The dynamics of excitons and trions in resonant tunneling diodes
I. Campsa, S.S. Maklerb,*, A. Vercikc, Y. Galvao Gobatoa, G.E. Marquesa,
M.J.S.P. Brasild
aDepartamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos-SP, BrazilbInstituto de Fısica, Universidade Federal Fluminense, Niteroi-RJ, Brazil
cFaculdade de Zootecnia e Engenharia de Alimentos, Departamento de Ciencias Basicas, Universidade de Sao Paulo, Pirassununga-SP, BrazildGrupo de Propriedades Opticas, Instituto de Fısica Gleb Wataghin, Universidade de Campinas, Campinas-SP, Brazil
Received 24 January 2005; received in revised form 13 April 2005; accepted 23 April 2005 by A. Pinczuk
Available online 11 May 2005
Abstract
The aim of this work is to study the dynamic formation and dissociation of trions and excitons in double barrier resonant
tunneling diodes. We propose a system of rate equations that takes into account the formation, dissociation and annihilation of
these complexes inside the quantum well. From the solutions of the coupled equations, we are able to study the modulation of
excitons and trions formation in the device as a function of the applied bias. The results of our model agree qualitatively with the
experiments showing the viability of these rate equations system to study the dynamics of complex systems.
q 2005 Elsevier Ltd. All rights reserved.
PACS: 78.55.Km; 78.67.Kn; 78.66.Kw
Keywords: A. Excitonic complexes; A. Trions; D. Resonant tunneling; D. Charge buildup
The Coulomb interaction between electrons and holes may
form complexes with two, three or more particles in high
quality semiconductor crystals. These excitonic complexes
were predicted by Lampert [1] in 1958. Since then, many
authors have studied their ground-state energies and wave
functions [2,3] in the presence of electric [4] and magnetic
fields [5] and also the influence of such atomic-like particles on
the optical [6–9] and on the transport [10–14] properties of
localized and resonant states in semiconductor devices.
The neutral exciton (X0) is the well known hydrogen-like
Coulomb binding of an electron to a hole. If one third
particle, such as an electron (hole), becomes bounded, via
this interaction, then a negatively (XK), (positively (XC))
charged exciton is formed and these particles are commonly
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2005.04.026
* Corresponding author. Address: Universidade Federal Flumi-
nense, Campus da Praia Vermelha, Niteroi 24210340, Brazil. Tel.:
C55 21 27142249; fax: C55 21 26295887.
E-mail address: [email protected] (S.S. Makler).
referred as trions. One of the most suitable devices to study
the formation of excitons and trions is the double barrier
resonant tunneling diode (DBRTD) where is easy to control
externally the carrier populations via the applied bias to the
structure. In this work, we have developed a set of
phenomenological coupled rate equations to explain the
experimental results for photoluminescence and transport
measurements on n-i-n DBRTD sample. For this structure,
the minority carriers (holes) are photo-generated by laser
excitation at one side thus only the negative trions are
significantly present. The formation of XC and neutral bi-
excitons (X2), would require a larger population of holes,
they can be observed, for example, in specially designed p-i-
n doped samples. As will be shown later, the model is
capable to reproduce qualitatively the optical and transport
experimental results.
Some of the mechanisms governing the population of
carriers considered in this work are: (i) rates for electron
(hole) injection to and the escape from the well,
Solid State Communications 135 (2005) 241–246
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I. Camps et al. / Solid State Communications 135 (2005) 241–246242
respectively; (ii) the formation of excitons due to electron–
hole interaction and due to the scattering mediated trion
non-radiative decay; (iii) the formation of trions via
electron–exciton interaction and free two-electron–one-
hole interaction; (iv) the exciton and trion decays with
emission of a photon (luminescence).
The easier way to form negative trions is through the
capture of an extra electron by neutral excitons. A simple
chemical law of mass action [15,16] can be used to obtain
the populations of particles involved in the process. The
relation between these populations, at a given temperature
T, can be estimated as
nenX0
nXK
ZKðTÞZ4mekBT
pZ2
� �eKðEXK
B =kBTÞ (1)
Observe that the determination of one population
species, say trions ðnXKÞ with binding energy EXK
B , requires
the knowledge of both exciton ðnX0 Þ and electron (ne)
concentrations. Furthermore, the expression (1) is a
consequence of a temperature dependent chemical equili-
brium between the reacting elements. This makes the study
of the kinetics in a system with different populations a very
difficult task, especially when other creation and annihil-
ation mechanisms between these particles are considered.
To account for these mechanisms governing the charge
buildup in the DBRTD and the kinetics of the different
particles, we are proposing a system of rate equations that
describe the balance between the constituents. This idea is
widely used to study the dynamics of different physical
systems [17,14,18–20] and was already suggested by Ramon
et al. [21] to be used in a similar system. The main populations
considered in this work are described as follows.
An exciton can be formed, when an electron (e) and a
hole (h) interact via Coulomb forces, and this process can be
represented by the chemical reaction, X0%eCh. The
population balance rate between free electrons, free holes
(free meaning not bounded to any excitonic complex) and
excitons, for this reaction, can be written as
Ta ZaðnX0 KnenhÞ (2)
The parameter a represents the creation/annihilation rate for
the exciton population ðnX0 Þ whereas the first (second) term
accounts for the dissociation (formation) of excitons. The
other parameters, ne and nh, are the free electron and hole
populations present at given instant and bias.When the X0
lifetime, inside the quantum well layer, is larger than the
recombination time, the formed excitons can further interact
with the free electrons (holes) to form negatively (posi-
tively) charged trions. In this symmetric n-i-n structure used
here, the electrons are the majority carriers ðne[nhÞ thus,
the probability to form negative trions is much larger than
for positive trions or for bi-excitons. Thus we will be only
considering here the formation of XK, and the reaction for
this process can be written as, XK%eCX0. From charge
and particle conservation, the populations of trions ðnXKÞ,
excitons and free electrons can be related through the
following expression
Tb ZbðnXK KnenX0 Þ (3)
where b represents the creation or annihilation rate of trions
due to the interaction between free electrons and X0. As
before, the first (second) term in expression (3) represent the
trion dissociation (formation).
Finally, if two free electrons interact with one free hole,
via Coulomb forces, the XK can be formed. Similarly, if two
free holes interact with one electron, the result is a XC. The
chemical reaction representing the trion formation can be
sketched as XK%eCeCh, and the balance rate governing
these particle populations can be written as
Tg ZgðnXK Kn2enhÞ (4)
Again, the parameter g represents the creation/annihila-
tion rate of trions formed from to two-electron–one-hole
interaction, and the first (second) term in Eq. (4) represents
the annihilation (creation) in the process shown as right
(left) arrow in the chemical reaction.
The exciton and trion creation/annihilation rates a, b and
g already include the e–e and e–h correlation effects [21].
Therefore, the exciton and trion populations appearing in the
expressions (2)–(4) can be understood as ‘effective’ values
that implicitly have taken into account, at least, the
stationary effects produced by e–e and e–h correlations.
When bias is applied to the structure, the electrons
(holes) accumulated on the emitter (collector) interface are
injected into the quantum well layer. To take into account
this process, we define the electron (hole) injection rate as
Ge(Gh). These two type of charge built inside the well may
either interact to form the excitonic complexes, as described
above, or may be ejected (escape) through the respective
opposite barriers. We account for these escape rates as Rene
and Rhnh for electrons and holes, respectively. Finally, the
last processes are the exciton and trion decay rates, given by
RX0nX0 and (RX)–nXK, respectively, that measure the
population change during the recombination of excitons
and trions with the emission of a photon. Each trion
recombination leaves a free electron. They form two
tunneling channels for the injection processes.
By taking into account these processes just described, the
system of rate equations that governs the dynamics of all
particle populations can be written as:
dne
dtZGe CTa CTb C2Tg KRene C ðRXKÞnXK;
dnh
dtZGh CTa CTg KRhnh;
dnX0
dtZKTa CTb KRX0nX0 ;
dnXK
dtZKTb KTg K ðRXKÞnXK
(5)
Fig. 1. Electron (solid and dotted lines) and hole (dashed line)
injection rates obtained from the experimental measurements (see
text for details).
I. Camps et al. / Solid State Communications 135 (2005) 241–246 243
It is worthwhile to observe that the populations obtained
from the solutions of this system of coupled equations
should be viewed as mean values calculated at TZ0 K. This
implies that Eq. (1) cannot be obtained by taking the limit
t/N in the time-dependent solutions. Moreover, we have
not considered quantum fluctuations on the population of
free carriers, which are referred as shot-noise [22]. Thus, we
want to solve the coupled equations for the populations ne,
nh, nXK and nX0 under steady state or stationary condition.
The bias dependence of the escape rates Re and Rh have
been calculated by using a tight-binding approach [20]. The
voltage dependence of the rates a, b, and g appearing in
expressions (2)–(4), were considered as having similar
functional dependence as the calculated in Ref. [21]. These
rates were calculated by using the scattering theory as
functions of the exciton and trion energies, depending on the
applied voltage and on the carrier densities. Finally, in order
to compare theoretical and experimental results, the electron
and hole injection rates, (Ge, Gh), as well as the decay rates,
(RXK and RX0 ), were taken from the experimental results.
The electron injection rate was obtained from the measured
current–voltage characteristics, as GeZI/e. The same
procedure has been used for hole injection rate extracted
from the photocurrent measurements [23].
Our simulation were performed on a symmetric n-i-n
GaAs–Al0.35Ga0.65As DBRTD grown by molecular beam
epitaxy, with 10 nm barriers and 5 nm well widths,
respectively. The system is enclosed by 60 nm undoped
GaAs layers and 300 nm Si-doped nC-GaAs (1018 cmK3)
layers grown on both sides of the structure. Annular contacts
on 500!600 mm2 mesas allow optical measurements under
applied voltage. The samples were mounted in a Janis close
cycle cryostat. The spectra were recorded by a Spex 500M
single spectrometer. In order to get PL emission, extra
minority holes needed to be created by optical excitation
near the contact. A coherent ArC ion laser was used as
excitation source and the PL signal was detected by a
photocounting system connected to a thermoelectrically
cooled R5108 Hamamatsu photomultiplier.
Observe, in Fig. 1, that the electron injection rate (solid
line) with the laser on is strongly dependent on voltage and
displays two peaks (P1 and P2) around 320 and 430 mV,
respectively. The origin of the peaks will be discussed
below. Also, the electron injection rate (dotted line) in the
dark (laser off) and the hole injection rate (dashed line) are
plotted. In the voltage range used in the measurements, the
deformation in the valence band is such that the holes are
seeing only one barrier and their injection rate increases
linearly with increasing bias [23].
The values for the decay rates, RXK and RX0 , were
extracted from the photoluminescence spectra shown in Fig.
2. These asymmetric spectra were fitted with Gaussian
functions for the XK and X0 recombination channels, with
the decay rates been taken from the full-width-half-
maximum of each fitted peak. The trion (left peak) and the
exciton (right peak) populations were obtained from the
resulting integrated intensities (area under the peak). The
inset shows very strong voltage dependence for both
populations.
The populations of free electron and hole were obtained
from the solution of the system (5), under stationary regime,
and they are shown in Fig. 3. Observe that electron
population has two peaks with slightly different intensities,
in contrast with the I–V curve shape (solid line in Fig. 1).
Therefore, even when the device is biased out of resonance
(the main resonance voltage is close to w430 mV), the free
electron population inside the quantum well displays
another well defined peak. The origin of this extra peak is
related to the dissociation of trions when scattered by free
electrons [20]. When a trion is fully dissociated via this
mechanism, the process increases the free electron popu-
lation. In order to become efficient this process requires a
minimum critical density of electrons (sample dependent
critical voltage) that is shown as this second peak in the I–V
characteristics. Above this critical voltage, the scattering
decreases population of trions.
In order to compare experimental and theoretical results
from the set (5) we have analyzed the dependence of nXK=
nX0 as function of bias. We are plotting, in Fig. 4, the
experimental [24] (symbols) and the calculated (line)
population ratios. As was shown before (see the inset in
Fig. 3), the trion population has a peak near 320 mV
whereas the exciton population has a minimum. Their
opposite behavior shows that scattering is interchanging
their population. Furthermore, we have used a monotonic
voltage dependence for the creation and annihilation rates a,
b and g. In the present case, this assumption was unable to
reproduce the experimental results. However, using non
monotonic bias dependence for a, b and g, the simulations
show a fairly good agreement with the experiment. There-
fore, we are lead to believe that the cause for the transition in
the nXK=nX0 ratio is the non linear voltage dependence for the
creation/annihilation rates a, b and g.
Summarizing, in this work we have proposed a system of
Fig. 2. Photoluminescence spectra for different applied bias from Ref. [24]. In the inset, the trion and exciton populations obtained from the
fittings of the spectra at each voltage.
I. Camps et al. / Solid State Communications 135 (2005) 241–246244
rate equations to describe the dynamics of trion and exciton
formation in DBRTD as functions of the applied voltage.
The main ingredients used are the processes associated to
excitons and trions formation/dissociation via the electron–
Fig. 3. Free electron and hole populations, obtained from the
hole interaction and via the exciton–electron interaction as
well as radiative and scattering processes together with the
injection (escape) of carriers into (from) the quantum well.
The steady state solution has given important information
stationary solution of system (5), as a function of bias.
Fig. 4. Experimental (circles) and calculated (solid line) normalized trion-exciton ratios.
I. Camps et al. / Solid State Communications 135 (2005) 241–246 245
on the population rates that reproduce the experimental
result. The fairly good agreement between theoretical and
experimental results has demonstrated the viability of the
proposed rate equations to study the dynamics of composed
systems with different simultaneous mechanisms governing
their population balance. Furthermore, in the case that the
values of the exciton and trion populations could be known,
it is possible to use the system (5) to obtain the particles
creation annihilations rates.
Acknowledgements
The authors are grateful to the Brazilian agencies
FAPESP and CNPq for financial support.
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