Near Surface Vortex Lattice in NbSe2 studied with low energy beta - NMR by Dong Wang A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University Of British Columbia August, 2006 © Dong Wang , 2006 Abstract I n t h i s thes is , the m a g n e t i c f ie ld d i s t r i b u t i o n i n the v o r t e x s ta te of NbSe2 is inves t i ga ted w i t h the t e chn ique of d e p t h reso lved /3-detected N M R us i ng a b e a m of h i g h l y p o l a r i z e d 8Li p r o d u c e d at the T R I U M F I S A C fac i l i t y . T h e /3-NMR l i neshape is a d i rec t measure of the l o ca l m a g n e t i c f i e ld d i s t r i b u -t i o n we igh t ed a c c o r d i n g t o the i m p l a n t a t i o n prof i l e of the 8Li b e a m . B y v a r y i n g the i m p l a n t a t i o n energy be tween 1 ~ 3 0 k e V , one c a n c o n t r o l the av -erage i m p l a n t a t i o n d e p t h co r responds i n a range be tween 5 ~ 1 3 6 n m . A b o v e Tc = 7.OK a r e l a t i v e l y n a r r o w resonance is observed whose w i d t h is a t t r i b u t e d to m a g n e t i c d i p o l a r b r o a d e n i n g f r o m the 93Nb nuc l ea r m o m e n t s . B e l o w Tc a m u c h b roade r a s y m m e t r i c l i neshape is observed , w h i c h is cha rac t e r i s t i c of a t r i a n g u l a r m a g n e t i c v o r t e x l a t t i ce . M o d e l i n g the m a g n e t i c field d i s t r i b u t i o n a l lows one to d e t e r m i n e b o t h the effect ive i n p l ane p e n e t r a t i o n d e p t h Ay a n d the effect ive i n p l ane coherence l eng th £|| or v o r t e x core r ad iu s . . In a m a g n e t i c field of 3 0 2 m T , we o b t a i n Ay = 2 7 9 ( 3 0 ) n m a n d £|| = 1 2 ( l ) n m . In a sma l l e r m a g n e t i c field of 1 0 . 8 4 m T , the effect ive coherence l e n g t h i n -creases d r a m a t i c a l l y t o a va lue of 77 (10 )nm. T h i s is more t h a n a n order of m a g n i t u d e la rger t h a n the expec t ed f r o m the coherence l e n g t h i n NbSe2-T h e o r i g i n of the g iant vor t i ces is d i scussed . W e propose a n e x p l a n a t i o n i n v o l v i n g the m u l t i b a n d n a t u r e of NbSe2. i i Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements x 1 Introduction 1 1.1 Superconductivity and the Vortex Lattice 1 1.2 /3-NMR 5 1.3 Niobium diselenide NbSe2 9 2 Theory 11 2.1 London equations and Ginzburg and Landau equations . . . . 11 2.2 Modified London equation 15 2.2.1 sLi+ stopping distributions PE(Z) 19 2.2.2 Cutoff factors 20 3 Experimental 24 3.1 Production of 8Li 24 3.2 Polarizer [19, 28, 39] 25 3.3 Spectrometer [28] 30 iii Table of Contents 4 Results and Analysis 34 4.1 Results in the normal state 34 4.2 Results in the vortex state 35 4.2.1 Results at B=302 mT , T=3.5 K 35 4.2.2 Results at B=10.84 mT , T=3.5 K 36 4.3 Discussion and Conclusion 37 Bibliography 44 iv List of Tables 1.1 E x a m p l e s of i sotopes su i t ab l e for (3NMR 8 1.2 Cohe rence l e n g t h £ a n d p e n e t r a t i o n d e p t h A of 2H — NbSe^ m e a s u r e d b y va r ious expe r imen t s . X\\ a n d £y are pa rame te r s i n the a-b p lane , a n d Ax a n d £j_ are pa r ame te r s p e r p e n d i c u l a r to the a-b p l ane 10 4.1 T h e f i t t ed pa rame te r s w i t h three d i f ferent cutof fs are s u m m a -r i z e d . T h e G a u s s i a n cuto f f has the lowest % 2 35 v List of Figures 1.1 The mean-field phase diagram. A normal state at high fields and temperatures, separated by the upper critical field line Hc2(T) from the mixed state, which in turn is separated by the lower critical field line Hci(T) from the Meissner state at low temperatures and fields 2 1.2 STM image of triangular vortex lattice at 1.8K in a magnetic field of IT. The scan range is about 6000 A. From reference [26] 4 1.3 An angular representation of the emittance amplitude with respect to the nuclear spin (arrow) 6 1.4 The geometry of a /5-NMR experiment. The 8 L i beam is nu-clear spin polarized along the axis of the static magnetic field Ho- Two detectors are used to collect the emittied B particles. An alternating magnetic field Hi cos(tot) is applied perpendic-ular to Ho. The resonance ioccurs when u is matched to the Larmor frequency in the local magnetic field at the site of the 8Li 7 1.5 Geometry of 2H - NbSe2(0001) 9 vi List of Figures 2.1 From London equation, the solid line represents the exponen-tial decay of a constant magnetic field applied parallel to the surface of an infinite superconductor. In the region x < £, the dashed line represents the effect of the order parameter from Ginzburg and Landau theory, assuming £ is zero at the surface. XL is the characteristic length on which the magnetic field decays, £ is the coherence length 13 2.2 The reduced sampling zone because of the symmetry of the vortex lattice indicated by the shade Also showed the Saddle point (S) and the minimum point(m) 17 2.3 The magnetic field distribution of vortex lattice. The mag-netic field is the maxim (M) at the center of the vortex core, and gives a high-field cutoff in n(B), the minimum (m) in n(B) occurs at the center of the triangle and gives the low field cutoff. The saddle point (s) which is halfway between the adjacent vortices gives a Van Hove singularity in n(B) for an ideal vortex lattice 18 2.4 Sample Stopping distribution curves of ten thousand sLi+ ions at the very near surface of NbSe2 single crystal under different implantation energies 2kV, 5kV, lOkV and 28kV simulated by TRIM.SP and the corresponding curves fitted by the Beta distribution function 19 2.5 The calculated curves using three different cutoffs at low field (10 mT) with £ = lOnm or £ = 80nm. All the other parame-ters are the same 22 2.6 The calculated curves using three different cutoffs at higher field (300 mT) with £ = lOnm or £ = 80nm. All the other parameters are the same 23 List of Figures 3.1 The Ta target tube 25 3.2 Details of the collinear laser pumped polarizer 26 3.3 optical pumping of 8 Li 27 3.4 Sample tuning peak by measuring the polarization with the polarimeter (6 decay asymmetry for both laser helicities) . . . 28 3.5 The 5 keV beam spot viewed by a CCD camera through the beam access window of the cryostat (8mm x 8mm). The target is a plastic scintillator 29 3.6 Longitudinal field j3 — NMR spectrometer for condensed mat-ter study at ISAC. Courtesy of G.D. Morris 32 3.7 (a) The spectrum with both helicities. (b) The final spectrum, result of the combination of both helicities 33 4.1 The j3 — NMR resonance in NbSe2 as a function frequency for different fields and orientations. The top two scans were taken with the field parallel to the c — axis but at two very different fields; whereas, the bottom scan is with the field perpendicular to the c — axis. The temperature is 10K in all cases 39 4.2 (a) The /3NMR resonance in the normal state of NbSe2 at 10K in a magnetic field of 300mT. The beam energy of 30 keV corresponds to a mean implantation depth < z >= 136nm. (b) The /3NMR resonance in the vortex state of NbSe2 at 3.5K in the same magnetic field of 300mT. The solid line is fitted using a Gaussian cutoff 40 List of Figures 4.3 T h e / 3 N M R resonance i n NbSe2 i n a low m a g n e t i c f i e ld 1 0 . 8 4 m T . (a) I n the n o r m a l s ta te at 1 0 K at a b e a m energy o f 2 k e V co r -r e sponds to a m e a n i m p l a n t a t i o n d e p t h < z >= 8nm. (b) I n the v o r t e x s ta te of NbSe2 at 3 .5K w i t h a m e a n i m p l a n t a t i o n d e p t h < z >= 8nm. (c) In the v o r t e x s ta te of NbSe2 at 3 .5K w i t h a m e a n i m p l a n t a t i o n d e p t h < z >= 84nm 41 4.4 T h e S T M measu remen t of NbSe2 i n a low m a g n e t i c f i e ld 3 0 m T at 1.8K. F r o m reference [26] 42 4.5 (a) S o l i d p o i n t s are the cusp f r equency i n s u p e r c o n d u c t i n g NbSe2 r e l a t i ve to the n o r m a l s ta te f requency , whereas the so l i d l ine is a m o d e l p r e d i c a t i o n of A c w i t h a d e p t h i n d e -penden t A = 1 5 5 n m a n d £ = 8 0 n m . ( b ) S o l i d po in t s are h i g h f ie ld cu to f f f r equency re l a t i ve to the n o r m a l s ta te f requency , whereas the so l i d l ine is a m o d e l p r e d i c a t i o n of A„ w i t h a d e p t h i ndependen t A = 1 5 5 n m a n d £ — 80nm r e spec t i ve l y a n d the d a s h l ine is the p r e d i c a t i o n of A„ w i t h a d e p t h i n d e p e n d e n t A = 1 5 5 n m a n d £ = l O n m 43 Acknowledgements First of all I would like to thank my thesis supervisor Rob Kiefl for his support, patience and guidance. Special thanks to Zaher Salman for his all kinds of helps. I would like to thank Andrew MacFarlane for his guidance. I would also like to thank my colleagues and coworkers G.D. Morris, K.H. Chow, Md Hossain, T.A. Keeler, T.J. Parolin, R.I. Miller, and Ff. Saadaoui for experiments. I would like to thank the expert technical support of R. Abasalti, B. Hitti, D. Arseneau, S. Kreitzman, P. Levy and S. Daviel. Finally I would like to thank Janet Johnson, Tony Walters and my parents . x Chapter 1 Introduction 1.1 Superconductivity and the Vortex Lattice The most well known characteristic of a superconductor is that it exhibits zero electrical resistance when it is cooled below a critical temperature Tc. A wide variety of elements, compounds, and alloys exhibit superconductivity. Perfect diamagnetism is a closely related characteristic property of a super-conductor. This means an externally applied magnetic field can not penetrate into the interior of a superconductor below its critical temperature. If the material is zero field cooled (ZFC), and then placed in an external magnetic field, the flux will be excluded from the superconductor; If the material is field cooled (FC), this means it is cooled in the presence of an external magnetic field, the flux will be expelled from the interior of the material when the tem-perature falls below the critical temperature. The former is a consequence of the perfect conductivity whereas the latter effect (so called Meissner effect ) is an additional characteristic of the superconducting ground state. The so called Meissner effect was discovered by Meissner and Ochsenfeld in 1933. Generally there are two types of superconductors. Type I superconductors exclude an applied external magnetic field up to certain critical value Hc. Below Hc the magnetic field penetrates into the surface of the superconduc-tor decaying over a characteristic length scale A, called magnetic penetration depth. Above Hc the material becomes a normal metallic conductor.Type II 1 Chapter 1. Introduction s u p e r c o n d u c t o r s exc lude the ex t e rna l m a g n e t i c f ie ld be low a ce r t a i n va lue Hci, b u t m a i n t a i n s u p e r c o n d u c t i v i t y u p u n t i l a u p p e r c r i t i c a l f i e ld Hc2. T h e mean-f ie ld phase d i a g r a m of a T y p e II s u p e r c o n d u c t o r s is s h o w n i n F i g . 1.1. 0 T Tc F i g u r e 1.1: T h e mean-f ie ld phase d i a g r a m . A n o r m a l s ta te at h i g h f ie lds a n d t empe ra tu r e s , sepa ra ted b y the u p p e r c r i t i c a l f i e ld l ine Hc2(T) f r o m the m i x e d s ta te , w h i c h i n t u r n is separa ted by the lower c r i t i c a l f i e ld l ine Hci(T) f r o m the M e i s s n e r s ta te at l ow t empe ra tu r e s a n d f ie lds. In the range Hci < Hext. < Hc2, the s u p e r c o n d u c t o r is i n a m i x e d s ta te i n w h i c h the m a g n e t i c f l ux penet ra tes t he s u p e r c o n d u c t o r i n the f o r m of q u a n t i z e d vor t i ces . T h i s is a resu l t of m i n i m i z a t i o n of t o t a l G i b b s free e n -ergy dens i t y of s u p e r c o n d u c t i n g a n d n o r m a l s ta te w i t h respect to a s p a t i a l l y dependen t c o m p l e x o rder p a r a m e t e r </>. <f> is zero above Tc a n d increases c o n -t i n u o u s l y as the t e m p e r a t u r e fells be low Tc. E a c h vo r t ex has a core reg ion w h i c h is " n o r m a l " . T h e m a g n e t i c f ie ld decreases i n an e x p o n e n t i a l m a n n e r 2 Chapter 1. Introduction away from the core over a characteristic length scale A. The order parameter 4> goes to zero at the center of each vortex and is close to a constant bulk value between vortices. The physical meaning of the order parameter is that 10|2 is proportional to the density of super-electrons. \(f>\2 has a zero value at the vortex center and approaches to its maximum value over a distance scale £, which is called the coherence length. A and £ are the basic characteristic length scales of superconductivity. The ratio of the penetration depth to the coherence length is called the Ginzburg-Landau parameter K = j. The surface tension ons = J dz[—\b\(f)\A+ ^u0M2] [16] is the difference of free energy per unit area between a homoge-neous phase, which could be either all normal or all superconducting and a mixed phase [16] If ons > 0, the system will remain in the homogeneous phase whereas if uns < 0, the superconductor will be in a mixed state. Ginzburg and Landau found that ons — 0 when K = This critical value separates Type I superconductors from Type II superconductors with Type I super-conductors for K < and Type II superconductors for K > Many Type II superconductors, form a triangular vortex lattice due to isotropic repulsive interactions between vortices. The triangular vortex lat-tice has been confirmed by Bitter pattern decoration experiments [10, 17, 24] and by Scanning Tunneling Microscopy (STM) [26]. The first vortex lattice STM image is shown in Fig. 1.2 by H.F. Hess. [26]. Other forms of vortex lattice such as a square lattice are also possible depending on the nature of the interaction between vortices. Generally, the behavior of A and £ is considered to be the same in both the Meissner state and vortex state [43]. They are both temperature dependent and field dependent . For extreme Type II superconductors(/t >> 1), If BapPi < Bc2/4:, the magnetic field distribution can be described by the simple London theory with adequate precision [13]. If the vortices are well separated and interact isotropically, one expects a perfect triangular vortex lattice with 3 Chapter 1. Introduction W . * « « * * * * * * * * 6000 A FIG. 2. Abrikosov flux lattice produced by a l -T magnetic field in NbSe 2 at 1.8 K. The scan range is about 6000 A. The gray scale corresponds to dl/dV ranging from approximately 1 x 10~ 8 mho (black) to 1 .5xl0~ 9 mho (white). Figure 1.2: STM image of triangular vortex lattice at 1.8K in a magnetic field of IT. The scan range is about 6000 A. From reference [26]. no preferred orientation with respect to the crystalline axes [27]. In a conventional superconductor, an energy gap at the Fermi surface is a third characteristic property of superconductors. In (Bardeen, Cooper and Schrieffer) BCS theory, electrons near the Fermi surface can be paired up (Cooper pairs) through a virtual phonon-induced attractive interaction. This pairing lowers the total energy of the superconducting ground state compared to the normal state of all unpaired electrons and leads to an energy gap A f c which is temperature dependent. For a so-called s-wave superconductor, the energy gap Afc is isotropic in k-space. Since the lowest energy required to 4 Chapter 1. Introduction excite an electron-hole pair from the superconductor ground state is 2A, the binding energy for a Cooper pair is 2A. When T << Tc, the deviation of A(T) from its zero temperature value, AA(T) = X(t) — A(0), by s-wave BCS theory, is given by Eq. 1.1 [8], where A(0) = 1.76/csT in the weak coupling limit. Then Eq. 1.1 reduce to, For non-s-wave superconductors, the line or point nodes in the energy gap A—> allows quasiparticle excitations to occur for an infinitesimal thermal energy. For example, Annett, Goldenfeld, and Renn show that for supercon-ductors with tetragonal or orthorhombic symmetry and a Fermi surface with spherical or cylindrical topology have line nodes in the gap and the tempera-ture dependence in penetration depth gives to A(T) oc T[4]. Such is the case for the high Tc superconductor YBa2Cu30r[25i\. Nuclear magnetic resonance (NMR) has been used widely as a probe of elec-tronic and magnetic properties in condensed matter physics. Classically, any nucleus with nonzero angular momentum has a magnetic moment p. When it is placed in an external magnetic field, it will experience a torque and will precess at the Larmor frequency proportional to p. This spin precession can be measured by detecting the induced electromagnetic field (emf) in a pick up coil. To generate a good NMR signal, conventional NMR needs about 1018 nuclear spins, and thus it is mostly a bulk probe of matter. The properties (1.1) (1.2) 1.2 P-NMR 5 Chapter 1. Introduction 8 L i ™^ 8Be + e- + v 6 Figure 1.3: An angular representation of the emittance amplitude with respect to the nuclear spin (arrow). of the resonance and spin lattice relaxation rate are a sensitive probe of local electronic and magnetic properties of the material. One can perform a variant of NMR with spin polarized radioactive nuclei. The nuclear polarization is monitored through the nuclear decay properties in particular /3-decay. Like muon spin resonance (p,SR), /3-detected NMR is based on the parity-violating weak decay of a spin-polarized nucleus. The properties of the resonance and spin lattice relaxation rate are a sensitive probe of local electronic and magnetic properties. In such decays the direc-tion of the outgoing energetic particle (beta electron) is correlated with the nuclear spin direction at the moment of decay. Fig. 1.3 shows the angular variation of emission probability with respect to the nuclear spin. Note the 6 Chapter 1. Introduction B a c k w a r d De tec tor F o r w a r d De tec tor Figure 1.4: The geometry of a /3-NMR experiment. The 8Li beam is nuclear spin polarized along the axis of the static magnetic field H0. Two detectors are used to collect the emittied (3 particles. An alternating magnetic field Hi cos(w£) is applied perpendicular to H0. The resonance ioccurs when to is matched to the Larmor frequency in the local magnetic field at the site of the 8Li. highest emission probability is along the opposite direction of nuclear spin in the case of 8 L i /3-detected NMR is best performed using an intense(> 108/s) highly po-larized (80%) low energy beam of radioactive nuclei generated at facilities such as ISOLDE and ISAC. At these dedicated facilities, it is possible to enhance the signal to noise ratio and control the implantation depth over an interesting range scale from 4-400 nm. The geometry of a /3- NMR experi-ment is shown in Fig. 1.4. Since the time revolution of the spin polarization is detected through 7 Chapter 1. Introduction Isotope Spin Ti/2(s) l(MHz/T) 8 decay Asymmetry Productionrates(s 1) 8Li 2 0.8 6.2 0.33 108 nBe 1/2 13.8 0.33 107 Table 1.1: Examples of isotopes suitable for 8NMR the anisotropic decay properties of the nucleus, this method requires about 10 orders of magnitude fewer spins compared to conventional NMR. Conse-quently, /3NMR is well suited to study dilute impurities, small structures , interfaces where there are few nuclear spins. As in the case of pSR, 8 NMR may also be suitable for studying magnetic properties of superconductors in superconducting state. Compared with pSR, which is used to study bulk property of superconductors, 8 NMR can be used to probe the supercon-ducting properties within a few hundred nanometers from the surface. The science applications are similar to what is possible with recent studies using low energy pSR. Some examples of isotopes suitable for 8NMR are shown in Table 1.1 [28]. For specific applications in condensed matter, the proper probe is chosen. 8 L i is the lightest alkali and can be produced at high rates (108/s). Futhermore, it can be easily polarized by a collinear laser method since the D l optical transition 8Li 2s 2S , 1/ 2 —• 2p2P1/2 is at a convenient wave length of 673 nm. The properties of 8Li, like the gyromagnetic ratio, make the resonance narrow and relaxation rate slow compared to heavier nuclei. 8Li is a spin 2 nucleus with a small quadrupole moment Q — 33mB. An-other suitable probe is 11 Be. It is the lightest spin 1/2 probe suitable for BNMR and can be polarized as a positive ion. Since it is spin 1/2, it is a pure magnetic sensor with no quadrupolar interaction. Soon we expect to have a polarized beam of 11 Be. However this thesis is concerned solely with 8Li. 8 Chapter 1. Introduction 1.3 N i o b i u m diselenide NbSe2 NbSe2 is a type II superconductor. There are two crystalline forms of NbSe^-2H — NbSe2 and AH — NbSe2 • The number 2 or 4 indicates the number of NbSe2 layers in a unit cell; The letter H indicates the hexagonal crystal symmetry. Each layer is a 'sandwich' of two layers of Se atoms with a layer of Nb atoms between them. Within a sandwich, the Nb and Se atoms are bonded by covalent forces, and form a 2D-hexagonal lattice. The separation between the Nb sheets is 6.3A and the dimension of the unit cell is 12.6 A. The layers are weakly coupled by van der Waals forces, and it is very easy to cleave along a plane parallel to layers like mica or graphite. The resulting atomically smooth surface is ideal for surface studies of the vortex lattice by STM, /3-NMR et al. The structure of a 2H - NbSe2 single crystal in (0001) direction is shown in Fig. 1.5[1]. f 0 N b O Se Figure 1.5: Geometry of 2H - NbSe2{0001). The 2H — NbSe2 single crystal used for the experiment was grown by a standard vapour transport technique that is discussed in Ref. [18] with a Chapter 1. Introduction i^i (A) a (A) A|| (A) A X (A) K \ \ K± Measurement Method 77 23 690 2300 301 91 specific heat [44] 902 272 specific heat [22] 88/251 25/91 specific heat [47] 110 37 1600 4800 421 141 specific heat [34] 85 21 1150 541 13.51 specific heat [37] 76 26 specific heat [35] 84 ATM [36] 2500 uSR [30] 1913,2824 13233,14364 6.93,5.14 uSR [42] 725 uSR [31] Table 1.2: Coherence length £ and penetration depth A of 2H — NbSe2 measured by various experiments. A|| and £|| are parameters in the a-b plane, and Aj_ and £j_ are parameters perpendicular to the a-b plane 1. Measured at Tc 2. Measured at 1.2K 3. Measured at 0.33TC 4. Measured at 0.6TC 5. Measured at 400mK near zero-field Tc = 7.0K and a transition width < 0.1/v"[43]. The two basic characteristic lengths, penetration depth A and coherence length £ of the 2H — NbSe2 have been measured by various experiments, see Table. 1.2. The wide variation in measured values of A and £ reflects the difficulty in making reliable measurements of these quantities. It is interesting to make a systematic study of A(T, B), £(T, B). The experiments summarized here are an attempt to do this using BNMR. 10 Chapter 2 Theory 2.1 London equations and Ginzburg and Landau equations For Type I superconductors, any external magnetic field HexL < Hc is ex-pelled from the bulk of the superconductor. This is called the Meissner effect. The flux expulsion occurs on a length scale A near the surface. Fritz and Heinz London proposed equations to explain the effect and predict the penetration depth of the external static magnetic field. A superconductor will generate supercurrent of carries of mass m and charge -e to counteract the external magnetic field. In an electric field i?, a carrier will experience a force by Newton's Law, F* = m ^ = -et (2.1) dt where ~v*s is the supercurrent carrier velocity. The supercurrent density is, X = -ensvt (2.2) where ns is the density of superconducting carriers at T = OK. From Eq. (2.1) and Eq. (2.2), we get the First London equation [16], 2 - M o A | ^ (2.3) where XL — J ''l 2 is the London penetration depth. Taking the curl 11 Chapter 2. Theory on both sides of Eq. (2.3) we get, V x E) = poX2LV x ^ (2.4) at Using Maxwell equation V x ~E) = — we get, d{u0X\V xX + ^) = Q dt which implies [16], (2.5) i? = -p0X2LV xX + Const. (2.6) where the constant is set to zero. Eq. (2.6) is called the Second London equation. Using Maxwell's equation V x ~B* = po~?s, which is valid in the absence of magnetization and displace-ment current, and V • B* = 0, we get, V x ( V x ^ ) = V ( v J ) - V ^ = - V 2 ^ = ~ (2.7) or, V B = Similarly we can also get r , 2 ^ _ (2.8) V 2 J : = £ (2.9) Eq. (2.8) and Eq. (2.9) correspond to the Helmholtz differential equation. The general solution in one dimension is, ~B*(x) = c?exp(^) + / ? e x p ( - ^ ) (2.10) XL X L where "d? and /? are parameters to be determined by the boundary condi-tions. For example, assume the surface of an infinite superconductor occurs 12 Chapter 2. Theory at x = 0 . If a field Bo is applied parallel to the vacuum-superconductor interface, then condition B(0) = B0 and B(+oo) — 0 leads to a = 0 and B = B(0). We then get [ 1 6 ] , B(x) = B0exp(—£-) x>0, ( 2 . 1 1 ) A L The distance dependencies of Eq. 2 . 1 1 in a bulk superconductor is shown by the solid line in Fig. 2 . 1 . It is an exponential decay. For vortex state, the radius dependence of field decay from the core is very similar to this. Figure 2 . 1 : From London equation, the solid line represents the exponential decay of a constant magnetic field applied parallel to the surface of an infinite superconductor. In the region x < £, the dashed line represents the effect of the order parameter from Ginzburg and Landau theory, assuming £ is zero at the surface. A ^ , is the characteristic length on which the magnetic field decays, £ is the coherence length. 1 3 Chapter 2. Theory For 0 < T < Tc, the temperature dependence of A L ( T ) can be approxi-mated in terms of a two-fluid model due to Gorter and Casimir, i.e., a mixed interpenetrating but noninteracting fluid of normal electron density nn(T) and superconducting electron density ns(T). The total conduction electron density n = nn(T) + ns(T), where ns(T = OK) = n; nn(T > Tc) = n. They assumed that na(T) = n[l — (T/Tc)4] and found good agreement with early experiments. This leads to the following temperature dependence of \i{T), X l { T ) = [1 - {T/Te)*]W ( 2 1 2 ) TO where A L (0) = For Type II superconductors in an applied external magnetic field between Bci < Bext, < Bc2, the magnetic flux forms a lattice of magnetic vortices each carrying an elementary quantum of flux <E>0. Ginzburg and Landau proposed a phenomenological explanation of this in 1950, by introducing of a complex order parameter 4> = |<^ |e*e. In their theory, the supercurrent is composed of carriers with mass m* = 2m, charge e* = 2e and density n* = | n s , ( \4>\ is the modulus and Q is the phase of the order parameter) m, e, ns are mass, charge, and density of normal electrons. \<f>\2 is proportional to the density of super electrons. The London equations,Eq. (2.3),Eq. (2.6) can be derived from GL theory. Assume a space in which the space for (x > 0) is filled with a superconductor and the left half-space (x < 0) is vacuum. We have the same distance dependence of magnetic field as Eq. (2.11), Bz{x) = B0 exp(-^-) £ << x < oo, (2.13) A L where B0 = — • This simple exponential solution is valid in the range x » £. When 0 < x < £, the magnetic field decays more slowly compared to the range £ << x as shown in dashed line in Fig. 2.1. 14 Chapter 2. Theory The critical fields Bc\ and Bc2 of Type II superconductors are given by, _ $ 0ln/c 5 c l - ( 2 ' 1 4 ) B c 2 = ^ (2.15) 2.2 Modified London equation To determine the magnetic field distribution near the surface in the vortex state of the superconductor, R.N. Goren and M. Tinkham developed a simple treatment [33, 38]. Consider a space that the right half-space (z > 0) is filled with a superconductor and the left half-space (z < 0) is vacuum with an external magnetic field applied along the z direction.For superconductors with large GL parameter K = | >> 1 at moderate magnetic fields Bext « Bc2, we may assume that inside the superconductor the London equation with source terms representing the vortices is sufficiently accurate to describe the spatial distribution Bz(x,y,z); and on the outside, Bz obeys Laplace's equation, d2Bx d2By d2B B * o ^ A , - » , 9 1 f i > - ^ - ^ - ^ - + V Q { Z ) = P ^ ( r ( 2 ' 1 6 ) where is a two dimensional vector in x-y plane, it vectors are the vortex positions, and 9 is the step function which is unity for z > 0 and zero for z < 0. A is Xab, the penetration depth in NbSe2 layers for our geometry. In the superconductor (z > 0), we can get B(x,y,z) by performing the Fourier sum of Eq. (2.19), as, Bz{x, y,z) = Y. Bz0*>z) cos(^ • ~&) ( 2 - 1 7 ) 15 Chapter 2. Theory The Fourier transform in the x-y plane Bz(k,z), where k is the re-ciprocal lattice vector of the vortices. Matching the continuous boundary condition at the surface and the finite value at infinity, one finds, -txfyrW*" * < 0 ' < 2 1 8 ) where A 2 = k2 + and B0 is the average field. Since the source term used in the Eq. (2.16)is a delta function, the cores of the vortices are infinitely small in this model and Eq. (2.17) diverges at r = 0 as k goes to infinity. A cutoff factor g(k, £) can be introduced to take into account the finite size of the vortex. This cutoff factor acts to suppress higher k components and at the same time remove the divergence of B (r=0). At reduced fields b = BQ/BC2 < 0.25, Brandt derived a smooth cutoff factor exp(—l/2k2^2) by approximately solving the isotropic Ginzburg-Landau (GL) equations for high K » 1 [12]. He also replaced A and £ by A / \ / l — b and — b to account for the field dependence of the parameters in the GL theory, respectively. Later A. Yaouanc et al. proposed an improved cutoff approximation exp(—V2k^) at low fields b << 1 [48]. Vortices which interact isotropically should form a triangular lattice. An approximate solution of the GL equations for homogeneous type II supercon-ductors very near Bc2 was first found by Abrikosov [3]. He predicted that for K » 1 a square lattice solution is stable just above Z?cl. Subsequently W.H. Kleiner et al. [29] found that a triangluar lattice solution of the GL equation has a lower free energy. The triangular lattice was observed experimentally in Bitter decoration experiments, which use very small magnetic particles to sense the magnetic field at the surface of a magnetic material, and by STM [26], by measuring the tunneling conductance dl/dV on Type II supercon-ductors like NbSe2 [24], YBa2Cu307 [10, 17] etc. 16 Chapter 2. Theory Figure 2.2: The reduced sampling zone because of the symmetry of the vortex lattice indicated by the shade Also showed the Saddle point (S) and the minimum point (m). Consider a magnetic field distribution n(B) from a triangular lattice of mag-netic vortices. The maximum magnetic field occurs (M) at the center of the vortex core, and gives a high-field cutoff in n(B), the minimum (m) in n(B) occurs at the center of the triangle and gives the low field cutoff. There is also a saddle point(s) halfway between adjacent vortices which gives rise to a Van Hove singularity in n(B) for an ideal vortex lattice with no other source of broadening. The magnetic field distribution is shown in Fig.2.3. Because of symmetry, the magnetic field distribution of the vortex lattice is the same as | of a triangular area formed by adjoining three vortices as shown in Fig. 2.2. 17 Chapter 2. Theory B Figure 2.3: The magnetic field distribution of vortex lattice. The magnetic field is the maxim (M) at the center of the vortex core, and gives a high-field cutoff in n(B), the minimum (m) in n(B) occurs at the center of the triangle and gives the low field cutoff. The saddle point (s) which is halfway between the adjacent vortices gives a Van Hove singularity in n(B) for an ideal vortex lattice. Since the 8Li+ probe ions have an implantation distribution in the super-conductor, the experimental line shape is obtained as a convolution of sum of the theoretical field distribution weighed by the depth distribution of 8Li+. In addition disorder effects in the triangular vortex lattice are accounted for by an additional Gaussian broadening function. 18 Chapter 2. Theory 1500 Depth (nm) Figure 2.4: Sample Stopping distribution curves of ten thousand 8Li+ ions at the very near surface of NbSe2 single crystal under different implantation energies 2kV, 5kV, lOkV and 28kV simulated by TRIM.SP and the corre-sponding curves fitted by the Beta distribution function. 2.2.1 8Li+ stopping distributions PE{Z) The stopping distributions of 8Li+ depends on implantation energy and tar-get material mainly via density but also on Z (the atomic number) and A (the atomic mass ) or the constituent atoms. The stopping distribution profiles are calculated by TRIM.SP, a numerical Monte Carlo program. For an energy from 1 kV to 30 kV, the implantation profiles were simulated with ten thousand 8Li+ ions. The resulting profiles 19 Chapter 2. Theory were fitted by a Beta Distribution function 2.20 , where C is the factor proportional to the number of stopping 8Li+ ions at z, a and B are two free parameters, T(pi) is the complete gamma function and zmax(E) is the maximum implantation depth of 8Li+ ions at a certain implantation energy. PE(Z) satisfies the following equation, / -^jLdz = C (2.21) Jo zmax{E) Some fitted stopping distribution curves are shown in Fig. 2.4 There was reasonable agreement between the simulation generated by the program and tests in other experiments on thin metallic film Al and Au[6]. 2.2.2 Cutoff factors As shown in the previous section, one of the most significant features on the magnetic field distribution of the vortex state is the high field cutoff at point M. The calculated field at M in turn depends sensitively on the vortex core cutoff function. For this reason, we include a detailed discussion here on the cutoff functions used in this work. The term Bz(lc,z) in Eq. (2.19) is multiplied by a cutoff factor g(k,£) in order to avoid divergence as lz goes to infinity in Eq. (2.17). And the Eq. (2.17) is modified to, Bz(x, y,z) = J2 B,(t, z)g{k, £) cos{t • 7?) (2.22) Generally, the cutoff factor g(k, £) can not to be a sharp one [23]. Commonly used cutoff factors are Gaussian-like exp(—ak2£2), where a is a number, e.g. 1/4 [48], 1/2 [12] and 2 [14]; Lorentzian cutoff factors exp(-/3k£),e.g. B = \[2 20 Chapter 2. Theory [14]; and Bessel cutoff factors gKi(g), where g is a function of A and £. The values of a, 8 and the form of function g depends on the magnitude of the external field, which controls the spacing between vortices. Three different cutoff factors were considered to test the effects on the parameters of interest (Aab and £ab, which are in plane of NbSe2 layers): 1. Gaussian cutoff factor exp(—k2£lb/2(l — 6)) 2. Bessel cutoff factor gKi(g) 3. Lorentzian cutoff factor exp(—\/2kl;ab/V1 — b) Accounting for the 8 L i + distribution PE(Z), the field distribution for each implantation energy (E) was modified from Eq. 2.22 into, The integral is performed by sampling 30 depths of the 8 L i + distribution and A is assumed to be the same for all depths. To compare with the experimental data, a large number of points in a unit cell (using Eq. 2.23 ) are sampled. This is used to generate a field distribution P(B) which is then convoluted with a Gaussian G(B) to account for the broadening from lattice disorder. Our fitting function is then, f (^-baseline, Bfactor, A, £, a) — Adeline ~ BfactarP(B) (^) G(B) (2.24) where P(B)<g>G(£) = P{8)G{B-8)dB, G(B—B) = -L_ e x p ( - A ( ^ The field sampling is performed in the reduced unit cell (shaded area) shown in Fig. 2.2 instead of the larger triangular area formed by three adjoining vortices. A total of 10,000 sampling points was sufficiently large that the parameters were insensitive to this number. The effects of these three cutoff factors on the calculated curves are shown in Fig. 2.5 for small fields (10 mT ), and in Fig. 2.6 for higher fields ( 300 mT ). f %max — v BE(X, y)= {£Bz(k, z)g(k, A, Z)\pE{z)dz J surface —$ k (2.23) 21 Chapter 2. Theory Gaussian cutoff Bessel cutoff Lorentzian cutoff X=150nm^=10nm, a=3kHz,Depth=150nm Gaussian cutoff Bessel cutoff Lorentzian cutoff ?t=150nm^=80nm, o=3kHz,Depth=150nm 60 80 100 120 140 Frequency (kHz) Figure 2.5: The calculated curves using three different cutoffs at low field (10 mT) with £ — lOnm or £ = 80nm. All the other parameters are the same. 22 Chapter 2. Theory 0.5 0.4 |-0.3 0.2 oo 0.1 0.0 0.8 ^ 0-6 r E 0.4 h 0.2 h 0.0 Gaussian cutoff Bessel cutoff Lorentzian cutoff ^=150nm^=10nm, a=3kHz,Depth=150nm Gaussian cutoff Bessel cutoff Lorentzian cutoff X=150nm^=80nm, a=3kHz,Depth=150nm 1850 1900 1950 Frequency (kHz) 2000 Figure 2.6: The calculated curves using three different cutoffs at higher field (300 mT) with £ = lOnm or £ = 80nm. All the other parameters are the same. 23 Chapter 3 Experimental 3.1 Production of 8Li TRIUMF has the world's largest diameter cyclotron which produces an in-tense proton beam. This is used to generate other beams of particles like neutrons, pions and muons are created by making protons collide with dif-ferent fixed targets. The production of polarized radioactive ion beam like 8Li for BNMR experiment begins with 500 MeV protons striking very thin Ta foils in a tube as shown in Fig.3.1[2]. The targets are heated up to 2300 °C in order to speed up the diffusion of 8Li in target. This is necessary in order that the 8Li diffuses out of the target in a time comparable to its natural lifetime of 1.2s. The 8Li atoms are surface ionized before they leave the target with a small energy spread of about 1- 2 eV. They are then accelerated up to an energy of 30 keV to form a beam with low emittance. The beam passes through a high resolution mass spectrometer to remove unwanted isotopes, so that a pure 8Li+ beam can be delivered to the target. Ta foil stacks heated to 2000 °C are usually used as the target for the durability and efficiency with the production rate about 8.3 x 107/s. The nearly monoenergetic high intensity, low emittance 8Li+ beam made from a surface ionization source is necessary to generate the large nuclear polarization required for /3-NMR. 24 Chapter 3. Experimental F i g u r e 3.1: T h e T a target t u b e 3.2 Polarizer [19, 28, 39] 8Li is the l ightes t a l ka l i a t o m su i t ab l e for BNMR w i t h nuc lea r s p i n 1 = 2, g y r o m a g n e t i c r a t i o 7 = 6.30l8MHz/T, a n d m e a n l i f e t ime T = 1.2s. A l s o i t c an be eas i l y p o l a r i z e d by o p t i c a l p u m p i n g us i ng a co l l i nea r laser t u n e d to the D l o p t i c a l t r a n s i t i o n of the 8 L i a t o m . W h e n the s L i + i o n is i m p l a n t e d i n to c r y s t a l l i n e ma te r i a l s , i t u sua l l y occup ies h i g h s y m m e t r y c r y s t a l l i n e sites a n d e x h i b i t s n a r r o w resonances. A l s o the s p i n l a t t i c e r e l a x a t i o n rates are s low c o m p a r e d w i t h heav ier nuc le i . These p rope r t i e s m a k e the p o l a r i z e d 8Li+ b e a m at ISAC we l l su i t ed for BNMR. T h e p o l a r i z e r is s h o w n i n F i g 3.2. F r o m left t o r igh t , a l o n g the b e a m d i -r e c t i on , the c o m p o n e n t s are a N a v a p o u r ce l l , a n e u t r a l a t o m dr i f t r eg ion , a n d a H e l i u m gas ce l l . Be fo re p o l a r i z a t i o n , the 8 L i + b e a m is n e u t r a l i z e d 25 Chapter 3. Experimental ' nlatpq (Jul ian iy monitor P l a t e s restriction F i g u r e 3.2: D e t a i l s of the co l l i nea r laser p u m p e d p o l a r i z e r b y pass ing i t t h r o u g h a N a v a p o u r ce l l a n d the u n n e u t r a l i z e d pa r t is re -m o v e d by de f lec t ion p la tes . T h e n e u t r a l pa r t t h e n d r i f t s 1.7 m i n a s m a l l (1 m T ) l o n g i t u d i n a l m a g n e t i c f ie ld w h i l e be i ng p u m p e d w i t h a c i r c u l a r l y p o l a r -i z ed dye laser co l l i nea r w i t h the b e a m ax is . B y reve rs ing the laser he l i c i t y , the nuc l ea r p o l a r i z a t i o n of the b e a m c an be eas i l y f l i p p e d . T h e resonance a m p l i t u d e o f (3NMR s p e c t r a t h e n c o u l d be n o r m a l i z e d to the fu l l B decay a s y m m e t r y b y u s i n g r i gh t-hand he l i c i t y a n d le f t-hand h e l i c i t y laser one af ter ano the r . T h e o p t i c a l p u m p i n g scheme for p o l a r i z e d 8Li is s h o w n i n F i g . 3.3. In theory , the degeneracy of e l ec t ron i c levels of a t o m s i n a weak m a g n e t i c f ie ld are sp l i t b y Zeeman effect interaction. T h e r e s u l t i n g sp l i t levels order c o r r e s p o n d i n g to the a z i m u t h a l q u a n t u m n u m b e r mp for the t o t a l angu l a r m o m e n t u m F, where F = I + J is the s u m of t o t a l e l ec t ron angu l a r momen-26 Chapter 3. Experimental turn J plus nuclear spin I. Providing the external field is weak, F is still a good quantum number. The DI atomic transition of 8Li 2s 2 Si/ 2 —» 2p 2P 1/ 2 occurs at 671 nm. Since the ground state of 8Li has two hyperfine levels split by 382MHz, both levels are pumped to achieve the highest polarization. An electro-optic modulator(EOM) is used to split the laser frequency by 382 MHz. The Doppler broadening of the 8Li beam, which is about 100MHz, is caused by the initial energy spread of the beam and collisions with Na atoms in Na vapor cell. The Doppler width decreases with beam energy. To achieve high polarization, the optical pumping light bandwidth should match the energy broadening of the 8Li beam. Tuning the resonance is done by fine adjustments to the beam energy(and resulting Doppler shift) by varying the deceleration potential on the Na va-por cell between 2 and 200 V, keeping the laser frequencies fixed. The laser frequencies are stabilized to within ±3MHz per day by a frequency stabiliz-ing He-Ne laser system. The tuning peak could be found by measuring the 27 Chapter 3. Experimental 0 .04 —O— right helicity ?\Q —O— left helicity 0 .00 \-0) -0.04 -E E < -0.12 -0.16 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 Voltage (V) Figure 3.4: Sample tuning peak by measuring the polarization with the po-larimeter {B decay asymmetry for both laser helicities) polarization with the polarimeter as shown like in Fig.3.4. By the selection rules for electromagnetic radiation from atomic tran-sition, only transitions with Am = +1 are allowed if using circular polar-ized light with positive helicity, whereas for decay all transitions satisfying: Am = +1,0, —1 are allowed. The lifetime for the spontaneous decay (27 ns) is short compared to the transit time through the polarizer. Consequently there is time for many cycles of excitation and spontaneous decay. This process pumps the atoms into the state with the highest quantum number mp = 5/2 and associated nuclear spin mi — 2. At a laser power of lOOmW, 80% nuclear polarization of 8Li can be achieved. The final stage is to strip 28 Chapter 3. Experimental Li Be am spot 8 mm Figure 3.5: The 5 keV beam spot viewed by a CCD camera through the beam access window of the cryostat (8mm x 8mm). The target is a plastic scintillator. off the valence electron by passing it through a He gas cell. The resulting scattering in both the Na and He cell leads to a beam emittance to which is slightly bigger than the initial beam. The reionized polarized Li+ ion beam is then guided with two 45° electrostatic bends which preserve the longitudinal polarization. To tune the beam spot on the sample, a CCD camera is used to check the spot on a plastic scintillator placed at the sample position and the beam spot is controlled by three Einzel lenses, three adjustable collimators, and various steering elements. The beam spot as shown in Fig. 3.5 finally has a diameter about 3 mm, in range of magnetic fields used in this study. The beam spot also increases slightly at very lowest implantation energy (< 5keV) [6]. The sample is held in a ultrahigh vacuum (UHV) environment (10 - 1 0 Torr) in order to avoid gases condensing on the sample at low temperatures. 29 Chapter 3. Experimental 3 . 3 S p e c t r o m e t e r [ 2 8 ] The schematic of the spectrometer for high field BNMR experiments is shown in Fig. 3.6. All the elements, the magnetic bore, the cryostat, Einzel lenses, and detector housings are held in UHV conditions achieved by two large cryopumps. The spectrometer is designed to be able to measure the 3 decay anisotropy from the implanted polarized ions in the sample either in time domain by measuring the decaying time after implantation, or in frequency domain by measuring the asymmetry as a function of RF frequency. The nuclear polarization is then in the same direction with respect to the beam and the high homogeneity superconducting solenoid. The spectrometer was originally designed for high fields 1 — 9T. However with the addition of neutral beam monitors, it is able to record good resonances in a field as low as 0.01 T, although the event rate at fields below 1 Tesla is substantially reduced since the small magnetic field is too weak to focus the emitted B electrons onto the detector. To change samples, the cryostat is mounted on large bellows so that it can be moved along the beam axis and when it is moved from the magnet bore. A load lock system on top of the main vacuum chamber is used to change the sample. 8Li beta decays to 8J3e (8Li -^>8Be+ve + e~)with a end point energy 13 MeV. Plastic scintillators are held in re-entrant housings with thin stainless-steel windows to detect B electrons. The average energy of the betas is about 6 MeV, which is sufficient to penetrate thin stainless-steel windows on the housings. Thin stainless-steel windows are needed so that the betas from 8Li can reach the plastic scintillator detectors which are outside the UHV vacuum. The 8Be decays within a few ps to 2 energetic alpha particles (E 1.5MeV). The spectrometer and all ancillary equipment( magnet, power supplies, cryostat, 30 Chapter 3. Experimental t h e r m o m e t r y , etc.) sit on a p l a t f o r m e l e c t r i c a l l y i so l a t ed f r o m t he g r o u n d . B y a p p l y i n g a h i g h vo l t age b ias to the p l a t f o r m , t he i m p l a n t i n g energy of ions t o be con t ro l l ed over a w ide range of energies 1-30 k e V . F o r 8Li, t h i s co r r e sponds to a n average i m p l a n t a t i o n d e p t h of be tween 4 ~ 1 3 6 n m i n NbSe2, a c c o r d i n g to the T R I M . S P s i m u l a t i o n s respec t i ve ly . In o rder to detect 3 e lec t rons i n the b a c k w a r d d i r e c t i o n ( o p p o s i t e t o the b e a m d i r e c t i on ) i n h i g h m a g n e t i c f ie lds, the back de tec to r is p l a c e d ou t s ide the magne t s ince the 6 e lec t rons are s t r o n g l y con f i ned t o b e a m ax i s ins ide the m a g n e t bore . T h e f o r w a r d a n d b a c k w a r d (3 de tec tors are p l a c e d at d i f ferent d i s t ance to the samp le b u t have s i m i l a r 0 d e t e c t i on eff ic iencies. A s s h o w n i n F i g . 1.4, a large s t a t i c m a g n e t i c f i e ld is a p p l i e d a l o n g the p o l a r i z a t i o n d i r e c t i o n of the i n c o m i n g 8Li+ a n d the c — axis of the NbSe2 samp le . A s m a l l R F m a g n e t i c f ie ld is a p p l i e d p e r p e n d i c u l a r to t he s t a t i c m a g n e t i c f ie ld a n d i n the a-b p l ane of the samp le . W h e n the f requency of the R F ma tches the L a r m o r f r equency of the 8Li nuc leus , t he nuc lea r p o -l a r i z a t i o n precesses l e a d i n g t o a r e d u c t i o n i n the t i m e averaged a s y m m e t r y . T h e b e t a decay a s y m e t r y is o b t a i n e d as a f u n c t i o n of R F f r equency w h i l e a con t i nuous b e a m of p o l a r i z e d 8Li is i m p l a n t e d i n to t he samp le . T h e p o -l a r i z a t i o n d i r e c t i o n c a n be fliped b y c h a n g i n g the laser he l i c i t y . F o r b o t h he l i c i t i es , the s p e c t r u m looks l i ke i n F i g . 3.7(a). T o reduce s y s t em i c e r ror , the f i na l s p e c t r u m is the resu l t of the c o m b i n a t i o n of the s p e c t r u m of b o t h he l i c i t i es w h i c h a c c u m u l a t e d for 3 0 m i n . - 6 0 m i n . as s h o w n i n F i g . 3.7(b). T h e s p e c t r u m cons is ts of a b o u t 150 p o i n t s . 31 Chapter 3. Experimental Figure 3.6: Longitudinal field B — NMR spectrometer for condensed matter study at IS AC. Courtesy of G.D. Morris 32 Chapter 3. Experimental -0.76 p -0.78 --0.80 -s -0.82 -s >> -0.84 -< -0.86 --0.88 a) o L helicity • R helicity 1880 1900 1920 1940 Frequency (kHz) 1960 1880 1900 1920 1940 Frequency (kHz) 1960 Figure 3.7: (a) T h e spectrum with both helicities. (b) T h e final spectrum, result of the combination of both helicities. 33 Chapter 4 Results and Analysis 4.1 Results in the normal state In the normal state, the spectra in F ig . 4.1 are well fitted to Gaussian line shape. Since NbSe-i is hexagonal, one expects the resonance to be split by an electric quadrupolar interaction at a non-cubic site. The unresolved nuclear electric quadrupole spl itt ing implies that the electric field gradients are 10-100 times smaller than in other non-cubic crystals. In the top panel of F ig . 4.1, at a static magnetic field B = 3T, the fitted line width defined as the F W H M equals 3 .5( l )kHz and is weakly dependent on the R F power level, indicating that the measured line width is close to the intrinsic width. The observed line width is attr ibuted to the nuclear moments of 9 3 A r 6 (100%5 = 9/2) and 77 Se(7.6%S = 1/2) plus the unresolved quadrupolar spl itt ing. A t a much smaller static magnetic field B = lOmT, the lineshape is almost identical. Th is confirms there is no large contribution to the line broadening in high field from a spead in Knight shifts or chemical shifts since in this case one would expect a decrease in the line width at low field. In the bottom panel of F ig . 4.1, the line width decreases to 1.5 kHz when the magnetic field is oriented perpendicular to the c — axis Th is is consistent wi th L i occupying a site in the Van der Waals gap since then one would expect line broadening from N b moments and quadrupolar splittings to be greatest when the field is parallel to the c — axis. [46] 34 Chapter 4. Results and Analysis G B L X(nm) 279 259 242 ((nm) 12 10 6.7 x2 234 299 322 Table 4.1: The fitted parameters with three different cutoffs are summarized. The Gaussian cutoff has the lowest x2 4.2 Results in the vortex state In a high field such as 302mT, the fitted parameters are similar to what is observed in bulk measurement of NbSe2 using muon spin rotation. The coherence length is about 12 nm. However, in a smaller field of 10.84 mT, the coherence length is greatly enhanced, about 80 nm, 7 times bigger than in high field. 4.2.1 Results at B=302 mT, T=3.5 K The fitting is performed with five free parameters using Eq. 2.24. The results for A and £ are summarized in Table 4.1. Fig. 4.2a shows the /?NMR resonance in the normal state of NbSe2 in a magnetic field of 302 mT at an implantation energy 30 keV, corresponding to a mean depth of 136 nm. The observed width is attributed to nuclear dipolar broadening from the 93Nb nuclear moments [46]. Below Tc, in Fig. 4.2b, the line broadens significantly and is very asymmetric, these are characteristic features of a triangular vortex lattice. The observed peak, or cusp field, shifts down A c from the normal state frequency. To quantify this we define A c as the shift fnor — fcusp- There is also a high field tail which corresponds to the magnetic field distribution near the vortex core. The frequency at the high field cutoff, is defined as A„ = / M — fnor, where / M is the theoretical high field 35 Chapter 4. Results and Analysis frequency at the core (M). Subsequent measurements at this field on a freshly cleaved sample have shown a somewhat broader line and correspondingly smaller value of A. This indicates the field distribution is sensitive to surface quality. 4.2.2 Results at B=10.84 mT, T=3.5 K Fig. 4.3 shows the /3NMR resonance in a magnetic field of 10.84mT. There is strong dependence both on temperature and implantation depth < z >. The normal state resonance in Fig. 4.3a is very similar to what is observed at 300mT. It is relatively narrow and independent of implantation energy. Without any modeling or fitting, several important features are evident in the spectra. Firstly, A c increases as the mean implantation depth changes from 8 nm (Fig. 4.3b) to 84 nm (Fig. 4.3c). Right at the surface the distri-bution is half as wide as it is deep inside the superconductor. Deep inside the superconductor the lineshape is characteristic of the bulk superconduc-tor; whereas, well outside the superconductor, any line broadening from the vortex lattice must vanish. In a London model only A and the magnetic field determine A c as described in Equation 4.1. / — fcusP oc Bo — Bcusp oc (4.1) In real superconductors where the vortices have finite size, A c also depends on £ and other factors which contribute to the effective vortex size such as vortex structure, delocalized quasiparticles, zero-point motion and thermal motion of the vortices [5, 15, 41]. Second, there is remarkable similarity between the resonance lineshapes in Fig. 4.2b and Fig. 4.3b even though the magnetic fields differ by a factor of 27. Typically Av can only be observed in much higher magnetic fields such as in Fig. 4.2b where the vortex density is high and the lineshape is significantly narrowed on the high frequency side. 36 Chapter 4. Results and Analysis The clear observation of A„ in the low magnetic field can only be explained by a very extended vortex core on the order of 80 nm. In Fig. 4.5, the solid lines represent the best fit to the whole energy scan with a depth independent A = 155nm and £ = 80nm. Overall A c and A„ vary in rough agreement with the simple London model. In particular A c decreases by about a factor of two as one approached the surface. Note however there are small deviations from the simple London model close to the surface. X-ray [32] and Helium scattering studies [7] report an anomaly in the uppermost layers. The most striking feature in the data is the large value of £. If one use a typical value for £ = lOrom and A = 155nm, one obtains a value of A„ = 150kHz or about 6 or 7 times larger than what is observed as shown in Fig. 4.5b. It is interesting to compare with the STM measurement of core size of NbSe2. From Fig. 4.4 in reference [26], at T = l&K and B = 30mT, the core size looks as big as 25nm. 4 . 3 D i s c u s s i o n a n d C o n c l u s i o n Recent studies of thermal conductivity and heat capacity shows that NbSe2 [9], like MgB2 [11, 40], is a multiband superconductor. The fermi surface con-sists of parts from several different bands crossing the fermi level. The energy gap A can thus be a function both of direction (Is) and of the band index n. There is a superconducting energy gap of magnitude A ~ lmeV derived from Nb 4d orbitals and a gap derived from Se 4p orbitals with a upper limit A ~ 0.2 ± O.lmeV in NbSe2 [45]. In a single band superconductor £ ~ 1/A, so it is natural to expect the value £ = Ylnm in higher magnetic field 300 mT is associated with a bigger energy band gap and the value £ = 80nm in lower magnetic field 10.84 mT is associated with a much smaller energy band gap. The simple relation £ ~ 1/A is only valid in the limit of zero interband coupling for a multiband superconductor [15, 49], NbSe2 seems to have weak 37 Chapter 4. Results and Analysis interband coupling at this temperature. For MgB2, in low magnetic field, large vortex cores about 50 nm have been observed on the 7r bands which have the smaller energy band gap compared with the larger 5 band value by scanning tunneling spectroscopy [20]. Previous /iSR work on bulk NbSe2 at temperatures T = 0.6TC and T = 0.33TC show that £ becomes larger as the magnetic field decreases [42], the extrapolated low field value is about 22nm at the lowest measured field, but the experiment data didn't reach the low magnetic field regions like 10 mT. Recent //SR work on bulk NbSe2 at an extremely low temperature T = 2t)mK, shows that the core size is around 6 — lOnm for both energy gaps [15]. They argue that, at this temperature, the thermal excitations are totally frozen out, so the quasiparticles in the core gives a smaller ex-pansive contribution to the core size, compared with previous /iSR measure-ments. The simple model £ ~ 1/A should not be applicable because of finite interband coupling. Otherwise the ratio of the two band gaps should be lOnm/Qnm = 1.67. Yokoya found a ratio of the larger band gap and the smaller band gap no smaller than 5/1 at 5.3 K. Boaknin reports a ratio about 3/1 at 2.4K. Measurements with a radiofrequency tunnel diode oscillator show that below 1.2K down to lOOmi^, "the gap on the Se sheet is at least as large as that on the Nb sheets", the ratio range is about 1.6 — 2 below 2.4K [21]. The smaller band gap seems open up at 5K, and continues up as the temperature goes down. So the results of surface measurements like 0NMR, STM are consistent with estimates of the vortex core using using the BCS expression £ ~ 1/A. Thus the multiband nature of NbSe2 may be a major factor responsible for the large vortex core reported here. After all, the most important result is the large vortex cores observed in low field about (B=10mT). 38 Chapter 4. Results and Analysis Figure 4.1: The B — NMR resonance in NbSe2 as a function frequency for different fields and orientations. The top two scans were taken with the field parallel to the c — axis but at two very different fields; whereas, the bottom scan is with the field perpendicular to the c — axis. The temperature is 10K in all cases. 39 Chapter 4. Results and Analysis Figure 4.2: (a) The 0NMR resonance in the normal state of NbSe2 at 10K in a magnetic field of 300mT. The beam energy of 30 keV corresponds to a mean implantation depth < z >= 136nm. (b) The /3NMR resonance in the vortex state of NbSe2 at 3.5K in the same magnetic field of 300mT. The solid line is fitted using a Gaussian cutoff. 40 Chapter 4. Results and Analysis 40 60 80 100 Frequency (kHz) Figure 4.3: The /3NMR resonance in NbSe2 in a low magnetic field 10.84mT. (a) In the normal state at 10K at a beam energy of 2 keV corresponds to a mean implantation depth < z >= 8nm. (b) In the vortex state of NbSe2 at 3.5K with a mean implantation depth < z >= 8nm. (c) In the vortex state of NbSe2 at 3.5K with a mean implantation depth < z >= 84nm. 41 Chapter 4. Results and Analysis > "O ;t *".V 0 500 D I S T A N C E (A) 1000 F IG . 5. Differential conductance dl/dV (arbitrary scale) at zero bias (lower curve) and at 1.3 mV (upper curve) as a func-tion of position. Figure 4.4: The STM measurement of NbSe2 in a low magnetic field 30mT at 1.8K. From reference [26]. 42 Chapter 4. Results and Analysis 0 210 180 Si 50 W 40 20 0 A=155 nm Z= 80 nm (b) A=155nm Z= 10nm 1 •A=155nm l= 80nm 0 50 100 150 <z> (nm) Figure 4.5: (a) Solid points are the cusp frequency in superconducting NbSt2 relative to the normal state frequency, whereas the solid line is a model pred-ication of A c with a depth independent A = 155nm and £ = 80nm.(b) Solid points are high field cutoff frequency relative to the normal state frequency, whereas the solid line is a model predication of Av with a depth independent A = 155nm and £ = 80nm respectively and the dash line is the predication of Av with a depth independent A = 155nm and £ = lOnm. .„ Bibliography [1] Crystal structure of 2H - NbSe2. 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Scanning-Tunneling-Microscope Observation of the Abrikosov Flux Lattice and the Density of States near and inside a Fluxoid. PRL, 62:214, (1989). [27] B. Keimer, W. Y. Shih, R. W. Erwin, J. W. Lynn, F. Dogan, and I. A. Aksay. Vortex Lattice Symmetry and Electronic Structure in YBa2Cu307. PRL, 73:3459, (1994). 46 Chapter 4. Results and Analysis [28] R. F. Kiefl, G. D. Morris, P. Amaudruz, R. Baartman J. Behr, J. H. Brewer, J. Chakhalian, S. Daviel, S. Dunsiger J. Doornbosa, S. R. Kre-itzman, T. Kuo, C. D. P. Levy, R. Miller, M. Olivo, R. Poutissou, G. W. Wight, and A. Zelenski. Complementarity of low-energy spin polarized radioactive nuclei and muons . Physica B, 289-290:640, (2000). [29] W. H. Kleiner, L. M. Roth, and S. H. Autler. Bulk Solution of Ginzburg-Landau Equations for Type II Superconductors: Upper Critical Field Region. PR, 133:A1226, (1964). [30] L . P L E , G.M.LUKE, B.J.STERNLIEB, W.W.WU, Y.J.UEMURA, JW.BRILL, and H.DRULIS. 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Near surface vortex lattice in NbSe₂ studies with low energy beta - NMR Wang, Doug 2006
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Title | Near surface vortex lattice in NbSe₂ studies with low energy beta - NMR |
Creator |
Wang, Doug |
Publisher | University of British Columbia |
Date Issued | 2006 |
Description | In this thesis, the magnetic field distribution in the vortex state of NbSe₂ is investigated with the technique of depth resolved β-detected NMR using a beam of highly polarized ⁸Li produced at the TRIUMF ISAC facility. The β-NMR lineshape is a direct measure of the local magnetic field distribution weighted according to the implantation profile of the ⁸Li beam. By varying the implantation energy between 1~ 30keV, one can control the average implantation depth corresponds in a range between 5 ~ 136nm. Above Tc = 7.OK a relatively narrow resonance is observed whose width is attributed to magnetic dipolar broadening from the ⁹³Nb nuclear moments. Below Tc a much broader asymmetric lineshape is observed, which is characteristic of a triangular magnetic vortex lattice. Modeling the magnetic field distribution allows one to determine both the effective in plane penetration depth λıı and the effective in plane coherence length ξıı or vortex core radius. In a magnetic field of 302mT, we obtain λıı = 279(30)nm and ξıı = 12(1)nm. In a smaller magnetic field of 10.84mT, the effective coherence length increases dramatically to a value of 77(10)nm. This is more than an order of magnitude larger than the expected from the coherence length in NbSe₂. The origin of the giant vortices is discussed. We propose an explanation involving the multiband nature of NbSe₂. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085246 |
URI | http://hdl.handle.net/2429/18344 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2006-11 |
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UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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