Dispersive Transient Photocurrent in Amorphous Silicon at High and

Transient photocurrent in hydrogenated amorphous silicon is studied in all relevant time regimes ..... two hole current slopes is -5 in crystalline si...
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J. Phys. Chem. B 2000, 104, 3924-3929

Dispersive Transient Photocurrent in Amorphous Silicon at High and Low Trap-State Occupations† M. H. Chu and C. H. Wu* Department of Electrical Engineering, UniVersity of MissourisRolla, Rolla, Missouri 65401 ReceiVed: October 13, 1999

Transient photocurrent in hydrogenated amorphous silicon is studied in all relevant time regimes following the illumination of a pulse of light at one end of the sample. When both electron and hole transport are taken into consideration, we find that, at a low occupation level of trap states or at a low intensity of illumination, there are five well-defined current slopes. The first three are located at a short-time range, which has not been probed experimentally, and they are due to the electron transport. The last two slopes originate from the well-known phenomenon of hole transport. Each bend from a current slope change has a particular physical meaning and is interpreted. At a high illumination intensity, all five current slopes become less well-defined, and the two current slopes that are due to hole transport can change drastically because of significant spacecharge effects. In particular, one of the hole slopes can even change from negative to positive, and the result is quite different from the well-known dispersive transport theory in disordered semiconductors.

I. Introduction Transient transport of photogenerated carriers in disordered semiconductors has been investigated by numerous authors1-20 since the pioneering work of Scher and Montroll,3,4 who used a random-walk-theory approach. A review of such an approach to dispersive transport has been published by Scher, Shlesinger, and Bendler.14 In the theory, the path of photogenerated holes transversing from one end to the other of a disordered semiconductor is equivalent to that of a random walker who has a pausing time distribution ψ(t) with an asymptotical long tail. Thus, if ψ(t) ∼ t-(1+R) with 0 < R < 1, then at a time before a significant portion of the hole carriers reaches the sample’s boundary, the transient photocurrent, I(t), is proportional to t-(1-R). At a later time, after the mean hole carrier has reached the boundary, I(t) is proportional to t-(1+R). Hence, there exist two well-defined negative-value current slopes of -(1 R) and -(1 + R) on a log I(t) vs log t plot, and the sum of those two hole current slopes has the value -2. It is also important to take note of the location of this current slope bend. The location of the bend of the two current slopes is the transit time of hole carriers, or the time when the mean portion of hole carriers arrives at the boundary. In the random-walk theory, trapped holes and free holes are handled together, having a single probability distribution function. The peak location of hole carriers does not move toward the boundary with increasing time, but is fixed at the location of the applied light pulse. This extreme result is due entirely to the form of the waiting-time distribution function, ψ(t), used in the calculation. In contrast, if ψ(t) ∼ e-λt, or a short-time decaying waiting-time distribution function is used, then a pure Gaussian or drift-diffusion transport for crystalline materials is achieved. Thus, between the two extremes, there are many different waiting-time distribution functions such that the behavior of hole transport will be of a nature between those of a pure dispersive and a pure Gaussian transport. One deficiency of the random-walk theory is that only hole transport is considered. Electron transport and space-charge †

Part of the special issue “Harvey Scher Festschrift”.

effects are not considered. Therefore, the validity of the randomwalk theory for transient photocurrent calculations is restricted to times within a couple orders of magnitude about the hole transit time, which is the range in which experimental data are normally gathered, and to low trap-state occupation levels only. Experimental transient photocurrent data for certain disordered materials can be fit to Scher-Montroll theory quite well, except at the short-time regime.14 Dispersive hole transport can be appropriately described using a drift-diffusion equation for holes and a trap-state occupation function. Any deviation from Gaussian transport is due to the time derivative of the trapstate occupation function. In disordered semiconductors, such as hydrogenated amorphous silicon (a-Si:H), there exists a finiteenergy mobility gap with well-defined acceptor-like and donorlike trap-state distributions in the gap. The free carriers are trapped and released according to Simmon-Taylor statistics.21 The finite trapping time implies that the peak location of trapped holes can move toward the direction of the applied electric field, in contrast to the result from Scher-Montroll theory. In addition, hole transport alone is not adequate. If both electrons and holes are considered, then each type of carrier will be transported according to its proper drift-diffusion equation and will be trapped according to the trapping dynamics described by Simmons-Taylor statistics. Furthermore, space-charge effects must also be considered if the concentration of photogenerated electron-hole pairs is so large that carrier transport can be evaluated accurately only in a self-consistent local electric field. In this work, we investigate the transient photocurrent in a-Si:H when both electron and hole transport and when the selfconsistent electric field are taken into account. We show that, rather than two well-defined hole current slopes, there exist five well-defined negative-value slopes at low illumination level. Interpretations of those current slopes are also presented. At high illumination intensity, current slopes are no longer welldefined. The value for a hole current slope can even become positive, and there is an additional broadening effect of the hole current at transit time. This phenomenon is described in section III. The sample geometry, the trap-state distributions of a-Si:

10.1021/jp993656r CCC: $19.00 © 2000 American Chemical Society Published on Web 03/29/2000

Transient Photocurrent in Amorphous Silicon

J. Phys. Chem. B, Vol. 104, No. 16, 2000 3925 be written as

∂n 1 - ∇‚J ) Gn - Tn ∂t q n

(1)

∂p 1 + ∇‚J ) Gp - Rp ∂t q p

(2)

and

Figure 1. Device geometry used for the simulation. V is the applied voltage, L is the channel length, d is the light illumination width, and W is the channel thickness. In our calculation, L ) 10 µm, d ) 0.5 µm, and W ) 0.1 µm. Two ohmic contact terminals are marked with S and D for convenience.

Note that the electron current, Jn(x, y, t), and the hole current, Jp(x, y, t), contain drift and diffusion components. Gn(Gp) and Rn(Rp) are, respectively, the corresponding rates of generation and recombination for electrons (holes). The donor-like trap charge, N+ dt(E), at trap-state energy E is related to the donor-like trap-state density, Ddt(E), and its occupation probability, f(E), through the relation

N+ dt(E) ) (1 - f)Ddt(E)

(3)

Similarly, the acceptor-like trap charge, Nat (E) at trap-state energy E is related to the corresponding quantities by

Nat (E) ) f‚Dat(E)

(4)

The occupation function of trap states, f(x, y, E, t), is a function of our two-dimensional geometry and trap-state energy level E at time t. This is the central part of our calculation using the Simmons-Taylor statistics. The occupation function f satisfies

Figure 2. Densities of donor-like and acceptor-like trap-state distributions in amorphous silicon that were used in the simulation. (a) High density of trap-state distributions. (b) Medium density of trap-state distributions. (c) Low density of trap-state distributions from ref 22.

H, and our computational method are described in section II. Our summary of and conclusions from this investigation are provided in section IV. II. Sample Geometry, Trap-State Distributions, and the Method for the Transient Photocurrent Calculation In our calculation of transient photocurrent, a simple twodimensional rectangular geometry of an a-Si:H material is used. Such a sample is shown in Figure 1, with an external applied voltage V. A thin-film sample of length L ) 10 µm and width W ) 0.1 µm is used. Two ohmic contacts are imposed at the boundaries, as indicated (S and D terminals), and a pulse of light with an electron-hole pair generation rate of Go is illuminated at one end of the boundary with a tiny strip width of d ) 0.5 µm for a duration of 10-13 s. We choose the value of Go such that the concentration of photogenerated electronhole pairs is in the range of 1014-1018 cm-3. Note that the carrier transport is essentially one-dimensional along the length L. Thus, a small value of W is used to simulate uniform carrier generation over the entire illuminated cross-sectional area of Wd. Typical values for the density of states (DOS) of acceptor-like, Dat(E), and donor-like, Ddt(E), trap states of a-Si:H, as determined on the basis of available data,22 are shown in Figure 2. Curve a indicates a high trap-state density, curve b, a medium density, and curve c, a low density. The electron concentration, n(x, y, t), and the hole concentration, p(x, y, t), satisfy their proper continuity equations and can

∂f ) rn - rp ∂t

(5)

rn(E) ) υσnn(1 - f) - en‚f

(6)

rp(E) ) υσppf - ep(1 - f)

(7)

where

and

σn (σp) is the electron (hole) trap-state capture cross section, en (ep) is the emission rate from the trap-state energy E to the conduction (valence) band, and υ is the thermal velocity. Note that the first term of eq 6 is the electron capture rate, and the second term is the electron emission rate at trap-state energy E. Similarly, the corresponding terms for holes are included in eq 7. The electron recombination rate, Rn, in eq 1 is related to rn(E) of eq 6 through

Rn )

∫EE rn(E)[Dat(E) + Ddt(E)] dE c

(8)

V

Similarly, the hole recombination rate, Rp, in eq 2 is related to rp(E) of eq 7 through

Rp )

∫EE rp(E)[Dat(E) + Ddt(E)] dE c

(9)

V

The total space charge, F, for the Poisson equation is given by F ) q[p - n + N+ d - Na +

∫EE N+dt(E) dE - ∫EE N-at(E) dE] c

V

c

V

(10)

where N+ d and Na are the usual ionized dopants from donors and acceptors, respectively.

3926 J. Phys. Chem. B, Vol. 104, No. 16, 2000

Figure 3. (a) S-Terminal current as a function of time for crystalline silicon. Here, V ) 200 V. Itot is the total current, In is the electron current component, Ip is the hole current component, and Idis is the displacement current component. Five current slope bends are marked from 1 through 5. The location of bend 1 indicates the time at which the peak of electron concentration arrives at the S-terminal boundary. At the location of bend 2, most of the electrons (99%) are removed from the channel. Note that, at time t ) 1.26 × 10-11 s, the displacement current changes the direction of flow from negative to positive. Before t ) 1.26 × 10-11 s, In and Idis are of opposite sign. Curve A indicates the total current when no electron transport is considered. (b) Corresponding D-terminal current as a function of time. At the location of bend 3, a small fraction of holes just reaches the D-terminal boundary. At the location of bend 4, the peak of the hole concentration is located at the D terminal. At bend 5, most of the holes (99%) are removed from the channel. Note that, at time t ) 10-10 s, Idis changes sign from positive to negative, and after t ) 10-10 s, In and Idis are of opposite sign. Positions 1-5 are located at 1.26 × 10-12, 2 × 10-11, 2 × 10-11, 1.26 × 10-10, and 5 × 10-10 s, respectively. The four transient photocurrent slopes have values of 0, -0.3, 0, and -5. Light intensity is such that electron-hole pairs of density 1015 cm-3 are generated.

We note that there are three time derivatives, ∂n/∂t, ∂p/∂t, and ∂f/∂t, from eqs 1, 2, and 5, respectively. Here, ∂n/∂t and ∂p/∂t determine the transit times for electrons and holes, respectively, and ∂f/∂t determines the trap-filling or trapemptying times. As we have shown earlier in our investigation of a-Si:H,23-27 it is the value of ∂f/∂t that is dominant when compared to that of ∂n/∂t or ∂p/∂t. Therefore, when two Gaussian transport equations (eqs 1 and 2) are combined with the trapstate occupation equation (eq 5), each type of carrier transport becomes dispersive or quite non-Gaussian. As a result, it is the time derivative of the occupation function that dominates the time dependence of n and p. Essentially, we compute the freeelectron and free-hole concentrations from the photogeneration and from the dark concentrations. Carriers drift and diffuse along the length L, as governed by the two continuity equations. In the process, carriers will go through multiple-trapping processes,

Chu and Wu

Figure 4. (a) D-Terminal current as a function of time at low illumination intensity for a-Si:H using high trap-state density from curve a of Figure 2. Idis changes sign at time t ) 6.31 × 10-8 s. There are now five well-defined current slopes. Five current slope bends are marked from 1 through 5. (b) Corresponding S-terminal current as a function of time. Idis changes sign at time t ) 5.01 × 10-10 s. The location of bend 1 is when the peak electron concentration reaches the S terminal. Bend 2 occurs when the acceptor-like trap charge reaches its maximum value at the boundary and when most of the free electrons (99%) are removed from the channel. Bend 3 occurs when a small fraction of holes just reaches the D terminal. Bend 4 occurs when the peak of the hole concentration reaches the D terminal. Bend 5 occurs when most of the holes (99%) are removed from the channel.

as governed by the dynamics of the occupation function of eq 5. This, in turn, changes the local electric field, as governed by the corresponding Poisson equation, which, in turn, changes the local free-electron and free-hole concentrations, as specified by the two continuity equations. The numerical values of n and p, the electric field, and the occupation function f are solved iteratively from the four coupling differential equations at a given time. The results presented here were obtained through the use of Semicad.27 We have tested this program against programs that we wrote ourselves, for which we published our results on thin-film transistors.23-26 The agreement is quite good, and the Semicad program is numerically more efficient than our own, self-written programs. III. Results and Discussion of Dispersive Transient Photocurrents Transient photocurrent in semiconductors can provide physical insights into carrier transport. When all trap states are absent, as in the case of a crystalline silicon material, the transient photocurrent will already have four distinctive transient current slopes if both electron and hole transports are considered. This case is shown at low illumination intensity in Figure 3a for the current at the S terminal and in Figure 3b for the current at the

Transient Photocurrent in Amorphous Silicon

Figure 5. (a) Acceptor-like trap-charge distribution as a function of time. Although L ) 10 µm, only the left side of a 2-µm enlarged section is shown. Each curve is for a different time: curve a, 10-13 s; b, 10-12 s; c, 10-11 s; d, 10-10 s; e, 10-9 s; and f, 10-8 s. (b) Free-hole distribution as a function of time. Each curve is for a different time: curve a, 10-13 s; b, 10-10 s; c, 10-9 s; d, 10-8 s; e, 10-7 s; f, 10-6 s; g, 10-5 s; and h, 10-4 s. Band positions 1-5 of Figure 4a are located at times t ) 3.16 × 10-12, 3.98 × 10-11, 3.98 × 10-9, 1 × 10-7, and 2.51 × 10-5 s, respectively. The five transient current slopes have values of 0, -2.6. -0.2, -0.21, and -2.4, respectively.

D terminal. They are identical in the total transient current, but different decompositions are shown here for comparison purposes. Electron, hole, and displacement components are indicated, and the location at which the displacement current changes sign is marked by an arrow. Note that, if only hole transport is considered, the total current is given by curve A in Figure 3a, and there is an appearance of only two current slopes. In fact, there are actually four distinctive current slopes with four slope bends marked by 1, 2 (or 3), 4, and 5 in the figure. The location of bend 1 is when the free-electron concentration is at a maximum, which occurs at the S-terminal boundary. The location of bend 2 is when most of the free electrons are removed from the sample, and that of bend 3 is when the holes start to arrive at the D terminal (Figure 3b). Those two locations turn out to be the same in this case, but will be different when a disordered semiconductor is used. Note that the location at which the displacement current changes sign is before bend 2. The location of bend 4 is when the majority of holes have arrived at the D terminal. At bend 5, the current has more or less reached the dark-current level. The values of the four current slopes are 0, -0.3, 0, and -5, respectively. Thus, the sum of two hole current slopes is -5 in crystalline silicon material. If a high-trap-state a-Si:H material, such as one with a trapstate density corresponding to curve a in Figure 2, is used, then the corresponding transient current is shown in panels a and b of Figure 4 for the D and S terminals, respectively. By

J. Phys. Chem. B, Vol. 104, No. 16, 2000 3927

Figure 6. Transient photocurrents as a function of time. Each curve is for a different trap-state distribution: curve a, high density of trapstate distributions from curve a of Figure 2 is used; curve b, medium density of trap-state distributions from curve b of Figure 2 is used; curve c, low density of trap-state distributions from curve c of Figure 2 is used. Bend positions 1-5 are located, in curve a, at times t ) 3.16 × 10-12, 3.98 × 10-11, 3.98 × 10-9, 1 × 10-7, and 2.51 × 10-5 s, respectively; in curve b, at times t ) 3.16 × 10-12, 3.98 × 10-11, 2.51 × 10-9, 3.16 × 10-8, and 1.58 × 10-6 s, respectively; and in curve c, at times t ) 3.16 × 10-12, 3.98 × 10-11, 1.58 × 10-9, 2.51 × 10-8, and 1.58 × 10-7 s, respectively. The values of the five current slopes for curve a are 0, -2.6, -0.2, -0.21, and -2.4, respectively; for curve b, 0, -2.6, -0.09, -0.22, and -2.2, respectively; and for curve c, 0, -2.6, -0.03, -0.19, and -3.8, respectively. As the density of acceptor-like trap states decreases, the current slope between bend 2 and bend 3 also decreases. The current slopes between bends 3 and 5 depend on the donor-like trap-state distribution. Because those trap states are the highest in curve b and the lowest in curve c, the current slopes between bends 3 and 4 reflect this fact accordingly.

comparing those curves with panels a and b of Figure 3, we note that there is a change in the number of current slopes, which is now five, as well as in the locations of slope bends, which are now delayed. The values of the five slopes are 0, -2.8, -0.2, -0.21, and -2.4, respectively. The location of bend 1 indicates the arrival of the peak of free electrons at the S terminal (the electron transit time). After the time of bend 1, the sample has been emptied of its free electrons, causing a steep drop in the transient current. However, photogenerated electrons are also trapped into the acceptor-like trap states. The concentration of those trapped electrons reaches its maximum value at the boundary of the S terminal at the time of bend 2. To show this point clearly, we plot the concentration of trapped electrons as a function of time in the very vicinity of the S terminal. This is shown in Figure 5a for the sequence of a gradual increase and decrease of those trap charges. From this time on, the transient current stops its steep decrease because, at this point, the trapped electrons are being emitted and are being removed by the S-terminal ohmic contact. The current slope is thus smaller, with a value of -0.2, until the time of bend 3. The location of bend 3 is when the tail of the photogenerated holes arrives at the D terminal. From that time on, a surge in the concentration of free holes at the terminal occurs, causing a rather steady current slope of -0.21 until a majority of the free holes are removed from the terminal. This occurs at the time of bend 4. To support this assertion, the variation in the free-hole concentration as a function of time is shown in Figure 5b to indicate the hole arrival time at bend 3 and the hole transit time at bend 4. After bend 4, the free and trapped holes are gradually released, causing a steep drop in the current with a slope of -2.4. This current drop continues until the current has decreased to the dark-current level at bend 5. We note that, when trap-state density is large, as in curve a of Figure 2, the appearance of only one current

3928 J. Phys. Chem. B, Vol. 104, No. 16, 2000

Chu and Wu

Figure 8. Transient photocurrent as a function of time at different illumination intensities. Each curve is for a different intensity of light. The corresponding electron-hole pair generation is, for curve a, n ) p ) 1013 cm-3; for curve b, n ) p ) 1015 cm-3; for curve c, n ) p ) 1017 cm-3; and for curve d, n ) p ) 1018 cm-3. Note that the location of bend 4, the time at which the hole current peak arrives at the D terminal, is unchanged with respect to the variation of light intensities.

Figure 7. Occupation function as a function of time at the channel under the illuminated area. (a) Low intensity of light with a generation of n ) p ) 1015 cm-3 electron-hole pairs. (b) Medium intensity of light with a generation of n ) p ) 1017 cm-3 electron-hole pairs. (c) High intensity of light with a generation of n ) p ) 1018 cm-3 electronhole pairs. Each curve is for a different time: curve a, 10-13 s; b, 10-9 s; c, 10-8 s; d, 10-7 s; e, 10-6 s; f, 10-4 s; g, 10-2 s; and h, 10-1 s.

slope between bends 2 and 4 is quite deceptive to experimentalists, as well as to the extension of the Scher-Montroll randomwalk theory to the short-time regime. In contrast to the case for the transient photocurrent in crystalline silicon, the locations of bends 2 and 3 are now separated, indicating the existence of an additional range of time during which trapped electrons are gradually released before the arrival of free holes at the D terminal. This separation is also obvious in panels a and b of Figure 5. To show that there are actually two distinctive current slopes between bends 2 and 4, we reduced the trap-state density from the high value of curve a in Figure 2 to the medium value of curve b and the low value of curve c, also in Figure 2. The results are shown in Figure 6 for the same low illumination intensity. Note that, when the trap-state density is varied, the free-electron current transient remains essentially unchanged because of the small disordered region that the electrons traversed. Because holes are being transported to the other end

Figure 9. Transient photocurrent as a function of time at a high illumination intensity and with a generation of 1017 cm-3 electronhole pairs. Note that, because there are more holes in the channel than in Figure 5, there is a broadening effect of the hole current and a delay in the time at which the displacement current changes sign. The net result is that, in this case, there is a positive current slope before bend 4. Bend positions 1-5 are located at times t ) 2.51 × 10-12, 3.98 × 10-11, 3.16 × 10-9, 1 × 10-7, and 1 × 10-3 s, respectively. The values of the five current slopes are 0, -2.2, -0.4, +0.1, and -2.8, respectively.

of the sample, the higher the trap-state density, the smaller the value of the sum of the slopes of the two hole currents. This fact can be observed in Figure 6. Thus, the sum of the slopes of the two hole currents, with a value of -2 from the ScherMontroll theory, is the limiting case for a highly disordered material only. In Figure 6, we clearly show that the sum of the slopes of the two hole currents is actually varying with trapstate density, as well as with illumination intensity, as we will show later. At a low illumination intensity, the electric field is almost constant, and the occupation level of trap states is low. If the light intensity is increased, the occupation level of trap states also increases. This is shown in Figure 7, panels a-c, at generation rates such that 1014, 1015, and 1017 cm-3 of electronhole pairs have been generated. The partial-filling and partialemptying sequences of trap states follow the two different

Transient Photocurrent in Amorphous Silicon

J. Phys. Chem. B, Vol. 104, No. 16, 2000 3929 intensities are plotted in panels a and b of Figure 10 for comparison. Note that, in Figure 10b, the electric field varies by more than one order of magnitude over the entire sample. In the region where the light pulse is applied, the electric field is negative, so that holes are actually removed by both terminals. This is the same condition as in space-charge-limited transport. IV. Conclusion The physical interpretation of phototransient current in disordered semiconductors, with respect to the transient current slopes and the slope bends, is well-defined only at a low illumination level. In this work, we extend our investigation to cover the entire relevant time regime and light-intensity range. In addition, electron transport and the space-charge effect are included properly. We show that the appearance of two welldefined hole transient current slopes from the Scher-Montroll theory (whose sum is -2) is valid at low light intensity, at high trap-state density, and when electron transport and space-charge effects are not taken into consideration. If electron transport is considered, there are three additional transient current slopes, corresponding to a free-electron transit time at the first slope bend, a maximum concentration of trapped electrons, occurring at the ohmic contact edge, at the second slope bend, and the arrival of hole carriers at the other edge at the third bend. At a high trap-state density and a low illumination intensity, the two slopes between the second and fourth bends can appear, deceptively, as one slope. These two slopes can be clearly distinguished when the trap-state density is decreased. At a high illumination intensity, the space-charge effect can produce a positive hole current slope before the hole carrier transit time, as well as a broadening of the hole current transient. Acknowledgment. We are grateful to Dr. James Drewniak for generously providing his time during the course of this work.

Figure 10. Electric field as a function of time. (a) Low intensity of light with a generation of n ) p ) 1015 cm-3 electron-hole pairs. Each curve is for a different time: curve a, 10-13 s; b, 10-12 s; c, 10-11 s; d, 10-10 s; e, 10-9 s; f, 10-8 s; g, 10-7 s; and h, 10-6 s. (b) Medium intensity of light with a generation of n ) p ) 1017 cm-3 electronhole pairs. Each curve is for a different time: curve a, 10-13 s; b, 10-12 s; c, 10-11 s; d, 10-9 s; e, 10-8 s; f, 10-7 s; and g, 10-6 s. Note that, in (a), the electric field is almost constant at 2 × 105 V/cm, whereas in (b), the electric field varies with time from a positive value to a negative value, specifically, from 3.2 × 105 to -1.0 × 104 V/cm.

mechanisms described by the occupation dynamics of eq 5. Those two sequences are indicated in Figure 7. The corresponding transient currents are plotted together in Figure 8. Note that, in our calculation, the electron-hole pair generation is increased successively from 1013 cm-3 to 1015, 1017, and 1018 cm-3 in going from curve a, to curves b-d, respectively. We want to point out that, when electron-hole pair generation reaches the 1017-1018 cm-3 level, the space-charge effect becomes very important and there are no longer two well-defined hole current slopes as shown in Figure 8. Instead, the current slope before bend 4 becomes positive, and the value of current slope after bend 4 also decreases as the light intensity is increased. In Figure 9, we have plotted the hole component of the current, Ip(t), to indicate that there is a broadening effect of the Ip(t) component at high illumination intensity. The time at which the displacement current changes sign is also delayed. The combination of the above two effects results in a positive current slope before bend 4. As electron-hole generation is increased further from 1017 to 1018 cm-3, the two free-electron current slopes at bend 1 also become less well-defined. The space-charge effect produces a very nonuniform electric field in this high-illumination situation. The electric fields for low and high illumination

References and Notes (1) Pfister, G.; Scher, H. Phys. ReV. B 1977, 15, 2062. (2) Gill, W. D. J. Appl. Phys. 1972, 43, 5033. (3) Montroll, E.; Scher, H. J. Stat. Phys. 1973, 9, 101. (4) Scher, H.; Montroll, E. Phys. ReV. B 1975, 12, 2455. (5) Shlesinger, M. J. Stat. Phys. 1974, 10, 421. (6) Schmidlin, F. Phys. ReV. B 1977, 16, 2362. (7) Noolandi, J. Phys. ReV. B 1977, 16, 4466. (8) Orenstein, J.; Kastner, M. Phys. ReV. Lett. 1981, 46, 1421. (9) Tiedje, T.; Rose, A. Solid State Commun. 1981, 37, 49. (10) Silver, M.; Ba¨ssler, H. Philos. Mag. Lett. 1987, 56, 109. (11) Street, R. A. Appl. Phys. Lett. 1982, 41, 1060. (12) Madan, A.; Shaw, M. The Physics and Applications of Amorphous Semiconductors; Academic Press: Boston, MA, 1988. (13) Abraham, M.; Halpern, V. Philos. Mag. B 1989, 60, 523. (14) Scher, H.; Shlesinger, M.; Bendler, J. Phys. Today 1991, 44, 26. (15) .Montroll, E.; West, B. On an Enriched Collection of Stochastic Processes. In Fluctuation Phenomena; Montroll, E., Lebowitz, J., Eds.; Elsevier/North-Holland: New York, 1979; Chapter 2. (16) Naito, H.; Ding, J.; Okuda, M. Appl. Phys. Lett. 1994, 64, 1830. (17) Shen, D. S.; Wagner, S. J. Appl. Phys. 1996, 79, 794. (18) Shen, D. S.; Conde, J. P.; Chu, V.; Liu, J. Z.; Maruyama, A.; Aljishi, S.; Wagner, S. Appl. Phys. Lett. 1988, 53, 1542. (19) Shen, D. S.; Wagner, S. J. Appl. Phys. 1995, 78, 278. (20) Crandall, R. S.; Balberg, I. Appl. Phys. Lett. 1991, 58, 508. (21) Simmons, J.; Taylor, G. Phys. ReV. 1971, 13, 1541. (22) Curve a from Nishida, S.; Fritzsche, H. Proceedings of the Spring Meeting of MRS, 1994. Curve c from Huang, J. S.; Wu, C. H. J. Appl. Phys. 1993, 74, 5231. Curve b from the Semicad program in ref 27. (23) Bullock, J. N.; Wu, C. H. J. Appl. Phys. 1991, 69, 1041. (24) .Huang, J. S.; Wu, C. H. J. Appl. Phys. 1993, 74, 5231. (25) Huang, J. S.; Wu, C. H. J. Appl. Phys. 1994, 76, 5981. (26) Chu, M. H.; Wu, C. H. Proceedings of the 1995 International Semiconductor Device Research Symposium, Charlottesville, VA, 1995; p 707. (27) Semicad Program from Dawn Technologies, 491 Macara Avenue, Suite 1002, Sunnyvale, CA 94086. (28) Chu, M. H.; Wu, C. H. J. Appl. Phys. 1997, 81, 6461.