Random Electrical Noise: A Literature Survey

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Random Electrical Noise: A Literature Survey Research Comments from Ciphers By Ritter

Terry Ritter

INTRODUCTION Commen ts on th e production , processin g, detection an d con version to un iform distribution . Pr oducin g Electr on ic Noise If we wish to gen er ate an d use electrical n oise, we h ave two main sources: Th ermal an d Sh ot. Both are fun damen tal sources of "wh ite" n oise, mean in g th at we h ave a deep statistical un derstan din g of h ow th ese sources beh ave. Un fortun ately, th is may n ot be particularly useful if th e n oise we h ave is actually due to variable pr ocessin grelated problems. Th ermal or Joh n son n oise results from th e Brown ian motion of ion ized molecules with in a resistan ce. Th ermal n oise is en tirely fun damen tal an d can n ot be elimin ated (alth ough th e effect can be reduced by reducin g or coolin g th e resistan ce). Carbon -composition resistors may give more n oise th an expected; th is added n oise is from device fabrication , varies widely in production , an d is n ot n ecessarily "wh ite." Th us, carbon -film, metal-film or wirewoun d resistors are more satisfactory th ermal n oise sources. To verify th at a n oise source is in deed producin g th ermal n oise, it may be useful to "sh ort out" th e source resistan ce an d verify a mark ed reduction in resultin g n oise (h opefully to un der 1/10 of th e origin al value). Th is of course implies an ability to quan tify th e mean amplitude of th e n oise sign al. Sh ot n oise typically results from th e flow of electron s th rough a h igh ly-ch arged field (lik e a vacuum tube or semicon ductor jun ction ) th at mak es th e exact motion of each electron in depen den t. Ultimately, electron flow is th e movemen t of discrete ch arges, an d surroun din g th e mean flow rate is a distribution related to th e laun ch time an d momen tum for in dividual ch arge carriers en terin g th e ch arged field.

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Sh ot n oise is fun damen tal, so n o true zen er can be n oiseless, an d an y especially "n oisy" zen er must be producin g someth in g beyond sh ot n oise. Sin ce oth er n oise sources (especially con tact n oise) are typically related to device fabrication an d are n ot n ecessarily "wh ite," th is "extr a" n oise sh ould be avoided. We sh ould th us seek th e lowest-n oise zen ers for n oise sources. Sin ce zen er n oise levels will vary with curren t an d temperature, some form of automatic gain con trol (AGC) may be n ecessary. As a curren t, sh ot n oise is proportion al to square-root curren t. But as a voltage across a semicon ductor jun ction , sh ot n oise is proportion al to th e inverse of square-root curren t (see Haitz an d Voltmer an d Vergers). For small curren ts, effects oth er th an sh ot n oise may domin ate. To verify th at a semicon ductor jun ction is producin g sh ot n oise, it may be h elpful to in crease th e curren t by 100x an d measure th e resultin g n oise voltage at 1/10 th e origin al value. Th is of course implies an ability to quan tify th e mean amplitude of th e produced n oise sign al. Th e various oth er properties we migh t measure -- such as th e time between zeros (or an y oth er level) does n ot seem to give us an y particular distribution advan tage. Possibly we could sh ow th at "an y" sor t of n oise is sufficien t for some sort of samplin g to produce on e un iformly distributed bit, but th is h as n ot been establish ed. Post-Pr oduction An alog Pr ocessin g It will be n ecessary to greatly amplify th e n oise in a lin ear broadban d man n er. Th is is h arder th an it soun ds, because common self-compen sated op-amps will h ave a 6 dB/octave rolloff for stability, an d we may n eed 60 dB total amplification flat to perh aps MHz frequen cies. (Th e ban dwidth will defin e th e width of th e min imum pulse an d th e maximum rate at wh ich th e n oise can be sampled.) It would seem th at producin g ideal n oise from a fun damen tal source is of little h elp if we modify th e result prior to detection . On th e oth er h an d, actual experimen ts with sh ot n oise sampled in th e audio ran ge by a PC soun d system sh ow an un expected degree of autocorrelation between samples. Essen tially, th is is th e ability to partially predict future values based on k n own past values, an d th us represen ts a much lower en tropy th an we migh t oth erwise th in k . Experimen tally, it appears th at usin g th e differen ce between sample values, in stead of th e raw values th emselves, reduces th e problem. Sin ce th is may be equivalen t to a digital h igh -pass filter, a n on -flat h igh -pass respon se may be more desirable th an we migh t h ave previously th ough t. Th e n oise output fr om zen er diodes ten ds to vary th rough time an d especially temperature. Even with th ermal n oise, some form of automatic gain con trol is probably n ecessary over production an d time, an d will imply some amoun t of sh ort-term amplitude correlation . Detection an d Con ver sion to Un ifor m Distr ibution A wh ite n oise source sh ould h ave a gaussian or n ormal distribution of in stan tan eous amplitude, an d I assume th at th e best-quality distribution defin es th e best-quality source. However, th ere are man y differ en t gaussian distribution s, as parameterized

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by th e mean an d stan dard deviation . Wh ich of th e possibilities we actually get depen ds upon th e n oise source an d samplin g mach in ery, so th is is n ot an easy gen eral way to produce values in a particular n ormal distribution . An easy way to con vert to th e flat distribution is to h ash th e full sample values in to a h ash result, wh ich is th en used as a flat ran dom value. T h e h ash does n ot h ave to be "cryptograph ic," sin ce even a simple h ash is n on -reversible, provided much more in formation is h ash ed th an used. Nor is a supposed in ability to con struct particular h ash values useful in th is application . Because a simple CRC can be deeply un derstood with out assumption s, it is superior in th is application to th e usual ad-h oc cryptograph ic h ash wh ich on ly fun ction s as we expect if various un proven assumption s are tr ue. On th e oth er h an d, suppose we adjust detection amplitude so th at th e n oise sign al is above detection exactly h alf th e time (th is probably implies some sort of automatic con trol, wh ich also implies a small amoun t of sh ort-term amplitude correlation ). If we th en sample at ran dom times (with a wide ran dom period between samples), we can produce on e un iformly-distributed bit per sample. However, it is difficult to guaran tee th at an y repetitive process will sample "at ran dom times." An d adjacen t close samples may support un wan ted correlation s. If we can detect in dividual n oise pulses, we can assume th at th e n umber of pulses wh ich arrive in a particular time is Poisson distributed. By in creasin g th e detection amplitude un til few pulses are detected (on average) an d coun tin g th e pulses in a given time, we can get a Poisson distr ibution of pulse-coun t values. (Th ere will always be some pulses too close togeth er to discrimin ate an d coun t separately, but we can reduce th is effect with wide ban dwidth , h igh -speed detection , an d large coun ts.) Th en we can output th e parity of th e curren t coun t to get on e (almost) un iformly distributed ran dom bit. We n ote th at th is meth od does n ot require ran dom samplin g times, someth in g difficult to require of a repetitive mach in e. Ver ification A serious problem with man y n oise-based gen erators is th at th e an alog n oise is buried deep in side an d can n ot be seen or measured by th e user. Th is is a pr oblem because wh at we wan t from such a gen erator is a guaran tee th at th e output depen ds upon un predictable quan tum even ts. If we were satisfied with ran dom source th at merely passed tests, we could easily use an y on e of th e man y determin istic statistical ran dom n umber gen erators (RNG's) design ed to pass such tests. Wh at we wan t an d expect is beyon d wh at can be tested extern ally. Wh at is n eeded is th e ability to turn off th e quan tum source, an d see th e output ch an ge. If we can n ot do th at, we can n ot be sure th at th e particular device we h ave really does depen d upon quan tum in formation . To verify correct operation of th e n oise source we migh t collect an d verify eith er or both Gaussian amplitude an d Poisson pulse-coun t distribution s durin g n ormal operation . (Th is is in addition to some h ardware ch eck to verify th at th e detected n oise is produced by th e expected sour ce.)

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Contents 1944 Rice gives th e th eoretical formulas for expected zeros an d expected maxima. 1948 MacDon ald describes th e Brown ian n ature of th ermal or Joh n son n oise. 1955 Burgess discusses sh ot an d avalan ch e n oise. 1956 Pierce discusses electrical n oise in gen eral, an d th ermal n oise in particular. 1962 Ragazzin i an d Ch an g discuss th e assumed Gaussian n ature of n oise. 1964 Gray et. al. discuss avalan ch e multiplication an d Zen er break down . 1965 Oliver discusses statistics related to both th ermal an d sh ot n oise. 1966 Th orn ton et. al. discusses statistics related to both th ermal an d sh ot n oise. 1968 Haitz an d Voltmer report results from measurin g semicon ductor avalan ch e n oise at microwave frequen cies. 1969 Gray an d Searle describe avalan ch e an d Zen er break down . 1971 Joh n son (th e discoverer of th ermal n oise) describes th ermal n oise. 1972 Millman an d Halk ias describe avalan ch e an d zen er break down , tun n elin g, an d Joh n son an d sh ot n oise. 1976 Ott describes both th ermal an d sh ot n oise, an d practical n oise measuremen t. 1980 Malvin o describes avalan ch e an d zen er effects. 1983 Warn er an d Grun g describe reverse bias break down . 1984 Zan ger describes zen er an d avalan ch e break down . 1987 Vergers describes sh ot n oise in p-n jun ction s. 1989 Horowitz an d Hill describe zen er n oise, th ermal n oise, sh ot n oise, an d n oise measuremen t.

1944 -- Rice Rice, S. 1944. Math ematical An alysis of Ran dom Noise. Bell System Technical Journal. 23: 282-332. 24: 46-156.

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Expected Zer os Per Secon d "For an ideal ban d-pass filter wh ose pass ban d exten ds fr om fa to fb th e expected n umber of zeros per secon d is" 2 [1/3 (fb3 - fa3) / (fb - fa)]1/2 "Wh en fa is zero th is becomes 1.155 fb, an d wh en fa is very n early equal to fb it approach es fb + fa." Expected Maxim a Per Secon d "For a ban d-pass filter th e expected n umber of maxima per secon d is" [3/5 (fb5 - fa5) / (fb3 - fa3)]1/2 "For a low-pass filter wh ere fa = 0 th is n umber is 0.775 fb." "Th e expected n umber of maxima per secon d lyin g above th e lin e I(t) = I1 is approximately, wh en I1 is large," 2

e -I1

/2 psi0

"wh ere psi0 is th e mean squar e value of I(t)."

1948 -- MacDonald MacDon ald, D. 1948. Th e Brown ian Movemen t an d Spon tan eous Fluctuation s of Electricity. Res. Appl. in Industry. 1: 194-203. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 7-16. "In 1827 th e biologist Robert Brown was studyin g un der h is microscope th e pollen grain s, some 0.0002 in ch in len gth , of th e plan t Clarckia pulchella. 'Wh ile examin in g th e form of th ese particles immersed in water, I observed man y of th em very eviden tly in motion . Th ese motion s were such as to satisfy me . . . th at th ey arose n eith er from curren ts in th e fluid, n or from its gradual evaporation , but belon ged to th e particle itself.'" ". . . it became clear th at th e effect was en tirely fun damen tal, an d A. Ein stein , in a series of classical papers, was th e first to provide a clear an alysis of th e problem as arisin g from con tin uous an d ran dom molecular bombardmen ts . . . ." ". . . von Nageli in 1879 h ad con sidered th e possibility of molecular bombardmen t but h ad con cluded because th e impulse due to on e collision was so min ute, th at th is could n ot be th e cause; for h e opin ed th at sin ce all direction s in space are equally lik ely th e cumulative effect of man y ran dom collision s could on ly be of th e same magn itude. Th e error is a common on e an d arises essen tially from implicitly regardin g a ran dom process as made up of regularly altern atin g favourable an d un favourable even ts; 5 of 26

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such a process is, h owever, a h igh ly or dered on e an d it is th ose very 'run s' of favourable (or un favourable) even ts, wh ich we sometimes regard as again st th e 'laws of ch an ce', wh ich ch aracterize a ran dom process an d give rise to th e relatively large fluctuation s observed." "Th e essen tials are well illustrated by th e fun damen tal 'ran dom walk ' problem, as first posed by Karl Pearson in 1905. A man (presumably very drun k ) tak es steps of equal len gth , l, from a startin g poin t O on after th e oth er in successively r an dom direction s. Wh ere is h e lik ely to be after n steps? Lord Rayleigh an swered th e problem immediately wh ere n is large; th e probability th at h e is at a distan ce between r an d r+dr from h is startin g poin t is p(r)dr = (2r/n l 2)e-(r

2

/nl 2)

dr.

"His average distan ce is th erefore [ the integral from 0 to infinity of r * p(r) dr, or ] (pi 1/2)/2 * (n l)1/2 "an d th us in creases with th e square r oot of th e time for wh ich h e con tin ues th e walk . . . ." [ It appears that the Rayleigh distribution models noise in the sense of peak amplitude over time./tfr ]

1955 -- Burgess Burgess, R. 1955. Electrical fluctuation s in semicon ductors. British J. Appl. Phys. 6: 185-190. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 59-64. SHOT NOISE "Th e term 'sh ot n oise' was origin ally applied to th e fluctuation s of curren t in a saturated vacuum diode due to th e ran domn ess of electr on emission from th e cath ode." "At low frequen cies such th at th e electron tran sit time t is small compared with (1 / w), th e [ Fourier ] tran sform F(f) ~ e an d th e spectral den sity assumes th e simple form (2 e I). Th e con cept of ran domn ess of rate of emission implies th at th e process is determin ed by a station ary Poisson distribution ." "An oth er importan t in stan ce of sh ot n oise arises in th e oth er extreme from a un iform semicon ductor, n amely th e motion of carriers across a h igh -field tran sition region , e.g. at a metallic con tact or at a p-n jun ction . Normally th e carrier velocities in such a region would be of th e order of 10 7 cm/s an d th e width of th e region would lie in th e ran ge of 10 -5 to 10 -3 cm so th at th e tran sit time would be n egligible except at th e h igh est microwave frequen cies. Furth ermore it may be r eadily sh own th at sin ce th e ch an ge in quasi-Fermi level for th e carriers across th e tr an sition region is very n early equal to th e applied voltage, th e effect of each electron tran sit is effectively to in duce a curren t impulse ed(t), an d th us full sh ot n oise may be attributed to th e 6 of 26

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flow." AVALANCHE NOISE "Wh en a barrier region is subjected to reverse bias th e electric field may reach th e order of 10 5 V/cm or greater, an d at th ese fields th ere occur ph en omen a wh ich cause a rapid in crease of curren t an d even tual break down ; it h as furth ermore been observed th at th e curren t is 'n oisy' in th is region , becomin g in creasin gly impulsive as break down is approach ed." "In silicon jun ction s McKay14 observed th at at th e on set of break down th ere appears a distin ctive form of impulsive n oise con sistin g of a ran dom sequen ce of rectan gular curren t pulses of variable duration but con stan t amplitude." "It is possible th at th e in evitable in h omogen eity of th e semicon ductor in th e n eigh bourh ood of th e jun ction gives rise to small region s (or 'weak spots') in wh ich break down occurs for lower applied voltage th an elsewh ere an d th is localized break down will switch from an 'off' to an 'on ' con dition an d back again , triggered by ran dom fluctuation ."

1956 -- Pierce Pierce, J. 1956. Ph ysical Sources of Noise. Proc. IRE. 44: 601-608. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 51-58. "Sh ot n oise, due to th e discrete n atur e of electron flow, is gen erally distin ct from Joh n son n oise, alth ough in some electron devices th e expression for th e n oise in th e electron flow h as th e same form as th at for Joh n son n oise. Wh en th e n oise in th e electron flow is greater or less th an pure sh ot n oise, th e motion s of th e electron s must be in some way correlated. In an electron stream of low n oise, th e ran dom in terception of a fr action of th e electr on flow can reduce th e correlation an d in crease th e n oise. Joh n son n oise an d sh ot n oise h ave a flat frequen cy spectrum." "Man y sorts of electrical sign als are called n oise." ". . . man y en gin eers h ave come to regard an y in terferin g sign al of a more or less un predictable n ature as n oise." "Th e th eory of n oise presen ted h ere is n ot valid for all sign als or ph en omen a wh ich th e en gin eer may iden tify as n oise." "Th e th eory of n oise is best adapted to h an dlin g sign als wh ich origin ate in truly ran dom processes, such as th e emission of electron s from a ph oto-surface or a h ot cath ode, or th e th ermal agitation of ch arges in a resistor. Wh en a cath ode emits electron s at so slow a rate th at we observe th eir effects in a circuit as separate pulses, we h ave impulse noise, an d th e th eory of n oise h as someth in g to say about th is. Wh en electron s are ran domly emitted so rapidly th at th e pulses th ey produce in th e circuit overlap, th e statistics of large n umbers applies, an d th e th eory of n oise tells us a great deal th at must be true of a large class of n oise sign als, despite differen ces in th e exact n ature of th eir sources." Joh n son Noise

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"Th e first source of n oise wh ich we con sider is Joh n son n oise, th e th ermal n oise from a resistor. Th e en gin eerin g fact is th at a resistor of resistan ce R acts lik e a n oise gen erator."

V2 = 4 k T R B I2 = 4 k T G B = 4 k T B / R R G B k T

= = = = =

resistance, ohms 1/R = conductance, mhos bandwidth, Hz Boltzmann's constant, 1.380E-23 joules / deg. K temperature in deg. K, or deg. C + 273

"Wh at is th e source of Joh n son n oise? In an ordin ary resistor, it is a summation of th e effects of th e very sh ort curren t pulses of man y electron s as th ey travel between collision s, each pulse in dividually h avin g a flat spectrum. In th is case th e n oise is a man ifestation of th e Brown ian movemen t of th e electron s in th e resistor." Sh ot Noise "Electricity is n ot a smooth fluid; it comes in little pellets, th at is, electron s. Th e flow of electron s in a vacuum tube is accompan ied by a n oise of th e same n ature as th e patter of rain on a roof. Sch ottk y, wh o first in vestigated th is ph en omen on , called it Schroteffekt (from sh ot); it is n ow usually called simply shot noise." "Lik e Joh n son n oise, sh ot n oise h as a flat spectrum. Th is is really wh at we sh ould expect of a ran dom collection of very sh ort pulses, each of wh ich h as a flat spectrum."

I2 = 2 e I0 B e = electron charge, 1.60E-19 coulombs I0 = dc current, Amps B = bandwidth, Hz Noise with a 1 / f Spectr um "A n oise made up of a ran dom sequen ce of sh ort pulses, or impulses, as are Joh n son n oise an d sh ot n oise, h as a flat spectr um. A n oise made up of a ran dom sequen ce of step fun ction s would h ave a 1 / f 2 spectrum. Th is is because a step is th e in tegral of an impulse, an d th e amplitude of an y frequen cy compon en t of th e step is 1 / ( 2 pi f ) times th at impulse. If th e amplitude varies as 1 / f, th e power will vary as 1 / f 2.

1962 -- Ragazzini and Chang Ragazzin i, J an d S. Ch an g. 1962. Noise an d Ran dom Processes. Proc. IRE.

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50: 1146-1151. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 25-30. "In most systems wh ich are of in terest to th e en gin eer an d th e design er, ran dom processes are assumed to be Gaussian or at least assumed to be satisfactorily approximated by such a distribution ." ". . . a ran dom sign al or n oise is usually gen erated by a lar ge n umber of in depen den t even ts . . . ." "It follows from th e cen tral limit th eorem in pr obability th at . . . th e amplitude of such a sign al [ is ] n ormally distributed . . . ."

1964 -- Gray, et. al. Gray, P., D. DeWitt, A Booth royd an d J. Gibbon s. 1964. Physical Electronics and Circuit Models of Transistors. Semicon ductor Electron ics Education Committee, Volume 2. Joh n Wiley an d Son s. 4.4 JUNCTION BREAKDOWN "In all real diodes th ere is a limitin g value of reverse voltage beyon d wh ich th e reverse curren t in creases greatly with out sign ifican t in crease of reverse voltage." (p. 63) "Th e abrupt break down of silicon an d well-cooled german ium types h as a useful n on -destructive ran ge [...]. Such diodes are widely used as voltage regulator s, an d devices in ten ded for th is service are called Zener diodes or breakdown diodes. (p. 64) "Th ere are two electron ic break down mech an isms in th e bulk semicon ductor wh ich can cause a voltage-saturated break down --Zen er break down an d avalan ch e break down . Zen er break down is a dir ect disruption of in teratomic bon ds in th e space-ch arge layer by very h igh electric fields (greater th an 10 6 volts/cm), wh ich produces mobile h ole-electron pairs. It is th e mech an ism of break down in good crystallin e in sulators an d it occurs in abrupt jun ction s between h igh ly doped region s. Avalan ch e break down occurs wh en th e acceleration of carriers in th e spacech arge region is gr eat en ough to cause ion izin g collision s with atoms, th us producin g mobile h ole-electron pairs. Sin ce avalan ch e multiplication can occur at electric fields appr eciably lower th an th ose required for Zen er break down , avalan ch e break down will occur befor e th e Zen er voltage can be reach ed, except in diodes with very large impurity con cen tration s. Silicon voltage-regulator diodes wh ich break down above 8 volts probably use th e avalan ch e mech an ism, wh ereas th ose wh ich break down below 5 volts work by Zen er br eak down . Between 8 an d 5 volts th e domin atin g mech an ism depen ds on th e exact impurity distribution at th e jun ction . Both mech an isms can be presen t in th e same diode. Note th at th e term Zener diode is often used with out regard to th e mech an ism to iden tify a diode in ten ded to operate at break down ." (p. 65) 4. 4. 1 Theory of Avalanche Multiplication

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"Avalan ch e multiplication occurs wh en th e electric field in th e space-ch arge layer is large en ough so th at carriers traversin g th e region acquire sufficien t en er gy to break covalen t bon ds in th eir collision s with th e crystal structure. Every such ion izin g collision produces a h ole an d an electron , each of wh ich is acceler ated by th e field an d h as a possibility of producin g an oth er ion izin g collision before it leaves th e space-ch arge r egion . Neglectin g r ecombin ation in th e layer, all th e car riers produced will con tribute to th e total reverse curren t." (p. 65) ". . . th e average rate at wh ich pairs are produced by impact ion ization depen ds n ot on ly on th e electric field, but also on th e distan ce th e carrier travels. Th us, wh en th e impurity con cen tration is in creased, th e width of th e space-ch arge layer decreases [...] an d th e peak electric field at break down decreases." (pp. 67-68) 4. 2. 2 Zener Breakdown

"Diodes with very large values of impurity con cen tration h ave n arrow space-ch arge region s an d develop h igh fields at low applied voltage. As th e reverse voltage in creases, both th e electric field an d th e width of th e space-ch arge layer in crease. Th e field reach es a large en ough value to cause Zen er br eak down before th e width of th e space-ch arge region is sufficien t to permit avalan ch e break down . Zen er break down is th e direct disruption of covalen t bon ds by th e electric field force, an d does n ot require acceleration of a primary carrier by th e field. Hen ce, th e Zen er break down voltage depen ds on ly on th e maximum field an d n ot on th e len gth of th e path in th e depleted region . Zen er br eak down occurs at fields of th e order of 10 6 volts/cm, wh ich ar e reach ed in abrupt jun ction s in silicon wh en th e dopin g is about 10 18 atoms/cm3 an d th e reverse bias is about 5 volts. In such a diode, both Zen er an d avalan ch e break down occur simultan eously. As th e dopin g is furth er in creased, th e path len gth at break down drops, an d on ly th e Zen er mech an ism prevails." (pp. 68-69) "A very reason able question at th is poin t is wh eth er [don or an d acceptor dopin g levels] can be in creased to th e poin t wh ere th e field associated with th e built-in poten tial barrier [...] alone will cause Zen er break down , so th at th e diode will be a sh ort circuit with n o applied voltage. Wh ile we h ave n ot studied th e th eory n eeded to un derstan d such devices in detail, th ey do in deed exist. Diodes wh ich break down at V = 0, or at sligh t r everse bias, are called backward diodes because th ey con duct by Zen er break down with small reverse voltage, but do n ot begin to sh ow much n ormal diode forward con duction un til an appreciable fraction of a volt of forward bias is applied. Hen ce, over a small voltage r an ge, th ey appear lik e diodes wh ich con duct wh en th e n-type side is made positive. More h eavily doped diodes wh ich are still in break down with some forward bias are called Esaki or tunnel diodes. Un til en ough forward bias is applied th ey con duct well, as th ough still in break down . As forward bias in creases, a poin t is reach ed at wh ich [th e differen ce between th e poten tial barrier an d th e applied voltage] n o lon ger is large en ough to provide th e con ductin g mech an ism an d forward curren t falls with in creasin g forward bias. Con sequen tly, th ere is a n egative resistan ce region . Th e curren t falls to a low value an d th en rises as th e diode en ters th e region of n ormal con duction ." (p. 69)

1965 -- Oliver

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Oliver, B. 1965. Th ermal an d Quan tum Noise. Proc. IRE. 53: 436-454. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 129-148. Statistics of Th er m al Noise ". . . an y lin early filtered or amplified th ermal n oise wave h as a Gaussian distribution of in stan tan eous amplitude." "Usually th e Gaussian amplitude distr ibution of th ermal n oise is developed on th e basis of a model source con tain in g a very large n umber of in depen den t gen erators each of wh ich produces an in fin itesimal con tribution to th e resultan t amplitude. For example, in a resistor, each con duction ban d electron as it is buffeted about produces a ran dom curren t wave. Th e total cur ren t is th en sh own to h ave a Gaussian distribution by th e Cen tral Limit Th eorem." ". . . th e en velope, A(t), [ that is, the peak amplitude ] of an y th ermal n oise wave h as a Rayleigh distribution . . . ." Sh ot Noise "Wh en ever discrete particles arrive at ran dom times th er e will be fluctuation s in th e rate of arrival. It is th ese fluctuation s th at con stitute sh ot n oise." "Let us assume th at a particle is equally lik ely to arrive at an y time, an d th at th e average rate of arr ival is r." "Un der th ese con dition s th e n umbers of arrivals in a given len gth of time are distributed accordin g to th e well-k n own Poisson distribution ." ". . . wh en th e aver age n umber of arrivals durin g th e observin g time is large, th e fluctuation s will approach a Gaussian distribution about th e mean with sigma = n 1/2."

1966 -- Thornton et. al. Th orn ton , R., D. DeWitt, E. Ch en ette an d P. Gray. 1966. Characteristics and Limitations of Transistors. Semicon ductor Electron ics Education Committee, Volume 4. Joh n Wiley an d Son s.

4 Noise 4.0 INTRODUCTION "'Noise' is a term used to sign ify extran eous sign als wh ich do n ot con vey an y useful in formation for th e problem at h an d, an d wh ich can on ly be described by th eir statistical properties." (p. 134) 4.1 ANALYSIS INVOLVING NOISE SOURCES

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4. 1. 1 Introduction

"Noise sources are un predictable in th e sen se th at in stan tan eous waveforms can n ot be predicted over an y sign ifican t in terval of time. On e can , h owever, descr ibe n oise sources in statistical terms, such as probabilities, mean -square values, an d correlation fun ction s." (p. 135) 4. 1. 3 Spectral Density

"Th e voltage wave from a n oise source v(t) will con tain a great man y frequen cy compon en ts. To in dicate h ow th ese compon en ts are distributed as a fun ction of frequen cy, we may plot wh at is called th e spectral density, a graph versus fr equen cy of mean-square noise voltage per unit bandwidth. Th is graph migh t be obtain ed by feedin g th e source in to a filter wh ich passes un atten uated all frequen cies in a ban d delta-f cen tered at f0, an d completely rejects all oth er frequen cies." "Because th e ordin ary Fourier tran sform for a ran dom sign al is n ot defin ed (because it does n ot, in gen eral, yield con vergen t in tegrals), we can n ot use con ven tion al Fourier tech n iques to fin d th e spectral den sity of th e n oise. In stead, we must first form th e autocorrelation function of v(t) is defin ed as

R(tau) = limit as T -> infinity of 1/(2 T) x integral from -T to T of v(t)v(t + tau)dt

(4.3)

Th e spectral den sity is th en by defin ition th e Fourier tr an sform W(f) of th e autocorrelation fun ction R(tau). Th at is

W(f) = integral from -infinity to +infinity of -(j)(2pi)(f)(tau)

R(tau) e

(4.4a)

d(tau)

For real time fun ction s, R(tau) is a real, even fun ction of tau. Th us, W(f) can be written as

W(f) = integral from -infinity to +infinity of R(tau) cos(2pi)(f)(tau) d(tau)

(4.4b)

"It is clear . . . th at because R(tau) is real, W(f) is a real, even fun ction of f; th at is, W(f) = W(-f)." (pp. 136-137) "Th e spectral den sity W(f), defin ed above, is a 'two-sided' represen tation ; th at is, it is defin ed in terms of both positive an d n egative frequen cy. However, by con ven tion in n oise an alysis, th e spectral den sity is defin ed in terms of positive frequen cy on ly (i.e., 'on e-sided'). Because of th e fact th at W(f) = W(-f), as sh own above, con version from th e two-sided represen tation to th e on e-sided spectral den sity, design ated h ere as S(f), in volves n oth in g more th an multiplyin g by a factor of two to accoun t for th e con tribution s of W(f) at n egative frequen cies." (p. 137) 4.2 NOISE IN A pn JUNCTION DIODE

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4. 2. 1 Shot Noise in Reverse Bias

"If a pn jun ction is several volts back -biased, th en , rough ly speak in g, all th e min ority carriers with in on e diffusion len gth of th e jun ction will move by diffusion in to th e space-ch arge region an d be attracted across it by th e h igh electr ic field. Th ese carriers create a curren t; but, because of th e discr ete n ature of th e electron ic ch arge, th e curren t will appear to a first approximation as a series of impulses . . . . Th e time at wh ich an y on e carrier tr averses th e jun ction is statistically in depen den t of th e time th at an y oth er carrier traverses th e jun ction , an d th us th e curr en t pulses can be assumed to be completely in depen den t of on e an oth er. As stated in Sec. 4.1.3, we fin d th e spectral den sity of th is curren t waveform by fin din g first th e autocorrelation fun ction , defin ed in th is case as" (p. 138)

R(tau) = limit as T -> infinity of 1/(2 T) x integral from -T to T of i(t) i(t + tau)dt "Th e spectral den sity of th e curren t, S i (w), will be th e Fourier tran sform of th is R(tau)." (p. 140) "In actual fact, th e curren t waveform will n ot con sist of impulses. Wh en an electron crosses th e space-ch arge layer, it in duces a curren t waveform more closely resemblin g a squar e pulse." "Th e . . . width of th e pulse is determin ed by th e time it tak es th e electron to move across th e space-ch arge region . Typically, th e field is large en ough so th at th e carriers reach a saturation velocity in th e order of 10 7 cm/sec. Th us th ey move with a con stan t velocity an d th e curren t pulse is approximately square . . . . Th e exact sh ape of th ese pulses is n ot of prime importan ce h ere, but it is sign ifican t th at the pulses have a width on the order of the transit time Tt through the space-charge layer." (p. 140) ". . . except for th e dc compon en t wh ich does n ot con cern us h ere, th e spectral den sity h as a relatively con stan t value of 2nq2 out to a frequen cy f = 1/10Tt. Th us we can con sider th e spectral den sity of th e n oise to be flat for all frequen cies of in terest in tran sistor circuits, i.e., below fT ." "On th e basis th at th e dc curren t flowin g th rough th e diode is i(t) = I0 = nq [with n being the average number of pulses per second], we led to th e importan t con clusion th at th e spectral den sity of sh ot n oise in a reverse-biased pn jun ction is virtually con stan t at a value

Si(f) = 2 q I0 for f above 0." (p. 141) 4. 2. 2 Shot Noise for Forward Bias

"For a forward bias, we can resolve th e diode curren t in to two ph ysically distin ct compon en ts. On th e basis of th e diode equation

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I = I0 (eq

V/kT

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- 1)

th ese compon en ts are

I1 = -I1 I2 = I0eq

V/kT

wh ere I1 is th e reverse saturation curren t arisin g from th ermally-gen erated carrier s on each side, an d I2 arises from th e diffusion of majority carriers again st th e poten tial barrier on each side. Wh ereas th e average values of th ese curren ts ten d to oppose each oth er in th e total curren t, th e n oise compon en ts associated with th ese curren ts, bein g un correlated, will add in a mean -square sen se." (p. 142) "We th us con clude th at a stron gly for ward-biased pn jun ction exh ibits th e full sh ot n oise associated with its average curr en t. "In zero bias, an d th erefore with th e jun ction in thermal equilibrium, th e two curren ts I1 an d I2 . . . can cel on th e average, but th eir n oise compon en ts are still in depen den t, an d add in a mean square sen se." "So . . . we fin d . . . for low en ough frequen cies th e spectral den sity

Si(f) = 4 q I0 . Th us, th e n oise is iden tical to sh ot n oise associated with a dc curren t twice as large as th e reverse saturation curren t of th e diode. Sin ce, in fact, th ere is n o dc curren t flowin g un der th e assumed zero-bias con dition s, it is con ven ien t to elimin ate I0 . . . by rewritin g in ter ms of th e in cremen tal diode con ductan ce. On differen tiatin g . . . we obtain

g0 = dI / dV at V = 0 . Th us

Si(f) = 4 k T g0 ." (p. 143)

1968 -- Haitz and Voltmer Haitz, R. an d F. Voltmer. 1968. Noise of a Self-Sustain in g Avalan ch e Disch arge in Silicon : Studies at Microwave Frequen cies. J. Appl. Phys. 39: 3379-3384. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 327-332. "Th e studies of avalan ch e n oise repor ted by Haitz are exten ded to frequen cies up to an d above th e avalan ch e frequen cy wa. It is foun d th at th e open -circuit spectral

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voltage den sity is flat with in +/- 5% from less th an 100 Hz up to frequen cies approach in g wa." "Durin g th e studies of low-frequen cy avalan ch e n oise it became eviden t th at special precaution s h ave to be tak en in order to preven t th e gen eration of excessive n oise resultin g from n on un iform break down . Th e combin ation of both a guard rin g to preven t edge break down an d a small break down area to reduce material n on un iformities h ave led to satisfactory results." "At curren ts below 2.5 mA th e measur ed n oise is larger th an th e n oise predicted . . . . Th is discrepan cy, wh ich is typical for avalan ch e diodes at low curren t den sities, is n ot serious. It is caused by extremely small n on un iformities of th e break down voltage." Fig. 2 is a graph of "Open Circuit Spectral Voltage Den sity" in n V/SQRT(Hz) versus "Curren t" in mA. At 1 mA th e graph sh ows about 100 (n V), at 10 mA about 40, an d at 0.1 mA th eory in dicates about 400, but th e actual device sh ows much more. For curren ts of 2.5 - 20 mA, th eoretical an d experimen tal avalan ch e n oise th us decrease with in creased cur ren t. Across th e gr aph ran ge, th eoretical n oise appears to decrease by a factor of 10 with a curr en t in crease of 100.

1969 -- Gray and Searle Gray, P., an d C. Searle. 1969. Electronic Principles: Physics, Models, and Circuits. Joh n Wiley & Son s. 6. 5. 1 Breakdown Diodes

"All jun ction diodes exh ibit a region of beh avior in th e r everse direction in wh ich large reverse curren ts can flow if th e reverse voltage exceeds a value referr ed to as th e reverse breakdown voltage. (p. 231) "Reverse break down in pn jun ction s may arise from eith er of two mech an isms . . . . (p. 232) "On e mech an ism th at causes reverse break down in pn jun ction s is avalanche multiplication. Th e carriers th at con stitute th e n or mal reverse curren t of a jun ction flow across th e space-ch arge layer from th e region s wh ere th ey are in th e min ority to th e region s wh ere th ey are in th e majority. Th us th ey move down th e poten tial barrier at th e jun ction an d are accelerated between collision s by th e field th ere. If th e field is large en ough (in th e ran ge of 2 x 10 5 volts/cm) th e en er gy th at th ese carriers acquire from th e field between collision s is sufficien t to produce a h ole-electron pair wh en th e en ergy is tran sferred to th e crystal durin g a collision . Th us a sin gle carr ier can produce an oth er pair of carriers, wh ich in turn flow out of th e space-ch arge layer an d con tribute to th e reverse curren t. Th ese secondary carriers can produce oth er pairs or tertiary carriers th rough th eir own collision s with th e lattice. In th is way th e rever se curren t is multiplied an d can become quite large." (p. 232) "Th e secon d mech an ism of jun ction break down is called Zener breakdown. If th e

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electric field in th e space-ch arge layer is stron g en ough (in th e ran ge of 5 x 10 5 volts/cm) th e force th at it exerts on boun d or valen ce electron s is sufficien t to strip some of th ose electron s away from th e valen ce bon ds, th ereby creatin g h ole-electron pairs th at con tribute to th e reverse curren t. Th ere is n o multiplication effect in volved in th is mech an ism; th e pairs are produced directly by th e field an d n ot th rough th e action of a primary carrier. "Silicon jun ction diodes th at are relatively ligh tly doped h ave break down voltages in th e ran ge of ten s or h un dreds of volts. In such diodes th e break down curren t is produced by avalan ch e multiplication . Diodes th at are more h eavily doped h ave lower break down voltages; th e space-ch arge layer is th in n er an d th e electric field is larger for th e same applied voltage, an d avalan ch e multiplication sets in at lower voltages. Diodes th at are very h eavily doped h ave break down voltages as small as on e or two volts. In such diodes th e break down curren t is pr oduced by th e Zen er mech an ism; th e electric field is very h igh , an d th e space-ch arge layer is so th in th at carriers spen d too little time in th e space-ch arge layer to produce sign ifican t n umbers of secon dary carrier pairs. Diodes th at break down for reverse voltages in th e ran ge of 6 to 8 volts h ave both mech an isms operatin g simultan eously. "Note th at alth ough both avalan ch e an d Zen er mech an isms are described as "break down ph en omen a," n eith er is, of itself, destructive or irreversible. Wh en th e reverse voltage is r educed below th e critical level, th e br eak down mech an ism subsides, an d th e jun ction beh aves n ormally on ce again . Of course, th e large curren ts an d h igh voltages associated with reverse break down can easily cause th e jun ction to overh eat, an d th is can lead to irreversible destruction of th e diode owin g to excessively h igh temperatures." (p. 233) 6. 5. 2 Tunnel Diodes

"In jun ction s th at are very h eavily doped, th e ph en omen on of Zen er break down can occur at very small reverse voltages. It can , in fact, occur at zero bias. A jun ction th at is in Zen er br eak down at zero bias will support lar ge curren ts for a r everse voltage (wh ich mak e th e field bigger) an d will gradually revert to n ormal operation as th e applied voltage is made positive (wh ich reduces th e electric field). A forward voltage of on e or two ten th s of a volt may be en ough to elimin ate th e Zen er break down mech an ism an d to reduce th e jun ction curren t. Furth er in creases in forward voltage pr oduce min ority-car rier in jection so th at th e curren t on ce again rises." "Devices beh avin g in th is man n er are called tunnel diodes. Th e n ame origin ates from a quan tum-mech an ical explan ation of th e Zen er break down mech an ism. Tun n el diodes are useful as circuit compon en ts because th ey h ave a region of negative in cremen tal con ductan ce, th at is, a region in wh ich th e I-V ch aracteristic h as n egative slope." (p. 236)

1971 -- Johnson Joh n son , J. B. 1971. Electron ic Noise: th e first two decades. IEEE Spectrum. 8: 42-46. Also reprin ted in : Electrical Noise: Fundamentals and Sources. 1977. M. Gupta, Ed. IEEE Pr ess. 17-21. 16 of 26

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"In th e 1918 paper , Dr. Sch ottk y eviden tly assumes th at th e grosser curren t fluctuation s produced by faulty tube structures . . . h ave been , or can be, elimin ated, an d h e is left with two sources of n oise th at are of a much more fun damen tal n ature. On e h e calls th e 'Warmeefek t,' in En glish n ow common ly n amed 'th ermal n oise.' Th is is a fluctuatin g voltage gen erated by electrical curren t flowin g th rough a resistan ce in th e in put circuit of an amplifier, n ot in th e amplifier itself. Th e motion of ch arge is a spon tan eous an d ran dom flow of th e electrical ch arge in th e con ductor in respon se to h eat motion in its molecules." "In th e case of th e 'th ermal n oise' . . . "th e electric ch ar ge is in effect h eld in lon g bags with walls relatively impervious to electron s at low temperature. Th e mass tran sport of ch arge alon g th e bag, or wires, un der th e in fluen ce of h eat motion , sets up th e poten tial differen ces th at gen erate th e fluctuatin g output of th e amplifier." "Wh en n ow on e en d of th e con ductor, th e 'cath ode' of th e tube, is h eated to in can descen ce, electron s can be emitted from th e cath ode surface to travel across th e vacuum toward th e an ode. Th e electron s are emitted at ran dom times, in depen den t of each oth er, an d th ey travel at differen t velocities, depen din g on in itial velocity an d voltage distribution for electron passage. In th e case of a small electron emission , a small n early steady flow of curren t results, with a superimposed smaller altern atin g curren t wh ose amplitude can be calculated from statistical th eory. Th is small curren t flowin g th ough th e amplifier gen erates th e 'Sch roteffek t,' or sh ot effect, in th e amplifier." ". . . for frequen cies above certain values, th e n oise power is con stan t up to very h igh frequen cies. For th ermal n oise th is con stan t power exten ds also to low values, wh ile for sh ot n oise th er e are man y exception s an d variation s."

1972 -- Millman and Halkias Millman , J. an d C. Halk ias. 1972. Integrated Electronics: Analog and Digital Circuits and Systems. McGraw-Hill. 3-11 BREAKDOWN DIODES "Diodes wh ich are design ed with adequate power-dissipation capabilities to operate in th e break down region may be employed as voltage-referen ce or con stan t-voltage devices. Such devices are k n own as avalanche, breakdown, or Zener diodes." (p. 73) Avalanche Multiplication

"Two mech an isms of diode break down for in creasin g reverse voltage are recogn ized. Con sider th e followin g situation : A th ermally gen erated carrier [...] falls down th e jun ction barrier an d acquires en ergy from th e applied poten tial. Th is carr ier collides with a crystal ion an d imparts sufficien t en ergy to disrupt a covalen t bon d. In addition to th e origin al carrier, a n ew electron -h ole pair h as n ow been gen erated. Th ese carriers may also pick up sufficien t en ergy from th e applied field, collide with an oth er crystal ion , an d create still an oth er electron -h ole pair . Th us each n ew carrier may, in turn , produce addition al carriers th rough collision an d th e action of disruptin g bon ds. Th is cumulative process is referred to as avalanche

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multiplication. It results in large reverse curren ts, an d th e diode is said to be in th e region of avalanche breakdown. (p. 74) Zener Breakdown

Even if th e in itially available carrier s do n ot acquire sufficien t en ergy to disrupt bon ds, it is possible to in itiate break down th rough a direct rupture of th e bon ds. Because th e existen ce of th e electric field at th e jun ction , a sufficien tly stron g force may be exerted on a boun d electron by th e field to tear it out of its covalen t bon d. Th e n ew h ole-electron pair wh ich is created in creases th e reverse curren t. Note th at h is process, called Zener breakdown, does n ot in volve collision s of carriers with th e crystal ion s (as does avalan ch e multiplication ). (p. 75) "Th e field in ten sity in creases as th e impurity con cen tration in creases, for a fixed applied voltage. It is foun d th at Zen er break down occurs at a field of appr oximately 2 x 10 7 V/m. Th is value is reach ed at voltages below about 6 V for h eavily doped diodes. For ligh tly doped diodes, th e break down voltage is h igh er, an d avalan ch e multiplication is th e predomin an t effect. Neverth eless, th e term Zener is common ly used for th e avalanche, or breakdown, diode even at h igh er voltages." (p. 75) 3-12 THE TUNNEL DIODE "A p-n jun ction diode of th e type discussed in Sec. 3-1 h as an impurity con cen tration of about 1 part in 10 8. With th is amoun t of dopin g, th e width of th e depletion layer, wh ich con stitutes a poten tial barrier at th e jun ction , is of th e order of a micron . Th is poten tial barr ier restrain s th e flow of carriers from th e side of th e jun ction wh ere th ey con stitute majority carriers to th e side wh er e th ey con stitute min ority carriers. If th e con cen tration of impurity atoms is greatly in creased, say, to 1 part in 10 3 (correspon din g to a den sity in excess of 10 19 cm-3), th e device ch aracteristics are completely ch an ged. Th is n ew diode was an n oun ced in 1958 by Esak i, wh o also gave th e correct th eoretical explan ation for its volt-ampere ch aracteristic. The Tunneling Phenomenon

Th e width of th e jun ction barrier var ies in versely as th e square root of th e impurity con cen tration [Eq. (3-21)] an d th erefore is reduced to less th an 100 An gstroms (10 -6 cm). Classically, a particle must h ave an en ergy at least equal to th e h eigh t of a poten tial-en ergy barrier if it is to move from on e side of th e barrier to th e oth er. However, for barriers as th in as th ose estimated above in th e Esak i diode, th e Sch rodin ger equation in dicates th at th ere is a large pr obability th at an electron will pen etrate through th e barrier. Th is quan tum-mech an ical beh avior is refer red to as tunneling, an d h en ce th ese h igh -impurity-den sity p-n jun ction devices are called tunnel diodes." (p. 77) 12-12 NOISE Thermal or Johnson, Noise

Th e electron s in a con ductor possess varyin g amoun ts of en ergy by virtue of th e

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temperature of th e con ductor. Th e sligh t fluctuation s in en ergy about th e values specified by th e most probable distribution are very small, but th ey are sufficien t to produce small n oise poten tials with in a con ductor. Th ese ran dom fluctuation s produced by th e th ermal agitation of th e electron s are called th e thermal, or Joh n son , n oise. Th e rms value of th e th ermal-resistan ce n oise voltage V n over a frequen cy ran ge fH - fL is given by th e expression

Vn

2

= 4 k T R B

(12-51)

wh ere

k T R B

= = = =

Boltzmann constant, 1.380E-23 J/deg K resistor temperature, deg K = deg C + 273 resistance, ohms fH - fL = bandwidth, Hz

It sh ould be observed th at th e same n oise power exists in a given ban dwidth regardless of th e cen ter frequen cy. Such a distribution , wh ich gives th e same n oise per un it ban dwidth an ywh ere in th e spectrum, is called white noise." (p. 401) "If th e con ductor un der con sideration is th e in put resistor to an ideal (n oiseless) amplifier, th e in put n oise voltage to th e amplifier is given by Eq. (12-51). An idea of th e order of magn itude of th e voltage in volved is obtain ed by calculatin g th e n oise voltage gen erated in a 1-M resistan ce at room temperature over a 10-k Hz ban dpass. Equation (12-51) yields for V n th e value of 13uV." (p. 402) Shot, or Schottky, Noise

Sh ot n oise is attributed to th e discrete-particle n ature of curren t carriers in semicon ductors. Normally, on e assumes th at th e curren t in a tran sistor or FET un der dc con dition s is a con stan t at every in stan t. Actually, h owever, th e curren t from th e emitter to th e collector con sists of a stream of in dividual electron s or h oles, an d it is on ly th e time-average flow wh ich is measured as th e con stan t curr en t. Th e fluctuation in th e n umber of carriers is called shot noise. Th e mean -square sh ot-n oise curren t in an y device is given by

In

2

= 2 q Idc B

wh ere

q = electronic charge, 1.60E-19 coulombs Idc = dc current, Amps B = bandwidth, Hz If th e load resistor is RL, a n oise voltage of magn itude InRL will appear across th e load." (p. 402)

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1976 -- Ott Ott, H. 1976. Noise Reduction Techniques in Electronic Systems. Joh n Wiley & Son s. THERMAL NOISE "Th ermal n oise comes from th ermal agitation of electron s with in a resistan ce, an d it sets a lower limit on th e n oise presen t in a circuit. Th er mal n oise is also referred to as resistan ce n oise or 'Joh n son n oise' (for J. B. Joh n son , its discoverer.)" "He sh owed th at th e open -circuit rms n oise voltage produced by a resistan ce is V t = ( 4 k T B R )1/2 wh ere, k = Boltzman n 's con stan t (1.38 x 10 -23 joules / deg. Kelvin ), T = Absolute temperature (deg. Kelvin ), B = Noise ban dwidth (Hz), R = Resistan ce (Oh ms)." (p. 198) "Alth ough th e rms value for th ermal n oise is well defin ed, th e in stan tan eous value can on ly be defin ed in terms of probability. Th e in stan tan eous amplitude of th ermal n oise h as a Gaussian , or n ormal, distr ibution ." (p. 203) "Th e crest factor of a waveform is defin ed as th e ratio of th e peak to th e r ms value." ". . . a crest value of approximately 4 is used for th ermal n oise." (p. 204) SHOT NOISE "Sh ot n oise is associated with curren t flow across a poten tial barrier. It is due to th e fluctuation of curren t aroun d an average value resultin g from th e ran dom emission of electron s (or h oles). Th is n oise is pr esen t in both vacuum tubes an d semicon ductors. In vacuum tubes, sh ot n oise comes from th e ran dom emission of electron s from th e cath ode. In semicon ductors, sh ot n oise is due to ran dom diffusion of carriers th rough th e base of a tran sistor an d th e ran dom gen eration an d recombin ation of h ole electron pairs." "Th e sh ot effect was an alyzed th eoretically by W. Sch ottk y in 1918. He sh owed th at th e rms n oise curr en t was equal to: Ish = ( 2 q Idc B )1/2 wh ere q = Electron ch arge (1.6 x 10 -19 coulombs), Idc = Average dc curr en t (A), B = Noise ban dwidth (Hz)." "Th e power den sity for sh ot n oise is con stan t with frequen cy an d th e amplitude h as a Gaussian distribution . Th e n oise is wh ite n oise an d h as th e same ch aracteristic as previously described for th ermal n oise." ". . . by measurin g th e dc curren t th r ough th e device, th e amoun t of n oise can be very accurately determin ed." "A diode can be used as a wh ite n oise source. If sh ot n oise is th e predomin an t n oise source in th e diode, th e r ms value of th e n oise curren t can be determin ed simply by measurin g th e dc curren t th rough th e diode."

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(pp. 208-209) [ Note that the above expression for shot noise current is not the same as shot noise voltage across a semiconductor junction./tfr ] MEASURING RANDOM NOISE "An oscilloscope is an often overlook ed, but excellen t device for measurin g wh ite n oise." "Th e rms value of wh ite n oise is approximately equal to th e peak -to-peak value tak en from th e oscilloscope, divided by eigh t. Wh en determin in g th e peak -to-peak value on th e oscilloscope, on e or two peak s th at are con siderably greater th an th e r est of th e waveform sh ould be ign ored. With a little experien ce, rms values can be accurately determin ed by th is meth od." (p. 212)

1980 -- Malvino Malvin o, A. 1980. Transistor Circuit Approximations, Third Edition. McGraw-Hill. Avalanche and zener effects

"Break down is cause by eith er of two effects: avalan ch e or zen er. Wh en a diode is reverse-biased, min ority carriers flow in th e reverse direction . For h igh er r everse voltages, th ese min ority carriers can reach sufficien t velocities to k n ock valen ce electron s out of th eir sh ells. Th ese released electron s become free electron s an d can attain sufficien t velocity to dislodge more valen ce electron s. Th e resultin g avalan ch e of free electron s pr oduces a large reverse bias curren t. "Zen er effect is differen t. Th e electric field across th e jun ction can become in ten se en ough to pull valen ce electron s directly out of th eir sh ells. Th is produces a large reverse curren t. Zen er effect is sometimes called high-field emission because it is th e electric field th at produces free electr on s. "Wh en a diode break s down , eith er avalan ch e or zen er effect predomin ates. Below 6 V, th e zen er effect is more importan t. Above 6 V, th e avalan ch e effect tak es over. Diodes with break down voltages greater th an 6 V sh ould be called avalan ch e diodes. But th e gen eral pr actice in th e in dustry is to refer to diodes exh ibitin g eith er effect as zener diodes." (p. 37)

1983 -- Warner and Grung Warn er, R., an d B. Grun g. 1983. Transistors: Fundamentals for the Integrated-Circuit Engineer. Joh n Wiley & Son s. 6-5 BREAKDOWN PHENOMENA IN REVERSE-BIASED JUNCTIONS "For most jun ction s [...] th ere exists a critical voltage above wh ich reverse curren t in creases with voltage, sometimes ver y sh arply. Such beh avior is called breakdown." (p. 470) 21 of 26

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6-5. 1 Tunneling

" . . . it is possible th at an electron can 'pen etrate' a poten tial barrier, even a barrier of great h eigh t, provided th e dimen sion X [barrier th ick n ess 'of th e same order as th e exten t of th e wave fun ction '] is small en ough . Such pen etration is called tunneling." (p. 471) 6-5. 2 Avalanche-Breakdown Fundamentals

"For h istorical reason s, voltage regulatin g diodes of an y sort are k n own commercially as Zen er diodes, n amed for Claren ce Zen er wh o offered an early description of th e direct excitation of electron s from th e valan ce ban d in to th e con duction ban d. (p. 476) "Th ere are two gen eral requiremen ts for avalan ch e break down . Th e electric field must be large, an d it must exist over a sufficien tly exten sive region ." "Fin ite space is required for th e carrier-multiplication to tak e place." (p. 477) "It turn s out also th at th e details of field profile are n ot very importan t." (p.477) ". . . X T [depletion-layer thickness] is th e best lon e cr iterion for predictin g avalan ch e-break down voltage V B." (p. 478) 6-5. 3 Avalanche-Breakdown Theory

"Th e experimen tal results given in th e previous section can be explain ed on ly rough ly by th eory sin ce th e ph ysical in teraction s are very complicated. An y avalan ch e-break down th eory must begin with a study of th e ion ization coefficien ts for electron s an d h oles, wh ere th ese coefficien ts represen t th e ability of en ergetic electron s an d h oles to produce addition al carriers in pairs. Th en , th e th eory must con tin ue with a calculation of th e con dition s for break down of a simple on e-dimen sion al step jun ction . Fin ally, it must address man y complex issues such as th e effects of jun ction curvature." (pp. 479-480)

1984 -- Zanger Zan ger, H. 1984. Semiconductor Devices and Circuits. Joh n Wiley an d Son s. 2-4. 3 "Zener" and "Avalanche" Breakdown

"Figure 2-17 sh ows a poin t, in th e reverse bias portion of th e graph , at wh ich th e reverse curren t in creases sh arply--th e 'zen er' or 'avalan ch e' region . Th e sudden in crease in reverse curren t is th e result of a very large r everse electric field." (p. 50) "Th e h igh reverse field accelerates th e con duction ban d (free) electron s. As th ese collide with valen ce electron s, th e en ergy of collision may be sufficien t to tr an sfer th e valen ce electron s in to th e con duction ban d. Th is in creases th e n umber of min ority carriers, in tun in creasin g th e reverse (min ority) curren t. As th e field is in creased, th is "multiplication " in min ority carriers is accelerated an d very h igh 22 of 26

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reverse curren ts ar e produced. Th is pr ocess is called 'avalan ch e.'" (p. 50) "Th e min ority carr ier con cen tration may be in creased by directly 'tearin g' electron s out of th e valen ce ban d (removin g th em from th e atomic orbit) in to th e con duction ban d." [...] "Th e pr ocess is called 'zen er break down .' At very h igh dopin g levels th e zen er break down occurs at lower fields th an th e avalan ch e break down . It is n ot surprisin g, th erefore, th at th e diodes design ed to exh ibit break down ch aracteristics at lower voltages are h eavily doped an d th e process in volved is th e zen er br eak down . (For voltages from 2.4 to 6 V th e process is zen er break down , an d from 6 V up th e process is avalan ch e break down .) Th ese processes are n ot n ecessarily destructive, th at is, th e diode can operate in th is region with n o damage to th e diode, provided oth er parameters, such as power dissipation , diode temperature, etc., are k ept with in boun ds." (p. 51)

1987 -- Vergers Vergers, C. 1987. Handbook of Electrical Noise. TAB Book s, Blue Ridge Summit, PA. SHOT NOISE "Th e term 'sh ot n oise' arose from th e study of ran dom variation s in th e emission of electron s from th e cath ode of a vacuum tube. If th ese var iation s are amplified an d listen ed to with a pair of h eadph on es or a loudspeak er, th ey soun d lik e 'lead sh ot' h ittin g a con crete wall. Sh ot n oise h as a flat spectral den sity lik e th ermal n oise. Th erefore, sh ot n oise can be con sider ed a 'wh ite n oise' process." (p. 96) Sh ot Noise in PN Jun ction s "Th e sh ot n oise gen erated in a pn jun ction h as th e same math ematical form as th at of th e temperature limited vacuum diode. Th e n oise seems to be gen erated by a n oise curren t gen erator in parallel with th e dyn amic resistan ce of th e diode." (p. 108) Ins = ( 2 e Idc B )1/2

e = Electron charge (1.6E-19 coulombs) I = current in amperes B = Bandwidth in Hertz "Lik ewise we may determin e th e sh ot n oise voltage by applyin g Oh ms Law." [ E = IR ] E ns = ( 2 e Idc B r d2 )1/2 [ or ] E ns = r d ( 2 e Idc B )1/2 wh ere r d is th e dyn amic resistan ce of th e jun ction ." (p. 109)

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"From electron ic ph ysics it is k n own th at th e dyn amic r esistan ce of a pn jun ction depen ds on temper ature an d th e direct curren t flowin g th rough th e jun ction . Th e dyn amic resistan ce represen ts th e ratio of a small ch an ge in diode voltage to a correspon din g ch an ge in diode curren t." r d = k T / e Idc [ we substitute and get ] E ns = ( 2 k T B r d )1/2 "It is obvious . . . th at th e sh ot n oise voltage across th e pn jun ction h as an equation very similar to th at of a th ermal n oise process." [ we can substitute again and get ] [ E ns = k T ( 2 B / e Idc )1/2 ]

k = Boltzmanns constant (1.38E-23 Joules/deg. Kelvin) T = Temperature in degrees Kelvin B = Bandwidth in Hertz rd = junction dynamic resistance e = Electron charge (1.6E-19 coulombs) "Th ere is a rath er in terestin g relation between th e sh ot n oise voltage across th e jun ction an d th e dyn amic resistan ce rd. Sin ce rd is in versely propor tion al to direct curren t, th e dyn amic resistan ce falls as direct curren t in creases. Th is causes th e sh ot n oise voltage across th e jun ction to decrease." ". . . sh ot n oise curren t is proportion al to th e square root of direct curren t wh ere dyn amic resistan ce is in versely proportion al to direct curren t. We fin d th at if direct curren t in cr eases, dyn amic resistan ce falls more quick ly th an sh ot n oise curren t rises. Th e result is th at sh ot n oise voltage becomes in ver sely related to" [ the square root of ] "direct curren t." (p. 110)

1989 -- Horowitz and Hill Horowitz, P. an d W. Hill. 1989. The Art of Electronics. 2n d Ed. Cambridge Un iversity Press. 6.14 Zen er Diodes "Zen er diodes can be very n oisy, an d some IC zen ers suffer from th e same disease. Th e n oise is related to surface effects, h owever, an d buried (or subsurface) zen er diodes are con sider ably quieter." [Since shot noise is a fundamental effect, there can be no zener diode which does not produce this noise. But if some zeners are especially "noisy," the extra noise does not come from the fundamental effect and so must have a suspicious statistical distribution./tfr]

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Figure 6.22 is titled: "Voltage n oise for a low-n oise zen er referen ce diode similar to th e type used in th e 723 regulator" an d plots en voltage n oise (uV/Hz 1/2) versus zen er curren t (mA). Th e graph sh ows somewh at less th an 0.1 uV n oise for 0.1 mA, an d somewh at less th an 0.01 uV n oise for 10 mA. In th is graph , th e n oise is in ver sely related to th e square root of th e curren t. (p. 335) Joh n son n oise "An y old resistor just sittin g on th e table gen erates a n oise voltage across its termin als k n own as Joh n son n oise. It h as a flat frequen cy spectrum, mean in g th at th ere is th e same n oise power in each h ertz of frequen cy (up to some limit of course). Noise with a flat spectrum is also called 'wh ite n oise'." (p. 430) ". . . a 10k resistor at room temperature h as an open -cir cuit rms voltage of 1.3uV, measured with a ban dwidth of 10k Hz . . . ." (p. 431) Sh ot n ose An electric curren t is th e flow of discrete electric ch arges, n ot a smooth fluidlik e flow." "If th e ch arges act in depen den t of each oth er, th e fluctuatin g curren t is given by Inoise(rms) = In R = ( 2 q Idc B )1/2 wh ere q is th e electron ch arge (1.60 x 10 -19 coulomb) an d B is th e measuremen t ban dwidth . For example, a 'steady' cur ren t of 1 amp actually h as an rms fluctuation of 57n A, measured in a 10k Hz ban dwidth ; i.e., it fluctuates by about 0.000006%." "Th e sh ot-n oise for mula . . . assumes th at th e ch arge car riers mak in g up th e curren t act in depen den tly. Th at is in deed th e case for ch arges crossin g a barrier, as for example th e curren t in a jun ction diode, wh ere th e ch arges move by diffusion ; but it is n ot true for th e importan t case of metallic con ductors, wh ere th ere are lon g-ran ge correlation s between ch arge carriers." (p. 432) Measur in g th e n oise voltage "Th e most accurate way to mak e n oise measuremen ts is to use a true rms voltmeter." "If you use a true r ms meter, mak e sure it h as respon se at th e frequen cies you are measurin g . . . ." "T rue rms meters also specify a 'crest factor' . . . ." "For Gaussian n oise, a crest factor of 3 to 5 is adequate." "Y ou can use a simple averagin g-type ac voltmeter in stead . . . ." "To get th e rms voltage of Gaussian n oise, multiply th e 'rms' value you read on an averagin g ac voltmeter by 1.13 (or add 1dB)." "A th ird meth od . . . con sists of look in g at th e n oise waveform on an oscilloscope: Th e rms voltage is 1/6 to 1/8 of th e peak -to-peak readin g . . . . It isn 't very accurate, but at least th ere's n o problem gettin g en ough measuremen t ban dwidth ." (p. 454) Terry Ritter, his current address, and his top page.

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Last updated: 2004-01-14 (from 2003-12-08, 1996-08-15)

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