Kinetics of defect creation in amorphous silicon thin film transistors

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 9 1 MAY 2003 

Kinetics of defect creation in amorphous silicon thin film transistors 

R. B. Wehrspohn a) 

Max-Planck-Institute of Microstructure Physics, Weinberg 2, D-06120, Halle, Germany 

M. J. Powell and S. C. Deane 

Philips Research Laboratories, Redhill, Surrey, RH1 5HA, United Kingdom 

�Received 25 September 2002; accepted 12 February 2003� 

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 0) (��1) , with the key result that ��3�. The time constant t0 is 

also found to depend on band-tail carrier density. Both results are in excellent agreement with 

experimental data. The t0 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. © 2003 American Institute of Physics. �DOI: 10.1063/1.1565689� 

I. INTRODUCTION 

A. Threshold voltage instability 

Amorphous silicon thin film transistors �TFT� were first 

proposed more than 20 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 insta- 

a� Electronic mail: wehrspoh@mpi-halle.de 

bility 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 0 log�1�t/t 0� �1� 

with V 0 the initial gate bias over threshold, V g�V ti , and 

t 0�� c �1 exp(EA /kT) 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 

0021-8979/2003/93(9)/5780/9/$20.00 5780 

© 2003 American Institute of Physics 

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J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Wehrspohn, Powell, and Deane 

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 C�V t�q�N D(t) with C the 

capacitance of the TFT. The first fit for defect state creation 

kinetics was based on a stretched exponential 6 

0 

�N D�t��N BT�1�exp��t/t0� 

� �� 2� 

�1 0 

with ��T/T 0 , t0�� c exp(EA /kT), and NBT the initial 

band-tail carrier density. This two-parameter fit (E A ,kT 0)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 �D�2 N�kN, �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 

k�D��t� ��1 . �4� 

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. 10–12 In particular, 

the dependence of the defect creation rate dN D(t)/dt 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 

dND�t� dt �� dNBT dt �k�N t��1 

� 

BT� � 

t0 with � in the range of 1.5 to 1.9, k�const, and t 0 

�� c �1 exp(EA /kT), 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 1��2 which yields a ‘‘stretched hyperbola’’ of the 

form 

1 

0 

�N D�t��N BT� 

1� 

�1��t/t 0� 1/� �6� 

�� 

with ��1. We have shown 12 that the three-parameter 

(E A ,T 0 ,�) stretched hyperbola fit is an improvement compared 

to the commonly used two-parameter stretched expo- 

nential fit, 6 since it takes into account the superlinear bias 

� 

dependence NBT �Eq. �5��. Jackson already proposed a similar 

kind of equation in 1990. 13 He realized in his 1989 

article 7 

that there is a problem with the standard 

�5� 

FIG. 1. Typical threshold voltage shift �V t of a bottom gate TFT S30L 

during bias stress (V g�30 V) for T�383 K. The SiN thickness is 300 nm 

and the initial threshold voltage is V ti�3 V; thus the bias over threshold is 

V 0�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 1990 article, 13 he proposed 

therefore a carrier-density-dependent hydrogen diffusion 

coefficient to modify the analytical approximation �Eq. 

�4��, 

D�D 0�N BT� � , �7� 

where D0 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 t0 on the initial 

0 

number of band-tail electrons NBT . 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 

5781 

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 


5782 J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Wehrspohn, Powell, and Deane 

FIG. 2. Threshold voltage shift �V t as a function of gate bias over initial 

threshold voltage V 0 for two samples ��: ��0.2 cm 2 V �1 s �1 ; �: � 

�1.3 cm 2 V �1 s �1 ). The constant-current stressing time was 1000 s at 

350 K. 

silicon bottom gate TFT under 30 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 300 °C on a 

crystalline silicon wafer. The preparation and measurement 

conditions were reported elsewhere. 15 The TFT was annealed 

to 500 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 0,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 �. 10–13 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 0 . 16 Two different 

TFT’s have been chosen, one with a very low mobility � 

of 0.2 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 0 

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 0 and thereby E A 

on the applied gate bias over initial threshold V0 , i.e., the 

0 14 

initial band-tail carrier density NBT. For example, stressing 

a TFT with a ten times higher V0 does not result in exactly 

ten times faster defect creation. In Fig. 3, a similar device as 

in Fig. 1 is stressed with 10, 20, and 30 V gate biases. The 

defect creation kinetics do not exactly scale, but the higher 

0 

the initial band-tail carrier density NBT , the more easily defects 

are created. Or in other words, t0 is shifted to lower 

values with increased gate bias V0 . 

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 ND(t); 7 superlinear bias dependence of 

ND(t); 10–13 0 14 

and dependence of t0 on initial bias NBT. FIG. 3. �a� Threshold voltage shift for bottom gate TFT S30L for different 

gate biases V g of 10, 20, and 30 V measured for different temperatures in the 

range from 300 to 390 K. The stressing times and temperatures have been 

unified by the thermalization energy E th�kT ln(� ft) with ��10 10 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 

0 

density NBT . 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 t0 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. 



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*/kT 0) 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 0 . 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→D�D, �8� 

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 20,21 

SiHHSi→Hi�SiHD �10� 

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 

5783 

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 

� dNA dt �R fN A 

�11� 

with NA the number of precursor bonds in state A and R f the 

forward reaction rate, defined as 

R f�� f exp��E*/kT� �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 kT0 �Fig. 4�. 

At the energy E M the density of states N0 equals that at the 

mobility edge. Thus NA *(E*) reads 

NA *�E*��N 0 exp��E M�E*�/kT0�. �13� 

The driving forces for the forward reaction are band-tail 

carriers, 13 so Eq. �12� has to be modified to 

R f�� f exp��E*��/kT�, �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�kT ln�n BT�, �15� 

where Eform is the formation energy of the neutral defect 

level �Fig. 5� and nBT is the normalized band-tail carrier 

density NBT /N c with Nc the density of states at the mobility 

edge. 29 Here we implicitly assume that nBT 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 



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 

dt � dN D 

dt �� f exp� ��E*�� 

kT 

� 

�� N0 exp ��E M�E*� 

�N 

kT D� . �16� 

0 

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 2�. 

2. Boltzmann approximation 

To solve Eq. �16�, one has to integrate over all times dt 

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 ND 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 Eth , 

Eth�kT ln�� ft��, �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 0kT 0 exp� � E M�E th 

kT 0 

� � ��Eth t dN � dtdE*. �18� 

0 dt 

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 kT�kT0 . 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 Eth �the first term� all precursors have converted to 

defect states whereas above Eth �second term� only the Boltzmann 

fraction converts to defect states. Solving Eq. �18� 

�see the Appendix�, one obtains for the defect creation rate 

dND dt �� fN 0� kT�2kT0�kT� kT � 

0�kT 

�exp� � E M�E form � ��1 

kT � nBT��t� �19� 

0 

with ��kT/kT0 . Equation �19� represents the defect creation 

rate if we assume an exponential distribution of barriers, 

a barrier lowering energy �, and the Boltzmann approxi- 

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�kT 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. 

mation 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 0.4 to 0.5 �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 0.8 to 1, still not in line 

with the experiments. Moreover, E M has no dependence on 

0 

the initial band-tail carrier density NBT, in contradiction to 

the observed band-tail carrier dependence of t0 . 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 nBT 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: kT�30.4 meV, kT 0�62 meV, 

t�1000 s. 



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 /dt for an energy barrier E* is 

dN D�E*,t� 

dtdE* �R f�E*,t�N A�E*,t��R b�E*,t�N D�E*,t� 

�20� 

with R f and Rb 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*�� 

kT � �� fn BT exp� � E*�E form 

kT � . 

�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* 

�kT ln(� ft)�� have completely equilibrated and defects 

with E*�kT ln(� ft)��kT 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 

kT � , �22� 

where � b is the attempt frequency for the backward reaction 

rate. Inserting Eqs. �13�, �21�, and �22� in Eq. �20�, 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* 


dtdE* �� f exp� � E*�E form 

kT 

�� nBTN0 exp� E*�E M 

kT0 � 

�N D�E*,t�� n BT� � b 

� f�� . �23� 

We numerically modeled Eq. �23� for typically 30 energy 

levels for a constant-voltage stress situation, i.e., NBT(t) 0 30 

�N BT��i�1 

ND(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 1000 s, we obtain ��2� by varying 

0 

the initial band-tail carrier density NBT �Fig. 7� and we ob- 

0 

serve no dependence of EA on NBT . Thus, these results are 

� 

TABLE I. Default parameters used in the numerical and analytical calcula- 

0 

tion. N0 has been normalized to 1. NI is the normalized value of NBT . 

Parameter Model III D Model III E Stretched hyperbola fit 

��T/T 0 0.53 0.53 0.53 

E A or E M 1.045 eV (E M) 1.045 eV (E M) 1.015 eV (E A) 

� f (�� b) 10 10 Hz 10 10 Hz 10 10 Hz 

E form 0 0 0 

N 0 1 1 1 

N I N I�N 0 N I�N 0 N I�N 0 

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 

Rb�t�� b exp� � E*�2E form�� 

kT � 

�1 

�� bn BT exp� � E*�E form 

kT � . �24� 

Inserting Eqs. �13�, �24�, and �22� in Eq. �20�, and considering 

that the total number of sites NA is constant �N A * 

�N D(t)�N A(t)�, one obtains for the rate of defect creation 

per energy barrier E* 


dtdE* �� f exp� � E*�E form 

kT 

�� nBTN0 exp� E*�E M 

kT0 �N D�E*,t�� nBT� �b . �25� 

� fn BT�� 

We have numerically modeled Eq. �25� for 30 energy 

levels for a constant-voltage stress situation, i.e., NBT(t) 0 

�N BT��i�1 

30 ND(E i * ,t). The parameters used in the calcu- 

lation 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 Eth with increasing E M . 

In a next step, we calculated � from the threshold voltage 

shift for different biases in the short time stressing regime 

(t�1000 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 satu- 

0 

ration as a function of Eth �Fig. 9�. With increasing NBT ,we 

Downloaded 20 Jul 2004 to 195.37.184.165. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 

� 

� 

5785


FIG. 8. Numerical simulation of the defect creation kinetics for different 

barrier heights E M�0.95, 1, 1.05, and 1.1 eV based on Eq. �25�. Parameters: 

kT�30.4 meV, kT 0�62 meV. The increase of E M leads to a parallel shift 

of the defect creation kinetics. 

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. 

FIG. 10. Comparison of experimental threshold voltage shift during gate 

bias stressing at V g�30 V at T�383 K, empirical stretched hyperbola fit 

�Eq. �6��, and numerically calculated data �Eq. �25��. For parameters, see 

Table I. 

�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 �/kT0 and � is proportional to kT ln(nBT) �Eq. �15��. 

Thus, the impact of nBT on the defect creation rate is modified 

by the power ��T/T 0 . 

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 bandtail 

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 0�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 ND(t), 0 

and the t0 dependence on initial bias NBT . For example, in 

Fig. 10, 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- 



derlining the numerical fit. Moreover, for a high-mobility 

TFT with kT0 of typically 50 meV, 15 we obtain for kT 

�30 meV a power ��3��1.8, whereas for a low-mobility 

TFT with typically kT0�60 meV, a power ��3��1.5 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 0 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. �6��. 12 

Nevertheless, as shown in Fig. 10 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 bandtail 

carrier density. Neglecting this dependence leads to differences 

in the fit parameter E A and kT 0 for different initial 

biases over threshold V 0 . 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 0 . For example, for the 

values of Fig. 3, the difference between stressing the TFT 

with 10 or 30 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 0 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 kT 0 , the characteristic energy for 

the exponential distribution of energy barriers, to be related 

to the characteristic energy of the valence band tail E v0 .In 

the absence of any network strain, kT 0 would be 2E v0 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. 20 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 v0 . 20 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 v0 to 2E v0 . Experimentally, 

we find kT 0 to be in the range (1 – 1.5)kT 0 . 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. 20,26 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* 

0 

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 10 10 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 0kT 0 exp� � E M 

kT 0� exp� � � 

kT 0� �� ft� kT/kT 0 

� t dN 

�� dtdE*. �A1� 

Eth 0 dt 

Differentiating N D yields 


5787


dN D 

dt �N 0kT 0 exp� � E M 

kT 0� exp� � � 

kT 0� �� ft� ��1 

� dN �� dE*. �A2� 

Eth dt 

For partially converted defects, one obtains for the integral 

of Eq. �A2� 

or 

� dND �Eth dt dE*�� 

� 

� fN 0 exp� � 

Eth 

E M�E* 

kT0 �exp� � E*�� 

kT � dE* �A3� 

� dND �Eth dt dE*�� 

� 

� fN 0 exp� � 

Eth 

E M 

kT0� exp� � 

kT� 

�exp� �� E* 

��1��dE*. �A4� 

kT� 

Integrating over all energy states E* reads 

dND dt �� fN 0� 1 

�1 1 

� 

kT kT0� exp� � E M 

kT0� �exp� � 

�� 

kT 

ft� 0� (�1��) . �A5� 

Inserting Eq. �A5� into Eq. �A2� and replacing � by n BT 

�Eq. �15�� yields 

� 

dN D 

dt �� fN 0� kT�2kT0�kT� kT0�kT �exp� � E M�E form 

kT � �n BT� 

0 � �� ft� (�1��) . �A6� 

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29 The band-tail carrier density is defined as nBT�exp��(E c�E F /kT)�. 


�

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