- Email: [email protected]
Structural memory and defect relaxation
J O U R N A L OF
ELSEVIER
Journal of Non-CrystallineSolids 190 (1995) 142-150
Structural memory and defect relaxation P.A. Fedders a,, H o w a r d M. Branz b a Department of Physics, Box 1105, Washington University, 1 BrookingsDrive, St. Louis, MO 63130-4899, USA b National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO 80401, USA
Abstract
To explain the unusual temperature- and sample-independent kinetics of transient capacitance carrier emission, a model of structural memory in a-Si:H is proposed. The rate-limiting step to carrier emission is an effectively one-dimensional random walk in configuration space where the final, activated, step of emission is very rapid. The model naturally explains the power-law emission decays as well as the scaling law that equates emission time to pulse time for a wide variety of samples and experimental conditions.
I. Introduction
Experimental studies of hydrogenated amorphous silicon (a-Si : H) require the introduction of non-equilibrium carrier concentrations. However, recent experiments [1-6] show that non-equilibrium carriers perturb the structure and alter the density of states. Thus one does not measure the equilibrium density of states directly and a non-trivial interpretation is necessary in order to deduce meaningful results. Further, complete relaxation of the structure of a disordered solid in response to changes in the charge states of the defects occurs on extremely long timescales (seconds at room temperature) compared with the rapid relaxations (picoseconds) thought to follow changes in the charge states in crystalline solids. This slow relaxation greatly complicates the interpretation of experiments.
* Corresponding author. Tel: + 1-314 935 6272. Telefax: + 1314 935 6219. E-mail: [email protected].
In this paper we introduce a new model we call 'structural memory' that explains the unusual relaxation kinetics observed at the a-Si:H dangling bond defect. In response to charge capture or emission, the defect must relax to a new equilibrium structural configuration. However, for short times after the charge changes, any partial relaxation remains reversible. These partial relaxations are reversible because certain crucial changes in the equilibrium positions of nearby atoms requires slow, coordinated, motions of many atoms. The fixed neighbors can be said to retain 'memory' of the defect configuration that existed before the charge changed. During the time interval for which structural memory is retained, one observes non-activated carrier emission kinetics from the defect determined principally by the coordinated atomic motions. Cohen et al. [1] have studied the ( - / 0 ) electronic level of the dangling bond defect (D) in a-Si:H by transient capacitance measurements. They show that the emission time, %, in n-type a-Si:H increases slowly with the time of the filling pulse, tp, between
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00267-7
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150 102
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Fig. 1. Transient capacitance experimental results for 7e versus tp, taken from Ref. [1].
100 ~s and 1 s at 290 K. The dependence of ~'e upon t o and T is shown in Fig. 1. For temperature-activated emission with a rate prefactor of 1013 s -1, the emission times of Fig. 1 would correspond [1] to a level deepening from about 0.6 to 0.85 eV below the conduction band edge, E c. The long time required for the deepening suggests that the structure determining the emission time requires seconds to completely relax at room temperature. A simplified description of the transient capacitance experiments is as follows. The sample is at equilibrium at a certain bias. The bias is changed during a filling pulse of duration tp and then the bias is returned to its original value. The filling pulse injects trapped charge into the depletion layer. For the shortest pulses of Fig. 1 and for t > tp, the emission of trapped charge, Q(t), is described by the power-law equation
Q(t) = ( Oo/2)( t/tp )-C.
(1)
Here c is approximately 0.5 and t is measured from the end of the injecting pulse. Eq. (1) holds for a number of samples, both n- and p-type [l,2], and is independent of temperature. Except for an overall magnitude, Qo, the decay is controlled entirely by the length of the filling pulse. In other words, it appears that the scaling laws governing the emission do not depend upon the electronic or material properties of the sample. We attempt to explain these results without assuming a priori that a defect's energy level falls logarithmically in time during the time of occupation and that this fall is proportional to the temperature.
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In previous papers [7-9] we examined the energetics of defect relaxation and showed that the equilibrium configuration of D must change upon charge capture. It is the purpose of this paper to offer a natural explanation for the observed relaxation kinetics of the D states. Ultimately, we believe the microscopic reason for the results is that amorphous or glassy materials have an almost infinite number of nearly degenerate states separated by energy barriers and that, because of this, relaxation takes a very long time and proceeds along a very convoluted path in configuration space. However, in this paper, we do not attempt a microscopic derivation, but merely a phenomenological one based on a few physically reasonable assumptions. Our major assumption is that the relaxation process is dominated by motion in configuration space that is analogous to a one-dimensional (1D) random walk or particle diffusion problem. The crucial nature of the 1D topology for the random walk is twofold. One aspect is that a diffusing or hopping particle (i.e., defect configuration) in one dimension starting at the origin will return to the origin again and again. In fact, even after waiting for any length of time, the particle will still return and keep returning. This is manifest in the t -1/2 time dependence of emission as will be explained below. The second aspect of one-dimensional diffusion is that a barrier on the relaxation path cannot be circumvented; the relaxation must ultimately take the particle over that barrier no matter how long it takes. Diffusion in higher dimensions does not, of course, share this physics. Thus in our picture, the 1D aspect is important because, in relaxing, even the best path through configuration space contains bottlenecks or barriers that the system must pass through. It turns out that this explains the exponent of c --- 0.5 in Eq. (1), and the dependence on only the filling pulse duration as well as the irrelevance of the materials properties and the temperature. That is, the actual hopping or diffusion rate in configuration space is irrelevant to the observed kinetics. Some more explanation of the relaxation process is useful at this point. The relaxation of a defect that changes its charge state involves many atoms because of the amorphous nature of a-Si:H and because the total energy is normally lowered, even a little, only by the concerted motion of a large num-
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P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
ber of atoms. Further, there are an enormous number of nearly degenerate configurations for any given energy and these are separated by energy barriers. If one considers a configuration of N atoms around a defect, then the relaxation takes place in a 3N-dimensional configuration space. However, we believe that there are numerous critical adjustments or bottlenecks in this large-dimensional space that must be overcome if the relaxation is to proceed and this gives the relaxation its one-dimensional character. An analogy in two dimensions is a mountainous terrain. Although the space is two-dimensional, there may be certain bottlenecks or passes that must be crossed in traversing the terrain. Although in principle there is the full two-dimensional freedom of lateral movement, the motion is essentially one-dimensional. Obviously this one-dimensional (or effectively one-dimensional) path will be very convoluted in configuration space and does not correspond to a straight line, although it is topologically equivalent to a straight line.
2. Analytical results for ID random walks In order to better describe the basic idea behind our theory, we review some of the properties of random walk diffusion that will be useful to us. A few of the results are well known and we have derived others. However, our derivations are somewhat long and tedious and will be published elsewhere. Initially, we discuss the one-dimensional hopping problem with all barriers identical. Consider a single particle (i.e., defect configuration) hopping on a 1D chain with a hopping rate or probability per unit time, F , for a particle to hop to either neighboring site. As is well known, if the particle starts at the origin (n = 0) at time t = 0, then the probability that the particle is at the site n at time t is exactly P(n, t) where P ( n, t) = I , ( z ) e -z,
(2)
where I, is the Bessel function of imaginary argument and z = 2 F t . At long times, z > > n 2, one obtains the square-root power law:
P-* 1/(4~rFt) 1/2.
(3)
The t - 1 / 2 dependence is a consequence of the onedimensionality. This same power law pervades all 1D random walk calculations. Next consider a slight refinement on the well known result above. Here we start the particle out at time t = 0 at the site n and we have a trap at the origin, n = 0. This corresponds to the beginning of the emission phase of a transient capacitance experiment. We let Pr(n, t) be the probability that the particle has not reached the origin yet at time t. The survival probability is dx
Pr(n, t) = f4r - - e - x ' sin (nO)/rr J0
(4)
x
where 0= sin-l((4Fx
- x 2 ) 1/2/2 F ).
(5)
For long times we obtain
P~ -~ n/(.rrFt) 1/2. The relevance of this i s as follows. We imagine the origin is the only defect configuration where there is a significant chance of emitting and the probability of emitting there is quite high. Then the probability that the defect has not yet emitted is given by Pr. We can also prove that, if a particle starts at the origin at time t = 0 and then walks away for a time tp, it will have a probability of precisely 0.5 of returning to the origin after an additional time of tp. Finally we must consider the possibility that there is a tendency or bias for the particle to hop in one direction over another. That is, at each step the configuration change is more likely to reduce the energy than increase it. Further, in the long run it will certainly decrease. The important question here is how long can the system behave as if that bias did not exist. In order to study this analytically we let F + T be the probability per unit time (hopping rate) for hopping to a site further from the origin and F-T be the rate for hopping to a site nearer the origin. In this case the exact analytic expression is very complicated and will not be reproduced here. In the physical case where F >> T and at long times, the probability that it will be at the origin at time t if it starts from site n at t = 0 is Pr --- ( 1 / 4 " r r I ' t ) 1/2
exp(-(ny/F)
- (T2t/F)).
(6)
P.A. Fedders, H.M. Branz/Journal of Non-Crystalline Solids 190 (1995) 142-150
Thus the bias towards moving toward the origin is unimportant for times t << F / 3 , 2. We have also performed a number of other exact calculations with particles starting somewhere and eventually reaching the origin or some other specific site. For long times, all probabilities decay as t -1/2. However all of these calculations involve systems where all sites are the same or, at most, a uniform gradient in hopping rates. In order to consider cases where there is a distribution of sites, we have performed simulations. These are described in Section 4, after we have described the assumptions required for the structural memory theory to apply to detect relaxation.
3. Requirements of structural memory We now turn to a more detailed exposition of the structural memory theory. Based upon the unusual kinetics of the transient capacitance short-pulse regime, we propose that a reversible 1D random walk determines the time for carrier emission from D. Because of the scaling laws inherent to 1D random walks, our theory reproduces in a natural way both the temperature-independent emission time and power-law decays observed in the short-pulse regime. In this section, we introduce the hypotheses that connects the 1D random walks discussed above to the configuration space of neighbor-atom positions at D. Our theory of the structural memory regime rests upon four principal assumptions about configurations at D before and after carrier capture: (a) the pure emission time from the pre-capture configuration is short compared with tp, the duration of the filling pulse; (b) the pure emission rate decreases rapidly as the configuration moves away from the precapture configuration; (c) the configuration space traversed by the defect D and its neighbors is effectively one-dimensional, at least for short times; (d) the energy surface for the configurational changes is relatively fiat, i.e., the energy dissipation per configurational step is small, at least for short times. With assumptions (a)-(d), we can understand the principal features of the transient capacitance short-
145
pulse kinetics. By definition, the 'pure' emission rate does not include time spent searching for the emitting configuration. The pure emission rate is the usual temperature-activated function of the carrier emission energy, i.e., the difference between the relevant electron energy eigenvalues of D and the band-edge state. Assumptions (a) and (b) together ensure that emission is most likely to take place from a configuration near the precapture configuration, but that the rate-limiting step to emission is not the 'pure' emission of the carrier itself. This final step of carrier emission requires a time short compared with the experimentally observed emission time in the shortpulse regime. Assumptions (c) and (d) enable the defect to return to a configuration closely related to its precapture configuration and emit. Assumption (c) summarizes the discussion of Section 1; the defect will only return in 1D. In higher dimensions, defects making random walks in configuration space are unlikely to pass their initial configuration a second time. Assumption (d) is also crucial to permit reversibility of the configurational changes. If too much energy is dissipated during steps away from the precapture configuration, the walk will tend 'downhill' and the defect will move irreversibly away from the initial configuration. If either of assumptions (c) or (d) breaks down, only 'normal' temperature-dependent emission will be observed. This corresponds to the temperature-dependent saturation regime that Cohen observes for long tp (Fig. 1) and is discussed in more detail elsewhere [8]. While assumptions (a)-(d) hold, structural memory determines the short-pulse emission kinetics. During the filling pulse, the defect makes a 1D random walk away from its precapture configuration for a time tp. During the emission phase, the random walk continues until the defect returns to this configuration and quickly emits the carrier. Random walks reproduce the unusual transient capacitance kinetics quite naturally. The characteristic time to return to the origin after a 1D random walk of time tp is T-independent and approximately tp. In fact, the return time is exactly tp when there are equal energy barriers between each configurational step and is slightly longer than tp when a random distribution of barriers determines the step times. Further, the distribution of return times decays
146
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
0.0
as a power law with c = 0.5 for equal barriers and c = 0.5 for a distribution of barriers or a distribution of dead-end waiting times, as we discuss in the next section.
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4. Further calculations
To investigate random walks in cases where an analytical solution is impossible, we have performed Monte Carlo computer simulations of particle motions on a 1D grid of positions, p. These positions are separated by barriers of height E b ( p ) that are generated at random but follow a user-determined energy distribution. The time to hop over a barrier is t o exp[Eb(p)/kT], where 1 / t o is the attempt frequency. Each particle starts at the position, p = 0, that corresponds to the precapture defect configuration. The probabilities of moving right and left on the grid are calculated at each point from the hop times to the right and left over the barriers. During its 'pulse' (or filling pulse) phase lasting tp, we allow a particle to random walk on the one-dimensional-lattice to a position, pp. During its 'emission' phase, a particle begins a new random walk from pp, subject to a fixed probability of emission at each point p. We restart the clock at the beginning of the emission phase and measure the total elapsed time to emission, for each particle. We obtain distributions of emission times by starting many particles at p = 0, each with the s a m e tp. For random distributions of barriers, we compute a new set of barriers for each particle. We confirm convergence of the emission time distribution by increasing the number of particles in certain cases. As a check on our Monte Carlo simulations, we first set all barriers on the grid equal, i.e., hop times to the left and to the right are equal at all points, p. During its emission phase, a particle emits with probability 1 at p = 0 and does not emit from any other location. For this delta-like emission function, a given particle's emission phase therefore ends with its emission the first time it returns to its point of origin. In Fig. 2, we plot N(t), the number of particles not yet emitted, against time (measured from the beginning of the emission phase) for the simple random walk. The results are shown for three
I
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~
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Fig. 2. Random walk emission decays for a uniform 1 izs hop time between sites. One-half decay and t -1/2 lines are given for reference and arrows indicates tp
different pulse times and a hop time of 1 txs. The results correspond to the analytic solutions presented in Section 2. The long-time decay is exactly powerlaw with c = 0.5. The time to emit one-half the particles, t~, is exactly equal to tp. N(t) and t~ are independent of T and scale only with tp. Of course, T does determine the hop time and pp but these are not measured in the experiments. Fig. 3 shows simulation results for random walks through a Gaussian distribution (half-width of 0.1 eV) of random energy barriers for four different pulse times. For a pulse time of 1 ns, we simulated 0.0
., "-';'::.'::.'~ ;'-.1................
:::::: :-'--"-::~:... -0.5
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Fig. 3. Random walk emission decays for a random distribution of hop barriers of half-width 0.1 eV. One-half decay line is given for reference and arrows indicate tv.
P~A.Fedders,ll.M. Branz/Journal of Non-CrystallineSolids 190 (1995) 142-150 the random walk at both 282 and 310 K. Emission is again by a delta function at the origin. The apparent power-law decay exponent, c, depends only weakly upon tp and r e varies between tp and 10 tp. The asymptotic regime is very far out in time because of the distribution of barriers. We interpret the result that ~'e > tp as follows. The highest barrier surmounted during the pulse phase is approximately E b = kT ln(tp/t o) and therefore the time to return to the origin over the highest barrier is approximately tp. However, the return time is slightly greater than tp because sometimes the particle surmounts an even higher barrier to move away from the origin during the emission phase. In the experiments of Carlen et al. [2], on p-type material, the power-law exponent c = 0.3 rather than 0.5. A distribution of barriers to D + relaxation might account for this value. Alternatively this can be explained by assuming that the relaxation by random walk in configuration space is dispersive as well as 1D. That is, at any stage in the relaxation the system may go off on a path in configuration space into a 'dead end' line segment. Further, if the distribution of energies to get out of these dead ends and back on the main path follows the exponential form necessary for dispersive transport, then one obtain a timedependent diffusion constant, D(t)
D ( t ) = D o / ( t o t ) ~,
(7)
where to and a depend on T and the details of the distribution [10]. The exponent, c, of the emission power law is then given by the equation c=(1
- o~)/2.
(8)
The conditions necessary for dispersive transport as described by Eq. (7) is that a diffusing particle (or defect configuration in our case) falls into a pit or dead end from time to time and that the distribution of dead ends is exponential in energy. In any case, this assumption along with the 1D random walk analogy for relaxation explains Carlen's data and other small deviations from c = 0.5. We also performed Monte Carlo simulations with the delta-function emission assumption relaxed. Instead, the emission probability per step decays exponentially away from the origin. We obtain similar power-law decays when (1) the minimum emission rate at the origin is within an order of magnitude of
147
the hop time between sites and (2) the decay of the emission rate with p is sufficiently rapid. We note that all the results reported above are unchanged if we restrict the random walks to the positive half of the grid. Finally, we have also investigated by Monte Carlo techniques the bias or tilted configurations discussed earlier. We introduce a position-dependent particle energy, E(p), d E / d p is the difference in total energy between positions p and p + 1. The bias per step to the walk is d E / d p ; i.e., the barrier to stepping right at p is actually E b + d E / d p and the barrier to stepping left at p + 1 is is E b - d E / d p . We confirm the analytical results described in Section 2, and if we set the bias, d E / d p , too high, the particles never return to the origin for emission. In this case N(t) does not decay until sufficient time passes for activated emission from a final configuration. When the number of steps taken is small, the tilt is irrelevant and most particles return to the origin as though the walking surface were fiat. Our analysis shows that the maximum number of steps that a particle can take and still expect to return is
N < ( k T ) Z / ( d E / d p ) 2.
(9)
Eq. (9) sets an upper bound to the amount of stored energy that can be dissipated per configuration step away from the origin during the structural memory regime. The possibility of returning to the precapture configuration persists for a time that is inversely proportional to the square of the energy change per step during the random walk. One upper limit to the time of the structural memory regime is certainly reached when N exceeds the limit allowed by Eq. (9); the random walk then becomes irreversible. We discuss microscopic aspects of the energy dissipation in the final section below. We emphasize that, while the number of steps in the interval tp may depend strongly on the temperature and material properties of the sample, the actual experimental results do not. These results depend only on taking a random walk from the origin and returning in the same amount of time, not on how large the configuration change is in that time. On the other hand, if the temperature or other properties of the system change during the experiment, our results
148
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
would be changed considerably. In fact, this is observed experimentally [6].
5. Microscopic considerations We refer the reader to our other papers [7-9] for a more detailed discussion of some of the relevant microscopic considerations of the structural memory model and structural relaxation. In these papers we have reviewed quantum mechanical calculations of the dependence of the D energy levels upon bond angle, 0. These calculations form a basis for understanding the relevance of assumptions (a) and (b). In this section we provide an overview of our microscopic picture of structural memory. Our emphasis is on those attributes of a-Si:H that explain slow relaxation. We rely frequently upon our experience with molecular dynamics (MD) simulation on a-Si : H. Relaxation in response to charge-state change is slower in a-Si : H than in crystalline semi-conductors. The long timescales are intimately connected to the disorder. A bond-angle change at a localized defect in a solid will apply forces to many neighboring atoms. In a crystal, the energy surface is simple and each of these atomic forces points toward the new energy minimum in configuration space. In an amorphous material, the energy surface is multi-dimensional, complicated, and has a large number of degenerate (or nearly degenerate) minima separated by energy barriers. As a result, the forces on the individual atoms do not necessarily point in the direction of the global energy minimum. A change of charge state at a defect therefore precipitates a slow search in the structural configuration space for a new local energy minimum. We suppose that relaxing the defect's structure to a new configuration often requires several atoms to move simultaneously to energetically unfavorable locations, i.e., these atomic motions would raise the total energy if undertaken independently. Only by cooperative motion do these position changes finally lower or conserve the total energy. The size of the barrier generally increases with the number of atoms that must move simultaneously. Further, the chances of several events occurring almost simultaneously [8] is very small and relatively independent of temperature. The effect of
simultaneous atomic motions can be modelled as hops over energy barriers between discrete configuration states. This corresponds to the Monte Carlo simulations described above. The effective one-dimensionality of the random walk in configuration space (assumption (c)) is essential to the structural memory effect. From a given configuration, there are numerous barriers leading to new metastable configurations, yet only one pathway is followed as the defect takes configurational steps towards and away from its precapture configuration. As viewed by the defect, the high dimensionality of the configuration space collapses to 1D because of the high probability of passing only over the lowest energy barriers at each step. This is rather like a canyon threading through complicated mountain topography and forcing effectively 1D travel upon most visitors to the 2D surface. Our recent molecular dynamics studies give some credence to the assumption of 1D relaxation. That is, there is preliminary evidence that the paths along which the configurations of atoms in our supercells change are roughly the same for simulation runs at different temperatures. Theoretical calculations [11,12] show that the energy eigenvalues of D depend roughly linearly on the first neighbor bond angle, 0, while the equilibrium total energy depends quadratically upon 0. These dependences are shown schematically in Fig. 4. Because one would expect the bond angle to relax very quickly to its new equilibrium position after a change in charge state at D, it may be surprising that assumption (d) holds long enough to permit observation of structural memory at timescales up to 0.1 s at room temperature. It appears as though energy dissipation is turned off at short times enabling a return to the initial configuration. The local energy as a function of 0 is then a constant as indicated by the dashed line in Fig. 4. First of all, from recent MD (molecular dynamics) simulations, the energy eigenvalue of a given dangling bond depends on angle roughly linearly as predicted by equilibrium calculations, even during dynamical simulations [13]. However, the dangling bond energy eigenvalue depends on a number of other factors and this dependence can be at least as large as the dependence on angle. For example, in
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
I start here
.... shoa time
b) I 95
-I i 100 105 bond angle (o)
infinite time I 110
Fig. 4. Schematic diagrams of the dependence upon D bond angle of (a) the electron energy eigenvalue and (b) the defect total energy. In (b), the solid line represents the thermodynamic (infinite-time) limit and the dashed line represents the non-dissipative (short-time) limit.
recent MD supercell simulations [13], we have found that dangling bond energy eigenvalues can vary by several tenths of an eV in different surroundings but with virtually identical angles. Some of the contributing factors are second-neighbor effects, potential fluctuations, differing charge transfers, and other rather long-ranged environmental factors. These factors could relax very slowly because of cooperative and hierarchical effects. In some cases, when we changed the charge state of a dangling bond (by moving the Fermi level) the amorphous network had difficulty in reaching a new steady-state configuration when it was quenched. Thus the fast relaxation of a-Si:H when a dangling bond changes charge state is far from certain. Further, the total energy of the samples or supercells does not correlate well with the single-particle energy or energy eigenvalue of the dangling bonds! We have recently studied the total energy and energy eigenvalues for a number of isolated dangling bonds in a supercell where all dangling bonds except the one under consideration were passivated by H. We found that dangling bond total energies varied by
149
about 1 eV and that there was very little correlation of the total energy of the supercell with the energy eigenvalue of the dangling bond in question. We have also found no discernible correlations between the fluctuations in the energy eigenvalue of the dangling bond and the temperature of the system [13]. This despite the fact that in our supercell 0.3 eV fluctuations in the dangling bond energy eigenvalue should correspond to an easily observed 50 ° fluctuation in the temperature of the system. We have also performed a number of dynamical studies on supercells with dangling bonds. We find pairs of Si atoms occasionally fluctuating !13,14] so that they are separated by more than 2.8 A and one might consider this a dangling bond pair. However, never with these fluctuations do we see any associated states in the gap or localized states near the broken bond. Thus, on the timescales of femtoseconds up to several picoseconds, MD simulations supply evidence that the total energy does not change with 0. This is shown schematically in Fig. 4. We think it likely that energy is easily taken up in strain fields as long as neighbors are constrained to their precapture positions. The MD results suggest that assumption (d) may also be valid still at longer times and this tends to give credence to the structural memory model. In conclusion, we have presented a phenomenological theory of the structural memory effect that reproduces all of the salient features of observed transient capacitance experiments. The heart of the model is the observation that a series of barriers in configuration space that cannot be passed over will lead to an effective 1D random walk in configuration space. This, along with the possibility of an exponential distribution of dead ends, leads to an explanation of the experimental facts. This includes the t -¢ time dependence where c---0.5 and the temperature and materials independence of the experiments. The late Marvin Silver inspired and encouraged this work by his insistence that something was amiss in the accepted picture of the a-Si:H density of states. He welcomed the discovery of new physics in new data, but was also content sometimes to admit that he did not understand things. Marvin was a good friend and colleague and will be missed. The authors
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P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
also thank Mathieu Kemp, Eric Schiff, Martin Carlen, Richard Crandall and Thomas Unold for many stimulating discussions that contributed to the development of these ideas. This work was supported in part by the US DOE under contract DE-AC0283CH10093 and in part by the NSF under grant DMR 93-05344.
References [1] J.D. Cohen, T.M. Leen and R.J. Rasmussen, Phys. Rev. Lett. 69 (1993) 3358. [2] M.W. Carlen, Y. Xu and R.S. Crandall, Phys. Rev. B51 (1995) 2173. [3] D. Han, D.C. Melcher, E.A. Schiff and M. Silver, Phys. Rev. B48 (1993) 8658.
[4] W. Graf, M. Wolf, K. Leihkamm, J. Ristein and L. Ley, J. Non-Cryst. Solids 164-166 (1993) 195. [5] F. Zhong and J.D. Cohen, Phys. Rev. Lett. 71 (1993) 597. [6] A. Gardner and J.D. Cohen, in: Amorphous Silicon Technology, 1994, Mater. Res. Soc. Symp. Proc. 336 (1994) 207. [7] H.M. Branz, Phys. Rev. B39 (1989) 5107. [8] H.M. Branz and P.A. Fedders, in: Amorphous Silicon Technology, 1994, Mater. Res. Soc. Symp. Proc. 336 (1994) 129. [9] H.M. Branz and E.A. Schiff, Phys. Rev. B48 (1993) 8667. [10] J. Kakalios, R.A. Street and W.B. Jackson, Phys. Rev. Lett 59 (1987) 1037. [11] Y. Bar-Yam and J.D. Joanopoulos, Phys. Rev. Lett. 56 (1986) 2203. [12] P.A. Fedders and A.E. Carlsson, Phys. Rev. B39 (1989) 1134. Tight-binding calculations yield a linear dependence of energy eigenvalue on bond angle for the ( 0 / + ) transition of D. [13] P.A. Fedders, Phys. Rev. B, in press. [14] P.A. Fedders, Y. Fu and D.A. Drabold, Phys. Rev. Lett. 68 (1992) 1888.
ELSEVIER
Journal of Non-CrystallineSolids 190 (1995) 142-150
Structural memory and defect relaxation P.A. Fedders a,, H o w a r d M. Branz b a Department of Physics, Box 1105, Washington University, 1 BrookingsDrive, St. Louis, MO 63130-4899, USA b National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO 80401, USA
Abstract
To explain the unusual temperature- and sample-independent kinetics of transient capacitance carrier emission, a model of structural memory in a-Si:H is proposed. The rate-limiting step to carrier emission is an effectively one-dimensional random walk in configuration space where the final, activated, step of emission is very rapid. The model naturally explains the power-law emission decays as well as the scaling law that equates emission time to pulse time for a wide variety of samples and experimental conditions.
I. Introduction
Experimental studies of hydrogenated amorphous silicon (a-Si : H) require the introduction of non-equilibrium carrier concentrations. However, recent experiments [1-6] show that non-equilibrium carriers perturb the structure and alter the density of states. Thus one does not measure the equilibrium density of states directly and a non-trivial interpretation is necessary in order to deduce meaningful results. Further, complete relaxation of the structure of a disordered solid in response to changes in the charge states of the defects occurs on extremely long timescales (seconds at room temperature) compared with the rapid relaxations (picoseconds) thought to follow changes in the charge states in crystalline solids. This slow relaxation greatly complicates the interpretation of experiments.
* Corresponding author. Tel: + 1-314 935 6272. Telefax: + 1314 935 6219. E-mail: [email protected].
In this paper we introduce a new model we call 'structural memory' that explains the unusual relaxation kinetics observed at the a-Si:H dangling bond defect. In response to charge capture or emission, the defect must relax to a new equilibrium structural configuration. However, for short times after the charge changes, any partial relaxation remains reversible. These partial relaxations are reversible because certain crucial changes in the equilibrium positions of nearby atoms requires slow, coordinated, motions of many atoms. The fixed neighbors can be said to retain 'memory' of the defect configuration that existed before the charge changed. During the time interval for which structural memory is retained, one observes non-activated carrier emission kinetics from the defect determined principally by the coordinated atomic motions. Cohen et al. [1] have studied the ( - / 0 ) electronic level of the dangling bond defect (D) in a-Si:H by transient capacitance measurements. They show that the emission time, %, in n-type a-Si:H increases slowly with the time of the filling pulse, tp, between
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-3093(95)00267-7
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150 102
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,
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UJ
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~
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350K [] 330K zx 31OK 0 290K o
~ i L I I I I I 10"5 I0-4 10-a 10"2 10-I I0O t01 I02 FILLING PULSE WIDTH (SECONDS)
Fig. 1. Transient capacitance experimental results for 7e versus tp, taken from Ref. [1].
100 ~s and 1 s at 290 K. The dependence of ~'e upon t o and T is shown in Fig. 1. For temperature-activated emission with a rate prefactor of 1013 s -1, the emission times of Fig. 1 would correspond [1] to a level deepening from about 0.6 to 0.85 eV below the conduction band edge, E c. The long time required for the deepening suggests that the structure determining the emission time requires seconds to completely relax at room temperature. A simplified description of the transient capacitance experiments is as follows. The sample is at equilibrium at a certain bias. The bias is changed during a filling pulse of duration tp and then the bias is returned to its original value. The filling pulse injects trapped charge into the depletion layer. For the shortest pulses of Fig. 1 and for t > tp, the emission of trapped charge, Q(t), is described by the power-law equation
Q(t) = ( Oo/2)( t/tp )-C.
(1)
Here c is approximately 0.5 and t is measured from the end of the injecting pulse. Eq. (1) holds for a number of samples, both n- and p-type [l,2], and is independent of temperature. Except for an overall magnitude, Qo, the decay is controlled entirely by the length of the filling pulse. In other words, it appears that the scaling laws governing the emission do not depend upon the electronic or material properties of the sample. We attempt to explain these results without assuming a priori that a defect's energy level falls logarithmically in time during the time of occupation and that this fall is proportional to the temperature.
143
In previous papers [7-9] we examined the energetics of defect relaxation and showed that the equilibrium configuration of D must change upon charge capture. It is the purpose of this paper to offer a natural explanation for the observed relaxation kinetics of the D states. Ultimately, we believe the microscopic reason for the results is that amorphous or glassy materials have an almost infinite number of nearly degenerate states separated by energy barriers and that, because of this, relaxation takes a very long time and proceeds along a very convoluted path in configuration space. However, in this paper, we do not attempt a microscopic derivation, but merely a phenomenological one based on a few physically reasonable assumptions. Our major assumption is that the relaxation process is dominated by motion in configuration space that is analogous to a one-dimensional (1D) random walk or particle diffusion problem. The crucial nature of the 1D topology for the random walk is twofold. One aspect is that a diffusing or hopping particle (i.e., defect configuration) in one dimension starting at the origin will return to the origin again and again. In fact, even after waiting for any length of time, the particle will still return and keep returning. This is manifest in the t -1/2 time dependence of emission as will be explained below. The second aspect of one-dimensional diffusion is that a barrier on the relaxation path cannot be circumvented; the relaxation must ultimately take the particle over that barrier no matter how long it takes. Diffusion in higher dimensions does not, of course, share this physics. Thus in our picture, the 1D aspect is important because, in relaxing, even the best path through configuration space contains bottlenecks or barriers that the system must pass through. It turns out that this explains the exponent of c --- 0.5 in Eq. (1), and the dependence on only the filling pulse duration as well as the irrelevance of the materials properties and the temperature. That is, the actual hopping or diffusion rate in configuration space is irrelevant to the observed kinetics. Some more explanation of the relaxation process is useful at this point. The relaxation of a defect that changes its charge state involves many atoms because of the amorphous nature of a-Si:H and because the total energy is normally lowered, even a little, only by the concerted motion of a large num-
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P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
ber of atoms. Further, there are an enormous number of nearly degenerate configurations for any given energy and these are separated by energy barriers. If one considers a configuration of N atoms around a defect, then the relaxation takes place in a 3N-dimensional configuration space. However, we believe that there are numerous critical adjustments or bottlenecks in this large-dimensional space that must be overcome if the relaxation is to proceed and this gives the relaxation its one-dimensional character. An analogy in two dimensions is a mountainous terrain. Although the space is two-dimensional, there may be certain bottlenecks or passes that must be crossed in traversing the terrain. Although in principle there is the full two-dimensional freedom of lateral movement, the motion is essentially one-dimensional. Obviously this one-dimensional (or effectively one-dimensional) path will be very convoluted in configuration space and does not correspond to a straight line, although it is topologically equivalent to a straight line.
2. Analytical results for ID random walks In order to better describe the basic idea behind our theory, we review some of the properties of random walk diffusion that will be useful to us. A few of the results are well known and we have derived others. However, our derivations are somewhat long and tedious and will be published elsewhere. Initially, we discuss the one-dimensional hopping problem with all barriers identical. Consider a single particle (i.e., defect configuration) hopping on a 1D chain with a hopping rate or probability per unit time, F , for a particle to hop to either neighboring site. As is well known, if the particle starts at the origin (n = 0) at time t = 0, then the probability that the particle is at the site n at time t is exactly P(n, t) where P ( n, t) = I , ( z ) e -z,
(2)
where I, is the Bessel function of imaginary argument and z = 2 F t . At long times, z > > n 2, one obtains the square-root power law:
P-* 1/(4~rFt) 1/2.
(3)
The t - 1 / 2 dependence is a consequence of the onedimensionality. This same power law pervades all 1D random walk calculations. Next consider a slight refinement on the well known result above. Here we start the particle out at time t = 0 at the site n and we have a trap at the origin, n = 0. This corresponds to the beginning of the emission phase of a transient capacitance experiment. We let Pr(n, t) be the probability that the particle has not reached the origin yet at time t. The survival probability is dx
Pr(n, t) = f4r - - e - x ' sin (nO)/rr J0
(4)
x
where 0= sin-l((4Fx
- x 2 ) 1/2/2 F ).
(5)
For long times we obtain
P~ -~ n/(.rrFt) 1/2. The relevance of this i s as follows. We imagine the origin is the only defect configuration where there is a significant chance of emitting and the probability of emitting there is quite high. Then the probability that the defect has not yet emitted is given by Pr. We can also prove that, if a particle starts at the origin at time t = 0 and then walks away for a time tp, it will have a probability of precisely 0.5 of returning to the origin after an additional time of tp. Finally we must consider the possibility that there is a tendency or bias for the particle to hop in one direction over another. That is, at each step the configuration change is more likely to reduce the energy than increase it. Further, in the long run it will certainly decrease. The important question here is how long can the system behave as if that bias did not exist. In order to study this analytically we let F + T be the probability per unit time (hopping rate) for hopping to a site further from the origin and F-T be the rate for hopping to a site nearer the origin. In this case the exact analytic expression is very complicated and will not be reproduced here. In the physical case where F >> T and at long times, the probability that it will be at the origin at time t if it starts from site n at t = 0 is Pr --- ( 1 / 4 " r r I ' t ) 1/2
exp(-(ny/F)
- (T2t/F)).
(6)
P.A. Fedders, H.M. Branz/Journal of Non-Crystalline Solids 190 (1995) 142-150
Thus the bias towards moving toward the origin is unimportant for times t << F / 3 , 2. We have also performed a number of other exact calculations with particles starting somewhere and eventually reaching the origin or some other specific site. For long times, all probabilities decay as t -1/2. However all of these calculations involve systems where all sites are the same or, at most, a uniform gradient in hopping rates. In order to consider cases where there is a distribution of sites, we have performed simulations. These are described in Section 4, after we have described the assumptions required for the structural memory theory to apply to detect relaxation.
3. Requirements of structural memory We now turn to a more detailed exposition of the structural memory theory. Based upon the unusual kinetics of the transient capacitance short-pulse regime, we propose that a reversible 1D random walk determines the time for carrier emission from D. Because of the scaling laws inherent to 1D random walks, our theory reproduces in a natural way both the temperature-independent emission time and power-law decays observed in the short-pulse regime. In this section, we introduce the hypotheses that connects the 1D random walks discussed above to the configuration space of neighbor-atom positions at D. Our theory of the structural memory regime rests upon four principal assumptions about configurations at D before and after carrier capture: (a) the pure emission time from the pre-capture configuration is short compared with tp, the duration of the filling pulse; (b) the pure emission rate decreases rapidly as the configuration moves away from the precapture configuration; (c) the configuration space traversed by the defect D and its neighbors is effectively one-dimensional, at least for short times; (d) the energy surface for the configurational changes is relatively fiat, i.e., the energy dissipation per configurational step is small, at least for short times. With assumptions (a)-(d), we can understand the principal features of the transient capacitance short-
145
pulse kinetics. By definition, the 'pure' emission rate does not include time spent searching for the emitting configuration. The pure emission rate is the usual temperature-activated function of the carrier emission energy, i.e., the difference between the relevant electron energy eigenvalues of D and the band-edge state. Assumptions (a) and (b) together ensure that emission is most likely to take place from a configuration near the precapture configuration, but that the rate-limiting step to emission is not the 'pure' emission of the carrier itself. This final step of carrier emission requires a time short compared with the experimentally observed emission time in the shortpulse regime. Assumptions (c) and (d) enable the defect to return to a configuration closely related to its precapture configuration and emit. Assumption (c) summarizes the discussion of Section 1; the defect will only return in 1D. In higher dimensions, defects making random walks in configuration space are unlikely to pass their initial configuration a second time. Assumption (d) is also crucial to permit reversibility of the configurational changes. If too much energy is dissipated during steps away from the precapture configuration, the walk will tend 'downhill' and the defect will move irreversibly away from the initial configuration. If either of assumptions (c) or (d) breaks down, only 'normal' temperature-dependent emission will be observed. This corresponds to the temperature-dependent saturation regime that Cohen observes for long tp (Fig. 1) and is discussed in more detail elsewhere [8]. While assumptions (a)-(d) hold, structural memory determines the short-pulse emission kinetics. During the filling pulse, the defect makes a 1D random walk away from its precapture configuration for a time tp. During the emission phase, the random walk continues until the defect returns to this configuration and quickly emits the carrier. Random walks reproduce the unusual transient capacitance kinetics quite naturally. The characteristic time to return to the origin after a 1D random walk of time tp is T-independent and approximately tp. In fact, the return time is exactly tp when there are equal energy barriers between each configurational step and is slightly longer than tp when a random distribution of barriers determines the step times. Further, the distribution of return times decays
146
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
0.0
as a power law with c = 0.5 for equal barriers and c = 0.5 for a distribution of barriers or a distribution of dead-end waiting times, as we discuss in the next section.
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4. Further calculations
To investigate random walks in cases where an analytical solution is impossible, we have performed Monte Carlo computer simulations of particle motions on a 1D grid of positions, p. These positions are separated by barriers of height E b ( p ) that are generated at random but follow a user-determined energy distribution. The time to hop over a barrier is t o exp[Eb(p)/kT], where 1 / t o is the attempt frequency. Each particle starts at the position, p = 0, that corresponds to the precapture defect configuration. The probabilities of moving right and left on the grid are calculated at each point from the hop times to the right and left over the barriers. During its 'pulse' (or filling pulse) phase lasting tp, we allow a particle to random walk on the one-dimensional-lattice to a position, pp. During its 'emission' phase, a particle begins a new random walk from pp, subject to a fixed probability of emission at each point p. We restart the clock at the beginning of the emission phase and measure the total elapsed time to emission, for each particle. We obtain distributions of emission times by starting many particles at p = 0, each with the s a m e tp. For random distributions of barriers, we compute a new set of barriers for each particle. We confirm convergence of the emission time distribution by increasing the number of particles in certain cases. As a check on our Monte Carlo simulations, we first set all barriers on the grid equal, i.e., hop times to the left and to the right are equal at all points, p. During its emission phase, a particle emits with probability 1 at p = 0 and does not emit from any other location. For this delta-like emission function, a given particle's emission phase therefore ends with its emission the first time it returns to its point of origin. In Fig. 2, we plot N(t), the number of particles not yet emitted, against time (measured from the beginning of the emission phase) for the simple random walk. The results are shown for three
I
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~
I 2
,~ t ''~
I 3
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",~
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Fig. 2. Random walk emission decays for a uniform 1 izs hop time between sites. One-half decay and t -1/2 lines are given for reference and arrows indicates tp
different pulse times and a hop time of 1 txs. The results correspond to the analytic solutions presented in Section 2. The long-time decay is exactly powerlaw with c = 0.5. The time to emit one-half the particles, t~, is exactly equal to tp. N(t) and t~ are independent of T and scale only with tp. Of course, T does determine the hop time and pp but these are not measured in the experiments. Fig. 3 shows simulation results for random walks through a Gaussian distribution (half-width of 0.1 eV) of random energy barriers for four different pulse times. For a pulse time of 1 ns, we simulated 0.0
., "-';'::.'::.'~ ;'-.1................
:::::: :-'--"-::~:... -0.5
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l
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Fig. 3. Random walk emission decays for a random distribution of hop barriers of half-width 0.1 eV. One-half decay line is given for reference and arrows indicate tv.
P~A.Fedders,ll.M. Branz/Journal of Non-CrystallineSolids 190 (1995) 142-150 the random walk at both 282 and 310 K. Emission is again by a delta function at the origin. The apparent power-law decay exponent, c, depends only weakly upon tp and r e varies between tp and 10 tp. The asymptotic regime is very far out in time because of the distribution of barriers. We interpret the result that ~'e > tp as follows. The highest barrier surmounted during the pulse phase is approximately E b = kT ln(tp/t o) and therefore the time to return to the origin over the highest barrier is approximately tp. However, the return time is slightly greater than tp because sometimes the particle surmounts an even higher barrier to move away from the origin during the emission phase. In the experiments of Carlen et al. [2], on p-type material, the power-law exponent c = 0.3 rather than 0.5. A distribution of barriers to D + relaxation might account for this value. Alternatively this can be explained by assuming that the relaxation by random walk in configuration space is dispersive as well as 1D. That is, at any stage in the relaxation the system may go off on a path in configuration space into a 'dead end' line segment. Further, if the distribution of energies to get out of these dead ends and back on the main path follows the exponential form necessary for dispersive transport, then one obtain a timedependent diffusion constant, D(t)
D ( t ) = D o / ( t o t ) ~,
(7)
where to and a depend on T and the details of the distribution [10]. The exponent, c, of the emission power law is then given by the equation c=(1
- o~)/2.
(8)
The conditions necessary for dispersive transport as described by Eq. (7) is that a diffusing particle (or defect configuration in our case) falls into a pit or dead end from time to time and that the distribution of dead ends is exponential in energy. In any case, this assumption along with the 1D random walk analogy for relaxation explains Carlen's data and other small deviations from c = 0.5. We also performed Monte Carlo simulations with the delta-function emission assumption relaxed. Instead, the emission probability per step decays exponentially away from the origin. We obtain similar power-law decays when (1) the minimum emission rate at the origin is within an order of magnitude of
147
the hop time between sites and (2) the decay of the emission rate with p is sufficiently rapid. We note that all the results reported above are unchanged if we restrict the random walks to the positive half of the grid. Finally, we have also investigated by Monte Carlo techniques the bias or tilted configurations discussed earlier. We introduce a position-dependent particle energy, E(p), d E / d p is the difference in total energy between positions p and p + 1. The bias per step to the walk is d E / d p ; i.e., the barrier to stepping right at p is actually E b + d E / d p and the barrier to stepping left at p + 1 is is E b - d E / d p . We confirm the analytical results described in Section 2, and if we set the bias, d E / d p , too high, the particles never return to the origin for emission. In this case N(t) does not decay until sufficient time passes for activated emission from a final configuration. When the number of steps taken is small, the tilt is irrelevant and most particles return to the origin as though the walking surface were fiat. Our analysis shows that the maximum number of steps that a particle can take and still expect to return is
N < ( k T ) Z / ( d E / d p ) 2.
(9)
Eq. (9) sets an upper bound to the amount of stored energy that can be dissipated per configuration step away from the origin during the structural memory regime. The possibility of returning to the precapture configuration persists for a time that is inversely proportional to the square of the energy change per step during the random walk. One upper limit to the time of the structural memory regime is certainly reached when N exceeds the limit allowed by Eq. (9); the random walk then becomes irreversible. We discuss microscopic aspects of the energy dissipation in the final section below. We emphasize that, while the number of steps in the interval tp may depend strongly on the temperature and material properties of the sample, the actual experimental results do not. These results depend only on taking a random walk from the origin and returning in the same amount of time, not on how large the configuration change is in that time. On the other hand, if the temperature or other properties of the system change during the experiment, our results
148
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
would be changed considerably. In fact, this is observed experimentally [6].
5. Microscopic considerations We refer the reader to our other papers [7-9] for a more detailed discussion of some of the relevant microscopic considerations of the structural memory model and structural relaxation. In these papers we have reviewed quantum mechanical calculations of the dependence of the D energy levels upon bond angle, 0. These calculations form a basis for understanding the relevance of assumptions (a) and (b). In this section we provide an overview of our microscopic picture of structural memory. Our emphasis is on those attributes of a-Si:H that explain slow relaxation. We rely frequently upon our experience with molecular dynamics (MD) simulation on a-Si : H. Relaxation in response to charge-state change is slower in a-Si : H than in crystalline semi-conductors. The long timescales are intimately connected to the disorder. A bond-angle change at a localized defect in a solid will apply forces to many neighboring atoms. In a crystal, the energy surface is simple and each of these atomic forces points toward the new energy minimum in configuration space. In an amorphous material, the energy surface is multi-dimensional, complicated, and has a large number of degenerate (or nearly degenerate) minima separated by energy barriers. As a result, the forces on the individual atoms do not necessarily point in the direction of the global energy minimum. A change of charge state at a defect therefore precipitates a slow search in the structural configuration space for a new local energy minimum. We suppose that relaxing the defect's structure to a new configuration often requires several atoms to move simultaneously to energetically unfavorable locations, i.e., these atomic motions would raise the total energy if undertaken independently. Only by cooperative motion do these position changes finally lower or conserve the total energy. The size of the barrier generally increases with the number of atoms that must move simultaneously. Further, the chances of several events occurring almost simultaneously [8] is very small and relatively independent of temperature. The effect of
simultaneous atomic motions can be modelled as hops over energy barriers between discrete configuration states. This corresponds to the Monte Carlo simulations described above. The effective one-dimensionality of the random walk in configuration space (assumption (c)) is essential to the structural memory effect. From a given configuration, there are numerous barriers leading to new metastable configurations, yet only one pathway is followed as the defect takes configurational steps towards and away from its precapture configuration. As viewed by the defect, the high dimensionality of the configuration space collapses to 1D because of the high probability of passing only over the lowest energy barriers at each step. This is rather like a canyon threading through complicated mountain topography and forcing effectively 1D travel upon most visitors to the 2D surface. Our recent molecular dynamics studies give some credence to the assumption of 1D relaxation. That is, there is preliminary evidence that the paths along which the configurations of atoms in our supercells change are roughly the same for simulation runs at different temperatures. Theoretical calculations [11,12] show that the energy eigenvalues of D depend roughly linearly on the first neighbor bond angle, 0, while the equilibrium total energy depends quadratically upon 0. These dependences are shown schematically in Fig. 4. Because one would expect the bond angle to relax very quickly to its new equilibrium position after a change in charge state at D, it may be surprising that assumption (d) holds long enough to permit observation of structural memory at timescales up to 0.1 s at room temperature. It appears as though energy dissipation is turned off at short times enabling a return to the initial configuration. The local energy as a function of 0 is then a constant as indicated by the dashed line in Fig. 4. First of all, from recent MD (molecular dynamics) simulations, the energy eigenvalue of a given dangling bond depends on angle roughly linearly as predicted by equilibrium calculations, even during dynamical simulations [13]. However, the dangling bond energy eigenvalue depends on a number of other factors and this dependence can be at least as large as the dependence on angle. For example, in
P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
I start here
.... shoa time
b) I 95
-I i 100 105 bond angle (o)
infinite time I 110
Fig. 4. Schematic diagrams of the dependence upon D bond angle of (a) the electron energy eigenvalue and (b) the defect total energy. In (b), the solid line represents the thermodynamic (infinite-time) limit and the dashed line represents the non-dissipative (short-time) limit.
recent MD supercell simulations [13], we have found that dangling bond energy eigenvalues can vary by several tenths of an eV in different surroundings but with virtually identical angles. Some of the contributing factors are second-neighbor effects, potential fluctuations, differing charge transfers, and other rather long-ranged environmental factors. These factors could relax very slowly because of cooperative and hierarchical effects. In some cases, when we changed the charge state of a dangling bond (by moving the Fermi level) the amorphous network had difficulty in reaching a new steady-state configuration when it was quenched. Thus the fast relaxation of a-Si:H when a dangling bond changes charge state is far from certain. Further, the total energy of the samples or supercells does not correlate well with the single-particle energy or energy eigenvalue of the dangling bonds! We have recently studied the total energy and energy eigenvalues for a number of isolated dangling bonds in a supercell where all dangling bonds except the one under consideration were passivated by H. We found that dangling bond total energies varied by
149
about 1 eV and that there was very little correlation of the total energy of the supercell with the energy eigenvalue of the dangling bond in question. We have also found no discernible correlations between the fluctuations in the energy eigenvalue of the dangling bond and the temperature of the system [13]. This despite the fact that in our supercell 0.3 eV fluctuations in the dangling bond energy eigenvalue should correspond to an easily observed 50 ° fluctuation in the temperature of the system. We have also performed a number of dynamical studies on supercells with dangling bonds. We find pairs of Si atoms occasionally fluctuating !13,14] so that they are separated by more than 2.8 A and one might consider this a dangling bond pair. However, never with these fluctuations do we see any associated states in the gap or localized states near the broken bond. Thus, on the timescales of femtoseconds up to several picoseconds, MD simulations supply evidence that the total energy does not change with 0. This is shown schematically in Fig. 4. We think it likely that energy is easily taken up in strain fields as long as neighbors are constrained to their precapture positions. The MD results suggest that assumption (d) may also be valid still at longer times and this tends to give credence to the structural memory model. In conclusion, we have presented a phenomenological theory of the structural memory effect that reproduces all of the salient features of observed transient capacitance experiments. The heart of the model is the observation that a series of barriers in configuration space that cannot be passed over will lead to an effective 1D random walk in configuration space. This, along with the possibility of an exponential distribution of dead ends, leads to an explanation of the experimental facts. This includes the t -¢ time dependence where c---0.5 and the temperature and materials independence of the experiments. The late Marvin Silver inspired and encouraged this work by his insistence that something was amiss in the accepted picture of the a-Si:H density of states. He welcomed the discovery of new physics in new data, but was also content sometimes to admit that he did not understand things. Marvin was a good friend and colleague and will be missed. The authors
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P.A. Fedders, H.M. Branz /Journal of Non-Crystalline Solids 190 (1995) 142-150
also thank Mathieu Kemp, Eric Schiff, Martin Carlen, Richard Crandall and Thomas Unold for many stimulating discussions that contributed to the development of these ideas. This work was supported in part by the US DOE under contract DE-AC0283CH10093 and in part by the NSF under grant DMR 93-05344.
References [1] J.D. Cohen, T.M. Leen and R.J. Rasmussen, Phys. Rev. Lett. 69 (1993) 3358. [2] M.W. Carlen, Y. Xu and R.S. Crandall, Phys. Rev. B51 (1995) 2173. [3] D. Han, D.C. Melcher, E.A. Schiff and M. Silver, Phys. Rev. B48 (1993) 8658.
[4] W. Graf, M. Wolf, K. Leihkamm, J. Ristein and L. Ley, J. Non-Cryst. Solids 164-166 (1993) 195. [5] F. Zhong and J.D. Cohen, Phys. Rev. Lett. 71 (1993) 597. [6] A. Gardner and J.D. Cohen, in: Amorphous Silicon Technology, 1994, Mater. Res. Soc. Symp. Proc. 336 (1994) 207. [7] H.M. Branz, Phys. Rev. B39 (1989) 5107. [8] H.M. Branz and P.A. Fedders, in: Amorphous Silicon Technology, 1994, Mater. Res. Soc. Symp. Proc. 336 (1994) 129. [9] H.M. Branz and E.A. Schiff, Phys. Rev. B48 (1993) 8667. [10] J. Kakalios, R.A. Street and W.B. Jackson, Phys. Rev. Lett 59 (1987) 1037. [11] Y. Bar-Yam and J.D. Joanopoulos, Phys. Rev. Lett. 56 (1986) 2203. [12] P.A. Fedders and A.E. Carlsson, Phys. Rev. B39 (1989) 1134. Tight-binding calculations yield a linear dependence of energy eigenvalue on bond angle for the ( 0 / + ) transition of D. [13] P.A. Fedders, Phys. Rev. B, in press. [14] P.A. Fedders, Y. Fu and D.A. Drabold, Phys. Rev. Lett. 68 (1992) 1888.