Superconductivity at high magnetic field

Transcription

1 Superconductivity at high magnetic field How can we achieve superconductivity at very high magnetic fields? What sort of materials should we choose to look at? Theory - introduction to superconductivity for the non-specialist - the central equation ORDINARY SUPERCONDUCTORS Orbital and Paramagnetic limitation Larkin-Ovchinnikov-Fulde-Ferrel modulated state EXTRAORDINARY SUPERCONDUCTORS Beating the Paramagnetic limit Beating the orbital limit 1 Andrew Huxley Scottish Universities Physics Alliance University of Edinburgh

2 plan [1] How to detect superconductivity at high field [2] Theoretical background for calculating critical magnetic field for superconductivity [3] Pauli limit [4] Orbital limit [5] Modulated states [6] Break down of quasi-classical approximation and superconductivity in very high fields Scottish Universities Physics Alliance 2 Andrew Huxley University of Edinburgh

3 Introduction Some high field properties of superconductors 3

4 A simple superconductor: CeRu2 Heat capacity 4

5 CeRu 2 CeRu2 simple, but not so simple... magnetisation Phase Diagram illusion of rich phase diagram Talapurker et al Physica C

6 High Temperature Superconductors YBCO H//c H c H. Nakagawa et al. / Physica B 246±247 (1998) 429<ETH>432 liquid LOFF From magneto-resistance orbital only para only J. L. O Brien et al. PRB (2000) 6

7 High Tc s Underneath Hc2 A Clean High Tc SC Image: C. Marcenat CEA-Grenoble 7

8 Example of data for FLL melting 16 a 1 12!s/k B [C(H) C(0)]T 1 (mj mol 1 K 2 ) T (K) T (K) Example of thermodynamic signature for FLL melting from Bouquet et al Nature (2001) : Fields are in Tesla 8

9 Clean-Dirty-Moving FLL in high Tc s Clean Dirty Moving Image: C. Marcenat CEA-Grenoble 9

10 what is best way to measure Hc2? standard techniques resistivity (difficulty if melting) ac susceptibility (difficult if melting) magnetisation (pinning & small signal if κ>>1) heat capacity (difficult if large background - high Tc) less standard techniques Nernst coefficient (sensitive to fluctuations) 10

11 What is the best way to measure Hc2? dt dx Vy eg. Nernst effect which is sensitive to presence of vortices is also sensitive to superconducting fluctuations Nernst Coeff. resistance Amorphous films of Nb.15Si.85 a b (µv K 1 T 1 ) ν R square (Ω) T (K) 0 1 T 0.25 T B = 0 T 0 T 2 T S tan θ 2,000 T c = 0.38 K T (K) A. Pourret et al Nature Physics c

12 Measurement of Hc2 conclusion Different techniques for different situations 12

13 part II Theoretical background for calculation of upper critical field 13

14 condensation - bosons macroscopic occupation of a quantum state off diagonal long range order (rigidity of wave function) 14

15 fermions NOT possible multiple occupation of a single state is not possible for fermions. ie we cannot factor the expectation value of a product of two fermion operators (into numbers), a consequence of fermion anti-commutation. 15

16 fermions: 16

17 Warm up problem Gap equation 17

18 Central Equation for superconductivity For T=0 ie GAP equation with normal state electron energy eg k 2 /2m 18

19 method 1 Variational wavefunction translational invariance singlet pairing ck are constants to be optimized further reading Quantum Liquids by A.J.Leggett Oxford Graduate Texts

20 method II: standard many body approach want 20

21 standard many body approach (k+q/2) s (k+q/2) s (k 1 +q/2) s = = V V V V... (-k+q/2) s (-k+q/2) s (-k 1 +q/2) s (k+q/2) s (-k+q/2) s = (k+q/2) s Key = =! (-k+q/2) s (k+q/2) s V (-k+q/2) s = (k+q/2) (k s 1 +q/2) s! = = V! (-k+q/2) s (-k 1 +q/2) s is potential scattering 21

22 Gap function (k+q/2) s (-k+q/2) s k 1 (k+q/2) s! = = V! (-k+q/2) s at Tc k = (iωn,k) ωn = π(2n+1)kbt 22

23 Gap equation in zero field for simplicity we will work at q=0 23

24 Calculation of Tc at Tc Math expanding contour to infinity in above 24

25 Calculation of Tc potential isotropic 25

26 Central Equation for superconductivity and in real space assuming spin independent isotropic short range interaction 26

27 End of warm up Now for magnetic fields Effect on electron spin Effect on orbital motion 27

28 Effect of magnetic field (I) Paramagnetic Effects (II) Orbital Effects flux-quantum 28

29 Part III Paramagnetic limit 29

30 SPIN : Pauli limitation B B 30

31 magnetisatic polarisation energy Pauli Limitation Clogston Chandrasekhar limit = condensation energy BCS g=2 for free electrons for g=2 31

32 Pauli limited - Gap equation Note effect is to shift ωn by igμb 32

33 Pauli Limiting only 1 st order Hc2/Tc[T/K] calculation assuming 2 nd order transition T/Tc 33

34 Pauli limitation in practice Clogston Chandraseka limit increased by: spin-orbit scattering from impurities self-energy interactions (eg electron-phonon, but not necessarily contributing to strong coupling) strong coupling 34

35 V3Si: Pauli limiting at work Hp 30 T dirty clean dirty clean Hp lower because Tc lower and (dhc2/dt)tc larger Orlando PRB (1979) Hc2 vs T Hc2/(dHc2/dT)Tc vs T/Tc 35

36 Clever ways to get to high fields H.W.Meul et al. PRL in 1984 B/Tc 5 B [Tesla] Eu 0.75 Sn 0.25 Mo 6 S 7.2 Se 0.8 B exchange field Eu H=B+<B ex > Eu moment aligned // B Exchange field cancels applied field B/T c 1 superconductivity Eu M B=0,<B ex >=0 Eu moment random orientation No net exchange field T [K] B ex 36

37 A More complex example of compensating fields 2D organic salt T. Konoike et al PRB 9514 (2004) shading is result of simple theory for paramagnet J-P effect including Canted Anti-ferromagnetism explains data 37

38 What is really needed to get to high fields? Pair Equal Spins 38

39 UPt3: An example of parallel spin pairing 2.5 normal phase 2 Magnetic Field (Tesla) C A + > 0.5 B Temperature (Kelvin) 39

40 UPt3: An example of symmetric spin pairing paramagnetic limitation for H//d no para. limitation for H d S=1 order parameter d-vector gives direction of zero spin projection 40

41 Equal spin pairing UPt3 (& also Sr2RuO4) have parallel spin pairing + > What about equal-spin pairing > and > How do we get this? 41

42 [ ] Look at interactions between electrons spin spin Opposite spins occupying the same site are strongly repelled leaving a +ve charge. Same spin electrons are attracted = Exchange Interaction origin of magnetism Hex =Jij σi.σj 42

43 In what metals will these interactions be strongest? Want strong exchange interaction - also give magnetic order Want interaction to be long range as found close to a continuous (2 nd order) transition Want above at low temperatures Fluctuations close to a magnetic transition become long range 43

44 URhGe: Quantum critical points direction of magnetic moment QCP s µ 0 H c [Tesla] st order transition lines µ 0 H [Tesla] b

45 Apply field along hard axis Lévy et al Nature Physics 2007 (& this afternoon) H // a [Tesla] URhGe superconductivity T c H//b [Tesla] T c magnetic transition 45

46 Part IV Orbital limit 46

47 Physical picture of orbital limit field induces circulation of electrons that form Cooper pairs B F vf centripetal force = vf e B E slope ħvf Δ q 1/ξ0 47

48 Physical picture of orbital limit field induces circulation of electrons that form Cooper pairs Orbit area = size over which wavefunction can change = πξ 2 area density of orbits available 1/ πξ 2 Each orbit can accommodate one quantum of flux, Φ0 Bc2 = Φ0 / πξ 2 Exact Result is Bc2 = Φ0 / 2πξ 2 48

49 microscopic derivation of orbital limit Central Equation for superconductivity and in real space We need real space formula since even for uniform field A is not translationally invariant eg A=Bz(0,x,0) Orbital limit given simply by replacing G with G in field 49

50 Orbital limit Usual Abrikosov-Gor kov theory assumes Semi-classical approximation (Gor kov 1958) assumes field is slowly varying with respect to wave functions. 50

51 Real space Green s function Math eg see Helfand & Werthamer PRL (1964) 51

52 Semi-classical approx n for Greens function in a magnetic field units of flux (Stokes) cgs SI 52

53 Orbital limit Recall in zero magnetic field Becomes in magnetic field - a fancy way of writing a Taylor expansion 53

54 Orbital limit want to express A term similarly substitute 54

55 Orbital limit use infinite order differential equation 55

56 Orbital limit 56

57 Orbital Limit z term separates from only get terms with same number of π+ and π- 57

58 Orbital limit for uniform field this is Harmonic Oscillator Problem Lowest Landau level solution gives highest T Can solve this equation to give Hc2[T] 58

59 Hc2[T]/slope at Tc vs T/Tc is universal Hc2[T]/Tc(dHc2/dT)Tc T/Tc Hc2[0]=0.73 (dhc2/dt)tc Tc 59

60 orbit & paramagnetic limiting include paramagnetic limitation plus many other effects impurity scattering spin orbit strong coupling... 60

61 Part V modulated LOFF states 61

62 orbit & paramagnetic limiting + modulated states include paramagnetic limitation interference with modulation term Can stabilise modulation when paramagnetic term is strong and scattering weak 62

63 LOFF state Fermi surface Fermi surface Q cannot pair time reversed states to give singlet - > Shift Fermi surface by Q Can now pair reversed states at least for some directions 63

64 LOFF state helped if nesting Fermi surface Fermi surface Q cannot pair time reversed states Shift Fermi surface by Q Can now pair reversed states over most of surface 64

65 Larkin Ovchinnikov Fulde Ferrel state 2 nd order Hc2/Tc[T/K] LOFF 1 st order transition T/Tc 65

66 FFLO Gruenberg & Gunther PRL 16, 996 (1966) 66

67 Beyond the simple LOFF - mix higher Landau levels FFLO extreme limit H - mix higher Landau levels T/T C A.1. Butdin. J.P. Brison/Physics Letters A 218 (1996)

68 FFLO state in κ-(bedt-ttf)2cu(ncs)2? R. Lortz, A. Demuer, Y. Wang SEE TALK TOMORROW by Albin Demeur 68

69 Part VI Breakdown of classical limit and superconductivity in very high fields! Theory only 69

70 High magnetic fields - Assumption of semi-classical approximation E Landau Levels different kz clean T[K] >> 1.34 H[Tesla] or 70 dirty

71 Failure of semi-classical approximation Semi classical approx n valid when number of electron Landau levels occupied at H is large: n(h=hc2[0]) (Ef/Tc) 2 >>1 typically well obeyed in most metals corrections to Gorkov s phase approximation give rise to small quantum oscillations of Hc2 Semi-classical approx n fails for small Fermi surface (but this also means strong coupling) Rasolt & Tešanović Rev. Mod. Phys (1992) 71

72 Breakdown of semi-classical approx n Effect of electron orbit quantisation Lowest Landau Level only H>HQL HQL=Hc2[0] (Ef/Tc) 2 deviations below T*=Tc0(Tc0/Ef) SC suppressed here by dirt Rasolt & Tešanović Rev. Mod. Phys (1992) 72

73 For a singlet state QL+LOFF Fermi-surface is essentially one dimensional therefore LOFF state is much more efficient 73

74 The end Conclusion Lots of interesting new things potentially to be discovered, particularly at very high magnetic fields 74