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Far-infrared magneto-conductivity of a YBa₂Cu₃O₇₋₈ superconducting thin film Matz, Daniel Johann 1995

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F A R - I N F R A R E D M A G N E T O - C O N D U C T I V I T Y O F A Y B a 2 C u 3 0 7 . 5 S U P E R C O N D U C T I N G THIN F I L M by DANIEL JOHANN MATZ B.A.Sc, The University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ENGINEERING PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1995 © Daniel Johann Matz, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date 7UK4> ' /??<>' DE-6 (2/88) 11 Abstract The far-infrared reflectivity of a superconducting film of YBa2Cu3C>7-s has been measured in a magnetic field up to 3.5 T. From this a Kramers-Kronig analysis has yielded the magneto-optic conductivity. In fitting these results to a theoretical model involving vortex core motion it was possible to fit the shape but not the magnitude of the frequency-dependent conductivity. Table of Contents Abstract ii Table of Contents iii List of Figures v List Of Tables vii Acknowledgement viii 1 Introduction 1 1.1 Magneto-Optics of Type-II Superconductors 1 1.2 Frequency Dependent Conductivity of Vortex Cores in Type-II Superconductors 2 1.3 Recent Far-Infrared Transmission Studies 3 1.4 New Reflectivity Measurements 4 1.5 Thesis Outline 5 2 Fourier Transform Spectroscopy 6 2.1 Introduction 6 2.2 Interferometer 6 2.3 FT-IR Advantages 9 2.4 Fourier Transformation 11 2.4.1 Picket Fence Effect and Zero Filling 14 2.4.2 Aliasing and Filtering 14 2.4.3 Leakage and Apodization 17 iv 2.4.4 Phase Correction • 18 3 Experimental Apparatus and Techniques 21 3.1 Introduction 21 3.2 Bruker 113V Spectrometer 21 3.3 Magnetic Field Apparatus 24 3.4 Reflectivity Module 30 3.5 Calculation of Optical Properties from Reflectivity Measurement 32 4 Experimental Results and Analysis 36 4.1 Far-Infrared Reflectivity 36 4.2 Far-Infrared Magneto-Conductivity of YBa2Cu307-s 41 5 Conclusion 48 Bibliography 49 V List of Figures Figure 1-1: Transmission ratios at 2.2 K for the YBCO/Si sample in unpolarized light in reference [1] 4 Figure 2-1: Schematic of a basic Michelson interferometer 7 Figure 2-2: Examples of spectra and their corresponding interferograms 12 Figure 2-3: Sampled interferogram and its computed spectrum 17 Figure 2-4: Apodization functions and their 'instrumental lineshapes' 19 Figure 3-1: Optical path of the basic Bruker IFS 113V interferometric spectrometer 22 Figure 3-2: Experimental apparatus 25 Figure 3-3: Resistance of carbon-glass resistor 29 Figure 3-4: Reflectivity module : 31 Figure 3-5: Reflectivity of aluminum 32 Figure 4-1: Ratio of normal-incidence power reflectivity of a superconducting YBa 2Cu 307.8 film in a magnetic field B to that in no field covering the range from 50 cm"1 to 400 cm"1 37 Figure 4-2: Ratio of normal-incidence power reflectivity of a superconducting YBa 2Cu 307.s film in a magnetic field B to that in no field covering the range from 30 cm"1 to 140 cm"1 38 Figure 4-3: The power reflectivity of a superconducting YBa2Cu307.5 film at room temperatre and at 20 K 39 Figure 4-4: The power reflectivity from 30 cm"1 to 1000 cm"1 of the SrTi0 3 substrate 40 Figure 4-5: The power reflectivity below 400 cm'1 of a superconducting YBa 2Cu 30 7 .6 film at 4.2 K as a function of magnetic field 42 Figure 4-6: The conductivities for the two lowest field values 43 Figure 4-7: The conductivities for 2.1 T and maximum value of 3.5 T 44 vi Figure 4-8: Comparison of the calculated and experimental conductivities at the two lowest field values 46 List Of Tables Table 3.1: Optical Elements of Bruker 113V 24 Table 3.2: Magnetic Field-Dependent Temperature Errors for Carbon-Glass Resistor 28 vm Acknowledgement I would like to thank my supervisor, Dr. J.E. Eldridge, for his patient and knowledgeable supervision. Special thanks are due to Dr. J. Carolan for kindly donating the magnet used for the experiments. I am also very grateful to Dr. Martin Dressel, who built a large portion of the experimental apparatus, and Bruno Gross, who helped to complete the apparatus and assisted in running the majority of the experiments. I am very grateful the Dr. Q.Y. Ma for providing all the samples used in the experiments. I would also like to thank Dr. W.N. Hardy and many others from the low temperature lab for their advice and assistance in cryogenic matters and those related to the operation of a superconducting magnet. 1 Chapter 1 Introduction 1.1 Magneto-Optics of Type-II Superconductors There are several important concepts in the far-infrared (FIR) magneto-optics of type-II superconductors: cyclotron resonance, vortex excitations, and vortex dynamics [1]. The cyclotron resonance concept is described by Kohn's theorem, which states that for an interacting electron system in a uniform applied magnetic field, H, the only excitation produced by a uniform electrodynamic field is the cyclotron resonance at frequency tub, where: eH (1.1) ( ° c = — ' m*c and ra* is the bare band mass [2]. This concept is certainly relevant for extreme type-II superconductors when the fields are sufficiently high such that the vortex lattice spacing is small compared to the London penetration depth XL SO that the magnetic field is nearly uniform. It has also been suggested that the Kohn theorem may be more general and apply to the low-field case as well [1]. On the other hand, type-II superconductors in the mixed state contain vortices whose cores support quantized quasiparticle states with approximate energy separation AE, given by: A 2 h2 (1.2) *£2 ' that are small compared to the energy gap A [3,4]. EF is the fermi energy, % is the superconducting coherence length, and the BCS relation is used between £, and A. For the case of a rigidly pinned vortex a strongly allowed resonance is predicted in the cyclotron active mode of circularly polarized light [5,6]. This transition, known as the vortex resonance, is expected at a frequency 2 n2m*c AE h (1.3) corresponding to the quasiparticle energy level spacing in the vortex core. In conventional superconductors hQ0 is small, -IO"4 meV, so that the quasiparticle levels effectively form a continuum. For the high-7; cuprate superconductors, however, where the coherence lengths are much smaller, this vortex core resonance is expected to be at M 2 0 = 10 meV, which corresponds to FIR frequencies. These two FIR magneto-optical concepts must also be related to the ideas of vortex dynamics, which are widely used to discuss the electrodynamic properties of type-II superconductors at low (DC or microwave) frequencies [7,8]. Conventional theories of vortex dynamics, however, are semi-classical hydrodynamic models that do not take the level of quantization of the vortex core into account [9,10]. The recent development of a theory by Hsu [11] based on the microscopic BCS theory which includes the electronic structure of the vortex cores has put vortex dynamics on firmer ground. 1.2 Frequency Dependent Conductivity of Vortex Cores in Type-H Superconductors The conductivity is calculated from the following equations of motion for the velocity of the vortex core \ L derived for the clean limit [12]: J = ne[\s+$(\L-\S)], (1.4) (1.6) V J is the spatially averaged current density, n is the carrier density, vs is the uniform background superfluid velocity, <I> = cot/Q0 ~ H/Hc2 (where cob and Q 0 are as previously 3 defined) is the magnetic field, S is the spatially averaged electric field, Nv is the areal vortex density, T v is the vortex quasiparticle lifetime, and a is the pinning frequency. These three equations in four unknowns ( J , £, \ L , \s) may be used to eliminate \ L and \s leaving J = o£ with . . ine2 c T ± ( C o ) = i m * c o o)(co + fl0)-i-(l-4))(ia)/Tv - a 2 ) [ ( c o + i 2 0 ) ( a ) ± t o c ) - i - i ( o / t v - a 2 J ' (1.7) where cj±=axx.± ia^ and Q 0 = (1 -<J>)£20. The + refer to cyclotron active (-) or inactive (+) modes of circularly polarized light. 1.3 Recent Far-Infrared Transmission Studies Recent measurements Karrai et al. [13] through a superconducting thin film of YBa 2 Cu 3 0 7 (YBCO) in a strong magnetic field have shown an increase in transmission below -125 cm"1 with increasing field. This is attributed to the excitation of the vortex core states. Subsequent measurements by the same group have shown optical activity with circular polarized light due to cyclotron resonance [14]. In a later paper Drew et al. [1] compare their results with the predictions of the Hsu model [11]. These results are reproduced below as Figure 1-1. The increased transmission below -35 cm"1 is reproduced by the theory, but the broad resonance at 65 cm"1, which the authors attribute to the vortex core resonance, does not appear from the calculation without artificially increasing the magnitude of <I> above its theoretical value of = oVQo- Both of these features grow with increased magnetic field. There are additional weaker features at 120 and 170 cm"1 which are not explained by the theory. In investigating other samples and different substrates it was found that the magnetic-field-induced enhanced transmission at low frequencies is always present, but the strength and line shape of the resonance feature near 65 cm"1 was sample dependent. The weakness of this feature in the T*IT spectra obtained using polarized light indicate that it is essentially non-4 0 50 100 150 200 a; ( c m ) Figure 1-1: Transmission ratios at 2.2 K for the YBCO/Si sample in unpolarized light in reference [1]. Thick solid line: measured data. Dotted line: best fit to conductivity function given by Equation (1.7) with best fit parameters a = 40 ± 2 cm"1, 1/Xv = 50 ± 5 cm"1, and QQ taken as 60 cm"1. Thin solid line: best fit obtained when <I> is taken as an independent parameter. Best fit parameters for this case are £2 0 = 52 ± 4 cm"1, a = 44 ± 2 cm"1, <X> = 0.094. chiral. The authors conclude that the observed vortex resonance at 65 cm"1 is induced by effects outside the Hsu model. 1.4 New Reflectivity Measurements In this thesis work, the ratio of normal-incidence reflectivity of a Y B C O film on a SrTi0 3 substrate in a magnetic field to that in no field has been measured. This ratio was used to convert the absolute reflectivity measured in a separate experiment to that appropriate for the different field strengths. The frequency-dependent magneto-conductivity was then 5 calculated directly using a Kramers-Kronig analysis. The conductivity shows a strong broad resonance between approximately 50 and 250 cm"1, whose intensity is strongly magnetic-field-dependent. No outstanding feature was found at 65 cm"1 for these samples. There is some structure in the resonance, presumably due to phonons. It is believed that the increase in transmission at low frequencies in references [1,13] is mainly due to the observed decrease in reflectivity. 1.5 Thesis Outline In Chapter 2 the principles and advantages of infrared Fourier tranform spectroscopy are discussed. The Bruker IFS 113V interferometer and other experimental apparatus is described in Chapter 3. The relationship between the various optical constants is introduced. The experimental results for the reflectivity of the Y B C O film in a magnetic field is presented in Chapter 4 and compared to the published transmission results [1,13] The frequency-dependent magneto-conductivity that is calculated from these results is compared to that predicted by the Hsu model [11] and it is shown that there are many features in the spectra that are not accounted for in the theory. The conclusions of the thesis are reviewed in Chapter 5 and some suggestions for future experiments are presented. 6 Chapter 2 Fourier Transform Spectroscopy 2.1 Introduction Infrared spectroscopy is routinely used by condensed matter physicists and analytical chemists to probe the optical properties of matter. Fourier Transform Infrared Spectroscopy (FT-IR) is the most advanced form of IR-spectroscopy and has many advantages over conventional IR-spectroscopy. However, with increased usefulness comes increased complexity, which must be understood in order to make full and proper use of a FT-IR instrument. In a conventional grating instrument the spectral trace is generated relatively straightforwardly by setting the appropriate knobs to control slit widths, scanning speed, etc. On the other hand, using a FT-IR instrument involves a lot of mathematical manipulations, such as Fourier transformations, phase correction, and apodization. Thus the FT-IR spectrometer consists of essentially two parts: • an optics bench, housing an interferometer, a sample chamber, and various detectors to cover the different frequency ranges, and • a computer with special software to carry out data acquisition and all the various mathematical manipulations. In order to successfully use a FT-IR spectrometer it is important to understand not only the optics, but also the mathematical manipulations carried out by the computer software to generate the final spectrum from the raw interferogram. The important considerations will be discussed in the following sections, and are described in more detail in references [15,16,17]. 2.2 Interferometer The heart of the optics bench in a FT-IR spectrometer is the interferometer. An idealized Michelson interferometer is shown in Figure 2-1. Infrared radiation is directed 7 Figure 2-1: A) Michelson Interferometer. S: source. D: Detector. Mi : fixed mirror. M 2 : moveable mirror. X: mirror displacement. B) Signal measured by detector D. This is the interferogram. C) Interferogram pattern of a laser source. Its zero crossings define the positions where the interferogram is sampled (dashed lines). towards the beamsplitter (so called because it transmits some the radiation while reflecting the rest). The reflected beam continues towards mirror Mu where it is reflected and returns to the beamsplitter, traversing a total distance of 2L. Meanwhile the transmitted beam travels towards and is reflected from mirror Mi. This mirror, however, is not fixed, but can be precisely displaced about L by a distance x. Thus the total path of this beam is 2(L+x). Consequentiy when the two portions of the incident beam recombine at the beamsplitter they 8 will exhibit a path difference, or optical retardation, of Ix. Hence they will be out of phase, which will lead to interference. From here the recombined beam passes through the sample chamber and is eventually focused on the detector D. Thus the signal measured by the detector is the intensity I(x) of the recombined beam as a function of the displacement x of the moving mirror M2. This is called the interferogram and is shown in Figure 2-1 B. The interferogram resulting from a monochromatic source is shown Figure 2-1 C. The detector signal is maximized when the partial beams interfere constructively, which occurs when the optical retardation, 2x, is an exact multiple of the wavelength X. On the other hand the detector signal is minimized when the beams interfere destructively, which occurs when 2x is an odd multiple of X/2. Thus the complete dependence of I(x) on x is given by: 7(JC) = S(V)[COS(2JIVX) + 1], (2.1) where v is the wavenumber, which is related to the wavelength X by: _ 1 (2.2) and S(v) is the intensity of the monochromatic line at wavenumber "v. In the case of a polychromatic source the resulting interferogram is the integration of the right hand side of Equation (2.1): / (x) = C" S ( V)[COS(2TTVX) + l]rfv, ( 2 ' 3 ) where and v m a x are the minimum and maximum wavenumber range of the polychromatic source. Equation (2.1) is very useful for practical measurement since it allows for precise tracking of the moving mirror M 2 . All modern FT-IR instruments use the interference pattern of a monochromatic He-Ne laser to control the movement of the scanning mirror and thus accurately control the optical path difference 2x. As illustrated in Figure 2-1 C the TR 9 interferogram is precisely sampled at the laser interferogram zero crossings. The sample spacing accuracy, Ax, between two zero crossings is determined solely by the laser wavelength precision. Since the spectrum sample spacing Av is inversely proportional to Ax, the error in Av is of the same order as in Ax. This results in a high precision (of order 0.01) cm"1) built-in wavenumber calibration and is known as the Connes advantage [15]. 2.3 FT-IR Advantages In addition to its high wavenumber accuracy FT-IR has other features which make it superior to conventional IR. Most of these arise from two major concepts known as the Fellgett and Jacquinot advantages [17]. The Jaquinot, or throughput (entendue) advantage arises from the fact that a FT-IR instruments can have a large circular source at the input or entrance aperture of the instrument with no strong limitations on the resolution. Furthermore it can be operated with large solid angles at both the source and the detector. In a grating instrument, however, the resolution depends linearly on the instrument's slit width, and the detected power depends on the square of the area of equal slits. The long and narrow slits required can never have the same area for the same resolving power as the interferometer. In addition, high resolution requires large radii for the collimating mirrors, which in turn necessitates small solid angles. Thus for the same resolution the FT-IR spectrometer can collect much more energy than a conventional grating instrument. In a conventional grating instrument the spectrum 5(v) is measured directly by recording the intensity at different monochromatic settings v . But in an FT-IR instrument the entire spectral range emanating from the source impinges on the detector simultaneously. This accounts for the so-called multiplex or Fellgett advantage. The practical significance of this can be explained by the following example. Suppose one is interested in obtaining the broad spectrum between wavenumbers Vj and v 2 with resolution 6v. The number of spectral elements in the broad band would then be given by: 10 w ( V 2 - V i ) _ A v M ~ ov " 6 v - (2-4) In a grating or prism instrument each small band of width 5v would be observed for a time T/M, where T is the total time to required to complete the scan over the range of interest, Av . The integrated signal, S, received in the band Sv is proportional to T/M. On the other hand, assuming it is random, the noise, N, would be proportional to -yJT/M . Thus the signal to noise ratio in the band ov is given by: '£] TiM - fL (2-5) In an interferometer, however, the entire signal from all the bands is received simultaneously in time T. Assuming random noise again, the signal to noise for the interferometer is given by: 7i) 'IT*- {Z6) ™ / Interferometer AM Thus the multiplex advantage of an interferometer over a grating machine is: (V"Interferometer ^ A/T = (2-7) Hence, given the same measurement time, an interferometer will have a higher signal to noise ratio over a grating instrument by a factor of 4M, or alternatively, for the same signal to noise ratio an interferometer will perform the same measurement in less time by a factor of M. Finally, the Fellgett and Jacquinot advantages allow the construction of interferometers which have much higher resolving power than dispersive instruments. Additional advantages are described in the literature, such as in the book by Bell [17]. 11 2.4 Fourier Transformation Formally the Fourier transform, 3 , and the inverse Fourier transform, 3"1, are defined where 7(x) is the intensity as a function of optical path difference and 5(v)is the desired spectrum as a function of wavenumber. Note that this requires a continuous interferogram over all values of optical path difference to compute the resulting spectrum. In practice, however, one has a discretely sampled interferogram over a limited range of path difference corresponding to the desired resolution. In this case the integral is replaced by a series: where the continuous variables x and v have been replaced by nAx and kAv, respectively. This is known as the discrete Fourier transform (DFT), as opposed to the continuous Fourier transform given by the integrals above. The spectrum spacing Av is related to Ax by: by: S(y) = £/(x)exp(2jcrv;c)<flc = 3{/(x)}, (2.8) I(x) = J2s(v)exp(-2ravx)<fv = 3"'{5(v)}, (2.9) (2.10) Av = 1 (2.11) NAx Similarly one can also write the inverse Fourier transform: (2.12) These Fourier transforms are easily illustrated by considering the simple case of a spectrum with one or two monochromatic lines, as shown Figure 2-2 A and Figure 2-2 B. For a few special cases, such as the Lorentzian in Figure 2-2 C, the Fourier transform is 12 4000 2000 WflVENUMBERS CM-1 X 1^ Figure 2-2: Examples of spectra (on the left)and their corresponding interferograms (on the right). A) One monochromatic line. B) Two monochromatic lines. C) Lorentzian line. D) Broadband spectrum of polychromatic source. known. For the case of experimentally determined data, however, the Fourier transform must be calculated numerically. While the precise shape of a spectrum cannot be easily visualized from studying the interferogram, it is still helpful to understand two simple trading rules for an approximation 13 between the interferogram, I(nAx), and the spectrum, S(kAv). In Figure 2-2 C one can extract the general qualitative rule that a finite spectral line width (as is always present in any real sample) is due to dampening in the interferogram. Stronger damping results in increased line width. By comparing the widths at half height (WHH) another related rule can be inferred. The WHH of a "hump-like" function is inversely proportional to that of its Fourier transform. This explains why the interferogram in Figure 2-2 C due to a broad band source has a very narrow peak around the zero path difference position at x = 0, while the wings, which contain most of the useful information, have a very low amplitude. Upon closer examination of Equation (2.12) one notes that for n = 0 the exponential term equals unity. Hence the intensity 7(0) measured at the interferogram centerburst is equal to the sum over all N spectral intensities divided by N. Thus the height of the centerburst is a measure of the average spectral intensity. In practice Equation (2.10) is rarely used directly because it is highly redundant. For each of the N points in the interferogram a summation of N terms must be evaluated, thus the number of required calculations is of order N2. On the other hand, for a so-called fast Fourier transform (FFT), the most common of which is the Cooley-Tuckey algorithm, the number of calculations is of order Nj~N . Thus for a typical spectrum of 1000 points the computation time reduced by a factor of -30. The small price to pay for the increased speed is that the number of interferogram points, N, cannot be chosen freely. For the Cooley-Tuckey algorithm, which is used by most FT-IR spectrometers, N must be a power of two. One must keep in mind that the DFT given by Equation (2.10) only approximates the continuous FT of Equation (2.8). Used with care it is a very good approximation; however, used blindly can lead to three well-known spectral artifacts: the picket-fence effect, aliasing, and leakage. 14 2.4.1 Picket Fence Effect and Zero Filling The picket fence effect is evident when the spectrum contains features which do not coincide with frequency sampled points ifcAv. If, in the worst case, one of these features lies exactly halfway between two sampled points, an erroneous signal reduction of 36% can occur [15]. In effect it appears as if one is viewing the true spectrum through a picket fence and those features lying behind the pickets are clipped. In practice the problem is usually less extreme since the spectral features are generally broad enough to be spread out over several sampling positions. Fortunately there is a solution to overcome this deficiency of the DFT. The picket-fence effect is overcome by adding zeros to the end of the interferogram before performing the DFT, thereby increasing the number of points per wavenumber in the spectrum. Thus zero filling has the effect of interpolating the spectrum and reducing the errors. It also has a cosmetic effect of smoothing out sharp discontinuities in the spectrum. As a general rule one should always at least double the original interferogram size for practical measurements by zero filling. It should also be noted that the procedure does not introduce errors since the instrumental line shape is not changed. Hence it is superior to polynomial interpolation procedures in the spectral domain. 2.4.2 Aliasing and Filtering Aliasing is another artifact that comes about from using the discrete Fourier transform. The sampled interferogram Is(x) can be related to the continuous interferogram Ic(x) by: Is(x) = \li(±)lc(x), (2.13) where LLI(x) is the shah function defined by: ni(x)=±o(x-n), ( 1 1 4 ) 15 where 6(x-n) is a Dirac delta function and n is an integer. Thus the shah function allows non-zero values only for integers and the interferogram in sampled at x = 0, ±Ax, ±2Ax, ±3Ax, . . . . In order to relate the sampled spectrum Ss(y) to the complete spectrum Sc(v) one must consider some of the mathematical properties of the shah function. The first property to note is that the shah function is periodic to any integer change in JC; hence, W\(x + m) = \ll(x). (2-15) One can also derive a scale rule for the shah function: \ft°x) = fa)t*(*-n), ( 2 ' 1 6 ) where a is some number. Finally, the Fourier transform of the shah function is given by: 3[[[\(ax)]=(^m)- ( 2 J 7 ) Thus the sampled spectrum Ss (v) is given by inverse Fourier transform of the sampled interferogram: 5J(v) = 3[QJfe)/cW]. (2-18) Using the convolution theorem, which states that the Fourier transform of a product of two functions is equal to the convolution of the Fourier transforms of each function, i.e., 3[F(v)G(v)] = 3[F(v)]®3[G(v)], (2.19) where the convolution integral is defined by: F(x) 0 G(x) = £ G(y)F(x - y)dy , ( 1 2 0 ) 16 allows one to one to relate the sampled spectrum to the complete spectrum: S,(v)= X^c(v)[v-nAv], (2.21) where (2.22) Thus upon computing the spectrum from the sampled interferogram one obtains an infinite number of complete spectra, with the first duplicate spectrum starting at nAv. This repetition of spectra is known as aliasing. If the repeated spectra do not overlap there is no problem. Whether or not this occurs depends on the magnitude of Av . If Ax is very small and Av is therefore very large, the spectra will be separated and no overlap will occur. If the reverse is true, the computed spectra will be the sum of positive and negative frequency contributions, as shown in Figure 2-3. In order to avoid this one must make sure that the maximum frequency contribution of the positive v spectrum does not overlap with the imaged negative v spectrum. This is accomplished by requiring that where v f is known as the folding or Nyquist wavenumber [16]. Thus the finer the interferogram spacing Ax is, the further apart the aliases are and the lower the danger of alias overlap. However, a smaller Ax also means an increased number of interferogram points N, which leads to longer computation times. Thus the goal is to choose the maximum sample spacing for which no overlap occurs. <v f = 1 (2.23) v. max 2 Ax ' 17 Ax Figure 2-3: A) Sampled interferogram. B) Computed spectrum showing positive v (solid line) and negative v (dashed line) spectra. C) Total computed spectrum, which is sum of positive and negative components in B. 2.4.3 Leakage and Apodization Unlike the picket fence effect and aliasing, leakage is not due to using the discrete form of the Fourier transform. Rather, it is caused by truncating the interferogram at finite optical path difference. Mathematically, this is accomplished by multiplying the infinite interferogram I(x) by a window function. Alternatively, by use of the convolution theorem, one can also convolve the spectrum S(y) with the Fourier transform of the window function or instrumental lineshape (ILS). Several different window functions and their corresponding 18 instrumental lineshapes are shown in Figure 2-4, the simplest of which is the 'boxcar' or square window function. The ILS of this function is the well known sine function. In addition to the main peak there are many sides peaks, called lobes or 'feet', which cause 'leakage' of the spectral intensity. Since these sidelobes, the largest of which is 22% of the main lobe amplitude, do not correspond to any actual spectral intensity but are merely an artifact due to the abrupt truncation of the interferogram, it is desirable to reduce their amplitude. This attenuation of the spurious 'feet' is known as 'apodization' (from the Greek word OCTCOS, which means 'removal of the feet'). The solution to this problem is to truncate the interferogram less abruptly. This is equivalent to finding an apodization function which has fewer sidelobes. Several such functions are shown in Figure 2-4. It can be seen that the feet of these functions are smaller than that of the sine function; however, one must also keep in mind that the width of the main lobe increases, which translates into a decrease in resolution. 2.4.4 Phase Correction The final step in converting the interferogram into a spectrum is phase correction. This is necessary because the FT of a measured interferogram generally yields a complex spectrum C(v) rather than a real spectrum 5(v) as obtained from a conventional spectrometer. The complex spectrum can be expressed in the following way: C(v) = R(y) + il(v) = 5(v) exp[/(j)(v)], (2.24) where S(y) is the desired spectrum and 0 is the phase error. The reason for getting a complex spectrum from doing the FT of the interferogram is due to the asymmetry of the interferogram about the point x = 0, which originates from the following sources : 19 Figure 2-4: Several Apodization Functions (left) and their 'Instrumental Lineshapes' (right). • None of the sampling positions coincides exactly with the zero path difference position, which is usually the case and causes a phase error linear in v . • Misaligned interferometer optics, which causes a wavenumber dependent phase error. Thus the aim of the phase correction procedure is to extract the amplitude spectrum S(v) from the complex output C(v) of the FT. This can be done by either taking the square root of the power spectrum P(y) = C(v) • C * (v): S(y) = [C(v) • C * (v)]V2 = [*2 (v) + 72 (v)]V 2, 20 (2.25) or by multiplying C(v) by the inverse of the phase exponential and taking the real part of the result: 5(v) = Re{ [C(v)exp[H(t)(v)]}, (2.26) where (t>(v) = tan_ 1 /(v) (2.27) In the absence of noise both procedures will yield the same result. However, if noise is present, as is always the case with real data, the noise contributions computed from Equation (2.25) are always positive, and in the worst case a factor of V2 larger than the correctly signed noise amplitudes computed from Equation (2.26). The latter procedure, given by Equation (2.26), is known as the "multiplicative phase correction", or "Mertz-method" [18]. 21 Chapter 3 Experimental Apparatus and Techniques 3.1 Introduction All the reflectivity measurements reported here were performed with a Bruker 113V spectrometer. For the magnetic field measurements the sample was placed in an external apparatus and the optical signal analyzed by the Bruker. In addition, zero-field measurements were performed in a reflectance module which was placed inside one of the Bruker sample chambers. Each of these systems, individually and how they work together, will be described in the following sections. 3.2 Bruker 113V Spectrometer The Bruker 113V spectrometer is a Genzel-type Michelson interferometer [19] operating under vacuum, which helps minimize unwanted water absorption and preserves thermal stability. It can be used from the far infrared (-10 cm"1) well into the near infrared (-15800 cm"1) with a maximum resolution of -0.03 cm"1. Figure 3-1 shows the ray diagram of a Bruker TPS 113V optical bench. The spectrometer incorporates an on-board acquisition processor (AQP) which performs data acquisition and fast Fourier transformations as well as controlling all the motor-driven optical components such as filters, beamsplitters, apertures, and mirrors switching between the various sample chambers, sources, and detectors. The AQP in turn is linked to a PC-based computer system whose software allows the operator to control the sample measurement and allows one to display and manipulate the resulting spectra. The unique design of the Genzel interferometer, in which the beams are focused on the beamsplitters, allows the latter to be quite small (about 2 cm in diameter as opposed to 12-20 cm in a conventional Michelson interferometer). Such large beamsplitters vibrate slightly, 22 Figure 3-1: Optical path of the basic Bruker IFS 113V interferometric spectrometer. (I) Source Chamber: a) glowbar, mercury arc lamp, Tungsten/Halogen/Quartz lamp; b) automated aperture. (II) Interferometer Chamber: c) optical filters, d) automatic beamsplitter changer, e) double-sided scarming mirror, f) control interferometer, g) reference laser, h) remote control alignment mirror, (m) Sample Chamber: i) sample focus, j) reference focus. (IV) Detector Chamber k) far-infrared DTGS (deuterated triglycerine sulfate) detector, mid-infrared M C T detector, near-infrared InSb and silicon diode detectors. which results in diffusion of the beam and increased spectral noise. This is known as the "drum-head" effect and is greatly reduced in the smaller beamsplitters of the Genzel interferometer. Another advantage is that the angle of incidence at the beamsplitter is only 14°, which results in increased light throughput and decreased polarization effects. Furthermore, their small size allows six to be mounted on rotatable carrier, which allows them to be changed while the instrument remains under vacuum. In addition to the six beamsplitters there is a choice of three sources, four apertures, four optical filters, and four 23 detectors. The bolometer is an external detector and is described in detail in Chapter 3.3. All of the optical elements are remotely selected by computer. Thus the entire instrumental range can be covered without breaking the vacuum and destroying the thermal equilibrium, which usually takes approximately two hours to achieve. The two beams from the beamsplitter are incident on opposite sides of a double-sided scanning mirror which is supported on a dry nitrogen gas bearing and driven by a linear induction motor. Translation of this mirror gives twice the path difference that is obtained in a conventional Michelson interferometer. Thus only half the mirror travel is required to obtain a given resolution. The scanning mirror positions are determined by fringe counting using the He-Ne laser reference interferometer. A white light source is used to determine the mirror position corresponding to the centerburst, or zero-path difference, as already discussed. The various optical components are summarized in Table 3.1. The combination of different sources, beamsplitters, and detectors results in three overlapping ranges covering the entire infrared spectrum. The thin mylar beamsplitters give rise to Fabry-Perot fringes whose spacing Av is given by: Av = J - , <W 2nd where n is the refractive index (1.6 for mylar) and d is the thickness of the beamsplitter. Thus to cover all of the far-infrared range a series of beamsplitters with thickness 50 um, 23 um, 6 urn, and 3.5 um are used. They have ranges of 20-60 cm"1, 30-120 cm"1, 100-400 cm"1, and 150-650 cm"1, respectively. The primary purpose of the optical fdter is to block ultraviolet radiation which could damage the thin mylar beamsplitters. For the detector used one must choose an appropriate scanning speed v. At a given scanning mirror velocity v (in cm/s) the frequency of the alternating electronic signal received by the detector is given by: 24 Table 3.1: Optical elements of Broker 113V Range (cm"1) Source Beamsplitter Detector Optical Filter 20 - 650 Hg arc Mylar Bolometer, DTGS Polyethylene 400 - 5000 Globar Ge/KBr M C T None 2000 - 9000 Tungsten Si/CaF2 InSb None / v=4vv , (3.2) where / 7 i s the modulation frequency in Hz and v is the wavenumber in cm*1. The Bruker rapid-scan uses high-pass and low-pass electronic filters to avoid aliasing (as opposed to optical filters used by slow-scanning spectrometers) since each electronic fv corresponds to a specific optical wavenumber v . This relation is also important in determining noise features caused by mechanical vibrations, which can be observed to move in the spectrum as the scanning mirror velocity v is changed. Typical scanning speeds are such that each scan takes several seconds. Hundreds of such scans are averaged to increase the signal to noise ratio before the Fourier transform is performed. The apodization function used is the 'three-term Blackmann-Harris* function: W(x„) = 0.42323 + 0.49755cos^j+0.07922cos^ where n = 1, 2, 3 , N , and A" is the number of interferogram sample points. 3.3 Magnetic Field Apparatus The experimental apparatus for the reflectivity measurements in the magnetic field is shown in Figure 3-2. This arrangement is somewhat more complicated than is usual for reflectivity measurements since the sample must be placed well inside the cryostat at the 25 Figure 3-2: Experimental Apparatus. A) Bruker DPS 113V, B) infrared beam rotated by 90° for illustrative purposes, C) toroidal mirror, D) polished brass light pipe, E) beam divider, F) polished brass light cone, G) doped Ge bolometer, H) liquid helium cryostat, I) stand, J) polished brass 45° reflector, K) vacuum window, L) stainless steel chamber pumping port, M) liquid helium dewar pumping port, N) thin walled stainless steel chamber, O) sample, heater, and carbon glass thermometer, P) superconducting magnet. 26 center of a superconducting magnet, as opposed to inside the spectrometer, which is usually the case for zero-field measurements. Additional optics are required to take the infrared beam out of the spectrometer, to the sample, and finally to the detector. The cryostat consists of two glass dewars, one fitting inside the other. The outer dewar is filled with liquid nitrogen to reduce the radiative heat load on the inner liquid helium dewar. Prior to conducting an experiment the entire system must be cooled to liquid nitrogen temperature before the inner dewar is filled with liquid helium. This requires a minimum of twelve hours. In order to cool the experimental apparatus within the inner dewar, the vacuum jacket is filled with helium gas to allow for heat transfer throught it. The outer dewar is kept full of liquid nitrogen during this time by an automatic level controller. This controller consists of two sensors placed inside the dewar and some electronics which control a solenoid to regulate the flow of liquid nitrogen into the dewar such that the liquid level is maintained between the positions of the two sensors. The superconducting magnet, P, is situated near the bottom of the liquid helium dewar. This magnet was homemade by J.C. Carolan of the UBC Physics Department. It was wound from single stranded copper clad 0.016" wire with a 0.010" NbTi core. The magnet has an outer diameter of 3.635", is 4" long and has a 2" bore. It has a resistance of 304 Q and an inductance of 2 H at room temperature. At its maximum rated current of 70 A it generates a central field of 4.9 T. Initial testing of the magnet showed the helium boil-off rate to be unacceptably high at 0.5 l/hr with no current and 0.7 l/hr at 50 A. This was attributed to poorly designed current leads. The original leads were fashioned from circular copper rods. This design was bad both in material selection and shape. The heat load introduced into a cryostat by current leads is due to two sources: conduction down the lead from room temperature, and resistive heating of the lead when current is flowing. Thus at first glance the choice of copper appears good due to its low electrical resistivity. However, using a pure metal can lead to a problem called "run-away". This comes about from the temperature dependence of the resistivity. When current is flowing 27 in such a conductor there are always localized regions where the resistivity is higher than that of the surrounding material. This results in localized heating. Due to the temperature sensitivity of the resistivty this results in a large increase in resistivity. This in turn causes more heating, and so on. Thus a relatively large portion of the conductor becomes hot and a large heat load on the cryostat. Alloys, whose resistivities change httle with temperature, do not suffer from this problem. In any croystat it is always important to fully utilize the cooling capacity of the liquid helium — not only the latent heat of vapourization but also the change in enthalpy of the gas as it warms up to room temperature. While a heat leak of 1 J will vapourize 48 mg of liquid helium, a further 74 J is required to warm this mass of gas to room temperature. Current leads are therefore made like heat exchangers such that the gas boiled away by the lead is constrained to flow past the electrical conductor. Thus a large portion of the heat generated within the lead or conducted along it is carried away by the gas stream and prevented from reaching the liquid helium. Clearly a lead with circular cross section is the worst possible shape for this application. Maximum heat transfer results from a shape in for which the perimeter to cross sectional area is maximized — such as a thin strip. While there are analytical methods for optimizing the shape of a lead for a given current and material [20,21], an empirical approach is often more expedient. The basic idea is to choose a sufficiently large cross sectional area to rninimize the resistive heating without introducing an unduly large conductive heat leak. With this in mind and some expert advice [22] it was decided to construct the leads from 1" x 0.020" brass strips. Resistive heating can be further reduced by tinning the leads with solder, which is superconducting at liquid helium temperature. A 50:50 alloy of PbSn was chosen for this purpose because its T c of 7.75 K is the highest of all the readily available commercial solders [23]. Subsequent testing of these leads showed a helium boil-off rate of 0.25 1/hr at 0 A and 0.35 1/hr at 70 A. This rate was low enough that a persistent switch was not recquired. 28 The stainless steel sample chamber passes through the center of the magnet. The sample holder, O, is mounted at the bottom of a polished brass light pipe such that the reflecting surface is at the exact center of the magnet. A heater, which can be used to raise the sample temperature above 4.2 K, is built into the bottom of the sample holder. The temperature is monitored by a carbon-glass resistor, whose resistance vs. temperature dependence is shown in Figure 3-3. As can be seen in the figure, the resistor is very sensitive near 4.2 K and thus the temperature can be monitored very accurately. Furthermore, the resistor is relatively insensitive to magnetic fields, as shown in the table below. Table 3.2: Magnetic Field-Dependent Temperature Errors, |A7|/r (%), for Carbon-Glass Resistor, taken from the Lakeshore Cryogenics catalogue [24]. T(K) 2.5 T 8 T 14 T 2.1 0.5 1.5 4 4.2 0.5 3 6 15 <0.1 0.5 1.5 35 <0.1 0.5 1 11- <0.1 0.5 1.5 The bolometer was removed from the vicinity of the magnetic field by the light pipe arrangement shown in Figure 3-2. The infrared beam, B, is extracted from the spectrometer by the a toroidal mirror at C (position j in Figure 3-1) into a polished brass light pipe, D. At E the beam encounters a beam divider which is simply a mirror covering half of the light pipe as shown in the figure. The radiation that passes by the mirror is absorbed by some microwave absorption material placed behind the mirror. In this manner half of the beam is reflected towards J, from where it is reflected towards the sample at O. Half of the returning beam continues to the bolometer at G. This was found to be the most efficient means (compared to using a beamsplitter) of reflecting the beam and having it continue on to the detector. Figure 3-3: Resistance of Carbon-Glass Resistor. 30 This detector is a commercial Infrared Laboratories1 4K composite bolometer of Gallium-doped Germanium on a sapphire sheet with a thin bismuth film. It is mounted in its own cryostat as shown in the figure. There are three filters which allow only the far-infrared radiation to impinge on the detector. One filter is at room temperature and the other two are cooled. The room temperature filter, which blocks visible radiation, consists of wedged white polyethylene disc with a black polyethylene film melted on it. The cold filters at 77 K and 4.2 K are wedged white polyethylene with 15 nm diamond dust thermally embedded on both sides of the filter. The diamond-dust scattering filters were constructed using techniques referred to in the literature [25]. The scattering filters are low pass filters which transmit only radiation below -650 cm"1, thus blocking room-temperature and mid-infrared radiation that would heat the bolometer and lower its sensitivity. The bolometer is electrically isolated from the Broker spectrometer. The bolometer signal goes through an Infrared Laboratories low-noise preamplifier before it reaches the Broker amplifier board. Since it is microphonic it is also vibrational^ isolated. 3.4 Reflectivity Modu le The reflectivity of the film at zero field was also measured in the reflectivity module shown in Figure 3-4, which fits inside one of the Broker sample chambers (position i in Figure 3-1). The sample and reference mirror are mounted on a sample holder such that both reflecting surfaces are in the same plane. They can be exchanged by moving the sample holder from outside in the direction designated by M. The incident beam from the interferometer is focused on the rectangular aperature F which can be adjusted to match the sample size. The first toroidal mirror has an external adjustment to allow for a change in sample height once the instrument is evacuated. The detector signal is maximized through a ^frared Laboratories, Inc., 1808 E.17* Street, Tuscon, Arizona, U.S.A., 85170. 31 1—I Figure 3-4: Reflectivity Module: A,B) reference mirror and sample, C) radiation shield, D) vacuum shroud, E) optional polarizer, F) adjustable rectangular aperture, G) plane mirrors', H) toroidal mirrors, I) chopper, J) external rod to move chopper, K) extension to sample chamber, L) plexiglass lid, M) illustration of translational degree of freedom, N) back of sample chamber. 32 combination of rotating the sample holder and adjusting the second toroidal mirror with the external adjusters. A chopper is brought into the beam path during this procedure to produce the A C signal required by the detectors. A detailed description of the reflectance module can be found in reference [26]. The reflectivity obtained from this apparatus is the ratio of the sample reflectivity to that of the aluminum reference mirror. By comparing this ratio to the published values of the aluminum reflectivity [27], which is shown in Figure 3-5, the absolute reflectivity of the film at zero field was obtained. Multiplying this by the ratio obtained from the magnetic field measurements yielded the absolute reflectivity of the film at the different magnetic field strengths. 3.5 Calculation of Optical Properties from Reflectivity Measurement Given that one has measured the reflectivity of a material it is then possible to calculated any of the response functions used in electromagnetic theory, such as the complex index of refraction (n=n + ik), the dielectric function (E =E1 + ie2), or the complex conductivity (6=o,+; 'c 2 ) . The complex reflectivity of a material is defined as: r=jRe», (3.4) where R is the power reflectivity and 9 is the phase difference between the incident and reflected waves. For large samples with high-quality surfaces there are a number of techniques for measuring 6 directly, such as ellipsometry, dispersive reflection spectroscopy, and asymmetric interferometry. These techniques cannot be used with small samples and only the power reflectivity can be measured directly. In this case the phase can be calculated from the Kramers-Kronig relation [28]: 33 1.00 I ' 1 <— 1 1 7 0.99 " 0.98 -0.97 " 0.96 " £ 0.95 " 0.94 " 0.93 " 0.92 " 0.91 " -0.90 ' 1 1 1 1 ' >- ' ' 1 — 0 2000 4000 6000 8000 10000 Wavenumber (cm"1) Figure 3-5: Reflectivity of Aliiminum, from reference [27]. For low frequencies the reflectivity is smoothly extrapolated to unity at zero frequency using the R «= V? relationship appropriate for metals. 34 TT JC ton f~ln[/?(co)]-ln[/?(co0)] 0/J<fco COQ -CO (3.5) The major drawback of this procedure is that one needs to know the power reflectivity over the entire spectral range. This problem is usually overcome by appending other researchers' data to ones own and making appropriate extrapolations when necessary. For light normally incident on a reflecting surface in vacuum the relationship between complex reflectivity and the complex index of refraction is given by: „ _ n - 1 (3.6) n + 1 which leads to: R = (l-n)2+k-(\ + n)2+k: (3.7) and 6 = tan" 2k .n2 + k2-l (3.8) The complex index of refraction is obtained by inverting the above two equations: 1-fl (3.9) n = 1 + R - 2>[R cos0 ' and k = 24H sine 1 + R - 2-jR COS6 ' (3.10) The complex dielectric function is related to the complex index of refraction by: e=n2 =(n + ik)2. (3.11) 35 Equating the real and imaginary parts gives £1 and £2 in terras of n and k, and also in terms of R and 6 from Equations (3.9) and (3.10): £ = k i = ( l - * ) 2 - 4 / ? s i n 2 e (3.12) 4v^(l-/?)sine (3-13) £2 — Ink • (l + R-2-jRcosQ)2 The real part of the conductivity, ai, is related to the imaginary part of the dielectric function by: VE^ (3.1.4) 1 " 60 ' where the factor of 60 in the denominator is required to give d units of (Qcm)"1. 36 Chapter 4 Experimental Results and Analysis 4.1 Far-Infrared Reflectivity The films used in these studies were grown by laser ablation at 770 °C on a circular SrTi0 3 substrate to a thickness of -1500 A . Typically they had T c values of 90 K and a resistivity of 30 |iQ-cm at 100 K. Figure 4-1 shows the ratio of R(B)/R(0) for a range of different magnetic field strengths obtained with a 6 |im thick beamsplitter, which covers the frequency range from 50 to 400 cm"1. Even at the modest maximum field of 3.5 T the effect is seen to be quite large, and the signal-to-noise ratio in excess of 200 allows clear differentiation between the different fields. The crossover point near 100 cm"1 separates the region of decreasing reflectivity with magnetic field from that of increasing reflectivity. This agrees with the transmission results of reference [13], where the strongly increasing transmission below 100 cm"1 must be mainly due to this reflectivity behaviour. Some of the smaller features in the spectra are a result of incomplete cancellation of instrumental features in the two spectra, but most are believed to due to phonons (see later conductivity spectra). Figure 4-2 shows the results obtained with a 23 am thick beamsplitter which is more sensitive in the cross over region from 35 to 140 cm"1. Figure 4-3 shows the power reflectivity of the Y B C O film at room temperature and at 20 K to 9000 cm"1. These were performed as separate experiments in the spectrometer using the reflectivity module described in Chapter 3.4. The accuracy of these data is between 1 and 2%. Figure 4-4 shows the reflectivity from 30 to 1000 cm"1 of an uncoated SrTi0 3 substrate using the same reflectivity module. The characteristic dip near 500 cm"1 in this spectrum, which agrees with the published data [27], is completely absent from the Y B C O reflectivity data in Figure 4-3, indicating that the substrate is not contributing to the signal. 37 o at m 04 1.04 1.02 1.00 0.98 0.96 0.94 0.92 0.90 \ v. 2.8 T / / / / 3.5 T / Y B C O on SrTi0 3 ) T = 4.2 K 0 100 200 300 400 Wavenumber (cm ) Figure 4-1: The ratio of normal-incidence power reflectivity of a superconducting YBa 2Cu 307-s film in a magnetic field B to that in no field, using the apparatus in Figure 3-2, and a 6 am thick mylar beamsplitter, which covered the range from 50 cm"1 to 400 cm"1. The resolution was 4 cm"1. 1.05 Y B C O on SrTi0 3 , T = 4.2 K 1.00 o 0.95 2.8T \ y y 3.5 T 0.90 0.85 I . L 20 40 60 80 100 120 140 Wavenumber (cm ) Figure 4-2: As for Figure 4-1 but with a 23 urn thick mylar beamsplitter, which increased sensitivity in the range from 30 cm"1 to 140 cm"1. The resolution was 4 cm"1. 39 Figure 4-3: The power reflectivity of a superconducting YBa2Cu307.5 film at room temperatre and at 20 K measured in the reflectivity module described in Chapter 0. The resolution was 4 cm"1. 40 41 Figure 4-5 shows the reflectivity of the five field strengths resulting from the application of the measured R(B)/R(0) ratios to the measured reflectivity at 20 K in zero field. The data in Figure 4-2 indicates that R(B)/R(0) is increasing below 35 cm"1 but the signal-to-noise ratio was poor at these low energies and the curves in Figure 4-5 were smoothly extrapolated to unity at zero frequency. There is also a problem with the high frequency exrapolation required for the Kramers-Kronig analysis. Figure 4-5 shows clearly that R(B) is greater than R(0) for large fields above 100 cm"1 and beyond 400 cm"1. Exrapolating R(B) to smoothly meet R(0) at high frequencies results in negative conductivities from the K K analysis. Clearly the reflectivity following a resonance should dip at some higher frequency but that point is outside the current experimental range. Arbitrarily extrapolating R(B) to 0.8 R(0) at 350 000 cm"1, which causes the 3.5 T curve to cross the zero field curve at -550 cm"1, reduces but does not eliminate the negative conductivity problem. 4.2 Far-Infrared Magneto-Conductivity of YBa 2Cu 307.s Figure 4-6 shows the conductivities resulting from the Kramers-Kronig analyses for the two lowest fields as well as zero field. While there is some structure in the latter case, the application of a field produced a broad response between 50 and 250 cm'1. Tic-marks have been placed near the 0.7 T curve at the wavenumbers of observed low energy infrared active phonons [29], i.e., at 150, 192, 286, and 317 cm"1 even those these were seen in c-axis films and would not be expected in these a-b films. A tic-mark has also been placed at 102 cm"1, the frequency calculated for the lowest energy ab-plane polarized phonon [30], and another at 230 cm"1, the frequency of an observed raman active phonon [31], allowing for the possibility of activation by coupling to this magneto-conductivity peak. Figure 4-7 shows the conductivities for the two highest fields, with the phonon tic-marks near the 2.1 T curve. Because the 3.5 T reflectivity in Figure 4-5 approaches unity to closely (e.g. 0.99 at 170 cm"1) the Kramers-Kronig analysis becomes extremely sensitive to the accuracy of the reflectivity 42 1.00 /W 0.98 0.96 £ 0 9 4 0 flj 0 1 OT K\\^ T../-r// •. '.2:1 T / / / WI J •'. / 3.5 T / vy 0.92 0.90 0.88 v. ^  x ' ' \ YBCO on SrTi03, 4.2 K 0.86 0 100 200 300 400 Wavenumber (cm" ) Figure 4-5: The power reflectivity below 400 cm"1 of a superconducting YBa 2Cu 307. 5 film at 4.2 K as a function of magnetic field. A smooth extrapolation to unity at zero frequency has been applied below 35 cm"1 to the field dependent values. Figure 4-6: The conductivities resulting from the Kramers-Kronig analyses of the reflectivity data at zero field and for the two lowest field values. The tic-marks indicate phonon frequencies reported elsewhere. 44 Figure 4-7: The conductivities resulting from the Kramers-Kronig analyses of the reflectivity data at 2.1 T and maximum value of 3.5 T. the near-unity value of the 3.5 T reflectivity at two points in Figure 4-5 produce a large uncertainty in the values of the dip and peak conuctivities (see text). The tic-marks indicate phonon frequencies reported elsewhere. 45 both in this region and outside. The negative dip and high positive peak therefore have large errors associated with them due to the small errors in the reflectivity. These results have been compared to that predicted by the Hsu model [11] with pinning considered. From Equation (1.7) one finds for the real part of the conductivity: With m* = 3.1 trie [1] cot is only 1.05 cm"1 at the maximum field of 3.5 T. Taking a value of 120 T (c axis) for Hc2 gives, from Equation (13), Q 0 = 15 cm"1, n is taken as 0.25 holes/unit cell, or 1.42 x 1021 cm"3. With these values the 1.4 T curve in Figure 4-8 was fitted to (cr+ + o_) (to compare to the unpolarized results), allowing a and xv to vary. The best fit was obtained for a = 133 cm"1 and 1/Tv = 197 cm"1. However, the calculated conductivity had to be scaled up by a factor of 44 to agree with the experimental result. This scaling factor has been included in the calculated curves in Figure 4-8, but its requirement demonstrates a clear discrepancy between the theory and these experimental results. (4.1) 46 12000 Wavenumber (cm ) Figure 4-8: Comparison of the calculated (solid curves) and experimental (dashed and dotted curves) conductivities at 1.4 T and 0.7 T. The fit was performed on the 1.4 T curve and yielded a = 133 cm"1 and l/xv = 197 cm 1 . A scaling factor of 44 was also required. The same values were used to calculate the 0.7 T curve which lies underneath the experimental curve by an almost constant background value. 47 Chapter 5 Conclusion In conclusion, the far-infrared reflectivity of a superconducting film of YBa2Cu307-8 has been measured measured in a magnetic field up to 3.5 T and was found to be very sensitive to the application of a magnetic field. Given such sensitivity, this technique is much more direct than transmission measurements, since the latter depend both on absorption in the film and reflection from multiple interfaces. If the reflectivity is affected by the magnetic field by the amounts measured, then the increased transmission below 100 cm'1 observed by other groups must be in large part due to this, and not solely due to a change in absorption. Thus the conductivty obtained from the Kramers-Kronig analysis leads to a conductivity peak at a much higher wavenumber than that inferred from the tranmission experiments. These preliminary results show a field dependent conductivity as predicted by theory [11]. While the shapes of the low field conductivity results can be fit to the theory, there is a clear discrepancy in the magnitude. It is possible that the observed effect has a completely different origin and may be related to film quality. Another explanation may involve the large zero-field conductivity and a possible redistribution of spectral weight upon application of a magnetic field. The possibility that others have misinterpreted their data should not be ruled out. In order to properly perform the Kramers-Kronig analysis , more measurements, of high quality film grown on high quality substrates, extended into the mid and near-infrared are required. 48 Bibliography [1] H.D. Drew, E. Choi, K. Karrai, Physica B 197, 624 (1994). [2] W. Kohn, Phys. Rev. 123, 1242 (1961). [3] C. Caroli, P.G. de Gennes and J. Maricon, Phys. Lett. 9, 307 (1964); J. Bardeen, R. Kiimmel, A .E . Jacobs and L. Tewordt, Phys. Rev. 187, 556 (1969). [4] F. Gygi and M . Schluter, Phys. Rev. B 41, 822 (1990). [5] Y.D. Zhu, E C . Zhang and H.D. Drew, Phys Rev. B 47, 596 (1993). [6] B. Janko and J.D. Shore, Phys Rev. B 46,9270 (1992). [7] N.C. Yeh, Phys Rev. 5 43,523 (1991). [8] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67, 386 (1991). [9] J. Bardeen and M.J. Stephens, Phys. Rev. 140, A1197 (1965). [10] P. Nozieres and WF. Vinen, Phil Mag, 14, 667 (1966). [11] T.C. Hsu, Phys Rev. B 46, 3680 (1992); T.C. Hsu, J. Phys. C 213, 305 (1993). [12] E.J. Choi, H.T.S. Lihn, H.D. Drew, and T.C. Hsu, Phys. Rev. B 49,13271 (1994). [13] K. Karrai, E.J. Choi, F. Dunmore, S. Liu.H.D. Drew, Qi Li , D.B. Fenner, Y.D. Zhu, and Fu-Chun Zhang, Phys. Rev. Lett. 69, 152 (1992). [14] K. Karrai, E. Choi, F. Dunmore, S. Liu, Z. Ying, Qi Li , T. Venkatesan, H.D. Drew, and D.B. Fenner, Phys. Rev. Lett. 69, 355 (1992). [15] W. Herres and J. Gronholz, Comp. Appl. Lab 2, 216 (1984). [16] J. Gronholz and W. Herres, Instruments & Computers 3,10 (1985). [17] R.J. Bell, Introductory Fourier Transform Spectroscopy, Academic Press, New York (1972). [18] L. Mertz, Infrared Physics 7, 17 (1967). [19] L. Genzel and J. Kuhl, Infrared Physics 18, 113 (1978). 49 [20] Martin N. Wilson, Superconducting Magnets, Oxford University Press, New York, (1983). [21] R. Wesche and A . M . Fuchs, Cryogenics, 34,145 (1994). [22] W.N. Hardy, private communication. [23] W.H Warren, Jr. and W.G. Bader, Rev. Sci. Instr., 40, 180 (1969). [24] "Installation and Application Notes for Cryogenic Sensors", Lakeshore Cryogenics, Inc. [25] H.R. Collard and R.C. Hogan, Infrared Physics, 26, 51 (1986). [26] J.E. Eldridge and C C . Homes, Infrared Physics, 29,143 (1989). [27] E.D. Palik, Handbook of Optical Constants of Solids, (Academic Press, New York, (1985), p. 369. [28] F. Wooten, Optical Properties of Solids, (Academic Press, New York, 1972). [29] C C . Homes, T. Timusk, R. Liang, D.A. Bonn, and W.N. Hardy, Phys. Rev. Lett. 71, 1645 (1993). [30] F.E. Bates and J.E. Eldridge, Solid State Commun. 64,1435 (1987). [31] R.M. Macfarlane, H.J. rosen, E .M. Engler, R.D. Jacowitz, and V . Y Lee, Phys. Rev. B 38, 284 (1988). 

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