1.
Introduction
The threshold or flat-band voltage shift (ΔVth or ΔVfb) is widely accepted as the unique parametric evaluation of the level of negative bias temperature instability (NBTI) degradation in device and is applied in the circuit aging simulation module of EDA tools[1–6]. It is rarely mentioned that mobility degradation is inevitable with generated interface states under NBTI stress. During long-term circuit operation, the mobility degradation may be more serious due to the generation of the interface states[7]. Ignoring mobility degradation probably results in overestimation of the level of the ON-state current in the device and lifetime in circuit. Therefore, an accurate mobility degradation model should be embedded in the existing NBTI model and applied to the circuit aging simulation module.
Traditionally, the reliability mechanisms (e.g. NBTI, hot carrier effect, self-heating effect) are considered individually. However, during circuit operation, the interactions between the mechanisms should not be ignored because they share critical parameters (current, field, temperature etc.). The self-heating effect (SHE) is normally observed in nanoscale like FinFET and results in the temperature increasing from room temperature to more than 60 °C[8, 9]. Such a significant temperature increment might result in more serious NBTI degradation because of strong temperature dependence, especially in the long-term region[10]. As a result, the drain current decreases and reduces the dissipation energy due to the SHE. It is very easy to simulate the aging degradation of devices caused by each reliability mechanism alone. However, it probably causes overestimation of the aging rate. Therefore, a simulation method considering the mutual influence of the reliability effects is established in this paper.
For the purpose of high precision circuit aging simulation, a mobility degradation model is proposed based on the universal NBTI model. Unlike the traditional NBTI simulation, which focuses on the threshold voltage shift only, the NBTI induced mobility model is implemented into the circuit aging simulation module, and the coupling effects of various mechanisms (e.g. NBTI-HCI and NBTI-SHE) are recalculated to avoid overestimation in long term aging simulation.
2.
Modeling of NBTI induced mobility degradation
It is well-known that NBTI is dominated by interface state generation and oxide hole-trapping[11, 12]. During fabrication, large numbers of silicon dangling bonds exist at the oxide-substrate interface. Hydrogen is usually used for passivation for the purpose of obtaining high performance. As illustrated in Fig. 1, under negative stress bias, silicon-hydrogen bonds are broken down and positively charged silicon dangling bonds (Pb center, origin of interface state) are generated. Hydrogen atoms are probably captured by oxygen vacancies located in the oxide rather than traveling towards the gate electrode. When stress is removed, the hydrogen travels successfully towards the gate electrode. On the other hand, holes injected into the gate oxide can possibly be captured by deep-level traps and it is difficult for them escape without being driven by the opposite electrical field.
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Figure1.
(Color online) Mechanisms of NBTI degradation, including oxide hole-trapping and interface state generation.
Fig. 2 shows the transfer characteristic shift of p-MOSFET under negative gate stress bias (Vg_str = ?2.6 V). The sample is of bulk p-MOSFET with W = 20 μm and L = 0.3 μm. In the sub-threshold region, the threshold voltage shifts to a larger absolute value with increased stress time. Although such a ΔVth leads to a reduction of the ON-state current, carrier mobility plays a more significant role in the linear region (Ids at Vgs = 1.5 V), which results in ON-state current degradation with increased stress time (e.g. up to 1000 s).
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Figure2.
(Color online) Flat band voltage shift and ON-state current degradation with stress bias Vg_str = ?2.6 V, and stress time of 100 and 1000 s.
To reproduce the measured NBTI degradation in practice, both of the above two mechanisms must be involved in the prediction model as independent components:
$Delta {V_{ m th_nbti}} = Delta {V_{ m th_operatorname{int} }} + Delta {V_{ m th_ox}} = qDelta {N_{ m T}}/{C_{ m ox}},$  | (1) |
$Delta {V_{ m th_operatorname{int} }} = left( {q/{C_{{text{ox}}}}} ight) {R_{ m str}} log (1 + t/{tau _{ m str}}),$  | (2) |
$Delta V_{{text{th_ox}}}^{} = left( {q{N_0}/{C_{{text{ox}}}}} ight) {t^{1/6}}.$  | (3) |
Here ΔVth_int, ΔVth_ox and ΔVth_nbti represent the threshold voltage shifts due to the interface state generation, oxide hole-trapping and the observable measurement result, respectively. τstr is the interface state generation time constant, Rstr is the interface state reaction rate coefficient, and N0 is the density of holes injected into oxide.
There are three factors, coulomb scattering, phonon scattering and surface roughness scattering (corresponding to μc, μp and μrs)[13–16], which influence the mobility together. In detail, the mobility is expressed as[17]
${mu _C} = frac{{{mu _1}Q_{ m inv}^{alpha_ 1}}}{{{Q_{ m dep}} + beta {Q_{ m inv}}({N_{ m T0}} + Delta {N_{ m T}})}};,$  | (4) |
$;{mu _1} = {mu _{1 m a}} - {mu _{1 m b}}T,$  | (5) |
${mu _{ m PH}} = alpha {T^{ - gamma }}E_{ m eff}^{alpha _2};,$  | (6) |
${mu _{ m SR}} = {mu _3}E_{ m eff}^{alpha_ 3},$  | (7) |
$frac{1}{{{mu _{ m eff}}}} = frac{1}{{{mu _{ m C}}}} + frac{1}{{{mu _{ m PH}}}} + frac{1}{{{mu _{ m SR}}}},$  | (8) |
$E_{ m eff} = frac{1}{{varepsilon_{ m si}}}left( {Q_{ m dep} + frac{{Q_{ m inv}}}{3}} ight),$  | (9) |
where Qinv and Qdep indicate the inversion charge density and depletion charge density, Eeff is the effective field and μeff is the total of mobility. μc, μp and μsr mobility are the coulomb, phonon, and surface roughness scattering terms, respectively, which are sensitive to ionized species, temperature and effective field Eeff. Other coefficients are fixed at μ1a = 264.9, μ1b = 0.4, α1 = 1, α2 = ?0.3, α3 = ?1.2, α = 1.46 × 105, μ3 = 61, β = 2.24 × 10?13, γ = 1.3. T is the temperature, which should be no larger than 650 °C for reasonable calculation of Eq. (5) in simulation or data characterization.
By substituting Eqs. (1)–(3) into Eq. (4), the mobility degradation due to the NBTI effect is described and shows good agreement with the measured data[17], as shown in Fig. 3. The sample is fabricated using the plasma nitridation (PNO) process with W = 15 μm and L = 0.16 μm. Traditionally, mobility is considered to include three components, coulomb, phonon and surface roughness scattering. As observed in Fig. 3, the interface state generation combined with the mobility model shows good agreement with the measured data. Therefore, among the mobility components, coulomb scattering due to the interface states is revealed as being mainly responsible for the mobility degradation.
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Figure3.
(Color online) Time evolution of mobility degradation reproduced based on the universal NBTI model.
3.
Coupling effect in aging simulation considering mobility degradation
Among the large number of reliability models, most of them are based on single experimental conditions. The purpose of single experimental conditions is to ignore the effect of other reliability problems. However, in actual circuits, various reliability problems will exist at the same time and interact with each other. For example, in the process of establishing and debugging the NBTI model, the source and the drain of device should be grounded. However, source voltage and drain voltage do exist in actual circuits. The effect of NBTI will be weakened, but the hot carrier effect is enhanced. Both effects are mutually restricted. Obviously, it is necessary to consider all kinds of reliability effects in circuit simulation and to consider the mutual interaction between them.
According to a study, the reliability issues in MOSFETs are not isolated, but are associated by shared parametrical shift (e.g. electrical field, drain current or temperature). The threshold voltage shift and mobility degradation play important role in interconnecting various reliability issues. In the circuit simulation, through these parameters, we can establish a simulation method which can consider the mutual influence of reliability effects.
Based on this idea, we can consider the mutual interaction among reliability effects, e.g. NBTI effect coupling with HCI and SHE. In the circuit aging simulation, the mobility degradation may significantly impact the aging calculation by changing the drain current, threshold voltage and temperature, especially in the long term region. In the circuit aging simulation, ignoring coupling effect results in mistaken assessment. Furthermore, threshold voltage and temperature are deduced by physical rules. Thus, the coupling simulation method could be of use in a larger range of voltages and temperatures, for example, in the temperature range of the device during normal operation and in the voltage range between operating voltage and critical breakdown voltage.
3.1
NBTI and HCI coupling simulation
With the co-effects of the gate and drain stress bias on p-MOSFETs, the channel carriers are accelerated and injected into the gate oxide[18, 19]. The threshold voltage shift (ΔVth_hci) is caused by the interface state generation close to the drain. Traditionally, the lucky electron model is used to describe ΔVth_hci as a function of the drain current (Ids) and maximum lateral channel electrical field (Emax).
$Delta {V_{ m th_hci}} = {C_1}{left[ {frac{{{I_{ m d}}}}{W} exp ( - frac{{{varphi _{ m it}}}}{{qlambda {E_{max }}}}) t} ight]^n},$  | (10) |
where φit is the critical energy for trap generation, and Emax is the effective and maximum field in oxide and channel. The mobility degradation leads to the reduction of Ids, which cannot be ignored since other reliability problems are also sensitive to the drain current. The HCI enhanced NBTI effect can be described by adding the threshold voltage shifts induced by both mechanisms[20].
Fig. 4 shows the flow chart of the coupling effect between NBTI and HCI. The NBTI results in carrier mobility degradation due to the interface state generation. Furthermore, the carrier mobility degradation results in reduction of Ids. The reduction of Ids does not contribute to NBTI but shows an evident influence on HCI. On the other hand, the carriers injected into gate oxide due to HCI also result in additional degradation of mobility and threshold voltage.
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Figure4.
The flowchart of NBTI and HCI coupling simulation. NBTI induced mobility degradation impacts the ON-state current, thus the HCI degradation.
The simulation results of NBTI-HCI coupling is shown as Fig. 5. A dynamic bias waveform is applied on the gate and Ids is monitored for the ΔVth calculation. As shown in Fig. 5(b), if the mobility degradation induced by NBTI is ignored, Ids will be overestimated. As shown in Fig. 5(c), the mobility degradation does not directly affect the threshold voltage degradation induced by NBTI. However, as shown in Fig. 5(d), without considering the mobility degradation, the final ΔVth will probably be overestimated, because mobility degradation affects the current Ids and the HCI is closely related to the current. Considering mobility degradation can effectively avoid overestimation.
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Figure5.
(Color online) NBTI induced mobility degradation reduces ON-state current and ΔVth in dynamic NBTI and HCI coupling simulation. (a) Stress waveform. (b) ON-state current. (c) ΔVth induced by NBTI only. (d) ΔVth induced by coupling of NBTI and HCI.
3.2
NBTI and SHE coupling simulation
In SOI MOSFET, because of the existence of a buried oxide layer with low thermal conductivity, the self-heating effect of the device becomes serious. SHE can easily result in the temperature of device increasing by 60 °C because the SiO2 substrate and short channel cannot dissipate heat in time[9]. SHE is power dissipation resulting from electricity. Hence, the carrier mobility degradation and threshold voltage degradation results in a reduction of Ids. In this situation, the effect of SHE will be reduced. At the same time, the temperature is closely related to the carrier mobility degradation and threshold voltage degradation. If the NBTI and the SHE are isolated in circuit simulation, the degree of aging of the circuit would obviously be miscalculated.
As shown in Fig. 6, the self-heating effect is modeled with the thermal network. In detail, the self-heating effects model is expressed as[21]
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Figure6.
The thermal network modeling the self-heating effect.
$frac{{Delta T}}{{R_{ m th}}} + {C_{ m th}} frac{{{ m d}Delta T}}{{{ m d}t}} = {P_{ m th}},$  | (11) |
where Cth is thermal capacitance, Rth is thermal resistance and Pth is thermal dissipation.
The flowchart of NBTI and SHE coupling simulation is shown in Fig. 7. Firstly, the initial flat-band voltage Vfb,i and initial temperature Ti are obtained from the initial network table. Next, the oxide electric field Eox,i is calculated through the formula
m ox,i}} = ({V_{
m g}} - {V_{{
m fb,i}}} - {varPhi _{
m s,i}})/{t_{ox}}$
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Figure7.
The flowchart of NBTI and SHE coupling simulation.
The impacts of mobility degradation on the threshold voltage shift (ΔVth) and temperature increment (ΔT) are shown in Fig. 8. There is an approximate 37% overestimation of ΔVth and a 29 K overestimation of ΔT when mobility degradation is not taken into account. The simulation results indicate that besides considering coupling effects, an accurate reliability model, such as the NBTI induced mobility degradation model, is also required for device and circuit level aging prediction.
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Figure8.
(Color online) Coupling simulation of NBTI and SHE. Lacing of NBTI induced mobility degradation induces (a) 37% overestimation of ΔVth and (b) 29 K overestimation of temperature increment.
As shown in Fig. 9, if mobility degradation is ignored, the degree of current degradation will be underestimated. Furthermore the degree of current degradation becomes more serious under conditions of rising drain voltage or gate voltage. Hence, taking carrier mobility degradation into consideration can avoid overestimation of circuit performance in circuit simulation.
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Figure9.
(Color online) Coupling simulation of NBTI and SHE in drain voltage and gate voltage.
4.
Conclusion
Besides focusing on the threshold voltage shift, NBTI induced mobility degradation also plays an important role in impacting the ON-state current. The mobility degradation is dominated by coulomb scattering due to interface state generation and is modeled on the basis of the universal NBTI model. The mobility model is then implemented in the circuit aging simulation module. It is revealed that, as one of the key parameters shared by various reliability mechanisms, the mobility degradation is indispensable in evaluating coupling effects (e.g. NBTI-HCI and NBTI-SHE) to avoid overestimation in circuit aging simulation. This aging simulation method, which considers shared parameters, can be extended to various interdependent reliability aging simulation problems to avoid errors in simulation.
