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Tunneling resistance of a one dimensional random lattice Carvalho, Isabel Cristina Dos Santos 1984

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TUNNELING RESISTANCE OF A ONE DIMENSIONAL RANDOM LATTICE ISABEL CRISTINA DOS SANTOS^CARVALHO .Sc., Universidade Federal Do Rio De Janeiro, B r a s i l , 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT . OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ®Isabel C r i s t i n a dos Santos Carvalho, 1984 • By MASTER OF SCIENCE i n September, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of ? \\ V S i c %  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date (DcJLsJU^ 13 ABSTRACT The r e s i s t i v i t y of a one-dimensional l a t t i c e consisting of randomly d i s t r i b u t e d conductivity and i n s u l a t i n g s i t e s i s considered. Tunneling resistance of the form R D i e^* i s assumed for a c l u s t e r of i adjacent i n s u l a t i n g s i t e s . Three d i f f e r e n t ensembles are considered and compared. In the f i r s t ensemble the number of i n s u l a t i n g "atoms" i s fixed and d i s t r i b u t e d i n a l i n e a r chain; i n the second one there exists a fixed p r o b a b i l i t y p of having an in s u l a t o r "atom" occupying a s i t e i n a l i n e a r chain, and f i n a l l y the third one consists of a l i n e bent into a c i r c l e and the p r o b a b i l i t y p i s considered. I t i s observed that i n the thermodynamic l i m i t , the average ensemble resistance per s i t e diverges at the c r i t i c a l f i l l i n g f r a c t i o n p c = e~^, while the variance of the resistance 2 diverges at the lower f i l l i n g f r a c t i o n p = p . Computer simulations of c i c large but f i n i t e systems, however, ex h i b i t a much weaker divergence of the resistance per s i t e at p c and no divergence of the variance at P C l « - i i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES . v DEDICATION v i i ACKNOWLEDGEMENTS v i i i CHAPTER ONE: INTRODUCTION 1 1.1 Motivation for the Work 1 1.2 Description of the Model 2 CHAPTER TWO: STATISTICS OF THE MODEL 6 2.1 Fixed Number of INSULATOR "ATOMS" Dist r i b u t e d i n the Chain 6 2.2 P r o b a b i l i t y p of having an Insulator Occupying a S i t e 10 2.3 Ensemble with Fixed P r o b a b i l i t y p and Line Bent in t o C i r c l e 17 2.4 Comparison Between Ensembles 23 CHAPTER THREE: COMPUTER SIMULATIONS 28 CHAPTER FOUR: CONCLUSION 35 REFERENCES 37 APPENDIX A: Ca l c u l a t i o n of < K± > for Fixed Number of Insulator "Atoms" Distributed i n the Chain...................... 38 APPENDIX B: Calcu l a t i o n of < K± Kj > for Fixed Number of Insulator "Atoms" Distrbuted i n the Chain 41 - i v -Page 2 APPENDIX C: Calcu l a t i o n of < R > for Fixed Number of Insulator "Atoms" Distributed i n the Chain 46 APPENDIX D: Calcu l a t i o n of < K ± > for Fixed P r o b a b i l i t y p of having an Insulator Occupying a Site i n a Line. 48 APPENDIX E: Ca l c u l a t i o n of < K ± K, > for Fixed P r o b a b i l i t y p of having an Insulator Occupying a Site i n a Line 50 APPENDIX F: Ca l c u l a t i o n of < R 2 > for Fixed P r o b a b i l i t y p of having an insula t o r occupying a s i t e i n a l i n e . . 55 APPENDIX G: Ca l c u l a t i o n of < K ± > for Fixed P r o b a b i l i t y p of having an Insulator Occupying a Site i n a Line Bent into a C i r c l e 58 APPENDIX H: Ca l c u l a t i o n of < YL± > for Fixed P r o b a b i l i t y p of having an Insulator Occupying a Site i n a Line Bent i n t o a C i r c l e 60 APPENDIX I: C a l c u l a t i o n of < R2 > for Fixed P r o b a b i l i t y p of having an Insulator Occupying a S i t e i n a Line Bent into a C i r c l e 65 - v -LIST OF FIGURES Figure Page 1 Diagram of a trapezoidal b a r r i e r i n an i n s u l a t i n g f i l m between two si m i l a r electrodes 3 2 Diagram of a l i n e a r chain of metal-insulator-metal junctions 4 3 a Dependence of the ensemble average resistance per s i t e on the f r a c t i o n nj/N for N = 100. Ensembles with fixed n^ and fixed p r o b a b i l i t y p i n a l i n e 8 b Dependence of the ensemble average resistance per s i t e on the f r a c t i o n n^N for N = 200. Ensembles with fixed ^ and fixed p r o b a b i l i t y p i n a l i n e 8 4 Dependence of the variance « R 2 > - < R > 2)/N 2R 2 on the f r a c t i o n n^/N. Ensemble with fixed n^ 11 2 2 2 2 5 Dependence of the variance (<R > - < R > ) / N R on the p r o b a b i l i t y p. Ensemble with fixed p r o b a b i l i t y p i n a l i n e 18 6 Dependence of the variance (< R 2 > - < R > 2)/N 2R 2 on the array s i z e . Ensemble with fixed p r o b a b i l i t y p i n a l i n e 19 7 Dependence of the ensemble average resistance per s i t e on the p r o b a b i l i t y p. Ensemble fixed p r o b a b i l i t y p and l i n e bent into a c i r c l e 21 8 Dependence of the variance « R 2 > - < R > 2)/N 2R 2 on the p r o b a b i l i t y p. Ensemble fixed p r o b a b i l i t y p and l i n e bent into a c i r c l e 24 - v i -Dependence of value of < RI >/NRQ, the contribution to < R > from a l l islands of size I, on the isl a n d s i z e . Ensembles fixed n\ and fi x e d p r o b a b i l i t y p i n a l i n e (a) P = 0.45, ni = 45 N = 100 (b) P = 0.48, ni = 48 N = 100 (c) P = 0.50, n l = 50 N = 100 (d) P = 0.52, n l = 52 N = 100 Sample mean resistance per s i t e for M runs versus the p r o b a b i l i t y p. Ensemble with fixed p r o b a b i l i t y p i n a l i n e Computer simulation geometric mean over M runs versus £n(N). Ensemble with fixed p r o b a b i l i t y p i n a l i n e . . . Computer simulation cumulative average resistance per s i t e versus the number of runs. Ensemble with f i x e d p r o b a b i l i t y p i n a l i n e Histogram of the logarithm of the sample mean resistance per s i t e for M runs. Ensemble with f i x e d p r o b a b i l i t y p i n a l i n e - v i i -DEDICATION To My Son - v i i i -ACKNOWLEDGEMENTS I would l i k e to thank my supervisor, Professor R. Barrie, for h i s support and guidance throughout this work. I would also l i k e to thank Professor B. Bergersen and Dr. P. Palffy-Muhoray for their help with the computer. F i n a l l y , I wish to thank the funding through CAPES, B r a s i l , and NSERC. - 1 -CHAPTER ONE INTRODUCTION 1.1 Motivation f o r the Work The properties of composite materials of metals and insulators have been the subject of many studies (ABE75). These materials have three d i f f e r e n t e l e c t r o n i c conduction regimes: m e t a l l i c regime, d i e l e c t r i c regime and t r a n s i t i o n regime. The o r i g i n a l motivation f o r . t h i s work was a study of the mechanism of conduction i n these granular metals i n the d i e l e c t r i c regime. In this regime metal p a r t i c l e s are dispersed i n a d i e l e c t r i c continuum. Abeles (ABE75), i n his work, calculates the lo w - f i e l d conductivity, o^, i n the d i e l e c t r i c regime of granular metals taking into consideration the tunneling through an i n s u l a t o r , from one i s o l a t e d m e t a l l i c grain to the next one. He takes into account the charging energy, E C, that i s necessary to remove an electron from one neutral metal grain and place i t on another neutral metal grain. Therefore, the expression for the t o t a l l o w - f i e l d conductivity, OL» i s the sum of products of mobility, charge and number density of charge c a r r i e r s over a l l possible percolation paths: where B(s) i s the density of percolation paths associated with the value s and s i s the grain separation. The exponent e""^:s i s the tunneling p r o b a b i l i t y wi th x = (2mcT/'n2) m the electron mass, <|> the bar r i e r - 2 --E°/2KT height and h Planck's constant. The exponent e i s the Boltzmann f a c t o r , which i s proportional to the number density of a l l charge c a r r i e r s with charging energy E°. Aft e r making some approximations, Abeles evaluates the expression and obtains: - 2 ( C / K T ) 1 / 2 oT = a e L o where O q i s a constant independent of temperature and C = 3(sE°. In order to i s o l a t e the role of tunneling i n this work, one can look at zero temperature and lo w - f i e l d conductivity, removing the idea of a charging energy f a c t o r . Thus one can consider the conduction to be through a one dimensional series of metal-insulator-metal junctions. 1.2 Description of the Model The tunneling e f f e c t i n junctions of metal-insulator-metal has been widely discussed (DUK69). In the present work, the regime of low voltage and the tunneling between two metals of the same kind at zero temperature w i l l be considered. Since our i n t e r e s t i s i n the low voltage ohmic regime, the p o t e n t i a l b a r r i e r w i l l be considered as rectangular. The expression for the tunneling current, for the above case, was obtained by Sommerfeld and Bethe (SOM33) and i s reproduced i n the paper by Holm (H0L51). Figure (1) shows the diagram of a trapezoidal b a r r i e r i n an i n s u l a t i n g f i l m between two s i m i l a r metal electrodes. In the ohmic regime the b a r r i e r can be considered as a rectangular one. - 3 -WORK FUNCTION FERMI ENERGY \ \ TUNNEL CURRENT OCCUPIEO LEVEL APPLIED VOLTAGE As-* METAL METAL INSULATOR Figure 1: Diagram of a trapezoidal b a r r i e r i n an i n s u l a t i n g f i l m between two s i m i l a r electrodes. The equation for the tunneling resistance per unit cross section area where h = Planck's constant q = electron charge As = thickness of the junction m = electron mass cj> = work function For our purposes we w i l l write the b a r r i e r thickness as As = i£ Q» where £ Q i s the unit of "atomic" spacing and i corresponds to the number, of s i t e s with unit length £ Q . The resistance R of a single i n s u l a t i n g c l u s t e r of i s i t e s due to tunneling w i l l then be of the form: i s (H0L51): R = ( h 2 As/q 2(2m4>) 1 / 2) exp [ (4T r As/h) (2m(j>) 1 / 2]; (1) R = R i e 1 o (2) where R q = h 2 Hja} (Imfy)1^ and b = (4 TT £ Q/h) (2m<)))1/'2. To obtain equation ( 1 ) , the tunneling p r o b a b i l i t y for a rectangular b a r r i e r was calculated and this was done by the evaluation of the transmission c o e f f i c i e n t D ( E X ) using the WKB approximation. As a model of the one dimensional granular metal, consider a l i n e a r chain of length N £ Q of metal and i n s u l a t o r "atoms" d i s t r i b u t e d randomly. Each s i t e of length iQ can be occupied by a metal or an i n s u l a t o r "atom".A clu s t e r of i consecutive i n s u l a t i n g u n i t s , each of unit length lQy w i l l form an i s l a n d of size i , Figure (2) i = 2 — M I I M M I M o N£ 0 Figure 2: Diagram of a l i n e a r chain of metal-insulator-metal junctions. In the future we w i l l talk about an array' of N s i t e s . The resistances of the metal islands are going to be considered equal to zero and the resistance of an i n s u l a t i n g i s l a n d w i l l be given by equation (2). In other words, the l i n e a r array w i l l be composed of a series of metal-insulator-metal junctions. A t y p i c a l junction of metal-insulator-metal, 1/2 with unit length (£ ) equal to 5A, would have b = 5.125<|> . Considering, - 5 -f o r instance, the case of SiU2(ABE75), b would be of the order of 10. In order to i l l u s t r a t e the study of the s t a t i s t i c s of the problem, the value of b w i l l be taken to be Zn2. This choice w i l l become clear l a t e r (See P. 9). - 6 -CHAPTER TWO STATISTICS OF THE MODEL The present model w i l l be concerned with the ohmic regime and zero temperature. Consider a l i n e a r chain of s i t e s that i s randomly occupied, either by an i n s u l a t o r "atom" or a metal "atom". The end two s i t e s of the chain are f i l l e d with metal. Each arrangement of metal-insulator-raetal w i l l form a ju n c t i o n . We consider the l i n e a r chain to be formed of a set of independent junctions. Take n^ as the number of d i e l e c t r i c atoms, (n2 - 2) the number of metal atoms such that N = n^ + n2 i s the number of s i t e s i n the l i n e a r chain. The c a l c u l a t i o n of the resistance and variance for d i f f e r e n t ensembles, i n this model, i s described below. 2.1 Fixed Number of Insulator "Atoms" Distributed In the Chain In the case of a fixed number of insulator "atoms" d i s t r i b u t e d i n a chain with metal ends, the t o t a l ensemble average of the resistance w i l l be given by: n l r b i < R > = R l i e < K, >, o . , i ' i=l where < > i s the average number of d i e l e c t r i c c l u s t e r s of s i z e i ( i = 1, 2 , . . . , n j ) . - 7 -In analogy with the d i s t r i b u t i o n of runs (WIL62), the p r o b a b i l i t y of having d i e l e c t r i c clusters of size i ( i = 1, 2, n^) i s : n 2 - 1 K! \ K PCK.f^}) - K I v ! .. .K ! /N - 2 1 2 n. 1 1 n x n l n l where fc. } = {K, , K~ , ... K }, I K. = K, £ i K, = n, and K i s the 1 i=l i=l 1 1 t o t a l number of metal-insulator-metal junctions. The value of K ranges from 0 to ni and the value of n\ from 0 to N-2. The expression for < > can then be evaluated < K^> = Jo (A ztn T^W K* I t i s shown i n Appendix A that: t n l The t o t a l resistance w i l l then be given by: < R > = R Q i e TN-TTy 1^  n x - i ' ( 5 ) We have not been able to express this sum i n any s i g n i f i c a n t l y simpler fashion. Figures (3a) and (3b) show the graph of the resistance per s i t e versus the f r a c t i o n n^/N for values of N = 100 and N = 200. - 8 -Figure 3; Dependence of the ensemble average resistance per site on the fraction n^/N. Ensembles with fixed n^  and fixed probability p in a l ine , (a) N = 100 (b) N = 200 - 9 -In the l i m i t of N, n\ and n2 tending to i n f i n i t y , such that n l n2 — = p and -JJ— = (1 - p), the expression for the resistance per s i t e i s : T T * « ' - » 2 7 - ^ 2 « > o (1 - pe ) • Therefore, the average resistance per s i t e diverges for an i n f i n i t e chain, when p = P c = e ^. The value of b i s then chosen to be equal to Zri2 such that the f i l l i n g factor p c = 1/2 and i s at the centre of the i n t e r v a l from p = 0 to p = 1. In order to determine < >, the moment < Kj > has to be calculated by using the following expression: v2"1 < K. K. > = I I — /.T K ' K. K., (7) with I K± = K and I K± i = ^ . i i From Appendix B, the expression for < Kj > i s : (n - D ( n 2 - 2) / 4 , < K. K. > = - 2 j ( 4 - 4 J + 6 i j < K i > ' <*) i t i s to be noted that the f i r s t term i s zero unless i + j < n^ This leads to: < R2 > = R2 11 i j e b ( i + J } < K K. >. (9) o . . J i j The average of the square of the resistance i s calculated i n - 10 -Appendix C to be: < R2 > - R2 r ( n 2 " 1 ) ( n 2 " 2 ) y 1 (4 m3 - m) bm /N - m - 4\ < R > _ R ° 1 ' N n " 2 j Jo 1 2 6 V n 2 " 4 ' x ( n 2 " 1 } V 1 ,2 2bi /N - i - 3\ i Again, we have not been able to express these sums i n any simpler form. In the l i m i t of N, n^, and n 2 tending to i n f i n i t y , such that n^/N = p and n 2/N = (1 - p), the expression for the variance i s < r Z > - < R > 2 • m - P) 2 p e 2 " a + ffi R (1 - pe ) Therefore, the variance diverges for an i n f i n i t e chain when -2b p = p = e 1 In order to evaluate the variance, the < R > and < R^  > are calculated numerically. The behavior of the variance with the r a t i o n^/N i s shown i n Figure (4). 2.2 Probability p of Having an Insulator Occupying a Site In this ensemble, there i s a fi x e d p r o b a b i l i t y p of having an i n t e r i o r s i t e occupied by an in s u l a t o r , and p r o b a b i l i t y (1-p) for such a s i t e to be occupied by a metal. The two end exterior s i t e s are occupied by metal. The p r o b a b i l i t y (M0041) of having xx\ d i e l e c t r i c "atoms" d i s t r i b u t e d i n Ki clu s t e r s of size i ( i = 1, 2,...,n]_), i s : r - 11 -- 12 -p < v K ; { K I } ) = P n i (i - p ) v 2 ^ K i , ^ ^ , y n2 " *\ (11) n 1 n x w i t n I K. - K, I i K. = n, and n, taking values from zero to (N - 2), i=l i=l 1 The evaluation of < > is performed on Appendix D, and i t gives: < K ± > = Y I 1 1 , P(n ; K; {K }) K p" 1 (1 - p)" 2 2 (12) n 1 =0 K=l {K 1} 1 1 1 with I K = K and I K, i = n,. i i 1 1 Therefore, j P ^ l " P){(1 " p) (N - i - 3) + 2}, i < N - 2 ^ > = I i = N - 2 The total resistance is then evaluated N-3 , < R > = R Q { I i p x ( l - p) {2 + (1 - p)(N - 3 - i) } e b l i=l + (N - 2) (pe b) N" 2} (14) After performing the sum over the island sizes: < R > / R Q = (1 - p) [2 +(1 - p)(N - 3)] [1 -(N - 2)(pe b) N" 3 + (N - 3) (pe b) N"" 2] x p e b b 2 - (1 - P ) 2 [l + p e b - ( N - 2) 2 ( p e b ) N ~ 3 + (2N 2 - ION + 11) (1 - pe ) x ( p eb)N-2 _ ( N _ 3 ) 2 (^bjN-lj Pe b + ( N _ 2 ) ( p eb ) N-2 ( 1 5 ) (1 - p e V - 13 -The graph of the ensemble average resistance versus the f i l l i n g f a c tor p i s plotted i n Figures (3a) and (3b). In the l i m i t of N •*• °°, the geometric series converges only for P < Pc = e _ b * At the p r o b a b i l i t y p c = e - b , the expression for the average resistance i s : N-3 N-3 9 9 < R > = R { I (1 - p)[2 + (1 - p)(N - 3)] i - I il (1 - P r + (N - 2)} C i=l i=l (16) Therefore, > < R C > = R q { ( 1 - p)[2 + (1 - p)(N - 3)] (N - 3)(N " 2) ( 1 - p ) 2 ( N - 3 ) ( N - 2 ) ( 2 N - 5) + ( N _ 2 ) } 6 At the thermodynamic l i m i t , of N + » and at the c r i t i c a l p r o b a b i l i t y p c , the average resistance per s i t e i s : < R c > 1 2 2 The c a l c u l a t i o n of the moment < Kj >, i s performed i n Appendix E, by performing the sums: N-2 n l /n - 1\ n „ -2 K ' 1 2 I K ± K. p \ l - p) 2 K l K i > = ^ £ , * K J n 1 =0 K=l {K } K ! K2J....K ! \ with I K, = K and I K. i = n,. (19) i i - 14 -Therefore, 1 < KA K. > = { (N - i - j - 4) (N - i - j - 5) (1 - p ) 2 + 6(N - i - j - 4) (1 - p) + 6} p i + j (1 - p ) 2 .8(1 + j , N - 3) + 2 ( l - p ) pN"3 6 ( i + j > N _ 3 ) + ^ [ ( N - 1 - 3) (1 - p) + 2] (1 - p) p 1 6(1, N - 2) + P N" 2 6 l j 6i,N-2 <20> | 1 i f m < n where 8(m, n) = I (_0 otherwise Using the above expression, the average of the square of the resistance can then be evaluated. This is done in Appendix F, and gives: ,2 N _ D2 rv V b ( i + j ) i j Therefore, 2 2 < R2 > - ( 1 1 2 P ) [(4A - C) S 3 + BS2 - 4BS4 + 4CS5 - AS^ ^o + I ( p e b ) N _ 3 (1 - p ) (N - 2) (N - 3) (N - 4) + (1- p ) {[2 + (N - 3) (1 - p ) ] T 2 - (1 - p ) T3} + (N - 2) 2 ( p e 2 b ) N ~ 2 (21) w h e r e B = 6(1 - p ) + (2N - 9) (1 - p ) 2 H l o o d a p p e a r s t o h a v e a n e r r o r i n < Kj >. - 15 -A = 6 + 6(N - 4) (1 - p) + (N - 4) (N - 5) (1 - p)' ,2 s v = I fflk ( p e b ) m ; C - (1 - p) N-4  m=0 T, = . I i k ( p e 2 1 3 ) 1  k i=0 S l " pe {1 - (N - 3) (p e b ) N ~ 4 + ( N - 4 ) (peV"""} (1 - P e b ) 2 bxN-3 (22) S2 = {pe b - ( p e b ) N - 3 (N - 3 ) 2 + ( p e V " 2 [2(N - 4) / , K 3 (1 - pe ) + 2(N - 4) - 1] - ( p e b ) N - 1 (N - 4 ) z + ( p e V } K 2 (23) S3 " pe {l - ( p e b ) N 4(N.- 3 ) J + ( p e 0 ) " J [3(N - 4 ) J + 6(N - 4 ) Z - 4] (1 - pe ) KN-3 + ( p e b ) N 2 [ -3(N - 4 ) 3 - 3(N - 4 ) 2 + 3(N - 4) - 1 ] + ( p e b ) N _ 1 (N - 4 ) J + 4(peD) + ( p e V } bx2- (24) S4 " pe 4 , KN-4 (1 - pe ) {l - (N - 3 T (peD) + (pe b ) N _ 3 [ 4(N - 4 ) 4 + 12(N - 4 ) 3 + 6(N - 4 ) 2 - 12(N - 4) - 11] - (pe b ) N _ 2 [ 6(N - 4 ) 4 + 12(N - 4 ) 3 - 6(N - 4 ) 2 - 12(N - 4) + 11] + ( p e b ) N _ 1 [4(N - 4 ) 4 + 4(N - 4 ) 3 - 6(N - 4 ) 2 + 4(N - 4) - 1] - ( p e b ) N (N - 4) H + l l ( p e D ) + l l ( p e V + ( p e V } bN2 bx3 (25) - 16 -(1 - pe ) + (pe b) N - 3[5(N - 4) 5+ 20(N - 4) 4+ 20(N - 4 ) 3 - 50(N - 4) -26 -20(N -4) 2] + (pe b) N~ 2[10(N - 4 ) 5 + 30(N - 4 ) 4 - 60(N - 4 ) 2 + 66] + (pe b) N - 1[10(N - 4 ) 5 - 20(N - 4) 3+ 20(N - 4 ) 4 - 20(N - 4) 2+ 50(N - 4)-26] + (pe b) N[5(N - 4 ) 5 + 5(N - 4 ) 4 - 10(N - 4) 3+ 10(N - 4 ) 2 - 5(N - 4) + 1] + ( p e b ) N + 1 ( N - 4 ) 5 + 26(peb) + 66(pe b) 2 + 26(pe b) 3 + (pe b) 4 } (26) „ 1 r 2b , 2KN-2 ... o N2 T 2 = 2 b 3 t p e " ( p e } ( N " 2 ) (1 - pe ) + ( p e 2 b ) N _ 1 [ 2 ( N - 3 ) 2 + 2(N - 3) - 1] - ( p e 2 b ) N (N - 3 ) 2 + (pe 2 b) 2} (27) (1 - pe ) + ( p e 2 b ) N ~ 2 [3(N - 3 ) 3 + 6(N - 3 ) 2 - 4] + ( p e 2 b ) N _ 1 [-3(N - 3) 3 - 3(N - 3 ) 2 + 3(N - 3) - 1] + ( p e 2 b ) N (N - 3) 3 + 4 (pe 2 b) + (pe 2 b) 2} (28) In the limit of N •* °°, the geometric series a l l converge only for p < e~2b. Consequently, the variance w i l l have a divergence at the probability p c^ equal to e - ^ . Considering the leading terms in the variance, at the thermodynamic 2 —2b limit and at p c = p c = e , the variance diverges as: < R2 > - < R >2 •*• N4 (1 - p ) 2 IZ Cj^ (29) - 17 -The behavior of the variance with the p r o b a b i l i t y p i s shown i n Figure (5) and the pl o t of the variance versus the array size i s i n Figure(6). 2.3 Ensemble with Fixed Probability p and Line Bent into a Circle This ensemble consists i n taking the l i n e a r array, with fixed p r o b a b i l i t y p, and bending i t into a c i r c l e . The case of n^ = 0 ( n 2 = N) and n^ = N (n^ = 0) w i l l be excluded. A s i m i l a r ensemble i s described by David and Barton (DAV62). Therefore, the p r o b a b i l i t y of having n^ d i e l e c t r i c s d i s t r i b u t e d i n {K.£ } c l u s t e r s of size i i s given by: »(n i ; K , { K . } ) = p" 1 (1 - p ) " 2 | ^ n 2 - IV K , K - 1 / K , ! K ' . . . K ! * / 1 2 n x With the above expression, the moment < KJ; >.can be calculated. This i s done i n Appendix G. The f i n a l expression for < K-^ > i s : < K > = i ' N p 1 (1 - p ) 2 for i N< N ' 2 UP*" 1 ( i -P) F O R I = N ~ U (31) The t o t a l resistance i s then evaluated - N(l - P) 2 Y K p e V + N(N - 1)(1 - p)(pe b ) N 1  R o i=l Therefore: - 18 -- 1 9 -Figure 6 : Dependence of the variance (< R2 > - < R >2)/N2R^ on the array size for three different values of p. Ensemble with fixed probability p in a l ine . - 20 -< j p = N ( 1 _ p )2 Pe b {1 - (N - 1) (peV" 2 + (N - 2) (peV" 1} o (1 - pe ) + (1 - p) N(N - 1) ( p e b ) N - 1 . (32) Again, as N •*• 0 0 the average resistance diverges at a c r i t i c a l probability Pc = e" b. Analysing the average resistance at p c one obtains: N-2 < R > = N(l - p) 2 I i + N(N - 1) (1 - p) R A 1 o i=l Therefore, <-|2- N ( i - p ) 2 <N - 2><N - " + N (N - 1) (1 - P) (33) o Figure (7) shows the graph of the ensemble average resistance versus the probability p. At the thermodynamic limit of large N, the average resistance per site goes as: %|^4N 2 (1 - P) 2. (34) The calculation of the moment < Ki Kj > is done in Appendix H. The expression for < Kj > is then given by: < K± K > = N p i + j (1 - p ) 3 {(1 - p) (N - i - j - 3) + 2} 0(i + j , N - 2) + N p i + J (1 - p) 2 6 ( ± + ^ N _ 2 ) + N p i ( l - p) 2 9(1, N-l) 6 ± J  + N (1 - p) p 1 « ( ± f N _ ^ 6 ± J (35) - 21 -Figure 7 Dependence of the ensemble average resistance per site on the probability p. Ensemble with fixed probability p and line bent into a circ l e (N = 100). - 22 -The average of the square of the resistance can then be evaluated. And the f i n a l expression which i s i n Appendix I i s : 2 < R2 > = f j ( 1 " P ) 3 {4[2 + (N - 3) ( 1 - p)] S3' R0 - [2 + (N - 3) ( 1 - p)] Sj} + ( 1 - p ) 3 { s ^ ( l - p) - 4 ( 1 - p) S^} + N ( p e b ) N " 2 ( 1 - p ) 2 (N - D ( N - 2)(N - 3) 6 + N(l - p ) 2 T 2 + (N - l ) 2 ( p e 2 b ) N - 1 N(l - p) (36) N-3 where = I m" (pe ) m ; / m=l Tl = I i 2 ( P e 2 b ) X i=l The value of each sum being equal to: S1 " ^ h ? t l - ^ - 2 ) ( P e b ) N - 3 + (N - 3 ) ( P e b ) N " 2 } 1 ( 1 - P e b r S* = ^ b 4 { 1 - ( P e b ) N " 3 (N - 2 ) 3 + ( 1 - pe ) + ( p e b ) N ~ 2 [3 (N - 3 ) 3 + 6(N - 3 ) 2 - 4] + ( p e b ) N _ 1 [-3 (N - 3 ) 3 - 3(N - 3 ) 2 + 3(N - 3) - 1 ] + ( p e b ) N (N - 3 ) 3 + 4pe b + ( p e b ) 2 } - 23 -S4' = P e t ? b 5 {1 - (N - 2 ) 4 ( p e b ) N - 3 (1 - pe ) + ( p e b ) N ~ 2 [4(N - 3 ) 4 + 12(N - 3 ) 3 + 6(N - 3 ) 2 - 12(N - 3) - 11] - ( p e b ) N _ 1 [6(N - 3 ) 4 + 12(N - 3 ) 3 - 6(N - 3 ) 2 - 12(N - 3) + 11] + ( p e b ) N [4(N - 3 ) 4 + 4(N - 3 ) 3 - 6(N - 3 ) 2 + 4(N - 3) - 1] - ( p e b ) N + 1 (N - 3 ) 4 + l l ( p e b ) 2 + l l ( p e b ) + ( p e b ) 3 } „, 1 r 2b , 2b NN-l ,„ n 2 x T 2 = ~ 2bTT t p e " ( p e > (N - 1) + (1 - pe ) + ( p e 2 b ) N [2(N - 2 ) 2 + 2(N - 2) - 1] - ( p e 2 b ) N + 1 (N - 2 ) 2 + ( p e 2 b ) 2 } Once more, as N + » the variance w i l l diverge at the p r o b a b i l i t y ~~2 b 21} p = e . The variance at p = e and at the l i m i t of large N, w i l l C l C l tend to the l i m i t i n g value: < R 2 > - < R > 2 1 ( .2 ..2 2~1 ' V 3 ( 1 " P c ) N * N R 1 Using the expressions for < R^  > and < R >, the variance was calculated numerically. The r e s u l t of this c a l c u l a t i o n i s shown i n Figure ( 8 ) . 2.4 Comparison Between Ensembles Figures (3a) and (3b) shows the t o t a l resistance per s i t e plotted against the p r o b a b i l i t y p for the approaches i n sections 2.1 and 2.2. The two ensembles are related by the condition that (n^/N) = p. For small p r o b a b i l i t y p, the two ensembles agree. / - 24 -10", {0 - f 1 1 r 1 1— 0.2O 0.22 0.24 0.26 0.28 0.30 PROBABILITY P Figure 8 : Dependence of the variance (< R2 > - < R > 2)/N 2R 2, on the probability p. Ensemble with fixed probability p and line bent into a c i rc le (N = 100). - 25 -The numerical value of the contribution to the resistance per s i t e due to each is l a n d size was calculated for the two approaches. The r e s u l t of i t i s shown i n Figures (9a), (9b) (9c), (9d), for an array of size N = 100. D i f f e r e n t values of p were considered. For f i n i t e N and large p the two approaches disagree. This discrepancy arises because the ensemble with fixed p r o b a b i l i t y p allows d i e l e c t r i c c l usters larger than the corresponding n^, of the ensemble with fixed n ^ The contribution of these large islands i s dominant i n the t o t a l resistance. Comparing the variances of approaches i n Sections 2.1 and 2.2, i t was observed that the variance for the ensemble with fixed p r o b a b i l i t y p has a higher value than the one with f i x e d n\. Following we quote the numerical values of the variance for the two ensembles for N = 100. Variance Ensemble f i x p Ensemble f i x n 1 nj_/N = 0.27 1.1 x 10 5 4.5 nj/N = 0.25 4.6 x 10 2 2.3 n x/N = 0.24 5.9 x 10 1.7 This behaviour i s again a r e s u l t of the contribution of the large i s l a n d s . The exact way that the resistance diverges, for the ensemble with fixed n i , was not obtained. The reason i s that i t was not possible to perform the sum over the i s l a n d sizes i n the expression for the resistance. Therefore, the thermodynamic l i m i t was taken before the sum was done. The ensembles of approaches 2.2 and 2.3 allow the comparison of two ensembles with f i x e d p but d i f f e r e n t boundary conditions. Comparing equations (13) and (31), i t can be seen that the way of putting an i s l a n d - 26 -• » «o *c te ae c n .0 H ISLAND SI7C B L A N D SJ7C ( C ) o . ae z 40 t0 B L A N D size o ac z .e u B L A N D S17C Figure 9: Dependence of the value of (< RI >/NRQ) on the island s i z e . < RI > i s the contribution to < R > from a l l the islands of size I. Ensemble with fixed n i (dashed l i n e ) and fixed p r o b a b i l i t y p ( s o l i d l i n e ) i n a (a) p = 0.45, nj (b) P = 0.48, nj (c) p = 0.50, nj (d) p = 0.52, rxi l i n e (N = 100). = 45 N = 100 = 48 N = 100 = 50 N = 100 = 52 N = 100 - 27 -of s i z e i i n a l i n e i s d i f f e r e n t from the one i n a c i r c l e . Equations (13) and (31) d i f f e r because the embedding of an island of size i i n a l i n e of length N i s ( N - i + 1 ) / N whereas i n a c i r c l e i t i s 1 (embedding of an is l a n d of size i i s the number of ways i t can be put i n divided by the t o t a l number of s i t e s ) . In the l i m i t of N * 0 0 these two embeddings are both unity. However, i n order to obtain the average value of the r e s i s t -ance, a summation over i has s t i l l to be performed. If this summation i s done before the l i m i t N •*• 0 0 i s taken, then these two ensembles produce d i f f e r e n t r e s u l t s . In p a r t i c u l a r , we have noted that for large N the l i n e problem gave the r e s u l t (18) « R > / N R « (1/6)(1 - p ) 2 N 2 ) whereas o c the c i r c l e problem gave the r e s u l t (34) ( < R > / N R q •> (1/2)(1 - p £ ) 2 N 2 ) . A s i m i l a r difference occurs for the variance. I t i s to be remembered that equation (13) i s for f i n i t e i . Thus by considering a l l three ensembles we have demonstrated, for this p a r t i c u l a r problem, the differences that occur for fixed n^ and fixed p and for d i f f e r e n t boundary conditions. - 28 -CHAPTER THREE COMPUTER SIMULATIONS I t transpires that for the computer simulations i t makes l i t t l e d i f f e r e n c e which boundary conditions, metal ends or c y c l i c boundary conditions, are used nor whether the randomness i s generated according to the ensemble of paragraph 2.1 or 2.2 of the previous chapter. We quote here the r e s u l t s for metal ends and f i l l i n g according to a fixed p. Figure (10) shows < R >^/NR0, the sample mean resistance per s i t e for M computer simulations. Computer simulations for d i f f e r e n t N were done; they herald a Jtn 2^ divergence of < R >M at p = p c as N becomes larger (PAL84). The value of < R >M at p c i s orders of magnitude smaller than the corresponding ensemble value obtained by approaches 2.1 or 2.2 of the previous sections. The question then arises as to the nature of the r e l a t i o n between the computer simulation of this chapter and the ensemble theory of the previous one. A previous work (PAL84) describes the problem and what should be the representative ensemble for the computer simulations. A summary of the r e l a t i o n between the computer simulation and the ensemble theory i s described i n the following paragraph. What the computer program does i s generate repeatedly a set {K^ } N v 2 b i and then calculates R = l i e K.. I t then averages this R over M i=l 1 computer runs. I t i s found that for a given M the computer each time produces zero for K i for i larger than some i m a x « In the computer - 29 -60 - i 50 40 A A T. AS v 30 H 20 10 0.1 0.2 0.3 0.4 PROBABILITY P 0.5 0.6 Figure 10: Sample mean resistance per s i t e < R >j^/NR0 for M runs versus the p r o b a b i l i t y p i n a l i n e (M - 1000, N = 1000). For each p the 1000 runs were done 3 times. - 30 -averaging over M runs, islands of size greater than i m a x w i l l not provide t h e i r very large contribution to the average resistance. An acceptable an a l y t i c d e s c r i p t i o n of the computer simulations i s to cut off the sum over i i n the ensemble approach, at some imax* I n this way, the e f f e c t of rare events, islands bigger than i m a x » a r e n o t taken i n t o account. The an a l y t i c description approaches the ensemble one when the number of runs i s approximately equal to M = eN-£nN (pAL84). In systems l i k e t h i s , where rare events dominate the ensemble arithmetic mean, we do not expect the sample mean to agree with the ensemble arithmetic mean; since as we found the sample mean does not pick up the rare events. The question then arises as to how i n the ensemble approach of Chapter 2 we are to predict t h e o r e t i c a l l y what experiment (ei t h e r computer simulation or laboratory measurements) should expect. I t has been suggested that the ensemble geometric mean, since i t i s not so dominated by rare events might be more comparable to the experiment geometric sample mean. Unfortunately, by approach of Chapter 2 we were not able to calculate the ensemble geometric mean. However, as a check on the relat e d suggestion that the computer simulation geometric means w i l l be less divergent than the arithmetic means, we i n fac t obtained also the computer simulation geometric mean. Figure (11) shows the geometric mean over M runs, of the resistance per s i t e against the logarithm of the array s i z e . I t appears that the geometric mean grows at p = p c as j£n(N), which i s a slower growth than the one of the arithmetic mean at p = p c which grows as In2 (NM) (PAL84). By comparing three groups of 1000 runs the geometric mean shows smaller f l u c t u a t i o n s i n comparison with the arithmetic mean. The pl o t of the cumulative average resistance per s i t e against the - 31 -25 - i 20 -15 -i 10 -i A A 6 A A A A A A A —1— 7.5 4.5 5.5 6 6.5 InN 8 5 Figure 11 : Computer simulation geometric mean over M runs versus l n ( N ) . Ensemble with fixed p r o b a b i l i t y p i n a l i n e . p = 0.5 and M = 1000, for each point such that the f l u c t u a t i o n can be observed. - 32 -number of runs for a set of 1000 runs i s shown i n Figure (12). From the p l o t of the cumulative average resistance per s i t e against the number of runs, the behavior of the average value of the resistance per s i t e can be analysed. I t can be seen that the mean of the resistance per s i t e exhibits sudden jumps followed by decreases. These jumps are caused by the appearance of large islands i n a c e r t a i n computer simulation. Then as the number of runs increases the importance i n the average of those large islands decreases. These decreases w i l l continue u n t i l another jump occurs. A s i m i l a r behavior has already been observed by Montroll and Shlesinger (MON83). In discussing income d i s t r i b u t i o n s , Montroll and Shlesinger (MON83) introduce the process of a m p l i f i c a t i o n , a m p l i f i c a t i o n of a m p l i f i c a t i o n , etc. S t a r t i n g with a basic d i s t r i b u t i o n , they can then construct a new d i s t r i b u t i o n which allows for the p o s s i b i l i t y of continuing le v e l s of a m p l i f i c a t i o n . Each amplifier class has the same basic d i s t r i b u t i o n function but the mean value of the quantity measured i s amplified N times with a small p r o b a b i l i t y A. Analogously, i n the cumulative average resistance per s i t e the mean of the resistance per s i t e suffers amplifications or jumps equal to e*5 and p r o b a b i l i t y X = p. This behavior i s also r e f l e c t e d i n the histogram of &n(R/NR0), Figure (13), where for large values of the resistance per s i t e the histogram has spikes at constantly spaced i n t e r v a l s . A histogram was done with a d i f f e r e n t value of b and i t was checked that the distances between the spikes were equal to the new value of b. In other words, each considerable increase i n the resistance i s mainly me^, where m i s an integer. - 33 -p=0.50 N=1000 NUMBER OF RUNS Figure 1 2 Computer simulation cumulative average resistance per s i t e for M runs. Ensemble with fixed probabilty p i n a l i n e . - 34 -HISTOGRAM OF In OF RESISTANCES PER SITE 8 0 n In O F R E S I S T A N C E S PER SITE Figure 13: Histogram of the logarithm of the sample mean resistance per s i t e for M runs. Ensemble with fixed p r o b a b i l i t y p i n a l i n e (M = 1000, N = 1000, p = 0.5). In order to be able to exhibit the whole histogram on one diagram, the horizontal scale i s in fa c t x with x - U n « R >M/NR0) - (-0.5)]/0.1 + 1. - 35 -CHAPTER FOUR CONCLUSION The s t a t i s t i c s of the problem of a random series of metal-insulator-metal junctions was studied considering three d i f f e r e n t ensembles. I t was found that the e f f e c t of large insu l a t o r c l u s t e r s was dominant i n the behavior of the ensemble average resistance and v a r i ance. Therefore, the main difference between the ensembles with fixed p r o b a b i l i t y p and fixed nj_ > d i s t r i b u t e d i n a l i n e a r arrangement, was due to the l i m i t a t i o n on the second ensemble to pick up larger island s i z e s . When comparing two ensembles with fixed p r o b a b i l i t y p, of having an i n s u l a t o r occupying a s i t e , but one i n a l i n e a r arrangement and the other one i n an arrangement of a l i n e bent into a c i r c l e , a discrepancy i s observed i n the way the ensemble average resistance diverges for N -»• 0 0 at p = p c. This f a c t i s due to the d i f f e r e n t embeddings i n the two ensembles. If the l i m i t N * 0 0 i s taken before the summation of i s l a n d contributions for d i f f e r e n t size i s l a n d s , then the two approaches agree. The divergence of the ensemble average resistance and variance at the l i m i t N •*• 0 0 was observed for the three ensembles at p = p c = e - b and p = p c^ = e - 2 ^ , r e s p e c t i v e l y . This behavior i s also t y p i c a l of the s o - c a l l e d G r i f f i t h s s i n g u l a r i t i e s (GRI69). In these, as i n our case, i n the l i m i t N ->• 0 0 the coherence length ( i n our case, the average d i e l e c t r i c i s l a n d size) does not go to i n f i n i t y u n t i l p = 1, i n one-dimensional systems. Before that, there occurs a divergence i n the resistance at p c« - 36 -As a suggestion for future work, the temperature dependence In the tunneling r e s i s t i v i t y i n a randomly-filled chain problem, should be considered. I t may be that at non-zero temperatures the behavior of the divergence i s s i g n i f i c a n t l y modified. Another i n t e r e s t i n g problem i s the extension of the present problem to a two-dimensional one, which would be a better approximation to the simulation of granular metals. In conclusion we would l i k e to point out that this project has made a s i g n i f i c a n t contribution to the study of a one-dimensional random system which displays a s e n s i t i v i t y to s t a t i s t i c a l l y rare events. - 37 -REFERENCES ABE75 Abeles, B., P. Sheng, M.D. Coutts, and A. A r i e . Advances i n Physics, 24_, 407 (1975). DAV62 David, F.N. and D.E. Barton. Combinatorial Chance, Charles G r i f f i n , London (1962). DUK69 Duke, C.B. Tunneling i n Solids, S o l i d State Physics, Supplement 10, Academic Press. New York, London (1969). GRA Gradshteyn, S./I.M. Ryzhlk. Table of Integrals Series and Products, p. 4, equation KR64 (71.1). GRI69 G r i f f i t h s , R.B. Phys. Rev. L e t t . 23, 17 (1979). GUN66 Gundlach, K.H. So l i d State Electron. j)» 9 4 9 (1 966). H0L51 Holm, R. J . Appl. Phys. 22, 569 (1951). MON83 E l l i o t t , W. Montroll and Michael F. Shlesinger. Journal of S t a t i s t i c a l Physics, V o l . 32, No. 2, 1983. M0041 Mood, A.M. Annals of Mathematical S t a t i s t i c s , V o l . 11-12, 1940-1941. PAL84 P. Palffy-Muhoray, R. Barrie, B. Bergersen, I. Carvalho and Freeman, M. Journal of S t a t i s t i c a l Physics, V o l . 35, Nos. 1/2, 1984. SOM33 Sommerfeld, A. and H. Bethe. Handbuch der Physik von Geiger and Scheel ( J u l i u s Springer-Verlag, B e r l i n , 1933), V o l . 24/2, p. 450. WIL62 Wilks, S.S. Mathematical S t a t i s t i c s , p. 144, Princeton, 1962. - 38 -APPENDIX A CALCULATION OF THE MOMENT < K± > FOR FIXED NUMBER OF INSULATOR "ATOMS" DISTRIBUTED IN THE CHAIN ru, - 1 < K i > i ) | i l K 1 , K ^ . . . S l | M - S ^ 1 . with 0 < n x ^  N - 2, O ^ K ^ r i j , O ^ i ^ n ^ {K± } = {K ±, K 2 > .. .K } I i K. = n, and I K. = K. i 1 1 i 1 But: I H P 1 K ' K. ! K„ ! .. .K ! V K J \ {K.} V * 2 l — \ -K T (K - 1)! ( n2 " 1] ' • K,!...(K,-!)!...K n ! \ K / 1 { K- } ^ i . . . ^ . -with I i K, = n, and I K. = K. i 1 i 1 Making the change of va r i a b l e s : K' = K - 1, K'. = J K! / S " *\ K_ = K ± - 1 i f j = i .K. otherwise K,! K0!...K ! I K j i Tl, \ K V KM / n2 " 1 r ,f, K\!...K!1...K* ! V K ' 1 1 l n, v {K!} V 1 with {K|} = {K{, K»,...,K^}, I K' = K - 1 and £ i K' = ^  - i . - 39 -Making use of the procedure given i n Wilks (Wil62), the above sum can be written as: Y K'! _ ( N 1 - 1 - 1 ) ! r£,i K!! K!!...K' ! (K - 2)! (n. - i - K + 1)! ' ( A 2 ) with I KI = K - 1 and I i K! = n. - i . i 1 i 1 Therefore I t i s worth n o t i c i n g that: - The range of K i s from K = 1 to the smaller - 1 and n^ - i + 1; - If n^ =0, then K = 0 which corresponds to < > =0; - For a given n^ ^ 1, the term K = 1 i s zero unless i = n^ .. Consequently, f o r a given nj and i < n^, the f i r s t moment i s given by: Settin g K" = K - 2 and not i c i n g that K" ^ n 2 - 3; the sum over K then reduces to: - 40 -i /N - a.\ K i Q U " 3 - K 7 V K " ~ "2)s i n c eq I 0 ( : ( p - % = n + p > ( G R A 6 5 ) ; where m is a natural number. It follows that ( n2 ~ X ) /n - -i - ^ < K l > - 7 H i T i r - N ( n . - i ) - l « * l < » - 3 In the case of i = n^, the smaller of (n 2 - 1) and (n^ - i + 1) is (n^ - i + 1), unless n 2 = 2 when they are equal. Therefore, < K > = _ 1 f n2 " M /-A ( n2 " 1 } /N - 2\ V 1 / \ - U " /N - 2 J * I n , The two cases, i < n^ and i = n^  can be combined as: < K. > = 2\ V - ' i ^ w i t h 1 < N - 3 <A4> Notice that i f nj = N - 2, < K± > = 0 unless i = N - 2 when < Ki > = 1. - 41 -APPENDIX B CALCULATION OF THE MOMENT < K ± R\. > FOR FIXED NUMBER OF INSULATOR "ATOMS" DISTRIBUTED IN THE CHAIN n 2 - 1 with I K = K, ^ i K = n p 0 < KV< n and 0 v< i s< n,. i i 1 1 Separating out the diagonal and off-diagonal terms i n (BI), the sum over {K^} can then be written as: £ K! £ K! 2 { K ± } K X ! K 2 ! . . . K N I ! K i K j = 6 i j { K ±} K l ! ! . . .Kn ! K i + ( 1 " V K ] [ ! V . . . K N ! K i K r <B2> with I K = K and £ i K , = n.. i i 1 1 Consider the diagonal term i n equation (B2) I K ! -> - K ! K -{Kj} K X ! ^ . . . ( K , - ! ) ! . . . ^ ! with £ K = K and £ i K = n.. i i 1 1 - 42 -Making the change of variables: K1 = K - 1 f K - 1 i f j = i K'. = j J J L K, o t h e r w i s e ' J e q u a t i o n (B3) i s e q u a l to: K' ! (K1. + 1) KM K' v y i t7 y 1 t.'llK!!...K!I...R' ! r , i KM .. .K! !.. .K' ! t K i l 1 1 " x 1 i n v y KM r M x rv,S K.M ...KM ...K' ! with I KJ = K - 1 and T i K'. = n, - i . i 1 f 1 1 Perform once more, the change of variables in the f i r s t equation of the right-hand side (B4) such as: K" = K' - 1, K" = j f K'. - 1 i f j = i J . KI otherwise. Equation (B3) is then written as: v(v - n V K"! y KM V ; K;,!...K"!...K" ! r f, KM ...KM ...K' ! » t ^ l 1 i ^ {K{1 1 i n x wi th I K^ = K - 1; £ i K| = ^ - i , i i £ KV = K - 2 and 5! 1 K £ = ,n1 - 2 i . From Appendix A, the value of the sum over {K^  } can then be obtained. And equation (B3) reduces to: - 43 -y K 2 K! fn - 21 - l \ /n - i - 1\ {K±} V V T T T " < K ( K " l ) I K - 3 J + K l K - 2 ) ( B 5 ) Analogously, the off-diagonal term i n (B2) i s : ( 1 - 6 , . ) I - K. K. 2 {K.} K.! K.I...K ! J 1 i J 1 2 n^ /n - i - j - l \ = K ( K - 1 ) ( K _ 3 (B6) With (B5) and (B6), the expression for (Bl) i s then: < K K > = {I K(K - 1) I " 1 K \ n, - 21 - 1\ / n 2 - 1' K - 3 / V K n i ,'n. - i - j - 1\ /n„ - l \ (1 - 6. .) • i ) (B7) with 0 ^ K < n j . Consider, f i r s t , the sum over K for the off-diagonal term i n (B7): n - 44 ->9 1) (B8) Sett i n g K" = K - 2 and rearranging terms, then equation (B8) i s given by: " I - 1 " ! /n - i - j - A /„ - 3 (°2 " '><»2 " 2> I ( „ - i - j - r ( V , K"=0 X V with the condition 0 ^ K" ^ n^ - i - j . By r e l a t i o n (A3), the f i n a l expression for (B8) i s /n, - i - j - 4\ = ( n 2 - l ) ( n 2 - 2) f j (B9) Consider now, the sums over K for the diagonal terms i n (B7). In analogy with equation (B9), the f i r s t term i s given by: n l In - 21 - l \ / n - 1 \ J o K ( K - 1 } v K _ 3 ; v * ; /N - 21-4 = ( n 2 - l ) ( n 2 - 2 ) ^ ^ j (BIO) - 45 -The second sum over K, i n equation (B7), can be s i m p l i f i e d as: ^ o ^ ' H * 1 * - 2 ' 1 ) 0 ^ - 0 ( b u ) Setting K' = K - 1 i n ( B l l ) and using r e l a t i o n (A3), equation ( B l l ) i s given by: Therefore, X k ( ? * - "2') C 2 * " ' ) = < n 2 • n f V - " 1 ' ) ( B u > With the re l a t i o n s (B9), (BIO) and (B12), the expression for < K ± Kj > i s : < h «j > - «° 2 - » ( N - 4 : T 4 ) - ( N ^ - V ) ! 013) - 46 -APPENDIX C CALCULATION OF < R 2 > FOR FIXED NUMBER OF INSULATOR "ATOMS" DISTRIBUTED IN THE CHAIN The average of the square of the resistance i s given by: < R 2 > - R20 I I U e b ( i + j ) < K ±K >, (Cl) with 0 < i + j < n j . From equation (B13), the < R2 > i s given by: < R 2 > _ ( n 2 1 ) ( n 2 " 2 ) n b(i+j) / N - i - j - 4 ' 2 / N - 2 \ L L ±3 e n - 4 R I I n • 1 j V 2 . ( n 2 " l ) v .2 2bi /N - i - 3\ ._ 0. + /N - 2^ I 1 6 [ n 2 - 3 1 ( C 2 ) n l Rearranging terms and making a change of variables i n (C2) such that: 9 i + j = m and i - j = n, the expression for < R > can be written as: . Jl ^ (n„ - l ) ( n 0 - 2) , 2 2 N , /„ ,\ < R > _ 2 2 _ y y (m - n ) bm / N - m - 4 D 2 " | N - 2 \ 1 1 4 e l n - 4 RQ ^ ^ 1 \ 2 (n 2 1) „ 2bi I N - i - 3 "l wi th - m ^  n ^  m and 0 ^  m ^  n^ 47 -Performing the summation over n, one obtains: < R > = ( n 2 ~ 1 ) ( n 2 " 2 ) r (4 m3 - m) bm ( N - m - 4 R o ( n x J \ 1 / . ( n 2 " 1 } v .2 2 b i / N - i - 3\ + i n - 2^  I 1 e I n - i ) ( C 3 )  n l w i t h m = i + j , O^m^n^ and 0 ^  i ^  n±, - 48 -APPENDIX D CALCULATION OF < > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE The moment < > i s given by: < h>- I I j ^ r r r f b T \ p 1 a - p> 2 . nj K {Kt} 1 V K2 n l with [ K. = K, I i K ± = n L, 0 < K < n x , i i 0 ^  i ( nj and 0 ( ii j s< N - 2 (Dl) The sum over {%} and the subsequent sum over K i n equation (Dl) has already been performed i n Appendix A, equations (A2) and (A4). Therefore: < K. > = ^  ( n 2 - 1) IN -n*-*l p" 1 (1 - p ) " 2 ' \ ( D 2 ) with 0 < nl ^ N - 2 and i ^  N - 3. Equation (D2) can be rewritten as: n, N-n,-2 / . . \ < K i > = I p ( l - p ) (N - i - 3) ( ~ V i nl \• 1 1 / n, N-n,-2 / „ . ~ \ + 2 I p 1 (1 - p) 1 / N l^SiJl (D3) - 49 -with i ^  N - 3 and 0 ^  nx ^ N-2. Making the change of variables in (D3), nj = - i and rearranging / n _ i _ A the limits in the sum over n! such that . 1 * 0 and 1 V ni I 'N - 1 - 3^  , 1 * 0 , one obtains nj 1 ' i m • / N-i-4 n, / „ . , < K. > = (1 - p) 2 p l ( l - p) 1*" 1- 4 I ( r E - ) 1 N " ^ 4 (N - i - 3) nj=0 + 2(1 - p) p ^ l - p ) ^ 1 " 3 I ( r ^ - ) N-i-3 _ n{ / N _ ± _ 3 nj=0 with i < N - 3. (D4) By the Binomial series definition the sums in (D4) can be calculated. Then: < K ± > = p 1 (1 - p) {(1 - p) (N - i - 3) + 2}, for i < N - 2. (D5) For the case of i = N-2, < K^ > is equal to p N - 2 . APPENDIX E CALCULATION OF < K. K. > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE r r r K' f °2 " 1 \ H l n 2 ~ 2 < K K > = I I J K 1 K I V K K i K i P ( 1 " P ) ; KM...K ! \ / J 1 L l J 1 wi th 0 ^ n. « N - 2, 0 ^  K < n , I K = K, i 1 i K ± = n l f 0 ^ i < n 1 and {K± } = {Kj, K 2 K N }. ( E l ) The expression for the sums over {K^ } and K can be obtained from Appendix B. Equation (El) can then be written as: < K 4 K. > - | • « „ 2 - 1) („ 2 - 2, I « " L-J- ") • + o. . (n„ - 1) _ i j 2 V 3 \ n, n 0-2 J } P 1 (1 " P) 2 , with 0 < n x ^ N - 2. (E2) Taking i n t o consideration that: "1=0 i f eit h e r r > n or r < 0; equation (E2) i s rewritten as: - 51 -< K. K. > = {I ( n 2 - 1) ( n 2 - 2) ^ " ^'J' ^ p" 1 (1 - p)" 2"'} x 9 ( i + j , N - 3 ) + 2 ( l - p ) p N " 3 6 ( 1 + j ) N _ 3 ) + + I ( n 2 - 1) (^N l^.'/j P1"1 (1 " P ) " 2 2 Q(i> N - 2) + 6 i j 6i,N-2 P N" 2' <E3> 1 i f m < n w i t h O s < n i < N - 2 and 9 ( m , n) - | Q o t h e r w i s e . Each sum over n^ , i n equation (E3), w i l l be evaluated separately i n the next paragraphs. The f i r s t sum over n j , i n equation (E3) can be rewritten as: X P (1 " P) (*2 " 1) ( n 2 - 2) ^ ^ J 4 j = n i N - n i ~ 2 / m • _ • I P (1 - P) {(N - i - j - 4)[(N - i - j - 5) ( I _\ n l \ 2 with 0 ,< ni O - 2 -Consider the f i r s t term i n the right-hand side of equation (E4). After making the change of variables n'^  = n^ - i - j and rearranging the l i m i t s of the sum, such that the binomial c o e f f i c i e n t i s not equal to zero one obtains: - 52 -(N - 1 - j - 4) (N - i - j - 5) (1 - p , « " ? (» - . ^ . V . - J 6 ' (N - i - J - 4) <N - 1 - ] - 5) ( 1 - p ) N " 2 f r ^ - p - ) 1 4 = (N - i - j - 4) (N - i - j - 5) p i + j (1 - p ) 4 (E5) Using the same procedure used to evaluate expression (E5), the others sums i n the right-hand side of equation (E4) are given by: 6(N - i - j - 4) (1 - p ) 3 p i + J ; (E6) and Considering equations (E5), (E6) and (E7), equation (E4) can then be evaluated, being equal to: I p 1 (1 - P) 1 <n2 - 1) ( „ 2 - 2 ) t" - ' 'J' 4 n^ \ 2 {(N - i - j - 4) [(N - i - j - 5) (1 - p ) 2 + 6(1 - p)] + 6} (1 - p ) 2 p i + j (E8) - 53 -Now, the second sum i n the expression for < Kj >, equation (E3), w i l l be calculated. Therefore v , . /N - i - 3\ n l "2 " 2 I ( n2 - 1 } ( n 2 - 3 ) P ( 1 " P ) V1 P j \ "1 i ' + 2 2 with n* = n - i . ( 1 - P) N 3 1 1 Therefore v . /N - i - 3\ n l n ^ J ( N 2 - 1 } ( n 2 - 3 J P ( 1 " P ) [(N - i - 3) (1 - p) + 2] (1 - p) p 1 , (E9) with O ^ n ^ ^ N - 2 . With equations (E8) and (E9), equation (E3) can then be evaluated. I t i s given by: - 54 -< K ± K > = {(N - i - j - 4) [(N - i - j - 5) (1 - p ) 2 + 6(1 - p)] + 6} (1 - p ) 2 p i + j 6(1 + j , N - 3) + 2 ( l - p ) p N - 3 « ( 1 + J > H . 3 ) + [(N - i - 3) (1 - p) + 2] (1 - p) p 1 6 6(1, N - 2) + 6 l j 6i,N-2 P N-2 (E10) APPENDIX F CALCULATION OF < R 2 > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE < R2 > - R2D {I I l j e b ( i + J } < K K >}, i j with O ^ i ^ N - 2 and <*L> - Hi i e b ( i + J > {p i +J (1 - p ) 2 [6 + 6<N - ± - j - 4) (1 - p) + R o 1 J (N - i - j - 4) (N - i - j - 5) (1 - p) 2} 9 ( i + j , N - 3) + 2 p N " 3 (1 - p) 6 ( . + j j N _ 3 ) + p 1 (1 - p){2 + (N - i - 3) (1 - p)} 6 ±. 9(1, N - 2 ) + 6^ p N ~ 2 6 i ) N_ 2} ( F l ) In the next paragraphs each term i n equation (Fl) w i l l be considered separately. a) I I i j ( p e b ) i + j (1 - p ) 2 {6 + 6(N - i - j - 4) (1 - p) i j + (N - i - j - 4) (N - i - j - 5) (1 - p ) 2 } , with i + j < N - 3. (F2) Performing a change of variables such that m = i + j ; n = i - j and rearranging the terms and the l i m i t s of the sums; equation (F2) i s then written as: - 56 -I ( p e b ) m (1 - p ) 2 {A - 6m (1 - p) - (2N - 9) mC + m2c} £ ^  " ^ = m n ( 11 2 P ) 2 I ( p e b ) m {(4A - C) m3 - 4Bm4 + Bm2 + 4Cm5 - Am} = m ( 11 2 ~ P ) {(4A - C) S 3 + BS 2 - 4BS 4 + 4CS 5 - ASj }, (F3) where: -m^n^m, O ^ m ^ N - 4 ; A = 6 + 6(N - 4) (1 - p) + (N - 4) (N - 5) ( 1 - p ) 2 ; B = 6(1 - p) + (2N - 9) (1 - p ) 2 ; C = (1 - p ) 2 N-4 b m K  s v = I (pe ) m . K m=6 b) I I i j ( p e V " 3 2 (1 - p ) 6 i + j ) N _ 3 = 2 ( p e b ) N " 3 (1 - p) I H (N - 3 ) 2 - *f [(N - 3) - 2 j ] 2 } = j=0 j=0 i ( p e b ) N ~ 3 (1 - p) (N - 2) (N - 3) (N - 4) (F4) c) I I i j p 1 e b ( 1 + j ) (1 - p){2 + (N - i - 3)(1 - p) } 6 x 9(1, N - 2) = i j N-3 I i 2 ( p e 2 1 5 ) 1 (1 - p) {2 + (N - i - 3) (1 - p)} = 1-0 (1 - p) [2 + (N - 3) (1 - p)] T 2 - (1 - p ) 2 T 3, (F5) - 57 -where T = £ i ( p e 2 " ) 1 . K i=0 d) X I i j e b< i +J> P ( N " 2 ) 6 6. - (N - 2 ) 2 ( p e 2 b ) N - 2 (F6) i j ' With equati ons (F3), (F4), (F5) and (F6) the f i n a l expression for < R 2 > i s 2 2 < R2 > = ( 1 1 2 P ) {(4A - C) S 3 + BS 2 - 4BS 4 + 4CS 5 - ASj} o + | ( p e b ) N 3 (1 - p) (N - 2) (N - 3) (N - 4) + (1 - p) [2 + (N - 3) ( 1 - p)] T 2 - (1 - p ) 2 T 3 + (N - 2 ) 2 ( p e 2 b ) N _ 2 . (F7) and with s = I ( p e b ) m  K m=0 N-3 „ 0, . _ V - K / 2b.l K i=0 - 5 8 -APPENDIX G CALCULATION OF < K± > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO A CIRCLE r v v K! I n2 " M N n l n2 n x K {K } K l * h. n 1 with 1 < iij < N - 1, I 4 K 4. nlf {Ki} = |Kj, K 2,...K n }, I K ± = K, 1 i I i K. = n, and 1 < i < n, . (Gl) i The sum over {K-j_} can be performed i n the same way as i n Appendix A leading to: with 1 ^  N - 2 and 1 ^  K ( i i p (G2) Performing the change of variables i n equation (G2) such that K' = K - 1 and rearranging the l i m i t s of the sums i n order that the binomial c o e f f i c i e n t i s not zero; equation (G2) then reduces to: < K i > - J |, N ( \ ^ ( n ! - i - K') ( 1 " *" 2> With 1 ^  n x ^ N - 1 and 1 ^  K' < n x - i . (G3) From equation (A3), expression (G3) i s given by: n2 1 - P) , (G4) - 59 -with 1 ^ rij ^  N - 1 and i ^  N - 2. Making the change of variables such that nj = n^ - i and rearranging the l i m i t s of the sum, equation (G4) i s then equal to: By the multinomial formula i t i s straightforward that: < K ± > = N p 1 (1 - p ) 2 for i ^ N - 2. A more general way of writing < > i s : N p 1 (1 - p ) 2 i f i ^ N - 2 < K i > = 1 ( N - n <G6> N p (1 — p) i f i = N - 1 and n 2 = 1. - 60 -APPENDIX H CALCULATION OF < K, K. > i J FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO CIRCLE K! „ „ n l , n2 N n2 ~ 1 with 1 ^  K ^ n p I ^  = K, I i K. = tij and 1 ^  i ^  n 1 (HI) Equation (HI) can be written as: < K K > - I I J K ! , K K p" 1 (1 - p ) " 2 1 [ A , (H2) with the same r e s t r i c t i o n s as i n equation (HI). Referring to Appendix B, the r e s u l t for the sum over {K^  } can be obtained leading to: < s K j > -11 p n i d - P)"2 i . (K(K - o - * : \ - i ) • wi th 1 ^  n\ 4 N ~ 1 a n d 1 < K < . In order to perform f i r s t the sum over K, consider now the f i r s t term at the right-hand side of equation (H3) and rearrange the terms. - 61 -Therefore: n 9 w / n 9 \ /n, - i - j - l \ K - 3 / K 2 N | ( n2 " 1 } ( K - 2 ) ( 1 K - 3 1 ) ' W l t h 1 < K < V ( H 4 ) Making the change of variables i n equation (H4), K" = K - 2 and using r e l a t i o n (A3); the f i n a l expression for equation (H4) i s then given by: K"=0 \ / \ K" - 1 / Now take i n t o consideration the second term i n the right-hand side i n equation (H3) and rearrange the terms. T n l „ , n 2 N F / M l n l i ' ( 1 " p ) ^ K U H r n n \ /n, - i - 1> K - 2 , n, ^ /a, - 1\ /tij - i - 1\ K - 1 j V K - 2 I I P X ( l - P) N K 2 - 1 ( V ) C 1 n„ / n„ - 1\ / n, - i - 1 K' - 1 > with 1 < K < n\ and K' = K - 1. Again using r e l a t i o n (A3), one gets: - 62 -n„ „ ITLA /n^ - i - l\ r n l n , n2 N „ ( r I p (1 - p) - K y ^ K _ 2 N p 1 (1 - p) 2 / N n J > r e s t r i c t e d to 1 < K ^ n ^ Considering equations (H5), (H6) and the l i m i t s at which the binomial c o e f f i c i e n t s are not equal to zero; the expression for < K i Kj >, equation (H3), reduces to: N-3 n n / < K. K. > = I p \l - p) N(n - 1) I 2 n =1 \ 1 , „ f _ . y _V_y) e(i + j , N - 2) + N p i + J ( 1 - p ) 2 6 ( ± + j ) N _ 2) + N f ^ p"1 (1 - P ) " 2 N ^N n ^ - ' i ^ 6 ± j ^ N " D + N(l - p) p 1 6 ± j 6 i > N_ 1 (H7) In order to evaluate the sum over n 1 at the f i r s t term i n equation (H7), perform the change of v a r i a b l e s , = - i - j • Therefore: I P (1 " p) N(N - n - 1) J_ n =1 \ n l 2 j n,' N-i-j-3 (H8) nj=0 With the r e l a t i o n n ! [ N i 7 J ~ 3 | = ( N - i - i - 3 ) ( N ~ i ~ J ~ 4 and using the d e f i n i t i o n of the binomial c o e f f i c i e n t s , equation (H8) reduces to: N( 1 - n ) N / P V+.J f(N - 1 - J - 1) _ K1-*) ( I - P ) ^ " 3 N-i-j-4 / \ n,"+l /„ . . . \ where n^ = nj - 1. Again using the d e f i n i t i o n of the binomial c o e f f i c i e n t s , one gets: N-3 n N-n / \ I P\l-P) 1 N(N- n - 1) N : 1 V ~ i 3 ) = n -1 \ n l 1 2 j N p i + j (1 - p ) 3 {(1 - p) (N - i - j - 3) + 2}. (H9) Consider now the second sum i n the right-hand side of equation (H7), and perform a change of var i a b l e s : nj = n^ - i . Therefore: N " f p" 1 (1 " P ) " 2 N ( N " 1 " 2 ) 6.. 9(1, N - 1) = Na-p)N i;rHnl+i ( b " 4 " 2 ) = - i ( i - p ) 2 Therefore, with (H9) and (H10), the expression for the moment < >, equation (H7), i s given by: - 64 -< K ± K. > = N p i + J (1 - p ) 3 {(1 - p) ( N - i - j - 3) + 2} 9(i + j , N - 2) + N-2 + N P — ( l - p ) ' 6 ( i + j > N _ 2) ( H l l ) + N p i ( l - p ) 2 6 ± j 0 ( i , N - 1) + N(l - p) p N 1 6 ± j 6 ± N - 1 - 65 -APPENDIX I CALCULATION OF < R 2 > FOR FIX PROBABILITY p OF HAVING AN INSULATOR OCCUPYING A SITE IN A LINE BENT INTO A CIRCLE < R2 > = R T I i j e b ( i + j ) < K. K. > (II) o I j 1 J i J Substitute equation ( H l l ) into equation ( I I ) . R 2 o I I i j ( p e b ) i + j N (1 - p ) 3 [2 +(N - 3)(1 - p)] 6(i + j , N - 2) -i j I I i j ( p e b ) i + J N (1 - p ) 4 ( i + j ) 9(1 + j , N - 2) + i j I I i j ( p e b ) i + j N (1 - p ) 2 6(1 + j , N - 2) + (12) i j H i j N p 1 e b ( ± + j ) (1 - p ) 2 6(1, N - 1) 6 ± + i j I I i j N(l - p) e b ( i + j ) p 1 6(1, N - 1) 6 i j The sums i n equation (12) are going to be evaluated separately i n the subsequent paragraphs. - 66 -a) I I i j ( p e b ) i + j 9(i + j , N - 2) = i j N-3 m 2 2 , N-3 , 3 I I ( P e b ) m = I ( p e b ) m = m=0 n=-m m=0 {4S^ - S'^ }; where i + j = m , i - j = n N-3 and S' = I ( p e V m\ (13) m=0 b) Analougously to the deduction of equation (13), i t is obtained that: I I i j ( p e b ) i + j ( i + j) 8(1 + j , N - 2) = L [4 S' - S£] (14) J N-2 ( p e b ) N 2 I i(N - 2 - i ) = I (N - 1) (N - 2) (N - 3) (pe b ) N ~ 2 (15) 1=0 d ) I I U P 1 e b ( i + j ) 6 9(1, N - 1) = N " f i 2 (pe 2 1 3 ) 1 = T' , (16) i j 3 1=0 1 N-2 where T' = \ ( p e 2 1 5 ) 1 i K . K i=0 - 67 -Consequently, with (13), (14), (15) and (16), equation (12) i s given by: 2 < R2 > = 72 ( ^ - S|) N(l - p) 3[2 +(N - 3)(1 - p)] -R o \j (4S> - S^) N(l - p ) 4 + N(l - p ) 2 i - (N - 1) (N - 2) (N - 3) ( p e b ) N ~ 2 + N(l - p ) 2 V2 + N(N - l ) 2 ( p e 2 b ) N _ 1 (1 - p) (17) N-3 . where S' = I ( p e b ) m mK; K m=0 T« - Y (pe 2V i K . K 1-1 

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