MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 1 8 512 Theory of Solids II Problem Set 8 Due April 13 2009 Type II Superconductor We begin with the Ginzburg Landau free energy F f dr 2 T Tc 2 1 4 2eA 2 Tc 2 i c 1 and consider T slightly below Tc 1 Calculate the free energy di erence between the superconducting state and the normal state and show that the thermodynamic critical eld Hc is given by Hc2 T 1 f 8 2 Tc T Tc 2 2 2 Now turn on a uniform magnetic eld H such that we are in the normal state 0 and gradually reduce H We look for an instability towards 0 The eld at which the instability happens is de ned as Hc2 T We calculate Hc2 T by the following steps a Using the condition F 0 under a variation of show that satis es 2 2eA i c 2 T Tc 2 0 Tc 3 b Near the critical point the last term in Eq 2 can be ignored and we have a linearized equation Instability 0 occurs when the linearized equation has a negative eigenvalue Notice that this equation has the same form as that of a single electron in a magnetic eld where the solution is known to be Landau levels Use this fact to show that Hc2 where 0 hc 2e Tc T 0 2 2 Tc 4 2 c Calculate the London penetration depth L T Express Hc in Eq 2 in terms of 1 2 the temperature dependent coherence length T TcT c T and the London penetration depth L T Show that the condition Hc2 Hc implies L T 1 T 2 Equation 5 is the condition for type II superconductivity 5
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