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Propagation of a Gunn wave under the influence of spatially homogeneous random perturbations
20 May 1996
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 2 14 (I 996) 30 l-306
Propagation of a Gunn wave under the influence of spatially homogeneous random perturbations F.G. Bass a Depurtmrnt b Semiconductor
oj’Physics, Physics
Received 8 December
Bur-lion Institute,
a,R. Bakanas b University. Gostuuto
52900
Rumut-Gun,
II. LT-2600.
1995; accepted for publication Communicated by L.J. Sham
Isrurl
Vilnius. Lithuuniu 21 February
1996
Abstract The propagation of Gunn layers (GL), i.e. the nonlinear waves of an electric charge distribution in a dissipative system, under the influence of random perturbations, which may be characterized as white Gaussian noise, is considered. The average characteristics of the perturbed GLs are analyzed and compared to those describing a “random walk” of kinks in a damped system [F.G. Bass, Yu.S. Kivshar, V.V. Konotop and Yu.A. Sinitsyn, Phys. Rep. 157 (1988) 631.
1. Introduction. The influence of small perturhations on a nonlinear wave
may be described following type, au/a7+
The propagation of nonlinear excitations under the influence of regular as well as random perturbations has been intensively investigated theoretically in the last few years, mainly in the case of conservative systems [l-3]. The nonlinear excitations in essentially dissipative systems, which can be described by nonlinear diffusion equations, are widely known in various fields of physics concerning superconductivity, solid state physics, plasma physics, biophysics etc. [4]. The analytical description of such excitations, spatially localized waves, under the influence of temporal as well as spatial perturbations is rather complicated even in the presence of small perturbations. In the simplest situations the evolution of such excitations under the influence of small perturbations 037%9601/96/$12.00 Copyright PII SO3759601(96)00192-2
by a diffusion
AL4= pf( y, 7).
where A is the nonlinear ii = -da2/ay2
equation
of the
(1) operator
+ p( u)a/ay
+ r( q
(2)
and the torque fly, T) describes a small ( p -=z 1) perturbation. By taking the explicit expressions of P(U), T(U) and allowing /3 = 0 in I!$. (1) one may obtain the unperturbed solutions uO( y) that describe the free waves. Here, y denotes the moving coordinate, y = x - CUT, where c0 is the velocity of the free wave. Thus, Eqs. (1) and (2) are written in the moving coordinates. To obtain the solution u( y, T) of Eq. (1) in the presence of a small disturbing torque f we apply a perturbation scheme that is based on the properties of the linear operator i
0 1996 Elsevier Science B.V. All rights reserved.
302
F.G. Buss, R. Bokunus / Physics Letters A 214 (I 996) 301-306
intimately related to the translational mode r(y) a du,(y)/dy,
ir=o, i=
-d-$+Q(y)&+U(y),
Q(Y)
=P[UO(Y%
U(Y)
=P'tY)dy
duetY)
+r'tyL
-k+t&
Q(y) -+O
if y+
4m.
-P(u,)](du,/d5+aAu/a5)
+ r( u0 + Au) - r( uO) + (ds/d+lAu/at - [ P’(UO) du,/d5+
A,Y,.
(5)
Owing to the fact that operator i is non-Hermitian it is possible to express the functions Y, through the eigenfunctions X, of the Hermitian operator Z?= S-‘( y)iS(y) [5,61,
(9)
In deriving (8) from (1) the relationship Au,,<5) = 0 has been taken into account. In the case of a small perturbation we expand the quantities S, Au and F in a power series of a small parameter p, +( 6, T) = c p”@“‘( 5, T), n= I
(6) In the presence of a small disturbing torque ( p -C 1) the solution of Eq. (1) u( y, T) may be presented in the following way, +Au(y,T),
By substitution of Eq. (10) into (8) the following evolution equation may be obtained in the p” order approximation,
duo(5)
Fds’“‘/dT+
a&&“‘(
where the parameter S(T) denotes the phase shift and Au( y, T) describes the distortion of the wave due to the presence of the perturbation f. Substituting (7) into (1) one rewrites Eq. (1) as follows,
T)
= F’“‘(
5,
T)/aT
T).
5,
(1’)
Expanding the functions Au(“) and F’“’ through the complete set of the eigenfunctions Y(I’ Au’“‘( 6, 7) = cT;“‘(T)Y,( (I F(“)( 5, T) =
Aueu,, (7)
4 = s, Au, F. (10)
+ LAu’“‘( 5,
u(~,T)=dy+s(T)]
+o)]Au.
(4)
Thus, the eigenfunctions Y, of the operator i make up a complete set and the corresponding eigenvalues h, are real, iv,=
(8)
where r= y -t S(T) is the comoving coordinate, i.e. the coordinate moving with the perturbed wave, and the additional “torque” g includes the nonlinear terms with respect to the small quantities ds/dT and AU, g= [~(uo+Au)
where the prime signifies the derivative with respect to uO, i.e. p’(y) = dp[u,( y)]/du,, etc. Here and in the following, an overline denotes a quantity intimately related to the translational mode. The existence of the translational mode evidently follows from the translational invariance of the operator R. We assume that
T>, S(T); u,(t)]>
&),
~F;“)(T)Y,( (I
(12)
5),
one rewrites Eq. (11) in the “L-representation” dT;“‘/dT+ =
(Y, 1du,/dc)
d.@‘)/dT+h,T,(“’
F:“)(T),
F:“‘(T)
= (Y(x)
1 F’“‘(
x,
T)).
(13)
Here, the following notation has been introduced, = F(Au,
s; uo; f),
(Y, IP>=
3cdxY,+(x)‘P(x, / --a
T),
(14)
F.G.
Bass. R. Bakunas/
where the dagger indicates the eigenfunction adjoint operator i’ = S- ‘Z?S, Y,‘(x)
=s-‘(x)x,*(x).
Physics
of the
(15)
Now, the evolution equations that describe the required functions, S(“)(T) and TA”), may be simply obtained from Eq. (13), if one takes into account that the contribution of the translational mode F(y) may be fully included into the phase shift S(“)(T). Thus, substituting d?“)/dr= 0 into (13), one obtains immediately S(~)(T) = (? Idu,/d5)-‘joid’F’“‘(r),
Letters
A 214 (1996)
of two types, GUM layers (accumulation and depletion waves) and GUM domains [7]. In describing the GWs by Eqs. (I), (2) and (17) we used the dimensionless units: the time T = r/rM is measured in the units of the dielectric relaxation time rM and the moving coordinate y = x/l, - co7 is scaled in the units of the characteristic length lo, which characterizes the electric field distribution over the specimen in the free GW case. The quantity I, denotes the size of the GW nucleus, i.e. it indicates the characteristic distance of the charge density localization, p(x) - du&x. t)/dx, in the free GW. The following dimensionless quantities are also introduced into Eqs. (l), (2) and (17): u = E/E,, w = u/u,, d= (1,/i,)‘,
T:“)(T)
=L’drFi”)(r)
exp[A,(t-T)].
( 16)
In deriving (16) from (13) we have supposed that the perturbing torque fly, T) has been “turned on” at the initial moment T= 0. Eq. (16) used in conjunction with (7), (10) and (12) enable us to describe the evolution of a nonlinear wave under small perturbations with the needed using (16) the stochastic accuracy. Additionally, characteristics of the wave may also be deduced in the case of a random torque fly, 7). The described perturbation scheme has been applied to the particular case of a Gunn wave influenced by random, spatially homogeneous perturbations.
2. The average characteristics
of a Gunn wave
In the case of a semiconductor sample with a N-type current-voltage characteristic Eq. (1) describes the evolution of the electric field distribution along the specimen u( y> [5-71. The disturbing torque f is induced by inhomogeneities of the doping profile in a sample or, in the case considered, originates from the current or voltage deviations in the external circuit. It is well known IS-71 that the functions p(u) and T(U) are simply related to the current-voltage characteristic w(u) of the specimen, ~(u)=hG~(U)=W(U)-Co.
303
301-306
f, =
(DTM)“‘,
hG
=
/&iv.
1,
=
TMup,
where the parameters E, and up correspond to the electric field and the electron drift velocity at the peak of the current-voltage characteristic and D is the diffusion coefficient. A simple theoretical model allowing the analytical description of free GWs has been proposed in Refs. [5,6], where the propagation of these waves under influence of the small, spatially localized perturbation has been examined. Analyzing the statistical characteristics of GLs influenced by the random perturbation we have used the model of a free GL. proposed in Ref. [5]. We suppose that the random perturbations are spatially homogeneous and are induced by current oscillations SJ(7) = J(T) - J, in the external circuit of the specimen. Here J, denotes the total current in the free GW case. We assume that J, = ca, thus. the “equal areas rule” holds when the perturbation SJ is absent. The disturbing torque f in our case is simply related to the current deviation 6J [5], f=
sJ(r),
(18)
and the perturbation scheme described above may be applied if the current deviations are small. In the case of Gaussian deviations 6J we write (6J(r))
= 0,
( 17)
The unperturbed solutions of Eq. (1) u& y> describe the free Gunn waves (GW), i.e. nonlinear excitations
Bjj=
(~J(T~)SJ(T~))
=
2~7~6(T~-
Tj),
(19)
304
F.G. Buss, R. Bakmns/Phy.sics
where r is an integer, C indicates the sum of correlatom Bjj covering all pair combinations of indices. The parameter u in (19) characterizes the external noise intensity and ( > denote the ensemble average, i.e. the average over all the possible realizations of
J(T). To describe the average characteristics of a GL distinctly one has to know both the eigenfunctions Y, and the corresponding eigenvalues A, explicitly. In addition, the solution of the unperturbed GL uO(y> is also required, as it follows from (7), (10) and (16). The needed expressions have been deduced in Ref. [5] with the current-voltage characteristic w(u) taken in the following form, W(U) =a+b
sin[y(u-F)],
(20)
where u, ,b, y, F are free parameters. With the help of (20), (1) and (2) the following field distribution u,(y) which describes the free GLs has been obtained in the strong diffusion case (I, 3 1,>, uO( y) = F + 2y-’
arctan sinh( *y),
(21)
where F=F+A/2 and A=27r/y=u,-u, denote the increment of the electric field in the GL. The quantities uM and u, (z+, > u,,,) indicate the extreme values 24, of the electric field in the GL. The signs + and - correspond to the accumulation and depletion wave, respectively. The function ?N y) = arctan sinh( k y) in (21) is defined as - 7r/2 < ‘SF(y) Q 7r/2. The characteristic distance describing the charge density distribution in the wave p(x) dZ?&x)/d x, i.e. the extent of the GL nucleus in the x-coordinates, is of the order of 1, = 1, AC, where A, = [ w’( ~,,)]-‘/~l~/l~ z=- 1. The needed functions Yg and the corresponding eigenvalues A, describe the continuous spectrum (a = k) and the single discrete eigenlevel X = 0 in this case, _?= 2-‘12 sech y, x,=
[2+*+
A=o,
l)] -“2(tanh
y-
ik) exp(iky),
h(k) = A,(/@ + l), S=exp[
+(yAo)-’
sech y] = 1,
(22)
where the parameter A, = by = w’!u,,) indicates the bottom of the continuous spectrum. To deduce the statistical characteristics of a GL influenced by a random perturbation SJ(T) explic-
Letters A 214 (1996) 301-306
itly one has to describe the statistical ensemble of GLs definitely, i.e. one needs to define the averaging procedure more exactly. Here we will be interested in the average characteristics of GLs that have “arrived” to a certain point of the specimen. Thus, following Kaup (see in Ref. [1]), the averaging “relative to the moving wave” has to be performed. The corresponding ensemble of GLs includes the waves of randomly distorted shapes exclusively. The averaging over irregular shifts of the phases S(T) is ignored in this case. Obviously, the “complete ensemble”, which implies the additional averaging over the random “displacements” of GLs S(T) too, describes an essentially different situation. It represents the infinite collection of the macroscopically identical specimens characterized by unique realizations of the random torque and, consequently, takes into account all irregular parameters of the wave. For instance, the averages over a “complete ensemble” have been analyzed in Refs. [8,9] to evaluate the statistical characteristics of KdV solitons under random perturbations. The average characteristics of a GL over a complete ensemble that describes the “random walk” of separate GLs will be described elsewhere. From Eq. (7) follows ~=U,(&)+=&&r),
(23)
where the overline denotes the averaged field, Z-&t, T) = (Au(~, 7)). Performing the averaging “relative to the moving GL” in (16) one gets with the help of Eqs. (18) (19) and (121, (rj”) = 0. Hence, m”(t, T) = 0, and the shape of the averaged field of the randomly perturbed GL lil( 5) = (u,( 6) + Au”‘( 5, T)) strictly follows that of the free GL, if the lowest approximation of the perturbation expansion is used. The obtained result shows that the realizations of the field of the GL wave. which is influenced by the random current deviations 6J, are symmetrically distributed around the field of the free GL u,,( &). Indeed, evaluating the probability distribution F,(T,> = (6(T, - T:')(T))) of the random parameter Tk one obtains with the help of Eqs. (16), (18), (19) and (12) FT(Tk) = [4”U;(T)]-“2
exp[-T,2/4&T)], (24)
F.G. Bass, R. Bukunas/Physics
where U;(T)
where =Ak’(Yk
Il>*a’[l
-exp(-2&(r)].
B(x)
Thus, the obtained distribution F7(Tk) is Gaussian and the dispersion c;(r) approaches the fixed value s;= A/i’(Yk ll>‘a2 in the long time limit r B 5-a = A;’ = l/w’(u,,). Here rR indicates the dielectric relaxation time. Moreover, from Eq. (16), used in conjunction with (18) and (19), follows that (S(‘)(T)) = 0. Hence, the average comoving coordinate ( 5, > = y strictly follows the nucleus center of the free GL. To evaluate the second-order average m2)( 5, 7) we write the disturbing torque FC2’(.$, 7) in the following way, P(
305
Letters A 214 (1996) 301-306
= (
1du,/dy)-’
dR( x)/dx
The function P,(x) in (28) is defined by Eq. (27), P,(x) = P(x; uyk = 1). Expression (28) describes the average field hU”’ in a stationary state that settles down at a relatively long time (7 B- TV) after the random perturbation ~.T(T) has been “turned on”. Substituting the explicit expressions for W(U), u,(y), Y,Cy) given by Eqs. (20). (21) and (22) into (28) and performing the needed integrations in (28) one obtains u’2’(
6) =
5. 7) x [27r’ + (2 - 7~ cash- ’ &)?I +2-‘w”[u,(5)]A(uc’))’
X tanh t/cash
(25) The expression (25) follows from Eqs. (8) and (9) if Eqs. (17) and (18) are taken into account. The small term, proportional to ho’ -SC1, is neglected in (25). Substituting the explicit expressions (201, (21) and (22) into Eq. (25) one gets after the averaging which has to be performed with the use of Eq. (19) ?F’ )= -(r*((Y
II>{?
I du,/dy)-’
+4%(5)]W&
dR( &)/dz$
r))3
(26)
where W=jx
--r
P[ 5; a&,]
dqr,(5),
=Ix--r
~q(5)=(Y,ll)Yy(5), dq/=
-*
dk(A,
+&-I
aqk(T) = 1 - exp[ - (A, + ~~)r].
(27)
Now. the average field uC2) may be evaluated from Eqs. ( 12), (16) and (13) used in conjunction with the explicit expressions (26) and (27). One gets from (12) by the assumption T B TV
G”‘(
4) = --a2
/
_:, dk A;‘(Yk
1 B)Y,(
&),
5.
(29)
Eq. (29) has been deduced from (28) by taking into account that the integral ( Yk I 1) is a relatively sharp function of k, localized in the interval 1 k 1 < 1. Thus, a sufficiently good approximation of the field huC2’( 5) is obtained if the function A, ’ in (28) is replaced by A;; 0 = A[ ‘. The integral ( Yk I 1) in (28) is simply evaluated by the method of contour integration in the complex plane if the explicit expressions for Yk, given by Eqs. (221, are employed. The closure condition of the eigenfunctions YW has also been employed in the derivation of Eq. (29). It is seen from Eq. (29) that the additional field a”( 5) is an odd function of 5, thus, the voltage drop over the specimen is not influenced by the random current deviations. Taking into account that the obtained field h”‘*’ is proportional to the intensity of the current noise (T’ and, hence, is sufficiently small if (T ’ -=z 1. one may roughly express the average GL field G= u,, + n2’ with the help of Eqs. (23). (21) and (28) in the following manner, $5)
=%(5/&(5)),
(30)
where Lo( [) = 1 + A;‘(&ry)2
(28)
27r2+(2-rrcosh-’
[)*I[-’
tanh 5.
306
F.G. Buss. R. Bakunos / Physics Letters A 214 (1996) 301-306
Thus, the field G( 5 1 is similar to that that describes the free GL of a slightly deformed nucleus. One can see from (30) that the “deformations” of the free wave induced by random perturbations 8J(r) are inhomogeneous and are more strongly pronounced in the region of the nucleus center of the free GL. The distortion of the nucleus A&.(c) = _&(&I - 1 is proportional to the noise intensity CT* and becomes negligible in the region outside the GL nucleus, i.e. A&(e)+0 when 5 + +w. Thus, the averaged field distribution along the specimen U( 5) in the outside region, 5 ZS- 1, practically coincides with that that describes the free GL. It is interesting to note that the average comoving coordinate ( t2) = y + (S(~)(T)), which has been evaluated with the help of (161, strictly follows the nucleus center of the free GL, i.e. ( t2) = y. Let us conclude by noting that the deviations of the average field of a GL influenced by a random, spatially homogeneous perturbation are mainly localized in the nucleus region of the free GL and are linearly increasing with the intensity of the noise. The average phase of the GL wave as well as the voltage drop along the wave are not influenced by the random perturbation. One can see that the main results which concern the “random walk” of GLs
are similar to those obtained for the “overdamped” sine-Gordon kink influenced by the random perturbations [l]. The main differences between these two cases concern the average phase shifts (s) and the (u,,) of the waves influaverage “amplitudes” enced by the random perturbations. The “conservation laws” concerning both the mean velocity of the GL ((3) = 0. (c) = c,> and the average “amplitudes” of GL ((u,, > = uex>, may be explained as conditioned by the fixed (in average) current regime, ( J ) = J, = ca, in our case.
References 111F.G. Bass, Yu.S. Kivshar, V.V. Konotop and Yu.A. Sinitsyn, Phys. Rep. 157 (1988) 63. and B.A. Malomed, Rev. Med. Phys. 61 ( 1989) 763. 131S.A. Gredeskul and Y.S. Kivshar, Phys. Rep. 216 (1992) I. [41 B.S. Kemer and V.V. Osipov, Autosolitons (Nauka, Moscow, 1991) (in russian). 151 R. Bakanas, F.G. Bass and V.V. Konotop, Phys. Status Solidi A I I2 (1989) 579. [61 R. Bakanas, Phys. Status Solidi A 128 (1991) 473. 171 M. Shur, GaAs devices and circuits (Plenum, New York, 1985). 181 M. Wadati, J. Phys. Sot. Japan 52 (1983) 2642. 191 T. Iizuka, Phys. Lett. A I81 (I 993) 39.
121Yu.S. Kivshar
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 2 14 (I 996) 30 l-306
Propagation of a Gunn wave under the influence of spatially homogeneous random perturbations F.G. Bass a Depurtmrnt b Semiconductor
oj’Physics, Physics
Received 8 December
Bur-lion Institute,
a,R. Bakanas b University. Gostuuto
52900
Rumut-Gun,
II. LT-2600.
1995; accepted for publication Communicated by L.J. Sham
Isrurl
Vilnius. Lithuuniu 21 February
1996
Abstract The propagation of Gunn layers (GL), i.e. the nonlinear waves of an electric charge distribution in a dissipative system, under the influence of random perturbations, which may be characterized as white Gaussian noise, is considered. The average characteristics of the perturbed GLs are analyzed and compared to those describing a “random walk” of kinks in a damped system [F.G. Bass, Yu.S. Kivshar, V.V. Konotop and Yu.A. Sinitsyn, Phys. Rep. 157 (1988) 631.
1. Introduction. The influence of small perturhations on a nonlinear wave
may be described following type, au/a7+
The propagation of nonlinear excitations under the influence of regular as well as random perturbations has been intensively investigated theoretically in the last few years, mainly in the case of conservative systems [l-3]. The nonlinear excitations in essentially dissipative systems, which can be described by nonlinear diffusion equations, are widely known in various fields of physics concerning superconductivity, solid state physics, plasma physics, biophysics etc. [4]. The analytical description of such excitations, spatially localized waves, under the influence of temporal as well as spatial perturbations is rather complicated even in the presence of small perturbations. In the simplest situations the evolution of such excitations under the influence of small perturbations 037%9601/96/$12.00 Copyright PII SO3759601(96)00192-2
by a diffusion
AL4= pf( y, 7).
where A is the nonlinear ii = -da2/ay2
equation
of the
(1) operator
+ p( u)a/ay
+ r( q
(2)
and the torque fly, T) describes a small ( p -=z 1) perturbation. By taking the explicit expressions of P(U), T(U) and allowing /3 = 0 in I!$. (1) one may obtain the unperturbed solutions uO( y) that describe the free waves. Here, y denotes the moving coordinate, y = x - CUT, where c0 is the velocity of the free wave. Thus, Eqs. (1) and (2) are written in the moving coordinates. To obtain the solution u( y, T) of Eq. (1) in the presence of a small disturbing torque f we apply a perturbation scheme that is based on the properties of the linear operator i
0 1996 Elsevier Science B.V. All rights reserved.
302
F.G. Buss, R. Bokunus / Physics Letters A 214 (I 996) 301-306
intimately related to the translational mode r(y) a du,(y)/dy,
ir=o, i=
-d-$+Q(y)&+U(y),
Q(Y)
=P[UO(Y%
U(Y)
=P'tY)dy
duetY)
+r'tyL
-k+t&
Q(y) -+O
if y+
4m.
-P(u,)](du,/d5+aAu/a5)
+ r( u0 + Au) - r( uO) + (ds/d+lAu/at - [ P’(UO) du,/d5+
A,Y,.
(5)
Owing to the fact that operator i is non-Hermitian it is possible to express the functions Y, through the eigenfunctions X, of the Hermitian operator Z?= S-‘( y)iS(y) [5,61,
(9)
In deriving (8) from (1) the relationship Au,,<5) = 0 has been taken into account. In the case of a small perturbation we expand the quantities S, Au and F in a power series of a small parameter p, +( 6, T) = c p”@“‘( 5, T), n= I
(6) In the presence of a small disturbing torque ( p -C 1) the solution of Eq. (1) u( y, T) may be presented in the following way, +Au(y,T),
By substitution of Eq. (10) into (8) the following evolution equation may be obtained in the p” order approximation,
duo(5)
Fds’“‘/dT+
a&&“‘(
where the parameter S(T) denotes the phase shift and Au( y, T) describes the distortion of the wave due to the presence of the perturbation f. Substituting (7) into (1) one rewrites Eq. (1) as follows,
T)
= F’“‘(
5,
T)/aT
T).
5,
(1’)
Expanding the functions Au(“) and F’“’ through the complete set of the eigenfunctions Y(I’ Au’“‘( 6, 7) = cT;“‘(T)Y,( (I F(“)( 5, T) =
Aueu,, (7)
4 = s, Au, F. (10)
+ LAu’“‘( 5,
u(~,T)=dy+s(T)]
+o)]Au.
(4)
Thus, the eigenfunctions Y, of the operator i make up a complete set and the corresponding eigenvalues h, are real, iv,=
(8)
where r= y -t S(T) is the comoving coordinate, i.e. the coordinate moving with the perturbed wave, and the additional “torque” g includes the nonlinear terms with respect to the small quantities ds/dT and AU, g= [~(uo+Au)
where the prime signifies the derivative with respect to uO, i.e. p’(y) = dp[u,( y)]/du,, etc. Here and in the following, an overline denotes a quantity intimately related to the translational mode. The existence of the translational mode evidently follows from the translational invariance of the operator R. We assume that
T>, S(T); u,(t)]>
&),
~F;“)(T)Y,( (I
(12)
5),
one rewrites Eq. (11) in the “L-representation” dT;“‘/dT+ =
(Y, 1du,/dc)
d.@‘)/dT+h,T,(“’
F:“)(T),
F:“‘(T)
= (Y(x)
1 F’“‘(
x,
T)).
(13)
Here, the following notation has been introduced, = F(Au,
s; uo; f),
(Y, IP>=
3cdxY,+(x)‘P(x, / --a
T),
(14)
F.G.
Bass. R. Bakunas/
where the dagger indicates the eigenfunction adjoint operator i’ = S- ‘Z?S, Y,‘(x)
=s-‘(x)x,*(x).
Physics
of the
(15)
Now, the evolution equations that describe the required functions, S(“)(T) and TA”), may be simply obtained from Eq. (13), if one takes into account that the contribution of the translational mode F(y) may be fully included into the phase shift S(“)(T). Thus, substituting d?“)/dr= 0 into (13), one obtains immediately S(~)(T) = (? Idu,/d5)-‘joid’F’“‘(r),
Letters
A 214 (1996)
of two types, GUM layers (accumulation and depletion waves) and GUM domains [7]. In describing the GWs by Eqs. (I), (2) and (17) we used the dimensionless units: the time T = r/rM is measured in the units of the dielectric relaxation time rM and the moving coordinate y = x/l, - co7 is scaled in the units of the characteristic length lo, which characterizes the electric field distribution over the specimen in the free GW case. The quantity I, denotes the size of the GW nucleus, i.e. it indicates the characteristic distance of the charge density localization, p(x) - du&x. t)/dx, in the free GW. The following dimensionless quantities are also introduced into Eqs. (l), (2) and (17): u = E/E,, w = u/u,, d= (1,/i,)‘,
T:“)(T)
=L’drFi”)(r)
exp[A,(t-T)].
( 16)
In deriving (16) from (13) we have supposed that the perturbing torque fly, T) has been “turned on” at the initial moment T= 0. Eq. (16) used in conjunction with (7), (10) and (12) enable us to describe the evolution of a nonlinear wave under small perturbations with the needed using (16) the stochastic accuracy. Additionally, characteristics of the wave may also be deduced in the case of a random torque fly, 7). The described perturbation scheme has been applied to the particular case of a Gunn wave influenced by random, spatially homogeneous perturbations.
2. The average characteristics
of a Gunn wave
In the case of a semiconductor sample with a N-type current-voltage characteristic Eq. (1) describes the evolution of the electric field distribution along the specimen u( y> [5-71. The disturbing torque f is induced by inhomogeneities of the doping profile in a sample or, in the case considered, originates from the current or voltage deviations in the external circuit. It is well known IS-71 that the functions p(u) and T(U) are simply related to the current-voltage characteristic w(u) of the specimen, ~(u)=hG~(U)=W(U)-Co.
303
301-306
f, =
(DTM)“‘,
hG
=
/&iv.
1,
=
TMup,
where the parameters E, and up correspond to the electric field and the electron drift velocity at the peak of the current-voltage characteristic and D is the diffusion coefficient. A simple theoretical model allowing the analytical description of free GWs has been proposed in Refs. [5,6], where the propagation of these waves under influence of the small, spatially localized perturbation has been examined. Analyzing the statistical characteristics of GLs influenced by the random perturbation we have used the model of a free GL. proposed in Ref. [5]. We suppose that the random perturbations are spatially homogeneous and are induced by current oscillations SJ(7) = J(T) - J, in the external circuit of the specimen. Here J, denotes the total current in the free GW case. We assume that J, = ca, thus. the “equal areas rule” holds when the perturbation SJ is absent. The disturbing torque f in our case is simply related to the current deviation 6J [5], f=
sJ(r),
(18)
and the perturbation scheme described above may be applied if the current deviations are small. In the case of Gaussian deviations 6J we write (6J(r))
= 0,
( 17)
The unperturbed solutions of Eq. (1) u& y> describe the free Gunn waves (GW), i.e. nonlinear excitations
Bjj=
(~J(T~)SJ(T~))
=
2~7~6(T~-
Tj),
(19)
304
F.G. Buss, R. Bakmns/Phy.sics
where r is an integer, C indicates the sum of correlatom Bjj covering all pair combinations of indices. The parameter u in (19) characterizes the external noise intensity and ( > denote the ensemble average, i.e. the average over all the possible realizations of
J(T). To describe the average characteristics of a GL distinctly one has to know both the eigenfunctions Y, and the corresponding eigenvalues A, explicitly. In addition, the solution of the unperturbed GL uO(y> is also required, as it follows from (7), (10) and (16). The needed expressions have been deduced in Ref. [5] with the current-voltage characteristic w(u) taken in the following form, W(U) =a+b
sin[y(u-F)],
(20)
where u, ,b, y, F are free parameters. With the help of (20), (1) and (2) the following field distribution u,(y) which describes the free GLs has been obtained in the strong diffusion case (I, 3 1,>, uO( y) = F + 2y-’
arctan sinh( *y),
(21)
where F=F+A/2 and A=27r/y=u,-u, denote the increment of the electric field in the GL. The quantities uM and u, (z+, > u,,,) indicate the extreme values 24, of the electric field in the GL. The signs + and - correspond to the accumulation and depletion wave, respectively. The function ?N y) = arctan sinh( k y) in (21) is defined as - 7r/2 < ‘SF(y) Q 7r/2. The characteristic distance describing the charge density distribution in the wave p(x) dZ?&x)/d x, i.e. the extent of the GL nucleus in the x-coordinates, is of the order of 1, = 1, AC, where A, = [ w’( ~,,)]-‘/~l~/l~ z=- 1. The needed functions Yg and the corresponding eigenvalues A, describe the continuous spectrum (a = k) and the single discrete eigenlevel X = 0 in this case, _?= 2-‘12 sech y, x,=
[2+*+
A=o,
l)] -“2(tanh
y-
ik) exp(iky),
h(k) = A,(/@ + l), S=exp[
+(yAo)-’
sech y] = 1,
(22)
where the parameter A, = by = w’!u,,) indicates the bottom of the continuous spectrum. To deduce the statistical characteristics of a GL influenced by a random perturbation SJ(T) explic-
Letters A 214 (1996) 301-306
itly one has to describe the statistical ensemble of GLs definitely, i.e. one needs to define the averaging procedure more exactly. Here we will be interested in the average characteristics of GLs that have “arrived” to a certain point of the specimen. Thus, following Kaup (see in Ref. [1]), the averaging “relative to the moving wave” has to be performed. The corresponding ensemble of GLs includes the waves of randomly distorted shapes exclusively. The averaging over irregular shifts of the phases S(T) is ignored in this case. Obviously, the “complete ensemble”, which implies the additional averaging over the random “displacements” of GLs S(T) too, describes an essentially different situation. It represents the infinite collection of the macroscopically identical specimens characterized by unique realizations of the random torque and, consequently, takes into account all irregular parameters of the wave. For instance, the averages over a “complete ensemble” have been analyzed in Refs. [8,9] to evaluate the statistical characteristics of KdV solitons under random perturbations. The average characteristics of a GL over a complete ensemble that describes the “random walk” of separate GLs will be described elsewhere. From Eq. (7) follows ~=U,(&)+=&&r),
(23)
where the overline denotes the averaged field, Z-&t, T) = (Au(~, 7)). Performing the averaging “relative to the moving GL” in (16) one gets with the help of Eqs. (18) (19) and (121, (rj”) = 0. Hence, m”(t, T) = 0, and the shape of the averaged field of the randomly perturbed GL lil( 5) = (u,( 6) + Au”‘( 5, T)) strictly follows that of the free GL, if the lowest approximation of the perturbation expansion is used. The obtained result shows that the realizations of the field of the GL wave. which is influenced by the random current deviations 6J, are symmetrically distributed around the field of the free GL u,,( &). Indeed, evaluating the probability distribution F,(T,> = (6(T, - T:')(T))) of the random parameter Tk one obtains with the help of Eqs. (16), (18), (19) and (12) FT(Tk) = [4”U;(T)]-“2
exp[-T,2/4&T)], (24)
F.G. Bass, R. Bukunas/Physics
where U;(T)
where =Ak’(Yk
Il>*a’[l
-exp(-2&(r)].
B(x)
Thus, the obtained distribution F7(Tk) is Gaussian and the dispersion c;(r) approaches the fixed value s;= A/i’(Yk ll>‘a2 in the long time limit r B 5-a = A;’ = l/w’(u,,). Here rR indicates the dielectric relaxation time. Moreover, from Eq. (16), used in conjunction with (18) and (19), follows that (S(‘)(T)) = 0. Hence, the average comoving coordinate ( 5, > = y strictly follows the nucleus center of the free GL. To evaluate the second-order average m2)( 5, 7) we write the disturbing torque FC2’(.$, 7) in the following way, P(
305
Letters A 214 (1996) 301-306
= (
1du,/dy)-’
dR( x)/dx
The function P,(x) in (28) is defined by Eq. (27), P,(x) = P(x; uyk = 1). Expression (28) describes the average field hU”’ in a stationary state that settles down at a relatively long time (7 B- TV) after the random perturbation ~.T(T) has been “turned on”. Substituting the explicit expressions for W(U), u,(y), Y,Cy) given by Eqs. (20). (21) and (22) into (28) and performing the needed integrations in (28) one obtains u’2’(
6) =
5. 7) x [27r’ + (2 - 7~ cash- ’ &)?I +2-‘w”[u,(5)]A(uc’))’
X tanh t/cash
(25) The expression (25) follows from Eqs. (8) and (9) if Eqs. (17) and (18) are taken into account. The small term, proportional to ho’ -SC1, is neglected in (25). Substituting the explicit expressions (201, (21) and (22) into Eq. (25) one gets after the averaging which has to be performed with the use of Eq. (19) ?F’ )= -(r*((Y
II>{?
I du,/dy)-’
+4%(5)]W&
dR( &)/dz$
r))3
(26)
where W=jx
--r
P[ 5; a&,]
dqr,(5),
=Ix--r
~q(5)=(Y,ll)Yy(5), dq/=
-*
dk(A,
+&-I
aqk(T) = 1 - exp[ - (A, + ~~)r].
(27)
Now. the average field uC2) may be evaluated from Eqs. ( 12), (16) and (13) used in conjunction with the explicit expressions (26) and (27). One gets from (12) by the assumption T B TV
G”‘(
4) = --a2
/
_:, dk A;‘(Yk
1 B)Y,(
&),
5.
(29)
Eq. (29) has been deduced from (28) by taking into account that the integral ( Yk I 1) is a relatively sharp function of k, localized in the interval 1 k 1 < 1. Thus, a sufficiently good approximation of the field huC2’( 5) is obtained if the function A, ’ in (28) is replaced by A;; 0 = A[ ‘. The integral ( Yk I 1) in (28) is simply evaluated by the method of contour integration in the complex plane if the explicit expressions for Yk, given by Eqs. (221, are employed. The closure condition of the eigenfunctions YW has also been employed in the derivation of Eq. (29). It is seen from Eq. (29) that the additional field a”( 5) is an odd function of 5, thus, the voltage drop over the specimen is not influenced by the random current deviations. Taking into account that the obtained field h”‘*’ is proportional to the intensity of the current noise (T’ and, hence, is sufficiently small if (T ’ -=z 1. one may roughly express the average GL field G= u,, + n2’ with the help of Eqs. (23). (21) and (28) in the following manner, $5)
=%(5/&(5)),
(30)
where Lo( [) = 1 + A;‘(&ry)2
(28)
27r2+(2-rrcosh-’
[)*I[-’
tanh 5.
306
F.G. Buss. R. Bakunos / Physics Letters A 214 (1996) 301-306
Thus, the field G( 5 1 is similar to that that describes the free GL of a slightly deformed nucleus. One can see from (30) that the “deformations” of the free wave induced by random perturbations 8J(r) are inhomogeneous and are more strongly pronounced in the region of the nucleus center of the free GL. The distortion of the nucleus A&.(c) = _&(&I - 1 is proportional to the noise intensity CT* and becomes negligible in the region outside the GL nucleus, i.e. A&(e)+0 when 5 + +w. Thus, the averaged field distribution along the specimen U( 5) in the outside region, 5 ZS- 1, practically coincides with that that describes the free GL. It is interesting to note that the average comoving coordinate ( t2) = y + (S(~)(T)), which has been evaluated with the help of (161, strictly follows the nucleus center of the free GL, i.e. ( t2) = y. Let us conclude by noting that the deviations of the average field of a GL influenced by a random, spatially homogeneous perturbation are mainly localized in the nucleus region of the free GL and are linearly increasing with the intensity of the noise. The average phase of the GL wave as well as the voltage drop along the wave are not influenced by the random perturbation. One can see that the main results which concern the “random walk” of GLs
are similar to those obtained for the “overdamped” sine-Gordon kink influenced by the random perturbations [l]. The main differences between these two cases concern the average phase shifts (s) and the (u,,) of the waves influaverage “amplitudes” enced by the random perturbations. The “conservation laws” concerning both the mean velocity of the GL ((3) = 0. (c) = c,> and the average “amplitudes” of GL ((u,, > = uex>, may be explained as conditioned by the fixed (in average) current regime, ( J ) = J, = ca, in our case.
References 111F.G. Bass, Yu.S. Kivshar, V.V. Konotop and Yu.A. Sinitsyn, Phys. Rep. 157 (1988) 63. and B.A. Malomed, Rev. Med. Phys. 61 ( 1989) 763. 131S.A. Gredeskul and Y.S. Kivshar, Phys. Rep. 216 (1992) I. [41 B.S. Kemer and V.V. Osipov, Autosolitons (Nauka, Moscow, 1991) (in russian). 151 R. Bakanas, F.G. Bass and V.V. Konotop, Phys. Status Solidi A I I2 (1989) 579. [61 R. Bakanas, Phys. Status Solidi A 128 (1991) 473. 171 M. Shur, GaAs devices and circuits (Plenum, New York, 1985). 181 M. Wadati, J. Phys. Sot. Japan 52 (1983) 2642. 191 T. Iizuka, Phys. Lett. A I81 (I 993) 39.
121Yu.S. Kivshar