and \(\sqrt {\langle \sigma _{x,y}^2 \rangle }\) is the mean square in plane displacement at T = 0 K of the boron atoms due to zero point motion measured by neutron scattering [132,143,204]. The energy splitting is ΔE = ±0.4 eV in MgB2 and it decreases a little going to AlMgB4 where there are only 0.03 σ holes per unit cell. Therefore, the quantum lattice fluctuations induce the electronic quantum fluctuations of the Γ point relative to the Fermi level, i.e., of the Lifshitz parameter z. This induces quantum charge fluctuations between the two σ bands and between the σ and π bands involving electronic states within the energy window of ±0.4 eV around the Fermi level. These charge fluctuations control the energy window of states involved in the pairing processes therefore we can estimate from Fig. 5 the expected width of the Feshbach shape resonance. The high Tc amplification by Feshbach shape resonance should occur in the range −350 < z < +350 meV where the 2D/3D Van Hove singularity sweeps through the Fermi level that is represented in Fig. 5 where the value z = 0 follows within the error bars of z.

The Lifshitz parameter z as a function of x for the Mg1−xAlxB2, Mg1−xScxB2 MgB2−xCx system has been calculated by De Coss et al. by band structure calculations described elsewhere [205], therefore, it has been possible to convert the variation of the critical temperature as a function of the number density of substituted ions x to the variation of Tc versus the universal Lifshitz parameter z for all doped magnesium diborides.

The universal scaling of the critical temperature Tc, of the σ and π superconducting gaps as a function of the Lifshitz parameter z is reported in Fig. 6, where the superconducting gaps in the π and in the σ band and critical temperature Tc obtained by aluminum [143], carbon [183,184] and scandium [177] substitutions are reported on the same scale. These results show that the two-gap behavior is present over all the range of the Lifshitz parameter of ±400 meV around z = 0. These results support the predictions that the doped materials remain in the clean limit for interband pairing although the large number density of impurity centers that reduce the intraband mean free path and it falsifies the predictions of the gap closing and reaching the standard single band value \(\smash{\frac{{2\Delta _0 }}{{K_{\rm B} T_{\rm c} }} = 3.52}\) at low doping. In order to identify the Feshbach shape resonance we have plotted in Fig. 7 the ratio Tc/TF (exp) for the aluminum doped case where TF is the Fermi temperature TF = ɛF/KB and ɛF = EAEF is the Fermi energy of the holes in the σ band, and Tc is the measured critical temperature. The Tc/TF ratio is a measure of the pairing strength (kFξ0)−1 where KF is the Fermi wavevector and ξ0 is the superconducting coherence length. In fact in the single band BCS theory this ratio is given by \(\smash{\frac{{T_{\rm c} }}{{T_{\rm F} }} = \frac{{e^\gamma /\pi ^2 }}{{k_{\rm F} \xi _0 }} = \frac{{0.36}}{{k_{\rm F} \xi _0 }}}\). In Fig. 7 we have plotted also the expected ratio Tc/TF (BCS) for the critical temperature in the single σ band of MgB2 calculated by the standard BCS approximations using the McMillan formula, the density of states and electron–phonon coupling obtained by band structure theory that gives for MgB2Tc (BCS)  =  20 K [148].
Fig. 6.

Panel (a): the critical temperature in aluminum, scandium and carbon doped magnesium diborides as a function of the Lifshitz parameter z = EΓEF; panel (b): the σ and π superconducting gaps; and panel (c): the ratio 2Δ/Tc in aluminium and carbon doped magnesium diboride as a function of the Lifshitz parameter z = EΓEF. The filled dots and squares are the direct measure of the two gaps by Tsuda et al. [182,183] in carbon doped MgB2.
Fig. 7.

The ratio Tc/TF (exp) (open squares) for the aluminium doped case where TF is the Fermi temperature TF = EF/KB for the holes in the σ band and Tc is the measured critical temperature for Al doped samples; The expected ratio Tc/TF(BCS) (open triangles) for the critical temperature in a BCS single σ band, The ratio\(\Delta T_{\rm c} /T_{\rm F} = (T_{\rm c} (\exp ) - T_{\rm c} ({\rm BCS}))/T_{\rm F}\) (solid dots) that measures the increase due to the interband pairing, for the Feshbach shape resonance, fitted with a asymmetric Lorentzian with Fano line-shape centered at z = 0, a half width Γ/2 = 300 meV and asymmetry parameter q = 4.

The increase of the critical temperature determined by the Feshbach shape resonance [125,162,180] of the interband pairing is given by \(\Delta T_{\rm c} /T_{\rm F} = (T_c (\exp ) - T_{\rm c} ({\rm BCS}))/T_{\rm F}\) that is plotted in Fig. 7 as a function of Lifshitz parameter z. It has been fitted with a asymmetric Lorentzian with Fano line-shape centered at z = 0, with the half width Γ/2 = 300 meV and an asymmetry parameter q = 4. This experimental curve provides the clear experimental evidence for the Tc enhancement driven by interband pairing shows a resonance centered at z = 0, as expected for Feshbach shape resonance.

In conclusion we have discussed the Feshbach shape resonances around a type (I), type (II), and type (III) ETT in heterostructures and in the particular case of MgB2 we have shown that the Feshbach resonance in interband pairing occurs in a superlattice of superconducting layers near a type (II) ETT in this particular multiband superconductor in the clean limit. The Feshbach shape resonance increases the critical temperature from very low Tc in all diborides to 40 K in MgB2 and it provides the clearest case of Tc amplification beyond the limit of 20 K. These results confirm that the Feshbach shape resonance in interband pairing could be a possible mechanism for high Tc, i.e., the quantum trick for suppressing decoherence effects at high temperature.

Finally we summarize the physical conditions to get a Feshbach shape resonance suppressing decoherence effects of temperature and impurity scattering driven by quantum superposition of pairs in states corresponding to different spatial locations. First, the material architecture is made of a superlattice of superconducting units (B) at atomic limit intercalated by a different material (Al, Mg) like in AlB2 diborides; second, the chemical potential is tuned in an energy window around a 2.5 Lifshitz electronic transition associated with the change of the topology of the Fermi surface of one of the subbands; third, the energy window of the resonance is controlled by quantum fluctuations driven by the zero point lattice vibrations that sweep the Van Hove singularity through the Fermi level.


This work is supported by MIUR in the frame of the project Cofin 2003 “Leghe e composti intermetallici: stabilità termodinamica, e reattività”

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© Springer Science+Business Media, Inc. 2005