Graded dots

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In semiconductors, we are used to think in terms
of quasi independent electronic and vibronic
degrees of freedom. However, we know that they
are interacting.
We need first to define what are the eigenstates
of each system and then depending on these
eigenstates to understand their interaction
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The limits of the Macro-atom model I the
coupling to the intrinsic degrees of freedom
In an atom for energies comparable to
the resonance transition (? eV) there exist only
electronic degrees of freedom the nuclei are
frozen. ? The atomic desexcitation is mostly
radiative .
In quantum dots there exist other elementary
excitations, vibrations, magnons,.
electrostatic fluctuations. The electrons are
coupled to them. ? The dot desexcitation is
not necessarily radiative.
Simple systems Long coherence time ?  easy 
manipulations
Complex systems Short coherences ?? Easy
manipulations??
Before carbon - copying on dots what we know
operative in atoms, it is necessary to ascertain
that the couplings between electrons and the
other excitations are indeed negligible. In
reality we shall show that this is possible only
at cryogenic temperatures.
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Energy relaxation
Quantum dot
Quantum well
ce
ch
tcap
E2
tcap
H2
trelax
trelax
Hdip.
H1
E1
Photon emission
Photon emission
In both cases one injects electron - hole pairs
in the continuum. Carriers have to be captured.
They have to relax to the lower possible
energy. Finally, they recombine. In dots the
capture and the relaxation involve discrete
levels. In wells the capture and the relaxation
involve 2D continuums.
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What is interesting in the previous example is
the vocabulary that we have used. We have
implicitly assumed that the electron and the
phonon worlds were totally independent as if
there was no way that both excitations could
interact so strongly that they would form a
single, entangled, world. Technically speaking,
we have reasoned in terms of factorized
quantum sates of the electron - phonon system.
These factorized states are the eigenstates of
Hel Hph but not of the actual HamiltonianHel
Hph Hel-ph. Thus, we need to examine in the
following how these factorized states evolve
with time and to find a criterion which could
justify our decoupling between the electronic and
vibrational degrees of freedom.
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So, in our small He-ph description we find that
the probability to leave the initial state is
proportional to the square of the strength of
the perturbation and at long time is
proportional to the square of the inverse of the
detuning
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FERMI RABI
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Sketch of the survival probability to be in the
initial state versus t for irreversible dilution
(Fermi) or for two levels systems (Rabi
oscillations)
The amplitude of the Rabi oscillations is the
larger when there is no energy detuning between
the two levels.
When the two levels system is coupled to a
thermostat, the Rabi oscillations become
damped.
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Irreversible dilution and Fermi golden rule
t 0
In bulk, quantum wells and quantum wires the
electrons ? phonons states form a broad
composite continuum. Electronic continuum ? broad
(? 1 eV) Acoustical phonon continuum ? narrow (?
20 meV) Optical phonon continuum ? very narrow (?
3-4 meV)
As a result the probability that the electron -
phonon system returns to its initial state
(survival probability) decays exponentially with
time
t
There is an irreversible dilution of the initial
state in the continuum of final states.
Equivalently, each factorized state ?nkgt??nqgt
acquires a finite lifetime given by the Fermi
golden rule
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Relaxation in QW s
In quantum wells the irreversible emission of
LO phonons is very efficient when energetically
allowed. Acoustical phonon emission is less
efficient but almost always energy allowed
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Energy loss rates
Once the Fermi golden rule established, one can
compute energy loss rate curves e. g. the power
given by the electron gas to the phonons is ltPgt
Snk f(enk) Pnk
energy conservation in the total electron -
phonon system
form factor (if dlt3)
final electron state is empty
electron - phonon shape
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Energy loss rate the triumph of the weak
coupling between electrons and phonons
Data adapted from J. Shah Hot carriers in
semiconductor nanostructures Physics ans
Applications. Acad. Press, 1992, p.290
By calculating the transition rates by means of
the Fermi golden rule, it is possible to compute
the Energy Loss Rate of excited carriers. The
comparison with experiments is excellent. For
dispersionless phonons (hwLO) and maxwellian
carriers, one finds ELR?(1/t)
hwLOexp(-hwLO/kTc) where Tc is the hot carrier
temperature
ENERGY LOSS RATE / HOLE (WATTS)
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