Open Access
Add to BibSonomyAbstract
Controlling the space charge distributions in a crystal is indispensable for controlling a KTa1-xNbxO3 (KTN) optical beam deflector. The space charge is built up by applying a voltage and injecting electrons into the KTN crystal. Although a homogeneous distribution is preferable, we observed experimentally that the injected electrons concentrated in the vicinity of the cathode and for some samples the concentration was much lower around the anode. We investigated the electron dynamics theoretically and found that such inhomogeneity was caused by a freezing effect where the motion was very slow considering the duration of the practical voltage application. The depth of the space charge spread or the electron penetration depth from the cathode was proportional to the applied voltage and the permittivity, and inversely proportional to the density of traps or localized states that bind electrons. We believe that the trap density was too large for the samples with inhomogeneous charge distributions.
© 2014 Optical Society of America
1. Introduction
Optical beam deflectors are used in various industrial areas. They can be applied to several kinds of laser displays. They may be essential components in light detection and ranging systems (LIDAR). Recently, a new type of optical beam deflector was proposed [1]. It is made of a block of electrooptic (EO) material such as potassium tantalate niobate (KTa1-xNbxO3, KTN) single crystal with a pair of film electrodes. A space charge is formed inside this material by injecting electrons (or holes) via the electrodes. The space charge first forms an electric field distribution and then a refractive index distribution via the EO effect. This index distribution bends a light beam. The deflector can easily achieve deflection angles exceeding several degrees, which are much larger than those obtained with conventional EO deflectors [2]. We have used a KTN deflector to develop a high-speed wavelength-swept light source for an optical coherence tomography (OCT) cross-sectional imaging system [3]. The speed of the KTN is attractive for the in-vivo observation of moving organic objects [4].
As described later, it is important for this deflector to control electron motions in the KTN crystal and thus to adjust the spatial distributions of the charge density. There have been several studies that discuss charge transport properties in relation to space charges in insulators [5–7]. However, their aim was mainly to analyze the space charge limited current (SCLC) and most interest is directed at I-V curve profiles. However, with a space charge controlled EO device, the key point is not the I-V curve but the spatially distributed charge density, which has not attracted so many researchers’ interest. There were no appropriate theories to explain the charge distribution phenomena observed in our KTN devices. This is why we have devoted our effort to analyzing the charge transport properties in dielectric materials such as KTN.
In this paper, we discuss the experimental spatial distributions of the charge and electric field for KTN single crystals and calculation methods that effectively simulate them. We observed electrons that were injected from a cathode, driven by an external electric field and then migrated toward the anode in the crystal. However, with some samples the electron motion stopped on the way to the anode. This phenomenon could not be simulated by steady state solutions of equations. Actually it was not a steady state but a frozen state that was moving very slowly. This idea was proven with numerical analyses. As also predicted with this theory, the depth of the electron penetration was experimentally observed to be proportional to the applied voltage, the permittivity and the reciprocal of the density of electron traps.
2. Beam deflection in relation to space charge
Here we explain the relationship between the deflection function and the space charge. Our space charge controlled beam deflector consists of a KTN single crystal block with a pair of film electrodes deposited on its two opposing faces. Figure 1 illustrates the KTN deflector. We assume that an electric field is perpendicular to the electrodes anywhere inside the block. Also, we take the x-axis parallel to the electric field and ignore any y- or z-dependences. When the length of the crystal block is L, the optical path length s is modulated because of EO index modulations Δn as
where n0 is the original refractive index and ε is the permittivity. The 2nd order EO coefficient g is g11 for an optical polarization parallel to E but g12 for a perpendicular polarization [8]. If E is spatially nonuniform, Δs is also nonuniform. Then the output light ray is bent according to Δs. The deflection angle ϕ is expressed as follows.The nonuniformity is produced by a space charge in the KTN block. The electric field E and charge density ρ are related by Gauss’s law as follows.As a result, ϕ is calculated as follows.Here ϕ is proportional to ρ. Without space charge ρ, no beam deflections occur. Thus controlling the space charge is a fundamental technique for this type of deflector. The space charge is formed by electrons injected into the KTN crystal through an electrode. In the next section, we introduce experimental charge distributions formed in such a manner.3. Experimentally observed charge density distributions
3.1 Experimental
When we apply a voltage to a KTN single crystal with a pair of electrodes, electrons are injected into the crystal from the cathode, and then drift toward the anode due to the electric field. They might transit the crystal and reach the anode. Also, they might be trapped by crystal defects somewhere in the crystal before reaching the anode. The trapped states are stable. Once the electrons are trapped, their charge distribution will be fixed and last for a few days at room temperature unless some driving force is applied such as an electric field [9]. The trapped electrons become mobile by absorbing photons with sufficient energy. To reset the crystal, we irradiate it with a 405-nm light from a light emitting diode (LED). The light excites the electrons to the conduction band of the crystal and the electrons diffuse out of the crystal to the electrodes.
We evaluated the charge distributions by measuring the distributions of retardation that had been formed by the charge distributions via the EO effect. Here the retardation R is measured as the difference between the optical path length for a light polarization parallel to the static electric field E(x) and that for a polarization perpendicular to the static electric field. From Eq. (1), we obtain
Here and are optical path length modulations induced by the EO effect for the two light polarizations. From R, we can calculate E(x) with this equation and then ρ can be deduced by using Eq. (3). The charge distribution measurement procedure has been reported in detail elsewhere [10]. The light source was a diode laser with a wavelength of 679 nm.We could also evaluate the time evolutions of the charge distributions by utilizing the trapping property described above. We applied a voltage to a sample for a short duration. When the voltage was switched off, electrons anywhere in the crystal are trapped there. Therefore, the distribution of the trapped electrons approximates the electron distribution at the time of switching off. We could measure the spatial charge distribution afterwards because the trapped states are stable as described above. We repeated this procedure with different voltage pulse widths and obtained a series of charge distributions as the time evolution of the distribution ρ(x, t). The bandwidth of the high voltage source was 500 kHz.
As samples, we used KTN single crystals grown by using the top seeded solution growth method [11]. We cut a crystal boule into blocks 1.0 mm x 3.2 mm x 4.0 mm in size. A pair of titanium electrodes was formed on the 3.2 mm x 4.0 mm faces. A KTN crystal undergoes a structural phase transition from a ferroelectric to a paraelectric phase at a temperature called the phase transition temperature Tc. We controlled the temperature of the block at a value above Tc in the paraelectric phase. In this phase, the permittivity of the crystal block can be changed by changing the temperature according to the Curie-Weiss law [11]. Tc was in the 30-40 °C range for our samples. Unless otherwise stated, the sample temperature was about 5 °C above Tc and the relative permittivity was 17,500. With this high permittivity, we could obtain a huge second order EO effect of 3.3 x 10−15 m2/V2.
3.2 Measurement results
Figure 2 shows spatial distributions of retardation and charge density for two deflector samples with approximately the same crystal composition. The left and right ends of each graph are the cathode and anode respectively. We acquired each curve with a constantly applied voltage and did not switch the power off until the measurement was finished. The charge density was negative because the carriers were electrons. As for sample A, the charge density was fairly uniform especially for 400 V except in small region near the cathode. The retardation at the cathode was almost zero and became even lower with distance from the cathode. However, with respect to sample B, the charged regions were restricted to the left hand side of the graph and the remainder was not charged. With this sample, it appears that all the electrons injected into the crystal were trapped by crystal defects and stopped before reaching the anode. As a result, in the right region, which was uncharged, the electric field did not vary and neither did the retardation. As the deflection angle of the deflector is approximated by the gradient of the retardation distribution, this fact means that the beam deflection was very weak in the right hand side region. As a beam deflector, it is preferable for the retardation gradient to be steep everywhere in the crystal as it is for sample A, but not for sample B. Although both samples were fabricated with the same processes, they showed different properties. We believe that this was due to a difference in the electron trap density and will discuss further in Section 5.
We examined the time evolutions of the charge distribution for another sample similar to sample B. Figure 3 shows an example. As described in the experimental section, we applied a pulse voltage to the sample and then measured the trapped charge density. The voltage was 300 V and the temperature was set at about 40°C so that εr was controlled at 17,500. The initial state was a state where the charge was neutral throughout the sample. To reset the sample to this uncharged state, we irradiated it with a light from a 405-nm LED. The light removed the trapped electrons from the crystal as described previously. Although the left end corresponds to the cathode as in Fig. 2, the horizontal axis is enlarged for this figure and the anode is not shown. The figure reveals that the negatively charged region spread from the cathode while voltage was being applied, in other words, electrons injected from the cathode drifted toward the anode induced by the electric field. However, it is also apparent that the motion was not linearly dependent on time as expected from the standard Ohmic law. We defined a factor “electron penetration depth” as the width at half maximum of the trapped electron density (half maximum of the absolute value of the charge density). Figure 4 shows the depth as a function of time for different voltages. It shows more clearly that the electron penetration depth was not proportional to time, and that the electron motion gradually slowed down with time and stopped. The final depth increased with increasing voltage. However, even at 400 V, the electron did not reach the anode that corresponds to the top end of Fig. 4.
4. Theoretically obtained charge distributions
4.1 Basic equations
In this section, we discuss analytical steady state solutions and time dependent numerical solutions. As the charged carriers, we consider only electrons and ignore holes. Then charge transport by electrons is analyzed with the following equations.
Equation (6-1) is Ohm’s law together with the diffusion term. Equation (6-2) shows the current continuity. Equation (6-3) is the one-dimensional Gauss’s law, which was rewritten from Eq. (3). Equation (6-4) is the rate equation. Finally, we must take into account the condition (6-5), which means the field integration is the voltage V. t is the time and x is the location ranging from 0 to d. The current density J, the free carrier density n, the density of trapped carrier nt, and the electric field E are functions of both x and t. e is the elementary charge. ε, μ and D are the permittivity, mobility, and diffusion coefficient of the material, respectively. n0 and nt0 are equilibrium values of n and nt without voltage application. Nt is the total trap density, that is, the density of occupied and unoccupied traps. In Eq. (6-4), the first term indicates the trapping of free carriers and the second term indicates carrier release from the traps. We ignored optical excitations.Here, Many made two simplifications [12]. First, he assumed that the diffusion term in Eq. (6-1) had little effect on the solution and ignored it. Second, he assumed that
This means that changes in the trapped electron density from the equilibrium value are much smaller than the unoccupied trap density, which, according to Many, holds for most dielectrics and situations of interest. These assumptions effectively simplify Eqs. (6-1)-(6-5) and also the analytical solution for the steady state. We followed the first simplification. However, as for the second assumption, for KTN, the densities of trapped electrons often become comparable to the trap density. Sometimes the traps are fully occupied by electrons. Therefore, we did not adopt this assumption.4.2 Steady state solutions
When in Eq. (6-2), the current density J becomes a constant. Then, ignoring the diffusion term in Eq. (6-1), the following common equation is derived.
From (6-3), (6-4) and this equation, we obtain the following equation to solve.Here we assume nt0 = 0. Then n0 is also zero. The solution isWe took the origin of the coordinate x at the cathode. E0 is the electric field at x = 0. θ0 and Ej are defined as follows. In most cases, the last logarithmic term on the right side of Eq. (10) can be ignored. In addition, when J is sufficiently small, the second linear term can also be ignored. Then x becomes a quadratic function of E and J becomes proportional to V2, which corresponds to Child’s law.E0 and Ej are not independent of each other. These parameters are related by the following equations which were derived with the help of Eq. (6-5).
Here d is the electrode gap and Ed is the electric field at x = d (the anode). When the voltage is fixed, E0, Ed and Ej are unknown and we cannot determine them all solely with Eqs. (13-1) and (13-2). We may add another condition that relates E0 and Ej concerning the cathode interface. However, such a condition has to introduce additional unknown factors such as interface state density. Therefore, from the experimentally obtained profiles, we simply assumed E0 = 0. Then, from a given V, we can fix remaining Ed and Ej and consequently, the current density J. Figure 5 shows examples of the solutions. As described, no = nto = 0.0. θ0 = 3 x 10−6 and εr = 17,500. From Fig. 2, we assumed that eNt was 250 C/m3 (Nt: 1.56 x 1021 m−3). No other parameters affect these profiles although μ is required for calculating J. The fact that the absolute values of the electric field and the retardation increase with the distance from the cathode is similar to the experimental observations. However, especially for sample B, the steady state solution Eq. (10) failed to simulate the experimental distributions. The difficulty is that the electrons stopped before reaching the anode for sample B as shown in Fig. 2. As mentioned, the last logarithmic term in Eq. (10) can be ignored. Then the coordinate x is a quadratic function of E. Thus does not converge to zero with a finite x. Even when we took the logarithmic term into account, we could not realize a plateau as shown on the right side of the retardation graph of sample B in Fig. 2. To simulate the distribution property, we must analyze the equation in a time dependent manner. Although it seems in Fig. 4 that the motion stopped in several seconds, the sample did not reach the final steady state.4.3 Numerically calculated spatial distributions
We cannot expect to obtain analytical time dependent solutions of the nonlinear partial differential Eqs. (6-1)-(6-5) and we must perform numerical calculations. We used a simple difference method. No approximations were made. For numerical calculations, there is no need to omit the diffusion term. Actually, however, the term is small compared with the first ohmic term in Eq. (6-1) at room temperature and has practically no effect on the solutions. Figure 6 shows calculated results of the electron penetration depths as a function of voltage application duration, which corresponds to the experimental results in Fig. 4. The permittivity and thickness of the KTN block were the same as those in Fig. 4. γ / e, β, eNt, and μ were 0.4 m3C−1s−1, 3 x 10−4 s−1, 250 C/m3, and 2 x 10−4 cm2V−1s−1, respectively. As in the previous section, we put n0 and nt0 at zero. The initial state was also an uncharged neutral state. The phenomenon whereby electrons injected from the cathode drifted toward the anode but stopped before reaching the anode was effectively simulated.
The spatial distributions at 10 s in Fig. 6 for the respective applied voltages are shown in Fig. 7.The conditions are the same as those for Fig. 5 except that Fig. 7 was obtained as time evolution results whereas Fig. 5 was obtained as steady state solutions. In addition, it seems that, in Fig. 6, the motions almost reached the final state. However, the profiles in Fig. 7 are different from the steady state profiles in Fig. 5. Electrons are confined in a limited region from the left end (the cathode) in Fig. 7 as they were for the experimental results in Fig. 2 whereas electrons reached the right end (the anode) in Fig. 5. Thus, although it appears that electron motion almost ceased at the right ends in Fig. 4, the experimental profiles in Fig. 2 are actually not the final steady states. They are “frozen” but are changing very slowly.
Intuitively the speed of motion should be closely related to the current density J. Figure 8 shows the current density as a function of the elapsed time. The conditions were the same as those for the 400 V curve in Fig. 7. The maximum current density 72 A/m2 at about 1 ms corresponds to 0.8 mA for our experimental samples with 3.4 mm x 3.2 mm electrodes. However it decreased rapidly until it was more than seven orders of magnitude smaller. This current attenuation almost completely suppresses the profile movement. Equation (6-1) indicates that the current density J vanishes when the free electron density n vanishes because both the ohmic current and the diffusion current vanish. Then the first term on the left side of Eq. (6-4) vanishes. However the trapped electron density nt does not vanish and then is determined by the nt term. When its coefficient β is large, the spatial profile quickly converges from those seen in Fig. 7 to those seen in Fig. 5. For Figs. 6 and 8, 1 / β was 3300 s, which was more than two orders longer than the time range of the calculations. Thus the spatial charge distributions were frozen as shown in Fig. 7. For the sample that we used to acquire the data in Fig. 4, 1 / β was assumed to be at least 100 s.
Time dependent currents with a constant voltage have been reported for KTN by Paul [13]. He did not mention the degree to which the absolute value of the current changed in his experiments. However, he waited for 10 minutes to obtain reproducible data and observed that the current was still varying. Therefore his sample might have a similar β to that for our Fig. 8.
5. Discussion
For use as an optical beam deflector, it is desirable that electrons injected from the cathode reach the anode so that the entire crystal region is charged. Figure 7 and the corresponding experimental result for sample B shown in Fig. 2 indicate that the electron penetration depth is proportional to the applied voltage. Actually the depth is also proportional to the permittivity. In addition, from numerical calculations, a greater penetration depth is deduced with a smaller total trap density Nt as shown in Fig. 9.Thus, to attain a uniform charge distribution, we should prepare a device with a sufficiently small trap density, control the device temperature so that the permittivity is kept at a high value, and apply a high voltage. The difference between samples A and B should be the difference in Nt because the permittivity and the voltage were the same.
The relationship between these three parameters in terms of effects on the penetration depth can be assumed from Gauss’s law (6-3). However Many’s normalization may be more convenient for a full understanding of it. He used the following units to eliminate the dimensions from the variables [12].
Here we denote his dimensionless variables with primes as follows.Then Eqs. (6-1)-(6-5) are transformed into the following equations. whereθ’0 is the value for θ0 defined by Eq. (11) in the limit of nt0 vanishing. In this framework, spatial profiles are defined in the range . Among the constants τ, N’t and θ’0, θ’0 becomes effective only after the profile is frozen as described previously. τ is the trapping time, which is the average lifetime of a free electron before it is trapped by a localized state. (Note that the unit of τ is tu.) When τ is small, ρ varies abruptly in a spatial profile and the E profile becomes sharp. When τ is large, ρ varies gently in a spatial profile and the E profile becomes dull (Fig. 10). The constant that primarily determines the penetration depth is N’t. As shown in Eq. (16-6) it is inversely proportional to . Thus the penetration depth is proportional to . The voltage V and the permittivity ε can be easily changed experimentally for a given sample. Figure 11 shows experimental penetration depths depending on εrV. The circles, squares and diamonds indicate measured results for different temperatures and thus for different relative permittivities εr. However, by taking εrV as the horizontal axis, all the plots are on one straight line, which validates the above consideration. In fact the penetration depth deviates slightly from its linear dependence on 1/Nt for a larger penetration depth, as can be seen on the right side of Fig. 9. Even for such conditions, its dependence on remains valid.
Fig. 10 Numerically calculated spatial distributions of charge density for different τ values. To change only τ and not N’t, we changed γ/e. Although θ’0 is simultaneously changed, it does not affect the frozen profiles.
There remains a difficulty regarding the mobility. Wemple et al. reported an electron mobility of the order of 1 cm2 V−1s−1 for KTN at room temperature [14]. However, we were not able to simulate the temporal changes shown in Figs. 3 and 4 with this mobility. As described above, we used a mobility of 2 x 10−4 cm2V−1s−1 for most of the calculation. The speed of the time evolution is basically determined by the mobility and the permittivity in our calculations. The permittivity of the samples was measured with an LCR meter. Thus the only uncertainty exists in the mobility. Wemple did not provide detailed information about his KTN sample other than the mobility. On the other hand Paul reported that his KTN sample was about 8 orders of magnitude more resistive than Wemple’s [13]. We have not succeeded in measuring the Hall mobility of our KTN crystals because of their high resistivity. Now we think the difference in mobility is due to the differences between samples.
6. Conclusion
We described the charge distributions in KTa1-xNbxO3 single crystals, which were formed by the application of voltage. We observed experimentally that the space charge of the electrons spread from the cathode toward the anode with elapsed time. However the motion of the space charge stopped before it reached the anode in some KTN samples whereas, for other samples, it reached the anode and a more homogeneous charge distribution was attained. We confirmed theoretically that such experimental charge distributions were actually a frozen state and that the space charge reaches the anode in the final state or in the steady state condition. For the frozen states, the electron penetration depth is proportional to the applied voltage, the permittivity and the reciprocal of the electron trap density. Therefore the difference among the samples should be the difference in the electron trap density. To obtain spatially homogeneous charge distributions, we should prepare a deflector with an appropriate trap density, keep the permittivity high and apply a sufficiently high voltage.
References and links
1. K. Nakamura, J. Miyazu, M. Sasaura, and K. Fujiura, “Wide-angle, low-voltage electro-optic beam deflection based on space-charge-controlled mode of electrical conduction in KTa1−xNbxO3,” Appl. Phys. Lett. 89(13), 131115 (2006). [CrossRef]
2. F. S. Chen, J. E. Geusic, S. K. Kurtz, J. G. Skinner, and S. H. Wemple, “Light modulation and beam deflection with potassium tantalate-niobate crystals,” J. Appl. Phys. 37(1), 388–398 (1966). [CrossRef]
3. S. Yagi, K. Naganuma, T. Imai, Y. Shibata, J. Miyazu, M. Ueno, Y. Okabe, Y. Sasaki, K. Fujiura, M. Sasaura, K. Kato, M. Ohmi, and M. Haruna, “Improvement of coherence length in a 200-kHz swept light source equipped with a KTN deflector,” Proc. SPIE 8213, 821333 (2012). [CrossRef]
4. Y. Okabe, Y. Sasaki, M. Ueno, T. Sakamoto, S. Toyoda, J. Kobayashi, and M. Ohmi, “High-speed optical coherence tomography system using a 200-kHz swept light source with a KTN deflector,” Opt. Photon. J. 03(02), 190–193 (2013). [CrossRef]
5. R. W. Smith and A. Rose, “Space-charge-limited currents in single crystals of cadmium Sulfide,” Phys. Rev. 97(6), 1531–1537 (1955). [CrossRef]
6. A. Rose, “Space-charge-limited current in solid,” Phys. Rev. 97(6), 1538–1544 (1955). [CrossRef]
7. M. A. Lampert, “Simplified theory of space-charge-limited currents in an insulator with traps,” Phys. Rev. 103(6), 1648–1656 (1956). [CrossRef]
8. J. E. Geusic, S. K. Kurtz, L. G. Van Uitert, and S. H. Wemple, “Electro-optic properties of some ABO3 perovskites in the paraelectric phase,” Appl. Phys. Lett. 4(8), 141–143 (1964). [CrossRef]
9. J. Miyazu, T. Imai, S. Toyoda, M. Sasaura, S. Yagi, K. Kato, Y. Sasaki, and K. Fujiura, “New beam scanning model for high-speed operation using KTa1-xNbxO3 crystals,” Appl. Phys. Express 4(11), 111501 (2011). [CrossRef]
10. T. Imai, J. Miyazu, and J. Kobayashi, “Measurement of charge density distributions in KTa1-xNbxO3 optical beam deflectors,” Opt. Mater. Express 4(5), 976–981 (2014). [CrossRef]
11. T. Imai, T. Imai, M. Sasaura, K. Nakamura, and K. Fujiura, “Crystal Growth and Electro-optic Properties of KTa1-xNbxO3,” NTT Tech. Rev. 5(9), 1 (2007), https://www.ntt-review.jp/archive/2007/200709.html.
12. A. Many and G. Rakavy, “Theory of transient space-charge-limited currents in solids in the presence of trapping,” Phys. Rev. 126(6), 1980–1988 (1962). [CrossRef]
13. R. Paul, “Raumladungsbegrenzte Ströme in halbleitenden K(Ta, Nb)O3-Mishkristallen,” Kristal und Technik 6(3), 405–416 (1971). [CrossRef]
14. S. H. Wemple, M. DiDomenico Jr, and A. Jayaraman, “Electron scattering in perovskite-oxide ferroelectric semiconductors,” Phys. Rev. 180(2), 547–556 (1969). [CrossRef]
