Fast Uncoupled Mode Space Approach for Nano-Scale Device
Simulation Based on Bridge-Function Pseudo-Spectral Method
Hiroki Fujikawa, Yuta Saitou, Satofumi Souma, and Matsuto Ogawa
Department of Electrical and Electronic Engineering, Kobe University, Japan
1-1 Rokkodai, Nada, Kobe, 657-8501, Japan
Email: 121t253t@stu.kobe-u.ac.jp / ogawa@eedept.kobe-u.ac.jp

1Introduction

Miniaturization of metal-oxide field-effect-transistors(MOSFETs) has been aggressively accelerated and they
have become as small as a few tens nm regime. It has
been very difficult to fabricate such ultra small devices
in small turn-around time within reasonable fabrication
cost. Therefore, it is essential to develop a device simulator which predicts the device performances with possible
quantum mechanical effects for the new era of nano-scale
devices, which allows us to develop the LSI technology
with conventional pace.
A non-equilibrium Green’s function (NEGF) is one of
the attracting methods for such quantum device simulations, since it can take into account both quantum mechanical effects and open boundary conditions. It is solved with
Poisson’s equation self-consistently. A real-space method(RSM) for solving the NEGF is considered to be suitable
for accurate simulations. However, the RSM requires quite
expensive computational costs and is difficult to apply forlarge-scale calculation physical model. On the other hand,in a mode space method (MSM), the transverse spatial
coordinate system is replaced by the corresponding eigenmode energies and only 2D Schrödinger equation has to besolved in each cross-section of the device along the channel
direction. The MSM for solving the NEGF transport equations can greatly save the computational resources. Although the MSM could reduce the computational time, inthe actual numerical simulations, conventional discretized
schemes, i.e., a finite difference method (FDM) or a finite
element method (FEM) are still inappropriate for largescale calculation without loss of accuracy.
In this paper, we have adopted a perturbative treatment, namely, a fast uncoupled mode space approach(FUMS)[1] into our previous Bridge-Function Pseudospectral Method (BPSM) [2] to reduce further the computational time. This method (FUMS-BPSM) enables us tocalculate much faster than our previous method, retaining high accuracy. We have affirmed that we can calculate
more accurately and faster than FDM or FEM by using
BPSM, CPU time for calculation by the BPSM is 60 times
faster than that of FDM.
22.1

TheoryBridge-Function-PseudoSpectral-Method(BPSM)

The BPSM is composed of the bridge-functions, connections of two Lagrange polynomials (Fig.1) and theGauss-Lobatto quandrature [3]. The continuity of elec-

trical flux density and probability current density, whichappear in Poisson’s equation and NEGF, respectively, arenaturally guaranteed by the bridge-function (solid line) at
any boundaries. The fact enables us to take the boundary conditions into account in much easier manner thanthe conventional PSMs. In addition, the FUMS is easily
implemented in the BPSM formalism.
2.2

Fast-Uncoupled-Mode-Space-approach(FUMS)

An uncoupled mode space approach (UMS) [1] requires
to solve the 2D Schrödinger equation-1D NEGF calculation repeatedly on each cross-section of the device underconsideration. Because of that, it is found that the computational costs and time rapidly increase as the numberof longitudinal mesh points increase. On the other hand,in the FUMS approach, the subband profile is approximated up to the first order by considering the change fromthe average of potential profile as the perturbation. Here,
we have only to solve the non-perturbative Schrödinger
equation once in the coupled 2D Schrödinger equation-1D
NEGF calculation, even when the number of longitudinalmesh points increase.
3Results and discussions

In order to show the superiority of the present method(FUMS-BPSM) over our previous BPSM (UMS-BPSM:
BPSM without perturbational treatment), we have applied
the FUMS-BPSM to the analysis of I-V characteristics ofa gate-all-around Si nano-wire MOSFET (SiNW FET) asshown in Fig. 2 [4]. In the analysis, Schrödinger-Poisson
equations and NEGF are solved simultaneously and selfconsistently. The orientation dependent effective masses
in the conduction band of Si are also taken into account.
Figure 3 and 4 show the local density of states andthe first five subband profiles along the channel calculatedby the UMS-BPSM (Fig.3) and FUMS-BPSM (Fig.4), respectively. In both figures, quantum interferences andtunneling effects can be observed. It should be noteworthy that the results of the FUMS-BPSM coincide quite
well with those of the UMS-BPSM, in spite of the factthat Schrödinger equation is solved only once in the lattermethod.
In Fig. 5, the comparison of the computational time isshown to obtain the subband profiles with increasing longitudinal nodal number Nx . The result of the UMS-BPSM(blue line) increases linearly as Nx increases, whereas theFUMS-BPSM (green line) keeps constant, practically. The[3] G.H. Golub and J.H. Welsch, “Calculation of Gaussquadrature rules,” Mathematics of Computation 23, p.
221, 1969.
[4] S. Bangsaruntip, A. Majumdar, and J.W. Sleight, “Gateall-around silicon nanowire,” 2010 Symposium on VLSITechnology Digest, p.21, 2010.

10x position [nm]

150.8
0.6
0.4
0.2

0.0
50Region 1Region 2

χ1, 1 (x)Region 3

χ3, 2 (x) χ3, 3 (x)xx13

2x14= x21 x2x23x3x24= x31 210x position [nm]15]140x33Fig. 1. 1D basis functions χi,m (x) (i = 3 and m = 3),
where i indicates the region, and m the basis. The solidlines are the bridge functions, which ensure the continuityof any physical quantities between the adjacent regions.

120
Computational time [sec]

x12

1e10
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
20 0.0Fig. 4. The computed local density of states and electronsubbands (solid lines) of SiNW FET (source region is located from x = 0 to 5 nm, channel is x = 5 to 15 nm,
drain is x = 15 to 20 nm) by FUMS-BPSM(Vg = 0.1 and
Vd = 0.1 V).

χ2, 1 (x)

χ1, 2 (x) χ1, 3 (x) χ2, 2 (x) χ2, 3 (x)3Fig. 3. The computed local density of states and electronsubbands (solid lines) of SiNW FET (source region is located from x = 0 to 5 nm, channel is x = 5 to 15 nm,
drain is x = 15 to 20 nm) by FUMS-BPSM(Vg = 0.1 and
Vd = 0.1 V).

][2] Y. Saito, T. Nakamori, S. Souma, and, M. Ogawa,“Bridge-Function Pseudospectral Method for Quantum
Mechanical Simulation of Nano-Scaled Devices,” Proc.
SISPAD 2011, 11-4, pp. 311-314, 2011.

503[1] J. Wang, E. Polizzi, and M. Lundstorm, “A threedimensional quantum simulation of silicon nanowire transistors with the effective-mass approximation,” J. Appl.
Phys. 96, 2192, 2004.

0.0[References0.2

Density of states m−

We have developed a fast-uncoupled mode-space bridgefunction pseudo-spectral method (FUMS-BPSM) for
nano-scaled device simulation where quantum effects are
fully taken into account.
As a result, it is found that the FUMS-BPSM reduces
the simulation time while keeping computational accuracy.
Therefore, the present FUMS-BPSM can be an efficientsimulation method for the nano-scale devices.

[0.4Conclusion

1e10
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
20 0.0Density of states m−

0.6Energy [eV]40.8

Energy [eV]present FUMS-BPSM remarkably shows that it is 20 times
faster than the previous UMS-BPSM on the same computational platform. This is mainly because, as mentioned
above, Schrödinger equation needs to be solved only one
time in the present method even when Nx increases.

1008060UMSFUMS

4020068101214161820

longitudinal nodal number NxFig. 2. Model structure of a SiNW MOSFET[4] underinvestigation. The channel orientation is along h100i.

Fig. 5. The computational time for solving Schrödingerequation at Vg = 0.1 V Vd = 0.1 V, increasing the longitudinal nodal number Nx .