Doubly doped crystal
Chapter 4 Chapter 4 System issues in two-center holographic recording
4.3 Effect of carrier mobility in holographic record-
4.3.1 Effect of carrier mobility in normal holographic record- mg
To study the effect of electron mobility (p) on M / # and S, we only need to consider the dependence of saturation hologram strength (Ao) and recording time constant (Tr) on p. We assume that recording and erasure time constants are approximately the same resulting in
M/#
Ao, ( 4.4)
where Te is the erasure time constant. Note that even if Tr and Te are not the same, they have similar variations with electron mobility. Therefore, the main parameter that represents the effect of p on M/# is the saturation hologram strength Ao. Note also that Ao is linearly proportional to saturation space-charge field E1,sat. Therefore, we can use
M/# ex Ao ex E1,sat (4.5)
to study the effect of p on M/#. The effect of p can be studied by using the formula for S in terms of E 1,sat and Tr as
S (4.6)
where hand L represent recording intensity and crystal thickness, respectively. Both hand L are independent of p.
The approximate formulas for E1,sat and Tr are
jphvl
+
jdifflepno (4.7)
(4.8)
where e and no are the electronic charge and the average (DC) electron concentra- tion in the conduction band, respectively. Furthermore, jphvl and jdiffl are the first Fourier components of bulk photovoltaic and diffusion current densities, respectively.
Furthermore, no can be approximately represented by
qFe SFeNieoI RO
IFe(NFe - N FeO ) , (4.9)
where qFeSFe and IFe represent the absorption cross section from Fe traps to the con- duction band and recombination coefficient of the Fe traps, respectively. Furthermore, N Fe , Nieo, and I RO represent total Fe concentration, electron concentration in the Fe traps, and average (DC) recording intensity, respectively. The formulas for Jphvl and
Jdiffl in a LiNb03:Fe crystal are
Jphvl
Jdiffl
(4.10) (4.11)
Here, K:Fe and hI in Equation (4.10) are bulk photovoltaic constant of the Fe traps at recording wavelength and the amplitude of sinusoidal recording intensity, respectively.
In Equation (4.11), kB, T, K , and nl represent Boltzmann constant, absolute temper- ature, amplitude of grating vector of the hologram, and the first Fourier component of the electron concentration in the conduction band. For holographic recording in congruently melting LiNb03:Fe crystals using transmission geometry, the bulk photo- voltaic current is dominant and the diffusion current density (jdiffl) in Equation (4.7) can be neglected resulting in
(4.12)
Replacing no in Equation (4.12) by its equivalent from Equation (4.9), we obtain
(4.13)
Putting Equations (4.13) and (4.8) into Equations (4.5) and (4.6), we obtain the dependence of M /
#
and S on J-l asM/#
S
1 J..l
C,
(4.14) (4.15 )
where C represents some constant independent of J-l. Note that Equations (4.14) and (4.15) are valid only in the regime of the domination of bulk photovoltaic current.
Therefore, these equations can be applied to congruently melting crystals. Equations (4.14) and (4.15) suggest that we can not increase sensitivity by increasing mobility, while we lose M/# by increasing mobility. This result might seem strange at the beginning since we know that the holograms are recorded faster at higher electron mobility (recording time constant becomes smaller at higher mobility). However, sensitivity depends on the ratio of saturation hologram strength (Ao) and recording time constant (Tr). When the bulk photovoltaic current is dominant, both Ao and Tr decrease with increasing J-l in a similar way resulting in approximate independence of sensitivity from J..l.
The situation is completely different in the regime of the domination of diffusion current. This is the case for stoichiometric LiNb03:Fe crystals, or in some cases, for congruently melting crystals in the 900 geometry. The saturation space-charge field in this regime can be represented by
E1,sat _ jdiffl ::::: - i f { kBT nl .
eJ-lno e no
Assuming unity modulation depth of recording intensity, we can use nl simplify Equation (4.16) as
kBT f{ .
e
(4.16)
no to
( 4.17)
Equation (4.17) suggests that in the regime of the domination of diffusion current, saturation space-charge field is approximately independent of J..l. Using this result,
we can summarize the dependence of 1111/# and 5 on /1 in this regime as
M/#
5
G'
/1 ,
( 4.18) ( 4.19)
where G' is a constant independent of /1. Equations (4.18) and (4.19) suggest that increasing mobility in the regime of the domination of diffusion current is a good idea for increasing sensitivity without affecting
M/#.
In this regime, the saturation hologram strength is independent of /1. Therefore, increasing /1 results in increasing sensitivity by reducing the recording time constant. The domination of diffusion current in LiNb03 occurs in near stoichiometric crystals or the crystals that are highly doped with MgO, or in some cases in the 90° geometry with small grating period (or high spatial frequency).As discussed above, we need to get to the regime of domination of diffusion current in LiNb03: Fe to start improving sensitivity by increasing /1. To do this, we need to increase /1 by using stoichiometric crystals, for example. However, we lose
NI/#
by a large factor in going from the regime of domination of photovoltaic current to that of the domination of diffusion current. This is clearly depicted in Figure 4.4 that shows the theoretical variation of the saturation hologram strength
(Ao
::=M/#)
and sensitivity (5) with electron mobility (/1) in a 0.85 mm thick LiNb03 crystal doped with 0.075 wt. % Fe203' The calculation is performed by solving Kukhtarev's equations [59] numerically using Fourier development explained before. We assumed that recording was performed by two red beams (wavelength 633 nm, intensity of each beam 250 mW/cm2). Although 633 nm is not the best wavelength for recording from Fe traps, we chose it to be the same as the recording wavelength in two-center recording discussed later. The two curves in each part of Figure 4.4 are calculated with and without considering diffusion current to show the regimes of the domination of the different components of current. As Figure 4.4 shows, we need to increase /1 by more than one order of magnitude from that of a typical congruently melting LiNb03 crystal to enter into the regime of the domination of the diffusion current
where sensitivity can be improved by increasing iJ further. During this process, M/#
is reduced by more than one order of magnitude before getting to the diffusion domi- nation regime where it becomes almost independent of iJ. It is important to note that the range of iJ shown in Figure 4.4 is not the practical range that can be obtained by changing the stoichiometry of LiNb03 crystals. It is not practically possible to increase electron mobility in LiNb03 by three orders of magnitude by simply changing the stoichiometry of the crystal or by doping it with MgO. Therefore, the usage of stoichiometric crystals or crystals doped highly with MgO in the transmission geom- etry is not a good idea to improve S. While sensitivity of these materials is similar to that of congruently melting crystals, their dynamic range (M / #) is much smaller than that of the congruently melting crystals.
Figure 4.5 shows the recording and read-out curve for a plane wave hologram recorded in a 0.85 mm thick LiNb03 crystal doped with 0.25 wt. % Fe203 and 4.3 wt. % MgO. Recording is performed in transmission geometry by two plane waves (wavelength 488 nm, intensity of each beam 15.5 mW/cm2, ordinary polarization) while reading is performed by one of the recording beams. The values of M /# and S calculated from Figure 4.5 are M/#