Note for Condensed Matter and Mterial Physics - CMMP By Salila Das

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Lecture 1 Superconductivity H. Kamerlingh onnes’, a Dutch scientist in 1911 observed that, certain metals and alloys exhibit infinite conductivity (i.e. almost zero resistivity) when they are cooled to sufficiently low temperature. This phenomenon is called superconductivity and the materials exhibiting this property are named as superconductors. He observed that when mercury was cooled down to below 4.2K, its resistivity suddenly drops to zero. Many metals like Pb , Al, Sn, Nb etc. show similar properties when cooled around 4K. Though many materials and alloys exhibit superconductivity but good conductors like Ag, Au, Cu, etc are not superconductors. Effect of magnetic field The main features which govern superconducting states of metal are (1) the temperature at which the specimen is kept Dr.Salila Das, Berhampur University, Odisha

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Lecture 1 (2) Magnetic field at which the specimen is placed If the temperature is increased above Tc, the superconductivity disappears. It also disappears if the magnetic field H is raised above the critical magnetic field 𝐻𝑐 ,where Η a function of 𝑇. For 𝑇 <𝑇𝑐 , Η=0 the specimen behaves as a superconductor and for 𝑇 >𝑇𝑐 , 𝐻𝑐 increases and superconductivity disappears. The variation of 𝐻𝑐 with temperature T is given by 𝑇 2 𝐻𝑐 (𝑇) = 𝐻𝑐 (0) [1 − (𝑇 ) ] 𝑐 𝐻𝑐 (0) : Critical magnetic field at 0 K Critical Field as a function of temperature The Meissner effect: If a weak magnetic field (Η < Η𝑐 ) is applied to a specimen below the transition temperature (i.e. 𝑇 <𝑇𝑐 ) no lines of magnetic induction exists inside the specimen. Magnetic lines of induction is pushed out of the specimen and B =0 This phenomenon was described by Meissner and Ochsenfeld in 1933 and is knows as flux exclusion or Meissner effect. When temperature is raised above 𝑇𝑐 , the lines of flux suddenly starts penetrating into specimen and the specimen returns back to normal state, the effect is reversible. For normal material 𝐵 =𝐵𝑎 +𝜇0 𝑀 Dr.Salila Das, Berhampur University, Odisha

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Lecture 1 Ba: external magnetic field and M is the magnetization produced inside the material. For superconducting material magnetic induction inside it is zero i.e. B=0 𝐵𝑎 = −𝜇0 Μ 𝑜𝑟 ⟹ 𝜆=− 1 4𝜋 Μ 𝐵𝑎 =− 1 𝜇0 𝐵𝑎 = −𝜇0 Μ = 𝜆 in SI system in CGS system This shows specimen is perfectly diamagnetic. Perfectly diamagnetic property and perfect conducting are two independent features of a superconductor. For a superconductor in a normal 𝜌 𝑖𝑠 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 𝑎𝑛𝑑 𝐽 𝑖𝑠 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦. state 𝐸 =𝜌𝐽 where From Maxwell’s electromagnetic equation ∇×𝐸 =− 𝜕𝐵 𝜕𝑡 For superconducting state 𝐸 = 0 ⟹ 𝜕𝐵 𝜕𝑡 = 0 ⇒ 𝐵 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 That means, on cooling the specimen to transition temperature, the magnetic flux passing through the specimen should not change. So when a perfect conductor is cooled in a magnetic field until its resistance becomes zero, the magnetic field inside the material gets frozen and cannot change irrespective of the applied field but Meissner effect is quite contradictory to it. According to Meissner effect, the essential property for defining superconductivity is perfect diamagnetism i.e. 𝐸 = 0 𝑎𝑛𝑑 𝐵= 0 . Thus behaviour of a superconductor is different from that of a perfect conductor. Superconducting state is a characteristic thermodynamic phase of a substance in which 𝐸 = 0 𝑎𝑛𝑑 𝐵= 0 Type 1 and Type 2 superconductors:- Dr.Salila Das, Berhampur University, Odisha

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Lecture 1 On the basis of diamagnetic response of the material; superconductors are of two types – type 1 and type 2. Type 1 superconductors are those for which complete Meissner effect is observed. 𝜆= Μ Β𝑎 =− 1 4𝜋 in CGS system If magnetic field on the superconductor increases, magnetisation M increases at 𝐻 = 𝐻𝑐 there is sharp transition from superconducting state to normal state takes place. Al, Zn, Hg and Sn are pure type 1 superconductors. Magnetisation versus applied magnetic field for bulk superconductors exhibiting (a) complete Meissner effect, i.e., perfect diamagnetism (Type 1) and (b) the intermediate or vortex state for fields between the upper 𝑯𝒄𝟐 and lower 𝑯𝒄𝟏 critical fields, i.e., imperfect diamagnetism (Type 2). For type 2 superconductors – when magnetic field is increased from𝐻 = 0 𝑡𝑜 𝐻 = 𝐻𝑐1 , the material behaves as pure superconductors and lines of magnetic field are rejected. If H is increased beyond 𝐻𝑐1 , the lower critical field, the lines of flux begin penetrating and the material is in mixed state up to 𝐻𝑐2 , the upper critical field. For 𝐻 > 𝐻𝑐2 the specimen return back to normal state. Meissner effect is incomplete in the region between 𝐻𝑐1 𝑎𝑛𝑑 𝐻𝑐2 . This region is known as vertex region. Zr and Nb are examples of type 2 superconductors. Type 2 superconductors are also known as hard Dr.Salila Das, Berhampur University, Odisha