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The coherence conundrum in BEDT-TTF superconductors; does interlayer transport die as temperature rises?. Paul Goddard , John Singleton , Arzhang Ardavan , Ross McDonald, Stan Tozer, Alimamy Bangura, John Schlueter

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  1. The coherence conundrum in BEDT-TTF superconductors; does interlayer transport die as temperature rises? Paul Goddard, John Singleton, Arzhang Ardavan, Ross McDonald, Stan Tozer,Alimamy Bangura, John Schlueter National High Magnetic Field Laboratory, Los Alamos National Laboratory, NM 87545, USA The Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK NHMFL Tallahassee, 1800 E Paul Dirac Dr, Tallahassee MSD, Argonne National Laboratory, Argonne, IL 60439 Supported by DoE, NSF, State of Florida and EPSRC (UK).

  2. Running order • What is interlayer coherence, and how might we make a system incoherent? • Model system to study: -(BEDT-TTF)2Cu(NCS)2. • Angle-dependent magnetoresistance oscillations (AMROs): how do we measure them? What do they mean? • Crank up the temperature: what do we destroy and how? • Summary. (For further reading: excellent background information on incoherence issues is given by McKenzie and Moses (Phys. Rev. B 1999, 2004), Osada (Physica E, 2003), Yakovenko (cond-mat, 2005), Lebed and Naughton, Phys. Rev. Lett. (2004).)

  3. Interlayer incoherence: the simplistic view (See Phys. Rev. Lett. 88 037001 (2002) and refs. therein.)

  4. Interlayer incoherence: disorder and temperature We showed that Weger’s criterion does not give a good representation of the situation at low T in BEDT-TTF salts; they have a 3D Fermi surface [Singleton, Phys. Rev. Lett. (2002), Goddard, Phys.Rev. B (2004)]. What about temperature?

  5. Fermi surface consists of a pair of Q1D sheets and a Q2D hole pocket; perspective view from Goddard et al., PRB (2004).

  6. Experimental: angle-dependent magnetoresistance oscillations (AMROs) • 4 probe measurement of interlayer resistance Rzz, proportional to rzz; • variable T: pumped 3He (0.5 K) - 50 K; • 2 axis rotator allows sample orientation to be changed continuously in magnetic field (high P also possible). Measurements performed at the NHMFL in Tallahassee: fields of up to 33 T (Bitter) or 45 T (Hybrid)

  7. Experiment involves rotating sample in a steady magnetic field; magnetoresistance is determined by detailed 3D Fermi-surface topology. -(BEDT-TTF)2Cu(NCS)2 T = 0.49 K We have a great deal of information here [see Phys. Rev. B 69, 174509 (2004)]: Shubnikov-de Haas oscillations from the  Q2D pocket and the  breakdown orbit; angle-dependent magnetoresistance oscillations (AMROs); the SQUIT (Suppression of QUasiparticle Interlayer Transport) when the field lies exactly in the sample’s conducting planes => definitive signature of a 3D Fermi surface [see Phys. Rev. Lett. 88, 037001 (2002)].

  8. Typical AMRO experiment; rotate sample in field for many  (rotation planes). B = 42 T (NHMFL hybrid magnet); T = 520 mK (Goddard, PRB 2004). Q2D Q1D Enlargement of SQUIT region for 3 different planes of rotation. -(BEDT-TTF)2Cu(NCS)2

  9. What causes AMROs and SQUITs? Consider evolution of quasiparticle orbits on Fermi surface due to field B [semiclassical interpretation- Blundell (1996)]. The origin of Q1D AMROs (“Lebed” oscillations) B B Quasiparticles are driven across Fermi surface by Lorentz force: velocities “rock” from side to side, decreasing the quasiparticle’s contribution to conductivity; along the axis of corrugation, velocities do not rock; conductivity increases. Magnetoresistance is determined by evolution of quasiparticle velocities (arrows); e.g. Chamber’s eqn.(vel.-vel. correlation fcn.):

  10. Yamaji oscillations The SQUIT The SQUIT occurs in exactly in-plane B: if the Fermi surface is 3D, orbits possible on the side. Two cases in -(BEDT-TTF)2Cu(NCS)2: B B These involve orbits about the Q2D Fermi-surface sections. At certain angles, periodic in tan, all orbits have same area; at such angles, the interlayer component of the velocity averages to 0;  peak in zz. (Goddard, Phys. Rev B (2004),)

  11. Fit SQUIT’s  dependence  interlayer transfer integral for -(BEDT-TTF)2Cu(NCS)2 [Goddard et al., PRB 04] We obtain ta= 0.065 meV or about 0.5 K. This is small- can easily access Anderson’s criterion using temperature.

  12. What might happen to the AMRO if system goes incoherent? If Fermi surface loses interlayer dispersion, we expect smooth variation of resistance, with either a maximum or a minimum for in-plane field (90o) (Osada, 2003). The changing shape of the MR will be a guide as to when we lose coherence.

  13. Raising the temperature, T B = 45 T (NHMFL Hybrid Magnet) Plane of rotation  = 90 degrees: AMRO from the 2D Fermi surface section (SQUIT plus “Yamaji” oscillations). As T is raised from 5.3 K to 40 K; the SdH oscs fade away; the AMRO fade away; and we are left with a broad peak, but NOT at 90 degrees. 45 T -(BEDT-TTF)2Cu(NCS)2

  14. = 90o (2D Fermi surface AMRO): fate of Yamaji oscillations as T rises. Q2D AMRO (Yamaji) fade: they have virtually gone by19.6 K and certainly gone by 24.5 K. Higher index AMRO (those closer to 90o) vanish more quickly than low-index ones. 45 T Increasing index 45 T Amp. of a selected Yamaji Osc. versus T.

  15. What does the SQUIT do as the temperature is raised? As T rises, the SQUIT remains the same angular width; slowly loses amplitude. It is just visible at 19.6 K (below) but not at 24.6 K. Expansion of lower T data close to 90o. SQUIT 45 T -(BEDT-TTF)2Cu(NCS)2

  16. SQUIT clearly survives to higher T than Yamaji oscillations in data recorded at 30 T (similar sample orientation): B = 30 T, Q2D AMRO SQUIT is there at 16.5 Kwhereas Yamajis have gone by 12.0 K.

  17. Change plane of rotation of field: AMRO due to quasi-1D Fermi-surface sections:  = 160o As in the case of the Q2D AMRO, the Q1D oscillations decrease with increasing T, until approx. sinusoidal dependence remains. Yet again, this is not centered on 90o. Higher index AMROs fade quicker with increasing T.  = 160o, B = 45 T -(BEDT-TTF)2Cu(NCS)2

  18. SQUIT In this plane of rotation, the SQUIT is also due to the Q1D Fermi surface sections. However, as before, it fades slowly with increasing T, remaining perceptible until 13 K. SQUIT DKC  = 160o, B = 45 T Over the same temperature range, most of the other AMRO features (Lebed-like and DKC) disappear.

  19. As the temperature is raised, the Q2D AMRO (Yamaji oscillations) and Q1D AMRO (“Lebed oscillations”) fade. The larger the index, the faster the oscillations fade. -(BEDT-TTF)2Cu(NCS)2 model Fermi surface 1) The Yamaji index increases in magnitude as  increases. As field tips further, each successive orbit meeting the Yamaji condition has a larger circumference. 2)Average k-space velocity will also decline with increasing tilt angle. Both 1) and 2) combine to increase the orbit transit time and thus decrease orbit angular frequency . 3) Criterion for observation of AMRO will be  > 1. 4)  high index AMROs, with slower , will be killed off more easily by scattering.

  20. What’s scattering the electrons? Phonons! • For small AMROs, amplitude   (Pippard limit- see Blundell 1996). • For each particular Yamaji oscillation  will be a constant; •  the amplitude will be determined by the temperature dependence of . 45 T data AMRO amplitude  1/T 5: looks like phonon scattering (see any old text book, e.g. mine) kills the AMRO. Similar conclusions apply to Q1D AMRO.

  21. In spite of the apparent “fragility” of the mechanism for the SQUIT, it survives to temperatures at least as high or higher than the Yamaji and “Lebed” AMRO. The SQUIT occurs in exactly in-plane B: if the Fermi surface is 3D, orbits possible on the side. Two cases in -(BEDT-TTF)2Cu(NCS)2: The SQUIT is caused by a large number of “Russian doll” orbits, each of which will have a different , and therefore decay with T at a different rate. As scattering renders successive orbits ineffective, there will always be another, smaller orbit still doing the job. SQUIT is observable until these become a negligible fraction of the velocity averaging. B B

  22. What might happen to the AMRO if system goes incoherent? We have killed the Yamaji oscillations and the SQUIT using phonons. Can we drive the system incoherent by going to still higher temperatures? If Fermi surface loses interlayer dispersion, we expect smooth variation of resistance, with either a maximum or a minimum for in-plane field (90o) (Osada, 1999).

  23. At higher temperatures, magnetoresistance becomes smoothly varying with , but peaks are not centred on 90o  = 90o: orbits on Q2D Fermi surface 45 T 45 T Note, however, that peak gets closer to 90o as T increases.

  24. At high T, for general planes of rotation: • MR is almost sine-like in ; • MR peak  is function of . This happens in other planes of rotation too:  = 160o, B = 45 T • At higher T, MR is: • almost sine-like in ; • NOT centred on 90o; • gets closer to 90o as T gets larger. 30 T, 30 K

  25. Tight-binding bandstructure; effective dimer model plus interlayer transfer integrals; Structure is monoclinic; transfer integral between layers lies along the a axis, at 110.7o to the conducting bc planes (Goddard, PRB 04). Even though the transfer integral is only ~0.5 K, the electrons “know” it is there and in which direction it points at 40 K.  MR peak position.

  26. Summary: AMRO in -(BEDT-TTF)2Cu(NCS)2 at high T As the temperature is raised, the Q2D AMRO (Yamaji) and Q1D AMRO (“Lebed”) fade. The larger the index, the faster they fade. Can understand in terms of k-space path length and phonon scattering ( T 5). The SQUIT survives to temperatures at least as high or higher than the Yamaji and “Lebed” AMRO. This is due to the nature of the orbits: a “Russian doll” collection with a range of transit times, some very short. For T > 25 K, the magnetoresistance becomes smoothly varying in , but NOT in general peaked at 90o (in-plane field). The peak tends towards 90o as the temperature rises further. This can be understood in terms of the 3D bandstructure of an anisotropic monoclinic system. In spite of the fact that kBT ~ 80 ta, all of these results can be understood in terms of a coherent system with a 3D bandstructure.

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