Running space charge wave instability in photorefractive crystals

1 January

1997

OPTICS COMMUNICATIONS

ELSEVIER

Optics Communications

133 (1997) 109- 115

Running space charge wave instability in photorefractive crystals N.A. Komeev, J.J. Sanchez Mondragon, S.I. Stepanov Institute Nucional de Astroj&u.

dptica y Electr&icu,

Apt. Postal 51 y 216, CP 72000, Puebia. Pue., Me.rico

Received 6 March 1996; revised version received

1 July 19%; accepted

5 July 1996

Abstract It is shown that in materials with a special type of photoconductivity temporal response, the running space charge wave is generated under uniform illumination when the external electric field exceeds the threshold value. The experimental evidence of the effect existence in semi-insulating GaAs is presented. The effect can possibly be used to obtain high two-wave mixing gain in photorefractive crystals.

1. Introduction Weakly damped waves of space charge trapped by impurities is a known phenomenon in photoconductors. Such waves play an important role in photorefractive grating formation when the external voltage is applied to the crystal (see, e.g. Refs. [ 1,2] and references therein). Below we demonstrate that in crystals with an “overshoot” in the photoconductivity response to the step-like change in light intensity (Fig. la>, weakly damped space charge waves can be pushed beyond the generation threshold and become self-sustained in the uniformly illuminated sample. Necessary conditions for this are the external electric field exceeding the threshold value and fast Maxwell relaxation (the numerical criteria are given in the body of the paper). The effect is predicted by the standard theory of the grating formation in photorefractive crystals for certain impurity level models. A number of photocurrent instabilities in semiconductors involving formation of moving high-field domains are known [3-51. Most of them have their origin in the negative differential resistance, as in the 0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science PII SOO30-4018(96)0045 l-8

Gunn effect [6]. But in the mechanism of the so called recombination instability discussed by Konstantinov and Perel’ [7], negative differential resistance is not required. In this case the instability results from dynamic properties of the electron-hole recombination process. It appears as a running wave of space charge in a crystal bulk when the external electric field is high enough. This “running wave” type is promising for photorefractive applications, because the appearance of the wave generated above the threshold means that below the threshold the externally induced wave with proper parameters undergoes strong amplification. The effect we discuss here is similar in some essential features to the recombination wave of Konstantinov and Perel’, but it has another microscopic origin, closely related to properties of photorefractive semiconductors.

2. The theory of the running wave formation The popular model of photorefractive crystal with one impurity level [8] does not predict generation,

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though it includes weakly damped waves. It also cannot explain the photoconductivity temporal response of the required type (Fig. la). A big number of two-level modifications can be found in the literature on photoconductivity and photorefractive effect. For recent work on two-level photorefractive models containing extensive bibliography one can see Ref. [91. The simple two-level modification explaining the photoconductivity response of interest to us is presented in Fig. 2. The physical idea of it is the following. The “photorefractive” level Ll produces photoelectrons which can be re-trapped by it. The generation rate g is supposed to be proportional to the light intensity and independent of the level population (most of traps are filled). The trapping time is

(a) 1

53

time

@I

5

time

Fig. 1. Two types of exponential photoconductivity relaxation after a step-like change in the light intensity; (a) “overshoot”, (b) “build-up”.

I33 (1997) 109-I 15 conduction

band

q-f% Ll

Fig. 2. Two-level photoconductor model. The level Ll produces electrons in the conduction band with a generation rate g under illumination. From the additional level L2 electrons are excited only thermally. Illumination transfers electrons from Ll to L2, thus making the carrier lifetime 7 smaller with time.

inversely proportional to the number of ionized donors. We suppose that from the additional level L2 the carriers can be excited only thermally (and not by the light). This level is nearly empty, so the generation rate is proportional to the number of filled traps and the trapping time is independent of the level population. The probability for an electron to be trapped by the level Ll is much higher than by the level L2. When the light is on, L2 starts to fill. The characteristic time of this process is much bigger than the electron lifetime T. This produces electron transfer from Ll to L2. The free carrier lifetime (determined by the number of ionized donors at Ll) becomes smaller and the free carrier concentration diminishes. The process stops when the electron concentration at L2 becomes high enough, so that the enhanced trapping due to the higher free carrier concentration is counterbalanced by the bigger thermal excitation. For the rest of the paper we will suppose that there exists a strong uniform illumination which results in the generation rate go, the free carrier concentration no, etc. We will be interested in the system reaction to the small uniform or nonuniform perturbation of generation rate g ‘(x, t), where x is the spatial coordinate and t is time. With some simplifying suppositions about generation and recombination rates the model gives the exponential decay of photoconductivity u ‘(t) after the small uniform light perturbation is on, a’(t)

= a,l,((

1 - a) + a exp( -t/TR)),

(1)

where rR is the time of photoconductivity relaxation, and (Y< 1 is a positive constant determining how much the photoconductivity diminishes. In the

NA. Korneeu et d/Optics

frequency function:

domain Eq. (1) corresponds

al(w)=c&

(

I--

Communicarionv

133 (1997) 109-115

to the transfer

1

l+;wT . R

relaxation times

Both (T ‘(t> and (T’(W) can be easily measured experimentally, the first one as the photocurrent pulse in a sample illuminated through the chopper, and the second as AC current component in the sample illuminated by the light with intensity sinusoidally modulated with frequency o. We also introduce carrier lifetime dependences on time and frequency, which for our model are given simply by

generation

(3)

and a similar equation for time dependence. Here e is the electronic charge, p is the carrier mobility, and g: is the amplitude of the uniform additional generation rate due to the amplitude modulation of light. Our purpose is to find the space charge grating formed by a small perturbation in light intensity (generation rate) of the running interference fringes type: g’( x, t) = Re( g’ exp(ii”(x + iwr)),

(4)

where K is the spatial frequency. The plus sign in Eq. (4) is chosen in order to use directly some of our previous results. Since the level structure is imposed, the space charge grating calculation for the low contrast of interference fringes is a routine procedure in photorefractive science. The approach which permits to do this directly from the known photoconductivity response was developed in our recent paper [lo], but the same result can be obtained in a more traditional way as well. For the complex amplitude of the running wave of the space charge electric field one has: E,,(w, K, .‘?f -a’(o)(E’+iE,,)

Kk,T/e

is the external electric field, k, constant, T is the temperature, is the diffusion field, Ok = E&,/U’

is the E, =

is the

region for spatial

frequency in inverse diffusion

15

20 electric

T(W) =++(eg:P),

Here E” Boltzman

III

25

30

35

40

field (drift length/diffusion

45

50

length)

Fig. 3. Generation region for Maxwell frequency and relaxation time product equal to 210 and a = 0.5 (Fig. la). Drift length is L, = PTI’? and diffusion length is L, = (/17kT/e)‘/*, with lifetime taken for high frequency. Negative sign of spatial frequency means that for electrons the wave runs in a positive direction, i.e. opposite to the drift of carriers.

Maxwell relaxation frequency, &co is the dielectric constant and a0 is the photoconductivity due to the strong uniform illumination. (T’(W) and T(W) are given by Eqs. (21, (3). The wave in space and time is given by an equation similar to Eq. (41, with the exception that E,, is the complex parameter now. i.e. it includes the phase difference between the running pattern of fringes and running electric field distribution. The familiar result describing weakly damped waves [I] follows from Eq. (5) by formally putting (Y= 0, which corresponds to immediate photoconductivity response. If there is no photoconductivity relaxation ((Y = 0) or for negative CY,formally corresponding to more usual relaxation of “build-up” type (Fig. lb), no denominator roots in Eq. (5) exist for real K and w. But for the case we consider here, Eq. (5) has poles for real o and K if the external electric field is high enough (Fig. 3). This means that any disturbance in light intensity having corresponding running component produces the running space charge wave with infinitely high amplitude. So, poles of Eq. (5) correspond to generation of the running wave. Of course, the low contrast approximation we used will be broken in this case, and nonlinear theory is needed here to determine the space charge electric field amplitude.

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Optics Communications 133 (1997) 109-I

Two parameters, product oi,,,ra, and CYdetermine whether the generation will develop or not. It can be shown that if CY(W~T~+ 1) < 1, no poles exist, whatever the external electric field is. If a( WITH + 1) > 1, two poles exist for external electric field higher than the critical one E&, and no poles exist for low electric field (E” < E& ). If the overshoot in the photoconductivity response is not pronounced (cr < OS), Maxwell relaxation must be faster than photoconductivity relaxation for generation to be developed. The region between the two branches in Fig. 3 corresponds to instability as well. Which waves from the possible interval will really be generated cannot be answered in the framework of me linearized theory we used. For comparison with the experiment, critical value of the electric field and corresponding w and K can be calculated from the measured photoconductivity temporal response. E” proves to be such to ensure carrier drift length approximately an order of magnitude higher than their diffusion length. Critical wave frequency has an order of the inverse photoconductivity relaxation time, and critical wavenumber K is close to the inverse diffusion length (Fig. 3). The form of our result Eq. (5) implies that it is the temporary response of the photoconductivity which matters, and not the specific level structure which gives this form of temporary response. This viewpoint is discussed in Ref. [lo]. Eq. (5) can easily be generalized for the case of a more complicated photoconductivity relaxation, since frequency dependence of photoconductivity is known whether from experiment or from theoretical considerations. To

110

T

CCD camera

laser

cl

msplitter

--

diaphragm

El photodetector

Fig. 4. Experimental setup for running space charge wave observation.

IS

obtain the space charge electric field it is sufficient to use proper (+‘( 01 and T(O) in Eq. (5). It can be shown that the general form of the photoconductivity relaxation with the “build-up” (Fig. lb): a(r) = (~(1 - p, exp(-t/rR,) & exp(-t/rR2) - . . . > with all p positive, does not give poles in Eq. (31, so the “overshoot” type is essential for generation. The theory presented is simplified in the sense that it does not include the influence of the second type of carrier and considers the simplest type of photoconductivity relaxation. But it demonstrates a new effect: in the crystal with the “overshoot” in the photoconductivity the dynamics of the weakly damped wave in the external electric field is modified in such way that it can become self-sustained.

3. Experiment

and discussion

To observe the running space charge wave we used a setup similar to that described in Ref. [ 111for the observation of electric field domains attributed to negative differential resistance (Fig. 4). The photorefractive sample with the external electric field applied was placed between crossed polarizers and illuminated with the light of a NdYAG laser (1064 nm). With due orientation of the crystal axes, the electric field changes become visible with a CCD camera because the field modulates the crystal birefringence. Our only crystal demonstrating photoconductivity response of the required type (Fig. la) was the semi-insulating GaAs sample grown in the Hughes Research Laboratories. It was also the only one with generation. Two other GaAs crystals and one CdTe crystal, all of them having a more usual photoconductivity temporal response (Fig. lb), did not demonstrate generation for fields up to 0.5 kV/mm. The crystal we worked with was sized at 4.5 X 4.5 X 6 mm3. Two silver paste electrodes were deposited on its faces 4.5 X 6 mm’. The typical photoconductivity response of the crystal to the square light pulse produced by a chopper was essentially of Fig. 1a type. Relaxation time (as taken at 1/e level) and the product of Maxwell relaxation frequency and relaxation time are presented in Fig. 5. With growing light intensity the characteristic time of relaxation

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Optics Communicarions 133 (1997) 109-I

becomes smaller, but the product WITH remains nearly constant because the frequency of the Maxwell relaxation grows. The parameter (Y is nearly 0.5 independently of the light intensity. The dependence of DC photocurrent on the external voltage is slightly superlinear and does not demonstrate any negative differential resistance up to the applied 1.2 kV. The generation is first seen as rather thin fringes (30-60 pm spacing, as estimated from the monitor screen) with poor contrast running in the direction from cathode to anode. Fringes first appear near the cathode where the electric field is higher because silver paste contacts are not ohmic. With a strong additional illumination of the cathode, which reduces the field near it, the region of generation moves to the crystal center. Near the threshold we could obtain generation only in a small part of the crystal (typically < 1 mm). The applied voltage necessary for fringe appearance is 350-400 V. With higher voltage the fringes become brighter and wider and spread to the whole crystal. By placing the photodetector with a point aperture in the image plane (Fig. 4) it is possible to investigate the frequency spectrum. For near-threshold electric field there is only one welldefined peak with relatively low amplitude (Fig. 6). For higher fields, other peaks appear, and for voltages higher than 1.5 kV the behavior seems to be fairly chaotic, though some wide peaks still exist in the spectrum. We could not find simple relations between the frequencies of the peaks, though in some cases there is a tendency to frequency doubling. We did not investigate the spatial frequency

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0

(4

50

100

150

200

400

600

6 C IO

I

0

a+

200

frequency(Hz) Fig. 6. Spectra of the photodetector signal (Fig. 4) for different values of external voltage; (a) 362 V, two smaller peaks are 60 and 120 Hz pick-up, (b) 418 V, and (c) 1715 V. Note the difference in scales. The light intensity is 13 mW.

n

0

0 lb

intensity (mW) Fig. 5. Photoconductivity parameters as function of the light intensity incident on the crystal surface.

spectrum for the moment, but it seems to include different components for high electric fields as well. The direction of the running wave implies that holes are the majority carrier. This is confirmed by measuring the sign of the nonstationary holographic current excited by the vibrating pattern of interfer-

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Communications

ence fringes [12] with no external voltage applied. From these measurements the diffusion length of the carriers can be estimated as well and proves to be equal to 4.8 + 0.8 pm. Calculations of Fig. 3 are made for experimentally observed values of ONTO and (Y.From the known diffusion length the theoretical critical value of electric field is E& = 115 &-20 V/mm, which agrees reasonably with the observed generation threshold 360 V/4.5 mm. The calculated frequency w/27r of near-threshold wave for relaxation time of 2.4 ms corresponding to the light intensity of 13 mW for which the spectra Fig. 6 were taken, is 185 Hz, the observed value is 95 Hz. The calculated critical wavelength is 9.6 f 1.6 pm, which is lower than the observed one (30-60 pm). According to Fig. 3, for 10% uncertainty in the electric field near the threshold, one has a four times difference between the lower and upper possible values of frequency and spatial frequency. Taking into consideration the nonuniformity of the electric field, we think that the experiment confirms the theory presented above. Eq. (5) describes not only the generation, but also the strong amplification for the space charge grating excited by a moving interference fringe pattern in the below-threshold regime. Theoretically, by making the parameters of the running wave close to the critical ones for an electric field a little lower than the generation threshold value, it is possible to obtain as high amplification (and, consequently, as high two-wave mixing gain) as needed. Strong two-wave mixing gain is desirable in a number of photorefractive applications. Practically, there are some limitations. The high gain can be realized only with a very low contrast of interference fringes (signal beam is much weaker than a pump one). Other factors to be considered are the high noise level near the threshold and the necessity to obtain an uniform external field. It is desirable to utilize a crystal with a higher value of the critical electric field than that found in our specimen. The resonance amplification of the space charge wave in the below-threshold regime can be observed by measuring the nonstationary holographic current produced by vibrating interference fringes with external electric field applied. Vibrating interference fringes can be represented as a sum of the stationary

133 (1997) 109-115

L

10-66

-I

10

100

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10000

frequency(Hz) Fig. 7. The AC electrical signal produced by vibrating interference fringes in the external chain as function of frequency for different external DC voltages. The load resistor is 9.2 k 0, light intensity 5 mW.

pattern and two waves running in opposite directions. When the frequency of vibration is near-resonance, one of these running components is amplified, producing a peak in the frequency response of the effect. The calculation based on Ref. [lo] demonstrates that such peak really appears in our case and its amplitude becomes infinitely high for a frequency giving a pole for given E and K (if such pole exists). The experimental arrangement for nonstationary current observation is described in Refs. [ 10,121. For the low fringe contrast we observed in the frequency region where generation develops a rather spectacular growth in the nonstationary current signal (Fig. 7). The peak seems to be “inhomogenously broadened” because of variations in a local electric field. The practical possibility to obtain strong two-wave mixing gain, predicted by the theory, as well as its detailed comparison with the experiment are under study now.

4. Conclusion In conclusion, the theory developed and the experiments with semi-insulating GaAs demonstrate the existence of a new mechanism of instability in photoconductors which originates from the photoconductivity temporal response and not from the

N.A. Korneev et al./ Optics Communications

negative differential resistance for DC current. The effect can possibly be utilized to obtain high twowave mixing gain in photorefractive crystals.

Acknowledgements The authors thank M. Klein and B. Wechsler from Hughes Research Laboratories for help with crystals and useful discussions.

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