Improving Electron Transport Simulation in Mesoscopic Systems by Coupling a Classical Monte Carlo Algorithm to a Wigner Function Solver
Because of its high switching speed, low power consumption and reduced complexity to implement a given function, resonant tunneling diodes (RTD’s) have been recently recognized as excellent candidates for digital circuit applications . Device modeling and simulation is thus important, not only to understand mesoscopic transport properties, but also to provide guidance in optimal device design and fabrication.
Several approaches have been used to this end. Among kinetic models, those based on the non-equilibrium Green function formalism  have gained increasing interest due to their ability to incorporate coherent and incoherent interactions in a uniffied formulation. The Wigner distribution function approach has been also extensively used to study quantum transport in RTD’s -. The main limitations of this formulation are the semiclassical treatment of carrier-phonon interactions by means of the relaxation time approximation and the huge computational burden associated to the self-consistent solution of Liouville and Poisson equations. This has imposed severe limitations on spatial domains, these being too small to succeed in the development of reliable simulation tools.
Based on the Wigner function approach, we have developed a simulation tool that allows to extend the simulation domains up to hundreds of nanometers without a significant increase in computer time . This tool is based on the coupling between the Wigner distribution function (quantum Liouville equation) and the Boltzmann transport equation. The former is applied to the active region of the device including the double barrier, where quantum effects are present (quantum window, QW). The latter is solved by means of a Monte Carlo algorithm and applied to the outer regions of the device, where quantum effects are not expected to occur.
Since the classical Monte Carlo algorithm is much less time consuming than the discretized version of the Wigner transport equation, we can considerably increase the simulation domains without paying a penalty in efficiency.
We have modeled this coupling by using the Monte Carlo distribution of carriers in k-space in the cells adjacent to the QW as boundary conditions for the step-bystep solution of the Liouville equation, while the Wigner distribution function at the edges of the QW dictates carrier injection to the classical regions.
By introducing in our tool a Poisson solver, necessary for self-consistency, we have simulated the I-V characteristic of RTD’s with typical physical parameters. Realistic simulation boxes of 300nm have been considered. These are much higher than those of previous works, while the times required to achieve convergence are similar. The main qualitative features of actual devices are reproduced by means of our tool, i.e., oscillatory behavior and current plateau in the negative differential resistance region.
By analyzing the potential profile and electron density distribution at various applied bias it is seen that charge accumulation in the well is maximum at resonance and no spurious or discontinuities have been found at the boundaries of the QW, which reveals that the coupling model is well-behaved.
We are currently comparing the simulation results to experimental data provided by other authors. Our preliminary results seem to indicate that allowing physical parameters to slightly vary from nominal values (parameter dispersion is unavoidable at the length scales we are dealing with), a reasonable fit between simulated and experimental results is possible.
The main conclusion of the work is that within the framework of the Wigner distribution function, we have developed a tool that provides improvement over previous simulators since realistic device dimensions can be considered without efficiency degradation. This allows to obtain more accurate simulated results and make the tool a potential candidate to aid in RTD device design and fabrication.
This work has been supported by the Dirección General de Enseñanza Superior under contract PB97-0182.