Chapter 1 Introduction and Motivation
1.4 An overview of the thesis
Chapter 2 outlines the GaN-based VCSEL. We fabricating a GaN-based VCSEL and overview the three main types of propertied GaN-based VCSELs. The reasons for using VCSEL structure with two dielectric mirrors in the thesis are presented. We presents the structure and designing issues of the GaN-based VCSELs with two dielectric mirrors.
The spectral reflectivity of the SiO2/TiO2 and SiO2/Ta2O5 DBRs are calculated by transfer matrix method. The standing optical field in the resonant cavity were also simulated in order to design the epitaxial structure for obtaining a optimal optical gain.
The fabrication steps of the two dielectric DBRs VCSELs are also demonstrated. Basic theory of the laser lift-off technique used in our work is reviewed and introduced. We have introduced a theory of the semiconductor microcavity and discussed the interaction between excitons and photons in the strong coupling regime. We also explore the Bose Einstein condensation characteristics. for polariton at the same time.
Chapter 3 shows the performance of the optically pumped GaN-based VCSELs. The threshold condition, cavity Q factor, characteristic temperature, polarization of the laser emission, divergent angle, gain and linewidth enhancement factor are characterized and discussed. The inhomogeneous indium composition in the InGaN/GaN MQWs are observed.
Chapter 4 present the frequency spacing between adjacent PL peaks decreases by almost a factor of five from 470 nm to 370 nm. We use the intrinsic material index dispersion and polariton dispersion to fit the experimental data, it shows that the latter fitting curve is much better than the former one. It is shown a very strong polariton dispersion in a multimode GaN surface emitting microcavity at room temperarure.
In the final chapter, chapter 5, an overview of this hesis and directions of future works are given and proposed.
InN
Figure 1.1 Lattice constant as s function of band gap energy of III-V nitride compounds.
(a) (b)
Figure 1.2 (a) Schematic structure of EEL. (b) Schematic structure of VCSEL.
10
o/30
o~10
oReference
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Chapter 2
Fundamental of GaN-based vertical cavity surface emitting lasers and semiconductor microcavities
2.1 Vertical cavity surface emitting lasers
2.1.1 Fundamental of VCSELsAs depicted in Figure 2.1, the typical structure of most VCSELs consist of two parallel reflectors which are distributed Bragg reflectors (DBRs), and a cavity including a multiple quantum wells (MQWs) served as active layer. Besides, diode lasers like the other types of lasers, diode laser contain three ingredients, including Gain medium, pumping source, and resonant cavity. The gain medium consists of a material which normally absorbs incident radiation over some wavelength range of interest. But, if it is pumped by inputting either electrical or optical energy, the electrons within the material can be excited to the higher, nonequilibrium energy level, so that the incident radiation can be amplified rather than absorbed by stimulating the de-excitation of these electrons along with the generation of additional radiation. If the resulting gain is sufficient to overcome the losses of some resonant optical mode of the cavity, this mode is said to have reached threshold, and relatively coherent light will be emitted. Pumping source provides the energy that can excite the electrons within gain medium at lower energy level to higher energy level. It could be either optical or electrical energy. The resonant cavity provides the necessary positive feedback for the radiation being amplified, so that a lasing oscillation can be established and sustained above threshold pumping level.
Therefore, the reflectivity necessary to reach the lasing threshold should normally be
higher than 99.9%. Corresponding to the ingredients of a laser, the active layer is the gain medium that amplify the optical radiation in the cavity; the top DBR, bottom DBRs and cavity form a resonant cavity where the radiation can interact with active region and have positive feedback.
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Light confined in the cavity reflect multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam. Recently, the Fabry-Perot cavity are employed in nearly all laser cavity. It is by using the interference of light reflected many times between two coplanar lightly-silvered mirrors. It is a high resolution instrument that has been used today in precision measurement and wavelength comparisons in spectroscopy. Also, the Fabry-Perot cavities can be used to ensure precise tuning of laser frequency. The thickness of the cavity and the position of the MQWs inside the MCs are two key features of the VCSEL devices. Thickness of a cavity decides the resonant wavelength that will lase above threshold condition according to
N n
L 2
= λ , where L is
cavity length, N is an integral, λ is resonant wavelength and n is the refractive index of the cavity. Usually the resonant wavelength of the cavity is chosen to be equal to the Bragg wavelength, such that the resonant wavelength can encounter the maximum reflectivity of the DBR. Typically, the cavity length of VCSELs is on the order of few half operating wavelengths. In such a short cavity device, the electromagnetic waves
would form standing wave patterns with nodes (electromagnetic wave intensity minima) and anti-nodes (electromagnetic wave intensity maxima) within the GaN microcavity.
The location of the InGaN/GaN MQWs with respect to the anti-modes can significantly affect the coupling of laser mode with the cavity field. As the MQWs are well aligned with the cavity field, the more electromagnetic wave interact with the MQWs result in more photon generated in the cavity, that is optical gain of resonant mode is increased.
The proper alignment of the MQWs region with the anti-nodes of the cavity standing wave field patterns will enhance the coupling and reduce laser threshold condition. As a result, the precise layer thickness control in the VCSEL fabrication is important.
Therefore, optical cavities are designed to have a large Q factor; a beam will reflect a very large number of times with little attenuation and the frequency linewidth of the beam is very small indeed compared to the frequency of the laser.
2.1.2 Distributed Bragg reflector
In the GaN-based VCSEL structure, a micro cavity with a few λ in the optical
thickness and a pair of high reflectivity (above 99%) distributed Bragg reflectors (DBRs) are necessary for reducing the lasing threshold. Recently, several groups have reported optically pumped GaN-based VCSELs mainly using three different kinds of vertical resonant cavity structure forms: (1) monolithically grown vertical resonant cavity consisting of epitaxially grown III-nitride top and bottom DBRs (epitaxial DBR VCSEL), (2) vertical resonant cavity consisting of dielectric top and bottom DBR (dielectric DBR VCSEL). (3) vertical resonant cavity consists of an epitaxially grown III-nitride top DBR and a dielectric DBR (hybrid DBR VCSEL).
In order to obtain high reflectivity DBRs to reduce the threshold condition of the
VCSELs, dielectric DBRs was used in the GaN-based VCSEL in this study. Distributed Bragg reflector (DBR) consists of an alternating sequence of high and low refractive index layers with quarter-wavelength thickness, as Figure 2.2 [2.1]. Therefore, it’s necessary to know the theory of quarter-wave layer before discussing the DBRs. Now, consider the simple case of a transparent plate of dielectric material having a thickness d and refractive index n
f, as shown in Figure 2.3. Suppose that the film is nonabsorbing and that the amplitude-reflection coefficients at the interfaces are so low that only the first two reflected beams (both having undergone only one reflection) need be considered. The reflected rays are parallel on leaving the film and will interference at image plane. The optical path difference (P) for the first two reflected beam is given by
P=nf[(AB)+(BC)]−n1(AD) (2.1)
The corresponding phase difference (δ) associated with the optical path length difference is then just the product of the free-space propagation number and P, that is, K0P.
If the film is immersed in a single medium, the index of refraction can simply be written
as n1=n2=n. It is noted that no matter nf is greater or smaller than n, there will be a relative phase shift π radians.
Therefore,
The interference maximum of reflected light is established when δ=2mπ, in other words, an even multiple of π. In that case, eq. (2.8) can be rearranged; and the interference minimum of reflected light when δ = (2m±1) π, in other words, an odd multiple of π. In that case eq. (2.9) also can be rearranged.
⎪⎪
Therefore, for an normal incident light into thin film, the interference maximum of reflected light is established when d = λ0/4n
f (at m=0). Based on the theory, a periodic structure of alternately high and low index quarter-wave layer is useful to be a good reflecting mirror. This periodic structure is also called Distributed Bragg Reflectors (DBRs). Therefore, the concept of DBR is that many small reflections at the interface between two layers can add up to a large net reflection. At the Bragg frequency the reflections from each discontinuity add up exactly in phase. Spectral-dependent of the reflectivity can be calculated by the transfer-matrix method [2.2]. Considering a layer if dielectric material b which is clad between two layers a and c. A transverse electromagnetic wave at normal incidence propagates throught the layer in z direction.
Taking the electric and magnetic (E and H) fields into consideration by Maxwell’s equation, a transmission matrix relating these fields can be written as
⎟⎟
In the equation, nb is the refractive index of layer b and ηo the impedance of free space, j is the unit imaginary number, kb is the phase propagation constant in layer b,
λ π
b
b n
k = 2 ,
where λ is the wavelength in free space. Here, the absorption was not considered in this discussion. For a multilayer, a matrix Mi is formed for each layer i of thickness di in the stack. By considering the effect of all layers with summation length of each layers L, a matrix M relates to input and output fields ca be obtained,
⎪⎪
where the Y is wave admittance and o and s refer to the incident and substrate respectively. If we have a layer of index n l between layer o and s of lower under, then the reflection from interface has a phase of π radians relative to the incident wave, because of the positive index step. If the thickness of the layer is a quarter wavelength the two
second reflection. For a stack with many alternate 1/4 wave (or (n/2+1/4) wave, n integral) layers of low and high index, all interfacial reflections will add in phase.
For a Bragg reflector made from quarter wavelength layers of indices n1 and n2,as shown in Figure 2.4, the maximum reflectivity R at resonant wavelength, also denoted as Bragg wavelength (λB), of a stack with m non-absorbing pairs can be expressed by:
2
The no and ns in the equation are the refractive indices of incident medium and substrate, respectively and m is pair numbers of the DBR. Layer thicknesses L1,2 have to be chosen as L1,2=λ /(4nB 1,2). The maximum reflectivity of a DBR therefore increases as the increasing difference in refractive indices and pair number of DBR. A broad spectral plateau of high reflectivity, denoted as a stop-band, appear around the Bragg wavelength, the width of which can be estimate as [2.3]
eff
. A wide stop-band provides a
larger tolerance between the designed λ and the main wavelength of the cooperated B MQWs, this is another important reason for us to use dielectric DBRs as mirrors in our VCSEL structure. When two such high-reflectance DBRs are attached to a layer with an optical thickness integer times ofλc/2 (λc ≈λ), a cavity resonance is formed at λc,
If(R1 ≈R2 =R), ) 1 2
(1− 2 ≤ ≤ R T
depending on φ. One characteristic parameter of the cavity quality is the cavity quality factor Q defines as:
2 length is λ/2, Q is the average number of round trips a photon travels inside the cavity before it escapes. Figure 2.5 shows an example of the reflection spectrum of a cavity.
The high-reflectivity or stop band of a DBR depends on the difference in refractive index of the two constituent materials, n △ (≣n1-n2). We can calculate by requiring the same optical path length normal to the layers for the DBR and the effective medium. The effective refractive index is then given by
1
For DBRs, the optical wave penetrates into the reflector by one or several quarter-wave pairs. Only a finite number out of the total number of quarter-wave pairs are effective in reflecting the optical wave. The effective number of pairs seen by the wave electric field is given by
For very thick DBRs (m→∞) the tanh function approaches unity and one obtains
2
Also, the penetration depth is given by
)
For a large number of pairs (m→∞), the penetration depth is given by
2
The factor of (1/2) in Eq. (2.20) is due to the fact that m applies to effective number of eff periods seen by the electric field whereas applies to the optical power. The optical power is equal to the square of the electric field and hence it penetrates half as far into the mirror. The effective length of a cavity consisting of two DBRs is thus given by the sum of the thickness of the center region plus the two penetration depths into the DBRs. So effective cavity length is extended in a semiconductor microcavity as
Lpen
DBR c
eff L L
L = +
2.2 Fundamental of semiconductor microcavities
2.2.1 polariton dispersion curve in the strong coupling regimeA typical structure of a semiconductor microcavity consisting of a λ/2 cavity layer sandwiched between two DBRs. The planar semiconductor MCs in the strong coupling regime have attracted a good deal of attention and controlled the interaction between photons and excitons, which leads to cavity polaritons, as shown in Figure 2.6. When the exciton state is strongly coupled to the cavity-photon mode, quasi-particle called cavity polaritons are produced with an anti-crossing dispersion relation, as shown in Figure 2.7.
Thus, the elementary excitations of the system are no longer exciton or photon, but a new type of quasi-particle called the polaritons. Polaritons, predicted theoretically by Hopfield [2.4] and Agranovich [2.5] in the end of 1950s, have been extensively studied in bulk semiconductor materials [2.6, 2.7], thin films [2.8, 2.9], quantum wells [2.10, 2.11], and quantum wires and dots [2.12, 2.13]. They can be interpreted as virtual exciton–photon pairs that propagate in the crystal because of a chain of processes of virtual absorption and emission of photons by excitons. The polariton states are true eigenstates of the system, so that once the polaritons are present there are no more pure excitons or photons.
The exciton-photon interaction are often described by linear dielectric dispersion model [2.14] or coupled oscillator model [2.15]. The dielectric dispersion is used to described transmissive or reflective spectral experiments. The coupled oscillator model on the other hand is more suitable for describing active PL experiment. In linear regime, the coupled oscillator rate equations for the interaction between a cavity mode a and exciton J are a i a igJ
where ω and Ω are photon and exciton frequencies, a and J are the respective decay rates, and g is the interaction constant. The formal solutions of polariton frequencies are
2
corresponds to the upper and lower branch polariton. We start from the Hamiltonian with the spacial dependence explicitly spelled out,