A. Carrier Interface Models in MINIMOS-NT

To calculate the carrier concentrations and the carrier temperatures at the interface of two semiconductor segments three different models are implemented in MINIMOS-NT. These are a model with continuous quasi Fermi level across the interface (CQFL), the thermionic emission model (TE), and the thermionic field emission model (TFE). The derivation of these models is described in [23]. The model used by the simulator can be specified in the input deck separately for electrons and holes for each semiconductor-semiconductor interface. Table A.1 shows the input deck keywords and their values.

To use a specific interface model at the interface of two semiconductor segments the appropriate value has to be assigned to the interfaceDD or interfaceHD keywords in the Electron or Hole subsection of the two segments within the Phys section of the input deck. The interface model is used only if the values of these keywords are the same for both segments. The "CQFL" model is the default model and is also used in case the specified values are not equal.

The following example shows how to specify the TFE model for electrons at the interface between the segments named Channel and UpperBarrier for drift-diffusion and hydrodynamic simulations. For the drift-diffusion interface model the effective tunnel length is modified from its default value to 5 nm.

Phys
{
   ...

   +Channel
   {
      Electron
      {
         interfaceDD = "TFE";
         interfaceHD = "TFE";

         InterfaceDD
         {
            TFE { d = 5e-9 m; }
         }
      }
   }
   +UpperBarrier
   {
      Electron
      {
         interfaceDD = "TFE";
         interfaceHD = "TFE";

         InterfaceDD
         {
            TFE { d = 5e-9 m; }
         }
      }
   }

   ...
}

Table A.2: TFE interface model keywords.

Keyword	Type	Values
interfaceDD	String	CQFL, TE, TFE
interfaceHD	String	CQFL, TE, TFE

Table A.1: Carrier interface model keywords.

Symbol	Keyword	Type	Unit
d	d	Quantity	m

In the following J denotes the current density, S the energy flux density, and $\Delta$E_$\nu$ the difference of band edges. The carrier concentration is denoted by $\nu$.

The following three carrier interface models are implemented in MINIMOS-NT:

• continuous quasi Fermi level model

  $$\nu \nu$$ (A1)

  $$\nu_{2}^{} \nu_{1}^{} \left(\vphantom{\frac{m_{\nu 2}}{m_{\nu 1}}}\right. {\frac{m_{\nu 2}}{m_{\nu 1}}} \left.\vphantom{\frac{m_{\nu 2}}{m_{\nu 1}}}\right)^{3/2}_{} \left(\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right. {\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}} \left.\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right)$$ (A2)

  $$\nu \nu {\frac{1}{\mathrm{q}}} \Delta \nu \nu$$ (A3)

  $$\nu \nu$$ (A4)

  
  

• thermionic emission model

  $$\nu \nu$$ (A5)

  $$\nu \left(\vphantom{v_{\nu 2}\cdot \nu_{2} - \frac{m_{\nu 2}}{m_{\nu1... ...\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu1}}\right)}\right. \nu \nu_{2}^{} {\frac{m_{\nu 2}}{m_{\nu 1}}} \nu \nu_{1}^{} \left(\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right. {\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}} \left.\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right) \left.\vphantom{v_{\nu 2}\cdot \nu_{2} - \frac{m_{\nu 2}}{m_{\nu1... ...\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu1}}\right)}\right)$$ (A6)

  $$\nu \nu {\frac{1}{\mathrm{q}}} \Delta \nu \nu$$ (A7)

  $$\nu \left(\vphantom{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 2}\cdot v... ...frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}\right)}\right. \nu \nu \nu_{2}^{} {\frac{m_{\nu 2}}{m_{\nu 1}}} \nu \nu \nu_{1}^{} \left(\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right. {\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}} \left.\vphantom{-\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right) \left.\vphantom{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 2}\cdot v... ...frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}\right)}\right)$$ (A8)

  
  

  with the thermionic emission velocity

  $$\nu \sqrt{\frac{2\cdot\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu i}}{\pi\cdot m_{\nu i}}}$$ (A9)

  
  

• thermionic field emission model

  $$\nu \nu$$ (A10)

  $$\nu \left(\vphantom{v_{\nu 2}\cdot \nu_{2} - \frac{m_{\nu 2}}{m_{\nu1... ...\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu1}}\right)}\right. \nu \nu_{2}^{} {\frac{m_{\nu 2}}{m_{\nu 1}}} \nu \nu_{1}^{} \left(\vphantom{-\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right. {\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}} \left.\vphantom{-\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right) \left.\vphantom{v_{\nu 2}\cdot \nu_{2} - \frac{m_{\nu 2}}{m_{\nu1... ...\frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu1}}\right)}\right)$$ (A11)

  $$\nu \nu {\frac{1}{\mathrm{q}}} \left(\vphantom{\Delta E_{\nu}- \delta E_{\nu}}\right. \Delta \nu \delta \nu \left.\vphantom{\Delta E_{\nu}- \delta E_{\nu}}\right) \nu$$ (A12)

  $$\nu \left(\vphantom{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 2}\cdot v... ...frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}\right)}\right. \nu \nu \nu_{2}^{} {\frac{m_{\nu 2}}{m_{\nu 1}}} \nu \nu \nu_{1}^{} \left(\vphantom{-\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right. {\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}} \left.\vphantom{-\frac{\Delta E_{\nu}- \delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}}\right) \left.\vphantom{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 2}\cdot v... ...frac{\Delta E_{\nu}}{\ensuremath{\mathrm{k_{B}}}\cdot T_{\nu 1}}\right)}\right)$$ (A13)

  
  

  with the thermionic emission velocity (A.9) and the barrier hight lowering

  $$\delta \nu \left\{\vphantom{\begin{array}{ll}\mathrm{q}\cdot E_{\perp 2}\cdot d, & E_{\perp 2} > 0\\ 0,& E_{\perp 2} \leq 0\end{array}}\right. \begin{array}{ll}\mathrm{q}\cdot E_{\perp 2}\cdot d, & E_{\perp 2} > 0\\ 0,& E_{\perp 2} \leq 0\end{array} \left.\vphantom{\begin{array}{ll}\mathrm{q}\cdot E_{\perp 2}\cdot d, & E_{\perp 2} > 0\\ 0,& E_{\perp 2} \leq 0\end{array}}\right.$$ (A14)

  
  

  For $\delta$E_$\nu$ = 0 the TFE model reduces to the TE model.