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qcl:photon-assisted_transport_and_gain_clamping [2020/11/19 16:20]
thomas.grange [Gain clamping]
qcl:photon-assisted_transport_and_gain_clamping [2022/03/30 17:19] (current)
thomas.grange
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 </​code>​ </​code>​
  
-Note that in the current version (2020-11-16), only a single EM mode is supported at a time.+Note that in the current version (2022-03-30), only a single EM mode is supported at a time (i.e. multi-mode lasing operation is not supported).
  
 The electric field in this EM mode can be either imposed (detection mode) or calculated self-consistently (gain clamping). The electric field in this EM mode can be either imposed (detection mode) or calculated self-consistently (gain clamping).
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 <​ElectricField unit="​V.m^-1">​1.0e6</​ElectricField>​ <​ElectricField unit="​V.m^-1">​1.0e6</​ElectricField>​
 </​EMmode>​ </​EMmode>​
 +</​EMfield>​
 </​code>​ </​code>​
  
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 The gain feature calculates the linear response to an a.c. incoming field. In this case, the d.c. current is not modified. ​ The gain feature calculates the linear response to an a.c. incoming field. In this case, the d.c. current is not modified. ​
 On the other hand, the photon-assisted transport is modeled through the use of a self-energy (self-consistent Born approximation) to describe both absorption and stimulated emission processes, and has an influence on the d.c. transport. ​ On the other hand, the photon-assisted transport is modeled through the use of a self-energy (self-consistent Born approximation) to describe both absorption and stimulated emission processes, and has an influence on the d.c. transport. ​
-In the case of the electron-photon self-energy,​ a gain is also calculated at the specified EM mode energy. However, the calculated gain slightly differ from the one calculated using linear response, even for small intensities,​ as broadening effects are not treated within the same approximations in the two cases. This is all the more the case when going to small photon energy (i.e. long wavelengths).+In the case of the electron-photon self-energy,​ a gain is also calculated at the specified EM mode energy. However, the calculated gain slightly differ from the one calculated using linear response, even for small intensities,​ as broadening effects are not treated within the same level of approximations in the two cases. This is all the more the case when going to small photon energy (i.e. long wavelengths). Hence this method can produce unreliable results in terahertz devices where broadening effects are comparable to the photon energy.
  
 ==== Gain clamping ==== ==== Gain clamping ====
-Gain clamping is relevant to the simulation of quantum cascade lasers above threshold. Indeed, when the gain surpasses ​the cavity losses, lasing starts and the gain is clamped to the cavity losses.+Gain clamping is relevant to the simulation of quantum cascade lasers above threshold. Indeed, when the modal gain $g_c$ reaches ​the cavity losses, lasing starts and the gain is clamped to the cavity losses, that is  
 +$$ g_c = \alpha$$ 
 +The modal gain (or cross-section gain), is defined as 
 +$$ g_c = \Gamma g$$ 
 +where $g$ is the material gain in the active region and $\Gamma$ the overlap factor between the cavity mode and the active region.
  
 To simulate gain clamping, the following command should be used: To simulate gain clamping, the following command should be used:
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        <​Cavity_Losses unit="​cm^{-1}">​2.76</​Cavity_Losses>​        <​Cavity_Losses unit="​cm^{-1}">​2.76</​Cavity_Losses>​
        <​GainClamping>​yes</​GainClamping>​        <​GainClamping>​yes</​GainClamping>​
 +       <​OverlapFactor>​0.9</​OverlapFactor>​
     </​Gain>​     </​Gain>​
 </​code> ​ </​code> ​
-Note that in this case, the EM electric field should be set to zero:+If not specified, the default value for the overlap factor $Gamma$ is assumed to be 1.0 (i.e. perfect overlap between the active region and the cavity mode). 
 + 
 +Note that in the case of gain clamping, the EM electric field value should be either not specified or set to zero:
 <​code>​ <​code>​
 <​EMfield>​ <​EMfield>​
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 </​code>​ </​code>​
  
-In this case, the electric field in the cavity is adjusted self-consistently so that the gain for this photon energy ​matchs ​the specified cavity losses. ​+Indeed, for gain clamping the electric field in the cavity is adjusted self-consistently so that the modal gain $g_c$ for this photon energy ​matches ​the specified cavity ​losses $\alpha$.  
 + 
 +=== Wall plug efficiency === 
 + 
 +The internal wall plug efficiency (WPE)  is calculated when gain clamping occurs. It is defined as 
 +$$ \eta_{\text{WPE}}^{internal} = \frac{P_{\text{stimulated emission}} - P_{\text{absorption}}}{UI} $$ 
 + 
 +If the front mirror losses are specified, the external WPE is output as 
 + 
 +$$ \eta_{\text{WPE}}^{external} = \eta_{\text{WPE}}^{internal} \frac{\alpha_{\text{fm}}}{\alpha_{\text{cavity}}}$$ 
 +where $\alpha_{\text{fm}}$ is the front mirror losses, which can be defined using the command: 
 +<​code>​ 
 +    <​Gain>​ 
 +       ... 
 +       <​Cavity_Losses unit="​cm^{-1}">​2.76</​Cavity_Losses>​ 
 +       <​FrontMirror_Losses unit="​cm^{-1}">​2.26</​FrontMirror_Losses>​ 
 +       ... 
 +    </​Gain>​ 
 +</​code>​ 
 + 
 +Note that the relation between the specified cavity losses and the front mirror one reads  
 +$$ \alpha_{\text{cavity}} =  \alpha_{\text{fm}} + \alpha_{\text{bm}} + \alpha_{\text{waveguide}} $$ 
 +where $\alpha_{\text{bm}}$ are the losses due to the back mirror and  $\alpha_{\text{waveguide}}$ are the waveguide ​losses. 
  
 +=== Example: Mid-infrared QCL  ===
  
 +The following animated gifs show the electron density, current density and output as the voltage is swept. Above the threshold bias (around 300 meV), the clamping of the gain to the cavity losses results a rapid increase in the internal electric field in the cavity. The photon-assisted transport results in a discontinuity in the slope of the current-voltage characteristics at the threshold bias/​current:​ above threshold, the current increases much faster as stimulated emission reduces the upper laser level lifetime.
 +{{:​negf:​bai2011_electrondensity.gif?​direct&​600|}}
 +{{:​negf:​bai2011_current.gif?​direct&​600|}}
 +{{:​negf:​bai2011_emission.gif?​direct&​600|}}
qcl/photon-assisted_transport_and_gain_clamping.1605802846.txt.gz · Last modified: 2020/11/19 16:20 by thomas.grange