The Theory of DBRs…

The DBRs are a simplest kind of periodic structure, which is made up of a number of quarter-wave layers with alternately high- and low- index materials.

Therefore, it’s necessary to know the theory of quarter-wave layer before discussing the DBRs.

Quarter-wave layer [21-22]

Consider the simple case of a transparent plate of dielectric material having a thickness d and refractive index nf, as shown in Fig 2.1. 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

(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, KoP. 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

The interference maximum of reflected light is established when δ=2mπ, in other words, an even multiple of π. In that case Eq. (2.9) can be rearranged to yield

The interference maximum of reflected light is established when δ=(2m±1)π, in other words, an odd multiple of π. In that case Eq. (2.9) can be rearranged to yield

Therefore, for an normal incident light into thin film, the interference maximum of reflected light is established when d = λ0/4nf (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).

Distributed Bragg Reflectors (DBRs) [23- 28]

DBRs serve as high reflecting mirror in numerous optoelectronic and photonic devices such as VCSEL. There are many methods to analyze and design DBRs, and the matrix method is one of the popular one. The calculations of DBRs are entirely described in many optics books, and the derivation is a little too long to write in this thesis. Hence, we put it in simple to understand DBRs. Consider a distributed Bragg

π

reflector consisting of m pairs of two dielectric, lossless materials with high- and low- refractive index nH and nL, as shown in Fig 2.2. The thickness of the two layers is assumed to be a quarter wave, that is, L1 =λB/4nH and L2 =λB/4nL, where theλB is the Bragg wavelength.

Multiple reflections at the interface of the DBR and constructive interference of the multiple reflected waves increase the reflectivity with increasing number of pairs.

The reflectivity has a maximum at the Bragg wavelength λB. The reflectivity of a DBR with m quarter wave pairs at the Bragg wavelength is given by

⎟⎟ where the no and ns are the refractive index of incident medium and substrate.The

high-reflectivity or stop band of a DBR depends on the difference in refractive index of the two constituent materials, △n (nH-nL). The spectral width of the stop band is given by

where neff is the effective refractive index of the mirror. It can be calculated 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

The length of a cavity consisting of two metal mirrors is the physical distance between the two mirrors. 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

(2.15)

For very thick DBRs (m→∞) the tanh function approaches unity and one obtains Also, the penetration depth is given by

)

For a large number of pairs (m→∞), the penetration depth is given by

L Comparison of Eqs. (2.17) and (2.19) yields that

) 2 (

1m L1 L2

L^pen= ^eff +

The factor of (1/2) in Eq. (2.20) is due to the fact that meff applies to effective number of periods seen by the electric field whereas Lpen 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.

2.2 Bragg Diffraction In 2D Triangular-Lattice Structure[29][³⁰]

In there, we consider the sample structure has three layers. The 2D triangular-lattice structure is in the middle layer. The real lattice and reciprocal lattice of 2D triangular-lattice structure is shown in Fig. 2.3 and 2.4, respectively. K1 and K2

are the Bragg vectors with magnitude |K|=2π/a, where a is lattice constant. We consider the TE mode in the 2D PC structure. The incident lightwave and diffracted lightwave must be satisfied the relationship,

kd=ki+q1K1+q2K2, q1,2=0, ±1, ±2, …

where kd is xy-component wave vector of diffracted lightwave, ki is xy-component wave vector of incident lightwave, q12 is order of coupling. In Eq. (2.22), ωd is the frequency of diffracted lightwave and ωi is the frequency of incident lightwave. The Eq. (2.21) and Eq. (2.22) represent the phase-matching condition and constant-freqency condition.

First, we consider the lattice point Γ in first Brillouin zones of the structure as shown in Fig. 2.5. Figure 2.5 shows that lightwaves are diffracted in five different Γ-Γ directions such as kd=ki+K1, kd=ki+K1+K2, kd=ki+2K2, kd=ki-K1+2K2, kd=ki-K1+K2, in addition, kd = ki＋K1 also satisfy the Bragg condition as shown in Fig. 2.6. So, in this case, the incident lightwave can couple with lightwave of the five different directions in in-plane which like a cavity and the coupling light also can emit normal to the sample surface due to the first Bragg diffraction.

Second, we consider the M point in first Brillouin zones of the structure as shown in Fig. 2.7. The diffraction condition is expressed as kd=ki+K1 and kd=ki-K1. Here, the kd, ki+K1 andkd=ki-K1 must be parallel to the Γ-M direction to satisfy the diffraction condition. In this case, the incident lightwave only coupled two waves that propagate in the forward and backward directions and the cavity can exist in each of the three different directions. This is similar to the 1D DFB laser.

Third, the K point in first Brillouin zones of the structure must be considered as shown in Fig. 2.8. In this case, the diffraction condition is expressed as, kd=ki+K1, kd=ki+K2, and the diffraction waves are not parallel to the Γ-K direction. It is different from the others. There are six directions in Γ-K direction and we can know that two different cavities exist simultaneously in the different Γ-K direction from Fig. 2.8.

The lasing mode is not clear due to the couple waves are emitted in the three different directions.

Finally, the lattice points of higher Brillouin zones of the structure are not described in here because those are similar to the lattice points of first Brillouin zones of the structure.

2.3 Spontaneous and Stimulated Transition [31]

In here, we can consider only a single level in both the conduction and valence bands in semiconductor. In Fig. 2.9, there are four basic mechanism of electron generation/recombination must be considered: (i) spontaneous emission, (ii) stimulated absorption, (iii) stimulated emission, (iv) nonradiative recombination. The solid circles represent electrons and the open circles represent holes.

The first case (Rsp) represents the electron in conduction band recombining with a hole in valence band spontaneously to generate a photon. If a lager number of photon is generated, the emission time and direction would be random, and the emission is not coherent radiation light. This is the primary mechanism of LED, in which photon feedback is not provided.

The second case (R12) represents the electron in valence band absorbs the energy of pumping source to generate the electron in conduction band while leaving a hole in the valence band.

The third case (R21) is the same as the second, only the sign of the interaction is reversed. An incident photon gets into the system, stimulate the electron and hole to recombine, and generate a new photon. This is an important mechanism because it is necessary for lasers.

The fourth case represents several nonradiative ways, a conduction band electron can recombine with a valence band hold without generating any photons such as nonradiative recombination centers and Auger recombination.

2.4 Operation Mechanism of Laser[29-30][31- 35]

The operation of a semiconductor lasers, can be understood by observing the flow of carriers into its active region, the generation of photons due to the recombination of some of these carriers, and the transmission of some of these photons out of the optical cavity. These dynamics can be described by a set of rate equations, one for the carriers and one for the photons in each of the optical modes.

Carrier density rate equation

The carrier density in the active region is governed by a dynamic process. In fact, we can compare the process of establishing a certain steady-state carrier density in the active region to that of establishing a certain water level in a reservoir which is being simultaneously filled and drained. This is shown schematically in Fig. 2.10. As we

proceed, the various filling (generation) and drain (recombination) terms illustrated will be defined. The current leakage illustrated in Fig. 2.8 contributes to reducing ηi

and is created by possible shunt paths around the active region. The carrier leakage, Rl, is due to carriers “splashing” out of the active region (by thermionic emission or lateral diffusion if no lateral confinement exists) before recombining. Thus, this leakage contributes to a loss of carriers in the active region that could otherwise be used to generate light.

For the DH active region, the injected current provides a generation term, and various radiative and nonradiative recombination process as well as carrier leakage provide recombination terms. Thus, we can write the carrier density rate equation,

(2.23)

rec

gen R

dt G

dN = −

where N is the carrier density (electron density), Ggen is the rate of injected electrons and Rrec is the ratio of recombining electrons per init volume in the active region.

Since there areηi I/q electrons per second being injected into the active region,

qV G_gen ηⁱI

= , where V is the volume of the active region. The recombination process is complicated and several mechanisms must be considered. Such as, spontaneous recombination rate, Rsp ~ BN², nonradiative recombination rate, Rnr, carrier leakage rate, Rl, (Rnr + Rl = AN+CN³), and stimulated recombination rate, Rst. Thus we can write Rrec = Rsp + Rnr + Rl +Rst. Besides, N/τ ≡ Rsp + Rnr + Rl, where τ is the carrier lifetime. Therefore, the carrier density rate equation could be expressed as

carrier density

rate equation ^st

i N R

Photon density rate equation

Now, we describe a rate equation for the photon density, Np, which includes the photon generation and loss terms. The photon generation process includes spontaneous recombination (Rsp) and stimulated recombination (Rst), and the main photon generation term of laser above threshold is Rst. Every time an electron-hole pairs is stimulated to recombine, another photon is generated. Since, the cavity volume occupied by photons, Vp, is usually larger than the active region volume

occupied by electrons, V, the photon density generation rate will be [V/Vp]Rst not just Rst. This electron-photon overlap factor, V/Vp, is generally referred to as the confinement factor, Γ. Sometimes it is convenient to introduce an effective thickness (deff), width (weff), and length (Leff) that contains the photons. That is, Vp=deffweffLeff. Then, if the active region has dimensions, d, w, and La, the confinement factor can be expressed as, Γ=ΓxΓyΓz, whereΓx = d/deff, Γy = w/weff, Γz = La/Leff. Photon loss occurs within the cavity due to optical absorption and scatting out of the mode, and it also occurs at the output coupling mirror where a portion of the resonant mode is usually couple to some output medium. These net losses can be characterized by a photon (or cavity) lifetime, τp. Hence, the photon density rate equation takes the form

photon density

rate equation _p

p

where βsp is the spontaneous emission factor. As to Rst, it represents the photon-stimulated net electron-hole recombination which generates more photons.

This is a gain process for photons. It is given by

Now, we rewrite the carrier and photon density rate equations carrier density

rate equation _p

p

Output power versus driving current

The characteristic of output power versus driving current (L-I characteristic) in a laser diode can be realized by using the rate equation Eq. (2.29) and (2.30). Consider the below threshold (almost threshold) steady-state (dN/dt = 0) carrier rate equation, the Eq. (2.29) is given by

( )

is above the threshold (I＞Ith), the carrier rate equation will be

From Eq. (2.31), the steady-state photon density above threshold where g = gth can be calculated as

steady state

photon density qv V

I

The optical energy stored in the cavity, Eos, is constructed by multiplying the photon density, Np, by the energy per photon, hν, and the cavity volume, Vp. That is Eos = NphνVp. Then, we multiple this by the energy loss rate through the mirrors, vgαm = 1/τm, to get the optical power output from the mirrors, P0 = vgαmNphνVp. By using Eq.

(2.30) and (2.32), and Γ=V/Vp, we can write the output power as the following equation

Now, by defining

m

differential quantum efficiency η

In fact, ηd is the differential quantum efficiency, defined as number of photons out per electron. Besides, dPo/dI is defined as the slope efficiency, Sd, equal to the ratio of output power and injection current. Figure 2.11 shows the illustration of output power vs. current for a diode laser. Below threshold only spontaneous emission is important;

above threshold the stimulated emission power increase linearly with the injection current, while the spontaneous emission is clamped at its threshold value.

Chapter 3

Fabrication of GaN-based 2D SEPC DFB Laser

3.1 Electron-Beam System [36][37]

The technique of EBL is using electron beam to generate patterns on a surface and the De broglie relationship (λ < 0.1nm for 10-50keV electrons) to avoid the diffraction limit. So the primary advantage of this technique is that it can beat the diffraction limit of light and create a pattern which only has few nanometers line-width without any mask. The first EBL machines, based on the scanning electron microscope (SEM), were developed in the 1960s.

The EBL system usually consists of electron gun, beam blanker for controlling the state of beam, electron lenses for focus the electrons beam, stage and computer control system as shown in Fig. 3.1.

Electron Sources

Electrons can be emitted from a conductor eitherby heating the conductor with a sharp point where the electrons obtain sufficient energy from the thermal source to overcome the work function of the conductor (thermal emission sources) or by applying a strong electric field where the electrons can tunnel through the work function (field emission sources). There are three key parameters of the electrons source such as virtual source size, brightness, and the energy spread of the emitted electrons.

The virtual source size determines the number of the lenses in order to form a small spot size at the target. Brightness is similar to the intensity of light optical, so the brighter the electron source has a higher current in the electron beam. The electrons beam with the wide energy spread is similar to white light which has one more wavelength while a beam with a narrow energy spread can be comparable to monochromatic light.

Table 3.1 shows the properties of common electrons sources. For many years the normal thermal emission source for lithography optics was formed by using loop of tungsten wire heated white hot by passing a current. Tungsten was chosen for it did not melt or evaporate at high temperature. Unfortunately, this source had a large energy spread caused by the very high operating temperature (2700K) and it was not very bright. Recently, lanthanum hexaboride has become the cathode because it has a very low work function and a high brightness. The electrons beam current for thermal emission source is dependent on operating temperature. Higher temperature can deliver greater electrons beam current (high bright) but the tradeoff is an

exponentially decreasing lifetime due to thermal evaporation of the cathode material.

Field emission source usually consist of a tungsten needle with a sharper point and the radius less than 1μm. This tip has sufficient electric field to pull electrons out of the metal. Although cold field emission source has been widely used in electron microscopes but it has some problem for EBL due to their instability with regard to short term noise as well as long term drift.

Although the field emission sources have some disadvantages for EBL, but now a technology is available to EBL is the thermal field emission source. It combines a field emission source with sharp tungsten and the heating of the thermal source.

Because the tip operates at 1800K, it is less sensitive to gases in the environment and can achieve stable operation for months at a time.

Electron Lenses

The electrostatic forces or magnetic forces can focus the electrons beam.

Although the electron lenses is similar to optical lenses in principle but there are some differences. Besides in some special cases, electron lenses just can be made to converge, not diverge. In addition, the quality of electron lenses in term of aberrations is poor compares with optical lenses. The field size and convergence angle that can be used is limited by the poor quality of electron lenses. The two types of aberrations critical to EBL are spherical aberrations and chromatic aberrations. The spherical aberrations mean the out zones of the lens focus more strongly than the inner zones.

The chromatic aberrations means electrons of slightly different energies get focused at different image planes. Both types of aberrations can be limited by reducing the convergence angle of the system, so the electron beam is confined to the center of the lenses.

Beam Blanker

The beam blanker can bank or turn the beam on or off. It usually consists of a pair of plates set up as a simple electrostatic deflector. One or both plates are linked to a blanking amplifier with a fast response time. A voltage is applied across the plates to turn the beam off. This is necessary when the electrons beam must move from one part of the sample to another.

Stage & Computer Control System

The stage can control the sample smoothly motion at high magnifications. It is important for EBL because the sample must move from one location to another seamlessly. The computer control system can control the stage moving to the position which is you want and can control the beam blanker to turn on or off. Also, the

compute control system can design any pattern by using GDS software.

3.2 Wafer Preparation

MOCVD grown structure and its reflectivity spectrum

The nitride heterostructure of GaN-based MCLED was grown by metal-organic chemical vapor deposition (MOCVD) system (EMCORE D-75) on the polished optical-grade c-face (0001) 2’’ diameter sapphire substrate, as shown in Fig. 3.2.

Trimethylindium (TMIn), Trimethylgallium (TMGa), Trimethylaluminum (TMAl), and ammonia (NH3) were used as the In, Ga, Al, and N sources, respectively. Initially, a thermal cleaning process was carried out at 1080°C for 10 minutes in a stream of hydrogen ambient before the growth of epitaxial layers. The 30nm thick GaN nucleation layer was first grown on the sapphire substrate at 530^oC, then 2µm thick undoped GaN buffer layer was grown on it at 1040^oC. After that, a 35 pairs of quarter-wave GaN/AlN structure was grown at 1040^oC under the fixed chamber pressure of 100Torr and used as the high reflectivity bottom DBR. Finally, the 5λ

Trimethylindium (TMIn), Trimethylgallium (TMGa), Trimethylaluminum (TMAl), and ammonia (NH3) were used as the In, Ga, Al, and N sources, respectively. Initially, a thermal cleaning process was carried out at 1080°C for 10 minutes in a stream of hydrogen ambient before the growth of epitaxial layers. The 30nm thick GaN nucleation layer was first grown on the sapphire substrate at 530^oC, then 2µm thick undoped GaN buffer layer was grown on it at 1040^oC. After that, a 35 pairs of quarter-wave GaN/AlN structure was grown at 1040^oC under the fixed chamber pressure of 100Torr and used as the high reflectivity bottom DBR. Finally, the 5λ