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Interaction of streamers and stationary corrugated ionization waves in semiconductors

A. S. Kyuregyan

All-Russia Electrical Engineering Institute, 12 Krasnokazarmennaya St., 111250 Moscow, Russian Federation

A numerical simulation of evolution of an identical interacting streamers array in semiconductorshas been performed using the diffusion-drift approximation and taking into account the impact andtunnel ionization. It has been assumed that the external electric field E0 is static and uniform, thebackground electrons and holes are absent, the initial avalanches start simultaneously from the nodes

of the plane hexagonal lattice, which is perpendicular to the external field, however the avalanchesand streamers are axially symmetric within a cylinder of radius R. It has been shown that undercertain conditions, the interaction between the streamers leads finally either to the formation of twotypes of stationary ionization waves with corrugated front or to a stationary plane ionization wave.A diagram of different steady states of this type waves in the plane of parameter E0, R has beenpresented and a qualitative explanation of the plane partition into four different regions has beengiven. Characteristics of corrugated waves have been studied in detail and discussed in the regionofR andE0 large values, in which the maximum field strength at the front is large enough for thetunnel ionization implementation. It has been shown that corrugated waves ionize semiconductormore efficiently than flat ones, especially in relatively weak external fields.

PACS numbers: 51.50.+v, 52.35.-g, 52.80.-s, 72.20.Ht

I. INTRODUCTION

The streamer mechanism of electric discharge is usedfor a long time for the description of pulse breakdownof various matters. A set of works is devoted to theo-retical and experimental studying of streamers, howeverbetween these two ways of research there is an essentialdiscrepancy. In vast majority of the theoretical worksperformed both by analytical and numerical methods,single streamers were studied. Meanwhile in practice thedischarge is usually carried out by a large number of in-teracting streamers. Such electrostatic interaction hasto be essential, in particular, in a pulse crown and in

a streamer zone of a long spark where the characteristicdistance between streamers is less than their lengths, butthere is more than their diameters [1, 2]. The excellentphotos especially visually illustrating this circumstance,were received for the last years (see, for example, [3, 4]).Modeling of such multistreamer discharges is a very com-plex three-dimensional problem, so as a first step towardsits solution a very simplified situation should be studied- the evolution of an identical streamers, which simul-taneously starts from nodes of one-dimensional (in thiscase, the model corresponds to experiments [5, 6] ) ortwo-dimensional periodic lattice.

As far as we know, the first attempt of this kind was

made in the work [7] whose author studied evolutionof the one-dimensional periodic array of the cylindricalstreamers propagating in air from a thin wire to the planeparallel to it. Interaction between the charged stream-ers was taken into account by the approximate analyt-ical solution of the corresponding electrostatic problemand introduction of so obtained amendments to the field

E-mail: ask@vei.ru

strength in a numerical model of a single streamer. Itwas shown that field strength before the front of eachstreamer in the array and the speed of their propaga-tion considerably decrease in comparison with a singlestreamer. This result is quite expected. However, theauthor [7] didnt manage to receive any additional in-formation, because in the framework of so-called 1.5D-dimensional numerical model which he used, the crosssizes and a form of each streamer are a priori set andtherefore arent subject to interstreamer interaction.

This shortage is devoided in the work [8], whose au-thors simulated the evolution of two-dimensional peri-odic array of negative streamers in gases in a uniformexternal field E0, using minimal model (taking into

account the drift, diffusion and impact ionization of elec-trons, background electrons are absent, additional mech-anisms of ionization are not taken into account [9]). Atthe initial stage, streamers develop independently fromeach other so that their length and front curvature ra-dius increase with constant velocities [10, 11]. But thenature of further evolution depends strongly on the dis-tance 2L between streamers. IfL is greater than somecritical value Lc(E0), the front of each streamer startsquickly becoming distorted as a result of transverse in-stability, described in [1215] for gases and in [11, 16] forsemiconductors. However, at L < Lc(E0) this instabil-ity is suppressed and eventually propagation of streamers

array becomes stable and self-similar: all of them travelwith constant velocityufand unaltered form of the front,which does not depend on the initial conditions and canbe described by a multi-valued function

yf(x) = 2

bL arccos

exp

2

x xfaL

, (1)

where xf = uft is a front position on the axis x paral-lel to the external field. This formula was obtained in[17] to describe the shape of the interface between two

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incompressible fluids with very different viscosities (forexample, water, which forces the glycerin) moving in thenarrow gap between two plates with width 2L (Hele -Shaw cell) [18, 19]. In this case the boundary velocityufin (1+ a/b) times more than velocityu0 of a viscous fluidfar ahead the border and therefore the matter conserva-tion law provides the relation

a+b= 1. (2)The formula (1) is also applicable to describe a number

of other physically different but mathematically equiva-lent processes, provided that the normal velocity of theinterface un at each point is proportional to the gradi-ent of some potential function (x, y), which satisfies theLaplace equation, and the interface itself is equipotential:

(x, yf) = const, un En |n(x, yf)|. (3)In experiments with Hele-Shaw cells (in this case, ispressure) it is always getb = 1/2, that is, after the estab-lishment of the steady state motion of the interface lessviscous liquid supplants exactly half of the cell width.This nontrivial selection phenomena was explained bythe influence of small, but finite deviation from the firstcondition in (3) because of the surface tension [18, 19] orkinetic undercooling [20].

Model of streamers satisfying to conditions (3) (in thiscase, is electric potential), was first used in the work[21] (see also the book [22]), but in fact they can be per-formed only very approximately. Violation of the formeris due to the finite conductivity of the plasma behind thefront and nonzero thickness of the front, and the latterdue to a more complicated dependence ofun on the fieldstrength En normal to front. Within the framework ofthe minimal model of streamers in gases, with some

reservations, it is possible to use the formula [9, 23]

un= u ve+ 2De, (4)

where =

vee/De is a parameter of exponential de-cay of the electron concentration n ahead of the front,ve, De and e are a drift velocity, a diffusion coefficientand an impact ionization coefficient, which locally de-pend on En. Usually in gases second term in the rightside of (4) is relatively small [9, 23], so at a constantelectron mobility e front velocity un eEn. This isa fact the authors of [8] deemed sufficient basis for ap-plicability of the formula (1) to interpret their resultsof numerical simulations, in particular, the approximateratio [41] EM En(xf, 0) = 2E0; if this is true, it wasignored that in the case of streamers there was no reasonfor equality (2).

The study of a similar problem with regard to semicon-ductors, is also very important. In practical terms, this isdue to the fact that multistreamer breakdown mechanismdetermines (at least in some regimes) [24, 25] the opera-tion of avalanche voltage sharpeners, which are commu-tators with unique characteristics [26, 27], but in the sci-entific terms this is due to features microscopic processes

in semiconductors. Among them there are particularlyimportant ones

- saturation of dependencies ve,h(E) in relatively verylow fields E Es 10 kV/cm, which, in particular,leads to a significant (relative to gas), increasing the ra-tio un/ve,h even within the framework of the minimalmodel of streamers [28, 29] and

- existence of tunneling ionization, which can also in-

crease the ratio ofun/ve,hby orders of magnitude [11, 3035].These features lead to the fact that un En only

at En Es, but in the usual range ofEn 10 103kV/cm functionun(En) varies from constant to exponen-tially strong. Therefore, the second of the conditions (3)is never executed, and the equation (2) can be satisfiedonly by chance at some relations between the parametersof the problem. Such a radical change in the classical for-mulation of the problem [18, 19] can lead to substantiallydifferent scenarios of evolution of the interface betweenthe phases in a highly non-equilibrium conditions.

In this paper, we tried to study this interesting problemby numerical simulation of the interaction between thestreamers in semiconductors. The main aim of this workis to obtain maximally detailed complete results for thesimplest case of a static uniform external field E0, whichcan provide the basis for further research.

II. MATHEMATICAL MODEL OF AN ARRAY

OF STREAMERS

Similar to the most works on the numerical simulationof streamers, the diffusion-drift approximation is used. Inthis approximation, the distributions of electrons, n(t, r),and holes, p (t, r), are described by the continuity equa-

tions, which we conveniently wrote in the form

(p+n)

t + (jh+ je) = 2(sg+ sr), (5)

(p n)t

+ (jh je) = 0, (6)

where the termssg,r describe all possible mechanisms ofgeneration and recombination, and the free carrier fluxdensities are given by the expression

je = ven (Den) , jh= vhp (Dhp) ,where the subscripts e and h correspond to electronsand holes, respectively. In this paper, we considered only

indirect-gap semiconductors such as Si, Ge or SiC, inwhich the rate of radiative recombination (and, hence,photoionization rate) is very small. Therefore the gen-eration of pairs occurs primarily due to the impact andtunnel ionization; consequently, the generation rate hasthe form

sg = (even+hvhp)h(n+p nth) +gt,where h(x) is the Heaviside unit step function, nth is acertain threshold density. It is introduced in order to

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exclude the appearance of nonphysical solutions due tothe impact ionization far ahead of the streamer front,where the densities of the electrons and holes generatedby the tunnel ionization in the external field are verylow (n+ p < nth) and the continual approximation iscertainly inapplicable [30, 36]. Under the assumptionthat the lifetime of charge carriers is much larger thanthe propagation time of the streamers, recombination is

disregarded; i.e.,sr = 0 is set. Moreover, the impact ion-ization coefficient e,h, drift velocities ve,h, and tunnelionization rategt are assumed to be specified instanta-neous and local functions of the field strength E(t, r),satisfying the Poisson equation

E= q

(p n) = , (7)

where, is electric potential, qis the elementary charge , is the permittivity of the semiconductor. The usualapproximations,

vh= ve= E, = vs/(E+Es),

e = h= exp(E/E),gt= gt(E/Et)

2 exp(Et/E),are used, and the dependence of the diffusion coeffi-cients De,h on E is disregarded, i.e., De = Dh = D =

const. The parameters vs, Es,, E, g and Et are de-termined by the band structure of semiconductors andelectron and hole scattering mechanisms. We used thesame typical values vs = 10

7cm/c, Es = 15 kV/cm,= 106cm1, E= 1, 5 MV/cm, g = 6, 7 1035cm3s1,Et = 22, 5 MV/cm, D = 20 cm

2c1 and = 11, 80, asin the [11].

The initial conditions for the system of equations (5) -

(7) have the form

(0, r) = E0x, (8)(0, r) = 0(r ri) , (0, r) = 0, (9)

whereE0 is the strength of the external field, which is di-rected alongx-axis, = q(p + n)/E, = q(pn)/Eare dimensionless concentration and the space chargedensity of electrons and holes, 0(r) is any quite stronglylocalized function satisfying the normalization condition

0(r) dr= 2q/E. The Gaussian distribution is used,

0(r) = 00exp

r2/r2, (10)

where 00 = 2q/3/2Er3. These initial conditions cor-respond to the appearance of one electronhole pair ateach point r = riat timet = 0. In accordance with whatwas said in the introduction, these points coincide withthe nodes of a planar, for example, hexagonal, lattice lo-cated in the plane x = 0. In this case, avalanches andgenerated by them streamers have the symmetry of regu-lar hexagonal prism. Therefore, in the cylindrical coordi-nate system r= {x,y,} (hereinafter y is distance fromthe point r to axis x, and is azimuthal angle), strictly

speaking, one should take into account the dependenceofn, p and on , that is necessary to solve the three-dimensional Cauchy problem with natural boundary con-ditions on the lateral faces of the prism, which requiresvery large computing resources. Meanwhile, the problemcan be considerably simplified if a prism with the widthof the lateral faces His approximate by a cylinder with

a radius R = H33/2 0.91H. In this case areaof the base of prism and cylinder are the same, and thedistance between their lateral surfaces does not exceed0.1H. Such a small difference between forms of the sidesurfaces should not have a significant impact on the pro-cesses of ionization and transport near the axis of sym-metry, which mainly determine the evolution of an arrayof streamers. To confirm the validity of this statementthe main parameters (the maximum field strength on thefrontEM, the concentration of electrons and holes in thexaxis behind the front , and the front velocityuf) ofplane, axially and hexagonal symmetrical streamers ob-tained by modeling with the same finite element meshesare shown in the Table. As can be seen, the parame-

ters of the plane streamer differ significantly from almostmatching parameters of axially and hexagonal symmetri-cal streamers. At the same time, the dimensionless com-puting speed xf/tvs (here t is the time spent onmodeling the process of promoting the front on the dis-tance xf) for a hexagonal streamers almost 100 timesless than for an axially symmetric one. Therefore, inthis paper we neglect the dependence ofn,p, (), weakat actual values of y

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The Cauchy problem (5) - (14) was solved by the finiteelement method with adaptive non-uniform mesh by theway described in [11].

III. RESULTS AND DISCUSSION

The calculations were performed for values E0 = (0.2

0.7)E and R = (20 8000)1. The simulation resultsare presented in Figures 1-13. They are very weaklydependent on the choice of the quantities r [34] andth = qnth/E [11]; in the present study, we used thevalues r = 2/and th= 10

8.As might be expected, avalanches and streamers de-

velop independently from each other as described in [11],while their length 2xf 2R. At xf = (2.5 3)R anelectrostatic interaction between them becomes signifi-cant (see Appendix). At first, it slows down the expan-sion of streamers (Fig. 1) and reduces the maximum fieldstrengthEMat the front (Fig. 2). Further results of theinteraction depend strongly on the values of the control

parameters E0 and R. It turned out that the plane of[1/E0, R] splits into four areas shown in Fig. 3, in whichthe character of streamers array evolution is qualitativelydifferent.

In regions 1 and 2 (i.e., at R > Rt(E0) and R0(E0) Rt, solid line: calculation for an isolatedstreamer, dashed line: the external fieldE0.

FIG. 3: Diagram of steady states of the hexagonal latticestreamers. Lines separate the plane (1/E0, R) into four quali-tatively different regions (see text). Solid squares: calculation

without tunneling ionization (with E0 >0.4E), open circles:calculation for the plane streamers array with period L, opensquares: the calculation for gases according to [8]), dashedline: calculation formulas (28),(29).

this approach lies in the fact that for largeR front veloc-ityuf is a lot more than the average directed velocity ofthe charge carriers. In this limit, their transport is notdirectly involved in the variation of , which is causedexclusively by the ionization as described by (16). Therole of the drift is reduced to the formation of the spacecharge ( see equation (17)), that suppresses the field be-hind the front according to Poisson equation (18), anddetermines the structure of a wave as a whole.

Dependencies yf(x) for such distant streamers shownin Fig. 5, are well described by a function (1) after re-

FIG. 4: Maximum field strength EM, front velocity uf,maximum and the average over the area av concentra-tions of charge carriers b ehind the front vs. parameter R atE0= 0.36E.

FIG. 5: Steady-front shapes of a hexagonal lattice of stream-ers with R= 8000 and various external field strengths E0.

The symbols indicate the positions of the points behind thefront, where the field strength on the results of the numericalsimulation is equal to 0.001E. Lines: approximation by theformula (1).

placement ofLtoR. However, for the reasons mentionedin the introduction, the function (1) should be consid-ered only as one of the suitable approximations of thesimulation results. Fitting parameters a, bincluded in it,depend onE0 essentially (see Fig. 6), at that a =b and(a+b)= 1. Dependencies of other front parameters onE0 are shown for this case in Fig. 7-9.

As expected, the maximum field strength at the frontEM increases with E0 (Figure 7); along withEMdimen-sionless timeuf/Rvs for which the front moves aheadon distance R (Fig. 7) and the concentration of, av(Fig. 8), all of which are independent on R, increasetoo. However, the ratioEM/E0 decreases (see Fig. 9).To explain this effects, it should be noted that with in-creasing (xf x) fieldEn and, hence ionization rate de-creases rapidly and becomes negligible at x < xi, whereEn < Ei En(xi) 0.1E. It is clear that the length(xf xi) of ionization region increases with EM andE0,

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FIG. 6: The parametersa (filled symbols) and b (open sym-bols), which determine the steady-front shapes of stream-ers array (see (1), vs. the external field strength E0 withR= 8000 (squares) and R= 4000 (circles).

FIG. 7: Maximum field strength EMand dimensionless timeof flightuf/ Rvs of the front distanceRvs. the external fieldstrengthE0 with R= 8000 (squares) R= 4000 (circles).

so radius bR of each streamer in the array must also in-crease. The parameter a, characterizing the degree ofsharpness of a streamer front [42], also increases withEM, but more slowly than b. In other words, this meansthat streamers have to become sharper, but thicker whenE0 rises. The results of numerical calculations ofEM forthe simplest model of the streamers (see Appendix) show

that the combined effect of these two factors should leadto falling dependence EM/E0 on E0, which is consistentwith the simulation results (see Fig. 9) in qualitativeterms. Some excess of the expected relations over ob-servable EM/E0 is due to the fact that the front of thestreamer has a finite thickness , and the ratio of /Rincreases with E0 [35].

Here we must mention an important result for practicalapplications: the average concentrationav of electronsand holes behind a corrugated ionization wave is much

FIG. 8: Maximum and the average over the areaav con-centrations of charge carriers behind the front vs. the externalfield strength E0 with R = 8000 (squares) and R= 4000(circles). Line: calculation of for a plane wave.

FIG. 9: RatioEM/E0 vs. the external field strength E0 withR = 8000 (squares) R = 4000 (circles). Filled symbols:the simulation results, open symbols: calculation formulas(32) using the values ofa and b, shown in Fig. 6.

more than behind the front of a plane wave (Fig. 8).The reason for this is that the reduction of the ionizationareab2 times in a corrugated wave compared with a flatone is compensated by an exponential increase of impact

ionization rate. This effect is particularly large in a rela-tively weak external field, where the dependence of(E)is very sharp.

With decreasingR it becomes apparent that the frontresembles not a cylinder with an oval tip, but a bad-minton shuttlecock (Fig. 1), and the slope of almostlinear section of yf(x) (shuttlecock feathers) decreasesapproximately inversely proportional toR (Fig. 10), andapproximation (1) is becoming less suitable for small R.To explain this effect, it should be noted that in the case

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FIG. 10: The derivative ofdyf/dx in the region of linearityof function yf(x) (open symbols) and the decay length 1/

of the field behind the front (filled symbols) vs. RwithE0=

0.36E. Symbols are obtained by processing the simulationresults, dashed line: calculation formulas (24) using the valuesofuf, shown in Fig. 4, solid line: calculation formulas (21)using the values ofb, shown in Fig. 6.

of stationary wave the kinematic relation

dyfdx

= unu2f u2n

(19)

holds, where un = un[En(x)] [43] and yf(xf) = 0 ,un(xf) = uf by definition. In the area of x < xi thefront itself ceases to be an ionization front, and it is justa thin layer of a space charge, moving mainly due to anelectron (if < 0 at the front) or holes (if > 0 atthe front) drift, so u

n = v[E

n(x)]. From this and the

scaling laws uf R, EM = EF(0, 0), it follows thatthe derivative ofdyf/dxtends to zero, and the streamersradius tends to a constant value of bR at R anduf/vs . It is easy to show that in this case the fieldEn behind the front decreases like

En(x) Eiexp[E(x xi)], (20)

where

E 1R

1.85

1 b 1

(21)

is a minimal positive root of the equation

J0(EbR)Y1(ER) = J1(ER)Y0(EbR), (22)

J and Yare Bessel functions of the first and secondkind of order . The substitution of (20) in (19) leadsafter integration to asymptotic dependence

yf(x) =bR vE

u2f v2ln

1 +

EiEs

eE(xxi)

, (23)

FIG. 11: The radial distribution of the charge carriers den-sity at different distances from the front with R = 100and E0 = 0.36E. Line: the simulation results, symbols: thecalculation formulas (27) with av = 0.0138,00= 0.059 and0R = 0.002.

which is valid under the condition that the second termis small in comparison with the first term. In semicon-ductors, it is usually Ei Es, so there is an area whereEs En Ei, v vs and

yf(x) yf(xi) vsu2f v2s

(x xi) (24)

according to the simulation results (see Fig. 1, 10). Theformula (24) should also be straight from (19), and there-fore, in contrast to (23), is valid even if the ratio ofuf/vsis not very large [44]. Consequently it is advisable to usean approximation

yf(x) = 2

bR arccos

exp

2

x xfaR

vsu2f v2s

(x xf), (25)

which is consistent with (24) in the region (x xf) >2aR at an appropriate choice of b, coincides with (1)at vs/uf 0 and well describes the shape of the frontat all ratiovs/ufas long as En> Es.

At more higher values of (xf x) the inequalityEnEsis satisfied, so it follows from (23), that functionyf(x)

seeks tobR exponentially:

yf(x) =bR 0EiEuf

eE(xxi), (26)

where0 = vs/Esis low-field mobility. In gasesEs Ei(that is = 0 in the topical range of fields), there-fore the range of linearity of yf(x) has to be absent,which is consistent with the results [8]. In a planar caseE=/2L(1 b), so that the formula (26) correctly de-scribes the asymptotic of function (1) at (xf x) ,

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FIG. 12: Distributions of the field strength E and chargecarriers concentration along the x-axis near the fronts ofthe plane (solid lines) and corrugated (dashed lines) waves

with R = 100 and E0 = 0.36E. Symbols: the calculationformulas (31) using the values of anduf, obtained in thesimulation of a plane wave. Concentration on the axis ofxis maximal at x = x R > Rc(E0), so it exists only atE0 < 0.4E. In strong external fields tunneling ionizationdoes not turn off and suppresses the instability even atR < Rc(E0).

With further decrease ofR curvature of a corrugatedfront and parameters EM,

are also reduced. Pri-mary avalanches begin to overlap until the avalanche-to-streamer transition and form a periodically perturbedstationary ionization front. The most long-wave har-

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monic of this perturbation is given by the boundary con-ditions of our problem and has a wave number k = /R.Perturbation amplitudes (in particular, the differenceEM E0) decrease monotonically with R as long as be-comes vanish at R = R0(E0). At still lowerR (in region3) perturbations generated by the primary avalanche de-cay, so that over time a plane impact ionization wavearises. This result is consistent with the conclusions of

linear theory of transverse instability of the impact ion-ization waves. For gases, such a theory was created bythe authors of [15], which showed that the perturbationswith k < k0 /4 had to increase with time, and themore shortwave perturbations had to decrease. This ap-proval is also true for semiconductors [16] if you use theappropriate to our case formula [28, 29]

=0

vsD0

12

+

1

4+

vsD0

, (28)

where0= (E0). This means that at

R < R0

4/ (29)

an array of avalanches should produce a plane impactionization wave [45] in accordance with the results ofmodeling. This wave propagates with the same rate

u =D

2

3 +

1 +

4vsD0

, (30)

for all R < R0 [28, 29] only due to drift, diffusion andimpact ionization, as the field strength at its front doesnot exceed E0 and is insufficient for tunneling ioniza-tion. The formula (29) describes well the dependence ofthe critical radiusR0 (and coinciding with it the criticalwidth L0 for an array of flat streamers) on E0, obtained

from the simulation (see Fig. 3).Another feature of the corrugated ionization waves wasfound, but not explained in the paper [8]. It consists inthe fact that the field behind the front tends to zero (incontrast to the isolated streamers [11, 34]), but decaysmuch more slowly than in the front of the plane waves.A typical example of such differences is presented in Fig.12. The paradox of this effect is the following. In the caseof stationary plane ionization waves, as it is easy to show,the field strength is attenuated by the law exp(x) with[46]

= uf

2D1 + 4vs

ED

Esu

2

f 1 (31)

in exact agreement with the simulation results (see Fig.12). The calculation of this formula for a corrugated wavealways gives a significantly (almost four times in the casecorresponding to Fig. 12) larger value of , but thesimulation results give significantly (approximately tentimes) smaller value of.

The reason for this discrepancy is a curvature of thefront. In a stationary wave conduction current is accu-rately compensated by displacement current, so that the

total current density is zero everywhere [47]. The situ-ation is completely different in corrugated waves. Nearx-axis ahead of the front conduction current and the dis-placement current coincide with the direction of wavepropagation, but displacement current is opposite di-rected and dominates near the surface y = R (wherethe concentration is very small). As a result, a vor-tex of current is formed at the front. The example of

such vortex is shown in Fig. 13. Near the surface y = Rfield strength, and with it the displacement current decayexponentially according to (20) with exponentE. Obvi-ously, the conduction current, providing the appearanceof ohmic field behind the front, decreases in the sameway near the axis ofx. It is clear that this ohmic fieldshould decrease approximately exponentially with the in-crement of E. This conclusion is confirmed by thesimulation results given in Fig. 10.

In conclusion of this section we will note two more cir-cumstances. Firstly, at givenE0 andR a stationary cor-rugated ionization wave is an attractor for a wide range ofinitial conditions: after sufficient time the same solutionis obtained

- for a point initial perturbation,- for an ellipsoidal initial perturbation with a trans-

verse semi-axis of order ofR/2,- as a result of restructuring a stationary corrugated

wave with control parameters{F0, R} after changing F0toF0,

- as a result of growth of small transverse perturbationsof the plane ionization, ifR < /kM, wherekMis a wavenumber of the most rapidly growing perturbations [16].

A similar (but less common) result was obtained forthe two-dimensional array of streamers in gases [8].

Secondly, apart from the above basic principles of evo-lution of interacting streamers more subtle effects werefound. In particular, at some values of control parametersE0 and R, small quasi-periodic variations of the quanti-ties behind the front and the maximum field strengthEM are observed, as well as small quasi-periodic devi-ations of the front shape from the linear function (24).A possible reason for such anomalies is the use of a toocoarse finite element mesh, the minimum size of which isdetermined by the resources of our computers. However,we can not exclude the physical reality of the observedeffects (see, for example [23]).

IV. CONCLUSION

In the present article the results of numerical model-ing of evolution of the two-dimensional periodic array ofidentical streamers in a constant and uniform field arestated for the first time. It turned out that the nature ofthe evolution of a streamers array in semiconductors wasmuch more complex in comparison with gases in a frame-work of minimal model [8]. The evolution of the arrayis completed in different ways, depending on the controlparameters of the problem - the external field E0 and the

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distance 2R between the streamers. For classification ofvarious scenarios of evolution a diagram of final state ofthe streamers array, dividing plane of [1/E0, R] for fourqualitatively various regions, is constructed and repre-sented on Fig. 3. In regions 1 and 2 interaction betweenstreamers leads over time to formation of two types of sta-tionary ionization waves with the corrugated front, differ-ing with ionization mechanisms. Specific characteristics

of fronts of these waves, caused by features of processesof ionization and charge transport in semiconductors, aredescribed in detail and explained on the basis of simplephysical reasons. In region 3, at enough small R, thearray of primary avalanches generates a planar impactionization wave. In weak external fields between regions1 and 2 there is region 4 in which stationary propagationof ionization waves is impossible because of developmentof transverse instability. This instability observed earlierat modeling of isolated streamers, can be called as localbecause it destroys (or doesnt destroy) fronts of eachstreamer in the array.

However, besides it global instability of array of

streamers (each of which is locally stable) is also possi-ble. In fact, in this work we considered that the primaryavalanches generating the array of streamers, started atthe same time from nodes of a ideal planar hexago-nal lattice. Meanwhile small deviations in time and/orin the provision of avalanches start will lead to emer-gence competition between streamers of the non-idealarray: some of them will appear in preferred positionand will develop quicker than others. Sooner or later suchstreamers will start reducing considerably a field strengthin the vicinity and to suppress propagation of neighbors.If distance d between the electrodes is sufficiently large,only the earliest and/or most remoted from neighborsstreamers will be able to overcome it, and the average

distance between such leading streamers should be of theorder of 3d (see Appendix). These reasons indicate theimportance of a global instability problem which will beanalyzed in a separate paper.

The author is grateful to A.V. Gorbatyuk for helpfuldiscussions. This work was supported by RFBR (grant13-08-00474).

Appendix A. The electrostatic interaction between

metal cylinders with ellipsoidal tips.

For a quick estimate of a maximum field strength EMof streamers in the array each of them can be representedin the form of a metal cylinder of length 2 l with a radiusbR and tips in the form of ellipsoids of rotation. Tointerpret the simulation results you need to know howthe array parameters affect the two value.

The first of these - the ratio of (R,b,l) EM(R,b,l)/EM(, b , l) - allows you to define the con-ditions under which the influence of the electrostatic in-teraction between streamers on EM becomes noticeable.These conditions depend little on the longitudinal axis

FIG. 14: Ratio of the maximum field strength at a singlecylinder and an array of cylinders with hemispherical caps(R,b,l) vs. their length 2l, radius bR and distance betweenthem 2RwithbR/l= 0.01 0.25 and R/l = 0.4 50.

FIG. 15: Ratio EM/E0 for an array of cylinders with ellip-soidal tips vs. the dimensionless length of the longitudinalae = a/2 and transverse b semi-axes with b = 0.2 0.6ae = 0.2 1.2.

of the ellipsoids aeR, therefore, when calculating itwould be possible to put ae = b for simplicity. Theresults of these calculations are shown in Fig. 14. Ascan be seen, for all relevant values ofb and R/l they are

well described by the function = tanh(5/2), where= R/l[1 + 3(bR/l)1/3]. Interaction becomes significantwhen is noticeably different from the one, that is at

8/10/2019 Em 11141

11/11

11

between the system cylinder+ellipsoid and the surfaceof revolution with forming a kind of (1) is minimal andfor all a, b do not exceed 5%. The results of such calcu-lations presented in Fig. 15, are well described by the

dependence

EM(a, b) = 0.88E0b4/5 (1 + 4a/5b) . (32)

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[40] A. S. Kyuregyan, Semiconductors, 41 (6), 761 (2007).[41] It is a good idea for those who study performed in [8]

array of streamers at the highest values ofL < Lc, when L and the deviation from the first condition (2) isminimal.

[42] The more a, so the streamer front is sharper at a fixedvalue bR .

[43] If the wave propagation is significantly affected by tun-nel ionization, velocity un depends not only on En, butalso on the whole field distribution along a force line in-tersecting the front [34, 35]. This makes the analytical

calculation of the whole front form quite so hopeless, buthas no effect on the properties of the function yf(x) atx < xi.

[44] It is interesting to note precise analogy between this ion-ization front and the shock waves produced by supersonicmotion of bodies in gases.

[45] A similar effect was observed earlier in the simulation offlame propagation in a tube [38]: reduction of its diam-eter to critical value corresponding of this process sup-pressed transverse instability and the flame front remainsflat.

[46] This formula is a bipolar analog of the formula (5.63) ofthe work [23].

[47] If the field ahead of the front of a stationary plane wave

is not uniform (as in p

+

n

n

+

-junction) then totalcurrent density is not zero, but is constant everywhere[40] and the field strength behind the front approachingto a finite value exponentially with exponent , that isalso determined by the formula (31).