nnp:1d_ingaas_laser_diode
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nnp:1d_ingaas_laser_diode [2020/02/20 15:50] takuma.sato [Recombination of carriers and emission spectrum] |
nnp:1d_ingaas_laser_diode [2024/01/03 15:17] stefan.birner removed |
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| ===== 1D InGaAs Multi-quantum well laser diode ===== | ===== 1D InGaAs Multi-quantum well laser diode ===== | ||
| Author: Takuma Sato, nextnano GmbH | Author: Takuma Sato, nextnano GmbH | ||
| + | |||
| + | **A newer version of this tutorial can be found here: | ||
| + | [[https://www.nextnano.com/manual/nextnanoplus_tutorials/1D/laser_diode.html|https://www.nextnano.com/manual/nextnanoplus_tutorials/1D/laser_diode.html]] | ||
| + | ** | ||
| In this tutorial, we simulate optical emission of a 1D InGaAs multi-quantum well laser diode grown on InP substrate. The blue region is the separate confinement heterostructure (SCH), which forms an optical waveguide in the transverse direction to confine the emitted light (red arrow). The multi-quantum wells and SCH are clad by InP on both sides. A voltage bias is applied to the gray edges. | In this tutorial, we simulate optical emission of a 1D InGaAs multi-quantum well laser diode grown on InP substrate. The blue region is the separate confinement heterostructure (SCH), which forms an optical waveguide in the transverse direction to confine the emitted light (red arrow). The multi-quantum wells and SCH are clad by InP on both sides. A voltage bias is applied to the gray edges. | ||
| Line 43: | Line 47: | ||
| **Radiative recombination** describes the recombination of electron-hole pairs at a position $\mathbf{x}$ by emitting a photon and is given by | **Radiative recombination** describes the recombination of electron-hole pairs at a position $\mathbf{x}$ by emitting a photon and is given by | ||
| $$ | $$ | ||
| - | R_{\mathrm{rad}}(\mathbf{x}) = C\left[n(\mathbf{x}) p(\mathbf{x}) – n_i^2\right], | + | R_{\mathrm{rad}}(\mathbf{x}) = C n(\mathbf{x}) p(\mathbf{x}), |
| $$ | $$ | ||
| - | where $n_i$ is the intrinsic density of the charge carriers. $C$ is a material dependent constant given in the database and has the unit of cm$^3$/s. $R_{\mathrm{rad}}(\mathbf{x})$ is written in ''emitted_photon_density.dat''. | + | where $C$ is a material dependent constant given in the database and has the unit of cm$^3$/s. |
| Since the radiative recombination process involves no phonons, this transition is vertical and therefore this contribution is only relevant for semiconductors with a direct band gap such as GaAs. | Since the radiative recombination process involves no phonons, this transition is vertical and therefore this contribution is only relevant for semiconductors with a direct band gap such as GaAs. | ||
| Line 60: | Line 64: | ||
| </figure> | </figure> | ||
| - | Emission spectrum is calculated when the following data is specified: | + | Energy-dependent charge density and emission spectrum are calculated when the following is specified (see [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/classical.htm|classical{} documentation]] for details): |
| <code> | <code> | ||
| classical{ | classical{ | ||
| + | energy_distribution{ | ||
| + | min = -1.5 # Integrate from | ||
| + | max = 0.5 # Integrate to | ||
| + | energy_resolution = 0.005 # Integration resolution | ||
| + | only_quantum_regions = yes # (default: no) | ||
| + | } | ||
| emission_spectrum{ | emission_spectrum{ | ||
| min = -1.5 # Integrate from | min = -1.5 # Integrate from | ||
| Line 72: | Line 82: | ||
| } | } | ||
| </code> | </code> | ||
| - | The mobility model and recombination models for the current equation are specified in ''currents{}'' as | + | The mobility model and recombination models for the current equation are specified in [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/currents.htm|currents{}]] as |
| <code> | <code> | ||
| currents{ | currents{ | ||
| Line 85: | Line 95: | ||
| </code> | </code> | ||
| - | The following flag specifies which equations to solve. | + | The [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/run.htm|run{}]] flag specifies which equations to solve. This is the main difference between ''LaserDiode_*_qm_nnp.in'' and ''LaserDiode_*_cl_nnp.in''. |
| <code> | <code> | ||
| + | # qm | ||
| run{ | run{ | ||
| solve_strain{} # solves the strain equation | solve_strain{} # solves the strain equation | ||
| Line 102: | Line 113: | ||
| } | } | ||
| } | } | ||
| + | |||
| + | # cl | ||
| + | run{ | ||
| + | solve_strain{} # solves the strain equation | ||
| + | solve_current_poisson{ # solves the coupled current and Poisson equations self-consistently | ||
| + | output_log = yes | ||
| + | } | ||
| + | } | ||
| + | |||
| </code> | </code> | ||
| In this case nextnano++ first solves the strain equation from the crystal orientation to decide the polarization charges (piezoelectric effect) and shifted bandedges. Then the program solves the coupled current-Poisson-Schroedinger equations in a self-consistent way (input file: ''LaserDiode_InGaAs_1D_qm_nnp.in''). For the classical calculation (''LaserDiode_InGaAs_1D_cl_nnp.in''), ''solve_quantum{}'' and ''outer_iteration{}'' are commented out to restrict the calculation to the current-Poisson equations only. | In this case nextnano++ first solves the strain equation from the crystal orientation to decide the polarization charges (piezoelectric effect) and shifted bandedges. Then the program solves the coupled current-Poisson-Schroedinger equations in a self-consistent way (input file: ''LaserDiode_InGaAs_1D_qm_nnp.in''). For the classical calculation (''LaserDiode_InGaAs_1D_cl_nnp.in''), ''solve_quantum{}'' and ''outer_iteration{}'' are commented out to restrict the calculation to the current-Poisson equations only. | ||
| Line 113: | Line 133: | ||
| </figure> | </figure> | ||
| - | For the bias $0.8$ V (Figure {{ref>bandedge2}}), in contrast, it lies above the red line, allowing electrons to flow into the quantum wells. An electron trapped in the quantum wells is likely to recombine with a hole in the valence band, emitting a photon. In the input file ''emitted_photon_density.dat'', one can see that the photons are emitted from this active region (not shown). Figure {{ref>emission}} shows the emission spectrum in this case. When the bias is too small, e.g. Figure {{ref>bandedge}}, the intensity is much smaller, as can be seen in Figure {{ref>IV}}. | + | For the bias $0.8$ V (Figure {{ref>bandedge2}}), in contrast, it lies above the red line, allowing electrons to flow into the quantum wells. An electron trapped in the quantum wells is likely to recombine with a hole in the valence band, emitting a photon. In the input file ''\Optical\emission_photon_density.dat'', one can see that the photons are emitted from this active region (not shown). Figure {{ref>emission}} shows the emission spectrum in this case. When the bias is too small, e.g. Figure {{ref>bandedge}}, the intensity is much smaller, as can be seen in Figure {{ref>IV}}. |
| <figure bandedge2> | <figure bandedge2> | ||
| Line 120: | Line 140: | ||
| </figure> | </figure> | ||
| - | With the input file ''LaserDiode_InGaAs_1D_qm_nnp.in'', we also obtain the electron- and hole-wavefunctions. They are stored in ''wf_probabilities_shift_quantum_region_Gamma_0000.dat''. | + | In the input file ''LaserDiode_InGaAs_1D_qm_nnp.in'', the single-band Schroedinger equation is coupled to the current-Poisson equation and solved self-consistently. The wave functions of electrons and holes along with eigenvalues are written in ''\Quantum\wf_probabilities_shift_quantum_region_Gamma_0000.dat'' and ''\Quantum\wf_probabilities_shift_quantum_region_HH_0000.dat'' (Figure {{ref>wf}} and {{ref>ev}}). The light hole and split-off states are out of the quantum wells and not of our interest here. |
| <figure wf> | <figure wf> | ||
| {{:nnp::laserdiode_tutorial_wf.png}} | {{:nnp::laserdiode_tutorial_wf.png}} | ||
| - | <caption>Probability distribution $|\psi(x)|^2$ of the localized modes of electrons and holes for the bandstructure {{ref>bandedge2}}. Horizontal lines are the corresponding eigenenergies. </caption> | + | <caption>Probability distribution $|\psi(x)|^2$ of the lowest localized modes of electrons and holes for the bandstructure {{ref>bandedge2}}. Horizontal lines are the corresponding eigenenergies. </caption> |
| </figure> | </figure> | ||
| - | + | ||
| - | The spontaneous and stimulated emission spectra are written in ''emitted_photon_spectrum.dat'' and ''stimulated_emission_spectrum.dat'', respectively. | + | <figure ev> |
| + | {{:nnp::laserdiode_ev.png}} | ||
| + | <caption>Eigenvalues of the Gamma-band and heavy-hole-band states in relation to bandedges. The Gamma band has single "miniband", whereas the heavy-hole band has three. </caption> | ||
| + | </figure> | ||
| + | |||
| + | The charge density with- and without quantum calculation shows different features due to the discretization of energy levels in quantum wells. In the output ''integrated_densities_vs_energy.dat'' we find the electron and hole densities as a function of energy (Figure {{ref>density}} and {{ref>density_Eresolution}}). This quantity is integrated over the device and has the unit [cm$^{-2}$eV$^{-1}$]. We observe a good correspondence between the spectra and relevant energies of the present structure. | ||
| + | |||
| + | <figure density> | ||
| + | {{:nnp::laserdiode_density.png}} | ||
| + | <caption>Electron (red) and hole (blue) densities integrated over the device as a function of energy. This figure illustrates the population inversion in stationary (quasi-equilibrium) state of the device under bias. Solid and dashed lines are for quantum and classical calculations, respectively. The black arrows mark the relevant energies of the structure {{ref>wf}} at bias of $0.8$ V. The hole density is shown in Figure {{ref>density_Eresolution}} with higher resolution.</caption> | ||
| + | </figure> | ||
| + | |||
| + | The hole density has been calculated with higher energy resolution in Figure {{ref>density_Eresolution}}. The ratchet-like shape within the wells results from the step-like density of states multiplied by (quasi) Fermi-Dirac distribution (cf. Figure 9.8 in [Chuang]). | ||
| + | |||
| + | <figure density_Eresolution> | ||
| + | {{:nnp::laserdiode_density_eresolution.png}} | ||
| + | <caption>Hole density integrated over the device from classical (dashed) and quantum (solid) calculation. The energy resolution has been increased by a factor of 10 from Figure {{ref>density}}.</caption> | ||
| + | </figure> | ||
| + | |||
| + | |||
| + | The spontaneous and stimulated emission spectra are written in ''\Optical\emission_spectrum_photons.dat'' and ''\Optical\stim_emission_spectrum_photons.dat'', respectively (Figure {{ref>emission}}). __**Please note that the stimulated emission calculation in nextnano++ assumes photon modes occupied by one photon each, i.e. takes into account neither energy-dependent photon density of states nor Bose-Einstein distribution. **__ | ||
| <figure emission> | <figure emission> | ||
| - | {{:nnp::laserdiode_tutorial_emission.png}} | + | {{:nnp::laserdiode_emission.png}} |
| - | <caption>Emission spectrum of the laser diode for the bias $0.8$ V.</caption> | + | <caption>Emission spectrum of the laser diode for the bias $0.8$ V. The peak is at around 0.7-0.8eV, which is consistent with the charge distribution in Figure {{ref>density}}. The stimulated emission does not occur above the quasi Fermi level separation, $E_{Fn}-E_{Fp}$.</caption> |
| + | </figure> | ||
| + | |||
| + | The absorption coefficient $\alpha(E)$ and gain (coefficient) $g(E)$ are essentially the same quantity with opposite signs, | ||
| + | \begin{equation} | ||
| + | \alpha(E)=-g(E). | ||
| + | \end{equation} | ||
| + | These are by definition independent of the initial photon population. __**Please note that the gain spectrum in nextnano++ is cut off where it is negative.**__. For details, see [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/classical.htm|classical{} documentation]]. | ||
| + | |||
| + | The spectrum changes its sign at the energy $E_{Fn}-E_{Fp}$, that is, the separation of the quasi Fermi levels. According to the output ''bandedges.dat'', this value is -0.0001-(-0.7702)=0.7701eV. The following result has been calculated classically. We also get qualitatively consistent results from quantum mechanical simulation. | ||
| + | |||
| + | <figure cl_gain> | ||
| + | {{:nnp::laserdiode_cl_gain.png}} | ||
| + | <caption>Classically calculated absorption coefficient and gain spectrum. The sign of the spectrum switches at the energy corresponding to the quasi Fermi-level separation in the active region. </caption> | ||
| </figure> | </figure> | ||
| Line 136: | Line 190: | ||
| <figure IV> | <figure IV> | ||
| - | {{:nnp::laserdiode_tutorial_IV.png}} | + | {{:nnp::laserdiode_IV.png}} |
| <caption>Charge current and photocurrent as a function of bias voltage (IV characteristics). This figure clearly shows the consequence of the difference in Figure {{ref>bandedge}} and {{ref>bandedge2}}.</caption> | <caption>Charge current and photocurrent as a function of bias voltage (IV characteristics). This figure clearly shows the consequence of the difference in Figure {{ref>bandedge}} and {{ref>bandedge2}}.</caption> | ||
| </figure> | </figure> | ||
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