3.6 Application to a HEMT

S	= - 2^. $\displaystyle \beta$ (e_b)^. $\displaystyle \left(\vphantom{\Lambda_{2} - \frac{m_{2}}{m_{1}}\cdot\Lambda_{1}\cdot\exp(-e_{\mathrm{b}})}\right.$ $\displaystyle \Lambda_{2}^{}$ - $\displaystyle {\frac{m_{2}}{m_{1}}}$ ^. $\displaystyle \Lambda_{1}^{}$ ^.exp(- e_b) $\displaystyle \left.\vphantom{\Lambda_{2} - \frac{m_{2}}{m_{1}}\cdot\Lambda_{1}\cdot\exp(-e_{\mathrm{b}})}\right)$ ,	(3.32)
$\displaystyle \Lambda_{i}^{}$	= k_B^.T_i^.v_i^.n_i,	(3.33)
$\displaystyle S_{2}$	$\displaystyle = S_{\perp}+ \frac{\ensuremath{\mathrm{k_{B}}}\cdot T_{1}}{\mathrm{q}}\cdot e_{\mathrm{b}}\cdot J_{\perp}.$	(3.34)

Similar to $\alpha$ in (3.12) the value of the factor $\beta$ depends on the energy barrier and the physical models taken into account.

**Figure 3.9:** Schematic cross section of the simulated delta-doped pseudomorphic double-heterojunction HEMT.
$\includegraphics[width=14cm]{eps/hemt.eps}$

The bias point was chosen in such a way that the carriers in the channel heat up and the effective barrier height is reduced considerably. Fig. 3.10 shows the electron temperature along a vertical cut across the heterojunction. Normally, the carrier concentration inside the channel is several orders of magnitude higher than in the upper barrier region. For the example shown the situation is reversed as the carrier temperature in the channel exceeds the temperature in the upper barrier region (3.31). The electron temperature is plotted in Fig. 3.11 for the same vertical cut as the electron concentration in Fig. 3.10. Fig. 3.12 shows that the electron concentration in the channel is lowest for the region between the drain sided end of the gate and the drain contact. This is the region where the electron temperature in the channel has a maximum. Therefore a large number of electrons has sufficient kinetic energy to surmount the barrier at the heterojunction and reach the upper barrier region (real-space transfer).

**Figure 3.10:** Electron concentration along a vertical cut across the channel of a pseudomorphic HEMT.
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**Figure 3.11:** Electron temperature along the same vertical cut as in Fig. 3.10.
$\begin{figure} \begin{center} \resizebox{14cm}{!}{ \psfrag{y [um]}[][]{$\display... ...udegraphics[width=14cm]{eps/hemt-electron-temp-cut.ps}} \end{center}\end{figure}$

**Figure 3.12:** Electron concentration in the channel below the gate.
$\begin{figure} \begin{center} \resizebox{11cm}{!}{ \includegraphics[width=8cm, angle=90]{eps/hemt-electron-conc-top-col.ps}} \end{center}\end{figure}$

**Figure 3.13:** Electron temperature in the channel below the gate.
$\begin{figure} \begin{center} \resizebox{11cm}{!}{ \includegraphics[width=8cm, angle=90]{eps/hemt-electron-temp-top-col.ps}} \end{center}\end{figure}$

Although the electron concentration in the channel is lower than in the supply a considerable amount of the current is conducted in the channel due to the much higher velocity in the channel.

Because of the strong variation of the electric field along the heterojunction interface the factors $\alpha$ and $\beta$ will also vary considerably. Together with the strong variation of the electron temperature in the channel this will cause the type of boundary condition to vary strongly. By using the new method there is no need to partition the interface and to apply separate treatment for the different regions at the interface to achieve convergence.