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TRANSPORT NOISE ARISING FROM DIFFUSION AND BULK OR SURFACE GENERATIONRECOMBINATION IN SEMICONDUCTORS BY HARSHAD MEHTA A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1981 To Dear mother and (late) father ACKNOWLEDGMENTS I am greatly indebted to Professor K.M. van Vliet for his most valuable guidance, suggestions and time; without his help this work would not have been possible. I am very thankful and highly obliged to Professor and Mrs. van Vliet and family for their generosity, kindness and greatness. I am very thankful to Professor A.D.Sutherland and Professor E.R.Chenette for all their help, guidance and helpful suggestions during this work. I am highly thankful to Professor Cherrington, Chairman, Electrical Engineering Department for his help from time to time. I am very thankful to Professor Neugroschel and Professor Ihas for serving on my advisory committee and their valuable sugges tions. I am very grateful to Professor A.D.Sutherland for his continued help and support whenever I needed. I am also very much thankful to my brother, Dr Jitu Mehta, for all his moral, social and financial support. I am also highly thankful to some of my best friends Kelly, Pankaj, Satish, Pramod and Kirit for their cooperation and help whenever I needed. I am highly thankful to the Electrical Engineering Department of the University of Florida and to the Centre de recherche de math6matiques appliques of the University of Montreal, for providing the financial support during this work. Last but not least I am highly thankful to Francine HouleMiller for her excellent work in typing this manuscript. TABLE OF CONTENTS ACKNOWLEDGEMENTS iii ABSTRACT vii Chapter Page 1 INTRODUCTION 1 1.1 Object 1 1.2 Flicker Noise 1 1.3 Transport Noise 2 1.4 Surface Noise 4 1.5 Embedded and Nonembedded Bodies 5 PART A: MATHEMATICAL METHODS AND PHYSICAL NOISE SOURCES 8 2 MATHEMATICAL METHODS FOR TRANSPORT NOISE 8 2.1 Methods 8 2.2 Langevin Equation and Source Methods 9 2.3 The Phenomenological Equation and 19 Correlation Methods 3 PHYSICAL NOISE SOURCES 28 3.1 Introduction 28 3.2 Volume Noise Sources 28 3.3 Surface Noise Sources 34 Chapter Page PART B: NOISE DUE TO VOLUME NOISE SOURCES 37 4 SYMMETRICAL EMBEDDED BODIES 37 4.1 Introduction 37 4.2 Survey of Older Work 37 4.3 Previous Work Using the Green's Function Method 39 4.4 Present Work: Carrier Fluctuations in Sphere 42 Due to Diffusion and GenerationRecombination 5 NONSYMMETRICAL EMBEDDED BODIES 52 5.1 Introduction 52 5.2 Voss and Clarke's 'P' Sources 52 5.3 Physical Diffusion Source 58 6 SYMMETRICAL NONEMBEDDED BODIES 68 6.1 Introduction 68 6.2 Summary of Previous Work 68 6.3 Present Work: Sphere 71 7 NONSYMMETRICAL NONEMBEDDED BODIES 90 7.1 Introduction 90 7.2 Linear Case 90 7.3 Rectangular Nonembedded Bar 92 PART C: NOISE DUE TO SURFACE NOISE SOURCES 98 8 NOISE DUE TO STOCHASTIC BOUNDARY CONDITIONS 98 8.1 Introduction 98 8.2 Summary of Previous Work 98 8.3 Noise Caused by Surface GenerationRecombination 99 APPENDIX 119 REFERENCES 128 BIOGRAPHICAL SKETCH 130 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRANSPORT NOISE ARISING FROM DIFFUSION AND BULK OR SURFACE GENERATIONRECOMBINATION IN SEMICONDUCTORS By Harshad Mehta March 1981 Chairman: Professor K.M. van Vliet Cochairman: Professor A.D.Sutherland Major department: Electrical Engineering The purpose of this thesis is to present a systematic study of the type of noise spectra in solids or solid state devices, which can arise from "transport noise," which stands for noise due to combination of causes, such as diffusion of carriers, generation and recombination (gr) processes of carriers in the bulk and at the surface. The techniques to be described apply equally well to the problem of heat diffusion, with heat transfer or reflection at the boundaries of the system; we have, however, mainly the carrier noise problem in mind. After an introduction we describe the various mathematical techniques, applicable to transport noise. The methods studied are (1) the eigen functions expansion method, (2) the Green's function method. Each of these methods can be applied to two physically distinct procedures, (a) the sources or Schottky method, (b) the correlation method. In chapter 3 we discuss the physical noise sources,viz. the diffusion source, gr source and surface source, as well as Clarke and Voss' P source. In the next few chapters we discuss: transport noise in symmetrical embedded bodies, in nonsymmetrical embedded bodies, in symmetrical non embedded bodies and in nonsymmetrical rnmmbedded bodies. The concept "embedded" body refers to a system which is part of a very large system, with which it exchanges particles or heat, such as a volume between probes on a large semiconductor. The concept embeddedd body" refers to a body inside which there is transport noise, with deterministic boundary conditions on the surface of the body. In chapter 4 we first review the older work on the circular patch and the cylinder (MacFarlane, Burgess, Richardson, van Vliet, Chenette). Next we solve for the noise of an embedded sphere, caused by diffusion and volume gr processes. For infinite carrier lifetime, the results are found to be in agreement with those of Fassett. For finite lifetime an ex tension of his results is obtained. The high frequency 3/2law is confirmed. In chapter 5 we discuss the results for a rectangular embedded bar, disc, or cube. Because of the lesser symmetry, the spectrum cannot be obtained in closed form, but only in a triple integral :form. The result has been numerically evaluated by computer techniques. Not all the ranges predicted by Clarke and Voss are visible, however. For the Psource the noise has also been evaluated. Contrary to Clarke vili and Voss' claim, no 1/f range is visible. Thus, their heat diffusion noise model cannot account for the observed 1/f noise in metal films. In chapter 6 symmetrical nonembedded bodies are discussed. Parti cular attention is given to the noise in a sphere, with boundary condi tions due to surface recombination. A closed form is obtained via an expansion of the Green's function in spherical polynomials. In chapter 7 the nonsymmetrical bar is discussed. The results given previously by Lax and Mengert and others were evaluated by computer. In chapter 8, finally, we discuss noise due to the surface gr source. This noise can be dominant due to the surface modulation factor; it has not been considered in all older theories. We discuss three examples: (a) a semiconductor single crystal sample with surface gr noise; (b) the noise in the channel of a MOSFET, caused by diffusion into the adjacent bulk and by gr (surface) processes near the oxide layer; (c) the same model but with special care for the alteration of the diffusion constant and mobility in the channel due to surface scattering. For models (b) and (c) it is shown that a realistic large 1/f range (104 hz up to 1011 hz) can occur. Thus this model can account for much of the high frequency 1/f noise in MOS devices. The other examples and models considered in this thesis firmly indicate that volume gr and diffusion processes give rather smooth spectra, but without a 1/f range. The high frequency asymptote is a3/2 2 always c (embedded case) or 1 (nonembedded case). CHAPTER 1 INTRODUCTION 1.1 Object The purpose of this study is to make a systematical investiga tion of diffusion and generationrecombination noise spectra. Such studies may aid in understanding the physical mechanisms underlying '1/f' noise. The noise spectra obtainable from transport processes are analyzed and the possibility of contributions to '1/f' noise is studied. It will be shown that most transport processes caused by volume sources do not give rise to 1/f noise. In addition, a detailed investigation has been undertaken of surface noise. This is noise caused by surface sources which give rise to stochastic boundary conditions at the surface. Surface noise contributions are studied in detail, since the older theories due to Hydel, Champlin2, LaxMengert3,and van Vliet4 only considered the volume noise contribution. This analysis gives for the first time a transport noise model which results in 1/f noise; however, the frequency range for which it occurs is often too high to ac count for the experimental data, unless very long lifetimes of the carriers can be shown to exist. 1.2 Flicker Noise The phenomenon of 1/f noise or 'flicker noise' started with the study of J.B.Johnson5 on noise in vacuum tubes in 1925. He noted, that besides shot noise, which has a white frequency spectrum, at low frequencies another noise component appears, which varies roughly inversely proportional to the frequency. The first interpre tation of this noise was given by W.Schottky6 in 1926. Schottky attributed the effect to fluctuations in the work function of the cathode material, caused by the random arrival and leaving of foreign atoms at the cathode surface. The spectrum computed by Schottky was not exactly of the form S(f) = A/f; rather, his effect results 2 2 in a Lorentzian spectrum, S(f) = B/(l+o T ). Since Schottky's mechanism failed to interpret Johnson's 1/f data, as well as the flicker noise later on observed in a large number of solids and devices, various authors have proposed a great variety of detailed studies to understand the physical mechanisms behind 1/f noise. In the subsequent chapters we shall summarize some of these efforts and extensively study the contribution of transport and surface noise to the origin of flicker noise. 1.3 Transport Noise Transport noise is caused by quantities subject to stochastic transport in the sample. The most notable examples involve particle diffusion and heat diffusion. Both are described by a linear partial differential equation. The noise can be obtained in two ways. In the first procedure, the 'sources' or 'Langevin' procedure, one adds noise source terms to the phenomenological transport equation, which then becomes of the form ( 2+ A)a (,t) = ((,t) (1.3.1) where A = DV2 and a = AT or An for heat or particle diffusion respectively, and is the volume noise source. The spectrum of the latter must be known. In the correlation procedure, on the other hand, one uses the ensembleaveraged stochastic equation, which by Onsager's7 prin ciple, is the phenomenological equation. Thus we must then solve t A) (aCr,t) aC(,0)) = 0 (1.3.2) where (<) denotes the ensemble average of a(E,t), conditional to a given value a(Zr,0) at time t = 0. In this case the covari ance function r Cz,') = (Aa(C,t)Aa(' ,t)) (1.3.3) must be known. The connection between both procedures is given by Atheorem due to van Vliet8, which connects the source spectra to the cova riance function CA ,+A )rCr,') = 2C,'). (1.3.4) To solve either (1.3.1) or (1.3.2) one can use two mathematical formalisms, viz.the eigenfunction expansion method and the Green's function description. Thus, in total, there are four methods to obtain transport noise. The equivalence of all these methods has been shown by van Vliet and Fassett9'S0. The details of these methods are described in the next chapter. 1.4 Surface Noise This is noise caused by stochastic boundary conditions at the surface. An example occurs when a semiconductor sample is subject to surface generationrecombination, with the surface re combination velocity S being unequal to zero or infinity. In the experiments on modulation of conductance by surface charges, it was found that only 10 percent of the induced charges were effective in changing the conductance. J.Bardeen11 proposed that the ineffective portion of the induced charges is lodged in states localized at the surface. Since the work of McWhorter12 and others, it is held that the traps in the oxide modulate the generationrecombination rate and this gives rise to flicker noise in the device. Hooge 13 explained the excess noise by an empirical model, though in his case the noise is a volume rather than a surface effect. Recently van Vliet, van der Ziel and Schmidt14 considered heat conduction in a thin film supported by a substrate and found that surface sources can give rise to 1/f noise. A sample with a small inversion layer near the surface which is coupled to larger bulk volume is considered in this thesis. The model is mathematically similar to that considered by van Vliet et al., 4 and, as for the thin film case, the surface noise gives rise to 1/f type of noise behavior with some theoretical restrictions. 1.5 Embedded and Nonembedded Bodies From the theory it emerges that one must distinguish between models involving embedded and nonembedded domains. In the former case, one supposes that the volume of interest for the noise, Vs, is part of a larger domain V. Fluctuations in the particle number or heat content of Vs arise due to sinks and sources in Vs (gr contributions) and due to transport across the boundary into the rest of V (diffusion drift contributions). An example is afforded by the case of field emission from a metal tip. Here ions within a, say circular, patch of the active area tip can modulate the electron emission; these ions therefore affect the emission, when they are within this circular patch Vs, which is a part of the larger cathode area V over which ions diffuse. The nonembedded case applies when we monitor the particle or heat content of a domain V which does not communicate with the ex s terior but on the boundary of which we have deterministicc boundary conditions', usually of the Dirichlet, Neumann or mixed type. An example is when we measure noise due to carrier fluctuations in a semiconductor sample Vs, subject to surface generationrecombination (mixed boundary conditions) on the free surface. This dissertation is divided into three parts. The first part, A, contains the mathematical and physical basis; in particular in chapters 2 and 3 we discuss the various mathematical methods for transport noise and the physical noise sources, respectively. The second part, B, gives the noise due to volume sources. In chapter 4 we discuss the noise for symmetrical embedded bodies. Chapter 5 is devoted to noise from nonsymmetrical embedded bodies. Chapters 6 and 7 consider the noise from symmetrical and nonsymmetrical non embedded bodies, respectively. Finally, in the third and last part, C, we consider the noise due to stochastic boundary conditions. In chapter 8 we study in detail the noise caused by surface generation recombination processes in bulk samples as well as in MOSFETs. PART A MATHEMATICAL METHODS AND PHYSICAL NOISE SOURCES CHAPTER 2 MATHEMATICAL METHODS FOR TRANSPORT NOISE 2.1 Methods As indicated in the introduction, there are basically two ways to attack the problem of transport noise. The first class of procedures considers (a) transport equation(s) in the sense of Langevin, that is, one adds source terms ((,t) which themselves have no memory, to the phenomenological equations) for the fluc tuating variables) aQ(,t); the variables ar,,t) are Markovian random of infinite dimensionality since C is a continuous parameter like a position in some domain V. The boundary conditions on the surface S of V may be deterministic or stochastic; in the latter case surface Langevin sources (,t) must be specified (o E S). The "Langevin procedure" or "sources procedure" is still the most versatile one. The stochastic Langevin equation can be solved with Green's functions, orthogonal or biorthogonal expansions or Fourier methods. The second class of procedures is based on the solution of the transport equation for the ensemble averaged transport quanti ties (a(r,t)a(r,0)> = a(E,t)>a(), in which the fluctuations Aa(r,t=O) are specified. The solution involves the "correlation procedure." The noise is obtained by the FourierLaplace trans formed Green's function of the transport equation, providing the covariance function, Fr(,,r') = (Aa(Z,t)Aa(Z',t)>s is known; here ()s denotes a stationary nonconditional average. This leads to the form suggested by van Vliet and Fassett9. This procedure is mathematically much faster than the Langevin procedure; however, it has the disadvantage that stochastic boundary conditions cannot be included. Nevertheless, if the transport equation is linear and if the surface sources are uncorrelated with the volume sources, the former can be separately considered with the sources method, due to the superposition principle. Also, despite the fact that the covariance function in the presence of surface sources is often not unique, it has been shown by van Vliet10 that the spectra resulting from the Green's function procedure are unique. We will discuss both classes of methods and their prospects for obtaining 1/f noise. 2.2 Langevin Equation and Source Methods The stochastic transport equation is supposed to be of the form La(r,t) = ( + A )a(r,t) = (,t), (2.2.1) where L is a linear differential or integral operator. For the case of diffusion: a = An (particle diffusion) A = DV 2, (2.2.2) a = AT Cheat diffusion) In the diffusion model A is a self adjoint operator which simplifies the mathematics considerably. The boundary conditions must be specified at the boundary S of V. The noise may concern the entire domain V or subdomain Vs c V. If V is infinite, the only condition is that aCx,t) goes sufficiently fast to zero. If on the surface S of V we have maCro,t) = 0, 'o E S, (2.2.3) m being a linear surface operator, the boundary conditions are deter ministic; if ma(~o,t) = C(o,t), o E S, (2.2.4) the boundary conditions are stochastic. 2.2.1 The Bulk Noise Because of the superposition principle, noise associated with F and ( can be separately considered. Thus first setting Y = 0, the boundary conditions are deterministic (2.2.3). Various methods will be given to solve from (2.2.1)(2.2.3). 2.2.1Ca) Eigenfunction expansion Let k( ) be the eigenfunctions of A subject to b.c.(2.2.3) that is Ask ) = Xk k(U; (2.2.5) let fk(z) be the eigenfunctions of the adjoint operator At, then At *k( = )1k4k). (2.2.6) The functions 4 and are biorthogonal, i.e. (Ck^ ) = f d3r k(,) (y) = 5k. (2.2.7) V Expanding aX,,t) and Cbt,t) in the functions k and denoting the expansion coefficients by ak(t) and Yk(t), then (2.2.1) gives Z[a(t)+kak(t)]4k = Z k (t)q( (2.2.8) k k Multiplying with 4/(r) and integrating over V and with (2.2.7) one obtains + X )ae(t) = e(t), (2.2.9) which is a standard one variable Langevin equation. In the usual . method of solution truncated Fourier series in the time interval (0,T) are used. Fourier integrals are not suitable since the Langevin equation (2.2.9) does not apply for negative T; thus .o in t a (t) = Z acne n = 2nn/T, n=0,l,..., (2.2.10) n= and similarly for z (t), one obtains (Ekn Ens (aknan) s = (2.2.11) (Ls +Xk) (iW+X*) (i+n ) (iYXn) Now the essence of the sources method is that the white spectra of (E(C,t)~((',t)), denoted as S(,,X'), are known. Thus according to the WienerKhintchine theorem, the sources correlation function is (r,t)('*X ',t')>s = SC,,Z')6(tt') (2.2.12) (E=C* is real), in which S must be known. Expanding in E.F., we have S(k(t)5e(t')> sk(9 ~ ) = S C,,,' )6 (tt'). (2.2.13) ke Multiplying this with ,C) \V,C('), integrating over d3rd3r', and using biorthogonality, it is found that 1 3 3 ( *k t) (t')>s = JJd3rd3r' SE !,,') k [ Ct ') 5(tt').(2.2.14) Representing the ('s by truncated Fourier series we further have, i(O tm t') 1 3 3 i (tt') Z (knem>s e = Tfd rd r'S ,') 'e n nm (2.2.15) from which we obtain (Eknm >s = ffd3rd3r'S( ')k '). (2.2.16) The spectrum is given by S a*(n) = lim 2T(ana s. (2.2.17) k t Tro Thus from (2.2.11) and (2.2.16) 1 3 3 Saa* = (iX+kk) i+X*) ffd rd r'S(r,')*kT)*t(,').(2.2.18) K z ('w+kk) (i Finally for the spectra of (a(g,t)a(c',t)> one obtains for the noise due to the bulk source : bulk S Cbu r',rc) = Z S (o)k )'), (2.2.19) a k ak Za or substituting from (2.2.18) one obtains Sbulk Zk 3 3 s (Xi)(i) ff d rld r2Sg C1 ( 2kr l) 2 .) a (X(+i)( i) .2.20) (2.2.20) To show the connection with the correlation procedure to be discussed in section 2.3, we write 1 = 1 f 1 1 . +i) + i (2.2.21) (Xk+ia)(w ia) k+j X*im ' For the covariance from (2.2.20) and (2.2.21) bulkk d ('. d) bulk ,Rb rbu lk,t') = j d( )ReSa ( w', ) 0 a 1 Y k C) ') d 3 3 2 (,ri 'Z2) *k (i) *k 1 ) = ~ k d rd r2S ( )1'2~ k1 C2Cr) (2.2.22) Operating on the LHS with A +A we note with (2.2.5) (A )bulk 1C ffd) 1 d 3 3r, (^ +,)b lk('') = 12 2 dk1 d 2S )1 T2 k1) /(X2) (2.2.23) Observing the closure property k 0 kC)l) = 6fr1) (2.2.24a) k 2 0Cr')QC2 = 6(r'2) (2.2.24b) and integrating out the delta functions we find (A +A ,)Fbulk ,z') = ). (2.2.25) This is the Atheorem due to van Vliet8. This theorem gives the connection between the sources spectra and the covariance function. Also it warns that sources spectra cannot be postulated ad hoc, if bulk ) is known from statistical mechanics. F (%, ') is known from statistical mechanics. Substituting (2.2.25) in equation (2.2.20) and using equation (2.2.21) one obtains bulk a hk C)Wfd') =2 k k & kt 1 x ffd3rld3r2 ( +A 2 bulk (, ';)2) (I 2) Introducing Green's theorem for the operators A and A : (Af,g)(f,Atg) = #C[f,g*].da, . (2.2.26) (2.2.27) where #Cdo is the bilinear concomitant involving a surface integral. Thus in (2.2.26) fA bulk 3, ATF (1"Z2 'k~ 1)d rl ,bulk t 3 jr C1', )Ar k C1)d rl ~1 +d2r0C[Frbulk Zr2 ) ] k 2 bulk = +kfd3rl rCbulk 1 2 1) +d2r 0C[rbulk rZO,;2) '* rO)]. Doing the same for and keeping in mind the closure property, we get bulk , Sa ( ' 1 3 =2 Zkk ) k+i fd k k k ,r) + hcj + surface terms, (2.2.28) AZ2' (2.2.29) where hcj stands for Hermitian conjugate, r r', io 4* io, X X*; the surface term involves the bilinear concomitant. The first part of (2.2.29) is called the volume contribution. It will be denoted by S When the bilinear concomitant is zero as is a often the case the volume contribution equals the bulk contribu tion (2.2.29). Often the interesting part is to know the spectra of the total value or spatial average of variable a(r,t) in V Thus let s A(t) = f a(r,t)d3r. V (2.2.30) The integration yields for (2.2.20) Sources form: bulk S (c) = A 3 3 \kW^W f f d rd r' F. k Z v v (V k+io) i) s s 3 3 x f f d r1d 3r2S (1~ 2)k1* IF/(l) 2) VV The correlation Let us suppose, (2.2.31) form, obtainable from (2.2.29) contains fd r'rT(,",e'). as is often the case, = [( (2.2.32) Then we find Correlation form: vol 4(AA2) 3 3 k S Vol =4 Re f f d3rd3r' Z k() (2.2.33) A V+= s V V k Xk+i+ s s In general \ = Pk+ivk is complex; for self adjoint case vk = 0 and (2.2.33) reduces to ol = 4AA2> pk fd3 rk( 12 S (W) = Z (2.2.34) a V 2 2 s k pk+o Finally, note that the eigenfunction method has several drawbacks. If the domain V is finite, the results occur in a series form which is not easily evaluated. If the eigenfunctions can be taken to be plane waves, then we end up with the integrals over ,space which are not trivial. We therefore prefer the Green's function method, whenever the Green's function can be evaluated in closed form, as is the case for suitable geometries (one dimensional bar, cylinder,sphere). The limiting spectral ranges can then be obtained. 2.2.1(b) Green's function procedure The Green's function procedure is much more direct. The Langevin's equation (2.2.1) is Fourier transformed with io t n 2nn aC(,t) = Z an () e ,n T, n=0,+l,+2,... (2.2.35a) n iW t (SC,t) = Z, %() e (2.2.35b) n This gives Cic +A )an( = n(.). (2.2.36) The Green's function G(Z,Z',in ) is defined by (in +A )GC,,',i n) = 65(I'), (2.2.37) and satisfies homogeneous boundary conditions corresponding to equa tion (2.2.3). Using (2.2.37), the solution of (2.2.36) is found by using the standard procedure and() = J drlG ,,' n) ). (2.2.38) V The spectrumis found by the result analogous to (2.2.17) bulk Sa (ug,',o) = lim 2T(an()a*((')> T = f d rld r2G R~1 i)G ( ,i)S (1 2) ; (2.2.39) VV bulk for S we find bulk 3 3 3 3 SbA k() = f / d rd r'f f d3rldr2Gc', ,io)G(', 'E i)S ( l'2) Sv V V V (2.2.40) These are the sources Green's function forms. It is well known that Green's functions can be expanded in a biorthogonal series G ',i = k k+i (2.2.41) When (2.2.41) is substituted in (2.2.39), the result (2.2.20) is recovered. 2.2.2 Surface Noise Now we consider the transport equation (2.2.1) with ( 0 and while we have stochastic boundary conditions (2.2.4); we also assume that and are uncorrelated. (If this is not the case there cross will be a contribution S .) 2.2.2(a) Eigenfunction expansion This method has not been tried due to the complexity of a series expansion in eigenfunctions; however a response method without Green's functions can be derived. 2.2.2(b) Green's function procedure Again, we make a Fourier analysis of the (homogeneous) Langevin equation and of the stochastic boundary condition (2.2.4) (ice +A )a (Z) = 0, (2.2.42) man () = n(). (2.2.43) The solution of these equations leads to an ) = fdaoH( ,i0 n o) (2.2.44) (do = d2 r ), where H is the surface Green's function. For the spectra we now obtain due to the surface sources ,Z )= jd0 H (2.2.45) surfa IL',W ) = fdao doHCC(,o,ii)H( ',o,ic)S(o ,) (2.2.45) surf while for S one obtains A Ssurf(] = f j d3rd3r'l f do do.H( ,jO,io)H(1' ,,iw)St oo). V V SS s s C2.2.46) In particular, let S = Zr)6(2) CT). Then surf,) = j 2r Z(C )I f d3rHCX,o,iw) 2. (2.2.47) S V s These are the sources Green's function forms. The covariance function associated with the surface noise is surf ) surf ,surf ') = Re Saurf, ). (2.2.48) 0 It can also be shown that rsurfr') satisfies the homogeneous Atheorem (Ar+Ar) surff) = 0 (,r' E V). (2.2.49) Total covariance r = bulk +surf, therefore,satisfies the Atheorem which is (A +A ,)r ') = ') = (,), (2.2.50) where E is the correlation strength of the volume sources. 2.3 The Phenomenological Equation and Correlation Methods The phenomenological equation corresponding to (2.2.1) is L(a,t)>a ,0) +( +A r)<(a(t)>a (O,) = 0, (2.3.1) a~r;, OP where >al,0) means a conditional ensemble average for systems ar,0) with fixed fluctuations at time t=0. The boundary conditions cor responding to (2.2.3) or (2.2.4) is m(a(x, 't)>a(,0o) = 0. (2.3.2) 2.3.1 The Volume Noise Since the surface sources average to zero, it is unlikely that the noise found by correlation method includes surface noise. Moreover, the correlation method does not give the complete bulk noise; in fact, it only gives what is identified as the volume noise in section 2.1. The reason for this is that the covariance func tion F, which figures centrally in this method, only represents the solution of the inhomogeneous Atheorem. The surface source and other boundary spectra give covariances which satisfy the homo geneous Atheorem. 2.3.1(a) Eigenfunction expansion The stochastic variable is once more expanded in eigenfunc tions 0k(r), which satisfy equation (2.3.2), this yields for for equation (2.3.1) ak (t)>{ak(O)} + k(ak(t)>{ak(0)} = 0, (2.3.3) which gives kt ak(t))>ak(O)} = and furthermore k(t) = <(a(t)a(0)>s = a(t)) Finally for the solution one obtains Svol aka. = 2 /dt[eict(ak(t)a*(O)>s + 0 =2( s 2(a ^ + itt))] e (2.3.4) We have used here the transposition property for the correlation function 4kt(t) = 'Sk(t). With an expression similar to (2.2.19) we find vol 1i Sa k t,z' ,) = 2Z i t/ ) aka s ). Next we consider the expansion for the covariance function r;E1,2) = = (aka sk l)1 (2). ke (2.3.5) (2.3.6) Multiplying with k(C ) 1 2) and integrating over all 1 and X2' one obtains (ak, a,> = ffd3rd 3r2 F(C1r2)4, l)d*, C2). (2.3.7) Substituting this into equation (2.3.5) we obtain Svol a =2 Id rd3r Vk d)) (1)) ( ) kt V V 1 1 X[Xk+ico 1 2). Xk~iwXCe (2.3.8) Carrying out the required integration, the result is obtained as Sa(,,',w) =2 Z fd3r"k )3(r") (E )F(",r') + hcj, (2.3.9) a k V k which is the same result as obtained previously via the sources method in equation (2.2.29). For the result of the noise of A(t) = f d3r a(,t), V using this relation one obtains (2.2.33) or (2.2.34). s 2.3.1(b) Green's function procedure For this procedure we need the solution of equation (2.3.1); forthis purpose we FourierLaplace transform equation (2.3.1), i.e. let 0 aCr, i) = J e (aFt))ad,.0) dt. (2.3.10) Then equation (2.3.1) yields Ci+Ar )a(,ia) = a(~,0). (2.3.11) The Green's function is given by Cic+A )G(X,,r',iw) = (6,'), (2.3.12) with the boundary condition mGxo,,r' ,i) = 0 corresponding to equation (2.3.2). By the standard procedure we find the solution as aC,id) = f d3r"G(,~",io)a(V",0). (2.3.13) V Therefore, by inversion of the FourierLaplace transform (aCt)>)a, ) = jideit fd3r"G(Z,,e,,i)aC(",0) (2.3.14) t))aO) 2 C V where c is a complex frequency and C is contour encircling all poles. For the correlation function we have (t ? 0) ,(I,.',t) = = dicot f d3r"G(C,",i )F 'bulk ,Z"). (2.3.15) C V For negative time (t) we use the transposition property C,*' ,t) = 4~(,',,X,t). For the spectrum the WienerKhintchine theorem then yields, W1 i(co)t 3 bulk Sol(,r,) = fdt dei()t fd3r"G,",i)bulkr',r") + hcj. a ,, 0 C V (2.3.16) Interchanging the time and contour integral we have f df dt ei(m )tG,", i) = f ds[i +68(Go)]G(,a",ic) C 0 C ) =2nG(,r,", io+0)(by Cauchy's theorem) (2.3.17) So now substituting the integral in equation (2.3.16) we get the result Svol ,', ) = 2 fd r"GC ,",i) ~b1','") + hcj. (2.3.18) a V This is vanVlietFassett form. The result is in full accord with equation (2.3.9) if we use the series form of the Green's function, equation (2.2.41). 2.3.2 The Boundary Noise The spectrum found by the above procedure does not include surface noise. However, it also does not comprise all the bulk noise, which is to some extent surprising. The reason is that the transport equation is not Markovian in the usual sense. The fact that the di mensionality is infinite means that the usual Markov property W2(a(Xt),a'(X',t')) = P(a(,,t) a'(',t'))Wl(a'(X',t')), (2.3.19) which is the basis for the evaluation of the correlation function in (2.3.15) must be doubted. In other words, the process cannot be Markovian unless boundary behaviour is also specified. The total noise can be broken up in two ways. The first way is Total = Sbulk + Ssurface (+Scross). (2.3.20) This breaking up is according to the considerations of section 2, the terms representing effects due to the bulk sources S.r,t), the surface sources C(;0,t) and due to their possible correlation. In the second way we evaluate volume noise according to section (2.3.1) based on (2.3.15). The remaining part to be evaluated in this section is called Sboundary. Thus Total = Svolume + Sboundary(+Scross) (2.3.21) To obtain Sboundary we start with the response expression (2.2.39). In the RHS we substitute for S the Atheorem, and obtain bulk 3 3 Sba uE ',l) = 2 / f/dd r d r2GC,,iw)GF(',,2,i)(A +Ar) F~,). VV 2 2.3 (2.3.22) We indicate by Fl,j12) here any (interior domain) particular solution of the A theorem, which may differ from physical covariance function by a solution of the homogeneous A theorem. The RHS of (2.3.22) is rewritten as RHS 2 d3 r G (r, X 3( RHS = 2 / drl G( 1,im) f d r2 G(r',2,im)(i Ar ) r(1' r2) V V 2 +2 f d3r2 G( ,', ,icw) J d3rl GC(br,ic)(; (c& )rl'r2) V V 1 3 3 ^ 2 f d rl Gg,'1, ico){f d r2 r2)Lr GCr',2,iw) V V 2 2 +# d roC V ixC l0),G('xg0, ic)]} S +2 f d3r2G( ',~2,i){f d rF(~ r1 2)L G(rr1,ic) V V 1 4+ d2r0CI r0F2) ,GCr0,iw)]}, (2.3.23) S where we have used Green's theorem in the form (L f,g)(f, Ltg) = (Af,g)(f tg) = d7 C[f,g*]. (2.3.24) According to the definitional equation for the Green's function t t t t* t* LX GC,,io) = Lr G* E1,,l ) = 5 1) (2.3.25b) Carrying out the delta function integration we are left with Sa ulk ') = 2 i dr" G(,r,",ici)rI,r ,r') + hcj a V +2fd3r" G,,r",im) f d2r C[F(r",ro),G(r',ro,ic)] + hcj. (2.3.26) S vol The first two terms comprise S The last two terms, together with S comprise ndary. The Green's function form for surf was given in (2.2.45). We then have a V S 3 3 2 2 + hcj + j dr d3 r' d2r0 d2r' H(,~0,iC)H(g', 0,ic). (2.3.27) V V SS s s We note that all terms in (2.3.27) are quadratic in the Green's function. Thus the division of S into Svol and Sbound separates the spectra in expressions linear and quadratic in G. The merit of this separation is that in many instances all terms of Sundary are zero for homogeneous systems and delta covariance function. For in homogeneous systems such as transitors, the boundary terms are of great importance, however;when Sboundary = 0, the evaluation of the noise by Svol as given in section (2.3.1(b)) is by far the fastest pro cedure. In equation (2.3.26) we used a particular solution r Another vol choice of this solution, i.e., r', leads to another division of Sv and Sboundary. However, the total result (2.3.26) is unique. Usually one chooses r such that the spectrum Sy vanishes, if possible. If on physical grounds it can be argued that there is no boundary spectrum contribution, then this choice of r is clearly the physical covariance function. 27 So far we have discussed the mathematical methods to obtain transport noise spectra. In the next chapter we shall discuss and briefly explain the different physical noise sources, their spectra and importance in calculating overall noise spectra of the physical system. CHAPTER 3 PHYSICAL NOISE SOURCES 3.1 Introduction We have seen in the previous chapter that a calculation of noise spectra with the Langevin or the sources method requires that the spectra of the Langevin noise source must be known. There are various physical noise sources with known spectra. Furthermore, we have also seen that there are two different types of noise, viz. first volume or bulk noise and second surface or boundary noise stemming from volume noise sources and surface noise sources, respectively. In this chapter we shall try to explain both types of noise sources as we encounter them in physical systems. 3.2 Volume Noise Sources15 3.2.1 Particle Diffusion Noise Source To explain the diffusion noise source let us assume that the fluctuations in number of particles (electrons or holes) are governed by the diffusion equation, say for electrons (An) + DV(An) = 0 (3.2.1) where An is the fluctuation in densityof electrons. where An is the fluctuation in densityof electrons. The Langevin noise source representation of this diffusion equation in the form of the standard diffusion noise source (,t) is given by a (An) + DV2An) = ) (,,t). (3.2.2) The source r follows from the stochastic current as follows. For (D)diff we have the randomness diff = qDV(An) + qT(C,t). (3.2.3) P0 diff ^ On a microscopic scale, '(3,t) represents the thermal scattering or Brownian motion velocity fluctuations of the carriers. This q source has a spectrum15 S = 4D(rTC.))68 ,X')I (3.2.4) where I is the unit tensor. We also write (n(Z)) = n0o). The conservation theorem for charge reads an 1 V d(AJ) = 0. (3.2.4a) at q ~ diff Substituting (3.2.3) we obtain n DV2 (n) = ..C,t). (3.2.5) Comparing (3.2.) with 3.2.2), we note Comparing (3.2.5) with (3.2.2), we note t) = C ,t) Hence for the spectrum S=(,'') = : SzCz,') (3.2.6a) = 4DV ., {no0)68C')}. (3.2.6b) This diffusion noise source is the proper noise source of sta tistical mechanics. As we can see by the Atheorem, due to van Vliet8, which connects Langevin's noise source spectra to the covariance function, 1 (A + )rCr,') = 2 E8 CX ') (3.2.7) the standard diffusion noise source gives for the covariance function a delta function: C,') = no(r)6(rr'). (3.2.8) All the previous calculations of diffusion noise by McFarlane16, Burgess Richardson 1, van Vliet and Fassett9, Lax and Mengert3,are based on the assumption that the spatial covariance function is a delta function given as constant x 6(zr'). This result can also be proven from first principles, see ref.9, page 329. 3.2.2 Heat Diffusion Noise Source The heat diffusion noise source can be explained by considering the relation between the heat conductivity a and the heat current K, which is K = aVT + 3Cr,t) (3.2.9) where 11 is a Langevin source. Heat conservation requires that aT cd a + V., = 0 (3.2.10) where c is specific heat per gram, and d is the density. We can find the stochastic heat diffusion equation by substituting (3.2.8) into (3.2.9) and by using T = (T>+AT, BAT 2 Vg~gt) aA_ CV (AT) = 1,t) = (,t) (3.2.11) at cd where a = a/cd is the thermal diffusion constant. The correlation strength or white noise spectrum, of the source Ti has been found by van Vliet, using the Boltzmann transport equation, to be S ,r') = 4Ik T() 2C )5rr'), (3.2.12) where T(C) is the time or ensemble averaged temperature at position x and I is the unit tensor. Clearly SC,') = (4k/c2d2) V .V Tr 2oar)6Fr')} (3.2.13) which is the spectrum of the standard heat diffusion noise source.14 3.2.3 Voss and Clarke's PSource19 Voss and Clarke introduced another noise source, which has a delta function type correlation function and a white spectrum given by SVci ( PO')= r' (3.2.14) where P0 is a constant. As we have seen previously noise sources cannot be chosen ad hoc due to limitations imposed by the Atheorem and statistical mechanics. Using the Atheorem the three dimensional spatial covariance function corresponding to the P noise source is found to be p2 rz') 16 D (rr (3.2.15) Such a longrange correlation function is highly unphysical. 3.2.4 Particle Volume GenerationRecombination Source Usually noise due to generation and recombination of carriers results in fluctuations in the resistance of the specimen under consideration. The noise can be described by the fluctuating number N of the carriers, say for electrons; then the equation governing the fluctuations in N is d = g(N) r(N) + Ag(t) Ar(t) (3.2.16) where g(N) and r(N) are generation and recombination rates and Ag(t) and Ar(t) are the randomness in these rates. Substituting N = N +AN, where NO is the equilibrium number of carriers and neglecting higherorder terms in AN yields the equilibrium condition g(N0) = r(N0) and the linearized Langevin equation, given by ddN AN AN dA + Ag(t) Ar(t) + + (t) (3.2.17) dt T gr where 1 dr d T (dN dN) NO The Langevin source gr(t) can be obtained by assigning shot noise to the random transitions Ag and Ar: SAg = 2g(N0) 2g0 (3.2.18) SA = 2r(N0) 2r0 = 2g0 (3.2.19) SAg,r = 0 (no correlation). Hence SEgr = S g + SAr 2SAg,Ar = 4gO. (3.2.20) Making a Fourier analysis of (3.2.17) one finds for the spectral connec tion 2 SNW) = .2 (3.2.21) SA(+l+oT ) l+CO Integrating over all frequencies, we find 2 1 go (AN2) = SAN() d = g [r'g C(3.2.22a) which is Burgess'20,21 grtheorem. In transport systems the spatial dependence must be added. Thus, for the density n(Z,t) subject to diffusion and generation recombination processes we have aAn(r,t) + DV2 An (, t) + An,t)t). at T diff('t) +agr 't)" The gr noise source now becomes S~gr ( ') = 4g056C'). (3.2.22b) gr 3.3 Surface Noise Sources 3.3.1 Particle Surface GenerationRecombination Source Just like generationrecombination processes in the bulk give rise to a bulk Langevin noise source, so generationrecombination processes at the surface can be represented by a surface noise source; this latter source causes surface transport noise, at the surface of the sample. For example, for the case of nonzero surface recombination velocity, the stochastic boundary condition can be written as DV 0(An) + SAn = (m(0,t) (3.3.1) where D is thediffusion constant for electrons V0 is the surface n ~0 concentration gradient and K(~0,t) is the surface stochastic Langevin noise source. For the threedimensional case this surface noise source has a spectrum S =4MaSn()5 (2) 0r) (3.3.2) and for the one dimensional case S = 4MaSn(X0)/A where M is the modulation factor > 1, S is the surface recombination velocity, A is the area of surface, a = (AN2>/N0 and (2)C0_) is the two dimensional delta function. 3.3.2 Heat Transfer Source The heat transfer source is a stochastic source due to ab sorption and emission of blackbody radiation at the surface of a body. For example, this occurs in the case of a thin film supported by a substrate with the front face of the body being blackened, so that it sees the blackbody radiation field. The boundary condition in this case becomes oCO) aAT gAT(O) = E(t). (3.3.3) axlx=0 Here (Ct) is the stochastic heat transfer source. The spectrum of this source is given by the generalized Nyquist's theorem S = 4kT2g = 16cBkT5 (3.3.4) where g = 4a T3 is the heat transfer conductance, TO is the tempera ture of the environment and the mean temperature of the body, and cB = 2 5k4/15h3 c2 is the StefanBoltzmann constant. PART B NOISE DUE TO VOLUME NOISE SOURCES CHAPTER 4 SYMMETRICAL EMBEDDED BODIES 4.1 Introduction In this chapter we shall study the noise spectra resulting from symmetrical geometries which are part of an infinite expanse i.e. the "embedded" case. We have already seen in a previous chapter that there are two volume noise sources, namely the standard diffusion noise source and Voss and Clarke's 'P' source. Further we have also seen that the 'P' source is highly unphysical. In this chapter we con sider only noise stemming from the diffusion noise source. 4.2 Survey of Older Work In this section we briefly summarize the results of previous work for obtaining noise spectra from symmetrical embedded bodies like the linear bar, the cylinder and the sphere, prior to the dis covery of the Green's function method. MacFarlane16 in 1950, studied the problem of the power density spectra of noise current resulting from contact noise in semiconductors or from emission noise in field emitter diodes. In these theories low frequency noise is attributed to the random movement of absorbed ions on the contact surface between two granules of a carbon resistor or on the emission tip of a field emitter tube. The current through the tube or the emission of electrons is assumed to take place only at localized patches on the surface and the absorbed ions are assumed to give rise to a dipole layer, which modulates the work function in such a way that AW is linearly related to the concentration of ions in that patch. This concentration fluctuation is due to diffusion of ions over the surface of which the conducting patch is a small part. MacFarlane found, by first calculating the autocorrelation function and then using the WienerKhintchine theorem, that the spectral power density of the noise current is given by RC)) {lexp(x1/4)[I0 ()x1/ 1(x/4)]cos(px) dx (4.2.1) 0 2 where p = wrOP, T = (2r) /D, r is the radius of the patch, and D is the diffusion constant of ions, and I's are the modified Bessel functions. In the limiting cases it was shown that, for a circular patch, (using probability theory results22.) R(c) = 0.2345 p3/4, p << 1 (4.2.2a) and R(o) = 0.1466 p9 p > 1 (4.2.2b) while for a long thin strip RC) = (2p)12, p << 1 (4.2.3a) and R(w) (2p)3/2 C4.2.3b) p >> 1. For the long thin strip this shows the occurrence of the 3/2 power law at high frequencies. This universal law was violated by the result C4.2.2b); Burgess showed MacFarlane's result to be in error; when properly integrated, a 3/2 power law at high frequencies also result for the circular patch. 4.3 Previous Work Using the Green's Function Method 4.3.1 Linear Bar Van VlietFassett9 considered similar problems, as discussed above, using the Green's function procedure. For the onedimensional diffusion equation A D a2 (4.3.1) at ax2 ax where Ap is a fluctuation variable and D is a diffusion constant, the Green's function is found by standard procedure to be GCx,ico,x') = pexpCiu0 xx'l) C4.3.2) 2u0D where u0 = Vw5U e3i/4. The spectrum for the fluctuations AP, where P is the total number of particles in a segment L is found to be SAp( ) = D Cle (coso+sin6)) (4.3.3) D8 where e = L() /2, in agreement with Burgess' solution17 of the integral given by MacFarlane16. The low frequency spectrum goes as 1/2 3/2 1/2 and the high frequency asymptote is again 3/2 The latter behavior is characteristic for all diffusion spectra. 4.3.2 Cylinder or Circular Patch Van Vliet and Chenette23 studied the noise spectra resulting from diffusion processes in cylindrical geometries. The spectral densities are derived from the relevant Green's functions. Also extensions to MacFarlane's16 spectrum for the circular patch were obtained. The noise spectrum for the fluctuating variable X, where X is the total number of particles in the cylindrical volume Vs, in terms of the relevant Green's function, is given by 4varX S M(c) = 4vX f G(Z,ic,X')dv dv' (4.3.4) s DD s s where Vs is the volume of domain Ds (cylinder or circle), which is part of an expanse V > Vs. The Green's function was found to be the Hankel function of the first kind of order zero; the spectrum is given by S C)= 8CyarX)a2 1 1 ii (1) S D) = aD Re I [ 1 + (xa)J1(xa)] Cxa) C4.3.5) where H and J are Hankel and Bessel functions, a is the radius of the cylinder, D is the diffusion constant and x = ivo . D It was found that for low frequencies the spectrum has loga rithmic behavior and at high frequencies the 3/2 behavior takes over. 4.3.3 Sphere The problem of three dimensional diffusion has been solved by van Vliet and Fassett9. Using the method of reciprocity, the 2 spectrum was found to be (for fluctuating variable AP, with AP = Po), 16TPR 5 GAP16P) o= 5 y 22+e [y Ccosy+siny) +4ycosy +2(cosysiny)]}, DY (4.3.6) where PO average number of particles in the sphere RO radius of sphere D diffusion constant and Y = 2ROVAC/2D) At low and high frequencies the spectrum was found to have the limiting values: 32TnPOR GA ) = 15D (4.3.7a) 00 P0(2D) 4nR G p( ) = 3/2 (4.3.7b) COW Again, at high frequencies the spectrum follows the 'universal' 3/2 power law. 4.4 Present Work: Carrier Fluctuations in a Sphere Due to Diffusion and GenerationRecombination In this section we investigate the noise spectra resulting from fluctuations in the carriersin a sphere, due to diffusion across the boundaries and due to sinks or sources inside the sphere. This problem has not been solved hitherto. We consider a sphere of radius 'a' (volume Vs) which is part of a very large expanse. The fluctuating variable is x, the concentra tion of particles in the spherical volume under consideration; the total number of particles in the sphere is X. The boundary conditions on the surface Ds of the sphere are 'fictitious' i.e. x and its derivatives are continuous across the boundary. We consider the equation ax 2 + ax DV x = 0 (4.4.1) at 1 where a = T is the volume lifetime and D is a diffusion constant. The Green's function g(r,tJl',t') satisfies the equation  (F,tr',t') + agC(,t[t',t') DV2gC,t l',t') = 6(tt')6r,'). (4.4.2) Assuming t' = 0 as initial time, with g(r,Or',0)=0, and taking the Laplace transform of equation (4.3.2) we obtain, SG(Z,s,r') + aGCr,s,r') DV2G(Cs,g') = 6(1,'). (4.4.3) This can be written as sG a ,sc') G(z,s,E,') + V2G(T,s, ') = 16') (4.4.4) where (x2+V2)GZs,') = ') x = i .F (4.4.5) (4.4.6) Since the boundary conditions on D are continuity of x and its derivative and the domain D is assumed to be very large, the ultimate boundary condition on the boundary of D is that G should behave properly at infinity, i.e. go sufficiently fast to zero. Adopting a spherical coordinate system we have 2 2 1 a 2 a 1 6 a 1 V r 2) + + ' (sine ).(4.4.7) 2 sr r 2 2 2 i 2 ae as . r r sin a(p r sinO Following Morse infinite domain only depends on with R = 0 as and Feshbach page 808, one can show that for the the Green's functions in spherical coordinates r' = R. So, evaluating V2 in spherical coordinates origin, we find 72 = 1 a CR2 ~a S2 aR l R R 2 a2 R aR aR2 R R Also we can write GC(,s,Z') = Gx(R). So, the Green's function equation (4.4.5) becomes S2Gx(R) 2 aGx(R) 1 (R) R + R R + x2G(R) = 16R R2 R aR x D 4R2 8R 4nR (4.4.8) (4.4.9) (4.4.10) Following Morse and Feshbach26, page 809, we need a solution for the above equation which goes as Sfor R 0. (4.4.11) R Therefore we require the solution to satisfy the following conditions 1 i) G (R) I as R 0. (4.4.12a) x In addition, the noise spectrum for high frequencies c * should go to zero, or ii) lim Gx(R) = 0. (4.4.12b) x '+im The only solution for equation (4.4.10), satisfying the above two conditions is ixR e Gx(R) = 4RD C4.4.13) The proof of the solution is as follows. +ixR +ixR a +ixeixR eixR a 1 x(R) = 4TRD D R C4.4.14) 2 .+ixR +ixR 2 2R) 2ixe ai 1 e x a2 (R) = x G + D R 4R 2 4R (4.4.15) aR aR +ixR +ixR 2 a 2ixei 2e a 1 4G (R) = (4.4.16) RaR 42 DR aR 4nR . The second term of (4.4.15) can be shown to not contain a delta function; it cancels the first term of (4.4.16). The third term of (4.4.15) and the second term of (4.4.16) are combined to yield +ixR 2 +ixR e a 2a 1 e 2 1 a + 7 ( C4.4.17) D a2 RaR 4R D 4KR aR For the latter we have +ixR +ixR ix 2 1 e ixR 6(R) D 41TR D 6(z') = 2.2 (4.4.18) 4wDR To prove (4.4.18) we must show R22 = 0 if R / 0. (4.4.19) and / V2 d3R1 = 4r. (4.4.20) small sphere The first statement follows from substitution of (4.4.8). For the latter statement we use Gansz' theorem: 213 1 f 2 1 d = grad r dS. (4.4.21) AV Trrl Ir'r ' Now grad(1/rr' ) = 1/R2 and dS = 4ffR2, which proves (4.4.20). We notice that the solutions +ixR GCR  x ) e4TRD satisfy the equation (V2+x2)G (R) = 1 (,'). Using the condition of equations(4.4.12a) and (4.4.12b), we find that the condition (4.4.12a) is satisfied by both solutions but using equation (4.4.12b) we find +ixR R(i) l e e lim RD (4.4.22) 4nRD 4nRD X ++~+a. and the condition is satisfied only by the solution +ixR GxR = eR (4.4.23) which is the required solution for the Green's function. which is the required solution for the Green's function. For the fluctuating variable X, the noise spectrum is given Sx (c) 4varX = vf G(r,i d,r') dV dV' s D D s s (4.4.24) The required double integration sought for the noise spectrum is I = f f d3r d3r' G (R). V V s s Further R can be written as R = Vr2+r'22rr'cose (4.4.25) (4.4.26) where 6 is the angle between r and r'. Using polar coordinates with r' as polar axis and integrating (4.4.25) first over r we need the integral (denoting by Il) ix r2 +r' 2rr'cose = fff r dr sine de dip 4nDVr+r' 2rr'cos (4.4.27) Further we write this integral as 12 2 SixVr +r' 2rr'cose = fr dr fdpfd(cosO) e 4nD r2+r'22rr'cose (4.4.28) Realizing that integration over cp contributes 2n and further that 2 2 d ixVr +r' 2rr'cose d(cose) e '2 2 ixVr +r' 2rr'cose ixrr' e 2' c2 r +r' 2rr'cos9 (4.4.29) we find that (4.4.28) reduces to S 1 a ix(r+r') 'ixl r',r I1 2xD r'i rdr [ei( e 0 (4.4.30) As to the double integration (4.4.25), we can write aa a r' a a f f f(r,r')dr dr' = f dr' f dr f(r,r') + f dr' f dr f(r,r') 0 0 0 0 0 r' and it can be shown that the second contribution equals the first one, i.e. aa a r' f f dr dr' f(r,r') = 2 f dr' f dr f(r,r') (4.4.31) 00 0 0 because of the symmetry of the integrand f(r,r'). We can thus restrict ourselves to r 5 r' and omit the absolute sign in (4.4.30). Realizing the fact that integration over the solid angle d2 con tributes 4n, the required double integration yields 8 a reix(r+r') eix(r+r'r)(r'r) ix(rr') r'8dr re + e re e r' I = f r'dr' + 2xDi rdr ix 2 ix 2 0 x x 0 (4.4.32) or 4TT r' 1 2ixr' r' 1 I = J r'dr' [( e2 + +e +( ]. (4.4.33) xDi ix 2 ix 0 x x Using the following relations due to integration by parts ir 2ixr' 2ixr' 2ixr' r'e e fr'dr' e = + (4.4.34) 2ix 2 4x and 2 2ixr' 2ixr' r2 2ixr' r' e r'e 1 2eixr' r' dr'e i 2 e, (4.4.35) 2x 4ix the required integration is finally found to be 41i a 2 2ixa ae2ixa e2ixa 3 a2 xDi 2 .3 4 3i (4.4.35) i 2x ix 2x 3i 2x 2x Using for the noise spectrum equation (4.4.24) and representing the AP l' P fluctuating variable as p, with = = p, the number of holes s s per unit volume, we get Sp(o)=Re64pOa [eP ( p24p+4)+( + 24)] (4.4.36) D(p where p = 2ixa. Equation (4.4.36) can be rewritten as 5 64np a 3 S Ap(c)=Re 5 (4e 4(pe +pe + 2 4]. (4.4.37) For the required noise spectrum we need to evaluate the real part of expression (4.4.37). We shall discuss two different cases. 4.4.1 Pure Diffusion (a=O) If no sinks or sources are present then a=0. In this case the required expression for the noise spectrum, taking the real part of equation (4.4.37), becomes 16ta p 2 SAp (c) = 5 _(2+y2+e [y2 (siny+cosy)+4ycosy+2(cosysiny)]}, DY (4.4.38) where y = 2a ' 2D The result (4.4.38) is the same as previously obtained by van Vliet and Fassett9 for the case a=O. The low and high frequency asymptotes are found to be (expanding sin and cos terms) 5 32npoa S (M) 15D (4.4.39a) and 1/2 2 p0(2D) 1/24a2 (4.4.39b) S p(c) 3/2 (4.4.39b) The plot for the noise spectrum is shown in the rn=0 curve of fig.1. Spectrum is constant at low frequencies and at high frequencies the spectrum goes as 3/2. The computer program used to plot the spectrum is given in the Appendix. (u=wTd, n=aTd). 4.4.2 Diffusion With Sinks (O0) When sinks are present a is not equal to zero. The required noise spectrum expression,taking real part of (4.4.37) becomes 16na5p0(p262) 16Tap0 K S Cp ) = 2 + (4.4.40) 3D(p2+6 2 D( 2 +6 5 where K = (10p2635p4655)[2p6+26e cos6(p+2)+e.sin6(62_ 244p)] (P5lOp 35+5p64 ) 252265e sin(p+2)+e=cos6(P 22+44p)4] (4.4.41) = 2a [v + ]a + (4.4.42) and  6 d 2a [ Zi + oJav+c ]. (4.4.43) Vff *U Fig. 1 Noise spectra for embedded sphere (n=0, r~0) Furthermore using the following dimensionless quantities Td = a /D, diffusion time u = Wrd d (4.4.44) and ) = "d/yv = d we get for p and 6 p = "TV + I+u2 (4.4.45a) and 6 = Vn+ V + (4.4.45b) If in the above expression q = 0, we have p = 6 = y and equation (4.4.40) reduces to equation (4.4.38). The general case a 0 has been considered for the first time as far as we know. The low and high frequency asymptotes are found to be the same as for a= 0 5 32Trp0a S ) 5 (4.4.46) AP 15D and P (2D) 1/24ra2 S () 2 (4.4.47) WAP W3/2 The noise spectrum has been plotted in fig.1 For low fre quencies the spectrum is constant and at high frequencies, the spectrum goes as c3/2. For q >> 1 if a range can be found such that Ts u < ) 3,then the spectrum goes as c2 and at higher frequencies W3/2 behavior takes over. The computer program used to plot the spectrum is given in the Appendix. CHAPTER 5 NONSYMMETRICAL EMBEDDED BODIES 5.1 Introduction We have seen in chapter 2 that for nonsymmetrical geometries Green's function cannot be found in closed form and therefore eigen function expansion methods should be used. In this chapter we shall study the noise spectra arising from nonsymmetrical embedded bodies for both VossClarke's19 'P' source and the physical diffusion noise15 source. 5.2 Voss and Clarke's 'P' Source 5.2.1 Nonsymmetrical Bar We consider the diffusion in an infinite vdimensional domain with fictitious boundary conditions i.e. embedded case. In all integration of chapter 2 d3r is replaced by dVr. The norm alized eigenfunctions of (2.2.2) are (Pk) = k /) e (5.2.1) with = (kl,...,k) and 2nn. ki = L (5.2.2) 1 v n. = 0,l,2,... and V = 7 L. ; (5.2.3) 1 i=l for embedded case V goes eventually to . The eigenvalues are 2 2 2 2 2 =k = = Dk where k kl+k2+...+k (5.2.4) 1 2 (.2.4 Since kvalues are dense, we can write Z fdk Z(~), with Z(J = (5.2.5) X (21Y) being the density of states for a domain of dimension v. Since all noise expressions are bilinear in (p or *, V cancels out for any sum Z in the limit V +o. Then, equation (5.2.5) is exact and the eigenfunction expansions simply represent Fourier integrals. Thus, (2.2.31) results in, with f dvk denoting integration over infinite v dimensional )space, . I vdr 2 bulk. 1 ( C dvk dvk s A (2 ,) (Dk2+ico)(Dk'2ico) A ( d1rV ri 2vr x fJ dVrl dVr2 S IC; Z2) e l (5.2.6) For Voss and Clarke's P source S(' = p56(1i ) (5.2.7) So for (5.2.6) we now obtain i. ik.r v 2 2 le e e d r P V bulk 0 v s bulk dvk (O) = (5.2.8) SA =2 fd k 4 2 (5.2.8) "A (2n) D2k +C2 L. L. Now considering V to be a rectangular bar of dimensions xi  2 2 bulk i=l,2,...,v, the result for SA becomes bulk 2 2 dvk v sin21 (iLi) A (o) PO(V s 2 4 2 2 (5.2.9) D k +w i=l (k.L.) 21 1 For a three dimensional rectangular bar v = 3, and 3 3 sin2 (.L.) bulk 2 2 d k 2 1 1 s () P(Vs) / ni A 0 s k i2 l 2 kiL) For a three dimensional 2 at = D/L., and for 1 1i' the following frequency a << e1, 1 << W < 02' '2 << << W3% COQ CO) rectangular bar the spectrum shows breakpoints the case L>> L2 >> L3 (nonsymmetrical) ranges are claimed by Voss and Clarke9, c .. ..1/2 1 SC() 2 c S 3/2 S O) C2 S (c)  e (5.2.11) Voss and Clarke1' noted in particular the 1/c range between limits 01 and c2. In an effort to find the breakpoints and the spectral frequency ranges the method of numerical integration was used. The complete analysis is given below and the listing of the computer program which uses the trapezoidal rule for numerical integration is given in the Appendix. Explicitly, for the three dimensional bar we have bulk SA (C) = const. f f J  dk dk dk x y z 2 2 2 2 2 D (k +k .+k ) +o x y z sin2 (kxL) sin2( L sin2(kL 1 2 1 2 1 2 =x k L x L xkL  SkxL1 2 L2 2 1 2 (5.2.10) (5.2.12) Let us call kL = k L2 k L3 = kz L3 Using the above transformation, we get as. integrand dt dr dr X L xL2xL 123 D 2 D 2 D 22 2 L L2 L 1 2 3 1 sin 2 x C ) 2 1 sin 2q x C) x 1 sin 2 ( 1 2> (5.2.14) Furthermore call S= D/L C1 = D/ (5.2.15) S= D/L, = D/L2 I2 = L2, C = D/L 3 3 O = O/V 12 3 , C O = F w 2 0 ) 3 _______ (5.2.16) "2= 2'1 3 = p/W 12 * We then obtain the integral in the following dimensionless form, with F(c) = const. S(o): (5.2.13) Fig. 2 Noise spectra for nonsymmetrical bar CVoss and Clarke's 'P' noise source) slope 1.5 slope 2.0  401 I 10' 10' 10"' I I0 100 + 40 + 30 * 20 + 10  20  30 0.3 ria \ 4 1 1. 1 F d dg d sinsin si2 1 2 n 2 F( f 2 2 22 1 1 (1 2 +3 +2 2 (5.2.17) Finally we choose as parameters L1 = 109m, L2 = 1cm and L = 0.1cm, D = 25cm2/sec (corresponding to mobility of 1000 cm /voltsec.) Then we have wl = 0.25rad, 2 = 25rad, w3 = 2500rad; and C1 = 102 2 = 1, and W3 = 10 These are the normalized break point frequencies. (Units for w's should read as Rad/sec.) Substituting all the above constants and using the symmetry property of the integrand we have for (5.2.17) 1 1 1 F d8 d df in n 2 sin 2 2 2 222 Cl17 0 0 0 (.01+ + lo +100o2 ) +c 22 (5.2.18) The plot of the spectrum is given in fig. 2. We have not been able to verify results as in (5.2.11). The spectrum has no 1/f range and goes much steeper at low frequencies; only the break points at C2 and c3 are visible. 5.3 Physical Diffusion Source We have seen in chapter 3 that the physical diffusion noise source has a spectrum S 4 = 4D V'n(X)6(X,'). (5.3.1) When this is substituted in equation (5.2.6), which is the general result for nonsymmetrical embedded diffusion, one finds for the spectrum 2 1 v 2 v sin2 (Li) bulk d k Dk2 sn2 i (5 S (m) = C 2 4 2 (5.3.2) "A D2k4+co i=l C.kiLi) which differs from equation (5.2.9) by the factor Dk2 in the numerator; C is a proportionality constant. (For particle diffusion C = 4n0/(2n) .) 5.3.1 Rectangular Bar For a three dimensional rectangular bar equation (5.3.2) becomes 2 1 3 2 3 sin2 (kLi) bulk ( d3k Dk2 i 2i (5..3) SA (C) = C f D2 n42 2 (5.3.3) A 24 2 1 2 SDk+Co i=1 (k.L.) Again for the case L1 >> L2 >> L3 (nonsymmetrical) Voss and Clarke claimed the spectral ranges as f<< fl S(o) = constant f1<< f f2, S(c) = log f f2 << f << f3, S(o) = f1/2 f >> f3, S(w) = f3/2 (5.3.4) Using the same transformation as in section 5.2, the additional factor Dk2 is found to be = D(k2+k2+k2) x y z D 2 D 2 D 2 L L L 1 2 3 1/3 2 2 2 1O2C3 ( 0 2 3n + ) (5.3.5) The overall noise spectrum is given as i2 2 2 M M 00 ddid&(wl +m 2 qCO3 F(c) = C f 2 2f 2 0 0 0 (1+2 +0 3 ) +m 1 1 1 in 2 sin 2 sin, 2 sin2 2) 2 2 2 2 (1 ) x ) x ( x ) (5.3.6) 2I For the values of L = 10cm, L = 1cm, L3 = O.lcm, and D = 25cm /sec, this becomes ddrddt (.012O +T2+100o 2) FC) = Cf f 2 2 2 2 2 00 (.012 +2 +100 ) +W2 1 1 1 sin( 1 2 sin2 (5.3.6a) x() x (1) x (1) (5.3.6a) The listing of the computer program is given in the Appendix. We have not been able to verify the results as claimed by Voss and Clarke's equation (5.3.4). The spectrum is given in fig. 3. The high frequency slope is 3/2 as predicted by Lax and Mengert3 Breakpoints 2 and w3 are visible and at low frequency (not shown) the spectrum levels off. Also there is no obvious 1/f range in the spectrum. Furthermore we can also see that in the noise spectrum for the nonsymmetrical bar for both the noise sources (namely 'P' source and diffusion source), figs.2 and 3, at the low frequencies there is a "wiggle" in the spectrum. The presence of such a "wiggle" must be attributed to the nonsymmetry of the volume under consideration. 20 10 slope0.38 slope.0.5  0 . 40 I I I II 20 0 30  I0' 102 10" 10O 10' 10' 10a Fig. 3 Noise spectra for nonsymmetrical bar CPhysical diffusion noise source) The "wiggle" appears on the noise spectrum towards the longest dimension of the specimen. As we shall see in the next subsection, for more sym metrical geometries like square bar and cube, the noise spectrum is smooth and the low frequency "wiggle" disappears (figs. 4 and 5 ). 5.3.2 Square Disc The noise spectrum arising from a square disc has been analyzed. In equation (5.3.6) the parameters have been changed. The longest dimen sion has been eliminated and a square disc of lcm2 cross section and thickness of 0.032cm has been considered. The noise spectrum is given as 2 2 2 F(c) = C f f d dt(0. 12+0.112+100l2) 2 2 2 2 0 (0.12+0.1 r2 +100t )+C 1 1 1 sin2 2 sinql sin 2 x () x () x ( ) (5.3.7a) 2 2 The computer program listing is given in the Appendix. The plot for the spectrum is shown in fig. 4. We can see that the spectrum varies smoothly and the low frequency "wiggle" disappears. Further at high frequency the spectrum goes as f3/2. Breakpoints atl=c 2 and C3 are visible and at low frequencies (not shown) the spectrum levels off. Fig. 4 Noise spectra for square disc and square bar CPhysical diffusion noise source) 10a 10 I 10 10 ITS 5.3.3 Square Rectangular Bar The noise spectrum resulting from a square rectangular embedded bar has been studied. The smallest dimension has been eliminated and a square bar of crosssection 0.lcm2 and length 3.16cm has been considered. Equation (5.3.6) for the noise spectrum is d~drjdt(olE2 2 2 F(C) = C f dgdd (0.02+102102 ) 0 o0.0t +10ri +102 )+2 1 1 1 sin2 2 sini 2 sin 2 S) 1 r2 (5.3.7) 2 2 2 The computer program listing is given in the Appendix. The plot for the spectrum is shown in fig.4 We see that the spectrum varies smoothly andagain the low frequency "wiggle" disappears. At high 3/2 frequencies the spectrum goes as Breakpoints at w=1 2 and &3 are visible and at low frequencies (not shown) the spectrum levels off. 5.3.4 Cube The noise spectra obtainable from a less asymmetrical embedded geometry like a cube has been studied. The longest and the smallest dimensions of the nonsymmetrical bar have been eliminated and we considered a cube of dimension 1cm3. The noise spectra is given by bulk 2 2Cf ddp 2 2 2 ssin2 2 sin2 2 sin 2 S(E +1 + C + 5.3.8) (5.3.8) The plot for the noise spectrum is shown in fig. 5. We can see that the spectrum is very smooth, the spectrum levels off at 1.6 low frequencies and the high frequency slope is f.6 The break point at W2 is quite visible and the roll off is much steeper than for the square bar. The computer program listing is given in the Appendix. Thus in this chapter we have analyzed the noise spectra obtainable from nonsymmetrical embedded bodies for 'P' source and nonsymmetrical as well as less asymmetrical embedded bodies for the diffusion noise source. We note that the spectral ranges claimed by Voss and Clarke are not verified. For the 'P' source their claim for a 1/f noise spectrum is far from true. Further we have seen that the noise spectra for the physical diffusion noise source results in f3/2 type of behavior at high frequencies. There may be a quasi frequency range where 1/f kind of behavior is observed, but this range is small. In the next chapters we shall study the effect of the boundary conditions on the noise spectra, both for the symmetrical and non symmetrical unembedded geometries. a \ 30 40 50 I I II 10"' Ifra 10" I 10' Fig. 5 (Slope indicated is per decade of frequency) Noise spectra for cube (Physical diffusion noise source) CHAPTER 6 SYMMETRICAL NONEMBEDDED BODIES 6.1 Introduction In this chapter we shall study the noise spectra resulting from symmetrical nonembedded bodies. We already have discussed in chapter 1 the difference between embedded and nonembedded cases, the latter being subject to boundary conditions at the surface of the body. We thus investigate the effects of the boundary condi tions on the noise spectra for symmetrical geometries. For example, in the case of surface generationrecombination, we have a mixed DirichletNeumann boundary condition for the fluctuating carrier density An, aAn + bv(An) = 0 (6.1.1) where a = s, surface recombination velocity and b = D, diffusion constant. The presence of boundary conditions results in (i) altering the shape of the spectra from that of the "infinite mediuid and (ii) correlation effects may occur. 6.2 Summary of Previous Work In this section we briefly review the results of the work done so far to obtain noise spectra from symmetrical nonembedded bodies such as linear bar and cylinder. 6.2.1 Linear Bar The problem of one dimensional diffusion in a finite domain was studied by van Vliet and Fassett9. The one dimensional diffusion in the region L < x f L with perfectly reflecting barriers at L and +L is considered, where L > d. The spectral density for the fluctuating variable AP, S p(o) in the subregion d 5 x f d is found to be 3 S ) P0(2L) sine+sinheF(a,be) (6.2. 9 (coshecose) where pO is the average number of particles per unit length, F(ae,be) = sinae coshbe + cosae sinhbe + sinbe coshae + cosbe sinhae and S= 2LV/2D, a = b = L L L Equation (6.2.1) reveals that at sufficiently high frequencies the 3/2 spectrum varies as 32, whereas at low frequencies the spectrum becomes constant (2D)3/2 S C2 = 3 3/2 (6.2.2a) (or (2L) 0 and d d2 (Ld)2 SAp (o) )= 4 (6.2.2b) So0 3L Thus the presence of reflecting boundaries limits the low frequency noise, whereas without walls, as in the embedded case the spec trum rises indefinitely as c * 0 according to 1/2 6.2.2 Cylinder The case of fluctuationsin the finite cylindrical domain with mixed DirichletNeumann conditions was investigated by van Vliet Chenette23 in 1965. Following Morse and Feshbach24, it was shown that the Green's function or consequently the total spectra can be written as total A Stta S + S (6.2.3) where S is the spectammas in the embedded case and SA is due to the presence of boundary conditions. Following this procedure the total spectra for the nonembedded cylinder was found to be 2 total A a St S + S = 4(varX)2 Re P(co) (6.2.4) x x D where 1 2a J1Ca) PCo) = {1 + 2o (xa)2 xa DxJ1 (xa) oJ (xa) J's are the Bessel functions x = i a is the radius and a is a constant;in the case of surface recombination, a is the surface recombination velocity. The spectrum is constant at low frequencies and at high fre quencies the spectrum goes as W2, with the presence of a diffusion 3/2 range 3/2) depending on related time constants. Thus due to the presence of boundary conditions the logarithmic low frequency behavior, as in the embedded case disappears and the spectrum becomes flat. 6.3 Present Work: Sphere In this section we shall consider the case that the domain p = V is a spherical domain of radius 'a' with homogeneous boundary conditions on the boundary of the domain an G ,s, ) + oGCa,s,. ) = 0 (6.3.1) a. where T is a surface position coordinate, n is the outer normal and a is the surface recombination velocity. The differential equa tion for the Green's function is d2G 2 dG 2 1 a a 1 82 1 2 dr + x Gsine a .2 ]G = '), dr r sin 0 acp (6.3.2) where Gx(,;') G(;,ico,r'), and x is the same as in equation (6.2.4) The polar axis we take along ;,'. Following the treatment as in Methods of Theoretical Physics, 24 vol.1, by Morse and Feshbach page 825, we expand the Green's function in terms of a complete set of functions involving all but one of the coordinates (0 and cp) with coefficients which are undeter mined functions of the uninvolved coordinate. Thus let Gx ') Z ,m(r,r')Pm (cose)eim(P') (6.3.3) S,m = 0,1,..., ; m = 4,...,+ . Substituting equation (6.3.3) in (6.3.2) for the first three terms on the left hand side,we get ,2 &,m ,m Sp x 2 dpx 2 E,m m ,im((p(p') 2 [d2 r dr + x Px ]p(cos@)e (6.3.4) Z,m dr and the remaining terms on the left hand side of (6.3.2) give (Quantum Mechanics, Schiff25, page 75) 2 1 1 8 8 1 a m im(P(P') 2 sin (sin + 2 ]p(cose)e r sin 8 a8p (5+1) 5 im((pp') (3 = 2 p (cose)e (6.3.5) r Now expanding the delta function on the right hand side of equation (6.3.2), we get 8 = C8r') 6(p(p')56(9') 2 sine (6.3.6) r Let 8e9 = Z Nm F(',cp')Pm(cos)eimp (6.3.7) m,e 2 25 where (Nm) is the normalization constant; according to25 (Quantum Mechanics, Schiff, page 73, equation (14.16)) N = 2 (6.3.8) Sm 4r (+ m )1 (6.3.8) Then by inversion from (6.3.7) we get F(e','p) = f 6(ee' N mP p(cose)eimed (6.3.9) where dS = sine dedp ;thus FC',(p') = NmP (cos')e im Hence we get _mI = Z m im(mipl') 6(rr') 6') = N2m Z Pm(cose)Pm(cose')e 2 (6.3.10) Sm m,t r Since the polar axis is along r', 9' = 0 and Pm(cose') = 1. (6.3.11) The result of the full substitution and equating term by term, yields 2 Z,m t,,m Px 2 dpx 2 t(t+1) Z,m 5(rr') d 2+dp2 + x2[l +l) ,m= (6.3.12) 2 r dr 2 x 2 2 ddr r Dr N2 tm The solutions of the homogeneous equation corresponding to (6.3.12) are given by spherical Bessel functions of first and second kind j,(xr) and y,(xr) respectively. The complete solution of equation (6.3.12) is given as Fw (r)wt (r'), r 5 r' N p,m Sm p = r2 (6.3.13) r D[W(w i,wi)] wi (r')w II(r), r > r' where W is the Wronskian evaluated at r=r' and we and wiI are two independent solutions which satisfy the boundary conditions at r=0 and r=a, respectively. For r=0 the solution must be regular. We thus have: wt = j (xr) (6.3.14) and w = Ajt(xr) + By(xr). (6.3.15) wSII satisfies the boundary condition (6.3.1) which is aw S II D ) = w (a). (6.3.16) ar a uII Substituting wiI in equation (6.3.16) we get aj (Cxr) Dy (xr)l DA[r ]ra DL = a[Aj(xa)+By (xa)] r=a or ay (xa) aj (xa) DB ar + aBye (xa) = DA r x Aj (xa). Substituting the recurrence relations, see ref. 26, ay (xa) ar = t(xa) xy+1(xa) and aj (xa) t r a jt(xa) xj+(xa) in equation (6.3.18) and simplifying, we obtain B D[a j(xa)xj,+,(xa)] +oj (xa) A a D[ y (xa)xyL+1(xa)] +ay (xa) A suitable choice for the coefficients A and B is: A = i(o+ a a) Dxy+1(xa)] and B = if(+ ) (xa) Dxj+1(xa)]. Substituting A and B in equation (6.3.15) we obtain for coI as S (xa) I a)] I W~I = j (xr)I(a a) y (xa)Dxiy 1(xa)] D iy (xr)[ (o+) j (xa) Dxj+ 1 (xa)] . =a ee+jx) (6.3.17) (6.3.18) (6.3.19) (6.3.20) (6.3.21) (6.3.22) (6.3.23) (6.3.24) Now, iy1(xa) = h )(xa) j (xa) (6.3.25) where he (xa) is the spherical Hankel function of the first kind. Thus we obtain D= (1) (1) (1) D. w1 = j (xr)[(C+ h (xaDxh+1 (xa)]h (xr)(a+ j)j (xa)Dxj+l1(xa)] (6.3.26) which can also be written as wmI = h(1 (xr) je(xr)Qe(xa) (6.3.27) where (1) Da (1) Dxh (xa) (a+)h xa) Q (Cxa) = D. (6.3.28) Dxj + 1 (xa) (+D) j (xa) The Wronskian as defined by wn1 wn, weI WeI' W = nII "mII' (' denotes first derivative) evaluated at r = r' is found to be W(wIw ii) = 2. (6.3.29) xr' Substituting for r f r' in equation (6.3.13) we get xiN2 emr m (1) pm (r,r') = m [j(xr)(h (xr') j Cxr)je(xr')Qe(xa)]. (6.3.30) Finally, substituting equation (6.3.30) and equation (6.3.8) in equation (6.3.2) we find for the Green's function, for r 5 r' SIm)!. (1) GxC 2 D (2 l)( I [jZ (Cxr)hh ( 'j (r)j(xr)j.tCxr')QE(xa)] S 4,m (6.3.31) For the noise spectrum we must evaluate the six fold integral S(o) = 2Re f// r dr d(cos9)dpr'2dr'd(cose')dp'GxC(,')) 6 fold reduced domain where the integral involves the reduced domain 0 < r < r', Integration over (p gives 2n for m=0; it gives zero for m#O. use the addition theorem, ref.25 which gives the relation ixR e i R (1) Z (2Z+l)j,(xr)h1 (xr)P (cose). t=0 (6.3.32) 0 5 r' I a. Next we (6.3.33) Substituting in equation (6.3.31) and separating the noise spectrum into the two parts, indicated by the [ ] of 6.3.31, we find for the first part ixR 2 2 e i S (c)= 4T Re f'"* r dr(dcosO)dcp' r' dr'd(cos9') 4w R 5 fold and (6.3.34) 4var X S () = V Is O). s The equation (6.3.34) for SI(w) is exactly the same as for the spectrum of the embedded sphere computed in section 4.3; this spectrum will now be denoted as SX(c). x Pm(cos9)eim(1 ) The second part of the spectrum has an integrand which is regular for r = r', the solution being the same for r r' as r > r'. Hence, we can at once integrate over the full domain 0 r < a, 0 < r' 5 a. Thus, 2xi 2 2 S (o) = Re 24D r2drd(cose)d(cose')r'2dr'd(p' r 5 fold, full domain x Z (21+l)j (xr)j (xr')Q Cxa)Pe(cose). (6.3.35) 0=O The integration over d(cos6')dp' gives 4v. Further due to the orthogonal property of Legendre functions, integration over d(cos6) contributes 2 only for 4=0: T 1 f P (cose)d(cos) = f Pt(z)PO(z)dz = 2650. 0 1 Thus we have 4xi 2 2 Si(co) =Re ff r dr r' dr' j0(xr)j0(xr')Qo(xa) (6.3.36) (Qo(xa), substituting &=0 in equation (6.3.28). We can separate the integration over r and r'; thus evaluating the integral fr2drJ(r) = fr2dr J1/2(xr) (6.3.37) In equation (6.3.37) we have used the relation which connects regular and spherical Bessel functions J and j, which is26 in(z) = Jn+/2 (z) (6.3.38) We can write equation (6.3.37) as xa 2 2 fr drj0(xri= f X d(xr)J/2(xr) (6.3.39) 0 2 x3 = xa (xr)3/2d(xr)J1/2(xr) (6.3.40) x 0 We can also use the derivative formula for theBessel function which is d[zVJ(z)] = zVJvl(z) (6.3.41) this gives for v = 3/2 the required integral as f z3/2J1/2(z)dz = z3/2 3/2(z) (6.3.42) which finally gives for the integral fr2drj(xr)= a3/2 J3/2(a) (6.3.43) (xa) And similarly we get for fr'2dr'j(xr') = J 3/2(xa). (6.3.44) (xa)3/2 2 3/2 Substituting equations(6.3.43) and (6.3.44) in equation (6.3.36) we get SI ()=Re ia3 2 J/(xa)] 2Q(xa) (6.3.45) Dx and for the spectrum we obtain 4varX SI() = SII() s 4 3 Taking V =s 4 Taking V a ,we obtain for the fluctuating variable X, 6 rivarX S () = Re ivar [J 3 xa) J2Q0(xa). (6.3.46) Dx Denoting the spectrum associated with second term by S (o), we can write the total spectrum for nonembedded sphere as total O a S (o ) = S=(;) + S A(). (6.3.47) Now we shall derive another form for S(co) in terms of Bessel A functions more suitable for combining with S (co). From equation (6.3.34) we can write 8varX 2 2 S = Re 8 r dr r' dr' d(cose)d(cos6') depdp' s 6 fold, reduced domain x x j,(xr)h t(xr')Pm(cos)eim(') (6.3.48) ,m DN (6..48) tm integration over dp gives 2n for m=0 only, integration over d(cose) gives 2 for Z=0 only and integration over d(cose')dp' gives 4r. Thus a r2 8varX 4nix a (1) r'2 SI = Re vT f D r' dr'h) (xr') f r2drj0(xr). (6.3.49) na 0 0 These integrals are similar as found in equation (6.3.37), thas r' 3/2  f rdrj0(xr) = 372 4 j3/ (xr'). (6.3.50) 0 x So we get for SI S8varX 4rwix n a (xr')2d(')(xr')3 ) x (61) S3 H xr')J d2(xr') (6.3.51) I Re Vs D 2 6r)3/2 xt 1/2 xr)3/2 8varX 4ni 1 Ta 3 a) = Re 8V D 15 2 (xr) d(xr)H 1/2 CxrJ Cxr). (6.3.52) s x 0 3/2 The required integral was calculated by Luke27 a =3 Cl (xa4(1) (4a 1) f (xr )3d(xr )H1 Cxr )J3(xr ) = (xa) J3a) + J (xa) H xa). 0 1T 3 2 2 2 2 2 2 And using25 (equation 9.1.27) which is (6.3.53) J5/2Cxa) = J1/2(xa) + x3a 3/2(xa), (6.3.54) and further using the Wronskian relation we found that J (xaH a) J/2(xa)H2(xa) = 2 (6.3.55) Substituting equations (6.3.54) and (6.3.55) we found that for equation (6.3.53) (also using (6.3.38)): (xa) 2i 6 (1) S6 (xa 4 + jl(xa)h1 (xa) (6.3.56) For SI we obtain finally S (w) = ReA [ a+ i j(a)h 1)(xa)] (6.3.57) where A 12(varX) (6.3.58) D (6.3.58) The spectrum for SII from equation (6.3.46) is S =Re 6 2ivar [J32(xa)] 2Q(xa) Dx = 12(varX) a2 (xa 2 x RDxa [2xa 3/2(xa)] Q0(xa) Ai 2 0(xa = Re[ l(xa)] 2Q0(xa) where Q0(xa) from equation (6.3.27) for Z = 0 Q Cxa) Dxh1l) (xa) ch01) (xa) Dxj(xa) (a) DxjI(xa)cj0(xa) For the total spectrum we now find total A 1 (1) .2 X (RKa) 3 3 i + 1(xa) 1 xa) j1(xa) xx) 3x)1 Dxhl (xa)Ch01)(xa) Dxj1 (xa)oj 0 (xa) (6.3.62) Solving for { } quantity first we get ah1 (xa)j0(xa)+oh0) xa)jl(xa) Dxj (xa)aj 0xa) Using ref. 25 (equation (10.1.31)) we have h0)(xa)j (xa) h(1 (xa)j0(xa) So we get for { } quantity i (a .2 (xa) S 1 j1 (xa) (xa)2 (Dxj (xa) o 0 (xa) Thus we get for the total spectrum: (6.3.59) (6.3.60) (6.3.61) i{ } = i 1Cxa) (6.3.63) i{ } (6.3.64) (6.3.65) total X (c) = ARe x 1 2 1 + 1a(xa) (xa)2 3 xa (Dxj1(xa)oj0(xa)) (xa) Further we know that xa = iaV (xa) = a 2i ). Introducing the following time constants 1 S = (volume lifetime) v a 2 a Td = (diffusion time) d D  a T = a S T (surface lifetime) , and also introducing the dimensionless quantities. u = Ord, T = Td/v S= /s, d we can write 2 (xa) = (iu+*). total We can rewrite equation (6.3.66) for SX as A total S (C) = ARe where = A 4(var X)a2 3 D 1 _ J,(xa) iu+[1 iu+r L+ j (xa ] (6.3.74) Dx2a (xa)(o/Dx)j(xa) Using the time constants and dimension less quantities defined in equations (6.3.69) to (6.3.72), we have Dx a 1 2 a(r~+iu) Dx a (6.3.66) (6.3.67) (6.3.68) (6.3.69) (6.3.70) (6.3.71) (6.3.72) (6.3.73) and (6.3.75) (6.3.76) Using equations (6.3.75) and (6.3.76), the spectrum finally becomes 2 total 4(var X)a ReP2() (6.3.77) SX ) = D where 1 3j1(ivi u) 1 P( C) = r+iu  (T'+iu)j l(iVT+iu) +iV+iu j0(iVrT+iu) J (6.3.78) Now we shall discuss various cases. 6.3.1 No Volume Sinks If there are no volume sinks present,then ri = 0. For this case 1 r 3jlCiuT) ] ReP(co) = Re j .. (6.3.79) hu i e iu)jt(iV1fu)+iVn je (iVonu)s fr The first term does not contribute. Using the relations for spherical Bessel functions26 we have sinz j0(z) = z z sinz cosz z) = 2 z z (6.3.80) (6.3.81) Thus we obtain Re P(c) = 3AD+BC) 4a 4(C2+D2) where a = 2 A = (sina cosha+a cosa cosha+a sina sinha) B = (cosa sinha + asina sinha ~ :osa cosha) (6.3.82) (6.3.83) (6.3.84) C = (cosa sinha+a sina sinhaa cosa cosha cosa sinha (6.3.85) D = (sina cosha+a cosa cosha+a sina sinha + Sifa cosha (6.3.86) At low frequencies, u << 1, and the spectrum reaches a limiting value. For finite domain a plateau is always reached. Expanding the trigonometric terms, we find canceling terms up to order a4: 1+55 Re P(c) = 15 (6.3.87) At high frequencies, u >> 1, and the spectrum reaches the asymptote 3 Re P(c) = (6.3.88) For a closer investigation we must distinguish between two cases. 6.3.1.(a) First, let t << 1, or Ts < td This option corresponds to the "diffusion limited" case. In equation (6.3.87) the terms in t are now irrelevant and the spectrum reaches the low 1 frequency limit Re P(0) = At high frequencies there is another 2 2 region, prior to the u2 region, in the range 1 << u < /2. 6.3.1.(b) The surface limited case occurs when >> 1, or Ts > .d In this case the diffusion process is practically instantaneous and the noise is only due to surface sinks. At high frequencies the limiting behavior is 2 for any u >> 1. However, tu2 this spectrum is also found at small u, since in the range 1 3 < u<< 1, we also found the limiting value of. the spectrum to be  1 ou This behavior occurs therefore for all u > . The turnover occurs 1 1 for u or co The plateau that then is reached has the Ts value  3. The full spectrum for various E has been plotted and is shown in fig.6. We see for >> 1, or for the surface limited case,the turnover frequency is u = 2/E. For E << 1, or for the diffusion limited case, for E = .01, the diffusion type behavior is observed, in the range 10 < u < 7x103. The computer program used to plot the noise spectrum is given in the Appendix. 6.3.2 No Surface Sinks aG If c = 0, = 0. This corresponds to E being infinite. an The spectrum despite the possible presence of microscopic diffusion exhibits only a regular relaxation spectrum. Re P(w) = T2 (6.3.89) T) +u or the spectrum S(o) = 4(var X) (6.3.90) 2 2 v This is the "volume limited case". 6.3.3 Both Surface and Volume Sinks Present When both surface and volume sinks are present we have that 710. We again use the relations for j and j0 as given in equa tions (6.3.80) and (6.3.81) and after much algebra the real part is found to be 86 .Lo (*10.0\ 8* I IO 10a 10o 10' 10 I Fig. 6 Noise Spectra for Nonembedded Sphere (n=0) (Slopes indicated are u3/2 and u2) Re P (w) = (MP+NQ)3(AM+BN) (6.3.91 V{(P P2p ) 2+(p2Q+ 2Q) 2 where S= + +Vu2 (6.3.92) 6 = / + v)+u (6.3.93) A = (sin6coshp+6cos6coshp+psin6sinhp) (6.3.94) B = (cos6sinhppcos6coshp+6snin5sinh) (6.3.95) C = A+ incosh (6.3.96) D = B cos6sinh D = B (6.3.97) P = (p262)C2p6D (6.3.98) Q = (p2 2)D+2p6C (6.3.99) M = C(2 )P 2PBQ (6.3.100) N = (262 )Q+2p6P. (6.3.101) At low frequencies in the above equation (6.3.91) p = v and 6 = 0. The low frequency plateau value for small n is found to be Re P(o) = 1 (6.3.102) as in the previous subsection. At high frequencies p = 6 = 5 The high frequency limiting value is found to be for any T 3 Re P(o) = (6.3.103) cwu 2u  u Fig. 7 Noise spectra for nonembedded sphere (r#O) For n = 0 equation (6.3.91) reduces to equation (6.3.82). The noise spectrum is plotted for different values of and r, and is shown in fig.7. We distinguish the following options. 6.3.3(a). For >> 1, or Ts d, we have a surface limited spectrum. The low frequency plateau is 3 Re P(0) . 2 And, at high frequencies the spectrum goes as u2 and the high frequency limiting value is found to be 3 Re P () w 2 as predicted. 6.3.3(b). For << 1 in addition to T << 1 we found a diffusion limited range. The low frequency limit is 1+55 Re P(co) = 15 15 1 And in the range 1 << u < 3 we see that the spectrum goes as u reflecting the typical diffusion behavior. 2 For high frequencies the spectra go as u The compute program used to plot the spectrum is given in the Appendix. 3/2 er CHAPTER 7 NONSYMMETRICAL NONEMBEDDED BODIES 7.1 Introduction In this chapter we shall analyze the noise spectra resulting from diffusion and generationrecombination in nonsymmetrical non embedded geometries. We have already seen in Chapter 1 the difference between embedded and nonembedded bodies,the latter being subjected to various boundary conditions. We shall use the boundary conditions stemming from a finite surface recombination velocity at either one pair or two pairs of surfaces of the semiconductor sample under consideration. A three dimensional case is of academic value only,since one always has an ohmic contact in one of the directions. 7.2 Linear Case The Green's function procedure has been used to find a closed form expression for the noise spectra resulting from samples being subject to a finite surface recombination velocity in the zdirection. D(CAP) = tSAp (7.2.1) z C where we have assumed that the sample is infinite in x and y direction. D is the diffusion constant, S is the surface recombination velocity and we have considered the sample dimension to be 2C in the zdirection;Ap being the fluctuation in the numberof particles, say holes. This problem finds application in noise resulting from buried layers as studied by Hsieh28. The diffusion equation is L(AP) = 5t(Ap) + a= 0 (7.2.2) az where T is the carrier lifetime; the Laplace transformed Green's function satisfies the equation G(z,s,z')(s+ 1) D Gzsz = (zz (7.2.3) az According to the standard techniques the Green's may be found by setting S u(z) u(z') z < z' G(z,zl) = DW(u,v) u(z')v(z) z z' (7.2.4) where W is the Wronskian of two independent solutions u(z) and v(z) which satisfy the left and right side boundary conditions, respectively, evaluated at z = z'. The solution is found to be [by Lax and Mengert3] I G(z,jo,z') = (2yDa) (coshyz+asinhyz)(coshyz'asinhyz') (7.2.5) for z 5 z' and 1 G(z,jc,z') = (2yTDa) (coshyz'+asinhyz')(coshyzasinhyz) (7.2.6) for z > z' where Y = V (l+jo'r)/DT (7.2.6) and sinyCi () coshyC a = (7.2.7) coshyC+ ) sinhyC Introducing the surface lifetime and diffusion time by Ts = C/S and Td = C2/D (7.2.8) 