UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1997-0581.pdf [ 8.72MB ]
Metadata
JSON: 831-1.0085088.json
JSON-LD: 831-1.0085088-ld.json
RDF/XML (Pretty): 831-1.0085088-rdf.xml
RDF/JSON: 831-1.0085088-rdf.json
Turtle: 831-1.0085088-turtle.txt
N-Triples: 831-1.0085088-rdf-ntriples.txt
Original Record: 831-1.0085088-source.json
Full Text
831-1.0085088-fulltext.txt
Citation
831-1.0085088.ris

Full Text

RESISTIVITY M E A S U R E M E N T S OF THIN FILMS OF BISMUTH: APPLICATIONS  FOR BUILDING BOLOMETRIC D E T E C T O R S By Christopher Grant Padwick  B . Sc. Hons (Physics) University of Regina, 1994  A  THESIS T H E  S U B M I T T E D  IN PARTIAL  R E Q U I R E M E N T S M A S T E R S  F U L F I L L M E N T  F O R T H E D E G R E E O F  O F  SCIENCE  in T H E  F A C U L T Y  D E P A R T M E N T  O F  O F  G R A D U A T E  PHYSICS  A N D  STUDIES  A S T R O N O M Y  We accept this thesis as conforming to the required standard  T H E  UNIVERSITY  O F  BRITISH  C O L U M B I A  Sept 1997 © Christopher Grant Padwick, 1997  O F  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 V 6 T 1W5  Date:  15  Abstract  T h e resistivity of t h i n bismuth films grown on sapphire substrates has been measured as a function of growth rate, thickness, and temperature. Seven different samples were measured. In all 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 i n the literature. T h e sheet resistance of the films at 4.2K was found to decrease w i t h increasing thickness. Based on the experimental data, a bismuth film of thickness 20uA should have a sheet resistance of 1 8 8 . 5 0 / D at helium temperatures. T h e 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 w i t h a layer of insulating B12O2, forming at the surface of the film.  The  sheet resistance and resistivity of a 5000A thick coating of vapor deposited a l u m i n u m was measured as a function of temperature in the range 300/^ — \.2K.  T h e resistivity  was found to be greater than that of the bulk material at all temperatures, and the sheet resistance at 8 0 / ^ was equal to R  n  = 0.051O.  n  Table of Contents  Abstract  ii  Table of Contents  iii  List of Tables  vi  List of Figures  vii  Acknowledgements 1  2  xi  Introduction  1  1.1  Introducing the Bolometer  1  1.1.1  1  Basics  1.2  Transmission Line Analogy  4  1.3  W h a t is a transmission line?  4  1.4  Transmission Line w i t h 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  Electrical Properties of Metals and T h i n F i l m s  14  2.1  Metals, Insulators, Semiconductors and Semi-metals  14  2.2  Semi-metals  17  2.3  F e r m i Energy and Fermi Surface  19  iii  3  2.4  Q u a n t u m Size Effect  22  2.5  Surface Scattering  24  2.6  G r a i n Boundary Scattering  25  2.7  Concluding Remarks  26  G r o w t h of T h i n F i l m s  29  3.1  Equipment  29  3.1.1  29  3.2  4  T h e r m a l Evaporation  Properties of T h i n F i l m s  31  3.2.1  31  T h i n F i l m Growth  3.3  Effect of Pressure on G r o w t h  37  3.4  Cohesion and Robustness  39  3.5  Substrate Cleanliness  41  3.6  Sample Preparation  41  Results  44  4.1  Temperature Dependence of T h e Resistivity  44  4.1.1  Surface Scattering  47  4.1.2  G r a i n Boundary Scattering  48  4.1.3  Q u a n t u m Size Effect  49  4.1.4  A New M o d e l  51  4.2  Low Temperature Resistivity  58  4.3  Effect of G r o w t h Rate on Resistivity  61  4.4  Resistivity R a t i o  63  4.5  Effect of Substrate  64  4.6  R o o m Temperature Resistivity  68  4.7  T h e Sheet Resistance at 4.2 K  69 iv  5  4.8  O x i d i z a t i o n of B i s m u t h F i l m s  70  4.9  Conclusions A b o u t T h i n F i l m s of B i s m u t h  70  Resistance of C o m p o s i t e A l u m i n u m  78  5.1  M A P Mirror  78  5.2  Experiment  78  5.3  Results  80  5.4  Conclusions  81  5.4.1  Vapor Deposited A l u m i n u m  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 agreement with experiment  2.2  4  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  4.3  18  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  4.4  45  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  4.5  The parameters for the fits shown in figure 4.16 . The fits are computed from equation 4.73  4.6  56  The data for the resistivity ratio, T, for films of various thicknesses. The data is plotted in figure 4.18  4.7  48  63  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  69  vi  List of Figures  1.1  A schematic diagram of a bolometer. See the text for an explanation of how the device operates  1.2  2  T w o 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  1.3  5  A transmission line w i t h a resistive impedance connected between the terminals. Consider the transmission line to be of infinite length  2.4  12  T h e distribution functions for electrons(/ ) and holes ( / / J , plotted for a e  material at 50 K w i t h a Fermi Energy of 0.027eV. A s the temperature tends toward absolute zero, the distributions become step-like i n nature. 3.5  21  A n S E M photo of a 2000A bismuth film grown on a sapphire substrate. T h e magnification i n this image is 10 000 times. T h e tick marks on the b o t t o m right indicate the scale of the photo  3.6  A n S E M photo of a 2000A bismuth film magnified 30 000 times.  34 The  crystallites are clearly visible, and the disordered nature of the film is apparent 3.7  35  A picture of a discontinuous bismuth film. T h e dark clumps are bismuth islands in the early stages of growth. The bright dots may be due to the formation of an oxide  3.8  36  A comparison between the lattice constants of sapphire and bismuth.  vn  . .  37  3.9  T h e resistance of a bismuth film was measured during and after deposition. T h e ambient pressure i n the bell jar was 1.7 x 1 0 ~ T o r ? \ 4  T h e film is  oxidizing inside the chamber, and thus its resistance is increasing w i t h time. 38 3.10  T h e resistance of a bismuth film during and after deposition. T h e actual evaporation only lasted for 1 minute. The ambient pressure i n the bell jar was 3.4 x 1 0  -6  torr. T h e decrease i n resistance with time is due to cooling  of the 3.11  film  39  A diagram of the sample used to measure the resistance of bismuth films. T h e 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. T h e thicknesses are: A  - 85A ; B - 87A ; C - 103 A ; D - 146A ; E - 500A ; G - 3005A. T h e resistivities of samples A, B, and C has been interpolated with straight lines i n the region between the data points 4.13 T h e grain boundary scattering applied to the data.  46 A marginal fit is  obtained for the room temperature resistivity data 4.14 Resistivity versus temperature for bismuth films of lpm  50 thickness.  The  data is from J i n et al.[2]. It is interesting to note that the resistivity of the b o t t o m curve is actually lower than that of the bulk, for temperatures greater than 100A''. T h i s fact was not noted by J i n et al., and indeed is hard to explain  54  4.15 T h e transmission coefficient and the Fermi Dirac distribution for a material at 200iv' are plotted on the same graph. T h e F e r m i - D i r a c distribution at absolute zero is also shown. A s the temperature increases, more particles w i l l be able to tunnel through the barrier, and the resistivity w i l l decrease.  vin  57  4.16 T h e data from J i n et al., B a b a et al., and sample D from this thesis. T h e 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 T h e resistivity at A.2K increases as the film thickness increases. This data disagrees w i t h that of B a b a et al.[3]  62  4.18 T h e resistivity ratios of the bismuth films increase as the thickness i n creases. T h e data point for sample F disagrees w i t h the other data.  . .  65  4.19 T h e resistivity of bismuth films grown on sapphire and glass, plotted as a function of inverse thickness. T h e solid lines are fits to the data, and the dashed line is the bulk resistivity at 295 K.  T h e values for the room  temperature resistivity from table 4.6 are plotted as well. T h e 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 T h e 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. T h e lines drawn i n show that a 198A film should have a sheet resistance of 188.5S7/• at 4.2K.  71  4.21 T h e residuals for the fit from figure 4.20. There is considerable scatter i n the sheet resistance of the thinnest  films  72  4.22 T h e resistivity was measured as a function of time for two samples. T h e thicknesses are E - 500A; C - 103A  IX  73  5.23 A voltage versus current curve for the M A P mirror sample at room temperature.  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 Fabr 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. 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 .  x  85  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 will be discussed. In addition, the physics of the transmission line will 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 i n the Cosmic Microwave Background R a d i a t i o n ( C M B R ) on the scale of approximately 1 degree.  T h e telescope operates i n the following manner.  Microwave light enters a two beam Fourier Transform Spectrometer after reflection from an a l u m i n u m mirror. After passage through the spectrometer, the light is directed onto two optical detectors called bolometers. These bolometers measure an interference pattern which is proportional to the difference i n 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 i n principle be any electrically conductive material. G o l d and bismuth are common choices for microwave absorbers. 1  Chapter  1.  2  Introduction  2) Substrate.  T h e substrate must be an electrical insulator, be mechanically rigid,  and have a very low heat capacity at low temperatures. T y p i c a 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 i n figure 1.1.  Liquid Helium  Bath  Figure 1.1: A schematic diagram of a bolometer. See the text for an explanation of how the device operates.  A bolometer functions i n 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 i n turn heats the thermistor, which changes its resistance. T h i s change i n 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. T h e bolometers i n the B A M telescope operate at a temperature of approximately 300mA'. Another requirement for successful bolometer operation is that the bolometer be i n 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 i n resistance would be measured. T h e basic thermodynamics of bolometer operation are easily derived from first principles. W e consider a body w i t h heat capacity C i n weak thermal contact with a cryogenic bath at temperature To.  T h e thermal conductance, G, of the link is defined i n the  following w ay, x  Ax  w  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 link. We consider only thermal conduction as contributing to the transport of heat. If the body, initially 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^^ dt  + G(T)AT  = W  (1.2)  where t is the time, and A T is the change i n temperature. The solution of equation 1.2 is, W t A T = - [ l - e " ] where r — C/G,  and is called the thermal time constant.  (1.3) In general, r is a function  of temperature since both the heat capacity of the body and the thermal conductance of the link are functions of temperature.  Equation 1.3 shows that the temperature of  the body w i l l rise i n an exponential fashion to an asymptotic value of T = W/G,  in a  Chapter  1.  Introduction  4  characteristic time 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 will respond slowly to any input of heat. Thus, one would like to minimize the heat capacity of the body i n order to get a device w i t h reasonably quick response.  W h e n 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. T h e specific heats of some common materials are listed i n table 1.1. Material aluminum copper gold bismuth sapphire  c(/j,J/molK) 405 200 165 40 0.05  Debye Temperature 428 343 165 119 1034  (K)  Table 1.1: T h e specific heats of some materials at a temperature of 0.3K. T h e values are computed from the Debye equation, which provides excellent agreement w i t h experiment.  1.2  Transmission Line A n a l o g y  T h e physics of the transmission line w i l l be developed i n the following sections. transmission line is directly analogous to many optical systems of interest.  The  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  W h a t is a transmission line?  A transmission line consists of 2 conductors i n a parallel orientation, separated by an insulator[5]. Transmission lines are very common i n 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.  ,  9V  dV = —dz dz n  ,  .  T  , di  = -Ldz—  at  ,  (1.4) '  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  , di , „ , dV dl = —dz = -Cdz—. dz ot T  (1.5)  E l i m i n a t i n g dz from these equations leaves two coupled differential equations 1.6 and 1.7. dV  dl  r  Differentiating 1.6 w i t h respect to z and differentiating 1.7 w i t h respect to t yields the following:  dV  dI  2  dz  2  r  -  2  L  dI  (1-8)  J ^ l  did  „d V  2  2  dzdt  ° 3t  (  2  1-9)  Direct substitution of equation 1.9 into 1.8 yields the following second order partial differential equation for the voltage:  dV  _  2  J ^ -  L  8V  _  2  C  1 dV 2  ~ d v - ^ ~ d ¥ -  (  1  -  1  0  )  E q u a t i o n 1.10 is readily recognized as the one dimensional wave equation. It describes the propagation of the voltage i n space and time down the transmission line. A similar equation may be derived for the current:  dI  _  2  dI  _  2  dz ~ ^-V W 2  LL  2  2  1 dI 2  ( L 1 1 )  Chapter  1.  Introduction  7  F r o m equations 1.10 and 1.11 we see that the voltage and the current down the transmission line w i t h speed v = 1/s/LC.  propagate  For two conductors i n 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. T h e inductance per unit length, L , is  L = ^Iri'b/a)  (1.13)  where p, is the magnetic permeability of the insulating material. T h e speed of propagation, v, is from 1.12 and 1.13 ,  1 VLC  '  ^'  1 4  '  )  E q u a t i o n 1.14 shows that i f the conductors are i n vacuum and separated only by free space, such that p = p  0  and e = e , then the speed of propagation w i l l be v = 0  l/^/p e , 0  Q  which is equal to the speed of light. T h e solution of equation 1.10 is composed of two traveling waves, one m o v i n g to the left and one moving to the right. The solution is of the following form:  V(z,t) = W^t  - z/v) + W (t + z/v)  (1.15)  2  Differentiating 1.15 w i t h respect to z yields  %  =  - z/v) + -W'{t + z/v) = -L -L l  d  (1.16)  Chapter  1.  8  Introduction  Integrating 1.16 w i t h respect to time yields the following equation for the current  I(z,t)  (Wi — W ) + 2  constant  where the constant comes from the integration and is not of interest for the traveling wave solution. T h e 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,  E q u a t i o n 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.  T h e current and voltage are waves which travel at speed v along the  transmission line i n the positive and negative z directions.  1.4  Transmission Line with a Resistive Load  In this section we w i l l consider an ideal transmission line with a resistive load attached between the outputs.  We w i 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 — — /. A s before the transmission line is ideal and has characteristic impedance Zn . ne  The traveling wave solution  developed i n section 1.2 consists of a wave moving to the right, and a wave moving to the left. T h e voltage across the load resistor will be given by the sum of the two traveling  Chapter  1.  9  Introduction  V(0,t)  = W (t)  + W (t)  1  = V  2  (1.19)  .  load  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)  - W (t))  = I  2  (1.20)  =  load  In these equations, W\ refers to the transmitted wave and W refers to the reflected 2  wave. We may calculate the reflection  coefficient,  p,  from 1.19 and 1.20 by taking the  ratio W (t) to Wi(i) as follows: 2  W 7—TT~-  P=  VV!  The transmission Vioad,  coefficient,  r,  ( L 2 1 )  =  ^load + Aline  can be calculated by taking the ratio of the load voltage,  to the transmitted wave W\, r = Vload  (1.22)  2 Zl ad Zload + Zn e 0  Wi  n  From equations 1.21 and 1.22 we can see that if Zi d oa  = Zn ,  p  ne  corresponds to no reflected wave, and the load impedance, Zi d,  is said to be  oa  matched  = 0 and r = 1. This impedance  to the transmission line. If Zi d — ° ° ; s would be the case for an open circuit, a  oa  1 ' Zu e —  r  lim  n  Zload  P=y—-z—-pf—  -.  ,-.  = 1,  . \  r r>  (1-23)  and we see that there is only reflection from the boundary and no transmission. In the case of a short circuit, Z\ i = 0, p — —1, and again there is only reflection. oa<  The power in the transmitted component of the wave is P  — I+W+  +  while the power in the reflected component of the wave is P_ = W /Zu 2  ne  reflected  power, P /, re  —  W /Zu ,  The  fractional  2  ne  is the ratio of P- to P , which is +  Prej  = ^  = p. 2  (1.24)  Chapter 1.  Introduction  10  F r o m conservation of energy, the power transmitted to the load resistor must be,  Ptrans = l-Pref  1.5  = 1-  P.  (1-25)  2  Sinusoidal Input Signal  Thus far the results obtained have been i n general applicable to any type of A C input signal. Special consideration of the case where the input signal is sinusoidal i n 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)  T h e voltage on the line w i l l again be the sum of two traveling waves. In phasor notation the voltage V(z,t) on the transmission line will be, V(z, t) = W e'  l(iz  x  +We  (1.27)  ipz  2  where W\ — \W\\ and W — \W \e , and 3 = u/v. The phase shift, 6 , could be applied l9p  2  P  2  to either W\ or W . T h e current on the line w i l l be given by, 2  I( ^t) = ^—(W e- -W e ). lpz  K  1  lPz  2  (1.28)  T h e 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. T h i s is the impedance seen by the generator(located at z — —I) looking down the transmission line at the load(located at z = 0). T h e 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.  11  Introduction  D i v i d i n g equation 1.27 by equation 1.28 one can obtain, z{-l,t)  =  z  llne  _ %_ .  (1-29)  m  The ratio W2/W1 is the reflection coefficient, p, from equation 1.21. Substituting for p i n equation 1.29 yields the following expression for the input impedance, ry,  , ,x  ry  Zi cos(3l)  +  oad  iZ sin(3l) lme  As a check of the result 1.30, the impedance at z — 0 should be just the load impedance, Zioad- Setting / = 0 i n 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 w i l l just be the load impedance, Zi d-  This makes sense since we have assumed that the  oa  transmission line is composed of ideal conductors. One can define a quantity called the admittance, and Yn  ne  — 1/Zu . ne  l/Zi d,  oa  oa  T h e n the admittance as seen from the generator is, W  7  +  'oadCOs(8l)  + iY sin(0l)  Yi cos(3l)  +  Y  X _ v  ine  1.6  by defining Yi d =  Y(z,t),  Une  •  iYi sin(3l) oad  Transmission Line W i t h a Resistive Short  Now we consider the situation depicted i n 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 i n parallel, with equivalent impedance Z  tot  ^L^Hne!{EL  + Ziine)-  =  T h e fractional power delivered to such a load w i l l be given by  equation 1.25, and is, p  One  -  =  1  - ( r a : )  2  -  ^  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.  12  Introduction  Figure 1.3: A transmission line with a resistive impedance connected between the terminals. Consider the transmission line to be of infinite length. The power dissipated in the resistive short will be PL = W /ZL 2  dissipated in the rest of the line will be Pu  = W /Zn . 2  ne  ne  and the power  Taking the ratio PL to  Pn , ne  we find, PL  Z  line  Pline  (  i  3  3  )  ZL  From equation 1.32, we notice that if ZL = Zu /2, ne  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  R e l a t i o n 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, dE  dE  2  2  x  dz  2  x  r  dt  2  ,  x  1.  Chapter  13  Introduction  which is the one dimensional wave equation. The solution of this equation is composed of two traveling waves, given by, E (z) x  = E e-  jkz  +  + E„e .  (1.35)  jkz  Similarly, the magnetic field is composed of traveling wave solutions of the form, H = ^[E e~'  kz  y  +  - E_e> ]  (1.36)  kz  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 = E, x  I = H, y  L = [i, C = e, and  =  ZQ. T h e 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 m e d i u m .  For example, the  characteristic impedance of free space is r) = ^J/j, /eo = 3770. 0  A l l the equations previously derived for the reflection coefficient, transmission coefficient, admittance, etc.  can be used for electro magnetic waves, provided we make  the substitutions described above. In particular, the transmission line w i t h 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. T h i s is exactly analogous to the situation described i n the previous section, except we make the substitution Zu  ne  = n = 3770. M a x i m u m power will be absorbed i n the resistive short when  ZL = 3 7 7 0 / 2 = 188.50, and thus we strive to produce an absorbing layer w i t h sheet resistance 1 8 8 . 5 0 / D at low temperatures.  Chapter 2  Electrical Properties of Metals and T h i n F i l m s  2.1  M e t a l s , Insulators, Semiconductors and Semi-metals  T h e electrical properties of many materials can be analyzed successfully i n terms of band theory. In this theory the interactions between the electrons i n a material and the crystal lattice cause the energy spectrum of the electrons to be broken up into discrete levels called bands. T h e electrons are only allowed to occupy certain discrete states w i t h i n these energy bands, and these states correspond to discrete values of the electron wave-vector k.  There are no available states i n between bands.  T h e gap in energy between these  allowed bands is called the band gap, and corresponds to forbidden zones. T h e number of available states i n k space i n each energy band can be calculated quite easily[6]. W e consider a one dimensional chain of atoms of length L. The spacing between adjacent atoms is a, and this is called the lattice constant.  T h e energy gaps occur for  values of A; = nir/a, where n is any positive or negative integer, and thus the w i d t h of an energy band is TT/U. T h e wave-vector can only take on discrete values inside an energy band.  These discrete values are |fc| — 2nn/L.  T h e number of available states i n an  energy band is ^ , where the negative values of the wave-vector have been accounted for. A l l o w i n g for spin degeneracy, since we are allowed to pack 2 electrons into every state k, gives # of states = 2 - = 2N, a  where the quantity N is the number of atoms per unit length. T h e extension to three  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. T h i s 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 w i l l be filled. T h e electrons i n a filled band have no empty states i n which to move, and an electric field applied to the crystal w i 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 w i l l only be half filled, and the substance will be a conductor.  T h i s 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. T h e behavior of a semiconductor can be understood i n 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 i n the lower energy band can be thermally excited to occupy states i n the higher energy band. Thus at higher temperatures, there are available states for the electrons to j u m p to, and the material is a conductor.  T h e conductivity of a  semi-conductor w i l l be governed by the distribution function of the electrons, which varies exponentially w i t h 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. T h e conduction i n semi-conductors is actually determined by the motion of two types of carriers. W h e n 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 = n q v h  h  + nqv.  h  e  e  (2.38)  e  If we treat the holes as particles which can undergo collisions, the average velocity in response to an electric field will be given by  = qhETh/rrih, and we can write  Vh  j = [+n v~k - n v ] e, h  e  (2.39)  e  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 =  n e"r h  h  m  h  +  n e"T e  e  (2.40)  E.  From the Free Electron Theory, one can derive [7] a value of the conductivity of a = ne r/m, 2  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 = e r/m. 2  Using this  definition, equation 2.40 becomes, j = [rihPh + n p ] E. e  e  (2.41)  Chapter 2. Electrical Properties  of Metals and Thin  Films  17  Thus, the current i n the material will be determined by the number densities and the mobilites of the carriers. T h e 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 i n the following way for a two carrier system, p -  -  v  -  1 np h  h  (2.42)  + n fi e  e  We conclude by stating that the resistivity depends inversely on the number densities and the mobilites of the carriers i n 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. T h e semi-metal elements are B i , A s , and Sb. O f 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. T h e 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 F e r m i Surface would be spherical, and these elements would exhibit a much lower resistivity. T h e slight distortion of the crystal lattice causes the Fermi Surface to become ellipsoidal, and causes the conduction and valence bands to overlap. T h e resulting F e r m i Surface consists of three small pockets of electrons, and one small pocket of holes. T h i s is the fundamental reason why the behavior of semi-metals differs from that of normal metals. T h e tremendous reduction of the carrier density compared to a good conductor such as a l u m i n u m causes the conductivity to be lower i n the semi-metals.  Chapter 2. Electrical  Properties  of Metals and Thin  Films  18  T h e conduction i n a semi-metal is highly anisotropic with respect to the crystal axes. T h i s 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], '0 E 2  m* = h  2  (2.43)  dk  2  where E is the energy and k is the wave-vector. T h e effective mass depends on the second derivative of the energy i n the band w i t h respect to the wave-vector. If this derivative is negative, then the effective mass w i l l be negative, and the carrier is no longer an electron, but a hole. T h e reason the effective mass is a useful quantity is because the carriers i n 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)  T h e effective masses for the carriers i n bismuth have been measured, and are i n general quite a bit lower than the mass of the free electron. T h e values for the effective masses in b i s m u t h are tabulated i n table 2.2, i n units of the free electron mass (mo). m  n  0.00113 0.000521  m 0.26 1.20 22  m 0.00443 0.0204 33  m -0.0195 23  -0.090  Table 2.2: T h e effective masses i n bismuth, after S m i t h et al. [1]. T h e top row are the masses at the bottom of the band, while the bottom row are the masses at the Fermi energy, t . F  T h e effective masses are positive i n three directions of the crystal, which correspond to conduction by electrons, and negative i n one direction, which corresponds to conduction by holes. T h e carrier density has also been measured by S m i t h et al.[l], and found to be Nbi = 2.75 x 1 0 c m , at room temperature. This carrier density is much lower than that 1 7  - 3  Chapter 2. Electrical Properties  of Metals and Thin  Films  19  of a normal metal like a l u m i n u m , which has a carrier density of N i = 18 x 1 0 c m ~ . In 2 2  3  a  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 and F e r m i Surface  T h e Fermi Surface is the boundary between occupied and unoccupied states at absolute zero. A l l the wave-vectors w i t h 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. T h e low carrier densities and the low effective masses i n the semi-metals combine to produce materials w i t h very low Fermi Energies. If the Fermi Surface of the material is spherical, the Fermi energy is given by the following equation[7],  T h e F e r m i energy of a metal is typically quite high, and this is due to the high carrier density. For example, the Fermi energy of a l u m i n u m is ep — 11.3eV (assuming a spherical Fermi energy surface). We may define the Fermi Temperature, T p , in the following way, T  F  =  (2.46)  k  B  where ks is Boltzmann's constant. m i n u m is Tp = 134 500K.  W i t h this formula, the Fermi Temperature of alu-  T h i s Fermi Temperature has nothing to do w i t h the actual  temperature of the electrons. It basically represents the ground state energy, and equivalent temperature, of the electrons i n 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 i n bismuth combine to give it a very low Fermi energy.  T h e Fermi energy of  bismuth has been measured by S m i t h et al.[l], and found to be ep — 0.027eV. F r o m this,  Chapter 2. Electrical Properties  of Metals and Thin  20  Films  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. W i t h 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], (2me ) / 1  A  ,  2  F  x  Approximating the effective mass of bismuth with m* = 0.01m and using the previously o  quoted value of the Fermi Energy yields a mean free path of A# = 10, 900A at room 8  Chapter 2. Electrical Properties  of Metals and Thin  Films  21  Figure 2.4: T h e distribution functions for electrons(/ ) and holes(/^), plotted for a mater i a l at 50K w i t h a Fermi Energy of 0.027eV. A s the temperature tends toward absolute zero, the distributions become step-like i n nature. e  Chapter 2. Electrical  temperature. 143A.  Properties  of Metals and Thin  Films  22  A similar calculation for a l u m i n u m yields a mean free path of AAI =  T h e long mean free path of the carriers i n bismuth as compared to the carriers  in a l u m i n u m is due to the tremendous differences (a factor of  10 !) 5  i n the carrier  concentrations.  2.4  Q u a n t u m Size Effect  The quantum size effect ( Q S E ) was first worked out by Sandomirskii[8] i n 1967. T h e argument is as follows. W h e n the dimensions of a material become comparable to the effective wavelength of the carriers at the Fermi Surface, the energy levels in the material w i l l be modified. T h e energy spectrum, which i n the bulk material consists of discrete bands, w i l l be further quantized by the reduction i n sample thickness. This quantization w i l l modify the density of states on the Fermi Surface, and will have an effect on the m o b i l i t y of the carriers. F r o m equation 2.41, we see that the conductivity of a material depends on the carrier density of holes and electrons, and on the mobility of each carrier. So, the Q S E w i l l have an effect on the conductivity of the material, and w i l l affect a l l quantities which depend on the mobility of the charge carriers. The Q S E is only observable i n 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 l u m i n u m is, A =  = 3.6A  (2.50)  where the value of 6p = 11.3eV has been used. T h e calculation shows that quantum size effects w i l l be observable i n a l u m i n u m films when the film thickness is on the order of 3 — AA. In practice, however, observation of Q S E i n a film of this thickness would be impossible, since a film of this thickness would certainly be discontinuous, and the Q S E would be masked by other effects. T h e situation is different for the semi-metals, which  Chapter 2.  Electrical Properties  of Metals and Thin  Films  23  have a low Fermi Energy, and thus a longer carrier wavelength. T h e carrier wavelength for b i s m u t h can be estimated from equation 2.50, using ep = 0.027eV, w i t h the result that A = 740A. A bismuth film of this thickness w i l l be continuous, and i n theory, the Q S E w i l l be readily observable. T h e Q S E model has some special predictions i n the case of a semi-metal. In the bulk material, the conduction and valence bands overlap by an amount 8. T h e Q S E causes the conduction and valence bands to break up into discrete sub-bands. system w i t h dimensions t x L  x L.  y  z  Consider a 3D  The dimensions i n the y and z directions are to be  considered infinite. The energy states of such a system are given by the following, E ,k ,k s  y  = e s + h (k 2  z  2  2  n  + k )/2m  (2.51)  2  y  z  n  where the subscript on the mass, ra , refers to the electron mass, and s is a positive n  integer. T h e wave-vectors i n the z and y directions are k — 27rs /L y  s  y  y  y  and k =  2ns /L ,  z  z  z  = s = 0, + — 1, +—2,... , and L and L are the lengths of the sides of the normalization z  y  z  rectangle. T h e quantity e„ i n 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. T h e energy states of the holes can be obtained from equation 2.51 by replacing the electron mass, m , by the mass of the hole, m . n  p  The Q S E  affects the energy states of the system i n only one component of the wave-vector. T h e band overlap, 5, is equal to the sum of the Fermi Energies of the electrons and the holes, 8 = e + c. n  p  (2.53)  A very intriguing prediction of the Q S E model is the so-called semi-metal to semiconductor. ( S M S C ) transition. T h e band overlap, 8, is a function of the thickness of the  Chapter 2. Electrical Properties  of Metals and Thin  Films  24  film, 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 critical thickness, t < a, the electron and hole bands w i l l no longer overlap. T h e result is a band gap of magnitude t  = S(a /t 2  g  — 1), and the material is no longer a semi-metal, but is a  2  semiconductor.  T h i s transition w i l l occur at a critical thickness which is equal to a = nh/\/2MS, M = m m l(m -\-m ). n  P  n  where  T h i s critical thickness is approximately 400A for bismuth. F i l m s  p  thinner than this are expected to exhibit semiconductor properties, while films thicker than this m a i n t a i n an overlap i n energy, and are semi-metals. T h e Q S E model makes several other quantitative predictions. According to the theory, the electrical conductivity of t h i n 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 w i t h a period of approximately 400/1, which is i n good agreement w i t h the theory[8]. T h e H a 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  T h e Fuchs-Sondhiemer model attributes the higher resistivity in t h i n films as compared to the bulk material to the scattering of the electrons off of the surfaces of the  film.  T h e Fuchs formula is obtained by solving the B o l t z m a n n transport equation using the relaxation time approximation[12]. The result is the following equation, — (Too  T h e conductivity  = 1-1.(1-p).  (2-54)  Oft  is the conductivity of the bulk material w i t h the same defect density  as the film, and k — t/X^, t is the thickness of the  where A ^ is the mean free path i n the bulk material and  film.  T h e 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 a l l values of A;[12]. W r i t i n g equation 2.54 i n terms of the resistivities, and using the b i n o m i a l expansion formula and keeping only first order terms, we find  1  =  +  (2.55)  Pc  M u l t i p l y i n g both sides of equation 2.55 by the thickness, t yields  (2.56)  pt = Poot + ^ f Z ( l - p ) .  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 i n chapters 4 and 5 of this thesis.  2.6  Grain Boundary Scattering  T h e grain boundary scattering model of Mayadas and Shatzkes [13] attributes the increase i n resistivity of a t h i n film as compared to the bulk material to scattering of electrons off of the grain boundaries. T h e model represents the grain boundaries as N parallel, partially reflecting planes, randomly distributed throughout the material. T h e average distance between the planes is d, and this can be taken as the average grain diameter. T h e B o l t z m a n n transport equation was solved w i t h 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;  - = -  +  ^  0  ^  ,  (2.57)  ' 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  film to the conductivity of the bulk material depends on the following equation, \  3  - \a  + a  2  2  - aHn (1 +  V  -  (2.58)  a  T h e parameter a i n equation 2.58 is defined i n the following manner,  where R is the grain boundary reflection coefficient, and Ao is the mean free path i n the bulk material. In terms of the resistivities, equation 2.58 becomes,  El = Po  I 3 [| - \a + a  2  =  - aHn ( l + ± ) ]  1  (2 60)  /(«)  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  T h e ratios of the resistivities can now be written as, ^• = 7 ^  Po  /(«)  (2.62)  = 1.390 + 1,  which should apply for larger values of a. T h e contribution of grain boundary scattering to the resistivity depends on three independent parameters, A , R, and d. Variations i n 0  any of the three parameters can cause a higher resistivity than that observed i n the bulk.  2.7  Concluding Remarks  T h e Fermi Temperature of metals is in general quite high due to the high carrier concentration. T h e physical result of this is that the energy distribution of the carriers is not changing appreciably w i t h temperature, in the temperature range 300A" — 4 . 2 A \ M a t h ematically this can be stated as follows: tp »  kT. Thus, the conductivity of a metal i n  Chapter 2. Electrical  Properties  of Metals and Thin  Films  27  this temperature range is governed almost exclusively by changes i n the mobilites of the carriers. T h e mobilites of the carriers in the bulk material are affected by thermal distortions of the lattice (lattice vibrations), scattering off of crystal dislocations, scattering off of lattice imperfections, and scattering off of impurities. T h e motion of the atoms i n the crystal lattice is due to thermal excitation, and the contribution to the scattering w i l l be temperature  dependent. T h e other scattering processes are temperature  independent.  Matthiessens's rule states that the total resistivity of the material will be the sum of the scattering contributions of both these terms, such that p = p (T) l  (2.63)  + p. 0  In general, the temperature independent term, p , dominates at low temperatures 0  while the temperature dependent term, pi, is an approximately linear function of T from room temperature to 77K,  then i n general varies as T  5  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 i n the film w i 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 a l u m i n u m , which appears i n chapter 5. T h e bulk data and t h i n film 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. E x p e r 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 w i t h decreasing temperature.  This is the same behavior as observed i n  metals, and indicates that the dominant scattering process at higher temperatures i n bulk b i s m u t h is electron-phonon scattering. However, the temperature dependence of the resistivity of t h i n films of bismuth is completely unlike that of the bulk material. In particular, the resistivity of a thin bismuth film increases as the temperature and becomes approximately constant at lower temperatures.  decreases,  A n t i c i p a t i n g our conclu-  sions somewhat, which are developed more fully i n 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 i n 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 i n terms of this model.  Chapter 3  G r o w t h of T h i n F i l m s  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 t h i n film on a substrate, but the films produced i n the present work were made by thermal evaporation. T h e method for producing the films w i l l be discussed, and the theory of thin film growth w i l l be briefly touched upon i n this chapter.  3.1.1  Thermal  Evaporation  T h e r m a l evaporation is a common and inexpensive technique for producing t h i n films. T h e substance to be evaporated is placed i n a resistance heater inside a vacuum chamber and heated until melting. Vapor atoms of the condensate collect on the surface of the substrate w i t h a speed depending on the location of the substrate i n 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 i n front of the substrate. T h e speed of deposition and deposition thickness are often measured using a commercially available crystal thickness monitor. T h i s device measures the fundamental frequency of oscillation of a quartz crystal, which changes slightly w i t h the amount of material deposited on it.  T h e crystal monitor is placed as close to the substrate as  possible during evaporation to accurately measure the thickness of the deposited film.  29  Chapter 3.  Growth of Thin  Films  30  Precision of a few angstroms is easily attainable with such a device. T h e 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 p u m p down, ionizing a l l the remaining air i n the chamber. This ionized air pulverizes everything i n the chamber and removes excess material from the surface of the substrate. T h e thermal evaporator is typically operated at pressures of 1 - 2 x l O  - 6  torr. This pressure is easily  attainable w i t h the use of a diffusion pump and a liquid nitrogen cooled cold trap. T h e substance to be evaporated is placed i n a tungsten boat, which is then placed between two poles of a transformer. Passing current through the boat causes it to heat up, which i n t u r n heats the material inside the boat. Tungsten is used because of its high melting temperature (3653K). If the substance to be evaporated is i n the form of solid pellets, like bismuth or i n d i u m , 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 i n the form of a wire, such as gold, then it may be twisted around a wire filament. W h e n 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. T h e 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 w i t h some precision.  Chapter 3.  3.2  Growth of Thin  Films  31  Properties of T h i n F i l m s  T h e physical properties of a thin film often depend heavily upon the procedure used to produce the film.  F i l m s produced by M B E are usually highly ordered and low defect  i n nature - those produced by thermal evaporation can be highly disordered with many defects. T h e deposition rate, substrate temperature, cleanliness of the substrate, purity of the evaporate, and quality of the vacuum can a l l have a major effect on the physical properties of the t h i n film. In this section the theory of thin film growth w i l l be briefly discussed.  3.2.1  T h i n F i l m Growth  Electron microscopy has been done i n situ during the growth of a t h i n film by a number of investigators(see the book by Chopra[15] for a comprehensive review). These experiments have revealed that thin film growth proceeds in a number of distinct steps. A t o m s 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. W h e n 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. T h e y are free to roam around on the surface of the substrate for a short time and if they do not lose any energy i n that time, then they w i l l re-evaporate and rejoin their companions i n the vapor phase. T h e probability that an atom w i l l re-evaporate is given by[12]  W = uexp'^r  (3.64)  1  where v is the vibrational frequency of the atom and Q  ad  is the energy of adsorption.  The atom w i 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 t h i n film growth  Chapter 3.  Growth of Thin  Films  32  begins. A t o m s w i l l also lose energy i n colliding with other atoms on the surface, and this begins the next stage of film growth. T h e atoms coalesce into small nuclei called "islands". These islands smatter the surface of the substrate and provide further growth centers for incoming atoms. T h e 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 u n t i l a critical radius is reached.  Islands larger than this radius w i l l break up  into two or more islands of smaller radius, and the process will begin again. T h e islands w i l l eventually become so numerous that contact between two or more islands w i l l be inevitable. T h e islands do not join together i n a contiguous manner, but j a m together i n 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 will eventually be covered to give a continuous thin film. Several factors can have an effect on the growth of a t h i n film. If the rate of incoming atoms is quite high, then an atom w i l l have less time to roam the surface of the substrate before it meets another atom. Island formation will occur sooner, and this process w i l l generally result i n a film w i t h a large number of small-sized grains. A film w i t h a slower growth rate has more of an opportunity to form large grains, and we expect to see fewer but larger grains i n a slowly deposited thin film. The substrate temperature can have a large effect on the properties of the et.  film.  A striking example is shown by K o m n i k  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 w i l l stick, and will produce a film w i t h very small grains. The result was a super-conducting film, w i t h a critical temperatures i n the range T — 4 — 8K. c  T h i s 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 m a t c h quite closely (typically within 1 percent), then epitaxial growth may occur. T h e 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. T h i s critical temperature is different for every pair of materials, and is determined largely by trial 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 w i t h a Scanning Electron Microscope ( S E M ) reveal the grain structure of the film quite nicely, and this is shown i n figures 3.5 and 3.6. T h e film i n these pictures has a thickness of 2000A, and was grown at a rate of lA/s.  The  substrate is sapphire. T h e 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 i n figure 3.7. T h e thickness of this film was 15A. The photo shows some small, bright dots smattering the surface of the substrate. These spots are presumably insulating i n 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 i n the early stages of growth.  Figure 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.  Chapter  3.  Growth of Thin  Films  35  Figure 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.  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  T h e longest dimension of the crystallites in figure 3.6 were measured, and the average was found to be x — 188 ± 50nm. This seems consistent w i t h the data of other investigators, whose results show that the average size of the grains i n bismuth scales roughly w i t h the thickness of the film[10]. These photos clearly show the disordered structure of the  film. It is not known i f bismuth grows epitaxially on sapphire.  T h e crystal structures  of the two materials are both hexagonal rhombic, although the space groups are not identical. T h e space group (SG) of bismuth is R3m(166) while the space group of sapphire(corundum) is R3c(167). T h e lattice constants match pretty closely, as shown i n the following table: Substance Bi Al 0 2  3  a 4.5459 4.7591 0  bo 4.5459 4.7591  c 11.8622 12.9894 0  Figure 3.8: A comparison between the lattice constants of sapphire and bismuth.  T h e greatest mismatch is between the lattice constants i n the c direction, which amounts to 8.7%. If the film is grown i n 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  Effect of Pressure on G r o w t h  T h e ambient pressure i n the vacuum chamber has a large effect on the growth of the film.  T h i s was discovered somewhat accidentally, as a film was deposited i n an ambient  pressure of 1.7 x 1 0 T o r r . T h e ambient pressure was high because one of the electrical _ 4  feedthroughs used i n the experiment developed a vacuum leak during pump down. W h e n the ambient pressure is around 1.0 x 1 0 T o r r , there is a mono-layer per second of air _ 6  molecules striking the surface of the substrate.  One can usually observe the pressure  Chapter 3.  Growth of Thin  Films  38  rising i n the bell jar when the evaporation is begun, which indicates that the evaporate source is heating up and outgassing inside the chamber. W h e n the quality of the vacuum is low, the film w i l l be subject to a large amount of impurities, and these impurities may adversely effect the electrical properties of the film. T h e effect of poor-vacuum is shown i n figure 3.9. T h e 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. W h e n 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 film. T h e resistance of another sample was measured during and after deposition for an R(D)  of Bi f i l m vs t i m e a f t e r T h e r m a l  Evaporation  Time(hrs)  Figure 3.9: T h e 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 T o r r . T h e film is oxidizing inside the chamber, and thus its resistance is increasing w i t h time. - 4  Chapter 3.  Growth of Thin  ambient pressure of 3.4 x 1 0  Films  - 6  39  Torr. This data is shown i n figure 3.10. T h e resistance  decreases w i t h time, and this is due to the cooling of the film. F r o m this data we can state a general rule of thumb : better films are grown i n lower ambient pressures. F r o m the data i n 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 i m i t i n g pressure for an o i l pumped chamber of 2 x 10~ Torr. 7  4—Wire Resistance of Bi F i l m vs T i m e After T h e r m a l  Evaporation  220  170  u 0  1  •  1  1 2  1  1  1  1  1  4 Time(hrs)  1  1  1 6  1  1  1  1 8  1  Figure 3.10: T h e resistance of a bismuth film during and after deposition. T h e actual evaporation only lasted for 1 minute. T h e ambient pressure i n the bell jar was 3.4 x 1 0 torr. T h e decrease i n resistance w i t h time is due to cooling of the film. - 6  3.4  Cohesion and Robustness  In principle any substance can be evaporated onto any other substance to give a t h i n film.  However, the cohesion between the two substances may be an issue. If the cohesive  Chapter 3.  Growth of Thin  Films  40  forces are p r i m a r i l y V a n Der Walls i n nature, which is a weak binding force, then the cohesion between the two substances w i l l be poor. T h e result w i l l be a film which can be easily scratched off. In general, a layer of oxide between the two substances w i l l improve cohesion enormously[15]. Such a layer is called a sticking layer, or adhesion layer. A n example of two substances w i t h poor cohesion is gold and sapphire. T h i s problem was directly encountered while preparing samples to measure the resistance of bismuth films for this thesis. T h e cohesion can be greatly increased if a sticking layer of c h r o m i u m is deposited between the gold layer and the sapphire. T h e robustness of a film can be quantified by applying a couple of simple tests. T h e first test is innocuously• named "the scratch test", i n which the surface of the film is scratched lightly w i t h a hard object. Visible flaking off of the film constitutes a fail. If the film 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 film, sticky side down. T h e 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 a l l the is removed by the sticky tape, this constitutes a fail.  film  If a film passes both tests, the  cohesion is quite good, and the film is said to be "robust". T h e 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 w i t h flying colors.  Chapter 3.  3.5  Growth of Thin  Films  41  Substrate Cleanliness  T h e cleanliness of the substrate has an effect on the cohesion of the film. A l l samples were cleaned using trichloroethelyne, acetone, and methanol i n an ultrasonic cleaner. Samples which weren't cleaned using the ultrasonic cleaner showed poor cohesion. T h i s seemed to be a necessary step i n the production of usable films.  3.6  Sample  Preparation  T h e 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 a l u m i n u m chip of dimensions 2.36 x 0.4 x 0.125 inches using Stycast 2850FT epoxy, set w i t h Catalyst #9. T h e dimensions of the a l u m i n u m chip were chosen for easy insertion into a l i q u i d h e l i u m storage dewar.  Before bonding the sapphire to the a l u m i n u m chip, the sapphire was cleaned  according to the following procedure: 5 mins i n hot trichloroethelyne bath and ultrasonic cleaner 5 mins i n hot acetone bath, and ultrasonic cleaner 5 mins i n hot methanol bath, and ultrasonic cleaner Rinse w i t h distilled water B l o w dry sample in dry nitrogen atmosphere D u r i n g cleaning the sample was handled by the edges only,using tweezers. T h e electrical leads consist of 4 individual strips of gold deposited by thermal evaporation lengthwise along the chip. T h e deposition of these electrical leads deserves some discussion.  In an earlier attempt to measure the resistance of the film, a mask was  constructed which allowed deposition of four gold circles i n a row near the edge of the substrate. T h e thought was to then deposit bismuth over the entire surface of the sapphire, and measure the resistance of the film using a four wire measurement. A l t h o u g h  Chapter 3.  Growth of Thin  Films  42  such a procedure w i 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 i n the film shows this more clearly. T h e conduction path i n such an arrangement is complex as the electric field lines are perpendicular at every point to the equipotent i a l lines. T h i s situation is resolved by depositing conductive leads lengthwise along the sapphire chip. T h e 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 trivial; if the film is rectangular i n shape and has dimensions N x M , where N is the w i d t h 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 t h i n (50 angstrom) layer of c h r o m i u m is deposited before the gold layer. T h e thickness of the gold layer is not critical, and a thickness of approximately 1000 angstroms was used successfully. 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 i n  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. T h e films easily resisted the so-called "scotch tape test", where a piece of scotch tape is placed on the surface of the film, sticky side down, and then quickly removed. T h e electrical connections to the gold strips were made by bonding 0.005 inch copper wires to them w i t h EpoTek silver epoxy. T h e other ends of these wires were bonded to a T 0 5 8 p i n header, again using silver epoxy. Ultrasonic wire bonding was investigated as a method to make the electrical connections, but it was found that the t h i n (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. A v a i l a b i l i t y of the ultrasonic bonder was also an issue, since our laboratory d i d not own one. The end result is shown i n figure 3.11.  Figure 3.11: A diagram of the sample used to measure the resistance of b i s m u t h 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 w i l l be presented. A number of interesting features are evident i n the data, and a separate section has been devoted to each.  4.1  T e m p e r a t u r e Dependence of T h e Resistivity  T h e most intriguing feature evident in the data is the anomalous temperature dependence of the resistivity. The resistivity of 6 thin films of bismuth has been characterized as a function of temperature i n the range 295K to A.2K. T h e resistivity at all temperatures was higher than that of the bulk material for every sample. T h e 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 w i t h temperature over the same temperature range.  T h e 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. T h e growth rate and thickness of each film, and the ambient pressure i n the bell jar at the time of evaporation are summarized i n table 4.3. T h e resistivity-temperature data for sample F is not plotted i n figure 4.12 for reasons given i n section three of this chapter. T h e films were a l 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 p u m p i n g , and  a further 28 hours of pumping d i d not lower the pressure any further. T h e inside of the 44  Chapter 4.  Results  45  j a r was meticulously cleaned w i t h 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 i m i t i n g pressure of the bell jar used Sample Label A B C D E F G  Thickness(A) 85 87 103 146 500 3000 3005  Pressure x 10 ( T o r r ) 3.8 4.2 3.8 4.0 4.0 4.4 4.4 6  Growth Rate(A/s) 0.25 30 0.33 1.2 0.33 1.0 17.0  Table 4.3: T h e thickness, growth rate, and pressure i n the bell jar just before evaporation. T h e ambient pressure was the pressure reached after approximately 20 hours of p u m p i n g .  in these experiments was approximately 3.8 x 10~ Torr. T h e pressure at evaporation of 6  each film was kept approximately constant, and as such each film is subject to the same amount of impurities present i n the bell jar. T h e temperature is measured w i t h a diode i n good thermal contact w i t h the sample. For a constant current, the voltage across the' diode varies linearly w i t h 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 i m i t . The points at 4.2AT are measured after contact w i t h the l i q u i d helium bath had been established. N o data is measured between 40A' and 4.2A^. T h e data for samples A, B, and C is interpolated w i t h a straight line between these points. T h e data for samples D , E, and G is not interpolated i n this region. T h e 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.  1  500  G  D  E  X  1  Results  46  1—I—I—r—l  1  1  l  l  l  I  I  r~|  \  \  400  £ 300 o c •!  1  •I  1  \  \\ «  c  t  -l->  \  B  .^\\  &  \  -|->  w  • i—i  S 200  A  ~~^V \\ \  #  v  \  B  —  :  \  I  I  I  I  I  I  I  I  \  \  \\  G  I  W  V  c  : 100  V^  ,  I  I  I  10  I  I  I  I  I  x  Vi_  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 i n the literature[3], which find that the resistivity plateaus at lower temperatures for thicker films. A feature that is evident i n all the data for samples A , B, and C is a m a x i m u m i n the resistivity at a temperature of approximately 65K, followed by a monotonic decrease w i t h temperature thereafter. T h e m a x i m u m 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  i n the data for the thicker films. This observation agrees w i t h published data i n the literature, which find that the resistivity m a x i m u m occurs only i n t h i n  films[ll].  T h e anomalous character of the temperature dependence of the resistivity of t h i n bismuth films has received much attention i n the literature. In most metals it is observed that the temperature dependent part of the resistivity of a metallic t h i n film is identical to the temperature dependent part of the resistivity of a bulk material[14]. T h e t h i n films always show a higher resistivity than the bulk, and this can be successfully interpreted i n 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 w i l l systematically apply each of the three models presented i n 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 t h i n bismuth films. i 4.1.1  Surface  Scattering  T h e 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. P l o t t i n g pt against t should yield a straight line. T h e data has been plotted for three different temperatures: 295A^, 60A^, and 4 . 2 ^ . T h e 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 i n table 4.4.  Chapter 4.  Results  48  Temperature(K)  Pbuik(pttcm)  p (pQ crn)  295 60 4.2  116 19.0 5.8  109.6 389.9 576.53  00  1  (l-p)Aoo(A) 126 -32.3 -52.8  Table 4.4: Parameters derived from analyzing the data i n terms of the Fuchs-Sondhiemer model. Notice that the specularity parameter must be greater than one to reproduce the values i n the last column of the table. The resistivity of a bulk sample with the same defect density as the film (poo) is seen to increase as the temperature decreases. This is an anomalous result, since the resistivity of bulk b i s m u t h decreases as the temperature decreases. Another anomaly i n the data are the y-intercepts for the fits to the 4.2K and 60K data. T h e 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 film than were incident i n the first place, and this can't be accounted for i n terms of the Fuchs - Sondhiemer model. These results indicate that the anomalous temperature dependence of the resistivity of t h i n bismuth films cannot be explained by non-specular reflection of the carriers from the film surfaces.  4.1.2  G r a i n B o u n d a r y Scattering  We can analyze the resistivity-temperature data i n terms of the grain boundary scattering model presented earlier. In this model, the mobility 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 i n general[10], then we can analyze the data at 295K i n terms of the grain boundary scattering model. If we plot the ratio of f i l m 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, a  35.0  (4.65)  Chapter 4.  Results  49  and thus A o — ^ = 35.0A. i — ti  (4.66)  Using the value A = 10900A, we find a reflection coefficient equal to R — 0.0032. 0  T h i s 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 i n these films at room temperature.  This data is plotted i n figure 4.13.  fails when we try to apply it to the data for the resistivities at 4.2K.  T h e model  T h e ratio of film  resistivity to bulk resistivity at 4.2K increases as a function of thickness, and this can't be explained solely i n terms of grain boundary scattering. It would require a negative value of the reflection coefficient, which doesn't have an interpretation i n terms of the grain boundary scattering model. T h e grain boundary scattering model doesn't account for the temperature dependence of the resistivity of thin bismuth films.  4.1.3  Q u a n t u m Size Effect  A quantitative prediction of the Q S E model is that the low temperature resistivity of a semi-metal film w i l l oscillate as a function of thickness, with a period of about 400A. T h e data presented i n 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 w i l l be observed i n bismuth for film thicknesses of about 400A. F i l m s 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(W /2kT) g  (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 W is the band overlap. A plot of log p vs 1/T should yield a straight line w i t h g  intercept b — log(A)  and slope m = W /2k. g  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 still 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. N o experimental group has claimed to observe the S M S C , and this may be evidence that the S M S C i n 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 i n t h i n bismuth films.  4.1.4  A New  Model  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 t h i n bismuth films. It is expected that these effects do play a role i n determ i n i n g the resistivity, but by themselves or i n combination they can't explain why the resistivity increases w i t h temperature, then plateaus to a constant value. In what follows, we w i l l explain the general features of a model which successfully explains many features i n the data presented here, and many features i n the data of other investigators. We note that although many groups have published data on the temperature dependence of t h i n 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 w i t h a phenomenological fit to the data based on our physical interpretation. Looking at figure 4.12, one is tempted to analyze the data i n terms of the mean free  Chapter 4.  Results  52  path of the carriers i n the material. However, this may lead to erroneous results. T h e resistivity of a material is determined by two important factors : the mobility and the number density of the carriers. Changes i n either one of these variables w i l l cause changes i n the resistivity of the material. Since the resistivity is affected by both variables, one cannot make general statements concerning the mobility of the carriers by only analyzing the resistivity curves. M a n y investigators who are interested i n the temperature dependence of t h i n bismuth films measure the H a l l coefficient, and the magneto-resistance coefficient. T h e H a l l effect occurs when a magnetic field is applied perpendicular to the current flowing i n a conductor. T h e 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 w i t h only one type of carrier, the H a l l coefficient is given by, (4.68)  RH = ~  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 H a 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 _  n x\ hi  nn  2  e  e  So, for a two component system, the H a l l coefficient probes the number density and mobility of the carriers. The sign of the H a l l coefficient indicates which carrier is playing the more dominant role. E x p e r i m e n t a l results reported i n the literature often give conflicting results for the H a l l coefficient i n t h i n 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  T h e size of the grains i n bismuth films has been shown by J i n et al.[2] to have a remarkable 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 i n figure 4.14. T h e result of this change i n sign of the T C R is that the resistivity of the film is observed to decrease w i t h decreasing temperature, which is the same behavior observed i n the bulk. T h e 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. T h e size of the grains i n 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 i n an unannealed sample. The conclusion reached from this work is that the mean free path of the carriers i n bismuth can be increased by annealing the film at a temperature close to the melting temperature (271C) of bismuth. In light of the data i n figure 4.14 from J i n et al., the size of the grains in bismuth films has an enormous effect on the resistivity at all temperatures. F r o m the analysis presented earlier, grain boundary scattering can not explain the anomalous temperature dependence of the resistivity. However, an implicit assumption i n 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. A s shown i n chapter two, the energy distribution of the carriers i n bismuth is a function of temperature, and this is a result of the low Fermi Energy of bismuth. T h e 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 i n the  film.  If we model the grain boundary as a potential barrier of height U and w i d t h L, the 0  transmission coefficient is given by the following equations[19],  Chapter  4.  Results  54  "i—i—r  200  •  ~i  Unannealed  i  r  i  i—i—i—r  • •  •  100 90 80 ^  £o 3  7  •  •  • • • •  •  0  60 z A n n e a l e d at 225C for 18h 50  >  40  CD  30  20  10 9 8  A n n e a l e d at 265C for 4 8 h J  I  I  I  I  L  J  L  10  100 T e m p e r a t u r e (K)  Figure 4.14: Resistivity versus temperature for bismuth films of lpm thickness. T h e 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.  55  Results  T =  4e{U  0  4 e ( [ / - e) + 0  T =  4c(e 4e(e - Uo) +  - e) Ugsinh (riL) 2  U) Q  U^sin?{kL)  o  (4.70)  e > Uo  (4.71) (4.72)  where n = y/Uo — e and fc = y/e — Uo- This represents the probability that a particle w i t h energy e w i l l be transmitted through the barrier. The transmission coefficient is plotted i n figure 4.15.  One can see that if e < <  Uo, the transmission coefficient is  essentially constant, and is small. W h e n 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 i n 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 w i l l be a step function, and the number of particles that can pass through the barrier easily w i l l be a constant. A s the temperature increases, the number of particles with energies greater than increases. However, i f tp «  tp  Uo, most of the particles will 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. A s the temperature increases further, a significant fraction of the particles may have energies which are comparable to the height of the barrier. A s a result, the number of particles passing through the barrier w i l l suddenly increase, since the transmission coefficient rises steeply when t ~ Uo- T h e F e r m i - D i r a c 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  critical value. This model reproduces many of the features seen i n the data if we postulate that the height of the barrier has some dependence on the size of the grains i n the film. If the barrier height decreases as the thickness of the film increases, then the temperature at which the resistivity becomes constant w i l l occur at a higher temperature for thinner films than for thicker films. This behavior is evident from the experimental data i n figure 4.12. Finally, we note that the resistivity-temperature curves can be fit approximately w i t h the following function,  p = P° + ^rjrzri  (- )  R  4  where T is the temperature, po is the bulk resistivity, and R and r are constants.  73  This  function has been applied to the data of B a b a et al. [3], J i n et al.[2], and the data for sample D, and this appears i n figure 4.16. In computing the fit for each curve, data from the annealed sample of J i n et al. has been used as the bulk resistivity, p . 0  This data is  shown as the b o t t o m curve i n figure 4.16. T h e parameters for the fits are listed i n table 4.5. F i l m ThicknessA 146 300 10000  R(pClcm)  325 1450 180  r(K) 140 210 48  Table 4.5: T h e parameters for the fits shown i n figure 4.16 . T h e fits are computed from equation 4.73.  F r o m figure 4.16 we see that the resistivity values for sample D(eind i n fact for a l l the samples i n this thesis) are i n disagreement w i t h the data of B a b a et al., and J i n et al. . A s can be seen from figure 4.12, the low temperature resistivity of the samples measured i n this work increases as the thickness increases, and this is the opposite trend to that observed by B a b a et al., and J i n et al. . Explanation of this is deferred until the next  Chapter 4.  Results  57  Figure 4.15: T h e 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. A s the temperature increases, more particles will be able to tunnel through the barrier, and the resistivity w i l l decrease.  Chapter 4.  Results  58  section i n this chapter. T h e function i n 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 film, and r roughly cooresponds to the temperature at which the resistivity starts to plateau. F r o m table 4.5, we see that for the data of B a b a 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. T h e data for the samples measured i n this thesis show this same general trend. In summary, the anomalous temperature dependence of the resistivity of t h i n b i s m u t h films is m a i n l y due to the fact that bismuth is a semi-metal, and as such has a low Fermi Energy. T h e energy distribution of the carriers is a function of temperature, for temperature variations i n the range 300 — 4.2/v. Scattering of the carriers from the grain boundaries causes the resistivity to become nearly constant at low temperatures. A s 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 i n the  4.2  film.  L o w Temperature Resistivity  T h e resistivity at 4.2K is found to increase as the thickness increases. The data is plotted i n figure 4.17. A data table appears i n table 4.7. This same behavior has been observed by C h u et a/.[20], but not by B a b a et a/.[3], or J i n et al.[2}. Referring to figure 4.16, which shows data from B a b a et al., J i n et al., and data for sample D, we note that the data presented i n this thesis show a different dependence on the film thickness than that  Chapter  4.  ~i  Results  1  59  1—i—i—r  t = 3 0 0 l , Data f r o m B a b a et.al  800 600 400 I  t=146A, Sample D  200 o  a  * i—i  'oo  t= 10,000A(Unannealed) Data f r o m J i n et.al.  100 80  60 40  20  10 8 6  E-  I  1  t=10,000A(Annealed), J  I  I  1  I  L  Data f r o m J i n et.al. J  L  J  I  10  I  I  L  J  L  100 Temperature(K)  Figure 4.16: T h e data from J i n et al., B a b a et al, and sample D from this thesis. T h e solid lines through the top 3 curves are fits computed from equation 4.73, and the parameters are listed i n 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 B a b a et al. . T h e low temperature resistivity of the films i n figure 4.12 increases w i t h increasing film thickness, while the data of B a b a et al. decreases w i t h increasing film thickness. T h e reason for the disagreement is not known, however, the effect may have something to do w i t h the quantum size effect ( Q S E ) . A s 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 a l l energy levels w i t h energies higher than the Fermi energy (ep) are empty, while a l l those below are filled. A s 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 i n relation to the crystal axes. C h u et al. [20] calculate that the 2D-3D transition w i 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. T h e orientation of the films grown i n this thesis is not known, but the thicknesses of the films A , B, and C are a l l less than the values listed above. T h e motion of the charge carriers i n 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 i n determining the m o b i l i t y 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 B a b a et al. were prepared on m i c a substrates, held at a temperature of 100C during evaporation. These films were epitaxial, as shown clearly by X - r a y diffraction studies[3]. The films presented here are most likely not epitaxial, judging by the large (8%) difference i n the lattice constants i n 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  Effect of G r o w t h Rate on Resistivity  T h e 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 i n thickness, but differ noticeably i n resistivity over the entire temperature range shown(see figure 4.12). Sample A was grown very slowly, w i t h a growth rate of 0.1 A/s,  while sample B was grown 300 times faster. T h e 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. T h e data for sample F seem to be i n disagreement w i t h that of sample G.  The  resistivity data for a l l the samples is tabulated i n table 4.6. T h e thicknesses of samples F and G are nearly identical, yet the resistivities do not agree at any temperature. T h e growth rate of sample F was l A / s , and the evaporation took nearly 50 minutes. T h e resistivity was measured as a function of thickness for this sample, and this is plotted i n figure 4.19. T h e data show that the resistivity reached a m i n i m u m 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  100  62  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 o f Baba et <d.[3].  Chapter  4.  63  Results  experiment. T h e tungsten boat can get quite hot during an evaporation, and the substrate may have been heated radiatively. For this reason, the resistivity-temperature data for this sample has not been plotted along with the other samples i n figure 4.12. T h i s 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 w i t h a higher resistivity than a film of comparable thickness but faster growth rate.  4.4  Resistivity Ratio  T h e resistivity ratio, T, is defined as follows, T = ^ i i  (4.74)  ^295  where p .2 is the resistivity at 4.2K, and p 4  295  is the resistivity at 295A^. T h e resistivity  ratios have been computed for seven samples, and the data appears i n figure 4.6. T h e Sample label A B C D E F G  Thickness(A) 85 87 103 146 500 3000 3005  P295(pQcm)  P4.2(p£lcm)  151.5 163.2 165.8 152.3 123.0 168.8 111.2  211.4 246.6 255.3 314.4 519.8 637.4 566.3  r  1.395 1.511 1.540 2.064 4.228 3.776 5.093  Table 4.6: T h e data for the resistivity ratio, T, for films of various thicknesses. T h e data is plotted i n 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 i n section 3 i n this chapter. A solid line has been drawn  Chapter 4.  Results  64  i n 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 w i t h a model. T h e usefulness of an exact fit is questionable as well i n light of the data from B a b a et al. [ 3 ] . T h e y have observed that films w i t h identical thicknesses grown simultaneously on m i c a substrates exhibit a scatter i n their resistivity ratios. Apparently, small differences i n the cleaving and the cleanliness of the substrate can have an effect on the electrical properties on the film.  4.5  Effect of Substrate  T h e resistivity was measured as a function of thickness for bismuth films grown on sapphire and on glass. T h e 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. T h e resistance of the film grown on the glass substrate was measured w i t h a two lead technique. T h e 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. T h e contact resistance was not measured for this sample, and thus the raw data appears i n figure 4.19. T h e 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 measured 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 B i s m u t h Films  100  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 w i t h a four lead dc technique, and the contact resistance i n this arrangement is not problematic since negligable current flows through the sensing leads. T h e resistivity is measured to be higher at all thicknesses for the film 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. T h e data is plotted i n figure 4.19. Most of the data points for each substrate seem to follow a straight line w i t h identical slope but unequal intercept i n 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: r i -|0.19  p = 414.0  pflcm,  It  p = 1096.0  -pO.19  t  sapphire substrate  (4-75)  pttcm, glass substrate  (4-76) (4.77)  F r o m figure 4.19 we see that the data points for sample F do not agree w i t h the other data points for bismuth grown on a sapphire substrate. The reason for this is explained in section 3 of this chapter. T h e data point for sample G lies below the bulk value of the resistivity. This gives an estimate of the systematic uncertainty present i n these experiments, since the resistivity of the thin film will never be lower than that of the bulk material. T h e 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 still dropping slowly up to 10 hours after the evaporation is completed. This indicates that the film cools very slowly i n vacuum, and 1 minute i n between data points is not sufficient to allow the film to come to thermal equilibrium with its surroundings.  Chapter 4.  Results  67  3000A  1 0 0 0 A  500A  300A  2 0 0 A 1 0 0 A  4 0 0  300  o  2 0 0  F  • I—I  > • I—I  w • I—I  00  100  O B i s m u t h on Glass  90  D  B i s m u t h on S a p p h i r e B u l k Value  80 70  J  0.0005  L  i  j  0.001  1  0.005  i  J  i i  I  0.01  l/Thicknessll- ; 1  Figure 4.19: T h e resistivity of bismuth films grown on sapphire and glass, plotted as a function of inverse thickness. T h e solid lines are fits to the data, and the dashed line is the bulk resistivity at 2 9 5 / \ . T h e values for the room temperature resistivity from table 4.6 are plotted as well. T h e 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  T h e 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 temperature bolometers. In the B A M experiment[4], the absorbing material is a thin coating of bismuth, which is deposited on a sapphire substrate. T h e resistance of the film is usually monitored i n 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 Temperature Resistivity  T h e r o o m temperature resistivity shows some anomalous characteristics. T h e resistivity is a function of thickness, but not a monotonic one. The data for the room temperature resistivity is tabulated i n figure 4.6.  T h e resistivity increases with thickness for the  thinner films, i.e. films w i t h thicknesses below 146A, but then decreases w i t h increasing thickness for films thicker than 146A. T h e reason for this behavior is not understood. It may be indicative of a transition i n terms of the mobility of the carriers inside the film.  If the m o b i l i t y 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 mobility 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.  4.7  Results  69  T h e 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 i n the B A M experiment[4] rely on this t h i n 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.  T h e sheet resistance of the  films was measured at 4.2K as a function of film thickness, and this data appears i n table 4.7. T h e relationship between the sheet resistance and the film thickness is clearer when F i l m Thickness(A) 85 87 103 146 500 3005  P4.2(/^Ocm)  Ro{tt)  211.4 239.4 247.8 317.4 519.8 566.3  248.7 279.8 240.6 217.4 104.0 18.9  Table 4.7: D a t a showing the sheet resistance and the resistivity at 4.2K for the b i s m u t h films prepared i n this thesis. The sheet resistance clearly decreases as the thickness increases. we plot the logarithm of Ra against the logarithm of the inverse thickness. T h e data is plotted i n figure 4.20. A quadratic fit gives an acceptable fit to the data, as evidenced by a plot of the residuals i n figure 4.21, although the sheet resistance of the thinnest films seems to be scattered considerably. N o improvement to the fit is observed for fitting the data to higher order polynomials. The equation of the fit plotted i n figure 4.20 is, 'is  log(R ) a  = 1.907 - 0.771log  - 0.279 log  where the thickness of the film, i , is i n angstroms.  i2  (4.78)  A calculation using equation 4.78  shows that a film w i t h a thickness of 198A will have a sheet resistance of 1 8 8 . 5 0 / D at  Chapter 4.  4.2K.  Results  70  T h i s is shown on the graph as the intersection between the vertical and horizontal  lines.  4.8  Oxidization of B i s m u t h  Films  The resistance of two of the samples, E and C, was measured as a function of time at room temperature.  T h e films were left sitting on the bench, exposed to the air, for a  period of 8 days. T h e resistance of each sample was measured at the same time each day. T h e data is plotted i n figure 4.22. T h e resistance of sample E is found to stay approximately constant over this time period, while the resistance of sample C is found to increase w i t h time. T h e 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 w i t h time. T h e growth of this insulating layer would tend to increase the resistivity, since the effective film thickness decreases. T h i s provides a plausible explanation for the data i n figure 4.22. T h e resistance of the thicker film is fairly insensitive to small changes i n the thickness caused by growth of the insulating layer, and thus stays basically constant over short time scales. T h e 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 A b o u t T h i n F i l m s of B i s m u t h  The m a i n motivation for investigating the electrical characteristics of t h i n bismuth films, for our laboratory, is the production of bolometeric detectors. T h e bolometers used i n the B A M experiment consist of small rectangular pieces of sapphire w i t h a thermistor  Chapter 4.  3000A  Results  71  1000A  500A  200A  100A  300  200 -  cd  ^ 100 g 80 fl70 w 60 CD  0)  50 40  CD  tj~l  30  20 0.001  0.01  l/Thicknessd- ) 1  Figure 4.20: T h e sheet resistance at 4.2/\ as a function of the film thickness. T h e solid line is a fit to the data, computed from equation 4.78. The lines drawn i n show that a 198A film should have a sheet resistance of 188.5fJ/D at 4 . 2 / \ .  Chapter 4.  Results  72  R e s i d u a l s vs Inverse T h i c k n e s s  10  a  0 h  -10  h  0.001  0.01  l/Thickness^- ) 1  Figure 4.21: T h e residuals for the fit from figure 4.20. There is considerable scatter i n the sheet resistance of the thinnest films.  Chapter  4.  Results  180  73  "i  i  i  I  i  •  1  i  1  1  1  r  "i  l  Sample C  •  160  o C  • i—i  >  • i—i -i-i  • i—I  GO QJ  PH  140  Sample E 120  J  0  1  i  2  i  i  i  i  i  i  i  4  i  6  i  i  i  8  Elapsed T i m e ( d a y s )  Figure 4.22: The resistivity was measured as a function of time for two samples. The thicknesses are E - 500A; C - 103A.  r  Chapter 4. Results  attached.  74  A t h i n film absorbing layer, which i n principle could be any conductive ma-  terial, is deposited on the surface of the sapphire. M a x i m u m power will be dissipated i n the absorber when the sheet resistance of the thin film is equal to 1 8 8 . 5 f i / n . A prim a r y result of this thesis is a method to consistently produce bismuth films w i t h a sheet resistance of 188.5Q/D at low temperatures. T h e method is summarized as follows: 1) Cleanliness of the substrate is important. A cleaning schedule similar to the one quoted i n chapter three should be applied. A n y other cleaning schedule may produce significantly different results than those obtained i n this thesis. 2) T h e thickness of the film should be approximately 200A. Reproducibility of the measurements has proved to be a problem, since small differences in the cleanliness or cleaving of the substrate cause noticeable fluctuations i n the resistivity. Thus, one is likely to get films of slightly different sheet resistances even though the thicknesses may be the same. 3) T h e growth rate is important. Slow growth, i.e. a rate which is less than l A per second, should be avoided. T h e 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. T h i s has been observed to affect the resistivity of the film unpredictably. A n extremely fast growth rate, i.e. 30A per second, was found to produce a film w i t h 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 w i t h a moderate growth rate, i n 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 l m s grown on glass and sapphire substrates with the same thickness were measured to have different resistivities. This indicates that the use of a glass witness slide  Chapter  4.  75  Results  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 S i O overlayer. The method above differs significantly from the current method employed to produce absorbing layers for bolometers. T h e m a i n 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 time 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 i n 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 accounts for the shape of the resistivity - temperature curves i n t h i n film bismuth. We have seen that the application of the standard models one invokes to explain the resistivity of a t h i n film a l l fail when applied to bismuth films. The anomalous temperature dependence of the resistivity of bismuth films arises primarily 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.  A s 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. T h e resistivity becomes approximately constant at low temperatures because most of the carriers have energies which are comparable to  Chapter  4.  Results  76  the F e r m i Energy tp, and are not energetic enough to tunnel through the grains easily. T h e reason this anomalous temperature dependence of the resistivity is not seen i n the b u l k m a t e r i a l is because the grains i n the bulk material are larger than the mean free p a t h of the carriers, and grain boundary scattering makes a negligible contribution to the resistivity. This is shown conclusively by the data of J i n et al. [2], who observe that the character of the resistivity-temperature curve can be completely changed by increasing the size of the grains i n the film by annealing it at a temperature close to its melting temperature. Properties like those of the bulk are observed in films w i t h grain sizes larger than the mean free path of the carriers i n bismuth, and this seems to indicate that grain boundary scattering makes a large contribution to the resistivity of thin b i s m u t h films. A final note; the grain size i n 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 w i t h decreasing thickness for films thinner than 2000A. T h e original motivation for this thesis was to change the temperature dependence of the resistivity of a t h i n b i s m u t h 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 r o o m 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 i n et al. for films of approximately 1 8 0 Q / 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 limited by the thickness of the film. W h e n one chooses to use thin films of bismuth as an absorbing layer for a bolometer, one is i n some sense stuck w i t h 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 A l u m i n u m  5.1  M A P Mirror  T h e Microwave Anisotropy P r o b e ( M A P ) is a satellite slated to fly i n the year 2000. Its goal is to measure the temperature anisotropics i n the entire microwave sky over a broad range of angular scales. M A P ' s microwave receivers are H i g h Electron M o b i l i t y Transistors ( H E M T ' s ) which are illuminated 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 S F - 7 0 A - 7 5 / R S - 3 spread fabric. T h e reflectivity of this material by itself is poor, and to make up for this shortfall a 5000A thick coating of a l u m i n u m has been vapor deposited on the surface. The emissivity of the mirror w i l l depend largely on the sheet resistance of the a l u m i n u m surface. T h e sheet resistance of the a l u m i n u m 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  T h e sample is cut to a rectangular shape w i t h dimensions 1.52" x 0.075". T h e sample is then glued onto the surface of an a l u m i n u m chip using M i l l e r Stevenson 907 Epoxy. This epoxy is an excellent electrical insulator, and electrically isolates the sample from the a l u m i n u m chip. Electrical leads are bonded to the sample i n a four wire measurement configuration using EpoTek silver epoxy.  The distance between the sensing leads is  78  Chapter 5.  Resistance  1.1780 inches.  of Composite  Aluminum  T h e room temperature resistance of this sample is 1.375 ohms.  79  With  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 w i t h a two wire resistance measurement. T h i s is not problematic, however, since the four wire resistance measurement is insensitive to the resistance of the measurement leads. T h i s is confirmed by measuring a V - I curve of the sample at room temperature. T h i s is plotted in figure 5.23. V - I Curve for MAP M i r r o r Sample  C u r r e n t (mA)  Figure 5.23: A voltage versus current curve for the M A P mirror sample at room temperature. T h e material shows ohmic behavior over the range of current shown. T h e error bars on each point are the same size as the plotting symbols. T h e 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  T h e temperature of the sample is measured w i t h a diode which is bonded to the backside of the sample holder. T h e diode is the load for a constant current source of 50 micro-amps, and as the temperature decreases, the diode voltage increases i n a linear fashion.  T h e sample is loaded into a cryogenic probe which is designed to fit inside  the neck of a liquid h e l i u m storage dewar. The probe is evacuated w i t h a mechanical vacuum pump and then backfilled w i t h helium gas prior to cooling. T h e temperature of the sample is changed by moving the probe to different vertical positions inside the neck of the dewar. T h e probe is initially positioned near the top of the dewar, and is lowered slowly into the liquid h e l i u m 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 i q u i d h e l i u m , the process is reversed using the same procedure. T h e total length of the experiment was 3 hours.  5.3  Results  T h e temperature and the resistance of the sample were each measured at five second intervals. T h e 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 i n units of fim. A graph of the resistivity for the temperature range 300/^ — 4.2K appears in figure 5.24. N o appreciable hysteresis was noticed between the cooling and warming data, and this indicates that the sample and the diode were i n good thermal equilibrium 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 i m i t . T h e data point at 4.2 K was obtained after contact  Chapter 5.  Resistance  of Composite  Aluminum  81  w i t h the liquid h e l i u m bath had been established. The accuracy i n the temperature is approximately +-0.5 K . Also shown i n figure 5.24 is data for bulk aluminum from the A m e r i c a n Institute of Physics Handbook.  5.4  Conclusions  5.4.1  Vapor Deposited Aluminum  A t all temperatures measured, the resistivity of the V D A coating is higher than that of the bulk material. C o m p a r i n g 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 i n resistance w i t h decreasing temperature and the existence of a temperature independent component of resistivity. T h e temperature independent part of the resistivity begins to dominate at a temperature of approximately 40A' i n 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. T h i s 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 i n terms of the models presented i n chapter 2, we find some surprising results. A t 295A', the values of resistivity derived from the graph are, p/u  m  4.3pftcm  and pbuik = 2.74pilcm.  —  A p p l y i n g 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 i n bulk a l u m i n u m at room temperature. T h i s negative value of the specularity parameter has no interpretation i n terms of the FuchsSondheimer model, and indicates that surface scattering is not the only contribution to  Chapter 5.  Resistance  of Composite  Aluminum  10  82  100  T e m p e r a t u r e (K) Figure 5.24: T h e resistivity of vapor deposited a l u m i n u m ( V D A ) versus temperature. T h e 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. W e can make a lot more progress i f we analyze the data i n 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)  U s i n g 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 i n the V D A sample. However, if we make an educated guess of approximately 2000A, which comes from the grain size evident i n 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 i n determining the resistivity of the V D A sample. T h e relevant parameter for microwave absorption i n this material is the sheet resistance. T h e sheet resistance of the V D A sample at a temperature of 80A' is, Ra(80K)  = 0.051O/D.  (5.81) = 1.35 x 1 0 , which is  T h e impedance of free space is ?? = 37717, and the ratio Ro/rj  - 4  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\ = 1 - — 2  V  ,  (5.82)  which yields a value for the percentage of reflected power of \p\ = 99.946%. Thus the 2  percentage of power absorbed i n this conductor w i l l be 1 — \p\ = 0.054%. 2  Scanning electron microscopy ( S E M ) of a cross section of the sample has been done by D r . L i q u i n W a n g of the Materials Engineering B r a n c h of the Goddard Space F l i g h t Center.  T h e S E M photos appear here with permission from M r . T i m V a n Sant of  Chapter  5.  Resistance  of Composite  Aluminum  84  the Materials Engineering Branch of the Goddard Space Flight Center. These pictures appear i n figures 5.25 and 5.26. These pictures clearly show the thread structure of the carbon fiber composite. T h e surface of the carbon fiber fabric was grit blasted prior to deposition of the a l u m i n u m , and this has resulted in a rough a l u m i n u m surface. T h i s is quite evident i n figure 5.26.  Figure 5.25: S E M micrograph of cross-section of V D A coated SF-70A-75 Spread Fabr i c / Y L A R S - 1 2 D . T h e V D A is the bright irregular line running horizontally through the m i d d l e of the micrograph. The magnification is 300 times.  Chapter  5.  Resistance  of Composite  Aluminum  85  Figure 5.26: The same sample as i n the previous figure, magnified 4000 times. T h e V D A is the bright irregular line running through the middle of the micrograph. T h e 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  5.4.2  5.  Resistance  of Composite  Aluminum  86  General Conclusions  Considerable progress has been made i n understanding the temperature dependence of resistivity of t h i n films of bismuth i n this work. We have come up w i t h a plausible model which explains the shape of the resistivity-temperature curve for t h i n bismuth films, and i  a phenomenological fit which describes the general shape of the curves. T h e resistivity of the b i s m u t h films can be explained i n terms of the semi-metal nature of bismuth, and a strong dependence on grain boundary scattering. The model presented here is unique, i n that no satisfactory explanation of the temperature dependence of the resistivity of t h i n film b i s m u t h has been published i n 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 t h i n bismuth films to the conditions under which they are grown, a mathematical model predicting the exact resistivity at a certain temperature is likely to find l i m i t e d use at best. W e have also found that a bismuth film of thickness 200A has a sheet resistivity of approximately 188.517/D at helium temperatures, and this is useful for building bolometeric i . . . ° detectors. T h e grain size of the grains i n films thinner than 2000.4. cannot reasonably exceed the thickness of the film, and thus annealing a t h i n film of bismuth is unlikely to increase the grain size. This means that the character of the temperature dependence of the resistivity of a t h i n bismuth film cannot be changed by annealing the film, and an improvement i n detector performance is not realized by such a strategy. T h e resistivity of vapor deposited a l u m i n u m has been characterized as a function of temperature, and was found to be higher at all temperatures than that of bulk a l u m i n u m . W e have found that t h i s c a n be successfully interpreted i n terms of scattering of carriers off of grain boundaries i n 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  w i t h resistivity equal to that of the bulk. T h e 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 likely make a suitable coating for the carbon fiber mirrors on the M A P sattelite.  Bibliography  G . E . S m i t h , G . A . Baraff, and J . M . Rowell. Effective g factor of electrons and holes i n 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 annealing on the transport properties of an epitaxial film of bismuth. Thin Solid Films, (110):29 - 36,  1983.  A k i r a K i n b a r a Shigeru B a b a , Hideaki Sugawara. Electrical resistivity of t h i n bism u t h films. Thin Solid Films, 31:329 - 335, 1976. Gregory S. Tucker, Herb P. Gush, M a r k Halpern, Ichiro Shinkoda, and B i l l Towlson. Anisotropy i n the microwave sky: Results from the first flight of the balloon-borne anisotropy measurement (bam). Astrophysical Journal, 475:L73-76, February 1997. Simon R a m o , John R . Whinnery, and Theodore V a n Duzer. Fields Communication  Electronics,  Second  L . Solymar and D . W a l s h . Lectures  Edition.  and Waves  in  J o h n W i l e y and Sons, 1965.  on the Electrical  Properties  of Materials.  Oxford  University Press, 1979. Charles K i t t e l . 1953.  Introduction  to Solid  State  Physics.  John W i l e y and Sons, Inc.,  V . B . Sandomirskii. Q u a n t u m size effect i n a semimetal film. Soviet Physics 25(1):101 - 106, J u l y 1967.  JETP,  V . P . Duggal, Raj R u p , and P. Tripathi. Quantum size effect i n t h i n bismuth  films.  Applied  Physics  Letters,  9(8):293 - 295, 1 September 1966.  R . A . Hoffman and D . R . F r a n k l . Electrical properties of t h i n bismuth Review B, 3(6):1825 - 1833, 15 M a r c h 1971.  films.  Physical  Y u . F . K o m n i k et. al. Features of temperature dependence of the resistance of t h i n bismuth films. Soviet Physics JETP, 33(2):364-373, August 1971. T . J . C o u t t s . Electrical  Conduction  in Thin  lishing Company, 1974.  88  Metal  Films.  Elsevier Scientific P u b -  Bibliography  89  [13] A . F . Mayadas, M . Shatzkes, and J.F.Janak. Electrical resistivity model for polycrystalline 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 a l u m i n u m , cobalt, nickel, palladium, silver and gold films. Thin Solid Films, (167):25 - 32, 1988. [15] K a s t u r i L . Chopra. Thin Film  Phenomena.  M c G r a w - 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 resistance of amorphous bismuth films. Soviet Journal of Low Temperature Physics, 6(10):629 - 635, October 1980. [17] C . Uher and W . P . P r a t t Jr. H i g h precision, ultra-low temperature resistivity measurements 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] A l b e r t Messiah. Quantum 1958.  Mechanics,  Volume 1. John W i l e y and Sons, New York,  [20] H . T . C h u , P . N . Henriksen, and J . Alexander. Resistivity and transverse magnetoresistance i n ultrathih films of pure bismuth. Physical Review B, 37(8):3900 - 3905, 15 M a r c h 1988. [21] J . L . C o h n and C . Uher. Electrical resistance and the time-dependent oxidation of semicontinuous bismuth films. Journal of Applied Physics, 66(5):2045 - 2048, 1 September 1989.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085088/manifest

Comment

Related Items