Vol.04,Special Issue 08, (EMDMSCBW-2019) December 2019, Available Online: www.ajeee.co.in/index.php/AJEEE
SUPER CONDUCTING PAIRING AND TWO‐BAND GAP NATURE OF AL AND C DOPED MgB2: PAIRING OF ELECTRON-ELECTRON AND ELECTRON-PHONE INTERACTION
Prof. Swati Bhatnagar
Assistant Professor (Department Of Physics), ISBA Institute of Professional Studies, Indore Abstract:- In this chapter a two‐band model for describing the longitudinal dielectric function for layered MgB2 and Al and C doped MgB2 have been developed to discuss the attractive pairing mechanism leading to superconducting state. A model dielectric function for and band carriers are set up that fulfils the appropriate sum rules on the electronic and ionic polarizabilities. The in‐layer electron‐phonon coupling strength and the Coulomb screening parameter for () band carriers are obtained from the residues of the dielectric function. It is noticed that the ‐ holes in the two‐dimensional boron planes via the screened phonon coupling is responsible for attractive interaction and the superconductor is in the strong coupling regime. Within this framework, the superconducting transition temperature Tcof MgB2 is estimated as 42 K and the pressure derivative of Tcis negative. The isotope effect, coherence length, magnetic penetration depth, and volume derivative of Tcare also estimated to be consistent with the earlier results. Within two‐band model of superconductivity in MgB2, we place particular emphasis on intraband channels, carriers either in or bands, as interband effects are negligible. In such a situation the superconductivity is originated from either or band carriers. The implications of the effective interactive potential with both and carriers for MgB2 and Al and C doped MgB2
and its analysis are discussed.
1. INTRODUCTION
The magnesium diborides, MgB2 have been due to two‐band superconductivity apart from the high‐transition temperature (Tc40 K), the preferred model systems in which broad range of theories and experimental methodologies and many physical properties have been unambiguously established soon after the discovery by Nagamatsu and collaborators [1].
Electron pairing via virtual exchange of vibrational excitations (phonons) may in principle occur in two‐bands of qualitatively different electronic structures crossing the Fermi level, leading to two different energy gap structures.
In MgB2 the boron planes are envisaged as conducting planes similar to that of CuO2‐ planes in cuprate superconductors [2]. Apart from high‐Tcand two‐band superconductivity, chemical substitution in the parent MgB2 system attracts increasing attention. A suppressed Tcof about 2‐3 K for even a very small substitution of 1‐ % ‐2 % Al in Mg1‐xAlxB2 is reported [3]. Carbon substitution for Boron in MgB2 reveals larger energy gap, , and distinctly small independent of the Tcand is interpreted in terms of band filling and inter band scattering effects [4].
The most direct experimental techniques like inelastic Neutron scattering measurement and Raman spectroscopy are powerful probe related to structural and vibrational modes, and electron‐phonon interaction in superconductors. Inelastic neutron scattering results in an acoustic phonon of energy of about 36 meV and highly dispersive optic branches peaking at 54, 78, 89, and 97 meV [5]. Band structure calculations reveal four important phonon modes: the E1u (41 meV) corresponds to an in plane oscillation of both Mg and B ions, the out of plane oscillation of Mg and B ions is from A2u (50 meV) and B1g (87 meV) corresponds to a tilting out of plane oscillation of B ions.
The E2g (67 meV) describes in‐plane B anharmonic oscillation [6]. These features in the phonon spectrum indicate the strong coupling of the E2g phonon to the holes as carriers in the boron plane. Raman studies revealed features corresponding to the phonon modes of E2g symmetry and energy gaps. The Raman spectra [7] of pure MgB2 is dominated by the broad feature centered at about 600 cm‐1 has been attributed to the E2g mode. The significant broadening of the Raman peak arises mainly from the exceptionally strong nonlinear electron‐phonon coupling of the E2g mode to partially occupied planar boron σ bands near the Fermi surface and its large anharmonicity [2].
However, the correlation between E2g mode and Tcis not clear so far. The hardening of E2g phonon mode in carbon substituted MgB2 is associated with suppression of electron‐phonon coupling [7]. Therefore, electron‐phonon interactions are expected to play
Vol.04,Special Issue 08, (EMDMSCBW-2019) December 2019, Available Online: www.ajeee.co.in/index.php/AJEEE
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an important role in the electron‐electron pairing mechanism in this material. Tunneling spectroscopy is known for the direct measurement of the electron‐ phonon interaction responsible for superconductivity. The interesting feature in MgB2 is the existence of two distinct gaps on different portions of the Fermi surface.
The two pair of carriers is only weakly interacting and the high Tcvalue is mainly due to a strong coupling between bending and stretching boron phonons and carriers of the Fermi surface. The two‐gap scenario in MgB2 is now commonly accepted and has independently been evidenced by several experimental techniques [8]. It was demonstrated from the point‐contact and tunneling measurements indicate that MgB2 shows two distinct energy gaps [9]. The 2D band indicate a large gap value of ∆σ7 ‐ 8 meV, whereas the 3D band has a small gap ∆π2 ‐ 3 meV.
However, both energy gap follow a conventional BCS like behavior and closing at the same critical temperature Tc= 39 K. The temperature dependence of the large gap ∆σ is qualitatively close to the BCS type but the ratio 2∆σ/kBTc= 5.3 surpass the BCS value. For MgB2 thin films, decreases almost linearly with decreasing Tcwhile is weakly affected in the range of Tcabove 30 K and can only be explained in terms of an increase in the interband scattering [10].
It is well documented that chemical substitution, boron isotope exchange and hydrostatic pressure lead in MgB2 to a reduction of Tc. To be specific, the phonon contribution to pairing can often be gauged from the shift in Tcdue to substitution of constituent elements by their isotopes. The effect of boron substitution on the Tcof superconducting MgB2 from magnetization measurement was about 0.26 relative to the one‐half BCS value [11]. In subsequent work, the isotope effect for both B and Mg has been reported and found a B isotope effect of 0.30 and a small αMg of about 0.02 [11].
Later on, Castro and collaborators [12] also derive nearly the same value from the change of the magnetization upon boron isotope exchange. The estimates of isotope exponent from magnetization measurement indicate that B phonons being involved in superconductivity, but Mg phonons apparently contribute little to the overall pairing. The total measured isotope effect was about 0.32, a value indicating a moderate‐to‐ strong coupling and points in a natural way towards a phonon based pairing.
Among the non‐conventional mechanisms, which have been invoked so far for high‐Tcmagnesium diborides, collective charge fluctuations leading to screening by collective excitations [13], bipolarons [14], electronic contribution within the resonating‐valence‐band (RVB) mechanism [15], electron‐hole asymmetry [16] are important. The charge fluctuation mechanism is emphasized in diboride superconductors by their strong anisotropy and by the presence of a layer stacking sequence.
Inelastic X‐ray scattering [7], Electron‐energy‐loss‐spectroscopy [18] and optical reflectivity measurement [19] have been successfully used to investigate the electronic structure of the MgB2 superconductor. The energy loss spectra for low momentum transfer predict free‐electron‐like plasmon just below 20 eV and low energy collective electron excitations around 3 eV.
2. THE MODEL
Let us start by giving a brief description of the layered structure of MgB2. Until now the MgB2 is known as a two‐gap superconductor with several anomalous properties originating from the existence of two separates sheets of the Fermi surface, one quasi‐ 2D (band) and second quasi‐3D (band). To be specific, two bands are formed by sp2 hybrid orbitals stretched along boron‐boron bonds and are two‐dimensional with hole type carriers.
While to that the two bonds are formed by pzorbitals of boron and are three‐dimensional with electron as carriers. Furthermore, band holes are strongly coupled with optical E2g phonon, while band electrons are weakly coupled with phonons. Both
and bands are different source of channels to superconductivity due to different parities.
Existence of two‐energy gap values points to weak or small inter band coupling in parent MgB2.
Vol.04,Special Issue 08, (EMDMSCBW-2019) December 2019, Available Online: www.ajeee.co.in/index.php/AJEEE
The stacking layer sequence of this system can be described by such layers Mg, B, Mg, B, Mg (See Fig.1).
Figure 2.1. Schematic representation of layered structure of MgB2.
The crystal structure of parent MgB2 is the so‐called AlB2 structure: honeycomb layers of boron atom alternate with hexagonal layer of Mg atoms. B atoms in different planes are on top of each other and the Mg atoms are at the center of the hexagons defined by the B atoms. In metal diboride the boron atoms accept electrons from the metal and the boron planes become negative charged. Holes then are the free carriers for 2D band in each boron plane.
Selective doping can affect scattering time in either or bands. It is well documented that there are two kinds of substitutions, which were frequently studied in MgB2: Carbon substitution for boron, which influences mostly intraband scattering in the 2D band, and Aluminum substitution for Magnesium, which influences mostly intraband scattering in a 3D band.
The effective mass of the carriers along the conducting boron plane is obtained from the electronic specific heat coefficient, , using the relation, m* = 3h2d/k 2. While estimating the effective mass we consider the interplanar distance from the structural data, d = 3.525 (3.338) [3.5164] Å and = 3.83 (1.25) [2.8] mJ/mol/K2 from the heat capacity measurement for MgB2 (Mg0.5Al0.5B2) [Mg(B0.9C0.1)2], respectively.Doping dependence of these is further listed in Table 1.
Askerzade et al. from magnetisation measurements, suggest that the effective mass ratio m*/m* is about 3. Hence, we write m* = mefor band electrons. It is noticed from the plot that a first pole at = 1 arises from the screened optical phonon, a second lies at =
2 and the third lies at = 3. Thus the effective potential is attractive in the ranges 1, A12 and A23 while it is repulsive in the domains 1A1 and 2.
Vol.04,Special Issue 08, (EMDMSCBW-2019) December 2019, Available Online: www.ajeee.co.in/index.php/AJEEE
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B c ,
Figure 2. Effective interaction potential V (q, ) at q = 2kFas a function of (eV) on a semi‐logarithmic scale.
In order to check our analysis quantitatively it would be important to explore the thermodynamical parameters governing the superconducting state as coherence length and penetration depth. The superconducting properties of two sets of bands in MgB2 are radically differed, due to the overlap of orthogonal and band functions. With this in mind, we evaluate thermodynamical parameters only for band carriers. MgB2 single crystals are usually described as fairly clean, with the band probably in the clean limit and the band probably in the dirty limit.
Model calculations reveal a smaller Fermi velocity v (2.7 x 107 cm sec‐1) relative to the values in conventional metals (1‐ 2 x 108 cm sec1). A small value of vFalong with a T of 42 K leads to a coherence length of 73 Å using the BCS expression = 0.15hv /k T . On the other hand, we can make use of more precise definition hvF/(0) with 2(0) = k T
as the value of critically depends on the strength of coupling. Here we find = 4.38 for
(0) = 7.9 meV as, the calculated value of electron‐phonon‐coupling constant is about 0.95, which leads to a larger value of relative to BCS value of 4.38.
3. CONCLUSIONS
Two types of bands crossing the Fermi level and with quantitatively different electronic structures leading to two different energy gap is recognized as important aspect of superconductivity in MgB2, Mg1‐xAlxB2, Mg(B1‐yCy)2 superconductors. To be specific, the electronic structure of MgB2 contains graphite‐type boron layers, which are separated by hexagonal close‐packed layers of magnesium. In parent MgB2 the Mg donates two electrons to the B planes, which can be viewed as an ionic solid containing well‐ separated B layers.
In this work we have treated the carrier‐carrier pairing (both and ) mechanism for superconductivity in such a layered material as arising from the in‐ layer Coulomb interactions screened by optical phonons (boron‐boron stretching or bending) and by charge fluctuations. The two‐band model for the superconductivity in MgB2, Mg1‐xAlxB2, Mg(B1‐yCy)2
superconductors thus works for either or band carriers. In the limit of small interband
Vol.04,Special Issue 08, (EMDMSCBW-2019) December 2019, Available Online: www.ajeee.co.in/index.php/AJEEE
scattering we have ignored the grouping of 2D band and 3D band carriers. Furthermore, for the sake of simplicity a single (longitudinal and transverse) optical phonon mode has been considered, with a flat dispersion relation.
The main focus has been on relating a two‐band model to physical parameters of the superconducting state, i.e. the transition temperature Tc, the energy gap parameter ∆σ, ∆π, the isotope‐ effect exponent , the pressure derivative dTc/dP and volume derivative dlnTc/dV. We should emphasize that our conclusions have been established only within the framework of screened phonon mechanism and weak interband scattering among parameter σ and π bands in MgB2, Mg1‐xAlxB2, Mg(B1‐yCy)2 superconductors.
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