Further insight into the oscillating photocarrier grating technique: influence of the oscillation amplitude

Abstract

Allowing for the simultaneous determination of the photocarriers drift mobilities and their small-signal recombination lifetime, the Moving photocarrier Grating Technique (MGT) is a useful characterization tool for photoconductive semiconductors. This technique is based on measuring the steady-state direct current induced by an interference pattern (IP) moving at a constant velocity between two coplanar ohmic contacts deposited on the semiconductor. The main drawback of the technique is the low level of the signal to be measured, which can be masked by noise. The Oscillating Photocarrier Grating technique (OPG), where the IP oscillates at a constant speed, has been proposed as an alternating current version of MGT, providing a higher signal-to-noise ratio. The IP oscillation is produced by the phase modulation of one of the interfering beams. Using the multiple trap** model we deduce the expression of the current density generated by OPG in a photoconductor. We observed theoretically and experimentally that OPG is not equivalent to MGT for the previously used amplitude of oscillation, especially when the IP moves at high speeds. However, we show that the desired equivalence between both techniques could be recovered by increasing the amplitude of oscillation. A phase modulator capable of achieving such amplitudes is required for the correct implementation of OPG.

Appendices

Appendix A

Deduction of the current density expression at the fundamental frequency

It is convenient to write the oscillation amplitude of the phase as A_pp = lπ, being l a positive number. It can be shown that when A_pp is a multiple of π (integer l), of the infinite terms that make up the generation rate (Eq. 7) no more than 6 influence the current induced at the fundamental frequency. These terms are given by:

$$g_{{ \pm \left( {l - 1} \right)\omega_{o} }} = \frac{{2lj\Delta G_{0} }}{{\pi \left( {2l - 1} \right)}},$$

(19)

$$g_{{ \pm l\omega_{o} }} = \frac{{\Delta G_{0} }}{2},$$

(20)

$$g_{{ \pm \left( {l + 1} \right)\omega_{o} }} = \frac{{ - 2lj\Delta G_{0} }}{{\pi \left( {2l + 1} \right)}}.$$

(21)

Note that for l = 1 there are only five terms since in Eq. 19 we have a single term g₀. When l is integer, the obtained complex amplitude of the average current density at the fundamental frequency is:

$$\Delta J_{OPG}^{{\omega_{o} }} = \frac{1}{2}\left[ {\begin{array}{*{20}c} {\Delta \sigma_{{l\omega_{o} }} \Delta \xi_{{\left( {l - 1} \right)\omega_{o} }}^{ *} + \Delta \sigma_{{ - l\omega_{o} }}^{*} \Delta \xi_{{ - \left( {l - 1} \right)\omega_{o} }} + \Delta \sigma_{{ - \left( {l - 1} \right)\omega_{o} }} \Delta \xi_{{ - l\omega_{o} }}^{ *} + \Delta \sigma_{{\left( {l - 1} \right)\omega_{o} }}^{*} \Delta \xi_{{l\omega_{o} }} } \\ { + \Delta \sigma_{{ - \left( {l + 1} \right)\omega_{o} }}^{*} \Delta \xi_{{ - l\omega_{o} }} + \Delta \sigma_{{\left( {l + 1} \right)\omega_{o} }} \Delta \xi_{{l\omega_{o} }}^{ *} + \Delta \sigma_{{l\omega_{o} }}^{*} \Delta \xi_{{\left( {l + 1} \right)\omega_{o} }} + \Delta \sigma_{{ - l\omega_{o} }} \Delta \xi_{{ - \left( {l + 1} \right)\omega_{o} }}^{ *} } \\ \end{array} } \right],$$

(22)

where the asterisk as superscript corresponds to the complex conjugation of the number. When ${\xi}_{ext}=0$, the symmetry between the generation rates ${g}_{\Omega}={g}_{-\Omega}$ implies the following relationships for the conductivity and the electric field amplitudes:

$$\Delta \sigma_{ - \Omega } = \frac{{g_{\Omega } }}{{g_{\Omega }^{*} }}\Delta \sigma_{\Omega }^{*} , \Delta \xi_{ - \Omega } = - \frac{{g_{\Omega } }}{{g_{\Omega }^{*} }}\Delta \xi_{\Omega }^{ *} .$$

(23)

In this case, Eq. 22 simplifies to:

$$ \begin{aligned}\Delta J_{OPG}^{{\omega_{o} }} &= \Delta \sigma_{{l\omega_{o} }} \Delta \xi_{{\left( {l - 1} \right)\omega_{o} }}^{ *} + \Delta \sigma_{{\left( {l - 1} \right)\omega_{o} }}^{*} \Delta \xi_{{l\omega_{o} }}\\ &\quad + \Delta \sigma_{{\left( {l + 1} \right)\omega_{o} }} \Delta \xi_{{l\omega_{o} }}^{ *} + \Delta \sigma_{{l\omega_{o} }}^{*} \Delta \xi_{{\left( {l + 1} \right)\omega_{o} }} .\end{aligned} $$

(24)

Note that under these circumstances, we only need to solve the general equations for three traveling waves with different values of Ω to obtain the current density induced at the fundamental frequency.

When A_pp is not a multiple of π (l is not an integer), the expression for the amplitude of the average current density induced at the fundamental frequency has infinite terms and is given by

$$ \begin{aligned}\Delta {J}_{OPG}^{{\omega }_{o}}&=\frac{1}{2}\sum_{i=1}^{\infty }\left[\Delta {\sigma }_{i{\omega }_{o}}\Delta {\xi}_{\left(i-1\right){\omega }_{o}}^{ *}+\Delta {\sigma }_{-i{\omega }_{o}}^{*}\Delta {\xi}_{\left(1-i\right){\omega }_{o}}\right.\\&\quad \left.+\Delta {\sigma }_{\left(1-i\right){\omega }_{o}}\Delta {\xi}_{-i{\omega }_{o}}^{ *}+\Delta {\sigma }_{\left(i-1\right){\omega }_{o}}^{*}\Delta {\xi}_{i{\omega }_{o}}\right].\end{aligned} $$

(25)

For A_pp = 3π/4, π/2 and π/4, the harmonic terms’ amplitudes of the generation rate are, respectively, given by:

$$g_{{ \pm i\omega_{o} }}^{l = 3/4} = \frac{{3\left( { - 1} \right)^{i} \Delta G_{0} }}{{\pi \sqrt 2 \left( {\frac{9}{4} - 4i^{2} } \right)}}\left( {1 + j\left( {1 + \left( { - 1} \right)^{i} \sqrt 2 } \right)} \right),$$

(26)

$$g_{{ \pm \left( {2i + 1} \right)\omega_{o} }}^{l = 1/2} = \frac{{2\sqrt 2 \Delta G_{0} }}{{\left( {4\left( {2i + 1} \right)^{2} - 1} \right)\pi }} , g_{{ \pm 2i\omega_{o} }}^{l = 1/2} = \frac{{ - j2\sqrt 2 \Delta G_{0} }}{{\left( {4\left( {2i} \right)^{2} - 1} \right)\pi }},$$

(27)

$$g_{{ \pm i\omega_{o} }}^{l = 1/4} = \frac{{\Delta G_{0} }}{{8\pi \left( {\frac{1}{16} - i^{2} } \right)}}\left( {\sqrt 2 \left( { - 1} \right)^{i} + j\left( {2 - \sqrt 2 \left( { - 1} \right)^{i} } \right)} \right),$$

(28)

where i = 0, 1, 2, ⋯. In these cases, is also verified that g_Ω = g_-Ω; therefore, in the absence of an external electric field (ξ_ext = 0), Eq. 23 are also verified. With these simplifications Eq. 25 reduces to:

$$ \begin{aligned}\Delta J_{OPG}^{{\omega_{o} }} &= \frac{1}{2}\sum_{i = 1}^{\infty } {\left( {1 - \frac{{g_{{i\omega_{o} }}^{*} g_{{\left( {i - 1} \right)\omega_{o} }} }}{{g_{{i\omega_{o} }} g_{{\left( {i - 1} \right)\omega_{o} }}^{*} }}} \right)}\\&\quad{\left( {\Delta \sigma_{{i\omega_{o} }} \Delta \xi_{{\left( {i - 1} \right)\omega_{o} }}^{ *} + \Delta \sigma_{{\left( {i - 1} \right)\omega_{o} }}^{*} \Delta \xi_{{i\omega_{o} }} } \right)} .\end{aligned} $$

(29)

Appendix B

Equivalence between MGT and OPG for A _pp multiple of ${\varvec{\pi}}$

In this appendix, we prove, for A_pp multiple of π, that Eq. 6 is strictly verified in the linear region of small frequencies. In the high-frequency regime we show that, the better the inequality A_pp ≫ π is fulfilled, the better Eq. 6 is satisfied.

In appendix A of Ref. [7] are solved the general transport equations for a generation rate consisting of an arbitrary harmonic travelling wave on a much greater uniform background. The harmonic amplitudes of the photoconductivity and electric field can be rewritten in the following way:

$$\Delta \sigma_{\Omega } = g_{\Omega } \Delta \sigma_{\Omega }^{0} , \quad \Delta \xi_{\Omega } = g_{\Omega } \Delta \xi_{\Omega }^{ 0} ,$$

(30)

where $\Delta \sigma_{\Omega }^{0}$ and $\Delta \xi_{\Omega }^{0}$ do not depends on the harmonic amplitude of the generation rate, g_Ω. The first-order Taylor expansion of $\Delta \sigma_{\Omega }^{0}$ and $\Delta \xi_{\Omega }^{0}$ at Ω = 0 are:

$$\Delta \sigma_{\Omega }^{0} \approx \Delta \sigma_{0}^{0} + \left( {\Delta \sigma_{0}^{0} } \right)^{\prime } \Omega , \Delta \xi_{\Omega }^{ 0} \approx \Delta \xi_{0}^{ 0} + \left( {\Delta \xi_{0}^{ 0} } \right)^{\prime } \Omega ,$$

(31)

where $\Delta \sigma_{0}^{0}$ and $\left( {\Delta \xi_{0}^{0} } \right)^{\prime }$ are real numbers, while $\left( {\Delta \sigma_{0}^{0} } \right)^{\prime }$ and $\Delta \xi_{0}^{0}$ are imaginary. Replacing Eqs. 30 in the expression of the MGT current density [7], it is obtained:

$$\Delta J_{MGT} = \frac{{\Delta G_{0}^{2} }}{2}Re\left\{ {\Delta \sigma_{\Delta \omega }^{0} \left( {\Delta \xi_{\Delta \omega }^{ 0} } \right)^{*} } \right\}.$$

(32)

Substituting Eqs. 31 in Eq. 32, discarding the higher-order terms and after some algebra, we get the expression of MGT current density in the low-frequency region:

$$\Delta J_{MGT} \approx \frac{{\Delta G_{0}^{2} }}{2}\Delta \omega \left[ {\Delta \sigma_{0}^{0} \left( {\Delta \xi_{0}^{ 0} } \right)^{\prime } - \left( {\Delta \sigma_{0}^{0} } \right)^{\prime } \Delta \xi_{0}^{ 0} } \right].$$

(33)

Replacing Eq. 30 in Eq. 24 and using Eqs. 19–21, we obtain the following formula for the current density induced by OPG at the fundamental frequency when A_pp is a multiple of π:

$$ \begin{aligned}\Delta J_{OPG}^{{\omega_{0} }} &= \frac{{ - lj\Delta G_{0}^{2} }}{\pi }\left[ {\frac{{\Delta \sigma_{{l\omega_{0} }}^{0} \left( {\Delta \xi_{{\left( {l - 1} \right)\omega_{0} }}^{ 0} } \right)^{*} + \left( {\Delta \sigma_{{\left( {l - 1} \right)\omega_{0} }}^{0} } \right)^{*} \Delta \xi_{{l\omega_{0} }}^{ 0} }}{{\left( {2l - 1} \right)}} }\right.\\&\qquad\qquad\left.{+ \frac{{\Delta \sigma_{{\left( {l + 1} \right)\omega_{0} }}^{0} \left( {\Delta \xi_{{l\omega_{0} }}^{ 0} } \right)^{*} + \left( {\Delta \sigma_{{l\omega_{0} }}^{0} } \right)^{*} \Delta \xi_{{\left( {l + 1} \right)\omega_{0} }}^{ 0} }}{{\left( {2l + 1} \right)}}} \right].\end{aligned} $$

(34)

Substituting Eqs. 31 in 34, dismissing the higher-order terms and after some algebra we get the equation for OPG current density in the low-frequency region:

$$\Delta J_{OPG}^{{\omega_{0} }} \approx \frac{{ - 2l\omega_{0} j\Delta G_{0}^{2} }}{\pi }\left[ {\Delta \sigma_{0}^{0} \left( {\Delta \xi_{0}^{ 0} } \right)^{\prime } - \left( {\Delta \sigma_{0}^{0} } \right)^{\prime } \Delta \xi_{0}^{ 0} } \right].$$

(35)

Taking the modulus of Eqs. 33 and 35, and using the relationship $\left| {\Delta \omega } \right| = l\omega_{0}$, we obtain Eq. 6.

The inequality A_pp ≫ π is equivalent to l ≫ 1. Discarding the $1$ in front of $l$ and $2l$ in Eq. 34, we get the formula of OPG current density in this case:

$$\Delta J_{OPG}^{{\omega_{0} }} = \frac{{ - j2\Delta G_{0}^{2} }}{\pi }Re\left\{ {\Delta \sigma_{{l\omega_{0} }}^{0} \left( {\Delta \xi_{{l\omega_{0} }}^{ 0} } \right)^{*} } \right\}.$$

(36)

Taking the modulus of Eqs. 32 and 36 and assuming $\left| {\Delta \omega } \right| = l\omega_{0}$, we get Eq. 6. Note that the inequality A_pp ≫ π or l ≫ 1, is also equivalent to a maximum displacement of the interference pattern much greater than half its spatial period (D_pp ≫ $\Lambda$/2).

Appendix C

Electro-optic modulator (EOM)

In a typical linear EOM, the refractive indexes in the main directions of the active crystal (perpendicular to each other) are directly proportional to the applied voltage. The proportionality constant is the same in both directions, although of opposite signs. In the absence of voltage applied to the modulator, a linearly polarized beam of light retains its initial polarization because the refractive indices are equal in both directions. Therefore, its electric field written in the main directions of the modulator is,

$$\vec{E} = E_{0} \cos (\theta )\cos (kz - \omega t)\hat{\text{x}} + E_{0} \sin (\theta )\cos (kz - \omega t)\hat{\text{y}},$$

(37)

where θ is the angle that forms the electric field of the incident beam with the main direction ${\hat{\text{x}}}$ of the EOM. By applying a potential difference to the modulator, there is an increase in the optical path for polarized light in one main direction and a decrease in the optical path of the same magnitude for polarized light in the other main direction. In this case, the electric field of the light beam at the output of the modulator is,

$$ \begin{aligned}\vec{E} &= \,E_{0} \cos \left( \theta \right)\cos \left( {kz - \omega t + \varphi \left( V \right)} \right){\hat{\text{x}}} \\&\quad+ E_{0} \sin \left( \theta \right)\cos \left( {kz - \omega t - \varphi \left( V \right)} \right){\hat{\text{y}}},\end{aligned} $$

(38)

Note that the EOM acts as a phase modulator of the light beam only when one of its main axes coincides with the direction of polarization of the incident light beam. On the other hand, the EOM acts as a polarization modulator capable of rotating 90° the initial direction of polarization only when the main axes of the modulator are at 45° from the polarization of the incoming light beam. In this case, Eq. 38 becomes,

$$\vec{E} = \frac{{E_{0} }}{\sqrt 2 }\cos \left( {kz - \omega t + \varphi \left( V \right)} \right){\hat{\text{x}}} + \frac{{E_{0} }}{\sqrt 2 }\cos \left( {kz - \omega t - \varphi \left( V \right)} \right){\hat{\text{y}}}{.}$$

(39)

When φ (V) = π/4, for example, we obtain a circularly polarized light at the output of the modulator, and for φ (V) = π/2 we obtain again linearly polarized light but rotated 90 degrees from the input direction of polarization.