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Resistivity measurements of thin films of bismuth : applications for building bolometric detectors Padwick, Christopher Grant 1997

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RESISTIVITY MEASUREMENTS OF THIN FILMS OF BISMUTH: APPLICATIONS FOR BUILDING BOLOMETRIC DETECTORS By Christopher Grant Padwick B. Sc. Hons (Physics) University of Regina, 1994 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R S O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Sept 1997 © Christopher Grant Padwick, 1997 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 Physics and Astronomy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 15 A b s t r a c t The resistivity of thin bismuth films grown on sapphire substrates has been measured as a function of growth rate, thickness, and temperature. Seven different samples were measured. In al l cases the resistivity of the films was completely unlike the resistivity of the bulk material, and each sample exhibited a negative temperature coefficient of resistance ( T C R ) . A model is presented which provides a physical interpretation of the shape of the resistivity - temperature curve of thin bismuth films. This model is a new contribution to this field of research, since there appears to be no satisfactory explana-tion for the anomalous temperature dependence of the resistivity of thin bismuth films published in the literature. The sheet resistance of the films at 4.2K was found to decrease with increasing thick-ness. Based on the experimental data, a bismuth film of thickness 20uA should have a sheet resistance of 188.50/D at helium temperatures. The resistance of the thinner films was found to increase as a function of time when exposed to the air, and this seems to be consistent wi th a layer of insulating B12O2, forming at the surface of the fi lm. The sheet resistance and resistivity of a 5000A thick coating of vapor deposited aluminum was measured as a function of temperature in the range 300/^ — \.2K. The resistivity was found to be greater than that of the bulk material at al l temperatures, and the sheet resistance at 80/^ was equal to Rn = 0.051O. n Table of Contents Abstract ii Table of Contents i i i List of Tables vi List of Figures vii Acknowledgements xi 1 Introduction 1 1.1 Introducing the Bolometer 1 1.1.1 Basics 1 1.2 Transmission Line Analogy 4 1.3 What is a transmission line? 4 1.4 Transmission Line wi th a Resistive Load 8 1.5 Sinusoidal Input Signal 10 1.6 Transmission Line W i t h a Resistive Short 11 1.7 Relation to Optics 12 2 Electrical Properties of Metals and T h i n Films 14 2.1 Metals, Insulators, Semiconductors and Semi-metals 14 2.2 Semi-metals 17 2.3 Fermi Energy and Fermi Surface 19 i i i 2.4 Quantum Size Effect 22 2.5 Surface Scattering 24 2.6 Gra in Boundary Scattering 25 2.7 Concluding Remarks 26 3 Growth of T h i n Fi lms 29 3.1 Equipment 29 3.1.1 Thermal Evaporation 29 3.2 Properties of T h i n F i lms 31 3.2.1 T h i n F i l m Growth 31 3.3 Effect of Pressure on Growth 37 3.4 Cohesion and Robustness 39 3.5 Substrate Cleanliness 41 3.6 Sample Preparation 41 4 Results 44 4.1 Temperature Dependence of The Resistivity 44 4.1.1 Surface Scattering 47 4.1.2 Gra in Boundary Scattering 48 4.1.3 Quantum Size Effect 49 4.1.4 A New Model 51 4.2 Low Temperature Resistivity 58 4.3 Effect of Growth Rate on Resistivity 61 4.4 Resistivity Rat io 63 4.5 Effect of Substrate 64 4.6 Room Temperature Resistivity 68 4.7 The Sheet Resistance at 4.2 K 69 iv 4.8 Oxidizat ion of Bismuth F i lms 70 4.9 Conclusions About T h i n Fi lms of Bismuth 70 5 Resistance of Composite A luminum 78 5.1 M A P Mir ror 78 5.2 Experiment 78 5.3 Results 80 5.4 Conclusions 81 5.4.1 Vapor Deposited A luminum 81 5.4.2 General Conclusions 86 Bibliography 88 v List of Tables 1.1 The specific heats of some materials at a temperature of 0.3K. The values are computed from the Debye equation, which provides excellent agree-ment with experiment 4 2.2 The effective masses in bismuth, after Smith et al. [1]. The top row are the masses at the bottom of the band, while the bottom row are the masses at the Fermi energy, cp 18 4.3 The thickness, growth rate, and pressure in the bell jar just before evapora-tion. The ambient pressure was the pressure reached after approximately 20 hours of pumping 45 4.4 Parameters derived from analyzing the data in terms of the Fuchs-Sondhiemer model. Notice that the specularity parameter must be greater than one to reproduce the values in the last column of the table 48 4.5 The parameters for the fits shown in figure 4.16 . The fits are computed from equation 4.73 56 4.6 The data for the resistivity ratio, T, for films of various thicknesses. The data is plotted in figure 4.18 63 4.7 Data showing the sheet resistance and the resistivity at 4.2K for the bis-muth films prepared in this thesis. The sheet resistance clearly decreases as the thickness increases 69 vi List of Figures 1.1 A schematic diagram of a bolometer. See the text for an explanation of how the device operates 2 1.2 Two parallel conductors with an ac voltage applied across the leads is all there is to a transmission line. The mutual capacitance and inductance between the two wires create some interesting effects 5 1.3 A transmission line with a resistive impedance connected between the terminals. Consider the transmission line to be of infinite length 12 2.4 The distribution functions for electrons(/ e) and holes ( / / J , plotted for a material at 50 K wi th a Fermi Energy of 0.027eV. As the temperature tends toward absolute zero, the distributions become step-like in nature. 21 3.5 A n S E M photo of a 2000A bismuth film grown on a sapphire substrate. The magnification in this image is 10 000 times. The tick marks on the bottom right indicate the scale of the photo 34 3.6 A n S E M photo of a 2000A bismuth film magnified 30 000 times. The crystallites are clearly visible, and the disordered nature of the film is apparent 35 3.7 A picture of a discontinuous bismuth film. The dark clumps are bismuth islands in the early stages of growth. The bright dots may be due to the formation of an oxide 36 3.8 A comparison between the lattice constants of sapphire and bismuth. . . 37 vn 3.9 The resistance of a bismuth film was measured during and after deposition. The ambient pressure in the bell jar was 1.7 x 10~ 4 Tor?\ The film is oxidizing inside the chamber, and thus its resistance is increasing wi th time. 38 3.10 The resistance of a bismuth film during and after deposition. The actual evaporation only lasted for 1 minute. The ambient pressure in the bell jar was 3.4 x 10 - 6 torr. The decrease in resistance with time is due to cooling of the film 39 3.11 A diagram of the sample used to measure the resistance of bismuth films. The finished sample is placed inside a thermal evaporator and bismuth is deposited over the entire sapphire surface 43 4.12 Resistivity versus temperature for bismuth films. The thicknesses are: A - 85A ; B - 87A ; C - 103 A ; D - 146A ; E - 500A ; G - 3005A. The resistivities of samples A, B, and C has been interpolated with straight lines in the region between the data points 46 4.13 The grain boundary scattering applied to the data. A marginal fit is obtained for the room temperature resistivity data 50 4.14 Resistivity versus temperature for bismuth films of lpm thickness. The data is from J in et al.[2]. It is interesting to note that the resistivity of the bot tom curve is actually lower than that of the bulk, for temperatures greater than 100A''. This fact was not noted by J in et al., and indeed is hard to explain 54 4.15 The transmission coefficient and the Fermi Dirac distribution for a material at 200iv' are plotted on the same graph. The Fermi-Dirac distribution at absolute zero is also shown. As the temperature increases, more particles wi l l be able to tunnel through the barrier, and the resistivity wi l l decrease. 57 vin 4.16 The data from J i n et al., Baba et al., and sample D from this thesis. The solid lines through the top 3 curves are fits computed from equation 4.73, and the parameters are listed in table 4.5. Note that the fits do not exactly reproduce the data points, but seem to describe the general shapes of the curves quite well 59 4.17 The resistivity at A.2K increases as the film thickness increases. This data disagrees with that of Baba et al.[3] 62 4.18 The resistivity ratios of the bismuth films increase as the thickness in-creases. The data point for sample F disagrees with the other data. . . 65 4.19 The resistivity of bismuth films grown on sapphire and glass, plotted as a function of inverse thickness. The solid lines are fits to the data, and the dashed line is the bulk resistivity at 295 K. The values for the room temperature resistivity from table 4.6 are plotted as well. The filled squares are data for sample F, which were taken during evaporation. It is evident that sample F behaves differently than the other samples, and radiative heating of the substrate during evaporation may have been a factor. . . 67 4.20 The sheet resistance at 4.2K as a function of the film thickness. The solid line is a fit to the data, computed from equation 4.78. The lines drawn in show that a 198A film should have a sheet resistance of 188.5S7/• at 4.2K. 71 4.21 The residuals for the fit from figure 4.20. There is considerable scatter in the sheet resistance of the thinnest films 72 4.22 The resistivity was measured as a function of time for two samples. The thicknesses are E - 500A; C - 103A 73 IX 5.23 A voltage versus current curve for the M A P mirror sample at room tem-perature. The material shows ohmic behavior over the range of current shown. The error bars on each point are the same size as the plotting symbols. The slope derived from this graph is 1.376 +- 0.005O which is in excellent agreement with measurements made using a four wire resistance technique 79 5.24 The resistivity of vapor deposited aluminum (VDA) versus temperature. The open squares are data taken while cooling the sample, and the filled squares are data taken while warming the sample 82 5.25 S E M micrograph of cross-section of V D A coated SF-70A-75 Spread Fab-r i c / Y L A RS-12D. The V D A is the bright irregular line running horizon-tally through the middle of the micrograph. The magnification is 300 times. 84 5.26 The same sample as in the previous figure, magnified 4000 times. The V D A is the bright irregular line running through the middle of the micrograph. The irregular surface of the V D A is evident from this picture. This is due to grit blasting of the carbon fiber spread before deposition of the V D A . 85 x Acknowledgements I would like to acknowledge several people who have helped me a great deal during my research and writing of this thesis. First, I would like to thank Dr. Mark Halpern of the U B C Experimental Cosmology Lab for many insightful discussions and suggestions. Working under Mark has been a great experience, and I have have found that he is always willing to help, any time of the day or night. Dr. Gregory Tucker has been a great source of knowledge, and has been very helpful thoughout the course of my grad studies. I would also like to thank Dr. Douglas Bonn for reading my thesis so promptly, and for making several very insightful comments. Finally, I would like to thank Colin Borys and Miranda Jackson for their Latex expertise, and for introducing me to new depths of insanity. XI Chapter 1 Introduction In this chapter, the basics of bolometer operation wi l l be discussed. In addition, the physics of the transmission line wi l l be discussed, with a focus on its relation to optics and applications for bolometers. 1.1 Introducing the Bolometer The B A M ( B a l l o o n Borne Anisotropy Measurement) instrument[4] is a telescope capable of measuring anisotropics in the Cosmic Microwave Background R a d i a t i o n ( C M B R ) on the scale of approximately 1 degree. The telescope operates in the following manner. Microwave light enters a two beam Fourier Transform Spectrometer after reflection from an a luminum mirror. After passage through the spectrometer, the light is directed onto two optical detectors called bolometers. These bolometers measure an interference pat-tern which is proportional to the difference in intensity between the two beams. The following discussion is directed towards the elementary physics of bolometer operation. 1.1.1 Basics A bolometer is a thermal device which measures small temperature changes. There are typically two distinct types of bolometers : composite and monolithic. A composite bolometer consists of three distinct parts: 1) A n absorbing layer. This can in principle be any electrically conductive material. Go ld and bismuth are common choices for microwave absorbers. 1 Chapter 1. Introduction 2 2) Substrate. The substrate must be an electrical insulator, be mechanically rigid, and have a very low heat capacity at low temperatures. Typica l choices of the substrate are sapphire and diamond. 3) Thermistor. This is a semiconductor device whose resistance is an extremely strong function of temperature, at low temperatures. Neutron Transmutation Doped ( N T D ) germanium is usually used. A schematic diagram of a bolometer is drawn in figure 1.1. Liquid He l ium Bath Figure 1.1: A schematic diagram of a bolometer. See the text for an explanation of how the device operates. A bolometer functions in the following manner. Electro magnetic radiation strikes the absorber, and the energy of the photon is converted to thermal energy. This thermal energy heats the substrate and in turn heats the thermistor, which changes its resistance. This change in resistance can be measured by a high gain amplifier and thus the electrical output of the device is proportional to the bolometeric intensity of the photon source, Chapter 1. Introduction 3 hence the name "bolometer". For the bolometer to operate effectively it must be operated at low temperatures. The bolometers in the B A M telescope operate at a temperature of approximately 300mA'. Another requirement for successful bolometer operation is that the bolometer be in poor thermal contact with its surroundings. If the thermal contact between the bolometer and its environment were good, then any power absorbed by the bolometer would be immediately transmitted to the cryogenic bath, and no change in resistance would be measured. The basic thermodynamics of bolometer operation are easily derived from first princi-ples. We consider a body with heat capacity C in weak thermal contact with a cryogenic bath at temperature To. The thermal conductance, G, of the link is defined in the following wxay, w Ax v 1 where k(T) is the thermal conductivity of the link, A is the cross-sectional area, and Ax is the length of the l ink. We consider only thermal conduction as contributing to the transport of heat. If the body, ini t ial ly at temperature To, is subjected to some external source of heat W , then the physics of the system can be described by the following differential equation, C ^ ^ + G(T)AT = W (1.2) dt where t is the time, and A T is the change in temperature. The solution of equation 1.2 is, W t A T = - [ l - e " ] (1.3) where r — C/G, and is called the thermal time constant. In general, r is a function of temperature since both the heat capacity of the body and the thermal conductance of the l ink are functions of temperature. Equation 1.3 shows that the temperature of the body wi l l rise in an exponential fashion to an asymptotic value of T = W/G, in a Chapter 1. Introduction 4 characteristic t ime r . One can see from equation 1.3 that if the heat capacity of the body is high, then the time constant w i l l be large, and the body wi l l respond slowly to any input of heat. Thus, one would like to minimize the heat capacity of the body in order to get a device wi th reasonably quick response. When designing a bolometer, one must consider the magnitudes of the heat capacity of both the absorber and the substrate, since these have a direct effect on the performance of the device. The specific heats of some common materials are listed in table 1.1. Mater ia l c(/j,J/molK) Debye Temperature (K) aluminum 405 428 copper 200 343 gold 165 165 bismuth 40 119 sapphire 0.05 1034 Table 1.1: The specific heats of some materials at a temperature of 0.3K. The values are computed from the Debye equation, which provides excellent agreement with experiment. 1.2 Transmission Line Analogy The physics of the transmission line wi l l be developed in the following sections. The transmission line is directly analogous to many optical systems of interest. A detailed analysis of the transmission line is useful because it provides an intuitive physical picture which can be extended exactly to many optical problems. 1.3 What is a transmission line? A transmission line consists of 2 conductors in a parallel orientation, separated by an insulator[5]. Transmission lines are very common in electronics; a regular bnc cable is an example. To understand the physics of the transmission line, consider two parallel ideal Chapter 1. Introduction 5 conductors (no internal resistance) as shown in figure 1.2. These conductors are connected by a resistive load ZL, and the circuit is driven by an A C signal. For simplicity assume that ZL — 0, and that the conductors are oriented along the z axis. The conductors, because of their proximity, are coupled by their mutual capacitance and their mutual inductance, and this gives rise to some interesting effects. Assume the conductors have mutual capacitance equal to C per unit length and mutual inductance equal to L per unit length. A thin slice dz through the conductors will have capacitance Cdz and inductance Ldz. The counter emf produced as a result of the changing current is proportional to the inductance, such that G-Figure 1.2: Two parallel conductors with an ac voltage applied across the leads is all there is to a transmission line. The mutual capacitance and inductance between the two wires create some interesting effects. n , 9V , . T , di , dV = —dz = -Ldz— (1.4) dz at ' The conductors are also coupled through their mutual capacitance, and the counter emf, dV, produces a current, dl, which is given by Chapter 1. Introduction 6 ,T di , „ , dV dl = —dz = -Cdz—. (1.5) dz ot El imina t ing dz from these equations leaves two coupled differential equations 1.6 and 1.7. dV rdl Differentiating 1.6 with respect to z and differentiating 1.7 with respect to t yields the following: d2V r d2I - L J ^ l (1-8) d z 2 did d2I „d2V dzdt ° 3t2 (1-9) Direct substitution of equation 1.9 into 1.8 yields the following second order partial differential equation for the voltage: d2V _ 82V _ 1 d2V J ^ - L C ~ d v - ^ ~ d ¥ - ( 1 - 1 0 ) Equation 1.10 is readily recognized as the one dimensional wave equation. It describes the propagation of the voltage in space and time down the transmission line. A similar equation may be derived for the current: d2I _ d2I _ 1 d2I dz2~LL^-V2W2 ( L 1 1 ) Chapter 1. Introduction 7 From equations 1.10 and 1.11 we see that the voltage and the current propagate down the transmission line with speed v = 1/s/LC. For two conductors in a coaxial configuration one may derive the capacitance per unit length C to be ln(b/a) where b is the radius of the outer conductor, a is the radius of the inner conductor, and e is the dielectric constant of the insulating material separating the conductors. The inductance per unit length, L , is L = ^Iri'b/a) (1.13) where p, is the magnetic permeability of the insulating material. The speed of propagation, v, is from 1.12 and 1.13 , 1 VLC ' ^ ' 1 4 ' ) Equation 1.14 shows that i f the conductors are in vacuum and separated only by free space, such that p = p0 and e = e0, then the speed of propagation wi l l be v = l/^/p0eQ, which is equal to the speed of light. The solution of equation 1.10 is composed of two traveling waves, one moving to the left and one moving to the right. The solution is of the following form: V(z,t) = W^t - z/v) + W2(t + z/v) (1.15) Differentiating 1.15 with respect to z yields % = - z/v) + l-W'{t + z/v) = -Ld-L (1.16) Chapter 1. Introduction 8 Integrating 1.16 with respect to time yields the following equation for the current where the constant comes from the integration and is not of interest for the traveling wave solution. The impedance of the transmission line can be derived by taking the ratio of V to / for either of the traveling waves. One ends up with the so called "characteristic impedance" of the line, which is defined by the following equation, Equation 1.18 shows that the impedance of an ideal transmission line whose ends are shorted(.Zx = 0) depends only on the mutual capacitance and mutual inductance between the conductors. The current and voltage are waves which travel at speed v along the transmission line in the positive and negative z directions. 1.4 Transmission Line with a Resistive Load In this section we wi l l consider an ideal transmission line with a resistive load attached between the outputs. We wi l l develop expressions for the transmission, and reflection of waves through a resistive boundary, as well as developing expressions for the power transmitted to and reflected from the load. These equations are identical to their optical counterparts. Referring to figure 1.2, the load, ZL, is attached at z = 0, and the A C generator is attached at the inputs which are located at z — — /. As before the trans-mission line is ideal and has characteristic impedance Znne. The traveling wave solution developed in section 1.2 consists of a wave moving to the right, and a wave moving to the left. The voltage across the load resistor will be given by the sum of the two traveling I(z,t) (Wi — W2) + constant Chapter 1. Introduction 9 V(0,t) = W1(t) + W2(t) = Vload. (1.19) The current through the load must be the sum of currents entering and leaving the load and is given by l(0,t) = -^—(W^t) - W2(t)) = Iload = (1.20) In these equations, W\ refers to the transmitted wave and W2 refers to the reflected wave. We may calculate the reflection coefficient, p, from 1.19 and 1.20 by taking the ratio W2(t) to Wi(i) as follows: P=W=7—TT~-  ( L 2 1 ) VV! ^load + Aline The transmission coefficient, r, can be calculated by taking the ratio of the load voltage, Vioad, to the transmitted wave W\, Vload 2 Zl0ad r = (1.22) Wi Zload + Znne From equations 1.21 and 1.22 we can see that if Zioad = Znne, p = 0 and r = 1. This corresponds to no reflected wave, and the load impedance, Zioad, is said to be impedance matched to the transmission line. If Zioad — °° ; a s would be the case for an open circuit, r 1 — ' Zune Zload -. ,-. r.r>\ lim P=y—-z—-pf— = 1, (1-23) and we see that there is only reflection from the boundary and no transmission. In the case of a short circuit, Z\oa<i = 0, p — —1, and again there is only reflection. The power in the transmitted component of the wave is P+ — I+W+ — W2/Zune, while the power in the reflected component of the wave is P_ = W2/Zune- The fractional reflected power, Pre/, is the ratio of P- to P+, which is Prej = ^ = p2. (1.24) Chapter 1. Introduction 10 From conservation of energy, the power transmitted to the load resistor must be, Ptrans = l-Pref = 1- P2. (1-25) 1.5 Sinusoidal Input Signal Thus far the results obtained have been in general applicable to any type of A C input signal. Special consideration of the case where the input signal is sinusoidal in nature warrants some attention, since the sinusoidal driving voltage is so commonplace. In particular, let us consider the case where the voltage applied to the transmission line varies according to the following equation, V(-l,t) = Vcos(ut). (1.26) The voltage on the line wi l l again be the sum of two traveling waves. In phasor notation the voltage V(z,t) on the transmission line wil l be, V(z, t) = Wxe'l(iz + W2eipz (1.27) where W\ — \W\\ and W2 — \W2\el9p, and 3 = u/v. The phase shift, 6P, could be applied to either W\ or W2. The current on the line wi l l be given by, I(K^t) = ^—(W1e-lpz-W2elPz). (1.28) The transmission and reflection coefficients for this system remain exactly the same as previously derived. Since we have assumed a definite form for the driving voltage, we may now calculate the input impedance. This is the impedance seen by the generator(located at z — —I) looking down the transmission line at the load(located at z = 0). The impedance at any point on the transmission line can be found by taking the ratio of V(z,t) and I(z,t). Chapter 1. Introduction 11 Div id ing equation 1.27 by equation 1.28 one can obtain, z{-l,t) = zllne _ %_m. (1-29) The ratio W2/W1 is the reflection coefficient, p, from equation 1.21. Substituting for p in equation 1.29 yields the following expression for the input impedance, ry, , ,x ry Zioadcos(3l) + iZlmesin(3l) As a check of the result 1.30, the impedance at z — 0 should be just the load impedance, Zioad- Setting / = 0 in equation 1.30 shows that this is indeed the case. It is worth noting that i f u = 0, as for a D C signal, then the input impedance as seen from the generator wi l l just be the load impedance, Zioad- This makes sense since we have assumed that the transmission line is composed of ideal conductors. One can define a quantity called the admittance, Y(z,t), by defining Yioad = l/Zioad, and Ynne — 1/Zune. Then the admittance as seen from the generator is, W 7 + X _ v Y'oadCOs(8l) + iYUnesin(0l) • Yiinecos(3l) + iYioadsin(3l) 1.6 T r a n s m i s s i o n L i n e W i t h a R e s i s t i v e S h o r t Now we consider the situation depicted in figure 1.3. The transmission line is shorted by a resistive impedance Zi. The resistive short and the rest of the line act like impedances in parallel, with equivalent impedance Ztot = ^L^Hne!{EL + Ziine)- The fractional power delivered to such a load wi l l be given by equation 1.25, and is, p - = 1 - ( r a : ) 2 - ^ One can see from equation 1.25 that if Zi = 0, the line is shorted, and no power is transmitted to the load. Taking the other extreme, if ZL — 00, the transmission line is no longer shorted, and all the incident power is transmitted to the rest of the line. Chapter 1. Introduction 12 Figure 1.3: A transmission line with a resistive impedance connected between the termi-nals. Consider the transmission line to be of infinite length. The power dissipated in the resistive short will be PL = W2/ZL and the power dissipated in the rest of the line will be Pune = W2/Znne. Taking the ratio PL to Pnne, we find, PL Zline ( i 3 3 ) Pline ZL From equation 1.32, we notice that if ZL = Zune/2, then the fractional transmitted power will be 75%, and the fractional reflected power will be 25%. Half as much power will be dissipated in the load impedance as in the line. 1.7 Relation to Optics Thus far we have only considered the transmission line in relation to the analysis of electrical circuits. However, it turns out that many optical problems can be formulated in terms of the transmission line analogy. To see this, we note that Maxwell's equations lead to the following equation for the electric field, d2Ex d2Ex , x dz2 r dt2 Chapter 1. Introduction 13 which is the one dimensional wave equation. The solution of this equation is composed of two traveling waves, given by, Ex(z) = E+e-jkz + E„ejkz. (1.35) Similarly, the magnetic field is composed of traveling wave solutions of the form, Hy = ^[E+e~'kz - E_e>kz] (1.36) Equations 1.35 and 1.36 have the same form as the equations derived previously for the voltage and the current on a transmission line(equations 1.27 and 1.28). In fact, one can transform exactly between the two sets of equations if one makes the following substitutions: V = Ex, I = Hy, L = [i, C = e, and = ZQ. The quantity "q = y ^ / e is the ratio of the electric field to the magnetic field, and has units of ohms. It is typically called the characteristic impedance of the medium. For example, the characteristic impedance of free space is r) = ^J/j,0/eo = 3770. A l l the equations previously derived for the reflection coefficient, transmission co-efficient, admittance, etc. can be used for electro magnetic waves, provided we make the substitutions described above. In particular, the transmission line with a resistive short provides a useful model of a bolometer. Consider light traveling through free space, incident on a resistive short of characteristic impedance ZL. This is exactly anal-ogous to the situation described in the previous section, except we make the substitu-t ion Zune = n = 3770. M a x i m u m power wi l l be absorbed in the resistive short when ZL = 3770/2 = 188.50, and thus we strive to produce an absorbing layer wi th sheet resistance 1 8 8 . 5 0 / D at low temperatures. Chapter 2 Electrical Properties of Metals and T h i n Films 2.1 Metals, Insulators, Semiconductors and Semi-metals The electrical properties of many materials can be analyzed successfully in terms of band theory. In this theory the interactions between the electrons in a material and the crystal lattice cause the energy spectrum of the electrons to be broken up into discrete levels called bands. The electrons are only allowed to occupy certain discrete states within these energy bands, and these states correspond to discrete values of the electron wave-vector k. There are no available states in between bands. The gap in energy between these allowed bands is called the band gap, and corresponds to forbidden zones. The number of available states in k space in each energy band can be calculated quite easily[6]. We consider a one dimensional chain of atoms of length L. The spacing between adjacent atoms is a, and this is called the lattice constant. The energy gaps occur for values of A; = nir/a, where n is any positive or negative integer, and thus the width of an energy band is TT/U. The wave-vector can only take on discrete values inside an energy band. These discrete values are |fc| — 2nn/L. The number of available states in an energy band is ^ , where the negative values of the wave-vector have been accounted for. Al lowing for spin degeneracy, since we are allowed to pack 2 electrons into every state k, gives where the quantity N is the number of atoms per unit length. The extension to three # of states = 2 - = 2N, a 14 Chapter 2. Electrical Properties of Metals and Thin Films 15 dimensions is straightforward, and gives the same result. Equation 2.37 shows that there are two allowed states for every atom in the crystal. This is an important result, since now we can predict the electrical qualities of a material given the number of valence electrons per atom. If a material has two valence electrons, then all available states wi l l be filled. The electrons in a filled band have no empty states in which to move, and an electric field applied to the crystal wi l l produce no net electron flow. Such a material is said to be an insulator. O n the other hand, i f there is only one valence electron per atom, the highest energy band wi l l only be half filled, and the substance wi l l be a conductor. This general rule applies very well to most of the elements in the periodic table, however, there are some very important exceptions. Silicon and germanium each have two valence electrons per atom, and should be electrical insulators. A t low temperatures this is true for very pure samples. A t higher temperatures, however, these substances exhibit a resistivity which is an exponential function of temperature. For this reason, these materials are called semi-conductors. The behavior of a semiconductor can be understood in terms of band theory. In an insulator, the band gap is large, and there is no way to introduce a transition from a lower energy band to a higher energy band. In a semi-conductor, the band gap is small enough that electrons in the lower energy band can be thermally excited to occupy states in the higher energy band. Thus at higher temperatures, there are available states for the electrons to jump to, and the material is a conductor. The conductivity of a semi-conductor wi l l be governed by the distribution function of the electrons, which varies exponentially with temperature. A t low temperatures, the electrons fill the lowest energy band, and since the electrons do not have enough thermal energy to make the transition to the higher band, the material is an insulator. The conduction in semi-conductors is actually determined by the motion of two types of carriers. When an electron makes the transition from a lower energy band to a higher Chapter 2. Electrical Properties of Metals and Thin Films 16 one, it leaves an unfilled state in the lower energy band. This unfilled state is not very surprisingly called a hole. Since the lower band now has an extra state, which other electrons may occupy, the conduction of the material increases. The hole in the lower energy band acts like a particle with a charge opposite to that of the electron, and will move in the opposite direction to an electron if an electric field is applied. Thus, the movement of the hole contributes to the current of the material. The total current will be the sum of the hole current and the electron current, j - jh + je = nhqhvh + neqeve. (2.38) If we treat the holes as particles which can undergo collisions, the average velocity in response to an electric field will be given by Vh = qhETh/rrih, and we can write j = [+nhv~k - neve] e, (2.39) since the holes and electrons have an opposite charge. The holes and electrons will move in opposite directions, and equation 2.39 can be written in the following way, j = From the Free Electron Theory, one can derive [7] a value of the conductivity of a = ne2r/m, where r is the average time between collisions, n is the number of electrons, e is the electronic charge, and m is the free electron mass. With this in mind, we see that the terms on the right hand side of equation 2.40 are just the conductivities of the holes and the electrons, respectively. Equation 2.40 is clearer is if we write the conductivity in terms of the mobility of the carriers, which is defined as p = e2r/m. Using this definition, equation 2.40 becomes, j = [rihPh + nepe] E. (2.41) nhe"rh nee"Te mh + E. (2.40) Chapter 2. Electrical Properties of Metals and Thin Films 17 Thus, the current in the material wi l l be determined by the number densities and the mobilites of the carriers. The resistivity is found by taking the ratio of the electric field to the current density. Doing this, we find that the conductivity and the resistivity are related in the following way for a two carrier system, We conclude by stating that the resistivity depends inversely on the number densities and the mobilites of the carriers in the material. 2 . 2 Semi-metals A semi-metal is a material which has metallic properties, but is distinguished from other metals by its low conductivity. The semi-metal elements are B i , As , and Sb. Of the three elements, bismuth has the highest resistivity. The reason the resistivity is so high for these elements is due to a couple of intriguing effects. The crystal structure of these three elements is the same, and is referred to as hexagonal rhombic. The hexagonal rhombic structure can be obtained from the simple cubic structure by making a couple of slight distortions. It is these distortions which are responsible for the strange behavior of semi-metals. If the crystal structure of these materials was simple cubic, the Fermi Surface would be spherical, and these elements would exhibit a much lower resistivity. The slight distortion of the crystal lattice causes the Fermi Surface to become ellipsoidal, and causes the conduction and valence bands to overlap. The resulting Fermi Surface consists of three small pockets of electrons, and one small pocket of holes. This is the fundamental reason why the behavior of semi-metals differs from that of normal metals. The tremendous reduction of the carrier density compared to a good conductor such as a luminum causes the conductivity to be lower in the semi-metals. p - - -v nhph + nefie 1 (2.42) Chapter 2. Electrical Properties of Metals and Thin Films 18 The conduction in a semi-metal is highly anisotropic with respect to the crystal axes. This is due to the anisotropy of the Fermi Surface. One refers to the effective mass of the carriers, which is defined by the following equation[7], m* = h2 '02E dk2 (2.43) where E is the energy and k is the wave-vector. The effective mass depends on the second derivative of the energy in the band with respect to the wave-vector. If this derivative is negative, then the effective mass wi l l be negative, and the carrier is no longer an electron, but a hole. The reason the effective mass is a useful quantity is because the carriers in a material, when under the influence of an electric field, move as if they had a mass defined by equation 2.43. In general the effective mass is a tensor quantity given by the following relation, ( - ) = 1 - ^ - (2 44) The effective masses for the carriers in bismuth have been measured, and are in general quite a bit lower than the mass of the free electron. The values for the effective masses in bismuth are tabulated in table 2.2, in units of the free electron mass (mo). m n m22 m33 m23 0.00113 0.000521 0.26 1.20 0.00443 0.0204 -0.0195 -0.090 Table 2.2: The effective masses in bismuth, after Smith et al. [1]. The top row are the masses at the bottom of the band, while the bottom row are the masses at the Fermi energy, tF. The effective masses are positive in three directions of the crystal, which correspond to conduction by electrons, and negative in one direction, which corresponds to conduction by holes. The carrier density has also been measured by Smith et al.[l], and found to be Nbi = 2.75 x 1 0 1 7 c m - 3 , at room temperature. This carrier density is much lower than that Chapter 2. Electrical Properties of Metals and Thin Films 19 of a normal metal like aluminum, which has a carrier density of Nai = 18 x 1 0 2 2 c m ~ 3 . In general, all the semi-metals have lower carrier densities and lower effective masses than normal metals do. 2.3 F e r m i E n e r g y a n d F e r m i Sur face The Fermi Surface is the boundary between occupied and unoccupied states at absolute zero. A l l the wave-vectors with magnitudes less than or equal to the Fermi wave-vector, kp, represent occupied states. A t absolute zero, all the electrons occupy the ground state. The low carrier densities and the low effective masses in the semi-metals combine to produce materials with very low Fermi Energies. If the Fermi Surface of the material is spherical, the Fermi energy is given by the following equation[7], The Fermi energy of a metal is typically quite high, and this is due to the high carrier density. For example, the Fermi energy of aluminum is ep — 11.3eV (assuming a spherical Fermi energy surface). We may define the Fermi Temperature, Tp, in the following way, where ks is Boltzmann's constant. W i t h this formula, the Fermi Temperature of alu-minum is Tp = 134 500K. This Fermi Temperature has nothing to do with the actual temperature of the electrons. It basically represents the ground state energy, and equiv-alent temperature, of the electrons in a material. In general, the Fermi Surface of most metals and semi-metals is non spherical, and equation 2.45 does not apply. The low carrier densities and low effective masses of the carriers in bismuth combine to give it a very low Fermi energy. The Fermi energy of bismuth has been measured by Smith et al.[l], and found to be ep — 0.027eV. From this, TF = kB (2.46) Chapter 2. Electrical Properties of Metals and Thin Films 20 one derives a Fermi Temperature of Tp = 320A', which is substantially lower than that of aluminum. The low value of the Fermi Energy means that the distribution of carriers can be changed substantially by temperature changes in the range of the Fermi Temperature. Since electrons and holes are fermions, the Fermi-Dirac distribution function applies. The distribution function for fermions is defined as follows, fe = r ^ - i (2.47) where e is the energy of the fermions, and T is the temperature. This distribution is a step function at absolute zero (T = 0), with the location of the step being at the Fermi Energy. The distribution of the holes created by thermal excitation of electrons in a material must be related to the electron distribution. In fact, the distribution function of holes is given by the following equation, which seems quite intuitive, h = l- fe- (2-48) Equation 2.47 shows that the distribution of electrons in a material is a function of the temperature of the system. The distribution functions for electrons and holes in bismuth are plotted in figure 2.4. Figure 2.4 shows that there are a substantial fraction of electrons with energies higher than the Fermi Energy, when the temperature of the system is 50 K. With a material like aluminum, which has a very high Fermi Energy, the distribution functions of the carriers are not modified appreciably by changes in the temperature of the system. The mean free path of the electrons on the Fermi Surface is a quantity of interest. It may be calculated from the following equation, after Kittel[7], A (2me F) 1/ 2 , x Approximating the effective mass of bismuth with m* = 0.01mo and using the previously quoted value of the Fermi Energy yields a mean free path of A#8 = 10, 900A at room Chapter 2. Electrical Properties of Metals and Thin Films 21 Figure 2.4: The distribution functions for electrons(/ e) and holes(/^), plotted for a mate-r ia l at 50K wi th a Fermi Energy of 0.027eV. As the temperature tends toward absolute zero, the distributions become step-like in nature. Chapter 2. Electrical Properties of Metals and Thin Films 22 temperature. A similar calculation for aluminum yields a mean free path of AAI = 143A. The long mean free path of the carriers in bismuth as compared to the carriers in a luminum is due to the tremendous differences (a factor of 105!) in the carrier concentrations. 2 . 4 Quantum Size Effect The quantum size effect (QSE) was first worked out by Sandomirskii[8] in 1967. The argument is as follows. When the dimensions of a material become comparable to the effective wavelength of the carriers at the Fermi Surface, the energy levels in the material wi l l be modified. The energy spectrum, which in the bulk material consists of discrete bands, w i l l be further quantized by the reduction in sample thickness. This quantization wi l l modify the density of states on the Fermi Surface, and wi l l have an effect on the mobil i ty of the carriers. From equation 2.41, we see that the conductivity of a material depends on the carrier density of holes and electrons, and on the mobil i ty of each carrier. So, the Q S E wi l l have an effect on the conductivity of the material, and wi l l affect al l quantities which depend on the mobil i ty of the charge carriers. The Q S E is only observable in thin films, since the dimensions of the film may be comparable to the wavelength of the carriers at the Fermi Surface. In normal metals, the carrier wavelength is very short. For example, the carrier wavelength of a luminum is, where the value of 6p = 11.3eV has been used. The calculation shows that quantum size effects wi l l be observable in aluminum films when the film thickness is on the order of impossible, since a film of this thickness would certainly be discontinuous, and the Q S E would be masked by other effects. The situation is different for the semi-metals, which A = = 3.6A (2.50) 3 — AA. In practice, however, observation of Q S E in a film of this thickness would be Chapter 2. Electrical Properties of Metals and Thin Films 23 have a low Fermi Energy, and thus a longer carrier wavelength. The carrier wavelength for bismuth can be estimated from equation 2.50, using ep = 0.027eV, with the result that A = 740A. A bismuth film of this thickness wi l l be continuous, and in theory, the Q S E wi l l be readily observable. The Q S E model has some special predictions in the case of a semi-metal. In the bulk material , the conduction and valence bands overlap by an amount 8. The Q S E causes the conduction and valence bands to break up into discrete sub-bands. Consider a 3D system wi th dimensions t x Ly x Lz. The dimensions in the y and z directions are to be considered infinite. The energy states of such a system are given by the following, Es,ky,kz = ens2 + h2(k2y + k2z)/2mn (2.51) where the subscript on the mass, ran, refers to the electron mass, and s is a positive integer. The wave-vectors in the z and y directions are ky — 27rsy/Ly and kz = 2nsz/Lz, sy = sz = 0, + — 1, +—2,... , and Ly and Lz are the lengths of the sides of the normalization rectangle. The quantity e„ in equation 2.51 is the quantized energy due to the Q S E , and is given by where t is the thickness of the film. The energy states of the holes can be obtained from equation 2.51 by replacing the electron mass, m n , by the mass of the hole, mp. The Q S E affects the energy states of the system in only one component of the wave-vector. The band overlap, 5, is equal to the sum of the Fermi Energies of the electrons and the holes, 8 = en + c p . (2.53) A very intriguing prediction of the Q S E model is the so-called semi-metal to semi-conductor. ( S M S C ) transition. The band overlap, 8, is a function of the thickness of the Chapter 2. Electrical Properties of Metals and Thin Films 24 fi lm, and decreases as the film thickness decreases. This is due to the movement of the electron band upwards and the hole band downwards. A t some crit ical thickness, t < a, the electron and hole bands w i l l no longer overlap. The result is a band gap of magnitude tg = S(a2/t2 — 1), and the material is no longer a semi-metal, but is a semiconductor. This transition wi l l occur at a crit ical thickness which is equal to a = nh/\/2MS, where M = mnmPl(mn-\-mp). This crit ical thickness is approximately 400A for bismuth. F i lms thinner than this are expected to exhibit semiconductor properties, while films thicker than this maintain an overlap in energy, and are semi-metals. The Q S E model makes several other quantitative predictions. According to the theory, the electrical conductivity of thin semi-metal films w i l l be an oscillatory function of the thickness. These oscillations have been reported by several groups[9] [10] [11]. The conductivity is found to oscillate with a period of approximately 400/1, which is in good agreement wi th the theory[8]. The Ha l l coefficient and the magneto-resistance coefficient should also display oscillatory behavior, and these have been observed by the same groups. 2.5 Surface Scattering The Fuchs-Sondhiemer model attributes the higher resistivity in thin films as compared to the bulk material to the scattering of the electrons off of the surfaces of the film. The Fuchs formula is obtained by solving the Bol tzmann transport equation using the relaxation t ime approximation[12]. The result is the following equation, — = 1-1.(1-p). (2-54) (Too Oft The conductivity is the conductivity of the bulk material with the same defect density as the film, and k — t/X^, where A ^ is the mean free path in the bulk material and t is the thickness of the film. The quantity p is called the specularity parameter, and is defined as the ratio of the number of electrons reflected at angle 8 to the number of Chapter 2. Electrical Properties of Metals and Thin Films 25 electrons incident at angle 8. Equation 2.54 is valid for values of k which are greater than one, which means for thick films. It has been shown, however, that equation 2.54 applies fairly well for al l values of A;[12]. Wri t ing equation 2.54 in terms of the resistivities, and using the binomial expansion formula and keeping only first order terms, we find 1 + = Pc (2.55) Mul t i p ly ing both sides of equation 2.55 by the thickness, t yields pt = Poot + ^ f Z ( l - p ) . (2.56) A plot of pt vs t should yield a straight line of slope m = p^ and intercept b = 3 A ° ^ (1 — p). This formula has been applied to the data appearing in chapters 4 and 5 of this thesis. 2.6 Grain Boundary Scattering The grain boundary scattering model of Mayadas and Shatzkes [13] attributes the increase in resistivity of a thin film as compared to the bulk material to scattering of electrons off of the grain boundaries. The model represents the grain boundaries as N parallel, partially reflecting planes, randomly distributed throughout the material. The average distance between the planes is d, and this can be taken as the average grain diameter. The Bol tzmann transport equation was solved with the assumption that the average time between collisions can be represented by a relaxation time, r . It is assumed that grain boundary scattering obeys Matthiessen's rule, and that the following equation holds; - = - + , (2.57) ^ ^ 0 ' j r o m s where To is the average time between collisions for the normal processes which give rise to the resistivity of a metal. W i t h these assumptions the ratio of the conductivity of the Chapter 2. Electrical Properties of Metals and Thin Films 26 fi lm to the conductivity of the bulk material depends on the following equation, (2.58) \ - \a + a 2 - aHn (1 + -3 2 V a The parameter a in equation 2.58 is defined in the following manner, where R is the grain boundary reflection coefficient, and Ao is the mean free path in the bulk material. In terms of the resistivities, equation 2.58 becomes, El = I = 1 (2 60) Po 3 [ | - \a + a 2 - aHn ( l + ± ) ] / ( « ) A considerable simplification of this result was noticed by De Vries[14]. For values of a > 3.5, the following formula provides only 5% deviation from the original function: ~ 1.39a + 1' ^ 2 ' 6 1 ^ The ratios of the resistivities can now be written as, ^ • = 7 ^ = 1.390 + 1, (2.62) Po /(«) which should apply for larger values of a. The contribution of grain boundary scattering to the resistivity depends on three independent parameters, A 0 , R, and d. Variations in any of the three parameters can cause a higher resistivity than that observed in the bulk. 2.7 Concluding Remarks The Fermi Temperature of metals is in general quite high due to the high carrier concen-tration. The physical result of this is that the energy distribution of the carriers is not changing appreciably with temperature, in the temperature range 300A" — 4 .2A\ Math -ematically this can be stated as follows: tp » kT. Thus, the conductivity of a metal in Chapter 2. Electrical Properties of Metals and Thin Films 27 this temperature range is governed almost exclusively by changes in the mobilites of the carriers. The mobilites of the carriers in the bulk material are affected by thermal dis-tortions of the lattice (lattice vibrations), scattering off of crystal dislocations, scattering off of lattice imperfections, and scattering off of impurities. The motion of the atoms in the crystal lattice is due to thermal excitation, and the contribution to the scattering wi l l be temperature dependent. The other scattering processes are temperature independent. Matthiessens's rule states that the total resistivity of the material wi l l be the sum of the scattering contributions of both these terms, such that p = pl(T) + p0. (2.63) In general, the temperature independent term, p0, dominates at low temperatures while the temperature dependent term, pi, is an approximately linear function of T from room temperature to 77K, then in general varies as T5 at low temperatures. As the dimensions of the material become comparable to the mean free path of the carriers on the Fermi Surface, additional scattering effects can contribute to the resistivity. In particular, scattering from the surfaces of the film and scattering from the boundaries of the grains in the fi lm wi l l increase the resistivity of the film compared to that of the bulk. These effects never change the character of the resistivity - temperature profile, meaning that the temperature dependent part of the resistivity is not affected by these additional scattering contributions. This can be seen from the data for vapor deposited aluminum, which appears in chapter 5. The bulk data and thin fi lm data essentially differ only by a constant offset on the y axis. This indicates that these additional scattering processes are not temperature dependent for metals. The situation for semi-metals is different. The resistivity of a semi-metal is higher than that of a normal metal, and this is due to the lower carrier concentration. Exper i -mentally, the temperature dependence of the resistivity of bulk bismuth is well described Chapter 2. Electrical Properties of Metals and Thin Films 28 by Matthiessens's rule, and the temperature dependent part of the resistivity decreases nearly linearly wi th decreasing temperature. This is the same behavior as observed in metals, and indicates that the dominant scattering process at higher temperatures in bulk bismuth is electron-phonon scattering. However, the temperature dependence of the resistivity of thin films of bismuth is completely unlike that of the bulk material. In particular, the resistivity of a thin bismuth film increases as the temperature decreases, and becomes approximately constant at lower temperatures. Anticipat ing our conclu-sions somewhat, which are developed more fully in chapter four, we surmise that this anomalous behavior may be due to a combination of scattering of the carriers off of grain boundaries, and changes in the energy distribution of the carriers as a function of temperature. Further discussion of this effect is deferred until chapter four, where the experimental data is interpreted in terms of this model. Chapter 3 Growth of T h i n Films 3.1 Equipment A major part of this thesis involved the growth of bismuth films on sapphire substrates. There are several different ways to grow a thin film on a substrate, but the films produced i n the present work were made by thermal evaporation. The method for producing the films wi l l be discussed, and the theory of thin film growth wi l l be briefly touched upon in this chapter. 3.1.1 Thermal Evaporation Thermal evaporation is a common and inexpensive technique for producing thin films. The substance to be evaporated is placed in a resistance heater inside a vacuum chamber and heated unti l melting. Vapor atoms of the condensate collect on the surface of the substrate with a speed depending on the location of the substrate in relation to the resistance heater, and the temperature of the resistance heater. Evaporation may be stopped immediately by use of a mechanical shutter which may be moved in front of the substrate. The speed of deposition and deposition thickness are often measured using a commercially available crystal thickness monitor. This device measures the fundamental frequency of oscillation of a quartz crystal, which changes slightly with the amount of material deposited on it. The crystal monitor is placed as close to the substrate as possible during evaporation to accurately measure the thickness of the deposited f i lm. 2 9 Chapter 3. Growth of Thin Films 30 Precision of a few angstroms is easily attainable with such a device. The substrate is cleaned inside the chamber during pump down by a device called a glow discharge. This is a high voltage rod which discharges electrically inside the chamber during pump down, ionizing al l the remaining air in the chamber. This ionized air pulverizes everything in the chamber and removes excess material from the surface of the substrate. The thermal evaporator is typically operated at pressures of 1 - 2 x l O - 6 torr. This pressure is easily attainable wi th the use of a diffusion pump and a l iquid nitrogen cooled cold trap. The substance to be evaporated is placed in a tungsten boat, which is then placed between two poles of a transformer. Passing current through the boat causes it to heat up, which in turn heats the material inside the boat. Tungsten is used because of its high melting temperature (3653K). If the substance to be evaporated is in the form of solid pellets, like bismuth or indium, a tungsten boat or crucible may be used. The advantage of the crucible over the boat is that the evaporate is kept at a more uniform temperature. If the substance to be evaporated is in the form of a wire, such as gold, then it may be twisted around a wire filament. When current is passed through the filament the evaporate melts and covers the entire length of the filament, thereby increasing the effective surface area of the evaporating material. The speed of deposition is controlled by the amount of current passing though the boat or filament. It is often not necessary to work out how much power is needed to bring a sample up to its melting temperature. B y slowly increasing the amount of current passing through the sample holder and watching the thickness monitor, the deposition speed can be controlled with some precision. Chapter 3. Growth of Thin Films 31 3.2 Properties of T h i n Films The physical properties of a thin film often depend heavily upon the procedure used to produce the f i lm. F i lms produced by M B E are usually highly ordered and low defect in nature - those produced by thermal evaporation can be highly disordered with many defects. The deposition rate, substrate temperature, cleanliness of the substrate, purity of the evaporate, and quality of the vacuum can al l have a major effect on the physical properties of the thin f i lm. In this section the theory of thin film growth wi l l be briefly discussed. 3.2.1 T h i n F i l m Growth Electron microscopy has been done in situ during the growth of a thin film by a number of investigators(see the book by Chopra[15] for a comprehensive review). These experiments have revealed that thin fi lm growth proceeds in a number of distinct steps. Atoms arriving on the surface of the substrate from the vapor phase are usually quite hot, as they have more or less the same temperature as the source from whence they came. When they encounter the surface of the substrate, which is presumably at a lower temperature than that of the atom source, they do not immediately lose all their energy. They are free to roam around on the surface of the substrate for a short time and if they do not lose any energy in that time, then they wi l l re-evaporate and rejoin their companions in the vapor phase. The probability that an atom wi l l re-evaporate is given by[12] W = uexp'^r1 (3.64) where v is the vibrational frequency of the atom and Qad is the energy of adsorption. The atom wi l l lose energy i f it collides with any point defects or crystal dislocations on the substrate, and S E M pictures reveal that it is these locations where thin film growth Chapter 3. Growth of Thin Films 32 begins. Atoms wi l l also lose energy in colliding with other atoms on the surface, and this begins the next stage of fi lm growth. The atoms coalesce into small nuclei called "islands". These islands smatter the surface of the substrate and provide further growth centers for incoming atoms. The size of these islands is something that is specific to each material being evaporated, and depends on a number of things e.g. the surface energy of the substrate and the energy of the incoming atoms. These islands grow larger and larger unt i l a cri t ical radius is reached. Islands larger than this radius wi l l break up into two or more islands of smaller radius, and the process wi l l begin again. The islands wi l l eventually become so numerous that contact between two or more islands wi l l be inevitable. The islands do not join together in a contiguous manner, but j am together in random orientations. This gives rise to the notion of a grain - the boundaries of the grains are simply the edges of all the other islands around it. In this way, the surface of the substrate wi l l eventually be covered to give a continuous thin film. Several factors can have an effect on the growth of a thin film. If the rate of incoming atoms is quite high, then an atom wi l l have less time to roam the surface of the substrate before it meets another atom. Island formation wi l l occur sooner, and this process wi l l generally result in a film with a large number of small-sized grains. A film with a slower growth rate has more of an opportunity to form large grains, and we expect to see fewer but larger grains in a slowly deposited thin fi lm. The substrate temperature can have a large effect on the properties of the film. A striking example is shown by Komnik et. al[16]. T h i n films of bismuth were grown on substrates at 4 K . Such a procedure is known as quenching the substrate. One expects that nearly every atom which strikes the surface wi l l stick, and wi l l produce a film with very small grains. The result was a super-conducting film, with a crit ical temperatures in the range Tc — 4 — 8K. This is quite striking indeed, since bulk bismuth is not super-conducting above temperatures of 10mfc[17]. Chapter 3. Growth of Thin Films 33 If the lattice constants of the substrate and those of the material being evaporated match quite closely (typically within 1 percent), then epitaxial growth may occur. The temperature of the substrate plays an important role in this process, as epitaxy is usually poor or non-existent i f the temperature is too low. This critical temperature is different for every pair of materials, and is determined largely by tr ial and error. It is known that bismuth grows epitaxially on mica, but only if the substrate is held at a constant temperature of 100 C[3][10]. Pictures of a bismuth film taken with a Scanning Electron Microscope ( S E M ) reveal the grain structure of the film quite nicely, and this is shown in figures 3.5 and 3.6. The film in these pictures has a thickness of 2000A, and was grown at a rate of lA/s. The substrate is sapphire. The temperature of the substrate was not controlled during the deposition. S E M photos of a very thin bismuth film were also taken, and these are shown in figure 3.7. The thickness of this film was 15A. The photo shows some small , bright dots smattering the surface of the substrate. These spots are presumably insulating in nature, since the brightness is due to the accumulation of charge at those points. These may be due to oxidation of the film. Also visible are small clumps, which seem to be bismuth islands in the early stages of growth. Figure 3.5: An S E M photo of a 2000A bismuth film grown on a sapphire substrate. The magnification in this image is 10 000 times. The tick marks on the bottom right indicate the scale of the photo. Chapter 3. Growth of Thin Films 35 Figure 3.6: An S E M photo of a. 2000A bismuth film magnified 30 000 times. The crys-tallites are clearly visible, and the disordered nature of the film is apparent. Chapter 3. Growth of Thin Films 36 Figure 3.7: A picture of a discontinuous bismuth film. The dark clumps are bismuth islands in the early stages of growth. The bright dots may be due to the formation of an oxide. Chapter 3. Growth of Thin Films 37 The longest dimension of the crystallites in figure 3.6 were measured, and the average was found to be x — 188 ± 50nm. This seems consistent with the data of other investi-gators, whose results show that the average size of the grains in bismuth scales roughly wi th the thickness of the film[10]. These photos clearly show the disordered structure of the film. It is not known if bismuth grows epitaxially on sapphire. The crystal structures of the two materials are both hexagonal rhombic, although the space groups are not identical. The space group (SG) of bismuth is R3m(166) while the space group of sap-phire(corundum) is R3c(167). The lattice constants match pretty closely, as shown in the following table: Substance a0 bo c 0 B i 4.5459 4.5459 11.8622 Al203 4.7591 4.7591 12.9894 Figure 3.8: A comparison between the lattice constants of sapphire and bismuth. The greatest mismatch is between the lattice constants in the c direction, which amounts to 8.7%. If the film is grown in the a-b plane of the substrate, then the lattice mismatch is 4%. This may be too great of a mismatch for epitaxial growth. 3 .3 E f f e c t o f P r e s s u r e o n G r o w t h The ambient pressure in the vacuum chamber has a large effect on the growth of the film. This was discovered somewhat accidentally, as a film was deposited in an ambient pressure of 1.7 x 1 0 _ 4 T o r r . The ambient pressure was high because one of the electrical feedthroughs used in the experiment developed a vacuum leak during pump down. When the ambient pressure is around 1.0 x 1 0 _ 6 T o r r , there is a mono-layer per second of air molecules striking the surface of the substrate. One can usually observe the pressure Chapter 3. Growth of Thin Films 38 rising in the bell jar when the evaporation is begun, which indicates that the evaporate source is heating up and outgassing inside the chamber. When the quality of the vacuum is low, the fi lm wi l l be subject to a large amount of impurities, and these impurities may adversely effect the electrical properties of the fi lm. The effect of poor-vacuum is shown in figure 3.9. The straight vertical line at t=0 hrs results from the use of the Hewlett-Packard digital multi-meter used to measure the resistance of the sample. When the meter is on the setting "4 wire resistance", and the sense leads are not connected, the reading on the multi-meter is 0.004O. As the evaporation proceeds, the sense leads become connected, and the multi-meter starts measuring the resistance of the evaporated fi lm. The resistance of another sample was measured during and after deposition for an R(D) o f B i f i l m v s t i m e a f t e r T h e r m a l E v a p o r a t i o n T i m e ( h r s ) Figure 3.9: The resistance of a bismuth film was measured during and after deposition. The ambient pressure i n t h e bell jar was 1.7 x 1 0 - 4 T o r r . The film is oxidizing inside the chamber, and thus its resistance is increasing with time. Chapter 3. Growth of Thin Films 39 ambient pressure of 3.4 x 1 0 - 6 Torr. This data is shown in figure 3.10. The resistance decreases wi th time, and this is due to the cooling of the fi lm. From this data we can state a general rule of thumb : better films are grown in lower ambient pressures. From the data in figure 3.10, it is evident that successful growth can be carried out at a pressure of 3.4 x 1 0 - 6 Torr. This pressure is not extremely low, and certainly nowhere near the l imi t ing pressure for an oi l pumped chamber of 2 x 10~ 7 Torr. 2 2 0 4 — W i r e R e s i s t a n c e o f B i F i l m v s T i m e A f t e r T h e r m a l E v a p o r a t i o n 1 7 0 u 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2 4 6 8 T i m e ( h r s ) Figure 3.10: The resistance of a bismuth film during and after deposition. The actual evaporation only lasted for 1 minute. The ambient pressure in the bell jar was 3.4 x 1 0 - 6 torr. The decrease in resistance with time is due to cooling of the film. 3 . 4 Cohesion and Robustness In principle any substance can be evaporated onto any other substance to give a thin film. However, the cohesion between the two substances may be an issue. If the cohesive Chapter 3. Growth of Thin Films 40 forces are pr imari ly Van Der Walls in nature, which is a weak binding force, then the cohesion between the two substances wi l l be poor. The result wi l l be a film which can be easily scratched off. In general, a layer of oxide between the two substances wi l l improve cohesion enormously[15]. Such a layer is called a sticking layer, or adhesion layer. A n example of two substances with poor cohesion is gold and sapphire. This problem was directly encountered while preparing samples to measure the resistance of bismuth films for this thesis. The cohesion can be greatly increased if a sticking layer of chromium is deposited between the gold layer and the sapphire. The robustness of a film can be quantified by applying a couple of simple tests. The first test is innocuously• named "the scratch test", in which the surface of the film is scratched lightly wi th a hard object. Visible flaking off of the fi lm constitutes a fail . If the fi lm is undamaged from the scratch test, the imaginative "scotch tape test" is the next step. True to its name, a piece of 3 M clear scotch tape is placed on the surface of the f i lm, sticky side down. The tape is pressed down using light pressure (as from your finger) and quickly removed. In an effort to quantify this procedure somewhat, N A S A spec. A S T M # specifies 3 M 810 tape with 10 oz applied pressure. If nearly al l the film is removed by the sticky tape, this constitutes a fail. If a film passes both tests, the cohesion is quite good, and the film is said to be "robust". The growth rate of the two layers was also found to have a large effect on the cohesion between gold and sapphire. If the chromium and gold layers were grown at rates which exceeded lOA/s, the cohesion between the layers was poor, and the gold film failed both the scratch test and the scotch tape test miserably. If the growth rate was kept at or below l A / s , the resulting films were extremely robust, and passed both the scratch test and the scotch tape test with flying colors. Chapter 3. Growth of Thin Films 41 3 .5 S u b s t r a t e C l e a n l i n e s s The cleanliness of the substrate has an effect on the cohesion of the f i lm. A l l samples were cleaned using trichloroethelyne, acetone, and methanol in an ultrasonic cleaner. Samples which weren't cleaned using the ultrasonic cleaner showed poor cohesion. This seemed to be a necessary step in the production of usable films. 3 .6 S a m p l e P r e p a r a t i o n The substrate used for these experiments was a rectangular piece of AI2O3 (sapphire) of dimensions 0.5 x 0.25 x O.OlOinches. This was bonded to an aluminum chip of dimensions 2.36 x 0.4 x 0.125 inches using Stycast 2850FT epoxy, set with Catalyst #9. The dimen-sions of the a luminum chip were chosen for easy insertion into a l iquid hel ium storage dewar. Before bonding the sapphire to the aluminum chip, the sapphire was cleaned according to the following procedure: 5 mins in hot trichloroethelyne bath and ultrasonic cleaner 5 mins in hot acetone bath, and ultrasonic cleaner 5 mins in hot methanol bath, and ultrasonic cleaner Rinse with distilled water Blow dry sample in dry nitrogen atmosphere During cleaning the sample was handled by the edges only,using tweezers. The electrical leads consist of 4 individual strips of gold deposited by thermal evap-oration lengthwise along the chip. The deposition of these electrical leads deserves some discussion. In an earlier attempt to measure the resistance of the f i lm, a mask was constructed which allowed deposition of four gold circles in a row near the edge of the substrate. The thought was to then deposit bismuth over the entire surface of the sap-phire, and measure the resistance of the film using a four wire measurement. Al though Chapter 3. Growth of Thin Films 42 such a procedure wi l l certainly measure the resistance of the film, this configuration does not allow easy estimation of the important quantity f i / D . A sketch of the equipotential lines in the fi lm shows this more clearly. The conduction path in such an arrangement is complex as the electric field lines are perpendicular at every point to the equipoten-t ia l lines. This situation is resolved by depositing conductive leads lengthwise along the sapphire chip. The equipotential lines are then lines parallel to the edge of the chip, and the electric field points perpendicular to these lines, so that the conductive path is perpendicular to the electrical leads. The estimation of the quantity f i / D is then t r ivia l ; if the film is rectangular in shape and has dimensions N x M , where N is the width of the sample and M is the sensing lead separation, then the sheet resistance is given by the following: where R is the measured resistance. To improve the cohesion between the sapphire substrate and the gold layer, a th in (50 angstrom) layer of chromium is deposited before the gold layer. The thickness of the gold layer is not cri t ical , and a thickness of approximately 1000 angstroms was used suc-cessfully. B o t h these layers are grown fairly slowly, with deposition rates never exceeding 1 angstrom/second. It was found that depositing either layer more quickly resulted in a film that would easily flake off if scratched. The films produced by slower deposition were extremely robust, and showed very good cohesion to the sapphire surface. The films easily resisted the so-called "scotch tape test", where a piece of scotch tape is placed on the surface of the f i lm, sticky side down, and then quickly removed. The electrical connections to the gold strips were made by bonding 0.005 inch copper wires to them wi th EpoTek silver epoxy. The other ends of these wires were bonded to a T 0 5 8 pin header, again using silver epoxy. Ultrasonic wire bonding was investigated as a method to make the electrical connections, but it was found that the thin (0.001") Chapter 3. Growth of Thin Films 43 gold wires would usually break when the sample was heated from room temperature to 270 degrees Celsius. This was most likely due to the thermal expansion of the Stycast. Avai labi l i ty of the ultrasonic bonder was also an issue, since our laboratory did not own one. The end result is shown in figure 3.11. Figure 3.11: A diagram of the sample used to measure the resistance of bismuth films. The finished sample is placed inside a thermal evaporator and bismuth is deposited over the entire sapphire surface. Chapter 4 Results In this chapter the results of the resistivity measurements on thin films of bismuth wi l l be presented. A number of interesting features are evident in the data, and a separate section has been devoted to each. 4.1 Temperature Dependence of The Resistivity The most intriguing feature evident in the data is the anomalous temperature depen-dence of the resistivity. The resistivity of 6 thin films of bismuth has been characterized as a function of temperature in the range 295K to A.2K. The resistivity at all tempera-tures was higher than that of the bulk material for every sample. The resistivity of each sample is observed to increase with decreasing temperature for thicknesses ranging from 87 — 3005A. This behavior is markedly different than that of the bulk resistivity, which decreases wi th temperature over the same temperature range. The data for the tem-perature dependence of the resistivity is shown in figure 4.12. The films were prepared by thermal evaporation in a diffusion-pumped bell jar. The growth rate and thickness of each film, and the ambient pressure in the bell jar at the time of evaporation are summarized in table 4.3. The resistivity-temperature data for sample F is not plotted in figure 4.12 for reasons given in section three of this chapter. The films were al l prepared under relatively poor vacuum conditions, and this is a result of the vacuum system used. A pressure of approximately 3.8 x 1 0 - 6 Torr was reached after 20 hours of pumping, and a further 28 hours of pumping did not lower the pressure any further. The inside of the 44 Chapter 4. Results 45 jar was meticulously cleaned with methanol, and most of the components inside the jar were replaced, but this had no effect on the final pressure. The presence of a leak was suspected, and the system was checked throughly using a helium spectrometer, but no leak was found. This lead to the conclusion that the l imi t ing pressure of the bell jar used Sample Label Thickness(A) Pressure x 10 6 (Torr) Growth Ra te (A/ s ) A 85 3.8 0.25 B 87 4.2 30 C 103 3.8 0.33 D 146 4.0 1.2 E 500 4.0 0.33 F 3000 4.4 1.0 G 3005 4.4 17.0 Table 4.3: The thickness, growth rate, and pressure in the bell jar just before evaporation. The ambient pressure was the pressure reached after approximately 20 hours of pumping. in these experiments was approximately 3.8 x 10~ 6 Torr. The pressure at evaporation of each fi lm was kept approximately constant, and as such each film is subject to the same amount of impurities present in the bell jar. The temperature is measured with a diode in good thermal contact wi th the sample. For a constant current, the voltage across the' diode varies linearly wi th temperature i n the range 295 —40A". Below 40A' the diode is no longer a good thermometer, and it is not used to infer the temperature below this l imi t . The points at 4.2AT are measured after contact wi th the l iqu id helium bath had been established. No data is measured between 40A' and 4.2A^. The data for samples A, B, and C is interpolated with a straight line between these points. The data for samples D , E, and G is not interpolated in this region. The results of other investigators indicate that the resistivity becomes constant at low temperatures, and the temperature at which the resistivity becomes constant is a function of the film thickness[11]. If we compare the data for samples D, E and G , it is evident that the resistivity of sample D is beginning gradually to plateau at 40/\", while the Chapter 4. Results 46 500 4 0 0 £ 3 0 0 o c -l-> • ! 1 • I 1 -|-> w • i—i S 200 100 1 1 1— I— I— r— l 1 1 l l l I I r~| G D E X \ \ \ \ \\ ct « .^\\ B & \ A # ~ ~ ^ V \ v\\ B \ V ^ : — V W c \ \ \ : \ \ G , x V-I I I I I I I I I I I I I I I I I i_ 10 100 T e m p e r a t u r e ( K ) Figure 4.12: Resistivity versus temperature for bismuth films. The thicknesses are: A -85A ; B - 87A ; C - 103 A ; D - 146A ; E - 500A ; G - 3005A. The resistivities of samples A, B, and C has been interpolated with straight lines in the region between the data points. Chapter 4. Results 47 resistivities of samples E and G clearly has not. This agrees with other measurements in the literature[3], which find that the resistivity plateaus at lower temperatures for thicker films. A feature that is evident in all the data for samples A , B, and C is a max imum in the resistivity at a temperature of approximately 65K, followed by a monotonic decrease wi th temperature thereafter. The maximum is more pronounced in the data for sample A (t = 85A) than for samples B(t = 87A) or C ( t = 103A). This feature is not found in the data for the thicker films. This observation agrees with published data in the literature, which find that the resistivity maximum occurs only in thin films[ll]. The anomalous character of the temperature dependence of the resistivity of thin bismuth films has received much attention in the literature. In most metals it is observed that the temperature dependent part of the resistivity of a metallic thin film is identical to the temperature dependent part of the resistivity of a bulk material[14]. The thin films always show a higher resistivity than the bulk, and this can be successfully interpreted in terms of the Fuchs-Sondh'iemer model and the grain boundary scattering model presented earlier. However, the situation for thin film bismuth is different. In what follows, we wi l l systematically apply each of the three models presented in chapter two to the resistivity - temperature data. We state at the outset that each model fails to adequately explain the temperature dependence of the resistivity of thin bismuth films. i 4.1.1 Surface Scattering The Fuchs-Sondhiemer model attributes the higher resistivity of the film (compared to the bulk material) to the scattering of electrons off of the surfaces of the film. Plot t ing pt against t should yield a straight line. The data has been plotted for three different temperatures: 295A^, 60A^, and 4 . 2 ^ . The data is well described by a linear relation for all three temperatures, and the slope and y-intercept of each line has been computed. These parameters appear in table 4.4. Chapter 4. Results 48 Temperature(K) Pbuik(pttcm) p00(pQ1crn) ( l-p ) A o o ( A ) 295 116 109.6 126 60 19.0 389.9 -32.3 4.2 5.8 576.53 -52.8 Table 4.4: Parameters derived from analyzing the data in terms of the Fuchs-Sondhiemer model. Notice that the specularity parameter must be greater than one to reproduce the values in the last column of the table. The resistivity of a bulk sample with the same defect density as the fi lm (poo) is seen to increase as the temperature decreases. This is an anomalous result, since the resistivity of bulk bismuth decreases as the temperature decreases. Another anomaly in the data are the y-intercepts for the fits to the 4.2K and 60K data. The specularity parameter, p, must take on values greater than one to explain the negative values of the y-intercepts at low temperatures. A value of p > 1 indicates that more electrons are bouncing off the surfaces of the fi lm than were incident in the first place, and this can't be accounted for in terms of the Fuchs - Sondhiemer model. These results indicate that the anomalous temperature dependence of the resistivity of thin bismuth films cannot be explained by non-specular reflection of the carriers from the film surfaces. 4.1.2 Gra in Boundary Scattering We can analyze the resistivity-temperature data in terms of the grain boundary scatter-ing model presented earlier. In this model, the mobili ty of the carriers is impeded by scattering off of the grain boundaries. If we assume that the average grain diameter has some relation to the film thickness, which seems to be true in general[10], then we can analyze the data at 295K in terms of the grain boundary scattering model. If we plot the ratio of f i lm resistance to bulk resistance pj/po against the thickness t, we find that a marginal fit to the data is obtained i f a is equal to the following, 35.0 a (4.65) Chapter 4. Results 49 and thus A o — ^ = 35.0A. (4.66) i — ti Using the value A 0 = 10900A, we find a reflection coefficient equal to R — 0.0032. This indicates that 99.68% of the particles incident on a grain boundary are transmitted through the boundary, which indicates that grain boundary scattering is a weak process in these films at room temperature. This data is plotted in figure 4.13. The model fails when we try to apply it to the data for the resistivities at 4.2K. The ratio of f i lm resistivity to bulk resistivity at 4.2K increases as a function of thickness, and this can't be explained solely in terms of grain boundary scattering. It would require a negative value of the reflection coefficient, which doesn't have an interpretation in terms of the grain boundary scattering model. The grain boundary scattering model doesn't account for the temperature dependence of the resistivity of thin bismuth films. 4 . 1 . 3 Quantum Size Effect A quantitative prediction of the Q S E model is that the low temperature resistivity of a semi-metal f i lm wi l l oscillate as a function of thickness, with a period of about 400A. The data presented in this thesis show no such oscillations. This is not conclusive, however, since the thicknesses of the films grown here were not varied at regular intervals to search for such oscillations. Another quantitative prediction of the Q S E model is that the S M S C transition wi l l be observed in bismuth for film thicknesses of about 400A. Fi lms thinner than this are predicted to have an energy gap, since the splitting due to the small film thickness has erased the band overlap. A semi-conducting material has a resistivity which is given by the following equation, 0 = Aexp(Wg/2kT) (4.67) Chapter 4. Results 50 Figure 4.13: The grain boundary scattering applied to the data. A marginal fit is obtained for the room temperature resistivity data. Chapter 4. Results 51 where Wg is the band overlap. A plot of log p vs 1/T should yield a straight line wi th intercept b — log(A) and slope m = Wg/2k. A plot of this for any of the data presented here fails to yield this type of relation. Another fact which should be noted is that the resistivity-temperature curves for thick samples (t = 500,3005A) has more or less the same shape as that of the thinner samples. According to the Q S E model, films thicker than 400A are st i l l semi-metals. This poses a problem, since the temperature dependence of the resistivity of a semi-metal and a semi-conductor are not the same. No experimental group has claimed to observe the S M S C , and this may be evidence that the S M S C in bismuth is being masked by other effects[18]. We conclude that the Q S E model is not able to explain the temperature dependence of the resistivity observed in thin bismuth films. 4.1.4 A New Mode l So far, we have seen that the surface scattering model, the grain boundary scattering model, and the quantum size effect can't explain the temperature dependence of the resistivity of thin bismuth films. It is expected that these effects do play a role in deter-mining the resistivity, but by themselves or in combination they can't explain why the resistivity increases wi th temperature, then plateaus to a constant value. In what follows, we wi l l explain the general features of a model which successfully explains many features in the data presented here, and many features in the data of other investigators. We note that although many groups have published data on the temperature dependence of th in bismuth films, no group has satisfactorily explained the physical mechanism responsible for such behavior. We provide a physical explanation for the temperature dependence of the resistivity and we have come up with a phenomenological fit to the data based on our physical interpretation. Looking at figure 4.12, one is tempted to analyze the data in terms of the mean free Chapter 4. Results 52 path of the carriers in the material. However, this may lead to erroneous results. The resistivity of a material is determined by two important factors : the mobility and the number density of the carriers. Changes in either one of these variables wi l l cause changes in the resistivity of the material. Since the resistivity is affected by both variables, one cannot make general statements concerning the mobili ty of the carriers by only analyzing the resistivity curves. M a n y investigators who are interested in the temperature dependence of thin bismuth films measure the H a l l coefficient, and the magneto-resistance coefficient. The H a l l effect occurs when a magnetic field is applied perpendicular to the current flowing in a conduc-tor. The resulting Lorentz Force causes an accumulation of charge on the sides of the conductor, which leads to a transverse voltage known as the H a l l Voltage. For a system wi th only one type of carrier, the Ha l l coefficient is given by, RH = ~ (4.68) ne where e is the charge on the electron, and n is the carrier concentration. Measuring the H a l l coefficient for such a system provides direct information about the carrier density, and the charge of the carriers(since the Ha l l Coefficient can be negative or positive). For a two component system, the formula (for isotropic carrier motion with respect to the crystal axes) is _ nhix\ - nen2e So, for a two component system, the Ha l l coefficient probes the number density and mo-bi l i ty of the carriers. The sign of the Ha l l coefficient indicates which carrier is playing the more dominant role. Experimental results reported in the literature often give conflicting results for the H a l l coefficient in thin bismuth films. This indicates that the conditions under which the film is grown have a noticeable effect on the electrical characteristics of the film. Chapter 4. Results 53 The size of the grains in bismuth films has been shown by J in et al.[2] to have a re-markable effect on the resistivity. In particular, they found that annealing thick epitaxial films (t — lpm) according to different schedules could completely change the character of the resistivity - temperature curve, even turning the temperature coefficient of resistance ( T C R ) from negative to positive. A plot of their data appears in figure 4.14. The result of this change in sign of the T C R is that the resistivity of the film is observed to decrease wi th decreasing temperature, which is the same behavior observed in the bulk. The size of the grains i n an unannealed sample of thickness \\im was measured (by J i n et al.) and was found to range from 0.5 — 5/im. The size of the grains in an annealed sample of the same thickness was found to range from 10 — 30/rm, which is approximately a factor of 20 larger than the grains in an unannealed sample. The conclusion reached from this work is that the mean free path of the carriers in bismuth can be increased by annealing the film at a temperature close to the melting temperature (271C) of bismuth. In light of the data in figure 4.14 from J in et al., the size of the grains in bismuth films has an enormous effect on the resistivity at all temperatures. From the analysis presented earlier, grain boundary scattering can not explain the anomalous temperature dependence of the resistivity. However, an implici t assumption in the derivation by Mayadas and Shatkes[13] of equation 2.60 is that the carrier density is constant, which is a valid assumption for materials with large Fermi Energies, namely metals. As shown in chapter two, the energy distribution of the carriers in bismuth is a function of temperature, and this is a result of the low Fermi Energy of bismuth. The anomalous temperature dependence of the resistivity can be explained as a result of the temperature dependence of the carrier energies, and scattering of the carriers off of grain boundaries in the film. If we model the grain boundary as a potential barrier of height U0 and width L, the transmission coefficient is given by the following equations[19], Chapter 4. Results 54 200 100 90 80 ^ 7 0 £ 60 o 3 50 40 z Annealed at 225C for 18h > CD 30 20 10 9 8 " i — i—r • Unannealed • ~i i i — i — i — r • Annealed at 265C for 48h J I I I I L J L i r • • • • • • • • 10 100 Temperature (K) Figure 4.14: Resistivity versus temperature for bismuth films of lpm thickness. The data is from J i n et al.[2]. It is interesting to note that the resistivity of the bottom curve is actually lower than that of the bulk, for temperatures greater than 100A^. This fact was not noted by J i n et a/., and indeed is hard to explain. Chapter 4. Results 55 T = 4e{U0 - e) (4.70) 4e([ / 0 - e) + Ugsinh2(riL) 4c(e - UQ) o T = 4e(e - Uo) + U^sin?{kL) e > Uo (4.71) (4.72) where n = y/Uo — e and fc = y/e — Uo- This represents the probability that a particle wi th energy e w i l l be transmitted through the barrier. The transmission coefficient is plotted in figure 4.15. One can see that if e < < Uo, the transmission coefficient is essentially constant, and is small. When e ~ Uo, the transmission probability increases quite quickly, and the particle can pass through the barrier without too much trouble. If the height of this potential barrier is comparable to the Fermi Energy of the carriers in bismuth, the number of particles able to pass through the barrier easily w i l l be a function of temperature. To see this, imagine that the Fermi Energy is less than the height of the potential barrier. A t absolute zero, the Fermi-Dirac distribution wi l l be a step function, and the number of particles that can pass through the barrier easily wi l l be a constant. As the temperature increases, the number of particles with energies greater than tp increases. However, i f tp « Uo, most of the particles wi l l have energies e < Uo, and the number of particles passing through the barrier w i l l be essentially unchanged, since the transmission coefficient is small. As the temperature increases further, a significant fraction of the particles may have energies which are comparable to the height of the barrier. As a result, the number of particles passing through the barrier wi l l suddenly increase, since the transmission coefficient rises steeply when t ~ Uo- The Fermi-Dirac distribution is plotted on the same graph as the transmission coefficient (see figure 4.15). In terms of the resistivity of such a material, this model predicts a constant resistivity at low temperatures, followed by a sudden decrease when the temperature exceeds some Chapter 4. Results 56 cri t ical value. This model reproduces many of the features seen in the data if we postulate that the height of the barrier has some dependence on the size of the grains in the f i lm. If the barrier height decreases as the thickness of the film increases, then the temperature at which the resistivity becomes constant wi l l occur at a higher temperature for thinner films than for thicker films. This behavior is evident from the experimental data in figure 4.12. Final ly , we note that the resistivity-temperature curves can be fit approximately with the following function, p = P° + R^rjrzri (4-73) where T is the temperature, po is the bulk resistivity, and R and r are constants. This function has been applied to the data of Baba et al. [3], J in et al.[2], and the data for sample D, and this appears in figure 4.16. In computing the fit for each curve, data from the annealed sample of J in et al. has been used as the bulk resistivity, p0. This data is shown as the bottom curve in figure 4.16. The parameters for the fits are listed in table 4.5. F i l m ThicknessA R(pClcm) r ( K ) 146 325 140 300 1450 210 10000 180 48 Table 4.5: The parameters for the fits shown in figure 4.16 . The fits are computed from equation 4.73. From figure 4.16 we see that the resistivity values for sample D(eind in fact for al l the samples in this thesis) are in disagreement with the data of Baba et al., and J i n et al. . As can be seen from figure 4.12, the low temperature resistivity of the samples measured in this work increases as the thickness increases, and this is the opposite trend to that observed by Baba et al., and J in et al. . Explanation of this is deferred unti l the next Chapter 4. Results 57 Figure 4.15: The transmission coefficient and the Fermi Dirac distribution for a material at 200K are plotted on the same graph. The Fermi-Dirac distribution at absolute zero is also shown. As the temperature increases, more particles wi l l be able to tunnel through the barrier, and the resistivity wi l l decrease. Chapter 4. Results 58 section in this chapter. The function in equation 4.73 is a phenomenological fit to the data. We note that the fit does not exactly reproduce all the data points, however, it seems to describe the general shape of each curve. Equation 4.73 has two free parameters, R and r. R corresponds to the low temperature resistivity of the fi lm, and r roughly cooresponds to the temperature at which the resistivity starts to plateau. From table 4.5, we see that for the data of Baba et al. and J i n et al., the value of r for thinner films is higher than the value of r for thicker films. The data for the samples measured in this thesis show this same general trend. In summary, the anomalous temperature dependence of the resistivity of thin bismuth films is mainly due to the fact that bismuth is a semi-metal, and as such has a low Fermi Energy. The energy distribution of the carriers is a function of temperature, for temperature variations in the range 300 — 4.2/v. Scattering of the carriers from the grain boundaries causes the resistivity to become nearly constant at low temperatures. As the temperature increases, so does the number of particles which are able to tunnel easily through the grains, and thus the resistivity drops as the temperature increases. The effect of annealing the film is to increase the average crystallite size, which increases the mean free path of the carriers in the film. 4.2 L o w T e m p e r a t u r e R e s i s t i v i t y The resistivity at 4.2K is found to increase as the thickness increases. The data is plotted in figure 4.17. A data table appears in table 4.7. This same behavior has been observed by Chu et a/.[20], but not by Baba et a/.[3], or J in et al.[2}. Referring to figure 4.16, which shows data from Baba et al., J i n et al., and data for sample D, we note that the data presented in this thesis show a different dependence on the film thickness than that Chapter 4. Results 59 ~i 1 1—i—i—r t = 3 0 0 l , Data f rom Baba et.al 800 600 400 I 200 o a 100 * i—i 80 60 'oo 40 20 E-10 8 6 1 t=146A, Sample D t= 10,000A(Unannealed) Data f rom Jin et.al. I t=10,000A(Annealed), Data f rom Jin et.al. J I I 1 I L J L J I I I L 10 100 Temperature(K) J L Figure 4.16: The data from J i n et al., Baba et al, and sample D from this thesis. The solid lines through the top 3 curves are fits computed from equation 4.73, and the parameters are listed in table 4.5. Note that the fits do not exactly reproduce the data points, but seem to describe the general shapes of the curves quite well. Chapter 4. Results 60 observed by Baba et al. . The low temperature resistivity of the films in figure 4.12 increases wi th increasing film thickness, while the data of Baba et al. decreases with increasing film thickness. The reason for the disagreement is not known, however, the effect may have something to do with the quantum size effect (QSE) . As the thickness of the film becomes comparable to the mean free path of the charge carriers, some interesting effects should begin to show themselves. A t absolute zero, the Fermi-Dirac distribution shows that al l energy levels with energies higher than the Fermi energy (ep) are empty, while al l those below are filled. As the film thickness decreases, it may occur that the only occupied levels for electrons and holes are those which correspond to zero longitudinal energies. If this is the case, the motion of the charge carriers may be considered to have become two dimensional. Such a transition depends on the direction of motion in relation to the crystal axes. Chu et al. [20] calculate that the 2D-3D transition wi l l be observed for the following film thicknesses: t = 255A, perpendicular to trigonal axis; t = 137A, perpendicular to binary axis; t — 134A, perpendicular to bisectrix axis. The orientation of the films grown in this thesis is not known, but the thicknesses of the films A , B, and C are al l less than the values listed above. The motion of the charge carriers in these films may very well be two dimensional, and this could account for the lower resistivities at low temperatures. If scattering off of the surfaces of the film is important in determining the mobil i ty of the charge carriers, the resistivity of a thicker film could be higher than the resistivity of a thinner film. This behavior would manifest itself at low temperatures since the states of lowest energy would be filled at low temperature. Another fact which may be important is that the films of Baba et al. were prepared on mica substrates, held at a temperature of 100C during evaporation. These films were epitaxial , as shown clearly by X-ray diffraction studies[3]. The films presented here are most likely not epitaxial, judging by the large (8%) difference in the lattice constants in the c direction between bismuth and sapphire. Perhaps the oriented nature of an Chapter 4. Results 61 epitaxial film gives rise to the disagreement between the data sets. 4 . 3 E f f e c t o f G r o w t h R a t e o n R e s i s t i v i t y The growth rate of the films was found to have an effect on the resistivity. In four of the samples, the thickness was kept constant while the growth rate of the film was varied. Samples A and B are very similar in thickness, but differ noticeably in resistivity over the entire temperature range shown(see figure 4.12). Sample A was grown very slowly, wi th a growth rate of 0.1 A/s, while sample B was grown 300 times faster. The data for sample B shows a resistivity which is higher at all temperatures than the resistivity of sample A, and this is presumably due to the tremendous difference in growth rate. One expects a rapidly grown film to have smaller, more numerous grains than a film which is grown slowly. If the transport properties of the charge carriers are affected by scattering off of these grain boundaries, a rapidly grown film would have more grains to scatter off of, and would presumably have a higher resistivity than a film of similar thickness but slower growth rate. The data for sample F seem to be in disagreement with that of sample G. The resistivity data for al l the samples is tabulated in table 4.6. The thicknesses of samples F and G are nearly identical, yet the resistivities do not agree at any temperature. The growth rate of sample F was l A / s , and the evaporation took nearly 50 minutes. The resistivity was measured as a function of thickness for this sample, and this is plotted in figure 4.19. The data show that the resistivity reached a min imum value at a thickness of 800A, then began to increase monotonically as the thickness decreased. The reason for this increase is unknown, but oxidization is suspected. Heating of the film may have been a factor as well, since the temperature of the substrate was not controlled during the Chapter 4. Results 62 100 500 1000 Thickness(A) Figure 4.17: The resistivity at 4.2K increases as the f i l m thickness increases. This data disagrees with that of Baba et <d.[3]. Chapter 4. Results 63 experiment. The tungsten boat can get quite hot during an evaporation, and the sub-strate may have been heated radiatively. For this reason, the resistivity-temperature data for this sample has not been plotted along with the other samples in figure 4.12. This data indicates that slow growth of a thick film may be a bad idea, since heating of the substrate and a higher concentration of impurities may lead to a film with a higher resistivity than a film of comparable thickness but faster growth rate. 4 . 4 R e s i s t i v i t y R a t i o The resistivity ratio, T, is defined as follows, T = ^ i i (4.74) ^295 where p4.2 is the resistivity at 4.2K, and p295 is the resistivity at 295A^. The resistivity ratios have been computed for seven samples, and the data appears in figure 4.6. The Sample label Thickness(A) P295(pQcm) P4.2(p£lcm) r A 85 151.5 211.4 1.395 B 87 163.2 246.6 1.511 C 103 165.8 255.3 1.540 D 146 152.3 314.4 2.064 E 500 123.0 519.8 4.228 F 3000 168.8 637.4 3.776 G 3005 111.2 566.3 5.093 Table 4.6: The data for the resistivity ratio, T, for films of various thicknesses. The data is plotted in figure 4.18. data from table 4.6 are plotted in figure 4.18. The general appearance of the data in figure 4.18 is similar to that of other investigators[3]. The data points show that the resistivity ratio is a function of thickness. The data point for sample F has been included in this plot, and it disagrees significantly with the other data. The reason for the large discrepancy has been discussed in section 3 in this chapter. A solid line has been drawn Chapter 4. Results 64 in a "connect the dots" style between the remaining data points. There is no theoretical framework from which to interpret this data, and thus no effort has been made to fit the data wi th a model. The usefulness of an exact fit is questionable as well in light of the data from Baba et al. [ 3 ] . They have observed that films with identical thicknesses grown simultaneously on mica substrates exhibit a scatter in their resistivity ratios. Apparently, small differences in the cleaving and the cleanliness of the substrate can have an effect on the electrical properties on the film. 4 .5 Effect of Substrate The resistivity was measured as a function of thickness for bismuth films grown on sap-phire and on glass. The thickness of the film, measured by a crystal thickness monitor located as close as possible to the substrate, was controlled by opening and closing a mechanical shutter. After each deposition the shutter was closed and the film was left to equilibrate for a period of about 1 minute before taking data. The resistance of the film grown on the glass substrate was measured with a two lead technique. The leads, which were brass wires, were bonded to gold pads on the substrate using Epo-Tek silver epoxy, and fed through a vacuum feed-through to a digital multi-meter. It was realized later that the contact resistance of the leads may not have been negligible. Contact resistances of approximately 20fi have been observed for leads improperly bonded to gold surfaces. The contact resistance was not measured for this sample, and thus the raw data appears in figure 4.19. The correction to the resistivity is expected to be small since the film resistance ranged from 1000f2 to 400O, which should be much larger than the contact resistance of the leads. In light of this, however, the resistivity could have been mea-sured much more accurately, and the data for the resistivity of bismuth grown on glass should be viewed as approximate only. The resistance of the film grown on the sapphire Chapter 4. Results 65 r vs Thickness(A) for Bismuth Films 1 0 0 1 0 0 0 Thickness(A) Figure 4.18: The resistivity ratios of the bismuth films increase as the thickness increases. The data point for sample F disagrees with the other data. Chapter 4. Results 66 substrate was measured wi th a four lead dc technique, and the contact resistance in this arrangement is not problematic since negligable current flows through the sensing leads. The resistivity is measured to be higher at all thicknesses for the fi lm grown on the glass substrate than for the film grown on the sapphire substrate. This could be due to the amorphous crystal structure of glass, but certainly the lead resistance is a factor. The data is plotted in figure 4.19. Most of the data points for each substrate seem to follow a straight line with identical slope but unequal intercept in these coordinates, which indicates a power law dependence on the thickness. The data is well described by the following two functions, which represent linear least squares fits: p = 414.0 r i -|0.19 It p = 1096.0 pflcm, sapphire substrate (4-75) pttcm, glass substrate (4-76) (4.77) -pO.19 t From figure 4.19 we see that the data points for sample F do not agree wi th the other data points for bismuth grown on a sapphire substrate. The reason for this is explained in section 3 of this chapter. The data point for sample G lies below the bulk value of the resistivity. This gives an estimate of the systematic uncertainty present in these experiments, since the resistivity of the thin film wil l never be lower than that of the bulk material. The data points for the film grown on the glass substrate are seen to deviate significantly from the fit at small thicknesses. The reason for this is most likely that the film was not given enough time to equilibrate between data points. Testing done since this run indicates that the resistance of the film is st i l l dropping slowly up to 10 hours after the evaporation is completed. This indicates that the film cools very slowly in vacuum, and 1 minute in between data points is not sufficient to allow the film to come to thermal equil ibrium with its surroundings. Chapter 4. Results 67 3000A 1000A 500A 300A 4 0 0 300 o • I—I > • I—I w • I—I 00 2 0 0 F 200A 100A 1 0 0 9 0 8 0 7 0 J L O B i s m u t h on Glass D B i s m u t h on Sapphire Bulk Value j i 1 i J i i I 0 . 0 0 0 5 0 . 0 0 1 0 . 0 0 5 l / T h i c k n e s s l l - 1 ; 0 . 0 1 Figure 4.19: The resistivity of bismuth films grown on sapphire and glass, plotted as a function of inverse thickness. The solid lines are fits to the data, and the dashed line is the bulk resistivity at 295/\ . The values for the room temperature resistivity from table 4.6 are plotted as well. The filled squares are data for sample F, which were taken during evaporation. It is evident that sample F behaves differently than the other samples, and radiative heating of the substrate during evaporation may have been a factor. Chapter 4. Results 68 The resistivity is found to be different for bismuth films grown on glass and sapphire substrates. This may have implications for the procedure used to produce low tempera-ture bolometers. In the B A M experiment[4], the absorbing material is a thin coating of bismuth, which is deposited on a sapphire substrate. The resistance of the film is usually monitored in situ using a glass witness slide, and the evaporation is terminated when the desired sheet resistance is reached. According to the results presented here, bismuth films grow differently on glass and sapphire substrates. This means that using a glass witness slide to monitor the sheet resistance may lead to an improperly thick coating of bismuth. This problem can be rectified by using a sapphire witness slide to monitor the sheet resistance, and/or using a thickness monitor to control the thickness of the evaporation. 4 . 6 R o o m T e m p e r a t u r e R e s i s t i v i t y The room temperature resistivity shows some anomalous characteristics. The resistivity is a function of thickness, but not a monotonic one. The data for the room temperature resistivity is tabulated in figure 4.6. The resistivity increases with thickness for the thinner films, i.e. films with thicknesses below 146A, but then decreases wi th increasing thickness for films thicker than 146A. The reason for this behavior is not understood. It may be indicative of a transition in terms of the mobili ty of the carriers inside the film. If the mobil i ty of the carriers is affected by the potential at the surface of the film, then the thickness would have an effect on the resistivity of the film. There could exist some thickness beyond which the mobili ty of the carriers is no longer as sensitive to the potential at the surface of the film. This may explain why the resistivity drops as the thickness increases for film thicknesses which are greater that 146A. Chapter 4. Results 69 4.7 The Sheet Resistance at 4.2 K The surface resistivity of the bismuth films at low temperatures is of interest because this is the relevant parameter for microwave absorption. It is important for our lab to ensure that the surface resistivity of the films is 1 8 8 . 5 0 / D at low temperatures, since the bolometeric detectors in the B A M experiment[4] rely on this thin coating to absorb microwaves. It has been possible to come up with an empirical model to predict the surface resistivity at 4.2K given the thickness of the film. The sheet resistance of the films was measured at 4.2K as a function of film thickness, and this data appears in table 4.7. The relationship between the sheet resistance and the film thickness is clearer when F i l m Thickness(A) P4.2(/^Ocm) Ro{tt) 85 211.4 248.7 87 239.4 279.8 103 247.8 240.6 146 317.4 217.4 500 519.8 104.0 3005 566.3 18.9 Table 4.7: Data showing the sheet resistance and the resistivity at 4.2K for the bismuth films prepared in this thesis. The sheet resistance clearly decreases as the thickness increases. we plot the logarithm of Ra against the logarithm of the inverse thickness. The data is plotted in figure 4.20. A quadratic fit gives an acceptable fit to the data, as evidenced by a plot of the residuals in figure 4.21, although the sheet resistance of the thinnest films seems to be scattered considerably. No improvement to the fit is observed for fitting the data to higher order polynomials. The equation of the fit plotted in figure 4.20 is, 'is i 2 log(Ra) = 1.907 - 0.771log - 0.279 log (4.78) where the thickness of the film, i , is in angstroms. A calculation using equation 4.78 shows that a film with a thickness of 198A wi l l have a sheet resistance of 1 8 8 . 5 0 / D at Chapter 4. Results 70 4.2K. This is shown on the graph as the intersection between the vertical and horizontal lines. 4 . 8 Oxidization of Bismuth Films The resistance of two of the samples, E and C, was measured as a function of time at room temperature. The films were left sitting on the bench, exposed to the air, for a period of 8 days. The resistance of each sample was measured at the same time each day. The data is plotted in figure 4.22. The resistance of sample E is found to stay approximately constant over this t ime period, while the resistance of sample C is found to increase with time. The reason for this increase could be due to the growth of an insulating layer of Bi^O^ on the surface of the film. Cohn and Uher[21]have observed growth of this type of layer on thin it < 90A) bismuth films as a function of time, and found that the sheet resistance increases with time. The growth of this insulating layer would tend to increase the resistivity, since the effective film thickness decreases. This provides a plausible explanation for the data in figure 4.22. The resistance of the thicker fi lm is fairly insensitive to small changes in the thickness caused by growth of the insulating layer, and thus stays basically constant over short time scales. The thinner film changes resistance quickly, and this is due to the reduction of the effective film thickness as a function of time. 4 . 9 Conclusions About T h i n Films of Bismuth The main motivation for investigating the electrical characteristics of thin bismuth films, for our laboratory, is the production of bolometeric detectors. The bolometers used in the B A M experiment consist of small rectangular pieces of sapphire with a thermistor Chapter 4. Results 71 3000A 1000A 500A 200A 100A 300 200 -cd ^ 100 g 80 fl 70 w 60 50 CD 0) 40 CD tj~l 30 20 0.001 l / T h i c k n e s s d - 1 ) 0.01 Figure 4.20: The sheet resistance at 4.2/\ as a function of the film thickness. The solid line is a fit to the data, computed from equation 4.78. The lines drawn in show that a 198A film should have a sheet resistance of 188.5fJ/D at 4.2/ \ . Chapter 4. Results 72 Residuals vs Inverse Th ickness 10 0 h a - 1 0 h 0.001 l / T h i c k n e s s ^ - 1 ) 0.01 Figure 4.21: The residuals for the fit from figure 4.20. There is considerable scatter in the sheet resistance of the thinnest films. Chapter 4. Results 73 1 8 0 "i i i I i 1 i 1 1 1 r "i l r 1 6 0 o C • i—i > • i—i - i - i • i—I GO QJ P H 1 4 0 • • Sample C Sample E 1 2 0 J 1 i i i i i i i i i i i i 0 2 4 6 Elapsed T ime ( d a y s ) 8 Figure 4.22: The resistivity was measured as a function of time for two samples. The thicknesses are E - 500A; C - 103A. Chapter 4. Results 74 attached. A thin film absorbing layer, which in principle could be any conductive ma-terial, is deposited on the surface of the sapphire. M a x i m u m power wi l l be dissipated in the absorber when the sheet resistance of the thin film is equal to 188.5f i /n . A pri-mary result of this thesis is a method to consistently produce bismuth films wi th a sheet resistance of 188.5Q/D at low temperatures. The method is summarized as follows: 1) Cleanliness of the substrate is important. A cleaning schedule similar to the one quoted in chapter three should be applied. Any other cleaning schedule may produce significantly different results than those obtained in this thesis. 2) The thickness of the film should be approximately 200A. Reproducibili ty of the measurements has proved to be a problem, since small differences in the cleanliness or cleaving of the substrate cause noticeable fluctuations in the resistivity. Thus, one is l ikely to get films of slightly different sheet resistances even though the thicknesses may be the same. 3) The growth rate is important. Slow growth, i.e. a rate which is less than l A per second, should be avoided. The reason is most likely due to a combination of radiative heating of the substrate inside the vacuum chamber, and buildup of oxides and other impurities. This has been observed to affect the resistivity of the fi lm unpredictably. A n extremely fast growth rate, i.e. 30A per second, was found to produce a film wi th higher resistance than a film of similar thickness but with a growth rate of O.lA per second. W i t h an extremely fast growth rate, it is hard to consistently produce films of the required thickness. Therefore, it is recommended that the films be grown wi th a moderate growth rate, in the range 1 — 5A per second. W i t h such a growth rate, radiative heating of the substrate is not a problem, since the temperature of the boat is not too high. Consistency of film thickness is easily attainable with the moderate growth rate. 4) F i lms grown on glass and sapphire substrates with the same thickness were mea-sured to have different resistivities. This indicates that the use of a glass witness slide Chapter 4. Results 75 to monitor the resistance of the film during evaporation is poor practice. A sapphire witness slide should be used. 5)The resistance of the films was measured to increase as a function of time, due to exposure to the air. This was observed to increase the resistivity of a thin film more than that of a thicker film. This increase in resistance might be stopped by coating the bismuth film with a SiO overlayer. The method above differs significantly from the current method employed to produce absorbing layers for bolometers. The main practical contribution of the results presented here is that the absorbing films may be produced by monitoring the thickness of the film, instead of using a witness slide to monitor the sheet resistance. This is a t ime saving step, since one is no longer required to produce a witness slide, which involves vacuum evaporation of gold leads on a glass substrate. A crystal thickness monitor can be used instead, and such a device is usually standard equipment in a thermal evaporator. Perhaps the most important result of this work is a physical interpretation of the temperature dependence of the resistivity of thin bismuth films. This is an important contribution, since there seems to be no model in the literature which successfully ac-counts for the shape of the resistivity - temperature curves in thin film bismuth. We have seen that the application of the standard models one invokes to explain the resistivity of a th in fi lm al l fail when applied to bismuth films. The anomalous temperature depen-dence of the resistivity of bismuth films arises primari ly because bismuth is a semi-metal. Semi-metals are characterized by low Fermi Energies, and a low carrier concentration. In such a material, the energy distribution of the carriers is a function of temperature, in the temperature range 300K — 4.2K. As the film is cooled from room temperature, fewer and fewer electrons and holes are able to tunnel through the grain boundaries easily, and the resistivity increases. The resistivity becomes approximately constant at low temperatures because most of the carriers have energies which are comparable to Chapter 4. Results 76 the Fermi Energy tp, and are not energetic enough to tunnel through the grains easily. The reason this anomalous temperature dependence of the resistivity is not seen in the bulk material is because the grains in the bulk material are larger than the mean free path of the carriers, and grain boundary scattering makes a negligible contribution to the resistivity. This is shown conclusively by the data of J in et al. [2], who observe that the character of the resistivity-temperature curve can be completely changed by increasing the size of the grains in the film by annealing it at a temperature close to its melting temperature. Properties like those of the bulk are observed in films with grain sizes larger than the mean free path of the carriers in bismuth, and this seems to indicate that grain boundary scattering makes a large contribution to the resistivity of thin bismuth films. A final note; the grain size in bismuth films has been shown by Hoffman et al.[10] to be practically independent of film thickness for thicknesses of 2000A and greater, and decreases wi th decreasing thickness for films thinner than 2000A. The original motivation for this thesis was to change the temperature dependence of the resistivity of a thin bismuth film by annealing it at a temperature near its melting temperature, as shown by J i n et al.(figure 4.14). If one could arrange it so that the sheet resistance of the film at low temperatures was equal to or lower than the sheet resistance at room temperature, then one could build a bolometer which has less bismuth and consequently, a lower heat capacity. Such a device would have a faster response than bolometers that are currently built . Several experiments were attempted to reproduce the results of J in et al. for films of approximately 180Q/D at low temperatures. These experiments all failed due to the following reason : the grain size can not reasonably exceed the thickness of the film, for a film thinner than 2000A. Thus, trying to anneal a bismuth film that is thinner than 2000A doesn't increase the grain size, since it is fundamentally l imited by the thickness of the film. When one chooses to use thin films of bismuth as an absorbing layer for a bolometer, one is in some sense stuck with the anomalous temperature dependence of Chapter 4. Results 77 the resistivity. Annealing such a thin film unfortunately doesn't change the temperature dependence of the resistivity. Chapter 5 Resistance of Composite Aluminum 5.1 M A P M i r r o r The Microwave Anisotropy P r o b e ( M A P ) is a satellite slated to fly in the year 2000. Its goal is to measure the temperature anisotropics in the entire microwave sky over a broad range of angular scales. M A P ' s microwave receivers are High Electron Mobi l i ty Transistors ( H E M T ' s ) which are i l luminated by 2 gregorian type mirrors. In order to reduce the payload weight, these mirrors have been fashioned out of a carbon fiber composite material, specifically SF-70A-75/RS-3 spread fabric. The reflectivity of this material by itself is poor, and to make up for this shortfall a 5000A thick coating of a luminum has been vapor deposited on the surface. The emissivity of the mirror w i l l depend largely on the sheet resistance of the aluminum surface. The sheet resistance of the a luminum coating has been measured at 80A', and the resistivity of the sample has been characterized as a function of temperature over the temperature range 300A' — A.2K. 5.2 Experiment The sample is cut to a rectangular shape with dimensions 1.52" x 0.075". The sample is then glued onto the surface of an aluminum chip using Mi l le r Stevenson 907 Epoxy. This epoxy is an excellent electrical insulator, and electrically isolates the sample from the a luminum chip. Electrical leads are bonded to the sample in a four wire measurement configuration using EpoTek silver epoxy. The distance between the sensing leads is 78 Chapter 5. Resistance of Composite Aluminum 79 1.1780 inches. The room temperature resistance of this sample is 1.375 ohms. W i t h such a low value of resistance, the contact resistance between the electrical leads and the sample surface produced noticeable effects when the connectivity was checked wi th a two wire resistance measurement. This is not problematic, however, since the four wire resistance measurement is insensitive to the resistance of the measurement leads. This is confirmed by measuring a V - I curve of the sample at room temperature. This is plotted in figure 5.23. V - I Curve for MAP Mir ror Sample Current (mA) Figure 5.23: A voltage versus current curve for the M A P mirror sample at room temper-ature. The material shows ohmic behavior over the range of current shown. The error bars on each point are the same size as the plotting symbols. The slope derived from this graph is 1.376 +- 0.005O which is in excellent agreement with measurements made using a four wire resistance technique. Chapter 5. Resistance of Composite Aluminum 80 The temperature of the sample is measured with a diode which is bonded to the backside of the sample holder. The diode is the load for a constant current source of 50 micro-amps, and as the temperature decreases, the diode voltage increases in a linear fashion. The sample is loaded into a cryogenic probe which is designed to fit inside the neck of a l iquid helium storage dewar. The probe is evacuated with a mechanical vacuum pump and then backfilled with helium gas prior to cooling. The temperature of the sample is changed by moving the probe to different vertical positions inside the neck of the dewar. The probe is ini t ia l ly positioned near the top of the dewar, and is lowered slowly into the l iquid helium bath. After each movement the probe is held stationary for a period of twenty minutes to allow the diode voltage to stabilize. Once the probe is submerged i n the l iquid helium, the process is reversed using the same procedure. The total length of the experiment was 3 hours. 5 . 3 R e s u l t s The temperature and the resistance of the sample were each measured at five second intervals. The resistivity has the following relation to the measured resistance: 0 075" P = RrneasJJf^; X 5 0 0 X 1 0 ~ V (5-79) which gives the result in units of fim. A graph of the resistivity for the temperature range 300/^ — 4.2K appears in figure 5.24. No appreciable hysteresis was noticed between the cooling and warming data, and this indicates that the sample and the diode were in good thermal equil ibrium during the experiment. The data points shown on the graph are the raw data for both the cooling and warming data sets. The diode is not an accurate temperature sensor for temperatures below 40 K , and was not used to infer the temperature below this l imi t . The data point at 4.2 K was obtained after contact Chapter 5. Resistance of Composite Aluminum 81 with the l iquid helium bath had been established. The accuracy in the temperature is approximately +-0.5 K . Also shown in figure 5.24 is data for bulk aluminum from the American Institute of Physics Handbook. 5.4 Conclusions 5.4.1 V a p o r D e p o s i t e d A l u m i n u m A t all temperatures measured, the resistivity of the V D A coating is higher than that of the bulk material . Comparing the measured data to the bulk data shows that the sample has some of the same characteristics as the bulk material, most notably a general decrease in resistance wi th decreasing temperature and the existence of a temperature independent component of resistivity. The temperature independent part of the resistivity begins to dominate at a temperature of approximately 40A' in both the V D A sample and the bulk material . One can see that the shapes of the curves for the V D A sample and the bulk material are quite similar, and seem to differ only by a constant offset on the y axis. This indicates that the temperature dependent parts of the resistivity of the thin film and the bulk material are about equal, but that the temperature independent parts of the resistivities are not. If we analyze the data in terms of the models presented in chapter 2, we find some surprising results. A t 295A', the values of resistivity derived from the graph are, p/um — 4.3pftcm and pbuik = 2.74pilcm. Apply ing equation 2.54 given in chapter 2, we find a value for the specularity parameter of p = —33.6, where we have used the value of A = 144A for the mean free path of the electrons in bulk aluminum at room temperature. This negative value of the specularity parameter has no interpretation in terms of the Fuchs-Sondheimer model, and indicates that surface scattering is not the only contribution to Chapter 5. Resistance of Composite Aluminum 82 10 100 Temperature (K) Figure 5.24: The resistivity of vapor deposited aluminum ( V D A ) versus temperature. The open squares are data taken while cooling the sample, and the filled squares are data taken while warming the sample. Chapter 5. Resistance of Composite Aluminum 83 the resistivity. We can make a lot more progress i f we analyze the data in terms of the grain boundary scattering model. Using the simplification noticed by De Vries[14], equation 2.61, we get the following equation for a : a = (5.80) Using the values for the resistivities of the sample and the bulk material at 295K gives a value of a = 0.409. We don't know the grain size of the grains in the V D A sample. However, if we make an educated guess of approximately 2000A, which comes from the grain size evident in the S E M pictures taken of bismuth (see chapter 3), the value of the reflection coefficient turns out to be R — 0.85. This value is only an estimate, however, it indicates that grain boundary scattering is a more dominant process than surface scattering in determining the resistivity of the V D A sample. The relevant parameter for microwave absorption in this material is the sheet resis-tance. The sheet resistance of the V D A sample at a temperature of 80A' is, Ra(80K) = 0.051O/D. (5.81) The impedance of free space is ?? = 37717, and the ratio Ro/rj = 1.35 x 1 0 - 4 , which is quite small . For a plane wave traveling through free space normally incident on such a conductor, we can calculate the reflected power from the following equation: \P\2 = 1 - — , (5.82) V which yields a value for the percentage of reflected power of \p\2 = 99.946%. Thus the percentage of power absorbed in this conductor wi l l be 1 — \p\2 = 0.054%. Scanning electron microscopy ( S E M ) of a cross section of the sample has been done by Dr . L iqu in Wang of the Materials Engineering Branch of the Goddard Space Fl ight Center. The S E M photos appear here with permission from M r . T i m Van Sant of Chapter 5. Resistance of Composite Aluminum 84 the Materials Engineering Branch of the Goddard Space Flight Center. These pictures appear in figures 5.25 and 5.26. These pictures clearly show the thread structure of the carbon fiber composite. The surface of the carbon fiber fabric was grit blasted prior to deposition of the a luminum, and this has resulted in a rough aluminum surface. This is quite evident in figure 5.26. Figure 5.25: S E M micrograph of cross-section of V D A coated SF-70A-75 Spread Fab-r i c / Y L A RS-12D. The V D A is the bright irregular line running horizontally through the middle of the micrograph. The magnification is 300 times. Chapter 5. Resistance of Composite Aluminum 85 Figure 5.26: The same sample as in the previous figure, magnified 4000 times. The V D A is the bright irregular line running through the middle of the micrograph. The irregular surface of the V D A is evident from this picture. This is due to grit blasting of the carbon fiber spread before deposition of the V D A . Chapter 5. Resistance of Composite Aluminum 86 5.4.2 General Conclusions Considerable progress has been made in understanding the temperature dependence of resistivity of th in films of bismuth in this work. We have come up with a plausible model which explains the shape of the resistivity-temperature curve for thin bismuth films, and i a phenomenological fit which describes the general shape of the curves. The resistivity of the bismuth films can be explained in terms of the semi-metal nature of bismuth, and a strong dependence on grain boundary scattering. The model presented here is unique, in that no satisfactory explanation of the temperature dependence of the resistivity of thin film bismuth has been published in the literature. We acknowledge that some of the details of this model have not been worked out in full mathematical rigor, however, due to the sensitive nature of the electrical properties of thin bismuth films to the conditions under which they are grown, a mathematical model predicting the exact resistivity at a certain temperature is l ikely to find l imited use at best. We have also found that a bismuth film of thickness 200A has a sheet resistivity of ap-proximately 188.517/D at helium temperatures, and this is useful for building bolometeric i . . . ° detectors. The grain size of the grains in films thinner than 2000.4. cannot reasonably exceed the thickness of the film, and thus annealing a thin film of bismuth is unlikely to increase the grain size. This means that the character of the temperature dependence of the resistivity of a th in bismuth film cannot be changed by annealing the film, and an improvement i n detector performance is not realized by such a strategy. The resistivity of vapor deposited aluminum has been characterized as a function of temperature, and was found to be higher at all temperatures than that of bulk aluminum. We have found that th i scan be successfully interpreted in terms of scattering of carriers off of grain boundaries in the film. The sheet resistance of the V D A has been measured at 80K, and found to be a factor of 10 larger than a sample of equivalent thickness, but Chapter 5. Resistance of Composite Aluminum 87 wi th resistivity equal to that of the bulk. The power absorbed by the V D A has been calculated to be approximately 0.054%, which is quite a small fraction of the incident power. Despite having a resistivity which is higher than the bulk resistivity, the V D A coating w i l l l ikely make a suitable coating for the carbon fiber mirrors on the M A P sattelite. Bibliography G . E . Smith , G . A . Baraff, and J . M . Rowell. Effective g factor of electrons and holes in bismuth. Phys. Rev. A, (135):A1118, 1964. B . Y . J i n , H . K . Wong, G . K . Wong, J . B . Ketterson, and Y . Eckstein. Effect of anneal-ing on the transport properties of an epitaxial film of bismuth. Thin Solid Films, (110):29 - 36, 1983. A k i r a Kinbara Shigeru Baba, Hideaki Sugawara. Electrical resistivity of thin bis-muth films. Thin Solid Films, 31:329 - 335, 1976. Gregory S. Tucker, Herb P. Gush, Mark Halpern, Ichiro Shinkoda, and B i l l Towlson. Anisotropy in the microwave sky: Results from the first flight of the balloon-borne anisotropy measurement (bam). Astrophysical Journal, 475:L73-76, February 1997. Simon Ramo, John R . Whinnery, and Theodore Van Duzer. Fields and Waves in Communication Electronics, Second Edition. John Wi ley and Sons, 1965. L . Solymar and D . Walsh. Lectures on the Electrical Properties of Materials. Oxford Universi ty Press, 1979. Charles K i t t e l . Introduction to Solid State Physics. John Wi ley and Sons, Inc., 1953. V . B . Sandomirskii . Quantum size effect in a semimetal film. Soviet Physics JETP, 25(1):101 - 106, July 1967. V . P . Duggal, Raj Rup, and P. Tripathi . Quantum size effect in thin bismuth films. Applied Physics Letters, 9(8):293 - 295, 1 September 1966. R . A . Hoffman and D . R . Frankl . Electrical properties of thin bismuth films. Physical Review B, 3(6):1825 - 1833, 15 March 1971. Y u . F . K o m n i k et. al . Features of temperature dependence of the resistance of thin bismuth films. Soviet Physics JETP, 33(2):364-373, August 1971. T. J . Coutts. Electrical Conduction in Thin Metal Films. Elsevier Scientific Pub-lishing Company, 1974. 88 Bibliography 89 [13] A . F . Mayadas, M . Shatzkes, and J.F.Janak. Electrical resistivity model for polycrys-talline films: The case of specular reflection at external surfaces. Applied Physics Letters, 14(11):345 - 347, 1 June 1969. [14] J . W . C De Vries. Temperature and thickness dependence of the resistivity of thin polycrystalline aluminum, cobalt, nickel, palladium, silver and gold films. Thin Solid Films, (167):25 - 32, 1988. [15] Kas tur i L . Chopra. Thin Film Phenomena. McGraw - H i l l Book Company, 1969. [16] Y u F . K o m n i k and B . I . Belevtsev. Temperature dependence of the electrical re-sistance of amorphous bismuth films. Soviet Journal of Low Temperature Physics, 6(10):629 - 635, October 1980. [17] C . Uher and W . P . Pratt Jr . High precision, ultra-low temperature resistivity mea-surements on bismuth. Physical Review Letters, 39(8):491-494, 22 August 1977. [18] M e i L u et. al . Low temperature electrical transport properties of single crystal bismuth films under pressure. Physical Review B, 53(3): 1609 - 1615, 15 January 1996. [19] Alber t Messiah. Quantum Mechanics, Volume 1. John Wiley and Sons, New York, 1958. [20] H . T . Chu , P . N . Henriksen, and J . Alexander. Resistivity and transverse magnetore-sistance in ultrathih films of pure bismuth. Physical Review B, 37(8):3900 - 3905, 15 March 1988. [21] J . L . Cohn and C. Uher. Electr ical resistance and the time-dependent oxidation of semicontinuous bismuth films. Journal of Applied Physics, 66(5):2045 - 2048, 1 September 1989. 

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