Essential Aspects of Negative Bias Temperature Instability (NBTI) Ahmad Ehteshamul Islama, Souvik Mahapatrab, Shweta Deorab, Vrajesh D. Mahetab, and Muhammad Ashraful Alamc a

Department of Materials Science and Engineering, University of Illinois at UrbanaChampaign, IL 61801, USA b Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India c Department of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907. USA Email: a [email protected], b [email protected], c [email protected] (Invited Paper)

We develop a comprehensive theoretical framework for explaining the key and characteristic experimental signatures of NBTI. The framework is based on an uncorrelated dynamics of interfacedefect creation/annihilation described by Reaction-Diffusion (R-D) theory and hole trapping/detrapping into/out-of oxide defects based on a generalized Shockley-Read-Hall model. The proposed theory can consistently explain the long-term stress-phase power-law time exponent, stress/relaxation-phase temperature dependence, characteristic feature of duty-cycle dependence, and universal feature of frequency independence - measured in DC and AC stress conditions over a wide variety of transistors. Thus, we confirm the general validity of R-D theory in explaining the universal features (irrespective of dielectric material) of both DC and AC NBTI. The non-universal features of NBTI have correlation with the amount of oxide defects within the dielectric and do not affect AC NBTI measurements at lower duty cycle. Decomposition of these (uncorrelated) universal and non-universal components is, therefore, essential before comparison with any theory.

1. Background Negative Bias Temperature Instability (NBTI) indicates a temperature accelerated degradation in MOS transistors when it is stressed with a negative gate voltage, i.e., with an oxide electric field that is directed from the channel towards the gate of a MOS transistor. In normal CMOS (which is the basis for today’s microprocessor) operation, only the PMOS transistors are subjected to such negative oxide electric field. Therefore, since its introduction in microprocessor in early 1970s [1, 2], NBTI-induced performance degradation in PMOS transistors (e.g., increase in threshold voltage, reduction in current drivability) has always been a concern in CMOS technology. Recent use of thin gate dielectric (a consequent increase in oxide electric field) and use of high-κ (oxynitride and Hf-based) dielectric materials have further enhanced NBTI degradation. As such, NBTI

is considered as one of the major reliability concerns in current CMOS technology [3-6]. This industry-wide reliability concern has encouraged numerous efforts (see reference [7] for a chronology of the number of papers being published in this area) to understand the essential aspects of NBTI degradation by using different types of transistor-based measurements and then develop appropriate theory to explain the experiments. However, there is a perception that a definitive interpretation of this PMOS-specific phenomenon remains elusive. Most of the literature on NBTI is focused on explaining isolated experimental signatures of NBTI degradation and recovery and, in the process, may sometimes overlook the fact that the proffered explanations have not been consistent with other characteristic features of NBTI. Such fragmented modeling efforts have made NBTI literature extremely confusing to the general audience. For capturing the experimental features, NBTI dynamics has sometimes been attributed to interface defect NIT [2, 8, 9], or to hole trapping into pre-existing oxide defects NHT [10-14], or to both NIT and NHT [15-22]. Recent introduction of Spin Dependent Recombination (SDR) experiment [23] have helped shed light on the nature of NBTI-specific defects, but the work is still in flux and have not been conclusive enough to settle the discussion. For example, even though many groups have historically attributed NBTI-related NIT generation to Pb-centers (by ESR and SDR experiments), recent SDR experiment [23] indicates that Pb-centers are apparently absent in transistors with high-κ oxynitride gate dielectric. These experiments have motivated several articles to explain NBTI from the point-of-view of bulk oxide defects, where NIT is either indirectly created via hole trapping into generated oxide defects NOT [22], or NIT is absent altogether [12, 13]. Unfortunately, however, the new theories have not always attempted to explain other well known features of NBTI experiments, i.e., (i) stress-phase power-law time exponent, (ii) stress and relaxationphase temperature dependence, (iii) duty cycle dependence, and (iv) universal frequency independence, and therefore the generic validity of such bulk-oxide defect oriented approaches remain questionable. In this manuscript, we develop a robust theoretical framework for explaining these key characteristic experimental features of NBTI, measured in DC and AC stress conditions for a wide variety of transistor technologies. The framework is based on an uncorrelated dynamics for Reaction-Diffusion (R-D) theory based interface defect creation/annihilation and Shockley-Read-Hall theory based hole trapping/detrapping into/out-of oxide defects. Our analysis suggests the necessity of decomposing NBTI measurements into a universal slow component associated with interface defect generation and a non-universal fast component associated with charge trapping/detrapping into/out-of oxide defects. The universal component can be consistently explained using R-D theory based interface defect dynamics and is directly relevant for product qualification. On the other hand, the non-universal component depends on the amount of oxide defect within the dielectric (and hence associated with the gate stack fabrication technology). Hence, this non-universal component is sample dependent and must be accounted only for high-fidelity experimental fitting in short-time scales. Therefore, one should decompose these two uncorrelated (universal and nonuniversal) components and then explain their respective features using respective theories. Any attempt to violate such decomposition will lead to proliferation of unphysical model parameters to capture the experimental trends.

1.1 Interface Defect in NBTI and Relevance of R-D Theory Historically, interface defects have always been a technology challenge for semiconductor industry. Unlike oxide defects, interface defects cannot be removed or reduced through purification (or gettering). At the silicon-dielectric interface of a MOS transistor, interface defects (dangling Si- bonds) are terminated or passivated (Si- + H Si-H) by using hydrogen compounds like silane (SiH4). Efficacy of such passivation techniques using atomic hydrogen (H) has been extensively studied since 1970s [1, 2429]. However, H-passivation of dangling bonds only provides a ‘time-zero’ (pre-use) solution from interface defects. After a period of transistor operation, mostly in the PMOS configuration, the interfacial Si-H bond starts to dissociate or depassivate (Si-H Si- + H) at normal operating condition in the presence of cold holes near the interface and reforms the dangling Si- bonds. Formation of these interface defects due to Si-H depassivation has been observed by using ESR and SDR experiments, especially on the Si/SiO2 interface [30, 31]. The same has also been confirmed using capacitance-voltage measurement and mobility degradation experiments [17, 32, 33]. Theory of interface defect formation has evolved around the (1) modeling of Si-H bond dissociation and (2) subsequent handling of the resultant hydrogen species. Si-H bond dissociation is either considered to be – (1a) cold-hole assisted [8, 9, 15, 34-36], (1b) dopant-activated [37, 38], or (1c) oxide-defect induced [22]. Resultant hydrogen species after Si-H bond dissociation is either considered to be – (2a) trapped within the oxide [10, 22, 33], or (2b) diffuse within the oxide/gate region [8, 15]. Mechanisms (1a) and (2b), or (1b) and (2b), constitute the basis of the so-called Reaction-Diffusion (R-D) framework. Note that the environment of Si-H bond (i.e., whether the resultant Si- bond is a Pb-center or a K-center [23]) or details of the dissociation process does not affect the characteristic predictions (i.e., power-law time exponent, frequency/duty cycle dependence, temperature dependence) of the R-D theory. Originally proposed in 1977 by Jeppson et al. [2] (and later refined by Alam et al. in early 2000s [8, 15, 34, 39, 40]) to interpret fractional kinetics of NBTI degradation, the implications of the R-D model has been explored in hundreds of papers through various generations of CMOS technology. R-D model has a major advantage over its counterparts in explaining interface defect formation, i.e., it provides a parameter-free interpretation of all the four broad universal features of NBTI degradation after nonuniversal and technology-specific hole trapping into oxide defect component is subtracted out [16, 41]. It can predict a long term time dependence of t1/6, long-term Arrheniusactivated temperature dependence, the frequency independence of degradation, and the characteristic shape of duty cycle dependence – regardless of the technology or operating condition. All these have made R-D theory (as discussed in section 3) the starting point for all the discussions of NBTI degradation. 1.2 Hole Trapping into Oxide Defects in NBTI Dynamics of trapping into oxide defects have been studied since 1970s’ [42-44], when Metal-Nitride-Oxide-Semiconductor (MNOS) structure became a popular memory element, because of its simple structure and nonvolatile nature. Detrapping of carriers from pre-existing charged defects in the nitride layers was identified as the main mechanism for limiting the retention time in these memories. Moreover, discharge time

was observed to increase with the increase in oxide thickness of the oxide layer, which proves that tunneling of carriers from charged defects within the nitride layer is gradually reduced with the increase in oxide thickness [42]. Similarly, trapping/detrapping of holes and electrons in/out-of pre-existing oxide defects has also been considered as a source of threshold voltage instabilities in MOS transistors during the same period [45-47]. Now-adays, the incorporation of high-κ materials within the dielectric of MOS structures have caused significant concerns from these hole trapping/detrapping issues during NBTI stress [10, 20, 48-56]. In addition, hole trapping/detrapping phenomena can also give rise to the observation of Random Telegraph Noise (RTN) and 1/f noise in modern transistors having high-κ gate dielectric [57-61]. Hole trapping is generally modeled by considering tunneling of channel carriers into oxide defects. These oxide defects can either be pre-existing [15, 18] or generated by electrical stress [20, 22, 62]. Pre-existing oxide defect is mainly an issue in overcoordinated high-κ (oxynitride or Hf-based) materials [63, 64]. Higher coordination number (i.e., number of nearest neighbors for a particular atom within the material) in high-κ materials makes it difficult to satisfy all the chemical bonding within the amorphous network. Thus, transistors with high-κ dielectric are prone to high density of pre-existing defects and generally suffer from BTI effects due to hole trapping (will be designated using NHT from now on). On the other hand, oxide defect generation (will be designated using NOT from now on) has always been a problem within the reliability community (irrespective of dielectric material) [62, 65-67] that leads to the well-known phenomena, called Time-Dependent Dielectric Breakdown or TDDB. These oxide defect formation has an empirically extracted to have universal nature [68] and is often characterized with a time dependence of t1/3 [69]. We will explain the theory of NHT and NOT in sections 4 and 5, respectively and therefore, show its relevance in explaining experimental features of NBTI that is presented in section 2.

2. Broad Empirical Features of NBTI 2.1 Summary of NBTI Measurements Being one of the major reliability concerns, NBTI has been studied by a large group of researchers all over the world. This breadth of experimental data provides us an opportunity to collect measurements from published reports across industry and academia (see Figure 1) and compare with our own measurements. The comparison is summarized as follows: n

Figure 1a shows the power-law time exponent (when ∆VT is expressed as At ) measured at long stress time (tSTS) in the industrial grade devices. Here, A is a voltage and temperature-dependent constant (see equation (13) for details on this voltage and temperature dependence). In spite of disparate sources, all devices unequivocally show an exponent of n ~1/6, independent of voltage and temperature. Figure 1b shows the stress-phase ultra-fast on-the-fly (UF-OTF) measurements (where time-zero delay [70] t0 is 1µs and ∆VT is estimated ignoring the mobility correction [71], i.e., ∆VT ~ ∆ID/ID0(VG-VT0)) at different temperature for different

oxynitride process splits. The measurements reveal the existence of a short-time, temperature independent fast component (that saturates within ~ms) in transistors having high %N for optimized (Type-A) plasma oxynitride dielectric. Decrease in %N within the dielectric reduces the contribution from this fast component and shows the existence of temperature activated slow component, even at short tSTS. Moreover, at relatively long tSTS > 1 s, all devices show some degree of temperature activation (less for high %N). Note that in un-optimized oxynitride process split (Type-B), the temperature-independence at short tSTS is even observed for low %N. The readers may wish to review reference [72] for an analysis of nitridation process flow on NBTI characteristics. Figure 1c shows NBTI relaxation measurements across different type-A oxynitride process splits (ultra-fast VT or UFV measurement of reference [50] is also shown for comparison). The important point to note here is the initiation of ~5% NBTI relaxation (tREC,start; where ∆VT(tREC)/ ∆VT(tSTS) ~ 95%) and the timedependence of NBTI relaxation. Though several studies [22, 50, 73, 74] on NBTI have reported the universality of log-t relaxation with tREC,start of ~ µs, our UFOTF measurements [16, 75, 76] demonstrate that tREC,start and the timedependence of NBTI relaxation depend on %N of the oxynitride high-κ gate dielectric, as well as on the difference between stress and recovery voltages (VSTS−VREC). In general, tREC,start is larger (~ ms) for low %N and smaller (VSTS VREC), very clearly indicating the non-universal nature of NBTI recovery. In addition, NBTI relaxation data also shows temperature independence at short tREC for high %N oxynitride transistors (Figure 2a,b). And similar to the stress-phase measurements of Figure 1b, the temperature independence at short tREC disappears with the reduction in %N. Figure 1d,e shows a comprehensive summary of duty cycle and frequency dependent NBTI measurement, obtained from a broad range of published reports across industry and academia. When normalized to DC, the duty cycle dependent measurement shows large spread in AC/DC ratio (= ∆VT(AC)/ ∆VT(DC)), although the frequency independence in AC/DC ratio at 50% duty cycle is generally observed (some older datasets show slight drop at higher frequency and most researchers now consider this droop to be a measurement artifact [77]).

0.2

TI data, IEDM'06 tSTR: 1-100Ks O O -25 C 45 C O O 105 C 145 C TSMC data, IRPS'05 O tSTR>1000hr, T=125 C 1/6

-1

∆ VT [Volts]

Value of n

0.3

10

EOX~8MV/cm O 55 C O 125 C 42.5% N

Type-A

-2

10

0.1 1.2

16.7% N

-3

(a) 1.8 2.4 |V [Volts] |VG,STS| |[Volts] STS

Type-B: 5.8% N

3x10 -6 10

-4

10

-2

0

10 10 tSTS [sec]

(b) 2

10

4

10

Open: ~23 %N Crossed: ~35 %N

∆ VT(tREC)/∆ VT(tSTS)

1.00 0.75

tREC,start

0.50

VSTS / VREC (V) -2.3 / -1.8 UF-OTF [24] -2.3 / -1.3 UF-OTF [24] -2.0 / 0.0 UFV

0.25 0.00 -12 10

-9

10

-6

-3

10

10

(c) 0

10

tREC/tSTS

Toshiba ST NS; Purdue/IIT: 10 %N

Infineon TUV

0.8 0.6 0.4 0.2

(d)

0.0 0

50 100 0 Duty cycle [%]

(e) 2

5

8

10 10 10 Frequency [Hz]

Figure 1. (a) Long-term power-law time exponent n (where, ∆VT ~ tn) for NBTI, collected from TSMC and TI measurements (taken from reference [78] and [40], respectively), indicates n ~ 1/6. Note that Freescale [79] and Infineon [80] data also shows similar time exponent. (b) Temperature dependent stress-phase NBTI measurement across different process split of oxynitride gate dielectric (type-A has optimum nitridation; and type-B has non-optimized nitridation) (c) Initiation of NBTI relaxation (tREC,start) varies with VREC and %N of the oxynitride high-κ dielectric (here, the UF-OTF are our measurements and ultra-fast VT or UFV measurement is taken from reference [50]). (d) AC/DC ratio (when ∆VT is measured at the end of AC cycles) vs. duty cycle, and (e) AC/DC ratio vs. frequency (at 50% duty cycle) plots for different technologies indicate wide spread in measured data. Measurements of (d-e) are taken from the following references: Toshiba [81], ST [21], IMEC [77], NUS [82], Infineon [80], TUV [11].

2.2 Modeling Challenge The broad scatter in NBTI measurements (as presented in Figure 1) makes NBTI modeling a considerable challenge. In an effort to address this challenge, NBTI researchers have taken widely different strategies: Some articles have used a wide distribution of capture and emission time constants (from 10-9-1014 s) for fitting each of the stress and relaxation phase measurements with independent distribution [80, 83]. Unfortunately, this approach results in proliferation of fitting parameters that cannot always be physically justified. Other articles have considered wide distribution of defects within the dielectric [12, 13] to capture the long-term nature of the time exponent. In addition, interaction and transfer of chemical species among finite numbers of energy wells [11, 22] is recently used to capture part of the experimental features of Figure 1. However, none of these approaches are comprehensive enough to consistently interpret the four characteristic features of NBTI degradation without using unphysical parameters, as summarized in Figure 1. In an attempt to explain the broad experimental features of NBTI, we ask the following questions: 1) Why is the long-term time exponent always ~1/6, regardless of the stress voltage or stress temperature, as shown in Figure 1a? Note that these results are usually obtained with small measurement delay (order of seconds), which however has insignificant effect when the stress time is very long. Obviously, the use of excessive measurement delay may lead to higher time exponent even at long stress time [84]. 2) Why does the early part of stress and relaxation experiments so sensitive to process details of dielectric material, as shown in Figure 1b,c? Why is process dependent part insensitive to temperature, especially at higher %N, as shown in Figure 1b and Figure 2a,b? And, in the same context, why tREC,start in Figure 1c has process dependence? 3) Why is there a sudden drop in ∆VT(AC)/ ∆VT(DC) in the range of 80-100% duty cycle (Figure 1d)? Why is the shape of duty cycle vs. ∆VT(AC)/ ∆VT(DC) universal (this is more evident, when Figure 1d is normalized with respect to 50% duty cycle value, as shown in Figure 2c) in the lower duty cycle regime? 4) Why is the measured NBTI at 50% duty cycle always frequency independent, irrespective of transistor technology (as shown in Figure 1e)? In this manuscript, we explain NBTI in a broader context by answering the aforementioned four questions, rather than focusing on a smaller subset. Therefore, we decompose measured ∆VT into three uncorrelated components: A. The first and major part of NBTI-induced ∆VT comes from the Reaction-Diffusion theory based interface defect (NIT) generation and relaxation. We will show how this component can explain: i) the universal observation of 1/6 power-law time exponent, ii) the slow and temperature dependent part of NBTI stress and relaxation, iii) the universal part of duty cycle dependence upto ~50% duty cycle, and iv) the universal observation of frequency independence.

Recovered ∆VT [Volts]

B. The second component of NBTI comes from hole trapping into pre-existing oxide defects (NHT). This component can explain: i) the fast and temperature independent part of NBTI stress and relaxation, ii) the sharply decreasing part of duty cycle dependence above ~50% duty cycle, and iii) the magnitude of ∆VT(AC)/ ∆VT(DC) at a particular duty cycle. C. The last component of NBTI comes from hole trapping into generated oxide defects (NOT). This, in addition to NHT, can explain the disparity of tREC,start in different nitrided transistors.

tSTS = 103 sec; VSTS = -2.3V 0.00

0.00

T

(a) 23%N

-0.04 -7 10

-4

10

-0.04

55

0C

85

0C

-1

Weak T dep.

-0.08

(b) 42.5%N

125 0C

-0.12 -7 10

2

10

Strong T dep.

10

-4

10

∆VT(AC)/∆VT(AC,d=50%)

tREC [sec] 3

IMEC ST TUV

2

-1

10 tREC [sec]

2

10

Infenion Ours Purdue/IIT

1 Line: R-D Theory

(c)

0 0

50

100

[%] Duty cycle, d (%)

Figure 2: (a-b) Similar to NBTI stress phase (Figure 1b), early relaxation phase is temperature independent. Such temperature independent early relaxation phase is cleanly observed for high %N transistors. (c) The universal shape of duty cycle dependence (upto d ~ 80%) can be nicely captured by R-D theory. Here, the universal shape of duty cycle is obtained by scaling the duty cycle dependent NBTI measurement with respect to the 50% duty cycle value (see section 7 for the justification). In other words, we express ∆VT using – ∆VT ( t ) = ∆VIT ( t ) + ∆VHT ( t ) + ∆VOT ( t ) q∆N IT (t ) ∫0 ox ∫E x ρ HT ( x , E , t ) dE dx ∫0 ox ∫E x ρOT ( x , E , t ) dE dx =α + + . C ox C oxTox C oxTox T

T

(1)

Here, ∆VIT, ∆VHT, and ∆VOT refer to the contributions to ∆VT from NIT, NHT, and NOT components, respectively; Cox is the oxide capacitance; Tox is the oxide thickness, α accounts for the fraction of donor type [85] NIT above the substrate Fermi-level that is contributing to NBTI; ρHT(x,E,t) represents trapped holes into the pre-existing oxide defects at location x (measured into the oxide from the poly/oxide interface) and at energy E at time t; and ρOT(x,E,t) represents trapped holes at generated oxide defects. 3. Theory of Interface Defect (NIT) In this section, we summarize the main features of R-D theory, considering both atomic (H) and molecular (H2) diffusion. Our goal is to show how the theory provides a parameter-free prediction of most of the experimental features of NBTI. R-D theory with H-H2 diffusion considers dissociation of Si-H bond and subsequent diffusion of hydrogen species, as governed by the following equations [15, 86]:

Si sub.

dN IT = k F ( N 0 − N IT ) − k R N IT N H(0) , dt (0) 2 δ dN H dN H(0) dN IT = DH + − δ k H N H(0) + δ k H 2 N H(0)2 , 2 dt dx dt (0) (0) 2 δ dN H 2 dN H 2 δ δ = DH 2 + k H N H(0) − k H 2 N H(0)2 , 2 dt dx 2 2 2 dN H d NH = DH − kH N H 2 + kH 2 N H 2 , 2 dt dx 2 dN H 2 d NH 2 1 1 = DH 2 + kH N H 2 − kH 2 N H 2 . 2 dt dx 2 2

Si

x

Si

x

Si

H H

(3) (4) (5) (6)

H2

H2 H

H

(2)

H2 H

H2

NH, NH2

~ 0

Dt x

Figure 3: Schematic of Si-H bond dissociation and consequent hydrogen diffusion, as considered in R-D theory based NIT dynamics. Equation (2) represents passivation/de-passivation effects of Si-H bond, where kF, kR, N0, NIT, NH(0) are defined as Si-H bond-breaking rate, Si-H bond-annealing rate, initial bond density available before stress, interface defect density, and hydrogen density at the Si/dielectric interface, respectively. Equations (3) and (4) correspond to the conservation

of fluxes of diffusing hydrogen species (H and H2) near the interface (along the x axis), whereas equations (5) and (6) describe diffusion (along the x axis) of H and H2. k H N H 2 and k H 2 N H 2 terms in equations (3)-(6) incorporate the H-H2 conversion within the generalized R-D framework. Among the symbols used in equations (3)-(6), kH and kH2 represent generation and dissociation rates of H2; DH and DH2 represent diffusion coefficients for H and H2; NH and NH2 represent the concentration of atomic and molecular hydrogen; δ represents the interfacial thickness (~1-2 Å). All parameters are greater than zero for the stress phase. In particular, kF has the following dependence of equation (7) that serves a physical way of explaining the oxide electric field Eox dependence of interface defect generation [15, 40, 87, 88]:

k F ~ ph * N 0 PT *exp(γ T Eox ) *exp ( aEox / kT ) ,

(7)

where ph is the hole concentration within the inversion layer, PT ~ exp(-√moxφbh) is the field-independent pre-factor for hole tunneling probability (mox: oxide effective mass and φbh: barrier height for hole tunneling), exp(γTEox) is the field-dependent factor for hole tunneling with field acceleration γT, and exp(aEox/kBT) is field-assisted Si-H bond dissociation enhancement factor (a: effective dipole moment and kBT: thermal voltage). Thus, field acceleration of NIT can be expressed as –

γ IT = γ T + a / k BT .

(8)

As shown in [15, 40], equation (8) explains the temperature dependence of NIT’s field acceleration (hence provides an experimental way of extracting a and γT). In addition, equation (7) can be used to explain the nitridation process dependence [87] and strain dependence [88] of NIT in NBTI measurements. Now, let us derive the key results of R-D model and show how it anticipates the key experimental features of NBTI. 3.1 Power-law time exponent of n ~ 1/6: Assuming N0 >> NIT and dNIT/dt ~ NIT/t, equation (2) simplifies to,

N H(0) =

kF N 0 − N IT t . kR N IT

(9)

Moreover, the numerical solutions indicate that for continuous NBTI stress, dNH(0)/dt and diffusion of H is negligible at all stress time [15], so that H2 diffusion part in equation (3) reduces to,

N IT = δ kH N H(0)2 − δ kH 2 N H(0)2 , t and the conservation of hydrogen species within the system suggests –

(10)

N IT ≈ N H(0)2 6 DH 2t .

(11)

Equation (11) requires that the extent of diffusion profile is larger than DH t , commonly used for approximating the complementary error function solution of diffusion equation [8, 34, 89]; which is more consistent with the use of 16 DH t / π in [90, 91]. Now, by eliminating NH(0) and NH2(0) from equations (9)-(11), we have –

N IT δ kH ( kF N 0 − N IT t ) δ kH 2 N IT − + = 0. t kR 2 N IT 2 6 DH 2t 2

(12)

Equation (12) is the (implicit) analytical solution of H-H2 R-D model, presented in equations (2)-(6), which compares very favorably with the detailed numerical simulation [15]. At long stress time (i.e., in the H2 diffusion limited regime), equation (12) simplifies to (when NIT/t becomes negligible) – 1/3

N IT

k = H kH 2

kF N0 kR

2/3

( 6 DH 2t )

1/6

≡ At n=1/6 .

(13)

Therefore, R-D theory anticipates the long-term time exponent of n ~ 1/6. In addition, R-D theory also establishes the fact that n should not depend on stress voltage and temperature. Both of these observations are consistent with the broad range of NBTI measurements (Figure 1a). 3.2 Long-term Temperature Dependence: Using appropriate activation energies for kF, kR and DH2 in equation (13) and assuming H-H2 conversion process has similar activation for forward and reverse reactions [92], the overall activation energy for ∆VIT = qNIT/Cox can be written as [15],

E A, IT ≡ nE A, H 2 + 2 3 ( E A, F - E A, R - aEox ) .

(14)

where EA,F, EA,R, EA,H2 are activation energies for kF, kR and DH2, respectively. Experimentally, one can estimate EA,IT ~ 0.1 eV by measuring NBTI in transistors having dominant interface defect generation (i.e., ∆VT ~ ∆VIT) [18, 32, 40, 50, 85, 88, 93, 94]. We have also shown that the variation of EA,IT due to aEox term comes within the error margin of activation energy estimation procedure [87]. This measured value of EA,IT and reported magnitude of EA,H2 ~ 0.6 eV [95] suggests that EA,F ~ EA,R. Thus, activation energy of molecular hydrogen diffusion governs the long-term temperature dependence of NBTI measurements. 3.3 Duty-cycle and Frequency Dependence: In order to obtain the duty cycle and frequency dependence of NIT, we derive the amount of NIT under AC NBTI stress condition. Here, we assume that the NBTI stress is

applied in a transistor for k (>> 1) cycles with 2k stress/relaxation steps, defined by the duty cycle d = tSTS/T; where T = tSTS + tREC is the period of the signal. We also define that the degradation at the end of k-cycles (with respect to degradation over a single stress cycle) using R2k ≡ NIT(kT)/NIT(tSTS) and R2k-1 ≡ NIT((k-1)T + tSTS)/NIT(tSTS) with R1 = 1, by definition. Following the analysis in Refs. [89] and [96], one can show that –

( R2k −1 )

1/ n

≈

1/ n k −1 k + ( R1 ) ≃ ≡ pk , 1 + ξ (1 − d ) 1 + ξ (1 − d )

(

(15)

)

or equivalently, R2k-1 ≈ R2k ≈ (pk)n, where p = 1/ 1 + ξ (1 − d ) . If two transistors are stressed at two different frequencies (f1 = 1/T1 and f2 = 1/T2) at same duty cycle, then for long (but same) integrated stress times (i.e., k1T1 = k2T2 or, k1tSTS,1 = k2tSTS,2), the ratio of net degradation for the two transistors is given by: n N IT , f1 ( k1T1 ) Rk1 N IT ( tSTS ,1 ) k1 tSTS ,1 = ≈ = 1; N IT , f1 ( k2T2 ) Rk 2 N IT ( tSTS ,2 ) k2 tSTS ,2 n

(16)

which is frequency-independent (as observed in Figure 1e). And similarly, the ratio of AC NBTI degradation at a given frequency for total duration of TT = k1T1 compared to the DC NBTI degradation for the same period of time is given by –

N IT , f1 ( k1T1 ) ( pk1 ) A (tSTS ,1 ) d ; = = n N IT , DC (TT = k1T1 ) A ( k1T1 ) 1 + ξ (1 − d ) n

n

n

(17)

which gives the characteristic duty cycle dependence of NBTI degradation (line in Figure 2c). We defer the discussion of the scaling algorithm till section 6.3, which is used in Figure 2c for the experimental data scaling. Thus, R-D theory can explain the universal observation of frequency independence (Figure 1d) and the characteristics trend of duty cycle dependence upto d ~ 80% (Figure 2c) for all NBTI experiments, measured in wide variety of process splits.

4. Theory of Hole Trapping into Pre-existing Oxide Defects (NHT) Let us now return to the second component of NBTI in equation (1) related to the charging of pre-existing oxide defects. Pre-existing defects are commonly observed in almost all high-κ dielectrics (like oxynitrides and Hf-based compounds), which is an inherent feature of such over-coordinated high-κ materials. Trapping into these preexisting defects leads to ∆VHT, independent of the generation of NIT or NOT. Here, the thinness of modern dielectric film (~ few nm) requires us to consider the effect of detrapping towards the transistor gate [15]. In addition, typical defect density (~ 1018-1019 cm-3, i.e., 1 trap in 100-1000 nm3) ensures that trap-to-trap transport is implausible.

Therefore, we only need to consider the processes involving single hop to and from the traps, i.e., dfT ( x, t ) dt

∞

= ∫ dET σ HT vth phTS →T (1- fT ) − nS TT → S fT − nGTT →G fT ,

(18)

−∞

where ph is the inversion layer hole density; vth is the thermal velocity; nS, nG are the concentration of detrapping states at substrate and poly-Si respectively; T’s are tunneling probabilities between substrate (S), defect/trap (T), and gate (G) of the MOS transistor; and σHT is the capture cross-section for the hole trapping/detrapping process. The first term in equation (18) represents hole trapping into pre-existing oxide defects from the substrate, whereas the second and third terms represent hole detrapping out of oxide defects towards substrate and gate, respectively. These hole capture (during trapping process) and release (during detrapping process) events can be elastic or inelastic, as discussed in detail in reference [97]. Stochastic nature of such hole capture and release events gives rise to random telegraph noise, as studied in references [14, 57-61]. In the following discussion, we focus on the implication of elastic/inelastic hole trapping into thin dielectric and study its implication in long-term NBTI degradation. The readers may wish to review reference [14] for understanding the short-stress features of hole trapping. Solution of equation (18) suggests (see reference [97] for details) the followings:

Time dependence: The role of detrapping towards the poly-gate – while negligible in thick films (as for the case in [42, 98]) – are fundamentally important in the trapping dynamics of thin films. In ultra-thin dielectrics of current CMOS technology, detrapping process limits the possibility of hole trapping in sites located near the gate. As a result, threshold voltage shift due to hole trapping into preexisting defects should saturate within orders of milliseconds, where the saturation time depends on the dielectric quality (i.e., nitridation) and thickness. As such, the corresponding threshold voltage shift can be approximated as –

β ∆VHT = AHT 1 − exp ( −t / τ ) ,

(19)

where AHT, τ and β are trapping parameters that depends on the dielectric quality and dielectric thickness. In general, τ and β are larger for larger dielectric thickness and AHT is larger for higher %N within the dielectric (i.e., for the dielectric having more NHT).

Voltage dependence: Voltage or field dependency of hole trapping arises from the relative position of the quasi-Fermi level EFS within the dielectric as a function of oxide electric field Eox. When Eox is reduced, more trapping sites below EFS are filled with electrons (see the change of trapping sites fT values from Figure 4b to Figure 4c), with corresponding reduction ∆VHT. As a result, hole trapping process shows significant voltage dependency.

Temperature dependence: Since tunneling is a temperature independent process, temperature dependency of hole trapping mainly depends on the temperature dependence of capture cross-section (σHT) and thermal velocity (vth). Recently, many hole trapping models [22, 59] consider structural relaxation as a part of the hole

trapping process. Structural relaxation causes temperature activation (which depends on the energy barrier that is required to overcome for achieving structural relaxation) in the trapping process and can effectively be incorporated within the σHT parameter of equation (18). As the hole trapping process is fast for modern transistors having thin gate dielectric, we can attribute the ultra-fast component of NBTI stress phase (in Figure 1b) to ∆VHT. The increase of this ultra-fast component with the increase in nitrogen within the silicon oxynitride (SiOxNy with 2x+3y=4 [99]) dielectric further confirms that this NBTI component is indeed related to pre-existing oxide defect. Such increase of oxide defect with nitrogen within the SiOxNy dielectric is routinely reported in literature [17, 18, 21, 100]. Moreover, our measurements suggest relatively weak temperature dependence for this ultra-fast ∆VHT component (Figure 1b), which indicates that hole trapping in our transistors have temperature independent σHT. Finally, as ultra-fast NBTI relaxation (Figure 1c) has similar features like the one for ultra-fast NBTI stress measurements, we expect initial part of NBTI relaxation to be related to an equivalent temperature independent, but nitrogen-dependent hole detrapping process. Therefore, like the stress phase, hole detrapping process in the relaxation phase will be complete within tREC = orders of ~ ms, depending on the nitrogen content within the oxynitride dielectric. VG - 2.1 V (a) - 1.0 V

2000 sec

1000 sec

1 fT

fT Substrate

Oxide

Gate

EFG 4

0.8 0.6 0.4

2 EFS 0.2

Substrate

5

Oxide

Gate

0.8 0.6

4

EFG

3 2 EFS

Empty of holes

1

0.4 0.2

(c)

(b)

0

6 ETrap [eV]

ETrap [eV]

6

Time

-20

0 xTrap [A]

20

0

0

-20

0

20

0

xTrap [A]

Figure 4: (a) Timing diagram used in the hole trapping simulation for a particular transistor having physical oxide thickness TPHY ~ 2.3 nm, effective oxide thickness EOT ~1.35 nm, and 39.22 %N2 dose (measured using XPS). The dielectric parameter (hole effective mass, oxide bandgap, and barrier height) for the simulation is calculated following the procedure stated in reference [87]. (b) Occupancy of defects or, hole trapping sites (assumed to have existence within the colored region only), after the transistor is kept at VSTS = -2.1 V for 1000 s. (c) Occupancy of hole trapping sites after the transistor is switched to VSTS = -1.0 V at 1000 s and kept at that voltage up to 2000 s.

5. Theory of Hole Trapping into Generated Oxide Defects (NOT) NBTI stress is also a TDDB stress in p-MOSFETs. Therefore, we also need to consider hole trapping/detrapping from newly created bulk oxide defects NOT, and corresponding ∆VOT, at NBTI stress condition. In stress phase, the trapping process follows the same equation (18). However, unlike NHT, in this case (slower) oxide defect generation is the rate-limiting process, rather than the hole trapping. Therefore, contrary to ∆VHT, ∆VOT is expected to show non-saturating behavior, proportional to the timedependent increase of bulk defect density. Indeed, Figure 5 confirms the existence of universal ∆VOT, even at NBTI stress conditions, as follows: Figure 5a indicates an increase in NBTI time exponent at higher VSTS, thus indicating a signature of oxide defect generation [62] or ∆VOT at higher VSTS and negligible ∆VOT at lower VSTS. Moreover, as the measurement scheme for [78] is slow, we expect the ∆VHT component in ∆VT to be negligible. Therefore, at lower VSTS, we can presume ∆VT ~ ∆VIT and calculate field acceleration for ∆VIT. Later, using the field acceleration for ∆VIT, we can calculate ∆VIT and thereby find ∆VOT = ∆VT – ∆VIT at higher VSTS (Figure 5b). The calculated ∆VOT for the measurements in [78] shows a decrease in time exponent for ∆VOT at higher VSTS, as also observed in [68]. Interestingly, the estimated ∆VOT at different VSTS scales universally and can be fitted (see Figure 5c) using a dispersive bulk defect generation model [68, 101] –

∆VT [a.u.]

VG5 VG4

field accelera tion of ΔVIT

VG3 VG2 VG1

(a)

∆VOT [a.u.]

Solid line – Dashed Line

n > 1/6

VSTS 2.3V 2.5V 2.7V 0

n ~ 1/6

(b) EOT~ 1.2 nm; T ~ 125 C

tSTS [a.u.]

∆VOT [a.u.]

tSTS [a.u.]

Fitting parameters: σ = 0.1 eV n = 3.38 -6 kf0 = 3x10

(c)

VSTS

Scaling

factor 2.3V 1.0 2.5V 10.0 2.7V 100.0

Scaled tSTS [a.u.]

Figure 5: (a) Time exponent of NBTI degradation (fitted solid lines) increases at higher NBTI stress voltage of VSTS ≥ VG4 [78]. (b) Extracted ∆VOT for the

measurements in [78] shows a decrease in time exponent for ∆VOT at higher VSTS. (c) Estimated ∆VOT at different VSTS scales universally (similar to the observation in [68]) and can be fitted using equation (20).

∫ n ( E ) dE , g ( E ) 1 − exp ( − k ( E ) * t ) ,

N OT = nOT ( E ) =

OT

F

k F ( E ) = k F 0 exp − ( E − E0 ) / kT , g (E) ~

1

exp ( E − E0 ) / σ OT

σ OT 1 + exp ( ( E − E ) / σ ) 2 OT 0

(20) ,

where E0 is the average energy of bond dissociation that is leading to oxide defects and σOT is its standard deviation. Therefore, ∆VOT is another component that we need to consider for NBTI stress phase. In the NBTI relaxation phase, trapped holes from the oxide defects will detrap in a manner similar to the detrapping from pre-existing oxide defects. So, hole detrapping in the NBTI relaxation phase both from the pre-existing and generated oxide defects can be handled by using a single detrapping formula, as done in reference [41].

6. Interpreting NBTI Measurements Our theoretical understanding of sections 3, 4 and 5 enables us to isolate the ∆VIT, ∆VHT and ∆VOT components of ∆VT. Note that without such decomposition, it is impossible to interpret and model the dynamics of ∆VT (and its components), measured during NBTI stress. Without performing such decomposition of interface and oxide defect components, recently several efforts [13, 14, 22, 51, 73, 102] have arraigned the RD theory to be inconsistent with NBTI relaxation (for an example of the purported inconsistency, see Figure 6a). Such alleged inconsistency has raised questions regarding the general validity of the R-D theory. Many alternative theories [12, 13, 21, 22] have also been proposed to explain ultra-fast NBTI relaxation, although their ability in predicting the broad features of NBTI remains questionable (see section 7 for further details). 6.1 Interpreting NBTI Stress Phase Measurements If we decompose ∆VT into its ∆VIT, ∆VHT and ∆VOT components [41] by calculating 0.3 ∆VIT using equation (12), ∆VHT using equation (19), ∆VOT using ~t [69], the voltage and temperature dependency of extracted components show remarkable consistency for transistors having wide variation in dielectric material. The analysis shows high-fidelity matching for stress and relaxation phase NBTI measurements and suggests that in the 1/6 range of tSTS ~ 1-1000 s, ∆VIT ~ t , ∆VHT ~ constant, and ∆VOT is quite small (Figure 6b). Based on these, we can approximate equation (1) in the range of tSTS ~ 1-1000 s with –

∆VT(tREC)/∆VT(tSTS)

1.0

16.7% N2; EOT ~ 2.35nm

Stress

VSTS/VREC=-2.9V/-1.3V

(b)

0.5

tSTS=10s

~ 5%

∆VIT

(a) ∆VHT

Data RD theory

0.0 -7 10

Relaxation ∆VT

∆VOT -4

10

-1

10 tSTS [sec]

2

10

tREC,start tNIT,start Time

Figure 6: (a) Naive (and unphysical) comparison of the time evolution of measured fractional NBTI relaxation at short tREC shows significant inconsistency with R-D solution. (b) Resolution between the measured ultra-fast ∆VT relaxation and comparatively slower NIT relaxation becomes possible, after decomposing ∆VT (solid line) into ∆VIT (dotted line), ∆VHT (dashed line) components; each having separate time-dynamics.

∆VT ≃ ∆VIT + ∆VHT = AIT t 1/ 6 + BHT .

(21)

We use equation (21) and estimate ∆VIT by subtracting a constant (saturated) ∆VHT ~ BHT from ∆VT for tSTS > 1 s in such a way that it provides time exponent n ~ 1/6 for ∆VIT at tSTS of 1-1000 s (Figure 7a). Here, the effect of mobility [71, 103] and electric fieldreduction [15] is taken into account in estimating ∆VT. Next, we repeat the decomposition at different temperature for same Eox and tSTS, so that we can extract the activation energy of the ∆VIT and ∆VHT components. Figure 7b shows the extracted ∆VIT and ∆VHT component at different temperature, which suggests EA for ∆VIT (EA,IT) is ~ 0.094 eV; which is expected for R-D model based interface defect generation with H2 diffusion (see section 3.2); on the other hand, EA for ∆VHT or EA,HT is ~ 0.04 eV, which is typically expected in any hole trapping process, involving tunneling and no structural relaxation. The signature of EA,IT > EA,HT is also evident from Figure 7c, which indicates an increase in the extracted ∆VIT/∆VT (i.e. decrease in ∆VHT/∆VT) with increase in temperature, at fixed tSTS. Identical procedure is followed to isolate ∆VIT and ∆VHT for the transistors of Figure 8, at different voltages and temperatures. The extracted EA,IT (supported by R-D theory of section 3) and EA,HT (supported by hole trapping theory of section 4) is similar for these transistors (see Figure 8a). Moreover, extracted ∆VHT for these nitrided transistors at a particular Eox, TSTS and tSTS (Figure 8b) is observed to increase significantly with the increase in %N (with a very rapid increase seen for %N > 30), which is indeed a signature of higher hole trapping for higher %N. On the other hand, ∆VIT only increases slightly with %N, which indicates that there is negligible change in the Si-H bond dissociation mechanism and hydrogen diffusion dynamics with the change in %N. Therefore, the extracted parameters for both ∆VIT and ∆VHT are consistent with the theoretical predictions.

EOT=1.4nm; 23 % Plasma N2 0 T=125 C VSTS= 2.1V ∆VT

∆VIT

6x10

∆VHT

NBTI Components [V]

NBTI Components [Volts]

-1

10

n=0.122+0.005 n~0.16

-2

10

(a) 0

10

1

2

10 10 tSTS [sec]

Fractions of ∆VT

1.0

EOT ~ 1.4nm; 23%N2 Dose ∆ VT

VSTS=-2.1V; tSTS=100s

∆VIT ∆VHT

EA=0.078eV

10

EA,IT=0.094eV

-2

EA,HT=0.04eV

3x10

3

-2

10

(b)

-3

25

30 35 -1 1/kBTSTS [eV ]

VSTS = -2.1V; tSTS = 103 sec

0.8 VIT fraction

0.6 0.4

VHT fraction

0.2 0.0

(c) (c)

50

80 110 0 Temperature [ C]

140

Figure 7: (a) Measured ∆VT and extracted ∆VIT and ∆VHT components for a transistor having optimized nitrided dielectric. (b) Temperature dependence and corresponding activation energies (EA) of ∆VT, ∆VIT and ∆VHT components for the same transistor. (c) Estimated ∆VHT/∆VT and ∆VIT/∆VT at different temperature (TSTS) indicate an increase (decrease) in ∆VIT/∆VT (∆VHT/∆VT) at higher temperature. Here, the error bars represent the noise in ID,lin0 measurement for OTF-ID,lin [104], which causes a ± 0.005 error in n for ∆VT and a ± 1mV error in estimated ∆VHT. 0.003

0.10

∆VIT

0.05

∆VHT

(a) 0.00 20

30 %N (atomic)

40

Degradation / EOT [a.u.]

Activation Energy [eV]

0.15

0.002

O

ttstress =100s, Tstress =125 C 0C TSTS = 125 STS = 100sec; EOX= 8MV/cm ∆VTT ∆V ∆VIT ∆V IT ∆VHT ∆V HT

0.001

0.000

(b) 20

30

40

%N (atomic)

Figure 8: (a) Activation energy for extracted ∆VIT and ∆VHT components indicates negligible %N dependence. (b) %N Dependence of measured ∆VT and extracted

∆VIT and ∆VHT components for a particular tSTS, TSTS and Eox. Lines are guide to the eye only. As shown in reference [41], consideration of ∆VOT in estimating ∆VIT and ∆VHT from ∆VT merely changes the signatures of ∆VIT and ∆VHT, presented in Figure 7 and Figure 8. In that case, estimated ∆VIT, ∆VHT, and ∆VOT shows good consistently with the theoretical expectations and suggests that ∆VOT is a component that one should consider at higher stress bias. 6.2 Interpreting NBTI Relaxation Phase Measurements Let us now apply the same NIT/NHT/NOT decomposition for NBTI relaxation measurements on the same transistor that we have previously analyzed through stressphase decomposition in Figure 7a, which also has the temperature independent NBTI relaxation for tREC < ms (see Figure 2a). Here, we further use the observation of Figure 9a, which suggests that for VSTS = -2.3 V, hole detrapping occurs (i.e., tREC,start shows sudden decrease) predominantly at VREC ≥ -1.8 V. Figure 9a also suggests that the amount of hole detrapping is similar from VREC = −1.3 V to −1.6 V for the minimum tREC ~ µs measured in this experiment.

∆VT(tREC)/∆VT(tSTS)

1.0

R-D: -2.0V

3

-1.6V -1.3V

0

TSTS = 125 C (a) VREC Dependence: ∆VT (a) -3

10

Gate

-1.8V

VSTS=-2.3V

0.0 -6 10

Oxide

0

10 tREC [sec]

Substrate Oxide

Gate

Hole trapping sites

R-D: -1.3V -2.0V

tSTS=10 sec

0.5

Substrate

EC

EC

EF

EF

EV

EV

(c)

(b)

Hole detrapping sites

3

10

Stress (VSTS)

Relaxation (VREC)

Figure 9: (a) NBTI relaxation experiments at different VREC show significantly different tREC,start. Since relaxation for VREC ≤ -1.8 V is very close the R-D theory, there is an additional relaxation mechanism for VREC > -1.8 V. Our observation suggests hole detrapping to be the additional mechanism. (b) Schematic (based on simulation within a Shockley-Read-Hall trapping-detrapping framework; section 4) for expected hole trapping sites (hatched region) at VSTS. (c) When gate bias is switched from VSTS to VREC, the hatched region will detrap the captured hole in a temperature independent manner. Thus, trapping sites within the quasi-Fermi levels at VREC = −1.8 V and −1.6 V (shown schematically by the hatched region in Figure 9c) will detrap all the holes that were captured by the oxide defects during stress within a timescale of ~ms. Considering such total hole detrapping at VREC= −1.3 V both from the pre-existing and generated oxide defects (here, ∆VOT is considered to be ~ 3 mV at tSTS =103 s, having the time dependence similar to Figure 5c), the resultant NIT component of the NBTI relaxation

experiment shows excellent consistency with R-D theory in terms of tNIT,start at all VREC (see Figure 10a). To understand the contribution of ∆VOT on tNIT,start, we redo the experiment-theory comparison by assuming ∆VOT ~ 0. Figure 10b suggests that consideration of finite ∆VOT increases tNIT,start by an order of magnitude. The remaining theory-experiment gap in terms of time dependence for tREC > tNIT,start reflects the inability of 1D diffusion formulation (considered in classical R-D model of section 3) to capture the details of an essentially 3D diffusion problem. Consideration of 3D diffusion ensures high-fidelity matching for the time dependence for tREC > tNIT,start [41]. Therefore, decomposing the contributions from interface and oxide defects enables us to explain the NBTI relaxation features in a theoretically consistent way.

0.5

0.0 -6 10

Lines: R-D Theory

1.0 -2.0V

3

tSTS= 10 sec VSTS= -2.3V

-1.8V

0

TSTS = 125 C

-1.3V

(a) VREC Dependence: ∆VIT, ∆VHT, VREC Dependence: ∆VIT ∆VOT(b) considered

10

-3

0

10 tREC [sec]

3

10

∆VIT(tREC)/∆VIT(tSTS)

∆VIT(tREC)/∆VIT(tSTS)

1.0

R-D: -2.0V R-D: -1.3V -2.0V

3

tSTS=10 sec

0.5

VSTS=-2.3V

-1.8V 0

TSTS = 125 C

-1.3V

(b) VREC Dependence: ∆VIT, ∆VHT (a) VREC∆VDependence: ∆VIT considered; OT ignored.

0.0 -6 10

10

-3

0

10 tREC [sec]

3

10

Figure 10: (a) (b) Consideration of ∆VIT, ∆VHT, and ∆VOT and their decomposition indicates excellent consistency of tNIT,start with R-D theory. (b) Neglecting ∆VOT reduces tNIT,start by an order of magnitude. Here, the error bar along the x-axis for VREC = -1.3 V is due to the ID,lin0 error in calculating ∆VIT/∆VT (see Figure 7). 6.3 Interpreting AC NBTI Measurements So far we have applied the decomposition procedure to identify the interface and oxide defect components of NBTI-induced ∆VT. Our analysis demonstrates that hole trapping and detrapping occur at similar time-scales (for example, compare the time-scale of the temperature independent hole trapping component in Figure 1b, and hole detrapping component in Figure 2b). Therefore, we expect total hole detrapping for a AC NBTI stress (measured at the end of OFF-state) with ≤ 50% duty cycle (Figure 11a). In other words (see Figure 11b), AC/DC ratio for ≤50% duty cycle in high %N transistors will measure ∆VT,AC/∆VT,DC ~ ∆VIT,AC/[∆VHT,DC + ∆VIT,DC] (considering ∆VOT component is quite small) and hence will always be less than the contribution from NIT’s component, AC/DC(NIT) = ∆VIT,AC/∆VIT,DC, predicted by R-D theory. Moreover, as ∆VHT decreases for smaller %N, low %N transistors will have ∆VT,AC/∆VT,DC ~ ∆VIT,AC/∆VIT,DC, and thus the

measured AC/DC ratio for low %N transistors should be consistent with the one obtained from the R-D theory. Time 1

∆VT ∆VIT

∆VHT

(a)

AC/DC Ratio

R-D: ∆VIT

VAC

Low %N High %N (b) 0 0%

50%

100%

Duty Cycle

Figure 11: (a) Since hole trapping and detrapping happens at similar time-scale, we expect total hole detrapping at the end of OFF-state during AC NBTI stress. Therefore, (b) AC/DC ratio (with ∆VT for AC NBTI measured at the end of AC cycles) will be consistent with the R-D theory only for transistors having lower ∆VHT or low %N. Indeed, our ∆VT,AC/∆VT,DC measurement on nitrided transistors show that for the lowest %N oxynitride dielectric (where ∆VT ~ ∆VIT), the predictions from R-D theory matches very well with the experimental data (Figure 12a). On the other hand, transistors with higher %N oxynitride dielectric have significant contribution from ∆VHT in DC NBTI stress, i.e., ∆VT,DC > ∆VIT,DC. As a result, ∆VT,AC/∆VT,DC is lower than ∆VIT,AC/∆VIT,DC (or the line predicted by the R-D theory) in such higher %N transistors. However, because AC NBTI stress at lower duty cycle predominantly reflects ∆VIT, the shape of ∆VT,AC/∆VT,DC vs. duty cycle plot is similar to the R-D’s prediction at lower duty cycle for all transistors. In other words, equation (17) is universal at lower duty cycle for any transistors, irrespective of %N within the dielectric. Such universal shape of ∆VT,AC/∆VT,DC vs. duty cycle dependence is evident in Figure 2c, where AC data at all duty cycle is scaled to d ~ 50% data to capture the intrinsic ∆VIT behavior (as total hole detrapping occurs at d ~ 50%). The scaling confirms a robust universality of the dutycycle data from diverse sources, even up to ~80% duty cycle. This indicates that, for the transistors of Figure 2c, time scale for hole-detrapping is much less than the time scale for hole trapping. At higher duty cycle, detrapping of holes is incomplete, and therefore the experimental data deviates from the prediction of R-D theory. Consideration of total hole detrapping for ~50% duty cycle also explains how R-D theory interprets the widely observed frequency independence in all NBTI measurements, irrespective of the type of dielectric, while overestimating the measured ∆VT,AC/∆VT,DC at different frequencies (Figure 12b and Figure 1e). Since ∆VHT will be absent for AC NBTI stress with ~50% duty cycle, the frequency independence of ∆VIT,AC/∆VIT,DC (following R-D theory; equation (16)) will also result in frequency independence of ∆VT,AC/∆VT,DC, irrespective of %N. However, for transistors with larger %N, the magnitude of ∆VT,AC/∆VT,DC will be lower than ∆VIT,AC/∆VIT,DC due to the presence of ∆VHT in ∆VT,DC. Therefore, ∆VT,AC/∆VT,DC measurement on nitrided transistor show frequency

independence, irrespective of %N (Figure 12b); while the magnitude of ∆VT,AC/∆VT,DC is only consistent with R-D theory for low %N transistors, when the hole trapping contribution is negligible. Thus, the co-existence of interface and oxide defects and their decomposition can explain both the duty cycle and frequency dependent NBTI experiments on nitrided transistors.

0

TSTS = 125 C

3

1.0

Shape similar to R-D

R-D 0.5

Higher %N

(a) 0.0 0

25

∆VT(AC)/∆VT(DC) @ tSTS=10 sec

3

∆VT(AC)/∆VT(DC) @ tSTS=10 sec

Low (10) %N 0

TSTS = 125 C

1.0

R-D Theory

0.5 Higher %N

0.0

(b) DC

10

0

10

3

10

6

10

9

Frequency [Hz]

Figure 12: (a) AC/DC ratio (when ∆VT is measured at the end of AC cycles) vs. duty cycle plot for different nitrided transistors. The experiments show remarkable consistency with the prediction of R-D theory (solid line) for low %N. (b) Though AC/DC ratio for any %N is always frequency independent, there is significant %N dependency due to the presence of ∆VHT in DC NBTI stress.

7. A Critique of Alternate NBTI Theories Contrary to the uncorrelated dynamics for interface and oxide defects that is used in this paper to explain the experimental features of NBTI, correlated model of these defects has also been recently used [22, 105] to explain the so-called universally observed temperature-dependence at arbitrary time scale. Actually, we observe such universal temperature dependence only in transistors having low %N [15, 16, 18, 19] (see Figure 1b), but not in all transistors as claimed in references [22, 105]. Moreover, the energywell theory [11] behind such correlated mechanism of NIT and NHT fails to explain the basic features of NBTI degradation, such as the existence of stress-phase power-law time exponent, frequency independence [106, 107] - features that has been universally observed in all SiON transistors (see Figure 1). Indeed, as shown in Figure 13, the powerexponent predicted by energy-well theory (or any of its recent versions [22]) is fragile and its range changes by orders of magnitude with small changes in the parameters like broadening of well-barrier σWB. In addition to this energy-well theory, some references consider NBTI (merely) as a trapping-detrapping problem into oxide defects [12, 13]. However, such theoretical analysis requires one to use a wide distribution of hole trapping sites (either spatially [12, 13] or energetically) for capturing the stress-phase power exponent and relaxation

dynamics. Experimental support for such wide distribution of hole trapping events (for explaining NBTI time dynamics over more than 15 decades) is still lacking in literature. 1.01

σWB

0.1 eV

0.9

0.7

Time Exponent

Time exponent

0.8

(a)

0.6

0.5 0.5

0.3 0.2 0.1

State 1

State 2

State 3

0.2 eV

0.3 eV

0.4

00

(b)

σWB = 0.4 eV -4

10-4 10

-2

10-2 10

0

2

10 0 102 10 10 Time [sec]

time [sec]

4

10 4 10

Figure 13: (a) Energy-well model (like the one presented in reference [11]) for explaining the dynamics of NBTI. Such energy-well model or its recent derivatives [22], in general, requires one to use a spread in barrier height (having a standard deviation of σWB) for capturing the NBTI dynamics. (b) Unless we use large σWB, it becomes difficult to obtain a robust power-law time exponent using such energywell model.

8. Conclusion In this paper, we have highlighted the importance of careful analysis of NBTI experiments, before comparing it with any theory. Our analysis demonstrates the uncorrelated dynamics of threshold voltage shift associated with interface and oxide defects can adequately explain the questions stated in section 2.2 of the manuscript. The universal features of DC and AC NBTI (i.e., long-term stress-phase power-law time exponent, long-term stress/relaxation-phase temperature dependence, characteristic feature of duty-cycle dependence, and universal feature of frequency independence) can be consistently explained using R-D theory based interface defect dynamics. The remaining non-universal features (i.e., short-term stress/relaxation-phase time dependence, short-term stress/relaxation-phase temperature independence, and the magnitude of AC/DC ratio at a particular duty cycle and frequency) are shown to be directly correlated to the amount of oxide defects within the dielectric. However, these non-universal features disappear for less than ~80% duty cycle and hence have no relevance for practical AC NBTI analysis. Therefore, a decomposition of these (uncorrelated) universal and non-universal components is established as a necessity before comparison with any theory. And without such decomposition, one will only be able to capture part (not all) of the experimental features of NBTI.

Acknowledgments The work is an accumulation of our last few years’ research on NBTI. It has been immensely benefited from the discussion with/contribution from D. Varghese, N. Kumar,

H. Kufluoglu, A. Krishnan, J. H. Lee, A. Oates, A. Jain, K. Ahmed, C. Olsen, E. Murakami, H. Aono, M. Masuduzzaman, G. Gupta, and H. Das. The financial support was obtained from Applied Materials, Taiwan Semiconductor Manufacturing Corporation, SRC-GRC, 2008 IEEE EDS PhD Fellowship, and 2009-2010 Intel Foundation PhD Fellowship.

References 1. B. E. Deal, M. Sklar, A. S. Grove and E. H. Snow, Journal of the Electrochemical Society, 114, 266 (1967). 2. K. O. Jeppson and C. M. Svensson, Journal of Applied Physics, 48, 2004 (1977). 3. K. Bernstein, D. J. Frank, A. E. Gattiker, W. Haensch, B. L. Ji, S. R. Nassif, E. J. Nowak, D. J. Pearson and N. J. Rohrer, IBM Journal of Research and Development, 50, 433 (2006). 4. D. K. Schroder, Microelectronics Reliability, 47, 841 (2007). 5. D. K. Schroder and J. A. Babcock, Journal of Applied Physics, 94, 1 (2003). 6. J. H. Stathis and S. Zafar, Microelectronics Reliability, 46, 270 (2006). 7. J. Campbell, in Tutorial of IEEE International Integrated Reliability Workshop, p. T4 (2009). 8. M. A. Alam and S. Mahapatra, Microelectronics Reliability, 45, 71 (2005). 9. S. Chakravarthi, A. Krishnan, V. Reddy, C. F. Machala and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 273 (2004). 10. V. Huard, M. Denais and C. Parthasarathy, Microelectronics Reliability, 46, 1 (2006). 11. T. Grasser, B. Kaczer and W. Goes, in IEEE International Reliability Physics Symposium, p. 28 (2008). 12. D. Ielmini, M. Manigrasso, F. Gattel and M. G. Valentini, IEEE Transactions on Electron Devices, 56, 1943 (2009). 13. P. M. Lenahan, in IEEE International Reliability Physics Symposium, p. 1086 (2010). 14. T. Grasser, H. Reisinger, P. Wagner, F. Schanovsky, W. Goes and B. Kaczer, in IEEE International Reliability Physics Symposium, p. 16 (2010). 15. A. E. Islam, H. Kufluoglu, D. Varghese, S. Mahapatra and M. A. Alam, IEEE Transactions on Electron Devices, 54, 2143 (2007). 16. A. E. Islam, S. Mahapatra, S. Deora, V. D. Maheta and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 733 (2009).

17. J. H. Lee and A. S. Oates, IEEE Transactions on Device and Materials Reliability, 10, 174 (2010). 18. S. Mahapatra, K. Ahmed, D. Varghese, A. E. Islam, G. Gupta, L. Madhav, D. Saha and M. A. Alam, in IEEE International Reliability Physics Symposium, p. 1 (2007). 19. S. Mahapatra, V. D. Maheta, A. E. Islam and M. A. Alam, IEEE Transactions on Electron Devices, 56, 236 (2009). 20. D. S. Ang, S. Wang, G. A. Du and Y. Z. Hu, IEEE Transactions on Device and Materials Reliability, 8, 22 (2008). 21. V. Huard, in IEEE International Reliability Physics Symposium, p. 33 (2010). 22. T. Grasser, B. Kaczer, W. Goes, T. Aichinger, P. Hehenberger and M. Nelhiebel, in IEEE International Reliability Physics Symposium, p. 33 (2009). 23. J. P. Campbell, P. M. Lenahan, A. T. Krishnan and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 503 (2007). 24. A. S. Grove, Physics and Technology of Semiconductor Devices, John Wiley and Sons (1971). 25. P. V. Gray and D. M. Brown, Applied Physics Letters, 8, 31 (1966). 26. Y. Nishi, Jpn J Appl Phys, 10, 52 (1971). 27. P. J. Caplan, E. H. Poindexter, B. E. Deal and R. R. Razouk, Journal of Applied Physics, 50, 5847 (1979). 28. K. L. Brower, Physical Review B, 38, 9657 (1988). 29. E. Cartier, J. H. Stathis and D. A. Buchanan, Applied Physics Letters, 63, 1510 (1993). 30. J. P. Campbell, P. M. Lenahan, A. T. Krishnan and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 442 (2006). 31. P. M. Lenahan, in Defects in Microelectronic Materials and Devices, D. Fleetwood, S. Pantolides and R. D. Schrimpf, Editors, p. 163, CRC Press (2008). 32. A. T. Krishnan, C. Chancellor, S. Chakravarthi, P. E. Nicollian, V. Reddy, A. Varghese, R. B. Khamankar, S. Krishnan and L. Levitov, in International Electron Devices Meeting (IEDM) Technical Digest, p. 688 (2005). 33. S. Tsujikawa, T. Mine, K. Watanabe, Y. Shimamoto, R. Tsuchiya, K. Ohnishi, T. Onai, J. Yugami and S. Kimura, in IEEE International Reliability Physics Symposium, p. 183 (2003). 34. M. A. Alam, H. Kufluoglu, D. Varghese and S. Mahapatra, Microelectronics Reliability, 47, 853 (2007).

35. S. Chakravarthi, A. T. Krishnan, V. Reddy and S. Krishnan, Microelectronics Reliability, 47, 863 (2007). 36. S. Mahapatra and M. A. Alam, IEEE Transactions on Device and Materials Reliability, 8, 35 (2008). 37. L. Tsetseris, X. J. Zhou, D. M. Fleetwood, R. D. Schrimpf and S. T. Pantelides, Applied Physics Letters, 86, 142103 (2005). 38. L. Tsetseris, X. J. Zhou, D. M. Fleetwood, R. D. Schrimpf and S. T. Pantelides, IEEE Transactions on Device and Materials Reliability, 7, 502 (2007). 39. M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 345 (2003). 40. A. E. Islam, G. Gupta, S. Mahapatra, A. Krishnan, K. Z. Ahmed, F. Nouri, A. Oates and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 329 (2006). 41. S. Mahapatra, A. E. Islam, S. Deora, V. D. Maheta and M. A. Alam, accepted in IEEE International Reliability Physics Symposium (2011). 42. L. Lundkvist, I. Lundstrom and C. Svensson, Solid-State Electronics, 16, 811 (1973). 43. R. A. Williams and M. M. E. Beguwala, IEEE Transactions on Electron Devices, 25, 1019 (1978). 44. G. L. Heyns and H. E. Maes, Applied Surface Science, 30, 153 (1987). 45. F. P. Heiman and G. Warfield, IEEE Transactions on Electron Devices, Ed12, 167 (1965). 46. P. Rossel, H. Martinot and D. Esteve, Solid-State Electronics, 13, 425 (1970). 47. I. Lundstrom, S. Christensson and C. Svensson, Phys. Status Solid (A), 1, 395 (1970). 48. V. Huard and M. Denais, in IEEE International Reliability Physics Symposium, p. 40 (2004). 49. C. R. Parthasarathy, M. Denais, V. Huard, G. Ribes, E. Vincent and A. Bravaix, in IEEE International Reliability Physics Symposium, p. 471 (2006). 50. H. Reisinger, O. Blank, W. Heinrigs, A. Muhlhoff, W. Gustin and C. Schlunder, in IEEE International Reliability Physics Symposium, p. 448 (2006). 51. C. Shen, M.-F. Li, C. E. Foo, T. Yang, D. M. Huang, A. Yap, G. S. Samudra and Y.C. Yeo, in International Electron Devices Meeting (IEDM) Technical Digest, p. 333 (2006). 52. T. Yang, C. Shen, M. F. Li, C. H. Ang, C. X. Zhu, Y. C. Yeo, G. Samudra, S. C. Rustagi, M. B. Yu and D. L. Kwong, IEEE Electron Device Letters, 26, 826 (2005).

53. M. F. Li, G. Chen, C. Shen, X. P. Wang, H. Y. Yu, Y. C. Yeo and D. L. Kwong, Jpn J Appl Phys, 43, 7807 (2004). 54. G. Bersuker, J. H. Sim, C. S. Park, C. D. Young, S. V. Nadkarni, R. Choi and B. H. Lee, IEEE Transactions on Device and Materials Reliability, 7, 138 (2007). 55. C. Shen, M. F. Li, X. P. Wang, H. Y. Yu, Y. P. Feng, A. T. L. Lim, Y. C. Yeo, D. S. H. Chan and D. L. Kwong, International Electron Devices Meeting (IEDM) Technical Digest, 733 (2004). 56. W. Goes, M. Karner, V. Sverdlov and T. Grasser, in Proceedings of International Symposium on the Physical & Failure Analysis of Integrated Circuits (IPFA), p. 249 (2008). 57. A. Ghetti, C. M. Compagnoni, A. S. Spinelli and A. Visconti, IEEE Transactions on Electron Devices, 56, 1746 (2009). 58. T. H. Morshed, S. P. Devireddy, Z. Celik-Butler, A. Shanware, K. Green, J. J. ChamberS, M. R. Visokay and L. Colombo, Solid-State Electronics, 52, 711 (2008). 59. D. Veksler, H. Park, C. Young, B. Taylor, G. Bersuker and R. Jammy, Solid State Technology, 53, 21 (2010). 60. H. D. Xiong, D. Heh, M. Gurfinkel, Q. Li, Y. Shapira, C. Richter, G. Bersuker, R. Choi and J. S. Suehle, Microelectronic Engineering, 84, 2230 (2007). 61. J. P. Campbell, J. Qin, K. P. Cheung, L. C. Yu, J. S. Suehle, A. Oates and K. Sheng, in IEEE International Reliability Physics Symposium, p. 382 (2009). 62. S. Mahapatra, P. B. Kumar and M. A. Alam, IEEE Transactions on Electron Devices, 51, 1371 (2004). 63. J. Robertson, Solid-State Electronics, 49, 283 (2005). 64. J. Robertson, Reports on Progress in Physics, 69, 327 (2006). 65. J. Sune, E. Farres, I. Placencia, N. Barniol, F. Martin and X. Aymerich, Applied Physics Letters, 55, 128 (1989). 66. M. Alam, B. Weir and P. Silverman, IEEE Circuits & Devices, 18, 42 (2002). 67. F. Crupi, R. Degraeve, G. Groeseneken, T. Nigam and H. E. Maes, IEEE Transactions on Electron Devices, 45, 2329 (1998). 68. D. Varghese, H. Kufluoglu, V. Reddy, H. Shichijo, D. Mosher, S. Krishnan and M. A. Alam, IEEE Transactions on Electron Devices, 54, 2669 (2007). 69. P. E. Nicollian, Microelectronics Reliability, 48, 1171 (2008). 70. A. E. Islam, H. Kufluoglu, D. Varghese and M. A. Alam, Applied Physics Letters, 90, 083505 (2007).

71. A. E. Islam, V. D. Maheta, H. Das, S. Mahapatra and M. A. Alam, in IEEE International Reliability Physics Symposium, p. 87 (2008). 72. V. D. Maheta, C. Olsen, K. Ahmed and S. Mahapatra, IEEE Transactions on Electron Devices, 55, 1630 (2008). 73. T. Grasser, B. Kaczer, R. Hehenberger, W. Gos, R. O'Connor, H. Reisinger, W. Gustin and C. Schlunder, International Electron Devices Meeting (IEDM) Technical Digest, 801 (2007). 74. B. Kaczer, T. Grasser, P. J. Roussel, J. Martin-Martinez, R. O'Connor, B. J. O'Sullivan and G. Groeseneken, in IEEE International Reliability Physics Symposium, p. 20 (2008). 75. S. Deora, V. D. Maheta, A. E. Islam, M. A. Alam and S. Mahapatra, IEEE Electron Device Letters, 30, 978 (2009). 76. S. Mahapatra, V. D. Maheta, S. Deora, E. N. Kumar, S. Purawat, C. Olsen, K. Ahmed, A. E. Islam and M. A. Alam, in Silicon Nitride, Silicon Dioxide, and Emerging Dielectrics, R. E. Sah, J. Zhang, J. Deen, J. Yota and A. Toriumi, Editors, p. 243, The Electrochemical Society Proceedings Series (2009). 77. R. Fernández, B. Kaczer, A. Nackaerts, S. Demuynck, R. Rodríguez, M. Nafría and G. Groeseneken, in International Electron Devices Meeting (IEDM) Technical Digest, p. 337 (2006). 78. C. L. Chen, Y. M. Lin, C. J. Wang and K. Wu, in IEEE International Reliability Physics Symposium, p. 704 (2005). 79. A. Haggag, G. Anderson, S. Parihar, D. Burnett, G. Abeln, J. Higman and M. Moosa, in IEEE International Reliability Physics Symposium, p. 452 (2007). 80. H. Reisinger, T. Grasser, W. Gustin and C. Schlunder, in IEEE International Reliability Physics Symposium, p. 7 (2010). 81. Y. Mitani, H. Satake and A. Toriumi, IEEE Transactions on Device and Materials Reliability, 8, 6 (2008). 82. S. Wang, D. S. Ang and G. A. Du, IEEE Electron Device Letters, 29, 483 (2008). 83. H. Reisinger, in NBTI Tutorial, IEEE International Integrated Reliability Workshop, p. T2 (2010). 84. V. Huard, C. Parthasarathy, C. Guerin, T. Valentin, E. Pion, M. Mammasse, N. Planes and L. Camus, in IEEE International Reliability Physics Symposium, p. 289 (2008). 85. A. T. Krishnan, S. Chakravarthi, P. Nicollian, V. Reddy and S. Krishnan, Applied Physics Letters, 88, 153518 (2006).

86. H. Kufluoglu and M. A. Alam, IEEE Transactions on Electron Devices, 54, 1101 (2007). 87. A. E. Islam, G. Gupta, K. Z. Ahmed, S. Mahapatra and M. A. Alam, IEEE Transactions on Electron Devices, 55, 1143 (2008). 88. A. E. Islam, J. H. Lee, W. H. Wu, A. Oates and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 107 (2008). 89. S. Kumar, C. H. Kim and S. S. Sapatnekar, in International Conference on ComputerAided Design, p. 6D.1 (2006). 90. J. Crank, The Mathematics of Diffusion, Oxford University Press (1980). 91. T. Grasser, W. Gos and B. Kaczer, IEEE Transactions on Device and Materials Reliability, 8, 79 (2008). 92. C. G. Van de Walle and B. R. Tuttle, IEEE Transactions on Electron Devices, 47, 1779 (2000). 93. D. Varghese, D. Saha, S. Mahapatra, K. Ahmed, F. Nouri and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 684 (2005). 94. G. Gupta, S. Mahapatra, L. Madhav, D. Varghese, K. Z. Ahmed and F. Nouri, in IEEE International Reliability Physics Symposium, p. 731 (2006). 95. M. L. Reed and J. D. Plummer, Journal of Applied Physics, 63, 5776 (1988). 96. M. A. Alam, Reliability Physics of Nanoelectronic Transistors, in EE 695-A lecture series, uploaded in http://cobweb.ecn.purdue.edu/~ee650/handouts.htm. 97. A. E. Islam, Ph. D. Dissertation, Theory and Characterization of Defect Formation and Its Implication in Variability of Nanoscale Transistors, Electrical and Computer Engineering, Purdue University, 2010. 98. T. L. Tewksbury and H. S. Lee, IEEE Journal of Solid-State Circuits, 29, 239 (1994). 99. S. V. Hattangady, H. Niimi and G. Lucovsky, Journal of Vacuum Science & Technology-A, 14, 3017 (1996). 100. K. Sakuma, D. Matsushita, K. Muraoka and Y. Mitani, in IEEE International Reliability Physics Symposium, p. 454 (2006). 101. K. Hess, A. Haggag, W. McMahon, B. Fischer, K. Cheng, J. Lee and J. Lyding, in International Electron Devices Meeting (IEDM) Technical Digest, p. 93 (2000). 102. H. Reisinger, O. Blank, W. Heinrigs, W. Gustin and C. Schlunder, IEEE Transactions on Device and Materials Reliability, 7, 119 (2007).

103. A. E. Islam, E. N. Kumar, H. Das, S. Purawat, V. D. Maheta, H. Aono, E. Murakami, S. Mahapatra and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 805 (2007). 104. V. D. Maheta, E. N. Kumar, S. Purawat, C. Olsen, K. Ahmed and S. Mahapatra, IEEE Transactions on Electron Devices, 55, 2614 (2008). 105. T. Grasser and B. Kaczer, IEEE Transactions on Electron Devices, 56, 1056 (2009). 106. A. E. Islam and M. A. Alam, unpublished results. 107. M. A. Alam, in IEEE International Integrated Reliability Workshop, p. 1 (2010).

Department of Materials Science and Engineering, University of Illinois at UrbanaChampaign, IL 61801, USA b Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India c Department of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907. USA Email: a [email protected], b [email protected], c [email protected] (Invited Paper)

We develop a comprehensive theoretical framework for explaining the key and characteristic experimental signatures of NBTI. The framework is based on an uncorrelated dynamics of interfacedefect creation/annihilation described by Reaction-Diffusion (R-D) theory and hole trapping/detrapping into/out-of oxide defects based on a generalized Shockley-Read-Hall model. The proposed theory can consistently explain the long-term stress-phase power-law time exponent, stress/relaxation-phase temperature dependence, characteristic feature of duty-cycle dependence, and universal feature of frequency independence - measured in DC and AC stress conditions over a wide variety of transistors. Thus, we confirm the general validity of R-D theory in explaining the universal features (irrespective of dielectric material) of both DC and AC NBTI. The non-universal features of NBTI have correlation with the amount of oxide defects within the dielectric and do not affect AC NBTI measurements at lower duty cycle. Decomposition of these (uncorrelated) universal and non-universal components is, therefore, essential before comparison with any theory.

1. Background Negative Bias Temperature Instability (NBTI) indicates a temperature accelerated degradation in MOS transistors when it is stressed with a negative gate voltage, i.e., with an oxide electric field that is directed from the channel towards the gate of a MOS transistor. In normal CMOS (which is the basis for today’s microprocessor) operation, only the PMOS transistors are subjected to such negative oxide electric field. Therefore, since its introduction in microprocessor in early 1970s [1, 2], NBTI-induced performance degradation in PMOS transistors (e.g., increase in threshold voltage, reduction in current drivability) has always been a concern in CMOS technology. Recent use of thin gate dielectric (a consequent increase in oxide electric field) and use of high-κ (oxynitride and Hf-based) dielectric materials have further enhanced NBTI degradation. As such, NBTI

is considered as one of the major reliability concerns in current CMOS technology [3-6]. This industry-wide reliability concern has encouraged numerous efforts (see reference [7] for a chronology of the number of papers being published in this area) to understand the essential aspects of NBTI degradation by using different types of transistor-based measurements and then develop appropriate theory to explain the experiments. However, there is a perception that a definitive interpretation of this PMOS-specific phenomenon remains elusive. Most of the literature on NBTI is focused on explaining isolated experimental signatures of NBTI degradation and recovery and, in the process, may sometimes overlook the fact that the proffered explanations have not been consistent with other characteristic features of NBTI. Such fragmented modeling efforts have made NBTI literature extremely confusing to the general audience. For capturing the experimental features, NBTI dynamics has sometimes been attributed to interface defect NIT [2, 8, 9], or to hole trapping into pre-existing oxide defects NHT [10-14], or to both NIT and NHT [15-22]. Recent introduction of Spin Dependent Recombination (SDR) experiment [23] have helped shed light on the nature of NBTI-specific defects, but the work is still in flux and have not been conclusive enough to settle the discussion. For example, even though many groups have historically attributed NBTI-related NIT generation to Pb-centers (by ESR and SDR experiments), recent SDR experiment [23] indicates that Pb-centers are apparently absent in transistors with high-κ oxynitride gate dielectric. These experiments have motivated several articles to explain NBTI from the point-of-view of bulk oxide defects, where NIT is either indirectly created via hole trapping into generated oxide defects NOT [22], or NIT is absent altogether [12, 13]. Unfortunately, however, the new theories have not always attempted to explain other well known features of NBTI experiments, i.e., (i) stress-phase power-law time exponent, (ii) stress and relaxationphase temperature dependence, (iii) duty cycle dependence, and (iv) universal frequency independence, and therefore the generic validity of such bulk-oxide defect oriented approaches remain questionable. In this manuscript, we develop a robust theoretical framework for explaining these key characteristic experimental features of NBTI, measured in DC and AC stress conditions for a wide variety of transistor technologies. The framework is based on an uncorrelated dynamics for Reaction-Diffusion (R-D) theory based interface defect creation/annihilation and Shockley-Read-Hall theory based hole trapping/detrapping into/out-of oxide defects. Our analysis suggests the necessity of decomposing NBTI measurements into a universal slow component associated with interface defect generation and a non-universal fast component associated with charge trapping/detrapping into/out-of oxide defects. The universal component can be consistently explained using R-D theory based interface defect dynamics and is directly relevant for product qualification. On the other hand, the non-universal component depends on the amount of oxide defect within the dielectric (and hence associated with the gate stack fabrication technology). Hence, this non-universal component is sample dependent and must be accounted only for high-fidelity experimental fitting in short-time scales. Therefore, one should decompose these two uncorrelated (universal and nonuniversal) components and then explain their respective features using respective theories. Any attempt to violate such decomposition will lead to proliferation of unphysical model parameters to capture the experimental trends.

1.1 Interface Defect in NBTI and Relevance of R-D Theory Historically, interface defects have always been a technology challenge for semiconductor industry. Unlike oxide defects, interface defects cannot be removed or reduced through purification (or gettering). At the silicon-dielectric interface of a MOS transistor, interface defects (dangling Si- bonds) are terminated or passivated (Si- + H Si-H) by using hydrogen compounds like silane (SiH4). Efficacy of such passivation techniques using atomic hydrogen (H) has been extensively studied since 1970s [1, 2429]. However, H-passivation of dangling bonds only provides a ‘time-zero’ (pre-use) solution from interface defects. After a period of transistor operation, mostly in the PMOS configuration, the interfacial Si-H bond starts to dissociate or depassivate (Si-H Si- + H) at normal operating condition in the presence of cold holes near the interface and reforms the dangling Si- bonds. Formation of these interface defects due to Si-H depassivation has been observed by using ESR and SDR experiments, especially on the Si/SiO2 interface [30, 31]. The same has also been confirmed using capacitance-voltage measurement and mobility degradation experiments [17, 32, 33]. Theory of interface defect formation has evolved around the (1) modeling of Si-H bond dissociation and (2) subsequent handling of the resultant hydrogen species. Si-H bond dissociation is either considered to be – (1a) cold-hole assisted [8, 9, 15, 34-36], (1b) dopant-activated [37, 38], or (1c) oxide-defect induced [22]. Resultant hydrogen species after Si-H bond dissociation is either considered to be – (2a) trapped within the oxide [10, 22, 33], or (2b) diffuse within the oxide/gate region [8, 15]. Mechanisms (1a) and (2b), or (1b) and (2b), constitute the basis of the so-called Reaction-Diffusion (R-D) framework. Note that the environment of Si-H bond (i.e., whether the resultant Si- bond is a Pb-center or a K-center [23]) or details of the dissociation process does not affect the characteristic predictions (i.e., power-law time exponent, frequency/duty cycle dependence, temperature dependence) of the R-D theory. Originally proposed in 1977 by Jeppson et al. [2] (and later refined by Alam et al. in early 2000s [8, 15, 34, 39, 40]) to interpret fractional kinetics of NBTI degradation, the implications of the R-D model has been explored in hundreds of papers through various generations of CMOS technology. R-D model has a major advantage over its counterparts in explaining interface defect formation, i.e., it provides a parameter-free interpretation of all the four broad universal features of NBTI degradation after nonuniversal and technology-specific hole trapping into oxide defect component is subtracted out [16, 41]. It can predict a long term time dependence of t1/6, long-term Arrheniusactivated temperature dependence, the frequency independence of degradation, and the characteristic shape of duty cycle dependence – regardless of the technology or operating condition. All these have made R-D theory (as discussed in section 3) the starting point for all the discussions of NBTI degradation. 1.2 Hole Trapping into Oxide Defects in NBTI Dynamics of trapping into oxide defects have been studied since 1970s’ [42-44], when Metal-Nitride-Oxide-Semiconductor (MNOS) structure became a popular memory element, because of its simple structure and nonvolatile nature. Detrapping of carriers from pre-existing charged defects in the nitride layers was identified as the main mechanism for limiting the retention time in these memories. Moreover, discharge time

was observed to increase with the increase in oxide thickness of the oxide layer, which proves that tunneling of carriers from charged defects within the nitride layer is gradually reduced with the increase in oxide thickness [42]. Similarly, trapping/detrapping of holes and electrons in/out-of pre-existing oxide defects has also been considered as a source of threshold voltage instabilities in MOS transistors during the same period [45-47]. Now-adays, the incorporation of high-κ materials within the dielectric of MOS structures have caused significant concerns from these hole trapping/detrapping issues during NBTI stress [10, 20, 48-56]. In addition, hole trapping/detrapping phenomena can also give rise to the observation of Random Telegraph Noise (RTN) and 1/f noise in modern transistors having high-κ gate dielectric [57-61]. Hole trapping is generally modeled by considering tunneling of channel carriers into oxide defects. These oxide defects can either be pre-existing [15, 18] or generated by electrical stress [20, 22, 62]. Pre-existing oxide defect is mainly an issue in overcoordinated high-κ (oxynitride or Hf-based) materials [63, 64]. Higher coordination number (i.e., number of nearest neighbors for a particular atom within the material) in high-κ materials makes it difficult to satisfy all the chemical bonding within the amorphous network. Thus, transistors with high-κ dielectric are prone to high density of pre-existing defects and generally suffer from BTI effects due to hole trapping (will be designated using NHT from now on). On the other hand, oxide defect generation (will be designated using NOT from now on) has always been a problem within the reliability community (irrespective of dielectric material) [62, 65-67] that leads to the well-known phenomena, called Time-Dependent Dielectric Breakdown or TDDB. These oxide defect formation has an empirically extracted to have universal nature [68] and is often characterized with a time dependence of t1/3 [69]. We will explain the theory of NHT and NOT in sections 4 and 5, respectively and therefore, show its relevance in explaining experimental features of NBTI that is presented in section 2.

2. Broad Empirical Features of NBTI 2.1 Summary of NBTI Measurements Being one of the major reliability concerns, NBTI has been studied by a large group of researchers all over the world. This breadth of experimental data provides us an opportunity to collect measurements from published reports across industry and academia (see Figure 1) and compare with our own measurements. The comparison is summarized as follows: n

Figure 1a shows the power-law time exponent (when ∆VT is expressed as At ) measured at long stress time (tSTS) in the industrial grade devices. Here, A is a voltage and temperature-dependent constant (see equation (13) for details on this voltage and temperature dependence). In spite of disparate sources, all devices unequivocally show an exponent of n ~1/6, independent of voltage and temperature. Figure 1b shows the stress-phase ultra-fast on-the-fly (UF-OTF) measurements (where time-zero delay [70] t0 is 1µs and ∆VT is estimated ignoring the mobility correction [71], i.e., ∆VT ~ ∆ID/ID0(VG-VT0)) at different temperature for different

oxynitride process splits. The measurements reveal the existence of a short-time, temperature independent fast component (that saturates within ~ms) in transistors having high %N for optimized (Type-A) plasma oxynitride dielectric. Decrease in %N within the dielectric reduces the contribution from this fast component and shows the existence of temperature activated slow component, even at short tSTS. Moreover, at relatively long tSTS > 1 s, all devices show some degree of temperature activation (less for high %N). Note that in un-optimized oxynitride process split (Type-B), the temperature-independence at short tSTS is even observed for low %N. The readers may wish to review reference [72] for an analysis of nitridation process flow on NBTI characteristics. Figure 1c shows NBTI relaxation measurements across different type-A oxynitride process splits (ultra-fast VT or UFV measurement of reference [50] is also shown for comparison). The important point to note here is the initiation of ~5% NBTI relaxation (tREC,start; where ∆VT(tREC)/ ∆VT(tSTS) ~ 95%) and the timedependence of NBTI relaxation. Though several studies [22, 50, 73, 74] on NBTI have reported the universality of log-t relaxation with tREC,start of ~ µs, our UFOTF measurements [16, 75, 76] demonstrate that tREC,start and the timedependence of NBTI relaxation depend on %N of the oxynitride high-κ gate dielectric, as well as on the difference between stress and recovery voltages (VSTS−VREC). In general, tREC,start is larger (~ ms) for low %N and smaller (VSTS VREC), very clearly indicating the non-universal nature of NBTI recovery. In addition, NBTI relaxation data also shows temperature independence at short tREC for high %N oxynitride transistors (Figure 2a,b). And similar to the stress-phase measurements of Figure 1b, the temperature independence at short tREC disappears with the reduction in %N. Figure 1d,e shows a comprehensive summary of duty cycle and frequency dependent NBTI measurement, obtained from a broad range of published reports across industry and academia. When normalized to DC, the duty cycle dependent measurement shows large spread in AC/DC ratio (= ∆VT(AC)/ ∆VT(DC)), although the frequency independence in AC/DC ratio at 50% duty cycle is generally observed (some older datasets show slight drop at higher frequency and most researchers now consider this droop to be a measurement artifact [77]).

0.2

TI data, IEDM'06 tSTR: 1-100Ks O O -25 C 45 C O O 105 C 145 C TSMC data, IRPS'05 O tSTR>1000hr, T=125 C 1/6

-1

∆ VT [Volts]

Value of n

0.3

10

EOX~8MV/cm O 55 C O 125 C 42.5% N

Type-A

-2

10

0.1 1.2

16.7% N

-3

(a) 1.8 2.4 |V [Volts] |VG,STS| |[Volts] STS

Type-B: 5.8% N

3x10 -6 10

-4

10

-2

0

10 10 tSTS [sec]

(b) 2

10

4

10

Open: ~23 %N Crossed: ~35 %N

∆ VT(tREC)/∆ VT(tSTS)

1.00 0.75

tREC,start

0.50

VSTS / VREC (V) -2.3 / -1.8 UF-OTF [24] -2.3 / -1.3 UF-OTF [24] -2.0 / 0.0 UFV

0.25 0.00 -12 10

-9

10

-6

-3

10

10

(c) 0

10

tREC/tSTS

Toshiba ST NS; Purdue/IIT: 10 %N

Infineon TUV

0.8 0.6 0.4 0.2

(d)

0.0 0

50 100 0 Duty cycle [%]

(e) 2

5

8

10 10 10 Frequency [Hz]

Figure 1. (a) Long-term power-law time exponent n (where, ∆VT ~ tn) for NBTI, collected from TSMC and TI measurements (taken from reference [78] and [40], respectively), indicates n ~ 1/6. Note that Freescale [79] and Infineon [80] data also shows similar time exponent. (b) Temperature dependent stress-phase NBTI measurement across different process split of oxynitride gate dielectric (type-A has optimum nitridation; and type-B has non-optimized nitridation) (c) Initiation of NBTI relaxation (tREC,start) varies with VREC and %N of the oxynitride high-κ dielectric (here, the UF-OTF are our measurements and ultra-fast VT or UFV measurement is taken from reference [50]). (d) AC/DC ratio (when ∆VT is measured at the end of AC cycles) vs. duty cycle, and (e) AC/DC ratio vs. frequency (at 50% duty cycle) plots for different technologies indicate wide spread in measured data. Measurements of (d-e) are taken from the following references: Toshiba [81], ST [21], IMEC [77], NUS [82], Infineon [80], TUV [11].

2.2 Modeling Challenge The broad scatter in NBTI measurements (as presented in Figure 1) makes NBTI modeling a considerable challenge. In an effort to address this challenge, NBTI researchers have taken widely different strategies: Some articles have used a wide distribution of capture and emission time constants (from 10-9-1014 s) for fitting each of the stress and relaxation phase measurements with independent distribution [80, 83]. Unfortunately, this approach results in proliferation of fitting parameters that cannot always be physically justified. Other articles have considered wide distribution of defects within the dielectric [12, 13] to capture the long-term nature of the time exponent. In addition, interaction and transfer of chemical species among finite numbers of energy wells [11, 22] is recently used to capture part of the experimental features of Figure 1. However, none of these approaches are comprehensive enough to consistently interpret the four characteristic features of NBTI degradation without using unphysical parameters, as summarized in Figure 1. In an attempt to explain the broad experimental features of NBTI, we ask the following questions: 1) Why is the long-term time exponent always ~1/6, regardless of the stress voltage or stress temperature, as shown in Figure 1a? Note that these results are usually obtained with small measurement delay (order of seconds), which however has insignificant effect when the stress time is very long. Obviously, the use of excessive measurement delay may lead to higher time exponent even at long stress time [84]. 2) Why does the early part of stress and relaxation experiments so sensitive to process details of dielectric material, as shown in Figure 1b,c? Why is process dependent part insensitive to temperature, especially at higher %N, as shown in Figure 1b and Figure 2a,b? And, in the same context, why tREC,start in Figure 1c has process dependence? 3) Why is there a sudden drop in ∆VT(AC)/ ∆VT(DC) in the range of 80-100% duty cycle (Figure 1d)? Why is the shape of duty cycle vs. ∆VT(AC)/ ∆VT(DC) universal (this is more evident, when Figure 1d is normalized with respect to 50% duty cycle value, as shown in Figure 2c) in the lower duty cycle regime? 4) Why is the measured NBTI at 50% duty cycle always frequency independent, irrespective of transistor technology (as shown in Figure 1e)? In this manuscript, we explain NBTI in a broader context by answering the aforementioned four questions, rather than focusing on a smaller subset. Therefore, we decompose measured ∆VT into three uncorrelated components: A. The first and major part of NBTI-induced ∆VT comes from the Reaction-Diffusion theory based interface defect (NIT) generation and relaxation. We will show how this component can explain: i) the universal observation of 1/6 power-law time exponent, ii) the slow and temperature dependent part of NBTI stress and relaxation, iii) the universal part of duty cycle dependence upto ~50% duty cycle, and iv) the universal observation of frequency independence.

Recovered ∆VT [Volts]

B. The second component of NBTI comes from hole trapping into pre-existing oxide defects (NHT). This component can explain: i) the fast and temperature independent part of NBTI stress and relaxation, ii) the sharply decreasing part of duty cycle dependence above ~50% duty cycle, and iii) the magnitude of ∆VT(AC)/ ∆VT(DC) at a particular duty cycle. C. The last component of NBTI comes from hole trapping into generated oxide defects (NOT). This, in addition to NHT, can explain the disparity of tREC,start in different nitrided transistors.

tSTS = 103 sec; VSTS = -2.3V 0.00

0.00

T

(a) 23%N

-0.04 -7 10

-4

10

-0.04

55

0C

85

0C

-1

Weak T dep.

-0.08

(b) 42.5%N

125 0C

-0.12 -7 10

2

10

Strong T dep.

10

-4

10

∆VT(AC)/∆VT(AC,d=50%)

tREC [sec] 3

IMEC ST TUV

2

-1

10 tREC [sec]

2

10

Infenion Ours Purdue/IIT

1 Line: R-D Theory

(c)

0 0

50

100

[%] Duty cycle, d (%)

Figure 2: (a-b) Similar to NBTI stress phase (Figure 1b), early relaxation phase is temperature independent. Such temperature independent early relaxation phase is cleanly observed for high %N transistors. (c) The universal shape of duty cycle dependence (upto d ~ 80%) can be nicely captured by R-D theory. Here, the universal shape of duty cycle is obtained by scaling the duty cycle dependent NBTI measurement with respect to the 50% duty cycle value (see section 7 for the justification). In other words, we express ∆VT using – ∆VT ( t ) = ∆VIT ( t ) + ∆VHT ( t ) + ∆VOT ( t ) q∆N IT (t ) ∫0 ox ∫E x ρ HT ( x , E , t ) dE dx ∫0 ox ∫E x ρOT ( x , E , t ) dE dx =α + + . C ox C oxTox C oxTox T

T

(1)

Here, ∆VIT, ∆VHT, and ∆VOT refer to the contributions to ∆VT from NIT, NHT, and NOT components, respectively; Cox is the oxide capacitance; Tox is the oxide thickness, α accounts for the fraction of donor type [85] NIT above the substrate Fermi-level that is contributing to NBTI; ρHT(x,E,t) represents trapped holes into the pre-existing oxide defects at location x (measured into the oxide from the poly/oxide interface) and at energy E at time t; and ρOT(x,E,t) represents trapped holes at generated oxide defects. 3. Theory of Interface Defect (NIT) In this section, we summarize the main features of R-D theory, considering both atomic (H) and molecular (H2) diffusion. Our goal is to show how the theory provides a parameter-free prediction of most of the experimental features of NBTI. R-D theory with H-H2 diffusion considers dissociation of Si-H bond and subsequent diffusion of hydrogen species, as governed by the following equations [15, 86]:

Si sub.

dN IT = k F ( N 0 − N IT ) − k R N IT N H(0) , dt (0) 2 δ dN H dN H(0) dN IT = DH + − δ k H N H(0) + δ k H 2 N H(0)2 , 2 dt dx dt (0) (0) 2 δ dN H 2 dN H 2 δ δ = DH 2 + k H N H(0) − k H 2 N H(0)2 , 2 dt dx 2 2 2 dN H d NH = DH − kH N H 2 + kH 2 N H 2 , 2 dt dx 2 dN H 2 d NH 2 1 1 = DH 2 + kH N H 2 − kH 2 N H 2 . 2 dt dx 2 2

Si

x

Si

x

Si

H H

(3) (4) (5) (6)

H2

H2 H

H

(2)

H2 H

H2

NH, NH2

~ 0

Dt x

Figure 3: Schematic of Si-H bond dissociation and consequent hydrogen diffusion, as considered in R-D theory based NIT dynamics. Equation (2) represents passivation/de-passivation effects of Si-H bond, where kF, kR, N0, NIT, NH(0) are defined as Si-H bond-breaking rate, Si-H bond-annealing rate, initial bond density available before stress, interface defect density, and hydrogen density at the Si/dielectric interface, respectively. Equations (3) and (4) correspond to the conservation

of fluxes of diffusing hydrogen species (H and H2) near the interface (along the x axis), whereas equations (5) and (6) describe diffusion (along the x axis) of H and H2. k H N H 2 and k H 2 N H 2 terms in equations (3)-(6) incorporate the H-H2 conversion within the generalized R-D framework. Among the symbols used in equations (3)-(6), kH and kH2 represent generation and dissociation rates of H2; DH and DH2 represent diffusion coefficients for H and H2; NH and NH2 represent the concentration of atomic and molecular hydrogen; δ represents the interfacial thickness (~1-2 Å). All parameters are greater than zero for the stress phase. In particular, kF has the following dependence of equation (7) that serves a physical way of explaining the oxide electric field Eox dependence of interface defect generation [15, 40, 87, 88]:

k F ~ ph * N 0 PT *exp(γ T Eox ) *exp ( aEox / kT ) ,

(7)

where ph is the hole concentration within the inversion layer, PT ~ exp(-√moxφbh) is the field-independent pre-factor for hole tunneling probability (mox: oxide effective mass and φbh: barrier height for hole tunneling), exp(γTEox) is the field-dependent factor for hole tunneling with field acceleration γT, and exp(aEox/kBT) is field-assisted Si-H bond dissociation enhancement factor (a: effective dipole moment and kBT: thermal voltage). Thus, field acceleration of NIT can be expressed as –

γ IT = γ T + a / k BT .

(8)

As shown in [15, 40], equation (8) explains the temperature dependence of NIT’s field acceleration (hence provides an experimental way of extracting a and γT). In addition, equation (7) can be used to explain the nitridation process dependence [87] and strain dependence [88] of NIT in NBTI measurements. Now, let us derive the key results of R-D model and show how it anticipates the key experimental features of NBTI. 3.1 Power-law time exponent of n ~ 1/6: Assuming N0 >> NIT and dNIT/dt ~ NIT/t, equation (2) simplifies to,

N H(0) =

kF N 0 − N IT t . kR N IT

(9)

Moreover, the numerical solutions indicate that for continuous NBTI stress, dNH(0)/dt and diffusion of H is negligible at all stress time [15], so that H2 diffusion part in equation (3) reduces to,

N IT = δ kH N H(0)2 − δ kH 2 N H(0)2 , t and the conservation of hydrogen species within the system suggests –

(10)

N IT ≈ N H(0)2 6 DH 2t .

(11)

Equation (11) requires that the extent of diffusion profile is larger than DH t , commonly used for approximating the complementary error function solution of diffusion equation [8, 34, 89]; which is more consistent with the use of 16 DH t / π in [90, 91]. Now, by eliminating NH(0) and NH2(0) from equations (9)-(11), we have –

N IT δ kH ( kF N 0 − N IT t ) δ kH 2 N IT − + = 0. t kR 2 N IT 2 6 DH 2t 2

(12)

Equation (12) is the (implicit) analytical solution of H-H2 R-D model, presented in equations (2)-(6), which compares very favorably with the detailed numerical simulation [15]. At long stress time (i.e., in the H2 diffusion limited regime), equation (12) simplifies to (when NIT/t becomes negligible) – 1/3

N IT

k = H kH 2

kF N0 kR

2/3

( 6 DH 2t )

1/6

≡ At n=1/6 .

(13)

Therefore, R-D theory anticipates the long-term time exponent of n ~ 1/6. In addition, R-D theory also establishes the fact that n should not depend on stress voltage and temperature. Both of these observations are consistent with the broad range of NBTI measurements (Figure 1a). 3.2 Long-term Temperature Dependence: Using appropriate activation energies for kF, kR and DH2 in equation (13) and assuming H-H2 conversion process has similar activation for forward and reverse reactions [92], the overall activation energy for ∆VIT = qNIT/Cox can be written as [15],

E A, IT ≡ nE A, H 2 + 2 3 ( E A, F - E A, R - aEox ) .

(14)

where EA,F, EA,R, EA,H2 are activation energies for kF, kR and DH2, respectively. Experimentally, one can estimate EA,IT ~ 0.1 eV by measuring NBTI in transistors having dominant interface defect generation (i.e., ∆VT ~ ∆VIT) [18, 32, 40, 50, 85, 88, 93, 94]. We have also shown that the variation of EA,IT due to aEox term comes within the error margin of activation energy estimation procedure [87]. This measured value of EA,IT and reported magnitude of EA,H2 ~ 0.6 eV [95] suggests that EA,F ~ EA,R. Thus, activation energy of molecular hydrogen diffusion governs the long-term temperature dependence of NBTI measurements. 3.3 Duty-cycle and Frequency Dependence: In order to obtain the duty cycle and frequency dependence of NIT, we derive the amount of NIT under AC NBTI stress condition. Here, we assume that the NBTI stress is

applied in a transistor for k (>> 1) cycles with 2k stress/relaxation steps, defined by the duty cycle d = tSTS/T; where T = tSTS + tREC is the period of the signal. We also define that the degradation at the end of k-cycles (with respect to degradation over a single stress cycle) using R2k ≡ NIT(kT)/NIT(tSTS) and R2k-1 ≡ NIT((k-1)T + tSTS)/NIT(tSTS) with R1 = 1, by definition. Following the analysis in Refs. [89] and [96], one can show that –

( R2k −1 )

1/ n

≈

1/ n k −1 k + ( R1 ) ≃ ≡ pk , 1 + ξ (1 − d ) 1 + ξ (1 − d )

(

(15)

)

or equivalently, R2k-1 ≈ R2k ≈ (pk)n, where p = 1/ 1 + ξ (1 − d ) . If two transistors are stressed at two different frequencies (f1 = 1/T1 and f2 = 1/T2) at same duty cycle, then for long (but same) integrated stress times (i.e., k1T1 = k2T2 or, k1tSTS,1 = k2tSTS,2), the ratio of net degradation for the two transistors is given by: n N IT , f1 ( k1T1 ) Rk1 N IT ( tSTS ,1 ) k1 tSTS ,1 = ≈ = 1; N IT , f1 ( k2T2 ) Rk 2 N IT ( tSTS ,2 ) k2 tSTS ,2 n

(16)

which is frequency-independent (as observed in Figure 1e). And similarly, the ratio of AC NBTI degradation at a given frequency for total duration of TT = k1T1 compared to the DC NBTI degradation for the same period of time is given by –

N IT , f1 ( k1T1 ) ( pk1 ) A (tSTS ,1 ) d ; = = n N IT , DC (TT = k1T1 ) A ( k1T1 ) 1 + ξ (1 − d ) n

n

n

(17)

which gives the characteristic duty cycle dependence of NBTI degradation (line in Figure 2c). We defer the discussion of the scaling algorithm till section 6.3, which is used in Figure 2c for the experimental data scaling. Thus, R-D theory can explain the universal observation of frequency independence (Figure 1d) and the characteristics trend of duty cycle dependence upto d ~ 80% (Figure 2c) for all NBTI experiments, measured in wide variety of process splits.

4. Theory of Hole Trapping into Pre-existing Oxide Defects (NHT) Let us now return to the second component of NBTI in equation (1) related to the charging of pre-existing oxide defects. Pre-existing defects are commonly observed in almost all high-κ dielectrics (like oxynitrides and Hf-based compounds), which is an inherent feature of such over-coordinated high-κ materials. Trapping into these preexisting defects leads to ∆VHT, independent of the generation of NIT or NOT. Here, the thinness of modern dielectric film (~ few nm) requires us to consider the effect of detrapping towards the transistor gate [15]. In addition, typical defect density (~ 1018-1019 cm-3, i.e., 1 trap in 100-1000 nm3) ensures that trap-to-trap transport is implausible.

Therefore, we only need to consider the processes involving single hop to and from the traps, i.e., dfT ( x, t ) dt

∞

= ∫ dET σ HT vth phTS →T (1- fT ) − nS TT → S fT − nGTT →G fT ,

(18)

−∞

where ph is the inversion layer hole density; vth is the thermal velocity; nS, nG are the concentration of detrapping states at substrate and poly-Si respectively; T’s are tunneling probabilities between substrate (S), defect/trap (T), and gate (G) of the MOS transistor; and σHT is the capture cross-section for the hole trapping/detrapping process. The first term in equation (18) represents hole trapping into pre-existing oxide defects from the substrate, whereas the second and third terms represent hole detrapping out of oxide defects towards substrate and gate, respectively. These hole capture (during trapping process) and release (during detrapping process) events can be elastic or inelastic, as discussed in detail in reference [97]. Stochastic nature of such hole capture and release events gives rise to random telegraph noise, as studied in references [14, 57-61]. In the following discussion, we focus on the implication of elastic/inelastic hole trapping into thin dielectric and study its implication in long-term NBTI degradation. The readers may wish to review reference [14] for understanding the short-stress features of hole trapping. Solution of equation (18) suggests (see reference [97] for details) the followings:

Time dependence: The role of detrapping towards the poly-gate – while negligible in thick films (as for the case in [42, 98]) – are fundamentally important in the trapping dynamics of thin films. In ultra-thin dielectrics of current CMOS technology, detrapping process limits the possibility of hole trapping in sites located near the gate. As a result, threshold voltage shift due to hole trapping into preexisting defects should saturate within orders of milliseconds, where the saturation time depends on the dielectric quality (i.e., nitridation) and thickness. As such, the corresponding threshold voltage shift can be approximated as –

β ∆VHT = AHT 1 − exp ( −t / τ ) ,

(19)

where AHT, τ and β are trapping parameters that depends on the dielectric quality and dielectric thickness. In general, τ and β are larger for larger dielectric thickness and AHT is larger for higher %N within the dielectric (i.e., for the dielectric having more NHT).

Voltage dependence: Voltage or field dependency of hole trapping arises from the relative position of the quasi-Fermi level EFS within the dielectric as a function of oxide electric field Eox. When Eox is reduced, more trapping sites below EFS are filled with electrons (see the change of trapping sites fT values from Figure 4b to Figure 4c), with corresponding reduction ∆VHT. As a result, hole trapping process shows significant voltage dependency.

Temperature dependence: Since tunneling is a temperature independent process, temperature dependency of hole trapping mainly depends on the temperature dependence of capture cross-section (σHT) and thermal velocity (vth). Recently, many hole trapping models [22, 59] consider structural relaxation as a part of the hole

trapping process. Structural relaxation causes temperature activation (which depends on the energy barrier that is required to overcome for achieving structural relaxation) in the trapping process and can effectively be incorporated within the σHT parameter of equation (18). As the hole trapping process is fast for modern transistors having thin gate dielectric, we can attribute the ultra-fast component of NBTI stress phase (in Figure 1b) to ∆VHT. The increase of this ultra-fast component with the increase in nitrogen within the silicon oxynitride (SiOxNy with 2x+3y=4 [99]) dielectric further confirms that this NBTI component is indeed related to pre-existing oxide defect. Such increase of oxide defect with nitrogen within the SiOxNy dielectric is routinely reported in literature [17, 18, 21, 100]. Moreover, our measurements suggest relatively weak temperature dependence for this ultra-fast ∆VHT component (Figure 1b), which indicates that hole trapping in our transistors have temperature independent σHT. Finally, as ultra-fast NBTI relaxation (Figure 1c) has similar features like the one for ultra-fast NBTI stress measurements, we expect initial part of NBTI relaxation to be related to an equivalent temperature independent, but nitrogen-dependent hole detrapping process. Therefore, like the stress phase, hole detrapping process in the relaxation phase will be complete within tREC = orders of ~ ms, depending on the nitrogen content within the oxynitride dielectric. VG - 2.1 V (a) - 1.0 V

2000 sec

1000 sec

1 fT

fT Substrate

Oxide

Gate

EFG 4

0.8 0.6 0.4

2 EFS 0.2

Substrate

5

Oxide

Gate

0.8 0.6

4

EFG

3 2 EFS

Empty of holes

1

0.4 0.2

(c)

(b)

0

6 ETrap [eV]

ETrap [eV]

6

Time

-20

0 xTrap [A]

20

0

0

-20

0

20

0

xTrap [A]

Figure 4: (a) Timing diagram used in the hole trapping simulation for a particular transistor having physical oxide thickness TPHY ~ 2.3 nm, effective oxide thickness EOT ~1.35 nm, and 39.22 %N2 dose (measured using XPS). The dielectric parameter (hole effective mass, oxide bandgap, and barrier height) for the simulation is calculated following the procedure stated in reference [87]. (b) Occupancy of defects or, hole trapping sites (assumed to have existence within the colored region only), after the transistor is kept at VSTS = -2.1 V for 1000 s. (c) Occupancy of hole trapping sites after the transistor is switched to VSTS = -1.0 V at 1000 s and kept at that voltage up to 2000 s.

5. Theory of Hole Trapping into Generated Oxide Defects (NOT) NBTI stress is also a TDDB stress in p-MOSFETs. Therefore, we also need to consider hole trapping/detrapping from newly created bulk oxide defects NOT, and corresponding ∆VOT, at NBTI stress condition. In stress phase, the trapping process follows the same equation (18). However, unlike NHT, in this case (slower) oxide defect generation is the rate-limiting process, rather than the hole trapping. Therefore, contrary to ∆VHT, ∆VOT is expected to show non-saturating behavior, proportional to the timedependent increase of bulk defect density. Indeed, Figure 5 confirms the existence of universal ∆VOT, even at NBTI stress conditions, as follows: Figure 5a indicates an increase in NBTI time exponent at higher VSTS, thus indicating a signature of oxide defect generation [62] or ∆VOT at higher VSTS and negligible ∆VOT at lower VSTS. Moreover, as the measurement scheme for [78] is slow, we expect the ∆VHT component in ∆VT to be negligible. Therefore, at lower VSTS, we can presume ∆VT ~ ∆VIT and calculate field acceleration for ∆VIT. Later, using the field acceleration for ∆VIT, we can calculate ∆VIT and thereby find ∆VOT = ∆VT – ∆VIT at higher VSTS (Figure 5b). The calculated ∆VOT for the measurements in [78] shows a decrease in time exponent for ∆VOT at higher VSTS, as also observed in [68]. Interestingly, the estimated ∆VOT at different VSTS scales universally and can be fitted (see Figure 5c) using a dispersive bulk defect generation model [68, 101] –

∆VT [a.u.]

VG5 VG4

field accelera tion of ΔVIT

VG3 VG2 VG1

(a)

∆VOT [a.u.]

Solid line – Dashed Line

n > 1/6

VSTS 2.3V 2.5V 2.7V 0

n ~ 1/6

(b) EOT~ 1.2 nm; T ~ 125 C

tSTS [a.u.]

∆VOT [a.u.]

tSTS [a.u.]

Fitting parameters: σ = 0.1 eV n = 3.38 -6 kf0 = 3x10

(c)

VSTS

Scaling

factor 2.3V 1.0 2.5V 10.0 2.7V 100.0

Scaled tSTS [a.u.]

Figure 5: (a) Time exponent of NBTI degradation (fitted solid lines) increases at higher NBTI stress voltage of VSTS ≥ VG4 [78]. (b) Extracted ∆VOT for the

measurements in [78] shows a decrease in time exponent for ∆VOT at higher VSTS. (c) Estimated ∆VOT at different VSTS scales universally (similar to the observation in [68]) and can be fitted using equation (20).

∫ n ( E ) dE , g ( E ) 1 − exp ( − k ( E ) * t ) ,

N OT = nOT ( E ) =

OT

F

k F ( E ) = k F 0 exp − ( E − E0 ) / kT , g (E) ~

1

exp ( E − E0 ) / σ OT

σ OT 1 + exp ( ( E − E ) / σ ) 2 OT 0

(20) ,

where E0 is the average energy of bond dissociation that is leading to oxide defects and σOT is its standard deviation. Therefore, ∆VOT is another component that we need to consider for NBTI stress phase. In the NBTI relaxation phase, trapped holes from the oxide defects will detrap in a manner similar to the detrapping from pre-existing oxide defects. So, hole detrapping in the NBTI relaxation phase both from the pre-existing and generated oxide defects can be handled by using a single detrapping formula, as done in reference [41].

6. Interpreting NBTI Measurements Our theoretical understanding of sections 3, 4 and 5 enables us to isolate the ∆VIT, ∆VHT and ∆VOT components of ∆VT. Note that without such decomposition, it is impossible to interpret and model the dynamics of ∆VT (and its components), measured during NBTI stress. Without performing such decomposition of interface and oxide defect components, recently several efforts [13, 14, 22, 51, 73, 102] have arraigned the RD theory to be inconsistent with NBTI relaxation (for an example of the purported inconsistency, see Figure 6a). Such alleged inconsistency has raised questions regarding the general validity of the R-D theory. Many alternative theories [12, 13, 21, 22] have also been proposed to explain ultra-fast NBTI relaxation, although their ability in predicting the broad features of NBTI remains questionable (see section 7 for further details). 6.1 Interpreting NBTI Stress Phase Measurements If we decompose ∆VT into its ∆VIT, ∆VHT and ∆VOT components [41] by calculating 0.3 ∆VIT using equation (12), ∆VHT using equation (19), ∆VOT using ~t [69], the voltage and temperature dependency of extracted components show remarkable consistency for transistors having wide variation in dielectric material. The analysis shows high-fidelity matching for stress and relaxation phase NBTI measurements and suggests that in the 1/6 range of tSTS ~ 1-1000 s, ∆VIT ~ t , ∆VHT ~ constant, and ∆VOT is quite small (Figure 6b). Based on these, we can approximate equation (1) in the range of tSTS ~ 1-1000 s with –

∆VT(tREC)/∆VT(tSTS)

1.0

16.7% N2; EOT ~ 2.35nm

Stress

VSTS/VREC=-2.9V/-1.3V

(b)

0.5

tSTS=10s

~ 5%

∆VIT

(a) ∆VHT

Data RD theory

0.0 -7 10

Relaxation ∆VT

∆VOT -4

10

-1

10 tSTS [sec]

2

10

tREC,start tNIT,start Time

Figure 6: (a) Naive (and unphysical) comparison of the time evolution of measured fractional NBTI relaxation at short tREC shows significant inconsistency with R-D solution. (b) Resolution between the measured ultra-fast ∆VT relaxation and comparatively slower NIT relaxation becomes possible, after decomposing ∆VT (solid line) into ∆VIT (dotted line), ∆VHT (dashed line) components; each having separate time-dynamics.

∆VT ≃ ∆VIT + ∆VHT = AIT t 1/ 6 + BHT .

(21)

We use equation (21) and estimate ∆VIT by subtracting a constant (saturated) ∆VHT ~ BHT from ∆VT for tSTS > 1 s in such a way that it provides time exponent n ~ 1/6 for ∆VIT at tSTS of 1-1000 s (Figure 7a). Here, the effect of mobility [71, 103] and electric fieldreduction [15] is taken into account in estimating ∆VT. Next, we repeat the decomposition at different temperature for same Eox and tSTS, so that we can extract the activation energy of the ∆VIT and ∆VHT components. Figure 7b shows the extracted ∆VIT and ∆VHT component at different temperature, which suggests EA for ∆VIT (EA,IT) is ~ 0.094 eV; which is expected for R-D model based interface defect generation with H2 diffusion (see section 3.2); on the other hand, EA for ∆VHT or EA,HT is ~ 0.04 eV, which is typically expected in any hole trapping process, involving tunneling and no structural relaxation. The signature of EA,IT > EA,HT is also evident from Figure 7c, which indicates an increase in the extracted ∆VIT/∆VT (i.e. decrease in ∆VHT/∆VT) with increase in temperature, at fixed tSTS. Identical procedure is followed to isolate ∆VIT and ∆VHT for the transistors of Figure 8, at different voltages and temperatures. The extracted EA,IT (supported by R-D theory of section 3) and EA,HT (supported by hole trapping theory of section 4) is similar for these transistors (see Figure 8a). Moreover, extracted ∆VHT for these nitrided transistors at a particular Eox, TSTS and tSTS (Figure 8b) is observed to increase significantly with the increase in %N (with a very rapid increase seen for %N > 30), which is indeed a signature of higher hole trapping for higher %N. On the other hand, ∆VIT only increases slightly with %N, which indicates that there is negligible change in the Si-H bond dissociation mechanism and hydrogen diffusion dynamics with the change in %N. Therefore, the extracted parameters for both ∆VIT and ∆VHT are consistent with the theoretical predictions.

EOT=1.4nm; 23 % Plasma N2 0 T=125 C VSTS= 2.1V ∆VT

∆VIT

6x10

∆VHT

NBTI Components [V]

NBTI Components [Volts]

-1

10

n=0.122+0.005 n~0.16

-2

10

(a) 0

10

1

2

10 10 tSTS [sec]

Fractions of ∆VT

1.0

EOT ~ 1.4nm; 23%N2 Dose ∆ VT

VSTS=-2.1V; tSTS=100s

∆VIT ∆VHT

EA=0.078eV

10

EA,IT=0.094eV

-2

EA,HT=0.04eV

3x10

3

-2

10

(b)

-3

25

30 35 -1 1/kBTSTS [eV ]

VSTS = -2.1V; tSTS = 103 sec

0.8 VIT fraction

0.6 0.4

VHT fraction

0.2 0.0

(c) (c)

50

80 110 0 Temperature [ C]

140

Figure 7: (a) Measured ∆VT and extracted ∆VIT and ∆VHT components for a transistor having optimized nitrided dielectric. (b) Temperature dependence and corresponding activation energies (EA) of ∆VT, ∆VIT and ∆VHT components for the same transistor. (c) Estimated ∆VHT/∆VT and ∆VIT/∆VT at different temperature (TSTS) indicate an increase (decrease) in ∆VIT/∆VT (∆VHT/∆VT) at higher temperature. Here, the error bars represent the noise in ID,lin0 measurement for OTF-ID,lin [104], which causes a ± 0.005 error in n for ∆VT and a ± 1mV error in estimated ∆VHT. 0.003

0.10

∆VIT

0.05

∆VHT

(a) 0.00 20

30 %N (atomic)

40

Degradation / EOT [a.u.]

Activation Energy [eV]

0.15

0.002

O

ttstress =100s, Tstress =125 C 0C TSTS = 125 STS = 100sec; EOX= 8MV/cm ∆VTT ∆V ∆VIT ∆V IT ∆VHT ∆V HT

0.001

0.000

(b) 20

30

40

%N (atomic)

Figure 8: (a) Activation energy for extracted ∆VIT and ∆VHT components indicates negligible %N dependence. (b) %N Dependence of measured ∆VT and extracted

∆VIT and ∆VHT components for a particular tSTS, TSTS and Eox. Lines are guide to the eye only. As shown in reference [41], consideration of ∆VOT in estimating ∆VIT and ∆VHT from ∆VT merely changes the signatures of ∆VIT and ∆VHT, presented in Figure 7 and Figure 8. In that case, estimated ∆VIT, ∆VHT, and ∆VOT shows good consistently with the theoretical expectations and suggests that ∆VOT is a component that one should consider at higher stress bias. 6.2 Interpreting NBTI Relaxation Phase Measurements Let us now apply the same NIT/NHT/NOT decomposition for NBTI relaxation measurements on the same transistor that we have previously analyzed through stressphase decomposition in Figure 7a, which also has the temperature independent NBTI relaxation for tREC < ms (see Figure 2a). Here, we further use the observation of Figure 9a, which suggests that for VSTS = -2.3 V, hole detrapping occurs (i.e., tREC,start shows sudden decrease) predominantly at VREC ≥ -1.8 V. Figure 9a also suggests that the amount of hole detrapping is similar from VREC = −1.3 V to −1.6 V for the minimum tREC ~ µs measured in this experiment.

∆VT(tREC)/∆VT(tSTS)

1.0

R-D: -2.0V

3

-1.6V -1.3V

0

TSTS = 125 C (a) VREC Dependence: ∆VT (a) -3

10

Gate

-1.8V

VSTS=-2.3V

0.0 -6 10

Oxide

0

10 tREC [sec]

Substrate Oxide

Gate

Hole trapping sites

R-D: -1.3V -2.0V

tSTS=10 sec

0.5

Substrate

EC

EC

EF

EF

EV

EV

(c)

(b)

Hole detrapping sites

3

10

Stress (VSTS)

Relaxation (VREC)

Figure 9: (a) NBTI relaxation experiments at different VREC show significantly different tREC,start. Since relaxation for VREC ≤ -1.8 V is very close the R-D theory, there is an additional relaxation mechanism for VREC > -1.8 V. Our observation suggests hole detrapping to be the additional mechanism. (b) Schematic (based on simulation within a Shockley-Read-Hall trapping-detrapping framework; section 4) for expected hole trapping sites (hatched region) at VSTS. (c) When gate bias is switched from VSTS to VREC, the hatched region will detrap the captured hole in a temperature independent manner. Thus, trapping sites within the quasi-Fermi levels at VREC = −1.8 V and −1.6 V (shown schematically by the hatched region in Figure 9c) will detrap all the holes that were captured by the oxide defects during stress within a timescale of ~ms. Considering such total hole detrapping at VREC= −1.3 V both from the pre-existing and generated oxide defects (here, ∆VOT is considered to be ~ 3 mV at tSTS =103 s, having the time dependence similar to Figure 5c), the resultant NIT component of the NBTI relaxation

experiment shows excellent consistency with R-D theory in terms of tNIT,start at all VREC (see Figure 10a). To understand the contribution of ∆VOT on tNIT,start, we redo the experiment-theory comparison by assuming ∆VOT ~ 0. Figure 10b suggests that consideration of finite ∆VOT increases tNIT,start by an order of magnitude. The remaining theory-experiment gap in terms of time dependence for tREC > tNIT,start reflects the inability of 1D diffusion formulation (considered in classical R-D model of section 3) to capture the details of an essentially 3D diffusion problem. Consideration of 3D diffusion ensures high-fidelity matching for the time dependence for tREC > tNIT,start [41]. Therefore, decomposing the contributions from interface and oxide defects enables us to explain the NBTI relaxation features in a theoretically consistent way.

0.5

0.0 -6 10

Lines: R-D Theory

1.0 -2.0V

3

tSTS= 10 sec VSTS= -2.3V

-1.8V

0

TSTS = 125 C

-1.3V

(a) VREC Dependence: ∆VIT, ∆VHT, VREC Dependence: ∆VIT ∆VOT(b) considered

10

-3

0

10 tREC [sec]

3

10

∆VIT(tREC)/∆VIT(tSTS)

∆VIT(tREC)/∆VIT(tSTS)

1.0

R-D: -2.0V R-D: -1.3V -2.0V

3

tSTS=10 sec

0.5

VSTS=-2.3V

-1.8V 0

TSTS = 125 C

-1.3V

(b) VREC Dependence: ∆VIT, ∆VHT (a) VREC∆VDependence: ∆VIT considered; OT ignored.

0.0 -6 10

10

-3

0

10 tREC [sec]

3

10

Figure 10: (a) (b) Consideration of ∆VIT, ∆VHT, and ∆VOT and their decomposition indicates excellent consistency of tNIT,start with R-D theory. (b) Neglecting ∆VOT reduces tNIT,start by an order of magnitude. Here, the error bar along the x-axis for VREC = -1.3 V is due to the ID,lin0 error in calculating ∆VIT/∆VT (see Figure 7). 6.3 Interpreting AC NBTI Measurements So far we have applied the decomposition procedure to identify the interface and oxide defect components of NBTI-induced ∆VT. Our analysis demonstrates that hole trapping and detrapping occur at similar time-scales (for example, compare the time-scale of the temperature independent hole trapping component in Figure 1b, and hole detrapping component in Figure 2b). Therefore, we expect total hole detrapping for a AC NBTI stress (measured at the end of OFF-state) with ≤ 50% duty cycle (Figure 11a). In other words (see Figure 11b), AC/DC ratio for ≤50% duty cycle in high %N transistors will measure ∆VT,AC/∆VT,DC ~ ∆VIT,AC/[∆VHT,DC + ∆VIT,DC] (considering ∆VOT component is quite small) and hence will always be less than the contribution from NIT’s component, AC/DC(NIT) = ∆VIT,AC/∆VIT,DC, predicted by R-D theory. Moreover, as ∆VHT decreases for smaller %N, low %N transistors will have ∆VT,AC/∆VT,DC ~ ∆VIT,AC/∆VIT,DC, and thus the

measured AC/DC ratio for low %N transistors should be consistent with the one obtained from the R-D theory. Time 1

∆VT ∆VIT

∆VHT

(a)

AC/DC Ratio

R-D: ∆VIT

VAC

Low %N High %N (b) 0 0%

50%

100%

Duty Cycle

Figure 11: (a) Since hole trapping and detrapping happens at similar time-scale, we expect total hole detrapping at the end of OFF-state during AC NBTI stress. Therefore, (b) AC/DC ratio (with ∆VT for AC NBTI measured at the end of AC cycles) will be consistent with the R-D theory only for transistors having lower ∆VHT or low %N. Indeed, our ∆VT,AC/∆VT,DC measurement on nitrided transistors show that for the lowest %N oxynitride dielectric (where ∆VT ~ ∆VIT), the predictions from R-D theory matches very well with the experimental data (Figure 12a). On the other hand, transistors with higher %N oxynitride dielectric have significant contribution from ∆VHT in DC NBTI stress, i.e., ∆VT,DC > ∆VIT,DC. As a result, ∆VT,AC/∆VT,DC is lower than ∆VIT,AC/∆VIT,DC (or the line predicted by the R-D theory) in such higher %N transistors. However, because AC NBTI stress at lower duty cycle predominantly reflects ∆VIT, the shape of ∆VT,AC/∆VT,DC vs. duty cycle plot is similar to the R-D’s prediction at lower duty cycle for all transistors. In other words, equation (17) is universal at lower duty cycle for any transistors, irrespective of %N within the dielectric. Such universal shape of ∆VT,AC/∆VT,DC vs. duty cycle dependence is evident in Figure 2c, where AC data at all duty cycle is scaled to d ~ 50% data to capture the intrinsic ∆VIT behavior (as total hole detrapping occurs at d ~ 50%). The scaling confirms a robust universality of the dutycycle data from diverse sources, even up to ~80% duty cycle. This indicates that, for the transistors of Figure 2c, time scale for hole-detrapping is much less than the time scale for hole trapping. At higher duty cycle, detrapping of holes is incomplete, and therefore the experimental data deviates from the prediction of R-D theory. Consideration of total hole detrapping for ~50% duty cycle also explains how R-D theory interprets the widely observed frequency independence in all NBTI measurements, irrespective of the type of dielectric, while overestimating the measured ∆VT,AC/∆VT,DC at different frequencies (Figure 12b and Figure 1e). Since ∆VHT will be absent for AC NBTI stress with ~50% duty cycle, the frequency independence of ∆VIT,AC/∆VIT,DC (following R-D theory; equation (16)) will also result in frequency independence of ∆VT,AC/∆VT,DC, irrespective of %N. However, for transistors with larger %N, the magnitude of ∆VT,AC/∆VT,DC will be lower than ∆VIT,AC/∆VIT,DC due to the presence of ∆VHT in ∆VT,DC. Therefore, ∆VT,AC/∆VT,DC measurement on nitrided transistor show frequency

independence, irrespective of %N (Figure 12b); while the magnitude of ∆VT,AC/∆VT,DC is only consistent with R-D theory for low %N transistors, when the hole trapping contribution is negligible. Thus, the co-existence of interface and oxide defects and their decomposition can explain both the duty cycle and frequency dependent NBTI experiments on nitrided transistors.

0

TSTS = 125 C

3

1.0

Shape similar to R-D

R-D 0.5

Higher %N

(a) 0.0 0

25

∆VT(AC)/∆VT(DC) @ tSTS=10 sec

3

∆VT(AC)/∆VT(DC) @ tSTS=10 sec

Low (10) %N 0

TSTS = 125 C

1.0

R-D Theory

0.5 Higher %N

0.0

(b) DC

10

0

10

3

10

6

10

9

Frequency [Hz]

Figure 12: (a) AC/DC ratio (when ∆VT is measured at the end of AC cycles) vs. duty cycle plot for different nitrided transistors. The experiments show remarkable consistency with the prediction of R-D theory (solid line) for low %N. (b) Though AC/DC ratio for any %N is always frequency independent, there is significant %N dependency due to the presence of ∆VHT in DC NBTI stress.

7. A Critique of Alternate NBTI Theories Contrary to the uncorrelated dynamics for interface and oxide defects that is used in this paper to explain the experimental features of NBTI, correlated model of these defects has also been recently used [22, 105] to explain the so-called universally observed temperature-dependence at arbitrary time scale. Actually, we observe such universal temperature dependence only in transistors having low %N [15, 16, 18, 19] (see Figure 1b), but not in all transistors as claimed in references [22, 105]. Moreover, the energywell theory [11] behind such correlated mechanism of NIT and NHT fails to explain the basic features of NBTI degradation, such as the existence of stress-phase power-law time exponent, frequency independence [106, 107] - features that has been universally observed in all SiON transistors (see Figure 1). Indeed, as shown in Figure 13, the powerexponent predicted by energy-well theory (or any of its recent versions [22]) is fragile and its range changes by orders of magnitude with small changes in the parameters like broadening of well-barrier σWB. In addition to this energy-well theory, some references consider NBTI (merely) as a trapping-detrapping problem into oxide defects [12, 13]. However, such theoretical analysis requires one to use a wide distribution of hole trapping sites (either spatially [12, 13] or energetically) for capturing the stress-phase power exponent and relaxation

dynamics. Experimental support for such wide distribution of hole trapping events (for explaining NBTI time dynamics over more than 15 decades) is still lacking in literature. 1.01

σWB

0.1 eV

0.9

0.7

Time Exponent

Time exponent

0.8

(a)

0.6

0.5 0.5

0.3 0.2 0.1

State 1

State 2

State 3

0.2 eV

0.3 eV

0.4

00

(b)

σWB = 0.4 eV -4

10-4 10

-2

10-2 10

0

2

10 0 102 10 10 Time [sec]

time [sec]

4

10 4 10

Figure 13: (a) Energy-well model (like the one presented in reference [11]) for explaining the dynamics of NBTI. Such energy-well model or its recent derivatives [22], in general, requires one to use a spread in barrier height (having a standard deviation of σWB) for capturing the NBTI dynamics. (b) Unless we use large σWB, it becomes difficult to obtain a robust power-law time exponent using such energywell model.

8. Conclusion In this paper, we have highlighted the importance of careful analysis of NBTI experiments, before comparing it with any theory. Our analysis demonstrates the uncorrelated dynamics of threshold voltage shift associated with interface and oxide defects can adequately explain the questions stated in section 2.2 of the manuscript. The universal features of DC and AC NBTI (i.e., long-term stress-phase power-law time exponent, long-term stress/relaxation-phase temperature dependence, characteristic feature of duty-cycle dependence, and universal feature of frequency independence) can be consistently explained using R-D theory based interface defect dynamics. The remaining non-universal features (i.e., short-term stress/relaxation-phase time dependence, short-term stress/relaxation-phase temperature independence, and the magnitude of AC/DC ratio at a particular duty cycle and frequency) are shown to be directly correlated to the amount of oxide defects within the dielectric. However, these non-universal features disappear for less than ~80% duty cycle and hence have no relevance for practical AC NBTI analysis. Therefore, a decomposition of these (uncorrelated) universal and non-universal components is established as a necessity before comparison with any theory. And without such decomposition, one will only be able to capture part (not all) of the experimental features of NBTI.

Acknowledgments The work is an accumulation of our last few years’ research on NBTI. It has been immensely benefited from the discussion with/contribution from D. Varghese, N. Kumar,

H. Kufluoglu, A. Krishnan, J. H. Lee, A. Oates, A. Jain, K. Ahmed, C. Olsen, E. Murakami, H. Aono, M. Masuduzzaman, G. Gupta, and H. Das. The financial support was obtained from Applied Materials, Taiwan Semiconductor Manufacturing Corporation, SRC-GRC, 2008 IEEE EDS PhD Fellowship, and 2009-2010 Intel Foundation PhD Fellowship.

References 1. B. E. Deal, M. Sklar, A. S. Grove and E. H. Snow, Journal of the Electrochemical Society, 114, 266 (1967). 2. K. O. Jeppson and C. M. Svensson, Journal of Applied Physics, 48, 2004 (1977). 3. K. Bernstein, D. J. Frank, A. E. Gattiker, W. Haensch, B. L. Ji, S. R. Nassif, E. J. Nowak, D. J. Pearson and N. J. Rohrer, IBM Journal of Research and Development, 50, 433 (2006). 4. D. K. Schroder, Microelectronics Reliability, 47, 841 (2007). 5. D. K. Schroder and J. A. Babcock, Journal of Applied Physics, 94, 1 (2003). 6. J. H. Stathis and S. Zafar, Microelectronics Reliability, 46, 270 (2006). 7. J. Campbell, in Tutorial of IEEE International Integrated Reliability Workshop, p. T4 (2009). 8. M. A. Alam and S. Mahapatra, Microelectronics Reliability, 45, 71 (2005). 9. S. Chakravarthi, A. Krishnan, V. Reddy, C. F. Machala and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 273 (2004). 10. V. Huard, M. Denais and C. Parthasarathy, Microelectronics Reliability, 46, 1 (2006). 11. T. Grasser, B. Kaczer and W. Goes, in IEEE International Reliability Physics Symposium, p. 28 (2008). 12. D. Ielmini, M. Manigrasso, F. Gattel and M. G. Valentini, IEEE Transactions on Electron Devices, 56, 1943 (2009). 13. P. M. Lenahan, in IEEE International Reliability Physics Symposium, p. 1086 (2010). 14. T. Grasser, H. Reisinger, P. Wagner, F. Schanovsky, W. Goes and B. Kaczer, in IEEE International Reliability Physics Symposium, p. 16 (2010). 15. A. E. Islam, H. Kufluoglu, D. Varghese, S. Mahapatra and M. A. Alam, IEEE Transactions on Electron Devices, 54, 2143 (2007). 16. A. E. Islam, S. Mahapatra, S. Deora, V. D. Maheta and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 733 (2009).

17. J. H. Lee and A. S. Oates, IEEE Transactions on Device and Materials Reliability, 10, 174 (2010). 18. S. Mahapatra, K. Ahmed, D. Varghese, A. E. Islam, G. Gupta, L. Madhav, D. Saha and M. A. Alam, in IEEE International Reliability Physics Symposium, p. 1 (2007). 19. S. Mahapatra, V. D. Maheta, A. E. Islam and M. A. Alam, IEEE Transactions on Electron Devices, 56, 236 (2009). 20. D. S. Ang, S. Wang, G. A. Du and Y. Z. Hu, IEEE Transactions on Device and Materials Reliability, 8, 22 (2008). 21. V. Huard, in IEEE International Reliability Physics Symposium, p. 33 (2010). 22. T. Grasser, B. Kaczer, W. Goes, T. Aichinger, P. Hehenberger and M. Nelhiebel, in IEEE International Reliability Physics Symposium, p. 33 (2009). 23. J. P. Campbell, P. M. Lenahan, A. T. Krishnan and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 503 (2007). 24. A. S. Grove, Physics and Technology of Semiconductor Devices, John Wiley and Sons (1971). 25. P. V. Gray and D. M. Brown, Applied Physics Letters, 8, 31 (1966). 26. Y. Nishi, Jpn J Appl Phys, 10, 52 (1971). 27. P. J. Caplan, E. H. Poindexter, B. E. Deal and R. R. Razouk, Journal of Applied Physics, 50, 5847 (1979). 28. K. L. Brower, Physical Review B, 38, 9657 (1988). 29. E. Cartier, J. H. Stathis and D. A. Buchanan, Applied Physics Letters, 63, 1510 (1993). 30. J. P. Campbell, P. M. Lenahan, A. T. Krishnan and S. Krishnan, in IEEE International Reliability Physics Symposium, p. 442 (2006). 31. P. M. Lenahan, in Defects in Microelectronic Materials and Devices, D. Fleetwood, S. Pantolides and R. D. Schrimpf, Editors, p. 163, CRC Press (2008). 32. A. T. Krishnan, C. Chancellor, S. Chakravarthi, P. E. Nicollian, V. Reddy, A. Varghese, R. B. Khamankar, S. Krishnan and L. Levitov, in International Electron Devices Meeting (IEDM) Technical Digest, p. 688 (2005). 33. S. Tsujikawa, T. Mine, K. Watanabe, Y. Shimamoto, R. Tsuchiya, K. Ohnishi, T. Onai, J. Yugami and S. Kimura, in IEEE International Reliability Physics Symposium, p. 183 (2003). 34. M. A. Alam, H. Kufluoglu, D. Varghese and S. Mahapatra, Microelectronics Reliability, 47, 853 (2007).

35. S. Chakravarthi, A. T. Krishnan, V. Reddy and S. Krishnan, Microelectronics Reliability, 47, 863 (2007). 36. S. Mahapatra and M. A. Alam, IEEE Transactions on Device and Materials Reliability, 8, 35 (2008). 37. L. Tsetseris, X. J. Zhou, D. M. Fleetwood, R. D. Schrimpf and S. T. Pantelides, Applied Physics Letters, 86, 142103 (2005). 38. L. Tsetseris, X. J. Zhou, D. M. Fleetwood, R. D. Schrimpf and S. T. Pantelides, IEEE Transactions on Device and Materials Reliability, 7, 502 (2007). 39. M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 345 (2003). 40. A. E. Islam, G. Gupta, S. Mahapatra, A. Krishnan, K. Z. Ahmed, F. Nouri, A. Oates and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 329 (2006). 41. S. Mahapatra, A. E. Islam, S. Deora, V. D. Maheta and M. A. Alam, accepted in IEEE International Reliability Physics Symposium (2011). 42. L. Lundkvist, I. Lundstrom and C. Svensson, Solid-State Electronics, 16, 811 (1973). 43. R. A. Williams and M. M. E. Beguwala, IEEE Transactions on Electron Devices, 25, 1019 (1978). 44. G. L. Heyns and H. E. Maes, Applied Surface Science, 30, 153 (1987). 45. F. P. Heiman and G. Warfield, IEEE Transactions on Electron Devices, Ed12, 167 (1965). 46. P. Rossel, H. Martinot and D. Esteve, Solid-State Electronics, 13, 425 (1970). 47. I. Lundstrom, S. Christensson and C. Svensson, Phys. Status Solid (A), 1, 395 (1970). 48. V. Huard and M. Denais, in IEEE International Reliability Physics Symposium, p. 40 (2004). 49. C. R. Parthasarathy, M. Denais, V. Huard, G. Ribes, E. Vincent and A. Bravaix, in IEEE International Reliability Physics Symposium, p. 471 (2006). 50. H. Reisinger, O. Blank, W. Heinrigs, A. Muhlhoff, W. Gustin and C. Schlunder, in IEEE International Reliability Physics Symposium, p. 448 (2006). 51. C. Shen, M.-F. Li, C. E. Foo, T. Yang, D. M. Huang, A. Yap, G. S. Samudra and Y.C. Yeo, in International Electron Devices Meeting (IEDM) Technical Digest, p. 333 (2006). 52. T. Yang, C. Shen, M. F. Li, C. H. Ang, C. X. Zhu, Y. C. Yeo, G. Samudra, S. C. Rustagi, M. B. Yu and D. L. Kwong, IEEE Electron Device Letters, 26, 826 (2005).

53. M. F. Li, G. Chen, C. Shen, X. P. Wang, H. Y. Yu, Y. C. Yeo and D. L. Kwong, Jpn J Appl Phys, 43, 7807 (2004). 54. G. Bersuker, J. H. Sim, C. S. Park, C. D. Young, S. V. Nadkarni, R. Choi and B. H. Lee, IEEE Transactions on Device and Materials Reliability, 7, 138 (2007). 55. C. Shen, M. F. Li, X. P. Wang, H. Y. Yu, Y. P. Feng, A. T. L. Lim, Y. C. Yeo, D. S. H. Chan and D. L. Kwong, International Electron Devices Meeting (IEDM) Technical Digest, 733 (2004). 56. W. Goes, M. Karner, V. Sverdlov and T. Grasser, in Proceedings of International Symposium on the Physical & Failure Analysis of Integrated Circuits (IPFA), p. 249 (2008). 57. A. Ghetti, C. M. Compagnoni, A. S. Spinelli and A. Visconti, IEEE Transactions on Electron Devices, 56, 1746 (2009). 58. T. H. Morshed, S. P. Devireddy, Z. Celik-Butler, A. Shanware, K. Green, J. J. ChamberS, M. R. Visokay and L. Colombo, Solid-State Electronics, 52, 711 (2008). 59. D. Veksler, H. Park, C. Young, B. Taylor, G. Bersuker and R. Jammy, Solid State Technology, 53, 21 (2010). 60. H. D. Xiong, D. Heh, M. Gurfinkel, Q. Li, Y. Shapira, C. Richter, G. Bersuker, R. Choi and J. S. Suehle, Microelectronic Engineering, 84, 2230 (2007). 61. J. P. Campbell, J. Qin, K. P. Cheung, L. C. Yu, J. S. Suehle, A. Oates and K. Sheng, in IEEE International Reliability Physics Symposium, p. 382 (2009). 62. S. Mahapatra, P. B. Kumar and M. A. Alam, IEEE Transactions on Electron Devices, 51, 1371 (2004). 63. J. Robertson, Solid-State Electronics, 49, 283 (2005). 64. J. Robertson, Reports on Progress in Physics, 69, 327 (2006). 65. J. Sune, E. Farres, I. Placencia, N. Barniol, F. Martin and X. Aymerich, Applied Physics Letters, 55, 128 (1989). 66. M. Alam, B. Weir and P. Silverman, IEEE Circuits & Devices, 18, 42 (2002). 67. F. Crupi, R. Degraeve, G. Groeseneken, T. Nigam and H. E. Maes, IEEE Transactions on Electron Devices, 45, 2329 (1998). 68. D. Varghese, H. Kufluoglu, V. Reddy, H. Shichijo, D. Mosher, S. Krishnan and M. A. Alam, IEEE Transactions on Electron Devices, 54, 2669 (2007). 69. P. E. Nicollian, Microelectronics Reliability, 48, 1171 (2008). 70. A. E. Islam, H. Kufluoglu, D. Varghese and M. A. Alam, Applied Physics Letters, 90, 083505 (2007).

71. A. E. Islam, V. D. Maheta, H. Das, S. Mahapatra and M. A. Alam, in IEEE International Reliability Physics Symposium, p. 87 (2008). 72. V. D. Maheta, C. Olsen, K. Ahmed and S. Mahapatra, IEEE Transactions on Electron Devices, 55, 1630 (2008). 73. T. Grasser, B. Kaczer, R. Hehenberger, W. Gos, R. O'Connor, H. Reisinger, W. Gustin and C. Schlunder, International Electron Devices Meeting (IEDM) Technical Digest, 801 (2007). 74. B. Kaczer, T. Grasser, P. J. Roussel, J. Martin-Martinez, R. O'Connor, B. J. O'Sullivan and G. Groeseneken, in IEEE International Reliability Physics Symposium, p. 20 (2008). 75. S. Deora, V. D. Maheta, A. E. Islam, M. A. Alam and S. Mahapatra, IEEE Electron Device Letters, 30, 978 (2009). 76. S. Mahapatra, V. D. Maheta, S. Deora, E. N. Kumar, S. Purawat, C. Olsen, K. Ahmed, A. E. Islam and M. A. Alam, in Silicon Nitride, Silicon Dioxide, and Emerging Dielectrics, R. E. Sah, J. Zhang, J. Deen, J. Yota and A. Toriumi, Editors, p. 243, The Electrochemical Society Proceedings Series (2009). 77. R. Fernández, B. Kaczer, A. Nackaerts, S. Demuynck, R. Rodríguez, M. Nafría and G. Groeseneken, in International Electron Devices Meeting (IEDM) Technical Digest, p. 337 (2006). 78. C. L. Chen, Y. M. Lin, C. J. Wang and K. Wu, in IEEE International Reliability Physics Symposium, p. 704 (2005). 79. A. Haggag, G. Anderson, S. Parihar, D. Burnett, G. Abeln, J. Higman and M. Moosa, in IEEE International Reliability Physics Symposium, p. 452 (2007). 80. H. Reisinger, T. Grasser, W. Gustin and C. Schlunder, in IEEE International Reliability Physics Symposium, p. 7 (2010). 81. Y. Mitani, H. Satake and A. Toriumi, IEEE Transactions on Device and Materials Reliability, 8, 6 (2008). 82. S. Wang, D. S. Ang and G. A. Du, IEEE Electron Device Letters, 29, 483 (2008). 83. H. Reisinger, in NBTI Tutorial, IEEE International Integrated Reliability Workshop, p. T2 (2010). 84. V. Huard, C. Parthasarathy, C. Guerin, T. Valentin, E. Pion, M. Mammasse, N. Planes and L. Camus, in IEEE International Reliability Physics Symposium, p. 289 (2008). 85. A. T. Krishnan, S. Chakravarthi, P. Nicollian, V. Reddy and S. Krishnan, Applied Physics Letters, 88, 153518 (2006).

86. H. Kufluoglu and M. A. Alam, IEEE Transactions on Electron Devices, 54, 1101 (2007). 87. A. E. Islam, G. Gupta, K. Z. Ahmed, S. Mahapatra and M. A. Alam, IEEE Transactions on Electron Devices, 55, 1143 (2008). 88. A. E. Islam, J. H. Lee, W. H. Wu, A. Oates and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 107 (2008). 89. S. Kumar, C. H. Kim and S. S. Sapatnekar, in International Conference on ComputerAided Design, p. 6D.1 (2006). 90. J. Crank, The Mathematics of Diffusion, Oxford University Press (1980). 91. T. Grasser, W. Gos and B. Kaczer, IEEE Transactions on Device and Materials Reliability, 8, 79 (2008). 92. C. G. Van de Walle and B. R. Tuttle, IEEE Transactions on Electron Devices, 47, 1779 (2000). 93. D. Varghese, D. Saha, S. Mahapatra, K. Ahmed, F. Nouri and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 684 (2005). 94. G. Gupta, S. Mahapatra, L. Madhav, D. Varghese, K. Z. Ahmed and F. Nouri, in IEEE International Reliability Physics Symposium, p. 731 (2006). 95. M. L. Reed and J. D. Plummer, Journal of Applied Physics, 63, 5776 (1988). 96. M. A. Alam, Reliability Physics of Nanoelectronic Transistors, in EE 695-A lecture series, uploaded in http://cobweb.ecn.purdue.edu/~ee650/handouts.htm. 97. A. E. Islam, Ph. D. Dissertation, Theory and Characterization of Defect Formation and Its Implication in Variability of Nanoscale Transistors, Electrical and Computer Engineering, Purdue University, 2010. 98. T. L. Tewksbury and H. S. Lee, IEEE Journal of Solid-State Circuits, 29, 239 (1994). 99. S. V. Hattangady, H. Niimi and G. Lucovsky, Journal of Vacuum Science & Technology-A, 14, 3017 (1996). 100. K. Sakuma, D. Matsushita, K. Muraoka and Y. Mitani, in IEEE International Reliability Physics Symposium, p. 454 (2006). 101. K. Hess, A. Haggag, W. McMahon, B. Fischer, K. Cheng, J. Lee and J. Lyding, in International Electron Devices Meeting (IEDM) Technical Digest, p. 93 (2000). 102. H. Reisinger, O. Blank, W. Heinrigs, W. Gustin and C. Schlunder, IEEE Transactions on Device and Materials Reliability, 7, 119 (2007).

103. A. E. Islam, E. N. Kumar, H. Das, S. Purawat, V. D. Maheta, H. Aono, E. Murakami, S. Mahapatra and M. A. Alam, in International Electron Devices Meeting (IEDM) Technical Digest, p. 805 (2007). 104. V. D. Maheta, E. N. Kumar, S. Purawat, C. Olsen, K. Ahmed and S. Mahapatra, IEEE Transactions on Electron Devices, 55, 2614 (2008). 105. T. Grasser and B. Kaczer, IEEE Transactions on Electron Devices, 56, 1056 (2009). 106. A. E. Islam and M. A. Alam, unpublished results. 107. M. A. Alam, in IEEE International Integrated Reliability Workshop, p. 1 (2010).