Download from stationary to moving domains

Noise-induced current oscillations in superlattices: from
stationary to moving domains
Eckehard Schöll∗ , Johanne Hizanidis∗ , Alexander Balanov∗,† and Andreas
Amann∗,∗∗
∗
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
†
School of Physics and Astronomy, University of Nottingham, UK
∗∗
Tyndall National Institute, Lee Maltings, Cork, Ireland
Abstract. We provide a detailed understanding of the transition between stationary and moving domains in semiconductor
superlattices. We show that it corresponds to a global bifurcation (saddle-node infinite period bifurcation on a limit cycle or
SNIPER). The dynamics close to such a bifurcation is moreover very sensitive to noise. We show that front motion, and hence
current oscillations, can be induced by shot noise of the tunneling processes even if the deterministic system exhibits a stable
stationary field domain.
Keywords: superlattices, global bifurcations, noise-induced oscillations
PACS: 05.45.-a, 73.21.Cd, 05.40.-a
INTRODUCTION
Superlattices are well known to exhibit complex dynamic
behavior if they are driven by a high dc bias into the
regime of negative differential conductivity. We consider
a sequential tunneling model [1], which assumes that
electrons are localized in one particular well and only
weakly coupled to the adjacent wells. The resulting tunneling current density Jm→m+1 (Fm , nm , nm+1 ) from well
m to well m + 1 depends only on the electric field Fm between both wells and the electron densities nm and nm+1
in the respective wells (in units of cm−2 ). The dynamic
equations are then given by the continuity equations:
e
dnm
= Jm−1→m − Jm→m+1 ,
dt
(1)
where e < 0 is the electron charge. The applied voltage
between emitter and collector gives rise to a global constraint (d is the superlattice period):
N
U =−
∑ Fm d,
(2)
m=0
where N is the number of wells in the superlattice. The
electron densities and the electric fields are coupled by
the discrete Poisson equation:
εr ε0 (Fm − Fm−1) = e(nm − ND ),
m = 1, ..., N,
FIGURE 1. Bifurcation diagram in the (σ ,U) plane. Dark
and white regions correspond to moving and stationary fronts,
respectively. The border of the dark region marks the bifurcation line where a saddle-node bifurcation on a limit cycle takes
place. Numbers denote the positions of the stationary accumulation front in the device.
(3)
where εr and ε0 are the relative and absolute permittivities, ND is the doping density, and F0 and FN are the
fields at the emitter and collector barrier, respectively.
We use Ohmic boundary conditions, J0→1 = σ F0 and
JN→N+1 = σ FN NnND where σ is the contact conductivity.
FROM STATIONARY TO MOVING
DOMAINS
Depending on the applied bias U and the contact conductivity σ , either stationary or moving electric field domains may occur [2, 3]. The former lead to multistability
due to multiple branches in the current-voltage characteristics, while the latter are associated with current oscillations, and with moving electron accumulation and
depletion fronts forming the boundaries of the field domains.
By varying σ and U, a sawtoothlike structure is observed in the (σ ,U) plane, where stationarity is periodically interrupted by tongues of moving fronts (Fig 1). At
the transition from stationary to moving fronts a global
bifurcation takes place, namely a saddle-node bifurcation on a limit cycle. In such a bifurcation, a stable and
NOISE-INDUCED MOVING FRONTS
Such excitable behavior close to a deterministic bifurcation described in the previous section, is also sensitive
to noise. We therefore consider the effect of shot noise
which for large current density can be approximated
by Gaussian white noise of the form D(Jm−1→m )ξm (t)
[4], with hξm (t)i = 0, hξm (t)ξm′ (t ′ )i = δ (t − t ′ )δmm′ , and
a current dependent noise intensity D(J) = (eJ/A)1/2
which increases with decreasing current cross section A.
This yields the Langevin equations e dndtm = Jm−1→m +
Dξm (t) − Jm→m+1 − Dξm+1 (t), where a stochastic force
is added to Eq. (1). The system is prepared at a stable accumulation front denoted by the cross in Fig. 1 and noise
is switched on. The behavior of the system changes dramatically with increasing noise intensity (Fig. 2(b,c)): after a time during which the accumulation front remains
stationary, a pair of a depletion and another accumulation
front is generated at the emitter.
Due to the global voltage constraint (2) the growing
dipole field domain between the injected depletion and
accumulation fronts requires the high field domain between the stationary accumulation front and the collector
to shrink, and hence that accumulation front starts moving towards the collector. For a short time two accumulation fronts and one depletion front coexist in the sample and form a tripole until the first accumulation front
reaches the collector and disappears. When the depletion front reaches the collector, the remaining accumulation front must stop moving because of the global constraint (2), and this happens at the position where the first
accumulation front was initially localized. After some
time noise generates another dipole at the emitter and
100
well # well # well #
an unstable fixed point lying on the same closed invariant manifold collide giving birth to a limit cycle. The
stable fixed point is associated with a stationary front
and a constant current density. Upon collision with a
saddle-point, it is replaced by a limit cycle of approximately constant amplitude and increasing frequency, corresponding to tripole oscillations in the spatiotemporal
picture. The frequency of these oscillations vs the bifurcation parameter U, obeys the characteristic square-root
scaling law that governs a saddle-node bifurcation on a
limit cycle. Approaching the critical point, the frequency
of the oscillations tends to zero. This corresponds to an
infinite period oscillation and therefore this bifurcation
is also known as saddle-node infinite period bifurcation
(SNIPER). Physically, it is associated with the successive
injection of dipole domains at the emitter, which traverse
the system, and interact with the stationary charge accumulation front which forms the boundary of the stationary field domain, thus triggering a tripole oscillation.
(a)
0
100
(b)
0
100
0
0
(c)
time [ns]
100
FIGURE 2. Noise-induced front motion: Space-time plots
of the electron density for (a) D=0 (no noise), (b) D =
0.5As1/2 /m2 , (c) D = 2.0As1/2 /m2 . Light and dark shading
corresponds to electron accumulation and depletion fronts, respectively. The emitter is at the bottom. U = 2.99V and σ =
2.0821012488Ω−1 m−1 . Superlattice parameters as in Ref.[3].
the above scenario is repeated. Therefore, noise induces a
global change in the dynamics of the system forcing stationary fronts to move through the entire device. These
moving fronts are associated with current oscillations in
the form of spiking with two disctinct timescales: one is
associated with the time needed for a new depletion front
to be triggered at the emitter, and the other is connected
to the time the depletion front takes to travel through the
device. The former is highly controlled by the level of
noise whereas the latter is rather insensitive to it. The interaction between the two timescales is moreover responsible for the effect of coherence resonance in the system
([5]) i.e., there is an optimal level of noise at which the
regularity of front motion is enhanced. Concluding, the
high sensitivity to changes in the noise intensity and in
the distance from the bifurcation point yield superlattices
promising for applications as fast noise sensors.
ACKNOWLEDGMENTS
This work was supported by DFG in the framework of
Sfb 555. We thank N. Janson for stimulating discussions
and help in the stability analysis.
REFERENCES
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2. E. Schöll, Nonlinear spatio-temporal dynamics and chaos
in semiconductors, Cambridge University Press, 2001.
3. A. Amann and E. Schöll, J. Stat. Phys. 119, 1069-1138
(2005).
4. Y. M. Blanter and M. Büttiker, Phys. Rep. 336, 1-166
(2000).
5. J. Hizanidis, A. G. Balanov, A. Amann and E. Schöll,
Phys. Rev. Lett. 96, 244104 (2006).