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 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 = 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\ < *« 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: 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 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: 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 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 " 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 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 I— Q o o • o • < 10 _ 10 _ 10 -1 10 10 10 10 FREQUENCY (HZ) 10 F i g u r e 4-25 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 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 a n d thus a temperature at which the con d u c t i v i t y is d o m i n a t e d by interact i o n effects. S e t t i n g Oint/ai = 1 this temperature, Tc, can be estimated. F r o m E q . A-3 we have: CHAPTER 6: APPENDIX 100 T y p i c a l values for le and b can be obtained f r o m Table-I. T h e value of D is estimated f r o m the equation D = a/e2N(€f), and the density of states at e/ is evaluated f r o m the free-electron f o r m u l a N(ej) = T h e value for n is obtained f r o m H a l l measurements. W i t h le ~ 5 x l 0 ~ 1 0 m, b ~ 5 x l 0 ~ 4 mK 2, a ~ 3 x l 0 5 ( f l m ) - 1 and n = 3 x l 0 2 8 m - 3 , a value of Tc ~ 12 K is obtained. 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