Running gratings in photoconductive materials

Transcription

1 Kukhtarev et al. Vol. 22, No. 9/ September 2005/J. Opt. Soc. Am. B 1917 Running gratings in photoconductive materials N. V. Kukhtarev and T. Kukhtareva Department of Physics, Alabama A&M University, Huntsville, Alabama S. F. Lyuksyutov and M. A. Reagan Departments of Physics, Chemistry, and Polymer Engineering, University of Akron, Akron, Ohio P. P. Banerjee Department of Electrical and Computer Engineering, University of Dayton, Dayton, Ohio P. Buchhave Department of Physics, Technical University of Denmark, Lyngby, 2800, Denmark Received December 16, 2004; revised manuscript received March 28, 2005; accepted April 4, 2005 Starting from the three-dimensional version of a standard photorefractive model (STPM), we obtain a reduced compact set of equations for an electric field based on the assumption of a quasi-steady-state fast recombination. The equations are suitable for evaluation of a current induced by running gratings at small-contrast approximation and also are applicable for the description of space-charge wave domains. We discuss spatial domain and subharmonic beam formation in bismuth silicon oxide (BSO) crystals in the framework of the smallcontrast approximation of STPM. The experimental results confirming holographic current existence in BSO crystal are reported Optical Society of America OCIS codes: , , INTRODUCTION The study of the influence of photoinduced gratings modulation on electric current in semiconductors and ferroelectric materials was of a great deal of interest in the 1970s, 1 3 resulting in what later became known as the band transport model, or the standard photorefractive model (STPM), also known as photorefractive equations. 4 The idea was that photoinduced gratings generate holographic voltages or transient holographic currents in a broad class of photoconductive materials. 3 6 After a substantial period of academic research, the methods, based on transient photocurrent, are now in a device-building stage 7 with potential applications in vibrometry for nondestructive testing. Different types of space-charge waves become more pronounced under large-magnitude electric fields, leading to high-frequency current oscillations, such as the Gunn effect in GaAs, for which the free carriers form moving electric field domains. Although originally formulated for semiconductors, STPM has been useful for different physical models in which the concept of photogeneration recombination and drift are justified. Several examples include the application of STPM for description of a grating recording in the photorefractive polymers and liquid crystals, ferroelectric crystals, semiconductor materials, and paraelectric electro-optic crystals. 2 5 The main goal of this paper is to formulate a general form of the starting equations describing known and stillexpected results stemming from the STPM. Taking into account growing interest in the spatiotemporal patterns in different fields of modern science, we write the equations in appropriate three-dimensional (3-D) form. Sections 2 and 4 discuss this goal. An unexpected result of this formulation suggests that the running gratings produce an electric holographic current. An additional goal of this paper is to verify this prediction for a onedimensional case in a photorefractive bismuth silicon oxide (BSO) crystal, as presented in Section 5. Evidence of the holographic current generation in photorefractive BSO will be presented and compared with theory. Section 3 presents detailed calculations for the case of smallcontrast approximation. The interpretation of the experimental results related to subharmonic beams and 8 photorefractive domains 9,10 in photorefractive materials will be discussed in Section BASIC EQUATIONS The STPM is defined as a drift diffusion recombination (DDR) model in a generalized approach, as suggested below. The system of equations for photogenerated mobile charged carriers with concentration n, photosensitively fixed in space-ionized centers N, and electric field E has the following form: 0 Ē t + e nē + ed n = J, 0 = e N N A n, /05/ /$ Optical Society of America

2 1918 J. Opt. Soc. Am. B/ Vol. 22, No. 9/ September 2005 Kukhtarev et al. N t = g + N 0 N rnn. 1 t i n 1 = n 0 m i E 1 E 1 E k E M Here, e is the effective charge of the carrier, is the mobility, D is the diffusion coefficient of the mobile carriers, N 0 is the total concentration of photosensitive centers, N A is the concentration of the compensating centers (acceptors), g is the optical generation rate, is the thermal generation rate, r is the recombination coefficient, 0 is the dielectric permittivity of vacuum, and is the relative dielectric constant. The total current J and the electric field Ē obey the following equations: J =0, Ē =0. This is an important assumption for considering a 3-D case. The photogalvanic current term on the left-hand side of the first equation in Eq. (1) for the materials without center inversion must be added: j i p = ijk F j F k *. Here, is the photogalvanic tensor and F j,k are the electric field components of the laser irradiation. Equations (1) and (2) present the basic material equations for description of photoinduced gratings formation in photoconductive materials. 3. SMALL-CONTRAST APPROXIMATION The above equations could be simplified for the case of a small-contrast interference pattern: I x,t = I 0 1+ m 2 exp ikx i t + c.c, n 1 1+ E D E M + i E 0 E M, M t i E 1 = ie D E 0 n 1 n 0 E 1, where = d / p = /SI 0, d is the dark conductivity, p is the photoconductivity, M = 0 / d + p, = rn A 1 are the recombination times, E k =en A N N A 0 kn 1 is the limiting space-charge field, E D = D/ k is the diffusion field, E M = k 1, and g 0 =SI 0. The total current for the one-dimensional case can be expressed similar to that in Refs. 2 and 3: J = n 1 1 e E E 7 n t. The notation determines spatial averaging. Equation (7) accounts for the contributions from both the steadystate current through the first term and the transient (electromotive force) current 11 through the second term. Both currents could be used for the crystal s characterization. The expression for this current can be simplified using the low-contrast approximation 12 * E J = e n 0 1 m 2 n m n t + m E n * 1, t where m n =2n 1 /n 0. The steady-state solution of Eq. (6) can be written as n 0 = g 0 N N A 1+, n 1 = m ie D i M 2 1+ a + ib n 0, 6 8 where m is the modulation index (intensity contrast), k is the grating vector, I 0 is the average spatial intensity, and is the frequency detuning between laser beams. For the one-dimensional case the carrier concentration n and the electric field E inside the crystal can be written as n = n 0 + n 1 exp iu + c.c, E = E 0 + E 1 exp iu + c.c, E 1 = m ie D E a + ib, with the coefficients a and b given by a =1+ E D E k + M E 0 E M, b = E 0 E k M 1+ E D E M u = kx t. The average electric field E 0 (for a short circuit) is equal to applied voltage divided by sample thickness. Assuming a linear recombination case we derive from Eqs. (1) (5) the following equations for n 0, n, and E: n 0 t = g 0 1+ N N A n 0, 5 The expression for the total current then can be written as J = e n 0 E 0 m2 E 0 + M E D a 2 + b. 2 Equation (11) suggests that the running photoinduced grating must produce an electric holographic current even without an external electric field. The sign of this current will depend on the direction of the grating. This is an im-

3 Kukhtarev et al. Vol. 22, No. 9/ September 2005/J. Opt. Soc. Am. B 1919 portant prediction in the framework of the STPM model considered in Section SUBHARMONIC BEAMS, PHOTOREFRACTIVE DOMAINS, AND SPACE-CHARGE WAVES IN PHOTOREFRACTIVE BSO The phenomenon of generation of subharmonic diffracted beams 8 between two frequency-detuned plane-wave beams incident on BSO crystal have been reported and studied by the Oxford Group, the RISOE Group, 16,17 and the group at the Technical University of Denmark. 10,18 20 This effect has been explained as the instability of space-charge waves (eigenmodes) naturally generated in the photoconductive crystals and enhanced through the application of the external electric field. These eigenmodes can be amplified further through the motion of the light interference pattern. 10 The spatial structure of the subharmonic beams appears through the domain pattern 9,10 in the near-field transversal cross section. The domains move from one side of the crystal to another parallel to the electric field vector, suggesting a complex dynamics of aggregation and mutual destruction. The domains move with a velocity equal to the velocity of the running grating but with an opposite sign. It was explained that the domain velocity is equal to the group velocity of spatial space-charge waves. 20 The 3-D version of the STPM is required for description of these spatial temporal dynamic patterns of the domain motion. At the quasi-steady-state approximation, valid for the case when the recombination of carriers is much faster than Maxwell s space-charge relaxation and the linear recombination, we can exclude N from Eq. (1). In this case the reduced equations for n and E would have the following form: 0 Ē nr N A + = G N 0 N A 1 + e e G t 0 Ē, 0 t + e ˆ n Ē + ed n = J, 12 where G g+. Excluding n we finally arrive with one reduced equation for the total electric current: E J = 0 t + e Ē + D 0 N A G + 0 Ē t eg N r en A + 0 Ē. 13 This equation for the current J along with two Maxwell s equations [Eq. (2)], compose the starting fundamental system for the 3-D case of subharmonic domain description. For the important special case of weak quasineutrality when en A 0 Ē, Eq. (13) can be simplified by neglecting quadratic terms of E: J = 0 Ē t + e Ē + D r G N N A e 1 G N 0 N A + t 0 Ē. 14 The nonlinear system of Eqs. (2) and (13) additionally includes parametric generation terms (with respect to G) that are useful from the point of view of nonlinear dynamics. This system describes parametrically generated subharmonics and space-charge waves. Recent experiments of laser light scattering in photorefractive materials reveal another interesting effect defined as self-organization of near-field scattering in the regular arrays (hexagon, squares, rolls, etc.) Here we emphasize one innovative view of the near-field pattern formation in transparent materials with phase (excluding absorption) gratings. The transversal modulation of the refractive index associated with phase gratings of a photorefractive crystal may be adequately mathematically described through the formation of optical waveguides or channels in a crystal s volume. These transversal modulations of the refractive index are visualized in the near field by optical channeling. 23 The concept of channeling is applicable for the domain pattern visualization observed at the near field. In the case of a subharmonic beam propagating between recording beams, the geometry is favorable for channeling that is provided when the modulation of the refractive index is large enough for wave guiding. The fundamental grating runs at the resonance velocity V r = r k 1 while the domain moves into the opposite direction at the same velocity. This interesting result is the manifestation of a specific dispersion relation for the space-charge waves for E k E 0 E M when it obeys a simple form = M 3 E 0 k 1, where M and are Maxwell and recombination times and is the photocarrier mobility. When inversed with respect to the k dispersion law for the amplitude of the wave package, formed by the two subharmonic waves, the group velocity for the wave package has the following form: V g = / k= /k, as confirmed experimentally. 10,20 5. EXPERIMENTAL OBSERVATION OF HOLOGRAPHIC CURRENT An experimental setup for holographic current measurements is presented in Fig. 1. A solid-state cw laser operating at power range mw and at a wavelength of 532 nm was used as a light source. The intersection angle between the beams varied to produce interference-pattern (IP) spacing between 15 and 20 m. The beams were polarized perpendicularly to the plane of incidence. A BSO crystal with dimensions mm 3 was placed inside a Faraday cage to prevent unwanted electrical noise from outer electrical circuits. Two electrodes were attached to the sides of the BSO crystal and connected to a 6040 Keithley picoammeter. In certain experiments a dc voltage between 2 and 7 kv was applied along the (001) axis. The analysis of the digitized electric current signal was performed using C script in the MATLAB environment on an IBM ThinkPad laptop.

4 1920 J. Opt. Soc. Am. B/ Vol. 22, No. 9/ September 2005 Kukhtarev et al. Fig. 1. Experimental setup for holographic current measurement. Radial diffraction holographic grating (RDHG) is presented in the inset. The major element used to produce a running grating through the motion of the IP spacing inside the BSO crystal was a radial diffraction holographic grating (RDHG) 18 presented in the inset of Fig. 1. This element is very flexible in generating a running IP with velocities up to 200 mm s 1. Although the RDHG produces nine visible diffraction orders, only the first and negative-first orders are used to generate the IP. The first-order diffracted beams are Doppler shifted by an equal but opposite frequency, plus and minus, as a result of the rotation of the RDHG. These two frequency-shifted beams are recombined to form a running IP. The velocity is determined as a function of the applied dc voltage to the motor driving the RDHG. The motor is specifically designed to operate at a slow but constant angular velocity, since any change in angular velocity will have a substantial effect on the velocity of the running IP. To measure the velocity of the IP, the BSO and the electrical equipment were replaced with a 25 objective and an additional mirror (Fig. 2) to expand the beams so that the fringes were visible to the naked eye with approximately 1 2 cm between the maximums. A photodetector with a narrow slit of about 0.1 mm was placed in the path of expanded beam and aligned parallel to the running fringes. The signal from the photodetector was acquired by a Hewlett Packard oscilloscope. The periodic IP shows up as a sinusoidal waveform on the oscilloscope trace, thus determining the time between adjacent maximums. The velocity of the IP can be determined, providing the IP spacing is measured (in most experiments, the spacing ranged between 15 and 25 m). Originally, the holographic current was measured without the application of the external electric field. The results of the current dependence on the IP velocity are presented in Fig. 3(a) for the laser intensity 12 mw cm 2 and the modulation factor (contrast) The IP spacing was measured to be 22.5 m. The current was in the range between 5 and 6.5 na and was found to depend strongly on the contrast m, m=2 I1 I 2 / I 1 +I 2. The results of calculations of the current based on Eq. (11) are presented in Fig. 3(b) for different values of photoconductivity [0.8 to m 1 ] and donor concentration (3.5 to m 3 ). It appears that the best fit occurs for p = m 1 and N= m 3. These values are in a good agreement with those quoted in the literature. 24,25 An experimentally measured current (filled dots) dependence on the IP velocity and calculation of the current (solid line) according to Eq. (11) with an applied electric field are presented in Fig. 4. The current increases with the velocity by a factor of approximately 2 with respect to that for the IP at rest. The typical value of the current for incident laser intensity 17.5 mw cm 2 and applied field 6kVcm 1 was in the range 1 10 A, 3 orders of magnitude higher than the current without voltage. The modulation factor was A qualitative agreement between experiment and calculation is evident for the IP s velocity equal to or greater than 10 3 ms 1. The discrepancy between the experimental data and theory for the IP s velocities smaller than 10 3 ms 1 could be associated with space-charge wave self-excitation 19,20 contributing into the holographic current, for which current theory does not explicitly account. For cases both with and without applied electric field, the holographic current was found to be dependent quasi-linearly on the modulation factor (IP contrast) DISCUSSION AND SUMMARY We have compared our results obtained for dielectric BSO crystal to similar calculations for semiconductors accomplished in small-contrast approximation. The concept of Fig. 2. Experimental setup for measurement of the velocity of running holographic gratings. The interference pattern is expanded by a 25 objective and reflected from a mirror to increase the path length so that individual fringes may be observed on a screen. A photodetector with a small slit opening is placed in the path of an expanded beam. As the fringes move, the current from the photodetector appears as a sinusoidal pattern on an oscilloscope, and the frequency is determined. Velocity is calculated by multiplying the frequency by the grating wavelength.

5 Kukhtarev et al. Vol. 22, No. 9/ September 2005/J. Opt. Soc. Am. B 1921 Fig. 4. Electric current experimentally measured as a function of velocity with an applied electric field 6 kv cm 1 forahighcontrast approximation m=0.98, represented by filled dots. The light intensity was 17.5 mw cm 2, and the current was 5 A in its maximum. Calculation of the electric current with respect to the grating velocity according to Eq. (11) is presented as a solid curve. The best fit was achieved for the following set of parameters. An applied electric field was E 0 =6.5 kv cm 1, the modulation index was m=0.98, the thickness of the BSO crystal was 1 cm, and the relaxation times were m = s and m = s. The following characteristic fields were selected: E k = kv cm 1, E D =0.11 kv cm 1, and E M =0.8 kv cm 1. Fig. 3. Holographic current as a function of velocity without an applied electric field. (a) Experimental dependence presented as filled circles; solid curve presents the result of calculation using Eq. (11) with the following parameters: N A = m 3, N= m 3, ª10 5 m 2 V 1 s 1, r = m 3 s 1, =56, I 0 =0.012 W cm 2, s=10 5 m 2 J 1, m =0.97, D =10 13 m 1, P = m 1, D= m 2 Cs 1 ; (b) theoretical curves of holographic current without applied electric field for four different values of photoconductivity and donor concentration, P = m 1, N= m 3 (solid curve); P = m 1, N= m 3 (dotted curve); P = m 1, N= m 3 (dashed curve); P = m 1, N= m 3 (dashed dotted dotted curve); P = m 1, N= m 3 (dashed dotted curve). the trap recharging waves in semiconductors was pioneered in the work in Ref. 26. The Erlangen group has reported on the moving-grating technique 27 applied for the study of bipolar semiconductors. The technique covers running gratings without an applied electric field in undoped a-si: H. Their calculations of the group describe the total current in terms of the grating velocity, relaxation time, and the mobility of both positive and negative charge carriers. Our results are in good agreement with that reported in Ref. 27 in the sense that the general trend of the holographic current dependence on the grating s velocity is the same. The methods of dc generation based on an oscillating interference pattern 28,29 have been suggested and compared with the moving-grating method presented in Ref. 30. In this paper we have formulated an approach and derived general equations valid for a 3-D case and for a high-contrast approximation. Starting with the DDR model, Maxwell s equations, and assuming a linear recombination case, we derived equations for n 0, n, and E. Using the steady-state solution of these equations and the low-contrast approximation, we derived an expression for the total current J. This expression suggests that a running grating produces an electric holographic current whose sign depends on the direction of the grating, even without application of an external electric field. We discovered a reduced equation for the total electric current and used this in conjunction with two of Maxwell s equations as the starting point for a 3-D description of spatial temporal dynamic patterns of the domain motion in pho-

6 1922 J. Opt. Soc. Am. B/ Vol. 22, No. 9/ September 2005 Kukhtarev et al. torefractive BSO crystals. This description describes parametrically generated subharmonics and space-charge waves. We mathematically described the transversal modulation of the refractive index through optical waveguides or channels inside the crystal volume and obtained results for the near-field intensity, which describes the experimentally observed phenomenon of contrast inversion. We have verified experimentally the theoretically predicted S-type dependence on the frequency detuning for the total current J without an external electric field. This curve was found to have a strong dependence on photoconductivity and number density of fixed-space photosensitive ionized centers N. The best-fit curve corresponds to photoconductivity of m 1 and donor s concentration N= m 3. S. F. Lyuksyutov is the corresponding author and can be reached at sfl@nebula.physics.uakron.edu. REFERENCES 1. J. Amodei and D. 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