Approximate H-transformation for numerical stabilization of a deterministic Boltzmann transport equation solver based on a spherical harmonics expansion☆,☆☆
Introduction
The drift-diffusion model, which has superior numerical stability [1], [2], has made great contributions to the theoretical study of semiconductor devices over the past decades and is still the workhorse of today’s TCAD (Technology Computer-Aided Design). However, it is also well known that the drift-diffusion model has some limitations when a sharp change in the electric field appears in the device [3], [4]. Although the carrier mobility is frequently calibrated to reduce the error due to the model deficiency, its calibration should be performed again when a different device is simulated.
The Boltzmann transport equation is the semi-classical equation to describe the carrier transport in the phase space. By solving the Boltzmann transport equation with an increased computational cost, carrier transport can be described more accurately. Among various ways to solve the Boltzmann transport equation, deterministic Boltzmann transport equation solvers for the three-dimensional electron gas [5], [6], [7], [8] have been actively studied. In these studies, the electron distribution function in the three-dimensional momentum space is calculated based on the spherical harmonic expansion. By replacing the dependence on the angle with harmonic coefficients, computational efficiency can be improved.
In the deterministic Boltzmann transport equation, a stabilization method that ensures a positive distribution function is required even under a strong electric field. The H-transformation [5], where the Boltzmann transport equation is written in the total energy space, provides improved numerical stability because the derivative with respect to the energy variable is completely eliminated. In the maximum entropy dissipation scheme [7], an exponential function considering position-dependent total energy is multiplied to improve the numerical stability. For a multi-subband Boltzmann transport equation solver, where the Schrödinger equation is additionally solved to obtain the subband structure, other stabilization schemes can be found. A hybrid method that uses both the phase (inelastic) space and the trajectory (elastic) space according to the scattering mechanism has been proposed [9]. It is noted that numerical simulations directly in the phase space [10] have been also reported.
Recently, the transient simulation results using an implicit time marching technique have been reported in [11]. In [11], the maximum entropy dissipation scheme was adopted. Although the H-transformation is widely adopted in the steady-state simulation, the scheme is not very convenient to be used in the transient simulation, because the distribution functions at previous time instances must be interpolated for the present potential profile. Therefore, the transient simulation capability that is compatible with the H-transformation has not been reported yet.
In this study, in order to overcome the difficulties originated from the stabilization scheme, an alternative stabilization scheme, the approximate H-transformation, is proposed. The organization of this manuscript is as follows. In Section 2, the proposed simulation methodology is briefly described. In Section 3, numerical results are shown. In Section 4, the research direction to which the proposed methodology can be applied is discussed. Finally, the conclusion is made in Section 5.
Section snippets
Approximate H-transformation
It is assumed that the energy variable is uniformly discretized with a spacing, . For the sake of notational simplicity, the dependence on the position variable is omitted whenever possible. For more complete notations, readers are referred to [12].
Let us consider a case where an implicit time marching scheme is adopted to discretize the time derivative term. At a given kinetic energy, , the time derivative of (a product of the density-of-states, , and the distribution function, ) at a
Numerical results
The proposed method has been implemented into our in-house deterministic Boltzmann transport equation solver. The physical models are the same with those in [11]. Three different stabilization schemes (the maximum entropy dissipation scheme, the conventional H-transformation, and the approximate H-transformation) are available.
Fig. 1 shows the structure under simulation. Starting from a 1200-nm-long resistor, a scaled structure can be generated by increasing a scaling factor, . As shown
Outlook
The approximate H-transformation employs a special . is close to the band minimum energy and proportional to an integer-multiple of the discrete energy spacing. The approximate H-transformation is a suitable method for transient simulations because it is not affected by the time-varying potential energy. This method may be applied to the two-dimensional/one-dimensional electron gas (multi-subband Boltzmann transport equation) as well as the three-dimensional electron gas. However, when the
Conclusion
In conclusion, a novel stabilization scheme has been successfully implemented into the deterministic Boltzmann transport equation solver. For the three-dimensional electron gas, the distribution functions at past time instances can be directly accessible without any interpolation procedure and the stability of the conventional H-transformation can be shared. The proposed method can be readily applied to the transient simulation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Cited by (0)
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The review of this paper was arranged by Francisco Gamiz.
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This research was supported by the National Research Foundation of Korea (NRF), Republic of Korea grant funded by the Korea government (NRF-2020M3H4A3081800).