As semiconductor technology progresses to smaller and smaller device dimensions, the effect of quantum confinement becomes more and more important. The density-gradient theory provides a computationally efficient method to include quantum confinement in the conventional drift-diffusion formulation commonly used for semiconductor device physics simulation. In the first post of a two-part blog series, we will briefly review the theory, with a focus on the equations implemented in the Semiconductor Module. Interested readers may consult Ref. 1 for a comprehensive review.
Electrostatics and Charge Carrier Conservation
We start by reviewing the fundamental equations common to the conventional drift-diffusion theory and the density-gradient (DG) theory. In the next section, we will discuss the difference between them.
Semiconductor device physics simulation concerns the transport of charge carriers driven by the electric field (drift) and the carrier concentration gradient (diffusion). The electric field (V/m) is given by the electrostatic equations under the quasistatic assumption:
(1)
where is the permittivity (F/m),
is the charge density (C/m3), and
is the electric potential (V).
The charge density, , is given by the contributions from the hole, electron, and ionized donor and ionized acceptor concentrations (1/m3) (
,
, and
and
, respectively).
(2)
where is the elementary charge (C).
The electron and hole concentrations ( and
) are governed by conservation laws given in the form of current continuity equations:
(3)
where is the time (s);
and
are the electron and hole current densities (A/m2); and
are the recombination and generation rates for the electrons and holes (1/m3/s).
The electron and hole current densities, and
, can be expressed in terms of the quasi-Fermi levels,
and
(V):
(4)
where and
are the electron and hole mobilities (m2/V/s),
and
are the conduction band and valance band edges (V),
and
are the electron and hole nonequilibrium contributions to the thermal diffusion coefficient (m2/s), and
is the temperature (K).
The conduction band edge, , and the valance band edge,
, are related to the electric potential,
; electron affinity,
; and band gap,
:
(5)
Note that in the implementation for the Semiconductor Module, all of the energy level variables () are scaled by the elementary charge to take on the unit of electric potential (V).
The Equations of States for the Drift-Diffusion and DG Theories
Both the conventional drift-diffusion theory and the DG theory obey the electrostatics and current conservation laws listed above. The difference between them lies in the equation of states of the electron and hole gases.
In the drift-diffusion theory, the charge carrier concentrations and
are related to the quasi-Fermi levels
and
via the following equations:
(6)
where and
are the conduction band and valance band effective density of states (1/m3),
is the Fermi–Dirac integral, and
is the Boltzmann constant (J/K).
In contrast, the DG theory adds a contribution from the gradients of the concentrations to the equation of states via the quantum potentials and
(V):
(7)
where the quantum potentials, and
, are defined in terms of the density gradients:
(8)
The DG coefficients, and
(V m2), are given by the inverse of the DG effective mass tensors,
and
(kg):
(9)
where is the Planck constant.
Solution Strategy
For the conventional drift-diffusion formulation, it is possible to explicitly insert the equation of states (6) into the fundamental equations, (1)~(5), and solve for the three dependent variables: ,
, and
.
For the DG formulation, the equation of states (7)~(8) forms implicit relations for the charge carrier concentrations. In this situation, it is necessary to introduce new dependent variables to solve the implicit equations. Following Ref. 1, we use the Slotboom variables and
(V) as the additional dependent variables:
(10)
The quantum potentials can now be expressed in terms of the Slotboom variables:
(11)
where is the inverse function of
.
The DG formulation then solves for the five dependent variables, ,
,
,
, and
, using the fundamental equations (1)~(5) and the equation of states (7)~(11). It is evident that this approach does not add an excessive computational burden as compared to other, more sophisticated quantum mechanical methods. Therefore, the DG theory provides an efficient alternative for engineering purposes.
Recombination Rates
Because the gradient of the carrier concentrations enters the equation of states in the DG theory, the equilibrium concentrations are no longer only a function of the equilibrium Fermi level, thus the recombination rates involve more complex evaluations (Ref. 2).
For explicit traps, the electron and hole recombination rates and
(1/m3/s) are now computed with the formula based on the quasi-Fermi level differences:
(12)
where and
are the average capture probabilities for the charge carriers (m3/s),
is the trap density (1/m3),
is the trap occupancy (1), and
is the quasi-Fermi level of the trap (V).
The trap occupancy is given by Fermi–Dirac statistics:
(13)
where is the degeneracy factor (1) and
is the trap energy level (V).
For the direct, Auger, and Shockley–Read–Hall (SRH) recombinations, the rate expressions may look familiar; however, the underlying definitions of the various parameters are more complex. For example, the SRH recombination rates and
(1/m3/s) are:
(14)
where and
are the electron and hole lifetimes (s).
The electron and hole equilibrium concentrations, and
(1/m3), now become:
(15)
where is the equilibrium Fermi level (V).
Note the appearance of the quantum potentials and
in the expressions.
The parameters and
(1/m3) are no longer constants, even in simple cases, and are computed using the original definitions:
(16)
Note the dependence on the carrier concentrations and
, which implicitly depend on the concentration gradients.
Boundary Conditions for the Slotboom Variables
In most cases, the boundary conditions for the Slotboom variables, and
, are simple natural boundary conditions (Ref. 3):
(17)
In the case of a boundary representing an abrupt potential barrier (for example, a silicon-oxide interface), Jin et. al. in Ref. 4 suggested using the Wentzel–Kramers–Brillouin (WKB) approximation to obtain the boundary condition:
(18)
where is the
coefficient in the oxide (V m2) and
is the penetration depth into the oxide (m), given by:
(19)
where and
are the effective masses in the oxide (kg) and
is the potential barrier height (V).
Options for Heterojunctions
Currently, in the conventional drift-diffusion formulation, the Semiconductor Module offers two options for heterojunctions: Continuous quasi-Fermi level and Thermionic emission.
The former can be easily extended to the DG formulation: Just let the quasi-Fermi level and the Slotboom variable be continuous across the heterojunction, which is automatic for Lagrange shape functions. This mimics the continuous nature of the quantum mechanical wave function, although it should be treated as phenomenology at best (Ref. 1).
The latter option assumes the thermionic emission process dominates and allows the quasi-Fermi level and the Slotboom variable to be discontinuous across the heterojunction. The same formula as the drift-diffusion theory is used for the thermionic current density, yielding similar results.
Final Remarks
The discussion above gives an overview of the DG formulation available in the Semiconductor Module. It should be emphasized that only the DG confinement theory is implemented, not the DG tunneling theory (Ref. 1). This implementation provides a computationally efficient option to include the effect of quantum confinement in device physics simulations for engineering purposes. In a follow-up blog post, we will use three example models to demonstrate the power of this simulation approach.
References
- M.G. Ancona, “Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices,” Journal of Computational Electronics, vol. 10, p. 65, 2011.
- M.G. Ancona, Z. Yu, R.W. Dutton, P.J. Vande Voorde, M. Cao, and D. Vook, “Density-Gradient Analysis of MOS Tunneling,” IEEE Transactions On Electron Devices, p. 2310, Vol. 47, No. 12, December 2000.
- M.G. Ancona, D. Yergeau, Z. Yu, and B.A. Biegel, “On Ohmic Boundary Conditions for Density-Gradient Theory”, Journal of Computational Electronics 1: 103–107, 2002.
- S. Jin, Y.J. Park, and H.S. Min, “Simulation of Quantum Effects in the Nano-scale Semiconductor Device,” Journal of Semiconductor Technology and Science, vol. 4, no. 1, p. 32, 2004.
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