Kinetics of defect creation in amorphous silicon thin film transistors

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 9 1 MAY 23 Kinetics of defect creation in amorphous silicon thin film transistors R. B. Wehrspohn a) Max-Planck-Institute of Microstructure Physics, Weinberg 2, D-612, Halle, Germany M. J. Powell and S. C. Deane Philips Research Laboratories, Redhill, Surrey, RH1 5HA, United Kingdom Received 25 September 22; accepted 12 February 23 We have developed a theoretical model to account for the kinetics of defect state creation in amorphous silicon thin film transistors, subjected to gate bias stress. The defect forming reaction is a transition with an exponential distribution of energy barriers. We show that a single-hop limit for these transitions can describe the defect creation kinetics well, provided the backward reaction and the charge states of the formed defects are properly taken into account. The model predicts a rate of defect creation given by (N BT ) (t/t ) (1), with the key result that 3. The time constant t is also found to depend on band-tail carrier density. Both results are in excellent agreement with experimental data. The t dependence means that comparing defect creation kinetics for different thin film transistors can only be done for the same value of band-tail carrier density. Normalization of bias stress data on different thin film transistors made at different band-tail densities is not possible. 23 American Institute of Physics. DOI: 1.163/ a Electronic mail: wehrspoh@mpi-halle.de I. INTRODUCTION A. Threshold voltage instability Amorphous silicon thin film transistors TFT were first proposed more than 2 years ago, 1 and soon after it was reported that these TFT s showed a threshold voltage instability. 2 The application of a gate voltage to the TFT for a prolonged period of time results in a shift of the TFT threshold voltage. This phenomenon has been widely investigated and is the subject of this article. Experimental measurements of the threshold voltage shift are usually carried out under one of two experimental conditions. Most commonly, there is constant bias stress, where a constant gate voltage is applied to the TFT and the threshold voltage shift with time is measured. During the bias stress, the TFT oncurrent decreases, and this affects the further rate of threshold voltage shift. Alternatively, experiments can be performed under constant current stress, where the applied gate bias to the TFT is continually adjusted in time to keep the TFT on-current constant. Keeping the TFT on-current constant is equivalent to keeping the band-tail electron density constant, during the stressing experiment. This means that to first approximation the Fermi level at the semiconductor gate insulator interface remains constant. Constant current stress can give additional information on the kinetics of the underlying mechanism. From a practical point of view, this situation occurs when a constant current is switched for example during active-matrix addressing of light emitting diodes. Reasons for the threshold voltage shift have been discussed in literature: Initially, charge trapping in the insulator was proposed as the instability mechanism. 2 However, measurements on ambipolar TFT s, 3 and the fact that the instability does not depend on the insulator (a-si:n x :H), 4 have unambiguously shown that the threshold voltage shift in the low voltage stress region is due to metastable defect creation in the amorphous silicon layer. For high biases typically larger than 2 MV/cm or low-quality SiN gate insulators, charge trapping dominates. 4 Metastable defect creation has been studied extensively for light-induced defects Staebler- Wronski effect. However, there is still no consensus about the nature of the light-induced defects and the exact description of the kinetics. 5 Two main differences exist between carrier-induced defect creation and light-induced defect creation. First, one type of carrier is present only either electrons or holes depending on the type of the device. Second, a thermal barrier for defect creation exists. For light-induced defect creation the recombination of an electron-hole pair provides the bond breaking energy and the defect creation process is essentially temperature independent, providing no information about the energy barrier. These two additional characteristics facilitate the modeling of carrier-induced defect creation irrespective of the exact knowledge of the microscopic defect creation reaction. B. Semiempirical models for threshold voltage shifts The first publications fitted the threshold voltage shift V t by a logarithmic behavior 2 V t V log1t/t 1 with V the initial gate bias over threshold, V g V ti, and t 1 c exp(e A /) where E A is the activation energy and c the attempt frequency. This model was based on the assumption that charge injection into the silicon nitride gate insulator is the dominant mechanism for defect creation. 3 However, as mentioned above, it turned out that additional defect state creation dominates for moderate biases in high-quality TFT s. 4 In the following, we suppose that we work only /23/93(9)/578/9/$ American Institute of Physics

2 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane 5781 under moderate bias where state creation is dominant. Then, the threshold voltage shift V t is proportional to the number of created defects N D (t) in a TFT under gate bias stress due to the capacitor equation CV t qn D (t) with C the capacitance of the TFT. The first fit for defect state creation kinetics was based on a stretched exponential 6 N D tn BT 1expt/t 2 with T/T, t 1 c exp(e A /), and N BT the initial band-tail carrier density. This two-parameter fit (E A, )is currently the most used fit to describe threshold voltage shifts in a-si:h TFT s. Jackson et al. 7 argued that a stretched exponential time behavior is obtained for the diffusion of a particle in a medium with randomly distributed traps. These systems are generally described by a Smoluchowski-type reaction-diffusion equation: 8 N t D2 NkN, 3 where k is the reaction rate constant and D the diffusion constant. This kind of rate equation also applies to defect creation in a-si:h if one assumes that hydrogen motion is the precursor for defect creation 7 and k is the rate constant for defect creation. Assuming a time-dependent diffusion constant, the diffusion term in Eq. 3 can be neglected and the change in the flux of hydrogen arriving at the defect creation site is integrated in the rate constant k, reading kdt 1. This is called dispersive transport and has been experimentally observed during hydrogen diffusion experiments in amorphous silicon. 9 Solving Eq. 3 in combination with Eq. 4, one obtains a stretched exponential time behavior. However, several groups have observed a significant deviation of this stretched exponential time dependence In particular, the dependence of the defect creation rate dn D (t)/ on the number of band-tail carriers, N BT, was not correctly described by Eq. 2. Recently, we presented an improved, semiempirical kinetic equation for defect creation: 12 dn D t dn BT t1 kn BT t with in the range of 1.5 to 1.9, kconst, and t 1 c exp(e A /), where E A is related to the most probable energy barrier for defect creation and c is the attempt frequency for defect creation. An analytic solution is possible for 12 which yields a stretched hyperbola of the form 1 N D tn BT 1 1t/t 1/ 6 with 1. We have shown 12 that the three-parameter (E A,T,) stretched hyperbola fit is an improvement compared to the commonly used two-parameter stretched exponential fit, 6 since it takes into account the superlinear bias dependence N BT Eq. 5. Jackson already proposed a similar kind of equation in He realized in his 1989 article 7 that there is a problem with the standard 4 5 FIG. 1. Typical threshold voltage shift V t of a bottom gate TFT S3L during bias stress (V g 3 V) for T383 K. The SiN thickness is 3 nm and the initial threshold voltage is V ti 3 V; thus the bias over threshold is V 27 V. Smoluchowski-type reaction-diffusion equation Eq. 3. It is normally solved for static traps. In the case of defect creation, the first arriving hydrogen atom neutralizes the trap and creates a defect. In the Jackson 199 article, 13 he proposed therefore a carrier-density-dependent hydrogen diffusion coefficient to modify the analytical approximation Eq. 4, DD N BT, 7 where D is the hydrogen diffusion prefactor. Inserting Eq. 7 into the time-dependent rate constant Eq. 4, one obtains a similar stretched hyperbola relationship as in Eq. 6. Note that this hydrogen diffusion model takes into account the forward reaction rate only. So far, all models are based on at least some empirical observation and are not ab initio models. There is no physical explanation for the parameter. Furthermore, the rate equations involving Eqs. 3 7 consider only the forward reaction rate. The backward reaction has been neglected. It will become evident in Sec. III that the backward reaction is crucial to the understanding of the physics of defect creation. It is the key to understanding the high value of. It is also the key to understanding the dependence of t on the initial number of band-tail electrons N BT. This latter experimental result cannot be modeled with any of the semiempirical models published up to now. 13,14 In the following we summarize the most relevant experimental results on threshold voltage shifts in amorphous silicon TFT s, which any model has to consider. We then develop an ab initio physical model, which fits all the key experimental results and which gives insight into the bias stressing. II. EXPERIMENTS Experimentally, the defect creation kinetics are strongly nonexponential and even nonstretched exponential. 12 Figure 1 shows the threshold voltage shift V t (t) of an amorphous

3 5782 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane FIG. 2. Threshold voltage shift V t as a function of gate bias over initial threshold voltage V for two samples :.2 cm 2 V 1 s 1 ; : 1.3 cm 2 V 1 s 1 ). The constant-current stressing time was 1 s at 35 K. silicon bottom gate TFT under 3 V gate bias stress, which corresponds to a field of about 1 MV/cm over threshold. The bias stress experiments were carried out on n-type, silicon nitride gate insulator a-si:h TFT s deposited at 3 C on a crystalline silicon wafer. The preparation and measurement conditions were reported elsewhere. 15 The TFT was annealed to 5 K for 1 h before the stress experiment. The second important characteristic of the threshold voltage shift V t (t) is its nonlinear behavior on the effective gate bias over threshold V,eff (t)v g,eff (t)v ti, i.e., nonlinear dependence on the number of band-tail carriers N BT (t). This is manifested in Eq. 5 by the power Experimentally, the power can be accurately determined by an experiment under constant current stress (N BT const) for which the kinetics have a power-law behavior for short and medium stressing times Eq. 5. Figure 2 shows V t (t) during constant current stress for short stressing times as a function of the applied gate bias over initial threshold V. 16 Two different TFT s have been chosen, one with a very low mobility of.2 cm 2 V 1 s 1, the other with a high mobility of 1.3 cm 2 V 1 s 1. For both TFT s, the threshold voltage shift V t (t) dependence on the gate bias over initial threshold V is superlinear, with the power lying typically between 1.5 and 1.9. A third important feature, which has not been considered in detail up to now, is the dependence of t and thereby E A on the applied gate bias over initial threshold V, i.e., the initial band-tail carrier density N BT. 14 For example, stressing a TFT with a ten times higher V does not result in exactly ten times faster defect creation. In Fig. 3, a similar device as in Fig. 1 is stressed with 1, 2, and 3 V gate biases. The defect creation kinetics do not exactly scale, but the higher the initial band-tail carrier density N BT, the more easily defects are created. Or in other words, t is shifted to lower values with increased gate bias V. In summary, any model has to account for these three important experimental characteristics of defect creation kinetics during gate-bias stress: nonexponential time behavior of defect creation N D (t); 7 superlinear bias dependence of N D (t); 1 13 and dependence of t on initial bias N BT. 14 FIG. 3. a Threshold voltage shift for bottom gate TFT S3L for different gate biases V g of 1, 2, and 3 V measured for different temperatures in the range from 3 to 39 K. The stressing times and temperatures have been unified by the thermalization energy E th ln( f t) with 1 1 s 1. b Derivative of a with respect to E th is plotted. It can be observed that the maximum of the probability distribution for defect creation shifts to lowerenergy E th with increased gate bias V g, i.e., increased band-tail carrier density N BT. III. MODELING A. Introduction We develop our model in three stages. We first recalculate carefully the previous approach for describing defect creation in a-si:h TFT s, i.e., considering the forward reaction rate only and solving for the defect density in steady state. We show that, taking this approach, one does not obtain the observed superlinear band-tail carrier dependence, but a sublinear band-tail carrier dependence. In a second step, we include the backward reaction during defect creation and show numerically that taking into account the backward reaction increases the band-tail carrier dependence to about 1. However, this does not lead to the observed superlinear behavior of defect creation on the initial band-tail carrier density N BT. Only when we take into account the charge states of the barrier and the final states are all features of the defect creation kinetics well described. In simple terms this is due to the quenching of the backward reaction by the increased number of band-tail carriers. Our model predicts a superlinear dependence of the defect creation rate on the initial band-tail carrier density and a stretched hyperbola time dependence for the total density of created defects. We show that the stretched hyperbola characteristic time constant t is also found to depend on the band-tail carrier density, in agreement with experimental results. Comparing defect creation kinetics for different TFT samples can only be done for the same value of band-tail carrier density. Normalization for different band-tail densities is not possible.

4 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane 5783 FIG. 4. Configurational diagram of the defect creation model. A is the initial state, A* the barrier state, B* an intermediate defect state, and B the final defect state. It is assumed that the transition from B* to B is not rate limiting and that the transmission probability is sufficiently low so that it does not affect the kinetics. We assume in model III C an exponential barrier distribution exp(e*/ ) and a carrier-dependent forward reaction activation energy E*. In addition, we assume in model III D a backward reaction activation energy of E*E form and in model III E a backward reaction activation energy of E*2E form. AtE M the density of barrier states is N. The situation for model III E is shown. B. Microscopic models for defect creation Similarly to other authors, we apply an exponential barrier model for carrier-induced defect creation. For all three stages of our model discussed in this article, the exact microscopic reactions are not important since the defect kinetics holds for most possible rate-limiting steps. In particular, breaking of a silicon-silicon bond, or emission of hydrogen out of a single hydrogenated silicon bond Si H, a double hydrogenated silicon bond SiHHSi or H 2 *) will exhibit the same defect creation kinetics. The only two ingredients required are an exponential distribution of barrier states and a barrier lowering due to the band-tail carriers. We therefore refer to the initial bond as the precursor bond. Depending on the specific microscopic reactions, this precursor bond might be a a Si Si bond 14,16 18 Si Si DD, where D is a silicon dangling bond; b a Si H bond 19 Si H H i D, 9 where H i is a mobile hydrogen interstitial; or c a SiHHSi or H 2 * bond 2,21 SiHHSi H i SiHD 1 where SiHD represents neighboring silicon dangling and silicon-hydrogen bonds. C. Forward reaction rate and Boltzmann approximation 1. Forward reaction rate We define a general three-state configuration coordinate diagram of defect creation Fig. 4. State A is the initial state, A* is the barrier state, and B* is the final state, i.e., the 8 FIG. 5. Effect of charge state on the defect formation energy: schematic diagram of formation energy for defect creation vs Fermi-level position Ref 22. E form is the formation energy for the neutral defect and the barrier lowering energy due to the formation of charged defects. defect state. According to the Eyring theory, the forward reaction rate of an activated reaction from a state A to a state B* over a barrier A* is dn A R f N A 11 with N A the number of precursor bonds in state A and R f the forward reaction rate, defined as R f f expe*/ 12 with E* being the energetic barrier for the forward reaction rate and f the attempt frequency for the forward reaction. The energy needed to break a strong precursor bond located in the extended states is defined as E M. We assume that the binding energy lowering of a weak precursor bond in the band tails is proportional to its one-electron energy lowering, i.e., the energy from the mobility edge to the weak bond energy. Therefore, we assume an exponential barrier distribution of states A* with a characteristic energy Fig. 4. At the energy E M the density of states N equals that at the mobility edge. Thus N A *(E*) reads N A *E*N expe M E*/. 13 The driving forces for the forward reaction are band-tail carriers, 13 so Eq. 12 has to be modified to R f f expe*/, 14 where is the carrier-induced lowering of the barrier for defect creation. is defined in line with the defect pool model for positive bias 13,22 Fig. 5 as E form lnn BT, 15 where E form is the formation energy of the neutral defect level Fig. 5 and n BT is the normalized band-tail carrier density N BT /N c with N c the density of states at the mobility edge. 29 Here we implicitly assume that n BT is spatially constant in a TFT. Due to the band bending near the Si/SiN interface, a spatially inhomogeneous distribution would be more realistic. However, defect pool modeling has shown

5 5784 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane that a constant Fermi level describes reasonably well the equilibrium defect distribution. 23 We therefore assume in the following that this is also the case for nonequilibrium defect creation. Inserting Eqs. 13 and 14 in Eq. 11 and considering that the total number of sites N A is constant N A *N D (t) N A (t), we obtain for the forward reaction rate per energy interval de* dn A dn D f exp E* N exp E ME* N D. 16 This equation is correct for one-electron barrier lowering, i.e., is due to one electron. If one assumes two-electron barrier lowering, has to be replaced by Boltzmann approximation To solve Eq. 16, one has to integrate over all times and all energy barriers de*. However, these two integrals are not separable since the energy an electron needs to overcome the barrier depends on the time t it attempts to escape. In order to solve this problem analytically, we neglect the depletion of precursor states N D on the right side of Eq. 16 and approximate the steady-state distribution by applying the Boltzmann distribution, taking carefully into account the barrier lowering. Then, all weak bonds which are below the thermalization energy E th, E th ln f t, 17 have been completely converted to defects. Note that in the case of electron accumulation the energy barrier is lowered by ; thus the thermalization energy has to be raised by the amount of barrier lowering. All weak bonds that are above E th convert only partially, approximately the Boltzmann fraction of the weak bonds. Thus, the total number of defects is N D N exp E ME th t dn de*. 18 Here we have set the upper limit of the integral over E* to infinity since the Boltzmann fraction of broken strong precursor bonds is negligible for. In a schematic picture, the total number of weak precursor bonds converted to defects after a time t is illustrated in Fig. 6. Energetically below E th the first term all precursors have converted to defect states whereas above E th second term only the Boltzmann fraction converts to defect states. Solving Eq. 18 see the Appendix, one obtains for the defect creation rate dn D f N 2 exp E ME form n BT t 1 19 FIG. 6. Density of precursor states N(E) that contribute to defect creation. The schematic diagram shows the density of converted dangling bonds for a thermalization energy E th ln(t). Energetically below E th, all precursors have completely converted to defect states, whereas above E th only the Boltzmann fraction converts to defect states. with /. Equation 19 represents the defect creation rate if we assume an exponential distribution of barriers, a barrier lowering energy, and the Boltzmann approximation for the steady-state situation. The last but one factor of Eq. 19 shows that the band-tail carrier dependence varies as a power of. Thus, taking into account only the forward reaction and the Boltzmann approximation for steady state leads to a sublinear band-tail carrier dependence with typically.4 to.5 Fig. 7, in contradiction to the observed dependence in the range of 1.5 to 1.9 Fig. 2. Even if one considers two electrons involved in the barrier lowering, is 2 and therefore in the range of.8 to 1, still not in line with the experiments. Moreover, E M has no dependence on the initial band-tail carrier density N BT, in contradiction to the observed band-tail carrier dependence of t. Note that previous calculations based on the forward reaction rate yield only 1, 24,25 due to a mistake in the lower bound of the first integral of Eq. 18, which we show depends on n BT due to Eq. 17. FIG. 7. Numerical simulation of the normalized number of defects N D as a function of the gate bias for model III C based on Eq. 19, for model III D based on Eq. 23 and for model III E based on Eq. 25. The band-tail carrier dependence of the defect creation rate varies as the power of shown next to the curves. Parameters: 3.4 mev, 62 mev, t1 s.

6 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane 5785 D. Forward and backward reactions To include the backward reaction, the reaction from B* over the barrier A* to the state A has also to be considered in the three-state configuration coordinate diagram of defect creation Fig. 4. We assume the same situation as in the previous stage: exponential barrier distribution of the states A* Eq. 13 and an activation energy lowering for the forward reaction rate Eq. 15. The site B* is also lowered by since the defect stays negatively charged. The rate of creation of defects dn D / for an energy barrier E* is dn D E*,t R de* f E*,tN A E*,tR b E*,tN D E*,t 2 with R f and R b are the forward and backward reaction rates, respectively. The forward reaction rate R f for the energy barrier E* is given by the activation energy, taking into account the barrier lowering see also Eq. 15: R f t f exp E* f n BT exp E*E form. 21 If there were no barrier distribution, then the defect creation kinetics would be dominated for short times by the forward reaction rate only. However, since there is an exponential distribution of barriers, the backward reaction is important for all times. For example, defects with barriers E* ln( f t) have completely equilibrated and defects with E* ln( f t) have a significant component of the backward reaction. The backward reaction rate is related to the activation energy from the defect site B* to the activated complex A*. We assume for the site B* a single barrier lowering similar to the barrier state: R b t b exp E*E form, 22 where b is the attempt frequency for the backward reaction rate. Inserting Eqs. 13, 21, and 22 in Eq. 2, and considering that the total number of sites N A is constant N A * N D (t)n A (t), one obtains for the rate of defect creation per energy barrier E* dn D E*,t de* f exp E*E form n BT N exp E*E M N D E*,t n BT b f. 23 We numerically modeled Eq. 23 for typically 3 energy levels for a constant-voltage stress situation, i.e., N BT (t) N BT 3 i1 N D (E i *,t). The key parameters are shown in Table I. We set f b based on Eyring theory. For short time stressing of about 1 s, we obtain 2 by varying the initial band-tail carrier density N BT Fig. 7 and we observe no dependence of E A on N BT. Thus, these results are TABLE I. Default parameters used in the numerical and analytical calculation. N has been normalized to 1. N I is the normalized value of N BT. Parameter Model III D Model III E Stretched hyperbola fit T/T E A or E M 1.45 ev (E M ) 1.45 ev (E M ) 1.15 ev (E A ) f ( b ) 1 1 Hz 1 1 Hz 1 1 Hz E form N N I N I N N I N N I N still in contradiction to the experimental observations 2 and 3 in Sec. II. In the next stage, the effect of the charge state of the created defect is included. E. Charge states The defect creation event is triggered by the band-tail carrier density because the activation energy for the forward reaction rate is lowered by. It is rather unlikely that two electrons will exit on one precursor bond, so that the activation energy has only once the barrier lowering energy. However, in the final state, one precursor bond will always create two defects, which under electron accumulation are charged. Thus, the site B* is lowered twice 2. Notice that one defect creation site, B*, corresponds to two electronically active states. Thus, we obtain for the backward reaction R b t b exp E*2E form b n 1 BT exp E*E form. 24 Inserting Eqs. 13, 24, and 22 in Eq. 2, and considering that the total number of sites N A is constant N A * N D (t)n A (t), one obtains for the rate of defect creation per energy barrier E* dn D E*,t de* f exp E*E form n BT N exp E*E M N D E*,t n BT f n BT. 25 We have numerically modeled Eq. 25 for 3 energy levels for a constant-voltage stress situation, i.e., N BT (t) N BT 3 i1 N D (E i *,t). The parameters used in the calculation are summarized in Table I. To check the calculation, we first varied only E M based on Eq. 25 Fig. 8. As expected, we obtain a linear shift of the defect creation kinetics toward higher thermalization energy E th with increasing E M. In a next step, we calculated from the threshold voltage shift for different biases in the short time stressing regime (t1 s). We obtain a superlinear band-tail carrier dependence for defect creation of 1.5 Fig. 7, in excellent agreement with the experimental data. In addition, we calculated for different inital biases the whole kinetics until saturation as a function of E th Fig. 9. With increasing N BT,we b

7 5786 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane FIG. 8. Numerical simulation of the defect creation kinetics for different barrier heights E M.95, 1, 1.5, and 1.1 ev based on Eq. 25. Parameters: 3.4 mev, 62 mev. The increase of E M leads to a parallel shift of the defect creation kinetics. FIG. 1. Comparison of experimental threshold voltage shift during gate bias stressing at V g 3 V at T383 K, empirical stretched hyperbola fit Eq. 6, and numerically calculated data Eq. 25. For parameters, see Table I. clearly observe a shift of the maximum of the probability distribution of defect creation, which is related to E A, toward lower thermalization energies. Simultaneously, the distribution broadens slightly. Both facts are in very good agreement with the experimental data Fig. 3. IV. DISCUSSION In the last section, we discussed the three stages of our model for defect creation. First, we considered the forward reaction only and included the backward reaction by steadystate considerations based on the Boltzmann approximation FIG. 9. a Numerical simulation of the defect creation kinetics for different initial band-tail carrier concentrations based on Eq. 25. A shift of the kinetics toward lower thermalization energies is observable for a relative increase of the band-tail carrier density by a factor of 1, 2, and 3, similar to Fig. 3. b Derivative of a with respect to the thermalization energy. Note that the maximum of the probability distribution of defect creation decreases with decreasing band-tail carrier density in a similar way as the experimental data in Fig. 3. Sec. III C. This allowed us to make an analytical solution for the defect creation kinetics. In terms of the configuration diagram Fig. 4, the barrier lowering and the steady-state Boltzmann approximation lead to an activation energy lowering of the forward reaction rate and backward reaction, i.e., a barrier lowering for the state A* Fig. 4. Note that for a single barrier the steady-state defect density would not change. However, due to the exponential distribution of barriers, a lowering of A* does lead to an increased defect density. The defect creation rate is proportional to (n BT ) due to the Boltzmann approximation. The barrier lowering in Eq. 19 is / and is proportional to ln(n BT ) Eq. 15. Thus, the impact of n BT on the defect creation rate is modified by the power T/T. In the second stage Sec. III D, we included the forward and backward reactions numerically and assumed that the barrier state A* and the final state B* are lowered by. Thus, the forward reaction rate is increased whereas the backward reaction is unchanged Fig. 4. This leads to an increased dependence of the defect creation rate on the banail carrier density. The defect creation rate is then proportional to (n BT ) 2, in line with our modeling Fig. 7. In the third stage Sec. III E, we assume one barrier lowering for the state A* and twice the barrier lowering for the final state. Thus, the forward reaction rate is increased and the backward reaction is quenched by the band-tail carrier density. A possible microscopic reaction representing these kinetics would be Si Si (A) D D (A*) D D (B*). This increases the efficiency of the band-tail carrier density on the defect creation rate to a superlinear value (n BT ) 3. Only modeling of the third stage is in line with all key experimental observations, namely, the stretched hyperbola time dependence, the superlinear bias dependence of N D (t), and the t dependence on initial bias N BT. For example, in Fig. 1, the experimental threshold voltage shift, the numerical data based on Eq. 25, and a stretched hyperbola fit Eq. 6 are shown. Table I shows the parameters used for the numerical fit and the stretched hyperbola fit. There is agreement between these three curves within a few percent, un-

8 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane 5787 derlining the numerical fit. Moreover, for a high-mobility TFT with of typically 5 mev, 15 we obtain for 3 mev a power 31.8, whereas for a low-mobility TFT with typically 6 mev, a power 31.5 is obtained, in very good agreement with our experimental data Fig. 2. In summary, there is a superlinear dependence of the band-tail carrier density on defect creation rate due to the double impact of band-tail carriers: lowering of the forward reaction rate and quenching of the backward reaction. Due to the quenching of the backward reaction rate by the band-tail carrier density, the probability distribution of defect creation barriers is also changed, thus leading to a lower t with N BT Fig. 9. This is in very good agreement with the experimental data in Fig. 3. This effect is not included in the steadystate approach described in Sec. III C and also not in Sec. III D and not in the stretched hyperbola equation Eq Nevertheless, as shown in Fig. 1 and Ref. 16, the stretched hyperbola is a useful fit. The effect of the backward reaction could be implemented by a dependence of E A on the banail carrier density. Neglecting this dependence leads to differences in the fit parameter E A and for different initial biases over threshold V. Therefore, it has to be taken care that the parameters extracted by the stretched hyperbola fit are comparable only if different TFT s are stressed with the same initial bias over threshold V. For example, for the values of Fig. 3, the difference between stressing the TFT with 1 or 3 V yields a shift of 4% in E A obtained by the stretched hyperbola fit. In units of V t, this corresponds to a difference of 25% in V t /V at E th 1 ev. Finally, the impact of the findings above on the configurational diagram in Fig. 4 is discussed. The activation energy gave us information on the state A* and the band-tail carrier dependence allowed us to gain information on state B*. This is a unique possibility for TFT s in contrast to the Staebler- Wronski effect in solar cells, where information neither on A* nor B* can be gained. Considering the three possible microscopic models in Sec. III B, we might expect, the characteristic energy for the exponential distribution of energy barriers, to be related to the characteristic energy of the valence band tail E v.in the absence of any network strain, would be 2E v for reaction a the breaking of a Si Si bond but zero for reactions b and c breaking of Si H bonds. In reality, we expect network strain to play a significant role. 2 This will reduce the characteristic energy for reaction a but increase the characteristic energy for reactions b and c. In particular, for reaction c we expect a characteristic energy of 2E v. 2 For reaction b, we expect the characteristic energy to be much lower. For reaction a, we expect the characteristic energy to be in the range of E v to 2E v. Experimentally, we find to be in the range (1 1.5). This favors reaction a, although this is probably not sufficient to distinguish between different models. The state B* represents two charged dangling bonds. The defect pool model has shown that the final thermal equilibrium state is two charged SiHD defects. 2,26 The state B* may not be the final state of the reaction. There can be a transmission from state B* to state B. In particular, state B* can be two charged dangling bonds (D D ), originating from one broken Si Si bond, while state B can be two isolated charged SiHD defects, formed as a result of subsequent hydrogen motion. Possible microscopic reactions for this process are discussed in detail elsewhere. 27 If the transmission probability from state B* to state B is low, then the kinetics of defect creation will be dominated by the reaction A to B*. For the forward and backward reactions, A B*, we have set the attempt frequencies f and b to be equal. This is correct in the single-hop limit of the Eyring theory. This is in contrast to the attempt frequency for any reactions from state B, which can be quite different. These are the defect annealing reactions. Recently, we reported that the attempt frequencies for defect creation and defect annealing differ by three orders of magnitude, 12,14 which is in line with earlier measurements. 13 This is consistent with the defect creation reaction being the breaking of a Si Si bond, while the defect annealing reaction is due to the breaking of a Si H bond. 28 The low attempt rate of 1 1 Hz of the defect creation reaction is due to the modulation of the Si Si phonon frequency by the probability that the bond is occupied by an electron and by the probability of a Si H bond switching event from a nearby SiHHSi site reaction B* to B). Defect annealing from state B) does not influence the kinetics of defect creation, under normal experimental conditions, while the backward reaction from state B* A plays a significant role in modifying the defect creation kinetics. V. CONCLUSION We have modeled the defect creation kinetics of amorphous silicon thin film transistors during gate-bias stress based on an exponential barrier model. The single-hop limit over an exponentially distributed barrier can describe the defect creation kinetics reasonably well, provided that the backward reaction and the charge states are taken into account. As a consequence of including these terms, the experimentally observed superlinear band-tail carrier dependence for defect creation and the band-tail carrier dependence of the barrier height are in good agreement with experimental observations. The two-parameter stretched hyperbola approximation, which was recently proposed by us to describe defect creation kinetics, is a reasonable approximation of the numerical results provided that the initial band-tail carrier density is the same in all devices being compared. VI. APPENDIX Inserting the thermalization energy E th Eq. 17 into Eq. 18 yields N D N exp E M exp f t / t dn de*. Differentiating N D yields A1

9 5788 J. Appl. Phys., Vol. 93, No. 9, 1 May 23 Wehrspohn, Powell, and Deane dn D N exp E M exp f t 1 dn de*. A2 dn D f N 2 exp E ME form n BT f t (1). A6 For partially converted defects, one obtains for the integral of Eq. A2 or dn D de* f N exp E ME* exp E* de* A3 dn D de* f N exp E M exp exp E* 1dE*. Integrating over all energy states E* reads dn D f N exp f t (1). exp E M A4 A5 Inserting Eq. A5 into Eq. A2 and replacing by n BT Eq. 15 yields 1 A. J. Snell, K. D. Mackenzie, W. E. Spear, P. G. LeComber, and A. J. Hughes, Appl. Phys. Lett. 24, M. J. Powell, Appl. Phys. Lett. 43, C. van Berkel and M. J. Powell, Appl. Phys. Lett. 51, M. J. Powell, C. van Berkel, and J. R. Hughes, Appl. Phys. Lett. 54, M. Stutzmann, Mater. Res. Soc. Symp. Proc. 467, W. B. Jackson and M. D. Moyer, Phys. Rev. B 36, W. Jackson, C. C. Tsai, and R. Thomson, J. Non-Cryst. Solids 114, P. Grassberger and I. Procaccia, J. Phys. Chem. 77, R. A. Street, Physica B 17, Y. Kaneko, A. Sasano, and T. Tsukada, J. Appl. Phys. 69, F. R. Libsch and J. Kanicki, Appl. Phys. Lett. 62, S. C. Deane, R. B. Wehrspohn, and M. J. Powell, Phys. Rev. B 58, W. B. Jackson, Phys. Rev. B 41, R. B. Wehrspohn, S. C. Deane, I. D. French, and M. J. Powell, J. Non- Cryst. Solids , R. B. Wehrspohn, S. C. Deane, I. D. French, I. G. Gale, M. J. Powell, and R. Bruggemann, Appl. Phys. Lett. 74, R. B. Wehrspohn, S. C. Deane, I. D. French, I. G. Gale, J. Hewitt, M. J. Powell, and J. Robertson, J. Appl. Phys. 87, M. Stutzmann, Philos. Mag. B 56, K. Morigaki, Jpn. J. Appl. Phys., Part 1 27, H. M. Branz, Phys. Rev. B 59, M. J. Powell and S. C. Deane, Phys. Rev. B 53, N. Kopidakis and E. A. Schiff, J. Non-Cryst. Solids 266, H. M. Branz, Phys. Rev. B 39, M. J. Powell, C. van Berkel, A. R. Franklin, S. C. Deane, and W. I. Milne, Phys. Rev. B 45, R. S. Crandall, Phys. Rev. B 43, Y. F. Chen, S. F. Huang, and W. S. Chen, Phys. Rev. B 44, M. J. Powell and S. C. Deane, Phys. Rev. B 48, M. J. Powell, S. C. Deane, and R. B. Wehrspohn, Phys. Rev. B 66, R. B. Wehrspohn, M. J. Powell, S. C. Deane, I. D. French, and P. Roca I Cabarrocas, Appl. Phys. Lett. 77, The band-tail carrier density is defined as n BT exp(e c E F /).