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, ATLAS 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.

 

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 Qeff 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 fi is the occupation factor for the i-th state and Ri is the modulus of the 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 rl is the position of the l-th point of the mesh, dlj is the distance between a point rj and rl, _lj in 3D is the area of the face of the Voronoi cell separating point l and j. Moreover, mlj represents the effective mass tensor, Nc(r) is the 3D density of states and F±1/2 represents the Fermi-Dirac integral of order ±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 NA = 1018 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 VG=1 V obtained with the BQP model and with the SP approach.

 

Figure 2. Same as figure 1 for VG=1.5 V.

 

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 NA = 1018 cm-3. Quantum confinement of electrons is strong both in x and y direction as shown with the isosurface of 1019 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 SiO2 layer. Also shown is the polysilicon gate deposited on the structure (source and drain electrodes are hidden for a better lisibility)

 

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 1019 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

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

Download pdf Version of this article