Gunn instability in finite samples of GaAs

dp/de=

[(1 + e 4 ) p - j e 3 ] / p e

3.

(2.15)

The phase plane of this equation is represented in fig. 4. T h e r e is only one trajectory that leaves the origin p = 0, e = 0, and has p ~ 0 as e -* + tangent to the nullcline of (2.15). This trajectory matches (AI.1) with a n o t h e r region where we can again ignore the diffusivity in (2.1). W e omit the details since they are standard. Notice that this stationary state is unique which immediately implies 1(8) 4= 1. Let us show 1(3) < 1 and therefore that the stationary solution is stable. Only the region where (2.15) holds contributes to 1(8). Numerical evaluation of this contribution shows that it grows monotonically from 0 (at j = 0) to 0.5 when j ~ + ~ (the value 0.5 is obtained from an asymptotic analysis valid in the limit j -~ + oo. See appendix 2). T h e contribution of the right b o u n d a r y layer (A1.2) is always smaller than 0.4 which shows the steady state to be stable for all these J ( 1 ( 8 ) < 0.9 < 1). T h e r e f o r e no bifurcations exist when B = eY(64/s).

Fig. 4. Phase plane of(2.15).

2.3. B

=

(~(8 2)

T h e r e are two reasons to study the limit B = ~(62). First of all, there are multiple steady states in this limit whereas we did not find them in the other cases for 8 $ 0. Secondly, the situations here described may persist for finite 8 and reasonable B. Notice that B is typically 0.01 for n - G a A s at ambient temperature. Thus controlling temperature and concentration of d o n o r impurities N 0 we may find a larger 8 ~ 0.1 in (1.11). T h e n our findings for B = ~ ( 6 2 ) may be of practical applicability in spite of the bad appearance of this limit. Let us therefore consider B = 8 2 / a 2, with a = G(1), in (2.14). W e find

dp/de = [(l+e4)(p+a)-aje3]/pe j = J / B s/4

3, (2.16)

T h e r e are many possible cases according to the values of a and j in (2.16). T h e limit a $0 corresponds to B s/4 << 6 << B 1/2, and there is an a c below which we have a situation similar to that found in section 2.2, with only one (stable) steady state. Above ac, it is possible to find multiple steady states for different values of the b o u n d a r y field e c--- pBj. We shall describe in detail what h a p p e n s for a = 1, which is representative of the situation for a = G(1) > a c. The phase plane corresponding to (2.16) is depicted in figs. 5 a - 5 d . For j
466

L.L. Bonilla, F.J. Higuera /Gunn instability in finite Ga,4s samples

P

/

\

i.

e

e

(a)

(b)

P

P

I e

@

(c)

(d)

Fig. 5. (a) Phase plane of (2.16) for Jl~ < J
1.7548, t h e r e are no critical p o i n t s on the axis p = 0. F o r j >J0, t h e r e a r e two critical p o i n t s e 2 and e3, e 2 < e 3. e 3 is always a saddle, while e 2 is an u n s t a b l e n o d e for a < ac ~ 0.285. F o r a > a c, t h e r e exist two values o f j, Jl a n d J2, Jl J2 (fig. 5d), a n d an u n s t a b l e spiral p o i n t for Jl < J < J 2 (fig. 5c; j~-~ 1.8217 a n d j 2 - ~ 4 0 . 8 3 for a = 1). O n l y o n e t r a j e c t o r y r e a c h e s t h e p-axis at finite p ( p = - 1 ) . F o r J0 < J 0 for /1 a~ ~- 0.3, b e l o w a~, j'~ =j~). N o t i c e that the first critical point, E~, P = 0 d o e s not a p p e a r in (2.16) with the scaling (2.13), since it is always E~ = ~ ( 1 ) as 6 $0.

F r o m fig. 5 we can see w h a t are t h e s t e a d y states c o r r e s p o n d i n g to the b o u n d a r y c o n d i t i o n s e = e c at x = 0 , l. W e have d e p i c t e d the p l a n e - p ( 0 ) versus l c o r r e s p o n d i n g to a = 1 in fig. 6. O u r results a r e as follows: (1) F o r j e 3, a n d with a single m a x i m u m if e 2 < e c ~ e 3.

(3) Let j] < j
L.L. Bonilla, F,J. Higuera / Gunn instability in finite GaAs samples

467

-p@

Y (a)

(b)

- p(o)

-p(O)

(c)

(d)

- p(o) - P(O)

(e)

(f)

Fig. 6. (a) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s 1 w h e n e c < exv (b) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s l w h e n e~l < e c < e 2. (c) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s l w h e n 0 < e 2 - e c << e 2. (d) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s l w h e n e 2 < e c < ex2. (e) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s l w h e n ex2 < e c < e 3. (f) C h a r g e at t h e c a t h o d e - p ( 0 ) v e r s u s l w h e n e 3 < e c.

which the trajectory going from the

n o d e to

which m a t c h e s the nullcline of (2.4) a n d it ap-

(0, - 1) crosses the e-axis; exl < e 2 < ex2. T h e n t h e r e is a u n i q u e stable steady state with a single

proaches the saddle p o i n t ( E l , 0 ) as closely as n e e d e d before leaving the n u l l c l i n e in a n a r r o w

e x t r e m u m for all l > 0 if e c < ex~ (fig. 6a), exl < e c < e 2 (e c n o t very close to e 2, fig. 6b), a n d ex2 < e c. T h e e x t r e m u m is a m a x i m u m for ex2 < e c < e 3 , a m i n i m u m otherwise (figs. 6e a n d 6f). F o r e c < e 2 (figs. 6a a n d 6b) there is a c e r t a i n length l x above which p(0) does not c h a n g e with l. This corresponds to the trajectory j o i n i n g (0, - 1) a n d (e 2, 0)

b o u n d a r y layer n e a r x = l. F o r e c very close to e 2 (e 2 - e c < e 2 the i n t e r m e d i a t e steady state with p(0) > 0 has two extrema. All o t h e r states have a

468

L.L. Bonilla, F,J. Higuera / Gunn instabifity in finite GaAs samples

-P(0)

(a)

with p > 0, cross the e-axis at exl a n d e×~, e×~ < e~2, respectively. F o r e~ < e ~ , e×2 < e~ < e~ and e 3 < e c, we have the situations d e p i c t e d in figs. 6a, 6e a n d 6f, respectively, as in (3). F o r e 2 < ec < e×2 t h e r e is a unique stable s t e a d y state with a single m a x i m u m for all 1 > 0 (fig. 6e). F o r e~l < e~ < e 2 the situation is similar to fig. 6d c h a n g i n g - p ( 0 ) to p(0) there: a stable state with a single minim u m exists for all l > 0, t h e r e are a stable state with a single m a x i m u m a n d an u n s t a b l e state with two e x t r e m a for all l > lm. T h e region of m u l t i p l e s t e a d y states occurs for very small B when 6 $0 a n d the fields at the c a t h o d e are very large, close to the m i n i m u m o f r(E). As 6 grows, the electric fields at the catho d e diminish in the region o f multiplicity, the c a t h o d e fields b e c o m e closer to the m a x i m u m of l,(E).

(b) Fig. 7. (a) Same as fig. 6 for il
single m i n i m u m if p(0) < 0 a n d a single m a x i m u m if p ( 0 ) > 0. (4) F o r Jl < J 0 a n d the t r a j e c t o r y e n t e r i n g (0, - 1) first cross the e-axis at e×l a n d e×2, e×j < e×2, respectively. F o r e c < e x l , ex2 < e c < e 3 a n d e 3 < e~, we have the situations d e p i c t e d in figs. 6a, 6e a n d 6f, respectively, as in (3). F o r e×l < e c < e 2 t h e r e is always a stable state with a single minim u m for all l > 0. In a d d i t i o n , for l > I m two new s t e a d y states a p p e a r : a stable state with a single m a x i m u m and an u n s t a b l e state which has a single m a x i m u m if p ( 0 ) > 0 at l = l m and two e x t r e m a if p(0) = 0 instead. T h e r e is also a finite s e q u e n c e of lengths link, I m l,, k two new u n s t a b l e solutions with several e x t r e m a a p p e a r . W h e n e~ 1" e2, K ~ cc (fig. 7a). F o r e 2 < e c < e×2 the situation is a n a l o g o u s to: the o n e just d e s c r i b e d c h a n g i n g p(0) to - p ( 0 ) , as can be seen in fig. 7b. (5) F o r j >J2 (fig. 5d), the s e p a r a t r i x e n t e r i n g e 3 a n d the e x c e p t i o n a l t r a j e c t o r y leaving e 2, b o t h

3. Voltage bias

3.1. Stability of stationary states under l~oltage bias W e use o u r previous results for c u r r e n t bias c o n d i t i o n s to e x a m i n e the effect of t h e bias (2.3). N o t i c e t h a t for each solution E ( x ; J ) of (2.1), (2.2), t h e r e is a voltage 0 such that E ( x ; J ) is a solution of (2.1)-(2.3). T o analyze the l i n e a r stability of s t e a d y states u n d e r (2.3), we linearize (1.9)-(1.12) a b o u t the state E(x; J(cb)). W e k e e p ~ constant, so t h a t J c h a n g e s to J ( O ) + ] ( t ) , and i n s t e a d of (2.7), (2.8) we o b t a i n

E, + ,,(E) E, + ,,,'(E) (Ex + 1)E - aExx = J ( t ) , (3.1) E,+E/p=J(t)

at x = O , I ,

f~i

E ( x, t ) d x + ] ( t ) R = 0.

T h e substitutions (2.9) a n d ] ( t ) =

(3.2) (3.3)

exp(~rt) trans-

469

L.L. Bonilla, F.J. Higuera /Gunn instability in finite Gads samples

form (3.1)-(3.3) into H$+~$=exp

[~v(E)dx)

-j0

~

'

(3.4)

To prove this result we need the following

I4~- - 6 ~ . + W(x,6)6,

W(x,6)

as in (2.11),

(3.5)

0(0) = 1/(~ +p-l),

O(l) =~O(O)exp(- fc:v(E)dx/26) (p~+ 1 4:0),

(3.6)

Z(o~) + R = O,

z(~) - "~j0E(x' t)

(3.7) dx

= f,i (x)exp{f,;

sgn[Im Z(iw)] = - s g n w and P* = 1 (the corresponding steady state has a unique unstable mode under current bias).

)dx.

Lemma. Re Z(iw)

and Im Z(iw) are even and odd functions of oJ E ~ respectively, and Z(o0 - , 0 as cr ~ ~c.

Proof. From eqs. (3.4) to (3.7), it follows that Z ( ~ ) is equal to the complex conjugate of Z(~r). Therefore, Re Z(iw) and Im Z(ioJ) are even and odd functions of w ~ • respectively. That Z(o0 -~ 0 as cr ~ ~ follows from the formula

(3.8)

Notice that Z ( 0 ) = ~'(J): the differential impedance at zero frequency coincides with the derivative of the voltage at the steady state with respect to the current. Clearly, the steady state ;~ stable if all the solutions of (3.7) have negative real parts, and unstable if there is at least one solution with Re o-> 0. The main result of this section is

Theorem. Let E(x; J)be a solution of (2.1), (2.2), and 4' the corresponding voltage. Let P* be the number of eigenvalues - o - with ~, > 0 of H (with $ = 0 at x = 0, I), and let N* be the number of solutions of (3.7) with positive real parts. Let lm Z(iw) 4 0 all real nonzero w. If $ ' ( J ) > 0, N* = P*. As a consequence, let Jc be the current at which E(x; J) loses stability under current bias (~0 = 0 if J = J c), and let $c be the voltage at which E(x; J(ch))loses stability under voltage bias (Re o-0 = 0 and all other solutions of (3.7) have negative real parts if $ =~bc). &c and J~ are related by (2.3). Let now 4 ~ ' ( J ) + R < 0. Then N* = P* + sgn[w Im Z(iw)]. As a consequence, a steady state with $ ' ( J ) + R < 0 is always unstable under the bias (2.3) if s g n I m Z ( i w ) = s g n ~ o , whereas it may be stable under (2.3) if

×(($~'exp(-~]

v ( E) ) d2x 6 )

+ ~,(0) - ~ ( l ) exp

× ~

o-+p-

/(o'-

o',,).

J0

28

(3.9)

Here - o ' , and 0 , ( x ) are eigenvalues and eigenfunctions of the operator H in (3.5) with zero-data boundary conditions; ( f , g ) = f~z)f(x)g(x) dx. (3.9) is obtained by solving (3.4) by an eigenfunction expansion and then inserting the result in (3.8). A similar formula was first used by Knight and Peterson [12], to discuss stability about the moving solitary wave solution. We now prove the theorem. First of all, let us consider the case Z(0) = cfl'(J)> 0 (e.g. the steady states of sections 2.1 and 2.2 verify this condition). Let C,~ be the positively oriented boundary of the right half-plane Re g > 0. Let Cf be the corresponding curve in the plane f(~r)=Z(~r)+R. Because of the lemma, Cf is invariant against reflection with respect to the real axis: if a point

470

L.L. Bonilla, F.J. Higuera / Gunn instability infinite GaAs samples

belongs to C t, so does its complex conjugate in the complex f-plane. Assume s g n I m Z ( i w ) = - s g n w , w e ~ . The lemma implies that Cj. is a positively oriented curve that joins the points R and Z(0) + R on the positive real axis, and it does not contain f = 0 inside. Then the principle of the argument implies that f(o-) has an equal number of zeros and poles inside C~, i.e. with positive real parts. The poles of f ( ~ ) are all the ~r,,'s plus ~r= _ p - 1 < 0. Similarly, when 4"(J) < 0 so that R + Z(O) < O, Cr is a negatively oriented curve that encloses jr'= 0. Then the winding number of the image curve Cf with respect to the origin is - 1 , and therefore the number of positive poles of Z(~r) surpasses that of zeros of Z(~r)+R with positive real part in one unit (principle of the argument). When sgn Im Z(iw) = sgn w, the orientation of C r is inverted in the previous arguments. Repeating them, we find N * = P * when Z ( 0 ) > 0 and N* = P* + 1 when Z(0) + R < 0, which completes the proof of the theorem.

Remark. The claim in ref. [2] that sgn Im Z(i6o) = - sgn w was based upon considering the operator ( H 2 + w 2) with zero-data boundary conditions to be positive. Although this is true when H is a matrix, it is not necessarily so for the present general case. Thus it is necessary to consider all possible signs of w Im Z(i~o) in the theorem. It is also possible that Im Z ( i w j ) = 0 for one or more

wj>O. Nonvanishing zeros of Im Z(i6o) may yield additional ways of destabilizing the steady states, as follows from numerical calculations of Z(io~) for piecewise linear velocity functions u ( E ) (ref. [20], chapter 6). The idea is as follows. For small enough voltage (correspondingly, small enough current), R = 0 , and Ec
zeros of Im Z(ito) appear for ~o > 0. Fig. 8 shows that the motion of one zero of Im Z(iw) through the imaginary axis increases in two units the winding number of the image curve C~ with respect to the origin. The principle of the argument then implies that two zeros of f(o-) cross the imaginary axis into the right half plane. If the scenario depicted in fig. 8 is realized when the voltage surpasses a critical value d~*, the two zeros that cross the imaginary axis at ~b = 6 " are either or= 0 (a double zero), or (generically) Z(_+ i~o0) = 0, ( H o p f bifurcations). See figs. 6-18 and 6 - 1 6 of ref. [20] for numerical realizations.

3.2. Voltage-current characteristics Here we exploit our results for current bias and those of the previous subsection to discuss how the stationary states change with voltage, and to conjecture the form of the voltage-current characteristic diagram (which can be considered our bifurcation diagram. For time-periodic solutions, we represent the current given by the average of the left side of (1.9) over one period versus the voltage). First of all, we shall consider only steady states and show that small regions with negative differential resistance are possible under voltage bias. Thus it is possible to have current-voltage characteristics resembling c ( E ) of fig. la for long enough samples. It is important to note that we do not try to cover all possible cases; instead, we only characterize possible shapes of J versus based upon our knowledge of the direct problem (current bias). - F o r very small J, ~ / l ~ E~ and the steady solution of the inverse problem corresponds to stable stationary solution of the direct problem on the upper horizontal branch of fig. 6c. J grows with 6- A simple calculation shows Im Z ( i w ) ~ - w l / ( 1 +w2), so that this steady state is also stable under voltage bias according to the theorem of section 3.1. - F o r moderate l and growing 6, the point representing the stationary solution under voltage bias leaves the horizontal branch in fig. 6c before

47l

L.L. Bonilla, F.J. Higuera / Gunn instability in finite GaAs samples Imf

Imf

Ref

Ref

(a)

(b) Imf

Re f

(c) Fig. 8. (a) Curve Cf for Z(0) > 0 , s g n l m Z ( k o ) = - s g n w : N*=P*. (b) Same as in (a). Two zeros of I m Z ( i w ) are created: N* = P * . (c) As in (b), but now the two zeros cross to the right plane: N* - P * = 2.

multiplicity of I(A) appears for the first time. O n the horizontal branch, ~b'(J) > 0. W h e n ~b (or J ) grows further, our stationary point will be on the lower b r a n c h of fig. 6c for E c < E 2 ( J ) and on the lower b r a n c h of fig. 6d for E c > E2(J). In both cases, ~ b ' ( J ) > 0, and since the transition is con-

tinuous, we are led to a m o n o t o n e increasing c u r r e n t - v o l t a g e characteristic. - F o r larger values of l, the direct problem has multiple steady states when J is such that E c and E 2 are close e n o u g h (cf. section 2.3). We now

472

L.L. Bonilla, F.J. Higuera / Gunn instability in finite GaAs samples

P(0)

Jo

the inverse problem corresponds to a point on the lower branch of fig. 6c.

Jl > Jo J2 ~ J1

J~J2

Fig. 9. Charge at the cathode - p ( 0 ) versus / for different values of the current. The vertical line corresponds to the sample length.

show how it is possible that the inverse problem may have a region of negative differential resistance, that is, a region where the slope of the current-voltage characteristic is negative. Clearly, the point representing the stationary solution of the inverse problem will still be on the upper horizontal branch of fig. 6c for 4~/1 small. There O ' ( J ) > 0. Above a certain value of da/l (corresponding to J = J2 in fig. 9, when the leftmost turning point of fig. 6c is reached at l = length of the sample, the stationary solution of the inverse problem moves to the intermediate branch of fig. 6c. Let us show that while the steady state solution is on this branch, we have 4 " ( J ) < 0. For current bias a new pair of steady states now appear corresponding to the two lower branches in fig. 6c. Except at the turning point, these solutions have larger values of 4~ than the solution on the upper branch of fig. 6c. The stationary solution of the inverse problem moves from the leftmost to the rightmost turning point of the intermediate branch of fig. 6c as c~/l grows. From section 2.3, we find that both turning points in fig. 6c move downwards to the right when J grows (since the flight time becomes longer for the trajectories corresponding to them in the phase plane). Then J diminishes from J2 t o J l in fig. 9. We have thus shown that J is a decreasing function of 4~ (l fixed) when the steady solution of the inverse problem is on the intermediate branch of fig. 6c. This implies Z(0) = &'(J) < 0. Similarly, we can show that Z ( 0 ) > 0 when the solution of

We now use several well-known facts about the differential impedance for piecewise linear velocity curves [20], to guess how the bifurcation diagram might be when the branches of timeperiodic solutions are included. In ref. [20], Shaw et al. perform a linear stability analysis of the steady state of the inverse problem. They find that the voltage at which the steady state becomes oscillatorily unstable corresponds to a current slightly smaller than the solution of t,(pJ) = J (or, equivalently, E~ = E 2, see eq. 6-83 of ref. [20]). On the other hand, numerical simulation of the complete inverse problem [20], and our asymptotic analysis of paper II indicate that there is a branch of oscillatory solutions caused by solitary wave dynamics [2], starting at a voltage corresponding to E c = E 2 (asymptotically as voltage and sample length tend to infinity). Let us assume that the branch of time-periodic solutions caused by motion of domains (solitary wave dynamics) starts as a H o p f bifurcation from a steady state. Given the asymptotic character of our results (they hold as l >> 1), a detailed calculation of the H o p f bifurcation is needed to decide whether it is subcritical or supercritical. In either case, it is clear that the bifurcation voltage and the beginning of oscillatory solutions (analyzed in p a p e r II) are almost coincident, as it was already remarked by earlier authors (ref. [20], page 142). Thus the range of dynamic hysteresis between oscillations and steady states (which plausibly exists only if the H o p f bifurcation is subcritical) is exceedingly small for long samples, in sharp contrast with some theoretical predictions [24]. Predictions of dynamic hysteresis were based on the possibility of having solitary waves of vanishingly small amplitude (existing for currents J up to t' M in fig. 1) on the whole real line {24]. The presence of contacts (necessary to implement voltage bias) forbids the current to raise much above the solution of t~(pJ)=J for the oscillatory states. This usually precludes the observation of small ampli-

L.L. Bonilla, F.J. Higuera /Gunn instability in finite GaAs samples

steady

state

oscillatory

state

Fig. 10. Conjectured bifurcation diagram of the inverse problem (current-voltage characteristic under voltage bias): mean current (averaged over one period) versus d~/l.

tude solitary waves of a semiconductor with contacts (see ref. [20] and also paper II). We have depicted in fig. 10 a plausible bifurcation diagram considering the arguments written above. We depict an increasing current-voltage characteristic for the steady states because regions with negative differential resistance are rare in the parameter space. The vertical H o p f bifurcating branch of oscillatory states could be either supercritical or subcritical in its fine structure, depending on detailed calculations not performed here.

4. D i s c u s s i o n

We have determined the stationary states of the usual model for the Gunn instability with realistic boundary conditions at the contacts and different biases. To complete a bifurcation diagram, we need to discuss the possibility of other relevant solutions. U n d e r current bias, we do not expect any more stable solutions such as timeperiodic or chaotic attractors. This follows from a general result of Hirsch's on strongly monotone flows [10], of which, that defined by (1.9) and (1.10) with constant current is a particular example. Roughly speaking, Hirsch's result shows the only attractors to be stationary solutions. Under the global constraint of voltage bias, Hirsch's result does not hold, and time-dependent attractors are possible. In fact, they will be analyzed in p a p e r II.

473

Hirsch's result shows that the present model cannot explain Gunn's observation of random firing of domains from the cathode to the anode of a semiconductor sample under current bias [8]. Gunn observed random firing of domains under voltage bias conditions (low impedance in the external circuit) for very large samples ( L > 0.2 mm) and also for shorter samples under current bias conditions (high impedance in the external circuit) [8]. In all cases very large electric fields at the cathode were necessary conditions for domains to be formed. Our results suggest that the phenomenon of random firing of domains under current bias conditions may be caused by fluctuations of the current [7], for a sample so large that multiple stableStationary states are possible. A possible mechanism follows: - M o v i n g J means moving the field at the contacts, thereby allowing a sudden loss of stability of, say, a low-field steady state in favor of a high-field state (consider, e.g. fig. 6c). - T h e r e is numerical evidence that such a disturbance travels from cathode to anode. If the high-current fluctuation lasts enough, a domain will be created after the current returns to the lower values where the low-field state is stable. We plan to test this scenario in future work. Consider now voltage bias (or a bias with a nonzero resistance R). Our results, together with the standard linear stability analysis of steady states of the model with piecewise linear L,(E) [20], suggest that H o p f bifurcation may be responsible for the Gunn effect in long samples (fig. 10). For short samples, or contacts with small resistivity, or curves c(E) with a short region of negative mobility, only one stationary state is possible under either current or voltage bias (see ref. [20] for a minimal length criterion based on linear stability analysis). This may lead us to think that no other stable solutions of the inverse problem are possible. On the contrary, numerical evidence shows the possibility of time-periodic solutions corresponding to the creation, motion and annihilation of solitary waves (or of traveling fronts, i.e. monotonic waves joining two plateaus.

474

L.L. Bonilla, F.J. Higuera / Gunn instability in finite GaAs samples

They are f o u n d when the contact resistivity is smaller than a critical value).

Acknowledgements The authors thank A. Lififin for valuable discussions and J.M. Vega for valuable discussions and for pointing out the results of ref. [10]. This work has b e e n s u p p o r t e d by D G I C Y T grants E S P 1 8 7 / 9 0 and PB89-0629 and by N A T O traveling grant C R G 900284.

Expressions (AI.1) and (A1.2) are also valid in the cases J > t" M and 0 < J < v m. Even when J is slightly larger than t'M, and l is large enough, we can use (AI.1). T h e r e is a large region in x, of width • ( [ J VM] j/z) w h e r e E - E M = G ( [ J - UM]1/2) and E x = ~ ' ~ ( J - UM). We now c o m p u t e the C o u r a n t - W e y [ integral (2.12). Notice that for B = G(1), V2/43 = (e2(1/(3), so that W(x; 3) in (2.11) remains positive unless t,'(E) E , < 0 and of order 1/6. Since v'(E) = ~ ( 1 ) (even as E ~ m), we need E x = ~(1/a). This is possible only in the b o u n d a r y layer (A1.2), where

[- W(x,3)/31l/2 dx Appendix 1. Stationary state for B = ~f(1) as ~ L 0 and its stability Let p > 0 (the case p = 0 was treated in ref. [2]). As 6 $0, we approximate (2.1), (2.2) in 0 < x < l by means of

v(E)(E~+I)=J,

1( ~

~

-2v

,

(E)fJ EEI v ( s ) d s __/,2(E)) 1/2

t:Ev(s) ds,

64

3 $0.

Then, for E between E c and E~,

E(O)=Ec=pJ.

This is equivalent to

1

I(6) ~ ~w fw
(A1.1)

E~

Let us consider the richest possible case, u m < J E3, E(x) in (AI.1) monotonically decreases from E c to E(l) which a p p r o a c h e s E 1 or E 3, respectively, as l--* ~. 1 b o u n d a r y layer is n e e d e d for E to grow from E(l) to E c. Let E l = E(l). In all cases the solution at the b o u n d a r y layer is

f~

v( s ) ds.

(11.3)

Notice that for E c > E 3, v ' ( E ) E , > O in the b o u n d a r y layer, so that 1 ( 3 ) = 0, and the steady state is stable. W h e n e v e r v ~ E ( E / s m a l l enough, for Ec < El), evaluation of (A1.3) yields

1(6) ~ ½{~/2 - ( 2 / w ) × t a n - ' ( ~ f 2 cot s i n - l [ EE/(~/2

El)l) ) ~ l,

~ v ( s ) ds ~ 6E,, (A1.4)

and~-fEds/fSv(r) Ec

/

dr. El

(A1.2) and the steady state is stable. Numerical evalua-

L.L. Bonilla, F.Z Higuera / Gunn instability in finite Go.As samples tion of 1(3) for E c < E 1 yields a smaller value than (A1.4). T h e same thing h a p p e n s w h e n E, <

ceases to hold. This corresponds to

E c < E 2 and w h e n E 2 < E c < E s. Two contributions to 1(3) instead of one occur when J > t , M and E c < E M if l is large e n o u g h ( E l close to E3): v'E, < 0 for E c < E < E M and for E m < E < E t < E 3 (Et close to E3). Numerical evaluation still yields 1(3) < 0.5 < 1. F r o m this analysis we conclude that the stationary state (AI.1), (A1.2) is always linearly stable if B = G(1) as 6 $ 0. If we want to destabilize the steady state, we n e e d further contributions to 1(3). This is possible if B is so small that v 2 / 4 6 b e c o m e s c o m p a r a b l e to t"E, even o u t s i d e . t h e b o u n d a r y layer.

( 6 j B _ 5 / 4 ) 1 / 2 ~ [ 2 B _ l / 2 ( X m _ Xm) ]1/2

A p p e n d i x 2. Stationary state for B = ~ ( ~ 4 / 5 )

or ( x tm --Xm) ~ 516 J B -3/4 << 1.

475

(A2.3)

Beyond x m (A2.1) is again valid, so that

x - x ' = ~ v ( s) / [ J - v( s)l E~ - B-'/4e(~'m).

(A2.4)

Finally, near x = l, the usual right b o u n d a r y layer is to be inserted. O f all the above m e n t i o n e d stages only the right b o u n d a r y layer and the stage (2.15) contribute to (2.12). Insertion of the scaling (2.13) in (2.12) yields, after a change of variable in the integral,

as 6 $ 0 and its stability

I,(j) = (2"rr) - I fr~ < 0 J 2 l / Z e - 3 p - ' d e ' Let

us

assume J > VM, E c < E 3 and B = ~(6 4/5) as a $0. T h e diffusivity may be ignored for 0 < x < x m, where

X m ~ f~mu(s)/[J

-- U(S)] ds.

(A2.1)

for

0
(A2.5)

which is the contribution to I(6) of the stage given by (2.15). Numerical evaluation of p(e) and of (A2.5) shows I,(j) to be a m o n o t o n e increasing function. Some values are

W h e n E is near the m i n i m u m of v ( E ) , the scaling (2.13) yields the distinguished limit where (2.15) holds. We need to match the solution of

v( E) ( E x + 1) = J

g~ = - 2 p ( e 4 - 3)e 2 - (1 + e 4 ) 2,

(A2.2)

to that of (2.15). T h e only trajectory of (2.15) that has p $0 as e $0 and as e---)oo is the one that leaves the origin and b e c o m e s tangent to the nullcline p =je3/(1 +e 4) as e--+ ~ (fig. 4). As ~ = - ( X - X m ) / B '/2--+ - ~ , e tends to 0, and as ~ + ~, e ~ + ~. O n the nullcline, p = e~ ~ j/e, which implies e ~ (2jsc) 1/2, as sc--+ + ~ . Notice that after a certain S~=S~m, j / e ~ 1 and (2.15)

•,(3) = 0.2273, I,(100) = 0.4395,

I,(10) = 0.3558, 11(500 ) = 0.4638.

Next we show that I , ( j ) T 0 . 5 as j--+ +0o. T h e n 0 < I,(j)<_ 0.5 for all j > 0. Let us evaluate p(e) for large j. For large e,

d p / d e ~ ( pe - j ) / p , or dZe/d~ :z ~ e d e / d s ~ - j which implies d e / d s ~ ~ leZ - j s ~.

(A2.6)

(A2.6) may be r e d u c e d to Airy's equation by means of the substitution e = - 2 q J q . The solution is e = - ( 4 j ) ' / 3 A i ' ( s ) / A i ( s ) , where s = ( j / 2 ) 1 / 3 ~ . AS e,l, 0, p-+j2/3po, where P0 is a positive constant of o r d e r 1. Since P0 =~ 0, we

L.L. Bonilla, F.J. Higuera / Gunn instability in finite GaAs samples

476

n e e d to insert a n o t h e r stage w h e r e e b e small. Then

dp/de

~ e -3,

or p =j2/3p11 _ , ye 2 ,

[7]

(A2.7)

which m a t c h e s the p r e v i o u s stage as e ~ +oc. Notice t h a t p = 0 w h e n e = e o =-j 1/3(2po)-1/2. F o r e < e 0, p b e c o m e s a s y m p t o t i c to je 3. In fact, p = j e 3 for 0 < e < e 0 since d p / d e = e ~ ( j 1/3)<< e -3 - j / p = el(j) as j ~ + ~ . B e t w e e n (A2.6) and (A2.7) we n e e d to insert a c o r n e r layer which d o e s not c o n t r i b u t e to l~(j). In fact, only the stage (A2.7) c o n t r i b u t e s to ll(j),

[8]

[9]

[10] [11] [12]

( 6 e 2 p o - 4) 1/2 d e e2po

= 2 1,

asj~

+,c.

12

[13]

e (A2.8)

N u m e r i c a l evaluation of the boundary layer contribution to (2.12) shows it to be smaller than 0.4. Thus 1 ( 6 ) < 1, and the stationary state built in this appendix is stable.

References

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[15]

[16]

[17] [18]

[1] K. Aoki and K. Yamamoto, Firing wave instability of the current filaments in a semiconductor. An analogy with neurodynamics, Phys. Lett. A 98 11983) 72-78. [2] L.L. Bonilla, Solitary waves in semiconductors with finite geometry and the G u n n effect, SIAM J. Appl. Math. 51(3) (1991). [3] L.L. Bonilla, F.J. Higuera and J.M. Vega, Stability of stationary solutions of extended reaction diffusion-convection equations on a finite segment, Appl. Math. Lett. 4 (1991) 41-44. [4] L.L. Bonilla and S.W. Teitsworth, Theory of periodic and solitary space charge waves in extrinsic semiconductors, Physica D 511 (199l) 545-559. [5] P.N. Butcher, The G u n n effect, Rep. Prog. Phys. 3{1 (1967) 97-148. [6] M. Bfiniker and H. Thomas, Bifurcation and stability of

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dynamical structures at a current instability, Z. Phys, B 39 (19791 301-311. A. Diaz-Guilera and J.M. Rubi, On fluctuations about nonequilibrium steady states near G u n n instability. 1. Density and electric field fluctuations, Physica A 135 (1986) 180-199; 11. Voltage fluctuations, Physica A 135 (19861 200 212. J.B. Gunn, instabilities of current and of potential distribution in GaAs and lnP, in: Proc. Symp. on Plasma Effects in Solids (Dunod, Paris. 19651 pp. 149 207. J.B. Gunn, Electronic transport properties relevant lo instabilities in GaAs, J. Phys. Soc. Japan (Suppl.) 21 ( 19661 505-507. M.W. Hirsch, Stability and convergence in strongly monotone flows, preprint (19841. B.W. Knight and G.A. Peterson, Nonlinear analysis of the G u n n effect, Phys. Rev. 147 (1966) 617-621. B.W. Knight and G.A. Peterson, Theory of the Gunn effect, Phys. Rev. 155 (1967)393-404. H. Kroemer, Nonlinear space-charge domain dynamics in a semiconductor with negative differential mobility, IEEE Trans. Electron Devices 13 (1966)27-4t/. D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Pergamon Press, Oxford, 1984). K.M. Mayer, J. Parisi, J. PeJnke and R.P. Huebener, Resonance imaging of dynamical filamentary current structures in a semiconductor, Physica D 32 (19881 306-317. J.D. Murray, On the G u n n effect and other physical examples of perturbed conservatkm equations, J. Fluid Mech. 33 (19701 315 346. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4 (Academic Press, New York, 19791. K. Seeger, Semiconductor Physics, 4th Ed. (Springer, Berlin, 1989). D.G. Seller, C.L. Littler and R.J. Justice, Nonlinear oscillations and chaos in n-lnSb. Phys. Lett. A 11"18(19851 462 467. M.P. Shaw, H.L. Grubin and P.R. Solomon, The G u n n - H i l s u m Effect (Academic Press, New York, 1979). S.M. Sze, Physics of Semiconductor Devices, 2nd Ed. (Wiley, New York, 19811. P. Szmolyan, An asymptotic analysis of the G u n n effect, preprint (1989). S.W. Teitsworth, 3"he physics of space charge instabilities and temporal chaos in extrinsic semiconductors, Appl. Phys. A 48 (1989) 127-136. A.F. Volkov and Sh.M. Kogan, Physica p h e n o m e n a in semiconductors with negative differential conductivity, Soy. Phys. Usp. 11 (1969) 881 903.