# A New Efficient Quantum Method: The Bohm Quantum Potential Model

This article presents a new approach to model the quantum confinement
of carriers in MOSFET or heterostructure. SILVACO has already included in its
device simulator ** ATLAS**, a Schrödinger-Poisson
solver and Density-Gradient model. The Schrödinger-Poisson (SP) solver
is the most accurate approach to calculate the quantum confinement in semiconductor
but it cannot predict the currents flowing in the device. To overcome this limitation,

**provides a Density Gradient (DG) model [1]. It allows the user to predict both the quantum confinement and the drift-diffusion currents along with the Fermi-Dirac statistics for a 2D structure. However this model exhibits poor convergence in 3D and with the hydrodynamic transport. Therefore, in collaboration with the University of Pisa, SILVACO has introduced in**

*ATLAS***, a new approach called Effective Bohm Quantum Potential (BQP) model. This model presented at SISPAD 2004 conference [2] exhibits many advantages. It includes two fitting parameters which ensure a good calibration for silicon or non-silicon materials, planar or non-planar devices. It is numerically stable and robust, and independent of the transport models used. Therefore it has been successfully implemented and tested in**

*ATLAS***. The table below summarizes the different models available in**

*ATLAS***related to the dimensionality and the transport models.**

*ATLAS*

X: available | |

S-P: Schrödinger-Poisson | DD: drift-diffusion |

DG: Density Gradient | EB: energy balance |

BQP: Bohm Quantum Potential | NEB: non-isothermal energy balance |

Historically, the definition of an effective quantum potential is Bohm’s interpretation of quantum mechanics [3], and has generated other more recent derivations based on a first order expansion of the Wigner equation [4], or on the so-called density gradient approach [5]. The BQP model has a few advantages: it does not depend on the transport model (drift-diffusion or hydrodynamic); Fermi-Dirac statistics can be straightforwardly included; it provides two parameters for calibration, whereas the Density Gradient has only one fitting parameter; finally, it exhibits very stable convergence properties.

The definition of the effective quantum potential *Q _{eff}*
is derived from a weighted average of the Bohm quantum potentials seen by all
single particle wavefunctions, that can be expressed as:

(1)

where *M* represents the effective mass tensor, *n*
is the electron density per unit volume, _ and _ depend on the eigen-functions
of the confined system, and the operator ,
which reduces to at equilibrium,
allows to treat non-equilibrium problem and has been defined as:

(2)

where *f _{i}* is the occupation factor for the
i-

*th*state and

*R*is the modulus of the i-

_{i}*th*single particle wavefunction. The main advantage of these expressions is that the effective quantum potential does not depend on the transport model and, in the case of Fermi-Dirac distribution, can be discretized on a Delauney mesh as:

(3)

where r* _{l }*is the position of the

*l*-th point of the mesh,

*d*is the distance between a point

_{lj }*r*and

_{j}*r*in 3D is the area of the face of the Voronoi cell separating point

_{l, _lj}*l*and

*j*. Moreover,

*m*represents the effective mass tensor,

_{lj}*N*is the 3D density of states and

_{c}(r)*F*represents the Fermi-Dirac integral of order ±1/2.

_{±1/2}We have treated _ and _, in Equation (3), as fitting parameters avoiding to solve the solution of the Schrödinger equation. With two fitting parameters, instead of one like in DG model, the BQP model allows an additional degree of freedom for calibration.

Below are presented the results of simulations performed on
simple structures of general interest in order to understand and verify the
accuracy of the proposed model. The first device is a MOS-capacitor, a device
in which the quantum confinement of electrons is predominant along one direction.
The MOS-C has a 2 nm oxide thickness, a p-type concentration in the bulk *N _{A}*
= 10

^{18}cm

^{-3}, and a metal gate. In Figures 1, 2 and 3, we show the electron density profiles for different applied gate voltages and the C-V curve obtained with the two different methods (SP and BQP). For the BQP model, the best values for the fitting parameters are _=0.46 and _=1.11.

As can be observed in Figures 1 and 2, the BQP method gives accurate height and shape of the carrier density profile as the applied voltage is varied. The method is also capable to well reproduce the carrier density per unit area obtained with the SP approach as shown in Figure 3.

Figure 1. Electron density below
the gate for a voltage applied to the gate of V1
V obtained with the BQP model and with the SP approach. _{G}= |

Figure 2. Same as figure 1 for
V=1.5 V. _{G} |

Figure 3. C-V curve obtained with
the BQP model and SP approach. |

The second device, in which electrons are confined in a quasi-one
dimensional wire, is a FinFET depicted in Figure 4. The oxide thickness is 3
nm and the acceptor concentration is N* _{A}* = 10

^{18}cm

^{-3}. Quantum confinement of electrons is strong both in x and y direction as shown with the isosurface of 10

^{19}cm

^{-3}in Figure 5 (for Vdrain=0.1 V and Vgate=1.15 V).

Figure 4. Representation of the
FinFET composed by a silicon fin embedded in a SiO
layer. Also shown is the polysilicon gate deposited on the structure (source
and drain electrodes are hidden for a better lisibility)_{2} |

Figures 5 and 6 show clearly the confinement of electrons under the gate oxide in the top corners of the silicon fin.

Figure 5. Representation of the
10 ^{19} cm^{-3} electron concentration isosurface inside
the silicon fin. |

Figure 6. 2D cross-section made
in the middle of the finFET of Figure 5. Electron concentration is shown. |

In conclusion, ** ATLAS** proposed a new
approach for quantum correction with the BQP model. The model is available both
for drift-diffusion and energy transport models. With respect to previously
published density gradient methods [1], the main advantages of this model are
represented by the addition of one degree of freedom, which may provide better
fit and improved calibration, by the possibility of including both Maxwell-Boltzmann
and Fermi-Dirac statistics, and by its independence of the transport model adopted.
SILVACO gratefully acknowledges support from University of Pisa.

References

- A. Wettstein, A. Schenk, W. Fichtner, IEEE Trans. Electron Devices 48, 279 (2001).
- G. Iannaccone, G. Curatola, G. Fiori, “Effective Bohm Quantum Potential for device simulators based on drift-diffusion and energy transport”, SISPAD 2004
- D. Bohm, Phys. Rev. 85, 166 (1952), 85, 180 (1952).
- G. J. Iafrate, H. L. Grubin, and D. K. Ferry, J. Phys. C, 42, C10-307 (1981).
- M. G. Ancona and G. J. Iafrate, Phys. Rev. B 39, 9536 (1989).