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 � � = ���

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) Isotope effect – � ∝ � ∝ � 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 � ∆ = ������� ��������� � = �ℎ��� SC Gap & Transition Temperature

∆0 ∝ exp (�) ∆0 ∝ ��

∆0 ∝ 1/� ∆0 ∝ ln (�) Summary: Theory Superconductivity

London Theory Ginzburg-Landau Theory

Penetration depth � Penetration depth � Length scale for magnetic field

� Coherence length ξ Length scale for variation of � = superconducting order ��� Φ parameter |�| � = ns= number of “super electrons” 2��

BCS Theory: Key Results Macroscopic Wavefunction: � = Δ � ℏ� � = 2∆0 = 3.5�� �Δ

“super electrons” are Cooper Pairs BCS Theory – Predictive Power

BCS Theory: Key Results Macroscopic Wavefunction: � = Δ � ℏ� � = 2∆0 = 3.5�� �Δ

“super electrons” are Cooper Pairs

� = 1.13 ℏ�exp(-1/�)

2 � = �������� − �ℎ���� �������� �������� � [��� �� ]

� = 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. Resistivity and Hall effect experiments were carried out—using a commercial Quantum Design physical property measurement system (PPMS)—in magnetic fields up to 9 T. The magnetic field and temperature were stabilized before measuring. Reversal of the field direction was used to eliminate contributions originating from contact misalignment.

III. RESULTS

In Fig. 1 the longitudinal resistivity ρxx is shown as a function of temperature and magnetic field perpendicular to the film. The zero-field resistivity curve yields Tc 14.96 K, = defined by the temperature with the largest derivative dρxx/dT . Notice that the sharpness of the transition allows determination of Tc with 20 mK precision. When a magnetic field B is applied perpendicular to the film, the transition temperature is gradually suppressed, as indicated in the inset of Fig. 1. (b) Raw Hall resistivity (ρxy)isothermsareshowninFig.2(a). Well above the superconducting transition, the negative Hall response ρxy scales linearly with magnetic field strength. Essentially no magnetoresistance is observed in ρxx and RH ρxy/B.Thisisconsistentwithasinglebandpicture = where the Hall coefficient is given by RH 1/(ne). Our 23 = −3 film has a carrier density n 4.2 10 cm− and hence 2 1/3 1 = × kF (3π n) 2.3 A˚ − .Theelectronicmeanfreepath = 2 = ℓ hk¯ F/ne ρ 2.0 AconfirmsthatourNbNfilmbelongsto˚ = = the dirty regime with a Ioffe-Regel parameter kFℓ 4.6. Being in the dirty regime, we use the Werthamer-Helfand-Hohenberg= dBc2 Paraconductivity relation [32] Bc2(0) 0.69Tc to estimate the zero- = − dT temperature upper critical field Bc2(0) 18 T (see the inset of ≈ Fig. 1). This implies a zero-temperature coherence length ξ0 1/2 = (%0/[2πBc2(0)]) 43 A,˚ where %0 is the flux quantum. = As ourSUPERCONDUCTING measurements are FLUCTUATIONS taken near the superconductingIN A THIN NbN . . .FIG. 2. (a) Hall resistivity PHYSICALρxy isotherms REVIEW for T BT95c,15.3and,224501(2017) 20.0 K. The inset displays the high-field linear field≈ dependence of ρxy.(b)ThenonlinearHallresistance&ρxy obtained by subtracting (a) the linear high-field dependence for each of the respective isotherms. Solid lines are guides to the eye.

transition temperature where the coherence length ξ and the penetration depth [33]diverge,thesuperconductinglength scales are generally larger than the film thickness. Our system thus displays two-dimensional superconductivity whereas the electrons sense a three-dimensional environment due to their short mean free path. PHYSICAL REVIEW B 95,224501(2017) Hall effect isotherms taken near the superconducting transition Tc display a sign change from negative to positive Superconducting fluctuations in a thin NbN film probed by thevalues Hall effect at low magnetic fields [Fig. 2(a)]. This sign change is observed in a narrow temperature window of 0.3 K above Daniel Destraz,1,* Konstantin Ilin,2 Michael Siegel,2 Andreas Schilling,1 and Johan Chang1 (a) 1Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, SwitzerlandTc.Deviationsfromlinearlow-fielddependenceare,however, 2Institut für Mikro- und Nanoelektronische Systeme (IMS), Karlsruher Institut für Technologie, D-76131observed Karlsruhe, up Germany to 1Kabovethesuperconductingtransition. (Received 14 December 2016; published 2 June 2017) We thus analyze∼ the isotherms in terms of a negative normal FIG.We 1. present In-plane a comprehensive resistivity study of of a 119 how superconductingA˚ thin NbN film fluctuations as a function in the normal statestate contribute contribution to the ρxy B and a positive response with a non- conductivity tensor in a thin (119 A)˚ film of NbN. It is shown how these fluctuations drive a sign change in ∝ of temperature for magnetic fields in steps of 1 T. The magnetic field linear field dependence. To investigate the positive response, the Hall coefficient RH for low magnetic fields near the superconducting transition. The scaling behaviors as is applieda function perpendicular of distance to the to transition the filmϵ plane.ln(T/T Solid) of the lines longitudinal are guides (σ )andtransverse( to theσ negative)conductivity linear normal state component is subtracted, = c xx xy n theDaniel eye.are found The to upper beDestraz consistent critical withet al., field GaussianBc2( fluctuationT ) (shown theory. in the Moreover, inset) excellent is defined quantitativei.e., agreement&ρxy betweenρxy ρxy.AsshowninFig.2(b),thepositive theory and experiment is obtained without any adjustable parameters. Our experimental results thus provide= a − by the point with the steepest slope on the respective transitions. The Hall effect response &ρxy decreases rapidly with increasing PRB 2017 case study of the conductivity tensor originating from short-lived Cooper pairs. red line in the inset is a linear fit used to evaluate Bc2(0)—see the temperature. In fact, it vanishes below the detection limit about mainDOI: text10.1103/PhysRevB.95.224501 for further explanation. 1KaboveTc.

I. INTRODUCTION served as a model system224501-2 for studies of out-of-equilibrium dynamics of superconductivity [27–30]. Here, we use a NbN Superconducting fluctuations in the normal state have film just outside the pseudogap regime to carry out a combined long been known to influence bulk properties such as con- paraconductivity and Hall effect study of the normal state ductivity and magnetization. Generally, a stronger response superconducting fluctuations. The sign of the contribution to fluctuations is expected for lower dimensions [1]. Many from superconducting fluctuations to the Hall conductivity studies have therefore been carried out on thin films or is defined by the derivative κ d ln(T )/dµ,whereµ layered compounds, such as the high-temperature cuprate c is the chemical potential [23,31=].− For most conventional superconductors [2]. Of particular(b) interest are systems that (b) superconductors, including NbN, κ < 0, and hence the Hall host a Bardeen-Cooper-Schrieffer (BCS) regime to Bose- coefficient due to Gaussian fluctuations is expected to be Einstein condensate (BEC) crossovers [3,4]. The BCS regime positive [R (SC) > 0]. It is also known that charge transport is characterized by conventional Gaussian fluctuations [1,5,6] H in NbN films is governed by electronlike carriers [24], whereas phase fluctuations are expected in the BEC regime. FIG. 3. Hall conductivity due to superconductingwhich generate fluctuations. a negative normal state quasiparticle (QP) In the cuprates, paraconductivity, torque magnetization, and FIG. 4. (a) The low-field value νH,0 !σxy/B for (B 0) Hall coefficient [RH(QP) < 0]. In NbN, short-lived Cooper = → Nernst effect experiments—insideThe subtraction of the the pseudogap normal state phase— response is described in the text. pairs and quasiparticles are thus expected(left axis) to contribute and the with paraconductivity !σxx (right axis) as a function have been interpreted as evidence for phase fluctuations Isotherms of !σxy shown in (a) are comparedopposite to Gaussian sign to fluctuation the Hall effect. Nearof ϵ.T Dashed, but still lines within are the predicted dependencies from Gaussian of superconductivity [7–10]. The topic, however, remains c theory (solid lines) explained in the text. Thethesame normal data state represented (T>T), we indeed find a sign reversal controversial as the same techniques have also produced c fluctuation theory without any adjustable parameters— see the text of the Hall effect response. This sign reversal facilitates results consistentas withνH Gaussian!σxy fluctuation/B vs B theoryare shown [11–17 in]. To (b). For magnetic fields lower for a detailed explanation. (b) The ghost critical field as a function of = the disentanglement of the Hall signal from quasiparticles make progress,than one wayB∗ forward(indicated is to by study arrows), superconducting the isotherms of νH become constant and short-lived Cooper pairs and henceϵ, obtained enables us from to study the isotherms shown in Fig. 3.Thedashedline fluctuations of related systems. Recently, a pseudogap phase at values νH,0. For the isotherm at 15.0 Kthe the response constant from plateau superconducting is corresponds (SC) fluctuations to B to∗ the Bc2ϵ/2 with Bc2 18 T obtained from the has been identified in disordered films of NbN and TiN, and not reached at the lowest measurable fieldsHall and effect. hence Although the flat the line Hall conductivity $σ generated= = it has been conjectured that it stems from phase fluctuating resistivity dataxy shown in Fig. 1. The dotted line corresponds to the by SC fluctuations is generally highly nonlinear, it does superconductivityindicates [18–20]. a In lower this context,bound. Thecareful gray studies bars below the arrows show the mirror image of the red line in the inset of Fig. 1. Vertical error bars scale with magnetic field B in the limit B 0. Consistent of the normal stateestimated fluctuations uncertainty are called of for.B Recently,and all solid the lines are guides to the eye. ∗ with Gaussian fluctuation theory, $σcorrespond/B →ϵ 2 toscales the with gray bars below the arrows in Fig. 3.Horizontal sister compound TaN, for which no pseudogap has been xy − ϵ ln(T/T )thatforϵ 1isameasureofthedistanceerror bars∝ in (a) and (b) correspond to an uncertainty in T of 20 mK. identified, has been studied, and it was demonstrated that c c to= the superconducting transition≪ (T T )/T .Furthermore, superconducting fluctuations manifest themselves in the Hall c c from the normal state Hall conductivity− isotherms we extract coefficient [21,22]—consistent with predictions of Gaussian Next, to compare with theoretical predictions,aghostcriticalfield the contribu-B ϵ.This,combinedwithapara- fluctuation theory [23]. ∗ dependent. As shown in Fig. 4(a), νH,0 drops almost two conductivity that scales as∝$σ ϵ 1,makesaconvincing Due to its promisingtion from potential superconductivity for applications to such the as conductivity tensor is being xx − case for Gaussian fluctuations in NbN.∝ orders Furthermore, of magnitude excellent by heating just half a Kelvin above single-photon detection and hot-electron bolometers, NbN is isolated. As NbN displays essentiallyquantitative no magnetoresistance agreement between Gaussianthe superconducting fluctuation theory transition. The onset of this low-field one of the best characterized superconducting films. Both the n and ρxx ρxy,thenormalstateconductivityand the experimentσ is, is found in the without any adjustable parameters. normal state metallic and the superconducting properties have xx plateau defines a field scale B∗ that scales with ϵ ln(T/Tc) ≫ Ourn study therefore provides an experimental demonstration = been widely studiedtemperature [20,24–26]. regime Perhaps for of this interest, reason, given it has by σxx 1/ρxx(9 T). The [Fig. 4(b)]. of how= Gaussian fluctuations of superconductivity contribute paraconductivity, shown in Fig. 4(a),isthengivenby!σxx n to the conductivity tensor. = σxx σxx, where σxx 1/ρxx(0 T). The Hall conductivity * − = IV. DISCUSSION [email protected] to superconductivity is extracted in a similar fashion,II. METHODS n n n n 2 n 2 !σxy σxy σ ,whereσ ρ /[(ρ ) (ρ ) ] and Published by the American Physical Societyxy under the termsxy of the xy AthinfilmofNbN(xx xy Tc 14.96 K) wasWe grown now on a discuss sapphire !σxx, !σxy,andtheghostcriticalfield = − 2 2 = − + = Creative Commonsσxy Attributionρxy 4.0/[ Internationalρxx ρxy].license. In Fig. Further3, isothermssubstrate using of dc!σ reactivexy and magnetronB∗.Generally,theparaconductivity sputtering of a pure !σxx scales with the distribution of this work= must− maintain attribution+ to the author(s) Nb target in a− mixture of Ar and N gases. The average 2 νH !σxy/B for temperatures just above the superconducting correlation2 length ξ (T )thatdivergesasT Tc. Gaussian and the published article’s= title, journal citation, and DOI. thickness d 119(2) Awasmeasuredwithastylusprofiler.˚ → transition Tc are shown. In the limit B 0, νH=(B)saturates fluctuations lead to a power-law divergence of the corre- → 1/2 2469-9950/2017/95(22)/224501(6)at νH,0 and becomes essentially 224501-1 independent of magnetic Published field. by thelation American length Physicalξ Society(T ) ϵ .Bycontrast,phasefluctuations ∝ − The amplitude of the plateau (νH,0)isstronglytemperature are expected to have an exponentially diverging correlation

224501-3 LITERATURE CLUB - II

(1) 2015 Discoveries of superconductivity

Nature 525, 73–76 (2015) – 203 K Superconductivity in HS2 Nature Materials 14, 285–289 (2015) – 103 K Superc. in FeSe

(2) Anti-ferromagnetic excitations Phys. Rev. Lett. 108, 177003 (2012) – in Sr2IrO4 (RIXS) Phys. Rev. Lett. 105, 247001 (2010) – in La2CuO4 (Neutron scattering)

(3) Unconventional superconductivity Science 336, 1554-1557 (2012) – Penetration depth @ QCP Nature Physics 11, 17–20 (2015) – SC fluctuations in URu2Si2

(4) Ferromagnetism & Skymions Science 323, 915-919 (2009) – Skymions in reciprocal space (MnSi) Nature 465, 901–904 (17 June 2010) – Skymions in real space (Fe0.5Co0.5Si)