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: sergio@if.uff.br (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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