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On fluctuations about non-equilibrium steady states near Gunn instability
Physica 135A (1986) 180-199 North-Holland, Amsterdam
ON FLUCTUATIONS
ABOUT NON-EQUILIBRIUM
STEADY STATES NEAR
GUNN INSTABILITY 1. DENSITY
AND ELECTRIC
A. DiAZ-GUILERA Departamenr de Termologia.
FIELD
FLUCTUATIONS
and J.M.
RUBi
Universitar Autcinoma de Barcelona.
Received
24 June
Bellurerrrr (Barcelona),
Spuirl
1985
Fluctuations about homogeneous and inhomogeneous non-equilibrium steady states near Gunn instability arc studied. Density and electric field autocorrelation functions in Fourier and real spaces are calculated in the framework of generalized Landau-Lifshitz fluctuating hydrodynamics. Some properties of correlation functions as critical slowing down and long-range bchaviour, common for systems away from equilibrium, are obtained. Critical behaviour is also investigated.
1. Introduction Fluctuation theory concerning equilibrium states’) step in its establishment is the Onsager regression fluctuations
decay,
on the average,
following
laws. Correlation functions can be computed tion theorems. They reduce to the value Einstein’s
fluctuation
theory
when
the
is well founded. A crucial hypothesis in which local
non-equilibrium
thermodynamic
starting from fluctuation-dissipaof the moments calculated from
continuum
limit
is removed
(delta
functions are replaced by (AV) ‘, AV being the volume of the cell in which fluctuations take place). On the other hand, light scattering experiments show the validity of the assumptions made because the measured intensity of scattered light agrees with the intensity computed from density correlation functions. As a matter of fact the essential feature leading to the compactness of the theory is the existence of well founded equilibrium ensembles. The situation changes drastically when fluctuations around non-equilibrium steady states occur’). Although non-equilibrium ensembles are not so well established, a generalized Onsager regression law and a generalized Einstein relation can be formulated. However, some properties valid at equilibrium, as time reversal symmetry or translational invariance, are in general lost. Moreover some other properties, as long-range behaviour of correlation functions, appear. In the 0378-4371/86/$03.50
0
Elsevier
Science
Publishers
B.V.
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
181
vicinity of critical points some correlation functions diverge, moreover they show critical slowing down behaviour. This common feature holds in some physical problems as Binard instability3), ballast resistor4) or Gunn instability5-7), for example. The Gunn instability is characterized by the appearance of a negative differential conductivity. When energy is supplied to a n-type GaAs sample, in the form of an increasing electric field, electrons go from a state with lower energy and high mobility to a state with higher energy but lower mobility. This fact is visualized in the velocity-field characteristic (fig. 1); at the extrema, the differential mobility becomes zero and the region between them is an unstable region of negative differential conductivity. Therefore it is expected that at these points the system presents critical behaviour, since a transition occurs from a state with a field distribution that is essentially uniform to a state which is highly non-uniform. Physical interest on Gunn diodes comes from the appearance of microwave oscillations when the voltage applied to the sample reaches a critical value, observed initially by Gunn8). It is important to note that the threshold voltage for the appearance of microwave oscillations coincides with the threshold value for negative differential conductivity under special conditions only’). In this paper we are not interested in discussing this fact. Our attention will be focussed to the behaviour of fluctuations near the threshold for negative differential conductivity.
v(E)
Fig. 1. Velocity-field
characteristic
(schematic).
182
A. DiAZ-GUILERA
Fluctuations
around
exhibiting
negative
Thomas’)
generalizing
by Buttiker stochastic
non-equilibrium
differential
Special
formula
who introduce attention
J.M.
steady
conductivity
the Nyquist
and Thomas”), forces.
AND
RUBi
states
in GaAs
have been
semiconductors
analyzed
to a non-equilibrium free parameters
by Pytte
and
situation,
and
in the variance
has been paid by Keizer’)
to explain
of
critical
fluctuations at the threshold for the appearance of microwave oscillations observed experimentally by Kabashima et al.“‘). Other measurements were made by Matsuno”), whereas Nakamura”) studied statistical properties of the Gunn instability in terms of mode-coupling Our aim in this paper is to analyze
theory. electron
density
and
electric
field
fluctuations around homogeneous and inhomogeneous steady states. In section 2 inhomogeneous steady state distributions are obtained by linearizing around homogeneous
steady
states.
Hysteresis
behaviour,
typical
of a first order
phase
transition, is observed. Correlation functions of fluctuations around nonequilibrium steady states and their critical behaviour are extensively analyzed in sections 3 and 4.
2. Fluctuations
around steady states
Our aim in non-equilibrium
this section is to introduce basic equations derived from thermodynamics”) as well as fluctuations around non-equilib-
rium steady states following tuating hydrodynamicslJ). 2.1.
an
extension
of
Landau-Lifshitz
fhtc-
Basic equations
Let us first formulate density
$
of the electrons,
the basic equations n, obeys
governing
the diffusion
the system.
+V.J=O,
where the total diffusion J=nv-DVn.
The number
equation
(2.1) flux J splits in convective
and diffusive
parts in the form (2.7-j
with v the velocity of the electrons and D the diffusion coefficient. This coefficient is in principle a function of the field, but usually it is assumed to be constant and equal to an effective value. This fact simplifies largely the treatment
FLUCKJATIONS NEAR GUNN INSTABILITY
and agrees with experimental the Poisson equation
183
results. The electric field E evolves according to
V*E=z(n-n,),
(2.3)
where e is the charge of the electrons, E the dielectric constant and ~~ the density of ionized donors, asumed to be constant along the wire. Taking the time derivative of (2.3) and employing (2.1) and (2.2) one arrives, after integration in space, to C?E = DVV. E - ; qlu(E)
at
- u(E)V.
E + i $ ,
(2.4)
where Z/A is an external current per unit area which appears as an integration constant and therefore may depend on time. In (2.4) we have written the velocity of the electrons as a function of the electric field only. The justification of this assumption lies in the fact that the intra- and inter-valley transfer relaxation times are very small and therefore electrons follow immediately any change of the field. By means of (2.1) and (2.3) it is also possible to arrive to Z
-
A
= en,u(E)
- V(eDn) + E $
,
(2.5)
which reflects the fact that the sum of convection, diffusion and displacement currents is precisely Z/A. Eq. (2.4) describes the evolution of the electric field when an electrodiffusive process takes place. In order to be complete one should specify the dependence of the velocity on the field. Since the differential mobility is the first derivative of the velocity with respect to the field, it is obvious to conclude that the stability of the stationary states is linked to the form of the velocity-field characteristic. In order to analyze briefly the stability of the system we can define a potential function 4(E) that plays the same role as the potential energy of a particle in 15.16 classical mechanics ), 4(E) = i 1 (s
- en,,u(E))
dE .
(2.6)
From (2.4) it can be seen that fD(V. E)* IS ’ analogue to the kinetic energy whereas u(E)V* E is a “dissipative” term. The potential defined above has an extremum when en,u(E) = Z/A. Thus depending on the value of Z and due to the form of the velocity-field
184
A. DiAZ-GUILERA
characteristic potential corresponds state.
when
to an unstable
Finally
this state
the potential
appears
has one, the
two or three
differential mobility
to the instability
extrema.
mobility
state while a positive
when the differential
corresponds
AND J.M. RUBi
A minimum
is negative;
thus
value is associated
vanishes
of the this case
to a stable
the state is metastable
and
point.
Steady states
2.2.
Homogeneous steady state solutions are obtained from (2.3) and (2.4) in the case II, = n,,. Such solutions are generated from the equation
(2.7)
which ignores the boundary conditions we will introduce below. As we will see later on, E, is related to the voltage externally applied to the sample (bias voltage). Inhomogeneous solutions are obtained from (2.4) setting the left-hand side zero. Then one obtains a non-linear differential equation which can be solved only by numerical methods. In this paper and for simplicity’s sake we will consider steady states which are close to the homogeneous solution E,, obtained directly from the external current through (2.7). The velocity can be developed in powers of the difference E, - E,, in the form
4%) = 4E,,) + (E, - 4,) - V,@) h-E,,+ . . where only terms linear in the difference Inserting (2.8) in (2.4) one arrives to
of (2.9)
- EU(E,,)V. E
(2.9)
is
4 = 4, + A exp(w) + B exp(w) , where
(2.8)
E, - E,, have been kept.
0 = EDVV. E - en,(E, - E,,) * VEu(E)j,=,,, In 1 - d the solution
3
(2.10)
we have introduced
(2.11)
and the definitions
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
(2.12)
’
E=E,,
185
In (2.10) A and B are integration constants which can be computed from boundary conditions. In this paper we will consider Dirichlet* boundary conditions for the field. Using the “virtual cathode” assumption”“) the field is taken to be zero at x = -L/2. At the other end the field is also taken to be zero usually9~18), due to charge conservation. However, some authors”) consider that the field takes a higher value at the anode than at the cathode. Although this is not a very important question both possibilities can be considered. In the case of zero field at the anode (ZFA) the integration constants are written
A
1 - exp(dJ ZFA= exp( p2L) - exp( pu,L) exp(I-G/~) , (2.13a)
B
exp( hL)
- 1
ZFA = exp(CL2L) - exp(piL)
exp( pzL12) ,
and assuming a finite value, equal to E,,, of the field at the anode (FFA) one obtains
A FFA
exP(kLL)w-G4~4 =
exrh4
-
exp(kL)
’
(2.13b) exp( kL)
exp( hL/2) B FFA = exp( pu,L) - exp( pi L)
’
At the steady state the density of electrons is easily obtained from (2.3) and (2.10). One has
n, = no +
z[E”,Aexp(w)
+ 1-4
exp(w)l
y
(2.14)
where no is also an homogeneous solution of the steady state density, independent of E,. In figs. 2 and 3 we plot the field distribution along the sample with respect to the homogeneous field for the boundary conditions (2.13). It can be seen that those boundary conditions are responsible for the fact that the steady state electric
* Notice that due to Poisson equation (2.3) Dirichlet boundary conditions for the density are Neumann for the field. In the same way, and due to the definition of the electric voltage, Dirichlet boundary conditions for the field correspond to Neumann for the voltage.
A. DiAZ-GUILERA
186
AND
J.M.
RUBi
W” \ W” I W”
0 distance Fig. 2. Steady of the
state electric
homogeneous
field
field distribution
when the field vanishes
E,,. (a) E,, = 1 V/cm;
(b)
E,, =2kV/cm;
at each end for different (C) E,, =3kVicm;
3.1 kV/cm.
0
0
\” 4 I W”
-:
0 dLstance Fig. 3. The same
as in fig. 2 but now the field is equal
to E, at the anode.
values
Cd) EC1=
FLUCTUATIONS NEAR GUNN INSTABILITY
187
field differs from the value E,, the solution of (2.7). At equilibrium the intensity I is equal to zero and the electric field is homogeneous and equal to zero too. In a state that is close to it (for example, E, = 1 V/cm, curves a in figs. 2 and 3) the field is not homogeneous but it can be practically considered as homogeneous, since the difference is only important in the ends of the wire. When E, increases, the stationary solution goes away from the homogeneous one until it reaches the instability point (E, = 3.11 kV/cm, in our model (2.15)) where the steady state field has a nonsensical distribution, since it is identically equal to zero for (2.13a) and equal to zero almost everywhere for (2.13b). This fact shows the importance of considering inhomogeneous solutions of the steady state electric field distribution, although at the instability point our linearization (2.8) breaks up. As will be seen in the next section, in order to compute correlation functions, we need the electron density distribution instead of the electric field. With the boundary conditions (2.13) the steady state electron density differs from its homogeneous solution n, only in a few per cent, although the electric field holds of the homogeneous value, and in particular one has at the instability point n, = it,, identically, for the ZFA boundary conditions (2.13a). It does not mean that near the instability point the electron distribution can be taken strictly homogeneous since in this case the linearization (2.8) is not correct. If other boundary conditions were considered, the result should be quite different. For example, assuming the electron density to be zero at each end the distribution would lead to a solution similar to that of the field given in fig. 2, or if one assumes 11,(x = -L/2) = 0 and 11,(x = L/2) = n,,, it should correspond to fig. 3. In these cases the electron density differs appreciably from the homogeneous solution. For this reason, when fluctuations are studied, the integration constants will be considered in general. The results plotted in figs. 2 and 3 have been calculated with the expression proposed by Kroemer”) for the GaAs velocity-field characteristic u(E) =
PoE[l+ w4Yl 1+ (EIE,)k
’
(2.15)
where p0 is the lower valley mobility and B is the ratio of upper valley mobility to lower valley mobility. The values of the parameters involved in (2.15) that show the best agreement with the experimental results are taken from ref. 12 and they are cc, = 8000 cm2/Vs, E, = 4000 V/cm, k = 4 and B = 0.05. Other parameters used in the calculation of the electric field distribution are, for GaAs, D = 200 cm2/s and E = 12 12). On the other hand there are parameters which characterize the sample as the length and the donor density. Acccording to usual samples on which experiments are performed”‘,~‘) we consider L = 10 microns and n0 = 1Or5cmm3.
188
A. DiAZ-GUILERA
A large number Gunn
devices
of papers have
been
on exact solutions publishedlX).
calculations,
but we need an analytical
the electron
density
states.
distributions
This is the reason
AND J.M. RUBi
of the electric
However, expression
in order
of either
to analyze
of our linearization
they
field distribution involve
the electric
fluctuations
in
numerical
around
field or steady
(2.8).
We have described the steady state variables, as electric field and electron density, in terms of an homogeneous field E,, related to the intensity through (2.7). Sometimes an electrical system is described externally. In our problem we can define the voltage
by the voltage applied across the sample as
I. / 2
Es(x)dx
U=
(2.16)
1
mf.!2 which is a function defined in (2.11).
u,FA
=
E
.
of the homogeneous electric field E,, only through Using (2.10) and (2.13) in (2.16) one arrives at
L
+
(cxp(/*,L)
0
- l)(exP(@zl)
exp(pzL)
- 1)
-ev(kL)
1 _ 1 E,) ) 1, ? PI 1
11, and ~~
(2 ITa)
(2.17b)
Eqs. (2.17) are slightly different but they can be analyzed in general since they lead to the same physical result. The first term on their right-hand side is linear in the field and corresponds to the voltage due to an homogeneous electric field. The second
terms come from the inhomogeneous
corrections
of the steady
state electric field. These terms are small enough unless near the instability point where they become dominant, in such a way that just at this point the voltage across the sample
(2.17)
is equal
to zero.
In order
to clarify this fact we plot in
fig. 4 the voltage for ZFA boundary conditions in terms of the homogeneous electric field E,,. The voltage in the unstable region has not been plotted since (2.17) has negative values and our treatment is not correct in this region. Within our linearized analysis of the steady state some physical information can be gained from fig. 4. An increase from equilibrium of the bias voltage produces a linear increase in the homogeneous field, E,, = U/L. But when the bias voltage reaches a critical value, the field E,, changes abruptly to a higher value lying on the high field stable branch of the velocity-field characteristic. If this process is reversed, a similar fact occurs, but now the change in the field is from the minimum of the velocity-field characteristic to a lower field. The process represents an hysteresis cycle, typical of a first order phase transition
FLUCTUATIONS
NEAR GUNN INSTABILITY
189
/ UZFA
Fig. 4. Representation of the voltage along the sample versus homogeneous electric field for ZFA boundary conditions. The hysteresis cycle is denoted by arrows.
that has been analyzed by Nakamura”) and observed by Kabashima et al.“). Nakamura’*) obtained a non-linear Langevin equation for the unstable mode responsible for the critical behaviour and it enables him to obtain a thermodynamic potential, over which hysteresis behaviour was analyzed. Although this potential appears in a natural form in an equation identical to the rotating-wave van der Pol equation for a laser oscillator, it has a very complicated structure in contrast to that expressed in (2.6) from which the hysteresis behaviour can also be inferred. 2.3. Fluctuations We will incorporate fluctuations in former scheme by adding a stochastic current JR to (2.3). Then the total fluctuation diffusion flux writes J=nv(E)-DVnf
JR.
(2.18)
Therefore and as we outlined above, from (2.1), (2.3), (2.9) and (2.18) one arrives at SE -=DVV.6E-y8E-fWiE-~JR, at
(2.19)
A. DiAZ-GUILERA
190
where
electric
extension following
field fluctuations
of fluctuating
AND J.M. RUBi
as 6E = E - E,. According to the the stochastic current satisfies the
are defined
hydrodynamics,
properties:
(JYr, t)),, = 0 ,
(2.20)
(Jy(r, t)JT(r’,t’))., = 2Dn,(r)6,,6(r where
( . . ),,
the symbol
stands
-
for non-equilibrium
r’)tS(t
-
t’) ,
averages.
(2.21)
Henceforth
we will omit
NE.
Eq. (2.19)
can be Fourier
in (k, co). Then
transformed
if we define
6E(k, w) = j dr 1 dt e’(w’mk’r)i5E(r, t) one arrives
(2.22)
at
6E(k, w) = - f [(y - iw)l + Dkk + $k]-‘JR(k,
(2.23)
w)
where p is the three-dimensional counterpart of (2.12a), y = en,,p/e and 1 is the unit matrix. Assuming the system to be isotropic, one has rlu,(E)/aE,I, =t,, = @,, , I_Lbeing the differential mobility. Notice that in order to introduce integrals in the Fourier infinite. This
transformation can be done
(2.22) we have tacitly if collective excitations
assumed that our system decay within a distance
is 1
shorter than the dimension of the system L 14). Since I is of the order of a characteristic velocity times a characteristic time the condition is v(E,,)/y < L, where v(E,,) - lo7 cm/s is not only the velocity at which electrons move but it also represents the velocity of propagation of fluctuations as will be shown later time’7,“‘) on. Moreover y - 10” s --’ is the inverse of the dielectric relaxation and will be also interpreted as the relaxation time of fluctuations. Therefore the condition is L 9 1O-5 cm and in Gunn devices the typical length is around 10-‘-10-7 Density
cm, as was outlined fluctuations
around
above;
therefore
the homogeneous
our
assumption
solution
is justified.
can be studied
also
from the equation a&? -=DV26n-P.Vn-ySn-V.JK, at where Sn = n - no. Eq. (2.24) is derived transforming (2.24) one arrives at 6n(k, w) = -i
(2.24) from (2.1),
km JR@, w> Dk’ + y - i(w - /3 - k) ’
(2.3)
and (2.18).
Fourier
(2.25)
FLUCTUATIONS NEAR GUNN INSTABILITY
191
Instead of transforming over k - owe can only transform in k. One arrives at the following equation: dSn(k, t) at
= -(Dk*
+ y + i/3 *k)Sn(k, t) - ik. JR.
Taking the non-equilibrium (Sn(k, t)) = exp[-(ip
(2.26)
average in (2.26) and using (2.20) one arrives at - k + Dk’ + y)t](Sn(k,
t = 0)) .
(2.27)
In view of this last expression, the k = 0 mode grows exponentially in time when the differential mobility is negative. Then (2.27) shows the enhancement of fluctuations in the unstable branch. The parameter y -’ represents the relaxation time of fluctuations. This parameter is also defined by some authors”,*‘) as the dielectric relaxation time or the time in which the charge redistributes in the semiconductor. Then when it is negative, the system goes away from its equilibrium state.
3. Density correlation
functions
The equations derived in the previous section are now applied to compute correlation functions for fluctuations around homogeneous and inhomogeneous steady states. At a first step we are going to analyze the density correlation function for an arbitrary dimension. From (2.25) and the Fourier transformation of (2.21) one arrives at (Sn(k, w)Sn(k’, w’))
= -(2r)df’
2Dn,ka k’6(k + k’)S(w + w’) [Dk* + y _ if@ _ p. k)][Dk” + y _ i(w’ - p. k’)]
’
(3’1)
where we have assumed that the steady state is homogeneous and its density equal to no. Fourier inversion of (3.1) leads, at equal time, to ,
(Sn(t, t)Sn(r’, t)) = n,6(r - r’) - “$$)
1 k2e~‘~~~~)
dk .
(3.2)
Now it is easily seen that the behaviour of the second term in the right-hand side of (3.2) depends strongly on the dimension of the system. Then one has for each
192
A. DiAZ-GUILERA
AND
J.M.
RUBi
dimension*
(Srr(x, t)Sn(x’, r)) = n,,qx -
x’)
5
-
e
-(‘\ -k”,c’ ,
d=
1, (3.3)
(Sn(r, t)Sn(r’, f)) = n,,6(r - r’) - f+
K,,( (r - r’( is) ,
d=2, (3.4)
(Sn(r,
t)) = n,,6(r-
t)Sn(r’,
4Tcl;l:‘_ em”r-r”‘t’-
r’) -
r,l
ci=3. (3.5)
where
K,, is the modified
Bessel function
of zero order.
Eqs. (3.3)-(3.5)
involve
the correlation length 5 = (D/y)“’ which is nothing but the Debye length. This length is also the length in which the charge rearranges through diffusion. At equilibrium this length becomes infinite and the non-equilibrium correlation functions tend to the equilibrium value n,,6(r - r’). Moreover non-equilibrium corrections exhibit Ornstein-Zernike behaviour and are of the same type to those encountered in the Debye model of plasmas”). the difference arising from the fact that for plasmas the correlation length 5 appears always as a finite parameter and therefore has a different meaning. At the instability point. where y is equal to zero, the correlation length diverges but non-equilibrium corrections to the correlation functions vanish. Minus signs in non-equilibrium corrections charge conservation, if at time r a positive fluctuation fluctuations
should, on the are correlated.
It is important evaluated appearing
to note
average,
take
that the correlation
correspond fluctuation place
at
to the fact that due to occurs at r, a negative any
functions
point
(3.3)-(3.5)
r’
in which have been
assuming y to be positive since in the opposite case the integral in (3.2) diverges. The same will be assumed in the calculation of
correlation
functions
A quantity
in the r-t
representation.
that is also interesting to analyze is the correlation function in the It can be obtained from (3.1) by Fourier inversion over w has
k-t representation.
and o ‘. One
(Sn(k, t)Sn(k’.
r’)) = (2+%,,6(k x exp{ -i/3
* Densities
appearing
in the correlation
+ k')
Dy; y
* k(t - t’) - (Dk’ + y)lf - t’l} ,
functions
must be interpreted
in each dimension
(3.6)
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
193
where we can see that j3 plays the role of a velocity of the propagation of density fluctuations. It can be also observed that the relaxation time of the k = 0 mode diverges as y -’ when the field approaches the instability point: critical slowing down behaviour. These results were also obtained in refs. 5 and 6. From (3.6) one can also see that the density correlation function in the k-f representation vanishes in the k = 0 limit, unless at the instability point, y = 0, in which (3.6) does not vanish and the density fluctuations are delta correlated as we saw in (3.3)-(3.5). In the k = 0 case both the relaxation time and the correlation function (3.6) are finite if the system is at the stable branch (y > 0). But if y is negative there will always exist one mode, in our continuum representation, such that its correlation time diverges and makes the density correlation function be divergent. Notice that in a discrete representation, imposing periodic boundary conditions on the fluctuations, the wave number k is given by k = (27rlL)m, where m is an integer different from zero. In that case the density correlation function (3.6) does not diverge at y = 0 but it does at y = -(27rlL)“D. This fact gives a criterion of stability of the fluctuations, as Butcher’) and Ridley22) do for a perturbation from the steady state, showing that diffusion stabilizes waves with sufficiently short wavelength. A stability criterion relating the differential mobility and the length of the sample appears commonly in the literature about the Gunn effect7’9”7 ). For GaAs the stability criterion of the waves involving the diffusion coefficient is most important for very short devices (0.5 microns) with a donor density around 1016cm-3 ” ), whereas in other cases different criteria should be used. Density fluctuations around inhomogeneous stready states (2.14) can be analyzed also. In k-o representation and in 1 - d one has @n(k,
w)&z(k’, co’)) = -47r
Dkk’G(w + w’)n,(k [Dk2 + y - i(w - @k)][Dk’*
+ k’)
+ y - i(w’ - pk’)]
’
(3.7) where n,(k) is the k-Fourier transform of the steady state electron density (2.14) that follows from the n,(x) appearing in the covariance of the random current (2.21). By inverse Fourier transform of (3.7) and integrating over w and w’ the equal-time correlation function is (sn(x, t)sn(x’, t)) =
& j-j-dk dk’ D(k -cc
-2Dkkrn,(k
+
kf)ei(kx+k’x’)
+ k’)’ - 2Dkk’ + 2y + iP(k + k’) ’ (3.8)
194
A. DiAZ-GUILERA
These
integrals
(%2(x, t)6n(x’,
are evaluated
t)) = n,(x)s(x
AND
J.M.
in the appendix
- x’) - n,,
RUBi
and its final result
is
em“ -““’ 25
(3.9) 4 being
a dummy
variable
and F(x, x’, 4) an auxiliary
function
defined
in the
appendix. Notice that the first term in the right-hand the corresponding term in eqs. (3.3)-(3.S), that now long-range
the coefficient contributions
of the to (3.9)
delta come
side of (3.9) is of the same kind as the difference arising from the fact
function depends on position. The from F(x, x’, q) and from the term
corresponding to the homogeneous but non-equilibrium correction studied in (3.3). In view of the definition of F(x, x’, q), (A.9), one concludes that in l-d the contribution is exponential. Moreover it is possible to see that translational invariance valid at equilibrium is not satisfied now since a transformation x+ x + a implies a change n,(x) + 11,(x + a) and F(x, x’, CJ)-+ exp(iqa)F(x, x’, 4). In eqs. (3.3)-(3.5) we saw that at the instability point (y = 0) the nonequilibrium corrections of the density correlation functions vanish in an homogeneous steady state. A detailed analysis of (3.9) shows that the inhomogeneous part of the steady state solution (2.14) makes the nonequilibrium density correlation function diverge. Although this result is independent of the integration conditions the divergence the instability
4. Electric
point
constants is slightly
A.,,
tends
field correlation
A and different
B obtained from the boundary in the two cases (2.13) since near
to zero as y whereas
AFFA remains
finite.
functions
Apart from the density correlation functions we are also interested in the expression of the electric field correlation function. It can be obtained in a way similar to that employed in the derivation of the density correlation function. Hence from (2.23) one obtains the electric field correlation function in the k-w representation in l-d: (6E(k, w)GE(k’, w’)) 4nDe2
S(w + w’)ns(k
= T[Dk2
y - i(o - /3k)][Dk”
+ k’) + y - i(w’ - pk’)]
t
(4.1)
195
FLUCTUATIONS NEAR GUNN INSTABILITY
By inverse Fourier transform, integrating over w and w’ and inserting the value of n,(k + k’) obtained in the appendix, the first contribution to the equal-time correlation function reads
(SE(x, t)SE(x', t))l =
$ [+jdk dk’
‘(’ +k;$,+;+^”
.
(4.2)
--m When the integral is performed one must take into account that y has a positive value in order the integral be convergent. One arrives at (%5(x, t)SE(x’,
t))
1
=
$
5 e-‘xpx”‘s .
Second contribution gives, following the same procedure the calculation of the density correlation function, (SE(x,
t)SE(x’,
t)), = %
e-icllJ’JqF(x, x’, q)lqzo .
Adding the three contributions
(Slqx, t)SE(x’, t)) =
1-
+ wIA 2E
emiplJiJq
$ +
(4.3) as that employed in
(4.4)
one finally arrives at 5
-w2B 2E
e-Ix-x’l/5
e-ir2J’Jq F(x, x’, q)lqzo .
1
(4.5)
This last expression can be written in a more compact form observing the dependence of n,(x) on the spatial coordinate (2.14) and the value of F(x, x’, q) at q = 0 from (A.9),
(S-W, t)SE(x’, t)) =
$ n, (4 +-)F(x, x’, q)lqzo.
(4.6)
From (4.3) one can notice that the homogeneous part of the electric field correlation function diverges as the correlation length 5 does when the system approaches the instability point. In that limit the density correlation functions (3.3)-(3.5) re d uce to the term containing the delta function. This is not a new result since Pytte and Thomas5) and Buttiker and Thomas6) arrived at a similar conclusion for long times. But when fluctuations around inhomogeneous steady states are studied both density and electric field correlation functions diverge at the instability point, although they do it in a different way.
196
A. DiAZ-GUILERA
In the k - t representation contribution
(E(k,
and taking
in a one-dimensional
t)GE(k’, t’)) =
AND J.M. RUBi
into
analysis
2n-Dn,,e’ E2
account
only the homogeneous
the field correlation
exp{ -ipk(t
function
looks
- t’) - (Dk’ + r)lt - t’l} Dk’ + y
x 6(k + k’) ,
(4.7)
where we can see that it is finite for the k = 0 mode, i.e. it has a finite range and diverges at the instability point in contrast to the density-density correlation function (3.6). From electric field fluctuations one can study voltage fluctuations that can be compared
with experiments.
This will be the subject
of a forthcoming
paper”).
Acknowledgements This paper has been partially supported by a grant of the Comision Investigation Cientifica y Ticnica of the Spanish Government.
Asesora
de
Appendix Fourier
transforming
equation
(2.14)
at ti
+r
+L
P,AE
th\ e e"' dx + ~
e A’ dx + p
n,(k) = n,,
one arrives
e"?‘dx
~7
-x
(A.1) The first integral is a delta function same structure and for this reason exponential
is Taylor
expanded
6(k). The second and third integral have the they are evaluated in a general form. If the
we get
(A .2)
FLUCI-UATIONS
Thus (A.l)
NEAR
GUNN
INSTABILITY
197
becomes
n,(k) = 2~,6(k)
n$O $
+ ?
a(k)(Ap;+’
$
+ I$;+‘)}.
(A.3)
For the sake of clarity we will analyze separately the three different contributions to n,(k). The first one corresponds to the homogeneous steady state electron density n,, and has been obtained in eq. (3.3). Inserting the second contribution into (3.8) one obtains
c@$ /+f
(6n(x, t)&z(x’, t))2 = g
dk dk’
n-0
_ ’
D(k
+
Ic’)~
2~kk -
--m
eWx+k’x’)
.
(A.4)
6(q)
(A.5)
Dq2 + 2D( q’ - q)q’ + 2y + iPq ’
G4.6)
a”
2Dkk’ + 2y + iP(k + k’) d(k + k’)”
6(k + k')
After a change of variables, k + k’ = q, k’ = q’, it reads
(Sn(x, t)&z(x’, t))2 =
$$ ; @$ l+f dq dq' no
.
_qq
X
and integrating
_
--ic
qt)qt
ei(q-q’b
eiq’*’
d”
Dq2 - 2D( q - q’)q’
+ 2y + i@q aq”
by parts 12times in q,
(&2(x, t)ifin(x’, t)), =
nzo v $6
E
eiqx
+oC 2D(q!
X i
dq’
-cc
_
q)q’
e-iq’(x-x’)
The integral appearing in (A.6) is +-= 2D(q!
i -m
dq’
_
q)qt
e-iq’(x-x’)
Dq2 + 2D( q’ - q)q’ + 2y + ipq eidx’-x)/2
_
= 27rS(x - x’)
1 (; +$ +g”2,x_xI,)
exp
_
n(Dq’+ 2~ + i&l 20 (
64.7)
A. DiAZ-GUILERA
198
AND
J.M.
RUBi
Then one obtains (Sn(x, t)Sn(x’, t))
2
-pifc e/*l\qx
=
-
x’)
r x,,,
(-ip,)”
P&
-
~
e
0”
,I_11
x (&I’ + 2~ + iPqJF(x, x’, 4)ly-,,,
(A.81
where F(x, x’, q) is an auxiliary function defined by lq(r*.l’)/7
F(x, x’. q) =
ez
$+%+E iPq (A.9) Rearranging
(A.B) it can be written as
(6n(x, t)Sn(x’, t)), =
-‘“lat e”“6(x x -f n-0
(-ip,)“’ n!
- x’) -
d”
j-g
F(x, x’, q)l,=,,
(A.10)
The third contribution is equal to the second one changing p, by pz and A by B. Adding the three contributions to the equal-time density correlation function, taking into account the expression of the steady state electron density (2.14) and the values of p, and pI (2.11), one finally obtains e~~‘H’1/5 (&2(x, t)tin(x’, t)) = n,(x)ti(x
- & xe
which corresponds
- x’) - n,,
25
{ /_L,A(y + 2&,)
-“zb’dq}F(x,
x’, q&:,,
e~‘F1”!a’ + /+B(Y + 2pp.z) ,
(A.ll)
to (3.9)
References 1) J.M. Rubi, Fluctuations Around Equilibrium, in: Recent Developements in Non-equilibrium Thermodynamics, J. Casas-Vazquez, D. Jou and G. Lebon, eds., Lecture Notes in Physics (Springer, Berlin, 1984). 2) A.-M.S. Tremblay, Theories of Fluctuations in Non equilibrium systems, in: Recent Developements in Non-equilibrium Thermodynamics, J. Casas-Vazquez, D. Jou and Q. Lebon, eds. Lecture Notes in Physics (Springer, Berlin, 1984).
FLUCTUATIONS
NEAR GUNN INSTABILITY
199
3) V.M. Zaitsev and MI. Shliomis, Soviet Physics JETP 32 (1971) 866. T.R. Kirkpatrick and E.G.D. Cohen, J. Stat. Phys. 33 (1983) 639. 4) R.A. Pasmanter, D. Bedeaux and P. Mazur, Physica WA (1978) 151. 5) E. Pytte and H. Thomas, Phys. Rev. 179 (1969) 431. 6) M. Buttiker and H. Thomas, Solid-State Electron. 21 (1978) 95. 7) J. Keizer, J. Chem. Phys. 74 (1981) 1350. 8) J.B. Gunn, Solid State Commun. 1 (1963) 88. 9) P.N. Butcher, Rep. Prog. Phys. 30 (1967) 97. 10) S. Kabashima, H. Yamazaki and T. Kawakubo, J. Phys. Sot. Jpn. 40 (1976) 921. 11) K. Matsuno, Phys. Lett. 31A (1970) 335. 12) K. Nakamura, J. Phys. Sot. Jpn. 38 (1975) 46. 13) S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962). 14) A.-M.S. Tremblay, M. Arai and E.D. Siggia, Phys. Rev. A23 (1978) 1451. 15) D. Bedeaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355. 16) B. Kadomtsev, Phenomenes Collectifs dans les Plasmas (Mir, Moscow, 1979). 17) J.E. Carroll, Hot Electron Microwave Generators (Elsevier, New York, 1970). 18) CM. Snowden, Rep. Prog. Phys. 48 (1985) 223; and references quoted therein. 19) H. Kroemer, IEEE Trans. Electron Devices ED13 (1966) 27. 20) B. Knight and G.A. Peterson, Phys. Rev. 147 (1966) 617. 21) N.G. van Kampen, Statistical Mechanics of Classical Plasmas, in: Fundamental Problems in Statistical Mechanics, E.G.D. Cohened. (North-Holland, Amsterdam, 1968). 22) B.K. Ridley, Proc. Phys. Sot. 86 (1965) 637. 23) A. Diaz-Guilera and J.M. Rubi, Physica 135A (1986) 200, subsequent article.
ON FLUCTUATIONS
ABOUT NON-EQUILIBRIUM
STEADY STATES NEAR
GUNN INSTABILITY 1. DENSITY
AND ELECTRIC
A. DiAZ-GUILERA Departamenr de Termologia.
FIELD
FLUCTUATIONS
and J.M.
RUBi
Universitar Autcinoma de Barcelona.
Received
24 June
Bellurerrrr (Barcelona),
Spuirl
1985
Fluctuations about homogeneous and inhomogeneous non-equilibrium steady states near Gunn instability arc studied. Density and electric field autocorrelation functions in Fourier and real spaces are calculated in the framework of generalized Landau-Lifshitz fluctuating hydrodynamics. Some properties of correlation functions as critical slowing down and long-range bchaviour, common for systems away from equilibrium, are obtained. Critical behaviour is also investigated.
1. Introduction Fluctuation theory concerning equilibrium states’) step in its establishment is the Onsager regression fluctuations
decay,
on the average,
following
laws. Correlation functions can be computed tion theorems. They reduce to the value Einstein’s
fluctuation
theory
when
the
is well founded. A crucial hypothesis in which local
non-equilibrium
thermodynamic
starting from fluctuation-dissipaof the moments calculated from
continuum
limit
is removed
(delta
functions are replaced by (AV) ‘, AV being the volume of the cell in which fluctuations take place). On the other hand, light scattering experiments show the validity of the assumptions made because the measured intensity of scattered light agrees with the intensity computed from density correlation functions. As a matter of fact the essential feature leading to the compactness of the theory is the existence of well founded equilibrium ensembles. The situation changes drastically when fluctuations around non-equilibrium steady states occur’). Although non-equilibrium ensembles are not so well established, a generalized Onsager regression law and a generalized Einstein relation can be formulated. However, some properties valid at equilibrium, as time reversal symmetry or translational invariance, are in general lost. Moreover some other properties, as long-range behaviour of correlation functions, appear. In the 0378-4371/86/$03.50
0
Elsevier
Science
Publishers
B.V.
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
181
vicinity of critical points some correlation functions diverge, moreover they show critical slowing down behaviour. This common feature holds in some physical problems as Binard instability3), ballast resistor4) or Gunn instability5-7), for example. The Gunn instability is characterized by the appearance of a negative differential conductivity. When energy is supplied to a n-type GaAs sample, in the form of an increasing electric field, electrons go from a state with lower energy and high mobility to a state with higher energy but lower mobility. This fact is visualized in the velocity-field characteristic (fig. 1); at the extrema, the differential mobility becomes zero and the region between them is an unstable region of negative differential conductivity. Therefore it is expected that at these points the system presents critical behaviour, since a transition occurs from a state with a field distribution that is essentially uniform to a state which is highly non-uniform. Physical interest on Gunn diodes comes from the appearance of microwave oscillations when the voltage applied to the sample reaches a critical value, observed initially by Gunn8). It is important to note that the threshold voltage for the appearance of microwave oscillations coincides with the threshold value for negative differential conductivity under special conditions only’). In this paper we are not interested in discussing this fact. Our attention will be focussed to the behaviour of fluctuations near the threshold for negative differential conductivity.
v(E)
Fig. 1. Velocity-field
characteristic
(schematic).
182
A. DiAZ-GUILERA
Fluctuations
around
exhibiting
negative
Thomas’)
generalizing
by Buttiker stochastic
non-equilibrium
differential
Special
formula
who introduce attention
J.M.
steady
conductivity
the Nyquist
and Thomas”), forces.
AND
RUBi
states
in GaAs
have been
semiconductors
analyzed
to a non-equilibrium free parameters
by Pytte
and
situation,
and
in the variance
has been paid by Keizer’)
to explain
of
critical
fluctuations at the threshold for the appearance of microwave oscillations observed experimentally by Kabashima et al.“‘). Other measurements were made by Matsuno”), whereas Nakamura”) studied statistical properties of the Gunn instability in terms of mode-coupling Our aim in this paper is to analyze
theory. electron
density
and
electric
field
fluctuations around homogeneous and inhomogeneous steady states. In section 2 inhomogeneous steady state distributions are obtained by linearizing around homogeneous
steady
states.
Hysteresis
behaviour,
typical
of a first order
phase
transition, is observed. Correlation functions of fluctuations around nonequilibrium steady states and their critical behaviour are extensively analyzed in sections 3 and 4.
2. Fluctuations
around steady states
Our aim in non-equilibrium
this section is to introduce basic equations derived from thermodynamics”) as well as fluctuations around non-equilib-
rium steady states following tuating hydrodynamicslJ). 2.1.
an
extension
of
Landau-Lifshitz
fhtc-
Basic equations
Let us first formulate density
$
of the electrons,
the basic equations n, obeys
governing
the diffusion
the system.
+V.J=O,
where the total diffusion J=nv-DVn.
The number
equation
(2.1) flux J splits in convective
and diffusive
parts in the form (2.7-j
with v the velocity of the electrons and D the diffusion coefficient. This coefficient is in principle a function of the field, but usually it is assumed to be constant and equal to an effective value. This fact simplifies largely the treatment
FLUCKJATIONS NEAR GUNN INSTABILITY
and agrees with experimental the Poisson equation
183
results. The electric field E evolves according to
V*E=z(n-n,),
(2.3)
where e is the charge of the electrons, E the dielectric constant and ~~ the density of ionized donors, asumed to be constant along the wire. Taking the time derivative of (2.3) and employing (2.1) and (2.2) one arrives, after integration in space, to C?E = DVV. E - ; qlu(E)
at
- u(E)V.
E + i $ ,
(2.4)
where Z/A is an external current per unit area which appears as an integration constant and therefore may depend on time. In (2.4) we have written the velocity of the electrons as a function of the electric field only. The justification of this assumption lies in the fact that the intra- and inter-valley transfer relaxation times are very small and therefore electrons follow immediately any change of the field. By means of (2.1) and (2.3) it is also possible to arrive to Z
-
A
= en,u(E)
- V(eDn) + E $
,
(2.5)
which reflects the fact that the sum of convection, diffusion and displacement currents is precisely Z/A. Eq. (2.4) describes the evolution of the electric field when an electrodiffusive process takes place. In order to be complete one should specify the dependence of the velocity on the field. Since the differential mobility is the first derivative of the velocity with respect to the field, it is obvious to conclude that the stability of the stationary states is linked to the form of the velocity-field characteristic. In order to analyze briefly the stability of the system we can define a potential function 4(E) that plays the same role as the potential energy of a particle in 15.16 classical mechanics ), 4(E) = i 1 (s
- en,,u(E))
dE .
(2.6)
From (2.4) it can be seen that fD(V. E)* IS ’ analogue to the kinetic energy whereas u(E)V* E is a “dissipative” term. The potential defined above has an extremum when en,u(E) = Z/A. Thus depending on the value of Z and due to the form of the velocity-field
184
A. DiAZ-GUILERA
characteristic potential corresponds state.
when
to an unstable
Finally
this state
the potential
appears
has one, the
two or three
differential mobility
to the instability
extrema.
mobility
state while a positive
when the differential
corresponds
AND J.M. RUBi
A minimum
is negative;
thus
value is associated
vanishes
of the this case
to a stable
the state is metastable
and
point.
Steady states
2.2.
Homogeneous steady state solutions are obtained from (2.3) and (2.4) in the case II, = n,,. Such solutions are generated from the equation
(2.7)
which ignores the boundary conditions we will introduce below. As we will see later on, E, is related to the voltage externally applied to the sample (bias voltage). Inhomogeneous solutions are obtained from (2.4) setting the left-hand side zero. Then one obtains a non-linear differential equation which can be solved only by numerical methods. In this paper and for simplicity’s sake we will consider steady states which are close to the homogeneous solution E,, obtained directly from the external current through (2.7). The velocity can be developed in powers of the difference E, - E,, in the form
4%) = 4E,,) + (E, - 4,) - V,@) h-E,,+ . . where only terms linear in the difference Inserting (2.8) in (2.4) one arrives to
of (2.9)
- EU(E,,)V. E
(2.9)
is
4 = 4, + A exp(w) + B exp(w) , where
(2.8)
E, - E,, have been kept.
0 = EDVV. E - en,(E, - E,,) * VEu(E)j,=,,, In 1 - d the solution
3
(2.10)
we have introduced
(2.11)
and the definitions
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
(2.12)
’
E=E,,
185
In (2.10) A and B are integration constants which can be computed from boundary conditions. In this paper we will consider Dirichlet* boundary conditions for the field. Using the “virtual cathode” assumption”“) the field is taken to be zero at x = -L/2. At the other end the field is also taken to be zero usually9~18), due to charge conservation. However, some authors”) consider that the field takes a higher value at the anode than at the cathode. Although this is not a very important question both possibilities can be considered. In the case of zero field at the anode (ZFA) the integration constants are written
A
1 - exp(dJ ZFA= exp( p2L) - exp( pu,L) exp(I-G/~) , (2.13a)
B
exp( hL)
- 1
ZFA = exp(CL2L) - exp(piL)
exp( pzL12) ,
and assuming a finite value, equal to E,,, of the field at the anode (FFA) one obtains
A FFA
exP(kLL)w-G4~4 =
exrh4
-
exp(kL)
’
(2.13b) exp( kL)
exp( hL/2) B FFA = exp( pu,L) - exp( pi L)
’
At the steady state the density of electrons is easily obtained from (2.3) and (2.10). One has
n, = no +
z[E”,Aexp(w)
+ 1-4
exp(w)l
y
(2.14)
where no is also an homogeneous solution of the steady state density, independent of E,. In figs. 2 and 3 we plot the field distribution along the sample with respect to the homogeneous field for the boundary conditions (2.13). It can be seen that those boundary conditions are responsible for the fact that the steady state electric
* Notice that due to Poisson equation (2.3) Dirichlet boundary conditions for the density are Neumann for the field. In the same way, and due to the definition of the electric voltage, Dirichlet boundary conditions for the field correspond to Neumann for the voltage.
A. DiAZ-GUILERA
186
AND
J.M.
RUBi
W” \ W” I W”
0 distance Fig. 2. Steady of the
state electric
homogeneous
field
field distribution
when the field vanishes
E,,. (a) E,, = 1 V/cm;
(b)
E,, =2kV/cm;
at each end for different (C) E,, =3kVicm;
3.1 kV/cm.
0
0
\” 4 I W”
-:
0 dLstance Fig. 3. The same
as in fig. 2 but now the field is equal
to E, at the anode.
values
Cd) EC1=
FLUCTUATIONS NEAR GUNN INSTABILITY
187
field differs from the value E,, the solution of (2.7). At equilibrium the intensity I is equal to zero and the electric field is homogeneous and equal to zero too. In a state that is close to it (for example, E, = 1 V/cm, curves a in figs. 2 and 3) the field is not homogeneous but it can be practically considered as homogeneous, since the difference is only important in the ends of the wire. When E, increases, the stationary solution goes away from the homogeneous one until it reaches the instability point (E, = 3.11 kV/cm, in our model (2.15)) where the steady state field has a nonsensical distribution, since it is identically equal to zero for (2.13a) and equal to zero almost everywhere for (2.13b). This fact shows the importance of considering inhomogeneous solutions of the steady state electric field distribution, although at the instability point our linearization (2.8) breaks up. As will be seen in the next section, in order to compute correlation functions, we need the electron density distribution instead of the electric field. With the boundary conditions (2.13) the steady state electron density differs from its homogeneous solution n, only in a few per cent, although the electric field holds of the homogeneous value, and in particular one has at the instability point n, = it,, identically, for the ZFA boundary conditions (2.13a). It does not mean that near the instability point the electron distribution can be taken strictly homogeneous since in this case the linearization (2.8) is not correct. If other boundary conditions were considered, the result should be quite different. For example, assuming the electron density to be zero at each end the distribution would lead to a solution similar to that of the field given in fig. 2, or if one assumes 11,(x = -L/2) = 0 and 11,(x = L/2) = n,,, it should correspond to fig. 3. In these cases the electron density differs appreciably from the homogeneous solution. For this reason, when fluctuations are studied, the integration constants will be considered in general. The results plotted in figs. 2 and 3 have been calculated with the expression proposed by Kroemer”) for the GaAs velocity-field characteristic u(E) =
PoE[l+ w4Yl 1+ (EIE,)k
’
(2.15)
where p0 is the lower valley mobility and B is the ratio of upper valley mobility to lower valley mobility. The values of the parameters involved in (2.15) that show the best agreement with the experimental results are taken from ref. 12 and they are cc, = 8000 cm2/Vs, E, = 4000 V/cm, k = 4 and B = 0.05. Other parameters used in the calculation of the electric field distribution are, for GaAs, D = 200 cm2/s and E = 12 12). On the other hand there are parameters which characterize the sample as the length and the donor density. Acccording to usual samples on which experiments are performed”‘,~‘) we consider L = 10 microns and n0 = 1Or5cmm3.
188
A. DiAZ-GUILERA
A large number Gunn
devices
of papers have
been
on exact solutions publishedlX).
calculations,
but we need an analytical
the electron
density
states.
distributions
This is the reason
AND J.M. RUBi
of the electric
However, expression
in order
of either
to analyze
of our linearization
they
field distribution involve
the electric
fluctuations
in
numerical
around
field or steady
(2.8).
We have described the steady state variables, as electric field and electron density, in terms of an homogeneous field E,, related to the intensity through (2.7). Sometimes an electrical system is described externally. In our problem we can define the voltage
by the voltage applied across the sample as
I. / 2
Es(x)dx
U=
(2.16)
1
mf.!2 which is a function defined in (2.11).
u,FA
=
E
.
of the homogeneous electric field E,, only through Using (2.10) and (2.13) in (2.16) one arrives at
L
+
(cxp(/*,L)
0
- l)(exP(@zl)
exp(pzL)
- 1)
-ev(kL)
1 _ 1 E,) ) 1, ? PI 1
11, and ~~
(2 ITa)
(2.17b)
Eqs. (2.17) are slightly different but they can be analyzed in general since they lead to the same physical result. The first term on their right-hand side is linear in the field and corresponds to the voltage due to an homogeneous electric field. The second
terms come from the inhomogeneous
corrections
of the steady
state electric field. These terms are small enough unless near the instability point where they become dominant, in such a way that just at this point the voltage across the sample
(2.17)
is equal
to zero.
In order
to clarify this fact we plot in
fig. 4 the voltage for ZFA boundary conditions in terms of the homogeneous electric field E,,. The voltage in the unstable region has not been plotted since (2.17) has negative values and our treatment is not correct in this region. Within our linearized analysis of the steady state some physical information can be gained from fig. 4. An increase from equilibrium of the bias voltage produces a linear increase in the homogeneous field, E,, = U/L. But when the bias voltage reaches a critical value, the field E,, changes abruptly to a higher value lying on the high field stable branch of the velocity-field characteristic. If this process is reversed, a similar fact occurs, but now the change in the field is from the minimum of the velocity-field characteristic to a lower field. The process represents an hysteresis cycle, typical of a first order phase transition
FLUCTUATIONS
NEAR GUNN INSTABILITY
189
/ UZFA
Fig. 4. Representation of the voltage along the sample versus homogeneous electric field for ZFA boundary conditions. The hysteresis cycle is denoted by arrows.
that has been analyzed by Nakamura”) and observed by Kabashima et al.“). Nakamura’*) obtained a non-linear Langevin equation for the unstable mode responsible for the critical behaviour and it enables him to obtain a thermodynamic potential, over which hysteresis behaviour was analyzed. Although this potential appears in a natural form in an equation identical to the rotating-wave van der Pol equation for a laser oscillator, it has a very complicated structure in contrast to that expressed in (2.6) from which the hysteresis behaviour can also be inferred. 2.3. Fluctuations We will incorporate fluctuations in former scheme by adding a stochastic current JR to (2.3). Then the total fluctuation diffusion flux writes J=nv(E)-DVnf
JR.
(2.18)
Therefore and as we outlined above, from (2.1), (2.3), (2.9) and (2.18) one arrives at SE -=DVV.6E-y8E-fWiE-~JR, at
(2.19)
A. DiAZ-GUILERA
190
where
electric
extension following
field fluctuations
of fluctuating
AND J.M. RUBi
as 6E = E - E,. According to the the stochastic current satisfies the
are defined
hydrodynamics,
properties:
(JYr, t)),, = 0 ,
(2.20)
(Jy(r, t)JT(r’,t’))., = 2Dn,(r)6,,6(r where
( . . ),,
the symbol
stands
-
for non-equilibrium
r’)tS(t
-
t’) ,
averages.
(2.21)
Henceforth
we will omit
NE.
Eq. (2.19)
can be Fourier
in (k, co). Then
transformed
if we define
6E(k, w) = j dr 1 dt e’(w’mk’r)i5E(r, t) one arrives
(2.22)
at
6E(k, w) = - f [(y - iw)l + Dkk + $k]-‘JR(k,
(2.23)
w)
where p is the three-dimensional counterpart of (2.12a), y = en,,p/e and 1 is the unit matrix. Assuming the system to be isotropic, one has rlu,(E)/aE,I, =t,, = @,, , I_Lbeing the differential mobility. Notice that in order to introduce integrals in the Fourier infinite. This
transformation can be done
(2.22) we have tacitly if collective excitations
assumed that our system decay within a distance
is 1
shorter than the dimension of the system L 14). Since I is of the order of a characteristic velocity times a characteristic time the condition is v(E,,)/y < L, where v(E,,) - lo7 cm/s is not only the velocity at which electrons move but it also represents the velocity of propagation of fluctuations as will be shown later time’7,“‘) on. Moreover y - 10” s --’ is the inverse of the dielectric relaxation and will be also interpreted as the relaxation time of fluctuations. Therefore the condition is L 9 1O-5 cm and in Gunn devices the typical length is around 10-‘-10-7 Density
cm, as was outlined fluctuations
around
above;
therefore
the homogeneous
our
assumption
solution
is justified.
can be studied
also
from the equation a&? -=DV26n-P.Vn-ySn-V.JK, at where Sn = n - no. Eq. (2.24) is derived transforming (2.24) one arrives at 6n(k, w) = -i
(2.24) from (2.1),
km JR@, w> Dk’ + y - i(w - /3 - k) ’
(2.3)
and (2.18).
Fourier
(2.25)
FLUCTUATIONS NEAR GUNN INSTABILITY
191
Instead of transforming over k - owe can only transform in k. One arrives at the following equation: dSn(k, t) at
= -(Dk*
+ y + i/3 *k)Sn(k, t) - ik. JR.
Taking the non-equilibrium (Sn(k, t)) = exp[-(ip
(2.26)
average in (2.26) and using (2.20) one arrives at - k + Dk’ + y)t](Sn(k,
t = 0)) .
(2.27)
In view of this last expression, the k = 0 mode grows exponentially in time when the differential mobility is negative. Then (2.27) shows the enhancement of fluctuations in the unstable branch. The parameter y -’ represents the relaxation time of fluctuations. This parameter is also defined by some authors”,*‘) as the dielectric relaxation time or the time in which the charge redistributes in the semiconductor. Then when it is negative, the system goes away from its equilibrium state.
3. Density correlation
functions
The equations derived in the previous section are now applied to compute correlation functions for fluctuations around homogeneous and inhomogeneous steady states. At a first step we are going to analyze the density correlation function for an arbitrary dimension. From (2.25) and the Fourier transformation of (2.21) one arrives at (Sn(k, w)Sn(k’, w’))
= -(2r)df’
2Dn,ka k’6(k + k’)S(w + w’) [Dk* + y _ if@ _ p. k)][Dk” + y _ i(w’ - p. k’)]
’
(3’1)
where we have assumed that the steady state is homogeneous and its density equal to no. Fourier inversion of (3.1) leads, at equal time, to ,
(Sn(t, t)Sn(r’, t)) = n,6(r - r’) - “$$)
1 k2e~‘~~~~)
dk .
(3.2)
Now it is easily seen that the behaviour of the second term in the right-hand side of (3.2) depends strongly on the dimension of the system. Then one has for each
192
A. DiAZ-GUILERA
AND
J.M.
RUBi
dimension*
(Srr(x, t)Sn(x’, r)) = n,,qx -
x’)
5
-
e
-(‘\ -k”,c’ ,
d=
1, (3.3)
(Sn(r, t)Sn(r’, f)) = n,,6(r - r’) - f+
K,,( (r - r’( is) ,
d=2, (3.4)
(Sn(r,
t)) = n,,6(r-
t)Sn(r’,
4Tcl;l:‘_ em”r-r”‘t’-
r’) -
r,l
ci=3. (3.5)
where
K,, is the modified
Bessel function
of zero order.
Eqs. (3.3)-(3.5)
involve
the correlation length 5 = (D/y)“’ which is nothing but the Debye length. This length is also the length in which the charge rearranges through diffusion. At equilibrium this length becomes infinite and the non-equilibrium correlation functions tend to the equilibrium value n,,6(r - r’). Moreover non-equilibrium corrections exhibit Ornstein-Zernike behaviour and are of the same type to those encountered in the Debye model of plasmas”). the difference arising from the fact that for plasmas the correlation length 5 appears always as a finite parameter and therefore has a different meaning. At the instability point. where y is equal to zero, the correlation length diverges but non-equilibrium corrections to the correlation functions vanish. Minus signs in non-equilibrium corrections charge conservation, if at time r a positive fluctuation fluctuations
should, on the are correlated.
It is important evaluated appearing
to note
average,
take
that the correlation
correspond fluctuation place
at
to the fact that due to occurs at r, a negative any
functions
point
(3.3)-(3.5)
r’
in which have been
assuming y to be positive since in the opposite case the integral in (3.2) diverges. The same will be assumed in the calculation of
correlation
functions
A quantity
in the r-t
representation.
that is also interesting to analyze is the correlation function in the It can be obtained from (3.1) by Fourier inversion over w has
k-t representation.
and o ‘. One
(Sn(k, t)Sn(k’.
r’)) = (2+%,,6(k x exp{ -i/3
* Densities
appearing
in the correlation
+ k')
Dy; y
* k(t - t’) - (Dk’ + y)lf - t’l} ,
functions
must be interpreted
in each dimension
(3.6)
FLUCTUATIONS
NEAR
GUNN
INSTABILITY
193
where we can see that j3 plays the role of a velocity of the propagation of density fluctuations. It can be also observed that the relaxation time of the k = 0 mode diverges as y -’ when the field approaches the instability point: critical slowing down behaviour. These results were also obtained in refs. 5 and 6. From (3.6) one can also see that the density correlation function in the k-f representation vanishes in the k = 0 limit, unless at the instability point, y = 0, in which (3.6) does not vanish and the density fluctuations are delta correlated as we saw in (3.3)-(3.5). In the k = 0 case both the relaxation time and the correlation function (3.6) are finite if the system is at the stable branch (y > 0). But if y is negative there will always exist one mode, in our continuum representation, such that its correlation time diverges and makes the density correlation function be divergent. Notice that in a discrete representation, imposing periodic boundary conditions on the fluctuations, the wave number k is given by k = (27rlL)m, where m is an integer different from zero. In that case the density correlation function (3.6) does not diverge at y = 0 but it does at y = -(27rlL)“D. This fact gives a criterion of stability of the fluctuations, as Butcher’) and Ridley22) do for a perturbation from the steady state, showing that diffusion stabilizes waves with sufficiently short wavelength. A stability criterion relating the differential mobility and the length of the sample appears commonly in the literature about the Gunn effect7’9”7 ). For GaAs the stability criterion of the waves involving the diffusion coefficient is most important for very short devices (0.5 microns) with a donor density around 1016cm-3 ” ), whereas in other cases different criteria should be used. Density fluctuations around inhomogeneous stready states (2.14) can be analyzed also. In k-o representation and in 1 - d one has @n(k,
w)&z(k’, co’)) = -47r
Dkk’G(w + w’)n,(k [Dk2 + y - i(w - @k)][Dk’*
+ k’)
+ y - i(w’ - pk’)]
’
(3.7) where n,(k) is the k-Fourier transform of the steady state electron density (2.14) that follows from the n,(x) appearing in the covariance of the random current (2.21). By inverse Fourier transform of (3.7) and integrating over w and w’ the equal-time correlation function is (sn(x, t)sn(x’, t)) =
& j-j-dk dk’ D(k -cc
-2Dkkrn,(k
+
kf)ei(kx+k’x’)
+ k’)’ - 2Dkk’ + 2y + iP(k + k’) ’ (3.8)
194
A. DiAZ-GUILERA
These
integrals
(%2(x, t)6n(x’,
are evaluated
t)) = n,(x)s(x
AND
J.M.
in the appendix
- x’) - n,,
RUBi
and its final result
is
em“ -““’ 25
(3.9) 4 being
a dummy
variable
and F(x, x’, 4) an auxiliary
function
defined
in the
appendix. Notice that the first term in the right-hand the corresponding term in eqs. (3.3)-(3.S), that now long-range
the coefficient contributions
of the to (3.9)
delta come
side of (3.9) is of the same kind as the difference arising from the fact
function depends on position. The from F(x, x’, q) and from the term
corresponding to the homogeneous but non-equilibrium correction studied in (3.3). In view of the definition of F(x, x’, q), (A.9), one concludes that in l-d the contribution is exponential. Moreover it is possible to see that translational invariance valid at equilibrium is not satisfied now since a transformation x+ x + a implies a change n,(x) + 11,(x + a) and F(x, x’, CJ)-+ exp(iqa)F(x, x’, 4). In eqs. (3.3)-(3.5) we saw that at the instability point (y = 0) the nonequilibrium corrections of the density correlation functions vanish in an homogeneous steady state. A detailed analysis of (3.9) shows that the inhomogeneous part of the steady state solution (2.14) makes the nonequilibrium density correlation function diverge. Although this result is independent of the integration conditions the divergence the instability
4. Electric
point
constants is slightly
A.,,
tends
field correlation
A and different
B obtained from the boundary in the two cases (2.13) since near
to zero as y whereas
AFFA remains
finite.
functions
Apart from the density correlation functions we are also interested in the expression of the electric field correlation function. It can be obtained in a way similar to that employed in the derivation of the density correlation function. Hence from (2.23) one obtains the electric field correlation function in the k-w representation in l-d: (6E(k, w)GE(k’, w’)) 4nDe2
S(w + w’)ns(k
= T[Dk2
y - i(o - /3k)][Dk”
+ k’) + y - i(w’ - pk’)]
t
(4.1)
195
FLUCTUATIONS NEAR GUNN INSTABILITY
By inverse Fourier transform, integrating over w and w’ and inserting the value of n,(k + k’) obtained in the appendix, the first contribution to the equal-time correlation function reads
(SE(x, t)SE(x', t))l =
$ [+jdk dk’
‘(’ +k;$,+;+^”
.
(4.2)
--m When the integral is performed one must take into account that y has a positive value in order the integral be convergent. One arrives at (%5(x, t)SE(x’,
t))
1
=
$
5 e-‘xpx”‘s .
Second contribution gives, following the same procedure the calculation of the density correlation function, (SE(x,
t)SE(x’,
t)), = %
e-icllJ’JqF(x, x’, q)lqzo .
Adding the three contributions
(Slqx, t)SE(x’, t)) =
1-
+ wIA 2E
emiplJiJq
$ +
(4.3) as that employed in
(4.4)
one finally arrives at 5
-w2B 2E
e-Ix-x’l/5
e-ir2J’Jq F(x, x’, q)lqzo .
1
(4.5)
This last expression can be written in a more compact form observing the dependence of n,(x) on the spatial coordinate (2.14) and the value of F(x, x’, q) at q = 0 from (A.9),
(S-W, t)SE(x’, t)) =
$ n, (4 +-)F(x, x’, q)lqzo.
(4.6)
From (4.3) one can notice that the homogeneous part of the electric field correlation function diverges as the correlation length 5 does when the system approaches the instability point. In that limit the density correlation functions (3.3)-(3.5) re d uce to the term containing the delta function. This is not a new result since Pytte and Thomas5) and Buttiker and Thomas6) arrived at a similar conclusion for long times. But when fluctuations around inhomogeneous steady states are studied both density and electric field correlation functions diverge at the instability point, although they do it in a different way.
196
A. DiAZ-GUILERA
In the k - t representation contribution
(E(k,
and taking
in a one-dimensional
t)GE(k’, t’)) =
AND J.M. RUBi
into
analysis
2n-Dn,,e’ E2
account
only the homogeneous
the field correlation
exp{ -ipk(t
function
looks
- t’) - (Dk’ + r)lt - t’l} Dk’ + y
x 6(k + k’) ,
(4.7)
where we can see that it is finite for the k = 0 mode, i.e. it has a finite range and diverges at the instability point in contrast to the density-density correlation function (3.6). From electric field fluctuations one can study voltage fluctuations that can be compared
with experiments.
This will be the subject
of a forthcoming
paper”).
Acknowledgements This paper has been partially supported by a grant of the Comision Investigation Cientifica y Ticnica of the Spanish Government.
Asesora
de
Appendix Fourier
transforming
equation
(2.14)
at ti
+r
+L
P,AE
th\ e e"' dx + ~
e A’ dx + p
n,(k) = n,,
one arrives
e"?‘dx
~7
-x
(A.1) The first integral is a delta function same structure and for this reason exponential
is Taylor
expanded
6(k). The second and third integral have the they are evaluated in a general form. If the
we get
(A .2)
FLUCI-UATIONS
Thus (A.l)
NEAR
GUNN
INSTABILITY
197
becomes
n,(k) = 2~,6(k)
n$O $
+ ?
a(k)(Ap;+’
$
+ I$;+‘)}.
(A.3)
For the sake of clarity we will analyze separately the three different contributions to n,(k). The first one corresponds to the homogeneous steady state electron density n,, and has been obtained in eq. (3.3). Inserting the second contribution into (3.8) one obtains
c@$ /+f
(6n(x, t)&z(x’, t))2 = g
dk dk’
n-0
_ ’
D(k
+
Ic’)~
2~kk -
--m
eWx+k’x’)
.
(A.4)
6(q)
(A.5)
Dq2 + 2D( q’ - q)q’ + 2y + iPq ’
G4.6)
a”
2Dkk’ + 2y + iP(k + k’) d(k + k’)”
6(k + k')
After a change of variables, k + k’ = q, k’ = q’, it reads
(Sn(x, t)&z(x’, t))2 =
$$ ; @$ l+f dq dq' no
.
_qq
X
and integrating
_
--ic
qt)qt
ei(q-q’b
eiq’*’
d”
Dq2 - 2D( q - q’)q’
+ 2y + i@q aq”
by parts 12times in q,
(&2(x, t)ifin(x’, t)), =
nzo v $6
E
eiqx
+oC 2D(q!
X i
dq’
-cc
_
q)q’
e-iq’(x-x’)
The integral appearing in (A.6) is +-= 2D(q!
i -m
dq’
_
q)qt
e-iq’(x-x’)
Dq2 + 2D( q’ - q)q’ + 2y + ipq eidx’-x)/2
_
= 27rS(x - x’)
1 (; +$ +g”2,x_xI,)
exp
_
n(Dq’+ 2~ + i&l 20 (
64.7)
A. DiAZ-GUILERA
198
AND
J.M.
RUBi
Then one obtains (Sn(x, t)Sn(x’, t))
2
-pifc e/*l\qx
=
-
x’)
r x,,,
(-ip,)”
P&
-
~
e
0”
,I_11
x (&I’ + 2~ + iPqJF(x, x’, 4)ly-,,,
(A.81
where F(x, x’, q) is an auxiliary function defined by lq(r*.l’)/7
F(x, x’. q) =
ez
$+%+E iPq (A.9) Rearranging
(A.B) it can be written as
(6n(x, t)Sn(x’, t)), =
-‘“lat e”“6(x x -f n-0
(-ip,)“’ n!
- x’) -
d”
j-g
F(x, x’, q)l,=,,
(A.10)
The third contribution is equal to the second one changing p, by pz and A by B. Adding the three contributions to the equal-time density correlation function, taking into account the expression of the steady state electron density (2.14) and the values of p, and pI (2.11), one finally obtains e~~‘H’1/5 (&2(x, t)tin(x’, t)) = n,(x)ti(x
- & xe
which corresponds
- x’) - n,,
25
{ /_L,A(y + 2&,)
-“zb’dq}F(x,
x’, q&:,,
e~‘F1”!a’ + /+B(Y + 2pp.z) ,
(A.ll)
to (3.9)
References 1) J.M. Rubi, Fluctuations Around Equilibrium, in: Recent Developements in Non-equilibrium Thermodynamics, J. Casas-Vazquez, D. Jou and G. Lebon, eds., Lecture Notes in Physics (Springer, Berlin, 1984). 2) A.-M.S. Tremblay, Theories of Fluctuations in Non equilibrium systems, in: Recent Developements in Non-equilibrium Thermodynamics, J. Casas-Vazquez, D. Jou and Q. Lebon, eds. Lecture Notes in Physics (Springer, Berlin, 1984).
FLUCTUATIONS
NEAR GUNN INSTABILITY
199
3) V.M. Zaitsev and MI. Shliomis, Soviet Physics JETP 32 (1971) 866. T.R. Kirkpatrick and E.G.D. Cohen, J. Stat. Phys. 33 (1983) 639. 4) R.A. Pasmanter, D. Bedeaux and P. Mazur, Physica WA (1978) 151. 5) E. Pytte and H. Thomas, Phys. Rev. 179 (1969) 431. 6) M. Buttiker and H. Thomas, Solid-State Electron. 21 (1978) 95. 7) J. Keizer, J. Chem. Phys. 74 (1981) 1350. 8) J.B. Gunn, Solid State Commun. 1 (1963) 88. 9) P.N. Butcher, Rep. Prog. Phys. 30 (1967) 97. 10) S. Kabashima, H. Yamazaki and T. Kawakubo, J. Phys. Sot. Jpn. 40 (1976) 921. 11) K. Matsuno, Phys. Lett. 31A (1970) 335. 12) K. Nakamura, J. Phys. Sot. Jpn. 38 (1975) 46. 13) S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962). 14) A.-M.S. Tremblay, M. Arai and E.D. Siggia, Phys. Rev. A23 (1978) 1451. 15) D. Bedeaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355. 16) B. Kadomtsev, Phenomenes Collectifs dans les Plasmas (Mir, Moscow, 1979). 17) J.E. Carroll, Hot Electron Microwave Generators (Elsevier, New York, 1970). 18) CM. Snowden, Rep. Prog. Phys. 48 (1985) 223; and references quoted therein. 19) H. Kroemer, IEEE Trans. Electron Devices ED13 (1966) 27. 20) B. Knight and G.A. Peterson, Phys. Rev. 147 (1966) 617. 21) N.G. van Kampen, Statistical Mechanics of Classical Plasmas, in: Fundamental Problems in Statistical Mechanics, E.G.D. Cohened. (North-Holland, Amsterdam, 1968). 22) B.K. Ridley, Proc. Phys. Sot. 86 (1965) 637. 23) A. Diaz-Guilera and J.M. Rubi, Physica 135A (1986) 200, subsequent article.