Alexey Shorokhov1, Timo Hyart2, Kirill Alekseev3

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Parametric amplification in superlattices

• Alexey Shorokhov1, Timo Hyart2, Kirill Alekseev3

1Mordovain State University, Saransk,
Russia 2University of Oulu, Oulu,
Finland 3Loughborough University, Loughborough, UK
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Energy in the tight-binding approximation
In the ballistic regime
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Dissipative regime
Bloch oscillations between scattering events if
W0t gt1
If we have a weak perturbation
VI-characteristic of a SL demonstrates NDC due to
the Bloch oscillations of electrons if W0tgt 1
(Esaki and Tsu, 1971 )
NDC
The static NDC should be accompanied by a
small-signal gain for frequencies from zero up to
several THz.
NDC leads to INSTABILITIES (electric domains)!
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How we can avoid domains? To use ac-pump instead
of dc-bias.
In this case we have a gain without domain in
some cases!
What is the physical nature of this effect?
Parametric resonance due to Bragg reflections!
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Recall parametric resonance in mechanics
Periodic variation of energy storing parameter
with frequency
Parametric resonance is instability phenomenon!
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Recall parametric resonance in electricity
Parametric circuit oscillations
A variable capacitance diode is used to vary the
capacitance in resonance circuit at twice the
resonance frequency
Oscillator signal (upper trace) and the
capacitance voltage with a parametrically excited
component at half the oscillatory frequency
(lower trace)
Decreasing the oscillatory frequency (upper
trace) increases the amplitude of the parametric
oscillations
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Differential equation
Here the mcoswt term acts as an energy source
and is said to parametrically excite the system
The Mathieu equation describes many physical
systems to a sinusoidal parametric excitation
such as an LC Circuit
In the case of strong dissipation and small m
y?0 and we have no resonance
Parametric amplifiers (paramp)
A parametric amplifier is basically a mixer with
gain.
We assume that the damping is sufficiently strong
that, in the absence of the driving force, the
amplitude of the parametric oscillations does not
diverge
Let the external driving force is at the mean
resonance frequency w0 then
for w0w1w/2 and f0
As m approaches the threshold 2g, the amplitude
diverges
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Parametric effects in superlattices Ballistic
case
Pure ac pump field
Message energy e(t) and mass mm0/e in band
vary with frequencies
Let also applied Epr(t) E2 cos(w2tf) (mode of
resonant cavity) with E2?0. Parametric resonance
at
satisfied if w2 odd or even harmonics of w1!
Note If a bias Edc is also included to the pump
field, e(t) oscillates with two combinations of
frequencies W0weeven and W0weodd, where
weoddsw1 with s1,3,5
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Balance equations
The balance equations are equivalent to the
Boltzmann equation in the single relaxation time
approximation and in the tight-binding
approximation
In the case t?8 the balance equations can be
transform to the Bloch-Ziner equations
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Ballistic case
energy
absorption
dc-curent
Definition of the dc-current
Definition of the absorption
Parametric term!
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Ballistic case
But what can we say about the additional terms in
dc-current and absorption?
They arise because the drive contains an initial
bias in phase!
Take the oscillatory driven dynamics
If we have only phase dependent probe field
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The differential inductance
For a small signal we can expand the velocity V(t)
Or in terms of a current
is the differential inductance corresponding to
one superlattice period.
Thus in the case of ballistic transport the
parametric effects are generated by an
oscillating inductance caused by the oscillating
effective mass of the electrons.
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Parametric effects in superlattices Dissipative
case

• Analytic solution of Boltzman equation

• Calculation of averaged with f(p,t) velocity
  V(t) and energy e(t).

• Calculation of absorption of probe A ltV (t)
  Epr(t)gtt

Results
If w2mw1 (m is the odd number), the absorption
is a sum of two terms
Acoh describes the parametric gain, Aincoh
describes the free carrier absortion
If parametric gain is more than the free carrier
absorption then Alt0, and
we have the amplification of the probe field
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Parametric amplification
The amplitude of coherent absorption Bgt0 and
b2E2/w2.
Gain has maximum at an optimal phase fopt
The variables B and fopt can be represented in
terms of the specific harmonics of energy and
out-of-phase component of electron velocity Vsin
VsinIsin
The physical origin of the parametric resonance
is a periodic variation of effective electron
masses in miniband and, at high THz frequencies,
also a variation of specific quantum inductance.
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• out-of-phase components of velocity describe
  inductive response of inertial miniband electrons
  to ac field in the limit w1tgtgt1
• interaction of miniband electrons with THz
  fields has quantum nature

Remarkably, the pump could suppress the free
carrier absorption!
Ainc gt 0 guarantees electric stability
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Absorption of the 4 harmonic
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Two schemes of paramp in SL without corruption
from generated harmonics
Single miniband and ac pump (w1), amplification
of weak signal (w2) If Edc 0 gain at w2
2w1, 4w1, . . . param resonance weeven/2w2
If Edc ? 0 gain at w2 w1/2, 3w1/2, . . .
param resonance weodd/2w2
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Results Threshold for amplification at even
harmonics, no dc bias
ffopt
Regions above the marked curves correspond to
gain at w2 2w2, 4w2, 6w2. Red areas correspond
to NDC (In SSL electric instability).
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Results Amplification at half-harmonics, with dc
bias (low threshold!)
Inset Magnitude of gain (A) vs pump amplitude E1
for w1t 0.25.
Amplification at w2 w2/2 for Edc Ec and f
fopt. Marked region between curves corresponds to
gain, red area corresponds to NDC.
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Direct current
Direct current due to harmonic mixing
Direct current due to applied bias
Parametric term
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Unbiased superlattice
No current for odd harmonics!
(energy has only even harmonics of the pump
field)
Direct current generation is the parametric
effect.