Diffusion-controlled bimolecular recombination of electrons and holes in a-Si:H

]OURNA

ELSEVIER

I~ O F

Journal of Non-CrystallineSolids 190 (1995) 1-8

Section 1. Recombination

Diffusion-controlled bimolecular recombination of electrons and holes in a-Si" H E.A. Schiff

*

Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA

Abstract

The fluctuation-dominated and mean-field models for diffusion-controlled bimolecular recombination are compared with picosecond domain transient optical experiments on a-Si:H. Using no fitting parameters, the fluctuation-dominated model gives a good account for the optical decay times as a function of laser intensity. A lower bound for the thermalization radius of a photogenerated electron-hole pair, r o > 12 nm, is inferred, a-Si :H is a felicitous material for exhibiting the fluctuation-dominated effects since it combines a modest electron mobility with an unexplained but large thermalization radius. However, the role of the electrostatic potential fluctuations due to the charged electrons and holes is not understood.

I. Introduction

Marvin Silver delighted in posing scientific riddles for his colleagues. Of these perhaps his favorite was a mobility paradox [1]. In brief, direct measurements of carrier drift mobilities in hydrogenated amorphous silicon ( a - S i : H ) yield fairly low values: the electron drift mobility is at most 2 c m 2 / V s near room temperature [2,3], and holes are far less mobile. For such low mobilities the recombination of electrons and holes should be diffusion-controlled, which means that the recombination rate is limited by the time required for electrons and holes to diffuse close enough to annihilate each other. There are two straightforward consequences expected for diffusion-controlled recombination [4]: geminate (or Onsager) recombination of an electron and hole pair immediately following their photogeneration, and

* Corresponding author. Tel: + 1-315 443 3908. Telefax: + 1315 443 9103. E-mail: [email protected].

recombination for electrons and holes which manage to escape the geminate process. Both of these consequences are well documented in a variety of low-mobility materials. The paradox is that neither of these consequences of low mobilities is found in experiments on a-Si : H. Geminate recombination is essentially negligible near room temperature [5], and estimates of the nongeminate, bimolecular recombination rate for electrons and holes are much lower than expected from the Langevin theory [6]. These problems cannot be explained away by adding new recombination mechanisms such as Auger processes: the geminate and Langevin recombination processes in principle represent lower bounds to recombination rates, and it is extremely disturbing to find that measurements yield recombination rates orders of magnitude lower than expected. Silver and his collaborators suggested several possible mechanisms which might resolve both aspects of the mobility paradox [1,4]. The fundamental mobility of photocarriers might be much higher than the

Langevin

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E.A. Schiff/Journal of Non-Crystalline Solids 190 (1995) 1-8

drift mobility measurements indicate, and thus recombination might not be diffusion-limited. The internal potential fluctuations due to the disorder of a-Si : H might be so large as to require modifications to the standard low-mobility calculations. There may be barriers which impede the final recombination process for electrons and holes. In the present paper I shall comment on the second, bimolecular recombination aspect of the mobility paradox for a-Si:H. In particular the analyses of bimolecular recombination in a-Si : H have apparently all been restricted to mean-field theories, which (by definition) largely ignore the role of fluctuations in the electron and hole densities. Incorporating such fluctuations can lead to qualitatively different conclusions regarding bimolecular recombination, as has become increasingly clear in theoretical work on diffusion-controlled bimolecular recombination in the past decade [7-10]. It seems surprising that the results in a-Si : H have not (to my knowledge) been compared with the predictions of the fluctuation-dominated model. Kang and Redner suggested that this comparison be done in their 1984 paper on these recombination models [9]. I shall show here that these models do indeed provide a more satisfactory account for several picosecond domain experiments on a-Si:H than the mean-field theories. Incorporating fluctuation effects into the analysis of bimolecular recombination in a-Si:H does not explain the surprisingly small magnitude for geminate recombination in a-Si:H near room temperature. However, fluctuation-dominated models assume an initial spatial distribution of electrons and holes which is random to some degree. In particular I estimate that the initial thermalization radius, r 0 (the typical separation of a photogenerated electron and hole following their photogeneration as a pair), must obey r 0 > 12 nm. This magnitude for r 0, although difficult to explain, is consistent with the small magnitude for geminate recombination. a-Si : H appears almost unique as a material which satisfies two criteria for observation of the fluctuation-dominated regime: it has a modest electron mobility, so that diffusion control is plausible, but it also has a reasonably large thermalization radius, which is necessary to produce observable fluctuations in the first place. On the other hand, charged

pair systems such as electrons and holes have, in the past, been considered unlikely candidates for observation of fluctuation-dominated behavior because of their large electrostatic interactions; I shall return to this problem at the end of this paper.

2. Langevin recombination The standard rate equation for bimolecular recombination in a bulk, homogeneous semiconductor is dN(t) dt = -bRN(t) P(t),

(1)

where N ( t ) is the density of electrons and P ( t ) is the density of holes, b R is the recombination coefficient governing the rate of bimolecular recombination. This rate equation is an example of a mean-field theory for recombination: we neglect correlations between the electron and hole densities and random fluctuations in the densities. This approach seems completely reasonable in crystalline semiconductors, for which the charge carriers have long mean free paths: the fluctuations and correlations are smoothed over length scales of the order of the mean free path. For materials in which both the electrons and holes have low mobilities, the assumption that the mean densities of electrons and holes are uncorrelated is clearly invalid: the Coulomb attraction of the electrons and holes modifies the mean electron density, N(r), near the hole. Assuming for simplicity that holes are immobile, one can solve for N ( r ) and for the recombination current, j(r), as a boundary condition problem involving the drift/diffusion equation:

j ( r) = - txE( r ) g ( r) - D V N ( r ) .

(2)

D and tx are the diffusion coefficient and mobility of the electrons, respectively. We refer the interested reader to Refs. [1,4] for further mathematical details. Assuming validity of the Einstein relation, D = (kT/e)tx, the remarkable consequence is that the recombination coefficient, b R, has the simple, Langevin form: bL = e~/Eeo,

(3)

where ee o is the dielectric constant of the material.

E~A.Schiff/Journal of Non-CrystallineSolids190 (1995) 1-8 Because the recombination coefficient, bL, is proportional to the mobility (and hence the diffusion constant), Langevin recombination is an example of diffusion-controlled recombination. Astonishingly, all the real complexities of a low-mobility material - the nature of the electron and hole states and the character of their transport - - are irrelevant to recombination except as they affect the carrier mobilities.

Both for this experiment and for experiments on a-Si:H, the densities of electrons and holes are equal. The initial densities are equal, since they are generated pairwise by the laser pulse in far larger densities than are present in the dark. The densities remain equal because each recombination event destroys one electron and one hole. Substituting N = P into the bimolecular equation, one obtains the solution:

N(t) = N 0 / ( 1 +bRNot ) 3. Hughes' experiment: bimolecular recombination in PVK:TNF

In this section I shall briefly describe just one experiment which elegantly illustrates Langevin recombination in an amorphous semiconductor; additional discussion of this topic may be found in the review by Silver and Shaw [11]. The purpose of this review is to make it clear that Langevin recombination has been a very successful model for low-mobility materials. The particular experiment I shall describe was done by Hughes [12] in the photoconducting polymer poly-N-vinylcarbazole:trinitrofluorenone (PVK:TNF). Photocarriers were generated uniformly throughout the bulk of a thin film of PVK:TNF using a pulsed laser. An external electric field was also applied across the film, so that a given carrier can suffer either of two fates: (i) it can be swept through the film and be collected onto the electrode on one face of the film, or (ii) it can recombine with a counter charge. Since the drift velocity of a carrier is v = / x E , the characteristic transit time, ttr , required to sweep the carriers apart is of the order ttr =

d2/2/zV,

(4)

where d is the thickness of the thin film and V is the applied voltage. For simplicity we assume that one carrier is much more mobile than the other; tt is the mobility of the faster carrier. The use of uniformly absorbed illumination reduces the mean distance traveled by carriers to reach the electrodes by a factor of 2. For times shorter than this transit time, the density of carriers is governed by the bimolecular recombination noted above:

dN(t)/dt = -bRNP

( t < ttr ).

(5)

3

( t < ttr),

(6)

where N O is the initial density generated by the laser pulse. Any carriers which recombine at times t < ttr will not be collected. We can (roughly) estimate the areal particle density, Nc (units are cm-2), which is collected during the experiment as Nc cx N(ttr)d. Using the Langevin value for bR, we obtain Nc = N( ttr ) d = Nod~(1 + No(d2/2tzV)(etz/eeo) ),

(7) and hence

No/Nod=

1/(1

+Nod%/2,,oV).

(8)

The point here is that the fraction of the photogenerated carriers which are ultimately collected (i.e., Nc/N od) is determined only by the geometrical and dielectric properties of the material - - even the mobility becomes irrelevant. This conclusion is also reached in the more exact calculation given by Hughes. Experimentally, Hughes found that the absolute values of NJNod measured in PVK:TNF yielded a dielectric constant as a fitting parameter which agreed within a few per cent with the known value - - a very satisfactory demonstration of the value of the Langevin theory.

4. Picosecond domain studies of bimolecular recombination in a-Si : H

Optical techniques using pulsed lasers are almost ideal for exploring bimolecular recombination in aSi:H. A very short ' p u m p ' pulse (typically 1 ps or less) from the laser photogenerates electrons and holes. The evolution of the density is then probed after a short delay (typically 1-1000 ps) by measur-

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E.A. Schiff/Journal of Non-Crystalline Solids 190 (1995) 1-8

a-Si:H

I0-10 • .

~

•

10 -11

10-12

-.... 1017

101a

1019

1020

NO (cm"3) Fig. 1. The points indicate measurements of the decay time for the changes in the optical properties of a-Si:H due to a short laser pump pulse. The decay times are correlated with the density, No, of absorbed photons from the pump pulse. O, Devlen et al. [18]; II, Esser et al. [17]; A, Fauchet et al. [20]. The solid line is the prediction of the fluctuation-dominatedbimolecular recombination model; the dotted line is the prediction of the Langevin mean-field model.

ing the transmittance or reflectance for a second optical pulse. Since the delay between the pulses can be varied readily, one measures the time-dependent change in the material's optical properties induced by the pump pulse. For a - S i : H these optical pump and probe experiments have now been done by several groups [ 1 3 20]. A consensus has emerged [20,21] that the signals are primarily sensitive to the time-dependent decay of the density of electrons (and holes) as they recombine. In Fig. 1 the results from several of these experiments at room temperature are summarized. Specifically, the characteristic response times, tl/2 (essentially the time for the optical response to decay to half its initial value), have been plotted as a function of the initial volume excitation density, N O. The measurements of Gustafson et al. [15] are not included because these authors did not report an absolute excitation density, N O. It is clear that there are substantial quantitative differences between the various measurements. I consider it unlikely that these differences are due to the use of differing samples of a - S i : H . The most reliable aspect of the measurements is probably the functional dependence of tl/2 upon N o . The estimation of N O is quite difficult and

is less reliable. The statistical uncertainty in tl/2 is quite modest (at least on logarithmic scales); however, the estimates of tl/2 from the different groups are based on differing treatments of photothermal signals and other optical corrections. These have been reviewed recently by Fauchet et al. [20] and by Tauc and Vardeny [21]. The dotted line in Fig. 1 is the prediction of the Langevin model described in Section 2, for which tl/2 = l / b E N 0 = e % / e t x N o,

(9)

The values in Fig. 1 correspond to /~ = 2 c m E / V s, which is the measured value for the electron drift mobility in a - S i : H in the picosecond domain [2,3]. Hole motion is neglected, since the hole mobility is much less than that of the electron [22]. It is clear from Fig. 1 that Langevin recombination provides a poor account for the measured response times. It is important to note that the measured times are longer than predicted. A recombination channel which competed with Langevin recombination would decrease tl/2 below the Langevin value. In addition to the difficulty in accounting for the magnitudes of tl/2 with Langevin recombination, the scaling of tl/2 with N O is weaker than the expected dependence N O 1. The discrepancy has generally been attributed to competition between a monomolecular recombination process (dominating at low intensities) and a bimolecular process at high intensities, as proposed by Jackson [16].

5. A fluctuation-dominated model In Fig. 1 I have also shown a prediction based upon a fluctuation-dominated, bimolecular recombination model, which apparently agrees better with the measurements than the mean-field form. The characteristic result of the fluctuation-dominated models is that, for longer times, the density of carriers decays as N ( t)

o~NJ/2( Dt) -d/4,

(10)

where d < 4 is the dimensionality of the system and D is the diffusion coefficient of the mobile carrier [7-10,23]. This form should be compared with the

E.A. Schiff/Journal of Non-CrystallineSolids 190 (1995) 1-8

5

prediction of the mean-field, diffusion-controlled model:

fore assume that the fluctuation-dominated form at all times can be approximated using

N ( t ) ~ ( D t ) -1,

N(t) = N0/(1 +

(11)

where it is important to emphasize that, unlike the fluctuation-dominated result, this long-time form is independent of the initial density, N O, and of the dimensionality, d. The reader is referred to the references for reviews and derivations of these fluctuation-dominated models. Here I shall briefly summarize a scaling argument which yields Eq. (10); the purpose is to introduce some of the essential physical ideas which will be needed for the next section on geminate recombination. If the initial distributions of electrons and holes are completely random, spatial fluctuations of the net charge density exist on all length scales. It is fairly straightforward to estimate the fluctuations. The volume associated with some length scale, l, is V = l d. Initially such a volume contains No la electrons and holes. The standard deviation is ( N Ola) 1/2 (the standard result for Poisson statistics). Again, assuming that the electrons and holes are independent, each such volume will contain an excess of either electrons or holes with rms density

6( N - P ) = [2N0 lg] 1/e/ld = (2No)1/2l -a/2.

(No/2)l/2(ot)d/4).

(14)

This assumption is purely an ansatz with the virtue that its short-time and long-time limits are correct. The response time, tl/2, is then obtained from (No/2)l/Z(Dt1/2)d/4 --- 1, leading to

tl/2 = ( N 0 / 2 ) - 2 / d I D .

(15)

This expression was used to prepare the fluctuationdominated curve for Fig. 1. The diffusion constant D = 0.05 cmZ/s was used, corresponding (via the Einstein relation) to an electron mobility /x = 2 c m Z / V s at room temperature. It is clear that this form gives a much better account for the measurements on a-Si:H than the mean-field form. The magnitudes of the measurements agree as well with the fluctuation model as could possibly be expected given the variations between the several groups. It seems amazing that a theory which uses only a single electrical mobility measurement should account this well for the decay times reported as a function of intensity in a purely optical experiment.

6. Temperature dependence and dispersion (12)

Now consider the time evolution of these initial fluctuations. In a time, t, density fluctuations on length scales less than (Dt) 1/2 will be erased by carrier diffusion processes. Within a volume (Dt) d/z, electrons and holes will have recombined in pairs, leaving only the excess density 6 ( N - P ) due to fluctuations. Substituting l = (Dt) 1/2 into Eq. (12), we obtain

U( t) = (2Uo)l/2( Dt) -a/4,

(13)

as promised earlier. These results are actually long-time or largelength-scale limits; they certainly do not apply to length scales smaller than the initial interparticle distance, N O 1/d. However, the optical measurements have a limited dynamic range, and their analysis requires an estimate valid at early times. We there-

A major issue in analyzing carrier transport in non-crystalline materials over the last twenty years has been dispersion: the magnitude of the mobility depends upon the timescale on which it is measured [25]. The electron mobility (and presumably the diffusion constant) in a-Si : H are dispersive at temperatures significantly below room temperature [26,27], although there is no evidence in direct transport measurements for significant dispersion at room temperature and above in modern specimens [2,3]. For the present analyses of recombination transients in a-Si:H, this temperature-dependent dispersive behavior can be treated by simply substituting a time-dependent diffusion coefficient based on drift mobility measurements into Eqs. (14) or (15). The form D ( t ) = D0(/,t) -(1 -TIT°) is an adequate representation of the measurements [26,27]. T is the specimen temperature and v (the attempt frequency),

6

E.A. Schiff/Journal of Non-Crystalline Solids 190 (1995) 1-8

D O (the microscopic diffusion constant) and TO are fitting parameters. The form itself emerges from an analysis in which carriers are trapped and released by localized bandtail states just below the conduction or valence bandtail edges. The bandtail states are assumed to have an exponential energy distribution, where the energy kbT o is the width of the exponential bandtail. With this substitution Eq. (14) becomes

N,,

N(t) =

1 + ( No/2)1/2[( D o ~ v ) ( v t ) r/ro]

(T< T0).

d/4

(16)

This form has the long-time decay t -(T/T°)(d/4). The most complete data on the temperature-dependent form for the optical transients are those of Gustafson et al. [15]. These data were interpreted using the dispersive D(t) above, but (in essence) substituting into the mean-field form for N(t) given in Eq. (6). The corresponding long-time decay is t r/ro. This form gave a satisfactory fit to the transient decays as a function of temperature. For the seven higher-quality specimens measured by the authors the mean estimate was To -~ 400 K. This magnitude of To is problematic for a - S i : H ; To is usually estimated to be less than 300 K in electron drift mobility measurements [2,3,25,26]. Using the fluctuation-dominated form, Eq. (16), with long-time decay t -~r/r°)(d/4), decreases the estimate of To by d/4. The resulting value, TO = 300 K, is consistent with the electrical measurements.

from temperature and field-dependent measurements in a-Si : H. An estimate of the thermalization radius, r 0, plays a crucial role in applying the fluctuation-dominated model. Fluctuations of the density ( N - P ) are only present for length scales smaller than r 0 [8]; they are cut off for length scales larger than the initial pair separation. One therefore anticipates a return to mean-field form for the recombination kinetics when Dt > r 2. A 50 nm thermalization radius corresponds to a return to mean-field kinetics at times greater than about 0.5 ns for a-Si:H. Inspecting Fig. 1 again, the agreement of the fluctuation-dominated model with measurements is quite good for response times tl/2 < 30 ps. We can infer a lower limit r 0 > 12 nm. Although observation of fluctuation-dominated behavior in a - S i : H on picosecond time scales is consistent with the large thermalization radius inferred from geminate recombination arguments, it is nonetheless mystifying that the electron and hole could separate this far within a thermalization time of order one picosecond. There has been no evidence in picosecond mobility measurements for any extraordinary behavior on early timescales. Diffusion of 40 nm within 1 ps requires a diffusion constant of about 16 cm 2 s - - about 300 times larger than the value used to prepare Fig. 1. This might seem unlikely - - except that recent ambipolar diffusion measurements for large laser intensities do in fact indicate very large diffusion coefficients (about 10 c m 2 / V s) in a - S i : H [24] under some conditions.

7. Geminate recombination in a - S i : H The quantum efficiency for generating a free electron and hole (and avoiding geminate recombination) is given by Onsager's theory as 7/= exp( - rc/ro),

(17)

where r c = e2/4~re%kBT and r 0 is the thermalization radius. At room temperature in a-Si:H, r c - - 5 nm. The thermalization radius, r o, which is the initial separation of the electron and hole immediately following photogeneration and thermalization to the band edges) must be inferred from experiment. The weakness of geminate recombination near room temperature indicates that r o is several times larger than r c. Carasco and Spear [5] estimated r o > 40 nm

8. Endnotes The role of the electric forces between regions of high electron density and regions of high hole density has been neglected here. Roughly speaking, the effect of the carrier drift in response to electric forces is to smooth potential fluctuations with amplitude greater than kBT/e faster than occurs by simple diffusion. The crossover time, tc, after which drift is dominant can be estimated using a dimensional analysis argument (cf., Ref. [23]) to be

Dt c cz 1/NZr 4, where r c = e2/4zre%kBT.

(18)

E,4. Schiff /Journal of Non-Crystalline Solids 190 (1995) 1-8

In Fig. 1 this crossover time intersects the fluctuation model at N O = 1019 cm -3. The fluctuationdominated model should fail if this crossover time occurs before the recombination time, or equivalently when the initial density N O > 1019 cm -3. There is no experimental evidence for such an effect apparent in Fig. 1. The effects anticipated from the electrostatic attraction of electrons and holes also fail to appear in electrical measurements at longer times in at least two experiments. Ju~ka et al. [6] have estimated the parameter, b~ (cf. Eq. (1)), of the mean field, m o n o m o l e c u l a r plus bimolecular recombination model [16] using subnanosecond resolution electrical measurements of photoinduced transient space-charge limited currents. Most of their measurements are sensitive to the long-time tail of recombination, since for higher intensities most carriers recombine on the picosecond scale. The authors obtain reasonable agreement with the mean-field m o d e l ' s expected independence of intensity at long times and high intensities (cf. Eq. (11)). A mean-field form seems reasonable, since both the thermalization radius and Coulombic effects lead to crossover from fluctuation-dominated to meanfield behavior at long times. However, the authors conclude that recombination is slower than that predicted by the Langevin model by some orders of magnitude. A similar conclusion can also be reached using the measurements of Conrad and Schiff [28], who reported the transient photocurrent in coplanar electrode structures as a function of laser intensity at still longer times than Ju~ka et al. Langevin, bimolecular recombination has a staggeringly simple long-time prediction for the photoconductivity: trp(t) = N ( t ) e/x = e t x / b R t = e e o / t .

(19)

At 1 txs, this relationship predicts O-p---10 -6 f~ 1 cm -1, which is nearly 1000 times smaller than the observed photoconductivities for high laser intensities in this work. On the other hand, the prediction of a t -1 decay and of temperature-independence are approximately confirmed in the measurements. As Marvin Silver had surmised over ten years ago, the effects expected from the Langevin and Onsager views on diffusion-controlled recombination incorporating electrostatic interaction of electrons and

7

holes are not found in a - S i : H . Either of the two assumptions of the analysis - - diffusion-control, or electrostatic attraction - - might fail in a-Si : H. However, it would appear from the picosecond measurements that a diffusion-controlled model does apply for a - S i : H - - at least if fluctuation effects are included. This statement is the main conclusion that can be drawn from the present work. However, accepting the validity of diffusion control leaves us with a conundrum, since the remaining assumption -the electrostatic attraction of electrons and holes - - would seem to be beyond challenge. If I were allowed one final speculation, it would be to suggest that intrinsic potential fluctuations in a - S i : H may affect the recombination physics as significantly as the extrinsic photocarrier density fluctuations discussed here. Marvin Silver educated me about classical diffusion-controlled recombination and its implications for a-Si : H. I am grateful to Marvin for the education on this subject and on others, for his encouragement of and interest in my work, and for sharing with me his joy in science and in life. I thank D. Ben-Avraham and Z. Vardeny for discussions related to this paper. This work was supported by subcontract XAN-4-13318-06 with the National Renewable Energy Laboratory.

References

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E~A. Schiff/Journal of Non-Crystalline Solids 190 (1995) 1-8

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[22] Q. Gu, Q. Wang, E.A. Schiff, Y.-M. Li and C.T. Malone, J. Appl. Phys. 76 (1994) 2310 and references therein. [23] T. Ohtsuki, Phys. Lett. 106A (1984) 224. [24] M. Petrauskas, J. Kolenda, A. Galeckas, R. Schwarz, F. Wang, T. Muschik, T. Fischer and H. Weinert, in: Amorphous Silicon Technology, 1992, ed. M. Thompson, Y. Hamakawa, P.G. Le Comber, A. Maoan and E.A. Schiff, (Materials Research Society, Pittsburgh, 1992) p. 553. [25] H. Scher, M.F. Shlesinger and J.T. Bendler, Phys. Today 44 [1] (1991) 26 and references therein. [26] T. Tiedje, in: Hydrogenated Amorphous Silicon If, ed. J.D. Joannopoulos and G. Lucovsky (Springer, New York, 1984) p. 261. [27] Q. Wang, H. Antoniadis, E.A. Schiff and S. Guha, Phys. Rev. B47 (1993) 9435. [28] K.A. Conrad and E.A. Schiff, Solid State Commun. 60 (1986) 291.