Strong pinning - pinned regular vortex lattice; Weak pinning - triangular depinned vortex lattice. Intermediate regime - ? Aim: to study the structure of flux-line lattice and its properties as a function of pinning strenght
2. Phase diagram Pinning energy density (a is the period of the pinning centers array) : The energy of the pinned vortex lattice : where U(0,0) is the pinning potential in the center of the pinning site, and is the energy of the flux-line lattice with a given symmetry
The energy of the deformed triangular lattice. Elasticity theory: where is the energy of an ideal triangular flux-line lattice, is the pinning energy, is the energy related to the lattice distortion.
where the integration is performed over the area S of the sample and is the pinning force acting on a vortex. A.I. Larkin, Zh. Eksp. Teor. Fiz. 58, 1466 (1970).
Vortex-pinning site interaction: where is amplitude of interaction, r is distance between vortex and the pinning site, and is length scale of potential. The total pinning potential in the sample:
The structure of the triangular vortex lattice, deformed by the square pinning array of Gaussian potential wells at.
The transition between the two vortex phases:. The pinning efficiency is controlled by parameter:
Vortex lattice phase diagram in the plane of the vortex-pinning site interaction amplitude and the pinning potential length-scale at sub-matching fields not exceeding, except and (H 1/8, H 1/4, H 3/8, H 1/3, H 2/3 ). Below curve 1 the deformed triangular lattice has the lowest energy, above –pinned phase.
The structure of the intermediate phase at (a) and (b). Black dots and open circles denote the positions of the vortices and the pinning sites. Dashed lines show the symmetry of the vortex lattice.
The energy of the intermediate phase: where is the energy of the vortex configuration. The boundary between the deformed triangular lattice and the intermediate state: The boundary between the intermediate phase and the square pinned lattice:
The parabolic pinning well: where r is the distance from the center of the well. The boundaries between the triangular lattice and intermediate phase, intermediate phase and square lattice, triangular lattice and square lattice:
Vortex lattice phase diagram in the plane of vortex-pinning site interaction amplitude and the pinning potential length-scale at. Fig (a) corresponds to the Gaussian potential of the vortex- pinning site interaction, (b) – to the parabolic one. The similar phase diagrams are also valid for.
3. Critical current The energies and of interstitial and pinned vortices: The stability criteria:
Vortex phase diagrams of the intermediate phase in the plane: applied current – amplitude of the vortex-pinning site interaction at and. In Fig. (a) the current flows along the rows of pinned vortices, in Fig. (b) – in a perpendicular direction. Fig. (b): Two-stage depinning. Spontaneous symmetry breaking
G.S. Mkrtchyan and V.V. Shmidt, Zh. Eksp. Teor. Fiz. 61, 367 (1971) [Sov. Phys. JETP 34, 195 (1972)]. H. Nordborg and V. M. Vinokur, Phys. Rev. B 62, (2000).
4. Conclusions The stable vortex configurations in a regular array of weak pinning sites at matching fields not exceeding H 1 were studied. The existence of different vortex phases is revealed: a deformed triangular lattice, the regular lattice of pinned vortices, and the regular intermediate phase. The phase diagrams of the vortex lattices are calculated in the plane of two coordinates: the amplitude of vortex-pinning site interaction and the pinning potential well size. We found that for some types of the potential the triple point can exist on the phase diagram, where all three phases coexist. The critical current in the intermediate phase is calculated. We found that it is highly anisotropic. When current flows parallel to the pinned vortices rows the depinning occurs when the Lorentz force suppresses the effective potential wells, in which interstitial vortices are trapped. When current flows perpendicular to the pinned vortices rows, first these effective caging potential wells are destroyed, and interstitial vortices jump to the neighboring vacant pinning sites. The phase diagrams for the stable vortex lattice in the plane of applied current-pinning potential amplitude are also obtained. W.V. Pogosov, A.L. Rakhmanov, and V.V. Moshchalkov "Vortex lattice in presence of a tunable periodic pinning potential", Phys. Rev. B 67, (2003).