Novel vortex lattice transition in d-wave superconductors

We study the vortex state in a magnetic field parallel to the c axis in the framework of the extended Ginzburg Landau equation. We find the vortex acquires a fourfold modulation proportional to cos(4φ) where φ is the angle r makes with the a-axis. This term gives rise to an attractive interaction between two vortices when they are aligned parallel to (1, 1, 0) or (1,−1, 0). We predict the first order vortex lattice transition at B = Hcr ∼ κ Hc2(t) from triangular into the square lattice tilted by 45 from the a axis. This gives the critical field Hcr a few Tesla for YBCO and Bi2212 monocrystals at low temperatures (T ≤ 10K). Typeset using REVTEX 1 The d-wave superconductivity in the hole-doped high Tc cuprate superconductors now appears to be universally established [1,2]. Then in a magnetic field parallel to the c axis we expect a square vortex lattice tilted 45 from the a axis at least in the vicinity of B = Hc2(T ) and below t(≡ T/Tc) ≤ 0.81 [3,4]. Indeed such vortex lattices though elongated in the a direction have been seen by small angle neutron scattering (SANS) [5] and scanning tunnelling microscope (STM) [6] both in twinned YBCO monocrystals at low temperature (T ∼ 4K) and in a small magnetic field (B ∼ a few Tesla). The object of this letter is to determine theoretically the corresponding vortex lattice transition at a small magnetic field which is experimentally relevant. Also in order this vortex lattice transition is to be observable experimentally, it should take place in a magnetic field smaller than the one where vortex lattice melting is established [7,8]. Further we shall show that the transition is exthermal and of the first order. Earlier the related transition has been analyzed in terms of the generalized London equation, which may be obtained from the phenomenological Ginzburg Landau equation for d-wave superconductivity [9] or from the rhombic term arising from the normal state properties [10]. However these authors concentrated on the modification of the magnetic interaction energy due to the quartic term in the kinetic energy. We believe, on the contrary, that the important modification lies in the vortex core structure [11]. Nevertheless these authors obtained the vortex lattice transition from a triangular lattice in small magnetic field to a rhombic lattice in a higher field. The result is similar to ours except for two crucial differences. First we find that the critical field Hcr where this transition takes place behaves like κHc2(t) where κ is the Ginzburg Landau parameter and Hc2(t) the upper critical field. This means Hcr ∼ 10 Hc2(t) in high Tc cuprates. Second the transition is of the first order with a sharp negative divergence in the magnetization. Within the weak-coupling model for d-wave superconductor [12] the extended GinzburgLandau equation is given by [11]