## Abstract

Vertical-cavity surface-emitting lasers emitting at 808 nm with unstrained GaAs/Al_{0.3}Ga_{0.7}As, tensilely strained GaAs_{x}P_{1-x}/Al_{0.3}Ga_{0.7}As and compressively strained In_{1-x-y}Ga_{x}Al_{y}As/Al_{0.3}Ga_{0.7}As quantum-well active regions have been investigated. A comprehensive model is presented to determine the composition and width of these quantum wells. The numerical simulation shows that the gain peak wavelength is near 800 nm at room temperature for GaAs well with width of 4 nm, GaAs_{0.87}P_{0.13} well with width of 13 nm and In_{0.14}Ga_{0.74}Al_{0.12}As well with width of 6 nm. Furthermore, the output characteristics of the three designed quantum-well VCSELs are studied and compared. The results indicate that In_{0.14}Ga_{0.74}Al_{0.12}As is the most appropriate candidate for the quantum well of 808-nm VCSELs.

© 2011 OSA

## 1. Introduction

Vertical-cavity surface-emitting lasers (VCSELs) have been proved as strong competitors to edge-emitting semiconductor lasers because of its significant advantages such as circular output beam, low threshold current, low cost and easy fabrication in two-dimensional arrays to scale up the power [1–3]. These advantages, together with its high-speed modulation, have made it widespread application in short-distance parallel fiber-optic interconnects and high-power laser source [4–6]. VCSELs emitting at many different wavelengths have been extensively studied and some 980, 850, and 780 nm devices have been commercialized into various lightwave systems [7–9]. Meanwhile, blue–ultra-violet GaN, red AlGaInP, and 1300–1550 nm long-wavelength devices are now being developed and have achieved a considerable performance [8,10,11].

808 nm is a wavelength of great interest for high-power laser sources. Such laser sources are used to pump solid state lasers (Nd:YAG or Nd:YVO4) for end uses such as material cutting, light welding, marking and printing [12,13]. However, we find very few reports on 808-nm VCSELs after an extensive literature search, because it is only actively studied in recent years. VCSEL emitting at 808 +/− 1 nm is first presented in an outlook with the output power of 25 mW [14]. Also, the emitting wavelength of 803.3 nm is achieved for a VCSEL with the output power of 0.3 W [15]. In addition, the high power of more than 120 W has been demonstrated for two-dimensional VCSEL arrays emitting about 808 nm [12]. Yet, these literatures mainly discussed the fabrication and output results. Until now, to our knowledge, there is no report about the structure design and material optimization for the quantum-well active region of 808-nm VCSELs.

In this paper, we design an unstrained GaAs, a tensilely strained GaAs_{x}P_{1-x} and a compressively strained In_{1-x-y}Ga_{x}Al_{y}As quantum-well active regions in order to emit near 808 nm at operating temperature of VCSELs. First, we present a comprehensive model for the calculation of the bulk bandgap, effective mass, band offset and critical thickness of these quantum-well systems. Then, the relationship between the emission wavelength and the width of quantum wells (QWs) with an arbitrary composition is established. Also, the PICS3D (Photonic Integrated Circuit Simulator in 3D) [16] simulation program is employed to calculate the material gain of these quantum-well systems at different temperatures with variant carrier densities. Furthermore, the threshold current density versus the number of QWs is plotted. Finally, the output power of the three designed quantum-well VCSELs is studied in considering the self-heating effect.

## 2. Parameters and physical model

To obtain the numerical parameters for both the GaAs_{x}P_{1-x} and In_{1-x-y}Ga_{x}Al_{y}As material systems, a linear interpolation between the parameters of the relevant binary semiconductors is utilized except for the unstrained bandgap energies. For physical parameter P, the interpolation formulas are given as [17]

The material parameters of the binary semiconductors [17–19] can be found in Table 1 . From Eqs. (1) and (2), we can obtain most physical parameters for GaAsP and InGaAlAs in the calculation, and the exceptional unstrained bandgap energies are calculated by following expressions

#### 2.1 Impact of strain on quantum-well band

Based on the strain theory [20,21], the bulk bandgap of strained quantum well is calculated. In the case of (001) oriented (z axis) growth, the strain has the following components:

*a*is the lattice constant of the quantum well,

*a*

_{0}is the lattice constant of the substrate,

*C*

_{11}and

*C*

_{12}are the elastic stiffness constants. For the quantum well layer, the conduction band is shifted by the energyand the valence bands are shifted by

*a*and

_{c}*a*are the conduction-band and valence-band hydrostatic deformation potentials, and

_{v}*b*is the valence-band shear deformation potential.

For tensilely strained GaAs_{x}P_{1-x} and compressively strained In_{1-x-y}Ga_{x}Al_{y}As material systems, the strained bandgaps can be expressed as

*E*

_{g}_{0}can be gotten from Eqs. (3) and (4),

*δE*

_{c}_{,}

*δE*

_{lh}_{,}and

*δE*can be respectively gotten by substituting Eqs. (5) and (6) into Eqs. (7), (8) and (9).

_{hh}During the calculation of the energy levels in the valence band of strained quantum well, the hole effective mass can be taken as

#### 2.2 Band offset

The band offset is the relative position of the band edges of the semiconductors constituting the QW. There are few experimental data for the band offsets of these strained semiconductor alloys with various compositions, so the theoretical model for the calculation of the conduction and valence band edges become important in designing and modeling stages. Here we present model-solid theory [17,18] for the band alignment of these systems.

The valence band position of potential well is given by

*E*is the average valence subband energy and Δ is the spin-orbit split-off band energy. The conduction band position may be calculated by simply adding the strained bandgap energy to the valence band position. Then the band offset ratio of the conduction can be given by

_{ν,αν}*E*w v and

*E*b v obtained by Eq. (13) are respectively the valence band position in the potential well and potential barrier materials, and

*E*w g and

*E*b g respectively correspond to the strain adjusted band gaps for the potential well and potential barrier materials.

#### 2.3 Critical thickness

To the strained material of lattice mismatched system, the maximum thickness *h _{c}* must be taken into account in order not to cause misfit dislocation. If the epitaxial layer is thin enough, as well as the mismatch rate is not more than 7-9%, the system can maintain the energy of elastic strain is less than the energy in forming dislocation. By using mechanical equilibrium model, Matthews gives the expression for the critical thickness

*h*as [22]

_{c}#### 2.4 Output power

For the uniform gain structure (UGS) VCSEL with quantum-well active layer, the threshold current can be written as [23]

*n*and

_{w}*L*are respectively the number and width of QWs,

_{w}*r*and

*R*are respectively the radius and reflectivity of VCSEL,

*B*is the effective recombination constant,

_{eff}*L*is the effective cavity length,

*α*is the total internal loss,

_{in}*a*is the gain coefficient, and

_{N}*N*is the carrier concentration at transparency. The differential quantum efficiency can be written as [24,25]where

_{t}*η*is the internal quantum efficiency. Considering the self-heating effect of VCSEL [26], the experience formula of the device output power can be expressed as [23,27]wherewhere

_{i}*T*is the cutoff temperature,

_{off}*R*is the series resistance,

_{d}*V*

_{0}is the turn-on voltage,

*I*is the injection current, and

*λ*is the thermal conductivity listed in Table 1. The other parameters used in the calculation of output power can be found in Table 2 [12,23]. It should be noted that the gain coefficient and the transparency carrier concentration are obtained by PICS3D simulator.

_{c}## 3. Results and discussion

In order to investigate the effect of strain on InGaAlAs QWs, we did an extensive literature search to find that the strain of InGaAlAs when less than 1.5% is the bigger the better [28]. However, the early epitaxial process indicated the epitaxial nucleation on the substrate is very difficult when the lattice mismatch of the two materials is more than 1%. Based on these two points, we determine the In content of InGaAlAs 0.14. In this case, the strain of InGaAlAs is about 1% with the critical value of nucleation.

Applying Eq. (1) to Eq. (15), we theoretically calculate the effective mass, strain, bulk bandgap, band offset and critical thickness of unstrained GaAs, tensilely strained GaAs_{x}P_{1-x} and compressively strained In_{1-x-y}Ga_{x}Al_{y}As quantum-well systems. The calculation results are listed in Table 3
. As can be seen, the compressively strained InGaAlAs has larger band offset than the tensilely strained GaAsP, which predicts to result in better carrier confinement in InGaAlAs quantum-well system [17].

#### 3.1 Energy levels

In the quantum well structure, the energy levels of square potential well can be estimated by the Kronig-Penney Model, and the distribution of energy levels can be gotten from the following equation [29]

*L*and

_{w}*L*are respectively the width of potential well and potential barrier,

_{b}*m*and

_{w}*m*are respectively the effective mass of potential well and potential barrier,

_{b}*V*is the band offset of the conduction band or valence band, and

*E*is the energy level.

After the calculations of band offset and effective mass listed in Table 3, the relationship between the quantum energy levels and the quantum-well width can be obtained by substituting Eq. (22) into Eq. (21). The simulation shows that the energy level diagrams of InGaAlAs with different compositions are similar, so we only list a representative case. Figure 1
shows the energy levels of (a) the unstrained GaAs, (b) the tensilely strained GaAs_{0.87}P_{0.13}, and (c) the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As potential wells with the Al_{0.3}Ga_{0.7}As barrier as a function of well width. In these three sub-graphs, the left one of each graph is the electronic energy levels which are calculated from the bottom of the conduction band, and the right one is the hole energy levels which are calculated from the top of the valence band. We can see that the hole energy levels of the unstrained GaAs is the most compact as shown in Fig. 1(a), while the tensilely strained GaAsP is the most decentralized as shown in Fig. 1(b), and the compressively strained InGaAlAs has a moderate situation as shown in Fig. 1(c). The first hole subband is the heavy hole band for the unstrained GaAs and the compressively strained InGaAlAs wells, while it is the light hole band for the tensilely strained GaAsP well.

#### 3.2 Emission wavelength

The emission wavelength of semiconductor quantum-well lasers is mainly determined by the transitions between the first subband electrons in the conduction band and the first subband holes in the valence band. The photon energy of transitions can be expressed as [30]

*E*and

_{c-lh}*E*can be respectively gotten from Eqs. (10) and (11), and

_{c-hh}*E*

_{c}_{1},

*E*

_{v}_{1},

*E*

_{lh}_{1}and

*E*

_{hh}_{1}can be gotten from Eq. (21).

By solving Eq. (23), we determine the composition and width of these three quantum wells under the condition of emitting a particular wavelength. Figure 2
plots the emission wavelength of (a) the unstrained GaAs, (b) the tensilely strained GaAsP, and (c) the compressively strained InGaAlAs potential wells with the Al_{0.3}Ga_{0.7}As barrier as a function of well width. Because the emission wavelength becomes red shifted with the increasing temperature of active region [26,31], the wavelength of our designed VCSELs should be about 800 nm at room temperature in order to emit about 808 nm at operating temperature. Taking into account the thickness error by MOCVD growth, the width of QWs should be taken as integer nanometer. Based on these two points, we can obtain the width of GaAs, In_{0.14}Ga_{0.74}Al_{0.12}As, In_{0.14}Ga_{0.72}Al_{0.14}As, and In_{0.14}Ga_{0.71}Al_{0.15}As wells is respectively 4 nm, 5 nm, 6 nm, and 7 nm with the emission wavelength of 800 nm as shown in Fig. 2(a) and Fig. 2(c). When the mismatch is less than 0.5%, the growth quality of materials will be good. Thus, we tend to select GaAs_{0.87}P_{0.13} as potential well because its strain is just less than 0.5% as seen in Table 3. In addition, its width is 12 nm with the emission wavelength of 800 nm as shown in Fig. 2(b).

#### 3.3 Material gain

The material gain is defined as the ratio of the additional photon flux to the total photon flux in per unit length. The gain spectral function can be expressed as [30]

*M*(

_{ji}*E*)|

_{cv}^{2}is the transition matrix element at transition energy

*E*,

_{cv}*ρ*is the reduced density of states,

_{r,ji}*n*is the background refractive index. Based on Eq. (24), the material gain is calculated in the PICS3D software.

_{av}After many simulation tests, we found that the gain peak wavelength of the unstrained GaAs well with width of 4 nm, the tensilely strained GaAs_{0.87}P_{0.13} well with width of 13 nm and the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As well with width of 6 nm is just near 800 nm at 300 K. Figure 3
shows the simulation results which are basically consistent with the theoretical results. The injection carrier density linearly changes from 1 × 10^{18} cm^{3} to 5 × 10^{18} cm^{3}, and the material gain increases with the increase of the injection carrier density. As can be seen, the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As has the highest peak material gain as shown in Fig. 3(c), while the unstrained GaAs has the lowest value as shown in Fig. 3(a), and the tensilely strained GaAs_{0.87}P_{0.13} has a moderate value as shown in Fig. 3(b).

To explore the effect of temperature on the peak material gain, we will discuss the gain peak of the three QWs described in Fig. 3. When the injection carrier density is 5 × 10^{18} cm^{3}, the peak material gain versus operating temperature for GaAs well with width of 4 nm, GaAs_{0.87}P_{0.13} well with width of 13 nm and In_{0.14}Ga_{0.74}Al_{0.12}As well with width of 6 nm is plotted in Fig. 4
. It is shown that GaAs_{0.87}P_{0.13} has the lowest peak material gain when the operating temperature is smaller than 295 K, while GaAs has the lowest value when the operating temperature is bigger than 295 K. We can also see that the tensilely strained GaAs_{0.87}P_{0.13} has the best temperature stability of the peak material gain, while the unstrained GaAs has the worst, and the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As has a moderate value. In particular, the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As has the highest peak material gain at all operating temperature.

When the injection carrier density is 5 × 10^{18} cm^{3}, the gain peak wavelength versus operating temperature for the three QWs is plotted in Fig. 5
. As can be seen, the gain peak wavelength becomes red shifted with the increase of temperature. At 300K, the gain peak wavelength of the three QWs is near 800 nm as described in Fig. 3. The working temperature of VCSELs is about 30-40 K higher than the room temperature [26]. At this time, the gain peak wavelength of these three QWs is near 808 nm as shown in Fig. 5. We can also see that the gain peak wavelength shift with temperature for the GaAs/Al_{0.3}Ga_{0.7}As QW is about 0.278 nm/K, while for the GaAs_{0.87}P_{0.13}/Al_{0.3}Ga_{0.7}As QW, it is about 0.245 nm/K, and for the In_{0.14}Ga_{0.74}Al_{0.12}As/Al_{0.3}Ga_{0.7}As QW, it is about 0.196 nm/K. That is to say, the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As has the best temperature stability of gain peak wavelength.

#### 3.4 Output characteristics

The threshold current density of the three designed quantum-well VCSELs can be obtained by substituting Eq. (17) divided by the area into Eq. (19), and the parameters used in the calculation can be found in Table 2. Figure 6
shows the threshold current density versus the number of the three QWs described in Fig. 3. As can be seen, the threshold current density decreases first and then increases with the increase of the number of QWs. When the number of QWs is three, the GaAs_{0.87}P_{0.13} and In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSELs have the lowest threshold current density. Meanwhile, the total thickness of these two QWs is smaller than the critical thickness listed in Table 3. When the number of QWs is four, the GaAs quantum-well VCSEL has the lowest threshold current density. In order to compare, we also choose the number of GaAs QWs three. At this time, the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL has the lowest threshold current density, while the unstrained GaAs has the highest, and the tensilely strained GaAs_{0.87}P_{0.13} has a moderate value which is a little higher than the lowest point.

So far, we can present the structure of these three quantum-well active regions of 808-nm VCSELs and Table 4 shows the detailed information. The AlGaAs spacer layers are employed so that the cavity is one wavelength in length. The Al content of gradual layer linearly changes from 0.3 to 0.6, which can limit the carriers and reduce the series resistance at the same time.

Considering the self-heating effect, the device output power can be obtained by substituting Eqs. (16), (17), (18) and (20) into Eq. (19). Figure 7
shows the relationship between the output power and the injection current under different apertures of the three designed quantum-well VCSELs. The injection current linearly changes from 0 to 500 mA. As can be seen, the output power increases first and then decreases with the increase of the injection current, which can be attributed to the self-heating effect of VCSELs. At the same time, the output power varies with the aperture size of VCSELs. The maximum output power of GaAs quantum-well VCSEL is next to 80 mW when the device diameter is 60 μm as shown in Fig. 7(a), while for GaAs_{0.87}P_{0.13} quantum-well VCSEL, it is next to 140 mW when the device diameter is 120 μm as shown in Fig. 7(b), and for In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL, it exceeds 150 mW when the device diameter is 160 μm as shown in Fig. 7(c).Thus, we can draw that the In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL has the largest output power. When the aperture is 80 μm, the slope efficiency of GaAs quantum-well VCSEL is about 57% as shown in Fig. 7(a), while for GaAs_{0.87}P_{0.13} quantum-well VCSEL, it is about 60% as shown in Fig. 7(b), and for In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL, it is about 65% as shown in Fig. 7(c). Thus, we can draw that the In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL has the largest slope efficiency. It is also shown that when the aperture size is the same, the In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL has the lowest threshold current, while the GaAs quantum-well VCSEL has the highest, and the GaAs_{0.87}P_{0.13} quantum-well VCSEL has a moderate value. This result is consistent with the result described in Fig. 6.

## 4. Conclusion

An unstrained GaAs/Al_{0.3}Ga_{0.7}As, a tensilely strained GaAs_{x}P_{1-x}/Al_{0.3}Ga_{0.7}As and a compressively strained In_{1-x-y}Ga_{x}Al_{y}As/Al_{0.3}Ga_{0.7}As quantum-well active regions are designed in order to emit near 808 nm at operating temperature of VCSELs. The numerical simulation shows that the gain peak wavelength is near 800 nm at room temperature for GaAs well with width of 4 nm, GaAs_{0.87}P_{0.13} well with width of 13 nm and In_{0.14}Ga_{0.74}Al_{0.12}As well with width of 6 nm. Furthermore, the output characteristics of the three designed quantum-well VCSELs are studied and compared. The number of QWs is chosen three due to the lowest threshold current density. The maximum output power of In_{0.14}Ga_{0.74}Al_{0.12}As quantum-well VCSEL exceeds 150 mW when the device diameter is 160 μm, while for GaAs_{0.87}P_{0.13} quantum-well VCSEL, it is next to 140 mW when the device diameter is 120 μm, and for GaAs quantum-well VCSEL, it is next to 80 mW when the device diameter is 60 μm. The simulations indicate that the compressively strained In_{0.14}Ga_{0.74}Al_{0.12}As is the most appropriate candidate for the quantum well of 808-nm VCSELs due to its better carrier confinement, higher temperature stability, the lowest threshold current density, and the highest material gain, slope efficiency and output power.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China (NNSFC) under Grant No. 60636020, 60706007, 10974012, 60876036, 90923037, 11074247, and 61006054.

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