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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 By Christopher P a u l B i d i n o s t i B . Sc. (Physics) B r a n d o n University  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF S C I E N C E  in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  December 1995 (c) Christopher P a u l B i d i n o s t i , 1995  In  presenting this  degree at the  thesis  University  in  partial  fulfilment  of British Columbia,  of  the  requirements  for  permission for extensive  copying of  this thesis for scholarly purposes may be granted by the  department  or  his  or  her  representatives.  It  is  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  advanced  I agree that the Library shall make it  freely available for reference and study. I further agree that by  an  head of my copying  or  my written  Abstract  T h e development of a high sensitivity ac susceptometer is outlined i n this thesis. T h e intended use of this device is for the measurement of the magnetic field dependent penetration depth of high temperature superconductors i n the Meissner state. T h e nature of the field depence may be useful i n 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 penetration depth and the vortex melting transition of the high temperature superconductor Y B a C u 0 .95. 2  3  6  ii  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 C o m p l e x Magnetic Susceptibility  3  2.2  E x p e r i m e n t a l Technique  4  2.3  C a l i b r a t i o n 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  T h e A C C o i l Set  20  3.3  The Electronics  25  4 Performance of the Apparatus  30  iii  4.1  4.2  4.3  5  6  7  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 i e l d Dependence  32  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 i e l d Dependence  36  Determining the Resolution of the Apparatus  39  4.3.1  Measuring A A ( T ) of Y B a C u 0 . 9 5  39  4.3.2  C a l c u l a t i n g the Resolution from A A ( T )  40  2  3  6  T h e Nonlinear Meissner Effect: A(H) of Y B a C u 0 . 9 5  42  5.1  42  2  3  Theory  6  5.1.1  Conventional G a p  43  5.1.2  Unconventional (d 2_ 2) G a p  44  5.1.3  Some Estimates  47  x  y  5.2  Experiment  47  5.3  Results and Discussion  48  T h e V o r t e x M e l t i n g T r a n s i t i o n in Y B a C u 3 0 . 2  6  9 g  55  6.1  Experiment  56  6.2  Results and Discussion  57  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  D i a g r a m of the cryostat  15  3.4  D i a g r a m of the ac susceptometer  17  3.5  T h e calculated axial field profile for the superconducting dc magnet. . . .  20  3.6  D i a g r a m of the ac coil sets  22  3.7  Construction of niobium secondary coils  25  3.8  T h e susceptometer circuit  26  3.9  T h e low temperature compensation circuit  28  3.10 T h e performance of the decade transformers  29  4.11 T h e temperature dependent backgound signal of the sapphire sample plate. 32 4.12 T h e 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 T h e dc field dependent backgound signal of the version II ac coil set.  . .  37  4.16 A A ( T ) for the Y B a C u 0 .95 sample Y l q  41  5.17 T h e d 2_ 2 gap function  45  2  x  3  6  y  5.18 Averaged data for magnetic field response of Y B a C u 0 . 9 5 sample Y l q 2  and background  3  6  51  vi  5.19 Corrected data for YBa Cu306.95 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  2  6.23 T h e vortex melting transition i n the YBa Cu306.95 sample U p 2 at various 2  dc  fields  59  6.24 T h e vortex melting transition in the Y B a C u 3 0 . 9 2  drive  6  5  sample U p 2 at various  fields  60  6.25 T h e vortex melting transition i n the Y B a C u 0 6 . 9 5 sample U p 2 at various 2  drive frequencies  3  61  vii  Acknowledgement  I would like to thank my family and a l 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 N a t u r a l Science and Engineering Research C o u n c i l for its financial support.  viii  Chapter 1  Introduction  For nearly a decade now there has been intensive research into high temperature superconductors. 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? , while X(H) for a superconductor with lines of nodes 2  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  superconductors.  2  M a n y other experiments have provided strong evidence to suggest  d 2_ 2 pairing, but the evidence is not conclusive [4]. Observation, then, of the nonlinear x  y  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 temperature superconductor YBa2Cus06.95 for fields below H \. Presently, measurements using c  ac susceptometers have become quite popular in the field of high temperature superconductivity. M a n y of the interesting features of superconductors - the Meissner effect, the critical temperature, vortex nucleation and dynamics - can be viewed macroscopically through changes i n the bulk magnetization of a sample. A s a result an ac susceptometer lends itself very well to the study of such materials. T h e ac susceptometer itself is a very simple device. Typically, it consists of a dc magnet, a solenoid used to generate an ac field, and a pair of detection coils. T h e m u t u a l 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. T h e 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. T h i s voltage is proportional to the sample's magnetic moment.  If the sample is a superconductor i n the Meissner  state, its moment w i l l be proportional to the volume of the region magnetically shielded by supercurrents.  T h i s 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). allows a measurement of AX(T,H)  = X(T,H)  reference temperature and field respectively.  - X(T ,H ) 0  0  where T  0  and H  0  This  are some  Chapter 2 AC Susceptibility and the AC Susceptometer  T h e magnetic susceptibility % of a material relates its magnetization M to an applied external magnetic field H. For linear media i n a dc field this relationship is simply M  ,  For nonlinear media or in the case where one applies a time dependent field a more relevant definition is dM  * = ddHi r where x is  2.1  n  o  w  <-> 22  the differential or ac susceptibility respectively.  The Complex Magnetic Susceptibility  Consider a sample i n a periodic external magnetic field H{t)  = H  dc  + His'mwt.  In  response to the field, the sample w i 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 (x  sin nwt - Xn cos nwt),  n  (2.3)  n=l  where I m ( ) denotes the imaginary part of the complex magnetization [5]. T h e coefficient Xn — X ~ n  iXn  l s  t  n  e  complex magnetic susceptibility of the n t h harmonic, where X  n  =  —FT / 7Tiii J=2-  3  M(t)smnwtdt,  Chapter 2.  AC Susceptibility  and the AC  Susceptometer  4  (2.4) "  v„ N  A s defined here, the Xn  w l |  ~~  w  =  i  fv-,*/\  -—— /  -KHI J=S.  Mit) cos nwt dt. V  ;  l typically be a functions of both H  dc  and Hi.  (This definition  is equivalent to that i n equation 2.2. See A p p e n d i x A . ) The fundamental component of the magnetization M(t) has a real part in-phase w i t h the applied field H(t) out-of-phase w i t h H(t).  = Hi(x\ s'mwt + x'i cos tut)  and an imaginary part that is 90°  T h i s 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) i n the sample. T h e higher harmonics are generally associated w i t h hysteresis and nonlinearity of magnetization [5]. In the limit of Hi —> 0, the higher harmonics w i l l become negligible and x'i ~^ jfj" — XdcFor example, a superconductor in the complete Meissner state w i l l have a magnetization field that is of equal amplitude but opposite direction to that of the applied field. E q u a t i o n 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 intermediate 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 i n Figure 2.1; a l 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. T h e only measured quantity i n 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. T h e secondary coils are made as identical as possible and are connected i n opposition. In the absence of a sample, the induced voltage of the coils w i l l be equal i n magnitude but 180° out of phase resulting i n an observed null signal. Specifically, from Faraday's law, s = -d$/dt,  the  induced voltage is (t)  where M\ and M  2  = 0  = (M -M )^  v  1  2  (2.5)  are the mutual inductances between the primary and each of the  secondary coils, and I  is the sinusoidal drive current. T o the extent that the coils are  v  identical M equals M and the minus sign reflects the fact that they are connected i n x  2  opposition (or counterwound), and hence v(t) = 0. In practice, however, it is not a t r i v i a l matter to achieve this balance. Even i f great care has been taken i n winding a l l the coils and the secondaries are placed symmetrically i n the ac field, a residual voltage is still very likely.  A d d i t i o n a l methods of compensation are usually employed to cancel this  mismatch and obtain the desired level of sensitivity and stability i n the susceptometer. The direct addition of a compensating voltage v (t) of appropriate magnitude and phase c  is a common technique. Here v(t) = ( M - M )^ x  2  + (t) Vc  = 0.  (2.6)  W i t h the system nulled, the introduction of a magnetic sample into one of the coils (the pick-up coil) w i l l now cause an imbalance in the signal. T h i s voltage is due solely to the sample. T o see this, consider the three voltage sources i n the secondary circuit, which sum to d<J>i dt =  d$ '  VoNA  2  dt dH(t)  + M{t) dt  dH{t) dt  + v {t), c  (2.7)  Chapter 2.  AC Susceptibility  and the AC  Susceptometer  6  Reference Input  dc Power Supply  Lock-in Amplifier  Voltage Compensation Circuit  Function Generator  dcCoil Primary Coil Secondary coils in series opposition Sample  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 i n each coil, and A is their cross sectional area. T h e first term in this equation is the induced voltage i n the pick-up coil, the second term is the induced voltage i n the empty (compensation) coil, and the t h i r d term, v (t), is again the applied c  compensation voltage. In light of equation 2.6, this simply reduces to v(t) =  dM(t) p NAdt 0  (2.8)  Chapter  2. AC Susceptibility  and the AC  Susceptometer  7  a signal voltage proportional to the rate of change of the sample magnetization. Substit u t i n g the definition of M{t) from equation  2.3 into equation  2.8 gives  oo  v{t)  —  ^2 (X n=l  n, wNAHi  n  0  oo  =  C  0  o y i n(x  v  n  c  o  t  s nu)  t  + Xn  S nw  n  + Xn  s  m  s  t)  m  nw  nwt),  (2-9)  n=l  with v  = HowNAHx.  0  T h e signal voltage is an infinite s u m of cosine a n d sine wave  harmonics.  2.3  Calibration of the Susceptometer  T h e expression for v above is only strictly true for a uniformly magnetized sample that 0  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 a n d the pick-up coil.  This  information is contained i n a calibration constant, a. It is the constant of proportionality for the experimentally relevant relation v(t) oc dM(t)/dt  = x H, w  a n d is related to the  mutual inductance between the sample and the pick-up coil, a can be calculated v i a the following consideration. T h e field due to the sample is characterized by internal currents of volume density V x M a n d surface density M x n , where M is the sample magnetization a n d 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 N turns s  each carrying current I [8]. T h i s implies a magnetization of magnitude M = NJ/l„  (2.10)  where l is the length of the sample. s  T h e induced voltage i n the pick-up coil is v{t) = - M  s  p  ^,  (2.11)  Chapter 2. AC Susceptibility  where M  sp  equation  and the AC  Susceptometer  8  is the mutual inductance between the sample and the coil.  Rearranging  2.10 and substituting gives  "« = ~ >W  '  M  (2 12)  or (t)  v  =  -cx wH. X  (2.13)  ° = ^  »  (2-14)  T h e calibration constant then is just  the mutual inductance per sample t u r n multiplied by the length of the sample. To calculate the mutual inductance, it is almost always easiest to consider the voltage induced i n an inner coil (sample) due to a current i n an outer coil (pick-up.) So, for a coil much longer t h a n the sample the mutual inductance is  M = M  ps  where N  p  A  s  = M  u N ^ N  =  sp  S  A  S  ,  and l are the number of turns and length of the pick-up coil respectively and p  is the cross sectional area of the sample. T h i s gives  a = ^V  ip  a  (2.15)  i  a calibration constant proportional to the sample volume V and independent of the s  parameter A ^ , as it must be. T h i s solution for a was of course obtained using the long coil approximation for the pick-up coil, which may not be appropriate i n 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. T h i s method is accurate for cylindrical samples w i t h magnetization vector parallel to both their axis and the pick-up coil axis. T o 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 i=l j=l where B'(r, z^) is the magnetic field per unit current of the i  th  loop of the pick-up coil  calculated at the distance r from the centre axis i n the plane of the j  th  solenoid loop  (see Figure 2.2.) T h e integration is over the cross sectional area of the solenoid loops, and can be computed numerically by d i v i d i n g the area into K x L finite equally sized elements AA  S  = A /KL. S  T h e value B'(r , z ) fc(  {j  at the centre of each AA  S  is taken to be  the value over the entire area element. T h i s gives  N N K L p  t=i  and  s  )AA  (2.17)  j=i k=i i=i  therefore 1V IS. N N K LLi JV  a =  N KL S  Notice that assuming B'(r ,z ) kl  i:j  S  s  £i £ £ £ * ' (  (2.18)  i=i j=i k=i i=i  = 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  End view of sample:  10  K  - AA = A /KL. - r —-— S  S  kl  Pick-up Coil  Compensation Coil cooooooooooooooocooo  OfOOOOOOOCOCCOOOOOOO  oooocoaco  ccccccmco solenoid loop o*ocxxxxxxxxoooaxxx) i  th  oooooooocxxcoooooooo  pick-up loop  Figure 2.2: Schematic for numerical calculation of a for a uniformly magnetized rectangular sample. It is shown approximated as a solenoid of equivalent dimension. T h e end view shows cross sectional area of sample split into K x L elements AA . S  of the magnetic field (for greater accuracy) and use a computer program to calculate a. T h e summations of equation  2.18 are easily translated into four programming loops.  T h e degree of precision i n the program is determined by the size of the model parameters N ,K, S  and L. T h e best accuracy can be had i f one uses 2  B'{r,z ) ij  = B /I z  = 2  n  y/(a + r)  2  + z<  9  9  K + a — r* — z E (a — r ) -I- z 2  2  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. of precision is determined by the size of the model parameters N ,K, S  1  T h e degree  and L used i n 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, if the compensation coil is i n close proximity to the sample. Under such conditions, there w i l l be an induced voltage i n this coil which is no longer negligible. T h e true calibration constant then is just (2.19)  Ol — Olpick—up Ot-compensationi  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. T h i s is a common technique. It is an attractive method i n 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. T h e 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 i n 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. T h e lock-in behaves as a voltmeter w i t h a very high Q bandpass filter at the input. It w i l l measure a signal of specific frequency only. T h i s 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. T h e reference phase of the lock-in must also be set so that its time base is synchronous w i t h the drive field H(t) = Hi sin wt. T h i s allows for the separation of the v(t) into its in-phase (x ) and n  Chapter 2.  AC Susceptibility  and the AC  out-of-phase (x' ) components, where the n  Susceptometer  th  n  12  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 i n the lock-in amplifier's ability 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 i n series w i t h the primary circuit. T h i s voltage w i l l be i n phase w i t h the p r i m a r y current and therefore in phase w i t h the applied field H{t).  T h e reference  phase setting is adjusted until the signal appears in only one channel (ie. the 0° or 90° setting.) T h i s channel is now set to measure the signal proportional to x'n- T h e channel in quadrature w i 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. F o r 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. H a v i n g the sample voltage go through the pre-amp (at the same settings) as well w i 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. T h i s 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 i n zero field and kept at sufficiently low temperatures w i l l not have any loss associated w i t h a small drive field (ie. Hi «  H .) Cl  T h e sample temperature  Chapter 2. AC Susceptibility  and the AC  Susceptometer  13  is changed w i t h i n this region of no loss. T h e resulting signal is due to the temperature dependence of the penetration depth, which is a purely inductive response. A g a i n the reference phase is set such that the resulting change i n signal appears i n only one channel. T h i s 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 limit 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. T h e cryostat and the electronics then w i l l be described separately from the susceptometer i n this chapter. The development of this apparatus was already a work i n progress of D r . W . Hardy, when I joined the project.  As a result, I will detail only those parts of the apparatus  that are specific to this experimental technique, and give particular highlight to those efforts i n 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 w i t h stainless steel vertical tubes (three of them, each 3/8" outer diameter) that connect to the susceptometer.  T h e 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 i n . T h e 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  15  Apparatus  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 t h i r d 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 w i t h i n the cap of the susceptometer.  Knob to turn drive shaft For He transfer  -fl •<  Brass flange  Vacuum line  Stainless steel tube  Copper baffle  1  ac susceptometer  Figure 3.3: D i a g r a m of the cryostat.  Chapter 3. Development  3.2  of the  Apparatus  16  The AC Susceptometer  A cross section of the ac susceptometer is shown i n Figure  3.4. In this diagram the  susceptometer is disassembled, as it would appear when one is changing the sample. T h e sample holder or pot is mated w i t h the m a i n body of the susceptometer w i t h the aid of two guide pins (not shown) that are permanently fixed to the dc coil form. T h e two pieces are held together by twelve 4 — 40 hex cap screws w i t h indium wire forming the vacuum seal. T h e dc coil form is bolted to the susceptometer cap i n 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 i n Figure 3.6. T h e ac coil set has a copper flange, which is fastened to the dc coil form using six 4 — 40 hex cap screws. T h e sample is mounted on a very t h i n , very pure sapphire plate w i t h a small amount of vacuum grease. T h e sample plate is held to the sapphire mounting block of the pot also w i t h a small amount of vacuum grease. W h e n 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 i n more detail below. T h e voltage compensation circuit, which is also shown in Figure 3.4, w i 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 w i t h 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  FT!  Quartz  7  I  I Sapphire  [XI i__ j  Coil Windings ac coil set I or II (detailed below)  Figure 3.4: D i a g r a m of the ac susceptometer. T h e sample holder or pot is disassembled from the dc coil form. T h e orientation of the p r i m a r y 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 t h i n , 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 susceptometer accurately constant. To do this requires a separate bath heater, thermometer and temperature controller. T h e total thermal power delivered to the bath is the sum of the power from the sample heater P and bath heater Pb. If Pb is i n i t i a l l y set (with s  P = 0) to a value just greater than the anticipated m a x i m u m power P s  max  to be used  by the sample heater over the course of an experiment, then Pb can always be adjusted later such that P + P = P , s  b  max  a constant, for any setting of P . s  T h e sample thermometer is a piece of a Lakeshore carbon glass resistor potted in Stycast 2 8 5 0 F T epoxy to a strip of copper foil. T h e foil is held to the mounting block w i t h G E varnish. T h e 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. M a n u a l or automated control of the sample temperature is done w i t h a Conductus L T C - 2 0 temperature controller. T h e bath heater, a 200O metal glaze resistor, is mounted to the base of the pot w i t h Stycast 2 8 5 0 F T epoxy. T h e bath thermometer, an A l l a n Bradley carbon composition resistor (2000 nominal room temperature resistance) is located at the side of the susceptometer cap. T h e temperature controller, built in-house, allows only manual adjustment of the set point.  19  Chapter 3. Development of the Apparatus  3.2.2  The Superconducting DC Magnet  T h e magnetic field along the central axis of a t h i n solenoid of finite length / and radius a is given by the equation  JJ  _ »o  NI  21  f  1  - l 2z  •  l  \[(2a/Z) + ( 2 z / 7 - l ) ] 2  2  1 / 2  1  +  2 Z  /  }  1  [(2a/Z) .+ ( 2 ^ + l ) ] 2  2  1 / 2  (  3  2  0  )  J'  where N is the number of turns, I is the current, and z is the distance from the centre of the solenoid [12]. U s i n g this equation a computer program was written to help select an appropriate design for the dc magnet.  T h e program returned only those w i n d i n g  configurations that would produce a symmetric magnet w i t h 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. T h e 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 i n the program the following design constraints: • T h e size of the superconducting ( N b - T i ) wire - 0.014 inches i n diameter. • T h e number of layers of windings i n the magnet - 30 layers each being separated by 0.002 inches of mylar. T h i s l i m i t set by the amount of available space. • T h e size and shape of the magnet - cylindrical, 2.0 inches i n length, w i t h a cent r a l notch. T h e notched design is simplest configuration to w i n d that can give a reasonably homogeneous  field.  T h e final design can be seen as a part of Figure 3.4. T h e 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.) E a c h 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 -2.0  •  '  -1.0  '  '  0.0  1  1  1.0  •  1  2.0  D i s t a n c e from centre of m a g n e t (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  C o m m o n 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, i n 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. T h e shield is electrically and thermally grounded to the copper flange. T h e faraday shield is composed of a single layer of closely packed insulated copper wires that are parallel to the coil axis. T h i s 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 i n 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. T h i s is then wrapped i n 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 i n pressure contact w i t h 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 w i t h silver epoxy. Here the washer is simply butted to the flange and held in place w i t h 1266 epoxy. T h e 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". T w o 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. T h e 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 i n each of the coils. T h e primary coil is wound w i t h #34 formvar insulated copper wire. T h e faraday shield was made from this same size wire. There are four layers of windings in the primary. T h e first and t h i r d layers are complete. T h e 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  Inchesfromcentre of primary coil  Version II  HI  Copper  B  Phenolic linen  ^  Phenolic Paper  I  -0.625  -0.225  0.0  0.225  0.625  Central axis  | Sapphire  j§|  Primary coil windings  1X1  Secondary coils  Inches from centre of primary coil  Figure 3.6: D i a g r a m of the ac coil sets. T h e faraday shielding is butted against the flange in version I, in version II it is siver epoxied into a groove i n the flange. T h e 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 p r i m a r y 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 discovered to be unsuitable for magnetic measurements.  E x p l i c i t l y , the problem w i t h 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.  T h e mag-  netism of the various construction materials of the coil set were tested w i t h a dc S Q U I D magnetometer. These tests did not reveal a definite source for the signal. T h e goal then in building a second version was to limit the amount of known magnetic impurities i n the vicinity of the secondary coils. T h i s was to be achieved in two ways: (1) using high purity construction materials and (2) by reducing the amount of material i n the vicinity of the secondary coils. A custom ground sapphire tube [13], w i t h an outer diameter 0.250", inner diameter of 0.198" and length 1.250", was used for the primary coil form i n this design. A p a r 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, w i 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. T h e y were wound separately on teflon forms, potted in 1266 epoxy, and then removed from the form once the epoxy had cured. T h i s process, which had an ~ 80% success rate of producing an intact coil, is described further in Figure 3.7. T h e 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 i n a slightly smaller coil.) T h e coils are then epoxied (with 1266) in place inside the sapphire tube. T w o 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 liquid nitrogen. T h i s 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 N b wire insulated w i t h formvar [15]. N i o b i u m was chosen for its superconducting properties. It has a T = c  9.2K,  so at liquid helium temperatures it would shield any possible ferromagnetic impurities inside its bulk from the magnetic fields. N i o b i u m was chosen over other superconducting metals such as P b or P b S n , because it has a high critical field H \ = 1980 gauss. T h i s c  is much larger than fields to be used i n the experiment, which ensures that the coil windings should always remain i n the Meissner state and not give rise to a signal due to a transition into the mixed state. Also, there w i l l be no losses and therefore no heating associated w i t h a type II superconductor in an oscillating field above  H \. c  The faraday shield and the primary coil were made from 0.003" diameter 99.99% pure copper wire insulated w i t h formvar [14]. T h e primary was wound using the same scheme as the one described above. T h e 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.) T h i s 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 i n this ac coil set, a l l connections were made on fixed terminal posts. T h e five posts, 2 - 5 6 threaded brass rod, are held i n 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 w i t h 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 i n 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. T h e n i o b i u m leads are first cleaned w i t h 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  ^0.200'-^  Figure 3.7: Construction of n i o b i u m secondary coils: T h e teflon form fits the 8 m m collet of coil w i n d i n g machine. T h e wound coil is potted i n 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. T h e entire piece is then immersed i n liquid N after which the solid coil is easily removed. Because the teflon does stretch during winding, the final length of the coil is ~ 0.23". 2  3.3 The Electronics T h e electronic set up for this apparatus resembles that shown i f Figure 2.1, w i t h the inclusion of a preamplification stage before the lock-in detector. T h e preamp is a Princeton A p p l i e d Research ( P A R ) M o d e l 119 Differential Preamplifier input to a P A R M o d e l 114 Signal C o n d i t i o n i n g Amplifier.  T h e preamp is operated i n the differential input  mode (and almost exclusively i n the 1 0 0 : 1 transformer mode) and has a single output to the Stanford Research Systems M o d e l SR850 D S P Lock-In Amplifier. T h e function generator, a Hewlett Packard 3325A, is coupled to the p r i m a r y circuit v i a an isolation transformer to break its ground loop. T h e schematic of the susceptometer circuitry, including the voltage compensation, is shown i n F i g u r e  3.8. T h e 100Q dropping resistor, i n the p r i m a r y circuit, is much  larger than the impedance (at / = 10 k H z ) of the rest of the circuit when the primary coil is at liquid helium temperatures.  A s a result, under the conditions of operation,  the combination of the dropping resistor and function generator behaves as an ac current source. T h e voltage drop across the l f i resistor can be used to set the phase of the lock-in  Chapter 3. Development  of the  amplifier. T h e capacitor C  res  Apparatus  26  i n the secondary circuit is used to achieve series resonance  w i t h the inductive component ( l a r g e l y dominated by the secondary coils) of the circuit. T h i s is done to optimize the source impedance for the P A R M o d e l 119 preamp operating in the transformer mode at 10 k H z . A t resonance the source impedance is the purely resistive 3.50 of the secondaries, which means the M o d e l 119 w i l l be operating w i t h i n 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 C  res  was not needed.  Inside cryostat. Figure 3.8: T h e susceptometer circuit. Coarse compensation of the out-of-phase voltage is done at low temperatures w i t h the variable mutual inductance. A l l fine compensation is done at room temperature v i a decade transformers.  T h e voltage compensation circuitry has two distinct parts: (1) the coarse compensation done at low temperatures by a homemade variable m u t u a l inductance and (2)  Chapter 3. Development  of the  Apparatus  27  the fine compensation done at room temperature by a set of Electro Scientific Industries Dekatran D T 7 2 A decade transformers capable of supplying six decades of voltage resolution. T h e low temperature compensation is used exclusively to null out-of-phase signals, while the room temperature circuit is used for b o t h 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.) T h e advantage to the low temperature nulling is the lower thermal noise. T h e variable inductor of (1) is a set of equal t u r n (but counter wound) coils in series w i t h the primary circuit that straddle a pick-up coil in series w i t h 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 i n Section 3.1 and shown i n F i g u r e 3.4 of this chapter. T h e skin depth of copper at 4.2K and 10 k H z 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 cutting the pick-up coil. T h i s means the magnitude and sign of this induced compensating voltage can be set by the position of the copper slug. T h e phase of this induced voltage is ~ 90° out of phase w i t h the drive current. Certainly losses from the copper and the phenolic paper are present, but they appear to be negligible. T h i s can be seen i n the plot of induced voltage vs. position of copper slug, displayed i n Figure 3.9. O n l y 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  (a)  of the  Apparatus  28  '  (b)  Figure 3.9: T h e low temperature compensation circuit, (a) D i a g r a m of the circuit itself, (b) T y p i c a l response of the circuit. T h e position of the t u n i n g slug is plotted as turns of the key from the extreme counter clockwise setting. In practice, one would null the imbalanced signal w i t h this circuit to near one of the m i n i m a indicated by the arrows. Fine adjustment of the compensation is then done w i t h the decade transformers. decade transformers, and the resulting voltages are then injected into the secondary circuit. A l l transformer ratios given i n Figure 3.8 are w i t h respect to the top inductor, and represent the actual number of turns i n each case. T h e phase difference between the primary current and output voltage from the transformer across the 1.3Q, resistor was experimentally determined to be 0.97°. T h e phase of the output voltage from the transformer i n series w i t h the primary circuit was measured to be 89.95°. T h e orthogonality of these two voltages allows one to quickly set the decade transformers to give a null signal. T h e i r performance over several decades of voltage transformation is shown in Figue 3.10.  Chapter  3.  Development  of the  Apparatus  29  0.060  ? & - - -a Out-of-phase 0.040  J  0.020  o.ooo \- & - - = = = = § 10"'  i<r  e- ©  11 n i  10"'  o  i  10"'  D e c a d e transformer ratio Figure 3.10: T h e performance of the decade transformers. There is negligible phase shift in the output from the decade transformers. T h e orthogonality of the input voltages is preserved.  Chapter 4 Performance of the Apparatus  T h i s apparatus was designed such that one could "see" a lA change i n 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 v = ^ V  s  2.13,  X  w H ,  (4.21)  lp  the induced voltage in a pick-up coil due to a sample of volume V and susceptibility s  X- For a superconductor i n 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 , to estimate 2  its change i n apparent volume due to a lA change i n A one multiplies the area by 2A. T h i s gives A V " = 2 x 1 0  - 1 6  m . For the first version of the pick-up coil N 2  p  = 1360, and  l = 0.2" = 5.08 x 10~ m . T a k i n g w = 2ir x 10 radHz and u. H = 10~ T (ie. 10 gauss), 3  4  3  p  0  gives a signal of about v = 3.4 x 10~ volts. 9  T h e intrinsic noise from the secondary coils at 1.2K is v  =  V4:kTRAv  «  ^ 4 ( 1 . 3 8 x 10- )(1.2)(3.5)(1)  «  1.5 x 1 0 "  23  1 1  volts.  The amplifier noise when operating at 10 k H z w i t h i n 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 w i t h a reasonable signal to noise ratio does seem quite possible. 30  Chapter 4. Performance  of the  Apparatus  31  T h e actual behavior of the ac susceptometer is the subject of this chapter.  The  resolution along w i t h measurements on the temperature and magnetic field response of the device w i l l be presented here. different susceptometers.  T h e two ac coils sets essentially represent two  T h e y 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 C  res  shown i n Figure  3.8. T h e true  resonant frequency was determined experimentally to be 10.7 k H z . T h i s frequency for the drive field was used for a l l experiments in this thesis except where explicitly stated.  4.1.1  Sample Holder Temperature Dependence  T h e dependence of the apparatus on the sample holder temperature T was measured s  w i t h the helium bath i n regulation at 1.2K, which is the lowest temperature reachable w i t h our p u m p i n g system. O n l y the sapphire sample plate was present inside the pick-up coil. T h e results for the out-of-phase signal are shown i n Figure 4.11. T h e signal is very small, and it rapidly becomes constant w i t h increasing temperature. A l t h o u g h there are very few data points, the susceptibility roughly fits a 1 / T dependence suggestive of a Curie t e r m due to paramagnetic impurities[17].  T h i s sample plate was later replaced w i t h a much higher purity sapphire plate, which when tested gave no detectable temperature dependent signal.  It appears then that  impurities i n the former plate were responsible for the paramagnetic observations in Figure 4.11.  Chapter 4. Performance  0.0  5.0  of the  Apparatus  10.0  Temperature (K)  32  15.0 0.0  0.2  0.4  0.6  0.8  1/Temperature (K" )  1.0  1  Figure 4.11: T h e temperature dependent backgound signal of the sapphire sample plate. / = 10.7 k H z . Hi = 8 gauss. X K ' J O C I / T appears to indicate paramagnetic behavior.  4.1.2  DC Magnetic Field Dependence  T h e field dependence of the background is shown i n Figure 4.12. A g a i n only the sapphire sample plate was present inside the pick-up coil. T h e data plotted is for the out-of-phase signal. T h e 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 i n excess of  100X the desired resolution. To make matters worse, the signal is also hysteretic. T h e dc magnet current was supplied by a H P 6632A programable power supply. T h i s 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. T h e presence of hysteresis i n general, however, is absolutely unmistakable.  A systematic study was undertaken to determine the magnetic characteristics of the  Chapter 4. Performance  of the  -1.0  Apparatus  33  0.0 1.0 DC magnet current (Amps)  2.0  Figure 4.12: T h e dc field dependent backgound signal. / = 10.7 k H z . Hi = 8 gauss. Inset shows more clearly the hysteresis. construction materials that are in the vicinity of the secondary coils. T h i s , it was hoped, would reveal which of the substances was responsible for the large background signal. T h e measurements were done w i t h a Q u a n t u m 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 i n the p r i m a r y coil though of a smaller gauge) were studied. Hysteresis loops over ± 2 0 0 0 gauss at 5 K for a l l materials are shown i n Figure 4.13. T h e data is normalized w i t h respect to mass by plotting the magnetic moment i n units of emu per mass of sample i n grams. T h e results show that a l l the materials are magnetic,  each w i t h a susceptibility \x\ = \M/H\ ~ 5 x 1 0  - 7  e m u g " g a u s s . O n l y the 1  -1  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  -2000.0  1  i  -1000.0  i  i  0.0  i  Applied field (gauss)  i  1000.0  i  2000.0  Figure 4.13: Hysteresis loops at 5 K for ac coil set construction materials measured w i t h 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 w i n d i n g imbalance. B y mass imbalance, meant an inhomogeneity i n the spatial distribution of a given material w i t h respect to the secondary coils. If a magnetic material is evenly distributed, then the voltage it would induce i n 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. T o remedy the problem, it was decided at this point to build another ac coil set. Great effort and expense was put into procuring the high purity materials for its construction. T h e design was altered to l i m i t the mass i n close proximity to the secondary coils.  Chapter 4. Performance  4.2  of the  Apparatus  35  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 k H z , so it was decided to forgo the use of the capacitor C . res  4.2.1  T h e effect on the noise was negligible.  Sample Holder Temperature Dependence  The dependence of the apparatus on the sample holder temperature T was again meas  sured w i t h the helium bath i n regulation at 1.2K, and w i t h only the high purity sapphire sample plate present inside the pick-up coil. A s expected, there was no detectable signal at low temperatures. However, as the sample plate was taken to higher temperatures a signal was observed. T h e onset of this high temperature response was at about 3 0 K . This experiment was repeated on a separate cool down of the apparatus, the sample plate having been removed, cleaned and replaced. T h e 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 i n 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 i n good contact w i t h the sapphire tube (the coils pulled away from the tube at low temperature as mentioned i n 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 T  s  along  w i t h a plot of ln(v) vs. l n ( T ) for b o t h trials. These show the signal voltage varying w i t h temperature as T  5 8  . W h i l e not strictly consistent w i t h the Stefan-Boltzman law  (radiative power oc T ) , any mechanism w i t h such a temperature dependence almost 4  certainly involves a radiative process. T h e temperature dependence of the emissivity of the sapphire plate may account for the deviation from T  4  behavior.  Chapter 4. Performance  of the  Apparatus  36  T h e experimental implications of this thermal effect w i l l be discussed later.  CD CD  e-  eg  - • First trial  "o >  - o S e c o n d trial  c co *  o 0.0  B — e — f r -  0.0  20.0  . -a- -cr  40.0  60.0  Temperature (K)  80.0  100.0  3.6  3.8  4.0  4.2  4.4  In(Temperature)  4.6  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 i a 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  —I ,x ma  to the dc magnet. T h e other  power supply, the H P 6632A, connected i n parallel, was programmed to step through current values from 0 to 2I . max  T h e 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 i a 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. T h e 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). T r i a l 1 is from a separate cool down from Trials 2 and 3. In T r i a l 3 the sample holder temperature was set at 100K, it was 1.2K for the other two.  i  -0.6  -0.3  1  1  0.0  1  1  0.3  Trial 1 - Sample temperature = 1.2K 1 1 I ^-i 1 1  0.6 0.0  100.0  r  1  200.0  300.0  400.0  Trial 3 - Sample temperature = 100K  Current (amps)  jj  m e  (seconds)  Figure 4.15: T h e dc field dependent backgound signal of the version II ac coil set for / = 10.7 k H z and H = 8 gauss. T h e out-of-phase voltage signal is plotted i n real time and as function of the magnet current. T h e drift (in T r i a l 1) was eliminated by thermally isolating the electronics. T h e background signal is much larger at T i = 100K x  samp  There are many notable features i n this data.  e  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 still 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 T i samp  e  = 1.2K data is roughly 1 0 X less. To give a true comparison of the  backgrounds however, one needs to m u l t i p l y 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.) T h i s 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. A l s o present i n the T r i a l 1 data is a fluctuation i n the voltage signal baseline. T h i s is easily seen in the time plot. It affects the hysteresis plots by smearing them i n the voltage axis direction. T h e source of this drift was discovered to be changes in the air temperature of the laboratory.  In particular the  fluctuations  i n an otherwise constant background  signal were found to be syncronized w i t h 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 a l u m i n u m plate and isolated inside a styrofoam box. Water circulation from a temperature controlled bath kept the temperature of the a l u m i n u m plate constant. T h e improvements i n performance were significant. T h e hysteresis loops in Trials 2 and 3 lie almost directly on top of one another.  T h i s repeatability, coupled  w i t h 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. T h i s 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 C u 0 6 . 9 5 . T h i s background is still too large to allow for direct measurement of the 2  3  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 w i l l allow improvement i n 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 k H z , so it appears transparent to the magnetic fields. In theory then, it may be possible to measure the background by t a k i n g the sample above T = 9 3 K . In c  practice however, the results of T r i a l 3 rule this out. T h e field dependent background of the apparatus w i t h sample plate temperature at 100K is significantly larger than at 1.2K. T h e 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 C u 0 . 5 2  3  6  9  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. T h e linear temperature dependence of A(T) at low T , a signature of superconductors w i t h nodes i n the gap function, was first observed here using this technique [18]. Repeat of this experiment were done w i t h the ac susceptometer. set were used to measure A A ( T ) = A ( T ) — \(1.2K)  measurements  B o t h versions of the ac coil  for the sample Y l q , a high purity,  twinned YBa2Cu30g.95 single crystal. T h e crystal is a t h i n slab of thickness t ~ 27 /im. T h e broad face of the crystal, the ab plane, is ~ 1.5 x 1.5 m m . A s i n the microwave 2  measurements, Hi is applied parallel to the ab plane. Measurements were taken over the range 0 — 2 0 K , well below temperatures where the thermal effects i n 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 d a t a 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- T h e 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  V ,  measuring the out-of-phase signal,  in situ  for each sample by  that results from t a k i n g the sample from 1.2K to  SjH  100K. In the normal state, the crystal is i n the t h i n limit 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 = 2V /t. SyJl  For version I, k = 2(27 x 1 0 ) / ( 1 . 6 3 " ) = 0.83 A / n V . 4  4  T  = 2(27 x 10 )/(0.4898~ ) = 2.8 A / n V .  For version II, k  4  H  T h e value  kn/ki —  4  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.  N /N T  n  = 2.7. T h i s new value means that the improvement i n the field  dependent background i n version II was actually just a factor of 3, not 3.8.  4.3.2  Calculating the Resolution from AA(T)  T h e resolution of the apparatus i n angstroms is calculated by m u l t i p l y i n g half the peakto-peak voltage noise (measured off a strip chart recorder that is output from the lock-in) by the calibration constant. were 12 x 1 0 ~  10  and 5 x 1 0  T y p i c a l values for the noise during the A ( T ) experiments - 1 0  volts peak-to-peak for version I and II respectively. T h i s  gives a resolution of 0.5 A for version I and 0.7 A for version II. B o t h coil sets exceed the desired 1 A resolution set out as the design m i n i m u m at the beginning of this chapter.  Chapter 4. Performance  of the  Apparatus  80.0 i) A  • cavity perturbation (f=961 MHz) o ac susceptometer I (f=10.7 kHz) A ac susceptometer II (f= 10.7kHz)  60.0  []  A O  CO  E o  -— i »  w  CO  c  40.0  < < 8  20.0  0.0 0.0  -flB-  5.0  10.0 Temperature (K)  15.0  Figure 4.16: AA(T) for the YBa Cu 06.95 sample Ylq. 2  3  20.  Chapter 5 The Nonlinear Meissner Effect: A ( H ) of YBa Cu 0 2  3  5.1 Theory T h e electrodynamics of a superconductor i n the Meissner state (H < H i) c  obey the  L o n d o n equations  ^  V  3  8  =  1 f  -^j . s  —*  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 superconductor. For high temperature superconductors, which are T y p e II superconductors, the characteristic length of this decay is the L o n d o n penetration depth, A = Here, p (T) s  ^m/(p p (T)e ). 2  0  s  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 = -ep (T)v , s  s  (5.22)  where v is the supercurrent velocity. Y i p and Sauls derive a nonlinear velocity term in s  42  The Nonlinear Meissner Effect: X(H) of Y B a C u 0 . 9 5 2  j. s  3  43  6  (In their model they consider the supercurrent to be largely confined to two dimen-  sions.) T h e y then re-solve the L o n d o n equation, now nonlinear w i t h the inclusion of the —*  higher order term i n j , and from this derive an effective penetration depth that depends s  explicitly on H. T h e y do this for both conventional and unconventional superconductors. T h e i r results are summarized below.  5.1.1  Conventional Gap  Here the supercurrent is given by j =  -ep (T)v  s  s  1  s  -a  (5.23)  T h e second term is a correction to the supercurrent of 5.22, which is reduced due to pair breaking in finite fields at nonzero temperatures. defined as A(T)/pf d(T)  Here v  s  «  v, c  the critical velocity,  (the energy gap divided by the Fermi momentum.) T h e coefficient  is always positive and tends to zero as T —» 0 , as a consequence of the existence of  the gap. Substituting this j  s  into the L o n d o n equation gives 1 _ v  Y2  where the term ct(T)  = 0.  s  (5.24)  < < 1 has been ignored in the L a p l a c i a n term. T h i s nonlinear  L o n d o n equation is solved in closed form for H, satisfying the boundary condition 1  dv  _e  s  ~r~  U=o — ~  dz  M  -  c  A n effective penetration depth is defined from the i n i t i a l decay rate of H inside the superconductor. It is given as X (T,H) eff  =  A(T)  1  1AT)  /  H  MT))  V ^ 1  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 C u 0 j 2  3  44  6  (5.25)  1 + 1<*{T)  A(T)  where A ( T ) is the zero field London penetration depth and H (T) 0  =  |eo (T)jeX(T) c  is a constant of the order of the thermodynamic critical field. T h e change i n A / / is e  proportional to H  2  and thus as expected X ff  is larger than A ( T ) . T h e effect of the field  e  therefore is to increase the effective penetration depth. B y pair breaking, the supercurrent density is reduced, and so too is the superconductor's ability to screen the field. Experimentally, one can probably expect a smaller correction to A ( T ) than what is predicted above. Here A / / was defined from the i n i t i a l decay rate, and so it is only e  truly applicable i n 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 X ff leads to a correction i n A ( T ) e  that is ~ 3 X too large. A l t h o u g h it is not explicitly stated in [2], this same definition for X ff  is most likely used in the case of unconventional gap as well. A s a result, the  e  corrections to A ( T ) given below sould also be considered as too large.  5.1.2  Unconventional  (d 2_ 2) x  y  Gap  A cross section of the Fermi tube for the  pairing state is shown i n Figure 5.17.  d 2_ 2 x  y  There are four nodes in the gap occurring at the positions  . A t T = 0, quasipar-  t i d e occupation at the nodes w i l l remain nonzero. T h i s gives rise to a quadratic v term s  —*  in the supercurrent j  s  that is independent of temperature. Because of the anisotropy of  the gap, however, no single form for the  supercurrent can be written. F o r the case  d 2_ 2 x  y  where v is along a node (see Figure 5.17), the supercurrent is given by the equation s  Is =  -ep (T)v s  s  1  \Vs\ Vo  (5.26)  The Nonlinear Meissner Effect: \(H)  of Y B a C u 0 . 2  where v = 2 A / i > / is a characteristic scale and A 0  0  3  0  6  45  l  is the gap m a x i m u m at an antinode.  The second term can be viewed as the quasiparticle backflow from the node opposite v , s  that reduces the supercurrent.  T h i s backflow is calculated over the wedge of occupied  states about the node.  ky  Figure 5.17: T h e d 2_ 2 gap function. T h e supercurrent velocity is along a node. O p p o site this node is the wedge of occupied quasiparticle states that constitute the backflow current. x  y  For the case where v is along an antinode (ie. 4 5 ° to a node, along the direction of s  the gap m a x i m u m ) , the supercurrent is j  s  =  The only difference w i t h equation term. lines.  s  6  1  -  1  \v  \/2  v  s  (5.27)  0  5.26 is the factor I / A / 2 that appears i n the backflow  T h i s factor is just the cos(45°) for the new projection of j  s  along the nodal  T h i s anisotropy w i l l appear in the resulting nonlinear L o n d o n equation and in  its solution as w e l l . 2  -ep (T)v  2  T h e factor of l / \ / 2 of course shows up i n the equations for the  Again the details of the calculation are given in [2].  The Nonlinear Meissner Effect: X(H) of Y B a C u 0 2  46  3  effective penetration depth, where hopefully it can be exploited experimentally to probe the anisotropy of the gap i n high T superconductors. c  The pertinent equations are X (T,H) eff  = A(T)  H || n o d e ,  (5.28) or  X (T,H) eff  ~ A(T)  • 1  2H  3K  +  and X (T,H) eff  = A(T)  H || antinode,  (5.29) or  X (T,H) eff  ~ A(T)  _L_2_ff +  V23H~  0  where H = (y /X{T)){c/e) is also of order the H , thermodynamic critical field. T h e ob0  0  c  servation of this linear field dependence i n the penetration depth would strongly support the existence of unconventional pairing. T h e subsequent observation of an anisotropy of 1/V2 i n X ff, for field orientations along a node and antinode, would be very strong e  evidence for One  d i_ i. x  y  should keep i n mind that the thermal excitations, responsible for the H de2  pendence of X ff e  i n the conventional superconductor, must necessarily be present i n  unconventional superconductors as well. However, for sufficiently small T a n d large H, the linear term w i l l dominate. Experimentally, the magnitude of H is limited by the critical field H \, the superconductor must be i n the Meissner state for meaningful rec  sults, so one must work at low temperatures. A t any finite T however, there w i l l remain a cross-over field below which the quadratic contribution w i l l become significant. If care is not taken, this could obscure detection of the linear behavior.  The Nonlinear  5.1.3  Meissner Effect: X(H) of YBa2Cu 0 , 3  47  6  Some Estimates  How low must the temperature be to allow for the observation of the linear behavior in the field range 0 —> H {! c  Y i p and Sauls argue that the cross-over field and temperature  w i l l be related by -jf- ~ jr. A s s u m i n g H  ~ 10 gauss, T ~ 100K, and H i ~ 250 gauss 4  0  c  c  for a typical cuprate superconductor [3] and taking H = f f / 1 0 implies a temperature c l  of 0.25K. T h i s is roughly 1/4 of the lowest temperature presently achievable w i t h our apparatus. A t I K then, the cross-over field is ~ 100 gauss. T h i s a very large fraction of H i,  but possibly the linear term w i l l still be observable over the remainder of the field  c  range. How large is the nonlinear Meissner effect expected to be? suggest the change i n Ay ( i f ) is roughly 2 A | | i f / i i c l  X u , Y i p and Sauls [2]  ~ 30 A for an ac measurement. For a  0  crystal of the size of Y l q (~ 1.5 x 1.5 x 0.027 m m ) and using the version II ac coil set, 3  this translates into a ~ 11 n V signal. T h i s is ~ 4 0 X larger t h a n the voltage noise of the version II ac coil set. However, i n terms of the field dependent signal of the apparatus itself, 30 A is only ~ | of the background at 250 gauss. T h i s 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 w i t h the version II ac coil set on the YBa2Cu 0 . 3  6  9 5  sample Y l q , a large crystal which w i l l  improve resolution. O n l y the field dependence of A w i l l be tested for now. T h e experimental set up was the same as that described i n section 4.2.2. T h e probe was cooled w i t h magnetic shielding around the dewar w i t h the shield left i n place for all measurements. F i e l d sweeps were made at different sample temperatures (1 and 10K in that order) for three different m a x i m u m dc current levels, corresponding to fields of  The Nonlinear Meissner Effect: \{H)  of Y B a C u 0 , 2  3  48  6  100, 200, and 240 gauss. T h e sample was cooled i n zero field before each set of trials for the three m a x i m u m 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. B a t h temperature was held constant at 1.2 K , and the drive field was kept at the same magnitude (8 gauss) and frequency (10.7 k H Z ) for a l l trials. T h e 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 w i t h this coil set, the background response could not be measured by heating the sample above T , which requires c  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 , m a k i n g this approach completely unacceptable. T h e only alternative was to measure the background, before and after the sample measurement, by actually removing the sample from the susceptometer. T h i s 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 i n the data analysis.  T h e 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.  T h e background measurement was repeated,  but this anomalous  T h e source of this problem has not yet been  response persisted.  The Nonlinear  identified.  Meissner Effect: X(H) of YBaaCugOe.,  Figure  49  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 still unknown. T h e results from the 240 gauss trials are shown below. T h e data for each t r i a l has been averaged over a l l of its consecutive hysteresis loops. T h e averaging is not sensitive to the direction i n which the current is stepped through the loop. T h i s was done deliberately to remove the effects of any slow monotonic drift in the baseline. T h e 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. T h e results are very unclear, and without proper knowledge of the background very little can be concluded about X(H) for Y B a C u 0 . 9 2  3  6  5  .  There is a large asymmetry associated w i t h 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 i n the magnitude of the corrected data. T h i s is also seen i n 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 i n the proximity of the pick-up coils. T h e fact that a l 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 ~ H i.  If one  c  considers the 4 n V spread between the background measurements at this field level to represent the uncertainty i n the background, then L\X(H)  =  16±llAati7.~  H. cl  This  The Nonlinear  Meissner Effect: X(H) of Y B a C u 0 , 2  3  50  6  would strongly suggest that the A A ( / f ) ~ 30 A predict by X u , Y i p and Sauls represents c l  an upper l i m i t for the nonlinear Meissner effect i n YBa2Cu 0 .95 . 3  6  T h e only real conclusion that can be drawn is that more work on the background w i 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. T h i s would certainly entail b u i l d i n g another ac coil set.  However, reliable measurements might still be possible w i t h 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. T h i s would eliminate the problem of the field dependent background being a function of the sample temperature as well.  T h i s in t u r n would allow for the i n situ determination of the background by  heating the Y B a C u 0 . 9 2  3  6  5  sample above T c .  The Nonlinear  60.0  co  2  40.0  3  6  0- - o Ylq at T =1.2K • • YlqatT =10.0K A A Background at T =1.2K s  \  s  \ \ A \  > CD  51  Meissner Effect: X(H) of YBa Cu 0 .95  \  o  >  "cC c D) CO  20.0 -  0.0 -0.6  \  -0.3 0.0 0.3 DC magnet current (amps)  Figure 5.18: Averaged data for magnetic field response of YBa2Cu 0 .9 sample Ylq and background. 3  6  5  Meissner Effect: X(H) of YBa Cu 0 ,  The Nonlinear  2  3  6  7.0 6.0  & - -o Ylq (at T=1.2K) - Background o- - -o Ylq (at T=10.0K) - Background s  s  5.0  > <D O) CO  o > "co c 00  4.0 3.0 JD  2.0 r / 0 /  1.0 -  0  0.0 -1.0 -0.6  0  0=. - n -  -0.3  ' 0'  -Q' "  '  0.0  0.3  Dc magnet current (amps)  Figure 5.19: Corrected data for YBa Cu30 .95 sample Ylq. 2  6  0.6  The Nonlinear Meissner Effect: X(H) of Y B a C u 0 , 2  3  6  53  Figure 5.20: Background measurements from separated cool downs, not done under identical conditions. T h i s data is consistent w i t h an observable magnetic effect i n Y B a C u 0 .95 • 2  3  6  The Nonlinear  Meissner Effect: X(H) of  -0.6  2  3  54  6 1  Typical background response of ac coil set II  12.0  o >  YBa Cu 0 .  -0.3  0.0  0.3  0.6 0.0  100.0  200.0  300.0  400.0  300.0  400.0  Anomalous behavior of ac coil set II  -0.6  -0.3  0.0  0.3  Current (amps)  0.6 0.0  100.0  200.0  Time (seconds)  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  YBa Cu30 5 2  6 9  In this chapter I w i l l present some preliminary results from studies of the vortex melting transition i n YBa2Cu 0 .9 done using this ac susceptometer w i t h the version I coil set. 3  6  5  T h i s transition was believed to be of first order, and so should be accompanied by a discontinuity i n the first derivatives of the free energy G. Recent observations at the University of B r i t i s h C o l u m b i a by L i a n g , B o n n a n d H a r d y [19] of a discontinuity in the magnetization (M = —dG/dH)  of a Y B a C u 3 06.95 single crystal provided the first direct 2  proof that the vortex melting transition was indeed first order. The sample, of superior quality, h a d exceptionally low pinning, as characterized by an irreversibility line well below its vortex melting transition. T h i s allowed for the very clear measurements. In the case where there is large amounts of pinning, the transition from a vortex liquid to a vortex solid would be much more broad and therefore escape detection. T h i s is precisely why there had never been a previous observation of this effect i n YBa2Cu30 . 6  95  .  T h e measurements of L i a n g et a l . 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. ac measurement? approximate ^  W h a t w i l l the transition look like for an  If one takes the real part of the complex susceptibility t o closely i n the limit that Hi «  Hd , then x'i would be expected to diverge c  at the discontinuity i n M. See Figure 6.22. T h i s hypothesis is the same whether the experiment is done as a function of dc field or temperature.  55  56  The Vortex Melting Transition in YBa2Cu 0 .95 3  6  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  YBa2Cu 0 .95 3  6  sample as above (labeled Up2), measurements were done  in a fixed field (Hd = 10, 20kOe) as a function of temperature (T = 100 — > 60 K), a c  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~ volts) due to the sample holder over the temperature range 4K-> 8  100 K.  The Vortex Melting Transition in Y B a C u 0 . 9 5 2  6.2  3  57  6  Results and Discussion  Figure 6.23 shows the voltage signal as a function of temperature for two field settings, 10 and 20 kOe. T h e drive voltage was 1 . 0 V _ ; the drive field Hi = 0.8 gauss. There P  P  is clearly a step in b o t h sets of data m a r k i n g the possible presence of the vortex lattice melting transition. T h e steps are present in both the i n 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 m a r k i n g the transition as would also be expected i f x'i is t r u l y the derivative of the dc magnetization. Is this the vortex melting transition then? Comparison w i t h the dc data [19] strongly suggests that it 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 k O e 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 . T h e sample thermometer i n 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. T h e T of this crystal was measured w i t h the susceptometer i n zero c  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 i n the field. A s a result, one does not measure a sharp transition. T h e crystal is i n 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 YBa2Cu 0 .95 3  58  6  However, the data in Figure 6.24 does show a small peak on the 1 . 0 V _ 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. D a t a for the in-phase (loss) signal is shown i n Figures 6.23 and 6.25. There is a step at the transition, much like the step i n the out-of-phase signal. It is obviously associated w i t h the latent heat of the first order transition. Also present is a peak in the data occurring at temperatures below the transition point. T h i s peak has been associated w i t h the irreversibility line [21]. It is clear from Figure 6.25 that the data is consistent w i t h this idea. Here the peak is seen to shift to lower temperature as the frequency drive field reduced. T h e dc measurement of the irreversibility line at 20 k O e is 75 K [19].  The Vortex Melting  Transition  in Y B a 9 . C u 3 O 6 . 9 5  0.4  ooooo  0  v  )<*°°0  0  <>  In-phase signal.  0  06,. = 20 kOe oBdc = 10kOe  O  o  CO  o >  & 0.2 CO  -*—*  o >  "CO  c  CT>  CO  0.0 81.0  83.0  85.0 87.0 Temperature (K)  0.0  89.0  oocrxccxxxDcaxxxxxx^  ooocco^cp o  o > o cu  g> o >  Out-of-phase signal, o Bd0 = 20 kOe oBdc = 10kOe  -0.4  "CO  c  CT)  CO  -0.8 84.0  85.0  86.0 87.0 88.0 Temperature (K)  89.0  90.0  Figure 6.23: The vortex melting transition in the YBa Cu 0 .95 sample Up2 at dc fields. 2  3  6  The Vortex Melting Transition in Y B a C u 3 0 . 9 2  6  60  5  0.0  BB BB  O O ' O O CS  < > > t> > > > ^  o  3  -0.1  < <<^o  < << <  Out-of-phase signal, o 1.0 V„ p-p  o 500 mV^p-p < 250 mV,p-p > 125 mV,p-p  o oo© Q  -0.2 86.0  87.0 88.0 Temperature (K)  89.0  Figure 6.24: T h e vortex melting transition i n the Y B a C u 0 . 9 5 sample U p 2 at various drive fields. 2  3  6  The Vortex Melting  Transition  in  YBa Cu30 . 2  61  6 95  3.00  80.0  82.0  84.0  86.0  88.0  Temperature (K)  Figure 6.25: The vortex melting transition in the YBa Cu 0 .95 sample Up2 at various drive frequencies. 2  3  6  Chapter 7 Conclusion  T h e primary goal of this thesis was to develop a high sensitivity ac susceptometer to measure the nonlinear Meissner effect i n the high temperature superconductor Y B a C u 3 0 . 9 . 2  6  5  In this regard the work to date must be considered unsuccessful and the goal presently unfulfilled.  T h e 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 w i t h the sample i n situ requires heating the sample above T = 93 K . A t this temperature the response of the pick-up coil is significantly altered by c  some heating effect that makes this approach unsuitable. T h e 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) w i t h any degree of precision. In fact, the only quantitative result that could be gathered about the nonlinear Meissner effect i n Y B a C u 0 . 9 5 is that A\(H ) 2  3  6  cl  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 C u 0 . 9 5 , however, present quite a different view of the 2  3  6  susceptometer. A t a fixed field, very precise, state-of-the-art measurements can be made w i t h this device. Ideally an ac susceptometer can be used to make measurements along  62  Chapter 7. Conclusion  63  any path i n 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. T h i s 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 still pertinent, and so the motivation to make further improvements to the design of the ac susceptometer still 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 w i n d i n g  the secondaries on teflon forms would be greatly reduced using a rigid form again.  Bibliography  [1] J . B u a n , B . Stojkovic, N . E . Israeloff, A . M . G o l d m a n , 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] D a v i d Pines preprint, 1995. To be published i n High T Superconductivity Ceo Family, eds. T . D . Lee and H . C . R e n , Gordon and Beach, 1995. c  and the  [5] T . Ishida and R . B . Goldfarb, Physical Review B 41, 8937 (1990). [6] M . Nikolo, A m e r i c a n Journal of Physics 63, 57 (1995). [7] A . F . K h o d e r and M . Couach, Cryogenics 31, 763 (1991). [8] R . B . Goldfarb and J . V . M i n e r v i n i , Review of Scientific Instruments 55, 761 (1984). [9] L . D . L a n d a u and E . M . Lifshitz, Electrodynamics Press, Oxford, 1960), pp. 124 and 125.  of Continuous  Media  (Pergamon  [10] T . Ishida and H . M a z a k i , Journal of A p p l i e d Physics 52, 6798 (1981). [11] A . Shaulov and D . D o r m a n , A p p l i e d Physics Letters 53, 2680 (1988). [12] M a r t i n A . Plonus, Applied Electromagnetics York, 1978), pp. 335 and 336.  ( M c G r a w - H i l l B o o k Company, New  [13] T h e sapphire tube was supplied by Insaco Inc. of Quakertown, P a . The sapphire was of optical quality. Cost was $780. [14] The copper wire was supplied by California F i n e W i r e Company of Grover Beach, C a . Cost was $160/500'. [15] T h e niobium wire was supplied by Supercon Inc. of Shrewsbury, M a . Included assay of the N b indicated < 100 p p m of ferromagnetic impurities. Cost was $290/300'. [16] F r o m the catalog Lock-in Amplifiers. Corporation.  Copyright © 1 9 7 7 Princeton A p p l i e d Research  64  Bibliography  65  [17] Neil W . Ashcroft and N . D a v i d M e r m i n , Solid State Physics (Harcourt Brace College Publishers, Orlando, 1976), pp. 656. [18] W . N . Hardy, D . A . B o n n , D . C . M o r g a n , R u i x i n g L i a n g , and K u a n Zhang, Physical Review Letters 70, 3999 (1993). [19] R i u x i n g L i a n g , D . A . B o n n and W . N . Hardy preprint, [20] R i u x i n g L i a n g , private communications,  1995.  1995.  [21] V . B . Geshkenbein, V . M . V i n o k u r , R . Fehrenbacher, P h y s i c a l Review B 43, 3748 (1991).  Appendix A Complex Susceptibility  1. It can be shown that the definition of definition x =  from equation 2.4 is equivalent to the  X  dM/dH.  Assume a general expression for the sample magnetization oo  M(t)  =  '£M e  i n w t  n  ,  n=0  where the M  n  are independent of t.  For an applied field H(t)  = H^c + H\ e  twt  evaluate the expression dM  _  dH  ~  dM{t)  dt  dt  dH(t)  oo  =  Y, n^ ~ /H . nM  {n  l)Wt  1  (A.l)  n=l  T h e 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,Xie - i=i i{l  1)wt  (A-2)  C o m p a r i n g terms of A . 2 and A . l gives nM  n  Xn = ~ ^ - ,  66  (A.3)  Appendix  A.  Complex  Susceptibility  67  for n = 1, 2,3 • • • oo. Redefining Xn = X n /  (which seems to be convention [1-2]) returns the result  n  Chapter 1. T h e dc field was explicitly included here. 2. Throughout this thesis the applied magnetic field was defined as H(t) = HJm{e )  = H sin  iwt  x  There is, of course, no reason why it could not be defined as H(t) = H Re(e )  = H ca&wt  iwt  x  x  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(K cosnwt  w i t h the complex susceptibility n = n — iK n  K  /  + K sinnwt),  U  =  n  W  n  fw  —rr-  TVH\ X  /  -  r  l  N  (A.4)  given by  .  Mit) cos nwt  at,  J=2UJ  (A.5) ^ M(t) s i n nwt at. To compare this definition of the susceptibility w i t h that of equation  2.4, it is  easiest to rewrite A . 4 as 00  M(t)  =  £  lm(iK e ),  71=1  inwt  n  (A.6)  Appendix  A. Complex  Susceptibility  68  and then rewrite 2.3 i n a new temporal variable r oo n=l  which is equivalent to oo  M(t)  = #i £  Im( „ e X  i  n  ^ ) ) ,  (A.7)  n=l  as WT = wt 4- \ is the transformation between the two time scales. F r o m A . 6 and A . 7 one gets the relation Xn = i*>n e  - M M r  / , for all n = 1,2,3,... . W r i t i n g i — (—1)^ 2  and recognizing that for this sequence the quantity e  _ m 7 r  /  = (—l) (z) = (—l) / ,  2  n  n  3 n  2  this can be transposed into the more appealing form Xn = (-lf ^K .  (A.8)  n+1  n  T h e definition of the fundamental susceptibility remains unchanged. T h i s not true in general for the higher harmonics, though for a l l n , \x \ — |«n|- T h e t h i r d harn  monic (which is often studied [10, 11]) changes sign under the alternate definition of H(t), and the real a n d imaginary parts of even harmonics are interchanged. Heeding Ishida and Goldfarb [5], one must keep these relations terlaboratory calculations.  comparisons  of the harmonic  susceptibilities  in mind for in-  as well as for theoretical  

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