1.1 Semiconductor microcavities
1.1.1 Microcavity quality factor and photon lifetime
An ideal optical microcavity would confine light indefinitely and would have resonant frequencies at precise values. Deviation from this ideal condition is described by the cavity quality factor Q, which could identify the performance of a micorcavity.
The definition of cavity quality factor can be expressed as λ/Δλ, where λ is the emission wavelength from cavity mode and Δλ is the linewidth (FWHM) of cavity mode. A simple way to experimentally probe the Q value is to perform the measurement of reflectivity spectrum at normal incidence. A sharp dip could be observed in the reflectivity spectrum if the cavity has precise design and high quality factor. Fig. 1.2 presents different optical microcavities classified by different confinement methods used. The corresponding cavity Q values are shown as well.
Another interpretation of a cavity Q value represents a measure of how many oscillations taking place inside the cavity before the excited photon energy dissipates out of the cavity. It means that the photon energy for narrow linewidth will be trapped in the cavity for longer period of time if we have a high-Q cavity. Based on this definition, we can express the cavity Q factor as
dt
where W is the energy stored in the system at resonance, T is the resonance period, and ω0 is the resonance frequency. Therefore, the energy evolution as function of time can be expressed as
We can also have Q related to the photon lifetime τp
ω0
τp = Q . (1.3)
This means that the exponentially decaying photon number has a lifetime given by τp.
Fig. 1.2 Optical Microcavities are organized by column according to the confinement method used and by row according to high Q and ultrahigh Q. (After [1])
Another important property for embedding an emitter inside a microcavity is the Purcell effect. It indicates that the rate of spontaneous emission can be modified by placing the light source in a resonant cavity [5]. The enhancement is given by the Purcell factor
eff
where n is the refractive index of the cavity, Veff is the effective volume of the cavity mode, and λC is the cavity mode emission wavelength. The crucial ration Q/Veff controls the emission rate, dependent on different spectral and spatial overlaps.
1.1.2 Planar microcavities for wide-bandgp materials
In this thesis, we will focus on the investigation of planar semiconductor microcavities due to the easy fabrication and measurements. Nevertheless, for wide-bandgap semiconductor materials, the fabrication of high-quality planar microcavities is very difficult due to the lack of lattice-matched substrate and lattice-matched DBRs. Therefore, the possible design of wide-bandgap semiconductor microcavities can be classified into three major types, as shown in Fig. 1.3. The first one is monolithically grown vertical resonant cavity consisting of epitaxially grown top and bottom DBRs [Fig. 1.3(a)]. The advantage of the fully epitaxial microcavity is the controllable cavity thickness which is beneficial to fabricate microcavity structure.
However, a high-Q cavity requires extremely high-reflectivity DBRs. The fully epitaxial wide-bandgap microcavity is very difficult to achieve this requirement due to the difficulty in growth of high-reflectivity nitride-based DBRs, which are usually employed as the reflectors for active layers constructed by wide-bandgap materials.
The second one is vertical resonant cavity consisting of dielectric top and bottom DBRs [Fig. 1.3(b)]. The double dielectric DBR microcavities can exhibit high cavity Q factors because of the high-reflectivity DBR, which are relatively easy to fabricate. The large refractive index contrast in dielectric materials can make high-reflectivity and large-stopband DBR with less number of pairs. The drawback of the double dielectric
DBR microcavities is the difficulty of controlling the cavity thickness precisely and the complicated fabrication process due to the employment of laser lift-off technique [6]. In addition, the thickness of the cavity should keep as thick as possible to avoid the damage of the active layer during the laser lift-off process. Such thick cavity length could increase the cavity mode volume and reduce the microcavity effect.
Fig. 1.3 Three typical kinds of wide-bandgap semiconductor microcavity structures.
(a) Fully epitaxial microcavities. (b) Double dielectric DBR microcavities. (c) Hybrid DBR microcavities.
The third one is the microcavity structure combining an epitaxially grown DBR and a dielectric type DBR which compromises the advantages and disadvantages of the above two microcavity structures [Fig. 1.3(c)]. The hybrid DBR microcavity can eliminate the complex process and keep the feasibility of coplanar contacts with dielectric DBR mesas for the future electrically pumped microcavity applications. The major requirement for the fabrication of hybrid DBR microcavity for wide-bandgap materials is to grow high-reflectivity and high-quality nitride-based DBRs. The relevant study about the issue of growing nitride DBRs will be discussed in Chapter 3.
1.1.3 Strong and weak coupling regimes
When we consider an excited atom in an ideal mocricavity, which means a microcavity without any optical loss, and the dipole transition frequency is resonant with the cavity mode, the excited atom (dipole) and cavity mode will couple, leading to a quantum of energy shifting back and forth between the atom and the cavity mode.
This fundamental dynamic of the atom-field system is reversible as long as the system is isolated and the process of the energy exchange between the atom and cavity mode is called Rabi oscillations. In general, we term this strong light-matter interaction a strong-coupling regime. Nevertheless, in reality the cavity will have a finite photon lifetime (finite cavity Q value) that will prevent the reversible process by allowing the energy to leak irreversibly into the continuum. Specifically, the atomic transition will couple to continuum radiation modes and thereby experience spontaneous decay of its population as well as polarization dephasing. This irreversible process means the system is in a weak-coupling regime. In spite of the reality of energy dissipation, strongly coupled systems can be observed even if the Rabi dynamic can exist only briefly. Strong coupling occurs when the atom-field coupling strength is faster than any underlying dissipative rate. Under conditions of strong coupling, weak optical probing near the microcavity resonant frequency reveals two spectral transmission peaks giving the energies of new eigen states, which are now entangled states of the atom and cavity field. This is one of the experimental techniques to probe the strong coupling regime [7].
The typical optoelectronic devices with microcavity structures operating under weak coupling regime are resonant-cavity light-emitting diodes (RCLEDs) and VCSELs which have been developed for a long time and commercialized in many applications.
As for the microcavity devices operating under strong coupling regime, they are an important hot topic in recent years and will be discussed in the following sections.
1.1.4 Semiconductor microcavity polaritons
Considering the situation when we have a semiconductor microcavity structure consisting of an active layer embedded between two high-reflectivity DBRs, this structure will be very similar to the configuration of atom-field interaction, as mentioned in section 1.1.3. Therefore, by exciting the active layer using laser sources or electrical injection, the free electron-hole pairs will be created. Furthermore, a quasi-particle, termed as exciton, is created if an excited electron-hole pair bound by the Coulomb interaction. This exciton has a similar function as the excited atom in a microcavity. When the exciton strongly interacts with the confined optical field of the semiconductor microcavity, it is possible for this system to be in the strong-coupling regime.
Fig. 1.4 Schematic sketch of the semiconductor microcavity consisting of an active layer embedded between two high-reflectivity DBRs.
Fig. 1.4 shows the schematic sketch of the microcavity configuration. If the rate of energy exchange between the cavity field and the excitons becomes much faster than the decay and decoherence rates of both the cavity photons and the excitons, an excitation in the system is stored in the combined system of photon and exciton. Thus the excitations of the system are no longer exciton or photon, but a new type of quasi-particles called the microcavity polaritons.
1.2 Dispersion of cavity polaritons
It is well-known that the optical properties of semiconductor materials are strongly dependent on their energy band structures. The energy band dispersion curves (E-k diagram) will dominate the semiconductor effective mass and density of states that determines the carrier population with optical pumping or electrical injection. Therefore, the controllable optoelectronic properties such as polarization and threshold condition can be demonstrated by means of band structure engineering. A typical example of band structure engineering is strained quantum wells employed in semiconductor lasers [8].
The modified valence band dispersion by strain in quantum wells decreases the density of states near the valence band edge. In addition, the strain effect induces the valence band splitting leading to the higher polarization degree in spontaneous emission.
Similarly, the dispersion of cavity polaritons also plays an important role in the optical properties. We will further describe the possible conditions causing different polariton dispersion curves in the following sections.
1.2.1 Dispersion curves of cavity photon and exciton
The planar microcavity confines the photon field in the direction of epitaxial growth but not in plane. The photon energy dispersion can be expressed as
2 //
2 k
n k E c
cav
cav = h ⊥ + , (1.5)
where k// is the photon wave vector parallel to the microcavity plane, k⊥ is the photon wave vector perpendicular to the microcavity plane and ncav is the effective refractive index of cavity. For lowest order mode of cavity, k⊥ = π/Lcav, where Lcav is the effective cavity length. In the region k// « k⊥, we have
2
Since the in-plane dispersion takes this parabolic form for small k//, the cavity photon acquires an effective mass of ncavπћ/(cLcav) ~ 10−5 m0 [9].
Neglecting electron-hole exchange, the bare exciton dispersion can be expressed as
)
is the effective Rydberg energy, Eg is the energy gap, and mr is the reduced mass. For the 1s exciton state with m = 1, the dispersion of exciton shows a parabolic curve and the effective mass of exciton is about 10−1 m0.
1.2.2 Cavity polariton dispersion curves
Considering a two-level system composed of the lowest order mode of cavity and the ground state of exciton, The Hamiltonian of the cavity polaritons is given by the following 2×2 matrix:
where Ω is the exciton-photon interaction energy. The energies of the polaritons, which are the eigen energies of the Hamiltonian, are deduced from the diagonalization procedure as
When the un-coupled exciton and photon are at resonance, Ecav = EX, lower and upper polariton energies have the minimum separation EUP − ELP = Ω, which is often called the Rabi splitting in analogy to the atomic cavity Rabi splitting. The dispersion curves with higher and lower energies are termed as upper polariton branch (UPB) and lower polariton branch (LPB), respectively. Fig. 1.5 shows a typical polariton dispersion curves including UPB and LPB when the photon and exciton energies are the same at the in-plane wave vector of zero. Under this condition, the exactly half-matter half-light system is formed at k// = 0. If the energy difference between photon and exciton is increased, we will only probe the polariton branch with higher fraction of photon based on the condition shown in Fig. 1.5.
Fig. 1.5 Typical polariton dispersion curves including UPB and LPB.
1.2.3 Effective mass and density of states
It has been mentioned that the cavity photon has an effective mass ~ 10−5 m0 and the exciton has an effective mass ~ 10−1 m0. Since the polaritons are mixed states of excitons and photons, the effective mass of polariton can be referred to as a reduced mass between exciton and photon, and is dependent on their fractions. The exciton and photon fraction in lower polariton are given by the amplitude squared of Xk// and Ck//
which are termed as the Hopfield coefficients [10], and satisfy
Specifically, the polariton effective mass is the weighted harmonic mean of the mass of its exciton and photon components:
cav of its center of mass motion, and mcav is the effective cavity photon mass. Since mcav is much smaller than mX,
The very small effective mass of lower polariton shows many different properties as
compared with micorcavities operating under weak-coupling regime. At high k// states, ΔE(k//) » Ω, dispersions of the lower and upper polaritons converge to the exciton and photon dispersions, respectively. Hence, it is noteworthy that the effective mass of lower polariton changes by four order of magnitude from k// ~0 to high k// states. This sharp difference in effective mass has directly corresponding changes in density of states of lower polaritons. This feature plays an important role in the energy relaxation dynamics of polaritons, which will be discussed in chapter 6.
1.2.4 Exciton-photon detuning
The typical polariton dispersion curves shown in Fig. 1.5 is the condition of the same photon and exciton energies at k// = 0. We could define the energy difference δ between photon and exciton energies at zero in-plane wave vector. This energy difference is generally termed as detuning δ. Typically, when the photon energy is larger than exciton energy at k// = 0, it is called positive detuning (δ = Ecav − EX). On the contrary, when the photon energy is smaller than exciton energy at k// = 0, it is called negative detuning. In addition, if the photon energy is exactly equal to the exciton energy as shown in Fig. 1.5, we call this condition as zero detuning. Fig. 1.6 presents the in-plane polariton dispersion curves of two exciton-polariton modes in semiconductor microcavity for different detunings between exciton and photon modes:
(a) positive detuning, (b) zero detuning, and (c) negative detuning. The horizontal dashed line shows un-coupled exciton mode and the curve dashed line shows un-coupled photon mode. The corresponding Hopfield coefficients are presented as well.
In Fig. 1.6, it can be found that the polariton dispersion curves are significantly different based on varied exciton-photon detunings. In the three cases, the negative detuning shows the most obvious variation as compared with the un-coupled photon and exciton modes. Besides, an evident anticrossing can be observed in the condition of negative
detuning that will be one of the important signatures of strong coupling when we try to probe the characteristics of microcavity polaritons.
Fig. 1.6 In-plane polariton dispersion curves of two exciton-polariton modes in semiconductor microcavity for different detunings between exciton and photon modes: (a) positive detuning, (b) zero detuning, and (c) negative detuning. The horizontal dashed line shows un-coupled exciton mode and the curve dashed line shows un-coupled photon mode. The corresponding Hopfield coefficients are presented as well.
1.2.5 Experimental probe of cavity polariton dispersion
In reality, the finite lifetime of the cavity photon and exciton should be taken into
account in the strong coupling system. As a linear superposition of an exciton and a photon, the lifetime of the polaritons is directly determined by
cav the non-radative decay rate of exciton. In typical semiconductor microcavities, we have γcav = 1 ~ 10 ps and γX ~ 1 ns. Therefore, the polariton lifetime is mainly dominated by
Polariton decays in the form of emitting a photon with the same k// and totoal energy ћω
= ELPB,UPB. The one-to-one correspondence between the internal polariton mode and the external out-coupled photon mode provides great convenience to experimental access to the strong coupling system. The external emitted photons carry direct information of the internal polaritons, such as the energy dispersion, population per mode, and statistics of the polaritons. Consequently, we could probe the internal properties of polaritons by collecting the emitted photons. From eq. (1.5) and consider the lowest order mode of cavity
where E0 represents the photon energy for k// = 0. Each in-plane photom mode couples
only with an exciton state with the same k//, which is related to the external angle of incidence θ via k// = (Ecav/ ћc)sinθ and hence a particular k// can be selected by varying θ that can be expressed as
2
The polariton dispersion and associated phenomena can thus be studied in angular-dependent experiments.
810 820 830 840 850 860
X~ 832 nm
Fig. 1.7 Typical angle-resolved reflectivity spectra. It can be found that the cavity mode wavelength shifts toward shorter wavelength (higher energy) with increasing angle, and the cavity mode crosses the exciton mode when the angle is about 42°.
From the viewpoint of experiments, the negative exciton-photon detuning is a good choice because of the obvious characteristic of anticrossing. A common experimental technique is to perform the angle-resolved photoluminescence (PL) or reflectivity spectra. By increasing the detection angle, the cavity photon energy increases as well, and it may cross the exciton energy due to the condition of negative detuning. If the microcavity system is in strong coupling regime, the polariton anticrossing will be observed by the angle-resolved measurements. Fig. 1.7 shows typical angle-resolved
reflectivity spectra. It can be found that the cavity mode wavelength shifts toward shorter wavelength (higher energy) with increasing angle, and the cavity mode crosses the exciton mode when the angle is about 42°. Therefore, the angle-resolved measurements may exhibit the signature of anticrossing by tuning the cavity mode energy.
In addition to the method of tuning the cavity mode energy by angle-resolved measurement, the temperature-dependent measurements can change the exciton mode due to the temperature dependence of energy bandgap. Nevertheless, the cavity photon mode will be slightly influenced as the temperature changes. With increasing temperature, the cavity photon energy decreases due to the temperature dependence of the refractive index. In the meantime, the exciton energy decreases due to the reduction of bandgap energy. The amount of the redshift of both energies is different, since the exciton energy is much more strongly affected by the temperature change than the cavity mode energy. Therefore, the detuning between exciton and cavity modes changes with temperature. If the change of exciton energy can cross the cavity mode energy by varying temperature, the anitcrossing may be observed. Consequently, the temperature-dependent measurement is an important technique for mainly tuning the exciton energy.
1.3 Historical review of microcavity polaritons
The development of semiconductor microcavity is reviewed in this section from the pioneering work of Weisbuch et al. in 1992 [11]. The realization of Bose-Einstein condensation (BEC) in semiconductor microcavities will be discussed. In addition, it is systemically reviewed for the semiconductor microcavities consisting of various materials, such as III-V and II-VI compounds, and organic materials. Finally, we will focus on the discussion about the wide-bandgap semiconductor microcavities due to
their potential in the realization of room-temperature (RT) polariton devices.
1.3.1 Bose-Einstein condensation
Historically, studies of semiconductor microcavities have been started by the first observations of exciton-polaritons in GaAs-based resonators at low temperature from the work of Weisbuch et al. [11]. By probing different points on the wafer, the cavity mode energy can be changed due to the non-uniformity in cavity thickness. Fig. 1.8(a) shows 5-K reflectivity spectra from a GaAs-based microcavity structure. Various detuning conditions between cavity and exciton modes are obtained by choosing various points on the wafer. Fig. 1.8(b) shows the reflectivity peak positions as a function of cavity detuning. A clear anticrossing is observed.
Fig. 1.8 (a) 5-K reflectivity spectra from a GaAs-based microcavity structure.
Various detuning conditions between cavity and exciton modes are obtained by choosing various points on the wafer. (b) Reflectivity peak positions as a function of cavity detuning. A clear anticrossing is observed. (After [11])
Various detuning conditions between cavity and exciton modes are obtained by choosing various points on the wafer. (b) Reflectivity peak positions as a function of cavity detuning. A clear anticrossing is observed. (After [11])