Chapter 2 Related theories of resonant cavities and Bragg distribution
2.2 Resonant cavity
2.2.1 Cavity modes
The amplitude distribution of an optical wave along the axis of the cavity is illustrated schematically in Fig 2.1.
Constructive interference of the waves within the cavity is required for laser action. The result is the condition of resonance: light waves are amplified strongly if, and only if, they satisfy the equation (2.1),
2nL=Nλ, (2.1) where L is the cavity length, n is the refractive index of the laser medium, nL is the so-called optical path, N is an integer and λ denotes the wavelength.
The integer N cannot be an arbitrary number. The gain of laser of the laser medium is also a function of the wavelength, G(l). Laser oscillation can only take place when the gain is large enough to maintain the resonance. Consequently, as shown in Fig.
2.2, the actual profile of wavelength emitted by a laser is the product of the envelope of longitudinal oscillation modes and the gain profile.
The distance between the modes can be obtained from equation (2.2).[11]
2nL
2 Mode
λ = λ
∆ (2.2) Considering dispersion effect, the equation is transformed into
L
ne = − is effective index of refraction including dispersion. And the frequency difference between each longitudinal mode is
L c is the light speed. From mode spacing we can calculate the cavity length of laser diode.
2.2.2 Laser conditions
An optical cavity is required to achieve laser action. The cavity causes the amplification of the stimulated emission process in the active medium. The degree of amplification is measured as gain and is expressed as
x
where I is the light intensity and x is the distance of light travel.
The intensity after the light beam travels the distance x can be written as eGx
I x
I( )= 0 (2.6) where I0 is the initial light beam at x=0. The intensity increases exponentially as the light beam propagates through the amplification medium.
Assume that the length of the laser medium within an optical cavity is La; the cavity length is Lc; absorption of material inside the cavity is α; the reflectances of the two mirrors are R1 and R2, as shown in Fig 2.3. Then, the intensity of the light after To maintain the amplification of the stimulated emission, it is required that
I0
I ≥ . (2.8) This leads to the condition for lasing
1 )
2.2.3 Finesse factor and quality factor
The Q factor or quality factor is a measure of the "quality" of a resonant system.
Quality factor is defined as
cycle
High quality factor indicates high ratio of energy stored to energy dissipated in the cavity and the low lasing threshold.
Assume that the average time of photo in cavity (photo lifetime) is tc and stored energy in cavity is ε. Then, energy loss per time is expressed as
tc
dt dε =−ε
(2.11) Using this definition of quality factor and equation (2.10), quality factor can be given by
where ν is resonant frequency and ω is angular frequency.
Assume that Δν1/2 is transmittance full-width at half-maximum in units of frequency. The relation betweenΔν1/2 and tc given by Then, we rewrite equation (2.12) by using equation (2.13) as
2 The cavity finesse is defined as the ratio of the transmittance peak separation to the transmittance full-width at half-maximum.
2
where ltotal is distance of one round trip in cavity. Inspection of equation (2.14) and equation (2.15), Q factor is related to finesse of the cavity by
l F
The threshold pumping power of lasers is a function of the operating temperature. The threshold pumping power changes with the temperature, a semi-empirical relation between them, can be expressed as:
) 0
( TT
th T P e
P = × (2.17) where P is a constant, Pth(T) is the threshold optical pumping power, T is the operating temperature and T0 is characteristic temperature. So we can get
0 = [ln( )]−1 P T P
T th (2.18) According to equation (2.17), the high characteristic temperature indicates the variation of threshold condition is insensitive to the variation of ambient temperature.
2.2.5 The spontaneous emission factor (β) The spontaneous emission factor β is defined as
=
∑
i
Ai
A0
β (2.19)
where Ai is the spontaneous emission rate of the active material into mode i and index 0 indicates the optical mode which will eventually lase.
The pump current as a function of the photon number can be expressed as ] where I is the injection current, p is the photo number, q is the electron charge, τsp is the spontaneous emission lifetime, τnr is the nonradiative recombination lifetime, γ is the cavity decay rate and ξ is a dimensionless parameter defined by
sp
V N
γτ
ξ = 0β (2.21)
N0 is the transparency carrier concentration of gain material and V is the volume of the active material.
The injection current versus output power curves were drawn, as shown in Fig.
2.4. The parameters of γ, τsp, N0, V and λ are 1012s-1, 10-9s, 1018 cm-3, 10-12 cm-3 and 1000nm, respectively.[13]
If the micro-cavity laser supports only one mode within the its gain bandwidth, and if non-radiative recombination is negligible, then photon emission is the only means of power dissipation and the quantum efficiency of the device must be unity both below and above threshold. In other words, the spontaneous emission factor is unity.
Inspection of Fig. 2.4, the factor could be generally obtained from the difference between the heights of the emission intensities on a logarithmic scale before and after lasing.
2.3 Theory of Bragg distribution reflectors (DBR)
Low threshold gain indicates that lasers can easily achieve the condition of lasing at low threshold pumping power. Inspection of equation (2.9), the threshold gain of laser is able to be reduced by increasing the reflectivity of reflectors.
2.3.1 Transfer matrix method
n0
low refractive indices (Fig. 2.5). The reflected waves from each interface will be in phase at the first incident surface and the amplitude of all reflected waves will be added to get high reflectance.[14]
The principle of DBR can be explained with transfer matrix method (TMM).
Consider the boundary conditions of normal incidence of a light wave on a single dielectric layer as in Fig.2.6
In Fig. 2.6, the amplitude of the electric vectors of the incident beam, the reflected beam, and the transmitted beam are E0, E0’ and ET respectively. The electric field amplitudes in the film are E1 and E1’ for the forward and backward traveling waves, respectively.
The boundary conditions require that the electric and magnetic fields be continuous at each interface. These conditions are expressed as follows,
Above equations can be simplified as:
1 +( E0’ / E0 ) = [cos kl – i(nT / nT)sin kl]ET /E0 (2.22)
n0 - n0 (E0’ / E0 ) = [-in1sin kl + nT cos kl]ET /E0 (2.23) Rewrite equations (2.22) and (2.23) in matrix form:
(2.24)
Simplify equation (2.24) as:
(2.25)
In multi-layers situation, we can extend equation (2.25) to N layers :
First Interface Second Interface
Electric field E0 + E0’ = E1 + E1’ E1eikl + E1’e-ikl = ET
Magnetic field H0 – H0’ = H1 – H1’ or n0E0 - n0 E0’ = n1E1 - n1 E1’
H1eikl + H1’e-ikl = HT or n1E1eikl - n1E1’e-ikl = nTET
T
Thus, the reflectance and transmittance of the N layer films are:
2.3.2 Comparison of simulated and experimental reflectivity of DBRs In this section, the experimental reflectivity and simulated reflectivity of DBRs were compared.
The 6 pairs TiO2/SiO2 DBRs were deposited on a GaN layer. The thickness of TiO2 and SiO2 layers were about 45nm and 75nm, respectively. Fig 2.7 shows the experimental and simulated reflectivity of 6 pairs TiO2/SiO2 DBRs. The experimental reflectivity and stop band were smaller than simulated reflectivity and stop band due to thickness fluctuation of DBRs and scattering at surface.
Figures of chapter 2
Figure 2.1: Light wave resonance in a cavity.
Mirror 1 Mirror 2
L
Figure 2.2: Several resonant modes can fit within the gain profile.
Gain profile
Cavity modes
Wavelength
Figure 2.3 Structure of resonant cavity.
Mirror 2 La
Lc
Gain medium Mirror 1
R1 R2
nH nL
dH dL
dH=λ/4nH dL=λ/4nL
Fig. 2.5: Schematic of a DBR
Fig. 2.4 The injection current versus output power curves are drawn. The parameters of γ, τsp, N0, V and λ are 1012s-1, 10-9s, 1018 cm-3, 10-12 cm-3 and 1000nm, respectively.
350 400 450 500 550 600 0
20 40 60 80 100 120
simulated reflectivity experimental reflectivity
Reflectivity(%)
Wavelength(nm)
Fig. 2.7 The experimental and simulated reflectivity of 6 pairs TiO2/SiO2 DBRs.
Fig. 2.6 Scheme of wave vector normal incident on single dielectric layer
n0 n1 nT
kT
ET
k1
k0
E1
E0
E0’ E1’ k1’ k0’
l Reflected
wave
Incident wave
Transmitted wave
Chapter 3 Experimental processes and measurement instruments
3.1 Experimental processes 3.1.1 Processes of VCSELs
The epitaxial structure of the GaN-based VCSEL was first grown on a (0001)-oriented sapphire substrate by metal organic chemical vapor deposition system.
The structure consists of a 30-nm nucleation layer, 4-µm undoped GaN, a multiple quantum-well (MQW) composed of 10 periods of 5-nm GaN barrier and 3-nm In0.1Ga0.9N well, and 280 nm undoped GaN as shown Fig.3.1. The original epitaxial wafer was cleaved to a size of 1.5×1.5 cm2. The backside of the sapphire substrate was polished using diamond slurries in order to reduce scattering of KrF excimer laser during the laser lift-off process. Then a dielectric DBR consisting of 6 pairs of SiO2/TiO2 was evaporated on the top of the grown structure to form a SiO2/TiO2
DBR/InGaN MQW/GaN/sapphire structure. The structure has a reflectivity of 98.3%
at 414 nm measured by a n&k analyzer, as shown in Fig. 3.2. Then, an array of disk-like SiO2/TiO2 DBR mesas with 60 µm in diameter was formed by standard photolithography process and buffer oxide etcher (BOE). The patterned SiO2/TiO2
DBR/InGaN MQW/GaN/sapphire structure was then mounted onto a host fused silica substrate by epoxy bonding processes. The mounted sample was then subjected to a laser lift-off (LLO) process similar to the process we reported earlier [15-16]. A KrF excimer laser at 248 nm was incident on the sapphire substrate to cause the deposition of GaN into gaseous nitrogen and gallium droplets. The average energy density of KrF excimer laser was approximately 600 mJ/cm2. Then, the sapphire was separated from the epitaxilly grown structure to form a GaN/InGaN MQW/SiO2/TiO2
DBR/silica substrate configuration. The transferred sample was dipped into HCl solution to remove the residual Ga on the n-GaN. After the residual Ga were removed, the mean surface roughness of the GaN surface measured by atomic force microscopy (AFM) was about 15 nm over a scanned area of 5×5 µm2, as shown in Fig. 3.3.
Scanning electron microscope images of the GaN surface before and after HCl dip were shown in Fig. 3.4(a) (b). The GaN surface of the lifted-off structure was then lapped and polished by diamond slurries to assure smooth surface for deposition of the second dielectric DBR. After the polish process, a smoother GaN surface with a mean surface roughness about 1nm over a scanned area of 10×10 µm2 was obtained,
than 3nm.
Finally, the second DBR consisting of 8 pairs of SiO2/Ta2O5 was deposited on the top of the polished GaN surface. The reflectivity of the SiO2/Ta2O5 DBR at 414 nm is 97.2% measured by a n&k analyzer, as shown in Fig. 3.6. The complete structure of the GaN VCSEL with two dielectric DBRs is shown in Fig. 3.7(a). Fig.
3.7(b) shows the microscopic top view image of the VCSEL array, the circular disk areas are the location of VCSELs with DBR cavity. Fig. 3.8(a)-(f) show the fabrication steps of the optically pumped GaN- based blue-violet vertical cavity surface emitting laser using wafer bonding and LLO techniques.
3.1.2 Issues of processes for reduction of cavity lengths
The numbers of cavity mode and threshold gain are related to cavity length.
Therefore, the cavity length is demanded to be controlled. ICP etching is a good method to control cavity lengths more exactly than lapping with diamond slurries.
The interface between the GaN layer and sapphire substrate is named as LLO surface when the sapphire is removed from bulk GaN layer by a laser lift-off technique. Fig. 3.9(a) shows the LLO GaN surface after ICP etching without pre-polished by diamond slurries and the etching was conducted under a gas mixture condition of Cl2/Ar = 50/30 standard cubic centimeter min (sccm), the 400W of ICP source power, 40W of bias power and 0.66Pa of chamber pressure for a 1 min etching time. In Fig. 3.9(b), the LLO GaN surface has been lapped by diamond slurries before ICP etching with the same ICP etching recipe. The right part of Fig. 3.9(b) is the region after ICP etching and the left part is the region before ICP etching.
Inspection of Fig. 3.9, the smoother surface (RMS~1nm) of LLO GaN surface with pre-polished was obtained after ICP etching. This is partly because the mean surface roughness (RMS~15nm) of a LLO GaN surface after wet etching is rougher than that (RMS<1nm) of an epitaxial GaN surface. In addition, the LLO GaN surface is a highly defective region due to a GaN buffer layer grown on a sapphire substrate at the low temperature.[17] The existence of defect would cause the higher etching rate of the region near defects and result a rough GaN surface after ICP etching.[18] In order to obtain a smooth GaN surface, it is necessary to remove the GaN buffer layer and to smooth the LLO GaN surface by pre-polished before ICP etching.
In conclusion, after ICP etching, the surface morphology of the sample with
pre-polished is better than that of the sample without pre-polished.
3.2 Laser lift-off technique 3.2.1 GaN Decomposition
In a report by Sun et al. [19] the thermal decomposition of MOCVD grown GaN on r-plane sapphire was found to occur at a temperature of 1000°C in a hydrogen ambient. They reported that the surface of the GaN thin film was totally decomposed leaving only a residual Ga droplet surface, following the equation:
)
Fig. 3.10 shows the equilibrium pressure temperature (P-T) curve for GaN under N2 ambient, determined experimentally by Karpinski et. al.[20] In the recent report [21-23], the critical temperature of GaN decomposition was estimated to be about 1000℃. The recently report [24-25] also shows the calculated P-T curve for GaN as show in Fig. 3.11. The decomposition of GaN→Ga(l)+N2(g) was occurred at a critical temperature of ~1000℃ at 1 atm. The GaN sample after laser irradiation tends to show some material residues such as Ga, and Ga oxide. These residues were then clean up by dilute acid solution such as HCl or H2SO4/H2O2. Besides, structural damage and chemical intermixing resulting from laser processing was minimal and that was confined to approximately the first ≅ 50nm of the resulting material.[26]
3.2.2 KrF excimer laser setup
Fig. 3.12 shows the schematic diagram of the setup for conducting the LLO experiment. A KrF excimer laser (Lambda Physick LPX210) at wavelength of λ=248 nm with pulse width of 25 ns was used for LLO technique. The maximum laser output energy was about 700 mJ. The frequency of laser can be varied from 1 Hz to 100Hz.
The LLO processing beam passed through a optical projection system, and then focuses onto the sample with a square spot size of 1.2×1.2 mm2. The samples were placed on the top of working station which can be moved by hand. The decomposition of GaN→Ga(l)+N (g) was occurred at the interface between a GaN layer and a
substrate was easily remove from the LEDs structure by heating the irradiated sample at a Ga melting point of about 30℃.
3.3 Optical measurement instruments 3.3.1 μ-PL systems
Fig. 3.13 shows the schematic diagram of the setup for measuring the photoluminescence (PL) spectrum of samples using a He-Cd laser. The He-Cd laser output power at wavelength of λ=325 nm was about 24mW under continuous wave operation. The laser beam passed through some optical lenses and mirrors, and then was normally incident onto the sample surface with a focused spot size of about 2 µm in diameter. The light emission was collected into a spectrometer (Jobin-Yvon Triax 320) with a resolution of 0.1 nm. Besides, the emission images of samples and the position where the laser beam located on samples could be measured using a charge couple device (CCD).
3.3.2 Optical pumping systems
Fig. 3.14 shows the schematic diagram of the setup for optical pimping by a Nd:
Yttrium-Vanadium-Oxide (YVO4) laser at 355 nm, with a repetition rate of 1 kHz and pulse width of 0.5 ns. The maximum of average laser output power was about 17mW.
The laser beam with a focused spot size of about 40 µm in diameter was normally incident on the sample surface and the light emission from the sample was collected using an imaging optic into a spectrometer (Jobin-Yvon Triax 320) with a resolution of 0.1 nm, and measurement by a charge couple device (CCD). The setup of optical pumping system is similar to the setup of a µ-PL system.
Figures of chapter 3
4um n-GaN
Sapphire 280nm GaN cap layer
InGaN/GaN MQW 10pairs
Fig.3.1. The epitaxial structure of the GaN-based wafer consists of a 30-nm nucleation layer, 4-µm undoped GaN, a multiple quantum-well (MQW) composed of 10 periods of 5-nm GaN barrier and 3-nm In0.1Ga0.9N well, and 280 nm undoped GaN
Fig.3.2. The reflectivity of 6 pairs TiO2/SiO2 DBR measured by a n&k analyzer.
350 400 450 500 550 600
0 20 40 60 80 100 120
6 pairs TiO2/SiO2 DBRs
Reflectance(%)
Wavelength(nm)
Ga
(a) (b)
Fig.3.4. Scanning electron microscope images of the LLO GaN surface (a) before and (b) after HCl dip
RMS : 15~20 nm
Fig.3.3. After the residual Ga were removed, the mean surface roughness of the GaN surface measured by atomic force microscopy (AFM) was about 15 nm over a scanned area of 5×5 µm2
Fig.3.5. The mean surface roughness of the polished GaN surface measured by atomic force microscopy was about 1 nm over a scanned area of 10×10 µm2
Fig.3.6. The reflectivity of the 8 pairs SiO2/Ta2O5 DBR measured by a n&k
350 400 450 500 550 600
0 20 40 60 80 100 120
8 pairs Ta2O5/SiO2 DBRs
Reflectance(%)
Wavelength(nm)
GaN film
Glass substrate DBR
Epoxy
(a) (b)
Fig.3.7. (a) The complete structure of the GaN VCSEL with two dielectric DBRs (b)the microscopic top view image of the VCSEL array
DBR
GaN film
DBR mesas by wet etching (c) epoxy bonding process (d) laser lift-off process (e) polishing using diamond slurries (f) deposition of 8 pairs Ta2O5/SiO2 DBR
Figure. 3.10: The equilibrium pressure temperature curve for GaN under N2 ambient, determined experimentally by Ref [3.4].
(a) (b)
Non-etched
etched
Figure. 3.9: (a)A SEM image of GaN surface after ICP etching without pre-polish (b) a SEM image of GaN surface with pre-polish. The right part is the region after ICP etching and the left part is the region before ICP etching.
Figure 3.11: Pressure-temperature curve for GaN. Shaded region is area where GaN decomposes at 1250 K.[3.8]
Mirror
Lens
Work station
KrF excimer laser
Figure 3.12: The schematic diagram of a laser lift-off process.
He-Cd laser(325nm)
Fig.3.13. The schematic diagram of the setup to measuring the photoluminescence (PL) spectrum by a He-Cd laser.
Fiber
410 412 414 416 418 420
0
Fig.3.14. The schematic diagram of the setup for optical pimping by a Nd:
Yttrium-Vanadium-Oxide (YVO4) laser at 355 nm, with a repetition rate of 1 kHz and pulse width of 0.5 ns.
Chapter 4 Results and discussions
4.1 Characteristics of a GaN-based microcavity 4.1.1 Photoluminescence spectra
The photoluminescence (PL) spectra of the VCSELs were measured using a µ-PL system. The pumping source was a He-Cd laser with the wavelength of 325nm.
Prior to the fabrication of GaN-based microcavity, we measured the PL spectra of as-grown GaN-based wafers and it showed the emission peak centered at 414nm with FWHM of 18nm, as shown in Fig. 4.1. The inset in Fig 4.1 is the structure of measured wafers.
Fig. 4.2 shows the PL spectra of cavity with only one side high reflectivity DBR, which was named as structure I, and the inset is the layer structure of structure I.
A clearly modification of the emission peak caused by cavity effect was observed. It shows that the DBR would not be degraded in bonding and laser lift-off processes.
The PL intensity (between 410-420nm) of a multiple quantum-well (MQW) layer was smaller than that (at 365nm) of a GaN layer due to scattering and absorption of the pumping source (a He-Cd laser) by the GaN layer.
Estimated effective refractive index (n-dn/dλ) of about 3.4 was obtained by substituting known experimental data (cavity length=4μm; peak wavelength=410nm;
free spectral range=6.5nm) shown in Fig 4.2 into equation (4.1), where L is cavity length, λ is peak wavelength, λFSR is free spectral range (wavelength spacing between the longitudinal cavity modes) and n is refractive index. The value of 3.4 generally consists with the value used in previous reports. [27]
d L
The Fig. 4.3 shows the PL spectra of the complete VCSEL structure, which was named as structure II, and the inset is the layer structure of structure II. The FWHM of cavity modes around 450nm was about 1nm and the value of quality factor (Q factor)
The Fig. 4.3 shows the PL spectra of the complete VCSEL structure, which was named as structure II, and the inset is the layer structure of structure II. The FWHM of cavity modes around 450nm was about 1nm and the value of quality factor (Q factor)