## 1.

## Introduction

Optically pumped vertical-external-cavity surface-emitting lasers (VECSELs), which replace the top distributed Bragg reflector of vertical-cavity surface-emitting lasers by a external cavity mirror and use optical pumping instead of electrical pumping, are attractive and under intensive study in recent years due to their power scalability, beam quality, and flexible spectral coverage.^{1} High output power and good beam quality 1-μm VECSELs have many applications in frequency conversion,^{2} energy utilization,^{3} and optical spectroscopy.^{4}

To obtain a more efficient or higher output power VECSEL, the elaborate design of quantum wells (QWs) is primary and necessary. Moloney^{5} and Hader^{6} have presented overviews of a novel first-principle quantum approach to design semiconductor QW for target wavelength. In this paper, we demonstrate a structural optimization of QWs that is mainly based on the analysis of material gain. We optimize the single-well structure according to the width, depth, and configuration dependences of the material gain, and then the double well (first introduced in a VECSEL by Fan^{7}) and triple well (have been experimentally used by Yoo^{8} and Schulz^{9}) structures are compared and optimized.

The theoretical model is quite straightforward. First, we use the famous *k·p* method to calculate the band structures. The Hamiltonian of electrons in a conduction band (CB) is

## Eq. 1

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} H^c (k) = \frac{{\hbar ^2 }}{2}\left( {\frac{{{\bm k}_t^2 }}{{m_{\rm e}^t }} + \frac{{{\bm k}_z^2 }}{{m_{\rm e}^z }}} \right) + V_{\rm e} (z) + E_{hc},\end{equation}\vskip-6pt\end{document}$${H}^{c}\left(k\right)=\frac{{\hslash}^{2}}{2}\left(\frac{{\mathit{k}}_{t}^{2}}{{m}_{\mathrm{e}}^{t}}+\frac{{\mathit{k}}_{z}^{2}}{{m}_{\mathrm{e}}^{z}}\right)+{V}_{\mathrm{e}}\left(z\right)+{E}_{hc},$$**is interpreted as a differential operator –**

*k**i*∇ and the transverse wave vector

*k**=*

_{t}

*k*_{x}+

*k*_{y}, [TeX:] $m_{\rm e}^t $${m}_{\mathrm{e}}^{t}$, and [TeX:] $m_{\rm e}^z $${m}_{\mathrm{e}}^{z}$ are the electron effective masses perpendicular (

*t*) and parallel (

*z*) to the growth direction,

*V*

_{e}(

*z*) is the potential energy of the unstrained conduction band edge. The CB structure can be determined by solving

*H*

^{c}ϕ

_{n}(

*z*) = [TeX:] $E_n^c $${E}_{n}^{c}$

*ϕ*

*(*

_{n}*z*) at

*k**= 0, then the parabolic approximation is used to obtain the whole CB structure and the envelope functions.*

_{t}The 6 × 6 Luttinger–Kohn Hamiltonian,^{10} in which the heavy-hole, light-hole, and spin-orbit coupling are taken into account, is used to obtain the valence band (VB) structures of the QW. The envelope function and the subband energy of VB in a QW can be obtained by solving the multiband effective mass equation,

## Eq. 2

[TeX:] \documentclass[12pt]{minimal}\begin{document}\vspace*{-6pt}\begin{equation} \sum\limits_m {\left[ {H^{vv^\prime} \left( {{\bm k}_t, - i\frac{d}{{dz}}} \right) + V_{\rm h} (z)} \right]g_m^{v^\prime} ({\bm k}_t,z) = E_m^v } g_m^v ({\bm k}_t,z),\end{equation}\vskip-8pt\end{document}$$\sum _{m}\left[{H}^{v{v}^{\prime}}\left({\mathit{k}}_{t},-i\frac{d}{dz}\right)+{V}_{\mathrm{h}}\left(z\right)\right]{g}_{m}^{{v}^{\prime}}({\mathit{k}}_{t},z)={E}_{m}^{v}{g}_{m}^{v}({\mathit{k}}_{t},z),$$*V*

_{h}(

*z*) is the potential of the hole and

*V*

_{e}(

*z*)

*= V*

_{h}(

*z*)

*+E*

*. For bandgap*

_{g}*E*

*, the increasing temperature caused bandgap reduction is*

_{g}*E*

*(*

_{g}*T*) =

*E*

*(0) – αΘ{[1 + (2*

_{g}*T*/Θ)

^{p}]

^{1/}

^{p}-1}/2, where

*E*

*(0) is the bandgap under 0 K, α is the slope parameter, Θ and*

_{g}*p*are the average phonon temperature and the phonon dispersion parameter. The Coulomb interaction is taken into account through the bandgap renormalization formula derived from many-body theory,

^{11, 12}

## Eq. 3

[TeX:] \documentclass[12pt]{minimal}\begin{document}\vspace*{-6pt}\begin{equation} \Delta E_g = - C\left[ {\frac{{\varepsilon _{{\rm st}}^5 }}{N}\left( {m_0 \frac{{m_c + m_v }}{{m_c m_v }} + BT^2 \frac{{\varepsilon _{{\rm st}} }}{N}} \right)} \right]^{ - 1/4},\end{equation}\vskip-6pt\end{document}$$\Delta {E}_{g}=-C{\left[\frac{{\varepsilon}_{\mathrm{st}}^{5}}{N}\left({m}_{0}\frac{{m}_{c}+{m}_{v}}{{m}_{c}{m}_{v}}+B{T}^{2}\frac{{\varepsilon}_{\mathrm{st}}}{N}\right)\right]}^{-1/4},$$_{st}is the static dielectric,

*m*

*and*

_{c}*m*

*are effective masses,*

_{v}*T*and

*N*are temperature and carrier density,

*C*and

*B*are two fit parameters. This simplification was found in good agreement with reported experimental results

^{13}and is suitable for the optimization of the QW structure, which is rather qualitative and not as rigorous as accurate design demands.

Then, the material gain of a QW is^{10, 14}

## Eq. 4

[TeX:] \documentclass[12pt]{minimal}\begin{document}\vspace*{-6pt}\begin{equation} g(\hbar \omega) = \frac{{\pi q^2 }}{{n_w c\varepsilon _0 m_0^2 \omega L_w }}\sum {\int {| {\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}} \over e} \cdot M_{nm} ({\it k}_t)}|^2} } \frac{{\left[f_n^c ({\bm k}_t) - f_m^v ({\bm k}_t)\right]\gamma /\pi }}{{\left[E_{nm}^{cv} ({\bm k}_t) - \hbar \omega\right]^2 + \gamma ^2 }}\frac{{{\bm k}_t d{\bm k}_t }}{{2\pi }},\end{equation}\vskip-6pt\end{document}$$g\left(\hslash \omega \right)=\frac{\pi {q}^{2}}{{n}_{w}c{\varepsilon}_{0}{m}_{0}^{2}\omega {L}_{w}}\sum \int |\genfrac{}{}{0pt}{}{{\scriptscriptstyle \rightharpoonup}}{e}\xb7{M}_{nm}\left({\mathit{k}}_{t}\right){|}^{2}\frac{[{f}_{n}^{c}({\mathit{k}}_{t})-{f}_{m}^{v}({\mathit{k}}_{t})]\gamma /\pi}{{[{E}_{\mathrm{nm}}^{\mathrm{cv}}({\mathit{k}}_{t})-\hslash \omega ]}^{2}+{\gamma}^{2}}\frac{{\mathit{k}}_{t}d{\mathit{k}}_{t}}{2\pi},$$*q*and

*m*

_{0}are the electron charge and rest mass,

*c*and ɛ

_{0}are the velocity of light and the permittivity in free space,

*n*

*and*

_{w}*L*

*are the refractive index and the width of the QW, γ is the half linewidth of the Lorentzian function,*

_{w}*M*

*(*

_{nm}

*k**) is the momentum matrix element, [TeX:] $f_n^c $${f}_{n}^{c}$(*

_{t}

*k**) and [TeX:] $f_m^v $${f}_{m}^{v}$(*

_{t}

*k**) are the Fermi distribution functions, and [TeX:] $E_{nm}^{cv} $${E}_{nm}^{cv}$(*

_{t}

*k**) is the transition energy.*

_{t}For strained QWs, the strain-induced hydrostatic deformation potential (*E** _{hc}* in CB and

*E*

*in VB) and the tetragonal distortion energy (*

_{hv}*E*

*) are included, the band offsets are calculated using model-solid theory, and all parameters of ternary alloy*

_{t}*ABC*are obtained by the interpolation of related binary material

*AB*and

*AC*as

*P*(

*AB*

*C*

_{x}_{1–x}) =

*xP*(

*AB*) + (

*1 – x*)

*P*(

*AC*) –

*x*(

*1 – x*)

*b*

*, where*

_{ABC}*b*

*is the bowing parameter of ternary alloy*

_{ABC}*ABC*, which indicates the deviation from the linear interpolation. Details for the numerical computation and parameters used in the calculation can be found in Refs. 10, 11, 13.

## 2.

## Optimization of a Single-Well Structure

In the following, we will discuss the influences of the width, depth, and configuration of the QWs on the material gain. In the calculation, the QWs are all In_{0.132}Ga_{0.868}As for operating near 1-μm wavelength, the carrier density is assumed to be a representative number of 3.6×10^{24} m^{−3} and the temperature to be 300 K except we have a special note.

## 2.1.

### QW Width

Figure 1 shows the width dependence of material gain of a single-well structure (with Al_{0.06}Ga_{0.94}As barrier). It can be seen that, when the width is <8 nm, although the narrower width brings stronger quantum confinement, thus, more intensive envelope function, the overlapping integration in transition matrix element still decreases because of the decreased integral domain. This leads to a significant reduction of material gain. When the width is >10 nm, the quantum confinement becomes weaker; the second confined level of electrons appears, and the gain spectrum becomes wider and with a subpeak.

When the composition of In in InGaAs QW is changed, the corresponding emission wavelength and peak gain are also changed. We plot the peak gains as functions of the emission wavelengths of QWs with different width in Fig. 1. It is quit clear that the optimal width is from 8 to 10 nm because the peak gain of QW of <8 nm is much smaller, and the peak gain of QW of >10 nm begins to decrease.

Figure 1 shows the relationship between the peak gain and temperature. As can be seen, the wider wells have better temperature stability. Because wells of <8 nm are very sensitive to the temperature and the improvement of the stability is not clear when the width exceeds 10 nm, the optimal width is 8–10 nm, when the temperature stability is considered.

## 2.2.

### QW Depth

The depth-dependence of material gain of a single-well structure is shown in Fig. 2 (the well width is assumed to be 8 nm, and the composition of Al in AlGaAs barrier is chosen to characterize the depth). The influence of the depth on the emission wavelength is obvious: the emission wavelength blueshifts of ∼20nm when the Al composition changes from 0 to 0.24. However, the affect of the depth on the peak gain is not clear: the peak gain increases only ∼2% when the Al composition varies from 0 to 0.24.

Changing Al composition of AlGaAs barrier, we plot the curves of peak gain versus emission wavelengths in Fig. 2. It shows that QWs with AlGaAs barriers have bigger gain than with GaAs (i.e., zero Al composition) barrier, and 0.1 composition of Al is enough because the contribution of more Al composition to the increase of gain is not clear.

## 2.3.

### QW Configuration

As shown in Fig. 3, we consider six different QW configurations, which are generally used in InGaAs strained QW VECSELs. Figure 4 shows the gain spectra of the above six

different QWs (the carrier density is assumed to be 4.6×10^{24} m^{−3}). We can see that the fourth QW has the largest peak gain and the shortest emission wavelength because of its deepest well depth, while the last QW has the secondary peak gain and the longest emission wavelength.

Figure 4 indicates the peak gains of six different single-well structures versus carrier density. Except for the fourth QW, which always has bigger peak gain, the contrast of the peak gains of the other five QWs is not evident when the carrier density is small. However, when the carrier density is increased, differences of the other five QWs are expanded, and the third and last QWs excel the rest, obviously, and this is in good agreement with reported experiment.^{15} Considering that GaAsP layer in the fourth QW may block the carriers diffusing to the well, we recommend the third and last QWs for high-power VECSELs.

## 3.

## Optimization of a Double-Well Structure

The gain spectra of double-well structure with different width are shown in Fig. 5 (the width means the width of every individual well; the spacer is 6 nm GaAs_{0.94}P_{0.06}). It can be concluded from Fig. 5 that the optimal width of a double-well structure is from 6 to 8 nm.

Generally, the spacer between two separated wells can be GaAs, AlGaAs, or GaAsP. When AlGaAs or GaAsP is used as the spacer, the Al/P composition will affect the gain. Figure 6 shows the influence of Al/P composition on the peak gain of a 6-nm double-well structure. We can see from Fig. 6 that AlGaAs/GaAsP spacer is better than GaAs spacer (i.e., zero Al/P), and the optimal compositions of Al or P are about 0.08 and 0.2, respectively.

## 4.

## Optimization of a Triple-Well Structure

The width dependence of gain of a triple-well structure is shown in Fig. 7 (the meaning of the width is same as in double-well structure). Because excessively high spacer will block the carriers diffusing to the middle well, the spacers are assumed to be 6 nm GaAs layer. It can be seen from Fig. 7, the optimal width of a triple-well structure is also between 6 and 8 nm.

Now we compare the single-, double-, and triple-well structures with each other. Figure 8 shows the band structure of them under same conditions: GaAs barriers, 6-nm well width, and 6-nm GaAs spacer. The inserts are their energy profiles and envelope functions of the first confined level in CB. As shown in Fig. 8, single-well structure has one confined level in CB and three confined levels in VB. For double- and triple-well structures, because of the coupling, every band splits into two and three levels, respectively, and the split separations between higher levels are clearer. This level split can be used to explain why there is no subpeak in the gain spectra of double- and triple-well structure when the width increases up to 12 nm.

Figure 9 shows the gain spectra of the three QW structures (6-nm width). It can be seen that a double-well structure has the largest gain, and this can partly been interpreted by the insets in Fig. 8: for the overlapping integration in the transition matrix element, single-well structure has the largest amplitude of envelope function but the smallest integral domain, while triple-well structure has the smallest amplitude of envelope function but the largest integral domain. In comparison, double-well structure has moderate amplitude of envelope function and integral domain, thus produce the largest gain.

Figure 9 shows the normalized confinement factors as functions of laser wavelengths. The designed operating wavelength is 1 μm. Obviously, single-well structure has the poorest tolerance: when the laser wavelength deviates from 1 μm to ∼10 nm, the confinement factor is decreased by 5%, and this may result in rollover of a VECSEL.

Figures 9 and 9 show that within a 20-nm deviated wavelength range, double-well structure has not only a higher gain than triple-well structure, but also a larger confinement factor than triple-well structure. Thus, if the tolerance is the main factor that needs to be considered in the design, then double-well structure seems to be enough for a 1-μm VECSEL. If taking into account the thickness of the active region, or for a long-wavelength VECSEL, triple-well, even quadruple-well structures may be an alternative.

## 5.

## Conclusions

In conclusion, on the basis of the analysis of material gain, we have presented a comprehensive structural optimization of the QWs used in a 1-μm VECSEL. For a single-well structure, the optimal width lies between 8 and 10 nm, the optimal depth is QW with ∼0.1 Al composition in AlGaAs barrier, and the optimal configurations are graded-index QW and QW with AlGaAs barrier and GaAs buffer layer. The optimized width of a double- or triple-well structure lies between 6 and 8 nm. For a double-well structure, using AlGaAs with ∼0.08 Al composition or GaAsP with ∼0.2 P compositions as the spacer can produce the maximum peak gain. Compared to its single- and triple-well counterparts, double-well structure provides higher gain and has enough tolerance to the wavelength deviation.