A new robust non-local algorithm for band-to-band tunneling simulation and its application to Tunnel-FET

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Abstract

A new non-local algorithm for accurately calculating the band-to-band tunneling current suitable for TCAD semiconductor simulators is proposed in this paper. The proposed algorithm captures the essential physics of multi-dimensional tunneling in a 2D structure, and is designed to be robust and to achieve independence on the mesh grid. The new algorithm enables accurate modeling of T-FET and investigation of its device physics.

Research highlights

► Caclulate band-to-band tunneling current in 2D device structures. ► Proposed algorithm based on multi-dimension extension of WKB approximation. ► Physical and robust simulation of tunneling FET is achieved with the new algorithm.

Introduction

The modeling of band-to-band tunneling (BTBT) is an important aspect of semiconductor device simulation and modeling. In the mainstream CMOS technology, BTBT has been traditionally considered as a second order effect that contributes to the leakage current. The simulation of BTBT recently attracted interest as researchers attempt to realize tunneling transistor (T-FET) that provides steep switching characteristics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

Since T-FET derives its drain current from the band-to-band tunneling at the source–channel junction, the transfer characteristics is not bound by the 60 mV/decade limit in MOSFETs. In T-FET, band-to-band tunneling is the principal device physics to be modeled. Additionally, as MOSFET scales into deca-nanometer scale, BTBT has become one of the dominating components in off-state leakage current, and requires more detailed modeling.

In both T-FET and MOSFET, the direction of the band-to-band tunneling current depends on the location. This makes the analytical modeling of BTBT in these devices very difficult, and numerical simulation in 2- or 3-dimensions is needed for accurate modeling.

Since band-to-band tunneling is a quantum mechanical phenomenon, a quantum transport formalism is required to simulate the BTBT current more rigorously. The NEGF method is a commonly used approach [15], and has been applied to T-FET in several studies [16], [17]. However, the NEGF method is computationally very expensive in 2- or 3-dimensional structures.

On the other hand, in the semi-classical drift-diffusion transport model used in TCAD simulators, tunneling can be included as a carrier generation mechanism. In TCAD simulators, analytical models are needed to express the carrier generation rate due to band-to-band tunneling.

BTBT has been studied extensively since the late 1950s. Keldysh and Kane derived an expression for BTBT by calculating the inter-band matrix element with stationary-phase method [18], [19]. Harrison and Stratton, on the other hand, used WKB approximation to treat the BTBT problem [20], [21], [22]. The Airy-function method can also be used to calculate the tunneling probability [23]. More recently, Schenk derived another expression using the Kubo formalism for tunneling conductivity [24], taking into account the effect of phonon-assisted tunneling more carefully. However, all the above analytical expressions are derived for a 1-dimensional geometry only. The experimental data of band-to-band tunneling current, that can be used to calibrate the above models, are mostly for 1D structures only [25], [26].

For 2D or 3D devices like MOSFET and T-FET, TCAD simulators use various algorithms to extend the 1D formulae for BTBT generation. Unfortunately, the BTBT algorithms in popular Technology CAD (TCAD) device simulators have several limitations, which make the reliable simulation of BTBT current in advanced device a difficult task. In order to use TCAD simulators to study the device physics of T-FET with reasonable confidence, the authors believe that a more robust algorithm for calculating BTBT current is necessary.

In this manuscript, we shall first discuss the criteria for a well-behaved BTBT algorithm in TCAD simulators, and summarize the limitations of existing BTBT algorithms. Following that, we shall describe a new algorithm of BTBT current calculation suitable for TCAD device simulation under the criteria. The proposed algorithm has been reported in Ref. [27] with the focus on its application to tunneling transistors. In this manuscript, the physical background and implementation details of the algorithm are documented.

Section snippets

Existing algorithms and limitations

A good BTBT calculation algorithm for TCAD simulation, in the authors’ opinion, should capture the following physics. First of all, the BTBT process is non-local, in that its rate depends on quantities at spatially separated locations. The carrier generation/recombination rate due to BTBT depends not only on the tunneling probability, but also on the carrier concentration on both ends of the barrier. Under thermal equilibrium, for example, the carrier quasi-Fermi levels are constant throughout

Tunneling path

The general problem of tunneling across a 2D or 3D barrier is difficult. However, it can be greatly simplified under certain assumptions [31], [32]. Huang derived from the WKB approximation that, if we assume all electrons strike on the tunnel barrier in the normal direction, the most probable tunneling path can be easily found using Newton mechanics in inverted potential and energy [31].

We shall not repeat the full derivation here, but it is necessary to outline some key steps to clarify the

Application on Tunneling-FET device physics

The tunneling FET structure shown in Fig. 6 is simulated using the newly proposed algorithm. The total body thickness is 40 nm, while only half of the structure is simulated due to the symmetry. The band-to-band tunneling occurs dominantly in the region enclosed by the dotted line near the P+ source. The source has a doping concentration of 1 × 1020 cm−3 and a lateral Gaussian straggle of σ = 2 nm. The gate oxide thickness is 0.8 nm. The identified tunneling paths at various energy levels are also

Extension to 3D mesh

In a 3-dimensional structure, the Ec and Ev fronts are curve surfaces, while the tunneling paths should define 1D pipes that connects the two surfaces. In the simple case with tetrahedral mesh elements, the Ec and Ev fronts each intercepts with the mesh on a triangulated surface.

Suppose one starts tracking the tunneling path from the Ec front. One first constructs a tessellation of the Ec surface, which is straightforward on a triangulated surface. From each cell of the tessellation, a

Conclusion

A new algorithm of calculating BTBT current in TCAD device simulator was proposed and implemented. The proposed non-local algorithm is based on the tunneling path approach, and unlike previous work, it identifies the most probable tunneling paths automatically. The paths are found using a 2D extension to the WKB method, and carries well-defined physics. Under the simplifying assumption of normal incidence, the tunneling paths coincides with the classical trajectory in the inverted potential.

Acknowledgement

The authors acknowledge a research grant (Award number NRF-RF2008-09) from the National Research Foundation (NRF) of Singapore.

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