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The development of a high sensitivity ac susceptometer and its application to the study of high temperature… Bidinosti, Christopher Paul 1995

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THE DEVELOPMENT OF A HIGH SENSITIVITY AC SUSCEPTOMETER AND ITS APPLICATION TO THE STUDY OF HIGH TEMPERATURE SUPERCONDUCTORS B y Christopher Pau l Bidinos t i B . Sc. (Physics) Brandon University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF G RADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1995 (c) Christopher Pau l Bidinost i , 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The development of a high sensitivity ac susceptometer is outlined in this thesis. The intended use of this device is for the measurement of the magnetic field dependent pene-tration depth of high temperature superconductors in the Meissner state. The nature of the field depence may be useful in discerning the superconducting pairing state in these materials. Test measurements on the completed ac susceptometer show that its field dependent background is too large to extract any meaningful results from this experiment. However, the device has been used successfully to measure the temperature dependent penetra-tion depth and the vortex melting transition of the high temperature superconductor Y B a 2 C u 3 0 6.95. i i Table of Contents Abstract ii List of Figures vi Acknowledgement viii 1 Introduction 1 2 AC Susceptibility and the AC Susceptometer 3 2.1 The Complex Magnetic Susceptibility 3 2.2 Experimental Technique 4 2.3 Cal ibrat ion of the Susceptometer 7 2.4 Signal Detection 11 2.5 Setting the Phase 12 3 Development of the Apparatus 14 3.1 The Cryostat 14 3.2 The A C Susceptometer 16 3.2.1 Thermometry 16 3.2.2 The Superconducting D C Magnet 19 3.2.3 The A C C o i l Set 20 3.3 The Electronics 25 4 Performance of the Apparatus 30 i i i 4.1 A C C o i l Set - Version I 31 4.1.1 Sample Holder Temperature Dependence 31 4.1.2 D C Magnetic F ie ld Dependence 32 4.2 A C C o i l Set - Version II 35 4.2.1 Sample Holder Temperature Dependence 35 4.2.2 D C Magnetic F ie ld Dependence 36 4.3 Determining the Resolution of the Apparatus 39 4.3.1 Measuring A A ( T ) of Y B a 2 C u 3 0 6 . 9 5 39 4.3.2 Calculat ing the Resolution from A A ( T ) 40 5 The Nonlinear Meissner Effect: A(H) of Y B a 2 C u 3 0 6 . 9 5 42 5.1 Theory 42 5.1.1 Conventional Gap 43 5.1.2 Unconventional (dx2_y2) Gap 44 5.1.3 Some Estimates 47 5.2 Experiment 47 5.3 Results and Discussion 48 6 T h e Vortex Melt ing Transition in Y B a 2 C u 3 0 6 . 9 g 55 6.1 Experiment 56 6.2 Results and Discussion 57 7 Conclusion 62 Bibliography 64 Appendices 66 iv A Complex Susceptibility List of Figures 2.1 General ac susceptometer design 6 2.2 Schematic for numerical calculation of the calibration constant a 10 3.3 Diagram of the cryostat 15 3.4 Diagram of the ac susceptometer 17 3.5 The calculated axial field profile for the superconducting dc magnet. . . . 20 3.6 Diagram of the ac coil sets 22 3.7 Construction of niobium secondary coils 25 3.8 The susceptometer circuit 26 3.9 The low temperature compensation circuit 28 3.10 The performance of the decade transformers 29 4.11 The temperature dependent backgound signal of the sapphire sample plate. 32 4.12 The dc field dependent backgound signal 33 4.13 Hysteresis loops at 5 K for ac coil set construction materials 34 4.14 Dependence of the susceptometer signal as a function of the sample holder temperature 36 4.15 The dc field dependent backgound signal of the version II ac coil set. . . 37 4.16 A A ( T ) for the Y B a 2 C u 3 0 6.95 sample Y l q 41 5.17 The dx2_y2 gap function 45 5.18 Averaged data for magnetic field response of Y B a 2 C u 3 0 6 . 9 5 sample Y l q and background 51 vi 5.19 Corrected data for YBa2Cu306.95 sample Y l q 52 5.20 Background measurements from separated cool downs 53 5.21 Anomalous background measurement 54 6.22 Predicted ac response of the vortex melting transition 56 6.23 The vortex melting transition in the YBa2Cu306.95 sample Up2 at various dc fields 59 6.24 The vortex melting transition in the Y B a 2 C u 3 0 6 . 9 5 sample Up2 at various drive fields 60 6.25 The vortex melting transition in the Y B a 2 C u 3 0 6 . 9 5 sample Up2 at various drive frequencies 61 v i i Acknowledgement I would like to thank my family and al l my friends. I would like to thank my supervisor, D r . Walter Hardy. He is a great teacher. I would like to thank the Natural Science and Engineering Research Counci l for its financial support. v i i i Chapter 1 Introduction For nearly a decade now there has been intensive research into high temperature super-conductors. The attraction to the study of these materials is fueled as much by their underlying physics as by their promise to be of great practical and commercial use. In terms of the physics there is a still great deal that is unknown. In fact, there is at present no theory that can satisfactorily explain the superconducting mechanism of high temperature superconductors. Many clues to solving this mystery rest in the knowledge of the pairing state, and a variety of experiments have been pursued with this in mind. One experiment that has been proposed, but not yet successfully carried out (at least in YBa2Cu306.95 [1, 2]), is the measurement of the magnetic field dependence of the Meissner state penetration depth. Measurements of critical fields and the mapping of the H — T phase diagram have always been of interest in the study of superconductors. There has been little interest, however, in precision measurements of X(H) in the Meissner state until the work of Yip and Sauls on the nonlinear Meissner effect [3]. Their work (detailed further in Chapter 5) revealed that the nature of the field dependence of A is sensitive to the pairing state. In particular, X(H) for a superconductor with a conventional isotropic gap function should vary as i? 2 , while X(H) for a superconductor with lines of nodes should be linear in H. Furthermore, the theory suggests that the field response may be used to locate the nodal lines in the case of an unconventional gap. This may help answer the long outstanding question of what the pairing state is in the high temperature 1 Chapter 1. Introduction 2 superconductors. Many other experiments have provided strong evidence to suggest dx2_y2 pairing, but the evidence is not conclusive [4]. Observation, then, of the nonlinear Meissner effect may supply some of the strongest evidence yet to answering this question. The reason for building the ac susceptometer to be described in this thesis is to be able to measure the magnetic field dependence of the penetration depth of the high tempera-ture superconductor YBa2Cus06.95 for fields below Hc\. Presently, measurements using ac susceptometers have become quite popular in the field of high temperature supercon-ductivity. Many of the interesting features of superconductors - the Meissner effect, the crit ical temperature, vortex nucleation and dynamics - can be viewed macroscopically through changes in the bulk magnetization of a sample. A s a result an ac susceptometer lends itself very well to the study of such materials. The ac susceptometer itself is a very simple device. Typical ly, i t consists of a dc magnet, a solenoid used to generate an ac field, and a pair of detection coils. The mutual inductance between each of the detection coils and the solenoid is the same, so when the two are connected in series the induced voltage across the pair is zero. The sample is placed inside one of the detection coils. Its presence w i l l alter the mutual inductance of this coil and now the net voltage across the coils is nonzero. This voltage is proportional to the sample's magnetic moment. If the sample is a superconductor in the Meissner state, its moment w i l l be proportional to the volume of the region magnetically shielded by supercurrents. This volume, always smaller then the actual volume of the sample because of the finite penetration of the magnetic field, is proportional to X(T,H). This allows a measurement of AX(T,H) = X(T,H) - X(T0,H0) where T0 and H0 are some reference temperature and field respectively. Chapter 2 AC Susceptibility and the AC Susceptometer The magnetic susceptibility % of a material relates its magnetization M to an applied external magnetic field H. For linear media in a dc field this relationship is simply M , For nonlinear media or in the case where one applies a t ime dependent field a more relevant definition is *=d i r <2-2> dM dH where x is n o w the differential or ac susceptibility respectively. 2.1 The Complex Magnetic Susceptibility Consider a sample in a periodic external magnetic field H{t) = Hdc + His'mwt. In response to the field, the sample wi l l develop a periodic magnetization M(t), which can be expanded as the Fourier series M(t) = H^MXne™*) n-l oo = Hi ^2 (xn sin nwt - Xn cos nwt), (2.3) n=l where I m ( ) denotes the imaginary part of the complex magnetization [5]. The coefficient Xn — Xn~ iXn l s t n e complex magnetic susceptibility of the nth harmonic, where Xn = —FT / M(t)smnwtdt, 7Tiii J=2-3 Chapter 2. AC Susceptibility and the AC Susceptometer 4 (2.4) " ~~w fv-,*/\ i v„ = -—— / Mit) cos nwt dt. N -KHI J=S. V ; A s defined here, the Xn w l | l typically be a functions of both Hdc and Hi. (This definition is equivalent to that in equation 2.2. See Appendix A . ) The fundamental component of the magnetization M(t) = Hi(x\ s'mwt + x'i cos tut) has a real part in-phase wi th the applied field H(t) and an imaginary part that is 90° out-of-phase wi th H(t). This attaches very obvious physical meaning to the fundamental susceptibility Xi- Xi corresponds to the inductive response (energy storage) of the sample, x" corresponds to losses (energy dissipation) in the sample. The higher harmonics are generally associated wi th hysteresis and nonlinearity of magnetization [5]. In the l imit of Hi —> 0, the higher harmonics wi l l become negligible and x'i ~^ jfj" — Xdc-For example, a superconductor in the complete Meissner state wi l l have a magne-tization field that is of equal amplitude but opposite direction to that of the applied field. Equation 2.4 gives x'i = —1 and x'i = 0 reflecting the perfect diamagnetism of the material. In the normal state (neglecting core diamagnetism and the real conductivity), the external field w i l l completely penetrate the sample, and x'i — Xi = 0- A t intermedi-ate temperatures x'i "will be a negative number (> —1), and x'i w U l D e a small positive number reflecting ac losses [6]. 2.2 Experimental Technique The general design of an ac susceptometer is shown in Figure 2.1; al l coils are coaxial. A n ac current is passed through the primary or drive coil to produce an alternating magnetic field. A steady state magnetic field can also be superimposed by applying a dc current to the outermost coil . The only measured quantity in this technique is the induced voltage across the series combination of the secondary coils, which of course is Chapter 2. AC Susceptibility and the AC Susceptometer 5 due to the time variation of the magnetic flux through these coils. The secondary coils are made as identical as possible and are connected in opposition. In the absence of a sample, the induced voltage of the coils w i l l be equal in magnitude but 180° out of phase resulting in an observed null signal. Specifically, from Faraday's law, s = -d$/dt, the induced voltage is v(t) = (M1-M2)^ = 0 (2.5) where M\ and M 2 are the mutual inductances between the primary and each of the secondary coils, and Iv is the sinusoidal drive current. To the extent that the coils are identical M x equals M 2 and the minus sign reflects the fact that they are connected in opposition (or counterwound), and hence v(t) = 0. In practice, however, it is not a t r iv ia l matter to achieve this balance. Even i f great care has been taken in winding al l the coils and the secondaries are placed symmetrically in the ac field, a residual voltage is st i l l very likely. Addi t iona l methods of compensation are usually employed to cancel this mismatch and obtain the desired level of sensitivity and stability in the susceptometer. The direct addition of a compensating voltage vc(t) of appropriate magnitude and phase is a common technique. Here v(t) = ( M x - M2)^ + Vc(t) = 0. (2.6) W i t h the system nulled, the introduction of a magnetic sample into one of the coils (the pick-up coil) wi l l now cause an imbalance in the signal. This voltage is due solely to the sample. To see this, consider the three voltage sources in the secondary circuit, which sum to d<J>i d $ 2 = VoNA dt ' dt dH(t) + M{t) dH{t) dt dt + vc{t), (2.7) Chapter 2. AC Susceptibility and the AC Susceptometer 6 dc Power Supply Reference Function Generator Input Lock-in Amplifier Voltage Compensation Circuit dcCoil Primary Coil Sample Secondary coils in series opposition Figure 2.1: General ac susceptometer design. where M(t) and H(t) are the sample magnetizaion and applied field respectively, N is the number of turns in each coil , and A is their cross sectional area. The first term in this equation is the induced voltage in the pick-up coil , the second term is the induced voltage in the empty (compensation) coil , and the third term, vc(t), is again the applied compensation voltage. In light of equation 2.6, this simply reduces to dM(t) v(t) = p0NA-dt (2.8) Chapter 2. AC Susceptibility and the AC Susceptometer 7 a signal voltage proportional to the rate of change of the sample magnetization. Substi-tut ing the definition of M{t) from equation 2.3 into equation 2.8 gives oo v{t) — n,0wNAHi ^2 n(Xn C 0 S nwt + Xn s m nwt) n=l oo = vo y i n(xn c o s nu)t + Xn s m nwt), (2-9) n = l with v0 = HowNAHx. The signal voltage is an infinite sum of cosine and sine wave harmonics. 2.3 Calibration of the Susceptometer The expression for v0 above is only strictly true for a uniformly magnetized sample that completely fills the coil . Such an arrangement is not typical. In general v(t) must depend upon the geometry and spatial orientation of the sample and the pick-up coi l . This information is contained in a calibration constant, a. It is the constant of proportionality for the experimentally relevant relation v(t) oc dM(t)/dt = xwH, and is related to the mutual inductance between the sample and the pick-up coil, a can be calculated v ia the following consideration. The field due to the sample is characterized by internal currents of volume density V x M and surface density M x n , where M is the sample magnetization and n is the unit vector normal to its surface. A cylindrical sample of uniform axial magnetization ( V x M = 0) can be equivalently modeled as a solenoid of the same size having Ns turns each carrying current I [8]. This implies a magnetization of magnitude M = NJ/l„ (2.10) where ls is the length of the sample. The induced voltage in the pick-up coil is v{t) = - M s p ^ , (2.11) Chapter 2. AC Susceptibility and the AC Susceptometer 8 where Msp is the mutual inductance between the sample and the coil . Rearranging equation 2.10 and substituting gives "« = ~M>W (2'12) or v(t) = -cxXwH. (2.13) The calibration constant then is just ° = ^ » (2-14) the mutual inductance per sample turn multiplied by the length of the sample. To calculate the mutual inductance, it is almost always easiest to consider the voltage induced in an inner coil (sample) due to a current in an outer coil (pick-up.) So, for a coil much longer than the sample the mutual inductance is u N M = Mps = Msp = ^ N S A S , where Np and lp are the number of turns and length of the pick-up coil respectively and As is the cross sectional area of the sample. This gives a = ^ V a i (2.15) ip a calibration constant proportional to the sample volume Vs and independent of the parameter A ^ , as it must be. This solution for a was of course obtained using the long coil approximation for the pick-up coil , which may not be appropriate in many cases. Other particular sample/coil arrangements also lend themselves to rather simple determinations of a [7, 8], but these too might prove unsuitable approximations for many sample sizes and shapes. I therefore Chapter 2. AC Susceptibility and the AC Susceptometer 9 present the following as a precise method for the numerical calculation of a for a uniformly magnetized sample. This method is accurate for cylindrical samples wi th magnetization vector parallel to both their axis and the pick-up coil axis. To the extent that one can ignore edge effects, this method w i l l also be a very good approximation for bar or slab shaped samples that are magnetized along one its lengths parallel to the pick-up coil axis. (Most superconducting samples to be studied using the susceptometer of this thesis are slab shaped, hence the motivation for this exercise.) There are no restrictions on the length of the sample. The fact that the sample is uniformly magnetized allows one to approximate it as a solenoid and use the definition of a from equation 2.14. It also allows one to write the mutual inductance as calculated at the distance r from the centre axis in the plane of the jth solenoid loop (see Figure 2.2.) The integration is over the cross sectional area of the solenoid loops, and can be computed numerically by dividing the area into K x L finite equally sized elements AAS = AS/KL. The value B '(r f c (, z{j) at the centre of each AAS is taken to be the value over the entire area element. This gives i=l j=l where B '(r, z^) is the magnetic field per unit current of the ith loop of the pick-up coil Np Ns K L )AA (2.17) t=i j=i k=i i=i and therefore JV 1VS IS. Li £ £ £ £ * ' ( N Ns K L a = NSKL i (2.18) i=i j=i k=i i=i Notice that assuming B'(rkl,zi:j) = B is constant over the entire sample recovers the solution for a of 2.15. O f course, the idea here is to assume a more complicated form Chapter 2. AC Susceptibility and the AC Susceptometer 10 End view of sample: - AAS = AS/KL. - rkl —-— K Pick-up Coil OfOOOOOOOCOCCOOOOOOO oooocoaco Compensation Coil cooooooooooooooocooo ccccccmco solenoid loop o*ocxxxxxxxxoooaxxx) oooooooocxxcoooooooo ith pick-up loop Figure 2.2: Schematic for numerical calculation of a for a uniformly magnetized rectan-gular sample. It is shown approximated as a solenoid of equivalent dimension. The end view shows cross sectional area of sample split into K x L elements AAS. of the magnetic field (for greater accuracy) and use a computer program to calculate a. The summations of equation 2.18 are easily translated into four programming loops. The degree of precision in the program is determined by the size of the model parameters NS,K, and L. The best accuracy can be had i f one uses B'{r,zij) = Bz/I = K + 2 9 9 a — r* — z (a — r) 2 -I- z2 E 2 n y/(a + r)2 + z< the general expression for the field per unit current of a circular current loop [9]. K and E are complete elliptic integrals of the first and second kind respectively. 1 The degree of precision is determined by the size of the model parameters NS,K, and L used in the program. :Some mathematical packages such as Mathematica or Maple have built in routines for calculating elliptic integrals. Chapter 2. AC Susceptibility and the AC Susceptometer 11 One further refinement needs to be made, i f the compensation coil is in close proximity to the sample. Under such conditions, there wi l l be an induced voltage in this coil which is no longer negligible. The true calibration constant then is just Ol — Olpick—up Ot-compensationi (2.19) The terms are calculated separately by employing equation 2.18 for each coil . In lieu of calculating a mathematically, calibration can also be done experimentally using a standard of known susceptibility. This is a common technique. It is an attractive method in that it allows one to make a simple determination of a without needing to know the exact geometries of the secondary coils. There is one drawback however. The resulting calibration constant, though possibly very precise, w i l l only be strictly valid for samples having the same dimensions and the same position inside the susceptometer as the standard. 2.4 Signal Detection The voltage v(t) induced in the secondary coils can be measured using an ac voltmeter or an oscilloscope. However, these devices can detect only the entire voltage sum of 2.9, and a great deal of information about the system is lost. Using a lock-in amplifier allows one to selectively measure both the in-phase and out-of-phase components of a given harmonic, revealing much more detail of the sample magnetization. The lock-in behaves as a voltmeter wi th a very high Q bandpass filter at the input. It w i l l measure a signal of specific frequency only. This frequency is determined by the frequency of a reference input. Usually this input is the sync out from the function generator providing the primary current. The reference phase of the lock-in must also be set so that its time base is synchronous wi th the drive field H(t) = Hi sin wt. This allows for the separation of the v(t) into its in-phase (xn) and Chapter 2. AC Susceptibility and the AC Susceptometer 12 out-of-phase (x'n) components, where the nth harmonic of w was used for the reference. A dual phase lock-in amplifier w i l l display both signals. W i t h devices having only one output, the phase must be changed by 90° to obtain the out-of-phase signal. Clearly, the real power of ac susceptometry resides in the lock-in amplifier's abil i ty to selectively measure both the real and imaginary parts of any Xn-2.5 Setting the Phase The phase can be set by using, as the input signal to the lock-in, the voltage drop across a pure resistance that is in series wi th the primary circuit. This voltage wi l l be in phase wi th the primary current and therefore in phase with the applied field H{t). The reference phase setting is adjusted unti l the signal appears in only one channel (ie. the 0° or 90° setting.) This channel is now set to measure the signal proportional to x'n- The channel in quadrature wi l l measure the x'n-It is important that the voltage sampled off the resistor have the same electrical path to the lock-in as would the signal voltage from the secondary coils. For example, the signal voltage may be stepped up by a pre-amp transformer before the lock-in. Ideally the transformer would not affect the phase. In practice however, one is almost certain to pick up a phase shift as most real transformers are only nearly ideal. Having the sample voltage go through the pre-amp (at the same settings) as well wi l l allow for an accurate phase setting. If one is studying superconductors, another method of setting the phase employs the properties of the sample itself. This method is convenient because it can be performed without altering the experimental set up. Here one takes advantage of the fact that a superconductor cooled in zero field and kept at sufficiently low temperatures w i l l not have any loss associated wi th a small drive field (ie. Hi « HCl.) The sample temperature Chapter 2. AC Susceptibility and the AC Susceptometer 13 is changed wi th in this region of no loss. The resulting signal is due to the temperature dependence of the penetration depth, which is a purely inductive response. Aga in the reference phase is set such that the resulting change in signal appears in only one channel. This w i l l be the out-of-phase channel. Chapter 3 Development of the Apparatus The real heart of an ac susceptometer is of course the trio of solenoids comprised of the primary and two secondary coils. To describe them as a single entity, which is often useful, I have coined the term ac coil set. Also, for the purposes of this chapter only, I w i l l l imi t the use of the term (ac) susceptometer to refer only to that part of the apparatus containing the dc magnet, ac coil set, and sample holder. The cryostat and the electronics then wi l l be described separately from the susceptometer in this chapter. The development of this apparatus was already a work in progress of Dr . W . Hardy, when I joined the project. As a result, I wi l l detail only those parts of the apparatus that are specific to this experimental technique, and give particular highlight to those efforts in development that were my own. It should be duly noted that the design of the apparatus was D r . Hardy's, and that the cryostat, the susceptometer (less the dc magnet) and one version of the ac coil set had already been built . I built the magnet and a second coil set. 3.1 The Cryostat The cryostat is of standard design. See Figure 3.3. It consists of a brass flange wi th stainless steel vertical tubes (three of them, each 3/8" outer diameter) that connect to the susceptometer. The flange is 0.350" thick and has an outer diameter of 5.375" to cover the 4" inner diameter helium dewar that the susceptometer sits in. The flange also contains a 3/8" hole and O-ring seal for a helium transfer tube (or safety blow-off valve, 14 Chapter 3. Development of the Apparatus 15 after transfer is complete), and two multi-lead connectors (not shown in diagram), one for the temperature control electronics, the other for the superconducting dc magnet leads. Connected to the stainless steel tubes are five copper disks, of diameter just less then 4", that act as radiation baffles. Leads for the primary and secondary coils are run down the centre of two of the tubes. The third tube is used as a vacuum line to evacuate the susceptometer; it also contains a drive shaft allowing mechanical adjustment of a voltage compensation circuit that is housed within the cap of the susceptometer. For He transfer Brass flange Knob to turn drive shaft -fl •< Vacuum line Stainless steel tube Copper baffle 1 ac susceptometer Figure 3.3: Diagram of the cryostat. Chapter 3. Development of the Apparatus 16 3.2 The AC Susceptometer A cross section of the ac susceptometer is shown in Figure 3.4. In this diagram the susceptometer is disassembled, as it would appear when one is changing the sample. The sample holder or pot is mated with the main body of the susceptometer wi th the aid of two guide pins (not shown) that are permanently fixed to the dc coil form. The two pieces are held together by twelve 4 — 40 hex cap screws wi th indium wire forming the vacuum seal. The dc coil form is bolted to the susceptometer cap in a similar way,with the exception of the guide pins which are not needed here. Because two different versions of the ac coil set were built , Figure 3.4 shows just a general template for its arrangement inside the susceptometer. B o t h versions are shown in greater detail in Figure 3.6. The ac coil set has a copper flange, which is fastened to the dc coil form using six 4 — 40 hex cap screws. The sample is mounted on a very thin, very pure sapphire plate wi th a small amount of vacuum grease. The sample plate is held to the sapphire mounting block of the pot also wi th a small amount of vacuum grease. When the ac coil set and the pot are fully assembled, the sample sits at the centre of the first coil of the secondary pair, ie. the pick-up coil . Other pertinent aspects of the susceptometer are discussed in more detail below. The voltage compensation circuit, which is also shown in Figure 3.4, wi l l be described in the next section of this chapter. 3.2.1 Thermometry It is essential to be able to change the temperature of the sample during an experiment without interferring with the ac measurements on the sample. For this reason the sample heater and thermometer are connected to a sapphire mounting block situated well away from the sample and the ac coil set. Sapphire has a very high thermal conductivity and Chapter 3. Development of the Apparatus 17 Drive shaft Bath thermometer dc coil form Superconducting dc magnet (inner notch formed by copper collar) Sample mounted to sapphire plate HH Brass Hi Copper FT7! Quartz I I Sapphire [XI Coil Windings i__ j ac coil set I or II (detailed below) Figure 3.4: Diagram of the ac susceptometer. The sample holder or pot is disassembled from the dc coil form. The orientation of the primary and secondary coils is shown only schematically. Chapter 3. Development of the Apparatus 18 can be made very pure, so it is also used for the sample mounting plate. Quartz on the other hand has much lower thermal conductivity and is therefore used to thermally isolate yet mechanically connect the mounting block to the rest of the pot. Provided then that the layer of vacuum grease between the block and the plate, and the plate and the sample is very thin, this system w i l l allow for very quick, effective and non-intrusive regulation of the sample's temperature. It is often desirable to keep the temperature of the helium bath surrounding the sus-ceptometer accurately constant. To do this requires a separate bath heater, thermometer and temperature controller. The total thermal power delivered to the bath is the sum of the power from the sample heater Ps and bath heater Pb. If Pb is in i t ia l ly set (with Ps = 0) to a value just greater than the anticipated maximum power Pmax to be used by the sample heater over the course of an experiment, then Pb can always be adjusted later such that Ps + Pb = Pmax, a constant, for any setting of Ps. The sample thermometer is a piece of a Lakeshore carbon glass resistor potted in Stycast 2850FT epoxy to a strip of copper foil. The foil is held to the mounting block wi th G E varnish. The sample heater is a 200S2 chip resistor attached to the underside of the sapphire block, and cannot be seen in the view of Figure 3.4. Manual or auto-mated control of the sample temperature is done with a Conductus L T C - 2 0 temperature controller. The bath heater, a 200O metal glaze resistor, is mounted to the base of the pot with Stycast 2850FT epoxy. The bath thermometer, an A l l a n Bradley carbon composition resistor (2000 nominal room temperature resistance) is located at the side of the suscep-tometer cap. The temperature controller, built in-house, allows only manual adjustment of the set point. Chapter 3. Development of the Apparatus 19 3.2.2 The Superconducting DC Magnet The magnetic field along the central axis of a thin solenoid of finite length / and radius a is given by the equation JJ _ »oNI f 1 - 2zll • 1 + 2 Z / 1 } ( 3 2 0 ) 21 \ [ (2a /Z) 2 + ( 2 z / 7 - l ) 2 ] 1 / 2 [(2a/Z) 2.+ ( 2 ^ + l ) 2 ] 1 / 2 J ' where N is the number of turns, I is the current, and z is the distance from the centre of the solenoid [12]. Using this equation a computer program was writ ten to help select an appropriate design for the dc magnet. The program returned only those winding configurations that would produce a symmetric magnet wi th a field homogeneity of better than 1 part per thousand over the length ( ~ 1 — 2mm) of a typical sample positioned at the centre of the pick-up coil . The secondary coils, separated by 0.25", are symmetrically placed about the geometrical centre of the magnet. If the magnet is wound accurately, the field w i l l be the same at each coil . It was necessary to also include in the program the following design constraints: • The size of the superconducting (Nb-Ti) wire - 0.014 inches in diameter. • The number of layers of windings in the magnet - 30 layers each being separated by 0.002 inches of mylar. This l imi t set by the amount of available space. • The size and shape of the magnet - cylindrical , 2.0 inches in length, wi th a cen-t ra l notch. The notched design is simplest configuration to wind that can give a reasonably homogeneous field. The final design can be seen as a part of Figure 3.4. The notch is formed from a copper collar epoxied (with Stycast 1266) to the magnet coil form. There are 14 layers of windings to the top of the notch, each of layer having 40 turns (20 turns on either side of the copper collar.) Each of the 16 complete layers above the notch has 136 turns on it. Chapter 3. Development of the Apparatus 20 The magnet calibration was determined theoretically using equation 3.20 and the average values of its turns density and layer thickness. The axial field profile of the magnet is shown in Figure 3.5. The magnitude of the field in the region of the susceptometer is calculated to be 468 gauss/amp. 500.0 O.o l • ' ' ' 1 1 • 1 -2.0 -1.0 0.0 1.0 2.0 Distance from centre of magnet (inches) Figure 3.5: The calculated axial field profile for the superconducting dc magnet. The inset highlights the region of interest containing the ac coil set. The positions of the secondary coils are shown centred at ± 0 . 2 2 5 inches. 3.2.3 The AC Coil Set The two versions of the ac coil set are shown in Figure 3.6. Only the top half of the cross section of each piece is shown; the axial field strength of the respective primary coil is superimposed on this view. The reason a second, slightly different version of the ac coil set had to built was that the first design proved to be completely unsuitable for the type of dc field scan measurements to be done. (This will be discussed further in Chapter 3.) Chapter 3. Development of the Apparatus 21 Common to both designs, however, is the copper flange used to mount the ac coil set to the main body of the dc coil form. Also, in both cases, phenolic linen washers are used as a lateral form for the primary coil , and a copper faraday shield is epoxied (with Stycast 1266) to the outer diameter of the primary coil's radial form. The shield is electrically and thermally grounded to the copper flange. The faraday shield is composed of a single layer of closely packed insulated copper wires that are parallel to the coil axis. This provides effective shielding of static electric fields, but doesn't support induced tangential currents that would otherwise shield the desirable magnetic drive field. To build the shield, the copper wire is first wound on a teflon form and potted in 1266 epoxy. Before the epoxy has fully cured this rather malleable coil is removed from the teflon and cut to form a sheet of parallel wires. This is then wrapped in place around the radial coil form and the epoxy is allowed to completely cure. In the version I coil set, the bare ends of the faraday shield are folded against the back of the phenolic linen washers and held in pressure contact with the copper flange; the washer is bolted to the flange. In version II, these ends fit into a groove in the flange that is filled wi th silver epoxy. Here the washer is simply butted to the flange and held in place wi th 1266 epoxy. The coil form of version I was machined out of phenolic paper rod. It has an outer diameter of 0.250", an inner diameter of 0.125", and an overall length of 1.250". Two square grooves of length 0.200" and inner diameter 0.167" act as forms for the secondary coils. These grooves, 0.250" apart, are positioned symmetrically about the centre of the form. The secondary coils are counter wound out of a single length of #44 copper wire (insulation unknown) and potted in Stycast 1266 epoxy. There are 1359 turns in each of the coils. The primary coil is wound with #34 formvar insulated copper wire. The faraday shield was made from this same size wire. There are four layers of windings in the primary. The first and third layers are complete. The second and fourth layers have Chapter 3. Development of the Apparatus 22 Version I Central axis -0.625 -0.225 0.0 0.225 0.625 Inches from centre of primary coil Version II Central axis H I Copper B Phenolic linen ^ Phenolic Paper I | Sapphire j§| Primary coil windings 1X1 Secondary coils -0.625 -0.225 0.0 0.225 0.625 Inches from centre of primary coil Figure 3.6: Diagram of the ac coil sets. The faraday shielding is butted against the flange in version I, in version I I it is siver epoxied into a groove in the flange. The field profiles of each primary coil is superimposed on the figure. only ten turns at either end; these compensation turns homogenize the field over the length of the coil . A spacer made of mylar tape fills the remainder of the second layer. The primary coil has a constant field strength of 127 gauss/amp over the length of the secondary coils, as calculated numerically using equation 3.20. A s mentioned before a second coil set had to be built , because this first one was dis-covered to be unsuitable for magnetic measurements. Expl ic i t ly , the problem wi th this version of the ac coil set was that it had a huge background signal dependent upon the applied dc magnet field. To make matters worse, this signal was hysteretic suggesting Chapter 3. Development of the Apparatus 23 the presence of ferromagnetic impurities in the apparatus itself. Obviously, i f one wants to make measurements as a function of the field, this can not be tolerated. The mag-netism of the various construction materials of the coil set were tested with a dc S Q U I D magnetometer. These tests did not reveal a definite source for the signal. The goal then in building a second version was to l imit the amount of known magnetic impurities in the vicini ty of the secondary coils. This was to be achieved in two ways: (1) using high purity construction materials and (2) by reducing the amount of material in the vicinity of the secondary coils. A custom ground sapphire tube [13], wi th an outer diameter 0.250", inner diameter of 0.198" and length 1.250", was used for the primary coil form in this design. Apar t from its high purity, which should ensure negligible contribution to the field dependent background, the sapphire also has another advantage. Its high thermal conductivity, wi l l quickly remove any heat (generated by the primary, radiated by the sample, etc.) from the ac coil set to the helium bath. There is no permanent coil form for the secondary coils of this coil set. They were wound separately on teflon forms, potted in 1266 epoxy, and then removed from the form once the epoxy had cured. This process, which had an ~ 80% success rate of producing an intact coil , is described further in Figure 3.7. The completed coils have 512 turns and nominal dimensions: O D = 0.188" and length= 0.23". (The O D of the two coils differ by ~ 0.002 — 0.003". Inadvertently, some of the coil forms were made of a softer teflon, which resulted in a slightly smaller coil.) The coils are then epoxied (with 1266) in place inside the sapphire tube. Two slits ground into the sapphire helped facilitate the application of the epoxy and allowed for the passage of the pick-up coil leads around the outside of the compensation coil . (It should be noted that the layer of epoxy between the coils and the sapphire appeared to go cloudy after being cooled in l iquid nitrogen. This apparent cloudiness is presumed to be due to the coils contracting away from the Chapter 3. Development of the Apparatus 24 sapphire.) The secondary coils were wound out of 0.003" diameter pure Nb wire insulated wi th formvar [15]. Niob ium was chosen for its superconducting properties. It has a T c = 9.2K, so at l iquid helium temperatures it would shield any possible ferromagnetic impurities inside its bulk from the magnetic fields. Niobium was chosen over other superconducting metals such as P b or PbSn , because it has a high critical field Hc\ = 1980 gauss. This is much larger than fields to be used in the experiment, which ensures that the coil windings should always remain in the Meissner state and not give rise to a signal due to a transition into the mixed state. Also, there wi l l be no losses and therefore no heating associated wi th a type II superconductor in an oscillating field above Hc\. The faraday shield and the primary coil were made from 0.003" diameter 99.99% pure copper wire insulated wi th formvar [14]. The primary was wound using the same scheme as the one described above. The wire is of a smaller diameter so a greater number of turns, 18 on either side, is required in the compensation layers (ie. the second and fourth layers.) This primary coil has a constant field strength of 254 gauss/amp over the length of the secondary coils, as calculated using equation 3.20. To protect the very fine wire leads used in this ac coil set, a l l connections were made on fixed terminal posts. The five posts, 2 - 5 6 threaded brass rod, are held in a lucite ring that is fastened around the copper flange of the coil set inside the susceptometer cap. There is a pair of posts for the primary coil connections wi th the power leads from the top of the cryostat, a second set for the secondary leads to the preamp, and a fifth post where the secondary coils are connected in series opposition. A l l lead ends from the ac coil set are placed inside tubes of 99.95% pure copper foil, which are then crushed between washers on the 2 - 5 6 bolt. The niobium leads are first cleaned wi th H F acid. Solder connections are easily made to the copper foil or to the brass posts themselves. Chapter 3. Development of the Apparatus 25 8mm ^ 0 . 2 0 0 ' - ^ Figure 3.7: Construction of niobium secondary coils: The teflon form fits the 8mm collet of coil winding machine. The wound coil is potted in epoxy, which is allowed to fully cure. A t this point the pilot hole at the end of the form is extended as the dashed line, and the extremity of teflon is pulled away. The entire piece is then immersed in liquid N2 after which the solid coil is easily removed. Because the teflon does stretch during winding, the final length of the coil is ~ 0.23". 3.3 The Electronics The electronic set up for this apparatus resembles that shown i f Figure 2.1, wi th the inclusion of a preamplification stage before the lock-in detector. The preamp is a Prince-ton Appl ied Research ( P A R ) Model 119 Differential Preamplifier input to a P A R Model 114 Signal Condit ioning Amplifier. The preamp is operated in the differential input mode (and almost exclusively in the 100:1 transformer mode) and has a single output to the Stanford Research Systems Model SR850 D S P Lock-In Amplifier. The function generator, a Hewlett Packard 3325A, is coupled to the primary circuit v i a an isolation transformer to break its ground loop. The schematic of the susceptometer circuitry, including the voltage compensation, is shown in Figure 3.8. The 100Q dropping resistor, in the primary circuit, is much larger than the impedance (at / = 10 kHz) of the rest of the circuit when the primary coil is at l iquid helium temperatures. As a result, under the conditions of operation, the combination of the dropping resistor and function generator behaves as an ac current source. The voltage drop across the l f i resistor can be used to set the phase of the lock-in Chapter 3. Development of the Apparatus 26 amplifier. The capacitor Cres in the secondary circuit is used to achieve series resonance with the inductive component ( largely dominated by the secondary coils) of the circuit. This is done to optimize the source impedance for the P A R Model 119 preamp operating in the transformer mode at 10 kHz. A t resonance the source impedance is the purely resistive 3.50 of the secondaries, which means the Model 119 wi l l be operating within the O.bdB noise figure contour at this frequency [16]. It should be noted, that the second version of the ac coil set has a much smaller impedance (due to the smaller self inductance of the secondary coils) and the capacitor Cres was not needed. Inside cryostat. Figure 3.8: The susceptometer circuit. Coarse compensation of the out-of-phase voltage is done at low temperatures wi th the variable mutual inductance. A l l fine compensation is done at room temperature v i a decade transformers. The voltage compensation circuitry has two distinct parts: (1) the coarse compen-sation done at low temperatures by a homemade variable mutual inductance and (2) Chapter 3. Development of the Apparatus 27 the fine compensation done at room temperature by a set of Electro Scientific Indus-tries Dekatran D T 7 2 A decade transformers capable of supplying six decades of voltage resolution. The low temperature compensation is used exclusively to null out-of-phase signals, while the room temperature circuit is used for both in-phase and out-of-phase compensation. Coarse compensation is not needed for any in-phase imbalance in the signal. It is quite small compared to the out-of-phase imbalance, which is due to the mismatch of the secondary coils and the inductive shielding of the sample (if present.) The advantage to the low temperature null ing is the lower thermal noise. The variable inductor of (1) is a set of equal turn (but counter wound) coils in series wi th the primary circuit that straddle a pick-up coil in series wi th secondary circuit, see Figure 3.9. These coils are wound on a 5/16" diameter phenolic paper tube. A 1/4" diameter copper slug sits inside the tube; its position can be varied from outside the cryostat by the drive shaft described in Section 3.1 and shown in Figure 3.4 of this chapter. The skin depth of copper at 4.2K and 10 kHz is ~ 0.002 — 0.003", so essentially the entire volume of the slug is shielded under these conditions, and it can therefore be used to effectively set the magnitude and sign of the magnetic flux cutt ing the pick-up coil . This means the magnitude and sign of this induced compensating voltage can be set by the position of the copper slug. The phase of this induced voltage is ~ 90° out of phase wi th the drive current. Certainly losses from the copper and the phenolic paper are present, but they appear to be negligible. This can be seen in the plot of induced voltage vs. position of copper slug, displayed in Figure 3.9. Only the out-of-phase component of the signal voltage imbalance is affected by this circuit. For fine tuning, two reference voltages (one in-phase, the other out-of-phase) are picked off the primary circuit, their magnitudes are appropriately transformed by the Chapter 3. Development of the Apparatus 28 (a) ' (b) Figure 3.9: The low temperature compensation circuit, (a) Diagram of the circuit itself, (b) Typica l response of the circuit. The position of the tuning slug is plotted as turns of the key from the extreme counter clockwise setting. In practice, one would null the imbalanced signal wi th this circuit to near one of the min ima indicated by the arrows. Fine adjustment of the compensation is then done wi th the decade transformers. decade transformers, and the resulting voltages are then injected into the secondary cir-cuit. A l l transformer ratios given in Figure 3.8 are wi th respect to the top inductor, and represent the actual number of turns in each case. The phase difference between the primary current and output voltage from the transformer across the 1.3Q, resistor was experimentally determined to be 0.97°. The phase of the output voltage from the transformer in series wi th the primary circuit was measured to be 89.95°. The orthogo-nality of these two voltages allows one to quickly set the decade transformers to give a null signal. Their performance over several decades of voltage transformation is shown in Figue 3.10. Chapter 3. Development of the Apparatus 29 0.060 0.040 J 0.020 ? & - - -a Out-of-phase o.ooo \- & - - = = = =§ e - © o 11 n i i 10"' i<r Decade transformer ratio 10"' 10"' Figure 3.10: The performance of the decade transformers. There is negligible phase shift in the output from the decade transformers. The orthogonality of the input voltages is preserved. Chapter 4 Performance of the Apparatus This apparatus was designed such that one could "see" a lA change in the penetration depth,A, of a typical sized Y B C O single crystal. W h a t is the sensitivity required for such resolution? To answer this consider equation 2.13, v = ^ V s X w H , (4.21) lp the induced voltage in a pick-up coil due to a sample of volume Vs and susceptibility X- For a superconductor in the Meissner state, X = X i = —1- (The signal is purely inductive.) The area of a typical high quality Y B C O crystal is ~ 1 x 1 m m 2 , to estimate its change in apparent volume due to a lA change in A one multiplies the area by 2A. This gives A V " = 2 x 1 0 - 1 6 m 2 . For the first version of the pick-up coil Np = 1360, and lp = 0.2" = 5.08 x 10~ 3 m. Taking w = 2ir x 10 4 radHz and u.0H = 10~ 3 T (ie. 10 gauss), gives a signal of about v = 3.4 x 10~ 9 volts. The intrinsic noise from the secondary coils at 1.2K is v = V4:kTRAv « ^4(1.38 x 10- 2 3)(1.2)(3.5)(1) « 1.5 x 1 0 " 1 1 volts. The amplifier noise when operating at 10 kHz within the 0.5 d B noise contour is ~ 0.09 n V for a 3.50 input. Theoretically, measuring a signal of 3.4 n V wi th a reasonable signal to noise ratio does seem quite possible. 30 Chapter 4. Performance of the Apparatus 31 The actual behavior of the ac susceptometer is the subject of this chapter. The resolution along with measurements on the temperature and magnetic field response of the device wi l l be presented here. The two ac coils sets essentially represent two different susceptometers. They are discussed separately and in chronological order to highlight how the unacceptable field dependence of the first coil set became the impetus for construction of the second. 4.1 AC Coil Set - Version I Here the secondary circuit includes the capacitor Cres shown in Figure 3.8. The true resonant frequency was determined experimentally to be 10.7 kHz . This frequency for the drive field was used for al l experiments in this thesis except where explicitly stated. 4.1.1 Sample Holder Temperature Dependence The dependence of the apparatus on the sample holder temperature Ts was measured wi th the helium bath in regulation at 1.2K, which is the lowest temperature reachable with our pumping system. Only the sapphire sample plate was present inside the pick-up coil . The results for the out-of-phase signal are shown in Figure 4.11. The signal is very small , and it rapidly becomes constant wi th increasing temperature. Although there are very few data points, the susceptibility roughly fits a 1/T dependence suggestive of a Curie term due to paramagnetic impurities[17]. This sample plate was later replaced wi th a much higher purity sapphire plate, which when tested gave no detectable temperature dependent signal. It appears then that impurities in the former plate were responsible for the paramagnetic observations in Figure 4.11. Chapter 4. Performance of the Apparatus 32 0.0 5.0 10.0 Temperature (K) 15.0 0.0 0.2 0.4 0.6 0.8 1/Temperature (K"1) 1.0 Figure 4.11: The temperature dependent backgound signal of the sapphire sample plate. / = 10.7 kHz . Hi = 8 gauss. X K ' J O C I / T appears to indicate paramagnetic behavior. 4.1.2 DC Magnetic Field Dependence The field dependence of the background is shown in Figure 4.12. Again only the sapphire sample plate was present inside the pick-up coil . The data plotted is for the out-of-phase signal. The size of the signal is very large. Even at an applied current of 0.5 amps (ie. 0.5 amps x 468 gauss/amp = 234 gauss « Hi of Y B C O ) the signal is in excess of 100X the desired resolution. To make matters worse, the signal is also hysteretic. The dc magnet current was supplied by a H P 6632A programable power supply. This power supply is of single polarity only, so one had to switch the leads to the magnet to measure the reverse field. There was sufficient interruption in data taking, during this process, to make uncertain the true pattern of the hysteresis loop around zero current. The presence of hysteresis in general, however, is absolutely unmistakable. A systematic study was undertaken to determine the magnetic characteristics of the Chapter 4. Performance of the Apparatus 33 -1.0 0.0 1.0 DC magnet current (Amps) 2.0 Figure 4.12: The dc field dependent backgound signal. / = 10.7 kHz. Hi = 8 gauss. Inset shows more clearly the hysteresis. construction materials that are in the vicini ty of the secondary coils. This , it was hoped, would reveal which of the substances was responsible for the large background signal. The measurements were done wi th a Quantum Design dc S Q U I D magnetometer. Samples of phenolic paper, 1266 epoxy, #44 copper wire (secondary coils), and #36 heavy formvar copper wire (similar to the wire in the primary coil though of a smaller gauge) were studied. Hysteresis loops over ± 2 0 0 0 gauss at 5 K for al l materials are shown in Figure 4.13. The data is normalized wi th respect to mass by plott ing the magnetic moment in units of emu per mass of sample in grams. The results show that a l l the materials are magnetic, each wi th a susceptibility \x\ = \M/H\ ~ 5 x 1 0 - 7 e m u g " 1 gauss - 1 . Only the 1266 epoxy does not show signs of hysteresis. It is impossible to tell from this data which of the materials are responsible for the Chapter 4. Performance of the Apparatus 34 -1.2 -i 1 i i i i i i -2000.0 -1000.0 0.0 1000.0 2000.0 Applied field (gauss) Figure 4.13: Hysteresis loops at 5 K for ac coil set construction materials measured with a dc S Q U I D magnetometer. background signal. (We do know that the 1266 is not responsible for the hysteresis.) It is also impossible to say whether the signal arises from a mass imbalance, or a winding imbalance. B y mass imbalance, meant an inhomogeneity in the spatial distribution of a given material wi th respect to the secondary coils. If a magnetic material is evenly distributed, then the voltage it would induce in each coil would be the same and no signal would be observed. O f course, i f imperfections exist between the windings of the coils (and this is quite likely over 1360 turns) then there w i l l be a signal regardless of balanced mass. These two effects cannot be distinguished from each other. To remedy the problem, it was decided at this point to bui ld another ac coil set. Great effort and expense was put into procuring the high purity materials for its construction. The design was altered to l imi t the mass in close proximity to the secondary coils. Chapter 4. Performance of the Apparatus 35 4.2 AC Coil Set - Version II The inductance of each of the new secondary coils was ~ 1 0 - 3 H giving a combined reactance for the pair of just 1250 at 10 kHz , so it was decided to forgo the use of the capacitor Cres. The effect on the noise was negligible. 4.2.1 Sample Holder Temperature Dependence The dependence of the apparatus on the sample holder temperature Ts was again mea-sured wi th the helium bath in regulation at 1.2K, and wi th only the high purity sapphire sample plate present inside the pick-up coil . As expected, there was no detectable signal at low temperatures. However, as the sample plate was taken to higher temperatures a signal was observed. The onset of this high temperature response was at about 30K. This experiment was repeated on a separate cool down of the apparatus, the sample plate having been removed, cleaned and replaced. The same behavior was observed again; the magnitude of the effect was almost twice as large this time. It is believed that the plate was positioned slightly farther inside the pick-up coil in the second experiment. This strongly suggests that the effect may be due to differential heating of the secondary coils caused by thermal radiation from the sample plate. It is also known that the coils are not in good contact wi th the sapphire tube (the coils pulled away from the tube at low temperature as mentioned in Chapter 2), which makes this idea seem reasonable. Figure 4.14 shows the raw data of the induced out-of-phase voltage as a function of Ts along wi th a plot of ln(v) vs. ln(T) for both trials. These show the signal voltage varying wi th temperature as T 5 8 . Whi le not strictly consistent wi th the Stefan-Boltzman law (radiative power oc T 4 ) , any mechanism with such a temperature dependence almost certainly involves a radiative process. The temperature dependence of the emissivity of the sapphire plate may account for the deviation from T 4 behavior. Chapter 4. Performance of the Apparatus The experimental implications of this thermal effect w i l l be discussed later. 36 e-- • First trial - o S e c o n d trial 0.0 B — e — f r - . -a- -cr 0.0 20.0 40.0 60.0 80.0 Temperature (K) CD CD eg "o > c co * o 100.0 3.6 3.8 4.0 4.2 4.4 4.6 In(Temperature) Figure 4.14: Dependence of the susceptometer signal as a function of the sample holder temperature for the version II ac coil set. Taking the natural log of both variables gives a near linear relationship. A t high temperature the respective slopes for each t r ia l are 5.6 and 6.0, an average of 5.8. 4.2.2 DC Magnetic Field Dependence To facilitate proper hysteresis measurements, two power supplies were used here. One, an Anatek 6007, was used to supply a constant current —Ima,x to the dc magnet. The other power supply, the H P 6632A, connected i n parallel, was programmed to step through current values from 0 to 2Imax. The total dc magnet current as measured by an H P 3478A multimeter and the signal voltage from the SR850 lock-in was logged by the control computer v ia the I E E E 488 bus. W i t h this method, many continuous hysteresis loops can be measured in succession. Field-dependent-background measurements over several loops are shown in Figure 4.15. The data is plotted as a function of the dc magnet current (shown on the left) and as Chapter 4. Performance of the Apparatus 37 a function of real time (shown on the right). Tr ia l 1 is from a separate cool down from Trials 2 and 3. In Tr ia l 3 the sample holder temperature was set at 100K, it was 1.2K for the other two. Trial 1 - Sample temperature = 1.2K i 1 1 1 1 1 1 I ^- i 1 1 1 r -0.6 -0.3 0.0 0.3 0.6 0.0 100.0 200.0 300.0 400.0 Trial 3 - Sample temperature = 100K Current (amps) j j m e (seconds) Figure 4.15: The dc field dependent backgound signal of the version II ac coil set for / = 10.7 kHz and Hx = 8 gauss. The out-of-phase voltage signal is plotted in real t ime and as function of the magnet current. The drift (in Tr ia l 1) was eliminated by thermally isolating the electronics. The background signal is much larger at Tsampie = 100K There are many notable features in this data. Most notable, of course, is the fact that the background of the version II coil set is also field dependent. Despite a huge Chapter 4. Performance of the Apparatus 38 effort to eliminate the problem, the field dependence st i l l persists and it is hysteretic. Compared to the first coil set at an applied current of 0.5 amps (see Figure 4.12) the signal for the Tsampie = 1.2K data is roughly 10X less. To give a true comparison of the backgrounds however, one needs to mult iply this figure by the ratio of turns of the two pick-up coils (this is essentially the ratio of the sensitivities of the two coil sets.) This means that the field dependent background of the second coil set was reduced by a factor of 10 x 511/1360 ~ 3.8 over the first design. A reasonable improvement, but not quite the 100X reduction required to bring this background down to the level of the resolution. Also present in the Tr ia l 1 data is a fluctuation in the voltage signal baseline. This is easily seen in the time plot. It affects the hysteresis plots by smearing them in the voltage axis direction. The source of this drift was discovered to be changes in the air temperature of the laboratory. In particular the fluctuations in an otherwise constant background signal were found to be syncronized wi th the automated room air-conditioner. To fix this problem, the room temperature electronics of Figure 3.8 (less the decade transformers) were thermally anchored to an aluminum plate and isolated inside a styrofoam box. Water circulation from a temperature controlled bath kept the temperature of the aluminum plate constant. The improvements in performance were significant. The hysteresis loops in Trials 2 and 3 lie almost directly on top of one another. This repeatability, coupled wi th the use of the computer (for data collection and current control), allows one to make a very large number of continuous traces over the hysteresis loop. This w i l l be useful as a form of signal averaging when it comes to measuring the nonlinear Meissner effect in Y B a 2 C u 3 0 6 . 9 5 . This background is s t i l l too large to allow for direct measurement of the effect, but it may be possible to subtract background data from sample data to give a reliable measure of the sample's field response. Averaging over many subsequent current loops for both sets of data wi l l allow improvement in the signal to noise. In the normal state, the thickness of a Y B C O crystals is much smaller than its skin Chapter 4. Performance of the Apparatus 39 depth at 10 kHz , so it appears transparent to the magnetic fields. In theory then, it may be possible to measure the background by taking the sample above Tc = 93K. In practice however, the results of Tr ia l 3 rule this out. The field dependent background of the apparatus wi th sample plate temperature at 100K is significantly larger than at 1.2K. The two would have to be the same for this technique to work. 4.3 Determining the Resolution of the Apparatus 4.3.1 Measuring A A (T) of Y B a 2 C u 3 0 6 . 9 5 Precision measurements of the temperature dependence of the penetration depth of Y B C O have been done at U B C using microwave cavity perturbation. The linear tem-perature dependence of A(T) at low T , a signature of superconductors wi th nodes in the gap function, was first observed here using this technique [18]. Repeat measurements of this experiment were done wi th the ac susceptometer. Bo th versions of the ac coil set were used to measure A A ( T ) = A(T) — \(1.2K) for the sample Y l q , a high purity, twinned YBa2Cu30g .95 single crystal. The crystal is a thin slab of thickness t ~ 27 /im. The broad face of the crystal, the ab plane, is ~ 1.5 x 1.5 m m 2 . As in the microwave measurements, Hi is applied parallel to the ab plane. Measurements were taken over the range 0 — 20K, well below temperatures where the thermal effects in the version II coil set become noticeable. Figure 4.16 shows A A ( T ) for both susceptometer measurements as well as the microwave measurements for the same crystal. A l l three sets of data are in good agreement, which establishes the ability of the ac susceptometer to do precision, state-of-the-art measurements. In the cavity perturbation method, the change in penetration depth is related to a shift in the resonant frequency of the cavity [18]. For the ac susceptometer, A A ( T ) is proportional to real part of the susceptibility x'i- The calibration for penetration depth Chapter 4. Performance of the Apparatus 40 measurements is exceptionally simple; one does not need to resort to any of the techniques described in Chapter 2. Here the susceptometer is calibrated in situ for each sample by measuring the out-of-phase signal, VSjH, that results from taking the sample from 1.2K to 100K. In the normal state, the crystal is in the thin l imit so there is essentially complete penetration of the drive field. A t 1.2K the penetration depth is negligible compared to the crystal thickness t. To a very good approximation then the transition from 1.2K to 100K corresponds to a A A ( T ) = t/2, and the calibration constant is just k = 2VSyJl/t. For version I, kT = 2(27 x 10 4 ) / (1.63" 4 ) = 0.83 A / nV . For version II, kH = 2(27 x 10 4)/(0.4898~ 4) = 2.8 A / nV . The value kn/ki — 3.4 is a precise measure of the sensitivity ratio between the two ac coil sets. In the section above, this was simply estimated by the turns ratio of the pick-up coils ie. NT/Nn = 2.7. This new value means that the improvement in the field dependent background in version II was actually just a factor of 3, not 3.8. 4.3.2 Calculating the Resolution from AA(T) The resolution of the apparatus in angstroms is calculated by mult iplying half the peak-to-peak voltage noise (measured off a strip chart recorder that is output from the lock-in) by the calibration constant. Typica l values for the noise during the A(T) experiments were 12 x 10~ 1 0 and 5 x 1 0 - 1 0 volts peak-to-peak for version I and II respectively. This gives a resolution of 0.5 A for version I and 0.7 A for version II. Bo th coil sets exceed the desired 1 A resolution set out as the design minimum at the beginning of this chapter. Chapter 4. Performance of the Apparatus 80.0 60.0 CO E o -i—» w CO c < < 40.0 20.0 i) A [ ] • cavity perturbation (f=961 MHz) o ac susceptometer I (f=10.7 kHz) A ac susceptometer II (f= 10.7kHz) A O 8 0.0 -flB-0.0 5.0 10.0 Temperature (K) 15.0 20. Figure 4.16: AA(T) for the YBa2Cu306.95 sample Ylq. Chapter 5 The Nonlinear Meissner Effect: A ( H ) of YBa 2Cu 30 5.1 Theory The electrodynamics of a superconductor in the Meissner state (H < Hci) obey the London equations ^ 1 f V 38 = -^js. —* It is a well known result from these equations that a magnetic field B decays inside a semi-infinite superconducting slab as B(z) = Be~z/\ (satisfying the boundary condition B(Q) = B) where z is the distance into the supercon-ductor. For high temperature superconductors, which are Type II superconductors, the characteristic length of this decay is the London penetration depth, A = ^m/(p0ps(T)e2). Here, ps(T) is the superfluid density; m and e are the electron mass and charge respec-tively. Under most conditions, as it was above, the supercurrent density is sufficiently defined by the linear relation I = -eps(T)vs, (5.22) where vs is the supercurrent velocity. Y i p and Sauls derive a nonlinear velocity term in 42 The Nonlinear Meissner Effect: X(H) of Y B a 2 C u 3 0 6 . 9 5 43 js. (In their model they consider the supercurrent to be largely confined to two dimen-sions.) They then re-solve the London equation, now nonlinear wi th the inclusion of the —* higher order term in js, and from this derive an effective penetration depth that depends explicit ly on H. They do this for both conventional and unconventional superconductors. Their results are summarized below. 5.1.1 Conventional Gap Here the supercurrent is given by js = -eps(T)vs 1 - a (5.23) The second term is a correction to the supercurrent of 5.22, which is reduced due to pair breaking in finite fields at nonzero temperatures. Here vs « vc, the crit ical velocity, defined as A(T)/pf (the energy gap divided by the Fermi momentum.) The coefficient d(T) is always positive and tends to zero as T —» 0 , as a consequence of the existence of the gap. Substituting this js into the London equation gives 1 _ Y2vs = 0. (5.24) where the term ct(T) << 1 has been ignored in the Laplacian term. This nonlinear London equation is solved 1 in closed form for H, satisfying the boundary condition dvs _e ~r~ U=o — ~ M -dz c A n effective penetration depth is defined from the ini t ia l decay rate of H inside the superconductor. It is given as 1 / H V1 ^ Xeff(T,H) = A(T) 1 AT) MT)) details of this calculation are given by Xu, Yip and Sauls [2]. They do not appear in the original paper on the nonlinear Meissner effect[3]. The Nonlinear Meissner Effect: X(H) of Y B a 2 C u 3 0 6 j 44 (5.25) A(T) 1 + 1<*{T) where A(T) is the zero field London penetration depth and H0(T) = |eo c(T)jeX(T) is a constant of the order of the thermodynamic crit ical field. The change in A e / / is proportional to H2 and thus as expected Xeff is larger than A(T) . The effect of the field therefore is to increase the effective penetration depth. B y pair breaking, the supercurrent density is reduced, and so too is the superconductor's abili ty to screen the field. Experimentally, one can probably expect a smaller correction to A(T) than what is predicted above. Here A e / / was defined from the in i t ia l decay rate, and so i t is only truly applicable in describing the magnetic field near the surface of the superconductor. However, any measurement w i l l be sensitive to the entire shielding range and not just the interface. We have estimated that this definition of Xeff leads to a correction in A(T) that is ~ 3 X too large. Al though it is not explicitly stated in [2], this same definition for Xeff is most likely used in the case of unconventional gap as well. As a result, the corrections to A(T) given below sould also be considered as too large. 5.1.2 Unconventional ( d x 2 _ y 2 ) Gap A cross section of the Fermi tube for the dx2_y2 pairing state is shown in Figure 5.17. . A t T = 0, quasipar-There are four nodes in the gap occurring at the positions t ide occupation at the nodes wi l l remain nonzero. This gives rise to a quadratic vs term —* in the supercurrent js that is independent of temperature. Because of the anisotropy of the gap, however, no single form for the dx2_y2 supercurrent can be written. For the case where vs is along a node (see Figure 5.17), the supercurrent is given by the equation Is = -eps(T)vs 1 \Vs\ Vo (5.26) The Nonlinear Meissner Effect: \(H) of Y B a 2 C u 3 0 6 . l 45 where v0 = 2 A 0 / i > / is a characteristic scale and A 0 is the gap maximum at an antinode. The second term can be viewed as the quasiparticle backflow from the node opposite vs, that reduces the supercurrent. This backflow is calculated over the wedge of occupied states about the node. ky Figure 5.17: The dx2_y2 gap function. The supercurrent velocity is along a node. Oppo-site this node is the wedge of occupied quasiparticle states that constitute the backflow current. For the case where vs is along an antinode (ie. 45° to a node, along the direction of the gap maximum), the supercurrent is js = -eps(T)v6 The only difference with equation 5.26 is the factor I / A / 2 that appears in the backflow term. This factor is just the cos(45°) for the new projection of js along the nodal lines. This anisotropy wi l l appear in the resulting nonlinear London equation and in its solution as we l l . 2 The factor of l / \ / 2 of course shows up in the equations for the 2 Again the details of the calculation are given in [2]. 1 - 1 \vs \/2 v0 (5.27) The Nonlinear Meissner Effect: X(H) of Y B a 2 C u 3 0 46 effective penetration depth, where hopefully it can be exploited experimentally to probe the anisotropy of the gap in high T c superconductors. The pertinent equations are Xeff(T,H) = A(T) or Xeff(T,H) ~ A(T) and Xeff(T,H) = A(T) or Xeff(T,H) ~ A(T) • 2H 1 + 3K H || node, (5.28) H || antinode, (5.29) _L_2_ff + V23H~0 where H0 = (y0/X{T)){c/e) is also of order the Hc, thermodynamic crit ical field. The ob-servation of this linear field dependence in the penetration depth would strongly support the existence of unconventional pairing. The subsequent observation of an anisotropy of 1/V2 in Xeff, for field orientations along a node and antinode, would be very strong evidence for dxi_yi. One should keep in mind that the thermal excitations, responsible for the H2 de-pendence of Xeff in the conventional superconductor, must necessarily be present in unconventional superconductors as well. However, for sufficiently small T and large H, the linear term wi l l dominate. Experimentally, the magnitude of H is l imited by the cri t ical field Hc\, the superconductor must be in the Meissner state for meaningful re-sults, so one must work at low temperatures. A t any finite T however, there wi l l remain a cross-over field below which the quadratic contribution wi l l become significant. If care is not taken, this could obscure detection of the linear behavior. The Nonlinear Meissner Effect: X(H) of YBa2Cu 30 6 , 47 5.1.3 Some Estimates How low must the temperature be to allow for the observation of the linear behavior in the field range 0 —> Hc{! Y i p and Sauls argue that the cross-over field and temperature wi l l be related by -jf- ~ jr. Assuming H0 ~ 10 4 gauss, Tc ~ 100K, and Hci ~ 250 gauss for a typical cuprate superconductor [3] and taking H = f f c l / 1 0 implies a temperature of 0.25K. This is roughly 1/4 of the lowest temperature presently achievable wi th our apparatus. A t I K then, the cross-over field is ~ 100 gauss. This a very large fraction of Hci, but possibly the linear term wi l l s t i l l be observable over the remainder of the field range. How large is the nonlinear Meissner effect expected to be? X u , Y i p and Sauls [2] suggest the change in Ay ( i f ) is roughly 2 A | | i f c l / i i 0 ~ 30 A for an ac measurement. For a crystal of the size of Y l q (~ 1.5 x 1.5 x 0.027 mm 3 ) and using the version II ac coil set, this translates into a ~ 11 n V signal. This is ~ 40X larger than the voltage noise of the version II ac coil set. However, in terms of the field dependent signal of the apparatus itself, 30 A is only ~ | of the background at 250 gauss. This is certainly measurable, if an accurate background signal can be subtracted from the total signal. 5.2 Experiment Measurements of the induced voltage as a function of applied dc current were made wi th the version II ac coil set on the YBa2Cu3 0 6 . 9 5 sample Y l q , a large crystal which wi l l improve resolution. Only the field dependence of A w i l l be tested for now. The experimental set up was the same as that described in section 4.2.2. The probe was cooled wi th magnetic shielding around the dewar wi th the shield left in place for a l l measurements. F ie ld sweeps were made at different sample temperatures (1 and 10K in that order) for three different maximum dc current levels, corresponding to fields of The Nonlinear Meissner Effect: \{H) of Y B a 2 C u 3 0 6 , 48 100, 200, and 240 gauss. The sample was cooled in zero field before each set of trials for the three maximum current settings; this was done by heating the sample to 100 K , turning off the dc magnet and the drive solenoid, then allowing the sample to cool. Ba th temperature was held constant at 1.2 K , and the drive field was kept at the same magnitude (8 gauss) and frequency (10.7 kHZ) for a l l trials. The number of consecutive sweeps through the current loop was typically 50 for the sample trials, while for the background trials it was 100. Background measurements were made at 1.2 K only, for each of the three dc field levels. Because of the temperature dependent signal associated wi th this coil set, the back-ground response could not be measured by heating the sample above T c , which requires a sample temperature ~ 100 K . It is known (see section 4.2.2) that the field dependent background at this temperature is much larger than it is at 1.2 K , making this approach completely unacceptable. The only alternative was to measure the background, before and after the sample measurement, by actually removing the sample from the suscep-tometer. This requires the susceptometer to be brought to room temperature and opened up between the subsequent runs. 5.3 Results and Discussion In the end only one set of background measurements (the one taken prior to the sample measurements) could be used in the data analysis. The background measured after the sample was removed was markedly different, both in magnitude and shape of its hysteresis loop, from any field response previously observed for this coil set. See Figure 5.21. Fearing contamination, the sample plate was removed and cleaned, and N 2 gas was blown through the susceptometer. The background measurement was repeated, but this anomalous response persisted. The source of this problem has not yet been The Nonlinear Meissner Effect: X(H) of YBaaCugOe., 49 identified. Figure 5.20 shows three typical backgrounds (including the first one from this experiment) for the apparatus. They are very similar, but as they were not taken under identical conditions the repeatability of the background for this method is st i l l unknown. The results from the 240 gauss trials are shown below. The data for each t r ia l has been averaged over a l l of its consecutive hysteresis loops. The averaging is not sensitive to the direction in which the current is stepped through the loop. This was done deliberately to remove the effects of any slow monotonic drift in the baseline. The hysteresis is also averaged when using this approach, so the signal appears as a unique function of each current setting. Figure 5.18 displays the averaged data for a l l measurements. Figure 5.19 shows the sample data corrected for the background. The results are very unclear, and without proper knowledge of the background very l i t t le can be concluded about X(H) for Y B a 2 C u 3 0 6 . 9 5 . There is a large asymmetry associated wi th the direction of the magnet current. The sample data for 1.2 and 10.0 K is very nearly the same for negative current values, but is substantially different for positive current. There is also an asymmetry in the magnitude of the corrected data. This is also seen in the background measurements, but had it been an effect of the background only then it should have disappeared from the corrected data. It did not. Possibly, this is an indication that the magnetic field might be distorted by some permanent magnetism in the proximity of the pick-up coils. The fact that al l background measurements (see Figure 5.20) differ from the sample data by the same sign does suggest that some field effect of the Y B a 2 C u 3 0 6 . 9 5 crystal was truly detected. A signal voltage of 6 n V was measured at —0.51 amps (ie. —240 gauss), and i f it is to be believed then this constitutes a AX(H) = 16 A for H ~ Hci. If one considers the 4 n V spread between the background measurements at this field level to represent the uncertainty in the background, then L\X(H) = 1 6 ± l l A a t i 7 . ~ Hcl. This The Nonlinear Meissner Effect: X(H) of Y B a 2 C u 3 0 6 , 50 would strongly suggest that the A A ( / f c l ) ~ 30 A predict by X u , Y i p and Sauls represents an upper l imi t for the nonlinear Meissner effect in YBa2Cu 30 6.95 . The only real conclusion that can be drawn is that more work on the background wi l l have to be done before a proper measurement of the \(H) can be made. It would be best if the background could be reduced further. This would certainly entail building another ac coil set. However, reliable measurements might s t i l l be possible wi th the present apparatus i f a better method for determining the background can be found. One solution might be to place a thermal shield inside the secondary coils. This would eliminate the problem of the field dependent background being a function of the sample temperature as well. This in turn would allow for the in situ determination of the background by heating the Y B a 2 C u 3 0 6 . 9 5 sample above Tc . The Nonlinear Meissner Effect: X(H) of YBa2Cu306.95 51 60.0 > CD co o > "cC c D) CO 40.0 20.0 -0.0 \ \ \ A \ -0.6 \ 0- - o Ylq at Ts=1.2K • • YlqatTs=10.0K A A Background at T =1.2K \ -0.3 0.0 0.3 DC magnet current (amps) Figure 5.18: Averaged data for magnetic field response of YBa2Cu306.95 sample Ylq and background. The Nonlinear Meissner Effect: X(H) of YBa2Cu306, > <D O) CO o > "co c 00 7.0 6.0 5.0 4.0 3.0 2.0 1.0 -0.0 --1.0 -0.6 & - -o Ylq (at Ts=1.2K) - Background o- - -o Ylq (at Ts=10.0K) - Background JD 0=. - n - - Q ' " ' ' 0 ' r / 0 0 / 0 -0.3 0.0 0.3 Dc magnet current (amps) 0.6 Figure 5.19: Corrected data for YBa2Cu306.95 sample Ylq. The Nonlinear Meissner Effect: X(H) of Y B a 2 C u 3 0 6 , 53 Figure 5.20: Background measurements from separated cool downs, not done under identical conditions. This data is consistent wi th an observable magnetic effect in Y B a 2 C u 3 0 6 .95 • The Nonlinear Meissner Effect: X(H) of YBa2Cu306.1 54 12.0 Typical background response of ac coil set II o > -0.6 -0.3 0.0 0.3 0.6 0.0 100.0 200.0 Anomalous behavior of ac coil set II -0.6 -0.3 0.0 0.3 0.6 0.0 Current (amps) 100.0 200.0 300.0 300.0 Time (seconds) 400.0 400.0 Figure 5.21: Anomalous background measurement for the version II coil set. The origins of the problem are not yet known. Chapter 6 The Vortex Melting Transition in Y B a 2 C u 3 0 6 9 5 In this chapter I wi l l present some preliminary results from studies of the vortex melting transition in YBa2Cu306.95 done using this ac susceptometer wi th the version I coil set. This transition was believed to be of first order, and so should be accompanied by a discontinuity in the first derivatives of the free energy G. Recent observations at the University of Br i t i sh Columbia by Liang, Bonn and Hardy [19] of a discontinuity in the magnetization (M = —dG/dH) of a Y B a 2 C u 3 06.95 single crystal provided the first direct proof that the vortex melting transition was indeed first order. The sample, of superior quality, had exceptionally low pinning, as characterized by an irreversibility line well below its vortex melting transition. This allowed for the very clear measurements. In the case where there is large amounts of pinning, the transition from a vortex l iquid to a vortex solid would be much more broad and therefore escape detection. This is precisely why there had never been a previous observation of this effect in YBa2Cu30 6 . 9 5 . The measurements of L iang et al . were done using dc S Q U I D magnetometer, but certainly similar tests on the same crystal using the ac susceptometer should also reveal the existence the vortex melting transition. What wi l l the transition look like for an ac measurement? If one takes the real part of the complex susceptibility to closely approximate ^ in the l imit that Hi « Hdc, then x'i would be expected to diverge at the discontinuity in M. See Figure 6.22. This hypothesis is the same whether the experiment is done as a function of dc field or temperature. 55 The Vortex Melting Transition in YBa2Cu306.95 56 dc measurement -M(T,H) - ac measurement (real part of magnetization) H, T Figure 6.22: Predicted ac response of the vortex melting transition. The real part of the complex susceptibility (dashed line) is expected to follow the derivative of the dc magnetization. The measurement can be made as a function of temperature of magnetic field. 6.1 Experiment Using the same YBa2Cu306.95 sample as above (labeled Up2), measurements were done in a fixed field (Hdc = 10, 20kOe) as a function of temperature (T = 100 —> 60 K), a region of operation suitable for the version I coil set. The dc magnet current was supplied by an HP6260B power supply. A custom computer program was written to control the sample temperature via the Conductus LTC-20, and collect data from the SR850 lock-in. The sample was held perpendicular to the magnetic field on the face of a sapphire block that was epoxied to the end of a standard sapphire sample.plate. This orientation is favorable for vortex formation in the sample. There was no background signal (within the resolution ±10~8 volts) due to the sample holder over the temperature range 4K-> 100 K. The Vortex Melting Transition in Y B a 2 C u 3 0 6 . 9 5 57 6.2 Results and Discussion Figure 6.23 shows the voltage signal as a function of temperature for two field settings, 10 and 20 kOe. The drive voltage was 1 .0V P _ P ; the drive field Hi = 0.8 gauss. There is clearly a step in both sets of data marking the possible presence of the vortex lattice melting transition. The steps are present in both the in and out-of-phase data. O n either side of the step the out-of-phase components do behave as predicted, however there is no spike marking the transition as would also be expected i f x'i is t ruly the derivative of the dc magnetization. Is this the vortex melting transition then? Comparison wi th the dc data [19] strongly suggests that i t is. Transitions occur at 87.90 K and 88.25 K at 10 kOe, and at 89.80 K and 90.20 K at 20 kOe for the ac susceptometer and dc magnetometer measurements respectively. One would expect the two measurements to be identical, since Hi « Hdc and so only represents a very small perturbation to the dc condition. A t both field values however, the difference between the two types of measurement is 0.4 K . The sample thermometer in the ac susceptometer was shown to change by only 0.06 K over an applied field of 20 kOe at 85 K , so this is not the problem. The T c of this crystal was measured wi th the susceptometer in zero field (Hi = 0.2 gauss) and found to be 92.7 K . It is given in [19] as 93.1 K . Clearly, the temperature discrepancy between the two devices is systematic. Possibly the sample thermometer in the susceptometer was not properly calibrated in this temperature range. It has been suggested [20] that due to hysteresis about the transition, the crystal, under the low drive fields present here, cannot respond quickly enough to the changes in the field. As a result, one does not measure a sharp transition. The crystal is in a mixture of the two states at the expected transition point and as a result one just measures an average magnetization. Unfortunately, no drive fields larger than 0.8 gauss ( 1 . 0 V P _ P ) were studied at the time of the experiment, so this hypothesis has not yet been tested. The Vortex Melting Transition in YBa2Cu306.95 58 However, the data in Figure 6.24 does show a small peak on the 1 .0V P _ P data (just at the low temperature end of the transition) that is not present for the lower drive fields. Possibly this is remnant of the much larger peak that was expected at the transition. Da ta for the in-phase (loss) signal is shown in Figures 6.23 and 6.25. There is a step at the transition, much like the step in the out-of-phase signal. It is obviously associated wi th the latent heat of the first order transition. Also present is a peak in the data occurring at temperatures below the transition point. This peak has been associated wi th the irreversibility line [21]. It is clear from Figure 6.25 that the data is consistent wi th this idea. Here the peak is seen to shift to lower temperature as the frequency drive field reduced. The dc measurement of the irreversibility line at 20 kOe is 75 K [19]. The Vortex Melting Transition in Y B a 9 . C u 3 O 6 . 9 5 0.4 CO o > & 0.2 CO -*—* o > "CO c CT> CO 0.0 )<*°°00 ooooo0 v <>0 In-phase signal. O 06,. = 20 kOe o oBdc = 10kOe 81.0 83.0 85.0 87.0 Temperature (K) 89.0 0.0 o > cu g> -0.4 o > "CO c CT) CO -0.8 o ooocco^cp oocrxccxxxDcaxxxxxx^ o Out-of-phase signal, o Bd0 = 20 kOe oBdc = 10kOe 84.0 85.0 86.0 87.0 88.0 89.0 90.0 Temperature (K) Figure 6.23: The vortex melting transition in the YBa2Cu306.95 sample Up2 at dc fields. The Vortex Melting Transition in Y B a 2 C u 3 0 6 . 9 5 60 0.0 -0.1 -0.2 86.0 < > > t> > > > ^  o < < < < o o o © Q 3 < < < ^  o B B B B O O ' O O CS Out-of-phase signal, o 1.0 V„ p-p o 500 mV^  < 250 mV, > 125 mV, p-p p-p p-p 87.0 88.0 Temperature (K) 89.0 Figure 6.24: The vortex melting transition in the Y B a 2 C u 3 0 6 . 9 5 sample Up2 at various drive fields. The Vortex Melting Transition in YBa2Cu306.95 61 3.00 80.0 82.0 84.0 86.0 88.0 Temperature (K) Figure 6.25: The vortex melting transition in the YBa2Cu306.95 sample Up2 at various drive frequencies. Chapter 7 Conclusion The primary goal of this thesis was to develop a high sensitivity ac susceptometer to mea-sure the nonlinear Meissner effect in the high temperature superconductor Y B a 2 C u 3 0 6 . 9 5 . In this regard the work to date must be considered unsuccessful and the goal presently unfulfilled. The susceptometer has a field dependent background that is significantly larger than the effect to be measured. It should be possible to subtract the background signal from the sample data, but this requires a reliable determination of the background which was never obtained. Measuring the background wi th the sample in situ requires heating the sample above Tc = 93 K . A t this temperature the response of the pick-up coil is significantly altered by some heating effect that makes this approach unsuitable. The only option was to measure the background by removing the sample, a very poor technique that was not reproducible to the desired level of resolution. So, not having sufficiently accurate knowledge of the background rendered the ac susceptometer useless for measuring X(H) wi th any degree of precision. In fact, the only quantitative result that could be gathered about the nonlinear Meissner effect in Y B a 2 C u 3 0 6 . 9 5 is that A\(Hcl) is less than the 30 A estimated by X u , Y i p and Sauls. Experiments on the vortex melting transition and the temperature dependence of the penetration depth of Y B a 2 C u 3 0 6 . 9 5 , however, present quite a different view of the susceptometer. A t a fixed field, very precise, state-of-the-art measurements can be made with this device. Ideally an ac susceptometer can be used to make measurements along 62 Chapter 7. Conclusion 63 any path in the H — T plane. It can be said then that a susceptometer was developed having a far greater sensitivity when measuring along field contours in the H—T plane. This is a more balanced conclusion to this thesis. It acknowledges the merits of the device, but does not hide its failures. It should be noted, that an experiment to measure the nonlinear Meissner effect is st i l l pertinent, and so the motivation to make further improvements to the design of the ac susceptometer st i l l exists. For the present ac coil set it may be possible to thermally shield the secondary coils from the sample holder, thereby allowing for the in situ determination of the background. If the coil set is to be rebuilt, the design should include a sapphire coil form that replaces the phenolic paper coil form of the version I coil set. Here the secondary coils would be wound onto the form and good thermal contact at low temperatures would be assured. Also the inaccuracies that arose from winding the secondaries on teflon forms would be greatly reduced using a rigid form again. Bibliography [1] J . Buan , B . Stojkovic, N . E . Israeloff, A . M . Goldman, 0 . Vails , J . L i u , and R . Shelton, Physical Review Letters 72, 2632 (1994). [2] D . X u , S. K . Y i p and J . A . Sauls, Physical Review B 51, 16233 (1995). [3] S. K . Y i p and J . A . Sauls, Physical Review Letters 69, 2264 (1992). [4] David Pines preprint, 1995. To be published in High T c Superconductivity and the Ceo Family, eds. T . D . Lee and H . C . Ren, Gordon and Beach, 1995. [5] T . Ishida and R . B . Goldfarb, Physical Review B 41, 8937 (1990). [6] M . Nikolo, American Journal of Physics 63, 57 (1995). [7] A . F . Khoder and M . Couach, Cryogenics 31, 763 (1991). [8] R . B . Goldfarb and J . V . Minerv in i , Review of Scientific Instruments 55, 761 (1984). [9] L . D . Landau and E . M . Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1960), pp. 124 and 125. [10] T . Ishida and H . Mazak i , Journal of Appl ied Physics 52, 6798 (1981). [11] A . Shaulov and D . Dorman, Appl ied Physics Letters 53, 2680 (1988). [12] Mar t i n A . Plonus, Applied Electromagnetics (McGraw-Hi l l Book Company, New York, 1978), pp. 335 and 336. [13] The sapphire tube was supplied by Insaco Inc. of Quakertown, Pa . The sapphire was of optical quality. Cost was $780. [14] The copper wire was supplied by California Fine Wire Company of Grover Beach, Ca . Cost was $160/500'. [15] The niobium wire was supplied by Supercon Inc. of Shrewsbury, M a . Included assay of the N b indicated < 100 ppm of ferromagnetic impurities. Cost was $290/300'. [16] From the catalog Lock-in Amplifiers. Copyright © 1 9 7 7 Princeton Appl ied Research Corporation. 64 Bibliography 65 [17] Nei l W . Ashcroft and N . David Mermin , Solid State Physics (Harcourt Brace College Publishers, Orlando, 1976), pp. 656. [18] W . N . Hardy, D . A . Bonn, D . C . Morgan, Ru ix ing Liang , and K u a n Zhang, Physical Review Letters 70, 3999 (1993). [19] Riux ing Liang, D . A . Bonn and W . N . Hardy preprint, 1995. [20] R iux ing Liang, private communications, 1995. [21] V . B . Geshkenbein, V . M . Vinokur , R . Fehrenbacher, Physical Review B 43, 3748 (1991). Appendix A Complex Susceptibility 1. It can be shown that the definition of X from equation 2.4 is equivalent to the definition x = dM/dH. Assume a general expression for the sample magnetization oo M(t) = ' £ M n e i n w t , n=0 where the Mn are independent of t. For an applied field H(t) = H^c + H\ etwt evaluate the expression dM _ dM{t) dt dH ~ dt dH(t) oo = Y,nMn^{n~l)Wt/H1. ( A . l ) n = l The left hand side of this equation must necessarily be a periodic function, call it x(t) and write it as the expansion k=0 or by relabelling the terms as the equivalent series oo m = Y,Xiei{l-1)wt- (A-2) i=i Comparing terms of A . 2 and A . l gives nMn X n = ~ ^ - , (A.3) 66 Appendix A. Complex Susceptibility 67 for n = 1, 2,3 • • • oo. Redefining Xn = X n / n (which seems to be convention [1-2]) returns the result Chapter 1. The dc field was explicitly included here. 2. Throughout this thesis the applied magnetic field was defined as H(t) = HJm{eiwt) = Hx sin There is, of course, no reason why it could not be defined as H(t) = HxRe(eiwt) = Hx ca&wt where Re( ) denotes the real part. Redefining H(t) only shifts the time origin and does not change the physical reality of the sample's magnetization. Here the magnetization would be expressed as OO M{t) = HiY.MKnJ™*) n = l oo = Hi ^2(KUcosnwt + KNsinnwt), (A.4) with the complex susceptibility nn = nn — iKn given by / W fw - r l . K = —rr- / Mit) cos nwt at, TVH\ J=2-X UJ (A.5) ^ M(t) s in nwt at. To compare this definition of the susceptibility wi th that of equation 2.4, it is easiest to rewrite A .4 as 00 M(t) = £ lm(iKn einwt), (A.6) 71=1 Appendix A. Complex Susceptibility 68 and then rewrite 2.3 in a new temporal variable r oo n = l which is equivalent to oo M(t) = # i £ I m ( X „ e i n ^ ) ) , (A.7) n = l as WT = wt 4- \ is the transformation between the two time scales. From A.6 and A . 7 one gets the relation Xn = i*>n e - M M r / 2 , for all n = 1,2,3,... . Wr i t ing i — (—1)^ and recognizing that for this sequence the quantity e _ m 7 r / 2 = (—l) n (z ) n = ( — l ) 3 n / 2 , this can be transposed into the more appealing form Xn = (-lfn+1^Kn. (A.8) The definition of the fundamental susceptibility remains unchanged. This not true in general for the higher harmonics, though for al l n , \xn\ — |«n|- The third har-monic (which is often studied [10, 11]) changes sign under the alternate definition of H(t), and the real and imaginary parts of even harmonics are interchanged. Heeding Ishida and Goldfarb [5], one must keep these relations in mind for in-terlaboratory comparisons of the harmonic susceptibilities as well as for theoretical calculations. 

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