Another model is adopted in  to study the behaviour of devices in the quan- tum ballistic limit and requires strong assumptions on the wavefunctions in the metal leads. All of these models provide a very accurate and complete physical information on the quantum mechanical phenomena occurring in the device. However, a certain lack of numerical robustness and the intensive computational cost make these models still unsuitable for routine industrial semiconductor device simulation.
These con- siderations strongly prompt towards investigating the approach ii. This latter has the advantage to exploit all the benefits arising from the well established mathematical and numerical experience on the basic DD model, allowing at the same time to design a state-of-the-art simulation tool. The QDD model emanates from a self-consistent derivation of a generalized equation of state for the electron gas which includes a dependence on the gradient of electron density. This, in turn, allows to incor- porate quantum phenomena description into the classical DD model by means of a quantum correction to the electric potential, the so called Bohm potential .
In this paper, we show that the modification of the density of states in the SPDD model can indeed be re- garded as a suitable quantum correction of the electric potential in the DD current relation. This latter family of models thus represents a suitable generalization of the classical drift-diffusion DD system, each particular model being identified by the constitutive relation for the quantum-correction to the electric potential. Following the derivation of a unified framework for QCDD transport models, we propose to apply a functional iteration technique, which is customarily and successfully used in standard DD-based semicon- ductor device simulation, to the decoupled numerical solution of the nonlinear boundary value problems deriving from both the SPDD and QDD models.
The iteration procedure is a generalized Gummel algo- rithm [13—15] and, to our knowledge, represents the first fully decoupled functional iteration procedure applied to the QDD model see also [16,17] for the use of other partially decoupled iterative maps. The advantage of adopting a Gummel-type iteration is twofold: on the one hand, it considerably saves compu- tational effort and memory storage at each step, which is of relevant importance in multidimensional sim- ulations; on the other hand, it leads to successively solving elliptic quasi-linear and linear boundary value problems for which efficient and stable discretization methods can be employed [18,19].
Moreover, the performance of the iterative procedure can be properly improved by resorting to suitable acceleration tech- niques, for example, Newton—Krylov subspace iterations [20,21]. Then, in Section 4, we introduce a functional iteration to construct a decoupled algorithm for the iterative solution of the SPDD and QDD systems, and in Section 5, we dis- cuss the finite element discretization of the various differential subproblems obtained after decoupling. The detailed description of the numerical algorithms used in the computations is discussed in Section 6.
Finally, in Section 7, we illustrate and compare the performance of the proposed algorithms and models on the numerical simulation of nanoscale devices in two spatial dimensions. Some concluding remaks and perspec- tives on future work are addressed in Section 8.
Theory of Semiconductor Quantum Devices
For ease of presentation, we develop in detail the derivation of the models only for electron carriers, as a similar treatment holds for hole carriers. In each of these two re- gions, we consider the material to be homogeneous and isotropic; this implies, in particular, that the elec- tric permittivity e is a scalar piecewise constant quantity over X. In 1 , the first two equations represent the continuity equations for electrons and holes n and p, while the third equation is the Poisson equation for the electrostatic potential u.
The issue of a suitable set of boundary conditions for 1 will be addressed in Section 3. In 2 , un describes the deviation of the distribution function from its equilibrium value and it reduces to the con- stant Fermi potential un0 at equilibrium when no current flow is expected. Let x be a fixed point in XSi. We defer the discussion of these two assumptions and of their associated physical consequences to the end of this section, where we will also extend the presentation below to situations in which these assump- tions do not hold.
Rather, its spatial variation is given by the wavefunctions wi x which are non- local functions of u x. Also notice that the factor 2 in 9 accounts for the spin degeneracy of the energy eigenstates. To solve this inconsistency, the classical expression 6 is forced to hold in regions of the device where quantum effects are assumed to be neg- ligible.
Passar bra ihop
To properly account for the resulting splitted consti- tutive relation for n, a quantum correction factor is defined as follows. A quantitative analysis of this issue is carried out in Section 3. In most real-life applications both these conditions are too restrictive, so that it is convenient to extend the SPDD model to the case where neither of the two conditions a and b holds. As for assumption a , we notice that the computation of the eigenvalues of the discretized Hamiltonian is a rather intensive numerical task.
A representation of the device subdvision into the several physical subdomains described in this section is shown in Fig.
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The QDD model The QDD model was introduced in  and originally named Density Gradient model because it was obtained by allowing the electron gas equation of state to relate the quasi-Fermi potential un not only to the electron density but also to its gradient. Notice that the QDD expressions for the current densities are formally iden- tical to 16 and The scaling has the advantage to emphasize the singularly perturbed nature of the equations, which can be used as in  to perform an a priori qualitative analysis of their solutions.
In Table 1, we summarize the non-dimensional coefficients for electrons and holes appearing in the QDD and SPDD equations, providing their numerical values in the case of a nanoscale device. Unified framework for quantum-corrected DD models We have shown in the preceeding sections that the QDD and SPDD models can be regarded as gener- alizations of the classical DD model, only differing by the choice of the constitutive relation for the quan- tum corrections Gn and Gp to the electric potential.
These latter models can be derived by an appropriate definition of the constitutive relations for the quantum correction poten- tials Gn and Gp. For sake of completeness, we consider both electron and hole contributions to charge trans- port.
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For ease of notation, we will use henceforth for any scaled quantity the same symbol as in the unscaled case. A constitutive relation for this term in quantum-corrected models is not well established. An example of a recombination model which fits 26 and respects the latter condition is proposed in  as a QDD extension of the classical Shockley—Read—Hall theory . Each particular model is characterized by the constitutive relations 25 6 and 25 7 for the correction terms Gn and Gp.
More precisely, in the QDD model this non-locality effect is obtained through a dependance of un on the gradient of the concentrations, while in the SPDD model it is obtained through a more detailed physical description of the density of states. A cross-validation and mutual com- parison among the three models discussed in this section will be the object of thorough investigation in the numerical experiments shown in Section 7.
Boundary conditions In this section, we define the proper boundary conditions for the QCDD models introduced in Section 3. On the one hand, in a quantum model, one would expect the carrier densities to become very small in the vicinity of the very high potential barrier given by the gate oxide, and in the limit of an infinite barrier one should predict that both carrier densities tend to zero. On the other hand, the condition of zero free charge carriers at the interface is incompatible with the mod- ified Maxwell—Boltzmann statistics for electrons and holes, because it would require Gn and Gp to tend to Fig.
To circumvent this problem, one can set the interface densities equal to a small but non-zero value cI, which might be estimated by a priori 1D computations with a model includ- ing tunneling of free carriers through the oxide barrier . Furthermore, at the interface we impose the nor- mal component of the current densities to vanish and the normal component of the electric displacement vector to be continuous, i. In the case of the SPDD model, the mathematical formulation of the boundary-value problem requires a further subdivision of X as anticipated in Section 2.
The choice of XSchr is the result of a careful trade-off. On the one hand, XSchr should be large enough to ensure that the closed boundary conditions 11 for Eq. On the other hand, the choice of a too wide XSchr can greatly increase the computational cost of the eigenvalue problem.
Conditions 32 — 35 still hold for the SPDD model.