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Measurement of the London penetration depth in the Meissner state of NbSe₂ using low energy polarized… Hossain, Md Masrur 2006

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Measurement of the London Penetration Depth in the Meissner State of NbSe2 using Low Energy Polarized 8 L i . by . Md Masrur Hossain 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 April 28, 2006 © Md Masrur Hossain 2006 i i Abstract In this thesis, the Meissner state of NbSe2 was investigated using low energy beam of spin polarized 8 L i . The 8 L i nuclear spin relaxation rate ijr was mea-sured as a function of temperature and magnetic field. The spin relaxation rate is sensitive to low frequency nuclear spin dynamics of the host Nb spins and is strongly field dependent. This is used to determine the reduction in the magnetic field upon cooling into the Meissner state. Using a calculated implantation profile and a model field distribution, one can extract a mea-sure of the absolute value of the London penetration depth A in Meissner state. In addition, a model field distribution, assuming a suppression of order parameter near surface, was developed. In this case, we can extract another length scale which is related to the "coherence" length £. The value of A depends on the model field distribution but is significantly longer than that obtained previously in the vortex state using pSR. From the measured internal magnetic field distribution, London penetration XL is extracted as a function of temperature. There is also evidence of the coherence peak in ijr of host nuclear spins. XL(T) follows the two-fluid model of superconduc-tivity. Depending on the model for internal field distribution, A L ( 0 ) varies in the range ( 1 7 9 5 - 2 4 3 4 ) A . M d Masrur Hossain i i i Contents Abstract i i Contents " i List of Tables v List of Figures v i Acknowledgements be I Thes is 1 1 Introduction 2 2 Superconductivity in London and Ginzburg-Landau Theo-ries 7 2.1 London Theory 7 2.2 Ginzburg-Landau Theory 10 2.3 Zero field case near superconducting boundary 11 2.4 B C S penetration depth 17 3 T ime Dependence of Polarization and Relaxation 21 3.1 Evolut ion of 8 L i Spin Polarization in NbSe2 in the Absence of Relaxation 21 3.2 Korr inga Relaxation & Knight Shift 26 3.3 Low Fie ld Spin Relaxation from Fluctuat ing Dipolar Fields . 31 4 Experimental 34 4.1 Beamline Properties and Spectrometer 34 4.2 The sample 37 4.3 Measurement of Polarization in Low Fie ld 41 4.3.1 P(t) in short pulse method 41 Contents iv 4.3.2 P(t) in Long pulse method 41 5 Measurements and Results 44 5.1 Normal State in High Magnetic Field 44 5.1.1 Korringa Relaxation 44 5.1.2 Dipolar Broadening of the Resonance 44 5.2 Low Field Measurements 49 5.2.1 Stopping Distribution 52 5.2.2 Analysis and fitting to determine XT, and £ 52 6 Summary & Conclusions 67 Bibliography 69 A Low Field Spin Relaxation from Fluctuating Dipolar Fields 73 V List of Tables 5.1 Penetration depth AL (0 ) and coherence length £(0) at T = 0 61 5.2 Average magnetic field, calculated using parameter in two models 64 5.3 Penetration depth Ar, and coherence length £ at temperature T = OK, using a common ini t ia l amplitude for fitting 64 5.4 Average magnetic field, in two models, using parameters from table 5.3 64 vi List of Figures 1.1 Critical temperature history 4 2.1 Magnetic field, as it enters a superconducting sample, accord-ing to the London model 9 2.2 (a) Numerical solution ( B n g o i ) and it's phenomenological fit (Bpjjgjj as given in Eq. 2.41) of the magnetic field in G-L theory, (b) Difference between them. £ = 150A and \L = 21001 in both figures 16 2.3 Typical density of states in a superconductor. The dashed line represents the normal state while the solid line represents the superconducting state 18 2.4 Temperature dependence of the superconducting energy gap in the weak coupling limit of BCS interaction 20 3.1 Time integrated polarization, as a function of magnetic field, showing resonant dips at level crossings with a single neigh-boring Nb 24 3.2 Time integrated polarization in low magnetic fields 25 3.3 Functions f(E), 1 - f(E), and f(E)[l - / ( £ ) ] . The thicker bell-shaped curve shows f(E)[l - f(E)] ' 30 3.4 Typical spectral density plot 33. 4.1 A schematic of the experimental layout. The 30 KeV 8 L i + ion beam is neutralized in the Na cell and then reionized in the He cell. In between, the beam is optically pumped with a laser tuned to the DI optical transition of the 8 L i atom. The resulting polarized beam is guided to /3-LCR spectrometer. . 35 4.2 Optical Pumping scheme for polarizing 8 L i 36 List of Figures v i i 4.3 A schematic of the spectrometer for /3-detected nuclear res-onance. The spin polarization is perpendicular to the beam direction. The principal axis of the electric field gradient at the 8 L i stopping site must have a component along z in order for a signal to be detected at zero applied field 38 4.4 (a) Polarization PZ(B) for both helicities (b) Normalized asym-metry is found by subtracting "down" helicity from "up" he-licity to remove the background effect 39 4.5 Top: NbSe2 cross-section (in 1120 plane). Bot tom: Three dimensional structure of 2H-NbSe2 40 4.6 (a) Beam is on between time (0,4)s. (b) Schematic polariza-tion P(t) as a function of time 43 5.1 The time evolution of normalized spin polarization 8 L i in NbSe2 in a magnetic field of 3T applied along the c-axis. The time differential measurements were done in short pulse mode. The solid lines are fits to a single exponential without any background 45 5.2 Comparison between spin relaxation of L i in A g and NbSe2 as a function of temperature. The applied field along c-axis is 3T 46 5.3 The /3 -NMR resonance in NbSe2 as a function of field and ori-entation. The top two scans were taken wi th the field parallel to the c-axis but at very different fields; whereas, the bottom scan is wi th the field perpendicular to the c-axis. The tem-perature is 10K in al l cases and so there is no line broadening due to 8 L i motion 47 5.4 Three different field spectra 50 5.5 Dependence of relaxation rate on magnetic field at T = 8K. . 51 5.6 Monte Carlo calculated stopping distribution p(x). M a x i m u m depth 8 L i reaches, at energy 30 K e V , is « 4100A arid the profile is centered at « 1360A 53 5.7 Three time integrated spectra at B = 125G. Solid lines rep-resent fitted baseline. Resonance line shape changes due to the asymmetrical field distribution in the vortex state 57 5.8 Comparison of relaxation between the vortex state and the Meissner state 58 5.9 Penetration depth as a function of temperature for an expo-nential model for B(x) 59 List of Figures v i i i 5.10 Penetration depth and coherence length as a function of tem-perature 60 5.11 Penetration depth as a function of temperature for an expo-nential model for B(x) wi th average overall amplitude 62 5.12 Penetration depth and coherence length as a function of tem-perature wi th average overall amplitude, for "£ model". . . . 63 5.13 Time differential spectra at three temperatures, 3.75K, 6.15K and 6.4K, in long pulse method 66 A . I Typica l correlation function 77 i x Acknowledgements I would like to thank my supervisor, Professor Rob Kief l , for his guidance and insight. I would also like to thank Professor Andrew MacFarlane for his guidance and thanks to my colleagues and coworkers D . Wang, Z . Salman, K . H . Chow, S. Daviel, T . A . Keeler, G . D . Morr is , R . I . Mi l le r , T . J . Parol in and H . Saadaoui at / 3 - N M R group in T R I U M F . Special thanks to R . I . Mil ler for helping me wi th the M I N U I T fitting routine. Also thanking lATfiX C T A N lion drawing by Duane Bibby; thanks to www.ctan.org Part I Thesis 2 Chapter 1 Introduction Historically superconductivity has played an important role in condensed matter physics. Before the discovery of the phenomena of superconductivity, it was known that the resistivity of a metal drops wi th decreasing temper-ature. Resistivity in metals is generally attributed to electron-phonon scat-tering, the rate of which is proportional to the thermally excited phonons. However, the number of thermally excited phonons is always finite above absolute zero and thus the resistivity should always be finite at any finite temperature. Consequently, K . Onnes' discovery of vir tual absence of re-sistivity in Mercury below 4.15K, in 1911 [1] was rather surprising. Soon after, in 1913, Lead was found to be superconducting below 7.2K and af-ter 17 years of this discovery, niobium was found to be superconducting at 9.2K. The vir tual absence of resistance in superconductor has been demon-strated by experiments wi th persistent currents in superconducting rings. Such currents have a decay time of magnitude of 10 5 years. Applications of superconductivity include very high-current transmission lines, high-field magnets and magnetic levitation. In 1954, the first successful supercon-ducting magnet was made using N b wire, which produced a field of 0.7T at 4.2K. In 1960, persistent current in a solenoid was used to provide the mag-netic field for a solid state maser. This was probably the first commercial application of superconductivity. The other important characteristic beyond zero resistivity is the phe-nomenon of the Meissner effect in which magnetic field is expelled [2] out of a sample when it's cooled below the so called critical temperature Tc. The phenomenon of the Meissner effect is different from perfect diamag-netism. In perfect diamagnetism, currents are generated to oppose any change in applied field. However, if the sample already had non-zero mag-netic flux through it, cooling through Tc wouldn't make any change in the field whereas, in the Meissner effect, the field would be expelled from the sample when cooled below Tc. This phenomenon of the Meissner effect led London brothers [3] to pro-pose equations to predict how the field is excluded from the sample and in particular, the field penetration near the surface. Chapter 1. Introduction 3 London's theory was later (1950) derived from the theory of Ginzburg and Landau [4] ( G L ) , who described superconductivity in terms of a macro-scopic complex order parameter (j> which roughly dictates the extent to which a system is ordered. In the case of superconductivity, the amplitude of order parameter is proportional to superconducting electron density. Al though the phenomenological G L theory had been successful, the mi-croscopic theory only came in 1957 from J . Bardeen, Leon Cooper and John Schrieffer [5]. The carriers of supercurrents were shown to be a pair of elec-trons ("Cooper pairs" [6]) wi th opposite spin and momentum. In 1986, J . G . Bednorz and K . A . Mul ler [7] discovered superconductivity in L a 2 - x B a I C u 0 4 at 35K, thus initiating the era of high-temperature superconductivity. A l -though met with ini t ial skepticism, the observations were validated when Uchida et. al. and C h u et. al reproduced original results in 1987. The same year, scientists produced Lanthanum compound L a 2 _ x S r x C u 0 4 which went superconducting ~ 4 0 K . In subsequent years, remarkable progress has been made in increasing the crit ical temperature as shown in F i g 1.1. Besides having a crit ical temperature Tc, superconductors also have crit-ical magnetic fields associated wi th them, above which their properties change. In this respect, superconductors are classified in two broad cat-egories, i) Type I, in which the material becomes normal above a critical magnetic field H c i . ii) Type II, in which the material has two crit ical mag-netic fields Hc\ and HC2. In type II, at H < i 7 c i , , t he sample remains in Meissner state and at Hc\ < H < Hci-, magnetic field penetrate sample in quantized vortices and for H > it becomes normal. Two other pa-rameters characterize superconductivity in general, namely the coherence length £ and the magnetic penetration depth A. The coherence length £ is the distance over which order parameter (j> varies appreciably and penetra-tion depth A is the depth over which shielding currents circulate to expel the applied external field. A and £ are two fundamental length scales in superconductivity. Other parameters of interest such as Ginzburg-Landau parameter K = j, two critical fields Hc\, thermodynarnical critical field Hc may be derived from them. Niobium compounds such as NbsSn and N b s T i have dominated research in conventional superconductivity since they have the high crit ical temper-ature values required for superconducting magnets. NbSe2 belongs to a transition metal dichalcogenides which have received considerable attention for their very interesting physical properties, such as superconductivity and existence of a charge density wave transition. Due to the anisotropy, the magnetization depends on the angle which the applied field makes wi th the c-axis. In a layered structure such as NbSe2, the c-axis is generally perpen-140 r 120 \-100 r-80 r-60 r-40 r-Critical Temperature in °K 125K, 1988 Tl Sr Ba Cu O X X X X XI 10K, 1987 92 K / BiCaSrCu2Og 1986 YBa 2 Cu 3 0; 23 K 1973 Nb3Ge ^35 K 1986 La0 Ba CuO,, 2-x x 4 2 0 I" 4.2 K 1911 Hg 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 Year Figure 1.1: Critical temperature history. Chapter 1. Introduction 5 dicular to the plane. N b S e 2 is a quasi 2D crystal where bonding wi thin the layers is strong and the bonding in between the layers is weak. U n t i l now, the penetration depth has been measured in the vortex state via muon spin rotation [8] and using microwave techniques [9, 10, 11, 12]. In vortex state measurement, Sonier et al. used a G L model for magnetic field distribution to extract A as a function of applied magnetic field. However, it was mentioned that Aa& measured is an effective penetration depth which is model dependent. Consequently, one may expect some difference in A measured in the Meissner state where there are no vortices. The microwave techniques used in [9, 10, 11, 12] reported London pen-etration depth for a number of high-T c superconductors. The microwave techniques are well-suited to measuring temperature dependence of A but generally not very sensitive to the absolute value of A. W i t h the microwave techniques, one obtains some averaged macroscopic penetration depth. The present experiment is sensitive to the absolute values of the penetra-tion depth and the coherence length in the Meissner state. The relaxation rate ^ , of the 8 L i probe, is dependent on the magnetic field that the 8 L i sees inside NbSe2 - In a sense, our method [13][14] is a direct way of measuring A(T) and £(T) since the functional dependence of magnetic field on depth and stopping distribution of probe 8 L i inside the superconductor uniquely determines the relaxation rates, thereby giving A(T) and in some circum-stances, £(T). The penetration depth is important, since, for example, it is often used to distinguish between different types of superconductivity since, at low temperatures, A(T) reflects change in the superfluid density. For ex-ample, A(T) is exponentially dependent on temperature in a conventional s-wave B C S superconductivity, whereas, in a d-wave superconductivity A(T) is linear in T. Al though the pairing mechanism of NbSe2 is thought to be that of s-wave B C S type, other possible pairing states involving complicated gap functions, have been suggested [15] [16] [17] [18] for other materials such as Y B a 2 C u 3 0 7 _ 5 . Thus an accurate determination of A(T) is one way to probe the symmetry of the pairing state. Recently, it has been suggested that there are two energy gaps in NbSe2 -This phenomenon has emerged as an explanation for anomalous proper-ties [19] of some s-wave superconductors. Interest in the possibility of a double gap is enhanced by the peculiar properties [20] of the 39K supercon-ductor M g B 2 . Since N b S e 2 has a similar planar quasi 2D crystal structure, it was speculated that similar characteristics may also be present in NbSe2 -Evidence for a second gap in NbSe2 has since been reported [21] [22] [23]. In this thesis, we measure a reduction of magnetic field B(x) as it enters the sample, v ia the change of nuclear relaxation rate >jr. For an exponen-Chapter 1. Introduction 6 tially decaying magnetic field, we have only one free parameter, namely the penetration depth A^. However, with imperfections on surface, there may be suppression of the order parameter cp near the surface thereby introducing a different form for B(x). The suppression is thought to be dominant on the range of 'coherence' length £. In our model, <p varies appreciably within a distance of 2\/2£. In this case, there are two free parameters A and £. In chapter 2, there is an introduction to London and Ginzburg theory and the functional dependence of magnetic field on A and £. This will be followed by a discussion on BCS superconductivity as it pertains to NbSe2-Chapter 3 contains a discussion of the various mechanisms which lead to time dependence of the nuclear polarization. This includes 8 Li -Nb nuclear Korringa relaxation which is dominant in high magnetic field. Chapter 4 contains a discussion on experimental setup. This will be followed by a discussion on NbSe2 structure and on polarization as function of time in two methods of measurement. The experimental results of measurements will be presented in the chap-ter 5 . I will show that the Korringa relaxation in NbSe2 is an order of magnitude smaller compared to Ag. I will then discuss the ^ measure-ments in the vortex state and in the Meissner state. The values of A and £ will be extracted from these measurements according to two models for the internal field distribution. I shall then show that the extracted A favors a model with the order parameter is suppressed at the surface. Chapter 6 contains a brief discussion of the results. 7 Chapter 2 Superconductivity in London and Ginzburg-Landau Theories 2.1 London Theory We consider the penetration depth in the Meissner state of a type II super-conductor. Below H c i , the London equations provide a good description of the electromagnetic properties. The relevant Maxwell 's equation is V x £ = - i f . (2.1) c dt In the classical Drude model of electrical conductivity, we have F = -m- -eE = rr£, (2.2) r dt where v is the average velocity of the electrons, m is the mass of an electron, E is the electric field the electrons are in and r is the relaxation time, i.e, roughly the time required to bring the drift velocity to zero if electric field was suddenly set to zero. In a normal metal, the competition between the scattering and the acceleration in E q . 2.2 leads to a steady state average velocity v = ^ . (2.3) m Assuming n conduction electrons per unit volume, we get the electric current density v ia Ohm's Law, J = nev = (^^j E = oE. (2.4) To describe superconductivity, London assumed that a certain density of electrons ns experience no relaxation i.e., letting TS in E q . 2.2 go to infinity. Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 8 This leads to f - f £ ) * M where n s is density of the superconducting carriers. Taking curl on both side of the E q . 2.5, we get m f V x ^ ] = V x £ (2.6) nse2 V dt Substituting Maxwel l E q . 2.1 in 2.6, we obtain the second London equation Interchanging the order of differentiation wi th respect to space and time in E q . 2.7, London postulated ^ ( v x / s ) + B = 0. (2.8) Assuming no time varying electric field, another Maxwel l equation connects Ja wi th B wi th the equation J s = h ^ x s) ^ Substituting E q . 2.9 into E q . 2.8, we get A 2 t ( V x V x B ) + B = 0, X2LV2B + B = 0, (2.10) where * = (2.11) XZL mcz In a vacuum-superconductor interface (which is also the case in our experi-ment), the solution of E q . 2.10 is given by 5 ( x ) = 2 ? ( 0 ) e x p ( - - ^ ) (2.12) and is schematically shown in F i g 2.1 The quantity Ax, is known as London penetration depth and Xj2 cx ns (i.e, superfluid density). The most im-portant success of the London Eqs. 2.9 and 2.10 is that a static magnetic Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 9 Figure 2.1: Magnetic field, as it enters a superconducting sample, according to the London model. Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 10 field is screened from the interior of a bulk superconductor over a charac-teristic penetration depth A ^ . A s one approaches the crit ical temperature Tc, ns'—> 0 continuously and as a consequence, A L ( T ) diverges as T —> Tc, according to E q . 2.11. The E q . 2.12 is true for T = 0. Gorter and Casimir [24] found that good agreement wi th early experiments could be obtained if one assumes, what is now known as two-fluid model, ns(T) = n 1 - . (2.13) B y substituting E q . 2.13 in E q . 2.11, we get the penetration depth as AL (0 ) with A L ( T ) = -AL (0 ) = (0 Anne2 i • 2 (2.14) (2.15) 2.2 G i n z b u r g - L a n d a u T h e o r y A phenomenological approach to superconductivity by Ginzburg and Lan-dau(GL, hereafter) is discussed in this section. We shall derive the functional form of the magnetic field as it penetrates the sample, in the Meissner state. Order parameter: G L theory assumes that the super-electrons(holes) of mass m*, charge e* and density n* are connected by relationships m e* * = 2m, = ±2e, 1 (2.16) wi th their hole (electron) counterparts m , ± e , and ns respectively. The G L theory is formulated in terms of complex order parameter 4>(r) which may be written as <p{r) - \<P(r)\ei@. (2.17) where G is the phase and \(f>(r)\ is the modulus of order parameter <p{r). <f>(r) plays a role in superconductivity similar to the role of wavefunction in quantum mechanics. The superelectron density is given by nl = | 0 ( r ) | 2 . (2.18) Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 11 and |<z>[ increases from zero as we go below the crit ical temperature T c . Ginzburg and Landau (G-L) assumed that close to the transition (from superconducting to normal) temperature T C ) the Gibbs free energy per unit volume G3 ((f)) may be expanded as a local functional as G s [ 4 > ] = G n + -L-X K - ^ V + e ^ J ^ l ' + ^ + a l ^ + ^ l 4 , (2.19) where Gn is free-energy density of normal state, ' i f ' is the applied field and 'a ' and '6' are functions of temperature only. To get <f>(r), Gs[4>] is minimized wi th respect to variations in the order parameter <t>(f). Taking derivative wi th respect to <j>* wi th <f> constant, the first G L equation gives |-*ftV + e*Xf <j>-a(j>- b\<t>\24> = 0. (2.20) The free energy is also a minimum wi th respect to variations in vector po-tential A and we get the second G L equation V x ( V x A) + ihe*{<j)*V<t> - G>VG6*) + —;A\<j>\2 = 0. (2.21) The Eqs. 2.20 and 2.21 are the two coupled differential equations which can be solved to determine the properties of superconductors. In the later sections, we wi l l see that the constants in E q . 2.20 and E q . 2.21 naturally lead to spatial dependence of the order parameter on the scale of £ and a dependence of the magnetic field on the scale of XL-2.3 Zero field case near superconducting boundary In this section we consider a superconducting material having a vacuum boundary. To determine the functional dependence of order parameter <f>(r) on depth, one must impose some boundary conditions. Assuming that no current flows into the surface, according to T inkham [25] and de Gennes [26], fir* e* r \ ± - V A)(j) Ki c ) 0. (2.22) A can be set to zero in the absence of a magnetic field since B = V x A. Then, from the second G L E q . 2.21, we obtain, -V2<t> + a<t> + b\<t>\2<j> = 0. (2.23) 2m Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 12 We can choose 4> to be real as the phase is constant. We assume that the right half-space (x > 0) is filled with superconductor and that the left side half-space is vacuum and thereby <j) is only function of x and V has only x component and the Eq. 2.23 may be written as - ^ S + « * + ^ = ° - <2-24> Changing variables by setting <j) = (^jf^ 2 f and setting u = | where the Eq. 2.24 becomes 0 + / ( l - / 2 ) = O. (2.26) To solve the Eq. 2.26, one needs to impose boundary conditions on / . There is some uncertainty here over the exact boundary condition in a vacuum-superconductor interface. One such condition, proposed by Poole [27] is that the order parameter (and thereby / ) is zero at surface and takes its full value deep inside the superconductor, i.e, yielding where / f a = 0) = 0, / f a = oo) = 1 (2.27) / = tanh , (2.28) rfoo-^)*. (2-29) giving <t> = 0ootanh ^ - | ^ . (2.30) An alternative assumption mentioned by Tinkham [25] is that the order parameter remains at its maximum value throughout the sample, even at boundary, i.e, / = 1 x > 0, (2.31) giving <t> = too- (2-32) Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 13 I wi l l use both forms (Eq. 2.30 and 2.32) to extract £ and \L. Even though the above <j>{r) has been formulated in absence of magnetic field, the functional form of <f>(r) is valid even in presence of an applied field wi th £ depending on field and temperature. We can use E q . 2.30 and E q . 2.32 to determine the magnetic field inside the superconductor. For our semi-infinite geometry and for a constant mag-netic field B0 outside superconductor (x < 0), the vector potential may be written as A = Ay(x)j (2.33) where Ay(x) = xB0 + AQ, x < 0 (2.34) and A0 is a constant for continuity of equation at x = 0. C . Poole as-sumed [27] that the phase of the order parameter is constant everywhere throughout the superconductor such that there is no current flowing into the superconductor. The constant phase may thereby conveniently be set equal to 0. It follows from the current density equation that J y { x ) = - e * 2 ^ 2 A y { x ) . (2.35) The second G L equation yields <PM*) _ Poe*2\<j>(x)\2 ~d^~ ~ m* Ay{X>- { 2 - 6 b ) Now, for the order parameter 4>(r) is given in E q . 2.30, the E q . 2.36 may be written as, d2Ay{x) _ dB(x) dx2 dx where A£ = ™ . | 2 - (2.38) coo B y differentiating E q . 2.37 wi th respect to x and substituting Ay(x) wi th the expression from E q . 2.37, we get a differential equation in magnetic field B(x) as <PB(X) 1 , , / x \ n ( , 2v/2 , / y/2x\ dBix) l n n n . = A i t a n h m ) {x)+~r v r ) (2-39) Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 14 The exact solution of Eq. 2.39 is too complicated for any algebraic manipu-lation and it was solved numerically for various values of parameters £ and A L with the boundary conditions B(x = 0) = B0, B(x = oo) = 0. (2.40) Mathematically, we set B(x = 20AL) = 0 since at a depth of 20AL, B(x) = e - 2 0 if field were purely exponential. In our case, B(x) becomes purely exponential at x > 2\/2£. A phenomenological function, that agrees well with numerical solutions, is found to be B(x) x exp >f% tanh ( ^ ) A L 1 > tanh V2Z (2.41) The difference between numerical solution of Eq. 2.39 and its phenomeno-logical fit is < 1% of applied field suggesting a fairly close approximation to the actual solution. Very close to the sample surface, i.e x <C £, tanh(x) « x, exp[x] « 1 + x, and Eq. 2.41 may be written as B(x) = B0exp 1 + c c2 I , , (2.42) (2.43) where c = yj£ A L ' x V2£ (2.44) The ratio of the second and third term in Eq. 2.43 is always less than 1 when 3c 2y < 1, Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 15 3 c > 1, x > 3 — . A L (2.45) For a specific case of £ = 100A and XL = 1500A, the E q . 2.45 yields x > 20A. So, the phenomenological function increases slightly (<C 1%) within range 0 < x < 3f£ but drops gradually as expected from the numerical solution. Alternatively, for large x tanh V2£ and B(x) may be written as where B(x) = B0 [1 + A ] exp A = exp A L x i . (2.46) (2.47) (2.48) Thus, B(x) is an exponentially decaying function but wi th a higher ampli-tude than B0 if extrapolated to x = 0. This is reasonable since at x > 2 \ /2£ , order parameter reaches its bulk value and exponential decay of field starts from there and very close to surface, B(x) varies slowly i.e, almost flat. One such numerical solution along wi th its phenomenological fit is shown in figure 2.2. As we observe from E q . 2.41, the order parameter <j>(r) attains it's bulk value inside a superconductor on the distance of the order £. W i t h the constant order parameter, given in E q . 2.32, the E q . 2.36 yields d?Ay(x) _ since, dx2 (PBjx) dx2 dAy(x) dx ^2 A/C2")' = B(x). (2.49) (2.50) W i t h the boundary condition (Eq. 2.40), the E q . 2.49 has the solution B(x) = B0e~^ (2.51) Bo th magnetic field expressions in Eqs. 2.41 and 2.51 wi l l be used to extract £ and A L -Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 16 I i i i i i i i i 0 2000 4000 6000 8000 Depth (°A) Figure 2.2: (a) Numerical solution ( B n s o j ) and it's phenomenological fit ( B p j i e n as given in E q . 2.41) of the magnetic field in G - L theory, (b) Dif-ference between them. £ = 150A and A L = 2100A in both figures. Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 17 2.4 B C S penetration depth The basic idea for B C S superconductivity is that an attractive interaction between electrons, regardless of their strength, can bind the electrons into pairs [6]. We consider a case for only two electrons added to the Fermi sea. The first electron attracts positive ions and these ions, in turn, attract the second electron giving rise to an effective attractive interaction between electrons. Due to the movement of ion cores, phonon waves are generated and the interaction between electrons is thereby phonon mediated. The total energy of the electron system is minimized when there are Cooper pairs compared to a Fermi gas with no correlation. The center of mass of a Cooper pair is zero since the electrons tend to have opposite momenta and spin \hk, T) and | — hk, i). Due to this opposite momenta and spin, it is labeled s-wave pairing since the relative angular momenta of the two electrons is zero. Al though NbSe2 is thought to be a conventional s-wave superconductor, recent works suggest that an other form of pairing involving non-zero angular momenta (eg. d-wave wi th L = 2, S = 0) may be involved in the superconductivity other materials such as YBa2Cu307. However, the macroscopic phenomenology of the resulting superconducting state, treated earlier by Ginzburg-Landau equations, is basically the same. One important consequence of the B C S theory is that the presence of a momentum dependent energy gap at the Fermi surface so that an amount of 2 A ^ energy is required to break a Cooper pair. The energy gap is schematically shown in Figure 2.3. The gap is opened at the Fermi energy as the temperature is lowered below the critical temperature. In the weak coupling l imit , where the gap A is much smaller than the characteristic phonon energy hwo, ^ = 3 . 5 2 . (2.52) KBJ-C The numerical factor 3.52 is well tested in experiments and found to be reasonable, in purely B C S type interactions. It is interesting to note that in NbSe2, it has been suggested that the upper energy band behaves largely like B C S 1 t h e constant factor in E q . 2.52 being « 3.9 but the smaller energy gap follows non-BCS behavior [22]. A ( T ) remains fairly constant unti l the phonon energy becomes enough to thermally excite the quasiparticles. Near the transition temperature Tc, A ( T ) varies as t § H - » ( - £ ) ' - T ~ T > Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 18 Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 19 and is graphically shown in F i g 2.4. Finally, we mention that the electronic properties of NbSe2 are anisotropic. This is largely due to the layered structure of the material. The degree of anisotropy is expressed in terms of a anisotropy parameter given by where ||a6(||c) indicates the field H perpendicular (parallel) to the c-axis of the sample and m* ,A ,£ and HC2 are G L effective mass, penetration depth, coherence length and upper crit ical field, respectively. (2.54) Chapter 2. Superconductivity in London and Ginzburg-Landau Theories 20 0.0 0.2 0.4 0.6 0.8 1.0 TIT c Figure 2.4: Temperature dependence of the superconducting energy gap in the weak coupling limit of BCS interaction. 21 Chapter 3 Time Dependence of Polarization and Relaxation In this chapter, I shall discuss three important processes that lead to time dependence of the nuclear polarization. • Coherent transfer of polarization between 8 L i and N b nuclei, through the magnetic dipolar interaction, but no relaxation of 8 L i . • Korr inga relaxation mechanism at high magnetic field due to scattering of conduction electrons off the nuclear spin. • Low field relaxation from fluctuating nuclear dipolar interaction be-tween the 8 L i and, the host, N b nuclear spins. 3.1 Evolution of 8 L i Spin Polarization in NbSe2 in the Absence of Relaxation For simplicity, we consider the time evolution of spin polarization in a sit-uation where the 8 L i is coupled to a single N b spin through the magnetic dipolar interaction. 8 L i nucleus interacts wi th a few host N b nuclei. How-ever, interaction of one pair (involving one 8 L i and one N b nuclei) is almost independent of the interaction of other pairs and it w i l l be sufficient to dis-cuss one pair only. The applied magnetic field direction is defined to be z direction Only that component of spin polarization is of interest since that is the observed quantity. The evolution of 8 L i polarization, in the z direction, is given by where p is the density operator (with p0 being its ini t ia l value) of the sys-tem and Iz is the ^-component of the 8 L i spin operator and H is the spin Pz(t) = (Iz) = Tr{p.Iz] (3.1) Chapter 3. Time Dependence of Polarization and Relaxation 22 Hamiltonian. Considering the case where we ignore S spin dynamics and any diffusion of 8 L i , at 10K and below, the effective Hamiltonian may be written as H = HI + HS + HIS, (3.2) where I and S are 8 L i and host (Nb) spins respectively. Hi and Hs are the Zeeman plus quadrupolar parts of Hamiltonian and His is the dipolar interaction between I and S. For simplicity, we treat 8 L i to be of pseudo spin- j nuclei and thereby having no quadrupolar interaction. The gyromag-netic ratio of this pseudo-spin ^ particle would st i l l be that of 8 L i . Nb would be treated wi th it 's real spin | . The Hamiltonian may be written as, in frequency units, j = -iLihB - lnJzB + vqJ2 + /3His, (3.3) where TLJ = 6 . 3 M H z / T and 7 r a = l O M H z / T are gyromagnetic ratios respec-tively and Ug = 1.325MHz [28] is the quadrupolar frequency of N b , /3 is the strength of dipolar interaction. For simulation of polarization, /3 was cho-sen to be 0.49 kHz since that is a rough estimate of the dipolar interaction strength (see chapter 5). His is the classical dipolar Hamiltonian and can be written as [29], His = A + B + C + D + E + F, (3.4) where, A = IZJz(l-3cos2d), B = - i (I+J- + r j + ) ( i - zcos2e), C = ~(I+JZ + IZJ+) ainOcoade-**, D =-\ (rjz + IZJ~) aindcoaOe**, E =--^I+J+)sin2ee-2it F =-h<I-J-)8in20e2i't>, (3.5) wi th I+ and I~ being spin raising and spin lowering operators, respectively, for 8 L i while J+ and J~ are spin raising and spin lowering operator, re-spectively, for N b , 9 and <j> are the are the polar and azimuthal angle for the angle between the vector connecting 8 L i and N b nuclei and* applied field Chapter 3. Time Dependence of Polarization and Relaxation 23 direction. The effect of raising and lower operators on eigenstates are given by the relationships J+\j,m) = hyjj{j + 1) - m ( m + m + 1), J-\j,m) = hjj(j + l)+m{m-l)\j,m-l), (3.6) where j is the spin quantum number for operator J. Polarization Pz(t) from equation 3.1 may be written for a spin-5 probe as [30] p Pz(t) = -TrJ2\(m\az\n)\2exp(iumnt) m,n = 5 Z { I H ^ I « > | 2 + 2 \{m\az\n)\2cos{unmt)} (3.7) where P is the total polarization at t = 0 and N is the number of states available, |m), \n) are the eigenkets of TL wi th energies Tvujm,fluin. B y averaging E q . 3.7 over L i t h i u m lifetime as, PZ(B) = - / Pz(t)e~rdt, (3.8) T Jo where r = 1.2s is the 8 L i Ufetime, we may get polarization along z direction PZ{B) which varies wi th magnetic field, as the resonance frequencies ujmn vary according to magnetic field. PZ(B) for equation 3.7 is shown in F i g 3.1 and in F i g 3.2. Note that the polarization at zero magnetic field is almost zero as the energy levels of 8 L i and that of N b are degenerate and flip-flop process in spin states can go on without any energy expense. The polarization is near 1 except near the magnetic fields where the Zeeman and quadrupolar interaction energies of 8 L i and N b match. Also, it may be noted from F i g 3.2 that the polarization takes nearly its full value over a range of 0-20G. The range scales wi th the dipolar interaction strength /?. Al though we have considered an interaction of a spin-^ nucleus wi th N b , a density matrix, wi th 8 L i spin being 2, wi l l cause changes in the amplitudes but not in the frequencies (see Ref.[31] and references therein). In particular, the ratio between time dependent and time independent parts of polarization wi l l depend on the ini t ia l density matrix. Chapter 3. Time Dependence of Polarization and Relaxation 24 00 d bO .3 to O JB CO d o bO el • t-H I O rjQ a O CI o a CN d o d (a)d o ••§ (S3 '§ f—H O ft . <S bO & .g •a 3 S "53 EH rt . . bO '-J Pi CO °5S Chapter 3. Time Dependence of Polarization and Relaxation 26 3.2 Korringa Relaxation & Knight Shift In metals, there is a magnetic coupling between nuclear spins and the con-duction electrons. If the nuclear and electronic moments are far apart, their magnetic interaction is given by the dipole interaction, H = Pe.pn _ 3(pe.r)(pn.r) where r is radius vector from nucleus to electron and pn and pe are nu-clear and electronic moments respectively. There is also a Fermi contact interaction between electron and nucleus given by Hhf = -y 7 n7e<5(r) / .5 , (3.10) which depends on the electron density at the site of the nucleus. Generally, this is much larger than the dipolar interaction. As the conduction electrons move through the crystal, a specific nucleus experiences a magnetic coupling wi th many electrons and an effective inter-action is found by averaging the expression in equation 3.10. In the absence of an external field, there is no preferential direction for electron moments and thereby zero average coupling wi th the nucleus. In contrast, application of magnetic field polarizes the electrons slightly, giving a small but finite av-erage hyperfine field, through s-state wave function coupling to the nuclei. Since the s-state interaction leads to the nucleus experiencing a magnetic field parallel to the electron spin polarization direction, the effective field at the nucleus is increased. This translates into an increase of the resonance frequency in a metal, compared to non-metallic insulators and is given by ojm = u>0 + Aoj = ^(H0 + AH), (3.11) where um is the observed Zeeman frequency in metal, u0 is the corresponding frequency in a non-metallic compound, H0 is the applied field and AH is the increase in magnetic field at the nucleus due to its interaction wi th polarized conduction electrons. is the fractional change in the resonance frequency and is called the Knight shift. In a resonance measurement, 8 L i is inserted into a sample and a radiofre-quency(RF) magnetic field is applied perpendicular to the applied static field H0. A s H0 is stepped through different values, the 8 L i polarization is unaf-fected unless the R F frequency is close to ujm causing a loss of polarization of 8 L i . O n or near resonances, the 8 L i spin polarization wi l l precess about Chapter 3. Time Dependence of Polarization and Relaxation 27 an effective field which is counted with respect to the applied field. Since, electrons are ejected preferentially in the direction of 8 L i spin, a dip (i.e resonance) in the beta decay asymmetry is observed at ujm. The shift in the resonance frequency is proportional to the degree of electronic polarization which scales with the magnetic field. In a normal metal, where the Pauli spin susceptibility is T-independent, the fractional shift in frequency ^ is temperature independent. We now discuss the magnetization for the nuclei ( 8Li) after they are inserted into the sample. Initially, the 8 L i spins are highly polarized and thereby out of temperature equilibrium with the lattice. Eventually, they relax to a common temperature of the lattice. We consider the relaxation of a system of nuclear spins whose Hamiltonian H has eigenvalues En with an occupation probability pn. If a system of N identical spins ( 8 Li in our case) is in thermal equilibrium with the lattice (Nb) at temperature T, then the rate of 8 L i making downward transitions in energy would be equal to the rate of 8 L i making upward transitions. We also assume that the transitions between every pair of energy levels are in equilibrium, i.e, VmWmn = PnWnm, (3.12) where Wmn the probability per second that the lattice induces a transition of the system from \m) to \n). Put in another way, the frequencies of transition in either direction of equilibrium, between any two energy states, are equal. This is also known as the principle of detailed balance [32]. Under these conditions, the rate of change of temperature of the 8 L i system may be written as [24] dA = ik^l, (3.13) dt Tx v ' where Wmn(Em — En) T . - 2 ' < 3 ' 1 4 ) with /3L = the thermal equilibrium temperature assuming that the transitions occur in 8 L i being thermal equilibrium with the lattice and is the relaxation rate of 8 L i polarization. We also assume that transitions are rapid enough to guarantee thermal equilibrium and after each lattice transition, the nuclei readjust among their approximate energy levels so that the lattice is once again in thermal equilibrium. For coupling to conduction electrons, the process may be viewed as being some electron with state |k, s) scattering off a 8 L i nucleus, with energy state Chapter 3. Time Dependence of Polarization and Relaxation 28 |m), to |k',s') while the 8 L i nuclei makes it's transition to state |re). The transition rate per second may be written as W m k s , n k v = y \(mks\V\nk's')\2 S(Em + Eks - E n - £ k V ) , (3.15) where the 8 L i - e _ wavefunction is \mks) = |m) |s)u k ( r )e i k r , (3.16) and V is the interaction potential given by equation 3.10. The delta func-tion in equation 3.15 makes sure that the total energy is conserved in the transition. The transition rate for 8 L i may be found by summing over all the possible electron configurations, W m n = ^ Wmk,s;nk',s'PksO- -Pk's'), (3-17) k,s;k',s' with p k s being unity if ks is occupied and zero otherwise. Writing the equation 3.17 in terms of energy and replacing pks by the Fermi function, Wmn may be written as 1 Wmn = ^^hz^lYJ{rn\Ia\n){n\Ia\m) y ct x J(\uU0)\2)2Ep2(E)f(E)[l-f(E)]dE, (3.18) where 7 e and jn are gyromagnetic ratios for electrons and 8 L i nucleus, re-spectively, la's are the three spin components of 8 L i , p(E) is the density of states, (|u k(0)| 2)£ is the average density of electrons, with energy E, at the nuclei position and f{E) is the Fermi occupation probability where f(E) = - J E ^ • (3-19) e kT +1 The Fermi function f(E) and it's two derivatives l-f(E) and f(E)[l-f{E)] are shown in Fig 3.3. As f(E)[l — f(E)] peaks up within a width of kT of Fermi energy Ep, it may be approximated as f(E)[l - f(E)] ~ kT6(E - EF). (3.20) 'For a detailed discussion, see C P . Slichter, Principles of Magnetic Resonance (Springer-Verlag, New York, 1990), p 151-156. Chapter 3. Time Dependence of Polarization and Relaxation 29 B y util izing equation 3.20, the equation 3.14 may be written as _ L = ^ f t s ( 7 n 7 e ) 2 < l « 2 ( o ) l > i y W f c r . (3-21) It may be noted from the E q . 3.21 that the Korr inga relaxation rate is dependent on the electron density at the nuclear position and on the density of states available at the Fermi energy and on the lattice temperature. Since the energy exchange between the 8 L i and conduction electron is very small compared to kT, most electrons can't take part in this interaction since they have no empty states nearby to make transition into. Thus, only electrons within fcgT of the Fermi surface take part in such process. Note that the quantity ( 1 ^ ( 0 ) 1 ) ^ in E q . 3.21 is also involved in the Knight shift, *"(\ul(0)\)EFX%, (3-22) H0 3 where xi is the total spin susceptibility of the electrons defined in terms of total z-magnetization of the electrons, TiJ v ia the equation, Wz = XSeH0. (3.23) B y using the equation 3.22 and the expression for a Fermi gas of noninter-acting gas, Xi = ^fp(EF), (3.24) the E q . 3.21 may written as, 2 TTi fAH\ \H0J One can define a Korr inga ratio as, s where IC = ^ (3-26) 47rA; The Korr inga ratio in E q . 3.26 is close to 1 for a perfect metal. Note that, the Korr inga relaxation is the only dominant relaxation mech-anism at high applied magnetic fields (H > IT). A t low fields, dipolar in-teraction between host nuclei Nb and probe 8 L i turns out to be far more important than the hyperfine interaction and is discussed in the next sec-tion. Chapter 3. Time Dependence of Polarization and Relaxation 31 3.3 Low F ie ld Sp in Relaxat ion from F luctuat ing D ipo lar Fields A t low fields, the interactions between 8 L i and N b are dipolar in nature. Random thermal fluctuations of N b nuclei create random fluctuating ef-fective field causing 8 L i polarization to relax. For a randomly fluctuating dipolar field, we may assume an effective interaction Hamiltonian, « l ( * ) = - 7 n f i £ HS)h- (3-28) q=x,y,z We assume that the fields Hqs take only two values ±hq, w i th an autocor-relation function [29], Hq(t)Hq,{t + T) = h2qe~^, (3.29) where r 0 is the "correlation time" which roughly estimates the time for Hq(r) to jump from ±hq to Thq. For simplicity, we assume 8 L i to be a pseudo spin- j system. For such a system, the relaxation rate ^ is given by - J - = 2Wi i . (3.30) T l 2~2 K 1 In general, the transition rate Wkm is given by [see appendix A for a detailed discussion] Wkm = J m f c ( g ~ f c ) , (3.31) where the spectral density Jmk(uj)du gives the interaction strength of 7Yi(r) over a frequency range u> + dw. A typical density Jmk{^) is shown in F i g 3.4. Jmk{u) has the Fourier transformation /oo Gmk(T)e-^Tdr, (3.32) -oo where Gmk{r) is known as the "correlation function" of Hi(t) and is given by [29] Gmk(r) = (m\ni(t)\k)(k\Hi(t + T)\m) (3.33) Gmk{r) gives us the functional dependence of Hi(t) wi th Hi(t + r ) . For a randomly fluctuating field as given in E q . 3.28, Wmk(r) i n equation 3.31 may be written as Wkm = ^J2fmk(m-k). (3.34) n Q Chapter 3. Time Dependence of Polarization and Relaxation 32 Using equations 3.28 and 3.33, the equation 3.32 may be written as /oo Hq(t)Hq>(t + r)e-^dr. (3.35) -oo Using the exponential correlation given in equation 3.29, one may obtain from equation 3.35, fmki") = 7 ^ 2 | ( m | / g | f c ) | 2 ^ — ^ (3.36) l ~r u/ t 0 and transition rate Wkm may be written as 2TQ m Y.llhl\{m\Iq\k)? L q 1 + (m - fc)2T(2 ' (3.37) For a strong static field along z, non-zero components in expectation values of (m\Iq\k) are \(m\Iq\k)\ = i , q = x,y \{m\Iz\k)\ = 0. (3.38) Due to the randomness of fluctuating field, we assume that al l the compo-nents of the field are equal, i.e, hx = hy = hz = h0, giving hi = \hl Using equation 3.30, one may obtain, where A = \J\'ynh0 and u0 = m — k = jH0 is the Larmor frequency at applied field H0, for a spin-^ system. Equation 3.39 is an important relationship between and applied field. The temperature dependence of r 0 induces temperature dependence into relaxation rate. Equation 3.39 wi l l be used to extract values of A and r Q in a later chapter. 34 Chapter 4 Experimental 4.1 Beamline Properties and Spectrometer In our experiment, a beam of radioactive nuclei [33] is given a significant polarization and then implanted onto the sample. In low fields and low tem-peratures, 8 L i loses its nuclear polarization primarily by exchanging energy with host nucleus Nb. Loss in polarization is detected via the /?-decay 4.1 8 L i 8 Be + ue + e~ (4.1) where e~ is emitted preferentially in the direction of 8 L i spin at the time of its decay. The asymmetry (i.e directional dependence) in the decay of nuclear polarization contains information about the local electronic and magnetic environment of the sample. The isotope separator and accelerator (ISAC) delivers a continuous beam low energy (« 28KeV) 8 L i at a rate « 10 7/s. 8 L i is a spin 2 nucleus with a mean lifetime T = 1.21s, a gyromagnetic ratio 8 7 = 630.15Hz/G and a small electric quadrupole moment Q = +33 mB. The unpolarized 8 L i is polarized as it passes through the "optical pumping region", shown in the figure 4.1. The ion beam is first neutralized as it passes through a Na vapor cell. During it's passage through this region, the 8 L i atoms are excited by a dye laser (A « 671 nm) tuned to the DI atomic transition 2 Si / 2 —*2 P\/2 of 8 Li(F ig. 4.2), where the outer shell electron is excited from / = 0 to I = 1. The ground state and its first excited state energies are further split by hyperfine interaction between electron orbital angular momentum (Z = 0,1) with total angular momentum = (2 ± ^) = §, | . For circular polarized light with positive helicity A m f = +1 for excitation whereas for spontaneous decay Amp = 0, ± 1 . After 10-20 cycles of absorption and spontaneous emission, a highly polarized state with F = ^,m,F = § i.e, a well-defined nuclear spin state with spin 2 is obtained. The polarization may be obtained as high as 80% [34]. The polarized beam is then passed through the He vapor cell to knock off one electron and thereby reionizing a fraction of it, so that it can be guided electrostatically into one of the two experimental stations. The fraction of p - N Q R spectrometer He re-ionizer optics cell bench Figure 4.1: A schematic of the experimental layout. The 30 KeV 8 L i + ion beam is neutralized in the Na cell and then reionized in the He cell. In between, the beam is optically pumped with a laser tuned to the DI optical transition of the 8 L i atom. The resulting polarized beam is guided to /3-LCR spectrometer. 00 Chapter 4. Experimental 37 the beam that does not get reionized by the He vapor cell, goes straight (i.e, not electrostatically bent) to the "neutral beam monitor". The neutral beam provides an independent measurement of overall asymmetry, which remains constant for a steady 8 L i ion beam. The polarization direction remains unchanged by these bends. Two of the stations, labeled as "low-field region" and "high-field spectrometer" in figure 4.1 are used for research in /3-NQR and /3 -NMR, respectively. In both spectrometers, a small Helmholtz coil can be used to introduce either a C W or a pulsed R F magnetic field Hi that is perpendicular to static magnetic field H0. The maximum value of H0 are 9T at high-field spectrometer and 20mT at the low-field spectrometer. The time evolution of 8 L i polarization is monitored using fast plastic scintillators placed forward and backward wi th respect to the ini t ia l spin direction. A typical frequency spectrum is shown in F i g 4.4 A schematic diagram of /3-NQR spectrometer (where most of the data were taken) is shown in Figure 4.3. A coil in an approximate Helmholtz configuration is used to apply a small oscillating magnetic field perpendic-ular to static magnetic field. Electrons emitted from 8 L i pass through thin stainless steel windows out of U H V chamber and reach detectors labeled as L and R . Detector telescopes consist of a pair of plastic scintillators wi th dimensions 10cm x 10cm x 0.3cm and are located outside of U H V chamber. A set of three coils are utilized to apply a static magnetic field (0-20mT) along ini t ia l polarization direction or to zero at the field within 0.005mT. The energy of the beam was set to be 28KeV corresponding to an average implantation depth of about 2000A. However, it has been demonstrated that a beam energy as low as lOOeV or less is achievable [35]. 4.2 The sample The crystal structure for NbSe2 is shown in F i g 4.5. The polytype is 2H-NbSe2 where the integer 2 stands for the number of layers in a unit cell and hexagonal crystal symmetry [36] is indicated by H . The NbSe2 layers are weakly coupled by Van der Waals interaction whereas, wi th in the layers, Nb and Se atoms are covalently bonded. Due to the weak coupling between the layers, it is easy to cleave the sample along a plane parallel to the layers. In our experiment, a freshly cleaved sample was used for measurements. It was exposed to air for only a short period of time ( « 30 minutes) before being loaded into the vacuum. Figure 4.3: A schematic of the spectrometer for /3-detected nuclear resonance. The spin polarization is perpen-dicular to. the beam direction. The principal axis of the electric field gradient at the 8 L i stopping site must have a component along i in order for a signal to be detected at zero applied field. Chapter 4. Experimental 39 • Down helicity o Up helicity i i i i i 18900 18910 18920 Frequency (KHz) Figure 4.4: (a) Polarization PZ(B) for both helicities (b) Normalized asym-metry is found by subtracting "down" helicity from "up" helicity to remove the background effect. Chapter 4. Experimental 40 Figure 4.5: Top: NbSe2 cross-section (in 1120 plane). Bottom: Three di-mensional structure of 2H-NbSe2-Chapter 4. Experimental 41 4.3 Measurement of Polar izat ion in Low F ie ld In this section, we discuss the results of measurements at low applied mag-netic fields (15 G - 45 G) at T = 8 K . The same NbSe2 sample was used to measure polarization decay. Methods for measuring P(t) are discussed briefly in the next two sections. 4.3.1 P(t) in short pulse method In this method, 8 L i beam was kept on for a shorter time period (0.5s) and the polarization p(t) was measured after beam goes off for about 10s. The functional dependence of p(t) is given by where T\ is the nuclear spin lattice relaxation time. From an experimental perspective, this method has the drawback that significant amount (> 90%) of beam time is lost and since the number of 8 L i is decreasing when the beam goes off the statistical errors gets larger as time goes on. The "long pulse" method was developed to improve the quality of data. 4.3.2 P(t) in Long pulse method In this method, the 8 L i beam was on for 4s and went off for 8s and the asymmetry was measured over this whole time range. This method is advantageous since the spectra contains about eight times the number of decay events compared to the short pulse method. Thus it is better for measuring slow relaxation times. Furthermore, it is also better for measuring short relaxation times since one observes polarization back to earlier times. The only disadvantage is that the P(t) is convolved with the pulse shape. Let Rodt' (R0 is the constant 8 L i incoming rate) be the number of 8 L i arriving in the sample at time interval (t',t! + dt') and surviving unt i l time t is The number of 8 L i that haven't decayed in the target at time t is given by: p(t) =p0exp - — (4.2) (4-3) N(t) = TR0 [1 - e x p ( - t y r ) ] . (4-4) Chapter 4. Experimental 42 Similarly, the average polarization at time t is given by: R o P o f* expj-t'Mexpl-t'/^dt' m = W) ' T 1 — exp[—t/T\ where Thus, the average polarization is time dependent for times on the scale of T'. The equation 4.5 is used throughout our calculation to fit the experimentally observed normal state polarization. As one would expect, the polarization starts off at its maximum P o and relaxes towards its equilibrium value of P0-7 on the time scale of r ' . If the beam goes off, the polarization wi l l relax from that value wi th a relaxation time T\. Thus, in a T\-measurement with a beam pulse width A , we expect the following form: P p u l s e O = PstepW> f o r 0 < t < A , ( t - A ) P s t e p ( A ) e x P for t > A . (4.7) Chapter 4. Experimental 43 Figure 4.6: (a) Beam is on between time (0,4)s. (b) Schematic polarization P(t) as a function of time. 44 Chapter 5 Measurements and Results 5.1 Normal State in High Magnetic Field 5.1.1 Korringa Relaxation Due to its layered crystal structure, NbSe2 shows an array of interesting properties. The atoms are metallically and covalently bonded wi thin a layer but experience only weak Van der Waals force between adjacent layers. This produces strong anisotropy in al l the electronic and mechanical properties such as penetration depth, coherence length and effective mass of carriers. Measurements of relaxation, in high field, were done using short pulses 0.5s) of 8 L i and polarization was measured as a function of time after the pulse. No R F field was present during the experiments. Fi ts to a single expo-nential were obtained as shown in F i g 5.1. The relaxation rate was observed to be a linear function of temperature as expected from Korr inga relaxation. The anomalously small proportionality constant 9(1) x 1 0 _ 5 K - 1 s _ 1 (see F ig . 5.2 )is 10 times smaller when compared to relaxation of 8 L i in A g [37]. This suggests that 8 L i in NbSe2 occupies a site in the Van der Waals gap where overlap wi th the conduction band is small. The extrapolated fit gives a non-zero relaxation rate at T = OK which we attribute to residual effects from dipolar interaction, which dominates Korr inga relaxation at low field but are highly suppressed at high field. A s we may observe from F i g 5.2 that the lowest relaxation time measured is ~ 100s whereas the 8 L i lifetime is s=s 1.2s and thereby we are in the l imit of our measuring capacity. This also contributes to the non-zero relaxation rates near OK. 5.1.2 Dipolar Broadening of the Resonance Nuclear resonance was observed in an applied static field H0 v ia the detec-tion of time averaged polarization as a function of applied frequency. The position and shape of spectra gives information about the local magnetic field and is attributed primarily to the nuclear dipolar interaction wi th N b . Typica l spectra are shown in F i g 5.3. Chapter 5. Measurements and Results 45 Figure 5.1: The time evolution of normalized spin polarization 8 L i in NbSe2 in a magnetic field of 3T applied along the c-axis. The time differential measurements were done in short pulse mode. The solid lines are fits to a single exponential without any background. Temperature (K) Figure 5.2: Comparison between spin relaxation of Li in Ag and NbSe2 as a function of temperature. The applied field along c-axis is 3T. Chapter 5. Measurements and Results 47 tmmmmmmm B= 3T Bparc -15 B=10mT Bparc B=10mT Bperpc 10 15 v-vQ (kHz) Figure 5.3: The /J-NMR resonance in NbSe2 as a function of field and ori-entation. The top two scans were taken with the field parallel to the c-axis but at very different fields; whereas, the bottom scan is with the field per-pendicular to the c-axis. The temperature is 10K in all cases and so there is no line broadening due to 8 L i motion. Chapter 5. Measurements and Results 48 Gaussian fits of the spectra indicate a F W H M (Full W i d t h at Half M a x -ima), i.e, the dipolar width to be ~ 2.4kHz, in case of B ± c. The width is very weakly dependent on the radio frequency (RF) power level which indicates that the measured linewidth is close to the intrinsic width. The linewidth also indicates the strength of dipolar interaction i.e. the energy of interaction would be order of HA. Due to the hexagonal crystal structure of 2H-NbSe2 and small electric quadrupole moment of +33 m B , we would have expected the resonance line to be split by the quadrupolar interaction present at any non-cubic site. The absence of resolved split t ing indicates that the electric field gradient at the 8 L i site is at least 10-100 times smaller than observed in most other non-cubic structures [38]. The l inewidth is attributed mainly to nuclear dipolar broadening plus some unresolved quadrupolar splitting. The small asymmetry in the op-posite helicities in F ig . 4.4(a) gives an upper l imit of quadrupolar strength (< lkHz) . The Se (77% abundance) should have a small dipolar interaction wi th 8 L i compared to the interaction wi th Nb since the magnetic moment for Nb is 6.8/XAT and the moment for Se is 0 .9 /XJV , where piy is the neutron magnetic moment. The top and middle panel of the F i g 5.3 are almost identical, confirming that there is no significant contribution to the line broadening in high field as such an effect would scale with magnetic field. When the field is parallel to the a-b plane (bottom panel of F i g 5.3), the width is reduced to 1.5kHz. This is consistent wi th the fact that 8 L i occupies a site in Van der Waals gap since we would expect greatest line broadening from N b moments and quadrupolar splittings when field is parallel to c-axis. In this case, the secular term IZSZ in E q n 3.5 depend on a term involving (1 — 3cos2d) where 6 is the angle between the applied field and the L i - N b interaction direction. Chapter 5. Measurements and Results 49 5.2 Low Field Measurements The sample was then cooled to (practical base) a temperature ~ 3.4K, in zero magnetic field (to avoid any flux trapping) and then the 30G field was turned on. Data 5.4 were taken as the temperature was increased gradually. The F i g 5.5 shows the field dependence of relaxation rate at T = 8K. Da ta were fitted using a single exponential relaxation model and ^ vs 5 was fitted using a semi-phenomenological form given in E q . 3.39, ± ~ ^ (5 1) A ! - ^ (5.2) where the two free parameters are A , an effective nuclear dipolar field strength, and T c , the correlation time for this field to fluctuate. A s may be noted from E q . 5.2, the 8 L i relaxation rate is dependent on the local magnetic field 8 L i is in , a measurement of relaxation rate in the Meissner state wi l l thus be a sensitive probe of magnetic field and the parameters that control flux expulsion from the superconductor, namely \L and £. A t high field, ^BTC 1 and the equation 5.1 becomes A 2 Tx ( 7 B ) 2 T C (5.3) wi th the only fitting parameter being To extract both A and r c , a slightly modified form(equation 5.2) was used for fitting. The fitted values of the parameters are A 2 1 0.71 ± 0 . 0 1 kHz , 2.96 ± 0 . 1 kHz , (5.4) yielding A = (0.49 ± 0 . 0 4 ) kHz , TC = (0.34 ± 0 . 0 1 ) ms. (5.5) The fitted line is shown in F i g . 5.5. The values of the fitted parameters are reasonable since dipolar fluctuation time is in the order of ms [29]. 0 2000 4000 6000 8000 Time (ms) Figure 5.4: Three different field spectra. Chapter 5. Measurements and Results 52 5.2.1 Stopping Distribution To properly interpret our measurements, we also require knowledge of the 8 L i range and range distribution inside the sample. We calculated the depth profile of 8 L i using Numerical Monte Carlo program T R I M . S P [39, 40]. Pro-grams like T R I M . S P are largely untested at low energy, nevertheless, there has been been work done to test the code using /3 -NMR technique [41]. These results indicate that reliable values for implantation depth of 8 L i in NbSe2 may be obtained using T R I M . S P . The stopping distribution is shown in F i g 5.6. A phenomenological function involving a Gaussian and a beta function of form x <xm (5.6) , where N is a normalization constant, was used to fit the simulation. The fitted parameters are found to be a = 0.51, B = 3.54, a = 2490A, xm = 411 lA . (5.7) Although, theoretically it 's possible that a few of 8 L i stop beyond xm, the probability is vanishingly small. It is clear that equation 5.6 provides a very good approximation to the T R I M . S P result and has the advantage of greatly speeding up the fitting procedure. The normalized stopping distribution p(x) determines the probability per unit depth of 8 L i stopping at a certain depth x and was used to fit the observed spectra in the superconducting state and the normalization constant is chosen such that C oo p(x)dx = 1, (5.8) f Jo / o to ensure that the sum of probabilities of 8 L i stopping somewhere inside the sample is 1. 5.2.2 Analysis and fitting to determine XL and £ In this section, / 3 -NMR measurements of London penetration depth Ax, and coherence length £ are presented. To fit the experimentally observed P(t ) , we need the functional dependence of the relaxation rate ( ^ r ) on the magnetic field B and the magnetic field's (B(x)) dependence on depth x . Analysis is done using two models for the internal magnetic field. 0 1000 2000 3000 4000 5000 Depth (Angstrom) Figure 5.6: Monte Carlo calculated stopping distribution p(x). Maximum depth 8 L i reaches, at energy 30 K e V , is s=a 4100A and the profile is centered at « 1360A. GO Chapter 5. Measurements and Results 54 • In the simple exponential model, the order parameter |</>(r)| is assumed to be at it's full value everywhere in the superconductor. In particular, there is no suppression of \(j>{r)\ near the surface. In this case, the magnetic field is given by B{x) = .B0exp x L 'AI (5.9) • In the more complex model, |</>(r)| is allowed to vary near the surface. In particular, we assume a boundary condition <f> = 0 at x = 0. Then B(x) is approximated by, B{x) = B0 1 + < exp V2t tanh (-^) exp t a n h ( ^ ) A L - 1 ^ tanh ^  (5.10) In both analysis, stopping distribution of 8 L i is assumed to be the same as given in equation 5.6. It is necessary to account for the temperature dependence of the relax-ation rate at constant magnetic field due to changes in T c with temperature. Ideally, this should be done at higher field in the vortex state using time differential measurments of In the vortex state, the average field is al-most unchanged so that any T-dependence in ^ is due to change in the T c and not from changes in the magnetic field due to screening by the su-perconductor. However, without such measurements, it was necessary to use time integrated measurements of ^ in the vortex state. Fig 5.7 gives time-integrated polarization, as a function of radiofrequency, for three tem-peratures with an applied magnetic field H — 125G. We are essentially interested only in the "baseline asymmetry" which becomes smaller as tem-perature goes from 3.9K to 5.5K but rises up again as temperature goes up from 5.5K. A comparison of the average relaxation rate of 8 L i between the vortex state and the Meissner state is shown in Fig. 5.8. The vortex state relaxation rates were extracted from the time-integrated asymmetry where the "baseline asymmetry" encodes the information about relaxation rate — since the polarization in absence of any RF field is given by 5T' l r°° Z{T) = - Pz(T,t)exp T Jo Chapter 5. Measurements and Results 55 1 foo - / F 0 e x p T JQ t ' "Ti. exp C T Po-where 1 1 (5.11) (5.12) P0 is the ini t ia l asymmetry, T is the 8 L i lifetime and T\ is the relaxation time, ijr may be written as 1 Po (5.13) where, P0 is adjusted to match the relaxation rate we measure from time-differential measurement at T = IK. The 7 K temperature is chosen to be the point of reference as the zero-field critical temperature is sa 7.2K. A s can be seen from the figure that time-differential and time-integrated relaxation rates match quite reasonably (see F ig . 5.8) except at T = 8K. However, relaxation from the neutral beam monitor 1 indicates that the 8 L i + beam was steady. It is possible that the beam spot on the sample moved from its center position yielding lower count rates and thereby reducing in i -t ial asymmetry yielding a higher ^ - than we would have expected. The time-differential runs were analyzed using a single exponential relaxation. The fits yield an "average" relaxation rate of 8 L i in NbSe2 since 8 L i experi-ences a range of fields in the Meissner state. The temperature dependence of the vortex state has a peak in ^ at about 0.8TC which we identify as the Hebel-Slichter coherence peak [42]. However, a peak usually occurs in a conventional superconductor « 0.9TC [43] for a conventional s-wave su-perconductor. Recent microwave conductivity measurements in MgB2 [44] found a coherence peak at « 0.6TC, which is attributed to be the result of a second smaller energy gap. The temperature dependence of the relaxation rate, in vortex state, is accounted for via a phenomenological function s(T) where _ P(T) S(T) - P(T = 7 K y and the relaxation rate may then be written as Ti ~ g(B) x s(T), (5.14) (5.15) 'See Ch. 4 for a discussion Chapter 5. Measurements and Results 56 where B0, A L , £ , g(B) are applied magnetic field, London penetration depth and coherence length, field dependent relaxation rate(equation 5.1) respec-tively. The assumption that ijr may be written as a product of g{B) and s(T), is strictly valid only at high magnetic fields since r c shouldn't vary wi th magnetic field. However, for small changes in temperature (3.4K -7 K ) , E q . 5.15 may sti l l be considered as a good approximation. Now, we have the polarization function P(^,t) as function of relaxation rate jr and ijr is functionally dependent on magnetic field B which, in turn, depends on depth from surface x. The function P(^,t) may be symboli-cally written as P(jr(B(x)),t). The polarization at time t is the average polarization over the stopping distance of 8 L i , namely roo P(t)= P{x,t)p(x)dx, (5.16) Vo where p(x) is the normalized stopping distribution of 8 L i inside NbSe2 and is shown in F ig . 5.6. The function P(t) does not have a closed form time dependence since it is composed of a series of exponentials. Computer code was generated to evaluate P(t) and nonlinear least squares fitting algorithm, based on M I N U I T [45] was used to fit P(t) to the observed asymmetries of eight time-differential measurements (each representing a different temperature) wi th shared ini t ia l asymmetry and common ratio of penetration depth and coherence length. F i g 5.9 shows the fitted (with shared ini t ia l asymmetry) penetration depth and it's fit according to the empirical model for XL{T) using the "Exponential field" model for magnetic field B(x). F i g 5.10 gives the fitted XL{T) and £(T) according to the " f model". The fitted XL(T) and £(T) were further fitted wi th the T 4 model for XL(T) where, (5.17) The fitted results are shown in table 5.1. A s can be seen from the table 5.1, that the penetration depth in the case of "exponential B(x )" is larger than that from "£ model" . This is reasonable since the magnetic field in the second model doesn't change significantly up to a dep ths £ and the change in relaxation rate in this region is small. This in turn leads to a smaller value of A L since the average relaxation rate must 0.020 r 0.015 N *-> (D E I 0.010 1-CO < 33 S 3 • • O 3.9 K 5.5 K 6.5 K 0.005 V u 40 50 60 70 Frequency (KHz) 80 90 Figure 5.7: Three time integrated spectra at B = 125G. Solid lines represent fitted baseline. Resonance line shape changes due to the asymmetrical field distribution in the vortex state. 0.0040 r-0.0035 r-t^o 0.0030 |-E t 0.0025 1 0.0020 0.0015 • Vortex State O Meissner state, T<7K Polynomial fit, Vortex state 5 7 T(K) 8 10 11 Figure 5.8: Comparison of relaxation between the vortex state and the Meissner state. Figure 5.9: Penetration depth as a function of temperature for an exponential model for B(x). Chapter 5. Measurements and Results 60 Figure 5.10: Penetration depth and coherence length as a function of tem-perature. Chapter 5. Measurements and Results 6 1 Model for B(x) A L ( 0 ) Tc Exponential 2 4 3 4 ( 1 7 ) - 6 . 6 ( 1 ) £ model 1 7 9 5 ( 2 1 ) 1 4 6 ( 1 ) 6 . 8 7 ( 2 ) Table 5 . 1 : Penetration depth A L ( 0 ) and coherence length £ ( 0 ) at T = 0 remain about the same. The global fits in two models yield DF = 1 . 3 5 3 5 exponential B(x) - 1 . 3 2 6 8 £ model ( 5 . 1 8 ) with D F being the degree of freedom in the fit. Even though the 'goodness of fit' of both models are nearly same, the empirical T 4 model for A L fits significantly better in the "£ model", —— = 1 2 . 0 exponential B(x) Db = 0 . 8 4 £ model ( 5 . 1 9 ) in particular. This is evidence that the second model is more realistic as the In-dependence is expected to be robust. In other words, the superconducting order parameter is reduced at surface compared to the bulk. For consistency, one would expect that the average field, calculated from the expression 5 . 2 0 and using the fit parameters from table 5 . 1 , Jo B(x)p(x)dx ( 5 . 2 0 ) to be equal to each other and also be equal to the magnetic field correspond-ing to average depth /•OO x = xp(x)dx ( 5 - 2 1 ) Jo However, the average magnetic fields in the two models differ by about a Gauss. This is due to the fact that the two models fit to two different ini t ia l amplitudes differing by ~ 5 % . Refitting the data wi th an average of two amplitudes from the previous fits, the resulting A L ( T ) and £ (T) are shown in F i g 5 . 1 1 and F i g 5 . 1 2 . Fit ted A L ( 0 ) and £ ( 0 ) are shown in table 5 . 3 The previous fit of the "£ model" yields higher overall amplitude than that from the "exponential Figure 5.11: Penetration depth as a function of temperature for an exponential model for B(x) with average overall amplitude. Chapter 5. Measurements and Results 63 o XL (measured) I i i i i i i i i i i i i i 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Temperature (K) Figure 5.12: Penetration depth and coherence length as a function of tem-perature with average overall amplitude, for "£ model". Chapter 5. Measurements and Results 6 4 T (B) ( £ model) (B) (Exponential B(x)) 3 . 7 5 1 6 . 6 1 7 . 4 6 . 5 2 4 . 9 2 5 . 8 Table 5 . 2 : Average magnetic field, calculated using parameter in two models. model". When fitted with the average amplitude from the two previous fits, the "£ model" yields lower TC to account for the low relaxation rate since a higher magnetic field would yield lower jr. A s a consequence, TC is suppressed compared to the previous fit where fits were done independently for two models, as may be noted from table 5 . 3 . Model for B(x) A L ( 0 ) TC Exponential 2 3 4 1 ( 6 ) - 6 . 6 ( 1 ) £ model 2 0 0 5 ( 1 4 ) 1 6 9 ( 1 ) 6 . 7 9 ( 1 ) Table 5 . 3 : Penetration depth XT, and coherence length £ at temperature T = OK, using a common ini t ia l amplitude for fitting. The average magnetic fields computed- from the fitted parameters shown in Table 5 . 3 in two models are shown in table 5 . 4 T (B) ( £ model) (B) (Exponential B(x)) 3 . 7 5 1 7 . 0 3 1 7 . 0 6 6 . 5 2 5 . 4 2 5 . 2 3 Table 5 . 4 : Average magnetic field, in two models, using parameters from table 5 . 3 The average magnetic fields are equal to each other within « 2 % . The 2 'goodness of fit', i.e, ^ gets slightly worse (approximately twice the previ-ous fit) for both fits since the average amplitude isn't the best fit amplitude for any of them. However, one feature that remains similar wi th the previ-ous fit is the crit ical temperature of corresponding models. TC in '£-model ' wi th the fixed ini t ia l asymmetry fit is ~ 1 . 2 % lower than that from varying ini t ia l asymmetry fit. This is reasonable when compared to a corresponding drop in ini t ia l asymmetry by ~ 2 . 5 % , as explained above. A set of three spectra corresponding to temperatures 3 . 7 5 K , 6 . 1 5 K and 6 . 4 K is shown in F i g 5 . 1 3 The amplitude at t = As decreases by 5 0 % as the Chapter 5. Measurements and Results 65 temperature is lowered from 6.4K to 6.15K, whereas a similar decrease in amplitude requires a change in temperature from 6.15K to 3.75K. This is an indication of the fact that relaxation rate jr (and thereby A£,(T), £ (T) ) doesn't vary significantly unti l near the critical temperature. 0 2000 4000 6000 8000 10000 Time (ms) Figure 5.13: Time differential spectra at three temperatures, 3.75K, 6.15K and 6.4K, in long pulse method. 67 Chapter 6 Summary & Conclusions We have investigated the normal and the Meissner state of NbSe2 using spin polarized 8 L i . In the normal state of NbSe2, at high field, the relaxation rate increases with temperature, a result we identify with Korringa relaxation, where the dominant interaction is via the conduction electrons scattering off 8 L i nuclei. The Korringa constant is an order of magnitude smaller compared to that in Ag, indicating 8 L i , with a small hyperfine coupling, occupying a site in the Van der Waals gap where the overlap with the conduction band is small. Absence of resolved line splittings in frequency measurements indicate that electric field gradient in 8 L i sites is also very small compared to other non-cubic crystals. In low fields, the fluctuating dipolar fields from nuclear spin dynamics, dominates the relaxation process. ^ shows a Lorentzian behavior as a func-tion of applied magnetic field B. The equation for was used to extract the dipolar field strength A and correlation time r c, in the normal state at T — 8K. The values of A and r c are reasonable since they are on the order of kHz and ms, respectively. From the vortex state measurements, we obtain the temperature dependence of r c. The temperature dependence of T c combined with the field dependence of jr was then used to extract the coherence length £ and London penetration depth Ax,, depending on the model for magnetic field distribution inside NbSe2. The two models for magnetic field distribution inside NbSe2 gives similar x 2 /DF . Although absolute value of A L ( T = 0) is slightly different depending on the assumed magnetic field distribution model inside in Meissner state, the temperature dependence of AL (T ) fits much better to the two-fluid temperature depen-dence model, when there is a suppression of order parameter <f)(r), near the surface. As our sample was cleaved in air, it's possible that oxidation on surface had an effect on suppressing the order parameter near surface. One way of improving the measurement would be to cleave the sample in vac-uum and perhaps cap it with Ag. Since, the average magnetic field in the two models give nearly same value even with independent amplitudes, one way of distinguishing between models would be to do measurements at dif-ferent implantation energies so that the average implantation depth x and Chapter 6. Summary & Conclusions 68 the corresponding average magnetic field (B) would be different from one another. 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Greene, and R . F . Kief l . Depth-controlled /3 -NMR of 8 L i in a thin silver film. Phys. Rev Lett, 93(157601), 2004. [38] Z . Salman. Elsevier Science, 2005. [39] J . F . Ziegler and J . M . Manoyan. Nuclear Instruments and Methods in Physics Research B, 35:215-228, 1988. [40] E . Morenzoni, H . Glckle, T . Prokscha, R . Khasanovand H . Luetkens, M . Birke, E . M . Forgan, C h . Niedermayer, and M . Pleines. Implan-tation studies of kev positive muons in thin metallic layers. Nuclear Instruments and Methods in Physics Research Section B, 192(3) :254-266, M a y 2002. [41] T . R . Beals, R . F . Kief l , W . A . MacFarlane, K . M . Nichol, G . D . Morris , C . D .P . Levy, S.R. Krei tzman, R . Poutissou, S. Daviel , R . A . Baartman, and K . H . Chow. Range straggling of low energy 8 L i + in th in metallic films using Q - N M R . Physica B, 326:205-208, 2003. [42] L . C . Hebel and C . P. Slichter. Phys. Rev., 113(1504), 1959. [43] R . F . Kief l et. al. Coherence peak and superconducting energy gap in rb^ceo observed by muon spin relaxation. Phys. Rev. Lett., 70:3987-3990, 1993. [44] B . B . J in , T . Dahm, A . I. Gub in , E u n - M i Choi , H y u n Jung K i m , Sung-I K Lee, W . N . Kang , and N . K l e i n . Anomalous coherence peak in the microwave conductivity of c-axis oriented m g b 2 thin films. Phys. Rev. Lett, (127006), 1991. [45] Minui t - function minimization and error analysis. CERN Program Library entry, D506, 1994-1998. 73 Appendix A Low Field Spin Relaxation from Fluctuating Dipolar Fields This discussion is taken from C P . Slichter, Principles of Magnetic Reso-nance , Second Edi t ion , "Springer-Verlag", 1980, Ber l in ; Germany. In this section ^ due to randomly fluctuating magnetic field is discussed. Density matrix formalism is ideally suited for our system as large number of atoms are interacting at the same time and the observables are averages over al l atoms. We discuss general time evolution of state |V>) of our system in the presence of a general time dependent interaction Hamiltonian H(t). If our system is described by state then it may be expressed as linear combination of orthogonal energy states M = $ > | n > . (A.1) n Expectation value of an operator 0( i .e an observable) is given by ^\6^\) = Y.c*mCn(m\6\n). (A.2) 71,771 Equation A .2 may be conveniently expressed as multiplication of two matrices <</>|6» = Y(n\p\m){m\6\n), = Tr{p,6}, (A.3) where p is the density matrix wi th components Pmn = (m\p\n). (A.4) In other words, knowing of the density matrix is equivalent to knowing the state However, we often wish to compute average expectation value of Appendix A. Low Field Spin Relaxation from Fluctuating Dipolar Fields 74 an operator in an ensemble of systems, i.e., W) = E^(m\6\n). (A.5) Only pnm varies from system to system as the wavefunction varies. In sub-sequent discussion, the overline indicating ensemble average is omitted to simplify notation, i.e. p = p. Assuming the Hamiltonian H to be identical for al l V>'s in ensemble, the time dependence of p is given by | - i [ p . W ] . (A.6) When H is independent of time (eg. static magnetic field) the solution of equation A .6 is given by p{t) = e-intp(0)e^nt. (A.7) In our case, the Hamiltonian consists of a large time-independent interaction H0 (the Zeeman interaction —jH0Iz ) and a smaller time-dependent term H\{t). Equation of motion for p then becomes i - i f c H + WJ. (A.8) To have a solution of equation A . 8 , a quantity is defined p*(t) such that p(t) = e-*ntp*(t)e*nt. (A.9) Substitution equation A . 9 in equation A .8 one may get - i[H0,p] + e - ^ ^ l e i n t = * r p i W o + W l ( t ) ] > (A.10) which yields ^ f i - ^ . « I W l (A.U) where Hl{t) = e - * W o * W i ( i ) e * W o * (A.12) Using a second order iteration, the equation A . 11 may be written as dp* jt) dt = ^ * ( 0 ) , « J ( i ) ] + [ f f j\[p*(0),Hl(t%Hl(t)}dt'. (A.13) Appendix A. Low Field Spin Relaxation from Fluctuating Dipolar Fields 75 Since p*(0) = p(0), a knowledge of p(0) and Hi would lead to an expression for p*(t). Assuming that init ial ly only eigenstate |fc) of Ho is occupied, $l(m\p\m) is the probability per second of a transition. Thus, (re|p*|m) = (n|p|m) = 0 unless n = m = k. (A.14) Then jt(m\p*(t)\m) = ±(m\p(t)\m) + (ff j\m\[Hl(t')p*(0)Hm + H*{t)p*{<S)H*{t')]\m)dt' (A.15) Using the matrix elements, (m\Hl(t)\n) = e^Em-E^\m\Hi{t)\n), (A.16) and adopting a convenient notation y ^ r n , (A.17) The equation A.15 may be written as ft(m\p(t)\m) = ^^ [ (mlW^fJ I f cX fc lWxWIm)^^- * )^ - * ) + (m|Wi(t)|fc><fc |Wi(« ,) |m)e*( m- f c^ t- t ' )]dt ' . (A.18) A s H\(t) varies from ensemble to ensemble, we take the average of the expression in equation A.18 as jt(m\p(t)\m) = -^J* [(m\Hi(t')\k)(k\Hi(t)\m) e ^ m ~ k ^ + (m|7 i 1 ( r ) | i t ) (A; |W 1 ( i ' ) |m)e i ( m - f e ) ( t - t ' ) ld t , . (A.19) For sake of simplicity, it may be assumed that the perturbation is stationary on the basis that the temperature of the system is steady and thereby (m|Wi(r)|fc)(A;|Wi(f)|m). depending on t and t' only through their difference T = t — t' and the energy levels m and A;. The dependence of r , m and k is summarized by defining Appendix A. Low Field Spin Relaxation from Fluctuating Dipolar Fields 76 it 'correlation function" G m fc ( r ) as Gmfc(r) = (m\Hi(t)\k)(k\Hi(t + T)\m) = (m\Hi(t + T)\k){k\Hi(t)\m) — Gmk(-T) (A.20) where the last equality follows from the fact that is stationary. Gmfc("r) tells how the Hamiltonian ri\(t) at time t is correlated with the Hamilto-nian T^i (£ + r ) at time t + T. Generally, the thermal movements of nuclear moments is negligible for times less than some characteristic time r c called the "correlation time", so that A schematic diagram of G m fc ( r ) is given in F i g A . l the equation A.19 may be written as For our measurement, resolution time t is greater than a few r c and inte-gration l imit may be replaced by oo yielding Hi(t) « Wi(t + r ) . (A.21) di - t ( m - f c ) T ^ T _ (A.22) dt i r°° i(m-k)Tdj. = W k (A.23) Figure A . I : Typical correlation function. 

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