# 2D Simulation of Pseudomorphic Heterojunction Devices Using The Fully Coupled Carrier Energy Balance Model

Introduction

The drift-diffusion approximation can be used to model many semiconductor device situations successfully. However, higher order approximations of the Boltzmann transport equation provide increased accuracy in those devices where non-localized transport effects (such as velocity overshoot) play a significant role. Devices in which non-localized effects are important include short channel MOSFETs, HBTs and HEMTs. Structures based on GaAs and other III-V materials exhibit especially pronounced velocity overshoot. This is the result of the combination of the electron intervalley transfer process and relatively long energy relaxation times. Strong velocity overshoot leads to experimental current - voltage characteristics that differ significantly from the characteristics predicted by the drift-diffusion model.

The general purpose 2D semiconductor device simulator
* ATLAS/Blaze *can predict the characteristics of III-V
devices,

*including the effects of velocity overshoot*. Silvaco recently developed a fully-coupled six equation solver for

*[1], [2]. This new solver accounts for both non-localized transport and nonisothermal lattice heating. It therefore provides a unique framework for obtaining fast, robust and accurate solutions to a very broad range of device simulation problems.*

**ATLAS**In this article, the simulation of a 0.7µm channel length InGaAs based pseudomorphic HEMT is described. The I-V characteristics of the device are calculated using the drift-diffusion approximation and using an Energy Balance model. The Energy Balance model demonstrates results that are in good agreement with those seen experimentally, but the drift-diffusion results exhibit systematic discrepancies.

Physical Models

Inter-valley electron transfer in III-V materials can lead to steady-state velocity field characteristics that display negative differential mobility (i.e. a velocity peak at relatively low fields). This behavior can be modeled using a semi-empirical expression, with parameters derived from experimentation and Monte Carlo simulations. The following expression for electron mobility is widely used:

where E is the electric field, µo is the low field mobility, Vsat is the saturated carrier velocity and Eo is the critical field. Velocity overshoot occurs when the mean energy of carriers is less than the value that carriers would have in the local field under homogenous steady-state conditions. In small geometry devices, carriers do not have enough time to acquire the steady-state energy associated with the local field. In order to predict the behavior of III-V devices it is necessary to account for both negative differential mobility and velocity overshoot.

Nonlocal transport phenomena can be modeled using
Monte Carlo methods, or using higher order moments of the Boltzmann
transport equation. The computation times required by these approaches
can be unacceptably high. This has lead to the development of intermediate
level approximations known as Simplified Hydrodynamic and Energy
Balance models [1]. Both of these models have been incorporated
into * Blaze*, but the latter produces velocity overshoot
characteristics that are in better agreement with those calculated
by Monte Carlo simulations (see [1]). The continuity equation for
electron carrier temperature has the following form:

where Sn is the electron energy flux density, Tn is the electron temperature, To is the lattice temperature, ten is the electron energy relaxation time, n is the electron concentration and Ec is the conduction band energy.

The electron energy flux density is given by:

where n is a transport coefficient and Kn is the thermal conductivity of electrons (which is itself dependent on electron temperature). Additionally, for the simulation of heterojunction devices, modifications must be made to the standard current continuity equations to account for a position-dependent band structure [2]. Further details of the Energy Balance Model can be found in references [1] and [2].

Simulation Results

To illustrate the importance of the physics described above in simulating III-V heterojunction devices, an AlGaAs/InGaAs/GaAs pseudomorphic HEMT has been simulated using both the drift-diffusion and Energy Balance models. The source and drain areas of the device are heavily implanted with n-type dopants, and the InGaAs channel is not intentionally doped. A schematic diagram of the device structure is given in Figure 1. Figure 2 shows the heterostructure energy band diagram through the gate with a gate bias of -1.0V.

**Figure 1. Cross section of
the HEMT structure.**

**Figure 2. Energy band diagram
through the gate at Vg=-1.0V.**

Figure 3(a) shows the drain current as a function of gate voltage at a drain bias of 2.0V, calculated using the drift- diffusion and Energy Balance models. The same data is given in logarithmic form in figure 3(b). Differences between the results are clearly seen in two distinct areas: the Energy alance results exhibit (i) a higher mid range drain current (as a result of velocity overshoot) and (ii) a reduction in drain current at higher gate biases (as the effects of a forward biased Schottky gate become apparent). The use of the Energy Balance model provides significantly more realistic transconductance predictions, as illustrated in Figure 4. The peak in transconductance predicted by the Energy Balance is observed in experimental results. The drift-diffusion results do not predict this peak, and so the predictions of the drift-diffusion model are qualitatively misleading at the higher gate voltages.

**Figure 3a. Comparsion of calculated
drain subthreshold currents at Vd=2.0V for
the drift diffusion and energy balance modles (linear scale).**

**Figure 3b. Comparison of drain
subthreshold currents (log scale)
at Vd=2.0V for drift diffusion and energy balance models.**

**Figure 4. Comparison of transconductance
predidicted
by the drift diffusion and energy balance models.**

Figures 5(a) and 5(b) show the predicted gate leakage current as a function of gate bias at a drain bias of 2.0V, in linear and logarithmic form respectively. These plots show the dramatic increase in gate current predicted by the Energy Balance model. This increase is also observed experimentally (for example [3]). The Energy Balance model predicts higher gate leakage currents because carriers acquire energy in the gate field, which leads to increasing numbers of electrons with sufficient energy to surmount the gate barrier. The drift-diffusion model does not capture this effect.

**Figure 5a. Gate leakage currents
predicted by the drift
diffusion and Energy Balance modles (linear scale).**

**Figure 5b. Gate leakage currents
predicted by the
drift-diffusion and Energy Balance models (log scale).**

Conclusion

* ATLAS/Blaze *can accurately simulate
III-V devices because it includes non-localized carrier transport
phenomena, including velocity overshoot. Additionally, the use of
fully coupled numerical techniques for solving systems of up to
six coupled equations provides a robust and accurate solver for
performing simulations, and allows self consistent inclusion of
coupling between carrier and lattice temperature. The differences
in results obtained using Energy Balance as opposed to drift- diffusion
are quite dramatic - as further illustrated in Figure 6, which shows
predicted terminal currents for the two models as a function of
gate bias (at a drain bias of 2.0V).

**Figure 6a. Terminal currents
predicted
by the drift-diffusion model.**

Figure 6b. Terminal currents
predicted

by the Energy Balance model.

References

[1] Y. Apanovich, E. Lumkis, B. Polsky, A.
Shur and P. Blakey,

"Steady State and Transient Analysis of Submicron Devices
Using Energy Balance and Simplified Hydrodynamic Models,"

IEEE Trans. CAD, vol. 13, pp. 1151-1159, 1994.

[2].Y. Apanovich, P. Blakey, R. Cottle, E.
Lyumkis, B. Polsky, A. Shur and A. Tchernaiev,

Numerical Simulation of Submicrometer Devices Including
Coupled Nonlocal Transport and Nonisothermal Effects",

IEEE Trans. Electron Devices, vol. 42, pp. 890-898, 1995.

[3].C.D. Wilson, A.G. O'Neill, S. Baier and
J. Nohava,

"A Complementary III-V Heterojunction FET Technology for
High Temperature Integrated Circuits,"

Mat. Sci. & Eng. B, vol. 29, 1995.