Chapter Microscopic Theory

Transcription

1 TT1-Chap4-1 Chapter 4 4. Microscopic Theory 4.1 Attractive Electron-Electron Interaction Phonon Mediated Interaction Cooper Pairs Symmetry of Pair Wavefunction 4.2 BCS Groundstate The BCS Gap Equation Ground State Energy Energy Gap and Excitation Spectrum 4.3 Thermodynamic Quantities 4.4 Determination of the Energy Gap Specific Heat Tunneling Spectroscopy

2 4. BCS Theory TT1-Chap4-2 after discovery of superconductivity, initially many phenomenological theories have been developd London theory (1935) macroscopic quantum model of superconductivity (1948) Ginzburg-Landau-Abrikosov-Gorkov theory (early 1950s) problem: phenomenological theories do not provide insight into the microscopic processes responsible for superconductivity impossible to engineer materials to increase T c if mechanisms are not known superconductivity originates from interactions among conduction electrons theoretical models for description of interacting electrons are required - very complicated - go beyond single electron (quasiparticle) models - not available at the time of discovery of superconductivity

3 4. BCS Theory TT1-Chap4-3 development of BCS theory by J. Bardeen, L.N. Cooper and J.R. Schrieffer in 1957 key element is attractive interaction among conduction electrons 1956: Cooper shows that attractive interaction results in pair formation and in turn in an instability of the Fermi sea 1957: Bardeen, Cooper and Schrieffer develop self-consistent formulation of the superconducting state: condensation of pairs in coherent ground state paired electrons are denoted as Cooper pairs general description of interactions by exchange bosons k 1 +q,s 1 k 2 -q,s 2 Bardeen, Cooper and Schrieffer identify phonons as the relevant exchange bosons suggested by experimental observation T c 1/ M ω ph isotope effect k in general, detailed nature of exchange boson 1,s 1 does not play any role in BCS theory possible exchange bosons: magnons, polarons, plasmons, polaritons, spin fluctuations,.. k 2,s 2

4 log (T c / K) 4. BCS Theory TT1-Chap isotope effect Sn data from: E. Maxwell, Phys. Rev. 86, 235 (1952) B. Serin, C.A. Reynolds, C. Lohman, Phys. Rev. 86, 162 (1952) Maxwell Lock et al. Serin et al. T c 1/M 1/2 J.M. Lock, A.B. Pippard, D. Shoenberg, Proc. Cambridge Phil. Soc. 47, 811 (1951) log M (arb. units) in general: T c 1/M b*

5 4.1 Attractive Electron Electron Interaction TT1-Chap4-5 intuitive assumption: superconductivity results from ordering phenomenon of conduction electrons problem: - conduction electrons have large (Fermi) velocity due to Pauli exclusion principle: 10 6 m/s 0.01 c - corresponding (Fermi) temperature is above K - in contrast: transition to superconductivity occurs at 1 10 K ( mev) task: - find interaction mechanism that results in ordering of conduction electrons despite their high kinetic energy - initial attempts fail: Coulomb interaction (Heisenberg, 1947) magnetic interaction (Welker, 1929)..

6 4.1.1 Phonon Mediated Interaction TT1-Chap4-6 known fact since 1950: - T c depends on isotope mass conclusion: - lattice plays an important role for superconductivity - initial proposals for phonon mediated e-e interaction (1950): H. Fröhlich, J. Bardeen static model of lattice mediated e-e interaction: - one electron causes elastic distortion of lattice: attractive interaction with positive ions results in positive charge accumulation - second electron is attracted by this positive charge accumulation: effective binding energy intuitive picture, but has to be taken with care wrong suggestion: - Cooper pairs are stable in time such as hydrogen molecule - pairing in real space

7 4.1.1 Phonon Mediated Interaction dynamic model of lattice mediated e-e interaction: - moving electrons distort lattice, causing temporary positive charge accumulation along their path track of positive charge cloud positive charge cloud can attract second electron - important: positive charge cloud rapidly relaxes again dynamic model electron 1 electron important question: How fast relaxes positive charge cloud when electron moves through the lattice? characteristic time scale τ: frequency ω q of lattice vibrations (phonons): τ = 1/ω q ω q /s (maximum frequency: Debye frequency ω D < /s) + + TT1-Chap4-8

8 4.1.1 Phonon Mediated Interaction resulting range of interaction (order of magnitude estimate) - how far can a second electron be to attracted by the positive space charge before it relaxes - characteristic velocity of conduction electrons: v F few 10 6 m/s interaction range: v F τ µm (is related to GL coherence length) important fact: - retarded reaction of slow ions results in large interaction range retarded interaction - retarded interaction is essential for achieving attractive interaction without any retardation: - short interaction range - Coulomb repulsion between e dominates retarded interaction has been addressed during discussion of screening of phonons in metals retarded interaction potential: q-dependent plasma frequency screened Coulomb potential 1/k s = Thomas-Fermi screening length negative for overscreening TT1-Chap4-9

9 4.1.2 Cooper-Pairs Question: How can we formally describe the pairing interaction? starting point: free electron gas at T = 0 - all states occupied up to E F = ²k²/2m Gedanken experiment: - add two further electrons, which can interact via the lattice - describe the interaction by exchange of virtual phonon virtual phonon: is generated and reabsorbed again within time Δt 1/ω q wave vectors of electrons after exchange of virtual phonon with wave vector q: total momentum is conserved: note: - since at T = 0 all states are occupied below E F, additional states have to be at E > E F - maximum phonon energy: ħω q = ħω D (Debye energy) interaction takes place in a spherical shell with radius k F and thickness Δk mω D /ħk F for given K only specific wave vectors k 1, k 2 are allowed for interaction process TT1-Chap4-10

10 4.1.2 Cooper-Pairs TT1-Chap4-11 K > 0 K = 0 possible phase space for interaction possible phase space is complete spherical shell important conclusion: available phase space for interaction is maximum for K = 0 or equivalently k 1 = - k 2 Cooper pairs with zero total momentum: (k, - k)

11 4.1.2 Cooper-Pairs wave function of Cooper pairs and corresponding energy eigenvalues: - two-particle wave function is chosen as product of two plane waves - since pair-correlated electrons are permanently scattered into new states pair wave function = superposition of product wave functions k F < k < k F + Dk, since restriction to energies E F < E < E F + ħω D probability to find pair (k, - k) note: - electron with k < k F cannot participate in interaction since all states are occupied - we will see later that superconductor overcomes this problem by rounding off the Fermi distribution even at T = 0 superconductor first have to pay (kinetic) energy for rounding off f(e) energy is obtained back by pairing interaction (potential energy) net energy gain TT1-Chap4-12

12 4.1.2 Cooper-Pairs TT1-Chap4-13 wave function of Cooper pair and corresponding energy eigenvalues: - we assume that pairing interaction only depends on relative coordinate Schrödinger equation: - insert and multiply by - integration over sample volume W: - we use scattering integral (gives probability for scattering process k k )

13 4.1.2 Cooper-Pairs TT1-Chap result: problem: we have to know all matrix elements!!! - simplifying assumption to solve the problem: - further simplification by summing up over all k using and - we introduce pair density of states (D(E) = DOS for both spin directions)

14 4.1.2 Cooper-Pairs TT1-Chap integration and solving for E yields: - for weak interaction ( ) we obtain approximation: important result: energy of interacting electron pair is smaller than 2E F bound pair state (Cooper pair) binding energy depends on V 0 and maximum phonon energy w D note: in Gedanken experiment we have considered only two additional electrons above E F - in real superconductor: interaction of all electrons in energy interval around E F - electron gas becomes instable against pairing - instability causes transition into new ground state: BCS ground state

15 4.1.2 Cooper-Pairs TT1-Chap4-16 typical interaction range: µm corresponds to size of Copper pair very large number of other pairs within volume of single pair: typically > 1 Mio. strong overlap of pairs coherent state

16 4.1.3 Symmetry of Pair Wavefunction TT1-Chap4-17 Cooper pair consists of two Fermions total wavefunction must be antisymmetric spin wavefunction: center of mass motion we assume K = 0 orbital part spin part antisymmetric spin wavefunction symmetric orbital function: L = 0, 2, (s, d,.) symmetric spin wavefunction antisymmetric orbital function: L = 1, 3, (p, f,.)

17 4.1.3 Symmetry of Pair Wavefunction isotropic interaction: interaction only depends on k in agreement with angular momentum L = 0, (s wave superconductor) corresponding spin wavefunction must by antisymmetric spin singlet state (S = 0) resulting Cooper pair: - L = 0, S = 0 is realized in metallic superconductors - higher orbital momentum wavefunction in cuprate superconductors (HTS): L = 2, S = 0 (d wave superconductor) spin triplet Cooper pairs (S = 1): - realized in superfluid 3 He: L = 1, S = 1 - evidence for L = 1, S = 1 also for some heavy Fermion superconductors (e.g. UPt 3 ) TT1-Chap4-18

18 4.1.3 Symmetry of Pair Wavefunction L = 0, S = 0 L = 2, S = 0 k y k y k x k x Superconductivity gets an iron boost Igor I. Mazin Nature 464, (11 March 2010) TT1-Chap4-19

19 4.1.3 Symmetry of Pair Wavefunction TT1-Chap4-21 Fe pnictogens (As, P) iron-based superconductors iron pnictides spacer Fe k y chalcogens (Se or Te) k x (a) s-wave, e.g. in aluminium (b) d-wave, e.g. in copper oxides (c) two-band s-wave with the same sign, e.g. in MgB 2 (d) an s ± -wave, e.g. in iron-based SC

20 4.1.3 Symmetry of Pair Wavefunction phase diagram of UPt 3 f-wave (E 2u ) Cooper pair wavefunction in three-dimensional momentum space phase A phase B Michael R. Norman, Science 332, (2011) TT1-Chap4-24

21 4.2 The BCS Ground State TT1-Chap4-26 interaction mechanism and Cooper pair formation already discussed how does the ground state of the total electron system look like? we expect: pairing mechanism goes on until the Fermi sea has changed significantly pairing energy goes to zero, pairing stops detailed theoretical description is complicated, we discuss only basics

22 TT1-Chap The BCS Ground State second quantization formalism (>1927, Dirac, Fock, Jordan et al.): 2 nd quantization formalism is used to describe quantum many-body systems quantum many-body states are represented in the so-called Fock state basis Fock states are constructed by filling up each single-particle state with a certain number of identical particles. 2 nd quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states. 2 nd quantization formalism is also known as the canonical quantization in quantum field theory, in which the fields are upgraded into field operators analogous to 1 st quantization, where the physical quantities are upgraded into operators creation and annihilation operators: fermionic creation operator c k,σ fermionic annihilation operator c k,σ anti-commutation relations particle number operator Pauli exclusion principle

23 4.2 The BCS Ground State TT1-Chap4-28 operator describing scattering from to two particle interaction potential (see exercise for derivation) pair creation and annihilation operators pair creation and annihilation operators obey the commutator relations the last two operators of the first term on the r.h.s. can be moved to the front by a even number of permutations sign is preserved

24 4.2 The BCS Ground State TT1-Chap4-29 some of the commutator relations of the pair operators are similar to those of Bosons although the pair operators consist only of electron (fermionic) operators P k, P k 0 but not equal to δ kk as expected for bosons, depend on k and T pair operators do not commute but are no bosonic operators powers of pair operators = 0 = 0 antisymmetry of fermionic wavefunction requires that powers of the pair operators disappear

25 4.2 The BCS Ground State TT1-Chap4-30 basic definitions and assumptions: 1. weak isotropic interaction: 2. pairing (Gorkov)amplitude: statistical average 3. Pauli principle pairing amplitude is anti-symmetric for interchanging spins and wave-vector: 4. spin part allows to distinguish between singlet and triplet pairing: 5. pairing potential: statistical average of pairing interaction

26 4.2 The BCS Ground State Hamilton operator: how to solve the Schrödinger equation? particle number operator most general form of N-electron wave-function: N/2 particles on M sites: problem: huge number of possible realizations, typically mean field approach: occupation probability of state k only depends on average occupation probability of other states Bardeen, Cooper and Schrieffer used the following Ansatz (mean-field approach): probability that pair state probability that pair state is not occupied is occupied TT1-Chap4-31

27 4.2 The BCS Ground State TT1-Chap4-32 How to guess the BCS many particle wavefunction? assume that the macroscopic wave function ψ r, t by a coherent many particle state of fermions = ψ 0 r, t e iθ(r,t) can be described coherent state of bosons discussed first by Erwin Schrödinger in 1926 when searching for a state of the quantum mechanical harmonic oscillator approximating best the behavior of a classical harmonic oscillator E. Schrödinger, Der stetige Übergang von der Mikro- zur Makromechanik, Die Naturwissenschaften 14, (1926). transferred later by Roy J. Glauber to Fock state R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131, (1963). Nobel Prize in Physics 2005 "for his contribution to the quantum theory of optical coherence", with the other half shared by John L. Hall and Theodor W. Hänsch.

28 4.2 The BCS Ground State Fock state representation of coherent state of bosons coherent state α is expressed as an infinite linear combination of number (Fock) states φ n = 1 a n n! 0 boson creation operator vacuum state Schrödinger (1926) normalization α = α e iφ is complex number probability for occupation of n particles is given by Poisson distribution - expectation value of number operator: N = α 2, ΔN = α = N 1 - relative standard deviation: ΔN N = 1 N 1 (as N 1) - uncertainty relation ΔN Δφ 1 2, Δφ 1 application: coherent photon state generated by laser TT1-Chap4-33

29 4.2 The BCS Ground State TT1-Chap4-34 Fock state representation of coherent state of fermions coherent bosonic state in analogy - we make use of the fact that higher powers of fermionic creation operators disappear due to Pauli principle (key difference to bosonic system): - normalization: satisfied if all factors = 1

30 4.2 The BCS Ground State TT1-Chap4-35 BCS ground state as coherent state of fermions coherence factors coherent superposition of pair states only average pair number is fixed ΔN = N 1 ΔN N = 1 N 1 ΔN Δφ 1 2 Δφ 1 u k and v k are complex probability amplitudes u k 2 : pair state with wave vector k is unoccupied v k 2 : pair state with wave vector k is occupied

31 4.2 The BCS Ground State TT1-Chap4-36 some expectation values (1): (see exercise sheets for detailed derivation)

32 4.2 The BCS Ground State TT1-Chap4-37 some expectation values (2): (pairing or Gorkov amplitude) (see exercise sheets for detailed derivation)

33 4.2 The BCS Ground State TT1-Chap4-38 determination of the probability amplitudes: task: determine probability amplitudes u k ² and v k ² in a self-consistent way by minimizing mean field BCS Hamiltonian minimization of expectation value by variational method: kinetic energy interaction energy single particle energy relative to µ:

34 4.2 The BCS Ground State TT1-Chap4-40 Bogoliubov-Valatin transformation rewriting of pair creation and annihilation operaors pair amplitude: insert into Hamiltonian and consider only terms linear in δg k make use of pair potential

35 TT1-Chap The BCS Ground State we use due to finite Δ k, Δ k, Hamiltonian describes interacting electron gas with new quasiparticles consisting of superposition of electron and hole states derive excitation energies by diagonalization of Hamiltonian Bogoliubov-Valatin transformation define new fermionic operators α k, β k and α k, β k by unitary transformation (rotation)

36 4.2 The BCS Ground State TT1-Chap4-42 use unitary matrix to rotate the energy matrix into eigenbasis of Bogoliubov quasiparticle spinors energy matrix appropriate unitary matrix to make transformed energy matrix diagonal: spinors of Bogoliubov quasiparticle operators

37 4.2 The BCS Ground State TT1-Chap4-43 inverse transformation

38 4.2 The BCS Ground State TT1-Chap4-44 replace operators by Bogoliubov quasiparticle operators resulting Hamiltonian: expressions in brackets must disappear we have to set expressions marked in yellow to zero to keep only diagonal terms and (quasiparticle number operators)

39 4.2 The BCS Ground State and multiply by Δ k /u k 2 (Δ k /u k 2 ), solve the resulting quadratic eqn. for Δ k v k /u k (Δ k v k /u k ) with the relative phase of u k and v k must be fixed and must be the phase of Δ k we can choose u k real and use the phase of v k corresponds to that of Δ k negative sign is unphysical corresponds to solution with maximum energy note that the phases of u k, v k and Δ k (u k, v k and Δ k ), although arbitrary, are related, since the quantity on the r.h.s. is real moreover TT1-Chap4-45

40 4.2 The BCS Ground State with v k 2 : probability that k is occupied probability v k 2 is smeared out around Fermi level even at T = 0: increase of kinetic energy smearing is required to allow for pairing interaction: reduction of potential energy > increase of kinetic energy v k 2 f(t = T c ) TT1-Chap4-47

41 4.2.1 The BCS Gap Equation BCS gap equation: we insert into Hamiltonian mean-field contribution contribution of spinless Fermion system with two kind of quasiparticles described by operators differs from the normal state value by the condensation energy (see below) and excitation energies TT1-Chap4-49

42 4.2.1 The BCS Gap Equation TT1-Chap4-50 determination of D by minimization of free energy: Hamiltonian has two terms: grand canonical partition function: constant term H 0 with no fermionic degrees of freedom term of free Fermi gas composed of two kind of fermions with energy ±E k partition function of an ideal Fermi gas: (since F = Nk B T ln Z ) solve for free energy: minimize free energy regarding variation of Δ k :

43 4.2.1 The BCS Gap Equation pairing susceptibility: ability of the electron system to form pairs we use: BCS gap equation - set of equations for variables Δ k - equations are nonlinear, since E k depends on Δ k - solve numerically TT1-Chap4-51

44 4.2.1 The BCS Gap Equation TT1-Chap4-52 energy gap D and transition temperature T c : trivial solution: requires for and for intuitive expectation for normal state non-trivial solution: we use approximations and we use pair density of states to integration and change from summation simple solutions for (i) T 0 (ii) T T c

45 4.2.1 The BCS Gap Equation (i) T 0: solve for D: (weak coupling approximation) compare to expression derived for energy of two interacting electrons: factor 2 in exp since we have assumed that the two additional electrons are in interval between E F and E F + w D and not between E F - w D and E F + w D TT1-Chap4-53

46 4.2.1 The BCS Gap Equation TT1-Chap4-54 (ii) T T c : with integral gives with and (Euler constant) critical temperature is proportional to Debye frequency w D 1/ M explains isotope effect!!

47 4.2.1 The BCS Gap Equation TT1-Chap4-55 compare key result of BCS theory considerable deviations for strong-coupling superconductors: is no longer a good approximation

48 4.2.1 The BCS Gap Equation D / D(0) (iii) 0 < T < T c : numerical solution of integral good approximation close to T c : Pb Sn In BCS theory close to T c : (characteristic result of mean-field theory) T / T c TT1-Chap4-56

49 4.2.2 Ground State Energy TT1-Chap4-57 determine condensation energy (for, ) calculate expectation value at T = 0 subtract mean energy of normal state at T = 0: making use of symmetry around µ no quasiparticle T = 0 see appendix G.3 in R. Gross, A. Marx, Festkörperphysik, 2. Auflage, De Gruyter (2014)

50 4.2.2 Ground State Energy TT1-Chap4-58 replace summation by integration. after some algebra: D(E F ): DOS for both spin directions number of Cooper pairs: average energy gain per Cooper pair: compare to

51 4.2.2 Ground State Energy TT1-Chap4-59 condensation energy per volume: with average condensation energy per electron is of the order of k B T c 2 /E F

52 4.2.3 Energy Gap and Excitation Spectrum dispersion of excitations (quasiparticles) from the superconducting ground state: mixed states of electron- and hole-type single particle excitations (reason: single particle excitation with k can only exist if there is hole with k, if not, Cooper pair would form) 4 T = 0 3 E k / D k 2 1 break up of Cooper pair requires energy 2E k D represents energy gap for excitation from ground state, minimum excitation energy k / D k equal superposition of electron with wavevector k and hole with wavevector -k TT1-Chap4-60

53 4.2.3 Energy Gap and Excitation Spectrum D s / D n (E F ) D s / D n (E F ) density of states: - conservation of states on transition to sc state requires - close to E F : T = 0 (a) 4 (b) 4 Pb/MgO/Mg 3 3 D = 1.34 mev T = 0.33 K 2 2 I. Giaever, Phys. Rev. 126, 941 (1962) E k / D k E / D TT1-Chap4-61

54 4.3 Thermodynamics Quantities occupation probability of qp-excitation given by given by D, which is contained in entropy of electronic system, which is determined only by occupation probability, is fixed by D hole-like electron-like derive specific heat from expression for entropy: after some math results from redistribution of qp on available energy levels results from T-dependence of energy gap TT1-Chap4-62

55 4.3 Thermodynamics Quantities T << T c : we have use approximations, hence there are only a few thermally excited qp for exponential decrease of C s with decreasing T C s exp ( - D(0) / k B T ) TT1-Chap4-63

56 4.3 Thermodynamics Quantities T << T c : exp (-1.5 T c / T) exponential decrease of C s with decreasing T C s exp (-D(0)/k B T) C s / T c 0.1 Vanadium Zinn T c / T TT1-Chap4-65

57 4.3 Thermodynamics Quantities TT1-Chap < T/T c < 1: D(T) decreases rapidly, hence there is a rapid increase of the number of thermally excited qp C s is getting larger than C n T T c : D(T) 0, we therefore can replace E k by k normal state specific heat finite for T < T c, zero for T > T c jump of specific heat

58 4.3 Thermodynamics Quantities jump of specific heat at T c : we use and approximation further key prediction of BCS theory compare: (good agreement with experiment) Rutgers formula TT1-Chap4-68

59 molare Wärmekapazität (mj/ mol K) C / T (mj/k 2 mol) TT1-Chap Thermodynamics Quantities 3 2 Al C s DC N.E. Phillips, Phys. Rev. 114, 676 (1959) C n T (K) T c 8 Vanadium (B ext = 0) Vanadium (B ext > B cth ) T (K 2 ) M. A. Biondi et al., Rev. Mod. Phys. 30, (1958)

60 4.3 Thermodynamics Quantities TT1-Chap4-70

61 TT1-Chap Determination of the Energy Gap energy gap determines excitation spectrum of superconductor we can use quantities that depend on excitation spectrum to determine D 1. specific heat 2. tunneling spectroscopy 3. microwave and infrared absorption 4. ultrasound attenuation 5... we concentrate on tunneling spectroscopy (specific heat already discussed in previous subsection)

62 4.4.2 Tunneling Spectroscopy TT1-Chap4-72 tunneling of quasiparticle excitations between two superconductors separated by thin tunneling barrier SIS tunnel junction: I SC 1 SC 2 tunnel junction fabrication by thin film technology and patterning techniques by shadow masks ( mm) by optical lithography ( µm) sketch: R I top view: V SC 1 oxide (2nm) SC 2 substrate 1 µm...1mm

63 4.4.2 Tunneling Spectroscopy tunneling processes result in finite coupling of SC 1 and SC 2, decribed by tunneling Hamiltonian tunnel matrix element descirbes the creation of electron ks> in one SC and the annihilation of electron qs> in other SC tunneling into state k> only possible if pair state empty resulting tunneling probability for each state k> there exists a state k > with but with resulting tunneling probability total tunneling probability does not depend on coherence factors semiconductor model for qp tunneling TT1-Chap4-73

64 4.4.2 Tunneling Spectroscopy TT1-Chap4-74 elastic tunneling between two metals (NIN): net tunneling current: for ev << µ and µ E F we can use

65 4.4.2 Tunneling Spectroscopy I ns / (G nn /e) D(0) elastic tunneling between N and S (NIS): E µ 1 2D ev µ NIS D 1n N E S D 2s T > T c 0 T T c T = eu / D(0) µ 1 2D ev D 1n N S µ 2 D 2s TT1-Chap4-75

66 4.4.2 Tunneling Spectroscopy TT1-Chap4-77 differential tunneling conductance (NIS): d-function peaked at E = ev for T 0 determination of superconducting DOS

67 I ss / (G nn /e) D(0) Tunneling Spectroscopy elastic tunneling between S and S (SIS): E µ 1 2D 2D µ 2 4 SIS ev S S 3 D 1s E D 2s 2 µ 1 2D T > T c 1 0 T T c T = eu / D(0) ev D 1s S 2D S µ 2 D 2s TT1-Chap4-78

68 4.4.2 Tunneling Spectroscopy TT1-Chap4-80 interpretation for T = 0 single electron tunnels from left to right: before tunneling k k occupied and empty pair state after tunneling k k two quasiparticles energy balance: e moves from left to right generation of two qp required voltage: minimal voltage: = 2D for D 1 = D 2

69 4.4.2 Tunneling Spectroscopy D / D(0) Pb Sn In BCS theory 0.2 measured D(T) dependence I. Giaever, K. Megerle, Phys. Rev. 122, (1961) T / T c TT1-Chap4-81

70 4.4.2 Tunneling Spectroscopy TT1-Chap4-82 measured D(T) dependence:

71 4.4.2 Tunneling Spectroscopy TT1-Chap4-83 current-voltage characteristics at finite temperatures (pa.. A) I NIN T=0 2D(T) 2D 0 ev ( few mev)