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AN IMPACT IONIZATION MODEL FOR CURRENT CONTROLLED NEGATIVE RESISTANCE AND g-r
INDUCED NONEQUILIBRIUM PHASE TRANSITIONS
E. Schöll
To cite this version:
E. Schöll. AN IMPACT IONIZATION MODEL FOR CURRENT CONTROLLED NEGATIVE RE- SISTANCE AND g-r INDUCED NONEQUILIBRIUM PHASE TRANSITIONS. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-57-C7-62. �10.1051/jphyscol:1981706�. �jpa-00221642�
JOURNAL DE PHYSIQUE
Colloque C7, supplément au n°10, Tome 42, octobre 1981 page. C7-57
AN IMPACT IONIZATION MODEL FOR CURRENT CONTROLLED NEGATIVE RESISTANCE AND g-r INDUCED NONEQUI LIBRIUM PHASE TRANSITIONS
E. Scholl
Institut f. Theoret. Phystk B, RWTH Aaehen, D-51 Aachen, FRG
Résumé. - Nous présentons dans ce papier un modèle simple de généra- tion-recombination basé sur un mécanisme d'ionisation des donneurs par impacts à deux étapes. Ce modèle conduit à une caractéristique Courant- Tension de type S et à des transitions de phases hors d'équilibre entre les branches hautes et basses de la résistivité. Nous avons montré qu' une règle de "surfaces égales" est une condition nécessaire pour la formation de filaments ou de couches de courant.
Abstract. - A simple generation-recombination (g-r) model based upon a two-step impact ionization mechanism of donors is presented. It leads to an S-type current-voltage characteristic and to nonequilibrium phase transitions between its high and low resistivity branches. The homogeneous steady state corresponding to the falling branch is shown to be unstable against the formation of high and low current layers and filaments, and an equal area rule for their coexistence is established.
1. Introduction
Current controlled negative differential resistance associated with an S-type current-voltage characteristic and with threshold switching is
1) 2) a wide-spread phenomenon in amorphous and crystalline semiconduc- tors, and insulators for which thermal as well as electronic mecha- nisms have been proposed. Here we consider a generation-recombination
(g-r) model which is based upon the impact ionization of both ground state and excited donors by hot electrons. It involves an n-type semiconductor with N_ donors, partially compensated by N- <. Nn fully occupied acceptors, at low temperature. Other g-r models with impact ionization of deeply trapped electrons and holes were studied previous- iy4)-
2. The model
The g-r processes considered are shown schematically in Fig. 1.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981706
JOURNAL DE PHYSIQUE
I
c o n d u c t i o n band1 4 A ,I
"*
1
donorsD
T: X: T* X* X; X1 r a t e c o n s t a n t s
F i g . 1: Schematic g-r p r o c e s s e s i n v o l v e d i n t h e model.
The r a t e c o n s t a n t s depend upon t h e e l e c t r i c f i e l d E . I t i s assumed t h a t i n t h e regime of n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e t h e f i e l d de- pendence of t h e impact i o n i z a t i o n c o e f f i c i e n t s dominates t h a t of a l l o t h e r r a t e c o n s t a n t s and of t h e m o b i l i t y .
The t r a n s p o r t e q u a t i o n s a r e t h e f o l l o w i n g :
The g-r r a t e s a r e g i v e n by
f ( n , n D , n D * ) = X: n + - T S p n + X
D 1 D nD n + X * 1 n ~ * n
g ( n , n D , n D * ) = - x * n + T* nD+ - X I no n D
n c o n c e n t r a t i o n of f r e e e l e c t r o n s
nD c o n c e n t r a t i o n of e l e c t r o n s i n donor ground s t a t e s nD" c o n c e n t r a t i o n of e l e c t r o n s i n donor e x c i t e d s t a t e s pD=ND-nD-no" c o n c e n t r a t i o n of unoccupied d o n o r s
N D t o t a l donor c o n c e n t r a t i o n
NA < N D compensating a c c e p t o r c o n c e n t r a t i o n
A e l e c t r o n c u r r e n t d e n s i t y
p e l e c t r o n m o b i l i t y D d i f f u s i o n c o n s t a n t T a b l e I: L i s t of symbols
3. Homogeneous steady states
The homogeneous solutions of (1)- (5) are determined by
A = f(n,nD) nD = g(n,nD) and nD* = N -N -n-nD (local neutrality) D A .
Depending upon the values of the rate constants, (8) admits up to three steady states, two of which are stable against small fluctuations
g n, dnD, while the third is not. The two stable steady states
(nl < n3) correspond to high (nl) and low (n3) resistivity states. The
carrier concentrations depend upon the applied static electric field Eo = V/L via X I and X 7 rt , where V is the sample voltage, and L the sample length. Thus j = n (Eo) e p E o yields an S-type static current- voltage characteristic, which is-plotted in Fig. 2 for the particular 3 -1 it =10-6exp(-1 .5/Eo) field dependence XI = 5xl0-~ exp (-6/Eo) cm s
3 -1 X1
cm s .
10 0.5 2 1
Fig. 2: Current-voltage characteristics for ND=l .5xlO cm-3.
NA/N = I /3. TS ~ =c m 3 s I , T" =10~s-'. The other parameters are in p<!
la) !: = 0.5.1.2.10; X" =i; (b) X* = I ,I. 1 ,I .5,2; :X = 0.5.
Switching between the high and the low resistivity branches of the current-voltage characteristic can be regarded as a nonequilibrium phase transition of first order between two steady states with over- lapping stability range, brought about by variation of Eo taken as a control parameter. More generally, Eo is determined from the load line of the external circuit with impedance R and applied voltage Uo. This includes in particular voltage controlled (R-0) and current con- trolled ( R + a ) conditions.
By appropriate irradiation the ionisation coefficient X: can be increased, leading to smaller and eventually vanishing bistability
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ranges (Fig. 2a) . Similarly, with increasing X * (e. g. increasing temperature, Fig. 2b) or decreasing compensation ratio N ~ / N ~ the bi- stability vanishes. This agrees well with the qualitative behaviour known from experiments 5 .
4. Inhomogeneous steady states
For small deviations S+ : = ( d E, bn, gnD, bnDIc ) from the unstable homogeneous steady state (go, n,nDl nD* ) ( 1 ) - (5) can be linearized.
With the ansatz g() N exp i (k.5 - t) for periodic boundary conditions and fixed current we find the dispersion law
where r : =
r
is the dielectric relaxation frequency,&
2, > 0 > h 2 are the eigenvalues of the linearized rate equations
( a ) , and e , A are positive constants depending upon the rate
coefficients, and hence upon Eo. The spectrum given by (10) contains the damped dielectric relaxation mode o = -ir and three coupled relaxation - recombination - diffusion modes. For transverse pertur- bations (k, b E J. Eo) we obtain in the long wavelength limit an undamped recombination-diffusion mode
-
2= i ( h f - D k ) (11)
with a dressed diffusion constant
, .
4 2 + a , e + A
D = D = I
( a l + r ) (a1-A2)
This leads to the bifurcation of a family of inhomogeneous stationary solutions.
To investigate the full nonlinear equations (1)-(5) for a plane geometry, we assume that the concentrations and the field
E = E e +Elex depend upon the coordinate x transverse to the static - 0 -Z
field go only, and that the transverse current vanishes. In the steady state (1)-(5) can then be reduced to
-
with the dimensionless coordinate 3 = x /LD and transverse field
& = (eLDEA ) / (kT) , where LD: = [ ( &D) /(4meP(ND-NA)] '12 . The
charge density 9 (n) is found from (3) - (5) by eliminating nD, nDf and is a nonlinear function.of n. It depends upon Eo via X I , X F , and we
assume Ell& Eo. A first integral of (13) is
with 4 (n) : =
l(
g (n) / n) dn.Eq. (14) includes inhomogeneous oscillatory, solitary and run-away solutions (Fig. 3) . The homogeneous steady states (nl ,n2,n3) are the extremal points of 4 ( n ) . A solitary solution corresponding to the coexistence of nl and n3, i.e. to plane low and high current layers, exists if and only if the condition
is satisfied. This is analogous to the Maxwell construction obtained for equilibrium and chemical nonequilibrium phase transitions6) or the equal area rule known from the Gunn effect7) and from other switching
1 ) models .
For
Fig.3: Phase portrait of the inhomogeneous steady states given by (14) with (15). The parameter varying along the phase trajectories is the transverse coordinate 7 .,The
saddle-to-saddle separatrlces represent the solitary solu- tions. The numerical parameters are those of Fig. 2 (aj , with
X: = 0 - i r s , Eo 8 . 5 Y/cm.
I . . . . . . . . . I ,,
a cylindrical geometry (13) is replaced by
where r is the dimensionless radial coordinate. The equal area rule (15) has to be modified by a surface term
and leads to cylindrical high or low current filaments with mean radius ro.
5. Conclusion
The mechanism leading to bistability in the above model can be inter- preted as a coupled two-step impact ionization process:
The donor ground levels are depleted by impact ionization (XI), which results in an enhancement of the excited donor population n D k r by (7).
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Simultaneously, the excited donors are impact ionized, with a rate
xl* nD*. n increasing thus stronger than linearly with n. In the bi- stability regime the impact ionization rate is negligible at 1owni.e.
in the high resistivity state, where recombination is balanced by the t simple generation process X: . Due to the superlinear increase in the impact ionization rate, a second stable steady state is possible at large n , i.e. the low resistivity state.
The model is applicable for materials where the dominant g-r processes are those of Fig. 1. More so--phisticated g-r models, e.g.
including Auger recombination, yield the same qualitative behaviour for suitable ranges of the rate constants.
There is a close analogy of the above switching transition with phase transitions in equilibrium and in other nonequilibrium systems 8 ,
such as critical slowing down, soft mode instability of the homo- geneous steady state against long wavelength perturbations, bifur- cation of inhomogeneous solutions, and phase coexistence when certain equal area rules are satisfied.
References
(1) Adler D., Shur M.S., Silver M., Ovshinsky S.R., J. Appl. Phys.
51, 3289 (1980).
-
(2) Bonch-Bruevich V.L., Zvyagin I.P., Mironov A.G., Domain Electri- cal Instabilities in Semiconductors.
Consultant Bureau (New York) 1975.
( 3 ) Klein N., Thin Solid Films 50, 223 ( 1 978) .
(4) Landsberg P.T., Pimpale A * , J. Phys. C?, 1243 (1976).
Landsberg P.T., Robbins D.J., Scholl E., Phys. stat. sol. (a) 50,
423 (1978).
Robbins D.J., Landsberg P.T., Scholl E., Phys. stat. sol. (a) 65., 353 (1981).
7
Scholl E., Landsberg P.T., Proc. Roy. Soc. A 365, 495 (1979).
(5) Khosla R.P., Fischer J.R., Burkey B.C., Phys- Rev. 2551 (1973).
(6) Schlogl F., Z. Phys. 253, 147 (1972) .
(7) Shaw M.P., Grubin H.L., Solomon P., The Gunn-Hilsum Effect.
Academic Press (New York) 1979.
(8) Biittiker M., Thomas H., 2. Physik B 33, 275 (1979), B 2, 301, (1979).
Haken H., Synergetics. Springer (Berlin) 1977.
