## Nanoelectronic Modeling Lecture 18: Introduction to RTDs - Quantum Charge Self-Consistency (Hartree)

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#### Abstract

The previous lectures showed that a semi-classical charge distribution results in a vast improvement in the realism of the electrostatic potential over a linear potential drop approximation. In this semi-classical charge and potential model the quantum mechanical simulation is performed once and the quantum mechanical charge is in general not identical to the semi-classical charge. In the high doping contacts that is a reasonable assumption, however close to heterointerfaces the quantum mechanical nature of the carriers reflects itself in the charge profile significantly and a semi-classical charge profile is not necessarily the best approximation.

Simple quantum charge-selfconsistent potential (Hartree) calculations feed the quantum charge into the potential calculation and iterate until the two do not change anymore from one iteration step to the next. The quantum mechanical charge calculations predict a charge accumulation inside the central RTD which increases the charge inbalance in the undoped region, which in turn forces the electrostatic potential to resist that charge inbalance. The resonances therefore do not get pulled down in voltage linearly anymore. The charge & potential interactions introduce a non-linear device behavior which stretches out the voltage axis, linearizes the I-V curve, and increases the peak current.

**Learning Objectives:**

- Semi-classical charge and quantum charge differ significantly at the interfaces and inside the RTD.
- The electrostatic potential based on a semi-classical charge is a much better approximation to the Hartree-self-consistent charge, compared to the linear potential drop assumption.
- Resonance energies are no longer simple linear functions of bias

• non-linear - Hartree charge self-consistent calculations stretch out the voltage axis at the current peak and linearize the I-V curve.
- The current peak is increased.
- Even symmetric RTDs show a significant charge accumulation at the current peak which is highly out-of equilibrium.

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Università di Pisa, Pisa, Italy