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An apparatus for the measurement of the surface resistance of high temperature superconducting thin films Knobel, Robert 1995

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AN APPARATUS RESISTANCE  FOR T H E MEASUREMENT OF T H E  SURFACE  OF HIGH TEMPERATURE SUPERCONDUCTING FILMS  By  .  Robert Knobel B.Sc. (Honours, Engineering Physics), Queen's University  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of  PHYSICS  We accept this thesis as conforming to the required standard  *  THE UNIVERSITY OF BRITISH COLUMBIA  1995 © Robert Knobel, 1995  THIN  In  presenting  this  thesis in partial  degree at the University of  fulfilment of  the  requirements  for  an advanced  British Columbia, I agree that the Library shall make it  freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of department  or  by  his  or  her  representatives.  It  is  understood  that  copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Abstract  A n apparatus to measure the millimeter-wave surface resistance of thin films of high temperature superconductors has been b u i l t . T h e apparatus consists of a flow cryostat, an open resonator and a swept-frequency millimeter-wave source/detector pair. T h e flow cryostat is used to cool the experiment to cryogenic temperatures (either 2 K with liquid helium, or 70 K with liquid nitrogen), while keeping a very short distance between the experiment and room temperature.  T h e cryogenic fluid is drawn through  the cryostat, and circulates through heat exchangers, absorbing incident heat. T h e open resonator allows sensitive measurement of the surface resistance of the film at high frequency. T h e resonator built operates at 117 G H z , and has a Q for copper at 77 K of 39000. The millimeter-wave source and detector are taken from a fixed frequency apparatus, and were modified to provide a frequency sweep capability. T h e source and detector operate in a band between 116.8 and 117.6 G H z . Measurements were performed on both metallic and superconducting samples. T h e metallic samples were used as calibration of the geometric factors of the resonator, in order to extract the value of the surface resistance. Four thin films of high temperature superconductors were measured, as a demonstration of the capabilities of the apparatus.  n  Table of Contents  Abstract  n  List of Tables <  vi  List of Figures  vii  Acknowledgement 1  2  3  ix  Introduction  1  1.1  Surface Impedance  3  1.2  Electrodynamics of Superconductors  1.3  Motivation  . . ,  6  • . ... : .  9  The Flow Cryostat .2.1  Introduction  2.2  Cryogenic Theory  2l3  Design . . . . . . . . . . .  14 '.......".*.•....•..'.,.  1*7 ...  . . . . .  . . . . . . . . . . . . '.  Theory of the Open Resonator 3.1  Introduction  3.2  History  3.3  Beam-Wave Theory  3.4  Cavity Losses  3.5  Beyond the Beam-Wave Theory  14  .,  21  •)  v  33 33  • • •  33. 35  .  :  ;  39 46  iii  \  4  5  Millimeter-Wave System  50  4.1  Introduction  50  4.2  Millimeter-Wave System  4.3  Resonator Design  4.4  Coupling  4.5  Microwave measurements  •.  ,.  50  :  • •  54  • • •  56 59  Experimental Method  62  5.1  Introduction  62  5.'2  Metallic Samples  5.3  Measurement Procedure  • •  •.  .63  '.:..-.  .64  6 Data 6.1  Introduction  6.2  Data  6.3 7  76 : . . . . . . .  76  : ...  77  6.2.1  Furukawa Electric Bi2Sr2CaiCu208 Thick Film . . . . . . .  6.2.2  U . B . C . Y B a C u 0 film on LaAlOg  6.2.3  U.B-.C. Y B a C u 0  6.2:4  McMaster Y B a C u 0 _ s l.OOOOA film.on LaAlOg . . . . . .  6.2,5  Comparison  2  3  2  3  2  . .'  79  7  film on S r T i 0  7  3  3  . . . . . . . .  . .... . .;". \  7  ......./...•....•.....  Finite Thickness Effects  . .  . .  ••. . . . . . . .  Introduction  7.2  80 81 84  .  .  Conclusions and Discussion 7.1  77  86 91  : . . . .  91  Equipment Performance  .  91  7.3' Surface Resistance Data  . .  92  7.4  :  Future Work  93  iv  97  Bibliography  v  List o f Tables  2.1  5.1  Heats and temperatures for the numerical model of the final flow cryostat design. The flow of liquid helium is 0.5 L / h r , and the pressure is 0.1 atm.  29  Fitting parameters for various functions .  73  vi  .  List of Figures  2.1 .2.2  Photograph of the flow cryostat  .  15  Photograph of the flow cryostat with shields removed  16  2.3  Schematic cut-away drawing of the flow cryostat  20  2.4  Isometric cut-away drawing of the flow cryostat.  2.5  Heat conduction as a function of inner pipe length  25  2.6  Temperatures as functions of inner pipe length  26  2.7  Heat conduction as a function of outer pipe length . .'  27  2.8  Temperatures as functions of outer pipe lengths  2.9  Temperature of inner pipes in final model .  •  23  '  27 29  2.10 Temperature of gas in the exchanger in the final model  30  2.11 Temperature of gas in the outer pipes in the final model  30  3.1  Terminology and geometry of open resonators  34  3.2  The semi-planar hemispherical geometry of the open resonator as used in the experiment  38  3.3  Field amplitude numerical simulation  40  3.4  Relative intensities of the cartesian components of the electric field in the complex point-source theory  48  4.1  Block diagram of the millimeter-wave system . .  51  4.2  Schematic diagram of waveguide circuit  53  4.3  The design of the resonator and sample positioning apparatus  .4.4  Mode chart for the resonator as built vii  . . . . . .  55 57  4.5  Numerical simulation of Q for copper at 77K  . . .  58  4.6  Numerical simulation of the resonator Q as a function of sample loss . . .  4.7  Lumped-element equivalent circuit of a resonator connected between a ,  59  generator and a detector.  60  5.1.  Surface resistance of copper at 117 GHz calculated from d.c. resistivity data 64  5.2  Comparison of the detected signal amplitude for direct transmission (ie. no resonator) for standard flanges and choke flanges.  . . . . . . . . . . . .  67  5.3  Q vs. temperature for brass and copper . . '  69  5.4  Resonant frequency as a function of temperature for brass and copper.  5.5  F i t t i n g of resonant peak  5.6  F i t t e d residuals for a sample resonant peak  5.7  Q vs. temperature for various fitting functions  5.8  Rs of copper and brass at 117 GHz as a function of temperature . . . . .  75  6.1  Furukawa Electric B S C G O thick film surface resistance  77  6.2  U . B . C . Y B C O on L a A l O substrate thin film surface resistance  79  6.3  U . B . C . Y B C O on S r T i O substrate thin film surface resistance  81  6.4  M c M a s t e r Y B C O on L a A l O substrate thin film surface resistance  6.5  Comparison of surface resistance measurements on two M c M a s t e r films at  .  70 71  different frequencies  -  72  '  74  . . . .  .  6.6  Comparison of the surface resistance of superconducting films. . . . . . .  6.7  Comparison of the surface resistance of superconducting films from the  82  83 84  literature.  89  6.8  Effective surface resistance for infinite substrate  90  6.9  Effective surface resistance for a 1mm thick substrate backed by copper. .  90  vm  Acknowledgement  I would like to thank, first of all, my supervisor Walter Hardy. His guidance and expertise have made this project possible. I ' m indebted to h i m for his patience and advice that have helped me grow as a scientist and a person. I would like to thank as well Doug B o n n , who has been a leader by example in the laboratory as well as a friend.  The  example of what a scientist should be presented by these two w i l l be a goal to which I will strive. The high temperature superconductivity group as a whole has helped me immensely, through encouragement, help and by making the lab an enjoyable place in which to work. Thanks, go to Dave Morgan, K u a n Zhang and Saeid K a m a l who have helped with friendship and scientific insight since I began, and to Chris B i d i n o s t i , A n d r e Wong who have helped since they began. Thanks to Dave Baar, R u i x i n g Liang and P i n d e r Dosanjh who have shown me the entrepreneurial side of science - encouraging me to open my m i n d to opportunities. Q . Y . M a and A n d r e Wong are to be thanked for develloping the collaboration of film growth and film characterization which is the center of this project. I am very much indebted to the U . B . . C . physics machine shop. T h e whole staff have helped through advice and conscientious work. I owe a special debt of gratitude to George Babinger, who built most of the cryostat, and provided essential advice for the rest of the project. T h a n k s to Pinder Dosanjh for great technical help throughout the project, and (with Levi Waldron) for measuring the D . C . resistivity of metals. I owe a huge thanks to W e n d y B r a h a m , whose support and faith in me never failed through the hard times. Thanks to Andrew H i l l , who has been a good friend and roommate during the last year. Thanks to my parents and sisters.who have always encouraged  ix  me, and, despite the miles apart, I've always-counted ori.them. Thank you to the Natural Sciences and.Engineering Research Council, for the financial support of the P G S scholarship.' The U . B . C . physics department also gets my thanks, for financial support thrqugh scholarships and teaching assistantships.  x  Chapter 1  Introduction  In 1986, Bednorz and Muller discovered superconductivity in lanthanum barium copper oxide at 35 K [1]. This discovery opened a new chapter and renewed interest in the study of superconductivity. Kamerlingh Onnes in Leiden discovered the first superconductor, mercury, in 1911 shortly after he developed the techniques of liquifying helium [2]. Many other metals were subsequently found to undergo a transition to superconductivity within a few degrees of absolute zero. The phenomenon has been studied both theoretically and experimentally in the intervening years and was thought to be a well understood phenomenon until the recent discoveries of high temperature superconductivity and other exotic superconductors. The theory of superconductivity developed in 1957 by Bardeen Cooper and Schrieffer [3] (called the B C S theory) has been a remarkable success, both in terms of the intuitive understanding of the mechanism that it imparts, as well as in its predictions of experimental results. This theory, with extensions by many other researchers, explains very well the behaviour of so-called "conventional superconductors" (circularly defined as those materials that are well described by the theories). The discovery of superconductivity in the LaBaCuO system was soon followed by the discovery of other cuprate compounds with superconducting transition temperatures (Tc) even higher ( Y B a C u 0 7 has a Tc of 93 K for example). With T^'s higher than 2  3  the boiling point of liquid nitrogen (77 K), the refrigeration needed to achieve superconductivity has become relatively inexpensive, compared to the cost of the liquid helium needed to cool conventional superconductors.  1  This latter requirement has limited the  Chapter 1. Introduction  2  applications where superconductors can be cost effective. W i t h high temperature superconductors, the possibility exists for many applications which, up u n t i l now, have been far too expensive to pursue. It m a y soon be possible to have magnetically levitated trains, low loss powerlines, and high speed superconducting computers. However the new cuprate high temperature superconductors are not understood very well, and are difficult to fabricate due to the complicated chemical structure. T h e simplest method of fabrication, to simply m i x and bake the ingredients, is capable of producing a superconductor, but the material is polycrystalline w i t h very poor properties. T h e materials are brittle ceramics, which cannot be simply drawn to form wires. These material problems have l i m i t e d the insight into the fundamental properties given by experiments, since many extrinsic effects mask the true intrinsic nature of the superconducting mechanism. Single crystals of superconductors are being made successfully [4], but are small and difficult to grow. They are currently useful for determining the basic properties of the materials, but not for applications. Most applications of high Tc superconductors use t h i n films.  These are made by depositing layers of the superconducting material (by  various methods including sputtering, laser ablation, chemical vapour deposition, etc.) on a dielectric substrate material whose lattice parameters closely m a t c h those of the superconducting crystal. These films can be patterned using methods similar to those used i n the semiconductor industry, and a variety of devices can be made. These processing methods are becoming very highly developed, and have enabled some very complicated devices to be envisioned. M a n y different types of transistor are being devised, and S Q U I D elements for magnetic sensing as well as Josephson junction switching elements have a l l been produced using t h i n films of superconducting materials. Perhaps the most likely prospect for the first commercially successful applications of high Tc superconductors is i n passive microwave devices [5]. These filters, delay lines,  Chapter 1. Introduction  3  interconnects, etc. all can be made to have superior properties if the conductors used have low loss.  Good quality high Tc superconducting thin films at 77 K have lower  loss than any standard conductor at microwave frequencies [6] (though not zero loss, cf. section 1.2). The quantity quoted most often for the electromagnetic loss at microwave frequencies is the surface resistance Rs-  This quantity is an effective measure of the  "quality" of the film, both in terms of its direct application in microwave use, as well as a general indication of the number of defects in the film (grain boundaries, lattice strain etc.). Such defects are common in films, and must be eliminated through tuning of the many parameters used in film growth. The purpose of the project described in this thesis is to build an apparatus to measure the surface resistance of films of high temperature superconductors. Such measurements are very commonly made [6], and have been made in this laboratory. However, the plan for this project is to have a device which is particularly suitable for thin films, and allows rapid, non-destructive testing for film-growth optimization.  1.1  Surface I m p e d a n c e  The concept of surface impedance is important for normal metals as well as superconductors. It will be instructive to outline where this quantity comes from in both cases. The microwave loss in the best normal metals and the best high temperature superconductors (at 77 K) is about the same at 100 GHz — though the mechanism is quite different. In this section I will define the concept and look at normal metals. Consider a good conductor for which Ohm's law applies (this derivation follows that in [7]):  3 = *E.  (1.1)  Chapter 1. Introduction  4  We can use M a x w e l l ' s equations (assuming harmonic time variation of the fields e ): 3Ujt  V  D  p  (1.2)  V  B = 0  (1.3)  E = -juB  (1.4)  V x  V x H  =  =  J+juD.  (1.5)  Inserting equation 1.1 into equation 1.5, we get  V x H = ( a + ju>e)E.  (1.6)  Equations 1.3 a n d 1.1 i m p l y p = 0. If we can neglect the displacement current (cr  cue,  which is a good approximation up to optical frequencies i n metals), then we can write (in various ways):  V E  =  jufiaE  (1.7)  V H  =  ju^o-H  (1.8)  jup.aJ.  (1.9)  2  2  or V J 2  =  T h e solutions to these equations depend on the particular boundary conditions, the simplest of which is to consider a semi-infinite slab of conductor of infinite depth (z > 0). T h i s case, though artificial, is actually very important because at high frequencies most bodies are m u c h larger than the depth of penetration of the electromagnetic fields. For a uniform field pointing along the plane of the conductor (say i n the x direction), the differential equation becomes: d Er 2  = junaE  = (3 E 2  x  X  (1.10)  Chapter 1.  Introduction  5  Which has the general solution E ^ C i e - ^ + cV*  (1.11)  Where the propagation constant 8 is most often written as p =  1  - ^  1  (  S=J ^ .  L  1  2  )  (1.13)  For E to remain finite as z —> oo, C2 = 0. Writing C i as i£ , the field at z = 0, we can x  0  write: = £ e  _ / 3 2  = E e0  0  z / s  e-  (1.14)  l z / s  with similar equations for both the magnetic field H and current density J. The quantity <5 is the skin depth in the material, and is in general a complex quantity (since a is a complex quantity). However, in a normal metal the real part of the conductivity dominates at millimeter-wave frequencies, and thus S is, to a very good approximation, real. The value of 6 is the depth at which the fields have decayed to 1/e their value at the surface. This description is valid, and the quantity of the skin depth is appropriate, for the situations where: a ^> u>t; where the dimensions of the sample are large in comparison to the skin depth; and where the mean free path of the electrons in the conductor is short in comparison to the skin depth (cf. chapter 5). The total amount of current carried in such a semi-infinite conductor (of unit width) is given by integrating the current density (given by the equivalent equation to 1.14): I  x  = r Jo  J dz x  =  r  Jo  Joe-t+M'Mdx  = ~~——~:• 1 +J  (1.15)  The surface impedance Zs is defined as the ratio of the electric field on the surface to the current carried in the conductor (here for unit width):  Chapter 1. Introduction  6  The surface impedance is often separated into its real and imaginary parts Z  s  =  R +jX  =  H . r l ± i )  s  (1.17)  s  +  (1.18)  I m f l ± i )  which, for a n o r m a l metal where Im(<5) w Im(cr) ~ 0, gives  Zs  -_ + L Rs  jXsK  s  +]  L _Jf ,Jf r  a+  ,  a  (  19)  where the real part Rs is the surface resistance and the imaginary part Xs is called the surface reactance. T h e surface resistance and surface reactance are measured i n the same units as resistance (Ohms). T h e surface impedance measured across two opposing edges of a square is independent of the size of the square. For this reason, the units used for surface reactance and impedance are often Cl/O, or Ohms per square. For more general geometry, where the above definition isn't obvious, the surface impedance is defined as [8] n x E = Z n x H x fi  (1.20)  5  along a surface w i t h n o r m a l n, where Zs w i l l i n general be a complex-valued tensor for anisotropic materials (where a is a tensor).  T h e power dissipated i n the conductor is  proportional to the surface resistance  P = \R Z  1.2  f I |n x H x n\ dS. J J Surface 2  s  (1.21)  Electrodynamics of Superconductors  In this section a brief overview of the electrodynamics of superconductors w i l l be given. This work has been the subject of m u c h effort over the years.  For a more complete  discussion of particular points, see the references which w i l l be cited throughout the  Chapter 1. Introduction  text.  7  For more complete summaries, see the reviews of T i n k h a m , W a l d r a m , or K l e i n  ([9][6][10]). There are a number of theoretical models used to describe the electrodynamic behaviour of superconductors, both microscopic and empirical [11] [12] [13].  T h e model  described here is a "generalized two-fluid m o d e l " , based on a number of papers from this lab [14] [15], but the basic idea was originally proposed, i n a more specific way, by Gorter and C a s i m i r [16] and by London [17].  T h e basis of this model is that the conduction  electrons i n a superconducting material can be thought of as being i n two types: the normal electrons and the superconducting electrons. T h e normal electrons behave like electrons do i n a non-superconducting material: dissipating energy b y scattering. T h e superconducting electrons (often referred to as the superfluid) however do not dissipate any energy. T h e dynamic behaviour of these two electron types can be described through transport equations [18]:  m  ^ L  dt  +  m  ^ T  =  N  =  n  e  2  E  ( L 2 2  )  (1.23)  NeE 2  s  dt  where JV and N are the number density of normal and superconducting electrons ren  s  spectively, r the scattering time of the normal electrons, and J are the current densities (for electron drift velocities v  n  = i V e v , J = N ev  n  n  and v ). s  n  s  s  s  T h e total electron  density, N = N + N , is constant. W h e n there is only a d.c. field, the superconducting s  n  electrons carry a l l the current. However, for an a.c. field, the inductive response of the superconducting electrons (caused by their inertia) allows some response by the normal electrons which dissipate energy. and  1.23 to  be:  a  = a + a, n  T h e conductivity can be derived from equation  1.22  Chapter 1. Introduction  8  Ner  . I  2  n  m ( l +LO T ) 2  =  ~  2  J  N e ur 2  Ne\  2  2  n  \m(l+u T ) 2  2  s  +  7^7/  o-i-jo-2  (1.25)  where o~i and <T are the real and imaginary parts of the conductivity respectively, and are 2  the fundamental quantities related to theories. T h e solution of the differential equation for the electric field given i n equations 1.10 and 1.11 hold as well for a superconductor as for a n o r m a l metal. However, i n this case, the approximation of real conductivity is no longer valid. T h e solution of the equation can thus be w r i t t e n as:  E w i t h 8 = yfjupLa,  o~\  — J<T , 2  = E e-  (1.26)  pz  x  0  as before. Using the separation of the complex conductivity as <r =  8 can be rewritten as:  8 = yjjujj.ai  + ufia .  (1-27)  2  The penetration depth of a superconductor, A, is defined through ( r°° H\\(z)  1  \  In a superconductor at millimeter-wave frequencies, a2 ^> 01 [it.  most of the current  is carried by the superfluid). Thus the fields at the surface of the superconductor are suppressed over a length scale set by A. A t a depth of A from the surface, the fields w i l l be 11e their value at the surface. Thus the values for the surface resistance and reactance m a y be derived i n this twofluid m o d e l as:  Rs  «  iorVcnA  X  =  tofxX.  s  3  (1.29) (1.30)  Chapter 1. Introduction  9  This derivation assumes local electrodynamics, where the penetration depth (scale over which fields change i n the material) is m u c h greater than both the mean free path of the electrons, as well as their coherence length [9]. B o t h of these conditions are well satisfied for high temperature superconductors [6]. T h e surface resistance Rs is directly proportional to the real part of the conductivity, and to the cube of the penetration depth.  T h e surface reactance is directly proportional to the penetration depth.  The  temperature dependence of the penetration depth is a very important quantity, since it can be related directly to the fraction of superconducting electrons [18]:  N _m 1  X  s  '  N ~ A2(T)'  1  j  In terms of practical uses for superconductors i n the millimeter-wave regime, an i m portant difference occurs between equations 1.19 and 1.29. For n o r m a l metals, the surface resistance increases w i t h frequency according to a* / , whereas for superconductors it goes 1  2  as to . There is therefore a crossover regime where the losses are comparable for metals 2  and superconductors at the same temperature. For the high temperature superconductors currently used, this crossover frequency (at 77K) is approximately at 100-150 G H z .  1.3  Motivation  There are a number of ways of measuring the surface resistance of a conductor. T h e most common way of m a k i n g these measurements is to form a resonant circuit, and measure the w i d t h of the resonance i n frequency. T h e Q of the resonance Q = (  Resonant frequency  \  y F u l l w i d t h at half m a x i m u m y is related to the surface resistance of the material forming the resonator (c/. chapters 3 and 4).  Chapter 1. Introduction  10  There are three general ways i n which the surface resistance of high  temperature  superconductors m a y be measured by resonant techniques: • patterning a stripline resonator out of t h i n films of the material. • creating a resonant cavity where some or a l l of the body is made from the material. • inserting a piece of the material into the interior of a resonant cavity. T h e first method [19] is of the most practical use, since it most resembles the final form of microwave circuit elements. A thin film of the material deposited on a substrate is patterned into a resonant structure, and the Q of the resonator is evaluated subject to other variables (ie. power, temperature, material properties). destructive technique, since the material must be patterned.  T h i s is, however, a  It is not very useful as a  diagnostic tool. A s well, it is unclear what effects the patterning has on the materials. It would be better to measure the films non-destructively beforehand. T h e t h i r d method is the one used most often i n this laboratory [14] [15] [20] [18]. Here a cavity resonator is constructed out of a material w i t h low loss (a conventional superconductor such as lead for best results) w i t h an opening through which the sample of interest is inserted. A g a i n , the change i n Q with various variables can be measured, giving the surface impedance of the material. This method can be very accurate, since the  filling  factor can be made quite high w i t h proper choice of resonator geometry [14]. A l t h o u g h it is an appropriate technique for single crystals, it is not generally suitable for thin  films.  T h e films must be made of the correct size to fit i n the resonator yet give high filling factor — which might mean breaking the film into small pieces. T h i n films are grown on substrates whose electromagnetic properties (loss tangent and dielectric constant) w i l l affect the measurement.  Finally, at millimeter-wave frequencies, the resonant cavities  become very small, and are difficult to fabricate and work w i t h .  Chapter 1. Introduction  11  The most common way to implement the second method is to replace one wall of a resonant cavity (normally a cylindrical T E o n cavity) w i t h a piece of the material ([21] [22] and many others). T h i s is a convenient method, although the results obtained are not very accurate i n general. This is because the loss due to the superconductor is only part of the loss i n the cavity, and separation of the two terms can be difficult.  The  superconductor must be i n close contact w i t h the rest of the cavity, and so the cavity as a whole must be subject to changes i n the same external variable(s) as the film (ie. temperature, magnetic field, etc.). T h e cavity material w i l l have some dependence i n its loss w i t h temperature, and w i l l be non-trivially included i n the measurement. Forming the cavity body from conventional superconducting material would give no advantage in terms of resolution, since most measurements w i l l be made above the Tc of any conventional superconductor. T h e film itself must be of the same size as the resonator, and the contact w i t h the resonator m a y i n itself damage the film. A very successful technique is the parallel plate resonator [23]. This technique has yielded some of the most sensitive measurements of surface resistance around 10 G H z . Here two films of the material are pressed together, separated by a t h i n teflon sheet. T h e assembly forms a resonator, whose Q m a y be measured as a function of temperature. The difficulty w i t h this technique is that two films are required, and their shape is quite important. For film characterization, it is desirable to have a technique not dependent on having two similar films. T h e objections raised for this m e t h o d apply as well to dielectric resonators [24] [25], where a bulk dielectric crystal forms the body of the resonator. The method described i n this thesis is the open resonator. This method is particularly appropriate for the measurement of surface resistance of t h i n films of high temperature superconductors at millimeter-wave frequencies.  It is not a m e t h o d invented here, it  has been described i n numerous papers [26] [27] [28], and it was the advantages of this method as described i n these papers that first attracted our interest. T h e open resonator  Chapter 1. Introduction  12  technique w i l l be described i n detail further on i n the thesis (cf. chapters 3 and 4), but a quick overview w i l l be appropriate here. T h e open resonator consists of a spherical mirror separated from the sample under test by a distance less than the radius of curvature of the m i r r o r (see figure 2.2). T h e electromagnetic field is focussed by the mirror on to the sample surface. T h e resonator is open to the environment, unlike conventional closed cavity resonators, this allows the resonator to be a larger size than would be possible for other geometries. A s well, the sample is physically separated from the rest of the resonator, so the temperature of the sample can be varied independently of the rest of the apparatus. T h i s apparatus is ideally suited for the measurement of thin films because only the top surface is i n contact w i t h the electromagnetic field, avoiding dependence on the substrate material. A s well there is no dependence on the shape of the superconducting sample (as long as it is larger than some m i n i m u m size), accommodating any shape of sample. In fact, the focus of the resonator can be scanned across the surface, giving information on the surface resistance over the area of the sample [28]. Such an apparatus (working at 77 K ) is being sold commercially by Conductus[29]. T h e open resonator is the only method capable of doing this non-destructively. Though this project has drawn on the ideas of other researchers i n terms of using an open resonator, there are problems which have not been dealt w i t h before i n this context: • T h e frequency used is quite high (117 G H z ) , involving complicated source and detector techniques (cf. chapter 4). • T h e samples measured are quite small, forcing the resonator to be used at its l i m i t in terms of spot size (cf. chapter 3). • T h e temperature dependence is being investigated, and a novel cooling technique has been developed particularly for this apparatus (cf. chapter 2).  Chapter 1. Introduction  13  T h i s thesis describes the development of the apparatus, as well as giving background on the theory. There is some data described, though it is secondary, since no exhaustive study has been done.  W i t h o u t systematic study of t h i n films under various growth  conditions, subject to various variables, the information is more qualitative i n nature.  Chapter 2  T h e Flow  2.1  Cryostat  Introduction  T h e cooling of the experiment becomes problematic when one wishes to use millimeterwaves.  A t these frequencies, the signal is carried through waveguides, which consist  of hollow rectangular tubes. Standard waveguides become quite lossy as the frequency increases, w i t h losses of about 1.5 d B per foot near 100 GHz[30]. A s well, the standing waves that arise from discontinuities i n the circuit can seriously degrade  measurement  accuracy, especially for long transmission lines. Thus, one wants to have as short a length of waveguide as possible. O n the other hand, standard cryostats involve the lowering of the experimental apparatus into a long dewar filled w i t h l i q u i d helium. Here the heat conducted along the supports and electrical connections (wires, coaxial cable, waveguides, etc.) is reduced by having long lengths of material w i t h low thermal conductivity. T h i s is exactly the opposite of the ideal m i Hi meter-wave system having short, highly conductive connections. For this experiment, we have designed a flow cryostat that avoids the long transmission lines (figures 2.1 and 2.2). In this apparatus, cryogenic fluid (liquid h e l i u m or l i q u i d nitrogen) is drawn up from a storage dewar and circulated through a series of heat exchangers, thus cooling the experiment. This contrasts w i t h standard immersion cryostats, where the experiment is immersed i n a large volume of l i q u i d . T h e present design allows for a very short length of waveguide (~5 cm) to go from r o o m temperature  14  Figure 2.1: T h e outer view of the flow cryostat. T h e l i d has been removed to show the heat shield. T h e plates on the sides are for adjusting the choke flanges (see chapter 4)  Chapter 2. The Flow Cryostat  16  Figure 2.2: T h e flow cryostat w i t h both the vacuum and heat shields removed. T h e two heat exchangers are clearly visible, w i t h the resonator and sample-holder attached to the inner one.  Chapter 2. The Flow Cryostat  17  to low temperature (~2 K w i t h l i q u i d helium, ~ 7 0 K w i t h l i q u i d nitrogen). A s well, it allows r a p i d thermal cycling of the whole experiment (~ 1/2 day) and can accommodate a variety of experiments. However, unlike the standard cryostat, where many researchers have refined the techniques [31], the design of the flow cryostat requires quite careful study and modeling. In this chapter I outline the basis for the design, and show the theoretical performance of the final setup.  2.2  Cryogenic Theory  In designing a cryostat, it is important to determine how the heat w i l l be transferred to the experimental chamber. For a flow cryostat i n an evacuated container, heat can be transferred to the experimental chamber by radiation across the vacuum, or by conduction along the solid materials either used for support of the apparatus, or used i n the experiment (eg. wires, waveguides, adjusting screws, etc.). B y m i n i m i z i n g the heat transfer through both of these mechanisms, the amount of cryogenic fluid needed can be reduced. T h e rate at which a surface of area A at temperature T emits thermal radiation Q is given by the Stefan-Boltzmann equation [32]:  Q = o-eAT . 4  T h e value cr = 5.67 x 1 0  - 8  Wm~ K 2  - 4  (2.1)  is the Stefan-Boltzmann constant. T h e emissivity  e is the ratio of the radiative energy emitted by an object at temperature T divided by the radiative energy emitted by a black body at the same temperature. It is a simplified model, i n which one uses an average of the frequency dependent emissivities that exist i n real materials, but should be adequate for the present use. T h e net rate of radiant energy exchange between surface 1 of area A at temperature T i and surface 2 at the (hotter)  Chapter 2. The Flow Cryostat  18  temperature T is 2  Q  = aEA(T -T ) 4  rad21  (2.2)  4  2  1  where E is a factor involving the emissivities of the two surfaces, and depends upon whether the reflections are diffuse or specular.  W e have polished the surfaces to give  specular reflection, since this gives lower transfer of heat. In this case  E=  (2.3) '  e + (l-e )ei 2  v  2  For the materials used i n this apparatus (aluminum, copper and brass), the emissivities ei and e are approximately 0.04-0.06 at room temperature and w i l l decrease w i t h de2  creasing temperature [32]. Thus E = 0.025 w i l l give a conservative estimate for the rate of radiative heat transfer i n the design. To reduce heat transfer by radiation, one can insert radiation shields. B y inserting a shield at an intermediate temperature T between T\ and T m u c h of the radiation s  2  can be intercepted. This shield m a y be floating, ie. thermally disconnected, or actively cooled. Such shielding is very important i n the flow cryostat, since the experiment w i l l be surrounded by material at room temperature, not l i q u i d nitrogen temperature as i n a standard cryostat. The rate of heat conduction Q  along a solid material of thermal conductivity k  cond  dT dT and cross sectional area A w i t h a temperature gradient —— is given by the relation [33] dx • Qcond  , dT =  ,  k  ^T-  fcA  .  (-) 2  4  B y using materials w i t h poor thermal conductivity and small cross-sectional area (eg. thin wall stainless steel) this conduction can be reduced. However, the thermal conduction can be further reduced, and perhaps made insignificant, by using the flowing gas to cool the supports.  T h e rate of heat absorption by a warming gas of heat capacity C  v  Chapter 2. The Flow Cryostat  19  flowing at a rate of h moles/second is  dQ = tiC dT p  (2.5)  gas  or,  dQ te  dT  gas  =  ( 2  -  6 )  if one can use distance as the independent variable (as i n gas traveling along a pipe). T h e approximation that C is constant w i t h temperature w i l l be used without significant v  reduction i n accuracy. Heat is transferred between the wall of the containing vessel and the gas it contains w i t h a temperature difference of AT over an area A according to  Q = hATA.  (2.7)  T h e value heat transfer coefficient, h, depends upon whether the flow of fluid is laminar or turbulent. T h e turbulence of a flow of fluid at velocity v w i t h viscosity 77 and density p through a pipe of diameter d is described by a dimensionless quantity called the Reynolds  number Re: Re =  ~.  P  (2.8)  A flow w i t h Reynolds number above 2300 is turbulent. For laminar flow, the heat transfer coefficient for a fluid of thermal conductivity k  gas  is given by [34]:  h oc k /d.  (2.9)  h - T ^ o ^ -  (2-10)  gas  For turbulent flow [35]  nC where the dimensionless P r a n d t l number is Pr = 4—  E  "'gas  and G = vp.  Chapter 2. The Flow Cryostat  20  Heat Shield  Access Ports  Needle Valve  Vacuum Can. sOuter Heat Exchanger  Needle Valve Control  Figure 2.3: Schematic cut-away drawing of the flow cryostat.  Chapter 2. The Flow Cryostat  2.3  21  Design  A s described earlier, the flow cryostat, instead of using a large reservoir of cryogenic l i q u i d surrounding the experiment, draws a small constant flow from a separate storage dewar. This flow of liquid, which rapidly becomes gas, is brought through a series of heat exchangers, absorbing incident heat through b o t h the latent heat and the heat capacity of the gas. T h e chief advantage of this design is that it allows short distances from the cryogenic region to the outside. W i t h a standard cryostat, this is difficult, and would contribute a large heat conduction path. E x t e n d i n g into the storage dewar is a small capillary (of t h i n wall stainless steel) which enters into the flow cryostat (see figure 2.3). Cryogenic l i q u i d is drawn up this line and enters a heat exchanger through the control of a needle valve. T h e heat exchanger consists of a hollow copper block, w i t h copper plates arranged inside so that the must take a circuitous path.  fluid  U p o n entering this chamber, the l i q u i d boils — at the  e q u i l i b r i u m temperature for the pressure set. B y lowering the pressure (increasing the p u m p i n g rate on the gas lines), the temperature at which the l i q u i d boils can be lowered. This temperature becomes the temperature of the experiment, since the experimental apparatus is attached to the top of this heat exchanger. It is important that as m u c h of the l i q u i d changes to gas here as possible, since small droplets of l i q u i d are not as efficient i n heat transfer as is the gas. To accomplish this, a finely corrugated disk of copper is inserted into the first stage of the exchanger, ensuring very good thermal contact immediately. T h e gas, upon leaving the inner exchanger, is carried through three t h i n wall stainless steel tubes to another, larger exchanger (see figure 2.4).  This heat exchanger, which  works i n a manner very similar to the first one, is connected to a radiative heat shield surrounding both the experiment and the inner exchanger. B y cooling this exchanger w i t h  Chapter 2. The Flow Cryostat  22  the flowing gas, the radiative heat load on the experiment is reduced. A s a compromise between simplicity of design and efficiency, it was decided to use only two heat exchangers (and one heat shield). T h e gas leaving the exchanger is carried through three more t h i n - w a l l stainless steel tubes, exiting the vacuum shield through ports on the underside. These tubes physically support the experiment, and since they conduct gas, give a reduced conductive heat load to the apparatus. T h e base temperature of the experiment is controlled through two variables: the p u m p i n g rate on the exit tubes, and the flow impedance provided by the needle valve at the inner exchanger.  B y tuning these two variables, we attempt to m i n i m i z e the  amount of coolant required. This is of course dependent upon proper design of the heat exchangers and connecting tubes. T h e flow cryostat was designed to be used w i t h l i q u i d helium as the coolant. However all of the work described i n this thesis uses liquid nitrogen as the coolant. T h e design was based on the assumption of l i q u i d h e l i u m both because it is quite expensive and because its low boiling temperature imposes more stringent design criteria. L i q u i d nitrogen, though it has different properties and thus the cryostat w i l l not operate at peak efficiency, is relatively cheap. Thus the design criteria for the flow cryostat are: • It should have the ability to reach 2 K w i t h 1 L / h r of l i q u i d H e flow. • T h e experimental chamber should have a m i n i m u m volume of approximately 7cm diameter by 5cm high. • T h e experimental chamber should have the ability to accommodate various experiments. • T h e flow cryostat should work on top of a storage dewar (either l i q u i d helium or  Chapter 2. The Flow Crvostat  23  Figure 2.4: Isometric cut-away drawing of the flow cryostat. liquid nitrogen). • There should be only a short distance from room temperature to cryogenic temperature. Computer modeling was used to determine the design of the cryostat. Initially, each portion of the design was examined separately, w i t h assumed values for important parameters. Once the design was set (which was affected at least as much by utility and ease of fabrication as it was by efficiency of the design), a complete simulation was used to check the performance. Conservative values were used initially for the design, since it is difficult to predict what sort of experimental apparatus might eventually be used i n this cryostat.  Chapter 2. The Flow Cryostat  24  T h e modeling was done by numerically solving the coupled differential equations describing the flow of the gas, and the heat exchange associated w i t h the gas. For the flow of gas through the pipes (between the exchangers and exiting the apparatus), the temperature of the two ends of the pipes, as well as the i n i t i a l temperature of the gas, were assumed. T h e differential equations were integrated using a R u n g e - K u t t a method u n t i l the computed final temperature of the pipe matched the set final temperature (essentially a "shooting method"[36]).  T h e flow was assumed to be turbulent, and any pressure  change along the length of the pipe was ignored (after assuring that both assumptions were justified). T h e heat capacity of the gas was assumed constant, and the heat exchange between the gas and the pipe was taken from equation 2.10. In figures 2.5 and 2.6, the dependence of the heat conduction and temperature of the gas vs. the length of pipe connecting the two heat exchangers is shown. A s can be clearly seen, the amount of heat transported down the pipes drops off strongly as the length is increased. It was decided that a length of 2-3 c m was sufficient to make the heat load along these pipes negligible. Similarly, i n figures 2.7 and 2.8, the dependence on the length of pipes exiting the heat exchanger is shown. Here, any length of pipe longer than 5 c m gives essentially zero heat conducted to the outer exchanger. In fact, even at 5 c m , the heat load along the exit pipes should be even smaller than calculated here. This is due to fact that the outer tube temperature is m u c h less than the assumed 300 K , and i n fact becomes frosted w i t h ice, even w i t h a moderate flow rate. T h e design of the heat exchangers involves m a x i m i z i n g the heat transfer between the gas and the body without imposing too large a pressure gradient. To increase the heat transfer, the surface area to which the gas is exposed should be m a x i m i z e d . T h e heat exchangers use plates of copper w i t h i n a copper block to cause a circuitous flow: the first plate has a hole i n the center; the next plate has no hole i n the center but has a gap at its outside edge (see figure 2.3). T h i s sequence is repeated, and provides a large increase  Chapter 2. The Flow Cryostat 0.20  25  0.20  0.10 h  0.05 0.00 Figure 2.5: Heat conducted (in Watts) along the pipes joining the inner heat exchanger w i t h the outer heat exchanger, as a function of the length of pipes (in cm). T h e 3 pipes are assumed to be 3/8" diameter, .012" wall thickness stainless steel. T h e inner exchanger temperature is assumed to be 4.2 K , and the outer exchanger temperature 30 K . T h e gas flow is set to be that corresponding to a flow of 1 L / h r of liquid helium.  Chapter 2.  The Flow Cryostat  26  30  3  4  5  6  Figure 2.6: Temperature of the helium gas and the pipe wall i n K e l v i n as a function of pipe length i n cm. A l l parameters are as assumed i n the previous figure.  Chapter 2. The Flow Cryostat  27  Figure 2.7: Heat conducted (in Watts) along the pipes joining the outer heat exchanger w i t h the vacuum shield, as a function of the length of pipes. T h e 3 pipes are assumed to be 3/8" diameter, .012" wall thickness stainless steel. T h e heat exchanger temperature is assumed to be 30 K , and the outside temperature to be 300 K . T h e gas flow is set at  Figure 2.8: Temperature of the h e l i u m gas and the pipe wall as a function of pipe length. A l l parameters are as assumed i n the previous figure.  Chapter 2. The Flow Cryostat  28  i n gas contact area over a simple pipe. M o d e l i n g of the heat exchangers proved to be simpler than for pipes, i n that the thermal conductivity of copper is so high that even .020" thick copper plates are essentially a thermal "short" to the heat flows encountered. Thus the problem reduces to determining the best design for the gas to absorb the required amount of heat while passing through a constant temperature body, which involves integrating a set of coupled differential equations. U n l i k e the pipe problem, there is no fixed end-point that must be satisfied, so no "shooting method" is required. T h e heat absorbed is m a x i m i z e d by having turbulent flow of the gas which is accomplished by throttling the gas at the plates.  A s well, the pressure must not drop significantly  through the exchanger or we w i l l not have good control over the temperature at the interior of the cryostat. In the end, it was determined that having three plates i n the exchangers, separated by approximately 1/10" gives negligible pressure drop, and allows the gas to reach thermal e q u i l i b r i u m w i t h the exchanger body before exiting (see figure 2.10). Note that a number of simplifying assumptions have been made here chiefly i n terms of assuming symmetric geometry, and uniform turbulent flow.  0  I  ,  ,  ,  0  ,  1  1  ,  ,  ,  ,  I  ,  ,  ,  ,  2  1  3  Distance along Tubes (cm) Figure 2.9: Temperature of gas and pipe for the inner pipes of the flow cryostat i n a self-consistent numerical model. T h e pressure is 0.1 a t m , and the flow rate 0.5 L / h r of liquid nitrogen. T h e inset shows the amount of heat (in m W ) flowing along the pipes.  Temperatures  Heats  Inner exchanger Heat Shield Gas at inner exch. Gas at end of inner pipe Gas at beg. of outer pipe Gas at exit R a d i a t i o n on inner chamber Conduction to inner exch. R a d i a t i o n on shield Conduction to shield Absorption i n inner exch. Absorption i n outer exch.  2.488 K 16.263 K 2.488 K 2.618 K 16.262 K 39.985 K 9.7 l O " W 0.00961 W 0.586 W 0.709 W 0.00961 W 1.273 W 7  Table 2.1: Heats and temperatures for the numerical model of the final flow cryostat design. T h e flow of liquid helium is 0.5 L / h r , and the pressure is 0.1 a t m .  Chapter 2.  The Flow Cryostat  30  20  5  10  15  Distance along Exchanger (cm) Figure 2.10: Temperature of the gas i n the outer exchanger of the final design of the flow cryostat for pressure of 0.1 atm and flow rate of 0.5 L / h r , along the circuitous path inside the exchanger. T h e shield temperature reaches an equilibrium temperature of 16.26 K . The inset shows the integral of the heat absorbed by the gas (in W). 300  g  200  4-> d i-i  o.o  <D  J  0.0  1.0  I  I  I  1  2.0  L_l  3.0  I  4.0  I  5.0  l_  I  6.0  7.0  CL,  S  H  100  o Tube Wall Temperature ° Gas Temperature  o1  ^ • • • • • • • • • • • • • • • • • • • • • • • • n  0  0  _i 1  2  i  i_ 3  n  n  n  D  D  D  D  [  j  4  5  :  i  n  a  i  D  G  n  n  D  D  i_  6  7  Distance along Tubes (cm) Figure 2.11: Temperature of the gas and pipe for the outer pipes of the flow cryostat i n its final configuration. T h e inset shows the amount of heat conducted along the pipe (in W). See the previous two figures.  Chapter 2. The Flow Cryostat  31  T h e final design of the cryostat is shown i n the figures. T h e full performance of the set design was modeled i n a self-consistent solution for various flow rates and pressures (figures 2.9 to 2.11 show a particular set of parameters). This modeling entailed setting the flow rate of l i q u i d , pressure of the exit gas, and external temperature; unlike for the i n i t i a l models, where each aspect of the cryostat was examined independently. T h e solution of these coupled differential equations was done iteratively, ending when a selfconsistent solution was found. For the condition shown i n the figures, 0.5 L / h r of l i q u i d h e l i u m flow at 0.1 a t m , the latent heat of the liquid is greater than the amount of heat incident on the inner exchanger, meaning that not a l l the l i q u i d boils before leaving the exchanger.  T h i s neglects the heat conduction along waveguides, etc., and thus is  an underestimate of the temperatures reached i n various portions of the cryostat. To partially account for this, it was assumed that any remaining liquid is transformed into gas by the excess heat (a crude approximation). As can be seen from figure 2.11, the exiting gas is still very cold (about 50 K ) . This suggests two things: first of a l l , that a slower flow of gas could probably be used. T h i s is true up to the l i m i t that the inner heat exchanger should reach a low enough temperature (as determined by the needs of the experiment). Secondly, that one would probably want to use another heat exchanger/heat shield combination. These exchangers are (theoretically) very efficient, as can be seen from figure 2.10, where the gas reaches equilibrium quite well. Unfortunately, another level i n the cryostat would entail greater complication i n b o t h the use and fabrication of the apparatus, and was not included i n the design for that reason. T h e modeling done i n this chapter has been solely for the steady state — once the experiment has reached operating temperature. In fact, a significant amount of cryogenic liquid would be needed to cool the apparatus to the operating point. For this reason, it is expected that i n practice the cryostat w i l l be pre-cooled w i t h liquid nitrogen before  Chapter 2. The Flow Cryostat  the liquid helium cooling is beg  Chapter 3  Theory of the Open Resonator  3.1  Introduction  U n l i k e closed cavity resonators, which have been a key tool for microwave engineers and physicists for decades, open resonators have been exploited only relatively recently. W h i l e closed resonators can be regarded as an extension to higher frequency of l u m p e d element R L C circuits, open resonators resemble more a scaling down i n frequency of optical mirrors. In this chapter I w i l l outline the theory pertaining to the existence and characteristics of stable resonant modes of such structures.  3.2  History  In the late 1950's, Fabry-Perot structures were shown to be useful as resonators at optical frequencies for use i n lasers.  T h e Fabry-Perot resonator consists of a pair of plane-  parallel mirrors facing each other, between which light reflects over several passes, storing electromagnetic energy.  These structures, initially analysed using optical techniques,  were later studied i n terms of the resonant modes of the electromagnetic field [37]. In related work, G o u b a u and Schwering showed that an electromagnetic field can be described i n terms of a complete set of cylindrical waves. If these waves are propagating largely along one axis, then they can be resolved into an elementary set of beams. These beams can be confined along the axis through a repeated series of guiding structures (ie. lenses). [38]  33  Chapter 3. Theory of the Open Resonator  Plane-Parallel, or F a b r y - P e r o t  H a l f - C o n f ocal  34  Confocal  Hemispherical  Concentric  Figure 3.1: Terminology and geometry of open resonators. B o y d and Gordon[39] developed the confocal resonator, i n which the plane parallel mirrors are replaced by spherical mirrors separated by their common radius of curvature (figure 3.1). This geometry has the advantage of requiring less precision i n the positioning and machining of the reflectors to give low loss. A s well, the losses due to diffraction are significantly m i n i m i z e d . Subsequent work generalized the solutions to reflectors of different radius of curvature, and an arbitrary separation. In an important review paper, Kogelnik and Li[40] reviewed the properties of such resonators, incorporating the beamwave theory of Goubau and Schwering. I w i l l m a i n l y follow their development of the theory i n this chapter. A s well, I will make use of the work of B u c c i and D i Massa[8] who use the eigenvalue approach of Kurokawa[41]. These methods a l l involve approximations which may not be valid for the experiment described i n this thesis. Cullen and coworkers, i n a series of papers[42] [43] [44], have attempted to determine the accuracy of these approximations, as well as to develop a more precise theory. These aspects w i l l be described later i n this chapter.  Chapter 3. Theory of the Open Resonator  3.3  35  Beam-Wave Theory  T h e behaviour of the electric and magnetic fields i n a resonator can be obtained by solving the wave equation for the magnetic field H or the electric field E  V H + fc H = 0  (3.1)  V E + A; E = 0  (3.2)  2  2  2  2  subject to the particular boundary conditions of the problem. Here k = 2ir/A is the propagation constant for wavelength A i n the m e d i u m , and is the eigenvalue of the differential equation. T h e field components i n cartesian coordinates of such a coherent wave satisfy the scalar wave equation V u + fc u = 0. 2  (3.3)  2  For a wave traveling i n the positive z direction, w i t h a harmonic time dependence, we can put u(x,y,z)  where the function  (3.4)  = ^(x,y,z)e-^e^  t  represents all deviations from a plane wave. Substituting  ip(x,y,z)  equation 3.4 i n equation 3.3, we get d j> dx  dV>  2  g-fc^-O  2  2  dy  2  dz  dz  2  '  If most of the variation of u w i t h z is taken up by the exponential, ie. tp is slowly ying i n the zz direction then we can neglect ^ ~ i n comparison w i t h k^-, and equation varying oz  l  dz  3.5 becomes [8]  T h i s is the parabolic or paraxial approximation often used for the solution of the open resonator problem. T h e validity and accuracy of this approximation w i l l be discussed later.  Chapter 3. Theory of the Open Resonator  36  T h e differential equation 3.6 has a form similar to the time dependent Schroedinger equation, suggesting that one can t r y a solution i n cartesian coordinates of the form [40]  k  tp = g(x,z)h(y,z) exp < -j  27O  (x + y 2  (3.7)  where g is a function of x and z, h is a function of y and z. P(z) represents a complex phase shift along the axis, and 'j(z) a complex beam parameter, which describes the intensity and curvature of the beam. Substituting we find that  d dz  dP dz  1  — = 1 and —— 7  (3.8)  For convenience we define 1  A  7  R(z)  (3.9)  KW (Z)  J  2  where one can see that R(z) is the radius of curvature of the wavefront (surface of constant phase) at z, and w(z) is a measure of the decrease of the field amplitude w i t h distance from the axis. Since this decrease is Gaussian i n nature, w(z) is the distance from the axis at which the amplitude is 1/e times that on the axis — often called the or  beam radius  spot size. A t the point where the beam contracts to its m i n i m u m diameter, called the  beam waist, the beam parameter is purely imaginary 7o =  (3.10)  JZQ  w here _ 7Twl  z = 0  A  kwl  2  (3-11)  Here w is the m i n i m u m spot size, still to be determined. Measuring z from this point, 0  we have 7 = jzo + z = — — S i + z.  A  (3.12)  Substituting equation 3.12 into equation 3.9, we get  w (z) = w l + l 2  2  (3.13)  Chapter 3. Theory of the Open Resonator  37  and  R(z)  (3.14)  z  T r y i n g functions of the form g (  ) and h I  \w(z)J  ] i n the differential equation 3.6  \w{z)J  gives equations of the following form for both functions  dg 2  j  2  -2x^-  + 2mg = 0  (3.15)  which defines the H e r m i t e polynomial of order ra [45] g •h = H  m  (V2^-)  H (V2 V  w(z)  w(z)  (3.16) /  For the fundamental mode (ra = p = 0), one can just integrate  dP(z) dz  =• ~Jh  to get  the phase shift, although it is more complicated for the higher order modes, yielding  P(z) = (ra + p + 1) t a n  - 1  ( — ) - j ln \Z J V w  (3.17)  0  C o m b i n i n g a l l these factors, and suppressing the time dependence, we get WQ  w(z)  w(z)  exp  exp < -j  t  kp kz - <S> (z) + 2R(z)\ 2  w  mp  (3.18)  where we define $  m p  = (ra + p + 1) tan  1  (3.19)  p =x + y. 2  2  2  (3.20)  A n t i c i p a t i n g later normalization following B u c c i and D i Massa[8], we write for mirror separation D (see figure 3.2) mp(x,y,z) =  u.  2  /  1  w(z)\j  Dir2 +Pmlp\ m  H V2 m  w(z)  t  kp p ( - J kz - $ (z) + 2R{z)\ 2  e x  P  fziT^l w (z) 2  t  e x  mp  (3.21)  Chapter 3. Theory of the Open Resonator  38  a  T  —1  Figure 3.2: T h e semi-planar hemispherical geometry of the open resonator as used i n the experiment. For the semi-planar geometry used i n this experiment (figure 3.2), the curvature of the beam at the spherical mirror is given by the curvature of the m i r r o r . (3.22)  R(z)U=D = Ro  A t the planar m i r r o r (the sample), the beam should have no curvature R(0) = oo. These geometrical constraints give the values for the constants z and w^: 0  zo = ^D(R  - D)  0  wl = l^D(R  0  (3.23)  - D).  (3.24)  In a resonator, neglecting any losses due to the size of the mirrors or finite conductivities, a standing wave w i l l be developed of the form  ^mpq = mpq ^ mpq u  (3.25)  U  where the (+) and (—) indicate waves traveling i n the positive and negative z direction. If u is a transverse component of the electric field (say u = e ), y  then the boundary  Chapter 3. Theory of the Open Resonator  39  condition that must be satisfied for perfectly conducting mirrors is ^ | Mirror surface  — 0.  (3.26)  T h i s boundary condition is approximate, and corrections to i t w i l l be discussed i n the following section. Using equations 3.25 and 3.21 this condition gives the propagation constant i n the lossless case  ^ ^±l±l -^(£) 1^p.  k =  tm  (3.27)  +  where the label q refers to the number of nodes along the z axis. T h e standing wave field can thus be w r i t t e n V (x,y,z) mpg  =  ,„, , H  : J W ^ i  w(z)\l  D7r2 +Pm\p\ m  m  {V^^r  m  \  w(z)J  v  ) H (v^^r p  \  p  v  w{z))  y  1 exp'  P  \w (z) 2  "  t  (3.28)  sin  T h e multiplicative constants have been chosen such that, following B u c c i and D i Massa, the eigenfunctions ^  m  v  q  of the wave equation are normalized (to the same order as the  approximation used i n its derivation) w i t h 1 i f (mpq) = (nst)  r r r  Jj j  ^mp^nstdv  =  (3.29) 0  Cavity  otherwise.  T h e mode of interest i n this experiment is the mode ^>oo , often called T E M o o since, q  ?  to a first approximation, both the electric and magnetic fields are transverse.  3.4  C a v i t y Losses  T h e important quantities of the resonance for microwave measurements are the resonant frequency and the Q, or quality factor of the resonator. For the lossless  derived  above, the resonant frequency of the mode mpq is given by -m + p + 1 ^mpq — ckmpq — C  D  _!  A  t  a  n  _  1  (D\ , 7T ( - ) + ^ ( ? + l)]  (3-30)  40  Chapter 3. Theory of the Open Resonator  z (m) Figure 3.3: T h e amplitude of the transverse electric field (e ) using the beam-wave approximation for the mode [m,n,p] = [0,0,14] and dimensions as used i n the experiment. y  and the Q is infinite, since it is defined as 1 — = Q  Total Power Loss  P =  u (Total Stored Energy)  (3.31)  LOU  and we have neglected a l l power losses up to now. W h e n the losses of the resonator are introduced, the resonant frequencies of the modes w i l l shift, and the resonances w i l l develop a finite w i d t h . These losses come from: • finite conductivity of the m i r r o r and sample surfaces. • diffraction from having non-infinite mirrors (ie. some of the beam is radiated away).  Chapter 3. Theory of the Open Resonator  41  • coupling between the resonator modes and the modes of the feeding waveguide. 1/Q can be written as a sum of contributions to the total loss of the cavity: 1  1  1  1  o 7)— n =  +  1  - )  +  sample  ^mirror  ,  7T-—~ n  +  (3  diffraction coupling  32  A n y electromagnetic field inside the cavity, including the losses, can be expressed as an eigenfunction expansion of the modes of the lossless case (except possibly at a finite number of points[41]):  E  =  £  m,p,q  V ' mpq^mpq m  ^ ] fmpq^mpq-  •H  (3.33)  m,p,q  For the y-polarized mode, let  e (x,y,z) mpq  =  W  ^  ^  j  J  v  ^  j  HJy/2-Z-r)  -  w{z)y DK2™+V m\p\"my " w{z))  sin  km  p q  z - $m  p  4+  k  m  p  q  p  2  2R(z)  +  4-  These modes form a complete orthogonal set.  2  " \ "w{z)J p v  j.  ^  2 '  exp - f  ^ '\w (z) t  2  i  (3.34)  Since the differential equation used is  scalar, there are two degenerate modes corresponding to the x and y polarisations. A n y inhomogeneity without rotational symmetry i n the xy plane of the resonator w i l l tend to split the frequency of these modes.  T h e resonator as designed has no method of  distinguishing the two modes. Following the argument of Kurokawa[41], V x e  mpq  field, since V x E =  is a function similar to the magnetic  —jupH.. Let us the define the functions h  mpq  such that  V X d pq — k q\l q. m  mp  (3.35)  mp  T h i s gives  W\Z)  cos  k m p q Z  pq  _ d>  mp  m p  + hmP  2R(z)  2  + tJ'/l TL  2  I  (3.36)  Chapter 3. Theory of the Open Resonator  42  for the same polarization. B y substituting equation 3.35 into V x h  mpq  we get the  symmetric result that  —k  V X hmp7 and V  T h e constants I  mpq  i n the expansion are so named due to their similarity to  mpq  mpq  (3.37)  G .  rnpq  the current and voltage i n transmission line equations: [41] V  = J j J E • e dv  mpq  (3.38)  mpq  Impq = J J J B.- h dv.  (3.39)  mpq  We can expand the function V x E i n terms of the functions h  mpq  for the same reasons  as given above, V xE= £  h  -  P  9  m,p,q  / / / ( V x E ) • h dv.  (3.40)  mpq  JJ J  E x p a n d i n g the integral, using vector calculus identities and Gauss' theorem, V xE =  Y  =  h  n  »w/ / / ( '«P9  V  •(  Ex  h  " w ) + E • ( V x h ))dv mpq  ( / y n • ( E x h )dS  JJj  +k  mpq  mpq  E • e dvj mpq  .  (3.41)  Similarly, we can expand V x H , V xH =  ™ qkmpq m,p,q e  P  / / / H • e dv  (3.42)  mpq  JJ J  where the surface integral vanishes because fi x e  = 0 on the surfaces of the cavity.  mpq  These eigenfunction expansions can be then substituted into Maxwell's equations: V xE =  -jw//H  V x H = jueE  Y^  hmp?  n  . • (  =  E  x  (3.43)  JJJ  h q)dS + k  -juft  mp  mpq  Yl  h  ™p? /  /  /  H  '  E • e dv mpq  h  m  p ^  u  Chapter 3. Theory of the Open Resonator  43  ' h pg dv = jue  ^ , ^mpqkmpq Iff  m  / / / E • e dv. mpq  m,p,q  m,p,q  JJJ  Since the functions h  and e  mpq  are orthogonal, we can equate the coefficients of each  mpq  vector, and solve for the constants I  mpq  I m  P*  ™  Vm  Here k  mpq  =  (3.44)  JJ J  and V : mpq  L 11 m  JUJ  U2  •(  h  Pq  =  n  x E)dS  (3.45)  J J m , • (n x E)dS.  (3.46)  h  mpq  P  is the propagation constant for the lossless mode labeled (mpq), and k is the  propagation constant for the actual field. T h e surfaces over which the integration is done all have different boundary conditions on the field, so j j h  m p g  • (n x E)dS  =  J J h  mpq  • (n x E)dS  Sample + J Jh  m p g  • (n x E)dS  +  Curved Mirror  J J h  mpq  • (n x E)dS +  Couphng Apertures  J J h  mpq  • (n x E ) d &  Everywhere Else  E a c h of these contributions is related to the terms i n the expansion of — i n equation 3.32. The expression for the electric field i n equation 3.33 is not correct up to the cavity boundary, since i t forces the tangential component of E to zero. For a conductor w i t h finite conductivity, we can use the boundary condition of surface impedance (cf. Introduction) [8]. n x E = Z h x H x ii.  (3.48)  s  So for the m i r r o r and sample surfaces, JJ h  mpq  • (n x E) dS  = =  Z jJ s  h  mpq  • n x H x hdS  X ] Im'p'q'Zs / / h  mpq  m'p'q'  • h i i idS. m p q  (3.49)  Chapter 3. Theory of the Open Resonator  44  T h i s total stored electromagnetic energy can be expressed i n terms of either the electric or magnetic fields:  U  = e/2JJjE-E*dv  = p/2 J J  Jn-Wdv 12 mpq |  (3.50)  L  mpq  mpq  The power loss on the surface is P  = Re =  Z J J H - H*dS s  2^2 2\2  Re  / / " m p g • iqidS hmip.  Irnpql i > ' m  m.v.a mm',p',q' l n< i m,p,g  p  g  (3.51)  •*  n  If losses are low, then we would expect the field present i n the cavity to be similar to the field i n a lossless cavity. Thus, the dominant t e r m i n the sum w i l l be for (mpq) = (m'p'q'), and we can rewrite the equations for power energy and ohmic power loss as: P  mpg  = R e [Z ] U pq m  JJ  \I p \  2  s  m  =  q  |h  nl'l\I pq\  (3.52)  | dS 2  mp9  (3.53)  .  m  Thus, we can write expressions for the Q of the cavity due to losses on the curved mirror and on the sample as  Y^—  = R e [ Z w ] A [ f\h pq\ dS 2  m  T^—  (3.54)  U[1 J JS  QSample  = MZs J— mtrr  Mirror  Ii  \h \ dS. 2  mpq  (3.55)  As written, this assumes that h  UljJL J JM  mpq  is parallel to both m i r r o r surfaces — which is not  true for the curved mirror. This is another instance where the paraxial approximation has been invoked. The fourth integral i n equation 3.47 deals with the losses due to diffraction.  This  term can be dealt w i t h by assuming that the field is an outgoing locally-plane wave, so  Chapter 3. Theory of the Open Resonator  45  that n x E  = (  0  n x H x n  (3.56)  where the surface of integration is to simply extend the mirror surfaces u n t i l they intersect [8] and Co is the impedance of free space. This solution would be very rough, since the approximations and assumptions made are quite extravagant. However, this term should be small for the fundamental mode if the mirror and sample have a diameter greater than twice the spot size. This follows from an empirical discussion by C u l l e n [44] based on analytic results by Weinstein [46] for the confocal geometry. Weinstein expresses the loss due to diffraction as Q = kD/A  (3.57)  where A is the loss per pass, given empirically by A ~ 200exp(-4a /u; ) 2  2  (3.58)  where a is the radius of the mirror (not radius of curvature) and w is the spot size of the beam at the mirror. For the geometry of the experiment, this gives Q's on the order of 10  8  — m u c h higher than the Q's due to ohmic losses. The area of the coupling apertures is small, and can be neglected i n many cases. How-  ever by using the boundary condition that the fields must be continuous w i t h the feeding waveguides, one can determine the amount of energy coupled into the resonant mode [8] [47].  It is likely, however, that any coupling determined theoretically w i l l disagree w i t h  the actual values, due to the cumulative effect of approximations and non-idealities i n the geometry (eg. non-circular coupling holes, finite thickness of the coupling holes, etc). A more empirical approach to the coupling w i l l be discussed i n the following chapter (cf. experimental data).  Chapter 3. Theory of the Open Resonator  46  T h e resonant frequency of the cavity, including losses, can be found by a variational formula [41]  J J J [(V x E ) + ( V • E ) ] 2  2  k —  dv-2J  | f i x E - V x EdS  7 7 7  /// ' E  •  (3.59)  E<to  T h e shift i n frequency due to the change of the penetration depth of the superconductor with frequency is an important quantity measured i n this laboratory [20] [15], however it is inaccessible to measurement i n this apparatus since the resonant frequency is not stable enough i n general.  3.5  Beyond the Beam-Wave Theory  T h e beam-wave theory, as developed i n the preceding section, depends upon the validity of the parabolic approximation  7T7 «  (3-«0)  OZ OZ which can be shown to be equivalent to saying  «  2  = ^  »  1.  (3.61)  T h i s constraint is definitely satisfied for the laser resonators for which this theory was developed and for which the resonator dimensions are m u c h greater than the wavelength of the light, however its validity is less obvious for millimeter-wave systems. T h e approxi m a t i o n is even less valid i n the experiment described here, since the objective is to have a spot size of the same order of magnitude as the wavelength of the radiation. A l t h o u g h Cullen has shown that the theory is actually more accurate than this i n terms of the resonant frequency (to approximately 0 ( o T ) [42]), the expressions for the fields are not 4  accurate to this order and a more accurate theory is desirable.  Chapter 3. Theory of the Open Resonator  47  A second problem w i t h the theory is that, as can be seen from equations 3.21 and 3.14, the surfaces of equal phase are not spherical, but have a parabolic shape.  This  difference w i l l be small near the axis, however, and can be treated w i t h perturbation techniques [42] [48]. A t h i r d remaining defect w i t h the theory lies i n the fact that it was developed using a scalar method. Thus the boundary condition that u = 0 on the mirror surfaces is not true for a l l cartesian components of e. W h a t actually is involved is ^tani  the tangential component of the electric field. This w i l l be close to a combination  of e  and e for large radius of curvature, but w i l l strictly involve e as well. A vector  x  y  z  theory, incorporating the relative intensities of the components, but satisfying the actual boundary conditions, is important for improved accuracy. A vector field theory of the fundamental mode of open resonators has been developed by C u l l e n and Y u [43]. T h e derivation of this theory w i l l be sketched here, and behaviour of the fields outlined, though most of the calculations from the preceding section w i l l not be repeated. The starting point of this theory is to consider an infinitesimal electric dipole located at the origin and directed along the x-axis. T h e vector potential of such a system is  A =  (3.62)  i ( l / r ) exp(-jkr)  where (3.63)  r = ^Jx + y + z . 2  2  2  If this dipole is shifted along the z axis a distance ZQ the change i n the field is t r i v i a l . If, however, one considers a shift of the field  —jz , 0  the interpretation is less obvious, but the  mathematics still simple (since Maxwell's equations w i l l be satisfied i n the same way). Thus r = \J x + y + {z + j z ) . 2  2  2  0  (3.64)  Chapter 3. Theory of the Open Resonator  4  48  r  s  0 . 001  0 .002  0 . 003  0 . 004  0 .005  Figure 3.4: Relative intensities of the cartesian components of the field along the y = x line using the complex point-source theory. Solid: \e \, Dashed: A^-iOole^, Dotted: &u;o|e |. T h e parameters were chosen for the geometry used i n the experiments of this thesis. y  2  If we now assume that z is large and expand equation 3.62 binomially, we find, except 0  for a constant factor  x + y 2  i  z + z  exp 0  -jkz — jk  2  (3.65)  "2(z+jz )_ 0  which is identical to equation 3.18 for m = p = 0 and the substitutions for R and w  0  [44]. T h e physical interpretation is now more clear, i n that the solution to Maxwell's equations i n the paraxial approximation, corresponds to a Gaussian beam. However, without m a k i n g the paraxial approximation, we have a solution that satisfies Maxwell's equations exactly, and that is an inherently vector solution (all six components of the fields can be found explicitly). These fields can be written down exactly[43], although they are quite complicated. Using a consistent order of approximation, 0(a~ ), Cullen 4  Chapter 3. Theory of the Open Resonator  49  and Y u give tractable equations for the fields i n an open resonator of even mode number 9-  e  y  =  too  f \\ 2  — exp - ,,2 w \ w  • (,  -  kp>\  2  . /,  ^  kp'  z  •  e,  I  o  ,  sin \kz - 3 $ +  -  Q  sin ^  - 4$ +  (3.67)  =  A n d the h field can be derived from this. A g a i n , this theory does not take spherically shaped mirrors for boundary conditions, but this can be corrected through perturbation theory, and the correction to the resonant frequency is 0 ( a ) , and can be ignored [43]. - 4  These equations can be used i n the same way that the beam-wave theory equations are used i n the development of Q, but the extra accuracy obtained is not useful i n most applications. T h i s is because the 1/Q contribution from the loading of the resonator by the waveguide circuit is likely to mask any deviation from the simple beam-wave picture. The complex-point source theory does give some idea of the relative intensities of the components of the field. It shows that the approximation that b o t h the electric and magnetic fields are transverse is only good to order 1/a (since e ~ x  ae ). z  Chapter 4  Millimeter-Wave System  4.1  Introduction  One of the important results from the preceding chapter is the expression for the spot size on the sample (equation 3.24):  w = l^D(R 2  0  To measure small samples, a small w  0  0  - D).  is required. This quantity scales inversely w i t h  the frequency, of course, so we w i l l want to work with high frequencies for good spatial resolution.  T h e highest frequency millimeter-wave source and detector pair available  i n the lab is at ~ 117 G H z , which is the frequency used i n our experiments. A s well, the decision to work w i t h a flow cryostat, and the final design of this cryostat, impose restrictions on the size of the resonator to be used. In this chapter, the design of the resonator and the millimeter-wave detector/source pair are outlined.  4.2  Millimeter-Wave System  T h e source and detector for the experiment are a modification of a 115 G H z spectrometer previously used for experiments on atomic hydrogen and deuterium [49]. T h e millimeterwave source is a klystron (Varian V R T 2 1 2 3 A 7 ) which operates nominally at 114.5 G H z . T h e klystron is actually quite old, and the lowest operable frequency has increased w i t h age. It is currently operated at about 115.6 G H z . This frequency is stabilized by locking to the 8  th  harmonic of a reference signal provided by the H P 83620A synthesizer (JREF = 50  Chapter 4. Millimeter-Wave System 386 Computer  _ _  _  T  51  48B BUS  ~1 HP 8663A Synth 0.B-1.B GHz +8 dBm  A/D Converter  10 MHz Kef.  amplitude But  PAR 5204 L o o k i n Amplifier Signal I D  200_MHz  +  D/A Converter  300.05 MHz  2DD.05 MHz  + Sweep Control  PED 619A Klystron Power Supply V„ = 2430V V„ = ~260V  Input  f  DC  Bias  Weinschel 438A  + 200.05 MHz  1-2 GHZ Microwave Systems MOS-5 "Lock-Box"  Klystron 115.80868 GHz  Pre-Amp  Sweep] Input  Mixer Diode  'EEF  *8 Harmonic Mixer  HP B349B Amplifier  HP 83620A Synth. +8.6 dBm 14.471085 GHz  AND KLY  Detector Diode  HOD  RESONATOR  KLY  MOD  Figure 4.1: Block diagram of the millimeter-wave system  Chapter 4. Millimeter-Wave System  52  14.471085 G H z for a l l work described here). This locking is accomplished by controlling the reflector voltage using a Microwave Systems M O S - 5 frequency stabilizer (or "lockbox") which keeps the klystron operating at 40 M H z from JREFIKLY = 8 * f F RE  ± 4 0 M H z = 115.80868GHz.  (4.1)  This is a stable baseline frequency from which the measurements are based, however it is desirable to have a swept frequency source for accurate measurements of Q [50]. T h e original spectrometer operated at a fixed frequency, m i x i n g the stabilized klystron signal w i t h 1480 M H z from a S S B generator, and using a heterodyne system [49]. Since there is no easy way to sweep the klystron frequency, we have used a modification of this system, the m a i n difference being that we scan the intermediate frequency, providing a l i m i t e d sweep capability. A swept frequency from the H P 8663A synthesizer (fsYN — 0.8 to 1.8 G H z ) is m i x e d w i t h both a 50 k H z source and a 200 M H z source (figure 4.1) giving a number of harmonics. In the original spectrometer, the unwanted sidebands were filtered out. In the current design, because /SYN is swept, such filtering is i m p r a c t i c a l . To obtain a clean single frequency to m i x with the klystron, a separate 1-2 G H z source (Weinschel 438A) is locked to the desired sideband. This locking is accomplished by setting the frequency of the oscillator roughly to the desired point through the sweep input of the Weinschel, and fine, fast adjustments through the F M input. A sweep control circuit has been designed so that if lock is lost, the frequency is swept rapidly until lock is regained. B y keeping the locking time shorter than the frequency step rate, erroneous measurements are avoided. This clean, variable frequency /MOD, is fed to a millimeter-wave diode which also receives a large signal from the klystron. T h e diode operates i n the non-linear region, so it mixes the two signals, sending a combination of /KLY, IKLY+IMOD  and JKLY—/MOD  to  the experimental resonator. T h e open resonator acts as a narrowband filter, so essentially  Chapter 4. Millimeter-Wave System  ^  Varian VRT2123A7 Klystron Control Data TRG Mixer  Cooling "Water  Un  To =] Modulation  To Lockbox] and Synth.  Control Data TRG Dir. Coupler  Hughes 44346H-310 Dir. Coupler  Hughes 47436H-1000 Harmonic Mixer  45° Twist  =fll=H  Tee See below for continuation  Variable Attenuator)  To ^ Demodulator|  Avantek 21-102 Pre-Amp  TRG Mixer Diode  Variable Attenuator  flHZHE*Tee  Figure 4.2: Schematic diagram of waveguide circuit  Chapter 4. Millimeter-Wave System  54  only one of these sidebands w i l l be near the resonant frequency /RES, the others w i l l be filtered  out. This signal is detected through the reverse of the input heterodyne system,  finally  giving a 50 k H z signal which is detected w i t h a P A R 5204 two-phase lock-in  amplifier. This complicated heterodyne system is used because the signal to noise ratio is quite low, and coherent phase sensitive detection must be used. A s the experiment currently stands, the phase information is not used i n determining the Q of the resonator, however the modifications to do so would certainly be possible. It turned out to be difficult to convert the fixed frequency S S B spectrometer to a swept source, but the final configuration yields an adequately level signal over the range of 116.8 G H z to 117.6 G H z (see the following chapter). This frequency is obtained by taking the positive polarity for a l l m i x i n g frequencies, the mode used for a l l measurements described here.  4.3  Resonator Design  Once the frequency of the source and the size of the experimental chamber have been set, it remains to determine the actual design of the resonator. T h e criteria are essentially as follows: 1. spot size wo as small as possible (limited by number 3 below). 2. diffraction minimized: keep the diameter of the mirror a > 2wi, the spot size on the curved mirror. 3. keep the assumption of k w^ ^> 1 satisfied. 2  To satisfy these constraints we would like to operate i n the regime where D ~ R — the hemispherical geometry. Other researchers using open resonators for Rs measurements are split on which geometry to use: either half-confocal or hemispherical [26] [27].  Chapter 4. Millimeter-Wave System  55  Vertical Positioning Screw  Quartz Glass Tube LJ7 Copper -jr4^-----'Mounting Tj Block  II I II  Sample Stainless ^ Steel Plates  /  Mirror Surface  II I  Y  Heat Exchanger  Waveguide Trough  Figure 4.3: T h e design of the resonator and sample positioning apparatus. for a photograph.  See also 2.2  For large samples or very high frequency, the semi-confocal setup is preferable, since diffraction losses are m i n i m i z e d . In this experiment, the total size of the resonator, and the size of the sample make the hemispherical resonator the ideal choice. The mirror to sample separation i n the apparatus, D, must be variable for two reasons. Firstly, samples are attached to a support facing the curved mirror (see figure 4.3 ), so any variation of substrate or sample thickness w i l l change the effective resonator dimensions, and thus its electromagnetic characteristics. In moving this block vertically D can be kept constant. Secondly, because of the l i m i t e d frequency sweep range, and the fact that theoretical calculations of the resonant frequency are approximate, the mirror separation must be variable to ensure that the /RES lies w i t h i n this accessible range. Additionally, as the cryostat cools, the mirror separation distance changes due to thermal contraction which changes /RES- Thus D must be variable at low temperature.  Chapter 4. Millimeter-Wave System  56  This is accomplished by having the sample-mounting block attached to two parallel flexible  stainless steel plates which allow vertical travel. T h e height is set by pushing  down from above w i t h a finely threaded screw. A retractable rod engages this screw so that there is no heat conduction when disengaged (figure 4.3). T h e sample mounting block is formed of copper (about 1 c m ) and is cooled v i a 3  conduction down the stainless steel plates. To this block is attached a thermometer and heater, to control the temperature of the sample. In early versions of the apparatus, the block was solid copper, but temperature changes were found to cause changes i n D due to thermal expansion and thus /RES- B y inserting a quartz glass tube i n the m a i n part of the block (see figure 4.3), problems w i t h thermal expansion were reduced to a more acceptable level. T h e actual frequency drift w i t h temperature w i l l be discussed i n the following chapter, but is now stable to w i t h i n 5 M H z over the course of an experiment. T h e resonator as fabricated has radius of curvature of R = 2.08 c m and radius of the m i r r o r aperture of a = 1.91 c m (see figure 2.2). T h e theoretical performance of a resonator w i t h these characteristics is shown i n the accompanying figures. These calculations neglect all losses other than the ohmic resistance of the sample and mirror.  4.4  Coupling  T h e microwave energy used for the experiment must be coupled both into and out of the resonator, i n order that measurements be made. We have decided to work w i t h a transmission mode cavity (in which energy is coupled i n at one spot, and out another) since this configuration is better suited to situations where the loss has a wide variation with temperature and sample. Coupling is accomplished by connecting waveguides to small apertures near the axis of the resonator on the curved mirror. Because there must be two holes, slightly off center, the transmission method does have some disadvantages  Chapter 4. Millimeter-Wave System  1.40  1.60  1.80 Mirror Separation(cm)  57  2.00  2.20  Figure 4.4: M o d e chart of the resonator as built. These modes depend on the beam-wave theory and are approximate. A s well losses are ignored. Hence, this chart is only a guide as to the behaviour of /RES w i t h mode number and D. compared to reflection methods.  These two imperfections i n the curved mirror cause  scattering and lower the Q of the resonator. It is difficult to determine a priori what size of holes are needed to give adequate signal strength.  There are theoretical and phenomenological models [47] [8], however  the assumptions made often negate their utility i n this problem. In this apparatus, the size of the coupling holes was determined empirically, by starting w i t h small holes and increasing their size u n t i l an adequate signal strength was observed. T h e final size of the holes is ~ 0.025" diameter, which is not negligible i n comparison to the wavelength of radiation, and thus w i l l lower the Q of the resonator.  Chapter 4. Millimeter-Wave System  58  160000 \-  150000 -  o 140000 -  130000 \-  I—i  0.016  1 1 1——•-—i  0.017  1 1 . 1 1 1 . . i i  0.018  0.019  . i i i •  0.02  Mirror Separation (m) Figure 4.5: N u m e r i c a l simulation of the resonator Q for copper at 77 K as a function of the m i r r o r separation i n the complex-point-source model (cf. Chapter 3). T h e thick line corresponds to the mode TEM0013 d the t h i n line to T E M i 4 . T h e data on this graph becomes increasingly inaccurate as D approaches R = 2.08cm, due to neglect of diffraction losses which dominate i n this regime. a  n  0 0  A related matter concerns the problem of bringing the microwave power to the resonator without a large heat load. T h i s is accomplished by having low thermal conductivity stainless steel waveguides, as well as non-contacting choke flanges. These flanges have been used i n microwave work for years for other reasons ([51] and other volumes of the M.I.T. Radiation Laboratory series], and also have been exploited i n low temperature work [52]. B y having a A / 4 shorted stub around the waveguide, the gap between waveguide sections is m i n i m i z e d as a discontinuity for the fields. R a d i a t i o n from the joint, as well as reflection down the waveguides are m i n i m i z e d . In the experiment, these flanges are used on both the input and output waveguides. T h e i r performance w i l l be discussed in the following chapter.  Chapter 4. Millimeter-Wave System  59  300000 250000 200000 • 150000 • 100000 • 50000 -  "  I i 0.01  ,  ,  , i , 0.05  • • • •  0.1  ,  .  ,  i  0.5  , i  , .i 1  •1 5  Sample Surface Resistance Figure 4.6: N u m e r i c a l simulation of the resonator Q at 117 G H z and 7 7 K , as a function of Rs of the sample i n the complex-point-source model. T h i s corresponds to a mirror separation of 1.986 c m .  4.5  Microwave measurements  Following the derivations of the previous chapter, the basic relation i n the measurement of a microwave resonator is: -!- oc ^2 All Losses.  (4.2)  W h e n measuring the Q of a resonator, one must have some method of coupling energy into and  out of the resonator, w i t h a circuit including source and detector. This contributes  to the losses for the resonator. Near a particular resonant mode of the resonator, it is possible to represent the energy stored i n the fields of the resonator by the energy stored i n the l u m p e d parameters of an equivalent circuit [53]. T h e equivalent circuit of a resonator can be described by three parameters:  Chapter 4. Millimeter-Wave System  60  Qo =  0  (4.4)  R  (4.5)  = LOQLQO  Ro  where LO is the resonant frequency, Q 0  uL  is the unloaded quality factor of the resonator  0  (called simply Q u n t i l this point) and R is the shunt resistance of the resonator . W h e n 0  joined to the rest of the system through coupling, the equivalent circuit can be represented as shown i n figure 4.7 [50].  Figure 4.7: Lumped-element equivalent circuit of a resonator connected between a generator and a detector. T h e method i n which the coupling is represented can be shown to have no effect on the final result, and is shown here as two ideal transformers. T h e loaded Q of the system, Qi, can be found by examining this circuit, giving: LO L 0  QL  =  (4.6)  R + nlZr + njZ  2  where Z\ and Z are the characteristic impedances of the input and output transmission 2  lines respectively [50]. Defining the coupling constants 2 Z\  (4.7)  and similarly for 8 , the relation between the unloaded Q and the loaded Q of the system 2  can be written  QL  QO  Chapter 4. Millimeter-Wave System  61  Qo  <3o  = i + i +i Qo  Ql  (4.8)  0,2  where Q i and Q2 are the coupling Q's. These two terms are simply two additional terms to equation 4.2. Thus the measured Q of a resonator w i l l be lower t h a n the Q if it were isolated from the circuit. To make good measurements of the Q of the resonator, it is desirable to work w i t h very weak coupling (small /3's). There is no way of modifying the coupling of the system as it is currently constructed, and hence no method of determining Q on its 0  own. T h e loss due to coupling, as well as due to all factors except the sample, should be constant between experiments and during an experiment at different temperatures. Thus the relation between the Q of the cavity and the surface resistance of the sample can be written, using equations 4.2 and 3.55 as: = Rs Q  , — S a m p l e  L  ff  \h\ dS + constant x (Other Losses).  (4.9)  2  Up J JSample*  '  V  J  \  J  In this equation the relationship between the surface resistance and the measured quantity of the Q of the resonator is clearly noted. T h e m a i n idea of the experiment is that this equation can be rewritten simply as (writing Q instead of QL again)  ^ = <*Rs.  anpU  +P  (4.10)  where a and 8 are two constants, independent of the sample. T h o u g h b o t h of these constants are related to values from the theory (in the case of a the theoretical value is shown above), the approximations made i n their derivation would make the final result suspect. In the following chapter the values of these constants are found experimentally.  Chapter 5  Experimental Method  5.1  Introduction  In this chapter, I will describe the techniques used in taking, calibrating and analyzing the surface resistance data as measured by the apparatus previously described. Measurement of the surface resistance of normal metals will be described here, although the techniques would be identical for superconducting thin films. The basic formula used in analyzing the data is the relation Q =  l aR  s  + /3  where Q, the resonator quality factor, is the measured quantity, and Rs, the surface resistance of the sample, is the desired quantity. This equation comes from the theoretical discussion in chapter 3, as well as the section on microwave measurements in chapter 4. The losses of the resonator each contribute independently to l/Q, allowing a simplification of the form of equation 5.1. The parameter a relates the fraction of the loss of the resonator due to the surface resistance of the sample. The parameter 8 encompasses all other losses in the resonator (surface resistance of the curved mirror, scattering losses, etc.). It is hoped that the product aRs is not small in comparison with 8 for accurate measurement. Although it is possible to make estimates for the values of a and 8 from theory (chapters 3 and 4), such values are only approximations, and ignore such losses as the finite size of the mirrors and the effect of coupling holes. Therefore, the parameters will 62  Chapter 5. Experimental Method  63  be determined empirically by measuring the surface resistance of known samples.  5.2  M e t a l l i c Samples  Since the surface resistance of high Tc superconductors at 117 G H z and ~ 77K is not much different from that of metals (such as copper) at similar temperatures, it is convenient to use good metals as references for calibration of the resonator. It is, however, important to realize that the simple relation for the surface resistance of a conventional metal given in the introduction (equation 1.19) does not necessarily apply for a good metal at cryogenic temperatures and high frequencies. This is due to the  anomalous  skin effect, where the mean free path in the material is comparable to the classical skin depth. Hence conduction is no longer a local phenomenon, and more complicated effects are introduced [54]. The criterion for these effects to be important is given by the product COT, the frequency of the radiation times the scattering time for electrons in the metal. When LOT <C 1, then the standard formula applies, Rs = —-. In Dingle's two papers [55] [56], (TO  the quantitative theory of Reuter and Sondheimer [54] is evaluated numerically, and the results for various metals are tabulated. The surface resistance for copper as a function of temperature at 117 GHz is shown in figure 5.2. The surface resistance of brass is used as a calibration as well. Here, the literature is not very much help when finding the resistivity, since there are many different alloys, all given the name brass. The d.c. resistivity of a small sliver of the same material used in the open resonator experiment was measured directly [58]. From this the surface resistance was calculated as Rs = 7. Both the copper and brass samples used for surface resistance 0  measurements were 1 x 1cm . They were mechanically polished to ~ 5//m grit, then 2  chemically polished (using a solution of 1/3 each of nitric acid, orthophosphoric acid and  Chapter 5. Experimental Method 0.10 i  0.00  64  1  i  0  50  100  150  200  •  •  i  250  • I  300  Temperature (K) Figure 5.1: Surface resistance of copper at 117 G H z calculated from d.c. resistivity data from the literature [57], using the standard skin depth formula (squares); and using a numerical solution of the full integral equations accounting for the anomalous skin effect [56]. glacial acetic acid at 70 d e g C [59]) to remove damage from the mechanical polishing. The brass surface became noticeably shinier after the chemical polish, though results were only mediocre w i t h copper, despite repeated trials w i t h various methods (including electrochemical polishing [59] [60]).  5.3  Measurement Procedure  The general procedure followed for a l l measurements w i l l be described here. T h e procedure was the same whether the sample was superconducting or metallic. The sample was attached to the underside of the mounting block of the sample gantry i n the cryostat (see figure 4.3) w i t h a small amount of vacuum grease. T h e cryostat was p u m p e d to ~ 1 0  - 6  torr and mounted onto a liquid nitrogen storage dewar. T h e cryostat  Chapter 5. Experimental Method  65  was cooled initially by opening the needle valve and drawing l i q u i d nitrogen through the heat exchangers rapidly by p u m p i n g w i t h a large mechanical p u m p . After about 45 minutes, when the cryostat had reached approximately 77 K , the flow of the nitrogen was reduced by closing the needle valve and regulating the pressure of the escaping nitrogen gas.  This controlled the temperature of the upper exchanger (and hence the curved  mirror of the resonator) to w i t h i n 69-69.6 K (for a l l measurements used here). Having three independent controls on the temperature (pressure of the exit gas, throttle valve outside the cryostat, and needle valve inside the cryostat) meant that there were more controls than actually needed, and the same temperature could be reached for various flow rates. In most runs, the m i n i m u m flow rate was used. The temperature of the sample was measured and controlled using a resistance bridge (made by the U . B . C . physics electronics shop) connected to a Lakeshore C E R N O X resistance temperature sensor. T h e thermometer, and another resistor used as a heater, were attached to the sides of the sample mounting block. T h e temperature gradient between this block and the rest of the experiment was maintained across the two stainless steel plates supporting the block. There were no noticeable temperature gradients between the sample and the block, which was verified by mounting a diode on the sample block i n the same manner as a sample. T h e temperature as measured by the diode corresponded closely to the temperature of the Lakeshore thermometer. Once the apparatus had cooled, the frequency of the source was set to a constant value (normally JMOD ~ 1200 M H z ) and the vertical position of the sample was varied u n t i l a resonant mode was found (by noting both a rapid change i n phase and an increase i n the amplitude of the received signal by the lock-in amplifier). It was difficult to determine i n advance what setting of the sample-mirror separation would give a resonant frequency w i t h i n the b a n d w i d t h of the apparatus.  This is due to the samples having varying  thicknesses, and due to the thermal contraction of the gantry. Once a resonant mode was  Chapter 5. Experimental Method  66  found, the choke flanges were adjusted to give the best signal. In the current apparatus, it is difficult to do this prior to cooling, since the transmitted signal for a high temperature superconductor sample at room temperature was undetectable (the Q was so low). T h e procedure was repeated, scanning for other modes of the cavity, once the choke flanges have been adjusted to give good transmission. The here.  same mode of the resonator has been used for a l l the experiments described It is identified by various methods: its "distance" from other modes (both i n  frequency and m i r r o r separation), its peak signal strength (largest of any mode observed), and  the behaviour w i t h respect to changing the frequency components m a k i n g up the  feeding signal (since I use the sideband where /RES = 8 X fsYN + JMOD, then increasing fsYN should lower /MOD at resonance, and vice versa). It is, nevertheless, difficult to make a precise identification w i t h the modes shown i n figure 4.4 for two reasons: firstly, because the measurement of the separation between the sample and the curved mirror is relatively imprecise; secondly, the equations defining the resonant frequency do not take into account such perturbations as the finite size of the surfaces, the effect of the coupling holes or ohmic losses of the mirrors on the resonant frequency. T h e result is that one usually doesn't know the exact resonator mode. Using the resonator at room temperature and probing the field with a point dielectric perturbation (a small drop of epoxy on a monofilament thread), we were able to show that there were no nodes i n the transverse field profile for the mode used. F r o m the evidence, we judge that the resonatoris most likely operating i n the T E M o i 4 or T E M 0 0 1 5 mode. A more conclusive study 0  of the mode pattern could be accomplished by scanning the resonator across the surface of an object w i t h a clearly defined transition i n surface resistance, such as a bi-metallic (brass / copper) surface. This was attempted early on, but the results were inconclusive due to l i m i t e d resolution (the resonator was at room temperature, and the experimental techniques not yet refined).  Chapter 5. Experimental Method  1.0 I —  — i — ' — i — • — i —  1  0.8 -  O  0.6 "  /  /  Cu  /  0  — i —  1  —  /  Q4 _ / /  j  — i —  1  ^  3  0.2  67  — i —  1  ~  — i —  1  — i —  ~  \.  / /  1  1  — i  1  —  — N  J  -  With Standard Flanges With Choke Flanges  /  o ' '  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  '116.9 117.0 117.1 117.2 117.3 117.4 117.5 117.6 117.7 117.8 117.9  Frequency (GHz) Figure 5.2: Comparison of the detected signal amplitude for direct transmission (it. no resonator) for standard flanges and choke flanges. A comparison of the signal obtained w i t h no resonator or choke flanges (direct waveguide connection) against the signal obtained w i t h no resonator but w i t h choke flanges is shown i n figure 5.3. A s can be seen, the transmission amplitude is quite flat, w i t h only small ripples as a function of frequency. T h e choke flange transmission is considerably poorer, and is very sensitive to misalignment. T h i s is something that must be remedied in future versions of the apparatus. Once the proper mode of the resonator was found and the flanges aligned, the Q of the resonator is found as a function of sample temperature.  T h e vertical adjustment  is set so that, at the lowest temperature (~ 70 K ) , the resonant frequency /MOD is 1210 to 1215 M H z . T h e same approximate frequency was used to avoid any sytematic errors which may occur when operating at different regions w i t h i n the b a n d w i d t h of the  Chapter 5. Experimental Method  68  millimeter-wave system. T h e source frequency is swept over the resonant frequency under computer control in 50 to 100 steps w i t h the amplitude signal from the lock-in amplifier being recorded at each point. It was found that the sweep time (how long the computer paused w i t h each data point) affected the value received for the Q. Fast sweeps gave low Q's, which increased as the sweep t i m e lengthened.  T h e experiment was operated i n the regime  where the Q's received were asymptotically close to the l i m i t i n g value. T h i s effect was due to the slow response time of the lock-in amplifier. T h o u g h a slow scan of frequency as performed here w i l l be more susceptible to drift i n the apparatus (due to changing temperature etc.), this effect was not noticeable w i t h i n one scan of the frequency when the nitrogen pressure was regulated (though there is a significant drift i n the frequency over the course of an experiment - see later). T h e data was collected on an Intel 286 computer (as described i n chapter 4) and transferred to a Sun Sparc workstation for analysis. T h e Q of the resonator was found by fitting the amplitude vs. frequency data to an appropriate function through a least-squares method. If we were measuring the power transmitted through the resonator, we would expect a Lorentzian relation between power and frequency about the resonant frequency. In this experiment, however, one is measuring the amplitude of the microwave signals which are proportional to -v/Power. There was a significant background noise associated w i t h the signal; including random fluctuations, as well as directly transmitted signal (leakage through isolators, direct transmission between coupling holes i n the cavity, and/or leakage from lower frequency portions of the circuit). In addition, the gain of the source/detection circuit was not necessarily uniform over the frequency range. Thus the amplitude vs. frequency was fitted to a function of  Chapter 5. Experimental Method 40000  69 -i—I—I—r-  39000 4F- * O  o  =F  o Copper  38000 37000 36000  J  i_i  .  •  i  30000  29000  28000 h  27000  80  85 90 95 Temperature (K)  100  105  110  Figure 5.3: Q vs. temperature for brass and copper. T h e lines j o i n the points i n the order the data was taken. the form:  V  A + Bv  'i +  (^y  + D + Ev.  (5.2)  T h i s was intended to account for both the background (linear) as well as variation i n the gain source/detector pair w i t h frequency (linear as well). The fitted results for the resonant frequency and Q of the resonator w i t h copper and brass as a function of temperature are shown i n figures 5.3 and 5.4. In both cases, the lines join points taken i n chronological order. Thus one can follow the drift i n frequency over t i m e i n figure 5.4, although there is a m u c h smaller corresponding drift i n Q. T h e error bars shown are from the least-squares fitting, assuming equal error for each data  Chapter 5. Experimental Method  70  G—©Copper  117.224  • — • Brass  N  g  117.222  o fl cr  &  117.220  fl c3 fl O  *  117.218  117.216  70  75  80  85 90 95 Temperature (K)  100  105  110  Figure 5.4: Resonant frequency as a function of temperature for brass and copper. A g a i n , lines j o i n points chronologically, and show clearly the drift i n frequency w i t h t i m e of about 5 M H z . Error bars from the fits are smaller than the symbols on this graph. point [36]. Various functional forms were compared i n the fitting b o t h for a single resonant peak, and also over a full experiment (see figures 5.5, 5.6, 5.7 and table 5.1). T h e final function chosen has the lowest y  2  i n most cases. Perhaps even more importantly, the Q displays  little change w i t h frequency (drifting over time), unlike the other functions (see figure 5.7).  T h i s is due to the asymmetric gain i n the fitting function corresponding more  closely to the actual behaviour of the experiment. Clearly, it would be better to have a flat gain over the b a n d w i d t h of the system, but modeling the asymmetry of the gain is an i n t e r i m solution. T h e earlier version of the experiment where thermal expansion effects were rather large produced results of dubious validity.  Chapter 5. Experimental Method  71  0.5  o Measured Data — Const. Background | — Linear Bkgd. j — Quadratic Bkgd. j Lin. Gain + Const Bkgd. f — Lin. Gain * (Sqrt(Lorentzian) + Const. Bkid.) Lin. Gain + Lin. Bkgd. p — Lin. Gain * (Sqrt(Lorentzian) + Lin. Bkgd.)  Q0 I i 1.1699e+ll  i  i  i  I  i  i  I  1.1701e+ll 1.1703e+ll Frequency (Hz)  i  i  I 1.1705e+ll i  Figure 5.5: Comparison of the various fitting functions w i t h the measured data. T h e data in question is for copper at 70 K and represents a "good" peak — where the amplitude and Q are both quite high. T h e functions are described in more detail i n table 5.1. T h e data i n figure 5.3 are fit to the equation 1  1  T)  7j  tyBrass  ^SCopper  =  a  i S  - RScopper)  R  B r a s s  (5-3)  to get the proportionality constant a. T h e result is then inserted into equation 5.1 to find the sample-independent loss 3. T h e result of the fit is shown i n figure 5.8.  The  values of the fitting constants received are a = (7.45 ± 0 . 3 2 ) x 1 0  - 5  fT  1  (5.4)  and, 6 = (2.334 ± 0.024) x 1 0 ~ . 5  (5.5)  Chapter 5.  Experimental Method  72 O Const. Background • Linear Bkgd.  -0.010  1  1  1.17e+ll  1  1  1.1701e+ll  ^ 1.1702e+ll 1  1 • 1.1703e+ll  1 1 1.1704e+ll  Frequency (Hz) Figure 5.6: Residuals of the fitted peak for the various functions used. T h e curve used for data analysis is shown w i t h the points joined. T h e uncertainty i n these values includes both the uncertainty i n the measurement of Q for each point, and the uncertainty i n fitting the two experiments to the d. c. data. N o systematic error is included i n the uncertainties.  Chapter 5. Experimental Method  Function 1  +  [  V  -  U  0  y 2  /  V  2  i+^-^) /r 2  A  ,  2  / P 2  +  B  +  +  Dv  2  p  -°  2  0  (,-twr +-g + ^ ) 2  1 +  p  Cv ,  2  Cv  + B + Cv  2  A+Vv  1  + 1  i+(^ ) /r ( +^)(  ^0  V B  A  73  x  2  Q  117.0231090 * 10  9  0.0003902  38481 ± 194  117.0229941 * 10  9  0.003456  37759 ± 5 6 0  117.0229815 * 10  9  0.002665  37759 ± 499  117.0229815 * 10  9  0.002665  37999 ± 496  117.0229809 * 10  9  0.003364  37792 ± 554  117.0229941 * 10  9  0.003456  37758 ± 554  117.0229941 * 1 0  9  0.003456  37759 ± 564  Table 5.1: Comparison of (some of) the fitting parameters for various functions, as shown in the accompanying figures. This is for a particular scan of copper at 71 K . T h e first line of the table shows the function used i n all subsequent work.  Chapter 5. Experimental Method  74  42000 40000  38000 36000  42000 40000 38000  42000  36000  40000 38000 36000  42000 -3 40000 38000  42000  36000  40000 38000 36000  42000 40000 38000  42000  36000  40000 38000 36000  75  80  85 90 95 Temperature (K)  100  105  110  F i gure 5.7: Comparison of Q vs. temperature for copper using various fitting functions, w i t h the points connected i n the order the data was measured. T h e functional form used is i n the top graph, the other graphs use the functions i n the same order as table 5.1. Note how the uncertainty is m u c h smaller, and the drift w i t h time is reduced i n the first graph.  Chapter 5. Experimental Method  75  0.20 -1—1—I—I-  0.15  a  OBrassfrommeasurement OCopperfrommeasurement — CopperfromLiterature - Brass fromD.C. meas.  a = (7.45+.32)10 O.  0.10  P = (2.334±.024)10'  s  Pi  0.05  0.00  70  75  80  85  90  95  100  105  110  Temperature (K) Figure 5.8: T h e cavity perturbation data measurements of R of copper a n d brass at 117 G H z fitted to values obtained from the literature and d.c. measurements. T h e fitted values of the constants a and 8 as determined from this data are shown i n the figure, and their uncertainty has been added to the error bars as shown. s  Chapter 6 Data  6.1  Introduction  In this chapter the measurements of surface resistance performed to date using the open resonator w i l l be presented. These samples were available to our lab i n November and December 1994, and are actually a good survey of the capabilities of the apparatus. There are, however, no "state of the art" t h i n films yet available i n the group, which might push the limits of the resolution of the apparatus.  In fact the problem is the opposite: the  surface resistances of the films are high enough that the Q of the resonator is quite low, and the transmission through the resonator very small. Measurements above the transition temperature are lacking for two reasons: first of all the loss of a high temperature superconducting thin film i n the n o r m a l state is quite large, which reduces the Q of the resonator, and the transmitted signal becomes too small to measure. A related problem, of the enhancement of loss when the skin depth is large, w i l l be discussed later i n the chapter. T h e second reason is that the sample thermometer had been calibrated incorrectly, and when the sample was thought to be above To, it was i n fact 4 K lower. T h i s problem has since been corrected, and the existing data re-analysed, however no new data has been taken.  76  Chapter 6.  Data  77  80  85  90  Temperature (K) F i gure 6.1: T h e temperature dependence of the surface resistance of a 6 /ITTI thick B S C C O thick film on an M g O substrate. T h e surface resistance of copper and brass are included for comparison.  6.2 6.2.1  Data Furukawa Electric B i S r C a i C u 2 0 8 Thick F i l m 2  2  Figure6.1 shows the surface resistance vs. temperature of a thick film of B i 2 S r 2 C a i C u 2 0 s . The film is deposited through a silk-screen-like method on an M g O substrate by the Furukawa Electric Company. Unlike the other films studied here, this film is not of Y B C O , but of B S C C O , a material which hasn't been studied as extensively in the microwave/millimeter-wave region. T h e film is also quite thick, at 6 fim, and has visible patterns on the surface, suggesting multiple grains as well as roughness. To avoid measuring this very rough layer, which would scatter the radiation and lower the resolution,  Chapter 6. Data  78  the sample was placed w i t h the film side towards the mounting block (ie. away from the curved mirror). A s a consequence the fields must penetrate through the substrate. This change w i l l have a number of effects: • the dielectric constant of the M g O (e = 16 at 1 0 G H z [6]) w i l l cause the effective r  size of the resonator to be larger, possibly shifting modes. • the dielectric loss of the substrate (tan S = 6.2 x 10~ [6]) w i l l lower the Q of the 6  resonator. • reflection w i l l occur from the substrate-air interface as well as from the substratefilm interface. T h e effective resonator length for the substrate-air surface w i l l not be resonant, and so w i l l scatter the energy and lower the measured Q. • any change of these parameters w i t h temperature w i l l give a contribution to the Q which w i l l be indistinguishable from properties of the superconductor itself. To avoid, or at least account for, these problems the substrate should be measured with a known surface below (such as copper). This has not yet been done. The surface resistance as measured is very high. This may be due to the substrate, but it is very likely due to the properties of the film itself. To have low surface resistance, the crystallinity of the superconductor needs to be very high. T h i s measurement is a non-local probe of the material, so a film which shows zero d.c. resistivity may only have small connected superconducting sections, and show high surface resistance. T h e appearance of the film, and the manner i n which it was fabricated, suggest problems of this sort.  Chapter 6. Data  79  0.5  -3TT  0.4 0.3  a oi  0.2 h 0.1 Las® a  s  0  0.0 70  80  75  o U.B.C. Y B E L , C U — Brass - - Copper  3  CV  0 ± _»_<L » »-t_t- - t-*70 75  5  on LaA10  3  -I- -  t  80  -r- ~  85  90  Temperature (K) Figure 6.2: The temperature dependence of the surface resistance of a thin film of YBa2Cu 07_5 on an LaA103 substrate. 3  6.2.2  U . B . C . Y B a C u 0 film on L a A 1 0 2  3  7  3  This YBa2Cu307_5 film was grown by Andre Wong in the U . B . C . physics department using pulsed laser ablation ablation [61] (film AW1107). This technique produces highly epitaxial films on a number of substrates, in this case on LaA103. There are a large number of variables and procedures which must be optimized to give this good epitaxy and crystallinity of the film. The films measured here which were grown at U . B . C . have not reached optimal properties, and show a Tc of approximately 85-88 K . This film shows a rapid decrease in the surface resistance with temperature, reaching 0.1 ri surface resistance at 71 K . This is the lowest surface resistance of the four films measured here, though it is still about twice the loss of copper at the same temperature. As mentioned before, this film was not measured over a wide enough temperature range  Chapter 6.  Data  80  to find the transition temperature — though the resolution of the apparatus was very poor at ~ 8 6 K where the measurements stopped. It is encouraging that the surface resistance was still decreasing when the lowest temperature was reached, but its high value and the low transition temperature suggest further work must be done to improve the film growth procedure. The thickness of this film has not been measured, but the growth time and conditions used give a value of about 3500A. This thickness is comparable to the penetration depth of the superconductor in the temperature range of the experiment. The derivation of the surface resistance in chapter 1 assumed that the thickness superconductor was much greater than the penetration depth, and thus is not appropriate in this case. The effective surface resistance of such thin films will be discussed in section 6.3.  6.2.3  U . B . C . Y B a C u 0 film on S r T i 0 2  3  7  3  This sample is another thin film of YBa2Cu307_§ (figure 6.3) grown by Andre Wong at U . B . C . using pulsed laser ablation. This sample, however, was grown on a substrate of S r T i O s , which probably accounts for the unusual behaviour observed for the temperature dependence of the surface resistance. Strontium Titanate (SrTiOs), is a ferroelectric material, which exhibits change in the relative permittivity between 300 at room temperature and 20000 at 4.2 K [62]. If any radiation is transmitted through the film then resonances can be established between the film and the copper mounting block. This will be discussed in section 6.3. It is, therefore, difficult to estimate the actual surface resistance of this sample. For this reason superconducting films on S r T i 0 are not often used for high frequency applications. This 3  is unfortunate, since S r T i 0  3  is a substrate with very good lattice match to Y B C O , and  does not exhibit twinning (unlike LaA103)[6].  Chapter 6.  Data  81  -1  1—  A U.B.C. YBa Cu 0 on SrTi0 — Brass - - Copper 2  3  75  3  g  11  o70  75  80 Temperature (K)  85  Figure 6.3: T h e temperature dependence of the surface resistance of a t h i n film of YBa CU 07_5 an SrTi03 substrate. o  2  6.2.4  n  3  McMaster Y B a C u 0 _ 10000A film on L a A 1 0 2  3  7  5  3  This is another film of YBa2Cus07_5 grown by pulsed laser ablation on LaAlC>3 (see figure 6.4). This film was grown at M c M a s t e r University through a process which has been well optimized to produce films of high epitaxy and crystallinity. This film, however was quite thick (1 /«m), and there were problems associated w i t h this. This film shows a sharp drop i n Rs below Tc, but levels out before 8 0 K at a level of about 0.15 ft. The growth of thicker films is currently the subject of substantial research, since thicker films are able to carry a larger amount of current, useful for high power applications. A l l substrates currently i n use have a lattice parameter that is slightly different from that of Y B a C u 3 0 7 _ 5 . A s the film thickness is increased, the strain that is induced 2  by this lattice m i s m a t c h causes dislocations, grain boundaries and other defects i n the  Chapter 6. Data  82  0.5  0.4 0.3 0.2  a  y1  s  2 *  .0.1 0.0  70  75  80  85 1  V McMaster YB a^Cu^ on LaA10 — Brass - - Copper _m 3  5 0  70  75  5  -  80  |* * 85  90  95  Temperature (K) Figure 6.4: T h e temperature dependence of the surface resistance of a 10000A thick film of YBa CU307_i grown at M c M a s t e r University on an S r T i O s substrate. 2  film. These defects tend to increase the surface resistance. Work is progressing both i n the development of new substrates w i t h a reduced lattice m i s m a t c h , as well as i n the techniques needed to grow films of a thickness of about 1 fxm. The surface resistance of this film was measured by D r . Doug B o n n at U . B . C . as well, using a 3.7 G H z resonant cavity method [14]. Unfortunately, the surface resistance showed a strong dependence on the power of the excitation, suggesting weak links i n the sample and precluding further measurements. T h e presence of weak links is consistent with current problems in growing thick films, as described above. Unfortunately, the open resonator described here does not have the capability of examining the power dependence of the surface resistance, i n order to determine if such weak links exist. Another film grown at M c M a s t e r University by the same process was measured by  Chapter 6. Data  83  2.0  -1  1  1  r—  1.5 I  f 1  1.0  o c CT (D  1  V-l  0.5 D3  0.0  70  80  90  100  Temperature (K) Figure 6.5: T h e temperature dependence of the surface resistance of two M c M a s t e r films (solid squares: 3000A thick, open circles: 10000A thick). T h e thi nner film was measured at 3.7 G H z and the thicker at 117 G H z using the open resonator. T h e data are scaled by the frequency squared, according to the two-fluid model. This scaling is not appropriate for the n o r m a l state, which accounts for the discrepancy near TcD r . B o n n . This film had a thickness of 3000 A, and showed a very low surface resistance (approximately 1 0  -4  Cl below 70K at 3710 M H z ) . A comparison of this film w i t h the  data i n figure 6.4 is shown i n figure 6.5. T h e data are scaled by the frequency squared, to account for the dependence of R$ on frequency.  T h e data are quite comparable,  w i t h the (scaled) Rs of the t h i n film only slightly lower than that of the 10000A film. T h e frequency scaling may actually be weaker than u; , since the two-fluid model used 2  to derive it is only approximate. Further study is needed to determine the frequency response of b o t h films and crystals. The M c M a s t e r film measured by the open resonator, does however show a clear Tc,  Chapter 6.  Data  84  6  O Furokawa BiSrCaCuO on MgO — Brass - - Copper OU.B.C. YBa Cu 0 . on LaA10 V McMaster Y B a j C ^ O ^ on LaA10 2  3  7  5  3  3  g Pi  o 70  75  v  80 85 Temperature (K)  90  95  Figure 6.6: T h e surface resistance of the superconducting films described earlier shown on the same plot for comparison. T h e surface resistance of the film grown on S r T i O s is not included, both for clarity and because of the systematic substrate problem. unlike the others. This is most likely due to the fact that the film is thick, thicker i n fact than the skin depth of the material i n the n o r m a l state. This allows measurements above the transition temperature to be made w i t h acceptable accuracy. In fact, it was the i n i t i a l measurement of a transition at 9 6 K that alerted us to the mis-calibration of the sample thermometer.  6.2.5  Comparison  Chapter 6.  Data  85  As mentioned earlier, the l i m i t e d data obtained so far cannot be considered to be the basis of an exhaustive study. Trends i n the data can be seen quite clearly, however: • T h e surface resistance of the thick B S C C O film is very high, and this film is not suitable for microwave work. • T h e films grown on LaA103 show the best microwave performance. • T h e effect of the changing dielectric constant of S r T i 0 makes its use as a substrate 3  for high-frequency superconducting circuits l i m i t e d . • T h e surface resistance of the M c M a s t e r film is lowest at high temperature, but reaches a l i m i t i n g value that is quite high (.15 fl) at about 75 K . • T h e surface resistance of the U . B . C . film grown on L a A 1 0  3  drops rapidly around  85K, and continues to drop below the temperature range accessible i n the measurement. The continuing decrease i n the surface resistance of the U . B . C . film on L a A 1 0 sug3  gests that there are less defects present than i n the M c M a s t e r film, though its low Tc may  be due to reduced oxygen concentration.  It is for measuring properties such as  these that this apparatus has been designed, and a systematic study would be fruitful. The work done by many researchers i n improving the crystallinity of thin films i n order to reduce the surface resistance contrasts w i t h the studies done i n this lab suggesting a certain level of defects reduce the surface resistance [63]. T h e cause of this discrepancy, that the lowest Rs is obtained i n films by striving to remove all defects, while i n crystals by adding defects, is currently unknown. It is possibly due to the nature of the defects; that dislocations and twins increase the surface resistance, while substitutional disorder can lower it.  Chapter 6. Data  86  Included here for comparison is a graph showing the surface resistance of t h i n films of Y B C O at 7 7 K vs. frequency (figure 6.7). This graph was taken from a paper published in 1992 [6], w i t h much of the data older than this, however data from more recent publications is quite similar.  6.3 The  Finite Thickness Effects surface resistance of the films as measured is likely to be dominated by extrinsic  effects such as defects. There is, however, an extrinsic effect for which we can correct: the effect of the finite thickness of the film. T h i n films of superconductors are grown to a few thousand angstroms thick — about the same size as the penetration depth i n the material. Thus the assumption of infinite thickness i n the derivation of the surface resistance i n chapter 1 is not justified. Thus the effect of the reflection of the electromagnetic energy off the backside of the film, as well as of transmission completely through the film, can no longer be ignored. The  effective surface impedance of a film of thickness d for penetration depth A on a  substrate of infinite thickness of relative p e r m i t t i v i t y e is given by [62]: r  #efr  =  R f{d/\)  X  =  X coih(d/\)  =  coth A)  e f f  + R trans  s  (6.1) (6.2)  s  where f X) W  _ t r a n s  ~  W  1/2  r  (ay*A)  Z  +  2  -  ^  1  smh (d/X) 2  0  where Z is the impedance of free space (377 fl). These equations are derived by con0  sidering both the reflection and transmission from both the top a n d b o t t o m of the film. This is done conveniently through impedance transformations, as described thoroughly  Chapter 6. Data  87  in the reference [62]. T h e function f(d/\)  is the enhancement of the loss i n the film.  T h i s occurs because the current density excited i n a film of finite thickness are greater than for an infinitely thick superconductor. T h e transmitted energy is described by the term  i?t  r a n s  .  T h i s term, i n most cases, is smaller than the enhancement due to increased  current. T h e situation is actually improved significantly by having the substrate backed by metal, as is the case i n this apparatus (see figure 6.9). Here, almost almost a l l of the transmitted energy is reflected back by the metallic backing, rendering the  -Rtrans  term  negligible. W h e n the thickness of the substrate (in between the film and the metallic support) resonates, the transmission which is normally negligible can have a significant effect. Such a resonance occurs when the thickness of the substrate D satisfies [62]: D = n —^  ,n = 1,2,3...  (6.3)  where A is the free-space wavelength of the radiation, and e ( T ) is the relative permit0  r  tivity of the substrate at temperature T. For most substrates, this would be a problem at only isolated frequency/thickness combinations. However, for S r T i O s the temperature dependence of the p e r m i t t i v i t y is dramatic. Here a series of resonances can occur w i t h i n a temperature sweep. Off resonance, the transmitted radiation is reflected back into the resonator, as described above. A t resonance, the radiation w i l l bounce back and forth between the conducting surfaces u n t i l it is dissipated. T h i s explains the peaks and valleys observed i n the temperature dependence of Rs of the Y B C O film grown on strontium titanate (see figure 6.3). K l e i n et. al. [62] calculate the relative p e r m i t t i v i t y of S r T i 0 using the positions of the m i n i m a of these resonances.  3  T h i s is only really possible at  temperatures above Tc of the film, since the rapid change i n Rs below Tc w i l l distort the curve. Near, and above Tc, where the penetration depth becomes large (and is replaced by  Chapter 6. Data  88  the skin depth), the finite thickness effect becomes more pronounced. T h e skin depth for a high temperature superconductor i n its n o r m a l state is given by  as shown i n equation 1.13.  A typical d.c.  resistivity i n the n o r m a l state just above  T c of p = 100/iOcm gives a skin depth of about 15000 A. This is thicker than most thin films, causing the spurious increase i n surface resistance noted before. It is for this reason that the apparent surface resistance becomes unmeasurably high i n thin films of superconductor above Tc- T h i s l i m i t a t i o n is inherent i n the measurement, and precludes accurate n o r m a l state measurements on thin films i n this apparatus.  Chapter 6.  Data  89  f {GHz] Figure 6.7: T h e frequency dependence of R s of untextured (circles) and c-axis textured (triangles) polycrystalline bulk or thick film samples, as well as for epitaxial t h i n films (squares) and single crystals (rhombuses) of Y B a C u 0 7 at 77K. This plot is taken from reference [6]. T h e cross i n the upper right corner is the data for the M c M a s t e r film from this thesis. 2  3  Chapter 6.  Data  90  Figure 6.8: T h e effective surface resistance as a function of the intrinsic surface resistance for a film 3500A thick at 117 G H z assuming A = 2600A and e = 16 [62]. r  Figure 6.9: The calculated effective surface resistance for various thicknesses of film. This figure is reproduced from [62], and assumes / = 87 G H z , A = 2600A and the substrate w i t h e = 16 is 1 m m thick, backed by copper. r  Chapter 7  Conclusions and  7.1  Discussion  Introduction  T h e project described i n this thesis has been successful i n creating an apparatus to routinely evaluate the surface resistance of superconducting films. These measurements can be used as feedback to those involved i n the deposition of these films, w i t h the goal of lowering the microwave loss of the materials.  7.2  E q u i p m e n t Performance  T h e apparatus as constructed, consisting of the cryostat, the millimeter-wave source / detector pair, and the resonator assembly, has been tested under a variety of conditions. T h e cryostat's performance has been numerically modeled thoroughly for liquid hel i u m cooling, and was designed based on this data.  T h e cryostat has been operated  numerous times using l i q u i d nitrogen cooling, w i t h good control of b o t h the sample and base temperature from 70 to 120 K . T h i s control was achieved using pressure regulation on the exit gas, as well as active control of the sample temperature using electronic regulation. A swept millimeter-wave source and detector have been assembled using a previously built fixed-frequency spectrometer.  T h e resulting system has a flat transmitted power  over a 500 M H z frequency range around 117 G H z , w i t h a peak detected signal of 200 m V , and a background noise level (due to leakage or random noise) of < 1 m V .  91  Chapter 7. Conclusions and Discussion  The  92  resonator as designed has an unloaded Q w i t h copper at 70K of 39000, and the  system can handle a m i n i m u m Q of ~ 2000. T h e resonant frequency is stable to w i t h i n 4 M H z over 50 K changes i n the temperature; this stability has required m i n i m i z a t i o n of the effect of thermal contraction. W i t h a data acquisition and analysis system that accounts for m u c h of the asymmetry present i n the gain of the millimeter-wave system, the Q and resonant frequency are fitted w i t h good precision.  7.3 The  Surface Resistance Data apparatus has been used to measure the surface resistance of superconducting films  as well as n o r m a l metals. T h e surface resistance of these n o r m a l metals has been used as a calibration of the geometrical factors and of the parasitic losses of the resonator. T h e fitted results show scatter, but well w i t h i n the error bars as given by the x from the fit 2  done to extract the Q. The  measurements on superconducting thin films give the variation of surface resis-  tance vs. temperature from 7 0 K to Tc of the  film.  Measurement of the n o r m a l state  properties is only possible for thick films where the skin depth of the material i n the normal state is less than the thickness of the  film.  The films measured show a steep drop i n Rs w i t h temperature below the superconducting transition.  This drop levels out i n a l l the films below 8 0 K , though Rs does  continue to decrease slowly. None of the films measured have surface resistance lower than copper at the same temperature, as is expected at this frequency. None of the films measured show outstanding performance i n comparison w i t h the best of the literature data, indicating that further development work is needed. The The  films  grown on L a A 1 0 3 show the lowest surface resistance of those measured.  Y B a C u 3 0 7 _ 5 films, which were epitaxially grown, show m u c h better microwave 2  Chapter 7. Conclusions and Discussion  93  performance than the thick B i 2 S r C a C u 0 8 film. T h e film grown on S r T i O a shows a 2  1  2  complicated effect due to the temperature dependence of the p e r m i t t i v i t y of the substrate causing resonances i n between the film and the copper mount. T h i s dependence makes such films difficult to use for high frequency applications. T h e effect of the finite thickness of the films is to increase the effective surface resistance depending on the ratio d/X (the thickness of the film divided by the penetration depth). T h i s effect is due to reflection off the rear surface of the film increasing the loss in the film. T h i s effect is magnified near Tc where the penetration depth increases, and can lead to an effectively broadened transition. There is a small amount of transmission through the film, but it is negligible unless resonance occurs (as above for S r T i O s ) . T h e thickness of the film is also an important quantity in terms of growth: since thinner films seem to have better crystallinity and thus lower microwave loss. Thicker films have an inherently higher current carrying ability, and thus are desirable for high power applications. T h e improvement of film growth techniques to have thick films with low Rs is a goal of many researchers.  Currently, the surface resistance of films has a  m i n i m u m as a function of thickness, w i t h the Rs enhancement l i m i t i n g the u t i l i t y of the thinnest films, and non-epitaxy l i m i t i n g the utility of the thickest films.  7.4  Future W o r k  Future work on this apparatus w i l l be greatly affected by a new millimeter-wave vector network analyzer which is expected to be delivered to the lab very soon. T h i s network analyzer w i l l allow more accurate measurements over a much wider range of frequencies. This should avoid any systematic shift i n the Q w i t h frequency due to asymmetry of the gain of the source/detector pair. However, many of the problems w i t h the current  Chapter 7. Conclusions and Discussion  94  apparatus are due to the poor performance of the choke flanges, these should be redesigned for better uniformity of transmission over frequency, as well as for greater ease of use. A new sample-mounting gantry should be built to provide: • A smaller thermal conduction to the base, so that higher temperatures can be reached simply.  This could also incorporate active sample cooling to reach the  lowest temperatures (using l i q u i d helium) without this large conduction currently present. • A carrousel, so that multiple samples can be measured during the same run. This w i l l allow b o t h reference samples, and m u l t i p l e unknown samples to be measured rapidly. This is important for routine measurements. • Very small thermal contraction effects. This property of the resonator was an afterthought i n the current design, and hasn't been fully perfected. W i t h very small thermal contraction, systematic errors i n measurement can be reduced. The current apparatus has not been used at liquid h e l i u m temperatures, due to the unsuitability of the current sample gantry. Such measurements are not expected to be done very often, since the high temperature (70-90 K ) properties are most important for applications of films. Studies of fundamental properties, however, w i l l require a greater temperature range than currently used. T h e use of a conventional superconducting spherical m i r r o r could increase the sensitivity of measurements made at all temperatures (at 4.2 K and 117 G H z , the surface resistance of N b is about 0.001 O, while for copper it is 0.04 Cl) [64]. T h e flow cryostat w i l l never be as efficient as a standard cryostat, though improvements to the current design can be made (chiefly i n the addition of more radiation shielding). These changes may be made i n the interest of using less helium.  Chapter 7. Conclusions and Discussion  95  Since the new network analyzer w i l l be capable of working over a number of modes of the resonator, a study of the frequency dependence of Rs w i l l be possible.  Such a  study can probe how well the prediction of Rs oc u> holds, as predicted by the simple 2  two-fluid model. A s well, a device to allow the study of the spatial dependence surface resistance of large area films at 77K can be constructed.  A simple version of this was  built previously, but suffered from instability due to temperature drifts. T h i s w i l l allow the routine measurement of the homogeneity of large-area films, which is required for many applications such as microwave filters. It is hoped that studies of fundamental properties of superconductors w i l l be possible w i t h the new millimeter-wave network analyzer.  Since higher frequencies and greater  resolution w i l l be possible, a goal of measuring single crystals is feasible. Currently single crystals of Y B a C u 0 7 _ 5 are too small for measurement, but the l i m i t a t i o n due to the 2  3  spot size approaching the wavelength of the radiation becomes more forgiving at higher frequencies. Comparison of the behaviour of single crystals of Y B a 2 C u 0 7 _ 5 , and other 3  materials, w i t h films is important work for both practical, as well as fundamental reasons. T h e effect of defects on the electromagnetic properties of these materials gives insight into the still-unknown mechanism for the superconductivity. Defects are currently suspected to give the electromagnetic properties of t h i n .films their strikingly different behaviour from single crystals. Investigation of these differences could aid i n i m p r o v i n g the  films  for practical purposes. There are a number of materials which currently show the promise of becoming useful for microwave work, though most work has focussed i n the past on Y B a C u 0 7 _ 5 . Studies 2  3  of the mercury, t h a l i u m and bismuth copper oxides w i l l help i n the development of these for commercial purposes, as well as i n the understanding of the whole family of high temperature cuprate superconductors. These materials w i l l require the same systematic study as has been done i n Y B a C u 0 7 _ 5 both for o p t i m i z a t i o n , as well as to determine 2  3  Chapter 7. Conclusions and Discussion  any similarities and differences i n their properties.  Bibliography  [1] J . G . Bednorz and K . A . M u l l e r . Z. Phys., B 6 4 , 1986. [2] H . K a m e r l i n g h Onnes. Leiden Comm., (120b, 122b, 124c), 1911. [3] J . Bardeen, L . N . Cooper, and J . R . Schrieffer. Physical Review, 108, 1957. [4] R u i x i n g L i a n g , P . Dosanjh, D . A . B o n n , D . J . Baar, J . F . Carolan, and W . N . Hardy. Physica C, 195, 1992. [5] G l o r i a B . L u b k i n . Applications of high-temperature superconductors approach the marketplace. Physics Today, M a r c h 1995. [6] N . K l e i n . E l e c t r o d y n a m i c properties of high temperature superconductor films. In J . J . Pouch, S. A . Alterowitz, and R . R . 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