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Physical modeling and numerical simulation of heat and carrier transport for high-power semiconductor devices semiconductor devices

Collaborator: H. Gajewski , H. Stephan

Cooperation with: G. Wachutka, W. Kaindl (Technische Universität München)

Supported by: DFG: ``Physikalische Modellierung und numerische Simulation von Strom- und Wärmetransport bei hoher Trägerinjektion und hohen Temperaturen'' (Physical modeling and numerical simulation of current and heat transport at high carrier injection and high temperatures) [2]

Description:

In 2002 our model [1], describing heat and carrier transport for semiconductor devices, has been advanced to silicon carbide (SiC). SiC is used in different crystal configurations (6H-SiC, 4H-SiC, 3C-SiC). Each of these materials possesses promising properties as basic materials for high-power, high-temperature and high-frequency electronics. The reason for this are special physical characteristics, which distinguish SiC from conventional semiconductor materials such as silicon. Those are first of all:

Deriving the system of nonlinear partial differential equations for the heat and carrier transport in semiconductor devices, we abided by the following physical principles:

1.
postulation of the free energy density;
2.
calculation of the equilibrium according to the entropy maximum principle;
3.
definition of the thermodynamic potentials near the equilibrium state;
4.
postulation of the evolution equations;
5.
calculation of the currents and right-hand sides according to the second law of thermodynamics;
6.
derivation of the heat equation.

The postulated system of equations

\begin{eqnarray*}
% latex2html id marker 729
-\nabla \cdot (\varepsilon \nabla \...
 ...artial t} + \nabla \cdot J_u & = & 0. ~~
\text{(energy equation)}\end{eqnarray*}

describes electron, hole and energy transfer, which is nonlinearly coupled by the electrostatic potential $\psi$ via Poisson's equation. Here, n, p and u are electron, hole and power density, Jn, Jp and Ju the appropriate currents, D the dopants and G the generation-recombination rate. (The ODEs describing the dynamics of electron/hole traps were described in detail in the WIAS Annual Research Report 1998.)

The determination of the equilibrium as state of maximal entropy by Lagrange's method suggests the Lagrange multipliers $\frac{\varphi_n}{T}$, $\frac{\varphi_p}{T}$ and $-\frac{1}{T}$to be thermodynamic potentials. Their gradients are the driving forces for the currents. That leads to the following current, under consideration of Onsager's principle:

\begin{eqnarray*}
\left(
\begin{array}
{c}
\!\!J_n\!\! \ [1.2em]
\!\!J_p\!\! \...
 ...\!\displaystyle{-\nabla \frac{1}{T}}\!\!\ \end{array}\right)\; .\end{eqnarray*}

In the case of the anisotropic SiC, $\sigma_{p}$,$\sigma_{n}$ and $\sigma_{np}$, and anu, anp and au are $3\times 3$ matrices. From the second law of thermodynamics (entropy S increasing in time)

\begin{eqnarray*}
\dot{S} = \int\limits_\Omega \dot{s} \, dx = 
- \int\limits_\Omega \nabla J_s \, dx +
\int\limits_\Omega d \, dx \geq 0\end{eqnarray*}

(here d denotes the dissipation rate) it follows for the currents

\begin{eqnarray*}
J_n & = & -\boldsymbol{\sigma_n} z_n-\boldsymbol{\sigma_{np}}(...
 ...bol{\kappa_L} \nabla T -(P_nT-\varphi_n)J_n+(P_pT+\varphi_p)J_p, \end{eqnarray*}

with

\begin{eqnarray*}
z_n = T\left(\nabla \frac {\varphi_n}{T}
+(P_nT-\varphi_n)\nab...
 ... \frac {\varphi_p}{T}
-(P_pT+\varphi_p)\nabla \frac {1}{T}\right)\end{eqnarray*}

and the energy carrier interaction terms ($\kappa_l$ is the heat conductivity)

\begin{eqnarray*}
\boldsymbol{a_{nu}} &=& \boldsymbol{\sigma_n} (\varphi_n - P_n...
 ...varphi_p + P_p T)(\varphi_n - P_n T) + \boldsymbol{\kappa_L} T^2.\end{eqnarray*}

For this model, thermodynamically consistent algorithms were developed and implemented into our program system WIAS-TeSCA .

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 ...agesonly}
\addtocounter{projektbild}{-1}\end{imagesonly}\end{figure}\endminipage

As an example we show a 6H-SiC DIMOS transistor--a typical high-power device ([4]). The crystal is oriented in such a way that the electron mobility in horizontal direction is five times higher than in the vertical direction. Figure 2 shows the electron flow for a gate voltage of 12 V and a drain voltage of 30 V. In comparison, Figure 3 shows the simulation result for isotropic mobility (e.g., in Si).


Fig. 2: Anisotropic case

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Fig. 3: Isotropic case

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\addtocounter{projektbild}{-1}\end{imagesonly}

References:

  1. G. ALBINUS, H. GAJEWSKI, R. HÜNLICH, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 16 (2002), pp. 367-383.
  2. H. GAJEWSKI, G. WACHUTKA, DFG-Antrag zum Projekt ``Physikalische Modellierung und numerische Simulation von Strom- und Wärmetransport bei hoher Trägerinjektion und hohen Temperaturen'', 1996, 1998, 2000 (applications for a DFG project).
  3. H. GAJEWSKI, Analysis und Numerik des Ladungsträgertransports in Halbleitern, GAMM-Mitteilungen, 16 (1993), pp. 35-57.
  4. M. LADES, Modelling and simulation of wide bandgap semiconductor devices: 4H/6H-SiC, Shaker Verlag, Aachen 2002.
  5. G. WACHUTKA, Rigorous thermodynamic treatment of heat generation and conduction in semiconductor device modeling, IEEE Trans. on CAD, CAD-9 (1990), pp. 1141-1149.


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5/16/2003