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Electron localisation in non-stoichimetric films of aluminum nitride produced by reactive sputtering Fortier, Normand 1986

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ELECTRON LOCALISATION IN NON-STOICHIOMETRIC FILMS OF ALUMINUM NITRIDE PRODUCED BY REACTIVE SPUTTERING by NORMAND FORTIER B.A.Sc, Ecole Polytechnique De Montreal, 1980 M.Sc, University Of B r i t i s h Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s thesis as conforming to the required standard j THE UNIVERSITY OF BRITISH COLUMBIA June 1986 © Normand F o r t i e r , 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ^///g"//CC  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 3-6 (3/81) A B S T R A C T i i Sputtering is a very versatile process for the fabrication of thin solid Gims. The subject of this thesis concerns the study of thin 61ms of non-stoichiometric aluminum nitride fabricated by voltage-controlled reactive sputtering. Using the cathode voltage on the sputtering target, the relative arrival rates of nitrogen and aluminum atoms on a substrate are precisely controlled. The method allows the deposition of films over a wide range of composition. This deposition technique is shown to be an extremely useful tool in the study of electron localisation in solids. This thesis will provide a contribution toward a more complete understanding of these localisation effects. As the composition of the deposited films is gradually varied from essentially pure aluminum to nearly stoichiometric aluminum nitride, the structural disorder of the resulting films is smoothly varied. This technique has permitted the observation of three distinct transport regimes. These regimes are: The Boltzmann regime, the regime of moderate disorder, and finally the regime of strong disorder. The results obtained in this work are consistent with the interpretation that an enhancement of the structural disorder is accompanied by a change in the nature of the wave functions. To each of the three transport regimes observed is attributed a specific type of wave function. In the Boltzmann regime the wave functions are the familiar Bloch waves. These wave functions are extended. In the regime of moderate disorder the wave functions are thought to be power-law localised, the envelope of the wave function decaying as a power law. Finally in the strong disorder limit the wave functions are thought to be exponentially localised. A model proposed by Kaveh and Mott in which the electron wave function is assumed to be power-law localised is shown to accurately describe the regime of moderate disorder. A simple extention of the model is shown to account for the observed free-electron behavior of the Hall effect and the thermoelectric power. A regime of strong disorder in which the electronic wave functions are thought to be exponentially localised is observed. In this regime the conductivity proceeds by variable-range-hopping: Precise measurements of the temperature dependence of the conductivity support the relation: According to the variable-range-hopping model, the form of the exponential tem-perature dependence should be a consequence of the relatively constant density of states at the Fermi level. The results of optical and thermoelectric power measure-ments support this hypothesis. iv T A B L E OF C O N T E N T S Page A B S T R A C T i i T A B L E O F C O N T E N T S iv L I S T O F T A B L E S v i i L I S T O F F I G U R E S v i i i A C K N O W L E D G E M E N T x C H A P T E R 1 I N T R O D U C T I O N 1 1.1 M o t i v a t i o n f o r this Work. 1 1.2 A p p r o a c h used. 3 1.3 O r g a n i s a t i o n of the Thesis. 4 C H A P T E R 2 E L E C T R O N L O C A L I S A T I O N I N S O L I D S 6 2.1 T h e B o l t z m a n n regime. 6 2.2 T h e weak localisation regime. 9 2.21 T h e de conductivity. 9 2.22 T h e thermoelectric power. 14 2.23 T h e H a l l effect. 15 2.24 Electron-electron interactions. 17 2.3 T h e regime of strong disorder. 18 2.31 T h e resistor network. 18 2.32 T h e percolation approach. 20 2.33 Variable-range-hopping. 22 2.34 Temperature dependence of the prefactor, 23 2.35 Frequency dependence of the conductivity. 25 2.36 T h e mechanism of polarisation. 25 V 2.37 A simplified m odel of ac conductivity. 26 2.38 A more realistic m odel of ac conductivity. 29 2.39 Temperature dependence of a(u>t T). 30 C H A P T E R 3 E X P E R I M E N T A L T E C H N I Q U E S 33 3.1 V o l t age-controlled reactive sputtering. 33 3.2 T h e pr e p a r a t i o n of the samples. 37 3.3 T h e de c o n d u c t i v i t y measurements. 37 3.4 T h e ac c o n d u c t i v i t y measurements. 38 3.5 T h e H a l l effect measurements. 38 3.6 T h e thermoelectric power measurements. 40 3.7 T h e o p t i c a l measurements. 40 3.8 D e t e r m i n a t i o n of the film structure. 45 C H A P T E R 4 R E S U L T S A N D D I S C U S S I O N S 46 4.1 F i l m structure. 46 4.2 T h e B o l t z m a n n regime. 53 4.3 T h e weak localisation regime. 55 4.31 T h e dc conductivity. 55 4.32 T h e ac conductivity. 65 4.33 T h e thermoelectric power and H a l l effect. 68 4.34 T h e low temperature behavior of S(T). 72 4.4 T h e strong localisation regime. 72 4.41 T h e dc conductivity. 72 4.42 T h e nature of the localised states. 79 4.43 T h e thermoelectric power measurements. 87 4.44 T h e ac conductivity. 91 vi C H A P T E R 5 C O N C L U S I O N S 96 C H A P T E R 6 A P P E N D I X 99 6.1 E v a l u a t i o n of the magnitude of <rtnt. 99 R E F E R E N C E S 101 LIST OF T A B L E S Values of kf, le and /,-. Values of a, N(ef) and ir€ij/hvta. viii LIST OF FIGURES F i g u r e Page 2-1 Temperature dependence of p{T) 13 2-2 T h e polar i s a t i o n mechanism 27 2-3 R a n d o m d i s t r i b u t i o n of i m p u r i t y sites 31 3-1 Cross section of the sp u t t e r i n g system 34 3-2 T h e H a l l effect measurements 39 3-3 T h e thermoelectric power measurements 41 3-4 T h e evaluation of T(X) - 42 3-5 T h e evaluation of or(A) 44 4-1 X-ray diffraction s p e c t r u m of AINX 47 4-2 E l e c t r o n transmission photograph, art = 1.5xl0 4 {Qcm)~l 48 4-3 E l e c t r o n diff r a c t i o n measurements, art = 10 4 (Qcm)-1 50 4-4 E l e c t r o n diffraction measurements, aTt = 600 ( Q c m ) " 1 51 4-5 E l e c t r o n diff r a c t i o n measurements, aTt = 200 ( f i c m ) - 1 52 4-6 p(T) versus T, ( B o l t z m a n Regime) 54 4-7 a(T) versus Vf, art = 7 7 4 0 ( f i c m ) _ 1 56 4-8 a(T) versus Vf, art = 3 4 0 0 ( f l c m ) ~ 1 57 4-9 a(T) versus T, art = 7 7 4 0 ( f i c m ) _ 1 58 4-10 a(T) versus T, arl = 3 4 0 0 ( n c m ) - 1 59 4-11 a{T) versus Vf, arl = 280{Qcm) 1 and aH = 210(ncm) 1 61 4-12 a(T) versus Vf, art = 3 3 ( n c m ) _ 1 63 4-13 cr(u) versus u, UJ < 10 8 Hz 66 4-14 c(uj) versus u, visible a n d u l t r a violet 67 4-15 S(T) versus T 69 4-16 n versus arl 71 4-17 S/T versus T2 73 4-18 (1 — R) versus s 75 4-19 lnp{T) versus 7 1" 1/ 4, G r l = 0.5 ( n c m ) - 1 76 4-20 \np{T) versus T - 1 / 4 , art = 0.5 ( f l c m ) - 1 77 4-21 T{\) versus A for 4/JV x 85 4-22 a(A) versus A for AINX 86 4-23 versus T 90 4-24 lnp(o;, T) versus T"""1/4 92 4-25 er(oj, T) versus w 93 4-26 T ) versus T* 94 A C K N O W L E D G E M E N T X I would like to thank Dr. R.R. Parsons for his continuous support d u r i n g the course of this work. I also whish to thank Dr. R. Barrie, J o h n Affi n i t o , a n d M i k e B r e t t for the many i l l u m i n a t i n g discussions we have had. F i n a l l y I wish to thank the Uni v e r s i t y of B r i t i s h C o l u m b i a and Dr.R.R. Par-sons for financial support. CHAPTER 1: INTRODUCTION 1 C H A P T E R 1 I N T R O D U C T I O N 1.1 Motivation For This Work A considerable amount of work has been devoted recently, b o t h experimen-tal l y 1 _ 1 5 a n d t h e o r e t i c a l l y , 1 6 - 3 9 to the study of transport properties i n disordered materials. In materials e x h i b i t i n g li t t l e disorder the t r a d i t i o n a l approach has been to assume that the scattering of the c o n d u c t i o n electrons by a r a n d o m po t e n t i a l (e.g. impurities) causes the electron wave functions ( B l o c h waves) to lose phase co-herence on the length scale of their mean free path. Nevertheless, the wave functions r emain extended. T h e electronic motion has a ballis t i c character, the interaction of the c o n d u c t i n g electrons with electric field, temperature gradient etc., being de-scribed by semi-classical equations of motion ( B o l t z m a n n approach). O n the other hand, i n strongly disordered materials, A n d e r s o n 8 0 has pointed out that the wave functions can become localised, with t h e i r envelopes decaying exponentially f r o m their l o c a l i s a t i o n points i n space. T h e c o n d u c t i o n i n these materials is expected to proceed by nearest-neighbor hopping and/or variable-range-hopping. M o r e recently, A b r a h a m s et al. 8 1 have proposed a scaling theory of the conductivity, which encompasses the l i m i t of weak and strong disorder but also reveals the existence of an intermediate regime of moderate disorder leading to an CHAPTER 1: INTRODUCTION 2 unexpected correction to the B o l t z m a n n c o n d u c t i v i t y (<xn). F o r three-dimensional systems at T = 0 K this correction to the co n d u c t i v i t y (a) is given by: 1 9 MO) (k,uf \ h) K ' le is the elastic mean free path, L a length scale related to the size of the sample, and kf the F e r m i wave vector. A t finite temperature L must be replaced by the /. . s 1/2 inelastic diffusion length Z/,- = ( - ^ J , where is the inelastic mean free p a t h . 3 A r g u i n g that the electronic m o t i o n i n moderately disordered m a t e r i a l is dif-fusive, K a v e h and M o t t 1 7 > 1 8 have reproduced the scaling result of A b r a h a m s et al. using diffusion equations. T h e y have also shown 1 9 that the correction to the B o l t z -m a n n c o n d u c t i v i t y can be obtained using wave functions of the f o r m ^ £ oc ^ LEXI/R2> where ^EXT is a n extended wave function, and ca l c u l a t i n g the c o n d u c t i v i t y using the Kubo-Greenwood formalism. 3 0 T h r e e qu a l i t a t i v e l y different transport regimes seem to emerge f r o m these the-oretical treatments, each one characterised by a different wave function. In the weak disorder l i m i t , the wave functions are extended a nd the transport properties well described by the semi-classical B o l t z m a n n equation. In the moderate disorder l i m i t the wave functions are considered to be power-law localised (\P£ oc ^LEXT/r2) and the c o n d u c t i v i t y should have a characteristic temperature dependence, as described by K a v e h and M o t t . 3 1 In the l i m i t of strong disorder, the wave functions are ex-pected to be exponentially localised and therefore the c o n d u c t i v i t y determined by nearest-neighbor h o p p i n g and/or variable-range-hopping. A l t h o u g h there is much evidence i n the literature of phenomena that can be a t t r i b u t e d to electron localisation effects ( m a x i mum i n (T(T), 8 4 <x{T) oc T, 1,8 a{T) oc y/T, 8 c{T) oc exp - ( ^ ) 1 / 4 , 8 3 ' 8 8 ' 8 5 , e t c ) , no complete pi c t u r e is available. In most cases because a narrow range of sample c o n d u c t i v i t y (or narrow range of sample disorder) is examined, on l y a single transport regime is observed. Moreover, CHAPTER 1: INTRODUCTION 3 theories of electron localisation have been m a i n l y concerned with the evaluation of the conductivity. M u c h less attention has been given to the study of other transport properties, and consequently li t t l e experimental work has been devoted to such measurements. T h e present work represents an effort i n establishing a more complete p i c t u r e of electron localisation i n solids, and the experimental observation of the three t r a n s p o r t regimes mentioned previously are reported. In the case of the moderate disorder the dc and ac conductivity, the H a l l effect, a nd the thermoelectric power have a l l been measured. T h e p r i n c i p a l m o t i v a t i o n was to verify whether the model of weakly localised states for the conductivity, proposed by K a v e h and M o t t , can be extended to predict the correct behavior for the thermoelectric power and the H a l l effect. In the strong disorder l i m i t , precise measurements of the temperature depen-dence of the dc and ac c o n d u c t i v i t y have been performed. Here the m o t i v a t i o n has been to establish which of the present theories of variable-range-hopping 8 6 - 8 8 best describes the experimental data. T h e p a r t i c u l a r f o r m of variable-range-hopping ob-served i n the samples fabricated in this work suggest that the density of states at the F e r m i level is nearly constant. T h e thermoelectric power of these samples has, therefore, been measured to verify this. F i n a l l y , measurements of the reflectance and the transmittance have been performed in order to give some insight into the nature of the localised states which give rise to variable-range-hopping. 1.2 A p p r o a c h U s e d T h e observation of the evolution of the transport properties f r o m a Boltz-m a n n regime to a regime of moderate and strong disorder requires the f a b r i c a t i o n of samples spanning a wide range of s t r u c t u r a l disorder, preferably f r o m a polycrys-taline structure to an amorphous one. T h i s requirement was achieved by f a b r i c a t i n g samples of non-stoichiometric a l u m i n u m n i t r i d e (AINX) by voltage-controlled reac-tive sputtering. T h i s combination of deposition technique and m a t e r i a l was ideally CHAPTER 1: INTRODUCTION 4 suited for the purpose of observing localisation effects, since AINX can be f a b r i c a t e d by this m e t h o d over a wide range of composition and s t r u c t u r a l disorder. 1.3 Organisation of the Thesis T h e organisation of the thesis wi l l be as followes: In C h a p t e r 2 the theoretical results needed for this work are presented. Since the experimental and theoretical results can n a t u r a l l y be d i v i d e d into three transport regimes ( B o l t z m a n n regime, moderate and strong disorder regimes), three subsections are devoted to their de-scription. In C h a p t e r 3 the deposition technique used in the f a b r i c a t i o n of the sam-ples, reactive sputtering, and the experimental techniques employed in the mea-surements of their transport properties, are described. T h e transport properties that have been measured are: 1) T h e temperature dependence of the ac (up to 5 x l 0 7 H z ) and the dc c o n d u c t i v i t y between 10 K and 300 K. 2) T h e temperature dependence of the thermoelectric power between 10 K and 300 K. 3) T h e H a l l effect, which was measured at r o o m temperature. 4) F i n a l l y the reflectance and transmittance of t h i n films of AINX between 800 n m and 200 n m wavelength; these measurements were obtained at r o o m temperature. C h a p t e r 4 constitutes the major chapter of the thesis. A l l the experimen-ta l results obtained i n the course of this work are presented i n this chapter. T h e first section (4.1) deals with the morphology of the deposited films. T h e results of x-ray diffraction, electron diffraction, and electron transmission micrographs are discussed. T h e s e results are used in the following sections to show that the lo-calisation effects observed i n these films can be a t t r i b u t e d to s t r u c t u r a l disorder. T h e experimental results on the transport properties have been d i v i d e d into three sections, and follows the same organisation that is used in the presentation of the theory of electron localisation. In these sections (4.2, 4.3, 4.4), careful comparison between theory and experiment is undertaken. CHAPTER 1: INTRODUCTION F i n a l l y , the m a i n results and the or i g i n a l findings of this thesis are assembled C h a p t e r 5 and the aspects that are not very well understood are presented. CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 6 C H A P T E R 2 T H E O R Y OF E L E C T R O N LOCALISATION IN SOLIDS 2 . 1 The Boltzmann Regime T h e ions i n a perfect c r y s t a l being arranged i n a regular array, we are led to consider the one-electron H a m i l t o n i a n : H0 = -~V2 + V0(T), (2.1) where the p o t e n t i a l V 0 ( r ) has the pe r i o d i c i t y of the lattice. T h e solutions to this equation are the f a m i l i a r B l o c h waves: |$(r) >= u(r)e t K r, (2.2) where the functions u(r) also have the symmetry of the Bravais lattice. T h ese wave functions are extended, consisting of plane waves mod u l a t e d by a perio d i c f u n c t i o n of the lattice. Since a wave i n a peri o d i c array of scatterers can propagate without attenuation, because of coherent constructive interference of the scattered waves, such a H a m i l t o n i a n has no mechanism for the dissipation of energy. T h e co n d u c t i v i t y of a perfect crystal is therefore infinite. T o give the c o n d u c t i v i t y a finite value, electron scattering mechanisms must be introduced. T h r e e such mechanisms come to m i n d immediately. 1) Thermal vibrations; 2) Impurities or crystal defects; 3) electron-electron interactions. T h e CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 7 first two mechanisms effectively destroy the perfect p e r i o d i c i t y seen by the conduct-in g electrons. These scattering mechanisms are introdu c e d i n the f o r m of a sma l l p e r t u b a t i o n V(T) added to the or i g i n a l H a m i l t o n i a n H0 of the perfect crystal. T h e resulting H a m i l t o n i a n is: H = #o + t/(r), (2.3) T h e wave functions that are solutions of this H a m i l t o n i a n c an be cal c u l a t e d using p e r t u r b a t i o n theory. T h e y have the general form: it»-1*. > + E < y ^ > !•»•>• (") D u r i n g a scattering event an electron discontinuously changes its wave vector f r o m a value k to a value k\ A scattering rate r _ 1 can thus be defined, r represents the average time an electron remains in a given state characterised by a wave vector k without encountering a scattering. U s i n g Fermi's Golden Rule the scattering rate can be writte n as: 8 9 T _ 1 = 12 Y\ < *«<lV(r)l*k+q > f W+q ~ E*)> (2;5) q T h e c o n d u c t i v i t y a can now be expressed i n terms of T. S t a r t i n g with the current density: 4 0 J = envav, (2.6) where e is the electronic charge, n the number of electrons per c m , a n d v a r the average electronic velocity; v a t , averages to zero i n the absence of an electric field. However, in the presence of an electric field E there is a non-zero electronic velocity CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 8 which is given by v a t , = ( c E / m ) r , where m is the electronic mass. T h e current density is then: It follows that the c o n d u c t i v i t y is given by: n e 2 r oB = , 2.8) m T h i s expression for the c o n d u c t i v i t y w i l l be referred as the B o l z m a n n c o n d u c t i v i t y and denoted <7n. T h e existence of a scattering rate allows the definition of a mean free p a t h 1, which is the average distance travelled by a conduction electron between two scattering events. O b v i o u s l y I is given by 1 = vavT. Since the conduction i n a metal takes place at the F e r m i energy, v o r = v y , where V y is the F e r m i velocity. U s i n g the free-electron expressions 4 0 n = ^ a ^ d V/ = jj^k/, E q . 2.8 can be expressed in terms of 1 as: T h e properties of a n wi l l be determined by the behavior of 1. In p a r t i c u l a r the tem-perature dependence of the c o n d u c t i v i t y will depend on the dominant scattering mechanism of electrons at a given temperature. Except at the lowest temperature where electron-electron interactions may become significant, the m a i n scattering mechanisms of electrons are due to impurities and lattice vibrations. T h e effec-tiveness of either mechanism is expressed by the values of their mean free p a t h l e (elastic) and L; (inelastic) respectively. U s i n g the Matthiessen's rule, the resulting mean free p a t h 1 is given by: / _ 1 = l~l + / r 1 . aB can thus be expressed as: 1 e\ 2 lj{ (2.10) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 9 T h e mean free p a t h 1,- is temperature dependent, h a v i n g the high temperature f o r m L; a T~x. T h i s result c an be obtained f r o m E q . 2.5. O n the other hand, the mean free p a t h i n t r o d u c e d by the impurities is temperature i n d e p e n d e n t . 4 1 T h u s at very low temperature L; » l e and therefore: °B * ^ Ik"le' (2-11} A s the temperature is lowered the c o n d u c t i v i t y tends toward a constant value. T h i s effect is observed i n most metals containing a small amount of impurities. A t high temperature l e 3>1«- and thus: GQ varies as T~l at high temperature. 2.2 T h e W e a k L o c a l i s a t i o n R e g i m e 2.21 T h e d c C o n d u c t i v i t y In the previous section, scattering mechanisms were introduced in the f o r m of a small p e r t u r b a t i o n v(r) i n the o r i g i n a l H a m i l t o n i a n Ha to give the c o n d u c t i v i t y a finite value. T h i s approach can be used to calculate the influence of a very s m a l l concentration of impurities i n an otherwise perfect crystal. B u t clearly this approach has to break down. It w i l l do so when the p o t e n t i a l t;(r) i n t r o d u c e d by the r a n d o m d i s t r i b u t i o n of impurities can no longer be treated as a sma l l p e r t u r b a t i o n of the ori g i n a l H a m i l t o n i a n Hc. K a v e h and M o t t 1 9 introduce instead a H a m i l t o n i a n of the form: H = -£-V* + V0(r), (2.13) (2.12) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS where Va = V(r — r,) . T h e vectors r, indicate the positions of the scattering sites (impurities) which are di s t r i b u t e d r a n d o m l y i n the material. T h e p o t e n t i a l seen by the c o n d u c t i n g electrons is therefore assumed t o be a completely r a n d o m f u n c t i o n of r, instead of a per i o d i c function of r onto which s m a l l r a n d o m fluc-tuations have been superimposed. A n approximate solution of the Schrodinger equation for this H a m i l t o n i a n has been given by K a v e h and M o t t . 1 9 T h e y showed that the eigenfunctions s o l v i n g the above H a m i l t o n i a n are of the form: constant. T h e wave f u n c t i o n Ve contains two terms, an extended t e r m equal to fu n c t i o n is said to be weakly localised or power-law localised. T h e wave functions ^ f thus contain power-law as well as extended terms. A c c o r d i n g to K a v e h and Mo t t , 1 9 these weakly localised states are responsible for the correction t e r m to the B o l t z m a n n c o n d u c t i v i t y appearing i n E q . 1.1. U s i n g wave functions of the f o r m 2.14, the c o n d u c t i v i t y can be calculated using the Kubo-Greenwood 2 0' 4 3 formalism: (2.14) kf is the Ferm i wave vector, l e the elastic mean free path, a n d A a nume r i c a l 2 m 2 Dt N(e)2-f(e) (2.15) where (2.16) and a = / c{c)de, (2.17) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 11 N{e) is the density of states, /(e) the Fermi-Dirac d i s t r i b u t i o n , and D the volume of the system considered. T h e other symbols have already been defined. K a v e h and M o t t 1 9 have shown that the use of wave functions of the f o r m 2.14 2 leads to a value of the m a t r i x element given by: = 1 ( W (2.18) where L, 1/2 B denotes the value of the m a t r i x element that is o b t a ined using extended wave functions. S u b s t i t u t i n g E q . 2.18 into E q . 2.17 one obtains: — ( 1 - e)N{e) fef{e)de, m' (2.19) A s s u m i n g that the density of states remains essentially free-electron-Uke as locali-sation effects set in, E q . 2.19 can be written as: a = (7n(l - c) = aB 1 -( W (2.20) One aspect of the theory of weakly localised states that can be verified experimen-ta l l y is the temperature dependence of the c o n d u c t i v i t y predicted by Eq. 2.20. T h e temperature dependence of the inelastic mean free p a t h /,• fixes the temperature dependence of the conductivity. A s s u m i n g that electron-phonon interactions are the dominant scattering mechanisms of conduction electrons, then at temperatures well above the Debye temperature wi l l have the form 1,- = aT~l. It follows that E q . 2.20 can be written as: <r(T) = aB{0) 1 - + **(0) ^{le/af2VT-(le/a)T (2.21) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 12 where o-p(O) is given by E q . 2.11. T h e c o n d u c t i v i t y has been expressed as a sum of a temperature independent t e r m and a temperature dependent one. T h e theory of weakly localised states therefore predicts the existence of a co n d u c t i v i t y m a x i m u m at a temperature T m given by: (2.22) E q . 2.21 moreover predicts t h a t f o r temperatures well above T m the c o n d u c t i v i t y is d o m i n a t e d by the t e r m linear i n T while for temperatures well below T m it is domina t e d by the t e r m i n \/T. F o r temperatures well below the Debye temperature, /,• should have the low temperature f o r m /, = bT~2. Electron-phonon interactions i n disordered materials do not have the same T - 3 temperature dependence found i n crystalline materials. It results f r o m a relaxation of the selection rules and to a non-conservation of the m o m e n t u m in electron-phonon c o l l i s i o n s . 4 8 ' 4 4 T h e above temperature dependence of leads to a temperature dependence of the c o n d u c t i v i t y i n disordered materials given by: a(T) = *B{0) 1 -(Me)' 3V/3 // e\ 1 / 2 (Me)' ( l ) 1 / 2 r - ( i ) r 2 ] ' ^ A c o n d u c t i v i t y m a x i m u m is also predicted at a temperature Tm given by: 3V3b 2kf\z T™ = ^-TT3, (2-24) Well below Tm the c o n d u c t i v i t y should increase linearly w i t h T while well above Tm it should decrease as T2. Fig. 2-1 indicates schematically the temperature dependence predicted by Eq. 2.21. A l s o shown on the figure is the temperature dependence expected i n the CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 13 0 50 100 150 200 250 300 T (KELVIN) F i g u r e 2-1 T e m p e r a t u r e dependence of the conductivity, continuous line: B o l t z -m a n n conductivty. Dotted line: M o d e l of K a v e h and M o t t ( E q . 2.21). CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS B o l t z m a n n regime. T h i s aspect of the theory of weakly localised states is discussed in section 4.31. 14 2.22 The Thermoelectric Power and Peltier Coefficient In the previous subsection a theoretical m odel of the conductivity, applicable to highly disordered materials, was presented. T h e m odel was shown to lead to significant departures f r o m the B o l t z m a n n conductivity. T h e resulting c o n d u c t i v i t y was evaluated using E q . 2.19 and assuming that the density of states appearing i n this equation retains a free-electron behavior. T h e free-electron e x p r e s s i o n : 4 6 * ( « / ) ( * • » ) where N(e/) is the density of states at the F e r m i level and n the electron density, was assumed to be appropriate for highly disordered materials. T h e v a l i d i t y of this assumption can be tested by examining its consequences on transport properties other t h a n the conductivity. T h e situation is especially interesting i n the case of the Pe l t i e r coefficient and the thermoelectric power. R e t u r n i n g to E q . 2.18, it is observed that the correction to the m a t r i x element 2 Dt is independent of the electronic energy e. It follows that the departure of the c o n d u c t i v i t y at a given energy, <x(e), f r o m its B o l t z m a n n value, crp(e), is also energy independent. It is also noted that the same correcting factor applies to the t o t a l c o n d u c t i v i t y a. T h u s as localisation effects set in. a{e) and a depart f r o m their B o l t z m a n n values cr#(e) and (TQ by the same amount. T h u s the relative c o n t r i b u t i o n to the t o t a l conduction at a given energy, remains unchanged th a t is: = . If the density of states does indeed remain free-electron like, this suggests that no corrections to the Peltier coefficient are expected since it is given by: 4 6 n = -- f(e-€f)^de, (2.26) t J <y CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 15 T h e Pel t i e r coefficient can be viewed as the energy carried by the electrons p er unit charge. T h i s energy is measured relative to the F e r m i energy. E a c h electron contributes to IT in p r o p o r t i o n to its relative c o n t r i b u t i o n to the t o t a l conduction, Since this r a t i o is unchanged by localisation effects, the Peltier coefficient should remain free-electron-like. T h e same argument applies to the thermoelectric power which is related to the Pel t i e r coefficient by S = Therefore: 4 6 T h e v a l i d i t y of the above argument can be verified experimentally by measuring the dependence of the thermoelectric power on n and T. If the thermoelectric power retains its free-electron form, the f u n c t i o n a l dependence of S on n and T should be given by: 4 6 ic2k\ 2m T 6e ^ ( 3 , 2 ) 2 / 3 ^ 7»' ( 2 ' 2 8 ) T h e temperature dependence of the thermoelectric power of samples for which the c o n d u c t i v i t y obeys E q . 2.20 have been measured. These results are discussed i n section 4.33. 2.23 The Hall Effect T h e verification of the dependence of the thermoelectric power on n requires an independent measurement of n. T h i s can be done in principle using the H a l l effect. Nevertheless it has to be verified whether electron localisation effects m o d i f y the usual free-electron f ormula used i n the evaluation of n. E x t e n d i n g the model of the c o n d u c t i v i t y i n the weakly localised regime to the H a l l effect indicates that the free-electron f o r m u l a : 4 6 H I , N = ^ V ( 2 ' 2 9 ) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 16 is s t i l l v a l i d . Here / is the current flowing i n the sample, Vg the measured H a l l voltage, d the thickness of the sample, H the magnetic field, a nd c the speed of light. T h e correctness of E q . 2.29 can be demonstrated in the following way: T h e m odel presented i n section 2.21 assumes that u p o n g r a d u a l l y increasing the disorder of a h y p o t h e t i c a l system, the observed reduction of the c o n d u c t i v i t y is m a i n l y due to a change i n the nature of the electronic wave functions. T h e density of carriers n is ce r t a i n l y affected by the enhancement of the disorder but is thought to be only a m i n o r effect. T h e influence of the disorder on the c o n d u c t i v i t y is thus thought to be a mobility effect. T h e correction to the c o n d u c t i v i t y induced by the disorder can thus be expressed i n terms of a corrected lifetime T*. T h e r e s u l t i n g value of the c o n d u c t i v i t y can therefore be written as: ( E q . 2.20) 1 -a = ne2T m neV a = m (2.30) In the evaluation of the H a l l v o l t a g e , 4 6 we are led to consider an equation of the form: (2.31) where jx is the current density i n the x-direction, and Ey and Jy the transverse electric field a nd current density i n the y-direction respectively. T h e H a l l field Ey is d e t ermined by the requirement that there is no transverse current. S e t t i n g Jy = 0, E q . 2.31 leads to: CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS Since a = (n e 2 r * ) / m , the quantity T* cancels out and a l l i n f o r m a t i o n on electron loc a l i s a t i o n disappears f r o m E q . 2.32. W i t h Jx = I/(Wd) and Vg = —EyW where W is the w i d t h of the sample, the free-electron formula ( E q . 2.29) is recovered f r o m E q . 2.32. T h e values of n of samples for which the c o n d u c t i v i t y obeys E q . 2.20 were obtained using the H a l l effect. T hese values of n were then compared with those deduced f r o m thermoelectric power measurements. These results are discussed i n section 4.33. 2.24 Electron-Electron Interactions T h e electron localisation theory presented i n section 2.21 is a single particle theory in which electron-electron interactions are neglected. A l t s h u l e r a nd A r o n o v 4 7 have proposed recently a theory t a k i n g into account these interactions in disor-dered solids. T h e y showed that these interactions reduce the density of states at the F e r m i level, leading to a correction to the B o l t z m a n n conductivity. Neglecting localisation effects, the c o n d u c t i v i t y is given by: 4 7 l eVV \ ' / 2 , S '""vTKm '  (2- 33) D is a diffusion constant given by D = y A crossover f r o m a conduction regime i n which the c o n d u c t i v i t y is d o m i n a t e d by localisation effects to a low temperature regime i n which the c o n d u c t i v i t y is d o m i n a t e d by interaction effects has been observed by Howson and G r e i g 1 i n dis-ordered CuioZr^Q and CU$QHfoo. T h e tr a n s i t i o n temperature i n these materials is around 10 K. Electron-electron interactions are therefore a low temperature effect and are thought to be neglegible i n the temperature range examined (10 K to 300 K ) in this work. In A p p e n d i x 6.1 it is shown that the crossover temperature i n the samples studied is at around 12 K. CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 18 2.3 The Regime of Strong Disorder A s discussed i n section 1.1, i n the l i m i t of very strong disorder the wave functions are expected to become exponentially localised. T h e wave functions do not extend throughout the system but are highly localised near some t r a p p i n g sites. C o n d u c t i o n takes place v i a h o p p i n g of electrons f r o m a t r a p p i n g site c o n t a i n i n g an electron w i t h energy e,- to an empty site where the electron w i l l have an energy ey. Such a con d u c t i o n process requires the assistance of phonons to ensure energy conservation and at T=0 K the conduction must vanish. T h e evaluation of the macroscopic c o n d u c t i v i t y of a ma t e r i a l i n which the c o n d u c t i o n takes place by h o p p i n g can be d i v i d e d into two steps. In the first step, the t r a n s i t i o n p r o b a b i l i t y of electrons hopping between localised states is calcu-lated, allowing the definition of equivalent resistors l i n k i n g the various sites. T h e second step consists of the evaluation of the macroscopic c o n d u c t i v i t y of a sample now viewed as a r a n d o m network of interconnected resistors. In section 2.31 the equivalent resistors are calculated. In section 2.32 the macroscopic c o n d u c t i v i t y is deduced f r o m the resistor network, using percolation theory. 2.31 The resistor network T h e t r a n s i t i o n p r o b a b i l i t y 7,-y f o r an electron h o p p i n g f r o m a site i to an empty site j is a fun c t i o n of the distance between the sites, r,y, and the energy difference between the sites, e,- — ey. T h e h o p p i n g process being a tunell i n g one i n which the i n i t i a l a n d final states have different energies, the dominant dependence of the t r a n s i t i o n p r o b a b i l i t y on ryy w i l l be exponential, 4 8 7,-y oc e~2AR'K Here a - 1 is the B o h r radius. If the final energy ey is larger t h a n the i n i t i a l energy e,-, a phonon of energy ey — e,- must be absorbed. 7,-y must be p r o p o r t i o n a l to the p r o b a b i l i t y of f i n d i n g a phono n w i t h the correct energy. T h i s p r o b a b i l i t y is p r o p o r t i o n a l to the number of phonons present i n the system at a temperature T and with the energy CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS ej - e,-. T h i s number - £y), is given by 4 9 n?(e,- - ey) = {exp[{ej - e , ) / f c 6 T ] } - 1 . T h e r e f o r e for £y greater t h a n £, and (ey — €i)/k},T » 1, 7,y oc e-V*Tn+{ii-u)lhT\^ (2.34) Similarly, if e, is greater than ey a phonon must be emitted i n the t r a n s i t i o n process and thus 7,y oc e~(2ar>iK M i l l e r and A b r a h a m 6 0 have calculated precisely this t r a n s i t i o n p r o b a b i l i t y between localised i m p u r i t y sites. A s s u m i n g hydrogenic wave functions for the lo-calised states they obtained: Hi = 7fye _ 2 o r°»7g(ey - €,•), (2.35) where _ 1 6 r ^ 2 ( e y - e.) ( 2 * a \ 2 . ,2 L M e y - ^ V - 4 (2.36) Here E\ is the deformation potential, e,- and ey the electron energy at sites i and j respectively, d the density, vt the speed of sound, a the B o h r radius, KK0 the dielectric constant, and r,y the intersite distance. T h e t e r m / 7 r ( e y - € t ) y V havt J is assumed to be much smaller than 1 and is n o r m a l l y neglected. F r o m 7,-y a t r a n s i t i o n rate can be calculated: (2.37) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 20 /,• and fj are the time averaged o c c u p a t i o n numbers of the sites i and j respectively. T h e current between site i and j is thus: Jij = -e(r,y - Yji), (2.38) In the absence of an electric field «7,y averages to zero. However, the presence of an electric field redistributes the electrons over the localised states, m o d i f y i n g the time averaged values of the o c c u p a t i o n functions f. A net current results between the site i and j , allowing the definition of an equivalent resistance between the two sites. T h e value of the resistance is given by: 4 8 Rij = R?jef'i, (2.39) where and €i3- = 2aryy + - £y| + |e,| + | €y|], (2.41) T h e h o p p i n g c o n d u c t i v i t y is thus reduced to that of c a l c u l a t i n g the c o n d u c t i v i t y of a r a n d o m network of resistors. 2.32 The Percolation Approach T h e resistances Rij depend exponentially on the distance and energy separa-tion between the sites. T h e values of 72,-y f o r m i n g the resistor network therefore span many orders of magnitude. Nevertheless the macroscopic resistance of the network is d o m i n a t e d by a l i m i t e d number of resistors h a v i n g definite values. T h e v a l i d i t y of this last statement can be seen in the following way. O n e can imagine removing all the resistors f r o m the network, leaving only the vertices (impurity sites). T h e n one c o u l d r e t u r n the resistors to the network one by one, i n ascending order, s t a r t i n g CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 21 with the resistor of the lowest value. T h e network c a n sustain a current only when a continuous p a t h of resistors span its entire length. In i t i a l l y resistors of low values are dispersed throughout the network but do not f o r m a continuous p a t h accross i t . T h e c o n d u c t i v i t y of the network is zero. A s the resistors are gradually added to the network, always in ascending order, a continuous p a t h wi l l suddenly be created, g i v i n g the c o n d u c t i v i t y a finite value ac. T h e percolation threshold of the network has been reached. T h e macroscopic c o n d u c t i v i t y of the network is determined by the value of the last resistor added, Rc. A l l the previous resistors do not contribute appreciably to the t o t a l resistance of the network, being much smaller t h a n Rc. T h e resistors added after the percolation threshold leave <rc essentially unchanged since these large resistors are shunted by the smaller one. We thus have: <rc « {lcRc)~\ (2.42) T h e value of the length scale lc has yet to be evaluated. T h i s w i l l be discussed i n section 2.34. Us i n g E q . 2.39, oc can be w r i t t e n as: "< = ^ f e " " ' <2-43> It can be shown that for a r a n d o m d i s t r i b u t i o n of localised states h a v i n g a tr a n s i t i o n p r o b a b i l i t y of the f o r m of Eq. 2.35 that ec = (T0/T)1^. 4 8 Thus, e2 y> /rp \ 1/4 T h e f o r m of the exponential temperature dependence of the c o n d u c t i v i t y is char-acteristic of a cond u c t i o n mechanism known as variable-range-hopping. A more int u i t i v e description of this conduction mechanism is given i n the following section. In section 2.34 the p h y s i c a l meaning of lc w i l l be discussed as well as the temperature dependence of the preexponential t e r m of E q . 2.44. CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 2.33 Variable-Range-Hopping W h e n the local i s a t i o n of the wave f u n c t i o n is p a r t i c u l a r l y strong (high value of a) the tra n s i t i o n p r o b a b i l i t y 7,y is maximised when ryy is as sm a l l as possible. T h e electrons do not hop fur t h e r t h a n the nearest empty localised states. If it is assumed that the localised states are d i s t r i b u t e d u n i f o r m l y throughout the volume of the sample and that the density of states near the F e r m i energy is constant, the average energy difference between nearest-neighbor localised states is: Ae = 3/[47rr,/iV(e/)], (2.45) Therefore 7,-y oc e-(2«r,->+Ae/fc 6r) > ( 2 4 6 ) and thus c{T) oc e~Ae/kbT. Nearest-neighbor h o p p i n g is characterised by a single activation energy independent of temperature. W hen the localisation of the wave functions is not as strong as i n the previous case, the tr a n s i t i o n p r o b a b i l i t y may be minimised when the electrons hop to more distant sites, because with more sites to choose from, the energy difference may be made smaller. U s i n g E q . 2.45, the transit i o n p r o b a b i l i t y can be written as: 7,-yoce V ' '"O-'VWV, (2.47) T h e exponential t e r m is minimised for hopping distances given by: ( 9 Yfi  fav = [snaN{ef)kbT) ' { 2 A 8 ) T h e variable-range nature of the conduction mechanism is clearly seen i n this last equation. A s the temperature is lowered the electrons hop to more distant sites. CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 23 T h e average h o p p i n g energy is evaluated by su b s t i t u t i n g E q . 2.48 into E q . 2.45, giving: A s the temperature is lowered, the average h o p p i n g distance increases while the average h o p p i n g energy diminishes. S u b s t i t u t i n g Eqs. 2.48 and 2.49 into E q . 2.46 one obtains 7,7 oc exp - (T0/T)l/\ (2.50) where 1 8 a 3 To~mjjjk~b (2*51) T h i s unusual temperature dependence is one of the most characteristic feature of variable-range-hopping. Such a temperature dependence has been observed in a wide variety of amorphous semiconducors 8 3> 8 8> 8 5 as well as in many d o p e d s e m i c o n d u c t o r s . 6 1 2.34 The Temperature Dependence of the Prefactor. T h e length lc was introduced i n E q . 2.42 to make the units consistent. Nev-ertheless lc has a physical significance which wil l now be discussed. R e t u r n i n g to the r a n d o m resistor network one can imagine cubes of side 1 di s t r i b u t e d w i t h i n the network. L e t us remove a l l the resistors w i t h i n the cubes and replace them i n ascending order u n t i l there is percolation between opposite faces. L e t e rj characterise the largest resistor ( E q . 2.39) that has to be added to reach p e r c o l a t i o n i n a given cube 1. T h e c o n d u c t i v i t y of the c u be is given by: e 2 i° (2.52) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 24 T h e values of w i l l fluctuate f r o m one cube to another, depending on the cube's lo c a t i o n i n the network. It can be shown 5 3 that these fluctuations varies w i t h 1 as: (2.53) where v « 0.9. T h u s as 1 increases the fluctuations i n the c o n d u c t i v i t y of the various cubes diminish. A t a certain value of 1 the fluctuations become so s m a l l that a/ essentially ceases to depend on 1. A t this point 07 is equal to the macroscopic c o n d u c t i v i t y of the whole network. T h i s length is given by: 5 a (2.54) It is called the correlation radius. S u b s t i t u t i n g this length scale i n t o E q . 2.44, the f o r m of temperature dependence of the c o n d u c t i v i t y can be evaluated. T h e following result i d obtained: <r(T) = BoT-Wezp - , (2.55) Such a temperature dependence has been obtained by Shklovskii 8 8 a n d Pollack 8 7 using a sim i l a r percolation approach. O t h e r temperature dependences of the prefactor may be fo u n d i n the literature. A l l e n et al. 8 5 use as a length scale the average hopping distance rav ( E q . 2.48) instead of the correlation radius LQ ( E q . 2.54). T h i s different length scale leads to: a[T) = A0T-l/2exp - {^J , (2.56) F i n a l l y , M o t t 8 6 assumes that the macroscopic c o n d u c t i v i t y varies as: a = e2N{ef)raviij, (2.57) CHAPTER S: THEORY OF ELECTRON LOCALISATION IN SOLIDS 25 and therefore he obtains: a{T) = C0T^exp - ^ J (2.58) E q . 2.57 is o b t a i n e d by w r i t i n g the c o n d u c t i v i t y as a — nfie with n = N(ef)ki,T, and assuming an E i n s t e i n m o b i l i t y of the form: Mott's expression for 7,y is also slightly different f r o m the one used by S h k l o v s k i i , 8 8 Pollack 8 7 a n d A l l e n . 8 6 T h e expression n = N(ef)k{,T is i n my view incorrect. T h e h o p p i n g process takes place w i t h i n a temperature dependent b a n d w i d t h centered a round the F e r m i energy. T h i s b a n d w i d t h is given by eav rather t h a n k(,T. It can be verified t h a t s u b s t i t u t i n g n = N(ef)eav into E q . 2.57 leads to the temperature dependence proposed by A l l e n et al. 8 5 ( E q . 2.56) T h e temperature dependence of the prefactor i n samples of AINX e x h i b i t i n g variable-range-hopping has been deduced f r o m precise c o n d u c t i v i t y measurements. These results are discussed i n section 4.41. 2.35 The Frequency Dependence of the Conductivity In this section a highly disordered m a t e r i a l i n which a l l the states near the F e r m i level are localised is again considered. Its c o n d u c t i v i t y i n the presence of an electric field of frequency u is calculated. A n intuitive description of the s i t u a t i o n is first given, followed by a more rigorous treatment of the present theories of h opping co n d u c t i o n i n an alternating electric field. 2.36 The Mechanism of Polarisation F o r an ac current b e i n g p r o p o r t i o n a l to the rate of change of the polarisa-ti o n , J = the electric field must induce time v a r y i n g dipoles i n the material. CHAPTER S: THEORY OF ELECTRON LOCALISATION IN SOLIDS 26 O n e possible mechanism is i l l u s t r a t e d i n F i g . 2-2 i n the case of a compensated semiconductor. T h e study of the ac c o n d u c t i v i t y i n compensated semiconductors is quite relevant to the p r o b l e m at h a n d (non-stoichiometric AINX) since at very sponse to the applied electric field, the figure shows an electron h o p p i n g f r o m a neutral donor site to a charged (unoccupied) donor site. T h e d ipole has changed b o t h in magnitude and orientation with respect to the o r i g i n a l one, i n d u c i n g an ac current. M o d e l s of ac c o n d u c t i v i t y based on s i m i l a r p o l a r i s a t i o n mechanisms have been developed m a i n l y by Pollak. 6 8' 6 4 A simplified version of these models is first presented i n the next section. 2.37 A S i m p l i f i e d M o d e l o f a c C o n d u c t i v i t y . T h e c o n d u c t i v i t y of a cube containing rj identical h o p p i n g centers is now considered. E a c h center consists of a p a i r of d onor sites separated by an energy 6e, a distance r, and h a v i n g the same orientation with respect to the applied electric field ( F i g . 2-2). T h e indices i and j refer to the two sites f o r m i n g a pair. T h e current of such a cube is given by: ft is the volume of the cube, z,- the projection along the field d i r e c t i o n of the separation of the ith donor site f r o m the nearest acceptor site, a nd the p r o b a b i l i t y low temperature t h e i r dc c o n d u c t i v i t y proceeds by variable-range-hopping. In re-dP J = — dt (2.60) where (2.61) CHAPTER S: THEORY OF ELECTRON LOCALISATION IN SOLIDS 27 _£ X; DONOR ACCEPTOR e o DONOR DONOR ACCEPTOR DONOR o Figure 2-2 Figure i l lustrating the polarisation mechanism considered. CHAPTER S: THEORY OF ELECTRON LOCALISATION IN SOLIDS of o c c u p a n c y of the ith donor site given by: (2.62) r,y and Tji are the transi t i o n rates of electrons f r o m site i to j a nd j to i respectively ( E q . 2.37). A s s u m i n g that each p a i r is independent then /,• = 1 — fj and the current can be w r i t t e n as: J = ner cos(e)^/,, (2.63) (see F i g . 2-2). T h e f o r m of the f u n c t i o n /,- after a sudden ap p l i c a t i o n of a steady electric field E is o b tained by solving E q . 2.62 w i t h the boundary conditions fi(t = 0) = Sf 1 + eW - l (2.64) and fi(t = oo) = -6c (2.65) eEr cos(8) where f = e ^ . U s i n g the solution of /,(£) the current J ( t ) c an be calculated: r/ ^ 1 - i 1 erE cos(Q) =± J{t) = -rjer cos{B)q — e < , 4 ~»* 2 (sfer) 6 (2.66) c 1 = r,y+ry,-. T r a n s f o r m i n g into a frequency dependent f o r m using Laplace's method, the c o n d u c t i v i t y is given by: = — rjr cos 1 C — — — +*: , (2.67) CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 2.38 A M o r e R e a l i s t i c M o d e l o f a c C o n d u c t i v i t y A more realistic m odel should take into account the fact t h a t pairs are not i d e n t i c a l but are r a n d o m l y arranged. E q . 2.67 should be averaged over 6, r and Se. A s s u m i n g Se ~ k^T, the t e r m cosh2 (^^f) 1S c^ o s e t° u n i t y and may be replaced by unity. D e f i n i n g a d i s t r i b u t i o n f u n c t i o n dg(r, Se) as being the number pairs of spacing r and energy separation Se} and assuming a ra n d o m d i s t r i b u t i o n of sites we have: dg(r, Se) = 4nNDr2dr, (2.68) T h e real part of the c o n d u c t i v i t y can be written as: '-iM'&lTrrry* (2-69) 3 * V J 1 + [wc(r)] <; fa ^ phe~2ari uph is a phonon frequency, the density of acceptor states, and Nj) the density of donor states. T h e m a i n c o n t r i b u t i o n to the above integral comes f r o m values of r for which UT(T) » 1, thus for r w = ( 2 a ) ' 1 In ( ^ ) , (2.70) A s s u m i n g that the f u n c t i o n — ^ — T « \ over a width around wc « 1, E q . 2.69 can be wri t t e n as: T h e above model predicts an almost linear frequency dependence of the c o n d u c t i v i t y as well as a temperature dependence v a r y i n g as y . CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 2.39 T h e T e m p e r a t u r e d e p e n d e n c e o f t h e a c C o n d u c t i v i t y T h e m o d e l discussed i n section 2.38 predicts a temperature dependence of the ac c o n d u c t i v i t y which is not observed either i n compensated crystalline semicon-ductors at low temperature or i n amorphous semiconductors, for which the m o d e l is presumably applicable. T h e p r o b l e m lies i n the assumption that the number of dipoles c o n t r i b u t i n g to the ac c o n d u c t i v i t y is temperature independent. A remedy proposed by P o l l a k 6 S is now discussed. F i g . 2-3 represents schematically a r a n d o m d i s t r i b u t i o n of i m p u r i t y sites i n a compensated semiconductor. T h e minus signs represent negatively charged acceptor sites and the plus sign, ionised donor sites. It is assumed that the number of donor sites is much larger t h a n the number of acceptor sites (ND » NA). T h e circles drawn around each ionised acceptor site represent a cross section of a sphere of radius (2.72) Fo r a given set of values of NA, vvh, and w, two l i m i t i n g cases describing the h o p p i n g of electrons can be defined; r w <!C rq and r w » rq. In the first case the h o p p i n g takes place w i t h i n a given sphere. T h e sample can thus be d i v i d e d into NA such spheres, each one c o n t a i n i n g a charge carrier. T h e number of charge carriers p a r t i c i p a t i n g in the c o n d u c t i o n is independent of temperature and equal to NA- A charge carr i e r can hop to any other donor site w i t h i n a given sphere as long as the energy difference between i n i t i a l a n d final states is of the order of k(,T. T h u s the number of allowable receiving states w i t h i n a sphere increases with temperature and is p r o p o r t i o n a l to Nj)k(,T. T h e t o t a l number of pairs that contribute to the c o n d u c t i v i t y is therefore p r o p o r t i o n a l to NANDHT. S u b s t i t u t i n g NANQ i n E q . 2.71 by NANDH^T, the CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS © O o * 0 O © o o © . o O V " o : o o • © IONISED DONOR © © IONISED ACCEPTOR o NEUTRAL DONOR F i g u r e 2-3 F i g u r e representing a d i s t r i b u t i o n of i m p u r i t y sites. CHAPTER 2: THEORY OF ELECTRON LOCALISATION IN SOLIDS 32 resulting c o n d u c t i v i t y is p r o p o r t i o n a l to: a oc (2.73) and is temperature independent. T h e case r w <g rq represents a highly correlated s i t u a t i o n i n which the occ u p a t i o n functions /,• w i t h i n a given sphere are not inde-pendent f r o m one another. In the second case, » rq, the ho p p i n g takes place between spheres of radius rq. T h i s s i t u a t i o n is u n c o r r e c t e d since the occupation of a donor site w i t h i n a given sphere is determined p r i m a r i l y by the occupation of the other d o n o r sites w i t h i n that sphere. In this case the number of charge carriers c o n t r i b u t i n g to the c o n d u c t i v i t y is temperature dependent. T h i s temperature dependence arises because it is conceivable that the energy of a charge carrier w i t h i n a given sphere may differ so much f r o m the average energy of the donor sites i n the other spheres that it cannot contribute to the c o n d u c t i v i t y unless it can acquire sufficient t h e r m a l energy. T h e number of charge carriers that can contribute to the c o n d u c t i v i t y is thus p r o p o r t i o n a l to N^k^T. T h e number of receiving states is s t i l l p r o p o r t i o n a l to NjykbT. F o r this uncorrelated case, the c o n d u c t i v i t y is p r o p o r t i o n a l to: T h e frequency dependence as well as the temperature dependence of the ac conduc-t i v i t y of samples e x h i b i t i n g variable-range-hopping i n the limit of w —* 0 have been examined. These results are presented and discussed i n section 4.44 (2.74) CHAPTER S: Experimental Techniques 33 C H A P T E R 3 Experimental Techniques 3.1 V o l t a g e - C o n t r o l l e d R e a c t i v e S p u t t e r i n g T h e samples studied i n this work were pr o d u c e d by sputtering. In this section a d e s c r i p t i o n of the s p u t t e r i n g equipment a nd process is presented. F i g . 3-1 shows a schematic cross section of the apparatus used. It consists of a v a c u u m chamber which can be evacuated to w 10-7 T o r r using a 10 c m diameter diffusion pump. A freon cold t r a p located between the diffusion p u mp and the chamber prevents its c o n t a m i n a t i o n f r o m o i l of the diffusion pump, and also acts as a cryopump. Various gases ( t y p i c a l l y argon, nitrogen, and oxygen) can be admitt e d into the chamber thro u g h independent leak valves (Granville-Phillips model 203). T h e flow of each gas is measured by independent flow meters (Hasting' H-5 model All-5). T h e pressure inside the chamber is regulated by adjusting the p u m p i n g speed of the diffusion pump using a t h r o t t l i n g valve located between the cold t r a p and the diffusion pump. T h e pressure is measured using a capacitance manometer (MKS Baratron). A t one extremity of the chamber is located a target made of the m a t e r i a l to be deposited. T h e target consits of a 0.70 c m thick c i r c u l a r plate, 15 cm i n diameter, firmly clamped by a metal r i n g to a water cooled backing plate. In this work 99.999 per cent pure A l was used for the target (Varian, Speciality Metal Division). A plasma discharge can be created near the target by a p p l y i n g N. CHAPTER 9: Experimental Techniques 84 Moss spectrometer rf power capacitance monometer T to diffusion pump throttle valve F i g u r e 3-1 Schematic cross-section of the s p u t t e r i n g system. CHAPTER S: Experimental Techniques 35 a difference of p o t e n t i a l between the target a c t i n g as a cathode, and the grounded chamber. T h e discharge is powered by a 5 kW, unfiltered, f u l l wave rectified, constant current, d c power supply ( P i a s m a Therm MDS-500D, 0-lOA, 0-1000V). T h e p l a s m a discharge is confined i n a torus d i r e c t l y i n front of the target using an electromagnet. In its simplest f o r m the sp u t t e r i n g process is a non-reactive one i n which a single inert gas, t y p i c a l l y argon, is admitt e d i n the chamber. U p o n i g n i t i n g the discharge, ions of argon are r a p i d l y accelerated toward the target. E v e n t u a l l y they collide with i t , a n d thr o u g h transfer of momentum, atoms of the target are ejected. Soon the entire chamber becomes coated by a t h i n layer of the target material. A substrate-holder/shutter arrangement accommodating six 2.5 x 5 c m glass substrates is located 10 c m away f r o m the target and well outside the discharge itself. B y c o n t r o l l i n g the shutter each substrate can be exposed i n d i v i d u a l l y to the flux of m e t a l particles emanating f r o m the target. C o m p o u n d materials can be deposited by a d m i t t i n g into the chamber a second gas, the reactive gas. T h e deposition of a compound hinges on the fact that u p o n creati n g a sufficiently high p a r t i a l pressure of reactive gas, the target becomes coated by a t h i n layer of an i n s u l a t i n g compound consisting of the chemical c o m b i n a t i o n of the atoms of the me t a l target with the gas species of the reactive gas. Intuitively one might t h i n k that the molecules of the compounds are then ejected f r o m the target, resulting in their accumulation on the substrates and the walls of the chamber. Nevertheless this is not the case. T h e f o r m a t i o n of an i n s u l a t i n g c ompound on the target plays the role of a regulator, actually c o n t r o l l i n g the flux of particles being ejected f r o m i t . I nsulating compounds hav i n g a much lower sputter y i e l d (number of particles ejected f r o m the target per incident ion) t h a n most m e t a l s , 6 8 the coverage of the target by an in s u l a t i n g c o m p o u n d reduces the flux of m e t a l atoms emitted f r o m the target. T h i s i n t u r n affects the composition of the m a t e r i a l being deposited since its composition is to a large extent determined by the relative a r r i v a l rates CHAPTER S: Experimental Techniques 86 of the metallic atoms of the target a n d the reactive gas species (molecules, atoms, ions) at the substrate. A low flow of reactive gas wil l i n h i b i t the formation of a com p o u n d on the target if the removal rate of atoms f r o m the target by the incident ions is higher t h a n the rate of c o m p o u n d for m a t i o n on the target. T h e surface of the target w i l l remain metallic a n d the sp u t t e r i n g y i e l d high. A large flux of metallic atoms w i l l cover the walls of the chamber. T h r o u g h getter p u m p i n g f r o m this fresh metallic layer, the p a r t i a l pressure of the reactive gas w i l l be further reduced. T h i s positive feedback ensures that the target remains uncovered a nd a very stable s i t u a t i o n develops. If the p a r t i a l pressure of the reactive gas is gradually increased, the rate of comp o u n d for m a t i o n on the target w i l l at one point become greater t h a n the removal rate, leading to a partial coverage of the target. T h e concept of p a r t i a l coverage should not be seen as a static s i t u a t i o n but a d y n a m i c one i n which m a t e r i a l is constantly being removed f r o m the target while a certain p r o p o r t i o n of its surface remains covered by an in s u l a t i n g layer. T h e p a r t i a l coverage of the target n o r m a l l y leads to a runaway transition and to a more stable situation i n which the target is f u l l y covered. T h i s r a p i d t r a n s i t i o n is driven by an enhancement of the reactive gas p a r t i a l pressure, caused by a reduction of the gettering due to a smaller flux of metallic particles ejected f r o m the p a r t l y covered target. Soon a sit u a t i o n is reached where the target is completely covered by an ins u l a t i n g layer of c o m p o u n d material. T w o mechanisms are responsible for the formation of a compound on the sur-face of a sp u t t e r i n g t a r g e t : 5 7 Chemisorption of the reactive neutral species of the spu t t e r i n g gas; Ion plating of the reactive gas species f r o m the sputteri n g current. T h e relative importance of either mechanism depends on the target-reactive gas com b i n a t i o n used. W h e n chemisorption dominates the reaction, stable operation can only be achieved if the target is either completely bare or completely c o v e r e d . 5 7 A n y intermediate s i t u a t i o n is unstable and leads to a r a p i d t r a n s i t i o n to either CHAPTER S: Experimental Techniques stable o p e r a t i n g point (covered or uncovered target). W h e n ion p l a t i n g is the dom-inant mechanism, any degree of target coverage constitutes a stable o p e r a t i n g p o i n t if the discharge voltage is c o n t r o l l e d . 6 7 F o r a given value of the p a r t i a l pressure of the reactive gas any degree of target coverage can be obtained by simply a d j u s t i n g the discharge voltage. B y c o n t r o l l i n g the target coverage it is possible to regulate the relative a r r i v a l rates of metallic and reactive gas species on a substrate. Since this ratio determines the composition of the deposited material, voltage-controlled reactive sputtering allows the deposition of compounds over a wide range of com-position. In this work, t h i n films of non-stoichiometric a l u m i n u m n i t r i d e (AINX) were deposited using this technique. 3.2 Preparation of the Samples C h r o m i u m / g o l d electrodes were evaporated successively on C o r n i n g 7059 glass substrates which h a d previously been washed i n trichlorethylene and isopropanol Following this first deposition, non-stoichiometric films of a l u m i n u m n i t r i d e of a thickness of « 500 n m were deposited using the m e thod discussed i n section 3.1. T h e thickness of the films was measured using a profilometer. It was decided to have the contact electrodes u nder the films instead of on their surfaces because of the r a p i d i t y with which the surface of non-stoichiometric films of a l u m i n u m nitr i d e oxidizes. A t r o o m temperature and pressure, a layer of fa 10 n m of A^Oz forms w i t h i n a p e r i o d of 24 h o u r s . 6 7 Following the deposition of the films, gold wires were cold welded to the electrodes using s m a l l amount of indium. 3.3 The dc Conductivity Measurements T h e c o n d u c t i v i t y of the samples was measured between 10 K and r o o m tem-perature. A cold finger s u p p o r t i n g a substrate was immersed i n a dewar filled w i t h h e l i u m gas. C o p p e r tubings welded i n an helicoidal pattern around the outside wall of the dewar were used to control the temperature inside i t , by c o n t r o l l i n g a flow CHAPTER S: Experimental Technique! 88 of l i q u i d h e l i u m i n the tubing. T h e h e l i u m gas inside the dewar acted as a heat ex-changer, and ensured a u n i f o r m temperature at the cold finger. T h e measurements of the c o n d u c t i v i t y were performed i n a four-point probe configuration using an electrometer as a current source. C u r r e n t s between 1 0 ~ 8 amp and 10 amp were used depending u p o n the resistivity of the samples being measured. In a l l cases the linearity of the resistance with the current was verified. T h e substrates were secured to the cold finger using v a c u u m grease. T h e temperature was measured using a semiconducting diode epoxied to the cold finger. T h e temperature of the film was assumed to be that measured at the cold finger. 3.4 T h e a c C o n d u c t i v i t y M e a s u r e m e n t s T h e real part of the ac c o n d u c t i v i t y was measured between 10 4 H z and 5 x l 0 7 Hz using a Q-meter. T h e s e measurements were taken at different temperatures between 10 K and 300 K using the same techniques described i n section 3.3. 3.5 T h e H a l l E f f e c t M e a s u r e m e n t s T h e H a l l measurements were performed at room temperature in a magnetic field of 10 3 Gauss. T h e experimental arrangement used is shown schematically i n F i g . 3-2. In the absence of a magnetic field, an ac current of 1 0 - 4 amp mis at a frequency of 290 Hz was sent t h r o u g h the sample v i a the electrodes a and b . T h e differential voltage appearing between the points c and d was applied to a potentiometer. T h e voltage at the output of the potentiometer was then adjusted to n u l l the voltage at point e. U p o n t u r n i n g on the magnetic field, a H a l l voltage would appear between point e and the output of the potentiometer. T h e advantage of this technique relies on the independence of the measured H a l l voltage on the posi t i o n of the contact electrodes. CHAPTER 3: Experimental Techniques 89 POTENTIOMETER F i g u r e 3-2 E x p e r i m e n t a l arrangement used i n the H a l l measurements. CHAPTER S: Experimental Techniques 3.6 The Thermoelectric Power Measurements T h e thermoelectric power was measured between 10 K and r o o m tempera-ture. T h e experimental arrangement used is shown i n Fig. 3-3. O n e extremity of the glass substrate onto which a sample had been deposited was secured to a cold finger using v a c u u m grease. T h e average temperature of the sample was taken as that measured at the cold finger. T h i s temperature was measured using a semicon-d u c t i n g diode. A t the other extremity of the glass substrate was suspended a small resistive heater with which a temperature gradient could be applied along the length of the sample. T e m p e r a t u r e gradients between 1 K and 5 K were used. T h e y were measured using thermocouple wires (chromel/alumel/chromel) mounted i n a differ-ential configuration. T h e thermocouple wires were cold welded with a very s m a l l amount of i n d i u m onto two evaporated chromium/gold electrodes located 0.5 m m away fr o m b o t h extremities of the sample. T h e measurements of the thermolectric power were repeated at various values of AT and proved to be independent of the value of the temperature gradient. T h e thermoelectric power was calculated f r o m the Seebeck relation S = — AV /AT. T w o gold wires attached to the extremities of the sample were used to measure the voltage AV appearing across the sample. T h e same gold wires were used i n a l l the measurements. T h e i r c o n t r i b u t i o n to the thermoelectric power was evaluated by first measuring the thermoelectric power of a gold / l e a d / g o l d thermocouple. U s i n g the results of Roberts 6 9 for the thermoelectric power of lead, the c o n t r i b u t i o n of the gold wires was deduced. 3.7 The optical Measurements T h e absorption coefficient of non-stoichiometric films of a l u m i n u m nitride was o b t a i n e d by measuring the reflectance and t ransmittance of the films using a spectrophotometer. T h e samples used i n these measurements were deposited on quartz substrates of commercial grade, and h a d a thickness of « 300 nm. T w o different approaches were used to evaluate the absorption coefficient depending on CHAPTER S: Experimental Techniques 41 F i g u r e 3-3 E x p e r i m e n t a l arrangement used i n the thermoelectric power measure-ments. CHAPTER S: Experimental Techniques 42 its magnitude. W h e n it was sufficiently high that i n t e r n a l reflections could be neglected, the following m e t h o d was used: F i g . 3-4 depicts the substrate/sample c o m b i n a t i o n a nd the various rays considered. r olo 1 -ad loO - g e • -ad lo (We) (l-^) e F I L M loO-feMi-'.xi-oe ad SUBSTRATE F i g u r e 3-4 Schematic of the various rays considered i n the evaluation of the trans-mittance. A simple analysis shows th a t the resulting transmittance is given by: T(A) = (1 - Ri)(l - Jfe)(l - R0)e-aW, (3.1) CHAPTER S: Experiment al Techniques 43 a(A) is the absorption coefficient and d the thickness of the film. T h e values of T"(A), -Ri(A), and i ? 0 ( A ) were measured succesively between 200 n m and 800 nm. T h e values of R2(X) given by: 6 0 (n, + ng) +{kt) where the indices s and g refer to the sample and substrate respectively, were calc u l a t e d f r o m E q . 3.2 by neglecting kt. T h e values of nt needed to evaluate Ri(X) were deduced f r o m the measured values of Ri(X). k, was again neglected i n the expression of R\{\) given by: (n,+ 1) +{kt) A f t e r s o l v i n g for a(A) using E q . 3.1, the values of kt were estimated using the expression "(A) = (3.4) A is the wavelength of the incident l i g t h . In a l l cases the values o b t a i n e d for ks j u s t i f i e d the a p p r o x i m a t i o n used in ^ i ( A ) a n d ^ ( A ) . A different approach was used to evaluate the absorption coefficient of films e x h i b i t i n g interference fringes. T h e method, depicted g r a p h i c a l l y i n F i g . 3-5 is due to Manifacier. 6 1 T h e sucessive m a x i m a and m i n i m a are linked together by a continuous curve as shown i n F i g . 3-5. T h e s e curves are referred as Tmax(X) and Tm{n(A) respectively. It can be shown that for weakly absorbing films the absorption coefficient can be expressed i n terms of Tmax and T m f n as: ^ N [ U A ) / U > f | « \ i + [r„„(A)/rmin(A)]1/2 J CHAPTER S: Experimental Technique* 44 CO 1.0 0.8 - 0.6 I 0.4 0.2 0 L 200 300 400 500 600 LAMBA (nm) 700 800 Figure 3-5 G r a p h i l l u s t r a t i n g the gra p h i c a l method used to evaluated the absorp-tion coefficient of films e x h i b i t i n g interference fringes. where and c 2 = (1 + ng) {n, + ngy K - 1 ) {ng-nsy (3.6) (3.7) T h e absorption coefficient of nearly stoichiometric a l u m i n u m n i t r i d e films as well as non-stoichiometric films have been obtained i n order to ascertain the presence CHAPTER S: Experimental Techniques 45 of localised states i n the b a n d gap of AINX. T h e results are discussed i n section 4.42. 3.8 Determination of the Film Structure T h e x-ray diffraction s p e c t r u m of non-stoichiometric films of a l u m i n u m ni-tride were obtained using the 0.154 n m Cu-Ka radiation. T h e s e films were deposited without contact electrodes and had a thickness of » 500 nm. E l e c t r o n diffraction measurements and transmission electron micrographs were obtained by depositing the films on very t h i n c a r b o n coated copper grids. T h e electron diffr a c t i o n pho-tographs shown i n 4-3 through 4-5 were obtained using the same magnification. T h e exposure times were somewhat different due to slight differences i n the thick-ness of the samples. T h e i r t y p i c a l thickness was « 30 nm. T h e s e measurements were made i n the metallurgy department at U.B.C. CHAPTER 4: RESULTS AND DISCUSSIONS C H A P T E R 4 RESULTS A N D DISCUSSIONS 4.1 F i l m S t r u c t u r e F i g . 4-1 shows the x-ray diffraction s p e ctrum of non-stoichiometric films of alu m i n u m n i t r i d e o b t a ined using 0.154 n m Cu — Ka r a d i a t i o n . T h e value of the r o o m temperature c o n d u c t i v i t y of each film is given on the figure. T h e x-ray s p e c t r u m of films w i t h r o o m temperature c o n d u c t i v i t y above 1200 ( f i c m ) - 1 exhibit sharp a l u m i n u m lines. B r o a d e r and weaker a l u m i n u m nit r i d e lines are also observed. T h e s e results suggest that films w i t h a r o o m temperature con-d u c t i v i t y above 1200 ( 1 7 c m ) - 1 are composed of very s m a l l particles of a l u m i n u m ni t r i d e dispersed i n an a l u m i n u m matrix. T h i s interpretation is s u p p o r t ed by elec-t r o n diffraction measurements and electron transmission photographs. F i g . 4-2 shows an electron transmission photograph of a sample h a v i n g a r o o m temperature c o n d u c t i v i t y equal to 1.5xl0 4 ( f l c m ) - 1 . T h i s photograph reveals the existence of a colum n a r m i c r o s t r u c t u r e which is characteristic of v a c u u m deposited coatings at low substrate temperature. T h e average column diameter observed i n this photograph is « 200 n m and corresponds to the size of the a l u m i n u m crystallites. T h e size of the a l u m i n u m n i t r i d e inclusions was estimated to be « 10 n m f r o m the line broadening observed i n the x-ray spectrum. T h i s value was obtained f r o m the Sherrer relation 63 t = 0.9X/Wcos(9b) where t is the particle size, A the r a d i a t i o n wavelength, W the CHAPTER 4: RESULTS AND DISCUSSIONS 47 44 42 40 38 36 34 32 30 2 X THETA (DEGREES) F i g u r e 4-1 X-ray diffraction s p e ctrum of non-stoichiometric a l u m i n u m nitride. CHAPTER 4: RESULTS AND DISCUSSIONS Figure 4-2 E l e c t r o n transmission photograph of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 1.5xl04(ncm) . CHAPTER 4-RESULTS AND DISCUSSIONS 49 w i d t h of the s p e c t r a l line, a nd Of, the B r a g g angle. T h i s number should only be seen as an lower estimate since the influence of non-uniform s t r a i n on the line broadening is neglected i n this analysis. Fig. 4-3 shows an electron diffraction p h o t o g r a p h of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 10 4 ( f t c m ) - 1 . W h i l e the alu-m i n u m n i t r i d e rings have a continuous "appearance, the a l u m i n u m r i n g pattern is discontinuous. T h e s e results also indicate that the structure of this film consists of a p o l y c r y s t a l l i n e a l u m i n u m m a t r i x i n which are embeded very s m a l l grains of a l u m i n u m nit r i d e . R e t u r n i n g to Fig. 4-1, it is observed that the reduction of the film conduc-t i v i t y f r o m 1200 ( f t c m ) - 1 to 700 ( f t c m ) - 1 is accompanied by a reduction of the a l u m i n u m line intensities and an increase of their linewidths, i n d i c a t i n g a tendency of the a l u m i n u m crystallites to break u p into smaller units. T h e intensities of the a l u m i n u m n i t r i d e lines increase slightly while their linewidths remain broad, sug-gesting that the number of a l u m i n u m nit r i d e particles is increasing while t h e i r sizes remain quite s m a l l ( < 10 nm). T h e a l u m i n u m rings observed i n an electron diffrac-tion p h o t o g r a p h of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 600 ( f t c m ) - 1 have a continuous appearance (Fig. 4-4) as opposed to the spotty r i n g structure ob-served i n films of higher c o n d u c t i v i t y (Fig. 4-3). T h i s also confirms the a l u m i n u m crystallites are breaking u p into smaller units as the film c o n d u c t i v i t y decreases. F i l m s w i t h r o o m temperature c o n d u c t i v i t y between 400 ( f t c m ) - 1 and 200 ( f t c m ) - 1 can be best described as being amorphous. T h e i r x-ray s p e c t r a reveal only b r o a d lines of low intensities (Fig. 4-1). These films appear completely structureless i n the electron transmission photographs. In electron diffraction measurements the films y i e l d only broad, diffuse rings (Fig. 4-5). F r o m the above observations i t can be concluded that the reduction of the film c o n d u c t i v i t y f r o m 1.5xl0 4 ( f t c m ) - 1 to 200 ( f t c m ) - 1 is accompanied by an increase i n the s t r u c t u r a l disorder present i n these films. CHAPTER 4: RESULTS AND DISCUSSIONS Figure 4-3 E l e c t r o n diffraction photograph of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 10* ( f t c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS Figure 4-4 Electron diffraction photograph of a f i lm having a room temperature conductivity of 600 ( Q c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS Figure 4-5 Electron transmission photograph of a film having a room temperature conductivity of 200 (ftcm) - 1. CHAPTER 4: RESULTS AND DISCUSSIONS 53 A s the film c o n d u c t i v i t y is reduced below 200 ( f t c m ) - 1 the a l u m i n u m nitr i d e lines of the x-ray s p e c t r u m gain i n strength and their linewidths decrease, i n d i c a t i n g an increase i n the volume fraction and size of the a l u m i n u m n i t r i d e particles. T h e a l u m i n u m x-ray lines remain very weak a n d are not observed i n films w i t h r o o m temperature c o n d u c t i v i t y below 10 ( f l c m ) - 1 . Nevertheless, the high value of the r o o m temperature c o n d u c t i v i t y of these films clearly suggests that they contain a l u m i n u m atoms i n excess. It is concluded that the a l u m i n u m atoms are either finely dispersed i n an a l u m i n u m nitr i d e m a t r i x (acting as dopant) or that they f o r m small metallic islands with an amorphous structure. S i m i l a r results have been observed i n non-stoichiometric films of zinc oxide: while these films were known to contain zinc atoms in excess, no zinc lines were observed i n the x-ray s p e c t r u m of the films. 6 8 4.2 T h e B o l t z m a n n R e g i m e Samples with r o o m temperature c o n d u c t i v i t y above 10 4 ( f t c m ) - 1 exhibit a metallic behavior well described by the B o l t z m a n n E q . 2.8. F i g . 4-6 which is a plot of the resistivity as a f u n c t i o n of temperature, shows a t y p i c a l result o b t ained with a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 3 x l 0 4 ( f i c m ) - 1 . A b o v e 50 K the resistivity increases linea r l y with temperature as predicted by Eq. 2.12. Below that temperature the resistivity tends towards a value independent of temperature. T h i s residual resistivity is a t t r i b u t e d to i m p u r i t y scattering as predicted by Eq . 2.11. T h e value of the elastic mean free p a t h le can be deduced from F i g . 4-6 by e x t r a p o l a t i n g the resistivity to T = 0 K (Eq . 2.11). T h e value of kf is estimated f r o m the value of n obta i n e d f r o m the H a l l measurements using the free-electron formula 1 /3 kf = (3fl" 2n) . A value of le = 2.7 n m is thus obtained. T h i s elastic mean free p a t h is already much shorter than that of pure a l u m i n u m (/e = 11 nm). T h e presence of a l u m i n u m n i t r i d e particles dispersed i n the a l u m i n u m m a t r i x is u n d o u b t a b l y CHAPTER 4: RESULTS AND DISCUSSIONS 5 10 'O T" X t E £ o £ E eo •£ UJ o 0 50 100 150 200 250 300 T (KELVIN) F i g u r e 4-6 p(T) versus T of a f i l m h aving a roo m temperature c o n d u c t i v i t y of 3xl0 4(ncm) - 1. T h e temperature independent residual resistivity below « 50 K is clearly observed. CHAPTER 4: RESULTS AND DISCUSSIONS 55 responsible for this shorter elastic mean free path. Nevertheless [kfle) S> 1 and the B o l t z m a n n approach s t i l l holds. 4.3 T h e W e a k L o c a l i s a t i o n R e g i m e 4.31 T h e d c C o n d u c t i v i t y A s discussed in section 2.21. the presence of power-law localised wave func-tions of the f o r m $fe oc c , k r / r 2 should lead to significant departures f r o m the B o l t z -m a n n conductivity. T h e model presented i n that section, developed by K a v e h a nd Mott, makes very specific predictions concerning the temperature dependences of the dc conductivity. In this section it is shown that a l l the temperature dependence predicted by this m odel ( E q . 2.21) is indeed observed p r o v i d e d that an inelastic mean free p a t h of the f o r m /,• = a T - 1 is assumed over the whole temperature range examined (10 K to 300 K ) . Moreover it is shown that the m o d e l predicts the correct magnitude of the conductivity. T h e temperature dependence of the c o n d u c t i v i t y of samples with r o o m tem-perature c o n d u c t i v i t y between 10 3 ( f t c m ) - 1 and 10 4 (Clem)-1 is characterised by a b r o a d m a x i m u m at a temperature Tm ( F i g . 4-7 t h r o u g h 4-10), as predicted by E q . 2.21. It is also observed that the value of Tm is a func t i o n of the mag-nitude of the c o n d u c t i v i t y ( F i g . 4-7 t h r o u g h 4-10). Samples with low r o o m temperature c o n d u c t i v i t y have a high value of Tm while samples with high r o o m temperature c o n d u c t i v i t y have a low value of Tm. T h e results of section 4.1 in d i -cate t h a t the r e d u c t i o n of the r o o m temperature c o n d u c t i v i t y of the deposited films f r o m 1.5xl0 4 ( f l c m ) - 1 to 200 ( f i c m ) - 1 is accompanied by an increase i n s t r u c t u r a l disorder. T h i s should i n t u r n lead to a shorter elastic mean free path. T h e f o r m of Tm given by E q . 2.22 is therefore consistent with the experimental observation that the smaller the r o o m temperature c o n d u c t i v i t y of a film, the higher its value CHAPTER 4: RESULTS AND DISCUSSIONS 56 O o 2, 8 10 12 14 16 18 1/2 1/2 T (KELVIN) F i g u r e 4-7 T e m p e r a t u r e dependence of the dc c o n d u c t i v i t y of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 7740 ( f t c m ) - 1 . A y/f behavior is observed below the c o n d u c t i v i t y m a x i m u m at 80 K. CHAPTER 4: RESULTS AND DISCUSSIONS 57 >-O o o X 3425 3405 16 18 T V 2 (KELVIN)172 F i g u r e 4-8 T e m p e r a t u r e dependence of the dc c o n d u c t i v i t y of a film h a v i n g a r o o m temperature c o n d u c t i v i t y of 3400 ( f l c m ) - 1 . A \ f f behavior is observed below the c o n d u c t i v i t y m a x i m u m at 110 K. CHAPTER 4: RESULTS AND DISCUSSIONS 58 7820 >-> O z> Q Z o o o • X g. 7670 0 5 0 100 150 200 250 3 0 0 T (KELVIN) F i g u r e 4-9 T e m p e r a t u r e dependence of the dc c o n d u c t i v i t y of a f i l m h a v i n g a r o o m temperature c o n d u c t i v i t y equal to 7740 ( f i c m ) - 1 . A linear behavior is observed above the c o n d u c t i v i t y m a x i m u m at 80 K. CHAPTER 4: RESULTS AND DISCUSSIONS 59 I-••••• O • => 5 Q X 3425 3405 L c> 3385 3365 3345 3325 50 100 150 200 250 300 T (KELVIN) F i g u r e 4-10 Temperature dependence of the dc c o n d u c t i v i t y of a film h a v i n g a ro o m temperature c o n d u c t i v i t y of 3400 ( f t c m ) - 1 . A linear behavior is observed above the c o n d u c t i v i t y m a x i m u m at 110 K. CHAPTER 4: RESULTS AND DISCUSSIONS of Tm. It is also observed that w i t h increasing temperature the c o n d u c t i v i t y first increases as Vf u n t i l the m a x i m u m in c o n d u c t i v i t y is reached ( F i g s . 4-7 a n d 4-8). A b o v e that temperature the c o n d u c t i v i t y decreases linearly with T, as predicted by Eq . 2.21 ( F i g s . 4-9 a n d 4-10). Samples w i t h c o n d u c t i v i t y between 10 3 (Clem)-1 and 100 ( f t c m ) - 1 do not exhibit a c o n d u c t i v i t y m a x i m u m in the temperature range examined ( F i g . 4-11). Instead the c o n d u c t i v i t y increases continuously as y/T as the temperature is in-creased between 10 K and 300 K. Such a temperature dependence is also predicted by E q . 2.21 for samples having s m a l l enough value of their elastic mean free p a t h to shift Tm to temperatures well above 300 K. A l l the temperature dependence predicted by Eq. 2.21 is therefore observed experimentally. F r o m F i g s . 4-7 t h r o u g h 4-11 and E q . 2.21, the values of le and /,• for each sample have been evaluated a n d the results are summarized in T a b l e I. A g a i n the values of kf were estimated using the free-electron formula kf = (37r 2n) and the values of n determined f r o m H a l l measurements. T h i s analysis shows that the values of le, /,• and kf needed to recover the correct magnitude of the c o n d u c t i v i t y are physically reasonable. Moreover the derivation of Eq . 2.21 is v a l i d only for (kfle) greater t h a n ~ 3. It is f o u n d that a l l samples with r o o m temperature c o n d u c t i v i t y much greater than 100 ( f ) c m ) _ 1 , for which E q . 2.21 pro-vides an accurate description of their temperature dependence, do yi e l d values of (kfle)2 greater t h a n 3. O n the other hand, deviations f r o m the \/T behavior are observed in samples with r o o m temperature c o n d u c t i v i t y smaller t h a n 1 0 0 ( f l c m ) - 1 2 2 ( F i g . 4-12). Es t i m a t e s of (kfle) for these samples y i e l d (kfle) ?» 3, thus at the lim i t of the v a l i d i t y of E q . 2.21. It is concluded that the c o n d u c t i v i t y of the films p roduced i n this work, w i t h r o o m temperature c o n d u c t i v i t y between 10 4 (Ucm)~l and 100 ( Q c m ) - 1 is domi-nated by electron localisation effects and that the m odel of K a v e h and M o t t provides CHAPTER 4: RESULTS AND DISCUSSIONS 61 300 h a x oo 0 2 4 6 8 10 12 14 16 18 T V 2 (KELVIN)1 2 F i g u r e 4-11 T e m p e r a t u r e dependence of the dc c o n d u c t i v i t y of two films showing a y/T behavior over the entire temperature range between 10 K and 300 K. Triangle: crl = 280 ( f t c m ) - 1 . Circle: art = 240 ( f l c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS T A B L E 1 a ( 3 0 0 i r ) ( f l c m ) - 1 *{0) ( f i c m ) " 1 (cm)"1 u (cm) ( M e ) ' (") a (cmT) 3 . 1 x l 0 4 4 . 3 x l 0 4 1.38xl0 8 2.7x10" -7 1.4xl0 3 1.8x10" -4 1.9xl0 4 2.41xl0 4 1.30xl0 8 1.7x10" -7 4 . 8 x l 0 2 1.4x10" -4 4.1xl0 3 3 . 4 x l 0 3 1.0x10 s 4.3x10" -8 21 2.6x10" -5 1.3xl0 3 l . l x l O 3 8.8xl0 7 3.0x10" -8 7 3.2x10--4 l . l x l O 3 9.9xl0 2 8.7xl0 7 2.9x10" -8 6.5 4.3x10' -4 2 . 9 x l 0 2 1.5xl0 2 6.6xl0 7 2.8x10" -8 3.5 1.4x10" -3 2 . 3 x l 0 2 10 6.2xl0 7 2.8x10" -8 3.0 3.4x10--4 T a b l e - I Values of kf, le, and a ( where /,• = aT~l) obtained f r o m the experimental results shown i n F i g s . 4-7 t h r o u g h 4-11. CHAPTER i: RESULTS AND DISCUSSIONS *-> I , > mmwm O "O c o O E o E 15 _ 8 10 12 14 16 18 -T (KELVIN) 1 2 F i g u r e 4-12 Temperature dependence of the dc c o n d u c t i v i t y of a f i l m h a v i n g a roo m temperature c o n d u c t i v i t y less t h a n 100 ( f l c m ) - 1 (art = 32 ( f i c m ) - 1 ) . A devi a t i o n f r o m the \ f f behavior is observed. CHAPTER 4: RESULTS AND DISCUSSIONS 64 a good description of the magnitude and temperature dependence of the conduc-tivi t y . T h e results of section 4.1 make it clear that electron localisation effects are due to s t r u c t u r a l disorder induced by the presence of al u m i n u m n i t r i d e particles dispersed i n the a l u m i n u m matrix. It is interesting to note that a good description of the temperature dependence of the c o n d u c t i v i t y is o btained using E q . 2.20 provided that one assumes an inelastic mean free p a t h of the form = aT~l over the whole temperature range examined. It is nevertheless well known that such a temperature dependence of /,- is expected only for temperatures well above the Debye temperature (0£>), which is equal to 428 K for bulk aluminum. T h i s apparent lowering of the Debye temperature i n granular materials has been noted by many workers. Howson 2 has observed an inelastic mean free p a t h p r o p o r t i o n a l to T - 1 down to ~ ©z>/3 i n CUIQTI^Q and Ti^Be^ZriQ-Below 0 j ? / 3 , /, was observed to crossover to a low temperature f o r m /, a T~2. Saub et al. 8 have also observed the same phenomenon in glassy Zrioo-xCux alloys. T h e evaluation of the Debye temperature of granular a l u m i n u m films (mixture of Al and A/2O3) have been obtained from specific heat measurments by Greene et al.. 6 4 T h e i r measurements indicate an important lowering of the Debye temperature by as much as 27 per cent i n the samples examined. T h e y a t t r i b u t e d the lowering of the Debye temperature to a softening of the lattice which is presumably induced by the presence of a l u m i n u m oxide grains embeded i n the a l u m i n u m matrix. T h e mismatch of the chemical bonds at the interface of the al u m i n u m and a l u m i n u m oxide grains would lower the force constant of the a l u m i n u m atoms, thus lowering the Debye temperature. Since the non-stoichiometric films of a l u m i n u m n i t r i d e s t u d i e d have also a granular structure (section 4.1), it is believed that a simi l a r m echanism is responsible for the observed temperature dependence of the inelastic mean free p a t h a nd the conductivity. CHAPTER 4: RESULTS AND DISCUSSIONS 65 4.32 T h e a c C o n d u c t i v i t y T h e frequency dependence of the c o n d u c t i v i t y i n the weakly localised regime has also been examined. These measurements have been performed only at r o o m temperature. A c c o r d i n g to K a v e h a nd Mott, 2 1 the frequency dependent conduc-t i v i t y i n the weakly localised regime should be given by: l + (wTe) I (kfl) 1 J J It is noted that this equation has a structure very similar to o{T) ( E q . 2.20). T h e derivation of <x(w) is i n fact very s i m i l a r to that of E q . 2.20, the inelastic dif-fusion length Li being replaced by L w = {D/u)1/2, where D is a diffusion constant. Li and Z w have also a sim i l a r meaning, representing the t y p i c a l distances an elec-tron diffuses before being either inelastically scattered or before the electric field changes direction. T h e interesting aspect of Eq. 4.1 is that it predicts an increase i n the c o n d u c t i v i t y with frequency up to u = Te~l. re is the elastic scattering time approximately equal to Vf/le. A t frequencies well above r e _ 1 the c o n d u c t i v i t y is domina t e d by the usual free-electron t e r m ^1 + (wr e) 2j . Eq. 4.1 therefore pre-dicts a m a x i m u m i n cr(u) at w ~ r e - 1 1 . Such a m a x i m u m has been observed by Rosenbaum et al. 5 i n the far-infrared at around r e - 1 = 2 x l 0 1 5 H z (100 c m - 1 ) . T h i s value of r e - 1 correlated well with their estimated value of le obtained f r o m d c co n d u c t i v i t y measurements. Values of le for samples of non-stoichiometric a l u m i n u m n i t r i d e i n the weakly localised regime are approximately equal to 5 x l 0 - 8 c m (T a b l e 1), which correspond to r e _ 1 « 4 x l 0 1 5 Hz. A max i m u m i n a(u) is therefore expected at that frequency. F i g . 4-13 shows the results obtained for a(u) at frequencies smaller than 10 8 Hz. A s expected no frequency dependence is observed since for such values of w, ure <§; 1. F i g . 4-14 shows the behavior of <r(w) i n the visible and u l t r a violet ( 1 0 1 7 CHAPTER 4: RESULTS AND DISCUSSIONS 66 F i g u r e 4-13 Beh a v i o r of (r(u) for frequencies below 10 8 Hz. Curve a and b: crrt =3.8x 10 4 ( f t c m ) - 1 and art = 2 . 4 x l 0 4 ( f t c m ) - 1 respectively. T h e temperature dependence of the c o n d u c t i v i t y of these samples is well described by the B o l t z m a n n equation ( E q . 2.12). A s expected the c o n d u c t i v i t y is frequency independent. Curve c and d: arl = 3 . 9 x l 0 3 ( f t c m ) - 1 and aTt = 2 . 9 x l 0 2 ( f t c m ) - 1 respectively. T h e tem-perature dependence of these two samples is well described by the m o d e l of K a v e h and M o t t . A g a i n the c o n d u c t i v i t y is frequency independent. CHAPTER 4: RESULTS AND DISCUSSIONS 67 8 E o o. 5 -(FREQUENCY)12 (HZ) 1 2 X 10 8 8 F i g u r e 4-14 B e h a v i o r of cr(w) i n the visi b l e a n d u l t r a violet. T h e r o o m temperature c o n d u c t i v i t y of this sample is 300 ( f i c m ) - 1 . Its temperature dependence is well described by the m o d e l of K a v e h and M o t t . Below 6 x l 0 1 7 H z the c o n d u c t i v i t y slowly increases w i t h u/. T h e sharp increase of the c o n d u c t i v i t y observed above 6 x l 0 1 7 Hz is a t t r i b u t e d to electronic t r a n s i t i o n across the b a n d gap. CHAPTER 4: RESULTS AND DISCUSSIONS 68 Hz to 8 x l 0 1 7 Hz). <T{U) was evaluated f r o m the measured absorption coefficient, a(uj), using the relation: cr(u) = K0 c a(u), (4.2) K0 is the dielectric p e r m i t i v i t y of free-space and c the speed of light. c*(w) itself was deduced f r o m reflectance and transmittance measurements, using the methods described i n section 3.7. In the visible and u l t r a violet (w Te~l) <r(w) should be a decreasing f u n c t i o n of frequency (free-electron behavior). Nevertheless cr(u) is observed to increase. Since u 3> r e - 1 , electron localisation effects cannot be respon-sible for the anomalous behavior of the conductivity. Instead it is a t t r i b u t e d to the presence of the sm a l l a l u m i n u m n i t r i d e grains dispersed i n the a l u m i n u m m a t r i x (section 4.1). T h i s is not an unrealistic assumption since it has been observed (sec-tion 4.42) that closely stoichiometric a l u m i n u m nit r i d e absorbed i n this frequency range. It is therefore concluded that the granular structure of the samples i n this c o n d u c t i o n regime prevents the observation of localisation effects i n a{u). T h e weak enhancement of cr(w) due to localisation effects is being masked by the much stronger enhancement induced by the presence of the a l u m i n u m n i t r i d e grains. 4.33 T h e T h e r m o e l e c t r i c p o w e r a n d H a l l e f f e c t T h e results presented i n section 4.31 indicate that the m o d e l of K a v e h and M o t t provides an accurate description of the temperature dependence and magni-tude of the c o n d u c t i v i t y of samples with r o o m temperature c o n d u c t i v i t y between 100 ( Q c m ) - 1 and 10 4 ( f l c m ) - 1 . However, the model assumes that the s t r u c t u r a l disorder responsible for the electron localisation effects leaves the density of states free-electron like. It was discussed i n section 2.22 that the measurement of the thermoelectric power co u l d help verify the va l i d i t y of this assumption. F i g . 4-15 shows the temperature dependence of the thermoelectric power of samples for which the c o n d u c t i v i t y is characterised by a \/f behavior between 10 CHAPTER 4: RESULTS AND DISCUSSIONS 69 cc 111 §-£ Q. > cc ±: H O O > m CO II 'O U J ZZ, o cc Lit 100 150 200 250 300 T (KELVIN) F i g u r e 4-15 T e m p e r a t u r e dependence of the thermoelectric power of samples for which the dc c o n d u c t i v i t y exhibits a \fT behavior between 10 K and 300 K. Square: 10 3 ( f t c m ) - 1 , Triangle: 315 (ilcm)~\ Circle: 25 ( f t c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS K and 300 K, that is, d o m i n a t e d by localisation effects over the temperature range examined. A b o v e w 100 K the thermoelectric power decreases linearly w i t h T as expected for a m e t a l e x h i b i t i n g l i t t l e or no disorder. T h e dependence of the ther-moelectric power on T is therefore free-electron like and supports the assumption that the density of states is unaffected by localisation effects. A s s u m i n g that the free-electron expression for the thermoelectric power 4 6 ( E q . 2.28) c m = 2m T holds, the value of n for different samples can be extracted f r o m the slopes of S(T). These values of n are plotted as a function of the "room temperature c o n d u c t i v i t y i n F i g . 4 16. O n t h a t same figure are p l o t t e d the values of n obtained f r o m H a l l measurements on a different set of samples. O n this figure a d o t t e d line has been drawn between the points as a v i s u a l a i d to suggest that b o t h sets of measurements yi e l d values of n which are consistent with one another. T h ese experimental results suggest that the temperature dependence of the thermoelectric power between « 100 K and r o o m temperature is free-electron-like. T h e values of n deduced f r o m the thermoelectric power measurements are consistent with those obtained from the H a l l measurements, and indicate that the thermoelec-tri c power a n d H a l l effect can be described using free-electron expressions (Eq. 2.28 and 2.29 respectively). T hese results strongly suggest that the density of states is indeed unaffected by localisation effects. T hese results are moreover in agree-ment w i t h those obtained by T h o m a s et a l . 6 5 u p o n comparing the dependence of the c o n d u c t i v i t y and electronic specific heat on n. T h e y found that while the dependence of the c o n d u c t i v i t y on n was dominated by loc a l i s a t i o n effects, the de-pendence of the electronic specific heat on n remained free-electron like, s u p p o r t i n g the idea that the density of states is esentially unaffected by localisation effects. CHAPTER 4: RESULTS AND DISCUSSIONS 71 10' >-> O o 2 P, O O 10^  10J 10' 10' 10^  10 .-1 I I — 1 — A — / A' / / m / / — / •/ /* i i •/ - / / I i i i i 0 1 2 3 4 x i o 2 2 N (CM-3) F i g u r e 4-16 C h a r g e carrier density p l o t t e d as a function of the r o o m tempera-ture conductivity. Square: A s determined f r o m thermolectric power measurements. Triangie: A s determined f r o m H a l l measurements. CHAPTER 4: RESULTS AND DISCUSSIONS 72 4.34 T h e Low Temperature Behavior of the Thermoelectric Power T h e low temperature behavior of the thermoelectric power is tentatively at-t r i b u t e d to phonon drag. T h i s effect usually dominates the temperature dependence of the thermoelectric power of metals at low temperature. A crude estimate of the p honon d r a g c o n t r i b u t i o n to the thermoelectric power is given by 6 6 Sg w Cv/ne where C „ is the lattice specific heat. T h e t o t a l thermoelectric power S, i n c l u d i n g the electronic (Se) and phonon-drag contributions Sg, can then be a p p r o x i m a t e d by 66 S = Cv/ne + Ce/ne, (4.3) where C e is the electronic specific heat. At low temperatures, Ce<xT and Cv oc Tz and Eq. 4.3 is often written as S/T = aT2 + b. If phonon d r a g dominates the thermoelectric power at low temperatures, S/T varies as T2 and the extrapolated value of S/T at T = 0 K ° yields the slope of the electronic component of the thermoelecric power (Se/T). T h e results of S/T p l o t t e d as a func t i o n of T2 are shown in F i g . 4-17. W h i l e the accuracy of the d a t a is not sufficient to convincingly show that S/T is indeed p r o p o r t i o n e l to T 2 , the extrapolated value of S/T at T = 0 K matches up with the high temperature d a t a where the phonon-drag component is quenched. 4.4 T h e Strong Localisation Regime 4.41 T h e dc Conduct ivi ty As discussed i n sections 1.1 and 2.3, i n the l i m i t of very strong disorder the wave functions are expected to become exponentially localised and the conductiv-ity to proceed by nearest-neighbor-hopping or variable-range-hopping. T h e former leads to a temperature dependence of the f o r m <r(T) oc exp — (^f-), while the latter CHAPTER 4: RESULTS AND DISCUSSIONS 73 S E / T V 0 2 4 6 8 10xio 3 T 2 (KELVIN)2 F i g u r e 4-17 S/T p l o t t e d as a fun c t i o n of T2. T h e extrapolated value of S/T to T = 0 K yields the slope of the electronic component of the thermoelectric power (Se/T), i n d i c a t i n g that the low temperature behavior of the thermoelectric power might be a t t r i b u t e d to phonon drag. Square: <rrt = 10 3 ( f i c m ) - 1 , Triangle: orl = 315 ( 1 7 c m ) - 1 , Circle: ori = 25 ( Q c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 74 leads to a more p e c u l i a r one, cr(T) oc exp — {Ift)1^. It was also discussed i n sec-tion 2.34 that although theories of variable-range-hopping agree on the exponential dependence of the c o n d u c t i v i t y a n d predict the same value of Ta ( E q . 2.51), they disagree on the exact f o r m of the prefactor preceeding the exponential term. In t e r m of the resistivity, p, the temperature dependence of the resistivity has the general form: p(T) = C0T?exp(?±yi (4.4) A t t e m p t s were made to establish experimentally: 1) W h e t h e r s = 1/4 provides the best description of the data; 2) T h e value of the exponent p. It was observed that the temperature dependence of samples with r o o m tem-perature c o n d u c t i v i t y smaller than « 2 ( f i c m ) - 1 was very strong and attempts were made at f i t t i n g the experimental d a t a between 10 K and r o o m temperature to an equation of the general f o r m of E q . 4.4. A s a first approximation, the exponent p was set to zero and the correlation coefficient R 6 7 was calculated for various values of s. A value of R = 1 corresponds to a perfect fit, while a value of R = 0 indicates that the d a t a cannot be fitted by a function of the f o r m of E q . 4.4. T y p i c a l results are shown in F i g . 4-18 and indicate that a value of s = 0.25 provides the best fit to the experimental data. O t h e r attempts were made at f i t t i n g the high and low temperature regions independently. None of these attempts were successful. U s i n g a value of s = 0.25, l i m i t s on the value of p were then established. It was f o u n d th a t the best fits were obtained w i t h values of p between 0.3 < p < 0.9. Deviations were clearly observed when using values of p outside this interval ( F i g . 4-19). In F i g . 4-20 is shown the temperature dependence of two samples obtained using p = 0.5 and s = 0.25. F o r these values of s and p, no deviations f r o m the tem-perature dependence predicted by E q . 4.4 are observed between 10 K and r o o m temperature. CHAPTER 4: RESULTS AND DISCUSSIONS 10* 1 0 1 0 1 0 ' _ 1 0 1 0 10-E X P O N E N T S F i g u r e 4-18 (1 — R), where R is the correlation coefficient, p l o t t e d as a f u n c t i o n of the exponent s, showing that the value s = 0.25 provides the best fit to the experimental d a t a of the c o n d u c t i v i t y between 10 K and 300 K. T h e sample used for this figure h a d a r o o m temperature c o n d u c t i v i t y of 2 ( Q c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 76 a i I— • • mum > "</> <D CC _> CD • E o B | o 10 10^  10 10" 10" 10' 10 10 10 1d2 id 3 / • / / P=1/3 • / / / / / / / / / / P=1 / .240 .290 .340 .390 .440 .490 .540 f / 4 (KELVIN)14 Figure 4-19 Temperature dependence of the dc resistivity of a film with a room temperature conductivity of 2 (fkm) - 1, plotted using the two limits established on P-CHAPTER 4: RESULTS AND DISCUSSIONS 77 CM \ ^ T . E T > ^ •* ••5 E « E cc o 10 10' 10e 10" 8 10' 10J 1tf w 10 10l 10' 10" .230 .280 .330 .380 .430 .480 .530 .580 f / 4 .71/4 ( K E L V I N ) F i g u r e 4-20 Temperature dependence of the dc resistivity of two samples using p = 0.5 and s = 0.25. These results indicate that the model of Al l e n et al. provides the best description to the experimental data. Triangle: cri =5.7xl0~ 2 ( f t c m ) - 1 . Circle: art = 2 ( f t c m ) " 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 78 O n l y the theory of A l l e n et al. 8 5 ( E q . 2.56), with a value of p = 0.5, is consistent with the limits established experimentally on the value of p. T h e theories of S h k l o v s k i i 8 8 (p = 0.25), Pollak 8 7 (p = 0.25) and M o t t 8 6 (p = -0.25) predict values of p which are not observed experimentally i n our samples. T h e m odel of M o t t 8 6 ( E q . 2.58) with p = —0.25 can easily be discarded since the negative value of p falls well outside the experimental l i m i t s established on p. A s discussed i n section 2.34, the m odel can also be discarded on technical grounds, the expression for the number of carriers n = N(cf)kt,T, being inappropriate for variable-range-hopping. T h e m odel of Pollak 8 7 and Shklovskii 8 8 (Eq. 2.55) predict a value of p (p = 0.25) which falls just outside the l i m i t s established on p. Nevertheless it is clear that the experimental d a t a do not follow such a temperature dependence ( F i g . 4-19). T h e m o d e l of A l l e n et al. 8 6 (Eq. 2.56) predicts a value of p (p = 0.5) which falls i n the m i d d l e of the l i m i t s established on p. T h i s m odel provides the best description of the temperature dependence of the measured co n d u c t i v i t y ( F i g . 4-20). These results suggest that the choice of rav as the length scale for the c o n d u c t i v i t y is a better one t h a n L0. A s discussed in section 2.34, the choice of La as the length scale for the c o n d u c t i v i t y was based on the argument that cubes of size L\, dispersed i n the r a n d o m resistor network, sample a large enough volume of the network that they can be considered i d e n t i c a l to one another. T h e macroscopic c o n d u c t i v i t y is then identified w i t h the c o n d u c t i v i t y of those cubes. A t r o o m temperature LQ is t y p i c a l l y of the order of ~ 100 nm. O n the other hand, variable-range-hopping theories are based on the existence at a given temperature of an average h o p p i n g distance rav ( E q . 2.48) between the localised states that participate i n the c o n d u c t i o n process. A t r o o m temperature and pressure rav is t y p i c a l l y ft* 6 nm. T h e existence of an average h o p p i n g distance, which is corroborated by the observation of variable-range-hopping itself, suggests that the resistor network is homogeneous on a scale much smaller than Lc. T h i s may explain why rav seems to constitute a better length scale. CHAPTER 4: RESULTS AND DISCUSSIONS E s t i m a t e s of a and N(ef) have been obtained for various samples f r o m the experimental values of C0 and Ta ( E q . 4.4), by solving the equations, 79 3 6 r r 3 m 2 ( dv> \ (kbN(e,)\ ^ (where KK0 ( E q . 2.36) has been replaced by KK0 = (nme2/k2a) and 18a3 , . These expressions have been obtained f r o m the model of A l l e n et al. 8 6. T h e parameters d, ve and E\ were assumed to be constant for a l l the samples a nd the values d = 10 3 kg/m?, vg = 4 x l 0 3 m/sec and E\ = 1 . 6 x l 0 - 1 8 J were used. It was verified that i n al l cases the cond i t i o n ireij/hvsa <IC 1 (where c,-y « kT(T0/T)1^) was met. T h e results of this analysis are summarized i n T a b l e 2. T h e y indicate that the theory of A l l e n 8 5 predicts values of a and N(ej) that are ph y s i c a l l y acceptable. O n e notes nevertheless that the c o n d i t i o n neij/hvga 1 is barely met. T h i s t e r m was neglected i n Eq. 2.36. If it is not neglected, it introduces a strong temperature dependence in the prefactor of the conductivity, which is not observed experimentally. T h e above experimental results therefore indicate that it is correct to neglect this term. T h e fact that it is aproximately equal to 0.25 instead of being much smaller t h a n 1 might be a t t r i b u t e d to the a r b i t r a r y choice of the value of the speed of sound vs used i n the above calculation. 4.42 The Nature of the Localised States in AINX So f a r n o t h i n g has been said of the nature of the localised states g i v i n g rise to variable-range-hopping conductivity. T h i s important question is now being addressed. CHAPTER 4: RESULTS AND DISCUSSIONS T A B L E 2 p{300K) C0 T0 a N(ef) (flcmr 1 (ncm)/y/K) K cm~l eV^cm-3 (-) 31 5.6x10" 10 7.2xl07 3.8xl07 1.6xl020 0.3 9 1.8x10" -7 l.lxlO 7 1.2xl07 3.2xl019 0.7 0.34 3.2x10" -8 9.3xl06 1.6xl07 lxlO 2 0 0.5 T a b l e - 2 Values of a, N(€f), and ir€ij/hvsa estimated f r o m experimental results ( F i g . 4-20), using the model of A l l e n et al.. CHAPTER 4: RESULTS AND DISCUSSIONS 81 In section 4.1 it was noted that the X-ray s p e c t r u m of films with r o o m temper-ature c o n d u c t i v i t y between 10 ( f t c m ) - 1 and 1 0 - 3 ( f t c m ) - 1 do not exhibit a l u m i n u m lines. Instead their s p e c t r u m only reveal sharp a l u m i n u m n i t r i d e lines. T h e r o o m temperature c o n d u c t i v i t y of pure A1N being » 1 0 - 1 3 ( f t c m ) - 1 , the observed conduc-t i v i t y of these samples suggest that they contain an important excess of a l u m i n u m atoms or alternatively a lack of nitogen atoms. T h r e e possible microstructures c o u l d account for the observed c o n d u c t i v i t y of these samples as well as their X-ray spectrum. (1) A crystal l i n e a l u m i n u m n i t r i d e m a t r i x i n which s m a l l a l u m i n u m islands of an amorphous structure are embeded. (2) A highly defective, non-stoichiometric, a l u minum n i t r i d e m a t r i x of a crys-talline nature c o n t a i n i n g a large number of defect states. (3) A highly defective, non-stoichiometric, a l u m i n u m nitr i d e m a t r i x of an amorphous structure i n which crystallites of stoichiometric a l u m i n u m n i t r i d e are embeded. T h e first m i c r o structure proposed is refered as a cermet (ceramic-metal) mi-crostructure. T h e electrical c o n d u c t i o n in cermets is usually explained i n terms of electrons h o p p i n g between the metallic islands dispersed i n the matrix. T h e m a t r i x itself does not p a r t i c i p a t e d i r e c t l y i n the conduction process. T h e c o n d u c t i v i t y of cermets has a temperature dependence of the form: 6 8 (T \1/2 a(T) cx ezp - ( ^ j , (4.7) Such a temperature dependence has been observed in a large number of composite materials and is now well e s t a b l i s h e d . 6 8 - 7 * T h e r e are nevertheless some questions concerning the exact mechanism leading to E q . 4.7. T w o models have been widely used to describe the c o n d u c t i v i t y of those materials. T h e first one, developed by A beles and Sheng, 6 8 is based on the existence of a charging energy needed to CHAPTER 4: RESULTS AND DISCUSSIONS create the charge carriers c o n t r i b u t i n g to the conductivity. T h i s charging energy is basic a l l y the energy required to remove an electron f r o m a neutral island a nd put it on another one located at a distance S away f r o m the i n i t i a l island. A c c o r d i n g to this model, the charging energy is approximately given by: S is the distance separating the two islands, D the diameter of the islands, and KKQ the dielectric constant of the i n s u l a t i n g matrix. Abeles and Sheng 8 8 also show that the charging energy and the p a r t i c u l a r microstructure of cermets conspire to y i e l d a temperature dependence of the f o r m of Eq. 4.7. However, to obtain the correct temperature dependence they have to make very specific assumptions concerning the microstructure of cermets. T h e r e is some evidence i n the literature that these assumptions might not be c o r r e c t . 7 8 , 7 4 T h e second approach, proposed by Efr o s and S h k l o v s k i i , 7 6 - 7 8 involves the existence of a gap i n the density of states at the F e r m i level. T h e y used the variable-range-hopping model presented i n section 2.33, and assume a density of states of the form: Such a density of states used i n conjunction w i t h the variable-range-hopping model, leads to a temperature dependence of the c o n d u c t i v i t y of the f o r m of E q . 4.7. T h e temperature dependence of the c o n d u c t i v i t y observed i n samples with r o o m temperature c o n d u c t i v i t y between 2 ( f l c m ) - 1 and 1 0 - 3 ( i l c m ) - 1 varies as: (4.8) N(e) oc (e - ef)2, (4.9) (4.10) CHAPTER 4: RESULTS AND DISCUSSIONS 83 It therefore suggests that the c o n d u c t i v i t y does not proceed by electronic h o p p i n g between metallic islands . Moreover the r o o m temperature c o n d u c t i v i t y of cermets is n o r m a l l y about 5 orders of magnitude smaller t h a n the c o n d u c t i v i t y observed in these f i l m s . 6 8 - 7 2 T h i s first p o ssibility can therefore be safely rejected. T h e second microstructure proposed can also be rejected on the ground that it seems unlikely that a m a t e r i a l containing « 10 2 0 defect states per e V per c m 3 ( T a b l e II) c o u l d retain a cry s t a l l i n e structure as t h e i r X-ray spectrum indicate. T h e X-ray sp e c t r a of a f i l m w i t h a r o o m temperature c o n d u c t i v i t y of 2 ( f t c m ) - 1 is i n fact very s i m i l a r to that of a film with a r o o m temperature c o n d u c t i v i t y of « 1 0 - 1 3 ( f i c m ) - 1 . T h u s their X-ray s p e c t r u m seem independent of the conductivity. T h e t h i r d microstructure thus seems the most likely. T h e c o n d u c t i v i t y would proceed v i a numerous localised defect states i n the amorphous and non-stoichiometric a l u m i n u m n i t r i d e matrix. T h e a l u minum n i t r i d e crystallites embeded i n the m a t r i x would however give rise to sharp X-ray diffr a c t i o n lines. T h e c o n d u c t i v i t y would therefore be determined by the nature of the m a t r i x itself and would be essentially unaffected by the presence of the a l u m i n u m n i t r i d e crystallites. T h e presence of s t r u c t u r a l defects would introduce a large number of localised states i n the b a n d gap of the m a t e r i a l f o r m i n g the matrix. A s indicated i n T a b l e II, the density of states of these films at the F e r m i level, calculated f r o m the variable-range-hopping model, is of the order of « 1 0 2 0 e V - 1 c m - 3 . T h e presence of such a large number of defect states i n the m a t r i x is suported by the physical appearance of the film3 themselves. T h e y appear completely black, a feature very reminiscent of amorphous materials i n which the b a n d gap is p o p u l a t e d by a large number of defect states. In order to verify this interpretation, transmission and reflection measurements in the v i s i b l e and the ultra-violet have been performed i n an attempt to detect the presence of defect states i n the b a n d gap of the c o n d u c t i n g m a t r i x i n a more direct way. CHAPTER 4: RESULTS AND DISCUSSIONS 84 F i g . 4-21 shows the transmittance of a non-stoichiometric film e x h i b i t i n g variable-range-hopping, as well as the transmittance of a closely stoichiometric alu-m i n u m ni t r i d e film. T h e temperature dependence of the c o n d u c t i v i t y of this latter film has not been examined because of its already very low r o o m temperature c o n d u c t i v i t y (~ 1 0 - 1 3 ( f t c m ) - 1 ) . T h e oscillations observed in the transmittance of the closely stoichiometric film between 350 n m and 800 n m are due to interference between the multiple reflections of the incident rays w i t h i n the film itself. A r a p i d decrease of the transmittance is observed between 300 n m and 200 n m as the energy of the incident photons become comparable to the b a n d gap of stoichiometric a l u m i n u m nitride, which is of the order of 6.2 eV. 5 7 T h e non-stoichiometric film of a l u m i n u m nitri d e on the other hand has a much lower transmittance over the whole s p e c t r a l range studied, and does not exhibit any oscillations. T h i s indicates that this film has a much higher absorption coefficient. F i g . 4-22 shows the absorption coeffi-cient c a l c u l a t e d for the same two films. T h e absorption peak observed at 4.8 e V (260 nm) i n the closely stoichiometric a l u m i n u m n i t r i d e film has been a t t r i b u t e d to transitions between localised states i n the b a n d gap a n d the extended states of the valence or conduction band. 6 7 T h e localised states would be the results of either a slight excess of a l u m i n u m atoms or a lack of nitrogen atoms. T h i s re-sult indicates the difficulty of p r o d u c i n g exactly stoichiometric a l u m i n u m nitride by voltage-controlled sputtering. T h e non-stoichiometric film on the other h a n d has a much higher absorption coefficient over the whole spectr a l range examined. T h i s result suggests that a much larger f r a c t i o n of the b and gap is populated by localised defect states. T h e enhancement of the absorption coefficient is also presumably the result of electronic transitions between the localised states and the valence and/or co n d u c t i o n bands. T h e c o n d u c t i v i t y by variable-range-hopping observed in this sample and i n s i m i l a r samples is a t t r i b u t e d to h o p p i n g of electrons between these localised states. T h e fact that the absorption coefficient is enhanced over the entire CHAPTER 4: RESULTS AND DISCUSSIONS 85 U J u E C O 1.0 0.8 0.6 0.4 _ 0.2 200 300 400 500 600 700 800 WAVELENGTH (nm) F i g u r e 4-21 Triangle: T r a n s m i t t a n c e of a non-stoichiometric f i l m of a l u m i n u m ni t r i d e e x h i b i t i n g variable-range-hopping. T h i s sample has a r o o m temperature c o n d u c t i v i t y of 2 ( f i c m ) - 1 . Square: T r a n s m i t t a n c e of a closely stoichiometric f i l m of a l u m i n u m ni t r i d e . T h e r o o m temperature c o n d u c t i v i t y of this sample has not been measured. It is beleived to be close to that of pure A1N, which is of the order of 1 0 " 1 3 ( f i c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 86 PHOTON ENERGY (ev) F i g u r e 4-22 Triangle: A b s o r p t i o n coefficient of a non-stoichiometric film of alu-m i n u m nitride. Square: A b s o r p t i o n coefficient of a closely stoichiometric film of alu m i n u m nitr i d e . T h e s e absorption coefficients have been calculated f r o m the tra n s m i t t a n c e measurements shown in F i g 4-21. CHAPTER 4: RESULTS AND DISCUSSIONS 87 s p e c t r a l range examined suggests that numerous types of defects are simultaneously present i n the c o n d u c t i n g matrix. T h e most probable are: 1) Excess of a l u m i n u m atoms, either i n t e r s t i t i a l l y or s u b s t i t u t i o n a l ^ . 2) Nitrogen vacancies. 3) B o n d and angle deformations of the a l u m i n u m n i t r i d e lattice. 4.43 T h e r m o e l e c t r i c P o w e r M e a s u r e m e n t s A s described i n section 2.33, conduction mechanisms i n v o l v i n g electronic h o p p i n g between localised states lead to an exponential temperature dependence of the c o n d u c t i v i t y of the form: In the case of variable-range-hopping the value of the parameter s depends on the f o r m of the density of states i n the neighborhood of the F e r m i energy. P o l l a k 7 9 has shown that for densities of states of the form: T h e temperature dependence of the c o n d u c t i v i t y of samples fabricated i n this work which exhibit variable-range-hopping is best described using s = 1/4. T h i s re-sult suggests that the density of states i n the neighborhood of the F e r m i energy is constant (/ = 0). T h e energy width over which the density of states is presum-ably constant can be estimated f r o m the average h o p p i n g energy ( E q . 2.49). Since this energy is an increasing f u n c t i o n of temperature, it should be evaluated at the highest temperature used i n the dc c o n d u c t i v i t y measurements, thus at T = 3 0 0 K. (4.11) N(e) = N 0 e - e f ,/>0, (4.12) that the parameter s is given by the expression: (4.13) CHAPTER 4: RESULTS AND DISCUSSIONS eav(300K) should give a lower b o u n d to the ban d w i d t h over which the density of states is constant. U s i n g E q . 2.49 and t y p i c a l values of a and N(ef) deduced f r o m the c o n d u c t i v i t y measurements (Table II), the average h o p p i n g energy at 300 K is f o u n d to be of the order of 6 0 x l 0 - 3 eV. T h e c o n d u c t i v i t y measurements indicate that the density of states should be constant over ± 3 0 x l 0 - 3 e V aro u n d the F e r m i energy (taken as be i n g zero). It is also important to point out tha t since the low-est temperature used i n the c o n d u c t i v i t y measurements is of the order of 10 K, corresponding to a h o p p i n g energy of 6 x l 0 ~ 3 eV, the f o r m of the density of states w i t h i n ± 3 x l 0 ~ 3 e V around ey is not p r o b e d by these measurements. It is therefore more accurate to conclude that the c o n d u c t i v i t y measurements indicate a constant density of states i n the interval [ ± 3 x l 0 - 3 , ± 3 0 x l 0 ~ 3 ] e V around the F e r m i energy. T h e r m o e l e c t r i c power measurements were undertaken to probe i n an inde-pendent manner the f o r m of the density of states. Kosarev 8 0 and Z v y a g i n 8 1 have obtained an approximate expression for the thermoelectric power when conduc t i o n proceeds by variable-range-hopping. T h e i r result, expressed i n terms of the density of states N(e) and the average hopping energy eav, takes the form: + « o v / 2 f eA"(eU 8 - 1 eT - £ < z v / 2 F o r a constant density of states i n the interval [—eat,/2,+eot,/2] E q . 4.14 yields a thermoelectric power which is identically zero. A s s u m i n g a sm a l l asymetry of the density of states around ey: N{e) « AT(ey) + edN{e) de (4.15) K o s a r e v and Z v y a g i n obtain: CHAPTER 4: RESULTS AND DISCUSSIONS 5 = 1 eav2 1 dN(ef) 3e T N(ef) de U s i n g E q . 2.49 for eav, the temperature dependence of S can be deduced: (4.16) ^ t lr\ I II. / W I §- I (4.17) c ( T \ - nkb (T W/2 1 ^ ( € ) 5(r) - g—(T0T) N M ^ where T0 is given by E q . 2.51 and g is a numerical constant. T h e m o d e l of K o s a r e v and Z v y a g i n predicts i n this case a decreasing thermoelectric power as the temper-ature decreases. S(T) eventually vanishes at T = 0 K. T h e exact temperature dependence of the thermoelectric power depends on the energy dependence of the density of states near ey. Nevertheless it can be shown that the temperature dependence of S(T) is q u a l i t a t i v e l y very different if the density of states i n the neighborhood of ej vanishes or not. U s i n g a density of states s i m i l a r to that of E q . 4.15 with the a d d i t i o n of a ba n d gap of magnitude A centered around ey, S^TJbecomes p r o p o r t i o n a l to: 1 1 / e 3 - A 3 \ Obv i o u s l y for A = 0 the result of Kosarev and Z v y a g i n is recovered. A t temperature for which eav <£L A , then S(T) a T - 1 . F o r e a„ » A , the result of K o s arev and Z v y a g i n is again recovered. T h e c o n d u c t i v i t y measurements between 10 K and 300 K suggest a nearly constant density of states i n the energy interval [ ± 3 x l 0 - 3 , ± 3 0 x l 0 - 3 ] e V around ey. A c c o r d i n g to the model of Kosarev and Zvyagin, the thermoelectric power in the same temperature range should be zero. F i g . 4-23 shows the temperature dependence of the thermoelectric power between 125 K and 275 K. T h e various CHAPTER 4: RESULTS AND DISCUSSIONS 90 0 to " 5 v . - • A • I -10 O > -15 -20I 125 150 175 200 225 250 275 T (KELVIN) F i g u r e 4-23 T e m p e r a t u r e dependence of the thermoelectric power of a fi l m exhibit-i n g variable-range-hopping. T h e r o o m temperature c o n d u c t i v i t y of this sample is 2 (flcm) - 1. CHAPTER 4: RESULTS AND DISCUSSIONS 91 symbols used in this figure indicate measurements obtained using different temper-ature gradients. T h e thermoelectric power is s m a l l but non-zero, i n d i c a t i n g t h a t the density of states is not constant as the c o n d u c t i v i t y measurements would sug-gest, but has a slight asymmetry a r r o u n d e/. O n the other h a n d the thermoelectric power is observed to decrease as the temperature is reduced, suggesting that there is no gap in the density of states. T hese measurements are therefore consistent with the c o n d u c t i v i t y measurements which indicate that the density of states does net vanish at €y. T h e temperature range as well as the accuracy of the d a t a are not sufficient to enable the determination of the temperature dependence of S(T) and thus the det e r m i n a t i o n of the weak energy dependence of N(e) around cy. 4.44 T h e a c c o n d u c t i v i t y F i g . 4-24 shows the temperature dependence of the dc c o n d u c t i v i t y as well as the temperature dependence of the c o n d u c t i v i t y measured at 3 different frequencies between 10 s Hz and 3. 4 x l 0 7 Hz. T h e s e results are p l o t t e d according to E q . 4.4, with s = 1/4 and p = 0.5. De f i n i n g the frequency dependent co n d u c t i v i t y as a(u,T) = o~MEAS — o~DC> where ( J M E A 8 is the measured c o n d u c t i v i t y at a given frequency and temperature, and o~dc is the value of the dc c o n d u c t i v i t y at the same temperature, it is observed that cr(u>, T) increases with frequency as a(u), T) oc u', with s = 0.9±0.05 ( F i g . 4-25), as predicted by Eqs. 2.73 and 2.74. It is also observed ( F i g . 4-26) that w i t h i n the experimental errors, o~(u, T) is temperature independent between 10 K and 300 K, suggesting the presence of strong electron correlation effects ( E q . 2.73). These results are seen as su p p o r t i n g the interpretation that the conduction i n these samples proceeds by variable-range-hopping between localised states at the Fe r m i level: 1) C o n d u c t i o n i n v o l v i n g extended electronic states leads to a frequency CHAPTER 4: RESULTS AND DISCUSSIONS 92 CM \ CM >• z t > i § o o o ^  o 10' 10 10 l 10 r1 10" 10 1<T 3.4X10 HZ 5 X 1 0 HZ 10 HZ V \ \ \ \ D.C. \ \ \ .25 .30 .35 .40 .45 T" 1 / 4(KELVIN)" V 4 F i g u r e 4-24 T e m p e r a t u r e dependence of the dc c o n d u c t i v i t y a nd the ac conduc-t i v i t y measured at three different frequencies. T h e r o o m temperature c o n d u c t i v i t y of this sample is 1 ( Q c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 93 > I— Q o o • o • < 10 _ 10 _ 10 -1 10 10 10 10 FREQUENCY (HZ) 10 F i g u r e 4-25 <T(UJ, T) versus log(o;). T h e value of the slope is 0.9±0.05. T h e r o o m temperature c o n d u c t i v i t y of this sample is 1 ( f i c m ) - 1 . CHAPTER 4: RESULTS AND DISCUSSIONS 94 > ZD O Q • Z 2 O I O Q • • < 30 25 20 0.5 0.4 0.3 0.2 3.4 X 1 0 ? HZ 5.0X10 HZ • • • 0 5 0 100 150 2 0 0 250 3 0 0 T(KELVIN) Figure 4-26 <T(UJ, T) versus T. No temperature dependence is observed in the temperature range examined. The room temperature conductivity of this sample is 1 (ftcm) - 1. CHAPTER 4: RESULTS AND DISCUSSIONS 95 dependence of the c o n d u c t i v t y of the form: where T - 1 is of the order of an o p t i c a l frequency. A t the frequencies used i n this work (below 5 x l 0 7 Hz), a(uj,T) should be essentially frequency independent if extended states were involved i n the c o n d u c t i o n process. 2) A s mentioned i n section 2.38, it is the averaging of E q . 2.66 over distances and energies between the localised states that leads to a frequency dependence of cr(w,T) of the f o r m <r(w,T) oc w', w i t h s smaller t h a n 1. Such a frequency dependence is observed experimentally (Fig. 4-25) and suggests t h a t the c o n d u c t i v i t y proceeds by variable-range-hopping. 3) Co n d u c t i o n mechanisms i n v o l v i n g carriers excited into localised states at the edges of the valence or c o n d u c t i o n bands w i l l also display a frequency dependence of the f o r m CT((JJ,T) OC W*. Nevertheless cr(u,T) w i l l have the same temperature depen-dence as the dc c o n d u c t i v i t y 4,7. <r(u;,T) is f o u n d t o be temperature independent and it indicates that the localised states are located near the F e r m i energy. It is worth n o t i n g t h a t while the temperature independence of a(u},T) also indicates the presence of strong electron correlation effects, such effects are not observed i n the l i m i t of the dc conductivity. S t r o n g electron correlations are expected to y i e l d a temperature dependence of the c o n d u c t i v i t y of the f o r m a^c oc exp — {T0/T)ll2 instead of the usual variable-range-hopping f o r m 7 6 <7<fc oc exp — (T0/T)1/* , which is observed experimentally i n our films. T h i s latter effect c o u l d be u n d e r s t o o d if rav 3> r 9 S> r w. U n f o r t u n a t e l y if the f o r m of rav a n d r w, given by Eqs. 2.48 and 2.70 are correct, then rav w r w i n our films. T h i s point is therefore not well u n d e r s t o o d at the moment. CHAPTER 5: CONCLUSIONS C H A P T E R 5 CONCLUS IONS F i l m s of non-stoichiometric a l u m i n u m nitr i d e have been fabricated by voitage-controlled reactive sputtering.57 B y co n t r o l l i n g the relative a r r i v a l rates of nitrogen and a l u m i n u m atoms on a substrate the m e t h o d allows the deposition of films over a wide range of compositions. T h i s deposition technique has proven to be a very useful t o o l i n the study of electron localisation i n solids. A s the composition of the deposited films is gradually varied f r o m nearly pure a l u m i n u m to nearly stoichiometric a l u m i n u m nitride, the s t r u c t u r a l disorder of the resulting films is smoothly varied. T h i s technique has p e r m i t t e d the observation of three distinct transport regimes. These regimes are: The Boltzmann regime, the regime of moderate disorder, and finally the regime of strong disorder. T h e results obtained i n this work are consistent with the interpretation that an enhancement of the s t r u c t u r a l disorder is accompanied by a change in the nature of the electronic wave functions. T o each of the three transport regimes observed is a t t r i b u t e d a specific type of wave function. In the Boltzmann regime the wave functions are the fa m i l i a r Bloch waves. These wave functions are extended. In the regime of moderate disorder the wave functions are thought to be power-Jaw localised, the envelope of the wave functions decaying as a power-law. F i n a l l y i n the strong disorder limit the wave functions are thought to be exponentially localised. CHAPTER 5: CONCLUSIONS 97 A m o d e l proposed by K a v e h and M o t t i n which the electronic wave func-tions are assumed to be power-law localised has been found to give an accurate description of the regime of moderate disorder. A l l aspects of the temperature dependence of the c o n d u c t i v i t y predicted by the model has been observed exper-imentally. T hese results indicate that the Debye temperature of these samples is subst a n t i a l l y lower than that of bulk aluminum. T h e lowering of the Debye tem-perature is a t t r i b u t e d to a softening of the lattice due to the granular structure (Al-AIN) of the material. A simple extention of the m o d e l has shown that it can account for the ob-served behavior of the thermoelectric power and H a l l effect. M o r e specifically it is found that the thermoelectric power and H a l l effect seem to remain free-electron like even though the wave functions are no longer thought to be extended. These measurements support the assumption made by K a v e h and M o t t that the density of states remains free-electron like i n the regime of moderate disorder. 1 9 T h e increase of the c o n d u c t i v i t y with frequency has been a t t r i b u t e d to the granular structure of the films rather than to localisation effects. It is thought that the presence of a l u m i n u m n i t r i d e grains dispersed i n the a l u m i n u m m a t r i x is instead responsible for the enhancement of the c o n d u c t i v i t y with frequency. A regime of strong disorder i n which the electronic wave functions are thought to be exponentially localised has been observed. In this regime the c o n d u c t i v i t y is found to proceed by variable-range-hopping. Precise measurements of the temper-ature dependence of the c o n d u c t i v i t y has shown that the relation: best described the experimental results. A c c o r d i n g to the variable-range-hopping model, this f o r m of the exponential temperature dependence (s = 1/4) should be a consequence of a constant density of states i n the neighborhood of the F e r m i energy. (5.1) CHAPTER 5: CONCLUSIONS 98 T h e f o r m of the density of states was thus p r o b e d by measuring the thermoelectric power of samples e x h i b i t i n g variable-range-hopping. T h e i r thermoelectric power was f o u n d to decrease as the temperature decreases, and the magnitude of the effect was comparable to what is observed in metals. A c c o r d i n g to the m odel of K o s a r e v and Zvyagin, the observed temperature dependence of the thermoelectric power indicate that the density of states does not vanish at the F e r m i level. T h i s result is therefore i n agreement with the c o n d u c t i v i t y measurements which also indicate that the density of states is finite at ej. O n the other h a n d the fact the thermoelectric power of these samples is non-zero indicate that the density of states is not constant but has a s m a l l energy dependence around ej. T h e temperature dependence of the prefactor preceeding the exponential t e r m in the expression of the c o n d u c t i v i t y suggests that the average hopping distance rav may constitute a better length scale of the c o n d u c t i v i t y t h a n the correlation radius L0. T h e l a t t e r should y i e l d a temperature dependence of the prefactor p r o p o r t i o n a l to 7 1 - 1/ 4 instead of the T - 1 / 2 which is observed experimentally i n these samples. It is thought that the length scale LQ overestimates the volume (L0Z) required to o b t a i n a representative sampling of the whole resistor network. These results indicate that the c o n d u c t i v i t y of a cube of volume (r a r 3) is already equal to the macroscopic conductivity. T h e ac c o n d u c t i v i t y is found to increase with frequency as <7(UJ,T) oc w* with s = 0.9 ± .05. T h i s result is seen as s u p p o r t i n g the interpretation that the c o n d u c t i v i t y proceeds by variable-range-hopping between localised states at the F e r m i level. No temperature dependence is observed in <r(w,T) between 10 K and 300 K in d i c a t i n g that electron correlation effects play a important role i n the ac conductivity. O n the other hand the dc c o n d u c t i v i t y is f o u n d to obey a f o r m of variable-range-hopping i n which correlation effects have been neglected. It is not clear at the present why correlation effects play a more import a n t role i n the ac c o n d u c t i v i t y t h a n i n its dc counterpart. CHAPTER 6: APPENDIX C H A P T E R 6 A P P E N D I X 6.1 Evaluation of the magnitude of c,n t. A t very low temperature E q . 2.20 can be approximated by: 1 e 2 1 here Z,, = ^ a Q d ' t = bT~2. 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