Modeling of Hot Carrier Degradation Using a Spherical Harmonics Expansion of the Bipolar Boltzmann Transport Equation M. Bina ⋄,• , K. Rupp •,† , S. Tyaginov • , O. Triebl • , and T. Grasser • ⋄ Christian Doppler Laboratory for Reliability in Microelectronis at the • Institute for Microelectronics, TU Wien, Austria † Institute for Analysis and Scientific Computing, TU Wien, Austria Abstract Recent studies have clearly demonstrated that the degrada- tion of MOS transistors due to hot carriers is highly sensitive to the energy distribution of the carriers. These distributions can only be obtained in sufficient detail by the simultaneous solu- tion of the Boltzmann transport equation (BTE) for both carrier types. For predictive simulations, the energy distributions have to be thoroughly resolved by including the fullband structure, impact ionization (II), electron electron scattering (EE), as well as the interaction of minority carriers with the majority carriers. We demonstrate that this challenging problem can be efficiently tackled using a deterministic approach based on the spherical harmonics expansion (SHE) of the BTE. Introduction While the first hot-carrier degradation (HCD) models were based around the channel electric field as the driving force, it has long been realized that the phenomenon is energy- rather than field-driven [1]. In order to obtain the energy distribution, the BTE has to be solved, which is challenging in its own right. Unfortunately, as HCD is highly sensitive to the high-energy tail of the distribution, therefore the mod- eling of the scattering operator requires special attention. In particular, impact ionization scattering as well as electron electron interactions have to be incorporated. For example, it has been shown that the adequacy of the BTE solution ignoring electron electron scattering can be seriously hampered [2]. Furthermore, it has been shown that the majority carriers can significantly contribute to the damage, requiring a coupled solution of the BTE for electrons and holes [3]. Finally, since an accurate resolution of the energy distribution at high energies is required, information about the full band structure has to be included into the model. Traditionally, this complicated problem has been approached by using the Monte Carlo method (MC) [4], which is computationally- and time-intensive, particularly when the high-energy tails of the distribution function have to be resolved in detail [5]. In this work we demonstrate a time-efficient SHE solution of the bipolar BTE, which is applied to the investigation of HCD in n-channel MOSFETs. Method We solve the Poisson equation and the bipolar BTE self- consistently on unstructured grids using the higher-order spherical harmonics expansion (SHE) simulator, ViennaSHE [6–8]. Full-band effects in silicon are accounted for using the method suggested by [9–11], cf. Fig 1. The scattering mechanisms considered are acoustical and optical phonon scattering, impurity scattering, impact ioniza- tion (II) [4] with secondary carrier generation and electron 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 Velocity [10 6 m/s] Energy [eV] 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 DOS (10 28 eV -1 m -3 ) Velocity for Electrons Velocity for Holes DOS for Electrons DOS for Holes Fig. 1: The density of states (DOS) and group velocity for relaxed silicon used for the solution of the bipolar BTE with SHE expansion techniques. The DOS is efficiently incorporated into the SHE of the BTE using the approximation put forward by [9, 10]. electron scattering (EE) [12]. To assess the damage caused by hot carriers, the acceleration integral (AI) defined as σ 0  f (ǫ)Z (ǫ)  ǫ - E th 1 eV  p v g (ǫ)dǫ, (1) has to be calculated, where σ 0 is the capture cross section, p = 11, f (ǫ) is the distribution function, Z (ǫ) the density of states (DOS), v g (ǫ) the group velocity and ǫ is the carrier energy [13, 14]. The AI is the kernel of the hot carrier degradation model and is used to describe single- and multiple-carrier bond dissociation processes [3, 13, 15]. To simulate the device degradation, measured as a relative decrease in I d, lin , we use the acceleration integrals for electrons and holes in our detailed degradation model [3]. Using this approach, two 2D n-channel MOSFETs with 250 nm and 25 nm channel lengths subjected to hot carrier stress at high oxide (≈ 8 MV/cm) and lateral electric fields (≈ 1 MV/cm) are investigated to assess the numerical and physical properties of the distribution function and acceleration integral. Interface states generated at the semiconductor-oxide interface during HCD disturb the electrostatics of the device and effect the carrier mobility. To incorporate these effects in a self-consistent manner, the AI was evaluated and used within our degradation model [3] to calculate the interface state density N it at each simulation step. Additionally in every step the obtained N it was used for the self-consistent treatment of trapped charges using Shockley-Read-Hall (SRH) theory [8]. These trapped charges act as coulomb scattering centers, thereby degrading the charge carrier mobility. Since it is not yet clear whether interface or oxide traps generated during HCD are governed by SRH trapping dynamics [16], the framework of ViennaSHE allows for the inclusion of arbitrary defect models. 30.5.1 IEDM12-713 978-1-4673-4871-3/12/$31.00 ©2012 IEEE