On the stability of wavefronts and solitary space charge waves in extrinsic semiconductors under current bias

Volume 156, number 3,4

PHYSICS LETTERS A

10 June 1991

On the stability of wavefronts and solitary space charge waves in extrinsic semiconductors under current bias Luis L. Bonilla Departainento Estructura y Conslituyenles de la Materia, Universidad Barcelona, Diagonal 647, 08028 Barcelona, Spain

and José M. Vega E. T.S.J. Aeronduticos, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Received 5 February 1991; accepted for publication 5 April 1991 Communicated by A.R. Bishop

Stability of both wavefronts and solitary waves under current bias is studied for a standard rate equation model of electrical conduction in extrinsic semiconductors which includes effects of field-dependent impurity impact ionization. We prove that monotonic traveling wavefronts are linearly stable whereas solitary space charge waves are unstable on the infinite one-dimensional line.

In a recent paper [1], Bonilla and Teitsworth have constructed different traveling wave solutions (periodic and solitary waves and monotone wavefronts) to a standard rate equation model of electrical conduction in extrinsic semiconductors which includes effects of field-dependent impact ionization [21. In ref. [1] it is proved that the periodic wavetrains are unstable under current bias, whereas instability ofthe solitary waves (domains) is only conjectured. Here we prove this conjecture and something else: monotonic wavefronts are stable under current bias. This property converts monotonic wavefronts into possible candidates for explaining current instabilities under large enough dc voltage bias in long enough semiconductor samples, although the mechanism responsible for current instabilities seems to be different in experiments with the currently available samples [3]. The model equations are 8a~/8T=ya0+p(ica0—ra.~), a0 + a,1,

a ( = const),

eôE/ôT=J~01(T)—e(pvd —D8p/ÔX), 0375-960l/9l/$ 03.50 © 1991

—

(lb)

(ic)

Here E denotes the electric field, and a0 a a~,a~ and p represent the neutral acceptor, ionized acceptor, and free hole concentrations, respectively. The total concentration of acceptor impurities is a constant a. In (1 a), ya0 describes the generation of ionized from neutral acceptors by thermal and farinfrared radiation (FIR) ofappropriate wavelength; y is proportional to the total photon flux. The rate of impact ionization of neutral acceptors is pica0, and the rate of hole recombination onto ionized acceptors is pram. (lb) is Ampere’s law, describing the balance between the hole current e(pvd—Dôp/ÔX) and the displacement current eOE/ÔT. J~0~ is the total current, equal to the current flowing through the cxternal circuit because the displacement current in a metallic wire is negligible. Because of this identification, .J~01is sometimes called the external current. Finally, (1c) is Poisson’s law for the electric field, in which the electric charge is given by the free hole, compensating donor (d), and ionized acceptor concentrations while e is the semiconductor permittivity [21. The hole diffusivity D is taken to be indepen—

.

(1 a)

—

~8E/ôX=e(p+d—a~).

Elsevier Science Publishers B.V. (North-Holland)

.

.

.

.

179

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dent of the electric field and approximately determined by the Einstein relation D=kBTO/2O/e, where 1Lo=dvd/d~is the mobility at E=0, and T0 is the lattice temperature. The drift velocity of the free holes, Vd, and the recombination and impact ionization coefficients, r and ic, are all functions of the local electric field E [1,2]. In ref. [1] a theory of waves traveling on an infinite one-dimensional sample (with constant current bias) was obtained by using the large separation between the dielectric relaxation time and the characteristic time of ionization of the impurities: the impurity ionization is a much slower process than dielectric relaxation. Ifwe nondimensionalize eq. (1) so that time is measured in the slow scale ofimpurity ionization, the following approximate equation can be derived for the evolution of the nondimensional electric field [11, 2E/aia~—(J/V)’aE/a~ a 2 —J[ (CIV)’ + k/ V] ôE/ô~ = Ca ~E/~ +Jk(J—pV)/V 2 (2a) .

Here the characteristic electric field is chosen to be v~/j~,the ratio between the saturation velocity of the carriers (the limit of vd as ~—~oo)and the zero-field mobility (dv~/dE),~o: (2b)

~

The independent variables are the time ~ and the moving coordinate ~=x—Ct, C> 0 being the wavespeed. The coefficients V, k andpare nonlinear functions of the electric field E(~, t) uniquely determined by Vd, rand ic [11, V(E)=vd/vS, (2c)

10 June 1991

[1]. Thus a necessary condition for the waves here described to exist is that the curve J=p(E) V(E) has a negative slope (the well-known negative differential resistance, NDR). Since the slope ofthe velocity curve is always positive, (2e) implies that the NDR is either caused by a negative differential impact ionization, dic/d~<0, or by a positive differential recombination coefficient, dr/d.E> 0 [11. For stationary solutions the left side of (2a) is zero and they correspond to traveling waves of the dcctric field. Three kinds of such solutions were found in ref. [1]:periodic wavetrains, solitary waves and monotone wavefronts. In the phase plane (E, dE/d~) they correspond to closed orbits, homoclinic and heteroclinic connections, respectively. We proved in ref. [1] that periodic wavetrains (closed orbits in the phase plane) are unstable solutions of (2) and conjectured orbits in the phase that planesolitary joiningwaves either (homoclinic (E 1, 0) or (E3, 0) with are alsopaper unstable. We prove ment itself) in the present and also prove this that statemonotone wavefronts (heteroclinic connections in the phase plane joining (E 1, 0) and (E3, 0), which only exist for unique values ofJ and C) are linearly stable solutions of (2). To proceed, let E(~)be either a solitary wave or a monotone wavefront. Let then .

E(~,r)=E(~)+eê(~)exp(AT), <
The linearized equation for

e(c~)is

.

(3)

Cê~+ [A+ft (~)Ie~ + [Af2( ~)+f3 ( ~) ] ê~0~ (4a) f, (~) J[ (CIV)’ + k/ V~, (4b) 2, (4c) —

f2(~)_=JV’/V f 2]’}, 3(~)_=J{[(C/V)’+k/V]’E~—[k(J—pV)/V

k(E)=(ic+r)/r 0,

p(E)=—l+aic/d(ic+r).

(2d) (2e)

In (2a), V’ (E) means dV/dE, etc. It is important to keep in mind that V(E) is monotone increasing, k(E)>.0 forfield, E>0,soand pVis an interval N-shapedoffunction of the thatthat there is an currents J, (Jm, JM), for which three constant solutions of (2) exist. We denote them by E~(J),j= 1, 2, 3, with E 1 (J)
(4d) Here subscripts mean differentiation with respect to the corresponding variable. Let ~be the spectrum (=the set of eigenvalues of finite multiplicity plus that the essential spectrum) of eq. 2(F). Notice A=O is an eigenvalue of (4) in L E~(which decays exponentially to zero as (4), with çe~~* ±oo) as an associated eigenfunction. If A =0 is a simple eigenvalue of (4) and the remainder of the spectrum has real part smaller than fl< 0, then we say that the wave E(~) is linearly stable. If, instead, the

Volume 156, number 3,4

PHYSICS LETTERS A

real part of a portion of the spectrum is strictly positive, then we say that the wave E(~) is linearly unstable. First of all, we prove the following Lemma 1. Let A be a complex number such that

10 June 1991

Q(c~)=[J(CV’ —kV)/2CV2]2 —f[(CV’ —kV)/2CV2]’E~ —f[k(f—pV)/V2]’/C.

(8d)

Because of (6) and (7), lemma 1 and a similar result for eq. (8) imply the existence of a one-to-one

Re A f

2 ( ±cc) +f3 ( ±cc) >0.

(5)

Then (i) A does not belong to the essential spectrum of (4), and (ii) ifA is an eigenvalue of (4) satisfying (5), then the associated eigenfunctions satisfy

e( c~)exp{

—

[A+ f~ ( ±cc)]~/2C}

(exponentially)

as

~—+

0

—~

±cc.

Proof By standard results from 2spectral theory with [4], (F) coincides the essential spectrum of (4) in L that of the equation obtained by replacingf ( ~)(1= 1, 2, 3) instead off(~)in eq. (4a), withf(t~)=J)(+cc) if t~?0and f(c~)=f(—cc)if t~<0.By e.g. ref. [5], lemmata 1 and 2, pp. 137—139, the essential spectrum of the modified equation is readily seen to satisfy property (i). Property (ii) follows straightforwardly from examination of the characteristic equation associated with (4) in the limits ±cc.

correspondence between the eigenvalues of (4a) satisfying (5) and the corresponding eigenvalues of (8a). Thus the stability properties we will obtain by analyzing (8a) will be directly applicable to the physically relevant equation (4a). (8a) is a somewhat unusual Schradinger equation since the eigenvalue enters quadratically in it. Although we cannot guarantee the usual properties (such as all the eigenvalues have to be real) to hold, we can prove Lemma 2. Eigenvalues of (8) with Re A> a (see 8c) are necessarily real. —

Proof Let us multiply (8a) by the complex conjugate of its solution, ~P”,and the complex conjugate of (8a) by ~P,subtract both equations and integrate over ~ from cc to + cc. The result is (A _A*) [A +A* + ] = 0, (9a) —

~—+

where Since from (4c) and (4d) we see that f(±cc)>0, j=2,3,

(6)

lemma 1 implies that the essential spectrum of (4a) plays no role on the stability of the wave E( ce). To analyze the eigenvalues of (4) it is convenient to perform a change of variable that eliminates the first derivative:

e(~)= W(~) xexp[~(At~+J$ [(C/V)’+k/V]dt~)].



$

WIWI2d~/

$

I~I2d~>0.

(9b)

The lemma follows immediately from eqs. (9). Since all unstable eigenvalues are real positive if they exist at all, according to lemma 2 we can restrict our study of stability to the case of real A. Consider the following auxiliary equation,

(7) (10)

Insertion of (7) in (4a) yields the eigenvalue problem, —

!P~+[A2+ w(5)A +Q(~)] !P=0,

(8a)

where, for real A, p(A) is the lowest eigenvalue of (10) inL2(P).Noticethat (l0)isastandardSchrödinger equation whose eigenvalues are necessarily real,

(8b)

that ~i(A) is a simple eigenvalue of (10), and that the associated eigenfunction does not vanish in —cc<~.’c+cc,seeref. [6].Whenlemmalistaken

where

w(t~)=J(CV’+kV)/CV2>-a>0,

(8c)

into account, we find that E(i~)is nearly unstable if there is a strictly positive zero of,u(A). Similarly, if 181

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no strictly positive zero of,u (A) exists, then the corresponding traveling wave is linearly stable. The shape of u(A) is given in the following

10 June 1991

Proof The function A-+u(A) is readily seen to be differentiable if A ~ 0 (redefine (10) in an appro-

Since ~ (A) is monotone increasing and ji (A) cc as A—~+ cc (consider the asymptotic behavior ofthe cigenfunctions of (10) as ~—+±cc), the ground state of (10), ~ (A), has one positive zero, and the solitary wave is unstable. For monotone wavefronts, u (0) =0 (because E~ does not vanish), and all possible zeros of the ~(A ) ‘s are negative. Thus monotone wavefronts are linearly stable under current bias.

priate abstract setting and apply the implicit function theorem). Also, ji’ (A) satisfies the following equation, obtained upon differentiation of (10) with respect to A, 2 Q A ~

We are indebted to Professor S.W. Teitsworth for useful comments. This work has been supported in part by the DGICYT under grants ESP187-90 and PB89-0629 and by the NATO traveling grant CRG

Lemma 3. d~/dA= 2A + ,so that j~(A)is monotone increasing for positive A.

—

4~+[A +w(~)A+ (~~)—it()]

900284

,i

=[it’(A)—2A—w(~)]W. (11) Let us multiply (10) by ~ (11) by W, subtractboth equations and integrate the result from cc to + cc. We then find the formula written in the lemma. —

From lemmata 2 and 3 the instability of solitary waves and the stability of wavefronts can immediately be proven. In fact, at A=0 an eigenfunctionof (8a) associated with the zero eigenvalue is /

W0 ( ~) E~exp(

—

~(f/C) jr [(Ci

J.

V) + k/ V] d~ /

For a solitary wave, E~has one zero, so that ~ corresponds to the first excited state of (10) for A = 0. Thus there is a ground state of (10) with ~t(0)<0.

182

—~

References [1] L.L. Bonilla and S.W. Teitsworth, Theory of periodic and solitary space charge waves in extrinsic semiconductors. Physica D (1991), in press. [2] S.W. Teitsworth, Appl. Phys. A 48 (1989) 127. [3] A.M. Kahn, D.J. Mar and R.M. Westervelt, Phys. Rev. B (1991), to be published. [4] M. Schechter, Spectra ofpartial differential operators (NorthHolland,Amsterdam, 1986). [5] D. Henry, Lecture notesin mathematics, Vol. 840. Geometric theory of semilinear parabolic equations (Springer, Berlin, [6] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV. Analysis ofoperators (Academic Press, New York, 1978).