In the last few decades, gallium arsenide (GaAs) alloys and III–V semiconductors were the most commercialized materials for laser diodes (LDs)^{[1, 2]}. However, although GaAs technology is mature and GaAs-based lasers are well known for their high efficiency in converting electricity into monochromatic light, the emission of coherent light is limited in the red and infrared regions of the optical spectrum^[3-7]. Therefore, due to the small bandgap of these materials, III–V semiconductors are unsuitable for the fabrication of short-wavelength optical devices^[8-15].

An important challenge in the field of optoelectronics is the development of short wavelength lasers which can emit coherent light in green, blue, violet, and ultraviolet (UV) regimes. In particular, III-nitride and IV wide bandgap materials have received a great attention for the promising realization of several devices^[16-24]. These devices are able to operate at high power levels and high temperatures, due to their mechanical hardness and larger band offset^[25-32]. Recently, GaN-based high efficiency short wavelength light emitting diodes (LEDs) and LDs have been commercialized by achieving apparent improvements in terms of crystal quality and conductivity control in solid-state lighting and high-density optical storage systems. In addition, one of the most interesting features of GaN is the realization of the wurtzite-type crystal for semiconductor lasers. Although zinc-blend GaN has also been studied, it still does not have as good qualities as the wurtzite crystal^{[33, 34]}.

In this paper, the optical gain and threshold current density of a compressive-strained AlGaN/GaN quantum well (QW) laser, which emits in the UV regime, have been investigated in a wide range of temperatures by means of the Atlas-Silvaco physical device simulator. The optical gain spectra have been calculated by using a simple parabolic band model and also a multiband formalism based on the perturbation theory (k.p model) that is particularly suited for crystals with wurtzite symmetry. In the first case, we have assumed the electron and hole wavefunctions calculated by taking into account transitions that occur only between the first levels of the conduction band and valence band (band-to-band model) similarly to Ref. [35]. In the second case, we have considered the intervalence band absorption effects from different energy levels, namely heavy-hole (HH), light-hole (LH), and split-off (SO), computed via a 6 × 6 k.p model in the QW region^{[36, 37]}. This model, accounting for an intermixing of states in the valence subbands, provides a much more accurate coupling with the conduction band for an improved laser diode optical response. The calculated optical gain is around 5000 cm^–1 for a QW width of 40 Å and an Al molar fraction of 0.1 at room temperature. The dependence of the laser threshold current density from temperature has been also investigated.

For a single QW with a width L_w, the threshold current density can be written as^[44]

The optical gain can be defined as the fractional increase in photons per unit of length in response to a given photon flux^[33]. The optical gain expression as a function of the photon energy hω is in the form^[38]

$\begin{array}{l} {G_{{\rm{opt}}}}\left( {h\omega } \right) \!=\! {\dfrac{{{{π}}{{{e}}^2}}}{{{n_{\rm{r}}}c\omega m_0^2{{{\varepsilon }}_0}}}} \displaystyle \mathop \sum \nolimits_{{{n}},{{m}}} \displaystyle \int {\dfrac{{{m_{\rm{r}}}}}{{{{π}}{h^2}{L_{\rm{w}}}}}} \dfrac{{\varGamma /\left( {2{{π}}} \right)}}{{{{\left( {E - h\omega } \right)}^2} + {{(\varGamma /2)}^2}}}\\ \qquad\qquad\;\;\times \left( {f\,_{\rm{c}}^n - f\,_{\rm{v}}^m} \right){\left| {{I_{m,n}}} \right|^2}{\left| {{M_{\rm{b}}}} \right|^2}{\rm{d}}E, \end{array}$  (1)

$ {J_{{\rm{th}}}} = q{L_{\rm{w}}}{n_{{\rm{th}}}}/{\tau _{{\rm{th}}}}, $  (24)

where ε₀ is the free-space dielectric constant, n_r is the material refractive index, h is Planck's constant, c is the light velocity, e is the electron charge, E is the transition energy, I_m,n is the overlap integral of electron and hole wavefunctions, Γ is the line-width due to Lorentzian broadening, L_w is the well width, M_b is the polarization-dependent bulk momentum matrix element, m₀ is the carrier mass in free-space, and m_r = (m_c⁻¹ + m_v⁻¹)⁻¹ is the reduced effective mass which depends on the valence (m_v) and conduction (m_c) band effective masses. Finally, in Eq. (1), the sum is taken all over the transitions between the electron and hole subband energy levels, labelled in the following as E_n and E_m, and the Fermi-Dirac distribution functions fⁿ_c and f^m_v are described by using

where n_th and τ_th are the threshold carrier density and lifetime, respectively. By increasing the temperature, the intrinsic losses of the laser diode tend to increase while the device internal efficiency factor tends to decrease. The variation of the threshold current with temperature is assumed in the form^[45]:

$ f\,_{\rm{c}}^n\left( E \right) = \frac{1}{{1 + {\rm{exp}}\left\{ {\left[ {{E_n} + \left( {{m_{\rm{r}}}/{m_{\rm{c}}}} \right)\left( {E - {E_{mn}}} \right) - {E_{{F_{\rm{c}}}}}} \right]/kT} \right\}}}, $  (2)

$ {J_{{\rm{th}}}}\left( T \right) = {J_{{\rm{th}}}}{\rm{ex}}{{\rm{p}}{\frac{{T - {T_0}}}{{{T_0}}}}}, $  (25)

where J_th is the reference value at a given temperature, and T₀ is a specific parameter that determines the effective degradation of the diode conduction capabilities and therefore the resulting increase of the laser threshold current. This temperature parameter was calculated for AlGaN-based laser diodes by fitting experimental data on J_th in Ref. [14]. In particular, T₀ was estimated to be 132 K in the low temperature range and it drops down to 89 K for T higher than about 395 K.

$ f\,_{\rm{v}}^m\left( E \right) = \frac{1}{{1 + {\rm{exp}}\left\{ {\left[ {{E_m} - \left( {{m_{\rm{r}}}/{m_{\rm{v}}}} \right)\left( {E - {E_{mn}}} \right) - {E_{{F_{\rm{v}}}}}} \right]/kT} \right\}}}, $  (3)

where E_mn = E_n − E_m, and E_Fc and E_Fv are the electron and hole quasi-Fermi levels, respectively. To calculate the optical gain, therefore, we have to calculate the QW subband energies and the relative electron and hole wavefunctions in the device active region, namely the GaN layer situated between the two AlGaN barrier layers. The strain effect and the strong built-in field effect due to the piezoelectric polarization of III-nitride materials have been carefully taken into account in the calculations of the laser optical gain.

A schematic diagram of the energy band structure assumed for modelling and simulations is shown in Fig. 1.

By using the perturbation theory in considering the valence intersubband transitions, the hole energy states have been computed via a 6 × 6 diagonalized k.p Hamiltonian matrix in the form of^{[36, 37]}:

$ H_{6 \times 6}^\nu \left( K \right) = \left[ {\begin{array}{*{20}{c}} {H_{3 \times 3}^U\left( K \right)} & 0\\ 0 & {H_{3 \times 3}^L\left( K \right)} \end{array}} \right], $  (4)

where and are 3 × 3 matrices given by

$ {H^U}\left( K \right) = \left[ {\begin{array}{*{20}{c}} F & {{K_t}} & { - i{H_t}}\\ {{K_t}} & G & {\Delta - i{H_t}}\\ {i{H_t}} & {\Delta + i{H_t}} & \lambda \end{array}} \right], $  (5)

$ {H^L}\left( K \right) = \left[ {\begin{array}{*{20}{c}} F & {{K_t}} & {i{H_t}}\\ {{K_t}} & G & {\Delta + i{H_t}}\\ { - i{H_t}} & {\Delta - i{H_t}} & \lambda \end{array}} \right]. $  (6)

These matrix elements, which contain the general expressions for a strained semiconductor, can be expressed as follows:

$ F = {\varDelta _1} + {\varDelta _2} + \lambda + \theta , $  (7)

$ G = {\varDelta _1} - {\varDelta _2} + \lambda + \theta , $  (8)

$ {\lambda } = \frac{{{\hbar ^2}}}{{2{m_0}}}\left( {{A_1}k_z^2 + {A_2}k_t^2} \right) + {D_1}{\varepsilon _{zz}} + {D_2}\left( {{\varepsilon _{zz}} + {\varepsilon _{yy}}} \right), $  (9)

$ {{\theta }} = \frac{{{\hbar ^2}}}{{2{m_0}}}\left( {{A_3}k_z^2 + {A_4}k_t^2} \right) + {D_3}{\varepsilon _{zz}} + {D_4}\left( {{\varepsilon _{zz}} + {\varepsilon _{yy}}} \right), $  (10)

$ {K_t} = \frac{{{\hbar ^2}}}{{2{m_0}}}{A_5}k_z^2, $  (11)

$ {H_t} = \frac{{{\hbar ^2}}}{{2{m_0}}}{A_6}{k_t}{k_z}, $  (12)

$ \varDelta = \sqrt 2 {\varDelta _3}, $  (13)

where A_1–6 are hole effective mass parameters, Δ_1–3 are split energies, D_1–4 are shear deformation potentials, k_t and k_z are wave vectors along the device x–y basal plane and z-direction, respectively. In addition, the strain tensors relative to the lattice constant mismatch between the QW and barrier layers, i.e. ε_xx, ε_yy, and ε_zz, are modelled through the expressions:

$ {\varepsilon _{xx}} = {\varepsilon _{yy}} = \frac{{{a_1} - {a_0}}}{{{a_0}}}, $  (14)

$ {\varepsilon _{zz}} = \frac{{{c_1} - {c_0}}}{{{c_0}}}, $  (15)

where a_i and c_i are lattice constants in the device structure.

The polarization modelling includes both the spontaneous polarization (P_sp), which is strain independent, and the piezoelectric polarization along the growth direction <0001> given by ^{[39, 40]}:

$ {P_{{\rm{pz}}}} = 2{d_{\rm{pz}}}\left( {{C_{11}} + {C_{12}} - 2\frac{{C_{13}^2}}{{{C_{33}}}}} \right){\varepsilon _{xx}}, $  (16)

where the terms C_ij are elastic constants and d_pz is an opportune coefficient.

The reference parameters of the binary compounds GaN and AlN (wurtzite crystals) are summarized in Table 1^[41]. Here, α and β are the coefficients that describe the nonlinear temperature dependence of the bandgap energy E_g according to the expression^[42]

$ {E_{\rm{g}}}\left( T \right) = {E_{\rm{g}}} - {\frac{{{{\alpha }}{T^2}}}{{\beta + T}}} . $  (17)

In addition, for a specific Al_xGa_1–xN ternary alloy, the variation of E_g due to the effective Al molar fraction x was calculated by using^{[42, 43]}

$ {E_{\rm{g}}}\left( {x,T} \right) = xE_{\rm{g}}^{{\rm{AlN}}}\left( T \right) + \left( {1 - x} \right)E_{\rm{g}}^{{\rm{GaN}}}\left( T \right) - 0.6x\left( {1 - x} \right), $  (18)

where the so-called bowing parameter fixed to 0.6 accounts for the deviation from a linear interpolation between the values of E_g^AlN(T) and E_g^GaN(T) calculated by considering that the coefficients α and β are also dependent on x as follows:

$ \alpha \left( x \right) = {\rm{}}\left( {1 - x} \right){\alpha _{{\rm{GaN}}}} + x{\alpha _{{\rm{AlN}}}} - 2.15x\left( {1 - x} \right), $  (19)

$ \beta \left( x \right) = {\rm{}}\left( {1 - x} \right){\beta _{{\rm{GaN}}}} + x{\beta _{{\rm{AlN}}}} - 1561x\left( {1 - x} \right). $  (20)

Finally, Adachi's refractive index model aids to calculate the Al_xGa_1–xN x-dependent refractive index as^[38]

$ {n_{\rm{r}}}\left( \omega \right) = {\sqrt {A{{\left( {\frac{{h\omega }}{{{E_{\rm{g}}}}}} \right)}^{ - 2}}\left( {2 - \sqrt {1 + \frac{{h\omega }}{{{E_{\rm{g}}}}}} - \sqrt {1 - \frac{{h\omega }}{{{E_{\rm{g}}}}}} } \right) + B} } , $  (21)

where the material parameters A and B are in the form

$ A\left( x \right) = 9.827 - 8.216x - 31.59{x^2}, $  (22)

$ B\left( x \right) = 2.736 + 0.842x - 6.293{x^2}. $  (23)

The investigated device structure consists of a GaN active layer situated between two 80-nm-thick Al_xGa_1–xN barrier regions. By fixing x = 0.1 and L_w = 40 Å as entry data for simulations, the optical gain spectrum is firstly calculated as a function of the wavelength for different carrier densities in the range 1 × 10¹⁸–1 × 10¹⁹ cm^–3 as shown in Fig. 2. The value n = 1 × 10¹⁸ cm^–3 can, in fact, be assumed as the threshold carrier density (transparency density) for the considered device. As we can see, the increase of the injection level in the active region determines the increase of the maximum optical gain for both the band-to-band and 6-band k.p models. These behaviours are due to the filling of high states in the conduction band and valence band with the resulting increase of the emission phenomena as reported for similar laser structures^[46,47]. In addition, the observable shift of the emission spectrum towards the shorter wavelengths is explained on the basis of the quantization effects that tend to increase the transition energy of the bound states and therefore the emission wavelength decreases following the standard expression λ = hc/E^[48].

The curves in Figs. 2(a) and 2(b) are, however, quite different each other. In fact, the optical gain values in Fig. 2(b) are strongly increased if compared to those reported in Fig. 2(a) as a consequence of the intermixing of available states in the valence subbands. In addition, the gain spectrum calculated using the band-to-band model is rather wide as a consequence of the large energy difference between the quasi-Fermi levels which results from the model assumptions.

The optical gain dependence on the Al molar fraction for n = 1 × 10¹⁹ cm^–3 is shown in Fig. 3.

It is worth noting that in Al_xGa_1–xN/GaN QW lasers an increasing value of x meaningfully increases the maximum optical gain moving the gain spectrum towards the shorter wavelengths. This shift is due to the increase of the Al_xGa_1–xN bandgap, calculated as wide as 3.896 eV for x = 0.2, and the change of the quantization energy. At the same time, the increase of the maximum gain value is mainly due to the increase in the optical confinement because the refractive index of the barrier layers tends to become higher than that in the GaN active region.

A fundamental geometrical parameter in the design of the considered device is the width of the GaN well (L_w). The plot of the maximum optical gain calculated for different values of L_w in the range 30 to 60 Å is shown in Fig. 4. A wider QW penalizes increasingly the optical gain and L_w = 40 Å should be assumed as a limit value in this study. The wider the QW, the stronger electrons and holes are separated by the piezoelectric field and therefore the optical gain reduces. In other words, when the QW width increases, the density of states in this region decreases.

The temperature effects on the optical gain behaviour are shown in Fig. 5. Here, it is evident that the temperature has only a limited impact on the maximum gain value. In more detail, the optical gain tends to decrease as the temperature increase in accordance with the variation of the Fermi-Dirac distributions in Eqs. (2) and (3). Meanwhile, by increasing the temperature, the GaN bandgap shrinks and carriers can scatter to other subbands leading to a change of the spectrum range that tends to shift toward the lower photon energies.

The temperature-dependent GaN bandgap profile is shown in Fig. 6. The interatomic spacing increases when the amplitude of the atomic vibrations increases due to an increased thermal energy. This effect causes a dilatation of the lattice and leads to a displacement of the conduction and valence band edges.

Finally, the laser threshold current density variation with temperature in the range 260–340 K is shown in Fig. 7.

By using the k.p model, the increase of the threshold current density with temperature is stronger than the simple band-to-band behaviour. This increase is on the order of a factor 5 at T = 340 K. The minimum J_th value is around 60 A/cm² for both models at very low temperatures.

In this paper, we have investigated the optical response of a strained wurtzite GaN/Al_xGa_1–xN quantum well laser. The effects of the carrier density, quantum well width, Al concentration in the barrier layers, and temperature have been taken into account by using two different band energy models in the device structure. In particular, the optical gain increases by increasing the carrier density as well as the Al concentration whereas it decreases with the quantum well width and temperature. The simulation analysis based on the k.p model, with a more accurate calculation, provides better results in terms of optical gain reducing also the dependence of the threshold current density on temperature. Our analysis suggests that laser diodes based on a GaN/AlGaN QW can emit in the UV with a significant optical gain and a low threshold current density.