Journal of Superconductivity and Novel Magnetism https://doi.org/10.1007/s10948-018-4586-y ORIGINAL PAPER Numerical Simulation of Phase Transitions in Type-II Annular Superconductor Using Time-dependent Ginzburg-Landau Equations Hasnain Mehdi Jafri 1 · Xingqiao Ma 1 · Congpeng Zhao 1,2 · Houbing Huang 1 · Tauseef Anwar 3 · Zhuhong Liu 1 · Long-Qing Chen 4 Received: 8 January 2018 / Accepted: 24 January 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract Time-dependent Ginzburg-Landau equations were solved by finite element method in two-dimensional space for order parameter and energy components of the annular superconducting sample in steady magnetic fields. Vortices preferred to penetrate from the inner surface of the annulus due to lesser energy required at the concave surface. A transition magnetic field strength was observed in spatial averages of carrier concentration and energy components, showing small bumps and abrupt variations, indicating phase transition from a non-vortex to vortex state. These effects were observed to repeat with every subsequent entry of a set of vortices into the sample; transition magnetic field strength was found to depend inversely on the annular width of the sample. The present work gives a better understanding of energy variations during phase transition from non-vortex to vortex state and predicts that vortex state can be avoided by tuning the wire thickness in practical applications, e.g., superconducting electromagnets. Keywords Ginzburg-Landau model · Abrikosov vortices · Type-II superconductor 1 Introduction In mesoscopic superconductors (dimensions of the order of coherence length ξ or penetration depth λ) surface effects are responsible for properties which are not present in bulk samples [1–5]. As a result, the vortex configurations in mesoscopic samples are strongly sample geometry-dependent [4] contrary to triangular vortex pattern observed in bulk samples [6].  Xingqiao Ma xqma@sas.ustb.edu.cn 1 Department of Physics, University of Science and Technology Beijing, Beijing 100083, China 2 ASIC, China Center for Information Industry Development, Beijing 100083, China 3 Energy Research Centre, COMSATS Institute of Information Technology, Lahore, Pakistan 4 Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA Appearance of two different states, i.e., multi-vortex state [7, 8] and giant vortex state [9–11] at mesoscopic size, have prompted studies with surprising results such as the paramag- netic Meissner effect [12], symmetry-induced antivortices [13], giant [11] and fractional-flux [14] vortices, and vortices trapped in blind holes [15]. According to famous Bardeen-Cooper-Schrieffer (BCS) theory, the ground state of condensate consists of oppo- site spin electron pairs (spin singlet state) bound by phonon interactions [16]. Ginzburg pointed out that this fragile state is destroyed by the development of homogeneous fer- romagnetic order of spins if its corresponding magnetic field exceeds thermodynamical critical magnetic field of that superconductor. Ginzburg-Landau (GL) theory [17] is probably the most successful macroscopic description of superconductivity. The proposed macroscopic quantum theory by Ginzburg and Landau makes use of a quartic potential and the Higgs mechanism of spontaneous symme- try breaking [18–21] to generate a local mass term of vector potential. It successfully describes the Meissner-Ochsenfeld effect [20, 22–25] for magnetic flux expulsion, the phe- nomenology associated with phase transition [26–28], and