# Feshbach Shape Resonance in Multiband Superconductivity in Heterostructures

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- First Online:

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

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## Abstract

We report experimental data showing the Feshbach shape resonance in the electron doped MgB_{2} where the chemical potential is tuned by Al, Sc, and C substitutions. The scaling of the critical temperature *T*_{c} as a function of the Lifshitz parameter *z* = *E*_{Γ}−*E*_{F}, where *E*_{F} is the chemical potential and *E*_{Γ} 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 *T*_{c}(*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 *T*_{c} in particular nanostructured architectures.

### KEY WORDS:

feshbach resonanceshape resonancedoped diborides## INTRODUCTION

The macroscopic quantum coherence in a superconducting condensate appears below a critical temperature *T*_{c} because it is suppressed by *decoherence* effects at high temperature. Understanding high *T*_{c} 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.

*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,

*k*and spin up. The construction of the BCS wavefunction has been inspired by the configuration interaction theory developed in nuclear [4–6 ] and atomic [7] physics.

In anisotropic superconductivity one has to consider configuration interaction between pairs, in an energy window Δ*E* 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 [10–42], for doped cuprate perovskites [43–114], for magnesium diboride [115–184] and for few other materials as Nb doped SrTiO_{3} [185], Sr_{2}RuO_{4} [186–188] YNi_{2}B_{2}C, LuNi_{2}B_{2}C [189], and NbSe_{2} [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 *v*_{F} 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.

*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

*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.

## FESHBACH SHAPE RESONANCES

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.

*T*

_{c}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 [75,76,80,81,87–90,93–97,102,109,110] 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

*T*

_{c}/

*T*

_{F}[195].

*L*where the chemical potential crosses the bottom

*E*

_{n}of the

*n*th 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

*E*

_{F}is tuned near the critical energy

*E*

_{F}=

*E*

_{n}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

*E*

_{F}=

*E*

_{n}. Therefore, the Feshbach shape resonance occurs by tuning the Lifshitz parameter

*z*=

*E*

_{F}−

*E*

_{n}around

*z*= 0 where in the Blatt proposal

*E*

_{n}is tuned by changing the film thickness. The prediction of Blatt and Thompson of the oscillatory behavior of

*T*

_{c}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.

*T*

_{c}via a Feshbach shape resonance in a different class of systems: superlattices of metallic units [75,76,80,81,87–90,93–97,102,109,110]. The idea is that high

*T*

_{c}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 CuO

_{2}planes of doped cuprate perovskites; (v) Superlattices of boron graphene monolayers intercalated by hcp metallic monolayers forming the composite AlB

_{2}crystallographic structure.

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 *T*_{c}. *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 *E*_{F} > *E*_{c} to a two-dimensional (2D) topology, for *E*_{F} < *E*_{c} and it is characteristic of superlattices of metallic layers.

*E*

_{F}=

*E*

_{c}. 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, [93–96,102] 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 λ

_{p}, amplitude

*V*

_{b}and width

*W*along the

*y*direction and constant in the

*x*direction expressed for

*x*= constant as

*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

_{p}, 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

*N*

_{b}solutions for \(E_n (k_y )\), with 1 ≤

*n*≤

*N*

_{b}, for each

*k*

_{y}in the Brillouin zone of the superlattice giving a dispersion in the

*y*direction of the

*N*

_{b}subbands with

*k*

_{x}= 0. The superlattice with its characteristic wavevector

*q*= 2π/λ

_{p}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

*n*and

*n*′ are the subband indexes.

*k*

_{x}(

*k*

_{x′}) is the component of the wavevector in the stripe direction (or longitudinal direction) and

*k*

_{y}, (

*k*

_{y′}) 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 Δ

_{n}(

*k*

_{y}) around the Fermi surface in a range \(\hbar \omega _{\rm o}\) depending from the subband index and the superlattice wave vector

*k*

_{y}. The self consistent equation, for the ground state energy gap Δ

_{n}(

*k*

_{y}) is

*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

*k*

_{y}. The structure in the interaction gives different values for the gaps Δ

_{n}giving a system with an anisotropic gaps in the different segments of the Fermi surface. The critical temperature

*T*

_{c}of the superconducting transition can be calculated by iterative method

_{n}(

*k*) = ɛ

_{n}(

*k*)−μ. These calculations show that the interband pairing enhances

*T*

_{c}[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.

## FESHBACH SHAPE RESONANCES IN DIBORIDES

MgB_{2} provides the simplest high *T*_{c} superconductor therefore it could play a key role for understanding high *T*_{c} superconductivity as atomic hydrogen for quantum mechanics. There is now growing evidence that MgB_{2} is a practical realization of the proposed *T*_{c} amplification process driven by Feshbach shape resonances in interband pairing [116,119,120,124,125,132,143,162,176,180].

_{2}(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 (AlB

_{2}) in 1935 [199] magnesium diboride (MgB

_{2}) 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 MgB

_{2}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

*T*

_{c}superconductors. The AlB

_{2}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.

The σ band is usually fully occupied in non superconducting diborides where there are no holes in the σ band, so these diborides, as AlB_{2}, have a Fermi surface of the type shown by the case *E*_{F} > *E*_{A} shown in Fig. 4. In MgB_{2} the σ band is partially unoccupied due to the electron transfer from the boron layers to the magnesium layers. In fact the chemical potential *E*_{F} in MgB_{2} is at about 750 meV below the energy *E*_{A} of the top of the σ band. Moreover, the chemical potential *E*_{F} in MgB_{2} is also at about 350 meV below the energy of the Γ point in the band structure (*E*_{Γ} > *E*_{F}). Therefore, the σ Fermi surface of MgB_{2} where *E*_{F} < *E*_{Γ} < *E*_{A} has the corrugated tubular shape with a two-dimensional topology of the type shown in Fig. 4 for the case *E*_{F} < *E*_{Γ}. Going from below (*E*_{F} < *E*_{Γ} < *E*_{A}) to above the energy of Γ point the σ Fermi surface becomes a closed FS with 3D topology like in AlMgB_{4} that belongs to the Fermi surface type shown in Fig. 4 for the case *E*_{Γ} < *E*_{F} < *E*_{A}. Therefore, by tuning the chemical potential *E*_{F} by electron doping the σ subband it is possible to reach the point where *E*_{F} 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 *E*_{F} = *E*_{Γ}, therefore, this transition will be studied here as a function of the Lifshitz parameter *z* = *E*_{Γ}−*E*_{F}. The influence of the proximity to a type (II) electronic topological transition on the anomalous electronic and lattice properties of MgB_{2} is shown by the anomalous pressure dependence of the *E*_{2g} phonon mode and *T*_{c} [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 MgB_{2} up to reach zero σ holes at the top of the σ band where *E*_{F} = *E*_{A}. We report in Fig. 5 the variation of the Lifshitz parameter *z* = *E*_{Γ}−*E*_{F} as a function of the number of holes in the σ subband for the case of Al doped Mg_{1−}_{x} Al_{x}B_{2}. 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 E_{2g} symmetry near the zone-center, split the partially occupied σ bands. The energy splitting of the two degenerate σ bands is given by Δ*E* = *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 Δ*E* = ±0.4 eV in MgB_{2} and it decreases a little going to AlMgB_{4} 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 *T*_{c} 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 Mg_{1−}_{x}Al_{x}B_{2}, Mg_{1−}_{x}Sc_{x}B_{2} MgB_{2−}_{x}C_{x} 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 *T*_{c} versus the universal Lifshitz parameter *z* for all doped magnesium diborides.

*T*

_{c}, 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

*T*

_{c}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

*T*

_{c}/T

_{F}(exp) for the aluminum doped case where

*T*

_{F}is the Fermi temperature

*T*

_{F}= ɛ

_{F}/

*K*

_{B}and ɛ

_{F}=

*E*

_{A}−

*E*

_{F}is the Fermi energy of the holes in the σ band, and

*T*

_{c}is the measured critical temperature. The

*T*

_{c}/T

_{F}ratio is a measure of the pairing strength (

*k*

_{F}ξ

_{0})

^{−1}where

*K*

_{F}is the Fermi wavevector and ξ

_{0}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

*T*

_{c}/T

_{F}(BCS) for the critical temperature in the single σ band of MgB

_{2}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 MgB

_{2}

*T*

_{c}(BCS) = 20 K [148].

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 *T*_{c} 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 MgB_{2} 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 *T*_{c} in all diborides to 40 K in MgB_{2} and it provides the clearest case of *T*_{c} 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 *T*_{c}, 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 AlB_{2} 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.

## ACKNOWLEDGMENTS

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