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Direct currents Zero resistance implies no voltage drop across a superconductor Therefore irrespective of length no power is generated! This is really true only if the current is dc A superconductor can be considered as a mixture of two “fluids” superelectrons normal electrons Temperature At T=0 all electrons are superelectrons, for T>Tc all electrons are normal, with superelectrons converting to normal electrons as Tc is approached The dc current must be carried by the superelectrons, and there must be no electric field, otherwise the superelectrons would continue to accelerate and the current would increase The normal electrons are effectively “shorted out” Lecture 2 Superconductivity and Superfluidity Alternating currents If an ac voltage is applied across a superconductor there will be a time varying electric field Superelectrons, like normal electrons, have mass and hence inertia So, the supercurrent lags the electric field and therefore produces an inductive impedence An inductive impedence in turn implies that there is an electric field present, so the normal electrons also carry some current The superconductor is therefore resistive, and appears as a perfect inductance in parallel with a resistance The inductive component is small (~10-12 that of normal resistance) at 100kHz and only 10-6 of the total current is carried by normal electrons But…... At higher (optical) frequencies (~1011Hz) the superconductor appears entirely normal …..to be discussed later!! Lecture 2 Superconductivity and Superfluidity Some definitions In free space: H dl I B dl oI H is “magnetic field” in A/m B is magnetic flux density measured in Tesla N turns/unit length BA I I Flux density in empty infinitely long solenoid, by Ampere’s law, is |B| = o NI (B = oH) Flux density in solenoid containing infinitely long sample with a magnetisation per unit volume of Mv is B = o(H + Mv) Lecture 2 (Mv has units of A/m) Superconductivity and Superfluidity Susceptibility For most materials (except ferromagnets, and paramagnets in very high magnetic fields and low temperatures) Mv Mv H with Mv = H paramagnet where is the (dimensionless) susceptibitity so: B = o H(1 + ) For most paramagnetic materials ~10-3, for diamagnets ~ -10-5 diamagnet H If a superconductor always maintains B=0 within its interior, then = -1 Perfect diamagnet/superconductor A superconductor can therefore be described as (a) a “perfect diamagnet” or (b) having screening currents flowing at the surface producing a field of magnitude MV equal and opposite to H Note that B=0 but H 0 within the superconductor Lecture 2 Superconductivity and Superfluidity Demagnetisation N turns/unit length, carrying current i F A B C E D With a superconducting sample in the solenoid Around ABCDEF H dl Ni and H dl AB Hi dl BCDEFHe dl 1 But with the sample removed from the solenoid Around ABCDEF H dl Ni Ha = field applied to sample, H’e = external field without sample Lecture 2 and H dl AB Ha dl BCDEFHe dl 2 Hi = internal field within sample, He= external field with sample Superconductivity and Superfluidity Demagnetisation N turns/unit length F A E B X So this term... Equating 2 and 1 Y ...is always greater than or equal to this term C D AB Ha dl BCDEFHe dl AB Hi dl BCDEFHe dl At X, screening currents cause He to be less than He’ But at Y, the effects of the screening currents are negligible, and He = He’ Therefore and Hi Ha Lecture 2 and the field inside the superconductor can exceed the applied field! Superconductivity and Superfluidity Demagnetisation corrections In general we write to axis Hi = Ha - HD n For the special case of an ellipsoid, the field is uniform throughout the body and Hi = Ha - nMv where n is the demagnetising factor For a superconductor Mv < 0, so Hi>Ha or Mv = Hi = -Hi so Hi (1-n) = Ha and Ha Hi (1 n) to axis 1.0 to axis 0.8 nx+ ny+ nz=1 to axis 0.6 0.5 0.4 0.2 sphere 0 0 1 This will be needed later! Lecture 2 2 3 4 5 6 Length/diameter ratio Superconductivity and Superfluidity The London Model An important consequence of flux exclusion in superconductors is that If magnetic flux density must remain zero in the bulk of a superconductor, then any currents flowing through the superconductor can flow only at the surface However a current cannot flow entirely at the surface or the current density would be infinite The concept of “penetration depth” must be introduced In 1934 F and H London proposed a macroscopic phenomenological model of superconductivity based upon the two-fluid model The London model introduced the concept of the (London) penetration depth and described the Meissner effect by considering superconducting electrodynamics Lecture 2 Superconductivity and Superfluidity Some electrodynamics Consider a perfect conductor in which the current is carried by n electrons current density J nev 1 and in an electric field mv eE 2 so the rate of increase of current density is Maxwell’s equations give J ne2 E m curlH J D 4 curlE B 0 and (1 ) 1 Assuming the displacement current D and 3 5 curlB o J 6 Then 5 and 3 give m curl J ne2 B 7 and 6 and 7 give m B curl curlB 2 one 8 equation 4 gives Lecture 2 Superconductivity and Superfluidity Some more electrodynamics m B curl curlB can be simplified using the standard identity 2 one (div B 0, div B 0) curl curlB grad div B 2 B So equation 8 becomes B B 2 with m one2 In one dimension this is simply 2 B B 2 x To which the solution is B A B A exp x B ( x ) B A exp x So, for x ~10-6cm, ie inside a perfect conductor B does not change ( B ( x ) 0 ) when BA changes x B decreases exponentially when we move into the material Lecture 2 Superconductivity and Superfluidity The London penetration depth 0 but also B 0 within a superconductor Experiment had shown that not only B B F and H London suggested that not only B B 2 but also B 2 To which the solution is B( x ) BA exp x L where L m onse2 L is BA BA exp x known as the London penetration depth x It is a fundamental length scale of the superconducting state Lecture 2 Superconductivity and Superfluidity Surface currents Working backwards from the London equation 2 B gives B m curl Js ns e2 B to equation 7 So, for a uniform field parallel to the surface (z-direction) the “new” equation 6 becomes B o Jy x B B and as A exp( x L ) x L B Jy A exp( x L ) o L or Jy JA exp x L BA Jy JA exp x L x So current flows not just at the surface, but within a penetration depth L Lecture 2 Superconductivity and Superfluidity