Superconductivity - Overview Last week (20-21.11.2017) This week (27-28.11.2017) Classification of - Theory Summary Superconductors - Josephson Effect - Paraconductivity Reading tasks Reading tasks Kittel: Kittel: Chapter: Superconductivity Chapter: Superconductivity Next - Next week (4-5.11.2017) 13.11 Guest Lecture: Marta Gibert Oxide electronics 14.11 Guest Lecture: Christof Aegerter Multiple scattering of light Superconductivity – “Course strategy” This course -- Experimental phenomenology -- London Theory -- Ginzburg-Landau “Theory” Condensed Matter Theory Course Spring 2018 – Prof. Titus Neupert Bardeen-Cooper-Schrieffer (BCS) theory 10 Superconductivity 285 From Kittel Summary London Theory Penetration depth � Type I1 superconductor 6 � � = 5 �1�3� ns= “super electrons” Figure 18 Variation of the magnetic field and en- ergy gap parameter A(x) at the interface of super- conducting and normal regions, for type I and type I1 superconductors. The energy gap parameter is a measure of the stabilization energy density of Ginzburg-Landau Theory 0 the soperconducting state. The field will extend out from the normal core a distance h into the super- conducting environment. The flux thus associated with a single core is d2HC1, Penetration depth � Length scale for magnetic field and this must be equal to the flux quantum a,, defined by (27). Thus HC1= @&rh2 . (30) Coherence length ξ Length scale for variation of superconducting order parameter |This is the field for nucleation of a single fluxoid.� | At H,, the fluxoids are packed together as tightly as possible, consistent with the preservation of the superconducting state. This means as densely as the coherence length 5 will allow. The external field penetrates the specimen almost uniformly, with small ripples on the scale of the fluxoid lattice. Each core is responsible for carrying a flux of the order of ?rt2H,, which also is quantized to @,. Thus gives the upper critical field. The larger the ratio All, the larger is the ratio of H,, to Hcl. From Kittel 10 Superconductivity 285 Summary Type I1 superconductor Figure 18 Variation of the magnetic field and en- ergy gap parameter A(x) at the interface of super- conducting and normal regions, for type I and type I1 superconductors. The energy gap parameter is a measure of the stabilization energy density of 0 the soperconducting state. The field will extend out from the normal core a distance h into the super- Ginzburg-Landau Theory conducting environment. The flux thus associated with a single core is d2HC1, and this must be equal to the flux quantum a,, defined by (27). Thus Penetration depth � Length scale for magnetic field HC1= @&rh2 . (30) This is the field for nucleation of a single fluxoid. Coherence length ξ Length scale for variation of superconducting order parameter |At H,, the fluxoids are packed together�| as tightly as possible, consistent with the preservation of the superconducting state. This means as densely as the coherence length 5 will allow. The external field penetrates the specimen almost uniformly, with small ripples on the scale of the fluxoid lattice. Each core is responsible for carrying a flux of the order of ?rt2H,, which also is quantized to @,. Thus gives the upper critical field. The larger the ratio All, the larger is the ratio of H,, to Hcl. Superconductivity – Classification Low-Tc Superc. High-Tc Superc. Tc < 30 K Tc > 30 K Type 1 Type 2 � < � � > � Clean Superconductor Dirty Superconductor Electron mean free path ℓ≫� Electron mean free path ℓ ≪ � Conventional Unconventional BCS theory Beyond BCS theory Type –II Superconductors Vortex lattice vs Vortex Liquid VORTEX LATTICE MELTING AND Hc2 IN ... PHYSICAL REVIEW B 86,174501(2012) 200 50 180 45 YBa2Cu3O6.45 160 40 YBa2Cu3O6.56 140 35 YBa2Cu3O6.67 120 30 milliohms 100 Tesla 25 80 m 20 B 60 15 Resistance 40 10 20 5 0 0 https://0 2 journals.aps.org4 6 8 10 12 14 16 18 20/prb22 24 /p26 28 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Magnetic Field Tesla df/10.1103/PhysRevB.86.1745 Reduced Temperature T Tc 30 120 YBa2Cu3O6.75 25 01 100 YBa2Cu3O6.86 Vortex Tesla 20 YBa2Cu3O6.92 80 Solid 15 Tesla 60 m B 10 40 Vortex Melting Field 5 20 0 0 0 5 10 15 20 25 30 35 40 45 50 55 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Temperature Kelvin Reduced Temperature T Tc FIG. 2. (Color online) Top: A magnified plot of the resistive https://journals.aps.org/prb/pdf/10.1103/PhysRevB.86.174501 FIG. 3. (Color online) The vortex lattice melting transition for transitions shown above in Fig. 1. The red dots are where the several different oxygen concentrations. The temperature axis has resistance is 1/100 of its value at 60 T (extrapolated for high been scaled by T , and the lines are best-fit lines to Eq. (2). All of temperatures where resistance was not measured to the highest fields). c the data points were acquired in the same manner as described in the Bottom: The same data points as highlighted in red in the top panel, caption of Fig. 2. now plotted as a function of temperature. The black line is a fit to Eq. (2), using the known parameters given in Table I,andgives µ0Hc2(0) 28 0.3T,andcL 0.37. values for the penetration depth were used when the exact = ± = doping values were not available. The penetration depth values and the interpolation are shown in the upper panel of Fig. 4. YBa Cu O from oxygen content 6.45 to 6.92, with T s 2 3 y c With λc, λab,andTc experimentally determined, the data ranging from 44.5 to 93.5 K, and identify trends that arise at each doping can be fit using only two parameters: cL and as a function of doping. Characteristic curves for several other Hc2(0). The fits in the lower panel of Fig. 2 and in Fig. 3 clearly dopings are shown in Fig. 3,allwithanupwardscurvature, show that three-dimensional vortex melting describes the in- although that shape becomes less pronounced for the higher T c field resistive transition in YBa2Cu3Oy from y 6.45 to 6.92. = samples. The penetration depth anisotropy, γ λc ,changesfrom 50 Equation (2) can be expanded about T and solved for B = λab ∼ c m at 6.45 to 16 at 6.92; this results in decreased curvature of the as shown in Blatter et al.,13 but if the full temperature range melting line∼ as oxygen content (and hole doping) increases. from 1.5 K to T is to be used then it is more accurate to c This is the same behavior seen in several different cuprates fit to the full implicit expression for B .TheuseofEq.(2) m of varying anisotropy, reported in Ando et al.6 The c and requires both the in-plane and out-of-plane zero-temperature L H (0) values extracted this way are given in Table I for all of penetration depths, as well as the T :thesevaluesarealso c2 c the dopings measured. The fact that the Lindemann number listed along with the hole doping (estimated using Liang remains relatively constant as a function of doping means that et al.20) in Table I.Thein-planepenetrationdepthvalues,λ , ab the shape of the melting curve is determined primarily by come from electron-spin resonance (ESR) measurements21 the penetration depths, which are becoming less anisotropic and from muon-spin rotation experiments,22 both performed as hole doping increases. The Lindemann number and the on comparable YBa2Cu3Oy crystals grown at the University c2 penetration depths appear only as the ratio L in Eq. (2), and of British Columbia (UBC). In the case of the ESR values, λabλc the geometric mean of λ and λ was taken. Out-of-plane we plot this ratio in the lower panel of Fig. 4.Theincrease a b 2 cL penetration depth values, λc,comefrominfraredreflectance of with hole doping is what is controlling the changing λabλc measurements,23 also performed on UBC crystals. Interpolated curvature as a function of doping. With this parameter setting 174501-3 Superconducting Magnets Ideal: Large upper critical field Hc2 Large upper critical current Strong vortex pinning (avoidance of vortex liquid) GH Isotope effect – �C ∝ �E ∝ � The smoking gun experiment � ∼ 0.5 TD Debye temperature M = atomic mass A. Bussmann-Holder and H. Keller Journal of Physics: Condensed Matter, Volume 24, Number 23 http://ee.sharif.edu/~varahram/hts-course/coop.htm BCS-Theory of Superconductivity Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) � = Δ �GOP ∆ = ������� ��������� � = �ℎ��� SC Gap & Transition Temperature ∆0 ∝ exp (�C) ∆0 ∝ �a�C ∆0 ∝ 1/�C ∆0 ∝ ln (�C) Summary: Theory Superconductivity London Theory Ginzburg-Landau Theory Penetration depth � Penetration depth � Length scale for magnetic field 6 � Coherence length ξ Length scale for variation of � = 5 superconducting order �1�3� Φ1 parameter |�| �C5 = ns= number of “super electrons” 2��5 BCS Theory: Key Results GOP Macroscopic Wavefunction: � = Δ � ℏ� � = j 2∆0 = 3.5�a�C �Δ “super electrons” are Cooper Pairs BCS Theory – Predictive Power BCS Theory: Key Results GOP Macroscopic Wavefunction: � = Δ � ℏ� � = j 2∆0 = 3.5�a�C �Δ “super electrons” are Cooper Pairs �C = 1.13 ℏ�Eexp(-1/�) 2 � = �������� − �ℎ���� �������� �������� � [��� �� ] �E = Debye Frequency Two-band superconductors Example of multi-sheet Fermi surface doi:10.1209/0295-5075/82/47011 Josephson Effect https://mappingignorance.org/2015/04/30/ho w-to-measure-tiny-temperature-differences- using-a-josephson-junction/ Superconducting Quantum Interference Device (SQUID) https://www.asme.org/engineering-topics/articles/ bioengineering/a-mini-sensor-for-brain-scanning http://hyperphysics.phy- astr.gsu.edu/hbase/Solids/Squid.html Paraconductivity Published in: Junyi Ge; Shixun Cao; Shujuan Yuan; Baojuan Kang; Jincang Zhang; Journal of Applied Physics 2010, 108, DOI: 10.1063/1.3481096 DESTRAZ, ILIN, SIEGEL, SCHILLING, AND CHANG PHYSICAL REVIEW B 95,224501(2017) Six gold contacts with Hall bar geometry were deposited onto (a) the film.