5. Superconductivity. R(T) = 0 for T < T c, R(T) = R 0 +at 2 +bt 5, B = H+4πM = 0,

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1 5. Superconductivity In this chapter we shall introduce the fundamental experimental facts about superconductors and present a summary of the derivation of the BSC theory (Bardeen Cooper and Schrieffer). This chapter is kept short due to the existence of the parallel lectures in this semester dedicated exclusively to the superconductivity. 5. Experimental results 5.. Zero resistance In9KamerlinghOnnes(NobelPrize93)observedthattheresistance of Hg shows an abrupt drop at T c = 4.5 K (Fig. 5.). Usually the resi- the T contribution originates from the electron-electron interaction. For a superconductor (as shown in Fig. 5.) R(T) = for T < T c where T c is the critical temperature. This behavior is independent of impurities. Thus a superconductor behaves like an ideal metal without impurities which means that a system in the superconducting phase can conduct without dissipation loss. Such behavior is technologically desirable and since decades a lot of effort has been devoted to the search for superconductors with high T c. Before 986 the highest T c was T c = 3 K for Nb 3 Ge. In order to reach this low temperatures one has to use expensive liquid helium for cooling. In 986 Bednorz and Müller discovered that by hole doping copper oxide La x Ba x CuO 4 they could get the material superconducting at T c 35 K. Following this route of doping copper oxides in 987 YBa Cu 3 O 7 x was shown to become superconducting at T c 9 K. This set a hallmark for technical purposes since in this case one can use cheap liquid nitrogen for cooling. The highest T c reached so far with doped cuprates is T c 33 K. In 8 a new class of high-temperature superconductors which are based on Fe was discovered. These superconductors were presented in one of the topical seminars. The superconducting state is not only characterized by zero resistance but also by other important features which we discuss in the following. 5. Meißner effect Abbildung 5.: Resistance of Hg at a superconducting transition. stance of normal metals like Cu Ag or Na shows the following temperature dependence: R(T) = R +at +bt 5 where R is proportional to the concentration of impurities in the system. The T 5 contribution is due to electron-phonon scattering processes and 99 A superconducting material expels completely any applied external field thus behaving like a perfect diamagnet. This property of a superconductor is named the Meißner effect. There is though a difference between a perfect diamagnet and the Meißner effect. In a superconductor the magnetic field is expelled independently of the cooling procedure either first cooling and then applying B or first applying B and then cooling (Fig. 5.). The superconducting state is a new state of matter. Inside the superconductor the magnetic induction B disappears: B = H+4πM = where H is the external magnetic field.

2 Abbildung 5.: Superconductor versus ideal conductor in a magnetic field. Abbildung 5.3: The H T phase diagram and the H-dependece of magnetization M for either type I or type II superconductor. S labels the superconducting state N the normal state and M the mixed state. This results in a discontinuity of the specific heat at T c (Fig. 5.4). Critical magnetic field If the external magnetic field is too strong then the superconducting state is destroyed. The dependence of the critical magnetic field H c on temperature is given by [ ( ) ] T H c (T) = H c () T c where H c () is the value of the critical magnetic field at zero temperature. 5.3 Entropy and specific heat In the normal state specific heat depends linearly on T c n v γt whereas in the superconducting state the dependence is exponential: c s v e k BT. Abbildung 5.4: Specific heat jump near the superconducting transition. Such behavior indicates the existence of an energy gap for thermal excitations in superconductors. The entropy given by S(T) S() = T dt c(t ) T

3 fulfills the relation: S n (T c ) = S s (T c ) which is characteristic of second order phase transitions. In the following sections we shall show that (i) the electron-phonon interaction in a metal can induce an attractive electron-electron interaction and (ii) two electrons at the Fermi sea in the presence of such an attractive interaction can build an energetically favorable bound state the Cooper pair state. 5.4 Cooper pair 5.4. Retarded pair potential In the superconducting state the attractive interaction between electrons at the Fermi surface leads to the formation of Cooper pairs. In the traditional superconductors the attractive electron-electron interaction is mediated by phonons via the following mechanism: The ionic motion deforms the lattice with a time scale τ given by τ π ω D 3 s where ω D is the Debye frequency. During this time a conduction electron moves a distance v F τ: v F τ 8 cm/s 3 s Å with v F being the Fermi velocity. A second electron can feel the retarded attraction of the first electron without the Coulomb repulsion playing a role Cooper instability of the Fermi sea We consider two electrons near the Fermi surface in the presence of an attractive potential V( r r ). The Schrödinger equation for this problem then reads h m ( + ) Ψ( r r )+V( r r )Ψ( r r ) = (ǫ+e F )Ψ( r r ) (5.) For a disappearing potential V = the binding energy ǫ = and the two-particle wavefunction is given by Φ V= ( r r ) = L 3/ei r L 3/ei r = L 3ei ( r r ). (5.) Note that here for simplification we are not considering the symmetrized form of the wavefunction. We also assume = =. We now consider a small V( r r ) and assume that Ψ( r r ) can in this case be taken as a linear combination of the basis functions (5.): Ψ( r r ) = L 3 g()e i ( r r ). (5.3) The -summation is here limited to a region near the Fermi surface: i.e. E F < h k m < E F + hω D g() = for { k < kf k > m(e F + hω D )/ h. The Debye frequency ω D is much smaller than the typical Fermi energy. Self-consistent equation By Fourier transforming the Schrödinger equation (5.) with the ansatz (5.3) we obtain h k m g( )+ L 3 g( )V k = (ǫ+e F )g() (5.4) 3 4

4 where V k = V( r)e i( ) r d 3 r describes the scattering of pairs of electrons from ( ) to ( ). We approximate V k by an attractive constant: { V for E V k = F < h k m h k m < E F + hω D otherwise. Then Eq. (5.4) assumes the following form ( h k ) m +ǫ+e F g() = V g( ) A. (5.5) L We insert g() A g() = h k m +ǫ+e F into Eq. (5.5) and get V A L 3 h k m ǫ E = A F k = V L 3 k h k m ǫ E F. (5.6) Out of this equation we can determine the binding energy ǫ of the electron pair. Binding energy We substitute in Eq. (5.6) the sum over k by an integral over the energy with the help of the density of states N(E F ): EF + hω D de = V N(E F ) = ( ) ǫ hω E F E ǫ E F V D N(E F )ln. ǫ Solving this equation for ǫ gives ǫ = hω D e /(V N(E F )) hω D e /(V N(E F )) <. 5 For V E F we obtain an exponentially small energy win for all V > which is not analytical in V. Therefore the previous result cannot be obtained from perturbation theory. The result of Cooper shows that in the presence of a (passive) Fermi sea and an infinitesimally small attractive interaction two electrons form a bound state (see Fig. 5.5). Abbildung 5.5: The Cooper pair picture. The attractive phonon-induced interaction is in real space isotropic and spin-independent. This leads to a pair wavefunction that is symmetric with respect to the spatial coordinates and antisymmetric (singlet) in the spin sector. Equivalently the pairing in the momentum space is between electrons with and. 5.5 BCS theory The Cooper problem deals with the case of two electrons near the Fermi surface. Since electrons are indistinguishable one has to consider a manybody wavefunction in order to describe all the conduction electrons in the superconducting state Hamilton operator in second quantization Theenergyofacrystallatticecanbedescribedintermsofacertainnumber of harmonic oscillators which have an equidistant energy spectrum: H ph = ( hω( qs) a qs a qs + ) qs 6

5 with a qs and a qs being creation and annihilation bosonic operators: [a qs a q s ] = δ q q δ ss [a qs a q s ] = [a qs a q s ] =. A highly excited state of a harmonic oscillator corresponds to a larger amount of phonons with energy hω(qs). In the Feynman representation a phonon can be pictured as a wavy line with an arrow (Fig. 5.6). Then e is He is ( is ) S H (+is ) S = H+i[HS] [[HS]S]+O(S3 ) = H +H I +i[h S]+i[H I S] [[H S]S]+O(S H I). We choose S so that the following equation i[h S] = H I Abbildung 5.6: Feynman s representation of a phonon. We consider the total Hamiltonian written in a second quantization form that describes both electrons and phonons as well as the electron-phonon interaction: H = ǫ()c c + q hω q a q a q + q M q (a q +a q)c + q c where ǫ() is the kinetic energy of electrons M q is the electron-phonon interactionc andc (a q anda q) are the fermionic (bosonic) creation and annihilationoperators.thefermionicoperatorsobeytheanti-commutation relations: {c kσ c σ } = δ δ σσ {c σ c σ } = {c σ c σ } =. We consider now a canonical transformation H T = e is He is with S being an hermitian operator. We divide the Hamiltonian into a non-interacting part H and an interacting part H I : H = H +H I. 7 is fulfilled. With this condition the term linear in H I disappears in the canonically transformed Hamiltonian. Thus we obtain e is He is H +i[h I S]+ i [H IS] = H + i [H IS]. By considering the explicit form of the matrix elements in the basis of H it can be shown (see for instance Czycholl Theoretical solid state physics ) that after the canonical transformation the original Hamiltonian results in an effective Hamiltonian of the following form H = H +H IT with H IT = M( q) hω q (ǫ k+ q ǫ q) ( hω q ) σσ c + qσ c σ c qσ c σ q describing an effective electron-electron interaction. The matrix elements become negative and the effective interaction becomes attractive when ǫ k+ q ǫ < hω q which indicates that there is a probability of an attractive electron-electron interaction in a region near the Fermi surface mediated by interchange of virtual phonons (Fig. 5.7). We can simplify H IT by 8

6 Abbildung 5.7: Interchange of a virtual phonon. Abbildung 5.8: BSC approximation for the virtual phonon interchange. (i) neglecting the q dependence of V( q): V( q) = M( q) hω q (ǫ k+ q ǫ q) ( hω q ) V; then H IT = V c + qσ c qσ c σ c kσ σσ with q ǫ k ǫ + q = ǫ q ǫ hω q hω D. (ii) by taking into account only interaction between electrons with opposite momentum and spin (Fig. 5.8). With that we obtain the BCS Hamiltonian: H = σ ξ k = ǫ E F. ξ k c σ c σ V c c c c This Hamiltonian can be alternatively solved by considering () a variational wavefunction and () the mean-field theory. Here we will only give a brief summary of both BCS wavefunction The BCS wavefunction [Bardeen Cooper and Schrieffer Phys. Rev (957)] consists of a linear combination of unoccupied and doubly occupied states: Ψ = ( ) u k +v c c. Here isan emptyfermisphere u k andv arevariationalparameters. v k /u g( ) provides the link to the Cooper ansatz. TheBCSwavefunctionhasnodefiniteparticlenumberbutithasadefinite phase. The normalization condition Ψ Ψ = u +v = is fulfilled through the parameterization u = ( + ξ ) v E = ( ξ ) k E k where E k is a free parameter related to the elementary excitations to the BCS ground state. u corresponds to the probability that a pair of states withoppositeandσisunoccupiedwhilev correspondstotheprobability that a pair of states with opposite and σ is occupied. The condition of N = const can be implemented either variationally or by means of the Bogoliubov transformation.

7 5.5.3 Mean-field treatment of the Hamiltonian The variational treatment of the BCS Hamiltonian is equivalent in this case to the mean-field treatment. Let us define the following decoupling of the interacting terms in the Hamiltonian: with H H eff = σ H eff = σ ǫ k c σ c σ V c c c c ξ k c σ c σ V c c c c + V c k c c c ; c c c c + V = V c c (5.7) = V c c being the order parameter of the superconducting phase. This effective Hamiltonian contains anomalous terms i.e. it is not bilinear with respect to a creation operator and an annihilation operator: [H eff N]. In order to diagonalize the Hamiltonian we consider the Bogoliubov transformation: α k = u c v c β k = u c +v c with u k + v k = ; {α k β } = {α k α } = {β β } = δ. By choosing u k and v as u = + ξ k ξ }{{ + } E k and v = ξ k ξ + H eff isensuredtobediagonal.thesevaluesarealsoobtainedbyconsidering the variational BCS wavefunction H eff = E k (α α +β β )+ (ξ k E )+ V where α k and β are new quasiparticles which correspond to excitations to the BCS ground state. Gap equation We perform statistical averaging on Eq. (5.7): = V ( ) ξ + f(e ) with f(e k ) = e βe + ; = V E k = V = V = V hω D E k ( tanh βe ) e βe + β tanh ξ + ξ + dǫn(ǫ) hω D ǫ + tanhβ ǫ +. We now focus on three limiting cases.

8 Case : T. In this case we make use of the fact that tanh β ǫ + and introduce (T = ). T Then the gap equation simplifies to We get hω D dǫ = VN(E F ) ǫ + = hω D sinh V N(E F ) hω D e V N(E F) = VN(E F )arcsinh ǫ hω D = VN(E F )arcsinh hω D. where N(E F ) is the density of states per spin. If the total density is considered then hω D e V N(E F). Case : T T (high temperature). Now and tanh β ǫ + β ǫ + hω D = VN(E F ) dǫ β ǫ + ǫ + = βn(e D F) hω. For T there is no trivial solution of the BCS equation (the previous equation goes into an inconsistency). This indicates that there must be a critical temperature T c. 3 Case 3: T = T c. In this case (T c ) = per definition. Then hω D = VN(E F ) = VN(E F ) hω D k B Tc dǫ tanh ǫ k B T c ǫ dx tanhx. x Since typically hω D 3 K and T c K we can assume that hω D. k B T c Then [ VN(E F ) ln hω D dx lnx ( lntanh x )] k B T c ( = VN(E F ) ln hω D ln π ) k B T c 4e γ where γ is the Euler constant. = V N(E F )ln.36 hω D k B T c k B T c =.36 hω D e V N(E F) () k B T c =.36 =.764. This is a universal BCS relation. The condition for weak coupling: () hω D. Since hω D M / T c M which is named the isotope effect. At T and T T c the order parameter behaves as (see also Fig. 5.9) T : (T) = πk B T c e /k B T 8 T T c : (T) = k B T c π T. 7ξ(3) T c 4

9 Abbildung 5.9: The order parameter as a function of temperature. All thermodynamic properties of a BCS superconductor can be obtained from the presented BCS theory. The comparison with experiment is excellent. 5