a r X i v :0805.3836v 1 [c o n d -m a t .m t r l -s c i ] 25 M a y 2008
Impact ionization fronts in miconductors:superfast propagation due to
“nonlocalized”preionization
Pavel Rodin 1*,Andrey Minarsky 2and Igor Grekhov 1
sasi
1
Ioffe Physicotechnical Institute,Politechnicheskaya 26,194021,St.-Petersburg,Russia,
2
Physico-Technical High School of Russian Academy of Science,Khlopina 8-3,194021,St.-Petersburg,Russia
(Dated:May 25,2008)
We discuss a new mode of ionization front passage in miconductor structures.The front of avalanche ionization propagates into an intrinsic miconductor with a constant electric field E m in p
rence of a small concentration of free nonequilibrium carriers -so called preionization.We show that if the profile of the initial carriers decays in the direction of the front propagation with a characteristic exponent λ,the front velocity is determined by v f ≈2βm /λ,where βm ≡β(E m )is the corresponding ionization frequency.By a proper choice of the preionization profile one can achieve front velocities v f that exceed the saturated drift velocity v s by veral orders of magnitude even in moderate electric fields.Our propagation mechanism differs from the one for well-known TRAPATT fronts.Finally,we discuss physical reasons for the appearance of preionization profiles with slow spatial decay.
PACS numbers:85.30.-z,72.20.Ht,71.55.-i
Propagation of impact ionization fronts in micon-ductor structures reprents a spectacular nonlinear effect 1,2,3,4which has important applications in pul power electronics.5In rever-biad p +-n -n +diode structures ionizing fronts propagate faster than the satu-rated drift velocity v s .1,2,3,4Such superfast propagation is possible due to the prence of small concentrations n 0,p 0of free electrons and holes in the depleted region.The free carriers which initiate an avalanche multiplication are often coined as “pre-ionization”of the medium.1,6Ac-cording to the conventional concept of ionization fronts in TRAPATT (TRAped Plasma Avalanche Triggered Tran-sit)diodes 7,8th
e avalanche multiplication occurs within the ionization zone of length ℓf =εε0(E m −E b )/qN d where electric field exceeds the effective threshold of im-pact ionization E b (Fig.1,curve 1).This length is fi-nite due to the slope of the electric field in the n ba dE/dx =qN d /εε0which depends on the doping level N d (note that n 0,p 0≪N d ).The finiteness of the ionization zone ℓf prevents a uniform avalanche multiplication in the whole n ba and thus ensures the existence of the traveling front mode of avalanche breakdown.However,this concept is not applicable to p -i -n structures with in-trinsic (N d =0)ba (Fig.1,curve 2)as well as to short overvoltaged structures becau in both cas E >E b in the whole n ba (Fig.1,curve 3).On the basis of TRAPATT-like front concept one would expect that in the two cas pre-ionization of the high-field region trig-gers quasiuniform breakdown ruining the traveling front mode.
In this paper we argue that superfast impact ionization fronts are nevertheless possible in p -i -n structures where E >E b everywhere in the high-field region providing the concentration profile of initial carriers n 0(x ),p 0(x )decays in the direction of front propagation.The propagation mechanism of such front is completely different from the conventional TRAPATT-like front.We find the front ve-locity analytically and show that it is controlled by the
FIG.1:Electric field profiles E (x )in the traveling ionization front.Profile 1corresponds to the conventional TRAPATT-like front with a finite size of impact ionization zone ℓf .Profile 2corresponds to p -i -n structure (N d =0).Profile 3corre-sponds to p +-n -n +structure with low n ba doping.
slope of pre-ionization profile,and that it can exceed v s by veral orders of magnitude.once是什么意思
We consider a planar impact ionization front and de-scribe it by the standard drift-diffusion model 7,8which consists of continuity equations for electron and holes concentrations n and p and the Poisson equation for the electric field E .For a lf-similar front motion with con-stant velocity v f the equations can be simplified by introducing new variables σ≡n +p ,ρ≡p −n .8For intrinsic miconductor (N d =0)the equations for σ,ρand E in the co-moving frame z =x +v f t become
d
dz 2
=2α(E )v (E )σ,(1)
v (E )σ+v f ρ−D dρ
dz
=
q
2
0.0
0.3
0.60.9
0.00
0.05
0.10
0.15
σp l / σ0
E m / E 0
FIG.2:Concentration of electron-hole plasma σpl ,normal-ized by σ0=2εε0α0E 0/q ,behind the front as a function of electric field E m for Townnd’s approximation of the impact ionization coefficient α(E )=α0exp(−E 0/E ).
where v (E )is the drift velocity,D is the diffusion coeffi-cient and α(E )is the impact ionization coefficient.Here we neglect recombination and assume that electrons and holes are identical in a n that α(E )=αn (E )=αp (E )and v (E )=v n (E )=v p (E ).For an infinite domain the boundary conditions are E →E m ,σ,ρ→0for z →−∞and E,ρ→0,σ→σm for z →+∞.
In the simplified
ca of
D =0we u Eq.(2)to ex-clude ρfrom Eqs.(1)and (3).This yields
d
v f
σ
=2α(E )v (E )σ,(4)
dE
εε0
v (E )
q
v 2f
q
E m
α(E )dE (8)
acodepends only on the electric field E m (Fig.2).The de-pendences p (E )and n (E )in the traveling front
p (E ),n (E )=εε0
v f ±v (E ) E m
E α(E )dE.(9)
follow directly from Eqs.(6,7)and are shown in Fig.3.Remarkably,a traveling front solution exists for any v f ≥v s .Within the approximation D =0the slowest solution corresponds to the shock front (discontinious at E =E m )that travels with a saturated drift velocity v f =v s .
0.0
0.10.20.30.40.5
0.00
0.05
0.10
0.150.20
0.2532
1
4,5
n / σ0
E / E 0
0.0
0.10.2
0.30.40.5
0.000
0.0020.0040.0060.008
0.0105
2431
p / σ0
E / E 0
FIG.3:Concentrations of electrons n and holes p ,normal-ized by σ0=2εε0α0E 0/q ,in the traveling front as func-tions of electric field E for different front velocites v f /v s =1,1.1,1.2,5,10(curves 1,2,3,4,and 5,respectively)and ap-proximations α(E )=α0exp(−E/E 0),v (E )=v s /(E +E s );E s /E 0=0.
01.The applied electric field is E m =E 0/2=50E s .Note that for the chon v (E )approximation v (E m )=0.98v s ,hence v f >v s for all shown curves.
The above analysis does not allow to lect a physi-cally relevant solution and hence to find the actual front velocity v f .The lection problem remains in the ca of D =0.This is a general feature of fronts propagat-ing into linearly unstable state [e Ref.9and references therein].Ionizing fronts belong to this class since the state (E =E m ,σ=0)is unstable:due to E m >E b any amount of free carriers leads to avalanche multiplication.It has also been suggested and confirmed by numerical simulations that in gas 10,11and miconductors 12ion-izing fronts are so called pulled fronts.For a pulled front,the dynamics in the part of the front where avalanche multiplication and screening are esntially nonlinear is subordinated to the linear dynamics of the front tip which fully determines the propagation velocity.9The dynam-ics of the front tip is described by the linearized (near the state E =E m ,σ=0)version of equations (1,2)with constant coefficients v (E )=v s and α(E )=α(E m )=αm
v f
dσ
dz −D d 2σ
dz =0,αm ≡α(E m ).
(11)
3
2
46
12345
6v
*
λ
*αm D/v s = 0.1
αm D/v s = 0.5
αm D/v s = 1
v f / v s
λ / αm
mmdbFIG.4:The dispersion relation v f (λ).
Here we take into account that in ionizing fields v (E )=v s =const.Solutions of the linear equations are expo-nential functions σ(z ),ρ(z )∼exp(λz ),where the disper-sion relation v f (λ)remains to be found.
It is known from the theory of pulled fronts that the actual front velocity strongly depends on the type of ini-tial conditions.9All possible initial conditions split in two class that lead to qualitatively different dynamics.Lo-calized conditions correspond to initial profiles σ(x,t =0)that are steeper than the profile exp(λ⋆x )with a cer-tain characteristic exponent λ⋆:σ(x )<C exp(λ⋆x )for x →−∞,where C is an arbitrary constant.In this ca the front profile eventually becomes smoother and asymptotically reaches the profile σ(z )∼exp(λ⋆z )at z →−∞that propagates with linear marginal stability velocity v ⋆=v f (λ⋆).9Any initial profile that is strictly equal to zero for sufficiently small value of x also rep-ren
ts a localized initial condition.Nonlocalized initial conditions correspond to profiles σ(x,t =0)with slow spatial decay that do not meet the above mentioned con-dition σ(x )<C exp(λ⋆x )for x →−∞and hence are smoother than exp(λ⋆x ).In this ca the front velocity is fully determined by σ(x,t =0):for the initial profile exp(λ0x )with λ0<λ⋆the front velocity is given by the dispersion relation v 0=v f (λ0).
The dispersion relation v f (λ)follows from the charac-terictic equation of Eqs.(10)and (11)
ℓ2λ3−2ℓ v f
v s 2
−1+2αm ℓ λ
−2αm
v f2426
v s (12)and is explicitly given by (e Fig.4)
英语关键词v f (λ)
λ+
λ
2
drum+ℓλ.(13)
The critical steepness λ⋆and the velocity v ⋆correspond to the minimum point 9and are given by
0.1
1
1
pear怎么读10
λ*
/ αm
v *
/ v s
v *
/ v s , λ*
/ αm
αm D/v s FIG.5:Linear marginal velocity v ⋆and the steepness λ∗.
λ⋆1
2
+A
,
(14)
v ⋆
妙笔作文
1+5αm ℓ−
(αm ℓ)2
(αm ℓ)2/4+αm ℓ.
The right branch of the v f (λ)(λ>λ⋆)corresponds to fronts who velocity increas with steepness λdue to diffusion.According to the concept of localized initial conditions,9the fronts are unstable:their steep pro-files eventually relax to the profile with exponential tip exp[λ⋆z ]that propagates with the velocity v ⋆.The left branch of the v f (λ)dependence (λ<λ⋆)corresponds to stable fronts who velocity decreas with λ.The fronts correspond to nonlocalized initial conditions which are in the focus of our interest.
Charateristic values of the dimensionless parameter αm ℓ≡αm D/v s are 0.1for Si and 1for GaAs devices.As follows from Fig.5,λ⋆>αm in the relevant interval.Physically it means that the front propagating with linear marginal stability velocity v ⋆is so steep that the valid-ity of the drift-diffusion approximation is questionable.This problem disappears for much smoother fronts that correspond to nonlocalized initial conditions (left branch in Fig.4)if λ<λ⋆.
Eq.(13)leads to a simple relation v f /v s =2αm /λfor the ionizing front velocity in ca of preionization with decay exponent λ<λ⋆.For such fast fronts the effect of diffusion is negligible;in particular,Eq.(4,5,6,7,8,9)are fully applicable.Although v f increa with αm and hence with electric field E m ,it is the ratio αm /λwhich actu-ally counts.This ratio can be arbitrarily large resulting in front velocities that exceed the saturated drift veloc-ity by many orders of magnitude.It means that a proper choice of slowly decaying preionization profile gives the possibility to achieve fast front propagation in even mod-erate (with respect to E b )electric fields.However,the concentration of electron-hole plasma generated by the front passage increas with E m (Fig.2).Generally,the
4
electromagnetic limitation v f<c,where c is the velocity of light,may be important:for v f comparable to c the full t of Maxwell equations shall substitute the Pois-son equation in the model becau the feedback from a nonstationary electromagneticfield created by the front passage on the front dynamics becomes esntial.
Pre-ionization profiles with slow spatial decay can appear due to photoionization by photons from den electron-hole plasma behind the front.In this ca,λ−1can be roughly identified as the light ab
sorption length.This mechanism is efficient in direct-band ma-terials and can be relevant for planar fronts in GaAs diode structures4as well asfinger-like streamers in direct-band bulk miconductors.6Another mechanism is re-lated tofield-enhanced ionization of deep centers in Si p+-n-n+structures ud in pul power applications.3 The high-voltage structures posss“hidden”deep lev-els–process-induced defects–with low recombination activity.13Preionization of the high-field space charge region can be due tofield-enhanced ionization of the deep centers embedded in the n ba.14This ionization is more efficient near the p+-n junction where the electric field is stronger.Hence the profile of initial carriers de-creas along the n ba.The characteristic decay length λ−1is expected to be a fraction of the n ba width W∼100µm.At the same time for low doped n ba the electricfield can be above the ionization threshold E b ev-erywhere.This may result in front velocities that exceed v s by veral orders of magnitude.Numerical simulations of such triggering process will be prented elwhere. Acknowledgements.–We are indebted to U.Ebert and V.Kachorovskii for enlightening discussions.This work was supported by the Programm of Russian Academy of Sciences,“Power miconductor electronics and pul technologies”.P.Rodin is grateful to A.Alekev for hospitality at the University of Geneva and acknowledges the support from the Swiss National Science Foundation.
*Electronic address:rodin@mail.ioffe.ru
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