Determination of oscillator strength of confined excitons in a semiconductor microcavity
We have achieved a significant experimental Rabi-splitting (3.4 meV) for confined polaritons in a planar semiconductor λ microcavity for only a single quantum well (SQW) of GaAs (10~nm) placed at the antinode. The Rabi-splitting phenomena are discussed in detail based on the semiclassical theory, wh...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Determination of oscillator strength of confined excitons in a semiconductor microcavity / E.A. Cotta, P.M.S. Roma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23702:1-11. — Бібліогр.: 30 назв. — англ. |
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irk-123456789-1534522019-06-15T01:26:18Z Determination of oscillator strength of confined excitons in a semiconductor microcavity Cotta, E.A. Roma, P.M.S. We have achieved a significant experimental Rabi-splitting (3.4 meV) for confined polaritons in a planar semiconductor λ microcavity for only a single quantum well (SQW) of GaAs (10~nm) placed at the antinode. The Rabi-splitting phenomena are discussed in detail based on the semiclassical theory, where two coupled harmonic oscillators (excitons and photons) are used to describe the system. In this way, we can obtain the dispersion curve of polaritons, the minimum value for the cavity reflectance and the oscillator strength to reach the strong coupling regime. This approach describes an ensemble of excitons confined in a SQW and includes a dissipation component. The results present a weak coupling regime, where an enhanced spontaneous emission takes place, and a strong coupling regime, where Rabi-splitting in the dispersion curve can be observed. The theoretical results are confronted with experimental data for the reflectance behavior in resonant and off-resonant conditions and present a great accuracy. This allows us to determine the oscillator strength of the confined excitons in the SQW with great precision. Нами досягнуто значного експериментального розщеплення Рабi (3.4 меВ) для обмежених поляритонiв у плоскому напiвпровiдниковому λ мiкрорезонаторi для одиночної квантової ями GaAs (10 нм), розмiщеної в антивузлi. Явище розщеплення Рабi детально обговорюється на основi напiвкласичної теорiї, коли для опису системи використовуються два зв’язанi гармонiчнi осцилятори (екситони i фотони). В такий спосiб можна отримати дисперсiйну криву поляритонiв, мiнiмальне значення для коефiцiєнта вiдбивання резонатора i силу осцилятора для досягнення сильнозв’язаного режиму. Цей пiдхiд описує ансамбль екситонiв обмежених одною квантовою ямою i враховує дисипацiю. Результати представляють як слабозв’язаний режим з посиленням спонтанної емiсiї, так i сильнозв’язаний режим, коли спостерiгається розщеплення Рабi на дисперсiйнiй кривiй. Теоретичнi результати порiвнюються з експериментальними даними для поведiнки коефiцiєнта вiдбивання в резонансних i нерезонансних умовах i є дуже точними. Це дозволяє з високою точнiстю визначити силу осциляторiв обмежених однiєю квантовою ямою екситонiв. 2014 Article Determination of oscillator strength of confined excitons in a semiconductor microcavity / E.A. Cotta, P.M.S. Roma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23702:1-11. — Бібліогр.: 30 назв. — англ. 1607-324X arXiv:1407.2430 DOI:10.5488/CMP.17.23702 PACS: 78.67.De, 42.25.Hz, 42.50.Ct, 42.50.Pq, 42.55.Sa, 42.70.Qs http://dspace.nbuv.gov.ua/handle/123456789/153452 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| description |
We have achieved a significant experimental Rabi-splitting (3.4 meV) for confined polaritons in a planar semiconductor λ microcavity for only a single quantum well (SQW) of GaAs (10~nm) placed at the antinode. The Rabi-splitting phenomena are discussed in detail based on the semiclassical theory, where two coupled harmonic oscillators (excitons and photons) are used to describe the system. In this way, we can obtain the dispersion curve of polaritons, the minimum value for the cavity reflectance and the oscillator strength to reach the strong coupling regime. This approach describes an ensemble of excitons confined in a SQW and includes a dissipation component. The results present a weak coupling regime, where an enhanced spontaneous emission takes place, and a strong coupling regime, where Rabi-splitting in the dispersion curve can be observed. The theoretical results are confronted with experimental data for the reflectance behavior in resonant and off-resonant conditions and present a great accuracy. This allows us to determine the oscillator strength of the confined excitons in the SQW with great precision. |
| format |
Article |
| author |
Cotta, E.A. Roma, P.M.S. |
| spellingShingle |
Cotta, E.A. Roma, P.M.S. Determination of oscillator strength of confined excitons in a semiconductor microcavity Condensed Matter Physics |
| author_facet |
Cotta, E.A. Roma, P.M.S. |
| author_sort |
Cotta, E.A. |
| title |
Determination of oscillator strength of confined excitons in a semiconductor microcavity |
| title_short |
Determination of oscillator strength of confined excitons in a semiconductor microcavity |
| title_full |
Determination of oscillator strength of confined excitons in a semiconductor microcavity |
| title_fullStr |
Determination of oscillator strength of confined excitons in a semiconductor microcavity |
| title_full_unstemmed |
Determination of oscillator strength of confined excitons in a semiconductor microcavity |
| title_sort |
determination of oscillator strength of confined excitons in a semiconductor microcavity |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/153452 |
| citation_txt |
Determination of oscillator strength of confined excitons in a semiconductor microcavity / E.A. Cotta, P.M.S. Roma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23702:1-11. — Бібліогр.: 30 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT cottaea determinationofoscillatorstrengthofconfinedexcitonsinasemiconductormicrocavity AT romapms determinationofoscillatorstrengthofconfinedexcitonsinasemiconductormicrocavity |
| first_indexed |
2025-07-14T04:36:51Z |
| last_indexed |
2025-07-14T04:36:51Z |
| _version_ |
1837595687896743936 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23702: 1–11
DOI: 10.5488/CMP.17.23702
http://www.icmp.lviv.ua/journal
Determination of oscillator strength of confined
excitons in a semiconductor microcavity
E.A. Cotta1,2, P.M.S. Roma1
1 Departamento de Física, Universidade Federal do Amazonas, Manaus, Brazil
2 Instituto Nacional de Ciência e Tecnologia em Nanodispositivos Semicondutores (INCT-DISSE), Brazil
Received February 22, 2014
We have achieved a significant experimental Rabi-splitting (3.4 meV) for confined polaritons in a planar semi-
conductor λmicrocavity for only a single quantum well (SQW) of GaAs (10 nm) placed at the antinode. The Rabi-
splitting phenomena are discussed in detail based on the semiclassical theory, where two coupled harmonic
oscillators (excitons and photons) are used to describe the system. In this way, we can obtain the dispersion
curve of polaritons, the minimum value for the cavity reflectance and the oscillator strength to reach the strong
coupling regime. This approach describes an ensemble of excitons confined in a SQW and includes a dissipa-
tion component. The results present a weak coupling regime, where an enhanced spontaneous emission takes
place, and a strong coupling regime, where Rabi-splitting in the dispersion curve can be observed. The theo-
retical results are confronted with experimental data for the reflectance behavior in resonant and off-resonant
conditions and present a great accuracy. This allows us to determine the oscillator strength of the confined
excitons in the SQW with great precision.
Key words:microcavity, Rabi-splitting, polariton, oscillator-strength, strong coupling, reflectance
PACS: 78.67.De, 42.25.Hz, 42.50.Ct, 42.50.Pq, 42.55.Sa, 42.70.Qs
1. Introduction
Ideally, a microcavity is a system in which a light-emitting material can interact with a single cavity-
resonant-mode or no interactive electromagnetic modes within the material transition width. Thus, en-
hanced or suppressed spontaneous emission can be seen in this system, and in a cavity with a very high
quality factorQ , an even spontaneous oscillatory emission can be induced [1, 2]. The couplings between
electronic excitations and optical modes have been of considerable interest in atomic systems, which al-
lows us tomake a direct connectionwith optical cavities as well as with optically active semiconductors in
semiconductor microcavities. The emission and reflection spectra of atoms in optical cavities are known
to exhibit splitting due to the coupling of their dipole transitions with the excitations of the vacuum-
radiation field [3, 4]. These coupled modes are called vacuum-field Rabi splitting, and are one of the
basics of quantum electrodynamics in a cavity. In the case of semiconductor microcavities, a strong field-
matter interaction occurs between the optical modes and excitons, whose modes are relatively sharp, so
the exciton-photon modes are often called cavity polaritons [5, 6]. These couplings are important for un-
derstanding such effects as optical bistability [7] and laser action [8], besides developing optoelectronic
and photonic devices [9].
Cavity polaritons are quasi-particles that are created due to a strong exciton-photon coupling in a
semiconductor microcavity. The cavity photon mode has a nearly parabolic in-plane dispersion and can
be described as the one having an effective mass which is usually a few times smaller than the exciton
mass. However, in the strong coupling regime, at resonance between the cavity mode and the exciton
state, the eigenmodes present an anticrossing between the exciton and confined photon modes resulting
in the appearance of two polariton branches (called Upper and Lower Polariton Branches), spectrally
separated by the Rabi splitting energy ħΩ.
© E.A. Cotta, P.M.S. Roma, 2014 23702-1
http://dx.doi.org/10.5488/CMP.17.23702
http://www.icmp.lviv.ua/journal
E.A. Cotta, P.M.S. Roma
Very high-Q optical micro- and nano-cavities present an extremely low photon loss rate in a signif-
icant small cavity mode volume, offering surprisingly rich physics, spanning many areas of research
including nonlinear optics, quantum optics, and device physics [13, 14]. Different microcavity architec-
tures presenting a lateral field confinement such as post, disk or toroidal microcavities, have been pre-
sented over the last decade. In all of them, the main objective is to optimize the miniaturization on a
subwavelength-scale, and minimize the diffraction-limit and the loss contributions such as surface scat-
tering, radiation and finite cavity mirror reflectance. Moreover, the use of different materials based on
CdxMn1−xTe, ZnSe, GaN, InxAs1−xP have been used to optimize solid-state carrier properties such as os-
cillator strength, exciton binding energy, among others. The combination of these efforts aims to produce
devices capable of emitting a singlemode laser at room temperature, where the photons present the same
quantum properties that have been registered at low temperatures. The use of multiple quantum wells
and quantum dots in traditional structures or in photonic crystals permitted the generation of samples
with a very highQ [15], a very low laser threshold and a very high Rabi splitting [16].
In principle, the Rabi-splitting is proportional to the square root of the exciton oscillator strength
f multiplied by the overlap of the exciton wave function with the electromagnetic field. The oscillator
strength is related to the probability of the transition from the crystal ground state to the exciton state
and is proportional to the dimensionality of the system. Due to the confinement of excitons, the binding
energy has a monotonous increase from the bulk value to 2D value, and reaches a maximum when the
well width is about the exciton Bohr radius [17]. This effect leads to more stable exciton states and to
much more prominent excitonic effects in absorption and photoluminescence. Moreover, despite the
Rabi splitting increase with f , it decreases with the cavity length.When polariton effects can be neglected
(weak-coupling regime), the oscillator strength per unit area can be related to the absorption coefficient
integrated over the absorption peak, as measured by an absorption experiment. However, in a strong-
coupling regime, due to the changes made in the cavity dispersion curve by polaritons, the reflectance
spectrum does not make the distinction between absorption and a change in the reflectivity-transmission
balance. Thus, a new treatment is needed to determine f .
In this paper, we present a comprehensive theoretical and experimental study of cavity-polariton dis-
persion in a single quantumwell (SQW) of GaAs (100 Å) confined by a low barrier potential (Al0.3Ga0.7As).
Using the linear semiclassical theory, we derive analytic equations for cavity-mode dispersion and cavity-
polariton eigenfrequencies. A careful analysis of transmittance spectra, using the semiclassical model to
describe the weak and strong coupling between the confined photons and excitons, produces a good
agreement between theoretical and experimental results. Thus, we show that the strong coupling regime
can be observed in a very simple sample with heavy-hole exciton transition. As a consequence, we can
directly measure the excitonic oscillator strength.
2. The sample and experimental setup
In our sample, a SQW is placed at the antinode of a λ cavity formed by two diffracted Bragg reflector
(DBR) mirrors (AlAs/Al0.2Ga0.8As) and kept at 10K in a cold finger cryostat. The SQW is surrounded by
Al0.3Ga0.7As barriers forming the spacer layer of the cavity [figure 1 (a)]. The sample was grown using
molecular beam epitaxy technique (MBE) on GaAs[100] substrates. It rotates during the growth of the
DBR mirrors and the SQW, but stops at a specific angle for the growth of the spacer layer on each side
of the SQW, generating a thickness gradient across the sample [see figure 1 (b)]. This permits to perform
a cavity-detuning when the sample is excited in different positions on the surface. The quality factor of
the cavity was measured, obtainingQ ≈ 3000. The structure is designed so that the energy of the lowest-
energy heavy-hole exciton transition is the same as that of the cavity length. Since the quantum well is
strained, only the lowest-energy heavy-hole exciton is observed.
The reflectance spectrum was obtained using an unpolarized white light source, focusing it normally
on the sample surface. The measured spot size on the sample is ∼ 40 µm. The spatial selection of the
backscattered light is provided by imaging the sample on the entrance slit of the analysing spectrometer
with a spectral resolution of 0.5 Å using a 50%−50% non-polarizing beam-splitter.
23702-2
Obtaining the oscillator strength by reflectance spectra
Figure 1. (a) (Color online) Scheme of the sample and (b) experimental measurement of cavity spacer
layer, mapping the reflectance spectrum resonance in a weak-coupling regime.
3. Theoretical framework
3.1. Exciton-Polaritons
Microcavity polaritons are mixed modes formed from cavity photons and dielectric excitations such
as excitons. When the microcavity contains one or several QWs, the emission from the confined elec-
tronic states (excitons) is strongly modified by the presence of the cavity. In semiconductors, wave-vector
conservation leads to the formation of quasi-stationary exciton-polariton states, since an exciton with a
given crystal momentum interacts with one photon mode with the same wave-vector. The breakdown of
crystal-momentum conservation along the growth direction leads to an intrinsic radiative lifetime of free
excitons with an in-plane wave-vector smaller than the light wave-vector.
Due to the formation of exciton-polaritons in semiconductor microcavities, two regimes can occur: (1)
the coupling of the electromagnetic field to the crystal excitation is smaller than the width of the cavity
mode (weak-coupling regime), where irreversible decay occurs, but the emission process is modified in
relation to the free-space case; (2) the light-matter coupling is larger than the decay rate (strong-coupling
regime), the situation that can only occur for excitons, where no irreversible decay happens. In the latter,
the energy oscillates between the exciton and photon modes (Rabi oscillations), and Rabi splitting occurs
in the frequency domain. The mixed exciton-cavity modes in the strong-coupling regime can be viewed
as two-dimensional polaritons, whose radiative decay rates are determined by the photon lifetime within
the cavity.
To describe the dynamics and the radiative recombination process of confined polaritons in a SQW,
we obtain the dispersion curve by the Hamiltonian
H =ħωc,k a†
k ak +ħωx,k b†
k bk +ħΩ
(
a†
k bk +b†
k ak
)
, (3.1)
where a†
k and b†
k are creation operators for photons and excitons, respectively, with momentum k. The
last term in equation (3.1) shows the linear exciton-photon interaction, with the coupling energy of
ħΩ=
√
Nqw(ħe)2
2εm0Leff
f . (3.2)
Here, Nqw is the number of QW’s within the cavity, ε is the permittivity of the semiconductor (m0 and e
is the mass and charge of the electron, respectively). The cavity is formed by two DBR mirrors with an
effective length Leff = Lc +λ0/2nc[nH nL/(nH −nL)], a refractive index nc and resonant in λ0. The DBR
mirrors consist of alternate layers, with high nH (Al0.2Ga0.8As) and low nL (AlAs) refractive indexes. The
additional length for the cavity in Leff, comes from the energy dependence of the DBR phase, that is often
much larger than Lc, and, therefore, produces an important Rabi-splitting reduction [18].
23702-3
E.A. Cotta, P.M.S. Roma
The Hamiltonian in equation (3.1) can be diagonalized using transformations to polariton basis oper-
ators pk = Xk bk +Ck ak and qk = Xk ak −Ck bk , where
Xk =
∆c,k +
√
∆2
c,k + (ħΩ)2
2
√
∆2
c,k + (ħΩ)2
1/2
, (3.3)
Ck =
(ħΩ)2
2
√
∆2
c,k + (ħΩ)2
1[
∆c,k +
√
∆2
c,k + (ħΩ)2
]
1/2
(3.4)
are the Hopfield coefficients [19], in which∆c,k = Ec(k)−Ex(k) is the cavity-exciton detuning. On this new
basis we can verify that an exciton-polariton is a linear superposition of an exciton and a photon state
with the same in-plane wave-number, where the probability of finding the polariton in any of these two
states is given by |Xk |2 and |Ck |2, that is cavity detuning dependent [see figure 2 (a)].
Thus, the Hamiltonian in equation (3.1) can be expressed as follows:
H =ħωUPq†
k qk +ħωLPp†
k pk . (3.5)
The interaction terms between qk (upper polariton branch— UPB) and pk (lower polariton branch—
-0.4 -0.2 0.0 0.2 0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
|X
k | 2
C0
X0
|C
k|2
-30 -20 -10 0 10 20 30
1.550
1.555
1.560
1.565
1.570
UPB
Cavity
Exciton
LPB
Degree)
-30 -20 -10 0 10 20 30
1.555
1.560
1.565
1.570
1.575
1.580
c > 0
En
er
gy
(e
V)
Degree)
UPB
Cavity
Exciton
LPB
(c)
-30 -20 -10 0 10 20 30
1.550
1.555
1.560
1.565
1.570
LPB
En
er
gy
(e
V)
Degree)
UPB
Exciton
Cavity
Figure 2. (a) Hopfield coefficients as a function of cavity-exciton detuning (∆c) given by equation (3.4) for
ħΩ= 3.4meV and Ex = 1.555 eV. (b) Dispersion curve for confined exciton-polariton in a 100 Å GaAs QW
with Al0.3Ga0.7As barrier in resonant condition. (c) the same as ∆c,k=0 > 0, and in (d) for ∆c,k=0 < 0. The
solid lines are the cavity dispersion curve (black) given by equation (3.7) and the exciton energy (grey),
that is approximately constant due to the narrow wave-vector range in which we analyse our data. In
the strong-coupling regime, the cavity mode splits into two branches (dotted lines): the upper (black) and
lower (grey) polariton branches.
23702-4
Obtaining the oscillator strength by reflectance spectra
LPB) are neglected. This Hamiltonian gives us the following eigenvalues for the polariton energy
EUP,LP = Ec +Ex
2
±
√
ħ2Ω2 +
∆2
c,k
4
, (3.6)
where EUP and ELP are the UPB and LPB energies. In the resonance ∆c,k=0 = ∆c = 0 [figure 2 (b)], the
difference of energy between LPB and UPB is given by ħΩ. In this case, the exciton-polariton can be
generated for excitation in resonance with the cavity only at the normal angle of incidence (θ = 0). For
∆c > 0 [figure 2 (c)] we cannot obtain a resonance condition for any k||, and the exciton-polariton is not
generated. In the case where ∆c < 0 [figure 2 (d)], the excitation energy must be higher than the cavity
mode, out of the narrow range of the angle of incidence (θ &±7◦) to generate a polariton with k|| , 0.
3.2. Semiclassical description of collective Rabi-splitting
The electromagnetic field stored in an optical resonator is such that the electric field E0 corresponding
to the ground state energy of the resonator mode is extremely large [20]. The value of E0 is obtained by
taking the electromagnetic energy ε0E2
0V stored in the volumeV of the cavitymode as equal to the energy
of half a photon, ħω/2. Thus, we obtain E0 =
√ħω/(2ε0V ). Consequently, the coupling energy ħg = |D·E0|
between an excitonic dipoleD and the field E0 can also be very large. Here, the relevant scale for a “large”
coupling g is set by the rates at which the energy dissipates from the system, either by spontaneous
emission from the exciton (transverse damping rate γ), or by leaking out of the cavity (damping rate κ).
The promised land in cavity quantum electrodynamics (CQED), often called the “strong-coupling” regime,
begins, therefore, when g is much larger than both γ and κ.
The incident wave-vector k0 can be decomposed in perpendicular kz and parallel k|| components in
relation to the QW plane (x−y plane). In the growth direction z, the excitons are confined, and therefore,
are found in discrete states. In the QWplane, the excitonic states are found as in bulk, but due to quantum
confinement, the degeneracy of the heavy-hole and the light-hole bands at the center of the band gap is
broken. Thus, k0 = k||+kz , where k2
|| = k2
x +k2
y = k0 sinθ (θ being the angle between the incident wave-
vector and the normal in relation to the surface of the sample) and |kz | = 2πnc/Lc (for a λ cavity). In this
case, the cavity dispersion is given by
Ec = ħc
neff
|k| = ħc
neff
√√√√ω2
0
c
sin2θ+ π2
L2
eff
, (3.7)
where neff =
√
n2
QW − (κ/γ) is the effective refractive index.
In a semiclassical framework, in order to describe the reflectance spectra of the microcavity, with an
ensemble of N polaritons confined in a SQW, coupled to a single cavity mode, we must begin considering
a driven damped interaction between two coupled oscillators. In this model, the system is driven by an
external radiation field with frequency ω. This field can interact with complex mixed-mode frequencies
close to a resonant mode, in which two states composed of a cavity-photon (with frequency ωc − iκ) and
a 1S heavy-hole exciton (with frequency ωx − iγ) are coupled with a matrix element g . The damping
Γ0 = (1/4πε)(π/nc)(e2/m0c) fqw is given by the decay rate of the exciton amplitude at k = 0 in a SQW,
where the exciton has an oscillator strength fqw on the quantum well plane [21].
Thus, the master equation obtained from the Maxwell-Bloch equation approach, that relates the phe-
nomenological parameters above to the intra-cavity (γ) and mirror (κ) losses is [22]
[ω−ωx + i(ωs +γ)][ω−ωc + i(ωs +κ)] = g 2
(3.8)
for the eigenvalue frequency ωs of the coupled system. The matrix element g makes explicit the coupling
between the cavity-photon and the exciton, and characterizes the oscillatory exchange of excitation be-
tween the exciton and the cavity field . The mirror loss κ (as well as g ) is dependent on the reflectivity of
23702-5
E.A. Cotta, P.M.S. Roma
the cavity mirrors R = p
R1R2 (composed of two mirrors with individual reflectance R1 and R2),
g =
√
1+ p
Rp
R
cΓ0
ncLeff
, (3.9)
κ = 1− p
Rp
R
c
ncLeff
. (3.10)
The eigenvalues ωs of (3.8) are given by
ω±
s = 1
2
[
γ+κ+ i(ωx +ωc −2ω)
]
± 1
2
√
(κ−γ)2 + (ωx −ω)Φx + (ωc −ω)Φc −4g 2 , (3.11)
whereΦx =ωx+3ωc−4(ω+ iκ) andΦc =ωc+3ωx−4(ω+iγ). These solutions describe the normal modes
formed by the intracavity field and the collective excitonic polarization. In equation (3.11) we identify
ω+
s as the excitonic eigenvalue, which expresses the well-known process of cavity-enhanced spontaneous
emission [23], and ω−
s as the cavity eigenvalue, which indicates the lesser known companion process of
exciton-inhibited cavity decay.
At the limit of the weak intracavity field for a coincident pump frequency ω, cavity resonance fre-
quency ωc and excitonic transition frequency ωx, the eigenvalues obtained for the coupled system are as
follows:
ω±
s =−1
2
[
(γ+κ)±
√
(γ−κ)2 −4g 2
]
. (3.12)
When g is small compared to both γ and κ, precisely 2g < |γ−κ|, the square root is purely real. This
corresponds to the so-called “weak-coupling” regime, which is usually not considered as CQED. However,
when γ¿ g 2/κ¿ κ, where Im[ω±
s ] is still zero, this is often called the “bad-cavity” limit of CQED. At this
limit, the cavity decay rate is the dominant one, but g 2/κ, which is the excitonic damping induced by
exciton-photon coupling, is larger than the excitonic damping γ itself. This means that excitons are more
likely to decay in the cavity mode, rather than in another mode outside the cavity. Since the light can be
coupled with excitons outside the cavity with great efficiency, this regime is interesting for applications
where the field must be extracted from the cavity with great efficiency.
However, when Im[ω±
s ] , 0, precisely 2g > |γ−κ|, one reaches the “strong-coupling limit” domain,
where g À (γ,κ), and ωs exhibit a normal-mode splitting. In the time domain, this limit corresponds to
the appearance of a coherent exchange of photons, at a rate g , between the cavity field and excitons
(Rabi-oscillations). In this context, the Rabi-splitting Ω=ω+
s −ω−
s = 2|Im[ωs]| is given by
Ω=
√
4|g |2 − (γ−κ)2 . (3.13)
In this case, we can verify that Ωmay be less than 2g due to the broadening.
The real and imaginary parts of ω±
s can be analysed in figure 3 on the resonance (ω = ωc = ωx).
The splitting in ω±
s occurs in the real part when the system presents it in the weak-coupling regime,
and in the imaginary part when it is in the strong-coupling regime, in which both widths coincide, and
are given by (κ+ γ)/2. The crossover occurs approximately at the point where the enhanced exciton
decay rate crosses the free cavity-mode decay rate. The critical reflectivity is approximately given by
Rc = 1−4
√
2(ncLeffΓ0/c) and is found to be R ≈ 84% with the parameters presented in figure 3 (a). In
figure 3 (b), we found that a material with an oscillator strength higher than fc = 2.77× 1012
cm
−2
is
necessary to construct a semiconductor microcavity that presents a strong-coupling regime. In this case,
our microcavity is capable of producing Rabi-splitting, since we are using a SQW of GaAs [24], where the
fqw ∼ 4.8×1012
cm
−2
.
The transmittance spectrum for the steady-state coupled system is related to solutions for ωs by [25]:
T (ω) =
∣∣∣∣κ[γ+ i(ωx +ωc −2ω)]
(ω+
s + iδc)(ω−
s + iδc)
∣∣∣∣2
, (3.14)
23702-6
Obtaining the oscillator strength by reflectance spectra
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0
10
20
30
40
50
st
ro
ng
-c
ou
pl
in
g
re
gi
m
e
Rc = 84%
Reflectance (%)
Im
[
s ] (TH
z)R
e[
s]
(T
H
z)
+
s
-
s
weak-coupling regime
(a)
1 2 3 4 5
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
R
e[
s]
(T
H
z)
Oscillator Strength (cm-2) x1012
-0.9
-0.6
-0.3
0.0
0.3
0.6
0.9
strong-coupling regime
weak-coupling regime
+
s
-
s
Im
[
s ] (TH
z)
fc = 2.77x1012cm-2
(b)
Figure 3. Behavior of the real (dotted lines) and imaginary (solid lines) parts of ω±
s (grey line to ω
−
s and
black line to ω+
s ) for the structure presented in figure 1 (a). The vertical dashed line limits the strong and
weak coupling regimes, where the critical reflectance and oscillator strength are Rc ≈ 84% [in figure (a)]
and fc ≈ 2.77×1012
cm
−2
[in figure (b)], respectively. The parameters set to produce this result are based
on a 100 Å GaAs SQW, where ε = 12.9, γ = 0.5 THz (obtained by the linewidth of the PL spectrum). The
DBR mirror properties are: nH = 3.527 and nL = 3.002; and nc = 3.458 (for λ0 = 800 nm).
here, δc = (ωc −ω) are the detuning between the cavity mode and incident frequency. The transmission
of the cavity shown in figure 4 (a) is similar to the empty cavity transmission, except for a double peak on
the cavity resonance, separated by Ω. This “Rabi-splitting” was observed experimentally in the semicon-
ductor microcavity, as will be discussed in the next section. In figure 4 (b) we can observe an anti-crossing
between the two peaks of the transmission spectra.
4. Results and analysis
Using the unpolarized white light source we make measurements of reflectivity at 10 K, in resonant
condition, that shows regions in the sample where the strong-coupling regime appears (see figure 5). This
regime can be distinguished from the typical cavity resonance by a double dip, ensuring the exciton-
polariton formation. Therefore, a direct measurement of the spectra gives us a Rabi-splitting energy of
ħΩ = 3.4 meV. A good fitting agreement in this figure was reached using the semiclassical coupled os-
cillator approach, and transfer matrix method, to determine κ and fqw. The “bare” exciton energy Ex
-10 -5 0 5 10
0
5
10
15
20
25 c = 0
c = -13meV
c = +13meV
Tr
an
sm
ita
nc
e
(%
)
x (THz)
(a)
-40 -20 0 20 40
1.53
1.54
1.55
1.56
1.57
1.58
En
er
gy
(e
V)
c (meV)
EUP
ELP
Ecav
Ex
(b)
Figure 4. (a) Theoretical results for transmission spectra using equations (3.11) and (3.14) for three differ-
ent conditions for cavity detuning. The minimum separation between the two peaks occur in ∆c = 0 and
(b) The dotted lines show the anti-crossing between upper (EUP =ħω+
s ) and lower (ELP =ħω−
s ) polariton
branches. The solid grey lines present the cavity-photon (Ec) and exciton (Ex) energies.
23702-7
E.A. Cotta, P.M.S. Roma
1.550 1.555 1.560 1.565
90
95
100
Ex
Energy (eV)
3.4meV
R
ef
le
ct
an
ce
(%
)
Ep
0.0
0.2
0.4
0.6
0.8
1.0
N
or
m
al
iz
ed
P
L
Figure 5. White light reflectance spectrum at normal incidence around the cavity resonance. The black
line is the experimental measurement and the dotted line is the theoretical fitting using the equation
(3.14). The parameters used in the fitting are: (κ;γ) = (5.0;5.0) THz, fqw = 6.1 × 1012
cm
−2
and Ec =
1.5551 eV. The grey line is the normalized photoluminescence (PL) spectrum of the microcavity, showing
the laser emission regime when excited quasi-resonantly by a Ti:Sapphire laser (Ep).
was fixed in all calculations, and was obtained experimentally by a PL spectrum of the sample using
a Ti:Sapphire pumping laser, in which we found Ex = 1.5554 eV (see the grey line in figure 5). Both
lines in the reflectance spectrum are well fitted by Lorentzian lines having the same FWHM linewidth
(full width at half maximum) of ∆E ≈ 3.5 meV, indicating a dominating homogeneous broadening. The
linewidth of the UPB corresponds to the nonradiative broadening of the heavy-hole excitons and the ho-
mogeneous broadening mechanisms for the polaritons, e.g., their radiative decay, and exciton dephasing
due to photon scattering and free-carrier scattering. The cavity linewidth becomes smaller under reso-
nant conditions for high fineness cavities, because the coupled-mode linewidth can be smaller than the
natural excitonic linewidth [26, 27] due to QW interaction.
In the strong-coupling regime, the dependence of the coupling factor 2g ≈ Ω on the QW oscillator
strength fqw (through exciton radiative rate Γ0), permits the use of the fitting procedure, presented in
figure 5, as the most accurate and reliable measurements of fqw. In our sample, the value obtained from
the fitting to fqw is 6.1× 1012
cm
−2
, which is in good agreement with the literature [28, 29] taking into
consideration an increase of fqw as the temperature decreases and the confinement effect.
The reflectance spectra for cavity detuning were registered while exciting the sample in different
positions on the surface, as can be viewed in figure 6. In this figure, we can observe the change from
strong to weak coupling regimes for displacements of about 500 µm along a line with steps of about
90 µm. Comparing figures 2 (a) and 6, we can analyse the behavior of the Hopfield coefficients |X0|2
and |C0|2 (exciton and photon behavior of the polariton), and identify their weight in the reflectance
spectrum in the strong-coupling regime. For ∆c > 0 (grey lines) we can verify that the right peak (higher
energy) survives, which corresponds to exciton-like behaviour for the polariton. For ∆c < 0 (black lines)
the left peak (lower energy) survives, which corresponds to predominantly photon-like behaviour. If
the vacuum-field Rabi-splitting exceeds the original broadening of the exciton line at the anticrossing
condition, the tails of the excitonic distribution weakly couple with the light remaining between the two
split modes and do not effect the reflection. The dotted line presented in this figure is an eye guide to
characterize the anti-crossing presented in figure 4 (b); as theoretically expected, this result provides
direct evidence of quantized cavity photon number.
We also varied the sample temperature to obtain the temperature-dependent reflectance measure-
ments, at low white light pump intensities (see figure 7). In this figure we can verify the presence of the
strong-coupling regime up to the critical temperature of Tc ≈ 40 K. The thermal energy for this temper-
ature (kBTc) is about equal to Rabi energy (ħΩ), so that above this temperature the thermal energy is
sufficient to break the exciton-photon coupling, dissociating the polaritons. The exciton binding energy
23702-8
Obtaining the oscillator strength by reflectance spectra
1.545 1.550 1.555 1.560 1.565
3.4 meV
495 m
0 m
R
ef
le
ct
an
ce
Energy (eV)
Figure 6. Formation of cavity polariton mapped by reflectance spectra of the microcavity around the cav-
ity resonance for different detunings between exciton and cavity modes. The position of each spectrum
was changed in equal steps of 90 µm along a line. The black line is for ∆c É 0 and the grey line for ∆c > 0.
The dotted line is an eye guide showing the anti-crossing theoretically expected in figure 4 (b).
for a SQW 100 Å GaAs with a barrier of Al0.3Ga0.7As is about 8 meV [17], which is much higher than kBTc.
Thus, the free-carrier is not broken, but the resonance in the reflectance spectra that survives for T > Tc
is due to the photon behavior.
All these microcavity concepts and results should influence the conventional design of important op-
tical and electro-optical devices such as horizontal and vertical-cavity emitting lasers, electro-optic mod-
ulators, and nonlinear optical etalons. In VCSEL’s (vertical cavity surface emitting lasers), a major impact
will occur only if laser action is based on exciton recombination. However, it is well known that, at least
at room temperature, exciton dissociation and carrier interactions are much faster than the exciton ra-
diative lifetime (in ns range), so that they usually decay in electron-hole pairs. The existence of rapid Rabi
oscillation might drastically change this state of affairs, as the oscillation could be faster than the disso-
ciation time, leading to an efficient radiative recombination whenever a coupled exciton-photon mode
escapes the cavity (polariton lasers). For nonlinear optical devices which rely on unrelaxed excitation,
excitons are still very important at room temperature and are the root of the unsurpassed performance
of QW heterostructure based systems [30]. In the near future they should lead to strongly improved de-
vice performances.
1.550 1.555 1.560 1.565
R
ef
le
ct
an
ce
Energy (eV)
10K
50K
Figure 7. Reflectance spectra around the cavity resonance at different temperatures in resonant condi-
tion. The temperature was changed in equal steps from 10 K to 50 K.
23702-9
E.A. Cotta, P.M.S. Roma
5. Conclusions
In conclusion, we have directly observed exciton-polariton Rabi-splitting of 3.4 meV by reflectance
characterization for only one GaAs single quantumwell. A good description of the asymmetries observed
in reflectivity spectra indicating a homogeneous broadening of the free-carrier is described within a sim-
ple semiclassical approach. Compared with the conventional absorption technique, this method allowed
us to study the oscillator strength of excitons confined in a SQWwith a higher accuracy. The experimental
results provide a very good agreement with theoretical purpose in which reflectance has been used as an
optical probe.
The semiclassical approach allows us to determine the critical reflectance and oscillator strength
necessary to achieve the strong-coupling regime for a general semiconductor microcavity based on
AlxGa1−xAs technology. Finally, we expect that these experiments will find an interesting application
in new coherent light sources.
Acknowledgement
The authors would like to thank F.M. Matinaga of Federal University of Minas Gerais (UFMG) for the
use of his cryostat and for fruitful discussions of the sample. We would also like to thank the Instituto
Nacional de Ciência e Tecnologia— Dispositivos Semicondutores (INCT-DISSE) and the National Research
Council (CNPq) for their financial support.
References
1. Morin S.E., Wu Q., Mossberg T.W., Opt. Photonics News, 1992, 3, No. 8, 8; doi:10.1364/OPN.3.8.000008.
2. Haroche S., In: Les Houches 1990, Session LIII: Fundamental Systems in Quantum Optics, Dalibard J., Rai-
mond J.M., Zinn-Justin J. (Eds.), North-Holland, Amsterdam, 1992, 767–940.
3. Agarwal G.S., J. Opt. Soc. Am. B, 1985, 2, 480; doi:10.1364/JOSAB.2.000480.
4. Zhu Y., Gauthier D.J., Morin S.E., Wu Q., Carmichael H.J., Mossberg T.W., Phys. Rev. Lett., 1990, 64, 2499;
doi:10.1103/PhysRevLett.64.2499.
5. Weisbuch C., Nishioka M., Ishikawa A., Arakawa Y., Phys. Rev. Lett., 1992, 69, 3314;
doi:10.1103/PhysRevLett.69.3314.
6. Gutbrot T., Bayer M., Forchel A., Reinecke T. L., Rudin S., Knipp P.A., Reithmaier J.P., Phys. Rev. B, 1998, 57, 9950;
doi:10.1103/PhysRevB.57.9950.
7. Cotta E.A., Matinaga F.M., Phys. Rev. B, 2007, 76, 073308; doi:10.1103/PhysRevB.76.073308.
8. Butov L.V., Nature, 2007, 447, 540; doi:10.1038/447540a.
9. Cotta E.A., ECS Transactions, 2009, 23, 545; doi:10.1149/1.3183762.
10. Jaynes E.T., Cummings F.W., Proc. IEEE, 1963, 51, 89; doi:10.1109/PROC.1963.1664.
11. Agarwal G.S., Phys. Rev. Lett., 1984, 53, 1732; doi:10.1103/PhysRevLett.53.1732.
12. Agarwal G.S., Puri R.R., Phys. Rev. A, 1986, 33, 1757; doi:10.1103/PhysRevA.33.1757.
13. Vahala K.J., Nature, 2003, 424, 839; doi:10.1038/nature01939.
14. Noda S., Fujita M., Asano T., Nature Photon., 2007, 1, 449; doi:10.1038/nphoton.2007.141.
15. Martínez L.J., Prieto I., Alén B., Postigo P.A., J. Vac. Sci. Technol. B, 2009, 27, 1801; doi:10.1116/1.3151832.
16. Yoshie T., Scherer A., Hendrickson J., Khitrova G., Gibbs H.M., Rupper G., Ell C., Shchekin O.B., Deppe D.G., Nature,
2004, 432, 200; doi:10.1038/nature03119.
17. Greene R.L., Bajaj K.K., Phelps D.E., Phys. Rev. B, 1984, 29, 1807; doi:10.1103/PhysRevB.29.1807.
18. Savona V., Hradil Z., Quattropani A., Phys. Rev. B, 1994, 49, 8774; doi:10.1103/PhysRevB.49.8774.
19. Hopfield J.J., Phys. Rev., 1958, 112, 1555; doi:10.1103/PhysRev.112.1555.
20. Haroche S., Kleppner D., Phys. Today, 1989, 42, 24; doi:10.1063/1.881201.
21. Andreani L.C., In: Confined Electrons and Photons: New Physics and Applications, Nato Science Series B, Vol. 340,
Burstein E., Weisbuch C. (Eds.), Plenum Press, New York, 1995, 57–112.
22. Carmichael H.J., Phys. Rev. A, 1986, 33, 3262; doi:10.1103/PhysRevA.33.3262.
23. Purcell E.M., Phys. Rev., 1946, 69, 674; doi:10.1103/PhysRev.69.674.2.
24. Houdré R., Stanley R.P., Oesterle U., Ilegems M., Weisbuch C., Phys. Rev. B, 1994, 49, 16761;
doi:10.1103/PhysRevB.49.16761.
25. Kimble H.J., Berman P.R., Cavity Quantum Electrodynamics, Academic Press, San Diego, 1994.
23702-10
http://dx.doi.org/10.1364/OPN.3.8.000008
http://dx.doi.org/10.1364/JOSAB.2.000480
http://dx.doi.org/10.1103/PhysRevLett.64.2499
http://dx.doi.org/10.1103/PhysRevLett.69.3314
http://dx.doi.org/10.1103/PhysRevB.57.9950
http://dx.doi.org/10.1103/PhysRevB.76.073308
http://dx.doi.org/10.1038/447540a
http://dx.doi.org/10.1149/1.3183762
http://dx.doi.org/10.1109/PROC.1963.1664
http://dx.doi.org/10.1103/PhysRevLett.53.1732
http://dx.doi.org/10.1103/PhysRevA.33.1757
http://dx.doi.org/10.1038/nature01939
http://dx.doi.org/10.1038/nphoton.2007.141
http://dx.doi.org/10.1116/1.3151832
http://dx.doi.org/10.1038/nature03119
http://dx.doi.org/10.1103/PhysRevB.29.1807
http://dx.doi.org/10.1103/PhysRevB.49.8774
http://dx.doi.org/10.1103/PhysRev.112.1555
http://dx.doi.org/10.1063/1.881201
http://dx.doi.org/10.1103/PhysRevA.33.3262
http://dx.doi.org/10.1103/PhysRev.69.674.2
http://dx.doi.org/10.1103/PhysRevB.49.16761
Obtaining the oscillator strength by reflectance spectra
26. Raizen M.G., Thompson R.J., Brecha R.J., Kimble H.J., Carmichael H.J., Phys. Rev. Lett., 1989, 63, 240;
doi:10.1103/PhysRevLett.63.240.
27. Yamamoto Y., Machida S., Björk G., Surf. Sci., 1992, 267, 605; doi:10.1016/0039-6028(92)91209-T.
28. Zhang B., Kano S.S., Shiraki Y., Ito R., Phys. Rev. B, 1994, 50, 7499; doi:10.1103/PhysRevB.50.7499.
29. Andreani L., Pasquarello A., Phys. Rev. B, 1990, 42, 8928; doi:10.1103/PhysRevB.42.8928.
30. Weisbuch C., Vinter B., Quantum Semiconductor Heterostructures, Academic Press, Boston, 1991.
Визначення сили осциляторiв обмежених екситонiв
у напiвпровiдниковому мiкрорезонаторi
E.A. Котта1,2, П.М.С. Рома1
1 Факультет фiзики, Федеральний унiверситет штату Амазонас,Манаус, Бразилiя
2 Нацiональний iнститут науки i технологiї напiвпровiдникових нанопристроїв, Бразилiя
Нами досягнуто значного експериментального розщеплення Рабi (3.4 меВ) для обмежених поляритонiв у
плоскому напiвпровiдниковому λмiкрорезонаторi для одиночної квантової ями GaAs (10 нм), розмiщеної
в антивузлi. Явище розщеплення Рабi детально обговорюється на основi напiвкласичної теорiї, коли для
опису системи використовуються два зв’язанi гармонiчнi осцилятори (екситони i фотони). В такий спосiб
можна отримати дисперсiйну криву поляритонiв, мiнiмальне значення для коефiцiєнта вiдбивання резо-
натора i силу осцилятора для досягнення сильнозв’язаного режиму.Цей пiдхiд описує ансамбль екситонiв
обмежених одною квантовою ямою i враховує дисипацiю. Результати представляють як слабозв’язаний
режим з посиленням спонтанної емiсiї, так i сильнозв’язаний режим, коли спостерiгається розщеплення
Рабi на дисперсiйнiй кривiй. Теоретичнi результати порiвнюються з експериментальними даними для
поведiнки коефiцiєнта вiдбивання в резонансних i нерезонансних умовах i є дуже точними. Це дозволяє
з високою точнiстю визначити силу осциляторiв обмежених однiєю квантовою ямою екситонiв.
Ключовi слова: мiкрорезонатор, розщеплення Рабi, сила осцилятора, сильний зв’язок, коефiцiєнт
вiдбивання
23702-11
http://dx.doi.org/10.1103/PhysRevLett.63.240
http://dx.doi.org/10.1016/0039-6028(92)91209-T
http://dx.doi.org/10.1103/PhysRevB.50.7499
http://dx.doi.org/10.1103/PhysRevB.42.8928
Introduction
The sample and experimental setup
Theoretical framework
Exciton-Polaritons
Semiclassical description of collective Rabi-splitting
Results and analysis
Conclusions
|