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Unified Presentation of l l f Noise inElectronic Devices: Fundamental l/f NoiseSourcesALDERT VAN DER ZIEL,FELLOW, IEEEThis review represents l / f noise in electronic devices in terms ofthe Hooge parameter aH of the devices. A generalized schematicis given for expressing the noise spectrum S,(f) in the external circuit in terms of distributed noise sources of the nonuniformand so one can evaluate aHfrom S,(f). Thedevices in terms of a, ;results can then be compared with Handel’s predictions for awDespite the fact that there are several objections to Handel‘s derivation of a ,,it seems that his final result usually agrees with experiment; apparently the results are not sensitive to the details of the(Bremsstrahlung)photon-electron interaction (Appendix I).Collision-free devices (pentodes, vacuum photodiodes, secondary emission multiplier stages, etc.) can always be represented byfundamental l / f noise sources after spurious noise sources havebeen eliminated or discriminated against. Collision-dominateddevices can show fundamental normal collision l / f noise, Umklappl / f noise, intervalley scattering l / f noise (if there are intervalleys),Umklapp l / f noise and, in long devices, coherentintervalleystate or Hooge-type l / f noise. Most of these processes occur, exceptpure intervalley l / f noise, which is replaced by intervalley Umklapp Ilf noise. Such devices include Schottky barrier diodes,n -p diodes, p-i-n diodes, n -p-n and p -n-p BITS, n-channel andp-channel Si-IFETs, and p-MOS devices operating under stronginversion. The schematic can also be applied to ballistic devices. I.INTRODUCTIONIt i s the aim of this review paper to present I l f noise insemiconductors, semiconductor devices, and collision-freedevices (like vacuum tubes) from a unified point of view,using an extended version of the Hooge equation [ I ] as avehicle. It i s then found that the Hooge parameter, introduced by this equation, can be used as a general measureof the noisiness of a system or device. It i s finally attemptedto correlate measured values of the Hooge parameter withthe values calculated from Handel’s quantum theory ofl l f noise [2], [3].The approach i s in itself not new: what is new, however,is i t s generalizationt o all systems and devices. Also, the program would already give full practical benefits if the invesManuscript received August 21,1987; revised November 6,1987.The submission of this paper was encouraged after review of anadvance proposal. The work performed in this paper was supported by NSF grants and ARO Grant DAAG-29-85-KO2356.The author is with the Department of Electrical Engineering,Universityof Florida,Cainesville, FL32611,andtheDepartmentof Electrical Engineering, University of Minnesota, Minneapolis, MN55455, USA.I E E E Log Number 8718798.tigation stopped after the measurement of the Hoogeparameter. But the comparison between the theory andexperiment opens up the possibilityof refuting or verifyingHandel’sformulas in a large number of cases,and will resultin a generalized framework in which all the experimentaldata can be placed.Section 11-A formulates and generalizes the Hooge equation to all collision-dominated systems involving mobility,diffusion, and cross-section fluctuations. It also applies tocollison-free processes involving vacuum tubes, Schottkybarrier diodes operating in the thermionic mode and indevices such as p-i-n diodes in which collision processesare not the determining factor. In those cases, the effectivenumber N of carriers i s better expressed in terms of thedevice current.Section 11-6 deals with Handel’s quantum theories ofI l f noise. The discussion does not imply validity of thoseequations, but simply states what the theories would predict;this istheonlywayinwhichanytheory, including Handel’s, can be verified or refuted by experimental data. Handel‘s theory i s based on the Hooge equation and gives anexpression for the Hooge parameter cyH. Theory and experiment thus deal with the same parameter. The main emphasis of the paper i s on the generalized framework.Section Ill discusses how S,(f) can be expressed in termsof cyH. To that end, Hooge’s equation for S,(f) i s replaced bythe spectrum of a distributed noise source. In the simplestcases one writes down the Langevin equation involving arandom source term H(x, t),linearizes this equation, solvesit, and expresses S,(f) in terms of integrals over the crossspectral intensity SH(x, x’, f) of H(x, t); the latter in turn, i sexpressed in terms of the Hooge equation. Sometimes thismethod i s inadequate and other methods must be used.These methods are by themselves not new, but are hereapplied systematically. Some of the applications are new.Section IV discusses several cases in which the noise doesnotobeythequantum Ilfnoisetheory. It isshownthat number fluctuation noise gives a l l f spectrum caused by a distributionoftimeconstants(McWhorter’s model).This isthecase when there are distributed traps in the surface oxide(MOSFETs, BJTs);it results in a current dependence that isdifferent from what i s expected from Hooge’s theory.Section V discusses measurements an many different0018-9219/88/0300-0233 01.0001988 IEEEPROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988233

devices. In most cases the predictions made by Handel’stheory are verified. This does not necessarily indicate thatthe mathematical derivation of these predictions are correct; this remains open to discussion.II. GENERALBACKGROUNDOF THE PROBLEMA. The Hooge Equation a n d the Hooge Parameter1) Collision-Limited Devices: W e first t u r n t o the Hoogeequation itself. W h e n a constant voltage V is applied t o asemiconductor resistor of resistance R, a fluctuating current I(t)isdeveloped. Thiscan o n l y c o m e a b o u t becausetheresistance R(t) of the device fluctuates. SinceV I(t)R(t) const.If R and 6R are independent of current, S,(f)/I2will be independent of current also. This result i s true for generationrecombination (g-r) spectra caused b y traps; they give Lorentzian spectra of the form const/(l w * I t) i.s thereforealso true for I l f spectra caused by a superposition of Lorentzian spectra, as i n McWhorter’s theory of I l f noise (Section IV). But, as w e shall see, the possibility must also beleft open that there are true l l f spectra, not caused by sucha superposition.Irrespective of the cause of the l l f noise, S,(f)/I2 may bewritten assmaller than 2 xfor sufficiently short resistors ( L 100pm) whereas the Hooge value of 2 xwas obtainedfor sufficiently long devices ( L 500 pm). A systematicexperimental study of the dependence of a,, u p o n thedevice length L , which has not been made so far, w o u l d bevery he1pf u I.Hooge gave n o proof of (2), b u t it i s easily seen that anequation like (2), w i t h constant aH,could be expected if thel / f noise i s generated b y N independent carriers. For in thatcase both l a n d Sl(f) w o u l d be proportional to N so that Sl(f)/I 2 w o u l dbe inversely proportional to N. This would be fundamental noise.Since the resistance R i s inversely proportional to theproduct pN, where p i s the carrier mobility, there can befluctuations 6p in p and (or) 6N in N, so that, since 6 p and6N are independentor Sl(f)12-If the fluctuation in p predominatesand the noise i s called mobility fluctuation l l f noise,whereasconstfand it may be implied that the noise is caused by resistancefluctuations. Clarke and Voss [4], [5] showed the presenceof such resistance fluctuations in a beautiful experiment.The question is n o w what other parameters enter into theconstant introduced by (la). Hooge suggested that for arectangular semiconductor the missing parameter was thenumber N of carriers of the sample and wrote the empiricalformula, n o w k n o w n as the Hooge equation,This equation neither proves anything nor predicts anything, but merely gives an operational definition of theHoogeparametera,,. It i s always valid, b u t i s only useful ifone can extract useful information o u t of the value of a,,.Since a rectangular semiconductor bar of length L andcross-sectional area A has a resistance R L2/(epN),wherep i s the carrier mobility, N follows from R, and hence ayHfrom(2). W h e n o n e d o e s t h i s for a n u m b e r of different semiconductor resistors of comparable length L , one can characterize the noisiness of the various materials by the parameter aH. Hooge [I] found i n that manner that for manysemiconductor samples a,, had a value of about 2 xnearly independent of the material. Hanafi e t al. [6] foundfor ten Hg, -,Cd,Teresistor bars w i t h different doping and(or) different values of x (but all made by similar techwith aniques), that aHhad an average value of 5 xspread of less than a factor 2. The near constancy of a,, suggests that this l l f noise is d u e t o a fundamentalmechanismof u n k n o w n origin; this is useful information that will befound t o be valid in other situations as well.Later it was found [7l, [8] that a,, could be considerably234(3)if the fluctuations in N predominate; the noise i s then callednumber fluctuation l l f noise. In principle either relationshipcan occur, b u t in practice mobilityfluctuation I l f n o i s epredominates in many cases. W e come back to that problem in Sections IV and V.Do (3a) and (3b) result in the Hooge equation? In orderthat this be the case, S,(f) and SN(f) must vary as l / f over awide frequency range and in addition S,(f)/p* and (or) SN(f)/N2must vary as 1lN.W e come back t o the spectral dependencies in Section IV, b u t wish to point o u t here that thelatter i s the case if SN(f) i s proportional t o N.W e shall n o w show that S,(f)lp2 always varies as 1/N. Inaddition, if each electron, in and by itself, produces Ilfnoise,the full Hooge equation (2) results.The proof i s simple, as Hooge [9] and van der Ziel et al.[IO] have demonstrated. W e introduce the short-termmobilityp,of the individual carriers. If N d o e s not fluctuate,and the p,’s are independentNPN 6p 1 lICl1c1 NPIF CLI(4)cI N6p,Nr l-so that S,(f)/p* varies as 1/N. I n addition SP,(f)/(pJ2is independent of N and was postulated to have a llfdependence.PROCEEDINGS OF THE IEEE, VOL.76, NO. 3,MARCH 1988

We may then writeWe thus see that for mobility fluctuations the Hoogeequation i s always valid and that aHi s defined as ftimes therelative mobility I l f spectrum S,,(f)l(Fi)2 of a single electron.This would then be fundamental l l f noise.Because of the Einstein relation eD kTp, mobility fluctuations correspond to fluctuations in the diffusion constant D. Consequently(5)A Hooge-typeequation may therefore also be expected forsolid-state devices governed by diffusion processes, suchas occur in p -n and n -p junction diodes, p -n-p andn -p-n BJTs, and Schottky-barrier diodes operating in thediffusion mode. Corrections may be neededfor degeneratesystems.Since FETs are bias-dependent semiconductor resistors,they should show I l f noise. For devices operating at nearzero drain bias the device i s a uniform semiconductor resistor, but for larger bias the resistor becomes nonuniformdue to channel pinch-off. For such nonuniform resistorsone must replacethe Hoogeequation by itsdifferential formholding for each section Ax at xwhere N(x) is the carrier density per unit length and l(x) thecurrent at x. It is thus possible to treat the Hooge equationasthespectrumof adistributed noisesourceH(x, t). Byevaluating the contributions of individual sections Ax to thespectrum Sl(f) of the total current I , one can express Sl(f)in terms of aHand other measurable device parameters, sothat aHcan be determined for all these devices and the relative noisiness of the various noise mechanisms can beestablished.The methodsfor solving such distributed noiseproblems are discussed in Section Ill. They work so longas Ax is larger than the free path length of the carriers.There i s one other further noise problem that requiresattention. In relatively long n -pdiodes part of the injectedcarriers disappears by recombination. In that case the lifetime 7 of the individual carriers fluctuates. It i s shown inSection Ill that for C 117 (C i s independent of x)Sdx, f) fN(x) Axc2(7)s o that this problem can be incorporated into the generalschematic.A related problem is the noise due to fluctuations in thecontact recombination velocity s, at an ohmic contact (sen Io’cmls). To that end consider a planar n -pdiode witha length wp of the p-region (wp L, short diode) whereL, (Dn7n)1/2i s the diffusion length of the electrons in thep-region. Then I, es,,N’(x),and, in analogy with (7)A similar effect can occur in the surface recombinationvelocitys in ajunction space-chargeregionor in the surfacerecombination velocity in the base region of a BJT. Here sis usually much smaller than s,(s, 6 x I O 6 cmls in a contact on n-type silicon, and s 1 cmls for a well-passifiedsurface on n-type silicon).Many papers have been written about samples of nonrectangular geometry. For references see Hooge et al.review paper [Ill.2) Collision-Free Devices: Up to here we discussed onlysemiconductor devices that were collision limited, so thataHwas determined by collision processes. We now turn todevices in which collisions either do not exist, as in vacuumtubes and in Schottky-barrier diodes operating in thethermionic mode, or to devices in which collision processesare notthedeterminingfactor,as in long p-i-n diodes.In that case a Hooge type equation of the form [12], [I31(8)describes the l l f noise. Here a,.,may have a different magnitude than in collision-dominateddevices, but N again i sthe number of carriers in the system. This is, e.g., the casefor vacuum tubes like space-charge-limitedvacuum diodes,triodes, and pentodes, or saturated vacuum photodiodes,and secondary emission multiplication stages, etc. It holdsfor any system in which the N carriers generate I l f noiseindependently (fundamental l l f noise).Since the current flow i s by injection, //e is the numberof carriers injected per second and N We is the numberof carriers present in the sample. Consequently, if we substitute for NIaHelS/(f) f7(8a)where 7 i s the carrier transit time. For an electron travelingbetween two parallel electrodes at a distance dZl,with negligible charge between them(8b)where v, and vl are the carrier velocities at the electrodes2 and 1, respectively. For space-charge-limitedcurrent flowbetween two parallel electrodes of distance dl7 3(dl -dm)(8c)V1where d, is the distance between the potential minimumand cathode, v1 (2e/m)”’(VV,)12’is the velocity withwhich the electrons arrive at the anode, V is the anodepotential, and V, the depth of the potential minimum infront of the cathode. For a long pi-n diode 7 is the time constant associatedwith the generation and recombination ofone hole-electron pair. In each case aHcan be determinedfrom S,(f) if 7 i s known.Not all noises in collision-free devices satisfy (8a).Whether or not they do, must be determined by comparingthe measured value of aHwith the theoretical values predicted in the next section. B. Handel’s Quantum Equations p], [3] where Nee 1/2 [N(O) N(wp)]wp i s the effective numberof minority carriers in the base region (see below).VAN DER ZIEL l/f NOISE IN ELECTRONICDEVICES1) Collision-Free Devices: We first start with a semiclassical consideration of collision-free devices. Since the only235

physical process present in such devices isacceleration, theobserved l l f noise must be associated with this acceleration. Now an accelerated electron generates low-frequencyBremsstrahlung; since its energy spectrum is independentof the quantum energy E for small E, and the number spectrum is found by dividing the energy spectrum by hf, it isobvious that this number spectrum varies as Ilf. The nearfield interaction of an electron with itsown Bremsstrahlungwill therefore give current l l f noise in the external circuitthat i s described by the Hooge parameter a,.,The. effect issemiclassical; to evaluate aHone needs wave mechanics.Handel uses a somewhat different model. He splits theelectron wave function into a large unperturbed part anda small part that is perturbed by the Bremsstrahlung emission. In the calculation the two parts beat with each otherand so give I l f noise back. Handel thus finds the following[14]:The first part is known as the Handelequation. Herec i s thevelocity of light, AVthe vectorial change in velocity alongthe electron path, and a the fine structure constant. Formotion between two parallel electrodesof distance d12,withterminal velocities v1 and v2, AV v2 - vl and 7 2d121(v2vl), as mentioned before. The main objection to thisapproach is against the beat process. For single electrons,in MKS units, where po 4 r X I O - ’ Hlm a a0 poce2I 137SAf) 2h4a A g e 1--.37r c2 f7(94But in some cases the current flows in charge conglomerates q. Since they are accelerated as a unit, they produceBremsstrahlung as a unit, and hence generate l l f currentnoise as a unit; consequently e2must be replaced by q2,ora a0( (9b)As afirst examplewe consider aspace-charge-limited vacuum diode. Here the shot noise i s space-charge-suppressed by a factor r2( 0.10 for normal operation) so thatS,(f) may be writtenS,(f) 2e1P 2(er2)/ 2ql(10)corresponding to shot noise of charges q, so that the effective charge is q er2. Hence the l l f noise may be writtens,(n r 3*c2 f7‘In vacuum diodes with oxide-coated cathodes the noise ismasked by l l f noise generated in the cathode coating, s othat (loa) is not verifiable.In vacuum photodiodesr4 1(no space-chargesuppression),Avismuch largerthan in thepreviouscase,and henceSif) becomes(12)when 7 i s the transit time between the secondary emissionelectrode (dynode) and the collecting electrode (anode). Insecondary emission pentodes this noise is usually maskedby the l l f noise, Sdf), of the primary current. The latter canbe suppressed satisfactorily by appropriate cathode feedback [12], [13], [15], [17]; in that case Sl(f) becomes measurable. Again, S,(f) varies as lV:”, where V, i s the potentialdifference between anode and dynode, and this can be verified (see Section V).In vacuum pentodes cathode I l f noise is distributedbetween screen grid and anode, whereas partition l l f noiseflows from screen grid to anode. The latter i s not spacecharge suppressed, whereas the former is. Nevertheless,the partition llfnoise i s maskedbycathodellfnoise, unlessthe latter is sufficiently reduced by a feedback resistor R,in the cathode lead. In that case the partition l l f noisebecomes measurable [12], [13], [I81 and i t s possible quantum character can be investigated (Section V).We have here discussed the predictions made by Handel’s quantum I l f noise theory for various vacuum tubes.By comparing the calculated spectra with the experimentaldata we may then be able to either refute or verify thesepredictions. The experimental data are independent ofHandel’s equations (9)-(12), and so can serve as independent checks of those equations.2) Collision-DominatedDevices: We now turn to quantum I l f noise in semiconductor devices. Here the devicesare collision-dominatedand (9) must be appropriately modified: in (9)A musJnowbe averaged over all collisions andbe replaced by A . Bremsstrahlung l l f noise is still considered the initiating process, however. Since the carriersare single electrons or single holes, they always have acharge *e; hence the fine structure constant (11always hasthe value a0 ll(137). Consequently, for a single scatteringprocess, (9) may be written as4a A AS2- 3.09 x IO- 37r c2C2(13)where the averaging must be performed in k-space over allscattering angles 8 and over the electron-velocity distribution [2], [3], [19].There are different scattering processes possible, eachhaving its associated mobility pi and Hooge parameter aHj.We then have according to Kousik and van Vliet [20](11)where 7 2d/(v2 v,) and AV (v2- vl). If

Electronic Devices: Fundamental l/f Noise Sources ALDERT VAN DER ZIEL, FELLOW, IEEE This review represents l/f noise in electronic devices in terms of the Hooge parameter aH of the devices. A generalized schematic is given for expressing the noise spectrum S,(f) in the external cir- cuit in terms of distributed noise sources of the nonuniform .

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