• Journal of Semiconductors
  • Vol. 41, Issue 6, 062301 (2020)
A. Menani1, L. Dehimi1、2, S. Dehimi3, and F. Pezzimenti4
Author Affiliations
  • 1Laboratory of Metallic and Semiconducting Materials (LMSM), Department of Electrical Engineering, Biskra University, Biskra, Algeria
  • 2Faculty of Science, Elhadj Lakhdar University, Batnal, Algeria
  • 3Research Centre in Industrial Technology (CRTI), Algiers, Algeria
  • 4DIIES – Mediterranea University of Reggio Calabria, Reggio Calabria, Italy
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    DOI: 10.1088/1674-4926/41/6/062301 Cite this Article
    A. Menani, L. Dehimi, S. Dehimi, F. Pezzimenti. Modelling and optical response of a compressive-strained AlGaN/GaN quantum well laser diode[J]. Journal of Semiconductors, 2020, 41(6): 062301 Copy Citation Text show less

    Abstract

    The effects of the quantum well (QW) width, carrier density, and aluminium (Al) concentration in the barrier layers on the optical characteristics of a gallium nitride (GaN)-based QW laser diode are investigated by means of a careful modelling analysis in a wide range of temperatures. The device’s optical gain is calculated by using two different band energy models. The first is based on the simple band-to-band model that accounts for carrier transitions between the first levels of the conduction band and valence band, whereas the second assumes the perturbation theory (k.p model) for considering the valence intersubband transitions and the relative absorption losses in the QW. The results reveal that the optical gain increases with increasing the n-type doping density as well as the Al molar fraction of the AlxGa1–xN layers, which originate the GaN compressive-strained QW. In particular, a significant optical gain on the order of 5000 cm–1 is calculated for a QW width of 40 Å at room temperature. In addition, the laser threshold current density is of few tens of A/cm2 at low temperatures.
    $\begin{array}{l} {G_{{\rm{opt}}}}\left( {h\omega } \right) \!=\! {\dfrac{{{{π}}{{{e}}^2}}}{{{n_{\rm{r}}}c\omega m_0^2{{{\varepsilon }}_0}}}} \displaystyle \mathop \sum \nolimits_{{{n}},{{m}}} \displaystyle \int {\dfrac{{{m_{\rm{r}}}}}{{{{π}}{h^2}{L_{\rm{w}}}}}} \dfrac{{\varGamma /\left( {2{{π}}} \right)}}{{{{\left( {E - h\omega } \right)}^2} + {{(\varGamma /2)}^2}}}\\ \qquad\qquad\;\;\times \left( {f\,_{\rm{c}}^n - f\,_{\rm{v}}^m} \right){\left| {{I_{m,n}}} \right|^2}{\left| {{M_{\rm{b}}}} \right|^2}{\rm{d}}E, \end{array}$ (1)

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    $ 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)

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    $ 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)

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    $ 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)

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    $ {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)

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    $ {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)

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

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

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    $ {\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)

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    $ {{\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)

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

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

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

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

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

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

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

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    $ {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)

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

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

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    $ {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)

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

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

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

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

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    A. Menani, L. Dehimi, S. Dehimi, F. Pezzimenti. Modelling and optical response of a compressive-strained AlGaN/GaN quantum well laser diode[J]. Journal of Semiconductors, 2020, 41(6): 062301
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