Transport and thermodynamic properties of high temperature superconductors (HTS) - an experimentalist's view (contd.)

Transcription

1 Transport and thermodynamic properties of high temperature superconductors (HTS) - an experimentalist's view (contd.) Powerpoint slides shown on Thursday 30 th April 2015 Notes on whiteboard Phase smearing factors for quantum oscillations as derived in Shoenberg s boo. Different inds of phase smearing all expressible in terms of Fourier transform, FT. FT of Fermi function gives R T slide 4 - thermal phase smearing. FT of Lorenztian - lifetime broadening - gives R D. Hence electron scattering rate but small angle scattering also effective. FT of 2 delta functions (spin splitting) - gives R S. But more simply from before: in magnetic field - spin up and spin down FS areas are slightly different, two values of area A and hence of dhva frequency F. Use identity sin(α+) + sin(α -) = 2sin α cos gives R s. Not affected by el-phonon interaction which does not alter spin susceptibility. Generally, torque = MxH Field independent anisotropic susceptibility = (1/2)(χ ab χ c )H 2 sin2 (when M= χh) Or general, possibly non-linear, quasi-2d case, = M c H sin where is angle between H and c axis (c is perpendicular to layers). M ab =0, M c = M c (Hcos ) [the c-axis magnetisation at a field Hcos ] Setch of FS reconstruction, slide 14, when triple unit cell in a and b directions. E.g. Q = (1/3,1/3, 0) 2π/a. Generates closed electron pocet from hole-lie FS arcs. Details for FS for OII YBCO and OD Tl-2201 not simply warped cylinder. Snae swallowed a chain shape. Relation between volume derivative of Helmholtz free energy and elastic constants (e.g. bul modulus) relevant for slide 19.

2 Theory of dhva effect D. Shoenberg, Magnetic oscillations in metals, Cambridge University Press, 1984 Allowed real space orbits (projected in plane perpendicular to B) contain integral number of flux quanta, h/e. Allowed -space orbits - Landau tubes areas(a) perpendicular to field B are quantised: a(, B ) = (n + )2eB/ electron energy, B projection of wave vector along B is constant since n is integer, constant = 1/2 for free electrons. d dt ev B Measured quantities, oscillate with frequency determined by extremal Fermi surface area, A. The lowest harmonic of the oscillatory magnetisation M osc is given by the Lifshitz-Kosevich (L-K) formula: And the oscillating torque by osc F M F 1 For quasi-cylindrical Fermi sheets the derivative is oscb dominated by the 1/cos dependence of A

3 Examples of Landau tubes for simple case of spherical Fermi surface and the more complicated case of an ellipsoidal one - from D. Shoenberg s boo. Extremal areas mared by blue arrows.

4 R D gives scattering rate 1/, R S gives Stoner enhancement factor of Pauli susceptibility. g is electronic g-factor, i is an integer, m S is enhanced by electron-electron interactions. So standard analysis procedure is: 1. Fourier transform osc vs. 1/B, get F i (). (Have to watch out for higher harmonics and combination frequencies caused by finite angular displacements of lever.) 2. Fit data to L-K formula over narrow field range to get R T at different temperatures T and hence m* values for various orbits. These are enhanced by a factor 1 +, and by possible electron-electron enhancement (*), over the band mass m B. Here is dimensionless electron-phonon enhancement factor, ( N(0)V in simplest form of BCS theory). 3. Obtain for each orbit by comparing m* with band-structure values m B. 2 da Here cyclotron band mass, m B 2 d (*) Shoenberg loc. cit., Wasserman and Springford, Adv. Phys. 45, 471 (1996)

5 Piezo-resistive cantilevers of type used in Atomic Force Microscopy (one method) Introduced for torque magnetometry by C. Rossel et al., J. Appl. Phys. 79, 8166 (1996) and in our lab. for measurements on single crystals of Tl 2 Ba 2 CuO 6+ C. Bergemann, A.W. Tyler, A.P. Macenzie, J.R. Cooper, S.R. Julian and D.E. Farrell, Phys. Rev. B57,14387 (1998), C. Bergemann (Ph.D thesis 1999). Boron implanted silicon lever approx. 150 x 50 x 3 m 3 resistance 2300 Ohms.

6 Use Wheatstone Bridge, currents 1-30 A, 77 Hz, loc-in detection. Sensitivity, bridge output noise typically 2 m, i.e. torque noise of 2 x Nm, or dyne-cms. (Limited by slow mechanical motion not by electrical noise, can be less noisy in vacuum). Torque = m x H. Very sensitive, especially at high fields, ~ 10-8 emu at 10T. Shown to be very effective for detection of dhva oscillations, i.e. Fermi surface studies, Sr 2 RuO 4 (Bergemann et al) and BEDT-TTF compound (Lupien et al). Particularly suitable for quasi-2d layered crystals, where often have m // c axis. Calibrate torque sensitivity, either from gravitational force or from shape anisotropy of superconductor in low fields, (for dhva wor need to now that it is T independent). Origin of MgB 2 wor, used same method for H irr (T) and M rev (T) of small single crystals of YBa 2 Cu 4 O 8 for comparison with thermal conductivity data (E.A. Yelland, Ph.D thesis, 2002) Then H c2 anisotropy of single crystals of MgB 2, prepared by high-pressure synthesis at ISTEC (Toyo). Later crystals also from J. Karpinsi and colleagues (ETH, Zurich) Cooled down to 1.4 K, 15T, very clear dhva oscillations at 3 frequencies! Initially two crystals : A 230 x 80 x 20 m 3 (Cambridge, K, 15T) B 230 x 200 x 40 m 3 (Bristol, K, 18T) National Magnet Lab. Tallahassee, (more dhva frequencies seen).

7 Picture of a x 0.08 x 0.02 mm 3 crystal of MgB 2 on a piezoresistive cantilever Crystal A From E.A. Yelland, PhD thesis 2002

8 E.A. Yelland et al., PRL 88, (2002)

9 MgB 2 Fermi Surface with dhva orbits J. Kortus et al, PRL 86, 4656 (2001) From A. Carrington et al. Physica C 456, (2007)

10 MgB 2 From A. Carrington et al. Physica C 456, (2007)

11 From: Doiron-Leyraud et al. Nature 447, 565 (2007)

12 B. Vignolle et al. Nature 455, 952 (2008)

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14 Later piezolever data in static fields up to 45 T Average m*/m e = 5.2+/-0.4 agrees well with heat capacity. Comparison with ARPES overall band narrowing.

15 Figures from review by: B. Vignolle et al., C. R. Physique 12 (2011) Quantum oscillations for YBa 2 Cu 3 O 7- TEP when superconductivity suppressed by large magnetic field Hall coefficient of YBa 2 Cu 3 O 7-d at high fields Possible phase diagram? 15

16 High energy X-ray diffraction Incommensurate CDW competing with superconductivity in YBa 2 Cu 3 O 6.67 CDW nesting vector (Q vector) Probable location of electron pocets 16

17 sqrt x10 Fit used Counts CDW delta 135K no PG CDW delta 270 no PG CDW delta 135 PG 340 CDW delta 270 PG (mj/g.at K 2 ) T(K) T(K) JRC (2012) Calculated changes in electronic entropy (S) caused by a CDW gap (a) in a flat quasi-2d band, and (b) for a density of states with a triangular pseudogap centred on Fermi energy. =C/T=dS/dT Case (b) much closer to experimental data for quenched polycrystalline UD67 YBCO Provisionally conclude that CDW develops out of PG state but presently repeating C/T measurements on single crystals with wellordered CuO chains 17

18 Photo of heat capacity sample holder showing two 31 mg YBCO crystals on 8x5 mm 2 sapphire plates connected by thermopiles to each other and to base of heat shield via a copper ring. E. Cavanna, (PhD student), I. Koanovic, JRC and J.W. Loram.

19 Bounding the pseudogap with a line of phase transitions in YBa 2 Cu 3 O 6+ A. Shehter et al Nature 498, 75 (2013) Pseudogap in YBa 2 Cu 3 O 6+ is not bounded by a line of phase transitions: Thermodynamic evidence. Phys. Rev. B (R) (2014) Blue squares neutron diffraction, red circles, resonant ultrasound, purple diamonds Kerr rotation. 19

20 E.M. Forgan et al cond_mat

21 Example of recent STM wor, Davis group.

22 Example of recent STM wor, Davis group.

23 Example of recent STM wor, Davis group

24 Important to compare STM and ARPES Fermi surface/fermi arcs data for Bi-2212 with bul measurements such as heat capacity and as shown here, London penetration depth from W. Anuool et al. Phys. Rev. B 80, (2009)

25 From Damascelli, Hussain and Shen, Rev. Mod. Phys. 75, 473 (2003)

26 Recent ARPES data for Bi-2212 Vishi et al., Proc. Nat. Acad. Sci., 109, 1831 (2012)

27 Recent ARPES data for Bi-2212 Vishi et al., Proc. Nat. Acad. Sci., 109, 1831 (2012)

28 Recent ARPES data for Bi-2212, Vishi et al., Proc. Nat. Acad. Sci., 109, 1831 (2012) cos( x )-cos( y ) is d-wave formula expected for actual Fermi surface. For cylindrical Fermi surface, cos( x )-cos( y ) = cos(cosφ) cos(sinφ) very similar to simple d-wave form Δ(φ) = Δ 0 cos(2φ) OD80 (p=0.205) is where slope of gap vs. cos( x )-cos( y ) at low T starts to fall from its value for UD crystals and also where gap vs. cos( x )-cos( y ) at low T becomes linear. OD65 (p= 0.22) is where gap is zero above T c and increases linearly with cos( x )-cos( y ) at low T.

29 Bi-2212, from Vishi et al., Proc. Nat. Acad. Sci., 109, 1831 (2012)

30 BCS theory: electrons attract via positive ions. QM lly phonon exchange. Lower energy if electrons near FS form Cooper pairs. 0 compare 0 ) ( FS i BCS c c c c v e u F E ) ( E u ) ( E v ' ' ' ' 2 V E BCS self-consistent gap equation (T=0): If V is a constant then gap parameter constant, s-wave superconductor i i c e c c e c v u v u New quasi-particles in s/c state + q

31 In classical superconductors, single particle tunnelling provides excellent confirmation of BCS theory. T is the matrix element for tunnelling from one side of the barrier to the other, varies as exp(-x), similar to overlap integral in tight binding theory. Fermi s golden rule: Prob./unit time 2 T 2 N(E), I 2T 2 e N R (E ev)n L (E)[ f R (E ev)-f L (E)]dE 7.6 Fig. 8.4 (Waldram) Densities of states, occupation and I-V tunnel characteristics at a low but finite temperature. (a) NIN, (b) SIN, (c) SIS. At low temperature di/dv of an SIN junction gives the DOS directly: I(V) 2 2T en R ev N 0 L (E)dE 7.7

32 Hence can obtain DOS directly from SIN curves. Also by using Eqn. 7.6 can determine (T). From I. Giaever and K. Megerle, Phys. Rev. 122, 1101 (1961) Other properties of classical superconductors such as microwave and far-infra red absorption, ultrasonic attenuation, thermal conductivity and NMR are also in good agreement with BCS theory. DOS vs. energy for Pb. Short dashed line, wea-coupling BCS, long dashed line experimental data, solid line full calculation. There are systematic deviations from BCS weacoupling theory for SIN tunnelling for certain strong-coupling superconductors such as Pb and Hg, where N(E F )V is no longer small (V is the attractive interaction between electrons). These can be analysed to obtain the product of the electronphonon interaction a 2 (w) and the phonon DOS F(w). This provides some of the most compelling evidence that s/c is induced by the electron-phonon interaction as shown on the next slide.

33 Upper part: Phonon density of states, F(w) in Pb as determined by neutron scattering. Lower part: the product a 2 (w) F(w) for Pb determined by analysing experimental tunnelling data. The striing similarity of the two plots is strong evidence in favour of superconductivity being caused by the electron-phonon interaction.

34 Intrinsic tunnelling of Bi-2212 mesas From T.M. Benseman et al. arxiv: v2

35 From T.M. Benseman et al. arxiv: v2

36 From T.M. Benseman et al. arxiv: v2

37 From T.M. Benseman et al. arxiv: v2

38 From T.M. Benseman et al. arxiv: v2

39 Brief overview of some recent papers on pnictide superconductors

40 Iron-based superconductors

41 BaFe2(As1-xPx)2

42 A. Carrington Cambridge seminar Dec. 2014

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44 Mass increase at AF QCP in CeRhIn5

45 P. Walmsley et al PRL 110, (2013)

46 I R Fisher, L Degiorgi and Z X Shen, Rep. Prog. Phys. 74, (2011)

47 Kuo et al, PRL,112, (2014)

48 I R Fisher, L Degiorgi and Z X Shen, Rep. Prog. Phys. 74, (2011)

49 Electronic specific heat of Ba 1 x K x Fe 2 As 2 from 2 to 380 K J.G. Storey et al., Phys. Rev. B 88, (2013)

50

51 Electronic specific heat of Ba 1 x K x Fe 2 As 2 from 2 to 380 K J.G. Storey et al., Phys. Rev. B 88, (2013)

52 Electronic specific heat of Ba 1 x K x Fe 2 As 2 from 2 to 380 K J.G. Storey et al., Phys. Rev. B 88, (2013)

53 Electronic specific heat of Ba 1 x K x Fe 2 As 2 from 2 to 380 K J.G. Storey et al., Phys. Rev. B 88, (2013)

54 Gaussian fluctuations and Nernst effect in hole doped cuprates

55 The Nernst effect above T c, Gaussian fluctuations or vortices (e.g. 2D Kosterlitz-Thouless-Berezinsi effects)? R.P. Huebener, Supercond. Sci. & Technol. 8, , (1995) 55

56 The vortex Nernst effect in a type II superconductor the circles represent vortices. Contours of constant (nv/k-t) Y. Wang, L. Li and N. P. Ong, Phys. Rev. B (2006) and earlier papers in Nature 56

57 I. Koanovic, JRC and M. Matusia, PRL, 102, (2009) Representative data 57

58 58 ab c Above T c s/c regions appear and disappear on time scale GL Free energy 2D ) ( ) ( 3D ) ( ) ( 2 3 T s T F T F T T F T F B s n B s n Superconducting Fluctuations ( c) B T T... ) ( 4 2 b T T a F F c n s Boltzmann factor ] / ) ( [ 2 T T T a Exp B c called Gaussian fluctuations s For introduction see e.g. Statistical Mechanics of Phase Transitions J.M. Yeomans Oxford University Press (1992)

59 Gaussian (wea) fluctuation formula Ussishin, Sondhi and Huse PRL,89, (2002) ab = 0 ab/(t/t c -1) 1/2 0 ab = ab / c, and for Ca (0.05) 0 ab and agree with earlier wor for YBCO. E.g. J.W. Loram et al. Phil. Mag. 65,1405 (1992) Lines fits to Gaussian formula 59

60 Electronic heat capacity C v /T for La 2-x Sr x CuO 4. x in% C from London penetration depth (low T) and room temperature resistivity anisotropy. ab (0) from analysis of C v. fluc B c ( ) 2 ab ( T Tc ) s (2 c ( T T ) 2 Typical fit to Gaussian formula for c fluc 60

61 No need to invoe special vortex physics to account for Nernst effect. Gaussian fluctuations plus measured changes in s(t) with x give trends observed. Evidence for above type of T*(p) line is wea. 61

62 Overall conclusions. 1. Cuprate superconductors studied by unprecedented number of research groups with tremendous variety of experimental techniques. 2. Basic transport properties show systematic patterns as the hole doping is increased. Also of practical importance because cannot tell strength of s/c simply from T c. They also led to the detection of quantum oscillations. 3. More empirical lins needed between these properties and sophisticated experiments such as STM, ARPES, neutron scattering and XRD. Perhaps they could be the cement for a more unified picture. 4. Intrinsic tunnelling is a bul probe and potentially powerful. May provide clear test regarding spin resonance mode as pairing boson but needs considerable theoretical input. 5. Pseudogap still mysterious. Comes in sharply around p = 0.19 at low temperature. But we still believe it arises from an energy scale not a well-defined temperature. 6. Future experiments? Heat capacity and magnetisation at high magnetic fields. CDW under pressure (simple test does plateau in T c vs x for YBCO x go away?) Magnetic impurities - ditto? Uniaxial stress e.g. does it introduce ab plane resistivity anisotropy in Bi-2212? High field transport of Bi-2212 crystals? Thermopower under pressure as probe of PG.