CMP Assignment 2 Updated
3
Alex Trinh Superconductivity Condensed Matter Physics 1 SUPERCONDUCTIVITY A. Trinh School of Physics, University of Sydney INTRODUCTION The phenomenon in which the electrical resistance of materials vanishes when it reaches a certain temperature is known as superconductivity. It was first discovered by physicist Heike Kamerlingh Onnes, three years after liquefying helium. This made sufficiently low temperatures available, which lead to the discovery of the vanishing of electrical resistance of certain metals. A few years later, Meissner and Ochsenfeld discovered that the superconductors were perfectly diamagnetic, in which an induced magnetic field is in a direction opposite to an external magnetic field, and perfectly conducting [1]. Early attempts were made to explain the behaviour due to superconductivity. The isotope effect, discovered by Maxwell and Reynolds, says that the critical temperature of a material is dependent on the isotopic mass. It gave support that superconductivity was due to electron- phonon interactions. This report details the physics behind superconductivity, from BCS theory to hole superconductivity. BCS THEORY The isotope effect was a good start as it gave support to electron-phonon interactions. To try and explain superconductivity, physicist Herbert Frohlich developed a model based on electron- phonon interactions but could not describe the properties of a superconductor. The successful explanation of superconductivity was developed by John Bardeen, Leon Cooper and J. Robert Schrieffer, known as BCS theory. The idea is that electrons in a superconductor form what is known as Cooper pairs, bound together by some attractive force. Electromagnetism suggests that electrons repel each other because of their like charges. The attraction comes from electron- phonon interactions. Consider the following scenario: An electron travels through the lattice and causes a distortion due to the electrostatic attraction between the electron and positive lattice. This distortion attracts an electron travelling in the opposite direction. The attraction only occurs if an electron is close enough before the ions return to their equilibrium position. The coupling of electrons to lattice vibrations is viewed as an exchange of phonons. This view can be treated quantum mechanically. Electrons don’t move through and collide with the lattice, rather they travel as Bloch waves that are diffracted by the lattice due to the existence of impurities. Consider a pair of electrons with wave vectors 1 and 2 with combined wavefunction, Ψ K 1 K 2 (r 1 ,r 2 )= 1 √2 [exp(iK 1 ∙r 1 )exp(iK 2 ∙ r 2 ) ± exp(iK 1 ∙r 2 )exp(iK 2 ∙r 1 )] (1) If 1 = - 2 , then substituting into Eqn. 1. will lead to a standing wave solution. Consider a scenario in which the wave is almost a standing wave, then the wave vectors are related by, 1 =− 2 + (2)
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Alex TrinhSuperconductivityCondensed Matter Physics 1 SUPERCONDUCTIVITY A.Trinh School of Physics, University of Sydney INTRODUCTION Thephenomenoninwhichtheelectrical resistanceofmaterialsvanisheswhenitreaches acertaintemperatureisknownas superconductivity.Itwasfirstdiscoveredby physicistHeikeKamerlinghOnnes,threeyears afterliquefyinghelium.Thismadesufficiently lowtemperaturesavailable,whichleadtothe discoveryofthevanishingofelectrical resistanceofcertainmetals.Afewyearslater, MeissnerandOchsenfelddiscoveredthatthe superconductorswereperfectlydiamagnetic,in which an induced magnetic field is in a direction oppositetoanexternalmagneticfield,and perfectlyconducting[1].Earlyattemptswere madetoexplainthebehaviourdueto superconductivity.Theisotopeeffect, discoveredbyMaxwellandReynolds,saysthat thecriticaltemperatureofamaterialis dependentontheisotopicmass.Itgavesupport thatsuperconductivitywasduetoelectron-phononinteractions.Thisreportdetailsthe physicsbehindsuperconductivity,fromBCS theory to hole superconductivity. BCS THEORY The isotope effect was a good start as it gave supporttoelectron-phononinteractions.Totry and explain superconductivity, physicist Herbert Frohlichdevelopedamodelbasedonelectron-phononinteractionsbutcouldnotdescribethe propertiesofasuperconductor.Thesuccessful explanation of superconductivitywas developed byJohnBardeen,LeonCooperandJ.Robert Schrieffer,knownasBCStheory.Theideais thatelectronsinasuperconductorformwhatis known as Cooper pairs, bound together by some attractiveforce.Electromagnetismsuggeststhat electronsrepeleachotherbecauseoftheirlike charges.Theattractioncomesfromelectron-phononinteractions.Considerthefollowing scenario:Anelectrontravelsthroughthelattice andcausesadistortionduetotheelectrostatic attractionbetweentheelectronandpositive lattice.Thisdistortionattractsanelectron travellingintheoppositedirection.The attractiononlyoccursifanelectronisclose enoughbeforetheionsreturntotheir equilibriumposition. Thecouplingofelectrons to lattice vibrations is viewed as an exchange of phonons.Thisviewcanbetreatedquantum mechanically. Electronsdontmovethroughandcollide withthelattice,rathertheytravelasBloch waves that are diffracted by the lattice due to the existenceofimpurities.Considerapairof electronswithwavevectors1and2with combined wavefunction, K1K2(r1, r2) =12[exp (iK1 r1)exp (iK2r2) exp (iK1 r2)exp (iK2 r1)](1) If1=-2,thensubstitutingintoEqn.1.will leadtoastandingwavesolution.Considera scenario in which the wave is almost a standing wave, then the wave vectors are related by, 1= 2+ (2) Alex TrinhSuperconductivityCondensed Matter Physics 2 where is a small value of the wave vector for allwavevectors.Thedifferenceintheenergies is given by = (
) =
(3) This energy shift must satisfy in order forthephotoninteractiontobenegative.This negativeinteractionmustdominateoverthe Coulomb interaction. The energy of the electron pairislowestifthecentreofmassenergyis zero. This means that the spins and momenta of the electrons are opposite. Their spins are also in oppositedirectionstomaximisetheprobability that the electrons are close together [2]. Cooperimprovedonthisbyassumingthe electronpairisinthegroundstate.Theground stateisalinearcombinationoftheoriginal statesoftheelectronsintheFermiseawith coefficientsthatareofthesamesign.The groundstateisusuallyformedbysatisfyingthe standingwavecondition.IfEqn.2.issatisfied, thenthisproducesanetcurrentflowwhich meansonaverage,oneoftheelectronsinthe pair will have a larger magnitude of momentum than its partner. The electron-phonon interaction doesntjustexistbetweenasinglepairof electrons.ACooperpaircangoontoproduce Cooperpairswithotherelectronsandalarge network of interactions is formed. If there are N electronsandN/2pairsareformed,theground statewavefunctionwillbetheproductofthe N/2identicalCooperpairwavefunctions.This wavefunction is symmetric thus alterations must bemadetosatisfythePauliexclusionprinciple [2].Forasinglepairofelectronsexcitedabove theFermisurface,anegativeinteractionina boundstateisalwaysformednomatterhow weak the interaction [3]. Acriticaltemperaturewasmentionedabove, anditisthetemperatureatwhichamaterial needs to be cooled to achieve superconductivity. Theexistenceofthistemperaturesuggeststhat thereisanenergygapinthedensityofstates, whichcharacterisesthebindingenergyofthe Cooper pairs. This energy gap is given by, = 2exp(((
))1) =72
(4) whereisthephononfrequency,givenby = (4/)1/2|sin (
2)|,(
)isthedensity ofstatesattheFermienergy,andisthe phonon-electroninteractionstrength.There exists a coherence length given by, 0= 2
/(5) where
istheFermivelocityandisthe velocityatwhichelectronsneartheFermi surfacetravelinthelattice,andistheenergy gap.Thecoherencelengthisthenominal distance between the electron pair. This length is usuallyhundredstothousandsofion separations.Ifthecoherencelengthiswithin, say, an atomic spacing, the Coulomb interaction will begin to dominate [4]. HOLE SUPERCONDUCTIVITY There is a class of superconductors that have high critical temperatures. Cooper pairs in these materialsarenotproducedbylattice deformationsassuggestedinBCStheoryas theircoherencelengthisusuallyveryshort, thereforeBCStheorycannotbeusedtoexplain their behaviour. [4] Apossibleexplanationforthehighcritical temperaturesistheinteractionbetweenholes andoutershellelectrons.Theirinteractionis muchstrongerthanelectron-phonon interactions,hencethecriticaltemperaturesare alsohigher.Thewholeconceptworkson polarizabilityofoutershellelectrons.Holesare producedwhenthereisavacantstatenearthe Fermienergy.ElectronsneartheFermisurface may accelerate to fill the empty state. The empty Alex TrinhSuperconductivityCondensed Matter Physics 3 stateappears to move as electrons fill an empty stateproducedbythepreviouselectron.Itcan beconsideredasanelectronwithnegative effectivemass.Becausethechargecarriersare positive,thereisacorrelationbetweenhigh temperaturesuperconductorsandpositiveHall coefficients, as seen in materials that contain Nb and Pb. Oneofthemostcommonhightemperature superconductorsareoxidesuperconductors. Their superconductivity is a result of holes at the oxygensitesofthematerial.Wecanproduce holes near these sites by doping the material.In general,itisassumedthathightemperature superconductorscanbecreatedbyusing elementstotherightoftheperiodictable.We canconsidertheholesasthechargecarriers insteadofthe7valenceelectrons.Their superconductivity will be a result of holes at the sitesofthoseelements,sameasintheoxides. Thisconceptcanalsobeappliedtoelements closertotheleftoftheperiodictablethoughit applies to a lesser extent. Hightemperaturesuperconductivityisalso observedinmaterialssuchasCuprateand cadmiumsulfideunderpressure[5].This suggeststhatifpressureisappliedtocertain materials,conductionoccursviaholes.Inthe aboveexample,holesareproducedaroundthe chloride and sulfide anions respectively. CONCLUSION Toconclude,thepropertyofcertainmaterials thatcausestheirresistancetovanishcanbe explained,toacertainextent,byBCStheory. The theory says that electrons travel through the latticeofamaterialasCooperpairsviaan exchange of phonons as a result of distortions in thelattice.Thetemperatureatwhichthe materiallosesallofitsresistanceisknownas thecriticaltemperature,andisaresultofan energygapinthedensityofstatesattheFermi energy.BCStheorycannotbeusedtoexplain hightemperaturesuperconductivityasitusesa differentdrivingmechanism.These superconductorshavehighcriticaltemperatures andapossibleexplanationisthat theyworkvia hole-electron cloud interactions which are much strongerthanelectron-phononinteractions.The holesaresituatedatthesitesofelementstothe rightoftheperiodictable,suchasoxygen. Researchintothesesuperconductorsarestill being performed. REFERENCES [1] J. R. Schrieffer, M. Tinkham, Superconductivity, Rev. of Mod. Phys. vol. 71, 1999 [2] M. S. Dresselhaus, Superconducting Properties of Solids, Solid State Physics Pt. IV [3]J.Bardeen,L.N.Cooper,J.R.Schrieffer,MicroscopicTheoryofSuperconductivity,Phys. Rev. 108, 1175 (1975) [4] The University of Sydney, n.d, Superconductivity, viewed 24th October 2014, http://www.physics.usyd.edu.au/~khachan/PTF/Superconductivity.pdf [5] J. E. Hirsch, Hole Superconductivity, Phys. Rev. vol. 134, 1989
