Characterization Techniques. M. Eisterer Atominstitut, TU Wien Stadionallee 2, 1020 Vienna, Austria

Transcription

1 Characterization Techniques M. Eisterer Atominstitut, TU Wien Stadionallee 2, 1020 Vienna, Austria EASIschool, September 7 th 2018

2 Outline Quantities of interest Intrinsic material parameters: T c, l, x; Extrinsic property: J c (flux pinning) Measurement Techniques Resistivity Magnetization Some others

3 Intrinsic Properties Transition Temperature, T c BCS: phonon frequency, DOS, coupling Unconventional SC: T c ~1/l 2 (empirical) Tuning normally difficult, e.g. pressure, doping Highest in clean materials: scattering is pair breaking for anisotropic energy gap (or multi-gap), only second order effects in s-wave SC

4 Intrinsic Properties Magnetic penetration depth, l Length scale for field changes in a superconductor Superfluid density: ~1/l 2 Increases with impurity scattering Coherence length, x Length scale for changes of the superconducting order parameter Size of a Cooper pair (clean limit) Decreases with impurity scattering Other intrinsic sc parameters can be calculated Condensation energy (E c ), critical fields (H c, H c1, H c2 ), depairing current density.

5 Importance of coherence length Upper critical field: B c2 = φ 0 2πξ 2 Efficient pinning centers should have a radius of about ξ (orthogonal to field). A small ξ potentially harms the inter-grain coupling A small ξ increases the condensation energy μ 0 H c 2 A small ξ favors high critical currents. Depairing current density J d = φ 0 3 3πμ 0 ξλ 2 2 : H c = φ 0 2π 2μ 0 λξ Maximum achievable J c by optimized pinning landscape: ~0.2 J d

6 Determination of coherence length From upper critical field: B c2 = φ 0 2πξ 2 Resistivity measurements Magnetization measurements Specific heat measurements Scanning Tunneling Microscopy (STM): Size of the vortex core Small angle neutron scattering

7 Importance of magnetic penetration depth Lower critical field: B c1 = φ 0 2πλ 2 ln(κ) A small l is a manifestation of a large superfluid density: n s = m e μ 0 λ 2 e 2 Condensation energy μ 0 H c 2 2 : H c = Depairing current density J d = φ 0 2π 2μ 0 λξ φ 0 3 3πμ 0 ξλ 2 (impacts on J c) Surface impedance: Z s = R s + ix s, X s = μ 0 ωλ Unconventional SC (cuprates, IBS): T c ~1/l 2

8 Determination of magnetic penetration depth From Meissner susceptibility From lower critical field: B c1 = φ 0 2πλ 2 ln(κ) Reversible magnetization B c from area below rev. magnetization curve: B c 2 Fit of (available parts of) M(H) to theory. Magnetic Force Microscopy (MFM): Field of flux-lines, surface currents Small-angle neutron scattering Muon spin rotation (μsr) Surface impedance, e.g., cavity perturbation method: change of Q of superconducting cavity 2μ 0

9 Ginzburg-Landau Theory Two important parameters: 1) Coherence length x: variation length of superconducting order parameter. 2) Magnetic penetration depth l: variation length of magnetic field. Ginzburg-Landau parameter κ λ ξ κ < 1 : type-i superconductors 2 κ > 1 : type-ii superconductors 2 GL-theory is in principle restricted to temperatures close to T c. Magnetic behavior does not change qualitatively at low temperatures.

10 Relation to BCS and London Theory BCS coherence length ξ 0 = ħv F πδ(0) ( size of a cooper pair ) ξ T = 0.74 ξ 0 l ξ T = t ξ 0 1 t (clean limit: ξ l) (dirty limit: ξ l mean free path) London penetration depth λ L 0 = λ T = 1 2 λ L (0) 1 t λ T = λ L (T) ξ l m e = 3 μ 0 n s e 2 2μ 0 e 2 v 2 F N 0 (clean limit) (dirty limit)

11 Extrinsic property: Critical current density J c Intrinsic: depairing current density J d = J c is given by flux pinning: J c = ηj d Extrinsic: Pinning efficiency 0 η 0.2 φ 0 3 3πμ 0 ξλ 2 Pinning centers: crystal defects, grain boundaries Pinning engineering: (artificial pinning) nano-inclusions

12 Measurements of Transition Temperature The superconductor should not be disturbed too much by the measurement: applied fields and currents etc. should be small. Most frequently used techniques: Resistive (I c ) Magnetic (H c1, H c, H c2 ) Specific heat (bulk probe!) Possible issues: surface effects, material inhomogeneities, thermal gradients

13 Jump in the specific heat at T c Raw data Difference to normal state behavior Jump is broadened by inhomogeneities T c -distribution: C. Senatore et al., 2007 IEEE Trans. Appl. Supercond

14 Resistive transitions Four point measurements: separate current and voltage contacts: Current reversal to get rid of non-dissipative components (thermal voltage, chemical potentials) Arbitrary evaluation criteria: 90 %, 50%, 10% (irreversibility line), tangent criteria, zero resistivity The real onset may be masked by a highly conductive sheath. Representative for the bulk?

15 MAGNETIZATION MEISSNER STATE

16 Perfect Diamagnetism Meissner state Infinite sample: B = μ 0 H + M = 0 H = M

17 AC Susceptiblity (SQUID) m (arb. un.) χ = M H eff ' (in-phase component) '' (out-of-phase component) Temperature (K) Applied ac-field: H 0 sin ωt Magnetization in Meissner state: H 0 sin ωt 1 D In-phase component of ac suceptibility (fourrier component of sin ωt ): χ = 1 Out-of-phase component (fourrier component of cos ωt ): χ = 0 Cose to T c : H c1 < H 0 Mixed state dissipation: χ > 0 (loss peak) DC susceptibility: χ = χ zero field cooled (zfc), H 0 <<H c1 χ < χ field cooled (fc) (flux pinning) Advantage of ac technique: Better resolution at the same H 0.

18 Perfect Diamagnetism Field enhancement caused by return-flux

19 Perfect Diamagnetism real type-i superconductor infinite geometry Effectively applied magnetic field below (1-D)H c : H eff = Ha + DM M = H eff H eff = H a (1 D) (along the equator of spheroids) m = DHaV, Meissner slope dm dha is universally DV.

20 Demagnetization factors D b=a b=a/2 b=a/10 D=1- h/2d Numerical calculation for cuboids a b h, a>b e.g. Chen et al., IEEE Trans. Magn. 41 (2005) 2077 Asymptotic behavior for thin disks (d h): D=1- h/2d a/h a/h*2a/(a+b), 4d/h 0.01 Empirical: a h d h 4d P ; d: larger lateral dimension, P: perimeter Field at the edges diverges with h 0. H eff = H a (1 D)

21 Thin plate of lead (vector magnetometer) magnetic moment (Am 2 ) 4.0x x x x x10-4 D ~0 parallel moment m H orthogonal moment m H H 0 M sf q H 0 sin(q) M sf H 0 cos(q) -8.0x10-4 D ~ angle between surface and applied magnetic field M sf = H 0 cos θ 1 D M sf = H 0 sin θ 1 D M H 0 = M sf cos θ + M sf sin(θ) M H 0 = M sf sin θ + M sf cos(θ) M H 0 = H 0 cos 2 θ 1 D H 0 sin 2 θ 1 D M H 0 = H 0 cos θ sin θ 1 1 D D

22 Shielding Fraction Experimental assessment of Meissner slope dm dha Calculation of shielded volume: V shielding = 1 D V shielding Shielding fraction: η = V sample h<1:sample is not entirely superconducting: cracks, secondary phases, etc. h=1:sample may be OK. Shielding fraction may differ from superconducting volume fraction! E.g. hollow sphere, powder Problematic: Thin samples: D is very sensitive to sample thickness Sample dimensions comparable to or smaller than magnetic penetration depth l.

23 Perfect Diamagnetism? B=0 only in the bulk of the superconductor. B penetrates the superconductor in a surface layer with thickness of the order of l. If l is not much smaller than the smallest sample dimension, h is reduced. Meissner susceptibility can be used to determine the magnetic penetration depth.

24 Meissner susceptibility MgB 2 powder MgB 2 bulk (cube) ac-susceptibility -0.1 ac-susceptibility Temperature (K) Temperature (K) Low shielding fraction Temperature dependence of susceptibility down to low temperatures Inter-grain coupling? Shielding fraction close to one below T c. Susceptibility hardly depends on temperature below 30 K. Material inhomogeneities or inter-grain (de-)coupling manifest near T c.

25 Meissner susceptibility Sintered Sm-1111 bulks 0.0 ac susceptibility, ' K 35 K sample LIC sample HIC B(T) Grains decouple with field in sample LIC

26 Thin plate parallel to field χ = M H 0 Krusin-Elbaum et al., PRL 62 (1989) 217 M = M 0 1 2λ h tanh h 2λ, M 0 = DH 0 Empirical two-fluid model: λ T = λ 0 1 t 4, t = T T c

27 Determination of l from Meissner susceptibility: experimental issues -1 at low temperature (unless sample is very thin): high experimental resolution is needed. D and V (superconducting sample volume) have to be known very accurately. Fit of D and l(0) to (T) for known temperature dependence of l: l(t) is influenced by strong coupling, symmetry of pairing, multi-bands, impurity scattering. Inhomogeneities influence (T) close to T c. Powder samples: geometry of grains, inter-grain coupling. Very thin samples: Misalignment between sample and applied field. λ T λ T λ 0 can be determined more easily/reliably.

28 Two-coil mutual inductance technique S. J. Turneaure et al. JAP 83 (1998) 4334

29 Self-Oscillating Tunnel-Diode Resonator (TDR) L C Resonance frequency f = 1 2π LC (e.g 14 MHz) Superconductor in coil changes L Change of resonance frequency Infinite slab (volume V s, width 2w): Δf f = V s 2V c 1 λ w tanh(w λ ) (V c : Volume of inductor) e.g. R. Prozorov and V. G. Kogan, RPP 74 (2011)

30 Summary: Meissner susceptibility M=-H eff only if λ R. (Hence, never near T c!) λ = λ(t) χ χ(t) Possibility for the determination of l. The demagnetization effect increases the effectively applied field (quite significantly in thin samples): H eff >H 0 is reduced by granularity (T) is changed by inhomogeneities (in particular near T c ).

31 MAGNETIZATION MIXED STATE

32 Vortices in type-ii superconductors real superconductor infinite geometry Vortices: normal conducting core (radius ~x), circular currents produce magnetic flux quantum φ 0.

33 0 M (T) Ginzburg-Landau Theory H H H c c1 c2 = 2 = = 2 0 ln 2 4 l x lx = 2 H c = H (T) Many important parameters can be obtained from the reversible magnetization curve. Unfortunately, flux pinning normally inhibits its measurement.

34 Magnetization measurements of H c2 1) M(T) at fixed H 0 T c H H c2 (T) T c (1 T) 2) M(H) at fixed T 0 H c2 (15 K)

35 Magnetization measurements of H c2 Possible issues: 1) Magnetic background (normal state properties, sample holder, vibrations in VSM) 2) Surface contamination (r n H c2 ) 3) Surface superconductivity H c3 =1.69(?)H c2 (resistive measurements) 4) Fluctuations, Ginzburg number

36 Estimation of large H c2 Large H c2 : superconductors with large. London theory is a good approximation for H<0.5H c2 : M lnh = φ 0 8πμ 0 λ 2 Plot m vs. ln(h). Extrapolate to m=0 to obtain H c2 (underestimation) Discussion of error: E. H. Brandt, Phys. Rev. B 68, (2003). Alternativ: Fit to alternative models for M(H)

37 Resistive transitions Arbitrary evaluation criteria: 90 %, 50%, 10% (irreversibility line, B irr ) T c B B c2 (T) (also specific heat) Cuprates: B c2 >>B irr

38 Magnetic determination of thermodynamic critical field Usually impeded by pinning! If pinning is weak, the reversible magnetic moment can be obtained by m rev = 1 2 m + + m m + (m ): magentic moment on increasing (decreasing) field branch

39 Summary: Mixed State Two important parameters: l, x. Determination of x from H c2. l s a very important parameter but hard to assess. Flux pinning normally impedes the determination of the thermodynamic (or reversible) magnetization curve.

40 FLUX PINNING CRITICAL CURRENTS

41 Critical State Model Flux pinning balances the Lorentz force: F L = J c0 B = Fmax p Current (or pinning) causes field gradients: B = μ 0 J Current results in a magnetic moment: m = 1 2 න r Jd3 r Bean Model: Current density in a superconductor is either 0 or J c. The direction of the current depends on geometry and history. If I < I c, J c does not flow in the entire sample volume.

42 Thermally activated depinning ν = ν 0 e U 0 k B T depinning rate U J = U 0 1 ± J J c0 Net forward jump rate: activation barrier J U 0 (1 e J ) c0 E J = 2ρ c J c0 e U 0 k B T sinh JU 0 J c0 k B T J U 0 1+ k B T e J c0 k B T = 2e U 0 k B T sinh JU 0 J c0 k B T 1) E J = ρ c J c0 e U 0 U 0 J k BT ek B T J c0, J Jc0 flux creep U 0 U 0 2) E J = 2ρ c k B T e k B T J =: ρ TAFF J, J J c0 thermally assisted flux flow J c J E crit < J c0

43 Thermally activated depinning U 0 =50k B T Flux Creep E/r c J c TAFF J/J c0 J c J E crit < J c0 The electric field criterion, E crit, is chosen arbitrarily or imposed by the experiment.

44 Thermally activated depinning E 10-5 krit (E c ) 3 flux flow E/r c J c TAFF U 0 /k B T=10 U 0 /k B T=50 E/r c J c x10-4 J/J c flux creep J/J c 8x10-5 E/r c J c 6x10-5 4x10-5 TAFF J J c0 exp c ( B, T ) J ( E, B, T ) = J ( E c ) J c0 2x J/J c

45 Typical electric field criteria Experimental E c (µv/cm) Depending on technique Resistive transport Your choice, sample size SQUID Sample size, J c, S (material, field, temperature) VSM (sweep mode) Sample size, field sweep rate VSM (step mode) Sample size, material, field, temperature AC-susceptibility <10-5 ->1 Sample size, amplitude and frequency, J c Scanning Hall probe Sample size, material, field, temperature Cuboid in a VSM: U E = 2(a + b) = 1 dφ 2 a + b dt = ab db 2 a + b dt Field ramp rate SQUID and SHP measurements: Electric field given by the relaxation of the magnetization.

46 SQUID vs. Resistive J c (A/m 2 ) Transport (1µV/cm) SQUID 77 K µ 0 H(T) Huge difference at high magnetic fields!

47 Assessment of J(E) in a wide range E = E c J J c n Combination of different techniques

48 CRITICAL CURRENT TRANSPORT MEASUREMENTS

49 I-V curves U Textbook I c I

50 I-V curves Textbook Kim -Anderson U U 0 =50k B T Flux Creep E/r c J c I c I TAFF J/J c0 Empirical: E = E c J J c n n-value (large n is desired)

51 J c -evaluation J = J = c exp c J ( E c ) Common electric field criteria: E c =0.1 or 1 µv/cm Simple intersection with I-V curve n J Fit of I-V curve by E = E c J c, n J c Slope of I-V curve in double logarithmic representation: n

52 Experimental Issues Heating/thermal voltages Strain: thermal expansion, Lorentz force Thermal instabilities: quench Transfer length (linear component) (minimum distance between current and voltage taps)

53 CRITICAL CURRENT MAGNETIC MEASUREMENTS

54 Bean Model m = 1 2 න r Jd3 r = J c 1 2 න r e Jd 3 r

55 Bean Model: Formulae Relation between J c and the Geometry irreversible component of m Cylinder of radius R and height h, axis parallel to H. Cylinder of radius R and height h, axis normal to H, h>2r J c = J c = 3m irr πr 3 h 3m irr 4R 3 h(1 3πRΤ16h) Cuboid with dimensions a b c, H parallel to c, a b. Cuboid with dimensions a b c, H parallel to c, b a. J c = J c = 4m irr ab 2 c(1 bτ3a) 4m irr a 2 bc(1 aτ3b) Sphere of radius R. J c = 8m irr π 2 R 4 For the fully penetrated state (all current loops have the same orientation)

56 Different Symmetry of Reversible and Irreversible Magnetization m = m irr + mrev m (arb. un.) m irr+ H = m irr H m irr+ H = m irr H m irr H = m H m + H 2 = m H + m + H 2 0 H (arb. un.) mrev H = mrev H mrev H = m H + m + H 2

57 Self field correction J c J c (H) J c = J c B! B = μ 0 H 0 + M rev B = μ 0 J + B self B norm Influence of aspect ratio on self-field d/h=1 d/h=0.1 d/h=10 d/h=1000 d/h= r/r B Infinite cylinder: z linear field profile, slope proportional to J c. r = μ 0J φ Calculation of self-field is normally based on the Biot-Savart law: H = H r = 0 = J c h 2 ln h2 + d 2 + d h Rough estimate: H = J c a 2 a: smallest sample dimension B self r = න J r r r 3 d3 r = J c න e J r r r 3 d3 r

58 Self-field correction Shift of measured points along x-axis: m(h) m(b) H B av B av = 1 N න V μ 0 (H 0 +M rev ) + B self r r 2 d 3 r m (arb. un.) J c = J c B! Iterative calculation: 1) Calculate J c (H) and M rev (H). 2) Calculate B av from B self (H,J c (H)) and M rev (H). 3) Calculate J c (B av ), M rev (B av ) from m + (B av ) and m - (B av ). 4) Re-iterate B av from B self (B av,j c (B av )), and M rev (B av ). 5) Continue with 3) until B av remains constant for all m. 6) J c (B)=J c (B av ) H(T) e.g. M. Zehetmayer, Phys. Rev. B 80 (2009)

59 Self-field correction M. Zehetmayer Phys. Rev. B 80 (2009) J c not assessable for small fields (B<~B * ) with transport nor magnetization measurements.

60 Sloppy Treatment in Literature J c (H)! Flux jumps H* not taken into account properly

61 Surface barriers/pinning m (arb. un.) Strongly asymmetric magnetization loop m (arb. un.) m + -m - /2 (m - +m + )/ H(T) Decomposition in reversible and irreversible moment Weak bulk pinning or decoupled grains (powder) Distinction of surface effects from significant contribution of M rev often difficult: Self field correction (including demagnetization). Estimation of M rev (H c1 ) from l and x (if known).

62 Granularity So far: perfectly homogeneous sample. Limit of sample consisting of decoupled (cubic) grains: Sum of contribution of all grains. Jc intra = N 6m intra irr 4 a g = 6m irr a 3 ag a g : grain size N = a 3 ag 3 : number of grains Weakly coupled grains: m irr = m intra irr + mirr inter m inter irr m intra = a irr ag Jc inter J c intra (cubic sample and grains) If this ratio is either small or large, Jc intra or Jc inter can be derived approximately.

63 Granularity (Scanning Hall Probe Microscopiy, SHPM) Local Jc intra and Jc inter y (mm) x (mm) B (T) Calculation of local currents by numerical inversion of the Biot- Savart law.

64 Granularity (SHPM)

65 Granularity (SHPM) Average Jc intra and Jc inter can be derived J. Hecher et al., SUST 29 (2016)

66 Irreversibility field 1.5x x10-3 m (Am 2 ) -1.0x x10-4 B irr -2.0x10-6 m (Am 2 ) x µ 0 H (T) -1.0x Transport (1µV/cm) SQUID -1.5x µ 0 H (T) J c (A/m 2 ) µ 0 H(T) Field where the magnetization curve becomes reversible J c 0. Reality: J(B, E crit ) < J crit

67 MAGNETOMETERS

68 Superconducting Quantum Interference Device (SQUID) The sample is moved through a pick-up coil system. The net flux is coupled to a SQUID sensor. Fit of the signal (U(z)) to the theoretical curve Typical resolution limit: Am 2 in zero field Am 2 at a few Tesla Even better in the VSM mode Second order gradiometer

69 Vibrating Sample Magnetometer (VSM) First order gradiometer or Mallison coil set. Sample vibrates and induces a voltage: U dφ dt = d dt ඵ B z x, y, z z 0 dxdy = dφ z z 0 dt = dφ z z 0 dz 0 dz 0 dt = Aω dφ z dz Typical resolution: 10-9 to 10-8 Am 2 Faster than a SQUID Field sweep mode Mallison coil set