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S. No. Temperature (K) SC parameter (?)= X x 10-22 J AFM parameter (? )= X x 10-22 J Specific heat(Ces) J/mol-K Free energy difference (J/mol) Critical field (kOe)

1 0.5 1.70 0.8229 0.00 -2.890 26.944

2 1.0 1.67 0.8334 2.20 -2.350 24.300

3 1.5 1.58 0.8170 19.20 -1.956 22.166

4 2.0 1.45 0.7465 50.06 -1.569 19.855

5 2.5 1.27 0.6366 83.48 -1.233 17.606

6 3.0 1.05 0.5032 119.72 -1.190 17.294

7 3.5 0.84 0.3758 127.09 -0.800 14.183

8 4.0 0.70 0.2839 131.43 -0.396 9.982

9 4.5 0.50 0.1822 132.37 -0.312 8.865

10 5.0 0.38 0.1204 127.16 -0.245 7.847

11 5.2 0.29 0.0000 123.09 -0.212 7.302

Table: SC Parameter (?), AFM Parameter (?), Specific Heat (Ces), Free energy difference, Critical field for HoNi2B2C

The conduction electrons in a metal cannot be both superconducting and anti-ferromagnetic ordered from the BCS theory. In this system, the antiferromagnetic transition takes place well below the superconducting state.

The superconductivity and long range antiferromagnetic order coexist below TN. Sahu et. al.[66], have taken the model Hamiltonian of two sublattices. They have proposed the ordered state and the magnetic moments of rare earth anti ferromagnets to investigate the interplay of SC and AFM in RNi2B2C.

The Hamiltonian of the system is mean field one and has been solved by using Green’s function techniques. SC and AFM order parameters are derived by considering the Fermi level lying in the middle of the antiferromagnetic band gap, (?_F=0).

The different physical quantities of the atomic subsystem are made dimensionless by dividing them by the nearest neighbor hopping integral 2t_0, with Wb=8t0 being the width of the conduction band.

The critical temperature and Neel temperature versus de Gennes factor are expressed as dG=(g-1)2J(J+1), where j is the lande factor and J is the total angular momentum of the R3+ ion.[67] (fig17).The phase diagram is almost linear, which agrees with a similar type of de Gennes scaling reported by Machida et al.

[68]. The enhancement of anti-ferromagnetism and the suppression of superconductivity are governed by exchange interaction of conduction electrons and rare earth f- electrons. The SC gap (z) and AFM gap (h) vs. temperature parameter (?) of HoNi2B2C for different external magnetic field parameter are plotted in Fig-18. When a magnetic field with respect to some quantized axis is applied, the SC gap and AFM gap remains unaffected, but the critical temperature and Neel temperature get reduced due to the application of magnetic field (fig.18).

Fig.19 shows the SC and AFM long range orders for the same external magnetic field for ErNi2B2C. They have observed that the SC gap is enhanced in the coexistence phase while AFM gap value is suppressed throughout the temperature range with the application of the magnetic field. In the normal state the anti-ferromagnetism (both Neel temperature and AFM order parameter) shows both increasing and decreasing trend on the application of external field[69], but the anti-ferromagnetism (both Neel temperature and AFM order parameter) shows both increasing and decreasing trend on the application of external field on the coexistence state of SC and AFM. It is due to the breaking of spin symmetry and formation of cooper pairs in ErNi2B2C. The effect of external magnetic field on SC and AFM is robust and their model could not solve SC and AFM self-consistently for TmNi2B2C and DyNi2B2C. The phase diagram shows that the critical temperature decreases and the Neel temperature increase with a de-Gennes factor which indicates the competition of SC and AFM. The effect of the external magnetic field is studied for HoNi2B2C and ErNi2B2C.

Similar types of graphs are obtained for DyNi2B2C, HoNi2B2C, ErNi2B2C. The U-shaped DOS explained the character of s-wave pairing as had been measured by point contact tunneling spectrum, suggesting the coexistence the SC and AFM order [57, 67]. With the onset of hybridization of the localized level with the conduction band, the density of state at Fermi level decreases with the decrease of the transition temperature parameter ?C . At any finite temperature, electrons from the Fermi level spill over to the conduction band due to thermal fluctuations in the absence of hybridization, thereby contributing to the Cooper pairing in the case of superconductivity (as the localized level lies at the Fermi level). Such excitation of electrons from localized f-level to the conduction band due to thermal fluctuation becomes an activated process in the presence of hybridization, due to the appearance of hybridization gap at the Fermi level. The hybridization gap competes with the gaps opening up at the onset of superconductivity. If the hybridization gap becomes too large, then the density of state at the Fermi level is suppressed. This prevents the occurrence of the other long-range order, resulting in re-entrant behavior.

Fig.25 shows the temperature variation of AFM order parameter (h) and N?el temperature (?N) for different values of the hybridization parameter (V). The N?el temperature (?N) decreases as hybridization increases. It confirms the fact that the effect of hybridization is more drastic on SC order parameter compared to AFM order parameter. The observed decrease in both the N?el temperature (?N) and superconducting transition temperature (?C) on doping (x) with non-magnetic impurity Lu in DyNi2B2C in the system Lu1-xDyNi2B2C for larger doping (x) [16] also resembles the suppression of ?N and ?C with increasing hybridization

In this chapter, we have given a theoretical calculation of superconducting and antiferromagnetic gap parameter by taking the model Hamiltonian of two sublattices as proposed by Fulde et.al. then Sahu et.al. considering both hybridization potential between localized electron with that of conduction electrons in presence of external magnetic field.

In the present model, the anti-ferromagnetism arises due to the staggered sub-lattice magnetization from the conduction electrons which are responsible for superconductivity. In addition to that, the localized f-electrons and itinerant conduction electrons hybridize near the Fermi level. In order to simulate AFM in the system, we considered two Ni sublattice. We have used a mean field theory for the itinerant electrons to study the effect of external magnetic field on antiferromagnetism in presence of hybridization and the localized f-electron. When a magnetic field is applied it splits the two-fold degenerate bands giving rise to quasiparticle energies. We have considered the effect by replacing ?_(k=) ?_k±??_BB.

? H?_C describes the conduction which represents the hopping of the quasi particles between the neighboring sites of the two sub-lattices.

? H?_c=?_k????_0 (k)(a_k?^ b_k?+h.c) ? —- (1)

a_k?^ (a_k? ) and ? b?_k?^ (b_k? )are the creation (annihilation) operators of electrons at site 1 and 2 of Ni respectively with momentum k and ?. Hopping takes place between nearest neighbor site of Ni with dispersion,

? ??_0 (k)=-2t_0 (cosk_x+cosk_y ) —————-(2)

where t_0 is the nearest neighbor hopping integral.

Strongly AFM correlation of conduction Ni d-electrons is simulated by a staggered magnetic field of strength h and this contributes to the Hamiltonian as;

? H?_h=(h/2)?_k????(? a_k?^ (a_k? )-? b?_k?^ (b_k? ))——– (3)

Effective hybridization between the f-electrons of the rare earth atom and the conduction electrons of Ni atom contributes to the Hamiltonian H_V given by

H_v=V?_k??(a_k?^ f_(1,k?)+b_k?^ f_(2,k?)+h.c) ————-(5)

Where f_ik?^ (f_ik? ) is the creation (annihilation) operator of the f-electrons and V is the strength of hybridization. Only on-site hybridization is included here.

The attractive part of intra-band interaction term leads to superconductivity. The mean field BCS Hamiltonian describing phonon mediated superconductivity is given by,

? H?_s=-??_k?[(a_(k?)^ a_(-k?)^+h.c)+(b_(k?)^ b_(-k?)^+h.c) ] ——- (6)

Here BCS type of phonon-mediated cooper pairing of conduction electrons of different Ni sites is taken into account. The inter sub-lattice pairing may be significant here but has not been taken into consideration for simplicity of numerical calculation. The superconducting order parameter? is given as

?=-?_K?V ?_k (? +?) ————-(7)

V_(k )is the strength of the attractive interaction between two electrons mediated by the phonons.

The intra f-electron Hamiltonian in the presence of external magnetic field B is described by

? H?_f=(?_f 1/2 g?_B B) ?_(i,ki=1,2)?f_(ik?)^ f_(ik?)+(?_f 1/2 g?_B B) ?_(i,ki=1,2)?f_(ik?)^ f_(ik?)—- (8)

g and ?_B are Lande g-factor and Bohr magnetron respectively.

The external magnetic field B contributes to the Hamiltonian H_(B,)written as,

? H?_B=1/2 g?_B B?_k??(a_k?^ a_k?+b_k?^ b_k? ) ——————-(9)

We calculated one electron Green’s function using the Hamiltonian given as above for the superconducting state and antiferromagnetic state of borocarbide system. We have

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