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Unifying the thermal–chemical and anode-hole-injection gate-oxide breakdown models
Microelectronics Reliability 41 (2001) 193±199
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Unifying the thermal±chemical and anode-hole-injection gateoxide breakdown models Kin P. Cheung * Bell Laboratories, Lucent Technologies, Room 1B217, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Received 21 March 2000; received in revised form 24 July 2000
Abstract Recent experimental data obtained from medium thickness, high quality thermal-oxide clearly indicates that the logarithmic oxide lifetime is linearly proportional to the oxide ®eld in the mid-to-low ®eld range. This strong support for the E model lends credibility to the thermal±chemical (TC) model proposed by McPherson and Baglee [IEEE International Reliability Physics Symposium (IRPS), 1985, p. 1]. In this work, the critical weakness of the TC model, namely the absence of the eect of electrons and holes on the breakdown lifetime is removed by integrating the TC model with the anode-hole-injection model. One single kinetic model is constructed and a single master equation is found for the new uni®ed model. This master equation for trap generation can successfully explain a wide range of experimental observations. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction For gate-oxide reliability assessment, accelerated tests for time to failure (TF) are essential and the model for TF extrapolation from the test condition to the operational condition is of critical importance. Since acceleration tests utilize elevated voltage or temperature or both, the extrapolation model must be able to handle both factors as well. For voltage acceleration, whether log(TF) is a linear function of the oxide electric ®eld E or of the inverse electric ®eld 1/E has been a heated debate or thicker. For oxides thinner than when oxides are 50 A 50 A, the model that predicts a 1/E dependent becomes applied voltage (V) dependent. Thus, both models predict E dependent for thin oxides. Such convergence in empirical behavior does not imply that the physics behind the two models has changed. Instead, the question of which one is the correct model to extrapolate accelerated test result to operational condition is more important then ever. The debate is no longer over E versus
*
Corresponding author. Tel.: +1-908-582-6483; fax: +1-908582-5980. E-mail address: [email protected] (K.P. Cheung).
1/E, rather, it is over which physical mechanism is correct. The physical model that can lead to a E dependent log(TF) was introduced by McPherson and Baglee [1]. It is commonly referred to as the thermal±chemical (TC) model. The physical model that can lead to 1/E dependent log(TF) was a combination of the hole trapping model introduced by Chen et al. [2] and the anode-holeinjection (AHI) mechanism proposed by Weinberg [3]. This combined model is commonly referred to as the AHI model. In their original form, the TC model and the AHI model have a very dierent view on how breakdown occur in oxide. Implicit in the TC model is the assumption that when defect density reaches a critical value, the oxide will break. For the AHI model, breakdown occurs when a critical density of positive charges is trapped in the oxide. As our understanding of the breakdown mechanism evolve, this dierence between the two models disappeared. It in now well accepted that breakdown occurs when a critical density of neutral traps is created in the oxide. The main dierence between these two models is now on the neutral trap generation mechanisms and therefore the generation rates. In the literature, there is at least one other model for trap generation that has extensive experimental support. This is the hydrogen model [4,5]. It is likely that no
0026-2714/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 0 0 ) 0 0 2 0 3 - 1
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K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
breakdown model is complete without a proper account on the eect of hydrogen. The hydrogen model has not been developed to the quantitative level for oxide TF projection. It is for this reason that it has largely been left out in the debate of TF extrapolation. Experimentally, 1/E dependent is often observed in very high oxide stress ®eld (>9 MV cm±1 ) range, while E dependent is exclusively observed in the lower stress ®eld range. This dierence is particularly clear in the oxide [6±8]. When oxide is much thickness range of 60±100 A impact ionization contributes sigthicker than 100 A, ni®cantly to trap generation. When oxide is much thin even the AHI model will lead to E ner than 60 A, dependent log(TF). Thus, for mechanistic study of trap generation in oxide, data in the thickness range of 60± are particularly important. These data open a 100 A window that allows us to separate the dierent physical forces that lead to oxide breakdown. The fact that both models predict E dependent log(TF) in ultra-thin oxide does not make them equivalent. If we do not have a reason to expect the trap generation physics to change, then the dierence observed in thicker oxide should still be operative in thinner oxide. This will lead to a dierent slope for the E dependent extrapolation. This dierence is extremely important because when oxide is ultra-thin, the intrinsic lifetime at operation condition is a limiting factor in technology scaling. From recently published results for oxides in the 60± ranges, it is now ®rmly established that neither 100 A model can explain all the data. It becomes clear that both models have merit and that they dominate at different stress ®eld region. Such realization prompted the proposal that oxide breakdown is a competing process between the two mechanisms, and that the trap generation rate is simply the sum of the two dierent trap generation rates [9]. Simply summing the two trap generation rates assumes, implicitly, that the two rates are independent of each other. Trap generation is a bond-breaking process. If the bonds being broken are the SiO bonds that form the normal oxide network, then even at breakdown, the concentration of available precursor sites hardly change. It is a reasonable assumption that the two trap generation processes remain independent of each other at all time. Experience tells us that this is not the case. It is well known that poorly grown oxide breaks much faster. Most people believe that the precursors for trap generation are defects themselves. If precursors can be depleted as was demonstrated recently [10], then the trap generation rate of competing mechanisms cannot be independent of each other. A kinetic model that includes both trap generation mechanisms must be constructed. Note that breakdown occurs when a critical density of traps is created in the oxide, any breakdown model, by necessity, must be a kinetic model.
To construct a uni®ed kinetic model, we must spell out clearly the reaction pathways for both mechanisms. The reaction pathway for the TC model is already well de®ned by McPherson et al. [6]. The same cannot be said about the AHI model since how do the holes get trapped and then subsequently converted into neutral trap has never been worked out. The physics of the E model is very simple and fundamental. It simply says that the dipole moment of a chemical bond can interact with the applied ®eld and be broken with ®nite probability. That this is a fact is indisputable. Unless the trap creation rate of this mechanism is negligible, any model of breakdown must include this mechanism. The quantitative success of the TC model in some of the reported cases is remarkable for such a simple model. It implies that the TC mechanism produces a non-negligible trap generation rate. It also implies that the trap generation rate at any condition is a combination of the TC rate plus whatever mechanism that dominates trap generation and that it cannot be slower than the TC rate. Thus we can be very con®dent about the choice of the TC model as the starting point. In the TC model, electron or hole does not play any role, which contradicts experience and is the main objection many researchers have against it. Intuitively, when a defect collides with a hole, the electric ®eld of the hole can also interact with the dipole to facilitate a conversion to trap. In close proximity, this ®eld is much higher than any applied ®eld. This reaction must exist in parallel with the one assumed in the TC model and, in situations where hole current is high, it should be the dominant reaction path. This proposed reaction allows the AHI model to share the same physics foundation as the TC model. A reaction pathway for the AHI model can be constructed from this proposal. Once both reaction pathways are clearly de®ned, we can construct a kinetic model that integrates both mechanisms. The result is a single master equation that provides a uni®ed framework for both high and low ®eld breakdown behavior.
2. The model The TC model assumes that precursor sites are converted into neutral traps by dipole±electric ®eld interaction k1
Vm ! N :
1
Here Vm denotes the mth precursor type, k1 is the reaction rate constant and N, the neutral defect formed by the reaction. Instead of taking oxygen vacancy as the only precursor like the original TC model, more than one type of precursors is assumed to be possible in the new model. This assumption is based on two facts. One
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
195
is the observed multi-precursors depletion behavior of SILC growth [10]. The other is the critical trap density to breakdown, according to the percolation model, is too high for the concentration of oxygen vacancy, the commonly accepted precursor. The reaction rate constant can be written as X Am e Ea1m =kT : 2 k1 m
The summation is over all the precursor types. The activation energy is, following McPherson and Baglee [1]: Ea1m DH0m
qm E:
3
Here Ea1m denotes the activation energy of the mth precursor in reaction (1). DH0 m is the enthalpy of activation (activation energy). qm is the dipole moment of the precursor and E is the oxide ®eld. The proposed reaction pathway for the AHI model ± the direct capturing of a hole by a precursor when a collision event occurs can be expressed as k2
k4
k3
e
Vm h ¢ P ! N :
4
Here P denotes a positively charged defect, which can capture an electron to become a neutral trap. A reverse reaction is included here to account for hole detrapping. The rate constant for the forward reaction (capturing of a hole) is X k2 Bm e Ea2m =kT : 5 m
The activation energies for the reaction is Ea2m DH0m
qm
q : 4pe2r
6
Here we used the electric ®eld of the hole to replace the oxide ®eld in reaction (1). Otherwise, the physics is the same as reaction (1), namely the barrier is lowered by the electric ®eld±dipole interaction energy. Since a hole is captured in this reaction, the result is not a neutral trap but a positively charged defect. Fig. 1 shows the electric ®eld of the hole as a function or less, the ®eld increases rapidly of distance. At 5 A beyond 15 MV cm±1 . At such high ®eld, the reaction barrier will be lowered to the point that reaction (2) will be limited only by the attempt rate (the pre-factor Bm ). Thus the capture cross-section (r) of a hole by defect sites is about 1 10 14 cm2 , similar to the value reported in Ref. [11]. Combining reactions (1) and (2), we can write down the set of rate equations for the model
Fig. 1. Electric ®eld of a point charge in a dielectric. When E increases beyond 15 MV cm±1 rapidly, implying a r < 5 A, hole capture cross-section of 110 14 cm2 .
dV k3 P k1 k2 Jh V ; dt dP k2 Jh V k3 k4 Je P ; dt dN k1 V k4 Je P : dt Here Jh and Je are hole current density and electron current density respectively. Here the unit of k1 and k3 is in s 1 while the unit of k2 and k4 are sJ 1 s 1 q cm 2 s 1 1 cm2 q 1 , with q being the unit charge. The solution to these equations is rather complicated. However, when hole detrapping is not signi®cant (i.e. k3 k2 Jh ), it can be reduced to a simpler form, which is the master equation of this model 1 N V 0 1 k4 Je e k1 k2 Jh t k2 Jh k4 Je 7 k 2 J h e k4 Je t : Here V 0 is the initial precursor concentration. 3. High ®eld limits When the last step of reaction (2), namely the capturing of an electron is very fast, the master equation can be further simpli®ed If k4 Je k1 k2 Jh : N V 0 1 e k1 k2 Jh t : 8 Eq. (8) is the basis for our discussion of unifying the E and 1/E models for gate-oxide breakdown. Before we do that, it is necessary to justify the approximation of k4 Je k1 k2 Jh to make sure that it is valid for the typical experimental condition where the E and 1/E dependent data are collected. It is well known that when stressing a not-too-thin oxide under high ®eld, rapid build-up and saturation of trapped holes occurs [12]. Using the set of rate equations
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K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
which means ln TF / 1=E: We thus also recovered the 1/E dependent of breakdown at high ®eld. Note that this is only valid for as long as Jh can be approximated with aJe . This approximation is not valid for thinner oxides.
4. Low ®eld limit
Fig. 2. Trapped positive charge concentration will saturate only if k4 Je is much greater than k1 k2 Jh . The actual magnitude of trapped hole concentration is shown in the insert.
and keeping the approximation of negligible hole detrapping (valid for not-too-thin oxide), we can calculate the accumulation of positively charged traps as a function of stress time for a number of relative values of k4 Je and k1 k2 Jh . Fig. 2 shows that rapid build-up and saturation of trapped holes can only happen if k4 Je is much larger than k1 k2 Jh . The approximation that leads to Eq. (8) is therefore well justi®ed. At high applied ®eld, if (this will be proved later) k2 Jh k1 , Eq. (8) becomes N V 0 1 e k2 Jh t : 9 In other words, trap generation is dominated by the hole capturing process (reaction (2)). Thus we recovered the experimental fact that the AHI model works better at high ®eld. According to the percolation model, breakdown is when trap density reaching a critical value. For measurements using ``identically made'' oxide samples, V 0 , should be the same. This means that k2 Jh t is a constant. Denoting t at breakdown as TF, we have k2 Jh t k2 Jh TF k2 QP const and QP const=k2 : From Eqs. (5) and (6), we know that k2 is also a constant that is independent of stress ®eld. We thus recovered the well-known AHI result of total hole-¯uence to breakdown is independent of stress ®eld. For not-too-thin oxides, Jh a Je , we have k2 Jh TF k2 aJe TF const: After rearrangement and writing out Je using the FN expression, we have TF
C k2 aGE2 e
H =E
At low ®eld, if k1 k2 Jh , Eq. (8) becomes N V 0 1e k1 t :
10
This is the original TC model. Again, using the idea of a critical trap density for breakdown from the percolation model, we can write k1 t k1 TF const: Combining with Eqs. (2) and (3) for k1 , we have ln TF / E: The E dependent is recovered. Clearly, the master equation has indeed uni®ed the high-/low-®eld breakdown behavior.
5. Quantitative evaluations We can estimate k2 using data from Ref. [2]. In Chen's work, very small (900 l2 ) capacitors were used, which leads to a high critical trap density according to the percolation model. Since our model is based on converting precursors into traps, precursor depletion can be expected when the trap density is very high. In a recent study, Lu et al. [10] showed that strong saturation of stress induced leakage current (SILC) growth can be observed when the capacitor size is very small. The saturation in SILC growth is a clear indication of precursor depletion. Chen's capacitor is somewhere between the two capacitor sizes used in Lu et al.'s. work, it is reasonable, therefore, to assume that the precursor depletion is only moderate. We therefore take N 0:1V 0 at breakdown for Chen's case. Once that is done, k2 can be estimated from QP , which was found to be 0.1 C cm2 [2]. We therefore estimate k2 to be 210 19 cm2 . 0:1 1
e
k2 Jh TF
k2 QP :
To estimate k1 , we can use Kimura's [7] data. In this case we assume [N]0.01[V]0 because that is more likely for larger size capacitor (Kimura did not specify). According to Eq. (10), we have 0:01 K1 TF:
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
Fig. 3. k1 and k2 Jh as a function of E. Con®rms that k1 dominates at low ®eld while k2 Jh can dominate at high ®eld.
197
Fig. 4. Data from various groups have wide range of slopes. This can only be explained that various oxides have dierent mix of precursor types.
In other words, k1 is related to TF and is a function of oxide, the oxide ®eld. Since Jh aJe , and for 100 A a 10 3 [13], we can compare k1 to k2 Jh for the full range of stress ®eld (Fig. 3). As can be seen, at low ®eld, k1 k2 Jh , while at high ®eld, k2 Jh dominates. The assumptions behind the two limiting cases are justi®ed. The commonly cited transition around 8±9 MV cm±1 is also reproduced. Note that the QP value from Chen et al.'s data must be over estimated by as much as an order of magnitude [14]. Thus the k2 value may be under estimated by an order of magnitude. If that were true, the cross over would be more prominent.
6. Further discussions Among the reports [6,7,15,16] that support the E model, the ®eld acceleration factor (slope of the log(TF) versus E curve) varies widely (Fig. 4). In a recent report, trap growth was found to be best modeled as the depletion of three types of precursor [10]. The literature on oxide has identi®ed many defect types [17]. It is therefore entirely reasonable for the new model to assume multiple precursor types. The actual mix of precursors depends on the growth and processing conditions of the gate oxide, and the ®eld acceleration factors as well as the dipole moments should naturally be dierent for dierent oxide. Thus it is entirely reasonable to ®nd that data from various groups have wide range of slopes. In addition to oxide quality dierence, it is also entirely possible that the precursor mix for the same oxide changes with distance from the interfaces. It is well known that strain in the oxide is highest at the SiO2 /Si interface. It is expected that even for the same defect type, the activation energy will be dierent. The precursor mix, which aects k1 , also aects the stress ®eld at which the transition from E to 1/E behavior occurs. For many reported cases such as those in
Fig. 5. When k1 is large, k2 Jh never dominates. All of these data show no evidence of 1/E dependent even at high ®eld, consistent with the model.
Fig. 4, the transition was not observed even at oxide ®eld as high as 12 MV cm±1 . When the k1 value of these experimental results are calculated, they indeed do not cross the hole capture dominated curve (k2 Jh ) (Fig. 5). (This comparison is not entirely valid. The value of k2 is expected to depend on the precursor type and therefore on oxide quality. One should only compare values from the same oxide.) 7. Master equation check Experiments that independently control Jh and Je are good tests for the master equation. For example, Satake et al. [18] found that, with a constant Jh and a small Je , QP decreases with increasing Je while QBD remains constant. These results can be explained with the master equation. Taking k4 Je k2 Jh , Eq. (7) becomes N V 0 1 e k4 Je t : 11
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K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
Eq. (11) suggests that when Je , is low, trap generation is controlled by the capturing of electrons. For the same capacitor type, the critical trap density for breakdown is ®xed. That means k4 Je TF is a constant regardless of the stress condition. If k4 were independent of stress condition (a reasonable assumption, see discussion later), a constant QBD would result, which is Satake et al.'s observation. When Je is low, conversion of positively charged trap into neutral trap by capturing an electron will be a bottleneck. Piling up of positive charges in the oxide can be expected (see insert of Fig. 2). Signi®cant precursor depletion will occur during most of the stress period. The hole trapping eciency is reduced over the same period. Since Jh is constant, QP becomes proportional to TF, which is controlled by Je . A higher Je leads to a smaller TF and a smaller Je leads to a longer TF. Thus Satake et al.'s observation that increasing Je leads to a decrease in QP and, increasing Jh leads to an increase in QP is easily explained. All of Satake et al.'s observations are in complete agreement with the master equation.
8. Ultrathin gate-oxide situations k3 cannot be When gate-oxide is thinner than 50 A, neglected and the set of rate equations can only be solved numerically. This is not too dicult once all the constants are pinned down. The detrapping of holes is a tunneling process (Fig. 6) with a known barrier height [19] and can be calculated. The one remaining unknown quantity, k4 , is likely to be smaller than k2 . The collision of an electron with a trapped hole is dierent from an electron interacting with a defect site. We proposed here that the electric
®eld of the hole could interact strongly with the dipole moment of a defect site. A natural question is why would not electrons do the same? If they do, then the much larger electron current density should make the rate of electron trapping by direct capture much higher than the rate of hole trapping. This is not experimentally observed. The key dierent between direct capturing of holes and electrons by defect sites is in the size of the waves associated with hole and electron. In SiO2 , the electrons traveling in the conduction band is far more delocalized than the hole traveling in the valence band. The light eective mass (therefore small local distortion of the network) and high velocity of an electron makes inelastic collision far less probable due to conservation of momentum. The probability of capturing an electron by a trapped hole is greatly enhanced by the strong local distortion of the network around the captured hole. This strong local distortion enhances the probability of multiple phonon emission, thus provide a way for momentum conservation. Nevertheless, the need to conserve the large momentum of a fast moving electron suggests that the magnitude of k4 , should be smaller than k2 . Careful experiments such as those done by Satake et al. [18] should help pin down the actual value.
9. Conclusion A physics based, uni®ed model for gate-oxide breakdown at both high and low ®eld has been developed. It has successfully account for a wide range of experimental observations. This new model treats the TC and AHI mechanisms as competing processes. The transition point from one to another will depend on the precursor mix and on the hole current density. For thicker oxides, the precursor mix is more important. For thinner oxides, the large direct tunneling electron current may cause the hole current density to be high enough [20] that the hole capture reaction continue to dominate at lower ®eld. That is, for thinner oxides, the AHI model may be valid for a wider range of ®eld. Whether this is indeed the case is not yet known at this point since there are no low ®eld data reported in the literature. This downward extension of the valid range of the AHI model is not expected to reach the normal operation ®eld. The rapid decrease of hole current density as a function of voltage ensures that trap creation rate due to AHI will be far below the trap generation rate due to TC at normal operation voltage.
References
Fig. 6. Hole detrapping by tuning out.
[1] McPherson JW, Baglee DA. Acceleration factor for thin gate oxide stressing. IEEE Inter Reliab Phys Symp (IRPS) 1985:1.
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199 [2] Chen IC, Holland S, Hu C. Oxide breakdown dependence on thickness and hole current-enhanced reliability of ultra thin oxide. IEDM 1986:660. [3] Weinberg ZA. Hole injection and transport in SiO2 ®lms on Si. Appl Phys Lett 1975;27(8):437. [4] Weinberg ZA, et al. Exciton or hydrogen diusion in SiO2 . J Appl Phys 1979;50(9):5757. [5] DiMaria DJ, Stasiak JW. Trap creation in silicon dioxide produced by hot electrons. J Appl Phys 1989;65(6):2342. [6] McPherson JW, Mogul HC. Disturbed bonding states in SiO2 thin-®lms and their impact on time dependent dielectric breakdown. IEEE Inter Reliab Phys Symp (IRPS) 1998:47. [7] Kimura M. Field and temperature acceleration model for time-dependent dielectric breakdown. IEEE Trans Electron Dev 1999;46(1):220. [8] Suehle JS, Chaparala P. Low electrical ®eld breakdown of thin SiO2 ®lms under static and dynamic stress. IEEE Trans Electron Dev 1997;44(5):801. [9] Hu C, Lu Q. A uni®ed gate oxide reliability model. IRPS 1999:47. [10] Lu Q, et al. A model of the stress time dependence of SILC. IRPS 1999:396. [11] Tzou JJ, Sun JY-C, Sah C-T. Field dependence of two large hole capture cross sections in thermal oxide on silicon. Appl Phys Lett 1983;43(9):861.
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[12] Nissan-Cohen Y, Shappir J, Frohman-Bentchkowsky D. High-®eld and current-induced positive charge in thermal SiO2 layers. J Appl Phys 1985;57(8):2830. [13] Lee J, Chen I-C, Hu C. Statistical modeling of silicon dioxide reliability. IEEE Int Reliab Phys Symp (IRPS) 1988:131. [14] Rasras M, et al. Photo-carrier generation as the origin of Fowler±Nordheim-induced substrate hole current in thin oxides. Int Electron Dev Meeting (IEDM) 1999: 465. [15] Suehle JS, Chaparala P. Low electric ®eld breakdown of thin SiO2 ®lms under static and dynamic stress. IEEE Trans Electron Dev 1997;44(5):801. [16] Wu EY, et al. Challenges for accurate reliability projections in the ultra-thin oxide regime. IEEE Int Reliab Phys Symp (IRPS) 1999:57. [17] Helms CR, Poindexter EH. The silicon-silicon-dioxide system: its microstructure and imperfections. Rep Prog Phys 1994;57:791. [18] Satake H, Takagi S-I, Toriumi A. Evidence of electronhole cooperation in SiO2 dielectric breakdown. IRPS 1997: 156. [19] Lu Y, Sah C-T. Thermal emission of trapped holes in thin SiO2 ®lms. J Appl Phys 1995;78(5):3156. [20] Bude JD, Weir BE, Silverman PJ. Explanation of stressinduced damage in thin oxides. IEDM 1998:179.
www.elsevier.com/locate/microrel
Unifying the thermal±chemical and anode-hole-injection gateoxide breakdown models Kin P. Cheung * Bell Laboratories, Lucent Technologies, Room 1B217, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Received 21 March 2000; received in revised form 24 July 2000
Abstract Recent experimental data obtained from medium thickness, high quality thermal-oxide clearly indicates that the logarithmic oxide lifetime is linearly proportional to the oxide ®eld in the mid-to-low ®eld range. This strong support for the E model lends credibility to the thermal±chemical (TC) model proposed by McPherson and Baglee [IEEE International Reliability Physics Symposium (IRPS), 1985, p. 1]. In this work, the critical weakness of the TC model, namely the absence of the eect of electrons and holes on the breakdown lifetime is removed by integrating the TC model with the anode-hole-injection model. One single kinetic model is constructed and a single master equation is found for the new uni®ed model. This master equation for trap generation can successfully explain a wide range of experimental observations. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction For gate-oxide reliability assessment, accelerated tests for time to failure (TF) are essential and the model for TF extrapolation from the test condition to the operational condition is of critical importance. Since acceleration tests utilize elevated voltage or temperature or both, the extrapolation model must be able to handle both factors as well. For voltage acceleration, whether log(TF) is a linear function of the oxide electric ®eld E or of the inverse electric ®eld 1/E has been a heated debate or thicker. For oxides thinner than when oxides are 50 A 50 A, the model that predicts a 1/E dependent becomes applied voltage (V) dependent. Thus, both models predict E dependent for thin oxides. Such convergence in empirical behavior does not imply that the physics behind the two models has changed. Instead, the question of which one is the correct model to extrapolate accelerated test result to operational condition is more important then ever. The debate is no longer over E versus
*
Corresponding author. Tel.: +1-908-582-6483; fax: +1-908582-5980. E-mail address: [email protected] (K.P. Cheung).
1/E, rather, it is over which physical mechanism is correct. The physical model that can lead to a E dependent log(TF) was introduced by McPherson and Baglee [1]. It is commonly referred to as the thermal±chemical (TC) model. The physical model that can lead to 1/E dependent log(TF) was a combination of the hole trapping model introduced by Chen et al. [2] and the anode-holeinjection (AHI) mechanism proposed by Weinberg [3]. This combined model is commonly referred to as the AHI model. In their original form, the TC model and the AHI model have a very dierent view on how breakdown occur in oxide. Implicit in the TC model is the assumption that when defect density reaches a critical value, the oxide will break. For the AHI model, breakdown occurs when a critical density of positive charges is trapped in the oxide. As our understanding of the breakdown mechanism evolve, this dierence between the two models disappeared. It in now well accepted that breakdown occurs when a critical density of neutral traps is created in the oxide. The main dierence between these two models is now on the neutral trap generation mechanisms and therefore the generation rates. In the literature, there is at least one other model for trap generation that has extensive experimental support. This is the hydrogen model [4,5]. It is likely that no
0026-2714/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 0 0 ) 0 0 2 0 3 - 1
194
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
breakdown model is complete without a proper account on the eect of hydrogen. The hydrogen model has not been developed to the quantitative level for oxide TF projection. It is for this reason that it has largely been left out in the debate of TF extrapolation. Experimentally, 1/E dependent is often observed in very high oxide stress ®eld (>9 MV cm±1 ) range, while E dependent is exclusively observed in the lower stress ®eld range. This dierence is particularly clear in the oxide [6±8]. When oxide is much thickness range of 60±100 A impact ionization contributes sigthicker than 100 A, ni®cantly to trap generation. When oxide is much thin even the AHI model will lead to E ner than 60 A, dependent log(TF). Thus, for mechanistic study of trap generation in oxide, data in the thickness range of 60± are particularly important. These data open a 100 A window that allows us to separate the dierent physical forces that lead to oxide breakdown. The fact that both models predict E dependent log(TF) in ultra-thin oxide does not make them equivalent. If we do not have a reason to expect the trap generation physics to change, then the dierence observed in thicker oxide should still be operative in thinner oxide. This will lead to a dierent slope for the E dependent extrapolation. This dierence is extremely important because when oxide is ultra-thin, the intrinsic lifetime at operation condition is a limiting factor in technology scaling. From recently published results for oxides in the 60± ranges, it is now ®rmly established that neither 100 A model can explain all the data. It becomes clear that both models have merit and that they dominate at different stress ®eld region. Such realization prompted the proposal that oxide breakdown is a competing process between the two mechanisms, and that the trap generation rate is simply the sum of the two dierent trap generation rates [9]. Simply summing the two trap generation rates assumes, implicitly, that the two rates are independent of each other. Trap generation is a bond-breaking process. If the bonds being broken are the SiO bonds that form the normal oxide network, then even at breakdown, the concentration of available precursor sites hardly change. It is a reasonable assumption that the two trap generation processes remain independent of each other at all time. Experience tells us that this is not the case. It is well known that poorly grown oxide breaks much faster. Most people believe that the precursors for trap generation are defects themselves. If precursors can be depleted as was demonstrated recently [10], then the trap generation rate of competing mechanisms cannot be independent of each other. A kinetic model that includes both trap generation mechanisms must be constructed. Note that breakdown occurs when a critical density of traps is created in the oxide, any breakdown model, by necessity, must be a kinetic model.
To construct a uni®ed kinetic model, we must spell out clearly the reaction pathways for both mechanisms. The reaction pathway for the TC model is already well de®ned by McPherson et al. [6]. The same cannot be said about the AHI model since how do the holes get trapped and then subsequently converted into neutral trap has never been worked out. The physics of the E model is very simple and fundamental. It simply says that the dipole moment of a chemical bond can interact with the applied ®eld and be broken with ®nite probability. That this is a fact is indisputable. Unless the trap creation rate of this mechanism is negligible, any model of breakdown must include this mechanism. The quantitative success of the TC model in some of the reported cases is remarkable for such a simple model. It implies that the TC mechanism produces a non-negligible trap generation rate. It also implies that the trap generation rate at any condition is a combination of the TC rate plus whatever mechanism that dominates trap generation and that it cannot be slower than the TC rate. Thus we can be very con®dent about the choice of the TC model as the starting point. In the TC model, electron or hole does not play any role, which contradicts experience and is the main objection many researchers have against it. Intuitively, when a defect collides with a hole, the electric ®eld of the hole can also interact with the dipole to facilitate a conversion to trap. In close proximity, this ®eld is much higher than any applied ®eld. This reaction must exist in parallel with the one assumed in the TC model and, in situations where hole current is high, it should be the dominant reaction path. This proposed reaction allows the AHI model to share the same physics foundation as the TC model. A reaction pathway for the AHI model can be constructed from this proposal. Once both reaction pathways are clearly de®ned, we can construct a kinetic model that integrates both mechanisms. The result is a single master equation that provides a uni®ed framework for both high and low ®eld breakdown behavior.
2. The model The TC model assumes that precursor sites are converted into neutral traps by dipole±electric ®eld interaction k1
Vm ! N :
1
Here Vm denotes the mth precursor type, k1 is the reaction rate constant and N, the neutral defect formed by the reaction. Instead of taking oxygen vacancy as the only precursor like the original TC model, more than one type of precursors is assumed to be possible in the new model. This assumption is based on two facts. One
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
195
is the observed multi-precursors depletion behavior of SILC growth [10]. The other is the critical trap density to breakdown, according to the percolation model, is too high for the concentration of oxygen vacancy, the commonly accepted precursor. The reaction rate constant can be written as X Am e Ea1m =kT : 2 k1 m
The summation is over all the precursor types. The activation energy is, following McPherson and Baglee [1]: Ea1m DH0m
qm E:
3
Here Ea1m denotes the activation energy of the mth precursor in reaction (1). DH0 m is the enthalpy of activation (activation energy). qm is the dipole moment of the precursor and E is the oxide ®eld. The proposed reaction pathway for the AHI model ± the direct capturing of a hole by a precursor when a collision event occurs can be expressed as k2
k4
k3
e
Vm h ¢ P ! N :
4
Here P denotes a positively charged defect, which can capture an electron to become a neutral trap. A reverse reaction is included here to account for hole detrapping. The rate constant for the forward reaction (capturing of a hole) is X k2 Bm e Ea2m =kT : 5 m
The activation energies for the reaction is Ea2m DH0m
qm
q : 4pe2r
6
Here we used the electric ®eld of the hole to replace the oxide ®eld in reaction (1). Otherwise, the physics is the same as reaction (1), namely the barrier is lowered by the electric ®eld±dipole interaction energy. Since a hole is captured in this reaction, the result is not a neutral trap but a positively charged defect. Fig. 1 shows the electric ®eld of the hole as a function or less, the ®eld increases rapidly of distance. At 5 A beyond 15 MV cm±1 . At such high ®eld, the reaction barrier will be lowered to the point that reaction (2) will be limited only by the attempt rate (the pre-factor Bm ). Thus the capture cross-section (r) of a hole by defect sites is about 1 10 14 cm2 , similar to the value reported in Ref. [11]. Combining reactions (1) and (2), we can write down the set of rate equations for the model
Fig. 1. Electric ®eld of a point charge in a dielectric. When E increases beyond 15 MV cm±1 rapidly, implying a r < 5 A, hole capture cross-section of 110 14 cm2 .
dV k3 P k1 k2 Jh V ; dt dP k2 Jh V k3 k4 Je P ; dt dN k1 V k4 Je P : dt Here Jh and Je are hole current density and electron current density respectively. Here the unit of k1 and k3 is in s 1 while the unit of k2 and k4 are sJ 1 s 1 q cm 2 s 1 1 cm2 q 1 , with q being the unit charge. The solution to these equations is rather complicated. However, when hole detrapping is not signi®cant (i.e. k3 k2 Jh ), it can be reduced to a simpler form, which is the master equation of this model 1 N V 0 1 k4 Je e k1 k2 Jh t k2 Jh k4 Je 7 k 2 J h e k4 Je t : Here V 0 is the initial precursor concentration. 3. High ®eld limits When the last step of reaction (2), namely the capturing of an electron is very fast, the master equation can be further simpli®ed If k4 Je k1 k2 Jh : N V 0 1 e k1 k2 Jh t : 8 Eq. (8) is the basis for our discussion of unifying the E and 1/E models for gate-oxide breakdown. Before we do that, it is necessary to justify the approximation of k4 Je k1 k2 Jh to make sure that it is valid for the typical experimental condition where the E and 1/E dependent data are collected. It is well known that when stressing a not-too-thin oxide under high ®eld, rapid build-up and saturation of trapped holes occurs [12]. Using the set of rate equations
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K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
which means ln TF / 1=E: We thus also recovered the 1/E dependent of breakdown at high ®eld. Note that this is only valid for as long as Jh can be approximated with aJe . This approximation is not valid for thinner oxides.
4. Low ®eld limit
Fig. 2. Trapped positive charge concentration will saturate only if k4 Je is much greater than k1 k2 Jh . The actual magnitude of trapped hole concentration is shown in the insert.
and keeping the approximation of negligible hole detrapping (valid for not-too-thin oxide), we can calculate the accumulation of positively charged traps as a function of stress time for a number of relative values of k4 Je and k1 k2 Jh . Fig. 2 shows that rapid build-up and saturation of trapped holes can only happen if k4 Je is much larger than k1 k2 Jh . The approximation that leads to Eq. (8) is therefore well justi®ed. At high applied ®eld, if (this will be proved later) k2 Jh k1 , Eq. (8) becomes N V 0 1 e k2 Jh t : 9 In other words, trap generation is dominated by the hole capturing process (reaction (2)). Thus we recovered the experimental fact that the AHI model works better at high ®eld. According to the percolation model, breakdown is when trap density reaching a critical value. For measurements using ``identically made'' oxide samples, V 0 , should be the same. This means that k2 Jh t is a constant. Denoting t at breakdown as TF, we have k2 Jh t k2 Jh TF k2 QP const and QP const=k2 : From Eqs. (5) and (6), we know that k2 is also a constant that is independent of stress ®eld. We thus recovered the well-known AHI result of total hole-¯uence to breakdown is independent of stress ®eld. For not-too-thin oxides, Jh a Je , we have k2 Jh TF k2 aJe TF const: After rearrangement and writing out Je using the FN expression, we have TF
C k2 aGE2 e
H =E
At low ®eld, if k1 k2 Jh , Eq. (8) becomes N V 0 1e k1 t :
10
This is the original TC model. Again, using the idea of a critical trap density for breakdown from the percolation model, we can write k1 t k1 TF const: Combining with Eqs. (2) and (3) for k1 , we have ln TF / E: The E dependent is recovered. Clearly, the master equation has indeed uni®ed the high-/low-®eld breakdown behavior.
5. Quantitative evaluations We can estimate k2 using data from Ref. [2]. In Chen's work, very small (900 l2 ) capacitors were used, which leads to a high critical trap density according to the percolation model. Since our model is based on converting precursors into traps, precursor depletion can be expected when the trap density is very high. In a recent study, Lu et al. [10] showed that strong saturation of stress induced leakage current (SILC) growth can be observed when the capacitor size is very small. The saturation in SILC growth is a clear indication of precursor depletion. Chen's capacitor is somewhere between the two capacitor sizes used in Lu et al.'s. work, it is reasonable, therefore, to assume that the precursor depletion is only moderate. We therefore take N 0:1V 0 at breakdown for Chen's case. Once that is done, k2 can be estimated from QP , which was found to be 0.1 C cm2 [2]. We therefore estimate k2 to be 210 19 cm2 . 0:1 1
e
k2 Jh TF
k2 QP :
To estimate k1 , we can use Kimura's [7] data. In this case we assume [N]0.01[V]0 because that is more likely for larger size capacitor (Kimura did not specify). According to Eq. (10), we have 0:01 K1 TF:
K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
Fig. 3. k1 and k2 Jh as a function of E. Con®rms that k1 dominates at low ®eld while k2 Jh can dominate at high ®eld.
197
Fig. 4. Data from various groups have wide range of slopes. This can only be explained that various oxides have dierent mix of precursor types.
In other words, k1 is related to TF and is a function of oxide, the oxide ®eld. Since Jh aJe , and for 100 A a 10 3 [13], we can compare k1 to k2 Jh for the full range of stress ®eld (Fig. 3). As can be seen, at low ®eld, k1 k2 Jh , while at high ®eld, k2 Jh dominates. The assumptions behind the two limiting cases are justi®ed. The commonly cited transition around 8±9 MV cm±1 is also reproduced. Note that the QP value from Chen et al.'s data must be over estimated by as much as an order of magnitude [14]. Thus the k2 value may be under estimated by an order of magnitude. If that were true, the cross over would be more prominent.
6. Further discussions Among the reports [6,7,15,16] that support the E model, the ®eld acceleration factor (slope of the log(TF) versus E curve) varies widely (Fig. 4). In a recent report, trap growth was found to be best modeled as the depletion of three types of precursor [10]. The literature on oxide has identi®ed many defect types [17]. It is therefore entirely reasonable for the new model to assume multiple precursor types. The actual mix of precursors depends on the growth and processing conditions of the gate oxide, and the ®eld acceleration factors as well as the dipole moments should naturally be dierent for dierent oxide. Thus it is entirely reasonable to ®nd that data from various groups have wide range of slopes. In addition to oxide quality dierence, it is also entirely possible that the precursor mix for the same oxide changes with distance from the interfaces. It is well known that strain in the oxide is highest at the SiO2 /Si interface. It is expected that even for the same defect type, the activation energy will be dierent. The precursor mix, which aects k1 , also aects the stress ®eld at which the transition from E to 1/E behavior occurs. For many reported cases such as those in
Fig. 5. When k1 is large, k2 Jh never dominates. All of these data show no evidence of 1/E dependent even at high ®eld, consistent with the model.
Fig. 4, the transition was not observed even at oxide ®eld as high as 12 MV cm±1 . When the k1 value of these experimental results are calculated, they indeed do not cross the hole capture dominated curve (k2 Jh ) (Fig. 5). (This comparison is not entirely valid. The value of k2 is expected to depend on the precursor type and therefore on oxide quality. One should only compare values from the same oxide.) 7. Master equation check Experiments that independently control Jh and Je are good tests for the master equation. For example, Satake et al. [18] found that, with a constant Jh and a small Je , QP decreases with increasing Je while QBD remains constant. These results can be explained with the master equation. Taking k4 Je k2 Jh , Eq. (7) becomes N V 0 1 e k4 Je t : 11
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K.P. Cheung / Microelectronics Reliability 41 (2001) 193±199
Eq. (11) suggests that when Je , is low, trap generation is controlled by the capturing of electrons. For the same capacitor type, the critical trap density for breakdown is ®xed. That means k4 Je TF is a constant regardless of the stress condition. If k4 were independent of stress condition (a reasonable assumption, see discussion later), a constant QBD would result, which is Satake et al.'s observation. When Je is low, conversion of positively charged trap into neutral trap by capturing an electron will be a bottleneck. Piling up of positive charges in the oxide can be expected (see insert of Fig. 2). Signi®cant precursor depletion will occur during most of the stress period. The hole trapping eciency is reduced over the same period. Since Jh is constant, QP becomes proportional to TF, which is controlled by Je . A higher Je leads to a smaller TF and a smaller Je leads to a longer TF. Thus Satake et al.'s observation that increasing Je leads to a decrease in QP and, increasing Jh leads to an increase in QP is easily explained. All of Satake et al.'s observations are in complete agreement with the master equation.
8. Ultrathin gate-oxide situations k3 cannot be When gate-oxide is thinner than 50 A, neglected and the set of rate equations can only be solved numerically. This is not too dicult once all the constants are pinned down. The detrapping of holes is a tunneling process (Fig. 6) with a known barrier height [19] and can be calculated. The one remaining unknown quantity, k4 , is likely to be smaller than k2 . The collision of an electron with a trapped hole is dierent from an electron interacting with a defect site. We proposed here that the electric
®eld of the hole could interact strongly with the dipole moment of a defect site. A natural question is why would not electrons do the same? If they do, then the much larger electron current density should make the rate of electron trapping by direct capture much higher than the rate of hole trapping. This is not experimentally observed. The key dierent between direct capturing of holes and electrons by defect sites is in the size of the waves associated with hole and electron. In SiO2 , the electrons traveling in the conduction band is far more delocalized than the hole traveling in the valence band. The light eective mass (therefore small local distortion of the network) and high velocity of an electron makes inelastic collision far less probable due to conservation of momentum. The probability of capturing an electron by a trapped hole is greatly enhanced by the strong local distortion of the network around the captured hole. This strong local distortion enhances the probability of multiple phonon emission, thus provide a way for momentum conservation. Nevertheless, the need to conserve the large momentum of a fast moving electron suggests that the magnitude of k4 , should be smaller than k2 . Careful experiments such as those done by Satake et al. [18] should help pin down the actual value.
9. Conclusion A physics based, uni®ed model for gate-oxide breakdown at both high and low ®eld has been developed. It has successfully account for a wide range of experimental observations. This new model treats the TC and AHI mechanisms as competing processes. The transition point from one to another will depend on the precursor mix and on the hole current density. For thicker oxides, the precursor mix is more important. For thinner oxides, the large direct tunneling electron current may cause the hole current density to be high enough [20] that the hole capture reaction continue to dominate at lower ®eld. That is, for thinner oxides, the AHI model may be valid for a wider range of ®eld. Whether this is indeed the case is not yet known at this point since there are no low ®eld data reported in the literature. This downward extension of the valid range of the AHI model is not expected to reach the normal operation ®eld. The rapid decrease of hole current density as a function of voltage ensures that trap creation rate due to AHI will be far below the trap generation rate due to TC at normal operation voltage.
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Fig. 6. Hole detrapping by tuning out.
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