Mark E. Snyder- Determination of the Unstable States of the Solid State Plasma in Semiconductor...

8/3/2019 Mark E. Snyder- Determination of the Unstable States of the Solid State Plasma in Semiconductor Devices

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D E T E R M I N A T I O N Q r T H E U N S T A B L E S T A T E S O F T.HE S O L I DS T A T E P L A S M A I N S E M I C O N D U C T O R D E V I C E S

b yM A R K E . S N Y D E R , B . S . , M . S .

A D I S S E R T A T I O NIN

P H Y S I C S

S u b m i t t e d t o t h e G r a d u a t e F a c u l t yo f T e x a s T e c h U n i v e r s i t y inP a r t i a l F u l f i l I m e n t o ft h e R e q u i r e m e n t s f o rt h e D e g r e e o fD O C T O R O F P H I L O S O P H Y

A p p r o v e d ^ 7

A c c e p t e d

M a y , 1 9 8 8


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n^id

ACKNGULEDGENENTSIt is difficult to pass along the proper regard to

everyone who has contributed to my education. Yet, I uiilJattempt to name my mentors, friends and colleagues withoutwhom my time here would have certainly been diminished.

The physics department has given me many opportunitieto make the best of my edi.cation. I have learned more abcteaching and obtained the chance to sharpen my teachingskills. The faculty has provided me with knowledge andexperience, which I hope I can continually improve upon.

I wish to thank the rrembers of my dissertationcommittee: Drs. Hyles, IshLhara, Peters and Lichti, who haoffered help and feedback r.hat contributed immensely to mydissertation preparation. flj committee represents theindividuals who have contributed the most to my education.

I wish to especially ",hank Dr. UJilliam Portnoy, whobeen a friend and mentor frj' many years. Or. Portnoy hasprovided a perspective on graduate school that has made thmeans to my educational ends more tolerable.

Finally, I wish to decicate this dissertation to thememory of my father, who unknouiingiy, lead me into thewonderful world of science.


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ft

TABLE Of' CONTENTS

ACKNaULEDGEHENTSABSTRACTLIST OF TABLES .LIST OF FIGURESCHAPTER

I. THE PROBLEn AND ITS pISTORICAL BACKGROUNDThe Physics of Device Failure: The Problem . . .

History of Research ResultsPreliminary Results of a FormalRedevelopment of DeviceFailure Phenomena. ,

Parallels Between Classical Plasmasand Solid State PlasmasRestatement of the Problem in Lightof Current Developments

II. GENERAL CHARACTERIZATION OF THE SOLIDSTATE PLASHA: PLASMA CRITERIAThe Classical PlasmaThe Solid State Plasma Criteria

III. ANALYSIS OF SPECIFIC NEGATIVE RESISTANCEIN SOLIDS PRGPOSED BY RIDLEYIn-depth Analysis of Us iSroots's Derivationof the Equations of Motion in IrreversibleProcesses of Continuous Systems

Analysis of Ridl?.y's Derivation

Ill


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Conditions of the Stationary State . . .Application of the Stationary Stateby Ridley to the Case of CCNR . . .Perturbation Under CCNR

IV. QUANTITATIVE ANALYSIS OF THE SOLIDSTATE PLASMA PINCHThe Basic Hydrodynamic Equations . . . .The Static PinchThe Hydrodynamic Stability of the Pinch.Solid State Plasma Parameters Necessaryto Initiate Pinching

V. APPLICATION OF THE PLASMA PARAMETERSTO SEMICONDUCTOR JUNCTIONSDescription of Semiconductor Junctions .Application of Plasma Pinch to aTransistor

Type A Second BreakdownType B Second BreakdownType C Second Breakdown

VI. GENERAL CONCLUSIONS ( ND FUTURE WORK. . .F u t u r e U l o r k

Application to Device Failure. . . .LIST OF REFERENCES

IV


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ABSTRACTUlhen a semiconductor device fails, a complicated set c

actions takes place to movp the device from a high voltage,low current state to a low voltage, high current state.Accompanying these changes are the collapse of isotropiccurrent flow to a small colijmn or filament of current whic^rapidly grows and heats the device to the melting point.These actions are collectivsiy called second breakdown.This work reviews past res;Its to show that second breakdowis the formation of a current controlled negative resistancCCCNR) regime which necessarily forces current flow to formcurrent filament. The current filament is modeled as a solstate plasma column undergoing a self-induced magnetic pineas a result of the CCNR. Ti'e dispersion relation is derivedto first order to show that che pinching leads to an unstabequilibrium that could lead to material failure.

The minimum set of parameters necessary to the formatiof second breakdown are determined to be satisfied by theBennett pinch criterion. Purthier application is tl-ien madea bipolar power transistoi- tnat has been driven into seconbreakdown to see how the derived criteria applies to asemiconductor device.

V


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LIST OF TABLES1. Type A second breakdown2. Type B second breakdown3 . Type C second breakdown

V I


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T T ^ i r1ST OF FIGURES

1. Experimental data polr-ts for the IN459 diode . . .2. A composite of data points for eight transistors .3. Failure power versus frequency of occurrence

for process 25, topology 13, configuration 1 . . .4. Failure power versus frequency of occurrencefor process 5, topology 38, configuration 2 . . .5. Failure power threshold variability versusmanufactures for SN43B9 transistors6. Example of the transitions for single carrierspace charge limited currents in an insulator. . .7. Forward current-voltage characteristics foralloyed germanium diodes8. Current density versus voltage characteristicsfor double injection i'.to lightly doped p-type silicon9. Current density versus reverse voltage characteristics for PIN diod&s at 300K10. General current densitij versus voltage forthe case of double injection in a semiconductor with recomhi''lation centers11. General current density versus voltage forthe double injection in a semiconductor withpartially filled recombination centers12. Plots of classical plasma parameters13. Relationships for the -lew plasma parameters . . .14. Plot of the concentration versus quantumdegeneracy parameter

V I 1


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15. Plot of the concentration versus theplasma parameter for classical coupling . . .16. Plot of the concentration versus the quantumdegeneracy parameter for germanium17. Plot of the concentration versus the quantumdegeneracy parameter for silicon18. Plot of the concentration versus the quantumdegeneracy parameter for gallium arsenide . . .19. Current-voltage characteristic exhibitingcurrent controlled negative resistance . . . .20. Geometrical interpretation of equation 95 ft ft21. Graph of the concentration versus the quantumdegeneracy parameter for pinch conditionsin germanium22. Graph of the concentration versus the quantumdegeneracy parameter for pinch conditionsin silicon23. Graph of the concentrar.ion versus the quantumdegeneracy parameter for pinch conditionsin gallium arsenide

ft 24. Schematic depiction of a CNPN; transistor25. Schematic depiction of 3 NPN transistorin cut-off26. Minority carrier density stored in the variousregions of a NPN transistor in various conduction states27. Doping profile for the Unitrode power transistor28. Graph of concentration versus quantumdegeneracy parameter for pinch conditionsin the Unitrode device29. Typical waveform of current and voltage fortype A second breakdown30. Typical waveform of current and voltage fortype B second breakdowf-i31. Typical waveform of current and voltage fortype C second breakdown

vi i i


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CHAPTER ITHE PROBLEM AND IiS HISTORICAL BACKGROUND

At first glance most scir?ntists would expect thatdetermining the conditions necessary to cause a transistor o;diode to fail would be related to a few simple factors.These factors would most cert;-jinly involve the current andvoltage. Failure would rher, he caused by literally meltingthe device with a large amount of current or forcing thedevice to arc over from a high voltage.

Indeed, tihere is very lirtle proof, considering thematerial these devices art- r iade of, that the above explanation of device failure is ur-true. Common experience showsthat when a diode is forced to pass several amperes ofcurrent, when it is rated for only a fraction of an ampere,something is going to melt. Similar situations can beconstructed for Just about ang semiconductor device availabletoday.

In the early 1960's t:he United States Air Force CUSAF)and several private firms attempted to determine exactly whatconditions of current and voltage caused semiconductordevices to fail. These first researchers took the obviouspath already described above --ini' fnund Bvidence that devicesfailed at levels very diffurenL tnan expected. In addition,


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the devices exhibited phenomena that no one could consistently explain.

This work will attempt to explain and determine theconditions under which a semiconductor device will exhibitsigns of impending failure due to an electrical overstress.Device failure will be linked to the onset of negativeresistance, which will be shown to force current in thedevice to collapse. This collapse will be modeled as a soliistate plasma that undergoes a self-induced magnetic pinch toan unstable equilibrium under negative resistance conditionsThe resultant instability will also be shown to be capable olinducing rapid device heating.

These results will lead to the quantitative developmentof a minimum pinch criteria that will be compared to a set ofmeasurements on a transistor that is forced into the conditions that are known to precede device failure. Theremainder of this chapter will detail the background ofdevice failure research and how experimental evidence ledthis author to conclude that device failure, and the formation of a negative resistance regime in the device, were oneand the same.

The Physics of Device Failure: The Problem

Historu of Research ResultsThe first published reports concerning device failure

were reported for diodes by Tauc and Abraham E13 in 1957.


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Failure of transistors was soon reported by Thornton andSimmons C2D in 1958. Thes-i; reports demonstrated experimentally that the failure state of a device was characterized ,a transition from a high voltage, low current state, to ahigh current, low voltage state. Soon after this transitioithe device failed as an open circuit or short. Autopsies offailed devices showed definitive signs of extensive heatingleading to melting of the semiconductor material.

For semiconductor devices made of silicon or germanium,this implied that temperatures were approaching the meltingpoint of the materials during failure. Many researchersconcluded that such temperatures were a result of ohmicheating of the device in this new regime Cof high current,low voltage) that occurred after avalanche breakdown. Thus,many researchers referred tc this regime, which occurredafter avalanche or first breakdown, as second breakdown.

Continued research soon broke into two approaches in aattempt to determine device failure levels C33. One groupapproached failure as a saltriy thermal event. A second grousought further information concerning why a device went intcthis new regime of operation called second breakdown. In t^early 1960's, a tremendous number of articles were publishedconcerning device failure, of which a partial listing is setforth in reference three.


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Two very general conclusion^3 came forth from these reports:1. Attempts to model device failure as a thermalrunaway event C4] had very limited success andsuffered from extreme problems of repeatability:5J .2. Devices that entered second breakdown suddenlyexhibited the collapse of isotropic current flow toa small current filament or column C6J. Thisfilament soon engulfed the device volume, meltingthe device as it grew C7I1 .

Such results were quite unexpected and continued to fuelfurther research along both lines of interest.

Researchers who followed conclusion one above madenumerous attempts to model device failure by utilizingextensive thermodynamic approaches coupled with devicegeometry . These initial ideas became the basis for furtherresearch and were developed into models by Uiunsch and BellCBJ in 1966 and Tasca LSI in 1970. These models generallyutilized a thermal diffusion model incorporating the increasing carrier concentration of a semiconductor with increasingtemperature driven by an external current. The feedbackmechanism provided by a semiconductor's increasing carrierconcentration, with temperature which, in turn, increases thcurrent, became the prime physical explanation of devicefailure in these models. The two models developed by UJunschand Tasca could not incorporate all aspects of device designnecessary for an accurate model, so a correction factor wasadded to each model by matching its results empirically toexperiment.


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Initially, results of î-ipplying these models to simplediodes, and transistors trented as diodes, loaked verypromising Csee figu res 1 and R). The only informationrequired to determi ne a devic e's failure level consisted ofthe material parameters and the appropriate device area.Further application of these two models to other diodes andtransisto rs showed very limited resul ts, and more importantly, the correction factor incorporated in the models wouldhave to be changed in an almnst. random fashion to bringpredicti ons closer to experimental results ClOD. Perusal ofthese studies show that some results were dissimilar due todifferences in the definiticî cf what constituted devicefai lur e. Variation of 'ohat rjonst i tuted the onset of failurealso presented problems in interpreting experimental approaches used by different n.^searchers.

The community of researchers truing to refine thermalmodels tried to clear up corlû'i]l.o'^^ by utilizing the onset ofsecorid breakdown as the critcrrirn for device faiiure.beverai r esearchers then trir?d to alter the old thermalmodels to explain second brear-.doiun as an electronic andthermal event. These electrothermal models L"il,12I] were, bythemselves, achi evem ents in modeling device dynamics whichoccurred on totally different time scales (thermal eventsoccur in microseconds, electrical events occur in


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1 0 .0 0 0 "! I I 1 I I i LX = Fl lur IO = No FIKjf

1 0 0 0

O

1 0 0

i O OT I M E I M i r r o M c o n d i )


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l o . o c x )

E

I ; C42)

and / is the total current density. Following De Groot'sprocedure. Just outlined to arrive at the entropy rateequation, we start with balance of forces

d v V - ^ - .,- V < - . C43)

The F k term contains the external non-electrical forcesThe energy equation becomes

d (I _ . .,p -V V^ d t V2 / ^ -. - V -[P v\J,

C44)

V A- y^^k '^kPk d tk

^ P 2_^ ejbc;:


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80Gibb's equation, with electrical contributions, becomes

d t d t d t

d c^ f i k - r r - ) .^kCkd tk kd( / )d t

C45)

Following De Groot's procedure we arrive at

P-d U"d t P V 77 - V X

+ ^ i ^ - j : - \ kCk d 4>d tC 4 6 )

Substituting equation 4B and the convective derivativefor the concentration into Gibb's equation results in thefollowing form

dS_ _

C47):r. - V .,/.-;- r.


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and

jlk = F k-TV^^] =Xk-ek^X^

Xk---^-Fk-ekV

This is expanded further, using equation 48 as

8 1

C 4 8 )

dSH V f'-'q ~ YlkPkJkTJq A^ 4" JK X}

7-l

C 4 S )

which is the form arrived at previously. The forces andfluxes derived for electrical conduction are invariant undera linear transformation of ^^ .^^ ^^fj, ,Analysis of Ridley's Derivation

We examine Ridley's derivation to insure that hisprocedure did not yield a result different then De Groot'sresults. Ridley's derivation starts with Gibb's equationwhere the variables have been expressed as per unit volume


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82

T S_ dUd t " d t E Pk dp kd t C50)For a solid, the volume does not change under currentconduction, which allows us to drop the work due to pressureand volume. The use of pk comes from multiplying De Groot'sprevious form of Gibb's equation by p to change the variablesto per unit volume.

Ridley introduces the additional electrical contributions of the chemical potential and internal energy as

U ^ U -T- ^ekpk(p

C51)P^k + ek4>

Which is the same as De Grnot's formulation. The energybalance equation can be written as

dU d , . .\ k

V - ( j , - ^ X ^ e f c J " ; < ^ j + T . ^ : - ^ P k - ^

C5e5


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83This form matches De Greets formulation for the change ininternal energy .

Ridley substitutes equstions 51 and 52 into Gibb'sequation to obtain a form similar to De Groot's formulation

d S_d t V . ( : ^ ^ - ^ ^ i ^ M ^^{Jq-Xq- ^flkJk-XqT T

CSS)d p)+^./^^r,./,_^,,^)

whereF =f =

Xq^Xk--

-- -V(/ ) ,Efc^fc"^^ .

Xy^ ^= --V/XA: .Ridley has broken the ohmic term F -J out explicitly in

equation 53, and has allowed for local fluctuation of themass density, an aspect De Groot did not allow. The formulation still demonstrates that the change in entropy is a sumof the negative divergence of flow and a positive productionrate .


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84Conditions of the Stationaru State

We can now apply the conditions of the stationary stateto our entropy rate equations to determine the action of thestate variables in the solid.

Prigogine C62D describes a stationary state as a condition where the state variables do not depend on time. Yet,the net mass flow is zero, and entropy production does notvanish. This is considered a non-equilibrium state of anopen system as opposed to an equilibrium state of an isolatedsystem where there is no entropy production Cbecause allforces vanish) .

In a sense, the non-equilibrium stationary state issimilar to a steady state cf the system with the conditionthat the entropy production rate is not zero.

Prigogine C63] characterizes the stationary state as asituation where we allow the transfer of matter and energybetween two phases of a syst.em at different temperatures suchthat the entropy production rate is positive, or

d S~d t = Jt^-^-^th + J-rnXm > 0. C54)

Onsager's equations allow us to view the fluxes andforces in terms of linear relations such that

Jth -" ^-LlXth + Z'12-V,. . CSS)


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85and

J - L21 Xih + L-nXra- CSS)

For the stationaru state, the mass flow is zero

Jr n L-lXth + LllXm = 0. C56)

O n s a g e r ' s r e c i p r o c a l r e l a t i o n s r e q u i r e L12 = L'li ^ t h a t

= U,Xl^ i- 2L2iX,KXm + L^2Xl > 0. C5 7)dtI f we t a k e t h e d e r i v a t i v e w i t h r e s p e c t t o A'"^ a t c o n s t a n t

Xth uie a r r i v e a t

_ A _ f ^ ) :=.2[L2lX.tk + L22Xm) = 2J^ = 0 . c 5 8 )dXm \ dt J \ ^

Therefore, the two conditions

d fdS\ .^ - ^ ' '" -dxZVTtl^^ ^53)

are equivalent as long as Onsager's equations are of linearform for the system.

The procedure followed above, where the stationarystates are characterized by the conjugate flows being zero.


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86is similar to the analogous mechanical situation where theconjugate momentums, of the forces, vanish. The stationarystate defines a minimum for the entropy production rate andis referred to as the state of minimum entropy production.If a stationary state exisLs, then the entropy is independentof time and the entropy production rate must be balanced bythe outward flow of entropy.

Application of the Stationary Stateby Ridley to the Case of CCNRIn this section we look at how Ridley applied the

concept of the stationary state to his entropy rate equation.Ridley applies the idea of the stationary state to thesituation of CCNR and VCNR by using the following argu~mentC70D .

In the steady state "j? I the total entropy change]is zero, as are the overall rates which appear inequations like

dtf _ r , v ^ d(f)k

and

"^P^ _ V f. ' ^d t d t

and in each case the production is balanced by thedivergence of the flow. At equilibrium both theproduction and flow are zero. Since it is to beexpected that the steady state will be as near the


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87equilibrium state as the various constants willallow, we expect the entropy production rate willbe as small as possible. This plausible argumentis at the basis of the principle of least entropyproduction.The equilibrium Ridley talks about is a thermostatic

equilibrium; all the driving forces vanish. This is not thecase for the situation he wishes to portray. Why he hasstated this is not clear and it may have contributed to latercriticisms of the paper.

Following Prigogine's arguments, Ridley assumes we havea stationary state; no mass flow and a balance of entropyflow. Ignoring heat flows, the remaining term for the totalchange of entropy contains an entropy production rate whichshould be at a minimum value for a stationary state.

From equation 53, the remaining term for the entropyproduction rate is

l-ikJk Xa F J2 = - y t JiiAi _|.Tk TuC 6 0 )

The Tk term allows different carrier species to havedifferent temperatures. In a stationary state no mass flowoccurs so Jk is zero. But, since the charge flow is infiiamentsCfor CCNR) and layersCfor VCNR) the mass flow is not


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88not zero for these structur>-:'S. Furthermore, Ridley assumesthat there is no temperature or chemical potential gradientsin these structures. This implies that thedriving forces X^and X. ^ are zero. Since Ridley assumes these structures arestationary states, the rate of change of thestructures attheir interfaces must be equal, but opposite.

These arguments leave the joule heating term as thesolesource of entropy production in thestructures or

TE=.F-J^F-Y,ekJ. C61)k K

Perturbation Under CCNRFor the case of CCNR Ridley explains the curve in figure 19as a single filamentary structure with cross-section, a ,which is a percentage of the total cross-section.

T' = EiJi{l - a) + E2J2d, C62)

where

/i(l -a) + J2a = Jo, C53)


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8 9


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so

>tnzLUCh-zL Ua:DU

~P '-MTM t-=. Eo

ELECTRIC FIELDEx


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and91

6Ji{l - a) + 5J2a= 0. C64)

We now apply a perturbation through the field variablesas required by Prigogine's results for equation set 59.

Fl = Fo + 3 E-,, Fo = Fo + 6E2 C65)

dJ . 1EJi = JQ -T -.--SE- J2 = JQ -\- ( F2, C66)

to arrive at

TSE 1 -^-) JJ_^^2a dE C67)

When 4i ^^ negative, as seen in experiment, theproduction rate decreases and filamentary structures arefavored. This is in accord with Prigogine's statement thatthe contribution of the rate of change of forces to the totalentropy production is negative or zero in a stationary state.But, Prigogine C64II states that this is true only under thefollowing conditions.


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921. Linear phenomenolcgical laws apply.2. Onsager's reciprocal relations are valid.3. Phenomenolcgical coefficients can be treated asconstants.Violation of these conditions are the basis for criti

cism of Ridley's paper by Volkov and Kogan C593, and Takeyamaand Kitahara C60I] . Let us examine Ridley's arguments to seeif he has violated Prigogine's conditions.

An examination of condition one, in light of thecurrent-voltage characteristic of figure 19 certainly seemsto be a flaw in Ridley's reasoning based on the fact thatCCNR and VCNR are inherently non-linear in their entirety.But, examination of Ridley's work shows that his analysisapplies to the negative difLerential region only. Experimental evidence at that time, and later do not dispute that thisregion is linear in a numberr of cases. Thus, Ridley's paperdoes not fall into dispute due to the first condition.

If the negative diffei^antial region is linear, and noextensive heat build-up is allowed, then, quantities dependent on heating are not changing. If we proceed further intothe positive resistance regjon, heat build-up is quitelikely, from Ridley's development, and linearity is lost.Thus, Ridley's derivation satisfies the second and thirdconditions required hy Prigogine's arguments.

It is interesting to observe that the two papers citedalso try to utilize Prigogirie's methods by linearizing theparticular phenomena they believe is causing negative


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93resistance in a material. Both papers show that CCNR occurs,but only if the generation and recombination mechanismspresent are linear Ca fact that is not clear from experimental evidence). Ridley's arguments are far more general.In his view, what ever the mechanism, it must be linear inthe negative resistance regime or the negative resistanceregime cannot occur Cit would violate the total entropyproduction rate for a stationary state) . Any proposedmechanism would have to be demonstrated to be linear in orderto apply it to the stationary state conditions.

As shown earlier in chapter II, and other studies C65!],negative resistance can be caused by a transition from asingle carrier current regime to a double carrier currentregime. As double carrier current begins to dominate currentflow, the negative resistance region disappears.

Therefore, as the CLjrrent transitions via CCNR, we cantreat the negative resistance regime as a stationary state.This allows us to utilize the general results of Ridley'swork as we model current flow under CCNR as a solid stateplasma.


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LhAPTER IVQUANTITATIVE ANALYSIS OF THE SOLID

STATE PLASMA PINCH

In chapter III, we determined that the conditionssurrounding CCNR place limits of action on certain macroscopic variables dealing with driving forces that influencecarrier transport.

If we model the solid state plasma under these limitations, it should be possible to determine a set of parametersthat describe the plasma's motion under CCNR. Application ofa perturbation to the plasma's motion under CCNR will allowus to determine the plasma's stability.

The motion of a plasm?., under the influence of electromagnetic fields, are best nandJed using the equations ofmotion for a fluid and Maxwell's equations. Utilizing thisinformation, attention in this chapter will first be given tothe similarity between diffusive fluid flow and diffusion ofm.agnetic fields in preparation to solve for the equilibriumcondition between the thermal and magnetic forces that occurin and around the solid state plasma under CCNR conditions.

This same problem will then be solved using a perturbation to determine if the magnetohydrodynamic equations arestable under CCNR.

34


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95The Basic iLdrodiinamic quations

The basic moment equationrj for conservation of massmomentum and energy are

Oi

^ Dt c C S S )

l^i ' ' -")=1^'-" l


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Jk = o-(F + ),

^ ^ ^dBV x F - ,c dt

^ - 4?r -r 47rcr / - U x BV X B = J = F +c c \ c

T a k i n g t h e c u r l o f t h e l a s t e q u a t i o n y i e l d s

a n d u s i n g V B = 0 we o b t a i n

c' dt c

By defining the magnetic viscosity as

c2

w can rearrange equation 71 to obtain

96

C63)

V xV xB = l-V x(U xB) . C70)c \c c dt

A - K O - dB ^-y :^ 47rO- , -; -,V"F + -T-Vx(C/ xF). C71)

/ = - , C72)A.Tca

^ =. r7V F ^ V X (C/ X F ) , C73)dt

The first term on the right hand side is a diffusion term and


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9:the sec ond term is a conv octi on term. This equati on issimilar in form to the hydrodynamic equation of flow past anobject.

If the fluid is at rest, the convec tive term v anishesand we arri ve at a solut ion simil ar to Stoke 's equation C66:

1

di C74)

Equation 74 . like Stoke's equation , relates the diffusion ofphysical qu ant iti es. From a fluid context, the diffusionequation states that the rate of fluid density changes by thetransfer of molecules from one region to another Ci.e., theprocess of diffusion). For- the hydromagnetic cas e, the rateof change of the magnetic field changes by diffusing throughthe electri cally conducting fluid.

Continuing the similarities between fluid and hydromagnetic di ff usi on, we can observe that the magnetic viscositycan be related to how fast the field diffuses in some time rCsimilar to a lifetime) into the fluid or

1,2 47ro-.L2T = C75)

w here L is a cha rac ter ist ic length Cor penetra tion depth).Conv ecti on can dominate dif fusion in the same manner asinert ial f orce s in a fluid can domi nat e viscous forces . A


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SBmeasure of the dominance ot' convection to diffusion is foundin the ratio of these values that yields the Reynclds numberC673

Prri. V C7B)

The conditions of the stationary state Cfrom chapterIII) cause diffusive forces to vanish. This implies thesolid state plasma is highly conductive compared to thesemiconductor surrounding it, or. the Reynold's number of thesolid state plasma is much smaller than unity under CCNRconditions. We can utilize this information to solve for thebalance of forces in the plrdsma .

The Static PinchThe similarities between fluid and hydromagnetic

descriptions allow us to look at the effect of a magneticfield on the motion of a highly conductive fluid. Inparticular, we can examine the effect of the pinching term,

under the conditions where the fluid is at rest withthe magnetic field lines. This does not mean the fluid isnot flowing in the material, merely that the magnetic fieldis frozen into the fluid and moves along with it Cas a consequence of the fluid's high conductivity).

The procedure will sta-t-t with the hydrodynamic equationfor conservation of energy in inviscid flow. Equations


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99relating the velocity of waves parallel and perpendicular tcthe field lines in the fluid will be used to arrive at asimple relation between the outward thermal pressure and theinward magnetic pressure. This relation will be found to bethe Bennett pinch criterioTi.

Under the assumption of a small Reynold's number wehave what is called steady, creeping flow. Under theseconditions the velocity, U , is inviscidCindependent oftime). From reference 74 the hydrodynamic equation forconservation of energy under- inviscid flow is

V{P + ~^pU ')-^p{uxU) = Q, C77)

where the vorticity is defined as CJ = V x U E x p a n d i n gU't h e g r a d i e n t o f ^ we g e t

UV ( ) =U x ( V x , [ / + ( [ / - V ) [ 7 ) , C 7 a )

S i m p l i f y i n g e q u a t i o n 77 y i e l d s

VP - p{U -VjU := 0. C7S)

Using conservation of momentum with the rate of changeof velocity being zero yields


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1 0 0

p{U \/)U = - V P +A / x B= - V P + ( '


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lOEmagnetostatic situation, where the magnetic force is balancedby the pressure gradient, .and the fluid velocity is zero,conservation of momentum, equation 68, allows us to write

p{U -VyJ --= -VP (V x F) X F47r CSS)

Where the left hand side cf equation 85 vanishes as aconsequence of the stationary state. From our Stoke's likeequation 74 we have

pd_ -qV^B = 0. C86)

We can rewrite the second term on the right hand side ofequation 85 for the static case as

47r 47r ^ ST T V 47r V-?1ST T c a 7 )And in tensor notation.

( V x F ) X B\ d /B,Bk\ d / F ^47r JL dri.\ 47r / 5rj;. \ STT ^L k C8B)

The Maxwell stress tensor is defined in Jackson C58] as

S. ^ I r? 2 TD lT,k = [E,E-,. r B,Bk - -^(E' - B')]47r C89)


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103Dropping the electric field stress term we arrive at a formequivalent to that obtained for equation 88 Cwe are notdiscounting the electric field at this point).

^ _ B,Bk B\ C90)

ord ^ d (B,Bk\ d 'B\ X

OTk UTk V 47r / OTk ^o-JT Jiy xB)xB

4^? C91)

Our static equation B'rL now reads as

d Tk d T^ k C92)

where F ^k is defined as the total stress tensor equal to

B'-ST T ./ 47r C93)

This can be reduced to diagonal form via transformationto the principal axes

I F , , - 5 , , A | = 0 . C94)


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This reduces, after some alaebra to104

\A DF -8 K

A . = P . ^S T T '

C 9 5 )

A3 = P - F^STT

o r

F, ^ + \

00

0P - 4 - ^- ST T

0

00

pC 9 6 )

A geometric interpretation of this matrix shows that A3is parallel to the field and A-, , A2 are perpendicular to thefield as shown in figure 2C^.

Under static conditions, the self magnetic force takeson the form

V P J X F CS7)


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105


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10 6

T U B E O F F O R C E

B

B


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10or

B --^P r= J. -p = 0. C98)

This demonstrates further that the magnetic field lines andthe current lie on lines cf symmetry as drawn in figure 20.For a straight column these equations give the appearance ofa shellCor shells) of current and their accompanying fieldlines nested around the cer^ter of the pinch column.

We can solve for the ^^eif magnetic force of the plasmacolumn in cylindrical coordinates to obtain

dPdr

_B d4:7rr dr[TB). CSS)

Multiplying by r^ and integrating out to the radius ofthe pinch Cinitially at r^ ) yields

dP f-^orl-d r -- y (rB)dirB),

ClOO)r'^P 2 I rPdr

Jd STT {rB'Ur.

If the thermal pressure var.ishes at the surface, and theenergy of holes and electrons are the same, then


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' KBT j n2rdr = ~(rBY'0 STT

N -2 irrndr = carriers/ unit length,%/ 0

/I2 A r A ' B r = ( i ) ( . B ) U , .

Solving for the current related to the field

or

9 r o(rF),=,, ^ - / J27rrdT^ Jo

108

ClOl)

Id 47r- [rB) = J , C102)r dr c

C103)

Solving equation 103 we arrive at the Bennett pinch criterion

/ 22 N K B T - or I' = 4:NKBTC\ C104)


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lOSThis result shows how the macroscopic variables, such

as, the current, temperatur-e and carrier density are relatedto achieve a balance between thermal forces and magneticforces under stationary stat:s conditions in the solid stateplasma .

This result is also utilized in classical gaseous plasmadischarges for determining the minimum conditions necessaryto pinch the column.

At this point, objecticns may be raised concerning theprevious derivations which lead to the Bennett pinch criterion. In chapters I and II, it was stated that semiconductorsundergoing second breakdown have very intense electric fieldsand heating present. Also, it is unclear that the situationin the device can be approximated by a magnetostatic caseduring second breakdown.

These criticisms are valid if one considers the entiresecond breakdown current chrnracteristic. Yet, Chapter IIIdemonstrated that a station-iry state is linear in characterduring the CCNR regime. Thi^s, over times that are shortcompared to the total time uihen CCNR occurs, the field doesapproximate a steady state, and under CCNR the plasma columnwill reach a static pinch.

The Hudromaanetic Stability of the PinchUnder CCNR a solid statp plasma column appears to obtain

an equilibrium condition described before as a staric pinch.


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110We now relax the conditions on the solid state plasma by

allowing all of our pertinent variables to vary slightly fromthe equilibrium situation. If the static pinch is in a stateof stable equilibrium, a perturbation of the pinch willreturn to its equilibrium condition. If the pinch is notstable upon perturbation, the pinch represents an unstableequilibrium. This can be determined by solving for thedispersion relationship for the pinch column.

To determine if an instability occurs we will resolvethe magnetohydrodynamic eqLjations Cequation set 68) under asmall perturbation. Variables with an o subscript are firstorder quantities and all perturbations are noted with primes.

P ^^ Po i- p\

u ^Uo-^ rl' .= u\

P = P, 4- p' C105)

B -=R.^B'T -^


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IllWe will also maintain our result from chapter III that

the plasma column has a conductivity much higher than thesurrounding semiconductor. Ignoring products of perturbations we find that the zeroth order quantities for equationset 68 are related as

VPo=: -{J x F ) ,c

PcPc '^ = c o n s t a n t , C 1 0 6 )

V ;-. BQ "= J o '

The first order results are

do'-^---^[Poin = ^.{jt

Po^ = -V.P'' + -(


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112

d Bdt- ='V X {U' X Bo),

C108)

c

Using equation sets 107 and 108, we want to arrive at aforce equation involving some small displacement of theplasma column in terms of zeroth order quantities aftereliminating the first order results. Eliminating p' and J'from the first order equations yields

| f + TV.(pl7') = 0, C103)

for the first order continui.ty equation. The displacement ofthe plasma can be defined as

g(7'o.."0 '"= T^-^o,

CllO)DC DrDt Dt

Integrating the continuity equation in equation set 107yields


121/183


122/183


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115where primed quantities are the results of perturbations suchthat V X V X A' vanishes .

The tangential field can be found from the image forceof the plasma at the plasma's surface as

!,_-!,nx(F; + -([/' + F,,0)

C117)nx(. FL + -(S^'xF,exO).c

where quantities with an i subscript are at or near theplasma column surface Cas a consequence of the plasmacolumn's conductivity bein.j much higher than the surroundingsemiconductor) . We can then conclude that

F; + --( C7' X F;0 = 0. CllS)

This allows us tc wrice equation 117 with equation 115as

^ d A' ^ fd( -n X ^- - n X - X noextd t \ d t CllS)


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116Integrating equation 119

n X A' = n X{C X Boext) = -[n C)Boext' C 1 2 0 )

which is arrived at by having the plasma column parallel tothe outer surface of the semiconductor. This is also aconsequence of the static pinch results. The plasma columninitially filled the semiconductor sample, and as a result ofbalance of forces, retreated a constant distance from thesample surface. No isotropies in position are allowedinitially under stationary state conditions.

Hydrodynamic considerations for conservation of energyrequire the total pressure to be continuous.

' M 2{Po + f ^Po) + p ' + v: [(ô,^ + ( e - v ) F o , + F ; ) ]STT C 1 2 1 )

1 . . ^ ; T 2STT

C 1 2 2 )

Using our equilibrium condition

Tr?2 p 2

ô,i ôextSTT STT C 1 2 3 )


125/183


126/183

lieand

Fo,x - (0,0, Fo. = constant). C126)

To further simplify the derivation, we will assume thatno pressure gradient exists radially in the plasma column, asrequired from the stationary state. This may appear to be anoversimplification, when it seems quite probable that aconcentration gradient may exist under a perturbation. Yet,there is no experimental evidence describing what type ofconcentration gradients exist, if any. Also, the stationarystate does not allow any ir^itial concentration gradientsCconcentration gradients woLjld cause diffusive forces, whichvanish under the stationary state).

Placing these conditinns into equation 124 results in

Po^H= iPoVi'v 0 + [v X V X (f X F ; , ) I X F ; , , .47 r C127)

For cylindrical perturbations we have

^(f) ~ ^[r'^,exp{im9 4- ikz)

= {Ur)Ae'r).C.{r))exp{zme ^ ikz).C12B)


127/183

119and since we are using very low orders of approximation forour static pinch, we will only utilize m equal zero. Resultsfor the classical plasma pinch indicate m equal zero corresponds to inducing a sausage instability and m equal one akink instability CBSJ. Research documented by Snyder C70]suggests that semiconductor devices could exhibit plasmainstabilities resembling a sausage or kink instabilitiesdepending on the actual geometry of current flow. Expanding123 with m = 0 in equation 128 yields

Or

Bo.47r I _%|(.,,)-.^.., C12S)

and

-pLo Co = k'Bo,


128/183

1 ,1 =2 \ l / 2 120

47rpy for the Alfv6n velocity, we can simplify theseequations to

( fc^^-c^) = ( .?+ .^) . i d , , + ike d i," - S . ^dr

(k'vl-co')(e = 0, C131)

ik^ci-. ) = c^ ik d. r dr ir(r)

These equations relate the displacement of the pinch to theperturbation initially started with equation set 105.Eliminating Cr from the above equations to arrive at thedisplacement in terms of the z component yields, after somealgebra,

(j.2 J, dj. ^,2^2^,2 _ ^2 ^^2 _|_ .y2^ C132)

This equation is in the form of a Bessel equation, andthe solutions are modified Bessel functions, ^^ = I^[kr) . TheI^ are not solutions at r equals zero. Our argument for theBessel function is rather complicated and takes the extendedform


129/183

^ 2 ^ 0:'cl-u')ik'vl-u;')hJclvl -a;2(c2 - vl)

121

= k' 1 + (frC^-^'a -(f) W+-a)

C133)

and solving for the z component of the displacement

ir = -TFikclik'vl 'j-)-uj vl

: K P ^ , ^ - ^ O' I' C 1 3 4 )

I f we use F' =VX ( X F o) for F-. the z c o m p o n e n t i s

r ar C 1 3 5 )

Placing the z components into our perturbation result,equation 124, weget

- B, J-7Pexp->-.*'V. -- v:('- .) =Boe47rr dr 47r( i F o e l C 1 3 5 )


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From nx A'=nx{Cx B'oe) =-{n-OBoe we s e e t h a t122

A' -^ Cr Bez-\- af, C137)

where a is some arbitrary function so that

V X v4' = da d- dz dr (rBe)

C138)

and

V X V X A' = f ^2 O ad: + zS&'a d'

.drdz dr"^ UrBe) = 0. C139)

These last two equations imply that

d ad: fir) a n d dc:t ddz dr {irBe) = g{z). C 1 4 0 )

B u t , s i n c e CrBe i s a f u n c t i o n o f r o n l y g{z) m u s t b e ac o n s t a n t a n d

V X .4' - g{z)e = 0, C 1 4 1 )

since F' is zero initially. We can now rewrite equation 136to get closer to the dispersion relation


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S u b s t i t u t i n g i n for 6 and (r

{cW-K\k'-c]-u^))Io[Hr)

S i m p l i f y i n g f u r t h e r we a r r i v e a t

1 2 3

W pi. . Bj{ec^-u)')(, BU,ikc]. Ai-Kikc'l 47rr C 1 4 2 )

Bl . FBl ' vl-I'^{H r)oz

clik^vl - a.') - u^'vl^ 2 ^ 2 _ ^ 2

C 1 4 3 )

, ,2 F, 2 /:(^-)F 2 , ^ r / ^ ( F r ) ' C 1 4 4 )

2 ^ i 'A i ?1 . [^


132/183

124 This situation is stable if an axial magnetic field is

present such that

F oz > B,'.r\H r r ^[ H r )

Jo{Hr) C147)

We can expand the dispersion relation further byapproximating the Bessel functions. Standard expansions ofthese functions are

x^ x^W - l ^ ^ + + ---,

x^ x'^ X'C148)

Io'{x) ^ I^{x) --= -{+- +2 16 384

If we expand them to first order only, our dis persi onrela tio n Cequ ation 144) simn lifi es to

L O = k'v. . 2 B'^e \{k'cl-u,')ik'vl-u;')4 7 r p . k"^clvl - Lv^cl ^ vl) C143)

Utilizing the definition for the speed of sound in a materialas

^p \fc, C150)


133/183

125where

k ^ - Debye wavenumber,^P = plasma frequency,

and eliminating the axial magnetic field leaves our firstorder dispersion equation as

or

Bl ik'^dLO 4-KpCO '

B^4:7rp k'~kH^ LO p

C151)

LO-f'-k'4 7 r p

1 - - ^ ^1 T i T r p c ; 14 7 r p

B^k^4 7rpu;2 C152)


134/183

126For this solution to be stable would require

p2---^- > 1/ 2 '4/Tpcj

C153)

which is not a physically admissable solution in the material .

These results show a static plasma pinch is unstableunder the conditions of CCNR. The growing oscillations ofthe pinch would certainly result in anistropies in carrierconcentration in the column, resulting in a movement awayfrom the stationary state.

This action allows thermal and electrical driving forcesto reappear and the rate of entropy production would rapidlyincrease, and possibly surpass the flow of entropy from thesample. This suggests rapid ohmic heating from an decreasein the conductivity of the plasma column. Growing oscillations at the column surface would certainly be limited byphonon dispersion which would transfer the kinetic energy ofplasma oscillations to the lattice, aiding heating of thedevice. An exact determination of this type of activity isbeyond the scope of this wcrk, hut, is consistent withresults that show devices exhibit signs of thermal shockafter the initiation of second breakdown.


135/183

127Solid State Plasma ParametersNecessaru to Initiate Pinching

The results of the last section indicate attempts todetermine when a device will fail after second breakdownoccurs is an attempt to model an inherently non-linear event.

Therefore, it seems more amenable to determine when aCCNR regime occurs under device operation. The Bennett pinchcriterion could be utilized to determine a minimum condition,beyond which, self-induced magnetic forces overcome thermalforces and produce unstable pinching. This can be related tothe quantum degeneracy parameter to yield a set of conditionsthat are inherently unstable under CCNR conditions.

Our starting point is the Bennett pinch criterion, asapplied to a situation where equilibrium is initiated. Inthis case, the thermal forces Just balance the magneticforces. Further increases in the magnetic force initiatesunstable pinching. This can be quantified as

I''A ^ ^ B F < - - - ^ , C154).o -KC-r'

w ere the current is in statamps and r is the column radius.

i"2-rrc2r"AB


136/183

128This value can be substituted into our relations for thequantum degeneracy parameter, or the plasma coupling parameter, to determine a set of maximum current curves for amaterial.

Results are shown for the materials of interest onfigures 21, 22 and 23 using material parameters from 1164].Values of N are graphed as concentrations per cubic centimeter and all results are for a plasma column of radius 10"^centimeters.

For the lowest current shown on the figures, theequivalent current density is greater than 3 A/cm?

To determine the current required to support an unstableplasma pinch, the concentration of carriers in the pinch isfollowed horizontally across the graph until it intersectsthe appropriate isotherm. These values are placed intoequation 155 to determine the maximum current that can flowbefore magnetic forces destabalize the plasma. For example,on figure 2 1, a concentration of around 6.3 x lO^^cm"^ at BOOK,requires about 10~^ amperes to became unstable with a columnradius of 10~ ^cm Additionally, the quantum degeneracyparameter at this point indicates the plasma is classical innature, which reaffirms our results which indicated pinchresults similar to those foi.jnd in classical gaseous plasmas.

These results also indicate that for a given concentration, temperature and current, more than one filament couldform. The smaller filamenti- would each have to posses va.lues


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129


138/183

iEOC

uCDa

20

IS

,

16

'

.~

130

14

121-^

- 3 .0 1.0LOGxo A


139/183

131


140/183

132

- 3 .0 - 1 . 0 0 .0 1 .0LOGxo A


141/183

133


142/183

2 0

1 8 -

i^ 16 -

C Da J

14

12

/ .y

- 3 . 0 - 1 . 0 0 . 0 _J1 .0LOGxo A


143/183

135below the maximum allowable values for the Bennett pinchcriterion, and could be treated as a set of current carryingwires with appropriate repulsive forces. As the current inthe material increases, filaments would coalesce back into asingle filament which would become unstable once it achievesthe maximum allowable current from the Bennett pinch criterion

Further results are indicated on figures 22 and 23 forsilicon and gallium arsenide respectively.


144/183

CHAPTER VAPPLICATION OF THE PLASMA PARAMETERS TO

SEMICONDUCTOR JUNCTIONS

In this chapter, the Bennett pinch criterion, derivedfor a solid state plasma in a CCNR regime, will be applied toa transistor that is placed into second breakdown andrecovered safely before damage can occur. These results forthe transistor are the only known experimentally repeatableresults of second breakdown and provide a means to determineif second breakdown in this device coincides with theconditions for unstable solid state plasma pinch.

As an introduction to these results, a short discussionon the current transport mechanisms in solid state devicesCe.g., diodes and bipolar transistors) will be presented.The results of this discussion will be used to determinewhether or not the transistor to be examined in a CCNRregime. Measurements of the collector current at secondbreakdown will be used to determine the carrier concentrationavailable for a plasma whose initial radius is equal to thedevice's radius.

136


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137Description of Semiconductor Junctions

When two dissimilar semiconductors are brought intocontact, a built-in potential is set up due to the differencein Fermi energies. This built-in potential sweeps freecarriers from the general area of contact, in both materials,until enough charge has been displaced to counteract thebuilt-in potential. The remaining charge free zone is calleda depletion region.

Application of an external potential changes theelectrical character of the depletion region depending onwhether the potential aids Ccalled reverse-bias) or counteracts Ccalled forward-bias) the depletion region's built-inpotential .

A single junction between two semiconductors is usuallycalled a diode. In reverse-bias a diode passes very littlecurrent until the external potential is large enough toinitiate impact ionization in the depletion region. Asreverse-bias increases, the electrical character of thedepletion region changes, from an insulator, to a semiconductor, when impact ionization occurs Csee chapter I I). Inforward-bias the built-in potential is decreased by theexternal potential allowing a large number of carriers to beinjected and a large current to flow. Under forward-biasconditions the depletion region is small enough that currentflow resembles semiconductor injection Csee chapter II ).


146/183

138If three dissimilar semiconductors are brought into

contact, with the middle semiconductor relatively thin, twodepletion regions occur separated by a very small distance.Injection activity at either junction can influence injectionconditions at the other Junction, creating what is called,the transistor effect.

Under different polarities and degrees of bias, theseJunctions can be used as an electronic switch, also known asa transistor Csee figure 24). When the transistor is on,both Junctions are forward-biased Cthe transistor is inquasi-saturation) or the base-emitter is forward-biased andthe collector-base is slightly reverse-biased Cthe transistoris active). By reverse-biasing both junctions, neitherJunction will pass current and the transistor is off Cin thecut-off state). Both situations are portrayed in figures 24and 25.

These effects are attributed to the control of charge inthe base region by the emitter's bias, and the degree of basedrive used to drain or replenish stored charge in the base.

Under active, quasi-saturated or cut-off conditions, thequantity of stored charge varies. As shown in figure 26, theability of a transistor to switch Cfrom an on to an offstate) is controlled by how quickly charge is drained out ofthe base Cor the collector) to achieve cut-off conditions.This is complicated if the external bias on the collector oremitter continues to pour charge into the base region.


147/183


148/183

EMITTER BASE

140

COLLECTOR

DEPLETION REGTONR (un biased state)

EMITTER BASE COLLECTOR

N-*-

%

;

I A

y

{AbI

D E PL ET IO N R EG IO N S ( q u a s i s a t u r a t e d s t a t e )


149/183

141


150/183

1 42

E M I T T E R

.^''N -"

B A S E^

,/i .


151/183

143


152/183

1 4 4

EMITTERCutoff \ i '"'^rj=rrrrr=.-r.rr

BASE COLLECTOR


153/183

145For example, from the results of chapter II, a transis

tor could be modeled as a diode and PIN diode connected inseries. Special consideration must be taken into account inmodeling the intervening base region, but current flowsoutside the base will stili meet the same primary obstacle ofspace charge limited current flow.

Conditions can be created where the collector andemitter potentials force their respective depletion regionsto extend deep into the base until they coalesce Cknown as apunch-through condition). Once this occurs a transition fromspace charge limited, single carrier to double carriercurrent flow occurs and continued current flow cannot berestricted.

If conditions are created during transistor turn offwhere carriers are depleted from the base at the expense ofcarriers in the collector, space charge limited current flowin the collector can eventually increase the electric fieldat the I-N+ interface Csee figure 25) to avalanche conditions. Avalanche creates a large number of electron-holepairs which neutralize the space charge and leads to doublecarrier current flow and CCNR 11713 .

These situations provide a useful test of the conclusions reached in chapter IV. Knowing the current levelsavailable at the collector prior to second breakdown, thecollector area, the charge carrier concentration, and themaximum thermal velocity of carriers in the collector region


154/183

146we can see if the minimum predicted conditions for solidstate plasma pinch coincide with experimental evidence ofsecond breakdown initiation.

The only repeatable experiments driving transistors into second breakdown, without damage, have been performed byPortnoy C72,73J. In the next section we will discuss theseresults and apply our pinch criterion to the tested devices.

Application of Plasma Pinch Criterionto TransistorsPortnoy's experimental set-up examines transistors in

the common emitter configuration as they are switching undera large inductive load C72I1 . When second breakdown occurs,the inductive load is quickly switched, from the device, toan energy dump, preventing device failure. We will use theresults of testing, listed in reference 73, for a transistormanufactured by Unitrode, with general characteristicsdescribed in figure 27.

Results of this study demonstrate that three types ofsecond breakdown occurred dependent on the device's initialcondition and the degree of reverse base drive. These typesof second breakdown have been called types A, B, and C C723by Portnoy .

Utilizing information derived in chapter IV, a graph ofthe necessary pinch conditions for the device were calculatedfor a column radius of 0.28 centimeters Csee figure 2 8).


155/183

1 4 :


156/183

148 148

DopingDensity

Atoms/CM"

EMITTER

COLLECTORBASE

-v 1 X 10-

v 1 . 5 X 10 1 4

10 ' 7 45 2 0X


157/183

149


158/183

150

1 5 . 0 0

1 4 . 5 0 -fOIEOcH

c07-i

CDO. J

1 3 . s o

l s . 0 0 -

- 1 .50 - 1.00 - 0 . 5 0LOGxo A


159/183


160/183


161/183

153


162/183

154slightly in reverse-biased. In either case, the depletionregions are not very large and do not extend across theentire base region initially. The bias of the Junctionscreates an excess of carriers in the base from the forward-biased emitter Cand collector). If the collector is slightlyreverse-biased, a small number of these excess carriers willbe drained off by the collector.

As the base drive is reversed to turn the transistoroff, the inductor senses the bias change on the transistorand begins to de-energizes across the transistor forcing thecollector into reverse-bias and trying to keep the emitterforward-biased .

General trends indicate that for type A breakdown thecollector-emitter voltage remains constant and rises Justbefore second breakdown. The collector current decreaseslinearly to zero and presumably becomes negative, and thebase-emitter voltage decreases more or less linearly, thendrops rapidly near second breakdown Csee figure 23) .

In terms of the device's switching dynamics, the emittercannot become reverse-biased until all the excess charge inthe base region is removed by the base drive and the collector. If this is correct, then second breakdown would notoccur until all the base charge is removed. Once emptied,punch-through or avalanche at the I-N+ interface can occur.


163/183

155Calculations of the minimum carrier concentration in the

collector are based on a maximum thermal velocity ofResults are listed in table 1.

The table lists the initial device conditions, in termsof the forward base drive, and the conditions as the transistor is being turned off by the reverse base drive. The timeto second breakdown is the time after initiation of thereverse base drive until a rapidly increasing current flow isdetected, and current is shunted away from the transistor.This is the transition from a high voltage, low currentstate, to a low voltage, high current state, or secondbreakdown.

The tabulated results show the collector current exceedsthe maximum pinch criterion for the minimum concentration ofcarriers in the collector.

Type B Second BreakdownThe state of the transistor for this type of second

breakdown is quasi-saturation. Typical current and voltagecurves with time are shown in figure 30.

Figure 30 shows collector current is constant afterapplying reverse base drive with an initially positiveemitter-base voltage and a small collector-emitter voltage.Near second breakdown the emitter base voltage tends to zeroand the collector-emitter voltage rises rapidly to a sustaining voltage.


164/183

156a>-Bi c; O)E =>=3 O^ ^

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165/183

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166/183

158 58 58 58


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168/183

160CU-S3:o = cz< C CL)i1 "5 ; - ^^ ^^ S.ECO (n C O C O C O C O

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169/183

161


170/183

162


171/183

163If the emitter-base becomes reverse-biased due to a high

reverse base drive then the collector's stored charge Cwhichis much smaller than the base's stored charge) will quicklybe dissipated allowing the collector to move into reverse-bias and punch-through cr avalanche conditions more quicklythan in type B breakdown.

Table 3 shows once again, the collector current farexceeds the maximum allowable pinching current.

From the quantitative point of view taken for theseresults, the criterion for solid state plasma pinch appear tobe satisfied in all cases, and the devices present characteristics that indicate the presence of a CCNR regime.


172/183

154 "

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173/183

165 65a>-B5O -;=-=- c:


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CHAPTER VIGENERAL CONCLUSIONS AND FUTURE UORK

This work has established the phenomena seen duringsecond breakdown Cin particular, current filamentation) in asemiconductor device is a consequence of the solid stateplasma pinching under CCNR conditions.

To accomplish pinch, certain requirements in thesemiconductor must be met. These requirements can begenerally determined through the use of the carrier concentration, quantum degeneracy parameter, current, and temperature as dictated by the Bermett pinch criterion.

The solid state plasma pinch was calculated to be anunstable event, under first order approximation, which leadsto growing oscillations ana non-linear behavior.

Future UlorkThe calculations for stability of the static pinch could

be solved to higher order modes with different distributionsof carrier temperatures. This is not a simple task, and atsome point, approximation, 'jr advanced numerical techniques,will have to be utilized.

lo7


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168Yet, it seems inevitable that this task could be greatly

simplified if repeatable experimental information on thesolid state plasma pinch were available.

This may be accomplished by utilizing microwave inter-ferometry on a semiconductor sample undergoing plasma pinch.

This technique would utilize comparison of the phaseshifted microwave signal passing through the solid stateplasma, to an unchanged reference signal, to find theplasma's index of refraction. This informatinn will detailthe concentration of the plasma in real time as the plasmagrows.

Applications to Device FailureThe information developed in this effort was initially

prompted by the desire to predict or explain device failure.Certainly, the information detailed here-in has shown

testing devices to thermal failure cannot accurately predictindividual device failure. A semiconductor device iselectrically unstable before thermal failure and going beyondthis initial instability is extremely difficult to model.

Application of the plasma parameters developed in thiseffort could possibly determine the minimum requirements forplasma pinch in some device'3, but, some device designs wouldinitiate a kink instability before a sausage instability.Thus further work is needed to determine a Bennett-likecriterion for the kink instability in a semiconductor.


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