# QUANTUM:

Quantum Mechical Simulator Released in Atlas

Introduction

Silvaco has recently released QUANTUM, a 2D quantum mechanical solver coupled into the poisson, drift diffusion and energy balance solvers already available in ATLAS. The physics of quantum mechanics have been included to model the behavior of heterojunction devices which exhibit properties such as quantized channel confinement and transmission or reflection at potential barriers. The model has been implemented for both electrons and holes. This allows more accurate physically based simulations of devices such as pseudomorphic HEMTs, as illustrated in Figure 1. An additional feature arising from this work has been the ability to simulate quantized MOS channels and shifted CV curves for thin oxide sub-quarter-micron devices.

Figure 1. Delta doped PHEMT - cutline through
the gate

showing smearing of the electron, concentration in the

quantum model as compared to classical.

Implementation

The two techniques that have been studied at Silvaco for the inclusion of quantum mechanical effects are a coupled Schrodinger-Poisson approach and a coupled Quantum moments technique.

Initial investigations centered around the implementation of a Schrodinger-Poisson system - in this technique, the Schrodinger equation is solved in two sections using the Sturm sequence approach to calculate the required eigen values and corresponding eigen vectors [1]. The eigen values represent discrete energy level subbands, in which electrons can reside. The eigen vectors give the probability of occupation of a given energy level for the particular bias conditions applied. This provides useful information in terms of how quantum mechanics is affecting the electron distribution within the device, and where the sub-bands occur. The implementation of this system is included in the December 1996 issue of the Simulation Standard. The main problem with this type of approach for a device simulator is that the quantum solution does not take into account the impact of quantum mechanics on the transport equations.

An alternative method of incorporating the effects of quantum behavior within a device simulator is to use the Quantum Moments approximation technique [2]. This technique uses an additional term on the transport equations which allows for the repositioning of carriers under the influence of quantization, using a quantum temperature, as illustrated in equation (1):

where and m* are the reduced planck constant and effective mass, n is the electron concentration and Uq is the quantum term incorporated into the transport equations.

The quantum moments equation, which provides explicit quantum corrections, is based upon the second moments of the Wigner function [3].

Hetrojunction Applications

The main application area for this type of model is in III-V heterojunction device structures, and to illustrate this, examples of a HEMT, PHEMT and heterojunction diode will be given.

Figure 2 shows a typical HEMT type structure with AlGaAs / GaAs layers. A cutline is shown through the gate illustrating the electron concentration for both the classical and quantum solutions. The quantum smoothing out of the classical concentration peak in the channel is clear from this figure, as is the increase of the carriers on the other side of the barrier.

Figure 2. Quantum and classical carrier

concentration for a HEMT structure.

Figure 1 (on the previous page) shows a PHEMT with an AlGaAs / InGaAs / GaAs structure. Again a cutline has been taken through the channel - this device has a delta (pulse) doping in the AlGaAs layer. The effects of the quantum model can be seen in the differences between the doping concentration and the electron concentration. Clearly the delta pulse has been 'smeared out' and the carriers are located in the channel potential well. Figure 3 shows an Id/Vd curve for this device showing the standard types of result that can be produced with a quantum based model.

Figure 3. Id/Vd curve at Vg=0V for a HEMT
using QUANTUM.

Note the correct prediction of conductor current levels

A simple heterojunction diode is shown in Figure 4, including the conduction and valance bands from anode to cathode. Figure 5 shows the electron concentration within this device for both classical and quantum structures. Again, the shift from sharp classical peaks can be seen in the quantized results, with a higher carrier concentration on the other side of the barrier. In Figure 6, the difference that this makes to the reverse current can be seen - the quantum case gives a higher leakage current in the device.

Figure 4. AlGaAs/GaAs heterojunction diode.

Figure 5. Electron concentration across the junction in reverse bias.

Figure 6. Heterojunction diode forward and
reverse bias currents.

Extra current in quantum case is due to smoothing of electron

concentration across the hetrojunction.

MOS Applications

Another set of applications are for ultr thin gate MOS structures. QUNATUM accounts for quantization effects in a MOS channel. This leads to changes in CV curves with thin gate oxides. Note that in this type of simulation we are using a physical description of what exactly is happening to the carrier, and therefore modeling the behavior of this phenomena accurately. This contrasts to other approximate methods for including quantum effects which use an empirical adjustment to Poisson equation. Figure 7 illustrates the device simulated and the classical verses quantum carrier concentration through the gate. Note the reduction in carrier concentration towards the surface of the device - this is typical of a quantized device and leads to the shift in CV characteristics shown in Figure 8 as the inversion layer is less inverted than in the classical case. Additionally, Figure 9 shows the classical and quantum ni - illustrating the surface dip due to quantization. The quantization of holes is possible in QUANTUM which allows the simulation of both nmos and pmos structures.

Figure 7. MOS capacitor structure with 30Å
gate oxide.

Channel quantization reduces the surface carrier concentration.

Figure 8. CV curves illustrating a shift due
to quantization.

The shift leads to erroneous Tox measurements of ultra-thin gate oxides.

Figure 9. Ni for classical and quantum cases

showing non-uniform Ni in quantized regions.

Conclusion

Silvaco has released QUANTUM, a quantum model within the framework of our existing 2D device simulator ATLAS. This is based on a quantum moments approximation derived from the second order moments of the Wigner function. The incorporation of this type of model allows more accurate, physically based simulation of quantized channel devices, in particular III-V HEMTs and heterojunction diodes, but is also applicable to sub quarter-micron MOS devices.

**References**

[1] R. Drury and C. Snowdon, "A Quasi-2D HEMT Model for Microwave CAD Applications," IEEE Trans. Electron Devices, vol. 42 no. 6, pp 1026, 1995

[2] Jing-Rong Zhou and David Ferry, "Simulation of Ultra-Small GaAs MESFETs Using Quantum Moment Equations," IEEE Trans. Electron Devices, vol. 39 no. 3, pp 473, 1992

[3] E. Wigner, "On the Quantum correction for thermal dynamic equilibrium,"
Phys. Rev., vol. 40 no. 5, pp. 749-754, 1932